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[
"Axion Dark Matter Detection with CMB Polarization",
"Axion Dark Matter Detection with CMB Polarization"
] | [
"Michael A Fedderke \nStanford Institute for Theoretical Physics\nDepartment of Physics\nStanford University\n94305StanfordCAUSA\n\nBerkeley Center for Theoretical Physics\nDepartment of Physics\nUniversity of California Berkeley\n94720BerkeleyCAUSA\n\nTheory Group\nPhysics Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA\n",
"Peter W Graham \nStanford Institute for Theoretical Physics\nDepartment of Physics\nStanford University\n94305StanfordCAUSA\n",
"Surjeet Rajendran \nBerkeley Center for Theoretical Physics\nDepartment of Physics\nUniversity of California Berkeley\n94720BerkeleyCAUSA\n\nDepartment of Physics & Astronomy\nThe Johns Hopkins University\n21218BaltimoreMDUSA\n"
] | [
"Stanford Institute for Theoretical Physics\nDepartment of Physics\nStanford University\n94305StanfordCAUSA",
"Berkeley Center for Theoretical Physics\nDepartment of Physics\nUniversity of California Berkeley\n94720BerkeleyCAUSA",
"Theory Group\nPhysics Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA",
"Stanford Institute for Theoretical Physics\nDepartment of Physics\nStanford University\n94305StanfordCAUSA",
"Berkeley Center for Theoretical Physics\nDepartment of Physics\nUniversity of California Berkeley\n94720BerkeleyCAUSA",
"Department of Physics & Astronomy\nThe Johns Hopkins University\n21218BaltimoreMDUSA"
] | [] | We point out two ways to search for low-mass axion dark matter using cosmic microwave background (CMB) polarization measurements. These appear, in particular, to be some of the most promising ways to directly detect fuzzy dark matter. Axion dark matter causes rotation of the polarization of light passing through it. This gives rise to two novel phenomena in the CMB. First, the late-time oscillations of the axion field today cause the CMB polarization to oscillate in phase across the entire sky. Second, the early-time oscillations of the axion field wash out the polarization produced at last-scattering, reducing the polarized fraction (TE and EE power spectra) compared to the standard prediction. Since the axion field is oscillating, the common (static) 'cosmic birefringence' search is not appropriate for axion dark matter. These two phenomena can be used to search for axion dark matter at the lighter end of the mass range, with a reach several orders of magnitude beyond current constraints. We set a limit from the washout effect using existing Planck results, and find significant future discovery potential for CMB detectors searching in particular for the oscillating effect.CONTENTSAcknowledgments 17A. Derivation of the polarization rotation effect 17B. Stokes parameters 18References Overwhelming gravitational evidence for the existence of dark matter (DM) is to be found over a wide range of astrophysical and cosmological scales (see Refs. [1, 2] for reviews). Elucidating the non-gravitational properties of the dark matter is one of the most pressing open problems in particle physics. A particularly intriguing class of potential DM candidates is supplied by light bosonic degrees of freedom, of which axions and axion-like particles[3][4][5], and dark photons [6-9] are prototypical.Apart from being well-motivated degrees of freedom from the standpoint of UV theory (e.g., Refs. [10, 11]), the canonical QCD axion is independently extremely well-motivated as a solution to the strong CP problem[12][13][14][15], in certain regions of parameter space. More generally, much attention in the literature has also recently been directed toward axion-like particles (hereinafter, 'axions'), 1 which interact with the Standard Model via similar couplings to the QCD axion, but which populate a much broader region of mass-coupling parameter space. In particular, extremely light axions, with astrophysically macroscopic de Broglie wavelengths, can act as 'fuzzy dark matter' (FDM) (see Ref.[16] for a review), which may provide a solution to a number of potential small-scale structure anomalies[17].Light bosonic DM also provides an interesting counterfoil to the long-dominant weakly interacting massive particle (WIMP) paradigm [18] for particle dark matter. Indeed, the absence of definitive evidence for the WIMP in the face of ever-advancing experimental sensitivities[19][20][21][22][23][24][25][26][27][28][29]is becoming a strong motivation to cast a wider net in the search for the identity of the dark matter. The very high phase-space occupancy numbers required for light bosons to constitute all of the dark matter [16], 2 resulting in effective classical-field-like behaviour of these candidates, gives rise to a broad array of novel effects (see, e.g., Ref.[39]), requiring experimental approaches quite distinct from the canonical WIMP search techniques. A plethora of such approaches have recently been proposed[5,[40][41][42][43][44][45][46][47].One of the more exotic consequences that arises within the context of light bosonic dark matter is the effect of a pseudoscalar dark-matter axion background field on the propagation of linearly polarized light. It has long been known that the breaking of parity associated with coupling of any generic non-stationary or non-uniform background pseudoscalar to electromagnetism gives rise to a 'birefringence' for the propagation of opposite-helicity photons, manifesting itself as a rotation of the plane of linear polarization of the light by an angle proportional to the difference in the values of the pseudoscalar field at the emission and absorption of the photon[48][49][50][51].3An extensive literature exists on this subject. Existing searches for the rotation effect when the pseudoscalar varies on temporal or spatial scales inaccessible at Earth-or local space-based facilities 4 exploit the fact that a variety of astrophysical and cosmological sources emit polarized light, which travels over very long baselines to reach Earth, allowing observable net rotations to accumulate over the long travel times. In the context of polarized emission from astrophysical sources (radio galaxies, pulsars, protoplanetary disks, etc.), searches have encompassed the rotation arising from DM axions[57][58][59][60][61], cosmologically slowly varying pseudoscalars [49-51, 57, 62-70], and the case where the related Chern-Simons term gives rise to the rotation[48]. More germane to the topic of the present work, the rotation of the polarized fraction of the cosmic microwave background (CMB) has been considered in the case where the pseudoscalar field varies slowly on cosmological timescales[50,[67][68][69]; more exotic scenarios have also been considered[73,77,79,82,[84][85][86][95][96][97][98][99][100][101]. Searches by CMB experimental collaborations for either isotropic or anisotropic static 'cosmic birefringence' are standard[92,[102][103][104][105][106].Relatively fewer works have discussed the rotation of the polarization of the CMB due to DM axions[107][108][109][110], which necessarily vary rapidly on cosmological timescales given that small-scale structure measurements (e.g., Lymanα measurements[111,112], as well as dwarf-galactic structure [17]) constrain such axions to have oscillation periods of at most O(1 yr).II. EXECUTIVE SUMMARYIn this work, we revisit the photon polarization rotation effect arising from a dark-matter axion on the polarized fraction of the CMB. We point out a number of important phenomena that arise in this context that appear to have either escaped notice or been under-appreciated in previous analyses.Firstly, we re-emphasize a fundamental point that has long been known in the literature [50]: the rotation angle of the plane of polarization that arises from the birefringence induced by the axion-photon coupling is proportional to 1 Such a particle in general has no fixed relationship between its mass and couplings. We avoid the more elaborate 'axion-like particle' nomenclature in favour of 'axion' as we refer almost exclusively to such particles in this work; where necessary, we distinguish the 'axion' from the 'QCD axion', whose mass is fixed once its QCD coupling is specified. 2 Over wide regions of parameter space, there are plausible production mechanisms for both axions and dark photons; see, e.g., . 3 This rotation effect is distinct from the axion-photon mixing effects that occur in an external magnetic field background, which also give rise to a rotation effect[52,53]. 4 On the other hand, interferometric searches for the rotation effect have been recently proposed [54-56] for the case of a DM axion in the mass range m φ ∼ 10 −14 -10 −9 eV, which varies on a timescale amenable to a laboratory setting.12 J 0 (x) ≈ 1 − x 2 /4 + · · · for x 1. | 10.1103/physrevd.100.015040 | [
"https://arxiv.org/pdf/1903.02666v1.pdf"
] | 119,269,040 | 1903.02666 | 41d5eb86a23d53615d4d080f0aa6ceb67794d802 |
Axion Dark Matter Detection with CMB Polarization
Michael A Fedderke
Stanford Institute for Theoretical Physics
Department of Physics
Stanford University
94305StanfordCAUSA
Berkeley Center for Theoretical Physics
Department of Physics
University of California Berkeley
94720BerkeleyCAUSA
Theory Group
Physics Division
Lawrence Berkeley National Laboratory
94720BerkeleyCAUSA
Peter W Graham
Stanford Institute for Theoretical Physics
Department of Physics
Stanford University
94305StanfordCAUSA
Surjeet Rajendran
Berkeley Center for Theoretical Physics
Department of Physics
University of California Berkeley
94720BerkeleyCAUSA
Department of Physics & Astronomy
The Johns Hopkins University
21218BaltimoreMDUSA
Axion Dark Matter Detection with CMB Polarization
(Dated: March 6, 2019)
We point out two ways to search for low-mass axion dark matter using cosmic microwave background (CMB) polarization measurements. These appear, in particular, to be some of the most promising ways to directly detect fuzzy dark matter. Axion dark matter causes rotation of the polarization of light passing through it. This gives rise to two novel phenomena in the CMB. First, the late-time oscillations of the axion field today cause the CMB polarization to oscillate in phase across the entire sky. Second, the early-time oscillations of the axion field wash out the polarization produced at last-scattering, reducing the polarized fraction (TE and EE power spectra) compared to the standard prediction. Since the axion field is oscillating, the common (static) 'cosmic birefringence' search is not appropriate for axion dark matter. These two phenomena can be used to search for axion dark matter at the lighter end of the mass range, with a reach several orders of magnitude beyond current constraints. We set a limit from the washout effect using existing Planck results, and find significant future discovery potential for CMB detectors searching in particular for the oscillating effect.CONTENTSAcknowledgments 17A. Derivation of the polarization rotation effect 17B. Stokes parameters 18References Overwhelming gravitational evidence for the existence of dark matter (DM) is to be found over a wide range of astrophysical and cosmological scales (see Refs. [1, 2] for reviews). Elucidating the non-gravitational properties of the dark matter is one of the most pressing open problems in particle physics. A particularly intriguing class of potential DM candidates is supplied by light bosonic degrees of freedom, of which axions and axion-like particles[3][4][5], and dark photons [6-9] are prototypical.Apart from being well-motivated degrees of freedom from the standpoint of UV theory (e.g., Refs. [10, 11]), the canonical QCD axion is independently extremely well-motivated as a solution to the strong CP problem[12][13][14][15], in certain regions of parameter space. More generally, much attention in the literature has also recently been directed toward axion-like particles (hereinafter, 'axions'), 1 which interact with the Standard Model via similar couplings to the QCD axion, but which populate a much broader region of mass-coupling parameter space. In particular, extremely light axions, with astrophysically macroscopic de Broglie wavelengths, can act as 'fuzzy dark matter' (FDM) (see Ref.[16] for a review), which may provide a solution to a number of potential small-scale structure anomalies[17].Light bosonic DM also provides an interesting counterfoil to the long-dominant weakly interacting massive particle (WIMP) paradigm [18] for particle dark matter. Indeed, the absence of definitive evidence for the WIMP in the face of ever-advancing experimental sensitivities[19][20][21][22][23][24][25][26][27][28][29]is becoming a strong motivation to cast a wider net in the search for the identity of the dark matter. The very high phase-space occupancy numbers required for light bosons to constitute all of the dark matter [16], 2 resulting in effective classical-field-like behaviour of these candidates, gives rise to a broad array of novel effects (see, e.g., Ref.[39]), requiring experimental approaches quite distinct from the canonical WIMP search techniques. A plethora of such approaches have recently been proposed[5,[40][41][42][43][44][45][46][47].One of the more exotic consequences that arises within the context of light bosonic dark matter is the effect of a pseudoscalar dark-matter axion background field on the propagation of linearly polarized light. It has long been known that the breaking of parity associated with coupling of any generic non-stationary or non-uniform background pseudoscalar to electromagnetism gives rise to a 'birefringence' for the propagation of opposite-helicity photons, manifesting itself as a rotation of the plane of linear polarization of the light by an angle proportional to the difference in the values of the pseudoscalar field at the emission and absorption of the photon[48][49][50][51].3An extensive literature exists on this subject. Existing searches for the rotation effect when the pseudoscalar varies on temporal or spatial scales inaccessible at Earth-or local space-based facilities 4 exploit the fact that a variety of astrophysical and cosmological sources emit polarized light, which travels over very long baselines to reach Earth, allowing observable net rotations to accumulate over the long travel times. In the context of polarized emission from astrophysical sources (radio galaxies, pulsars, protoplanetary disks, etc.), searches have encompassed the rotation arising from DM axions[57][58][59][60][61], cosmologically slowly varying pseudoscalars [49-51, 57, 62-70], and the case where the related Chern-Simons term gives rise to the rotation[48]. More germane to the topic of the present work, the rotation of the polarized fraction of the cosmic microwave background (CMB) has been considered in the case where the pseudoscalar field varies slowly on cosmological timescales[50,[67][68][69]; more exotic scenarios have also been considered[73,77,79,82,[84][85][86][95][96][97][98][99][100][101]. Searches by CMB experimental collaborations for either isotropic or anisotropic static 'cosmic birefringence' are standard[92,[102][103][104][105][106].Relatively fewer works have discussed the rotation of the polarization of the CMB due to DM axions[107][108][109][110], which necessarily vary rapidly on cosmological timescales given that small-scale structure measurements (e.g., Lymanα measurements[111,112], as well as dwarf-galactic structure [17]) constrain such axions to have oscillation periods of at most O(1 yr).II. EXECUTIVE SUMMARYIn this work, we revisit the photon polarization rotation effect arising from a dark-matter axion on the polarized fraction of the CMB. We point out a number of important phenomena that arise in this context that appear to have either escaped notice or been under-appreciated in previous analyses.Firstly, we re-emphasize a fundamental point that has long been known in the literature [50]: the rotation angle of the plane of polarization that arises from the birefringence induced by the axion-photon coupling is proportional to 1 Such a particle in general has no fixed relationship between its mass and couplings. We avoid the more elaborate 'axion-like particle' nomenclature in favour of 'axion' as we refer almost exclusively to such particles in this work; where necessary, we distinguish the 'axion' from the 'QCD axion', whose mass is fixed once its QCD coupling is specified. 2 Over wide regions of parameter space, there are plausible production mechanisms for both axions and dark photons; see, e.g., . 3 This rotation effect is distinct from the axion-photon mixing effects that occur in an external magnetic field background, which also give rise to a rotation effect[52,53]. 4 On the other hand, interferometric searches for the rotation effect have been recently proposed [54-56] for the case of a DM axion in the mass range m φ ∼ 10 −14 -10 −9 eV, which varies on a timescale amenable to a laboratory setting.12 J 0 (x) ≈ 1 − x 2 /4 + · · · for x 1.
the difference in the values of the axion field at the emission and the absorption of the photon, and is independent of the details of the behaviour of the axion field along the photon trajectory. 5 Applying this understanding to the polarized fraction of the CMB, we find two main phenomenological implications: (1) there will be an AC oscillation, at the axion oscillation period, of the CMB polarization pattern on the sky as measured today, arising as a result of the local DM axion phase evolving over the total lifetime of a CMB experiment; and (2) given that a DM axion must oscillate many times during the CMB decoupling epoch, there is a necessity in the line-of-sight approach to computing CMB anisotropies [113] to average the axion-induced linear polarization rotation angle over all possible axion field phases explored during the decoupling epoch: this 'washes out' the polarization, leading to a reduction of the net polarization fraction of the CMB light as compared to the ΛCDM expectation. Relatedly, the cancellations inherent in having a fast-oscillating axion field at the decoupling epoch result in only a highly suppressed net rotation of the CMB linear polarization angle at any point on the sky as compared to a naïve estimate of the effect made taking into account only the axion field magnitude at decoupling.
The polarization washout effect is quadratic in the early-time axion field times the axion-photon coupling (i.e., g 2 φγ φ 2 * , a small parameter), and existing Planck CMB measurements constrain g φγ φ * 1.5 × 10 −1 (future, cosmicvariance-limited reach: g φγ φ * 5.7×10 −2 ). Normalizing the axion field φ * to be all the dark matter at the decoupling epoch, a m φ ∼ 10 −22 eV dark-matter axion is excluded for g φγ 1.4×10 −13 GeV −1 (future: g φγ 5.1×10 −14 GeV −1 ). The AC effect is linear the late-time axion field times the axion-photon coupling (i.e., g φγ φ 0 , a small parameter); taking an informed estimate for the present detectable amplitude of the AC oscillation to be 0.1 • (future assumed reach: 0.01 • ) allows a reach of g φγ φ 0 ∼ 3.5×10 −3 (future: g φγ φ 0 ∼ 3.5×10 −4 ). However, the local dark-matter density is smaller than that at decoupling, φ 0 /φ * ∼ 10 −2 , with the resulting sensitivity to a m φ ∼ 10 −22 eV dark-matter axion from the AC effect being g φγ ∼ 2.3 × 10 −13 GeV −1 (future: g φγ ∼ 2.3 × 10 −14 GeV −1 ), comparable to that of the washout effect given current sensitivities. Both effects have a reach a few orders of magnitude beyond existing bounds for the lightest possible fuzzy axion dark matter masses. The AC effect, being linear in the axion-photon coupling and not cosmic-variance limited, has better potential future reach than the washout effect.
Our work is complementary to, and not in conflict with, the many existing analyses cited in Sec. I that consider the distinct phenomenology that arises for pseudoscalars that vary slowly on cosmological timescales. The effects we note, arising from the faster-oscillating DM axions, are new. We defer a detailed comparison to previous work considering DM axions to the body of the paper.
In the remainder of this paper, we review how the axion-induced modifications to Maxwell's equations give rise to a photon polarization rotation effect (Sec. III), which we then employ in a series of increasingly realistic toy models (Sec. IV) designed to illustrate the resulting washout and oscillation phenomenology, building up to our analysis of the CMB (Sec. V) and main results (Fig. 2). After discussing past work (Sec. VI), we conclude (Sec. VII). Additional details are given in Appendices A and B.
III. AXION ELECTRODYNAMICS
We consider the action
S = d 4 x √ −g 1 2 (∇ µ φ)(∇ µ φ) − V (φ) − 1 4 F µν F µν − J µ A µ − 1 4 g φγ φF µν F µν ,(1)
where g is the metric determinant, A µ is the photon, F µν ( F µν ) is the (dual) field-strength tensor, J µ is the EM current, φ is the axion, and g φγ is the axion-photon coupling constant, which has mass-dimension −1; throughout this paper, we will assume that V (φ) = 1 2 m 2 φ φ 2 . It is well known that the axion-photon coupling gives rise to modifications to electrodynamics in an axion field background [114]. As we show in detail in Appendix A, if we specialize to a homogeneous, isotropic FLRW universe with scale factor a, working in the conformal-comoving co-ordinate system (η, x) where η is conformal time such that the line element is ds 2 = [a(η)] 2 dη 2 − dx 2 , and we assume that the axion background field varies along only the x 3 = z spatial direction, 6 φ(η, x) = φ(η, z), the photon equations of motion admit the following approximate plane-wave solution [50]:
A 0 = A 3 = 0 (2) A σ (η, z) = A σ (η , z ) exp −iω(η − η ) + ik(z − z ) + iσ g φγ 2 ∆φ(η, z; η , z ) (ω = k),(3)
where ∆φ(η,
z; η , z ) ≡ φ(η, z) − φ(η , z )(4)
is the difference in axion phase between absorption at (η, z) and emission at (η , z ), and where we have defined the opposite-helicity transverse photon degrees of freedom
A σ ≡ 1 √ 2 (A 1 − iσA 2 ) (σ = ±1).(5)
This approximate solution holds in the regime where the axion field varies in space and time much more slowly than the photon field; see Appendix A. The leading effect of the opposite-sign corrections to the two helicity modes A σ shown in Eq. (3) is to cause a rotation of the linear polarization of the electromagnetic field by an angle ∆θ ∝ g φγ ∆φ, where ∆φ ≡ φ(η abs. , x abs. ) − φ(η emit , x emit ) is the difference of the axion field values at photon absorption and photon emission.
Explicitly, the electric field E ⊥ ≡ (E x , E y ) t in the plane transverse toẑ undergoes a clockwise rotation as viewed by an observer looking back toward the source of the photon (i.e., an observer directing their view in the (−ẑ)-direction):
E i ⊥ (η) = a(η ) a(η) 2 exp [−iω(η − η ) + ik(z − z )] R ij g φγ 2 ∆φ E j ⊥ (η ) (ω = k),(6)
where
R(θ) ≡ cos θ sin θ − sin θ cos θ ,(7)
where the redshift factor correctly accounts for the fact that radiation redshifts in an FLRW universe as ρ ∝ E 2 ∝ a −4 , and in this expression for the fields we have neglected terms ∝ g φγ ∂ η φ/ω 1; we also sum over repeated indices assuming a 3-metric equal to the identity. The rotation effect is independent of the frequency of the light.
It is worth reiterating that, from Eq. (6), the E ⊥ -field rotates through a angular excursion ∆θ = g φγ 2 ∆φ(η abs. , x abs. ; η emit ,
x emit ) = g φγ 2 C ds n µ ∂ µ φ = g φγ 2 [φ(η abs. , x abs. ) − φ(η emit , x emit )] ,(8)
where C is the path of the photon in spacetime from the point of emission to the point of absorption, and n µ is the null tangent vector to C. This form of the result makes clear that it is indeed a cumulative integrated effect along the whole path of the local derivative axion-photon coupling in Eq. (1). 7 We emphasize strongly that the net rotation effect is independent of the details of the axion field configuration along the photon path at all points between emission and absorption: the net rotation depends only on the initial and final axion field values [50]. It is immediate from this understanding of the effect that treating the net rotation angle arising from a photon passing through multiple different coherently oscillating axion patches as a stochastic process of random jitters in the polarization angle, as has been done in number of recent works, is incorrect; see further discussion in Sec. VI.
IV. SIMPLE TOY MODELS
In this section we demonstrate the impact of the polarization rotation effect elucidated in Sec. III in the context of a series of three related, simplified toy models designed to bear broad similarity to an observer detecting polarized photons emitted from the CMB surface of last scattering. The third model essentially gives all the physics underlying our results for the effect of a DM axion on the CMB.
7 Recall: − 1 4 d 4 x √ −g g φγ φFµν F µν = 1 2 d 4 x √ −g g φγ (∂µφ)Aν F µν . x y z ϕ ϑn e ϕê θ e 1ê 2 ψ ψ Q < 0, U = 0 Q > 0, U = 0 U > 0 , Q = 0 U < 0 , Q = 0
FIG. 1. Axis and angle conventions.
A. Minkowski spacetime, source at one instant of time
To avoid a number of the initially distracting complexities of FLRW spacetime, we will first work in Minkowski spacetime with a ≡ 1 in this subsection. First consider an observer at x obs. = (t, 0), receiving photons from a distant source localized on the fixed-time surface x source = (t , D(t, t )n), wheren ≡ sin ϑ cos ϕx + sin ϑ sin ϕŷ + cos ϑẑ is the direction from observer to source, and the distance D(t, t ) ≡ t − t is such that x obs. and x source are light-like separated. Suppose that in an observation time δt π/m φ , the observer receives from the source at random times within that interval a total of (N + M ) photons, with N of those photons having been emitted as linearly polarized along theê 1 -direction at the source, and M of the photons having been emitted as linearly polarized along thê e 2 -direction at the source. We takeê 1 ≡ cos ψê ϑ (n) + sin ψê ϕ (n) andê 2 ≡ − cos ψê ϕ (n) + sin ψê ϑ (n), wherê e ϑ (n) ≡ cos ϑ cos ϕx + cos ϑ sin ϕŷ − sin ϑẑ andê ϕ (n) ≡ − sin ϕx + cos ϕŷ; see Fig. 1. 8 We are agnostic here to the mechanism generating the polarization, but assume it to be an immutable, constant characteristic of the source (in particular, we assume the fractions of photons of each polarization in any sized sample of photons do not change).
Assume further that this entire setup occurs in a near-homogeneous background axion field φ(t, x) (i.e., |∂ t φ| |∇φ|); this implies that φ obs. (t) ≡ φ(t, 0) ≈ φ 0 cos(m φ t+α) and φ source (t, t ) ≡ φ(t , D(t, t )n) ≈ φ 0 cos(m φ t +β(t−t )), with α and β(t−t ) being some phases, in general different by O(π) because we will assume that D is much larger than either π/m φ or the larger axion field coherence length ∼ π/(m φ v). Note that, as written here, β(t − t ) = β(D(t − t )) is also supposed to capture additional phase variation not made explicit in the m φ t term, due to the small spatial gradients of the axion field; as a result of the assumed near-homogeneity,
|β(t + π/m φ − t ) − β(t − t )|
π. Suppose our observer has a polarization-sensitive detector with two sensors, oriented such that one sensor is sensitive to polarization along theê ϑ (n)-direction, and the other is sensitive along the orthogonalê ϕ (n)-direction. The observer determines the Stokes parameters (see Appendix B) characterizing the incoming photons by performing an incoherent sum over the individual photons received from directionn during the observation time δt:
I(n) = N i=1 I i + M j=1 I j , V (n) = 0,(9)Q(n) = N i=1 Q i + M j=1 Q j , U (n) = N i=1 U i + M j=1 U j ,(10)
where I i,j , Q i,j , and U i,j are the Stokes parameters as defined in Appendix B, computed with the electric fields of each individual photon. At the source, E i (t , Dn) ∝ê 1 for i = 1, . . ., N , while E j (t , Dn) ∝ê 2 for j = 1, . . ., M . However, we know from Sec. III that the linear polarizations of each individual photon will rotate in a (counter-)clockwise direction when ∆φ > 0 (< 0), as viewed by an observer looking in then-direction, due to the axion oscillation between photon emission and photon absorption:
E i (t, 0) = E 0ê1 cos g φγ 2 ∆φ − E 0ê2 sin g φγ 2 ∆φ i = 1, . . ., N(11)= E 0 cos ψ + g φγ 2 ∆φ ê ϑ (n) + E 0 sin ψ + g φγ 2 ∆φ ê ϕ (n) (12) E j (t, 0) = E 0ê2 cos g φγ 2 ∆φ + E 0ê1 sin g φγ 2 ∆φ j = 1, . . ., M(13)
= −E 0 cos ψ + g φγ 2 ∆φ ê ϕ (n) + E 0 sin ψ + g φγ 2 ∆φ ê ϑ (n) (14) where ∆φ ≡ φ 0 [cos(m φ t + α) − cos(m φ t + β(t − t ))], and E 0 is the field amplitude at the observer, which we assume to be the same for every photon. Therefore, the observer will measure 9
I(n) = E 2 0 (N + M ), V (n) = 0,(15)(Q ± iU )(n) = I exp ±2i ψ + g φγ 2 ∆φ = exp ±2i g φγ 2 ∆φ (Q ± iU ) 0 (n),(16)
where we have defined ≡ (N − M )/(N + M ) as an intrinsic polarization asymmetry of the source, and (Q ± iU ) 0 (n) to be the values that would be measured in the axion decoupling limit g φγ → 0. From this setup we can immediately observe one interesting signal: suppose we made two sets of measurements of I(n), Q(n), U (n) at times t 1 and t 2 , separated by something on the order of |t 1 − t 2 | ∼ O(π/m φ ), using integration durations δt π/m φ for each measurement. I(n) is the same at both times and, by virtue of the fact that β varies slowly on scales ∼ π/m φ , we can write
∆φ 2 − ∆φ 1 ≈ φ 0 [cos(m φ t 2 + α) − cos(m φ t 1 + α)] .(17)
Therefore, Q(n) and U (n) will be observed to rotate into each other as a function of t 2 (for fixed t 1 ) as though the detector angle ψ were being oscillated with a period ∼ 2π/m φ and excursion amplitude g φγ φ 0 /2. In other words, we will see an axion-induced 'AC oscillation' of the Q(n) and U (n) Stokes parameters measured to be emitted from a light-like separated source located on a fixed-time surface in the past, assuming that the source properties are otherwise fixed over the relevant timescales, and the source has some intrinsic net polarization asymmetry.
B. Minkowski spacetime, source smeared in time
The toy model of Sec. IV A fails to account for a situation in which the source is not localized at one point in time on the past lightcone of the observer; we extend the model to that case in this subsection.
We continue in Minkowski spacetime with a = 1. Suppose now that our source is otherwise unchanged from the toy model of Sec. IV A, but that there is instead a smeared region along the past lightcone of the observer which contributes all the photons arriving at the observer within a time interval δt around the time t. That is, instead of being localized at x = rn = Dn at one fixed instant of time t , it is rather the case that our source has some unit-normalized probability density function (PDF) g(t ) to be spatially located at x k = r kn = D kn = (t − t k )n at the time t = t k when it emits each of the k = 1, . . ., N + M photons that the observer receives in the δt-interval around time t. 10 We will assume g(t ) has non-negligible support only in the regiont ∈ [t − T /2, t + T /2] where t − t T π/m φ : i.e., the duration T of the photon emission / source smearing is small compared to the distance to the source, but large compared to the axion period. 11 We will additionally assume that 2π|∂ t g(t)| m φ , so that g(t) is fairly constant on one axion period.
In this second case, the expressions for the observer-measured Stokes parameters are modified to account for fact that each individual photon has its own value of ∆φ:
I = E 2 0 (N + M ), V = 0,(18)(Q ± iU )(n) = E 2 0 N i=1 exp ±2i ψ + g φγ 2 ∆φ i − E 2 0 M j=1 exp ±2i ψ + g φγ 2 ∆φ j ,(19)
9 Note that a right-handed rotation of the axes (ê ϑ (n),êϕ(n)) by an angle ξ (i.e., a clockwise rotation of the axes (ê ϑ (n),êϕ(n)) as viewed in the right panel of Fig. 1) would send ψ → ψ − ξ, with the resulting transformation Q ± iU → e ∓2iξ (Q ± iU ). 10 Note that (x k − x observer ) 2 = (t − t k ) 2 − D 2 k = 0; the photons are emitted at slightly different times and distances, such that they all arrive at the same instant at the observer. 11 Had we instead assumed the hierarchy t − t π/m φ T , such that the axion field oscillates slowly during the period of photon emission, we would recover the result of Sec. IV A.
where
∆φ k ≡ ∆φ(t, t k ) ≡ φ 0 [cos(m φ t + α) − cos(m φ t k + β(t − t k ))] ,(20)
with t k the emission time for photon k. For fixed t, and assuming that we are in the limit where N and M are large enough that we may pass to the continuum limit (i.e., we have large photon statistics), we have
(Q ± iU )(n) = I dt g(t ) exp ±2i ψ + g φγ 2 ∆φ(t,t ) .(21)
Now,
2ψ + g φγ ∆φ(t,t ) = 2ψ + g φγ φ 0 cos(m φ t + α) − g φγ φ 0 cos(m φt + β(t −t )); (22) although β(t−t ) can vary by O(π) on timescales ∼ T , we still have the scale separation |m φ T | |β(t−t −T /2)−β(t− t + T /2)|. We can thus separate the integration domain into a large number of subdomainst ∈ [t n − δt /2,t n + δt /2], where δt = 2π/m φ , such that |β(t −t n − δt /2) − β(t −t n + δt /2)|
π, so that we may take β(t −t ) ≈ β(t −t n ) ≡ β n to be independent oft in each subdomain. Moreover, we have assumed that g(t ) varies slowly on the timescale δt , and so may approximate g(t ) ≈ g(t n ) ≡ g n in each subdomain:
(Q ± iU )(n) ≈ I n g n t n +δt /2 t n −δt /2 dt exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) − g φγ 2 φ 0 cos(m φt + β n ) (23) = I n g n δt exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) J 0 (g φγ φ 0 ) (24) = I dt g(t ) exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) J 0 (g φγ φ 0 ) (25) = J 0 (g φγ φ 0 ) exp ±2i g φγ 2 φ 0 cos(m φ t + α) (Q ± iU ) 0 (n),(26)
where (Q ± iU ) 0 are as before the values that would be measured in the axion decoupling limit g φγ → 0. We also used that g(t) is a unit-normalized PDF, and the integral representation of the Bessel function of the first kind, J 0 (x):
π −π dx exp [iA + iB cos(x + δ)] = 2πe iA J 0 (B). 12 We thus see again that there will be an 'AC effect' ascribable to the axion field oscillation at the observer: a time-varying rotation of Q(n) and U (n) into each other as though the detector were being rotated about then-axis through an angular excursion of amplitude g φγ φ 0 /2 at period 2π/m φ .
We also now find a new 'washout' effect ascribable to the axion field oscillating rapidly at the source: the 'smearing' of the source in time leads to a smearing of polarizations measured in any arbitrary fixed observer frame (provided the source is not varying), leading to an effective reduction in the polarized fraction interpreted to be produced at the source: → J 0 (g φγ φ 0 ). We note that this second effect is only valid in the regime in which the emission does not occur during a period rapid compared to the axion oscillation frequency (i.e., that we have T 2π/m φ ); in the opposite limit (T 2π/m φ ), the results of Sec. IV A would apply instead. Finally, we emphasize that, in this regime, there is no unsuppressed leading-order rotation of the polarization axis that arises from the axion field behaviour at the source [4,10], in sharp contrast to the usual treatment of static 'cosmic birefringence' which assumes a setup closer to our initial case from Sec. IV A, with T 2π/m φ . We do however note that we have in our above derivation neglected the 'last partially uncancelled oscillation' which would lead to a residual overall rotation from source to observer that is power-law suppressed by ∼ (m φ T ) −1 compared to the naïve estimate of an O(g φγ φ 0 ) rotation (note that we disagree on the size of the suppression compared to Ref. [10], which claims an exponential ∼ e −m φ T suppression).
If we imagined the extension of the preceding analysis to a collection of sources at different angular positionsn with respect to the observer, and all beaming photons toward the observer, and analyzed how the polarization from each location would be impacted, our conclusions would hold point-by-point: there is a Q(n) ↔ U (n) oscillation driven by the axion field at the location of the observer, and a second-order reduction in the Q(n) and U (n) values arising from the axion field oscillating many times during the emission from the smeared source.
C. FLRW spacetime, source smeared in time
Consider the third and final toy model, which will be taken to be similar to the model in Sec. IV B, with two exceptions: (a) the analysis will be conducted in an FLRW spacetime instead of in Minkowski spacetime, and (b) instead of assuming near-homogeneity of the axion field across our entire setup, the axion field at the source will be taken to be approximately the axion field as it would be at the epoch of CMB last scattering, while the axion field at the observer will be taken to be the axion field as it would be locally in the Milky Way (MW).
Since the analysis of Sec. III (or, more specifically, the detailed derivation in Appendix A) indicates that the polarization rotation effect operates in FLRW the same as it would in Minkowski, (a) is an almost trivial change: we simply locate the observer at x obs. = (η, 0) and have the source for each of the (N + M ) photons located at x source, n = (η n , D nn ), where D n ≡ D(η, η n ) ≡ η − η n is the co-moving distance between the observer and source, and η is conformal time. We assume that η n is drawn from the distribution g(z(η )) with g(z) the visibility function, which we will take to be localized around z = z * , where z * is the redshift of last scattering. 13 To evaluate the axion field at the source, we consider the axion field equation, Eq. (A1), ignoring the source term on the RHS, and assume that spatial gradients are irrelevant: |∇φ|/a |∂ t φ|. Moreover, we evaluate the axion equation of motion in the FLRW spacetime describing our universe:φ
(t) ≡ [a(t)] 3/2 φ(t)(27)∂ 2 tφ (t) + m 2 φ 1 + 6π m 2 φ M 2 Pl. p total φ (t) = 0,(28)
where we definedφ(t) to factor off the dominant scale-factor dependence of φ(t), and we used the Friedmann equations to trade out derivatives of a for the total pressure from all sources:
p total = p matter + p Λ + p R = p Λ + p R (29) = −ρ Λ + 1 3 ρ R (30) = ρ c −Ω 0 Λ + Ω 0 R 3 a −4 ,(31)
where Ω 0 i ≡ ρ i (t 0 )/ρ c are the present-day fractional energy densities. But then using Ω 0
R = Ω 0 M /(1 + z eq. ), we have 6π m 2 φ M 2 Pl. p total = 3H 0 2m φ 2 −Ω 0 Λ + Ω 0 M 3 (1 + z) 4 1 + z eq. .(32)
Using the latest Planck (TT,TE,EE+lowE+lensing+BAO) results [115], Ω 0 Λ = 0.6889 (56), Ω 0 M = 0.3111(56), z eq. = 3387 (21), z * = 1089.89 (21), and H 0 = 67.77 (42)
km s −1 Mpc −1 , we find that 6π m 2 φ M 2 Pl. p total ≈ −3.3 × 10 −24 + 2.1 × 10 −16 1 + z 1 + z * 4 × m φ 10 −21 eV −2 .(33)
As we will be assuming that z ∼ z * within ∼ 20% in this section, the [ · · · ] bracket in Eq. (28) is negligibly different from 1, and we can write
φ(t ,n) ≈ φ * (n) 1 + z 1 + z * 3/2 cos (m φ t + β) ,(34)
where φ * (n) is a normalization that in principle we allow to vary across the sky, β is a phase, and z = z(t ) is the redshift. If the axion constitutes a fraction κ of all the cold dark matter at the average redshift of decoupling, and we ignore the small matter perturbations at decoupling which would give rise to variations of φ * (n) across the sky, we have
m 2 φ φ 2 * = κρ c Ω 0 c (1 + z * ) 3 = (3κ/8π)M 2 Pl. Ω 0 c h 2 (H 0 /h) 2 (1 + z * ) 3 (35) ⇒ φ * (n) = 1.1 × 10 11 GeV × κ 1/2 × m φ 10 −21 eV −1 × Ω 0 c h 2 0.11933 1/2 ,(36)
where we used the Planck result Ω 0 c h 2 = 0.11933(91) [115]. In evaluating the axion field at the observer, we will use the axion field local to us in the MW:
φ(t) ≈ φ 0 cos(m φ t + α(t)),(37)
where α(t) is a phase approximately constant on timescales shorter than the axion coherence time t coh. ∼ 2π/(m φ v 2 0 ) with v 0 the MW virial velocity, and we will take φ 0 to be normalized such that the axion field constitutes a fraction κ of the local DM density, ρ 0 ≈ 0.3 GeV/cm 3 :
m 2 φ φ 2 0 = κρ 0 (38) ⇒ φ 0 = 1.5 × 10 9 GeV × κ 1/2 × m φ 10 −21 eV −1 × ρ 0 0.3 GeV/cm 3 1/2 .(39)
The reader will note that the axion field normalizations given at Eqs. (36) and (39) differ by only two orders of magnitude, which is understood by virtue of the fact that ρ ∝ φ 2 , our galaxy is locally overdense compared to the present-day critical density (ρ c ∼ 4.8 × 10 −6 GeV/cm 3 ) by roughly 5 orders of magnitude, and the dark matter in the universe around last scattering was ∼ (1 + z * ) 3 ∼ 10 9 times denser that the present-day critical density; the fact that these two normalizations, φ * and φ 0 , are not wildly dissimilar will be important.
Most of the analysis of Sec. IV B now goes through as before, with the exception that we need to additionally note that in a time period
δt = 2π/m φ around z = z * ,z ≡ z(t ) is such that we have δz ≡ z(t + δt ) − z(t )
z, so that at the step analogous to the subdivision of the integration domain just above Eq. (23), we may write 1 +z ≈ 1 +z n in each subdomain: 14
(Q ± iU )(n) ≈ I dt g(t ) exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) − g φγ 2 φ * (n) 1 +z 1 + z * 3/2 cos(m φt + β(t,t )) (40) ≈ I n g n t n +δt /2 t n −δt /2 dt exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) − g φγ 2 φ * (n) 1 +z n 1 + z * 3/2 cos(m φt + β n )(41)= I exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) n g n δt J 0 g φγ φ * (n) 1 +z n 1 + z * 3/2(42)= I exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) dt g(t ) J 0 g φγ φ * (n) 1 +z 1 + z * 3/2 (43) = I exp ±2i ψ + g φγ 2 φ 0 cos(m φ t + α) dz g(z) J 0 g φγ φ * (n) 1 +z 1 + z * 3/2(44)= J 0 (g φγ φ * (n)) exp ±2i g φγ 2 φ 0 cos(m φ t + α) (Q ± iU ) 0 (n),(45)
where (Q ± iU ) 0 are as before the values that would be measured in the axion decoupling limit g φγ → 0, and where for convenience we defined g(t ) ≡ g(z)(dz/dt ), as well as a visibility-function-weighted average axion field amplitude at decoupling:
J 0 (g φγ φ * (n)) ≡ dz g(z) J 0 g φγ φ * (n) 1 +z 1 + z * 3/2 ≈ J 0 (g φγ φ * (n)) ,(46)
where the latter approximate equality holds since g(z) is fairly strongly peaked aroundz = z * . Within the context of this most-developed toy model, we see the same overall effects as in Sec. IV B, with the magnitude of the AC oscillation effect now explicitly seen to be governed by the amplitude of the local axion field value, and the magnitude of the washout effect controlled by a visibility-function-weighted average of the axion field amplitude at last scattering. Finally, in the limit where g φγ φ 0, * 1, so that all the axion effects are small, Eq. (45) immediately tells us that the effect of φ * manifests itself only quadratically in the reduction of the Stokes Q(n) and U (n) parameter amplitudes (J 0 (x) ∼ 1 − x 2 /4 for small x), while the effect of φ 0 appears linearly in the amplitude of the oscillating Q(n) ↔ U (n) rotation at period 2π/m φ . Therefore, despite the hierarchy g φγ φ 0 ∼ 10 −2 g φγ φ * 1, it is incumbent upon us in what follows to keep track of both the washout and the AC effect, as it is not a priori clear how the magnitudes of (g φγ φ * ) 2 and g φγ φ 0 will compare in the relevant part of parameter space. In other words, even though the axion field is larger at last scattering than today, the washout effect is quadratic while the AC effect is linear in the axion field, so both effects could be important.
V. CMB OBSERVABLES AND LIMITS
By construction, the toy model results of Sec. IV C give a good approximation to the effects to be expected in the realistic CMB case: a polarization power washout on all (pristine; see comments below) scales, and an AC oscillation effect at period 2π/m φ , with the size of the effects given by Eq. (45). As the polarization rotation effect we are considering impacts only the propagation, and not (at least at leading order) the production, of polarized photons, it is not necessary at the level of accuracy required for the present work to modify any publicly available CMB codes (e.g., cmbfast [113], camb [118]) to explicitly track the washout effect through the decoupling epoch. 15 We judge that simply taking the usual output of unmodified CMB codes, and transforming the output polarizations in the manner indicated by Eq. (45) is sufficient, with one exception discussed below. Furthermore, note that the visibility function g(z) appears only in the average φ * , and our results are thus insensitive to its exact shape, so long as it is strongly peaked near z ∼ z * , but slowly varying on the timescale of one axion oscillation; in the relevant region of axion mass parameter space, this is always the case with any physically reasonable g(z).
While the washout effect can be constrained using current publicly available CMB power spectrum data (from, e.g., Planck [115]), a correct treatment of the AC oscillation effect necessarily requires access to non-public time-series CMB polarization data, and demands a more rigorous treatment of experimental systematics than is appropriate for the scope of the present work. Therefore, while a combined approach considering both effects simultaneously would be preferable, we will instead present two independent estimates for the reach of each effect: (a) the current reach for a washout-only search using the sensitivity of current Planck [115] CMB power spectra, as well as a projection for future reach; and (b) the approximate reach attainable by a detailed time-series analysis as informed by the statistical power of current DC searches for cosmic birefringence, as well as a minimal projection for possible future reach.
A. Washout of polarization
For the purposes of the washout analysis, we neglect the local oscillating axion field, and consider the washout effects via their impact on the CMB T T , T E, and EE auto-and cross-correlation power spectra. 16 In harmonic-space all-sky CMB analysis (see, e.g., Ref. [122]), the observed temperature maps T (n) are decomposed over a basis of scalar spherical harmonics, and the observed Q(n) and U (n) maps are decomposed over a basis of spin-weight-2 spherical harmonics:
T (n) ≡ l,m a T,lm Y lm (n)(47)(Q ± iU )(n) ≡ l,m a ±2,lm · ±2 Y lm (n),(48)
from which the E and B modes are constructed as [122] a E,lm ≡ − 1 2 (a +2,lm + a −2,lm ) 15 In connection with any such approach however, we do caution the reader that some care may be required when the axion field oscillates many (thousands of) times during the decoupling epoch, as an accurate evaluation of the time integrals over the polarization source functions (see, e.g., Ref. [113], and modifications in Refs. [74,87,107,108]) do require a much finer time-sampling than would ordinarily be required in the case of standard ΛCDM, in order not to alias over the axion oscillations. 16 In considering full-experiment combined maps produced by, e.g., Planck, we will be implicitly averaging over the local axion phase evolution too, as each point on the sky was revisited multiple times over the multi-year duration of the Planck experiment (see, e.g., footnote 34 of Ref. [121]) and such repeat observations were obviously combined without making any corrections for an actual on-the-sky variation of the CMB polarization arising in the manner we consider in this work. However, since the local axion field amplitude is noticeably smaller than the field amplitude at decoupling, φ 0 ∼ 10 −2 φ * , this second averaging / washout effect should not dramatically alter the prediction for the polarization reduction which we obtain considering only the axion field behaviour at the decoupling epoch. Reach for current and future searches for the washout and AC oscillation effects induced by an axion field of mass m φ constituting all of the dark matter (κ = 1, in the notation of Sec. IV C). We find that the green shaded region is excluded on the grounds that a reduction in the predicted polarization amplitude of at least 0.58% is in conflict at 95% confidence with Planck measurements [115]; see Sec. V A 2. The dashed green line indicates how the bound would improve if the 95%-confidence upper limit on the fractional reduction were improved to 0.082%, our projection for the sensitivity if all three of the power spectra C T T,T E,EE l were measured to the cosmic-variance limit out to lmax = 3000 (assuming the Planck sky coverage, f sky = 0.577); see Sec. V A 3. The dash-dotted red line indicates where the AC oscillation effect has an amplitude of rotation of at least 0.1 • ; based on the statistical power of the DC search conducted in Ref. [105], we estimate that, at a minimum, this region may be excludable at 95% confidence given current sensitivity of the Planck measurements, although this would require a dedicated time-series analysis to confirm. The dotted red line indicates the reach of the AC oscillation search if a 0.01 • rotation amplitude were detectable, as an example of future potential. For comparison, the reach for the AC search exceeds the current (respectively, future) washout reach for a rotation amplitude of 0.059 • (0.022 • ). Note that all of our bounds and projections are upward-sloping diagonal lines g φγ ∝ m φ because all the rotation effects are ∝ g φγ φ0, * , while m 2 φ φ 2 0, * ∝ const.; see Eqs. (35) and (38). The vertical grey line marked 'Lyman-α' denotes the smallest axion mass claimed to be consistent at 95% confidence with observed small-scale structure in the Lyman-α forest [112]. The graduated shaded grey region labelled 'small-scale structure' is where the axion de Broglie wavelength is large enough to cause tension with observed galactic small-scale structure [16]. The horizontal grey shaded region is the current 95%-confidence exclusion limit from CAST [119]. The horizontal dashed grey line is a limit arising from conversion to gamma-rays in galactic magnetic fields of axions emitted by SN1987A [120] (we take their '10.8M , Jansson and Farrar' limit).
a B,lm ≡ i 2 (a +2,lm − a −2,lm ) ,(49)
with the observed C XY,l auto-and cross-correlation power spectra being defined as
C obs. XY,l ≡ 1 2l + 1 m a * X,lm a Y,lm ,(51)
for X, Y ∈ {T, E, B}. Up to the limitations imposed by cosmic variance, the observed C obs. XY,l are compared to the true C XY,l ≡ C obs.
XY,l (with · · · denoting the average over the ensemble of possible statistical realizations of the sky), which are a prediction of the cosmology of interest. From this, it is clear that an axion-induced reduction in the amplitude of Q(n) and U (n) point-by-point on the sky will result in a reduction in the amplitudes of the C T E,EE,BB power spectra compared to the values expected for these quantities in a base ΛCDM cosmology with the amplitude of C T T held fixed. Quantitatively, the relevant effects are (assuming that φ * (n) ≈ φ * is independent ofn, except for small fluctuations):
(Q ± iU )(n) → J 0 (g φγ φ * ) (Q ± iU )(n) ⇒ a {E,B},lm → J 0 (g φγ φ * ) a {E,B},lm ,(52)
implying that 17
C T E,l → J 0 (g φγ φ * ) C T E,l ,(53)C EE,l → [J 0 (g φγ φ * )] 2 C EE,l ,(54)
where φ * is as defined at Eq. (46). For values of g φγ φ * such that J 0 (g φγ φ * ) = 1 − δ for some δ 1, it follows that the fractional reduction in the EE power spectrum, |(∆C EE,l )/C EE,l | ∼ 2δ, is approximately twice the fractional reduction in the T E power spectrum, |(∆C T E,l )/C T E,l | ∼ δ.
Note, however, that the approach outlined above is only correct for those 'pristine' CMB multipoles that have not been significantly re-processed since the decoupling epoch; this prescription fails at low-l, since the polarization power for l 20 is significantly contaminated by polarization generated by rescattering after reionization [125]. We will therefore mostly confine our attention to the high-l multipoles (l 50) of the CMB power spectra, which reflect the polarization imprinted at the last scattering surface.
In order to estimate the current constraining power of CMB data on the washout effect, we make use of the fullmission Planck [115] results for the measured power spectra, along with the current best-fit ΛCDM model (specifically, the Planck TT+TE+EE+lowE+lensing fit [115]).
Current bound: approximate analysis
We first perform an approximate analysis, considering only 50 ≤ l ≤ l max to avoid the low-l reionization bump (l max = 1996 is the highest available Planck polarization multipole [115]). In this multipole range, there is within ΛCDM models a single undetermined overall normalization, A ∝ A s e −2τ , which governs the C {T T,T E,EE},l spectra [116,117]; this is the most obvious ΛCDM parameter combination whose variation could partially compensate the polarization washout effect. 18 Therefore, we profile over the joint overall normalization A of the three power spectra in performing a one-parameter profile likelihood ratio test for the alternative hypothesis that assumes the existence of a non-zero polarization reduction parameter r ≡ g φγ φ * (m φ ) , with φ * (m φ ) ≈ φ * (m φ ) as in Eq. (36), against the null-hypothesis of no polarization reduction, r = 0. Quantitatively, we define 19
−2 ln L(r ≡ g φγ φ * (m φ ) ; A) ≡ XY ∈{T T,T E,EE} l=lmax l=50 C observed XY,l − A · f XY (r) · C theory XY,l u(C observed XY,l ) 2 (55) where f XY (r) ≡ 1 XY = T T J 0 (r) XY = T E [J 0 (r)] 2 XY = EE ,(56)
and where l max = 1996, C observed XY,l is the Planck observed power, C theory XY,l is the best-fit (TT+TE+EE+lowE+lensing) prediction of the base ΛCDM model, and u(C observed XY,l ) is the uncertainty in the C observed XY,l (all as quoted by Ref. [115]). We form the test-statistic
∆χ 2 (r) ≡ −2 ln L(r;Â(r)) L(r = 0;Â(r = 0)) ,(57)
17 Note that since the temperature and polarization maps are imperfectly correlated [123,124], it is appropriate to treat the effects at the lower of the power spectra, and in particular to incorporate cosmic-variance uncertainties on all the power spectra. 18 To develop an intuitive understanding for why this compensation can occur, take the crude approximation u(C l ) ∝ C l ; each term in the sum over l in Eq. (55) would then take the approximate form
∼ (1 − A) 2 + (1 − (1 − δ)A) 2 + 1 − (1 − δ) 2 A 2
where δ ≈ r 2 /4 for small r, if exact agreement between the observed and theory C l is additionally crudely assumed. If A = 1 is fixed, this term contributes an amount ∼ 5δ 2 to the sum; on the other hand, if A is profiled over and allowed to float to A = 1 + r, then the contribution to the sum is only ∼ 2δ 2 . Failure to profile over A would result in setting a bound on r a factor of ∼ (5/2) 1/4 ≈ 1.25 too strong (although note that this heuristic argument should not be taken seriously in any quantitative sense). 19 We make the approximation that we may ignore any off-diagonal elements in the covariance matrix in constructing this likelihood.
where hatted quantities are the best-fit values of parameters given a fixed value of r. 20 The one-parameter 95%confidence limit is then obtained as ∆χ 2 (r 95 ) = 3.84; or, expressed in terms of a bound on g φγ φ * (m φ ):
1 − J 0 (g φγ φ * (m φ )) ≤ 4.9 × 10 −3 (i.e., a 0.49% reduction in Q ± iU ) [approximate] (58) ⇒ g φγ 1.3 × 10 −12 GeV −1 × m φ 10 −21 eV × κ × Ω 0 c h 2 0.11933 −1/2 ,
[approximate] (59) where in the second line we have used the expansion of the Bessel function as the argument is necessarily small in the vicinity of the limit.
Current bound: full analysis
The exclusion limit at Eq. (59) from the analysis procedure of Sec. V A 1 is approximate. In order to give a rigorous exclusion limit from the current Planck data, we undertook a more complete analysis in which we profiled over the full six-dimensional ΛCDM base parameter set Θ ≡ (Ω b h 2 , Ω c h 2 , 100 θ mc , τ, ln(10 10 A s ), n s ). We form the test-statistic for the one-parameter profile likelihood ratio test on r as 21
∆χ 2 (r) ≡ −2 ln L(r;Θ(r)) L(r = 0;Θ(r = 0)) ,(60)
utilizing camb [118] to compute the necessary power spectra, and sampling the ΛCDM parameters to find the best fit parameters using Monte Carlo (MC) techniques. As this is computationally intensive, we were informed in our choices of the values of r to investigate by the results of our approximate analysis of Sec. V A 1. Moreover, excluding the low-l multipoles completely in this full analysis would lead to an almost unconstrained parameter degeneracy between A s and τ . In this part of our analysis we therefore retain all multipoles down to l = 2, but in order to not incorrectly wash out the polarization of the non-pristine multipoles at l 20, we replace the function f XY (r) as defined at Eq. (56)
by f XY (r) → (1 − h(l)) + f XY (r)h(l), where h(l) ≡ 1 2 [1 + tanh((l − l 0 )/δl)]
with l 0 = 20 and δl = 5; this approximate choice only turns the polarization washout effect on for l 20, and it breaks the parameter degeneracy in the same sense that the low-l reionization bump usually breaks this parameter degeneracy in base ΛCDM models.
This more rigorous analysis procedure yields a one-parameter 95%-confidence exclusion bound given the current Planck data of 1 − J 0 (g φγ φ * (m φ )) ≤ 5.8 × 10 −3 (i.e., a 0.58% reduction in Q ± iU ) (61)
⇒ g φγ 1.4 × 10 −12 GeV −1 × m φ 10 −21 eV × κ × Ω 0 c h 2 0.11933 −1/2 ;(62)
which is a weakening of the limit on g φγ at fixed m φ by 9% as compared to the approximate bound shown in Eq. (59). Eq. (62) gives the bound shown in the green shaded region in Fig. 2. Our full analysis thus validates the approximate approach of Sec. V A 1 within 10%, and also definitively settles the question as to whether any multi-parameter combinations of base ΛCDM parameters can completely mimic or compensate our effect: no such combinations exist.
Future reach
In order to give a perhaps optimistic estimate where this search could reach in the future, we assume that the T T , T E, and EE power spectra can all be measured at the cosmic-variance limit from l = 50 to l max = 3000 with the same sky coverage as Planck, f sky = 0.577 [115]. At large l, the results of Ref. [126] indicate that the likelihood for observing the dataĈ l ≡ (Ĉ T T l ,Ĉ T E l ,Ĉ EE l ) given a model predicting C l ≡ (C T T l , C T E l , C EE l ) is approximately a multivariate Gaussian with the covariance matrix 22
Σ l ≈ 1 (2l + 1)f sky 2(C T T l ) 2 2C T T l C T E l 2(C T E l ) 2 2C T T l C T E l C T T l C EE l + (C T E l ) 2 2C T E l C EE l 2(C T E l ) 2 2C T E l C EE l 2(C EE l ) 2 ,(63)
in the limit where cosmic variance dominates noise. For our estimate for future sensitivity, we revert to the approximate analysis procedure of Sec. V A 1, having verified in Sec. V A 2 that it is accurate within 10% for setting exclusion bounds. As we are setting a future projected exclusion bound on r ≡ g φγ φ * (m φ ) , we take as our 'data' the predictions of the Planck TT+TE+EE+lowE+lensing best fit [115] base ΛCDM model assuming no polarization washout:Ĉ l ≡ (C l ) Planck ΛCDM best fit (which we re-compute using camb [118]). 23 The 'theory' we fit to these data, C l , is taken to be the modified ΛCDM prediction with washout that was used at Eqs. (55) and (56):
C XY l ≡ A · f XY (r) · (C XY l )
Planck ΛCDM best fit . Our likelihood function for the one-parameter profile likelihood ratio test on r is thus
−2 ln L(r ≡ g φγ φ * (m φ ) ; A) ≡ lmax l=50 ∆ T l Σ −1 l ∆ l ,(64)
where
∆ l ≡ C l −Ĉ l ⇒ ∆ XY l ≡ (C XY l ) Planck ΛCDM best fit · [A · f XY (r) − 1] .(65)
Performing the same profile likelihood ratio test as contemplated at Eq. (57), we find that our estimate for a future 95%-confidence exclusion bound under these assumptions is given by
1 − J 0 (g φγ φ * (m φ )) ≤ 8.2 × 10 −4 (i.e., a 0.082% reduction in Q ± iU ),(66)⇒ g φγ 5.1 × 10 −13 GeV −1 × m φ 10 −21 eV × κ × Ω 0 c h 2 0.11933 −1/2 ,(67)
which is a factor of ∼ √ 7 better than the current limit given at Eq. (62). The green dashed line in Fig. 2 shows the result at Eq. (67). Of course, if higher multipoles can be probed at the cosmic-variance limit, this reach could potentially be improved further; however, significant foregrounds exist for the temperature maps above l ∼ 3000 (see, e.g., Ref. [128]), so sensitivity estimates are more complicated to make.
Further discussion
In all of the above, we have neglected the small fluctuations in the value of φ * (n) across different locations in the sky owing to fluctuations in the DM density; these would induce only a very small (fractionally ∼ 10 −5 ) modulation on top of the uniform amount by which the amplitudes of C T E,EE,BB are reduced from their base ΛCDM predicted values. Additionally, the approximation φ * ≈ φ * is valid to within (at worst) 10%, as the visibility function is sufficiently fairly sharply peaked around z ∼ z * .
Finally, as we noted in Sec. IV C, there is no unsuppressed net (DC) rotation of the polarization at any point on the sky induced by the axion field at last scattering; there will only be a residual 'last uncancelled period' effect (i.e., an incomplete cancellation of the positive and negative polarization rotation angle deviations averaged over the CMB decoupling epoch), and this will be suppressed by a factor of ∼ ∆t osc. /∆t decoupling 10 −5 . Thus, the usual birefringence searches for a conversion of CMB polarization E modes into B modes are not very useful to search for a dark-matter axion since the field oscillates rapidly at the decoupling epoch; see also comments in this regard in Refs. [107,108]. Of course, there is still an overall AC polarization rotation due to the late-time effect of the axion field today (at the position of the detector). We analyze this in the next subsection. 22 We have also checked that may approximate the covariance matrix by its diagonal entries with essentially negligible impact on the estimated future bound. Note however that, since the T E-T E diagonal entry of Σ l is not just proportional to (C T E l ) 2 , one must still always still run, e.g., camb [118] to obtain an accurate value for the ratio C T T
l C EE l /(C T E l ) 2 .
In considering the AC oscillation effect, we ignore the power reduction from the previous subsection. Searches for the Q(n) ↔ U (n) oscillation effect in Eq. (45) require dedicated analyses to be performed by CMB experimental collaborations as the real on-the-sky variation of the polarization pattern on timescales short or comparable (∼ a few hours to a few years) to the total observation times (∼ months to years) demands consideration of non-public time-series data; such an analysis is beyond the intended scope of this work.
However, we can provide a rough, minimal estimate for the reach of such a search. Most recently, Planck [105] has performed an analysis looking for an all-sky DC rotation of polarization that would give rise to a rotation of E modes into B modes, and have quoted null results (within uncertainties) for such searches that have statistical 68% uncertainties at the 0.05 • level on the all-sky rotation angle [105]; see, e.g., Refs. [102,103] for older analyses with lower precision. The systematic uncertainties on such searches are much larger, at the 0.3 • level [105], but are dominated by uncertainty in the absolute calibration of the relative orientation of the bolometers [105] (but see, e.g., Ref. [129] for possible improvements in absolute calibration possible for ground-based detectors), which is irrelevant for an AC oscillation search so long as a fixed reference frame for the experiment can be maintained.
We estimate that a search in, e.g., the Planck time-series data for an oscillation signal should be able to resolve an oscillation with an amplitude roughly approaching the statistical power of the searches for these all-sky DC effects, although we have not considered in detail the impact of possible confounding systematic effects for such a search. Based on this estimate, we assume a limit on the amplitude of the AC oscillation (g φγ φ 0 /2) of 0.1 • , at least, would be attainable at 95% confidence in an analysis of currently existing time-series data, leading to
g φγ φ 0 ≤ 3.5 × 10 −3 (68) ⇒ g φγ ≤ 2.3 × 10 −12 GeV −1 × m φ 10 −21 eV × κ × ρ 0 0.3 GeV/cm 3 −1/2 ;(69)
this gives the dash-dotted red line in Fig. 2. As an example of the possible reach of this technique, we consider the impact of improving the sensitivity to the AC oscillation amplitude to 0.01 • (g φγ φ 0 ≤ 3.5 × 10 −4 ); since the amplitude of the rotation ∝ g φγ , this would correspond to the factor-of-10 improvement in the coupling reach compared to Eq. (69) that is shown with the dotted red line in Fig. 2. For comparison, the reach for the AC search exceeds the current (respectively, future) washout reach for a rotation amplitude of 0.059 • (0.022 • ). Note that with the minimal assumed sensitivity of the AC oscillation effect, both the washout and AC effects have similar reach, but since the AC oscillation effect is proportional to g φγ , while the washout effect is proportional to √ g φγ and is almost cosmic-variance limited already, the AC effect holds more promise for increased future reach.
In particular, we do not attempt to estimate the ultimate reach of this technique here, though it could possibly be significantly better than the reach shown in Fig. 2. The minimal reach estimate we have used comes from what is already possible for measuring the static, absolute value of the polarization. However, the effect we are looking for should be easier to measure since many of the limitations of the static search do not apply to a time-oscillating signal.
While we have based our reach estimates here on Planck results, searches for the AC effect are of course possible with time-series data from any of the existing or proposed ground-based, polarization-sensitive CMB experiments (e.g., BICEP and the Keck Array [130][131][132], ACTpol [133], SPTpol [134], POLARBEAR-2 [135], Simons Observatory [136], etc.). 24 A dedicated experiment built to search for oscillating CMB polarization could also potentially have a significantly greater sensitivity to the AC effect than we plot in Fig. 2; we defer a detailed examination of this point to future work.
Aliasing and averaging effects will complicate setting experimental bounds for sufficiently short axion periods (high m φ ) compared to, respectively, the interval between successive measurements of the same point on the sky, and the time for 'one observation' during the overall integration time for any experiment. Although aliasing effects render an experiment blind to certain specific axion masses (i.e., frequencies), they do not generally imply a loss of sensitivity at other masses; they are also mitigated against by the existence of multiple CMB polarization experiments which all have distinct survey strategies. Moreover, with respect to the averaging effect, note that by 'one observation' we do not mean the time required to integrate to get a decent statistical uncertainty on the polarization, but rather the much higher data acquisition rate of the experimental apparatus. Given that the axion coherence time ∼ (m φ v 2 ) −1 ∼ 10 6 /m φ is much longer than the oscillation period ∼ 1/m φ , 25 it is possible through appropriate data analysis to extract an oscillatory AC signal even when the 'single observation data points' are individually quite noisy. However, as can be 24 K.D. Irwin and Z. Ahmed (2019). Private communication. 25 We exceed CAST bounds for masses below approximately m φ ∼ 10 −19 eV, corresponding to frequencies ∼ 3 × 10 −5 Hz, or periods ∼ 10 −3 yr. Thus, the coherence times for even the fastest-varying axions of interest to us are on the order of a thousand years. seen from Fig. 2, the relevant axion periods in the parameter range of interest to us are generally quite long, and certainly much longer than the data acquisition rates of modern CMB experiments; we have thus ignored complications from this effect in estimating the reach in our parameter range of interest.
VI. DISCUSSION OF PREVIOUS WORK
In this section we distinguish the present paper from prior relevant work in the literature in some detail, supplementing the comments in this regard that we have already made in Secs. I and II.
Ref. [107] 26 derives the effects on the CMB of a time-varying axion field (see also Refs. [69,74,97]). The authors of Ref. [107] note and examine numerically that the usual static rotation approximation (i.e., the usual treatment of 'cosmic birefringence' for cosmologically slowly varying fields) fails to accurately capture all relevant effects when the axion field oscillates rapidly; in particular, the comparison shown at their Fig. 6 between this full numerical evolution and results based on the static rotation approximation do hint at some of the polarization washout features we have examined in the present work. However, Ref. [107] does not give an explanation or derivation of the polarization washout effect, nor does it employ the numerical results hinting at it in setting limits on the axion.
Separately, a number of recent works (e.g., Ref. [110] in the CMB context, and Ref. [58] in an astrophysical context) have treated the net rotation angle accumulated by a photon traversing N different coherently oscillating axion DM patches by summing incoherently over the mean-square rotation angles from each patch, obtaining a 'random-walk' √ N enhancement. As we have shown, such a treatment is erroneous, as the net rotation is simply proportional to the difference of axion field values at photon emission and photon absorption, independent of the intervening behaviour of the axion field [50]. 27 One other related previous work of which we are aware, Ref. [109], looks for static anisotropic birefringence induced by axion dark matter at the time of last scattering; however, as we have discussed, axion dark matter does not actually exhibit an unsuppressed signal of this type, as the net rotation angle induced at the last scattering surface averages to approximately zero point-by-point on the sky.
Our bounds compare favourably with some recent analyses of polarization rotation induced by axion dark matter in astrophysical contexts: over the relevant range of masses, we exceed by a factor of a few the limits set in Ref. [59] from considering AGNs as the polarized source; 28 and our limits are comparable to the recently revised (see footnote 27) projections made in Ref. [60], and limits set in Ref. [61], 29 using pulsars as the linearly polarized source.
In summary, our work is to our knowledge the first to call attention to the AC oscillation effect in the CMB context (although see Refs. [58][59][60][61] for recent analyses in astrophysical contexts employing this idea), and the first to give a clear derivation of the polarization washout. Moreover, we urge caution in utilizing previous CMB-based birefringence bounds on the axion dark matter parameter space that have not considered the effects of the rapid time-variation of the field at the decoupling epoch.
VII. CONCLUSION
In this work, we proposed a new technique to search for axion dark matter in the lowest allowed mass range. In particular, this appears to be one of the most sensitive ways to directly detect (theoretically well-motivated) axiontype fuzzy dark matter [16]. We found that axion dark matter has two novel effects on the polarization of the CMB. First, a uniform reduction in polarization power from the standard ΛCDM expectation for all l 20. This arises from the axion field oscillating many times during the CMB decoupling epoch, resulting in a washout of imprinted polarization as compared to the net polarization expected to be imprinted by the local CMB temperature quadrupole polarization source at last scattering. Second, a real on-the-sky AC oscillation of the CMB polarization at the period of the axion field, which is amenable to experimental detection, but which requires detailed and dedicated time-series analyses by CMB experimental collaborations.
The washout effect is quadratic in a small number, the axion field times the axion-photon coupling (∼ g 2 φγ φ 2 ), but arises from early times when the axion field is larger. The AC effect is linear in the axion field times the coupling (∼ g φγ φ), but arises from late times when the axion field is smaller. Given the current and projected sensitivities as shown in our main results in Fig. 2, these two effects are seen to be of roughly comparable reach in constraining axion parameter space in the fuzzy dark matter region m φ ∼ 10 −19 -10 −22 eV. Indeed, we have used the washout effect to set a bound on axion dark matter using current Planck results that is several orders of magnitude beyond previous limits. This bound is already close to the ultimate cosmic-variance-limited reach for this effect; see Fig. 2.
Beyond setting limits, both effects have discovery potential. No CMB physics can mimic either effect, and the AC effect especially appears distinct from any cosmological background. Of course, as in any precision experiment, care must be taken to eliminate other backgrounds, but there are strong checks on a positive signal. For example, all CMB detectors must see the same effect; in particular, the signal for the AC effect must be in phase in the different detectors. This would be dramatic confirmation of a detection. For the future, the AC effect, though requiring a dedicated analysis, has the greater potential reach since it is linear in the coupling (and not limited by cosmic variance). We have drawn two representative curves in Fig. 2 for the AC analysis, but we suspect that the ultimate reach of this method could even be beyond either of these. Thus, undertaking these AC analyses is important as this is ultimately the better way to search for axion dark matter in the lowest mass ranges.
In general, there are multiple ways to directly search for axion dark matter in the lighter part of its mass range, both with terrestrial experiments (e.g., Refs. [138][139][140][141][142][143][144]) and astrophysical observations (e.g., Refs. [59][60][61]). CMB polarization appears to be one of the most promising approaches for the very lightest end of the axion mass range, and offers the exciting possibility of directly detecting fuzzy dark matter.
Consider the following solution ansatz for Eq. (A4):
A σ (η, z) = F σ (η, z) exp [−iω(η − η ) + ik(z − z ) + iG σ (η, z; η , z )] ,
where F σ and G σ are real functions. Substituting into Eq. (A4), we have
∂ 2 η ln F σ + (∂ η ln F σ ) 2 − ω 2 − (∂ η G σ ) 2 + 2ω∂ η G σ + i −2(ω − ∂ η G σ )∂ η ln F σ + ∂ 2 η G σ + −∂ 2 z ln F σ − (∂ z ln F σ ) 2 + k 2 + (∂ z G σ ) 2 + 2k∂ z G σ + i −2(k + ∂ z G σ )∂ z ln F σ − ∂ 2 z G σ = σg φγ (∂ z φ) [i∂ η ln F σ + ω − ∂ η G σ ] + σg φγ (∂ η φ) [−i∂ z ln F σ + k + ∂ z G σ ](A6)
Equating the real and imaginary parts, we have, using that ∂ 2 x ln A + (∂ x ln A) 2 = ∂ 2 x A/A for x ∈ {η, z},
k 2 − ω 2 + 2ω∂ η G σ + 2k∂ z G σ + ∂ 2 η F σ F σ − ∂ 2 z F σ F σ + (∂ z G σ ) 2 − (∂ η G σ ) 2 = σg φγ (∂ z φ) [ω − ∂ η G σ ] + σg φγ (∂ η φ) [k + ∂ z G σ ] (A7) − 2(ω − ∂ η G σ )∂ η ln F σ + ∂ 2 η G σ − 2(k + ∂ z G σ )∂ z ln F σ − ∂ 2 z G σ = σg φγ (∂ z φ) [∂ η ln F σ ] + σg φγ (∂ η φ) [−∂ z ln F σ ] .(A8)
In order to make progress, we introduce a formal small parameter by assuming that φ, F σ , and G σ are functions of 'slow' time (space) variables τ (ξ) which are considered to vary slowly on the scale of 1/ω (1/k):
τ ≡ ωη ⇒ ∂ η = ω∂ τ (A9) ξ ≡ kz ⇒ ∂ z = k∂ ξ .(A10)
Substituting into Eqs. (A7) and (A8), we obtain
n 2 0 − 1 + 2 ∂ τ G σ + 2 n 2 0 ∂ ξ G σ + 2 ∂ 2 τ F σ F σ − 2 n 2 0 ∂ 2 ξ F σ F σ + 2 n 2 0 (∂ ξ G σ ) 2 − 2 (∂ τ G σ ) 2 = n 0 σg φγ (∂ ξ φ) [1 − ∂ τ G σ ] + n 0 σg φγ (∂ τ φ) [1 + ∂ ξ G σ ] (A11) − 2 (1 − ∂ τ G σ )∂ τ ln F σ + 2 ∂ 2 τ G σ − 2n 2 0 (1 + ∂ ξ G σ )∂ ξ ln F σ − 2 n 2 0 ∂ 2 ξ G σ = 2 n 0 σg φγ (∂ ξ φ) [∂ τ ln F σ ] − 2 n 0 σg φγ (∂ τ φ) [∂ ξ ln F σ ] .(A12)
where we have defined n 0 ≡ k/ω. Consistent with these equations up to corrections at O( 2 ), we have n 0 = 1 (A13)
G σ = g φγ φ 2 + const. (A14) F σ = const.;(A15)
by appropriate choice of the constants, we immediately arrive at the desired result, Eq. (3). Note that we have not assumed that g φγ φ is small in the above derivation, only that φ varies slowly on the time or length scales of the photon field.
Appendix B: Stokes parameters
We define the Stokes parameters following the conventions of Ref. [145]. Suppose that the electric field can be written as (the real part of)
E ≡ e −ikx E 1ê 1 + E 2ê 2 ≡ e −ikx E +ê + + E −ê − (B1) whereê ± ≡ 1 √ 2 (ê 1 ± iê 2 ) , E ± ≡ 1 √ 2 E 1 ∓ iE 2 .(B2)
The Stokes parameters (I, Q, U, V ) defined with respect to these frames are I ≡ |ê 1 · E| 2 + |ê 2 · E| 2 = |E 1 | 2 + |E 2 | 2 (B3) ≡ |ê * + · E| 2 + |ê * − · E| 2 + = |E + | 2 + |E − | 2 (B4)
Q ≡ |ê 1 · E| 2 − |ê 2 · E| 2 = |E 1 | 2 − |E 2 | 2 (B5)
≡ 2 Re ê * + · E * ê * − · E = 2|E + ||E − | cos (ϕ − − ϕ + ) (B6) U ≡ 2 Re (ê * 1 · E) * (ê * 2 · E) = 2|E 1 ||E 2 | cos (ϕ 2 − ϕ 1 ) (B7) ≡ 2 Im ê * + · E * ê * − · E = 2|E + ||E − | sin (ϕ − − ϕ + ) (B8) V ≡ 2 Im (ê * 1 · E) * (ê * 2 · E) = 2|E 1 ||E 2 | sin (ϕ 2 − ϕ 1 ) (B9) ≡ |ê * + · E| 2 − |ê * − · E| 2 + = |E + | 2 − |E − | 2 (B10)
where ϕ X is defined by E X ≡ |E X |e iϕ X for any X ∈ {1, 2, +, −}.
A field which is linearly polarized has V = 0; V = 0 implies elliptical polarization, with V = ±I implying circular polarization of helicity ±1. A linearly polarized field with Q > 0, U = 0 is polarized along theê 1 -axis (E 1 = 0 and E 2 = 0); whereas a linearly polarized field with Q < 0, U = 0 is polarized along theê 2 -axis (E 2 = 0 and E 1 = 0). On the other hand, a linearly polarized field with U > 0, Q = 0 is polarized along the (ê 1 +ê 2 )-axis (E 1 = E 2 ); whereas a linearly polarized field with U < 0, Q = 0 is polarized along the (ê 1 −ê 2 )-axis (E 1 = −E 2 ). See Fig. 1, but note that we define the Stokes parameters there in terms of theê ϑ (n) andê ϕ (n) axes.
FIG. 2 .
2FIG. 2. Reach for current and future searches for the washout and AC oscillation effects induced by an axion field of mass m φ constituting all of the dark matter (κ = 1, in the notation of Sec. IV C). We find that the green shaded region is excluded on the grounds that a reduction in the predicted polarization amplitude of at least 0.58% is in conflict at 95% confidence with Planck measurements [115]; see Sec. V A 2. The dashed green line indicates how the bound would improve if the 95%-confidence upper limit on the fractional reduction were improved to 0.082%, our projection for the sensitivity if all three of the power spectra C T T,T E,EE
This statement applies generally to any pseudoscalar field coupled to a photon in a fashion similar to the axion.6 In this section, z = x 3 is the third spatial coordinate, not the redshift.
The sign convention for theê 1,2 system is chosen this way such that the triplet (p,ê 1 ,ê 2 ) forms a right-handed co-ordinate system, wherep ≡ −n is the direction of photon propagation from source to observer.
In contrast to Sec. III, here z is the redshift, not the third spatial co-ordinate, x 3 .
In our universe, g(z) has a FWHM spanning z ∼ 1000-1200[116,117], corresponding to a cosmic time interval T = ∆t ∼ 10 5 yrs, while an axion with mass m φ oscillates with a period T φ ∼ 0.1 yrs × (10 −21 eV/m φ ).
By not fixingÂ(r = 0) ≡ 1, we allow for the possibility in our procedure that the differences between our simplified version of the Planck likelihood in this approximate analysis and the full Planck analysis pipeline give rise to a small offset between the parameter point that our procedure deems to be the best-fit cosmology and the parameter point that the full Planck analysis pipeline selects as the best-fit cosmology. This offset is at the per mille level; i.e.,Â(r = 0) − 1 ∼ O(10 −3 ).21 We again permitΘ(r = 0) to float to account for differences in our analysis procedure and the full Planck analysis pipeline causing a small shift in what we deem the best-fit cosmology as compared to the Planck results.
This is the 'Asimov data set'[127] for this exclusion bound.
Ref.[108] is a related, shortened version of that paper by the same authors, to which we will omit further reference here for the sake of brevity.27 See also the 'Note added' in Ref.[59]. Similar erroneous treatments also appeared in v1 of the arXiv preprints of Refs.[60,61], but were corrected in revised versions (arXiv v2) as this work was being finalized.28 Although we note that results of Ref.[59] require knowledge of the dark-matter density near the centre of elliptical galaxies hosting such AGNs, which is naturally subject to some considerable uncertainty.29 The pulsar considered in Ref.[61], J0437-4715, is only D ∼ 156 pc distant from Earth[137], which is within an axion coherence length λ coh. ∼ 2π/(mav 0 ) ∼ 156 pc × (3.5 × 10 −22 eV/ma) × (220 km s −1 /v 0 ) for some part of the axion mass range of interest, and close enough that the axion field amplitude (i.e., DM density) at the pulsar would be similar to that at Earth. But then the difference in the axion field between emission at x = (t − D, Dn) and absorption atx = (t, 0) is ∆φ ≈ φ 0 [cos (mat) − cos (ma(t − D) + δ)], where δ ∼ O(πD/λ coh. ). We would thus expect to see some loss of sensitivity in the results of Ref.[61] at certain specific axion masses (maD mod 2π ≈ 0) when ma 3.5 × 10 −22 eV.
We acknowledge an abuse of notation on the RHS of Eq. (A3): g φγ is the axion-photon coupling; gνα is the metric.
ACKNOWLEDGMENTSWe thank Zeeshan Ahmed, Simone Ferraro, Kent Irwin, Uros Seljak, and Leonardo Senatore for useful conversations.Appendix A: Derivation of the polarization rotation effectThe classical field equation for the axion field arising from Eq. (1), assuming thatwe ignore the backreaction term on the RHS. The classical field equations for the photon are[114]In Lorenz gauge, g αβ ∇ α A β = 0, the field equations for the potentials in a source-free region become 30where we have used that ναλρ ∇ λ A ρ = ναλρ ∂ λ A ρ owing to the symmetries of the Levi-Civita tensor and Christoffel symbols. We specialize to a homogeneous, isotropic FLRW universe with scale factor a, and work in the conformal-comoving co-ordinate system (η, x) such that the line element is ds 2 = [a(η)] 2 dη 2 − dx 2 ; η is conformal time. Furthermore, we will seek solutions to the photon equations of motion which take the form φ = φ(η, z) and A µ = A µ (η, z) (i.e., solutions in which the variation in the fields is negligible the direction transverse to some selected direction, here chosen without loss of generality to be theẑ-direction). A solution consistent with the ν = 0, 3 photon equations of motion and the Lorenz gauge condition is A 0 = A 3 = 0, while the ν = 1, 2 equations of motion can be decoupled by defining the definite-helicity transverse field variables as at Eq. (5), in terms of which we have
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| [] |
[
"Size-dependent Correlation Effects in Ultrafast Optical Dynamics of Metal Nanoparticles Typeset using REVT E X 1",
"Size-dependent Correlation Effects in Ultrafast Optical Dynamics of Metal Nanoparticles Typeset using REVT E X 1"
] | [
"T V Shahbazyan \nDepartment of Physics and Astronomy\nVanderbilt University\nBox 1807-B37235NashvilleTN\n",
"I E Perakis \nDepartment of Physics and Astronomy\nVanderbilt University\nBox 1807-B37235NashvilleTN\n"
] | [
"Department of Physics and Astronomy\nVanderbilt University\nBox 1807-B37235NashvilleTN",
"Department of Physics and Astronomy\nVanderbilt University\nBox 1807-B37235NashvilleTN"
] | [] | We study the role of collective surface excitations in the electron relaxation in small metal particles. We show that the dynamically screened electronelectron interaction in a nanoparticle contains a size-dependent correction induced by the surface. This leads to new channels of quasiparticle scattering accompanied by the emission of surface collective excitations. We calculate the energy and temperature dependence of the corresponding rates, which depend strongly on the nanoparticle size. We show that the surface-plasmonmediated scattering rate of a conduction electron increases with energy, in contrast to that mediated by a bulk plasmon. In noble-metal particles, we find that the dipole collective excitations (surface plasmons) mediate a resonant scattering of d-holes to the conduction band. We study the role of the latter effect in the ultrafast optical dynamics of small nanoparticles and show that, with decreasing nanoparticle size, it leads to a drastic change in the differential absorption lineshape and a strong frequency dependence of the relaxation near the surface plasmon resonance. The experimental implications of our results in ultrafast pump-probe spectroscopy are also discussed. | 10.1103/physrevb.60.9090 | [
"https://arxiv.org/pdf/cond-mat/9902005v1.pdf"
] | 14,483,768 | cond-mat/9902005 | b7c7835651ade49572c1c0d25676743c55bd5fad |
Size-dependent Correlation Effects in Ultrafast Optical Dynamics of Metal Nanoparticles Typeset using REVT E X 1
arXiv:cond-mat/9902005v1 1 Feb 1999
T V Shahbazyan
Department of Physics and Astronomy
Vanderbilt University
Box 1807-B37235NashvilleTN
I E Perakis
Department of Physics and Astronomy
Vanderbilt University
Box 1807-B37235NashvilleTN
Size-dependent Correlation Effects in Ultrafast Optical Dynamics of Metal Nanoparticles Typeset using REVT E X 1
arXiv:cond-mat/9902005v1 1 Feb 1999
We study the role of collective surface excitations in the electron relaxation in small metal particles. We show that the dynamically screened electronelectron interaction in a nanoparticle contains a size-dependent correction induced by the surface. This leads to new channels of quasiparticle scattering accompanied by the emission of surface collective excitations. We calculate the energy and temperature dependence of the corresponding rates, which depend strongly on the nanoparticle size. We show that the surface-plasmonmediated scattering rate of a conduction electron increases with energy, in contrast to that mediated by a bulk plasmon. In noble-metal particles, we find that the dipole collective excitations (surface plasmons) mediate a resonant scattering of d-holes to the conduction band. We study the role of the latter effect in the ultrafast optical dynamics of small nanoparticles and show that, with decreasing nanoparticle size, it leads to a drastic change in the differential absorption lineshape and a strong frequency dependence of the relaxation near the surface plasmon resonance. The experimental implications of our results in ultrafast pump-probe spectroscopy are also discussed.
I. INTRODUCTION
The properties of small metal particles in the intermediate regime between bulk-like and molecular behavior have been a subject of great interest recently. [1][2][3][4] Even though the electronic and optical properties of nanoparticles have been extensively studied, the effect of confinement on electron dynamics is much less understood. Examples of outstanding issues include the role of electron-electron interactions in the process of cluster fragmentation, the role of surface lattice modes in providing additional channels for intra-molecular energy relaxation, the influence of the electron and nuclear motion on the superparamagnetic properties of clusters, and the effect of confinement on the nonlinear optical properties and transient response under ultrafast excitation. 1,2,4 These and other dynamical phenomena can be studied with femtosecond nonlinear optical spectroscopy, which allows one to probe the time evolution of the excited states with a resolution shorter than the energy relaxation or dephasing times.
Surface collective excitations play an important role in the absorption of light by metal nanoparticles. In large particles with sizes comparable to the wave-length of light λ (but smaller than the bulk mean free path), the lineshape of the surface plasmon (SP) resonance is determined by the electromagnetic effects. 1 In small nanoparticles with radii R ≪ λ, the absorption spectrum is governed by quantum confinement effects. For example, the momentum non-conservation due to the confining potential leads to the Landau damping of the SP and to a resonance linewidth inversely proportional to the nanoparticle size. 1,5 Confinement changes also non-linear optical properties of nanoparticles: a size-dependent enhancement of the third order susceptibilities, caused by the elastic surface scattering of single-particle excitations, has been reported. [6][7][8] Extensive experimental studies of the electron relaxation in nanoparticles have recently been performed using ultrafast pump-probe spectroscopy. [9][10][11][12][13][14][15][16] Unlike in semiconductors, the dephasing processes in metals are very fast, and nonequilibrium populations of optically excited electrons and holes are formed within several femtoseconds. These thermalize into the hot Fermi-Dirac distribution within several hundreds of femtoseconds, mainly due to e-e and h-h scattering. [17][18][19][20] Since the electron heat capacity is much smaller than that of the lattice, a high electron temperature can be reached during less than 1 ps time scales, i.e., before any significant energy transfer to the phonon bath occurs. During this stage, the SP resonance was observed to undergo a time-dependent spectral broadening. 11,13 Subsequently, the electron and phonon baths equilibrate through the electron-phonon interactions over time intervals of a few picoseconds. During this incoherent stage, the hot electron distribution can be characterized by a time-dependent temperature. Correlation effects play an important role in the latter regime. For example, in order to explain the differential absorption lineshape, it is essential to take into account the e-e scattering of the optically-excited carriers near the Fermi surface. 11 Furthermore, despite the similarities to the bulk-like behavior, observed, e.g., in metal films, certain aspects of the optical dynamics in nanoparticles are significantly different. 14,11,16 For example, experimental studies of small Cu nanoparticles revealed that the relaxation times of the the pump-probe signal depend strongly on frequency: the relaxation was considerably slower at the SP resonance. 11,16 This and other observations suggest that collective surface excitations play an important role in the electron dynamics in small metal particles.
Let us recall the basic facts regarding the linear absorption by metal nanoparticles embedded in a medium with dielectric constant ǫ m . We will focus primarily on noble metal particles containing several hundreds of atoms; in this case, the confinement affects the extended electronic states even though the bulk lattice structure has been established. When the particles radii are small, R ≪ λ, so that only dipole surface modes can be optically excited and non-local effects can be neglected, the optical properties of this system are determined by the dielectric function 1
ǫ col (ω) = ǫ m + 3pǫ m ǫ(ω) − ǫ m ǫ(ω) + 2ǫ m ,(1)
where ǫ(ω) = ǫ ′ (ω) + iǫ ′′ (ω) is the dielectric function of a metal particle and p ≪ 1 is the volume fraction occupied by nanoparticles in the colloid. Since the d-electrons play an important role in the optical properties of noble metals, the dielectric function ǫ(ω) includes also the interband contribution ǫ d (ω). For p ≪ 1, the absorption coefficient of such a system is proportional to that of a single particle and is given by 1
α(ω) = −9p ǫ 3/2 m ω c Im 1 ǫ s (ω) ,(2)
where
ǫ s (ω) = ǫ d (ω) − ω 2 p /ω(ω + iγ s ) + 2ǫ m ,(3)
plays the role of an effective dielectric function of a particle in the medium. Its zero, ǫ ′ s (ω s ) = 0, determines the frequency of the SP, ω s . In Eq. (3), ω p is the bulk plasmon frequency of the conduction electrons, and the width γ s characterizes the SP damping. The semiclassical result Eqs. (2) and (3) applies to nanoparticles with radii R ≫ q −1 T F , where q T F is the Thomas-Fermi screening wave-vector (q −1 T F ∼ 1Å in noble metals). In this case, the electron density deviates from its classical shape only within a surface layer occupying a small fraction of the total volume. 21 Quantum mechanical corrections, arising from the discrete energy spectrum, lead to a width γ s ∼ v F /R, where v F = k F /m is the Fermi velocity. 1,5 Even though γ s /ω s ∼ (q T F R) −1 ≪ 1, this damping mechanism dominates over others, e.g., due to phonons, for sizes R < ∼ 10 nm. In small clusters, containing several dozens of atoms, the semiclassical approximation breaks down and density functional or ab initio methods should be used. [1][2][3][4] It should be noted that, in contrast to surface collective excitations, the e-e scattering is not sensitive to the nanoparticle size as long as the condition q T F R ≫ 1 holds. 22 Indeed, for such sizes, the static screening is essentially bulk-like. At the same time, the energy dependence of the bulk e-e scattering rate, 23 γ e ∝ (E−E F ) 2 , with E F being the Fermi energy, comes from the phase-space restriction due to the momentum conservation, and involves the exchange of typical momenta q ∼ q T F . If the size-induced momentum uncertainty δq ∼ R −1 is much smaller than q T F , the e-e scattering rate in a nanoparticle is not significantly affected by the confinement. 24 In this paper we address the role of collective surface excitations in the electron relaxation in small metal particles. We show that the dynamically screened e-e interaction contains a correction originating from the surface collective modes excited by an electron in nanoparticle. This opens up new quasiparticle scattering channels mediated by surface collective modes. We derive the corresponding scattering rates, which depend strongly on the nanoparticle size. The scattering rate of a conduction electron increases with energy, in contrast to the bulk-plasmon mediated scattering. In noble metal particles, we study the SP-mediated scattering of a d-hole into the conduction band. The scattering rate of this process depends strongly on temperature, and exhibits a peak as a function of energy due to the restricted phase space available for interband scattering. We show that this effect manifests itself in the ultrafast nonlinear optical dynamics of nanometer-sized particles. In particular, our self-consistent calculations show that, near the SP resonance, the differential absorption lineshape undergoes a dramatic transformation as the particle size decreases. We also find that the relaxation times of the pump-probe signal depend strongly on the probe frequency, in agreement with recent experiments.
The paper is organized as follows. In Section II we derive the dynamically screened Coulomb potential in a nanoparticle. In Section III we calculate the SP-mediated quasiparticle scattering rates of the conduction electrons and the d-band holes. In Section IV we incorporate these effects in the calculation of the absorption spectrum and study their role in the size and frequency dependence of the time-resolved pump-probe signal.
II. ELECTRON-ELECTRON INTERACTIONS IN METAL NANOPARTICLES
In this section, we study the effect of the surface collective excitations on the e-e interactions in a spherical metal particle. To find the dynamically screened Coulomb potential, we generalize the method previously developed for calculations of local field corrections to the optical fields. 25 The potential U(ω; r, r ′ ) at point r arising from an electron at point r ′ is determined by the equation 26 U(ω; r, r ′ ) = u(r − r ′ ) + dr 1 dr 2 u(r − r 1 )Π(ω; r 1 , r 2 )U(ω; r 2 , r ′ ),
where u(r − r ′ ) = e 2 |r − r ′ | −1 is the unscreened Coulomb potential and Π(ω; r 1 , r 2 ) is the polarization operator. There are three contributions to Π, arising from the polarization of the conduction electrons, the d-electrons, and the medium surrounding the nanoparticles:
Π = Π c + Π d + Π m .
It is useful to rewrite Eq. (4) in the "classical" form
∇ · (E + 4πP) = 4πe 2 δ(r − r ′ ),(5)
where E(ω; r, r ′ ) = −∇U(ω; r, r ′ ) is the screened Coulomb field and P = P c + P d + P m is the electric polarization vector, related to the potential U as
∇P(ω; r, r ′ ) = −e 2 dr 1 Π(ω; r, r 1 )U(ω; r 1 , r ′ ).(6)
In the random phase approximation, the intraband polarization operator is given by
Π c (ω; r, r ′ ) = αα ′ f (E c α ) − f (E c α ′ ) E c α − E c α ′ + ω + i0 ψ c α (r)ψ c * α ′ (r)ψ c * α (r ′ )ψ c α ′ (r ′ ),(7)
where E c α and ψ c α are the single-electron eigenenergies and eigenfunctions in the nanoparticle, and f (E) is the Fermi-Dirac distribution (we seth = 1). Since we are interested in frequencies much larger than the single-particle level spacing, Π c (ω) can be expanded in terms of 1/ω. For the real part, Π ′ c (ω), we obtain in the leading order 25
Π ′ c (ω; r, r 1 ) = − 1 mω 2 ∇[n c (r)∇δ(r − r 1 )],(8)
where n c (r) is the conduction electron density. In the following we assume, for simplicity, a step density profile, n c (r) =n c θ(R − r), wheren c is the average density. The leading
contribution to the imaginary part, Π ′′ c (ω), is proportional to ω −3 , so that Π ′′ c (ω) ≪ Π ′ c (ω).
By using Eqs. (8) and (6), one obtains a familiar expression for P c at high frequencies,
P c (ω; r, r ′ ) = e 2 n c (r) mω 2 ∇U(ω; r, r ′ ) = θ(R − r)χ c (ω)E(ω; r, r ′ ),(9)
where χ c (ω) = −e 2n c /mω 2 is the conduction electron susceptibility. Note that, for a step density profile, P c vanishes outside the particle. The d-band and dielectric medium contributions to P are also given by similar relations,
P d (ω; r, r ′ ) = θ(R − r)χ d (ω)E(ω; r, r ′ ),(10)P m (ω; r, r ′ ) = θ(r − R)χ m E(ω; r, r ′ ),(11)
where χ i = (ǫ i −1)/4π, i = d, m are the corresponding susceptibilities and the step functions account for the boundary conditions. 27 Using Eqs. (9)-(11), one can write a closed equation for U(ω; r, r ′ ). Using Eq. (6), the second term of Eq. (4) can be presented as −e −2 dr 1 u(r − r 1 )∇ · P(ω; r 1 , r ′ ). Substituting the above expressions for P, we then obtain after integrating by parts
ǫ(ω)U(ω; r, r ′ ) = e 2 |r − r ′ | + dr 1 ∇ 1 1 |r − r 1 | · ∇ 1 [θ(R − r)χ(ω) + θ(r − R)χ m ] U(ω; r 1 , r ′ ) +i dr 1 dr 2 e 2 |r − r 1 | Π ′′ c (ω; r 1 , r 2 )U(ω; r 2 , r ′ ),(12)
with
ǫ(ω) ≡ 1 + 4πχ(ω) = ǫ d (ω) − ω 2 p /ω 2 ,(13)
ω 2 p = 4πe 2n c /m being the plasmon frequency in the conduction band. The last term in the rhs of Eq. (12), proportional to Π ′′ c (ω), can be regarded as a small correction. To solve Eq. (12), we first eliminate the angular dependence by expanding U(ω; r, r ′ ) in spherical harmonics, Y LM (r), with coefficients U LM (ω; r, r ′ ). Using the corresponding expansion of |r − r ′ | −1 with coefficients Q LM (r, r ′ ) = 4π 2L+1 r −L−1 r ′L (for r > r ′ ), we get the following equation for U LM (ω; r, r ′ ):
ǫ(ω)U LM (ω; r, r ′ ) = Q LM (r, r ′ ) + 4π [χ(ω) − χ m ] L + 1 2L + 1 r R L U LM (ω; R, r ′ ) +ie 2 L ′ M ′ dr 1 dr 2 r 2 1 r 2 2 Q LM (r, r 1 )Π ′′ LM,L ′ M ′ (ω; r 1 , r 2 )U L ′ M ′ (ω; r 2 , r ′ ),(14)
where
Π ′′ LM,L ′ M ′ (ω; r 1 , r 2 ) = dr 1 dr 2 Y * LM (r 1 )Π ′′ c (ω; r 1 , r 2 )Y L ′ M ′ (r 2 ),(15)
are the coefficients of the multipole expansion of Π ′′ c (ω; r 1 , r 2 ). For Π ′′ c = 0, the solution of Eq. (14) can be presented in the form
U LM (ω; r, r ′ ) = a(ω)e 2 Q LM (r, r ′ ) + b(ω) 4πe 2 2L + 1 r L r ′L R 2L+1 ,(16)
with frequency-dependent coefficients a and b. Since Π ′′ c (ω) ≪ Π ′ c (ω) for relevant frequencies, the solution of Eq. (14) in the presence of the last term can be written in the same form as Eq. (16), but with modified a(ω) and b(ω). Substituting Eq. (16) into Eq. (14), we obtain after lengthy algebra in the lowest order in Π ′′
c a(ω) = ǫ −1 (ω), b(ω) = ǫ −1 L (ω) − ǫ −1 (ω),(17)
where
ǫ L (ω) = L 2L + 1 ǫ(ω) + L + 1 2L + 1 ǫ m + iǫ ′′ cL (ω),(18)
is the effective dielectric function, whose zero, ǫ ′ L (ω L ) = 0, determines the frequency of the collective surface excitation with angular momentum L, 1
ω 2 L = Lω 2 p Lǫ ′ d (ω L ) + (L + 1)ǫ m .(19)
In Eq. (18), ǫ ′′ cL (ω) characterizes the damping of the L-pole collective mode by single-particle excitations, and is given by
ǫ ′′ cL (ω) = 4π 2 e 2 (2L + 1)R 2L+1 αα ′ |M LM αα ′ | 2 [f (E c α ) − f (E c α ′ )]δ(E c α − E c α ′ + ω),(20)
where M LM αα ′ are the matrix elements of r L Y LM (r). Due to the momentum nonconservation in a nanoparticle, the matrix elements are finite, which leads to the size-dependent width of the L-pole mode: 5,25
γ L = 2L + 1 L ω 3 ω 2 p ǫ ′′ cL (ω).(21)
For ω ∼ ω L , one can show that the width, γ L ∼ v F /R, is independent of ω. Note that, in noble metal particles, there is an additional d-electron contribution to the imaginary part of ǫ L (ω) at frequencies above the onset ∆ of the interband transitions.
Putting everything together, we arrive at the following expression for the dynamicallyscreened interaction potential in a nanoparticle:
U(ω; r, r ′ ) = u(r − r ′ ) ǫ(ω) + e 2 R LM 4π 2L + 1 1 ǫ L (ω) rr ′ R 2 L Y LM (r)Y * LM (r ′ ),(22)withǫ −1 L (ω) = ǫ −1 L (ω) − ǫ −1 (ω)
. Equation (22), which is the main result of this section, represents a generalization of the plasmon pole approximation to spherical particles. The two terms in the rhs describe two distinct contributions. The first comes from the usual bulk-like screening of the Coulomb potential. The second contribution describes a new effective e-e interaction induced by the surface: the potential of an electron inside the nanoparticle excites high-frequency surface collective modes, which in turn act as image charges that interact with the second electron. It should be emphasized that, unlike in the case of the optical fields, the surface-induced dynamical screening of the Coulomb potential is size-dependent.
Note that the excitation energies of the surface collective modes are lower than the bulk plasmon energy, also given by Eq. (19) but with ǫ m = 0. This opens up new channels of quasiparticle scattering, considered in the next section.
III. QUASIPARTICLE SCATTERING VIA SURFACE COLLECTIVE MODES
In this section we calculate the rates of quasiparticle scattering accompanied by the emission of surface collective modes. We start with the scattering of an electron in the conduction band. In the first order in the surface-induced potential, given by the second term in the rhs of Eq. (22), the corresponding scattering rate can be obtained from the
Matsubara self-energy 26 Σ c α (iω) = − 1 β iω ′ LM α ′ 4πe 2 (2L + 1)R 2L+1 |M LM αα ′ | 2 ǫ L (iω ′ ) G c α ′ (iω ′ + iω),(23)
where G c α = (iω − E c α ) −1 is the non-interacting Green function of the conduction electron.
Here the matrix elements M LM αα ′ are calculated with the one-electron wave functions ψ c α (r) = R nl (r)Y lm (r). Since |α and |α ′ are the initial and final states of the scattered electron, the main contribution to the Lth term of the angular momentum sum in Eq. (23) will come from electron states with energy difference E α − E α ′ ∼ ω L . Therefore, M LM αα ′ can be expanded in terms of the small parameter E 0 /|E c α − E c α ′ | ∼ E 0 /ω L , where E 0 = (2mR 2 ) −1 is the characteristic confinement energy. The leading term can be obtained by using the following procedure. 5, 25 We present M LM αα ′ as
M LM αα ′ = c, α|r L Y LM (r)|c, α ′ = c, α|[H, [H, r L Y LM (r)]]|c, α ′ (E c α − E c α ′ ) 2 ,(24)
where H = H 0 + V (r) is the Hamiltonian of an electron in a nanoparticle with confining
potential V (r) = V 0 θ(r − R). Since [H, r L Y LM (r)] = − 1 m ∇[r L Y LM (r)] · ∇,|E c α − E c α ′ | −1 , we obtain M LM αα ′ = c, α|∇[r L Y LM (r)] · ∇V (r)|c, α ′ m(E c α − E c α ′ ) 2 = LR L+1 m(E c α − E c α ′ ) 2 V 0 R nl (R)R n ′ l ′ (R)ϕ LM lm,l ′ m ′ ,(25)
with ϕ LM lm,l ′ m ′ = drY * lm (r)Y LM (r)Y l ′ m ′ (r). Note that, for L = 1, Eq. (25) becomes exact. For electron energies close to the Fermi level, E c nl ∼ E F , the radial quantum numbers are large, and the product V 0 R nl (R)R n ′ l ′ (R) can be evaluated by using semiclassical wave-functions.
In the limit V 0 → ∞, this product is given by 5
2 E c nl E c n ′ l ′ /R 3 ,
where E c nl = π 2 (n + l/2) 2 E 0 is the electron eigenenergy for large n. Substituting this expression into Eq. (25) and then into Eq. (23), we obtain
Σ c α (iω) = − 1 β iω ′ L n ′ l ′ C L ll ′ 4πe 2 (2L + 1)R E c nl E c n ′ l ′ (E c nl − E c n ′ l ′ ) 4 (4LE 0 ) 2 ǫ L (iω ′ ) G c α ′ (iω ′ + iω),(26)
with
C L ll ′ = M,m ′ |ϕ LM lm,l ′ m ′ | 2 = (2L + 1)(2l ′ + 1) 8π 1 −1 dxP l (x)P L (x)P l ′ (x),(27)
where P l (x) are Legendre polynomials; we used properties of the spherical harmonics in the derivation of Eq. (27). For E c nl ∼ E F , the typical angular momenta are large, l ∼ k F R ≫ 1, and one can use the large-l asymptotics of P l ; for the low multipoles of interest, L ≪ l, the integral in Eq. (27) can be approximated by 2 2l ′ +1 δ ll ′ . After performing the Matsubara summation, we obtain for the imaginary part of the self-energy that determines the electron scattering rate
ImΣ c α (ω) = − 16e 2 R E 2 0 L L 2 dE g l (E) EE c α (E c α − E) 4 Im N(E − ω) + f (E) ǫ L (E − ω) ,(28)
where N(E) is the Bose distribution and g l (E) is the density of states of a conduction electron with angular momentum l,
g l (E) = 2 n δ(E c nl − E) ≃ R π 2m E ,(29)
where we replaced the sum over n by an integral (the factor of 2 accounts for spin).
Each term in the sum in the rhs of Eq. (28) Consider now the L = 1 term in Eq. (28), which describes the SP-mediated scattering channel. The main contribution to the integral comes from the SP pole in ǫ −1
1 (ω) = 3ǫ −1 s (ω),
where ǫ s (ω) is the same as in Eq. (3). To estimate the scattering rate, we approximate Imǫ −1 s (ω) by a Lorentzian,
Imǫ −1 s (ω) = − γ s ω 2 p /ω 3 + ǫ ′′ d (ω) [ǫ ′ (ω) + 2ǫ m ] 2 + [γ s ω 2 p /ω 3 + ǫ ′′ d (ω)] 2 ≃ − ω 2 s ǫ ′ d (ω s ) + 2ǫ m ω s γ (ω 2 − ω 2 s ) 2 + ω 2 s γ 2 ,(30)
where ω s ≡ ω 1 = ω p / ǫ ′ d (ω s ) + 2ǫ m and γ = γ s + ω s ǫ ′′ d (ω s ) are the SP frequency and width, respectively. For typical widths γ ≪ ω s , the integral in Eq. (28) can be easily evaluated, yielding ImΣ c α (ω) = −
24e 2 ω s E 2 0 ǫ ′ d (ω s ) + 2ǫ m E c α 2m(ω − ω s ) (ω − E c α − ω s ) 4 [1 − f (ω − ω s )].(31)
Finally, using the relation e 2 k F [ǫ ′ d (ω s ) + 2ǫ m ] −1 = 3πω 2 s /8E F , the SP-mediated scattering rate, γ s e (E c α ) = −ImΣ c α (E c α ), takes the form γ s e (E) = 9π
E 2 0 ω s E E F E − ω s E F 1/2 [1 − f (E − ω s )].(32)
Recalling that E 0 = (2mR 2 ) −1 , we see that the scattering rate of a conduction electron is size-dependent: γ s e ∝ R −4 . At E = E F + ω s , the scattering rate jumps to the value 9π(1 + ω s /E F )E 2 0 /ω s , and then increases with energy as E 3/2 (for ω s ≪ E F ). This should be contrasted with the usual (bulk) plasmon-mediated scattering, originating from the first term in Eq. (22), with the rate decreasing as E −1/2 above the onset. 26 To estimate the size at which γ s e becomes important, we should compare it with the Fermi liquid e-e scattering rate, 23
γ e (E) = π 2 q T F 16k F (E−E F ) 2 E F . For energies E ∼ E F + ω s , the two rates become comparable for (k F R) 2 ≃ 12 E F ω s 1 + E F ω s 1/2 k F πq T F 1/2 .(33)
In the case of a Cu nanoparticle with ω s ≃ 2.2 eV, we obtain k F R ≃ 8, which corresponds to the radius R ≃ 3 nm. At the same time, in this energy range, the width γ s e exceeds the mean level spacing δ, so that the energy spectrum is still continuous. The strong size dependence of γ s e indicates that, although γ s e increases with energy slower than γ e , the SP-mediated scattering should dominate for nanometer-sized particles. Note that the size and energy dependences of scattering in different channels are similar. Therefore, the total scattering rate as a function of energy will represent a series of steps at the collective excitation energies E = ω L < ω p on top of a smooth energy increase. We expect that this effect could be observed experimentally in time-resolved two-photon photoemission measurements of sizeselected cluster beams. 28 We now turn to the interband processes in noble metal particles and consider the scattering of a d-hole into the conduction band. From now on we restrict ourselves to the scattering via the dipole channel, mediated by the SP. The corresponding surface-induced potential, given by the L = 1 term in Eq. (22), has the form
U s (ω; r, r ′ ) = 3e 2 R r · r ′ R 2 1 ǫ s (ω) .(34)
With this potential, the d-hole self-energy is given by
Σ d α (iω) = − 3e 2 R 3 α ′ |d αα ′ | 2 1 β iω ′ G c α ′ (iω ′ + iω) ǫ s (iω ′ ) ,(35)
where
d αα ′ = c, α|r|d, α ′ = c, α|p|d, α ′ /im(E c α − E d α ′ )
is the interband transition matrix element. Since the final state energies in the conduction band are high (in the case of interest here, they are close to the Fermi level), the matrix element can be approximated by the bulk-like expression c, α|p|d, α ′ = δ αα ′ c|p|d ≡ δ αα ′ µ, the corrections due to surface scattering being suppressed by a factor of (k F R) −1 ≪ 1. After performing the frequency summation, we obtain for ImΣ d
α ImΣ d α (ω) = − 9e 2 µ 2 m 2 (E cd α ) 2 R 3 Im N(E c α − ω) + f (E c α ) ǫ s (E c α − ω) ,(36)with E cd α = E c α − E d α . We see that the scattering rate of a d-hole with energy E d α , γ s h (E d α ) = ImΣ d α (E d α )
, has a strong R −3 dependence on the nanoparticle size, which is, however, different from that of the intraband scattering, Eq. (32).
The important difference between the interband and the intraband SP-mediated scattering rates lies in their energy dependence. Since the surface-induced potential, Eq. (34), allows for only vertical (dipole) interband single-particle excitations, the phase space for the scattering of a d-hole with energy E d α is restricted to a single final state in the conduction band with energy E c α . As a result, the d-hole scattering rate, γ s h (E d α ), exhibits a peak as the difference between the energies of final and initial states,
E cd α = E c α − E d α , approaches
the SP frequency ω s [see Eq. (36)]. In contrast, the energy dependence of γ s e is smooth due the larger phase space available for scattering in the conduction band. This leads to the additional integral over final state energies in Eq. (28), which smears out the SP resonant enhancement of the intraband scattering.
As we show in the next section, the fact that the scattering rate of a d-hole is dominated by the SP resonance, affects strongly the nonlinear optical dynamics in small nanoparticles. This is the case, in particular, when the SP frequency, ω s , is close to the onset of interband transitions, ∆, as, e.g., in Cu and Au nanoparticles. 1,11,13,15 Consider an e-h pair with excitation energy ω close to ∆. As we discussed, the d-hole can scatter into the conduction band by emitting a SP. According to Eq. (36), for ω ∼ ω s , this process will be resonantly enhanced. At the same time, the electron can scatter in the conduction band via the usual two-quasiparticle process. For ω ∼ ∆, the electron energy is close to E F , and its scattering rate is estimated as 28 γ e ∼ 10 −2 eV. Using the bulk value of µ, 2µ 2 /m ∼ 1 eV near the L-point, 29 we find that γ s h exceeds γ e for R < ∼ 2.5 nm. In fact, one would expect that, in nanoparticles, µ is larger than in the bulk due to the localization of the conduction electron wave-functions. 1
IV. SURFACE PLASMON NONLINEAR OPTICAL DYNAMICS
In this section, we study the effect of the SP-mediated interband scattering on the nonlinear optical dynamics in noble metal nanoparticles. When the hot electron distribution has already thermalized and the electron gas is cooling to the lattice, the transient response of a nanoparticle can be described by the time-dependent absorption coefficient α(ω, t),
given by Eq. (2) with time-dependent temperature. 30 In noble-metal particles, the temperature dependence of α originates from two different sources. First is the phonon-induced correction to γ s , which is proportional to the lattice temperature T l (t). As mentioned in the Introduction, for small nanoparticles this effect is relatively weak. Second, near the onset of the interband transitions, ∆, the absorption coefficient depends on the electron temperature T (t) via the interband dielectric function ǫ d (ω) [see Eqs. (2) and (3)]. In fact, in Cu or Au nanoparticles, ω s can be tuned close to ∆, so the SP damping by interband e-h excitations leads to an additional broadening of the absorption peak. 1 In this case, the temperature dependence of ǫ d (ω) dominates the pump-probe dynamics. Below we show that, near the SP resonance, both the temperature and frequency dependence of ǫ d (ω) = 1 + 4πχ d (ω) are strongly affected by the SP-mediated interband scattering.
For non-interacting electrons, the interband susceptibility,
χ d (iω) =χ d (iω) +χ d (−iω),
has the standard form 26
χ d (iω) = − α e 2 µ 2 m 2 (E cd α ) 2 1 β iω ′ G d α (iω ′ )G c α (iω ′ + iω),(37)
where G d α (iω ′ ) is the Green function of a d-electron. Since the d-band is fully occupied, the only allowed SP-mediated interband scattering is that of the d-hole. We assume here, for simplicity, a dispersionless d-band with energy E d .
Substituting G d α (iω ′ ) = [iω ′ − E d + E F − Σ d α (iω ′ )] −1 , with Σ d α (iω)
given by Eq. (35), and performing the frequency summation, we
obtainχ d (ω) = e 2 µ 2 m 2 dE c g(E c ) (E cd ) 2 f (E c ) − 1 ω − E cd + iγ s h (ω, E c ) ,(38)
where g(E c ) is the density of states of conduction electrons. Here
γ s h (ω, E c ) = ImΣ d (E c − ω)
is the scattering rate of a d-hole with energy E c − ω, for which we obtain from Eq. (36),
γ s h (ω, E c ) = − 9e 2 µ 2 m 2 (E cd ) 2 R 3 f (E c )Im 1 ǫ s (ω) ,(39)
where we neglected N(ω) for frequencies ω ∼ ω s ≫ k B T . Remarkably, γ s h (ω, E c ) exhibits a sharp peak as a function of the frequency of the probe optical field. The reason for this is that the scattering rate of a d-hole with energy E depends explicitly on the difference between the final and initial states, E c − E, as discussed in the previous section: therefore,
E c − E F < ∼ k B T .
Since the main contribution toχ ′′ d (ω) comes from energies E c − E F ∼ ω − ∆, the d-hole scattering becomes efficient for electron temperatures k B T > ∼ ω s − ∆. As a result, near the SP resonance, the time evolution of the differential absorption, governed by the temperature dependence of α, becomes strongly size-dependent, as we illustrate in the rest of this section.
In the numerical calculations below, we adopt the parameters of the experiment of Ref.
11, which was performed on R ≃ 2.5 nm Cu nanoparticles with SP frequency, ω s ≃ 2.22 eV, slightly above the onset of the interband transitions, ∆ ≃ 2.18 eV. In order to describe the time-evolution of the differential absorption spectra, we first need to determine the timedependence of the electron temperature, T (t), due to the relaxation of the electron gas to the lattice. For this, we employ a simple two-temperature model, defined by heat equations for T (t) and the lattice temperature T l (t):
C(T ) ∂T ∂t = −G(T − T l ), C l ∂T l ∂t = G(T − T l ),(40)
where C(T ) = ΓT and C l are the electron and lattice heat capacities, respectively, and G is the electron-phonon coupling. 31 The parameter values used here were G = 3.5 × 10 16
Wm −3 K −1 , Γ = 70 Jm −3 K −2 , and C l = 3.5 Jm −3 K −1 . The values of γ s and µ were extracted from the fit to the linear absorption spectrum, and the initial condition for Eq. (40) was taken as T 0 = 800 K, the estimated pump-induced hot electron temperature. 11 We then selfconsistently calculated the time-dependent absorption coefficient α(ω, t), and the differential transmission is proportional to α r (ω) − α(ω, t), where α r (ω) was calculated at the room temperature.
In Fig. 1 we plot the calculated differential transmission spectra for different nanoparticle sizes. Fig. 1(a) shows the spectra at several time delays for R = 5.0 nm; in this case, the SP-mediated d-hole scattering has no significant effect. Note that it is necessary to include the intraband e-e scattering in order to reproduce the differential transmission lineshape observed in the experiment. 11 For optically excited electron energy close to E F , this can be achieved by adding the e-e scattering rate 23
γ e (E c ) ∝ [1 − f (E c )][(E c − E F ) 2 + (πk B T ) 2 ]
to γ s h in Eq. (38). The difference in γ e (E c ) for E c below and above E F leads to a lineshape similar to that expected from the combination of red-shift and broadening.
In Figs. 1(b) and (c) we show the differential transmission spectra with decreasing nanoparticle size. For R = 2.5 nm, the apparent red-shift is reduced [see Fig. 2(b)]. This change can be explained as follows. Since here ω s ∼ ∆, the SP is damped by the interband excitations for ω > ω s , so that the absorption peak is asymmetric. The d-hole scattering with the SP enhances the damping; however, since the ω-dependence of γ s h follows that of α, this effect is larger above the resonance. On the other hand, the efficiency of scattering increases with temperature, as discussed above. Therefore, for short time delays, the in- In Fig. 2 we show the time evolution of the differential transmission at several frequencies close to ω s . It can be seen that the relaxation is slowest at the SP resonance; this characterizes the robustness of the collective mode, which determines the peak position, versus the single-particle excitations, which determine the resonance width. For larger sizes, at which γ s h is small, the change in the differential transmission decay rate with frequency is smoother above the resonance [see Fig. 2(a)]. This stems from the asymmetric lineshape of the absorption peak, mentioned above: the absorption is larger for ω > ω s , so that its relative change with temperature is weaker. For smaller nanoparticle size, the decay rates become similar above and below ω s [see Fig. 2(b)]. This change in the frequency dependence is related to the stronger SP damping for ω > ω s due to the d-hole scattering, as discussed above. Since this additional damping is reduced with decreasing temperature, the relaxation is faster above the resonance, compensating the relatively weaker change in the absorption.
This rather "nonlinear" relation between the time-evolution of the pump-probe signal and that of the temperature, becomes even stronger for smaller sizes [see Fig. 2(c)]. In this case, the frequency dependence of the differential transmission decay below and above ω s is reversed. Note, that a frequency dependence consistent with our calculations presented in Fig. 2(b) was, in fact, observed in the experiment of Ref. 11. At the same time, the changes in the linear absorption spectrum are relatively small.
V. CONCLUSIONS
To summarize, we have examined theoretically the role of size-dependent correlations in the electron relaxation in small metal particles. We identified a new mechanism of quasiparticle scattering, mediated by collective surface excitations, which originates from the surface-induced dynamical screening of the e-e interactions. The behavior of the corresponding scattering rates with varying energy and temperature differs substantially from that in the bulk metal. In particular, in noble metal particles, the energy dependence of the d-hole scattering rate was found similar to that of the absorption coefficient. This led us to a self-consistent scheme for the calculation of the absorption spectrum near the surface plasmon resonance.
An important aspect of the SP-mediated scattering is its strong dependence on size.
Our estimates show that it becomes comparable to the usual Fermi-liquid scattering in nanometer-sized particles. This size regime is, in fact, intermediate between "classical" particles with sizes larger than 10 nm, where the bulk-like behavior dominates, and very small clusters with only dozens of atoms, where the metallic properties are completely lost.
Although the static properties of nanometer-sized particles are also size-dependent, the deviations from their bulk values do not change the qualitative features of the electron dynamics. In contrast, the size-dependent many-body effects, studied here, do affect the dynamics in a significant way during time scales comparable to the relaxation times. As we have shown, the SP-mediated interband scattering reveals itself in the transient pumpprobe spectra. In particular, as the nanoparticle size decreases, the calculated time-resolved differential absorption develops a characteristic lineshape corresponding to a resonance blueshift. At the same time, near the SP resonance, the scattering leads to a significant change in the frequency dependence of the relaxation time of the pump-probe signal, consistent with recent experiments. These results indicate the need for a systematic experimental studies of the size-dependence of the transient nonlinear optical response, as we approach the transition from boundary-constrained nanoparticles to molecular clusters.
the numerator in Eq. (24) contains a term proportional to the gradient of the confining potential, which peaks sharply at the surface. The corresponding contribution to the matrix element describes the surface scattering of an electron making the L-pole transition between the states |c, α and |c, α ′ , and gives the dominant term of the expansion. Thus, in the leading order in
represents a channel of electron scattering mediated by a collective surface mode with angular momentum L. For low L, the difference between the energies of modes with successive values of L is larger than their widths, so that the different channels are well separated. Note that since all ω L are smaller than the frequency of the (undamped) bulk plasmon, one can replaceǫ L (ω) by ǫ L (ω) in the integrand of Eq. (28) for frequencies ω ∼ ω L .
for a d-hole with energy E = E c − ω, the dependence on the final state energy, E c , cancels out in ǫ s (E c − E) [see Eq. (36)]. This implies that the optically-excited d-hole experiences a resonant scattering into the conduction band as the probe frequency ω approaches the SP frequency. It is important to note that γ s h (ω, E c ) is, in fact, proportional to the absorption coefficient α(ω) [see Eq. (2)]. Therefore, the calculation of the absorption spectrum is a self-consistent problem defined by Eqs. (2), (3), (38), and (39). It should be emphasized that the effect of γ s h on ǫ ′′ d (ω) increases with temperature. Indeed, the Fermi function in the rhs of Eq. (39) implies that γ s h is small unless
crease in the absorption is relatively larger for ω > ω s . With decreasing size, the strength of this effect increases, leading to an apparent blue-shift [see Fig. 2(c)]. Such a strong change in the absorption dynamics originates from the R −3 dependence of the d-hole scattering rate; reducing the size by the factor of two results in an enhancement of γ s h by an order of magnitude.
FIGURESFIGFIG. 2 .
2Temporal evolution of the differential transmission at frequencies close the SP resonance for nanoparticles with (a) R = 5 nm, (b) R = 2.5 nm, and (c)
The authors thank J.-Y. Bigot for valuable discussions. This work was supported by NSF CAREER award ECS-9703453, and, in part, by ONR Grant N00014-96-1-1042 and by Hitachi Ltd.
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| [] |
[
"Physical geometry of the quasispherical Szekeres models",
"Physical geometry of the quasispherical Szekeres models"
] | [
"Robert G Buckley \nUniversity of Texas at San Antonio\n\n",
"Eric M Schlegel \nUniversity of Texas at San Antonio\n\n"
] | [
"University of Texas at San Antonio\n",
"University of Texas at San Antonio\n"
] | [] | The quasispherical Szekeres metric is an exact solution to Einstein's equations describing an inhomogeneous and anisotropic cosmology. Though its governing equations are well-known, there are subtle, often-overlooked details in how the model's functions relate to its physical layout, including the shapes and relative positions of structures. We present an illustrated overview of the quasispherical Szekeres models and show exactly how the model functions relate to the physical shape and distribution of matter. In particular, we describe a shell rotation effect that has not previously been fully understood. We show how this effect relates to other known properties, and lay out some mathematical tools useful for constructing models and picturing them accurately. | 10.1103/physrevd.101.023511 | [
"https://arxiv.org/pdf/1908.02697v3.pdf"
] | 199,472,828 | 1908.02697 | bdc2e7db87f9dd732aa14b363c9cfe838259a9d1 |
Physical geometry of the quasispherical Szekeres models
(Dated: August 8, 2019)
Robert G Buckley
University of Texas at San Antonio
Eric M Schlegel
University of Texas at San Antonio
Physical geometry of the quasispherical Szekeres models
(Dated: August 8, 2019)PACS numbers: 0420Jb, 9880-k Keywords: inhomogeneous universe models
The quasispherical Szekeres metric is an exact solution to Einstein's equations describing an inhomogeneous and anisotropic cosmology. Though its governing equations are well-known, there are subtle, often-overlooked details in how the model's functions relate to its physical layout, including the shapes and relative positions of structures. We present an illustrated overview of the quasispherical Szekeres models and show exactly how the model functions relate to the physical shape and distribution of matter. In particular, we describe a shell rotation effect that has not previously been fully understood. We show how this effect relates to other known properties, and lay out some mathematical tools useful for constructing models and picturing them accurately.
I. INTRODUCTION
As cosmological observations become ever more precise, it becomes insufficient to describe structures and their evolutions by approximate means, such as perturbation theory (which fails to capture nonlinear structure growth at late times) and N-body simulations (which do not typically fully incorporate general relativity). For this reason, interest is growing in inhomogeneous models based on exact solutions to Einstein's equations. Although these models have restrictions in the shapes of structures allowed, they are useful because they allow direct, exact calculations of observables such as lensing convergence, angular diameter distance, and redshift, free from approximations or ambiguities.
The most extensively studied inhomogeneous exact solution is the Lemaître-Tolman (LT) class of models [1][2][3]. 1 This describes a spherically symmetric configuration of pressureless dust matter, with an optional cosmological constant. It provides a highly idealized simulation of simple structures such as cosmic voids, allowing us to calculate their observational effects. Many studies have examined LT models in various configurations .
However, the symmetry of the LT metric severely constrains the types of structures that it can model. The real universe contains structures that are not so simple, consisting of complex arrangements of walls and voids. The asymmetry of these arrangements can qualitatively alter the evolution and observational effects in ways the LT models cannot capture. A more advanced metric capable of better representing such structures is the quasispherical subclass of the Szekeres metric [31]. This metric is a generalization to the LT metric, breaking the sym- * [email protected] † [email protected] 1 Some refer to these as Lemaître-Tolman-Bondi (LTB) models, though Bondi's contribution came fourteen years after the initial discovery. metry and allowing for much greater variety. Many studies have examined Szekeres models and employed them to investigate various cosmological observables (e.g., [32][33][34][35][36][37][38][39][40][41][42][43][44][45]), and others have explored their general mathematical and geometric properties (e.g., [46][47][48][49][50][51][52][53][54][55]). The mathematics involved in the models are well-developed enough to run simulations, generate data, and make predictions. Describing and visually depicting the actual shapes of structures in such simulations, though, remains a challenge. The mathematical complexity is a barrier to intuition.
The purpose of this paper is to clarify and illustrate the geometry of the Szekeres models, showing the connections between the model functions and the physical picture the models represent. Some of this is merely elaborating on already-known features, but we also describe a shell rotation effect that has not yet been fully understood. We will show how this fits in with other properties of the model. We will also provide some mathematical tools for building and adjusting models, as well as for handling calculations within these models.
This paper is organized as follows. In section II we lay out the metric and evolution equations defining the Szekeres models. We also explain the role of each function and how they relate to simpler models. We describe the two coordinate systems most commonly used in Szekeres models in section III, and show how they relate to one another. In section IV we briefly go over some of the physical restrictions on the model functions. In section V we go into greater detail explaining how the asymmetric shape and physical properties of the metric relate to the model-defining functions. In section VI we present special cases which result in partial symmetry, laying out the mathematical conditions and discussing some of the benefits of such models. In section VII we provide some useful equations, including geodesic equations and coordinate transformations, as well as outlines of numerical methods that can be used to visualize the model's spatial arrangement. We show an alternative picture in section VIII, in which the model is seen as a hypersurface embedded in a higherdimensional background manifold. Finally, we make our closing discussion in section IX.
II. BASIC MODEL DEFINITIONS
The Szekeres metric is a generalization of the LT metric, introduced in 1975 by P. Szekeres [31]. Like the LT metric, it contains only a comoving, irrotational, pressureless dust, and optionally a cosmological constant. Its constant-t hypersurfaces are conformally flat [46], and it has no gravitational radiation due to a lack of changing quadrupoles [47], but unlike LT, the Szekeres metric does have dipoles in its matter distribution. In general, it has no symmetry; there are no Killing vectors, except in special cases [48].
There are three subtypes of Szekeres models, each with different geometry: quasispherical, quasiplanar, and quasipseudospherical (or quasihyperbolic). They are described by the metric
ds 2 = −dt 2 + (R − R E E ) 2 − k dr 2 + R 2 E 2 (dp 2 + dq 2 ). (1)
Primes denote partial derivatives with respect to the radial coordinate r, and = +1, 0, or −1, corresponding to the quasispherical, quasiplanar, and quasipseudospherical cases respectively. The quasispherical subtype has attracted the most attention, as it is simplest to understand, includes LT as a special case (allowing for direct comparisons), and is best suited to describing localized structures. From here on, when we refer to the Szekeres models, we will mean the quasispherical subtype. The Szekeres metric is remarkably similar to the LT metric. Like LT, the Szekeres spacetime consists of a series of spherical shells labeled by the coordinate r, and the functions R = R(t, r) and k = k(r) play the same roles as in LT-that is, R represents the proper areal radii of the shells, and k is related to the local spatial curvature. The coordinates on the shell, p and q, relate to the more familiar θ and φ by a stereographic projection, as we will explain in section III; the shells are still perfectly spherical. The only real difference comes from the function E = E(r, p, q), which describes the departure from LT-unlike LT, the shells are not concentric, nor is matter distributed evenly across a given shell. The function E(r, p, q) is defined in terms of three arbitrary functions of r as E(r, p, q) = [p − P (r)] 2 + [q − Q(r)] 2 + S(r) 2 2S(r) .
(We will hereafter omit the factor, as we are focusing here on the = +1 case.) We will refer to the functions P (r), Q(r), and S(r) as the "dipole functions". Together, they describe a dipolar asymmetry that can change from shell to shell. The details of this asymmetry are subtle, and will be discussed in detail in section V. For now, we note that when S = P = Q = 0 (meaning E = 0 as well), the metric reduces to the LT metric. Einstein's equations applied to this metric reduce to two useful equations. The first describes the evolution of R over time:
1 c 2Ṙ (t, r) 2 = 2M (r) R(t, r) − k(r) + 1 3c 2 Λ R(t, r) 2 ,(3)
where a dot indicates a partial derivative with respect to t. The function M (r) arises as an integration constant, and it gives the total effective gravitational mass inside the shell. Λ is the usual cosmological constant. This equation has exactly the same form as the evolution equation for LT models, meaning that the asymmetries are arranged in such a way that they do not affect the evolution of the shells. (Apostolopoulos shows how this arises mathematically, through a decoupling of the evolution equations from the spatial divergence and curl equations [49].) In fact, a close comparison can further be drawn with the first Friedmann equation. R(t, r) functions much like the scale factor, only with the addition of r-dependence, and the mass and curvature functions M (r) and k(r) play similar roles to the density (times a factor of 4πG R 3 /3) and curvature of a pressureless Friedmann-Lemaître-Robertson-Walker (FLRW) universe. Each individual shell of constant r evolves just as if it was a slice of an FLRW universe, albeit a different FLRW universe for each shell. Therefore, the solution to Eq. (3) can be written as a generalized form of the solution to the Friedmann equation, as shown in appendix A. R evolves in a hyperbolic fashion where k < 0, parabolic where k = 0, and elliptic where k > 0.
The second relation from Einstein's equations gives the mass density:
4π G c 2 ρ(t, r, p, q) = M (r) − 3M (r) E (r,p,q) E(r,p,q) R(t, r) 2 R (t, r) − R(t, r) E (r,p,q) E(r,p,q) .(4)
From here on, we will use units in which the gravitational constant G and speed of light c both equal unity. Note that the function M (r) is different from the threedimensional integral of ρ (even though, as we will soon see, E /E averages to 0 across any shell), since Eq. (4) omits the curvature factor from the metric (1). Nevertheless, it is M (r) that guides the gravitational evolution (3). This is why we call M (r) the "effective gravitational mass" instead of the "total mass". Equation (4) reduces to the LT mass density in the case E /E = 0. The asymmetric modifications will be explained more intuitively as we build our understanding of the E function.
Integrating Eq. (3) with respect to t gives us another free function of r as an integration constant:
t − t B (r) = R(t,r) 0 d R 2M (r)/ R − k(r) + Λ R 2 /3 .(5)
The function t B (r) is called the "bang-time function," as it gives the time of the singularity R = 0 for any given shell. 2 Because the shells evolve independently from each other, they are mathematically allowed to begin their evolution at different times. That is, unlike the FLRW model which has all of space emerge from the big bang singularity simultaneously, Szekeres models (as well as LT models) can have different shells emerge at different times-outer shells can already be expanding while inner shells remain in the singularity, as shown in Fig. 1. While this picture is a conceptual departure from the Standard Model's homogeneous big bang, a nonsimultaneous big bang is actually the more generic case, and the Standard Model only avoids it with a great deal of tuning. Nevertheless, remember that LT and Szekeres models are only valid in regimes where the energy content is well-approximated as pressureless dust. If we look back too close to the big bang, we reach the point of matter-radiation equality, by which point this approximation fails. These models, then, can only realistically describe what happens later. Because the big bang itself is before the time of the models' validity, the bang-time function does not necessarily reflect the precise nature of the big bang. Rather, it encodes velocity and density variations near the beginning of the time when the Szekeres model applies. It can be used to simulate decaying modes of perturbations, which contribute extreme density contrasts near the big bang but are insignificant at late times [56][57][58]. The magnitudes of decaying modes are severely limited by CMB observations and standard models of inflation, so many authors take t B = 0 [23-27, 32, 37, 42, 43, 58-61], but others argue this assumption is not sufficiently justified [19,20,62].
We would appear to have six free functions of r: M , k, and t B shared with LT models, plus the three dipole functions. However, since r is merely a label, it carries with it a gauge freedom; all of the above equations are invariant under transformations of the form r = f (r) (as long as f is monotonic). This means that we can impose a definition of our choosing on one function of r, granting some welcome convenience. For instance, in some situations, it is advantageous to define M (r) = r 3 , or even M (r) = r. Alternatively, we could define R(t 0 , r) = r, where t 0 is the present time, which would make any one of the three free functions determined by knowledge of the other two by solving Eqs. (3) and (5). Any choice restricts the range of possible models somewhat-a choice of M (r) = r 3 cannot accommodate a model containing a vacuum over some range of r, and defining R(t 0 , r) = r rules out geometries in which R is not monotonic in r, such as a closed universe or wormhole topology. But wide ranges of models remain available given any choice, and every model has gauge choices available. Regardless of our gauge choice, we end up with five functional degrees of freedom with which to define a particular model.
III. COORDINATE SYSTEMS
There are two different basic coordinate systems that are useful for handling the dimensions along the spherical surfaces: projective coordinates and spherical coordinates. Each has advantages that make it useful in different situations.
A. Projective coordinates
These are the most commonly used angular coordinates for Szekeres models, and the ones we have used in the previous section. Usually labeled p and q, these coordinates map to the sphere by a simple Riemannian stereographic projection, as illustrated in Fig. 2. Lines emerge from a projection point at the "top" the sphere, and intersect both the sphere and a two-dimensional (2D) projection plane at one point each. The plane is a distance S(r) "below" the projection point (in arbitrary units), and it is marked by a Cartesian grid with its origin displaced by (−P, −Q). Because these are functions of r, the coordinates can map differently on different shells.
At the projection point, the coordinates diverge. There is in general nothing physically special about this region, only a coordinate singularity, but this can cause problems for numerical calculations. And as we will see in section V, the relative orientation of this point on different shells is not constant. The convenience of these projective coordinates centers on the simplicity of the metric, Eq. (1). It is diagonal in this form, with reasonably simple components. It results in simple geodesic equations and Riemann and Ricci tensors (see appendix B), convenient for calculations. However, these coordinates are somewhat opaque to intuition. The shape of the E function (2) on the sphere, for instance, is not immediately obvious. We can tell that it has a minimum at (p, q) = (P, Q), and increases away from this point (assuming S is positive) proportionally to the square of the distance in the projective plane, but it is less obvious how this maps to the sphere.
More important than E itself, though, is the combination E /E. In projective coordinates, it takes the form
E E = −2 P (p − P ) + Q (q − Q) − S S (p − P ) 2 + (q − Q) 2 + S 2 − S S .(6)
The solution to E /E = 0 traces a circle in the projective plane, centered at
(p c , q c ) = P − P S S , Q − Q S S ,(7)
with radius
L = S S P 2 + Q 2 + S 2 .(8)
In the special case S = 0, the solution is a line instead of a circle. In either case, the line or circle demarcates a boundary between a region in which E /E is positive and one in which it is negative. See [54] for further discussion.
A circle in a stereographic projection maps onto a circle on the sphere, and vice versa. The E /E = 0 circle is special, though-it is a great circle on the sphere. It will be useful to note some properties of great circles and their mappings to the projective plane. A great circle in the projective plane is identified by a relation between the circle's center point and its radius:
L 2 gc = (p c − P ) 2 + (q c − Q) 2 + S 2 ,(9)
where (p c , q c ) is the center of the circle in the projective plane. This does not, however, correspond to the center of the circle on the sphere. On the sphere, there are two center points. The great circle can be seen as the intersection of the sphere with a plane through the sphere's center, and a line orthogonal to the plane through the sphere's center intersects the sphere at these two points, with projective coordinates
(p o − P, q o − Q) = −S 2 ± S L gc (p c − P ) 2 + (q c − Q) 2 (p c − P, q c − Q).(10)
We will soon see that for the E /E = 0 great circle, these orthogonal points are the locations of the extrema of E /E. The relative positions of these points in the projective plane are illustrated in Fig. 3. It will also be useful to note that for any (p 1 , q 1 ), the antipodal point has coordinates
(p 2 , q 2 ) = (P, Q) − S 2 (p 1 − P, q 1 − Q) (p 1 − P ) 2 + (q 1 − Q) 2 .(11)
B. Spherical coordinates
We can bring the coordinates to a more familiar form with a simple transformation: 3
p − P = S cot θ 2 cos φ, (12a) q − Q = S cot θ 2 sin φ.(12b)
The coordinates θ and φ describe the same geometry, but in terms of latitude and longitude instead of stereographic projection, with the pole θ = 0 corresponding to the projection point, where p and q diverge. In these spherical polar coordinates, the metric is significantly more complicated and no longer diagonal: 4
ds 2 = −dt 2 + (R − R E E ) 2 1 − k + R 2 (1 − cos θ) 2 P 2 + Q 2 + S 2 S 2 − 2 1 − cos θ S S E E dr 2 + 2 R 2 sin θ 1 + cos θ E E − S S dr dθ + 2 R 2 sin 3 θ 1 + cos θ Q S cos φ − P S sin φ dr dφ + R 2 (dθ 2 + sin 2 θ dφ 2 ).(13)
Though this appears far more complicated, in some ways these coordinates provide greater clarity. For instance, note that if we set dr = 0, the spatial parts reduce to the metric of a 2-sphere, making it easier to see that surfaces of constant r are indeed spherical (with radius R) regardless of anisotropies induced by E. The E function itself takes a simpler form in spherical coordinates:
E(r, θ, φ) = S(r) 1 − cos θ .(14)
While simpler, this form does not offer much immediate insight. Recall, though, that the E function only affects the density (4) through E /E. The metric in spherical coordinates only contains E in the form of E /E as well. This important expression takes an evocative form in spherical coordinates:
E E = − S cos θ + (P cos φ + Q sin φ) sin θ S .(15)
This makes it clear that P /S defines an anisotropy in the direction of (θ, φ) = (π/2, 0), Q /S in the direction 3 A similar transformation using tan instead of cot is possible as well. This is equivalent, only with the projection point placed at θ = π. 4 The partial r derivatives here are still taken with p and q held constant, not θ and φ.
(π/2, π/2), and S /S in the direction θ = 0-what we would call the "x", "y", and "z" directions in rectangular coordinates. While these rectangular coordinates are valid only on a given shell, not globally across the model, it will nevertheless sometimes be useful to refer to a "local rectangular frame" (or LRF). 5 That is, in the LRF,
x ≡ sin θ cos φ, y ≡ sin θ sin φ, z ≡ cos θ. (16) We can also see from Eq. (15) that E /E has a dipolar shape over the sphere (see [52] for a detailed derivation), ranging from a maximum value of
E E max = (P ) 2 + (Q 2 ) + (S ) 2 S(17)
to the negative of this same value at the antipodal point. The directions of these angular extrema are given by
θ max = cos −1 − S P 2 + Q 2 + S 2 ,(18a)φ max = − sign Q cos −1 − P P 2 + Q 2(18b)
(taking S to be positive for all r). This is simply the set of spherical coordinates for the point (x, y, z) max = (P , Q , S )/ P 2 + Q 2 + S 2 in the LRF. The corresponding projective coordinates are
p max = P − P S /S + (E /E) max , (19a) q max = Q − Q S /S + (E /E) max ,(19b)
and p min and q min are related in the same way to
(E /E) min = −(E /E) max .
It is straightforward to see that these correspond to the great circle orthogonal points defined in Eq. (10) for the E /E = 0 great circle, Eqs. (7)- (8). We see that the geometry of E /E on a single 2-sphere shell is quite simple. It is fully described by a dipole, symmetric about the two extrema and vanishing on the great circle midway between them. The magnitude and orientation of this dipole can change from one shell to the next, though, so the model as a whole is generally not symmetric.
IV. PHYSICAL RESTRICTIONS
While the Szekeres model has five functional degrees of freedom, the domains from which these functions can be chosen are not unlimited. They must satisfy certain conditions to avoid singularities and other pathological behavior. A few such restrictions are straightforward:
• All of the free functions should be continuous.
• In order to maintain the Lorentzian signature of the metric, we must have k ≤ 1. Equality can only occur when R − R E /E = 0. Since this must hold for all (p, q), this means R = P = Q = S = 0, and subsequently M = 0 to keep the density finite. This scenario is a regular maximum or minimum (i.e. a belly or neck), which can occur in closed models or wormhole topologies. They are further discussed in [54].
• Since shells have positive size, R ≥ 0.
• S = 0, or else E diverges and the projective mapping fails. Since S must be continuous to keep S finite, this means that S cannot change sign. By convention, we choose S > 0.
A few others require a little more explanation.
A. Origin conditions
An origin in Szekeres models, as in LT, is a value of r = r 0 at which the size of the shell R(t, r 0 ) vanishes, besides the Big Bang or Big Crunch. A model with a wormhole topology has no origin [54], but for those that do, it is usually put at r = 0 for all t. A closed universe model has a second origin at some other r.
Maintaining regularity at the origin(s) requires the density and curvature (in terms of the Kretschmann scalar, see appendix B) to remain finite, except at the bang or crunch singularities. The time evolution at the origin should also smoothly match its surroundings. Hellaby and Krasiński [54] have studied these quantities in the limit r → r 0 to obtain regularity conditions on the model functions near the origin:
M ∼ R 3 , k ∼ R 2 , (S, P, Q) ∼ R n ,(20)
where n ≥ 0.
B. Non-negative mass density
Aside from a recent controversial proposal [63], cosmology typically does not involve negative masses, and it is unclear how negative mass would behave gravitationally if it did exist. It is therefore common to require that the mass density is non-negative everywhere. 6 In LT models, this simply means that M and R must have the same sign. In Szekeres models, this condition is still necessary, but no longer sufficient. The full condition is
M − 3M E E ≥ 0,(21)or ≤ 0 if R − RE /E ≤ 0 as well.
This can be translated into a direct restriction on the derivatives of the dipole functions. For Eq. (21) to hold across an entire shell, it suffices to satisfy the equation at the angular maximum of E /E, which is given in Eq. (17). We therefore have
(P ) 2 + (Q 2 ) + (S ) 2 S ≤ M 3M .(22)
C. Shell crossing
In LT models, shell crossing may occur when R = 0 at any r and t. Unless this is a regular extremum (requiring that both 1 − k(r 0 ) and M (r 0 ) equal zero), multiple shells are occupying the same space, causing the density to become infinite across the entire shell (unless M = 0 as well). This happens when one shell expands (or contracts) through another shell (and all shells in between). This is a physically unrealistic picture-in the real world, pressure would become non-negligible before this point, preventing the singularity from forming. Because the model does not include pressure, it loses validity in this situation.
In Szekeres models, shell crossing can occur even if R remains positive, if R − R E /E = 0 at any point in the full four-dimensional spacetime, besides at a regular extremum. Unlike in LT models, the singularity is not in general a full sphere, but rather begins as a point where the shells meet on one side. The surface of intersection grows from this point. On any given shell r, the shell crossing (if any) traces a circle parallel to the E = 0 great circle.
Because they are not physically realistic, we typically want to construct our models in such a way that no shell crossings occur, at least in the region of interest. The basic requirement is that there are no solutions to R − R E /E = 0, which means
(P ) 2 + (Q 2 ) + (S ) 2 S ≤ R R .(23)
For a more thorough discussion, including conditions on M , k, and t B which ensure Eq. (23) holds globally, see [54].
V. EFFECTS OF THE DIPOLE FUNCTIONS
We have seen that the three dipole functions are solely responsible for the anisotropy in the model, each defining anisotropies in orthogonal directions. Although we have laid out their effects on the metric and on the density at a surface level, we have been vague so far about what is actually going on-what sort of physical anisotropies the dipole functions create. In this section we will build a more complete intuitive understanding of their physical effects, to the point where we can draw physically accurate pictures of the model.
The primary effects of the dipole functions are threefold: they shift the shells relative to each other, rotate their axes, and redistribute matter. We briefly explained these effects previously in [45], but we shall examine them in greater depth here.
A. Shell shifting
The first effect is that the dipole functions displace the centers of the shells-that is, the shells are nonconcentric. Specifically, the shell at r + δr is shifted relative to the shell at r in the direction (θ, φ) = (π/2, 0) by a distance
δ x = R P /S √ 1 − k δr,(24)δR + Δ δR -Δ x x FIG. 4.
A simple illustration of how the dipole functions displace shells relative to each other. The geometric centers of the two shells are marked by 'x's. If the shifting between the two shells is all in the same direction, i.e. axially symmetric, the total shifting amount is ∆ = (R (E /E)max)/ √ 1 − k dr between the two shells.
in the direction (π/2, π/2) by
δ y = R Q /S √ 1 − k δr,(25)
and in the direction (0, 0) by
δ z = R S /S √ 1 − k δr,(26)
where the "x", "y", and "z" subscripts refer to the axes of the shell's LRF. This is most easily seen by examining the metric in projective coordinates, Eq. (1). Because the metric is diagonal in these coordinates, the proper distance along a line connecting an arbitrary point to a nearly-adjacent shell is minimized by holding p and q constant, and this distance can directly be seen in g rr . Without the dipole functions, this distance is simply R dr/ √ 1 − k. The dipole functions add only the term −R E /E to the numerator. From the shape of E /E discussed in section III, we can see that this gives a dipolar modulation about (θ max , φ max ), defined in Eq. (18). This corresponds to a relative displacement of the shells, as we have described. The shells are farther apart in the direction of the shifting, and closer together in the opposite direction. This is illustrated in Fig. 4.
This effect can be considered to be responsible for the E /E term in the denominator of the mass density func- tion, Eq. (4). Where the shells are compressed, the density increases proportionally, and likewise it decreases where the shells are stretched apart. The term in the numerator is a separate effect, indicating that the matter distribution of each shell is not held fixed as they are shifted around. This is further discussed in section V-C.
The shell shifting effect makes it necessary to be especially careful of shell crossing singularities. We already saw in section IV-C that Szekeres models run into shell crossings more easily than LT models. Now, we can more clearly picture why this is the case. Because the shells are non-concentric, smaller shells can intersect larger ones if we are not careful. The surface of intersection across all shells can form a non-trivial shape, an example of which is illustrated in Fig. 5.
B. Shell rotation
The second effect is more subtle. It has been noted by Hellaby [55] that the orientations of the coordinates appear to change from shell to shell, as evidenced by the rotation of the orthonormal tetrad along spatial paths. This property seems to have been overlooked by many authors. We have determined exactly how the shells' coordinates are rotated in terms of the dipole functions, which we briefly noted in [45], and expand upon here.
The dipole functions change the shells' spherical coordinate frames as follows: the shell r + δr is rotated relative to shell r by δr P /S about the point (π/2, −π/2) (the −y axis in the shell's LRF) and by δr Q /S about the point (π/2, 0) (the x axis). A derivation proving that this rotation, combined with the shell shifting, results in the given metric in spherical coordinates, Eq. (13), can be found in appendix C.
It may seem as though this is merely a coordinate effect, only present when using spherical coordinates. However, even in the projective coordinates, this effect is built into the behavior of the dipole functions. The directions along which the dipole functions act (the axes of the LRF) are rotated by this effect. For this reason, a model with only S varying with r (P and Q constant) is axially symmetric, whereas a model with only P or Q varying is not. For instance, if we set S = Q = 0, and P = 0, anisotropies will all be oriented along the "x" direction in each shell's LRF, but the shell rotation will cause the "x" direction to change from shell to shell, resulting in the structures being smeared over a range of angles. Figure 6 illustrates such a case, showing how both the shifting and rotation affect the physical layout of the model. Note that with shifting alone, the geodesic line appears very curved, but it is nearly straight when the rotation is taken into account. 7 The reason this rotation effect is necessary can be explained in terms of the relation between the projective and spherical coordinates. Because the metric in projective coordinates, Eq. (1), is diagonal, we know that the shortest line connecting a point on a shell to a nearby shell is one with constant p and q. This is not in general true for the spherical coordinates, with one important exception. The point θ = 0 serves as the projection point, and therefore always corresponds to the point where p and q diverge. Indeed, since the off-diagonal terms g rθ and g rφ both include a sin θ factor, the spherical metric is diagonal at this point. At θ = 0, then, the line of constant θ and φ must also correspond to the shortest connecting line between two shells. Due to the shell shifting, this forces one to be rotated relative to the other, as illustrated in Fig. 7.
Note that the magnitudes of shifting, Eqs. (24)-(26) have a factor of R(t, r), whereas the rotation magnitude does not. Unlike the shell shifting effect, the shell rotation is not time-dependent. The relative orientation of shells is maintained as they evolve. This missing factor also means that the shell rotation generally decreases in magnitude with increasing r, as P /S and Q /S are limited by the shell crossing condition, Eq. (23).
C. Matter distribution
The third effect is not on the metric itself, but rather on the matter distribution. We saw how the shell shift- incorporating the shell shifting. The true width of the dense wall structure is more clearly seen, narrower than it seemed before. The geodesic still does not appear straight. (c): incorporating shell rotation. The wall is distorted, and the geodesic is now very nearly straight. This is the most physically accurate method of plotting in a two-dimensional image. The same method was used in the figures in the rest of this paper and [45], and is explained in more detail in section VII-B.
ing effect is reflected in the denominator of the density function, Eq. (4), but this cannot explain the term in the numerator. This term represents a dipolar redistribution of matter on each shell, along the same direction as the shell shifting. This redistribution is necessary to preserve the spherical shape and FLRW-like expansion of the shells. Even though the density in a shell's interior is not symmetric, it must always be arranged in a particular way such that the total gravitational effect on the shell allows it to expand uniformly, maintaining its spherical shape. This peculiar arrangement is what makes it possible for the model to be mathematically tractable.
The total impact on the density across a shell depends on how that shell's density in the corresponding LT model, ρ LT , compares to the effective average density in its interior,ρ int . Specifically, if for a given shell
ρ LT ≡ M 4πR 2 R > 3M 4πR 3 ≡ρ int ,(27)
then the overall density will be greatest on the side where the shells are compressed, and least on the opposite side. If, instead, the inequality is reversed, the density will be less where the shells are compressed and greater where they are stretched apart. The former case commonly occurs when modeling a void with secondary structures in and around it. Because a void's density profile increases from the center, every shell has a greater density than the interior. Therefore, Szekeres shifting results in a sharp, thin overdense wall on one side (narrow because of the shell compression), with a broader, shallower underdensity on the opposite side.
Conversely, models with a central overdensity will see a perhaps less realistic arrangement of deep, narrow voids opposite thicker density bulges. Suppose we hold the LTB model functions constant while increasing the magnitudes of the dipole functions. As the shell shifting brings the shells close together on one side, the density contrast on the compressed side approaches positive or negative infinity, depending on whether the shell is overdense or underdense compared to the interior (or it stays constant, if ρ LT =ρ int ). Restricting the functions to being below the shell crossing limit, Eq. (23), imposes a limit on the density on the stretched side, equal to halfway between ρ LT andρ int : 8
ρ min/max = M 8πR 2 R + 3M 8πR 3 .(28)
Whether this is an upper limit or a lower limit depends on how the shell's density compares to the interior. In either case, though, this means that a shell that is (over/under)dense anywhere is (over/under)dense everywhere, relative to the interior. If the two quantities ρ LT andρ int are equal, then the density on the shell will be uniform regardless of the dipole functions. This means that if we begin with a homogeneous background LT model (equivalent to FLRW), we cannot create inhomogeneities through the dipole functions alone. This presents a challenge if we wish to describe a complex arrangement of structures while maintaining homogeneity on large scales. To produce an overdensity on a section of a shell, the density across the entire shell must be greater than the average interior density. This can still be achieved without a single dominant central void (or overdensity), for instance by using an oscillating radial density profile, as in Sussman's prescription of periodic local homogeneity (PLH) [64].
In any case, the angular extrema of the density (and other scalar quantities) always coincide with the angular extrema of E /E-the locations of maximum stretching and compression. Extending our view to the full threedimensional (3D) space, the radial positions of the extrema are more difficult to find; ρ = 0 is generally a difficult equation to solve, and some of the solutions are saddle points rather than maxima or minima (see [64] for further discussion). Furthermore, as this equation is time-dependent, the extrema are not generally comoving. Different scalars can have extrema at different positions, but all scalars have an extremum at the origin [64]. Expansion rates and density profile in a onedimensional slice through the center of a typical void-and-wall model.
D. Secondary effect: expansion rates and shear
In FLRW models, the expansion rate is characterized by a single Hubble value, the same in all locations and all directions.
θ FLRW = 3H FLRW = 3ȧ a(29)
In LT models, the expansion rate is modified not only by making it a function of the radial coordinate, but also by splitting it into two different rates: the transverse expansion H ⊥ (expansion in the two directions along the shell's surface) and the longitudinal expansion H (expansion between shells, along the line of sight of an observer at the center).
H ⊥,LT =Ṙ R ,(30)H ,LT =Ṙ R .(31)
In Szekeres models, the transverse expansion rate is unchanged, because each shell evolves the same as in the corresponding LT model. The longitudinal (or radial) expansion, however, is modified due to the shell shifting.
H =Ṙ −Ṙ E E R − R E E .(32)
These two expansion rates are plotted in Fig. 8 for an example void-and-wall model. The overall expansion is given by the trace of the gradient of the matter velocity field, θ = u α ;α (where a semicolon denotes a covariant derivative with respect to the following coordinate). In comoving coordinates (as we are using), the matter flow u α simply equals (1, 0, 0, 0). In terms of H and H ⊥ , the scalar expansion equals θ = 2H ⊥ + H .
This shows the overall rate of change of the size of a region. To see how the shape of a region changes, we look at the symmetric traceless component of the velocity field gradient, the shear, defined as σ α β = h αγ u (γ;β) − 1 3 θh α β , where h αβ = g αβ + u α u β . In Szekeres models,
σ α β = 1 3 (H − H ⊥ )diag(0, 2, −1, −1).(34)
The shear tells how space is stretched differently in different directions. The antisymmetric part of the velocity field gradient, the rotation, vanishes in Szekeres models:
ω α β = h αγ u [γ;β] = 0.(35)
This is what is meant by the models being "irrotational". Because of this property, the Szekeres metric is incapable of modeling virialized structures. This is one of its most important limitations, but it allows the mathematics to remain tractable.
VI. SYMMETRY CONDITIONS
Most choices for the dipole functions result in models with no global symmetry at all. Setting all three to constants, however, reduces the metric to the LT metric, which possesses full spherical symmetry. There are also special cases with intermediate levels of symmetry.
A. Axial symmetry
If the Szekeres dipole lies along the same direction at all r, the overall model will be axially symmetric. Such a model will have a Killing vector field reflecting its single continuous rotational symmetry.
In some cases, it is advantageous to construct a model to be axially symmetric. While this reduces the freedom of the model, it also reduces the number of variables that must be considered, both in the model's definition and in simulations of observations. One can generate a complete picture of an axial observer's sky by only varying one angle in the geodesics. Furthermore, radial geodesics along an axis of symmetry have the special property of always staying on the axis, greatly simplifying the geodesic equation calculations, as we will show in section VII-C. In fact, this is the only case in which a purely radial geodesic is possible in a Szekeres model [65]. 9 Because the P and Q functions rotate the frames of the shells relative to each other, a model with only P or only Q nonvanishing is not axially symmetric. A model with only S nonvanishing, however, is symmetric, with the axis of symmetry passing through θ = 0 and θ = π on all shells. This is the simplest kind of axially symmetric model, having no shell rotation. Unfortunately, the p and q coordinates diverge on half of the symmetry axis, which can be inconvenient for calculations.
Models symmetric about other axes are possible, though not as simple. As shells are rotated relative to each other, the relative shifting direction must change to compensate, in order to maintain a straight line. Along a general symmetry axis, the spherical coordinates do not hold constant. The projective coordinates, though, do hold constant on the symmetry axis. This can be inferred from the diagonality of the metric in projective coordinates, Eq. (1). From any given point, the shortest distance to a nearby shell is along the path satisfying dp = dq = 0. If this point is on the symmetry axis, we can deduce from symmetry arguments that this minimal distance is along the axis; if it were in any other direction, the symmetry would be broken. We will use this property of the projective coordinates to derive conditions on the dipole functions which result in axial symmetry, similar to the presentation in [67]. Georg and Hellaby [51] derive the same equations via the Killing equations.
The shifting direction on any shell corresponds to one of the two angular extrema of E /E, which were given in Eq. (19), but can also be found by solving
E E ,pr = E ,p E ,r , E E ,qr = E ,q E ,r .(36)
Commas in subscripts denote partial derivatives with respect to the following coordinate(s). If these equations hold for the same (p, q), which we will call (p 0 , q 0 ), across all r, the model is axially symmetric. (Such a model will also satisfy the equations for a second (p, q), which we will call (p 1 , q 1 ), for the half of the symmetry axis on the opposite side of the origin.) Expanded in terms of the three dipole functions, these equations become
2(p 0 − P )S S − 2(q 0 − Q)(p 0 − P )Q = (p 0 − P ) 2 − (q 0 − Q) 2 − S 2 P ,(37a)2(q 0 − Q)S S − 2(q 0 − Q)(p 0 − P )P = (q 0 − Q) 2 − (p 0 − P ) 2 − S 2 Q .(37b)
As mentioned previously, these are immediately solved by taking P = Q = 0, with P = p 0 and Q = q 0 . If Q = 0 but P = 0, the axial symmetry conditions reduce to
2(p 0 − P )S S = (p 0 − P ) 2 − (q 0 − Q) 2 − S 2 P ,(38a)(q 0 − Q)S S − (q 0 − Q)(p 0 − P )P = 0.(38b)
One way to satisfy Eq. (38b) is to take S S = (p 0 − P )P . But if we then plug this into Eq. (38a), we get either P = 0 (and therefore S = 0) or
(p 0 − P ) 2 + (q 0 − Q) 2 + S 2 = 0.(39)
The former case is simply an LT model, whereas the latter can only be satisfied for real-valued functions if S = 0. This would result in rampant division-by-zero singularities in the metric and density function, so it is not a valid solution. We therefore go back to Eq. (38b) and instead only consider the solution Q = q 0 . We can then easily solve Eq. (38a) by relating the dipole functions by
S 2 = C 2 (p 0 − P ) − (p 0 − P ) 2 ,(40)
where C 2 is an arbitrary constant. A similar solution is possible if P = 0 but Q = 0, by simply replacing p 0 and P with q 0 and Q. If both P and Q = 0, the axial symmetry conditions are satisfied by [67]
q 0 − Q = C 0 (p 0 − P ),(41a)S 2 = C 1 (p 0 − P ) − (C 2 0 + 1)(p 0 − P ) 2 .(41b)
We can see that the case in Eq. (40) is simply the special case when C 0 = 0. The symmetry axis passes through the origin, and (p 0 , q 0 ) covers only one side of it. The other side is antipodal to the first on each shell, so its coordinates can be found by applying Eq. (11). In the case P = Q = 0, we mentioned that the projective coordinates diverge on the other side, but in the other cases, it has the coordinates
p 1 = p 0 − C 1 C 2 0 + 1 , q 1 = q 0 − C 0 C 1 C 2 0 + 1 .(42)
For the special case of P = 0, we need only set C 0 = 0 and then swap p and P with q and Q. The angular coordinates of the symmetry axis are determined by the constants C 0 and C 1 according to
C 0 = cot φ ax ,(43a)C 1 = 2S sin θ ax cos φ ax .(43b)
We see that φ ax holds constant with r, while θ ax must vary as S(r) varies. We also note that Eq. (43a) has two solutions, corresponding to the two sides of the symmetry axis, which also means Eq. (43b) has two corresponding solutions for θ ax .
The angles θ ax cannot be constant because the shells rotate as r increases, depending on the direction and magnitude of the dipolar asymmetry. Specifically,
θ ax = E E ax sin θ ax .(44)
This means that as r increases, any anisotropy along the axis rotates the shells' frames in a way that drives θ = 0 towards the symmetry axis on the side where the shells are compressed, and θ = π towards the axis on the side where they are stretched apart, as shown in Fig. 9. With this illustration, we can see that even though the value of θ ax changes with r, it still traces a straight line across the model; the change in its value is due to the movement of the θ = 0 reference point.
Despite appearing curved in spherical coordinates, though, the symmetry axis is indeed a straight line. These models can be indirectly obtained by starting with a model with only S nonvanishing and performing a Haantjes transformation, as we will describe in section VII-D-6.
B. Bilateral symmetry
A less restrictive form of symmetry is bilateral symmetry, in which case the directions of shell shifting and rotation all lie on the symmetry plane. The simplest way to achieve this is by setting P or Q to 0. In this case, the symmetry plane is φ = ±π/2 or φ = {0, π}, respectively.
Slightly more generally, we can get bilateral symmetry by satisfying Eq. (41a), without Eq. (41b). This gives a symmetry plane passing through the poles and the point (p 0 , q 0 ), directed at angle φ = arctan C 0 .
We may wish to place the symmetry plane off of the pole θ = 0, though. This can be achieved by starting with a model satisfying Eq. (41a) and applying a Haantjes transformation, which is a rotation transformation we will detail in section VII-D-6. Remarkably, the end result matches the form of the equation for a great circle, Eq. (9). That is, any model for which a constant circle in the (p, q) plane represents a great circle on the sphere for all r exhibits bilateral symmetry. Moreover, this constant great circle marks the intersection of the symmetry plane with each shell.
But how does such a simple constraint keep the Szekeres dipole pointing in the same plane, when the shells are being rotated in different directions? To develop an intuitive understanding of how this works, we first recall that the dipole functions determine the (negative) displacement of the projection plane and its origin relative to the projection point, as shown in Fig. 2. Equation (9), then, identifies L gc as the distance between point (p c , q c ) and the projection point. If we hold p c , q c , and L gc constant with r, we can imagine the two points to be connected by a rigid rod. We are allowed to move the projection plane around as r increases, representing the dipole functions changing over r, but this motion is constrained to an imaginary sphere. This setup is illustrated in Fig. 10.
When we move the projection plane while moving outwards in r, we create a dipole anisotropy through the derivatives of the dipole functions. Because the motion of the plane is constrained, the direction of the dipole is constrained to the great circle parallel to the plane tangent to this imaginary sphere at the point (p c , q c ). This point is now at a different location on the sphere, so it would seem that the tangent plane has been rotated, but remember the rotation effect of the dipole functionsthe shell rotates in a way that exactly counteracts the rotation of the tangent plane! With respect to a single shell, the tangent plane does not rotate, so all of the extrema lie in the same plane for all r, as long as Eq. (9) holds. (The shell shifting effect is not a concern, because the shifts are all within the plane as well.)
This kind of symmetry does not confer most of the advantages of axial symmetry. It does not allow one to eliminate an entire coordinate from investigation; at best, it divides the number of directions that need to be considered by 2. It does allow for more complex arrangements of multiple structures than axially symmet-ric models, but for more than two structures it is too restrictive to be realistic. Perhaps the greatest advantage of bilateral symmetry is in graphical representation. Because paper and screens are two-dimensional, we can typically only clearly visualize a two-dimensional slice of the full model. If this slice is the symmetry plane, all of the structures will be neatly aligned within the page, making for a clear and faithful representation of the model. All of the figures in this paper showing examples of Szekeres models feature bilateral symmetry.
VII. METHODS AND TOOLS
A. Tracking shell shifting and rotation
There are situations in which we need to be able to understand the geometry of a model, including its shell shifting and rotation, such as in generating spatially accurate images, as in Figs. 6 and 12. In axially symmetric models with P and Q constant, this is a straightforward matter of numerically calculating the shifts by integrating Eq. (26). In general, though, care must be taken in calculating the orientations of each shell, as even the rotation axes change as the shells rotate. An example of a procedure for calculating the total displacements and rotations of each shell follows.
Begin at r = 0, with a 3 × 3 matrix A, set initially to the identity matrix. A(r) represents the orientation of the LRF of shell r relative to that of the innermost shells. That is, using the innermost shells' LRF as a basis, rows 1-3 of A(r) give the components of the x, y, and z unit vectors respectively of the LRF at shell r. Also create a three-component vector ∆ representing the total shifts, and initialize it to zero. Then, increment outwards in small steps of δr, up to a maximum r max , at each step doing the following:
Calculate a new axis matrix
A(r + δr) = R y P S δr R x − Q S δr A(r),(45)
where R x and R y are rotation matrices about the LRF's x and y axes, 10 defined as The shell rotation effect rotates the whole picture, as indicated by the curved arrows. When we account for this, we see that the dipole plane does indeed coincide with that of shell r1.
R x (ψ) = 1 0 0 0 cos ψ − sin ψ 0 sin ψ cos ψ ,(46)
Calculate new shifts according to 11
∆(t, r + δr) = ∆(t, r) + R(t, r) 1 − k(r) A T (r) P /S Q /S S /S (r).(48)
3. Append the new shift value and axis matrix to an array, to keep track of both at each step in r.
When finished, the array will hold all of the information about the displacement and orientation of shells up to r max . Note that the shifts calculated by this procedure are only valid at a single time slice. To plot at a different time, the shifts must be re-calculated. The axis matrix A(r), however, is invariant with time.
B. Plotting
Once we have the information about the shifts and rotations of each shell, we must still choose a mapping from the curved 3D space of the model to an image on flat 2D paper, if we wish to generate a plot like those in this paper.
First of all, reducing the model to two dimensions is far simpler if the model has bilateral symmetry, as mentioned previously. The symmetry plane contains all of 11 As we will explain in the next subsection, it is sometimes appropriate to omit the 1 − k(r) factor here, depending on how one chooses to map the curved space onto a flat image. the extrema of the density function, and its intersection with any shell is a great circle, so the size of the circle drawn on the page corresponds neatly to the shell's size.
Next, we must eliminate the curvature. It is of course impossible to do so without introducing some kind of distortion. The simplest method is to pretend that k(r) = 0. This must be taken into account when building the shifts in Eq. (48). We can then use the array of shifts and rotations from the previous subsection to map the Szekeres coordinates to Cartesian coordinates as
X Y Z (t, r, θ, φ) = R(t, r) A T (r) sin θ cos φ sin θ sin φ cos θ + ∆(t, r).(49)
(For 2D plots, we only use two of these coordinates.) With this mapping, each shell is drawn with radius R(t, r). Distances along a shell's surface are conveyed accurately, but distances between shells are distorted, depending on the true curvature function.
Another choice of mapping preserves distances along lines of constant (p, q), but distorts the sizes of shells and distances along their surfaces. We build the shift array with the curvature factor included, and then map the model to Cartesian coordinates as
X Y Z (t, r, θ, φ) = R(t, r) 1 − k(r) A T (r) sin θ cos φ sin θ sin φ cos θ + ∆(t, r).(50)
C. Calculating null geodesics
The light beams we observe follow null geodesic paths, defined by
k α ;β k β = 0,(51)
where k α = dx α /dλ is the tangent vector, and λ is the affine parameter. The non-vanishing Christoffel symbols for the Szekeres metric are given in appendix B. While the geodesic equations are fairly lengthy when fully written out, a simplification is possible through the definitions [68]
F = R 2 E 2 ,(52)H = (R − R E E ) 2 1 − k .(53)
With these compactified functions, the geodesic equations become
dk t dλ + 1 2 H ,t (k r ) 2 + 1 2 F ,t (k p ) 2 + (k q ) 2 = 0, (54a) H dk r dλ + dH dλ k r − 1 2 H ,r (k r ) 2 − 1 2 F ,r (k p ) 2 + (k q ) 2 = 0, (54b) F dk p dλ − 1 2 H ,p (k r ) 2 + dF dλ k p − 1 2 F ,p (k p ) 2 + (k q ) 2 = 0, (54c) F dk q dλ − 1 2 H ,q (k r ) 2 + dF dλ k q − 1 2 F ,q (k p ) 2 + (k q ) 2 = 0. (54d)
These equations can be numerically integrated to trace the path of a light beam through the model. A total of eight variables must be tracked (the position and tangent vector), using eight first-order differential equations-Eq. (54) and k α = dx α /dλ. Usually, we are interested in observed light beams, so we choose a location for the observer and a direction of observation to set the initial values for the eight variables, then propagate the equations backwards in time to the source. The redshift along the beam is readily obtained by
1 + z = (k α u α ) s (k α u α ) o ,(55)
where subscripts s and o denote the source and observer respectively, and u α is the four-velocity of the source or observer. We can set u α o = (1, 0, 0, 0) because the matter is comoving, and we can normalize the null geodesic tangent vector at the observer so that k t o = −1, so we are left with simply
1 + z = −k t s .(56)
In the case of axially symmetric models, there is a special case of geodesics which propagate along the symmetry axis, with k p = k q = 0 along their entire length. This simplifies the equations greatly. The null condition then gives a direct relation between k t and k r , and thus between t and r:
dt dr = k t k r = ± R − R E E √ 1 − k ,(57)
where the sign depends on which direction the geodesic is moving. Applying this to Eq. (54a), along with Eq. (56), we can deduce an integral formula for redshift:
d(1 + z) dλ =Ṙ −Ṙ E E R − R E E (1 + z) 2 ,(58a)1 1 + z d(1 + z) dt s = −Ṙ −Ṙ E E R − R E E , (58b) ln(1 + z) = − ts toṘ −Ṙ E E R − R E E dt,(58c)
where all quantities are evaluated on the geodesic, that is, with r = r(t).
We can then express the geodesic equations directly in terms of z [33]:
dt dz = − 1 1 + z R − R E Ė R −Ṙ E E , (59a) dr dz = ± 1 1 + z √ 1 − k R −Ṙ E E .
(59b)
D. Coordinate transformations
The labeling of coordinates in a Szekeres model has considerable flexibility. We have already mentioned the gauge freedom in the radial coordinate-defining a new coordinate r = f (r) (where f is monotonic) gives a new description of the same physical model, obeying the same basic equations.
Likewise, the transverse coordinates can be transformed while maintaining the form of the metric-not as freely as the radial coordinate, but in several specific ways.
Translation
The simplest transformation is a constant translation,
( p, q) = (p + p 0 , p + q 0 ).(60)
To maintain the form of the metric, we also transform the dipole functions, as ( P , Q, S) = (P + p 0 , Q + q 0 , S).
This transformation only moves the origin point for the projective coordinate labeling; it does not affect the spherical coordinates.
Scaling
It is also possible to perform a linear scaling transformation while maintaining the model's physical form. This is done by simply multiplying both projective coordinates and all three dipole functions by the same nonzero constant:
( p, q) = µ(p, q),(62)
( P , Q, S) = µ(P, Q, S).
Again, this does not affect the spherical coordinates.
Polar rotation
A third simple transformation consists of rotating the p and q coordinates:
( p, q) = (p cos ψ + q sin ψ, −p sin ψ + q cos ψ), (64) ( P , Q, S) = (P cos ψ + Q sin ψ, −P sin ψ + Q cos ψ, S).
This does affect the spherical coordinates, by φ = φ + ψ-a simple rotation about θ = 0 (with θ = π also fixed). While this axis is not in general the same for shells of different r, due to the shell rotation effect, the form of the metric and the physical arrangement of structures are preserved.
Swapping p and q
We can easily see that the metric is invariant under the substitution ( p, q) = (q, p),
( P , Q) = (Q, P ).
This amounts to a reflection across the φ = π/4 plane. Combined with polar rotations, this can reflect the coordinates across any polar plane.
Inversion
As we saw in section III, the projective coordinates diverge where θ = 0. This can cause problems for numerical calculations passing near this axis, even though there is nothing physically special happening there. When this occurs, a translation or scaling transformation cannot remove the infinity, but we can invert the coordinates so that the divergence occurs somewhere else, without changing the model's physical features or their positions relative to each other. This is done with the transformation ( p, q) = (p, q)
p 2 + q 2 ,(68)
( P , Q, S) = (P, Q, S)
P 2 + Q 2 + S 2 .(69)
Clearly, the point on any given shell where the projective coordinates previously diverged now has p = q = 0, and vice versa. Calculations can now proceed unimpeded through the region that was formerly problematic.
Because the angle θ = 0 corresponds to the divergence point of the projective coordinates, and this point has moved (relative to physical structures), we can see that this transformation does affect the spherical coordinates. The point θ = 0 corresponds to a new angle satisfying
cot φ = P Q , cot θ 2 = P 2 + Q 2 S .(70)
The new point at which θ = 0 corresponds to an old angle in a similar fashion. More generally, the spherical and projective coordinates over the entire shell have been reflected across the great circle defined by
p 2 + q 2 = P 2 + Q 2 + S 2 ,(71)
which crosses the point halfway between θ = 0 and p = q = 0. In spherical coordinates, this reflection surface is given by P sin θ cos φ + Q sin θ sin φ + S cos θ = 0.
That is, in terms of the LRF, the reflection surface is a plane through the origin and perpendicular to the vector (x, y, z) ref = (P, Q, S)/ P 2 + Q 2 + S 2 . It is then easy to see that we can adjust the orientation of this reflection surface by first performing a translation, as described above; such a translation simultaneously moves the point p = q = 0 and changes the LRF vector (x, y, z) ref .
Note, however, that the definition of this reflection surface is a function of r. It is therefore not a flat plane across the entire model. Each shell's coordinates are reflected in a different direction, but the metric keeps the same form, and the overall picture still follows the same rules of shifting and rotation between shells laid out in section V, just with different dipole functions, Eq. (69). The examples of inversion shown in Fig. 11 show how this works.
We can therefore avoid the θ = 0 axis when calculating geodesics by performing this inversion operation whenever p and q get large enough to adversely affect precision. We must take care to also transform the geodesic tangent vector accordingly:
( k p , k q ) = (k p , k q ) p 2 + q 2 − 2(p, q) pk p + qk q (p 2 + q 2 ) 2 .(73)
Haantjes transformation
A more general transformation allows us to rotate the coordinates about an arbitrary direction. This kind of transformation is called a Haantjes transformation.
In a quasispherical Szekeres model, a Haantjes transformation modifies the p and q coordinates as follows [67]:
p = p + D 1 (p 2 + q 2 ) τ , (74a) q = q + D 2 (p 2 + q 2 ) τ ,(74b)τ = 1 + 2D 1 p + 2D 2 q + (D 2 1 + D 2 2 )(p 2 + q 2 ),(74c)
where D 1 and D 2 are arbitrary constants. In order to preserve the form of the metric, we must also transform the dipole functions:
P = P + D 1 (P 2 + Q 2 + S 2 ) T , (75a) Q = Q + D 2 (P 2 + Q 2 + S 2 ) T , (75b) S = S T ,(75c)T = 1 + 2D 1 P + 2D 2 Q + (D 2 1 + D 2 2 )(P 2 + Q 2 + S 2 ).(75d)
This transformation amounts to a rotation about the points satisfying
p 2 + q 2 = P 2 + Q 2 + S 2 , (76a) D 1 p + D 2 q = D 1 P + D 2 Q,(76b)
by an angle of ψ = 2 arccos (P, Q, S) · (P + D 1 , Q + D 2 , S) (P, Q, S) (P + D 1 , Q + D 2 , S) ,
where (P, Q, S) = P 2 + Q 2 + S 2 . The rotation axis is restricted to the great circle halfway between θ = 0 and p = q = 0, but this great circle can be moved by first performing a coordinate translation. By exercising this freedom in combination with the free variables D 1 and D 2 , the rotation can be of any angle about any axis. The Haantjes transformation can be seen as a combination of three simpler transformations: an inversion, followed by a translation of (D 1 , D 2 ), followed by another inversion. The first inversion reflects the coordinates across the surface p 2 + q 2 = P 2 + Q 2 + S 2 , and the translation reorients this surface before the second inversion reflects the coordinates again. This sequence is illustrated in Fig. 11.
Haantjes transformations can be useful in model construction, as they provide a means of moving an axially symmetric anisotropy to any desired direction. One can create a model with P = Q = 0, meaning anisotropies only arise in the z direction as a result of S, and then one can apply an appropriate Haantjes transformation to reorient the anisotropy to any direction, with a mixture of all three dipole functions automatically satisfying the conditions for axial symmetry, Eq. (36). The (D 1 , D 2 ) values needed to transform the axis to the direction (θ, φ) (measured at the lowest r value of the anisotropy, r l ) are given by
D 1 = 1 − cos θ sin θ cos φ e −S(r l ) ,(78a)D 2 = 1 − cos θ sin θ sin φ e −S(r l ) .(78b)
Due to the shell rotation effect, the (θ, φ) values of the maximum anisotropy will not be constant with r after the transformation. The D parameters are related to the C coefficients of the axial symmetry equations (41) by
C 0 = D 2 D 1 ,(79a)C 1 = 1 D 1 . (79b)
The new symmetry axis has coordinates
p 0 = D 1 D 2 1 + D 2 2 ,(80a)q 0 = D 2 D 2 1 + D 2 2 ,(80b)
on one side, and (0, 0) on the other.
E. Example construction method: randomized series of structures
One of the key strengths of the Szekeres metric is its ability to simulate multiple structures. When designing such a model, though, we should be careful of how the shell rotation effect influences the structures' shapes. As we have seen in Fig. 6, overdensities that may appear symmetrical (i.e., with the ratios of P /S, Q /S, and S /S held constant over some range of r) are in fact smeared by the shell rotation, if they are not centered at θ = 0 or π. This can systematically distort our structures based on their orientation, which is undesirable if we wish them to have similar shapes.
We have devised a process for generating a randomized series of structures that are all the same kind of shape-individually axially symmetric-regardless of their orientation. This method involves performing Haantjes transformations in a piecewise manner over separate intervals of r. Because we do not apply the transformations globally, they are no longer simply coordinate transformations, but physical rearrangements of structures. The process goes as follows: r i+1 > r i . We introduce axially symmetric Szekeres anisotropies by making S a piecewise function, with S /S vanishing at all r i (to ensure continuity when we are done).
2. Then, we generate a list of random angles (θ i , φ i ), which will correspond to the directions of the maximum density contrast at the lower bound of each section. To move the anisotropies to these randomly chosen angles, we apply the Haantjes transformation separately to each interval, using Eq. (75). The D coefficients for each interval are given by Eq. (78), using (θ i , φ i ) for the interval r i < r < r i+1 .
3. The dipole functions will now be discontinuous at each r i , creating shell crossing singularities. To remedy this, we first address the discontinuities in S using a scaling transformation, Eq. (63). One interval at a time, starting with i = 1, we scale the dipole functions by a factor S i−1 (r i )/S i (r i ) (where S i is the S function in the interval r i < r < r i+1 ).
4. The S function is now continuous, but the P and Q functions are not. This can be fixed similarly with a series of shifts, using Eq. (61). Again starting at i = 1, we shift the p coordinate by P i−1 (r i )−P i (r i ), and likewise for q, and proceed through the remaining intervals one at a time.
We are left with a single model containing an arrangement of structures, each individually axially symmetric, but together completely asymmetric in a randomized fashion. An example is illustrated in Fig. 12. Each structure is oriented in the direction of (θ i , φ i ) at the lower boundary of the r interval; due to shell rotation within the interval, the angle will generally be different at the upper boundary, following Eq. (44). Nevertheless, if the angles were chosen uniformly over the sphere, there will be no correlation between the orientations of sequential structures.
FIG. 12. An example of a model constructed through randomized piecewise Haantjes transformations. The structure angles were chosen over the unit circle rather than the unit sphere, so that the model maintains a symmetry plane on which all of the maximum density contrasts fall, making for a better picture, but the construction process is fully capable of creating three-dimensional arrangements. This method of model construction results in a coarsegrained array of large, pancake-like structures. This can be done with any base LT model, for instance creating a series of overdense walls inside a giant void, as in [36]. But it fits particularly well with a model with a radially oscillating density, following the same series of r intervals. These kinds of models, called "Onion" models by some [24,42], are compatible with Sussman's prescription of periodic local homogeneity (PLH) [64].
VIII. EMBEDDING IN 4-DIMENSIONAL SPACE
As we mentioned previously, the Szekeres model's curved geometry makes it impossible to depict directly on flat paper without distortion. It is possible, though, to view a const-t slice as a curved hypersurface embedded within a 4-dimensional space. With such a description, we can examine unambigously how the curvature function and dipole functions interact to produce the Szekeres metric. This also gives confirmation that the shell shifting and rotation effects we have described do indeed fully account for the form of the metric.
If the curvature function is everywhere non-negative, an embedding is possible within a flat Euclidean background manifold. If the curvature function is negative anywhere, we can use a background manifold with constant negative curvature. We consider these two cases separately.
A. Non-negative curvature
Let the background space have coordinates (X, Y, Z, W ), with a line element ds 2 = dX 2 + dY 2 + dZ 2 + dW 2 .
The origin of the Szekeres model hypersurface coincides with the origin of the (X, Y, Z, W ) coordinates, and it extends initially outwards in the X-Y -Z hyperplane, which is aligned with the LRF's x, y, and z axes. We can then identify (X, Y, Z, W ) ≈ R(t 0 , r)(sin θ cos φ, sin θ sin φ, cos θ, 0) for sufficiently small r. (Because we are only looking at one spatial slice at a time, we will use t = t 0 everywhere.) In an LT model, a positive curvature function gives the surface a bowl shape, rising in the W direction as r increases. Each shell r is a sphere of radius R, parallel to the the X-Y -Z hyperplane. The slope of the bowl is given by
α w (r) = ± k(r) 1 − k(r) .(81)
This is what gives radial distances the extra curvature factor. Going from r to r + dr, we must move outwards by R dr, but also "upwards" in the orthogonal W direction by α w R dr, for a total distance of 1 + α 2 w R dr, or R dr/ √ 1 − k. Positively curved Szekeres models have the same sort of bowl geometry, but with a modification due to the dipole functions. We have seen how the dipole functions shift the shells relative to each other, so over a span δr, the bowl extends farther outwards on one side than the other. But because the curvature function in the metric modifies the entire term R − R E /E, not just R , the slope of the bowl must be the same everywhere around the shell. 12 This means that, from the viewpoint of this embedding, the dipole functions have an extra geometric effect not previously discussed: they not only shift and rotate shells, but also tilt them within the background manifold. Specifically, in terms of the LRF at shell r (which now has a fourth direction, "w", orthogonal to the shell's hyperplane), the shell at r + dr makes an angle P S α w dr in the x-w plane, Q S α w dr in the yw plane, and S S α w dr in the z-w plane. Because of this tilting, the "upwards" shifts due to the bowl's slope do not go in the background's W direction, but rather the LRF's w direction. 13 The resulting shape is illustrated in Fig. 13.
The approach for determining the orientation of each shell is similar to the one presented in section VII-A, with the addition of another dimension. We define a 4 × 4 matrix A(r) that relates a shell's LRF axes to the axes of the background manifold (with rows 1-4 giving the x, y, z, and w directions respectively). It is initialized as A ij (0) = δ ij for simplicity. 14 As r increases, it evolves according to
A (r) = 0 0 P S P S α w 0 0 Q S Q S α w − P S − Q S 0 S S α w − P S α w − Q S α w − S S α w 0 A(r). (82)
The Szekeres model's hypersurface, then, is defined by
X Y Z W (t 0 , r) = R(t 0 , r) A T (r) sin θ cos φ sin θ sin φ cos θ 0 + ∆(t 0 , r),(83)
where ∆ = ∆(t 0 , r) is a 4-component vector denoting the total displacement of the center of shell r relative to the origin. It satisfies
∆ (t 0 , r) = A T (r) R P /S R Q /S R S /S R α w (t 0 , r).(84)
We initialize it as ∆ i (t 0 , 0) = 0, for simplicity. Note that the shifts from the dipole functions do not have the curvature factor included in Eqs. (24)- (26), as that component of the displacement is accounted for in the fourth component.
B. Arbitrary curvature
If k(r) < 0 for some r, the quantity α w defined in Eq. (81) is not real, so an embedding in a 4-dimensional ordinates. 13 This is why we have called the slope αw instead of α W . 14 Any initialization is acceptable, as long as it is orthonormal. Euclidean manifold as above is not possible. We can, however, adapt the above embedding scheme for a 4dimensional background manifold of constant negative curvature.
The line element of the background manifold can be written in terms of a radial coordinate ρ and three angular coordinates (α, β, γ) as
ds 2 = 1 1 − k b ρ 2 dρ 2 + ρ 2 dα 2 + sin 2 α dβ 2 + sin 2 α sin 2 β dγ 2 , (85)
where k b is the curvature of the background. We will use χ(t, r, θ, φ) to denote the set of background coordinates (ρ, α, β, γ) corresponding to a point on the Szekeres hypersurface.
We consider the region local to a particular shell r 0 . We align the background coordinates to this shell, so that
χ(t 0 , r 0 , θ, φ) = R, π 2 , θ, φ .(86)
The only parts of the embedding in the non-negative curvature case that depended on the curvature were the shifts and tilts in the local w direction, through α w . We will assume for now that everything works as before, and we only need to find the new α w . For this, it is sufficient to consider a model in which P = Q = 0, so we only need to deal with S.
The shell at r 0 + dr has coordinates in the background manifold χ(t 0 , r 0 + dr, θ, φ) = χ(t 0 , r 0 , θ, φ)
+ R + R S S cos θ, −α w R R + S S cos θ , − S S sin θ, 0 dr.(87)
By taking χ(t 0 , r 0 +dr, θ 0 +dθ, φ 0 +dφ)−χ(t 0 , r 0 , θ 0 , φ 0 ) and plugging the results into the background metric, Eq. (85),
we can obtain the metric on the Szekeres hypersurface:
ds 2 = R + R S S cos θ 2 1 − k b R 2 + R 2 S 2 S 2 sin 2 θ + α 2 w R + R S S cos θ 2 dr 2 − 2R 2 S S sin θ dr dθ + R 2 dΩ 2 .(88)
From here, we can compare to the Szekeres metric in spherical coordinates, Eq. (13). Requiring g rr to match gives us an equation we can solve for α w :
1 1 − k b R 2 + α 2 w = 1 1 − k α w = ± 1 1 − k − 1 1 − k b R 2 = ± k − k b R 2 (1 − k)(1 − k b R 2 )(89)
This solution is only real if k b R 2 < k. This means that we require the background curvature to be more negative than the strongest negative curvature of the Szekeres model. In addition to the background manifold being non-Euclidean, note also that α w depends on R, which is a function of t. Unlike the flat-background case, here the slope of the embedding surface changes over time.
With this solution for α w , the local metric around the shell r 0 matches the Szekeres metric, as desired. We can envision extending this embedding for the entire Szekeres model by moving both inwards and outwards in incremental steps dr, at each step reorienting the background coordinates to match Eq. (86). We then have a global embedding, though it is not expressed as concisely as in Eq. (83). We can also confirm that this method works when P and Q are non-zero, though the equations get considerably more lengthy.
IX. DISCUSSION
Quasispherical Szekeres models are an anisotropic generalization to the spherically symmetric Lemaître-Tolman (LT) models. They have great potential as fully general-relativistic representations of cosmic structures. However, their mathematical complexity can be daunting, and renders them somewhat opaque to understanding. It is easy to misconstrue the physical picture behind the equations.
The dipole functions alter the matter distribution and the geometry of the models in multiple ways, which we have thoroughly explained. The shells of constant t and r are not only non-concentric, but also non-aligned. This relative rotation of the shells has often been overlooked, but as we have shown, it has deep connections to several other aspects of the model, such as the relationship between the two most common coordinate systems (projective and spherical), the conditions for axial symmetry, and geodesic paths. With this shell rotation effect properly accounted for, we can generate density plots that show the true shape of the model more accurately than before. We see that geodesics appear nearly straight, as they should, while structures defined by simple functions are sometimes skewed in ways not immediately obvious.
The conditions by which Szekeres models are axially symmetric are well-known, but with a better understanding of the overall geometry dictated by the model functions, we can understand the equations more intuitively. Moreover, we have shown that a less restrictive condition results in bilateral symmetry, which we have also illustrated in terms of the shell rotation effect. We have also reviewed the coordinate transformations that preserve the form of the metric, and used these in combination with the symmetry conditions to give an example of a model construction method that produces a randomized series of simple structures.
Finally, we have used our understanding of the geometry to build the spatial Szekeres metric as a 3-dimensional surface embedded in a 4-dimensional space. Because the metric on this surface fully matches the Szekeres metric, this confirms that the effects we have described tell the whole story. That is, the shell rotation effect is real, and there are no other hidden geometric effects waiting to be discovered.
We have sought to provide the reader with a firm understanding of the Szekeres models' basic properties, as well as some of the practical tools needed to work with them. This is not a comprehensive analysis of the properties of the Szekeres models. Other works have already explored certain aspects of them in greater detail. There is a great deal of potential work yet to be done, though, and it is important to hold a clear picture of what we are working with as we forge ahead.
The non-zero Christoffel symbols for the quasispherical Szekeres metric are
Γ t rr = (R ,r − R E,r E )(R ,tr − R ,t E,r E ) 1 − k Γ p rr = (R ,r /R − E ,r /E)(E ,rp E − E ,r E ,p ) 1 − k Γ t pp = Γ t qq = R ,t R E 2 Γ q rr = (R ,r /R − E ,r /E)(E ,rq E − E ,r E ,q ) 1 − k Γ r rt = Γ r tr = R ,tr − R ,
The remaining nonzero terms can be found by the Riemann tensor's symmetry properties:
R αβγδ = −R αβδγ = −R βαγδ = R γδαβ . (B3)
The Ricci scalar is simply
R = 8πρ + 4Λ (B4)
The Weyl curvature tensor is split into electric and magnetic parts as [33,54]
E α β = C α γβδ u γ u δ = M R − R M /3 R 3 (R − R E /E) diag(0, 2, −1, −1),(B5)H αβ = 1 2 αγµν C µν βδ u γ u δ = 0.(B6)
The Kretschmann scalar, which is useful for identifying real singularities, is [52]
K = R αβγδ R αβγδ = (8π) 2 4 3ρ 2 int − 8 3ρ int ρ + 3ρ 2 + 4 3 Λ(2Λ + 8πρ),(B7)
whereρ int is the "average" density inside shell r, defined in Eq. 27. This tells us that the spacetime singularities coincide with divergent densities, i.e. shell crossings and the big bang/crunch. The 3-spaces of constant time have non-zero Riemann tensor components [53]
3 R r prp = 3 R r qrq = k /2 − k E E R(R − R E E ) ,(B8)3 R p qpq = k R 2 ,(B9)
and a Ricci scalar [34] 3 R = 2 k R 2
k /k − 2E /E R /R − E /E + 1 ,(B10)
Note that the model is perfectly spatially flat if k = 0, even if there is significant inhomogeneity.
Appendix C: Proof of shell rotation in metric
We are interested in the geometry on a constant-t hypersurface. We begin with the LT metric in spherical co-ordinates: ds 2 = −dt 2 + (R ) 2 1 − k dr 2 + R 2 (dθ 2 + sin 2 θ dφ 2 ). (C1)
The shell shifting effect modifies the separation between shells in the radial direction (orthogonal to the shell surface) by replacing R by R − R E /E. Because this change is in the radial direction, the actual distance is affected by the curvature, and we have built this into our definition of the shifting. The shift also adds a transverse separation between points on different shells, though, even when the angular coordinates are held constant. Because this component of the distance is along the shell's surface, the curvature function does not play a role. Accounting only for the shifting effect, the points (r, θ, φ) and (r + dr, θ + dθ, φ + dφ) have a total separation in the θ direction of
R dθ + R P S cos θ cos φ + Q S cos θ sin φ − S S sin θ dr,(C2)
and in the φ direction of R sin θ dφ + R P S sin φ − Q S cos φ dr.
A simple diagram of this decomposition is shown in Fig. 14.
The total square separation is simply the sum of the squares of the components in these orthogonal directions (because space is approximately Euclidean on sufficiently small scales). This would give
ds 2 = −dt 2 + (R − R E E ) 2 1 − k + R 2 P S cos θ cos φ + Q S cos θ sin φ − S S sin θ 2 + R 2 P S sin φ − Q S
cos φ 2 dr 2 + 2R 2 P S cos θ cos φ + Q S cos θ sin φ − S S sin θ dr dθ + 2R 2 sin θ P S sin φ − Q S cos φ dr dφ + R 2 (dθ 2 + sin 2 θ dφ 2 ).
But this does not match the metric as given in Eq. (13). Shell shifting alone is not enough. The rotation described previously adds a further transverse separation between the two points. In the θ direction, this amounts to an additional
R − P S cos φ − Q S sin φ dr,(C5)
and in the φ direction, R − P S cos θ sin φ + Q S cos θ cos φ dr.
Incorporating both effects, the rr part of the metric becomes
g rr = (R − R E E ) 2 1 − k + R 2 P S (cos θ − 1) cos φ + Q S (cos θ − 1) sin φ − S S sin θ 2 + R 2 (1 − cos θ) 2 P S sin φ − Q S cos φ 2 = (R − R E E ) 2 1 − k + R 2
(1 − cos θ) 2 P 2 + Q 2 + S 2 S 2 + 2R 2 (1 − cos θ) P S sin θ cos φ + Q S sin θ sin φ + S 2 cos θ S 2 .
(C7)
FIG. 14. The separation between nearby points on different shells without accounting for shell rotation. The dotted arc shows where the outer shell would be without shifting; the red arrow shows how the individual points on the outer shell have moved. We have used dθ = 0 for illustrative purposes, so the red and blue dots in each set have the same θ, measured from the top. The blue lines coming from the blue dots show the radial and transverse (θ) directions, and the blue arrows and dashed lines show the components of the total separation in each direction. Two sets of points are shown to demonstrate the cos θ factor of the P /S term in Eq. C2 (in the limit δr → 0).
With some simplification, we can confirm that this matches the corresponding term in Eq. (13). Similarly, we can find g rθ from Eqs. (C2) and (C5):
g rθ = R 2 (cos θ − 1) P S cos φ + Q S sin φ − R 2 S S sin θ = R 2 1 − cos θ sin θ E E + R 2 cos θ − cos 2 θ sin θ − sin θ S S .(C8)
Again, this matches Eq. (13). Finally, the rφ term:
g rφ = R 2 sin θ(1 − cos θ) Q S cos φ − P S sin φ . (C9)
Thus, we see that the shell shifting and rotation effects fully encapsulate the differences between the LT and Szekeres metrics.
(but quickly), plateaus, and then ramps back down to 0.
This is plotted in Fig. 15(c). The (1 + r) −0.99 factor was taken from the model used in [36], and C P is an overall strength factor, where a value of 1 would push the shells very close to a shell crossing. In this case, we used C P = 0.5. To obtain P (r), we simply integrate. The geodesic shown in Fig. 6 was generated from the initial values
r 0 = 400,(D4a)
p 0 = P (400) + cot 5π 12 ,
q 0 = 0, (D4c) k t = −1,(D4b)k r = cos 7π 8 √ g rr ,(D4d)k p = − E(r 0 , p 0 , q 0 ) R(t 0 , r 0 ) sin 7π 8 ,(D4e)
FIG. 1 .
1An illustration of a non-simultaneous big bang, showing only one spatial dimension. Lines are surfaces of constant r. The thick red vertical line is the big bang singularity.
FIG. 2 .
2A 2D diagram showing how the projective coordinates map to the sphere. The dipole functions describe the negative offset of the projection plane from the projection point. The units are arbitrary, as long as the dipole functions and projective coordinates use the same units.
FIG. 3 .
3(a): How a great circle on the sphere appears in the projective plane. The large red dot is the center of the projective circle, whereas the two smaller blue dots are the two centers of the great circle on the sphere. (b): The same great circle on the sphere, with the same points marked. The green dot is the projection point.
FIG. 5 .
5An example of shell crossing in a Szekeres model. The shell crossing surface is outlined in red.
FIG. 6 .
6Three ways of plotting a cross-section of a void+wall Szekeres model, to illustrate the effects of the dipole functions on the coordinates. The only active dipole function is P (r)-the other two are constant. Detailed model definitions are given in appendix D. The blue line shows the path of an arbitrary null geodesic. Black circles mark shells of constant r in steps of 80 Mpc, and yellow dots mark the centers of these shells. The green dashed line marks θ = 0 (the LRF's +z axis at each shell), and the red dotted line marks θmax, where the density has its angular maximum. (a): plotted in "naïve" coordinates, as though it was an LT model. Notice that the geodesic is not a straight line. (b):
FIG. 7 .
7A 2-dimensional illustration explaining how the rotation effect arises. Blue lines connect points of constant θ. Double red lines connect points of constant p. Ticks on the outer shell mark where θ = 0 and θ = π would be without the rotation effect, for comparison, and the arrows show the rotation.
FIG. 8. Expansion rates and density profile in a onedimensional slice through the center of a typical void-and-wall model.
FIG. 9 .
9An illustration of how θax changes with r due to the shell rotation effect. The solid green line marks θ = 0 on the top half and θ = π on the bottom half. The dashed red line shows where these points would be without the shell rotation effect, for comparison. The dotted blue line marks θax.
FIG.10. A side view of the situation described in the text, regarding the symmetry plane's compensating response to shell rotation. (a): The projection plane (solid black line with arrows) for a shell r1 (solid circle) is positioned so that the point pc (red dot) is a distance Lgc from the projection point (green dot). The red dot can move as r increases, but only in the directions tangent to the dashed circle. The total dipole direction is parallel to this motion, so it must lie on the great circle parallel to the dashed tangent line, shown edge-on as a solid blue line. The blue ticks on the projection plane mark where the edges of the great circle map to. (b): At a later shell r2, the point pc has moved along the Lgc constraint sphere some distance. In the frame of shell r2, it appears that the dipole lies on a different plane than before (i.e. the blue line is at a different angle). (c):
1 .FIG. 11 .
111First, we divide the model into separate intervals of r, with boundaries r i , with i = 0..n, r 0 = 0, and Visualization of a series of coordinate transformations. Colors indicate the p coordinate, from blue at negative infinity to red at positive infinity (q is 0 in the displayed slice), and the dotted and dashed green lines show where it vanishes and diverges, respectively (the LRF's −z and +z directions). The purple line is the reflection surface defined in Eq. (71). (a): the initial state, an axially symmetric model with P (r) = Q(r) = 0. It then undergoes an inversion transformation, with the result shown in (b). (c) shows the effect of a translation of the p coordinate, and (d) is after a second inversion. The combined effect is a Haantjes transformation.
13. Three views of a slice of a positively-curved Szekeres model as an embedding in a 4-dimensional Euclidean manifold. The blue line shows the path of an arbitrary geodesic. Black circles mark shells of constant r in steps of 80 Mpc. The green dashed line marks θ = 0, and the red dotted line marks θmax, where the density has its angular maximum.
FIG. 15 .
15The density (a), curvature (b), and P (c) functions of the example model used inFig. 6.
E ,r E ,p /E 2 − E ,rp /E R ,r /R − E ,r /E Γ p pr = Γ p rp = Γ q qr = Γ q rq = E ,r E ,q /E 2 − E ,rq /E R ,r /R − E ,r /E (B1)The Riemann tensor can be summarized using the compactified functions defined in Eq. (52) by[42] ,p F ,p + F 2 ,r − H ,q F ,q ) ,p F ,p + F 2 ,r + H ,q F ,q )t
E,r
E
R ,r − R
E,r
E
Γ p
pp = −Γ p
qq = Γ q
pq = Γ q
qp = −
E ,p
E
Γ r
rr =
R ,rr − R ,r
E,r
E − R
E,rr
E + R
E,r
E
2
R ,r − R
E,r
E
−
1
2
k ,r
1 − k
Γ q
qq = −Γ p
qq = Γ p
pq = Γ p
qp = −
E ,q
E
Γ r
pp = Γ r
qq = −
R
E 2
1 − k
R ,r − R
E,r
E
Γ p
pt = Γ p
tp = Γ q
qt = Γ q
tq =
R ,t
R
Γ r
rp = Γ r
pr =
R ,r
R
−
E ,r
E
Γ r
rq = Γ r
qr =
R trtr = −
1
2
H ,tt +
H 2
,t
4H
R tptp = R tqtq = −
1
2
F ,tt +
F 2
,t
4F
R rprp = −
1
2
(H ,pp + F ,rr ) +
1
4
H ,t F ,t +
1
4H
(H ,r F ,r + H 2
,p ) +
1
4F
(H R rqrq = −
1
2
(H ,qq + F ,rr ) +
1
4
H ,t F ,t +
1
4H
(H ,r F ,r + H 2
,q ) +
1
4F
(−H R pqpq = −
1
2
(F ,pp + F ,qq ) +
1
4
F 2
,t −
1
4H
F 2
,r +
1
2F
(F 2
,p + F 2
,q )
It is possible to have collapsing solutions, in which case the RHS takes a negative sign and the function is better called the "crunchtime function", but this is simply a time reversal of the same situation.
We will refer to the LRF throughout this paper.
Szekeres models are mathematically capable of including negative densities, but such models would be considered unphysical.
Although the geodesic's path covers a range of times, and the density plot shows a constant-time slice, the size of the structure is small enough that it does not evolve appreciably in the time it takes the geodesic to traverse it.
In the case of an underdense shell, an even stronger limit is in place if we restrict the density to be everywhere non-negative: ρmax = 2ρ LT /(1 + ρ LT /ρ int ).
Further, Krasinski and Bolejko show that such radial geodesics are the only ones in Szekeres models that are "repeatable light paths", meaning two geodesics at different times can follow the same spatial path, resulting in no visible drift[66].
The rotations approximately commute because their arguments are small.
This slope is measured in terms of the LRF, not the background co-
ACKNOWLEDGMENTSWe would like to thank Dr. Charles Hellaby for very helpful comments and discussion. Graphics were generated and some computations were performed with the Wolfram Mathematica 10 software [70].We then introduce anisotropy through P (r), keeping Q(r) at 0 and S(r) at 1. We define P (r) as a piecewise function such that r P (r) starts at 0, ramps up smoothlyAppendix A: Solutions to the evolution equationsIn the case of Λ = 0, the solution to Eq. (3) is explicit in a parametric form, using a parameter η(t, r):These cases are hyperbolic, parabolic, and elliptic, respectively. In practice, the precision of a numerically calculated solution suffers near points where k(r) crosses 0, so it is sometimes necessary to use a series expansion to handle "near-parabolic" regions (see also[30], app. B):The precise conditions for this solution to apply depend on the model details, gauge choices, and code structure. One example condition is |k(r)| < ζ 0 |k (r)|r, where ζ 0 is a small constant. If Λ = 0, the solution requires an elliptic integral arising from Eq. (5).Appendix B: Christoffel symbols and curvature tensors Appendix D: Example model definitionHere we detail the function definitions used in the model shown inFig. 6.We begin with a background FLRW model with H 0 = 70 km s −1 M pc −1 , Ω Λ = 0.7, and Ω m = 0.3.We then define an LTB model that matches onto this background at large r. We use a radial coordinate scaled so that R(t 0 , r) = r in units of Mpc, and set t B (r) = 0. The density profile at t 0 is a modified version of the universal void density profile found by Hamaus et al.[69]ρ(r) =ρ 1 + δ c 1 − (r/r s )where δ c is the central density contrast, r v characterizes the size of the void, r s gives a scale radius at which the density equals the background, and α and β determine the inner and outer slopes of the void's wall, respectively. We have used δ c = −0.85, r v = 200, r s = 182, α = 2.18, and β = 9.482. The density function is plotted inFig. 15(a). From the density, we obtain M (r) by integrating:The curvature function k(r) is then fixed by Eq. (5). We solve this equation numerically at a series of r values, in increments of 1, and interpolate to obtain a smooth function, plotted inFig. 15(b).
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| [] |
[
"Parton shower effects in¯± production at NLO QCD *** *** 26-30 July 2021 *** *** Online conference, jointly organized by Universität Hamburg and the research center DESY *** * Speaker Parton shower effects in¯± production at NLO QCD",
"Parton shower effects in¯± production at NLO QCD *** *** 26-30 July 2021 *** *** Online conference, jointly organized by Universität Hamburg and the research center DESY *** * Speaker Parton shower effects in¯± production at NLO QCD"
] | [
"Manfred Kraus [email protected] \nPhysics Department\nFlorida State University\n32306-4350TallahasseeFLU.S.A\n",
"Manfred Kraus \nPhysics Department\nFlorida State University\n32306-4350TallahasseeFLU.S.A\n"
] | [
"Physics Department\nFlorida State University\n32306-4350TallahasseeFLU.S.A",
"Physics Department\nFlorida State University\n32306-4350TallahasseeFLU.S.A"
] | [
"The European Physical Society Conference on High Energy Physics (EPS-HEP2021)"
] | We present a selection of results from our recent study of →¯± production matched to partons showers at NLO QCD at the LHC. Theoretical predictions are obtained at perturbative orders O ( 3 ) and O ( 3 ), where the different contributions are studied first separately at the inclusive level before being combined within a realistic two same-sign lepton signature. We investigate in detail uncertainties originating from missing higher-order corrections and from the parton-shower matching scheme employed. | 10.22323/1.398.0482 | [
"https://arxiv.org/pdf/2110.12512v1.pdf"
] | 239,769,187 | 2110.12512 | e9d6ab5f2319ad165b6e6af3154efd65940828c5 |
Parton shower effects in¯± production at NLO QCD *** *** 26-30 July 2021 *** *** Online conference, jointly organized by Universität Hamburg and the research center DESY *** * Speaker Parton shower effects in¯± production at NLO QCD
24 Oct 2021
Manfred Kraus [email protected]
Physics Department
Florida State University
32306-4350TallahasseeFLU.S.A
Manfred Kraus
Physics Department
Florida State University
32306-4350TallahasseeFLU.S.A
Parton shower effects in¯± production at NLO QCD *** *** 26-30 July 2021 *** *** Online conference, jointly organized by Universität Hamburg and the research center DESY *** * Speaker Parton shower effects in¯± production at NLO QCD
The European Physical Society Conference on High Energy Physics (EPS-HEP2021)
24 Oct 2021Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/
We present a selection of results from our recent study of →¯± production matched to partons showers at NLO QCD at the LHC. Theoretical predictions are obtained at perturbative orders O ( 3 ) and O ( 3 ), where the different contributions are studied first separately at the inclusive level before being combined within a realistic two same-sign lepton signature. We investigate in detail uncertainties originating from missing higher-order corrections and from the parton-shower matching scheme employed.
Introduction
Top-quark pair production in association with a gauge boson is one of the rarest scattering processes in the Standard Model (SM). Nonetheless, it has received much attention recently as it constitutes a large background to many SM measurements and searches for physics beyond the Standard Model (BSM). Most notably, the →¯± process is the dominant background for SM measurements of the¯and of the four top-quark production process in the multi-lepton signatures. However, in both cases, recent measurements of the¯background contributions reveal tensions with the SM predictions [1].
In recent years a lot of progress has been made to improve the theoretical description of the →¯± process. For instance, NLO QCD and electroweak (EW) corrections for predictions based on stable top-quarks are already known for some time [2][3][4][5]. Furthermore, the impact of threshold resummation on total cross sections and differential distributions has been studied in Refs. [6][7][8][9][10]. In addition, also the NLO QCD corrections to the decay in the NWA has been studied [11] as well as the inclusion of off-shell effects and non-resonant contributions has been addressed for the first time in Refs. [12][13][14] for the dominant QCD production mode as well as for the full one-loop SM corrections in Ref. [15]. Furthermore, the process has been matched to parton showers [16][17][18] and effects from multi-jet merging have been studied [19][20][21] as well. Also the approximate inclusion of full off-shell effects in parton-shower matched calculations of on-shell has been investigated in Ref. [22]. In the following, we present results for theoretical predictions for the →¯± process at O ( 3 ) and O ( 3 ) matched to parton showers via the P -B framework. Figure 1: The structure of higher-order corrections for the production of the →¯± final state.
Outline of the calculation
LO NLO α 2 s α α s α 2 α 3 α 3 s α α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α 2 s α 2 α s α 3 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 α 4 Q C D E W Q C D E W Q C D E W
In Fig. 1 . The implementation of the dominant QCD corrections in the P -B is based on virtual amplitudes provided via NLOX [23,24]. We compare results obtained with the P -B to those obtained with MG5_ MC@NLO [25] as well as the S framework [26,27]. Spin-correlated top-quark decays are, depending on the framework employed, taken into account in the study of the two same-sign lepton signature according to Refs. [28][29][30]. Theoretical predictions based on the P -B and MG5_ MC@NLO are interfaced with the P 8 parton shower, while for S its own Catani-Seymour shower is used. Thus, our comparison allows to compare different parton-shower matching schemes as well as different parton showers. Further details on the specific setup of the comparison can be found in Ref. [18].
Inclusive¯production
First we compare the different Monte Carlo generators at the fully inclusive level with stable top quarks and gauge bosons. Jets are defined via the anti-jet algorithm with a separation parameter of = 0.4 and we require jets to fulfill the following constraints Transverse momentum of the hardest jet In Fig. 2 we highlight the transverse momentum of the leading jet for the QCD production mode of →¯± at O ( 3 ) and the rapidity of the leading jet for the EW production channel at O ( 3 ). Even though the presented observables have only leading-order accuracy they serve as a good representation of our findings also for inclusive NLO accurate observables. We refer to Ref. [18], where we have studied more distributions for both production channels. For the transverse momentum distribution we find very good agreement between all employed generators. All predictions including parton-shower effects align very well with the fixed-order NLO QCD result in the tail of the distribution. This is expected, since the hard matrix elements are reliable for hard emissions and shower corrections are small in this region. On the other hand, we find differences between shower based and fixed-order predictions at the level of 20% at the beginning of the spectrum, which is dominated by soft and collinear emissions. These differences can be attributed to the leading logarithmic resummation performed by the parton showers. Missing higher-order corrections dominate the uncertainties for most of the plotted range. Only at the beginning of the distribution matching uncertainties become comparable in size. On the contrary, we observe large differences in the EW predictions at O ( 3 ) as can be seen in the rapidity distribution on the right of Fig. 2. None of the parton-shower based predictions is in agreement with fixed-order calculation.
( ) > 25 GeV , | ( )| < 2.5 , jets ≥ 0 . (1) fixed-order NLO POWHEG-BOX MG5 aMC@NLO Sherpa inc. ttW ± QCD 10 −5 10 −4 10 −3 10 −2p T (j 1 ) [GeV] (σ/σ NLO ) matching fixed-order NLO POWHEG-BOX MG5 aMC@NLO Sherpa inc. ttW ± EW
To be specific, we find deviations of the order of 50% for the P -B , 100% for S and more than 200% for MG5_ MC@NLO. As indicated by the large matching uncertainties of the MG5_ MC@NLO prediction the resulting curve depends crucially on the choice of the initial shower scale.
Two same-sign lepton signature
Turning now to the comparison for a realistic two same-sign lepton signature at the fiducial level. Final-state particles are subject to the following phase space cuts
(ℓ) > 15 GeV , | (ℓ)| < 2.5 , ( ) > 25 GeV , | ( )| < 2.5 ,(2)
where jets are formed using the anti-jet algorithm with = 0.4. Additionally, we require exactly 2 same-sign leptons, at least 2 jets and at least 2 light jets. Predictions shown in the following correspond to the sum of¯+ and¯− processes as well as the combination of O ( 3 ) and O ( 3 ) contributions.
Representative of our full findings in Ref. [18] we show the transverse momentum distribution of the leading jet on the left and the invariant mass distribution of the two hardest light jets on the right of Fig. 3. The hardest jet predominantly originates from the top-quark decay and therefore allows for a comparison of the different approaches to model spin-correlated decays in the event generators. We find very good agreement between the various predictions with differences less than 5% for most of the plotted range. In the beginning of the distribution differences are slightly larger at the level of 10%. This, however, can be attributed to the different treatment of radiation from heavy quarks. Over the whole plotted range we also find that scale uncertainties dominate the theoretical uncertainties and are of the order of 10% − 20%. From the invariant mass distribution of the two hardest light jets, as depicted on the right of Fig. 3, we can draw multiple conclusions. First of all, we notice the that the peak of the distribution is located around the boson resonance, i.e. the leading jets originate from the hadronic decaying boson that is described at leadingorder accuracy for all generators. Furthermore, we see that the EW production mode starts as a +10% correction that increases towards the end of the spectrum up to +25%. Generally, the EW contribution becomes sizable if the considered observable is sensitive to forward jets. However, for most observables we studied the inclusion of the O ( 3 ) contribution constitutes a constant +10% correction at the differential level.
Summary
We presented results of our recent comparison [18] of the →¯± process for a two same-sign lepton signature. We find overall very good agreement between the various generators employed in the case of the QCD production mode at O ( 3 ). On the contrary, for the EW production of the →¯± final state we observe sizable differences between the generators.
Nonetheless, the less accurate modeling of the EW contribution has only a small impact once QCD and EW contributions are combined because the latter are generally a 10% effect on top of the dominant QCD contribution. Only in phase space regions dominated by forward jets the EW contributions becomes sizable. Furthermore, we investigated spin-correlation effects in the top-quark decay modeling, which can also modify the shape of leptonic observables at the 10% level. Further improvements in the realistic description is highly signature dependent. For instance, in multi-lepton signatures the NNLO QCD corrections to the →¯± production process are of utmost importance. For multi-lepton signatures also the matching of the full off-shell computation to parton showers will further improve the description of fiducial phase space volumes. However, for signatures involving the hadronic boson decays the inclusion of NLO QCD corrections in the decay is inevitable.
Figure 2 :
2Differential cross section distribution as a function of the transverse momentum of the hardest jet at O ( 3 ) (l.h.s) and of the rapidity of the hardest jet at O ( 3 ) (r.h.s).
Figure 3 :
3Differential cross section distribution as a function of the transverse momentum of the hardest jet (l.h.s) and of the invariant mass of the two hardest light jets (r.h.s).
Transverse momentum of the hardest jet dσ/dp T [pb/GeV] Transverse momentum of the hardest jet0.7
0.8
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1
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(σ/σ
NLO )
scales
0
100
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AcknowledgementsThe author acknowledges support by the U.S. Department of Energy under the grant DE-SC0010102.
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. E Bothmann, 10.21468/SciPostPhys.7.3.034arXiv:1905.09127Sherpa], SciPost Phys. 7334hep-phE. Bothmann et al. [Sherpa], SciPost Phys. 7, no.3, 034 (2019) doi:10.21468/SciPostPhys.7.3.034 [arXiv:1905.09127 [hep-ph]].
. S Frixione, E Laenen, P Motylinski, B R Webber, 10.1088/1126-6708/2007/04/081arXiv:hep-ph/0702198JHEP. 0481hep-phS. Frixione, E. Laenen, P. Motylinski and B. R. Webber, JHEP 04, 081 (2007) doi:10.1088/1126-6708/2007/04/081 [arXiv:hep-ph/0702198 [hep-ph]].
. P Artoisenet, R Frederix, O Mattelaer, R Rietkerk, 10.1007/JHEP03(2013)015arXiv:1212.3460JHEP. 0315hep-phP. Artoisenet, R. Frederix, O. Mattelaer and R. Rietkerk, JHEP 03, 015 (2013) doi:10.1007/JHEP03(2013)015 [arXiv:1212.3460 [hep-ph]].
. P Richardson, 10.1088/1126-6708/2001/11/029arXiv:hep-ph/0110108JHEP. 1129hep-phP. Richardson, JHEP 11, 029 (2001) doi:10.1088/1126-6708/2001/11/029 [arXiv:hep- ph/0110108 [hep-ph]].
| [] |
[
"STATUS OF CMS DARK MATTER SEARCHES IN 2011",
"STATUS OF CMS DARK MATTER SEARCHES IN 2011"
] | [
"Sezen Sekmen [email protected] \nDepartment of Physics\nFOR THE CMS COLLABORATION\nFlorida State University\n32306TallahasseeFloridaUSA\n"
] | [
"Department of Physics\nFOR THE CMS COLLABORATION\nFlorida State University\n32306TallahasseeFloridaUSA"
] | [
"Romanian Journal of Physics"
] | We present the status of dark matter searches performed by the Compact Muon Solenoid Experiment using 7 TeV pp data collected by the CERN Large Hadron Collider in 2010 and 2011. The majority of the results shown here were obtained using 1.1 fb −1 of data. We give highlights from analyses searching for candidates such as WIMPs, gravitinos, axinos and TeV scale particles. All observations so far were found to be consistent with the Standard Model predictions. The search results were used to set exclusion limits on various new physics scenarios. | null | [
"https://arxiv.org/pdf/1112.3518v1.pdf"
] | 59,148,583 | 1112.3518 | 417f7c8b34e96df5bf00d29db955922639e8a026 |
STATUS OF CMS DARK MATTER SEARCHES IN 2011
Sezen Sekmen [email protected]
Department of Physics
FOR THE CMS COLLABORATION
Florida State University
32306TallahasseeFloridaUSA
STATUS OF CMS DARK MATTER SEARCHES IN 2011
Romanian Journal of Physics
Compiled December 16, 2011Large Hadron ColliderCompact Muon Solenoiddark matter
We present the status of dark matter searches performed by the Compact Muon Solenoid Experiment using 7 TeV pp data collected by the CERN Large Hadron Collider in 2010 and 2011. The majority of the results shown here were obtained using 1.1 fb −1 of data. We give highlights from analyses searching for candidates such as WIMPs, gravitinos, axinos and TeV scale particles. All observations so far were found to be consistent with the Standard Model predictions. The search results were used to set exclusion limits on various new physics scenarios.
INTRODUCTION
The quest for new physics has entered a fascinating era with the start of the CERN Large Hadron Collider (LHC) [1] in 2009. As of November 2011, over 5 fb −1 of 7 TeV proton-proton collision data have been collected, and are being relentlessly analyzed to search for any hint of new physics that may lead to the replacement of the Standard Model (SM) of particle physics with a better theory. These searches are unique in that they are guided only by the unknowns-the puzzling deficiencies of the SM-and not by any concrete theory or experimental result that indicates where the new physics may dwell. Some guiding questions include: what is the mechanism of electroweak symmetry breaking, what is the origin of flavor, do all the forces unify and if so, how, and (assuming this is the correct question to ask) how should one construct a quantum theory of gravity?
Another outstanding mystery, which particle physics tries to unravel together with cosmology, is the nature of dark matter in the Universe, whose existence is suggested by the astrophysics observations. A great variety of new physics models have been devised to address these issues, and naturally, most of these models accommodate candidates for dark matter. The close link between particle physics and cosmology suggests that given a clue for new physics at the LHC, we are likely to extract some information also on the nature of dark matter. A gigantic effort is ongoing in all LHC experiments to discover such a clue. Here, we will focus on the Compact Muon Solenoid (CMS) experiment that leads a diverse program for new physics searches, and summarize the current studies which may particularly help our understanding of dark matter. After a short introduction to CMS, we will give the status of searches that look for a variety of dark matter candidates, ranging from weakly interacting massive particles to gravitinos and axinos to TeV-scale dark matter candidates. Except a few cases, the results we show were obtained using ∼1.1 fb −1 of LHC data. So far, all CMS measurements are consistent with the SM.
THE CMS DETECTOR
CMS is one of the two generic-purpose detectors at the LHC. Figure 1 shows a schematic drawing of the CMS detector. The central feature of the detector is a superconducting solenoid providing an axial magnetic field of 3.8 T. The bore of this solenoid is instrumented with several particle detection systems. Charged particle trajectories are measured within the field volume by a silicon pixel and strip tracker system, with full azimuthal (φ) coverage and a pseudorapidity (η) acceptance from −2.5 to +2.5. Here, η = − ln(tan(θ/2)) and θ is the polar angle with respect to the counterclockwise beam direction. The tracking volume is surrounded by a lead tungstate crystal electromagnetic calorimeter (ECAL) and a brass/scintillator hadron calorimeter that provide an η coverage from −3 to +3. The forward hadron calorimeter extends the calorimetric coverage symmetrically to |η| < 5. Muons are identified in gas ionization detectors embedded in the steel return yoke of the magnet. The CMS detector is nearly hermetic, which allows for momentum-balance measurements in the plane transverse to the beam axis. A more detailed description of the CMS detector can be found in [2].
THE CMS DARK MATTER SEARCH
CMS is an approximately hermetic instrument which can detect objects such as jets, b−jets, electrons, muons, taus and missing transverse energy ( / E T ) precisely, which makes it a general purpose detector capable of looking for a variety of dark matter particle candidates. Currently, the most commonly explored candidates are the weakly interacting massive particles (WIMPs) such as neutralinos or sneutrinos suggested by supersymmetry (SUSY), Kaluza-Klein particles suggested by models with universal extra dimensions (UED), or the lightest T-odd parity particles suggested by little Higgs models. There also exist searches for other supersymmetric candidates such as gravitinos or axinos through indirect hints from heavy stable charged particles (HCSPs). In addition to these, other analyses look for dark photons, which may point to TeV scale dark matter particles.
The road from pp collisions to any inference on the nature and properties of dark matter is long and arduous. First, CMS has to discover a deviation from the SM in one or more final states. Then, investigations will be made to determine if the discovered signal is due to new physics that includes a dark matter candidate. This may also yield a rough estimate of the dark matter particle's mass. Subsequently, the masses of all other particles related to the new physics will be measured based on the kinematics of the high branching ratio decays. This will be followed by precision measurements involving cross sections, branching ratios, angular distributions and rare decays that may need 14 TeV of center of mass energy and at least O(10 fb −1 ) of integrated luminosity to explore fully. All these investigations will eventually lead to calculations of quantities such as the dark matter relic density and various interaction cross sections involving the dark matter particles.
However, as noted earlier, CMS has yet to see any deviation from the SM predictions. Therefore, the current CMS activity is mainly focused on mapping this consistency with the SM into exclusions on candidate new physics models. So far, all such new physics interpretations that can be related to dark matter are done within the framework of supersymmetry (SUSY). In the following, we will summarize the results of searches for new physics related to dark matter, and the interpretation of these results. Current searches can be classified as follows: i) searches with missing energy final states; ii) searches for heavy stable charged particles and iii) searches for lepton jets.
SEARCHES WITH MISSING ENERGY FINAL STATES
The most conventional way to search for many classical SUSY scenarios with WIMP dark matter candidates like neutralinos or sneutrinos is to look for an excess of events with high missing energy. Such SUSY scenarios are typically characterized by dominant direct production of squarks and gluinos, and occasionally of charginos http://www.nipne.ro/rjp submitted to Romanian Journal of Physics ISSN: 1221-146X
and neutralinos at the LHC, followed by cascade decays involving leptons and jets into a pair of heavy, neutral, stable particles. All these lead to a diverse set of final states with missing energy, which are systematically explored by CMS. Table 1 lists the missing energy final states subject to CMS SUSY searches. Though there is no significant excess, there are some events at the high missing transverse momentum tail. An event display for a high missing transverse momentum event from the 2010 dataset is shown in Figure 3 in order to illustrate the kinematics of such events. The high missing transverse momentum in this striking event was checked to be not due to any detector effect. Furthermore, none of the jets were identified as originating from a heavy flavour quark, and none of the jet invariant mass combinations matches the W or top masses.
The most popular model that has been used for interpreting new physics searches with missing energy is the constrained MSSM (CMSSM) which is defined by four * Note however that this figure is only an illustration, and that CMS analyses generally quantify SM background yields based on data-driven estimation methods. free parameters and a sign (m 0 , m 1/2 , A 0 , tan β and sgn(µ)) at the GUT scale. Figure 4 shows the 95% CL exclusion curves obtained by various analyses using the 2010 and 2011 data on the m 0 − m 1/2 plane with fixed A 0 = 0, tan β = 10 and µ > 0. We see that, assuming neutralino dark matter, the low m 0 − m 1/2 bulk region and some parts of the stau coannihilation region (that give relic densities consistent with the WMAP measurements) are disfavored. A large part of the favored regions predict relic densities higher than the observed WMAP upper limit, which might indicate viability of non-WIMP candidates such as axions/axinos or gravitinos. CMS results from the analyses involving photons were interpreted within the context of general gauge mediation (GGM) models. Here, the lightest supersymmetric particle (LSP) and hence, the dark matter candidate is a light, O(∼MeV) gravitino, and the next-to-lightest supersymmetric particle (NLSP) is a neutralino (χ 0 1 ), which decays promptly to the gravitino plus a photon. Interpretations were made for two cases:
• for a bino-likeχ 0 1 LSP, plus a gluino (g) or squark (q) next-to-next-to-lightest supersymmetric particle (NNLSP) that decays to quarks + NLSP,
• for a purely winoχ 0 1 LSP, plus a wino-like chargino (χ ± 1 ) NNLSP which is almost mass degenerate with the neutralino and hence decays directly to W +G.
No signal was observed in the photon channels, and this is consistent with the constraints from Big Bang Nucleosynthesis (BBN) that disfavor SUSY models with a gravitino LSP and a neutralino NLSP. Figure 5 shows the lower 95% CL exclusion contours on the squark-gluino mass plane for GGM benchmark models with a fixed χ 0 1 mass of 375 GeV. Contours are shown for a bino-likeχ 0 1 , using results from the ≥ 2 photon + jets + / E T search (left), and for a wino-likeχ 0 1 , using results from the = 1 photon + ≥ 3 jets + / E T search (right).
SEARCHES FOR HEAVY STABLE CHARGED PARTICLES
CMS has performed other more exotic searches that looked for heavy stable charged particles (HSCPs). HSCPs are, as the name suggests, heavy and charged particles that can traverse the whole detector before they decay. Thus, HCSPs would look like "non-relativistic" muons in the detector, where their relatively low speed is due to their non-negligible mass. Discovering HCSPs would suggest non-WIMP dark matter such as gravitinos and axinos, which have relatively weak couplings to other sparticles. In cases where the gravitino or axino is the LSP and a charged slepton or squark is the NLSP, the weak gravitino coupling implies that the charged NLSP will have a long enough lifetime to traverse the detector before it decays.
Searches for HCSPs were conducted by selecting tracks reconstructed in the inner tracker detector that have a large ionization loss (dE/dx) and high p T . A further selection was also studied, which additionally requires that these tracks be identified in the muon system and that they have large time of flight [20]. Since no excess was observed, lower limits were set on the masses of various HCSP candidates. Figure 6 shows predicted theoretical cross sections and observed 95% CL upper limits on the cross section for gluinos and stops, for various nuclear interaction models. Results are shown for 1.1 fb −1 data, for the tracker-only selection (left) and tracker plus muon selection (right).
SEARCHES FOR LEPTON JETS
Recent astrophysical observations showing an excess of high energy positrons in the cosmic ray spectrum [21] have inspired various new models [22] in which this excess arises from the annihilation of TeV-scale dark matter particles in the galactic halo. Such models may also account for the observed discrepancies in direct searches for dark matter. Some realizations of these models typically accommodate an extra U (1) symmetry that couples weakly to the SM. Breaking of this U (1) results in a light vector boson with mass ∼O(1 GeV), called a "hidden" or a "dark" photon. These dark photons have a small kinetic mixing with the SM, which allows them to decay into collimated lepton pairs, or occasionally to hadrons. In supersymmetric extensions of such models, dark photons can be produced in the SUSY cascade decays. CMS has performed a search using 35 pb −1 of 2010 data for groups of collimated muon pairs-also called "lepton jets"-to investigate the existence of dark photons or other non-SM low mass resonances. However, again, the results are consistent with the SM [23]. Figure 7 shows the dimuon invariant mass distribution for events with a single dimuon with p T > 80 GeV compared with the background (left), where the consistency with the SM is evident. Fig. 7 -Dimuon invariant mass distribution for events with a single dimuon of p T > 80 GeV compared with the expected background. If new physics had been discovered in this channel, it would appear as a narrow peak above the background curve (with resolution similar to that of the J/ψ, which is the detector resolution) [23].
CONCLUSIONS
We have presented the current status of searches by the Compact Muon Solenoid collaboration for new physics involving dark matter, based on analyses of 7 TeV proton-proton collision data collected in 2010 and 2011 at the LHC. Most of the results shown here are based on 1.1 fb −1 of data. We gave highlights from several CMS analyses searching for dark matter candidates such as WIMPs, gravitinos, axinos and TeV scale particles. So far, no search has observed any trace of new physics. Consequently, we have, as yet, no direct clues of the nature of dark matter. The observed consistency with the SM has been used to set exclusion limits on various new physics models.
The LHC is still young-it has much more data to collect, and higher collision energies to attain-thus, the quest for new physics has only just begun. In the event of a discovery, a long series of rigorous interpretation studies will await us, which would necessarily involve the combination of collider and astrophysics results in order to reveal the true nature of dark matter.
Fig. 1 -
1Schematic drawing of the CMS detector http://www.nipne.ro/rjp submitted to Romanian Journal of Physics ISSN:
Figure 2
2shows a comparison of the distributions of data with various SM and SUSY Monte Carlo simulations as a function of the hadronic transverse momentum H T = i p jet i T (left) and missing hadronic transverse momentum / H T = i p jet i T (right). This illustrates the consistency of the LHC data with the SM predictions * .
Fig. 2 -
2Distributions of hadronic transverse momentum H T (left) and missing hadronic transverse momentum / H T (right) for data and various Monte Carlo simulation samples after all baseline selection cuts except those on H T and / H T respectively (from the jets + missing transverse momentum analysis[9]).
Fig. 3 -
3Event 70626194, in luminosity section 49 of run 148953. (r, φ) view of the highest missing transverse momentum event passing the event selection imposed in the 2010 jets + missing transverse momentum analysis [19]. http://www.nipne.ro/rjp submitted to Romanian Journal of Physics ISSN: 1221-146X
Fig. 4 -
495%CL exclusion curves obtained by various analyses using the 2010 and 2011 data on the m 0 − m 1/2 plane with fixed A 0 = 0, tan β = 10 and µ > 0.
Fig. 5 -
5Lower 95% CL exclusion contours on the squark-gluino mass plane for GGM benchmark models with a fixedχ 0 1 mass of 375 GeV. Contours are shown for a bino-likeχ 0 1 , using results from the ≥ 2 photon + jets + / E T search (left), and for a wino-likeχ 0 1 , using results from the = 1 photon + ≥ 3 jets + / E T search (right).[11].
Fig. 6 -
6Predicted theoretical cross sections and observed 95% CL upper limits on the cross sections for different combinations of models and scenarios considered in the analysis. Results are shown for the tracker-only selection (left) and tracker plus muon selection (right). The bands represent the theoretical uncertainties on the cross section values[20].
Table 1 .
1CMS 2011 SUSY searches using missing energy final statesFinal state
Ldt Ref
(fb −1 )
jets + /
E T (using the α T variable [3])
1.1
[4]
jets + /
E T (using the razor variable [5])
0.8
[6]
jets + /
E T (using the M T 2 variable [7])
1.1
[8]
jets + missing transverse momentum
1.1
[9]
jets +b−jets + /
E T
1.1
[10]
photons + jets + /
E T
1.1
[11]
single lepton + jets + transverse momentum
1.1
[12]
same-sign dileptons + jets + /
E T
0.98 [13]
opposite-sign dileptons + jets + /
E T
0.98 [14]
Z + /
E T
0.98 [15]
Z + /
E T + jets
2.1
[16]
0.191 [17]
multileptons + /
E T
2.1
[18]
Sezen Sekmen for the CMS Collaboration(c) 2011-2012 RJP
http://www.nipne.ro/rjp submitted to Romanian Journal of Physics ISSN: 1221-146X
Sezen Sekmen for the CMS Collaboration(c) 2011-2012 RJP
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| [] |
[
"On Uncertainty Relations in the Product Form",
"On Uncertainty Relations in the Product Form"
] | [
"Xiaofen Huang \nSchool of Mathematics and Statistics\nHainan Normal University\n571158HaikouChina\n",
"Tinggui Zhang \nSchool of Mathematics and Statistics\nHainan Normal University\n571158HaikouChina\n",
"Naihuan Jing \nSchool of Mathematics\nSouth China University of Technology\n510640GuangzhouChina\n\nDepartment of Mathematics\nNorth Carolina State University\nNC27695RaleighUSA\n"
] | [
"School of Mathematics and Statistics\nHainan Normal University\n571158HaikouChina",
"School of Mathematics and Statistics\nHainan Normal University\n571158HaikouChina",
"School of Mathematics\nSouth China University of Technology\n510640GuangzhouChina",
"Department of Mathematics\nNorth Carolina State University\nNC27695RaleighUSA"
] | [] | We study the uncertainty relation in the product form of variances and obtain some new uncertainty relations with weight, which are shown to be tighter than those derived from the Cauchy-Schwarz inequality. | 10.1088/1674-1056/27/7/070302 | [
"https://arxiv.org/pdf/2003.10696v1.pdf"
] | 125,953,553 | 2003.10696 | 876d609b4d98d40a463e5bc689d44ee402f276f7 |
On Uncertainty Relations in the Product Form
Xiaofen Huang
School of Mathematics and Statistics
Hainan Normal University
571158HaikouChina
Tinggui Zhang
School of Mathematics and Statistics
Hainan Normal University
571158HaikouChina
Naihuan Jing
School of Mathematics
South China University of Technology
510640GuangzhouChina
Department of Mathematics
North Carolina State University
NC27695RaleighUSA
On Uncertainty Relations in the Product Form
Uncertainty relationObservablesVariance-based PACS numbers: 0365Bz8970+c
We study the uncertainty relation in the product form of variances and obtain some new uncertainty relations with weight, which are shown to be tighter than those derived from the Cauchy-Schwarz inequality.
I. INTRODUCTION
The uncertainty relations have played a fundamental role in the development of quantum theory not only in the foundation and also in recent investigations of quantum information and quantum communication, in particular in the areas such as entanglement detection [1,2], security analysis of quantum key distribution in quantum cryptography [3], quantum metrology and quantum speed limit [4][5][6]. Usually the uncertainty relations are expressed in terms of the product of variances of the measurement results of two incompatible observables.
Other forms of the uncertainty relations include entropic uncertainty principle [7][8][9], the majorization technique [10][11][12] and the recent weighted uncertainty relation [13].
In 1927 Heisenberg [14] analyzed the observation of an individual electron with photons and obtained the famous uncertainty principle
(∆P ) 2 (∆Q) 2 ≥ ( 2 ) 2 ,(1)
where (∆P ) 2 and (∆Q) 2 are the variances of the position P and momentum Q respectively.
The variance or standard deviation of the observable A with respect to the state ρ is defined by (∆A) 2 = A 2 − A 2 , where A = trρA is the mean value of the observable A.
The inequality (1) shows that the uncertainty in the position and momentum of a quantum particle are inversely proportional to each other: a particle's position and momentum cannot be known simultaneously. Thus the accuracy of quantum measurement is limited by the uncertainty principle. This principle uncovers a fundamental and peculiar feature in the atomic world, and is considered as one of the cornerstones of quantum mechanics.
Robertson [15] formulated the uncertainty relation for arbitrary pair of non-commuting observables A and B (with bounded spectra):
(∆A) 2 (∆B) 2 ≥ 1 4 | [A, B] | 2 .(2)
where Robertson's uncertainty relation is further generalized by Schrödinger [16]:
(∆A) 2 (∆B) 2 ≥ 1 4 | [A, B] | 2 + 1 4 | {A, B} − A B | 2 .(3)
This relation is evidently stronger than Heisenberg's uncertainty relation, and it also shows that the commutator reveals incompatibility while the anticommutator encodes correlation between observables A and B.
The goal of this note is to give a family of generalized Schrödinger uncertainty relations using a stronger Cauchy-Schwarz inequality.
II. GENERALIZED UNCERTAINTY RELATIONS
We start with a quantum system in the quantum state ρ = |Ψ Ψ| on the Hilbert space and β = (β 1 , β 2 , . . . , β n ). We will not distinguish the two inner products as long as it is clear from the context.
The variance of observable A can be expressed as (∆A) 2
= A 2 − A 2 = Ā 2 = α|α , thus (∆A) 2 (∆B) 2 = α|α β|β = n i=1 |α i | 2 n i=1 |β i | 2 .
Theorem 1 For observables A and B, we have the generalized uncertainty relation in the variance-based product form given by
(∆A) 2 (∆B) 2 ≥ n i=1 |α i | 1+λ |β i | 1−λ n i=1 |α i | 1−λ |β i | 1+λ ,(4)
where λ is any real number ∈ [0, 1], α i = ϕ i |Ā|Ψ and β i = ϕ i |B|Ψ .
Proof: For any real number λ ∈ [0, 1] one has the following Callebaut inequality [17]:
( n i=1 a i b i ) 2 ≤ n i=1 a 1+λ i b 1−λ i n i=1 a 1−λ i b 1+λ i ≤ n i=1 a 2 i n i=1 b 2 i .(5)
where
{a i } n i=1 , {b i } n i=1
are two sequences of positive real numbers. Then
(∆A) 2 (∆B) 2 = n i=1 |α i | 2 n i |β i | 2 ≥ n i=1 |α i | 1+λ |β i | 1−λ n i=1 |α i | 1−λ |β i | 1+λ ,
so we get the inequality in the Theorem.
We remark that our generalized uncertainty relation is stronger than Schrödinger's uncertainty relation, as the generalized Cauchy inequality shows that | α|β | is smaller than the right-hand side of our uncertainty relation.
The uncertainty relation (4) can be tightened by optimizing over the sets of complete orthonormal bases. Then we can improve the uncertainty relation by the Callebaut inequality as in the proof of Theorem 1.
Theorem 2 For observables A and B, one has the following uncertainty relation
(∆A) 2 (∆B) 2 ≥ max {|ϕ i } n i=1 |α i | 1+λ |β i | 1−λ n i=1 |α i | 1−λ |β i | 1+λ = L 1 .(6)
where λ is any fixed number ∈ [0, 1], α i = ϕ i |Ā|Ψ and β i = ϕ i |B|Ψ .
Theorem 3 For observables A and B, we can get the uncertainty relation in the following
(∆A) 2 (∆B) 2 ≥ max {|ϕ i } n i=1 (|α i | 2 + |β i | 2 ) n i=1 |α i | 2 |β i | 2 |α i | 2 + |β i | 2 := L 2 ,(7)
where α i = ϕ i |Ā|Ψ and β i = ϕ i |B|Ψ .
Proof: Consider another generalized Cauchy-Schwarz inequality, the Milne inequality [18]:
( n i=1 a i b i ) 2 ≤ n i=1 (a 2 i + b 2 i ) n i=1 a 2 i b 2 i a 2 i + b 2 i ≤ n i=1 a 2 i n i=1 b 2 i .(8)
where
{a i } n i=1 , {b i } n i=1
are two sequences of real numbers. Therefore
(∆A) 2 (∆B) 2 = α|α β|β = n i |α i | 2 n i |β i | 2 ≥ n i=1 (|α i | 2 + |β i | 2 ) n i=1 |α i | 2 |β i | 2 |α i | 2 + |β i | 2 .
Remark For observables A and B, we can get the uncertainty relation in the following
(∆A) 2 (∆B) 2 ≥ max{L 1 , L 2 }.(9)
Both the Callebaut and Milne inequalities (5), (8) are stronger than the usual Cauchy-Schwarz inequality, thus our uncertainty relations are tighter than those uncertainty relations derived from the Cauchy-Schwaz inequality, for example, the uncertainty inequality in [19] about observables:
(∆A) 2 (∆B) 2 ≥ 1 4 ( n | [Ā,B ϕ n ] Ψ + {Ā,B ϕ n } Ψ |) 2 .(10)
whereB ϕ n = |ϕ n ϕ n |B. The following example shows the our bound is tighter than the bound given by Mondal-Bagchi-Pati.
Example:We plot the lower bound of the product of variances of two incompatible observables, A = L x , B = L y , two components of the angular momentum of spin one particle with a state |Ψ = cosθ|1 − sinθ|0 , where the state |1 and |0 are the eigenvectors of L z corresponding to eigenvalues 1 and 0 respectively. Take the angular momentum operators with = 1:
L x = 1 √ 2 0 1 0 1 0 1 0 1 0 , L y = 1 √ 2 0 −i 0 i 0 −i 0 i 0 , L z = 1 0 0 0 0 0 0 0 −1 (11)
III. CONCLUSIONS
Uncertainty relations play a central role in the current research in quantum theory and quantum information [20][21][22]. We have derived a family of new product forms of variancebased uncertainty relations, which are expected to help further investigate the uncertainty relation. (4) with the weight λ = 1 2 , the red one is with the weight λ = 1 3 , and the black one is for Mondal-Bagchi-Patis' lower bound in (10).
[A, B] = trρ[A, B], the expectation value of the commutator [A, B] = AB − BA.
with the inner product | and consider observables A and B. Define the operatorĀ = A − A I associated with a given operator A. Let {|ϕ i } be an orthonormal basis of the Hilbert space and writeĀ|Ψ = i α i |ϕ i ,B|Ψ = i β i |ϕ i so that ĀB = α|β , where α|β is the usual inner product (linear in the second argument) for the vectors α = (α 1 , α 2 , . . . , α n )
FIG. 1 :
1The uncertainty relation for observables L x , L y at state |Ψ : the blue curve is the lower bound in
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| [] |
[
"Static and dynamic structure factors in three-dimensional randomly diluted Ising models",
"Static and dynamic structure factors in three-dimensional randomly diluted Ising models"
] | [
"Pasquale Calabrese \nDipartimento di Fisica dell'Università di Pisa and INFN\nLargo Pontecorvo 2I-56127PisaItaly\n",
"Andrea Pelissetto \nDipartimento di Fisica dell'\nUniversità di Roma \"La Sapienza\"\nINFN\nPiazzale Aldo Moro 2I-00185RomaItaly\n",
"Ettore Vicari \nDipartimento di Fisica dell'Università di Pisa and INFN\nLargo Pontecorvo 2I-56127PisaItaly\n"
] | [
"Dipartimento di Fisica dell'Università di Pisa and INFN\nLargo Pontecorvo 2I-56127PisaItaly",
"Dipartimento di Fisica dell'\nUniversità di Roma \"La Sapienza\"\nINFN\nPiazzale Aldo Moro 2I-00185RomaItaly",
"Dipartimento di Fisica dell'Università di Pisa and INFN\nLargo Pontecorvo 2I-56127PisaItaly"
] | [] | We consider the three-dimensional randomly diluted Ising model and study the critical behavior of the static and dynamic spin-spin correlation functions (static and dynamic structure factors) at the paramagnetic-ferromagnetic transition in the high-temperature phase. We consider a purely relaxational dynamics without conservation laws, the so-called model A. We present Monte Carlo simulations and perturbative field-theoretical calculations. While the critical behavior of the static structure factor is quite similar to that occurring in pure Ising systems, the dynamic structure factor shows a substantially different critical behavior. In particular, the dynamic correlation function shows a large-time decay rate which is momentum independent. This effect is not related to the presence of the Griffiths tail, which is expected to be irrelevant in the critical limit, but rather to the breaking of translational invariance, which occurs for any sample and which, at the critical point, is not recovered even after the disorder average. | 10.1103/physreve.77.021126 | [
"https://arxiv.org/pdf/0711.3673v2.pdf"
] | 18,758,197 | 0711.3673 | f4f78fb1ec39a71fc11ab05e7bf68f8ebb988c5a |
Static and dynamic structure factors in three-dimensional randomly diluted Ising models
28 Feb 2008
Pasquale Calabrese
Dipartimento di Fisica dell'Università di Pisa and INFN
Largo Pontecorvo 2I-56127PisaItaly
Andrea Pelissetto
Dipartimento di Fisica dell'
Università di Roma "La Sapienza"
INFN
Piazzale Aldo Moro 2I-00185RomaItaly
Ettore Vicari
Dipartimento di Fisica dell'Università di Pisa and INFN
Largo Pontecorvo 2I-56127PisaItaly
Static and dynamic structure factors in three-dimensional randomly diluted Ising models
28 Feb 2008arXiv:0711.3673v2 [cond-mat.dis-nn]numbers: 6460F-7510Nr7540Gb7540Mg
We consider the three-dimensional randomly diluted Ising model and study the critical behavior of the static and dynamic spin-spin correlation functions (static and dynamic structure factors) at the paramagnetic-ferromagnetic transition in the high-temperature phase. We consider a purely relaxational dynamics without conservation laws, the so-called model A. We present Monte Carlo simulations and perturbative field-theoretical calculations. While the critical behavior of the static structure factor is quite similar to that occurring in pure Ising systems, the dynamic structure factor shows a substantially different critical behavior. In particular, the dynamic correlation function shows a large-time decay rate which is momentum independent. This effect is not related to the presence of the Griffiths tail, which is expected to be irrelevant in the critical limit, but rather to the breaking of translational invariance, which occurs for any sample and which, at the critical point, is not recovered even after the disorder average.
I. INTRODUCTION AND SUMMARY
The effect of disorder on magnetic systems remains, after decades of investigation, a not fully understood subject. It is then natural to investigate relatively simple models, to try to understand the common features of disordered systems. In this regard, randomly diluted spin systems are quite interesting. First, they represent simple models which describe the universal properties of the paramagnetic-ferromagnetic transition in uniaxial antiferromagnets with impurities 1 and, in general, the order-disorder transition in Ising systems in the presence of uncorrelated local dilution. Second, they give the opportunity for investigating general problems concerning the effects of disorder on the critical behavior. Indeed, several important results, theoretical developments, and approximation schemes found for these models have been later generalized to more complex systems like spin glasses, quantum disordered spin models, etc....
In this paper we consider three-dimensional randomly diluted Ising (RDI) systems. Their critical behavior has been extensively studied. 1,2,3 There is now ample evidence that the magnetic transition in these systems, if it is continuous, belongs to a unique universality class, and several universal properties are now known quite accurately. Beside the static critical behavior, we also investigate the dynamic critical behavior, considering a purely relaxation dynamics without conservation laws, the so-called model A, 4 which is appropriate for uniaxial antiferromagnets. We focus on the dynamic (time-dependent) spin-spin correlation function G(x 2 − x 1 , t 2 − t 1 ) ≡ σ(x 1 , t 1 ) σ(x 2 , t 2 ) ,
where σ(x, t) is an Ising variable, the overline indicates the quenched average over the disorder probability distribution, and · · · indicates the thermal average. From the function G(x, t), one obtains the static (equal-time) structure factor G(k) and the dynamic structure factor G(k, ω). They are physically relevant quantities, which can be measured in neutron or X-ray scattering experiments. 5 It is therefore interesting to study the effects of disorder on these physical quantities, and check whether disorder gives rise to qualitative changes with respect to pure systems. We investigate their scaling behavior close to the magnetic transition for T → T + c in the high-temperature phase. As we shall see, while the critical behavior of the static structure factor is very similar to that in pure Ising systems, the critical behavior of the dynamic structure factor is significantly different; in particular, the large-momentum behavior shows some new features.
Since the critical region in the paramagnetic phase, i.e. for T T c , is located in the Griffiths phase, 6,7 it is mandatory to discuss first the relevance of the so-called Griffiths singularities and Griffiths tails for the universal critical behavior of the correlation functions when T → T + c . In fact, one of the most notable features of randomly diluted spin systems is the existence of the so-called Griffiths phase for T c < T < T p , where T p is the critical temperature of the pure system. This is essentially related to the fact that, in the presence of disorder, the critical temperature T c is lower than T p , and therefore, in the temperature interval T c < T < T p , there is a nonvanishing probability to find compact clusters without vacancies (Griffiths islands) that are fully magnetized. They give rise to essential nonanalyticities in thermodynamic quantities. 6,7 Moreover, these clusters are responsible for a nonexponential tail in dynamic correlation functions. 8,9,10,11 In the case of RDI systems one can show that
G(x, t) ≈ G G (t) = B exp[−C(ln t) 3/2 ],(2)
for any finite x and t → ∞, which implies a diverging relaxation time. We should mention that these effects are quite difficult to detect, and there is still no consensus on their experimental evidence even in systems with correlated disorder, in which these effects are magnified (see, e.g., Ref. 7,12 and references therein).
Griffiths essential singularities give quantitatively negligible effects on thermodynamic quantities and on the static critical behavior. One can argue that also the Griffiths tail (2) is irrelevant in the critical limit: the nonexponential tail does not contribute to the critical scaling function associated with G(x, t). 8 This is essentially due to the fact that B and C that appear in Eq.
(2) are expected to be smooth functions of the temperature, approaching finite constants as T → T c . Thus, in the critical limit, t → ∞, T → T c at fixed tξ −z , where ξ ∼ (T − T c ) −ν is the diverging correlation length, the nonexponential contribution simply vanishes. To understand why, let us consider the simplified situation in which the contributions to the autocorrelation function G(x, t) due to the Griffiths islands and to the critical modes just sum as
G(t) ≈ G C (t) + G G (t) = aξ b exp(−ctξ −z ) + B exp(−C(ln t) 3/2 ),(3)
where we neglect all couplings between Griffiths and critical modes. Here, G C (t) is the critical contribution, while G G (t) is the nonexponential Griffiths tail, which dominates for t ≫ t * , where t * is the time at which the two terms have the same magnitude. In the critical limit we have t * ∼ ξ z (ln ξ) 3/2 . Since the critical limit is taken at fixed t/ξ z , the relevant quantity is t * /ξ z , which diverges as (ln ξ) 3/2 approaching the phase transition. This means that, for any fixed value of t/ξ z , the condition t ≪ t * is always satisfied sufficiently close to the critical temperature T c , i.e., the nonexponential tail is negligible.
In order to determine the scaling behavior of the spin-spin correlation function (1), we perform Monte Carlo (MC) simulations of a three-dimensional RDI model on a simple cubic lattice and perturbative field-theoretical (FT) calculations. In the following we briefly summarize our main results.
The high-temperature critical behavior of the static structure factor G(k) is substantially analogous to that of G(k) in pure Ising systems. 13 We consider the universal scaling function g(Q 2 ) ≡ G(k)/ G(0), where Q 2 ≡ k 2 ξ 2 and ξ is the second-moment correlation length. We find that g(Q 2 ) is very well approximated by the Ornstein-Zernike form (Gaussian free-field propagator) g(Q) OZ = 1/(1 + Q 2 ): deviations are less than 1% for Q 5 and increase to 5% at Q ≈ 50. At large momenta, for Q 10 say, the static structure factor follows the Fisher-Langer law; 14 in particular, g(Q 2 ) ≈ 0.92/Q 2−η with η ≈ 0.036 for Q 30.
At variance with the static case, the dynamic structure factor displays substantial differences with respect to the pure case, even though, as expected, the Griffiths tail turns out to be irrelevant in the critical limit. We consider the universal scaling function
Γ(Q 2 , S) ≡ lim T →T + c G(k, t) G(k, 0)(4)
in the critical limit t → ∞, k → 0, and T → T + c at fixed Q and S. Here S ≡ t/τ int , where τ int is the zero-momentum integrated autocorrelation time. Perturbative field theory shows that Γ(Q 2 , S) decays exponentially, as in the case of pure systems, and this is confirmed by the simulation results. In pure Ising systems 15 the large-S decay rate of Γ(Q 2 , S) depends on Q: the large-S behavior is very similar to that of Γ(Q 2 , S) in the noninteracting Gaussian model, i.e. Γ(Q 2 , S) ∼ exp[−κ(Q 2 )S], where κ(Q 2 ) = (1 + Q 2 ). This behavior drastically changes in the presence of random impurities; in particular, the large-S decay rate becomes independent of Q. MC simulations and FT perturbative calculations show that, for generic values of Q, Γ(Q 2 , S) has two different behaviors as a function of S. For small values of S, it decreases rapidly, with a rate that increases as Q increases, as it does in the pure system. 15 For large S instead, Γ(Q 2 , S), and therefore G(k, t), decreases with a momentumindependent rate. For large Q and S we find
Γ(Q 2 , S) ∼ S a Q −ζ e −κS ,(5)
where a and ζ are critical exponents, and κ does not depend on Q. We present a physical argument which relates the different large-S behavior compared to pure systems to the loss of translational invariance. Such a phenomenon is obvious for a given fixed sample, but at the critical point translational invariance is not even recovered after averaging over disorder, because of the absence of self-averaging. Note that this phenomenon is only related to disorder and thus, it is expected in all systems in which disorder is relevant.
In general, the perturbative calculations predict a scaling behavior of the form
Γ(Q 2 , S) ∼ Q −ζ f ζ (S),(6)
for large Q, where f ζ (S) is a function of S such that f ζ (S) ∼ S a e −κS for S → ∞. This behavior implies that G(x, t) is always nonanalytic for x = 0 and any t. Indeed, because of Eq. (6), the integral
d d Q Q 2n Γ(Q 2 , S)g(Q 2 ) ∼ d d k k 2n G(k, t)(7)
diverges for n ≥ n c ≡ (ζ + 2 − d − η)/2 (d is the spatial dimension). Since the moments of G(k, t) are directly related to the derivatives of G(x, t) with respect to x computed for x = 0, the n-th derivative of G(x, t) diverges for n ≥ n c ; hence G(x, t) is not analytic at x = 0. This implies that the scaling function F (Y 2 , S), defined as
G(x, t) = ξ −d+2−η F (Y 2 , S), Y 2 ≡ x 2 /ξ 2 ,(8)behaves as F (Y 2 , S) = f 0 (S) + f λ (S)|Y | λ + · · ·, where λ = ζ + 2 − d − η, for Y 2 → 0. MC
simulations indicate that ζ ≈ 2 in three dimensions, which implies λ ≈ 1. This phenomenon does not occur in pure systems, since in this case the decay rate κ depends on Q and guarantees the integrability of the integrand which appears in Eq. (7) for any n. Therefore, F (Y 2 , S) has an analytic expansion around Y 2 = 0, i.e., F (Y 2 , S) = f 0 (S) + f 2 (S)Y 2 + · · ·.
This nonanalyticity should be a general property of random models in which disorder is relevant, and not specific of RDI systems.
Finally, it is worth mentioning that in the case of three-dimensional randomly dilute multicomponent spin models, such as the XY and the Heisenberg model, the effects of disorder found in RDI models are expected to be suppressed in the critical limit T → T + c , and should only appear as peculiar scaling corrections. 16 Indeed, the asymptotic critical behavior of the correlation functions is expected to be the same as that in the corresponding pure model, because the pure fixed point is stable under random dilution (according to the Harris criterion, 17 dilution is irrelevant if the specific-heat exponent α of the pure system is negative).
The paper is organized as follows. In Sec. II we introduce the model that we study. In Sec. III we discuss the static structure factor in the high-temperature phase. In Sec. III A we define the quantities that are computed in the MC simulation, in Sec. III B we present some perturbative calculations, while in Sec. III C we discuss the MC results. In Sec. IV we discuss the dynamic structure factor. Again, we first define the basic quantities (Sec. IV A), then we present a one-loop perturbative calculation (Sec. IV B), and finally we report the MC results (Sec. IV C and IV D). In the appendices we report some details of the perturbative calculations.
II. THE MODEL
We consider the randomly site-diluted Ising model with Hamiltonian
H ρ = − <xy> ρ x ρ y σ x σ y ,(9)
where the sum is extended over all pairs of nearest-neighbor sites of a simple cubic lattice, and ω = 0.37(6) 19 ), while field theory predicts ω = 0.25(10), 22 ω = 0.32 (6). 26 For the next-to-leading exponent ω 2 , an appropriate analysis of the FT expansions gives 20 ω 2 = 0.82 (8), which is consistent with the MC results for improved models. 20 We consider the static (equal-time) two-point correlation function G(x) ≡ G(x, t = 0).
σ x = ±1
In the infinite-volume limit we define the second-moment correlation length ξ
ξ 2 ≡ − 1 χ ∂ G(k) ∂k 2 k 2 =0 ,(10)
where G(k) is the Fourier transform of G(x) and
χ ≡ x G(x) = G(0)(11)
is the magnetic susceptibility. It is also possible to define an exponential correlation length ξ exp . Given the infinite-volume G(x), we define
ξ exp ≡ − lim |x|→∞ |x| ln G(x) .(12)
In the critical limit ξ and ξ exp diverge. If t r ≡ (T − T c )/T c and T c is the critical temperature, for |t r | → 0 we have in the thermodynamic limit
ξ, ξ exp ∼ |t r | −ν ,(13)
where ν is a universal critical exponent. In the same limit, correlation functions have a universal behavior. For instance, the infinite-volume G(k)/χ becomes a universal function of the scaling variable
Q 2 ≡ k 2 ξ 2 ,(14)
i.e. we can write in the scaling limit k → 0, t r → 0 at fixed Q
χ −1 G(k) ≈ g(Q 2 ),(15)
where g(x) is universal. Moreover, the ratio ξ 2 /ξ 2 exp converges to a universal constant S M defined by
ξ 2 /ξ 2 exp ≈ S M .(16)
For a Gaussian theory the spin-spin correlation function shows the Ornstein-Zernike (OZ)
behavior
G OZ (k) = Z k 2 + r .(17)
It follows χ = Z/r, ξ 2 = 1/r, and
g OZ (Q 2 ) = 1 1 + Q 2 .(18)
Moreover, S M = 1.
Fluctuations change this behavior. For small Q 2 , g(Q 2 ) is analytic, so that we can write the expansion
g(Q 2 ) −1 = 1 + Q 2 + n=2 c n Q 2n ,(19)
where the coefficients c n parametrize the deviations from the OZ behavior. For large Q 2 , the structure factor behaves as
g(Q 2 ) ≈ C 1 Q 2−η 1 + C 2 Q (1−α)/ν + C 3 Q 1/ν + · · · ,(20)c 2 = −4(1) × 10 −4 ,(21)c 3 = 1.2(3) × 10 −5 ,(22)c 4 = −5(2) × 10 −7 .(23)
These results are fully consistent with those obtained in the √ ǫ expansion, in spite of the fact that in that case we have not applied any resummation and we have simply set ǫ = 1.
We can also compute S M . Since the coefficients c n are very small, we obtain
S M ≈ 1 + c 2 = 0.9996(1),(24)
where we used the estimate (21) of c 2 . As in the Ising case, 13,37 the coefficients c n show the pattern |c n | ≪ |c n−1 | ≪ ... ≪ |c 2 | ≪ 1 for n ≥ 3. (25) This is consistent with the expected analyticity properties of G(k). Since the complex-plane singularity in G(k) −1 that is closest to the origin is expected to be the three-particle cut located at k = ±3i/ξ exp , 38,39 the function g(Q 2 ) −1 is analytic up to Q 2 = −9S M . It follows that c n ≈ −c n−1 /(3 √ S M ), at least asymptotically.
The large-Q behavior can be investigated in the √ ǫ expansion. The three-loop calculation of the two-point function reported in App. A 1 allows us to determine the perturbative expansion of the coefficients C i appearing in Eq. (20). Setting ǫ = 1 in the expressions (A8),
we obtain C 1 ≈ 0.95, C 2 + C 3 = −0.96.
In order to compare with the experimental and numerical data it is important to determine g(Q 2 ) for all values of Q. For the pure Ising structure factor, several interpolations have been proposed with the correct large-and small-Q behavior. 13,38,39,40,41,42,43 The most successful one is due to Bray, 39 which incorporates the expected singularity structure of g(Q 2 ). In this approach, one assumes 1/g(Q 2 ) to be well-defined in the complex Q 2 plane, with a cut on the negative real Q 2 axis, starting at the three-particle cut Q 2 = −r 2 with r 2 = 9S M . Then, one obtains the spectral representation
H(Q 2 ) ≡ ∞ r du u 1−η F (u) Q 2 + u 2 , 1 g(Q 2 ) = 1 + Q 2 H(Q 2 ) S M H(−S M ) ,(26)
where F (u) is the spectral function, which must satisfy F (+∞) = 1, F (u) = 0 for u < r,
and F (u) ≥ 0 for u ≥ r.
In order to obtain an approximation one must specify F (u). Bray 39 proposed to use a spectral function that gives exactly the Fisher-Langer asymptotic behavior, i.e.
F B (u) = P 1 (u) − P 2 (u) cot 1 2 πη P 1 (u) 2 + P 2 (u) 2 ,(27)
where P 1 (u) = 1 + C 2 u p cos
πp 2 + C 3 u 1/ν cos π 2ν , P 2 (u) = C 2 u p sin πp 2 + C 3 u 1/ν sin π 2ν ,(28)
with p ≡ (1 − α)/ν. To obtain a numerical expression we fix ν = 0.683, η = 0.036, 20 and use the estimate (24) of S M . We must also fix C 2 and C 3 . Bray proposes to fix C 2 + C 3 to its ǫ-expansion value (in our case C 2 + C 3 = −0.96) and then to determine these constants by requiring F B (u = r) = 0. These conditions give C 2 = −8.04 and C 3 = 7.07. As a check,
we can compare the estimates of c n and C 1 obtained by using Bray's approximation g B (Q 2 ) with the previously quoted results. We obtain
C 1 = 2 sin πη/2 π S M H(−S M ) ≈ 0.92,(29)
and c 2 ≈ −4 · 10 −4 , c 3 ≈ 9 · 10 −6 , c 4 ≈ −4 · 10 −7 . These results are in very good agreement with those obtained before.
C. Monte Carlo results
In this section we study Hamiltonian (9) At each iteration we measure the correlation function G(x; β, L) and the structure factor G(k; β, L). Since rotational invariance is recovered in the critical limit, to speed up the Fourier transforms, we determine it as
G(k; β, L) = 1 3 x,y,z (e ikx + e iky + e ikz ) σ(0, 0, 0)σ(x, y, z) ,(30)
where the sum runs over the coordinates (x, y, z) of the lattice sites. Of course, on a finite lattice k can only assume the values 2πn/L, where n is an integer such that 0 ≤ n ≤ L − 1.
We also compute the second-moment correlation length ξ(β, L) defined by
ξ(β, L) 2 ≡ G(0; β, L) − G(k min ; β, L) k 2 min G(k min ; β, L) ,(31)
where k min ≡ 2π/L,k ≡ 2 sin k/2. For L → ∞, ξ(β, L) converges to the infinite-volume definition (10) with L −2 corrections. (right). We only report data corresponding to k ≤ k max = π/3. We also report an interpolating curve (dashed line), which is obtained by fitting all data at fixed Q reported on the right, as explained in the text.
In order to determine g(Q 2 ) we go through several different steps. First, for each β and L we interpolate the numerical data in order to obtain G(k; β, L) for any k in the range [0, π].
For this purpose we fit the numerical results for h(k; β,
L) ≡ G(0; β, L)/ G(k; β, L) to h(k; β, L) = 1 + nmax n=1 a nk 2n ,k = 2 sin k 2 .(32)
We increase n max until the sum of the residuals (χ 2 ) is less than half of the fitted points (those corresponding to 1 ≤ n ≤ L − 1), i.e. χ 2 < L/2 (note that the data are strongly correlated and thus it makes no sense to require χ 2 /DOF ≈ 1, where DOF = L − 1 − n max is the number of degrees of freedom of the fit). In most of the cases we take n max = 5, but in a few cases we had to take n max as large as 10.
Then, we investigate the finite-size effects. In the critical limit we expect
h(k; β, L) ≈ F Q ≡ kξ(β, L), ξ(β, L) L .(33)
Equivalently, one can also use
h(k; β, L) ≈ F Q ≡kξ(β, L), ξ(β, L) L ,(34)
wherek = 2 sin k/2. The two scaling forms are equivalent in the scaling limit k → 0, L → ∞, ξ(β, L) → ∞ at fixed Q (or Q) and ξ(β, L)/L; as a consequence, the function F (x, y) is the same in the two cases. Indeed,k = k + O(k 3 ), and thus, by keeping fixed Q or Q, one only changes analytic corrections decaying as L −2 . In particular, whatever choice is made, the structure factor g(Q 2 ) is equal to 1/F (Q, 0). Apparently, the corrections we are talking about here are less relevant than the nonanalytic corrections that should decay as L −ω 2 , ω 2 = 0.82 (8), and thus, a priori one would expect only small differences between the two approaches. Instead, as we show below, only by keeping Q fixed is one able to determine the structure factor in the infinite-volume limit.
In Fig. 1 In the two panels we also show the interpolation of the data at fixed Q. As expected, the data at fixed Q converge to this interpolation, but it is clear that no real information could have been obtained on the infinite-volume limit from the data in the left panel. In order to clarify why scaling at fixed Q is so much better than scaling at fixed Q, we consider the lattice Gaussian model with nearest-neighbor couplings. In this case, the spin-spin correlation function on a finite lattice is given by
G G (k) = Ẑ k 2 + r ,(35)
so that ξ 2 = 1/r and
h(k) = 1 + Q 2 .(36)
Thus, if we take the finite-size scaling limit at fixed Q there are no finite-size corrections:
the scaling is exact on any finite lattice. On the other hand, at fixed Q we obtain
h(k) ≈ (1 + Q 2 ) 1 − 1 12L 2 Q 4 (L 2 /ξ 2 ) 1 + Q 2 + · · · .(37)
In this case we have 1/L 2 corrections, which diverge as ξ/L → 0, exactly as we observe in our data. These corrections moreover increase with Q and thus make it difficult, if not impossible, to estimate the structure factor.
As a consequence of the above-reported discussion we consider below the finite-size scaling limit at fixed Q. In critical limit is obtained for k → 0, results close to the antiferromagnetic point k = π cannot have a good scaling behavior. Therefore, in the analysis we have only considered values of k such that k ≤ k max . If k max varies between π/4 and π/3, the final results are essentially independent of k max . The data reported in Figs. 1 and 2 scale as predicted by Eq. (34).
Within the precision of our results some corrections to scaling are only visible for Q = 5 and ξ(β, L)/L 0.2. They however die out fast in the interesting limit ξ(β, L)/L → 0. Note also that, as Q increases, the number of available points decreases and indeed we are not able to go beyond Q ≈ 50 with our data.
In order to determine the infinite-volume limit F (Q, 0), we have taken all data satisfying ξ(β, L)/L ≤ 0.5 and we have fitted them to
h(k; β, L)|k ξ= b Q = a 0 + jmax j=1 a j exp[−jL/ξ(β, L)].(38)
The fitting form (38) is motivated by theory, which predicts exponentially small finite-size corrections in the high-temperature phase. With the precision of our data it is sufficient to take j max = 2 to obtain χ 2 /DOF 1. The coefficient a 0 allows us to estimate g(Q 2 ):
g(Q 2 ) = 1/F (Q, 0) = 1/a 0 .
The results for k max = π/4 and π/3 are essentially identical within errors up to Q ≈ 40 (for k max = π/4 we do not have enough data to determine reliably F (Q, 0) for Q 40). In the following we take those corresponding to k max = π/3, which allow us to compute g(Q 2 ) up to Q = 50. In order to detect scaling corrections we We also report Bray's approximation, in which C 2 + C 3 is fixed to the √ ǫ value (Bray-ǫ), and the structure factor for the pure Ising model (Ising). The curve "interp" (solid line) corresponds to the interpolation g int (Q 2 ) reported in Eq. (40).
to ξ(β, L)/L = 0.158 andk = 0.989. For Q = 41 this lattice is no longer considered, since the corresponding k exceeds k max = π/3 (k max = 1). For Q = 41, the result with the smallest ξ(β, L)/L corresponds to ξ(β, L)/L = 0.236. The extrapolation to the infinite-volume limit is therefore much more imprecise.
For Q → ∞, g(Q 2 ) ≈ C 1 /Q 2−η , see Eq. (20). We fit the estimates of ln g(Q 2 ) reported in Fig. 3 (they correspond to integer values of Q between 1 and 50) to a + (η − 2) ln Q.
If we include only data with Q > Q min = 15 and 20, we obtain η = 0.032(1), 0.032(2), respectively. The error we quote here assumes that all data are independent, which is not the case. In order to determine the correct error bar, one should take into account the covariance among the results at different values of Q. This is not easy and therefore, in order to estimate the role of the statistical correlations, we use a more phenomenological approach. If g est (Q 2 ) is the estimate of g(Q 2 ) and σ(Q 2 ) the corresponding error, we consider new data g est (Q 2 ) − σ(Q 2 ) with the same error and we repeat the fit. We obtain η = 0.029 and η = 0.027 for Q min = 15 and 20. Analogously, if we consider g est (Q 2 ) + σ(Q 2 ), we obtain η = 0.035, 0.037. This simple analysis indicates that ±0.005 is a plausible estimate of the statistical error. Therefore, we quote η = 0.032(5) as our final result. This estimate is in good agreement with that reported in Ref. 20, η = 0.036(1), obtained from a finite-size scaling analysis of the susceptibility. In order to estimate C 1 , we consider g(Q 2 )Q 2−η , fixing η to η = 0.036(1). 20 For Q 20 this quantity is essentially constant: g(Q 2 )Q 2−η = 0.921(1), 0.920(1), 0.917(2), 0.919(3), for Q = 20, 25, 30, 35. We thus take
C 1 = 0.919(3)[3](39)
as our final estimate. The error in brackets gives the variation of the estimate as η varies by one error bar (±0.001). This estimate is close to the FT result C 1 ≈ 0.95 and in perfect agreement with the estimate (29) obtained by using Bray's approximation for the spectral function, C 1 ≈ 0.92. Indeed, as can be seen in Fig. 3, Bray's interpolation represents a very good approximation of the numerical data, deviations being quite tiny.
In Fig. 3 we also report the structure factor in pure Ising systems (we use the phenomenological approximation reported in Ref. 13, see their Eq. (30) with Q max = 15 and n max = 6).
In the pure case, deviations from the OZ behavior are larger: the addition of impurities has the effect of reducing the deviations from the OZ behavior.
Finally, we report a phenomenological interpolation which reproduces well our numerical data and is consistent with the large Q 2 behavior, g(Q 2 ) ≈ 0.919Q 0.036 /(1 + Q 2 ):
g int (Q 2 ) = (1 + 0.0227953Q 2 + 0.0000839355Q 4 ) 0.009 1 + Q 2 .(40)
IV. DYNAMIC STRUCTURE FACTOR IN THE HIGH-TEMPERATURE PHASE
In this section we consider the dynamic behavior of the Metropolis algorithm, which is a particular example of a relaxational dynamics without conservation laws, the so-called model A, as appropriate for magnetic systems. In Ref. 21 we computed the dynamic critical exponent, obtaining z = 2. 35(2). Here, we focus on the dynamic structure factor.
A. Definitions
To investigate the dynamic behavior we consider the time-dependent two-point correlation function (1) and its Fourier transform G(k, t) with respect to the x variable. Then, we define the integrated autocorrelation time
τ int (k) ≡ 1 2 ∞ t=−∞ G(k, t) G(k, 0) = 1 2 + ∞ t=1 G(k, t) G(k, 0) ,(41)
and the exponential autocorrelation time
τ exp (k) ≡ − lim |t|→∞ |t| ln G(k, t) ,(42)
which controls the large-t behavior of G(k, t). Here t is the Metropolis time and one time unit corresponds to a complete lattice sweep.
Beside τ int (k) and τ exp (k) we also define autocorrelation times τ int,x and τ exp,x . 21 In general,
given an autocorrelation function A(t) we define
I(s) ≡ 1 2 + 1 A(0) s t=1 A(t),(43)τ eff (s) ≡ n ln[A(s − n/2)/A(s + n/2)] ,(44)
for any integer s and any fixed even n. By linear interpolation these functions can be extended to any real s. Then, we define τ int,x and τ exp,x as the solutions of the consistency
equations τ exp,x = τ eff (xτ exp,x ),(45)τ int,x = I(xτ int,x ).(46)
These definitions have been discussed in Ref. 21. There, it was shown that they provide effective autocorrelation times with the correct critical behavior. For x → ∞, τ exp,x and τ int,x converge to τ exp and τ int , respectively.
As discussed in the introduction, for T c < T ≤ T p the correlation function G(x, t) does not decay exponentially for any finite value of x, but presents a slowly decaying tail, cf. Eq. (2).
Therefore, τ exp (k) diverges for all T c ≤ T < T p . As discussed in Ref. 21, this is not the case for the effective exponential autocorrelation time τ exp,x , which is finite for any finite x. Note that correlation functions decaying as in Eq.
(2) have a finite time integral and thus the integrated autocorrelation time is finite.
In the critical limit the autocorrelation times diverge. If t r ≡ (T − T c )/T c and T c is the critical temperature, for |t r | → 0 we have
τ int (k) ∼ τ exp,x (k) ∼ τ int,x (k) ∼ |t r | −zν ∼ ξ z ,(47)
where ν is the usual static exponent and z is a dynamic exponent that depends on the considered dynamics: ν = 0.683(2) and z = 2. 35(2) in the present case. 20,21 In the same limit, G(k, t)/ G(k, 0) becomes a universal function of the scaling variables
Q 2 ≡ k 2 ξ 2 , S ≡ t/τ int (0),(48)
i.e. we can write G(k, t)
G(k, 0) = Γ(Q 2 , S),(49)
where Γ(Q 2 , S) is universal, even in S, i.e., Γ(Q 2 , S) = Γ(Q 2 , −S), and satisfies the normalization conditions
Γ(Q 2 , 0) = 1, ∞ 0 Γ(0, S)dS = 1.(50)
The function G(k, 0) is the static structure factor whose critical behavior has been discussed in Sec. III A. Using Eq. (15) we can write G(k, t) = χg(Q 2 )Γ(Q 2 , S). Analogously, we have
τ int (k) τ int (0) ≡ f int (Q 2 ),(51)
where the scaling function f int (Q 2 ) is universal and satisfies f int (0) = 1.
It is important to note that Eq.
(2) does not necessarily imply that the scaling function Γ(Q 2 , S) decays nonexponentially. On the contrary, as argued in Sec. I, the Griffiths tail (2) becomes irrelevant in the critical limit. In view of that discussion it is natural to define a scaling function
f exp (Q 2 ) ≡ − lim |S|→∞ |S| ln Γ(Q 2 , S) ,(52)
which we call, rather loosely, the scaling function associated with the exponential autocor-
relation time. Indeed, if f exp (Q 2 ) is finite, for S → ∞ we have Γ(Q 2 , S) ∼ S a exp(−S/f exp (Q 2 )),(53)
where a is some critical exponent. In terms of quantities that are directly accessible numerically, we can define it as
f exp (Q 2 ) = lim x→∞ lim k→0;ξ→∞ τ exp,x (k) τ int (0) .(54)
Of course, the two limits cannot be interchanged.
The dynamic structure factor G(k, ω) is defined as
G(k, ω) = ∞ −∞ dt G(k, t)e iωt = 2 ∞ 0 dt G(k, t) cos ωt.(55)
In the scaling limit we introduce a new scaling function σ(Q 2 , w) defined by
σ(Q 2 , w) ≡ G(k, ω) τ int (0) G(k, 0) w ≡ ωτ int (0).(56)
The function σ(Q 2 , w) is essentially the ratio of the dynamic and static structure factors and is directly related to Γ(Q 2 , S):
σ(Q 2 , w) = 2 ∞ 0 dS Γ(Q 2 , S) cos wS.(57)
It is even in w and satisfies the normalization conditions:
σ(0, 0) = 2, ∞ −∞ dw 2π σ(Q 2 , w) = 1.(58)
Moreover, we have σ(Q 2 , 0) = 2f int (Q 2 ).
For a Gaussian theory the spin-spin correlation function is given by
G G (k, t) = Ze −Ω(k 2 +r)|t| k 2 + r .(59)
It follows τ int (k) = [Ω(k 2 + r)] −1 , so that
Γ(Q 2 , S) = e −(Q 2 +1)|S| , f int (Q 2 ) = f exp (Q 2 ) = 1 1 + Q 2 .(60)
Finally, we have σ(Q 2 , w) = 2(1 + Q 2 ) w 2 + (1 + Q 2 ) 2 .
(61)
B. Field-theory results
The dynamic structure factor can be computed in perturbation theory. The explicit one-loop calculation is reported in App. B. Two facts should be noted. First, perturbation theory predicts an exponential decay for Γ(Q 2 , S) for any Q 2 . This is consistent with the argument presented in the introduction, which predicted the absence of the Griffiths tail in the critical scaling functions. Second, one-loop perturbation theory predicts f exp (Q 2 ) to be independent of Q 2 . We wish now to argue that this result is exact and is related to the breaking of translational invariance in disordered systems. Indeed, consider the spin-spin correlation function for a given disorder configuration {ρ},
γ(x 1 , x 2 ; t 1 − t 2 ; {ρ}) ≡ σ(x 1 , t 1 )σ(x 2 , t 2 ) ρ ,(62)
and the corresponding Fourier transform
γ(k 1 , k 2 ; t 1 − t 2 ; {ρ}) = x 1 x 2 e ik 1 x 1 +ik 2 x 2 γ(x 1 , x 2 ; t 1 − t 2 ; {ρ}).(63)
In pure systems translational invariance implies that γ(k 1 , k 2 ; t 1 − t 2 ; {ρ}) vanishes unless k 1 = −k 2 . This is not the case in disordered systems, where translational invariance is lost.
The average of γ over disorder vanishes for k 1 = −k 2 [it indeed corresponds to G(k, t 1 − t 2 )], and thus translational invariance is somewhat recovered. However, this does not mean that the critical theory is translationally invariant. For instance, consider
| γ(k 1 , k 2 ; t 1 − t 2 ; {ρ})| 2 .(64)
It can be easily verified in perturbation theory that this quantity is not zero for any k 1 and k 2 . Note that this breaking of translational invariance survives in the infinite-volume limit only close to the critical point. In the paramagnetic phase, far from the critical transition, self-averaging occurs and thus also the quantity (64) vanishes for k 1 = −k 2 when L → ∞.
Let us now show that, if translational invariance (both for the Hamiltonian and the transition rates) holds, the decay rate is k dependent: modes corresponding to different momenta decouple. Indeed, following Refs. 9,44, let L be the Liouville operator associated with the dynamics, and λ a and ψ a be the corresponding eigenvalues and eigenvectors. Then, we have the spectral representation (t > 0)
G(k, t) = a e −λat | σ(k)|ψ a | 2 ,(65)
where the sum runs over all eigenstates with nonvanishing eigenvalue of L. Here we have introduced the inner product
f |g = α π α f * α g α ,(66)
where f and g are functions defined over the configuration space, π α is the equilibrium distribution, and the sum runs over all configurations α of the system. If the system is translationally invariant, L commutes with the generator T of the translations; hence, the eigenstates of L are also eigenstates of T . Thus, we have decoupled sectors corresponding
to different values of the momentum k and therefore we have where the sum runs over the eigenstates of momentum k. Hence, if λ 1 (k) is the smallest eigenvalue in each sector, we have G(k, t) ∼ e −κt with κ = λ 1 (k); hence, the decay rate is k dependent. If translational invariance is lost, all eigenfunctions contribute to each single value of k. Note, however, that this does not necessarily imply that the decay rate κ in Eq. (5) is Q independent. Indeed, one should average over the disorder distribution and this average could wash out the effect. We expect this to happen in the infinite-volume limit at fixed T , for T = T c . The perturbative results show that this is not the case at the critical point. Hence, all modes are coupled in the critical limit and κ is momentum independent.
G(k, t) = a e −λa(k)t | σ(k)|ψ a (k) | 2 ,(67)
This argument indicates that the Q-independence of κ is strictly related to the breaking of self-averaging at the critical point and thus, we expect a similar phenomenon to occur for the low-temperature critical dynamical structure factor.
In Fig. 4 we report Γ(Q 2 , S) as obtained by using Eqs. (B29) and (B32) and simply setting ǫ = 1. The behavior we observe is quite different from what is observed in the Gaussian model. In this case, Eq. (60) implies ln Γ(Q 2 , S) = −(1 + Q 2 )|S|. As a consequence, with a logarithmic vertical scale, the data fall on straight lines with increasing slope as Q 2 → ∞.
Here instead, Γ(Q 2 , S) first decreases rapidly and then bends so that the large-S decay is Q 2 -independent. This behavior is also very different from that observed in the pure Ising model, whose dynamical critical behavior is very close to that of the Gaussian model. 15 If f exp (Q 2 ) = f exp is independent of Q 2 , for S → ∞ we expect a behavior of the form
Γ(Q 2 , S) ≈ f (Q 2 )S a exp(−S/f exp ),(68)
where a is a critical exponent. At one loop, the calculations reported in App. B give a = 0 for Q 2 = 0, a = −1 for Q 2 = 0, and f (Q 2 ) ∼ Q −2 for Q → ∞. In general, we expect f (Q 2 ) to vanish with a nontrivial exponent in the large-Q limit and thus we write
f (Q 2 ) ∼ Q −ζ ,(69)
with a new exponent ζ.
Given G(k, t), one can compute G(x, t), which can be written in the scaling form
G(x, t) = ξ −d+2−η F (Y 2 , S), Y 2 ≡ x 2 /ξ 2 .(70)
Perturbation theory, see App. B, indicates that F (Y 2 , S) is not analytic for Y 2 → 0. It predicts a behavior of the form
F (Y 2 , S) = f 0 (S) + f λ (S)|Y | λ + · · · ,(71)
where λ is a new exponent that can be related to the exponent ζ which appears in Eq. (69): λ = ζ −1 −η (in d dimensions, as we discuss in App. B, λ = ζ + 2 −d −η). The exponent λ is positive (hence ζ must be larger than 1 + η), since G(x = 0, t) is always finite. The quantity f λ (S)|Y | λ represents a subleading nonanalytic correction to the leading term f 0 (S).
Finally, in Fig. 5 we report the one-loop perturbative expression of σ(Q 2 , w). Note that the width of σ(Q 2 , w) does not decrease with increasing Q 2 , as it does in the Gaussian model. This is a consequence of the large-S behavior of Γ(Q 2 , S), whose decay is independent of
Q 2 .
C. Simulation details
In this section we study the critical dynamics of Hamiltonian (9)
where the sum runs over the coordinates (x, y, z) of the lattice sites; the time t is expressed in units of Metropolis lattice sweeps.
Given G(k, t), we determine τ int (0). More precisely, we determine τ int,x (0) with x = 5, as defined by the self-consistent equation (46). As discussed above, this is a good autocorrelation time for any x; therefore, we use this quantity to obtain a high-temperature estimate of z. We have also determined τ int,x (0) with x = 8. The results for x = 5 and x = 8 are consistent within errors, indicating that we can take τ int,5 (0) as an estimate of τ int (0). We also consider the effective exponents τ exp,x (0) defined by Eqs. (44) and (45) with
A(t) = G(k = 0, t).
The results we quote correspond to n = 2.
Some results are reported in Table I (19) to ξ(β, L)/L ≈ 0.24, 0.12, and 0.06. No scaling corrections are observed in τ int (0) within the quoted errors, and thus, for each β, we assume that the estimate of τ int (0) for the largest lattices is an infinite-volume result. Also τ exp,1 (0) apparently does not show finite-size effects.
On the other hand, τ exp,2 (0) is clearly decreasing as L increases. This indicates that finitesize effects on G(k, t) increase with t, a result that we will check explicitly below, considering the correlation function.
D. Dynamic structure factor
We first use the estimates of the autocorrelation times to obtain an estimate of z. (8) is the next-to-leading correction-to-scaling exponent. Hence, τ (β) behaves as where ∆ 2 = νω 2 = 0.56 (6). Thus, we fit the data to For each Q, we report results corresponding to three different values of β and L.
τ (β) ≈ c(β c − β) −zν (1 + b(β c − β) ∆ 2 + · · ·),(73)ln τ (β) = −zν ln(β c − β) + a + b(β c − β) ∆ 2 ,(74)
where the error has been chosen conservatively, in order to include the result of fit (a) with its error. This result is very close to the one-loop FT estimate. Eq. (B32) gives f exp ≈ 1.168
for ǫ = 1.
Finally, we provide an interpolation of our numerical data. The curves reported in Fig. 6 are well fitted by a function of the form
Γ(0, S) = e −κS + 4 k=1 a k (cS) k 1 + (cS) 4 e κS .(78)
The constant κ has been fixed by using Eq. we observed for the static structure factor in Sec. III C. There, a good scaling behavior was only observed at fixed Q and not at fixed Q. Here instead, scaling corrections at fixed Q are quite small; the behavior at fixed Q is actually slightly worse.
An important prediction of the FT analysis is that Γ(Q 2 , S) decays with the same rate for all values of Q. To check this prediction we consider the ratio G(k, t)/ G(0, t), which converges to Γ(Q 2 , S)/Γ(0, S) in the scaling limit. For S → ∞, this quantity should behave as
S α exp −S(f exp (Q 2 ) −1 − f exp (0) −1 ) ,(79)
where α is some critical exponent. Field theory predicts f exp (Q 2 ) = f exp independent of Q, so that we expect Γ(Q 2 , S)/Γ(0, S) to behave as S α for large S, without exponential factors.
Thus, if field theory is correct, ln[ G(k, t)/ G(0, t)] should become constant as t increases, apart from possible slowly varying logarithmic corrections. In Fig. 9 we show this ratio for the lattice with β = 0.282, L = 128, which has been chosen because of its relatively small errors up to S ≈ 4. The results for β = 0.284, L = 128, which are more asymptotic and give access to larger values of Q, are more noisy. The plot shows that the MC data are consistent with the FT prediction. Note that the constant behavior is observed better for larger values of Q. This is in agreement with the FT results shown in Fig. 4 and can be understood qualitatively quite easily. Roughly, at one loop Γ(Q 2 , S) is the sum of two terms,
ae −S + be −(1+Q 2 )S(80)
(we neglect here additional powers of Q and S), so that the ratio we are considering corresponds to (a + be −Q 2 S )/(a + b). Thus, the ratio approaches a constant with corrections of order e −Q 2 S . For large Q 2 they die out fast, and thus a constant behavior is observed for small values of S.
While the decay rate of Γ(Q 2 , S) is independent of Q 2 , the amplitude decreases rapidly with Q 2 . For large Q 2 we expect the behavior Γ(Q 2 , S) ∼ S a Q −ζ exp(−κS), where κ = 1/f exp , see Eq. (68). We wish now to obtain a rough estimate of the exponent ζ. For this purpose we take the data that appear in Fig. 9 and we multiply them by Q ζ , trying to fix ζ in such a way to obtain a good collapse of the data. In Fig. 9 we report the scaled results corresponding to two different values of ζ. If we try to have a good collapse of the data corresponding to n = 4, 8, 12 the best result is obtained for ζ = 2.3. However, the data with n = 20 behave in a significantly different way. If we try to include also the data with n = 20, the quality of the collapse worsens and the best result is obtained for ζ = 1.9.
These results indicate that ζ ≈ 2 (but with a large error), so that Γ(Q 2 , S) behaves roughly as Q −2 S a exp(−κS). It is interesting to observe that this is exactly the behavior predicted close to four dimensions by perturbation theory.
Finally, we determine an interpolation formula for Γ(Q 2 , S). We find that all data are well fitted by taking
Γ(Q 2 , S) = a 0 e −κ 1 S + 4 k=1 a k (cS) k 1 + (cS) 4 e κ 1 S + 4 k=0 d k S k e −κ 2 S ,(81)
fixing κ 1 = 0.840. The results of the fits for a few chosen values of Q 2 are reported in Table II. We have not required a 0 + d 0 = 1, a condition that follows from Γ(Q 2 , S = 0) = 1, but we have verified that the results satisfy this condition quite precisely. By using a linear interpolation, the results we report should allow the reader to determine Γ(Q 2 , S) for any Q in the range 0 ≤ Q 2 ≤ 50 with reasonable precision. We stress that this interpolation formula only represents a compact expression that summarizes the numerical results. The chosen parametrization has indeed a purely phenomenological value.
From these expressions it is easy to determine the scaling function σ(Q 2 , w) related to the dynamic structure factor. In Fig. 11 we plot the scaling function σ(Q 2 , w) as obtained by integrating the interpolating function determined above. We report it for the same values of Q 2 that appear in Table II
H = d d x 1 2 (∂ µ ϕ) 2 + 1 2 rϕ 2 + 1 2 ψϕ 2 + 1 4! g 0 ϕ 4 ,(A1)
where ψ(x) is a spatially uncorrelated random field with Gaussian distribution
P (ψ) = 1 √ 4πw exp − ψ 2 4w . (A2)
Using the standard replica trick, it is possible to replace the quenched average with an annealed one. As a result of this procedure, one can investigate the static critical behavior of RDI systems by applying standard FT methods to the Hamiltonian 45
H replica = d d x i 1 2 (∂ µ φ i ) 2 + rφ 2 i + ij 1 4! (u 0 + g 0 δ ij ) φ 2 i φ 2 j ,(A3)
where i, j = 1, ...N and u 0 = −6w. RDI results are obtained by taking the limit N → 0.
√ ǫ-expansion results
The scaling function g(Q 2 ) can be determined by using the results reported in Ref. 37.
We obtain the expansion g(y) −1 = 1 + y − ǫ 1 53 ψ 2 (y) + 18 6/53 2809 [24 + 7ζ(3)]ǫ 3/2 ψ 2 (y) + O(ǫ 2 ) = 1 + y − 0.0188679ǫψ 2 (y) + 0.069887ǫ 3/2 ψ 2 (y)
+ O(ǫ 2 ) ,(A4)
where ψ 2 (y) is the two-loop contribution defined in Ref. 37. Note that the only relevant threeloop diagram contributes only at order ǫ 5/2 (hence, at five loops), since it is proportional
to 1 27 (u + 3v)(4u + 3v) 2 ,(A5)
and, at the fixed point, 4u * + 3v * is of order ǫ and not of order √ ǫ.
The expansion of ψ 2 (y) for small y can be found in Ref. 37. It allows us to obtain the expansions c 2 = 0.000141891ǫ − 0.000525567ǫ 3/2 + O(ǫ 2 ), c 3 = −3.62134 · 10 −6 ǫ + 0.0000134135ǫ 3/2 + O(ǫ 2 ), c 4 = 1.53623 · 10 −7 ǫ − 5.69021 · 10 −7 ǫ 3/2 + O(ǫ 2 ), c 5 = −8.28575 · 10 −9 ǫ + 3.06905 · 10 −8 ǫ 3/2 + O(ǫ 2 ). (A6)
The expansion of ψ 2 (y) for large values of y is 39 ψ 2 (y) = − 1 4 y log y + 2Q 0 y − 3 4 log 2 y + 2Q 1 + . . .
with Q 0 ≈ 0.507826 and Q 1 ≈ 0.1289. Matching the large-momentum expansion of Eq. (A4)
with the Fisher-Langer behavior (20) we obtain the expansions of the coefficients C i :
C 1 = 1 + 0.0191632ǫ − 0.0709809ǫ 3/2 + O(ǫ 2 ),C 2 = −1/2 − 0.757042ǫ 1/2 + 1.34297ǫ + c 2,3 ǫ 3/2 + O(ǫ 2 ),C 3 = −1/2 + 0.757042ǫ 1/2 − 1.35726ǫ + c 3,3 ǫ 3/2 + O(ǫ 2 ),(A8)
where c 2,3 + c 3,3 = 0.052964. Setting ǫ = 1, we obtain C 1 ≈ 0.95, C 2 + C 3 = −0.96.
Massive zero-momentum results
We have determined the low-momentum behavior of g(Q 2 ) −1 in the MZM scheme by using the perturbative results of Ref. 37. The four-loop expansions of the first few coefficients c n are:
c 2 = − 1 6480 u 2 − 1 2430 uv − 2 10935 v 2 − 0.0000120404u 3 − 0.0000481617u 2 v −0.0000481617uv 2 − 0.0000142701v 3 − 0.00000718972u 4 − 0.0000383452u 3 v −0.0000617074u 2 v 2 − 0.0000379116uv 3 − 0.00000842479v 4 , c 3 = 1 122472 u 2 + 1 45927 uv + 4 413343 v 2 − 0.00000281924u 3 − 0.0000112769u 2 v −0.0000112769uv 2 − 0.00000334132v 3 + 0.00000660026u 4 + 0.00000352014u 3 v +0.00000562891u 2 v 2 + 0.00000339371uv 3 + 0.000000754158v 4 , c 4 = − 1 1889568 u 2 − 1 708588 uv − 1 1594323 v 2 + 0.000000347995u 3 + 0.00000139198u 2 v +0.00000139198uv 2 + 0.000000412438v 3 − 0.000000141114u 4 − 0.000000752609u 3 v −0.00000119627u 2 v 2 − 0.000000740733uv 3 − 0.000000164607v 4 .(A9)
The MZM renormalized quartic couplings u and v are normalized so that at tree level The relaxational model-A dynamics is described by the stochastic Langevin equation 4 ∂ϕ(r, t) ∂t
= −Ω δH(ϕ) δϕ(r, t) + ζ(r, t),(B1)
where ϕ(r, t) is the order parameter, H(ϕ) is the Hamiltonian (A1), Ω is a transport coefficient, and ζ(r, t) is a Gaussian random field (white noise) with correlations ζ(r, t) = 0, ζ(r 1 , t 1 )ζ(r 2 , t 2 ) = Ωδ(r 1 − r 2 )δ(t 1 − t 2 ).
The correlation functions generated by the Langevin equation (B1) at equilibrium, averaged over the noise ζ and the quenched disorder ψ, can be obtained from the FT action 46
S(ϕ,φ) = d d x dtφ(∂ t ϕ − Ω∆ϕ − Ωφ + Ωrϕ) (B3) + Ωg 0 3! dtφϕ 3 + Ω 2 u 0 6 dtφϕ 2 ,
whereφ is the response field. In this framework, no replicas are introduced. 46 We consider the correlation function G(x, t) and the response function R(x, t) defined by
G(x 2 − x 1 , t 2 − t 1 ) = ϕ(x 1 , t 1 )ϕ(x 2 , t 2 ) ,(B4)R(x 2 − x 1 , t 2 − t 1 ) = φ(x 1 , t 1 )ϕ(x 2 , t 2 ) ,(B5)
and their spatial Fourier transforms G(k, t) and R(k, t). In equilibrium, they are not independent, but related by the fluctuation-dissipation theorem; for t > 0 they satisfy the
relation ∂ t G(x, t) = −ΩR(x, t)
. For a general introduction to the FT approach to equilibrium critical dynamics, see, e.g., Refs. 47,48. Some perturbative calculations can be also found in Refs. 49.
One-loop calculation
We first compute the response function R(k, t) and then use the fluctuation-dissipation theorem to derive the correlation function G(k, t). At one-loop we obtain in dimensional and with L u (k, t) the contribution of the graph on the left and on the right, respectively.
regularization
R(k, t) = R G (k, t) − g 0 2 L g (k, t) − u 0 3 L u (k, t) + O(u 2 0 , u 0 g 0 , g 2 0 ),(B6)
where
R G (k, t) = θ(t) exp[−Ω(k 2 + r)t](B7)
is the Gaussian tree-level response function, and L u (k, t) and L g (k, t) are the one-loop contributions, see Fig. 12. It is straightforward to obtain
L g (k, t) = Ω ∞ −∞ ds d d q (2π) d R G (k, t − s) R G (k, s) G G (q, 0) = − N d ǫ r 1−ǫ/2 Ωt R G (k, t),(B8)
where ǫ ≡ 4 − d, N d ≡ 2/[Γ(d/2)(4π) d/2 ], and G G (k, t) is the Gaussian tree-level correlation function (59). Analogously, we obtain
L u (k, t) = Ω 2 ∞ −∞ ds 1 ∞ −∞ ds 2 d d q (2π) d R G (k, t − s 2 ) R G (q, s 2 − s 1 ) R G (k, s 1 ) = N d ǫ (1 + ǫγ E /2)(Ωt) ǫ/2 (Ωtk 2 − 1) R G (k, t) + θ(t) 1 (4π) 2 e −Ωtr F (Ωtk 2 ) + O(ǫ), (B9) where F (x) ≡ −1 + e −x + e −x (x − 1)[Ei(x) − γ E − ln x],(B10)
and Ei(x) is the exponential integral function.
Renormalizing the response function in the MS scheme we obtain where r, Ω, u, g are renormalized parameters (note that we use the same symbols r and Ω for both the bare and the renormalized parameters, since no confusion can arise). The final expression is obtained by setting g and u equal to their fixed-point values:
R(k, t) = R G (k, t) − g 32π 2 Ωtr ln r R G (k, t) (B11) − u 48π 2 (γ E + ln Ωt)(Ωtk 2 − 1) R G (k, t) −
Given Eq. (B18), we can compute G(x, t) at the critical point. We expect the scaling behavior G(x, t) = (Ωt) −(d+η−2)/z F (X),
X ≡ x(Ωt) −1/z . (B20)
Using Eq. (B18) we obtain
F (X) = 1 4π 2 X 2 (1 − e −X 2 /4 ) − u * 48π 2 F 1 loop (X),(B21)
with
F 1 loop (X) = 1 4π 2 X ∞ 0 dK J 1 (KX) e −K 2 − 1 + K 2 e −K 2 Ei(K 2 ) − ln K 2 ,(B22)
where J 1 (x) is a Bessel function. In the derivation we have taken into account that u ∼ √ ǫ.
It is interesting to note that F 1 loop (X) is not regular for X → 0. Indeed, the explicit calculation gives
F 1 loop (X) = − 1 16π 2 ln X 2 4 + O(X 2 ln X 2 ).(B23)
This result indicates that F (X) is not regular for X → 0, but behaves as
F (X) ≈ a 0 + b 0 |X| λ + · · · (B24)
where λ is a critical exponent. Comparing this expression with Eqs. (B21) and (B23) we
obtain a 0 + b 0 = 1 16π 2 + O( √ ǫ), (B25) b 0 λ = u * 384π 4 + O(ǫ) = − 1 16π 2 6ǫ 53 + O(ǫ).(B26)
It is not possible to compute λ, a 0 , and b 0 separately at this order. A two-loop computation of the term proportional to ǫ ln 2 X is needed.
In the high-temperature phase, it is convenient to replace r and Ω with the correlation length ξ and the zero-momentum integrated autocorrelation time τ int (k = 0), defined in Sec. III A and IV A, respectively. A tedious calculation gives ξ −2 = r + 1 32π 2 (g + 2u/3)r ln r,
Ωτ int (k) = 1 k 2 + r − g 32π 2 r ln r (k 2 + r) 2 + u 48π 2 k 2 ln r (k 2 + r) 2 + u 48π 2 1 k 2 + r .
Using these results, we obtain the one-loop expression of the scaling function Γ(Q 2 , S):
Γ(Q 2 , S) = e −S(Q 2 +1) − u * 48π 2 S(Q 2 + 1)e −S(Q 2 +1) − u * 48π 2 (γ E + ln S)SQ 2 e −S(Q 2 +1) −
u * 48π 2 e −S(Q 2 +1) − e −S − u * 48π 2 Q 2 S − 1 Q 2 + 1 e −S(Q 2 +1) Ei(SQ 2 ) − γ E − ln SQ 2 − u * 48π 2 1 Q 2 + 1 Ei(−S) − e −S(Q 2 +1) (γ E + ln S) ,(B29)
where u * is the fixed-point value (B13) of u.
It is interesting to discuss the large-S and small-S behavior of the scaling function (B29).
For large S we obtain
Γ(Q 2 , S) ≈ e −S(1+Q 2 ) − u * 48π 2 1 + Q 2 (Q 2 ) 2 e −S S 2 − u * 48π 2 (1 + Q 2 − Q 2 ln Q 2 )Se −S(1+Q 2 ) .(B30)
For Q 2 = 0 the dominant term is the last one. In this case we can rewrite the large-S behavior as
Γ(Q 2 , S) ≈ exp −S 1 + u * 48π 2 ,(B31)
For Q 2 = 0, the dominant term is the second one, so that for any Q 2 the scaling function decays as e −S /S 2 . Thus, at this perturbative order, we obtain the result
f exp (Q 2 ) = 1 + O( √ ǫ).(B33)
The correlation function Γ(Q 2 , S) decays with the same rate for all values of Q. As we discuss in Sec. IV B, this is a consequence of the loss of translational invariance in dilute systems.
Equation (B33) should be contrasted with the result obtained for the integrated autocorrelation times. Using Eq. (B28), we obtain
τ int (k) τ int (0) = f int (Q 2 ) = 1 Q 2 + 1 ,(B34)
without one-loop corrections. This shows that τ int (k) decreases as Q increases as it does in the Gaussian model.
Let us now consider the limit S → 0. The scaling function has an expansion of the form Γ(Q 2 , S) = 1 + n=1 S n (a n + b n ln S).
(B35)
Note the presence of terms proportional to ln S. They should be generically expected, since in the critical limit (it corresponds to Q → ∞ and S → 0), the correlation function depends on k 2 (Ωt) 2/z ∼ Q 2 S 2/z ∼ Q 2 S 1 + u * 48π 2 ln S .
The presence of these logarithms implies that the function Γ(Q 2 , S) is not analytic for S → 0.
It is also important to discuss the large-momentum behavior of Γ(Q 2 , S). For Q 2 → ∞ the tree-level term vanishes exponentially as e −SQ 2 , while the one-loop term decays only algebraically, as 1/Q 2 . More precisely, for Q 2 → ∞ we have
Γ(Q 2 , S) ≈ − u * 48π 2 e −S S + Ei(−S) Q −2 .(B37)
The presence of these slowly decaying terms implies the singularity of the behavior of G(x, t) for x → 0 and for any t. In the critical limit we expect the scaling behavior (70), i.e.,
G(x, t) = ξ −d+2−η F (Y 2 , S), with Y 2 ≡ x 2 /ξ 2 . We obtain for Y → 0 F (Y 2 , S) ≈ 1 16π 2 e −S S + Ei(−S) 1 + u * 24π 2 ln Y + u * f corr (S) + · · · (B38)
for Y → 0, where f corr (S) is a function of S. The presence of a term proportional to ln Y implies that F (Y 2 , S) is not analytic as Y → 0, i.e. has a behavior of the form F (Y 2 , S) = f 0 (S)+f λ (S)|Y | λ +· · ·, where λ is the same exponent that appears in Eq. (B24).
Note that Eq. (B38) is apparently consistent with the assumption that f 0 (S) = 0. If this were the case, we would obtain
λ = u * 24π 2 = − 6ǫ 53 + O(ǫ).(B39)
This result would imply λ < 0 and thus F (Y 2 , S) would diverge as Y → 0 for any S, at least for ǫ small. This behavior is clearly unphysical; thus, f 0 (S) should be nonvanishing.
The critical limit is obtained by taking S → 0. Requiring the limiting function to be of the form (B20), we obtain
for S → 0. The S-dependent prefactor appearing in Eq. (B38) behaves as 1/S for S → 0, which is consistent with these expressions.
The nonanalytic behavior of F (Y 2 , S) as Y → 0, implies that Γ(Q 2 , S) should decay as a power of Q as Q → ∞. A simple calculation gives
Γ(Q 2 , S) ∼ f λ (S)Q 2−d−λ−η ,(B41)
for Q 2 → ∞. The exponent ζ defined in Eq. (69) is given by
ζ = λ + d + η − 2 = 2 + O( √ ǫ).(B42)
Finally, we report σ(Q 2 , w), cf. Eq. (56). A long calculation gives σ(Q 2 , w) = 2α α 2 + w 2 + u * 24π 2 α(α 2 − w 2 + 2αw 2 ) w(α 2 + w 2 ) 2 Arctan w + α(α 2 − w 2 − 2α) 2(α 2 + w 2 ) 2 ln(1 + w 2 )
+ 1 w 2 + 1 + (w 2 − α)Q 2 (w 2 + 1)(α 2 + w 2 ) − 2α 3 (α 2 + w 2 ) 2 ,(B43)
where α ≡ 1 + Q 2 .
For large w, σ(Q 2 , w) behaves as
σ(Q 2 , w) ≈ 2α w 2 1 + u * 48π 2 (1 − ln w) ,(B44)
which is compatible with the expected behavior w −(2−η+z)/z .
Note also that the singularities of σ(Q 2 , w) in the complex w-plane that are closest to the origin are w = ±i, independently of Q 2 . This is a direct consequence of the fact we have already noticed that the large-t behavior is momentum independent. As a consequence, the width of the structure factor does not decrease with Q 2 as it does in pure systems. 49, 435 (1977). 5 In Born approximation the static and dynamic structure factors are proportional to the elastic and inelastic cross section, respectively; k and ω are respectively proportional to the transferred momentum and energy.
a behavior predicted theoretically by Fisher and Langer 14 and proved in the FT framework in Refs. 35,36. B. Field-theory results We determined the coefficients c n by using two different FT approaches: the √ ǫ-expansion approach, in which the renormalization-group parameters are computed as series in powers of √ ǫ, ǫ = 4 − d, and the massive zero-momentum (MZM) approach, in which one works directly in three dimensions. We computed the first few coefficients c n to O(ǫ 3/2 ) in the √ ǫ expansion, and to four loops in the MZM scheme. The corresponding expansions are reported in App. A. Setting ǫ = 1 in the √ ǫ expansions (A6), we obtain c 2 = −4 × 10 −4 , c 3 = 1.0 × 10 −5 , c 4 = −4 × 10 −7 , and c 5 = 2 × 10 −8 . In the MZM approach, resumming the perturbative expansions (A9) as discussed in Ref. 22, we obtain
at p = 0.8 in the high-temperature phase, with the purpose of determining the infinite-volume spin-spin correlation function G(k). We perform simulations on lattices of size 32 ≤ L ≤ 256 in the range 0.275 ≤ β ≤ 0.2856 [note that 21 β c = 0.2857431(3)]. The number of samples varies with N, being of the order of 3000, 10000, 30000, 40000 for L = 256, 128, 64, 32. For each sample, we start from a random configuration, run 1000 Swendsen-Wang and 1000 Metropolis iterations for thermalization, and then perform 2000 Swendsen-Wang sweeps.
FIG. 1 :
1(Color online) Estimates of h(k; β, L) vs ξ(β, L)/L at fixed Q = 10 (left) and Q = 10
we show the numerical data for Q = 10 (left) and Q = 10 (right). On the left one observes very large size corrections which make impossible in practice the determination of the infinite-volume limit ξ(β, L)/L → 0. On the right instead, there are no significant scaling corrections and all data fall approximately on a single curve. Size corrections are small for ξ(β, L)/L 0.20 and the extrapolation to ξ(β, L)/L → 0 is feasible.
FIG. 2 :
2Fig. 2we show h(k; β, L) for Q = 5,20, 50. Sincek can be at most 2, for each Q we can only consider values of β and L such that ξ(β, L) ≥ Q/2. However, since the (Color online) MC results for h(k; β, L) vs ξ(β, L)/L at fixed Q for three different values of Q: 5 (top), 20 (middle), 50 (bottom). Only data satisfying k ≤ k max = π/3 are reported. The interpolation (dashed line) corresponds to a fit of the data with L ≥ 64 as described in the text.
FIG. 3 :
3have repeated the analysis including each time only data such that L ≥ L min . The results are essentially independent of L min . For instance, for Q = 5, one of the values we considered in Fig. 2 (in this case some scaling corrections are present for ξ(β, L)/L 0.20), we obtain a 0 = 25.881(6), 25.881(7), 25.882(9) for L min = 32, 64, 128, respectively. This is due to the fact that a 0 is determined by the results at small values of ξ(β, L)/L and in this range there are essentially no scaling corrections. In the following we choose conservatively L min = 64. Our final estimate of g(Q 2 ) is reported in Fig. 3. Deviations from the OZ behavior are quantitatively small and indeed at Q = 50 the relative deviation is only 0.05. It is important to note that the estimates of g(Q 2 ) at different values of Q are correlated since the estimates of G(k; β, L) for different values of k are statistically correlated. This explains the regularity of the results. Note also that the error changes rather abruptly in a few cases. For instance, this occurs between Q = 40 and Q = 41. This happens because at Q = 40, the estimate of a 0 is essentially determined by the result obtained for β = 0.2853, L = 256, which corresponds (Color online) Estimates of the scaling function g(Q 2 )(1 + Q 2 ) for integer values of Q.
FIG. 4 :
4(Color online) Scaling function Γ(Q 2 , S) as a function of S, as obtained in one-loop perturbation theory.
FIG. 5 :
5at p = 0.8 in the high-temperature phase, with the purpose of determining the time-dependent spin-spin correlation function G(k, t) and the related dynamic structure factor G(k, ω). We perform simulations on lattices of size L ≤ 128 in the range 0.275 ≤ β ≤ 0.284, corresponding to 4 ξ 16. For each disorder sample, we start from a random configuration, run 1000 (Color online) Scaling function σ(Q 2 , w) as a function of w ≡ ωτ int (0), as obtained in one-loop perturbation theory. Swendsen-Wang and 1000 Metropolis iterations for thermalization, and then N it Metropolis sweeps [typically, we took N it varying between 30τ int (0) and 100τ int (0)]. The number of samples varies between 5000 and 20000. We measure the second-moment correlation length ξ(β, L) defined in Eq. (31) and the correlation function G(k, t).As we did for the static structure factor, we determine G(k, t) as G(k, t) = 1 3 x,y,z (e ikx + e iky + e ikz ) σ(0, 0, 0; 0)σ(x, y, z; t) ,
Since the model is approximately improved, 20 the scaling corrections proportional to (β c − β) ων , ω = 0.29(3) are suppressed. Thus, the leading scaling corrections behave as (β c −β) ω 2 ν , where ω 2 = 0.82
FIG. 6 :
6(Color online) Scaling function G(0, t)/ G(0, 0) as a function of S, for different values of β and L.
FIG. 7 :FIG. 8 :
78setting β c = 0.2857431(3).20,21 If we fit τ int (0), including only the data satisfying β ≥ β min , we obtain zν = 1.64(3), 1.59(4), 1.62(8) for β min = 0.275, 0.278, 0.280. The results are stable with β min and allows us to estimate zν = 1.61(6) that includes all estimates with their error bars. If we now use 20 ν = 0.683(2), we obtainz = 2.36(9),(75)which is in perfect agreement with the estimate z = 2.35(2) obtained at the critical point.21 As a check we have repeated the analysis by using τ exp,1 (0). We obtainzν = 1.59(3), 1.56(5), 1.47(10) for β min = 0.275, 0.278, 0.280, which are essentially consistent with the estimates obtained above. Let us now consider G(k, t). Let us first focus on the case k = Q = 0. Numerical results are reported in Fig. 6 vs S ≡ t/τ int (0). Scaling and finite-size corrections are small and indeed all data fall approximately onto a single curve. Some deviations are only observed for S 3, indicating that finite-size corrections increase with S. Let us now consider the large-S behavior and let us estimate the universal ratio f exp (0). The data show a reasonably good exponential behavior so that we can assume that we are considering values of t that are much before the region in which G(0, t) shows the Griffiths tail. We perform fits of the (Color online) Estimates of [bτ int (0)] −1 , which converges to f exp (0) in the critical limit. The coefficient b is obtained by fitting G(0, t)/ G(0, 0), as described in the text, see Eq. (76). In fit (a) we consider together the data corresponding to (L = 64, β = 0.278), (L = 128, β = 0.281), and (L = 128, β = 0.282). In fit (b) we only consider the results obtained for L = 128, β = 0.281. time only data in the range S min ≤ S ≤ S max ≈ 5. The fit parameter b provides an estimate of f exp (0): in the critical limit f exp (0) = [bτ int (0)] −1 . Since finitesize corrections are important, we only consider data with small ξ/L. In Fig. 7 we report the results corresponding to two sets of data. In fit (a) we consider three sets of results: those corresponding to L = 64, β = 0.278 and those with L = 128 and β = 0.281, 0.282. Correspondingly, we have ξ(β, L)/L = 0.088, 0.061, 0.073, respectively. In fit (b) we only use the lattice with the smallest value of ξ/L available: L = 128 and β = 0.281. The results of fit (a) become independent of S min for S min 2.5 and give f exp (0) = 1.21(1). Fit (b) is less stable and a plateau is less evident. They hint at a lower value for the ratio, varying between 1.20 (at S min = 3) and 1.18 (at S min = 4), though with a large statistical error. We have also analyzed the data corresponding to lattices with larger ξ/L, finding larger values of [bτ int (0)] −1 . This indicates that this quantity decreases with decreasing ξ(β, L)/L and thus the difference obtained between fits (a) and (b) may be a real finite-size effect. For this reason our final result corresponds to fit (b). We quote f exp (0) = 1.19(3), (Color online) Scaling function Γ(Q 2 , S) as computed numerically for two values of Q 2 .
FIG. 9 :
9(77): we take κ = 0.840 ≈ 1/1.19. All other constants have been obtained by a fit of the data for 0 ≤ S ≤ 5. We obtain c = 1.69, a 1 = −0.23825, a 2 = 0.16430, a 3 = 0.13261, a 4 = −0.28028. As a check of this parametrization we verify the normalization conditions (50). The first condition is satisfied exactly, the second one to very good precision: the integral between 0 and infinity of Γ(0, S) as given by the parametrization (78) is equal to 1.0033.Let us now consider G(k, t) for k = 0. Again, let us first discuss the finite-size and scaling corrections. For this purpose, we must compare G(k, t) for different values of β and L, but at the same value of Q ≡ kξ. Since the momenta accessible on a finite lattice of size L are quantized and therefore estimates are obtained only for Q = 2πnξ/L, n integer, for each t we should interpolate the numerical data as we did in Sec. III C. However, by a fortunate accident, such an interpolation is not needed here. Indeed, the lattice with L = 64,(Color online) Ratio G(k, t)/ G(0, t) ≈ Γ(Q 2 , S)/Γ(0, S) obtained at β = 0.282, L = 128, versus S. Results correspond to different values of Q 2 = 4π 2 n 2 ξ 2 /L 2 . The values n = 2, 4, 8, 12, 20 correspond to Q 2 ≈ 0.84, 3.37, 13.5, 30.3, 84. β = 0.281 and that L = 128, β = 0.284 have both ξ(β, L)/L = 0.1237(1). Moreover, for L = 128, β = 0.281, ξ/L is exactly 1/2 (within the small statistical errors) of the previous value. Thus, results with the same k for the first two systems and those with 2k for the third one correspond quite precisely to the same value of Q. In Fig. 8 we report results corresponding to Q = 2π × 0.1237 ≈ 0.78 and Q = 20π × 0.1237 ≈ 7.8. All results fall again onto a single curve for both values of Q. Finite-size and scaling corrections are apparently negligible in this range of values of Q and S. This result should be compared with what
online) Ratio Q ζ G(k, t)/ G(0, t) ≈ Q ζ Γ(Q 2 , S)/Γ(0, S) obtained at β = 0.282, L = 128, versus S. Results correspond to different values of Q 2 = 4π 2 n 2 ξ 2 /L 2 . The values n = 2, 4, 8, 12, 20 correspond to Q 2 ≈ 0.84, 3.37, 13.5, 30.3, 84. The figure on the left corresponds to ζ = 1.9, that on the right to ζ = 2.3.
FIG
. 11: (Color online) Numerical estimate of the scaling function σ(Q 2 , w) as a function of w ≡ ωτ int (0), as obtained by integrating the interpolating expression (81). The values of the momenta are: Q 0 = 0,
. The qualitative behavior is in full agreement with the FT prediction, compare withFig. 5. Quantitatively, perturbation theory is also reasonably predictive. For Q = 0 and w < 5, the relative differences between the FT and the numerical expression are less than 2%. For larger values of Q 2 differences increase: for instance, for the values of Q that appear inFig. 11, field theory predicts σ(Q 2 , 0) ≈ 2, 1.09, 0.32, 0.091, 0.041, while we obtain numerically σ(Q 2 , 0) ≈ 2, 1.09, 0.33, 0.105, 0.072. These discrepancies are probably the fault of both field theory-after all, we are at one loop-and of the numerical results-for large Q the data are noisy and the estimates have a large error. In any case this comparison indicates that, up to Q = 5, errors are under control and the reported expressions are precise enough for all practical purposes. APPENDIX A: PERTURBATIVE RESULTS FOR THE STATIC STRUCTURE FACTOR The static behavior of Ising systems with random dilution can be studied starting from the Landau-Ginzburg-Wilson Hamiltonian 45
u
= u 0 /m and v = v 0 /m. Their fixed-point values are u * = −18.6(3) and v * = 43.3(2) (obtained by means of MC simulations 27 ), and u * = −13.5(1.8) and v * = 38.0(1.5) (obtained by resumming the six-loop β-function 22 ). APPENDIX B: ONE-LOOP CALCULATION OF THE RESPONSE AND COR-RELATION FUNCTIONS
FIG. 12 :
12The one-loop graphs contributing to the response function. We indicate with L g (k, t)
u 48π 2 θ(t)e −Ωtr F (Ωtk 2 ), (B12)
which gives for the exponential autocorrelation-time scaling function [see Eq. (52)]
f 0 (
0S) ∼ S (2−η−d)/z , f λ (S) ∼ S (−λ+2−η−d)/z ,
are Ising spin variables, and ρ x are uncorrelated quenched random variables, that their continuous transitions belong to a single universality class. The RDI universality class has been extensively studied by using FT methods and MC simulations. obtained by a finite-size analysis of MC data. These estimates are in good agreement with those obtained by using field theory. An analysis of the six-loop perturbativewhich are equal to 1 with probability p (the spin concentration) and 0 with probability
1 − p (the impurity concentration). For p s < p < 1, where p s is the site-percolation point
(p s = 0.3116081(13) on a simple cubic lattice 18 ), the model has a continuous transition with
a ferromagnetic low-temperature phase.
MC simulations of RDI systems have shown rather conclusively (see, e.g., Refs. 1,2,3,
19,20,21) At present, the most accurate estimates of the critical exponents are 20 ν = 0.683(2) and
η = 0.036(1), expansions in the three-dimensional massive zero-momentum scheme gives 22 ν = 0.678(10)
and η = 0.030(3). Note the good agreement between FT and MC results, in spite of the
fact that the perturbative FT series for dilute systems are not Borel summable. 23,24,25 Also
the correction-to-scaling exponents have been determined quite accurately. For the leading
exponent ω, MC simulations give 21 ω = 0.29(2) (older simulations gave ω = 0.33(3) 20
A full characterization of the types of disorder that lead to a transition in the RDI universality class is still lacking. For instance, RDI transitions also occur in systems that are,27 Beside the critical
exponents, also the equation of state, 28 some amplitude ratios, 28,29,30 the universal crossover
functions between the pure and the RDI fixed point 29 and between the Gaussian and the
RDI fixed point, 29,31 and the crossover exponent in the presence of a weak random magnetic
field 32 have been computed.
not ferromagnetic: this is the case of the Edwards-Anderson model (±J Ising model), which
is frustrated for any amount of disorder. 33 Nonetheless, the paramagnetic-ferromagnetic
transition line that starts at the pure Ising transition point and ends at the multicritical
Nishimori point belongs to the RDI universality class. 34
Beside the static behavior, we also consider the critical behavior of a purely relaxation
dynamics without conservation laws, the so-called model A, as appropriate for uniaxial
magnets. 4 The critical behavior of the model-A dynamics for RDI systems has been re-
cently studied numerically (for a critical review of the existing results, see Ref. 21). It has
been shown that the relaxational dynamics belongs to a single dynamic universality class, 21
characterized by the dynamic critical exponent z = 2.35(2).
For Hamiltonian (9) an accurate study of the dependence of the size of the corrections to
scaling on p is reported in Ref. 20. It turns out that the leading scaling corrections associated
with ω = 0.29(2) are suppressed for p = p * = 0.800(5), in agreement with the findings of
Refs. 19,27. For this reason we have performed our simulations at p = 0.8.
III. STATIC STRUCTURE FACTOR IN THE HIGH-TEMPERATURE PHASE
A. Definitions
. Since we are interested in infinite-volume quantities, we must be sure that finite-size effects are negligible. A detailed check is performed at β = 0.281, where we can compare simulation results at different values of L, corresponding
TABLE I :
IMC results for the randomly site-diluted Ising model at p = 0.8. We report the number of samples N s , the number of Metropolis iterations in equilibrium N it , the second-momentcorrelation length ξ, the zero-momentum integrated autocorrelation time τ int (0), and the effective
zero-momentum exponential autocorrelation times τ exp,x (0), for x = 1, 2.
β
L
N s
N it
ξ
τ int (0)
τ exp,1 (0)
τ exp,2 (0)
0.275
32
20000
30000
4.452(4)
36.66(27)
37.92(18)
39.9(5)
0.278
32
20000
5000
5.601(4)
62.25(23)
65.70(25)
71.3(6)
64
20000
5000
5.622(2)
61.98(21)
65.41(30)
69.4(9)
0.280
32
20000
8000
6.872(7)
102.1(6)
106.8(5)
118.2(1.2)
0.281
32
20000
10000
7.800(9)
139.4(7)
146.7(7)
164.1(1.7)
64
20000
5000
7.917(4)
139.7(8)
148.2(7)
158(2)
128
10000
5000
7.924(2)
138.8(1.0)
147.1(1.1)
157(3)
0.282
64
15000
20000
9.331(6)
207.0(1.0)
220(2)
238(3)
128
5000
20000
9.346(5)
205.5(1.6)
218.8(1.3)
232(4)
0.283
64
20000
20000
11.551(10)
342.8(2.1)
361.0(1.6)
402(4)
0.284
128
5000
50000
15.837(16)
716(6)
753(9)
842
TABLE II :
IINumerical values of the coefficients appearing in the interpolation formula (81).Q 2 = 0.842
Q 2 = 5.262
Q 2 = 21.05
Q 2 = 47.36
a 0
−2.30388
0.0972492
−0.1390160
−0.1482217
a 1
1.85440
−0.1867741
−0.0166780
0.0609449
a 2
−1.03426
0.2245075
0.0882145
−0.0108184
a 3
−1.04434
−0.0670895
−0.0483811
−0.0040681
a 4
2.29488
−0.0788428
0.1494524
0.1510844
d 0
3.30388
0.9027518
1.1390167
1.1482211
d 1
−1.87228
−0.5008092
−13.808512
−19.439356
d 2
2.03391
12.846119
114.32824
182.53584
d 3
−0.64676
−27.878988
−319.65479
−587.00842
d 4
0.13949
58.784027
519.97023
1021.6385
κ 2
1.95
9.08
10.46
12.40
c
2.000
1.124
1.087
1.282
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. G Schehr, R Paul, Phys. Rev. E. 7216105G. Schehr and R. Paul, Phys. Rev. E 72, 016105 (2005).
| [] |
[
"Microscopic theory of the insulating electronic ground states of actinide dioxides AnO 2 (An=U, Np, Pu, Am, and Cm)",
"Microscopic theory of the insulating electronic ground states of actinide dioxides AnO 2 (An=U, Np, Pu, Am, and Cm)"
] | [
"M.-T Suzuki \nCCSE\nJapan Atomic Energy Agency\n5-1-5277-8587Kashiwanoha, KashiwaChibaJapan\n\nDepartment of Physics and Astronomy\nUppsala University\nBox 516SE-751 20UppsalaSweden\n",
"N Magnani \nChemical Sciences Division\nActinide Chemistry Group\nLawrence Berkeley National Laboratory\n1 Cyclotron Road94720-8175BerkeleyCAUSA\n",
"P M Oppeneer \nDepartment of Physics and Astronomy\nUppsala University\nBox 516SE-751 20UppsalaSweden\n"
] | [
"CCSE\nJapan Atomic Energy Agency\n5-1-5277-8587Kashiwanoha, KashiwaChibaJapan",
"Department of Physics and Astronomy\nUppsala University\nBox 516SE-751 20UppsalaSweden",
"Chemical Sciences Division\nActinide Chemistry Group\nLawrence Berkeley National Laboratory\n1 Cyclotron Road94720-8175BerkeleyCAUSA",
"Department of Physics and Astronomy\nUppsala University\nBox 516SE-751 20UppsalaSweden"
] | [] | The electronic ground states of the actinide dioxides AnO2 (with An=U, Np, Pu, Am, and Cm) are investigated employing first-principles calculations within the framework of the local density approximation +U (LDA+U ) approach, implemented in a full-potential linearized augmented planewave scheme. A systematic analysis of the An-5f states is performed which provides intuitive connections between the electronic structures and the local crystalline fields of the f states in the AnO2 series. Particularly the mechanisms leading to the experimentally observed insulating ground states are investigated. These are found to be caused by the strong spin-orbit and Coulomb interactions of the 5f orbitals; however, as a result of the different configurations, this mechanism works in distinctly different ways for each of the AnO2 compounds. In agreement with experimental observations, the nonmagnetic ground states of plutonium and curium dioxide are computed to be insulating, whereas those of uranium, neptunium, and americium dioxides require additional symmetry breaking to reproduce the insulator ground states, a condition which is met with magnetic phase transitions. We show that the occupancy of the An-f orbitals is closely connected to each of the appearing insulating mechanisms. We furthermore investigate the detailed constitution of the noncollinear multipolar moments for transverse 3q magnetic ordered states in UO2 and longitudinal 3q high-rank multipolar ordered states in NpO2 and AmO2. | 10.1103/physrevb.88.195146 | [
"https://arxiv.org/pdf/1305.5627v2.pdf"
] | 119,195,150 | 1305.5627 | 4d02fb8cae6a2d79d4d9b41e5a8e368b74b81ca1 |
Microscopic theory of the insulating electronic ground states of actinide dioxides AnO 2 (An=U, Np, Pu, Am, and Cm)
24 May 2013
M.-T Suzuki
CCSE
Japan Atomic Energy Agency
5-1-5277-8587Kashiwanoha, KashiwaChibaJapan
Department of Physics and Astronomy
Uppsala University
Box 516SE-751 20UppsalaSweden
N Magnani
Chemical Sciences Division
Actinide Chemistry Group
Lawrence Berkeley National Laboratory
1 Cyclotron Road94720-8175BerkeleyCAUSA
P M Oppeneer
Department of Physics and Astronomy
Uppsala University
Box 516SE-751 20UppsalaSweden
Microscopic theory of the insulating electronic ground states of actinide dioxides AnO 2 (An=U, Np, Pu, Am, and Cm)
24 May 2013(Dated: May 27, 2013)arXiv:numbers: 7120Nr7127+a7530Et
The electronic ground states of the actinide dioxides AnO2 (with An=U, Np, Pu, Am, and Cm) are investigated employing first-principles calculations within the framework of the local density approximation +U (LDA+U ) approach, implemented in a full-potential linearized augmented planewave scheme. A systematic analysis of the An-5f states is performed which provides intuitive connections between the electronic structures and the local crystalline fields of the f states in the AnO2 series. Particularly the mechanisms leading to the experimentally observed insulating ground states are investigated. These are found to be caused by the strong spin-orbit and Coulomb interactions of the 5f orbitals; however, as a result of the different configurations, this mechanism works in distinctly different ways for each of the AnO2 compounds. In agreement with experimental observations, the nonmagnetic ground states of plutonium and curium dioxide are computed to be insulating, whereas those of uranium, neptunium, and americium dioxides require additional symmetry breaking to reproduce the insulator ground states, a condition which is met with magnetic phase transitions. We show that the occupancy of the An-f orbitals is closely connected to each of the appearing insulating mechanisms. We furthermore investigate the detailed constitution of the noncollinear multipolar moments for transverse 3q magnetic ordered states in UO2 and longitudinal 3q high-rank multipolar ordered states in NpO2 and AmO2.
I. INTRODUCTION
Recently, f electron materials containing actinide elements have drawn considerable interest, stimulated by observations of intriguingly ordered ground states forming at low temperatures. A striking example is the formation of the hidden order phase in URu 2 Si 2 , where the origin of the arising low-temperature electronic order could not unambiguously be disclosed even after a quarter century (see, e.g., Ref. 1). High-rank multipoles have been proposed recently as possible candidates for the order parameter in the hidden order phase. 2-7 Another example is the ordered ground state of NpO 2 , which, after many years could experimentally be established to be due to a high-rank multipolar order, in the absence of any dipolar moment formation. 8,9 The richness of 5f electron physics can be attributed to their multiple degrees of freedom where entangled spin and orbital moments may occur, activated through the strong spin-orbit (S-O) interaction in the actinide elements as well as on-site Coulomb interactions. These conditions are in particular met in the actinide dioxides, which have provided a treasure trove of a rich variety of multi-orbital physics over many years. [9][10][11][12] The actinide dioxides crystallize in the cubic fluorite structure, yet, the ground-state properties of the actinide dioxides are intriguingly diverse. The current knowledge of their ordered ground states stipulates that in UO 2 transverse 3q magnetic dipolar and 3q electric quadrupolar order is realized; 13,14 NpO 2 is characterized by a 3q ordered magnetic multipole of a high rank, in the ab-sence of any dipole moment formation. 8,9 AmO 2 undergoes a conspicuous phase transition at around 8.5 K. 15 While a peak structure in the magnetic susceptibility was found, 15 neutron-diffraction measurements could not detected antiferromagnetic order in agreement with the Mössbauer measurement. 16,17 PuO 2 is a simple nonmagnetic insulator, 18 whereas CmO 2 has an unexpected paramagnetic moment. 19 Surprisingly, each of these actinide dioxides is an insulator, in spite of the very different emerging orders and physical properties. This suggests that in these dioxides different gap-formation mechanisms are in fact operative. The origin of gap formation can only restrictedly be established through experiments. Conversely, first-principles electronic structure calculations are well suited to study the gap formation mechanisms in relation to the unusually ordered states, but only a few such first-principles calculations, accounting for multipolar order states have been performed. 20 A framework of first-principles calculations allows for a description of the bulk properties even if the investigated system contains some localized f electrons in an open shell. However, it is known that the localized character of f electrons is not reproduced with the basic approximations employed in first-principles calculations, like the local density approximation (LDA) or generalized gradient approximation (GGA) in which the two-body correlation between electrons in the open shell is not captured sufficiently. It was actually reported that the LDA approach led to states that are inconsistent with the CEF ground states determined ex-perimentally in the AnO 2 . 21,22 Moreover, the calculations predicted metallic ground states 21,23 whereas the AnO 2 compounds with light actinides are known to be large gap semi-conductors or insulators with energy gaps around 2 eV. [24][25][26] Therefore, first-principles calculations taking into account the strong Coulomb interaction have been applied and have provided insight in the detailed magnetic and electronic character of UO 2 , [27][28][29][30][31][32][33][34] of NpO 2 , 20,31,34 and of PuO 2 , 30,31,[34][35][36][37][38][39][40][41] although the f characters provided in some of these calculations are different from those of the experimental ground states, especially for the ordered states. As taking strong electron correlations into account is necessary to obtain the energy gap in the first-principles calculations, the insulating ground states of AnO 2 are referred to as Mott insulators. 29,42,43 Thus, to appropriately describe the ground states in the AnO 2 compounds, it is imperative to include the effects of strong Coulomb interaction, strong spin-orbit coupling, and the multiorbital character of the f electrons on equal footing. In this paper, we perform such first-principles calculations, focusing particularly on the mechanisms of insulating gap formation. We show that the insulating mechanisms differ substantially from one AnO 2 compound to another, depending on the involved f states in AnO 2 .
The LDA+U method with S-O interaction included has been successful in capturing the ground state properties of f -electron compounds. 20,[44][45][46][47][48][49][50][51] This method is especially suited to describe the local character of f electrons on the same footing with the electronic band description. Moreover, there are two specific properties of the LDA+U framework that make it suitable to study complex ordered phases in the AnO 2 : the spin-orbital dependence of the local potential, which is essential since appearing order parameters involve multiple spin and orbital degrees of freedom, and the ability to take into account noncollinearity of local order parameters, because the multipole moments on each An site can be expressed through the local potential only. 20 Here we have adopted the LDA+U method with spin-orbit interaction, combined with the framework of the full-potential linearized augmented plane wave (FLAPW) band-structure method. Since a large Coulomb U tends to increase the anisotropic character of the f states, the full-potential treatment is an important ingredient, too, to adequately reproduce the f states in the AnO 2 .
In the following, we provide a complete computational study of the electronic structures of the AnO 2 , focusing on the activated spin-orbital degree of freedom in the ground states. First, we give in Sec. II an outline of the known experimental properties of the AnO 2 compounds. In Sec. III, we describe the employed computational framework, and in Sec. IV, we report the obtained results and provide discussions separately for our calculations of the non-ordered states in the AnO 2 and for the ordered states computed for UO 2 , NpO 2 , and AmO 2 . (1) 6 , in D 3d symmetry.
II. PROPERTIES OF AnO2 COMPOUNDS
The actinide dioxides crystallize in the cubic fluorite crystal structure. They are characterized by an ionicity with An 4+ cations and O 2− anions, which is strong for the early 5f elements, but is reduced for the late actinide elements. 31 The dioxides with open 5f shell are further characterized by the strong localized character of the 5f electrons on the An atoms. Note that ThO 2 has no occupied f orbitals and is therefore not considered here.
The An-5f states multiplets are classified as a Γ 5 triplet, a Γ 8 quartet, and a Γ 1 singlet in UO 2 , 14,52-54 NpO 2 , 54-56 and PuO 2 , 54,57,58 respectively, based on the Russell-Saunders LS coupling scheme for their paramagnetic states. In a one electron description, the degenerate 5f orbitals split in j= 5 2 and j= 7 2 orbitals under the action of strong S-O coupling. The j= 5 2 orbitals, which have a lower orbital energy than the j= 7 2 orbitals, further split into a Γ 7 doublet and a Γ 8 quartet through the cubic crystalline electric field (CEF), see the schematic description in Fig. 1. Following a simple point charge model, the energy level of the |j = 5/2, Γ 8 quartet should be lower than that of the |j = 5/2, Γ 7 doublet due to the oxygen anions located in [1 1 1] directions from the actinide cations. If the CEF is large enough to neglect the higherlying Γ 7 orbitals, the f states can be described by states in which two, three, and four electrons occupy the Γ 8 orbitals for U 4+ , Np 4+ , and Pu 4+ ions, respectively. Although the simple one electron description qualitatively explains the local character of the f states on the An single ion in these compounds, 59,60 it is nonetheless necessary to analyze the electronic structure on the basis of first-principles calculations to reveal the bulk properties that involve the conduction as well as localized electrons, such as the insulating behavior of the AnO 2 . 20,30,31 UO 2 exhibits a clear phase transitions at T = 30.8 K. 61,62 The low temperature phase of UO 2 is associated with a transverse 3q antiferromagnetic structure, 13,63,64 which superseded the earlier suggestions of 1k (Ref. 65) or 2k (Ref. 66) structures, with a dipolar magnetic uranium moment of 1.74 µ B . 67, 68 The dipolar 3q magnetic order in UO 2 is accompanied by a 3q ordering of quadrupoles, which corresponds to a distortion of the 5f charge density along the direction of dipolar moment. 13,64 It has been proposed that the superexchange quadrupolar interaction is an important ingredient to stabilize the 3q magnetic ground state 64,69,70 as well as for the magnetization dynamical properties observed for UO 2 . 13,71 Also, recent first-principles studies based on an LDA+U method successfully showed the stability of the 3q magnetic ordered state in UO 2 . 29,33 The phase transition observed 61,72 at T = 25.4 K in NpO 2 has been an enigmatic issue until not too long ago. At an earlier stage of the research on this issue, the low temperature phase of NpO 2 had been speculated to arise from an antiferromagnetic order due to a similar behavior of the temperature dependent susceptibility to that of UO 2 . However, no dipole magnetic moments essential to characterize the phase transition have been observed in neutron scattering 55,56 and Mössbauer experiments. [73][74][75] Intensive experimental and theoretical efforts then were invested to identify the unusual ordered state. Crucial experiments to identify aspects of the ordered state were the muon spin rotation measurement which detected breaking of the time reversal symmetry 76 and resonant X-ray scattering which identified the electric symmetry of the low temperature phase. 77 Several experiments are consistent with, and have confirmed the noncollinear 3q multipole order (MPO) with Γ 5 symmetry multipoles. 55,[76][77][78][79][80][81][82][83][84][85][86][87][88][89] In principle, the Γ 5 octupole 90 moments could be detectable in resonant X-ray measurements, 91 however, direct observation of the octupole moments has not been successful so far. Several recent theories pointed out that a multipolar ordered state with triakontadipole (rank 5) primary order parameter would explain the small weight of the octupole moments. 8,9,20 The phase transitions to the 3q MPO states break translation symmetry of the face centered cubic (FCC) crystal structure but preserve a simple cubic (SC) symmetry containing a four-sublattice unit cell. 77,92,93 A symmetry analysis of the low temperature phases observed in UO 2 and NpO 2 has already been performed. 81,94,95 The space group F m3m is lowered to the P a3 and the P n3m groups in the ordered phases of UO 2 and NpO 2 , respectively. 77,94 Since the magnetic space group of the 3q-MPO states have no non-unitary part, both electric and magnetic multipoles which belong to the same symmetry can spontaneously appear in these ordered states. The four-sublattice unit cell contains one site of equivalent 8 oxygen atoms which has one Wycoff parameter in the P a3 and two nonequivalent fixed oxygen sites (cubic 2a site and tetragonal 6d site) in the P n3m space group. These facts are clearly reflected in the measured 17 O-NMR spectra. 84,96 Although the antiferromagnetically ordered state in UO 2 would allow the oxygen atoms to move with the Wycoff parameter, it was reported that the deformation of the oxygen position is considerably small. 68 In PuO 2 , the magnetic susceptibility is independent of temperature up to 1000 K. 97 The nonmagnetic property is understood from the CEF analysis for the Pu 4+ ion giving that the ground state is the nonmagnetic singlet and the first excited state, which is 123 meV above the ground state, is a triplet. 57,58,98 Transport measurements, furthermore, showed that PuO 2 is an insulator with a 1.8-eV activation gap, 24 while optical spectroscopy gave a 2.8-eV direct gap. 26 A recent Pu-NMR study confirmed the nonmagnetic character of the ground state of PuO 2 . 18 In AmO 2 a clear antiferromagnetic-looking phase transition is observed at 8.5 K, 15 however, no magnetic moment has been observed in Mössbauer nor in neutron scattering measurements. 16,17 The fingerprint of this dioxide is similar to the one obtained in earlier studies of NpO 2 , and hence, an AF-MPO state is expected in AmO 2 . The CEF ground states of the Am 4+ ion in AmO 2 have been believed to be a Γ 7 doublet. 15,99,100 However, the Γ 7 state has no degree of freedom for the higher rank multipoles and therefore seems to contradict the experiments which observe no magnetic dipole moments. A recent CEF analysis based on the j-j coupling method discussed an instability of the Γ 7 ground state and possibility of stabilization of the Γ 8 ground state, which can induce higher order multipoles without inducing a dipole moment. 101 Notably, there are many experimental challenges to distinguish the essential bulk contribution of AmO 2 due to the strong self-radiation damage caused by alpha decay. [102][103][104][105] For the next actinide dioxide, CmO 2 , only a few experiments have thus far been reported for the detailed constitution of its ground state. 19,106 Cm 4+ has six electrons in the 5f shell, producing a 7 F 0 ground state from Hund's rules. However, a paramagnetic moment has been observed in CmO 2 which is unexpected from the nonmagnetic ground state. 19 Niikura and Hotta explained the magnetic behavior of CmO 2 by assuming the existence of a magnetic excited state with an excitation energy smaller than Landé interval rule. 107
III. COMPUTATIONAL METHOD
The LDA+U method 108-111 provides the one electron Hamiltonian as
h LDA+U = h LDA + τ γγ ′ |τ ℓγ v τ ℓ γγ ′ τ ℓγ ′ | ,(1)
where τ and ℓ denote the atoms and angular momenta of the orbitals, respectively, for which the +U potentials are introduced. γ (γ ′ ) is an index related to an orbital m (m ′ ) and a spin s (s ′ ), or, alternatively, double-valued irreducible representations of the site symmetry of the An obtained through a unitary transformation. The local potentials are determined through the spin-orbital dependent density matrix, n τ ℓ γγ ′ = kb τ ℓγ|kb kb|τ ℓγ ′ , through a projection from the band states |kb to the local basis |τ ℓγ . The density matrix as well as the charge density are determined selfconsistently.
The calculations have been performed for non-ordered states in the series of AnO 2 and for ordered states in UO 2 , NpO 2 , and AmO 2 . In the calculations of nonordered states, we used the FCC unit cell with the space group F m3m (No. 225) and applied a relation 45,51 The ordered states are calculated with the four sub-lattice unit cell mentioned above. It is known that the large U introduced in the LDA+U method can induce some meta-stable states especially in calculations of ordered states and may lead to convergence to an electronic state that is inconsistent with the realistic ground state. 38,[112][113][114][115] To avoid this problem, we made use of experimental information concerning the CEF ground states and the order parameters to control the occupations for the initial density matrix and adapt the symmetry preserving the 3q structure of the order parameters observed for UO 2 and NpO 2 . Doing so, we could confirm that the calculations successfully stabilized the ordered states after convergence of the charge density and the density matrices on the An sites.
n iℓ −m−m ′ −s−s ′ =(−1) m+m ′ +s−s ′ n iℓ * mm ′ ss ′ , imposed by time- reversal symmetry.
The (noncollinear) magnetic multipole moments can be described through the local LDA+U potentials. The expectation values of the multipole operators O τ ℓ defined on the An atoms are calculated with the local basis set {|τ ℓγ } defined inside the muffin-tin spheres, following the expression
O τ ℓ = kb γγ ′ kb|τ ℓγ O τ ℓ γγ ′ τ ℓγ ′ |kb = γγ ′ O τ ℓ γγ ′ n τ ℓ γ ′ γ .(2)
Explicit expressions for the multipole operators have been listed by Kusunose. 11 We used the exchangecorrelation functional of Gunnarsson and Lundqvist for the LDA potential. 116 The Coulomb U parameter has been chosen as U = 4 eV and the exchange J in the range of 0−0.5 eV. These values have previously been shown to provide an accurate description of measured properties of actinide dioxides. 39,117,118 The double-counting term has been chosen as in the fully localized limit, 119 leaving out the spin dependency of the Hund's coupling part to adapt it for the nonmagnetic LDA part of Eq. (1).
In the basis set we have included the Np 5f , 6d, 7s, and 6p orbitals as valence states and Np 5d, 6s orbitals as semi-core states and for oxygen we treated the 2s and 2p states as valence states. The muffin-tin sphere radii were 1.4Å for An and 0.9Å for O. The plane-wave cut-offs used in the calculations were about 250 and 900 plane waves at the Γ point for calculations with the FCC normal cell and for the 4-sublattice SC unit cell in the multipolar calculations, respectively. In reciprocal space we used for the self-consistent convergence (charge density and density of states (DOS) calculation) 12×12×12 (24×24×24) k-points in the FCC unit cell and 6×6×6 (12×12×12) for the 4-sublattice SC unit cell.
IV. RESULTS AND DISCUSSION
A. Non-ordered state calculations
We first discuss the results obtained from the nonordered state calculations to examine how the anisotropic f character is reproduced with the LDA+U method. In Figures 2 and 3 we show the band structures and the DOS, respectively, calculated by both the LDA and LDA+U methods for the AnO 2 compounds. Figure 4 shows the FCC BZ of the non-ordered state. The LDA approach only correctly predicts that non-ordered CmO 2 would be an insulator. In the LDA calculations, the |j = 5 2 , Γ 7 and the |j = 5 2 , Γ 8 states are present in the same energy range around the Fermi level, leading to a more or less homogeneous f electron occupation for the j = 5/2 orbitals. The computed orbital occupations of the f states are given in Table I. As a consequence of the homogeneous occupation, the f states predicted by the LDA calculations do not have sufficient anisotropic character as would be expected from the CEF states. In other words, the LDA calculations fail to produce the CEF ground states in the AnO 2 series as also was previously pointed out, 21,22 whereas the correct CEF behavior for lighter actinide dioxides is reproduced if only the Conversely, the LDA+U calculations do lead to anisotropic f states, in which the |j = 5/2, Γ 8 quartet is dominantly occupied for An=U, Np, and Pu in AnO 2 due to a large splitting between the |j = 5/2, Γ 7 and |j = 5/2, Γ 8 orbitals. In PuO 2 , an insulator ground state is obtained with the |j = 5/2, Γ 8 orbitals being fully occupied. This state is consistent with the experimentally observed nonmagnetic insulator ground state with a singlet Γ 1 CEF state. 18,24 The anisotropic character of the f states can be visually seen from the f -charge distribution on the Pu sites shown in Fig. 5(a) and (b). It can be seen that the charge distribution is extended to the [1 0 0] and the equivalent axes, reflecting an f wave function with Γ 8 symmetry.
In UO 2 and in NpO 2 the strong ionization constrains the f states to containing 2 or 3 electrons by the strong CEF effect. As a result, the Γ 8 orbitals being occupied with 2 or 3 electrons lead to metallic electronic states in UO 2 and in NpO 2 in the non-ordered calculations. In AmO 2 , the Γ 7 doublet is populated with about two electrons and the Γ 8 quartet has some hole, rendering also AmO 2 metallic in non-ordered LDA+U calculations. These results obtained from nonmagnetic calculations imply that the non-ordered states in AmO 2 should exhibit a strong susceptibility due to the Γ 8 orbitals. The finding of an incompletely filled Γ 8 orbital is consistent with the recent CEF theory suggesting that multipole order is realized within the Γ 8 CEF ground state in AmO 2 . 101 Since the Γ 7 CEF ground states, which were suggested on the basis of experimental results, 15,99,100 could not bring about any higher order multipoles, our results support the Γ 8 CEF ground states in AmO 2 . Fur-thermore, our results purport that symmetry breaking of some sort is necessary to reproduce the experimentally observed insulator ground states in UO 2 , NpO 2 , and AmO 2 . Especially, the insulator ground states can evidently not be produced with the non-ordered state calculations for NpO 2 and for AmO 2 , since the FCC primitive cells containing an odd number of electrons will always lead to uncompensated metallic states when time-reversal symmetry is present.
For CmO 2 , both the LDA and the LDA+U calculations produce the insulator ground states in the nonmagnetic state calculations. We can speculate that a reason that the LDA approach is already a good approximation for CmO 2 comes from the weakly anisotropic character of the f states in the nonmagnetic ground state, which can be visually seen in Fig. 5. The Cm 5f charge distribution, seen from the [1 0 0] direction [ Fig. 5(c)], is much more isotropic than that of Pu [ Fig. 5(a)]. Assuming that CmO 2 has a tetravalent Cm 4+ ion just as the An ion in the other actinide dioxides, a singlet ground state with J = 0 is expected. In this regard, the nonmagnetic insulator ground states obtained in both calculations seem to be a natural consequence of the tetravalent ionized state in CmO 2 . Meanwhile, a neutron diffraction experiment reports detection of an effective paramagnetic moment µ eff ∼ 3.4 µ B , which would be consistent with the Curie-Weiss behavior observed for the magnetic susceptibility. 19 Niikura and Hotta investigated the possibility of having a magnetic excited state just above the nonmagnetic ground in the Cm 4+ ion with a small excitation energy, aiming to provide an understanding for the unexpected magnetic behavior in CmO 2 . 107
B. Ordered state calculations
Next, we have performed electronic structure calculations for the UO 2 , NpO 2 , and AmO 2 compounds allow-ing for self-consistent convergence to a symmetry-broken ordered ground state. Figure 4 shows the SC Brillouin zone (BZ) for the 3q ordered state, which displays a rela- tionship to the FCC BZ of the non-ordered state. In the following the results of the ordered-state calculations are discussed in detail for each of these actinide dioxides.
For UO 2 , we choose the initial density matrix to correspond to the non-ordered state with the Zeemantype field along [1 1 1] equivalent directions, keeping a transverse-3q structure as symmetry breaking term. The converged electronic structure corresponds to the transverse 3q ordered state with Γ 4 local multipoles. The obtained band structures and DOS are shown in Figs. 6 and 7, for U = 4 eV and two J values, 0 and 0.2 eV, respectively. The energy gaps upon convergence are generated through the splitting of the f states, which is in turn induced through the large U . In nonmagnetic UO 2 the plus and minus j z components of the j z =±5/2, ±3/2, ±1/2 orbitals are degenerate; because of the Coulomb U these orbitals split into the lower and upper Hubbard bands as seen in Fig. 7. For U = 4 eV the experimentally observed energy gap 25 of about 2 eV is well reproduced with the 3q magnetic order in the calculations.
In the transverse 3q magnetic order, the dipolar magnetic moments are induced along [1 1 1] axis for the U atom at (0, 0, 0), along the [-1 1 -1] axis for the U at (0, 1/2, 1/2), along the [-1 -1 1] axis for the U at (1/2, 0, 1/2), and along [1 -1 -1] for the U at (1/2, 1/2, 0). The local magnetic moments on the uranium sites are shown in Fig. 8 as a function of the Hund's coupling parameter J. As seen from the figure, the magnetic moment is dominated by the orbital moment and is gradually enhanced for increasing J. The calculated orbital and spin 1.93 µ B for J = 0.2 eV, providing good agreement with the experimental value 25,68 of 1.74 µ B . Our results show that the complex constitution of the f ground state, with contributions from the hybridization between U-5f and O-2p states as seen in Fig. 7, leads to a reduction of the magnetic moments expected from the Γ 5 CEF ground state, which is slightly higher than 2.0 µ B . 69 We further provide in Fig. 8 2) for the uranium f electrons. We find that the quadrupole moments also develop with increasing J, changing the sign around J = 0.03 eV. The finite contribution from the quadrupole moments is consistent with recent experiments. 13,64 In Figs. 9(a) and (b) the spatial shape of the uranium 5f -wave function is displayed by plotting its spin moment distribution projected to the [1 1 1] local axis on an isodensity surface of the calculated f -charge density. It is seen that an overall, dipolar magnetic moment exists along the [1 1 1] axis. Also, reflecting the local site symmetry of the ordered state, the calculated charge and spin distributions preserve the C 3i symmetry for UO 2 .
Next, we consider NpO 2 . The obtained band structures and DOS are shown in Figs. 10 and 11, for U = 4 eV and two J values, 0 and 0.2 eV, respectively. For this actinide dioxide the computed electronic structure corresponds to the longitudinal 3q ordered state with Γ 5 local
multipoles.
A schematic picture of how the one-electron f orbital levels will split, depending on the symmetry at the Np sites in NpO 2 , has been provided in Fig. 1. The Γ 5 multipole order lowers the O h local symmetry to the D 3d symmetry on the Np sites. Thus, the |j=5/2, Γ 8 quartet splits into the two singlets Γ 4 and Γ 5 and one doublet Γ
(1) 6 in the MPO states. The two singlets Γ 4 and Γ 5 are degenerated under time-reversal invariance but not in the magnetic MPO states, which break the time-reversal symmetry. The two doublets derived from the Γ 7 and Γ 8 orbitals belong to the same symmetry and hybridize with each other in the MPO states. A Γ − 5 multipole is expected to be produced with the three 5f electrons occupying the Γ 4 singlet and the Γ (1) 6 doublet, which are split from the Γ 8 quartet in the paramagnetic state. Accordingly, we thus choose, as initial density matrix, the one corresponding to the local f states in which the Γ 4 singlet and the Γ (1/2, 1/2, 0) ion. Reflecting the D 3d local symmetry, the Np-f orbital components in the DOS still keep degeneracy for the Γ 6 doublets in the exotic magnetic multipole ordered states as shown in Fig. 11.
The top-right panel of Fig. 12 shows the calculated J dependence of the multipole moments O normalized to the multipole moments for the initial f occupation, O 0 , as mentioned above. The noncollinear multipole moments in the ordered state of NpO 2 are strongly affected by the value of the Hund's coupling J, which also affects the occupation difference of f -orbitals, see the topleft panel. At J = 0, the Γ 4 singlet and the Γ (1) 6 doublet are fully occupied. As J increases, the f -electron on the Γ (1) 6 doublet starts to transfer to the Γ (2) 6 doublet through hybridization. This transfer strongly enhances the triakontadipole moment and suppresses the octupole moment in NpO 2 . Thus, the Γ 5 -triakontadipole moment can be the leading order parameter in NpO 2 . 20 The charge and spin distributions of the neptunium 5f -wave function are plotted in Figs. 9(c) and (d). In contrast to the equivalent distributions of the U 5f wave function in Figs. 9(a) and (b), the spin distribution corresponds to a vanishing atomic spin moment along the [1 1 1] local axis. The calculated charge and spin distributions preserve the D 3d symmetry of NpO 2 .
Next, we consider AmO 2 . The order parameter in the ground state of AmO 2 has not been determined experimentally yet. The recent theoretical study 101 suggested the Γ 8 CEF ground state to explain the experimental facts. 102 It is hence plausible to have order parameters which are similar to those of NpO 2 . We thus choose initial density matrixes, so as to have finite Γ 5 multipole moments, corresponding to the local 5f state which has filled Γ (1) 6 and Γ
(2) 6 doublets as well as the Γ 4 singlet in the calculations for ordered states in AmO 2 . Then, our calculations lead to the solution of the longitudinal 3q ordered state with Γ 5 -local multipoles. The obtained band struc- We have found that the multipolar ordering is responsible for stabilizing the insulating ground state in AmO 2 , like in NpO 2 ; the resulting gap is indicated by the shaded area in Fig. 13. Meanwhile, the constitution of the highrank multipoles in AmO 2 differs substantially from the one of NpO 2 , see Fig. 12. The Γ 5 -multipoles in AmO 2 are rather insensitive for small J values (bottom-right panel), whereas they decrease for J values larger than 0.1 eV. This striking difference of the J dependence of the multipoles in AmO 2 from those of NpO 2 stems from the fact that the |j = 5/2, Γ 7 orbitals are completely occupied in the f 5 state of the Am 4+ ion and there is no state available to couple to it through the Hund's coupling J. Furthermore, the calculated energy gap for AmO 2 is found to depend strongly on the Hund's coupling J, see Fig. 13. A similar sensitivity was not observed for UO 2 nor for NpO 2 . As can be seen from Fig. 13, the energy gap in AmO 2 decreases with increasing J. This implies that a large Hund's rule J can make the insulator solution unstable in the longitudinal 3q ordered state of Γ − 5 multipoles in AmO 2 . FIG. 14: (Color online) As Fig. 11, but for the multipolarordered phase of AmO2.
V. CONCLUSIONS
We have investigated the origin of the gap formation in the actinide dioxides. The origin of the insulating gap formation is found to lie in the strong on-site Coulomb repulsion of the 5f electrons and strong spin-orbit interaction in the relevant f orbitals in the AnO 2 compounds. LDA+U calculations for a non-long-range ordered state reproduce well the energy gaps following the singlet CEF ground states in PuO 2 and CmO 2 . On the other hand, the insulating ground states in UO 2 , NpO 2 , and AmO 2 are obtained only by allowing for the symmetry lowering that can give rise to the ordered states. Thus, the strong correlation is necessary to describe the anisotropic f ground states in AnO 2 . The energy gaps and magnetic properties are correctly reproduced within the accepted range of the parameters, U and J, by taking the proper magnetic space symmetries in the calculations. Especially, using values of J in the acceptable range reproduces the experimentally measured magnetic moment as well as the energy gap in UO 2 , and in addition the contribution from the electric quadrupole moments is enhanced with increasing J. We also showed that the active multipoles in these ordered states are closely related to the orbital occupation, and the higher rank Γ 5 multipole ordered states in AmO 2 have quite different constitution of the multipoles from NpO 2 . Whereas Hund's coupling J enhances the energy gap in NpO 2 together with changing the constitution of high-rank multipoles, it reduces the energy gap in AmO 2 without change in the qualitative constitution of the multipoles. Further experimental investigations are required and encouraged to verify the here-computed ground state properties of AmO 2 and CmO 2 .
FIG. 1 :
1Schematic description of the actinide one-electron j=5/2 orbitals in the crystal fields with O h and D 3d pointgroup symmetry, respectively. The Γ8 quartet in cubic O h symmetry splits in two singlets, Γ4 and Γ5, degenerate under time reversal symmetry, and one doublet state, Γ
FIG. 2 :
2(Color online) Calculated band structures of the actinide dioxides assuming the non-magnetic solution (see text). Left-hand panels give the AnO2 band structures calculated with the LDA approach, right-hand panels those computed with the LDA+U method (U = 4 eV).
FIG. 3 :
3(Color online) Calculated density of states (DOS) for the non-magnetic solution (see text) of the AnO2, calculated by the LDA (left-hand panels) and by the LDA+U method (right-hand panels). The presence of a gap at the Fermi energy is indicated by the grey shaded area.
FIG. 4 :
4(Color online) The first Brillouin zones of the FCC unit cell used for calculations of non-magnetic or nonmultipolar ordered actinide dioxides (red lines) and the SC Brillouin zone of the 4-sublattice unit cell for 3q-ordered states (blue lines). High-symmetry points are indicated.FIG. 5: (Color online) Charge distributions of the An-5f electrons in PuO2 ((a), (b)) and CmO2 ((c), (d)) computed with the LDA+U (using U = 4 eV and J = 0). (a), (c) are looked at from the [1 0 0] direction, and (b), (d) from the [1 1 1] direction. The lines on the outside of the spheres represent the contour lines for the charge distribution (cf. Ref. 120).
FIG. 6 :
6(Color online) Computed band structures and density of states in the transverse 3q Γ4u magnetic MPO state of UO2, calculated by the LDA+U method (U = 4 eV, J = 0 or J = 0.2 eV). The gap computed at the Fermi energy is depicted by the shaded area.
FIG. 7 :
7(Color online) The orbital-projected DOS components computed for the multipolar ordered phase of UO2, for J = 0 (left) and J = 0.2 eV (right).
FIG. 8 :
8(Color online) Top: computed magnetic dipole moments in UO2, as a function of the Hund's exchange J parameter. Shown are the spin moment S111 , orbital moment L111 , and total moment, M111 = L111 + 2 S111 (see text). Bottom: the quadrupole moments O111 = 1√ 3 [ Oyz + Ozx + Oxy ] in UO2 as a function of J for the f electrons.moments are L 111 = 1 √ 3 [| L x | + | L y | + | L z |] = 1.85 µ B and S 111 = 1 √ 3 [| S x | + | S y | + | S z |] = −0.24 µ B for J = 0 and L 111 = 2.58 and S 111 = −0.33 µ B for J = 0.2 eV. The total local magnetic moments are M = L 111 + 2 S 111 = 1.37 µ B for J = 0 and FIG. 9: (Color online) Charge and spin distributions of the An-5f electrons, computed by U = 4 eV and J = 0, in UO2 ((a), (b)), NpO2 ((c), (d)), and AmO2 ((e), (f)). The distributions in (a), (c), (e) are looked at from the [1 0 0] direction, and those in (b), (d), (f) from the [1 1 1] direction, corresponding to the threefold axis, for UO2, NpO2, and AmO2, respectively. The charge distributions are depicted by the isodensity surface, and the distributions of spin-moments (shown by the color code) are plotted as projections onto the [1 1 1] axis, which corresponds to the local threefold axis of the An sites.
FIG. 10 :
10(Color online) Computed band structures and density of states in the longitudinal 3q Γ5u magnetic MPO state of NpO2, calculated by the LDA+U method (U = 4 eV, J = 0 or J = 0.2 eV).
occupied by three f electrons. In the longitudinal 3q structure, the induced Γ 5 multipoles obey a relation, for instance, O yz = O zx = O xy for the Np (0, 0, 0) ion, O yz =− O zx =− O xy for the Np ion at (0, 1/2, 1/2), − O yz = O zx =− O xy for the Np (1/2, 0, 1/2) ion, and − O yz =− O zx = O xy for the Np
FIG. 11 :
11(Color online) The orbital-projected DOS components computed for the multipolar ordered phase of NpO2, for J = 0 (left) and J = 0.2 eV (right).
FIG. 12 :
12(Color online) Left: calculated one-electron orbital occupation numbers as a function of exchange J at U = 4 eV, for the MPO states of NpO2 and AmO2. Right: Calculated expectation values of the multipolar (quadrupolar, octupolar, hexadecapolar, and triakontadipolar) order parameters in NpO2 and AmO2 as a function of J. tures and DOS are shown in Figs. 13 and 14, for U = 4 eV and again two J values, 0 and 0.2 eV, respectively. The charge and spin distributions of the americium 5fwave function are plotted in Figs. 9(e) and (f). These are similar to the equivalent distributions computed for the Np 5f wave function in Figs. 9(c) and (d).
FIG. 13 :
13(Color online) Computed band structures and density of states in the longitudinal 3q Γ5u magnetic MPO state of AmO2, calculated by the LDA+U method (U = 4 eV and J = 0 or J = 0.2 eV).
TABLE I :
ICalculated 5f electron occupation numbers per one-electron orbital on the An sites in the AnO2 (An=U, Np, Pu,
Am, and Cm) fluorite-structure compounds. The calculations have been performed assuming the non-magnetically ordered
state, both with the LDA and with the LDA+U (U = 4 eV) approach.
LDA
LDA+U (U = 4 eV)
Orbital
UO2
NpO2
PuO2
AmO2
CmO2
UO2
NpO2
PuO2
AmO2
CmO2
j=5/2
Γ7 (2)
0.41
0.81
1.16
1.54
1.85
0.14
0.16
0.22
1.91
1.92
Γ8 (4)
1.59
2.18
2.81
3.34
3.80
2.00
3.09
3.86
3.44
3.93
Total
2.00
3.00
3.97
4.88
5.65
2.14
3.25
4.09
5.34
5.85
j=7/2
Γ6 (2)
0.11
0.11
0.12
0.15
0.19
0.06
0.06
0.08
0.07
0.13
Γ7 (2)
0.20
0.26
0.31
0.37
0.46
0.13
0.06
0.19
0.26
0.41
Γ8 (4)
0.23
0.27
0.35
0.44
0.60
0.11
0.29
0.20
0.13
0.29
Total
0.54
0.65
0.78
0.96
1.25
0.31
0.41
0.47
0.46
0.83
lower Γ 8 states are taken into account.
the J dependence of the quadrupole moments O 111 = 1 √ 3 [ O yz + O zx + O xy ], calculated from Eq. (
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| [] |
[
"Log-Sobolev inequality and proof of Hypothesis of the Gaussian Maximizers for the capacity of quantum noisy homodyning",
"Log-Sobolev inequality and proof of Hypothesis of the Gaussian Maximizers for the capacity of quantum noisy homodyning"
] | [
"A S Holevo \nSteklov Mathematical Institute, RAS\nMoscowRussia\n"
] | [
"Steklov Mathematical Institute, RAS\nMoscowRussia"
] | [] | In the present paper we give proof that the information-transmission capacity of the approximate position measurement with the oscillator energy constraint, which underlies noisy Gaussian homodyning in quantum optics, is attained on Gaussian encoding. The proof is based on general principles of convex programming. Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality. We hope that this method should work also for other models lying out of the scope of the "threshold condition" ensuring that the upper bound for the capacity as a difference between the maximum and the minimum output entropies is attainable. | null | [
"https://export.arxiv.org/pdf/2204.10626v3.pdf"
] | 248,366,251 | 2204.10626 | aedb812737168c8fa07814c6c5293d5ac72aba8d |
Log-Sobolev inequality and proof of Hypothesis of the Gaussian Maximizers for the capacity of quantum noisy homodyning
17 Aug 2022
A S Holevo
Steklov Mathematical Institute, RAS
MoscowRussia
Log-Sobolev inequality and proof of Hypothesis of the Gaussian Maximizers for the capacity of quantum noisy homodyning
17 Aug 2022
In the present paper we give proof that the information-transmission capacity of the approximate position measurement with the oscillator energy constraint, which underlies noisy Gaussian homodyning in quantum optics, is attained on Gaussian encoding. The proof is based on general principles of convex programming. Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality. We hope that this method should work also for other models lying out of the scope of the "threshold condition" ensuring that the upper bound for the capacity as a difference between the maximum and the minimum output entropies is attainable.
Introduction
Quantum Shannon theory was rapidly developing during the past decades. As distinct from the classical case, quantum channel is characterized by a whole variety of different capacities, depending on the type of transmitted information (classical or quantum) and on additional resources used during transmission. Among these capacities, the classical capacity, i.e. the ultimate rate of reliable transmission of classical data via a quantum channel, plays a distinguished role, both historically and because of its central importance for quantum communications. A long-standing problem is the classical capacity of bosonic Gaussian channels of various kinds. Hypothesis of Gaussian Maximizers (HGM) states that the full capacity of such channels is attained on Gaussian encodings. The conjecture turned out remarkably difficult. After a series of intermediate steps, a breakthrough was made in the papers [2], [3], [24], where HGM was proved for important class of multimode gauge co-or contra-variant channels (called phase-insensitive in quantum optics). In [12], [20] the solution was extended to a much broader class of channels satisfying certain "threshold condition", essentially ensuring that the upper bound for the capacity as a difference between the maximum and the minimum output entropies is attainable. In [12], [19] these findings were extended to Gaussian measurement (quantum-classical) channels. At the same time, HGM remains open for rather large variety of quantum and quantum-classical channels lying beyond the scope of the threshold condition [17]. At this point we want to mention that many authors (see e.g. [5], [25], [29], [22]) explored the maximum information restricting to the class of Gaussian encodings. In the absence of proof of HGM such results give only a lower bound for the full classical capacity of the Gaussian channel.
Among these models there are rather elementary Gaussian channels, such as approximate position measurement with the oscillator energy constraint, which essentially underlies noisy Gaussian homodyning in quantum optics. While this and related models were studied in earlier quantum communication papers (see e.g. [1], [6]), we could not find a proof of HGM for them in the literature. In the present paper we give such a proof which is based on general principles of convex programming. Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of log-Sobolev inequality [4], [23]-one of the highlights of Analysis in the past century. We think that this method should work also for other models out of the scope of the threshold condition, and hope to address it in further publications.
The classical capacity of quantum measurement
Let H be a Hilbert space of quantum system. Statistics of a quantum measurement with the outcomes y ∈ Y is described by a probability operatorvalued measure M = {M(dy)} (POVM) on H, that is M(dy) ≥ 0 and M(dy) = I (the unit operator on H). As shown e.g. in [15], under mild separability assumptions there exists a countably finite measure µ(dy) such that for any density operator ρ the distribution of the measurement outcomes TrρM(dy) is absolutely continuous w.r.t. µ(dy), thus having the probability density p ρ (y). The affine map M : ρ → p ρ (·) will be called the measurement channel.
An ensemble (encoding) E = {π(dx), ρ(x)} consists of probability measure π(dx) on a "alphabet" X and a measurable family of density operators (quantum states) x → ρ(x) on the Hilbert space H of the quantum system. The average state of the ensemble is the barycenter of this measurē
ρ E = X ρ(x) π(dx).
The classical Shannon information between the input x and the measurement outcome y is equal to
I(E, M) = π(dx)µ(dy)p ρ(x) (y) ln p ρ(x) (y) pρ E (y)
In what follows we will consider POVMs having (uniformly) bounded operator density, M(dy) = m(y)µ(dy), with m(y) ≤ b, so that the probability densities p ρ (y) = Tr ρm(y) are uniformly bounded, 0 ≤ p ρ (y) ≤ b. Moreover, by including b into µ(dy), we can assume without loss of generality that b = 1. Then the output differential entropy
h M (ρ) = − p ρ (y) ln p ρ (y)µ(dy)(1)
is well defined with values in [0, +∞] (see [15] for the detail). . Next we define the quantity (1):
e M (ρ) = inf E:ρ E =ρ h M (ρ(x))π(dx),(2)
which is an analog of the convex closure of the output differential entropy for a quantum channel [28]. Let H be a Hamiltonian in the Hilbert space H of the quantum system, E a positive number. Then the energy-constrained classical capacity of the measurement channel M is
C(M, H, E) = sup E:Trρ E H≤E I(E, M),(3)
where maximization is over the input ensembles of states E satisfying the energy constraint Trρ E H ≤ E. As shown in [20], proposition 1, it is indeed the capacity in the sense of information theory, i.e. the ultimate rate of the classical asymptotically reliable data transmission via the measurement channel. Note that the measurement channel is entanglement-breaking [14] hence its classical capacity is additive and is given by the one-shot expression (3).
If h M (ρ E ) < +∞, then I(E, M) = h M (ρ E ) − h M (ρ(x))π(dx).(4)
By using (4), (2), we obtain
C(M, H, E) = sup ρ:TrρH≤E [h M (ρ) − e M (ρ)] (5) ≤ sup ρ:TrρH≤E h M (ρ) − inf ρ h M (ρ).
There are important cases where the last inequality turns into equality thus giving the value of the capacity. This happens when the maximizer of the first term can be represented as a mixture of (pure) states minimizing h M (ρ) [10], [20]. In particular, all the instances where the Hypothesis of Gaussian Maximizer was proved for Gaussian channels so far refer to that case. In the present paper we propose a method allowing to prove this hypothesis for the first case where this condition is violated and the inequality in (5) is strict and hence becomes useless. In sec. 5 we illustrate it on the example of approximate position measurement (corresponding to noisy homodyning in quantum optics) where this violation happens in the most extreme form.
Reduction to a convex programming problem
In this section we propose a method of computation of the quantity e M (ρ) based on similarity of the optimization problem in the right side of (2) and general quantum Bayes estimation problem [8], [9], [11]. Consider a measurement channel given by the map M : ρ → p ρ (y) = Tr ρm(y), where m(y) is a uniformly bounded positive-operator-valued function of y ∈ Y, such that m(y)µ(dy) = I. Any ensemble E = {π(dx), ρ(x)} where x ∈ X , can be equivalently considered as a probability distribution π(dρ) on the whole set of quantum states S = S(H) (with the carrier concentrated of the states ρ(x), x ∈ X ). Another equivalent description of E is given by the positive (but not probability!) operator-valued measure Π(dρ) = ρπ(dρ) with values in S. The average state is then
ρ E = S ρπ(dρ) = Π(S),
and the minimized functional
F (E) = S h (p ρ ) π(dρ) = S Tr K(ρ)Π(dρ), where K(ρ) = − m(y) ln p ρ (y)µ(dy).(6)
By fixing an average state ρ, we arrive at the optimization problem
S Tr K(ρ)Π(dρ) −→ min Π(dρ) ≥ 0 Π(S) = ρ,
which is formally similar to one arising in the general quantum Bayes problem [9], [11]. The minimized functional is affine in E = {Π(dρ)} and the constraints are convex, so it is a convex programming problem. Under certain regularity condition the problem was investigated in [11], where the following necessary and sufficient conditions for optimality of an ensemble E 0 = {Π 0 (dρ)} were given, which we reproduce here formally: There exists Hermitian operator Λ 0 such that
(i) Λ 0 ≤ K(ρ) for all ρ ∈ S; (ii) [K(ρ) − Λ 0 ] Π 0 (dρ) = 0. Moreover, Λ 0 is the solution of the dual problem max {Tr ρΛ : Λ * = Λ, Λ ≤ K(ρ) for all ρ ∈ S} .(7)
By integrating (ii), we obtain the equation for determination of Λ 0
S K(ρ) Π 0 (dρ) = S K(ρ) ρ π 0 (dρ) = Λ 0 ρ.(8)
The sufficiency of the conditions (i), (ii) is easy to demonstrate formally (cf.
[8]): for any E = {Π(dρ)} F (E) = S Tr K(ρ)Π(dρ) (i) ≥ S Tr Λ 0 Π(dρ) = Tr Λ 0 ρ = S Tr Λ 0 Π 0 (dρ) (ii) = S Tr K(ρ)Π 0 (dρ) = F (E 0 ).
Coming back to the parametric representation E = {π(dx), ρ(x)} , we can write the condition (ii) as
[K(ρ(x)) − Λ 0 ] ρ(x) = 0, a.e. x ∈ X ,
which means that the equality holds a.e. with respect to the measure π 0 (dx). The equation (8) becomes
K(ρ(x)) ρ(x) π 0 (dx) = Λ 0 ρ.(9)
In the case of the measurement channel M : ρ → p ρ (y) = Tr ρm(y), this equation reduces to
− m(y) ln p ρ(x) (y) ρ(x) µ(dy) π 0 (dx) = Λ 0 ρ.
In specific applications, like HGM for bosonic Gaussian channel, the candidate for an optimal encoding usually can be found by optimizing in the class of Gaussian encodings. Then the condition (ii) for this candidate can be verified and the operator Λ 0 found, while a major difficulty may be the check of the operator inequality (i).
Gaussian systems
We will systematically use some notations and results from the books [13], [14]. Consider the finite-dimensional symplectic vector space (Z, ∆) with Z = R 2s and the canonical symplectic matrix
∆ = diag 0 1 −1 0 j=1,...,s .(10)
In what follows H will be the space of an irreducible representation z → W (z); z ∈ Z, of the canonical commutation relations (CCR)
W (z)W (z ′ ) = exp[− i 2 z t ∆z ′ ] W (z + z ′ ).(11)
Here W (z) = exp i Rz are the unitary Weyl operators with the generators
Rz = s j=1 (x j q j + y j p j ), z = [x j y j ] t j=1,...,s(12)
and R = [q 1 , p 1 , . . . , q s , p s ] is the row vector of the bosonic position-momentum observables, satisfying the canonical commutation relation
q j p k − p k q j ⊆ iδ jk I, j, k = 1, . . . , s.
In quantum communication theory q j , p j describe the relevant modes of the field on receiver's aperture (see, e.g. [26]). A number of analytical complications related to unboundedness of operators unavoidably arises in connection with Bosonic systems and CCR. In our treatment of CCR we focus on the algebraic aspects essential for applications while a presentation of the related analytical detail such as domains of definition, selfadjointness etc. can be found in the literature 1 The displacement operators D(z) = W (−∆ −1 z) satisfy the equation that follows from the canonical commutation relations (11)
D(z) * W (w)D(z) = exp iw t z W (w).(13)
The quantum Fourier transform of a trace class operator ρ is defined as
Tr ρD(w)
When ρ is a density operator, this is called the quantum characteristic function of the state ρ. The quantum Parceval formula holds [13]:
Tr ρσ * = Tr ρD(w) Tr σD(w) d 2s w (2π) s(14)
From now on we will consider states ρ with finite second moments:
s j=1 Trρq 2 j + Trρp 2 j ≡ TrρRR t < ∞.
The set of these states will be denoted S 2 . For such states the matrix of second moments is defined as
α (2) = Re TrR t ρR,(15)
and the covariance matrix as
α = Re Tr (R − m) t ρ (R − m) = α (2) − m t m ≤ α (2) ,
where m = TrρR is the row-vector of the first moments (the mean vector ). It is a real symmetric 2s × 2s-matrix satisfying [13] α
≥ ± i 2 ∆,(16)
The state is centered if m = 0. For centered states the covariance matrix and the matrix of second moments coincide and are equal to (15).
A Gaussian state ρ m,α is determined by its quantum characteristic function
Tr ρ m,α W (z) = exp im t z − 1 2 z t αz .(17)
Here α is the covariance matrix and m is the mean vector. For a centered state we denote ρ α = ρ 0,α . For ρ ∈ S 2 we have h M (ρ) ≤ h M (ρ α ) < +∞, where α is the matrix of the second moments of the state ρ by the maximum entropy principle. With the quadratic Hamiltonian
H = RǫR t ,(18)
where ǫ is real positive definite 2s × 2s-matrix, the energy constraint reduces
to 2 Sp α ǫ ≤ E.(19)
We denote the set of all states ρ with the fixed matrix of second moments α by S(α) and we will study the following α-constrained capacity
C(M; α) = sup E:ρ E ∈S(α) I(E, M) = sup ρ∈S(α) [h M (ρ) − e M (ρ)] .(20)
The energy-constrained classical capacity (5) A Gaussian measurement channel M in the sense of [14], [18] is defined via the operator-valued characteristic function of the form
e iz t w M(dz) = exp i R Kw − 1 2 w t βw ,(21)
where K is a scaling matrix, β is the measurement noise covariance matrix, β ≥ ± i 2 K t ∆K. The case K = I 2s (as well as of a general nondegenerate K) corresponds to the type 1 Gaussian measurement channel (with the multimode noisy heterodyning, see e.g [1], [26] as the prototype). However (21) includes also type 2 and 3 measurement channels (noisy and noiseless multimode homodyning) in which case K is a projection onto an isotropic subspace of Z (i.e. one on which the symplectic form ∆ vanish). The following theorem was proved in [17]: Theorem 1. Let M be a general Gaussian measurement channel. The optimizing density operator ρ in (20) is a (centered) Gaussian density operator ρ α :
C(M; α) = h M (ρ α ) − e M (ρ α ),(22)
and hence for a quadratic Hamiltonian
The theorem asserts that the average state of an optimal encoding for a Gaussian measurement is Gaussian but says nothing about the detailed structure of the ensemble. It is well known that a non-Gaussian ensemble can have Gaussian average state (a canonical example is ensemble of the Fock states with the geometric distribution).
Hypothesis of Gaussian Maximizers (HGM): Let M be an arbitrary Gaussian measurement channel. Then there exists an optimal ensemble for (2) and hence for (3) which is Gaussian, more precisely it consists of (properly squeezed) coherent states with the displacement parameter having Gaussian probability distribution.
For Gaussian measurement channels of the type 1 the minimum output differential entropy inf
ρ h M (ρ)(24)
is attained on a pure Gaussian (i.e. squeezed) state (this follows from the result of [3] applied to the entropy function and the complex structure associated with M). If in addition the Gaussian state ρ α satisfies the "threshold condition" which means that the covariance matrix α dominates the covariance matrix of the entropy-minimizing squeezed state, then ρ α can be represented as a Gaussian mixture of these squeezed states, thus giving the optimal ensemble. This implies the validity of the HGM and an efficient computation of the α-constrained capacity as
C(M, H, E) = h M (ρ α ) − min ρ h M (ρ)
see [20]. On the other hand, this does not work when the "threshold condition" is violated, and notably, for all Gaussian measurement channels of the type 2 (noisy homodyning), with the generic example of the energyconstrained approximate measurement of the position q = [q 1 , . . . , q s ] subject to Gaussian noise. In the section 6 we will apply the method from section 3 to prove the HGM in this case for a single mode system. Our strategy will be the computation of e M (ρ α ) with the optimality conditions of that section and relying on the formula (23). Remark 1. In the case of the oscillator-type Hamiltonian
H = s j,k=1 q ǫ q q t + p ǫ p p t ,
where q = [q 1 , . . . , q s ], p = [p 1 , . . . , p s ], the energy constraint is
Sp ǫ q α q + Sp ǫ p α p ≤ E.(25)
Then the maximization in (23) can be taken over only block-diagonal covariance matrices α = α q 0 0 α p . The argument relying upon concavity of the capacity as the function of the average state of the ensemble [27], is similar to one given in sec. 4 of [21] for the case of entanglement-assisted capacity.
The classical capacity of approximate position measurement
The approximate (unsharp) measurement of position q in one mode q, p (a mathematical model for noisy homodyning) is given by POVM M(dy) = m(y)dy, where
m(y) = 1 √ 2πβ exp − (q − y) 2 2β = 1 √ 2πβ D(y) exp − q 2 2β D(y) * ,(26)
where β > 0 is the power of the Gaussian noise, D(y) = exp (−iyp). The Gaussian measurement channel given by this POVM acts on a centered Gaussian state ρ α with the covariance matrix α = α q 0 0 α p by the formula
M : ρ α → exp − y 2 2 (α q + β) dy 2π (α q + β) .(27)
Take the oscillator Hamiltonian H = 1 2 (q 2 + p 2 ) . The problem is to compute the classical capacity
and the maximization is over the input ensembles of states E (encodings) satisfying the energy constraint Trρ E H ≤ E. Remark 1 above shows that we can restrict to ensembles with average state having the diagonal matrix α as above.
In other words, one makes the "classical" measurement of the observable
Y = q + ξ, ξ ∼ N (0, β),
with the quantum energy constraint Tr ρ(q 2 + p 2 ) ≤ 2E, aiming to transmit the maximum information. The difficulty is that one measures q, while imposing the constraint on the energy H = 1 2 (q 2 + p 2 ) , involving an implicit constraint on p which does not commute with q.
As we have mentioned, there is no general "Gaussian maximizer" result for C in such cases, therefore we will first find the maximum over (special) Gaussian ensembles. The final goal will be to prove the HGM showing thus that the found Gaussian ensemble is in fact a solution of the capacity problem (28) by checking the optimality conditions from sec. 3.
HGM for approximate position measurement [16]: the maximum is attained on the Gaussian ensemble E gauss = {π 0 (dx), ρ 0 (x)} where ρ 0 (x) = |x δ x| is the pure Gaussian (squeezed) state with the vector |x δ = D(x) |0 δ , and the squeezed vacuum has zero mean and the following second moments:
δ 0| q 2 |0 δ = δ, Re δ 0| qp |0 δ = 0, δ 0| p 2 |0 δ = 1 4δ
.
The distribution π 0 (dx) = 1 √ 2πγ exp − x 2 2γ dx.
For the fixed centered Gaussian state ρ α with the covariance matrix α = α q 0 0 α p , in order that the average state of the ensemble to be ρ(x)π 0 (dx) = ρ α the parameters should
satisfy 1 4δ = α p , δ + γ = α q , whence δ = 1 4α p , γ = α q − 1 4α p .(29)
This ensemble encodes the information solely into the displacement x of the position leaving the momentum intact. Similarly to (27)
p ρ 0 (x) (y) = 1 2π (β + δ) exp − (y − x) 2 2 (β + δ) .(30)
Using this and (27) we get the "Gaussian" values
h M (ρ α ) = 1 2 ln (α q + β) + 1 2 ln(2πe),(31)e M (ρ α ) = 1 2 ln 1 4α p + β + 1 2 ln(2πe),(32)
hence taking into account (23),
C gauss (M; α) = h M (ρ α ) − e M (ρ α ) = 1 2 ln α q + β 1 4αp + β .(33)
The Gaussian constrained capacity is thus
C gauss (M, H, E) = max αq+αq≤2E 1 2 [ln (α q + β) − ln (1/ (4α p ) + β)](34)
= max
αp 1 2 [ln (2E − α p + β) − ln (1/ (4α p ) + β)] ,
where in the second line we took the maximal value α q = 2E − α p . Differentiating, we obtain the equation for the optimal value α p :
4βα 2 p + 2α p − (2E + β) = 0,
the positive solution of which is
α p = 1 4β 1 + 8Eβ + 4β 2 − 1 ,(35)
whence
C gauss (M, H, E) = ln 1 + 8Eβ + 4β 2 − 1 2β .(36)
The parameters of the optimal Gaussian ensemble are obtained by substituting the value (35) into (29) with α q = 2E − α p 3 . The case of sharp position measurement (β = 0) formally corresponding to M(dy) = δ(q −y)dy, is not included in the discussion above. Yet for β = 0 the formula (36) gives δ = 1/4E and C(M, H, E) = C gaus (M, H, E) = ln 2E.
The last formula was obtained in the paper [6] where also a general upper bound
ln 1 + E − 1/2 β + 1/2 = ln 2(E + β) 1 + 2β(37)
for (28) was given (Eq. (28) in [6], see also Eq. (5.39) in [1]) .
Checking the optimality condition
Starting from this section it will be convenient to use natural logarithms; then one can return to the binary logarithms if necessary. Theorem 2. The Gaussian encoding E gauss is optimal for the approximate Gaussian position measurement M and the oscillator energy constraint as desribed in previous section. Its constrained capacity C(M, H, E) is equal to (36).
Proof. We use the method of sec. 3 and check the optimality conditions (i), (ii) for the Gaussian encoding E gauss = {π 0 (dx), ρ 0 (x)} . We start with computation of Λ 0 for E gauss .
By (6), (26), (30),
K(ρ 0 (x)) = 1 √ 2πβ exp − (q − y) 2 2β ln 2π (β + δ) + (y − x) 2 2 (β + δ) dy = c + (q − x) 2 + β 2 (β + δ)
, c = ln 2π (β + δ).
Hence
K(ρ 0 (x))ρ 0 (x) = c + (q − x) 2 + β 2 (β + δ) D(x) |0 δ x| = D(x) c + q 2 + β 2 (β + δ) |0 δ x| = D(x) c + β 2 (β + δ) + δ (β + δ) q 2 2δ + 2δp 2 − 2δ 2 p 2 (β + δ) |0 δ x| .
Taking into account that the squeezed vacuum |0 δ is the ground state for the corresponding oscillator Hamiltonian
q 2 2δ + 2δp 2 |0 δ = |0 δ ,
and that D(x) commute with p 2 , we have
K(ρ 0 (x))ρ 0 (x) = c + β + 2δ 2 (β + δ) − 2δ 2 p 2 (β + δ) |x δ x| .
Integrating over π 0 (dx), and taking into account that |x δ x| π 0 (dx) = ρ α , we obtain
K(ρ 0 (x))ρ 0 (x)π 0 (dx) = c + β + 2δ 2 (β + δ) − 2δ 2 p 2 (β + δ) ρ α .
Comparing with (9), we obtain
Λ 0 = c + β + 2δ 2 (β + δ) − 2δ 2 p 2 (β + δ) .(38)
By condtruction, this is Hermitian operator satisfying [K(ρ 0 (x)) − Λ 0 ] ρ 0 (x) = 0, i.e. the condition (ii) of section 3.
To check the condition (i) it is sufficient to prove
ψ| Λ 0 |ψ ≤ ψ| K(ρ) |ψ(39)
for arbitrary density operator ρ and a dense subset of ψ ∈ H. We can assume that ψ is a unit vector. Since for unit vectors ψ. With Λ 0 given by (38) it amounts to ψ| m(y) |ψ ln ψ| m(y) |ψ dy + ln 2π (β + δ) + β + 2δ 2 (β + δ)
≤ 2δ 2 (β + δ) |ψ′(x)| 2 dx.(41)
The proof of this inequality is the subject of the following section.
A generalization of log-Sobolev inequality
Let f (x) = |ψ(x)| 2 be a smooth probability density on R and
T t f (y) = 1 √ 2πt exp − (y − x) 2 2t f (x)dx.
Then the inequality we wish to prove, replacing β by t:
T t f (y) ln T t f (y)dy + ln 2πe (t + δ) + δ 2 (t + δ) ≤ 2δ 2 (t + δ) |ψ′(x)| 2 dx(42)
for t, δ ≥ 0. For t = 0, δ = 1 this is the logarithmic Sobolev inequality [4]. For t = 0, δ > 0 it can be obtained by a change of variable (see also (44) below).
Proof of (42). We start from the version of the log-Sobolev inequality in [23] (with dimensionality n = 1):
|ψ(x)| 2 ln |ψ(x)| 2 ψ 2 2 dx + ln a + 1 ≤ a 2 π |ψ′(x)| 2 dx.(43)
Let ψ 2 = 1 then f (x) = |ψ(x)| 2 is a probability density, f (x)dx = 1. Also take a = √ 2πδ, then (43) becomes
f (x) ln f (x)dx + ln √ 2πδ + 1 ≤ 2δ |ψ′(x)| 2 dx(44)
which is the same as (42) for t = 0. Denote
F (t, δ) = (t + δ) T t f (x) ln T t f (x)dx + (t + δ) ln 2πe (t + δ) + δ 2 − 2δ 2 |ψ′(x)| 2 dx.
We have to prove F (t, δ) ≤ 0; t, δ > 0.
We have just proved that F (0, δ) ≤ 0. If we prove that ∂ ∂t F (t, δ) ≤ 0, then (45) and hence (42) will follow. We have
∂ ∂t F (t, δ) = T t f (x) ln T t f (x)dx + (t + δ) [ln T t f (x) + 1] ∂ ∂t T t f (x) dx + ln 2πe (t + δ) + 1 2 .
Taking into account that ∂ ∂t T t f (x) = 1 2 ∂ 2 ∂x 2 T t f (x) and integrating by parts in the second integral, we can transform it as
[ln T t f (x) + 1] ∂ ∂t T t f (x) dx = − 1 2 ∂ ∂x [ln T t f (x) + 1] ∂ ∂x T t f (x) dx = −2 ∂ ∂x T t f (x) 2 dx.
Denote g(x) = T t f (x), then it is also a probability density, and denoting t + δ =δ we obtain
∂ ∂t F (t, δ) = g(x) ln g(x)dx + ln 2πδ + 1 − 2δ d dx g(x) 2 dx.
Hovewer by (44) this is nonpositive. Thus (45) and hence (42) follows. This also completes the proof of theorem 2.
8 Comment on the proof of the estimate for the convex closure of the output entropy
The sufficient conditions for optimality from section 3 were applied in our proof of theorem 6 to unbounded operators and thus require a corresponding refinement. While this can be done in general, here we wish point out that given the inequality (41), there is another, direct way to rigorous proof of theorem 6. Then the merit of the convex programming approach is in that it allowed to generate the conjectured inequality (41). First, we note that (41) can be extended to functions ψ from the Sobolev space H 1 (R), which are square-integrable along with its first generalized derivative ψ ′ . This space is the natural domain of definition of the momentum operator p.
To complete the proof of theorem 6, in view of (22) and (31), we have only to prove that e M (ρ α ) ≡ inf E:ρ E =ρα h M (ρ(ξ))π(dξ) = 1 2 ln 1 4α p + β + 1 2 ln 2πe,
where the infimum is taken over encodings E = {π(dξ), ρ(ξ)} satisfyingρ E = ρ α . The concavity of h M (ρ) implies that we can restrict to ensembles of pure states ρ(ξ) = |ψ ξ ψ ξ | , so that |ψ ξ ψ ξ | π(dξ) = ρ α , since we can always perform the convex decomposition for all density operators ρ(ξ) into pure states without changing the barycenter and without increasing the value of the minimized functional. It follows that pψ ξ 2 π(dξ) = Tr ρ α p 2 < ∞,
hence ψ ξ ∈ H 1 (R) for π−almost all ξ. Applying the inequality (41) to ψ ξ and rearranging terms, we get h M (|ψ ξ ψ ξ | ) ≥ ln 2π (β + δ) + β + 2δ 2 (β + δ) − 2δ 2 β + δ pψ ξ 2 .
Integrating with respect to π(dξ) and taking into account (47) we get h M (ρ(ξ))π(dξ) ≥ ln 2π (β + δ) + β + 2δ 2 (β + δ) − 2δ 2 β + δ α p .
With δ = 1 4αp we get the value at the right-hanf side of (46), which is attained for the encoding of theorem 6. This proves (46) and hence the theorem.
Similar comment applies to the proof of HGM for approximate joint position-momentum measurement channel (noisy heterodyning) in our subsequent e-print arXiv:2206.02133. which is ≥ 0 (≤ 0) if u ≥ v (u ≤ v) . Hence (50) follows. Also we have obtained that (42) is exact: it turns into equality for Gaussian (49) with a = δ.
M (ρ α ) − e M (ρ α )] .
Figure 1 :
1(color online) The Gaussian classical capacity (36) and the upper bound (37) (β = 1).
ψ| K(ρ) |ψ = − ψ| m(y) |ψ ln Trρm(y) = − ψ| m(y) |ψ ln ψ| m(y) |ψ + ψ| m(y) |ψ ln ψ| m(y) |ψ Trρm(y) ≥ − ψ| m(y) |ψ ln ψ| m(y) |ψ , due to nonnegativity of the relative entropy of the two probability densities. Thus (39) will follow if we prove ψ| Λ 0 |ψ ≤ − ψ| m(y) |ψ ln ψ| m(y) |ψ dy (40)
See e.g.[13],[14] for the detail of mathematical treatment of expressions with the unbounded operators related to R.
We denote Sp trace of s × s-matrices as distinct from trace of operators on H.
Notably, the expression (36) is of the same type as the one obtained in[5] by optimizing the information from applying sharp position measurement to noisy optimally squeezed states (the author is indebted to M. J. W. Hall for this observation).
AppendixLet us illustrate the inequality (42) for GaussiansThen (42) reduces toTo prove this inequality, notice it becomes equality for u = v. The derivative d/du is
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| [] |
[
"Theory of one-dimensional double-barrier quantum pump in two-frequency signal regime",
"Theory of one-dimensional double-barrier quantum pump in two-frequency signal regime"
] | [
"M M Mahmoodian \nInstitute of Semiconductor Physics\nSiberian Division\nRussian Academy of Sciences\n630090NovosibirskRussia\n",
"M V Entin \nInstitute of Semiconductor Physics\nSiberian Division\nRussian Academy of Sciences\n630090NovosibirskRussia\n"
] | [
"Institute of Semiconductor Physics\nSiberian Division\nRussian Academy of Sciences\n630090NovosibirskRussia",
"Institute of Semiconductor Physics\nSiberian Division\nRussian Academy of Sciences\n630090NovosibirskRussia"
] | [] | A one-dimensional system with two δ-like barriers or wells bi-chromaticaly oscillating at frequencies ω and 2ω is considered. The alternating signal leads to the direct current across the structure (even in a symmetric system). The properties of this quantum pump are studied in a wide range of the system parameters. PACS numbers: 73.50.Pz, 85.35.Be The quantum pump is a device that generates the stationary current under action of alternating voltage; it is a subject of numerous recent publications (for example,1,2,3,4,5,6,7,8,9,10. The quantum pump is essentially analogous to various versions of the photovoltaic effect studied in details from the beginning of the eighties11,12,13,14,15. The difference is that the photovoltaic effect is related to the emergence of a direct current in a homogeneous macroscopic medium (the only exception is the mesoscopic photovoltaic effect), while the pump is a microscopic object. From the phenomenological point of view, the emergence of a direct current in the pump is not surprising, since any asymmetric microcontact can rectify ac voltage. However, analysis of adiabatic transport in the quantum-mechanical object leads to new phenomena, such as quantization of charge transport 3 . Just this, analytically solvable, adiabatic approach was utilized in the most of studies of quantum pumps 4,5,6 . In the recent papers we have carried out the extensive study of the simplest model of the one-dimensional quantum pump, containing two delta-like harmonically oscillating barriers/wells. This model demonstrates rich behavior which is ruled by a variety of system parameters. The present paper deals with similar system to which alternating bi-chromatic voltages are applied. The system can be exemplified by a quantum wire with two narrow gates. The stationary bias between the source and the drain is supposedly absent.The system has a variety of regimes of the pump operation, depending on the system parameters, e.g. frequency and amplitudes. The effect is sensitive to the phase coherence of alternating signals and can exist even in symmetric systems. The stationary current is possible also in the case of different amplitudes of alternating fields. We have studied the system both analytically and numerically. The analytical approach is based on the perturbational (with respect to amplitudes (u ij ) of a.c. signal) consideration. The current contains independent contributions caused by u ij and an interference term. The elastic, absorption and emission channels participate in the process. The case of strong alternating signal was studied numerically.We mostly concentrate on the case of symmetric system as more interesting by its phase sensitivity.Basic EquationsThe considered model is described by the onedimensional time-dependent potential:U (x) = (u 11 sin ωt + u 12 sin 2ωt)δ(x + d) + (u 21 sin ωt + u 22 sin 2ωt)δ(x − d).(1)where t is the time, 2d is the distance between δ-barriers (wells); quantities u ij are measured in units ofh/md (m is the electron mass); p, E, and ω are the momentum, energy, and frequency measured in units ofh/d,h 2 /2md 2 , andh/2md 2 , respectively. The solution to the Schrödinger equation with the potential (1) is searched in the formHere, p n = p 2 + nω and p = √ E. The wave function (2) corresponds to the wave incident on the barrier from the left. (In the final formulas, we mark the directions of incident waves by the indices "→" and "←"). The form of solution (2) corresponds to absorption (for n > 0) or emission (n < 0) of n field quanta by an electron after the elastic process. Quantities t n and r n give the corresponding amplitudes of transmission (reflection). If the value of p n becomes imaginary, the waves moving away from the barriers should be treated as damped waves, so that Imp n > 0.The transmission amplitudes obey the equations: t n = | 10.1209/0295-5075/77/67002 | [
"https://arxiv.org/pdf/cond-mat/0610832v1.pdf"
] | 119,090,557 | cond-mat/0610832 | 45c387a4bfdb55ad6d47b4478584f3f2274ea46d |
Theory of one-dimensional double-barrier quantum pump in two-frequency signal regime
30 Oct 2006
M M Mahmoodian
Institute of Semiconductor Physics
Siberian Division
Russian Academy of Sciences
630090NovosibirskRussia
M V Entin
Institute of Semiconductor Physics
Siberian Division
Russian Academy of Sciences
630090NovosibirskRussia
Theory of one-dimensional double-barrier quantum pump in two-frequency signal regime
30 Oct 2006
A one-dimensional system with two δ-like barriers or wells bi-chromaticaly oscillating at frequencies ω and 2ω is considered. The alternating signal leads to the direct current across the structure (even in a symmetric system). The properties of this quantum pump are studied in a wide range of the system parameters. PACS numbers: 73.50.Pz, 85.35.Be The quantum pump is a device that generates the stationary current under action of alternating voltage; it is a subject of numerous recent publications (for example,1,2,3,4,5,6,7,8,9,10. The quantum pump is essentially analogous to various versions of the photovoltaic effect studied in details from the beginning of the eighties11,12,13,14,15. The difference is that the photovoltaic effect is related to the emergence of a direct current in a homogeneous macroscopic medium (the only exception is the mesoscopic photovoltaic effect), while the pump is a microscopic object. From the phenomenological point of view, the emergence of a direct current in the pump is not surprising, since any asymmetric microcontact can rectify ac voltage. However, analysis of adiabatic transport in the quantum-mechanical object leads to new phenomena, such as quantization of charge transport 3 . Just this, analytically solvable, adiabatic approach was utilized in the most of studies of quantum pumps 4,5,6 . In the recent papers we have carried out the extensive study of the simplest model of the one-dimensional quantum pump, containing two delta-like harmonically oscillating barriers/wells. This model demonstrates rich behavior which is ruled by a variety of system parameters. The present paper deals with similar system to which alternating bi-chromatic voltages are applied. The system can be exemplified by a quantum wire with two narrow gates. The stationary bias between the source and the drain is supposedly absent.The system has a variety of regimes of the pump operation, depending on the system parameters, e.g. frequency and amplitudes. The effect is sensitive to the phase coherence of alternating signals and can exist even in symmetric systems. The stationary current is possible also in the case of different amplitudes of alternating fields. We have studied the system both analytically and numerically. The analytical approach is based on the perturbational (with respect to amplitudes (u ij ) of a.c. signal) consideration. The current contains independent contributions caused by u ij and an interference term. The elastic, absorption and emission channels participate in the process. The case of strong alternating signal was studied numerically.We mostly concentrate on the case of symmetric system as more interesting by its phase sensitivity.Basic EquationsThe considered model is described by the onedimensional time-dependent potential:U (x) = (u 11 sin ωt + u 12 sin 2ωt)δ(x + d) + (u 21 sin ωt + u 22 sin 2ωt)δ(x − d).(1)where t is the time, 2d is the distance between δ-barriers (wells); quantities u ij are measured in units ofh/md (m is the electron mass); p, E, and ω are the momentum, energy, and frequency measured in units ofh/d,h 2 /2md 2 , andh/2md 2 , respectively. The solution to the Schrödinger equation with the potential (1) is searched in the formHere, p n = p 2 + nω and p = √ E. The wave function (2) corresponds to the wave incident on the barrier from the left. (In the final formulas, we mark the directions of incident waves by the indices "→" and "←"). The form of solution (2) corresponds to absorption (for n > 0) or emission (n < 0) of n field quanta by an electron after the elastic process. Quantities t n and r n give the corresponding amplitudes of transmission (reflection). If the value of p n becomes imaginary, the waves moving away from the barriers should be treated as damped waves, so that Imp n > 0.The transmission amplitudes obey the equations: t n =
A one-dimensional system with two δ-like barriers or wells bi-chromaticaly oscillating at frequencies ω and 2ω is considered. The alternating signal leads to the direct current across the structure (even in a symmetric system). The properties of this quantum pump are studied in a wide range of the system parameters. The quantum pump is a device that generates the stationary current under action of alternating voltage; it is a subject of numerous recent publications (for example, 1,2,3,4,5,6,7,8,9,10 . The quantum pump is essentially analogous to various versions of the photovoltaic effect studied in details from the beginning of the eighties 11,12,13,14,15 . The difference is that the photovoltaic effect is related to the emergence of a direct current in a homogeneous macroscopic medium (the only exception is the mesoscopic photovoltaic effect), while the pump is a microscopic object. From the phenomenological point of view, the emergence of a direct current in the pump is not surprising, since any asymmetric microcontact can rectify ac voltage. However, analysis of adiabatic transport in the quantum-mechanical object leads to new phenomena, such as quantization of charge transport 3 . Just this, analytically solvable, adiabatic approach was utilized in the most of studies of quantum pumps 4,5,6 . In the recent papers we have carried out the extensive study of the simplest model of the one-dimensional quantum pump, containing two delta-like harmonically oscillating barriers/wells. This model demonstrates rich behavior which is ruled by a variety of system parameters. The present paper deals with similar system to which alternating bi-chromatic voltages are applied. The system can be exemplified by a quantum wire with two narrow gates. The stationary bias between the source and the drain is supposedly absent.
The system has a variety of regimes of the pump operation, depending on the system parameters, e.g. frequency and amplitudes. The effect is sensitive to the phase coherence of alternating signals and can exist even in symmetric systems. The stationary current is possible also in the case of different amplitudes of alternating fields. We have studied the system both analytically and numerically. The analytical approach is based on the perturbational (with respect to amplitudes (u ij ) of a.c. signal) consideration. The current contains independent contributions caused by u ij and an interference term. The elastic, absorption and emission channels participate in the process. The case of strong alternating signal was studied numerically.
We mostly concentrate on the case of symmetric system as more interesting by its phase sensitivity.
Basic Equations
The considered model is described by the onedimensional time-dependent potential:
U (x) = (u 11 sin ωt + u 12 sin 2ωt)δ(x + d) + (u 21 sin ωt + u 22 sin 2ωt)δ(x − d).
where t is the time, 2d is the distance between δ-barriers (wells); quantities u ij are measured in units ofh/md (m is the electron mass); p, E, and ω are the momentum, energy, and frequency measured in units ofh/d,h 2 /2md 2 , andh/2md 2 , respectively. The solution to the Schrödinger equation with the potential (1) is searched in the form
ψ = n exp [−i(E + nω)t] δ n,0 exp ipnx d + r n exp − ipnx d , x < −d, a n exp ipnx d + b n exp − ipnx d , −d < x < d, t n exp ipnx d , x > d.(2)
Here, p n = p 2 + nω and p = √ E. The wave function (2) corresponds to the wave incident on the barrier from the left. (In the final formulas, we mark the directions of incident waves by the indices "→" and "←"). The form of solution (2) corresponds to absorption (for n > 0) or emission (n < 0) of n field quanta by an electron after the elastic process. Quantities t n and r n give the corresponding amplitudes of transmission (reflection). If the value of p n becomes imaginary, the waves moving away from the barriers should be treated as damped waves, so that Imp n > 0.
The transmission amplitudes obey the equations:
t n = u 12 u 22 g n−2 u 12 u 21 g n−2 + u 11 u 22 g n−1 −i u 12 e −2ipn−2 + u 11 u 21 g n−1 − i u 22 e −2ipn −u 12 u 21 g n−2 − i u 11 e −2ipn−1 − i u 21 e −2ipn − u 11 u 22 g n+1 −u 12 u 22 (g n−2 + g n+2 ) − u 11 u 21 (g n−1 + g n+1 ) + 2ip n e −2ipn −u 12 u 21 g n+2 + i u 11 e −2ipn+1 + i u 21 e −2ipn − u 11 u 22 g n−1 i u 12 e −2ipn+2 + u 11 u 21 g n+1 + i u 22 e −2ipn u 12 u 21 g n+2 + u 11 u 22 g n+1 u 12 u 22 g n+2 • T → n−4 T → n−3 T → n−2 T → n−1 T → n T → n+1 T → n+2 T → n+3 T → n+4 = 2ipe −ip δ n,0 ,(3)
and
u 12 u 22 g n−2 u 11 u 22 g n−2 + u 12 u 21 g n−1 −i u 22 e −2ipn−2 + u 11 u 21 g n−1 − i u 12 e −2ipn −u 11 u 22 g n−2 − i u 21 e −2ipn−1 − i u 11 e −2ipn − u 12 u 21 g n+1 −u 12 u 22 (g n−2 + g n+2 ) − u 11 u 21 (g n−1 + g n+1 ) + 2ip n e −2ipn −u 11 u 22 g n+2 + i u 21 e −2ipn+1 + i u 11 e −2ipn − u 12 u 21 g n−1 i u 22 e −2ipn+2 + u 11 u 21 g n+1 + i u 12 e −2ipn u 11 u 22 g n+2 + u 12 u 21 g n+1 u 12 u 22 g n+2 • T ← n−4 T ← n−3 T ← n−2 T ← n−1 T → n T ← n+1 T ← n+2 T ← n+3 T ← n+4 = 2ipe −ip δ n,0 .(4)
Here, g n = sin 2p n /p n . Provided that electrons from the right and left of the pump are in equilibrium, and that they have identical chemical potentials µ, the stationary current is
J = e πh dE n (|T → n | 2 − |T ← n | 2 )f (E)θ(E + nω),(5)
where f (E) is the Fermi distribution function, and θ(x) is the Heaviside step function. The current is determined by the transmission coefficients with real p n only. At a low temperature, it is convenient to differentiate the current with respect to the chemical potential:
G = e ∂ ∂µ J = G 0 n θ(µ + nω)(|T → n | 2 − |T ← n | 2 ) p=pF .(6)
Here G 0 = e 2 /πh is the conductance quantum and p F is the Fermi momentum. The resultant quantity G can be treated as a two-terminal photoconductance (the conductance for simultaneous change of chemical potentials of source and drain).
Perturbations theory
In low-amplitude limit the stationary current (its derivative G) is proportional to
G ∝ α 1 u 11 u 21 + α 2 u 12 u 22 + α 3 u 2 11 u 22 + α 4 u 2 21 u 12 .
Let's consider the case of symmetric system with u 11 = −u 21 = u, u 12 = −u 22 = v. The systems symmetry leads 2 from corresponding terms in the transmission coefficients T 0 , T ±1 , T ±2 :
T 0 ∝ 1 + A 0 u 2 v, T ±1 ∝ A ±1 u + B ±1 uv,(7)T ±2 ∝ A ±2 u 2 + B ±2 v.
The quantities A 0 , A ±1 , B ±1 , A ±2 , B ±2 depend on g ±1 , g ±2 which contains one-and two-photon singularities. The calculations support this dependence.
Numerical results
The figures 1-5 show typical plots of the derivative of the stationary current with respect to the Fermi energy ∂J/∂E F = G × 2e 2 /h as a function of the Fermi momentum (Figs. 1-3), the amplitude (Fig. 4) and the frequency (Fig. 5).
The figures 1-3 present the quantity G as a function of the Fermi momentum for u 21 = u 22 = −1, ω = 2 and u 11 = u 12 = 1 (Fig. 1), 2 (Fig. 2). The peaks of the curves correspond to the multi-photon threshold resonances with zero energy state. The increasing of u 11 = u 12 leads to the strengthening of multi-photon singularities.
The curve in the Fig. 1 corresponds to antipodal signals with same amplitudes (u 12 = −u 22 ), that demonstrate the importance of the phase coherency of the signals (in such conditions the stationary current in the same system under monochromatic voltage is vanishing 9 ). The current oscillates with the Fermi momentum due to the interference of the electron waves in the structure. Be-resonances with the zero energy state and their photon repetitions. Here u11 = u12 = 1, u21 = u22 = −1, ω = 2. The left deltafunction oscillates from the barrier to zero hight, the right delta-function corresponds to well whose depth changes from zero value The figure 3 shows the behavior of G for large enough frequency where the only threshold singularity exists in the concerned range of Fermi momentum values.
The figure 4 depicts G versus u 11 for the symmetric system (u 11 = u 12 = −u 21 = −u 22 ). These values are chosen so that there should be no current caused by harmonic signals only (if u 11 = −u 21 = 0, u 12 = u 22 = 0 or u 11 = u 21 = 0, u 12 = −u 22 = 0).
Besides, the current vanishes in the adiabatic approximation which is commonly used for consideration of quantum pumps. In the adiabatic regime the charge transfer per a cycle is proportional to the area covered by two parameters in the phase space. In our case the parameters are {u 1 (t), u 2 (t)}. If u 11 = u 12 = −u 21 = −u 22 the trajectory in the phase space {u 1 (t), u 2 (t)} is a straight line and covers zero area (see insert). Hence in the adiabatic (ω → 0) approximation G/ω → 0.
At low u ij G ∝ u 2 ω u 2ω . This behavior corresponds to coherent photovoltaic effect 16 . At large u ij the ampli- are determined by the commensurability of the characteristic wavelength of excited electrons with the distance between delta-functions. The curves for different frequencies demonstrate splitting and beating of oscillations. Their amplitude decays with u ij . The oscillations period changes with the frequency.
The figure 5 demonstrates the dependence of G on the frequency. The complicated structure of G in the low frequency region is explained by multi-photon resonances reducing for larger frequencies.
Conclusions
The alternating voltage produces the stationary current by pumping electrons between leads x < −d and x > d. The effect is sensitive to the phase coherence of alternating signals and can exist even in symmetric systems. The stationary current is possible also in the case of different amplitudes of alternating fields.
We have studied the system both analytically and numerically. The analytical approach is based on the per- consideration. The current contains independent contributions caused by u ij and an interference term. The elastic, absorption and emission channels participate in the process. The case of strong alternating signal was studied numerically.
Our calculations show that the stationary current is a sophisticated function of the parameters, that reflects the interference effects, presence of virtual states, and threshold singularities.
The motivation to consider the symmetric case is that in this case the stationary current caused by a single frequency is suppressed. Only two intercoherent frequencies can cause the stationary current. Hence, the system with specially symmetrized barriers can be used for heterodyning of signals with ω and 2ω frequencies and for frequency binding of similar laser sources. The controllability of the system permits to realize the symmetric conditions intentionally with good accuracy.
PACS numbers: 73.50.Pz, 73.23.-b, 85.35.Be
FIG. 1 :
1The derivative of the G versus the Fermi momentum.
FIG. 2 :
2The dependence of G on the Fermi momentum. We set u11 = u12 = 2, u21 = u22 = −1, ω = 2.
FIG. 3 :FIG. 4 :
34G versus the Fermi momentum; u11 = u12 = 2, u21 = u22 = −1, ω = 50 and different u11 = u12. The singularity at pF = 7.1 corresponds to the single-photon threshold pF = The dependence of G on the amplitudes u11 = u12 = −u21 = −u22 for pF = 0.7 and different signal frequencies.For uij → 0, G ∝ u3 11 . Insert: trajectory in the phase space {u1(t), u2(t)}. The enclosed area in the phase space is zero.
FIG. 5 :
5The dependence of G on the frequency for u11 = u12 = 1, u21 = u22 = −1, pF = 0.7.
* Electronic address: [email protected] † Electronic address: [email protected]. * Electronic address: [email protected] † Electronic address: [email protected]
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| [] |
[
"Scalable surface area characterisation by electrokinetic analysis of complex anion adsorption",
"Scalable surface area characterisation by electrokinetic analysis of complex anion adsorption"
] | [
"D A H Hanaor ",
"M Ghadiri ",
"W Chrzanowski ",
"& Gan ",
"Dorian A H Hanaor \nSchool of Civil Engineering\nUniversity of Sydney\n2006NSWAustralia\n",
"Maliheh Ghadiri \nFaculty of Pharmacy\nUniversity of Sydney\n2006NSWAustralia\n",
"Wojciech Chrzanowski \nFaculty of Pharmacy\nUniversity of Sydney\n2006NSWAustralia\n",
"Yixiang Gan \nSchool of Civil Engineering\nUniversity of Sydney\n2006NSWAustralia\n",
"D A H Hanaor ",
"M Ghadiri ",
"W Chrzanowski ",
"& Gan "
] | [
"School of Civil Engineering\nUniversity of Sydney\n2006NSWAustralia",
"Faculty of Pharmacy\nUniversity of Sydney\n2006NSWAustralia",
"Faculty of Pharmacy\nUniversity of Sydney\n2006NSWAustralia",
"School of Civil Engineering\nUniversity of Sydney\n2006NSWAustralia"
] | [] | 2014). Scalable surface area characterization by electrokinetic analysis of complex anion adsorption. Langmuir, 30(50), 15143-15152.Abstract:By means of in-situ electrokinetic assessment of aqueous particles in conjunction with the addition of anionic adsorbates, we develop and examine a new approach for the scalable characterisation of specific accessible surface area of particles in water. For alumina powders of differing morphology in mildly acidic aqueous suspensions, effective surface charge was modified by carboxylate anion adsorption through the incremental addition of oxalic and citric acids. The observed zeta potential variation as a function of proportional reagent additive was found to exhibit the inverse hyperbolic sine type behaviour predicted to arise from monolayer adsorption following the Grahame-Langmuir model. Through parameter optimisation by reverse problem solving, the zeta potential shift with relative adsorbate addition revealed a near-linear correlation of a defined surface-area-dependent parameter with the conventionally measured surface area values of the powders, demonstrating that the proposed analytical framework is applicable for the in-situ surface area characterisation of aqueous particulate matter. The investigated methods have advantages over some conventional surface analysis techniques owing to their direct applicability in aqueous environments at ambient temperatures and the ability to modify analysis scales by variation of adsorption cross-section. . (2014). Scalable surface area characterization by electrokinetic analysis of complex anion adsorption.Langmuir, 30(50), 15143-15152. | 10.1021/la503581e | [
"https://arxiv.org/pdf/2106.03411v1.pdf"
] | 4,697,498 | 2106.03411 | abb6339da8ad5d35e1ade591cd40cf1c0bbb6210 |
Scalable surface area characterisation by electrokinetic analysis of complex anion adsorption
D A H Hanaor
M Ghadiri
W Chrzanowski
& Gan
Dorian A H Hanaor
School of Civil Engineering
University of Sydney
2006NSWAustralia
Maliheh Ghadiri
Faculty of Pharmacy
University of Sydney
2006NSWAustralia
Wojciech Chrzanowski
Faculty of Pharmacy
University of Sydney
2006NSWAustralia
Yixiang Gan
School of Civil Engineering
University of Sydney
2006NSWAustralia
D A H Hanaor
M Ghadiri
W Chrzanowski
& Gan
Scalable surface area characterisation by electrokinetic analysis of complex anion adsorption
10.1021/la503581e1 2Surface areaelectrophoresisadsorptionzeta potential
2014). Scalable surface area characterization by electrokinetic analysis of complex anion adsorption. Langmuir, 30(50), 15143-15152.Abstract:By means of in-situ electrokinetic assessment of aqueous particles in conjunction with the addition of anionic adsorbates, we develop and examine a new approach for the scalable characterisation of specific accessible surface area of particles in water. For alumina powders of differing morphology in mildly acidic aqueous suspensions, effective surface charge was modified by carboxylate anion adsorption through the incremental addition of oxalic and citric acids. The observed zeta potential variation as a function of proportional reagent additive was found to exhibit the inverse hyperbolic sine type behaviour predicted to arise from monolayer adsorption following the Grahame-Langmuir model. Through parameter optimisation by reverse problem solving, the zeta potential shift with relative adsorbate addition revealed a near-linear correlation of a defined surface-area-dependent parameter with the conventionally measured surface area values of the powders, demonstrating that the proposed analytical framework is applicable for the in-situ surface area characterisation of aqueous particulate matter. The investigated methods have advantages over some conventional surface analysis techniques owing to their direct applicability in aqueous environments at ambient temperatures and the ability to modify analysis scales by variation of adsorption cross-section. . (2014). Scalable surface area characterization by electrokinetic analysis of complex anion adsorption.Langmuir, 30(50), 15143-15152.
Introduction:
The ability to meaningfully characterise particle interface structures in aqueous media is of importance in a range of high value industrial processes and applications including catalysis 1 , pharmaceutics 2 , water treatment 3 and across the broader field of chemical engineering.
In particulate material, as for natural surfaces in general, a unique value for specific surface area, in terms of area per unit mass, cannot be categorically defined. In similarity to the wellknown coastline paradox, this results from the scale variance of surface structures [4][5] . Thus over the past decades for the assessment of surface driven material functionality we commonly speak of the gas-accessible surface area, most frequently measured by N2 adsorption in conjunction with BET or Langmuir type isotherm interpretation for multi-layer or mono-layer gas adsorption [6][7][8][9] . While improved sensitivity and accuracy is achievable through the adsorption of heavier gases (Krypton or Argon), such methods nonetheless suffer from known limitations with respect to measurement scale and conditions [10][11] .
Conventional methods, such as BET, have the clear advantage of facilitating standardised comparative analyses using purpose-built commercially available analytical apparatus. However, there are known drawbacks to the use of such tools. Specifically gas adsorption methods are limited with respect to: (i) measurement scale -N2/O2/Kr/Ar exhibit molecular adsorption cross-sections in the range 0.14-0.23 nm 2 12 (ii) measurement temperature -BET analysis is most commonly undertaken at cryogenic temperatures (e.g. 77K for N2) (iii) measurement environmentsuch methods are generally applied to dry powder. The aforementioned scale variance of surface structures means that the assessment of surface area at a constant measurement resolution is problematic. Furthermore, in many applications including catalysis, environmental remediation and in chemical engineering in general, particulate materials are applied in an aqueous environment at ambient or high temperatures, thus motivating the adoption of surface area characterisation tools that can be applied in analogous conditions, with the aim of conducting target application relevant interface characterisation.
Among alternative adsorption based surface area analysis methods put forward over recent decades aqueous and organic suspension based methods feature prominently. Typically the analysis of specific surface area by adsorption in liquid media involves the selection of a material-and application-appropriate adsorbate compound, or 'molecular probe', and the intermittent analysis of an indicative parameter to characterize the presence of residual free adsorbate in inter-particle fluid 13 . This is typically achieved ex-situ using calorimetric, spectroscopic, titration-based or visual inspection of inter particle fluids [14][15][16][17] . This type of surface characterisation is encumbered by the need for parallelised analyses and the limitation to systems involving complete or near complete adsorption.
The importance of the hierarchical or fractal nature of particle interfaces with surrounding media has resulted in an increasingly wide range of adsorption-based studies addressing the description and measurement of surface area scale variance in particulate materials. Such research efforts were pioneered by studies by Avnir et. al. in a series of publications in the 1980s and 90s 5,[18][19][20][21][22] . Conventional nitrogen adsorption isotherms can interpreted to yield information regarding surface fractality using the Frenkel-Halsey-Hill Theory 20,23 . This method has limitations and its application to systems of unknown surface area is problematic. Further methods to probe scale variance of small aqueous particles, exhibiting high refractive indices, include laser light scattering interpreted using Rayleigh-Gans-Debye theory [23][24] .
More recently, electrochemical approaches to characterising roughness and fractal surface structures in electrodes have been reported using cyclic voltammetry, double layer capacitance analysis and diffusion limited current measurement 25-27 . While not utilised to gauge accessible surface area, these studies highlight the applicability of using multi-ionic interactions for scalable surface analyses.
Although the formation of multilayers of polyelectrolytes has been reported 28 , the adsorption of complex ions, i.e. molecular ionic species, at aqueous particle surfaces is best described by Langmuir isotherms (Type-I), appropriate due to the electro-sterically limited quasi-monolayer type adsorption exhibited [29][30] . Saturation is approached with increasing adsorbate surface density as the result of electrostatic repulsion of charged species limiting further surface ligation.
The electrokinetic behaviour exhibited by suspended inorganic particles is known to vary with the adsorption of surfactant molecules to particle surfaces. Recent studies reported the variation of zeta-potential (the electric potential at the shear surface between a particle and its suspending media) with adsorption of carboxylate anions to surfaces of TiO2 and ZrO2 [31][32] . These studies found that the double layer behaviour observed in suspensions was governed by parameters of adsorbate size and particle surface area.
In the present work we investigate the merit of electrokinetic analyses for the direct assessment of adsorbate-accessible surface area of aqueous granular materials in a recirculating suspension. By introducing a new methodology to interpret adsorption isotherms through indicative zetapotential variation we gauge the appropriateness of electrokinetic analysis in the analysis of surface structure and particlereagent interactions.
Methodology
Solute ionic adsorbates ligating to particle surfaces in suspension form a quasi-monolayer, the density of which exhibits an electro-steric limit towards steady state conditions, governed by the size and charge of adsorbates. Consequently this process can be described by a Langmuir type adsorption isotherm relating fractional surface coverage to adsorbate concentration 30,33 . Fractional surface coverage θf ∈ [0,1], is defined as the ratio of the areal density of surface adsorbed molecules NS to the total number of effective surface sites per unit area Ntot. For adsorption from solution to particles with a given total surface area this is expressed following the Langmuir form as:
C C N N tot S f 1 (1a)
Here proportional to the total number of effective surface sites and thus in monolayer chemisorption for a constant adsorbent mass (ma), the coefficient κ is proportional to the system volume V and inversely proportional to the available adsorbent surface area As, which in turn is the product of specific surface area (as) and ma. An expression to account for specific surface area scaling can be written as
1 1 1 1 1 1 1 1 1 , q a m CVa m CVa VA s a s a s f s (2)
where q=CV/ma corresponds to the quantity of adsorbate per gram of adsorbent (in mol/gram).
We can define a surface area dependant
adsorption coefficient K [mol/g]: s a K such that s a K 1 (3a)
Thus the expression given in Eq. 2. simplifies to the expression
1 1 1 Kq f (3b)
The adsorption of complex ions is associated with an increase in surface charge density σ with relation to the charge density of particle surfaces in the absence of adsorbate, σ0. The density of surface charge in turn governs electrokinetic parameters 34 .
On the basis of the Gouy Chapman theory, following the Grahame equation form, the Stern potential ψδ (at the plane of adsorbed species) and zeta potential ζ are generally related to the apparent surface charge, σ, of a suspended particle through an inverse hyperbolic sine relationship of the type given in Eq. 4.
2 1 M arcsinh M (4)
The parameter σ may refer to true surface charge of the solid phase or of observed Sternlayer charge and accordingly M1 is a system constant related to variables of temperature, permittivity and counter-ion concentration and speciation, while M2 is a constant dependant on temperature and shear plane separation 35- 37 .
It has been found that the adsorption of complex ions to particle surfaces in aqueous suspension manifests in a shift of zeta potential with relative adsorbate concentration following a sigmoidal form with electrokinetic parameters tending towards a quasi-steady state as adsorption capacity is approached 31 . For adsorption of anionic species to suspended particles exhibiting initially positive zeta potential values, the shift in zeta potential is expected to exhibit the relationship shown in Eq. 5 as a function of fractional surface
m M ' 2 M arcsinh 0 1 (5)
As the adsorption of ionic species in aqueous solution manifests in the formation of a charged quasi-monolayer, the generalised relationship given by the Grahame-Langmuir model describing the dependency of potential to adsorbate concentration can be written as:
1 0 2 1 1 ' arcsinh Kq M M m (6)
For the anionic adsorption to particles exhibiting an initially positive zeta potential, as studied in the present work, the proportional zeta potential (ζ', taken relative to the initial value of ζ(θ=0) ) exhibits the trend shown in Figure 1. as a function of surface coverage θ.
Here plots are shown for increasing (negative) adsorbate charge. In similarity to fractional surface coverage, we can define a readily measureable quantity of fractional zeta potential shift β ∈ [0,1] such that:
0 0 S(7)
Under given conditions β represents the double layer modification at conditions of zeta potential ζ relative to the surface-saturated state. Here the value of ζ0 corresponds to the zeta potential of suspended particles at the given pH level in the absence of adsorbate and ζS represents the zeta potential exhibited by suspended particles under conditions equivalent to monolayer coverage (ζS= ζ(θ=1)).
is a correction factor included to account for the shift of solution parameters, namely pH and ionic strength, of interparticle fluid (outside the Stern layer) with adsorbate addition. For a system of low solids loading where the ionic strength and pH are assumed to remain sufficiently stable so as not to impart significant ζ manipulation, we accept a value of =1 (and hence β(θ=1)= 1).
0 2 0 2 0 2 1 0 2 arcsinh ) ' ( arcsinh arcsinh 1 ' arcsinh M M M Kq M m m (8)
Consequently β is expected to vary following the plot shown in Figure 2. It can be seen that increasing specific surface area, linearly correlated to K, manifests in a shift of the sigmoidal relationship. By zeta potential analysis in conjunction with adsorbate addition, the parameter β and its variation can be measured experimentally in situ in suspensions or slurries containing particles of unknown surface structure to gauge accessible surface area per unit mass at tuneable scales.
Experimental procedures:
In order to evaluate the electrokinetic based surface characterisation of particles in aqueous suspension, calcined and ground high purity alumina powders (Al2O3, Baikowski, >99.9% purity) of varying particle size and specific surface area were chosen as characteristic adsorbent materials. Powder characteristics are summarised in Table 1. Morphology of powders was assessed by Scanning Electron Microscopy analysis at 5 kV acceleration by means of a Zeiss-Ultra SEM. Zeta potential measurements were achieved using a Malvern Nano ZS analyser with an automated peristaltic additive dispensing system with suspension recirculation. This apparatus utilises Phase Analysis Light Scattering (PALS) to assess the electrophoretic mobility of particles and thus facilitate the measurement of zeta potential insitu (in the recirculating suspension). Samples were suspended in deionized water to give 10 ml suspensions with solids loadings of 0.1 wt%. In order to impart positive initial zeta potential values and adequate deflocculation, suspensions were adjusted to pH= 4 with the dropwise addition of diluted HCl. Citric and Oxalic acids (99%, Univar) were used as anionic adsorbates, incrementally added in the form of dilute aqueous solutions to stirring alumina suspensions by means of automated dispensing. For each additive increment, three EK measurements were taken with one minute intervals separating the measurements.
Results and Discussion
SEM micrographs of the Al2O3 powders used are shown in Figure 3. A typical hierarchical microstructure is observed with hard agglomerates consisting of finer primary particles, with the size of primary particles and agglomerate fractality resulting in an increasing specific surface area from P1 to P6. In similarity to previous observations from ZrO2 and TiO2 suspensions [31][32] , the addition of dilute citric and oxalic acids to pH=4 alumina suspensions brings about a measurable shift of the electrokinetic properties of suspended particles through the surface adsorption of carboxylate anions. This is shown in Figure 4, where zeta potential is plotted against the relative adsorbate addition (in proportion to the mass of particles). Here dashed lines show mean values while vertical bars show the data spread. Zeta potential was found to vary from an initial value of ~45-60 mV in the HCl adjusted pH =4 suspension to a final value in the range 0-10 mV subsequent to significant carboxylate addition. From repeated measurement at various concentrations it was established that for the studied systems, the observable change in surface charge resulting from carboxylate adsorption reaches equilibrium in less than two minutes. For this reason, measurements involved an equilibration time under stirring before repeated zeta-potential analyses were carried out. It can be seen that curves resulting from the use of oxalic acid are shifted to higher relative concentration values (with respect to solids mass) in comparison with the results from citric acid addition owing to the smaller adsorption cross section of oxalate anions relative to citrate. Although being substrate and speciation dependant, these cross sections on oxide materials are reported varyingly in the region of ~0.6 and ~1.4 nm 2 for oxalate and citrate anions respectively [38][39][40][41] . Similarly, with increasing specific surface area (P1…P6) changes in electrokinetic behaviour are observed at higher additive levels. Figure 5 shows the measured relative shift in zeta potential (ζ' = ζ/ζ0) in comparison to that predicted from the Grahame-Langmuir relationship from Eq. 6. In addition to facilitating initial deflocculation, the pre-acidification of suspensions ensured that the observed electrokinetic behaviour was mediated by reagent interface interactions rather than by changes to suspension pH or ionic strength. This is further illustrated in Figure 6, showing values for ζ, pH and conductivity during the addition of carboxylic reagents to representative suspensions of P4. The slight decrease in pH would typically be expected to bring about an increase in ζ values rather than a decrease, while the moderate increase in suspension conductivity would too not be expected to significantly vary the electrokinetic behaviour of particles. An initial increase in pH is found in similar systems as the result of the displacement of surface hydroxyls with carboxylate adsorption 31 . The measureable differences in the indicative electrokinetic behaviour seen between substrate powders of different surface area and adsorbates of different size, as shown here, demonstrate the applicability of electrokinetic probing for in-situ surface area assessments of aqueous particles. Towards this end, parameter fitting by reverse problem solving is carried out in order to quantitatively evaluate this relationship. While BET measurements are generally limited to the gas accessible specific surface area relative to N2 at 77 K (although other gases are utilisable), the present approach gives an additional degree of freedom enabling scale specific surface characterisation at room temperature in aqueous media by using complex ionic adsorbates of varied effective size.
Identification of model parameters:
Averaged data for the relative shift in zeta-potential was interpreted using a parallelised least squares type fitting process to determine optimised values for the model parameters in Eq. 8. Simultaneous multiple parameter optimisations for all particle types were carried out separately for results from both oxalic and citric additives using an unbounded Levenberg Marquardt algorithm. Tolerance was set as 1x10 -14 and convergence was achieved after ~300 iterations using centred finite differences for curves for both oxalic and citric adsorbates. In this manner an optimised K value for each paring of adsorbent/adsorbate was determined, while optimal values for The close agreement of the fitted parameters and the experimental data suggest that under the conditions employed, adsorption behaviour followed approximately the monolayer form of the Grahame-Langmuir model. Alternative adsorption models, as may be appropriate for protein/polymer multilayer adsorption or other non-type-I forms 42-44 , can facilely be incorporated into the present approach through suitable substitution or adaptation of the Langmuir isotherm form with a more appropriate model for the evaluation of the θ/C relationship in the numerical framework used for parameter fitting.
From the data fitting carried out it was determined that for the adsorption of citrate anions the ratio of net negative change in apparent Stern plane charge relative to initial conditions, 0 ' m , was 2.04, while for oxalate adsorption this value was optimised at 1.9. These values are indicative of charge density formed by monolayer adsorption of these carboxylate anions and are dependant on parameters of ion speciation, adsorption /desorption rates and maximum surface coverage density. For a given system the optimised value for the parameter K (Kfit) is expected to exhibit linear dependence on scale-specific effective surface area, following Eq. 3. While no true single value of specific surface area can be defined, examining the validity of this model requires comparison with a standardised surface area metrology method. Therefore, to further assess the applicability of the current methods towards surface area analysis we examine the correlation between K values and the conventional N2 adsorption isotherm determined surface area. As shown in Figure 8, this comparison of Kfit values against BET derived surface area for the 6 powder types shows a trend supporting the linear correlation of fitted K values with specific surface area. Deviations from linear behaviour are expected to result from a combination of (a) fundamental discrepancies between room temperature anionic adsorption in aqueous media and gas adsorption at low temperatures and (b) experimental uncertainties in establishing and fitting the curves of electrokinetic variation. It is likely that the first parameter is of greater relevance for powders of higher surface area while the latter issue plays an important role for substrate materials of lower specific surface area, which exhibit greater sensitivity in their EK behaviour.
Kfit values are found to be approximately 1.6-2.0 times higher in results from incremental oxalic acid addition relative those obtained using citric additive, this is consistent with the larger adsorption cross section of the citrate anion as reported for various oxide substrates. It should be noted that as with gas adsorption methods 12 , the precise adsorptive cross section in terms of nm 2 per molecule is contingent also on the substrate material and consequent density of exchangeable surface groups. For this reason, in contrast to comparative analysis, meaningful quantitative analyses require a system specific calibration to establish a reference point for a known surface area under given conditions. Furthermore, the monolayer density is affected by the adsorbate speciation, which in turn is influenced by pH, meaning significant pH fluctuations would be detrimental to the accuracy of the analysis. For the conditions utilised here (pH=4) oxalate is expected to exhibit a speciation of 57% AHand 43% A 2-, while citrate is expected to speciate following 76% AH2 -13%AH 2and 11% AH3 (where A represents the fully deprotonated molecule) 31 . were obtained for this relationship fitted to pass through the origin Kfit (aS=0)=0. Owing to the fundamental discrepancies between BET methods and the adsorption methods explored here, it may be appropriate to additionally examine the correlation of results with surface area obtained by ex-situ analysis of adsorption (e.g. spectroscopic analysis of supernatant fluids) or by microanalytical methods (AFM, SEM or TEM).
The slope of K/as ( 1 ) in units of μmol m -2 is proportional to the density of monolayer adsorption. For the fitted data, The slopes of 2.13 μmol m -2 and 1.28 μmol m -2 for the adsorption of oxalate and citrate respectively are in good agreement with previously reported values for these species 31,45 The computational and experimental methods followed here, in regards to the analysis of ζ variation as a function of mass-relative reagent addition with parameter fitting to the Grahame-Langmuir form to facilitate surface area assessment, can be applied with the use of a range of alternative complex cationic and anionic species potentially including, amongst others, the use of ammonium salts, phosphates/sulphates, xanthates or polysorbates. The appropriateness of these surface-interacting reagents is coupled to various substrate and system characteristics and necessitates adaptation on a case-by case basis. Importantly, electrokinetic parameters, determined here by PALS analysis of electrophoresis, can also be measured using alternative techniques including electroosmosis, streaming potential or streaming current. Such methods may be more appropriate for conducting in-situ analyses of surface structure in more condensed particle-fluid systems such as slurries. This approach has relevance for the control and optimisation of industrial processes where the ability to assess surface structure of aqueous particles without lengthy ex-situ analysis is of value.
Conclusions
We have shown a numerical and experimental framework that can be applied to acquire surface structure information through means of electrokinetic characterisation. Specifically, the variation of zeta potential exhibited by adsorbent particles as a function of proportional adsorbate concentration follows the Grahame-Langmuir relationship for the monolayer-type adsorption of complex ionic species in aqueous media. The adsorption of carboxylate ions to alumina particles was used to demonstrate the merit of this behaviour for the quantitative and comparative characterisation of particle surfaces. A near linear relationship between the optimised value found from reverse-problem solving for the surface-area normalised adsorption coefficient K and the conventionally determined surface area indicates that the analysis of zeta potential variation (or other electrokinetic properties) in conjunction with incremental addition of cationic or anionic adsorbates is applicable for the controllable scale specific evaluation of accessible surface area. By further employing ionic species of known and controlled adsorption cross section with respect to the substrate material in question, this approach can be used in the assessment of scale variance of fractal surface structures in aqueous particulate matter.
Using an automated dispension-measurement system, the experiments carried out here involved in situ electrokinetic analysis based surface characterisation. Thus, PALS based measurements and suspension modification were performed on a single recirculating aqueous system. This approach is advantageous relative to the analysis of gas accessible surface using N2 at 77 K, as it allows application relevant room temperature analysis for aqueous particle-fluid systems in applications including water treatment, photocatalysis and industrial processing. Furthermore, such methods offer advantages in terms of rapidity and scalability relative to existing aqueous methods for surface area analysis which typically involve intermittent secondary analysis to assess levels of adsorption density or concentration of adsorbate in supernatant fluids 46 .
The broad range of acceptable ionic adsorbate compounds that can be employed to facilitate the measurement of surface area by the assessment, in situ or otherwise, of indicative electrokinetic or electrochemical parameters is in contrast to existing methods for surface characterisation by adsorption isotherm interpretation that are inflexible with respect to surface-interacting compounds and thus do not readily facilitate application-specific analyses and the assessment of scale variance. Although precise quantitative evaluation of surface area requires calibration with reference to specimen of known surface area to facilitate back calculation, the use of in-situ electrokinetic analyses can readily facilitate the comparative analysis between suspended powders of varying surface structures, at tuneable scales.
shift in observed double-layer charge (within the sphere defined by the slipping plane) brought about by monolayer adsorption (occupation of all effective sites)
Figure 1 .
1Relative zeta potential as a function of fractional surface coverage for anionic adsorption for a range of different adsorbate anion charge values following Eq. 5.
Figure 2 .
2Fractional zeta potential shift β ∈ [0,1], as s function of additive concentration for a given system volume.
Figure 3 .
3SEM micrographs of Al2O3 powders used (a)-(f) corresponding to P1-P6 in Table 1. Hanaor, D. A. H., Ghadiri, M., Chrzanowski, W., & Gan, Y. (2014). Scalable surface area characterization by electrokinetic analysis of complex anion adsorption. Langmuir, 30(50), 15143-15152.
Figure 4 .
4Variation of Zeta potential with additive ratio for powders P1-P6 with (a) citric acid and (b) oxalic acid adsorbates. Vertical bars show data range at each measurement point.
Figure 5 .
5Relative zeta potential shifts resulting from oxalate and citrate adsorption compared to the Grahame Langmuir model.
Figure 6 .
6Variation of ζ, pH and conductivity in P4 suspensions with (a) citrate and (b) oxalate adsorption.
M
were evaluated for each of the two adsorbate reagents optimised to yield the best combined fitting across all 6 curves in each dataset. The results of data fitting for each of the six powders are shown inFigure 7.
Figure 7 .
7Parameter optimisation for fractional ζ shift as a function of proportional additive ratio.
Figure 8 .
8Optimised values for surface-area normalised adsorption coefficient K plotted against BET measured surface area values.The approximate linearity of the Kfit /aS(BET) relationship is indicative of the applicability of electrokinetic analysis for the quantitative or comparative evaluation of surface area.
Where [A] is the concentration of adsorbate in solution and [AS] and [S] represent the surface densities of occupied and unoccupied sites on the adsorbent particles. [A] is inversely proportional to the system volume while [S] isC corresponds to the volumetric
concentration of adsorbate in the system, and
the coefficient κ corresponds to the ratio of
adsorption/desorption
for
a
given
adsorbate/adsorbent pair such that:
]
][
[
]
[
S
A
AS
(1b)
Table 1 .
1Characteristics of alumina powders used in the present workSample
Designation
Supplier
designation
Milling
method
Al2O3 Phases BET surface
area (m 2 g -1 )
Agglomerate
size (nm)
P1
CR1
Jet milled
α
3
1100
P2
CR6
Jet milled
α
6
600
P3
SMA6
Ball milled
α
7
300
P4
CR15
Jet milled
90% α 10%γ 15
400
P5
CR30F
Jet milled
80% α 20%γ 26
400
P6
CR125
Jet milled
γ
105
300
. Applying oxalic 1 values in Eq. 3a together with the Kfit values allows the evaluation of as values of 12.5, 12.6, 13.3, 19.6, 29.6 and 102.2 m 2 g -1 respectively for materials P1 to P6, while the citrate adsorption thus yields values of 3.9, 7.2, 10.6, 15.8, 44.2 and 100.1 m 2 g -1 for P1 to P6. It is important to note that these values are provisional as they are calculated on the basis of a linear fitting with relation to BET values. Extracting more definitive or meaningful numerical quantities would require the use of one or more reference samples of well characterised surface structure.
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| [] |
[
"Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings",
"Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings"
] | [
"Ali Girayhan Özbay \nDepartment of Aeronautics\nTurbulence Simulation Group\nImperial College London\nSW7 2AZLondonUK\n",
"Sylvain Laizet \nDepartment of Aeronautics\nTurbulence Simulation Group\nImperial College London\nSW7 2AZLondonUK\n"
] | [
"Department of Aeronautics\nTurbulence Simulation Group\nImperial College London\nSW7 2AZLondonUK",
"Department of Aeronautics\nTurbulence Simulation Group\nImperial College London\nSW7 2AZLondonUK"
] | [] | The usage of neural networks (NNs) for flow reconstruction (FR) tasks from a limited number of sensors is attracting strong research interest, owing to NNs' ability to replicate high dimensional relationships. Trained on a single flow case for a given Reynolds number or over a reduced range of Reynolds numbers, these models are unfortunately not able to handle flows around different objects without re-training. We propose a new framework called Spatial Multi-Geometry FR (SMGFR) task, capable of reconstructing fluid flows around different two-dimensional objects without re-training, mapping the computational domain as an annulus. Different NNs for different sensor setups (where information about the flow is collected) are trained with high-fidelity simulation data for a Reynolds number equal to approximately 300 for 64 objects randomly generated using Bezier curves. The performance of the models and sensor setups are then assessed for the flow around 16 unseen objects. It is shown that our mapping approach improves percentage errors by up to 15% in SMGFR when compared to a more conventional approach where the models are trained on a Cartesian grid, and achieves errors under 3%, 10% and 30% for pressure, velocity and vorticity fields predictions, respectively. Finally, SMGFR is extended to predictions of snapshots in the future, introducing the Spatio-temporal MGFR (STMGFR) task. A novel approach is developed for STMGFR involving splitting DNNs into a spatial and a temporal component. We demonstrate that this approach is able to reproduce, in time and in space, the main features of flows around arbitrary objects. | 10.1063/5.0087488 | [
"https://arxiv.org/pdf/2202.03798v2.pdf"
] | 246,652,197 | 2202.03798 | 0698c91faa104d930082c0cb6ab81ae1faace2a7 |
Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings
Ali Girayhan Özbay
Department of Aeronautics
Turbulence Simulation Group
Imperial College London
SW7 2AZLondonUK
Sylvain Laizet
Department of Aeronautics
Turbulence Simulation Group
Imperial College London
SW7 2AZLondonUK
Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings
(Dated: 6 April 2022)AIP/123-QED
The usage of neural networks (NNs) for flow reconstruction (FR) tasks from a limited number of sensors is attracting strong research interest, owing to NNs' ability to replicate high dimensional relationships. Trained on a single flow case for a given Reynolds number or over a reduced range of Reynolds numbers, these models are unfortunately not able to handle flows around different objects without re-training. We propose a new framework called Spatial Multi-Geometry FR (SMGFR) task, capable of reconstructing fluid flows around different two-dimensional objects without re-training, mapping the computational domain as an annulus. Different NNs for different sensor setups (where information about the flow is collected) are trained with high-fidelity simulation data for a Reynolds number equal to approximately 300 for 64 objects randomly generated using Bezier curves. The performance of the models and sensor setups are then assessed for the flow around 16 unseen objects. It is shown that our mapping approach improves percentage errors by up to 15% in SMGFR when compared to a more conventional approach where the models are trained on a Cartesian grid, and achieves errors under 3%, 10% and 30% for pressure, velocity and vorticity fields predictions, respectively. Finally, SMGFR is extended to predictions of snapshots in the future, introducing the Spatio-temporal MGFR (STMGFR) task. A novel approach is developed for STMGFR involving splitting DNNs into a spatial and a temporal component. We demonstrate that this approach is able to reproduce, in time and in space, the main features of flows around arbitrary objects.
I. INTRODUCTION
Most fluid dynamics experiments have access to only sparse measurements, due to the intrusive (i.e. flow-altering) nature of pitot tubes and pressure probes used for measurements. Although noninvasive methods to obtain full flow fields in experiments such as particle imaging velocimetry 1 (PIV), magnetic resonance velocimetry 2 (MRV) and laser doppler flowmetry 3 (LDF) exist, their usage can be limited by practicality, cost or safety constraints; for instance PIV systems often require 'Class IV' lasers that can gravely harm human eyes and cost thousands of dollars. Despite these practical limitations in experiments, knowledge of the full flow fields is often critical to understanding the dynamics of many complex fluid flows. Flow reconstruction (FR) methodologies can offer reliable estimation of a full flow field from only sparse measurements.
Callaham et al. 4 describe the FR task in terms of a high-and a low-dimensional state vector x ∈ R m , s ∈ R p , m >> p, where x represents the 'full' flow field and s are sparse sensor measurements.
The two state vectors are linked through measurement and reconstruction operators H : R m → R p and P : R p → R m such that
s = H(x)(1)
x = P(s)
The goal of the FR task is to find an approximation mapping R : R p → R m such that some measure of error, typically L 2 , betweenx = R(s) and x = P(s) is minimized. In practice, F is a statistical or deep learning algortihm with a parameter set w optimized to fit some dataset; i.e. R(s) = R(s, w).
Deviating from the formulation by Callaham et al. 4 to use some generic objective function L instead of the L 2 norm, the flow reconstruction task can be expressed as a minimization problem of the following form arg min w L(x, R(s, w))
Historically, methods such as Gappy PODs 5,6 and Linear Stochastic Estimation 7 are some of the main methodologies investigated for FR, but they are often unsuitable for multi-geometry FR (the reconstruction of the flow field generated past an arbitrary geometry), as detailed in Section II. Research in this field has recently intensified, and a new wave of studies -the vast majority focused on using neural networks (NNs) -have been published, see 4,[8][9][10] to name a few. As a starting point for neural network based flow reconstruction, these publications largely focus on obtaining models that work on a single fluid flow case, typically predicting vorticity fields for incompressible flow past a circular cylinder at a single (or over a narrow range) of Reynolds numbers. As a result, such approaches do not have the potential yet to be used in wind tunnel testing driven shape optimization, as training such NNs first requires collecting large datasets of the flow past specific objects.
Training NNs for multi-geometry FR from sparse sensors is not straightforward. Within the aforementioned setting of vorticity field reconstruction on Cartesian grids for two-dimensional (2D) incompressible flows past arbitrary objects, naively augmenting the dataset with multiple objects results in models that fail to reproduce key flow features; concentrations of vorticity in boundary layers and near stagnation points often disappear, and the objects themselves are engulfed by amorphous blobs of non-physical vorticity concentrations. The root cause of these issues is that, within these settings, the models lack information regarding the shape of the object they are making predictions on, as the need for such information is obviated by the single-shape nature of the datasets.
Overcoming these issues requires a representation of the flow field in a way that removes the necessity for the model to predict the shape of the object immersed in the fluid. An effective tool for this is a mapping, whereby all possible geometries are mapped to a single shape. For 2D cases, this can be achieved via the Schwarz-Christoffel (S-C) conformal mapping, which can be used to map any l-connected domain to a disc with l holes 11 . Thus, the fluid domain in any bluff body flow with a single object can be mapped to an annulus.
In this work, the Spatial Multi-Geometry Flow Reconstruction (SMGFR) task is introduced, with the objective of reconstructing pressure, velocity and vorticity fields surrounding randomly generated objects immersed in a fluid flow from sparse sensor measurements. S-C mappings are utilized to map the fluid computational domains surrounding the said randomly generated bluff bodies to annuli, and a dense field sampling approach based on grids uniformly spaced in angular and radial directions in the annular domains is developed. Over a comprehensive set of 24 experiments (different models and sensor arrangements), the mapping approach is compared to reconstruction based on uniformly spaced Cartesian grids, and the performance of different sensor setups and NN architectures (feed forward, U-Net 12 and Fourier Neural Operator 13 ) in SMGFR are investigated.
As a further step towards spatio-temporal reconstruction, a modified version of the SMGFR is proposed whereby the model is expected to construct snapshots at future times given sensor readings at a present time. In this task, dubbed Spatio-temporal Multi-Geometry Flow Reconstruction (STMGFR), a model obtained from the spatial-only SMGFR task (called the spatial model) is coupled with a second neural network model. This second model, called the temporal model, accepts the reconstructed dense field as its input and predicts the state of the full flow field k time-steps in the future. The resulting system, composed of the spatial and the temporal models, is thus able to reconstruct the full flow fields k time-steps in the future given current sensor measurements.
The work is organized as follows: first, a brief overview of recent and historical approaches to the FR task is provided in Section II. Subsequently, an introduction to the S-C mapping and details of its novel application to the FR task are presented in Section III. The dataset constructed to take advantage of the S-C mapping is described in Section IV, and the models chosen to fit this dataset as well as their training procedures are described in Section V. Finally, the results are showcased in Section VI and a summary of this work and plans for future investigations are provided in Section VII.
II. RELATED WORK
The FR task, as introduced in Section I, falls under the broad umbrella of inverse problems 14 .
Commonly encountered in a wide range of scientific and engineering fields including fluid dynamics, inverse problems often lack well-defined, unique solutions and rely on minimization of some objective such as the L 2 norm, as encountered e.g. in the Moore-Penrose pseudo-inverse for linear least-squares problems. Dubois et. al. 8 In comparison, the body of works investigating deep learning based FR for the reconstruction of the flow past arbitrary objects without re-training is small. One notable recent work in this area is by Chen et al. 34 , using graph convolutional neural networks (GCNNs) for reconstructing steady flow fields around random objects. Training a GCNN to predict the velocity and pressure fields around 1600 random objects generated via Bezier curves, they applied the model to predict the pressure and velocity fields around 400 test geometries, which permitted the estimation of drag and lift coefficients with very small mean percentage error levels. The present work differs from Chen et al. 34 as it focuses on the usage of a novel mapping approach to achieve the geometry invariance as opposed to graph convolutions which permits the usage of traditional NN architectures. Here the aim is to reconstruct instantaneous snapshots as opposed to steady fields, and to explore predicting future instantaneous snapshots from current measurements. Additionally, this work is conducted at a substantially higher Re = 300 as opposed to Re = 10 in Chen et al. 34 which leads to the emergence of unsteady rotational flows. To assess the performance of the models, the focus is on reconstructing the vorticity fields (which are difficult to predict from pressure and velocity sensors, as shown in Section VI A). Reconstructed data for the pressure and velocity fields are also briefly presented for completeness.
III. SCHWARZ-CHRISTOFFEL MAPPINGS
Conformal transformations have been used extensively in fluid dynamics, especially the wellknown Joukowsky and Kármán-Trefftz 35 (K-T) transformations which map the unit circle to airfoil shapes. However, the usefulness of these two transformations can be limited due to their inability to generalize to arbitrary shapes. A more flexible alternative is the Schwarz-Christoffel (S-C) mapping, which is a conformal transformation historically used to map polygonal simply connected domains to the unit disc. The S-C mapping has been extended in the recent decades to multiply connected domains. Although the existence of a conformal mapping between any given two l-connected domains is guaranteed 11 , the practical computation of such an S-C mapping typically requires the use of numerical methods to determine a number of parameters in the mapping expression, referred to as the Schwarz-Christoffel parameter problem. Numerically implementing the S-C mapping for doubly (or higher) connected domains is not a trivial undertaking. In this work, the DSCPACK code 36 , which is a Fortran package aimed at computing S-C mappings between doubly connected domains bounded by polygons to annuli, was utilized to solve the parameter problem.
A rigorous treatment of the methodology used in this package lies beyond the scope of this work, for which the reader is directed to established works in S-C mapping literature 11,37,38 , though the general strategy can be summarized as follows: denoting z as the complex coordinates in the original domain and w as the complex coordinates in the annulus domain, DSCPACK uses an expression of the form
z = g(w) = g(w c ) +C w w c W (w) dw,(4)W (w) = M ∏ q=1 σ µ, −w µw 0q α 0q −1 N ∏ r=1 σ µ, −µw w 1r α 1r −1 ,(5)σ (µ, w) = 1 + ∞ ∑ b=1 µ b 2 (w b + w −b ),(6)
as the mapping, where C is some complex valued constant; µ is the radius of the inner ring of the annulus in the w-domain; M, α 0q , w 0q and N, α 1r , w 1r are the number of vertices, the turning angles and prevertices 39 of the outer and the inner polygon, respectively. Of these variables, C, µ, w 0q and w 1r are unknowns ('accessory parameters' of the mapping) and must be computed by solving a series of nonlinear integral equations
z 01 − z 0M = C w 01 w 0M W (w) dw,(7)|z 0,q+1 − z 0q | = |C w 0,q+1 w 0q W (w) dw|, q ∈ [1, M),(8)z 11 − z 1N = C w 11 w 1N W (w) dw,(9)|z 1,r+1 − z 1r | = |C w 1,r+1 w 1r W (w) dw|, r ∈ [1, N),(10)z 1N − z 0M = C w 1N w 0M W (w) dw,(11)
where z 0q and z 1r denote the complex coordinates of the polygon vertices in the original domain.
The DSCPACK code solves this nonlinear system using a Newton iteration scheme. The resulting expression maps the outer ring (with unity radius) of an annulus to the outer polygonal boundary, while the inner ring is mapped to the inner polygon.
Once the forward mapping g is known, the inverse mapping f can be approximated for any
(fixed) z =z using Newton iteration:z − g(w) = 0 (12) w n+1 = w n +z − g(w n ) g (w n ) (13) g (w n ) = CW (w n ),(14)
where Equation 14 follows from the application of the fundamental theorem of calculus to Equation 4.
To ensure smooth interoperability of DSCPACK with modern machine learning packages, a set of Python bindings to a modified form of the code were developed, dubbed pydscpack. Furthermore, a number of enhancements to the original code were made to parallelize performance-critical sections with OpenMP. Figure 1 depicts an S-C mapping computed using pydscpack, for a geometry used in this study.
IV. DATASET
A. Geometries
B. Ground truth values
Using uniform Dirichlet velocity boundary conditions (u, v) = (1.0, 0.0) along the external edges of the domain, the flow around each object was computed for Re = uL m /ν = 300 (where ν is the kinematic viscosity) using the PyFR solver 41 , which is a flux reconstruction 42 based advectiondiffusion equation solver using the artificial compressibility approach to solve the incompressible Navier-Stokes equations. It was chosen for its Python interface and GPU acceleration capabilities.
The simulations were performed on two Nvidia V100 GPUs. Normalizing the physical time τ by the large eddy turnover time to obtain τ * = uτ/L m , t ∈ [0..600] snapshots (containing the pressure and velocity field components p t,i , u t,i and v t,i ) were recorded per geometry for a total of 48080 snapshots between τ * = 3.333 and τ * = 23.333.
Following the simulations, referring to the fluid domain around each G i as F i , the forward and inverse mappings g i and f i between F i and the corresponding annuli A i were computed using pydscpack. 64 × 256 grids uniformly spaced in the radial and angular directions with coordinates w A,i were generated for each A i . Subsequently, w A,i were mapped back to the original domains F i using the computed mappings g i to obtain the annular grid coordinates in the original domain z A,i = g i (w A,i ). The velocity, pressure and vorticity fields u t,i , v t,i , p t,i and ω t,i from the highfidelity simulation data were interpolated to z A,i to obtain the interpolated fieldsũ t,i ,ṽ t,i ,p t,i and ω t,i , which form the ground truth values of the Annular dataset.
This sampling strategy and grid resolution provide a high grid density near the object, scaling with a factor of 1/r based on distance to the center of the annulus. It ensures that the regions of the flow with high vorticity concentrations have enough grid points for a correct representation of the vortical structures. Additionally, as a baseline case, a further collection of ground truth values sampled naively on a 128 × 128 uniformly spaced Cartesian grid was also produced, with the same number of grid point as for the mapping approach. Note that these grids are used solely for the interpolation of flow variables, not to perform the fluid simulations.
C. Inputs
The inputs of the dataset are vectors of pressure and/or velocity values s t,i at a sparse number of sensor locations, obtained via interpolation of the PyFR solution to the sensor locations. The sensor setup to build the inputs of the dataset is split into two sensor types, chosen to represent a setup that can be practically implemented in a laboratory environment:
• Pressure: Placed on the surface of each G i , with equal angular spacing along the inner ring of each A i .
• Velocity: Positioned on a rectangular grid spanning a 2L m /3 × 4L m /3 region, the left edge of which is L m /6 units behind the rearmost point of each G i and the centroid of which is vertically level with each G i .
Based on this general template three setups with varying sensor quantities were considered, summarized in Table I. Figure 3 depicts the medium sensor setup for a sample geometry.
D. Normalization
Normalizing inputs and/or outputs plays an important role in obtaining good performance from deep learning algorithms, as it permits better conditioning of the gradients within the optimization landscape during training, by keeping the per-layer statistical distribution of the gradients 43 . A variety of data normalization methodologies, including mean centering, standardization and min-max scaling were tried in a preliminary study. Denoting X as the dataset inputs and T as the target values, X + and X − as the maximum and minimum values of X, respectively, and µ and σ as the mean and standard deviation, respectively, the three normalization methods can be summarized as follows:
Mean centering:
X = X − µ,T = T − µ,(15)Standardization:X = (X − µ)/σ ,T = (T − µ)/σ (16) Min-max scaling:X = X − X − X + − X − + X − ,T = T − X − X + − X − + X − (17)
Mean centering both inputs and ground truth values based on the ground truth mean values was chosen as the data normalization method, as it provides the results with the lowest validation loss levels for models trained using either the Cartesian and Annulus datasets.
V. EXPERIMENTAL SETUP
Two related tasks have been investigated, with the dataset detailed in Section IV. The first is identical to spatial flow reconstruction tasks from sparse sensors in previous literature 4,8,33,44 , but with the inclusion of snapshots taken from a multitude of geometries in the training and validation datasets. As a reminder the name Spatial Multi-Geometry Flow Reconstruction (SMGFR) is used to describe the FR task in this specific configuration. The second task is a generalization of SMGFR where target snapshots are in the future relative to the sensor measurements by a fixed amount of time ∆τ * , as opposed to SMGFR where the target snapshots are contemporaneous with the measurements, dubbed Spatio-temporal Multi-Geometry Flow Reconstruction (STMGFR).
For both tasks, the dataset from Section IV was split by randomly choosing the data associated with 64 geometries as the training set; the remaining 16 geometries constituted the validation set.
Training in all experiments was conducted using the Adam 45 algorithm using an initial learning rate (LR) of 10 −3 , reduced by 90% each time the loss values plateaued. A. Spatial multi-geometry flow reconstruction (SMGFR)
Using the notation in Section IV, the SMGFR task can be summarized as predictingp t,i ,ũ t,i , v t,i orω t,i given s t,i . The experiments investigate the performance of four different models, all implemented using Tensorflow 46 v2.5.1, with the parameter counts available in Table II. The latter two models are described schematically in details in Figure 4. Below is an overview of the models investigated, with some arguments to justify their use in the present study: ... in the parentheses indicate the shape of each block's output tensor; D1=64 and D2=256 for the Annulus dataset, and D1=D2=128 for the Cartesian dataset. In the SD-UNet, each convolution block is formed of two convolution layers, each preceded by a batch normalization layer and followed by a dropout layer, each deconvolution block is formed of a batch normalization layer followed by a deconvolution with stride 2.
Input (B,D1,D2) Padded 2D Convolution (C,D1,D2) Real-valued FFT (B,D1, ⌊D2/2 +1 ⌋ ) Complex Weight W (2,B,C,M1,M2) Pick first M1 x first M2 modes (B,M1,M2) Pick last M1 x first M2 modes (B,M1,M2) Stack (2,B,M1,M2) Tensor product ijkl,ijmkl->imkl (2,C,M1,M2) Place modes into 0-initialized array (C,D1, ⌊D2/2 +1 ⌋ ) Inverse real- valued FFT (C,D1,D2) Sum (C,D1,D2) Activation function (C,D1,D2) Result (C,
B. Spatio-temporal multi-geometry flow reconstruction (STMGFR)
The spatio-temporal multi-geometry flow reconstruction (STMGFR) task extends the purely spatial SMGFR task presented under Section V A. Whereas SMGFR focuses on obtaining a reconstructionx t,i of the (interpolated) target fieldx t,i (e.g. Training the temporal model is done in a supervised manner. Since the temporal model would be expected to perform well given reconstructed inputsx t,i from the spatial model in an inference scenario, in each epoch the input associated with every sample is randomly chosen to be either a reconstructed vorticity fieldx t,i or a ground truth fieldx t,i , with a 50% chance for each. Using separate spatial and temporal models with parameter counts P s and P t (versus a larger single model with parameter count P s + P t directly predictingx t+k,i from s t,i ) is highly computationally efficient as it permits easy re-training of multiple temporal models for different values of k. The results from the spatial model can be easily cached and re-used when training a new temporal model, which translates to computational speedups as well as model accuracy benefits owing to the possibility of using larger batch sizes given a fixed pool of memory.
VI. RESULTS
The results of a series of SMGFR and STMGFR experiments are detailed in this section, the setups of which are detailed in the previous chapters. First, to compare the accuracy and quality of our reconstruction methodology to the previous work by Chen et al. 34 , we briefly present results on reconstructing pressure and velocity fields in Section VI A. Additionally, we numerically demonstrate that the reconstruction of vorticity presents a greater challenge than the reconstruction of pressure and velocity from pressure and velocity sensors.
Subsequently, we proceed to detailed comparisons of vorticity reconstruction performance, where the differences between the various setups are visible more acutely. Section VI B details the results from the spatial reconstruction task with four models and three sensor setups, as detailed in Section IV and Section V A. With the best performing configuration for the spatial task identi- A. Spatial multi-geometry pressure and velocity reconstruction
In order to compare the quality of our reconstruction methodology with the previous multigeometry reconstruction work by Chen et al. 34 , we briefly present the results of training the k = 0 SD-UNet+FNO combination from Section V B on pressure and velocity data in Table III, using the large sensor setup.
Additionally, to demonstrate that a fairly complicated and non-linear reconstruction relationship P is present between the sensor measurements and the vorticity fields investigated in Sections Section V A and Section V B, we include two difficulty measures D and M in Table III. These measures are based on the Frobenius norms of the Spearman rank correlation coefficient matrix 49 (SRCC) and Mutual Information 50 (MI). Defining ψ t,i = [s t,i x t,i ] ∈ R p+m as a vector concatenating the sensor measurements s ∈ R p and the full field x ∈ R m (pressure, vorticity etc.) for a particular snapshot i at a particular time t, we construct a large matrix Ψ containing the entirety of the data in our dataset:
Ψ = ψ 0,0 ψ 0,1 . . . = s 0,0 x 0,0 s 0,1 x 0,1 . . . . . . (18)D = ||D s,x || 2 = ||D x,s || 2 M = ||M s,x || 2 = ||M x,s || 2(20)
The difficulty metrics clearly show the reason for lower performance when predicting ω with all setups. The D scores display greater correlation between the sensor inputs and pressure/velocity fields compared to the vorticity field. This translates to, on average, greater monotonicity in the relations mapping the sensor data to the full field data for the pressure and velocity fields compared to the vorticity field. M , meanwhile, demonstrates that the probability distributions of the sensor inputs and full field values are substantially more alike (i.e. have lower relative entropy). Both of these have profound effects on the accuracy of the neural networks, which manifests in the difference in the MAPE scores when predicting different target fields. Since the higher difficulty associated with predicting the vorticity field is illustrated, we move forward to comparing the quality of our pressure and velocity results with previous works. Chen et al. 34 reported reconstruction errors amounting to 7.70 × 10 − 3 in a similar setup, but at a substantially lower Re, predicting on flow cases within steady, laminar flow regimes only. Thus, considering the substantially higher Re in this work which results in the creation of unsteady vortical structures, the differences in data generation methodologies, and the different objective involving the prediction of instantaneous as opposed to steady fields, the error levels exhibited are in line with previous works. The MAPE levels, under 3% and 10% respectively for pressure and the velocity components, clearly demonstrate that our work is a clear step forward for SMGFR. A gallery of sample velocity and pressure predictions is provided in Appendix A.
B. Spatial multi-geometry vorticity reconstruction
Considering the higher difficulty of predicting the vorticity field from pressure/velocity sensors, and to push the boundaries of the neural networks for flow reconstruction tasks, a comprehensive set of 24 experiments for the vorticity SMGFR have been conducted to highlight the differences between the combinations of sensor setups, model architectures and sampling strategies. Table IV summarizes the performance of all combinations. To provide deeper insight into the performance of the models with the different sensor and sampling setups beyond the overall error values, a gallery of predictions is provided. Figures 6, 7 and 8 depict the predictions for a bluff body-like shape, dubbed 'Shape A', using the large, medium and small sensor setups, respectively. Furthermore, Figure 9 and Figure Moving forward to Shape B in Figure 9, the difference is most striking for the SD and SD-Large, where the Cartesian versions of the models predict a high concentration of negative vorticity entirely engulfing the object, which is highly non physical. Additionally, the location of the high positive vorticity blob near the 'leading edge' of the object is predicted as detached from the object surface. In contrast, the same models with annular sampling correctly predict these key The trends for the other shapes continue in the case of Shape C in Figure 10. Similar to both feedforward models for Shape B, the SD-Large model with Cartesian sampling spuriously predicts a large concentration of very high vorticity surrounding the object while also substantially underestimating the intensity of the downstream vortices. The SD-UNet with annular sampling performs the best but with Cartesian sampling the error is much higher, also due to an underestimation of the intensity of the two counter-rotating vortices behind the object. Finally, the SD-FNO is the trendbreaker, with Cartesian sampling managing a narrow quantitative win thanks to a better estimation of the vortex intensity. However, from a qualitative perspective, both SD-FNO predictions are noisy and do not accurately reproduce the smoothness of the vorticity field unlike the SD-UNet.
As a final remark, we draw attention to the presence of high error along the same contours for different sampling strategies and models among the images for each snapshot. This is due to the presence of very high percentage error (despite low absolute error) near the zero contours of the target field due to very small denominators. Visible in low vorticity areas across all geometries, it is ultimately caused by the objective function as explained above. It is also the main reason why the MAPE and HV-MAPE may appear high with values consistently between 20% and 60%.
C. Spatio-temporal multi-geometry vorticity reconstruction
Since the SD-UNet model, used alongside annular sampling and the large sensor setup, was identified as the best performing combination in Section VI B, this combination was chosen for the spatial model in this work's approach to the STMGFR task. The performance of the temporal model is summarized in Table V for different values of the temporal gap k (refer to Table IV As expected, the error declines as the temporal gap k is reduced and the model performs better when ground truth snapshots are provided as the inputs as opposed to reconstructed inputs. Surprisingly, however, MAPE levels for this task are substantially lower than results for the purely spatial task in Table IV despite Finally, Shape F in Figure 13 is a thick airfoil like shape, also set at a high incidence angle The good performance of the temporal model when given ground truth snapshots as inputs is largely consistent with the previous literature on the FNO 13 , where the capability of the FNO to time-march the vorticity field for turbulence-in-a-box settings were demonstrated with low error levels. The additional insight, however, is that these results demonstrate that the FNO model coupled with our training methodology (whereby 50% of the inputs are randomly replaced with spatial model predictions) is robust to noisy inputs, with a loss of accuracy of the order of 2% (as shown in Table V) despite average errors in the input approaching 40%. In fact, in two of the three cases displayed in the figures, the MAPE of the temporal model prediction from the spatial model reconstruction relative to the ground truth at time t + k is lower than the MAPE of the spatial model reconstruction relative to the ground truth at time t. In a physical experimental setting, where measurement noise is a real concern, this is a key capability as the impact of the measurement errors on the spatial reconstruction will not catastrophically degrade the accuracy of the temporal reconstruction.
D. Training and inference time
The final topic of discussion for comparing the relative merits of the different model architectures in the previous sections is the computational cost of training and using each model. Table VI outlines the wall-clock runtimes for running training (conducted on an IBM AC922 system with two 20-core POWER9 CPUs and two Nvidia V100 GPUs) and inference (conducted with an AMD EPYC 7443 CPU and a single Nvidia A100 GPU) using each model, using the single-precision floating point format. While the SD-UNet and SD-FNO consistently displayed better performance in terms of numer- The runtime costs for the (temporal) FNO model used as the temporal component of the approach to STMGFR are even higher than the spatial models, though it is likely that an implementationspecific issue was present as GPU power draw was observed to be substantially lower for this model during inference than the four spatial architectures despite high utilization statistics.
VII. CONCLUSION AND FUTURE WORK
This work introduced the Spatial and Spatio-temporal Multi-Geometry Flow Reconstruction (SMGFR, STMGFR) tasks for reconstructing dense contemporaneous or future vorticity fields of flows past arbitrary objects from current sparse sensor measurements, respectively, without geometry-specific training. To achieve optimal performance in these tasks, the use of Schwarz-Christoffel mappings to choose the sampling points of the dense fields was explored.
The performance of four different models was investigated on the SMGFR task, using datasets For the novel STMGFR task involving the prediction of future snapshots given current sensor measurements, an innovative approach separating the task to spatial and temporal components was developed, whereby a spatial model first reconstructs the current snapshot given the current measurements (equivalent to the SMGFR task) and subsequently a temporal model predicts the future snapshot given the predicted current snapshot. A stack of Fourier Neural Operator 13 layers acting as the temporal model was coupled with the best performing configuration from the SMGFR experiments used as the spatial model. The temporal model was trained to be robust to input noise caused by inaccuracies in spatial model predictions, by randomly providing it spatial model predictions or ground truth snapshots for each sample during training. Experimentation indicated that this approach is capable of accurately reconstructing future vorticity snapshots with mean absolute percentage error levels on the order of 30%. Furthermore, using a temporal gap of zero, the same two model setup can be used to improve accuracy of models used in SMGFR, also bringing their MAPE levels below 30%.
We hope to expand the investigations in this work in the future by:
• Developing models that perform well over a range of Reynolds numbers. The present work focused on predictions for flows at Re ≈ 300; the models presented are not expected to perform well for other Re and likely require re-training to adjust to different Reynolds numbers. Two potential ways of overcoming this are using a dataset containing snapshots from a range of Reynolds numbers, and using 'physics-informed' loss functions.
• Experimentation with more advanced neural network architectures. While this work focused on the relatively simple case of supervised training of feedforward and convolutional architectures to focus on investigating the mapping approach presented, techniques such as generative adversarial networks 54 and variational autoencoders 44 are gaining traction in FR literature. Adoption of such techniques may lead to higher accuracy in SMGFR and STMGFR tasks.
• Extending the methodology to 3D fluid flows. The mapping approach in this work relies on the Schwarz-Christoffel mapping, which is defined for the complex plane C only. A version of this work for R 3 will require an alternative mapping approach.
• Prediction of lift and drag coefficients. Although this work focused on the reconstruction of vorticity fields for the ease of comparison with previous works, changing the target fields to velocity and pressure can permit the prediction of the lift and drag coefficients. The mapping approach is especially conducive to this as, unlike Cartesian sampling, it eliminates the need to further process the vorticity and pressure fields to obtain values at the object boundary.
VIII. ACKNOWLEDGEMENTS
This work was supported by a PhD studentship funded by the Department of Aeronautics, Imperial College London and an Academic Hardware Grant provided by Nvidia. The authors would like to thank Sean Chai, Neil Ashton and Thomas Delillo for fruitful discussions at the start of this study.
IX. CODE REPOSITORIES
The pydscpack library for computing Schwarz-Christoffel mappings and the code for replicating the results in this work can be found on GitHub.
FIG
. 1. A random geometry (left) and its annular preimage (right). Blue and green contours depict the norm and argument in the w domain, respectively. The outer boundary of the domain is smaller than the ones used in the actual study for illustrative purposes.
A collection of 80 geometries G i , i ∈ [0, 79] were generated using random Bezier curves, via the bezier_shapes package by Viquerat et al.40 . The control points for the Bezier curves were chosen randomly in a square domain with characteristic length L m . Each geometry was placed in the center of a 40L m /3 × 40L m /3 square domain. The fluid domains were meshed with gmsh using a combination of triangular and tetrahedral elements, with c. 20000 elements per geometry.
Figure 2
2shows 12 of the geometries generated using this method.
FIG. 2 .
2Twelve geometries used in this study, among a total of 80.
1 .
1Shallow Decoder (SD) 4 : A 3 layer feedforward neural network with 40 units and ReLU activations in the intermediate layers, identical to the setup in Erichson et al. 4 . 2. SD-Large: A larger SD with 4 layers and 2048 units in each layer, included to assess whether the SD is sufficiently parametrized. Additionally, this model incorporates leaky ReLU activations and batch normalization 47 , motivated by the well-known positive impact of rectifier non-linearities 48 and activation normalization 43 on DNN performance. 3. SD-UNet: An SD model with 512 and 2048 units in the intermediate layers, followed by a reshape operation to a 2D grid and a four-level U-Net 12 model with 64 channels in the base level, leaky ReLU activations, batch normalization and p = 0.25 dropout. This model is included to assess the performance of a well-studied convolutional image-to-image translation model, as opposed to the fully connected SD architecture. 4. SD-FNO: An SD model identical to the one present in the SD-UNet, followed by four Fourier Neural Operator (FNO) 13 layers. Included due to remarkable performance of the FNO architecture in previous studies related to fluid flows.
FIG. 4 .
4Diagrams of the SD-UNet (top), an FNO layer (middle), and the SD-FNO (bottom). The values
FIG. 5 .
5x t,i =ω t,i ) given the measurements s t,i , STMGFR is a generalization to flow fields at k time-steps in the future given the current measurements; i.e. obtaining a reconstructionx t+k,i ofx t+k,i given s t,i . In this work, two values for the temporal interval are investigated: a short interval where k = 20 (equal to ∆τ * = 0.667, or 3.33% of the total simulation time), and a longer interval where k = 80 (equal to ∆τ * = 2.667 or approximately 13% of the total simulation time). This way it is possible to investigate temporal gaps both shorter and longer than the large eddy turnover time τ * = 1.0.An effective way to tackle this FR problem is first obtainingx t,i from s t,i using one of the models detailed in Section V A, dubbed the spatial model. Subsequently, a second model (the temporal model) is used to obtainx t+k,i fromx t,i , as summarized inFigure 5. A stack of six FNO layers was used as the temporal model, which have been demonstrated to be accurate when used to timemarch the two-dimensional Navier-Stokes equations13 . Every FNO layer has 64 channels and 32 modes per spatial dimension, while the convolutions use [1, 1] kernels, translating to 50,635,313parameters. The spatial model was chosen as the SD-UNet based on its high performance in the spatial task (as detailed in Section VI B, with identical weights), with a configuration identical to the one in Section V A. Likewise, the chosen sampling strategy is annular, due to its superior performance in Section VI B. Summary of the two-step spatio-temporal reconstruction approach.
fied in this section, Section VI C presents the efforts in combining the spatial-only configuration with a further time-marching model to predict future vorticity fields from current sensor measurements, as explained in Section V B. Section VI D compares the wall-clock runtimes of all models considered. To focus on the most rotational regions of the flows, the analyses are constrained to a region of the computational domain immediately surrounding and downstream of the objects investigated, encompassing a rectangular region with vertices [−L m , L m ], [−L m , −L m ], [4L m , L m ] and [4L m , −L m ].
Subsequently, we compute the SRCC and MI matrices D, M ∈ R (p+m)×(p+m) based on thecolumns of Ψ. Both are composed of four sub-matrices D s,s , M s,s ∈ R p×p ; D x,x , M x,x ∈ R m×m ; D s,x , M s,x ∈ R p×m ; D x,s , M x,s ∈ R m×p .The first two sub-matrices contain the SRCC values of the sensor measurements and full field values among themselves, while the latter two contain the SRCC values between the sensor measurements and the full field. The difficulty measures D and M are defined as the Frobenius norms of D s,x and M s,x , respectively:
Among the three major variables differentiating the experiments -the sampling method, model architecture and sensor setup -the factor most consistently leading to superior results is the sampling method. The annular sampling method enables substantially lower MAPE and HV-MAPE with all models and sensor setups tested, lowering MAPE and HV-MAPE by up to 15 and 21 percentage points respectively (translating to 26% and 40%), in the case of the SD model with the large sensor setup. The error-reducing effect of the annular sampling is especially prominent in the high vorticity regions of the flow, as evidenced by comparing the differences in HV-MAPE versus differences in MAPE when comparing the two sampling strategies; out of the 12 combinations of models and sensor setups, HV-MAPE has a larger absolute gap than MAPE between the two sampling methodologies in all cases except for the SD model using the Large sensor setup, with the differences in HV-MAPE being larger by about 3 percentage points on average.Next in the order of importance is the model architecture. The two architectures which more explicitly isolate the various spatial length scales, by the way of pooling-convolution-upsampling branches for the SD-UNet or convolution in Fourier space in the case of the SD-FNO, exhibit superior percentage error metrics for all sensor and sampling setups. This is not surprising given the established superiority of convolutional architectures in various computer vision tasks over fully connected networks 52 . However, the importance of the architecture varies by the sampling method. While MAPEs and HV-MAPEs are at most within a 5% and 3% range of each other (respectively) between models for a particular sensor setup in the case of annular sampling, this gap increases to as much as 14% with Cartesian sampling. Controlling for the number of parameters (as seen inTable II), the impact of the architecture declines even further, as the SD-Large model is consistently ahead of the standard SD model.The final variable to discuss is the sensor setup, which does not have a great impact on the accuracy of the models. The largest setup provides five times more information as the smallest, only leading to modest improvements in terms of the MAPE which do not exceed 5% for any model or sampling strategy. However, especially in the case of the SD and SD-Large models, taking advantage of the extra information is possible only in conjunction with the usage of annular sampling.The merit of having a large sensor setup is that, from a purely computational perspective, increasing the number of sensors is very cheap, as it leads only to an increase in the number of parameters and computational cost in the first layer of the model, opening up an avenue for modest accuracy improvements almost 'for free'. In an experimental setting, however, this may be counteracted by the burden of a significantly more labor-intensive physical sensor setup.
10 showcase the predictions for 'Shape B', an oval-like object set at a high incidence angle relative to the incident flow displaying separation over its upper surface, and 'Shape C', a thick flat plate-like object with characteristic counter-rotating vortices immediately downstream of the object, for the large sensor setup. Juxtaposing the models' performance over a range of flows showing a diverse range of dynamics, the three snapshots are chosen to illustrate different levels of relative performance between the sampling strategies and models; Shape A's snapshot is chosen to display the relative strengths of the top performing combination (SD-UNet with annular sampling), Shape B displays a case where Cartesian sampling wins in terms of MAPE (albeit losing by a far greater margin in terms of HV-MAPE), and in Shape C's case the top performers are closely matched. Starting with Shape A in Figures 6-8, the focus is on the differences between the Cartesian and annular sampling methods. The boundary layers stand out as the areas of greatest difference between the sampling methods. With Cartesian sampling, none of the models can accurately predict the existence of the high vorticity concentrations near the stagnation point, the highly positive concentration of vorticity along the upper surface of the object, or the location of the separation on the lower surface. In contrast, all three of these features are correctly predicted with the annular sampling, even by the relatively simple feedforward SD and SD-Large models, regardless of the sensor setup. The two sampling strategies are closer in terms of performance in the wake, as evidenced by the greater similarity in the error maps, but the Cartesian models have higher error downstream of the object.Additionally, comparing the predictions from different sensor setups for Shape A, the above conclusion regarding the limited impact of the setup on model performance is reinforced. Error levels do not show a strong tendency to decline with larger sensor setups, exemplified by the lowest errors for the SD-UNet being displayed for the medium sensor setup inFigure 7. Randomness in model training, caused by model initialization and stochastic model architecture features like dropout, has a substantially greater impact on performance in particular snapshots. This effect disappears when averaged over the whole dataset and many epochs of training, resulting in the paradoxical looking situations like predictions from models using fewer sensors being more accurate for specific samples despite higher overall error.
flow features, appropriately placing the concentrations of vorticity near the stagnation point, the separation near the leading edge, and the starting vortex-like formation near the rear of the object, though also performing poorly with respect to the vortex shed further downstream of the object.These issues are not as prevalent for the more complicated SD-UNet model, with the Cartesian sampling ekeing out a rare quantitative win in terms of the MAPE (by 5%) against annular for the SD-UNet owing to the high error with annular sampling far downstream of the object despite more accurate predictions near the object. Lower performance further away from the object in the case of annular sampling is caused by the same radial and angular spacing, which translates to a lower grid point density further away from the object (see Section IV B). As a consequence, as with the other snapshots, annular sampling displays greater performance nearer the object and achieves a decisive win versus Cartesian sampling once the low-vorticity regions far downstream of the object are filtered out, achieving a 15% lower HV-MAPE.
for the spatial model's performance). A set of results with k = 0, at which point the temporal model (FNO) effectively acts like an autoencoder since ground truth snapshots are provided as inputs, are included in addition to the k = 20 and k = 80 values mentioned in Section V B to provide greater insight into the temporal model's behavior.
its greater difficulty. The reason for this is clearly illustrated by the k = 0 results: the additional FNO model placed 'at the end' of the SD-UNet model, with a parameter count exceeding the entirety of the SD-UNet, serves as a denoising autoencoder 53 which substantially reduces the error of the SD-UNet output. Effectively acting like a model employed in SMGFR as opposed to STMGFR in this setting, examples of reconstructed snapshots from this k = 0 'SD-UNet + FNO' configuration are presented in Appendix B.As k increases, the denoising effect declines monotonically as the temporal model must simultaneously correct for the errors in its input and predict the time evolution of the input snapshot.Hence, to dissect the sources of error, we focus on results with k = 80 over a number of example snapshots; similar to Section V A,Figures 11 to 13provide example ground truth and predicted snapshots from three further shapes -dubbed Shapes D, E and F -chosen from the validation dataset for displaying a diverse range of flow dynamics.Shape D, inFigure 11, is a thin flat plate like object set at a high angle of attack relative to the incoming flow. The temporal model predictions replicate key flow features successfully, correctly predicting the location of the two shed vortices behind the object, demonstrating that the model is effective at capturing the mechanisms of advection in this flow. Some notable sources of error are the under-estimation of the intensity of these vortices and the missing concentration of high positive vorticity on the lower surface of the object, which is uncharacteristic given the good performance of SD-FNO predictions in Section V A in this regard.Shape E (Figure 12) is a bluff-body like object similar to Shape A (Figure 6). The temporal reconstruction in the figure has the lowest MAPE among the three snapshots presented in the figures. The phenomenon of note in this example is that, unlike the previous example where the most challenging aspect of the problem was predicting the advection of vortices, the model has to predict the formation of a vortex at time t + k from the snapshot at time t, in this instance immediately behind the object. This is executed very well by the model, as evidenced by the error map, which shows very low error (below 10% throughout) for the region corresponding to the vortex.
relative to the incoming flow. The challenge present in this example is essentially the combination of the those in Figures 11 and 12; there is no vortical structure present at time t, but by time t + k vortical structures have already formed and advected downstream from the object. This more complicated challenge translates to relatively higher quantitative error metrics for the snapshot depicted in the figure compared to the previous examples, but critical flow features are very well replicated at time t + k. The two shed counter-rotating vortices are predicted at the correct location with correct intensity, the shape of the interface between high and low vorticity regions immediately downstream of the object is correct, and the highly rotational intense separation regions past the leading and trailing edges of the object are present. Overall, despite the substantially more difficult challenge in this example, the model runs well and offers a good understanding of key fluid dynamics phenomena in this flow.
fields at time t (top) and t + k (bottom); the middle column depicts the reconstruction of the snapshot at t from sensors by the spatial model (top), the reconstruction of the snapshot at t + k given the ground truth snapshot by the temporal model (middle) and the reconstruction of the snapshot at t + k given the spatial model reconstruction (bottom); the right column displays the corresponding error maps. Vorticity colormap constrained to 5% of max(|ω t+k,i |).
FIG. 13 .
13STMGFR as applied to a snapshot from Shape F. The left column shows the ground truth vorticity fields at time t (top) and t + k (bottom); the middle column depicts the reconstruction of the snapshot at t from sensors by the spatial model (top), the reconstruction of the snapshot at t + k given the ground truth snapshot by the temporal model (middle) and the reconstruction of the snapshot at t + k given the spatial model reconstruction (bottom); the right column displays the corresponding error maps. Vorticity colormap constrained to 3% of max(|ω t+k,i |).
ical error metrics and qualitative factors such as correct predictions of locations and intensities of vortical structures in Section VI B, this improvement comes at a substantial cost in terms of training and model runtime. This can substantially hinder the relative usefulness of the SD-UNet and SD-FNO in a laboratory setting where real-time reconstruction of flow on lower-power devices is desired.
generated via both the novel mapping aided sampling strategy and a more traditional Cartesian sampling strategy, with three different sensor setups. The results showed that the mapping aided approach provides a substantial boost in accuracy for all model and sensor setup configurations, enabling percentage errors under 3%, 10% and 30% for reconstructions of pressure, velocity and vorticity fields, respectively. Improvements in terms of mean absolute percentage error exceed 15 percentage points in select cases for the challenging vorticity reconstruction tasks, while the impact of the size of the sensor setup is modest. The best performing model architecture was a convolutional architecture based on the U-Net 12 . Comparisons of snapshots from the different configurations revealed that the usage of the mapping approach substantially boosted the accuracy of the predictions in the immediate vicinity of the objects.
FIG. 14 .FIG. 15 .
1415Ground truth (left), predicted (middle) and percentage error (right) fields for pressure (top), uvelocity (middle) and v-velocity (bottom) for a validation snapshot. Examples using the spatial (SD-UNet) plus temporal (FNO) model architecture with k = 0 from Section V B. Left column contains the ground truth snapshots, middle column contains the predictions and the right column contains the percentage error maps.
divide the types of approaches to FR into three categories:1. Direct reconstruction: A set of parameters is optimized to learn an approximator R to the inverse operator G, as summarized in Equation 3.2. Regressive reconstruction:The unsupervised learning counterpart of direct reconstruction methods, these methods attempt to fit a series of modes 15 to the available flow data. Methods relying on the principal orthogonal decomposition (POD) fall within this category.3. Data assimilation: Dynamical systems based approaches such as Kalman filtering 16 are used to evolve high-dimensional systems in time based on sensor measurements Historically, some of the earliest works in FR originate from meteorology, dating back to the 1980s. Falling mostly within the third category of this classification, Gustafsson et al.17 provide a comprehensive survey of these methods. Framing the FR task in a variational setting[18][19][20] , they aimed to predict the time evolution of weather systems given past states and point observations from satellites and ground stations. Despite widespread adoption in weather prediction throughout the world, such approaches are unsuitable for the specific setting explored in this work (reconstruction of current and future vorticity fields from a single instantaneous measurement), due to the necessity of supplying high-quality initial guesses and the time evolution of sensor measurements over a long period of time, which are assumed to be unavailable.Interest in FR outside the meteorological community emerged in the late 1980s, with methods that fall within the second category of the above classification. Linear stochastic estimation(LSE) 7 is a prominent tool with roots in this era, which reconstructs flow fields based on the computation of a correlation matrix between the sensor inputs and the full flow fields. Recently, it has been applied to reconstruction of PIV-obtained fields in flows over flat plates from microphone measurements 21 , reconstruction of velocity fields in internal combustion engines from sensor measurements to identify locations of vortical structures 22 or estimation of all components of the dense velocity field in wall turbulence from hot-wire measurements of the horizontal velocity only 23 . A further early statistical technique for estimating dense flow fields given sensor measurements is the Gappy POD 5 method, originally developed for reconstructing images of human faces. Examples of the application of the Gappy POD method to FR have included predicting missing information in Direct Numerical Simulation (DNS) data 6 , estimation of the lift coefficient of a NACA0012 airfoil under plunging motion in a Mach 0.6 flow by reconstructing the simulated dense velocity and pressure fields from pressure sensors on its surface 24 , and filling in missing data points in PIV snapshots from gas turbine combustors 25 .The spectrum of statistical methods applied to FR contains several further less-investigated avenues. Notably, the sparse representation 26 technique, with origins in facial recognition, has been recently applied for FR of free-shear and mixing layer flows27 . New methods, such as the Sparse Fourier Divergence Free method 28 dedicated specifically to the investigation of incompressible flows, are being developed. Unfortunately, the linear unsupervised nature of the methods in this category lends itself to making predictions for flow datasets incorporating a single geometry only, and we are not aware of any previous works using such methods for multi-geometry datasets. The universal approximation capabilities of deep neural networks with nonlinear activation functions29 , belonging to the first category in the above classification, are therefore a better fit for multi-geometry FR as evidenced by their state-of-the-art performance in many generative Machine Learning (ML) tasks such as text30 and image 31 generation. The strength of NNs is evident in FR research as well, with a rapid rise in the number of works investigating the application of neural networks to FR.Still, most NN-based FR techniques focus on training models that work for a single flow connoisy sparse measurements from the object surfaces, finding that although neural models are not necessarily the best performing at low noise, they constitute the most robust option under high noise situations. Sun and Wang 10 investigated the usage of physics-constrained Bayesian NNsfiguration only (without retraining), some notable examples of which are provided: Erichson et
al. 4 reconstructed vorticity fields in the wake of a circular cylinder from measurements on its sur-
face using a 'shallow' NN (i.e. feedforward NN with few layers and units), while Kumar et al. 32
improved upon this technique with a recurrent autoencoder based architecture which can reduce
the error by over an order of magnitude when extremely sparse sensor setups involving only a
single sensor are used. Dubois et al. 8 investigated linear and nonlinear autoencoder NNs with and
without variational training, and found that non-linearities in NNs enable identification of domi-
nant flow modes, while variational training leads to higher robustness to noise at the cost of higher
error. Fukami et al. 9 compared the performance of feedforward NNs with three non-neural ML
techniques on reconstructing the wake behind a circular cylinder and a flapped airfoil based on
for flow reconstruction from sparse sensors in stenotic vessels and T-shaped geometries, which
permit significantly higher noise robustness compared to standard NNs. Usage of NNs in this con-
text have also been extended to experimental as opposed to simulated data, for example by Carter
et al. 33 who used feedforward NNs for the reconstruction of experimental PIV-obtained velocity
fields above the suction surface of a NACA0012 airfoil, achieving superior results compared to
non-neural methods.
TABLE I .
INumber of sensors by type and configuration # of pressure sensors # of velocity sensors FIG. 3. Illustration of the medium sensor setup for one of the geometries used in the study.Small
12
4
Medium
25
9
Large
50
25
L m
6
4L m
3
2L m
3
Pressure sensors
Velocity sensors
Object centroid
similar
TABLE II .
IINumber of parameters per model and sensor setup, broken down by layer type, for SMGFR.Only the first feedforward layer's parameter numbers are affected by the sensor setup.Large sensor setup
Medium sensor setup
Small sensor setup
Dense Conv/FNO
Total
Dense Conv/FNO
Total
Dense Conv/FNO
Total
SD
677,424
-
677,424
675,144
-
675,144
674,224
-
674,224
SD-Large 46,399,488
-
46,399,488 46,282,752
-
46,282,752 46,235,648
-
46,235,648
SD-UNet 34,683,392 7,711,301 42,394,693 34,654,208 7,711,301 42,365,509 34,642,432 7,711,301 42,353,733
SD-FNO 34,683,392 2,105,857 36,789,249 34,654,208 2,105,857 36,760,065 34,642,432 2,105,857 36,748,289
D1,D2)Shallow Decoder
+ Reshape
(1,D1,D2)
Sensor inputs
(n_sensors)
Fourier Neural
Operator
(16,D1,D2)
Padded 2D
Convolution
(16,D1,D2)
Fourier Neural
Operator
(16,D1,D2)
Fourier Neural
Operator
(16,D1,D2)
Fourier Neural
Operator
(16,D1,D2)
Padded 2D
Convolution
(128,D1,D2)
Padded 2D
Convolution
(1,D1,D2)
Dense flow field
(1,D1,D2)
TABLE III .
IIIMean absolute error (MAE) and mean absolute percentage error (MAPE) levels from the pressure and velocity reconstruction experiments using the k = 0 SD-UNet+FNO combination from Section V B. MI scores were computed using 50 × 50 bins via sklearn's 51 adjusted_mutual_info_score. ω results are identical to k = 0 results in Section VI C -note that MAE is lower with Cartesian sampling despite higher MAPE, as Annulus sampling concentrates grid points near the boundary of the object wherelarger vorticity concentrations are typically observed.
p
u
v
ω
D
(Higher is easier)
306.22
298.99
301.85
242.82
M
(Higher is easier)
302.72
296.37
269.81
207.75
Annular
MAE 1.18 × 10 −2 2.64 × 10 −2 1.22 × 10 −2 3.10 × 10 −1
MAPE 2.43%
8.26%
9.40%
28.89%
Cartesian
MAE 1.33 × 10 −2 3.32 × 10 −2 1.64 × 10 −2 4.95 × 10 −2
MAPE 3.32%
11.56%
15.61%
33.56%
TABLE IV .
IV% errors of the model predictions, averaged over the validation dataset, per sensor setup and sampling strategy. Two different values for the % errors are presented; the standard MAPE, and the High Vorticity MAPE (HV-MAPE) filtering out the data points with vorticity magnitude below 1% of the max ground truth vorticity. In both cases, sampling points with percentage errors above 200% were excluded from the MAPE calculation, due to error values approaching infinity near the ground truth zero vorticity field contours. The best performing sampling strategy-model architecture combinations for each sensor setup are highlighted with italics and underscores for the MAPE and HV-MAPE, respectively.Two different metrics of percentage error, as seen inTable IVare used in this section: the standard MAPE, and the High Vorticity MAPE (HV-MAPE) computed from points for which the vorticity magnitude exceeds 1% of the maximum absolute vorticity in a snapshot. Comparing the MAPE and HV-MAPE, it can be seen that HV-MAPEs are consistently lower than the MAPEs, which is not surprising given the choice of the MAE as the loss function which assigns a greater penalty to the regions of higher target field magnitude given constant percentage errors. As a consequence, the models implicitly prioritize lowering the percentage errors for areas with high vorticity, which is desirable since these are the most important features when analyzing the dynamics of fluid flows. Overall MAPEs are between 40% and 47% for the Annular sampling while they are between 46% and 60% for the Cartesian sampling. HV-MAPEs can go as low as 31% for the Annular sampling and 39% for the Cartesian sampling.Sensor setup Sampling
SD
SD-Large
SD-UNet
SD-FNO
MAPE HV-MAPE MAPE HV-MAPE MAPE HV-MAPE MAPE HV-MAPE
Large
Annular 44.29% 34.28% 43.80% 31.85% 39.92% 31.37% 40.83% 31.78%
Cartesian 59.88% 46.14% 57.52% 53.36% 47.64% 39.88% 46.56% 39.34%
Medium
Annular 46.86% 34.35% 45.45% 33.77% 42.61% 32.73% 43.66% 32.93%
Cartesian 55.41% 48.99% 58.65% 48.68% 48.49% 42.99% 48.17% 42.12%
Small
Annular 47.35% 34.72% 47.20% 34.06% 45.03% 33.17% 44.04% 33.77%
Cartesian 57.40% 50.89% 59.77% 48.59% 51.56% 44.47% 51.81% 45.81%
TABLE V .
VTable IV. Only the temporal model was re-trained for different values of k.% errors of the temporal model predictions, averaged over the validation set, given ground truth
snapshots and reconstructed snapshots (i.e. spatial model outputs) as inputs. Outliers filtered identically to
k (∆τ * )
0 (0.0)
20 (0.667)
80 (2.667)
MAPE HV-MAPE MAPE HV-MAPE MAPE HV-MAPE
From ground truth 19.76% 10.75% 23.40% 11.75% 29.58% 19.53%
From reconstruction 28.89% 17.86% 31.02% 17.86% 31.88% 21.97%
Ground truth at t MAPE: 27.41% FIG. 11. STMGFR as applied to a snapshot from Shape D. The left column shows the ground truth vorticity fields at time t (top) and t + k (bottom); the middle column depicts the reconstruction of the snapshot at t from sensors by the spatial model (top), the reconstruction of the snapshot at t + k given the ground truth snapshot by the temporal model (middle) and the reconstruction of the snapshot at t + k given the spatial model reconstruction (bottom); the right column displays the corresponding error maps. Vorticity colormap constrained to 5% of max(|ω t+k,i |). MAPE: 23.88% FIG. 12. STMGFR as applied to a snapshot from Shape E. The left column shows the ground truth vorticityReconstruction at t
% Error
4
2 0
2
4
0 20 40 60 80 100
MAPE: 28.54%
Ground truth at t + k
Reconstruction at t + k
% Error
From ground truth at t
MAPE: 27.28%
From Reconstruction at t
Ground truth at t
Reconstruction at t
% Error
4
2 0
2
4
0 20 40 60 80 100
MAPE: 24.25%
Ground truth at t + k
Reconstruction at t + k
% Error
From ground truth at t
MAPE: 22.83%
From Reconstruction at t
TABLE VI .
VIWall-clock runtimes for training and running the models in Sections VI B and VI C for the large sensor setup. Training runtime statistics are compiled for the training dataset (consisting of 601 × 64 = 38464 snapshots); inference statistics are for the validation dataset (consisting of 601 × 16 = 9616 snapshots). The training time column shows the total training time (equal to [Time/epoch] × [No. of epochs]). Model Batch size Batches/epoch Time/epoch (s) No. of epochs Training time (s) Inference time (s)Spatial
SD
500
77
5
813
4065
0.76
SD-Large
500
77
7
560
3920
1.52
SD-UNet
48
802
157
183
28731
8.98
SD-FNO
50
770
147
196
28812
6.91
Temporal
FNO
(k = 0)
100
372
187
248
46376
44
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| [] |
[
"Symmetrization and enhancement of the continuous Morlet transform",
"Symmetrization and enhancement of the continuous Morlet transform"
] | [
"Robert W Johnson [email protected] \nAlphawave Research Atlanta\nGAUSA\n"
] | [
"Alphawave Research Atlanta\nGAUSA"
] | [] | The continuous wavelet transform using the Morlet basis may be symmetrized by using an appropriate normalization factor. The cone-of-influence is addressed through a renormalization of the wavelet based on power. The power spectral density may be deconvolved with the wavelet response matrix to produce an enhanced wavelet spectrum. The enhanced transform provides maximum resolution of the harmonic content of a signal. | 10.1142/s0219691311004511 | [
"https://arxiv.org/pdf/0912.1126v2.pdf"
] | 17,421,998 | 0912.1126 | 27a1e3a7fab2f5f3dc16585229f310f338454f04 |
Symmetrization and enhancement of the continuous Morlet transform
6 Dec 2009 December 6, 2009
Robert W Johnson [email protected]
Alphawave Research Atlanta
GAUSA
Symmetrization and enhancement of the continuous Morlet transform
6 Dec 2009 December 6, 2009arXiv:0912.1126v1 [physics.data-an]MSC: 42C4065T60 kewords: continuous wavelet transformwavelet enhancementpower spectral densitydata analysis
The continuous wavelet transform using the Morlet basis may be symmetrized by using an appropriate normalization factor. The cone-of-influence is addressed through a renormalization of the wavelet based on power. The power spectral density may be deconvolved with the wavelet response matrix to produce an enhanced wavelet spectrum. The enhanced transform provides maximum resolution of the harmonic content of a signal.
Introduction
The continuous wavelet transform using the Morlet basis [1,2] has become quite popular for data analysis [3,4,5,6,7,8,9,10,11,12,13,14]. There is some variety in the literature as to the assignment of the normalization factors, and we propose a rearrangement so as to produce a symmetric forward and inverse transform pair. The cone-of-influence is addressed through a renormalization of the wavelet amplitude which keeps its power constant for a given scale. The renormalized power spectral density may then be enhanced by deconvolution with the wavelet response matrix, yielding the maximum spectral resolution of the harmonic content. We conclude by discussing the utility of these algorithms and point out a recent application.
The lack of a quantitative power spectral density has long hampered wider adoption of the continuous wavelet transform for data analysis. A mathematical engineer wants more than just a pretty picture-being able to give a numerical estimate to the power carried within a particular frequency band is of practical importance, and the use of a wavelet rather than Fourier transform allows that estimate to be time dependent. When the data is limited in duration, interesting features may be located in that region where wavelet truncation has become a significant effect. By renormalizing the power, the useful range may be extended beyond the cone-of-influence, and perfect reconstruction holds for all but the edge-most locations.
The breadth in scale of the wavelet response to a pure signal tone is a consequence of its localization in phase space, as the spectral and temporal resolutions are inversely related. Again, what a mathematical engineer wants is a precise identification of the frequency spectrum, in which features are maximally resolved. By treating the continuous wavelet transform as a theoretical apparatus acting upon a signal, one may essentially calibrate the device given some basic assumptions on the form of the signal components. Minimizing the discrepancy between the continuous instant wavelet power and the convolution of the calibration matrix with the estimate then yields the sharpest resolution of the time-varying harmonic content of a signal.
Normalization and central frequency
We first discuss the symmetrization of the transform and its effect on the central frequency employed. One may write the usual Morlet wavelet [6,13] at scale s = 1/f s = 2π/ω s and offset t with unity sample rate ∆t ′ ≡ 1 using the parameter η ≡ (t ′ − t)/s as the product of a scale dependent constant C, a normalized window Φ, and a normalized wave Θ,
ψ 0 s,t (t ′ ) ≡ C 0 s Φ 0 s,t (t ′ )Θ 0 s,t (t ′ ) = π −1/4 s −1/2 e −η 2 /2 e iω 1 η ,(1)
where ω 1 ≈ 2π is the central frequency of the mother wavelet at unity scale and zero offset, ψ 0 1,0 (t ′ ) = π −1/4 e −t ′2 /2 e iω 1 t ′ . The window has an extent of −⌊sχ to ⌊sχ and the wavelet a length of N t ′ = 2⌊sχ+1, where the parameter χ = 6 defines the resolution width. The mother wavelet is normalized to unit energy ∞ −∞ | ψ 0 1,0 (ω)| 2 dω = 1 so that the Fourier transform of ψ 0 (t ′ /s) is ψ 0 (sω) = √ 2πs ψ 0 1,0 (sω). Pulling over from the denominator of the inverse transform a factor of the scale s and including a factor √ 2 which represents the transform response at negative scales gives a normalization C s = √ 2 C 0 s /s which produces a transform with some very desirable properties,
ψ s,t (t ′ ) = √ 2 π −1/4 s −3/2 e −η 2 /2 e iω 1 η .(2)
The mother wavelet now has a norm of 2, which we interpret as including the response at negative scales to negative frequencies, and that of a scaled wavelet is now 2/s 2 . With appeal to the photon, the energy of a localized wave is proportional to its frequency E ν ∝ ν = s −1 ν , thus its power (energy per time) should be proportional to its energy over its period, P ν ∝ s −2 ν . The analysis of a (mean-subtracted) signal y(t) = k Re(A k e iω k t ) with duration N t proceeds by the symmetric forward and inverse transform pair for ψ * (η) = ψ(−η), with perfect reconstruction within the cone-of-influence and quantitative agreement between the estimated power and the sum of the squared amplitudes of the signal components (cf. Equations (6) and (9) by Frick, et al [6]). Use of logarithmic scale spacing requires retention of the factor ∆s. The root-mean-square power spectral density PSD(s, t) ≡ |CWT| 2 is normalized such that the integrated area of an isolated peak in the instant wavelet power IWP t (s) ≡ PSD(s, t) returns half the square of the amplitude A k of the signal component, P rms = k A 2 k /2. The margins of the PSD determine the mean wavelet power MWP(s) = N −1 t t PSD(s, t) and the integrated instant power IIP(t) = s PSD(s, t)∆s. The analysis of a test signal of duration N t = 300 time units with signal components of unit amplitude and periods of 5, 15, and 50 is shown in Figure 1. The coneof-influence is marked with a solid line in (a), and the more restrictive cone-of-admissibility denoting the first wavelet truncation at a given scale is marked with a dashed line. A trough appears at the scale of the signal duration N t in (b), beyond which we identify the extremely low frequency ELF region where s > N t . Apparent is the loss of transform response in (c), where the IIP falls below the rms power P rms = 1.5, and the loss of reconstruction in (d) at the signal edge.
CWT(s, t) = t ′ ψ * s,t (t ′ )y(t ′ ) ,(3)ICWT(t) = Re s t ′ ψ * s,t (t ′ )CWT(s, t ′ )∆s ,(4)
The central frequency given by Torrence and Compo [13] to unify the Fourier period and wavelet scale, λ 1 /s 1 = 4π/(ω 1 + 2 + ω 2 1 ) = 1 yielding ω 1 = 2π − 1/4π, is no longer appropriate for our normalization. Using the same test signal, we consider transforms with central frequencies 2π − 1/4π, 2π, and 2π + 1/4π and forward scalings of s −1/2 , s −1 , and s −3/2 appearing in the CWT. The top row in Figure 2 displays the instant wavelet power for a single central frequency at the center of the transform t = N t /2, and the bottom row shows its gradient for all three central frequencies in the vicinity of the central signal peak; similar graphs obtain for the other peaks, noting that the forward scaling of s −1 in (b) and (e) corresponds to that recently proposed by Liu, et al [15]. Only the transform with scaling s −3/2 produces peaks with an integrated area equal to half the sum of squared amplitudes, and we note that the locations of its peaks coincide with the signal periods for the central frequency of ω 1 = 2π+1/4π. The response of the symmetrically normalized CWT is that of a theoretical apparatus whose point spread function preserves the area of a Dirac distribution representing the power carried by a pure signal component of infinite duration with constant amplitude and period.
Renormalization
We next introduce a renormalization which compensates for the reduction in response outside the cone-of-influence. The cone-of-influence is defined by the e-folding time, t e = s √ 2 for the Morlet wavelet, indicating that region beyond which the response of the CWT is significantly affected by the wavelet truncation, which begins at the cone-of-admissibility. Various algorithms have been proposed for its rectification [5,6,8,16,12]; however, we have found that algorithms which alter the shape of the analyzing wavelet also affect its frequency response. Thus, we are led to proposing a simple renormalization such that for transform coefficients outside the cone-of-admissibility the wavelet is given a norm of 2/s 2 . For wavelets truncated by either edge of the signal, the window Φ τ is shifted by an offset τ relative to an unshifted window Φ 0 defining the time span t ′ . The length of a truncated wavelet ψ τ is defined to be the lesser of the raw wavelet length or the signal length, N τ = min(N t ′ , N t ). The offset τ (t) is determined from either the center of the signal or the location of the coneof-admissibility, and the algorithm to keep everything aligned gets a bit complicated: for
τ ′ = max(0, ⌊sχ − ⌊N t /2) and t ′ (k) = [−⌊sχ, ⌊sχ] indexed by k from 1 to N t ′ , if τ ≤ 0 then t ′ → t ′ [1, min(N t ′ , N t )] + τ ′ , else t ′ → t ′ [max(1, N t ′ − N t + 1), N t ′ ] − τ ′ .
The end result is simply to truncate either edge of the wavelet as necessary. Then for the amplitude of the truncated wavelet ψ τ = C s Φ τ Θ, with C s ← C s,t = C s 2/s 2 |ψ τ | 2 we define the renormalized continuous wavelet transform RCWT.
Using the same test signal as above, in Figure 3 we display the analysis using the RCWT; the reconstruction in (d) is noticeably improved, and the power estimation in (c) is not as affected near the signal edge. The apparent increase in the IIP over the rms value represents we feel an aliasing in time, rather than scale, of the total power, as the mean discrepancy from the rms power t (IIP − P rms )/N t = 5.6 × 10 −4 is small. In Table 1 we display the ratio of the mean instant power MIP = t smax s=2 ∆ s PSD/N t to the rms power for the CWT and RCWT, considering also an integration over scale which stops at the signal duration s max = N t rather than s max = 2N t . As pointed out by Frick et al [6], wavelet truncation also affects the admissibility condition. One commonly subtracts from Θ 0 ≡ e iω 1 η the DC component ψ 1 (0) ∝ e −ω 2 1 /2 ≡ d c ∼ 10 −9 of the mother wavelet so that the zero mean wave becomes Θ = Θ 0 −d c . Then, the adaptive wavelet transform is defined by Θ
= Θ 0 − d s,t , where d s,t ≡ Θ 0 τ = t ′ Θ 0 Φ τ / t ′ Φ τ
is the weighted mean of the remaining wave Θ 0 τ . Normalization as above with C s,t then defines the renormalized adaptive wavelet transform RAWT of Reference [16]. In practice, we have found that the RCWT neglecting admissibility outperforms the RAWT, shown in Figure 3(b) as a dashed line, by a small but noticeable margin; the troughs between peaks are slightly deeper, and the reconstruction is slightly better. The reason, we feel, is that the adaptive admissibility condition alters the shape of the wavelet, hence its frequency response.
The hallmark of wavelet analysis is its ability to track signal components with periods that vary in time. Considering now a test signal of the same duration, for periods 5, 15, and 50 we adjust the squared amplitudes to be 0.1, 1, and 0.5 respectively (rms power of 0.8) and impose independent sinusoidal variation to the periods on the order of the duration. In Figure 4 we display the RCWT analysis of such a signal, and in Figure 5 we compare the FFT power spectrum [17] with the MWP. The height of the peaks in the MWP appears to represent the relationship between the powers of the signal components better than the peaks in the FFT. Also, shown in Table 2 is the agreement between the mean squared amplitude of the signal and the sum of the FFT power spectrum, which are quite close numerically to the value of the integrated MWP-the discrepancy decreases with increasing resolution in scale. For comparison, the integrated MWP for the CWT analysis of this signal is 0.77325, which is significantly less than the rms power.
Enhancement
With our transform now responding like a theoretical apparatus for measuring a signal's power spectral density, one may apply the techniques of resolution enhancement common in the analysis of experimental data [18]. First one writes the point spread function as the response matrix R(s, s ′ ) defined by the integrated power of a wavelet of scale s convoluted with a signal component of period λ = s ′ . For this analysis we take the signal components to be Hann windowed cosine functions for the duration of the wavelet,
R(s, s ′ ) = t ′ ψ * s,0 (t ′ ) cos(2πt ′ /s ′ )H s (t ′ ) 2 ,(5)
which gave the best performance among the options. We note that here one is making an assumption on the form of the underlying signal elements whose composition represents the original signal. Then, for each IWP in the PSD, the enhanced instant power EIP is the solution to the equation
0 = s s ′ R(s, s ′ )EIP t (s ′ )∆s ′ − IWP t (s) 2 ∆s ,(6)
found in a least-squares sense with non-negativity constraints. Note that it is the redundancy in scale of the CWT which provides the resolution enhancement of the EIP. In practice, we apply the enhancement to the peak PSD using R → 2R. The effect is to replace broad peaks in the IWP with sharp spikes at the scale of the corresponding signal component, as shown in Figure 6 (a) for the IWP at the midpoint of the duration of the signal in the paragraph above. The reconstructed IWP (dashed) differs slightly from the original IWP (solid), sometimes exceeding it as no constraint has been placed on preserving the norm.
In general, one's wavelet response may extend beyond one's region of calculation for signal periods near either cutoff, and the enhancement procedure is capable of recapturing the lost (uncomputed) power. If one's application indicates the signal is bandwidth limited to that region well within the cone-of-influence yet far from the Nyquist scale, then enforcement of a norm-preserving constraint during the minimization is suggested. The enhanced power spectral density EPSD(s, t) is then defined simply as the collection of enhanced instant powers. Interestingly, one may effectively suppress the low frequency contribution by raising the tolerance of the minimization as in panel (b), where the EIP represents only the main signal components.
In Figure 7 we show the EPSD of our most recent test signal, as well as the mean enhanced power MEP and integrated enhanced power IEP, which exceeds the previous IIP by a significant margin for which we have not yet found a correction factor. The variation in scale of the signal periods is well-resolved within the cone-of-influence, and a video scanning through the EIPs is available as an online supplement. Reconstruction from the EPSD is not yet well-defined; however, one may attempt a reconstruction using the phase of the RCWT and the original renormalized basis as shown in (d), which might not be perfect but does faithfully represent the original signal. For signal components within the cone-of-influence, the EPSD provides the maximum resolution in scale available from the RCWT.
Conclusions
The utility of these algorithms should be apparent to anyone familiar with one dimensional data analysis and power spectrum estimation. The extension of the renormalization prescription to multi-dimensional wavelet analysis is straightforward; less so for the enhancement procedure. The symmetric normalization adopted here presents a power spectral density which behaves exactly as it should, where its margins may put on a physical basis, and provides perfect reconstruction without the introduction of an arbitrary factor. The mean wavelet power represents the relationship between signal components better than the fast Fourier transform, and the integrated instant power agrees with the rms power of the signal components.
A recent application of algorithms very similar to the ones presented here is found in [8,16]. In that work, by addressing the power spectral density of the historical sunspot record, a relation is found between the level of solar magnetic activity and the temperature observed in central England. One also may consider its application to signal encoding, manipulation, and compression, providing an alternate basis for reconstruction. For temporally resolved power spectrum estimation, the symmetric wavelet periodogram has become quite a useful tool indeed.
In summary, the continuous wavelet transform may be normalized to account for the response at negative scales, resulting in a symmetric forward and inverse transform pair with perfect reconstruction. It may then be renormalized to account for wavelet truncation by keeping a constant wavelet power across scale, where neglecting the admissibility condition results in better performance, extending the useful range beyond the cone-of-influence. By deconvolution with the wavelet response matrix, the enhanced power spectral density provides the maximum resolution in scale of the harmonic content carried by a signal.
Figure 1 :
1CWT power spectral density (a), mean wavelet power (b), integrated instant power (c), and reconstruction (d) for a test signal with components of unit amplitude and periods 5, 15, and 50 with rms power of 1.5. An unlabelled tick appears at the scale of the signal duration N t = 300. Overlaying the PSD in (a) are the cone-of-influence (solid) and the cone-of-admissibility (dashed).
Figure 2 :
2Instant wavelet power (top row) and its gradient at the central peak (bottom row) for forward transform normalizations labelled by column and central frequencies of 2π − 1/4π (dash-dot), 2π (dashed), and 2π + 1 4π (solid). The test signal has components of unit amplitude and periods 5, 15, and 50. Each peak in (c) has an area of 0.5, which is equal to the rms power of the signal component.
Figure 3 :
3RCWT power spectral density (a), mean wavelet power (b), integrated instant renormalized power (c), and reconstruction (d) for a test signal with components of unit amplitude and periods 5, 15, and 50. The MWP for the RAWT is shown in (b) as a dashed line. Reconstruction is maintained until the edge-most data points.
Figure 4 :
4RCWT power spectral density (a), mean wavelet power (b), integrated instant renormalized power (c), and reconstruction (d) for a test signal with components of timevarying periods around 5, 15, and 50 and squared amplitudes of .1, 1, and .5 respectively.
Figure 5 :
5The FFT power spectrum (dotted) compared to the RCWT mean wavelet power (solid). Locations of the discrete FFT values are marked ( * ).
Figure 6 :
6EIP (spikes) and IWP (solid) at the midpoint of the signal duration. The reconstructed IWP (dashed) slightly exceeds the original IWP. Panel (a) used a tolerance of 10 −8 , and panel (b) used a tolerance of 10 −4 to suppress the low frequency components.
Figure 7 :
7Enhanced power spectral density (a), mean enhanced power (b), integrated enhanced power (c), and reconstruction (d) for a test signal with components of time-varying periods around 5, 15, and 50. The increase in the IEP is attributed to the reconstructed IWP exceeding the original IWP.
Table 1 :
1Ratio of mean integrated power to P rms .PSD
CWT
RCWT
s max
600
300
600
300
MIP/P rms 0.95754 0.95697 1.002 1.0004
Table 2 :
2Comparison of FFT and RCWT rms power.t y 2 /N t
f FFT
600
s=2 ∆ s MWP
300
s=2 ∆ s MWP
0.79793
0.79793
0.80908
0.80777
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| [] |
[
"Reinforcement Learning for Decentralized Stochastic Control and Coordination Games",
"Reinforcement Learning for Decentralized Stochastic Control and Coordination Games"
] | [
"Bora Yongacoglu ",
"Member, IEEEGürdal Arslan ",
"Member, IEEESerdar Yüksel "
] | [] | [] | In the study of stochastic dynamic team problems, analytical methods for finding optimal policies are often inapplicable due to lack of prior knowledge of the cost function or the state dynamics. Reinforcement learning offers a possible solution to such coordination problems. Existing learning methods for coordinating play either rely on control sharing among controllers or otherwise, in general, do not guarantee convergence to optimal policies. In a recent paper, we provided a decentralized algorithm for finding equilibrium policies in weakly acyclic stochastic dynamic games, which contain team games as an important special case. However, stochastic dynamic teams can in general possess suboptimal equilibrium policies whose cost can be arbitrarily higher than a team optimal policy's cost. In this paper, we present a reinforcement learning algorithm and its refinements, and provide probabilistic guarantees for convergence to globally optimal policies in team games as well as a more general class of coordination games. The algorithms presented here are strictly decentralized in that they require only access to local information such as cost realizations, previous local actions, and state transitions. | null | [
"https://arxiv.org/pdf/1903.05812v1.pdf"
] | 119,157,875 | 1903.05812 | 5b89b9789a60c89e829b7e1a20f0f24dbf04450a |
Reinforcement Learning for Decentralized Stochastic Control and Coordination Games
14 Mar 2019
Bora Yongacoglu
Member, IEEEGürdal Arslan
Member, IEEESerdar Yüksel
Reinforcement Learning for Decentralized Stochastic Control and Coordination Games
14 Mar 2019arXiv:1903.05812v1 [math.OC] 1Index Terms-Stochastic teamsdiscounted cost optimizationdecentralized controlreinforcement learningmachine learning
In the study of stochastic dynamic team problems, analytical methods for finding optimal policies are often inapplicable due to lack of prior knowledge of the cost function or the state dynamics. Reinforcement learning offers a possible solution to such coordination problems. Existing learning methods for coordinating play either rely on control sharing among controllers or otherwise, in general, do not guarantee convergence to optimal policies. In a recent paper, we provided a decentralized algorithm for finding equilibrium policies in weakly acyclic stochastic dynamic games, which contain team games as an important special case. However, stochastic dynamic teams can in general possess suboptimal equilibrium policies whose cost can be arbitrarily higher than a team optimal policy's cost. In this paper, we present a reinforcement learning algorithm and its refinements, and provide probabilistic guarantees for convergence to globally optimal policies in team games as well as a more general class of coordination games. The algorithms presented here are strictly decentralized in that they require only access to local information such as cost realizations, previous local actions, and state transitions.
I. INTRODUCTION
Stochastic dynamic games are a significant generalization of both Markov decision processes (MDPs) and of repeated games, and provide a framework for studying cooperative decision making in a complex environment. Of particular interest are stochastic dynamic team games, in which all players incur the same stage costs, and coordination games in which there exist globally optimal joint policies. In finite state-action stochastic dynamic team games, at least one team optimal policy (that is, a joint policy that achieves the lowest expected discounted cost for every initial state) is guaranteed to exist. Despite the incentive to coordinate, the problem of finding and playing a team optimal policy is highly nontrivial due to incomplete information and decentralization. In this paper, we present a family of learning algorithms for team games and coordination games, and prove that these algorithms asymptotically achieve coordination on optimal policies. Furthermore, our algorithms are appropriate for decentralized control, in that they do not require control sharing among agents and require limited knowledge of the system before play.
When critical information about the other players, the cost function, or the transition probabilities is unavailable, standard tools used in classical stochastic control for recovering an optimal policy (e.g. dynamic programming, linear programming, and convex analytic methods) cannot be applied. Successful coordination on optimal policies requires identifying which policies are optimal, and in the event that there are multiple optimal joint policies, there is the further issue of selecting the one that will serve as the target for coordination. One viable alternative to analytical methods is to use learning algorithms, in which controllers simultaneously take actions and attempt to learn which policies are optimal.
A number of successful learning algorithms have been developed for single-agent environments such as Markov decision processes. Methods such as Q-learning [19] asymptotically learn optimal policies in spite of incomplete information. For the twin reasons that firstly multi-agent environments are ubiquitous in applications [9] and secondly that analytical approaches are not possible under incomplete information, there has been considerable interest in the study of reinforcement learning in games. The analysis of learning in games is much more difficult than that of single-agent learning: the presence of multiple decision-making agents, each actively learning and modifying their behaviour, results in a non-stationary environment, hence the convergence results from the singleagent literature at not guaranteed. In addition to the more complex, non-stationary environment an agent faces, there are also strategic considerations related to action selection. For discussion on the problem of teaching while learning, see [14] and the references therein.
Early works attempting to generalize modern reinforcement learning methods to multi-agent settings were experimental in nature: [16] studied Q-learning in a predator-prey game, [13] considered Q-learning as applied to specific block pushing task, and [3] considered both standard Q-learning and a modified joint action learner in repeated team games. Though conjectures were presented about the convergence of their methods, these early papers lacked rigorous results.
A number of rigorous results followed shortly after the experimental work, focusing primarily on zero-sum games and team games. [8] and its successor [7] developed the Nash Q-learning algorithm, in which each player maintains jointaction Q-factors for itself and for every other player, and computes Nash equilibrium at each step. Under uniqueness conditions, Nash Q finds equilibrium policies. Friend-or-Foe Q-learning (FFQ) is presented in [10]. FFQ is computationally inexpensive compared to Nash Q, and offers guarantees under weaker conditions, but requires common access both to the controls of all players as well as understanding of whether each other player is a 'friend' or 'foe'. [18] presents an algorithm for playing a stochastic game and proves that it converges to a team optimal equilibrium in any stochastic team game. More recently, [21] present an actor-critic algorithm for use in multiagent settings with some rigorous guarantees. The results of both [18] and [21] rely on common access to the past actions of all agents.
While these publications succeeded in finding optimal policies in spite of incomplete information, the methods they use require control sharing. That is, the actions taken by one player are required to be communicated to or observed by other players. In many interesting applications, for instance when controllers are in different physical locations and do not have communication channels between them, control sharing is either impossible or severely restricted. In such settings, the already difficult issue of multi-agent learning is further complicated. On the one hand, the evaluation of joint policies is complicated since individual agents cannot refer to the actions of the remaining agents, and on the other hand the set of all joint policies might not be effectively searched.
Another practical objection to the use of control sharing is that the number of joint actions grows exponentially in the number of players. If each player is required to keep a table of Q-factors for every state and joint action pair, the problem quickly becomes intractable. In contrast, while it lacks theoretical guarantees, direct application of standard Qlearning (without reference to joint actions) is sensible for at least this computational reason. Moreover, if players are oblivious to the existence of other players, then individual Q-learning is justifiable also from an individual rationality point of view, since the players may believe they are facing a Markov decision process.
In light of these considerations among others, there has been a steady interest in learning algorithms that allow for greater decentralization but still come with rigorous guarantees. [12] presents an algorithm for playing repeated games with guarantees for maximizing the sum of payoffs across agents. [11] provides three algorithms with provably desirable convergence properties for use in repeated games. The methods presented there require no control sharing, and rely instead on using the history of cost realizations to set an aspiration level that will be used to determine whether the current action choice is satisfactory. In the same vein, [2] provides an aspiration learning algorithm for playing a repeated coordination game. This method too does not use control sharing and has desirable convergence properties, but was designed for the relatively narrow class of repeated games rather than for stochastic games. Since the state does not change in repeated games, there are no long run considerations to account for when selecting actions, which allows the player to focus on the cost realizations alone when setting the aspiration levels. In contrast, stochastic games do have state transitions that impact the long run cost of a policy, hence cost "readings" are noisy and algorithm dependent, and the setting of the aspiration levels must account for that.
In [1], we introduced Decentralized Q-learning, a two timescale modification of Q-learning that provably finds meaningful Q-factors and leads to equilibrium in any weakly acyclic stochastic game, and in particular in any stochastic team game. The algorithm presented in [1] is decentralized in the sense that agents use only local information and do not share controls. However, the generality of this result comes at a cost: in stochastic team games, there are in general both team optimal equilibrium policies and suboptimal equilibrium policies, and the suboptimal equilibrium can perform arbitrarily worse than the optimal equilibrium. (A simple but illustrative example is offered in Section II). Consequently, guarantees of finding an equilibrium joint policy are not satisfactory in the context of decentralized control when cost minimization is a design goal.
In this paper, we present a decentralized learning algorithm and its refinements for playing stochastic coordination games, a class of games which model decentralized control problems and contain stochastic team games as a special case. We give formal guarantees of convergence to globally optimal policies. As observed above, prior work was either experimental with no rigorous results, or when rigorous required too much information sharing (in particular, control sharing). More recently, attempts to decentralize have found some success in the stochastic game setting, but have so far only guaranteed person-by-person optimality, while we guarantee global optimality.
The remainder of the paper is organized as follows: in section II, we specify the model of finite stochastic team games, give the problem statement, and provide the relevant background. In section III, we present our main algorithm and state a convergence result for stochastic teams, then extend the result to the broader setting of stochastic coordination games. Section IV relates the algorithm developed here in the team setting back to the more general setting of weakly acyclic games, and strengthens a result from [1]. Section V presents refinements of the main algorithm. These refinements leverage additional information that may be available a priori, and come with more powerful convergence results. Section VI contains numerical results from a simulation study, and the final section concludes. The proofs of our claims are contained in the appendices.
II. BACKGROUND AND PROBLEM STATEMENT A. Markov Decision Processes and Q-Learning
A (discounted) Markov Decision Process (MDP) is a discrete time process characterized by the following: 1) A finite set of states X 2) A random initial state x 0 ∈ X 3) A finite set of control actions U 4) A discount factor β ∈ (0, 1) 5) A cost function c : X × U → R 6) A Markov transition kernel P [·|x, u] for determining the next state given the current state-action pair At time t ∈ Z ≥0 , the system is in state x t ∈ X and the agent must select a control action u t ∈ U. The agent then incurs a stage cost c t (x t , u t ), and the system randomly transitions to the next state, x t+1 according to the probability distribution P [·|x t , u t ]. We assume that at time t = 0, the agent observes the state x 0 , and for times t ≥ 1, the agent has access to the following information:
I t = (x t , x τ , u τ , c(x τ , u τ ) : τ ∈ {0, 1, . . . , t − 1})
A policy is a rule for selecting control actions based on the history of observations. In principle, the agent can use any type of policy, provided their action selection is a (possibly random) function of their observations. However, we will focus on the set of stationary Markov policies, denoted ∆ := {γ : X → P(U)}, where P(U) is the set of probability distributions on U. A policy γ induces a probability measure on the sequence of states and actions, which in turn allows us to define the relevant cost criterion:
J x (γ) := E γ [ ∞ t=0 β t c(x t , u t )|x 0 = x]
The agent is interested in finding a policy γ * ∈ ∆ such that J x (γ * ) = inf γ∈∆ J x (γ) for every initial state x ∈ X. It is well-known (see, for example, [6]) that there exists a stationary deterministic policy π * : X → U that achieves this infimum. We will denote the set of all stationary and deterministic policies by Π := {π : X → U}. It is immediate that Π ⊂ ∆, and also that |Π|∈ N while ∆ is uncountable.
When the cost function and transition kernel is known, iterative methods such as value iteration can be used to recover an optimal policy. When the system's model is not known, model-free reinforcement learning techniques such as Q-learning [19] can be used to recover an optimal policy.
In standard Q-learning, the agent begins with an arbitrary vector of Q-factors Q 0 ∈ R X×U , and interacts with the system and updates the vector as feedback is received, according to the update rule
Q t+1 (x, u) = α t (x, u)[c(x t , u t ) + β min v∈U Q t (x t+1 , v)] +(1 − α t (x, u))[Q t (x, u)], if (x, u) = (x t , u t ) Q t (x, u), if (x, u) = (x t , u t )
where a t (x, u) is a step-size parameter. The algorithm is well-studied in the context of MDPs, and has been shown that if all state-action pairs are visited infinitely often and if the step-sizes are square summable but not summable almost surely, then Q t → Q * , where Q * is the fixed point of a Bellman-like update. See [20], [17] for details.
Once Q * is attained, one can recover the value function V * : X → R by taking V * (x) = min u∈U Q * (x, u), or one can recover an optimal policy π * by setting π * (x) = argmin u∈U Q * (x, u). Moreover, learned Q-factors can be exploited during play: [15] presents a method for playing a MDP in which the agent's action selection converges to that of an optimal policy.
The popularity of Q-learning in MDPs is justified: it is easy to implement, and comes with theoretical guarantees for asymptotically recovering optimal policies. However, it should be emphasized that these theoretical guarantees are predicated on stationarity of the system: when the agent visits a state-action pair, the feedback received-which comes in the form of the cost realization and the state transition-is always generated by the same Markovian source. If the system being controlled was not an MDP, then the theoretical guarantees do not apply. In particular, if multiple decision making agents are present and each agent is actively learning and modifying their policy, there is no immediate reason to believe individual Q-learning as presented should converge and be useful for recovering an optimal policy.
B. Stochastic Games and Decentralized Q-Learning
A finite stochastic game is a multi-agent generalization of a MDP, and is characterized by the following: 1) N ∈ N decision making agents, the i th denoted DM i 2) A finite set of states X 3) A random initial state x 0 ∈ X 4) For each DM i : A finite set of control actions U i A discount factor β i ∈ (0, 1) A cost function c i : X × U 1 × · · · × U N → R 5) A Markov transition kernel P [·|x, u] for determining the next state given the current state and current joint action u = (u 1 , . . . , u N ) ∈ U 1 × · · · × U N . At time t ∈ Z ≥0 , the system is in state x t , and each DM i chooses a control action u i t . While DM i only selects u i t , their incurred cost is given by c i (x t , u 1 t , . . . , u N t ). Following the play of this stage game, the system randomly transitions to state x t+1 according to P [·|x t , (u 1 t , . . . , u N t )]. We consider the situation in which DM i observes only the state variable and their own actions. At time t = 0, DM i observes x 0 , and at time t ≥ 1, we assume DM i has access to the following information:
I i t = (x t , u i s , x s , c(x s , u s ) : s ∈ {0, 1, . . . , t − 1})
In particular, DM i cannot see the past joint actions u j s for j = i and s ∈ N. This is in contrast to previous works such as [18], [7], [10] and [21].
In analogy to MDPs, the goal for DM i is to minimize the expectation of their discounted series of costs. Unlike the MDP setting, DM i 's costs are also affected by the control actions of the other decision makers. Policies are defined as before, and as before we will focus on stationary Markov polices. For DM i , the set of stationary Markov policies is denoted ∆ i := {γ i : X → P(U i )}, and the set of deterministic stationary policies is denoted Π i := {π : X → U i }. To refer to joint actions and joint policies, we denote the set of joint stationary Markov policies by ∆ := × N i=1 ∆ i , the set of joint stationary and deterministic policies by Π := × N i=1 Π i , and the set of joint actions is denoted by U := × N i=1 U i . We use the notational convention u −i to refer to the joint action of all agents except for i. That is, u −i = (u 1 , . . . , u i−1 , u i+1 , . . . , u N ), and we write u = (u i , u −i ). Similar shorthand will be used for policies.
A joint policy γ ∈ ∆ induces a probability measure on the sequence of states and joint actions, which we use in defining DM i 's cost criterion:
J i x (γ) := E[ k≥0 (β i ) k c i (x k , u k )|x 0 = x]
where u k ∈ U is the (random variable) joint action selected by the joint policy γ = (γ i , γ −i ) at time k. Then, DM i 's goal is to select a policy to minimize this cost.
Definition 1. Let γ −i ∈ ∆ −i be a stationary Markov joint policy. The policy γ * i is said to be a best response to γ −i (for DM i ) if J x ((γ * i , γ −i )) = min γ i ∈∆ i J x ((γ i , γ −i )), ∀x ∈ X Definition 2.
A joint policy γ * = (γ * 1 , . . . , γ * N ) ∈ ∆ is said to be a (Markov perfect) equilibrium if γ * i is a best response to γ * −i for each DM i ∈ {1, . . . , N }.
Every stochastic game has at least one equilibrium joint policy [5]. Since γ −i ∈ ∆ −i is stationary Markov, the environment that DM i faces is equivalent to a MDP. Hence, for any γ −i ∈ ∆ −i , there is always a stationary deterministic best response for player i. We denote this set of best responses by BR i γ −i := {π * i ∈ Π i : π * i is a best response to γ −i } = ∅. Moreover, we can describe the set BR i γ −i using Q i γ −i , the unique fixed point of Q-factor update for the equivalent MDP.
Q i γ −i ∈ R X×U i satisfies the fixed point equation Q i γ −i (x, u i ) = E γ −i (x) [c i (x, u i , u −i ) + β min v i ∈U i Q i γ −i (s(x), v i )]
where the expectation is taken with respect to the joint distribution of u −i determined by γ −i (x), and where s(x) denotes the state that follows x after (u i , u −i ) is played. Then, BR i γ −i can be expressed as
BR i γ −i = {π i ∈ Π i : Q i γ −i (x, π i (x)) = min v i ∈U i Q i γ −i (x, v i ), ∀x ∈ X}
Despite the existence of stationary deterministic best responses, the existence of a stationary deterministic equilibrium policy is not guaranteed in general. (When randomization is allowed, existence is guaranteed.) Repeated zero-sum games, such as matching pennies and rock-paper-scissors, offer simple examples of games without stationary and deterministic policies. However, when restricting to the class of games we are interested in, which model cooperative systems, we are guaranteed the existence of stationary and deterministic equilibrium policies. Definition 4. In a stochastic team game, a joint policy π * is said to be team optimal if
J x (π * ) = inf γ∈∆ J x (γ) ∀x ∈ X
Given a stochastic team game and supposing that the system was centralized, we can study an associated MDP with action set U = × N i=1 U i . Any joint policy in the stochastic team game is an admissible policy in the reduction (though the converse is not always true), so the minimum cost for a state in the reduction is no greater than the minimum cost achievable in the stochastic team game. Then, since the reduction is a MDP, there exists a stationary and deterministic optimal policy π * ∈ Π = {π : X → Π 1 × · · · × Π N }. We note that π * is an admissible joint policy in the corresponding team problem. By the previous observation that the reduction's optimal costs lower bound the achievable costs in stochastic team game, we have that π * is team optimal. Thus, in a stochastic team game, the set of stationary and deterministic team optimal policies Π OP T = {π ∈ Π : π is team optimal } is always non-empty.
Our objective as algorithms designers is the following: given a finite discounted stochastic team game without control sharing, we wish to provide the players with a learning algorithm for playing the game that provably converges to a team optimal policy. In [1], we presented Decentralized Q-learning and proved convergence to equilibrium policies almost surely in the larger class of weakly acyclic games. The algorithm instructs a player to use the same stationary policycalled a baseline policy-for a large number of consecutive stage games-the collection of which is called an exploration phase. At the end of an exploration phase (and never during one), players are allowed to update their baseline policies in a synchronized manner. In this way, the system is stationary for long enough for Q-learning to return meaningful Q-factors. The Q-factors acquired during an exploration phase can be used to construct a best response policy and the update process will eventually lead to players using baseline policies that form an equilibrium. 1 . After this is achieved, players will have no incentive to change their policies, and so play will settle. It is immediate that if π * ∈ Π OP T , then π * is an equilibrium joint policy. The converse is emphatically not true. One can construct stochastic team games for which suboptimal equilibrium joint policies exists. For an illustration of how poorly suboptimal equilibrium policies can perform, consider the repeated team game presented in Table I. Here, α > 1, |X|= 1, and the state dynamics are trivial. It is apparent that the joint policy of playing Up and Left in each period, denoted π sub , is an equilibrium joint policy, as is the joint policy of playing Down and Right in each period, denoted π * . We have that
J(π sub ) = k≥0 β k α = α 1−β while J(π * ) = k≥0 β k = 1 1−β .
Alarmingly, not only is J(π sub ) greater than J(π * ), but the performance gap, as measured either by J(π sub ) − J(π * ) or by J(π sub )/J(π * ), can be made arbitrarily large. Indeed, if α > 3, even the joint policy of independently selecting actions uniformly at random outperforms π sub . Consequently, the Decentralized Q-learning algorithm-which is guaranteed to find one of π * or π subdoes not achieve the design goal outlined above.
In the next section, we present a learning algorithm for playing stochastic team games. These algorithms build on the exploration phase technique from [1], but also exploit the following structural result on Q-factors in team games: Lemma 1. Consider a discounted stochastic team game as defined above. If π * ∈ Π OP T is team optimal andπ ∈ Π \ Π OP T is not team optimal, then for every DM i , we have that
x∈X Q i π * −i (x, π * i (x)) < x∈X Q ĩ π −i (x,π i (x))
Proof. Let V * be the value function for the corresponding reduction MDP for which the set of control
actions if U = × N i=1 U i . Let γ −i ∈ ∆ −i
be given and let V i γ −i be the value function for the MDP faced by DM i when the remaining players follow γ −i .
We have that
V * (x) = V i π * −i (x) ≤ V ĩ π −i (x)
and
V i π * −i (x) = min u i ∈U i Q i π * −i (x, u i ) = Q i π * −i (x, π * i (x)) V ĩ π −i (x) = min u i ∈U i Q ĩ π −i (x, u i ) ≤ Q ĩ π −i (x,π i (x))
From this, we can see that
x∈X Q i π * −i (x, π * i (x)) ≤ x∈X Q ĩ π −i (x,π i (x)) Suppose for some DM i we have Q ĩ π −i (x,π i (x)) = V * (x) for every x, then J x (π) = V * (x) = inf γ∈∆ J x (γ) for every state x.
That is,π is team optimal. This contradicts our choice ofπ, hence there must be some state x for which
Q ĩ π −i (x,π i (x)) > Q i π * −i (x, π * i (x))
, which completes the proof.
This fact provides for us an avenue for separating optimal policies from suboptimal ones by focusing on Q-factors.
III. NEW ALGORITHMS To motivate our algorithm, let us first consider a stochastic process on the set of joint policies Π that is induced by players updating their policies according to the following (unrealistic) procedure:
Idealized Update Procedure for DM i Set parameter γ i ∈ (0, 1), maps f i , h i : Π → P(Π i ) For exploration phase k ≥ 0 • If π k ∈ Π OP T : -π i k+1 = π i k , w.p. 1 − γ i ζ i k ∼ f i (π k ), w.p. γ i • Else (π k / ∈ Π OP T ) -π i k+1 = θ i k ∼ h i (π k ) (Increment k,K i ,K i > 0 such that min π∈Π,π i ∈Π i f i (π)(π i ) ≥ K i and min π∈Π,π i ∈Π i h i (π)(π i ) ≥K i .any ǫ > 0, there existsγ > 0 such that γ i ∈ (0,γ) for all i implies µ * f,h,γ (Π OP T ) ≥ 1 − ǫ Moreover, for any initial probability measure µ 0 on Π, we have lim n→∞ µ 0 A n f,h,γ = µ * f,h,γ . Proof. For notational ease, we proceed with µ * = µ * f,h,γ and A γ = A f,h,γ .
All players' following this procedure induces a time homogenous Markov chain on Π. For fixed γ = (γ 1 , · · · , γ N ) with γ i , K i ,K i > 0 for each i, we have that the chain is irreducible, hence there exists unique µ * such that µ * = µ * A γ . We use this to lower bound µ * (Π OP T ) = π∈Π OP T µ * (π) as follows:
π * ∈Π OP T µ * (π * ) = π * ∈Π OP T ( π∈Π µ * (π)A γ (π, π * )) = a∈Π OP T µ * (a)( π * ∈Π OP T A γ (a, π * )) + b / ∈Π OP T µ * (b)( π * ∈Π OP T A γ (b, π * )) ≥ [ N j=1 (γ i K i ) + N j=1 (1 − γ i )] a∈Π OP T µ * (a) + [|Π OP T | N j=1 (K i )] b / ∈Π OP T µ * (b) This implies that µ * (Π OP T ) ≥ F (γ), where γ = (γ 1 , . . . , γ N ), and F (γ) := |Π OP T | N j=1K j 1 − [ N j=1 (1 − γ j ) + N j=1 (γ j K j )] + |Π OP T | N j=1K j SinceK i > 0 for each i, F is continuous at γ = (0, · · · , 0) and F (0, . . . , 0) = 1. Hence, there exists some δ > 0 such that |γ|< δ implies F (γ) ≥ 1 − ǫ.
From this, the result follows immediately.
If our goal as algorithm designers is to ensure that the probability of players using team optimal policies is sufficiently high, the limiting behaviour of this idealized update process is desirable. However, this procedure clearly cannot be implemented: due to incomplete information about the game, Π OP T is unknown, and since players are not sharing controls, π −i k is not observed by DM i . Nevertheless, this motivates our algorithm, which is presented below.
Algorithm 1 (for DM i ) Set Parameters • Q i a compact subset of R X×U i • {T k } k≥0 , T k ∈ N for all k, the exploration phase lengths.
Common to all DM i -Define t 0 = 0, and in general t k+1 = t k + T k . tolerance for suboptimality when constructing best-response sets • d i > 0, a tolerance for sub-optimality when setting the aspiration level
• ρ i ∈ (0, 1) an experimentation probability • Functions f i , g i , h i : Π i × 2 Π i → P(Π i ) • δ i > 0 a• {α i n } n≥0 a sequence of step sizes such that α i n ∈ [0, 1] n≥0 α i n = ∞ n≥0 (α i n ) 2 < ∞ (e.g. a i n = 1/n r , for r ∈ (1/2, 1].) • γ i ∈ (0, 1)
, the probability of randomly updating baseline policy when aspiration level is satisfied • W ∈ N, a window size for constructing the aspiration levels. Common to all DM i .
• Select arbitrary π i 0 ∈ Π i and Q i 0 ∈ Q i • Receive the initial stage x 0 Iterate k ≥ 0 (k th exploration phase) For t = t k , t k + 1, . . . , t k+1 − 1 u i t = π i k (x t ) w.p. 1 − ρ i any u i ∈ U i w.p. ρ i /U i Receive cost c i (x t , u i t , u −i t ) Receive the next state x t+1 (chosen according to P [·|x t , u i t , u −i t ])
n i t = the number of visits to (x t , u i t ) in the k th exploration phase (up to and including t)
Q i t+1 (x t , u i t ) = (1 − α i n i t )Q i t (x t , u i t ) +α i n i t [c i (x t , u i t , u −i t ) + β i min v i ∈U i Q i t (x t+1 , v i )] Q i t+1 (x, u i ) = Q i t (x, u i ) for every (x, u i ) = (x t , u i t ) End (For) BR i k = {π i ∈ Π i : Q i t k+1 (x,π i (x)) ≤ min v i ∈U i Q i t k+1 (x, v i ) + δ i for every x ∈ X} S i k = x∈X Q i t k+1 (x, π i k (x)) If 0 ≤ k ≤ W − 1: Select π i k+1 uniformly at random from Π i Else: ∃B ∈ N such that B · W ≤ k ≤ BW + W − 1 Λ i k = min{S i (B−1)W , S i (B−1)W +1 , . . . , S i BW −1 } + d i If S i k < Λ i k π i k+1 = θ i k ∼ g i (π i k , BR i k ), w.p. 1 − γ i ζ i k ∼ f i (π i k , BR i k ), w.p. γ i Else (S i k ≥ Λ i k ) π i k+1 = ω i k ∼ h i (π i k , BR i k ) End End Reset Q i t k+1 to any Q i ∈ Q i (e.g. project Q i t k+1 onto Q i ) End k th exploration phase. (Go to exploration phase k + 1) Assumption 1. For any states x, x ′ ∈ X, there exists H ∈ N and joint actionsũ 0 , . . . ,ũ H such that P (x H+1 = x ′ |x 0 = x, u j =ũ j , ∀j ∈ {0, 1, . . . , H}) > 0
Assumption 1 is common to reinforcement learning methods. If it does not hold, then there exist transient states that will be visited a number of times and then never revisited. The play in these states affects the long run discounted cost, but there will be no opportunity for subsequent experimentation, and even cleverly designed learning algorithms will fail to reliably find optimal policies in such settings.
Assumption 2. For every player i = 1, . . . , N , we have 0 < δ i <δ and 0 < d i <d, whereδ,d are constants defined in the appendix that depend only on the game. There
Assumption 3. For each player i, we have (a) there ex- ists K i ,K i > 0 such that f i (π i , A i )(π i ) ≥ K i and h i (π i , A i )(π i ) ≥K i for all A i ∈ 2 Π i , π i ,π i ∈ Π i ; (b) if π i ∈ A i , then g i (π i , A i )(π i ) = 1.exists •γ(ǫ, (f i , h i ) N i=1 ) > 0 •ρ(δ i ,δ, d i ) > 0 •W (ǫ, (f i , h i , γ i ) N i=1 ) ∈ N • T ǫ (ǫ, (f i , h i , γ i , δ i , d i ) N i=1 W,δ) ∈ N such that if γ i ∈ (0,γ), ρ i ∈ (0,ρ) for every DM i , W ≥W , and T k ≥ T ǫ for all exploration phases k, then lim inf n P (π n ∈ Π OP T ) ≥ 1 − ǫ
Proof. The proof of Theorem 1 is given in the appendix.
The preceding algorithm was designed for stochastic team games, where c i = c j and β i = β j for all players i, j. However, the convergence result can be easily extended to a wider class of games that model cooperative systems.
Definition 5. A finite, discounted stochastic game is said to be a coordination game if there exists some set Π * ⊆ Π such that the following hold: We may call the set Π * the set of coordination policies. This definition is similar to what [2] calls a strict coordination game that satisfies hypothesis (H2), but is more general: firstly, it is applicable to stochastic games, and secondly it drops the requirement that all π ∈ Π\ (Π * ∪Π EQ ) are Pareto inefficient. In Appendix D, we present a comparison of Theorem 2 to the main result of [2].
(a) x∈X Q i π * −i (x, π * i (x)) < x∈X Q i π −i (x, π i (x)), ∀ DM i , π ∈ Π \ Π * , and π * ∈ Π * (b) x∈X Q i π * −i (x, π * i (x)) = x∈X Q i π −i (x, π i (x)), ∀ DM i , π, π * ∈ Π *
Using Lemma 1, it is immediate that a stochastic team game is a coordination game. Consider the cost minimization repeated game presented in Table II for an example of a coordination game that is not a team game. Here, Π * = (Down, Right). There exists
•γ(ǫ, (f i , h i ) N i=1 ) > 0 •ρ(δ i ,δ, d i ) > 0 •W (ǫ, (f i , h i , γ i ) N i=1 ) ∈ N • T ǫ (ǫ, (f i , h i , γ i , δ i , d i ) N i=1 W,δ) ∈ N such that if γ i ∈ (0,γ), ρ i ∈ (0,ρ) for every DM i , W ≥W , and T k ≥ T ǫ for all exploration phases k, then lim inf n P (π n ∈ Π * ) ≥ 1 − ǫ
Proof. The proof of Theorem 2 can be found in the appendix.
As is common in aspiration learning methods (see, e.g. [2]), Algorithm 1 largely relies on the principle of "win-stay, lose-shift." However, for reasons discussed below, we prescribe additional randomness during the policy update, with greater randomness when the current policy does not attain the aspiration level. This randomness ensures the effective search of the set of joint policies, prevents becoming stuck prematurely in suboptimal equilibrium joint policies, and facilitates analysis.
Algorithm 1 mimics the Idealized Update Procedure (IUP). The 'If' suite of IUP is replaced with a proxy for the question π k ∈ Π OP T : using experience up to time k, DM i will set an aspiration level Λ i k and a score S i k for the current policy. Verifying whether π k ∈ Π OP T is replaced by verifying whether S i k < Λ i k . Unlike previous aspiration learning methods such as [2], which largely focus on the repeated game setting, our algorithms are designed for stochastic team games. Due to the more complicated long run cost considerations in stochastic games, the issue of setting an aspiration level is harder than in the repeated game case, in which cost realizations can be used directly without reference to state transitions. Thus, absent major modifications, the prior methods cannot be used in stochastic games. In light of Lemma 1, a viable alternative is to use Q-factors to construct an aspiration level that separates optimal joint policies from suboptimal policies.
Relying on Q-factors to construct aspiration levels presents other problems. The first challenge is that the convergence of Q-factors is only asymptotic, and the learning process must be stopped at a finite time in order to update policies. Hence the obtained Q-factors will be random variables. This can be mitigated by allowing a longer exploration phase length. The second challenge is that if other players experiment and change policies erratically, the stationarity of the environment may be compromised and Q-learning may not converge even asymptotically. The two-timescale approach in [1] combined with low experimentation probabilities ρ i handles this issue.
The noise in the obtained Q-factors is the primary reason we opt to use long but finite memory for setting the aspiration level: we wish to discard unattainably good performances from memory. This finite memory is in turn one of the reasons for which we have introduced randomness into the policy update portion: to avoid the situation in which play has settled into a suboptimal equilibrium policy, and the aspiration level for each player is according to the suboptimal equilibrium rather than according to a team optimal policy.
IV. APPLICATION TO WEAKLY ACYCLIC GAMES
In this section we present a specific variant of Algorithm 1 with desirable convergence properties in weakly acyclic games.
Definition 6. Given a stochastic game as defined in Section II, a (possibly finite) sequence π 0 , π 1 , . . . in Π is said to be an improving path if for each k there is a unique player i k such that π i k k = π i k k+1 and furthermore π i k k+1 is a best response to π −i k k whereas π i k k is not. Definition 7. A stochastic game as defined in Section II is said to be weakly acyclic under strict best replies (responses) if there is an improving path starting at any π ∈ Π and ending at a deterministic equilibrium policy.
All team games are weakly acyclic games, though not all coordination games (as defined in Section III) are weakly acyclic-the repeated game presented in Table II, for example, is not weakly acyclic. In [1], it was shown that for any ǫ > 0, P (π n ∈ Π EQ ) ≥ 1 − ǫ could be achieved for n sufficiently large. However, if the weakly acyclic game being played was also a team game (or a coordination game), no optimality guarantees were provided. We now strengthen that result.
Algorithm 2 (for DM i ) Set Parameters • Q i a compact subset of R X×U i
• A sequence of positive integers {T k } k≥0 , the exploration phase lengths.
-Define t 0 = 0, and in general t k+1 = t k + T k . • ρ i ∈ (0, 1) an experimentation probability • λ i ∈ (0, 1) the probability of inertia • δ i > 0 a tolerance for suboptimality when constructing best-response sets • d i > 0 a tolerance for sub-optimality when evaluating a baseline policy • {α i n } n≥0 a sequence of step sizes such that
α i n ∈ [0, 1] n≥0 α i n = ∞ n≥0 (α i n ) 2 < ∞ • γ i 1 , γ i 2 ∈ (0, 1) such that γ i 1 + γ i 2 ≤ 1. • Common W ∈ N, a window size • Select arbitrary π i 0 ∈ Π i and Q i 0 ∈ Q i • Receive the initial stage x 0 Iterate k ≥ 0 (k th exploration phase) Iterate t = t k , t k + 1, . . . , t k+1 − 1 u i t = π i k (x t ) w.p. 1 − ρ i any u i ∈ U i w.p. ρ i /U i Receive cost c i (x t , u i t , u −i t ) Receive the next state x t+1 (chosen according to P [·|x t , u i t , u −i t ]) n i t = the number of visits to (x t , u i t )
in the k th exploration phase (up to and including t)
Q i t+1 (x t , u i t ) = (1 − α i n i t )Q i t (x t , u i t ) +α i n i t [c i (x t , u i t , u −i t ) + β i min v i ∈U i Q i t (x t+1 , v i )] Q i t+1 (x, u i ) = Q i t (x, u i ) for every (x, u i ) = (x t , u i t ) End BR i k = {π i ∈ Π i : Q i t k+1 (x,π i (x)) ≤ min v i ∈U i Q i t k+1 (x, v i ) + δ i for every x ∈ X} S i k = x∈X Q i t k+1 (x, π i k (x)) If 0 ≤ k ≤ W − 1:
Select π i k+1 uniformly at random from Π i Else:
∃B ∈ N such that B · W ≤ k ≤ BW + W − 1 Λ i k = min{S i (B−1)W , S i (B−1)W +1 , . . . , S i BW −1 } + d i If S i k < Λ i k w.p. γ i 1 , select π i k+1 ∈ Π i uniformly at random w.p. 1 − γ i 1 , select π i k+1 according to best respond- ing with inertia: If π i k ∈ BR i k , then π i k+1 = π i k . Else, then π i k+1 = π i k w.p. λ i π * i ∈ BR i k w.p. 1−λ i |BR i k | If S i k ≥ Λ i k w.p. γ i 1 +γ i 2 , select π i k+1 ∈ Π i uniformly at random w.p. 1 − (γ i 1 + γ i 2 )
, select π i k+1 according best responding with inertia:
If π i k ∈ BR i k , then π i k+1 = π i k .
Else, then π i k+1 =
π i k w.p. λ i π * i ∈ BR i k w.p. 1−λ i |BR i k | End End Reset Q i t k+1 to any Q i ∈ Q i (e.g. project Q i t k+1 onto Q i )
End k th exploration phase. (Go to exploration phase k + 1)
• γ(ǫ) > 0 •γ(ǫ, γ i 2 , γ) > 0 •ρ(δ i ,δ, d i ) > 0 •W (ǫ, γ i 1 , γ i 2 ) ∈ N • T ǫ (ǫ, γ i 1 , γ i 2 , W, δ i ,δ, d i ) ∈ N such that if for every DM i we have γ i 2 < γ, γ i 1 ∈ (0,γ), ρ i ∈ (0,ρ), if W ≥W , and T k ≥ T ǫ for all exploration phases k, then lim inf n P (π n ∈ Π EQ ) ≥ 1 − ǫ
Moreover, if the game is also a coordination game, then we can replace Π EQ by Π * and the inequality above holds.
Proof. The proof of Theorem 3 is in the appendix.
V. REFINEMENTS WHEN ADDITIONAL INFORMATION IS AVAILABLE
We return now to the team problem. Algorithm 1 was designed so that the aspiration level can be estimated during the course of play. In some cases, the aspiration level will be available a priori. If such information is available, stronger convergence results can be proved. In this section, we present two such algorithms. Let us define two quantities necessary for selecting an appropriate aspiration level:
OP T := x∈X Q i π * −i (x, π * i (x)), π * ∈ Π OP T N EXT := min i,π∈Π\Π OP T x∈X Q i π −i (x, π i (x))
Algorithm 3 (for DM i ) (Learning with Aspiration Threshold)
Set Parameters
• Q i a compact subset of R X×U i • A sequence of positive integers {T k } k≥0 , the exploration phase lengths.
-Define t 0 = 0, and in general t k+1 = t k + T k . • ρ i ∈ (0, 1) an experimentation probability • λ i ∈ (0, 1) the probability of inertia • δ i > 0 a tolerance for suboptimality when constructing best-response sets • Λ ∈ R, a threshold value for comparing to total cost • {α i n } n≥0 a sequence of step sizes such that α i n ∈ [0, 1] n≥0 α i n = ∞ n≥0 (α i n ) 2 < ∞ (e.g. a i n = 1/n r , for r ∈ (1/2, 1].) • γ i ∈ (0, 1), probability for randomly updating baseline policy • Select arbitrary π i 0 ∈ Π i and Q i 0 ∈ Q i • Receive the initial stage x 0 Iterate k ≥ 0 (k th exploration phase) Iterate t = t k , t k + 1, . . . , t k+1 − 1
u i t = π i k (x t ) w.p. 1 − ρ i any u i ∈ U i w.p. ρ i /U i Receive cost c i (x t , u i t , u −i t )
Receive the next state x t+1 (chosen according to P [·|x t , u i t , u −i t ]) n i t = the number of visits to (x t , u i t ) in the k th exploration phase (up to and including t)
Q i t+1 (x t , u i t ) = (1 − α i n i t )Q i t (x t , u i t ) +α i n i t [c i (x t , u i t , u −i t ) + β i min v i ∈U i Q i t (x t+1 , v i )] Q i t+1 (x, u i ) = Q i t (x, u i ) for every (x, u i ) = (x t , u i t ) End BR i k = {π i ∈ Π i : Q i t k+1 (x,π i (x)) ≤ min v i ∈U i Q i t k+1 (x, v i ) + δ i for every x ∈ X} S i k = x∈X Q i t k+1 (x, π i k (x)) If S i k < Λ π i k+1 = π i k Else S i k ≥ Λ π i k+1 = π i ∈ Π i w.p. γ i /|Π i | π i k w.p. (1 − γ i )λ i π * i ∈ BR i k w.p. (1 − γ i )(1 − λ i )/|BR i k | Reset Q i t k+1 to any Q i ∈ Q i (e.g. project Q i t k+1 onto Q i ) End k th exploration phase.
(and continue to exploration phase k + 1) Theorem 4. Consider a stochastic team game in which all players employ Algorithm 3, and for which Assumption 1 holds. Let Λ ∈ (OP T, N EXT ). Then there existsδ > 0,ρ > 0 such that if δ i ∈ (0,δ) and ρ i ∈ (0,ρ) for every i, and lim k→∞ T k = +∞, then lim k→∞ P (π k ∈ Π OP T ) = 1
Proof. See appendix
Next, we present an algorithm with ρ i = 0 for each i. That is, agents will not do any experimenting during the course of an exploration phase. Though we previously required ρ i > 0, this modification is easily handled.
Algorithm 4 (for DM i ) (Learning with Aspiration Threshold) Set Parameters
• V i a compact subset of R X • A sequence of positive integers {T k } k≥0 , the exploration phase lengths.
-Define t 0 = 0, and in general t k+1 = t k + T k . • Λ ∈ R, a threshold value for comparing to total cost • {α i n } n≥0 a sequence of step sizes such that α i n ∈ [0, 1] n≥0 α i n = ∞ n≥0 (α i n ) 2 < ∞ (e.g. a i n = 1/n r , for r ∈ (1/2, 1].) • γ i ∈ (0, 1), probability for randomly updating baseline policy • Select arbitrary π i 0 ∈ Π i and J i 0 ∈ V i • Receive the initial stage x 0 Iterate k ≥ 0 (k th exploration phase) Iterate t = t k , t k + 1, . . . , t k+1 − 1
u i t = π i k (x t )
Receive cost c i (x t , u i t , u −i t ) Receive the next state x t+1 (chosen according to P [·|x t , u i t , u −i t ]) n i t = the number of visits to x t in the k th exploration phase (up to and including t) The proof of Theorem 5 is presented in the appendix. In contrast to the statements the previous theorems, Theorem 5 does not have any parameter restrictions that depend on the game being played. In comparison with the previous algorithms, it is extremely simple and gives a powerful result, all predicated only on the condition that Λ ∈ (OP T, N EXT ) is available at the outset of play. Despite being a restrictive condition in general, it leads to a very desirable result. Since players do not experiment with non-policy actions, the results of Theorem 5 are stronger than those of Theorem 4: as the baseline policy converges, we also have convergence on the level of stage games.
J i t+1 (x t ) = (1 − α i n i t )J i t (x t ) +α i n i t [c i (x t , u i t , u −i t ) + β i J i t (x t+1 )] J i t+1 (x) = J i t (x) for every x = x t End S i k = x∈X J i t k+1 (x) If S i k < Λ π i k+1 = π i k Else S i k ≥ Λ Select π i k+1 ∈ Π i uniformly at random. Reset J i t k+1 to any J i ∈ V i (e.g. project J i t k+1 onto V i ) End k
VI. A SIMULATION STUDY: A TWO STATE COORDINATION
GAME We consider the following two player stochastic team game with U 1 = U 2 = X = {1, 2}. Decision makers wish to minimize the expectation of their series of discounted costs, but are unaware of the specifics of the game being played. The stage games for states 1 and 2 have the following costs:
State 1
State 2 1 2 1 +1 + 3 2 +3 + 1 1 2 1 +10 + 10 2 +10 + 13 State 1 is a low cost state and State 2 is a high cost state. The transition probabilities, given below, are constructed so that when players successfully coordinate their actions (on an action that is state-dependent) the system will transition with high probability to the low cost state. Otherwise, the system will transition with high probability to the low cost state.
P [x t+1 = 1|x t = u 1 t = u 2 t ] = 0.95 P [x t+1 = 1|¬(x t = u 1 t = u 2 t )] = 0.05
In particular, when in State 2, players are faced with the choice between on the one hand incurring a lower short term cost (+10) and likely remaining in the high cost state and on the other hand paying a higher short term cost (+13) with the hopes of transitioning to the low cost state and avoiding sustained high costs.
For sufficiently large discount factors, the unique team optimal joint policy is for both players to play action 1 when in state 1 and for both players to play action 2 when in state 2. However, there are three suboptimal equilibrium policies, namely (i) both players always play 1; (ii) both players always play 2, (iii) both players always choose the action with label opposite the label of the current state.
Using a discount factor of β = 0.8, we ran simulations of four Algorithms. Algorithms A and B were in the form of Algorithm 1 and used a finite memory of previous scores to construct an aspiration level, while C and D used an aspiration level that was provided a priori. In every algorithm, when facing S i k < Λ i k , DM i was instructed to update uniformly at random with probability γ 1 (common for both agents), and when facing S i k ≥ Λ i k , DM i was instructed to update uniformly at random with probability γ 1 + γ 2 (common to both agents). When not updating uniformly at random (with probability 1 − γ 1 , 1 − (γ 1 + γ 2 ) for S i k < Λ i k and S i k ≥ Λ i k , respectively), Algorithms A and C called for best responding with inertia, while Algorithms B and D called for no update.
In each case the empirical results confirmed the theoretical results above. The results are summarized below. The algorithms performed generally as we expected. The disparity across algorithms owes largely to the parameter selection: in the simulations of Algorithm B, we chose γ 2 = 1 − γ 1 , which lead to quickly shifting away from policies that were judged to be suboptimal. In contrast, we chose γ 2 = 0.2 for the simulations of algorithms C and D, and for the simulation of Algorithm A we chose γ 2 = 0.1. Furthermore, we length of the exploration phases varied (from 10,000 plays for Algorithm A, 7,500 for Algorithms C and D, and only 5,000 for Algorithm B), and the aspiration value used for algorithms C and D was chosen without extensive tuning. The window length W = 30 in Algorithm A, while W = 50 for Algorithm B. In this paper, we presented a learning algorithm for playing stochastic coordination games, and provided rigorous results on the convergence of baseline policies to team optimal or coordination policies. While previous studies have focused on the relatively narrow class of repeated games rather than the broader class of stochastic games, or otherwise used a large degree of control sharing among players to achieve their convergence results, we have provided a method for achieving coordination without any control sharing during play and with limited prior information about the game.
The proof methods used in this paper differ substantially from those used in our previous work [1]. Here, our analysis centres on using Markov chains to approximate the true process, and are more similar to the methods used in [2]. The algorithms presented are amenable to further variants and can be modified as needed, and the Markov chain analysis used for the convergence guarantees can likewise be easily modified.
APPENDIX A: PROOF OF THEOREMS 1 AND 2 This appendix contains the proofs of Theorems 1 and 2. Since team games are a special case of coordination games with Π * = Π OP T , the proof of Theorem 1 follows from the proof of Theorem 2, so we present the proofs in reverse order.
We will make use of the following lemma. Recall (see [4]) the definition of the Dobrushin coefficient of an n × n matrix P : δ(P ) := min{ n j=1 min{P (i, j), P (k, j)} : i, k ∈ {1, . . . , n}}.
Lemma 3. Consider a time homogenous Markov chain on the finite set X = {x 1 , . . . , x n } with transition matrix R such that δ(R) > 0. Let µ * be the unique stationary distribution. Let ∆ > 0 be given.
If µ 0 is any initial probability distribution on X and {M k } k∈N is a sequence of right stochastic matrices on X satisfying
|M k (x, y) − R(x, y)|< τ for all k ≥ 0 and x, y ∈ X, where τ < min{ δ(R) 2n , δ(R)∆ 4n }, then we have that lim sup k→∞ µ 0 M 0 · · · M k − µ * < ∆
Proof. First, we recall a useful inequality involving the Dobrushin coefficient (see, e.g. [4]): for any probability distributions ν, µ on X, we have νR − µR 1 ≤ (1 − δ(R)) ν − µ 1 .
If δ(R) > 0, then R is a contraction, since δ(R) ≤ 1 is guaranteed by R's being a transition matrix. By the Banach fixed point theorem, there exists a unique invariant measure µ * = µ * R, and furthermore for any initial distribution µ 0 , we have lim k→∞ µ 0 R k = µ * .
By assumption, |R(a, b)−M k (a, b)|< τ for every a, b ∈ X and every k ≥ 0. As a consequence, we have |δ(R)−δ(M k )|≤ nτ . Then, our choice of τ < δ(R)
2n implies that 1 − δ(M k ) ≤ 1− δ(R) 2 for every k. Thus, sup k≥0 {1−δ(M k )} ≤ 1− δ(R) 2 =: ρ ∈ (0, 1).
For each k ≥ 0, let us write E k := M k − R, and note that the matrix E k satisfies 0 ≤ |E k (a, b)|< τ for every a, b ∈ X.
µ 0 M 0 − µ * 1 = µ 0 M 0 − µ * R 1 by invariance of µ * = µ 0 M 0 − µ * M 0 + µ * E 0 1 ≤ µ 0 M 0 − µ * M 0 1 + µ * E 0 1 ≤ (1 − δ(M 0 )) µ 0 − µ * 1 +nτ ≤ ρ µ 0 − µ * 1 +nτ
By induction on k and using µ k = µ 0 M 0 · · · M k−1 , we have that
µ k − µ * 1 ≤ ρ k µ 0 − µ * 1 +nτ k−1 j=0 ρ j
Note that the term nτ
k−1 j=0 ρ j ≤ nτ ∞ j=0 ρ j = nτ 1−ρ = 2nτ δ(R) < ∆ 2 by our choice of τ < δ(R)∆ 4n . Hence, µ k − µ * 1 ≤ ρ k µ 0 − µ * 1 + ∆ 2
for every k. Then, since ρ ∈ (0, 1), we have lim k→∞ ρ k µ 0 − µ * 1 = 0 for any µ 0 . Thus, for sufficiently large k, we have ρ k µ 0 − µ * 1 < ∆ 2 and consequently µ k − µ * 1 < ∆.
PROOF OF THEOREM 2 Let ǫ > 0 be given. The theorem statement is trivial for ǫ ≥ 1, so we proceed with ǫ ∈ (0, 1).
For a coordination game, consider the Idealized Update Procedure introduced in Section III, replacing Π OP T by Π * . Suppose players use the fixed maps f i , h i :
Π i × 2 Π i → P(Π i ) with corresponding fixed lower bounds K i ,K i : min π i ,π i ∈Π i ,A i ⊆Π i f i (π i , A i )(π i ) ≥ K i > 0 and similarly for K i with h i replacing f i .
This induces a family of transition matrices A f,h,γ parametrized by the choice of γ i for each DM i , i = 1, . . . , N . By Lemma 2, there existsγ ǫ/2 such that
γ i ∈ (0,γ ǫ/2 ) for every i implies µ * f,h,γ (Π * ) ≥ 1−ǫ/2, where µ * f,h,γ is the invariant measure for A f,h,γ . Fix γ 1 , · · · , γ N < γ ǫ/2 .
Proof Program
For k ∈ N and π, π ′ ∈ Π, we define
µ k (π) := Prob(π k = π) C k (π, π ′ ) = Prob(π k+1 = π ′ |π k = π) ⇒ µ k+1 = µ 0 C 0 · · · C k Prob(π k ∈ Π * ) = π * ∈Π * µ k (π * )
The matrices C k are right stochastic matrices, so if we can ensure that |C k (π, π ′ ) − A f,γ (π, π ′ )|< τ for every π, π ′ ∈ Π and for all but finitely many exploration phases k, where
τ < min{ δ(A f,h,γ ) 2|Π| , δ(A f,h,γ )(ǫ/4) 4|Π| } = δ(A f,h,γ )ǫ 16|Π|
then the result follows Lemma 3 immediately due to our selection of τ, γ 1 , . . . , γ N :
lim sup k→∞ µ k − µ * f,h,γ 1 < ǫ/4 ⇒ ∃m ∈ N : µ k − µ * f,h,γ 1 < ǫ 2 , for every k ≥ m ⇒ µ k (Π * ) ≥ µ * f,h,γ (Π * ) − ǫ 2 ≥ 1 − ǫ, for every k ≥ m
So, we will argue that appropriately selected parameters guarantees that
|C k (π, π ′ ) − A f,h,γ (π, π ′ )|< τ(1)
for every π, π ′ ∈ Π and for all but finitely many exploration phases k.
In order to achieve (1), we define an event R k such that for all π, π ′ ∈ Π, k ≥ 0, we have P r(π k+1 = π ′ |π k = π, R k ) = A f,h,γ (π, π ′ ) Then we argue that parameter selection can ensure P r(R k |π k = π) ≥ 1 − τ for all but finitely many k.
Proof Details
For notational clarity, let us define S(i, π) := x∈X Q i π −i (x, π i (x)). Next, we define some objects necessary for our parameter restrictions:
δ := min{|Q i π −i (x, u) − Q i π −i (x, v)|: i, π −i ∈ Π −i , x ∈ X, u, v ∈ U i , |Q i π −i (x, u) − Q i π −i (x, v)| = 0} O i := min π∈Π S(i, π) N i := min{S(i, π) : π ∈ Π, S(i, π) > O i } d := min i N i − O i 4 W (τ, γ) = min{W ∈ N : W > ln( τ 4 N j=1 [γ j K j ∧K j ]) ln(1 − N j=1 [γ j K j ∧K j ]) } where a ∧ b := min{a, b}. Fix W ≥W (τ, γ).
δ is the minimum non-zero distance between Q-factors. It will be used for constructing best-response sets. Convergence of Q-factors to within 1 2 min{δ,δ − δ i } in every component will ensure that the best-response sets of each player are recovered without error.d will be used in setting the aspiration level. Assumption 2 requires 0 < d i <d and 0 < δ i <δ for all DM i .
By constructiond > 0 and O i is achieved by any π ∈ Π * . In the case of a stochastic team game O i = OP T and N i := N EXT for each i, where OP T and N EXT are as defined in Section V. In the case of stochastic team games, one can show OP T < N EXT . Let d * = min i {d i }.
For a fixed deterministic baseline policy π −i k ∈ Π −i , let us writeπ −i k ∈ ∆ −i to mean the stationary but nondeterministic policy actually used by DMs −i, in which DM j follows π j k with probability 1 − ρ j and randomizes over Π j with probability ρ j . Recall that Q i ω −i is a fixed, deterministic vector of Q-values for player i facing a stationary environment where DMs −i follow ω −i ∈ ∆ −i .
We now define an upper boundρ(δ i , d i ) > 0 on the experimentation probabilities ρ i . first, chooseρ 1 > 0 that satisfies the following: if ρ i ∈ (0,ρ 1 ) for each i ∈ {1, . . . , N },
then Q i π −i k − Q ī π −i k ∞ < 1 4 min i {δ i ,δ − δ i },
for each player i and time k. Second, chooseρ 2 > 0 that satisfies the following:
if ρ i ∈ (0,ρ 2 ) for each i ∈ {1, . . . , N }, then we have Q i π −i k − Q ī π −i k ∞ < d * 4|X|
for each player i and time k. Since the exploration phase portion of Algorithm 1 is identical to the exploration phase portion of Algorithm A presented in [1], and since Assumption 1 holds, we can invoke [1, Lemma 3] to conclude that suchρ 1 ,ρ 2 > 0 exist. Now we defineρ := min{ρ 1 ,ρ 2 }.
For B ∈ Z, let B(B) := {B · W, B · W + 1, . . . , (B + 1)W − 1}. For any j, B ≥ 0, we define the following events:
E j := {|S i j − x∈X Q i π −i j (x, π i j (x))|< d * 2 , ∀i} any(B) := {π j ∈ Π * : j ∈ B(B)} all(B) := ∩ j∈B(B) E j F j := { Q i tj+1 − Q i π −i j ∞ < 1 2 min i {δ i ,δ − δ i }, ∀i}
For k ∈ N : k ≥ W , we have that k ∈ B(B + 1) for some B ≥ 0. In this case, Λ i k = min{S i j : j ∈ B(B)} + d i . We have
P (π k ∈ Π * |E k ∩ any(B) ∩ all(B) ∩ {S i k < Λ i k }) = 1 P (π k ∈ Π * |E k ∩ any(B) ∩ all(B) ∩ {S i k ≥ Λ i k }) = 0
Moreover, conditional on F k also occurring, we have that the best-response set of each player is recovered without error.
That is, P (BR i k = BR i π −i k , ∀i|F k ) = 1. Then, it is clear that by taking R k := E k ∩ F k ∩ any(B) ∩ all(B), we have Pr(π k+1 = π ′ |π k = π, R k ) = A f,h,γ (π, π ′ )
We now invoke Lemma 2 from [1, Appendix B]: there exists L ∈ N such that if T j ≥ L for every j ∈ N, we have
P Q i tj+1 − Q ī π −i j ∞ ≤ Ξ, ∀i ≥ 1 − Ξ where Ξ = min{ d * 4|X| , 1 4 min i {δ i ,δ − δ i }, τ 4W N i=1 [γ i K i ∧ K i ]} > 0
With all of the preceding parameter restrictions in place, we claim that if T j ≥ L for every j ∈ N, then
P (R k ) ≥ 1 − τ N i=1 [γ i K i ∧K i ]
, for every k ≥ W By our choice ofρ ≤ρ 2 and the restriction ρ i ∈ (0,ρ),
we have that Q i π −i j − Q ī π −i j ∞ < d * 4|X|
for every player i and time j. Then, since Ξ ≤ d * 4|X| , applying the triangle inequality and summing over x ∈ X gives us the following for all j ≥ 0:
Q i tj+1 − Q ī π −i j ∞ ≤ Ξ, ∀i ⇒ |S i j − x∈X Q i π −i j (x, π i j )|< d * 2 , ∀i ⇐⇒ E j Thus, we have P (E C j ) ≤ τ 4W N i=1 [γ i K i ∧K i ]
for all j. In particular, this implies
P (E k ) ≥ 1 − τ 4 N i=1 [γ i K i ∧K i ], for k ≥ W, and P (∪ j∈B(B) E C j ) ≤ j∈B(B) P (E C j ) ≤ τ 4 N i=1 [γ i K i ∧K i ] ⇒ P (all(B)) ≥ 1 − τ 4 N i=1 [γ i K i ∧K i ]
Similarly, sinceρ ≤ρ 1 and Ξ ≤ 1 4 min i {δ i ,δ − δ i }, we have by the triangle inequality that
{ Q i t k+1 − Q ī π −i k ∞ < 1 4 min i {δ i ,δ − δ i }, ∀i} ⊂ F k Hence P (F k ) ≥ 1 − τ 4 N i=1 [γ i K i ∧K i ].
Since players update their policies independently, and DM i considers any policy with probability no less than
γ i K i ∧K i , we have that P (π j = π) ≥ N i=1 [γ i K i ∧K i ] for every π ∈ Π. In particular, P (π j ∈ Π * ) ≥ N i=1 [γ i K i ∧K i ].
Comparison to a binomial random variable with W i.i.d. trials and probability of success N i=1 γ i K i ∧K i , shows that
P (any(B) C ) ≤ (1 − N i=1 [γ i K i ∧K i ]) W ≤ τ 4 N i=1 [γ i K i ∧K i ]
where the second inequality holds by our choice of W . All together, the preceding implies that
P (R k ) ≥ 1 − τ N i=1 [γ i K i ∧K i ], for any k ≥ W(2)Then, since P (π j = π) ≥ N i=1 [γ i K i ∧K i ] > 0 for every π ∈ Π, inequality (2) in turn implies P (R k |π k = π) ≥ 1 − τ, for all π ∈ Π
Finally, recall that we argued A f,h,γ (π, π ′ ) = P (π k+1 = π ′ |π k = π, R k ). Using the law of total probability on C k (π, π ′ ) := P (π k+1 = π ′ |π k = π), we see that |C k (π, π ′ ) − A f,γ (π, π ′ )|< τ, for every π, π ′ ∈ Π, k ≥W We invoke Lemma 3 in the manner outlined at the beginning of the proof, and the result follows.
PROOF OF THEOREM 1 A finite, discounted stochastic team game is a coordination game with coordination set Π * = Π OP T . Hence the result follows from the proof of Theorem 2.
APPENDIX B: PROOF OF THEOREM 3 We begin with a lemma that will be used in the sequel. We use the matrix norm given by M ∞ := max i,j |M i,j |. We denote the n × n identity matrix by I n Lemma 4. Consider a time homogernous Markov chain {X t } t≥0 that takes values in the finite set X, with transition matrix D and initial distribution. Suppose that for some set A ⊂ X, there exists δ > 0 and H ∈ N such that:
(i) x∈X\A D(a, x) = 0 for every a ∈ A (ii) a∈A D H (x, a) ≥ δ for every x ∈ X Let ǫ > 0 be given. Then, there exists ξ > 0 such that for any sequence of right stochastic matrices
{M k } k≥1 satisfying M k − D ∞ < ξ for all k ≥ 1, we have lim inf n→∞ µM 1 · · · M k (A) ≥ 1 − ǫ
for any initial probability distribution µ on X.
Proof. Let ǫ > 0 be given. It is clear from hypotheses (i), (ii) that for any µ, we have that lim n→∞ µD n (A) = 1. In particular, it is true for the point mass at x ∈ X, µ = δ x , for any x ∈ X. Since a general probability vector µ on X can be expressed as a convex combination of the (finitely many) point masses, µ = x∈X µ(x)δ x , we choose N ∈ N such that δ x D k (A) > 1 − ǫ/2 for every x ∈ X and k ≥ N . By linearity, such N then also satisfies µD N (A) > 1 − ǫ/2 for arbitrary probability vector µ on X.
Suppose that a sequence of right stochastic matrices {M k } k≥1 satisfies M k −D ∞ < ξ for some ξ > 0 and for all k ∈ N.
Writing E k := M k − D for all k, we have E k ∞ < ξ. For k = 1, µM 1 − µD = µM 1 − µ(M 1 − E 1 ) = µE 1 .
In general, using the conventions that D 0 := I |X| , and M j+1 · · · M n := I |X| whenever j + 1 > n , we have that
µM 1 · · · M n − µD n = µ[ n j=1 D j−1 E j M j+1 · · · M n ]
This claim can be argued by induction, and is omitted. We note that if v is a 1 × |X| row vector and B is a |X|×|X| matrix, we have that
vB 1 ≤ |X| k=1 max i |v(i)|· |X| j=1 |B j,k | = max i |v(i)|· |X| k=1 |· |X| j=1 |B j,k |
From this, we see that for any n ∈ N, we have
µM 1 · · · M n − µD n 1 = µ[ n j=1 D j−1 E j M j+1 · · · M n ] 1 ≤ n j=1 µD j−1 E j M j+1 · · · M n 1 ≤ ξ|X|n
The final inequality can be deduced from the following two points: firstly, µD j−1 is a probability vector and E j ∞ < ξ, so max i |µD j−1 E j (i)|< ξ max ℓ µD j−1 (ℓ) ≤ ξ · 1. Secondly, M j+1 · · · M n is a right-stochastic matrix, i.e. it has nonnegative entries and all rows sum to one; hence, the sum of all its entries is the number of rows, |X|.
In particular, this holds for n = N , which was chosen so that µD N (A) ≥ 1 − ǫ/2 for any initial distribution µ. If we take ξ < ǫ 2|X|N , then it follows that
µM 1 · · · M N − µD N 1 < ǫ 2 , and a∈A µD N (a) ≥ 1 − ǫ/2
Together, we have
µM 1 · · · M N (A) > µD N (A) − ǫ/2 > 1 − ǫ
This holds for an arbitrary probability vector µ. For time index n ≥ N , we can write µM 1 · · · M n =μM n−N +1 · · · M n−N +N whereμ = µM 1 · · · M n−N is again a probability vector on X, hence the same argument applies and we again see that µM 1 · · · M n (A) > 1 − ǫ for all n ∈ N.
Recall best responding with inertia, as presented in [1]:
Best Responding with Inertia (for DM i ) Set λ i > 0, select π i 0 ∈ Π i (arbitrary) For k ≥ 0:
If π i k ∈ BR i π −i k : π i k+1 = π Else: ζ i k ∈ BR i π −i k selected uniformly at random π i k+1 = ζ i k , w.p. 1 − λ i k π i k , w.p. λ i ,
In a weakly acyclic game, the (idealized) stochastic process on Π obtained when all players use best responding with inertia (with access to their true best response sets) satisfies the hypotheses of Lemma 4, with Π EQ playing the role of the distinguished set A. The transition matrix of that Markov chain is parametrized by λ = (λ i ) N i=1 . Let us denote the transition matrix by D λ .
Proof of Theorem 3
Let ǫ > 0 be given. Suppose all players use Algorithm 2 to update their policies, with fixed λ i ∈ (0, 1) for each i. Denote by D λ the transition matrix for the corresponding Best Responding with Inertia process. By Lemma 4, there exists ξ such that C k − D λ ∞ < ξ for all but finitely many k implies lim inf n P (π n ∈ Π EQ ) ≥ 1 − ǫ/2. Fix such an ξ > 0. Then, choose γ > 0 such that (1 − γ) N ≥ 1 − ξ/2, and fix γ i 2 < γ for every DM i .
Note that Algorithm 2 is in the form of Algorithm 1 and satisfies Assumption 3: for each DM i f i (π i , A i ) (identically) is the uniform distribution on Π i (so K i ≥ 1/|Π i |> 0) and h i (π i , A i ) is a convex combination of the uniform distribution and some other probability distribution
(henceK i ≥ (γ i 1 + γ i 2 )/|Π i |≥ γ i 2 /|Π i |> 0.)
We restrict the γ i 1 parameters as follows: for m ∈ {1, 2, . . . , |Π|} writing γ = (γ 1 1 , . . . , γ N 1 ), we consider the function
F m (γ) := m N j=1 K j 1 − [ N j=1 (1 − γ j 1 ) + N j=1 γ j 1 K j ] + m N j=1
K j When its denominator is non-zero, F m is continuous. Since K j > 0 as noted above, it is continuous in particular when γ i 1 = 0 for all i. We have F m (0) = 1, so there exists γ * m > 0 such that if γ i 1 ∈ (0, γ * m ) for each DM i , then F m (γ) ≥ 1−ǫ/2. We produce such γ * m for each m ∈ {1, . . . , |Π|}, and takē γ := min m γ * m . If the game at hand is a coordination game, then 1 ≤ |Π * ≤ |Π|, and so our choice ofγ and γ i 1 ∈ (0,γ) for all i implies µ * f,h,γ (Π * ) ≥ 1 − ǫ/2. (Where µ * f,hγ is the invariant measure of A f,h,γ , the transition matrix for the Markov chain induced by all players' using the Idealized Update Procedure outlined in Section III.) If the game is not a coordination game, then nothing is gained or lost by the following restriction: for every i, fix γ i 1 ∈ (0, min{γ, γ − γ i 2 }). Next, we define the following objects, using the shorthand
S(i, π) = x∈X Q i π −i (x, π i (x)) O i = min{S(i, π) : π ∈ Π} N i = min{S(i, π) : π ∈ Π, and S(i, π) > O i } d = min i N i − O i 4
If the game is weakly acyclic but not a coordination game, then we do not wish to assign any special significance to these quantities. On the other hand, if it is a coordination game, then these quantities are analogous to those defined in the proof of Theorem 2.
We now repeat parts of the proof of Theorem 2: defininḡ δ as in Appendix A, Assumption 2 requires d i ∈ (0,d) and δ i ∈ (0,δ) for all i. We define d * = min i d i .ρ is defined as in Appendix A. Define
τ < ǫ 16|Π| N j=1 γ i 1
Note: since weakly acyclic games are in general not coordination games, unqualified references to the Idealized Update Process outlined in Section III and to the corresponding matrix A f,h,γ are spurious. However, if the game is indeed a coordination game, this choice of τ lower bounds the Dobrushin coefficient δ(A f,h,γ ): γ). Then, as in the proof of Theorem 2, we invoke [1, Appendix B, Lemma 2]: there exists L ∈ N such that T k ≥ L implies
δ(A f,h,γ ) = min π,π π ′ ∈Π min{A f,h,γ (π, π ′ ), A f,h,γ (π, π ′ )} ≥ |Π| N i=1 [γ i 1 /|Π i |] = N i=1 γ i 1 > 0 ⇒ τ < δ(A f,γ )ǫ 16|Π| Definẽ W (τ, γ) := min{w ∈ N : w > ln( τ 4 N j=1 γ j 1 /|Π j |) ln(1 − N j=1 γ j 1 /|Π i |) } Fix W ≥W (τ,P ( Q i t k+1 − Q ī π −i k ∞ ≤ Ξ) ≥ 1 − Ξ where Ξ := min{b 1 , b 2 , b 3 , b 4 } and b 1 = d * 4|X| , b 2 = 1 4 min i {δ i ,δ − δ i } b 3 = τ 4W N j=1 [γ i 1 /|Π j |], b 4 = ξ 2 N j=1 (γ j 1 /|Π j |)
These parameter restrictions are sufficient to achieve our result. Define N k to be the event in which no players randomize at the end of exploration phase k. i.e., when N k holds, all players use best responding with inertia to update their baseline policies. Define
A k := { Q i t k+1 − Q i π −i k ∞ < 1 2 min i {δ i ,δ − δ i }, ∀i}.
Given our parameter restrictions, A k guarantees all players have correctly recovered their true best response sets.. Then, we have for every π, π ′ ∈ Π
P (π k+1 = π ′ |{π k = π} ∩ N k ∩ A k ) = D λ (π, π ′ )
Next, we notice the following implication: if P (N k ∩ A k |π k = π) ≥ 1−ξ, then C k −D λ ∞ < ξ, which allows us to invoke Lemma 4 and conclude that lim inf n P (π n ∈ Π EQ ) ≥ 1 − ǫ. Next, we will prove that P (N k ∩ A k |π k = π) ≥ 1 − ξ.
Since the random variables determining when DM i updates randomly are independent of π k (and independent across players and time) by assumption, we have P (N k |π k = π) = P (N k ) ≥ On the other hand, since P (π k = π) ≥ N i=1 γ i 1 /|Π i | for any π ∈ Π, we have P (A k ) ≥ 1 − ξ 2 P (π k = π) by our choice of Ξ. In turn, this implies P (A k |π k = π) ≥ 1 − ξ/2. Together, P (N C k ∪ A C k |π k = π) ≤ ξ. Invoking Lemma 4 achieves lim inf n P (π n ∈ Π EQ ) ≥ 1 − ǫ, proving the first part of Theorem 3. Now suppose that the game is a coordination game. The parameter restrictions offered here satisfy those outlined in the proof of Theorem 2, hence we also have lim inf n P (π n ∈ Π * ) ≥ 1 − ǫ. This completes the proof.
APPENDIX C: PROOFS OF THEOREMS 3 AND 4
Lemma 5. Consider a Markov chain on a finite set X = {x 1 , . . . , x n } with transition matrix D. Suppose that there is a (maximal) absorbing set A ⊂ X such that b∈X\A D(a, b) = 0 for every a ∈ A and D(b, a) ≥ δ > 0, for some δ > 0 and for any b ∈ X \ A, a ∈ A.
Let µ 0 be any initial distribution, and let {M k } k≥0 be a sequences of n × n right stochastic matrices with entries indexed by element of X. Define µ k+1 = µ 0 M 0 · · · M k . If Proof. This can be proved using Lemma 4 in a sequential argument.
PROOF OF THEOREM 3
We begin by constructing a stochastic process to be used in comparison to the actual stochastic process we wish to consider. Update C (for DM i ) Select λ i , γ i ∈ (0, 1) Select π i 0 ∈ Π i uniformly at random from Π i . For k ≥ 0, repeat:
• if π k ∈ Π OP T -π i k+1 = π i k • else (π k / ∈ Π OP T ) -w.p. γ i , select π i k+1 uniformly at random from Π i . -w.p. 1 − γ i , select π i k+1
according to best responding with inertia:
* If π i k ∈ BR i π −i k , then π i k+1 = π i k * Else: select η i k ∈ BR i π −i k
uniformly at random, and
π i k+1 = π i k w.p. λ i η i k w.p. 1 − λ i Increment k (End Update C)
Update C induces a time homogenous Markov chain on Π, in which all players follow Update C and choose their random updates independently. In this case, the Markov chain is not irreducible: the set Π OP T is positive recurrent, and the set Π \ Π OP T is transient. Let the transition matrix associated to this Markov Chain be given by Γ γ,λ .
For OP T , N EXT as defined in Section V, let Λ ∈ (OP T, N EXT ). We recall the definition ofδ as the minimum non-zero separation between Q-factors, and impose δ i ∈ (0,δ) for each DM i . Next, we defineρ > 0 to satisfy the following: if ρ i ∈ (0,ρ) for each i, then
Q i π −i k − Q ī π −i k ∞ < min{a 1 , a 2 }
holds for each DM i and time k, where a 1 = 1 2 min i {δ i ,δ − δ i } and a 2 = 1 2|X| min{N EXT − Λ, Λ − OP T }. Since min{a 1 , a 2 } > 0, [1, Lemma 3] guarantees that such aρ > 0 exists.
For exploration phase index j ∈ N, we define
E j := {ω ∈ Ω : |S i j − x∈X Q i π −i j (x, π i j (x))| < min{N EXT − Λ, Λ − OP T }, ∀i} F j := {ω ∈ Ω : Q i tj+1 − Q i π −i j ∞ < min i {δ i ,δ − δ i }, ∀i}
If all players are using Algorithm C and choose their updates independently, we have the following:
P (π k+1 = π ′ |π k , E k ∩ F k ) = Γ γ,λ (π, π ′ )
wherever this conditional probability is defined. This is because when conditioning on E k ∩ F k , we have the equivalence
S i k < Λ ⇐⇒ x∈X Q i π −i j (x, π i j (x)) = OP T
Then, both conditionally and unconditionally, we have
x∈X Q i π −i j (x, π i j (x)
) ⇐⇒ π k ∈ Π OP T . We now define the probability vectors µ k and the matrices of conditional expectations C k : µ k (π) := P (π k = π) ∀π ∈ Π, k ∈ Z ≥0 C k (π, π ′ ) := P (π k+1 = π ′ |π k = π), if µ k (π) > 0 Γ γ,λ (π, π ′ ), if µ k (π) = 0 ⇒ µ k+1 = µ 0 C 0 · · · C k Given our choice of Λ ∈ (OP T, N EXT ), the parameter restrictions δ i ∈ (0,δ), ρ i ∈ (0,ρ) for each i, and lim k→∞ T k → +∞, we can now prove that lim k→∞ C k (π, π ′ ) = Γ γ,λ (π, π ′ ), for every π, π ′ ∈ Π Since we defined C k (π, π ′ ) := Γ γ,λ (π, π ′ ) for π with µ k (π) = 0 and argued that P (π k1 = π ′ |π k = π, E k ∩ F k ) = Γ γ,λ (π, π ′ ), it is sufficient to show that for any ξ > 0, there exists some time T ξ ∈ N such that k ≥ T ξ implies P (E k ∩ F k |π k = π) > 1 − ξ for every π ∈ Π with µ k (π) > 0.
We invoke [1,Lemma 4] to conclude that there exists T 1 ξ ∈ N such that T k ≥ T 1 ξ implies
P Q i t k+1 − Q ī π −i k ∞ < 1 2|X|
min{NEXT − Λ, Λ − OPT}, ∀i, k ≥ 1 − ξ 2
and then invoke [1, Lemma 1] to conclude that such a lower bound T ξ exists for any baseline policy, i.e. for any π k = π. Then, we have
|S i k − x∈X Q i π −i k (x, π i k (x))|
PROOF OF THEOREM 4 Update D (for DM i ) Select γ i ∈ (0, 1) Select π i 0 ∈ Π i uniformly at random from Π i . For k ≥ 0, repeat:
• If π k ∈ Π OP T π i k+1 π i k • Else (π k / ∈ Π OP T ) -π i k+1 = π i ∈ Π i w.p. γ i /|Π i | π i k w.p. 1 − γ i Increment k (End Update D) Update D induces a time homogenous Markov chain on Π, corresponding to all players using Update D and making their updates independently of one another and across time. For this process, Π OP T is positive recurrent and Π \ Π OP T is transient. Letting γ = (γ 1 , . . . , γ N ), we denote the transition matrix by D γ .
We define OP T and N EXT as above, and require that Λ ∈ (OP T, N EXT ). We define U i k (x) := {π i k (x)}. If DM i follows Algorithm D, then during exploration phase k, they are engaging in trivial Q-learning, where their action set in state x is given by the singleton U i k (x). By [17, Theorem 4], we have that for any ξ > 0, there exists T ξ ∈ N such that T k ≥ T ξ implies
P ( V i t k+1 −V i π −i k ∞ < 1 |X| min{N EXT −Λ, Λ−OP T }) ≥ 1−ξ
where V i · ∈ R X is the vector of Q-factors restricted to the entries (x, π i k (x)) of Q i · . We define the event E k :={ω ∈ Ω :
|S i k − x∈X V i π −i k (x)|< min{N EXT − Λ, OP T − Λ}, ∀i}
By set containment and summing over x ∈ X, it is clear that if T k ≥ T ξ , then P (E k ) ≥ 1 − ξ. The event E k guarantees that all players are correctly assessing whether the current baseline policy is optimal or not, hence we have that P (π k+1 = π ′ |π k = π, E k ) = D γ (π, π ′ ) for any π, π ′ ∈ Π for which this conditional probability is defined.
We define µ k (π) := P (π k = π) C k (π, π ′ ) := P (π k+1 = π ′ |π k = π), if µ k (π) > 0 D γ (π, π ′ ) else ⇒ µ k+1 = µ 0 C 0 · · · C k Since lim k T k = +∞, we have that lim k P (E k ) = 1. In turn, this implies that lim k→∞ C k (π, π ′ ) = D γ (π, π ′ ) for every π, π ′ ∈ Π We invoke Lemma 5, noting that Π OP T is the absorbing set, which completes the proof.
APPENDIX D: APPLICATION TO REPEATED GAMES
In this appendix, we consider Algorithm 1 when applied to repeated coordination games. Without long-run cost considerations, the body of the algorithm and the proof of convergence can be greatly simplified. As mentioned in Section III, a coordination game (as defined here) generalizes what [2] calls a strict coordination game that satisfies hypothesis (H2). We offer the game from Table II as an example of a coordination game (as defined here) that is not a coordination game as defined in [2]. This demonstrates that, even when restricted to repeated games, coordination games as presented here are a strictly larger class of games than strict coordination games satisfying (H2). We state our result in the cost minimization setting, and will use Π and U interchangeably, as they are equivalent when |X|= 1.
Algorithm 1 for Repeated Games (A1RG) (for DM i )
Set parameters
• The functions f i , g i : U i → P(U i ) • γ i ∈ (0, 1), the probability of randomly updating when aspiration level is satisfied.
• u i 0 ∈ U i (arbitrary) For k ≥ 0 (k th stage game) Play u i k ; receive c i k = c i (u 1 k , . . . , u N k ) Set Λ i k := min{c i k : 0 ≤ τ ≤ k} If c i k ≤ Λ i k -u i k+1 = u i k , w.p. 1 − γ i ζ i k ∼ f i (u i k ), w.p. γ i Else: -u i k+1 ∼ g i (u i k ) (Increment k)
Theorem 6. Consider a repeated coordination game in which all players use A1RG. Suppose there exist K i > 0 such that f i (u i ), g i (u i ) > K i holds for each i and u i ∈ U i . Then, for any ǫ > 0, there existsγ ǫ such that if γ i ∈ (0,γ ǫ ) for all i, then lim inf n→∞ P (π n ∈ Π * ) ≥ 1 − ǫ Proof. Let ǫ > 0 be fixed. Then, we invoke Lemma 2 (replacing Π OP T by Π * ) to recoverγ ǫ/2 such that γ i ∈ (0,γ ǫ/2 ) for each i gives µ * f,g,γ (Π * ) ≥ 1 − ǫ/2. Since c i is minimized by exactly those u ∈ U * (equivalently, those π ∈ Π * ), we have that the sequence {Λ i k } k≥0 is non-increasing and is constant after Π * is visited for the first time. Moreover, for times k after Π * is visited for the first time, c i k ≤ Λ i k ⇐⇒ π k ∈ Π * . We thus have P r(π k+1 = π ′ |{π k = π} ∩ (∪ k j=0 {π j ∈ Π * }) = A f,g,γ (π, π ′ ) We define C k (π, π ′ ) := P r(π k+1 = π ′ |π k = π) if P r(π k = π) > 0 A f,g,γ (π, π ′ ) otherwise Now note that lim k→∞ P (∪ k j=0 {π j ∈ Π * }) = 1, since K i > 0 for all i. Thus |C k (π, π ′ ) − A f,g,γ (π, π ′ )|→ 0. After discarding at most finitely many indices, we invoke Lemma 3, since δ(A f,g,γ ) > 0, and conclude that µ k (Π * ) ≥ 1 − ǫ for k sufficiently large.
The preceding result is very much in the spirit of the main result in [2]. The method in [2] uses additional randomness during the aspiration level update, while the method here uses additional randomness during the policy update. [2] presents an aspiration learning method for repeated coordination games satisfying one of hypothesis (H1) or (H2), and proves that play converges to a desirable set, while we present an aspiration learning method for stochastic coordination games and prove convergence to a desirable set. When restricted to repeated games, coordination games as defined here contain the set of strict coordination games satisfying (H2), but do not contain strict coordination games satisfying (H1). Conversely, strict coordination games satisfying (H1) do not contain coordination games as defined here.
ACKNOWLEDGMENTS
This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The authors are with the Department of Mathematics and Statistics, Queen's University, Kingston, ON, Canada, Email: {1bmy,[email protected]}
Definition 3 .
3A stochastic game is called a stochastic team game if c i = c j = c and β i = β j = β for all i, j ∈ {1, . . . , N } In this case, J i = J k for all DMs i, k, and we may drop the player index.
Theorem 1 .
1Consider a finite, discounted stochastic team game in which all players use Algorithm 1. Suppose Assumptions 1, 2, and 3 hold. Let ǫ > 0.
Theorem 2 .
2Consider a coordination game (as defined above) in which all players use Algorithm 1. Suppose Assumptions 1, 2, and 3 hold. Let ǫ > 0.
th exploration phase. (and continue to exploration phase k + 1) Theorem 5. Consider a stochastic team game in which all players employ Algorithm 4, and for which Assumption 1 holds. If Λ ∈ (OP T, N EXT ), and lim k→∞ T k = +∞, then lim k→∞ P (π k ∈ Π OP T ) = 1
N j=1 [1 − (γ j 1 + γ j 2 )] ≥ (1 − γ) N ≥ 1 − ξ/2,by choice of γ and the restrictions on γ i 1 , γ i 2 .
lim k→∞ M k (a, b) = D(a, b) for every a, b ∈ Xthen lim k→∞ a∈A µ k (a) = 1.
TABLE I REPEATED
IGAME WITH SUBOPTIMAL EQUILIBRIUMLeft
Right
Up
α
α + 1
Down
α + 1
1
begin next exploration phase.)Lemma 2. Consider a finite, discounted stochastic team
game as defined above. Suppose all players update their
policies according to the Idealized Update Procedure,
and that the random variables ζ i
k , θ i
k are mutually in-
dependent across players and time. Suppose there exist
Denote the transition matrix of the induced time homogenous Markov chain on Π by A f,h,γ , and denote its unique invariant measure by µ * f,h,γ . Then for
TABLE II REPEATED
IICOORDINATION GAMELeft
Middle
Right
Up
(10, 3)
(5, 7)
(20,20)
Centre
(5, 7)
(10, 3)
(20,20)
Down
(20,20)
(20,20)
(0,0)
That best response dynamics can lead to an equilibrium policy is a property of weakly acyclic games. Best response dynamics will not find an equilibrium joint policy in general.
By our definitions ofρ and a 2 , this implies that P (E k ) ≥ 1 − ξ 2 . Since our choice ofρ also gave us Q iHence, a similar invoking of [1, Lemma 4] allows us to recover the existence of T 2 ξ such thatAll together, we have that if T k ≥ T ξ := max{T 1 ξ , T 2 ξ }, then P (E k ∩ F k |π k = π) ≥ 1 − ξ for any π with P (π k = π) > 0. In turn, this means that if T k ≥ T ξ , then |C k (π, π ′ ) − Γ γ,λ (π, π ′ )|< ξ for all π, π ′ ∈ Π. Since T k → +∞, we have that lim k→∞ C n (π, π ′ ) = Γ γ,λ (π, π ′ ) for every π, π ′ ∈ Π. Thus, we can invoke Lemma 5. The absorbing set for Γ γ,λ is Π OP T , and the result of Theorem 3 is obtained.
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"A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems",
"A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems"
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"Nicola Demo \nMathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly\n",
"Marco Tezzele [email protected]‡[email protected]§[email protected] \nMathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly\n",
"Andrea Mola \nMathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly\n",
"Gianluigi Rozza \nMathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly\n"
] | [
"Mathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly",
"Mathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly",
"Mathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly",
"Mathematics Area\nInternational School of Advanced Studies\nmathLab\nSISSA\nvia Bonomea 265I-34136TriesteItaly"
] | [] | In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in[31,24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in[7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry -assuming the topology is inaltered by the deformation -, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water. * [email protected] † | null | [
"https://arxiv.org/pdf/1905.05982v1.pdf"
] | 155,092,446 | 1905.05982 | 110afe3c6cd064733f6295a593cdd8929092b823 |
A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems
May 16, 2019
Nicola Demo
Mathematics Area
International School of Advanced Studies
mathLab
SISSA
via Bonomea 265I-34136TriesteItaly
Marco Tezzele [email protected]‡[email protected]§[email protected]
Mathematics Area
International School of Advanced Studies
mathLab
SISSA
via Bonomea 265I-34136TriesteItaly
Andrea Mola
Mathematics Area
International School of Advanced Studies
mathLab
SISSA
via Bonomea 265I-34136TriesteItaly
Gianluigi Rozza
Mathematics Area
International School of Advanced Studies
mathLab
SISSA
via Bonomea 265I-34136TriesteItaly
A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems
May 16, 2019
In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in[31,24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in[7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry -assuming the topology is inaltered by the deformation -, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water. * [email protected] †
Introduction
The reduced basis method (RBM) [13,21] is a well-spread technique for reduced order modeling, both in academia and in industry [24,23,31,20], and consists in two phases: an offline phase that can be carried out on high performance computing facilities, and an online one that exploits the reduced dimensionality of the system to perform the parametric computation on portable devices. In the offline stage the reduced order space is created from full order complex simulations computed for certain values of the parameters. The selection of the reduced basis functions that span this new reduced space can be carried out by different techniques. In this work we employ the proper orthogonal decomposition (POD) [19,3], which is based on the singular value decomposition (SVD), on the set of high-fidelity snapshots. After the creation of such space, in the online phase a new parametric solution is calculated as a linear combination of the precomputed reduced basis functions. The creation of a reduced order model is crucial in the shape optimisation context where the optimiser needs to compute several high-fidelity simulations.
Novelty of this work is the creation of a reduced order space containing the manifold of admissible shapes by applying POD over the sampled geometries, in order to reduce the parameter space dimension and to enhance the order reduction of the output fields. To generate the original design space we employ the free form deformation (FFD) method, a well-known shape parametrisation technique. Another approach for reduced order modeling enhanced by parameter space reduction technique can be found in [30] where they propose a coupling between POD-Galerkin methods and active subspaces. After the creation of the reduced space for the admissible shapes, we can exploit this new parametric formulation for the construction of the reduced space for the output fields, using the non-intrusive technique called POD with interpolation (PODI) [5,6,10] for the online computation of the coefficients of the linear combination. We would like to cite [16] where they present the concept of a shape manifold representing all the admissible shapes, independently of the original design parameters, and thus exploiting the intrinsic dimensionality of the problem.
This work is organised as follows: after the presentation of the general setting of the problem, there is a brief overview of the FFD method, then we illustrate how the parameter space reduction is performed, and we present PODI for the reduction of the high-fidelity snapshots. Finally the numerical results are presented with the conclusions and some perspective.
The problem
Let Ω ⊂ R 3 be the reference hull domain. We define a parametric shape morphing function M as follows
M(x; µ) : R 3 → R 3 ,(1)
which maps Ω into the deformed domain Ω(µ) as Ω(µ) = M(Ω; µ), where µ ∈ D ⊂ R 5 represents the vector of the geometrical parameters. D will be properly defined in Section 4. Such map M can represent many different morphing techniques (not necessarily affine) such as free form deformation (FFD) [26], radial basis functions (RBF) interpolation [4,17,15], and the inverse distance weighting (IDW) interpolation [27,33,11,2], for instance. In this work we use the FFD, presented in Section 3, to morph a bulbous bow of a benchmark hull. We chose the DTMB 5415 hull thanks to the vast amount of experimental data available in the literature, see for example [18].
In Figure 1 the domain Ω and a particular of the bulbous bow we are going to parametrize and deform. The pipeline is the following: using geometrical FFD parameters we generate several deformed hulls; then we apply the POD on the coordinates of the points describing the deformed geometries, and the new parameters will be the POD coefficients of the selected modes; after checking for possible linear dependencies between these coefficients, we sample the reduced parameter space producing new deformed hulls upon which we are going to actually perform CFD simulations. Regarding the full order model, we use the Reynolds-averaged Navier-Stokes equations to describe the incompressible and turbulent flow around the ship. The Froude number has been set to 0.2 and we chose the k-ω SST model for the turbulence since it is one of the most popular benchmark for hydrodynamic analysis for industrial naval problems. In this way we reduce the parameter space, staying on the manifold of the admissible shapes, and reducing the burden of the output reduced space construction through PODI.
Free form deformation of the bulbous bow
Here we are going to properly define the deformation map M introduced in Eq. (1), which we employed for this work, and that corresponds to the free form deformation (FFD) technique. The original formulation of the FFD can be found in [26], for more recent works in the context of reduced basis methods for shape optimization we cite [14,22,28]. It has also been applied to naval engineering problems in [7,8,32], while for an automotive case see [25].
The FFD map is the composition of three maps described in the following, while for a visual representation we refer to Figure 2:
• the function ψ maps the physical domain to the reference one where we construct the reference lattice of points, denoted with P around the object to be morphed;
• the function T performs the actual deformation since it applies the displacements defined by µ FFD to the lattice P . It uses the B-splines or Bernstein polynomials tensor product to morph all the points inside the lattice of control points;
• finally we need to map back the deformed domain to the physical configuration through the map ψ −1 . Se we can define the FFD map M through the composition of the three maps presented above as
M(x, µ FFD ) := (ψ −1 • T • ψ)(x, µ FFD ) ∀x ∈ Ω.(2)
In Figure 3 it is possible to see the actual lattice of points we used, in green, for a particular choice of the FFD parameters. For an actual implementation of this method in Python, along with other possibile deformation methods, we refer to the open source package called PyGeM -Python Geometrical Morphing [1]. In order to reduce the parameter space dimension we apply the POD on a set of snapshots that depends on the FFD parameters. Each snapshot is the collection of all the coordinates of the points defining the stl file geometry. Since the generation of these snapshots does not depend on complex simulations but only on the particular FFD deformation, we are able to create a dataset with as many entries as we want. So we create a database of N train = 1500 geometrical parameters µ FFD ∈ D := [−0.3, 0.3] 5 sampled with a uniform distribution. Moreover we create the corresponding database of mesh coordinates u corresponding to these parameters, that is Θ = [u(µ FFD, 1 )| . . . |u(µ FFD, N train )]. Then we perform the singular value decomposition (SVD) on Θ in order to extract the matrix of POD modes:
Θ = ΨΣΦ T ,(3)
where with Ψ and Φ we denote the left and right singular vectors matrices of Θ respectively, and with Σ the diagonal matrix containing the singular values in decreasing order. The columns of Ψ, denoted with ψ i , are the so-called POD modes. We can thus express the approximated reduced mesh with the first N modes as
u N = N i=1 α i ψ i ,(4)
where α i are the so called POD coefficients. To compute them in matrix form we just use the database we created as follows
α = Ψ T Θ,(5)
and then we truncate to the first N modes and coefficients.
After the selection of the number of POD modes required to have an accurate approximation of each geometry, we end up with the first reduction of the parameter space, that is with 3 POD coefficients µ POD := α ∈ R 3 , we are able to represent all the possible deformations for µ FFD ∈ D. So we can express every geometry with 3 modes, but the coefficients can still be linearly dependent. We can investigate this dependance by plotting every component µ POD , we can constraint it to be inside the quadrilateral in Figure 4, on the right. So we are able to express every possible geometry described with the original 5 FFD parameters with only 2 new independent parameters. We can thus sample the full parameter space using a new reduced space, preserving the geometrical variability, and reducing the construction cost of the reduced output field space. This, as we are going to present, results in a faster optimization procedure.
Non-intrusive reduced order modeling by means of PODI
Proper orthogonal decomposition with interpolation is a non-intrusive datadriven method for reduced order modeling allowing an efficient approximation of the solution of parametric partial differential equations. As well as for the geometries, we collect in a database the high-fidelity solutions of several CFD simulations corresponding to different configurations, then we apply the POD algorithm to the solutions matrix -the matrix whose columns are the solutions -in order to extract the POD modes that span the optimal space which the solutions belong to. Thus the solutions can be projected onto the reduced space: we represent the high-fidelity solutions as linear combination of the POD modes. Similarly to Eq. (4), the modal coefficients of the i-th solution x PODI i -also called the reduced solutionare obtained as:
x PODI i = U T x i ∀i ∈ {1, . . . , M }(6)
where U refers to the POD modes and M is the number of high-fidelity solutions. We call N the number of POD modes and N the dimension of high-fidelity solutions then x PODI i ∈ V N and x i ∈ V N . Since in complex problems we have an high number of degrees of freedom, typically we have N N . The low-rank representation of the solutions allows to easily interpolate them, exploting the relation between the reduced solutions and the input parameters: in this way, we can compute the modal coefficients for any new parametric point and project the reduced solution onto the high dimensional space for a real-time approximation of the truth solution. This technique is defined non-intrusive, since it relies only on the solutions, without requiring information about the physical system and the equations describing it. For this reason it is particularly suited for industrial problem, thanks to its capability to be coupled also with commercial solvers. The downside is the error introduced by the interpolation, depending by the method itself, and the requirement of solutions with the same dimensionality, that can be a problem if the computational grid is built from scratch for any new configuration. Possible solutions are the projection of the solution on a reference mesh [7], or to deform the grid using the laplacian diffusion [29]. Moreover, we cite [12,25] for other examples of PODI applications. For this work, we employed the open source Python package EZyRB [9] as software to perform the data-driven model order reduction.
Numerical results
In this section we present the results for the application of the complete pipeline to the problem presented in Section 2.
First, we sample the full parameter space D extracting N POD = 100 parameters to construct the reduce order model without any further reduction, and we identify this approach with the subscript "POD". Then, as explained in Section 4, we compute the shape manifold with 1500 different deformations, and we extract the new coefficients describing the new reduced parameter space. We sample this 2-dimensional space uniformly and we collect N POD+reduction = 80 solution snapshots. We can compare the decay of the singular values of the snapshots matrix for the two approaches. In Figure 5 we can note how the proposed computational pipeline results in a faster decay and thus in a better approximation for a given number of POD modes. We underline that, despite the gain is not so big, the results do not involve further high-fidelity simulations. We only collected several different deformations at a negligible computational cost with respect to a single full order CFD simulation. Moreover the construction of the interpolator takes a huge advantage of the reduced parameter space since it counters the curse of dimensionality.
We can conclude that the proposed preprocessing step has sever benefits in terms of accuracy of the reduced order model at a small cost from a computational point of view.
Conclusions and perspectives
In this work we presented a complete data-driven numerical pipeline for shape optimization in naval engineering problems. The object was to find the optimal bulbous bow to minimize the total drag resistance of a hull advancing in calm water. First we parametrized and morphed the bulbous bow through the free form deformation method. Then we reduced the parameter space dimension approximating the shape manifold with the use of proper orthogonal decomposition and the investigation on linear dependance of the POD coefficients. We create the reduced order model sampling only the reduced two dimensional parameter space and with POD with interpolation we can compute in real time the outputs of interest for untried new parameters. Thus the optimizer can query the surrogate model and find the optimal shape.
Figure 1 :
1Complete hull domain representing the DTMB 5415 and, on the right, a zoom on the bulbous bow.
Figure 2 :
2Sketch of the FFD map M composition. The domain is mapped to a reference configuration, then the lattice of FFD control points induce the body deformation, and finally the morphed object is mapped back to the physical space.
Figure 3 :
3Example of FFD parametrisation and morphing of the DTMB 5415 hull. In green the lattice of control points that define the actual deformation.
4
Reduction of the parameter space through POD of the mesh coordinates
each other. As we can see from the plot on the left inFigure 4, we can approximate µ
Figure 4 :
4POD coefficients dependance. On the left we have µ
blue, and in red the linear regression to approximate one as a function of the other. On the right µ
blue, and in red the boundaries defining the quadrilateral in which the sampling is performed.
Figure 5 :
5POD singular values decay as a function of the number of modes. The blue line corresponds to the original sampling in the full parameter space, while the red dotted line, which identifies the POD+reduction approach, corresponds to the sampling in the new reduced parameter space.
AcknowledgementsThis work was partially performed in the context of the project SOPHYA -"Seakeeping Of Planing Hull YAchts", supported by Regione FVG, POR-
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| [] |
[
"Hierarchical Regression Network for Spectral Reconstruction from RGB Images",
"Hierarchical Regression Network for Spectral Reconstruction from RGB Images"
] | [
"Yuzhi Zhao \nCity University of Hong Kong\nHong Kong SARChina\n",
"Lai-Man Po \nCity University of Hong Kong\nHong Kong SARChina\n",
"Qiong Yan \nSenseTime Research\n\n",
"Wei Liu \nSenseTime Research\n\n\nHarbin Institute of Technology\nChina\n",
"Tingyu Lin \nCity University of Hong Kong\nHong Kong SARChina\n"
] | [
"City University of Hong Kong\nHong Kong SARChina",
"City University of Hong Kong\nHong Kong SARChina",
"SenseTime Research\n",
"SenseTime Research\n",
"Harbin Institute of Technology\nChina",
"City University of Hong Kong\nHong Kong SARChina"
] | [] | Capturing visual image with a hyperspectral camera has been successfully applied to many areas due to its narrowband imaging technology. Hyperspectral reconstruction from RGB images denotes a reverse process of hyperspectral imaging by discovering an inverse response function. Current works mainly map RGB images directly to corresponding spectrum but do not consider context information explicitly. Moreover, the use of encoder-decoder pair in current algorithms leads to loss of information. To address these problems, we propose a 4-level Hierarchical Regression Network (HRNet) with PixelShuffle layer as inter-level interaction. Furthermore, we adopt a residual dense block to remove artifacts of real world RGB images and a residual global block to build attention mechanism for enlarging perceptive field. We evaluate proposed HRNet with other architectures and techniques by participating in NTIRE 2020 Challenge on Spectral Reconstruction from RGB Images. The HRNet is the winning method of track 2 -real world images and ranks 3rd on track 1 -clean images. | 10.1109/cvprw50498.2020.00219 | [
"https://arxiv.org/pdf/2005.04703v1.pdf"
] | 218,581,433 | 2005.04703 | fe168113047577c9e30ce9c3f4efd666057d0099 |
Hierarchical Regression Network for Spectral Reconstruction from RGB Images
Yuzhi Zhao
City University of Hong Kong
Hong Kong SARChina
Lai-Man Po
City University of Hong Kong
Hong Kong SARChina
Qiong Yan
SenseTime Research
Wei Liu
SenseTime Research
Harbin Institute of Technology
China
Tingyu Lin
City University of Hong Kong
Hong Kong SARChina
Hierarchical Regression Network for Spectral Reconstruction from RGB Images
Capturing visual image with a hyperspectral camera has been successfully applied to many areas due to its narrowband imaging technology. Hyperspectral reconstruction from RGB images denotes a reverse process of hyperspectral imaging by discovering an inverse response function. Current works mainly map RGB images directly to corresponding spectrum but do not consider context information explicitly. Moreover, the use of encoder-decoder pair in current algorithms leads to loss of information. To address these problems, we propose a 4-level Hierarchical Regression Network (HRNet) with PixelShuffle layer as inter-level interaction. Furthermore, we adopt a residual dense block to remove artifacts of real world RGB images and a residual global block to build attention mechanism for enlarging perceptive field. We evaluate proposed HRNet with other architectures and techniques by participating in NTIRE 2020 Challenge on Spectral Reconstruction from RGB Images. The HRNet is the winning method of track 2 -real world images and ranks 3rd on track 1 -clean images.
Introduction
Hyperspectral (HS) imaging technology refers to the spectral signature is densely sampled to many narrow bands. It combines imaging technology with spectral technology to detect the two-dimensional geometric space and one-dimensional spectral information of the target to obtain continuous, narrow-band images with high spectral resolution. Normally, most of the civil cameras capture only three primary colors. However, HS spectrometers can obtain the spectrum of each pixel in the scene and collect the information into a set of images. To visualize HS images, a response function is adopted to transform HS images into RGB format. Conversely, we can acquire HS images from * Corresponding author: [email protected] the visible format by learning the inverse function. In this paper, we propose a general hierarchical regression network (HRNet) for spectral reconstruction from RGB images.
HS imaging technology has many advantages and particular characteristics. There have been many applications based on HS imaging technology, e.g, remote sensing technology [25], pedestrian detection [17,23], food processing [29], medical imaging [2]. However, in recent years, the development of HS imaging has encountered a bottleneck since it mainly depends on spectrometers. The traditional spectrometers saves images with huge volume and need long operation time, which restricts HS imaging technology applied to portable platforms and high-speed moving scenes [28]. Although researchers have continuously optimized the traditional pipeline [7,35], these hardware devices are still expensive and of high complexity. Thus, we present a low cost and automate approach only based on RGB cameras. To address the problem, we propose a HR-Net that learns the process of RGB images to corresponding HS projections.
In general, spectral reconstruction is an ill-posed problem. Moreover, there is unknown noise in environment leading to degraded RGB images. However, there is dense correspondence between RGB images and HS images, making it possible to exploit the correlation from many RGB-HS pairs. Since the information of RGB image is much less than HS image, there may be many reasonable HS image combinations corresponding to a same RGB image. The algorithm needs to learn a reasonable mapping function that produces high-quality HS images. With the development of deep convolutional neural network (CNN), it is eligible to learn the blind mapping for spectral construction.
The previous methods [32,21,33,6,36] mainly utilize an auto-encoder structure with residual blocks [14]. The network often performs convolution at low spatial resolution since the features are more compact and the computation is more efficient. However, as the network goes deeper, it fails to remain the original pixel information due to per-forming down-sampling by convolutions. To address this problem, we introduce a lossless and learnable sampling operator PixelShuffle [31]. To further boost the quality of generated images, we propose a hierarchical architecture that extracts the features of different scales. At each level, the input is obtained by the reverse PixelShuffle (PixelUnShuffle) that no pixel is lost. Moreover, we propose to use residual dense block and residual global block in HRNet for removing artifacts and noise and modelling remote pixel correlation, respectively.
In general, there are three main contributions of this paper:
(1) We propose a HRNet that utilizes PixelUnShuffle and Pixelshuffle layers for downsampling and upsampling without information loss. We also propose residual dense block with residual global block to enlarge perceptive field and boost generation quality;
(2) We propose a 8-setting ensemble strategy to further enhance the generalization of HRNet;
(3) We evaluate proposed HRNet on NTIRE 2020 HS dataset. The HRNet is winning method of track 2 -real world images and ranks 3rd on track 1 -clean images.
Related work
Hyperspectral image acquisition. Conventional methods for hyperspectral image acquisition often adopt spectrograph with spatial scanning or spectral scanning technology. There are several types of scanner utilized for capturing images including pushbroom scanner, whiskbroom scanner, and band sequential scanner. They have been widely used to many applications such as detector, environmental monitoring and remote sensor for decades. For instance, pushbroom scanner and whiskbroom scanner are used for photogrammetric and remote sensing by satellite sensors [28,5]. However, those devices need to capture the spectral information of single points or bands separately, then scan the whole scene to get a fully HS image, which is difficult to capture scenes with moving objects. In addition, they are too large physically and not suitable for portable platforms. In order to address the problems, many kinds of non-scanning spectrometers have been developed to adapt the application of dynamic scenes [10,7,35].
Hyperspectral image reconstruction from RGB images. Since the traditional methods for hyperspectral image acquisition are not portable or time-consuming for many applications, current methods attempt to reconstruct hyperspectral image from RGB image. By learning the mapping from RGB images to hyperspectral images on a big RGB-HS dataset, it is more convenient to obtain many HS images. Recent years have witnessed various studies including sparse coding and deep learning. In 2008, Parmar et al. [27] proposed a data sparsity expanding method to recover the spatial spectral data cube. Arad et al. [3] first lever-aged HS prior in order to create a sparse dictionary of HS signatures and their corresponding RGB projections. While Aeschbacher et al. [1] pushed the performance of Arad et al.'s method for better accuracy and runtime based on A+ framework [34].
Beyond the dataset provided by Arad et al. [3], many approaches proposed their own dataset. For instance, Yasuma et al. [37] utilized a CCD camera (Apogee Alta U260) to captured 31-band multispectral images (400700 nm, at 10 nm intervals) of several static scenes. Nguyen et al. [26] captured a dataset by Specim's PFD-CL-65-V10E (400 nm to 1000 nm) spectral camera and there were total 64 images. Chakrabarti et al. [8] explored a statistical model based on 55 HS images of indoor and outdoor scenes. With the improvement of the scale and resolution of natural HS dataset, the training of deep learning method becomes more feasible, a number of algorithms based on convolutional neural network were proposed [21,33]. Simon et al. [20] proposed a fully convolutional densely connected Tiramisu network with one hundred layers for semantic segmentation. Galliani et al. [11] enhanced it for spectral image super-resolution. Can et al. [6] improved it to avoid overfitting to the training data and obtain faster inference speed. Moreover, Xiong et al. [36] proposed a unified HSCNN framework for hyperspectral recovery from both RGB and compressive measurements. To boost the performance, they developed a deep residual network named HSCNN-R, and another distinct architecture that replaces the residual block by the dense block with a novel fusion scheme, named HSCNN-D, collectively called HSCNN+ [32].
Convolutional neural networks. The convolutional neural networks have been successfully applied in many low-level vision tasks, e.g. colorization [39,19], inpainting [18,38], deblurring [22], denoising [13,9], and demosaicking [9,40]. Hyperspectral reconstruction, as one of low-level task, has gained great improvement of performance recently by deep convolutional neural networks. In order to facilitate convergence and extract features effectively, many well-known basic blocks are utilized in those frameworks such as residual block and dense block. He et al. [14] proposed a residual network initially for image classification. It improves the accuracy obviously compared with traditional cascade convolutional structure. Then, the residual block has been widely used in image enhancement region for maintaining low-level features by the short connection. It was enhanced by densenet proposed by Huang et al. [16] to improve the feature fusion ability. Moreover, Hu et al. [15] strengthened them by a squeeze-and-excitation network including a feature attention mechanism. It was implemented by MLP layers for modelling connections of pixels in different spatial location. In general, our HRNet combines the advantages of above methods and provides a more effective and accurate solution for HS reconstruction.
Methodology
Dataset
We train our approach on the HS dataset provided by NTIRE Challenge 2020. This dataset consists of three parts: spectral images, clean RGB images (for track 1) and real world RGB images (for track 2). There are overall 450 RGB-HS pairs in training for both tracks involving different scenes. Each spectral image has the information of 31 bands in range of 400 nm to 700 nm. It is of 482 × 512 spatial resolution. To generate its corresponding RGB image, there is a fixed response function applied to HS bands. The rendering process can be defined as:
RGB = HS × ResponseF unc.(1)
The RGB images and HS images include 3 and 31 channels, respectively. The ResponseF unc maps each HS band to visible channel R, G, and B by 93 parameters. For clean RGB images, they are constructed by a known response function and saved as uncompressed format. However, the real world RGB images are acquired by unknown response function with additional blind noise and demosaicking operation. Some examples are illustrated in Figure 1 (e.g. 1st band approximately covers the 395-405 nm range).
HRNet architecture
Generally, we propose a 4-level network architecture for high-quality spectral reconstruction from RGB images, as shown in Figure 2. The PixelUnShuffle layers [31] are utilized to downsample the input to each level without adding parameters. Therefore, the number of pixels of input is fixed while the spatial resolution decreases. Conversely, the learnable PixelShuffle layers are adopted to upsample feature maps and reduce channels for inter-level connection. The PixelShuffle only reshapes feature maps and does not introduces interpolation like bilinear upsampling. It allows the network to learn upsampling operation adaptively.
For each level, the process is decomposed to inter-level integration, artifacts reduction, and global feature extraction. For inter-level learning, the output features of subordinate level are pixel shuffled, then concatenated to current level, finally processed by an additional convolutional layer to unify channel number. In order to effectively reduce artifacts, we adopt residual dense block [14,16], containing 5 dense-connected convolutional layers and a residual. Moreover, the residual global block [14,15] with short-cut connection of input is used to extract attention for every remote pixels by MLP layers.
Since the features are most compact in bottom level, there is a 1 × 1 convolutional layer attached to the last of bottom level in order to enhance tone mapping by weighting all channels. The two mid levels process features at different scales. Moreover, the top level uses the most blocks to effectively integrate features and reduce artifacts thus produce high-quality spectral images. The illustration of these blocks are in Figure 3.
Implementation details
We only use L1 loss in the training process, which is a PSNR-oriented optimization for the system. The L1 loss is defined as:
L 1 = E[||G(x) − y|| 1 ],(2)
where x and y are input and output, respectively. The G( * ) is the proposed HRNet. Note that, we utilize the local patches for efficient training. The input RGB image and output spectral images are cropped in same spatial region. For network architecture, all the layers are LeakyReLU [24] activated except output layer. We do not use any normalization in HRNet to maintain the data distribution. The reflect padding is adopted for each convolutional layer in order to reduce border effect. The weights of VCGAN are initialized by Xavier algorithm [12]. For training details, we use the entire NTIRE 2020 HS dataset (450 HS-RGB pairs for both tracks) at training. The whole HRNet is trained for 10000 epochs overall. The initial learning rate is 1 × 10 −4 and halved every 3000 epochs. For optimization, we use Adam optimizer with β 1 = 0.5 , β 2 = 0.999 and batch size equals to 8. The image pairs are randomly cropped to 256 × 256 region and normalized to range [0, 1]. All the experiments are implemented using 2 NVIDIA Titan Xp GPUs. It takes approximately 7 days for whole training process.
Ensemble strategy
Since the solution space of spectral reconstruction is often large, there may be multiple settings that achieve same performance on the training set. Therefore, a single network may lead to poor generalization performance since it tends to fall into local minima. However, we can minimize this risk by combining multiple network settings to enhance generalization and fuse the knowledge. In order to perform ensemble strategy, we use 4 other hyper-parameter settings and train HRNet from scratch for both tracks. These settings can be summarized as:
• Re-train the HRNet using baseline training setting.
• Exchange the position of residual dense block and residual global block in HRNet, and use baseline training setting.
• Train the network with different batch size (2 or 4) and keep other hyper-parameter settings, network architecture.
• Train the network with different cropping patch size (320 × 320 or 384 × 384) and keep other hyperparameter settings, network architecture.
Therefore, there are 8 kinds of training methods. All the methods used for ensemble are trained for 10000 epochs. We record the MRAE (mean absolute value between all bands of generated spectral images G(x) and ground truth y) every 1000 epochs, as shown in Table 1 and Figure 4.
Experiment
Experimental settings
We evaluate proposed HRNet by comparing with other network architectures and conducting ablation study on NTIRE 2020 HS dataset. For each track, there are 10 validation RGB images. The evaluation metrics are defined as:
• MRAE. It computes the pixel-wise disparity (mean absolute value) between all bands of generated spectral images G(x) and ground truth y. It explicitly represents the construction quality of network. It is defined as:
M RAE = 1 N N i=1 |G(x) i − y i | y i ,(3)
where N denotes the overall pixels of spectral images.
• RMSE. It computes the root mean square error be- tween the generated and ground truth spectral images with 31 bands. It is defined as:
RM SE = 1 N N i=1 (G(x) i − y i ) 2 .(4)
• Back Projection MRAE (BPMRAE). It evaluates the colorimetric accuracy of recovered RGB images from the generated and ground truth spectral images by a fixed camera response function. It is defined as:
BP M RAE = 1 N N i=1 |(R × G(x)) i − (R × y) i | y i ,(5)
where R denotes the function ResponseF unc.
Comparison with other architectures
We utilize two common network architectures for comparison: U-Net [30] and U-ResNet [30,14]. Both of them have been widely used in many previous low-level tasks [19,18,38,22,9,40]. The first convolutional layer and last convolutional layer utilize 7 ×7 convolution without changing spatial resolution. The training scheme for all methods are same. Other details are concluded as: (1) U-Net. The encoder layers perform convolution with stride of 2. The spatial resolution of bottom feature map equals to 1 × 1. There are short concatenations between each encoder layer and decoder layer with same resolution; (2) U-ResNet. The total number of encoder layers and decoder layers are half of U-Net. Instead, there are 4 residual blocks attached to the last layer of encoder. The concatenations are reserved.
We train both networks using same hyper-parameters of HRNet until convergence. There is no ensemble strategy used. We generate the reconstructed spectral images using the best epoch of them. The results are summarized in Table 2. We also visualize each method in Figure 5 and 6 by pseudo-color map. The first three rows show the data distribution of 3 methods and last row indicates ground truth. We recommend readers to compare textures of background.
There are two reasons that proposed HRNet outperforms other two methods. The first is that HRNet utilizes Pix-elShuffle to connect each level. Traditional nearest or bilinear upsampling will introduce redundancy information to features, which is unnecessary for feature extraction. However, by the combination of PixelUnShuffle and Pix-elShuffle, HRNet could process high-level features more efficiently. The second is that HRNet adopts two residualbased blocks, which facilitate convergence and assist each level to exploit different scales of features. Moreover, the blocks with residual learning helps remove artifacts. The residual global block enhances context information since it models correlation for every two pixels.
Ablation study
In order to demonstrate the effectiveness of both residual dense block (ResDB) and residual global block (ResGB), we replace them by plain convolution layers with similar FLOPs. The results in track 1 -clean images is shown in Table 3. The baseline of HRNet is shown in Table 1, which has better performance comparing with all ablation settings. If we delete all ResDB or ResGB in HRNet, the MRAE decreases the most, which demonstrates the combination of both blocks is significant for spectral reconstruction. We conduct another experiment that shrinks the HRNet model size by decreasing channels of each convolutional layer to half, one fourth, and one eighth of original numbers. It will compress model size greatly by sacrificing pixel fidelity. To better compare these settings, we conclude the multiplyaccumulate operation (MACs), total network parameters (Params), model size saved on machine (Weights) and 3 quantitative metrics results in Table 4
Testing result on NTIRE 2020 challenge
The proposed HRNet ranks 3rd and 1st on track 1 and track 2, respectively, of NTIRE 2020 Spectral Reconstruction from RGB Images Challenge [4]. The comparison results on testing set are summarized in Table 5 and 6. Moreover, the HRNet has better performance on track 2 since it adopts two effective blocks for removing artifacts while utilizes learnable PixelShuffle upsampling operator. The ensemble strategy works obviously on both tracks that improves the MRAE from 0.042328 to 0.039893 since it avoids the HRNet to fall into local minima. In conclusion, both HRNet architecture and ensemble strategy contribute to spectral reconstruction performance.
Conclusion
In this paper, we presented a 4-level HRNet for automatically generating spectrum from RGB images. For each level, it adopts both residual dense block and residual global block for effectively extracting features. While the Pix-elShuffle is utilized for inter-level connection. Then, we proposed a novel 8-setting ensemble strategy to further enhance the quality of predicted spectral images. Finally, we validated the HRNet outperforms the well-known low-level vision frameworks such as U-Net and U-ResNet on NTIRE 2020 HS dataset. Furthermore, we presented 3 types of compressed HRNets and analyzed their reconstruction performance and computing efficiency. The proposed HRNet is the winning method of track 2 -real world images and ranks 3rd on track 1 -clean images.
Figure 1 .
1Visualization of NTIRE 2020 HS dataset. For each group, from top to bottom and left to right, they represent clean RGB images, real world RGB images, HS images with 400 nm, 410 nm, 420 nm, 500 nm, 600 nm, and 700 nm channels, respectively.
Figure 2 .
2Illustration of the architecture of HRNet. Please visit the project web page https://github.com/zhaoyuzhi/ Hierarchical-Regression-Network-for-Spectral-Reconstruction-from-RGB-Images to try our codes and pretrained models.
Figure 3 .
3Illustration of the architecture of residual dense block (ResDB) and residual global block (ResGB).
Finally, we utilize the epoch with best MRAE value of 8 methods for computing average.
Figure 4 .
4The MRAE between ground truth spectral images and the generated images of different hyper-parameter settings for ensemble.
Figure 5 .
5Visualization of generated results from U-ResNet, U-Net, and proposed HRNet on NTIRE 2020 HS validation set track 1.
Figure 6 .
6Visualization of generated results from U-ResNet, U-Net, and proposed HRNet on NTIRE 2020 HS validation set track 2.
Table 5 .
5The final testing results of NTIRE 2020 Spectral Reconstruction from RGB Images Challenge track 1 -clean images.
. The MACs, Params, and Weights of baseline HRNet are 182.347 Gb, 31.705 Mb, and 123.879 Mb, respectively. Users can choose high-quality HRNet to obtain high pixel fidelity of spectral images (MRAE = 0.042328) or high-efficiency HRNet with small size (Weights = 2.410 Mb).
Table 6 .
6The final testing results of NTIRE 2020 Spectral Reconstruction from RGB Images Challenge track 2 -real world images.
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"S Dȃscȃlescu \nFaculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania\n",
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"S Dȃscȃlescu \nFaculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania\n",
"C Nȃstȃsescu \nFaculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania\n\nInstitute of Mathematics of the Romanian Academy\nPO-Box 1-764RO-014700BucharestRomania\n",
"L Nȃstȃsescu [email protected] \nFaculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania\n\nInstitute of Mathematics of the Romanian Academy\nPO-Box 1-764RO-014700BucharestRomania\n"
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"Faculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania",
"Faculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania",
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"Faculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania",
"Institute of Mathematics of the Romanian Academy\nPO-Box 1-764RO-014700BucharestRomania",
"Faculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania",
"Faculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania",
"Institute of Mathematics of the Romanian Academy\nPO-Box 1-764RO-014700BucharestRomania",
"Faculty of Mathematics and Computer Science\nUniversity of Bucharest\nStr. Academiei 14, Bucharest 1RO-010014Romania",
"Institute of Mathematics of the Romanian Academy\nPO-Box 1-764RO-014700BucharestRomania"
] | [] | We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H-comodule algebra, we uncover the connection between A and the smash product A#H * with respect to the Frobenius and symmetric properties. Mathematics Subject Classification: 16T05, 18D10, 16S40 | null | [
"https://arxiv.org/pdf/1603.06257v1.pdf"
] | 119,317,873 | 1603.06257 | 1c844c4a69eadcdda76f8f29028f50f8af061031 |
Symmetric algebras of corepresentations and smash products
20 Mar 2016
S Dȃscȃlescu
Faculty of Mathematics and Computer Science
University of Bucharest
Str. Academiei 14, Bucharest 1RO-010014Romania
C Nȃstȃsescu
Faculty of Mathematics and Computer Science
University of Bucharest
Str. Academiei 14, Bucharest 1RO-010014Romania
Institute of Mathematics of the Romanian Academy
PO-Box 1-764RO-014700BucharestRomania
L Nȃstȃsescu [email protected]
Faculty of Mathematics and Computer Science
University of Bucharest
Str. Academiei 14, Bucharest 1RO-010014Romania
Institute of Mathematics of the Romanian Academy
PO-Box 1-764RO-014700BucharestRomania
Symmetric algebras of corepresentations and smash products
20 Mar 2016arXiv:1603.06257v1 [math.RA]Hopf algebracomodulemonoidal categoryFrobenius algebrasymmetric alge- brasmash product
We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H-comodule algebra, we uncover the connection between A and the smash product A#H * with respect to the Frobenius and symmetric properties. Mathematics Subject Classification: 16T05, 18D10, 16S40
Introduction and preliminaries
We work over a basic field k. A finite dimensional algebra A is called Frobenius if A and its dual A * are isomorphic as left (or equivalently, as right) A-modules. Frobenius algebras arise in representation theory, Hopf algebra theory, quantum groups, cohomology of compact oriented manifolds, topological quantum field theory, the theory of subfactors and of extensions of C *algebras, the quantum Yang-Baxter equation, etc., see [13]. A rich representation theory has been uncovered for such algebras, see [14], [19]. It was showed in [1], [2] that A is Frobenius if and only if it also has a coalgebra structure whose comultiplication is a morphism of A, Abimodules. This equivalent definition of the Frobenius property has the advantage that it makes sense in any monoidal category. The study of Frobenius algebras in monoidal categories was initiated in [15], [20], [21], and such objects have occurred in several contexts, for example in the theory of Morita equivalences of tensor categories, in conformal quantum field theory, in reconstruction theorems for modular tensor categories, see more details and references in [11], [12], [16]; recent developments can be also found in [5].
The representation theory of Frobenius algebras uncovers several symmetry features, for example there is a duality between the categories of left and right finitely generated modules, and the lattices of left and right ideals are anti-isomorphic. Among Frobenius algebras there is a class of objects having even more symmetry. These are the symmetric algebras A, for which A and A * are isomorphic as A, A-bimodules. The category of commutative symmetric algebras is equivalent to the category of 2-dimensional topological quantum field theories, see [1]. Symmetric algebras appear in block theory of group algebras in positive characteristic, see [19, Chapter IV].
It is not clear how one could define symmetric algebras in an arbitrary monoidal category. Symmetric algebras in monoidal categories with certain properties were first considered in [10], as
Frobenius algebras and symmetric algebras of corepresentations
Let H be a Hopf algebra, and let A be a finite dimensional right H-comodule algebra, with H-coaction a → a 0 ⊗ a 1 . Then there exists an element i a i ⊗ h i ⊗ a * i ∈ A ⊗ H ⊗ A * such that a 0 ⊗ a 1 = i a * i (a)a i ⊗ h i for any a ∈ A; this element corresponds to the H-comodule structure map of A through the natural isomorphism A ⊗ H ⊗ A * ≃ Hom(A, A ⊗ H). A right H-comodule structure is induced on A * by a * → i a * (a i )a * i ⊗ S(h i ), for any a * ∈ A * .
If we consider the left H * -actions on A and A * associated with these right H-comodule structures, denoted by h * · a and h * · a * for h * ∈ H * , a ∈ A, a * ∈ A * , we have
(h * · a * )(a) = i h * (S(h i ))a * (a i )a * i (a) = a * ( i h * (S(h i ))a * i (a)a i ) = a * ((h * S) · a) so (h * · a * )(a) = a * ((h * S) · a)(1)
Moreover, A * ∈ M H A , with the usual right A-action; this means that the A-module structure of A * is right H-colinear. It is known (see [6,Theorem 2.4]) that the following are equivalent:
(1) A ≃ A * in M H A ; (2) There exists a nondegenerate associative bilinear form B : A × A → k such that B(h * · a, b) = B(a, (h * S) · b) for any a, b ∈ A, h * ∈ H * ; (3)
There exists a linear map λ : A → k such that λ(h * · a) = h * (1)λ(a) for any a ∈ A, h * ∈ H * , and Ker λ does not contain a non-zero right ideal of A; (4) There exists a linear map λ : A → k such that λ(h * ·a) = h * (1)λ(a) for any a ∈ A, h * ∈ H * , and Ker λ does not contain a non-zero subobject of A in M H A ; (5) A is a Frobenius algebra in the category M H . The connections between an isomorphism θ : A → A * as in (1), a bilinear map B as in (2) and a linear map λ as in (3) On the other hand, A * is also a left A-module in a natural way, but in general A * is not an object of A M H (with a similar compatibility condition for the A-action and H-coaction). However, A * is an object in A (S 2 ) M H , where A (S 2 ) is just the algebra A, with the H-coaction shifted by S 2 , i.e. a → a 0 ⊗ S 2 (a 1 ).
Assume now that H is a cosovereign Hopf algebra in the sense of [3], i.e. there exists a character u on H (in other words, u is a grouplike element of the dual Hopf algebra H * , or equivalently, an algebra morphism from H to k) such that S 2 (h) = u −1 (h 1 )u(h 3 )h 2 for any h ∈ H; this is the same with (S 2 ) * being an inner algebra automorphism of H * via u. Following [3], we say that u is a sovereign character of H. Then f : A → A (S 2 ) , f (a) = u −1 · a = u −1 (a 1 )a 0 , is an isomorphism of right H-comodule algebras, and it induces an isomorphism of categories
F : A (S 2 ) M H → A M H where for M ∈ A (S 2 ) M H , F (M )
is just M , with the same H-coaction, and A-action * given by a * m = f (a)m, for any a ∈ A and m ∈ M . By restriction, this induces an isomorphism of categories (and we denote it by F , too) Now we give equivalent characterizations of this property. The next result can be derived from [11, Proposition 4.6], using the structure of duals in a category of corepresentations. In our sketch of proof, explicit description is given for several ways to describe symmetry of algebras of corepresentations. Proposition 2.2 Let A be a right H-comodule algebra, where H is a cosovereign Hopf algebra. Keeping the above notation, the following are equivalent.
F : A (S 2 ) M H A → A M H A Now A * ∈ A (S 2 ) M H A , so then F (A * ) ∈ A M H A .
(1) A is (H, u)-symmetric.
(2) There exists a nondegenerate bilinear form B :
A × A → k such that B(b, ca) = B(bf (c), a), B(b, a) = B(f (a), b), and B(b, h * · a) = B((h * S) · b, a) for any a, b, c ∈ A, h * ∈ H * .
(3) There exists a linear map λ : A → k such that λ(ba) = λ(af (b)) and λ(h * · a) = h * (1)λ(a) for any a, b ∈ A, h * ∈ H * , and also Ker λ does not contain a non-zero right ideal of A.
and θ is right A-linear if and only if
B(b, ac) = B(cb, a) for any a, b, c ∈ A.(3)
We see that if (2) and (3)
Moreover, if (2) and (4) 2) More generally, if H is an involutory Hopf algebra, i.e. S 2 = Id, then H is obviously cosovereign with ε as a sovereign character. In this case, if A is a finite dimensional algebra in
M H , then A * ∈ A M H A , so F (A * ) is just A * , with the usual left and right A-actions. Thus A is (H, ε)-symmetric if and only if A * ≃ A in A M H A .
Remark 2.4
The definition of symmetry depends on the cosovereign character. Thus it is possible that a cosovereign Hopf algebra H has two sovereign characters u and v, and a right H-comodule algebra A is (H, u)-symmetric, but not (H, v)-symmetric. Indeed, let H = kC 2 , where C 2 = {e, g} is a group of order 2 (e is the neutral element), and the characteristic of k is = 2. Let A be a commutative C 2 -graded division algebra with support C 2 ; for example one can take A = kC 2 . Then H is involutory, so it is a cosovereign Hopf algebra with two possible sovereign characters ε = p e + p g and
u = p e − p g , where {p e , p g } is the basis of H * dual to the basis {e, g} of H. We have u 2 = ε, so u −1 = u.
It is easy to see that A is (H, ε)-symmetric, i.e. graded symmetric in the terminology of [6], for example by using the results of [7], where the question whether any graded division algebra is graded symmetric is addressed. On the other hand, A is not (H, u)-symmetric. Indeed, let λ : A → k be a linear map such that λ(ab) = λ(b(u · a)) for any a, b ∈ A. We have u · a = a for any a ∈ A e (the homogeneous component of degree e of A), and u · a = −a for any a ∈ A g . Then for b = 1 and a ∈ A g we get λ(a) = 0, thus λ(A g ) = 0. Also, for a, b ∈ A g we obtain λ(ab) = 0, and since A g A g = A e , this shows that λ(A e ) = 0. Thus λ must be zero.
Remark 2.5
It is obvious that a graded symmetric algebra is symmetric as a k-algebra. More general, if H is involutory, then a (H, ε)-symmetric algebra is symmetric as a k-algebra. However, for an arbitrary cosovereign Hopf algebra H with sovereign character u, if A is (H, u)symmetric, then A is not necessarily symmetric as a k-algebra, as we show in the following example.
Let H = H 4 , the 4-dimensional Sweedler's Hopf algebra. It is presented by algebra generators c and x, subject to relations
c 2 = 1, x 2 = 0, xc = −cx
The coalgebra structure is defined by
∆(c) = c ⊗ c, ∆(x) = c ⊗ x + x ⊗ 1, ε(c) = 1, ε(x) = 0. The antipode S satisfies S(c) = c, S(x) = −cx, thus S 2 (x) = −x.
Apart from ε, H has just one more character α, given by α(c) = −1, α(x) = 0; it acts on the basis elements of H by
α · 1 = 1, α · c = −c, α · x = x, α · (cx) = −cx (5)
H is cosovereign, and the only sovereign character is α. We show that the linear map λ : H → k, λ(1) = λ(c) = λ(cx) = 0, λ(x) = 1 makes H an (H, α)-symmetric algebra. Indeed, we first see by a straightforward checking that λ is right H-colinear (or equivalently, left H *linear). Next, an easy computation using (5) shows that any element of the form ba − a(α · b) lies in the span of 1, c and cx, thus λ(ba) = λ(a(α · b)) for any a, b ∈ H.
Finally, let I be a left ideal of H contained in Kerλ. Let z = δ1 + βc + γcx ∈ I, where δ, β, γ ∈ k. Then cz = δc + β1 + γx ∈ I ⊆< 1, c, cx >, so γ = 0. Then xz = δx − βcx ∈ I ⊆< 1, c, cx >, so δ = 0. Now cxz = −βx ∈ I ⊆< 1, c, cx >, so β must be zero, too. Thus z = 0, and H is (H, α)-symmetric. This will also follow from Proposition 4.4.
On the other hand, H is not symmetric as a k-algebra.
Indeed, if λ : H → k is a linear map such that λ(ab) = λ(ba) for any a, b ∈ H, then λ(cx) = λ(xc) = −λ(cx), so λ(cx) = 0, and λ(x) = λ(ccx) = λ(cxc) = −λ(x), so λ(x) = 0. Thus the two-sided ideal < x, cx > of H is contained in Kerλ,
showing that H is not symmetric. This can also be seen from a general result saying that a Hopf algebra is symmetric as an algebra if and only if it is unimodular (i.e. the spaces of left integrals and right integrals in H coincide) and S 2 is inner, see [17]; for H 4 the square of the antipode is inner, but the unimodularity condition fails.
Now we explain how examples of (H, u)-symmetric algebras in the category M H can be constructed, where H is a cosovereign Hopf algebra with sovereign character u. We recall that for any algebra A (in the category of vector spaces), and any left A, right A-bimodule M , one can construct an algebra structure on the space A ⊕ M with the multiplication defined by (a, m)(a ′ , m ′ ) = (aa ′ , am ′ + ma ′ ); this is called the trivial extension of A and M . The unit of this algebra is (1, 0). If M = A * with the usual A, A-bimodule structure, the trivial extension of A and A * is simply called the trivial extension of A, and it is a symmetric algebra, see [14,Example 16.60].
Proposition 2.6 Let A be a right H-comodule algebra, where H is cosovereign with sovereign character u. Then E(A) = A ⊕ F (A * ), with the direct sum structure of a right H-comodule, and the algebra structure obtained by the trivial extension of A and the A,
A-bimodule F (A * ), is a right H-comodule algebra which is (H, u)-symmetric. Proof The multiplication of E(A) is given by (a, a * )(b, b * ) = (ab, a * b * + a * b) = (ab, (u −1 · a)b * + a * b)
for any a, b ∈ A and any a * , b * ∈ A * .
We first see that
E(A) is a right H-comodule algebra. Indeed, the H-coaction on E(A) is ρ : E(A) → E(A) ⊗ H, given by ρ(a, a * ) = (a 0 , 0) ⊗ a 1 + (0, a * 0 ) ⊗ a * 1 Then ρ((a, a * )(b, b * )) = ρ(ab, a * b * + a * b) = ((ab) 0 , 0) ⊗ (ab) 1 + (0, (a * b * ) 0 ) ⊗ (a * b * ) 1 + (0, (a * b) 0 ) ⊗ (a * b) 1 = (a 0 b 0 , 0) ⊗ a 1 b 1 + (0, a 0 * b * 0 ) ⊗ a 1 b * 1 + (0, a * 0 b 0 ) ⊗ a * 1 b 1 = ( (a 0 , 0) ⊗ a 1 + (0, a * 0 ) ⊗ a * 1 )( (b 0 , 0) ⊗ b 1 + (0, b * 0 ) ⊗ b * 1 ) = ρ(a, a * )ρ(b, b * )
Let λ : E(A) → k be the linear map defined by λ(a, a * ) = a * (1) for any a ∈ A, a * ∈ A * . Then
λ(h * · (a, a * )) = (h * · a * )(1) = a * ((h * S) · 1) (by (1)) = a * ((h * S)(1)1) = h * (1)a * (1) = h * (1)λ(a, a * ) Now λ((a, a * )(u −1 · (b, b * ))) = λ((a, a * )(u −1 · b, u −1 · b * )) = λ(a(u −1 · b), (u −1 · a)(u −1 · b * ) + a * (u −1 · b)) = (u −1 · b * )(u −1 · a) + a * (u −1 · b) = b * ((u −1 S) · (u −1 · a)) + a * (u −1 · b) (by (1)) = b * (u · (u −1 · a)) + a * (u −1 · b) = b * (a) + a * (u −1 · b) = λ(ba, (u −1 · b)a * + b * a) = λ((b, b * )(a, a * ))
Finally, we see that Kerλ does not contain non-zero right ideals. Indeed, if λ((a, a * )E(A)) = 0, then b * (u −1 · a) + a * (b) = 0 for any b ∈ A, b * ∈ A * . If we take b * = 0, we get that a * (b) = 0 for any b, so a * = 0. Then b * (u −1 · a) = 0 for any b * , so u −1 · a = 0, showing that a = 0.
We conclude that λ makes E(A) an (H, u)-symmetric algebra.
⊓ ⊔
We note that the previous result shows that any finite dimensional algebra in the category M H , where H is cosovereign via u, is a subalgebra of an (H, u)-symmetric algebra, and also a quotient of an (H, u)-symmetric algebra. The construction in Proposition 2.6 helps us to provide more examples of algebras that are symmetric in categories of corepresentations with respect to certain characters, but which are not symmetric as k-algebras.
c · 1 = 1, c · X = −X, x · 1 = 0, x · X = 1.
The dual space A * has a left H-action given by (h · a * )(a) = a * (S(h) · a) for any h ∈ H, a * ∈ A * and a ∈ A. If we consider the basis {p 1 , p X } of A * dual to the basis {1, X}, this action explicitly writes
c · p 1 = p 1 , c · p X = −p X , x · p 1 = −p X , x · p X = 0.
On the other hand, A * has usual left A-module structure given by
1p 1 = p 1 , 1p X = p X , Xp 1 = 0, Xp X = p 1 ,
and usual right A-module structure given by
p 1 1 = p 1 , p X 1 = p X , p 1 X = 0, p X X = p 1 .
A is also a right comodule algebra over the dual Hopf algebra H * . Since H * is cosovereign with sovereign character c (via the isomorphism H ≃ H * * ), the associated left A-module structure on F (A * ) is given by a * a * = (c · a)a * for any a ∈ A, a * ∈ A * . Thus
1 * p 1 = p 1 , 1 * p X = p X , X * p 1 = 0, X * p X = −p 1
The we can consider the algebra E(A) = A ⊕ F (A * ), whose basis is {1, X, p 1 , p X }, and multiplication induced by
X 2 = p 2 1 = p 2 X = p 1 p X = p X p 1 = 0 X * p 1 = 0, X * p X = −p 1 , p 1 X = 0, p X X = p 1
If we denote u = X and v = p X , we can present E(A) by generators u, v, subject to relations
u 2 = v 2 = 0, vu = −uv. The left H-module structure of E(A) is given by c · u = −u, c · v = −v, x · u = 1, x · v = 0.
If we denote by {P 1 , P c , P x , P cx } the basis of H * dual to the standard basis {1, c, x, cx} of H, the right H * -comodule structure of E(A) is given by
u → u ⊗ P 1 + c · u ⊗ P c + x · u ⊗ P x + (cx) · u ⊗ P cx = u ⊗ (P 1 − P c ) + 1 ⊗ (P x + P cx ) v → v ⊗ (P 1 − P c )
Since the Hopf algebra H is selfdual, a Hopf algebra isomorphism being given by 1 → P 1 + P c , c → P 1 − P c , x → P x − P cx , cx → −P x − P cx , we can regard A as a right H-comodule algebra. Summarizing, E(A) is the algebra with generators u, v, relations
u 2 = v 2 = 0, vu = −uv
and H-comodule structure given by
u → u ⊗ c − 1 ⊗ cx, v → v ⊗ c By Proposition 2.6, E(A) is (H, α)-symmetric, where α = P 1 − P c is the distinguished grouplike element of H * .
On the other hand, E(A) is not symmetric as a k-algebra. Indeed, if λ : E(A) → k is a linear map such that λ(zz ′ ) = λ(z ′ z) for any z, z ′ ∈ E(A), then λ(uv) = λ(vu) = −λ(uv), thus λ(uv) = 0. But the 1-dimensional space spanned by uv is a two-sided ideal of E(A), so Kerλ contains a non-zero ideal.
Frobenius smash products
Let A be an algebra in M H , where H is a finite dimensional Hopf algebra. Then A is a left H * -module algebra and we can consider the smash product A#H * , which is an algebra with multiplication given by (a#h * )(b#g * ) = a(h * 1 · b)#h * 2 g * It is known that A is a Frobenius algebra if and only if so is A#H * , see [4].
On the other hand, A#H * is an algebra in the category M H * , with the H * -coaction induced by the comultiplication of H * , i.e. a#h * → a#h * 1 ⊗ h * 2 . The aim of this section is to discuss the connection between A being a Frobenius algebra in M H , and A#H * being a Frobenius algebra in M H * .
We We recall that a right (respectively left) integral in H is an element t ∈ H such that th = ε(h)t (respectively ht = ε(h)t) for any h ∈ H, and a left integral on H is an element T ∈ H * such that h * T = h * (1)T for any h * ∈ H * . Theorem 3.1 Let H be a finite dimensional Hopf algebra, and let A be a finite dimensional right H-comodule algebra which is a Frobenius algebra in the category M H . Then the smash product A#H * is a Frobenius algebra in the category M H * .
Proof Let λ : A → k be a linear map whose kernel does not contain non-zero right ideals of A, and such that λ(h * · a) = h * (1)λ(a) for any h * ∈ H * and a ∈ A. Let t be a non-zero right integral in H. Define a linear map λ : A#H * → k such that λ(a#h * ) = λ(a)h * (t) for any a ∈ A, h * ∈ H * We have that λ(h · z) = ε(h)λ(z) for any h ∈ H and z ∈ A#H * . Indeed
λ(h · (a#h * )) = λ(a#h * 2 (h)h * 1 ) = h * 2 (h)λ(a)h * 1 (t) = λ(a)h * (th) = λ(a)ε(h)h * (t) = ε(h)λ(a#h * )
We show that Ker(λ) does not contain non-zero subobjects of A#H * in the category M H * A#H * . Let I be a right ideal of A#H * which is also a right H * -subcomodule (or equivalently, invariant with respect to the induced left H-action on A#H * ) such that I ⊂ Ker(λ).
We know that (A#H * ) co H * = A#1 ≃ A and A#H * /A is a right H * -Galois extension, see [8,Example 6.4.8]. Then J = I co H * is a right ideal of A and the Weak Structure Theorem for Hopf-Galois extensions shows that the map
J ⊗ A (A#H * ) → I, m ⊗ z → mz
is an isomorphism in the category M H * A#H * , see [8,Theorem 6.4.4]. In particular I = (J#1)(A#H * ) = J#H * . Since λ(I) = 0, we see that λ(J) = 0, so J = 0. We conclude that I must be zero.
⊓ ⊔
Corollary 3.2 A finite dimensional Hopf algebra H is Frobenius in the category M H .
Proof k is a right H * -comodule algebra in a trivial way, and it is clear that k is Frobenius in M H * . By Theorem 3.1 we get that k#H * * is Frobenius in M H * * . Since H * * ≃ H as Hopf algebras, we see that H is Frobenius in M H .
⊓ ⊔
The following example shows that the converse of Theorem 3.1 is not true.
Example 3.3 Let A = A 0 ⊕ A 1 be a superalgebra, i.
e. a C 2 -graded algebra, which is Frobenius as an algebra, but not graded Frobenius (i.e. it is not a Frobenius algebra in the category M kC 2 of supervector spaces). In this example we use the additive notation for the operation of C 2 . Examples of such A are given in [6,Section 6]; the trivial extension associated to a finite dimensional algebra is one such example. Let µ : A → k be a linear map whose kernel does not contain non-zero left ideals of A. We define λ : A#(kC 2 ) * → k, λ(a#p x ) = µ(a) for any a ∈ A, x ∈ C 2
Then λ(y ⇀ (a#p x )) = λ(a#p x−y ) = µ(a) = ε(y)λ(a#p x ).
On the other hand, Kerλ does not contain non-zero left ideals of A#(kC 2 ) * . Indeed, assume that λ((A#(kC 2 ) * )z) = 0, where z = a#p 0 + b#p 1 . Since (c#p 0 )z = ca 0 #p 0 + cb 1 #p 1 , we have 0 = µ(ca 0 ) + µ(cb 1 ) = µ(c(a 0 + b 1 )) for any c ∈ A. Thus µ(A(a 0 + b 1 )) = 0, showing that a 0 + b 1 = 0, and then a 0 = b 1 = 0. Similarly, since (c#p 1 )z = ca 1 #p 0 + cb 0 #p 1 , we obtain a 1 = b 0 = 0. Thus z = 0. We conclude that λ makes A#(kC 2 ) * a Frobenius algebra in the category M (kC 2 ) * .
Symmetric smash products
The following shows that the good connection between a finite dimensional right H-comodule algebra A and the smash product A#H * being Frobenius does not work anymore for the symmetric property.
Example 4.1 Let C 2 =< c >= {e, g} be the cyclic group of order 2, and let A be a C 2 -graded algebra which is symmetric and such that the homogeneous component A e is not symmetric. For example one can take the trivial extension A = R ⊕ R * of a non-symmetric algebra R, with the grading A e = R, A g = R * . Then A is a right kC 2 -comodule algebra, so we can consider the smash product A#(kC 2 ) * . Denote by {p e , p g } the basis of (kC 2 ) * dual to the basis {e, g} of kC 2 . Then 1#p e is an idempotent in A#(kC 2 ) * and it is easy to check that In this section we discuss the connection between A being a symmetric algebra in M H with respect to some character of H, and A#H * being a symmetric algebra in M H * with respect to some character of H * (i.e. a grouplike element of H).
Let H be a finite dimensional Hopf algebra. Then there exists a character α ∈ H * such that th = α(h)t for any left integral t in H and any h ∈ H; α is called the distinguished grouplike element of H * , and it also satisfies ht ′ = α −1 (h)t ′ for any right integral t ′ in H and any h ∈ H, see [18,Section 10.5] or [8,Section 5.5].
Similarly, there exists a distinguished grouplike element g of H, such that T h * = h * (g)T for any left integral T on H and any h * ∈ H * . We note that in [18], g −1 is called the distinguished grouplike element of H; we prefer the way we defined because g will play the same role for H as α does for H * . It is showed in [18,Theorem 10.5.4] that for any left integral t in H
∆(t) = S 2 (t 2 )g −1 ⊗ t 1(6)
Applying this for H * we see that for any left integral T on H one has
∆(T ) = (T 2 S 2 )α −1 ⊗ T 1(7)
and then for any h ∈ H
T ↼ h = T 1 (h)T 2 = ((T 2 S 2 )α −1 )(h)T 1 = T 2 (S 2 (h 1 ))α −1 (h 2 )T 1 = α −1 (h 2 )(S 2 (h 1 ) ⇀ T )
Thus for any left integral T on H and any h ∈ H
T ↼ h = α −1 (h 2 )(S 2 (h 1 ) ⇀ T )(8)
Now if t is a left integral in H, then S(t) is a right integral in H and
∆(S(t)) = S(t 2 ) ⊗ S(t 1 ) = S(t 1 ) ⊗ S(S 2 (t 2 )g −1 ) ( by (6)) = S(t 1 ) ⊗ gS 2 (S(t 2 )) = S(t) 2 ⊗ gS 2 (S(t) 1 )
We conclude that for any right integral t in H
∆(t) = t 2 ⊗ gS 2 (t 1 )(9)
We also see that for a right integral t in H and h ∈ H
t 1 ⊗ ht 2 = ε(h 1 )t 1 ⊗ h 2 t 2 = S(h 1 )h 2 t 1 ⊗ h 3 t 2 = S(h 1 )(h 2 t) 1 ⊗ (h 2 t) 2 = α −1 (h 2 )S(h 1 )t 1 ⊗ t 2
Thus we showed that
t 1 ⊗ ht 2 = α −1 (h 2 )S(h 1 )t 1 ⊗ t 2(10)
Theorem 4.2 Let H be a finite dimensional Hopf algebra, and let g and α be the distinguished grouplike elements of H and H * . We assume that S 2 (h) = g −1 hg = α −1 (h 1 )α(h 3 )h 2 for any h ∈ H. Then a right H-comodule algebra A is (H, α)-symmetric if and only if A#H * is (H * , g)-symmetric.
Proof We note that
S −2 (h) = α(h 1 )α −1 (h 3 )h 2(11)
for any h ∈ H. Assume that A is (H, α)-symmetric, and let λ :
A → k such that λ(ba) = λ(a(α −1 · b)) = λ(aα −1 (b 1 )b 0 ), λ(h * · a) = h * (1)λ(a)
for any a, b ∈ A, h * ∈ H * , and also Ker λ does not contain a non-zero right ideal of A. Let t be a non-zero right integral in H and define λ : A#H * → k, λ(a#h * ) = λ(a)h * (t) as in the proof of Theorem 3.1. This makes A#H * a Frobenius algebra in the category M H * . In order to see that A#H * is symmetric in M H * , it remains to show that λ(zz ′ ) = λ(z ′ (g −1 ⇀ z)) for any z, z ′ ∈ A#H * , where g −1 ⇀ (a#h * ) = a#(g −1 ⇀ h * ). Indeed, we see that
λ((b#g * )(g −1 ⇀ (a#h * )) = λ((b#g * )(a#(g −1 ⇀ h * ))) = λ(b(g * 1 · a)#g * 2 (g −1 ⇀ h * )) = λ(bg * 1 (a 1 )a 0 )g * 2 (t 1 )h * (t 2 g −1 ) = λ(ba 0 )g * (a 1 t 1 )h * (t 2 g −1 ) = λ(a 0 α −1 (b 1 )b 0 )g * (a 1 t 1 )h * (t 2 g −1 ) = λ(a 0 b 0 )g * (a 1 b 1 S(b 2 )t 1 )α −1 (b 3 )h * (t 2 g −1 ) = λ(((S(b 1 )t 1 ) ⇀ g * ) · (ab 0 ))α −1 (b 2 )h * (t 2 g −1 ) = ((S(b 1 )t 1 ) ⇀ g * )(1)λ(ab 0 )α −1 (b 2 )h * (t 2 g −1 ) = g * (S(b 1 )t 1 )λ(ab 0 )α −1 (b 2 )h * (t 2 g −1 ) = λ(ab 0 )g * (S(b 1 )t 2 )α −1 (b 2 )h * (gS 2 (t 1 )g −1 ) by (9) = λ(ab 0 )g * (S(b 1 )t 2 )α −1 (b 2 )h * (t 1 ) = λ(ab 0 )g * (t 2 )α −1 (b 3 )h * (α −1 (S(b 1 ))S(S(b 2 ))t 1 ) by (10) = λ(ab 0 )g * (t 2 )α −1 (b 3 )h * (α(b 1 )S 2 (b 2 )t 1 ) = λ(ab 0 )g * (t 2 )h * (S 2 (α(b 1 )α −1 (b 3 )b 2 )t 1 ) = λ(ab 0 )g * (t 2 )h * (S 2 (S −2 (b 1 ))t 1 ) by (11) = λ(ab 0 )g * (t 2 )h * (b 1 t 1 ) = λ(ab 0 )g * (t 2 )h * 1 (b 1 )h * 2 (t 1 ) = λ(a(h * 1 · b))(h * 2 g * )(t) = λ(a(h * 1 · b)#h * 2 g * ) = λ((a#h * )(b#g * ))
Conversely, assume that A#H * is (H * , g)-symmetric, and let µ : A#H * → k be a linear map whose kernel does not contain non-zero right ideals of A#H * , and such that µ(h ⇀ z) = ε(h)µ(z) and µ(zz ′ ) = µ(z ′ (g −1 ⇀ z)) for any h ∈ H and any z, z ′ ∈ A#H * . Let T be a left integral on H and defineμ : A → k,μ(a) = µ(a#T ).
Let a ∈ A and h * ∈ H * . We note that g ⇀ T is a right integral on H, see [8,Proposition 5.5.4]. Let z = a#(g ⇀ T ) and z ′ = 1#h * . Then
zz ′ − z ′ (g −1 ⇀ z) = (a#(g ⇀ T ))(1#h * ) − (1#h * )(a#T ) = a#(g ⇀ T )h * − (h * 1 · a)#h * 2 T = a#h * (1)(g ⇀ T ) − (h * 1 · a)#h * 2 (1)T = h * (1)a#(g ⇀ T ) − (h * · a)#T Since µ(zz ′ ) = µ(z ′ (g −1 ⇀ z)), we get µ(h * · a) = µ((h * · a)#T ) = h * (1)µ(a#(g ⇀ T )) = h * (1)ε(g)µ(a#T ) = h * (1)μ(a)
If I is a subobject of A in A M H contained in Kerμ, then µ(I#T ) = 0. But I#T is a left ideal of A#H * , so it must be zero. Then I must be zero, too.
To show that A is symmetric in M H it only remains to check thatμ(ba) =μ(a(α −1 · b)) for any a, b ∈ A. This holds true sincẽ µ(ba) = µ(ba#T ) = µ((b#ε)(a#T )) = µ((a#T )(g −1 ⇀ (b#ε))) = µ((a#T )(b#(g −1 ⇀ ε))) Proof In order to use the notation we have already developed, it is more convenient to show that if H * is cosovereign by g, then H * is (H * , g)-symmetric. If t is a right integral in H, by the proof of Theorem 3.1 (when we take A = k and identify A#H * with H * ) we have that the linear map λ : H * → k, λ(h * ) = h * (t) is H-linear, and its kernel does not contain nonzero subobjects of H * in M H * H * . On the other hand, for any h * , g * ∈ H * λ(g * (g −1 ⇀ h * )) = g * (t 1 )h * (t 2 g −1 )
= µ((a#T )(b#ε)) = µ(a(T 1 · b)#T 2 ) = µ(aT 1 (b 1 )b 0 #T 2 ) = µ(ab 0 #(T ↼ b 1 )) = µ(ab 0 #α −1 (b 2 )(S 2 (b 1 ) ⇀ T )) by (8) = ε(S 2 (b 1 ))α −1 (b 2 )µ(ab 0 #T ) = ε(b 1 )α −1 (b 2 )µ(ab 0 #T ) = α −1 (b 1 )µ(ab 0 #T ) = µ(a(α −1 · b)#T ) =μ(a(α −1 · b)) ⊓ ⊔Remark
= g * (t 2 )h * (gS 2 (t 1 )g −1 ) by (9) = h * (t 1 )g * (t 2 ) = (h * g * )(t) = λ(h * g * ) so λ makes H * an (H * , g)-symmetric algebra. ⊓ ⊔
Passing to coinvariants
If A is a right H-comodule algebra which is Frobenius (respectively symmetric) as an algebra, it is a natural question to ask whether this property transfers to the subalgebra of coinvariants A coH . It is easy to see that such a transfer does not hold. Indeed, let A be the algebra from Example 4.1, which is symmetric. A is a kC 2 -comodule algebra, and its subalgebra of coinvariants is just A e , which is not even Frobenius. The following shows that a good transfer occurs if A is Frobenius in the category M H , provided H is cosemisimple.
Proposition 5.1 Let H be a cosemisimple Hopf algebra. If A is a right H-comodule algebra which is Frobenius in the category M H , then A coH is a Frobenius algebra. If moreover, H is involutory and A is (H, ε)-symmetric, then A coH is symmetric.
Proof Let i : A coH → A be the inclusion map, and let i * : A * → (A coH ) * be its dual. Since i is a morphism of A coH , A coH -bimodules, then so is i * . If A is Frobenius in M H , let θ : A → A * be an isomorphism in the category M H A . We show that i * θi : A coH → (A coH ) * is an isomorphism of right A coH -modules, i.e. A coH is Frobenius. In fact it is enough to show that i * θi is injective; since Im θi = (A * ) coH , this is the same with showing that i * |(A * ) coH is injective. The left H * -action on A * induced by the right H-coaction is (h * · a * )(a) = h * S(a 1 )a * (a 0 ), for any A ∈ A. Then a * ∈ (A * ) coH if and only if h * · a * = h * (1)a * for any h * ∈ H * , and this means that a * ( h * S(a 1 )a 0 − h * (1)a) = 0 for any a ∈ A and any h * ∈ H * . Since S is bijective (H is cosemisimple), we get that a * ∈ (A * ) coH if and only if a * vanishes on the subspace V =< h * (a 1 )a 0 − h * (1)a | h * ∈ H * , a ∈ A >
Since
Ker i * |(A * ) coH = {a * ∈ (A * ) coH | a * (A coH ) = 0 } we see that i * |(A * ) coH is injective if and only if V + A coH = A. But this is indeed true, since for a left integral T on H such that T (1) = 1, one has a = T ·a−(T ·a−T (1)a). Moreover, T ·a ∈ A coH , since h * · (T · a) = (h * T ) · a = h * (1)T a for any h * ∈ H * , and obviously T · a − T (1)a ∈ V .
For the second part we just have to note that i * θi is a morphism of A coH , A coH -bimodules since θ is an isomorphism of A, A-bimodules; now the proof of the first part works also in this case.
⊓ ⊔
, (4) are given by θ(a)(b) = B(a, b), λ(a) = B(1, a), B(a, b) = λ(ab).
( 4 )
4There exists a linear map λ : A → k such that λ(ba) = λ(af (b)) and λ(h * · a) = h * (1)λ(a) for any a, b ∈ A, h * ∈ H * , and also Ker λ does not contain a non-zero subobject ofA in M H A . More equivalent conditions can be added if we change right ideal with left ideal in (3), and M H A with A M H in (4). Proof We combine the proof of the equivalent characterizations of a symmetric algebra in the category of vector spaces, see [14, Theorem 16.54], and [6, Theorem 2.4], recalled above. Thus for (1) ⇔ (2), if θ : A → F (A * ) is a linear map, then let B : A × A → k be the bilinear map defined by B(a, b) = θ(b)(a). Then it is straightforward to check that θ is left A-linear if and only if B(b, ca) = B(bf (c), a) for any a, b, c ∈ A,
hold, then B(b, a) = B(b, a1) = B(bf (a), 1) = B(f (a), b), thus B(b, a) = B(f (a), b) for any a, b ∈ A,
hold, then B(b, ac) = B(f (ac), b) = B(f (a)f (c), b) = B(f (a), cb) = B(cb, a), so (3) holds. We have that θ is H-colinear if and only if B(b, h * · a) = B((h * S) · b, a) for any a, b ∈ A, h * ∈ H * , and θ is bijective if and only if B is non-degenerate, thus (1) ⇔ (2) is clear. For (1) ⇔ (3), λ and B determine each other by the relations λ(a) = B(1, a) for any a ∈ A, respectively B(a, b) = λ(ba) for any a, b ∈ A. ⊓ ⊔ Example 2.3 1) If H = kG, the group Hopf algebra of a group G, then S 2 = Id, so H is cosovereign with ε as a sovereign character. A right H-comodule algebra is just a G-graded algebra A, and A is (H, ε)-symmetric if and only if A is graded symmetric in the sense of [6, Section 5].
Example 2. 7
7Assume that k has characteristic = 2, and let H = H 4 be Sweedler's Hopf algebra. Let A = k[X]/(X 2 ), a 2-dimensional algebra, with basis {1, X}, and relation X 2 = 0. Let c and x be the endomorphisms of the space A such that c(1) = 1, c(X) = −X, x(1) = 0, x(X) = 1. Then c 2 = Id, x 2 = 0 and xc = −cx. Moreover, c is an algebra automorphism of A, and it is easy to check that x(ab) = c(a)x(b) + x(a)b for any a, b ∈ A, thus A is a left H-module algebra with the actions of c and x given by the endomorphisms above, i.e.
consider the usual left and right actions of H * on H, h * ⇀ h = h * (h 2 )h 1 and h ↼ h * = h * (h 1 )h 2 , and the usual left and right actions of H on H * , denoted by h ⇀ h * and h * ↼ h, where h ∈ H and h * ∈ H * . H also acts on A#H * by h ⇀ (a#h * ) = a#(h ⇀ h * ).
(
1#p e )(A#(kC 2 ) * )(1#p e ) = A e #p e ≃ A e = R Then (1#p e )(A#(kC 2 ) * )(1#p e )is not a symmetric algebra, so neither is A#(kC 2 ) * by[14, Exercise 16.25].
Definition 2.1 Let H be a cosovereign Hopf algebra with u as a sovereign character. A finite dimensional right H-comodule algebra A is a symmetric algebra in the category M H with respect to u if F (A * ) ≃ A in the category A M H A . In this case we simply say that A is (H, u)-symmetric.
4.3 (1) The conditions on H in Theorem 4.2 are satisfied if H is involutory and unimodular, and H * is unimodular. Indeed, in this case the distinguished grouplike elements are trivial, i.e. α = ε and g = 1. For example, this happens if H = kG, where G is a finite group. Thus a finite dimensional G-graded algebra is graded symmetric if and only if the smash product A#(kG) * is symmetric in M (kG) * with respect to 1. (2) In the case where the characteristic of k is 0, it is known that H is involutory if and only if H is semisimple, if and only if H is cosemisimple, see [18, Theorem 16.1.2], and in this situation H and H * are always unimodular. Thus Theorem 4.2 applies to any semisimple Hopf algebra in characteristic 0. (3) If k has positive characteristic, Theorem 4.2 applies to any semisimple cosemisimple Hopf algebra H. Indeed, it is known that any such H is involutory, see [9, Theorem 3.1].(4) A Hopf algebra satisfying the conditions of Theorem 4.2 is not necessarily involutory, and it may be not unimodular; take for example Sweedler's 4-dimensional Hopf algebra.As a consequence of Theorem 4.2 we obtain that if a Hopf algebra H is cosovereign by α and H * is cosovereign by g, then H is (H, α)-symmetric. In fact, we can prove that H is (H, α)-symmetric with less assumptions.Proposition 4.4 Let H be a finite dimensional Hopf algebra which is cosovereign with sovereign element α, the distinguished grouplike element of H * . Then H is (H, α)-symmetric.
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| [] |
[
"Robust Monitoring of Time Series with Application to Fraud Detection",
"Robust Monitoring of Time Series with Application to Fraud Detection"
] | [
"Peter Rousseeuw ",
"Domenico Perrotta ",
"Marco Riani ",
"Mia Hubert ",
"Peter Rousseeuw ",
"Domenico Perrotta ",
"Marco Riani ",
"Mia Hubert ",
"\nDepartment of Mathematics\nJoint Research Centre\nDepartment of Economics\nUniversity of Leuven\nIspraBelgium., Italy\n",
"\nDepartment of Mathematics\nUniversity of Parma\nItaly\n",
"\nUniversity of Leuven\nBelgium. Peter Rousseeuw and Mia\n"
] | [
"Department of Mathematics\nJoint Research Centre\nDepartment of Economics\nUniversity of Leuven\nIspraBelgium., Italy",
"Department of Mathematics\nUniversity of Parma\nItaly",
"University of Leuven\nBelgium. Peter Rousseeuw and Mia"
] | [] | Time series often contain outliers and level shifts or structural changes. These unexpected events are of the utmost importance in fraud detection, as they may pinpoint suspicious transactions. The presence of such unusual events can easily mislead conventional time series analysis and yield erroneous conclusions. A unified framework is provided for detecting outliers and level shifts in short time series that may have a seasonal pattern. The approach combines ideas from the FastLTS algorithm for robust regression with alternating least squares. The double wedge plot is proposed, a graphical display which indicates outliers and potential level shifts. The methodology was developed to detect potential fraud cases in time series of imports into the European Union, and is illustrated on two such series. | 10.1016/j.ecosta.2018.05.001 | [
"https://arxiv.org/pdf/1708.08268v4.pdf"
] | 88,516,031 | 1708.08268 | fec839c5aabf86477c2892ce801b28e3d2b3598a |
Robust Monitoring of Time Series with Application to Fraud Detection
20 May 2018 May 22, 2018
Peter Rousseeuw
Domenico Perrotta
Marco Riani
Mia Hubert
Peter Rousseeuw
Domenico Perrotta
Marco Riani
Mia Hubert
Department of Mathematics
Joint Research Centre
Department of Economics
University of Leuven
IspraBelgium., Italy
Department of Mathematics
University of Parma
Italy
University of Leuven
Belgium. Peter Rousseeuw and Mia
Robust Monitoring of Time Series with Application to Fraud Detection
20 May 2018 May 22, 2018arXiv:1708.08268v4 [stat.CO]alternating least squaresdouble wedge plotlevel shiftoutliers
Time series often contain outliers and level shifts or structural changes. These unexpected events are of the utmost importance in fraud detection, as they may pinpoint suspicious transactions. The presence of such unusual events can easily mislead conventional time series analysis and yield erroneous conclusions. A unified framework is provided for detecting outliers and level shifts in short time series that may have a seasonal pattern. The approach combines ideas from the FastLTS algorithm for robust regression with alternating least squares. The double wedge plot is proposed, a graphical display which indicates outliers and potential level shifts. The methodology was developed to detect potential fraud cases in time series of imports into the European Union, and is illustrated on two such series.
Introduction
When analyzing time series one often encounters unusual events such as outliers and structural changes, like those in Figure 1. Both series track trade volumes, and were extracted from the official trade statistics in the COMEXT database of Eurostat. This database contains monthly trade volumes (aggregated over several transactions, possibly involving different traders) of products imported in the European Union (EU) in a four-year period.
The plot titles in Figure 1 specify the code of the traded product in the EU Combined Nomenclature classification (CN code), the country of origin, and the destination (a member state of the EU). The CN code determines whether the volumes are expressed in tons of net mass and/or other units (liters, number of items, etc.), the rate of customs duty applied, and how the goods are treated for statistical purposes. The data quality is quite heterogeneous across countries and products, but some macroscopic outliers (manifest errors) have already been removed or corrected by statistical authorities and customs services. (b) import of sugars including chemically pure lactose, maltose, glucose and fructose, sugar syrups, artificial honey and caramel, from the Ukraine into Lithuania (P17049075-UA-LT).
Both of these time series exhibit a downward level shift. Knowing when such structural breaks occur is important for fraud detection. For instance, a sudden reduction in trade volume may coincide with an increase for a related product or another country of origin, which could indicate a misdeclaration with the intent of deflecting customs duties.
There are many products and countries of origin in the CN classification, but not all of these combinations occur and the number of products at risk of fraud is relatively small. Still, the number of relevant combinations of a product at fraud risk, a country of origin and a country of destination is around 16,000. As a result, every month around 16,000 time series need to be analyzed for anti-fraud purposes. This requires an automatic approach that is able to report accurate information on outliers and the positions and amplitudes of level shifts, and that runs fast enough for that time frame. The method proposed in this paper meets those objectives, and provides a graphical display that can be looked at whenever the automatic monitoring system detects a significant level shift.
Our method follows the approach which first computes a robust fit to the majority of the data and then detects outliers by their large residuals, as described in the review paper (Rousseeuw and Hubert, 2018).
A different statistical approach to monitor international trade data for fraud was proposed by Barabesi et al. (2016) who tested whether the distribution of trade volumes follows the Newcomb-Benford law. In the current paper we also take the time sequence of the trades into account. We will focus on a parametric approach to estimate level shifts, which differs from the nonparametric smoothing methods in Fried and Gather (2007) or robust methods for REGARIMA models (Bianco et al., 2001). A popular technique is the X13 ARIMA-SEATS Seasonal Adjustment methodology (Findley et al., 1998;U.S. Census Bureau, 2017).
X-13 is based on automatic fitting of ARIMA models and includes detection of additive outliers and level shifts. We will compare our results with those of X-13 in Section 5. See also Galeano and Peña (2013) for a review of robust modeling of linear and nonlinear time series.
Although this paper was motivated by the need to analyze many short time series of trade data, we will describe the methodology more generally so it can be applied to other types of time series that may be longer and can be modeled with more parameters.
The structure of the paper is as follows. In Section 2 we introduce our model and methodology for robustly analyzing a time series which contains a trend, a seasonal component and possibly a level shift in an unknown position, as well as isolated or consecutive outliers. In Section 3 we illustrate the proposed approach using the well-known airline data (Box and Jenkins, 1976), as well as contaminated versions of it in order to test the ability of the method to detect anomalies. In this section we also introduce the double wedge plot, which visualizes the presence of a level shift and outliers. In Section 4 we apply our methodology to the time series in Figure 1. Section 5 compares our results to those obtained by a nonparametric method and to X-13. The case where more than one level shift occurs is discussed in Section 6. Section 7 concludes, and the Appendix proves a result about our algorithm.
Methodology
The model
The time series y(t) = y t (for t = 1, . . . , T ) we will consider may contain the following terms:
1. a polynomial trend, i.e. A a=0 α a t a ;
2. a seasonal component, i.e.
S t = B b=1 β b,1 cos 2πb 12 t + β b,2 sin 2πb 12 t .(1)
When B = 1 this is periodic with a one-year period, B = 2 corresponds with a sixmonth period etc. We assume the amplitude of the seasonal component varies over time in a polynomial way, i.e. y t ∼ 1 + G g=1 γ g t g S t ;
3. a level shift in an unknown time point 2 δ 2 T , i.e. δ 1 I(t δ 2 ) with I(.) the indicator function.
The general model is thus of the form
y t = A a=0 α a t a + B b=1 β b,1 cos 2πb 12 t + β b,2 sin 2πb 12 t 1 + G g=1 γ g t g + δ 1 I(t δ 2 ) + ε t .(2)
One may assume that the irregular component ε t of the non-outliers is a stationary random process with E[ε t ] = 0 and σ 2 = V ar[ε t ] < ∞. Let us collect all unknown parameters in a vector θ = (α 0 , α 1 , . . . , β 1,1 , β 1,2 , . . . , γ 1 , γ 2 , . . . , δ 1 , δ 2 ) of length p. Then model (2) can be written as
y t = f (θ, t) + ε t with f (θ, t) = A a=0 α a t a + S t (1 + G g=1 γ g t g ) + δ 1 I(t δ 2 ) .
The model does not need to contain all of these components, as some coefficients can be zero.
The nonlinear LTS estimator
Model (2) is nonlinear in the parameters β b,1 , β b,2 , γ g and δ 2 . As there may be outliers in the time series, we propose to estimate θ by means of the nonlinear least trimmed squares (NLTS) estimator (Rousseeuw, 1984;Stromberg and Ruppert, 1992;Stromberg, 1993):
θ NLTS = argmin θ h j=1 r 2 (j) (θ)(3)
where T /2 h < T and r 2 (j) (θ) is the j-th smallest squared residual (y t
− f (θ, t)) 2 . Our default choice for h is [0.75 T ].
The √ n-consistency and asymptotic normality of NLTS were studied byČížek (2005,2008). To compute the estimator, we propose to combine ideas from the FastLTS algorithm for robust linear regression (Rousseeuw and Van Driessen, 2006) with the alternating least squares (ALS) method.
We first describe how we use the alternating least squares procedure. We temporarily assume that the estimated shift timeδ 2 is fixed, and that we want to solve (3) for a subset of the y t with at least p − 1 observations, at least one of which is to the left ofδ 2 and at least one of which is equal or to the right ofδ 2 . We denote the indices of the subset as
E ⊂ {1, 2, . . . , T } with #E p − 1 , where E must overlap with {1, .
. . ,δ 2 − 1} as well as {δ 2 , . . . , T }. These conditions are required to make the parameters in (2) identifiable from the subset y E = {y t ; t ∈ E}. We then go through the following steps:
1.
[Initialization] Set γ g = 0 for g = 1, . . . , G. Then a part of (2) drops out, leaving
y t = A a=0 α a t a + B b=1 β b,1 cos 2πb 12 t + β b,2 sin 2πb 12 t + δ 1 I(t δ 2 ) + ε t (4)
which is linear in the parameters α a , β b,1 , β b,2 and δ 1 . By applying linear LS to the subset y E , we obtain the initial estimatesα
(0) a ,β (0) b,1 ,β (0) b,2 andδ(0)
1 .
[Iteration]
For k = 1, 2, ... repeat the following steps:
• [ALS step A] Let S (k−1) t = B b=1 β (k−1) b,1 cos 2πb 12 t +β (k−1) b,2 sin 2πb 12 t in which the coefficientsβ (k−1) b,1 andβ (k−1) b,2
come from the previous step. Keeping S (k−1) t fixed yields the model
y t − S (k−1) t = A a=0 α a t a + S (k−1) t G g=1 γ g t g + δ 1 I(t δ 2 ) + ε t(5)
which is linear in the parameters α a , γ g , and δ 1 . We then apply LS using only the observations in the subset y E , yielding the estimatesα
(k) a ,γ (k) g andδ (k) 1 . • [ALS step B] Keeping the estimated coefficientsα (k) a ,γ (k) g andδ (k) 1
from the previous step fixed yields the model
y t − A a=0α (k) a t a −δ (k) 1 I(t δ 2 ) = B b=1 β b,1 cos 2πb 12 t + β b,2 sin 2πb 12 t 1 + G g=1γ (k) g t g + ε t (6)
which is linear in the parameters β b,1 and β b,2 . We then apply LS using only the observations in the subset y E , yielding the estimatesβ
(k) b,1 andβ (k)
b,2 . Then we go back to ALS step A.
Letθ k be the vector of coefficients after iteration step k. We repeat the above steps until ||θ k −θ k−1 ||/||θ k−1 || is below a threshold, or a maximal number of iterations (say 50) is attained. Here || · || is the Euclidean norm.
In words, ALS solves the nonlinear LS problem of fitting (2) to the data set y E by alternating between the solution of two linear LS fits, (5) and (6).
Our goal is to solve the nonlinear LTS problem (3). A basic tool for linear LTS is the C-step (Rousseeuw and Van Driessen, 2006) which we now generalize to the nonlinear setting.
[C-step] Start from a subset H (k) ⊂ {1, 2, . . . , T } to which we fitθ (k) obtained by applying ALS. Then compute the residuals r t = y t − f (θ (k) , t) for the whole time series, that is, for t = 1, . . . , T and not just for t ∈ H (k) . Next retain the h observations with smallest squared residuals, yielding the new subset H (k+1) . Then apply ALS to H (k+1) , yielding a new fitθ (k+1) . It is shown in the Appendix that the new fitθ (k+1) is guaranteed to have a lower objective function than the old fitθ (k) . It is possible to iterate the C-step until convergence, which will occur in a finite number of steps.
Using these building blocks, we now describe the entire algorithm to compute the NLTS fit to the model (2). Let t (1) , . . . , t (S) be the ordered indices of the possible positions δ 2 of the level shift, for example the set {u + 1, . . . , T − u} for some u > 0. The algorithm then consists of the following steps. This subset should contain the index t (s) , one observation y t with t < t (s) and p − 3 observations drawn at random from the whole time series. Note that we impose that t (s) belongs to E because the purpose of step 1 is to select the most suitableδ 2 = t (s) .
ii. Run the initialization and ALS steps described above on E, keepingδ 2 = t (s) fixed. Then take two C-steps. [Two C-steps is enough at this stage, in line with the results of Rousseeuw and Van Driessen (2006).] If a singular solution is obtained during the computations, restart without increasing m.
The choice of M is a compromise since the expected number of outlier-free subsets E is proportional to M but the computation speed is inversely proportional to M. In our experiments we found that M = 250 was sufficient to obtain stable results.
(c) Consider only the nbest elemental subsets (among the M that were tried) that yielded the lowest objective function so far. Apply C-steps to them until convergence and store these nbest solutions. In our examples we found that setting nbest to 10 worked well.
(d) If s > 1 also start from the nbest elemental sets found when investigating t (s−1) , but this time settingδ 2 = t (s) . Apply C-steps to them until convergence.
(e) Take the fit with the lowest objective among these 2 × nbest candidates, and denote it byθ (s) .
(f) Store the corresponding scaled residuals
r t (θ (s) ) = r t (θ (s) ) h t=1 r 2 (t) (θ (s) )/h for t = 1, . . . , T .(7)
2. Retain overall best solution. Among the fitsθ (s) for s = 1, . . . , S take the one with lowest objective function h t=1 r 2 (t) (θ (s) ) and denote it byθ opt . For estimating the scale of the error term we can use h t=1 r 2 (t) (θ opt ) . But since this sum of squares only uses the h most central residuals, the estimate needs to be rescaled. The variance σ 2 (h) of a truncated normal distribution containing the central h/T portion of the standard normal is
σ 2 (h) = 1 − 2T h Φ −1 T + h 2T φ Φ −1 T + h 2T
by equation (6.5) in (Croux and Rousseeuw, 1992). Therefore we computẽ
σ 2 = h t=1 r 2 (t) (θ opt )/(hσ 2 (h)) .(8)
Note that this makesσ 2 consistent, but not yet unbiased for small samples. Therefore we include the finite-sample correction factor from Pison et al. (2002) in our final scale estimateσ.
3. Locally improving the shift position estimate. The previous steps have yielded an estimateδ 2 of the position of the level shift, but it may be imprecise. For instance, it may happen that the h-subset underlyingθ opt does not itself contain the time pointŝ δ 2 orδ 2 + 1 . In order to improve the estimate we check in its vicinity as follows:
• Take a window W aroundδ 2 . For each t * in W , we replaceδ 2 by t * while keeping the other coefficients fromθ opt and the scale estimateσ. Compute the residuals r t from these coefficients and let f (t * ) = t∈W ρ(r t /σ) with ρ the Huber function
ρ(x) = x 2 /2 if |x| b b|x| − b 2 /2 if |x| > b
In our simulations and the analysis of international trade time series (of the kind given in Figure 1) the best results were obtained with b equal to 1.5 or 2. In our implementation the defaults are b = 2 and a window W of width 15.
• Our finalδ 2 is the t * in W with lowest f (t * ). If it is different from the estimate we had before, we recompute the scaled residuals.
4. Weighted step. We apply the univariate outlier detection procedure described in (Gervini and Yohai, 2002) and (Agostinelli et al., 2015) to the T scaled residuals (y t − f (θ opt , t))/σ . By default we use the 99% confidence level. Alternatively, one could use the thresholds obtained in Salini et al. (2015).
5. Final fit. We apply nonlinear LS to all the points that have not been flagged as outliers in the previous step, starting from the initial estimateθ opt and keepingδ 2 fixed. For this we can iterate ALS steps until convergence. The standard errors obtained in the last two ALS steps can be used for inference.
Note that h must be at least the number of parameters p in the model for identifiability.
When h is as low as T /2 this means T /p > 2. However, for stability it is often recommended that T /p > 5, see e.g. Rousseeuw and Leroy (1987). The Matlab code of the algorithm can be downloaded from /www.riani.it/rprh/ . In the following sections we will apply it to several data sets.
Airline data and the double wedge plot
The airline passenger data, given as Series G in Box and Jenkins (1976), has often been used in the time series analysis literature as an example of a nonstationary seasonal time series. It consists of T = 144 monthly total numbers of airline passengers from January 1949 to December 1960. Box and Jenkins developed a two-coefficient time series model of factored form that is now known as the airline model. In this section we will analyze these data using our method, and then contaminate the data in various ways to see how the method reacts. Uncontaminated data. We fit the data by model (2) the coefficients in the weighted step. Only a few regular observations received an absolute residual slightly above the cutoff value.
The top panel of Figure 3 is a byproduct of the algorithm, and is useful for visualizing the presence of (groups of) outliers and a level shift. The first step of the algorithm ranges over all potential positions t (1) , . . . , t (S) of a level shift. These tentative positions t (s) are on the vertical axis. For any t (s) we plot the absolute scaled residuals |r t (θ (s) )| given in (7), in all of the times t = 1, 2, . . . , T on the horizontal axis. The color in the plot depends on the size of that absolute residual and ranges from black (large residuals) over red and yellow to white (small residuals). The color scale is at the right of the plot. Scaled residuals larger than 50 are shown as if they were 50, so that even a very far outlier cannot affect the color coding. In the same spirit, uninformative scaled residuals smaller than 2.5 are shown as if they were 0, so in white. Of course the user can easily modify these default choices.
Outliers have a large absolute scaled residual from the robust fit, so in this plot isolated outliers will appear as dark vertical lines, and groups of consecutive outliers as dark vertical bands. In this example we clearly see the contamination. The regular observations with scaled residual slightly above 2.5 do not stand out as they are in light yellow.
Contamination 2. In the second contamination setting we introduce a persistent level shift and three isolated outliers, two of which lie in the proximity of the level shift which makes the problem harder. For this we added the value 1300 to all responses from t = 68 onward, at t = 45 the response is lowered by 800, at t = 67 by 600, while at t = 68 and t = 69 we added an additional 800. The bottom panel of Figure 4 shows the observed and fitted values. Again all inserted outliers are clearly detected, and a few regular observations have small crosses indicating that their scaled absolute residual was slightly above 2.5 .
The plot of the absolute scaled residuals |r t (θ (s) )| in the top panel of Figure 4 now looks more eventful with two dark triangles. Together these 'wedges' signal a level shift.
To understand this effect, let us assume that the true level shift is at position t * and the algorithm is in the process of checking the candidate t (s) = t * − r. Then the algorithm will treat the y t at t * − r + 1, . . . , t * − 1 as outliers and the resulting robust fit (still for that t (s) ) will show r − 1 consecutive outliers. Similarly, when the algorithm tries t (s) = t * + r to the right of t * , the best solutions will show r outliers. As a result, when approaching the true level shift position t * from the left the scaled residuals we are monitoring will form a dark upward-pointing wedge, and to the right of the true t * we obtain an analogous wedge pointing downward. In the top panel of Figure 4. Panel (a) of Figure 5 shows the boxplots of the objective function h t=1 r 2 (t) (θ (s) j ) attained by the 2 × nbest = 20 best solutionsθ j in step 1(e) of the algorithm. It is thus also a free byproduct of the estimation. If a level shift is present in the central part of the time series, this plot will typically have a U shape. In this example the lowest values of the trimmed sum of squared residuals occur in the time range 60-80. The continuous curve which connects the lowest objective value for each s reaches its global minimum at t (s) = 73.
However, the curve is quite bumpy in that region, with several local minima and a near-constant stretch on 67-70, so the position of the minimum is not precise. This kind of situation motivated the local improvement in step 3 of the algorithm. Panel (b) of Figure 5 shows f (t (s) ) = t∈W ρ(r t /σ) as a function of the tentative position t (s) (with ρ the Huber function with b = 2) on the interval 53-82. This curve has a much better determined minimum, in fact at t = 68, confirming the benefit of the local improvement step. Contamination 3. In the final contaminated dataset we inserted a level shift and a group of consecutive outliers following it. To complicate things even more, we also put in a stretch of contamination to the left of the level shift, as well as an isolated outlier. The bottom panel of Figure 6 shows the robust fit, which succeeded in recovering the structure and flagging the outliers. In the top panel of Figure 6 we see the typical double wedge pattern indicating a level shift. The two reddish bands flag the groups of consecutive outliers, whereas the single line corresponds to the isolated outlier at t = 90. In this example the thick end of the upper wedge is yellow, so the absolute scaled residuals are not as large there. This part corresponds to a tentative level shift of around 100, which is very far from the true one, and in such cases the fit may indeed be quite different.
Analysis of trade data
Our main goal is to analyze the many short time series of trade described in Section 1.
After trying several model specifications in the class (2) we found that the best results were obtained by using a linear trend, two harmonics, and one parameter to model the varying amplitude of the seasonal component, that is, A = 1, B = 2 and G = 1. Note that this yields p = 9 parameters including the position and height of a potential level shift, which is not too many compared to the length of the time series (T = 48). As an example we now apply our method to the time series in Figure 1. In this case the level shift might point to a different type of violation. The market of sugar and high-sugar-content products, such as CN code 17049075, is very restricted and regulated. The EU applies country-specific quotas for these products, with lower import duty for imports below the quota and a higher duty beyond this limit (tariff rate quotas).
Therefore, it would be in an exporter's interest to circumvent the quota by mislabeling this product as a somewhat related product that is not under surveillance. In this situation one would check for upward level shifts in related products from the same country.
Note that the t-values and p-values provided by LTS can help select a model. Table 1 fits model (2) with A = 1 (linear trend), B = 2 (two harmonics), and G = 1 (the amplitude varies linearly). If we increase A we find that a quadratic trend is insignificant, and the same for increasing G. The t-values indicate that there is enough evidence for a 6-month seasonal effect but are less clear on the question whether B should be increased further for these short time series. In any case the detection of the level shift turns out to be stable as a function of B here.
Comparison with other methods
We now compare our results with those obtained by the nonparametric method introduced by Fried (2004) and Fried and Gather (2007) for robust filtering of time series. For this we used the function robust.filter from the R package robfilter of Fried et al. (2012).
The robust fitting methods are applied to a moving time window of size width, which needs to be an odd number. Tables 2 and 3 widths giving rise to the detection of a level shift (width = 7 for P 12119085 KE GB and width = 11 for P 17049075 UA LT). In the first series we see that the level shift is detected well for the appropriate width, but the fit itself is not as tight. Also in the second series a reasonable level shift position is found but the fit is not that close to the series. This can be explained by the fact that a nonparametric method has no prior knowledge about the data as it has to work on any data set, whereas our parametric model benefits from knowledge about the typical behavior of trade time series. In that sense the comparison is not entirely fair.
We also run the well-known X-13 ARIMA-SEATS method (Findley et al., 1998 Figure 11 does the same for P17049075 UA LT.
The blue curves are the time series, and the fits are in black. In both cases X-13 does detect the level shift. It obtains the model (0 1 1) which only has a moving average and no seasonal component. As a result its forecast (shown in red) has no seasonal component either. Note that in Figure 10 the 90% tolerance band around the forecast is much wider for X-13 than for LTS.
Let us now return to the airline data with contamination 1 described in Section 3. The results of LTS were shown in Figure 3. We now apply X-13 to it. The R-code and output are available in Section A.1 of the Supplementary Material. The model found by X-13 is ARIMA with (1 1 0)(0 1 0) whereas for the uncontaminated airline data it was (0 1 1)(0 1 1). In this example the X-13 fit has 7 nonzero coefficients describing level shifts, and 1 nonzero coefficient for an AO outlier. The bottom panel of Figure 12 shows the time series and the X-13 fit which accommodates the outliers. On the other hand, the LTS fit in the top panel follows the pattern of the majority of the data, so the outliers have large residuals from it. Also the forecasts are quite different: those of LTS increase and have a narrow tolerance band, while those of X-13 slightly decrease and have wide tolerance bands. For the uncontaminated airline data (that is, without outliers) the forecasts and tolerance bands of LTS and X-13 were very similar.
Note that X-13 fits each set of consecutive outliers by a level shift at the start and a level shift afterward. That description is indeed equivalent to the consecutive outliers representation. Our point is that accommodating the outliers gives a close fit to the observed time series, but as we see here it can inflate the forecast band.
On the airline data without outliers, X-13 automatically log transforms the data before fitting it. On the airline data with contamination it does not, because the time series contains at least one negative value. (Other transformations would be possible, but in automatic mode X-13 only considers the log transform.) In this example the negative values were due to outliers, but in fact many trade time series in the EU database have at least one zero value, correctly reflecting that a certain product was not imported for a month, which will also prevent X-13 from transforming the data.
To investigate this issue further, we looked at two ways to make the contaminated airline data positive. The first was to add a constant so that the minimum of the contaminated time series becomes 1 (we also tried 5, 10 and 50). The second was to truncate the series from below at 1 (or 5, 10, 50) so the downward outliers of contamination 1 remain visible.
However, in none of these cases did X-13 carry out a logarithmic transform, indicating that its transformation criterion was affected by the outliers.
Also note that the outliers have a large magnitude in this example. In response to a referee request we also provide an example with a level shift that is smaller than the
Conclusions and outlook
We have introduced a new robust approach to model and monitor nonlinear time series with possible level shifts. A fast algorithm was developed and applied to several real and artificial datasets. We also proposed a new graphical display, the double wedge plot, which visualizes the possible presence of a level shift as well as outliers. This graph requires no additional computation as it is an automatic by-product of the estimation. Our approach thus allows to automatically flag outlying measurements and to detect a level shift, which is important in fraud detection as these may be indications of unauthorized transactions.
At the European Joint Research Centre, this methodology was validated by comparing its results to those of visual inspection of many trade series by subject-matter experts.
Supplementary Material
The supplementary material to this paper contains some R code and worked-out examples.
A.2 Effect of a small level shift
For this example the time series has length T = 150 and it is generated according to model (2) with A = 1, B = 3, G = 1. In particular α 0 = 1, α 1 = 1, β 1,1 , β 1,2 , β 2,1 , β 2,2 , β 3,1 , β 3,2 ] = [20, −20, 12, −12, 4, −4] and γ 1 = 8.88 . There is one level shift of height δ 1 = 13, 000 at time δ 2 = 40. The error term is generated with a signal to noise ratio of 20. All the components in Figure 13 are shown using the same vertical scale so their relative size can be seen. The trend is increasing but appears horizontal if we compare it to the magnitude of the other components. We see that the level shift is smaller than the seasonal component, and similar in size to the spread of the error term (called "irregular" in the plot). The time series will be made available on /www.riani.it/rprh/ . After generating this time series we also create a stretch of outliers by subtracting 29,000
from the values at times in [131,140]. The top panel of Figure 14 shows the result of the LTS procedure, with the time series (blue), the estimated trend including the estimated level shift (purple), the overall fit (black) and the forecast (red). The level shift is clearly visible, and the outliers stand out by their sizeable residual (look at y t −ŷ t in this plot) for t in [131,140].
The bottom panel shows the X-13 fit, which does not detect the level shift. Instead there is a mild increase in the X-13 trend where the level shift takes place, followed by a mild decrease in the vicinity of the stretch of outliers. The X-13 forecast is stationary, whereas the LTS forecast has increasing seasonal fluctuations in line with the underlying model. The tolerance band around the forecast is much wider for X-13 than for LTS.
A.3 More than one level shift
We now consider an example with two level shifts. Starting from the original airline data, we subtract 100 from the values at times in [1,30] and add 200 at the times in [100,144] which creates level shifts at times 31 and 100. Applying LTS to these contaminated data correctly detects the level shift at time 100, as seen in Figure 15 with the objective function and its local refinement (similar to Figure 5). The resulting double wedge plot in the top panel of Figure 16 actually reveals both level shifts. Interestingly, the LTS fit in the lower panel of Figure 16 flags the first 30 points as a stretch of outliers.
In the next step we undo the level shift that was found, by subtractingδ 1 = 194.47 from y t in all t δ 2 = 100. To this modified time series we again apply LTS, which now correctly detects the level shift at time 31 as seen in Figure 17. The resulting double wedge plot in Figure 18 now shows only this level shift (since the other one has been removed).
The final fit no longer shows any outliers. If we undo also the second level shift and run LTS again, no more level shifts are found.
We also ran examples where the level shifts had roughly the same size and with more than two level shifts, with similar results.
Figure 1 :
1Monthly trade volumes of two products imported in the European Union in a four-year period: (a) imports of plants used primarily in perfumery, pharmacy or for insecticidal, fungicidal or similar purposes, from Kenya into the UK (P12119085-KE-GB);
1 .
1Loop over all t (s) where s = 1, . . . , S and do: (a) Temporarily setδ 2 = t (s) . (b) Now loop over m ranging from 1 to the number of trial subsets M, and do: i. Construct an elemental subset E containing p − 1 different observations.
Figure 2 :
2Airline data: observed and fitted values based on model (2) with a quadratic trend, a quarterly seasonal component, and a quadratically varying amplitude.
Figure 3 :
3with A = 2, B = 4 and G = 2.This means that we assume a quadratic trend, a quarterly seasonal component, and a quadratically varying amplitude. The resulting NLTS fit (3) closely follows the data, as can be seen inFigure 2. In this example no data point has been flagged as outlying. From the standard errors (not shown) we conclude that all coefficients are significant except for the height of the level shift.Contamination 1. We now contaminate the series by adding three groups of outliers, yielding the blue curve in the bottom panel ofFigure 3. More precisely, the value 300 is subtracted from all responses in the interval[50, 55] while 300 is added on[122, 127] and 400 is subtracted on[130, 134] . The fitted values (dotted curve) from NLTS closely follow the observed values for the regular observations. The flagged outliers are indicated by red crosses, whose size is proportional to the absolute magnitude of their residual. We see that all the outliers we added are clearly recognized as such, and they were not used to estimate Airline data with contamination 1: double wedge plot (top) and observed and fitted values (bottom).
Figure 4 :
4Airline data with contamination 2.
Figure 5 :
5we observe two opposite wedges tapering off in the proximity of the true level shift position, around t = 68. In this region we observe a small rectangle (centered at position 68) bridging the two wedges. The rectangle is due to the two outliers in the proximity of the level shift. The isolated outlier at position 45 yields a single dark vertical line like those in Figure 3. Airline data with contamination 2: (a) boxplots of the 20 lowest objective function values attained at each t (s) ; (b) local improvement of the shift position estimate.
Figure 6 :
6Airline data with contamination 3.
Figure 7 :
7P12119085 KE GB: double wedge plot (top) and observed and fitted values (bottom). The robust fit to series P12119085-KE-GB (bottom panel of Figure 7) suggests three moderate outliers in positions 1, 9 and 15. The fit closely matches the level shift which is therefore well captured. The double wedge plot in the top panel of Figure 7 has two wedges which point to a level shift position around 27-28. The local refinement step selects position t = 27.Columns 2-4 ofTable 1show the coefficients of the final fit together with their tstatistics and p-values. Most coefficients are significant, and in particular the t-statistic of the height of the level shift is quite large with |t| = 14.7 . This drop looks anomalous because in the period considered, Kenya was the only country of the East African Community (EAC) paying high European import duties on flowers and related products including CN 12119085. On the other hand, Kenya is the third largest exporter of cut flowers in the world. One would therefore check for a simultaneous upward level shift in an EAC country not paying import duties, which could point to a misdeclaration of origin.
Figure 8 Figure 8 :
88shows the results for the second series, P17049075 UA LT. The double wedge plot indicates the presence of a level shift around position 35. The local refinement yields the position t = 34. Interestingly, there is a reddish line right before the level shift. This is due to an outlier in position 32 which gets a red cross in the bottom panel of the figure.The double wedge plot also reveals a yellow strip at positions 29 and 30, indicating two less extreme outliers. Finally, the plot also shows some small reddish areas that correspond to local irregularities, for instance observations 4, 5, 17 and 18 which are flagged as outliers in the bottom panel. Columns 5-7 ofTable 1list the coefficients of the final fit. P17049075 UA LT: double wedge plot (top) and observed and fitted values (bottom).
report the position of the level shift(s) and outlier(s) detected with all default options and various choices of window widths. We also tested different robust choices for the trend and scale estimation and some values for the adapt option which adapts the moving window width, with similar results. Figures 9(a) and (b) show the resulting fits obtained by the nonparametric filter, for
Figure 9 :
9Fits obtained by a nonparametric time series filter for (a) P12119085 KE GB; (b) P17049075 UA LT .
Figure 10 :
10; U.S. Census Bureau, 2017) on both trade time series, by means of the R package seasonal (Sax, 2017) which interfaces X-13. This method fits an ARIMA model with a seasonal component. In additional to the coefficients required for the ARIMA model, X-13 has T additional parameters for level shifts, one at each time point t = 1, . . . , T , plus T parameters for additive outliers (AO). Their coefficients are estimated by stepwise regression, so most of them remain zero. For detecting isolated outliers, i.e. outliers surrounded by non-outlying values, this approach works quite well. Trade series P12119085 KE GB: time series (blue), fit (black) and forecast (red) obtained by LTS (top panel) and X-13 (bottom panel).
Figure 10 shows
10the X-13 fit to the trade series P12119085 KE GB in the lower panel, with the LTS fit in the upper panel for comparison.
Figure 11 :
11Trade series P17049075 UA LT: time series (blue), fit (black) and forecast (red) obtained by LTS (top panel) and X-13 (bottom panel).
Figure 12 :
12Airline data with contamination 1: time series (blue), fit (black) and forecast (red) obtained by LTS (top panel) and X-13 (bottom panel).
Figure 13 :
13Components of the generated data set of Section A.2.
Figure 14 :
14Data of Section A.2: time series (blue), trend (purple), overall fit (black), and forecast (red) obtained by LTS (top) and X-13 (bottom).
Figure 15 :
15First estimation of a level shift in the data of A.3: (a) boxplots of the 20 lowest objective function values attained at each t (s) ; (b) local improvement of the shift position estimate.
Figure 16 :Figure 17 :Figure 18 :
161718First fit to the data of Section A.3. Estimation of a level shift in the data of A.3, after undoing the first level Fit to the data of A.3 after undoing the first estimated level shift.
Table 1 :
1Coefficient estimates, t-statistics and p-values for series P12119085 KE GB(columns 2-4) and P17049075 UA LT (columns 5-7).
P12119085 KE GB
P17049075 UA LT
Coeff t-stat p-values Coeff t-stat p-valueŝ
α 0
115.27
25.6
0 55.14
14.3
0
α 1
1.59
5.80
0
0.90
4.52
0
β 11
-2.83 -0.72
0.47 15.55
3.75 0.00056
β 12
-12.42 -2.65
0.012
3.61
0.85
0.40
β 21
-9.07 -1.95
0.059 -32.50 -7.64
0
β 22
-22.60 -4.80
0 -16.06 -3.72 0.00061
Table 2 :
2P 12119085 KE GB: Positions of level shifts and outliers detected by a nonparametric time series filter, using different window widths.Window width Level shift position(s) Outlier position(s)
3
-
[ 10, 16, 17, 37, 38, 46, 47 ]
5
-
[ 16, 17, 18, 47 ]
7
[ 27 ]
[ 17, 18 ]
9
[ 27, 37 ]
-
11
[ 27 ]
-
Table 3 :
3P 17049075 UA LT: Positions of level shifts and outliers detected by a nonpara-
metric time series filter, using different window widths.
Window width Level shift position(s) Outlier position(s)
3
-
[ 2, 15, 44 ]
5
-
[ 15, 16, 17, 28, 30, 31, 33, 39, 40, 41 ]
7
-
[ 30, 31, 33, 34 ]
9
-
[ 30, 31, 33, 34, 35 ]
11
[ 30 ]
[ 32 ]
seasonal component, in Section A.2 of the Supplementary Material. The case of several level shifts Our basic model (2) only covers the situation where at most one level shift occurs, which is a reasonable assumption for short time series. When several level shifts can occur, we first apply our approach to the original time series. If it detects a level shift we can modify the time series by undoing the break, that is, subtractδ 1 from all y t to the right of the level shift, after which one can search for the next level shift, and so on. A detailed example of this procedure is shown in subsection A.3 of the Supplementary Material.6
Combining (9)-(11) yieldsso the new h-subset H (k+1) has an objective function that is less than or equal to that of H (k) . Note that the only way to obtain equality is if no coefficients have changed, in which case the iteration stops.AppendixHere we prove that a C-step (as used in the first step of the NLTS algorithm) can only decrease the LTS objective function.Let H (k) be the current h-subset with its corresponding nonlinear LS coefficientsθ (k) =). Now consider H (k+1) , the h-subset which contains the h observations with smallest squared residual with respect toθ(k). Then by constructionThe ALS step A then yieldsθ (k+0,δ 2 ). Since it is the LS solution of the linear model(5),r 2 (t) (θ (k+1) ) t∈H (k+1) r 2 (t) (θ (k+0.5) ) .
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> library("seasonal"). > library("seasonal")
> library. > library("forecast")
. > > Y = Airpassengers, 50:55] = y[50:55]-300> y = AirPassengers > y[50:55] = y[50:55]-300
. > , 122:127] = y[122:127]+300> y[122:127] = y[122:127]+300
. > , 130:134] = y[130:134]-400> y[130:134] = y[130:134]-400
> out = seas(y, forecast.save = "forecasts. > out = seas(y, forecast.save = "forecasts")
> summary(out) Coefficients: Estimate Std. Error z value Pr(>|z|). > summary(out) Coefficients: Estimate Std. Error z value Pr(>|z|)
Obs, 144 Transform: none > forec = series. out, c("forecast.forecasts", "s12"))Obs.: 144 Transform: none > forec = series(out, c("forecast.forecasts", "s12"))
. > Forec, 1:144,1] = y > plot(forec[,1],ylim=c(-100,800),col="blue> forec[1:144,1] = y > plot(forec[,1],ylim=c(-100,800),col="blue")
> fit = trend(out) + forecast::seasonal(out). > fit = trend(out) + forecast::seasonal(out)
| [] |
[
"A simplistic approach to the study of two-point correlation function in galaxy clusters",
"A simplistic approach to the study of two-point correlation function in galaxy clusters"
] | [
"Durakhshan Ashraf Qadri [email protected] \nDepartment of Physics\nNational Institute of Technology Srinagar\nJammu and Kashmir -190006India\n",
"Abdul W Khanday †[email protected] \nDepartment of Physics\nNational Institute of Technology Srinagar\nJammu and Kashmir -190006India\n",
"Prince A Ganai \nDepartment of Physics\nNational Institute of Technology Srinagar\nJammu and Kashmir -190006India\n"
] | [
"Department of Physics\nNational Institute of Technology Srinagar\nJammu and Kashmir -190006India",
"Department of Physics\nNational Institute of Technology Srinagar\nJammu and Kashmir -190006India",
"Department of Physics\nNational Institute of Technology Srinagar\nJammu and Kashmir -190006India"
] | [] | We developed the functional form of the two-point correlation function under the approximation of fixed particle number densityn.We solved the quasi-linear partial differential equation (PDE) through the method of characteristics to obtain the parametric solution for the canonical ensemble. We attempted many functional forms and concluded that the functional form should be such that the two-point correlation function should go to zero as the value of system temperature increases or the separation between the galaxies becomes large . Also we studied the graphical behavior of the developed two-point correlation function for large values of temperature T and spatial separation r. The behavior of the two-point function was also studied from the temperature measurement of clusters in the red-shift range of 0.023 − 0.546. * Electronic address: | null | [
"https://arxiv.org/pdf/2206.15173v1.pdf"
] | 250,144,611 | 2206.15173 | 171f0490f4b28bef003f1145c86f482d597b416c |
A simplistic approach to the study of two-point correlation function in galaxy clusters
30 Jun 2022
Durakhshan Ashraf Qadri [email protected]
Department of Physics
National Institute of Technology Srinagar
Jammu and Kashmir -190006India
Abdul W Khanday †[email protected]
Department of Physics
National Institute of Technology Srinagar
Jammu and Kashmir -190006India
Prince A Ganai
Department of Physics
National Institute of Technology Srinagar
Jammu and Kashmir -190006India
A simplistic approach to the study of two-point correlation function in galaxy clusters
30 Jun 2022‡ Electronic address: [email protected] i
We developed the functional form of the two-point correlation function under the approximation of fixed particle number densityn.We solved the quasi-linear partial differential equation (PDE) through the method of characteristics to obtain the parametric solution for the canonical ensemble. We attempted many functional forms and concluded that the functional form should be such that the two-point correlation function should go to zero as the value of system temperature increases or the separation between the galaxies becomes large . Also we studied the graphical behavior of the developed two-point correlation function for large values of temperature T and spatial separation r. The behavior of the two-point function was also studied from the temperature measurement of clusters in the red-shift range of 0.023 − 0.546. * Electronic address:
I. INTRODUCTION
Galaxies in clusters serve as robust cosmological observatories and special astrophysical laboratories. Thus, they provide a veritable understanding about the Universe at large scale. The Universe exhibits the hierarchical behavior which exists at all scales, from the smallest quantum particles to the ultimately vast structures, with galaxy clusters occupying the top pyramid of the structure formation. Being the virialized structures in the universe,clusters of galaxies are the largest gravitationally bound objects in the universe. The distribution of the matter at large scales is dominated by the gravitational interaction and the gravitational clustering of galaxies plays an important role in the evolution of the Universe. The studies of gravitational clustering have the advantage that the rules do not change as the system evolves thus making them the useful cosmological probes. A linear theory of the development of the structure from the initial isothermal and adiabatic density perturbations to the present day observed cosmic web has been developed extensively [1]. The current observations of Galaxy clusters indicate that their motions have been strongly influenced by their mutual gravitational dynamics [2] .
Physical processes requiring a long and complicated sequence of events are responsible for the non-linear clustering phenomenon [3]. Because Clustering is a many-body gravitational problem involving billions of stars and thousands of galaxies, it cannot be solved in the same way as the standard two-body problem, so a statistical treatment is required, following an analogy between thermodynamics and statistical physics [4]. The concept of distribution function is fundamental to a statistical description of a dynamical system . The distribution and correlation functions define the overall clustering of galaxies. The implications of these ideas are also consistent with observations. The statistical mechanical study of the structure formation and distribution in the universe has been studied rigorously in [5][6][7][8][9][10]. The authors have employed various modified gravity theories to study the effects of these modifications on the statistical properties of large scale structures in the universe.
One of the most common approaches to study the origin of the Universe is to analyze correlation functions [12]. The correlation function is a general method for studying the distribution of galaxies in a cluster and is a ubiquitous tool for measuring the degree of clustering.
The two-point correlation function is defined through the conditional probability of finding a ii galaxy in some region dV 2 at distance r from a second galaxy in region defined by volume dV 1 , as [11]
P 2|1 =n(1 + ξ 2 (r))dV 2 ,(1)
where r = r 2 −r 1 is the spatial separation between the galaxies and ξ 2 is the two-point correlation function. In general, the two-point correlation function will depend on the absolute positions of the two galaxies i.e. r 1 and r 2 . However, if we average over all directions then ξ 2 becomes a function of r = r 2 − r 1 for a statistically homogeneous system. The nature of gravitational interaction being pairwise , it directly depends on lower order correlation functions such as two-point correlation rather than higher order correlations.
The development of functional form of the two-point correlation function has been attempted in [13]. The authors of [13] have developed the two-point correlation function with a variable number of the system particles. Similarly, for a single component system the the two-point correlation function has also been developed in [14]. Although this is a valid attempt to develop the function, yet the system is such that the appreciable change in the particle number takes place on a time scale larger than the relaxation time of the cluster of galaxies.
The development of the two-point correlation function keeping the particle number fixed is the aim of the present work. We focus on the variation of the correlation function with a changing system temperature T and inter-particle separation r keeping the particle number N fixed.
This paper has been arranged as follows. In section (II) we develop a differential equation of the two-point correlation function keeping the number densityn constant. In section (III)
we propose the functional form of the differential equation developed in section (II) and we also define the uniqueness of the functional form and therefore the behavior of clusters in the expanding universe. In section (IV) we study the graphical behavior of the function for varying system temperature T and spatial separation r. In section (V) we study the two-point function using data analysis. Finally, in section (VI) we make the discussion and conclusion.
iii
II. DEVELOPMENT OF DIFFERENTIAL EQUATION FOR TWO-POINT CORRELA- TION FUNCTION
We assume an infinite system of point galaxies having same mass m in order to keep the system uniform. For such a system of assumed point particles interacting gravitationally, the internal energy , U and pressure, P satisfy the standard statistical thermodynamic relations;
U = 3 2 N T (1 − b) (2) P = N T V (1 − 2b),(3)
where N represents the average number of particles in a canonical ensemble given by;
N =nV andn = N V ,
wheren represents the average number density of the system particles.
Equations ( 2) and (3) are the equations of State, and b is a dimensionless parameter, a measure of the ratio of the gravitational correlation energy and the kinetic energy due to peculiar velocities(K = 3 2 N T ) and is given by;( [14]).
b = − W 2K
As discussed above for ξ 2 to provide a good description, the system should be statistically homogeneous throughout, so that average for any sufficiently large subset of galaxies and that of a small subset in the same volume will be the same.
For a grand canonical ensemble, in which particle number changes, two-point correlation function ξ 2 depends onn, T and r i.e., ξ 2 = ξ 2 (n, r, T ) and the variation in ξ 2 for such a system can be written as;
dξ 2 = ∂ξ 2 ∂T dT + ∂ξ 2 ∂r dr + ∂ξ 2 ∂n dn(4)
iv It is a well established fact that the number of particles (galaxies) does not change appreciably in a cluster unless a collision occurs. Hence, it is justified to fix the particle number densityn.
It can also be seen from the behavior of chemical potential in a cluster of galaxies, see e.g., [6].
Thus we fixn such that;
∂ξ 2 ∂n = 0
. Under this assumption the differential equation for the two-point correlation function ξ 2 (r, T )
can be written as;
T ∂ξ 2 ∂T − r ∂ξ 2 ∂r = 0.(5)
Equation (5) is the reduced first order partial differential equation for two-point correlation function.
III. THE POSSIBLE FUNCTIONAL FORM FOR THE TWO-POINT CORRELATION FUNCTION ξ 2
The equation (5) is of the form of Quasi-linear Partial differential equation. Hence method of characteristics is an important tool for solving hyperbolic-type partial differential equations (PDEs). Through this method, special curves known as the Characteristics Curves are determined along which the partial differential equation becomes the family of Ordinary differential equations (ODEs). Once the ODEs are obtained, they can be solved along the characteristics curves to get the solutions and thus can be related to the solution of the original PDEs. While we attempt for the development of the functional form of ξ 2 , the following observed boundary conditions must be satisfied;
1. In a homogeneous universe, the gravitational clustering of galaxies requires ξ 2 to have be positive for some limiting values ofn , T and r.
2. For very low values of T and r the correlation function ξ 2 will increase for a constant value ofn . Similarly for large value of T and r, the corresponding correlation function ξ 2 will decrease for constant value ofn .
3. Due to virial equilibrium, galaxy clustering becomes more dominant as two-point correlation function ξ 2 increases, which implies that at low temperatures and high densities, v more and more clusters are formed.
4. The change in particle number becomes significant if a collision occurs, otherwise the particle number does not vary in a given volume.
The system of equations defined by equation (5) can be solved by the method of characteristics conveniently. We write our equation as;
T ∂ξ 2 ∂T − r ∂ξ 2 ∂r = 0 (6)
The equation (6) can further be written as;
dT dr = T −r (7)
After separation of variables and integrating, The equation (6) leads to;
ln |T | = − ln |r| + ln |C1|
=⇒ C 1 = T.r,(8)
where C 1 is some unknown function and a solution to the PDE.
Again, we set ;
∂ξ 2 ∂r = 0 =⇒ C 2 = ξ 2 ,(9)
where c 2 is another unknown function.
From equations (8) and (9) the functional form of ξ 2 has the following parametric structure;
ξ 2 (T, r) = f (T.r).(10)
vi Equation (10) will be used to obtain the exact functional form of the two-point correlation function.
In order to explain the uniqueness of the devised functional form (10), we test various combinations and finally choose the solution of equation (6) as
ξ 2 (T, r) = α 1 1 + α 2 T r ,(11)
where α 1 and α 2 are free parameters that should be fixed through the comparison of this solution with the observational data.
The evolution of the two-point function with the red-shift is also important and has been studied in [16]. The two-point correlation function satifies the following relation:
ξ 2 (r, z) = ξ 0 2 (1 + z) −(3+ ) .
In a similar fashion we can also study the evolution of the correlation function by combining the well know relationship r(t) = r 0 (1 + z) with equation 11, the two point correlation function 11 can be written as
ξ 2 = α 1 1 + α 3 T (1 + z) ,(12)
where ,α 3 = α 2 r 0 , is again a parameter. From the graph we see that the correlation function decreases with increasing system temperature, fig. (3). It is also observed that there are certain fluctuation at some red-shifts which may correspond to the clumpy distribution of galaxy clusters in some regions of space as seen in fig.(4) .
VI. DISCUSSION AND CONCLUSION
In this paper we studied the two-point correlation function for a fixed particle number densitȳ n. The approximation is valid because the change in the particle number in a cluster occurs on a very slow time scale. We treat our system in quasi-equilibrium and developed the functional form of the correlation function.After attempting many combinations , we choose the one in equation (11). We also visualized the graphical behavior of the correlation function for varying system temperature T and spatial separation r. We also studied the variation of ξ 2 with a changing temperature for various clusters at different red-shifts, 0.023 ≤ z ≤ 0.546. From fig. (3) we observed that the two-point function shows an appreciable decline with increasing system temperature that is predicted in equation 11. We also observed fluctuation at certain red-shifts whic can be possibly due to some unusual distribution in some regions of space as depicted in fig (4).
IV. GRAPHICAL REPRESENTATION OF THE TWO-POINT CORRELATION FUNCTIONThe behavior of the correlation function developed here can be visualized graphically from figure (1). It is clear from the graph that as the value of the parameter T.r increases the correlation decreases and tends to zero for relatively large values of T.r. This behavior is expected as we see that the galaxies far away from each other are less correlated and have minimum influence on one another. Similarly, the evolution of the correlation function with the gas temperature and with an increasing separation is also shown infig. (2). It can be seen that the function shows a steep dip with increasing values of T and r independently.vii FIG. 1: Variation of the correlation function ξ 2 as a function of system temperature T and mutual separation r for fixed values of α 1 and α 2 . FIG. 2: 3D visualization of the behavior of correlation function ξ 2 as a function of system temperature T and mutual separation r.V. STUDY OF THE VARIATION OF CORRELATION FUNCTION WITH TEMPER-ATURE THROUGH DATAThe variation of the two-point correlation function with system temperature shows an appreciable depreciation as can be seen from the graphical visualization,fig. (reffig1). In terms of the data measurements, the behavior of the correlation function with varying temperature of various clusters is also shown in the graph,fig.(3). We have used the cluster temperature viii values from the work[15]. The data includes measurements by Berkeley-Illinois-Maryland-Association(BIMA)and Owens Radio Observatory (OVRO). The values of cluster red-shifts, temperature and the corresponding values of the two-point correlation function is given in table (I).
ixFIG. 3 :
3Variation of the two-point correlation function with increasing temperature (kT(eV)).
FIG. 4 :
4(a-f) sky distribution in RA(deg) and DEC(deg) coordinates of galaxy clusters in various red-shift ranges xii
TABLE I :
ICluster red-shifts(z), gas temperature(kT(eV)) and corresponding ξ 2 valuesClusters
z kT (eV ) ξ 2
A1656
0.023 6.62 0.1105
A2204
0.152 8.12 0.0871
A1689
0.183 8.25 0.1059
A520
0.200 8.33 0.0909
A2163
0.202 9.59 0.081
A773
0.216 10.1 0.0602
A2390
0.232 10.13 0.0742
A1835
0.252 10.6 0.0685
A697
0.282 10.68 0.057
ZW3146
0.291 11.53 0.07
RXJ1347
0.451 13.45 0.0561
CL0016+16 0.546 13.69 0.0479
MS0451 -0305 0.550 16.18 0.0489
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| [] |
[
"A More Scalable Mixed-Integer Encoding for Metric Temporal Logic",
"A More Scalable Mixed-Integer Encoding for Metric Temporal Logic"
] | [
"Vince Kurtz ",
"Hai Lin "
] | [] | [] | The state-of-the-art in optimal control from timed temporal logic specifications, including Metric Temporal Logic (MTL) and Signal Temporal Logic (STL), is based on Mixed-Integer Convex Programming (MICP). The standard MICP approach is sound and complete, but struggles to scale to long and complex specifications. Drawing on recent advances in trajectory optimization for piecewise-affine systems, we propose a new MICP encoding for finite transition systems that significantly improves scalability to long and complex MTL specifications. Rather than seeking to reduce the number of variables in the MICP, we focus instead on designing an encoding with a tight convex relaxation. This leads to a larger optimization problem, but significantly improves branch-and-bound solver performance.In simulation experiments involving a mobile robot in a gridworld, the proposed encoding can reduce computation times by several orders of magnitude. | 10.1109/lcsys.2021.3132839 | [
"https://arxiv.org/pdf/2112.01326v1.pdf"
] | 244,798,731 | 2112.01326 | f47a25b9cab44e1ff9c564b97250a06f05af54b9 |
A More Scalable Mixed-Integer Encoding for Metric Temporal Logic
Vince Kurtz
Hai Lin
A More Scalable Mixed-Integer Encoding for Metric Temporal Logic
The state-of-the-art in optimal control from timed temporal logic specifications, including Metric Temporal Logic (MTL) and Signal Temporal Logic (STL), is based on Mixed-Integer Convex Programming (MICP). The standard MICP approach is sound and complete, but struggles to scale to long and complex specifications. Drawing on recent advances in trajectory optimization for piecewise-affine systems, we propose a new MICP encoding for finite transition systems that significantly improves scalability to long and complex MTL specifications. Rather than seeking to reduce the number of variables in the MICP, we focus instead on designing an encoding with a tight convex relaxation. This leads to a larger optimization problem, but significantly improves branch-and-bound solver performance.In simulation experiments involving a mobile robot in a gridworld, the proposed encoding can reduce computation times by several orders of magnitude.
I. INTRODUCTION AND RELATED WORK
Timed temporal logics like Metric Temporal Logic (MTL) and Signal Temporal Logic (STL) offer a compact means of expressing complex specifications with timing constraints. Efficient methods of enabling a system to satisfy logical specifications are of particular interest in robotics and Cyber-Physical Systems. For example, a mobile robot need to visit several types of waypoints before a deadline (see Fig. 1).
Early work on the synthesis problem focused primarily on automata-based methods [1], [2], which face severe scalability challenges. A prominent alternative is to encode the problem as a satisfiability problem (SAT/SMT) [3], [4]. These methods tend to scale well [5], but solutions are not globally optimal, even with convex optimization as a theory solver [6].
In this paper, we focus on the Mixed-Integer Convex Programming (MICP) approach to synthesis [7], [8]. In addition to being sound (any solution satisfies the specification) and complete (a solution will be found if one exists), MICP methods are guaranteed to find a globally optimal solution. Furthermore, the MICP paradigm generalizes naturally to systems with highdimensional dynamics [9] and allows for maximizing the STL robustness measure [10]. Existing MICP methods tend to be slower than SAT/SMT methods [5], however.
The primary drawback of the MICP approach is scalability [9]. Standard MICP encodings introduce a new binary variable for each timestep and each sub-formula in the specification. Since the worst-case complexity of MICP is exponential in The 1. A robot in a grid-world is tasked with visiting six types of waypoints (red, green, blue, yellow, purple and brown squares) while avoiding obstacles (black squares). A standard mixed-integer encoding takes 1 hour 25 minutes to find an optimal path, while our proposed approach takes only 26 seconds.
the number of binary variables, performance rapidly degrades for complex specifications and long time-horizons. For this reason, much research has focused on encodings with fewer binary variables and constraints. For example, [11] show that half as many constraints can be used if specifications are first re-written in Postive Normal Form (PNF), and [12] iteratively solve a sequence of smaller MICPs.
More recently, there has been a trend toward avoiding integer programming entirely. For certain fragments of STL and MTL (related logics that exclude certain operators or combinations of operators), synthesis can be done with convex programming directly [7]. Similar fragments have been used for synthesis based on Control Barrier Functions (CBFs) [13] and learning [14]. For STL, smooth approximations of the robustness score [15]- [18] have been used to find local solutions via gradient descent.
None of the above methods are sound and complete for the full syntax of MTL or STL, however. This limits them to relatively simple scenarios, and they may struggle to find a solution (or be heavily dependent on an initial guess) in the case of more complex specifications like the one in Fig. 1.
In this work, we revisit MICP for timed temporal-logic, focusing in particular on MTL. Taking inspiration from recent work on trajectory-optimization for piecewise-affine systems [19], we note that while the worst-case complexity of MICP depends primarily on the number of binary variables, performance in practice often depends more heavily on the tightness of the convex relaxation (a convex program where the binary variables in the original MICP are allowed to take continuous values), since modern MICP solvers rely heavily on the branch-and-bound algorithm [20].
Following [19], the basic idea behind our proposed encoding is to introduce binary variables for each possible transition at every timestep, rather than for each possible state. This results in more binary variables than a standard encoding, but a tighter convex relaxation and better performance in practice [21]. This approach holds notable similarities to mixed-integer encodings for various problems in graph theory, including the shortest path problem (SPP) [22] and traveling salesman problem (TSP) [23], which can be viewed as special cases of MTL.
Our primary contributions are summarized as follows: 1) We present a more scalable mixed-integer encoding for finite transition systems subject to MTL specifications. 2) Our proposed method is sound and complete.
3) The convex relaxation of our proposed encoding is at least as tight as that of a standard MICP encoding. 4) The convex fragment induced by our proposed encoding is strictly larger than that of standard MICP encoding. 5) In simulation experiments, our proposed approach outperforms standard MICP and SAT-based synthesis. The remainder of this paper is organized as follows: background and a formal problem statement is presented in Section II. We summarize the standard MICP approach in Section III and present our proposed encoding in Section IV, along with proofs of soundness, completeness, and convex relaxation tightness. We provide simulation examples in Section V and conclude with Section VI.
II. BACKGROUND
A. Metric Temporal Logic
Metric Temporal Logic (MTL) is an extension of Linear Temporal Logic (LTL) which allows for timing-related deadlines [24]. The syntax of MTL is defined as:
ϕ := | π | ¬ϕ | ϕ 1 ∧ ϕ 2 | ϕ 1 ∪ [t1,t2] ϕ 2(1)
where π ∈ AP is an atomic proposition, boolean operators "not" (¬) and "and" (∧) can be used to define disjunction (∨), and the temporal operator "until" (∪ [t1,t2] ) can be used to define "always" ( [t1,t2] ) and "eventually" (♦ [t1,t2] ). We assume bounded-time specifications, i.e., t 2 is finite:
Assumption 1 (Bounded-time Specification). There exists some 0 ≤ T < ∞ such that the satisfiability of ϕ can be uniquely determined in T timesteps.
MTL semantics are defined over words σ = σ 0 , σ 1 , σ 2 , . . . , where σ t ∈ 2 AP is the set of atomic propositions that hold at timestep t. We denote that σ satisfies the MTL formula ϕ with σ ϕ, and that the suffix σ t , σ t+1 , . . . satisfies ϕ with σ t ϕ. MTL semantics are defined recursively as follows:
σ t π ⇐⇒ π ∈ σ t σ t ¬ϕ ⇐⇒ σ t ϕ σ t ϕ 1 ∧ ϕ 2 ⇐⇒ σ t ϕ 1 and σ t ϕ 2 σ t ϕ 1 ∪ [t1,t2] ϕ 2 ⇐⇒ ∃t ∈ [t + t 1 , t + t 2 ] s.t. σ t ϕ 2 and ∀t ∈ [t, t − 1], σ t ϕ 1 σ ϕ ⇐⇒ σ 0 ϕ Remark 1.
MTL is closely related to Signal Temporal Logic (STL), though MTL is defined over discrete atomic propositions, whereas STL is defined over continuous-valued signals.
B. Mixed Integer Programming
MICP considers problems of the form
min x,z J(x, z) (2a) s.t. A x z ≤ c (2b)
where x is a vector of real-valued decision variables, z is a vector of binary-valued 1 decision variables, A and c are matrices of appropriate dimensions, and J(·, ·) is convex.
If the values of z are fixed, (2) can be solved rapidly with convex programming. But trying every possible z would be prohibitively expensive. Fortunately, there are several tricks that allow MICP solvers to mostly avoid this worst-case scenario [25,Chapter 1.2]. The most prominent such method is branch-and-bound.
The main idea behind branch-and-bound is to "branch" on some of the binary variables by fixing their values as 0 or 1. With these variables fixed, we have a smaller MICP with fewer binary variables. We then obtain a "bound" by solving a convex relaxation: all of the other binary variables are allowed to take continuous values in [0, 1]. This convex relaxation can be solved quickly with specialized convex optimization methods, and provides a lower bound on the optimal cost.
If the convex relaxation is infeasible, we know that the binary values that we fixed are incorrect, allowing us to rapidly eliminate all of the possible solutions with those values. Furthermore, any integer-feasible solution provides an upper bound on the optimal cost. If a given branch has a convex relaxation with a higher cost than this upper bound, we can similarly eliminate solutions in that branch.
Clearly, the tightness of the convex relaxation has a significant impact on the efficiency of the branch-and-bound algorithm. A tighter convex relaxation will allow the solver to eliminate suboptimal branches rapidly, while a loose convex relaxation is more likely to result in the worst-case scenario of fully exploring every possible branch. With this in mind, we define the relaxation gap: Definition 1 (Relaxation Gap). Given a MICP of the form (2), let J * be the optimal cost andJ * be the optimal cost of the convex relaxation. Then the relaxation gap is given by
r gap = (J * −J * )/J * .(3)
If the relaxation gap is zero, integer constraints are unnecessary and the problem can solved with convex programming directly. This is the case for a number of interesting problems, including the SPP in graph theory [22] and a convex fragment of timed temporal logic [7, Theorem 1]. Even for nonconvex problems (e.g., MTL and the TSP), the relaxation gap plays an important role in determining solver performance [19], [21].
C. Problem Formulation
In this paper, we consider synthesis over finite-state labeled transition systems:
Definition 2 (Transition System). A transition system T S = (S, s 0 , →, AP, L, C)
is a tuple consisting of the following elements:
• S is a finite set of states • s 0 ∈ S is an initial state • →⊆ S × S are transition relations • AP is a finite set of atomic propositions • L : S → 2 AP is a labeling function • C : (S × S) → R + is a cost function
Note that each transition in T S is associated with a cost. Our goal will be to find a minimum-cost path through T S that satisfies a given MTL specification.
We now provide several definitions to allow for more efficient discussion of transition systems. First, a path is merely a sequence of states that obeys the transition relations:
Definition 3 (Path). A sequence of states s = s 0 , s 1 , . . . , s T is a path of TS if (s t , s t+1 ) ∈→ ∀t ∈ [0, T − 1].
Note that we focus in this paper on finite-length paths. Every path is associated with a total cost:
Definition 4 (Path Cost). Given path s, J(s) = T −1 t=0 C(s t , s t+1 )
is the total path cost associated with s. In addition to the cost of any given path, we are also interested in the corresponding sequence of atomic propositions: Note that the trace is a word over which we can consider satisfaction of an MTL formula. With some liberty of notation, we write s ϕ if σ(s) ϕ. Finally, we define the set of adjacent states as those that can be transitioned to from the current state:
Definition 6 (Adjacent Set). Given state s ∈ S, the adjacent set of s is given by Adj(s) = {s ∈ S | (s, s ) ∈→}.
Note that if T S contains self-loops, i.e., (s i , s i ) ∈→, then s i ∈ Adj(s i ). We can now provide a formal problem statement: Problem 1. Given transition system T S and bounded-time MTL specification ϕ, find the minimum-cost path through T S that satisfies ϕ, i.e.,
min s J(s) (4a) s.t. s ϕ. (4b) III. STANDARD MIXED-INTEGER ENCODING
In this section, we present the standard method of encoding (4) as an MICP. Our presentation in this section is based primarily on [7] and [9], which consider STL specifications, but similar encodings for MTL are also popular [8], [12].
The basic idea is to introduce a binary variable b s (t) for each state s and each timestep t 2 . We pose the problem such that b s (t) = 1 means the optimal path visits state s at time t.
We start with the following dynamics constraints:
b s0 (0) = 1, (5a) s∈S b s (t) = 1 ∀t ∈ [0, T ],(5b)s ∈Adj(s) b s (t + 1) ≥ b s (t) ∀s ∈ S,(5c)
where (5a) establishes s 0 , (5b) ensures that only one state can be occupied at each timestep, and (5c) enforces transition relations.
To satisfy the specification, we add additional variables and constraints which are defined recursively. The main insight is to define new binary variables, z ϕ (t), such that z ϕ (t) = 1 only if ϕ is satisfied starting from time t. First, note that conjuction and disjunction can encoded as linear constraints as follows:
z n i=1 z i ⇐⇒ z ≤ z i ∀i and z ≥ 1 − n + n i=1 z i (6) z n i=1 z i ⇐⇒ z ≤ n i=1 z i and z ≥ z i ∀i(7)
This allows us to encode satisfaction of ϕ as follows:
π =⇒ z π = s∈S|L(s)=π b s (t),(8a)¬ϕ =⇒ z ¬ϕ = 1 − z ϕ , (8b) ϕ 1 ∧ ϕ 2 =⇒ z ϕ1∧ϕ2 = z ϕ1 ∧ z ϕ2 , (8c) ϕ 1 ∪ [t1,t2] ϕ 2 =⇒ (8d) t ∈[t+t1,t+t2] z ϕ2 (t ) ∧ t ∈[t,t −1] z ϕ1 (t ) .
With this in mind, we can write problem (4) as follows:
min bs,z J(b s ) (9a) s.t. Transition System Constraints (5),(9b)
MTL Constraints (8), (9c) z ϕ (0) = 1.
(9d)
We focus here on finding a minimum-cost path, but a (convex) cost function can also be designed for different purposes, such as maximizing the STL robustness score [10].
The scalability limitations of this standard MICP formulation are well-known [9], [15], [17], [18], [26]. In particular, this encoding is associated with rapidly increasing solve times in the case of long time horizons (which increase the number of binary variables) and complex specifications (which increase the complexity of the constraint structure).
IV. MAIN RESULTS
In this section, we exploit the fact that is often the tightness of convex relaxation, rather than the number of binary variables and constraints, that determines MICP scalability to propose a more efficient MICP encoding. Our main inspiration in this regard is [19], which presents a more efficient MICP for control of PWA systems by increasing the number of binary variables but tightening the convex relaxation.
We begin by constructing a directed graph G = (V, E), associated with T S. Each node i ∈ V corresponds to a state s ∈ S and a timestep t, i.e., i = (s, t). Each edge (i, j) ∈ E connects two nodes only if there is a corresponding transition:
(s, t), (s , t + 1) ∈ E ⇐⇒ s ∈ Adj(s).
Additionally, for each node i ∈ V we define the input set I i = {j | (j, i) ∈ E} and the output set
O i = {j | (i, j) ∈ E}.
Our basic idea is to introduce a binary variable for every edge in the graph (the standard encoding (9) introduces a binary variable for each node). This may seem counterintuitive, as there are many more edges than nodes, but similar formulations perform well for special cases of temporal-logic planning, including SPP [22], TSP [23], and PWA control [19].
More specifically, we define binary variables a ij for each edge, where a ij will take unit value only if the edge (i, j) is part of the optimal satisfying path. We can then implicitly define variables b i = b s (t) representing the total flow through each node as follows, since i = (s, t):
b s (t) = j∈Oi a ij if t = 0 j∈Ii a ji otherwise.(10)
These flow variables take binary values at optimality, and can be used to enforce MTL constraints following (8).
Our proposed MICP encoding can then be written as:
min aij ,bs,z J(a ij ) (11a) s.t. j∈Oi a ij − j∈Ii a ji = 1 if i = (s 0 , 0) 0 if t < T (11b)
Occupancy constraints (10) (11c)
MTL Constraints (8) (11d) z ϕ (0) = 1 (11e)
where (11b) establishes flow constraints and an initial state, and (11c-11e) enforce satisfaction of the MTL formula. The proposed encoding is sound and complete:
Theorem 1. Any solution to (11) satisfies the specification ϕ (soundness); and if a satisfying path exists which satisfies ϕ, then a solution to (11) exists (completeness).
Proof. The theorem follows from the inclusion of constraints (11c-11e) and [9, Theorem 1].
Furthermore, the relaxation gap of our proposed encoding is no greater than that of the standard encoding for all MTL specifications:
Theorem 2. Let r * gap be the relaxation gap associated with (11) and r gap be the relaxation gap associated with (9). Then r * gap ≤ r gap . Proof. First, note that the optimal (non-relaxed) cost for both (9) and (11) are the same.
Next, note that any solution of a convex relaxation of (11) is also a valid solution to a convex relaxation of (9). This is because the (non-binary) flow constraints (11b) are sufficient for enforcing the (non-binary) transition constraints (5a-5c). Now assume that we have r * gap > r gap . That would mean that the optimal cost associated with a convex relaxation of (11) is less than the optimal cost associated with a convex relaxation of (9). But this is a contradiction, since the (relaxed) solution to (11) is also a solution to (9).
Furthermore, the converse is not always true. Specifically, the transition constraint (5c) allows some "flow" to pass between non-adjacent nodes in a convex relaxation. Such nonadjacent flows are not allowed in (11). This means that it is often the case that r * gap r gap (see Section V).
Remark 2.
While the worst-case complexity of (11) is higher than that of (9), as there are more binary variables, the relative tightness of the convex relaxation leads to better scalability in practice, as shown in Section V.
Finally, we show that the relaxation gap is zero for a surprisingly large fragment of MTL:
Theorem 3. For MTL specifications ϕ belonging to the fragment
ψ := π | ψ 1 ∧ ψ 2 | ψ 1 ∨ ψ 2 ϕ := [t1,t2] ψ | ♦ [t2,t2] ψ | ψ 1 ∪ [t2,t2] ψ 2 | ϕ 1 ∧ ϕ 2(12)
the relaxation gap associated with (11) is zero.
Proof. For formulas over which only conjunction is used, the problem is convex and thus the relaxation gap is zero. This follows from [7,Theorem 1]. Note that because the interval associated with the temporal operators "eventually" and "until" is a single timestep, disjunctions in the MTL encoding (8) occur only between state formulas ψ.
With this in mind, consider the case where the formula contains at least one set of disjunctions over state formulas, i.e., k ψ k and there are some flows a ij ∈ (0, 1) that take non-binary values at optimality. We will show that in this case, there is always a binary solution that achieves the same cost.
Note that for the fragment (12), any a ij ∈ (0, 1) arise only due to several possible "paths" satisfying different possible state formulas ψ k . Furthermore, each of these paths must have equal cost. This is easily established by contradiction: if the paths do not have equal cost, a lower-cost solution can be obtained by following only the lower-cost paths. Therefore following any single path (a ij ∈ {0, 1}) results in the same cost. Thus the relaxation gap is zero and the theorem holds.
This means that for specifications in the fragment (12), the synthesis problem can be solved in polynomial time using linear programming. This fragment is significantly more expressive than the convex fragment associated with the standard MICP encoding (9), which only considers atomic propositions, conjunctions, and the "always" operator [7]. (12) is the same as that considered in [26], which presents a scalable but incomplete synthesis method. In contrast, our proposed encoding is sound and complete for all specifications, including those in this fragment.
Remark 3. The fragment
It may be somewhat surprising that disjunctions, which would seem to introduce some sort of inherently combinatorial aspect to the problem, can be included in a convex fragment. This sort of convexity despite the presence of disjunctions is a feature shared with the LP encodings of the SPP, where the solver must choose between several seeming disjointed paths, but a convex formulation is possible.
V. SIMULATION EXPERIMENTS
In this section, we demonstrate the scalability of our proposed encoding (11) on several robot motion planning problems. The transition system T S models a robot in an N × N grid-world, as shown in Fig. 1. Each state s ∈ S corresponds to a grid cell. Transitions to adjacent cells (including diagonals) are associated with cost 1 while transitions to the same cell have cost 0. No other transitions are allowed.
All experiments were performed on a laptop (i7 processor, 32GB RAM) using Gurobi [20] (version 9.0.3, default options) as the MICP solver. Drake [27] python bindings were used to interface with the solver.
In addition to comparing our proposed MICP encoding (11) with the standard MICP encoding (9), we consider an SATbased approach in which the constraints (9b-9d) are passed to the z3 SAT solver [4]. This method returns a non-optimal solution and tends to be faster than the standard MICP.
We first consider the simple reach-avoid scenario shown in Fig. 2a, where a robot must reach a goal (green) and avoid an obstacle (red). The atomic propositions for this scenario are AP = {goal, obstacle} and the specification is given by
¬obstacle ∪ [0,T ] goal,
where we chose T = 15.
The standard MICP encoding (9) introduces 1600 binary variables and finds a minimum-cost 3 satisfying path in 12.7s. The SAT-based approach is slightly faster, finding a solution in 12.3s. Our proposed encoding (11) introduces 11760 binary variables, but takes only 0.47s to find an optimal solution. The relaxation gap is 0.9999 for the standard approach and 0.94 for our proposed method. This supports the idea that even a modest reduction in the relaxation gap can have a significant impact on MICP performance in practice.
Remark 4. Note that this simple reach-avoid specification could also be solved as an SPP with LP. This raises the prospect that there may exist yet stronger MICP encodings that would reduce to LP for specifications like this one.
We now consider a standard scenario for synthesis from timed temporal logic: in addition to reaching a green goal and avoiding red obstacles, the robot must also visit one of two blue targets (Fig. 2b). The specification is given by
♦ [0,T ] (target one∨target two)∧♦ [0,T ] goal∧ [0,T ] ¬obstacle with T = 15.
SAT and standard MICP both take 25.9s to find a solution, while our proposed approach takes only 0.61s. The relaxation gap is 0.9996 for standard MICP and 0.8739 for our approach.
We also consider the simpler multi-target scenario shown in Fig. 3, where the specification is given by ♦ [T,T ] (target one ∨ target two), 3 Diagonal transitions have the same cost as horizontal/vertical transitions.
(a) Standard MICP convex relaxation (b) Proposed MICP convex relaxation Fig. 3. Convex relaxations of the standard encoding (9) and our proposed encoding (11) for a simple multi-target specification. The relaxation for our approach satisfies the specification directly, since the specification belongs to the convex fragment (12), but this is not the case for the standard encoding. and T = 5. Note that this specification belongs to the fragment (12) where our proposed encoding is convex, but not the convex fragment induced by the standard MICP encoding.
Since this problem is relatively small and simple, both MICP methods find the optimal solution rapidly (under 0.01s for both methods). What is more interesting is to consider the solutions to the convex relaxations of each encoding. These are shown in Fig. 3, where grid cells are shaded according to the net flow T t=0 b s (t). The relaxation of our proposed encoding (3b) provides a satisfying solution directly, since ϕ belongs to the fragment 12. The relaxation of the standard encoding (3a), however, does not respect the transition constraints. Instead, this solution requires the robot to occupy multiple cells at once.
The fact that our proposed encoding has a tighter convex relaxation suggests that our approach scale well to long time horizons and complex specifications. We verify this experimentally by considering a class of more complex, randomly generated scenarios. These scenarios consists of obstacles as well as several groups of targets. The robot is tasked with visiting at least one target in each group while avoiding obstacles. One example of such a specification is shown in Fig. 1.
We denote the number of target groups as N g , the number of targets in each group as N t , and the number of obstacles as N o . The specification is given by
[0,T ] ¬obstacle ∧ Ng k=1 ♦ [0,T ] Nt l=1 target l k ,(13)
where target l k denotes the l th target in group k. Randomly generating scenarios with a given number of obstacles and targets allows us to test scalability with respect to specification complexity. Specifically, we use the number of target groups, N g , as a proxy for specification complexity. As N g increases, it becomes more difficult for the robot to find the shortest path that visits each group.
Specifically, we set up an experiment where scenarios are randomly generated with the following parameters:
N t = 2, N o = 2N g , N = 5 + N g , T = 15.
10 trials with the above parameters were considered for several values of N g . We compared the resulting solve times of our proposed MICP encoding with the standard MICP method and SAT-based synthesis. The results are shown in Fig. 4a. The box plot for each trial shows the median (horizontal line), upper and lower quartiles (shaded box) and range (whiskers).
The standard MICP approach (rightmost bars, diagonally striped orange boxes) performs the worst, with solve times exceeding one minute for the most complex scenarios. The SAT approach (middle bars, blue boxes with vertical stripes) consistently outperforms the standard MICP, which makes sense given the fact that the SAT method finds any feasible solution, while the MICP approach searches for a globally optimal one. Our proposed MICP encoding (left bars, solid greed boxes) outperforms both of the other methods, with all solve times under 5s even for the most complex scenarios.
It may seem surprising that our proposed approach outperforms the SAT method while also finding a globally optimal solution. We believe that this superior performance is due, again, to the efficiency of branch-and-bound on our proposed encoding. Specifically, the MTL constraints (11c-11e) are all linear in the decision variables, meaning the MTL specification is in some sense always satistisfied (though transition constraints may not be) even for a convex relaxation. This allows the branch-and-bound algorithm to "hone in" rapidly on satisfying solutions. SAT solvers, on the other hand, do not have access to this sort of efficient heuristic.
Finally, we consider scalability with respect to specification length. This is known to be a significant limitation for existing MICP encodings [10], since the number of binary variables increases linearly with the time bound T . We consider the same class of randomly-generated scenarios, this time with the following parameters: N = 10, N g = 3, N t = 2, N o = 3, and various values of T. The results are shown in Fig. 4b. Our approach again consistently outperforms the standard MICP encoding, especially for long time horizons.
VI. CONCLUSION
We propose a new MICP encoding for finding an optimal path through a finite-state transition system subject to MTL specifications. By virtue of having a tighter convex relaxation, our proposed approach outperforms existing MICP and SATbased synthesis methods in terms of speed and scalability to long and complex specifications. Furthermore, this encoding allows specifications within a larger convex fragment to be solved using convex programming directly. Future work will focus on extensions to unbounded specifications, STL (where the advantages of MICP include applicability to systems with high-dimensional PWA physical dynamics), and probabilistic systems.
Definition 5 (
5Trace). The trace of path s is given by σ(s) = L(s 0 ), L(s 1 ), . . . , L(s T ).
Fig. 2 .
2Simple robot motion planning specifications in which the robot must navigate through a grid-world and reach a goal (green) and an intermediate target (blue) while avoiding obstacles (red)
Fig. 4 .
4Scalability tests with respect to specification complexity and length. Our proposed approach (solid green) consistently outperformed a SAT approach (striped blue) and a standard MICP encoding (striped orange).
authors are with the Departments of Electrical Engineering, University of Notre Dame, Notre Dame, IN, 46556 USA. {vkurtz,hlin1}@nd.edu This work was supported by NSF Grants CNS-1830335, IIS-2007949.Fig.
Note that in general z can take integer values. In most applications of MICP, however, only binary variables are considered. For this reason, we will refer to z as binary variables throughout this paper.
In the STL case, binary variables are introduced for each predicate and subformula, which correspond to states if the system is abstracted as a finitestate transition system.
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"Prospects of neutrino oscillation measurements in the detection of reactor antineutrinos with a medium-baseline experiment"
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"Mikhail Batygov \nDepartment of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA\n",
"Stephen Dye \nDepartment of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA\n",
"John Learned \nDepartment of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA\n",
"Shigenobu Matsuno \nDepartment of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA\n",
"Sandip Pakvasa \nDepartment of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA\n",
"¶ \nDepartment of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA\n",
"Gary Varner \nDepartment of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA\n"
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"Department of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA",
"Department of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA",
"Department of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA",
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"Department of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA",
"Department of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA",
"Department of Physics and Astronomy\nUniversity of Hawaii at Manoa\n96822HonoluluHawaiiUSA"
] | [] | Despite the dramatic progress made in neutrino oscillation studies recently, several fundamental neutrino parameters remain either unknown or poorly measured. We discuss in detail a method for their measurement by precision studies of oscillation-caused neutrino energy spectrum distortions, for which a large underwater inverse beta decay detector appears to be a perfect tool. Results determine optimal baselines and necessary exposures in the presence of systematic uncertainties and the unavoidable background from terrestrial antineutrinos. | null | [
"https://arxiv.org/pdf/0810.2580v2.pdf"
] | 117,530,036 | 0810.2580 | 796ebc9acdb369155e52cbe82f274d1978f39017 |
Prospects of neutrino oscillation measurements in the detection of reactor antineutrinos with a medium-baseline experiment
8 Nov 2008
Mikhail Batygov
Department of Physics and Astronomy
University of Hawaii at Manoa
96822HonoluluHawaiiUSA
Stephen Dye
Department of Physics and Astronomy
University of Hawaii at Manoa
96822HonoluluHawaiiUSA
John Learned
Department of Physics and Astronomy
University of Hawaii at Manoa
96822HonoluluHawaiiUSA
Shigenobu Matsuno
Department of Physics and Astronomy
University of Hawaii at Manoa
96822HonoluluHawaiiUSA
Sandip Pakvasa
Department of Physics and Astronomy
University of Hawaii at Manoa
96822HonoluluHawaiiUSA
¶
Department of Physics and Astronomy
University of Hawaii at Manoa
96822HonoluluHawaiiUSA
Gary Varner
Department of Physics and Astronomy
University of Hawaii at Manoa
96822HonoluluHawaiiUSA
Prospects of neutrino oscillation measurements in the detection of reactor antineutrinos with a medium-baseline experiment
8 Nov 2008numbers: 1460Pq2665+t2850Hw
Despite the dramatic progress made in neutrino oscillation studies recently, several fundamental neutrino parameters remain either unknown or poorly measured. We discuss in detail a method for their measurement by precision studies of oscillation-caused neutrino energy spectrum distortions, for which a large underwater inverse beta decay detector appears to be a perfect tool. Results determine optimal baselines and necessary exposures in the presence of systematic uncertainties and the unavoidable background from terrestrial antineutrinos.
INTRODUCTION
Neutrino flavor transformations are determined by the elements of the PMNS matrix [1,2] and the differences between the squares of neutrino mass eigenvalues. PMNS matrix represents the mixture between flavor and mass eigenstates of neutrinos and is conventionally decomposed as shown in (1). Neutrino oscillation experiments can yield the best estimations for some of those parameters, which has been demonstrated by SNO [3] and KamLAND [4].
U = U e1 U e2 U e3 U µ1 U µ2 U µ3 U τ 1 U τ 2 U τ 3 =
where s ij = sin θ ij , c ij = cos θ ij , δ is the phase factor (non-zero if neutrino oscillation violates CP symmetry). α 1 and α 2 Majorana phase factors (non-zero only if neutrinos are Majorana particles), to which neutrino oscillation experiments are not sensitive.
Besides the PMNS matrix, neutrino oscillations depend on mass eigenvalues or, more precisely, on the difference between the squared mass eigenvalues. If there are three neutrino mass eigenvalues, then there are only two independent differences, the third being either a sum or a difference of the other two.
Neutrinos studied in experiments are produced in certain flavor eigenstates with known abundances of each of them or, as an important special case, in only one flavor eigenstate. For example, neutrinos are generated in the atmosphere with the known (ν µ +ν µ )/(ν e +ν e ) ratio of about two for low energies; solar neutrinos and reactor antineutrinos are, initially, all ν e andν e , respectively.
Detector sensitivity is, generally, flavor dependent. In particular, the inverse beta decay, the primary method for detecting reactor antineutrinos since the very beginning of neutrino experiments [5], involves electron antineutrinos only. Therefore, the number of detected neutrino events can be different from the no-oscillation expectation. The deficit of observed neutrinos compared to no-oscillation prediction was first detected in a solar neutrino experiment [6]. However the rate information alone could not provide sufficient evidence to ascribe conclusively the phenomenon of neutrino "disappearance" to flavor oscillations.
The energy dependence of neutrino oscillations not only changes the neutrino event rate but also distorts the observed neutrino energy spectrum. The spectrum distortion provides more information about the PNMS matrix components and neutrino mass eigenstates than rate studies alone can.
The inverse beta decay method offers excellent energy sensitivity, which is very valuable for the oscillation studes. Recoil smearing present in this reaction is small compared to detector energy resolution, the latter being the main limiting factor in the accuracy ofν e energy measurement. Other advantages include a relatively large cross section of the reaction and, most importantly, very powerful background suppression due to the characteristic double-coincidence signature. The limitations of this method are theν e energy threshold of about 1.8 MeV and weak directionality.
The success of a neutrino oscillation experiment depends not only on the characteristics of the detector and on the neutrino source but also on the proper choice of the distance between the two (the baseline). There is no single baseline optimal for all neutrino oscillation studies. For example, the average baseline of KamLAND experiment, about 180 km, is fairly good for θ 12 and especially for ∆m 2 12 but not for θ 13 , ∆m 2 13 and ∆m 2 23 . Moreover, such parameters as detector resolution, the amount, the nature of the background and the a-priori information about its properties can affect the optimal baseline value. A tunable baseline experiment, which implies movable detector or source, may have a big advantage here.
These considerations, along with the interest in studying terrestrial antineutrinos, led to the idea of a big KamLANDlike underwater detector [7,8]. The potential of such a detector for neutrino oscillation parameter measurements was the primary motivation for the study presented here. However, the scope of the actual study is much wider and not limited to the Hanohano project. The results are in fact applicable to any similar medium-baselined experiment.
SPECTRUM DISTORTIONS DUE TO OSCILLATIONS
For baselines associated with current and near-future reactor based neutrino experiments (up to hundreds of kilometers), the matter effects [9,10] critical in solar neutrino studies are not significant, so the vacuum oscillation approximation can be used. As was mentioned above, the inverse beta decay detection is sensitive to electron antineutrinos only. Reactors produce exclusively electron antineutrinos as well, so the observable effect is the apparent "disappearance" of a fraction of reactor-produced electron antineutrinos. Theν e "survival" probability is given by the formula [11,12]:
P (ν e →ν e ) = 1 − cos 4 (θ 13 ) sin 2 (2θ 12 ) sin 2 ∆ 12 − sin 2 (2θ 13 ) cos 2 (θ 12 ) sin 2 ∆ 13 − sin 2 (2θ 13 ) sin 2 (θ 12 ) sin 2 ∆ 23 ,(2)
where ∆ ij = |∆m 2 ij |R 4Eν . Note that "atmospheric" mixing angle θ 23 does not affect the ν e survival and hence not measurable in electron neutrino disappearance experiments. Here, R is the "baseline", the distance between theν e source and the detector.
Given the evidence from solar neutrino experiments [13,14] that m 2 > m 1 , and the knowledge that ∆m 2 23 ≫ ∆m 2 12 from SuperK [15], K2K [16], MINOS [17] on the one hand and KamLAND [4] on the other, only two neutrino hierarchies out of possible six are allowed with currently available data. They are commonly referred to as "Normal Hierarchy, NH" (m 1 < m 2 < m 3 ) and "Inverted Hierarchy, IH" (m 3 < m 1 < m 2 ) (Fig. 1). The former implies that ∆m 2 13 > ∆m 2 23 , the latter that ∆m 2 23 > ∆m 2 13 , so the sufficiently precise measurement of the those squared mass differences should be enough to establish the neutrino mass hierarchy.
The measurement of ∆m 2 13 , ∆m 2 23 and the mass hierarchy with this approach is possible, in theory, only if θ 13 is finite and, in practice, if this mixing angle is large enough. Moreover, if the "solar" mixing is maximum (θ 12 = π/4), the ∆m 2 13 and ∆m 2 23 become mutually indistinguishable, thus still ruling out the mass hierarchy study, although their values may still be determined without knowing "which is which". This maximum mixing is strongly disfavored by KamLAND [4] and essentially excluded by solar eperiments [13,14]. Unfortunately, the same can not be said about the θ 13 since only the upper limit for this value exists today and there is no experimental evidence that it is not zero. If it is, then futureν e vacuum oscillation experiments are limited to probing θ 12 and ∆m 2 12 along with setting still better upper limits on the θ 13 itself. That said, global analysis shows a slight preference for non-zero θ 13 [18].
A typical reactorν e spectrum [19,20] multiplied by the inverse beta decay cross section [21] is shown in Figure 2, dotted. The antineutrinos are generated in β − decays of short living fission products of initial fissionable fuel isotopes: 235 U , 238 U , 239 P u, 241 P u. For this study, the ratio of the isotopes is taken the same as in [4]. Such a spectrum can be observed at very short-baselined experiments (baseline ≪ 1 km), where oscillation effects are negligible.
At much longer baselines (30 km and above), the "solar" oscillations governed by θ 12 and ∆m 2 12 lead to an energydependent deficit of the observedν e events. The effect of those oscillations alone is the "coarse" oscillatory pattern of event deficit over the spectrum (Figure 2, dashed) with a high amplitude (determined by sin 2 θ 12 ) and a relatively low frequency (determined by ∆m 2 12 ). The amplitude of oscillations driven by the squared mass differences ∆m 2 13 and ∆m 2 23 is proportional to sin 2 2θ 13 and much smaller than that of "solar" oscillations for any currently allowed value of this mixing angle. Because ∆m 2 13 and ∆m 2 23 are known to be larger than ∆m 2 12 , the frequency of those sub-dominant oscillations is higher. A typical ν e energy spectrum expected for a non-zero θ 13 is shown in Figure 2, solid.
The spectrum analysis approach has already been successfully used by KamLAND to set by now the best limits on ∆m 2 12 and to confirm SNO and SuperK values for θ 12 . The idea to measure the remaining three of the five oscillation parameters by precision measurement of the sub-dominant oscillation pattern in a reactorν e disappearance experiment has been already suggested and thoroughly examined [22,23,24].
In this paper, we examine the capacity of an intermeidate baseline (30-90 km) reactorν e experiment for measuring θ 12 , θ 13 , ∆m 2 12 , ∆m 2 13 , ∆m 2 23 and neutrino mass hierarchy. Although this study has been motivated by the project of a big underwater detector Hanohano ( [7,8]), we make no assertions specific for that particular choice. A special emphasis is placed on the systematic uncertainties and technical limitations present in any real experiment. In our study of the sensitivity to each of the oscillation parameters we take into account the impact of those uncertainties, as well as some detector parameters and the baselines on the resulting performance to formulate in a quantitative way the requirements to which such an experiment must conform.
THE SCOPE OF ANALYSIS
In this study, we consider the measurement of all the oscillation parameters to which suchν e disappearance experiments are sensitive: θ 12 , θ 13 , ∆m 2 12 , ∆m 2 13 , ∆m 2 23 . Three types of detector-related systematic uncertainties are considered which are present to some extent in any experiment and are capable of a non-trivial impact on the sensitivity to the target parameters. Although the success of the Borexino experiment [25] suggests that careful detector design can make the inverse-beta basedν e detection almost background-free, geologically produced antineutrinos [4,26] will technically remain a background source for a reactorν e study in the lower energy region. What makes this background especially significant is the lack of exact information about its overall intensity and the relative amounts of antineutrinos produced in the "Uranium-Radium" and "Thorium" decay series. This amounts to two more systematic uncertainties which have to be left unconstrained within geologically feasible models.
The following detector-related uncertainties were accounted for:
• The uncertainty in the predicted event rate. It is sensitive to the number of target protons (due to fiducial volume estimation error and uncertainty of the scintillator composition), the efficiency of coincidence selection cuts, and live time estimation error. For current similar experiments, this error tends to be on the order of 1 to 5%. Below we refer to this as "efficiency" error.
• The uncertainty in the detector energy resolution estimation. Although often ignored in current experiments, it can have a considerable effect on the measurement of the θ 13 mixing angle from medium baselines. Numerically, it can be quite big (about 10%) depending on the detector calibration options.
• The "linear" energy scale uncertainty. This is the uncertainty in the average number of photoelectrons produced by an 1 MeV event. The amount of this uncertainty depends on the detector calibration as well. Normally it can be made quite small (around 1%) but its impact on the resulting accuracy of the parameter estimation may still be noticeable.
The energy scale in scintillator-based detectors is in fact substantially non-linear, this non-linearity always producing additional systematic uncertainties which are often rather tricky to parametrize. However the study of this error is very detector-specific, requires extensive Monte-Carlo simulations with real calibration data feedback and considering it at this stage would be too speculative. Although, KamLAND internal studies indicate that the nonlinear energy scale uncertainty is less of an issue than the linear one which we can take into account now, Hanohano or any other future experiment will have to revisit this issue, once the real experimental feedback from the detector becomes available.
The geo-neutrinos yield two more systematic uncertainties:
• Total detectable terrestrial antineutrino flux, conventionally expressed in Terrestrial Neutrino Units (TNU) defined as the number of inverse-beta decay interactions per 10 32 free protons per year.
• The ratio ofν e originating from the 238 U decay chain to those coming from the 232 T h decay chain.
Although geological models do provide some guidelines for the expected geo-neutrino flux and KamLAND was able to produce the first experimental measurement of the flux, these data are of little use for the purpose of future experiments, including Hanohano, because the geo-neutrino estimation precision needed to produce an appreciable advantage over the "agnostic" approach is about one order of magnitude higher than available now.
STATISTICAL ANALYSIS PROCEDURE
Since we've included background and systematics in the analysis, the direct likelihood approach has been chosen over the combination of the matched digital filter and the Fourier transform of the spectrum employed in the earlier publications dedicated to or motivated by Hanohano project [8,27]. This approach facilitates the accommodation of the systematic uncertainties and the background. The likelihood method used here is the unbinned statistical analysis similar to the one employed by KamLAND experiment [4,26,28,29]. Instead of the real experimental data, a series of "experiments" can be simulated as sequences of "events" with energies distributed according to the spectra distorted by different oscillation parameters (including the background). The potential sensitivity is essentially the ability of the data analysis to distinguish between different hypotheses about the oscillation parameter sets.
This study is based on the "rate+shape" likelihood function defined for a real or simulated experiment as:
L( Eν e | η) = e −Nexp Nevents i=1 f (E ī νe | η),(3)
where Eν e = {E 1 νe ...E N νe } are the event energies, N -the number of observed events, N exp ( η) -the expected number of events (given the set of parameters η ≡ {∆m 2 12 , ∆m 2 13 , ∆m 2 23 , θ 12 , θ 13 }), f (E ī νe | η) -ν e energy spectrum normalized to N exp (after the distortion by the set of parameters η). Note that while both Eν e and η are denoted as vectors, these vectors are in different spaces. The Eν e has as many dimensions as the sum of the number ofν e and background events in the experiment, and is fixed for a given experiment. The η lies in the parameter space, its dimensionality being the number of unknown parameters to be fitted, and is variable.
The best fit is obtained by varying the parameter vector η to achieve the maximum value of L or its logarithm, the latter often being more convenient to calculate and handle. After the best fit point η 0 has been found, the general prescription to evaluate the sensitivity to some individual parameter η k is the following:
• Make a small increment (or decrement) ǫ to the η k from the "best fit" point η 0
k : η ′ k = η 0 k + ǫ.
• Find a new point of maximum likelihood by varying all the parameters η except for η k which is kept fixed at η ′ k . The new maximum L ′ over the subspace constrained by the requirement η k = η ′ k is not higher than the global maximum L 0 .
• Repeat the above steps with varying ǫ until the condition log L 0 − logL ′ = Q CL is met for both positive and negative increments of ǫ. The corresponding points η ′low k and η ′high k will limit the confidence range for the k-th parameter.
The value Q CL depends on the confidence level for which the range is to be determined. For an individual parameter variation and the confidence level equal to 1σ, Q 1 = 1 2 . In general, for a CL of nσ, Q n = 1 2 n 2 . When instead of a onedimensional confidence range, a multidimensional confidence region in the parameter subspace is required, the values Q CL will be different but the general procedure will not change. The same is true for the case of discriminating two discrete hypotheses, e.g. between the normal and the inverted neutrino mass hierarchies. More detailed information on the likelihood analysis can be found in [31].
For a simulated experiment, the experimental points Eν e do not exist in the first place and are generated according to some reasonable choice of parameters η. Except for this initial stage, the rest of the analysis is the same as described above. If the initial choice of the parameters to simulate the events is not too far off, this procedure will yield an accurate prediction for the sensitivity of the actual experiment.
Systematic uncertainties are introduced by adding "hidden" parameters to the parameter space and allowing them to vary during the search for the maximum likelihood as well. When some information about these values is available a "penalty" term is subtracted from log(L) to account for the fact that big deviations from the central values of those hidden parameters are unlikely. If the parameters of uncertainty are normally distributed around their central value and if all systematic uncertainty parameters are uncorrelated, the "penalty" term takes on the form:
1 2 NSP j=1 δη 2 kj σ 2 j ,(4)
where N SP is the number of systematic uncertainty parameters, k j is the index of the parameter corresponding to the j-th uncertainty, δη kj is the deviation of the k j -th parameter from its most probable value, and σ j is the value of systematic error ascribed to the j-th uncertainty. When some uncertainties are correlated, the penalty term becomes a more general positive definite quadratic form but for our current study that is not the case.
The full equation for the likelihood logarithm with systematic uncertainties takes on the form:
log L( Eν e | η) = −N exp + Nevents i=1 log(f (E ī νe | η)) + 1 2 NSP j=1 δη 2 kj σ 2 j(5)
In this work, we used the total of nine continuous parameters. These are four neutrino oscillation parameters: sin 2 (2θ 12 ), sin 2 (2θ 13 ), ∆m 2 12 , ∆m 2 13 . Note that for each of the two mass hierarchies, ∆m 2 23 is determined by the two other squared mass differences and has not to be introduced into the parameter space. Five parameters were dedicated to systematic uncertainties: two geo-neutrino parameters and three detector-associated systematic errors as described in the previous section. The geo-neutrino parameters are left unconstrained, which is equivalent to infinite σ in (4). The default values for the systematic errors in "efficiency", energy resolution estimation and energy scale were taken to be 2%, 8% and 1%, respectively, which is reasonably conservative for experiments of this kind. Additionally, two extreme cases were analyzed: the most "optimistic" one -with the corresponding parameters fixed at zero deviations as if they were known exactly, and the most "pessimistic" one -with those three uncertainties left unconstrained as well as geo-neutrino parameters. Although practically impossible, these limiting cases indicate how much sensitivity can be gained by improving the systematics and, conversely, how much would be lost if the systematic errors of the real experiment happen to be worse than expected.
"SOLAR" MIXING ANGLE θ12
The "solar" mixing angle has been fairly well constrained by SNO [3] and KamLAND [4]. Our computations suggest that there is still an opportunity for a significant improvement, though.
This measurement is moderately sensitive to detector-based systematic uncertainties but the terrestrial antineutrino background is much more troublesome (Fig 3). These background decrease the sensitivity by about a factor of two and drives the optimum baseline for this parameter to about 60 km, which conflicts with the goal of measuring θ 13 and ∆m 2 13 in the same experiment as well. Constraining the geo-neutrino flux would change the situation, but currently there seems to be no way of doing that. The geo-neutrino flux measurements made by KamLAND [4,26] are not nearly precise enough to improve the situation noticeably and, besides, not directly applicable to a future experiment located elsewhere, especially in the ocean.
Even with that background and the associated uncertainty, the sensitivity of the medium-baseline experiment is noteworthy. For a 10 KT detector at 50 km from a 6 GWt nuclear plant, an exposure of 300 gigawatt-kiloton-years (5 years) is required to achieve the one-sigma confidence range of 0.01 in sin 2 (2θ 12 ), which is about 5 times better than the current best estimation. At 60 km, the same precision can be achieved with just above one-third of this exposure.
Although higher energy resolution is always better, the θ 12 study does not exhibit appreciable dependence on this parameter and 0.05 × E vis [M eV ] is almost as good as 0.025 × E vis [M eV ]. Of the detector-associated systematics, the most significant is the "efficiency" uncertainty. The measurement of this parameter by KamLAND is more difficult to improve on. For example, the target sensitivity for Hanohano is 0.07 × 10 −5 eV 2 , which would be about three times better than the current best estimation. As our calculations suggest, this can be achieved in 300 gigawatt-kiloton-years at the 60 km baseline or in 450 gigawattkiloton years from 50 km.
Still longer baselines offer better sensitivity for this particular study (Fig 4) but would be clearly sub-optimal for all other oscillation parameters. The part (b) of the plot shows the dramatic effect of terrestrial neutrino background and, particularly, the uncertainty of this background. Without geo-neutinos, the same detector would be four times more efficient at 60 km and seven times at 50 km.
The fact that geo-neutrinos drive the optimum baseline towards longer distance may seem somewhat counterintuitive. The shorter the baseline, the higher the reactorν e rate, so the relative fraction of terrestrialν e background is smaller and should have a smaller effect. However, at shorter baselines, the reactorν e deficit due to oscillations appears mostly in the lower-energy zone where it is harder to separate from the variation in the terrestrial neutrino background.
Like the θ 12 measurement, this study is not demanding of detector energy resolution and not particularly sensitive to detector-associated systematics. 2 12 , as a function of baseline: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed).
MIXING ANGLE θ13
This is a very important oscillation parameter not only because of its theoretical significance but also because its value defines the amplitude of the sub-dominant high-frequency oscillations governed by ∆m 13 and ∆m 23 . Only if θ 13 is not zero (and not too small) is it possible to measure those mass squared differences inν e disappearance experiments. Currently, only an upper bound for this angle is known (from the CHOOZ experiment [30]): sin 2 2θ < 0.1. Several experiments are proposed or already under construction to set better limits and the Hanohano detector can contribute to those efforts.
The sensitivity profiles ( Figure 5) show that medium baselines (above 30 km) are not optimal for this study and much shorter ones are better from the statistical standpoint. However even at 50 km the absolute sensitivity can be quite impressive with a big detector. Except for the longest baselines (60 km and above) which are clearly suboptimal, this study is not severly affected by the geo-neutrino background and its uncertainties. The systematics of the detector itself, however, play a more important role here. At 50 km, the main systematic error is the uncertainty of energy resolution estimation, followed by the "efficiency" error. At shorter baselines the "efficiency" uncertainty dominates.
Although the medium baselines have a strong statistical disadvantage for θ 13 measurement, they also have the compelling feature that systematic uncertainties do not ruin the measurement. Unlike the shorter baseline experiments where relatively more information is obtained through the neutrino event rate, the spectrum shape distortion characteristic of medium baseline is not so easy to imitate by any of the detector systematic errors. This means that, in the long run when even better accuracy for θ 13 is required, medium baseline experiments may prove to be more robust. Figure 6 exhibits another important feature of this measurement: its energy resolution dependence. Although not as critical as for the hierarchy study (below), the effect of detector energy resolution is quite noticeable. Compromising this parameter to 0.05 × E vis [M eV ] from the 0.025 × E vis [M eV ] (as projected for Hanohano) will cost about 2.5 times the exposure. Unlike all previously considered parameters where the potential sensitivity of an experiment could be predicted more or less accurately based just on theν e exposure, this measurement depends on the value of θ 13 which is still unknown. Any quantitative sensitivity prediction makes sense only with some particular value of θ 13 in mind. The larger the mixing angle, the easier it is to determine ∆m 2 13 , ∆m 2 23 and neutrino mass hierarchy. It has been found that the sensitivity scales approximately as the square of sin 2 2θ 13 . In other words, getting the same sensitivity in ∆m 2 13 , ∆m 2 23 and neutrino mass hierarchy if sin 2 2θ 13 = 0.01 will take four times the exposure required if sin 2 2θ 13 = 0.02. In this paper we carried out calculations for two scenarios: sin 2 2θ 13 = 0.05, and sin 2 2θ 13 = 0.025.
Another ambiguity associated with the study of ∆m 2 13 and ∆m 2 23 follows from the closely related question of neutrino mass hierarchy. Depending on the actual value of the mixing angle θ 13 , the neutrino mass hierarchy may turn out unfeasible to establish at an adequate CL. At the same time, within each of the two possible hierarchies, stringent limits on both ∆m 2 13 and ∆m 2 23 may still be set with reasonable exposures. Since the ambiguity is at worst only a two-fold one, it makes sense to estimate the "known-hierarchy" sensitivity to either ∆m 2 13 or ∆m 2 23 . Figure 7 and 8 show that shorter baselines are better for this study, although this trend is not as pronounced as with θ 13 measurement and actually reverses below 25 km. It is clear that ∆m 2 13 study is not systematics-constrained, including the systematics from geo-neutrinos. On the other hand, the dependence on energy resolution for this measurement is even stronger than that for θ 13 (Figure 9 and 10).
NEUTRINO MASS HIERARCHY
Like the ∆m 2 13 /∆m 2 23 measurement, the hierarchy study depends on the actual value of θ 13 . Our calculations suggest that for any θ 13 it takes more statistics to make a high confidence level conclusion about the hierarchy than to measure ∆m 2 13 and ∆m 2 23 to an accuracy of 0.025 × 10 −3 eV 2 . This makes reliable hierarchy determination with a 10 kt detector feasible only if the mixing angle turns out to be quite high (sin 2 2θ 12 ≥ 0.05). The sensitivity dependence on this value is approximately quadratic in the exposure as well. The baseline profile of the sensitivity to the hierarchy is shown in Figure 11. After taking into account the geo-neutrino background and the uncertainties the optimum baseline remains in the same range as was found earlier with less comprehensive models [8,27] 13 within a given mass hierarchy, as a function of detector energy resolution: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dotdashed). 60 km baseline is assumed. more. Of the systematic errors, the most damaging is the geo-neutrino flux uncertainty, although its effect is not as decisive as for the solar parameter studies.
The hierarchy study proves to be the most demanding of the detector energy resolution (Figure 12). Even within the best values for that parameter of the detector achievable today the sensitivity dependence on the energy resolution enters its asymptotic 4-th power curve. In particular, this implies that between two detectors of the same photocathode area but different volumes, the bigger detector will offer inferior sensitivity: all other parameters being equal, a smaller relative photocathode coverage will lead to lower resolution which will prevail over the higher reactorν e statistics.
Theoretically, the mass hierarchy study is secondary to the measurement of ∆m 2 13 and ∆m 2 23 and is determined immediately after those mass differences are found. The analysis of the oscillatedν e energy spectrum yields all mass differences (provided, θ 13 = 0 and θ 12 = π/4). In practice, however,the squared mass differences can be measured with limited accuracy only and at a limited CL. This may not be sufficient to determine the hierarchy. Moreover, it has been found [35]that for any combination of ∆m 2 13 and ∆m 2 23 there exists another one (denoted below as ∆ ′ m 2 13 and ∆ ′ m 2 23 ) that delivers a similar oscillation pattern despite comprising the oppositie mass hierarchy. The similarity is never perfect and, given enough statistics, it is always possible to distinguish between the two spectra but it may take much more exposure to discriminate between those "conjugate" opposite hierarchy solutions than to constrain the squared mass differences within one of the solutions with a remarkable precision.
In Figure 13, the curves provide the measure of relative "unlikeliness" of an alternative hypothesis, assuming normal hierarchy and ∆m 2 13 = 2.4 × 10 −3 eV 2 , for which the experiment was simulated. Zero χ 2 means an indistinguishable hypothesis, the higher its value, the less statistics is needed to discriminate the hypothesis. Solid lines show the normal hierarchy and dotted lines show the inverted one. The dashed vertical lines through the centers of the dotted curves point to the ∆ ′ m 2 13 which combined with the inverted hierarchy provide the closest similarity to the simulated physical spectrum. Comparison of Figure 13(a) and Figure 13(b) explains why the 60 km baseline offers better hierarchy discrimination than 40 km, although the latter yields significantly better sensitivity to ∆m 2 13 /∆m 2 23 within each of the two hierarchies: the vertex of the quasi-parabolic dotted curve is located higher for 60 km.
The "conjugate" ∆ ′ m 2 13 for a given real ∆ ′ m 2 13 is not the same for different baselines, which implies that two measurements at different baselines may offer improved efficiency. For instance, the allowable values of ∆ ′ m 2 13 to which the 60 km baselined measurement is least sensitive are much better excluded at 40 km and vice versa. In case of a land-based detection, a multiple-detector configuration can be considered. Hanohano, additionally, can use the advantage of its movability and make two consecutive exposures instead of one twice as long. Indeed, as the comparison between Figure 13(c) and Figure 13(d) shows, the combination of 60 and 40 km baselined observations should provide a better hierarchy resolution than one twice as long at the practically optimal 50 km. Although the advantage is marginal, we considered only a two-baseline combination with equal exposures. A more systematic optimization with different exposures and possibly more than just two baselines should offer further gains.
CONCLUSION
With a single detector and a 300 kiloton-gigawatt-year exposure and a 5-6 GW thermal powerν e source, both ∆m 2 12 and θ 12 are expected to be measured with the accuracy of 1%, a three to five time improvement on the current best limits [4]. An experiment with a more powerful reactorν e source but the same exposure due to a smaller detector size or a shorter livetime will have a slight advantage because of the relatively smaller geo-neutrino effect.
Detector-associated systematic uncertainties do not appear to be a significant limiting factor in the resulting accuracy. These studies are not particularly sensitive to the detector resolution and 5 − 6% × E vis [M eV ] would be quite sufficient. However the expected sensitivity, especially that for ∆m 2 12 , will be severely handicapped by the presence of geo-neutrinos and in particular by the lack of accurate estimation for the intensity of this background source, effectively turning its value into yet another systematic uncertainty -the dominating one for this study. This is not a "hard" limitation since geo-neutrinos, having a different spectrum, can not mimic the reactorν e oscillaton pattern, but rather an efficiency impairment.
The study of the θ 13 mixing angle is different, in that the medium baselines that this study deals with are not optimal. On a per-event basis, the baseline dependence of sensitivity is rather flat, but short baselines gives quadratically higher event rate and hence lower statistical error. On the other hand, longer baselines offer more robustness with respect [35] This problem had been pointed to in [23] and [32]. Our sumulations confirm it, no matter whether Fourier transform is used in the data analysis or not. Assuming sin 2 2θ13 = 0.05: exposure necessary to discriminate the neutrino mass hierarchies to 66.8% CL, as a function of detector energy resolution: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed). 60 km baseline is assumed. to the "efficiency" systematic uncertainty. Given the careful design of Hanohano or any similar (considering the size and the baseline) detector and its accurate calibration, it is still possible to reach an accuracy of better than 0.02 in sin 2 2θ 13 evaluation, which is competitive with the dedicated experiments like Double Chooz and Daya Bay. The sensitivity to this mixing angle does not exhibit strong dependence on its own real value. In other words, setting the upper limit of 0.02 for sin 2 2θ 13 if the angle happens to be zero takes about as much exposure as setting the range between 0.03 and 0.07 if the value is really 0.05.
The measurement of ∆m 2 13 and ∆m 2 23 and the closely related question of neutrino mass hierarchy are common in their dependence on the actual value of θ 13 . If the angle turns out to be big enough (within currently allowed values), then some spectacular results are possible. If it is zero or very small, nothing interesting can be measured with either Haonhano or any other similar experiment. The necessary exposure is approximately inversely quadratic to the value of sin 2 2θ 13 in both cases, although the hierarchy measurement generally requires much more statistics. For a moderately optimistic scenario in which θ 13 = 0.025, Hanohano or a similar experiment can yield a very good estimation for the values of the squared mass differences but reliable mass hierarchy separation may call for prohibitively long exposures. Another feature that these measurements have in common is the requirement for the excellent energy resolution.
Generally, assuming either normal or inverted hierarchy, the problem of ∆m 2 13 and ∆m 2 23 becomes simpler, less demanding of the energy resolution and with higher chance of success for unfavorably small values of θ 13 . Even if the hierarchy question is not conclusively answered at that stage, the squared mass differences can still be measured, even though the remaining hierarchy ambiguity will split the allowable solutions into two groups. In such a case, the result of the ∆m 2 13 and ∆m 2 23 study will have the form: ∆m 2 13 = ∆ norm ± ∆δ norm ∆m 2 23 = ∆ norm − ∆m 2 12 ± ∆δ norm for normal hierarchy, and ∆m 2 13 = ∆ inv ± ∆δ inv ∆m 2 23 = ∆ inv + ∆m 2 12 ± ∆δ inv for inverted hierarchy, with one solution somewhat more favored over the other (i.e. should the hierarchy discrimination be achieved to marginal confidence levels). Here ∆ norm and ∆ inv are best fit values for the ∆m 2 13 for the normal and inverted scenarios, respectively, the ∆m 2 12 is expected to be found from the same experiment with superior accuracy and ∆δ norm and ∆δ inv are the error bars for both the ∆m 2 13 and ∆m 2 23 in the normal and inverted hierarchy, respectively.
The sensitivity properties for oscillation parameters in a single-baseline experiment are summarized in Table I and II. As was proposed in [33] and discussed in numerous later publications, neutrinos may have non-standard interactions which could affect the flavor content at the source and also the flavor content detected. For our case at hand, it means that the observed mixing angles θ 12 and θ 13 in general will differ from the "true" mixing angles. For example, the measured θ 13 can be larger than the true θ 13 [34]. Since the survival probability (2) depends on the effective θ 13 , the NSI have no adverse effect on the determination of ∆m 2 13 , ∆ 2 23 and the neutrino mass hierarchy. In fact an effective θ 13 larger than the real one will be advantageous for these studies.
At the same time, the effective θ 13 measured at different baselines are going to be different, should these interactions take place. This way, medium baseline experiments targeting θ 13 will become complementary to the short baseline ones in testing new physics.
The two most important qualitative conclusions from this study are the following:
• Medium-baselinedν e oscillation experiments are not systematics-constrained. This follows from the shapes of oscillated spectra. In particular, physically feasible systematic errors do not tend to imitate the spectral distortions characteristic of the neutrino oscillations.
• There is no single baseline optimal for all oscillation studies. The difference in sensitivity profiles is big enough to give an advantage to multiple detector or/and movable detector configurations. Even in individual studies where a pronounced baseline optimum exists, a multiple baseline configuration can outperform a single baseline configuration, as has been shown for the neutrino mass hierarchy discrimination case.
This work was partially supported by the U.S. Department of Energy grant DE-FG02-04ER41291 and the University of Hawaii.
FIG. 1 :FIG. 2 :
12Two neutrino mass hierarchies allowable with currently available data; all three combinations with m2 < m1 are excluded by solar experiments; the m1 < m3 < m2 order is excluded by the fact that ∆m 2 12 ≪ ∆m 2 23 ≈ ∆m2 13 . Typical reactorνe spectrum: non-oscillated (dotted), with θ13 = 0 (dashed), and with θ13 = 0.05 (solid).
"
SOLAR" SQUARED MASS DIFFERENCE ∆m 2 12
FIG. 3 :FIG. 4 :
34Exposure yielding the sensitivity of 0.01 in sin 2 2θ12, as a function of baseline: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed). Exposure yielding the sensitivity of 0.07 × 10 −5 eV 2 in ∆m
FIG. 5 :FIG. 6 :FIG. 7 :FIG. 8 :FIG. 9 :
56789Exposure yielding the sensitivity of 0.02 in sin 2 2θ13, as a function of baseline: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed). Detector energy resolution equal to 0.025 × p Evis[M eV ] is assumed.Energy resolution, sqrt(Evis[MeV]) 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Required exposure, kT*GWtExposure yielding the sensitivity of 0.02 in sin 2 2θ13 from the baseline of 60 km, as a function of the detector energy resolution: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed). Assuming sin 2 2θ13 = 0.05: exposure yielding the sensitivity of 0.025 × 10 −3 eV 2 in ∆m 2 13 within a given mass hierarchy, as a function of baseline: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed). Detector energy resolution equal to 0.025 × p Evis[M eV ] is assumed. Assuming sin 2 2θ13 = 0.025: exposure yielding the sensitivity of 0.025×10 −3 eV 2 in ∆m 2 13 within a given mass hierarchy, as a function of the baseline: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed). Detector energy resolution equal to 0.025 × p Evis[M eV ] is assumed.Energy resolution, sqrt(Evis[MeV])Assuming sin 2 2θ13 = 0.05: exposure yielding the sensitivity of 0.025 × 10 −3 eV 2 in ∆m 2 13 within a given mass hierarchy, as a function of detector energy resolution: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dotdashed). 60 km baseline is assumed.
FIG. 10 :
10Assuming sin 2 2θ13 = 0.025: exposure yielding the sensitivity of 0.025 × 10 −3 eV 2 in ∆m 2
FIG. 11 :
11Assuming sin 2 2θ13 = 0.05: exposure necessary to discriminate the neutrino mass hierarchies to 66.8% CL, as a function of baseline: with unconstrained detector systematics (dotted), with "default" detector systematics (solid), assuming no detector systematics (dashed), assuming no detector systematics and no geo-neutrinos (dot-dashed). Detector energy resolution equal to 0.025 × p Evis[M eV ] is assumed. Energy resolution, sqrt(Evis[MeV]) 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Required exposure, kT*GWt
. 13: χ 2 excess for an alternative hypothesis about the ∆m 2 13 and the mass hierarchy over the "real" one for which the events were simulated (∆m 2 13 = 2.4×10 −3 eV 2 , normal hierarchy). Solid line is for "correct" hierarchy, dotted -for the "wrong" one. The larger the χ 2 excess, the easier the alternative hypothesis to rule out. (a) single detector at 60 km; (b) single detector at 40 km; (c) two half-sized detectors, one at 60 km and another at 40 km; (d) single detector at 50 km.
TABLE I :
IParameter sensitivity properties for θ12, ∆m 2 12 and θ13 with Hanohano or similar detectors.Parameter
θ12
∆m 2
12
θ13
Detector systematics dependence
low
low
high
Geoνe dependence
high
high
low
νe energy resolution dependence
low
low
high
Optimal baseline for single detector, km 60-70
70-80
<20
Expected sensitivity
0.01 0.07 × 10 −5 eV 2 0.02
TABLE II :
IISensitivity properties for ∆m 2 13 (or ∆m2 23 ) and neutrino mass hierarchy.Parameter
∆m 2
13
M. H.
Detector syststematics dependence
low
low
Geoνe dependence
low
avḡ
νe energy resolution dependence
v. high extreme
Dependence on θ13
yes
yes
Optimal baseline for single detector, km <30
50
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| [] |
[
"Contextual Parameter Generation for Universal Neural Machine Translation",
"Contextual Parameter Generation for Universal Neural Machine Translation"
] | [
"Emmanouil Antonios Platanios \nMachine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n\n",
"Mrinmaya Sachan [email protected] \nMachine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n\n",
"Graham Neubig [email protected] \nMachine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n\n",
"Tom M Mitchell [email protected] \nMachine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n\n"
] | [
"Machine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n",
"Machine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n",
"Machine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n",
"Machine Learning Department\n‡ Language Technologies Institute Carnegie Mellon University\n"
] | [
"Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing"
] | We propose a simple modification to existing neural machine translation (NMT) models that enables using a single universal model to translate between multiple languages while allowing for language specific parameterization, and that can also be used for domain adaptation. Our approach requires no changes to the model architecture of a standard NMT system, but instead introduces a new component, the contextual parameter generator (CPG), that generates the parameters of the system (e.g., weights in a neural network). This parameter generator accepts source and target language embeddings as input, and generates the parameters for the encoder and the decoder, respectively. The rest of the model remains unchanged and is shared across all languages. We show how this simple modification enables the system to use monolingual data for training and also perform zero-shot translation. We further show it is able to surpass state-of-theart performance for both the IWSLT-15 and IWSLT-17 datasets and that the learned language embeddings are able to uncover interesting relationships between languages. | 10.18653/v1/d18-1039 | [
"https://www.aclweb.org/anthology/D18-1039.pdf"
] | 52,100,117 | 1808.08493 | a8a863e85a95919773868204d672f1260e0058ce |
Contextual Parameter Generation for Universal Neural Machine Translation
October 31 -November 4. 2018
Emmanouil Antonios Platanios
Machine Learning Department
‡ Language Technologies Institute Carnegie Mellon University
Mrinmaya Sachan [email protected]
Machine Learning Department
‡ Language Technologies Institute Carnegie Mellon University
Graham Neubig [email protected]
Machine Learning Department
‡ Language Technologies Institute Carnegie Mellon University
Tom M Mitchell [email protected]
Machine Learning Department
‡ Language Technologies Institute Carnegie Mellon University
Contextual Parameter Generation for Universal Neural Machine Translation
Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing
the 2018 Conference on Empirical Methods in Natural Language ProcessingBrussels; BelgiumOctober 31 -November 4. 2018425
We propose a simple modification to existing neural machine translation (NMT) models that enables using a single universal model to translate between multiple languages while allowing for language specific parameterization, and that can also be used for domain adaptation. Our approach requires no changes to the model architecture of a standard NMT system, but instead introduces a new component, the contextual parameter generator (CPG), that generates the parameters of the system (e.g., weights in a neural network). This parameter generator accepts source and target language embeddings as input, and generates the parameters for the encoder and the decoder, respectively. The rest of the model remains unchanged and is shared across all languages. We show how this simple modification enables the system to use monolingual data for training and also perform zero-shot translation. We further show it is able to surpass state-of-theart performance for both the IWSLT-15 and IWSLT-17 datasets and that the learned language embeddings are able to uncover interesting relationships between languages.
Introduction
Neural Machine Translation (NMT) directly models the mapping of a source language to a target language without any need for training or tuning any component of the system separately. This has led to a rapid progress in NMT and its successful adoption in many large-scale settings (Wu et al., 2016;Crego et al., 2016). The encoder-decoder abstraction makes it conceptually feasible to build a system that maps any source sentence in any language to a vector representation, and then decodes this representation into any target language. Thus, various approaches have been proposed to extend this abstraction for multilingual MT (Luong et al., 2016;Dong et al., 2015;Johnson et al., 2017;Ha et al., 2016;Firat et al., 2016a).
Prior work in multilingual NMT can be broadly categorized into two paradigms. The first, univer-sal NMT (Johnson et al., 2017;Ha et al., 2016), uses a single model for all languages. Universal NMT lacks any language-specific parameterization, which is an oversimplification and detrimental when we have very different languages and limited training data. As verified by our experiments, the method of Johnson et al. (2017) suffers from high sample complexity and thus underperforms in limited data settings. The universal model proposed by Ha et al. (2016) requires a new coding scheme for the input sentences, which results in large vocabulary sizes that are difficult to scale. The second paradigm, per-language encoder-decoder (Luong et al., 2016;Firat et al., 2016a), uses separate encoders and decoders for each language. This does not allow for sharing of information across languages, which can result in overparameterization and can be detrimental when the languages are similar.
In this paper, we strike a balance between these two approaches, proposing a model that has the ability to learn parameters separately for each language, but also share information between similar languages. We propose using a new contextual parameter generator (CPG) which (a) generalizes all of these methods, and (b) mitigates the aforementioned issues of universal and per-language encoder-decoder systems. It learns language embeddings as a context for translation and uses them to generate the parameters of a shared translation model for all language pairs. Thus, it provides these models the ability to learn parameters separately for each language, but also share information between similar languages. The parameter generator is general and allows any existing NMT model to be enhanced in this way. 1 In addition, it has the following desirable features: 1. Simple: Similar to Johnson et al. (2017) and Ha et al. (2016), and in contrast with Luong et al. (2016) and Firat et al. (2016a), it can be applied to most existing NMT systems with some minor modification, and it is able to accommodate attention layers seamlessly. 2. Multilingual: Enables multilingual translation using the same single model as before. 3. Semi-supervised: Can use monolingual data. 4. Scalable: Reduces the number of parameters by employing extensive, yet controllable, sharing across languages, thus mitigating the need for large amounts of data, as in Johnson et al. (2017). It also allows for the decoupling of languages, avoiding the need for a large shared vocabulary, as in Ha et al. (2016). 5. Adaptable: Can adapt to support new languages, without requiring complete retraining. 6. State-of-the-art: Achieves better performance than pairwise NMT models and Johnson et al. (2017). In fact, our approach can surpass stateof-the-art performance. We first introduce a modular framework that can be used to define and describe most existing NMT systems. Then, in Section 3, we introduce our main contribution, the contextual parameter generator (CPG), in terms of that framework. We also argue that the proposed approach takes us a step closer to a common universal interlingua.
Background
We first define the multi-lingual NMT setting and then introduce a modular framework that can be used to define and describe most existing NMT systems. This will help us distill previous contributions and introduce ours.
Setting. We assume that we have a set of source languages S and a set of target languages T . The total number of languages is L = |S ∪ T |. We also assume we have a set of C ≤ |S| × |T | pairwise parallel corpora, {P 1 , . . . , P C }, each of which contains a set of sentence pairs for a single source-target language combination. The goal of multilingual NMT is to build a model that, when trained using the provided parallel corpora, can learn to translate well between any pair of languages in S×T . The majority of related work only considers pairwise NMT, where |S| = |T | = 1.
NMT Modules
Most NMT systems can be decomposed to the following modules (also visualized in Figure 1).
Preprocessing Pipeline. The data preprocessing pipeline handles tokenization, cleaning, normalizing the text data and building a vocabulary, i.e. a two-way mapping from preprocessed sentences to sequences of word indices that will be used for the translation. A commonly used proposal for defining the vocabulary is the byte-pair encoding (BPE) algorithm which generates subword unit vocabularies (Sennrich et al., 2016b). This eliminates the notion of out-of-vocabulary words, often resulting in increased translation quality.
Encoder/Decoder. The encoder takes in indexed source language sentences, and produces an intermediate representation that can later be used by a decoder to generate sentences in a target language. Generally, we can think of the encoder as a function, f (enc) , parameterized by θ (enc) . Similarly, we can think of the decoder as another function, f (dec) , parameterized by θ (dec) . The goal of learning to translate can then be defined as finding the values for θ (enc) and θ (dec) that result in the best translations. A large amount of previous work proposes novel designs for the encoder/decoder module. For example, using attention over the input sequence while decoding (Bahdanau et al., 2015;Luong et al., 2015) provides significant gains in translation performance. 2 Parameter Generator. All modules defined so far have previously been used when describing NMT systems and are thus easy to conceptualize. However, in previous work, most models are trained for a given language pair, and it is not trivial to extend them to work for multiple pairs of languages. We introduce here the concept of the parameter generator, which makes it easy to define and describe multilingual NMT systems. This module is responsible for generating θ (enc) and θ (dec) for any given source and target language. Different parameter generators result in different numbers of learnable parameters and can thus be used to share information across different languages. Next, we describe related work, in terms of the parameter generator for NMT:
• Pairwise: In the simple and commonly used pairwise NMT setting (Wu et al., 2016;Crego et al., 2016), the parameter generator would generate separate parameters, θ (enc) and θ (dec) , for each pair of source-target languages. This re- Dong et al. (2015), Luong et al. (2016) and Firat et al. (2016a), the parameter generator would generate separate encoder parameters, θ (enc) , for each source language, and separate decoder parameters, θ (dec) , for each target language. This leads to a reduction in the number of learnable parameters for multilingual NMT, from O(ST ) to O(S +T ). On one hand, Dong et al. (2015) train multiple models as a one-to-many multilingual NMT system that translates from one source language to multiple target languages. On the other hand, Luong et al. (2016) and Firat et al. (2016a) perform many-to-many translation. Luong et al. (2016), however, only report results for a single language pair and do not attempt multilingual translation. Firat et al. (2016a) propose an attention mechanism that is shared across all language pairs. We generalize the idea of multiway multilingual NMT with the parameter generator network, described later. • Universal: In the case of Ha et al. (2016) and Johnson et al. (2017), the authors propose using a single common set of encoder-decoder parameters for all language pairs. While Ha et al. (2016) embed words in a common semantic space across languages, Johnson et al. (2017) learn language embeddings that are in the same space as the word embeddings. Here, the parameter generator would provide the same parameters θ (enc) and θ (dec) for all language pairs. It would also create and keep track of learnable variables representing language embeddings that are prepended to the encoder input sequence. As we observed in our experiments, this system fails to perform well when the training data is limited. Finally, we believe that embedding languages in the same space as words is not intuitive; in our approach, languages are embedded in a separate space.
In contrast to all these related systems, we provide a simple, efficient, yet effective alternativea parameter generator for multilingual NMT, that enables semi-supervised and zero-shot learning. We also learn language embeddings, similar to Johnson et al. (2017), but in our case they are separate from the word embeddings and are treated as a context for the translation, in a sense that will become clear in the next section. This notion of context is used to define parameter sharing across various encoders and decoders, and, as we discuss in our conclusion, is even applicable beyond NMT.
Proposed Method
We propose a new way to share information across different languages and to control the amount of sharing, through the parameter generator module. More specifically, we propose contextual parameter generators.
Contextual Parameter Generator. Let us denote the source language for a given sentence pair by s and the target language by t . Then, when using the contextual parameter generator, the parameters of the encoder are defined as θ (enc) g (enc) (l s ), for some function g (enc) , where l s denotes a language embedding for the source language s . Similarly, the parameters of the decoder are defined as θ (dec) g (dec) (l t ) for some function g (dec) , where l t denotes a language embedding for the target language t . Our generic formulation does not impose any constraints on the functional form of g (enc) and g (dec) . In this case, we can think of the source language, s , as a context for the encoder. The parameters of the encoder depend on its context, but its architecture is common across all contexts. We can make a similar argument for the decoder, and that is where the name of this parameter generator comes from. We can even go a step further and have a parameter generator that defines θ (enc) g (enc) (l s , l t ) and θ (dec) g (dec) (l s , l t ), thus coupling the encoding and decoding stages for a given language pair. In our experiments we stick to the previous, decoupled, form, because unlike Johnson et al. (2017), it has the potential to lead to an interlingua.
Concretely, because the encoding and decoding stages are decoupled, the encoder is not aware of the target language while generating it. Thus, we can take an encoded intermediate representation of a sentence and translate it to any target language. This is because, in this case, the intermediate representation is independent of any target language. This makes for a stronger argument that the intermediate representation produced by our encoder could be approaching a universal interlingua, more so than methods that are aware of the target language when they perform encoding.
Parameter Generator Network
We refer to the functions g (enc) and g (dec) as parameter generator networks. Even though our proposed NMT framework does not rely on a specific choice for g (enc) and g (dec) , here we describe the functional form we used for our experiments. Our goal is to provide a simple form that works, and for which we can reason about. For this reason, we decided to define the parameter generator networks as simple linear transforms, similar to the factored adaptation model of Michel and Neubig (2018), which was only applied to the bias terms of the output softmax:
g (enc) (l s ) W (enc) l s ,(1)g (dec) (l t ) W (dec) l t ,(2)
where l s , l t ∈ R M , W (enc) ∈ R P (enc) ×M , W (dec) ∈ R P (dec) ×M , M is the language embedding size, P (enc) is the number of parameters of the encoder, and P (dec) is the number of parameters of the decoder. Another way to interpret this model is that it imposes a low-rank constraint on the parameters. As opposed to our approach, in the base case of using multiple pairwise models to perform multilingual translation, each model has P = P (enc) + P (dec) learnable parameters for its encoder and decoder. Given that the models are pairwise, for L languages, we have a total of L(L − 1) learnable parameter vectors of size P . On the other hand, using our contextual parameter generator we have a total of L vectors of size M (one for each language), and a single matrix of size P × M . Then, the parameters of the encoder and the decoder, for a single language pair, are defined as a linear combination of the M columns of that matrix.
Controlled Parameter Sharing. We can further control parameter sharing by observing that the encoder/decoder parameters often have some "natural grouping". For example, in the case of recurrent neural networks we may have multiple weight matrices, one for each layer, as well as attentionrelated parameters. Based on this observations, we now propose a way to control how much information is shared across languages. The language embeddings need to represent all of the languagespecific information and thus may need to be large in size. However, when computing the parameters of each group, only a small part of that information is relevant. Let
θ (enc) = {θ (enc) j } G j=1 and θ (enc) j ∈ R P (enc) j ,
where G denotes the number of groups. Then, we define:
θ (enc) j W (enc) j P (enc) j l s ,(3)
where W (enc) j ∈ R P (enc) j ×M and P (enc) j ∈ R M ×M , with M < M (and similarly for the decoder parameters). We can see now that P (enc) j is used to extract the relevant information (size M ) for parameter group j, from the larger language embedding (size M ). This allows us to control the parameter sharing across languages in the following way: if we want to increase the number of per-language parameters (i.e., the language embedding size) we can increase M while keeping M small enough so that the total number of parameters does not explode. This would not have been possible without the proposed low-rank ap-proximation for W (enc) , that uses the parameter grouping information.
Alternative Options. Given that our proposed approach does not depend on the specific choice of the parameter generator network, it might be interesting to design models that use side-information about the languages that are being used (such as linguistic information about language families and hierarchies). This is outside the scope of this paper, but may be an interesting future direction.
Semi-Supervised and Zero-Shot Learning
The proposed parameter generator also enables semi-supervised learning via back-translation. Concretely, monolingual data can be used to train the shared encoder/decoder networks to translate a sentence from some language to itself (similar to the idea of auto-encoders by Vincent et al. (2008)). This is possible and can help learning because of the fact that many of the learnable parameters are shared across languages.
Furthermore, zero-shot translation, where the model translates between language pairs for which it has seen no explicit training data, is also possible. This is because the same per-language parameters are used to translate to and from a given language, irrespective of the language at the other end. Therefore, as long as we train our model using some language pairs that involve a given language, it is possible to learn to translate in any direction involving that language.
Potential for Adaptation
Let us assume that we have trained a model using data for some set of languages, 1 , 2 , . . . , m . If we obtain data for some new language n , we do not have to retrain the whole model from scratch. In fact, we can fix the parameters that are shared across all languages and only learn the embedding for the new language (along with the relevant word embeddings if not using a shared vocabulary). Assuming that we had a sufficient number of languages in the beginning, this may allow us to obtain reasonable translation performance for the new language, with a minimal amount of training. 3
Number of Parameters
For the base case of using multiple pairwise models to perform multilingual translation, each model has P + 2W V parameters, where P = P (enc) + P (dec) , W is the word embedding size, and V is the vocabulary size per language (assumed to be the same across languages, without loss of generality). Given that the models are pairwise, for L languages, we have a total of L(L − 1)(P + 2W V ) = O(L 2 P + 2L 2 W V ) learnable parameters. For our approach, using the linear parameter generator network presented in Section 3.1, we have a total of O(P M + LW V ) learnable parameters. Note that the number of encoder/decoder parameters has no dependence on L now, meaning that our model can easily scale to a large number of languages.
Experiments
In this section, we describe our experimental setup along with our results and key observations.
Setup. For all our experiments we use as the base NMT model an encoder-decoder network which uses a bidirectional LSTM for the encoder, and a two-layer LSTM with the attention model of Bahdanau et al. (2015) for the decoder. The word embedding size is set to 512. This is a common baseline model that achieves reasonable performance and we decided to use it as-is, without tuning any of its parameters, as extensive hyperparameter search is outside the scope of this paper.
During training, we use a label smoothing factor of 0.1 (Wu et al., 2016) and the AMSGrad optimizer (Reddi et al., 2018) with its default parameters in TensorFlow, and a batch size of 128 (due to GPU memory constraints). Optimization was stopped when the validation set BLEU score was maximized. The order in which language pairs are used while training was as follows: we always first sample a language pair (uniformly at random), and then sample a batch for that pair (uniformly at random). 4 During inference, we employ beam search with a beam size of 10 and the length normalization scheme of (Wu et al., 2016). We want to emphasize that we did not run experiments with other architectures or configurations, and thus this architecture was not chosen because it was favorable to our method, but rather because it was a frequently mentioned baseline in existing literature.
All experiments were run on a machine with a single Nvidia V100 GPU, and 24 GBs of system memory. Our most expensive experiment took about 10 hours to complete, which would cost about $25 on a cloud computing service such as Google Cloud or Amazon Web Services, thus making our results reproducible, even by independent researchers.
Experimental Settings. The goal of our experiments is to show how, by using a simple modification of this model, (i) we can achieve significant improvements in performance, while at the same time (ii) being more data and computation efficient, and (iii) enabling support for zero-shot translation. To that end, we perform three types of experiments:
1. Supervised: In this experiment, we use full parallel corpora to train our models. Plain pairwise NMT models (PNMT) are compared to the same models modified to use our proposed decoupled parameter generator. We use two variants: (i) one which does not use autoencoding of monolingual data while training (CPG*), and (ii) one which does (CPG). Please refer to Section 3.2 for more details. 2. Low-Resource: Similar to the supervised experiments except that we limit the size of the parallel corpora used in training. However, for GML and CPG the full monolingual corpus is used for auto-encoding training. 3. Zero-Shot: In this experiment, our goal is to evaluate how well a model can learn to translate between language pairs that it has not seen while training. For example, a model trained using parallel corpora between English and German, and English and French, will be evaluated in translating from German to French. PNMT can perform zero-shot translation in this setting using pivoting. This means that, in the previous example, we would first translate from German to English and then from English to French (using two pairwise models for a single translation). However, pivoting is prone to error propagation incurred when chaining multiple imperfect translations. The proposed CPG models inherently support zero-shot translation and require no pivoting.
For the experiments using the CPG model without controlled parameter sharing, we use language embeddings of size 8. This is based merely on the fact that this is the largest model size we could fit on one GPU. Whenever possible, we compare against PNMT, GML by Johnson et al. (2017), 5 and other state-of-the-art results.
Datasets. We use the following datasets:
• IWSLT-15: Used for supervised and lowresource experiments only (this dataset does not support zero-shot learning). We report results for Czech (Ch), English (En), French (Fr), German (De), Thai (Th), and Vietnamese (Vi). This dataset contains~90,000-220,000 training sentence pairs (depending on the language pair), 500-900 validation pairs, and~1,000-1,300 test pairs. • IWSLT-17: Used for supervised and zero-shot experiments. We report results for Dutch (Nl), English (En), German (De), Italian (It), and Romanian (Ro). This dataset contains~220,000 5 We use our own implementation of GML in order to obtain a fair comparison, in terms of the whole MT pipeline. We have modified it to use the same per-language vocabularies that we use for our approaches, as the proposed shared BPE vocabulary fails to perform well for the considered datasets. training sentence pairs (for all language pairs except for the zero-shot ones),~900 validation pairs, and~1,100 test pairs. Data Preprocessing. We preprocess our data using a modified version of the Moses tokenizer (Koehn et al., 2007) that correctly handles escaped HTML characters. We also perform some Unicode character normalization and cleaning. While training, we only consider sentences up to length 50. For both datasets, we generate a per-language vocabulary consisting of the most frequently occurring words, while ignoring words that appear less than 5 times in the whole corpus, and capping the vocabulary size to 20,000 words.
Results.
Our results for the IWSLT-15 experiments are shown in Table 1. It is clear that our approach consistently outperforms both the corresponding pairwise model and GML. Furthermore, its advantage grows larger in the low-resource setting (up to 5.06 BLEU score difference, or a 2.4× increase), which is expected due to the extensive parameter sharing in our model. For this dataset, there exist some additional published state-of-the-art results not shown in Tables 1 and 2. Huang et al. (2018) report a BLEU score of 28.07 for the En)Vi task, while our model is able to achieve a score of 29.03. Furthermore, Ha et al. (2016) report a BLEU score of 25.87 for the En)De task, while our model is able to achieve a score of 26.77. 6 Our results for the IWSLT-17 experiments are shown in Table 2. 7 Again, our method consistently outperforms both PNMT and GML, in both the supervised and the zero-shot settings. Furthermore, the results indicate that our model performance is robust to different sizes of the language embeddings and the choice of M for controllable parameter sharing. It only underperforms in the degenerate case where M = 1. It is also worth noting that, in the fully supervised setting, GML, the current state-of-the-art in the multilingual setting, underperforms the pairwise models.
The presented results provide evidence that our proposed approach is able to significantly improve performance, without requiring extensive tuning.
Language Embeddings. An important aspect of our model is that it learns language embeddings. In Figure 2 we show pairwise cosine distances between the learned language embeddings for our fully supervised experiments. There are some interesting patterns that indicate that the learned language embeddings are reasonable. For example, we observe that German (De) and Dutch (Nl) are most similar for the IWSLT-17 dataset, with Italian (It) and Romanian (Ro) coming second. Furthermore, Romanian and German are the furthest apart for that dataset. These relationships agree with linguistic knowledge about these languages and the families they belong to. We see similar patterns in the IWSLT-15 results but we focus on IWSLT-17 here, because it is a larger, better quality, dataset with more supervised language pairs. These results are encouraging for analyzing such embeddings to discover relationships between languages that were previously unknown. For example, perhaps surprisingly, French (Fr) and Vietnamese (Vi) appear to be significantly related for the IWSLT-15 dataset results. This is likely due to French influence in Vietnamese because to the occupation of Vietnam by France during the 19 th and 20 th centuries (Marr, 1981). 6 We were unable to find reported state-of-the-art results for the rest of the language pairs. 7 Note that, our results for IWSLT-17 are not comparable to those of the official challenge report (Cettolo et al., 2017), as we use less training data, a smaller baseline model, and our evaluation pipeline potentially differs. However, the numbers presented for all methods in this paper are comparable, as they were all obtained using the same baseline model and evaluation pipeline.
Implementation and Reproducibility
Along with this paper we are releasing an implementation of our approach and experiments as part of a new Scala framework for machine translation. 8 It is built on top of TensorFlow Scala (Platanios, 2018) and follows the modular NMT design (described in Section 2.1) that supports various NMT models, including our baselines (e.g., Johnson et al. (2017)). It also contains data loading and preprocessing pipelines that support multiple datasets and languages, and is more efficient than other packages (e.g., tf-nmt 9 ). Furthermore, the framework supports various vocabularies, among which we provide a new implementation for the byte-pair encoding (BPE) algorithm (Sennrich et al., 2016b) that is 2 to 3 orders of magnitude faster than the released one. 10 All experiments presented in this paper were performed using version 0.1.0 of the framework.
Related Work
Interlingual translation (Richens, 1958) has been the object of many research efforts. For a long time, before the move to NMT, most practical machine translation systems only focused on individual language pairs. Since the success of end-to-end NMT approaches such as the encoderdecoder framework (Sutskever et al., 2014;Bahdanau et al., 2015;Cho et al., 2014), recent work has tried to extend the framework to multi-lingual translation. An early approach was Dong et al. (2015) who performed one-to-many translation with a separate attention mechanism for each decoder. Luong et al. (2016) extended this idea with a focus on multi-task learning and multiple encoders and decoders, operating in a single shared vector space. The same architecture is used in (Caglayan et al., 2016) for translation across multiple modalities. Zoph and Knight (2016) flipped this idea with a many-to-one translation model, however requiring the presence of a multi-way parallel corpus between all the languages, which is difficult to obtain. Lee et al. (2017) used a single character-level encoder across multiple languages by training a model on a many-to-one translation task. Closest to our work are more recent approaches, already described in Section 2 (Firat et al., 2016a;Johnson et al., 2017;Ha et al., 2016), that attempt to enforce different kinds of parameter sharing across languages. Parameter sharing in multilingual NMT naturally enables semi-supervised and zero-shot learning. Unsupervised learning has been previously explored with key ideas such as back-translation (Sennrich et al., 2016a), dual learning (He et al., 2016), common latent space learning (Lample et al., 2018), etc. In the vein of multilingual NMT, Artetxe et al. (2018) proposed a model that uses a shared encoder and multiple decoders with a focus on unsupervised translation. The entire system uses cross-lingual embeddings and is trained to reconstruct its input using only monolingual data. Zero-shot translation was first attempted in (Firat et al., 2016b) who performed zero-zhot translation using their pre-trained multi-way multilingual model, fine-tuning it with pseudo-parallel data generated by the model itself. This was recently extended using a teacher-student framework . Later, zero-shot translation without any additional steps was attempted in (Johnson et al., 2017) using their shared encoderdecoder network. An iterative training procedure that leverages the duality of translations directly generated by the system for zero-shot learning was proposed by Lakew et al. (2017). For extremely low resource languages, Gu et al. (2018) proposed sharing lexical and sentence-level representations across multiple source languages with a single target language. Closely related is the work of Cheng et al. (2016) who proposed the joint training of source-to-pivot and pivot-to-target NMT models. Ha et al. (2018) are probably the first to introduce a similar idea to that of having one network (called a hypernetwork) generate the parameters of another. However, in that work, the input to the hypernetwork are structural features of the original network (e.g., layer size and index). Al-Shedivat et al. (2017) also propose a related method where a neural network generates the parameters of a linear model. Their focus is mostly on interpretabil-ity (i.e., knowing which features the network considers important). However, to our knowledge, there is no previous work which proposes having a network generate the parameters of another deep neural network (e.g., a recurrent neural network), using some well-defined context based on the input data. This context, in our case, is the language of the input sentences to the translation model, along with the target translation language.
Conclusion and Future Directions
We have presented here a novel contextual parameter generation approach to neural machine translation. Our resulting system, which outperforms other state-of-the-art systems, uses a standard pairwise encoder-decoder architecture. However, it differs from earlier approaches by incorporating a component that generates the parameters to be used by the encoder and the decoder for the current sentence, based on the source and target languages, respectively. We refer to this novel component as the contextual parameter generator. The benefit of this approach is that it dramatically improves the ratio of the number of parameters to be learned, to the number of training examples available, by leveraging shared structure across different languages. Thus, our approach does not require any extra machinery such as backtranslation, dual learning, pivoting, or multilingual word embeddings. It rather relies on the simple idea of treating language as a context within which to encode/decode. We also showed that the proposed approach is able to achieve state-of-theart performance without requiring any tuning. Finally, we performed a basic analysis of the learned language embeddings, which showed that cosine distances between the learned language embeddings reflect well known similarities among language pairs such as German and Dutch.
In the future, we want to extend the concept of the contextual parameter generator to more general settings, such as translating between different modalities of data (e.g., image captioning). Furthermore, based on the discussion of Section 3.3, we hope to develop an adaptable, never-ending learning (Mitchell et al., 2018) NMT system.
Figure 2 :
2Pairwise cosine distance for all language pairs in the IWSLT-15 and IWSLT-17 datasets. Darker colors represent more similar languages.
Thank you very much. Per-Language: In the case ofTrainable variables
Legend
Computed values (tensors)
L Language embedding size
W Word embedding size
P Number of parameters
text Example input
Intermediate
representa�on
that can be
decoded to any
target language
Encoder parameters
(e.g., LSTM weights)
Decoder parameters
(e.g., LSTM weights)
Embeddings
Embedding
Embedding
SOURCE
LANGUAGE
TARGET
LANGUAGE
ATTENTION
L
L
W
ENGLISH
VIETNAMESE
Vocabulary lookup
Cám ơn rất nhiều.
Encoder
Decoder
Decoupled
Pairwise
BLUE TITLES INDICATE
DIFFERENT OPTIONS
Coupled
SOURCE/TARGET
SOURCE
Parameters do not depend on
the language embeddings. They
are learnable variables.
This represents the typical
pairwise NMT se�ng where the
parameters are different for
each language pair.
TARGET
P
P
P
Parameter Generator
Figure 1: Overview of an NMT system, under our modular framework. Our main contribution lies in the parameter generator
module (i.e., coupled or decoupled -each of the boxes with blue titles is a separate option). Note that g denotes a parameter
generator network. In our experiments, we consider linear forms for this network. However, our contribution does not depend
on the choices made regarding the rest of the modules; we could still use our parameter generator with different architectures
for the encoder and the decoder, as well as using different kinds of vocabularies.
sults in no parameter sharing across languages,
and thus O(ST ) parameters.
•
Table 1 :
1Comparison of our proposed approach (shaded rows) with the base pairwise NMT model (PNMT) and the Google multilingual NMT model (GML) for the IWSLT-15 dataset. The Percent Parallel row shows what portion of the parallel corpus is used while training; the rest is being used only as monolingual data. Results are shown for the BLEU and Meteor metrics. CPG* represents the same model as CPG, but trained without using auto-encoding training examples. The best score in each case is shown in bold.BLEU
Meteor
PNMT
GML
CPG*
CPG
PNMT
GML
CPG*
CPG
En )Cs
14.89
15.92
16.88
17.22
19.72
20.93
21.51
21.72
Cs )En
24.43
25.25
26.44
27.37
27.29
27.46
28.16
28.52
En )De
25.99
15.92
26.41
26.77
44.72
42.97
45.97
46.30
De )En
30.93
29.60
31.24
31.77
30.73
29.90
30.95
31.13
En )Fr
38.25
34.40
38.10
38.32
57.43
53.86
57.42
57.68
Fr )En
37.40
35.14
37.11
37.89
34.83
33.14
34.34
34.89
En )Th
23.62
22.22
26.03
26.33
-
-
-
-
Th )En
15.54
14.03
16.54
26.77
21.58
21.02
22.78
23.05
En )Vi
27.47
25.54
28.33
29.03
-
-
-
-
Vi )En
24.03
23.19
25.91
26.38
27.59
26.96
28.23
28.79
100% Parallel Data
Mean
26.26
24.12
27.30
27.80
32.98
32.03
33.67
34.01
En )Cs
5.71
8.18
8.40
9.49
12.18
14.97
15.25
15.90
Cs )En
6.64
14.56
14.81
15.38
13.02
20.04
19.98
20.87
En )De
11.70
14.60
15.09
16.03
29.98
33.74
34.88
36.19
De )En
18.10
19.02
19.77
20.25
22.57
23.27
23.65
24.40
En )Fr
24.47
25.15
24.00
25.79
44.10
44.84
44.95
46.22
Fr )En
23.79
25.02
24.55
27.12
26.28
26.61
26.20
28.18
En )Th
7.86
15.58
18.41
17.65
-
-
-
-
Th )En
7.13
9.11
10.19
10.14
13.91
16.32
16.78
16.92
En )Vi
18.01
17.51
18.92
18.90
-
-
-
-
Vi )En
6.69
16.00
16.28
16.86
13.39
21.01
21.34
22.28
10% Parallel Data
Mean
13.01
16.47
17.04
17.76
21.93
25.10
25.38
26.37
En )Cs
0.49
1.25
1.57
2.38
4.60
6.24
6.28
8.38
Cs )En
1.10
1.76
1.87
4.60
6.29
7.13
7.08
11.15
En )De
1.22
4.13
4.06
6.46
12.23
18.29
17.61
23.83
De )En
1.46
3.42
3.86
7.49
7.58
8.79
8.95
13.73
En )Fr
2.88
7.74
7.41
12.45
13.88
21.29
21.80
30.36
Fr )En
4.05
5.22
5.06
11.39
9.58
9.86
9.83
16.34
En )Th
1.22
5.72
8.01
9.26
-
-
-
-
Th )En
1.42
1.66
1.65
3.37
6.08
7.22
5.89
8.74
En )Vi
5.35
5.61
5.48
8.00
-
-
-
-
Vi )En
2.01
3.57
3.64
6.43
7.86
8.76
8.48
12.04
1% Parallel Data
Mean
2.12
4.01
4.26
7.18
8.51
10.95
10.74
15.58
Table 2 :
2Comparison of our proposed approach (shaded rows) with the base pairwise NMT model (PNMT) and the Google multilingual NMT model (GML) for the IWSLT-17 dataset. Results are shown for the BLEU metric only because Meteor does not support It, Nl, and Ro. CPG 8 represents CPG using language embeddings of size 8. The "C4" subscript represents the low-rank version of CPG for controlled parameter sharing (see Section 3.1), using rank 4, etc. The best score in each case is shown in bold.BLEU
In fact, it could likely be applied in other scenarios, such as domain adaptation, as well.
Note that depending on the vocabulary that is used and on whether it is one shared vocabulary across all languages, or one vocabulary per language, the output projection layer of the decoder (which produces probabilities over words) may be language dependent, or common across all languages. In our experiments, we used separate vocabularies and thus this layer was language-dependent.
This is due to the small number of parameters that need to be learned in this case. To put this into perspective, in most of our experiments we used language embeddings of size 8.
We did not observe any "forgetting" effect, because we keep "re-visiting" all language pairs throughout training. It would be interesting to explore other sampling schemes, but it is outside the scope of this paper.
https://github.com/eaplatanios/symphony-mt 9 https://github.com/tensorflow/nmt 10 https://github.com/rsennrich/subword-nmt
AcknowledgmentsWe would like to thank Otilia Stretcu, Abulhair Saparov, and Maruan Al-Shedivat for the useful feedback they provided in early versions of this paper. This research was supported in part by AFOSR under grant FA95501710218.
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Extreme Adaptation for Personalized Neural Machine Translation. Paul Michel, Graham Neubig, Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics. the 56th Annual Meeting of the Association for Computational LinguisticsShort PapersPaul Michel and Graham Neubig. 2018. Extreme Adaptation for Personalized Neural Machine Trans- lation. In Proceedings of the 56th Annual Meet- ing of the Association for Computational Linguistics (Short Papers), pages 312-318.
Never-Ending Learning. Tom Mitchell, William Cohen, Estevam Hruschka, Partha Talukdar, Bishan Yang, Justin Betteridge, Andrew Carlson, Bhavana Dalvi, Matt Gardner, Bryan Kisiel, Jayant Krishnamurthy, Ni Lao, Kathryn Mazaitis, Thahir P Mohamed, Ndapa Nakashole, A Emmanouil, Alan Platanios, Mehdi Ritter, Burr Samadi, Richard Settles, Derry Wang, Abhinav Wijaya, Xinlei Gupta, Abulhair Chen, Malcolm Saparov, Joel Greaves, Welling, 10.1145/3191513Communications of the ACM. 615Tom Mitchell, William Cohen, Estevam Hruschka, Partha Talukdar, Bishan Yang, Justin Betteridge, Andrew Carlson, Bhavana Dalvi, Matt Gard- ner, Bryan Kisiel, Jayant Krishnamurthy, Ni Lao, Kathryn Mazaitis, Thahir P. Mohamed, Ndapa Nakashole, Emmanouil A. Platanios, Alan Ritter, Mehdi Samadi, Burr Settles, Richard Wang, Derry Wijaya, Abhinav Gupta, Xinlei Chen, Abulhair Saparov, Malcolm Greaves, and Joel Welling. 2018. Never-Ending Learning. Communications of the ACM, 61(5):103-115.
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On the Convergence of Adam and Beyond. J Sashank, Satyen Reddi, Sanjiv Kale, Kumar, International Conference on Learning Representations. Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar. 2018. On the Convergence of Adam and Beyond. In International Conference on Learning Represen- tations.
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Improving Neural Machine Translation Models with Monolingual Data. Rico Sennrich, Barry Haddow, Alexandra Birch, Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics. the 54th Annual Meeting of the Association for Computational LinguisticsRico Sennrich, Barry Haddow, and Alexandra Birch. 2016a. Improving Neural Machine Translation Models with Monolingual Data. In Proceedings of the 54th Annual Meeting of the Association for Com- putational Linguistics, pages 86-96.
Neural Machine Translation of Rare Words with Subword Units. Rico Sennrich, Barry Haddow, Alexandra Birch, Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics. the 54th Annual Meeting of the Association for Computational LinguisticsRico Sennrich, Barry Haddow, and Alexandra Birch. 2016b. Neural Machine Translation of Rare Words with Subword Units. In Proceedings of the 54th An- nual Meeting of the Association for Computational Linguistics, pages 1715-1725.
Sequence to Sequence Learning with Neural Networks. Ilya Sutskever, Oriol Vinyals, V Quoc, Le, Advances in Neural Information Processing Systems. Ilya Sutskever, Oriol Vinyals, and Quoc V. Le. 2014. Sequence to Sequence Learning with Neural Net- works. In Advances in Neural Information Process- ing Systems, pages 3104-3112.
Extracting and Composing Robust Features with Denoising Autoencoders. Pascal Vincent, Hugo Larochelle, Yoshua Bengio, Pierre-Antoine Manzagol, http:/doi.acm.org/10.1145/1390156.1390294Proceedings of the 25th International Conference on Machine Learning. the 25th International Conference on Machine LearningPascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. 2008. Extracting and Composing Robust Features with Denoising Au- toencoders. In Proceedings of the 25th Inter- national Conference on Machine Learning, pages 1096-1103.
. Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, Jeff Klingner, Apurva Shah, Melvin Johnson, Xiaobing Liu, Łukasz Kaiser, Stephan Gouws, Yoshikiyo Kato, Taku Kudo, Hideto Kazawa, Keith Stevens, George Kurian, Nishant Patil, Wei Wang, Oriol Vinyals. Greg Corrado, Macduff Hughesand Jeffrey Dean. 2016. Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation. CoRR, abs/1609.08144Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V. Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, Jeff Klingner, Apurva Shah, Melvin Johnson, Xiaobing Liu, Łukasz Kaiser, Stephan Gouws, Yoshikiyo Kato, Taku Kudo, Hideto Kazawa, Keith Stevens, George Kurian, Nishant Patil, Wei Wang, Cliff Young, Jason Smith, Jason Riesa, Alex Rudnick, Oriol Vinyals, Greg Corrado, Macduff Hughes, and Jeffrey Dean. 2016. Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation. CoRR, abs/1609.08144.
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[
"Mass-density and Phonon-frequency Relaxation Dynamics of Water and Ice at Cooling",
"Mass-density and Phonon-frequency Relaxation Dynamics of Water and Ice at Cooling"
] | [
"Chang Q Sun \nFaculty of Materials and Optoelectronics and Physics\nKey Laboratory of Low-Dimensional Materials and Application Technologies\nXiangtan University\n411105HunanChina\n\nSchool of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore\n",
"Xi Zhang \nSchool of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore\n",
"Xiaojian Fu \nDepartment of Materials Science and Engineering\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n\nCollege of Materials Science and Engineering\nChina Jiliang University\n310018HangzhouChina\n",
"Weitao Zheng \nSchool of Materials Science\nJilin University\n130012ChangchunChina\n",
"Jer-Lai Kuo \nInstitute of Atomic and Molecular Sciences\nAcademia Sinica\n10617TaipeiTaiwan\n",
"Yichun Zhou \nFaculty of Materials and Optoelectronics and Physics\nKey Laboratory of Low-Dimensional Materials and Application Technologies\nXiangtan University\n411105HunanChina\n",
"Zexiang Shen \nSchool of Physics\nNanyang Technological University\n639798Singapore\n",
"Ji Zhou \nDepartment of Materials Science and Engineering\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n"
] | [
"Faculty of Materials and Optoelectronics and Physics\nKey Laboratory of Low-Dimensional Materials and Application Technologies\nXiangtan University\n411105HunanChina",
"School of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore",
"School of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore",
"Department of Materials Science and Engineering\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina",
"College of Materials Science and Engineering\nChina Jiliang University\n310018HangzhouChina",
"School of Materials Science\nJilin University\n130012ChangchunChina",
"Institute of Atomic and Molecular Sciences\nAcademia Sinica\n10617TaipeiTaiwan",
"Faculty of Materials and Optoelectronics and Physics\nKey Laboratory of Low-Dimensional Materials and Application Technologies\nXiangtan University\n411105HunanChina",
"School of Physics\nNanyang Technological University\n639798Singapore",
"Department of Materials Science and Engineering\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina"
] | [] | Coulomb repulsion between the bonding electron pair in the H-O covalent bond (denoted by"-") and the nonbonding electron pair of O (":") and the specific-heat disparity between the O:H and the H-O segments of the entire hydrogen bond (O:H-O) are shown to determine the O:H-O bond angle-length-stiffness relaxation dynamics and the density anomalies of water and ice. The bonding part with relatively lower specific-heat is more easily activated by cooling, which serves as the "master" and contracts, while forcing the "slave" with higher specific-heat to elongate (via Coulomb repulsion) by different amounts. In the liquid and solid phases, the O:H van der Waals bond serves as the master and becomes significantly shorter and stiffer while the H-O bond becomes slightly longer and softer (phonon frequency is a measure of bond stiffness), resulting in an O:H-O cooling contraction and the seemingly "regular" process of cooling densification. In the water-ice transition phase, the master and the slave swap roles, thus resulting in an O:H-O elongation and volume expansion during freezing. In ice, the O-O distance is longer than it is in water, resulting in a lower density, so that ice floats. PACS numbers: 61.20.Ja, 61.30.Hn, 68.08.Bc Supplementary Information and a molecular dynamics movie are accompanied.The anomalous behavior of the density of water as it transitions to ice and its associated hydrogen bond (defined as the entire O:H-O) angle-length-stiffness cooling relaxation dynamics continue to baffle the field, despite the intensive investigations carried out in the past decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Established database[11]shows that both the liquid and the solid H 2 O undergoes the seemingly regular process of cooling densification at different rates. At the water-ice transition phase, volume expansion takes place and results in ice with a density 9% lower than the maximal density of water at 277 K [1,11]. The cooling densification is associated with a redshift of the high-frequency H-O phonons ω H (∼3000 cm −1 )[17,18]while the freezing expansion is accompanied with a blueshift of the ω H [19]. However, the understanding of the fundamental nature underlying the observed mass-density and phonon-frequency transition dynamics and their correlation still remains incomplete. Numerous models have been developed for explaining water's expansion upon freezing or at supercooling state. The elegant models include the thermal fluctuations in the mono-phase of tetrahedrally-coordinated structures[4,6]and the mixed-phase of low-and high-density fragments with thermal modulation of the fragmental ratios[9,20]. Little attention has been paid to the mechanism for the seemingly regular process of cooling den-sification in both the liquid and the solid phase. The Raman shift of the low-frequency O:H (ω L ∼ 200 cm −1 ) phonons in various phases has not yet been systematically characterized. Therefore, there is a great need for a consistent framework to understand the density and phonon-frequency transitions and the associated hydrogen bond angle-length-stiffness relaxation dynamics of H 2 O through its full set of states.In this letter, we show that we have been able to reconcile the measured mass-density [1,11]and Ramanfrequency transitions of water/ice based on the framework of O:H-O bond specific-heat disparity, Raman sectroscopy measurements, and molecular dynamics (MD) calculations of the hydrogen bond angle-length-stiffness relaxationin of water/ice over the full temperature range. The often overlooked Coulomb repulsion between the electron pair in the H-O covalent bond and the nonbonding electron lone pair of oxygen [21] and the specific-heat disparity of the O:H-O bond are shown to be the key to resolving the density puzzles.We consider the basic structural unit of O δ− :H δ+ -O δ−[21,22](also denoted as "O· · ·H-O") to represent the O δ− -O δ− interactions in H 2 O, except for H 2 O under extremely high pressures and temperatures[23]. The fraction δ represents the polarity of the H δ+ -O δ− polarcovalent bond. InFig.1a, the pair of dots on the left | 10.1021/jz401380p | [
"https://arxiv.org/pdf/1210.1634v3.pdf"
] | 118,655,142 | 1210.1634 | 7763e95b9a5a9a6e6f707e635be64605283943fb |
Mass-density and Phonon-frequency Relaxation Dynamics of Water and Ice at Cooling
2 Apr 2013
Chang Q Sun
Faculty of Materials and Optoelectronics and Physics
Key Laboratory of Low-Dimensional Materials and Application Technologies
Xiangtan University
411105HunanChina
School of Electrical and Electronic Engineering
Nanyang Technological University
639798Singapore
Xi Zhang
School of Electrical and Electronic Engineering
Nanyang Technological University
639798Singapore
Xiaojian Fu
Department of Materials Science and Engineering
State Key Laboratory of New Ceramics and Fine Processing
Tsinghua University
100084BeijingChina
College of Materials Science and Engineering
China Jiliang University
310018HangzhouChina
Weitao Zheng
School of Materials Science
Jilin University
130012ChangchunChina
Jer-Lai Kuo
Institute of Atomic and Molecular Sciences
Academia Sinica
10617TaipeiTaiwan
Yichun Zhou
Faculty of Materials and Optoelectronics and Physics
Key Laboratory of Low-Dimensional Materials and Application Technologies
Xiangtan University
411105HunanChina
Zexiang Shen
School of Physics
Nanyang Technological University
639798Singapore
Ji Zhou
Department of Materials Science and Engineering
State Key Laboratory of New Ceramics and Fine Processing
Tsinghua University
100084BeijingChina
Mass-density and Phonon-frequency Relaxation Dynamics of Water and Ice at Cooling
2 Apr 2013(Dated: May 3, 2014)
Coulomb repulsion between the bonding electron pair in the H-O covalent bond (denoted by"-") and the nonbonding electron pair of O (":") and the specific-heat disparity between the O:H and the H-O segments of the entire hydrogen bond (O:H-O) are shown to determine the O:H-O bond angle-length-stiffness relaxation dynamics and the density anomalies of water and ice. The bonding part with relatively lower specific-heat is more easily activated by cooling, which serves as the "master" and contracts, while forcing the "slave" with higher specific-heat to elongate (via Coulomb repulsion) by different amounts. In the liquid and solid phases, the O:H van der Waals bond serves as the master and becomes significantly shorter and stiffer while the H-O bond becomes slightly longer and softer (phonon frequency is a measure of bond stiffness), resulting in an O:H-O cooling contraction and the seemingly "regular" process of cooling densification. In the water-ice transition phase, the master and the slave swap roles, thus resulting in an O:H-O elongation and volume expansion during freezing. In ice, the O-O distance is longer than it is in water, resulting in a lower density, so that ice floats. PACS numbers: 61.20.Ja, 61.30.Hn, 68.08.Bc Supplementary Information and a molecular dynamics movie are accompanied.The anomalous behavior of the density of water as it transitions to ice and its associated hydrogen bond (defined as the entire O:H-O) angle-length-stiffness cooling relaxation dynamics continue to baffle the field, despite the intensive investigations carried out in the past decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Established database[11]shows that both the liquid and the solid H 2 O undergoes the seemingly regular process of cooling densification at different rates. At the water-ice transition phase, volume expansion takes place and results in ice with a density 9% lower than the maximal density of water at 277 K [1,11]. The cooling densification is associated with a redshift of the high-frequency H-O phonons ω H (∼3000 cm −1 )[17,18]while the freezing expansion is accompanied with a blueshift of the ω H [19]. However, the understanding of the fundamental nature underlying the observed mass-density and phonon-frequency transition dynamics and their correlation still remains incomplete. Numerous models have been developed for explaining water's expansion upon freezing or at supercooling state. The elegant models include the thermal fluctuations in the mono-phase of tetrahedrally-coordinated structures[4,6]and the mixed-phase of low-and high-density fragments with thermal modulation of the fragmental ratios[9,20]. Little attention has been paid to the mechanism for the seemingly regular process of cooling den-sification in both the liquid and the solid phase. The Raman shift of the low-frequency O:H (ω L ∼ 200 cm −1 ) phonons in various phases has not yet been systematically characterized. Therefore, there is a great need for a consistent framework to understand the density and phonon-frequency transitions and the associated hydrogen bond angle-length-stiffness relaxation dynamics of H 2 O through its full set of states.In this letter, we show that we have been able to reconcile the measured mass-density [1,11]and Ramanfrequency transitions of water/ice based on the framework of O:H-O bond specific-heat disparity, Raman sectroscopy measurements, and molecular dynamics (MD) calculations of the hydrogen bond angle-length-stiffness relaxationin of water/ice over the full temperature range. The often overlooked Coulomb repulsion between the electron pair in the H-O covalent bond and the nonbonding electron lone pair of oxygen [21] and the specific-heat disparity of the O:H-O bond are shown to be the key to resolving the density puzzles.We consider the basic structural unit of O δ− :H δ+ -O δ−[21,22](also denoted as "O· · ·H-O") to represent the O δ− -O δ− interactions in H 2 O, except for H 2 O under extremely high pressures and temperatures[23]. The fraction δ represents the polarity of the H δ+ -O δ− polarcovalent bond. InFig.1a, the pair of dots on the left
bond angle-length-stiffness relaxation dynamics and the density anomalies of water and ice. The bonding part with relatively lower specific-heat is more easily activated by cooling, which serves as the "master" and contracts, while forcing the "slave" with higher specific-heat to elongate (via Coulomb repulsion) by different amounts. In the liquid and solid phases, the O:H van der Waals bond serves as the master and becomes significantly shorter and stiffer while the H-O bond becomes slightly longer and softer (phonon frequency is a measure of bond stiffness), resulting in an O:H-O cooling contraction and the seemingly "regular" process of cooling densification. In the water-ice transition phase, the master and the slave swap roles, thus resulting in an O:H-O elongation and volume expansion during freezing. In ice, the O-O distance is longer than it is in water, resulting in a lower density, so that ice floats. The anomalous behavior of the density of water as it transitions to ice and its associated hydrogen bond (defined as the entire O:H-O) angle-length-stiffness cooling relaxation dynamics continue to baffle the field, despite the intensive investigations carried out in the past decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Established database [11] shows that both the liquid and the solid H 2 O undergoes the seemingly regular process of cooling densification at different rates. At the water-ice transition phase, volume expansion takes place and results in ice with a density 9% lower than the maximal density of water at 277 K [1,11]. The cooling densification is associated with a redshift of the high-frequency H-O phonons ω H (∼3000 cm −1 ) [17,18] while the freezing expansion is accompanied with a blueshift of the ω H [19]. However, the understanding of the fundamental nature underlying the observed mass-density and phonon-frequency transition dynamics and their correlation still remains incomplete. Numerous models have been developed for explaining water's expansion upon freezing or at supercooling state. The elegant models include the thermal fluctuations in the mono-phase of tetrahedrally-coordinated structures [4,6] and the mixed-phase of low-and high-density fragments with thermal modulation of the fragmental ratios [9,20]. Little attention has been paid to the mechanism for the seemingly regular process of cooling den-sification in both the liquid and the solid phase. The Raman shift of the low-frequency O:H (ω L ∼ 200 cm −1 ) phonons in various phases has not yet been systematically characterized. Therefore, there is a great need for a consistent framework to understand the density and phonon-frequency transitions and the associated hydrogen bond angle-length-stiffness relaxation dynamics of H 2 O through its full set of states.
In this letter, we show that we have been able to reconcile the measured mass-density [1,11] and Ramanfrequency transitions of water/ice based on the framework of O:H-O bond specific-heat disparity, Raman sectroscopy measurements, and molecular dynamics (MD) calculations of the hydrogen bond angle-length-stiffness relaxationin of water/ice over the full temperature range. The often overlooked Coulomb repulsion between the electron pair in the H-O covalent bond and the nonbonding electron lone pair of oxygen [21] and the specific-heat disparity of the O:H-O bond are shown to be the key to resolving the density puzzles.
We consider the basic structural unit of O δ− :H δ+ -O δ− [21,22] (also denoted as "O· · ·H-O") to represent the O δ− -O δ− interactions in H 2 O, except for H 2 O under extremely high pressures and temperatures [23]. The fraction δ represents the polarity of the H δ+ -O δ− polarcovalent bond. In Fig.1a, the pair of dots on the left represents the nonbonding lone pair ":" and the pair on the O atom on the right represents the bonding electron pair "-". The H atom serves as the point of reference in the O:H-O system. The lone pair on the left belongs to the sp 3 -orbit hybridized oxygen and the bonding pair is shared by the H-O and centred at sites close to oxygen. For completeness, we define the entire hydrogen bond to be O:H-O, the intra-molecular polar-covalent bond as the H-O bond and the inter-molecular van der Waals (vdW) bond as the O:H bond hereon. (Table I), the specific heat ηL of O:H rises faster towards the asymptotic maximum value than the ηH . Such a specific-heat decomposition results in three regions that correspond, respectively, to the phases of liquid (I), solid (III), and liquid-solid transition (II) with different specific-heat ratios. R is the gas constant. (The ηL in the solid phase differs from the ηL in the liquid, which does not influence the validity of the hypothesis).
As illustrated in Fig.1a, a hydrogen bond is comprised of the O:H bond (broken lines) and the H-O bond rather than either of them alone. The H-O bond is much shorter, stronger, and stiffer than the O:H bond (com-parison shown in Table I). The O:H bond breaks at the evaporating point (T v ) of water (373 K) [17]. However, the H-O bond is much harder to break as the bond energy of ≈4.0 eV [1] is twice that of the C-C bond in diamond (1.84 eV) [24].
Combined with the forces of the Coulomb repulsion, f q ∝(d O−O ) −2 , and resistance to deformation, f rx (x = H for the H-O and L for the O:H bond), the force of cooling contraction, f dx drives these two segments to relax in the same direction but by different amounts. The k x , which varies nonlinearly with the d x , approximates the second derivative of the inter-atomic potential at equilibrium. The magnitude of the f dx varies with the specific heat of the specific bond in a particular temperature range. The Coulomb repulsion between the electron pairs, as represented by the pairs of dots in Fig.1a, is the key to the O:H-O bond relaxation under excitation [1,21].
Generally, the specific heat is regarded as a macroscopic quantity integrated over all bonds, and is the amount of energy required to raise the temperature of the substance by 1 K. However, in dealing with the representative bond of the entire specimen, one has to consider the specific heat per bond that is obtained by dividing the bulk specific heat by the total number of bonds involved. In the case of the O:H-O bond, we need to consider the specific heat (η x ) characteristics of the two segments separately (see Fig.1b) because of the difference in their strengths. The slope of the specific-heat curve is determined by the Debye temperature (Θ Dx ) while the integration of the specific-heat curve from 0 K to the T mx [27] is determined by the cohesive energy of the bond energy E x . The specific-heat curve of the segment with a relatively lower Θ Dx value will rise to the maximum value faster than the other segment. The Θ Dx , which is lower than the T mx , is proportional to the characteristic frequency of the vibration (ω x ) of the segment. Thus, we have the following relations (see Table I):
ΘDL ΘDH ≈ 198 ΘDH ≈ ωL ωH ≈ 200 3000 ≈ 1 15 T mH 0 ηH dt T mL 0 ηLdt ≈ EH EL ≈ 4.0 0.1 ≈ 40
(1)
Based on this consideration, the maximal specific-heat ratio is estimated to be ηH ηL ≈8. Such a specific-heat disparity between the O:H and the H-O segments creates three temperature regions with different ηL ηH ratios, which should correspond to the phases of liquid (I), solid (III), and liquid-solid transition (II).
The consistency in the number of regions (i.e. phases I, II, III) of the proposed specific-heat curve (Fig.1b), the mass-density transition [11], and the O:H-O bond angle-length-stiffness relaxation dynamics ( Fig. 2 and Fig. 3) suggest that the segment with a relatively lower specific heat is thermally more active than the other segment. This thermally active segment serves as the "master" that undergoes cooling contraction while forcing the [25,26], stiffness ωx (vibration frequency), melting point Tmx, and the inter-atomic and inter-electron-pair interactions of the O:H-O bond compared with those of the C-C bond in a diamond [24].
Segment (x) dx (nm) Ex (eV) ωx (cm −1 ) ΘDx (K) Tmx (K)(η H < η L ) : (η L < η H ) : (η H = η L ) : f dH > (f dL + f rL + f rH ) f dL > (f dH + f rL + f rH ) f dH = f dL ⇒ ∆d O−O ⇒ (∆V ) 1/3 < > = 0(2)
Because of the strength difference of the two segments [21], iii) At the crossing points (Fig.1b), η H =η L and f dH =f dL . There is a transition between O-O expansion and contraction, corresponding to the density maximum at 277 K and the density minimum below the freezing point [11,12]. iv) Meanwhile, the repulsion increases the O:H-O angle θ and polarizes the electron pairs during relaxation. It has been shown that a segment increases in stiffness as it becomes shorter, while the opposite occurs as it elongates [21,29]. The Raman shift, which is proportional to the square root of bond stiffness, approximates the length and strength change of the bond during relaxation directly. Approximating the vibration energy of a vibration system to the Taylor series of the inter-atomic potential energy, u x (r), leads to:
∆ω x ∝ ∂ 2 u x (r) µ∂r 2 r=dx 1/2 ∝ E x /µ d x ∝ Y x d x (3)
The stiffness is the product of the Young's modulus (Y x ∝E x /d 3 x ) and the length of the segment in question [21]. The µ is the reduced mass of the vibrating dimer. Therefore, the Raman shift is a measure of the segmental stiffness.
In order to verify our hypotheses and predictions regarding the O:H-O bond angle-length-stiffness change, the specific-heat disparity, and the density and phononfrequency anomalies of water/ice, we have conducted Raman measurements and MD calculations as a function of temperature. Two MD computational methods were used in examining the mono-and the mixed-phase models. Details of the experimental and calculation proce- This mechanism differs completely from the mechanism conventionally adopted for the standard cooling densification of other regular materials in which only one kind of chemical bond is involved. In other materials, cooling shortens and stiffens all the inter-atomic bonds [28]. In contrast, in the transition phase II [11,13,14], the master and the slave swap roles. The O:H bond elongates more than the H-O bond shortens so that a net gain in the O-O length occurs. The θ angle widening (Fig.2b) could also contribute to the volume change. In the liquid phase I, the mean θ valued at 160 • remains almost constant. However, the snapshots of the MD trajectory in Fig.2c and the MD movie in the attached [30] show that the V-shaped H-O-H molecules remain intact at 300 K over the entire duration recorded, accompanied by high fluctuations in the θ and the d L in this regime, which indicates the dominance of tetrahedrally-coordinated water molecules [31]. In region II, cooling widens the θ from 160 • to 167 • , which contributes a maximum of +1.75% to the O:H-O bond elongation and 5.25% to volume expansion. In phase III, the θ increases from 167 • to 174 • and this trend results in a maximal value of -2.76% to the volume contraction.
The calculated temperature dependence of the O-O distance as shown in Fig.2d matches satisfactorily with that of the measured density profile [1,11]. Importantly, the O-O distance is longer in ice than it is in water, and therefore, ice floats.
The measured Raman spectra in Fig.3 show three regions: T>273 K (I), 273 ≥ T ≥ 258 K (II), and T<258 K (III), which are in agreement with the MD calculations [1] and our predictions: i) At T>273 K (I), abrupt shifts of the ω L from 75 to The MD-movie [30] shows that in the liquid phase, the H δ+ and the O δ− attract each other between the H δ+ :O δ− but the O δ− -O δ− repulsion prevents this occurrence. The intact O-H-O motifs are moving restlessly because of the high fluctuations and frequent switching of the H δ+ :O δ− interactions. Furthermore, the coupled cooling ω L blueshift and ω H redshift provide further evidence for the persistence of the Coulomb repulsion between the bonding and the nonbonding electron pairs in liquid. The presence of the electron lone pair results from the sp 3 -orbit hybridization of oxygen that tends to form tetrahedral bonds with its neighbors [1,31]. Therefore, the H 2 O in the bulk form of liquid could possesses the tetrahedrally-coordinated structures with thermal fluctuation [6,31,32]. Snapshots of the MD trajectory in [1] revealed little discrepancy between the mono-and the mixed-phase structural models.
The proposed mechanisms for: i) the seemingly regular processes of cooling densification of the liquid and the solid H 2 O, ii) the abnormal freezing expansion, iii) the floating of ice, and, iv) the three-region O:H-O bond angle-length-stiffness relaxation dynamics of water and ice have been justified. Agreement between our calculations and the measured mass-density [11] and phononfrequency relaxation dynamics in the temperature range of interest has verified our hypotheses and predictions: i) Inter-electron-pair Coulomb repulsion and the segmental specific-heat disparity of the O:H-O bond determine the change in its angle, length and stiffness and the density and the phonon-frequency anomalies of water ice.
ii) The segment with a relatively lower specific-heat contracts and drives the O:H-O bond cooling relaxation. The softer O:H bond always relaxes more in length than the stiffer H-O bond does in the same direction. The cooling widening of the O:H-O angle contributes positively to the volume expansion at freezing. iii) In the liquid and the solid phase, the O:H bond contracts more than the H-O bond elongates, resulting in the cooling densification of water and ice, which is completely different from the process experienced by other regular materials. iv) In the freezing transition phase, the H-O bond contracts less than the O:H bond lengthens, resulting in expansion during freezing. v) The O-O distance is larger in ice than it is in water, and therefore, ice floats. vi) The segment increases in stiffness as it shortens, while the opposite occurs as it elongates. The density variation of water ice is correlated to the incoporative O:H and H-O phonon-frequency relaxaion dynamics.
Special thanks to Phillip Ball, Yi Sun, Buddhudu Srinivasa, and John Colligon for their comments and expertise. Financial support received from NSF (Nos.: 21273191, 1033003, and 90922025) China is gratefully acknowledged.
Coulomb repulsion between the bonding electron pair in the H-O covalent bond (denoted by"-") and the nonbonding electron pair of O (":") and the specific-heat disparity between the O:H and the H-O segments of the entire hydrogen bond (O:H-O) are shown to determine the O:H-O
PACS numbers: 61.20.Ja, 61.30.Hn, 68.08.Bc Supplementary Information and a molecular dynamics movie are accompanied.
a, b) Forces and (c) the respective specific heats of the two segments in the O:H-O bond. Combined with the Coulomb repulsion fq between the electron pairs (pairs of dots in a) and the resistance to deformation frx (x = H for the H-O and L for the O:H segment), the cooling contraction force f dx drives the two segments to relax in the same direction but by different amounts. (c) Because of the difference in their Debye temperatures
the length of the softer O:H bond always relaxes more than that of the stiffer H-O bond: |∆d L |>|∆d H |. The two segments relax in the same direction because of the repulsion. Thus, we expect the O:H-O bond to relax in the following manners during cooling: i) In the transition phase II, η H <η L and f dH ≫f dL . The H-O bond contraction dominates. The stiffer H-O bond contracts less than the O:H bond elongates, resulting in ∆d O−O =∆d L − ∆d H >0. Therefore, a net O-O length gain and an accompanying volume expansion (∆V > 0) takes place. ii) In the liquid I and the solid III phase, η L <η H and f dL ≫f dH . The master and the slave swap roles. The softer O:H bond contracts significantly more than the H-O bond elongates, ∆d O−O =∆d H − ∆d L <0. Hence, a net O-O contraction results in a gain in the mass density.
dures are described in the supplementary information [1]. Fig.2 shows the MD-derived change of (a) the H-O bond and the O:H bond length, (b) the O:H-O bond angle θ, (c) the snapshots of the MD trajectory, and (d) the O-O distance as a function of temperature. As shown in Fig.2a, the shortening of the master segments (denoted with arrows) is always coupled with a lengthening of the slaves during cooling. The temperature range of interest consists of three regions: in the liquid region I and the solid region III, the O:H bond contracts significantly more than the H-O bond elongates, resulting in a net loss of the O-O length. Thus, cooling-driven densification of H 2 O happens in both the liquid and the solid phase.
FIG. 2 :
2(a) Segmental length change of the O:H-O bond in the phases of liquid (I), solid (III), and liquid-solid transition (II). Arrows denote the cooling contraction of the master segments, which are coupled with the expansion of the slaves. ∆T = T -Tmax with Tmax = 277 K is the maximal density temperature. (b) O:H-O bond angle widening driven by cooling also exhibits three regions. (c) Snapshots of the MD trajectory show that the V-shaped H-O-H molecues remain intact at 300 K because of the robustness of the H-O bond (∼4.0 eV/bond) with pronounced quantum fluctuations in the angle and in the dL in liquid phase. (d) The change of the O-O distance agrees with the measured three-region water and ice densities [1, 11]. In ice, the O-O distance is longer than that in water, which results in ice floating.
TABLE I :
ISummary of the segmental length dx, strength Ex (energy)[1], Debye temperature ΘDx
When the temperature drops from 209 to 30 K the ω H shift from 3253 to 3218 cm −1 . For clusters of 5 nm size or smaller, the ω H shifts by an addition of 40 cm −1 . iii) At 273∼258 K (II), the situation reverses. Cooling shifts the ω H from 3150 to 3170 cm −1 and the ω L from 220 to 215 cm −1 , see the shaded areas. Agreeing with the Raman ω H shift measured in the temperature range between 270 and 273 K [1, 19], the coupling of the ω H blueshift and the ω L redshift confirms the exchange in the master and the slave role of the O:H and the H-O bond during freezing.(a)
50 100 150 200 250 300 350 400
278
288
298
Intensity (a.u.)
98K
108
118
128
138
148
158
168
178
188
198
208
218
228
238
248
258
268
273
(cm
-1
(b)
3000
3200
3400
3600
288
298
273
278
Intensity )a.u.)
98K
108
118
128
138
148
158
168
178
188
198
208
218
228
238
248
258
268
(cm
-1
)
FIG. 3: Temperature dependent Raman shifts of (a) ωL (the
O:H phonon) and (b) ωH show three regions: T> 273 K (I),
273 K<T<258 K, and T<258 K (III) .
220 cm −1 and the ω H from 3200 to 3150 cm −1 indicate
ice formation. The coupled ω L blueshift and ω H redshift
indicate that cooling shortens and stiffens the O:H bond
but lengthens and softens the H-O bond in the liquid
phase, which confirms the predicted master role of the
O:H bond.
ii) At T<258 K (III), the trend of phonon relaxation
remains the same as it is in the region of T > 273 K
despite a change in the relaxation rates. Cooling from
258 K stiffens the ω L from 215 to 230 cm −1 and softens
the ω H from 3170 to 3100 cm −1 . Other supplementary
peaks at ∼300 and ∼3400 cm −1 are found to be insignif-
icant. The cooling softening of the ω H mode agrees with
that measured using IR spectroscopy [33] of ice clusters
of 8∼150 nm sizes.
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| [] |
[
"Locally anisotropic momentum distributions of hadrons at freeze-out in relativistic heavy-ion collisions",
"Locally anisotropic momentum distributions of hadrons at freeze-out in relativistic heavy-ion collisions"
] | [
"Maciej Rybczyński \nInstitute of Physics\nJan Kochanowski University\nPL-25406KielcePoland\n",
"Wojciech Florkowski \nInstitute of Physics\nJan Kochanowski University\nPL-25406KielcePoland\n\nThe H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nPL-31342KrakówPoland\n"
] | [
"Institute of Physics\nJan Kochanowski University\nPL-25406KielcePoland",
"Institute of Physics\nJan Kochanowski University\nPL-25406KielcePoland",
"The H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nPL-31342KrakówPoland"
] | [] | A spheroidal anisotropic local momentum distribution is implemented in the statistical model of hadron production. We show that this form leads to exactly the same ratios of hadronic abundances as the equilibrium distributions, if the temperature is identified with a characteristic transversemomentum scale. Moreover, to a very good approximation the transverse-momentum spectra of hadrons are the same for isotropic and anisotropic systems, provided the size of the system at freeze-out is appropriately adjusted. We further show that this invariance may be used to improve the agreement between the model and experimental HBT results. | 10.1088/0954-3899/40/2/025103 | [
"https://arxiv.org/pdf/1206.6587v1.pdf"
] | 119,180,584 | 1206.6587 | b89fb23b738d688afc46e61ec7b4f0ff2956cc46 |
Locally anisotropic momentum distributions of hadrons at freeze-out in relativistic heavy-ion collisions
28 Jun 2012
Maciej Rybczyński
Institute of Physics
Jan Kochanowski University
PL-25406KielcePoland
Wojciech Florkowski
Institute of Physics
Jan Kochanowski University
PL-25406KielcePoland
The H. Niewodniczański Institute of Nuclear Physics
Polish Academy of Sciences
PL-31342KrakówPoland
Locally anisotropic momentum distributions of hadrons at freeze-out in relativistic heavy-ion collisions
28 Jun 2012(Dated: June 28, 2012)numbers: 2575-q2575Dw2575Ld Keywords: relativistic heavy-ion collisionsparticle spectrafemtoscopyLHC
A spheroidal anisotropic local momentum distribution is implemented in the statistical model of hadron production. We show that this form leads to exactly the same ratios of hadronic abundances as the equilibrium distributions, if the temperature is identified with a characteristic transversemomentum scale. Moreover, to a very good approximation the transverse-momentum spectra of hadrons are the same for isotropic and anisotropic systems, provided the size of the system at freeze-out is appropriately adjusted. We further show that this invariance may be used to improve the agreement between the model and experimental HBT results.
I. INTRODUCTION
Thermal analyses of hadronic yields in heavy-ion collisions have become a popular tool for the interpretation of the data collected at the AGS [10][11][12], SPS [13][14][15][16][17][18][19], and RHIC energies [20][21][22][23][24][25]. Most often, the successful results of such analyses are understood as the evidence for high-level thermalization of the hadronic systems produced in such collisions.
If the single-freeze-out scenario is assumed [22][23][24], in addition to the hadron yields, the thermal approach may be used to describe other physical observables. In particular, it has been successfully used to reproduce the RHIC transverse momentum spectra [26,27], collective flow [25,28,29], femtoscopic observables [30][31][32], and charge balance functions [33,34].
In the thermal approach, one usually obtains very good fits with just a few thermodynamic parameters, such as the temperature, T , the baryon and strangeness chemical potentials, µ B and µ S , or in the extended approaches the quark fugacities, γ s and γ q , and additional parameters describing the system's geometry and expansion (flow).
Recently, the single-freeze-out model has been applied to the LHC data describing transverse-momentum spectra of hadrons produced in Pb+Pb collisions at the beam energy √ s NN = 2.76 TeV [35]. It has been shown that the model reproduces well the spectra of pions, kaons, and hyperons, however, it does not reproduce the proton spectra. This results agrees with an earlier finding that the ratios of all hadron abundances measured at the LHC, except for protons, may be described within the thermal model [36].
In this paper we show that the agreement of the thermal models with the data does not necessarily imply the * Electronic address: [email protected] † Electronic address: [email protected] fact that the system is locally equilibrated. We take into account possible differences in the local distribution of transverse and longitudinal momenta (with the directions defined with respect to the beam axis). Using the covariant Romatschke-Strickland form (RSF) for the anisotropic phase-space momentum distributions, we show that i) the ratios of hadron multiplicities are exactly the same as in the locally equilibrated system, if the temperature T is identified with a characteristic transversemomentum scale Λ, ii) the transverse-momentum spectra of hadrons are changed, to a very good approximation, only by an overall factor which depends on the momentum anisotropy parameter appearing in RFS -since this change may be easily compensated by a change of the parameters defining the size of the system, the transversemomentum spectra may be also treated as insensitive to the discussed momentum anisotropy, finally, iii) the discussed invariance of the spectra may be used to improve the model HBT results for pions.
Generally speaking, our results demonstrate robustness of the thermal framework against specific variations of the model assumptions. This and other similar studies (see, for example, [37]) show that the thermal approach is quite successful even in the situations where matter is out of equilibrium. Our results concerning the ratios of hadronic abundances agree qualitatively with the result of Ref. [38], where an approximate invariance of the ratios has been demonstrated with slightly different physical assumptions.
We stress that we consider the anisotropy between longitudinal and transverse momenta considered in the local rest frame of the fluid element. These two directions are defined always with respect to the beam axis. We do not consider the anisotropy of the transverse-momentum distributions in the (p x , p y ) plane, quantified by the harmonic flow coefficients v n .
For simplicity, in the discussion of the spectra we consider boost-invariant and cylindrically symmetric models that have been implemented in the Monte-Carlo gener-ator THERMINATOR [42,43]. We compare the results of the model calculations with the LHC data for Pb+Pb collisions at the beam energy √ s NN = 2.76 TeV.
II. COOPER-FRYE FORMALISM AND COVARIANT ROMATSCHKE-STRICKLAND FORM
In this paper we assume that primordial hadrons are emitted from the freeze-out hypersurface Σ and their spectra may be calculated with the help of the Cooper-Frye formula
E p dN d 3 p = dΣ µ (x)p µ f (x, p),(1)
where E p is the particle's energy, dΣ µ is the element of the freeze-out hypersurface, and f (x, p) is the phasespace distribution function of the emitted particles. For systems in local thermal equilibrium one assumes Fermi-Dirac (ǫ = −1) or Bose-Einstein (ǫ = +1) statistics and uses the distribution functions
f eq (x, p) = g exp p · U − µ T − ǫ −1 (2)
where g is the number of internal degrees of freedom, T is the freeze-out temperature, and µ is the chemical potential. Typically, the freeze-out conditions are defined by the fixed values of T and µ, hence these two quantities may be treated as constants in (2). The equilibrium distributions depend on the scalar product of the particle four-momentum p µ and the fourvelocity U µ of the fluid element. We use the standard parameterizations
p µ = E p , p ⊥ , p = (m ⊥ cosh y, p ⊥ , m ⊥ sinh y) (3) and U µ = γ(1, v ⊥ , v ),(4)where γ = (1 − v 2 ) −1/2 .
In order to study the effects of the anisotropic distributions, we modify Eq. (2) and use [39][40][41]
f RSF (x, p) = g exp (p · U ) 2 + ξ(p · V ) 2 Λ − ǫ −1 .
(5) Here ξ is the anisotropy parameter and Λ is a typical transverse-momentum scale characterizing the particles in the system. For local thermodynamic equilibrium ξ = 0 and Λ = T (from now on we set µ = 0).
The four-vector V µ appearing in (5) defines the direction of the beam and has the structure
V µ = γ z (v z , 0, 0, 1), γ z = (1 − v 2 z ) −1/2 .(6)
We note that the four-vectors U µ and V µ satisfy the normalization conditions
U 2 = 1, V 2 = −1, U · V = 0.(7)
In the local rest frame (LRF) of the fluid element, U µ and V µ have simple forms
U µ = (1, 0, 0, 0), V µ = (0, 0, 0, 1),(8)
and the distribution function (5) is reduced to
f RSF (x, p) = g exp m 2 + p 2 ⊥ + (1 + ξ)p 2 Λ − ǫ −1 .
(9) Neglecting the hadron mass (m = 0) and quantum statistics (ǫ = 0) we recover the commonly used RSF [39].
III. RATIOS OF HADRONIC YIELDS
Starting with the Cooper-Frye formula (1) we obtain the multiplicity of the hadron species i in the form
N i = dΣ µ (x) d 3 p E p p µ f i (x, p).(10)
If the distribution function is taken as RSF, we have
f i (x, p) = f i RFS (p · U, p · V )
, and the integral over the momentum in (10) yields the particle current
d 3 p E p p µ f i RFS (p · U, p · V ) = n i (Λ, ξ(x)) U µ .(11)
A few comments are in order now: i) we assume now that the freeze-out is defined by the fixed value of the hard scale Λ, hence, Λ is kept constant in (11), ii) we allow for the spacetime dependence of the anisotropy parameter ξ but we assume that ξ is the same for all hadronic species, iii) there is no term proportional to V µ on the right hand of Eq. (11) due to quadratic dependence of the distribution function on p · V . The calculation of the density n i (Λ, ξ(x)) may be done in the LRF and the result is
n i (Λ, ξ(x)) = n i,eq (Λ) 1 + ξ(x) .(12)
where n i,eq (Λ) is the particle density in equilibrium at the temperature defined by the parameter Λ. Using Eq. (12) in (10) we find
N i = n i,eq (Λ) dΣ µ (x)U µ (x) 1 + ξ(x)(13)
and for the ratios
N i N j = n i,eq (Λ) n j,eq (Λ) .(14)
We thus see that the ratios of hadronic multiplicities are exactly the same as the ratios obtained for equilibrium distributions at the temperature Λ.
We also note that for the boost-invariant systems we have
N i N j = dNi dy ∆Y dNj dy ∆Y = dNi dy dNj dy ,(15)
where ∆Y is the rapidity range. Hence, in this case, the ratios of rapidity densities are the same as the ratios obtained for equilibrium at the temperature Λ.
IV. TRANSVERSE-MOMENTUM SPECTRA
In this Section we calculate the transverse-momentum spectra in the thermal model with single freeze-out and compare our results with the LHC data. We use THERMINATOR 2 [42,43] and choose two options for the shape of the freeze-out hypersurface: i) the Cracow model, and ii) the blast-wave model. These two models have been recently discussed in Ref. [35]. In this work we follow closely the method presented in [35]. The main difference is the use of the modified distributions at freeze-out.
A. Cracow model
In the Cracow model the freeze-out hypersurface is defined by the condition that the invariant time (t 2 − x 2 − y 2 − z 2 ) 1/2 is fixed. The value of this time, τ 3f , and the transverse size of the system, r max , are the two geometric parameters of the model. The flow of matter at freeze-out has the Hubble form U µ = x µ /τ 3f . If the hadronic system is in equilibrium, the freezeout conditions are defined by the two extra parameters: the freeze-out temperature, T , and the freeze-out baryon chemical potential, µ. In this work we neglect the chemical potential and introduce the scale parameter Λ instead of T . In addition, the hadron distributions are taken as RSF with the anisotropy parameter ξ, which we keep constant in this Section. Thus, altogether we have four independent parameters: τ 3f , r max , Λ and ξ.
In Ref. [35] we analyzed Pb+Pb collisions at the beam energy √ s NN = 2.76 TeV for three centrality classes. We used three parameters: τ 3f , r max and T . The freeze-out temperature was always set equal to T 0 = 165.6 MeV, while the geometric parameters depended on the centrality class, we used: τ 0 3f = 9.0 fm and r 0 max = 11.4 fm for c = 0%-5%, τ 0 3f = 7.4 fm and r 0 max = 9.6 fm for c = 10%-20%, and finally τ 0 3f = 5.9 fm and r 0 max = 7.25 fm for c = 30%-40%.
We have added the superscript 0 to mark that the above values have been used in the equilibrium calculations. When switching from the equilibrium distributions (2) to the anisotropic distributions (5), we use the following scheme
Λ = T 0 τ 3f = τ 0 3f (1 + ξ) 1/6 , r max = r 0 max (1 + ξ) 1/6 .(16)
The model transverse-momentum spectra obtained for this choice of the parameters are shown in Fig. 1 as solid lines and compared to the data. The part (a) describes the results for ξ = 2.5, while the part (b) shows the results for ξ = −0.5. The results of the standard thermodynamic fit, obtained in Ref. [35], are represented by the dashed lines. We clearly see that the change introduced by the momentum anisotropy, induced by the term ξ(p · V ) 2 in the exponential function, is very well compensated by the renormalization of the size of the system according to Eq. (16). We note that the power 1/6 appearing in Eq. (16) implies that the volume of the system in the Cracow model scales as (1 + ξ) 1/2 , which compensates the same factor in the denominator of Eq. (12). Using the same rescaling for τ 3f and r max we find that the transverse flow profiles change very moderately (within 20%). Hence, the invariance of our results for the spectra with respect to the transformation (16) is only approximate. This property is explained in more detail in the Appendix.
B. Blast-wave model
In this Section we present our results obtained with the modified version of the blast wave model. This model differs from the Cracow model by the form of the freezeout hypersurface and the form of the transverse flow. The shape of the freeze-out curve in the Minkowski space is controlled by the parameter A, whereas the magnitude of the flow is controlled by the parameter v T . In the equilibrium version used in Ref. [35] we have used four parameters: A, T 0 , τ 0 2f = r 0 max , and v T . Their values are listed in Table 1 of Ref. [35].
Similarly to the case of the Cracow model, when switching from equilibrium to the anisotropic distributions, we make the following change of the parameters
Λ = T 0 τ 2f = τ 0 2f (1 + ξ) 1/6 , r max = r 0 max (1 + ξ) 1/6 .(17)
Our results are shown in Figs. 2 and 3. Similarly to the Cracow model we observe that the spectra are (almost) insensitive to the induced anisotropy provided the geometric parameters of the model are appropriately rescaled.
V. HBT RADII
In Ref. [35] we have used the Cracow and the blastwave model with different values of the parameter A to calculate the pion HBT radii. The main outcome of these calculations is that the radii are reproduced best in the blast-wave model with A = −0.5. We recall that the freeze-out conditions described by a negative value of A correspond to the situation where the outer parts of the system freeze out earlier than the system's interior. This is typical for more advanced hydrodynamic models.
In this Section we show the results of the calculations which are analogous to those presented in Ref. [35]. We choose the blast-wave model with A = −0.5 and incorporate the momentum anisotropy at freeze-out. For each value of the anisotropy parameter ξ we modify the geometric parameters according to the formula (17). As we have seen in the previous Section, the modification of the geometric parameters compensates the effect of the anisotropy in the transverse-momentum spectra. On the other hand, the change of the geometric parameters should affect the HBT radii, as they are connected with the space-time extensions of the system. Our results for the pion HBT radii obtained in the blast-wave model with A = −0.5 for several values of the anisotropy parameter ξ are shown in Fig. 4 (ξ = −0.5small dots, ξ = 0 -solid line, ξ = 0.5 -large dots, ξ = 1.0 -long dashes, ξ = 2.5 -short dashes). As expected, R side grows with increasing values of ξ. This radius has a simple geometric interpretation of the transverse size of the system, hence, the increase of R side reflects simply the growth of r max . We have found that the radius R out behaves in the similar way. On the other hand, the radius R long is practically independent of the anisotropy parameter ξ. The latter behavior may be understood as a net result of the two effects: a decrease of the longitudinal homogeneity length due to the decreased longitudinal pressure and an increase of this length due to the overall increase of the size of the system. Interestingly, the two effects almost completely cancel each other.
The results represented by the solid lines correspond to the local equilibrium studied in Ref. [35]. They agree reasonably well with the HBT data. However, Fig. 4 shows that the agreement with the data may be improved if we introduce a non-zero anisotropy. A much better agreement with the data is obtained for ξ = 1.0 than with ξ = 0.
The general conclusion from the HBT calculations presented in this Section is that the anisotropy of the momentum may be introduced as an additional characteristics of the system at freeze-out and its non-trivial value may be used to improve the agreement with the data.
VI. CONCLUSIONS
In this paper we have considered locally anisotropic momentum distributions of hadrons at freeze-out in relativistic heavy-ion collisions. We have taken into account the local anisotropy between the longitudinal and transverse momenta. Our results show that physical observables, such as the ratios of hadron abundances or the hadronic transverse-momentum spectra, are in practice indistinguishable from those obtained in an analogous equilibrium calculations -the effect of the momentum anisotropy may be compensated by an appropriate change of the geometric models of the system. We have also demonstrated that this freedom of the parameters may be used to improve the agreement of the model calculations with the measured HBT radii.
Our results indicate insensitivity of the thermal approach against specific variations of the model assumptions and show that it may be quite successful even in the situations where matter is out of equilibrium. 2011/01/D/ST2/00772.
VII. APPENDIX
In this Section, we explain the approximate scaling of the transverse-momentum spectra. As an example, we consider the Cracow model. In the case of the blast-wave model, the arguments leading to the approximate scaling are similar.
In the Cracow model, the spectra of primordial particles at zero rapidity are given by the integrals of the form
dN dyd 2 p ⊥ (18) = τ 3 3f (2π) 3 2π 0 dφ ∞ −∞ dη ϑ max ⊥ 0 dϑ ⊥ cosh ϑ ⊥ sinh ϑ ⊥ × [m ⊥ coshϑ ⊥ coshη − p ⊥ sinhϑ ⊥ cos φ] × exp [−β(m ⊥ coshϑ ⊥ coshη − p ⊥ sinhϑ ⊥ cos φ)] .
Here β = 1/T , φ is the azimuthal angle, η is the spacetime rapidity, and θ ⊥ is the transverse rapidity connected with the transverse distance, r = τ 3f sinh θ ⊥ . The momentum anisotropy is implemented by replacing T by Λ and by the change of the argument of the exponential function,
m ⊥ coshϑ ⊥ coshη − p ⊥ sinhϑ ⊥ cos φ → (m ⊥ coshϑ ⊥ coshη − p ⊥ sinhϑ ⊥ cos φ) 2 + ξm 2 ⊥ sinh 2 η.
The expression under the square root may be rewritten in the equivalent form as (p ⊥ cos φ cosh θ ⊥ − m ⊥ cosh η sinh θ ⊥ ) 2 +m 2 + p 2 ⊥ sin 2 φ + m 2 ⊥ (1 + ξ) sinh 2 η.
where cosh η is relatively suppressed by sinh θ ⊥ . Since the main contribution to the integral (18) comes from the region where η ≈ 0, the hyperbolic functions may be replaced by their approximations, cosh η ≈ 1 and sinh η ≈ η. Then, one can easily see that the factor 1 + ξ may be eliminated by the appropriate change of the integration variable, which leads to the overall change of the normalization of the spectra by the factor (1+ξ) −1/2 .
FIG. 1 :
1(Color online) Transverse-momentum spectra of positive pions (dots) and kaons (triangles) measured in Pb+Pb collisions at the beam energy √ sNN = 2.76 TeV[44] compared to the Cracow model results with ξ = 2.5 (a) and ξ = −0.5 (b). The anisotropic hadron distributions are used in the model calculations. The equilibrium parameters have been rescaled according to the formula(16). The experimental and model results are shown for the centrality class c = 0% − 5%.
FIG. 2 :
2(Color online) The transverse-momentum spectra obtained in the blast-wave model for three different values of the parameter A, ξ = −0.5 (solid lines) and ξ = 0 (dashed lines). The equilibrium parameters have been rescaled according to the formula(17). The data are taken from[44].
FIG. 3 :
3(Color online) The same asFig. 2but for ξ = 2.5 (solid lines) and ξ = 0 (dashed lines). The data are taken from[44].
FIG
. 4: (Color online) The pion HBT radii Rout (a), R side (b), and R long obtained in the blast-wave model with the freezeout slope parameter A = −0.5 for five different values of the local momentum anisotropy ξ. The model results are compared to the ALICE data[45].
AcknowledgmentsWe thank M. Strickland and R. Ryblewski for discussions and critical comments concerning the manuscript. This work was supported by the Polish Ministry of Science and Higher Education under Grant No. N N202 263438, and National Science Centre, grant DEC-
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| [] |
[
"Subradiance-protected excitation spreading in the generation of collimated photon emission from an atomic array",
"Subradiance-protected excitation spreading in the generation of collimated photon emission from an atomic array"
] | [
"K E Ballantine \nDepartment of Physics\nLancaster University\nLA1 4YBLancasterUnited Kingdom\n",
"J Ruostekoski \nDepartment of Physics\nLancaster University\nLA1 4YBLancasterUnited Kingdom\n"
] | [
"Department of Physics\nLancaster University\nLA1 4YBLancasterUnited Kingdom",
"Department of Physics\nLancaster University\nLA1 4YBLancasterUnited Kingdom"
] | [] | We show how an initial localized radiative excitation in a two-dimensional array of cold atoms can be converted into highly-directional coherent emission of light by protecting the spreading of the excitation across the array in a subradiant collective eigenmode with a lifetime orders of magnitude longer than that of an isolated atom. The excitation can then be released from the protected subradiant eigenmode by controlling the Zeeman level shifts of the atoms. Hence, an original localized excitation which emits in all directions is transferred to a delocalized subradianceprotected excitation, with a probabilistic emission of a photon only along the axis perpendicular to the plane of the atoms. This protected spreading and directional emission could potentially be used to link stages in a quantum information or quantum computing architecture.II. ATOM-LIGHT COUPLINGSingle excitation modelWe consider a 2D square array of N atoms in the yz plane, with the lattice spacing d, and a J = 0 → J = 1 arXiv:1909.01912v1 [cond-mat.quant-gas] | 10.1103/physrevresearch.2.023086 | [
"https://arxiv.org/pdf/1909.01912v3.pdf"
] | 202,538,435 | 1909.01912 | d1263330f66bbb291cfb8ec3b207f92c2113818b |
Subradiance-protected excitation spreading in the generation of collimated photon emission from an atomic array
K E Ballantine
Department of Physics
Lancaster University
LA1 4YBLancasterUnited Kingdom
J Ruostekoski
Department of Physics
Lancaster University
LA1 4YBLancasterUnited Kingdom
Subradiance-protected excitation spreading in the generation of collimated photon emission from an atomic array
(Dated: September 5, 2019)
We show how an initial localized radiative excitation in a two-dimensional array of cold atoms can be converted into highly-directional coherent emission of light by protecting the spreading of the excitation across the array in a subradiant collective eigenmode with a lifetime orders of magnitude longer than that of an isolated atom. The excitation can then be released from the protected subradiant eigenmode by controlling the Zeeman level shifts of the atoms. Hence, an original localized excitation which emits in all directions is transferred to a delocalized subradianceprotected excitation, with a probabilistic emission of a photon only along the axis perpendicular to the plane of the atoms. This protected spreading and directional emission could potentially be used to link stages in a quantum information or quantum computing architecture.II. ATOM-LIGHT COUPLINGSingle excitation modelWe consider a 2D square array of N atoms in the yz plane, with the lattice spacing d, and a J = 0 → J = 1 arXiv:1909.01912v1 [cond-mat.quant-gas]
I. INTRODUCTION
Collective optical interactions in cold trapped atomic ensembles are being actively studied experimentally where systems with increasing densities can now be achieved [1][2][3][4][5][6][7][8][9][10][11][12][13][14].
Since the light-mediated resonant dipole-dipole interactions depend on the average atomic separation, the close proximity of the atoms can lead to a correlated optical response [15,16] that no longer obeys continuous medium descriptions of electrodynamics [17,18]. The quest for systems where the collective optical response could potentially be easily manipulated has resulted in the studies of strong light-atom coupling -and coupling of light with other dipolar resonators -in regular arrays .
Control, storage and transmission of collective excitations could play a key role in modular quantum architecture [53][54][55], consisting of individual quantum elements with coherent links [36,56]. Subradiant states [57], which decay more slowly than an isolated emitter, couple weakly to external fields and have posed a long-standing experimental challenge, with observations first emerging in two-and few-particle systems [58][59][60][61][62][63], and now also in larger ensembles [11,42]. Due to the isolation from the environment, subradiant modes have been shown to be useful in transport of excitations [64][65][66][67]. Recent work has explored light transport for closely spaced atoms in arrays with topological edge states [68,69] and in a onedimensional (1D) chain or ring [70][71][72].
We show here how an initial localized excitation at the center of a planar array of cold atoms, which will emit radiation in all directions, can be transported across the lattice and converted into highly directional emission. The initial localized excitation will have an overlap with several of the collective radiative many-atom excitation eigenmodes of the system. These collective modes arise from the effective dipole-dipole interactions between atoms, and will each have a collective resonance shift and a collective linewidth that differs from that of a single atom. As these eigenmodes have different linewidths, they will decay at different rates, and interference of the modes leads to the spatial spreading of the excitation across the lattice. After some time, the excitation will be in a very subradiant collective mode delocalized over the entire array. In this case coherence across the array comes from the collective nature of the many-atom eigenmode, rather than any driving field. After we release the excitation by controlling the atomic levels, the result is coherent and highly directional emission of a photon from the array.
In our model, we assume that a two-dimensional (2D) square array of cold atoms is prepared with a single atom per lattice site and with a single-photon excitation initially localized at the center of the lattice. Such excitations could be produced by coupling to a nearby emitter or by selectively controlling the atomic excitation. By choosing the initial state and the lattice spacing appropriately, this local excitation is transferred to one of two delocalized subradiant target modes, a mode with uniform in-phase out-of-plane polarization, and a mode where the phase of the out-of-plane polarization varies by π between neighboring atoms. Once the delocalized mode is established, Zeeman splitting can couple the uniform subradiant out-of-plane mode to a uniform in-plane mode that is strongly radiating, allowing the excitation to decay. We calculate the far-field radiation pattern of this emission which is highly collimated in the direction normal to the plane. Collimated emission has been achieved in plasmonic planar arrays after spreading of an initial excitation [73], but without an analogous procedure of transferring the excitation between weakly (dark) and strongly (bright) radiating states. An off-resonant pair of atomic layers has been proposed as a phased-array antenna for photons [36]. atomic transition (Fig. 1). The full quantum dynamics of the atomic system for a given initial excitation and in the absence of a driving laser follows from the quantum master equation for the many-atom density matrix ρ;
ρ = i j,ν ∆ ν σ + jνσ − jν , ρ + i jlνµ(l =j) Ω (jl) νµ σ + jνσ − lµ , ρ + jlνµ γ (jl) νµ 2σ − lµ ρσ + jν −σ + jνσ − lµ ρ − ρσ + jνσ − lµ ,(1)
whereσ + jν = (σ − jν ) † = |e jν g j | is the raising operator to the excited state ν on atom j. The single atom Zeeman splittings, ∆ ν , are given relative to the m = 0 transition frequency ω, with ∆ 0 = 0 and ∆ − = −∆ + ≡ ∆ = ω−ω ± , where ω ± are the frequencies of the m = ±1 transitions, respectively. The diagonal terms of the dissipative matrix γ (jj) νν = γ = D 2 k 3 /(6π 0 ) correspond to the single atom resonance linewidth with the reduced dipole matrix element D and k = 2π/λ = ω/c. The off-diagonal elements in the dissipation and interaction terms are given by the real and imaginary parts of
Ω (jl) νµ + iγ (jl) νµ = ξG (jl) νµ ,(2)
where ξ = 6πγ/k 3 . Here, the dipole-dipole interaction between the atoms j and l with the orientations of the dipolesê ν andê µ at positions r j and r l , respectively, is determined by the dipole radiation kernel [74]
G (jl) νµ =ê * ν · G (r j − r l )ê µ ,(3)
where
G αβ (r) = ∂ ∂r α ∂ ∂r β − δ αβ ∇ 2 e ikr 4πr − δ αβ δ(r) ,(4)
and the contact interaction term between the atoms has explicitly been removed [75] G αβ (r) = G αβ (r) + δ αβ δ(r) 3 .
For ideal point dipoles this contact interaction is inconsequential for the physics [76], and by assuming hard-core atoms it vanishes as the atoms cannot overlap. We assume that the Zeeman level splitting ∆ of the J = 1, m = ±1 states, is controllable, as could be obtained, e.g., by magnetic fields or by AC-Stark shifts of lasers or microwaves [77]. For the majority of what follows, we take ∆ = 0 and study the transfer and decay of an initial excitation in the absence of Zeeman splitting. In Sec. II 3 we show how turning on the Zeeman splitting ∆ couples a subradiant state, with very slow decay, to a rapidly decaying bright mode, allowing the photon to radiate away. Instead of solving the general quantum master equation for the full many-atom dynamics, we restrict ourselves to single-excitation systems determined by the initial state of precisely one electronic excitation and study their evolution. The dynamics of the single-excitation subspace decouples to give
∂ tρ (jk) νµ = iH (jl) ντρ (lk) τ µ − iρ (jl) ντ H (lk) * τ µ ,(6)
whereρ (jk) νµ = G|σ − jν ρσ + kµ |G are the matrix elements of ρ corresponding to the single excitation, |G is the state with all atoms in the ground state, and
H (jk) µν = ∆ µ δ jk δ µν + Ω (jk) µν (1 − δ jk ) + iγ (jk) µν .(7)
While dissipation will mix the single-excitation subspace with the ground state, the dynamics within the single-excitation subspace are coherent. Since we always assume a pure initial state, the single-excitation part of the density matrix retains the form
ρ(t) = |Ψ(t) Ψ(t)| ,(8)
where |Ψ(t) is a single-excitation state whose norm | Ψ(t)|Ψ(t) | 2 is not conserved due to the dissipation. This state can be expanded in terms of the atomic excitations
|Ψ(t) = j,ν P (j) ν (t)σ + jν |G .(9)
Regarding single-particle expectation values, equations of motion can equivalently be written in terms of these single-excitation subspace amplitudes [78]. This is best expressed by rearranging the excitation amplitudes into a vector b 3j−1+ν = P (j) ν (ν = −1, 0, 1), and similarly arranging the evolution Hamiltonian into a 3N ×3N matrix
H 3j−1+ν,3k−1+µ = H (jk)
νµ . Then the time evolution is described byḃ
= iH b,(10)
where H is a non-Hermitian matrix. This matrix equation describing the dynamics of the amplitudes of a single excitation state is formally equivalent to the equations of motion of N classical linear coupled dipoles. Thus we can interpret the results in terms of the decay of a single photon, or the decay of a classical coherent dipolar excitation. For a dipole P (j) ν of the jth atom, the scattered field at r reads [74]
0 E (j) (r) = G(r − r j ) D νê ν P (j) ν .(11)
In order to understand the behavior of the system as it evolves away from the initial condition, we study the eigenvalues δ n + iυ n and eigenmodes v n of the coupling matrix H , where δ n = ω −ω n is the shift of the collective mode resonance from that of a single atom and υ n is the collective resonance linewidth.
Since H is not Hermitian, the eigenmodes will not be orthogonal, but they still form a basis. One can therefore uniquely express the excitation amplitudes as
b(t) = n c n (t)v n .(12)
The amplitudes c n of the collective modes evolve independently and each satisfy the equation of motioṅ
c n = (iδ n − υ n ) c n ,(13)
with solution
c n (t) = exp [t(iδ n − υ n )]c n (0).(14)
Excitation spreading
We now study numerically the evolution and decay of an initial localized excitation, examining how it spreads across the lattice and comes to occupy a target delocalized mode. In this section, we assume there is no Zeeman splitting (∆ = 0). Hence, the J = 0 → J = 1 transition is isotropic, with the symmetry broken only by the orientation of the plane of the lattice. In the following section we will look at the effect of Zeeman splitting in coupling different eigenmodes.
For simplicity, for our numerical analysis, we assume that the planar array of atoms is uniformly excited at time t = 0 at nine sites at the center of the array. While the excitation decays to zero in the long-time limit, it is clear from Eq. (14) that if there is appreciable initial excitation of a subradiant state with υ n γ, then this will dominate at intermediate times when other mode amplitudes have decayed. There are two collective eigenmodes of particular relevance for planar arrays, namely those with coherent uniform excitations of dipoles. The first is a uniform in-phase polarization which points perpendicular to the plane, along the x axis, and which we denote by P P . The second one, denoted P I , has a uniform in-phase polarization in-plane, here chosen to be along the y axis. The eigenmode P I directly couples to an incident plane wave propagating normal to the plane and is generally very easy to excite. The eigenmode P P can be directly coupled to P I by a symmetry breaking where induced Zeeman level shifts drive transitions between the modes [27,29]. Furthermore, these modes are useful in understanding the spectral response of the array, and can play the role of collective versions of dark and bright states [29] of a standard single-particle electromagnetic induced transparency (EIT) [79].
The eigenmode P P can be targeted with an appropriate initial excitation even when initially the system only exhibits a localized excitation. Here we also target another collective eigenmode, an antiferromagnetic excitation denoted by P AF , where each atom has a polarization in the normal direction along the x axis which is π out of phase with each of its nearest neighbors. This mode is of interest because it has quickly varying phase across the lattice, but can be reached from a localized initial excitation where the polarization of almost all the atoms is zero.
The resonance linewidth of each of these modes is shown in Fig. 2 as a function of lattice spacing for a 31 × 31 lattice. The uniform out-of-plane mode P P and the antiferromagnetic mode P AF can be strongly subradiant at the appropriate array spacings. The antiferromagnetic mode has maximum phase variation in the y and z directions, and so for periodic boundary conditions is located at the corner of the Brillouin zone with quasi-momentum q = (π/d, π/d). For small lattice spacing the quasi-momentum here is greater than the free-space wavevector, |q| > k, and the mode cannot decay [68]. While the linewidth is non-zero for a finite lattice, it is still much smaller than the linewidths of the uniform eigenmodes for d < ∼ λ/2. The uniform modes at q = 0 are not protected by momentum conservation; in the limit of an infinite array the uniform in-plane eigenmode only radiates exactly normal to the plane (the zeroth order Bragg peak) for any subwavelength lattice spacing [24,27,52]. For the P P mode, however, the dipoles oscillate along the x axis and so emit mainly in the lattice plane, where light is absorbed by other atoms and must undergo many scattering events to escape. This leads to this mode also being very subradiant for larger lattices, scaling with the number of atoms N as υ P /γ ≈ N −0.9 [27], with the mode becoming completely dark in the infinite lattice limit, lim N →∞ υ P = 0. The in-plane mode P I has a linewidth which is well described for d < λ by its large-N limit [24,29] υ I,∞ ≡ lim
N →∞ υ I = 3λ 2 γ 4πd 2 .(15)
If we choose the initial excitation to match one of these eigenmodes in the center of the lattice, and to be zero everywhere else, we expect it to have a non-zero overlap with the collective mode. If this mode is sufficiently subradiant, then, as other modes decay, it will come to dominate and the initial localized excitation will quickly evolve into a coherent, subradiant state extended across the entire lattice. However, since H is not Hermitian, the eigenvectors are not orthogonal in general. We therefore use the definition
L j (t) = |v T j b(t)| 2 i |v T i b(0)| 2 ,(16)
for a measure of the occupation of an eigenmode v j in the current state b which has been shown to describe accurately the contribution of the collective mode populations in the decay dynamics of radiative excitations [27]. This measure can also be used to determine a target eigenmode of a finite lattice which most closely matches an ideal mode with equal amplitude on every site. We take this measure to be normalized at t = 0. For a 31 × 31 square lattice with a lattice spacing of d = 0.75λ and ∆ = 0, the uniform mode P P is very subradiant, with a linewidth (5 × 10 −4 )γ, as shown in Fig. 2. To target this mode, we start from an initial excitation which has P (j)
x ≡ (P Fig. 3 (a), where the mode occupation of the uniform perpendicular mode, L P , is plotted along with the sum of the occupations of all other eigenmodes, denoted by L . The dynamics can be understood by looking at the initial mode occupations, plotted in Fig. 3 (b). While several modes are initially occupied, the target mode, indicated by the red square, is the most subradiant. As the less subradiant modes decay rapidly, the total excitation initially falls off quickly. However, because the target mode is very subradiant, the population of this mode decays much slower, and after some time it becomes the dominant mode. At this point, light is effectively stored in a very subradiant state delocalized across the entire lattice.
For an appropriate choice of the spacing d = 0.55λ, the antiferromagnetic eigenmode is more subradiant than the uniform perpendicular mode. Also the antiferromagnetic mode can be targeted by a matching initial excitation; here we take the alternating polarization P (j)
x = ±1/3 on each of the central nine atoms, where the sign varies between nearest neighbors, and zero everywhere else. The initial excitation and the time dynamics are shown in Fig. 3 (c) and (d). As in the case of the uniform mode, the initial localized excitation tends to a delocalized state.
The importance of subradiance in letting the excitation spread across the lattice is illustrated by comparing an attempt to target the uniform in-plane mode P I . The result is shown in Fig. 3 (e) and (f). Because the excited modes are not significantly subradiant, the occupations decay on a much faster timescale. Even within this short time, however, the occupation of the target mode is never higher than that of all other modes.
The spatial spreading is illustrated in Fig. 4, which shows the absolute value of the excitation amplitude (which always points normal to the plane) across the lattice at various times, for initial in-phase and antiferromagnetic excitation at the center. The excitation quickly spreads across the lattice as each eigenmode component of the initial excitation decays at different rates. After about t = 500/γ, the excitation has settled into the most subradiant target mode.
In the case of both the modes P P and P AF , while the initial excitation is successfully transferred to the target mode, the final overall occupation is L ≈ 0.02 (representing the probability of generating a single delocalized photon), much less than the initial occupation which is normalized to one. This is due to the low occupation of the target mode in the initial state, which is approximately 0.04 of the total, as the initial excitation is localized to ≈ 1% of the atoms. This could be improved by starting with a larger excitation. The occupation of the target mode rises to 0.07 for an initial excitation of the central 16 atoms for example, and to 0.1 for an initial excitation of the central 25 atoms.
Photon release
In the previous section we described how an initially localized excitation can be transferred to a delocalized subradiant eigenmode occupation, extending over the entire array of atoms. As the photon then is effectively stored in a subradiant state, it decays slowly and little light is emitted. To release the photon, the Zeeman splitting can be turned on at the desired time. This can be understood by considering the two uniform eigenmodes of the system P I and P p . When we introduce a non-zero Zeeman splitting, these two modes are no longer eigenmodes, but are coupled to one another. The dynamics in the presence of Zeeman splitting can be understood by considering a two-mode model [27,29], which for a driven system represents a linearized version of the EIT equations for bright and dark states [79]. The two-mode model qualitatively captures many aspects of the dynamics of the full lattice for sufficiently large arrays, since the phasematching conditions of the other modes are not satisfied. We find that it can also illustrate how the excitation is transferred from one uniform mode to the other in the present case. The relevant physics is captured by the simple coupled two-mode dynamicṡ
P P = (iδ P − υ P )P P − ∆P I ,(17a)P I = (iδ I − υ I )P I + ∆P P ,(17b)
where δ P,I is the collective resonance shift and υ P,I the collective linewidth of the uniform out-of-plane and inplane modes, respectively. These two modes capture much of the physics because once the dipoles are all oscillating in phase, they will continue to do so even when Zeeman splitting is turned on and the direction of the effective magnetic field breaks the isotropy of the J = 0 → J = 1 transition. As seen from Eqs. (17a) and (17b), in the presence of Zeeman splitting, P P is not an eigenmode, and the dipoles start to rotate towards the plane. Applying the The angle θmax such that 99% of the integrated far field intensity is between 0 < θ < θmax, where θ = arctan ( k 2 y + k 2 z /kx), as a function of the number of atoms N . To compare different lattice sizes, the excitation is released when the target mode accounts for 90% of the population. (e) Far-field radiation for a 71 × 71 lattice, released when 90% of the excitation was in the mode PP , showing highly directional emission of the stored excitation. The intensity I rel is plotted in relative units scaled to a maximum of one.
splitting for a short time, the excitation can then be transferred to the mode P I , where each atom has approximately uniform in-phase polarization in the y direction, with much faster emission rate, allowing the excitation to quickly radiate away. We calculate the emitted light, given by the sum of all the atomic contributions from Eq. (11), in the far-field limit (r d, λ, where r is the distance from the source to the observation point) [74]. Figure 5 shows the results of Zeeman splitting being turned on at t = 800/γ for a short time. As soon as the splitting is turned on, the occupation P P of the outof-plane mode falls rapidly, while the occupation of the in-plane mode P I shows a corresponding rise. Since this mode has a much larger linewidth, the occupation then begins to fall quickly and, within a time ≈ 5/γ, the photon has been emitted from the lattice.
While emission from the initial excitation is omnidirectional, emission from the delocalized mode is highly collimated along the direction of the x axis, perpendicular to the lattice. We quantify this by the angle θ max such that integrating between 0 ≤ θ ≤ θ max and 0 ≤ φ ≤ 2π gives 99% of the total integrated forward-scattered intensity, where θ = arctan ( k 2 y + k 2 z /k x ) is the polar angle of the ray to the k x axis and φ = arctan (k y /k z ) the azimuthal angle in the k y k z plane. For the initial excitation, this angle is θ max = 0.997(π/2), i.e. the light is not collimated at all. However, when the excitation is released from the delocalized mode, the photon emission is highly directional, with θ max = 0.05(π/2). In this case the emission is equal in the forward and backward directions. Forward-only scattering could be achieved using two arrays offset in the x direction with a suitable phase shift [36], analogously to the directed radiation of antennas.
The far-field radiation pattern of the delocalized mode is that of a 2D diffraction grating, dominated by the central zeroth order Bragg peak (the higher order Bragg peaks do not exist because of the subwavelength lattice spacing). For lattices with a higher number of atoms, this central peak becomes sharper. To compare different lattice sizes we start with the same initial excitation on the central nine atoms, and let it evolve until the target uniform mode accounts for 90% of the remaining mode occupation. Increasing the number of atoms leads to a sharp drop in θ max , as shown in Fig. 5 (d). For the largest lattice (71 × 71), we find θ max = 0.02(π/2) which represents a photon wave-packet that is highly localized in k-space. For an infinite array, the propagation reaches the precise 1D limit propagating only in the x direction.
The process described in this section is reminiscent of the procedure to slow and store light within an atomic cloud using the standard single-particle EIT [80,81]. In our case it is the collective modes of the atomic array which act as the bright and dark states. To release the photon, the Zeeman splitting couples these states, playing the role of the coupling laser which is usually used to restore transparency. This allows the photon to radiate away while preserving spatial coherence.
The effects of position fluctuations
The atoms may not be perfectly localized, but rather their positions will fluctuate due to the finite size of the trap. We account for this by taking many individual realizations with fixed atom positions drawn randomly from a harmonic oscillator ground-state probability distribution for each lattice site with root-meansquare width l, and stochastically averaging over these realizations [21]. This procedure has been shown to reproduce a full quantum model exactly [75,82]. The result of spatial disorder is shown in Fig. 6, where we plot the surviving amplitude P = j,µ |P (j) µ | 2 as a function of time for varying fluctuation lengths. For increasing disorder, the excitation decays more quickly, as also observed in other subradiance-protected excitation transfer studies [72]. However, in these cases the life-time is still much longer than the corresponding case of an in-phase in-plane localized excitation with no disorder. This can be understood by looking at the distribution of the eigenmodes, weighted by their initial occupation L, across many stochastic realizations. This is compared in Fig. 6 to the distribution of the most subradiant modes which contribute up to 30% of the initial excitation of each realization, shown in green. Although disorder means that each realization has far fewer very subradiant modes, these few modes are consistently wellrepresented in the initial excitation.
III. CONCLUDING REMARKS
Subradiant states are isolated from the environment and therefore difficult to excite. Standard field excitation typically only results in a very small fraction of the total population to notably subradiant modes [11]. Transferring a more substantial population to slowly radiating states typically requires first the breaking of the eigenmode symmetry before the excitation, followed by restoration of the symmetry, such that the modes are temporarily made to interact with radiation [27]. Here we have demonstrated a probabilistic conversion. We have shown how a localized single-photon excitation of an atomic lattice, with omnidirectional emission, can spread out into a subradiant state which is delocalized across the whole lattice and is protected from decay. For suitable lattice spacing, this initial excitation can evolve into either a uniform mode, with the polarization of each atom in phase, or an antiferromagnetic mode, with the polarization of each atom π out of phase with its nearest neighbors. In the case of the uniform mode, we have shown how turning on a Zeeman splitting allows the excitation to be released as highly collimated directional emission.
Such an operation could form part of a quantum information or quantum computing architecture [36,54,83], coupling via short-range interactions to an input state, and coherently converting this local excitation into directional emission, and effective 1D propagation. This output could then be transferred via free space to another stage. Future work could identify collective quantum effects in higher-order correlations beyond the singleexcitation limit, and study the effect of such correlations on the emitted collimated light.
ACKNOWLEDGMENTS
We thank Chris Parmee for reading and commenting on the manuscript. We acknowledge financial support from EPSRC.
FIG. 1 .
1(a) Illustration of planar array of atoms with an initial single-photon excitation localized on a small number of atoms at the center of the array. The arrows represent the direction of the atomic dipoles that are oriented normal to the plane and will propagate toward a uniform subradiant eigenmode. (b) The atomic level structure where the Zeeman splitting ∆ is only introduced for the release of the subradiant excitation.
FIG. 2 .
2Collective linewidth as a function of lattice spacing of the uniform out-of-plane eigenmode (υP ), the uniform-inplane eigenmode (υI ), and the antiferromagnetic out-of-plane eigenmode (υAF) with the phase of polarization varying by π between nearest neighbors. The dashed-dotted line shows the analytic infinite-lattice limit formula for υI given by Eq.(15).
the central nine atoms and 0 on all other atoms. The resulting time dynamics are shown in
FIG. 3 .
3Time evolution and initial eigenmode distribution of a localized excitation. (a) The occupation measure LP for the uniform out-of-plane mode and the sum L of the occupation measure of all other modes for an initial in-phase out-of-plane excitation localized on the central nine atoms of a 31 × 31 lattice, with lattice spacing d = 0.75λ. The delocalized uniform mode quickly grows to 100% of the remaining excitation. (b) The initial mode occupations for the initial excitation and parameters in (a), ordered by collective linewidth υi. The target uniform out-of-plane mode is illustrated by the red square. (c) Mode occupation LAF of the antiferromagnetic out-of-plane mode and the sum L of all other mode occupations for an initial out-of-plane excitation localized on the central nine atoms, the phase of which varies by π between nearest neighbors. Here the the 31 × 31 lattice has lattice spacing d = 0.55λ. (d) The initial mode occupations for the initial excitation and parameters in (c), where the red square now denotes the target antiferromagnetic delocalized mode. (e) Mode occupation LI of the uniform in-plane mode, and the sum of L of all other modes for an initial in-phase inplane excitation on the central nine atoms, showing that in the absence of subradiance the delocalization is not achieved. (f) Initial mode occupation for the excitation in (e), where the red square indicates the uniform in-plane mode.
FIG. 4 .
4Spatial distribution of the absolute value of polarization |P | (the polarization is oriented normal to the lattice plane) on each atom of a 31 × 31 lattice at varying times. (a-e) Evolution of an in-phase excitation (targeting uniform PP mode) with lattice spacing d = 0.75λ and (f-j) evolution of an antiferromagnetic excitation (targeting antiferromagnetic PAF mode) with lattice spacing d = 0.55λ at times γt = 0, 50, 100, 400, and 800 from left to right. Note that at each time slice |P | is normalized independently.
FIG. 5 .
5Highly collimated light emission from a uniform subradiant eigenmode delocalized across the entire array. (a) To release the light, Zeeman level splitting ∆ = 3.3γ is turned on at t = 800/γ for a short period of time 0.5/γ to transfer the excitation from the collective eigenmode where the dipoles are uniformly pointing normal to the lattice (with the occupation measure LP ) to the one where they are coherently oscillating in phase on the lattice plane (LI ). The sum of the occupations of all the other modes is L . (b) Image of the far-field radiation of the initial excitation at t = 0, and (c) the released radiation at t = 801/γ for a 31 × 31 lattice. (d)
FIG
. 6. (a) Decay of polarization |P | for a localized in-phase out-of-plane excitation for position fluctuations with standard deviation l around each site, and of a localized in-phase inplane excitation at fixed atom position for comparison. (b) A histogram of the occupation measure of the initial excitation of in-phase polarization on the central nine atoms with each eigenmode having a decay rate υ (dark blue), and the distribution of the most subradiant modes which contribute up to 30% of the excitation for each realizations (light green). Results are averaged over 5000 individual realizations of position fluctuations.
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| [] |
[] | [
"Michel Grabisch [email protected] \nChristophe LABREUCHE Thales Research & Technology Domaine de Corbeville\nUniversité Paris I -Panthéon-Sorbonne\n91404Orsay CedexFrance\n"
] | [
"Christophe LABREUCHE Thales Research & Technology Domaine de Corbeville\nUniversité Paris I -Panthéon-Sorbonne\n91404Orsay CedexFrance"
] | [] | The paper proposes a general notion of interaction between attributes, which can be applied to many fields in decision making and data analysis. It generalizes the notion of interaction defined for criteria modelled by capacities, by considering functions defined on lattices. For a given problem, the lattice contains for each attribute the partially ordered set of remarkable points or levels. The interaction is based on the notion of derivative of a function defined on a lattice, and appears as a generalization of the Shapley value or other probabilistic values. | 10.1007/s10472-007-9052-7 | [
"https://arxiv.org/pdf/0711.2115v1.pdf"
] | 373,223 | 0711.2115 | 738ed27027e6f20f9b34f4155d421ca037efd8cc |
14 Nov 2007
Michel Grabisch [email protected]
Christophe LABREUCHE Thales Research & Technology Domaine de Corbeville
Université Paris I -Panthéon-Sorbonne
91404Orsay CedexFrance
14 Nov 2007Derivative of functions over lattices as a basis for the notion of interaction between attributesinteraction indexShapley valuecapacitygamelatticediscrete derivative
The paper proposes a general notion of interaction between attributes, which can be applied to many fields in decision making and data analysis. It generalizes the notion of interaction defined for criteria modelled by capacities, by considering functions defined on lattices. For a given problem, the lattice contains for each attribute the partially ordered set of remarkable points or levels. The interaction is based on the notion of derivative of a function defined on a lattice, and appears as a generalization of the Shapley value or other probabilistic values.
The concept of interaction: an introduction
Let us consider a set N of criteria describing the preferences of a decision maker (DM) over a set X of objects, alternatives, etc. We assume that for any object x ∈ X, we are able to build a vector of scores (a 1 , . . . , a n ) describing the satisfaction of the DM for x, w.r.t. each criterion. For this reason, and in order to remain at an abstract level, we call this vector a tuple, which we identify with the object or alternative. We may suppose for the moment that scores are given on the real interval [0, 1], with 0 and 1 having the meaning of "unacceptable" and "totally satisfying" respectively.
We make the simplifying assumption that the preference of the DM is solely determined by binary tuples, i.e. whose scores are either 0 or 1 on each criterion, the preference for other tuples being more or less an interpolation between binary tuples. More precisely, denoting by (1 A , 0 A c ) the binary tuple having a score of 1 for all criteria in A ⊆ N, and 0 elsewhere, this amounts to assigning an overall score v(A) in [0, 1] to (1 A , 0 A c ). Doing this for all A ⊆ N, we have defined a set function v : 2 N −→ [0, 1].
Although this is not essential in the sequel, we may impose to v some natural properties. First, we may set v(∅) := 0 and v(N) := 1, since A = ∅ (resp. N) corresponds to a binary act having all its scores being equal to 0 (resp. 1). Second, considering A ⊆ B, this leads to two binary tuples of which one dominates the other, in the sense that on each criterion one is at least as good as the other. Then it seems natural to impose v(A) ≤ v(B). This is called isotonicity. A set function v satisfying these two conditions is called a capacity [3] (also called fuzzy measure [20]).
Let us now consider the case n = 2 in some detail. There are 4 binary tuples (0, 0), (0, 1), (1,0) and (1,1), and we know already that the first and last have overall scores 0 = v(∅) and 1 = v(N). What about the 2 remaining ones ? There are two extreme situations, under isotonicity.
• v({1}) = v({2}) = 0, which means that for the DM, both criteria have to be satisfactory in order to get a satisfactory tuple, the satisfaction of only one criterion being useless. We say that the criteria are complementary.
• v({1}) = v({2}) = 1, which means that for the DM, the satisfaction of one of the two criteria is sufficient to have a satisfactory tuple, satisfying both being useless. We say that the criteria are substitutive.
Clearly, in these two situations, the criteria are not independent, in the sense that the satisfaction of one of them acts on the usefulness of the other in order to get a satisfactory tuple (necessary in the first case, useless in the second). So we may say that there is some interaction between the criteria 1 .
What should be a situation where no interaction occurs, i.e. criteria act independently ? It is a situation where the satisfaction of each criterion brings its own contribution to the overall satisfaction, hence:
v({1, 2}) = v({1}) + v({2}).
Note that in the first situation, v({1, 2}) > v({1}) + v({2}), while the reverse inequality holds in the second situation. This suggests that the interaction I 12 between criteria 1 and 2 should be defined as :
I 12 := v({1, 2}) − v({1}) − v({2}) + v(∅).(1)
This is simply the difference between binary tuples on the diagonal (where there is strict dominance) and on the anti-diagonal (where there is no dominance relation). The interaction is positive when criteria are complementary, while it is negative when they are substitutive. This is consistent with intuition considering that when criteria are complementary, they have no value by themselves, but put together they become important for the DM.
In the case of more than 2 criteria, the definition of interaction is more tricky but follows the same idea (see below). In fact, when n > 2, we may define the interaction between 3, 4, . . . , n criteria as well. The general definition of interaction for capacities has been given in [8], and has been axiomatized in [12].
The above story for introducing interaction can be made fairly more general. Let us first take interval [−1, 1] instead of [0, 1] for expressing scores, and consider that 1 For further discussion on substitutive and complementary criteria, see Marichal [16].
for the DM, values −1, 0 and 1 are particular because they express respectively total unsatisfaction, neutrality and total satisfaction. Then we are led to consider ternary tuples (1 A , −1 B , 0 (A∪B) c ), whose overall score is denoted by v(A, B). It is convenient to denote by Q(N) := {(A, B) | A, B ⊆ N, A ∩ B = ∅}. Now v is defined on Q(N), and as for capacities, it seems natural to impose v(N, ∅) := 1, v(∅, ∅) := 0, and v(∅, N) := −1. Also using the dominance argument, we should have, if
A ⊆ A ′ , v(A, B) ≤ v(A ′ , B)
and v(B, A) ≥ v(B, A ′ ). Such a v is called a bi-capacity [10,9]. The interaction for bi-capacities, called bi-interaction in [9], has been defined accordingly, and follows the same principle. When n = 2, since we have 3 particular levels −1, 0 and 1, the square [−1, 1] 2 is divided into 4 small squares and has 9 ternary tuples. In each small square, we apply the same definition as with capacities, i.e. Eq. (1). Hence, we have four interaction indices to describe interaction with n = 2, namely (see Figure 1):
I {1,2},∅ := v({1, 2}, ∅) − v({2}, ∅) − v({1}, ∅) + v(∅, ∅) =: I({1, 2}, ∅) (2) I ∅,{1,2} := v(∅, ∅) − v(∅, {1}) − v(∅, {2}) + v(∅, {1, 2}) =: I(∅, ∅) I 1,2 := v({1}, ∅) − v(∅, ∅) − v({1}, {2}) + v(∅, {2}) =: I({1}, ∅) I 2,1 := v({2}, ∅) − v({2}, {1}) − v(∅, ∅) + v(∅, {1}) =: I({2}, ∅).
The notation I A,B means that criteria in A are positive, while criteria in B are negative. As it will become clear later, a better notation is I(A, B), where (A, B) is the ternary tuples corresponding to the upper right corner of the square in consideration (i.e. the best possible tuple in the square).
(0, 1) (1, 1) (0, 0) (1, 0) (0, −1) (1, −1) (−1, 1) (−1, 0) (−1, −1) v({1, 2}, ∅) v({2}, ∅) v({2}, {1}) v({1}, {2}) v(∅, {2}) v(∅, {1, 2}) v({1}, ∅) v(∅, {1}) v(∅, ∅)
Let us now take a general point of view. We consider n-dimensional tuples in X := X 1 × · · · × X n , where it is assumed that each X i is a partially ordered set, whose order relation is denoted by ≤ i . We consider that on each dimension X i , there exist reference levels r i 1 , . . . , r i q i , which for the problem under consideration, convey some special meaning of interest, describing e.g. some particular situation, and that these reference levels form a lower locally distributive lattice (L i , ≤ i ). Denoting by L := L 1 × · · · × L n the product lattice with the product order, we define a real function v : L −→ R, assigning a real value to any combination of reference levels on each dimension. Let us give some instances of this general framework.
voting games and ternary voting games: defining N := {1, . . . , n} as the set of voters, for each voter there exist two or three reference levels, which are: voting in favor, voting against (case of classical voting games), and abstention (case of ternary voting games [6]). For classical games, we have L i = {0, 1}, ∀i ∈ N, with level 1 corresponding to voting in favor, so that L = 2 n , and v(A) = 1 if the bill is accepted when A is the set of voters voting in favor, or v(A) = 0 if the bill is rejected. For ternary voting games, we have L i = {−1, 0, 1}, with 0 corresponding to abstention and −1 to voting against. Then L = 3 n , and it is convenient to denote an element of L by a pair (A, B), where A is the set of voters in favor, and B the set of voters against. As before, v(A, B) = 1 (the bill is accepted) or 0 (the bill is rejected). Note that here X i coincides with L i , ∀i ∈ N.
cooperative games and bi-cooperative games: we replace voters by players. Reference levels in the case of cooperative games are 0 and 1, corresponding to non participation and participation in the game. Hence L = 2 n , and v(A) is the asset that the coalition A of players will win if the game is played. For bi-cooperative games, L = 3 n , and v(A, B) is the asset that A will receive when coalition A plays against coalition B, the remaining players not taking part in the game. Classically, here also X i coincides with L i , although one may consider any degree of participation between full participation and non participation (fuzzy games), which leads to
X i = [0, 1].
multicriteria decision making: this corresponds to the framework given in the introduction. We have L i = {0, 1} for all i ∈ N if we consider only two reference levels "unacceptable" and "totally satisfying", which leads to capacities, and L i = {−1, 0, 1} if a neutral level is added, which leads to bi-capacities, as explained above. Let us remark that our general framework allows one to be much more general: one may have more than 3 levels, adding for example intermediate levels such as "half satisfactory", etc., or even introduce non comparable levels, provided the lattice structure is preserved. For example, the level "don't know" may be incomparable with "neutral", but smaller than "satisfactory" and greater than "unsatisfactory", thus leading to the lattice 2 2 . In addition, we may consider different L i for each criterion. The function v defines the overall score given to an tuple having various reference levels on criteria.
data analysis: the construction is the same as for multicriteria decision making, but the meaning conveyed by the dimensions and the reference levels can be much more general, depending on the kind of data, being for example "high", "medium", "low", etc. We do not even need to have numerical dimensions, so that ordinal data analysis can be done. The meaning of v(x) for x ∈ L depends on the aim of the analysis. We propose three main examples:
• evaluation of x. For example, x is some kind of prototypical product, and a user or consumer gives an evaluation of it, which defines v(x) (subjective evaluation).
• classification in some category. v(x) is the label of the category, or takes value 0 or 1 (does not belong or belongs to a given category: in this latter case we need as many functions v as the number of categories) (pattern recognition).
• the number of items identical or similar to x in the data set (data mining). Suppose we have a large set D of data with some distance defined on it.
x ∈ L defines a particular protopyical datum. Then v(x) is the cardinality of the set of data x ′ ∈ D within a given distance of x, or v(x) is the sum of the inverse distances from any x ′ ∈ D to x.
We propose in this paper a general definition of interaction, which can be applied to the above defined framework, and encompasses already existing definitions of interaction for capacities and bi-capacities. The precise meaning of interaction is governed by the meaning of the function v. In game theory, it describes the synergy between players or voters, the interest to forming or not forming certain coalitions. In multicriteria decision making, it tells which criteria play a key role (and how), which criteria are redundant (with which one) in the decision process. In data mining, when v is a counting function as above, the interaction has a statistical flavor close to correlation. Indeed, since the interaction index is roughly speaking a difference of the diagonal and anti-diagonal, a positive (resp. negative) interaction corresponds to a positive (resp. negative) correlation. In pattern recognition, interaction is very informative for feature selection (see an application of interaction in this topic in [7]). Clearly, the interaction is a key concept in knowledge discovery, and has a strong descriptive power. We detail its construction and properties in the sequel, after recalling classical results.
For simplicity, the cardinality of sets A, B, S, . . . will be denoted by the corresponding lower case a, b, s, . . ., and we will often omit braces for singletons. We put N := {1, . . . , n}.
2 Importance and interaction indices for L = 2 n and L = 3 n
We recall in this section the classical definition for L = 2 n , (which corresponds to capacities, or more generally set functions, pseudo-Boolean functions [14]), and the one for L = 3 n (bi-capacities, bi-cooperative games).
Let v : 2 N −→ R, with v(∅) = 0 (game). As it will become clear, the interaction index is a generalization of the power index or importance index φ v (i), i ∈ N, which expresses to what extent an element i ∈ N (attribute, dimension) has importance or power for the problem under consideration. The general form is:
φ v (i) = S⊆N \i α 1 s [v(S ∪ i) − v(S)],(3)α 1 s ∈ R.
The value of the coefficients α 1 s has to be determined by additional requirements. The most important example is the Shapley index [19], where
α 1 s = (n − s − 1)!s! n! , s = 0, . . . , n − 1,(4)
obtained by the following property:
n i=1 φ v (i) = v(N)
, expressing a sharing of the total value among all elements, according to their importance (efficiency axiom). Another classical example is the Banzhaf index [1], where α 1 s = 1 2 n−1 , s = 0, . . . , n − 1. The interaction index [8] expresses the interaction among a coalition (group) S ⊆ N of elements:
I v (S) = T ⊆N \S α s t ∆ S v(T ),(5)
where α s t ∈ R, and ∆ S v(T ) is the derivative of v w.r.t. S at T for S ⊆ N \ T , and defined recursively as follows:
∆ ∅ v(T ) := v(T ) ∆ i v(T ) := v(T ∪ i) − v(T ) ∆ S v(T ) := ∆ i (∆ S\i v(T )), |S| > 1. Observe that I v ({i}) ≡ φ v (i)
, hence an interaction index is a generalization of an importance index. It is possible to define recursively the interaction index from the importance index [12]. Then, choosing a particular importance index (hence the coefficients α 1 s ) defines uniquely the coefficients α s t . Let us introduce some notations, borrowed from game theory. The restricted game v N \K is the game v restricted to elements (players) in N \ K, hence v N \K (S) = v(S) for any S ⊆ N \ K, and is not defined outside. The reduced game v [K] is the game where all elements in K are considered as a single element denoted by [K], i.e. the set of elements is then
N [K] := (N \ K) ∪ {[K]}. The reduced game is defined by, for any S ⊆ N \ K: v [K] (S) = v(S) v [K] (S ∪ {[K]}) = v(S ∪ K).
The recursion axiom writes
I v (S) = I v [S] ([S]) − K⊆S,K =∅,S I v N\K (S \ K).(6)
Its meaning is simple when |S| = 2. Indeed, the formula can be written as
I v [i,j] ([i, j]) = I v N\i (j) + I v N\j (i) + I v (i, j).
It means that the importance of elements (e.g. players) i, j taken together is the sum of individual importances when the other is absent, and the interaction they have between them. Hence a positive interaction means that the overall importance of i, j is greater than the sum of their respective marginal importances (see [12] for another equivalent axiom). This axiom leads to the following formula for α t s (n), the argument indicating the number of players in the game α t s (n) = α 1 s (n − t + 1), ∀s = 0, . . . , n − t, ∀t = 1, . . . , n − 1.
When φ v is the Shapley index, we obtain the Shapley interaction index, whose coefficients are, using (7):
α s t := (n − s − t)!t! (n − s + 1)! .
We have generalized the above notions to the case of bi-capacities and bi-cooperative games [9,11], and given an axiomatization [11,15]. As explained in Section 1, we have to consider all combinations between positive and negative parts of the X i 's (see Eq.
(2)), and following the notation introduced there, we denote by I S,T , (S, T ) ∈ Q(N), the interaction among elements when S is the set of positive elements, and T is the set of negative elements. The Shapley index divides into two indices I {i,∅} and I {∅,i} , defined by:
I {i,∅} := S⊆N \i (n − s − 1)!s! n! ∆ i,∅ v(S, N \ (S ∪ i)) (8) I {∅,i} := S⊆N \i (n − s − 1)!s! n! ∆ ∅,i v(S, N \ S)(9)
where the derivatives are defined by:
∆ i,∅ v(S, T ) := v(S ∪ i, T ) − v(S, T ), (S, T ) ∈ Q(N \ i) ∆ ∅,i v(S, T ) := v(S, T \ i) − v(S, T ), (S, T ) ∈ Q(N), S ∋ i, T ∋ i. ∆ i,∅ v(S,I(i, ∅) + I(∅, i) = v(N, ∅) − v(∅, N).
As above, the derivative ∆ S,T can be defined recursively from these equations, and the definition of the Shapley interaction index is:
I S,T := K⊆N \(S∪T ) (n − s − t − k)!k! (n − s − t + 1)! ∆ S,T v(K, N \ (K ∪ S)).
Mathematical background and general framework for interaction
We try now to have a general view of previous definitions, thanks to results from lattice theory. We first introduce necessary definitions (see e.g. [2,4,13]). Let (L, ≤) be a lattice, we denote as usual by ∨, ∧, ⊤, ⊥ supremum, infimum, top and bottom (if they exist). If x and y in L are incomparable, we write x||y. Q ⊆ L is a downset of L if x ∈ Q and y ≤ x imply y ∈ Q. For any x ∈ L, the principal ideal ↓ x is defined as ↓ x := {y ∈ L | y ≤ x} (downset generated by x). For x, y ∈ L, we say that x covers y (or y is a predecessor of x), denoted by x ≻ y, if there is no z ∈ L, z = x, y such that x ≥ z ≥ y. (L, ≤) is lower semi-modular (resp. upper semi-modular ) if for all x, y ∈ L, x ∨ y ≻ x and x ∨ y ≻ y imply x ≻ x ∧ y and y ≻ x ∧ y (resp. x ≻ x ∧ y and y ≻ x∧y imply x∨y ≻ x and x∨y ≻ y). A lattice being upper and lower semi-modular is called modular. A lattice is modular iff it does not contain N 5 as a sublattice (see Fig. 2). A lattice is distributive when ∨, ∧ satisfy the distributivity law, and it is complemented when each x ∈ L has a (unique) complement x ′ , i.e. satisfying x ∨ x ′ = ⊤ and x ∧ x ′ = ⊥. A modular lattice is distributive iff it does not contain M 3 as a sublattice (see Fig. 2). A lattice is linear if it is totally ordered. A lattice is said to be Boolean if it has a top and bottom element, is distributive and complemented. When L is finite, it is Boolean iff it is isomorphic to the lattice 2 n for some n. (L, ≤) is said to be lower locally distributive if it is lower semi-modular, and it does not contain a sublattice isomorphic to M 3 . Equivalently, it is lower locally distributive if for any x ∈ L, the interval [ y≺x y, x] is a Boolean lattice (see [17] for a survey).
An element i ∈ L is join-irreducible if it cannot be written as a supremum over other elements of L and it is not the bottom element. When L is finite, this is equivalent to i covers only one element. Let us call J (L) the set of all join-irreducible elements of L.
In a finite distributive lattice, any element y ∈ L can be decomposed in terms of join-irreducible elements. The fundamental result due to Birkhoff is the following. We call η(x) the normal decomposition of x, we have
x = η(x).
The isomorphism says that x ≤ y iff η(x) ⊆ η(y), hence η(x ∨ y) = η(x) ∪ η(y) and so on. The decomposition of some x in L in term of supremum of join-irreducible elements is unique up to the fact that it may happen that some join-irreducible elements in η(x) are comparable. Hence, if i ≤ j and j is in a decomposition of x, then we may delete i from the decomposition of x. We call minimal decomposition the (unique) decomposition of minimal cardinality, denoted by η * (x). Atoms are join-irreducible elements covering ⊥. A lattice is atomistic if all join-irreducible elements are atoms. A finite distributive atomistic lattice is Boolean.
As shown by Dilworth [5], any x ∈ L has a unique join-irreducible minimal decomposition iff it is lower locally distributive.
A useful result is the following
↓ x = {y | η(y) = j∈K j, K ⊆ η(x)}.(10)
When L = 2 n , join-irreducible elements are simply atoms (i.e. singletons of N). When L = 3 n , join-irreducible elements are (i, i c ) and (∅, i c ), ∀i ∈ N.
Let (L, ≤) be a locally finite partially ordered set, and a function g : L −→ R. Consider the following equation
g(x) = y≤x f (y).(11)
There is a unique solution f : L −→ R to this equation, called the Möbius transform of g (see Rota [18]). Note that in a sense, f could be considered as the derivative of g.
As said in the introduction, our general framework for the definition of interaction will be to consider finite lower locally distributive lattices L 1 , . . . , L n , with top and bottom of L i denoted by ⊤ i , ⊥ i , i = 1, . . . , n, and the product lattice L := L 1 × · · · × L n with the product order. Sometimes, we will need in addition that the L k 's are modular (hence they are distributive). We set N := {1, . . . , n}. A vertex of L is an element x = (x 1 , . . . , x n ) of L where x i is either ⊤ i or ⊥ i , for i = 1, . . . , n. We denote by Γ(L) the set of vertices of L. Note that if L is a Boolean lattice, then L = Γ(L). For Q(N), vertices are of the form (A, A c ), A ⊆ N.
Since L i is finite and lower locally distributive, it can be represented by join-irreducible elements. Then join-irreducible elements of L are simply of the form
i = (⊥ 1 , . . . , ⊥ j−1 , i 0 , ⊥ j+1 , . . . , ⊥ n ),
for some j ∈ {1, . . . , n} and some i 0 ∈ J (L j ). Hence, there are n j=1 |J (L j )| joinirreducible elements in L.
Derivative of a function over a lattice
Let (L, ≤) be a finite lower locally distributive lattice, and f : L −→ R a real-valued function on it.
Definition 1 Let i ∈ J (L). The derivative of f w.r.t. i at point x ∈ L is given by:
∆ i f (x) := f (x ∨ i) − f (x). Note that ∆ i f (x) = 0 if i ≤ x. We say that the derivative ∆ i f (x) is Boolean if [x, x ∨ i]
is the Boolean lattice 2 1 , otherwise said x ∨ i ≻ x. Differentiating two times w.r.t two join-irreducible elements i, j such that i||j (i and j are incomparable) leads to:
∆ i (∆ j f (x)) = ∆ j (∆ i f (x)) = f (x ∨ i ∨ j) − f (x ∨ i) − f (x ∨ j) + f (x).
We call this quantity the second derivative w.r.t i, j or the derivative w.r.t i ∨ j, denoted by ∆ i∨j f (x). Note that allowing i ≤ j leads to ∆ i∨j f (x) = −∆ i f (x).
Using the minimal decomposition, the derivative w.r.t any element y can be defined.
Definition 2 Let x, y ∈ L, and y = ∨ n k=1 i k be the minimal decomposition of y into join-irreducible elements. Then the derivative of f w.r.t y at point x is given by:
∆ y f (x) = ∆ i 1 (∆ i 2 (· · · ∆ in f (x) · · · )).
The derivative is Boolean if [x, x ∨ y] is the Boolean lattice 2 n . The derivative is 0 if for some k, i k ≤ x. The following lemma gives practical equivalent conditions.
Lemma 1 Let x, y ∈ L. (i) The derivative ∆ y f (x) is 0 whenever η(x) ∩ η * (y) = ∅. (ii) The derivative ∆ y f (x) is Boolean iff η(x ∨ y) = η(x) ∪ η * (y).
Proof: (i) Let k ∈ η(x) ∩ η * (y). Since k ∈ η(x), all join-irreducible elements below k are also in η(x), hence η(k) ⊆ η(x). By Th. 1, this is equivalent to k ≤ x, which in turn implies that the derivative is 0 since k ∈ η * (y). (ii) Let us consider first y = i ∈ J (L), and suppose ∆ i f (x) is Boolean. Since x ∨ i ≻ x, by isomorphism, we have η(x ∨ i) ≻ η(x), which means that there exists some k ∈ J (L) such that η(x ∨ i) = η(x) ∪ {k}. Since η(x ∨ i) = η(x) ∪ η(i), k ∈ η(i), and all other j ∈ η(i) belong also to η(x). Hence k = i = η * (i) since η(i) = {j ∈ J (L) | j ≤ i}. The converse is clear. Applying recursively this result proves (ii).
As a consequence of (ii), the lattice [x, x ∨ y] is isomorphic to (P(η * (y)), ⊆). We express the derivative in terms of the Möbius transform of f .
Proposition 1 Let i be a join-irreducible element such that ∆ i f (x) is Boolean. We denote by m the Möbius transform of f . Then
∆ i f (x) = y∈[i,x∨i] m(y).
Proof: We have:
∆ i f (x) = y≤x∨i m(y) − y≤x m(y) = y∈↓(x∨i)\↓x m(y), since ↓ x ⊂↓ (x ∨ i). Using Lemma 1 (ii), we have η(x ∨ i) = η(x) ∪ {i}. Applying (10), we get ↓ (x ∨ i)\ ↓ x = {y | η(y) = j∈K j ∪ {i}, K ⊆ η(x)} = [i, x ∨ i]
since we get i for K = ∅, and x ∨ i for K = η(x), and the set is clearly an interval.
Based on this, we can show the general result:
Theorem 2 Let x, y ∈ L, such that ∆ y f (x) is Boolean.
Then
∆ y f (x) = z∈[y,x∨y] m(z).
Proof: We proceed by recurrence on |η * (y)|. The result is already shown for |η * (y)| = 1. Let us suppose it holds for some y, and consider y ′ = y ∨ i, with i ∈ η(y) and ∆ y ′ f (x) being Boolean. We have: m(z).
∆ y ′ f (x) = ∆ i (∆ y f (x)) = ∆ y f (x ∨ i) − ∆ y f (x) = z∈[y,Since [y, x ∨ y] = {z | η(z) = η(y) ∪ j∈J j, J ⊆ η(x)} and [y, x ∨ y ∨ i] = {z | η(z) = η(y) ∪ j∈J j, J ⊆ η(x) ∪ {i}} we get [y, x ∨ y ∨ i] \ [y, x ∨ y] = {z | η(z) = η(y) ∪ j∈J j ∪ {i}, J ⊆ η(x)} = [y ′ , y ′ ∨ x],
the desired result.
The close link between our derivative and Möbius transform is not surprising since the Möbius transform has already a meaning of derivative.
Let us apply these results to the case of usual capacities and bi-capacities. It suffices to check if formulas coincide for join-irreducible elements. For capacities, we have for
any i ∈ N, ∆ i v(A) := v(A ∪ i) − v(A)
, so that we recover the definition above. Note that this coincides with the notion of derivative for pseudo-Boolean functions [14]. For bi-capacities, we have
∆ (i,i c ) v(A, B) = v(A ∪ i, B) − v(A, B) = ∆ i,∅ v(A, B) ∆ (∅,i c ) v(A, B) = v(A, B \ i) − v(A, B) = ∆ ∅,i v(A, B),
which again coincides with the definition given above, although notation differs.
Interaction: the general case
As seen in Section 2, the definition of the derivative is the key concept for the interaction index. Using our general definition of derivative with new notation, let us express the interaction when L = 3 n using the notation ∆ (S,T ) . Imposing the same argument to I and ∆, we are led to:
I v (S, T ) = K⊆T (t − k)!k! (t + 1)! ∆ (S,T ) v(K, N \ (K ∪ S)),(12)
with the correspondence I v S,T = I v (S, (S∪T ) c ). Observe that these two notations precisely correspond to those introduced in Eqs. (2).
We remark that the derivative in the above expression is taken over some vertices of Q(N \ S). Also, the importance index corresponds to derivatives w.r.t. join-irreducible elements. Based on these observations, we are now in position to propose a definition using our general framework (see Section 3). Roughly speaking, the interaction index w.r.t. x ∈ L is a weighted average of the derivative w.r.t. x, taken at vertices of L "not related" to x. The weights can be determined recursively from the cases where x is a join-irreducible element, and the coefficients for these cases are determined by some normalization condition (e.g. efficiency-like condition in the case of the Shapley index).
Definition of interaction
We begin by defining the importance index, i.e. interaction index w.r.t. a join-irreducible element.
Definition 3 Let i = (⊥ 1 , . . . , ⊥ j−1 , i 0 , ⊥ j+1 , . . . , ⊥ n ) be a join-irreducible element of L.
The interaction w.r.t. i of v is any function of the form
I(i) := x∈Γ( Q j−1 k=1 L k )×{i 0 }×Γ( Q n k=j+1 L k ) α 1 h(x) ∆ i v(x),(13)
where i 0 is the (unique) predecessor of i 0 in L j , h(x) is the number of components of x equal to ⊤ l , l = 1, . . . , n, and α 1 k ∈ R for any integer k.
Observe that the constants α 1 h(x) do not depend on i. Also, the derivative is Boolean. Let us show that this definition encompasses the case of capacities and bi-capacities. For capacities, L k = {0, 1} for all k, with 1 as unique join-irreducible element, joinirreducible elements of L = 2 n are singletons, all elements in L are vertices, and h(x) is the cardinality of sets. Thus we get for a singleton j ∈ N:
I(j) = A⊆N \j α 1 |A| [v(A ∪ j) − v(A)]
as desired.
I(j, j c ) = A⊆N \j α 1 |A| ∆ (j,j c ) v(A, N \ (A ∪ j))
which has the required form. Let us examine now the case of (∅, j c ), which is, in vector form, (−1, . . . , −1, 0, −1, . . . , −1). This time Γ(3 j−1 ) × {−1} × Γ(3 n−j ) is Γ(L), after removal of vertices (A, A c ) with j ∈ A. In summary, we obtain:
I(∅, j c ) = A⊆N \j α 1 |A| ∆ (∅,j c ) v(A, N \ A)
which has again the required form. Let us generalize Def. 3 to a class of elements of L denoted byL and defined as follows:L := K⊆NL K , with
L K := {x ∈ L | ∀k ∈ K, ∃! i k ∈ L k such that ∀i ∈ η * (x k ), i ≻ i k , and x k = ⊥ k if k ∈ N\K}
In words, it is the set of elements whose coordinates are either bottom or such that the minimal decomposition covers a unique element. Observe that for the case where L k is a linear lattice or an atomistic one (i.e. practical cases of interest),L = L.
Definition 4 Let K ⊆ N, x ∈L K , and denote as above by i k , for all k ∈ K, the element covered by all i ∈ η * (x k ). The interaction w.r.t. x of v is any function of the form
I(x) := y|y k =⊤ k or ⊥ k if k ∈K,y k =i k else α |K| h(y) ∆ x v(y)(14)
where h(y) is the number of components of y equal to ⊤ l , l = 1, . . . , n.
The derivative is Boolean if in addition the L k 's are modular (and hence distributive), by application of the following Lemma.
Lemma 2 If L k is distributive, k = 1, . . . , n, then for any K ⊆ N, any x ∈L K , ∆ x v(y) is Boolean for any y such that y k = ⊤ k or ⊥ k , k ∈ K, and y k = i k , where i k is the element covered by all i ∈ η * (x k ).
Expression with the Möbius transform and efficiency
Let us express I(x) w.r.t the Möbius transform. First we recall the result for bi-capacities, which writes [9,11]:
I(S, T ) = (S ′ ,T ′ )∈[(S,T ),(S∪T,∅)] 1 t − t ′ + 1 m(S ′ , T ′ ).
We have the following general result.
Theorem 3 Let K ⊆ N, and assume distributivity holds for every L k , k ∈ K. The expression of the interaction index for x ∈L K in terms of the Möbius transform is given by:
I(x) = z∈[x,x] β |K| k(z) m(z),β |K| k(z) = n−k(z) l=0 n − k(z) l α |K| (k(z)−|K|+l)(15)
Proof: Since the derivative is Boolean by Lemma 2, we can apply Th. 2, and we get:
I(x) = y|y k =⊤ k or ⊥ k if k ∈K,y k =i k else α |K| h(y) z∈[x,y∨x] m(z).(16)
Then for any y such that y k = ⊤ k or ⊥ k if k ∈ K, and y k = i k else, (y ∨ x) k = x k when k ∈ K, other coordinates being ⊤ k or ⊥ k , in any combination. Hence for all possible such y, z takes any value in [x, (⊤ N \K , x K )], where (⊤ N \K , x K ) has coordinate ⊤ k when k ∈ K, and x k else. Denoting byx the right bound of this interval, we get
I(x) = z∈[x,x]
β z m(z).
It remains to express β z in terms of α |K| h(x) . Let us take a fixed z ∈ [x,x] and examine for which y's in (16) it belongs to [x, y ∨ x]. Note that z k = x k for all k ∈ K. Since y l , l ∈ K is either ⊥ l or ⊤ l , we must have y l = ⊤ l whenever z l = ⊥ l , the other coordinates not in K being free. The result is then:
β z = y|y l =⊤ l if z l =⊥ l ,l ∈K α |K| h(y) .
Denoting by k(z) the number of coordinates not equal to ⊥ l , we get
β z = n−k(z) l=0 n − k(z) l α |K| k(z)−|K|+l
Remarking that β z depends only on k(z) and |K|, we get the desired result.
Let us check if we recover the coefficients for bi-capacities and Shapley index. For (S, T ) = (i, i c ) and (∅, i c ), we have β (S ′ ,T ′ ) = 1 n−t ′ . We apply (15), noting that (S ′ , T ′ ) has n − t ′ coordinates different from bottom:
β (S ′ ,T ′ ) = t ′ l=0 t ′ l α 1 n−t ′ −1+l = t ′ l=0 t ′ l (n − t ′ − 1 + l)!(t ′ − l)! n! = t ′ l=0 t ′ !(n − t ′ − 1 + l)! l!n! .
In [11], the following combinatorial result was shown:
k i=0 (n − i − 1)!k! n!(k − i)! = 1 n − k .
Applying the above formula with i = t ′ − l, we get the desired result.
It is possible to find easily the β 1 k(z) coefficients if we consider a normalization condition as for the Shapley index. Let us define efficiency as i∈J (L)
I(i) = v(⊤) − v(⊥),(17)
and call Shapley interaction index the resulting interaction index. Applying Th. 3, we get:
i∈J (L)
I(i) = i∈J (L) z∈[i,ǐ] β 1 k(z) m(z).
Let us take m such as it is non zero only for a given z ∈ L, say z 0 , such that for all coordinates z l different from bottom, we have z l ∈ J (L l ). Since x∈L m(x) = v(⊤), we have necessarily m(z 0 ) = v(⊤) − v(⊥). Observe that z 0 belongs to all intervals [i,ǐ] such that z 0 ≥ i and z 0 ≤ǐ. Recalling that i = (⊥ 1 , . . . , ⊥ j−1 , i 0 , ⊥ j+1 , . . . , ⊥ n ) anď ı = (⊤ 1 , . . . , ⊤ j−1 , i 0 , ⊤ j+1 , . . . , ⊤ n ), if z has only coordinate z l = ⊥ l , then only i such that i l = z l is suitable. More generally, if z has only k coordinates different from bottom, then we have only k choices for i. Hence, for such z
β 1 k(z) = v(⊤) − v(⊥) k(z)[v(⊤) − v(⊥)] = 1 k(z) .
Let us apply this to the Shapley index for bi-capacities. We get:
β 1 (S ′ ,T ′ ) = 1 n − t ′ .
Suppose the β 1 k(z) 's are determined by some rule, as above. Since k(z) takes values in {1, . . . , n}, there are n coefficients β 1 k(z) , while for α 1 h(x) , h(x) ∈ {0, . . . , n−1}, so that there are also n coefficients. Th. 3 tells us that α 1 0 , . . . , α 1 n−1 can be computed from β 1 1 , . . . , β 1 n by solving the triangular linear system (15). Since there is no 0 on the diagonal of the matrix, there is always a unique solution to this system.
Applying this observation to the Shapley interaction index, we get the following result.
Theorem 4 When L k is distributive, for all k = 1, . . . , n, the coefficients α 1 0 , . . . , α 1 n−1 of the Shapley interaction index I(i), i ∈ J (L) (i.e. satisfying Def. 3 and (17)) are given by:
α 1 k = (n − 1 − k)!k! n! .
The recursion axiom for the linear case
Let us generalize the recursion axiom (6) to compute I(x), with the following additional restriction: all L k 's are linear lattices. Hence, all previous results apply. Also, all derivatives involved are Boolean. Let J ⊆ N, and consider x such that x k = ⊥ k if k ∈ J, and x k = i k else, for some i k ∈ J (L k ). We denote as before by i k the unique predecessor of i k . We introduce additional notations. For any K ⊆ J, K = ∅, J, the function v restricted to k∈N \K L k is denoted by v N \K x and defined by:
v N \K x (y) := v(y ′ ), with y ′ k := i k , if k ∈ K y k , else , ∀y ∈ k∈N \K L k . The function v reduced to x is a function v [x] defined on k∈N \J L k × {⊥ [x] , ⊤ [x] } by: v [x] (y) := v(φ [x] (y)), ∀y ∈ k∈N \J L k × {⊥ [x] , ⊤ [x] }, and φ [x] : k∈N \J L k × {⊥ [x] , ⊤ [x] } −→ L is defined by φ [x] (y) := y ′ , with y ′ k := i k , if k ∈ J and y [x] = ⊤ [x] i k , if k ∈ J and y [x] = ⊥ [x] y k , if k ∈ J.
We propose the following recursion formula:
I v (x) = I v [x] (⊥ N \J , ⊤ [x] ) − K⊆J,K =∅,J I v N\K x (x |N \K ),(18)
where ⊥ N \J stands for the vector (⊥ k ) k∈N \J , and x |N \K is the restriction of x to coordinates in N \ K.
Let us check if we recover (6) ) writes [S] in our case, so that the formula is recovered.
The following result holds.
Theorem 5 Denoting by α j k (n) the coefficients α j k involved into (14), the recursion formula (18) induces the following recursive relation:
α j k (n) = α 1 k (n − j + 1), ∀k = 0, . . . , n − j, ∀j = 1 . . . , n.(19)
Proof: We prove the result by recurrence on j := |J|. It is obviously true for j = 1, and let us assume it is true up to j − 1. Simplifying notations, the left term in (18) writes:
I v (x) = y N\J ∈Γ( Q k∈N\J L k ) y J =i J α j h(y) (n)∆ x v(y) = y N\J ∈Γ( Q k∈N\J L k ) α j h(y N\J ) (n)∆ x v(y N \J , i J )
where y A indicates the vector y restricted to coordinates in A, and i J is the vector with coordinates i k , k ∈ J. Using similar notations, the right term writes:
y N\J ∈Γ( Q k∈N\J L k ) y [x] =⊥ [x] α 1 h(y N\J ) (n − j + 1)∆ ⊤ [x] v [x] (y) − ∅ =K⊂J y N\J ∈Γ( Q k∈N\J L k ) y J \K =i J \K α j−k h(y N\J ) (n − k)∆ x N\K v N \K x (y) = y N\J ∈Γ( Q k∈N\J L k ) α 1 h(y N\J ) (n − j + 1) ∆ ⊤ [x] v [x] (y N \J , ⊥ [x] ) − ∅ =K⊂J ∆ x N\K v N \K x (y N \J , i J\K )
where equality comes from the recurrence hypothesis. Hence, Eq. (18) is equivalent to:
y N\J ∈Γ( Q k∈N\J L k ) α j h(y N\J ) (n)∆ x v(y N \J , i J ) − α 1 h(y N\J ) (n − j + 1) ∆ ⊤ [x] v [x] (y N \J , ⊥ [x] ) − ∅ =K⊂J ∆ x N\K v N \K x (y N \J , i J\K ) = 0
Since the equality holds for any v, we should have for any y 0 ∈ Γ( k∈N \J L k ):
α j h(y 0 ) (n)∆ x v(y 0 , i J ) − α 1 h(y 0 ) (n − j + 1) ∆ ⊤ [x] v [x] (y 0 , ⊥ [x] ) − ∅ =K⊂J ∆ x N\K v N \K x (y 0 , i J\K ) = 0
We are done if we prove that
∆ x v(y 0 , i J ) − ∆ ⊤ [x] v [x] (y 0 , ⊥ [x] ) + ∅ =K⊂J ∆ x N\K v N \K x (y 0 , i J\K ) = 0.(20)
The derivative ∆ x v(y 0 , i J ) is the sum of terms ±v(z), with z j = i j or z j = i j whenever j ∈ J. We may assume w.l.o.g. that J = {1, . . . , j}. We associate to each such z a set K ⊆ J containing the coordinates where z j = i j , and denote with some abuse of notation v(z) by v(K). Hence ∆ x v(y 0 , i J ) can be represented by the sum: Using the last 2 expressions, the right side of (20) writes:
K⊆J (−1) |K| v(J \ K) − v(J) + v(∅) + ∅ =K⊂J L⊆J\K (−1) |L| v(J \ (K ∪ L)) = K⊆J L⊆J\K (−1) |L| v(J \ (K ∪ L)) − v(J) = K ′ ⊆J v(J \ K ′ ) k ′ k=0 k ′ k (−1) k ′ −k − v(J) = 0.
Note that α j k (n) depends only on k and n − j. Using (19), we are now able to give the coefficients for the interaction index, which coincide with those of (12): α j k = (n − j − k)!k! (n − j + 1)! .
Concluding remarks
We end the paper by giving some interpretation of our definition of interaction, and indicate perspectives. Taking a particular combination of reference levels for dimensions in K ⊆ N, denoted by x in Def. 4, we compute the "difference with alternate signs" between the value of the function v at this point x and point i K , which is the combination of levels obtained by just removing one after the others the join-irreducible elements composing x. Now, for dimensions outside K, we consider only the combination of extreme values ⊥ k , ⊤ k , k ∈ K, instead of all possible combinations of reference levels, which would have been too much complicated. The interaction index I(x) is just the weighted average of all these "difference with alternate signs" between x and i K , computed over all possible combinations of ⊥ k , ⊤ k , for k ∈ K. To our opinion, this is the simplest possible way to define it, encompassing classical cases of L = 2 n and 3 n . Observe however that our definition cannot be applied for all x ∈ L, but only toL (see definition in Sec. 5). This restriction seems however of little effect, since it does not concern linear or atomistic lattices (which include, e.g., Boolean lattices and the partition lattice), the most useful cases in practice.
Results on the particular form of α 1 k remain simple and identical to the classical cases whenever the L k 's are distributive, since in this case derivatives become Boolean, hence the underlying structure of computation is identical to the classical case L = 2 n . For other cases, specific computations have to be done.
Lastly, the recursion axiom permits to derive all coefficients α j k from the α 1 k 's, provided all L k 's are linear. A further way of research would be to propose a more general formula, which seems however at first sight, difficult.
Figure 1 :
1Ternary tuples when n = 2
T ) is the contribution of element i when it acts as a positive element, while ∆ ∅,i v(S, T ) is the (negative) contribution of i when acting as a negative element. Hence the two above Shapley values are average contributions of an element when it acts as a positive or as a negative element.The coefficients are obtained through an efficiency axiom which reads: i∈N
Figure 2 :
2The lattices M 3 (left) and N 5 (right)
Theorem 1
1Let L be a finite distributive lattice. Then the map η : L −→ O(J (L)), where O(J ) is the set of all downsets of J , defined by η(x) := {i ∈ J (L) | i ≤ x} = J (L)∩ ↓ x is an isomorphism of L onto O(J (L)).
For bi-capacities, L k = {−1, 0, 1} for all k, with J (L k ) = {0, 1}. The height function is h(A, B) = |A|. Let us consider first the case where the join-
withx k := (⊤ k ) for k ∈ K,x k = x k else, and k(z) is the number of coordinates of z not equal to ⊥ l , l = 1, . . . , n. Moreover, the real constants β|K| k(z) are related to the α |K| h(x) 's by:
for capacities. Taking S ⊆ N, the restricted game v N \K S for ∅ = K ⊂ S, is defined by v N \K S (T ) = v(T ) if T ⊆ N \ K, and does not depend on S. The reduced game is defined over N \ S ∪ {[S]}, and φ(T ) = T if T ∋ [S], and T \ {[S]} ∪ S else. Now observe that (⊥ N \J , ⊤ [x]
)
|K| v(J \ K). Similarly, we have ∆ ⊤ [x] v [x] (y 0 , ⊥ [x] ) = v(J) − v(∅) by definition of v [x] , and ∆ x N\K v N \K x (y 0 , i J\K ) = L⊆J\K (−1) |L| v(J \ (K ∪ L)).
Proof:We have to prove that [y, x ∨ y] is isomorphic to 2 |η * (x)| , with y defined as above., which is atomistic. Since it is also distributive, it is Boolean and isomorphic to 2 |η * (x k )| .
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| [] |
[
"Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case",
"Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case"
] | [
"Yunyan Yang \nDepartment of Mathematics\nRenmin University of China\n100872BeijingP. R. China\n",
"Xiaobao Zhu \nDepartment of Mathematics\nRenmin University of China\n100872BeijingP. R. China\n"
] | [
"Department of Mathematics\nRenmin University of China\n100872BeijingP. R. China",
"Department of Mathematics\nRenmin University of China\n100872BeijingP. R. China"
] | [] | The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let (Σ, β) be a closed Riemann surface with a divisor β, and K λ = K + λ, where K : Σ → R is a Hölder continuous function satisfying max Σ K = 0, K 0, and λ ∈ R. If the Euler characteristic χ(Σ, β) is negative, then by a variational method, it is proved that there exists a constant λ * > 0 such that for any λ ≤ 0, there is a unique conformal metric with the Gaussian curvature K λ ; for any λ, 0 < λ < λ * , there are at least two conformal metrics having K λ its Gaussian curvature; for λ = λ * , there is at least one conformal metric with the Gaussian curvature K λ * ; for any λ > λ * , there is no certain conformal metric having K λ its Gaussian curvature. This result is an analog of that of Ding and Liu[14], partly resembles that of Borer, Galimberti and Struwe[3], and generalizes that of Troyanov [26] in the negative case.Keywords: Prescribing Gaussian curvature, conical singularity 2010 MSC: 58E30, 53C20The Gauss-Bonnet formula leads to Σ Ke 2u dv g = Σ κdv g = 2πχ(Σ).Note that the solvability of (1) is closely related to the sign of χ(Σ). If χ(Σ) > 0, then Σ is either the projective space RP 2 or the 2-sphere S 2 . In the case of RP 2 , it was shown by Moser | 10.5186/aasfm.2019.4411 | [
"https://arxiv.org/pdf/1706.02059v1.pdf"
] | 119,675,761 | 1706.02059 | c309f535c4e3dd53f6ca171553569950bb7d19e5 |
Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case
7 Jun 2017
Yunyan Yang
Department of Mathematics
Renmin University of China
100872BeijingP. R. China
Xiaobao Zhu
Department of Mathematics
Renmin University of China
100872BeijingP. R. China
Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case
7 Jun 2017Prescribing Gaussian curvatureconical singularity 2010 MSC: 58E3053C20
The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let (Σ, β) be a closed Riemann surface with a divisor β, and K λ = K + λ, where K : Σ → R is a Hölder continuous function satisfying max Σ K = 0, K 0, and λ ∈ R. If the Euler characteristic χ(Σ, β) is negative, then by a variational method, it is proved that there exists a constant λ * > 0 such that for any λ ≤ 0, there is a unique conformal metric with the Gaussian curvature K λ ; for any λ, 0 < λ < λ * , there are at least two conformal metrics having K λ its Gaussian curvature; for λ = λ * , there is at least one conformal metric with the Gaussian curvature K λ * ; for any λ > λ * , there is no certain conformal metric having K λ its Gaussian curvature. This result is an analog of that of Ding and Liu[14], partly resembles that of Borer, Galimberti and Struwe[3], and generalizes that of Troyanov [26] in the negative case.Keywords: Prescribing Gaussian curvature, conical singularity 2010 MSC: 58E30, 53C20The Gauss-Bonnet formula leads to Σ Ke 2u dv g = Σ κdv g = 2πχ(Σ).Note that the solvability of (1) is closely related to the sign of χ(Σ). If χ(Σ) > 0, then Σ is either the projective space RP 2 or the 2-sphere S 2 . In the case of RP 2 , it was shown by Moser
Introduction
The problem of prescribing Gaussian curvature on smooth Riemann surfaces has been well understood [19]. Let (Σ, g) be a closed smooth Riemann surface, χ(Σ) be its topological Euler characteristic, and κ : Σ → R be its Gaussian curvature. Ifḡ = e 2u g with a smooth function u, then the Gaussian curvature of (Σ,ḡ) satisfiesκ = e −2u (κ + ∆ g u), where ∆ g denotes the Laplacce-Beltrami operator with respect to the metric g. A natural question is whether for any smooth function K : Σ → R, there is a smooth function u such that the metric e 2u g has K its Gaussian curvature. Clearly this is equivalent to solving the elliptic equation [22] that the equation (1) has a solution u, provided that K ∈ C ∞ (S 2 ) satisfies sup Σ K > 0 and K(p) = K(−p) for all p ∈ S 2 . While the problem on S 2 is much more complicated and known as the Nirenberg problem, see for examples [19,4,5,6,7]. If χ(Σ) = 0, the problem has been completely solved by Kazdan-Warner [19]. While if χ(Σ) < 0, the problem was studied by Kazdan and Warner [19] via the method of upper and lower solutions. They proved that if K ≤ 0 and K 0, then (1) has a unique solution. Later, Ding and Liu [14] considered the case that K changes sign. Precisely, replacing K by K + λ in (1) with K ≤ 0, K 0, and λ ∈ R, they obtained the following conclusion by using a method of upper and lower solutions and a variational method: there exists a λ * > 0 such that if λ ≤ 0, then (1) has a unique solution; if 0 < λ < λ * , then (1) has at least two solutions; if λ = λ * , then (1) has at least one solution; if λ > λ * , then (1) has no solution. Recently, using a monotonicity technique due to Struwe [24,25], Borer, Galimberti, and Struwe [3] partly reproved the above results and obtained additional estimates for certain sequence of solutions that allow to characterize their bubbling behavior. Further analysis in this direction has been done by Galimberti [16], del Pino and Román [13].
The problem of prescribing Gaussian curvature can also be proposed on surfaces with conical singularities. Let Σ be a closed Riemann surface, p 1 , · · · , p ℓ be points of Σ and θ 1 , · · · , θ ℓ be positive numbers. Denoteχ
= 2πχ(Σ) + ℓ i=1 (θ i − 2π).
Then it was proved by Troyanov [26] that if 0 <χ < min{4π, 2θ 1 , · · · , 2θ ℓ }, then any smooth function on Σ, which is positive at some point is the Gaussian curvature of a conformal metric having at p i a conical singularity of angle θ i ; ifχ = 0, then a smooth nonconstant function K : Σ → R is the Gaussian curvature of a conformal metric having at p i a conical singularity of angle θ i if and only if Σ Kdµ < 0, where dµ is the area element of the original singular metric; ifχ < 0, then any smooth negative function on Σ is the Gaussian curvature of a unique conformal metric having at p i a conical singularity of angle θ i . As in the smooth Riemann surface case, the prescribing Gaussian curvature problem on the 2-sphere with conical singularity is most delicate. The case ℓ = 2 was studied by Chen and Li [8,9]. While the case ℓ ≥ 3 was considered by Eremenko [15], Malchiodi and Ruiz [21], Chen and Lin [10], Marchis and López-Soriano [12], and others.
In this paper, we focus on the negative case, namelyχ < 0. Precisely we shall prove an analog of the result of Ding and Liu [14], and thereby part of results of Borer, Galimberti, and Struwe [3]. Though we still use the variational method, which had been employed by Ding and Liu, we have to overcome difficulties in the presence of conical singularities. In particular, we have to establish the strong maximum principle, which is essential for the method of upper and lower solutions in our setting.
The remaining part of this paper is organized as follows: In Section 2, we give some notations for surfaces with conical singularities and state our main results; In Section 3, the maximum principle for the Laplace-Beltrami operator and the Palais-Smale condition for certain functional are discussed; In Section 4, following the lines of [14,3], we prove our main theorem.
Notations and main results
Let us briefly recall some geometric concepts from Troyanov [26]. In general, a closed Riemann surface Σ is defined to be a topological space with an atlas {φ i :
U i → C}, where if U i ∩ U j ∅, then the coordinate transformation φ i • φ −1 j is conformal, i.e.
, holomorphic or 2 anti-holomorphic. Two such atlases define the same structure on Σ if their union is still such an atlas. A conformal Riemannian metric is defined by g = ρ(z)|dz| 2 locally, where z is a coordinate on Σ and ρ is a positive measurable function. A divisor on a Riemann surface is a formal sum β = ℓ i=1 β i p i , where p i ∈ Σ and β i > −1, i = 1, · · · , ℓ. The set supp β = {p 1 , · · · , p ℓ } is the support of β, and the number |β| = ℓ i=1 β i is the degree of the divisor. A conformal metric g on Σ is said to represent the divisor β if g ∈ C 2 (Σ \ supp β) verifying that if z i is a coordinate defined in a neighborhood U i of p i , then there is some
u i ∈ C 2 (U i \ {p i }) ∩ C 0 (U i \ {p i }) such that g = e 2u i |z i − z i (p i )| 2β i |dz i | 2 .(2)
Under the circumstances, g is said to have a conical singularity of order β i or angle θ i = 2π(β i +1) at p i , i = 1, · · · , ℓ. The Euler characteristic of (Σ, β) is defined by
χ(Σ, β) = χ(Σ) + |β|,
where χ(Σ) is the topological Euler characteristic of Σ, and |β| = ℓ i=1 β i is the degree of β. Let κ : Σ \ supp β → R be the Gaussian curvature of g. If κ can be extended to a Hölder continuous function on Σ, then it was shown by Troyanov [26] that a Gauss-Bonnet formula holds:
Σ κdv g = 2πχ(Σ, β),(3)
where dv g denotes the Riemannian volume element with respect to the conical metric g. Let (Σ, β) be a closed Riemann surface with a divisor β = ℓ i=1 β i p i , and the metric g represents β with β i > −1, i = 1, · · · , ℓ. It follows from (2) that there exists a smooth Riemannian metric g 0 such that g = ρg 0 ,
where ρ > 0 on Σ, ρ ∈ C 2 loc (Σ \ supp β), and ρ ∈ L r (Σ) for some r > 1. Let W 1,2 (Σ, g) be the completion of C ∞ (Σ) under the norm u W 1,2 (Σ,g) = Σ (|∇ g u| 2 + u 2 )dv g
1/2
, where ∇ g denotes the gradient operator with respect to the metric g. It was observed by Troyanov [26] that W 1,2 (Σ, g) = W 1,2 (Σ, g 0 ). As a consequence, by the Sobolev embedding theorem for smooth Riemann surface (Σ, g 0 ) and the Hölder inequality, one has W 1,2 (Σ, g) ֒→ L p (Σ, g), ∀p > 1.
We now state the following:
Theorem 1. Let (Σ, β) be a closed Riemann surface with a divisor β = ℓ i=1 β i p i . Suppose that the Euler characteristic χ(Σ, β) < 0, K : Σ → R is a Hölder continuous function, max Σ K = 0 and K 0. Let K λ = K + λ, λ ∈ R.
Assume that a conformal metric g represents β. Let κ : Σ \ supp β → R be the Gaussian curvature of g, and κ can be extended to a Hölder continuous function on Σ. Then there exists a constant λ * > 0 such that (i) when λ ≤ 0, there exists a unique conformal metric on Σ with Gaussian curvature K λ , representing the divisor β; (ii) when 0 < λ < λ * , there exist at least two conformal metrics on Σ with the same Gaussian curvature K λ , representing the divisor β; (iii) when λ = λ * , there exists at least one conformal metric on Σ with Gaussian curvature K λ * , representing the divisor β; (iv) when λ > λ * , there is no function u ∈ W 1,2 (Σ, g) ∩ C 2 (Σ \ supp β) ∩ C 0 (Σ) such that e 2u g has the Gaussian curvature K λ .
3
Since the metric g has the Gaussian curvature κ, and the metric g λ = e 2u g has the Gaussian curvature K λ = K + λ. A standard calculation shows
∆ g u + κ − K λ e 2u = 0 on Σ \ supp β.(6)
Note that if u ∈ W 1,2 (Σ, g) is a distributional solution of the equation
∆ g u + κ − K λ e 2u = 0 on Σ,(7)
we have by elliptic estimates u ∈ C 2 (Σ \ supp β) ∩ C 0 (Σ), and thus (6) holds. Hence, in order to prove Theorem 1, it suffices to show the following:
Theorem 2.
Under the same assumptions as in Theorem 1, there exists a λ * > 0 such that (i) if λ ≤ 0, then (7) has a unique distributional solution; (ii) if 0 < λ < λ * , then (7) has at least two distributional solutions; (iii) if λ = λ * , then (7) has at least one distributional solution; (iv) if λ > λ * , then (7) has no distributional solution.
For the proof of Theorem 2, we follow closely Ding and Liu [14] by employing a variational method. In particular we use the upper and lower solutions principle and the strong maximum principle. In the remaining part of this paper, (Σ, g) will always denote a conical singular Riemann surface given in Theorem 1; we do not distinguish sequence and subsequence; moreover we often denote various constants by the same C, even in the same line.
Preliminary analysis
In this section, we prove maximum principle, Palais-Smale condition, upper and lower solutions principle, which will be used later. Compared with the smooth Riemann surface case, all the above mentioned things need to be re-established since the metric g has conical singularity.
Maximum principle
We first have a weak maximum principle by integration by parts, namely
Lemma 3 (Weak maximum principle). For any constant c > 0, if u ∈ W 1,2 (Σ, g) ∩C 0 (Σ) satisfies ∆ g u + cu ≥ 0 in the distributional sense, then u ≥ 0 on Σ.
Proof. Denote u − = min{u, 0}. Testing the equation
∆ g u + cu ≥ 0 by u − , one has Σ (|∇ g u − | 2 + cu −2 )dv g ≤ 0.
This leads to u − ≡ 0 on Σ.
Moreover, using the Moser iteration (see for example Theorems 8.17 and 8.18 in [17]), we obtain the following strong maximum principle.
Lemma 4 (Strong maximum principle). Let u ∈ W 1,2 (Σ, g) ∩ C 0 (Σ) satisfy that u ≥ 0 on Σ, and that for some positive constant c, ∆ g u + cu ≥ 0 in the distributional sense. If there exists a point
x 0 ∈ Σ such that u(x 0 ) = 0, then there holds u ≡ 0 on Σ. Proof. Step 1. If v ∈ W 1,2 (Σ, g) ∩ C 0 (Σ) satisfies v ≥ 0 on Σ, and ∆ g v − cv ≤ 0 (8)
in the distributional sense, where c is a positive constant, then there exists some constant C depending only on
(Σ, g) such that v L ∞ (Σ) ≤ C v L 2 (Σ,g) .(9)
Now we use the Moser iteration to prove (9). For any p ≥ 2, testing (8) by v p−1 and integrating by parts, we have
Σ |∇ g v p 2 | 2 dv g ≤ cp 2 4(p − 1) Σ v p dv g . Hence v p 2 W 1,2 (Σ,g) ≤ C p v p 2 L 2 (Σ,g) for some constant C. Then the Sobolev embedding (5) leads to v p 2 L 4 (Σ,g) ≤ C p v p 2 L 2 (Σ,g) , which is equivalent to v L 2p (Σ,g) ≤ C 2 p p 2 p v L p (Σ,g) . Taking p = p k = 2 k , k = 1, 2, · · · , we have v L p k+1 (Σ,g) ≤ C 2 p k p 2 p k k v L p k (Σ,g) ≤ C k j=1 2 1− j 2 k j=1 2 1− j j v L 2 (Σ,g) ≤ C v L 2 (Σ,g) .(10)
Letting k → ∞ in (10), we conclude (9).
Step 2.
Let u ∈ W 1,2 (Σ, g) ∩ C 0 (Σ) be a nonnegative distributional solution of ∆ g u + cu ≥ 0,(11)
where c is a positive constant. Then there exists some constant C such that
u L 2 (Σ,g) ≤ C inf Σ u.(12)
Without loss of generality, we assume u ≥ ǫ > 0, otherwise we can replace u by u + ǫ. We claim that that u −1 is a distributional solution of ∆ g u −1 − cu −1 ≤ 0. To see it, we recall that g = ρg 0 , where ρ : Σ → R is a positive function, ρ ∈ L q (Σ) for some q > 1, and g 0 is a smooth Riemannian metric. Then for any φ ∈ W 1,2 (Σ, g 0 ) with φ ≥ 0, we calculate
Σ ∇ g u −1 ∇ g φ − cu −1 φ dv g = Σ ∇ g 0 u −1 ∇ g 0 φ − cρu −1 φ dv g 0 = − Σ ∇ g 0 u∇ g 0 (φu −2 ) + 2φu −3 |∇ g 0 u| 2 + cρu(φu −2 ) dv g 0 ≤ − Σ ∇ g 0 u∇ g 0 (φu −2 ) + cρu(φu −2 ) dv g 0 = − Σ ∇ g u∇ g (φu −2 ) + cu(φu −2 ) dv g .
This together with (11) confirms our claim. Now we have by Step 1,
sup Σ u −1 ≤ C u −1 L 2 (Σ,g) , 5 which leads to inf Σ u ≥ C Σ u −2 dv g − 1 2 = C Σ u −2 dv g Σ u 2 dv g −1/2 Σ u 2 dv g 1/2
. Thus, to prove (12), it suffices to show there exists some constant C such that
Σ u −2 dv g Σ u 2 dv g ≤ C.(13)Let w = log u − γ, where γ = 1 Vol g (Σ) Σ log u dv g . We shall prove that Σ e 2|w| dv g ≤ C,(14)
which implies
Σ e 2(γ−log u) dv g ≤ C, Σ e 2(log u−γ) dv g ≤ C.
This immediately leads to (13). We are only left to prove (14). Testing the equation (11) by u −1 , we have
Σ (∇ g u −1 ∇ g u + c)dv g ≥ 0.
It follows that Σ |∇ g w| 2 dv g ≤ C.
Note that Σ wdv g = 0. In view of (15), we conclude from the Poincaré inequality that
w W 1,2 (Σ,g) ≤ C.(16)
Recall that the metric g represents the divisor β = ℓ i=1 β i p i with β i > −1, i = 1, · · · , ℓ. Denote b = min{1, 1 + β 1 , · · · , 1 + β ℓ }. Then the Trudinger-Moser inequality for surfaces with conical singularities [26] together with (16) implies that Σ e 2|w| dv g ≤ Σ e bw 2 w 2 W 1,2 (Σ,g)
+ 1 b w 2 W 1,2 (Σ,g) dv g ≤ C Σ e bw 2 w 2 W 1,2 (Σ,g) dv g ≤ C.(17)
Thus (14) holds and the proof of Step 2 terminates. One can easily see that the conclusion of the lemma follows from (12).
It is remarkable that only subcritical Trudinger-Moser inequality was employed in (17). Such inequalities are important tools in geometry and analysis. For more details, we refer the reader to recent works [1,20,23,27,28,29,11,18] and the references therein. 6
Palais-Smale condition
For any λ ∈ R, we define a functional E λ :
W 1,2 (Σ, g) → R by E λ (u) = Σ (|∇ g u| 2 + 2κu − K λ e 2u )dv g ,(18)
where κ : Σ → R is the Gaussian curvature of g, K λ = K + λ is defined as in Theorem 1.
Lemma 5 (Palais-Smale condition). Suppose that Σ − λ = {x ∈ Σ : K λ < 0} is nonempty for some λ ∈ R. Then E λ satisfies the (PS ) c condition for all c ∈ R, i.e., if u j is a sequence of functions in W 1,2 (Σ, g) such that E λ (u j ) → c and dE λ (u j ) → 0, then there exists some u 0 ∈ W 1,2 (Σ, g) satisfying u j → u 0 in W 1,2 (Σ, g).
Proof. Let (u j ) be a function sequence such that E λ (u j ) → c and dE λ (u j ) → 0, or equivalently
Σ (|∇ g u j | 2 + 2κu j − K λ e 2u j )dv g = c + o j (1),(19)Σ (∇ g u j ∇ g ϕ + κϕ − K λ e 2u j ϕ)dv g = o j (1) ϕ W 1,2 (Σ,g) , ∀ϕ ∈ W 1,2 (Σ, g),(20)
where o j (1) → 0 as j → ∞. Note that supp β = {p 1 , · · · , p ℓ } is a set of finite points. Σ − λ \ supp β must contain a domain Ω such that the closure of Ω is also contained in Σ − λ \ supp β. In view of (4), there would exist two positive constants C 1 and C 2 depending only on Ω such that
C 1 g 0 ≤ g ≤ C 2 g 0 on Ω.
Denote u + j = max{u j , 0}. Based on an argument of Ding and Liu ( [14], Lemma 2), where a mistake was corrected by Borer, Galimberti and Struwe ([3], Appendix), for another domain Ω ′ ⊂⊂ Ω, there exists a positive constant C depending only on C 1 , C 2 and dist g (Ω ′ , ∂Ω) such that
Ω ′ (|∇ g u + j | 2 + u + j 2 )dv g ≤ C.(21)
Taking ϕ ≡ 1 in (20), one has
Σ K λ e 2u j dv g − Σ κdv g = o j (1).
This together with the Gauss-Bonnet formula (3) gives
Σ K λ e 2u j dv g = 2πχ(Σ, β) + o j (1).(22)
Inserting (22) into (19), we conclude
Σ (|∇ g u j | 2 + 2κu j )dv g = c + 2πχ(Σ, β) + o j (1).(23)
We now claim that u j is bounded in L 2 (Σ, g). Suppose not, there holds u j L 2 (Σ,g) → ∞. We set v j = u j / u j L 2 (Σ,g) . Note that
Σ κ u j u j 2 L 2 (Σ,g) dv g = o j (1). 7
This together with (23) leads to
Σ |∇ g v j | 2 dv g = o j (1).(24)
Hence v j is bounded in W 1,2 (Σ, g) and (24) leads to v j → γ in W 1,2 (Σ, g) for some constant γ.
Since v j L 2 (Σ,g) = 1, we have γ 0. It follows from (23) that
Σ κv j dv g ≤ o j (1).(25)
Letting j → ∞ in (25), we obtain 2πχ(Σ, β)γ ≤ 0 by using the Gauss-Bonnet formula (3). Since χ(Σ, β) < 0 and γ 0, we have γ > 0. On the other hand, we conclude by (21) that
Ω ′ (|∇ g v + j | 2 + v + j 2 )dv g = o j (1),
which leads to γ ≤ 0. This contradicts γ > 0 and confirms our claim.
Since u j is bounded in L 2 (Σ, g), we have by (23) that u j is bounded in W 1,2 (Σ, g). Up to a subsequence, we can assume u j converges to u 0 weakly in W 1,2 (Σ, g), strongly in L s (Σ, g) for any s > 1. A Trudinger-Moser inequality for surfaces with conical singularities [26] implies that e 2u j is bounded in L s (Σ, g) for any s > 1. Hence e u j converges to e u 0 in L s (Σ, g) for any s > 1. This together with (20)
leads to Σ |∇ g u j | 2 dv g = Σ (−κu 0 + K λ e 2u 0 u 0 )dv g + o j (1) = Σ |∇ g u 0 | 2 dv g + o j (1).
This implies that u j → u 0 in W 1,2 (Σ, g).
Upper and lower solutions principle
Let f : Σ × R → R be a smooth function. u ∈ W 1,2 (Σ, g) ∩ C 2 (Σ \ supp β) ∩ C 0 (Σ) is defined to be an upper (lower) solution to the elliptic equation
∆ g u + f (x, u) = 0,(26)
if u satisfies ∆ g u + f (x, u) ≥ (≤) 0 in the distributional sense on Σ and point-wisely in Σ \ supp β.
Lemma 6 (Upper and lower solutions principle). Suppose that ψ, ϕ ∈ W 1,2 (Σ, g)∩C 2 (Σ\supp β)∩ C 0 (Σ) are upper and lower solutions to (26) respectively, and that ϕ ≤ ψ on Σ. Then (26) has a solution u ∈ W 1,2 (Σ, g)
∩ C 2 (Σ \ supp β) ∩ C 0 (Σ) with ϕ ≤ u ≤ ψ on Σ.
Proof. We follow the lines of Kazdan and Warner [19]. Let A be a constant such that −A ≤ ϕ ≤ ψ ≤ A. Since Σ is closed, one finds a sufficiently large constant c such that G(
x, t) = ct + f (x, t) is increasing in t ∈ [−A, A]
for any fixed x ∈ Σ. Define an elliptic operator Lu = ∆ g u + cu for u ∈ W 1,2 (Σ, g) ∩ C 2 (Σ \ supp β) ∩ C 0 (Σ). Now we define
ϕ 0 = ϕ, ϕ j = L −1 (G(x, ϕ j−1 )), ∀ j ≥ 1 ψ 0 = ψ, ψ j = L −1 (G(x, ψ j−1 )), ∀ j ≥ 1. 8
Here L −1 : L 2 (Σ, g) → W 1,2 (Σ, g) is well defined due to the Lax-Milgram theorem. This together with the definition of upper and lower solutions and the monotonicity of G(x, t) with respect to t leads to
Lϕ ≤ Lϕ 1 = G(x, ϕ) ≤ G(x, ψ) = Lψ 1 ≤ Lψ.
Then the weak maximum principle (Lemma 3) implies that
ϕ ≤ ϕ 1 ≤ ψ 1 ≤ ψ.
By induction, we have
ϕ ≤ ϕ j−1 ≤ ϕ j ≤ ψ j ≤ ψ j−1 ≤ ψ, j = 1, 2, · · · .
Clearly we can assume that ϕ j converges to u 1 and ψ j converges to u 2 point-wisely. By elliptic estimates, one concludes that the above convergence is in C 2 loc (Σ \ supp β) ∩ C 0 (Σ). Moreover, v = u 1 or u 2 is a distributional solution to Lv = G(x, v).
Proof of Theorem 2
In this section, we prove Theorem 2 by using variational method.
Unique solution in the case λ ≤ 0
Proof of (i) of Theorem 2. Assume max Σ K = 0 and K 0. If λ < 0, this has been proved by Troyanov ([26], Theorem 1). We now consider the general case λ ≤ 0. Let E λ be the functional defined as in (18), where K λ = K + λ. Claim 1. E λ is strict convex on W 1,2 (Σ, g).
It suffices to prove that for any u ∈ W 1,2 (Σ), there exists some constant C > 0 such that
d 2 E λ (u)(h, h) ≥ C h 2 W 1,2 (Σ,g) ∀h ∈ W 1,2 (Σ, g).(27)
Suppose not. There would be a function u ∈ W 1,2 (Σ, g) and a function sequence (h j ) ⊂ W 1,2 (Σ, g) such that h j W 1,2 (Σ,g) = 1 for all j and d 2 E λ (u)(h j , h j ) → 0 as j → ∞. One may assume up to a subsequence, h j converges to h ∞ weakly in W 1,2 (Σ, g), strongly in L p (Σ, g) for any p > 1, and almost everywhere in Σ. Since
d 2 E λ (u)(h j , h j ) = 2 Σ (|∇ g h j | 2 − 2K λ e 2u h 2 j )dv g
and K λ ≤ 0, we conclude Σ |∇ g h j | 2 dv g → 0 and Σ K λ e 2u h 2 j dv g → 0, which leads to h ∞ ≡ C 0 for some constant C 0 , and further
C 2 0 Σ K λ e 2u dv g = Σ K λ e 2u h 2 ∞ dv g = lim j→∞ Σ K λ e 2u h 2 j dv g = 0.
Clearly Σ K λ e 2u dv g < 0, and thus C 0 = 0. This contradicts h ∞ L 2 (Σ,g) = lim j→∞ h j L 2 (Σ,g) = 1. Hence (27) holds.
Claim 2. E λ is coercive.
Since for any ǫ > 0, there exists a constant C(ǫ) such that Σ κudv g ≤ ǫ u 2 W 1,2 (Σ,g) + C(ǫ), it suffices to find some constant C > 0 such that for all u ∈ W 1,2 (Σ, g), there holds
Σ (|∇ g u| 2 − K λ e 2u )dv g ≥ C u 2 W 1,2 (Σ,g) .(28)
Suppose not. There would exist a sequence of functions (u j ) satisfying
Σ (|∇ g u j | 2 + u 2 j )dv g = 1, Σ (|∇ g u j | 2 − K λ e 2u j )dv g = o j (1).
It follows that up to a subsequence, u j converges to u * weakly in W 1,2 (Σ, g) and strongly in L p (Σ, g) for any p > 1. One easily see that
0 < Σ (|∇ g u * | 2 − K λ e 2u * )dv g ≤ lim j→∞ Σ (|∇ g u j | 2 − K λ e 2u j )dv g = 0,
which is impossible. Hence (28) holds.
In view of Claims 1 and 2, a direct method of variation shows inf u∈W 1,2 (Σ,g) E λ (u) can be attained by some u 0 ∈ W 1,2 (Σ, g) and u 0 is the unique critical point of E λ .
Existence of λ *
When λ = 0, the equation (7) becomes
∆ g u + κ − Ke 2u = 0 on Σ.(29)
Let u be a solution of (29). The linearized equation of (29) at u reads ∆ g v − 2Ke 2u v = 0, which has a unique solution v ≡ 0. By the implicit theorem, there is a sufficiently small s > 0 such that for any λ ∈ (0, s), the equation (7) has a solution. Define λ * = sup s : the equation (7) has a solution for any λ ∈ (0, s) .
One can see that λ * ≤ − min Σ K. For otherwise K λ > 0 for some λ < λ * . Integrating (7), we obtain
0 > 2πχ(Σ, β) = Σ κdv g = Σ K λ e 2u dv g ≥ 0,
which is impossible. In conclusion, we have 0 < λ * ≤ − min Σ K. Further analysis (Subsection 4.4, Claim 2) implies that λ * < − min Σ K.
Multiplicity of solutions for 0 < λ < λ *
Proof of (ii) of Theorem 2. Fix λ, 0 < λ < λ * . We shall seek two different solutions of (7), one is a strict local minimum of the functional E λ , the other is of the mountain-pass type. The proof will be divided into several steps below.
Step 1. Existence of upper and lower solutions.
Take λ 1 with λ < λ 1 < λ * . Let u λ 1 ∈ W 1,2 (Σ, g) ∩ C 2 (Σ \ supp β) ∩ C 0 (Σ) be a solution of (7) at λ 1 . Set ψ = u λ 1 . One can see that ψ is a strict upper solution of (7), namely
∆ g ψ + κ − K λ e 2ψ > 0.(31)
Clearly the equation
∆ g η = −κ + 1 Vol g (Σ) Σ κdv g(32)
has a distributional solution η ∈ W 1,2 (Σ, g) ∩ C 2 (Σ \ supp β) ∩ C 0 (Σ). Let ϕ = η − s, where s is a positive constant. Obviously ϕ < ψ on Σ for sufficiently large s.
Since Σ κdv g = 2πχ(Σ, β) < 0, we have ∆ g ϕ + κ − K λ e 2ϕ = 1 Vol g (Σ) Σ κdv g − K λ e 2η−2s < 0,(33)
provided that s is chosen sufficiently large. Thus ϕ is a strict lower solution of (7).
Step 2. The first solution of (7) can be chosen as a strict local minimum of E λ .
Let f λ (x, t) = ct − κ + K λ e 2t . Fix a sufficiently large positive constant c such that f λ (x, t) is increasing in t ∈ [−A, A], where A is a constant such that −A ≤ ϕ < ψ ≤ A. Let F λ (x, u) = u 0 f λ (x, t)dt. It is easy to see that E λ (u) = Σ |∇ g u| 2 dv g + c Σ u 2 dv g − 2 Σ F λ (x, u)dv g − Σ K λ dv g . Define a functionf λ (x, t) = f λ (x, ψ(x)) when t > ψ(x) f λ (x, t) when ϕ(x) ≤ t ≤ ψ(x) f λ (x, ϕ(x)) when t < ϕ(x)
and a functionalÊ
λ (u) = Σ (|∇ g u| 2 + cu 2 )dv g − 2 ΣF λ (x, u)dv g − Σ K λ dv g , whereF λ (x, t) = t 0f λ (x, s)ds.
ObviouslyÊ λ is bounded from below on W 1,2 (Σ, g). Denote
a = inf u∈W 1,2 (Σ,g)Ê λ (u).
Taking a function sequence (u j ) ⊂ W 1,2 (Σ, g) such thatÊ λ (u j ) → a as j → ∞. It follows that u j is bounded in W 1,2 (Σ, g), and thus up to a subesequence the Sobolev embedding and the Trudinger-Moser inequality lead to u j converges to some u λ weakly in W 1,2 (Σ, g), strongly in L q (Σ, g) for any q > 1, almost everywhere in Σ, and e 2u j converges to e 2u λ in L 1 (Σ, g).
HenceÊ λ (u λ ) ≤ a.
Then by the definition of a, we concludê
E λ (u λ ) = inf u∈W 1,2 (Σ,g)Ê λ (u).
As a consequence u λ satisfies the Euler-Lagrange equation
∆ g u λ + cu λ =f λ (x, u λ ) (34) 11
in the distributional sense. By elliptic estimates, one has u λ ∈ C 2 (Σ \ supp β) ∩ C 0 (Σ).
Noting that f (x, t) is increasing with respect to t ∈ [−A, A], we have
∆ g ϕ(x) + cϕ(x) ≤ f λ (x, ϕ(x)) ≤f λ (x, u λ (x)) ≤ f λ (x, ψ(x)) ≤ ∆ g ψ(x) + cψ(x)
in the distributional sense. In view of (31), (33) and (34), one concludes by the strong maximum principle (Lemma 4) that
ϕ(x) < u λ (x) < ψ(x), ∀x ∈ Σ.(35)
ObviouslyÊ λ (u) = E λ (u) for all u ∈ W 1,2 (Σ) with ϕ ≤ u ≤ ψ. For any h ∈ C 1 (Σ), we define a function ζ(t) = E(u λ + th), t ∈ R. In view of (35), there holds ϕ ≤ u λ + th ≤ ψ and thuŝ E λ (u λ + th) = E λ (u λ + th), provided that |t| is sufficiently small. Since u λ is a minimum ofÊ λ on
W 1,2 (Σ, g), we have ζ ′ (0) = dE λ (u λ )(h) = 0 and ζ ′′ (0) = d 2 E λ (u λ )(h, h) ≥ 0. Therefore we have Σ (∇ g u λ ∇ g h + κh − K λ e 2u λ h)dv g = 0, ∀h ∈ C 1 (Σ),(36)Σ (|∇ g h| 2 − 2K λ e 2u λ h 2 )dv g ≥ 0, ∀h ∈ C 1 (Σ).(37)
Since C 1 (Σ) is dense in W 1,2 (Σ, g), (36) and (37) still hold for all h ∈ W 1,2 (Σ, g). We further prove that there exists a positive constant C such that
d 2 E λ (u λ )(h, h) ≥ C h 2 W 1,2 (Σ,g) , ∀h ∈ W 1,2 (Σ, g).(38)
For the proof of (38), we adapt an argument of Borer, Galimberti, and Struwe ([3], Section 2).
Since
d 2 E λ (u λ )(h, h) ≥ 0 for all h ∈ W 1,2 (Σ, g), we have Λ := inf h W 1,2 (Σ,g) =1 d 2 E λ (u λ )(h, h) ≥ 0.
Suppose Λ = 0. We claim that there exists some h with h W 1,2 (Σ,g) = 1 such that d 2 E λ (u λ )(h, h) = 0. To see this, we let h j satisfy h j W 1,2 (Σ,g) = 1 and d 2 E λ (u λ )(h j , h j ) → 0 as j → ∞. Up to a subsequence, we can assume h j converges to some h weakly in W 1,2 (Σ, g), strongly in L q (Σ, g) for all q > 1, and almost everywhere in Σ. It follows that
lim j→∞ Σ |∇ g h j | 2 dv g = Σ 2K λ e 2u λ h 2 dv g ≤ Σ |∇ g h| 2 dv g .
This leads to h j → h in W 1,2 (Σ, g) as j → ∞, and confirms our claim. Moreover, since the g); that is, h is a weak solution of the equation
functional v → d 2 E λ (u λ )(v, v) attains its minimum at v = h, it follows that d 2 E λ (u λ )(h, w) = 0 for all w ∈ W 1,2 (Σ,∆ g h = 2K λ e 2u λ h.(39)
Note that h is not a constant. For otherwise (39) yields
0 > 2πχ(Σ, β) = Σ K λ e 2u λ dv g = 0,
which is impossible. Multiplying (39) by h 3 , we get
d 4 E λ (u λ )(h, h, h, h) = −16 Σ K λ e 2u λ h 4 dv g = −24
Σ h 2 |∇ g h| 2 dv g < 0.
Since d 2 E λ (u λ + th)(h, h) attains its minimum at t = 0, we have d 3 E λ (u λ )(h, h, h) = 0, which together with the facts dE λ (u λ ) = 0 and d 2 E λ (u λ )(h, h) = 0 leads to
E λ (u λ + ǫh) = E λ (u λ ) + ǫ 4 24 d 4 E λ (u λ )(h, h, h, h) + O(ǫ 5 ) < E λ (u λ )(40)
for small ǫ > 0. Applying elliptic estimates to (39), we have h ∈ C 0 (Σ). Then there exists ǫ 0 > 0 such that if 0 < ǫ < ǫ 0 , then ϕ ≤ u λ + ǫh ≤ ψ on Σ, and thus by (40),
E λ (u λ + ǫh) = E(u λ + ǫh) < E λ (u λ ) =Ê λ (u λ ),
contradicting the fact that u λ is the minimum ofÊ λ . Therefore Λ > 0 and (38) follows immediately. As a consequence, u λ is a strict local minimum of E λ on W 1,2 (Σ, g).
Step 3. The second solution of (7) can be achieved by a mountain pass theorem.
Let u λ be as in Step 2. Since u λ is a strict local minimum of E λ on W 1,2 (Σ, g), there would exist a sufficiently small r > 0 such that inf u−u λ W 1,2 (Σ,g) =r
E λ (u) > E λ (u λ ).(41)
Moreover, a calculation of Ding and Liu ( [14], Page 1061) shows for any λ > 0, E λ has no lower bound on W 1,2 (Σ, g). In particular, there exists some v ∈ W 1,2 (Σ, g) verifying that
E λ (v) < E λ (u λ ), v − u λ W 1,2 (Σ,g) > r.(42)
Combining (41), (42) and Lemma 5, we obtain by using the mountain-pass theorem due to Ambrosetti and Rabinowitz [2] that the mini-max value
c = min γ∈Γ max u∈γ E λ (u)
is a critical value of E λ , where Γ = {γ ∈ C([0, 1], W 1,2 (Σ, g)) : γ(0) = u λ , γ(1) = v}. Equivalently there exists some u λ ∈ W 1,2 (Σ, g) satisfying E λ (u λ ) = c and dE λ (u λ ) = 0. Thus u λ is a solution of the equation (7) and u λ u λ . Finally elliptic estimates imply that u λ ∈ C 2 (Σ \ supp β) ∩ C 0 (Σ). (7) at λ * Proof of (iii) of Theorem 2. For any λ, 0 < λ < λ * , we let u λ be the local minimum of E λ obtained in the previous subsection. In particular, u λ is a solution of (7) and Σ (|∇ g φ| 2 − 2K λ e 2u λ φ 2 )dv g ≥ 0, ∀φ ∈ W 1,2 (Σ, g).
Solvability of
The remaining part of the proof will be divided into several claims as below. Claim 1. There exists some constant C such that u λ ≥ −C on Σ uniformly in λ ∈ (0, λ * ).
To see this, we let η satisfy (32) and ϕ s = η − s for s > 0. The analog of (33) reads ∆ g ϕ s + κ − Ke 2ϕ s < 0
with s chosen sufficiently large, say s ≥ s 0 . Equivalently ϕ s is a lower solution of (7) at λ = 0, provided that s ≥ s 0 . Clearly ϕ s is also a strict lower solution of (7) at λ ∈ (0, λ * ) for any s ≥ s 0 . We now prove that u λ ≥ ϕ s 0 , and consequently claim 1 holds. For otherwise, by varying s ∈ [s 0 , ∞), we find that for some s there holds u λ ≥ ϕ s on Σ, and u λ (x 0 ) = ϕ s (x 0 ) for some x 0 ∈ Σ. Then the strong maximum principle (Lemma 4) implies that u λ ≡ ϕ s on Σ, which is impossible.Suppose K λ * ≥ 0. Let g = e 2v g be a metric with constant Gaussian curvature −1, where v is a solution of ∆ g v + κ + e 2v = 0. In view of (i) of Theorem 2, such a function v uniquely exists. LetMultiplying the above equation by e −2w λ and integrating by parts, one hasThis together with K λ * ≥ 0 leads to K λ * ≡ 0, which contradicts the assumption that K λ * is not a constant.Claim 3.Let Ω and Ω ′ are two domains in Σ such that Ω ′ ⊂⊂ Ω ⊂⊂ Σ − λ * \ supp β. Then u + λ is bounded in W 1,2 (Ω ′ , g) with respect to λ ∈ (0, λ * ).Note that K : Σ → R is Hölder continuous. If λ ∈ (0, λ * ), then sup Ω K λ ≤ sup Ω K λ * ≤ −ǫ for some ǫ > 0 depending only on K, λ * and Ω. Similar to the proof of (21), we conclude Claim 3.(7)is solvable at λ * .Claim 4. The equationHaving Claims 1-3 in hand and arguing as Ding and Liu did in the proof of ([14], (c) of the main theorem), we conclude that both e 2u λ and u λ are bounded in L q (Σ, g) for all q > 1. By elliptic estimates, we have up to a subsequence, u λ converges to some u in W 1,2 (Σ, g), where u is a solution of ∆ g u + κ − K λ * e 2u = 0.By elliptic estimates, u ∈ C 2 (Σ \ supp β) ∩ C 0 (Σ). This gives the desired result.4.5.The equation(7)has no distributional solution when λ > λ * Proof of (iv) of Theorem 2. Suppose (7) has a solution u λ 1 at some λ 1 > λ * . Then for any λ, 0 < λ < λ 1 , u λ 1 is an upper solution of (7). Similar to (33), we can easily construct a lower 14 solution ϕ of (7) such that ϕ ≤ u λ 1 . In view of the upper and lower solutions principle (Lemma 6), there would exist a solution of(7), which contradicts the definition of λ * (see (30) above).Acknowledgements. This work is supported by National Science Foundation of China (GrantNos. 11171347, 11471014, 41275063 and 11401575).
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| [] |
[
"Extreme mass ratio inspirals: perspectives for their detection",
"Extreme mass ratio inspirals: perspectives for their detection"
] | [
"Stanislav Babak \nAlbert Einstein Institute\nAm Muehlenberg 1D-14476GolmGermany\n",
"Jonathan R Gair \nInstitute of Astronomy\nUniversity of Cambridge\nCB3 0HACambridgeUK\n",
"Robert H Cole \nInstitute of Astronomy\nUniversity of Cambridge\nCB3 0HACambridgeUK\n"
] | [
"Albert Einstein Institute\nAm Muehlenberg 1D-14476GolmGermany",
"Institute of Astronomy\nUniversity of Cambridge\nCB3 0HACambridgeUK",
"Institute of Astronomy\nUniversity of Cambridge\nCB3 0HACambridgeUK"
] | [] | In this article we consider prospects for detecting extreme mass ratio inspirals (EMRIs) using gravitational wave (GW) observations by a future space borne interferometric observatory eLISA. We start with a description of EMRI formation channels. Different formation scenarios lead to variations in the expected event rate and predict different distributions of the orbital parameters when the GW signal enters the eLISA sensitivity band. Then we will briefly overview the available theoretical models describing the GW signal from EMRIs and describe proposed methods for their detection. * | 10.1007/978-3-319-18335-0_23 | [
"https://arxiv.org/pdf/1411.5253v1.pdf"
] | 119,113,631 | 1411.5253 | 4e704831c9757b6feaf74f91c27de2b4d2a1d924 |
Extreme mass ratio inspirals: perspectives for their detection
19 Nov 2014 November 20, 2014
Stanislav Babak
Albert Einstein Institute
Am Muehlenberg 1D-14476GolmGermany
Jonathan R Gair
Institute of Astronomy
University of Cambridge
CB3 0HACambridgeUK
Robert H Cole
Institute of Astronomy
University of Cambridge
CB3 0HACambridgeUK
Extreme mass ratio inspirals: perspectives for their detection
19 Nov 2014 November 20, 2014
In this article we consider prospects for detecting extreme mass ratio inspirals (EMRIs) using gravitational wave (GW) observations by a future space borne interferometric observatory eLISA. We start with a description of EMRI formation channels. Different formation scenarios lead to variations in the expected event rate and predict different distributions of the orbital parameters when the GW signal enters the eLISA sensitivity band. Then we will briefly overview the available theoretical models describing the GW signal from EMRIs and describe proposed methods for their detection. *
Introduction
Extreme mass ratio inspirals (EMRIs) arise following the capture of a small compact object (CO) -a white dwarf, neutron star or stellar mass black hole -by a massive black hole (MBH) in the centre of a galaxy. The astrophysical processes that lead to the formation of EMRIs are described in detail in section 2. The inspiralling CO loses energy and angular momentum through emission of gravitational radiation, and the initially wide and very eccentric orbit gradually shrinks and becomes more circular. EMRIs are among the most interesting gravitational wave (GW) sources that could be observed by the proposed eLISA detector. eLISA (evolving Laser Space Interferometer Antenna) is a space-based gravitational wave detector which is scheduled for launch in 2034. It will be sensitive to GWs in the frequency range 0.1 − 100 milliHertz. Sources in this band include the mergers of massive black hole binaries, which will be observable up to a redshift z = 20, and numerous white dwarf binaries in the Milky Way, in addition to EM-RIs. We will discuss the event rate and the expected precision of parameter estimation for EMRI soures in section 2.
During an EMRI, the CO typically spends 10 5 − 10 6 orbital cycles in the eLISA band before plunging into the central MBH. We need to model the phase of GW signal from EMRIs with an accuracy of a fraction of a cycle in order to detect the signal and correctly extract the parameters of the binary system. This is a challenging problem, which has not yet been solved in full. Due to the extreme mass ratio, m/M ∼ 10 −4 − 10 −6 , we can treat the problem perturbatively, considering the field of the CO and the emitted GWs as a small perturbation of the background spacetime of the central MBH. At the leading orders in mass ratio the internal structure of a CO is not important and so the CO is conventionally treated as a delta-function. As often happens in such an approximation, the self-field is divergent at the position of the CO, and requires proper treatment (regularization) [1]. The resulting perturbation has the form of a tail expression, and depends on an integral over the entire past history of the CO's trajectory. In the limit that the mass ratio goes to zero, the motion is described by a geodesic. However, the mass of a CO is small but not zero and due to interaction of the self field of the particle with a background, the trajectory slowly deviates from a geodesic path [2]. This can be described effectively as the action of a force (self-force) on the inspiraling object. In practice, the geodesic trajectory is used to compute the tail integral entering the self-force, and the resultant force is used to update the geodesic trajectory accordingly. In section 3, we will summarize various ways to compute the GW signal from EMRIs and describe how the evolution of the orbital motion can be described using an osculating elements approach. The CO may also be spinning and this spin is coupled to the background curvature and alters the trajectory of the CO, forcing it to deviate from the corresponding geodesic of a non-spinning body. The trajectory of a spinning particle (in the limit of vanishing mass ratio) is described by the Mathisson-Papapetrou equation. Attaching a spin to a point particle is not uniquely defined, leaving a freedom to choose the dipole moment of a body (see [3] for a description of spinning objects in the weak field approximation). This freedom manifests itself through the need to specify a spin supplementary condition (SSC) in order to obtain a unique solution to the equations of motion. In order to understand these complications, we consider in subsection 3.3 the motion of a spinning particle in de Sitter space time. This space time possesses a non-trivial curvature but is still fully symmetric. For more details on the computation of the selfforce and on the Matthisson-Papetrou equations we refer to other articles in this issue.
Last, but not least we want to consider the question of detectability of GW signals from EMRIs. The GWs generated by an EMRI system are characterized by 14 parameters: two masses m, M , the dimensionless spin of the MBH, a, and its orientation, θ K , φ K ; six parameters describing the CO's position and velocity at some fiducial time or equivalently the instantaneous shape and phase of the orbit at that time (eccentricity e, inclination of the orbital plane to the spin of MBH ι, semi-latus rectum p and the initial phases φ r , φ θ , ϕ corresponding to the three coordinate degrees of freedom); the sky location of the source, θ, φ, and its luminosity distance, D L . Many of these parameters are highly correlated. The GW signal comprises a superposition of orbital harmonics, with the number of harmonics and their relative strength strongly dependent on the eccentricity and binary orientation. The strength of the signal observed in the detector varies with time as eLISA moves around the sun (amplitude modulation) and the relative motion of the detector and the source induces a time-dependent Doppler modulation of the phase. The main challenge in detecting EMRIs is the multi-modality of the likelihood. The likelihood can be seen as a hyper-surface embedded in the 14-dimensional parameter space. It has multiple strong maxima and the main challenge is to find the highest (global) maximum. In section 4 we describe algorithms to do this which were successfully demonstrated on the Mock LISA data challenges [4].
Throughout this paper we use geometrical units G = c = 1.
Astrophysics of extreme mass ratio inspirals
In this section we will consider possible channels leading to EMRI formation, the expected number of EMRI events that will be observed for eLISA and the likely accuracy with which eLISA will constrain their parameters. Then we will briefly summarize some of the potential impact of EMRI detections for astrophysics and fundamental physics.
Formation of EMRIs
The "extreme mass ratio" refers to the fact that the mass of the CO is of order of 1 − 10M ⊙ , while the mass of the central (capturing) object is in the range 10 5 − 10 7 M ⊙ . Current astrophysical observations indicate that massive compact objects of this kind are present in the nuclei of all sufficiently massive galaxies for which the central part can be resolved. The best example is the nucleus of the Milky Way, in which a few dozen bright O-B stars (so called S-stars) have been observed in Keplerian orbits around a central object with an estimated mass of ∼ 4 × 10 6 M ⊙ . In addition, the compactness of this object suggests that it must be a massive black hole. These massive objects in the centres of galaxies are typically surrounded by clusters of stars. In the "standard" picture of EMRI formation, the stars are spherically distributed around the MBH (which should be approximately true for sufficiently large distances) and dense enough for efficient 2-body relaxation, i.e., mutual gravitational deflection and contact collisions. The timescale for this process, the relaxation time t rlx , is defined as the time required to change the angular momentum of a star by an amount J c , where J c is the angular momentum of a star on a circular orbit with the same semimajor axis. A smaller t rlx implies that stars can be more easily deflected on to very eccentric orbits with a small periapsis passage. If a CO object on an initially wide orbit is perturbed onto such a trajectory, it will lose energy to GW bursts emitted near periapsis (r p ) and its orbit will gradually shrink. While the semi-major axis is very large, the CO can still efficiently interact with other stars at the apoapsis and could be either deflected onto a plunging orbit with r p < 8M or onto a wide orbit which does not emit appreciable GW radiation. To become an observable EMRI, the CO must remain on the highly eccentric orbit until its period becomes smaller than ∼ 10 3 − 10 4 s, at which point it is continually radiating GWs in the eLISA sensitivity band. While we will be primarily interested in such EMRIs here, the bursts of GWs produced during periapsis passages in the early stages of the process could also be potentially detected by eLISA if the event is in the nucleus of nearby galaxies [5].
When the stars interact gravitationally, they tend to divide the kinetic energy equally and, while equipartition is not reached in practice, this process causes more massive objects to sink deeper in the potential well of the MBH. This process is called mass segregation. As a result we expect stellar mass black holes to form a steep power-law density cusp around the MBH n(r) ∼ r −α with α ≃ 1.7 − 2.0, which dominates for r < 0.1pc. The lighter stellar species form shallower density profiles with α ≃ 1.3 − 1.5 [6]. The relaxation time is inverse proportional to the density of the CO and it should therefore be smaller for the stellar mass black holes.
In order for an object to become an EMRI, it should efficiently dissipate energy through GW emission, and have a sufficiently low probability to be deflected onto a different orbit. This condition implies that the time scale for orbital decay by GW emission, t GW , should be smaller than (1 − e)t rlx , where e is orbital eccentricity. Once the orbital period reaches P < 10 4 s, the CO completely decouples from the cusp, which happens for orbits with semi-major axis a EM RI ∼ 0.05pc.
For typical orbits around an MBH, the number of stars enclosed by the orbit is rather small and so the gravitational potential created by the "field" stars is not a smooth symmetric function. This gives rise to a torque acting on a CO on an orbit with semi-major axis a CO of τ ∼ √ N m * /a CO , where N is a number of field stars with mass m * inside the CO orbit. If the precession of the CO orbit is slow compared to the timescale over which the distribution of field stars changes significantly, the CO experiences a nearly constant torque over some time. This mechanism, known as resonant relaxation, changes the angular momentum of the CO, but not its energy. The characteristic time scale associated with resonant relaxation, t RR , is significantly smaller than t rlx and so this process can significantly boost the EMRI event rate. Resonant relaxation plays an important role for orbits with a CO ≤ a EM RI [7], [8], [9], [10]. However, for COs on eccentric orbits with small perhaps radii, the relativistic (GR) precession can be very high, which effectively destroys the resonant relaxation effect. The point at which this occurs is known as the "Schwarzschild barrier" [11]. The existence of this barrier means that resonant relaxation is not as effective at boosting EMRI rates as one might first think, although if the MBH has significant spin then the impact of the "Schwarzschild barrier" is somewhat diminished due to the lower value of the plunge periapsis for prograde orbits [12]. In this case, COs that would normally be considered as plunging and hence undetectable around a Schwarzschild MBH actually perform many cycles in the eLISA band and may contribute significantly to the event rate [13].
In this picture, the critical thing for having a high EMRI rate is to have compact objects in the "loss-cone" (orbits with impact parameter sufficiently small that they can be captured or tidally disrupted by the MBH). Several channels have been suggested that can replenish the loss-cone and thereby significantly boost the EMRI rate, including triaxiality of the potential (nonspherical galactic nuclei) [14] or the presence of massive perturbers (such as intermediate mass BHs, and/or molecular clouds) in the vicinity of the orbits [15].
The complex dynamics of this standard capture scenario for EMRI formation means that the astrophysical event rates are very uncertain. To estimate event rates we will use a current best guess of 400Gyr −1 for Milky Way-like black holes, dominated by EMRIs in which the CO is a black hole. This rate is taken from [16].
As well as this standard mechanism for EMRI formation, there are two other plausible channels. Tidal binary disruption. It is possible that within the radius of influence of a MBH there is a binary fraction of at least a few percent [17]. If a binary approaches the MBH it can be tidally disrupted and, if this happens, one star is ejected at very high velocity while the other star becomes tightly bound to the MBH. The captured CO is expected to end up on an orbit with a semi-major axis of a few hundred AU and a pericentre distance of a few to tens of AU, implying that it will circularise by the time it enters the eLISA frequency band [17,18]. This is a distinct feature of this formation channel, since in the standard scenario we expect the EMRIs to have a significant residual eccentricity even at plunge [19], e pl ∼ 0.1 − 0.3. We observe in the Milky Way so called "hyper-velocity" stars [20]. which are moving away from the galactic centre with large velocities. The best current explanation for the presence of short-lived S-stars in the vicinity of the Milky Way MBH is that they came there following the tidal disruption of binaries, while the observed hyper-velocity stars are the thrown away companions [21].
Formation of stellar remnants in a disk. Observation of active galactic nuclei suggests the presence of a circum-nuclear gaseous disk accreting onto the MBH. If the disk is thick and sufficiently massive, the outer part could fragment and form stars. If migration through the disk is sufficiently slow, stars formed in this way could evolve to form compact object remnants (neutron star or black hole) which subsequently spiral into the MBH as an EMRI in the equatorial plane (the accretion disk, at least its inner parts, is expected to be aligned with the MBH's equatorial plane [22]). The interaction with the gas is also likely to keep the orbit of a CO close to circular, so the distinct feature of this channel of EMRI formation is a circular orbit in the black hole equatorial plane.
By measuring the orbital parameters we will be able to say which of these three channels provides the most likely explanation for how the EMRI was formed. For more details on the dynamics of galactic nuclei we refer the reader to the comprehensive review [23].
Expected event rate estimation
In this section we follow [24] and briefly outline how the expected event rate of EMRIs observed by eLISA can be estimated. To make this estimation we require an intrinsic event rate R(M, a, µ), where M is the mass of the MBH, a its spin in units of M and µ = m/M is the mass ratio. The intrinsic event rate tells us how often EMRIs are formed (i.e., how often they enter the eLISA sensitivity band) per galaxy hosting a MBH with parameters M, a. The mass ratio parameter tells us the nature of a CO, i.e., whether it is a stellar mass BH, neutron star or white dwarf. As discussed in the previous subsection, due to mass segregation we expect stellar mass BHs to be the most likely candidate for EMRIs, so we choose a canonical value for the CO mass of m = 10M ⊙ . We will normalize the mass of the MBH by the mass expected for a Milky Way type galaxy M M W 3 × 10 6 M ⊙ . So far we do not have information about the distribution of the spin of MBHs of this mass. X-ray observations of some active galactic nuclei provide information about the spin of accreting MBHs in the centre, but those black holes are of higher mass > 10 7 M ⊙ and embedded in the gaseous circumnuclear disk. In addition all present estimations of the spin are heavily model dependent and could vary significantly depending on the underlying assumptions [25]. Therefore, here we assume a uniform distribution of the spin within its physical range a ∈ (−1, 1). The estimation of the intrinsic event rate is a very challenging task, as described above and in more detail in [23], which depends quite heavily on the underlying assumptions about the efficiency of mass segregation, the relative importance of different EMRI formation channels and the interplay between resonant relaxation and the "Schwarzschild barrier". Here we adopt the estimate derived in [16] which for stellar mass BHs is
R = 400Gyr −1 M 3 × 10 6 M ⊙ β (1) where β ≈ 0.19.
If the duration of EMRI signals was significantly shorter than the observation time, then the observed event rate would be determined by computing the distance at which the signal-to-noise ratio (SNR) equals some detection threshold ρ thr and then multiplying the rate per unit volume by the volume contained by that distance, assuming a uniform distribution of EMRIs in the local Universe. However, EMRIs are long-lived, and the SNR can be accumulated for as much of the inspiral as coincides with the eLISA observation. Fixing all the parameters of the EMRI, we can compute the SNR as a function of the time left to plunge, t pl . As we increase t pl from zero, the SNR first increases, then reaches a maximum before starting to decrease. There is a decrease of SNR for large t pl because the finite observation time means that we are ultimately only observing systems that are rather wide, with not very efficient GW emission, and with emission primarily at low frequencies where acceleration noise rises rather steeply. This means that if an EMRI is at all detectable, the SNR as a function of t pl intersects the line SNR= ρ thr at two times, t early , t late , and we can define the EMRI ob-servable lifetime as τ (λ i ) = t late (λ i ) − t early (λ i ), where λ i corresponds to all the other parameters of the EMRI (besides t pl ) which we have fixed. If EMRIs plunge at a rate R per year in a particular galaxy, then τ R gives the expected number of events from that galaxy (after appropriate averaging over parameters λ i ). Among all parameters describing the EMRI system the most important are M, a, and we denote the remaining parameters asλ i . We define N (M, a, z)dM da as the number of MBHs per comoving volume with mass M ∈ [M, M + dM ], spin a ∈ [a, a + da], and at redshift z. We make two further assumptions (i) that the mass and spin distributions are independent; and (ii) that the distribution of MBH mass and spin are independent of redshift. The first assumption reflects our level of ignorance, and the second assumption is reasonable given how far we can observe EMRIs (with eLISA we will able to see EMRIs up to z max ≈ 0.7). In this range we can ignore the evolution of masses and spins with z. Under these as-
sumptions N (M, a, z)dM da = (dn/d ln M )(M )d ln M p(a)da, where p(a)
is the probability distribution function for the spin p(a)da = 1. As described above, we assume this is uniform in our calculations, but we keep it here in the equation for completeness.
The expected event rate is then
N eLISA = ∞ z=0 dz M high M low d ln M 1 a=−1 daR(M, a)τ (M, a, z,λ i ) dn d ln M (M )p(a) dV c dz λ i(2)
Here (dV c /dz)dz is the comoving volume in the redshift range [z, z + dz].
The triangular brackets denote the averaging over other EMRI parameterŝ λ i . We note that in practice the intrinsic event rate could also depend on some parameters from the setλ i (depending on the channel of EMRI formation). The mass function (dn/d ln M ) can be deduced from measured galaxy luminosity functions using the observed L − σ, M − σ correlations. In the range of interest to eLISA, this functions approximately flat [26], so we adopt
dn d ln M = n 0 M 3 × 10 6 M ⊙ α (3)
with canonical values n 0 = 0.002 Mpc −3 , α = 0. If we assume these canonical values, with β = 0.19, a mission duration of 2 years and a detection threshold of ρ thr = 20, we estimate that eLISA would observe 25 to 50 events in two years [27,28]. This spread in the predicted number of events comes from uncertainties in the waveform model and system parameters, but a much larger uncertainty, which is not taken into account here, arises from the uncertainty in the true value of R.
Science return from observing GW signals from EMRIs
Detection of EMRIs and measurement of their parameters provides unique astrophysical data which cannot be obtained by any other means. We expect to be able to learn information about stellar populations in the centre of the Milky Way in the future by observing pulsars in the nuclear stellar cluster region using the SKA [29]. Inferring similar properties of other galaxies through observations of EMRIs will also us to compare the nucleus of the Milky Way with nuclei of other galaxies. The number of observed EMRI events and the mass distribution of the COs will tell us about the physics of mass segregation, the masses and spins of stellar mass compact objects and about the steepness of the stellar cusps in the centres of galaxies. In addition, EMRI observations will provide precise measurements of MBH masses and spins in a new mass range. EMRIs will probe galaxies containing black holes with masses 10 5 − 10 7 M ⊙ , and such galaxies tend to be of lower mass and not particularly luminous in the electromagnetic spectrum. Extracting information about the nuclei of those galaxies is therefore very challenging, if not impossible, using electromagnetic observations and eLISA therefore has tremendous potential to inform us about these systems. Observations show that the masses of black holes in galactic nuclei correlate with the mass, luminosity and the stellar velocity dispersion of their host galaxy [30]. These correlations imply that black holes evolve along with their hosts throughout cosmic time, but it is not yet known if this coevolution extends down to the lowest galaxy and black hole masses, since those systems may have differences in the accretion properties [31], dynamical effects [32], or cosmic bias [33]. eLISA observations of EMRIs will significantly improve our knowledge of the MBH mass function (e.g., inferring the parameter α in eqn. [3]), as well as allowing us to measure the intrinsic event rate (for example constraining the parameter β in eqn. [1]), determine the relative importance of different channels of EMRI formation and measure the spatial distribution (relative to the MBH) of different types of CO. This is made possible by the ultra-precise determination of EMRI parameters with GW observations. In Figure 1 we show how accurately we expect to measure the most important parameters: MBH mass (M ) and spin (a), CO mass (m), orbital eccentricity just before the plunge (end of inspiral) (e pl ). The last parameter, ∆Q, is a possible deviation in the MBH quadrupole moment away from the Kerr value, which will be discussed at the end of this subsection. In addition an observation of an EMRI will allow us to determine the luminosity distance to the source (D L ) with an accuracy of ≤ 1% and to localize the source on the sky to about 0.2 square degrees. Such a fantastic accuracy is achieved because the source is long lived -the CO spends 10 4 − 10 6 cycles in the close vicinity of a MBH. Using matched filtering we will be able to determine the phase of an EMRI to an accuracy of half a cycle, a fractional phase accuracy of 10 −6 -10 −4 . All information about the binary system is encoded in the GW phase and so we can expect to make measurements of the intrinsic parameters to this same fractional accuracy. Measurements of the extrinsic parameters, such as sky localisation, are not as precise since these measurements come not from the phase but from the modulation of the GW signal (in amplitude and in phase) caused by eLISA's orbital motion.
These precise phase measurements mean we can also use EMRIs to test the "no-hair" theorem: if the central massive compact object is indeed described by the Kerr metric, as general relativity predicts. The spacetime outside a stationary, axisymmetric object is fully determined by its mass, M l , and current, S l , multipole moments. Since these moments fully characterise the spacetime, the orbits of the smaller object and the gravitational waves it emits are determined by these multipole moments. The emitted GWs therefore encode a map of the spacetime structure and by observing these gravitational waves with eLISA we can precisely characterise the multipole structure of the central object. Extracting the moments from the EMRI waves is analogous to geodesy. If the central object is a Kerr black hole, then all multipole moments are determined by its mass and spin ("nohair" theorem):
M l + iS l = (ia) l M l+1
If we can measure the first three moments we can therefore check whether the central object is consistent with being a Kerr black hole. Figure 1 shows that we should be able to measure a deviation in the mass quadrupole moment from the Kerr value, Q = |M 2 − M Kerr 2 |, to a precision of δQ ≈ (10 −2 − 10 −3 )M 3 . EMRIs could therefore also serve as laboratories for testing fundamental physics. For more discussion on this topic we refer to [28,34]. the data is analysed using an inaccurate model there will also by systematic errors in the parameter estimates, which could be larger than the statistical errors from detector noise. It is therefore important to accurately model the GW signal coming from EMRIs, to ensure reliable estimation of parameters and improve the detectability, since a mismatch between the signal and template will cause a drop in the SNR and decrease in the observed volume by (SN R/SN R optimal ) 3 .
In this section we will describe currently available models for EMRI signal and discuss their effectualness (if they are able to recover the optimal SNR) and faithfulness (if the systematic errors in parameter estimation is below the statistical errors due to presence of the noise). Note that effectualness does not imply faithfulness: a model could recover a significant fraction of the SNR with a large systematic bias in the parameters. In other words, the shift in the parameters from the true values could (partially) compensate for inaccuracies in the model.
Waveform inventory
Unlike the inspiral of a comparable mass binary, the merger (here we call it plunge) of a CO with MBH and subsequent ring-down are suppressed by a factor of the mass-ratio and are therefore not observable by eLISA. We therefore only need to model the inspiral part of the signal up to a plunge. However, for the whole of the inspiral observable by eLISA, the CO is orbiting in the strong-field region close to the MBH and moving at ultra-relativistic speeds. This makes modelling an EMRI signal somewhat different from modelling GWs from a binary of two nearly-equal mass MBHs. Here we briefly outline some of the currently available models for GW signals from EMRIs. More detailed description can be found in other papers in this volume.
Post-newtonian expansion. The post-newtonian approach describes the GW signal as an expansion in velocity v. As mentioned above the CO in EMRI systems are fast moving and spend 10 4 − 10 6 cycles in a regime where v is large. The EOB approach [35,36] is the most suitable for modelling EMRIs by construction (the conservative dynamics reduces to the test-mass in the limit m/M → 0), however the dissipative part (fluxes) are needed to a very high post-newtonian order, which is not currently known. In addition, the analytic expressions for the fluxes are known only for nearly circular and nearly equatorial orbits, while we expect EMRI orbits to be both eccentric and inclined [37].
Analytic "kludge" waveforms. This model was introduced primarily to study detection rates and parameter estimation for EMRIs [38]. The main advantage of these waveforms is that they are fast to generate, so they are suitable for large Monte-Carlo simulations, and they were extensively used to develop detection algorithms (see section 4). This model is an extension of the work by Peters and Mathew [39], it represents emission from a CO in Keplerian orbit augmented by imposing (post-newtonian) relativistic precession of the orbital plane and the direction to perihelion. The dissipative evolution is taken from post-newtonian calculations. This model is not particularly accurate but it captures the main physical processes occurring in EMRIs.
Numerical "kludge" waveform, or semi-relativistic model. The idea of the numerical kludge waveforms is to combine an exact particle trajectory (up to inaccuracies in the phase space trajectory and conservative radiation reaction terms) with an approximate expression for the GW emission. By including the particle dynamics accurately, we hope to capture the main features of the waveform, even if we are using an approximation for the waveform construction. The idea was introduced in [40,41] and was further evolved with some modifications in [42,43].
The procedure to compute a numerical kludge waveform has two stages. Firstly, a phase-space inspiral trajectory is constructed, i.e., the sequence of geodesics that an inspiral passes through, by integrating prescriptions for the evolution of the six constants of the motion (energy, angular momentum, Carter constant and three initial phases). Initial work has used post-newtonian expressions (augmented by some consistency corrections and by fitting to solutions of the Teukolsky equations) to evolve these constants. This inspiralling trajectory is computed numerically thus the name "numerical kludge". Once the trajectory has been constructed a waveform is generated by identifying the Boyer Lindquist coordinates along the trajectory with spherical-polar coordinates in a flat space time and applying weak-field GW emission formulae, in particular the quadrupole-octupole approxima-
tion:h jk = 2 D L Ï jk − 2n iS ijk + n i ... M ijk | t ′ =t−D L ,(4)
where I ik , M ijk are the mass quadrupole and octupole moments and S ijk is the current quadrupole moment of the binary system, n i is a unit vector pointing from MBH to the position of a CO, and overdots denote time derivatives. These waveforms are somewhat slower to generate as compared with the analytic kludge due to the numerical integration of the orbital trajectory, but it is far more faithful up to the last month or less (semilatus rectum p ≈ 6M ) before the plunge.
Adiabatic inspirals based on Teukolsky formalism. The very first framework for black hole perturbation theory in a Kerr background was the Teukolsky formalism [44], which encapsulates all gravitational radiative degrees of freedom in a single "master" wave equation (the "Teukolsky equation") for the Weyl scalars, Ψ 0 and Ψ 4 . A key feature of this equation is that it admits separation of variables in the frequency domain, which effectively reduces it to a pair of ordinary differential equations. The Teukolsky equation has been solved in the frequency domain [45] and in the time domain [46], but both approaches assume the orbit that acts as the source of the perturbation is a geodesic. The rate of change in energy, angular momentum, Carter constant (averaged over several orbits) are evaluated from the gravitational wave field and then used to update the parameters of the geodesic in an adiabatic manner. This procedure misses the evolution of the other constants of motion (initial positions) as well as making the adiabatic assumption. As a result, the waveforms are not accurate on a very long time scale, but they are the most faithful model on time scales ∼ M 2 /m.
Self-force waveforms. An accurate description of the self-force and its derivation is given in other papers in this volume, so we only briefly mention it here. As mentioned above, the extreme mass ratio in an EMRI system allows the waveform to be determined using perturbation theory. The inspiralling object can be regarded as a small perturbation on the background spacetime of the central black hole, except very close to the small object. In the vicinity of the small object, the spacetime can be regarded as a Schwarzschild BH moving under the influence of an external tidal field due to the MBH. Matching these two regimes allows one to obtain an expression for the self-force acting on the CO. The self force can be seen to arise as a result of the interaction of the self field of the CO with the non-flat background geometry, which causes the lines of force to be bent and act back on the CO. The self-force can be conventionally split into two parts: non-time symmetric (dissipative) and time-symmetric (conservative). The former part causes the inspiral and dominates while the latter part can be eliminated by a redefinition of the orbital frequencies at each instance, which means it is effectively second-order in mass ratio. The adiabatic Teukolsky based waveforms take into account only the dissipative part of the self force, neglecting the conservative part, which defines the domain of its validity. The self force is computed assuming the CO is moving on a geodesic, then it is used to adjust the geodesic (inspiral) before the self-force is recomputed again. The computation of the self force is somewhat complicated as it treats the CO as a delta function in the background spacetime, which requires mathematical apparatus for regularization of some divergent inte-grals. It is possible to subtract the singular part from the field equations (by finding a singular solution valid in the vicinity of the CO) and the resulting equations are manifestly regular and contain on the right hand side a smooth effective source [47], which allows the field equations to be coupled to the equations of motion and integrated. This procedure can be written for a scalar field (representing a CO carrying a scalar charge and ignoring the gravitational part of the self-force) as [48]
(Φ r ) ;α ;α = S(x; z(τ ), u(τ )) (5) Du α dτ = q m(τ ) (g αβ + u α u β ) ▽ β Φ r (6) dm dτ = −qu β ▽ β Φ r ,(7)
where Φ is the scalar field, q is the scalar charge, m the mass of the CO, S is an effective source term and u α is the CO four-velocity. Greek indices are being used to indicate space-time components, and a semicolon denotes a covariant derivative with respect to the background spacetime, g αβ . A similar procedure can be applied to the gravitational field. So far only the self-force waveform for a Schwarzschild background has been computed, but recent progress has been rapid and so we expect the extension to Kerr to be completed within a few years. Numerical relativity waveforms. The ultimate goal would be to compute EMRI waveforms using numerical integration of the full GR field equations. State-of-the-art techniques have enabled the computation of waveforms for the last 20-50 cycles of the inspiral, merger and ring-down of comparable mass ratio binaries. The simulation of an EMRI requires the computation of a few orders of magnitude more cycles, plus the resolution of two very different spatial scales. This is far beyond the capability of current computational resources and techniques. In addition, the time step for explicit numerical integration is set by the smallest characteristic scale in the problem, which is the mass of the CO in this case. Numerical waveforms will be very useful for the calibration of current calculations based on perturbation techniques, but new numerical methods will have to be developed to handle EMRIs.
Evolving perturbed geodesic motion
In this subsection we will focus on how we can compute the evolution of the orbit. The orbital evolution is the key ingredient for creating numerical kludge waveforms and waveforms based on the self force. In fact this is the same problem, the main difference is in how the waveform is computed from the orbital trajectory. To compute the orbital evolution we must solve the forced geodesic equation:
u β u β ;α = a α ,(8)
where a α is the 4-acceleration. The acceleration is essentially the self-force, but the method we will describe here for solving this equation is also applicable to the case where a α represents some other kind of external perturbation. This perturbation could be caused by a second (intermediate) MBH (if the CO in the EMRI is inspiralling into an MBH that is in a wide MBH binary), a molecular cloud or disc, another star or compact objector basically anything that can cause a slow modification of the geodesic orbit. Here we assume that the acceleration has been derived in some other way and are only interested in the effect it has on the inspiral trajectory. The rest of this subsection summarizes results described in more detail in [49]. We use an osculating elements approach to evolve eqn. (8). If the perturbing force is small 1 , we can represent the perturbed trajectory at each instant by the unique geodesic passing through the same position with the same velocity and see the orbital evolution as a slow variation of the constants of these instantaneously-tangent geodesics. A general geodesic in Kerr spacetime is described by eight constants of motion: J = {m, E, L z , Q, ψ 0 , χ 0 , φ 0 , t 0 }, however two of them (CO mass and initial time m, t 0 ) are not truly dynamical, so we will work with the remaining 6: orbital energy (E), orbital angular momentum projected onto the spin of the MBH (J z ), Carter constant (Q) and three initial phases (ψ 0 , χ 0 , φ 0 ) describing the initial position of the CO on the orbit in r, θ and φ respectively. The osculating element for of the equation of motionr = f geo + δf , is
z α (τ ) = z α g (J A (τ ), τ ), → ∂z α g ∂J A ∂J A ∂τ = 0 (9) ∂z α ∂τ = ∂z α g ∂τ (J A (τ ), τ ), → ∂ż α g ∂J A ∂J A ∂τ = δf α .(10)
The first set of equations describes a "geodesic" motion with slowly changing orbital "constants", and the second set gives us the evolution of the orbital "constants" as a function of the perturbing force. The advantage of using the osculating elements approach is that we can use an adiabatic approximation (or, more generally, a two-time-scale expansion [50]) to evolve EMRIs, for which the radiation reaction time scale is much longer than the orbital time scale, allowing us to more easily study secular effects.
The osculating elements approach was first used in [51] to study Eq. 8 in Schwarzschild background, and was extended to Kerr in [49]. The authors in [49] wrote the osculating element equations on two different forms, using the Kinnersley tetrad or "Hughes" variables (i.e., in terms of the orbital constants and the total phase variables [45]). In both cases, the appearance of an apparent divergence in the osculating equations of motion at turning points is avoided. The techniques were applied to a toy problem in which an EMRI was evolving under the influence of a perturbing force due to drag from surrounding material. This "gas-drag" force was taken to be proportional to the velocity of the inspiralling compact object. The two different approaches were shown to give identical results, and the comparison of the exact and adiabatic solutions to the problem identified the domain of validity of the adiabatic approach. Although the gas-drag problem was considered only to illustrate the methods, it yielded interesting results. In particular, it was found that the influence of the drag force was to drive the inspiral of the object, but also to increase the eccentricity of the orbit and decrease the orbital inclination. A gravitational wave driven inspiral would tend to show a decreasing eccentricity and so these two types of perturbing force would be distinguishable in an EMRI observation.
Osculating elements were also used to generate inspirals in a Schwarzschild background under the influence of the gravitational self-force in [52]. The formalism developed for Kerr inspirals in [49] has not been used for any other studies so far, but this will be done once suitable models for perturbing forces are available. Another type of orbital perturbation, which can also be interpreted in terms of a perturbing force acting on a geodesic, is the influence of the spin of the CO on the trajectory. This will be discussed in detail in the next subsection.
Spinning particle in de Sitter space-time
In this subsection we will consider a spinning CO. There are several contributions to this proceedings which describe the motion of a spinning body in a given background in great detail. Here we will give only a brief summary, then show how we can formulate the motion in terms of the osculating elements approach described in the last subsection. To understand the motion of a spinning CO in the MBH spacetime, we will first consider a simpler problem. We will describe analytically the motion of a spinning test body in de Sitter spacetime.
The motion of a test mass in an arbitrary spacetime is governed by the Mathisson-Papapetrou equations
D τ p α = − 1 2 R µνβ α u β S µν (11) D τ S αβ = 2p [α u β] .(12)
The first complication is that the 4-momentum p µ and 4-velocity u µ are not parallel
p α = mu α + u β D τ S αβ .(13)
Here D τ denotes a covariant derivative with respect to the proper time, square brackets denote the anti-symmetric part, R µνβ α is the Riemann tensor of the background space time and S µν = −S νµ is the spin tensor. The difference between p α and u α means that there is an ambiguity in what we call the mass -we can define this as m = p α u α or M 2 = p α p α . The second complication is that there is not a sufficient number of equations to determine all of the unknowns. In order to close the system we need to introduce an additional "spin supplementary condition" (SSC). There is an arbitrariness in choosing the SSC, which is usually attributed to how we choose the representative word line of a test mass (this is equivalent to choosing a dipole moment of a spinning CO). The main reason that the SSC is needed is that there is an ambiguity in the definition of the spin tensor for a point mass. The point mass is an approximation of an extended body (for which the spin tensor is well defined) when the size is much less than the radius of curvature of the background spacetime. The most common SSCs are (i) p α S αβ = 0, (ii) u α S αβ = 0, (iii) w α S αβ = 0 (14) SSC (i) is usually referred to as the Tulczyjew condition [53], (ii) is the Frenkel-Pirani condition [54,55] and (iii) was first introduced in [56] and is referred to as the w-condition.
As mentioned above we want to write the Mathisson-Pappetrou equations as a set of first order equations using the osculating elements approach. To achieve this, we must first write the equations of motion in the form of a forced geodesic equation for a non-spinning particle:
u α = d 2 x α ds 2 + Γ ρσ α dx ρ ds dx σ ds = f α ,(15)
which we want to rewrite later in the form [10]. We denote the SSCs (i), (ii) and (iii) as "T" and "F" and "w", and consider first the "T" condition,
S ab p b = 0.
In that case, we have M = const, butṁ (u α ,u α ) =Ṡ αβu α u β . We can introduce a new time variable, λ, with dλ = mdτ and useũ α to denote the coordinate velocity in the new coordinatesũ α := dx α /dλ = u α /m. The equations then become dp α dλ
+ Γ ρσ α p ρũσ = − 1 2 S ρσũµ R ρσµ α , dS αβ dλ + Γ ρσ α S βσũρ + Γ ρσ β S ασũρ = 2p [αũβ] , u α = dx α dλ = 1 M 2 p α + 2S αβ S ρσ R βǫρσ p ǫ 4M 2 + S µβ S ρσ R µβρσ ,(16)
which now have no explicit dependence on m and so we can proceed to write them in osculating element form. In particular, we can differentiate the third equation with respect to λ and then use the first equation to get an equation for dũ α dλ + Γ ρσ αũβũγ that depends only on position and velocity, and not on derivatives ofũ α . The explicit expression for the covariant total derivative ofũ a is given by:
Dũ α dλ = m 2 ũ µ +ũ β DS µβ dλ d dλ H α µ + Γ ρσ αũρũσ −Γ ρσ µũρ m 2 ũ σ +ũ ν DS σν dλ H α µ − 1 M 2 δ α µ − 1 2 S ρσũβ R ρσβ µ H α µ − 1 M 2 δ α µ ,(17)
where we made use of the following abbreviation:
H α µ := 2S αβ S ρσ R ρσβµ 4M 4 + M 2 S ǫλ S κν R κνǫλ .
The third equation in (16) gives an implicit dependence of p α on the spin tensor and velocity (p α = p α (u β , S βγ )) which we can use to integrate the (second) equation for the spin tensor. We note, however, that the standard osculating element formulation of the equations implicitly imposes the condition thatũ αũ α = 1 and hencẽ u α f α = 0. This is no longer true after this change of variables. However, there is a way to put the equations into this standard osculating element form when there is an arbitrary force on the right hand side. To tackle this problem we can again make a change of integration variable to a new variable, q say. We then have (18) and the equations become
dx α dλ = dq dλ dx α dq d 2 x α dλ 2 = dq dλ 2 d 2 x α dq 2 + d 2 q dλ 2 dx α dqd 2 x α dλ 2 + Γ ρσ α dx ρ dλ dx σ dλ = f ′α = 1 (dq/dλ) 2 f α − d 2 q dλ 2 dx α dq .(19)
We can impose the orthogonality condition be solving
d 2 q dλ 2 = f α dx α /dq g αβ (dx α /dq)(dx β /dq)(20)
and the force becomes
f ′α = 1 (dq/dλ) 2 f α − f γ dx γ /dq g µν (dx µ /dq)(dx ν /dq) dx α dq .(21)
So, to compute the new force we need to know the value of dq/dλ. We can set this to one initially and then simultaneously integrate the equation d dq
dq dλ = 1 dq/dλ f α dx α /dq g µν (dx µ /dq)(dx ν /dq) .(22)
This is a somewhat complicated procedure, but the right hand sides of the new equations now do not depend on derivatives of velocity and so the problems identified above no longer apply. The (iii) SSC (w-condition) is the most suitable for the osculating elements approach. In this case, we use an arbitrary normalized time-like vector w α w α = 1 and impose the following conditions
w α S αβ = 0, D τ w α = 0.(23)
The vector field w α is parallel propagated along the world line of the test mass and these conditions imply p α = mu α and m is conserved. This SSC is the most suited for the osculating elements approach. Alternatively one can linearize the equations with respect to the spin S µν In this case the relation between the velocity and the 4-momentum takes the simple form
p α L = mu α ,(24)
and the supplementary conditions "T" and "F" coincide. The equations of motion are now given by:u
α L = − 1 2m S ρσ u β R ρσβ α ,(25)S αβ L = 0.(26)
As is apparent from (26), this form of the equations of motion is suitable for the osculating orbits method, yielding a perturbing force of the form
f α L = f α u α , S αβ .(27)
We will now stop considering a general background space time and focus on a particular choice: de Sitter. This is a spacetime with a constant curvature which is at the same time fully symmetric. This allows us to solve the equations of motion analytically and to gain better understanding of the trajectories and the role of the SSC.
The motion of spinning test particles in de Sitter spacetime has previously been investigated by [57] where it was found that under the Tulczyjew SSC, the trajectory is a geodesic with the parallel transport of an appropriately defined spin vector. In addition, under the Frenkel-Pirani SSC, it was found that the trajectory is perturbed about a geodesic by an oscillatory motion but the final solution for the trajectory was left as a numerical integration. We focus on this oscillatory motion in more detail and relate it to motion under the w-condition.
The first Mathisson-Papapetrou equation (11) simplifies in de Sitter spacetime to
D τ p α = 1 l 2 S αβ u β ,(28)
where l is a real constant, related to the Ricci scalar via R = 12/l 2 . At first glance, it might appear that the Frenkel-Pirani SSC will lead to the simplest trajectories, as D τ p α is identically zero in this case. However, due to the difference between 4-momentum and 4-velocity in (13), this generically leads to non-geodesic motion. We can write the equation of motion under both the Frenkel-Pirani and the w-condition in the same functional form, given by
D τ u α = ± ωη αβµν F β u µ S ν (F σ S σ ) 2 + (F σ F σ ) S 2 , D τ F α = 0, D τ S α = 0, u α u α = 1, F α S α = u α S α , F α F α = u α F α , S α S α = −S 2 ,(29)
where S α is a spin 4-vector constructed from the spin tensor such that the SSC is satisfied, S and ω are real constants, and η αβµν is the permutation symbol. Differentiating the equation for D τ u α , results in
D 2 τ u α = −ω 2 (u α − F α ) ,(30)
demonstrating that F α can be viewed as a forcing term for the oscillations. The frequency of oscillation ω and the forcing term F α are different for the two SSCs: for the Frenkel-Pirani case, we find
F α F = m M 2 p α ,(31)ω F = 2M S ;(32)
while under the w-condition,
F α w = (u σ w σ ) w α − u σ S σ S 2 S α ,(33)ω w = S 2M l 2 .(34)
As we have an explicit equation for D τ u α , we could now numerically integrate, using the method of osculating elements, to find the trajectory. Instead, it is possible to find a general analytic solution to (29) for the motion of spinning test particles in de Sitter spacetime. As a starting point, we note that the solution in Minkowski spacetime has been determined previously (see [56,58,59], for example). Under both the Tulczyjew and w-conditions, the particle follows a geodesic whilst under the Frenkel-Pirani condition, the particle undergoes purely circular motion, boosted along a central geodesic.
Since the de Sitter and Minkowski geometries are both maximally symmetric, it might be expected that a similar solution representing circular motion will be found in de Sitter spacetime. We are interested in the 16 components of the position x α , velocity u α , forcing term F α , and spin S α 4-vectors, using spherically symmetric static coordinates. Using the ten isometries of the de Sitter spacetime and the four constraints in (29), it is possible to show that a completely general solution to the equations of motion is given by
x µ (τ ) = t = u t τ, r, θ = π 2 , φ = u φ τ ,(35)u µ (τ ) = u t = 1 − r 2 /l 2 + ω 2 r 2 1 − 2r 2 /l 2 , u r = 0, u θ = 0, u φ = r 2 /l 2 + ω 2 (l 2 − r 2 ) l 2 − 2r 2 ,(36)F µ (τ ) = F t = − u φ u t l 2 F φ , F r = 0, F θ = 0, F φ = − u φ u t 2 ω 2 l 2 ,(37)S µ (τ ) = S t = − u φ u t l 2 S φ , S r = 0, S θ = ± 1 r S 2 + ω 2 l 4 (S φ ) 2 (u t ) 2 , S φ ,(38)
where r and S φ are free constants. This solution explicitly corresponds to circular motion about the origin at a frequency that tends to ω in the limit that l → ∞, consistent with the Minkowski result. In spacetimes with fewer symmetries than de Sitter, we do not anticipate that such an exact analytic solution for the trajectory can be found, although progress can still be made. Different classes of pole-dipole orbits have been identified in the equatorial plane of Kerr [60] and it has been shown numerically that the motion of spinning test particles in Schwarzschild is of a helical nature [61]. The existence of the exact de Sitter solution can be used to further our understanding of spinning test particle trajectories in these more physical spacetimes.
In addition, the similarity of the solutions in de Sitter under the wcondition and the Frenkel-Pirani SSC will hopefully lead to a better understanding of these SSCs. Particularly, we note here that the product of covariant frequencies, ω P ω w = 1/l 2 is dependent only on the curvature of de Sitter and not on the multipole moments of the test particle. If a similar fundamental link between the two SSCs exists in other spacetimes, it might allow us to infer properties of the Frenkel-Pirani trajectory by numerically integrating the simpler equation of motion under the w-condition.
Detecting GW signals from EMRIs
In the previous two sections we have described the formation of EMRI systems and how the gravitational waves they generate can be modelled. Both those problems are very hard and not yet solved in full, and those astrophysical and theoretical uncertainties in EMRI rates and in models of the GW signal are coupled to the data analysis challenges. Before we describe specific data analysis algorithms for extracting EMRI GW signals from the detector data we will give a general description of the signal and the problems we face in data analysis.
As mentioned earlier, an EMRI generates 10 5 − 10 6 gravitational waveform cycles in the eLISA band. We therefore need to model it very accurately if we want to avoid systematic biases in the inferred parameter estimates. The expected signal-to-noise ratio (SNR) from those systems is not very high (probably less than 50), but during the Mock LISA Data Challenges (MLDCs) successful extraction of EMRI signals with SNRs as low as 19 was demonstrated, using the same approximate EMRI model (the analytic kludge described earlier) for both injection and recovery. As described in Section 1, the EMRI signal depends on 14 parameters, if the spin of the CO is ignored, which is justifiable for mass ratios less than ∼ 10 −4 . It is convenient to describe the EMRI's dynamics in the frame fixed relative to the spin axis of the MBH. The spin direction is usually taken to be the z−axis, but we have full freedom in choosing the orientation of the x, y-axes, and this choice is degenerate with the initial azimuthal position of the CO. Since the signal is long lived (stays in band for the entire duration of observation) there is a significant modulation of the amplitude and the phase of the waveform caused by the orbital motion of the detector. This allows us to measure the source sky position with a precision of a few degrees for signals with SNR∼ 20 [4].
EMRIs are primarily GW sources for a future space-based detector like eLISA [28], and the data analysis discussion presented in this section is based on analysing data from such an instrument. Here we will always assume that the instrumental noise is Gaussian (but not white) and that EMRIs are the only GW sources in the data. These are not realistic assumptions for eLISA like data, but make the problem more tractable and the resulting algorithms are still likely to be effective when the assumptions are relaxed. For the purpose of developing data analysis algorithms and EMRI detection strategies we use somewhat simplified models of GW signal (in particular the analytic "kludge" model described in the subsection 3.1), which capture the main physical features present in the expected signal (periapsis and orbital precession, slow inspiral, Doppler modulation, multiple harmonics) and are also fast to generate numerically and so can be used for computationally expensive parameter estimation. The need to quickly evaluate hundreds of thousands of waveforms to perform data analysis is the main factor which prevents us from using more realistic models. If the data analysis algorithms do not use any model specific features, they can be easily ported to use the best GW signal model available at the time the data is analysed.
There are two data analysis challenges associated with the search for EMRI signals. The first one is to find a signal in the noise, in other words to test the null hypothesis that the observed data is consistent with noise only. This could be a problem for signals with SNR below 20, however we do expect to see a few dozen signals from EMRIs with SNR above 20, which should be detected with high statistical significance. Therefore we will concentrate on such reliably detectable signals. The situation will become more complex when other GW signals are present in the data (especially the foreground from Galactic white dwarf binaries) and/or with realistic instrumental noise. We do expect some environmental and instrumental artefacts to be present in the data and the LISA Pathfinder [62] measurements (scheduled for launch in July 2015) will allow us to simulate a more realistic eLISA data stream in the near future.
The second problem is what we will focus on in the rest of the current section. The large dimensionality of the parameter space of possible signals makes a grid-type search completely infeasible, so instead we will rely on (pseudo)-stochastic search methods, primarily based on Markov chain Monte-Carlo (MCMC) techniques. Various implementations of MCMC for searches for EMRIs signals are described in [63,4,64], but the basic idea is to construct a chain which moves predominantly in the direction of increasing likelihood. The complication is that the EMRI likelihood hyper-surface has numerous local maxima some of which could be as much as 70 − 80% of the global maximum and these local maxima are widely separated in the parameter space. The problem is similar to finding the tallest tree in a forest. A standard MCMC based search will reach a local maximum and get stuck there for a significant number of steps. Theoretically MCMC has a non-zero probability of exploring the whole parameter space and finding the global maximum, but in practice it can get stuck on a strong local maximum for a very long time. Since we consider here only clearly detectable signals, when we refer to a detection we will mean successfully finding the global maximum of the likelihood (which is near the true parameters of a simulated signal and by "near" we mean comparable to the expected statistical deviations due to the presence of detector noise).
In order to detect a GW signal from an EMRI we need an algorithm which can explore efficiently a large part of the parameter space and at the same time concentrate more on regions of high likelihood. Parallel tempering MCMC is one such algorithm and it was used in the MLDCs by N.
Cornish [65]. Here we describe two other methods which share the same core principle, based on understanding and exploiting the reason for the presence of local maxima. To understand this reason, we need to look carefully at the GW signal. The GW signal from an EMRI is a superposition of harmonics of three fundamental frequencies, which slowly evolve as the CO inspirals.
h(t) = l,m,n h lmn (t) = ℜ l,m,n A lmn (t)e i(lΦr +mΦ θ +nΦ φ ) (39)
These fundamental frequencies (instantaneously or for a geodesic motion) are associated with three degrees of freedom: the radial frequency is associated with eccentric motion from periapsis to apoapsis and back; the polar frequency (θ−motion) is associated with spin-orbital coupling and the resulting precession of the orbital plane around the spin axis of the MBH; and finally the frequency of azimuthal motion [66,45]. The frequencies evolve under radiation reaction (self-force) on a time scale associated with the mass ratio, which is for EMRIs significantly longer than the orbital time scale. As the CO spirals toward the MBH the overall amplitude of the signal is slightly increasing but the amplitude of individual harmonic depends on the instantaneous orbital parameters like eccentricity and inclination. Due to orbital circularisation under radiation reaction [39] the amplitude of some harmonics (high l) will decrease while that of some other (low l) harmonics will increase, but in all cases the amplitude of each harmonic is a smooth and slowly varying function of time. We can construct a periodogram of the EMRI signal, and it looks like a comb in the time-frequency plane, see Figure 2 as an example.
The global maximum corresponds to the case when two combs representing a signal and a search template coincide exactly in amplitude everywhere in the time-frequency plane. The reason for the local maxima is a partial overlap between the signal harmonics and the harmonics of a template. These might not be the same harmonics (the same set of l, m, n) and the strength of a given local maximum will depend on how long (in frequency and in time) the harmonics of the signal and template coincide.
In the search for a GW signal we use matched filtering which is an optimal detection technique in the presence of Gaussian noise and can be seen as an inner product of the data x(t) = n(t) + s(t) with a template h(t).
Here n(t) is the instrumental noise and the signal s(t) = s(t; λ) depends on the parameters of the source ( λ), which we are trying to estimate. The inner product is defined as Figure 2: The time-frequency plot of a typical GW signal from an EMRI, there are 30 clearly identifiable harmonics slowly evolving in time. The amplitude is colour coded. The time is in seconds.
(x, h) = 2ℜ ∞ 0x * (f )h(f ) S n (f ) df,(40)
where tilde denotes a Fourier transformed quantity and S n (f ) is the onesided power spectral density of the noise in the detector. If the signal is confined within a narrow frequency band around f 0 , so that we can treat S n (f 0 ) as almost constant, the inner product can also be written in the time domain in a simple form:
(x, h) ≈ 1 S n (f 0 ) T 0 x(t)h(t)dt,(41)
where T is the observation time (or duration of a template). The assumption that S n (f ) is approximately constant over the signal evolution is valid for signals of duration up to 2-5 months (dependent on the parameters). Since the amplitude of an EMRI signal is a slowly growing function of time, one can see from Eq. (41) that the SNR (SN R 2 = (s, s)) roughly grows as the square root of the observation time. We can use a maximum likelihood estimator to determine the GW parameters. The likelihood ratio is given by
Λ( λ) = P (x|h( λ) P (x|0) = e (x,h( λ))− 1 2 (h( λ),h( λ)) ,(42)
where P (x|h( λ) is the probability that the data x would be observed when a signal corresponding to the specified set of parameters is present in the data and P (x|0) is the probability that the data would be observed when no signal was present. Usually the likelihood (or log-likelihood) can be maximised over some parameters of the signal analytically, whereas maximisation over other parameters requires a numerical search. The analytically maximised likelihood is quite often referred as the F -statistic [67,64,63]. Based on the equations (40), (41) we can introduce a cumulative likelihood (or cumulative F -statistic) in the time and/or in the frequency domain by varying the upper limit of integration. If the template matches the signal exactly we expect to have steady growth of the cumulative F -statistic as a function of time or frequency (in other words it should be a monotonic and not decreasing function). In the case of a local maxima we will observe "bursts" of increase in the F -statistic around instances of time (or frequency) where one or more harmonics of the template and signal match. This is illustrated in Figure 3, the left panel shows schematically a harmonic of a template successively intersecting and overlapping with two different harmonics of the signal, one of which (in black) if stronger than the other. Figure 3: Cartoon showing two harmonics of a signal in green and black (black being stronger) and a harmonic of the template intersecting the signal at two instances (left plot). In the right plot we give the corresponding accumulation of the F -statistic in time. The two significant positive slopes (in pink) corresponds to two instances of overlaps between a signal and a template.
In the right panel of the same Figure, we show the corresponding accumulation of the F -statistic, and the instances of two intersections are clearly seen here as a rapid increase in the F -statistic. This illustrates nicely the reason for the presence of strong local maxima in the parameter space which we hit while constructing the Markov chain: harmonics of a signal can reproduce (overlap) one or a few strong harmonics of a signal for a span of time sufficient to accumulate a significant value of the detection statistic. This makes a "curse" into a "blessing": we can use the information of the locations of the local maxima to guide the search to find the global maximum of the likelihood. This is a key part of the search for EMRIs and the main basis for the two specific methods described in the following subsections. We find many local maxima by running multiple MCMC chains with different seeds, and then analyse the accumulation of the F -statistic to identify the parts (harmonics) of the signal that were found at each of those local maxima. Then we use this information to run a constrained MCMC (as described in subsection 4.1) or place them on the time-frequency plane and fit them with the harmonic tracks of a template by varying the source parameters (as described in subsection 4.2).
Constrained Markov Chain Monte Carlo search
In this subsection we will summarize the method which was successfully used to analyse the Mock LISA data challenge [4] and described in greater detail in [63]. In this method we split the data into 6-month long subsets and start by analysing each of them separately, before joining them together once we have started to lock onto the signal.
In the first step we perform a stochastic search: we randomly draw parameters from the prior range and evaluate their likelihoods. This is continued until multiple statistically significant points have been identified in the parameter space. Those points are then refined by running small MCMC chains seeded at those points. The local maxima are then analysed to find common harmonics (in time and frequency). These are identified as sections of harmonics of the true signal, although usually we do not know the associated harmonic indices.
In the second step we run a constrained MCMC. The sections of harmonics found in the first stage serve as constraints. We do realise that those constraints might not be exact, so we first run the MCMC with the frequency constraints and adjust the other parameters then we release the constraint and allow the code to adjust the constrained frequencies before fixing these again and repeating. This works very well in practice, even if the frequency of some of the (especially weak) harmonics was not determined very accurately initially. We also run several chains simultaneously to check for convergence to a global maximum.
In the third step we join the 6-month-long subsets of data together and let the chains adjust to match together the best found solutions in each subset. This method was used to analyse simulated data with a single relatively strong (SNR between 50 and 130) EMRI signal ( [63]). The identification of a signal was remarkably good with an ultra-precise recovery of the system parameters. The technique was also used to analyse the third Mock LISA data challenge data set, for which there was a single data set with five weak (SNR about 20) EMRI signals. The technique successfully identified two signals, while for the other three signals we identified that they were present but did not determine reliable estimates of their parameters before the challenge deadline.
Detection of EMRIs using a phenomenological template family
In this subsection we summarise the method described in [64]. The main idea of this approach was to detect GW signals from EMRIs in a model independent way using a minimal set of assumptions about the signal: (1) the orbital motion can be described by six slowly (on the radiation reaction time scale) changing quantities; and (2) the signal is represented by a set of harmonics of those (three) orbital frequencies with slowly changing amplitude. Those are rather mild constraints and should describe also "dirty" EMRIs where the orbital motion is perturbed either by the astrophysical environment or by a deviation in the spacetime geometry of the central BH [34,68]. We can use the assumption of slow frequency and amplitude evolution to decompose the phase and amplitude of each harmonic as a Taylor series and perform the search over the coefficients of the Taylor expansion. We call this a phenomenological EMRI template -the relationship between the Taylor series coefficients and the physical parameters depends on the specific model for the GW signal from an EMRI system. By searching over phenomenological parameters (Taylor coefficients) we do not restrict ourselves to any specific model within the framework of our assumptions above. The truncation of the Taylor series and the number of harmonics included depends mainly on the SNR of the signal: for weak signals we have to use a higher order expansion in order to match the signal for a longer time. Detection of EMRI signals in this model independent way allows us to relax stringent requirements on the accuracy of the theoretical model and to test alternatives to the assumption of a CO inspiral occurring in a pure vacuum Kerr spacetime.
Here we describe the simulations performed in [64]. Three month long data sets were simulated containing an EMRI signal (SN R = 50) using the numerical kludge as a model. Multiple MCMC searches using the phenomenological templates were carried out with different starting seeds. The results were collected and analysed for the presence of local maxima. For each identified maximum a patch of the signal harmonic which was found was extracted and placed on the time-frequency plane. The resulting map looks as presented in Figure 4. In this example the injected source was a strong signal and the method recovered 13 harmonics. In more realistic cases we would expect to recover 3-5 harmonics only. We note that the strong harmonics (at low frequency) are better recovered (through the full duration of the observation). Notice also that the last month of the data is recovered less well than the first two, which is due to the orbital motion of the detector -the antenna beam pattern during the last month is pointing away from the source.
In the second stage it is necessary to assume a certain EMRI model, so that the found harmonics can be identified and the physical parameters of the system recovered. In particular it is here that we can assume several alternatives: a CO spiralling toward a Kerr MBH, a CO spiralling into a massive boson star, a "dirty" Kerr black hole (a bumpy BH or a complex astrophysical environment). Once the model is assumed, we can find the set of parameters which give the best fit to the found set of harmonics (in amplitude and in their evolution). One can use a simple chi-square test of goodness of fit to estimate how well the assumed model describes the observed harmonic tracks and hence make a statement about the model. Results for the recovery of orbital parameters if the same model is used for recovery and signal generation were presented in [64].
Conclusion
In this article we have described one of the most interesting GW sources for the future space based gravitational wave observatory eLISA. We have briefly described the various channels for EMRI formation and expected event rates. Then we went through an inventory of available models for the GW signal generated by EMRIs. We also briefly discussed the osculating element approach for integration of the forced (under radiation reaction) motion of a CO in Kerr spacetime, and its application to the case of a spinning CO. One non-trivial question is the influence of the spin supplementary condition on the computed motion of a spinning CO and we have addressed this by looking at a simplified case: the motion of a spinning test mass in de Sitter spacetime. This should provide guidance on how to proceed in the case of a Schwarzschild or Kerr spacetime. Finally we have described the challenges which we will face in extracting GW signals generated by EMRIs from eLISA data. The main problem is to search for a global maximum of likelihood in the multidimensional parameter space, when multiple strong local maxima are also present. We have described how one can extract useful information about the signal from the locations of those local maxima in order to direct the search to the correct solution. In addition we have outlined the possibility that these methods can be used to verify that the central massive compact object is indeed described by the Kerr metric, as predicted by general relativity.
Figure 1 :
1Expected precision of parameter estimation from observed EMRI events, computed using the Fisher information matrix: MBH mass (M , dashed line), CO mass (m, solid line), MBH spin (a), orbital eccentricity before plunge (e pl ) and deviation of the MBH quadrupole moment from the Kerr value (Q).
Figure 4 :
4Recovered patches of the signal corresponding to a strong accumulation of the F -statistic.
Modelling the GW signal from EMRIsFor detection of EMRIs we will utilize matched filtering, this technique assumes that we can model the GW signal and then cross-correlate it with the data. The EMRI signal depends on 14 parameters (actually on 17 if we take into account the spin of CO), which we do not know a priori and need to infer from the measured data. To do this, we must generate many signals from a given model (templates) across the full, 14-D, parameter space to find the parameters that best fit the data (this set of parameters, which maximize the likelihood, are called "maximum likelihood estimators" of those parameters). We will describe the search procedure in detail in the next section.The presence of noise in the data stream causes the best-fit parameters to differ from the true parameters of the GW signal. The size of this difference can be estimated using the Fisher information matrix, as shown inFig. 1. If
In fact this formalism does not assume the force is small -there is a unique geodesic passing through any given point with a particular velocity and so any trajectory can be described as an osculating geodesic. However, the approach is most useful when the force is small since then the trajectory remains almost geodesic and parameterising it in terms of instantaneous geodesic motion is useful.
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| [] |
[
"SPARSE FILTERED NERVES",
"SPARSE FILTERED NERVES"
] | [
"Nello Blaser ",
"Morten Brun "
] | [] | [] | We give an elementary approach to efficient sparsification of filteredČech complexes and more generally sparsification of filtered Dowker nerves. For certain point clouds in high dimensional euclidean space the sparse nerves presented here are substantially smaller than the sparse Rips complex constructed by Sheehy in[12]. This means that is feasible to compute persistent homology of some point clouds and weighted undirected networks whose Rips complexes can not be computed by state of the art computational tools.Our construction is inspired by[12]. We view it as consisting of two parts. In one part we replace a metric by a Dowker dissimilarity whose nerve is a small approximation of theČech nerve of the original metric. In the other part we replace the nerve of a Dowker dissimilarity by a smaller homotopy equivalent filtered simplicial complex. This filtered simplicial complex is the smallest member of a class of sparsifications including the ones in [6] and[3]. | null | [
"https://arxiv.org/pdf/1810.02149v1.pdf"
] | 119,157,318 | 1810.02149 | 86f2b09327acc83cfabe8a3a2d3e2df44ae78f7c |
SPARSE FILTERED NERVES
4 Oct 2018
Nello Blaser
Morten Brun
SPARSE FILTERED NERVES
4 Oct 2018
We give an elementary approach to efficient sparsification of filteredČech complexes and more generally sparsification of filtered Dowker nerves. For certain point clouds in high dimensional euclidean space the sparse nerves presented here are substantially smaller than the sparse Rips complex constructed by Sheehy in[12]. This means that is feasible to compute persistent homology of some point clouds and weighted undirected networks whose Rips complexes can not be computed by state of the art computational tools.Our construction is inspired by[12]. We view it as consisting of two parts. In one part we replace a metric by a Dowker dissimilarity whose nerve is a small approximation of theČech nerve of the original metric. In the other part we replace the nerve of a Dowker dissimilarity by a smaller homotopy equivalent filtered simplicial complex. This filtered simplicial complex is the smallest member of a class of sparsifications including the ones in [6] and[3].
Introduction
Given a subset L of a metric space W , the ambientČech complex C t (L, W ) is the nerve of the set of t-balls in W centred at points in L considered as a cover of the union of such balls. Considering L as a metric space with the induced metric, we callČ t (L, L) the intrinsicČech complex of L. Up to homotopy, another way to describe the ambienť Cech complexČ t (L, W ) is as the filtration t part of the Dowker nerve of the function
Λ : L × W → [0, ∞]
obtained by restricting the metric d : W × W → [0, ∞) to the subset L × W of W × W . From the perspective of computer implementation these relativeČech complexes and their Dowker counterparts have the defect that they grow rapidly when the size of L increases. In order to mitigate this Sheehy, Botnan-Spreemann and Cavanna-Jahanseir-Sheehy proposed sparse approximations toČ t (L, W ) in the situation where W = R d equipped with a convex metric and L is a finite subset of W [12,4,6]. Inspired by their work, in [3], we constructed sparsifications of nerves of Dowker dissimilarities satisfying the triangle inequality. In this paper we construct sparsifications of arbitrary Dowker dissimilarities, that is, arbitrary functions of the form
Λ : L × W → [0, ∞].
Our construction is minimal among a class of sparsifications including the ones in [6] and [3]. In the situation where L and W are finite and all the values Λ(l, w) for (l, w) ∈ L × W are stored in memory these sparsifications can be implemented on a computer in a direct way. In the situation where W = R d with Euclidean metric and L is a finite subset of W we use the miniball algorithm [10] to implement sparsě Cech nerves.
The intrinsic and the ambientČech complexes are related by the inclusionsČ t (L, L) ⊆Č t (L, W ) ⊆Č 2t (L, L), so their corresponding persistent homologies are 2-interleaved. The ambientČech complex has homotopy type given by the sublevel filtration for the function f : W → [0, ∞) whose value on a point in W is its distance to L. If L is contained in a metric subspace N of W , then the persistent homologies ofČ(L, W ) andČ(N, W ) are additively interleaved with interleaving factor given by the Hausdorff distance d H (L, N) between L and N. Moreover, the persistent homologies of C(L, L) andČ(N, N) are interleaved with the factor 2d H (L, N), that is, two times the Hausdorff distance between L and N. Thus the persistent homologies of bothČ(L, L) andČ(L, W ) can be considered as an approximations to the intrinsicČech homology of N. In particular if W is a Riemannian manifold with distance given by the geodesic metric, then the persistent homologies ofČ(L, W ) andČ(W, W ) are additively interleaved with interleaving factor given by the Hausdorff distance d H (L, W ) between L and W . At filtration values up to the convexity radius of W the persistent homologyČ(W, W ) is isomorphic to the homology of W . (See e.g. [2,Section 6.5
.3])
This paper is organized as follows. Section 2 introduces the reader to the basic concepts used throughout the remaining sections. In [3] we did not explain how interleavings with respect to translation functions (see Definition 2.3) are related to matchings. Since this is crucial to the interpretation of persistence diagrams of sparse nerves we discuss this in Section 3. In Section 4, we introduce truncation of Dowker nerves and give a direct argument showing that the truncated Dowker nerve is interleaved with the Dowker nerve of the original Dowker dissimilarity. In Section 5, we sparsify Dowker nerves in a way that preserves homotopy type. In particular, persistent homology does not change under sparsification. This sparsification is obtained via a function R : L → [0, ∞] having certain properties. Functions with these properties, we call restriction functions. With the concept of restriction functions at hand we display the smallest restriction function relative to a parent function ϕ, the (Λ, ϕ)-restriction. In Section 6, we give a short description of details behind python package for computation of persistent homology of sparsified Dowker nerves.
Preliminaries
2.1. Filtratrations. We consider the interval [0, ∞] as a category with the underlying set of the interval as objects and with a morphism s → t if and only if s ≤ t.
Definition 2.1. Let C be a category. The category of filtered objects in C is the category of functors from [0, ∞] to C. That is, a filtered object in C is a functor C : [0, ∞] → C and a morphism f : C → C ′ of filtered objects in C is a natural transformation.
Recall that a function α :
[0, ∞) → [0, ∞) is order preserving if s ≤ t implies α(s) ≤ α(t).
Definition 2.2. Let β : [0, ∞) → [0, ∞) be an order preserving function with lim t→∞ β(t) = ∞. The generalized inverse of β is the order preserving function
β ← : [0, ∞) → [0, ∞), β ← (t) = inf{s ∈ [0, ∞) | β(s) ≥ t}
with the defining property β ← (t) ≤ s if and only if t ≤ β(s).
Definition 2.3. A translation function is an order preserving function
α : [0, ∞) → [0, ∞) with t ≤ α(t) for every t ∈ [0, ∞).
Definition 2.4. Given a filtered object C : [0, ∞] → C and a trans-
lation function α : [0, ∞) → [0, ∞), the pull-back of C along α is the filtered object α * C = C • α with (α * C)(t) = C(α(t)). The α-unit of C is the morphism α * C : C → α * C with α * C (t) = C(t → α(t)) : C(t) → (α * C)(t) = C(α(t)).
Definition 2.5. Let k be a field. The category of persistence modules over k is the category of filtered objects in the category of vector spaces over k.
Definition 2.6. Let k be a field and let α : [0, ∞) → [0, ∞) be a translation function. A persistence module V over k is α-trivial if the α-unit of V is trivial, that is, if α * V = 0.
Dowker Dissimilarities.
Our presentation of this preliminary material closely follows [3].
Definition 2.7 (Dowker [9]). The nerve of a relation R ⊆ X × Y is the simplicial complex
NR = { finite σ ⊆ X | ∃ y ∈ Y with (x, y) ∈ R for all x ∈ σ}.
The following definition is inspired by the concept of networks as it appears in [7]. Definition 2.8. A Dowker dissimilarity Λ consists of two sets L and W and a function Λ :
L×W → [0, ∞]. Given t ∈ [0, ∞], we let Λ t ⊆ L×W be the relation Λ t = {(l, w) ∈ L × W | Λ(l, w) < t}. Definition 2.9. Let Λ : L × W → [0, ∞] be a Dowker dissimilarity.
The Dowker Nerve NΛ of Λ is the filtered simplicial complex with vertex set L and the nerve NΛ t of the relation Λ t in filtration degree t ∈ [0, ∞].
Definition 2.10. A morphism C : Λ → Λ ′ of Dowker dissimilarities Λ : L × W → [0, ∞] and Λ ′ : L ′ × W ′ → [0, ∞] consists of a relation C ⊆ L × L ′
so that for every t ∈ [0, ∞] and for every σ ∈ NΛ t , the set
NC(σ) = {l ′ ∈ L ′ | (σ × {l ′ }) ∩ C is non-empty}
is non-empty and contained in NΛ ′ t . If C ⊆ L × L ′ and C ′ ⊆ L ′ × L ′′ are morphisms C : Λ → Λ ′ and C ′ : Λ ′ → Λ ′′ of Dowker dissimilarities, then the composition C ′ C : Λ → Λ ′′ is the subset of L × L ′′ defined by
C ′ C = {(l, l ′′ ) | there exists l ′ ∈ L ′ with (l, l ′ ) ∈ C and (l ′ , l ′′ ) ∈ C ′ }.
The identity morphism ∆ L : NΛ → NΛ is
∆ L = {(l, l) | l ∈ L} ⊆ L × L.
Proposition 2.11. The Dowker nerve is functorial in the sense that it induces a functor N from the category of Dowker dissimilarities to the category of functors from [0, ∞] to the category of topological spaces.
Proof. Let C ⊆ L × L ′ and C ′ ⊆ L ′ × L ′′ be morphisms C : Λ → Λ ′ and C ′ : Λ ′ → Λ ′′ of Dowker dissimilarities. Given t ∈ [0, ∞], the functions NC : NΛ t → NΛ ′ t and NC ′ : NΛ ′ t → NΛ ′′ t are order preserving. Thus, they induce morphisms of geometric realizations of barycentric subdivisions. The identity morphism ∆ L : Λ → Λ induces the identity function N∆ L : NΛ → NΛ. In order to finish the proof we show that NC ′ (NC(σ)) = N(C ′ C)(σ) for every σ ∈ NΛ t . If l ′′ ∈ NC ′ (NC(σ)), then there exists l ′ ∈ NC(σ) so that (l ′ , l ′′ ) ∈ C ′ . Since l ′ ∈ NC(σ) there exists l ∈ σ so that (l, l ′ ) ∈ C. We conclude that (l, l ′′ ) ∈ C ′ C and thus l ′′ ∈ N(C ′ C)(σ). Conversely, if l ′′ ∈ N(C ′ C)(σ), then there exists l ∈ σ so that (l, l ′′ ) ∈ C ′ C. By definition of C ′ C this means that there exists l ′ ∈ L ′ so that (l, l ′ ) ∈ C and (l ′ , l ′′ ) ∈ C ′ . We conclude that l ′ ∈ NC(σ) and that l ′′ ∈ NC ′ (NC(σ)).
Corollary 2.12. Let k be a field. The persistent homology H * (NΛ) of NΛ with coefficients in k is functorial in the sense that it is a functor from the category of Dowker dissimilarities to the category of persistence modules over k.
2.3.
Interleaving. Here we present a notion of interleaving inspired by Bauer and Lesnik [1].
Definition 2.13. Let C and C ′ be filtered objects in a category C and let α :
[0, ∞) → [0, ∞) be a translation function. (1) A morphism G : C → C ′ is an α-interleaving if for every t ∈ [0, ∞] there exists a morphism F t : C ′ (t) → C(α(t)) in C such that α * C (t) = F t • G(t) and α * C ′ (t) = G(α(t)) • F t .
(2) We say that C and C ′ are α-interleaved if there exists an αinterleaving G : C → C ′ .
Suppose we are in the situation that we have an inclusion K ⊆ L of filtered simplicial complexes and that we are able to compute the filtration value of simplices in L, but we have no constructive way of computing the filtration value of simplices in K. Then we can construct at filtered simplicial sub-complex K ′ of L with K ′ ∞ = K ∞ . If know that the inclusion K ⊆ L is an α-interleaving, then the following lemma implies that also the inclusion K ′ ⊆ L is an α-interleaving. This situation occurs for example when L is aČech complex.
Lemma 2.14. Let C, C ′ and C ′′ be filtered objects in a category C and let α : [0, ∞) → [0, ∞) be a translation function. Let G : C → C ′ and G ′ : C ′ → C ′′ be morphisms of filtered objects. Suppose that G ′ (t) : C ′ (t) → C ′′ (t) is a monomorphism for every t ∈ [0, ∞] and that the composition G ′ G : C → C ′′ is an α-interleaving. Then also G ′ : C ′ → C ′′ is an α-interleaving.
Proof. Let t ∈ [0, ∞], and pick E t : C ′′ (t) → C(α(t)) such that
α * C (t) = E t • (G ′ G)(t) and α * C ′′ (t) = (G ′ G)(α(t)) • E t . Defining F t = E t G ′ (t) : C ′ (t) → C(α(t)) the above relations imply that α * C (t) = E t • (G ′ G)(t) = (E t G ′ (t)) • G(t) = F t • G(t) and G ′ (α(t)) • α * C ′ (t) = α * C ′′ (t) • G ′ (t) = (G ′ G)(α(t)) • E t • G ′ (t) = G ′ (α(t)) • G(α(t)) • F t . Then also α * C ′ (t) = G(α(t))•F t , since G ′ (α(t)) is a monomorphism.
The following results are analogues of [ Lemma 2.16 (Triangle inequality). Let G : C → C ′ be an α-interleaving and let G ′ : C ′ → C ′′ be an α ′ -interleaving of filtered objects in a category C. Then the composition
G ′′ = G ′ G : C → C ′′ is an αα ′ - interleaving. Proof. Let α ′′ = αα ′ and let t ∈ [0, ∞]. By definition there exist morphisms F α ′ (t) : C ′ (α ′ (t)) → C(α(α ′ (t))) and F ′ t : C ′′ (t) → C ′ (α ′ (t)) so that α * C (α ′ (t)) = F α ′ (t) • G(α ′ (t)) and α * C ′ (α ′ (t)) = G(α(α ′ (t))) • F α ′ (t) . and α ′ * C ′ (t) = F ′ t • G ′ (t) and α ′ * C ′′ (t) = G ′ (α ′ (t)) • F ′ t . Let F ′′ t = F α ′ (t) F ′ t : C ′′ t → C αα ′ (t)
. The above relations imply that the right hand triangles in the diagram
C(t) C ′ (t) C ′′ (t) C(α ′ (t)) C ′ (α ′ (t)) C(αα ′ (t)) G(t) α ′ * C (t) α ′′ * C (t) G ′ (t) α ′ * C ′ (t) F ′ t G(α ′ (t)) α * C (α ′ (t)) F α ′ (t)
commute. The quadrangle in the above diagram commutes since G is a natural transformation and commutativity of the left hand triangle follows directly from the definition of the definition of the α ′′ -unit. We conclude that
α ′′ * C (t) = F ′′ t • G ′′ (t)
. The above relations also imply that the upper triangles in the diagram
C ′′ (t) C ′ (α ′ (t)) C(αα ′ (t)) C ′′ (α ′ (t)) C ′ (αα ′ (t)) C ′′ (αα ′ (t)) F ′ t α ′ * C ′′ (t) α ′′ * C ′′ (t) F α ′ (t) α * C ′ (α ′ (t)) G ′ (α ′ (t)) G(αα ′ (t)) α ′ * C ′′ (α ′ (t)) G ′ (αα ′ (t))
commute. The quadrangle in the above diagram commutes since G ′ is a natural transformation and commutativity of the left hand triangle follows directly from the definition of the definition of the α ′′ -unit. We conclude that
α ′′ * C ′′ (t) = G ′′ (α ′′ (t)) • F ′′ t .
We find that the following lemma justifies our definition of α-interleaving. Proof. Suppose first that G is an α-interleaving. Fix t and pick F t :
V ′ (t) → V (α(t)) so that α V * (t) = F t • G(t) and α V ′ * (t) = G(α(t)) • F t . Then v ∈ ker G(t) implies α ker G * v = α V * v = F t (G(t)(v)) = 0. Similarly, if v ′ + im G(t) ∈ coker G(t), then α coker G * (v ′ + im G(t)) = α V ′ * (v ′ ) + im G(α(t)) = 0 since α V ′ * (v ′ ) = G(α(t))(F t v ′ ) ∈ im G(α(t))
. Thus ker G and coker G are α-trivial. Conversely, suppose that ker G and coker G are α-trivial and fix t. Choose a basis e 1 , . . . , e a for ker G(t) and choose f 1 , . . . , f b so that e 1 , . . . , e a , f 1 , . . . , f b is a basis for C(t). Note that G(t)(f 1 ), . . . , G(t)(f b ) are linearly independent in C ′ (t) and choose g 1 , . . . , g c so that
G(t)(f 1 ), . . . , G(t)(f b ), g 1 , .
. . , g c is a basis for C ′ (t). We use this basis to define F t : C ′ (t) → C(α(t)) as follows: On basis elements of the form
G(t)(f i ) we define F t (G(t)(f i )) = α C * (t)(f i ).
Now consider basis elements of the form g i . Since α coker G * = 0 we know
that α C ′ * (t)(g i ) ∈ im G(α(t)). We choose c i ∈ C(α(t)) so that G(α(t))(c i ) = α C ′ * (t)(g i ) and define F t (g i ) = c i . Since α ker G = 0 we have α C * (t)(e i ) = 0 = F t (G(e i )), so F t G(t) = α C * (t).
On the other hand, the equation
G(αt)(F t (G(t)(e i ))) = G(αt)(α C * (e i )) = α C ′ * (G(t)(e i )) shows that α C ′ * (t) = G(αt)F t .
Matchings
Our presentation of matchings follows [11].
Definition 3.1. The set of persistence intervals is the set E of intervals in [0, ∞].
We write a for the closure of an interval a ∈ E. Note that a is determined by the end points of the interval a.
Definition 3.2. A persistence diagram consists of a set X and a function p : X → E from X to the set of persistence intervals. We refer to the elements of X as persistence classes.
Definition 3.3. A matching R of two persistence diagrams p : X → E and p ′ : X ′ → E consists of a relation R ⊆ X × X ′ with the property that the compositions π 1 : R → X, π 1 (x, x ′ ) = x and π 2 : R → X ′ , π 2 (x, x ′ ) = x ′with
the inclusion of R in X × X ′ and the projections to X and X ′ respectively are injective with p•π 1 = p ′ •π 2 . We say that a persistence class x ∈ X is matched by R if there exists a persistence class x ′ ∈ X ′ so that (x, x ′ ) ∈ R. Similarly we say that a persistence class
x ′ ∈ X ′ is matched by R if there exists a persistence class x ∈ X so that (x, x ′ ) ∈ R. Definition 3.4. Let J be a subset of [0, ∞]. The persistence module k(J) has values k(J)(t) = k if t ∈ J 0 otherwise
and structure maps equal to identity maps whenever possible.
Definition 3.5. Let α : [0, ∞) → [0, ∞) be a translation function and let p : X → E be a persistence diagram. We say that a persistence class x ∈ X is α-trivial if the persistence module k(p(x)) is α-trivial. Otherwise we say that x is α-nontrivial. Note that if p(x) has end points b < d, then p(x) is α-trivial if and only if d ≤ α(b).
Definition 3.6. Let V be a persistence module over a field k. We say that p :
X → E is a persistence diagram of V if there exists an isomorphism of the form V ∼ = x∈X k(p(x)).
Definition 3.7. The category of pointwise finite dimensional persistence modules over the field k is the full subcategory of the category of persistence modules V over k with V t finite dimensional for every t ∈ [0, ∞].
We restate the decomposition theorem for pointwise finite-dimensional persistence modules [8, Theorem 1.1] in our notation.
Theorem 3.8. Let k be a field. Every pointwise finite dimensional persistence module over k has a persistence diagram.
We now state the generalized induced matching theorem [1, Theorem 6.1] and [11,Theorem 3.2]. In order to do this we use the generalized inverse function of a translation function from Definition 2.2. Theorem 3.9. There exists a function χ : Mor(Pers) → Match from the set of morphisms of pointwise finite dimensional persistence modules over the field k to the set of matchings with the following properties: Let f : V → V ′ be a morphism of pointwise finite persistence modules and let χ(f ) be of the form
χ(f ) : (X p − → E) → (X ′ p ′ − → E),
that is,
χ(f ) ⊆ X × X ′ .
Assume that f is an α-interleaving and that (x,
x ′ ) ∈ χ(f ) with p(x) = [b, d] and p ′ (x ′ ) = [b ′ , d ′ ]
. Then the following holds:
(1) b ′ ≤ b < d ′ ≤ d and (2) b ≤ α(b ′ ) and (3) α ← (d) ≤ d ′ .
Moreover all α-nontrivial persistence classes of X and X ′ are matched by χ(f ).
In the above situation, if α is bijective, then (3) is equivalent to
d ≤ α(d ′ ).
If we further assume that x ′ is α-nontrivial, then α(b ′ ) < d ′ and the point (b, d) lies in the box with corners (b ′ , d ′ ) and (α(b ′ ), α(d ′ )). Conversely, if α is bijective and x is α-nontrivial, then α(b) < d and the point (b ′ , d ′ ) lies in the box with corners (α ← b, α ← d) and (b, d). Proof. It suffices, for every t ∈ [0, ∞], to find a map f t : NΛ t → NΓ α(t) so that the following diagrams commute up to homotopy:
Truncated Nerves
NΓ t NΛ t NΓ α(t) N Γ t≤α(t) ft and NΛ t NΓ α(t) NΛ α(t) . ft N Λ t≤α(t)
Fix t and choose a function f t : L → L so that for every l ∈ L with Λ(l, w) < t the inequalities Λ(f t (l), w) < α(t) and Λ(f t (l), w) < T (f t (l)) hold.
Below we first show that f t induces a simplicial map
f t : NΛ t → NΓ α(t) .
That is, we show that if σ ∈ NΛ t , then f t (σ) ∈ NΓ α(t) . Next we show that f t (σ) ∪ σ ∈ NΛ α(t) so that the lower of the above displayed diagrams commutes up to homotopy. We will finish by showing that if σ ∈ NΓ t , then f t (σ) ∪ σ ∈ NΓ α(t) so that also the upper of the above displayed diagrams commutes up to homotopy. Let σ ∈ NΛ t and pick w ∈ W so that Λ(l, w) < t for every l ∈ σ. Then, for every l ∈ σ we have Λ(f t (l), w) < α(t) and Λ(f t (l), w) < T (f t (l))) so in particular Γ(f t (l), w) = Λ(f t (l), w) < α(t). This implies both that f t (σ) ∈ NΓ α(t) and that f t (σ) ∪ σ ∈ NΛ α(t) . Finally, if σ ∈ NΓ t and we pick w ∈ W so that Γ(l, w) = Λ(l, w) < t for every l ∈ σ, then the above argument also implies that f t (σ)∪σ ∈ NΓ α(t) .
P (l ′ , l) = {Λ(l ′ , w) | w ∈ W with α(Λ(l, w)) ≤ Λ(l ′ , w)}.
Given a base point l 0 ∈ L the cover dissimilarity Λ α : L × L → [0, ∞] for Λ with respect to α by
Λ α (l ′ , l) = 0 if l = l ′ ∞
if l = l ′ and l ′ = l 0 sup(P (l ′ , l) ∪ {0}) if l = l ′ and l ′ = l 0 .
If L is finite and < is a total order on L with l 0 as minimal element, we define the α-insertion radius λ α (l) of l ∈ L as
λ α (l) = ∞ if l = l 0 sup k≥l inf l ′ <l Λ α (l ′ , k) if l = l 0
Given l ∈ L with l = l 0 and t ∈ [0, ∞] with α(t) > 0, pick l ′ ∈ L minimal with Λ α (l ′ , l) < α(t). (Such an l ′ exists since Λ α (l, l) = 0 and Λ α (l 0 , l) = ∞). Let w ∈ W with Λ(l, w) < t. Then either Λ(l ′ , w) > Λ α (l ′ , l) or Λ(l ′ , w) ≤ Λ α (l ′ , l) < α(t). If Λ(l ′ , w) > Λ α (l ′ , l), then Λ(l ′ , w) < α(Λ(l, w)) ≤ α(t). The insertion radius of l ∈ L with respect to the total order < is
Thus, Λ(l, w) < t implies α(t) > Λ(l ′ , w). Since ∞ = λ α (l 0 ) ≥ α(t) and λ α (l ′ ) = sup k≥l ′ inf l ′′ <l ′ Λ α (l ′′ , k) ≥ inf l ′′ <l ′ Λ α (l ′′ , l) ≥ α(t), the function λ α : L → [0, ∞] is an α-truncation function for Λ.λ(l) = ∞ if l = l 0 inf l ′ <l Λ(l ′ , l) otherwise.
For Λ as in Definition 4.5 a farthest point sample L = {l 0 < · · · < l n } can be produced recursively starting from an initial point l 0 . When l 0 , . . . , l k have been produced, we choose l k+1 so that inf l ′ ∈{l 0 ,...,l k } Λ(l ′ , l k+1 ) = sup
l ′′ / ∈{l 0 ,...,l k } inf l ′ ∈{l 0 ,...,l k } Λ(l ′ , l ′′ ).
Note that If αΛ(l, w) ≤ Λ(l ′ , w), then the triangle inequality for d implies that Λ(l ′ , w) ≤ Λ L (l ′ , l) + Λ(l, w) and therefore Λ(l, w) ≤ Λ L (l ′ , l)/(c − 1). This, together with the triangle inequality for d implies that
λ(l) = ∞ if l = l 0 sup k≥l inf l ′ <l Λ(l ′ , k) otherwise.Λ(l ′ , w) ≤ Λ L (l ′ , l) + Λ(l, w) ≤ Λ L (l ′ , l) + Λ L (l ′ , l) (c − 1) = cΛ L (l ′ , l) (c − 1) .
From this consideration we can conclude that
Λ α (l ′ , l) ≤ cΛ L (l ′ , l) (c − 1) and that λ α (l) ≤ cλ L (l) (c − 1) .
Since λ α is an α-truncation function of Λ, so is the function T (l) = cλ L (l)/(c − 1).
There exist many truncation functions for a given translation function α. We have not succeeded in finding a class of truncation functions for α that are practical to implement and produces a smallest possible simplicial complex under this constraint. We leave this as a problem for further investigation. If the goal is merely to construct a Dowker dissimilarity whose Dowker nerve is small the amount of possibilities is even bigger.
Sparse Filtered Nerves
Definition 5.1. Let Λ : L × W → [0, ∞] be a Dowker dissimilarity and let R : L → [0, ∞] and ϕ : L → L be functions. We say that l ∈ L is a slope point if R(l) > R(l ′ ) for every l ′ ∈ ϕ −1 (l).The (R, ϕ)-nerve of Λ is the filtered simplicial complex N(Λ, R, ϕ) with N(Λ, R, ϕ)(t) consisting of all σ ∈ NΛ t such that there exists w ∈ W satisfying:
(1) Λ(l, w) < t for all l ∈ σ.
(2) Λ(l, w) ≤ R(l ′ ) for all l, l ′ ∈ σ and (3) Λ(l, w) < R(l) for all slope points l in σ. (1) For all (l, w) ∈ L × W with Λ(l, w) < Λ(ϕ(l), w) we have Λ(ϕ(l), w) ≤ R(l).
(2) For every l ∈ L with we have R(ϕ(l)) ≥ R(l).
P (l, l ′ ) = {Λ(l ′ , w) | w ∈ W with Λ(l, w) < Λ(l ′ , w)} and define ρ(l, l ′ ) = sup P (l, l ′ ) if P (l, l ′ ) is non-empty 0 if P (l, l ′ ) = ∅. The (Λ, ϕ)-restriction R(Λ, ϕ) : L → [0, ∞] is defined in several steps. First define R ′ : L → [0, ∞] by R ′ (l) = ρ(l, ϕ(l)) if ϕ(l) = l ∞ if ϕ(l) = l.
Given l ∈ L, let D(l) be the set of descendants of l, that is, l ′ ∈ D(l) if and only if there exists m ≥ 0 so that l = ϕ m (l ′ ). Next, we define R(Λ, ϕ) : L → L by
R(Λ, ϕ)(l) = max l ′ ∈D(l) R ′ (l ′ ).
Then, for every l ∈ L we have R(Λ, ϕ)(ϕ(l)) ≥ R(Λ, ϕ)(l), and ϕ(l) = l implies R(Λ, ϕ)(l) = ∞. Also, Λ(l, w) < Λ(ϕ(l), w) implies Λ(ϕ(l), w) ≤ R ′ (l) ≤ R(Λ, ϕ)(l).
Proposition 5.5. Let Λ : L × W → [0, ∞] be a Dowker dissimilarity with L finite and let ϕ : L → L be a parent function. Then the (Λ, ϕ)restriction R(Λ, ϕ) is the minimal restriction function for Λ relative to ϕ: If R is another restriction function for Λ relative to ϕ, then R(Λ, ϕ)(l) ≤ R(l) for every l ∈ L.
Proof. In the notation of Definition 5.4 it suffices to show that ρ(l, ϕ(l)) ≤ R(l) for all l ∈ L. We can assume that ϕ(l) = l because otherwise ρ(l, ϕ(l)) = ∞ = R(l). Given l ∈ L, if there exists a w ∈ W with Λ(l, w) < Λ(ϕ(l), w)
we have Λ(ϕ(l), w) ≤ R(l). By construction of ρ, this implies that ρ(l, ϕ(l)) ≤ R(l).
If no such w ∈ W with Λ(l, w) < Λ(ϕ(l), w)
exists, then ρ(l, ϕ(l)) = 0 ≤ R(l).
Proposition 5.5 shows that the (Λ, ϕ)-restriction function is the minimal restriction function for Λ relative to ϕ. In the following two examples we show that the sparsifications from [3,12] also are (R, ϕ)-nerves and that therefore the (Λ, ϕ)-restriction results in smaller nerves.
Example 5.6. Let Λ : L × W → [0, ∞] be a Dowker dissimilarity with L finite. As in Definition 5.4, given l, l ′ ∈ L let
P (l, l ′ ) = {Λ(l ′ , w) | w ∈ W with Λ(l, w) < Λ(l ′ , w)} and define ρ(l, l ′ ) = sup P (l, l ′ ) if P (l, l ′ ) is non-empty 0 if P (l, l ′ ) = ∅. Define R 0 : L → [0, ∞] by R 0 (l) = inf{ρ(l, l ′ ) | l ′ ∈ L and l ′ = l}
and let < be a total order on L with minimal element l 0 so that R 0 (l ′ ) > R 0 (l) implies l ′ < l. Given l ∈ L let Q 0 (l) = {l ′ ∈ L | l ′ < l and ρ(l, l ′ ) = R 0 (l)}.
If Q 0 (l) is non-empty we define ϕ(l) = min Q 0 (l).
Otherwise, that is, if Q 0 (l) is empty, we let R 1 (l) = inf{ρ(l, l ′ ) | l ′ < l}. and Q 1 (l) = {l ′ ∈ L | l ′ < l and ρ(l, l ′ ) = R 1 (l)}. If l is the minimal element of L, then Q 1 (l) is empty and we define ϕ(l) = l. Otherwise Q 1 (l) is non-empty and we define ϕ(l) = min Q 1 (l).
Since ϕ(l) ≤ l for every l ∈ L and < is a total order on L, the function ϕ : L → L is a parent function. We define R : L → [0, ∞] to be the restriction function for Λ relative to ϕ constructed in Definition 5.4. Example 5.7 (Parent restriction). In [3] we constructed the sparse filtered nerve NΛ of a Dowker dissimilarity Λ : L × W → [0, ∞] with L finite. In this example we describe a function ϕ : L → L and a restriction R for Λ relative to ϕ so that the (R, ϕ)-nerve of Λ is equal to the sparse Dowker nerve in [3,Definition 38]. Given l ∈ L we let W l = {w ∈ W | Λ(l, w) < ∞} and τ (l) = sup{Λ(l, w) | w ∈ W l }.
Let l 0 ∈ L and define τ : L → [0, ∞] as the function
τ (l) = ∞ if {l = l 0 } τ (l) otherwise.
Given l ∈ L we let
Q(l) = {l ′ | τ (l ′ ) > τ (l)}.
If Q(l) is non-empty we pick l ′ ∈ Q(l) and define ϕ(l) = l ′ . Otherwise we define ϕ(l) = l 0 . The parent restriction R : L → [0, ∞] is defined by
R(l) = τ (ϕ(l)).
It is readily verified that the above structure satisfies is a sparsification function for Λ with respect to ϕ. The R-nerve N(Λ, R, ϕ) is the sparse Dowker nerve introduced in [3, Definition 38] taking t to β(t) is bijective. Note that this implies that β ← β = id.
Let λ : L → [0, ∞] be the canonical insertion function for Λ as defined in [3] and let ϕ : L → L be the associated parent function. That is, for L = {l 0 , . . . , l n }, the function λ : L → [0, ∞] is defined by
λ(l) = ∞ if l = l 0 sup w∈W inf l ′ ∈{l 0 ,...,l k−1 } Λ(l ′ , w) if l = l k
and ϕ(l) is the smallest element in L such that there exists w ∈ W with Λ(λ(l), w) = λ(l). Recall that the (λ, β)-truncation of Λ is the Dowker dissimilarity Γ :
L × W → [0, ∞] defined in [3] by Γ(l, w) = Λ(l, w) if Λ(l, w) ≤ αβ ← λ(l) and β(0) ≤ λ(l) ∞ otherwise.
Since elements l ∈ L with λ(l) < β(0) do not contribute to the Dowker nerve of Γ we assume without loss of generality that β(0) ≤ λ(l) for every l ∈ L.
This truncation is not a truncation of Λ as defined in Definition 4.2. However, the Dowker dissimilarity Γ ′ : L × W → [0, ∞] described by the formula
Γ ′ (l, w) = Λ(l, w) if Λ(l, w) < αβ ← λ(l) ∞ otherwise
is a truncated Dowker dissimilarity and is smaller than Γ. In our implementation we use this description. The Sheehy restriction function is
S : L → [0, ∞], S(l) = α 2 β ← λ(l)
Our assumption on l 0 implies that S(l 0 ) = ∞. The Sheehy parent function ϕ : L → L is defined as follows: first we define ϕ(l 0 ) = l 0 for the minimal element l 0 in L. Next, given l = l 0 , choose w ′ ∈ W so that (l, w ′ ) ∈ T and use that λ is a β-insertion function to choose l ′ ∈ L so that Λ(l ′ , w ′ ) ≤ βαβ ← λ(l) < λ(l ′ ), and define ϕ(l) = l ′ . Since
λ(l) ≤ ββ ← λ(l) ≤ βαβ ← λ(l) < λ(l ′ )
we have S(ϕ(l)) > S(l) for every l ∈ L with S(l) < ∞. Given w ∈ W with Γ ′ (l, w) < ∞ we have Λ(l, w) < αβ ← λ(l), so for l ′ and w ′ as above the triangle inequality gives Λ(l ′ , w) ≤ Λ(l ′ , w ′ ) + Λ(l, w) < βαβ ← λ(l) + αβ ← λ(l) = α 2 β ← λ(l).
The inequality Λ(l ′ , w) ≤ α 2 β ← λ(l) implies
βα ← Λ(l ′ , w) ≤ βα ← α 2 β ← λ(l) ≤ βαβ ← λ(l) < λ(l ′ ).
Since ϕ(l) = l ′ we can conclude that
Γ ′ (ϕ(l), w) = Λ(ϕ(l), w) ≤ α 2 β ← λ(l) = S(l)
for every l ∈ L. We conclude that Γ ′ (l, w) < ∞ implies Γ ′ (ϕ(l), w) ≤ S(l). Proof. Since R(l) = ∞ whenever ϕ(l) = l, the two complexes agree when when L is of cardinality 1, and thus the result holds in this case. Let t ∈ [0, ∞] and let n > 1. Below we will show that if Λ : L × W → [0, ∞] is a Dowker dissimilarity with L a set of cardinality n and R is a restriction function for Λ relative to ϕ so that the inclusion
ι : N(Λ, R)(t) → NΛ t
is not a homotopy equivalence, then there exists a Dowker dissimilarity Λ ′ : L ′ ×W → [0, ∞] with L ′ a set of cardinality n−1 and R ′ a restriction function for Λ ′ relative to a function ϕ ′ : L ′ → L ′ so that the inclusion
ι : N(Λ ′ , R ′ )(t) → NΛ ′ t
is not a homotopy equivalence. Negating this we obtain the inductive step implying that the result holds for all finite sets L.
As above, let Λ : L × W → [0, ∞] be a Dowker dissimilarity with L a set of cardinality n > 1 and let R be a restriction function for Λ relative to ϕ. Fix t ∈ [0, ∞] and pick l n ∈ L so that firstly R(l n ) ≤ R(l) for every l ∈ L and secondly l n is not in the image of ϕ : L → L. This is possible since L is finite and R(ϕ(l)) ≤ R(l) for every l ∈ L. Suppose that ι : N(Λ, R)(t) → NΛ t is not a homotopy equivalence. Then there exists l ∈ L with R(l) < t since otherwise the two complexes are obviously equal. In particular R(l n ) < t. Let L ′ = L \ {l n } and let Λ ′ : L ′ × W → [0, ∞] be the restriction of Λ to L ′ × W ⊆ L × W . Further, let R ′ : L ′ → [0, ∞] be the restriction of R to L ′ and let ϕ ′ : L ′ → L be the restriction of ϕ to L ′ . Since l n is not in the image of ϕ we can consider ϕ ′ as a function ϕ ′ : L ′ → L ′ . Clearly R ′ is a restriction function for Λ ′ relative to ϕ ′ .
We define f t : L → L ′ by
f t (l) = ϕ(l n ) if l = l n l otherwise.
Given σ ∈ NΛ t we claim that σ ∪ f t (σ) ∈ NΛ t . If l n / ∈ σ, then this claim is trivially satisfied. In order to justify the claim when l n ∈ σ we pick w ∈ W with Λ(l, w) < t for every l ∈ σ. If Λ(l n , w) ≥ Λ(ϕ(l n ), w) then Λ(ϕ(l n ), w) < t and σ ∪ f t (σ) ∈ NΛ t . Otherwise by part (1) of Definition 5.3 the inequalities Λ(l n , w) < t and Λ(l n , w) < Λ(ϕ(l n ), w) imply that Λ(ϕ(l n ), w) ≤ R(l n ) < t. We conclude that σ ∪ f t (σ) ∈ NΛ t also in this situation.
Next we claim that
σ ∈ N(Λ, R, ϕ)(t) implies σ ∪ f t (σ) ∈ N(Λ, R, ϕ)(t).
Again we only need to consider the case l n ∈ σ. We have already shown that σ ∪f t (σ) ∈ NΛ t . Pick w ∈ W with Λ(l, w) < t, Λ(l, w) < R(l) and Λ(l, w) ≤ R(l ′ ) for every l, l ′ ∈ σ. Note in particular that Λ(l n , w) < t. If Λ(l n , w) ≥ Λ(ϕ(l n ), w) then
R(l) ≥ R(l n ) > Λ(l n , w) ≥ Λ(ϕ(l n ), w)
for all l ∈ L, so σ∪f t (σ) ∈ N(Λ, R)(t). On the other hand, if Λ(l n , w) < Λ(ϕ(l n ), w) then by (1) of Definition 5.3 the inequality Λ(l n , w) < t implies Λ(ϕ(l n ), w) ≤ R(l n ) ≤ R(l) for all l ∈ L. If ϕ(l n ) is a slope point, then Λ(ϕ(l n ), w) ≤ R(l n ) < R(ϕ(l n )). We conclude that σ ∪ f t (σ) ∈ N(Λ, R, ϕ)(t).
We can now conclude that the function f t : L → L ′ defines simplicial maps
f t : NΛ t → NΛ ′ t and f t : N(Λ, R, ϕ)(t) → N(Λ ′ , R ′ , ϕ ′ )(t).
On the other hand, the inclusion ι : L ′ → L defines simplicial maps ι : NΛ ′ t → NΛ t and ι : N(Λ ′ , R ′ , ϕ ′ )(t) → N(Λ, R, ϕ)(t). Moreover the above claims imply that the compositions
NΛ t ft − → NΛ ′ t ι − → NΛ t and N(Λ, R, ϕ)(t) ft − → N(Λ ′ , R ′ , ϕ ′ )(t) ι
− → N(Λ, R, ϕ)(t) are contiguous to identity maps. Since f t ι is the identity this implies that geometric realizations of the inclusions
NΛ ′ t ι − → NΛ t and N(Λ ′ , R ′ , ϕ ′ )(t) ι − → N(Λ, R, ϕ)(t)
are homotopy equivalences. Since we have assumed that the geometric realization of the inclusion
N(Λ, R, ϕ)(t) ι − → NΛ t
is not a homotopy equivalence, we can conclude that the geometric realization of the inclusion
N(Λ ′ , R ′ , ϕ ′ )(t) ι − → NΛ ′ t
is not a homotopy equivalence, as desired.
Implementation
In this section we will explain how to implement the computation of persistent homology of sparse approximations to the ambient-and intrinsic filteredČech complexes of a finite subset of Euclidean space as well as general finite Dowker dissimilarities. Our implementation is available on github.
6.1. Interleaving Lines. Our approximations toČech-and Dowker nerves are interleaved with the originalČech-and Dowker nerves. As a consequence their persistence diagrams are interleaved with the persistence diagrams of the original filtered complexes. In order to visualize where the points may lie in the original persistence diagrams, we can draw the matching boxes from Theorem 3.9. However, this result in messy graphics with lots of overlapping boxes. Instead of drawing these matching boxes we draw a single interleaving line. Points strictly above the line in the persistence diagram of the approximation match points strictly above the diagonal in the persistence diagram of the original filtered simplicial complex. More precisely, the matching boxes of points above the interleaving line do not cross the diagonal, while the matching boxes of all points below the diagonal have a non-empty intersection with the diagonal. Given t ∈ [0, ∞], the Dowker nerve N t Λ can be described as the nerve of the set of s-balls in W centred at points in L for s ≤ t . TheČech complexČ t (L, W ) is the nerve of the set of t-balls in W centred at points in L. Since, every ball in W of radius s ≤ t centred at a point in L is contained in a t-ball in W centred at a point in L the geometric realization of the inclusionČ t (L, W ) ⊆ N t Λ is a homotopy equivalence [3,Example 7.11]. It follows that the persistent homologies ofČ(L, W ) and NΛ are isomorphic.
More generally, let the Dowker dissimilarity Λ, the insertion function λ and the order preserving function β : [0, ∞] → [0, ∞] be as required in Example 5.8. In particular Λ can be obtained from a metric as above, and λ can be a farthest point sampling as in Definition 4.5. We have implemented the truncated Dowker dissimilarity Γ : L × W → [0, ∞] given in Example 5.8. By Proposition 4.3 the Dowker nerve of Γ is α = id +β-interleaved with the Dowker nerve of Λ. Using the restriction function R l 0 Γ and the Sheehy restriction function S we obtain sparse approximations N(Γ, R Γ ) and N(Γ, S) to the Dowker nerve of Λ. Below we elaborate on the implementation of these.
6.3. Sparsification of finite Dowker dissimilarities. Let Γ : L × W → [0, ∞] be a Dowker dissimilarity. Suppose that L is finite and let R be a truncation function for Γ relative to ϕ : L → L.
The underlying simplicial complex NΓ ∞ of the Dowker nerve NΓ consists of those subsets σ of L whose filtration value inf w∈W max l∈σ Γ(l, w) is finite. If there exist (l, w) ∈ L × W with Γ(l, w) = ∞, then NΓ ∞ may not be full simplex, and it makes sense to ask for its maximal simplices.
Choose a total order ≤ on L so that ϕ(l) ≤ l for every l ∈ L. Every maximal simplex of NΓ ∞ will be of the form σ(l, w) = {l ′ ∈ L | Γ(l ′ , w) < ∞ and l ′ ≤ l)}.
This means that the maximal simplices of NΓ ∞ can be found by computing σ(l, w) for every (l, w) ∈ L×W with Γ(l, w) < ∞. This strategy can be elaborated to find the maximal simplices of N ∞ (Γ, R, ϕ). Recall that given t ∈ [0, ∞], the simplicial complex N t (Γ, R, ϕ) consists of all subsets σ of L such that there exists w ∈ W satisfying:
(1) Λ(l, w) < t for all l ∈ σ.
(2) Λ(l, w) ≤ R(l ′ ) for all l, l ′ ∈ σ and (3) Λ(l, w) < R(l) for all slope points l in σ.
Given l ∈ L let W l = {w ∈ W | Γ(l, w) < ∞} and σ(l) = {l ′ ∈ L | R(l ′ ) ≤ R(l)}.
Given w ∈ W we define σ(l, w) = {l ′ ∈ σ(l) | Γ(l ′ , w) ≤ R(l) and Γ(l ′ , w) < R(l ′ ) if l ′ is a slope point}.
Then every maximal simplex of N ∞ (Γ, R, ϕ) is of the form σ(l, w) for some (l, w) ∈ L×W with w ∈ W l . In our implementation we have used this strategy to construct the set of maximal simplices in N ∞ (Γ, R, ϕ). 6.4. Sparsification of ambientČech filtrations. Let now Λ be a Dowker dissimilarity of the form Λ : L × W → [0, ∞] obtained by restricting the Euclidean metric on W = R d to the subset L × W of W × W . We will modify the above strategy in order to obtain a sparse approximation of the Dowker nerve of Λ. Let the insertion function λ and the order preserving function β : [0, ∞] → [0, ∞] be as required in Example 5.8 and let Γ be the truncation of Λ as in Example 5.8. Let R be a restriction function for Γ relative to ϕ : L → L, Unfortunately, in general, we do not understand the geometry well enough to be able to construct the maximal simplices in N ∞ (Γ, R, ϕ). Instead we will construct a filtered simplicial complex N a (Λ, R, ϕ) such that N(Γ, R, ϕ) ⊆ N a (Λ, R, ϕ) ⊆ NΛ and such that we are able to construct the maximal simplices of N a ∞ (Λ, R, ϕ). Let α = β + id. Lemma 2.14 implies that since the inclusion N(Γ, R, ϕ) ⊆ NΛ is an α-interleaving, also the inclusion N a (Λ, R, ϕ) ⊆ NΛ is an α-interleaving. In order to construct N a (Λ, R, ϕ), we specify a simplicial complex N a ∞ (Λ, R, ϕ) containing N ∞ (Λ, R, ϕ) and define N a t (Λ, R, ϕ) = N a ∞ (Λ, R, ϕ) ∩ NΛ t for t ∈ [0, ∞]. Thus, we can use the miniball algorithm to compute filtration values in N a (Λ, R, ϕ). Working out the definition, we see that a subset σ of L is in N ∞ (Γ, R, ϕ) if and only if there exists w ∈ R d so that Λ(l, w) < αβ ← λ(l) and Λ(l, w) ≤ R(l ′ ) for all l, l ′ ∈ L and Λ(l ′ , w) < R(l ′ ) for all slope points l ′ in σ. Note that if σ ∈ N ∞ (Γ, R, ϕ) and w are as above and if l ′′ ∈ σ satisfies λ(l ′′ ) ≤ λ(l) for every l ∈ σ, then by the triangle inequality Λ(l, l ′′ ) ≤ Λ(l, w) + Λ(l ′′ , w) < β ← αλ(l) + min l ′ ∈σ β ← αλ(l ′ ) and Λ(l, l ′′ ) ≤ Λ(l, w) + Λ(l ′′ , w) ≤ 2R(l ′ ) for l, l ′ ∈ L. This leads us to define N a ∞ (Λ, R, ϕ) as the simplicial complex consisting of all σ ⊆ L so that there exists l ′′ ∈ L satisfying Λ(l, l ′′ ) < β ← αλ(l) + min l ′ ∈σ β ← αλ(l ′ ) and Λ(l, l ′′ ) ≤ 2R(l ′ ).
for every l, l ′ ∈ L. The above discussion shows that N ∞ (Γ, R, ϕ) ⊆ N a ∞ (Γ, R, ϕ) as desired. The parent function we use in praxis is obtained from a farthest point sampling.
Lemma 2 .
215 (Functoriality). Let C and C ′ be filtered objects in a category C, let α : [0, ∞) → [0, ∞) be a translation function and let H : C → D be a functor. If C and C ′ are α-interleaved, then the filtered objects HC and HC ′ in D are α-interleaved.
Lemma 2 . 17 .
217Let α : [0, ∞) → [0, ∞) be a translation function and let V and V ′ be persistence modules. A morphism G : V → V ′ of persistence modules is an α-interleaving if and only if both ker G and coker G are α-trivial.
Definition 4. 1 .
1Let Λ : L × W → [0, ∞] be a Dowker dissimilarity and let α : [0, ∞) → [0, ∞) be a translation function. We say that a function T : L → [0, ∞] is an α-truncation function for Λ if for all t ∈ [0, ∞] and all l ∈ L there exists l ′ ∈ L so that for all w ∈ M with Λ(l, w) < t we have that Λ(l ′ , w) < α(t) and Λ(l ′ , w) < T (l ′ ).
Definition 4. 2 .
2Let Λ : L × W → [0, ∞] be a Dowker dissimilarity, let α : [0, ∞) → [0, ∞) be a translation function and let T : L → [0, ∞] be an α-truncation function for Λ. The T -truncation of Λ is the Dowker dissimilarity Γ : L × W → [0, ∞] defined by Γ(l, w) = Λ(l, w) if Λ(l, w) < T (l) ∞ otherwise.
Proposition 4. 3 .
3Let Λ : L × W → [0, ∞] be a Dowker dissimilarity, let α : [0, ∞) → [0, ∞) be a translation function and let T : L → [0, ∞] be an α-truncation function for Λ. Let Γ be the T -truncation of Λ. Then the inclusion of the nerve NΓ of Γ in the nerve NΛ of Λ is an α-interleaving.
Example 4. 4 .
4Let Λ : L × W → [0, ∞] be a Dowker dissimilarity and let α : [0, ∞) → [0, ∞) be a translation function. Given l, l ′ ∈ L, let
Definition 4 . 5 .
45Given a Dowker dissimilarity of the form Λ : L × L → [0, ∞] with L finite a farthest point sample for Λ is a total order < on L with minimal element l 0 so that for l = l 0 we have inf l ′ <l Λ(l ′ , l) = sup l ′′ ≥l inf l ′ <l Λ(l ′ , l ′′ ).
Example 4 . 6 .
46Let d : W × W → [0, ∞] be a metric, let L be a finite subset of L and let Λ : L × W → [0, ∞] be the restriction of d to the subset L × W of W × W . Let Λ L : L × L → [0, ∞] be the restriction of Λ to the subset L × L of L × W and let L = {l 0 < · · · < l n } be a farthest point sampling for Λ L . We write λ L (l) = λ(l) for the corresponding insertion radius. Let c > 1 and let α : [0, ∞) → [0, ∞) be the translation function α(t) = ct.
Definition 5. 2 .
2A function ϕ : L → L is a parent function if ϕ n (l) = l for n > 0 implies ϕ(l) = l. Note that ϕ : L → L is a parent function if and only if the directed graph with L as set of nodes and E(ϕ) = {(ϕ(l), l) | l ∈ L, ϕ(l) = l} as set of edges is acyclic. Definition 5.3. Let Λ : L×W → [0, ∞] be a Dowker dissimilarity and let ϕ : L → L be a parent function. We say that a function R : L → [0, ∞] is a restriction function for Λ relative to ϕ if the following holds:
( 3 )
3If ϕ(l) = l, then R(l) = ∞. Definition 5.4. Let Λ : L×W → [0, ∞] be a Dowker dissimilarity and let ϕ : L → L be a parent function. Given l, l ′ ∈ L let
Example 5 . 8 (
58Sheehy restriction). Let Λ : L × W → [0, ∞] be a Dowker dissimilarity with Λ(l, w) < ∞ for all (l, w) ∈ L × W and satisfying the triangle inequality. The specific case we have in mind is L = W and d : L × L → (0, ∞) a metric. Let α : [0, ∞) → [0, ∞) be a translation function of the form α = id +β for an order preserving function β : [0, ∞) → [0, ∞) so that the function [0, ∞] → [β(0), ∞)
Theorem 5 . 9 .
59Let Λ : L × W → [0, ∞] be a Dowker dissimilarity and let R be a restriction function for Λ relative to ϕ : L → L. If L is finite, then for every t ∈ [0, ∞] the geometric realization of the inclusion ι : N(Λ, R, ϕ)(t) → NΛ t is a homotopy equivalence.
6. 2 .
2Truncation of Dowker dissimilarities. Let Λ be a Dowker dissimilarities of the form Λ : L × W → [0, ∞] obtained by restricting a metric d : W × W → [0, ∞) on W to the subset L × W of W × W .
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| [] |
[
"MIN-MAX THEORY FOR CELL COMPLEXES",
"MIN-MAX THEORY FOR CELL COMPLEXES"
] | [
"Lacey Johnson ",
"Kevin Knudson "
] | [] | [] | In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. In this paper we introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the Mountain Pass Lemma and give an alternate proof of a discrete Lusternik-Schnirelmann Theorem. | 10.1142/s100538672000036x | [
"https://arxiv.org/pdf/1811.00719v1.pdf"
] | 119,131,841 | 1811.00719 | 308d632a15b9c4b10914d700c643647665c14b43 |
MIN-MAX THEORY FOR CELL COMPLEXES
Lacey Johnson
Kevin Knudson
MIN-MAX THEORY FOR CELL COMPLEXES
In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. In this paper we introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the Mountain Pass Lemma and give an alternate proof of a discrete Lusternik-Schnirelmann Theorem.
Introduction
Given a smooth function f : M → R, where M is a smooth manifold, a collection of min-max data for f is a pair (H, S), where H is a collection of homeomorphisms of M containing functions that decrease the sublevel sets of f and S is a collection of subsets of M closed under the collection H. Given such a pair, the min-max principle produces critical values of the function f .
In this paper we introduce a discretized version of this theory. While we state results for simplicial complexes, our results hold on the category of regular cell complexes. After reviewing the basics of discrete Morse theory in Section 2, in Section 3 we define the notion of min-max data associated to a discrete Morse function f : K → R on a simplicial complex K. The collection H will not be a set of homeomorphisms; rather it consists of functions defined on the power set of the set of cells of K. There are various operators associated to the function f which yield the requisite property of decreasing the level sets, and so we are able to obtain a discrete min-max principle which allows us to identify critical values of f .
As applications of this, we deduce the following. First, we obtain a discrete version of the Mountain Pass Lemma: given two local minima for f , there is a critical edge e "between" them; see Section 4 (a proof of the corresponding smooth result may be found in [5], p. 89). We also give an alternate proof of a discrete Lusternik-Schnirelmann Theorem as first presented in [1]; this is the content of Section 5.
Morocco, July 2018. He thanks the organizers, Driss Bennis in particular, for the invitation and for an enjoyable and stimulating conference.
Discrete Morse Theory
Let K be any finite simplicial complex, where K need not be a triangulated manifold nor have any other special property [3]. When we write K we mean the set of simplices of K; by |K| we mean the underlying topological space. Let α (p) ∈ K denote a simplex of dimension p. Let α < β denote that simplex α is a face of simplex β.
If f : K → R is a function define U (α) = {β (p+1) > α | f (β) ≤ f (α)} and L(α) = {γ (p−1) < α | f (γ) ≥ f (α)}.
In other words, U (α) contains the immediate cofaces of α with lower (or equal) function values, while L(α) contains the immediate faces of α with higher (or equal) function values. Let |U (α)| and |L(α)| be their sizes.
Definition 2.1. A function f : K → R is a discrete Morse function if for every α (p) ∈ K, (i) |U (α)| ≤ 1 and (ii) |L(α)| ≤ 1.
Forman showed that conditions (i) and (ii) are exclusive -if one of the sets U (α) or L(α) is nonempty then the other one must be empty ( [3], Lemma 2.5). Therefore each simplex α ∈ K can be paired with at most one exception simplex: either a face γ with larger function value, or a coface β with smaller function value. We refer to conditions (i) and (ii) as the Morse conditions.
Definition 2.2. A simplex α (p) is critical if (i) |U (α)| = 0 and (ii) |L(α)| = 0.
A critical value of f is its value at a critical simplex.
Definition 2.3. A simplex α (p)
is regular if either of the following conditions holds: (i) |U (α)| = 1; (ii) |L(α)| = 1; as noted above these conditions cannot both be true ( [3], Lemma 2.5).
Note that regular simplices occur in pairs. Suppose there is a pair σ (p) < τ (p+1) of simplices in K such that σ has no other cofaces in K. Then K \ {σ, τ } is a simplicial complex called an elementary collapse of K. We say that K collapses to L if L can be obtained from K by a sequence of elementary collapses. We denote this by
K L. A complex K is collapsible if K {v} for some vertex v. Two Morse functions f, g : K → R are equivalent if for every pair σ (p) < τ (p+1) in K, f (σ) < f (τ ) if and only if g(σ) < g(τ )
. A proof of the following is embedded in the proof of Theorem 3.3 of [3]. Main results of DMT. Given c ∈ R, we have the level subcomplex K c = ∪ f (α)≤c ∪ β≤α β. That is, K c contains all simplices α of K such that f (α) ≤ c along with all of their faces. We have the following two combinatorial versions of the main results of classical Morse theory. The fundamental idea in their proofs is to collapse the sublevel complexes by removing pairs {σ < τ } in the discrete gradient vector field of the function f . Theorem 2.5 (DMT Part A, [3]). Suppose the interval (a, b] contains no critical value of f . Then K b is homotopy equivalent to K a . In fact, K b collapses onto K a .
The next theorem explains how the topology of the sublevel complexes changes as one passes a critical value of a discrete Morse function. In what follows,ė (p) denotes the boundary of a p-simplex e (p) . Theorem 2.6 (DMT Part B, [3]). Suppose σ (p) is a critical simplex with f (σ) ∈ (a, b], and there are no other critical simplices with values in (a, b]. Then K b is homotopy equivalent to attaching a p-cell e (p) along its entire boundary in K a ; that is,
K b = K a ∪ė(p) e (p) .
The associated gradient vector field. Given a discrete Morse function f : K → R we may associate a discrete gradient vector field as follows. Since any noncritical simplex α (p) has at most one of the sets U (α) and
L(α) nonempty, there is a unique face ν (p−1) < α with f (ν) ≥ f (α) or a unique coface β (p+1) > α with f (β) ≤ f (α)
. Denote by V the collection of all such pairs {σ < τ } (referred to as Morse pairs). Then every simplex in K is in at most one pair in V and the simplices not in any pair are precisely the critical cells of the function f . We call V the gradient vector field associated to f . We visualize V by drawing an arrow from α to β for every pair {α < β} ∈ V . Observe that V "points" in the direction of decrease for f ; as such it corresponds to the negative gradient of the function and we may write V = −∇f . Theorems 2.5 and 2.6 may then be visualized in terms of V by collapsing the pairs in V using the arrows. Thus a discrete gradient (or equivalently a discrete Morse function) provides a collapsing order for the complex K, simplifying it to a complex L with potentially fewer cells but having the same homotopy type.
The collection V has the following property. By a V -path, we mean a sequence
α (p) 0 < β (p+1) 0 > α (p) 1 < β (p+1) 1 > · · · < β (p+1) r > α (p) r+1 where each {α i < β i } is a pair in V . Such a path is nontrivial if r > 0 and closed if α r+1 = α 0 . Forman proved the following result.
Theorem 2.7 ( [3]). If V is a gradient vector field associated to a discrete Morse function f on K, then V has no nontrivial closed V -paths.
Ascending and descending regions. If g : M → R is a Morse function on a smooth compact manifold M and if p is a critical point of g, there are two subspaces associated to p: the ascending and descending discs. The former is the set of points q in M such that the integral curve of −∇g passing through q converges to p as t → ∞ while the latter is the set of points whose integral curves converge to p as t → −∞. If p has index λ, then the ascending disc is topologically a disc of dimension n − λ and the descending disc has dimension λ.
The discrete analogues of these objects were constructed by Jerše and Mramor Kosta in [4]. The idea is that descending discs should be unions of V -paths, just as in the smooth case (where descending discs are foliated by integral curves of a gradient-like vector field). Ascending discs are then constructed as descending discs for the dual vector field V * on the dual cell complex K * .
For our purposes, however, we need only consider ascending discs of local minima and for this it suffices to consider the basin of a minimum v, defined in [2]. This is the maximal subcomplex of K that collapses to v; we denote it by A(v).
Min-max data associated to a discrete Morse function
Let K be a finite simplicial complex and let f : K → R be a discrete Morse function (which we may assume injective). Recall that for a ∈ R, K a denotes the sublevel complex associated to a. We denote by L a the sublevel
set L a = {σ ∈ K : f (σ) ≤ a}. Note that K a is the closure of L a .
Denote by P(K) the power set of K. (2) S is a collection of subsets of K such that for all h ∈ H and all S ∈ S, h(S) ∈ S.
Mountain Pass Lemma
Our goal in this section is to prove the Mountain Pass Lemma in the setting of an arbitrary simplicial complex. Recall the classical smooth situation. Given a Morse function f : M → R on the compact manifold M , suppose f has two local minima, say at x 0 and x 1 with f (x 0 ) < f (x 1 ). Assuming M is connected, there is a path from x 0 to x 1 . The Mountain Pass Lemma asserts that the infimum over all such paths γ of the maximum value sup t {γ(t)} is a critical value of f .
The picture to have in mind is shown in Figure 1. Thinking of f as a height function, the minima at x 0 and x 1 represent depressions in the landscape. To get from one low point to another, one must cross over a ridge; the lowest point z on the ridge will be a saddle point; i.e. a critical point of f . While this result may seem obvious, it is not at all clear how one could prove it on a general n-manifold, let alone on a general simplicial complex. Our plan is to construct an appropriate collection of min-max data and then apply Theorem 3.2.
Suppose V is a discrete gradient vector field on K; say V = −∇f for some discrete Morse function f . Denote by −, − the inner product on the chains C p (K, Z) defined by making all cells orthogonal and define a map V : C p (K, Z) → C p+1 (K, Z) as follows. Suppose σ is a p-cell with fixed orientation. If there is a (p + 1)-cell τ with {σ < τ } ∈ V , then set
V (σ) = − ∂τ, σ τ,
where ∂ is the boundary map in K. If there is no such τ , (for example, if σ is critical), then set V (σ) = 0. We extend this linearly to a map on all of C p (K).
Definition 4.1. The discrete flow operator φ :
C p (K) → C p (K) is defined by φ = Id + ∂V + V ∂.
The intuition here is that if v is a vertex then either (a) v is critical and should be fixed by the flow, or (b) v is not critical and it should flow to the other vertex of the edge e is paired with by the gradient V ; that is,
V (v) = ±e = v + ∂V (v).
The following is Theorem 6.4 of [3]. (2) If i = j, then a ij ∈ Z. If i = j and a ij = 0, then f (σ j ) < f (σ i ).
For our purposes, we will need to consider the collection of cells that appear in the image of φ. Note that we may apply φ to sets of p-cells in K to obtain a map, denoted Φ : P(K) → P(K), defined by
Φ(A) = σ∈A φ(σ).
That is, for a cell σ we are using the notation φ(σ) to mean the collection of cells appearing in the chain φ(σ).
We also define a second map Φ on P(K) by
Φ(A) = subcomplex of K generated by Φ(A).
We give examples of Φ and Φ in Figure 2. The maps Φ and Φ will be useful for constructing min-max data. Consider H = {Φ}. By Proposition 4.2, condition (1) in the definition of min-max data for a discrete Morse function is satisfied. In fact, for any a, regular or not, Φ(K a ) ⊆ K a (see, e.g. Lemma 1 of [2]), but Φ only decreases the sublevel set for regular values. The same is true for the map Φ. In fact we have the following, which is a special case of Theorem 4 of [2]. Proof. Note that any critical simplex in K a is also in Φ(K a ), and hence so are its faces. So a simplex σ in K a \ Φ(K a ) must be regular and paired with some τ ∈ K a . Order such pairs by decreasing f -value; the maximal pair must be of the form {α < β} where α is a free face of β (note that if a is a regular value then this pair consists of the unique simplex in f −1 (a) and either a coface of lower value or face of higher value). Removing the pairs in decreasing order yields a collapsing sequence K a Φ(K a ).
Now suppose that v 0 and v 1 are critical vertices for the discrete Morse function f and assume f (v 0 ) < f (v 1 ). Let S be the collection of subsets of K consisting of sets of the form E = {v 1 , e 1 , e 2 , . . . , e r }, where the e i form an edge path beginning at v 1 and ending in the basin A(v 0 ). Writing e i =< u i−1 , u i >, we insist on the following condition: if u j belongs to the ascending disc of v 0 then f (e j ) > f (e j+1 ) > · · · > f (e r ). This condition is not strictly necessary, but it does prevent path bifurcation when acted upon by Φ. We may assume that such paths do not pass through another critical
vertex v = v 0 , v 1 . Note that if E ∈ S, Φ(E) = ∅. Indeed, since v 1 is a critical vertex, Φ(v 1 ) = v 1 . Theorem 4.4. If E ∈ S, then Φ(E) ∈ S.
Assuming this we see that (H, S) is a collection of min-max data on K. By Theorem 3.2 we see that
c = min E∈S max e∈E f (e)
is a critical value of f . This critical value must be greater than f (v 1 ) since any set E in S has the form E = {v 1 , e 1 , . . . , e r } with r ≥ 1 and f (e 1 ) > f (v 1 ). Thus, among all the edge paths from v 1 to v 0 , there is one passing through a critical edge e and the value of f (e) is minimal among all such paths. Thus, e may be thought of as the "saddle point" along the "ridge" between v 0 and v 1 .
We have thus proved the discrete version of the Mountain-Pass Lemma and we refer to the critical edges that result as mountain-pass edges. The Mountain-Pass Lemma implies that if a discrete Morse function has two vertices that are strict local minima, then it must admit a critical edge.
Another explanation of why there should exist a critical edge with function value greater than f (v 1 ) is to observe that the sublevel complex K f (v 1 ) is disconnected while the cell complex K is connected. The change in topological type in going from K f (v 1 ) to K can be explained by the presence of a critical edge with function value greater than f (v 1 ).
Remark 4.5. A slightly different version of this result was proved in [2]; the authors did not employ min-max theory to do so. We will evaluate this scenario as needed in what follows. We proceed by induction on r, the number of edges in a set E. The case r = 1 is essentially described above. Here we have E = {v 1 , e} for some edge e. The other vertex u of e lies in the ascending disc of v 0 . If e is critical we then have φ(e) = e + V (u) and so Φ(E) = {v 1 , e, V (u)} is another path ending in A(v 0 ). Since u is in A(v 0 ) it cannot be paired with e and so the second case above cannot occur. This leaves the final possibility that e is regular and paired with a 2-simplex < v 1 , u, w >. In this case we see that φ(e) =< v 1 , w > + < u, w > +V (u). It could be that V (u) = − < u, w >, which implies that w also belongs to A(v 0 ) and hence Φ(E) = {v 1 , < v 1 , w >} is another element of S. Otherwise, V (u) is another edge ending in A(v 0 ) and so Φ(E) = {v 1 , < v 1 , w >, < u, w >, V (u)} ∈ S. Now assume the result holds for all paths of length < r and suppose E = {v 1 , e 1 , . . . , e r } ∈ S. If {v 1 , e 1 , . . . , e r−1 } ends in A(v 0 ), then φ(e r ) also lands in A(v 0 ) and by the induction hypothesis, Φ(E) ∈ S. Now suppose that {v 1 , e 1 , . . . , e r−1 } does not end in A(v 0 ). This means that there is no gradient path u r−1 < 0 > w 1 < 1 > · · · < s > v 0 . However, there is a path
η 0 > u r < η 1 > ν 1 < η 2 > · · · < η t > v 0 .
It follows that e r = η 0 and f (e r ) > f (u r ). Note that this implies that u r−1 could not be paired with e r (else f (u r ) < f (e r ) which would imply that u r−1 ∈ A(v 0 )). Thus, e r is paired with a 2-simplex σ (whose faces are e r , e , and e ), or e r is critical. In the first case, φ(e r ) = e + e + V (u r ) + V (u r−1 ) simply diverts the path E around the edges of σ and so Φ(E) ∈ S. If e r is critical then φ(e r ) = e r + V (u r−1 ) + V (u r ) and so Φ(E) still lies in S. This completes the proof.
Remark 4.6. The reader might wonder why we used the map Φ and edge paths that do not include the vertices as opposed to the map Φ and 1dimensional subcomplexes. The issue is that arbitrary edge paths emanating from v 1 might contain vertices that are paired with edges not in the path. Applying Φ or Φ to such an object leads to subcomplexes that have extraneous edges jutting from the path. We would then be forced to expand the collection S to accommodate this, causing some technical difficulties.
A discrete Lusternik-Schnirelmann theorem
Suppose K is a simplicial complex and that L is a subcomplex. Recall that L is collapsible if it collapses to a vertex. The following definition appears in [1].
Definition 5.1. Suppose that L is a subcomplex of K. Then L has geometric precategory less than or equal to m in K, denoted dgcat K (L) ≤ m if there exist m + 1 subcomplexes U 0 , . . . , U m in K, each of which is collapsible in K, such that L ⊆ m i=0 U i . If dgcat K (L) < m we say dgcat K (L) = m. The discrete (geometric) category of L in K is then dgcat K (L) = min{ dgcat K (L ) : L L }.
For each k, let
Γ k = {L ⊂ K : ∃K a L with dgcat K (K a ) ≥ k − 1}.
Recall that if A ∈ P(K), we define Φ(A) to be the subcomplex of K generated by Φ(A). Proof. Suppose L ∈ Γ k . Then there is some a ∈ R with K a L. Note that Φ(K a ) Φ(L) and since Lemma 4.3 implies K a Φ(K a ) we see that Φ(L) ∈ Γ k . As a consequence of this, we obtain the following Lusternik-Schnirelmann theorem, originally proved in [1] (Theorem 26). Proof. By Corollary 5.3 each c k is a critical value of f . We may assume that the global minimum of f , which occurs at some vertex v is f (v) = 0. Then c 1 = 0 and K c 1 contains one critical simplex. Proceeding inductively, suppose K c k−1 has at least k − 1 critical simplices and consider K c k . Since f is excellent, c k−1 < c k and so f −1 (c k ) contains at least one new critical simplex. Thus K c k contains at least k critical simplices. So if c 1 < c 2 < · · · < c dgcat(K)+1 are the distinct critical values then K c dgcat(K)+1 ⊆ K contains at least dgcat(K) + 1 critical simplices. Therefore dgcat(K) + 1 ≤ m.
Lemma 2 . 4 .
24Let f : K → R be a discrete Morse function. Then there is an injective discrete Morse function g : K → R equivalent to f having the same critical simplices as f .
Definition 3. 1 .
1A collection of min-max data for the discrete Morse function f : K → R is a pair (H, S) satisfying the following conditions.(1) H is a collection of maps P(K) → P(K) such that for every regular value a of f there exist ε > 0 and h ∈ H such that h(L a+ε ) ⊂ L a−ε .
Theorem 3. 2 .Figure 1 .f
21If (H, S) is a collection of min-max data for the discrete Morse function f on K then the numberc = min S∈S max σ∈S f (σ)is a critical value for f . Proof. Suppose not; that is, assume that c is a regular value for f . Then there exist ε > 0 and h ∈ H such thath(L c+ε ) ⊂ L c−ε .By the definition of c there is an S ∈ S such that max σ∈S f (σ) < c + ε;that is, S ⊂ L c+ε . Then S = h(S) ∈ S and h(S) ⊂ L c−ε . It follows thatmax σ∈S f (σ) ≤ c − ε, Visualizing (σ) ≤ c − ε,contrary to the choice of c.
Proposition 4. 2 .
2Let σ 1 , . . . , σ r denote the p-cells of K, each with a chosen orientation. Write φ(σ i ) = j a ij σ j . Then (1) For every i, a ii = 0 or 1, and a ii = 1 if and only if σ i is critical.
Lemma 4 . 3 .
43For each a ∈ R, K a Φ(K a ).
Figure 2 .
2A complex K (left), Φ(K) (center), Φ(K) (right).
Proof of Theorem 4.4. We begin by computing the action of Φ on a single edge e =< v, w >. There are three cases.(1) Suppose e is critical.Then φ(e) = e + V ∂e = e − V (v) + V (w). It follows that Φ(e) = {e, V (v), V (w)}.(2) If e is regular and paired with one of its vertices (say v), then φ(e) = e + V ∂e = V (w). Thus Φ(e) = {V (w)}. (3) If e is regular and paired with the 2-simplex with edges e, e , e , then φ(e) = e + V ∂e + ∂V (e). Several things could happen in this case depending on how the vertices v and w are paired (or not) by V .
Theorem 5. 2 .
2For each k, ({Φ}, Γ k ) is a collection of min-max data.
Corollary 5 . 3 .
53For each k the real number c k = min L∈Γ k max σ∈L f (σ) is a critical value of f .
Corollary 5. 4 .
4Let f : K → R be a discrete Morse function with m critical values. Then dgcat(K) + 1 ≤ m.
Acknowledgements. The second author spoke about some of these results at the International Conference on Algebra and Related Topics in Rabat,
Lusternik-Schnirelmann category for simplicial complexes. S Aaronson, N Scoville, Illinois J. Math. 57S. Aaronson, N. Scoville, Lusternik-Schnirelmann category for simplicial complexes, Illinois J. Math. 57 (2013), 743-753.
Skeletonization and partitioning of digital images using discrete Morse theory. O Delgado-Friedrichs, V Robins, A Sheppard, IEEE Transactions on Pattern Analysis and Machine Intelligence. 37O. Delgado-Friedrichs, V. Robins, A. Sheppard, Skeletonization and partitioning of digital images using discrete Morse theory, IEEE Transactions on Pattern Analysis and Machine Intelligence 37 (2015), 654-666.
Morse theory for cell complexes. R Forman, Adv. Math. 134R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), 90-145.
Ascending and descending regions of a discrete Morse function. G Jerše, N Mramor Kosta, Computational Geometry. 42G. Jerše, N. Mramor Kosta, Ascending and descending regions of a discrete Morse function, Computational Geometry 42 (2009), 639-651.
E-mail address: [email protected] E-mail address: kknudson@ufl. L Nicolaescu, An invitation to Morse theory. Gainesville, FL 32611SpringerDepartment of Mathematics, University of Florida2 ed.L. Nicolaescu, An invitation to Morse theory, 2 ed., Springer (2011). E-mail address: [email protected] E-mail address: [email protected] Department of Mathematics, University of Florida, Gainesville, FL 32611
| [] |
[
"Cosmological simulations of the Santa Barbara cluster: the influence of scalar fields",
"Cosmological simulations of the Santa Barbara cluster: the influence of scalar fields"
] | [
"M A Rodríguez-Meza \nDepto. de Física\nCol. Escandón\nInstituto Nacional de Investigaciones Nucleares\nApdo. Postal 18-102711801MéxicoD.F\n"
] | [
"Depto. de Física\nCol. Escandón\nInstituto Nacional de Investigaciones Nucleares\nApdo. Postal 18-102711801MéxicoD.F"
] | [] | We present numerical N -body simulation studies of large-scale structure formation. The main purpose of these studies is to analyze the several models of dark matter and the role they played in the process of large-scale structure formation. We analyze in this work a flat cold dark matter dominated model known as the Santa Barbara cluster. We compare the results for this model using the standard Newtonian limit of general relativity with the corresponding results of using the Newtonian limit of scalar-tensor theories. An specific model is the one that considers that the scalar field is non-minimally coupled to the Ricci scalar in the Einstein-Hilbert Lagrangian. Comparisons of the models are done showing results of rotation curves, density profiles, and velocity dispersions for halos formed at z=0. We analyze, in particular, the Santa Barbara cluster and its possible equation of state. PACS numbers: 95.30.Sf; 95.35.+d; 98.65.-r; 98.65.Dx | 10.1063/1.3058567 | [
"https://arxiv.org/pdf/0810.0491v1.pdf"
] | 117,828,165 | 0810.0491 | 01461814daeb6c02da73a65de3326a87bb5082c2 |
Cosmological simulations of the Santa Barbara cluster: the influence of scalar fields
2 Oct 2008 (Dated: October 2, 2008)
M A Rodríguez-Meza
Depto. de Física
Col. Escandón
Instituto Nacional de Investigaciones Nucleares
Apdo. Postal 18-102711801MéxicoD.F
Cosmological simulations of the Santa Barbara cluster: the influence of scalar fields
2 Oct 2008 (Dated: October 2, 2008)
We present numerical N -body simulation studies of large-scale structure formation. The main purpose of these studies is to analyze the several models of dark matter and the role they played in the process of large-scale structure formation. We analyze in this work a flat cold dark matter dominated model known as the Santa Barbara cluster. We compare the results for this model using the standard Newtonian limit of general relativity with the corresponding results of using the Newtonian limit of scalar-tensor theories. An specific model is the one that considers that the scalar field is non-minimally coupled to the Ricci scalar in the Einstein-Hilbert Lagrangian. Comparisons of the models are done showing results of rotation curves, density profiles, and velocity dispersions for halos formed at z=0. We analyze, in particular, the Santa Barbara cluster and its possible equation of state. PACS numbers: 95.30.Sf; 95.35.+d; 98.65.-r; 98.65.Dx
I. INTRODUCTION
The Santa Barbara (SB) cluster model was introduced by Frenk et al. [1] in order to study in a systematic way a flat cold dark matter (FCDM) dominated universe using a variety of numerical codes. The main goal of this comparison was to asses the reliability of cosmological simulations of clusters in one of the simplest astrophysical relevant case. They compared the images and global properties of the cluster obtained by the different numerical codes. Heitmann et al. (2005) [2] analyze again the cluster with others new numerical codes and with a similar purpose and now it has become one of the standard cases of study. Given that this is a standard case to test we have decided to analyze it in the framework of the scalar-tensor theories (SST) of gravity.
Scalar fields have been considered as one of the best possible ways to modify gravity. The work by Nordström, published before general relativity, formulated a conformally flat scalar theory of gravity [3], and finally, the scalar field role in gravity has been stablished since the pioneering work of Jordan, Brans, and Dicke [4,5]. Nowadays they are considered as a mechanism for inflation [6]; the dark matter component of galaxies [7]; the quientessence field to explain dark energy in the universe [8]. The main goal of this work is to study the large scale structure formation where the usual approach is that the evolution of the initial primordial fluctuation energy density fields evolve following Newtonian mechanics in an expanding background [9]. The force between particles are the standard Newtonian gravitational force. Now, we will see that we can introduce the scalar fields by adding a term in this force. This force will turn out to be of Yukawa type with two parameters (α, λ) [10]. For so many years this kind of force, the so called fifth force, was thoroughly studied theoretically [11] and many experiments were done to constrain the Yukawa parameters [12]. We have also been studying, in the past years, the effects of this kind of force on some astrophysical phenomena [10,13,14,15] and in cosmological simulations [16,17]. The Yukawa force comes as a Newtonian limit of a scalar-tensor theory with the scalar field non-minimally coupled to gravitation [18] although other alternatives can be found [19].
Our general purpose is to find the role these scalar fields play on the large scale structure formation processes. In particular, in this work we present some results about the role scalar fields play on cosmological simulations that form the SB cluster.
We start by discussing a FCDM model and the general approach in N -body simulations (See Bertschinger [20] for details). Then, we present the modifications we need to do to consider the effects of a static scalar field and we show the results of this theory for the FCDM model that form the Santa Barabara cluster [1,2]. To perform the simulations we have modified a standard treecode the author has developed [21] and the Gadget 1 [22] (see also http://www.astro.inin.mx/mar) in order to take into account the contribution of the Yukawa potential. We finish this paper by discussing how we can obtain the equation of state for a dark matter halo in the framework of general relativity and its Newtonian limit.
II. EVOLUTION EQUATIONS FOR A CMD UNIVERSE
A. General Scalar-tensor theory
The Einstein equations for a typical scalar-tensor theory with a massive scalar field non-minimally coupled to the geometry are given by
R µν − 1 2 g µν R = 1 φ 8πT µν + 1 2 V g µν + ω φ ∂ µ φ∂ ν φ − 1 2 ω φ (∂φ) 2 g µν + φ ;µν − g µν φ ,(1)
and the scalar field equation
φ + φV ′ − 2V 3 + 2ω = 1 3 + 2ω 8πT − ω ′ (∂φ) 2 ,(2)
where () ′ ≡ ∂ ∂φ . Here g µν is the metric, R is the Ricci's scalar, R µν the Ricci's tensor, T µν is the energy-momentum tensor, and ω(φ) and V (φ) are arbitrary functions of the scalar field φ. We will not consider a cosmological constant contribution in this work.
B. Newtonian approximation of STT
The study of large-scale formation in the universe is greatly simplified by the fact that a limiting approximation of general relativity, Newtonian mechanics, applies in a region small compared to the Hubble length cH −1 (cH −1 0 ≈ 3000h −1 Mpc, where c is the speed of light, H 0 = 100h km/s/Mpc, is Hubble's constant at this epoch and h ≈ 0.7), and large compared to the Schwarzschild radii of any collapsed objects. The rest of the universe affect the region only through a tidal field. The length scale cH −1 0 is of the order of the largest scales currently accessible in cosmological observations and H −1 0 ≈ 10 10 h −1 yr characterizes the evolutionary time scale of the universe. Therefore we need to describe the STT theory in its Newtonian approximation, that is, where gravity and the scalar fields are weak (and time independent) and velocities of dark matter particles are non-relativistic. We expect to have small deviations of the metric with respect to Minkowski metric and of the scalar field around the background field, defined here as φ and can be understood as the scalar field beyond all matter. If one defines the perturbations φ = φ − φ and h µν = g µν − η µν , where η µν is the Minkowski metric, the Newtonian approximation gives [18]
R 00 = 1 2 ∇ 2 h 00 = G N 1 + α 4πρ − 1 2 ∇ 2φ ,(3)∇ 2φ − m 2 SFφ = −8παρ ,(4)
we have set φ = (1 + α)/G N and α ≡ 1/(3 + 2ω). We are considering that the influence of dark matter is due to a boson field of mass m SF governed by Eq. (4), that is the modified Helmholtz equation. Equations (3) and (4) represent the Newtonian limit of a STT with arbitrary potential V (φ) and function ω(φ) that where Taylor expanded around φ . The resulting equations are then distinguished by the constants G N (the local gravitational constant), α, and λ = h P /m SF c. Here h P is Planck's constant. Note that Eq. (3) can be cast as a Poisson equation for ψ ≡ (1/2)(h 00 +φ/ φ ),
∇ 2 ψ = 4π G N 1 + α ρ .(5)
The next step is to find solutions for this new Newtonian potential given a density profile, that is, to find the so-called potential-density pairs. General solutions to Eqs. (4) and (5) can be found in terms of the corresponding Green functions, and the new Newtonian potential is (see [10,14] for details)
Φ N ≡ 1 2 h 00 = − G N 1 + α dr s ρ(r s ) |r − r s | −α G N 1 + α dr s ρ(r s )e −|r−rs|/λ |r − r s | + B.C.(6)
The first term of Eq. (6), given by ψ, is the contribution of the usual Newtonian gravitation (without scalar fields), while information about the scalar field is contained in the second term, that is, arising from the influence function determined by the modified Helmholtz Green function, where the coupling ω(α) enters as part of a source factor.
To simulate cosmological systems, the expansion of the universe has to be taken into account. Also, to determine the nature of the cosmological model we need to determine the composition of the universe, i. e., we need to give the values of Ω i for each component i, taking into account in this way all forms of energy densities that exist at present.
If a particular kind of energy density is described by an equation of state of the form p = wρ, where p is the pressure and w is a constant, then the equation for energy conservation in an expanding background, d(ρa 3 ) = −pd(a 3 ), can be integrated to give ρ ∝ a −3(1+w) . Then, the Friedmann equation for the expansion factor a(t) is written as
H 2 ≡ȧ 2 a 2 = H 2 0 i Ω i a 0 a 3(1+wi) − k a 2(7)
where k characterizes the geometry of the universe (k = 0 for a flat universe), and w i characterizes the equation of state of specie i. Here,
Ω i ≡ ρ i /ρ c , with ρ c = 3H 2 /8πG N .
The most familiar forms of energy densities are those due to pressureless matter with w i = 0 (that is, nonrelativistic matter with rest-mass-energy density ρc 2 dominating over the kinetic-energy density ρv 2 /2) with Ω DM ≈ 0.22 and radiation with w i = 1/3 and Ω R ≈ 2 × 10 −5 . The density parameter contributed today by visible, nonrelativistic, baryonic matter in the universe is Ω B ≈ 0.04. There is also another main component, the energy density associated to a cosmological constant, Ω Λ ≈ 0.74 with an equation of state such that w i = −1. In this work we will consider a model with only one energy density contribution. One which is a pressureless and nonbaryonic dark matter with contribution given by Ω DM that does not couple with radiation. The above equation for a(t) in this case becomeṡ
a 2 a 2 = H 2 0 Ω DM a 0 a 3 − k a 2(8)
Here, we employ a cosmological model with a static scalar field which is consistent with the Newtonian limit given by Eq. (6). Thus, the scale factor, a(t), is given by the following Friedman model (see a more general case in [17]),
a 3 H 2 = H 2 0 Ω DM0 1 + α + 1 − Ω DM0 1 + α a(9)
where Ω DM0 is the dark matter density evaluated at present, respectively. We notice that the source of the cosmic evolution is deviated by the term 1 + α when compared to the standard Friedman-Lemaitre model. Therefore, it is convenient to define a new dark matter density parameter by Ω (α) DM ≡ Ω DM /(1 + α). This new density parameter is such that always Ω (α) DM = 1, which implies a flat dark matter dominated universe, and this shall be assumed in our following computations. For positive values of α, a flat cosmological model demands to have a factor (1 + α) more energetic content (Ω DM ) than in standard FCDM cosmology. On the other hand, for negative values of α one needs a factor (1 + α) less Ω DM to have a flat universe. To be consistent with the CMB spectrum and structure formation numerical experiments, cosmological constraints must be applied on α in order for it to be within the range (−1, 1) [23,24,25,26].
In the Newtonian limit of STT of gravity, the Newtonian motion equation for a particle i is written as
x i + 2 H x i = − 1 a 3 G N 1 + α j =i m j (x i − x j ) |x i − x j | 3 F SF (|x i − x j |, α, λ)(10)
where x is the comovil coordinate, and the sum includes all periodic images of particle j, and F SF (r, α, λ) is
F SF (r, α, λ) = 1 + α 1 + r λ e −r/λ(11)
which, for small distances compared to λ, is F SF (r < λ, α, λ) ≈ 1 + α 1 + r λ and, for long distances, is F SF (r > λ, α, λ) ≈ 1, as in Newtonian physics.
We now analyze the general effect that the constant α has on the dynamics. The role of α in our approach is as follows. On one hand, to construct a flat model we have set the condition Ω (α) DM = 1, which implies having (1 + α) times the energetic content of the standard FCDM model. This essentially means that we have an increment by a factor of (1 + α) times the amount of matter, for positive values of α, or a reduction of the same factor for negative values of α. Increasing or reducing this amount of matter affects the matter term on the r.h.s. of the equation of motion (10), but the amount affected cancels out with the term (1 + α) in the denominator of (10) stemming from the new Newtonian potential. On the other hand, the factor F SF augments (diminishes) for positive (negative) values of α for small distances compared to λ, resulting in more (less) structure formation for positive (negative) values of α compared to the FCDM model. For r ≫ λ the dynamics is essentially Newtonian.
III. RESULTS
A. Cosmological simulations of the Santa Barbara cluster
In this section, we present results of the cosmological simulations of a FCDM universe with and without SF contribution. The initial condition of the system corresponds to the well known Santa Barbara cluster data that we get from the Heitmann's Cosmic Data Bank web page (http://t8web.lanl.gov/people/heitmann/test3.html). The initial condition uses a box 64 Mpc size, 128 3 particles and start at redshift z = 63.
In 1999 Frenk et al. [1] reported results of an extensive code comparison project involving twelve different codes. The aim of the project was to compare different techninques for simulating the formation of a cluster of galaxies (by now widely known as the Santa Barbara cluster) in a flat cold dark matter universe and to decide if the results from different codes were consistent and reproducible. We have repeated this test but restricted ourselves only to the dark matter component of the test. The cosmological parameters used in the simulation are as follows. We used 128 3 particles with initial positions and velocities with Ω DM = 1, Ω Λ = 0, Ω B = 0, H 0 = 50 km/s/Mpc, box size = 32 Mpc/h, σ 8 = 0.9 for the present-day linear rms mass fluctuation in spherical top spheres of radius 16 Mpc. The FCDM universe contents is given in units of the critical density, ρ c . The initial fluctuation spectrum was taken to have an asymptotic spectral index, n = 1, and the shape parameter Γ = 0.25, the value suggested by observations of large-scale structure [27]. See [1] and Heitmann's web page above for full details. In the shown snaps at z = 0, Fig. 1, the big cluster near the center of the frame is the SB cluster.
The particle masses are ≈ 0.434 × 10 10 M ⊙ /h. The individual softening length was 50 kpc/h. These choices of softening length are consistent with the mass resolution set by the number of particles.
We now present results for the FCDM model previously described. Because the visible component is the smaller one and given our interest to test the consequences of including a SF contribution to the evolution equations, our model excludes gas particles, but all its mass has been added to the dark matter. We restrict the values of α to the interval (−1, 1) [23,24,25,26] and use λ = 1 Mpc/h, since this scale turns out to be an intermediate scale between the size of the clump groups and the separation of the formed groups.
In Fig. 1 we show x-y snapshots at redshift z = 0 of our FCDM model. Fig. 1 (a) presents the standard case without SF, i.e., the interaction between bodies is through the standard Newtonian potential. In (b) we show the case with α = 1, λ = 1 Mpc/h. In (c) α = −1/2, λ = 1 Mpc/h. In (d) α = −1/4, λ = 1 Mpc/h. One notes clearly how the SF modifies the matter structure of the system. The most dramatic cases are (b) and (c) where we have used α = 1 and α = −1/2, respectively. Given the argument at the end of last section, in the case of (b), for r ≪ λ, the effective gravitational pull has been augmented by a factor of 2, in contrast to case (c) where it has diminished by a factor of 1/2; in model (d) the pull diminishes only by a factor of 3/4. That is why one observes for r < λ more structure formation in (b), less in (d), and lesser in model (c). The effect is then, for a growing positive α, to speed up the growth of perturbations, then of halos and then of clusters, whereas negative α values (α → −1) tend to slow down the growth.
Next, we find the groups in the system using a friend-of-friend (FOF) algorithm and select one of the most massive ones. The chosen group is located approximately at the center in Fig. 1, the SB cluster. The group was analyzed by obtaining their density profiles ( Fig. 2(a)) and circular velocities ( Fig. 2(b)). The more cuspy case is for α = 1 and the less cuspy is for α = −1/2. The circular velocity curves where computed using v 2 c = G N M (r)/r. The case with α = 1 corresponds to higher values of v c , since this depends on how much accumulated mass there is at a distance r and this is enhanced by the factor F SF for positive values of α.
B. Equation of state of a dark matter halo
We finish this work by discussing a possible way to find a dark matter halo equation of state (EOS). The EOS may be useful to characterize more completely the state of a dark matter halo and could be a way to discriminate dark matter models.
In general relativity both density and pressure contribute to modify the space-time geometry. For a static and spherically symmetric system in equilibrium like a dark matter halo we can not a priori neglect the pressure contribution. Therefore, and in the Newtonian limit we must solve the equation (see Misner et al. [28])
1 r 2 d dr r 2 dΦ dr ≈ 4πG(ρ + p)(12)
together with the equation of hydrostatic equilibrium condition, dp dr = −(ρ + p) dΦ dr (13) where p(r) = p r (r) + 2p t (r) and we have assumed spherical symmetry. We also use the flatness condition on the circular velocity at large distances,
V 2 c0 = r dΦ N dr = constant(14)
to construct the boundary condition dp
dr r=RH = −(ρ(R H ) + p(R H )) V 2 c0 R H(15)
where R H is the size of the system. The other condition is
p(R H ) = 0(16)
Results for two density profiles are given in figure III B. The dashed line is for the isothermal density profile given by,
ρ ISO = ρ c 1 + (r/r c ) 2(17)
where ρ c is the core density and r c is the scale length of the matter distribution given by the isothermal profile. Whereas solid line is the Navarro-Frenk-White (NFW) density profile given by
ρ N F W = ρ cs (r/r s )(1 + r/r s ) 2(18)
where ρ cs is the NFW typical density and r s is the scale length of the matter distribution given by the NFW profile. We may observe that NFW EOS has two power laws, i.e., behavior p ∼ ρ n , for low densities n = 1.4 and for high densities n = 1 that is the behavior of the isothermal EOS. We may compare this results with the EOS of the SB cluster, shown in Fig. III B. Where we have assumed in order to do the calculations, that p = ρσ 2 r , here σ 2 r is the radial dispersion of velocities of the cluster. This numerical EOS is shown in figure III B. We may notice the power law behavior, n = 1 which corresponds to an isothermal density profile or to the high density case for the NFW EOS case.
IV. CONCLUSIONS
The theoretical scheme we have used is compatible with local observations because we have defined the background field constant < φ >= G −1 N (1 + α). A direct consequence of the approach is that the amount of matter (energy) has to be increased for positive values of α and diminished for negative values of α with respect to the standard FCDM model in order to have a flat cosmological model. Quantitatively, our model demands to have Ω/(1 + α) = 1 and this changes the amount of dark matter and energy of the model for a flat cosmological model, as assumed. The general gravitational effect is that the interaction including the SF changes by a factor F SF (r, α, λ) ≈ 1 + α 1 + r λ for r < λ in comparison with the Newtonian case. Thus, for α > 0 the growth of structures speeds up in comparison with the Newtonian case. For the α < 0 case the effect is to diminish the formation of structures. For r > λ the dynamics is essentially Newtonian.
Additionally we have found numerically and EOS for the SB cluster. However, we assume that p = ρσ 2 r , where ρ and σ r were obtain from SB cluster particle data. We leave for a future paper the numerical computation of p r and p t from the pressure tensor of the SB cluster and more detailed analysis. We have compared the power-law behaviour (p ∼ ρ n ) of the SB cluster EOS with the corresponding behavior of an EOS obtained solving Eqs. (12) and (13) for two density profiles, isothermal profile (17) and NFW profile (18). We see that the SB cluster is like the isothermal profile for some range of lower densities. Even more NFW profile has two power-law, for low densities behaves as n = 1.4 and for high densities as n = 1 that corresponds two the isothermal profile.
Acknowledgements: This work was supported by CONACYT, grant number I0101/131/07 C-234/07, IAC collab-
FIG. 1 :
1x-y snapshots at z = 0 of a FCDM universe. See text for details.FIG. 2: (a) Density profiles of the SB cluster at z = 0. The cluster is located at the center in Fig 1. Vertical scale is in units of ρ0 = 10 10 M⊙h −1 /(h −1 kpc) 3 . (b) The corresponding circular velocity. FIG. 3: Equation of state for two density profiles. Pseudo-isothermal profile which has a power law behavior, p ∼ ρ n with n = 1 and the NFW profile which has two power law behavior, one with n = 1.4 and another with n = 1. Pressure and density are in geometrical units
FIG. 4 :
4The SB cluster extracted from the numerical simulation of the CDM model using a FOF algorithm.FIG. 5: Equation of state for SB cluster (dots). p0 is the unit of pressure 6.77 × 10 −13 Pascals. The solid line is for an equation of state p ∼ ρ n with n = 1.
The simulations were performed in the UNAM HP cluster Kan-Balam. . The simulations were performed in the UNAM HP cluster Kan-Balam.
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| [] |
[
"QUOTIENTS OF BANACH SPACES WITH THE DAUGAVET PROPERTY",
"QUOTIENTS OF BANACH SPACES WITH THE DAUGAVET PROPERTY"
] | [
"Vladimir Kadets ",
"Varvara Shepelska ",
"Dirk Werner "
] | [] | [] | We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak * analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L1[0, 1] over an ℓ1subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pe lczyński in the negative. | 10.4064/ba56-2-5 | [
"https://arxiv.org/pdf/0806.1815v1.pdf"
] | 16,062,729 | 0806.1815 | f34aeef96575bd5686a43ca1eb45aadc888c40db |
QUOTIENTS OF BANACH SPACES WITH THE DAUGAVET PROPERTY
11 Jun 2008
Vladimir Kadets
Varvara Shepelska
Dirk Werner
QUOTIENTS OF BANACH SPACES WITH THE DAUGAVET PROPERTY
11 Jun 2008arXiv:0806.1815v1 [math.FA]
We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak * analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L1[0, 1] over an ℓ1subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pe lczyński in the negative.
Introduction
Throughout the paper X stands for a Banach space. Recall that X has the Daugavet property if the identity Id + T = 1 + T , (1.1) called the Daugavet equation, holds true for every rank-one operator T : X → X. (We shall find it convenient to abbreviate this by writing X ∈ DPr.) It is known that in this case (1.1) holds for the much wider class of so-called narrow operators. This class includes all strong Radon-Nikodým operators (which map the unit ball into a set with the Radon-Nikodým property) and in particular compact and weakly compact operators, operators not fixing a copy of ℓ 1 and linear combinations of the above mentioned types of operators [10]. Among the spaces with the Daugavet property are C(K)-spaces and vector-valued C(K)-spaces for perfect compact Hausdorff spaces K, L 1 (µ)and vector-valued L 1 (µ)-spaces for non-atomic µ, wide classes of Banach algebras, but also some exotic spaces like Talagrand's space from [14] (see [9]) or Bourgain-Rosenthal's space from [3] (see [11]). All the spaces with the Daugavet property are non-reflexive, moreover they cannot have the Radon-Nikodým property and necessarily contain "many" copies of ℓ 1 [9].
There are several results on the stability of the Daugavet property under passing to "big" subspaces or quotients over "small" subspaces. In particular, if X ∈ DPr, then X/E ∈ DPr for every reflexive subspace E ⊂ X [13]. A preliminary version of that theorem appeared in [9] for X = L 1 [0, 1]. In a private conversation after a talk by the third-named author on the results of [9], A. Pe lczyński asked whether the last result can be generalized for E ⊂ L 1 [0, 1] being not necessarily reflexive, but having the Radon-Nikodým property (RNP). The question appeared quite non-trivial for the authors of [9], maybe because the efforts were concentrated on attempts to prove that L 1 [0, 1]/E ∈ DPr if E has the RNP. This question was reiterated in [13] and [7]. In this paper we are now going to present a negative answer.
Our approach to Pe lczyński's question will be indirect. If the answer was positive, this would mean that every subspace E ⊂ L 1 [0, 1] with the RNP has the following "smallness" property, since the RNP is hereditary: L 1 [0, 1]/F ∈ DPr for every subspace F ⊂ E. We introduce this property formally in a general setting (we call it poverty) and characterise it geometrically. Then we give a description of poor subspaces of L 1 (Ω, Σ, µ), and using that description we present an ℓ 1 -subspace E of L 1 [0, 1] which is not poor. Since ℓ 1 has the RNP, this leads to a counterexample.
To do all this we use duality arguments, but in order to be able to apply these arguments we have to consider a generalisation of the Daugavet property. Let us recall that a subspace Y ⊂ X * is said to be norming (more precisely 1-norming) if x = sup{|y * (x)|: y * ∈ Y, y * ≤ 1}
for every x ∈ X. This is equivalent to saying that the closed unit ball of Y is weak * dense in the closed unit ball of X * . Throughout the paper Y denotes a norming subspace of X * . Definition 1.1. We say that X has the Daugavet property with respect to Y (X ∈ DPr(Y )) if the Daugavet equation (1.1) holds true for every rank-one operator T : X → X of the form T = y * ⊗ x, where x ∈ X and y * ∈ Y .
This generalisation of the ordinary Daugavet property was introduced in an equivalent form in [1]. It was motivated by the fact that the Daugavet property is not stable under passing to ultraproducts (this was proved in [11] and an open problem at the time when [1] was written), but the ultraproduct of spaces with the Daugavet property has the Daugavet property with respect to the ultraproduct of the dual spaces. The basic motivation for us in the present paper is that the Daugavet property does not generally pass to the dual, but it is obvious that if the original space X has the Daugavet property, then X * ∈ DPr(X).
The structure of the paper is as follows. Section 2 contains characterisations of the Daugavet property with respect to Y in terms of slices, similar to [9]. Our eventual aim is to study "small" subspaces Z ⊂ X of spaces with the Daugavet property, called poor subspaces; this will be done in Section 5. It will turn out that Z ⊂ X is poor if and only if Z ⊥ ⊂ X * enjoys a variant of the properties called richness and wealth in [10]. Such spaces are defined by means of a smallness property of the corresponding quotient map called narrowness. Narrow operators are studied in our context in Section 4. In order to prove that weakly compact operators on dual spaces are X-narrow we have included the technical Section 3 about convex combinations of slices. Finally, in Section 6 we characterise poor subspaces of C(K) and L 1 (µ) and derive that both spaces contain copies of ℓ 1 that are not poor provided they are separable, K is perfect and µ is atomless. This will lead to a negative answer to Pe lczyński's question mentioned above (see Theorem 6.10).
A reader who is interested in that result only might wish to skip Section 3 and Section 4 apart from Definition 4.3, and he or she might also wish to only consider the case X = Y * in Section 5.
Much of the paper follows the lines of [10], and we omit proofs if they don't differ much from those in that paper.
We use standard notation such as B X and S X for the unit ball and the unit sphere of a Banach space X, and we employ the notation
S U (x * , ε) = x ∈ U : x * (x) > sup u∈U x * (u) − ε
for the slice of a bounded convex subset U ⊂ X determined by x * ∈ S X * . In the case of U = B X we omit the index U in the notation above:
S(x * , ε) = {x ∈ B X : x * (x) > 1 − ε}.
For ε > 0 and x ∈ S X we consider the weak * slice of the dual ball B X * , i.e.,
S(x, ε) = {x * ∈ B X * : x * (x) > 1 − ε},
as a particular case of a slice. When we have a need to stress in what space the slice is considered we use notation like S(X, x * , ε) or S(X * , x, ε). The symbol ex C stands for the set of extreme points of a set C. In this paper we deal with real Banach spaces although our results extend to the complex case with minor modifications.
Basic descriptions of the generalised Daugavet property
In this section we collect generalisations of characterisations of the standard Daugavet property to the setting of DPr(Y ). Let us start with a simple lemma about slices; cf. [9, Lemma 2.1].
Lemma 2.1. The following statement holds true in any Banach space X for any norming subspace Y ⊂ X * :
Let y ∈ S X , x * 0 ∈ S Y and ε ∈ (0, 1). Assume that there is some x ∈ S(x * 0 , ε/8) such that x + y > 2 − ε/8. Then there is an x * 1 ∈ S Y such that S(x * 1 , ε/8) ⊂ S(x * 0 , ε) and e + y > 2 − ε for all e ∈ S(x * 1 , ε/8).
Proof. Since x + y > 2 − ε/8 there is an x * ∈ S Y such that x * (x + y) > 2 − ε/8. Then x * (x) > 1 − ε 8 and x * (y) > 1 − ε 8 . (2.1) Define x * 1 ∈ S X * by x * 1 = x * 0 + x * x * 0 + x * ; we remark that x * 0 + x * ≥ (x * 0 + x * )x > 2 − ε 4 .
Then for every e ∈ S(
x * 1 , ε/8) we have (x * 0 + x * )(e) > 1 − ε 8 2 − ε 4 , (2.2) so x * 0 (e) > 1 − 2 ε 8 − ε 4 + ε 8 ε 4 > 1 − ε 2 ,
i.e., e ∈ S(x * 0 , ε), and the inclusion S(x * 1 , ε/8) ⊂ S(x * 0 , ε) is proved. Further, (2.2) implies that x * (e) > 1 − ε/2 which together with (2.1) means that e + y ≥ x * (e + y) > 2 − ε.
The following result is the analogue of [9, Lemma 2.2].
Theorem 2.2. If Y is a norming subspace of X * , then the following assertions are equivalent.
(i) X has the Daugavet property with respect to Y .
(ii) For every x ∈ S X , for every ε > 0, and for every y * ∈ S Y there is some y ∈ S(y * , ε) such that
x + y ≥ 2 − ε. (2.3) (iii)
For every x ∈ S X , for every ε > 0, and for every y * ∈ S Y there is a slice S(y * 1 , ε 1 ) ⊂ S(y * , ε) with y * 1 ∈ S Y such that (2.3) holds for every y ∈ S(y * , ε 1 ).
Proof. The implication (iii) ⇒ (ii) is obvious; the implication (ii) ⇒ (iii) follows from Lemma 2.1. What remains to prove is the equivalence (i) ⇔ (ii).
Let us start with (i) ⇒ (ii). Fix some x ∈ S X , x * ∈ S Y and ε > 0 and consider the operator T : X → X, T := x * ⊗ x. According to (1.1), Id + T = 2, so there is a y ∈ S X such that y + T y ≥ 2 − ε/2 and x * (y) ≥ 0. Substituting the value of T y we obtain that
y + x * (y)x ≥ 2 − ε/2
which means that x * (y) ≥ 1 − ε/2 (i.e., y ∈ S(x * , ε)) and
y + x ≥ y + x * (y)x − |x * (y) − 1| ≥ 2 − ε,
which proves the implication. For the converse implication (ii) ⇒ (i) consider an operator T : X → X, T = x * ⊗ x, where x ∈ X and x * ∈ Y . Since the validity of (1.1) for T implies (1.1) for all operators of the form aT with a > 0, it is sufficient to consider the case of T = 1, and the representation T = x * ⊗ x can be taken in such a way that x ∈ S X and x * ∈ S Y . Due to (ii), for every ε > 0 there is a y ∈ S(x * , ε) satisfying (2.3). Then
Id + T ≥ y + T y = y + x * (y)x ≥ y + x − |x * (y) − 1| ≥ (2 − ε) − ε,
which by arbitrariness of ε means that Id + T = 2.
A useful tool: convex combinations of slices
This section deals with a technical device that will be useful in the proof of Theorem 4.8.
Definition 3.1. Let Y be a norming subspace of X * , and let U ⊂ X be convex and bounded. A subset V ⊂ U is called a quasi-σ U (X, Y ) neighbourhood if it is a finite convex combination of slices of U generated by elements of Y ; i.e., there are λ k ≥ 0, k = 1, . . . , n, with n k=1 λ k = 1 and slices S 1 , . . . , S n ⊂ U generated by elements of Y such that λ 1 S 1 +· · ·+λ n S n = V .
The following lemma is known for the ordinary weak topology (see [2,Lemme 5.3]; it was rediscovered in [13]). The σ(X, Y )-version proof coincides almost word-to-word with the original one.
Lemma 3.2. Under the conditions of the above definition every relatively
σ(X, Y )-open subset A ⊂ U contains a quasi-σ U (X, Y ) neighbourhood.
The next theorem and its corollary were essentially proved by Shvidkoy [13]. He considered the ordinary Daugavet property, but the proof in the general case is virtually the same. Theorem 3.3. Let Y be a norming subspace of X * and X ∈ DPr(Y ). Then for every ε > 0, every x ∈ S X and every quasi-
σ B X (X, Y ) neighbourhood V there exists an element v ∈ V such that v + x ≥ 2 − ε.
Proof. Let V = n k=1 λ k S k be a representation of V as a convex combination of slices. Using repeatedly (ii) of Theorem 2.2 one can construct
x k ∈ S k such that (x + j<k λ j x j ) + λ k x k ≥ x + j<k λ j x j + λ k − ε/n. Then v = n k=1 λ k x k will be the element of V we need. Corollary 3.4.
If Y is a norming subspace of X * and X has the Daugavet property with respect to Y , then the following is true: For every x ∈ S X , for every ε > 0, and for every
σ(X, Y )-open subset U ⊂ X intersecting B X there is some y ∈ U ∩ B X such that x + y ≥ 2 − ε.
Narrow operators with respect to a norming subspace
We will eventually study subspaces satisfying a certain smallness condition called "poverty"; this will be dual to the notion of a "rich" subspace from [10]. The latter class is defined by the requirement that the canonical quotient map is "narrow". This section deals with such operators.
First we will recall and modify some definitions from [10]. Let X, E be Banach spaces.
Definition 4.1. An operator T ∈ L(X, E) is said to be a strong Daugavet operator if for every two elements x, y ∈ S X and for every ε > 0 there is an element z ∈ S X such that z + x > 2 − ε and T z − T y < ε. We denote the class of all strong Daugavet operators on X by SD(X).
Corollary 3.4 shows that if X ∈ DPr(Y ) then every T ∈ L(X, E) of the form T = f ⊗ e, where e ∈ E and f ∈ Y , is a strong Daugavet operator, and conversely, thanks to Theorem 2.2, if every f ∈ Y ⊂ X * = L(X, R) is strongly Daugavet, then X has the Daugavet property with respect to Y .
There is an obvious connection between strong Daugavet operators and the Daugavet equation (cf. [10, Lemma 3.2]). Definition 4.3. Let X ∈ DPr(Y ). An operator T ∈ L(X, E) is said to be narrow with respect to Y (or Y -narrow for short) if for every x, e ∈ S X , ε > 0 and every slice S ⊂ B X generated by an element of Y and containing e there is an element v ∈ S such that x + v > 2 − ε and T v − T e < ε. We denote the class of all Y -narrow operators on X by NAR Y (X).
The notations SD(X) and NAR Y (X) do not mention the range space E because the corresponding definitions do not actually depend on the values of T , but only on the norms of those values, i.e., these are not properties of the operator T itself, but just of the seminorm x → T (x) on X. For more about this ideology see [10].
The following statement is a complete analogue of [10, Lemma 3.10(a)], so we omit the proof.
Lemma 4.4. Let T ∈ NAR Y (X). Let S 1 , .
. . , S n ⊂ B X be a finite collection of slices generated by elements of Y , and let U ⊂ B X be a convex combination of these slices, i.e., there are λ k ≥ 0 with n k=1 λ k = 1 such that λ 1 S 1 + · · · + λ n S n = U . Then for every ε > 0, every x 1 ∈ S X and every w ∈ U there exists an element u ∈ U such that u + x 1 > 2 − ε and T (w − u) < ε.
Let us recall an operation with operators that was introduced in [10]. For operators T 1 : X → E 1 and T 2 :
X → E 2 define T 1+ T 2 : X → E 1 ⊕ 1 E 2 , x → (T 1 x, T 2 x); i.e., (T 1+ T 2 )x = T 1 x + T 2 x . Remark 4.5. Let X, E be Banach spaces, Y ⊂ X * be a norming subspace, T ∈ L(X, E). If T+ y * ∈ SD(X) for every y * ∈ Y , then T ∈ NAR Y (X).
In the setting of Y = X * this was actually given as the definition of a narrow operator in [10], and our definition was given as an equivalent condition in Lemma 3.10 of [10].
We are now going to introduce a class of operators that turn out to be Y -narrow; they correspond to the strong Radon-Nikodým operators in the case Y = X * , which contain the weakly compact operators. We need two technical definitions. Definition 4.6. Let X, E be Banach spaces, F ⊂ E * be a norming subspace and ε > 0. A point e of a convex subset A ⊂ E is said to be an (F, ε)-denting point if there is a functional f ∈ S F and a δ > 0 such that e − a < ε whenever a ∈ A satisfies the condition f (a) > f (e) − δ. We say that A ⊂ E is F -dentable if for every ε > 0 the set A is contained in the closed convex hull of its (F, ε)-denting points. An operator T ∈ L(X, E) is said to be Fdentable if T (B X ) is F -dentable. An E * -dentable operator is called dentable.
Definition 4.7. An operator T ∈ L(X, E) is said to be hereditarily F - dentable if for every x * ∈ X * the operator T+ x * : X → E ⊕ 1 R, x → (T x, x * x) isF -dentable, whereF consists of all functionals (f, β): E ⊕ 1 R → R, with f ∈ F and β ∈ R, of the form (f, β) ((e, t)) = f (e) + βt.
Remark that every strong Radon-Nikodým operator and in particular every weakly compact operator is hereditarily dentable, by well-known geometric characterisations of sets with the RNP [4, Chap. 3].
Theorem 4.8. If X ∈ DPr(Y ), T : X → E is a hereditarily F -dentable operator and T * (F ) ⊂ Y , then T is Y -narrow.
Proof. According to Remark 4.5 it is sufficient to prove thatT = T+x * ∈ SD(X) for every x * ∈ Y . Fix x, z ∈ S X and ε > 0. By Definition 4.1, to prove the theorem we have to find an element v ∈ B X such that x + v > 2 − ε and T (z − v) < ε.
SinceT (B X ) isF -dentable there are λ k ≥ 0 with n k=1 λ k = 1 and (F , ε/2)-denting points e 1 , . . . , e n ∈ T (B X ) such that
T z − n k=1 λ k e k < ε/2.
By the definition of anF -denting point there are slices S k = ST (B X ) (f k , ε k ) of T (B X ) generated by elements ofF such that e k ∈ S k and the diameter of each of the S k is less than ε/2. Denote W := n k=1 λ k S k . Since dist(T z, W ) < ε/2 and diam W < ε/2 we have
T z − w < ε for every w ∈ W. (4.1) Denote y * k :=T * f k . By assumption, y * k ∈ Y . Consider slices V k = {v ∈ B X : y * k (x) > ε k } and the quasi-σ B X (X, Y ) neighbourhood V := n k=1 λ k V k . SinceT (V k ) ⊂ S k and consequentlyT (V ) ⊂ W , (4.1) implies that T z − T v < ε for every v ∈ V.
It remains to apply Theorem 3.3 to get a v ∈ V with x + v > 2 − ε.
Corollary 4.9. A weak * -weakly continuous operator on a dual space X * ∈ DPr(X) is X-narrow.
Proof. It is clear that such an operator is weakly compact. The hereditary dentability of a weakly compact operator T : X * → E has been mentioned in the above remark; it remains to observe that T * (E * ) ⊂ X if T is weak *weakly continuous.
Note that a weakly compact adjoint operator S * : X * → V * is weak *weakly continuous, i.e., S * * (V * * ) ⊂ X; cf. [6, Section VI.4].
Rich and poor subspaces
In [10] we introduced rich subspaces Z ⊂ X, building on work in [8] and [12]. We showed that this condition is equivalent to saying that every superspace Z ⊂Z ⊂ X has the Daugavet property; the latter property was called wealth in [10]. We now extend and dualise these ideas.
5.1.
Richness. The next proposition shows a kind of stability of the Daugavet property when one passes from the original space to a "big" subspace.
Lemma 5.1. Let X ∈ DPr(Y ). Then for every x ∈ S X , every ε > 0 and for
every separable subspace V ⊂ Y there is an x * ∈ S X * such that x * (x) ≥ 1−ε and x * + f = 1 + f for all f ∈ V .
Proof. Consider a dense sequence (f n ) ∞ n=1 ⊂ V such that every element is repeated infinitely many times in the sequence. Applying (v) of Theorem 2.2 to the slice S(x, ε) of B X * and to f 1 and then applying it step-by-step to f n and to the slices obtained in the previous steps, we construct a sequence of closed slices S(x, ε) ⊃ S(x 1 , ε 1 ) ⊃ S(x 2 , ε 2 ) ⊃ . . . with ε n < 1/n such that x * + f n ≥ 2 − ε n−1 for all x * ∈ S(x n , ε n ). By w * -compactness of all S(x n , ε n ), there is a point x * ∈ ∞ n=1 S(x n , ε n ) ⊂ S(x, ε). This is exactly the point we need.
Proposition 5.2. Let X ∈ DPr(Y ), and let Z ⊂ X be a subspace such that Z ⊥ is a separable subspace of Y . Then Z ∈ DPr(Y | Z ).
Proof. Let z ∈ S(Z), and let
S = {z * ∈ Z * = X * /Z ⊥ : z * ≤ 1, z * (z) ≥ 1 − ε} be a slice of B Z * . Fix a [z * ] ∈ S(Y /Z ⊥ )
. We have to prove the existence of an [x * ] ∈ S such that x * + z * = 2. Applying Lemma 5.1 with x = z and V = lin({z * } ∪ Z ⊥ ) we obtain an x * ∈ S X * such that x * (z) ≥ 1 − ε and
x * + f = 1 + f for all f ∈ V.
Then [x * ] ∈ S and
[x * + z * ] = inf f ∈Z ⊥ x * + z * + f = inf f ∈Z ⊥ (1 + z * + f ) = 1 + [z * ] = 2.
If a subspace Z ⊂ X satisfies the conditions of the proposition above then so do all the subspaces of X containing Z. Hence Z has the property that Z ∈ DPr(Y |Z ) for every subspaceZ ⊂ X containing Z. Let us formalise this property. Definition 5.3. Let X ∈ DPr(Y ). A subspace Z ⊂ X is said to be wealthy with respect to Y ifZ ∈ DPr(Y |Z ) for every subspaceZ ⊂ X containing Z.
Thus Proposition 5.2 can be rephrased by saying that Z ⊂ X is wealthy with respect to Y if Z ⊥ is separable and X ∈ DPr(Y ).
The main result of this subsection is a characterisation of Y -wealthy subspaces through Y -narrow operators, analogous to [10, Theorem 5.12].
Definition 5.4. Let X ∈ DPr(Y ). A subspace Z ⊂ X is said to be rich with respect to Y if the quotient map q: X → X/Z is a Y -narrow operator.
It turns out that the following theorem holds.
Theorem 5.5. Let X be a Banach space and Y be a norming subspace of X * such that X ∈ DPr(Y ). Then for a subspace Z ⊂ X the following properties are equivalent:
(i) Z is wealthy with respect to Y . (ii) Z is rich with respect to Y .
The proof is very similar to the proof of [10, Theorem 5.12].
5.2.
Poverty as a dual property to richness.
Definition 5.6. Let X ∈ DPr. A subspace Z ⊂ X is said to be poor if X/Z ∈ DPr for every subspaceZ ⊂ Z.
Our study of poor subspaces uses duality, so let us start with a very simple observation that we state as a proposition for easy reference.
Proposition 5.7. A Banach space X has the Daugavet property if and
only if X * ∈ DPr(X). Hence, a subspace Z of a space X with the Daugavet property is poor if and only if for every subspaceZ ⊂ Z its dual (X/Z) * = Z ⊥ has the Daugavet property with respect to X/Z. Now we are ready to give the basic characterisations of poverty.
Theorem 5.8. Let X ∈ DPr. For a subspace Z ⊂ X the following conditions are equivalent.
(i) Z is poor.
(ii) X/Z ∈ DPr for every subspaceZ ⊂ Z of codimension codim ZZ ≤ 2. (iii) Z ⊥ is a subspace of X * that is rich with respect to X. (iv) For every x * , e * ∈ S X * , ε > 0 and for every x ∈ S X such that e * (x) > 1 − ε there is an element v * ∈ B X * with the following properties: v * (x) > 1−ε, x * +v * > 2−ε and (e * −v * ) | Z < ε; that is, the quotient map from X * onto X * /Z ⊥ is narrow with respect to X.
Proof. (i) ⇒ (ii) follows immediately from the definition of poor subspaces. Let us prove (ii) ⇒ (i). According to Proposition 5.7, we have to prove that for every subspace Z 1 ⊂ Z, its dual Z ⊥ 1 has the Daugavet property with respect to X/Z 1 . Fix Z 1 ⊂ Z. Applying (ii) of Theorem 2.2 we see that for every
x * ∈ S Z ⊥ 1 , ε > 0 and every [x] ∈ S X/Z 1 we have to find y * ∈ S Z ⊥ 1 such that y * ([x]) ≥ 1 − ε and x * + y * ≥ 2 − ε. Since [x] ∈ S X/Z 1 , there exists z * ∈ S Z ⊥ 1 such that z * ([x]) = 1. DenoteZ = Z ∩ ker x * ∩ ker z * . Evidently,
Z is a subspace of Z of codim ZZ ≤ 2 and Z 1 ⊂Z. Also remark that
1 = [x] X/Z 1 ≥ [x] X/Z ≥ z * ([x]) = 1,
which implies [x] ∈ S X/Z . By our assumptionZ ⊥ has the Daugavet property with respect to X/Z, and hence for x * ∈ SZ ⊥ and [x] ∈ S X/Z there is y * ∈ SZ ⊥ such that y * ([x]) ≥ 1 − ε and x * + y * ≥ 2 − ε. Then y * ∈ SZ ⊥ ⊂ S Z ⊥ 1 , and it meets all the requirements. Now we will prove that (ii) ⇔ (iii). Theorem 5.5 implies that (iii) holds if and only if Z ⊥ is a subspace of X * that is wealthy with respect to X; and this is equivalent to the claim that for every x * , y * ∈ S X * the space W = lin(Z ⊥ ∪{x * , y * }) has the Daugavet property with respect to X/W ⊥ (cf. [10, Lemma 5.6(iii)]. But for a spaceẐ ⊃ Z ⊥ the existence of x * , y * ∈ S X * such that W = lin(Z ⊥ ∪ {x * , y * }) is equivalent to the existence of a spacẽ Z ⊂ Z such that W =Z ⊥ and codim ZZ ≤ 2. Thus we get that (iii) is equivalent to the claim thatZ ⊥ ∈ DPr(X/Z) for every subspaceZ ⊂ Z of codim ZZ ≤ 2, which is equivalent to (ii) according to Proposition 5.7.
The remaining equivalence (iii) ⇔ (iv) is just a reformulation of the definition of a rich subspace.
As a corollary we can give a proof of the following theorem of Shvidkoy [13].
Corollary 5.9. Let X ∈ DPr and let Z be a reflexive subspace of X. Then the quotient space X/Z also has the Daugavet property.
Proof. Since every subspace of a reflexive space is also reflexive, the statement of this corollary is equivalent to the claim that every reflexive subspace Z of X ∈ DPr is poor. According to Theorem 5.8, it is sufficient to prove that Z ⊥ is a subspace of X * that is rich with respect to X, i.e., that the quotient map q: X * → X * /Z ⊥ is an X-narrow operator. As X * /Z ⊥ is isometric to Z * , which is reflexive, this follows from Corollary 4.9.
6. Applications to the geometry of C(K) and L 1 For a compact Hausdorff space K denote by M (K) the dual space of C(K), i.e., M (K) is the Banach space of all (not necessarily positive) finite regular Borel signed measures on K. (In the sequel, all measures on K will be tacitly assumed to be finite regular Borel measures.) We are going to prove a theorem which gives a characterisation of operators on M (K) that are narrow with respect to C(K). For this theorem we will need the following lemma in which ∂A denotes the boundary of a set A ⊂ K. Lemma 6.1. Let K be compact, f ∈ C(K), and µ be some positive measure on K. Then for every ε > 0 there exists a step functionf = n k=1 β k χ A k on K such that µ(∂A k ) = 0 for k = 1, . . . , n, A 1 ∪ · · · ∪ A n = K and f −f ∞ < ε.
Proof. Since the image measure ν = µ•f −1 on R has at most countably many atoms, it is possible to cover f (K) by finitely many half-open intervals I k = (β k−1 , β k ] of length < ε such that ν({β 0 , . . . , β n }) = 0. Let A k = f −1 (I k ); thenf = n k=1 β k χ A k works. Theorem 6.2. Let K be a perfect compact Hausdorff space. An operator T on M (K) is narrow with respect to C(K) if and only if for every open subset U ⊂ K, for every two probability measures π 1 , π 2 on U and for every ε > 0 there is a probability measure ν on U such that T (ν − π 1 ) < ε and π 2 − ν > 2 − ε.
Proof. We first prove the "only if" part. By the definition of a narrow operator (Definition 4.3), for every x, e ∈ S M (K) , ε > 0 and every weak * slice S of B M (K) containing e, there exists v ∈ S such that x + v > 2 − ε and T (e − v) < ε. Fix ε 1 > 0 and let x = −π 2 and e = π 1 . Since U is open and π 1 (U ) = 1, we can find f ∈ C(K) taking values in [0, 1] with supp f ⊂ U and f dπ 1 > 1 − ε 1 . By Definition 4.3 there existsν ∈ S M (K) such that the following inequalities hold:
f dν > 1 − ε 1 , T (ν − π 1 ) < ε 1 , π 2 −ν > 2 − ε 1 .
Letν =ν + | U . Using the properties of f we have ν −ν < 2ε 1 and thus
1 − 3ε 1 < ν ≤ 1 + 2ε 1 , T (ν − π 1 ) < ε 1 (1 + 2 T ), π 2 −ν > 2 − 3ε 1 .
Hence for ν =ν/ ν we have ν −ν = 1 − ν < 3ε 1 and consequently
π 2 − ν ≥ π 2 −ν − ν −ν > 2 − 3ε 1 − 3ε 1 = 2 − 6ε 1 ,
and T (ν − π 1 ) ≤ T (ν − π 1 ) + T (ν − ν) < (1 + 5 T )ε 1 . Then taking ε 1 = min{ ε 6 , ε 1+5 T } completes the proof of the "only if" part. Now consider the "if" part. Given µ 1 , µ 2 ∈ S M (K) , ε > 0 and a weak * slice S of B M (K) containing µ 1 , we have to find ν ∈ S such that µ 2 + ν > 2 − ε and T (µ 1 − ν) < ε. Since one can wiggle the slice S a bit, there is, by Lemma 6.1, no loss of generality in replacing S by a slice generated by a function of the form f = n k=1 β k χ A k , where A 1 , . . . , A n are measurable sets with (|µ 1 | + |µ 2 |)( n k=1 ∂A k ) = 0. (Note that in general this new slice will not be relatively weak * open.) On the other hand, using the Hahn decomposition theorem we have K = 4 i=1 B i , where B 1 is a set on which µ 1 is positive and µ 2 is negative, B 2 is a set on which µ 2 is positive and µ 1 is negative, and B 3 (resp. B 4 ) is a set where both µ 1 and µ 2 are positive (resp. negative).
Fix ε 1 > 0 and let G 1 be an open set such that G 1 ⊃ B 1 and |µ i |(G 1 \B 1 ) < ε 1 (i = 1, 2). Define C k = G 1 ∩ A k and let U k = int C k , k = 1, . . . , n. Clearly C k \U k ⊂ ∂A k , so the U k are open sets with the following properties:
U k ⊂ C k and (|µ 1 | + |µ 2 |)(C k \ U k ) = 0.
Consider those U k for which µ 1 (U k ∩ B 1 ) = 0, µ 2 (U k ∩ B 1 ) = 0 and define two probability measures on U k by
µ i,k = µ i| U k ∩B 1 µ i (U k ∩ B 1 ) (i = 1, 2).
By assumption there exists a probability measureν k on U k such that
T (ν k − µ 1,k ) < ε 1 and µ 2,k −ν k > 2 − ε 1 . Define ν k = µ 1 (U k ∩ B 1 ) ·ν k . Then we have ν k = ν k (U k ) = µ 1 (U k ∩ B 1 ), µ 1| U k − ε 1 ≤ ν k ≤ µ 1| U k + ε 1 (6.1)
and
µ 2| U k + ν k = µ 2 (U k ∩ B 1 ) · µ 2,k + µ 2| U k \B 1 + µ 1 (U k ∩ B 1 ) ·ν k ≥ |µ 2 (U k ∩ B 1 )| · µ 2,k − |µ 1 (U k ∩ B 1 )| ·ν k − µ 2| U k \B 1 ≥ |µ 2 |(U k ) + |µ 1 |(U k ) − 4ε 1 (6.2)
and
T (ν k − µ 1| U k ) ≤ T (µ 1 (U k ∩ B 1 ) · (ν k − µ 1,k )) + T (µ 1| U k \B 1 ) ≤ ε 1 (1 + T ). (6.3) For U k with µ 1 (U k ∩ B 1 ) = 0 or µ 2 (U k ∩ B 1 ) = 0, the inequalities (6.1)-(6.3) hold with ν k = µ 1| U k ∩B 1 . Now define the measure µ 1 1 by µ 1 1 | U k = ν k , µ 1 1 | K\ S n k=1 U k = µ 1| K\ S n k=1 U k .
From (6.1), (6.2), and (6.3) we obtain the following properties of µ 1 1 :
µ 1 − nε 1 ≤ µ 1 1 ≤ µ 1 + nε 1 , f dµ 1 1 − f dµ 1 ≤ nε 1 (6.4) and µ 2| G 1 + µ 1 1 | G 1 ≥ n k=1 (µ 2| U k + ν k ) − (|µ 2 | + |µ 1 |) n k=1 C k \ U k ≥ n k=1 (|µ 2 |(U k ) + |µ 1 |(U k )) − 4nε 1 ≥ |µ 1 |(G 1 ) + |µ 2 |(G 1 ) − (4n + 2)ε 1 , (6.5) T (µ 1 1 − µ 1 ) = n k=1 T (ν k − µ 1| U k ) ≤ (n + n T )ε 1 . (6.6) Now defineB 2 = B 2 \ G 1 .
Notice thatB 2 is a set of negativity for µ 1 1 and a set of positivity for µ 2 . Following the same lines as above we define G 2 ⊃B 2 and construct µ 2 1 ∈ M (K) such that
|µ 2 |(G 2 \B 2 ) < ε 1 , |µ 1 1 |(G 2 \B 2 ) < ε 1 , µ 1 1 − nε 1 ≤ µ 2 1 ≤ µ 1 1 + nε 1 and f dµ 2 1 − f dµ 1 1 ≤ nε 1 , T (µ 2 1 − µ 1 1 ) ≤ (n + n T )ε 1 , µ 2| G 2 + µ 2 1 | G 2 ≥ |µ 1 |(G 2 ) + |µ 2 |(G 2 ) − (4n + 2)ε 1 .
From (6.4), (6.5), (6.6) and the above inequalities we obtain the estimates
1 − 2nε 1 ≤ µ 2 1 ≤ 1 − 2nε 1 , f dµ 2 1 − f dµ 1 ≤ 2nε 1 and T (µ 2 1 − µ 1 ) ≤ (2n + 2n T )ε 1 , (6.7) µ 2| G 1 ∪G 2 + µ 2 1 | G 1 ∪G 2 ≥ |µ 1 |(G 1 ∪ G 2 ) + |µ 2 |(G 1 ∪ G 2 ) − (8n + 10)ε 1 .
Finally, the definition of the sets B 3 and B 4 implies that µ 2 + µ 2 1 ≥ µ 1 + µ 2 − (8n + 10)ε 1 = 2 − (8n + 10)ε 1 . (6.8)
Hence for ε 1 small enough, the normalized signed measure ν = µ 2 1 / µ 2 1 satisfies all the required conditions, which completes the proof of the theorem.
Applying this theorem to the operator µ → µ | Z yields by Theorem 5.8:
Corollary 6.3. Let K be a perfect compact. A subspace Z ⊂ C(K)
is poor if and only if for every open subset U ⊂ K, for every two probability measures π 1 , π 2 on U and for every ε > 0 there is a probability measure ν on U such that ν − π 1 Z * < ε and π 2 − ν > 2 − ε.
For a closed subset K 1 of K denote by R K 1 the natural restriction operator R K 1 : C(K) → C(K 1 ). Note that for an operator S: E → F between Banach spaces the following assertions are equivalent, by the (proof of) the open mapping theorem: (i) S is onto; (ii) S(B E ) is not nowhere dense; (iii) 0 is an interior point of S(B E ). Corollary 6.4. Let K be a perfect compact Hausdorff space, K 1 ⊂ K be a closed subset with non-empty interior, and let Z be a poor subspace of C(K).
Then R K 1 (B Z ) is nowhere dense in B C(K 1 ) .
Proof. Apply Corollary 6.3 with U = int K 1 , π 1 = π 2 and a sufficiently small ε > 0 to see that R K 1 (B Z ) cannot contain a ball rB C(K 1 ) of radius r > 0.
We now deal with poor subspaces of L 1 . Let (Ω, Σ, µ) be a finite measure space. Denote by Σ + the collection of all A ∈ Σ with µ(A) > 0. Theorem 6.5. Let (Ω, Σ, µ) be a non-atomic finite measure space. An operator T on L ∞ := L ∞ (Ω, Σ, µ) is narrow with respect to L 1 := L 1 (Ω, Σ, µ) if and only if for every ∆ ∈ Σ + and for every ε > 0 there is g ∈ S L∞ such that g = 0 off ∆ and T g < ε. Moreover, in the statement above g can be selected non-negative.
Proof. First we prove the "if" part. By the definition of a narrow operator (Definition 4.3) for every x, y ∈ S L∞ , every f ∈ S L 1 such that f · y dλ > 1 − δ (i.e., y ∈ S(f, δ)) and every ε > 0 we have to find z ∈ S(f, δ) such that x + z > 2 − ε and T (y − z) < ε. By density of step functions we may assume without loss of generality that there is a partition A 1 , . . . , A n of Ω such that the restrictions of x, y and f to A k are constants, say a k , b k and c k respectively. Fix some ε 1 > 0. Since x = 1, there exists k such that |a k | > 1 − ε 1 . Let B ∈ Σ + be a subset of A k with µ(B) ≤ ε 1 and A k \ B ∈ Σ + . By our assumption there existsẑ ∈ S L∞ such that z ≥ 0, z is supported on B and T (z) ≤ ε 1 . Denotez = y + (sign(a 1 ) − b 1 )ẑ. It is easy to see that z = 1, x +z = 2 − ε 1 , T (y −z) < ε 1 andz ∈ S(f, δ − ε 1 ). To finish the proof of this part it is sufficient to repeat the reasoning from the end of the proof of Theorem 6.2. Now we consider the "only if" part. Since T is narrow with respect to L 1 , T is also a strong Daugavet operator. Hence as in [10, Theorem 3.5] we can get a functiong ∈ S L∞ which satisfies all the requirements, except being non-negative. To fix this we argue as in [8,Lemma 1.4] and finally get some non-negative g possessing all the properties listed above.
Again, specialising to the restriction operator g ∈ L ∞ = (L 1 ) * → g | Z ∈ Z * we obtain the following characterisation of poor subspaces. Corollary 6.6. Let (Ω, Σ, µ) be a non-atomic finite measure space. A subspace Z ⊂ L 1 (Ω, Σ, µ) is poor if and only if for every ∆ ∈ Σ + and for every ε > 0 there is g ∈ S L∞ such that g = 0 off ∆ and g Z * < ε. Moreover, in the statement above g can be selected non-negative.
For a subset A ∈ Σ + denote by Q A the natural restriction operator Q A :
L 1 (Ω, Σ, µ) → L 1 (A, Σ | A , µ | A ).
Corollary 6.7. Let (Ω, Σ, µ) be a non-atomic finite measure space, A ∈ Σ + and let Z be a poor subspace of L 1 (Ω, Σ, µ).
Then Q A (B Z ) is nowhere dense in B L 1 (A,Σ| A ,µ) .
Proof. Apply Corollary 6.6 with ∆ = A and a sufficiently small ε > 0 to see that Q A (B Z ) cannot contain a ball rB L 1 (A) of radius r > 0. The Corollaries 6.4 and 6.7 look very similar. The next definition extracts the significant common feature. Definition 6.8. Let X ∈ DPr. A subspace E ⊂ X is said to be a bank if E contains an isomorphic copy of ℓ 1 and for every poor subspace Z of X, q E (B Z ) is nowhere dense in B X/E (here q E denotes the natural quotient map q E : X → X/E). If E ⊂ X is a bank, then B X/E will be called the asset of E.
In this terminology a poor subspace cannot cover a "significant part" of a bank's asset. Theorem 6.9. Let X ∈ DPr and E ⊂ X be a bank with separable asset. Then X contains a copy of ℓ 1 which is not poor in X.
Proof. Let {e n } n∈N ⊂ 1 2 B E be equivalent to the canonical basis of ℓ 1 and let {x n } n∈N ⊂ B E be a sequence such that {q E (x n )} n∈N is dense in B X/E . Then, if one selects a sufficiently small ε > 0, the sequence of u n = e n + εx n ∈ B E is still equivalent to the canonical basis of ℓ 1 , and the image of this sequence under q E equals {εq E (x n )} n∈N , which is dense in εB X/E . This means that the closed linear span of {u n } n∈N is the copy of ℓ 1 we need.
The next theorem is an immediate corollary of Theorem 6.9.
Theorem 6.10. In every C(K)-space with perfect metric compact K and in every separable L 1 (Ω, Σ, µ)-space with non-atomic µ there is a subspace isomorphic to ℓ 1 that is not poor.
Proof. Corollary 6.4 implies that if K is a perfect compact and K 1 ⊂ K is a proper closed subset with non-empty interior, then C 0 (K \ K 1 ) := {f ∈ C(K): f (t) = 0 ∀t ∈ K 1 } is a bank with B C(K 1 ) being its asset. Corollary 6.7 implies that if (Ω, Σ, µ) is a non-atomic finite measure space and A ∈ Σ + , then L 1 (Ω\A) is a bank with B L 1 (A) being its asset. Separability of these assets follows from the separability of the spaces C(K) and L 1 (Ω, Σ, µ) considered. It is left to apply Theorem 6.9. Theorem 6.10 answers Pe lczyński's question mentioned in the introduction in the negative since it provides a non-poor ℓ 1 -subspace Z ⊂ L 1 [0, 1]. By definition this means that for some subspaceZ ⊂ Z, L 1 [0, 1]/Z fails the Daugavet property; but Z has the RNP and so does its subspaceZ. In fact, by Theorem 5.8 one can chooseZ of codimension ≤ 2, henceZ is isomorphic to ℓ 1 as well. Let us some up these considerations. Corollary 6.11. There is a subspace E ⊂ L 1 [0, 1] that is isomorphic to ℓ 1 and hence has the RNP, but L 1 [0, 1]/E fails the Daugavet property.
7. Some open questiions 1. Is it true that every separable space with the Daugavet property has an ℓ 1 -subspace which is not poor?
2. Can the separability condition in Theorem 6.9 be omitted? 3. Is it true that every subspace without copies of ℓ 1 of a space with the Daugavet property is poor? We don't even know the answer in the case of C[0, 1]. 4. Is it true that if X ∈ DPr and Y ⊂ X is a subspace with a separable dual, then the quotient space X/Y also has the Daugavet property? This question also appears in [13].
Lemma 4 . 2 .
42If T : X → X is a strong Daugavet operator, then T satisfies the Daugavet equation (1.1).
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| [] |
[
"Global-Supervised Contrastive Loss and View-Aware-Based Post-Processing for Vehicle Re-Identification",
"Global-Supervised Contrastive Loss and View-Aware-Based Post-Processing for Vehicle Re-Identification"
] | [
"Zhijun Hu [email protected] ",
"Yong Xu ",
"Jie Wen [email protected] ",
"Xianjing Cheng [email protected] ",
"Zaijun Zhang ",
"· Lilei ",
"Sun · ",
"Yaowei Wang [email protected] ",
"Corresponding Author)Zhijun Hu ",
"Yong Xu ",
"Jie Wen ",
"Xianjing Cheng ",
"Lilei Sun [email protected] ",
"Zaijun Zhang ",
"Yaowei Wang ",
"\nSchool of Mathematics and Statistics\nCollege of Computer Science and Technology\nGuangxi Normal University\n541004GuilinChina\n",
"\nCollege of Computer Science and Technology\nShenzhen Key Laboratory of Visual Object Detection and Recognition\nGuizhou University\n550025Guiyang, ShenzhenChina., China\n",
"\nKey Laboratory of Complex Systems and Intelligent Computing and School of Mathematics and Statistics, Qiannan Normal University for Nationalities\nGuizhou University\n550025, 558000Guiyang, DuyunChina., China\n",
"\nPeng Cheng Laboratory\n518055ShenzhenChina\n"
] | [
"School of Mathematics and Statistics\nCollege of Computer Science and Technology\nGuangxi Normal University\n541004GuilinChina",
"College of Computer Science and Technology\nShenzhen Key Laboratory of Visual Object Detection and Recognition\nGuizhou University\n550025Guiyang, ShenzhenChina., China",
"Key Laboratory of Complex Systems and Intelligent Computing and School of Mathematics and Statistics, Qiannan Normal University for Nationalities\nGuizhou University\n550025, 558000Guiyang, DuyunChina., China",
"Peng Cheng Laboratory\n518055ShenzhenChina"
] | [] | In this paper, we propose a Global-Supervised Contrastive loss (L GSupCon ) and a view-aware-based post-processing (VABPP) method for the field of vehicle re-identification. The traditional supervised contrastive loss (L SupCon ) calculates the distances of features within the batch, so it has the local attribute. While the proposed L GSupCon has new properties and has good global attributes, the positive and negative features of each anchor in the training process come from the entire training set. The proposed VABPP method is the first time that the view-aware-based method is used as a post-processing method in the field of vehicle re-identification. The advantages of VABPP are that, first, it is only used during testing and does not affect the training process. Second, as a post-processing method, it can be easily integrated into other trained re-id models. We directly apply the view-pair distance scaling coefficient matrix calculated by the model trained in this paper to another trained re-id model, and the VABPP method greatly improves its performance, which verifies the feasibility of the VABPP method.Keywords Vehicle re-identification · deep learning · view-aware · globalsupervised contrastive · post-processing. | 10.48550/arxiv.2204.07943 | [
"https://arxiv.org/pdf/2204.07943v1.pdf"
] | 248,227,585 | 2204.07943 | 4b7d281754fa9056ceff5106a0afcc6232e22ce2 |
Global-Supervised Contrastive Loss and View-Aware-Based Post-Processing for Vehicle Re-Identification
17 Apr 2022
Zhijun Hu [email protected]
Yong Xu
Jie Wen [email protected]
Xianjing Cheng [email protected]
Zaijun Zhang
· Lilei
Sun ·
Yaowei Wang [email protected]
Corresponding Author)Zhijun Hu
Yong Xu
Jie Wen
Xianjing Cheng
Lilei Sun [email protected]
Zaijun Zhang
Yaowei Wang
School of Mathematics and Statistics
College of Computer Science and Technology
Guangxi Normal University
541004GuilinChina
College of Computer Science and Technology
Shenzhen Key Laboratory of Visual Object Detection and Recognition
Guizhou University
550025Guiyang, ShenzhenChina., China
Key Laboratory of Complex Systems and Intelligent Computing and School of Mathematics and Statistics, Qiannan Normal University for Nationalities
Guizhou University
550025, 558000Guiyang, DuyunChina., China
Peng Cheng Laboratory
518055ShenzhenChina
Global-Supervised Contrastive Loss and View-Aware-Based Post-Processing for Vehicle Re-Identification
17 Apr 2022Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Vehicle re-identification · deep learning · view-aware · global- supervised contrastive · post-processing
In this paper, we propose a Global-Supervised Contrastive loss (L GSupCon ) and a view-aware-based post-processing (VABPP) method for the field of vehicle re-identification. The traditional supervised contrastive loss (L SupCon ) calculates the distances of features within the batch, so it has the local attribute. While the proposed L GSupCon has new properties and has good global attributes, the positive and negative features of each anchor in the training process come from the entire training set. The proposed VABPP method is the first time that the view-aware-based method is used as a post-processing method in the field of vehicle re-identification. The advantages of VABPP are that, first, it is only used during testing and does not affect the training process. Second, as a post-processing method, it can be easily integrated into other trained re-id models. We directly apply the view-pair distance scaling coefficient matrix calculated by the model trained in this paper to another trained re-id model, and the VABPP method greatly improves its performance, which verifies the feasibility of the VABPP method.Keywords Vehicle re-identification · deep learning · view-aware · globalsupervised contrastive · post-processing.
Introduction
Due to the vigorous development of deep learning, deep learning methods have penetrated into many research fields Banerjee et al. (2021); Huang et al. (2020); Sharma et al. (2021), so does as the field of vehicle re-identification. Vehicle re-identification aims to find the same vehicle as the query vehicle in non overlapping cameras Liu et al. (2016). Although the license plate can be used as the unique identification of the vehicle, it is usually difficult to capture the license plate correctly due to factors such as occlusion, illumination and camera distance. Vehicle re-identification has attracted more and more attention because it only uses vehicle appearance to identify vehicles.
The two biggest challenges faced by vehicle re-identification are, first, how to design a metric learning method to extract more discriminative vehicle features. Second, the visual differences of vehicle appearance are very large. The same vehicle may look very different from different views, while different vehicles of the same model and color produced by the same manufacturer may look very similar from the same view. For the first challenge, due to the good performance, triplet loss Schroff et al. (2015); Hoffer and Ailon (2015) has been used in the field of vehicle re-identification for a long time Zhao et al. (2021a,b). However, for triplet loss, an anchor only has one positive sample and one negative sample, which limits the performance of triplet loss. The advantage of L SupCon Khosla et al. (2020) is that for an anchor, it makes use of all positive samples and all negative samples in the batch, which increases the stability of training Khosla et al. (2020). Since references Khosla et al. (2020); Chen et al. (2020a); He et al. (2020) claimed that increasing the number of negative IDs can improve performance, we hope to continue to increase the number of negative IDs on the basis of L SupCon . We proposed a loss function named Global-SupCon Loss (L GSupCon ), for an anchor, the proposed L GSupCon extends the positives and negatives to the entire training set (Fig. 1).
For the second challenge, there have been many researches on view-awarebased method Khorramshahi et al. (2019); Zheng et al. (2021); Chu et al. (2019), which were all conducted by adding orientation suppression to the model parameters in the training process to make the model parameters learned the view-aware knowledge to improve the model performance. Such methods are difficult to be integrated into other methods. This paper proposes a viewaware-based post-processing method (VABPP). As far as we know, this is the first time that view-aware method is proposed as a post-processing method. The proposed VABPP aims to improve the experimental accuracy during testing by scaling the distance between the features of the images in the gallery set and the feature of the query image, and can be easily integrated into other methods. Fig. 2 shows the motivation of the VABPP method. Based on our baseline, Fig. 2 shows the matching results of a query image q matching in the gallery set. If all the rear view vehicles (blue boxes) are moved forward by the distance equal to the length of 19 images, so that the green boxes and the blue boxes are crosswise ranked, and we can find the last blue image is moved before the first red image, that is, all positive matches are moved before all negative matches, which significantly improves the matching accuracy. Note that for the sake of intuition, the above introduction is based on translation, but scaling is used later of this paper.
To sum up, the main contributions of this paper are as follows:
(1)We design a Global-Supervised Contrastive loss, 'all positive and negative features do not have the property of gradient, and the positive and negative features of each anchor are from the entire training set.
(2) We design a view-aware-based post-processing method to improve the test accuracy by stretching (or shrinking) the distances between the features of the images with the same view in the gallery set and the feature of the query image in a whole with a distance scaling coefficient. The distances between the features of the images of the same view and the anchor feature have the same scaling coefficient, while the distances between the images with different views and the anchor have different scaling coefficients. This method can be easily integrated into other methods without affecting the training process.
(3) The proposed method is verified on the three data sets widely used in vehicle re-identification, and our method achieves the state-of-the-art level.
The rest of this paper is arranged as follows. The second part reviews some works related to this paper. The third part introduces the main idea of the proposed method. The fourth part expounds the proposed method in detail. The fifth part analyzes the reasons why the proposed method is effective by experiments and compares it with the state-of-the-art methods. The sixth part summarizes the full paper.
2 Related work 2.1 Vehicle re-identification based on metric learning Metric learning method is usually packaged into the form of loss function, so that it can be used directly in the program. The loss functions commonly used in the field of vehicle re-identification include contrastive loss and triplet loss. The advantage of contrastive loss is that it can increase the distances between features of different classes while decrease the distances between features of the same class. Shen et al. Shen et al. (2017) utilized LSTM network to memorize path, and utilized Siamese neural network to regularize similarity scores for robust re-identification performance. Zhu et al. Zhu et al. (2018a) proposed using Siamese neural network structure to simultaneously extract the deep features of input vehicle image pairs under the supervision of joint identification and verification. Zhu et al. Zhu et al. (2018b) proposed a densely connected convolutional neural network for vehicle re-identification, which adopted the structure of Siamese neural network and included two deep feature learning branches with shared parameters, which effectively improved the feature learning ability. The disadvantage of contrastive loss is that, for an anchor, a positive pair and a negative pair are randomly selected for the batch, which weakens the training mechanism and the training speed is particularly slow. The advantage of triplet loss is that it has a hard mining mechanism which speeds up the training speed. Bai et al. Bai et al. (2018) proposed that in triplet network learning, by adding an intermediate representation "group" between the sample and each vehicle to model the intra-class variance, divided the samples of each vehicle into several groups, and establish multi granularity triplet samples between different vehicles and different groups in the same vehicle to learn fine-grained features. Kumar et al. Kuma et al. (2019) used triplet loss to solve the problem of vehicle re-identification, and introduced the formal evaluation of triplet sampling variants (batch samples) into the re-identification task. Lou et al. Lou et al. (2019a) proposed coupling re-id model to feature distance adversarial network, and designed a new feature distance adversarial scheme to generate hard negative samples online in feature space. Ghosh et al. Ghosh et al. (2021) introduced relationship preserving triplet mining (RPTM), which is a triplet mining scheme guided by feature matching to ensure that triples respect the natural subgroups in the object ID, and used this triplet mining mechanism to establish vehicle pose estimation to form a triplet cost function. The disadvantage of triplet loss is that for each anchor, there is only one positive sample and one negative sample, while ignoring other positive and negative samples. For the supervised contrastive loss, for each anchor, all the positive and all the negative samples in the batch were used. Huynh et al. Huynh (2021) formed a strong baseline by applying the supervised contrastive loss and network with multi head method for the field of vehicle re-identification.
vehicle re-identification based on view-aware methods
Compared with person re-identification, the biggest challenge of vehicle reidentification is that the appearance of vehicles change greatly with the change of viewpoint. In order to overcome this difficulty, scholars have proposed many view-aware based methods. The keypoint based attention model can judge the orientation information according to the keypoint information. Wang et al. Wang et al. (2017), Khorramshahi et al. Khorramshahi et al. (2019) and Zheng et al. Zheng et al. (2021) extracted local region features in different orientations based on the location of keypoints, and combines the global fea-tures to form orientation invariant features. However, the above methods required expensive keypoint annotations and were difficult to implement when the two vehicle images don't have any common visible area. So an effective approach is to employ a GAN network to generate vehicle images in invisible view. ; and Pan et al. Pan et al. (2020) utilized convolutional neural networks (CNN) and long short-term memory (LSTM) to learn the transition between different vehicle viewpoints, and can infer the vehicle features containing all view information from one view. Zhou et al. Zhou and Shao (2017) and Lou et al. Lou et al. (2019b) designed cross-view generative adversarial networks to efficiently infer cross-view images, combining the features of the original images with the features of the generated cross-view images to learn vehicle re-identification distance metric. However, at present, the generative adversarial network works have poor effect on objects with large parallax changes such as vehicles. The view-aware based embedding network can use the view label information to narrow the distance between features of images of the same vehicle with different views, while pushing the distance between features of images of different vehicles with the same view. Wang et al. Wang et al. (2021), Chu et al. Chu et al. (2019) and sun et al. Sun et al. (2020) designed different branches to extract the features of different viewpoints (or spaces) respectively, and use the triplet loss to narrow the distance between the features of images of the same vehicle, and simultaneously increase the distance between different vehicles. Teng et al. Teng et al. (2020) designed a multi-view and multi-branch network, each branch learned the features of each viewpoint, and combined a spatial attention model to enhance the discriminativity of features. Zhu et al. Zhu et al. (2020) first extracted the vehicle features and orientation features of the vehicle image, then the distance between orientation features was subtracted from the distance between vehicle features, so as to reduce the difference caused by orientation. Chen et al. Chen et al. (2020b) proposed a special Semantics-guided Part Attention Network (SPAN) to robustly predict the part attention masks for different views of vehicles, so as to achieve the purpose of adaptive view aware. Li et al. Li et al. (2020) generated potential view labels through clustering and considers view information to improve vehicle re-identification performance. Jin et al. Jin et al. (2021) first introduced several potential view clusters for a vehicle to simulate potential multi-view information. Each view cluster had a learnable center.
The main idea of the proposed method
In this section, we will introduce the idea of L GSupCon and VABPP.
The main idea of L GSupCon
The proposed L GSupCon has different properties from the traditional supervised contrastive loss (L SupCon ). L SupCon is essentially a local function related to batch samples. The positives and negatives of an anchor only come from the batch, so in this paper, the L SupCon is called local supervised contrastive loss, and can be denoted as L LSupCon . While in our proposed L GSupCon , by changing the gradient properties of the positive and negative features, the positives and negatives of each anchor can come from the entire training set, so that the entire training set contributes to the training of the anchor, thus greatly enhancing the global property of L GSupCon , which makes the training process easier to converge to a global stable point. In L LSupCon , the parameter optimization process makes each anchor feature move to the local optimal point in the batch. While in L GSupCon , the parameter optimization process makes each anchor feature move to the global optimal point in the entire training set, the anchor is close to the positive features of the entire training set and is also far from the negative features of the entire training set.
Reference Khosla et al. (2020); Chen et al. (2020a); He et al. (2020) has explained that increasing the negative IDs can improve the performance. If we want to increase the negative images in L LSupCon , we can only achieve this by increasing the batch size. During forward and backward of training, each layer of the deep model will store gradients generated related to the newly added batch samples, which will occupy a lot of memory and make the program unable to run. Masters et al. Masters and Luschi (2018) and Ge et al. Ge et al. (2015) have explained that, a smaller batch size can make the model better jump out of the local optimal solution during training, so increasing batch size is not conducive to model convergence. But if we increase the number of positives and negatives in the proposed L GSupCon , by doing this, we don't increase the batch size, and the program will only increase a small amount of memory to store these increased features, these memory increments have no relationship with the depth of the model, because we remove all the gradient attributes of all positive features and all negative features, during training process, the program will not allocate memory to store the gradients corresponding to these newly added features.
The main idea of VABPP
The ultimate purpose of the VABPP method is to reduce the matching difference caused by different views during the testing process, so in the following sections, all the introductions about this method, except the calculation of the view-pair distance scaling coefficient matrix ∆ V ×V in section 4.3 is under the training set, the rest are all based on the test set. The key step of the VABPP method is to find out the center of the distances between features of all positive images with the same view and the feature of the query image, for this query image, there is a maximum of V such centers, where V is the number of views of the dataset, and then scale all such centers to the same position with corresponding scaling coefficients, then find out all images in the gallery set with the view corresponding to each center, and scale the distances between the features of these images and the feature of the query image with the corresponding scaling coefficient.
In Fig. 2, we can see in the matching results, first, all positive images with the same view have a high probability of being compactly ranked together, second, all positive images with different views are probably not cross-ranked, and thirdly, the top-ranked false matches and the top-ranked correct matches have a high probability of having the same view as the query image (such as the front view in Fig. 2), the reason is that images with this view are easier to match for query images
The VABPP method hopes that in all correct matches, images with different views are cross-ranked. If the number of images of different views of a vehicle is the same, ideally, in the same interval, it is hoped that the same number of images of each view of this vehicle are included.
In order to achieve this purpose, we need to scale the distances between the features of the images with different views and the feature of query image with different scaling coefficients (i.e., multiply the distance by a scaling coefficient, for example, d = 0.5d means to change the distance to 0.5 times of the original distance, 0.5 is the scaling coefficient), and the scaling coefficients of the distances between the features of the images with the same view and the feature of the query image is the same (that is, the rankings between all images with the same view are unchanged), if all the positive images with a certain view are ranked relatively low (such as the rear view images in Fig. 2), then the rankings of all images with this view are moved forward as a whole (Fig. 3)). It is reflected in Fig. 2 that the distances between the features of the images with rear views and the feature of the query image is shrunk, so that all images with rear view are moved forward. It is easy to understand that the more difficult to match the images of a certain view, the lower the rankings (such as the rear view images in Fig. 2), the closer the required scaling coefficient to 0 to moved these images to specified position (scaling coefficient is usually a positive value greater than 0 and less than 1 ).And we stipulate that the distances between the features of the images with the same view as the query image and the feature of the query image is not scaled, that is, the scaling coefficient is 1.
The advantage of this is that the scaling coefficient of the distance between the feature of a negative image which is easy to be wrong matched and the feature of the query image is usually larger (closer to 1), while the scaling coefficient of the distance between the feature of a positive image that is difficult to match correctly and the feature of the query image is smaller (closer to 0), so after applying the VABPP method, the rankings of the positives that are difficult to match correctly move more forward than the rankings of the negative images that are more likely to be incorrectly matched, which improves the test performance (Fig. 3). ).
Traditional view-aware-based vehicle re-identification methods are used in the training process, so that these methods are difficult integrated into other non-view-aware-based methods. The advantages of the VABPP are that, first, Fig. 3: The effect of VABPP. The purpose of VABPP is to make c 1 , c 2 and c 3 on the left, after scaling, become c 1 , c 2 and c 3 on the right, and c 1 =c 2 =c 3 = c 1 , that is, all the distances in the above three lines, are scaled according to the formula d i = d i c1 ci (i = 1, 2, 3). Where c i and c i (i = 1, 2, 3) represents the average distances of the four green points in the i-th row on the left and the right, respectively, where the green points and red points represent the distances between the positive features and the query feature, and distances between the negative features and the query feature, respectively. The query image has view v 1 , The first row, the second row and the third row respectively represent the distance between the features all the images in the gallery set with views v 1 , v 2 and v 3 (v 1 , v 2 and v 3 are unequal to each other) and the feature of query image and the fourth row represents the schematic diagram of the distances from the first row, the second row, and the third row put together. Since the images represented in the first row and the query image have the same view, the distances of the first row do not change. The left and right represent the situation before and after applying the VABPP method, respectively. It can be seen that the fourth row on the right (compared to the left) has clearly separated the positives and negatives.
it does not affect the training process, and second, the VABPP can be easily integrated into other methods.
Proposed method
Below we introduce the details of the method proposed in this paper. We first use the proposed L GSupCon to train the re-id model, and then apply the VABPP on the trained model. Therefore, we first introduce L GSupCon , then introduce the details of VABPP, and then introduce the steps of integrating the VABPP method into other arbitrary trained model. Fig. 4 is a framework of the entire proposed method.
Global Supervised Contrastive loss
The formula of L LSupCon is as follows:
L SupCon = i∈I −1 |P (i)| p∈P (i)⊂I log exp(f i f p /τ ) a∈A(i)⊂I exp(f i f a /τ )(1)
Fig. 4: Our pipeline. Our pipeline includes the losses module and the viewaware-based post-processing module. The losses module shows the pipeline of our proposed L GSupCon method. The view-aware-based post-processing module shows the pipeline of our proposed VABPP method. The black arrow in the figure represents the training of the re-id network, and the blue arrow represents the calculation process of the view-pair distance scaling coefficient matrix ∆ V ×V . The red arrow represents the application of VABPP method in the test set. l, m and n in the figure represent the number of images in the training set, query set and gallery set respectively. '.' represents the distance and '*' represents the view-pair distance scaling coefficient.
Where I is the set of all images in the batch, A(i) = I\{i}, P (i) ≡ {p ∈ A(i) : y p = y i } is the set of all images in the batch that are different from image i and have the same vehicle ID as image i. y p and y i represent the ground truth labels of images p and i, respectively. |P (i)| is the cardinality. τ is a scalar temperature parameter. f i is an anchor feature, f p is a positive feature and f a is a feature of any image in the batch that is different from f i . The formula of our proposed L GSupCon is as follows:
L GSupCon = i∈I −1 | P (i)| p∈ P (i) log exp(f i f p /τ ) a∈T exp(f i f a /τ )(2)
where T is the training set, and P (i) ≡ {p ∈ T : y p = y i } is the set of all positive images of i in the training set. f i ∈ I is the anchor feature which has gradient attribute (i.e. the ∂L GSupCon ∂fi is generally not zero). f p is a positive feature without gradient attribute (i.e.
∂L GSupCon ∂ fi is always zero), f a is feature of a without gradient attribute, where a ∈ T , in the training process, we use a global dictionary to store all global features , and during each iteration, we use The difference between L LSupCon and L GSupCon . In L LSupCon , positive features and negative features have derivative property, while in L GSupCon , they do not have derivative property. L LSupCon is a local-to-local loss with two-way movement, while L GSupCon is a local-to-global loss with one-way movement.
the features in the batch to update the corresponding features in the global dictionary.
The parameter optimization process of L LSupCon has the property of Localto-Local two-way movement between features. That is, the parameter optimization process will make the anchor feature close to/away from the positive/negative features, but meanwhile, the positive/negative features are also close to/away from the anchor feature, and all these movements are conducted within batch features. The parameter optimization process of L GSupCon has the property of Local-to-Global one-way movement of only the feature in the batch (anchor) close to/away from the positive/negative features in the entire training set (Fig. 5). This can be observed by the gradients of L LSupCon and L GSupCon with respect to f i in equations 1 and 2, respectively (see appendix A for the solutions process of these two gradients):
∂L GSupCon ∂f i = 1 τ { p∈ P (i) f p [ Γ ip − 1 | P (i)| ] + n∈ N (i) f n Γ in } (3) ∂L LSupCon ∂f i = 1 τ { p∈P (i) f p [Γ ip − 1 |P (i)| ] + n∈N (i) f n Γ in } + 1 τ { p∈P (i) f p [Γ pi − 1 |P (p)| ] + n∈N (i) f n Γ ni } (4) Where Γ xy = exp(fx fy/τ ) a∈T exp(fx fa/τ ) and Γ xy = exp(fxfy/τ ) a∈A(x) exp(fxfa/τ ) .3, Γ ip − 1 | P (i)| → 0 and Γ in → 0
show that in the L GSupCon , the anchor feature is close to all the positive features of the entire training set and away from all the negative features of the entire training set. In equation 4, Γ ip − 1 |P (i)| → 0 and Γ in → 0 show that in the L LSupCon , the anchor feature is close to all the positive features in the batch and is far away from all the negative features in the batch. Meanwhile, Γ pi − 1 |P (p)| → 0 and Γ ni → 0 show that all the positive features in the batch are close to the anchor feature and all the negative features in the batch are far away from the anchor feature.
L GSupCon VS L LSupCon (1) The purpose of L GSupCon is to improve the performance by increasing the number of negative IDs, so generally speaking, the larger the training set, the better the performance.
(2) L LSupCon has the property of Local-to-Local two-way movement and L GSupCon has the property of Local-to-Global one-way movement. If L LSupCon and L GSupCon are combined together to train the model, this Local-to-Local two-way movement mode and Local-to-Global one-way movement mode will promote each other when training with a smaller training set, while L LSupCon will restrict the performance of L GSupCon when training with a larger training set.
Since the proposed L GSupCon is better for larger training set, we use the following weighted loss to train the re-id model for the dataset with a larger training set:
L = λ ID L ID + λ M etric L GSupCon(5)
The same as our baseline, we use the label smooth cross entropy loss as our ID loss, i.e. L ID in formula 5 is ID loss. And in formula 5, the same as our baseline, λ ID and λ M etric are calculated by Momentum Adaptive Loss Weight method Huynh (2021); Yu et al. (2021). While for smaller training set, we use the following weighted loss to train the re-id model:
L = λ ID L ID + λ M etric L GSupCon + λ M etric L LSupCon(6)
Overview of the VABPP method
After training the re-id model, we can use the VABPP method during testing process. The VABPP method can play the best performance in the steady state, and the more stable the state, the better the performance of the VABPP method. The steady state here refers to the statistical aspect, that is, the more data used for statistics, the better the performance of VABPP. We assume that the number of images per view is large enough for any vehicle and that most vehicles have images of all V views, here V is the total number of views in the dataset, so we make the following assumptions:
p∈P vs x 1 |P vs x | d(f x , f p ) k v,vs ≈ p∈P v t y 1 |P v t y | d(f y , f p ) k v,vt(7)
Where x and y are arbitrary two images with the same view v (also arbitrary), and f x , f y and f p are the features of x, y and p, respectively. v s and v t are any two views. P vs x and P vt y are the sets of vehicle images with the same vehicle ID as x and y respectively, and the views of the images in these two sets are v s and v t respectively. And |·| represents the cardinality. (i, j) represents a positive view pair when the view of the query image is i and the view of the positive image is j, k i,j ∈ K V ×V is used to measure the difficulty of matching the positive image with the query image when the view of the query image is i and the view of the positive image is j. If the matching is relatively easy (i.e. the distance between the query image feature and the positive feature is small), the k i,j is also relatively small, on the contrary, the k i,j is relatively large. we stipulate k i,i = 1, i = 1, 2, . . . , V .
In formula 7, let v s = v t , then the following formula is obtained:
p∈P vs x 1 |P vs x | d(f x , f p ) ≈ p∈P vs y 1 |P vs y | d(f y , f p )(8)
Since x and y have the same view, but not necessarily the same vehicle ID, so we have the following inference : We
denote c(V(x), v s ) = p∈P vs x 1 |P vs x | d(f x , f p ) as the distance center of the view pair (V(x), v s ), where V(x) represents the function of finding the view ID of image x. c(i, j) ∈ C V ×V , C V ×V is called view-pair distance center matrix.
Then let v = v t = i and v s = j in formula 8, we get the following formula:
c(i, i) c(i, j) ≈ 1 k(i, j)(9)
Denote
δ(i, j) = 1 k(i, j) ≈ c(i, i) c(i, j)(10)
We call δ(i, j) the view-pair distance scaling coefficient when the view of the query image is i and the view of the candidate image is j. Here δ(i, j) ∈ ∆ V ×V , and ∆ V ×V is the view-pair distance scaling coefficient matrix. Similarly there are δ(i, i) = 1, i = 1, 2, . . . , V . So far, according to formula 10, we can get the following formula:
c(i, i) ≈ c(i, 1) · δ(i, 1) ≈ c(i, 2) · δ(i, 2) ≈ · · · ≈ c(i, V ) · δ(i, V )(11)
This formula shows that δ(i, j) can scale c(i, j) to almost the same point as
c(i, i), (i, j = 1, 2, ...V ).
The inference process of formula 11 is based on the test set, but the key problem here is that the test set cannot participate in the calculation of the matrix ∆ V ×V , so we use the training set to calculate ∆ V ×V , and finally apply it to the test set. The Fig 6 shows the flow of the VABPP method. As can be seen in the Fig. 6(a), the VABPP method first uses the training set to calculate the viewpair distance center matrix C V ×V , and then uses C V ×V to find ∆ V ×V . The VABPP method can then be integrated into any other trained re-id model by directly using the matrix ∆ V ×V calculated in Fig. 6(a). As can be seen in the Fig. 6(b), if the VABPP method is directly applied to any other trained re-id model, we not only need the matrix ∆ V ×V , but also a orientation extraction model to extract the orientation labels of all the test set images. So we provide the training set orientation labels of all the three datasets 1 . We believe that if the orientation extraction model is not trained with the orientation labels we provided, the experimental results may be different and ∆ V ×V needs to be recalculated. Therefore, the rest of this section will first introduce the calculation process of ∆ V ×V , and then introduce how to directly apply the VABPP method into other methods.
The calculation of view-pair distance scaling coefficient matrix ∆ V ×V
Because the test set can not participate in the calculation of ∆ V ×V , we use the training set to calculate ∆ V ×V , and finally apply ∆ V ×V to the test set during the test process. Therefore, all the calculation of this section is based on the training set. As shown in Fig. 4, the specific method we calculate ∆ V ×V is as follows. Firstly we input all the images of the training set T = {t 1 , t 2 , . . . , t l } (l is the total number of images in the training set) into our trained re-id network to extract features, then we obtain the feature set
F T = {f T 1 , f T 2 , .
. . , f T l } of the training set. However, the training set is not divided into query set Q and gallery set G, so let Q = T and G = T . Then their corresponding feature sets are F Q = F T , and F G = F T . For each query image q ∈ Q, we calculate the distances between the feature of q and all its positive features in F G , and use these distances to calculate the view-pair distance center matrix C V ×V . Finally, we use C V ×V to calculate ∆ V ×V . Here, the calculation formula of element c(i, j) of C V ×V is as follows:
c i,j = q∈Q ( i) g∈G(j|q) d(f q , f g ) q∈Q(i) g∈G(j|q) 1(12)
Here, Q(i) = {q|q ∈ Q, V(q) = i} is the set of all images with view i in Q, G(j|q) = {g|y(g) = y(q), V(g) = j, Camid(g) = Camid(q)} is a set of all images in G that have the same vehicle ID as the query image q and have view j, but all images with the same camera as q are removed.f q and f g represent the features of image q and g, respectively. According to formula 10, the calculation formula of element δ(i, j)(i, j = 1, 2, . . . , V ) of is as follows:
δ i,j = 1, i = j c i,j /c i,i , i = j(13)
We can know from formula 13 that ∆ V ×V is a matrix whose main diagonal elements are all 1, indicating that before and after the application of VABPP method the distances between the features of the images with the same view as the query image (i.e. i = j) and the feature of the query image does not change.
Modify test distance
After calculating ∆ V ×V from the training set, we are not in a hurry to use it to modify the test distance (MTD), because the test set and training set have different distance distributions and we must unify the distance distributions between the test set and the training set. Fig. 7 shows the difference between the distance distribution of the training set and the test set of our baseline. As can be seen from Fig. 7, the distance distribution of the test set (i.e. d test ) is a smooth convex curve, and the distance distribution of the training set (i.e. d training ) is a curve from concave to convex. The reason is that baseline uses L LSupCon to pull the positive features closely and push the negative features away during training, but there is no such mechanism during testing. Since it can be observe from formula 7 we are more concerned about the positive features, in order to make the distances between the positive features and query features in the test set have similar distance distribution with the distances between the positive features and anchor features in the training set, in the test set, for any q ∈ Q and g ∈ G, we make the following distance modifications:
d unif ied (f q , f g |θ) = d γ (f q , f g |θ)(14)
Where γ is a hyper exponential parameter. d(f q , f g |θ) ∈ D m×n test , D m×n test is the matrix composed of the distance between the features of all images in the query set Q and the features of all the images in the gallery set G, d unif ied (f q , f g |θ) ∈ Fig. 7: Different distance distributions between training set and test set. All curves are drawn according to the formula d(r) = 1 |Q| q∈Q d α sorted (r|q). Where α is a hyper exponential parameter and Q is the query set, d sorted (r|q) is the r-th value of the distance between the query image q and all the images in gallery set G in ascending order. For the drawing of curve d training , we set Q = G = T , where T is the training set. All these curves are drawn under the condition of removing the images with the same ID and same camera as the query image q in the gallery set. It is obvious that d training , d 4 test and d 6 test are concave curves, while d test and d 2 test are convex curves.
D m×n unif ied , D m×n unif ied is the distance matrix composed of all elements of D m×n test applied by the UDD method. We do this for the following two reasons. First, by observing the distances distributions in the baseline (Fig. 7) , we find that the distances between positive pairs is far less than 1 in both the training set and the test set. Second, the distance between the positive pairs in the test set is generally larger than that in the training set. The value of γ can change the bending degree and bending direction of the distance curves in the test set.
Note: in Fig. 7, it seems that the curve of d 6 test is the most similar to that of d training , but the fact is not necessarily when γ = 6 is the best. The reasons are as follows:
(1) Several loss functions are used in the training of the training set, and the training accuracy is very high. Almost all the top ranking images are positive images.
(2) When drawing these curves, the distances between the feature of each query image and the features of all images in the gallery are sorted from small to large, then all such distances calculated by all query images are averaged at the corresponding position. However, the test accuracy is not as high as the training accuracy, and the test distances are far larger than the training distances. Therefore, the n-th power in Fig. 7 will be disturbed by the larger value.
(3) The number of positive images in the test set is different from that in the training set, so that the abscissa in Fig. 7 cannot be aligned in the test set and training set. Now, we start to modify the test distance. In Section 3.2, we analyze that for a query image q with view i, δ(i, j) can scale c(i, j) to the same point as c(i, i)(i, j = 1, 2, . . . , V ). But what we need is an overall scaling of the distances between all gallery image features with view j and the query image, so we need to modify all test distances. The method of modifying test distance is abbreviated as MTD. The goal of the MTD method is to use the following distance formula to conduct the final re-id similarity matching:
d scaled (f q , f g |θ) = d unif ied (f q , f g |θ) × δ(V(q), V(g))(15)
Where, q ∈ Q and g ∈ G, Q = {q 1 , q 2 , . . . , q m } is the query set, G = {g 1 , g 2 , . . . , g n } is the gallery set, and θ is the model parameter. d scaled (f q , f g |θ) ∈ D m×n scaled . D m×n scaled is a distance matrix composed of all elements of D m×n unif ied applied by the UDD method.
Steps for integrating VABPP into other trained re-id model
Using the calculated in this paper and the orientation labels of the training set given in this paper, the steps of integrating VABPP method into other methods are as follows:
Step1. Train a re-id model and a orientation extraction network with specific methods
Step2. Extract the vehicle features and view labels of the test set.
Step3. Normalize vehicle features and calculate the distances between all query image features and image features of all gallery sets to form a distance matrix D m×n test , where m and n represent the number of images in the query set and gallery set respectively.
Step4. Use formula 16 to unify the test distance distribution and training distance distribution to obtain the unified distance matrix D m×n unif ied of the test set.
Step5. Use the extracted test set view labels and ∆ V ×V to calculate the view-pair distance scaling coefficient matrix ∆ m×n test between all images of the query set and images of gallery sets.
Step6. Use D m×n unif ied • ∆ m×n test to replace the original D m×n test for similarity matching.
Experiments
Datasets
VeRi-776 dataset There are 20 cameras were used to take images of this dataset, and images of each vehicle are taken by 2-18 cameras. The training set contains 37778 images of 576 vehicles. The query set and gallery set contain the same 200 vehicle IDs, and the number of images are 1678 and 11579 respectively. There are eight orientations in this dataset, including front, rear, left, right, left front, left rear, right front and right rear.
VehicleID dataset This dataset has no camera information, and all images are taken from the front or rear. The training set consists of 113346 images of 13164 vehicles. The test set is divided into three sub sets: large, medium and small, including 19777, 13377 and 6493 images of 2400, 1600 and 800 vehicles respectively. Randomly select one image from each vehicle in these subsets to form the corresponding gallery set, and the rest form the corresponding query set.
VERI Wild dataset There are 174 cameras were used to take images of this dataset. The training set contains 277797 images of 30671 vehicles. The test set is divided into three subsets: large, medium and small. The query set of the three subsets contains 3000, 5000 and 10000 vehicle images (one image per vehicle), and the gallery set contains 38861, 64389 and 128517 images respectively. This dataset includes six orientations: front, rear, left front, left rear, right front and right rear.
Training configurations
We use Huynh Huynh (2021) as our baseline and resnext101 ibn a as the backbone. But we removed the mixstyle module of the baseline. We use the Momentum Adaptive Loss Weight method introduced in references Huynh (2021); Yu et al. (2021) to update the weights λ ID and λ M etrix . We resize the image size to 320× 320, and apply data enhancement methods such as color jitters, random flip, brightness and contrast adjustment, random erase and random cropping. The batch size is 64. We use ADAM optimizer with the cosine annealing scheduler, total training epoch is set to 24. For the VeRi-776 dataset, the batch is composed of 8 identities, each identity contains 8 images, and the initial learning rate is set to 3.5× 10-4. For the VehicleID dataset, the batch is composed of 16 identities, each identity contains 4 images, and the initial learning rate is set to 3.5× 10-5. For VERI Wild dataset, the batch is composed of 32 identities, each identity contains 2 images, and the initial learning rate is set to 10-4. The program is implemented on pytorch, and use a single NVIDIA GeForce RTX 3090 GPU.
Relationship between L LSupCon and L GSupCon
As mentioned in section 4.1, L GSupCon is better for larger datasets. For smaller datasets, we need to combine L LSupCon and L GSupCon to train the model, we use three datasets of different sizes to verify this conclusion. The VeRi-776 dataset contains 576 training IDs, the VehicleID dataset contains 13164 training IDs, and the VERI Wild dataset contains 30671 training IDs. Further Table 1: In the VeRi-776 dataset, VehicleID dataset and VERI Wild dataset, we use L LSupCon (L L ) and L GSupCon (L G ) and the combination of these two losses, and respectively use resnet50 ibn a,resnext101 ibn a and resnet152 as the backbones. r50, r101 and r152 respectively represent resnet50 ibn a,resnext101 ibn a and resnet152.
Backbone Method
VeRi-776 VehicleID VERI Wild small medium large small medium large mAP(%)r1(%)r1(%)r5(%)r1(%)r5(%)r1(%)r5(%)mAP(%)r1(%)mAP(%)r1(%)mAP(%)r1(%) more, in order to show the generalization of this conclusion, we carry out corresponding experiments on three backbone models: resnet50 ibn a,resnext101 ibn a and resnet152. TABLE 1 shows the experimental results.
VeRi-776 dataset In TABLE 1, we find that in the VeRi-776 dataset, the experimental results of L GSupCon are not significantly better than that of L LSupCon , but the results of L LSupCon + L GSupCon are significantly better than that of any one of the two losses used alone, indicating that the number of training IDs (or images) for this dataset is not sufficient for the proposed L GSupCon to perform well.
VehicleID dataset The number of training IDs in the VehicleID dataset is significantly larger than that in the VeRi-776 dataset. And we can find in TABLE 1, in this dataset, the results of using L GSupCon for the three backbones are better than those using L LSupCon . In the large sub test set, the rank-1 has been improved by 3% for the resnet152 backbone. And the experimental results of L LSupCon + L GSupCon are also significantly better than that of using either of the two losses alone. Compared with baseline (i.e. L LSupCon ), the improvement is more significant, and the rank-1 has been improved by 4.9% for the resnet101 ibn a backbone in the small sub test set, indicating that the number of training IDs (or images) for this dataset is large enough to make the performance of the proposed L GSupCon used alone to exceed that of the L LSupCon used alone.
VERI Wild dataset For this dataset, as the number of training IDs in this dataset reaches 30671, which is the largest in the three dataset, it can be seen that with the increase of the number of positive images and negative IDs, the mAPs of using L GSupCon in the three backbones than those of using L LSupCon (i.e. baseline) with an increase of almost all by 8%. Even in the large sub test set, for resnet101 ibn a backbone, the mAP of using L GSupCon has an increase of more than 10% compared with that of using L LSupCon . For this data set, the results of using L GSupCon is obviously better than those of using L LSupCon + L GSupCon , which shows that the number of training IDs of this dataset is large enough, the L GSupCon 's one-way movement mode which tends to move to the global optimal solution has played a good role, and this shows that when L LSupCon + L GSupCon is used to train the model, the local-to-local two-way movement mode of L LSupCon affects the performance of L GSupCon .
The ablation experiment of view-aware based post processing method
We use resnext101 ibn a as the backbone to conduct ablation experiments on MTD method and on the value of γ for UDD method. For the VeRi-776 dataset and VehicleID dataset, we use L LSupCon + L GSupCon as metric loss, while for the VERI Wild dataset, we only use L GSupCon as the metric loss. As can be seen in TABLE 2, when γ = 2, the result is the best, so the value of γ in subsequent experiments is 2. Since all the curves in Fig. 7 are the results of sorting the distances between the feature of each query image and the features of all images in the gallery from small to large, then all such distances calculated by all query images are averaged at the corresponding positions, while the training set is well trained, but the test set are not trained, although the bending direction and bending degree of the distance distribution curves of d 6 test (i.e. γ = 6) and the distance distribution curve of training set (i.e. d training ) are the closest in Fig. 7, however, this does not mean that the experimental result is the best when γ = 6. Fig. 7 can only show that the distance distributions between the test set and the training set are different. Comparing Fig. 8(b) and Fig. 8(c), we can see that some rankings of the negative images with the same view as the query image (front view), such as images with indexes 7481, 7482, 7454, 7480, have been moved back a lot in Fig. 8(c) compared with Fig. 8(b), because their distance scaling coefficient is the largest (first row of TABLE 5, δ(0, 0) = 1), and the distances between the features of images of other views and the feature of the query image become smaller, but the distances between the features of the images of the front view and the feature of the query image are invariable, so that the rankings of almost all images with front view become more rearward. Meanwhile, there are only 37 correctly matches in Fig. 8(b), but there are a total of 43 correctly matches in Fig. 8(c). It can be seen that the VABPP method is very helpful to improve the experimental accuracy. The main reason why VABPP method is effective is that it can make the rankings of positive images that are difficult to match higher. Fig. 8(d) and Fig. 8(e) show why the VABPP method is effective. It can be seen that although the overall rankings of Fig. 8(d) and Fig. 8(e) is different, the rankings within images of the same color (i.e. the same view) is fixed, and the VABPP method increases the cross-ranking between positives with different views.
(%) r1(%) r1(%) r5(%) r1(%) r5(%) r1(%) r5(%) mAP(%) r1(%) mAP(%) r1(%) mAP(%) r1(%) w/o
Application examples of VABPP method
We list all the ∆ V ×V s matrices calculated by our trained re-id model on the VeRi-776 dataset, VehicleID dataset, and VERI Wild dataset in TABLE 5, TABLE 6 and TABLE 7 in Appendix B, respectively. Recently, the most commonly used baseline in the vehicle re-id field is Bag-of-Tricks Luo et al. (2019). In order to verify that the VABPP method can be easily integrated into other methods, in the three datasets, we apply the VABPP method to the Bag-of-Tricks Luo et al. (2019). For this experiment, we don't need to recalculate ∆ V ×V . For these three datasets, we directly use the corresponding ∆ V ×V in the (2019) + VABPP are much higher than that of without VABPP method in all the three datasets. In the VeRi-776 dataset, the mAP of with VABPP is 1.48% higher than that of without VABPP. In the three sub test sets of the VehicleID dataset, CMC@1s of with VABPP are 3.23%, 5.03% and 4.15% higher than those of without VABPP, respectively. In the three sub test sets of the VERI Wild dataset, mAPs of with VABPP are 3.48%, 3.83% and 4.10% higher than that of without VABPP, respectively. It can be seen that such a great improvement is achieved at the condition of without affecting the training process and without recalculating matrix ∆ V ×V , indicating that the proposed method has a high practical value. The Bag-of-Tricks Luo et al. (2019) is a commonly used baseline before, so we believe that the VABPP method can be easily integrated into other methods by directly using the matrices ∆ V ×V s we calculated. It should be emphasized that in TABLE 2, γ = 2 is the best, while in TABLE 3, it is the best when γ = 4. The reason is that the relationship between the distance distributions of training set and test set are different when the model structures are different or training methods are different, so it is necessarily to re-verify the optimal value of γ. In addition, it is important to ensure that test features are normalized.
By analyzing the results in TABLE 2 and TABLE 3, we can also see that the performances of VABPP in VehicleID dataset and VERI Wild dataset are significantly better than that in VeRi-776 dataset. The reasons are as follows:
1)The total number of training images in the VeRi-776 dataset is 37778, and there are 64 values need to be counted (TABLE 5). While the total number of training images in the VehicleID dataset and VERI Wild dataset are 113346 and 277797 respectively, there are only 4 (TABLE 6) and 36 (TABLE 7) values need to be counted respectively, which makes the statistical values in the later two datasets are more general.
2)Although we divid the VeRi-776 dataset into 8 views, in fact, this dataset is far more than 8 views, because this dataset can be divided into 8 views in the horizontal position and can also be continuously divided in the vertical position. The reason is that some of the cameras in this dataset are in high positions and some are in low positions, which makes the division of only 8 views is significantly too few. The reason why we no longer continue to make more detailed division is that the total number of images in this dataset is too small, and the calculation of the ∆ V ×V matrix needs to be done in viewpairs of images within the same vehicle ID. If we continue to make detailed division, due to the small size of VeRi-776 dataset, the ∆ V ×V will be even less statistically significant. While the VehicleID dataset and VERI Wild dataset have only two and six orientations respectively, and all the cameras used to capture these two datasets are almost at the same horizontal position. 5.7 Compare with the state-of-the-art methods
We compare the proposed methods with some state-of-the-art methods in the VeRi-776 dataset, VehicleID dataset and VERI Wild dataset. These methods include:
(1) View-aware based method: include VVAER Khorramshahi et al. (2019), PCRNet Liu et al. (2020), VARID Li et al. (2020), VSCR Teng et al. (2021) and VAT Yu et al. (2022) (2) Metric learning based method: include GSTE Bai et al. (2018), FDA-Net Lou et al. (2019a), SAVER Khorramshahi et al. (2020), HRCN Zhao et al. (2021b), LCDNet+BRL+RR Fu et al. (2022).
As can be seen from TABLE 4, in all three datasets, our method is almost the best compared with those view-aware based methods or those metric learning based methods, and our method is also the best compared with the two papers published in 2022 (i.e. VAT Yu et al. (2022) and LCDNet+BRL+RR Fu et al. (2022)).
Discussion
Compared with the traditional supervised contrastive loss, the proposed globalsupervised contrastive loss has the advantage that in each iteration, the features of the entire training set can be used, but the disadvantage is that it will occupy more memory, but these extra memory has no relationship with the number of layers of the deep model, so it generally does not affect the running of the program. The proposed global-supervised contrastive loss performs better in datasets with larger training sets, while for smaller training sets, it needs to be combined with traditional supervised contrastive loss.
The advantages of the VABPP are that it is used during testing without affecting the training process, and the view pair distance scaling coefficient matrix ∆ V ×V s provided by us can be used directly, which makes our proposed VABPP method can be easy integrated into other methods. This method also has two shortcomings. One is that it needs to label the orientation information of the training set, and use the orientation information to train an orientation extraction network to extract test set image orientations. Second, since the view-pair distance scaling coefficient matrix ∆ V ×V is calculated by counting elements of the distance matrix between all training set features, the larger the dataset, the performance will be the better, because the statistical significance of the smaller data set is relatively smaller.
Summary
In this paper, we propose a global-supervised contrastive loss and a viewaware-based post-processing method to address two challenges in the field of vehicle re-id. The global-supervised contrastive loss has a good effect on the training set with a large number of training IDs. We verify this conclusion by using three different backbones in three datasets widely used in vehicle reidentification. The view-aware-based post-processing method does not affect the training process, because it is only used in testing. We provide the orientation labels of the training set of the three datasets, and also provide the calculated view-pair distance scaling coefficient matrices of the three datasets, which makes it easy to integrate the VABPP method into other methods, and we use experiments to integrate VABPP into the Bag-of-Tricks Luo et al.
(2019) as a baseline commonly used in the field of vehicle re-identification, which verifies the feasibility of this method. .
L GSupCon = i∈I −1 | P (i)| p∈ P (i) log exp(fi fp/τ ) a∈T exp(fi fa/τ ) = −1 | P (i)| p∈ P (i) log exp(fi fp/τ ) a∈T exp(fi fa/τ ) + k∈I,k =i −1 | P (k)| p∈ P (k) log exp(z k fp/τ ) a∈T exp(z k fa/τ ) = −1 | P (i)| p∈ P (i) {f i f p /τ − log[ a∈T exp(f i f a /τ )]} + k∈I,k =i −1 | P (k)| p∈ P (k) log exp(z k fp/τ ) a∈T exp(z k fa/τ ) = −fi τ | P (i)| p∈ P (i) f p − log[ a∈T exp(f i f a /τ )] + k∈I,k =i −1 | P (k)| p∈ P (k) log exp(z k fp/τ ) a∈T exp(z k fa/τ ) (16) L LSupCon = i∈I −1 |P (i)| p∈P (i) log exp(f i fp/τ ) a∈A(i) exp(f i fa/τ ) = −1 |P (i)| p∈P (i)
Appendix B: The universal ∆ V ×V s of the three datasets
The ∆ V ×V s in TABLEs 5 and 6 are obtained by training with L GSupCon + L LSupCon , while the ∆ V ×V in TABLE 7 is obtained by training with L GSupCon alone. Take the first row in TABLE 5 as an example to understand the meaning of the whole table. Let q ∈ Q and g ∈ G, where V(q) = 0. If V(g) = 0, because δ(0, 0) = 1, so in formula 15, we have d scaled (f q , f q |θ) = d unif ied (f q , f q |θ)×1. If V(g) = 1, because δ(0, 1) = 0.8930, so in formula 15, we have d scaled (f q , f q |θ) = d unif ied (f q , f q |θ) × 0.8930, and so on. We can see that the larger the value of δ(i, j), the easier query image with view i to be matched with the images in the gallery set with the view j. Generally speaking, the images with the same view is easier to match. We can note that in TABLE 5, we found that when the query image q is with the right view (the penultimate row), all the values are bigger than 1 except δ(5, 5) = 1, this situation is the opposite of the other rows. The reason is that there are too few images with the right view in the VeRi-776 dataset, and the statistics are not general.
Zhu, X., Luo, Z., Fu, P., and Ji, X. (2020). Voc-reid: Vehicle re-identification based on vehicle-orientation-camera. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 602-603.
Fig. 1 :
1Triplet loss, L SupCon and L GSupCon . The green dotted line represents samples in a batch, and the purple line represents the entire training set. Triplet loss only uses a positive and a negative in the batch. The L SupCon considers all positives and all negatives in the batch. The L GSupCon considers all positives and all negatives in the entire training set.
Fig. 2 :
2A matching result of the baseline model. The matching result is sorted in ascending order of the Euclidean distance between the features of the query image and the features of the images in the gallery set.
Fig. 5 :
5Fig. 5: The difference between L LSupCon and L GSupCon . In L LSupCon , positive features and negative features have derivative property, while in L GSupCon , they do not have derivative property. L LSupCon is a local-to-local loss with two-way movement, while L GSupCon is a local-to-global loss with one-way movement.
Inference 1 :
1If vehicle image x and y have the same view, then the average of the distances between features of all positives of x with view v and the feature of x and the average of the distances between features of all positives of y with view v and the feature of y are almost equal.
Fig. 6 :
6The flow of the proposed VABPP method.
Fig. 8 :
8Demonstration of the effect of the proposed method. The query images in (a), (b), (c), (d) and (e) are the same image. The number in each box represents the original index of the image in this box in the gallery set.
The training should converge, so we have finally we have each term of equations 3 and 4 should tend to 0. In equation∂L GSupCon
∂fi
→ 0 and
∂L LSupCon
∂fi
→
0,
Table 2 :
2The ablation experiments of VABPP method in the three datasets.Method
VeRi-776
VehicleID
VERI Wild
small
medium
large
small
medium
large
mAP
Table 3 :
3In Bag-of-TricksLuo et al. (2019), the comparison of before and after applying VABPP method. The ∆ V ×V in TABLEs 5, 6 and 7 in the Appendix B are directly used here. " †" in the table indicates the VABPP method is used, where γ = 4.Dataset
mAP
rank-1
rank-5
VeRi-776
77.1
95.4
98.2
VeRi-776 †
78.6
95.3
98.2
VehicleID
Small
89.0
83.6
96.0
Small †
92.3
87.8
97.9
Medium
84.8
78.7
93.2
Medium †
89.8
85.0
96.0
Large
83.6
77.7
91.6
Large †
87.8
82.6
94.4
VERI Wild
Small
77.0
92.1
97.5
Small †
80.5
92.3
97.6
Medium
70.9
89.4
95.8
Medium †
74.7
89.6
96.0
Large
62.6
85.3
93.4
Large †
66.7
85.5
93.6
Table 4 :
4Comparison between the proposed method and the state-of-the-art
methods in the three datasets. m(%) meas mAP(%).
TABLE 5 ,
5TABLE 6 and TABLE 7. The experimental results of Bagof-Tricks Luo et al. (2019) and Bag-of-Tricks Luo et al. (2019)+VABPP are shown in TABLE 3. As can be seen from TABLE 3, all the results of Bagof-Tricks Luo et al.
Table 5 :
5The universal ∆ V ×V in VeRi-776 dataset. The number of 0,1,2,3,4,5,6 and 7 represent front, rear, left, front left, rear left, right, front right and rear right, respectively.
Table 6 :
6The universal ∆ V ×V in VehicleID dataset. The number of 0 and 1 represent front and rear, respectively.0
1
0
1
0.4597
1 0.6455
1
Table 7 :
7The universal ∆ V ×V inVERI Wild dataset. The number of 0,1,2,3,4 and 5 represent front, rear, rear right, front left, rear left and front right.0
1
2
3
4
5
0
1
0.6556
0.6521
0.8721
0.6676
0.9675
1
0.6834
1
0.9303
0.6932
0.9146
0.6887
2
0.6658
0.9112
1
0.6982
0.9687
0.6972
3
0.8653
0.6597
0.6784
1
0.7247
0.9390
4
0.6203
0.8152
0.8815
0.6786
1
0.6383
5
0.9478
0.6473
0.6690
0.9272
0.6731
1
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| [] |
[
"On Index Coding in Noisy Broadcast Channels with Receiver Message Side Information",
"On Index Coding in Noisy Broadcast Channels with Receiver Message Side Information"
] | [
"Behzad Asadi ",
"Lawrence Ong ",
"Sarah J Johnson "
] | [] | [] | This letter investigates the role of index coding in the capacity of AWGN broadcast channels with receiver message side information. We first show that index coding is unnecessary where there are two receivers; multiplexing coding and superposition coding are sufficient to achieve the capacity region. We next show that, for more than two receivers, multiplexing coding and superposition coding alone can be suboptimal. We give an example where these two coding schemes alone cannot achieve the capacity region, but adding index coding can. This demonstrates that, in contrast to the two-receiver case, multiplexing coding cannot fulfill the function of index coding where there are three or more receivers. | 10.1109/lcomm.2014.020414.132589 | [
"https://arxiv.org/pdf/1401.7404v1.pdf"
] | 8,421,544 | 1401.7404 | bed1c493aa31724b804f836ec7ccbce239d1fa91 |
On Index Coding in Noisy Broadcast Channels with Receiver Message Side Information
29 Jan 2014
Behzad Asadi
Lawrence Ong
Sarah J Johnson
On Index Coding in Noisy Broadcast Channels with Receiver Message Side Information
29 Jan 2014arXiv:1401.7404v1 [cs.IT] 1
This letter investigates the role of index coding in the capacity of AWGN broadcast channels with receiver message side information. We first show that index coding is unnecessary where there are two receivers; multiplexing coding and superposition coding are sufficient to achieve the capacity region. We next show that, for more than two receivers, multiplexing coding and superposition coding alone can be suboptimal. We give an example where these two coding schemes alone cannot achieve the capacity region, but adding index coding can. This demonstrates that, in contrast to the two-receiver case, multiplexing coding cannot fulfill the function of index coding where there are three or more receivers.
I. INTRODUCTION
We consider additive white Gaussian noise broadcast channels (AWGN BCs) with receiver message side information, where the receivers know parts of the transmitted messages a priori. Coding schemes commonly used for these channels are superposition coding, multiplexing coding and index coding.
In this letter, we show that index coding is redundant where there are only two receivers, but can improve the rate region where there are more than two receivers.
A. Background
Superposition coding is a layered transmission scheme in which the codewords of the layers are linearly superimposed to form the transmitted codeword. Specifically, consider an AWGN channel where the transmitter sends messages M 1 and M 2 . A two-layer transmitted codeword is
u (n) (M 1 ) + v (n) (M 2 ) ,
where u (n) = (u 1 , u 2 , . . . , u n ) and v (n) = (v 1 , v 2 , . . . , v n ) are the codewords of the layers. This scheme was proposed for broadcast channels [1] and is also used in other channels, e.g., interference channels [2] and relay channels [3]. Superposition coding can achieve the capacity region of degraded broadcast channels [4], [5]. In broadcast channels with receiver message side information, superposition coding is used in conjunction with multiplexing coding [6] and index coding [7].
In multiplexing coding, two or more messages are bijectively mapped to a single message, and a codebook is then constructed for this message. For instance, suppose a transmitter wants to send messages M 1 ∈ {1, 2, . . . , 2 nR1 } and M 2 ∈ {1, 2, . . . , 2 nR2 }. The single message M m = [M 1 , M 2 ] is first formed from M 1 and M 2 , where [·] denotes a bijective map. Then codewords are generated for M m , i.e., x (n) (m m ) where m m ∈ {1, 2, . . . , 2 n(R1+R2) }. This coding scheme is also called nested coding [8] or physical-layer network coding This work is supported by the Australian Research Council under grants FT110100195 and DE120100246. [9]. Multiplexing coding can achieve the capacity region of broadcast channels where the receivers know some of the messages demanded by the other receivers, and want to decode all messages [10]. This scheme can also achieve the capacity region of a more general scenario where a noisy version of messages is available at the receivers, who also need to decode all messages [11].
The transmitter of the broadcast channel can utilize the structure of the side information available at the receivers to accomplish compression by XORing its messages. This is performed such that the receivers can recover their requested messages using XORed messages and their own side information. As an example, consider a broadcast channel where M 1 and M 2 are the two requested messages by two receivers, and each receiver knows the requested message of the other receiver. Then the transmitter only needs to transmit M x = M 1 ⊕ M 2 , which is the bitwise XOR of M 1 and M 2 with zero padding for messages of unequal length. Moduloaddition can also be used instead of the XOR operation [12]. This coding scheme is called index coding. It is also called network coding [13], as any index coding problem can be formulated as a network coding problem [14].
Separate index and channel coding is a suboptimal scheme in broadcast channels with receiver message side information. The separation, where the receiver side information is not considered during the channel decoding of the XOR of the two messages, leads to a strictly smaller achievable rate region in two-receiver broadcast channels [9], [10]. Separate index and channel coding has been shown to achieve within a constant gap of the capacity region of three-receiver AWGN BCs with only private messages [15].
Using a combination of index coding, multiplexing coding and superposition coding, Wu [16] characterized the capacity region of two-receiver AWGN BCs for all possible message and side information configurations. In this setting, the first receiver requests {M 1 , M 3 , M 5 } and knows M 4 , and the second receiver requests {M 2 , M 3 , M 4 } and knows M 5 . The transmitted codeword of the capacity-achieving transmission scheme is
u (n) (M 1 ) + v (n) (M mx ) ,(1)
where
M mx = [M 2 , M 3 , M 4 ⊕ M 5 ].
The combination of these coding schemes can also achieve the capacity region of some classes of three-receiver lessnoisy and more-capable broadcast channels where (i) only two receivers possess side information, and (ii) the only message requested by the third receiver is also requested by the other two receivers (i.e, a common message) [17].
B. Contributions
In this work, we first show that, in two-receiver AWGN BCs with receiver message side information, index coding need not necessarily be applied prior to channel coding if multiplexing coding is used. To this end, we show that index coding is a redundant coding scheme in (1), and multiplexing coding and superposition coding are sufficient to achieve the capacity region. We then derive the capacity region of a three-receiver AWGN BC. Prior to this letter, the best known achievable region for this channel was within a constant gap of the capacity region [15]. In this channel, index coding proves to be useful in order to achieve the capacity region; superposition coding and multiplexing coding cannot achieve the capacity region of this channel without index coding. Our result indicates that index coding cannot be made redundant by multiplexing coding in broadcast channels with receiver message side information where there are more than two receivers.
II. AWGN BC WITH SIDE INFORMATION
In an L-receiver AWGN BC with receiver message side information, as depicted in Fig. 1, the signals received by receiver i, Y
(n) i i = 1, 2, . . . , L, is the sum of the transmitted codeword, X (n) , and an i.i.d. noise sequence, Z (n) i i = 1, 2, . . . , L, with normal distribution, Z i ∼ N (0, N i ).
The transmitted codeword has a power constraint of n l=1 E X 2 l ≤ nP and is a function of source messages,
M = {M 1 , M 2 , . . . , M K }. The messages {M j } K j=1
are independent, and each message, M j , is intended for one or more receivers at rate R j . This channel is stochastically degraded, and without loss of generality, we can assume that receiver 1 is the strongest and receiver L is the weakest in the sense that
N 1 ≤ N 2 ≤ · · · ≤ N L .
To model the request and the side information of each receiver, we define two sets corresponding to each receiver; the wants set, W i , is the set of messages demanded by receiver i and the knows set, K i , is the set of messages that are known to receiver i.
III. WHERE INDEX CODING IS NOT REQUIRED
In this section, we consider the general message setting in two-receiver AWGN BCs with receiver message side information, where the knows and wants sets of the receivers are given by
Receiver 1: W 1 = {M 1 , M 3 , M 5 }, K 1 = {M 4 },Receiver 2: W 2 = {M 2 , M 3 , M 4 }, K 2 = {M 5 }.(2)
The capacity region of this channel has been derived by Wu [16]. It is achievable using a combination of index coding, multiplexing coding, and superposition coding, as in (1).
For the special case where M 1 = M 2 = M 3 = 0, Oechtering et al. [10] have shown that multiplexing coding alone can achieve the capacity region. In spirit of Oechtering et al., we now show that index coding is also not necessary for the general case (2).
Theorem 1: Multiplexing coding and superposition coding are sufficient to achieve the capacity region of two-receiver AWGN BCs with receiver message side information. Proof: We can use only multiplexing coding and superposition coding to achieve the capacity region of the two-receiver AWGN BC with the general message setting given in (2), i.e.,
X (n) Y (n) 1 Z (n) 1 . . . Y (n) 2 Z (n) 2 Y (n) L Z (n) Lx (n) = u (n) (M 1 ) + v (n) (M m ) ,(3)
where This result indicates that multiplexing coding can fulfill the function of index coding in two-receiver AWGN BCs with receiver message side information.
Remark 1: For two-receiver discrete memoryless broadcast channels with receiver message side information, we conjecture that index coding is not necessary. Using the same reasoning as above, we can always replace it with multiplexing coding without affecting the achievable rate region.
IV. WHERE INDEX CODING IMPROVES THE RATE REGION
In this section, we demonstrate that multiplexing coding cannot fulfill the role of index coding in AWGN broadcast channels with receiver message side information where there are more than two receivers. To this end, we establish the capacity region of a three-receiver AWGN BC where the wants and knows sets of the receivers are Receiver 1:
W 1 = {M 1 }, K 1 = ∅, Receiver 2: W 2 = {M 2 }, K 2 = {M 3 }, Receiver 3: W 3 = {M 3 }, K 3 = {M 2 },(4)
and show that multiplexing coding and superposition coding cannot achieve the capacity region of this channel without index coding.
A. Using Index Coding Prior to Multiplexing Coding and Superposition Coding
In this subsection, we establish the capacity region of the broadcast channel of interest, stated as Theorem 2.
Theorem 2: The capacity region of the three-receiver AWGN BC with the configuration given in (4) is the closure of the set of all rate triples (R 1 , R 2 , R 3 ), each satisfying
R 1 < C αP N 1 , (5a) R 2 < C (1 − α)P αP + N 2 , (5b) R 3 < C (1 − α)P αP + N 3 ,(5c)
for some 0 ≤ α ≤ 1, where C(x) = 1 2 log 2 (1 + x). Here, we prove the achievability part of Theorem 2; the proof of the converse is presented in the appendix.
Proof: (Achievability) Index coding and superposition coding are employed to construct the transmission scheme that achieves the capacity region. The codebook of this scheme contains two subcodebooks. The first subcodebook includes 2 nR1 i.i.d. codewords, u (n) (m 1 ) where m 1 ∈ {1, 2, . . . , 2 nR1 }, and U ∼ N (0, αP ) for an 0 ≤ α ≤ 1. The second subcodebook includes 2 n max{R2,
R3} i.i.d. codewords, v (n) (m x ) where m x = m 2 ⊕ m 3 , m x ∈ {1, 2, . . . , 2 n max{R2,R3} }, V ∼ N (0, (1 − α)P )
, and V is independent of U . Using superposition coding, the transmitted codeword over the broadcast channel is given by
x (n) = u (n) (M 1 ) + v (n) (M x ) .(6)
The achievability of the region in (5a)-(5c) using the transmission scheme in (6) can be verified by considering two points during the decoding. First, receivers 2 and 3 consider u (n) as noise. Since receiver 2 knows M 3 a priori, and receiver 3 knows M 2 a priori, we obtain (5b) and (5c) as the requirements for achievability. Second, receiver 1 decodes m x while treating u (n) as noise. This requires
max{R 2 , R 3 } < C (1 − α)P αP + N 1 ,(7)
for achievability. However, considering the inequalities in (5b) and (5c), this condition is redundant and can be dropped. Receiver 1 then removes v (n) from its received signal and decodes m 1 , which yields (5a) in the achievable region.
B. Not Using Index Coding Prior to Multiplexing Coding and Superposition Coding
In this subsection, we characterize the achievable rate region for the broadcast channel of interest when M
x = M 2 ⊕ M 3 in (6) is replaced with M m = [M 2 , M 3 ]
. This employs the same XOR-multiplexing substitution shown to be optimal in the tworeceiver case. This means that the messages are directly fed to multiplexing coding and superposition coding. The codebook of this transmission scheme also contains two subcodebooks in which only the second subcodebook is different from the scheme using index coding. The second subcodebook of this scheme includes 2 n(R2+R3) i.i.d. codewords, v (n) (m m ) where m m = [m 2 , m 3 ], m m ∈ {1, 2, . . . , 2 n(R2+R3) }, V ∼ N (0, (1 − α)P ) for an 0 ≤ α ≤ 1, and V is independent of U . The transmitted codeword of this scheme using superposition coding is given by
x (n) = u (n) (M 1 ) + v (n) (M m ) .(8)
The requirements for achievability concerning the decoding at receivers 2 and 3 are the same as (5b) and (5c). This is because the unknown information rates of both M x and M m are the same from the standpoint of each of these receivers. So, as far as these two receivers are concerned, multiplexing coding gives the same result as index coding.
However, decoding v (n) at receiver 1 while treating u (n) as noise requires
R 2 + R 3 < C (1 − α)P αP + N 1 ,(9)
for achievability which is not a redundant condition considering (5b) and (5c). The difference between (7) and (9) is because receiver 1 needs to decode the correct v (n) over the set of 2 n max{R2,R3} candidates for the former, but over the set of 2 n(R2+R3) candidates for the latter. After decoding v (n) , receiver 1 decodes m 1 which requires (5a) for achievability. When the messages are directly fed to multiplexing coding and superposition coding, an extra condition, given in (9), is required for achievability and, as a result, this scheme cannot achieve the capacity region. Even if receiver 1 does not decode v (n) and treats it as noise, it can only decode u (n) at rates up to R 1 < C αP (1−α)P +N1 , which is strictly smaller than (5a). Receiver 1 can also use simultaneous decoding [18, p. 88] to decode m 1 which requires (5a) and R 1 + R 2 + R 3 < C P N1 for achievability; in this case, the extra condition on the sum rate prevents this scheme from achieving the capacity region.
Note that alternative message combinations for multiplexing coding and superposition coding are also possible. The proof of their suboptimality is straightforward but tedious and repetitive.
V. CONCLUSION
In this work, we first showed that multiplexing coding can fulfill the function of index coding in two-receiver AWGN BCs with receiver message side information. We next established the capacity region of a three-receiver AWGN BC, where superposition coding and multiplexing coding alone cannot achieve the capacity region unless index coding is also used. This shows that index coding cannot be discharged by multiplexing coding in broadcast channels with receiver message side information where there are more than two receivers.
APPENDIX
Based on the proofs for the AWGN BC without side information [4], [18], we prove the converse part of Theorem 2 using Fano's inequality and the entropy power inequality (EPI). We also use the fact that the capacity region of a stochastically degraded broadcast channel without feedback is the same as its equivalent physically degraded broadcast channel [18, p. 444] where the channel input and outputs form a Markov chain, X → Y 1 → Y 2 → · · · → Y L , i.e.,
Y 1 = X + Z 1 , Y i = Y i−1 +Z i i = 2, 3, . . . , L,(10)
whereZ i ∼ N (0, N i − N i−1 ) for i = 2, 3, . . . , L.
Fig. 1 .
1The AWGN broadcast channel with receiver message side information, where W i ⊆ M is the set of messages demanded by receiver i, and K i ⊂ M is the set of messages known to receiver i a priori.
M m = [M 2 , M 3 , M 4 , M 5 ]. The only difference between (1) and (3) is that M 4 ⊕M 5 in M mx is replaced with [M 4 , M 5 ]. This scheme achieves the same rate region as (1), i.e., the capacity region. This is because from the standpoint of each receiver, the amount of uncertainty to be resolved in M 4 ⊕ M 5 is the same as that in [M 4 , M 5 ]. This uncertainty to be resolved in M 4 ⊕ M 5 (or [M 4 , M 5 ]) is M 5 by receiver 1, and M 4 by receiver 2.
Proof: (Converse) Based on Fano's inequality, we havewhere ǫ n , ǫ ′ n and ǫ ′′ n tend to zero as n → ∞. For the sake of simplicity we use ǫ n instead of ǫ ′ n and ǫ ′′ n as well in the rest. The rate R 2 is upper bounded aswhere (a) follows from the independence of M 2 and M 3 ,In this channel, R 3 is also upper bounded aswhere (f ) is the result of substituting from(15)andinto the conditional EPI[18, p. 22] for Ywhere we haveFinally, for R 1 , we havewhere (g) is the result of substituting from(15)andIn(17), (l) is due to h(Y From(14),(16),(17)and since ǫ n goes to zero as n → ∞, the proof of the converse for Theorem 2 is complete.
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| [] |
[
"Whitehead and Ganea constructions for fibrewise sectional category",
"Whitehead and Ganea constructions for fibrewise sectional category"
] | [
"J M García-Calcines "
] | [] | [] | We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibrewise sectional category is the analogue of the ordinary sectional category in the fibrewise setting and also the natural generalization of the fibrewise unpointed LS category in the sense of Iwase-Sakai. On the other hand the fibrewise pointed version is the generalization of the fibrewise pointed LS category in the sense of James-Morris. After giving their main properties we also establish some comparisons between such two versions. | 10.1016/j.topol.2013.10.023 | [
"https://arxiv.org/pdf/1302.3845v2.pdf"
] | 119,308,767 | 1302.3845 | b52d1f11a24b2108c33b3dc1373cc0755e9521e0 |
Whitehead and Ganea constructions for fibrewise sectional category
6 Mar 2013
J M García-Calcines
Whitehead and Ganea constructions for fibrewise sectional category
6 Mar 2013arXiv:1302.3845v2 [math.AT]
We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibrewise sectional category is the analogue of the ordinary sectional category in the fibrewise setting and also the natural generalization of the fibrewise unpointed LS category in the sense of Iwase-Sakai. On the other hand the fibrewise pointed version is the generalization of the fibrewise pointed LS category in the sense of James-Morris. After giving their main properties we also establish some comparisons between such two versions.
Introduction
The sectional category secat(p) of any map p : E → B is the least non negative integer k such that there exists a cover of B constituted by k + 1 open subsets on each of which p has a local homotopy section. When p is a fibration, then we can consider local strict sections in the definition, retrieving the usual notion of sectional category (or Schwarz genus) of p (see [22]). This is a lower bound of the Lusternik-Schnirelmann category of the base space and also a generalization as secat(p) = cat(B) when E is contractible. Apart from its usefulness in mathematical problems such as the computation of the roots of a complex polynomial, the embedding problem or the classification of bundles, the sectional category is also crucial for the notion of topological complexity of a space. The topological complexity of a space X, denoted as TC(X), is the sectional category of the evaluation fibration π : X I → X ×X, α → (α(0), α(1)). It was established by Farber [10,11] in order to face the motion planning problem in robotics from a topological perspective. The topological complexity is also interesting in algebraic topology itself, independently of its original purpose in robotics, as it is closely related to difficult tasks such as the immersion problem for real projective spaces [12]. Since its apparition, this relatively new numerical homotopy invariant has been of great interest for many researchers working on applied algebraic topology. It is remarkable the work of Iwase-Sakai [14] who related the topological complexity to what they call fibrewise unpointed LS category cat * B (−) in the fibrewise setting. Namely, if ∆ X denotes the diagonal map, pr 2 the projection onto the second factor, and d(X) ≡ X ∆X −→ X × X pr2 −→ X the induced fibrewise pointed space over X, then they proved that TC(X) = cat * X (d(X)). They also work with a pointed version in the fibrewise context and consider TC M (X) the monoidal topological complexity and cat B B (−) the fibrewise pointed LS category in the sense of James-Morris [17]. In this case the equality TC M (X) = cat X X (d(X)) holds.
The notion of fibrewise unpointed LS category of Iwase-Sakai, and the notion of fibrewise pointed LS category of James-Morris suggest a natural generalization, an analogue notion of sectional category in the category of fibrewise (pointed) spaces over a space B. The aim of this paper is to establish such a generalization, the fibrewise sectional category, denoted by secat B (−), as well as a Whitehead-Ganea approach of it. We also present its pointed version, secat B B (−), and some of their most important properties. In order to present our work we have divided the paper into four sections. In the first section we establish a Strøm-type model category and give some background about fibrewise homotopy such as fibrewise homotopy pullbacks, pushouts and joins. In the second section we introduce the main notion of the paper, the one of fibrewise sectional category, and give its main properties. Among them its Whitehead-Ganea approach (so it can be considered as an abstract sectional category in the sense of [18] or [5]) in which we will strongly use the results about fibrewise joins given in the previous section. This can be summarized in the following theorem. For details about its statement the reader is referred to the first sections of the paper:
Theorem 0.1. Let f : E → X be any fibrewise map between normal spaces, or a closed fibrewise cofibration with X normal. Then the following statements are equivalent:
(i) secat B (f ) ≤ n (ii) The diagonal map ∆ n+1 : X → n+1 B X factors, up to fibrewise homotopy, through the fibrewise sectional n-fat wedge
X / / ∆n+1 " " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ T n B (f ) jn n+1 B X (
iii) The n-th fibrewise Ganea map p n : G n B (f ) → X admits a fibrewise homotopy section.
It is also important the relationship of fibrewise sectional category with the ordinary sectional category. In this sense we have obtained the following result, which is not true in general. Here by a fibrant space over B we mean a fibrewise space over B in which the projection is a Hurewicz fibration.
Theorem 0.2. Let f : E → X be a fibrewise map between fibrant spaces over B. Then secat B (f ) = secat(f ).
The third section is dedicated to the fibrewise pointed case, where similar results are displayed and proved. Finally, in the last section we compare the unpointed and the pointed versions. As we will see, they are not so different as one might think. We present two results, being quite general the first of them.
Theorem 0.3. Let f : E → X be a fibrewise pointed map in Top w (B) between normal spaces, or a closed fibrewise cofibration with X normal. Then
secat B (f ) ≤ secat B B (f ) ≤ secat B (f ) + 1
The second one, which closes our paper, is more restrictive but also interesting. It is a generalization of a result given by Dranishnikov when he compares topological complexity and monoidal topological complexity.
Theorem 0.4. Let f : E → X a fibrewise pointed map between pointed fibrant and cofibrant spaces over B. Suppose that B is a CW-complex and X a paracompact Hausdorff space satisfying the following conditions:
(i) f : E → X is a k-equivalence (k ≥ 0); (ii) dim(B) < (secat B (f ) + 1)(k + 1) − 1. Then secat B (f ) = secat B B (f ).
Of course, in all these theorems we also obtain corollaries replacing in the corresponding statements fibrewise sectional category by fibrewise unpointed LS category or fibrewise pointed sectional category by fibrewise pointed LS category. Therefore, by Iwase-Sakai, we also retrieve known results about topological complexity and monoidal topological complexity.
1 Fibrewise homotopy theory.
We begin by giving some preliminary definitions and results on fibrewise homotopy theory that will be important throughout all the paper. For basic notation and terminology theory we have considered the reference [3].
Let B be a fixed topological space. A fibrewise space over B consists of a pair (X, p X ), where X is a topological space and p X : X → B a map from X to B. The map p X is usually called the projection. If there is not ambiguity we will denote X the fibrewise space (X, p X ). If X and Y are fibrewise spaces, then a fibrewise map (over
B) f : X → Y is just a map f : X → Y satisfying p Y f = p X X f / / pX ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ Y pỸ~⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ B
The corresponding category of fibrewise spaces and fibrewise maps over B will be denoted as Top B .
If X and Y are fibrewise spaces, then the binary fibrewise product of X and Y will be denoted by X × B Y = {(x, y) ∈ X × Y : p X (x) = p Y (y)}. We will denote by n B X the product of n copies of a given fibrewise space X. The fibrewise cylinder of a fibrewise space X is just the usual product space X×I (where I denotes the closed unit interval [0, 1]) together with the composite X × I pr −→ X pX −→ B as the projection p X×I . We will denote by I B (X) the fibrewise cylinder of X. The definition of fibrewise homotopy ≃ B between fibrewise maps comes naturally as well as the notion of fibrewise homotopy equivalence.
Axiomatic homotopy for fibrewise spaces
Now, for any fibrewise space X take the following pullback in the category Top of spaces and maps. Here X I (and B I ) denotes the free path-space provided with the cocompact topology, p I X is the obvious map induced by p X and c : B → B I is the map that carries any b ∈ B to the natural constant path c b in B I .
P B (X) / / X I p I X B c / / B I Thus, P B (X) = B × B I X I = {(b, α) ∈ B × X I : c b = p X α} being P B (X) → B
the obvious projection onto B. P B (X) is called the fibrewise cocylinder of X (or fibrewise free path space of X). The fibrewise cylinder and fibrewise cocylinder constructions give rise to functors
I B , P B : Top B → Top B
Associated to the functor I B there are defined natural transformations i 0 , i 1 : X → I B (X) and ρ : I B (X) → X (given by i ε (x) = (x, ε) and ρ(x, t) = x). Analogously, associated to P B there are defined natural transformations d 0 , d 1 : P B (X) → X and c : X → P B (X) (given by d ε (b, α) = α(ε) and c(x) = (p X (x), c x )). Moreover, we have that (I B , P B ) is an adjoint pair in the sense of Baues (see [1, p.29
]).
A fibrewise mapj : A → X is said to be a fibrewise cofibration (over B) if it satisfies the Homotopy Extension Property, that is, for any fibrewise map
X f / / i0 3 3 Y I B (X) H d d
As known, the fibrewise cofibrations are cofibrations in the usual sense. One has just to take into account that, for a given space Z, the product B × Z is a fibrewise space considering the canonical projection B × Z → B. Therefore, j : A → X is a fibrewise embedding, that is, j : A → j(A) is a fibrewise homeomorphism. Then we can consider, without loss of generality, that the fibrewise cofibrations are pairs of the form (X, A). Such pairs are also called fibrewise cofibred pairs.
An important characterization of fibrewise cofibred pair, slightly changed in [20,Prop 5.2.4] and also proved in [3] in the closed case, is given by what is called a fibrewise Strøm structure. Proposition 1.1. Let (X, A) be a fibrewise pair. Then (X, A) is fibrewise cofibred if and only if (X, A) admits a fibrewise Strøm structure, that is, a pair (ϕ, H) consisting of:
(i) A map ϕ : X → I satisfying A ⊆ ϕ −1 ({0}); (ii) A fibrewise homotopy H : I B (X) → X satisfying H(x, 0) = x, H(a, t) = a
for all x ∈ X, a ∈ A, t ∈ I, and H(x, t) ∈ A whenever t > ϕ(x).
If A is closed the ϕ can be taken so that A = ϕ −1 ({0}).
An interesting consequence of Proposition 1.1 is the following fact. Recall that given A, U subspaces of a topological space X, U is said to be a halo of A in X if there exists a map ϕ : We will be particularly interested in closed fibrewise cofibred pairs (or closed fibrewise cofibrations), which are closed fibrewise pairs (X, A) (i.e., A is a closed subspace of X) such that (X, A) is fibrewise cofibred. As in the classical topological case, this is not a very restrictive condition; for instance, if (X, A) is any fibrewise cofibred pair with X Hausdorff, then necessarily A is a closed subspace of X.
X → I such that A ⊆ ϕ −1 ({0}) and ϕ −1 ([0, 1)) ⊆ U.
A fibrewise fibration is a fibrewise map p : E → Y such that it verifies Homotopy Lifting Property with respect to any fibrewise space
Z f / / i0 E p I B (Z) H / / < < Y
If p : E → Y is any fibrewise map such that it is a Hurewicz fibration, then p is a fibrewise fibration. However, in general, the converse is not true. For instance, if X is a fibrewise space, then p X : X → B is always a fibrewise fibration, but p X need not be a Hurewicz fibration.
Denote by f ib B , cof B and he B the classes of fibrewise fibrations, closed fibrewise cofibrations (equivalently, closed fibrewise cofibred pairs) and fibrewise homotopy equivalences, respectively. It is not hard to check that Top B is an I-category and a P -category in the sense of Baues (see [1, p.31] for definitions). More is true, one can also check that the Relative Homotopy Lifting Property holds, i.e. if (X, A) is any closed fibrewise cofibred pair and p : E → Y any fibrewise fibration, then any commutative diagram in Top B of the form Following the reasonings given by Strøm in [21] we also have the following theorem, which should be compared with [20, Th 5.2.8]. 1.2 Homotopy pushouts, pullbacks and joins in the fibrewise setting.
X × {0} ∪ A × I / / _ E p I B (X) / / Y
In this subsection we will establish the Cube Lemma in Top B . This result will be crucial in order to deal with a Whitehead-Ganea type characterization for fibrewise sectional category. First we need to present the notions of fibrewise homotopy pullback and fibrewise homotopy pushout, which are simply the corresponding homotopy limis in the fibrewise axiomatic setting. Given a fibrewise homotopy commutative diagram
X g f / / Y h H Z k / / K ( * ) with a fibrewise homotopy H : hf ≃ B kg, there is another homotopy commu- tative diagram E h,k q p / / Y h G Z k / / K ( * * ) in which E h,k = {(y, (b, θ), z) ∈ Y ×P B (K)×Z ; h(y) = θ(0) , k(z) = θ(1)} with the natural projection E h,k → B, given by (y, (b, θ), z) → b.
Here p and q are the obvious restrictions of the projections and G is the fibrewise homotopy defined as G(y,
(b, θ), z, t) = θ(t). There is a fibrewise whisker map w : X → E h,k given by w(x) = (f (x), (p X (x), H(x, −)), g(x)) satisfying pw = f, qw = g and G(w × id) = H.
The homotopy commutative square (*) is said to be a fibrewise homotopy pullback whenever w is a fibrewise homotopy equivalence. Given h and k fibrewise maps there always exist their fibrewise homotopy pullback. The square (**) is called the standard fibrewise homotopy pullback.
There is the dual notion in the sense of Eckmann-Hilton. Given f : X → Y and g : X → Z fibrewise maps we can consider the quotient fibrewise space C f,g := (Y ⊔ I B (X) ⊔ Z)/ ∼ where ∼ is the equivalent relation generated by the elemental relations (x, 0) ∼ f (x) and (x, 1) ∼ g(x), for all x ∈ X. The induced fibrewise map w ′ : C f,g → K is called the fibrewise co-whisker map. If w ′ is a fibrewise homotopy equivalence then the square of ( * ) is called fibrewise homotopy pushout. Remark 1.5. As in the classical case, fibrewise homotopy pullbacks and fibrewise homotopy pushouts can be also characterized by the weak universal property of fibrewise homotopy limits and colimits or through factorization properties. The reader is referred to [19], [6] or [2].
A combination of fibrewise homotopy pullbacks and fibrewise homotopy pushouts is the fibrewise join of two fibrewise maps f : X → Z and g : Y → Z. Namely, the fibrewise join of f and g, X * Z Y, is the fibrewise homotopy pushout of the fibrewise homotopy pullback of f and g,
• / / Y { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ g X * Z Y # # X : : ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ f / / Z
being the dotted arrow the corresponding co-whisker map, induced by the weak universal property of fibrewise homotopy pushouts.
The next result relates fibrewise cofibrations and fibrewise fibrations and is one of the key tools in the proof of the Cube Lemma. Lemma 1.6. Consider a fibrewise pullback of the following form, where p is a fibrewise fibration over B
P j ′ / / p ′ E p A j / / X If j :
A → X is a closed fibrewise cofibration, then so is its base change j ′ : P → E.
Proof. We can suppose that (X, A) is a closed fibrewise cofibred pair, and j the natural inclusion, so that there exists (ϕ, H) a fibrewise Strøm structure. Moreover, P = A × X E = p −1 (A) with j ′ the inclusion and p ′ the corresponding restriction of p. Now take a lift in the diagram
E id / / i0 E p I B (E) H(p×id) / / H 7 7 X
Defining H(e, t) = H(e, min{t, ϕp(e)}), then one can easily check that ( H, ϕp) is a fibrewise Strøm structure for the pair defined by the inclusion j ′ . Theorem 1.7 (Cube Lemma). Given a fibrewise homotopy commutative cube
W ⑧ ⑧ ⑧ ⑧ / / X ⑧ ⑧ ⑧ ⑧ Y / / Z A ⑧ ⑧ ⑧ ⑧ / / B ⑧ ⑧ ⑧ ⑧ C / / D
in which the bottom face is a fibrewise homotopy pushout and all sides are fibrewise homotopy pullbacks, then the top face is also a fibrewise homotopy pushout.
Proof. Following an analogous reasoning to the one given in [19] we can suppose without loss of generality that the cube is strictly commutative in which:
• The arrows A → B and A → C are closed fibrewise cofibrations, and the bottom face is a fibrewise pushout;
• Z → D is a fibrewise fibration and all side squares are fibrewise pullbacks.
This way the top square is of the form
A × D Z / / B × D Z C × D Z / / Z
where, by Lemma 1.6 above, all the arrows are closed fibrewise cofibrations. Now consider P the fibrewise pushout of C × D Z←−A × D Z−→B × D Z and θ : P → Z the fibrewise map induced by the pushout property. A simple inspection proves that θ is a fibrewise isomorphism (compare with [7, 6.1] ) concluding that this square is a fibrewise (homotopy) pushout.
Remark 1.8. Note that, even satisfying the Cube Lemma, Top B is not a Jcategory in the sense of Doeraene [6] as it has no zero object.
2 Fibrewise sectional category.
And now we will establish the main notion of the paper in the fibrewise context, the one of fibrewise sectional category. First we give some background on fibrewise LS category and its unpointed version.
Let X be a fibrewise space. The fibrewise L.-S. category of X, cat B (X), is the minimal number n ≥ 0 such that there exists a cover {U i } n i=0 of X by n + 1 open subsets, each of them admitting a fibrewise homotopy commutative diagram in Top B of the form
U i / / pX |Ui ❅ ❅ ❅ ❅ ❅ ❅ ❅ X B si ? ? ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦
If there is no such n, then we say cat B (X) = ∞. This notion was given by James-Morris in [17] (see also [16] and [3]). Unfortunately, this invariant is not very manageable from the axiomatic point of view. Instead we will consider a certain variant of cat B (−), given by Iwase-Sakai in [14]. For this, we deal with fibrewise pointed spaces. By a fibrewise pointed space over B we mean a fibrewise space X together with a fibrewise map s X :
B → X (i.e., s X : B → X is a section of p X ).
If X is a fibrewise pointed space, then the fibrewise unpointed L.-S. category of X, cat * B (X), is the minimal number n ≥ 0 such that there exists a cover {U i } n i=0 of X by n + 1 open subsets, each of them fibrewise categorical, that is, admitting a fibrewise homotopy commutative diagram in Top B of the form
U i / / pX |Ui ❅ ❅ ❅ ❅ ❅ ❅ ❅ X B sX ? ? ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦
If there is no such n, then we say cat * B (X) = ∞. Obviously, cat B (X) ≤ cat * B (X) and the equality holds when X is vertically connected, that is, all possible sections s i : B → X are fibrewise homotopic to s X (see [17]). As James-Morris assert, when B is a CW-complex a fibre bundle over B with fibre F is vertically connected if dim(B) does not exceed the connectivity of F.
Open-like definition of fibrewise sectional category
We want to give the natural generalization of fibrewise unpointed LS category by considering the analogous notion of sectional category in the fibrewise setting.
U in / / s ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ X E f > > ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦
The fibrewise sectional category of f, denoted as secat B (f ) is the minimal number n such that X admits a cover {U i } n i=0 constituted by fibrewise sectional open subsets. If there is no such n, then secat B (f ) = ∞.
When f : E → X is a fibrewise fibration, then we may suppose in the definition of fibrewise sectional that the triangles are strictly commutative. On the other hand, from the definition of fibrewise sectional category it is clear the identity cat * B (X) = secat B (s X ).
We can generalize this fact. Observe that given a fibrewise homotopy commutative diagram in Top B of the form A fibrewise contractible space (or just a shrinkable space) is any fibrewise space having the fibrewise homotopy type of B. On the other hand if X and Y are fibrewise pointed spaces over B, by a fibrewise pointed map f : X → Y we mean a fibrewise map such that f s X = s Y . Applying the above comments to the commutative triangle f s E = s X we obtain Proposition 2.2. Let f : E → X be any fibrewise pointed map. Then
E λ / / f ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ E ′ f ′~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ Xsecat B (f ) ≤ cat * B (X)
If E is fibrewise contractible, then secat B (f ) = cat * B (X).
One can also check that the fibrewise unpointed LS category cat * B (−) is invariant by fibrewise pointed maps which are fibrewise homotopy equivalences. Proposition 2.3. If f : X → Y is a fibrewise pointed map such that it is a fibrewise homotopy equivalence, then
cat * B (X) = cat * B (Y ).
In particular, if X is a fibrewise pointed space over B, then cat * B (X) = 0 if and only if X is fibrewise contractible.
The axiomatic approach of fibrewise sectional category
Now we study the fibrewise sectional category secat B (−) from a Whitehead-Ganea approach. Let f : E → X be a fibrewise map. For each n ≥ 0 we consider the fibrewise sectional n-fat wedge as the fibrewise map j n :
T n B (f ) → n+1 B X inductively defined as follows: set T 0 B (f ) = E, j 0 = f : E → X, and define j n : T n B (f ) → n+1 B X as the join in Top B • / / X × B T n−1 B (f ) w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ idX ×B jn−1 T n B (f ) jn ' ' B × B n B X 8 8 q q q q q q q q f ×B id / / n+1 B X
On the other hand, the n-th fibrewise Ganea map of X, G n B (X) pn −→ X, is inductively defined as follows:
Set p 0 := f : E → X (so G 0 B (f ) = E). If p n−1 is already constructed, then G n B (f ) is the fibrewise join of G n−1 B (f ) pn−1 −→ X f ←− E,
and p n is the induced whisker map:
• / / E | | ② ② ② ② ② ② f G n B (f ) pn ! ! G n−1 B (f ) 9 9 t t t t t t t pn−1 / / X
The following result is a direct consequence of Theorem 1.7, the Cube Lemma in the fibrewise context. Its proof is similar to the classical case (see, for instance, [4]) and therefore is omitted and left to the reader. Lemma 2.4. Let f : E → X be a fibrewise map. Then, for each n ≥ 0, there is a fibrewise homotopy pullback
G n B (f ) pn / / T n B (f ) jn X ∆n+1 / / n+1 B X
where ∆ n+1 denotes the n + 1-st diagonal map.
Lemma 2.5. Let f : E → X be a closed fibrewise cofibration. Then, the fibrewise sectional n-fat wedge is, up to fibrewise homotopy,
T n B (f ) = {(x 0 , x 1 , ..., x n ) ∈ n+1 B X ; x i ∈ E for some i ∈ {0, 1, ..., n}} being j n : T n B (f ) ֒→ n+1 B X the canonical inclusion.
Proof. It is evident for n = 0. Next, we check that the following square is a fibrewise homotopy pullback:
E × B T n−1 B (f ) _ id×B jn−1 f ×B id / / X × B T n−1 B (f ) _ id×B jn−1 E × B n B X f ×B id / / n+1 B X
The standard fibrewise homotopy pullback of f × B id and id X × B j n−1 , call it L, is given by the elements
H(e,x, (b, γ), x,ȳ; t) = (e, γ 2 (t), (b, δ(t)), γ 1 (1 − t),ȳ),
being γ = (γ 1 , γ 2 ) and δ(t)(s) = (γ 1 (s(1 − t)), γ 2 ((1 − s)t + s)).
Finally, taking into account that
f × B id : E × B T n−1 B (f ) → X × B T n−1 B (X)
is a closed fibrewise cofibration, the fibrewise homotopy pushout of f × B id and the inclusion id × B j n−1 is just its honest fibrewise pushout. The result follows by induction. Remark 2.6. If f is any fibrewise map, then we can factor it through a closed fibrewise cofibration followed by a fibrewise homotopy equivalence
E f / / f ′ ❆ ❆ ❆ ❆ ❆ ❆ ❆ X X ′ ≃ > > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤
Therefore any fibrewise map can be considered, up to fibrewise homotopy, as a closed fibrewise cofibration. Moreover, as this factorization can be taken through the fibrewise mapping cylinder, if E and X are normal, then X ′ is also normal.
Theorem 2.7. Let f : E → X be any fibrewise map between normal spaces, or a closed fibrewise cofibration with X normal. Then the following statements are equivalent:
(i) secat B (f ) ≤ n (ii) The diagonal map ∆ n+1 : X → n+1 B
X factors, up to fibrewise homotopy, through the fibrewise sectional n-fat wedge
X / / ∆n+1 " " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ T n B (f ) jn n+1 B X (iii)
The n-th fibrewise Ganea map p n : G n B (f ) → X admits a fibrewise homotopy section.
Proof. Statements (ii) and (iii) are equivalent as a consequence of Lemma 2.4 and the weak universal property for fibrewise homotopy pullbacks. Now we check that statements (i) and (ii) are equivalent. Taking into account the above remark and the fact that statements (i)-(iii) are invariant by fibrewise homotopy equivalences, we can suppose without loss of generality that f is a fibrewise closed cofibration with X normal. X, where L i : I B (X) → X is the fibrewise map defined as follows:
L i (x, t) = x , x ∈ X \ B i H i (x, th i (x)) , x ∈ U i
Taking into account Lemma 2.5 and that {A i } n i=0 covers X it is straightforward to check that H : ∆ n+1 ≃ B j n ϕ, where ϕ : X → T n B (f ) is defined as ϕ(x) := L(x, 1).
Conversely, suppose a fibrewise map ϕ : X → T n B (f ) and a fibrewise homotopy L : ∆ n+1 ≃ B j n ϕ. Then, L = (L 0 , ..., L n ) and j n ϕ = (ϕ 0 , ..., ϕ n ) where L i : I B (X) → X and ϕ i : X → X for each i. Since f : E ֒→ X is fibrewise closed cofibration, by Remark 1.2 we have that E is a fibrewise strong deformation retract of an open neighborhood U in X. Take a fibrewise retraction r : U → E and a fibrewise homotopy H : I B (U ) → X such that H(x, 0) = x, H(x, 1) = f r(x) for all x ∈ U, and H(e, t) = e = f (e), for all e ∈ E and t ∈ I.
Defining U i = ϕ −1 i (U ) we obtain {U i } n i=0
an open cover of X and fibrewise homotopies G i :
I B (U i ) → X G i (x, t) = L i (x, 2t) , 0 ≤ t ≤ 1 2 H(ϕ i (x), 2t − 1) , 1 2 ≤ t ≤ 1 If s i : U i → E denotes the composite U i ϕi −→ U r −→ E, then G i (x, 0) = x and G i (x, 1) = f s i (x), for all x ∈ U i .
Remark 2.8.
Observe that taking f = s X in the above theorem we obtain the corresponding Whitehead-Ganea characterization of cat * B (−). Compare with [14].
Fibrewise sectional category and ordinary sectional category.
Recall that if f : X → Y is any map, then the sectional category secat(f ) can be defined as the least non-negative integer n such that Y admits an open cover {U i } n i=0 where each U i has a local homotopy section s i : U i → X of f. When f is a Hurewicz fibration, then in this definition we can take local strict sections, recovering the usual notion of sectional category (or Schwarz genus [22]) for fibrations.
On the other hand, one of the main motivations of the invariant cat * B (−) is the fact that it retrieves the notion of topological complexity of a space in the sense of Farber [10,11]. The topological complexity of a space X, denoted TC(X), is defined as the sectional category of π : X I → X×X, α → (α(0), α(1)). Of course, since the fibration π : X I → X × X is homotopy equivalent to the diagonal map ∆ X : X → X × X, then we can redefine TC(X) as secat(∆ X ). The product space X × X can be seen as a fibrewise pointed space over X with ∆ X the diagonal map as the section and pr 2 : X × X → X the projection. Denoting by d(X) such a fibrewise pointed space we have the following known equality (see [14]):
TC(X) = cat * X (d(X)) This fact can be generalized for the unpointed LS category. Under a not very restrictive condition on the fibrewise pointed space X we will be able to prove that cat * B (X) = secat(s X ). By a fibrant space over B we mean a fibrewise space over B, X, such that the projection p X : X → B is a Hurewicz fibration. A pointed fibrant space over B is a fibrant space over B which is also pointed over B.
E f / / λ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ X E p ? ? ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧
by a homotopy equivalence λ followed by a Hurewicz fibration p. We observe that this factorization also has sense in Top B considering in E the composite E p −→ X pX −→ B as the natural projection, which is a Hurewicz fibration. Being E and E fibrant spaces over B we conclude that actually λ : E → E is a fibrewise homotopy equivalence. Furthermore, secat(f ) = secat(p) and secat B (f ) = secat B (p). As an interesting and surprising consequence of Theorem 2.9 and its corollary we have that, in order to deal with secat B (f ) or cat * B (X) we may forget the projections and work in the non-fibrewise world, at least for weak fibrant spaces.
Remark 2.11. Observe that, in general, the above equality is not true. For example, consider X = ([0, 1] × {0, 1}) ∪ ({0} × [0, 1]) and B = [0, 1] with s X (t) = (t, 0) and p X (t, s) = t. If X were fibrewise contractible, then the fibre p −1 X ({t}) would have the same ordinary homotopy type of {t}, for all t ∈ [0, 1]; but by connectivity reasons this is not the case. Therefore, by Corollary 2.3 we have that cat * B (X) ≥ 1. But secat(s X ) = 0, since X is contractible in the ordinary sense.
The hypothesis of Theorem 2.9 can be relaxed. Recall that a Dold fibration (or weak fibration) is a map p : X → B which satisfies the weak covering homotopy property (see [8] for details). A Dold fibration is completely characterized by the fact that it is fibrewise homotopy equivalent to a Hurewicz fibration. Then we can consider weak fibrant spaces over B, i.e., fibrewise spaces over B in which the projection is a Dold fibration. It is immediate to check the equality secat B (f ) = secat(f ) when E and X are just weak fibrant spaces over B, and the equality cat * B (X) = secat(s X ) when X is a pointed weak fibrant space over B.
3 The fibrewise pointed case.
The category of fibrewise pointed spaces and fibrewise pointed maps will be denoted by Top(B). Observe that the space B together with the identity is the zero object so that Top(B) is a pointed category. Any subspace A ⊆ X containing the section (i.e., s X (B) ⊆ A) is a fibrewise pointed space. This way, the inclusion A ֒→ X is a fibrewise pointed map. These class of subspaces will be called fibrewise pointed subsets of X.
For any fibrewise pointed space X we can consider its fibrewise pointed cylinder as the following pushout
B × I pr / / sX ×id B X × I / / I B B (X)
with the obvious projection induced by the pushout property. This pointed cylinder functor gives the notion of fibrewise pointed homotopy between fibrewise pointed maps, that will be denoted by ≃ B B . We point out that giving a fibrewise pointed homotopy F : I B B (X) → Y is the same as giving a fibrewise homotopy
F ′ : I B (X) → Y such that F ′ (s X (b), t) = s X (b)
, for all b ∈ B and t ∈ I. The notion of fibrewise pointed homotopy equivalence comes naturally. On the other hand we can consider Any fibrewise pointed map which is a fibrewise cofibration is a fibrewise pointed cofibration. Also, any fibrewise fibration is a fibrewise pointed fibration. Unfortunately, as P. May and Sigurdsson assert in [20, p.82], on each case the converse is not true even for the simplest case, in which B is a point. Despite this fact, in order to deal with LS-type invariants in the homotopy category of Top(B), we still can achieve a reasonable structure on the category of fibrewise pointed spaces based one the fibrewise structure.
P B B (X) = B × B I X I = {(b, α) ∈ B × X I : c b = p X α}
A fibrewise well-pointed space is a fibrewise pointed space X in which the section s X : B → X is a closed fibrewise cofibration. Let Top w (B) denote the full subcategory of Top(B) consisting of fibrewise well-pointed spaces. One cannot expect Top w (B) to be a model category as it is not closed under finite limits and colimits. However the following proposition is enough for our purposes. First we need a technical lemma, whose proof is analogous to the ordinary case and therefore is omitted (see [21]). Lemma 3.1. Let j : A → X and i : X → Y be fibrewise maps such that i and ij are closed fibrewise cofibrations. Then j is also a closed fibrewise cofibration. Proof. As the cobase change of a fibrewise cofibration is a fibrewise cofibration, the second statement of the lemma is trivially true. Now suppose p : E → X and f : X ′ → X fibrewise pointed maps between fibrewise well-pointed spaces where p is a fibrewise fibration. Consider the following diagram of pullbacks
F i / / E ′ p ′ f ′ / / E p B s X ′ / / X ′ f / / X
As X ′ and X are well-pointed and p is a fibrewise fibration, by Lemma 1.6 i and f ′ i are closed fibrewise cofibrations. Considering Proposition 3.1 and the fact that (f ′ i)s F = s E is a closed fibrewise cofibration we have that F is well-pointed and therefore s E ′ = is F is a closed fibrewise cofibration.
A certain version of the following result also appears in [ (ii) If f is a closed map, then f is a fibrewise pointed cofibration if and only if f is a fibrewise cofibration;
(iii) f is a fibrewise pointed homotopy equivalence if and only if f is a fibrewise homotopy equivalence.
Proof. Part (i) is given in [3,Prop 16.3], while part (iii) is consequence of the abstract Dold's theorem in the cofibration category Top B ([1, p.96]). A simple inspection reveals that the reasonings given in [21] can be straightforwardly extrapolated to the fibrewise setting for the proof of part (ii).
These facts lead to a great simplification in this setting and allow us to prove directly the following result.
U in / / s ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ X E f > > ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦
The fibrewise pointed sectional category of f, denoted as secat B B (f ) is the minimal number n such that X admits an open cover {U i } n i=0 constituted by fibrewise pointed sectional subsets. When such n does not exist then secat B B (f ) = ∞.
When f is a fibrewise pointed fibration, having a local strict section of f is equivalent to having a local fibrewise pointed homotopy section of f. On the other hand it is immediate to check the identity
secat B B (s X ) = cat B B (X),
that is, the pointed fibrewise sectional category of s X is exactly the fibrewise pointed LS category in the sense of James-Morris [17].
A fibrewise pointed contractible space is any fibrewise pointed space having the same fibrewise pointed homotopy type of B. The proof of the following result is analogous to the fibrewise unpointed case. Proposition 3.6. Let f : E → X be any fibrewise pointed map. Then
secat B B (f ) ≤ cat B B (X)
If E is fibrewise pointed contractible, then secat B B (f ) = cat B B (X).
As a consequence of Theorem 3.4, by [5] or [18] we can define a manageable axiomatic notion of fibrewise pointed sectional category from two equivalent approaches: Whitehead's and Ganea's. Moreover, the fibrewise pointed n-fat wedge and the n-th fibrewise pointed Ganea fibration can be chosen as j n :
T n B (f ) → n+1 B
X and p n : G n B (f ) → X, the ones given in the unpointed case. The proof of the following theorem is completely analogous to the one given in Theorem 2.7 and therefore is omitted. For the particular case f = s X compare the equivalence of (i) and (ii) in our theorem with [17, Prop 6.1 and Prop 6.2] Theorem 3.7. Let f : E → X be a fibrewise pointed map in Top w (B) between normal spaces, or a closed fibrewise pointed cofibration with X normal. Then the following statements are equivalent:
(i) secat B B (f ) ≤ n (ii) The diagonal map ∆ n+1 : X → n+1 B
X factors, up to fibrewise pointed homotopy, through the fibrewise n-fat wedge
X / / ∆n " " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ T n B (f ) jn n+1 B X (iii)
The n-th fibrewise Ganea map p n : G n B (f ) → X admits a fibrewise pointed homotopy section. Remark 3.8. As in the unpointed case, observe that taking f = s X in the above theorem we obtain a Whitehead-Ganea characterization of cat B B (−). Compare with [14].
We conclude this section with some comments and remarks. Recall from [14] that the monoidal topological complexity of a space X, denoted TC M (X), is the least integer n such that X × X can be covered by n + 1 open sets {U i } n i=0 on each of which there is a local section s i of the free path fibration π X : X I → X × X, π X (γ) = (γ(0), γ(1)) satisfying that ∆ X (X) ⊂ U i and s i (x, x) = c x , for all x ∈ X.
As we have previously commented the product space X × X can be seen as a fibrewise pointed space over X, d(X), with ∆ X the diagonal map as the section and pr 2 : X × X → X the projection. We have the equality
TC M (X) = cat X X (d(X))
for any space X (see [14] for details). In particular TC M (X) = secat X X (∆ X ). We assert that the monoidal topological complexity can also be seen as TC M (X) = secat X X (π X ).
Indeed, considering X I as a fibrewise pointed space over X with section c : X → X I sending x → c x (c x is the constant path at x) and projection ev 1 : X I → X the evaluation map α → α(1), we have that in the following commutative
diagram in Top(X) X c / / ∆X # # • • • • • • • • • X I πX { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ X × X
the homotopy equivalence c : X → X I is actually a fibrewise pointed homotopy equivalence over X. Trivially ev 1 c = id X , and the formula H(α, t)(s) = α((1 − t)s + t) defines a fibrewise pointed homotopy over X between id X I and the composite c ev 1 .
4 Unpointed versus pointed fibrewise sectional category.
In this last section we will check how close secat B (−) and secat B B (−) are. Obviously, the inequality secat B (−) ≤ secat B B (−) always holds.
Theorem 4.1. Let f : E → X be a fibrewise pointed map in Top w (B) between normal spaces, or a closed fibrewise cofibration with X normal. Then
secat B (f ) ≤ secat B B (f ) ≤ secat B (f ) + 1
Proof. We can suppose without loss of generality that f : E ֒→ X is a closed fibrewise cofibration with X normal. Therefore, by Corollary 1.2, E is a fibrewise strong deformation retract of an open neighborhood N in X. Take r : N → E a fibrewise retraction and a fibrewise homotopy G :
I B N → X with G(x, 0) = x, G(x, 1) = f r(x)
, for all x ∈ N and G(e, t) = e, for all e ∈ E and t ∈ I. If we consider V i = U i \ E for i ∈ {0, ..., n} and V n+1 = N, then we have
that {V i } n+1
i=0 is an open cover of X, and by normality one can find {W i } n+1 i=0 a refined open cover of X such that W i ⊆ W i ⊆ V i , for all i ∈ {0, ..., n + 1}.
Now we define the open subset
N = N ∩ (X \ W 0 ) ∩ ... ∩ (X \ W n )
Obviously, N ∩ W i = ∅ and E ⊆ N , for all i ∈ {0, ..., n}.
If O i = W i ∪ N , for i ∈ {0, ..., n} and O n+1 = N , then {O i } n+1
i=0 is a cover of X constituted by n + 2 open fibrewise pointed categorical subsets (observe that W n+1 ⊆ V n+1 = N ). Indeed, it only remains to check that O i is fibrewise pointed sectional, for i = n + 1. In this case the following fibrewise pointed homotopy L i :
I B (O i ) → X proves this fact L i (x, t) = H i (x, t), x ∈ W i G(x, t), x ∈ N Remark 4.2.
Observe that Theorem 4.1 is also true when we consider fibrewise pointed embeddings f : E ֒→ X with X normal and such that f admits an open fibrewise pointed sectional subset U ⊆ X. For instance, when X is an ENR (Euclidean Neighborhood Retract) then ∆ X : X → X × X is known to satisfy this condition.
As a corollary of the above result we have that if X is any normal fibrewise well-pointed space over B, then
cat * B (X) ≤ cat B B (X) ≤ cat * B (X) + 1
In particular, if X is a locally finite simplicial complex (or more generally, an ENR), then TC(X) ≤ TC M (X) ≤ TC(X) + 1.
These two latter results have been already proved in [15] and [9].
In order to establish the statement of our second theorem we need the following lemma whose proof can be found in [20,Th 24.1.2]. Here the bracket [Z, K] B denotes the set of fibrewise homotopy classes of fibrewise maps Z → K over B. is a bijection if dim(X) < n and a surjection if dim(X) = n.
If X is a fibrewise pointed space over B, then we say that X is cofibrant when the section s X is a closed cofibration in Top. Observe the difference with the notion of being well-pointed, in which s X is a closed fibrewise cofibration. Every well-pointed fibrewise space is cofibrant but, in general, the converse is not true. Now we are ready for our result.
Theorem 4.4. Let f : E → X be a fibrewise pointed map between pointed fibrant and cofibrant spaces over B. Suppose, in addition, that B is a CW-complex, X is a paracompact Hausdorff space and the following conditions are satisfied:
(i) f : E → X is a k-equivalence (k ≥ 0);
(ii) dim(B) < (secat B (f ) + 1)(k + 1) − 1.
Then secat B (f ) = secat B B (f ).
Proof. Take a factorization of f in Top
E f / / λ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ X E p ? ? ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧
by a closed cofibration and homotopy equivalence λ followed by a Hurewicz fibration p. Then E is a fibrewise pointed space over B considering λs E as section and the composite p X p as a projection. Moreover λ is a fibrewise pointed homotopy equivalence as it is a homotopy equivalence between fibrant and cofibrant spaces. Therefore, we have the equalities secat B (f ) = secat B (p) = secat(p) and secat B B (f ) = secat B B (p).
Suppose secat(p) = n and take the iterated classical join of n + 1 copies of p, j p n : * n X E → X. Observe that this is a Hurewicz fibration which is a fibrewise pointed map over B. Then j n p admits a strict section σ : * n X E → X which, necessarily, must be a fibrewise map over B. However, σ need not be a fibrewise pointed map over B. In order to solve this problem we proceed as follows:
Taking into account that j n p is an ((n + 1)(k + 1) − 1)-equivalence over X between fibrant spaces over X, and that dim(B) < (n + 1)(k + 1) − 1, we have by Lemma 4.3 that (j n p ) * : [B, * n X E] X → [B, X] X is a bijection and, in particular, is injective. Now, if s * n X B : B → * n X B denotes the corresponding section for * n X B, then (j n p ) * ([σs X ]) = [j n p σs X ] = [s X ] = [j n p s * n X E ] = (j n p ) * ([s * n X E ]) Therefore there exists a fibrewise homotopy over X, H : σs X ≃ X s * n X B . As s X : B ֒→ X is a closed cofibration and j n p : * n X E → X a Hurewicz fibration, by the ordinary Relative Homotopy Lifting Property we can consider a lift in Top
X × {0} ∪ B × I (σ,H) / / _ * n X E j n p X × I H 5 5 prX / / X
We define σ ′ := Hi 1 : X → * n X E. Then one can straightforwardly check that σ ′ is a fibrewise pointed map over B such that j n p σ ′ = id X . From this fact one can easily check that there exists an open cover {U i } n i=0 of X in which U i contains s X (B) and there is s i : U i → E a strict local section of p, satisfying s i s X = λs E , for all i. But this implies that secat B B (p) ≤ n.
Corollary 4.5. Let B be a CW-complex over X. Suppose, in addition, that X is a paracompact Hausdorff pointed fibrant and cofibrant space over B satisfying the following conditions:
(i) s X : B → X is a k-equivalence (k ≥ 0);
(ii) dim(B) < (cat * B (X) + 1)(k + 1) − 1.
Then the equality cat * B (X) = cat B B (X) holds.
Specialising to topological complexity we obtain the following known result [9]. Corollary 4.6.
[9] Let X be a k-connected CW-complex such that dim(X) < (TC(X) + 1)(k + 1) − 1. Then TC(X) = TC M (X).
f : X → Y and any fibrewise homotopy H : I B (A) → Y such that Hi 0 = f j, there exists a fibrewise homotopy H : I B (X) → Y such that Hi 0 = f and HI B (j)
Corollary 1 . 2 .
12Given (X, A) any fibrewise cofibred pair, then A is a fibrewise strong deformation retract of an open neighborhood U in X. Such open subset U is a halo of A.
admits a lift. All these facts are summarized in the following propositionProposition 1.3. The category Top B together with the classes of cof B , f ib B and he B has an IP category structure in the sense of Baues. In particular, Top B is a cofibration category and a fibration category of Baues.
Theorem 1 . 4 .
14The category Top B together with the classes of cof B , f ib B and he B has a proper closed model category structure in the sense of Quillen.
Definition 2 . 1 .
21Let f : E → X be a fibrewise map over B and consider an open subset U of X. Then U is said to be fibrewise sectional if there exists a morphism s : U → E in Top B such that the following triangle commutes up to fibrewise homotopy
one has the inequality secat B (f ′ ) ≤ secat B (f ). Indeed, if U is an open fibrewise sectional categorical subset of X with local fibrewise homotopy section s : U → E of f, then s ′ = λs : U → E ′ is a local fibrewise homotopy section of f ′ .
γ(0) = (f (e),x) and γ(1) = (x,ȳ). Define ω : E × B T n−1 B (f ) → L by ω(e,ȳ) = (e,ȳ, (b, C (f (e),ȳ) ), f (e),ȳ) (where C (f (e),ȳ) denotes the constant path in (f (e),ȳ)), and ω ′ : L → E × B T n−1 B (f ) by ω ′ (e,x, (b, γ), x,ȳ) = (e,ȳ). Then, ω ′ ω = id and ωω ′ ≃ B id through the fibrewise homotopy
Assume that secat B (f ) ≤ n and consider {U i } n i=0 an open cover of X and H i :I B (U i ) → X a fibrewise homotopy satisfying H i (x, 0) = x and H i (x, 1) = f s i (x), for all x ∈ U i , where s i : U i → E is a fibrewise map. AsX is a normal space there exist, for each i, closed subsets A i , B i and an open subset Θ i such that A i ⊆ Θ i ⊆ B i ⊆ U i and {A i } n i=0 covers X. Now, by Urysohn characterization of normality, take h i : X → I a map such that h i (A) = {1} and h i (X \ Θ i ) = {0}. Then we obtain a fibrewise homotopy L = (L 0 , ..., L n ) : I B (X) → n+1 B
Theorem 2 . 9 .
29Let f : E → X be a fibrewise map between fibrant spaces over B. Then secat B (f ) = secat(f ) Proof. The inequality secat(f ) ≤ secat B (f ) is obvious so it only remains to check secat B (f ) ≤ secat(f ). First consider a factorization of f in Top
If U is any open subset of X with a strict local section s : U → E of p, then s is necessarily a fibrewise map over B and therefore U is also an open fibrewise sectional subset of X. The result follows taking open coverings.
Corollary 2 . 10 .
210Let X be a pointed fibrant space over B. Then cat * B (X) = secat(s X ).
which is simply the fibrewise space P B (X) together with the section (id B , cs X ) : B → P B B (X) induced by the pullback property. There are functorsI B B , P B B : Top(B) → Top(B)as well as induced natural transformations i 0 , i 1 :X → I B B (X), ρ : I B B (X) → X and d 0 , d 1 : P B B (X) → X, c : X → P B B . Moreover, (I B B , P B B )is an adjoint pair in the sense of Baues.Associated to these functors, there are defined the notions of (closed) fibrewise pointed cofibration and of fibrewise pointed fibration, which are characterized by the natural Homotopy Extension Property and the Homotopy Lifting Property in Top(B), respectively. Moreover, Top(B) has an I-category and a P -category structure in the sense of Baues ([1, p.31]).
Proposition 3 . 2 .
32Top w (B) is closed under the pullbacks of fibrewise pointed maps which are fibrewise fibrations. Similarly, Top w (B) is closed under the pushouts of fibrewise pointed maps which are closed fibrewise cofibrations.
Proposition 3 . 3 .
33Let f : X → Y be a fibrewise pointed map between fibrewise well-pointed spaces over B. Then, (i) f is a fibrewise pointed fibration if and only if f is a fibrewise fibration;
Theorem 3 . 4 .
34The category Top w (B) with the structure inherited by Top B is a J-category in the sense of Doeraene[6].Proof. Considering the above two propositions combined with Proposition 1.3 we have that Top B induces in Top w (B) a cofibration and fibration category structures in the sense of Baues. Observe that the factorizations of a fibrewise pointed map are taken through the fibrewise mapping cylinder or the fibrewise mapping track. Again, propositions 3.3 and 3.2 together with Theorem 1.7 show that Top w (B) satisfy the Cube Lemma.
Definition 3 . 5 .
35Let f : E → X be a fibrewise map over B and consider an open subset U of X containing s X (B). Then U is said to be fibrewise pointed sectional if there exists a morphism s : U → E in Top(B) such that the following triangle commutes up to fibrewise pointed homotopy
Suppose that secat B (f ) = n and take {U i } n i=0 an open cover of X, where each U i is fibrewise sectional with s i : U i → E the local fibrewise homotopy section of f. For each i ∈ {0, ..., n} we choose H i : I B U i → X a fibrewise homotopy satisfying H i (x, 0) = x and H i (x, 1) = f s i (x), for all x ∈ U i .
Lemma 4 . 3 .
43Let X be a CW-complex over B and e : Y → Z an n-equivalence over B between fibrant spaces. Thene * : [X, Y ] B → [X, Z] B ; [α] → [eα]
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| [] |
[
"Stellar populations of seven early-type dwarf galaxies and their nuclei ⋆",
"Stellar populations of seven early-type dwarf galaxies and their nuclei ⋆"
] | [
"S Paudel \nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\n69120HeidelbergGermany\n",
"⋆⋆T Lisker \nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\n69120HeidelbergGermany\n"
] | [
"Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\n69120HeidelbergGermany",
"Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\n69120HeidelbergGermany"
] | [
"Astron. Nachr. / AN"
] | Dwarf galaxies are the numerically dominating population in the dense regions of the universe. Although they seem to be simple systems at first view, the stellar populations of dwarf elliptical galaxies (dEs) might be fairly complex. Nucleated dEs are of particular interest, since a number of objects exhibit different stellar populations in their nuclei and host galaxy. We present stellar population parameters obtained from integrated optical spectra using a Lick index analysis of seven nucleated dwarf elliptical galaxies and their nuclei. After subtracting the scaled galaxy spectra from the nucleus spectra, we compared them with one another and explore their stellar populations. As a preliminary result, we find that the luminosity weighted ages of the nuclei slightly lower than those of galaxies, however, we do not see any significant difference in metallicity of the host galaxies and their nuclei. | 10.1002/asna.200911273 | [
"https://arxiv.org/pdf/0909.1269v1.pdf"
] | 13,943,236 | 0909.1269 | e00a98127d59bd5b2f3b8c5f84b092a06eaa9ad9 |
Stellar populations of seven early-type dwarf galaxies and their nuclei ⋆
Sep 2009. 2009
S Paudel
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
69120HeidelbergGermany
⋆⋆T Lisker
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
69120HeidelbergGermany
Stellar populations of seven early-type dwarf galaxies and their nuclei ⋆
Astron. Nachr. / AN
99988Sep 2009. 2009Received August 2009 Published online later/ DOI please set DOI!galaxies:dwarf -galaxies: evolution -galaxies: formation -galaxies: statistics
Dwarf galaxies are the numerically dominating population in the dense regions of the universe. Although they seem to be simple systems at first view, the stellar populations of dwarf elliptical galaxies (dEs) might be fairly complex. Nucleated dEs are of particular interest, since a number of objects exhibit different stellar populations in their nuclei and host galaxy. We present stellar population parameters obtained from integrated optical spectra using a Lick index analysis of seven nucleated dwarf elliptical galaxies and their nuclei. After subtracting the scaled galaxy spectra from the nucleus spectra, we compared them with one another and explore their stellar populations. As a preliminary result, we find that the luminosity weighted ages of the nuclei slightly lower than those of galaxies, however, we do not see any significant difference in metallicity of the host galaxies and their nuclei.
Introduction
Though diffuse elliptical galaxies represent the majority of the galaxy populations in dense regions of the nearby Universe like rich clusters, their origin and evolution remain still a matter of debate. Several recent and past studies show that these galaxies exhibit a great variety of kinematics and stellar population properties (Michielsen et al. 2008;van Zee, Skillman & Haynes 2004;Poggianti et al. 2001;Ma-teo1998;) posing a number of questions to the current understanding of the external and internal evolution of dwarf elliptical galaxies. A recent systematic study by Lisker et al. 2008, 07, 06 of dEs in the Virgo cluster revealed a striking heterogeneity of this galaxy class. They found different subclasses, with significantly different shapes, colors, and spatial distributions. Those dEs with a disk component have a flat shape and are predominantly found in the outskirts of the cluster, suggesting that they -or their progenitorsmight have just recently fallen into the cluster environment. In contrast, dEs with a compact stellar nucleus follow the classical picture of dwarf ellipticals: they are spheroidal objects that are preferentially found in the dense cluster center. Interestingly, though, the majority of dEs with disks also host a nucleus in their center. This would not seem too surprising if one assumed that the nuclei were already in place before the galaxy received its present-day configuration. However, several proposed mechanisms for nucleus formation are based on late nucleus formation, e.g. Van den Bergh (1986) proposed that the nuclei of dE,N could have formed from gas that sank to the centre of the more slowly ⋆ Based on observations made at the European Southern Observatory, Chile(ESO Programme ID 078.B-0178(A)) ⋆⋆ Corresponding author: e-mail: [email protected] rotating objects. Furthermore, rounder dEs with compact nuclei are predominantly present in highly dense environments like the centre of galaxy clusters. The pressure from the surrounding inter-galactic medium may allow dwarf galaxies to retain their gas during star formation and produce multiple generations of stars (Babul 1992, Silk et al. 1987, forming nuclei in the process. In both proposed scenarios, the stars constituting the nuclei are formed late, out of gas within the galaxy. In contrast, Oh & Lin 2000 suggested that the nuclei might have formed from globular clusters that migrated into the galaxy center, thus meaning that the nucleus' stars were formed separately and probably earlier than most stars of the host galaxy. Large globular clusters or super star clusters are candidates for becoming such nuclei of dEs. Given these different possibilities for the formation process of nuclei, and also of dEs itself, can the nuclei thus tell us something about the formation history of their host galaxies?
Motivated by these reasons for the importance of investigating stellar population differences between the nuclei and the host dEs outside of the central region, we have studied seven nucleated dEs of the Virgo cluster, using medium resolution spectroscopy.
Sample and Observation
The sample consists of seven early-type nucleated dwarf galaxies of the Virgo cluster, having different substructure or color characteristics: one has a blue central region ( dE(bc) ), two show faint signatures of disks (dE(di) ), and four others have no such features and are simply nucleated ( dE(N) ), according to Lisker et al. 2007 Table 1 The galaxies are identified by their number in the Virgo cluster catalog (VCC, Binggeli et al. 1985) ried out over six half nights between March 16 to 18, 2007, with the FORS2 instrument mounted on UT1 at ESO VLT, using the multi-object spectroscopy (MXU) mode. Integration times varied between 10 and 30 minutes, depending on the surface brightness of each galaxy. The galaxy properties and the final signal-to-noise ratio of the extracted spectra are listed in Table 1. The spectra are flux calibrated using spectrophotometric standard stars, which were observed during the run. A slit width of 1" and length of 40" were used, with the 300V grism providing a dispersion of 3.36Å / pixel. This setup provided a spectral resolution, as measured from the FWHM of the arc lines, of ∼11Å at ∼5000Å, which is somewhat below the Lick resolution (∼8.4Å at 5000Å).
We integrated over the interval 2"<r<7" along the spatial direction, to extract the spectrum of the underlying host galaxy, thereby considering that the nucleus light is negligible beyond the maximum size of the seeing disk (1.5") that occured for our sample. Since we know that the nucleus light sits on top of the underlying galaxy, we subtracted the galaxy light from the central nucleus spectrum. This process is illustrated in Fig. 1: first, we fitted the galaxy light to a smooth exponential profile beyond 2" from the centre, then we extrapolated the galaxy light inwards to the centre, and subtract this central galaxy light from the total light. Finally, we integrate over the central 3 pixels (0.75") along the spatial direction to extract one-dimensional nucleus spectra.
Result
The luminosity-weigthed age and metallicity, can be inferred from a comparison of selected line-strength indices with models of single stellar populations such as those of Bruzual & Charlot 2003 (BC03 here after). In the Fig 2, we show Lick/IDS index-index diagrams with SSP model grid taken from BC03 based on Padova 1994 isochrones. Note that, Our spectra are flux calibrated but the resolution is somewhat lower than model resolution, we therefore degrade the BC03 model spectra down to our resolution in order to match the resolution of the model and observed spectra. From Hβ -[MgFe]' diagram in the Fig 2, we can infer luminosity weighted ages and metallicity of nuclei and their host galaxies. Black squares represent the nuclei and filled gray circles the respective host galaxies. It is conspicuous that most of the nuclei are located upward of their respective host galaxies (see the number labels in the figure), indicating that their stellar population is younger than that of the host. Furthermore, the one dE with a blue central region, VCC 308, has the youngest nucleus, confirming the recent star-formation activity in its centre. In contrast to that, only one (VCC 1945) shows the opposite trend, namely that the galaxy is younger than its nucleus. We do not see any significant difference in metallicity between nuclei and host galaxies. Although the sample is quite small, we do not find any difference in the stellar populations of dEs and their nuclei with respect to different substructure properties.
The <Fe>-Mgb diagram (Fig. 3) gives an estimate of the α/Fe element ratio, relative to solar values of the model. The almost all the dEs and and their nuclei are consistent with solar abundance, however one VCC1348 behave differently, lying in the region of super solar abundance.
Conclusion
Although the numbers are small, we find that dE nuclei are slightly younger than their host galaxies. Such a difference in the age of nuclei and galaxies supports the theory of late nucleus formation, in the sense that a non-nucleated dE formed first, and the nucleus formed subsequently within the dE due to secondary star formation (e.g. Bothun & Mould 1988) or infalling gas (e.g. Silk et al. 1987) leading to continuous star formation activity in the nucleus.
Fig. 1
1Schematic view of the fitting of the light profile of the galaxy and nucleus. The crosses represent the distribution of total light (i.e galaxy + nucleus) and solid line represents the exponentially fitted light profile of the galaxy. The dashed line is the residual nucleus flux after the subtraction of the galaxy light.
Fig. 2
2The age-sensitive index Hβ as a function of the metallicity-sensitive index[MgFe]. Overplotted are the stellar population models of BC03. The solid lines are lines of constant age from 1 to 15 Gyr. The solid, almost vertical, lines stand for constant metallicity from [Z/H] = -1.64 to 0.55 dex for BC03. For both this plot andFig. 3, the nuclei are shown with black squares, while the host galaxies are shown by filled gray circles.
Fig. 3 <Fe>
3Vs Mgb diagram.
. The observations were car-S. Paudel & T. Lisker: Stellar populations of dEsName
RV
Mag Type S/N gal. S/N nuc
VCC0308 1850±39 13.19
bc
34
33
VCC0216 1281±26 14.35
di
34
44
VCC0490 1267±12 13.80
di
27
32
VCC0929 0910±10 12.60
n
43
53
VCC1254 1278±18 14.27
n
32
65
VCC1348 1968±25 14.33
n
27
42
VCC1945 1619±10 14.50
n
37
30
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.an-journal.orgc 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
AcknowledgementWe are supported within the framework of the Excellence Initiative by the German Research Foundation (DFG) through the Heidelberg Graduate School of Fundamental Physics (grant number GSC 129/1). S.P. acknowledges the support of the International Max Planck Research School (IMPRS) for Astronomy and Cosmic Physics at the University of Heidelberg.
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. D Michielsen, A Boselli, C J Conselice, E Toloba, I M Whiley, A Arag On-Salamanca, M Balcells, N Cardiel, A J Cenarro, J Gorgas, R F Peletier, A Vazdekis, MNRAS. 3851374Michielsen D., Boselli A., Conselice C. J., Toloba E., Whiley I. M., Arag on-Salamanca A., Balcells M., Cardiel N., Cenarro A. J., Gorgas J., Peletier R. F., Vazdekis A., 2008, MNRAS, 385, 1374
. M L Mateo, ARA&A. 36435Mateo M. L., 1998, ARA&A, 36, 435
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. B M Poggianti, T J Bridges, B Mobasher, D Carter, M Doi, M Iye, N Kashikawa, Y Komiyama, S Okamura, M Sekiguchi, K Shimasaku, M Yagi, N Yasuda, ApJ. 562689Poggianti B. M., Bridges T. J., Mobasher B., Carter D., Doi M., Iye M., Kashikawa N., Komiyama Y., Okamura S., Sekiguchi M., Shimasaku K., Yagi M., Yasuda N., 2001, ApJ, 562, 689
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| [] |
[
"Detection and estimation of spikes in presence of noise and interference",
"Detection and estimation of spikes in presence of noise and interference"
] | [
"Damien Passemier [email protected] \nDepartment of Electronic and Computer Engineering\nCEMSE Division King Abdullah University of Science and Technology\nAlcatel-Lucent Chair Supélec\nHong Kong University of Science and Technology\nGif-sur-YvetteFrance\n",
"Abla Kammoun [email protected] \nDepartment of Electronic and Computer Engineering\nCEMSE Division King Abdullah University of Science and Technology\nAlcatel-Lucent Chair Supélec\nHong Kong University of Science and Technology\nGif-sur-YvetteFrance\n",
"Mérouane Debbah [email protected] \nDepartment of Electronic and Computer Engineering\nCEMSE Division King Abdullah University of Science and Technology\nAlcatel-Lucent Chair Supélec\nHong Kong University of Science and Technology\nGif-sur-YvetteFrance\n"
] | [
"Department of Electronic and Computer Engineering\nCEMSE Division King Abdullah University of Science and Technology\nAlcatel-Lucent Chair Supélec\nHong Kong University of Science and Technology\nGif-sur-YvetteFrance",
"Department of Electronic and Computer Engineering\nCEMSE Division King Abdullah University of Science and Technology\nAlcatel-Lucent Chair Supélec\nHong Kong University of Science and Technology\nGif-sur-YvetteFrance",
"Department of Electronic and Computer Engineering\nCEMSE Division King Abdullah University of Science and Technology\nAlcatel-Lucent Chair Supélec\nHong Kong University of Science and Technology\nGif-sur-YvetteFrance"
] | [] | In many practical situations, the useful signal is contained in a low-dimensional subspace, drown in noise and interference. Many questions related to the estimation and detection of the useful signal arise. Because of their particular structure, these issues are in connection to the problem that the mathematics community refers to as "spike detection and estimation". Previous works in this direction have been restricted to either determining the number of spikes or estimating their values while knowing their multiplicities. This motivates our work which considers the joint estimation of the number of spikes and their corresponding orders, a problem which has not been yet investigated to the best of our knowledge. | null | [
"https://arxiv.org/pdf/1405.4846v1.pdf"
] | 88,520,248 | 1405.4846 | 261b888c78d2912ded2e439f415a652985ac3fe7 |
Detection and estimation of spikes in presence of noise and interference
Damien Passemier [email protected]
Department of Electronic and Computer Engineering
CEMSE Division King Abdullah University of Science and Technology
Alcatel-Lucent Chair Supélec
Hong Kong University of Science and Technology
Gif-sur-YvetteFrance
Abla Kammoun [email protected]
Department of Electronic and Computer Engineering
CEMSE Division King Abdullah University of Science and Technology
Alcatel-Lucent Chair Supélec
Hong Kong University of Science and Technology
Gif-sur-YvetteFrance
Mérouane Debbah [email protected]
Department of Electronic and Computer Engineering
CEMSE Division King Abdullah University of Science and Technology
Alcatel-Lucent Chair Supélec
Hong Kong University of Science and Technology
Gif-sur-YvetteFrance
Detection and estimation of spikes in presence of noise and interference
In many practical situations, the useful signal is contained in a low-dimensional subspace, drown in noise and interference. Many questions related to the estimation and detection of the useful signal arise. Because of their particular structure, these issues are in connection to the problem that the mathematics community refers to as "spike detection and estimation". Previous works in this direction have been restricted to either determining the number of spikes or estimating their values while knowing their multiplicities. This motivates our work which considers the joint estimation of the number of spikes and their corresponding orders, a problem which has not been yet investigated to the best of our knowledge.
I. INTRODUCTION
Detecting and estimating the components of a signal corrupted by additive Gaussion noise is a fundamental problem that arises in many signal and array processing applications. Considering a large number of received samples, one can easy see that their covariance matrix exhibit a different behaviour depending on the number of the components of the useful signal. In light of this consideration, first methods of signal detection like techniques using the Roy Test [14] or those using information theoretic criteria [15] have been based on the eigenvalues of the empirical covariance matrix. Recently, the advances in the spectral analysis of large dimensional random matrices have engendered a new wave of interest for the scenario when the number of observations is of the same order of magnitude as the dimension of the received samples, while the number of signal components remain finite. Such a model is referred to as the spiked covariance model [6]. It has allowed the emergence of new detection schemes based on the works of the extreme eigenvalues of large random Wishart matrices [9], [10], [8]. It is especially encountered in multi-sensor detection [12] and power estimation problems [16], which are at the heart of cognitive radio applications. This model has also found application in subspace estimation problems with a particular interest on the estimation of directions of arrival [5].
From a mathematical perspective, the focus has been either to detect the presence of sources and estimate their numbers This research has been partly supported by the ERC Starting Grant 305123 MORE (Advanced Mathematical Tools for Complex Network Engineering).
[7], [11] or to estimate their powers [1]. The general case where the objective is to extract as much as possible information has not been addressed to the best of our knowledge. This motivates our work which proposes an easy way to jointly estimate the number of sources, their powers and their multiplicities in the case where different sources are using the same power values.
II. SYSTEM MODEL
Consider the p-dimensional observation vector x i ∈ C p at time i:
x i = K k=1 √ α k W k s k,i + σe i where • (W k ) 1≤k≤K
is an orthogonal family of rectangular unitary p × m k matrices (i.e, for 1 ≤ k ≤ K, W k has orthogonal columns and W k W H j = 1 k=j I p ); • (α k ) 1≤k≤K are K positive distinct scalars such that α 1 > α 2 > · · · > α K ; • (s k,i ) 1≤k≤K ∈ C m k ×1 are independent random vectors with zero mean and variance 1; • e i ∈ C p×1 is complex Gaussian distributed (i.e. e i ∼ CN (0, I)) and represent the interference and noise signal; • σ 2 is the strength of the noise. Therefore, we consider K distinct powers α k , each of multiplicity m k . Gathering n observations x 1 , . . . , x n into a p × n observation matrix X = [x 1 , . . . , x n ], we obtain
X = [W 1 , · · · , W K ] √ α 1 I m1 0 . . . 0 √ α K I m K × s 1,1 · · · s 1,n . . . . . . . . . s K,1 · · · s K,n + σ [e 1 , · · · , e n ] .
or equivalently
X = Σ 1 2 Y (1)
where Y is a matrix of independent entries with zero mean and variance 1 and Σ is the theoretical covariance matrix of arXiv:1405.4846v1 [math.ST] 19 May 2014 the observations given by:
Σ = U (α 1 + σ 2 )I m1 0 . . . (α K + σ 2 )I m K 0 σ 2 I n−m U H
where m = K k=1 m k and U is an orthogonal matrix. Note that Σ has K distinct eigenvalues with multiplicities m 1 , . . . , m K and one eigenvalue equal to σ 2 with multiplicity p − m.
This model corresponds to the spiked covariance model [6]: here we allow spikes with multiplicities greater than one. It can be encountered as shown in [16] for power estimation purposes in cognitive radio networks. Another interesting application is met in the array processing field and in particular in the problem of the estimation of the angles of arrival. In this case, the received signal matrix is given by [5]:
X = A(θ)P 1 2 S + σN (2) where A = [a(θ 1 ), . . . , a(θ m )], a(θ i ) being the p × 1 steering vector, P = diag (αI m1 , . . . , α K I m K ), S is the m × n trans- mitted matrix of i.i.d Gaussian entries and N = [e 1 , · · · , e n ].
Note that in this case, A can be considered as unitary, since
A H A → I m when p → ∞.
Previous methods dealing with the estimation of directions of arrivals has so far assumed a prior estimation of the number of sources [5]. Such information is obviously not always available in practice. This motivates our paper, which proposes a method to jointly estimate the number of sources as well as their multiplicities.
III. ESTIMATION OF SPIKES' VALUES AND MULTIPLICITIES
The estimation technique relies on results about the asymptotic behavior of the covariance matrix. As shown in the following proposition proven in [2], the asymptotic spectral properties of the covariance matrix depend on the eigenvalues α 1 , · · · , α K of the matrix Σ. Proposition 1. Let S n be the sample covariance matrix given by:
S n = 1 n n k=1 x i x H i
Denote by λ n,1 > λ n,2 > · · · > λ n,p the p eigenvalues of S n arranged in decreasing order. Let
s i = i k=1 m k and J k the index set J k = {s k + 1, . . . , s k + m k }, k ∈ {1, . . . , K}. Let φ(x) = x + σ 2 + γσ 2 1 + σ 2 x for x = 0 and assume that γ n = p n → γ. Then, if φ (α k ) > 0 (i.e. α k > σ 2 √ γ) for any k ∈ {1, . . . , K}, we have almost surely λ n,j → φ(α k ), ∀j ∈ J k Remark 1. Under the condition φ (α k ) > 0 for all k ∈ {1, .
. . , K}, the empirical distribution of the spectrum is composed of K + 1 connected intervals: a bulk corresponding to the Marčhenko-Pastur law [3] followed by K spikes. To illustrate this, we represent in Figure 1, the empirical histogram of the eigenvalues of the empirical covariance matrix when K = 3, (α 1 , α 2 , α 3 ) = (7, 5, 3), (n, p) = (4000, 2000) and σ 2 = 1. Figure 1 provides us insights about an intuitive approach to estimate the multiplicities of spikes and their values given their number K. Actually, one needs to rearrange the eigenvalues and then detect the largest gaps that correspond to a switch from one connected interval to the next one.
This leads us to distinguish two cases whether K is either known or not. We will consider these cases in turn in the following.
A. K is known
In this case, we propose to estimate the eigenvalues by considering the differences between consecutive eigenvalues: δ n,j = λ n,j − λ n,j+1 , j ≥ 1.
Indeed, the results quoted above imply that a.s. δ n,j → 0, for j / ∈ {s i , i = 1, . . . , K} whereas for j ∈ {s i , i = 1, . . . , K}, δ n,j tends to a positive limit given by φ(α j ) − φ(α j+1 ). Thus it becomes possible to estimate the multiplicities from indexnumbers j where δ n,j is large. If K is known, we will take the indices corresponding to the K larger differences δ n,i . Denote by i 1 , . . . , i p the indices of the differences δ n,i such that δ n,i1 ≥ · · · ≥ δ n,ip . Then, the estimator (m 1 , . . . ,m K ) of the multiplicities (m 1 , . . . , m K ) is defined by
m 1 =min {i k , k ∈ {1, . . . , K}} m 2 =min {i k , k ∈ {1, . . . , K}\{m 1 }} −m 1 m j =min {i k , k ∈ {1, . . . , K}\{m 1 , . . . ,m j−1 }} − j−1 i=1m î m K =max {i k , k ∈ {1, . . . , K}} − K−1 i=1m i
The proposed consistent estimator of the number of the spikes is therefore given by the following theorem, for which a proof is omitted because of lack of space: Theorem 1. Let (x i ) 1≤i≤n be n i.i.d copies. of x which follows the model (1). Suppose that the population covariance matrix Σ has K non null eigenvalues (α i + σ 2 ) 1≤i≤K such that α 1 > · · · > α K > σ 2 √ γ with respective multiplicity (m k ) 1≤k≤K (m 1 + · · · + m K = m), and p − m eigenvalues equal to σ 2 . Assume that p/n → γ > 0 when n → ∞. Then the estimator (m 1 , . . . ,m K ) is strongly consistent, i.e (m 1 , . . . ,m K ) → (m 1 , . . . , m K ) almost surely when n → ∞.
B. K is not known
As Figure 1 shows, eigenvalues outside the bulk are organized into K clusters, where within each cluster, all eigenvalues converge to the same value in the asymptotic regime p, n → +∞ such that p/n → γ. If K is not estimated correctly, applying the previous method, will lead to either gathering two close clusters (K is under-estimated) or to subdividing the clusters corresponding to the highest spikes (K is over-estimated). Clearly, the second order results within each cluster seems to bring useful information which allows to discard these cases. In particular, in the sequel, we will rely on the following proposition which is a by-product of Proposition
in [2]:
Proposition 2. Assume that the settings of Theorem 1 holds. Let g k = s k j=s k−1 +1 λ n,j , the sum of the eigenvalues corresponding to the k-th cluster. Then, when n → ∞ such that p/n → γ > 0, g k verify
√ n (g k − m k φ(α k )) L − → N (0, 2m k v 2 k ) where v 2 k = 2α 2 k ((α k − 1) 2 − γ)/(α k − 1) 2 , α k = α k /σ 2 + 1 and L denotes the convergence in distribution.
Theorem 2 establishes that the sum of the eigenvalues within the k-th cluster behaves as a Gaussian random variable with mean and variance depending on the unknown value α k . One way to remove the uncertainty in the unknowns α k is to assume that they are random with a priori known distribution π (α 1 , . . . , α K |K). A possible case would correspond to the situation where they are uniformly distributed over a finite discrete set 1 .
Since the clusters are asymptotically independent [13], the likelihood function (distribution of g = [g 1 , · · · , g K ] under the underlying parameters α 1 , · · · , α K , m 1 , · · · , m K , K) is given by:
f (g|α 1 , . . . , α K , K) = K k=1 1 2πv 2 k e − 1 2v 2 k (g k −m k φ(α k )) 2
where the multiplicities m 1 , . . . , m K can be estimated in a consistent way given the number of classes K as it has been shown in section III-A. Hence, the maximum likelihood function f (g|K) is given by:
f (g|K) = E [f (g|α 1 , . . . , α K , K)](3)
where the expectation is taken over the a priori distribution π(α 1 , · · · , α K |K). The maximum likelihood estimator K is thus given by
K = max 1≤k≤Kmax E [f (g|α 1 , . . . , α K , K)] ,
where K max is a known upper bound of K. Once K is estimated, the multiplicities can be retrieved by using the method in Section III-A.
To sum up, when K is unknown, the estimation of the unknown parameters using the a priori π consists in the following steps : 1) Compute the consecutive differences of the ordered eigenvalues of the sample covariance matrix S n given by δ n,j = λ n,j − λ n,j+1 ; 2) For each k ranging from one to K max , calculate the corresponding estimator (m
IV. NUMERICAL EXPERIMENTS
We consider in our simulations the model described by (2)
given in section II with A(θ) = p −1/2 [exp (−iv sin(θ)π)] p−1 v=0 ,
where θ is chosen uniformly on [0, 2π). We assume that the set of the a priori spikes is E = {1, 3, 5, 7} and that the values α 1 , . . . , α K are uniformly distributed over this set.
In the sequel, we will display the empirical probability P(K = K) calculated over 500 independent realizations. For each iteration, we choose the "true" values of spikes uniformly in the set E, but with the same fixed proportion m i /m, i = 1, . . . , K.
We consider two different experiments: in the first one, we study the performance of our method for different level of noise variances whereas for the second one, we consider the impact of the number of spikes m for a fixed noise variance.
A. Performance of the proposed method with respect to the variance of the noise In this experiment, we consider the detection of the number of K = 3 different clusters of 500×1 ( p = 500 ) signals from n = 1000 samples. We assume that the unknown multiplicities are m 1 = 1, m 2 = 4, m 3 = 2. Since the minimum value of the spike is assumed to be 1, σ 2 has to be lower than 1/ √ c = 1.4142 in order to keep a gap betweenλ m and λ m+1 (see Theorem 1). The noise variance is expressed in dB 10 log 10 (σ 2 ). Table I illustrates the obtained results : Fig. 2. Empirical probability of P(K = K) as a function of (p, n), for Models A, B and C.
Our estimator performs well, especially for low noise variances. When σ 2 is getting close to the threshold 1.41 (i.e. 1.50 dB), the estimator becomes less accurate, which was expected sinceλ m is very close to the bulk.
B. Influence of the number of spikes m
We study in this experiment the impact of the number of spikes in the performance of the proposed estimation method. Similarly to the previous simulation settings, we set K = 3 and γ = p/n = 0.5. We consider the following three models: Figure 2 displays the frequency of correct estimation for these three models with respect to p. Note that these models keep the p/n and m i /m fixed except m/n which is different. In that way, only the impact of the variation of the number of spikes is visualized.
As expected, our estimator performs better in Model A than in Model B and C. In both cases, we observe the asymptotic consistency, but the convergence is slower for Model C. Remark 2. Once K was correctly estimated, we have noticed by simulations that the multiplicities are correctly estimated. This is in accordance with our Theorem 1.
V. CONCLUSION
The problem of signal detection appears naturally in many signal processing applications. Previous works used to deal with this problem partially by assuming extra knowledge about the number of spikes or their corresponding orders. This work is therefore an attempt to consider the general problem where the objective is to estimate all the unknown parameters. In particular, we show that when the number of different spikes is known, their multiplicities can be estimated consistently. In light of this consideration, we propose a Bayesian estimation method which jointly infer the number of spikes and their multiplicities. The experiments that we carried out support the performance of the proposed technique.
Fig. 1 .
1Histogram of the eigenvalues of the empirical covariance matrix.
Select K such that it maximizes the maximum likelihood function.
•
Model A: m = 4, with m 1 = 1, m 2 = 2, m 3 = 1; • Model B: m = 8, with m 1 = 2, m 2 = 4, m 3 = 2; • Model C: m = 12, with m 1 = 3, m 2 = 6, m 3 = 3;
TABLE I EMPIRICAL
IPROBABILITY OF P(K = K) AS A FUNCTION OF THE σ 2 .σ 2 (dB)
-50
-40
-30
-20
-10 -6.99 -5.223 -3.98 -3.01 -2.22
P(K = K)
0.992 0.978 0.988 0.986 0.984 0.978 0.978 0.980 0.964 0.974
SNR (dB)
-1.55 -0.97 -0.46
0
0.41 0.80 0.97 1.14 1.30 1.46
P(K = K)
0.972 0.954 0.960 0.968 0.942 0.926 0.896 0.850 0.694 0.476
100
200
300
400
500
0.0
0.2
0.4
0.6
0.8
1.0
p
Frequency of correct estimation
Model A
Model B
Model C
A discrete distribution for powers has been considered in[4].
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| [] |
[
"Self-Supervised Bernoulli Autoencoders for Semi-Supervised Hashing",
"Self-Supervised Bernoulli Autoencoders for Semi-Supervised Hashing"
] | [
"Francisco Ricardoñanculef s:[email protected] \nDepartment of Informatics\nDepartment of Computer Science and Engineering\nFederico Santa María Technical University\n8940572SantiagoChile\n",
"Mena \nDepartment of Informatics\nDepartment of Computer Science and Engineering\nFederico Santa María Technical University\n8940572SantiagoChile\n",
"Antonio Macaluso s:[email protected] \nUniversity of Bologna\n40136Bologna BOItaly\n",
"Stefano Lodi [email protected] \nUniversity of Bologna\n40136Bologna BOItaly\n",
"Claudio Sartori [email protected] \nUniversity of Bologna\n40136Bologna BOItaly\n"
] | [
"Department of Informatics\nDepartment of Computer Science and Engineering\nFederico Santa María Technical University\n8940572SantiagoChile",
"Department of Informatics\nDepartment of Computer Science and Engineering\nFederico Santa María Technical University\n8940572SantiagoChile",
"University of Bologna\n40136Bologna BOItaly",
"University of Bologna\n40136Bologna BOItaly",
"University of Bologna\n40136Bologna BOItaly"
] | [] | Semantic hashing is an emerging technique for large-scale similarity search based on representing highdimensional data using similarity-preserving binary codes used for efficient indexing and search. It has recently been shown that variational autoencoders, with Bernoulli latent representations parametrized by neural nets, can be successfully trained to learn such codes in supervised and unsupervised scenarios, improving on more traditional methods thanks to their ability to handle the binary constraints architecturally. However, the scenario where labels are scarce has not been studied yet.This paper investigates the robustness of hashing methods based on variational autoencoders to the lack of supervision, focusing on two semi-supervised approaches currently in use. The first augments the variational autoencoder's training objective to jointly model the distribution over the data and the class labels. The second approach exploits the annotations to define an additional pairwise loss that enforces consistency between the similarity in the code (Hamming) space and the similarity in the label space. Our experiments show that both methods can significantly increase the hash codes' quality. The pairwise approach can exhibit an advantage when the number of labelled points is large. However, we found that this method degrades quickly and loses its advantage when labelled samples decrease. To circumvent this problem, we propose a novel supervision method in which the model uses its label distribution predictions to implement the pairwise objective. Compared to the best baseline, this procedure yields similar performance in fully supervised settings but improves significantly the results when labelled data is scarce. Our code is made publicly available at https://github.com/amacaluso/SSB-VAE. | 10.1007/978-3-030-93420-0_25 | [
"https://arxiv.org/pdf/2007.08799v1.pdf"
] | 220,633,411 | 2007.08799 | 084defe86d31988c44c0617817ba65df62ce1900 |
Self-Supervised Bernoulli Autoencoders for Semi-Supervised Hashing
Francisco Ricardoñanculef s:[email protected]
Department of Informatics
Department of Computer Science and Engineering
Federico Santa María Technical University
8940572SantiagoChile
Mena
Department of Informatics
Department of Computer Science and Engineering
Federico Santa María Technical University
8940572SantiagoChile
Antonio Macaluso s:[email protected]
University of Bologna
40136Bologna BOItaly
Stefano Lodi [email protected]
University of Bologna
40136Bologna BOItaly
Claudio Sartori [email protected]
University of Bologna
40136Bologna BOItaly
Self-Supervised Bernoulli Autoencoders for Semi-Supervised Hashing
Semantic hashing is an emerging technique for large-scale similarity search based on representing highdimensional data using similarity-preserving binary codes used for efficient indexing and search. It has recently been shown that variational autoencoders, with Bernoulli latent representations parametrized by neural nets, can be successfully trained to learn such codes in supervised and unsupervised scenarios, improving on more traditional methods thanks to their ability to handle the binary constraints architecturally. However, the scenario where labels are scarce has not been studied yet.This paper investigates the robustness of hashing methods based on variational autoencoders to the lack of supervision, focusing on two semi-supervised approaches currently in use. The first augments the variational autoencoder's training objective to jointly model the distribution over the data and the class labels. The second approach exploits the annotations to define an additional pairwise loss that enforces consistency between the similarity in the code (Hamming) space and the similarity in the label space. Our experiments show that both methods can significantly increase the hash codes' quality. The pairwise approach can exhibit an advantage when the number of labelled points is large. However, we found that this method degrades quickly and loses its advantage when labelled samples decrease. To circumvent this problem, we propose a novel supervision method in which the model uses its label distribution predictions to implement the pairwise objective. Compared to the best baseline, this procedure yields similar performance in fully supervised settings but improves significantly the results when labelled data is scarce. Our code is made publicly available at https://github.com/amacaluso/SSB-VAE.
Abstract-Semantic hashing is an emerging technique for large-scale similarity search based on representing highdimensional data using similarity-preserving binary codes used for efficient indexing and search. It has recently been shown that variational autoencoders, with Bernoulli latent representations parametrized by neural nets, can be successfully trained to learn such codes in supervised and unsupervised scenarios, improving on more traditional methods thanks to their ability to handle the binary constraints architecturally. However, the scenario where labels are scarce has not been studied yet.
This paper investigates the robustness of hashing methods based on variational autoencoders to the lack of supervision, focusing on two semi-supervised approaches currently in use. The first augments the variational autoencoder's training objective to jointly model the distribution over the data and the class labels. The second approach exploits the annotations to define an additional pairwise loss that enforces consistency between the similarity in the code (Hamming) space and the similarity in the label space. Our experiments show that both methods can significantly increase the hash codes' quality. The pairwise approach can exhibit an advantage when the number of labelled points is large. However, we found that this method degrades quickly and loses its advantage when labelled samples decrease. To circumvent this problem, we propose a novel supervision method in which the model uses its label distribution predictions to implement the pairwise objective. Compared to the best baseline, this procedure yields similar performance in fully supervised settings but improves significantly the results when labelled data is scarce. Our code is made publicly available at https://github.com/amacaluso/SSB-VAE.
I. INTRODUCTION
Given a dataset D = {x (1) , x (2) , . . . , x (N ) }, with x ( ) ∈ X ∀ ∈ [N ], similarity search is the problem of finding the elements of D that are similar to a query object q ∈ X, not necessarily in D. This is a fundamental task in computer science, lying at the foundation of many algorithms for pattern recognition. The rapid increase in the amount of highdimensional data such as images, audio and text, has increased the interest for this type of search in the last years and raised the need for methods that can approach the task with reduced processing time and memory footprint.
If X is equipped with a similarity function s : X × X → R 1 and n is small, a simple approach to solve this problem is a linear scan: compare q with all the elements in D 1 the greater the value of s, the more similar are the objects.
and return x ( ) if s(x ( ) , q) is greater than some threshold θ. The value of θ can be given in advance, computed to return exactly k results or, more often, chosen to maximize information retrieval metrics such as precision and recall [1]. If X ⊂ R d , with small d, tree-based indexing methods such as KD-trees have been traditionally used to perform efficient scans when N is large. Unfortunately, if also d is large, the computational performance of these data structures quickly degrades. Besides, if the similarity function s determining the relevant results is not perfectly known, these methods cannot be used.
Semantic hashing deals with similarity search by learning a similarity-preserving hash function h(x) ∈ {0, 1} B that maps similar data to nearby positions in a hash table, preventing at the same time undesirable collisions. Items similar to a query q can be easily found by just accessing all the cells of the table that differ a few bits from h(q). As binary codes are storage-efficient, these operations can be performed in main memory even for very large datasets.
Although early hashing algorithms were randomized methods devised to preserve specific and well-known similarity functions (e.g. cosine) [8], it was soon realized that methods based on machine learning could significantly reduce the number (B) of bits required to preserve similarity by exploiting the fact that real data is often not uniformly distributed in X. One of the first methods of this type [19] used a deep probabilistic model to learn a manifold underlying the data distribution. Unfortunately, training this model was hard in practice and perhaps for this reason, most subsequent research on hashing preferred to adopt more shallow architectures or, slightly later, deterministic neural network models that were easier to train. Shallow algorithms that flowered in this period include Spectral Hashing [25], Iterative Quantization [6], Kernel Locality Sensitive Hashing [12] and LDA-Hash [21]. Popular examples of non-probabilistic deep learning methods include the Binary Autoencoder of [2], UH-BDNN [5], Deep Hashing [15], most of them based on deterministic autoencoders augmented with constraints.
In the last years, machine learning has seen a renewed interest in probabilistic graphical models parametrized by neural nets. A hallmark of this approach is the ability to back-propagate gradients through stochastic layers with low variance [10,9], which has permitted to scale these models to very large datasets and improve performance in many tasks. In particular, stochastic models taking advantage of variational autoencoders have shown to systematically outperform more traditional hashing algorithms [3]. It has been shown that this advantage can be further improved using Bernoulli latent representations that naturally encode the binary constraint underlying hash codes and thus reduce the quantization error arising from continuous representations [17]. Other recent contributions have shown that these models can be easily extended to leverage supervision, that is labels conveying information about the semantic content of the items to be indexed. Two different supervision schemes have been proposed. One of them, often termed pointwise supervision in earlier literature, augments the training objective to predict the label distribution of a training pattern [3]. The other method, often termed pairwise supervision in previous art, exploits the labels to define an additional objective in which pairs of items with the same label are required to have similar hash codes and pairs of items with different labels are enforced to have different hash codes. This approach yields state-of-the-art performance in [4] assuming that the labels of all the training examples are known.
As in many real-world tasks, obtaining labelled data is difficult and time-consuming, understanding the efficacy of the current methods to exploit annotations when they are scarce is a matter of significant importance in practice. In this paper, we show that even if using the ground-truth labels to introduce pairwise constraints can yield advantages when the number of labelled samples is large, this approach degrades quickly as the level of supervision decreases. We found that in many cases, it actually looses its advantage with respect to a model employing pointwise supervision only. To overcome this limitation, we propose to equip the variational autoencoder with a novel mechanism to exploit annotations in which the label distributions required for the pairwise loss are replaced by the model's own predictions. Ground-truth labels are still used to supervise the estimation of the label distributions but the pairwise constraints ask now for a consistency between the codes and the model's own beliefs about the class of the patterns. Experiments on text and image retrieval tasks show that this procedure is competitive to the best baseline in scenarios of label abundance, but it is more effective in scenarios of label scarcity.
II. RELATED WORK
The problem of representing high-dimensional data using binary codes that preserve their semantic content and support efficient indexing has been extensively studied in the literature. Traditionally, a first distinction is made between data dependent and data independent methods. Data independent methods are randomized techniques devised to preserve specific similarity functions (e.g. cosine) or distance metrics (e.g. Euclidean) [8]. They have good theoretical grounds, but usually they require long hash codes to achieve satisfactory performance. Data dependent methods can often obtain more effective and compact hash codes by learning the underlying structure of the dataset [25,19]. Many techniques to achieve this goal have been studied in the last years, including unsupervised, supervised, and semi-supervised approaches.
Unsupervised methods rely purely on the properties of the points to be indexed. For instance, Iterative Quantization (ITQ) [6] computes the codes by applying PCA followed by a rotation that minimizes the quantization error arising from thresholding. Spectral Hashing (SpH) [25] poses hashing as the problem of partitioning a graph that encodes information about the geometry of the dataset. Recently, more flexible unsupervised models based on autoencoders have been proposed. In [2], a classic (deterministic) autoencoder is trained for hashing by minimizing the reconstruction error with an explicit constraint to handle the quantization error. The constraint prevents the model to be trained with back-propagation and a mixed integer programming solver needs to be applied. The method in [5] employs an deeper architecture for the encoder but keeps the architecture of the decoder. Binary representations are enforced in the vein of [2], using constraints. In spite of their computational complexity, experimental results of [2] and [5] are encouraging and suggest that hashing based on autoencoders can outperform other deep learning methods such as [16]. Our work is more related to the methods in [3], [17] and [4], which make use of deep variational autoencoders for hashing. Unlike classic autoencoders that learn a one-toone map between the input space and the Hamming space, variational autoencoders learn the most likely region of the code space where an input pattern should be allocated. In order to reconstruct data from the latent representation, the model is constrained to place similar data in a similar region of the code space, reproducing in the training phase the mechanism that will be used later for search. [3] showed that this fundamental difference between classic and variational autoencoders is relevant for hashing and yields to significantly better results. Later, [17] demonstrated that the use of Bernoulli instead of Gaussian latent variables helps to reduce the quantization loss arising from the use of continuous representations. This idea is also used in [4] and extended to incorporate supervision.
Supervised hashing algorithms leverage information about the semantics of the items in the dataset to improve the hash codes. Pointwise methods such as [24] use labels or tags to implicitly enforce a consistency between the codes and the annotations. Pairwise methods assume that pairs of objects have been annotated as similar or dissimilar and attempt to explicitly preserve these similarities in the formulation. In [16] pairwise similarity relations are derived from class labels and integrated into the training objective of a neural net. Deep methods that leverage information from triplets or lists have also been proposed [13]. A different form of supervision that is often combined with deep learning techniques is selftaught hashing [28]. In this approach, a classic (often shallow) method is first used to find hash codes for a set of training examples. Then, a supervised (often deep) model is used to predict the codes of unseen data. A limitation of this approach is the sensitivity of the first step to the features used to represent the data, which may require to iterate the two steps.
Hashing methods tailored to semi-supervised scenarios, in which labelled and unlabelled samples are available, have started to be studied in the last years. They integrate supervised and unsupervised learning mechanisms. The method in [5] extends the objective function of a traditional (unsupervised) auto-encoder with an term based on pairwise supervision. [23] presents various methods based on linear projections, which combine pairwise supervision with an unsupervised learning goal inspired in information theory (max entropy). In [20], pointwise and pairwise supervision schemes and combined with spectral methods [25]. [18] uses pairwise and triplet-wise supervision to extend ITQ, a linear method for unsupervised hashing [6]. Building on the idea of self-training [22], a classic approach for dealing with partially labelled datasets, [27] has recently explored an iterative method in which label representations and hash codes are learned together in the model. By predicting the labels of data without annotations, the labelled dataset can be expanded and the model retrained. While fairly successful, this hashing model requires iterative re-training. Our method differs from this approach also in that it incorporates an explicit unsupervised learning mechanism -the supervised one being complementary. Other recent methods termed self-trained or self-supervised in the literature use actually a different approach. The method in [14] is based on learning label encodings that substitute standard one-hot vectors. The method in [26] is actually an application of self-taught hashing to cross-modality hashing in which the first stage exploits pairwise supervision. In computer vision, self-supervised hashing methods often uses domain information that cannot be easily generalized to other tasks. For instance [29] uses the different frames of a video to build similar pairs of examples and randomly picked frames to form dissimilar pairs. Image rotations or image patching have also been employed in other works.
Besides being able to learn compact and effective hash codes in a principled unsupervised way, variational autoencoders can be easily extended to exploit annotations. The seminal method in [3] has shown indeed that by training the model to learn both the data distribution and the label distribution, one can dramatically increase the efficacy of the hash codes in similarity search tasks. Building on this idea, [4] proposed to further extend the objective function of the model by using pairwise supervision. This approach yields state-of-the-art performance in [4] assuming that the labels of all the training examples are known. Our work extends these recent studies by considering a semi-supervised scenario in which a small set of instances have been annotated with class labels. We show that in this setting, the advantage of the pairwise method can significantly degrade when the number of annotated samples is small. Up to the best of our knowledge, the method we propose to deal with this issue has not been previously explored. Certainly, it can be connected with co-supervised methods [22] in which two or more learners iteratively teach each other to substitute the lack of annotations. These methods often used different and conditionally independent feature representations of the data. Our method does not use multiple training stages, resorts on a single feature representation, is used inside the same model and is specifically tailored to hashing with variational autoencoders.
III. METHODS
A. Generative Model
As in related works [17], we pose hashing as an inference problem, where the objective is to learn a probability distribution q φ (b|x) of the code b ∈ {0, 1} B corresponding to an input pattern x. This framework is based on a generative process involving two steps: (i) choose an entry of the hash table according to some probability distribution p θ (b), and (ii) sample an observation x indexed by that address according to a conditional distribution p θ (x|b). The parameters of this random process are learnt in such a way that it approximates the real data distribution.
B. Bernoulli Autoencoders
Following [10], the distribution q φ (b|x) is called the encoder, and the distribution p θ (x|b) the decoder. In the original construction, q φ (b|x) is chosen to be a Gaussian N (µ φ (x), σ 2 φ (x)) and binary codes are obtained by thresholding µ φ (x) around its empirical median [3]. In Bernoulli variational autoencoders (B-VAEs) in contrast, the encoder is chosen to be a multi-variate Bernoulli Ber(α(x)) with activation probabilities α(x). This choice permits to handle the binary constraint underlying hashing in an architectural way, creating an inductive bias that can significantly reduce the quantization loss incurred from thresholding Gaussian representations [17].
C. Parametrization by Neural Nets
To learn flexible non-linear mappings, the activation probabilities of the encoder α(x) can be represented using a neural net f (x; φ). The architecture of this model is chosen according to the dataset. In the simplest case, it is obtained as the composition of L fully-connected layers f 1 • . . . f L−1 • f L where f 1 : X → R n1 accommodates the input data (a feature vector) and f L : R n L → [0, 1] B produces the activation probabilities. The latter is usually obtained by using a layer of independent sigmoid neurons [7].
The architecture for the decoder depends also on the application. In regression problems with real data, p θ (x|b) is often implemented using a Gaussian N (µ θ (x), σ 2 ) where µ θ ((x) is predicted using a neural net g(b; θ) with linear output layer. In text and image retrieval applications it is more common to represent the data using normalized features x i ∈ [0, 1] (word frequencies or pixels). In this case g(b; θ) can be implemented using a net g 1 •. . . g L −1 •g L that ends with a layer of sigmoid neurons (other layers can use other activations, e.g. ReLU).
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I i O H G 5 K a R C / G C u f f Y v 0 s N X o + G 5 O R 4 o P E z H m 7 n b B n C 3 J Z v E E U p D B m m Z Q 0 j S 1 R M o Y + H Q r j b J 6 M 5 t N F c j 6 R G g q I m 6 P i H u J m 8 d b p U E a g I P a z E Y N M c Z P I D T 8 f 0 g b z c f w F c q D G v s u c j f 0 z T v h 0 d t c F v C R 9 K r m Y z E R Y o v u X e n C j g f d T M c G 8 j 0 0 C u P g F c N E H z q a O y y v C X G O T 4 Q i w p q n G G h T v V + c a 0 Y w K L 1 P M t 6 e N B 5 A O f C P N M d U T f x H O n 7 s S 5 5 V S I S Z j H o P h I o a u H y T p J F y 3 z U S A b V Q x J h p M l F 0 E b 0 j i P 7 Z O 2 n Q E c R m v h X U 3 p V z
V s a 9 a I f n 0 O f f T X q S W 6 U q 2 A s M S k / l Z F L I L c M U k 8 h x U 2 t q Q Q / J 2 z 9 n W V r g V k H E c O I 9 f k 8 Q E Z U l n 9 5 1 l z r 5 3 t u N s 2 / t c 0 Z m f L D G g T S 2 0 9 V L Z Q n + 2 F j r S L C U z x X O j y 4 r / I / 2 b W q 2 T n C p X 7 u B O a X g K t j d n Y C f 8 v Y P H G 6 3 Q 4 8 P N 2 m 4 w 2 8 Y V 9 B Q 9 R y 9 Q i L b R L v q A D t A R Y g j Q V / Q N f V / + W V m t P K 4 8 m V J v L M 3 O P E J z V n n 2 C 4 b c e d g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A H 3 U K U u 7 A 5 W D 4 1 Z P O q C I a V j m l o s = " > A A A E Z 3 i c h V N t a x N B E N 7 a q D W + t F U Q w S + r S U A k h r v S W v p B q F R E U G l r 0 h f I h r K 3 N 8 k t 3 b 0 7 d j d J w 7 K / w K / 6 4 / w J / g v 3 L q l t I u r A 3 T 3 M P D v z z O x N l A u u T R D 8 W L q x X L l 5 6 / b K n e r d e / c f r K 6 t P z z W 2 V A x O G K Z y N R p R D U I n s K R 4 U b A a a 6 A y k j A S X S + V 8 R P R q A 0 z 9 K O m e T Q k 3 S Q 8 j 5 n 1 H j X 4 c X Z W i 1 o B a X h P 0 E 4 A z U 0 s 4 O z 9 e U D E m d s K C E 1 T F C t u 2 G Q m 5 6 l y n A m w F U b Z K g h p + y c D q C r c 5 p y n f R s R C M Q C 0 G a 5 w I k Z T 3 L 0 3 x o I G W u e i 1 u q d R 6
I i O H G 5 K a R C / G C u f f Y v 0 s N X o + G 5 O R 4 o P E z H m 7 n b B n C 3 J Z v E E U p D B m m Z Q 0 j S 1 R M o Y + H Q r j b J 6 M 5 t N F c j 6 R G g q I m 6 P i H u J m 8 d b p U E a g I P a z E Y N M c Z P I D T 8 f 0 g b z c f w F c q D G v s u c j f 0 z T v h 0 d t c F v C R 9 K r m Y z E R Y o v u X e n C j g f d T M c G 8 j 0 0 C u P g F c N E H z q a O y y v C X G O T 4 Q i w p q n G G h T v V + c a 0 Y w K L 1 P M t 6 e N B 5 A O f C P N M d U T f x H O n 7 s S 5 5 V S I S Z j H o P h I o a u H y T p J F y 3 z U S A b V Q x J h p M l F 0 E b 0 j i P 7 Z O 2 n Q E c R m v h X U 3 p V z
V s a 9 a I f n 0 O f f T X q S W 6 U q 2 A s M S k / l Z F L I L c M U k 8 h x U 2 t q Q Q / J 2 z 9 n W V r g V k H E c O I 9 f k 8 Q E Z U l n 9 5 1 l z r 5 3 t u N s 2 / t c 0 Z m f L D G g T S 2 0 9 V L Z Q n + 2 F j r S L C U z x X O j y 4 r / I / 2 b W q 2 T n C p X 7 u B O a X g K t j d n Y C f 8 v Y P H G 6 3 Q 4 8 P N 2 m 4 w 2 8 Y V 9 B Q 9 R y 9 Q i L b R L v q A D t A R Y g j Q V / Q N f V / + W V m t P K 4 8 m V J v L M 3 O P E J z V n n 2 C 4 b c e d g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " v S k A k B 9 I l 0 h 6 4
F 6 K 0 P j B p o Z T J l Y = " > A A A E W H i c h V N d a x N B F J 2 2 U W t s t Q X x x Z f B J C B S w 2 6 h + i Q o F R F U W k 3 6 A Z l Q Z m d v s k N n Z p e Z 2 c Q w z B / w 1 V / n L / B v O L t J a Z O i D u z m c O / J v e e e 2 Z s U g h s b R b / W 1 j c a d + 7 e 2 7 z f f L D V 3 H 7 4 a G f r 1 O S l Z n D C c p H r 8 4 Q a E F z B i e V W w H m h g c p E w F l y e V j l z y a g D c 9 V 3 8 4 K G E o 6 V n z E G b U h d H y x 0 4 q 6 U X 3 w b R A v Q A s t z s X u x j F J c 1 Z K U J Y J a s w g j g o 7 d F R b z g T 4
Z o e U B g r K L u k Y B q a g i p t s 6 B K a g F h J 0 q I Q I C k b O q 6 K 0 o J i v n k j 7 6 g 0 Z i Y T j z u S 2 s y s 5 q r g 3 3 K j X F m z X I 3 J R P N x Z p e i g 3 4
8 d B W 5 b t 4 h G h R M W S 4 l V a k j W q Y w o q W w 3 h X Z Z L l c I p c L 6 V J A u j e p b i H d q 9 5 G l T I B D W n w R o x z z W 0 m 9 4 M / p A f 2 0 / Q b F E C t e 5 9 7 l 4 Z n m v G 5 d z c F v C A j K r m Y L U Q 4 Y k Z X e n C n g 4 + U m G E + w j Y D X H 0 A u J o D 5 / P A 1 R V h b r D N c Q L Y U G W w A c 1 H z a V B D K M i y B T L 4 x k b A K h x G G R v S s 0 s X I Q P / 7 s W F 5 R S I W Z T n o L l I o V B M J L 0 M 2 5 6 d i b A d Z o Y E w M 2 y b 9 H b 0 g W f l y b 9 O g E 0 j r f i t t + T r n u 4 1 5 2 Y / L 5 S x H c X q X W 5 W q 2 B s s y m w c v K t k V u G Y S e Q l a d f d l S d 4 d e t c 9 i A 8 i M k 0 j H / A r k t m o b u n d k X f M u w / e 9 b 3 r h Z i v J g v O E g v G t m L X r p W t z O d a s S d 7 t W S m e W F N 3 f F / p H 9 T m 2 1 S U O 3 D C s a r C 3 c b n O 5 3 4 4 C / R m g T P U X P 0 H M U o 9
f o L f q I j t E J Y i h F P 9 D P j d + N 7 c b j + a q u r y 1 2 d h c t n c a T P 4 f z d x Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " T M I N Q u I
x C 9 f k J A A f f p Z g b 9 i Q l A o = " > A A A E X H i c h V N d a x N B F J 2 2 U W u s t v V B B F 8 G k 4 B I D L u B 6 p O g V E R Q a W v S D 8 i E M j t 7 k x 0 6 s 7 v M z C Y N w / w C X / X H + R P 8 F 8 5 u U t q k q A O 7 O d x 7 c u + 5 Z / Z G u e D a B M G v t f W N 2 p 2 7 9 z b v 1 x 9 s P X y 0 v b O 7 d a K z Q j E 4 Z p n I 1 F l E N Q i e w r H h R s B Z r o D K S M B p d L F f 5 k 8 n o D T P 0 r 6 Z 5 T C U d J z y E W f U + N D R 5 f l O I + g E 1 c G 3 Q b g A D b Q 4 h + e 7 G 4 c k z l g h I T V M U K 0 H Y Z C b o a X K c C b A 1 V u k 0 J B T d k H H M N A 5 T b l O h j a i E Y i V J M 1 z A Z K y o e V p X h h I m a v f y F s q t Z 7 J y O G W p C b R q 7 k y + L f c K E u N X q 7 G Z K T 4 O D F L 0 U E / H N q S X D V v E Q U p T F k m J U 1 j S 5 S M Y U Q L Y Z z N
k 8 l y u U g u F 1 K F g L g 9 K e 8 h b p d v n R Y y A g W x 9 0 a M M 8 V N I r v e H 9 I D 8 3 n 6 D X K g x n 7 I n I 3 9 M 0 3 4 3 L u b A l 6 S E Z V c z B Y i L N G j K z 2 4 1 c I H q Z h h P s I m A V x + A r i c A 2 f z w N U V Y a 6 x y X A E W N N U Y w 2 K j + p L g 2 h G h Z c p l s f T x g N I x 3 6 Q 9 p T q m b 8 I 5 / 9 3 L c 4 r p U L M p j w G w 0 U M A 2 8 k 6 S d c 9 8 x M g G 3 V M S Y a T J R d B m 9 J 4 n 9 s k / T o B O I q 3 w i b b k 6 5 7 m N f d U L y 5 W v u 3 V 6 l V u U q t g L D E p N 5 L 0 r Z J b h m E n k B K u 1 0 Z U H e 7 z v b 2 Q v 3 A j K N A + f x a 5 K Y o G r p 7 I G z z N m P z v a d 7 f m Y K y f z z h I D 2 j R C 2 6 y U r c x n G 6 E j 7 U o y U z w 3 u u r 4 P 9 K / q f U m y a l y f g f D 1 Y 2 7 D U 6
6 n d D j o w B t o m f o O X q B Q v Q G v U O f 0 C E 6 R g w B + o 5 + o J 8 b v 2 v b t S f z b V 1 f W 6 z t Y 7
R 0 a k / / A A P J e I U = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " T M I N Q u I
x C 9 f k J A A f f p Z g b 9 i Q l A o = " > A A A E X H i c h V N d a x N B F J 2 2 U W u s t v V B B F 8 G k 4 B I D L u B 6 p O g V E R Q a W v S D 8 i E M j t 7 k x 0 6 s 7 v M z C Y N w / w C X / X H + R P 8 F 8 5 u U t q k q A O 7 O d x 7 c u + 5 Z / Z G u e D a B M G v t f W N 2 p 2 7 9 z b v 1 x 9 s P X y 0 v b O 7 d a K z Q j E 4 Z p n I 1 F l E N Q i e w r H h R s B Z r o D K S M B p d L F f 5 k 8 n o D T P 0 r 6 Z 5 T C U d J z y E W f U + N D R 5 f l O I + g E 1 c G 3 Q b g A D b Q 4 h + e 7 G 4 c k z l g h I T V M U K 0 H Y Z C b o a X K c C b A 1 V u k 0 J B T d k H H M N A 5 T b l O h j a i E Y i V J M 1 z A Z K y o e V p X h h I m a v f y F s q t Z 7 J y O G W p C b R q 7 k y + L f c K E u N X q 7 G Z K T 4 O D F L 0 U E / H N q S X D V v E Q U p T F k m J U 1 j S 5 S M Y U Q L Y Z z N
k 8 l y u U g u F 1 K F g L g 9 K e 8 h b p d v n R Y y A g W x 9 0 a M M 8 V N I r v e H 9 I D 8 3 n 6 D X K g x n 7 I n I 3 9 M 0 3 4 3 L u b A l 6 S E Z V c z B Y i L N G j K z 2 4 1 c I H q Z h h P s I m A V x + A r i c A 2 f z w N U V Y a 6 x y X A E W N N U Y w 2 K j + p L g 2 h G h Z c p l s f T x g N I x 3 6 Q 9 p T q m b 8 I 5 / 9 3 L c 4 r p U L M p j w G w 0 U M A 2 8 k 6 S d c 9 8 x M g G 3 V M S Y a T J R d B m 9 J 4 n 9 s k / T o B O I q 3 w i b b k 6 5 7 m N f d U L y 5 W v u 3 V 6 l V u U q t g L D E p N 5 L 0 r Z J b h m E n k B K u 1 0 Z U H e 7 z v b 2 Q v 3 A j K N A + f x a 5 K Y o G r p 7 I G z z N m P z v a d 7 f m Y K y f z z h I D 2 j R C 2 6 y U r c x n G 6 E j 7 U o y U z w 3 u u r 4 P 9 K / q f U m y a l y f g f D 1 Y 2 7 D U 6
6 n d D j o w B t o m f o O X q B Q v Q G v U O f 0 C E 6 R g w B + o 5 + o J 8 b v 2 v b t S f z b V 1 f W 6 z t Y 7
R 0 a k / / A A P J e I U = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " b a J 6 7 e 3 0 J y y N V j s
b 1 o L U h W 1 Y m 0 Y = " > A A A E Z 3 i c h V P t b t M w F P W 2 A q N 8 b A M J I f H H 0 F Z C q F T J p M E v p K E h h A R o G + 0 + p L q a H O e 2 s W Y n k e 2 0 q y w / A X / h 4 X g E 3 g I n 7 b S 1 E 2 A p y d G 9 x / e e e x x H u e D a B M G v l d W 1 2 q 3 b d 9 b v 1 u / d f / B w Y 3 P r 0 b H O C s X g i G U i U 6 c R 1 S B 4 C k e G G w G n u Q I q I w E n 0 f l e m T 8 Z g 9 I 8 S 3 t m m s N A 0 l H K h 5 x R 4 0 O H F 2 e b j a A T V A v f B O E c N N B 8 H Z x t r R 2 Q O G O F h N Q w Q b X u h 0 F u B p Y q w 5 k A V 2 + R Q k N O 2 T k d Q V / n N O U 6 G d i I R i C W k j T P B U j K B p
a n e W E g Z a 5 + L W + p 1 H o q I 4 d b k p p E L + f K 4 N 9 y w y w 1 e r E a k 5
H i o 8 Q s R P u 9 c G B L c t W 8 R R S k M G G Z l D S N L V E y h i E t h H E 2 T 8 a L 5 S K 5 W E g V A u L 2 u D y H u F 2 + d V r I C B T E 3 h s x y h Q 3 i d z 2 / p A u m M + T b 5 A D N f Z D 5 m z s n 0 n C Z 9 5 d F / C K D K n k Y j o X Y Y k e X u r B r R b e T 8 U U 8 y E 2 C e D y F 8 D l H D i b B S 6 P C H O N T Y Y j w J q m G m t Q f F h f G E Q z K r x M s T i e N h 5 A O v K D t C d U T / 1 B O L / v S p x X S o W Y T n g M h o s Y + t 5 I 0 k u 4 7 p q p A N u q Y 0 w 0 m C i 7 C N 6 R x H 9 s k 3 T p G O I q 3 w i b b k a 5 6 m N f d 0 L y 5 W v u 3 V 6 m V u U q t g L D E p N 5 L 0 r Z J b h i E n k O K u 1 s y 4 K 8 3 3 O 2 s x P u B G Q S B 8 7 j N y Q x Q d X S 2 X 1 n m b M
f n e 0 5 2 / U x V 0 7 m n S U G t G m E t l k p W 5 r P N k J H 2 p V k p n h u d N X x f 6 R / U + t N k l P l / B 0 M l 2 / c T X C 8 3 Q k 9 P g w a u 8 H 8 N q 6 j Z + g F e o l C 9 B b t o k / o A B 0 h h g B 9 R z / Q z 7 X f t Y 3 a k 9 r T G X V 1 Z b 7 n M V p Y t e d / A E e m e U Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "
A A n E L U X f G l w 5 W 7 L Y T b A v b v T W I K k = " > A A A E Z 3 i c h V P t b t M w F P W 2 A q N 8 b A M J I f H H 0 F Z C q F T J x O A X 0 t A Q Q g K 0 Q b s P q a 4 m x 7 l t r N l J Z D v t K s t P w F 9 4 O B 6 B t 8 B J O 2 3 t B F h K c n T v 8 b 3 n H s d R L r g 2 Q f B r Z X W t d u P m r f X b 9 T t 3 7 9 3 f 2 N x 6 c K S z Q j E 4 Z J n I 1 E l E N Q i e w q H h R s B J r o D K S M B x d L Z X 5 o / H o D T P 0 p 6 Z 5 j C Q d J T y I W f U + N D X 8 9 P N R t A J q o W v g 3 A O G m i + D k 6 3 1 g 5 I n L F C Q m q Y o F r 3 w y A 3 A 0 u V 4 U y A q 7 d I o S G n 7 I y O o K 9 z m n K d D G x E I x B L S Z r n A i R l A 8 v T v D C Q M l e / k r d U a j 2 V k c M t S U 2 i l 3 N l 8 G + 5 Y Z Y a v V i N y U j x U W I W o v 1 e O L A l u W r e I g p S m L B M S p r G l i g Z w 5 A W w j i b J + P F c p F c L K Q K A X F 7 X J 5 D 3 C 7 f O i 1 k B A p i 7 4 0 Y Z Y q b R G 5 7 f 0 g X z K f J N 8 i B G v s + c z b 2 z y T h M + + u C n h B h l R y M Z 2 L s E Q P L / T g V g v v p 2 K K + R C b B H D 5 C + B y D p z N A h d H h L n G J s M R Y E 1 T j T U o P q w v D K I Z F V 6 m W B x P G w 8 g H f l B 2 h O q p / 4 g n N 9 3 K c 4 r p U J M J z w G w 0 U M f W 8 k 6 S V c d 8 1 U g G 3 V M S Y a T J S d B 2 9 J 4 j + 2 S b p 0 D H G V b 4 R N N 6 N c 9 r E v O y H 5 / C X 3 b i 9 T q 3 I V W 4 F h i c m 8 F 6 X
s E l w y i T w D l X a 2 Z U H e 7 T n b 2 Q l 3 A j K J A + f x a 5 K Y o G r p 7 L 6 z z N k P z v a c 7 f q Y K y f z z h I D 2 j R C 2 6 y U L c 1 n G 6 E j 7 U o y U z w 3 u u r 4 P 9 K / q f U m y a l y / g 6 G y z f u O j j a 7 o Q e f 3 3 V 2 A 3 m t 3 E d P U H P 0 H M U o j d o F 3 1 E B + g Q M Q T o O / q B f q 7 9 r m 3 U H t U e z 6 i r K / M 9 D 9 H C q j 3 9 A 0 j m e U g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "
A A n E L U X f G l w 5 W 7 L Y T b A v b v T W I K k = " > A A A E Z 3 i c h V P t b t M w F P W 2 A q N 8 b A M J I f H H 0 F Z C q F T J x O A X 0 t A Q Q g K 0 Q b s P q a 4 m x 7 l t r N l J Z D v t K s t P w F 9 4 O B 6 B t 8 B J O 2 3 t B F h K c n T v 8 b 3 n H s d R L r g 2 Q f B r Z X W t d u P m r f X b 9 T t 3 7 9 3 f 2 N x 6 c K S z Q j E 4 Z J n I 1 E l E N Q i e w q H h R s B J r o D K S M B x d L Z X 5 o / H o D T P 0 p 6 Z 5 j C Q d J T y I W f U + N D X 8 9 P N R t A J q o W v g 3 A O G m i + D k 6 3 1 g 5 I n L F C Q m q Y o F r 3 w y A 3 A 0 u V 4 U y A q 7 d I o S G n 7 I y O o K 9 z m n K d D G x E I x B L S Z r n A i R l A 8 v T v D C Q M l e / k r d U a j 2 V k c M t S U 2 i l 3 N l 8 G + 5 Y Z Y a v V i N y U j x U W I W o v 1 e O L A l u W r e I g p S m L B M S p r G l i g Z w 5 A W w j i b J + P F c p F c L K Q K A X F 7 X J 5 D 3 C 7 f O i 1 k B A p i 7 4 0 Y Z Y q b R G 5 7 f 0 g X z K f J N 8 i B G v s + c z b 2 z y T h M + + u C n h B h l R y M Z 2 L s E Q P L / T g V g v v p 2 K K + R C b B H D 5 C + B y D p z N A h d H h L n G J s M R Y E 1 T j T U o P q w v D K I Z F V 6 m W B x P G w 8 g H f l B 2 h O q p / 4 g n N 9 3 K c 4 r p U J M J z w G w 0 U M f W 8 k 6 S V c d 8 1 U g G 3 V M S Y a T J S d B 2 9 J 4 j + 2 S b p 0 D H G V b 4 R N N 6 N c 9 r E v O y H 5 / C X 3 b i 9 T q 3 I V W 4 F h i c m 8 F 6 X
s E l w y i T w D l X a 2 Z U H e 7 T n b 2 Q l 3 A j K J A + f x a 5 K Y o G r p 7 L 6 z z N k P z v a c 7 f q Y K y f z z h I D 2 j R C 2 6 y U L c 1 n G 6 E j 7 U o y U z w 3 u u r 4 P 9 K / q f U m y a l y / g 6 G y z f u O j j a 7 o Q e f 3 3 V 2 A 3 m t 3 E d P U H P 0 H M U o j d o F 3 1 E B + g Q M Q T o O / q B f q 7 9 r m 3 U H t U e z 6 i r K / M 9 D 9 H C q j 3 9 A 0 j m e U g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A H 3 U K U u 7 A 5 W D 4 1 Z P O q C I a V j m l o s = " > A A A E Z 3 i c h V N t a x N B E N 7 a q D W + t F U Q w S + r S U A k h r v S W v p B q F R E U G l r 0 h f I h r K 3 N 8 k t 3 b 0 7 d j d J w 7 K / w K / 6 4 / w J / g v 3 L q l t I u r A 3 T 3 M P D v z z O x N l A u u T R D 8 W L q x X L l 5 6 / b K n e r d e / c f r K 6 t P z z W 2 V A x O G K Z y N R p R D U I n s K R 4 U b A a a 6 A y k j A S X S + V 8 R P R q A 0 z 9 K O m e T Q k 3 S Q 8 j 5 n 1 H j X 4 c X Z W i 1 o B a X h P 0 E 4 A z U 0 s 4 O z 9 e U D E m d s K C E 1 T F C t u 2 G Q m 5 6 l y n A m w F U b Z K g h p + y c D q C r c 5 p y n f R s R C M Q C 0 G a 5 w I k Z T 3 L 0 3 x o I G W u e i 1 u q d R 6
I i O H G 5 K a R C / G C u f f Y v 0 s N X o + G 5 O R 4 o P E z H m 7 n b B n C 3 J Z v E E U p D B m m Z Q 0 j S 1 R M o Y + H Q r j b J 6 M 5 t N F c j 6 R G g q I m 6 P i H u J m 8 d b p U E a g I P a z E Y N M c Z P I D T 8 f 0 g b z c f w F c q D G v s u c j f 0 z T v h 0 d t c F v C R 9 K r m Y z E R Y o v u X e n C j g f d T M c G 8 j 0 0 C u P g F c N E H z q a O y y v C X G O T 4 Q i w p q n G G h T v V + c a 0 Y w K L 1 P M t 6 e N B 5 A O f C P N M d U T f x H O n 7 s S 5 5 V S I S Z j H o P h I o a u H y T p J F y 3 z U S A b V Q x J h p M l F 0 E b 0 j i P 7 Z O 2 n Q E c R m v h X U 3 p V z
V s a 9 a I f n 0 O f f T X q S W 6 U q 2 A s M S k / l Z F L I L c M U k 8 h x U 2 t q Q Q / J 2 z 9 n W V r g V k H E c O I 9 f k 8 Q E Z U l n 9 5 1 l z r 5 3 t u N s 2 / t c 0 Z m f L D G g T S 2 0 9 V L Z Q n + 2 F j r S L C U z x X O j y 4 r / I / 2 b W q 2 T n C p X 7 u B O a X g K t j d n Y C f 8 v Y P H G 6 3 Q 4 8 P N 2 m 4 w 2 8 Y V 9 B Q 9 R y 9 Q i L b R L v q A D t A R Y g j Q V / Q N f V / + W V m t P K 4 8 m V J v L M 3 O P E J z V n n 2 C 4 b c e d g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A H 3 U K U u 7 A 5 W D 4 1 Z P O q C I a V j m l o s = " > A A A E Z 3 i c h V N t a x N B E N 7 a q D W + t F U Q w S + r S U A k h r v S W v p B q F R E U G l r 0 h f I h r K 3 N 8 k t 3 b 0 7 d j d J w 7 K / w K / 6 4 / w J / g v 3 L q l t I u r A 3 T 3 M P D v z z O x N l A u u T R D 8 W L q x X L l 5 6 / b K n e r d e / c f r K 6 t P z z W 2 V A x O G K Z y N R p R D U I n s K R 4 U b A a a 6 A y k j A S X S + V 8 R P R q A 0 z 9 K O m e T Q k 3 S Q 8 j 5 n 1 H j X 4 c X Z W i 1 o B a X h P 0 E 4 A z U 0 s 4 O z 9 e U D E m d s K C E 1 T F C t u 2 G Q m 5 6 l y n A m w F U b Z K g h p + y c D q C r c 5 p y n f R s R C M Q C 0 G a 5 w I k Z T 3 L 0 3 x o I G W u e i 1 u q d R 6
I i O H G 5 K a R C / G C u f f Y v 0 s N X o + G 5 O R 4 o P E z H m 7 n b B n C 3 J Z v E E U p D B m m Z Q 0 j S 1 R M o Y + H Q r j b J 6 M 5 t N F c j 6 R G g q I m 6 P i H u J m 8 d b p U E a g I P a z E Y N M c Z P I D T 8 f 0 g b z c f w F c q D G v s u c j f 0 z T v h 0 d t c F v C R 9 K r m Y z E R Y o v u X e n C j g f d T M c G 8 j 0 0 C u P g F c N E H z q a O y y v C X G O T 4 Q i w p q n G G h T v V + c a 0 Y w K L 1 P M t 6 e N B 5 A O f C P N M d U T f x H O n 7 s S 5 5 V S I S Z j H o P h I o a u H y T p J F y 3 z U S A b V Q x J h p M l F 0 E b 0 j i P 7 Z O 2 n Q E c R m v h X U 3 p V z
V s a 9 a I f n 0 O f f T X q S W 6 U q 2 A s M S k / l Z F L I L c M U k 8 h x U 2 t q Q Q / J 2 z 9 n W V r g V k H E c O I 9 f k 8 Q E Z U l n 9 5 1 l z r 5 3 t u N s 2 / t c 0 Z m f L D G g T S 2 0 9 V L Z Q n + 2 F j r S L C U z x X O j y 4 r / I / 2 b W q 2 T n C p X 7 u B O a X g K t j d n Y C f 8 v Y P H G 6 3 Q 4 8 P N 2 m 4 w 2 8 Y V 9 B Q 9 R y 9 Q i L b R L v q A D t A R Y g j Q V / Q N f V / + W V m t P K 4 8 m V J v L M 3 O P E J z V n n 2 C 4 b c e d g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A H 3 U K U u 7 A 5 W D 4 1 Z P O q C I a V j m l o s = " > A A A E Z 3 i c h V N t a x N B E N 7 a q D W + t F U Q w S + r S U A k h r v S W v p B q F R E U G l r 0 h f I h r K 3 N 8 k t 3 b 0 7 d j d J w 7 K / w K / 6 4 / w J / g v 3 L q l t I u r A 3 T 3 M P D v z z O x N l A u u T R D 8 W L q x X L l 5 6 / b K n e r d e / c f r K 6 t P z z W 2 V A x O G K Z y N R p R D U I n s K R 4 U b A a a 6 A y k j A S X S + V 8 R P R q A 0 z 9 K O m e T Q k 3 S Q 8 j 5 n 1 H j X 4 c X Z W i 1 o B a X h P 0 E 4 A z U 0 s 4 O z 9 e U D E m d s K C E 1 T F C t u 2 G Q m 5 6 l y n A m w F U b Z K g h p + y c D q C r c 5 p y n f R s R C M Q C 0 G a 5 w I k Z T 3 L 0 3 x o I G W u e i 1 u q d R 6
I i O H G 5 K a R C / G C u f f Y v 0 s N X o + G 5 O R 4 o P E z H m 7 n b B n C 3 J Z v E E U p D B m m Z Q 0 j S 1 R M o Y + H Q r j b J 6 M 5 t N F c j 6 R G g q I m 6 P i H u J m 8 d b p U E a g I P a z E Y N M c Z P I D T 8 f 0 g b z c f w F c q D G v s u c j f 0 z T v h 0 d t c F v C R 9 K r m Y z E R Y o v u X e n C j g f d T M c G 8 j 0 0 C u P g F c N E H z q a O y y v C X G O T 4 Q i w p q n G G h T v V + c a 0 Y w K L 1 P M t 6 e N B 5 A O f C P N M d U T f x H O n 7 s S 5 5 V S I S Z j H o P h I o a u H y T p J F y 3 z U S A b V Q x J h p M l F 0 E b 0 j i P 7 Z O 2 n Q E c R m v h X U 3 p V z
V s a 9 a I f n 0 O f f T X q S W 6 U q 2 A s M S k / l Z F L I L c M U k 8 h x U 2 t q Q Q / J 2 z 9 n W V r g V k H E c O I 9 f k 8 Q E Z U l n 9 5 1 l z r 5 3 t u N s 2 / t c 0 Z m f L D G g T S 2 0 9 V L Z Q n + 2 F j r S L C U z x X O j y 4 r / I / 2 b W q 2 T n C p X 7 u B O a X g K t j d n Y C f 8 v Y P H G 6 3 Q 4 8 P N 2 m 4 w 2 8 Y V 9 B Q 9 R y 9 Q i L b R L v q A D t A R Y g j Q V / Q N f V / + W V m t P K 4 8 m V J v L M 3 O P E J z V n n 2 C 4 b c e d g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A H 3 U K U u 7 A 5 W D 4 1 Z P O q C I a V j m l o s = " > A A A E Z 3 i c h V N t a x N B E N 7 a q D W + t F U Q w S + r S U A k h r v S W v p B q F R E U G l r 0 h f I h r K 3 N 8 k t 3 b 0 7 d j d J w 7 K / w K / 6 4 / w J / g v 3 L q l t I u r A 3 T 3 M P D v z z O x N l A u u T R D 8 W L q x X L l 5 6 / b K n e r d e / c f r K 6 t P z z W 2 V A x O G K Z y N R p R D U I n s K R 4 U b A a a 6 A y k j A S X S + V 8 R P R q A 0 z 9 K O m e T Q k 3 S Q 8 j 5 n 1 H j X 4 c X Z W i 1 o B a X h P 0 E 4 A z U 0 s 4 O z 9 e U D E m d s K C E 1 T F C t u 2 G Q m 5 6 l y n A m w F U b Z K g h p + y c D q C r c 5 p y n f R s R C M Q C 0 G a 5 w I k Z T 3 L 0 3 x o I G W u e i 1 u q d R 6
I i O H G 5 K a R C / G C u f f Y v 0 s N X o + G 5 O R 4 o P E z H m 7 n b B n C 3 J Z v E E U p D B m m Z Q 0 j S 1 R M o Y + H Q r j b J 6 M 5 t N F c j 6 R G g q I m 6 P i H u J m 8 d b p U E a g I P a z E Y N M c Z P I D T 8 f 0 g b z c f w F c q D G v s u c j f 0 z T v h 0 d t c F v C R 9 K r m Y z E R Y o v u X e n C j g f d T M c G 8 j 0 0 C u P g F c N E H z q a O y y v C X G O T 4 Q i w p q n G G h T v V + c a 0 Y w K L 1 P M t 6 e N B 5 A O f C P N M d U T f x H O n 7 s S 5 5 V S I S Z j H o P h I o a u H y T p J F y 3 z U S A b V Q x J h p M l F 0 E b 0 j i P 7 Z O 2 n Q E c R m v h X U 3 p V z
V s a 9 a I f n 0 O f f T X q S W 6 U q 2 A s M S k / l Z F L I L c M U k 8 h x U 2 t q Q Q / J 2 z 9 n W V r g V k H E c O I 9 f k 8 Q E Z U l n 9 5 1 l z r 5 3 t u N s 2 / t c 0 Z m f L D G g T S 2 0 9 V L Z Q n + 2 F j r S L C U z x X O j y 4 r / I / 2 b W q 2 T n C p X 7 u B O a X g K t j d n Y C f 8 v Y P H G 6 3 Q 4 8 P N 2 m 4 w 2 8 Y V 9 B Q 9 R y 9 Q i L b R L v q A D t A R Y g j Q V / Q N f V / + W V m t P K 4 8 m V J v L M 3 O P E J z V n n 2 C 4 b c e d g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A H 3 U K U u 7 A 5 W D 4 1 Z P O q C I a V j m l o s = " > A A A E Z 3 i c h V N t a x N B E N 7 a q D W + t F U Q w S + r S U A k h r v S W v p B q F R E U G l r 0 h f I h r K 3 N 8 k t 3 b 0 7 d j d J w 7 K / w K / 6 4 / w J / g v 3 L q l t I u r A 3 T 3 M P D v z z O x N l A u u T R D 8 W L q x X L l 5 6 / b K n e r d e / c f r K 6 t P z z W 2 V A x O G K Z y N R p R D U I n s K R 4 U b A a a 6 A y k j A S X S + V 8 R P R q A 0 z 9 K O m e T Q k 3 S Q 8 j 5 n 1 H j X 4 c X Z W i 1 o B a X h P 0 E 4 A z U 0 s 4 O z 9 e U D E m d s K C E 1 T F C t u 2 G Q m 5 6 l y n A m w F U b Z K g h p + y c D q C r c 5 p y n f R s R C M Q C 0 G a 5 w I k Z T 3 L 0 3 x o I G W u e i 1 u q d R 6
I i O H G 5 K a R C / G C u f f Y v 0 s N X o + G 5 O R 4 o P E z H m 7 n b B n C 3 J Z v E E U p D B m m Z Q 0 j S 1 R M o Y + H Q r j b J 6 M 5 t N F c j 6 R G g q I m 6 P i H u J m 8 d b p U E a g I P a z E Y N M c Z P I D T 8 f 0 g b z c f w F c q D G v s u c j f 0 z T v h 0 d t c F v C R 9 K r m Y z E R Y o v u X e n C j g f d T M c G 8 j 0 0 C u P g F c N E H z q a O y y v C X G O T 4 Q i w p q n G G h T v V + c a 0 Y w K L 1 P M t 6 e N B 5 A O f C P N M d U T f x H O n 7 s S 5 5 V S I S Z j H o P h I o a u H y T p J F y 3 z U S A b V Q x J h p M l F 0 E b 0 j i P 7 Z O 2 n Q E c R m v h X U 3 p V z
V s a 9 a I f n 0 O f f T X q S W 6 U q 2 A s M S k / l Z F L I L c M U k 8 h x U 2 t q Q Q / J 2 z 9 n W V r g V k H E c O I 9 f k 8 Q E Z U l n 9 5 1 l z r 5 3 t u N s 2 / t c 0 Z m f L D G g T S 2 0 9 V L Z Q n + 2 F j r S L C U z x X O j y 4 r / I / 2 b W q 2 T n C p X 7 u B O a X g K t j d n Y C f 8 v Y P H G 6 3 Q 4 8 P N 2 m 4 w 2 8 Y V 9 B Q 9 R y 9 Q
i L b R L v q A D t A R Y g j Q V / Q N f V / + W V m t P K 4 8 m V J v L M 3 O P E J z VD V v I F y k V A g + J c = " > A A A E b 3 i c h V N t b 9 M w E P a 2 A q O 8 b f C B D 0 j I o q u 0 o V I l 0 8 a 0 D 0 h D Q w g J 0 A b t X l B d T Y 5 z b a z Z S W S 7 7 S r L v 4 K v 8 M P 4 G f w D n L R j a x F w U p J H d 4 / v n j v n o l x w b Y L g x 8 L i U u X G z V v
L t 6 t 3 7 t 6 7 / 2 B l 9 e G x z g a K w R H L R K Z O I 6 p B 8 B S O D D c C T n M F V E Y C T q L z / S J + M g S l e Z a 2 z T i H r q T 9 l P c 4 o 8 a 7 v h A q 8 o S u X 2 y c r d S C Z l A a / h O E U 1 B D U z s 8 W 1 0 6 J H H G B h J S w w T V u h M G u e l a q g x n A l y 1 T g Y a c s r O a R 8 6 O q c p 1 0 n X R j Q C M R e k e S 5 A U t a 1 P M 0 H B l L m q t f i l k q t x z J y u C 6 p S f R 8 r H D + L d b L U q N n s z E Z K d 5 P z I y 3 0 w 6 7 t i C X x e t E Q Q o j l k l J 0 9 g S J W P o 0 Y E w z u b J c D Z d J G c T q Y G A u D E s 7 i N u F G + d D m Q E C m I / G 9 H P F D e J 3 P T z I S 0 w 7 0 e f I Q d q 7 J v M 2 d g / o 4 R P Z n d d w H P S o 5 K L 8 V S E J b p 3 q Q f X 6 / g g F W P M e 9 g k g I t f A R d 9 4 G z i u L w i z D U 2 G Y 4 A a 5 p q r E H x X n W m E c 2 o 8 D L F b H v a e A B p 3 z f S G F E 9 9 h f h / L k r c V 4 p F W I 8 4 j E Y L m L o + E G S d s J 1 y 4 w F 2 H o V Y 6 L B R N l F 8 I o k / m P X S I s O I S 7 j t X D N T S h X d e y L Z k g + f M z 9 t O e p Z b q S r c C w x G R + F o X s A l w x i T w H l T Y 3 5 Y C 8 3 n e 2 u R 1 u B 2 Q U B 8 7 j l y Q x Q V n S 2 Q N n m b N v n W 0 7 2 / I + V 3 T m J 0 s M a F M L 7 V q p b K 4 / W w s d a Z S S m e K 5 0 W X F / 5 H + T a 2 u k Z w q V + 7 g b m l 4 A n a 2 p m A 3 / L 2 D x 5 v N 0 O N P W 7 W 9 Y L q N y + g J e o b W U Y h 2 0 B 5 6 h w 7 R E W J I o q / o G / q + 9 L P y u P K 0 g i f U x Y X p m U d o x i o b v w C k w 3 z b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "
w N j h t / S M O P 1 Q 2 H D V v I F y k V A g + J c = " > A A A E b 3 i c h V N t b 9 M w E P a 2 A q O 8 b f C B D 0 j I o q u 0 o V I l 0 8 a 0 D 0 h D Q w g J 0 A b t X l B d T Y 5 z b a z Z S W S 7 7 S r L v 4 K v 8 M P 4 G f w D n L R j a x F w U p J H d 4 / v n j v n o l x w b Y L g x 8 L i U u X G z V v
L t 6 t 3 7 t 6 7 / 2 B l 9 e G x z g a K w R H L R K Z O I 6 p B 8 B S O D D c C T n M F V E Y C T q L z / S J + M g S l e Z a 2 z T i H r q T 9 l P c 4 o 8 a 7 v h A q 8 o S u X 2 y c r d S C Z l A a / h O E U 1 B D U z s 8 W 1 0 6 J H H G B h J S w w T V u h M G u e l a q g x n A l y 1 T g Y a c s r O a R 8 6 O q c p 1 0 n X R j Q C M R e k e S 5 A U t a 1 P M 0 H B l L m q t f i l k q t x z J y u C 6 p S f R 8 r H D + L d b L U q N n s z E Z K d 5 P z I y 3 0 w 6 7 t i C X x e t E Q Q o j l k l J 0 9 g S J W P o 0 Y E w z u b J c D Z d J G c T q Y G A u D E s 7 i N u F G + d D m Q E C m I / G 9 H P F D e J 3 P T z I S 0 w 7 0 e f I Q d q 7 J v M 2 d g / o 4 R P Z n d d w H P S o 5 K L 8 V S E J b p 3 q Q f X 6 / g g F W P M e 9 g k g I t f A R d 9 4 G z i u L w i z D U 2 G Y 4 A a 5 p q r E H x X n W m E c 2 o 8 D L F b H v a e A B p 3 z f S G F E 9 9 h f h / L k r c V 4 p F W I 8 4 j E Y L m L o + E G S d s J 1 y 4 w F 2 H o V Y 6 L B R N l F 8 I o k / m P X S I s O I S 7 j t X D N T S h X d e y L Z k g + f M z 9 t O e p Z b q S r c C w x G R + F o X s A l w x i T w H l T Y 3 5 Y C 8 3 n e 2 u R 1 u B 2 Q U B 8 7 j l y Q x Q V n S 2 Q N n m b N v n W 0 7 2 / I + V 3 T m J 0 s M a F M L 7 V q p b K 4 / W w s d a Z S S m e K 5 0 W X F / 5 H + T a 2 u k Z w q V + 7 g b m l 4 A n a 2 p m A 3 / L 2 D x 5 v N 0 O N P W 7 W 9 Y L q N y + g J e o b W U Y h 2 0 B 5 6 h w 7 R E W J I o q / o G / q + 9 L P y u P K 0 g i f U x Y X p m U d o x i o b v w C k w 3 z b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "
w N j h t / S M O P 1 Q 2 H D V v I F y k V A g + J c = " > A A A E b 3 i c h V N t b 9 M w E P a 2 A q O 8 b f C B D 0 j I o q u 0 o V I l 0 8 a 0 D 0 h D Q w g J 0 A b t X l B d T Y 5 z b a z Z S W S 7 7 S r L v 4 K v 8 M P 4 G f w D n L R j a x F w U p J H d 4 / v n j v n o l x w b Y L g x 8 L i U u X G z V v
L t 6 t 3 7 t 6 7 / 2 B l 9 e G x z g a K w R H L R K Z O I 6 p B 8 B S O D D c C T n M F V E Y C T q L z / S J + M g S l e Z a 2 z T i H r q T 9 l P c 4 o 8 a 7 v h A q 8 o S u X 2 y c r d S C Z l A a / h O E U 1 B D U z s 8 W 1 0 6 J H H G B h J S w w T V u h M G u e l a q g x n A l y 1 T g Y a c s r O a R 8 6 O q c p 1 0 n X R j Q C M R e k e S 5 A U t a 1 P M 0 H B l L m q t f i l k q t x z J y u C 6 p S f R 8 r H D + L d b L U q N n s z E Z K d 5 P z I y 3 0 w 6 7 t i C X x e t E Q Q o j l k l J 0 9 g S J W P o 0 Y E w z u b J c D Z d J G c T q Y G A u D E s 7 i N u F G + d D m Q E C m I / G 9 H P F D e J 3 P T z I S 0 w 7 0 e f I Q d q 7 J v M 2 d g / o 4 R P Z n d d w H P S o 5 K L 8 V S E J b p 3 q Q f X 6 / g g F W P M e 9 g k g I t f A R d 9 4 G z i u L w i z D U 2 G Y 4 A a 5 p q r E H x X n W m E c 2 o 8 D L F b H v a e A B p 3 z f S G F E 9 9 h f h / L k r c V 4 p F W I 8 4 j E Y L m L o + E G S d s J 1 y 4 w F 2 H o V Y 6 L B R N l F 8 I o k / m P X S I s O I S 7 j t X D N T S h X d e y L Z k g + f M z 9 t O e p Z b q S r c C w x G R + F o X s A l w x i T w H l T Y 3 5 Y C 8 3 n e 2 u R 1 u B 2 Q U B 8 7 j l y Q x Q V n S 2 Q N n m b N v n W 0 7 2 / I + V 3 T m J 0 s M a F M L 7 V q p b K 4 / W w s d a Z S S m e K 5 0 W X F / 5 H + T a 2 u k Z w q V + 7 g b m l 4 A n a 2 p m A 3 / L 2 D x 5 v N 0 O N P W 7 W 9 Y L q N y + g J e o b W U Y h 2 0 B 5 6 h w 7 R E W J I o q / o G / q + 9 L P y u P K 0 g i f U x Y X p m U d o x i o b v w C k w 3 z b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "
w N j h t / S M O P 1 Q 2 H D V v I F y k V A g + J c = " > A A A E b 3 i c h V N t b 9 M w E P a 2 A q O 8 b f C B D 0 j I o q u 0 o V I l 0 8 a 0 D 0 h D Q w g J 0 A b t X l B d T Y 5 z b a z Z S W S 7 7 S r L v 4 K v 8 M P 4 G f w D n L R j a x F w U p J H d 4 / v n j v n o l x w b Y L g x 8 L i U u X G z V v
L t 6 t 3 7 t 6 7 / 2 B l 9 e G x z g a K w R H L R K Z O I 6 p B 8 B S O D D c C T n M F V E Y C T q L z / S J + M g S l e Z a 2 z T i H r q T 9 l P c 4 o 8 a 7 v h A q 8 o S u X 2 y c r d S C Z l A a / h O E U 1 B D U z s 8 W 1 0 6 J H H G B h J S w w T V u h M G u e l a q g x n A l y 1 T g Y a c s r O a R 8 6 O q c p 1 0 n X R j Q C M R e k e S 5 A U t a 1 P M 0 H B l L m q t f i l k q t x z J y u C 6 p S f R 8 r H D + L d b L U q N n s z E Z K d 5 P z I y 3 0 w 6 7 t i C X x e t E Q Q o j l k l J 0 9 g S J W P o 0 Y E w z u b J c D Z d J G c T q Y G A u D E s 7 i N u F G + d D m Q E C m I / G 9 H P F D e J 3 P T z I S 0 w 7 0 e f I Q d q 7 J v M 2 d g / o 4 R P Z n d d w H P S o 5 K L 8 V S E J b p 3 q Q f X 6 / g g F W P M e 9 g k g I t f A R d 9 4 G z i u L w i z D U 2 G Y 4 A a 5 p q r E H x X n W m E c 2 o 8 D L F b H v a e A B p 3 z f S G F E 9 9 h f h / L k r c V 4 p F W I 8 4 j E Y L m L o + E G S d s J 1 y 4 w F 2 H o V Y 6 L B R N l F 8 I o k / m P X S I s O I S 7 j t X D N T S h X d e y L Z k g + f M z 9 t O e p Z b q S r c C w x G R + F o X s A l w x i T w H l T Y 3 5 Y C 8 3 n e 2 u R 1 u B 2 Q U B 8 7 j l y Q x Q V n S 2 Q N n m b N v n W 0 7 2 / I + V 3 T m J 0 s M a F M L 7 V q p b K 4 / W w s d a Z S S m e K 5 0 W X F / 5 H + T a 2 u k Z w q V + 7 g b m l 4 A n a 2 p m A 3 / L 2 D x 5 v N 0 O N P W 7 W 9 Y L q N y + g J e o b W U Y h 2 0 B 5 6 h w 7 R E W J I o q / o G / q + 9 L P y u P K 0 g i f U x Y X p m U d o x i o b v w C k w 3 z b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "
w N j h t / S M O P 1 Q 2 H D V v I F y k V A g + J c = " > A A A E b 3 i c h V N t b 9 M w E P a 2 A q O 8 b f C B D 0 j I o q u 0 o V I l 0 8 a 0 D 0 h D Q w g J 0 A b t X l B d T Y 5 z b a z Z S W S 7 7 S r L v 4 K v 8 M P 4 G f w D n L R j a x F w U p J H d 4 / v n j v n o l x w b Y L g x 8 L i U u X G z V v
L t 6 t 3 7 t 6 7 / 2 B l 9 e G x z g a K w R H L R K Z O I 6 p B 8 B S O D D c C T n M F V E Y C T q L z / S J + M g S l e Z a 2 z T i H r q T 9 l P c 4 o 8 a 7 v h A q 8 o S u X 2 y c r d S C Z l A a / h O E U 1 B D U z s 8 W 1 0 6 J H H G B h J S w w T V u h M G u e l a q g x n A l y 1 T g Y a c s r O a R 8 6 O q c p 1 0 n X R j Q C M R e k e S 5 A U t a 1 P M 0 H B l L m q t f i l k q t x z J y u C 6 p S f R 8 r H D + L d b L U q N n s z E Z K d 5 P z I y 3 0 w 6 7 t i C X x e t E Q Q o j l k l J 0 9 g S J W P o 0 Y E w z u b J c D Z d J G c T q Y G A u D E s 7 i N u F G + d D m Q E C m I / G 9 H P F D e J 3 P T z I S 0 w 7 0 e f I Q d q 7 J v M 2 d g / o 4 R P Z n d d w H P S o 5 K L 8 V S E J b p 3 q Q f X 6 / g g F W P M e 9 g k g I t f A R d 9 4 G z i u L w i z D U 2 G Y 4 A a 5 p q r E H x X n W m E c 2 o 8 D L F b H v a e A B p 3 z f S G F E 9 9 h f h / L k r c V 4 p F W I 8 4 j E Y L m L o + E G S d s J 1 y 4 w F 2 H o V Y 6 L B R N l F 8 I o k / m P X S I s O I S 7 j t X D N T S h X d e y L Z k g + f M z 9 t O e p Z b q S r c C w x G R + F o X s A l w x i T w H l T Y 3 5 Y C 8 3 n e 2 u R 1 u B 2 Q U B 8 7 j l y Q x Q V n S 2 Q N n m b N v n W 0 7 2 / I + V 3 T m J 0 s M a F M L 7 V q p b K 4 / W w s d a Z S S m e K 5 0 W X F / 5 H + T a 2 u k Z w q V + 7 g b m l 4 A n a 2 p m A 3 / L 2 D x 5 v N 0 O N P W 7 W 9 Y L q N y + g J e o b W U Y h 2 0 B 5 6 h w 7 R E W J I o q / o G / q + 9 L P y u P K 0 g i f U x Y X p m U d o x i o b v w C k w 3 z b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "
w N j h t / S M O P 1 Q 2 H D V v I F y k V A g + J c = " > A A A E b 3 i c h V N t b 9 M w E P a 2 A q O 8 b f C B D 0 j I o q u 0 o V I l 0 8 a 0 D 0 h D Q w g J 0 A b t X l B d T Y 5 z b a z Z S W S 7 7 S r L v 4 K v 8 M P 4 G f w D n L R j a x F w U p J H d 4 / v n j v n o l x w b Y L g x 8 L i U u X G z V v
L t 6 t 3 7 t 6 7 / 2 B l 9 e G x z g a K w R H L R K Z O I 6 p B 8 B S O D D c C T n M F V E Y C T q L z / S J + M g S l e Z a 2 z T i H r q T 9 l P c 4 o 8 a 7 v h A q 8 o S u X 2 y c r d S C Z l A a / h O E U 1 B D U z s 8 W 1 0 6 J H H G B h J S w w T V u h M G u e l a q g x n A l y 1 T g Y a c s r O a R 8 6 O q c p 1 0 n X R j Q C M R e k e S 5 A U t a 1 P M 0 H B l L m q t f i l k q t x z J y u C 6 p S f R 8 r H D + L d b L U q N n s z E Z K d 5 P z I y 3 0 w 6 7 t i C X x e t E Q Q o j l k l J 0 9 g S J W P o 0 Y E w z u b J c D Z d J G c T q Y G A u D E s 7 i N u F G + d D m Q E C m I / G 9 H P F D e J 3 P T z I S 0 w 7 0 e f I Q d q 7 J v M 2 d g / o 4 R P Z n d d w H P S o 5 K L 8 V S E J b p 3 q Q f X 6 / g g F W P M e 9 g k g I t f A R d 9 4 G z i u L w i z D U 2 G Y 4 A a 5 p q r E H x X n W m E c 2 o 8 D L F b H v a e A B p 3 z f S G F E 9 9 h f h / L k r c V 4 p F W I 8 4 j E Y L m L o + E G S d s J 1 y 4 w F 2 H o V Y 6 L B R N l F 8 I o k / m P X S I s O I S 7 j t X D N T S h X d e y L Z k g + f M z 9 t O e p Z b q S r c C w x G R + F o X s A l w x i T w H l T Y 3 5 Y C 8 3 n e 2 u R 1 u B 2 Q U B 8 7 j l y Q x Q V n S 2 Q N n m b N v n W 0 7 2 / I + V 3 T m J 0 s M a F M L 7 V q p b K 4 / W w s d a Z S S m e K 5 0 W X F / 5 H + T a 2 u k Z w q V + 7 g b m l 4 A n a 2 p m A 3 / L 2 D x 5 v N 0 O N P W 7 W 9 Y L q N y + g J e o b W U Y h 2 0 B 5 6 h w 7 R E W J I o q / o G / q + 9 L P y u P K 0 g i f U x Y X p m U d o x i o b v w C k w 3 z b < / l a t e x i t > b < l a t e x i t s h a 1 _ b a s e 6 4 = " H I U 0 P J u 2 U 2 D 0 y 8 5 H E k 1 i k q H 7 z s M = " > A A A F O X i c h V R b b 9 M w F M 6 g w C i 3 D R 5 5 s d Z W Q m h U S b U x 7 Q G p M I S Q A G 2 j u 0 l 1 N T n J S W P V T i L b v c n y L + A V f g 2 / h E f e E K / 8 A Z y 0 o 0 v R 4 E h O T s 7 5 f C 7 f c e x n j E r l u t 9 W r l 2 v 3 L h 5 a / V 2 9 c 7 d e / c f r K 0 / P J H p U A R w H K Q s F W c + k c B o A s e K K g Z n m Q D C f Q a n / m A v 9 5 + O Q E i a J k d q m k G P k 3 5 C I x o Q Z U 2 H / v l a z W 2 6 h a C / F W + u 1 J y 5 H J y v V z Z w m A Z D D o k K G J G y 6 7 m Z 6 m k i F A 0 Y m G o D D y V k J B i Q P n R l R h I q 4 5 7 2 i Q 9 s y U m y j A E n Q U / T J B s q S A J T v e T X k 6 K 9 s o 1 w K a f c N 6 j B i Y r l s i 8 3 X u W L 0 k T J c r S A + 4 L 2 Y 1 W y d o + 8 n s 7 B s 4 I E J D A O U s 5 J E m o s e A g R G T J l d B a P y t F 8 X o 4 z y o c S b u Z P m Q y 5 D w J C S x T r p 4 K q m L c s W b g D 6 t 3 4 I 2 R A l H 6 d G h 3 a N Y 7 p j M j L q Z / i i H D K p v P 0 G s v o o h L U a K D 9 h E 0 R j Z C K A e X n A e U N o H R m u J g X o h K p F P m A J E k k k i B o V C 2 1 I A P C b J m s 3 J h U V o G k b x v Z H B M 5 t R M w d h 8 n A 1 s 2 A 6 V A V P G i V D x h E K m C W C J E O u 6 2 e t 2 e b l Q R w j B R 7 c K G d q 1 o z N K k X w Z H l L G 2 0 T X P r p Y x p b h X o G e h F 9 8 4 x x S f l k H m E 4 E X G 8 x F 2 a k l R l R L 4 e 1 / w 9 h 0 T E N Q l I X Q t c c A H 8 V U d t S U w S y J B O W n E / c F j u 1 L 1 3 G H j C A s / D W v b m a Q B V n 6 W d P D 7 z 9 k 9 r A s Q 4 t w B V q A C m K V 2 o H m 3 O f K A o n 5 A E T S b P E h f r l n d H P b 2 3 b x O H S N 1 Z / j W L l F S q P 3 j Q 6 M f m P 0 k d E d a z P 5 e O z x w A q k q n m 6 X l S 2 1 F / O M d 4 s S g 4 E z Z Q s M v 4 P 9 G 9 o t Y 4 z I k x x q + w W g m b K z t Z c 2 f X + 3 C o n r a Z n 9 c O t W v v V / H 5 Z d R 4 7 G 8 4 T x 3 N 2 n L b z 1 j l w j p 3 A A e e T 8 9 n 5 U v l a + V 7 5 U f k 5 g 1 5 b m e 9 5 5 J S k 8 u s 3 w D X N u w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H I U 0 P J u 2 U 2 D 0 y 8 5 H E k 1 i k q H 7 z s M = " > A A A F O X i c h V R b b 9 M w F M 6 g w C i 3 D R 5 5 s d Z W Q m h U S b U x 7 Q G p M I S Q A G 2 j u 0 l 1 N T n J S W P V T i L b v c n y L + A V f g 2 / h E f e E K / 8 A Z y 0 o 0 v R 4 E h O T s 7 5 f C 7 f c e x n j E r l u t 9 W r l 2 v 3 L h 5 a / V 2 9 c 7 d e / c f r K 0 / P J H p U A R w H K Q s F W c + k c B o A s e K K g Z n m Q D C f Q a n / m A v 9 5 + O Q E i a J k d q m k G P k 3 5 C I x o Q Z U 2 H / v l a z W 2 6 h a C / F W + u 1 J y 5 H J y v V z Z w m A Z D D o k K G J G y 6 7 m Z 6 m k i F A 0 Y m G o D D y V k J B i Q P n R l R h I q 4 5 7 2 i Q 9 s y U m y j A E n Q U / T J B s q S A J T v e T X k 6 K 9 s o 1 w K a f c N 6 j B i Y r l s i 8 3 X u W L 0 k T J c r S A + 4 L 2 Y 1 W y d o + 8 n s 7 B s 4 I E J D A O U s 5 J E m o s e A g R G T J l d B a P y t F 8 X o 4 z y o c S b u Z P m Q y 5 D w J C S x T r p 4 K q m L c s W b g D 6 t 3 4 I 2 R A l H 6 d G h 3 a N Y 7 p j M j L q Z
/ i i H D K p v P 0 G s v o o h L U a K D 9 h E 0 R j Z C K A e X n A e U N o H R m u J g X o h K p F P m A J E k k k i B o V C 2 1 I A P C b J m s 3 J h U V o G k b x v Z H B M 5 t R M w d h 8 n A 1 s 2 A 6 V A V P G i V D x h E K m C W C J E O u 6 2 e t 2 e b l Q R w j B R 7 c K G d q 1 o z N K k X w Z H l L G 2 0 T X P r p Y x p b h X o G e h F 9
4 x x S f l k H m E 4 E X G x F 2 a k l R l R L 4 e 1 / w 9 h 0 T E N Q l I X Q t c c A H V U d t S U w S y J B O W n E / c F j u 1 L 1 3 G H j C A s / D W v b m a Q B V n 6 W d P D 7 z 9 k 9 r A s Q 4 t w B V q A C m K V 2 o H m 3 O f K A o n 5 A E T S b P E h f r l n d H P b 2 3 b x O H S N 1 Z / j W L l F S q P 3 j Q 6 M f m P 0 k d E d a z P 5 e O z x w A q k q n m 6 X l S 2 1 F / O M d 4 s S g 4 E z Z Q s M v 4 P 9 G 9 o t Y 4 z I k x x q + w W g m b K z t Z c 2 f X + 3 C o n r a Z n 9 c O t W v v V / H 5 Z d R 4 7 G 4 T x 3 N 2 n L b z 1 j l w j p 3 A
A e e T 8 9 n 5 U v l a + V 7 5 U f k 5 g 1 5 b m e 9 5 5 J S k 8 u s 3 w D X N u w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H I U 0 P J u 2 U 2 D 0 y 8
H E k 1 i k q H 7 z s M = " > A A A F O X i c h V R b b 9 M w F M 6 g w C i 3 D R s d Z W Q m h U S b U x 7 Q G p M I S Q A G 2 j u 0 l 1 N T n J S W P V T i L b v c n y L + A V f g 2 / h E f e E K / 8 A Z y 0 o 0 v R 4 E h O T s 7 f C 7 f c e x n j E r l u t 9 W r l 2 v 3 L h a / V 2 9 c 7 d e / c f r K 0 / P J H p U A R w H K Q s F W c + k c B o A s e K K g Z n m Q D C f Q a n / m A v 9 + O Q E i a J k d q m k G P k 3 C I x o Q Z U 2 H / v l a z W 2 6 h a C / F W + u 1 J y H J y v V z Z w m A Z D D o k K G J G y 6 7 m Z 6 m k i F A 0 Y m G o D D y V k J B i Q P n R l R h I q 4 7 2 i Q 9 s y U m y j A E n Q U / T J B s q S A J T v e T X k 6 K 9 s o 1 w K a f c N 6 j B i Y r l s i 8 3 X u W L 0 k T J c r S A + 4 L 2 Y 1 W y d o + 8 n s 7 B s 4 I E J D A O U s J E m o s e A g R G T J l d B a P y t F 8 X o 4 z y o c S b u Z P m Q y D w J C S x T r p 4 K q m L c s W b g D 6 t 3 4 I 2 R
A l H 6 d G h 3 a N Y 7 p j M j L q Z / i i H D K p v P 0 G s v o o h L U a K D 9 h E 0 R j Z C K A e X n A e U N o H R m u J g X o h K p F P m A J E k k k i B o V C 2 1 I A P C b J m s 3 J h U V o G k b x v Z H B M 5 t R M w d h 8 n A 1 s 2 A 6 V A V P G i V D x h E K m C W C J E O u 6 2 e t 2 e b l Q R w j B R 7 c K G d q 1 o z N K k X w Z H l L G 2 0 T X P r p Y x p b h X o G e h F 9
4 x x S f l k H m E 4 E X G x F 2 a k l R l R L 4 e 1 / w 9 h 0 T E N Q l I X Q t c c A H V U d t S U w S y J B O W n E / c F j u 1 L 1 3 G H j C A s / D W v b m a Q B V n 6 W d P D 7 z 9 k 9 r A s Q 4 t w B V q A C m K V 2 o H m 3 O f K A o n 5 A E T S b P E h f r l n d H P b 2 3 b x O H S N 1 Z / j W L l F S q P 3 j Q 6 M f m P 0 k d E d a z P 5 e O z x w A q k q n m 6 X l S 2 1 F / O M d 4 s S g 4 E z Z Q s M v 4 P 9 G 9 o t Y 4 z I k x x q + w W g m b K z t Z c 2 f X + 3 C o n r a Z n 9 c O t W v v V / H 5 Z d R 4 7 G 4 T x 3 N 2 n L b z 1 j l w j p 3 A
A e e T 8 9 n 5 U v l a + V 7 5 U f k 5 g 1 5 b m e 9 5 5 J S k 8 u s 3 w D X N u w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H I U 0 P J u 2 U 2 D 0 y 8
H E k 1 i k q H 7 z s M = " > A A A F O X i c h V R b b 9 M w F M 6 g w C i 3 D R s d Z W Q m h U S b U x 7 Q G p M I S Q A G 2 j u 0 l 1 N T n J S W P V T i L b v c n y L + A V f g 2 / h E f e E K / 8 A Z y 0 o 0 v R 4 E h O T s 7 f C 7 f c e x n j E r l u t 9 W r l 2 v 3 L h a / V 2 9 c 7 d e / c f r K 0 / P J H p U A R w H K Q s F W c + k c B o A s e K K g Z n m Q D C f Q a n / m A v 9 + O Q E i a J k d q m k G P k 3 C I x o Q Z U 2 H / v l a z W 2 6 h a C / F W + u 1 J y H J y v V z Z w m A Z D D o k K G J G y 6 7 m Z 6 m k i F A 0 Y m G o D D y V k J B i Q P n R l R h I q 4 7 2 i Q 9 s y U m y j A E n Q U / T J B s q S A J T v e T X k 6 K 9 s o 1 w K a f c N 6 j B i Y r l s i 8 3 X u W L 0 k T J c r S A + 4 L 2 Y 1 W y d o + 8 n s 7 B s 4 I E J D A O U s J E m o s e A g R G T J l d B a P y t F 8 X o 4 z y o c S b u Z P m Q y D w J C S x T r p 4 K q m L c s W b g D 6 t 3 4 I 2 R
A l H 6 d G h 3 a N Y 7 p j M j L q Z / i i H D K p v P 0 G s v o o h L U a K D 9 h E 0 R j Z C K A e X n A e U N o H R m u J g X o h K p F P m A J E k k k i B o V C 2 1 I A P C b J m s 3 J h U V o G k b x v Z H B M 5 t R M w d h 8 n A 1 s 2 A 6 V A V P G i V D x h E K m C W C J E O u 6 2 e t 2 e b l Q R w j B R 7 c K G d q 1 o z N K k X w Z H l L G 2 0 T X P r p Y x p b h X o G e h F 9
4 x x S f l k H m E 4 E X G x F 2 a k l R l R L 4 e 1 / w 9 h 0 T E N Q l I X Q t c c A H V U d t S U w S y J B O W n E / c F j u 1 L 1 3 G H j C A s / D W v b m a Q B V n 6 W d P D 7 z 9 k 9 r A s Q 4 t w B V q A C m K V 2 o H m 3 O f K A o n 5 A E T S b P E h f r l n d H P b 2 3 b x O H S N 1 Z / j W L l F S q P 3 j Q 6 M f m P 0 k d E d a z P 5 e O z x w A q k q n m 6 X l S 2 1 F / O M d 4 s S g 4 E z Z Q s M v 4 P 9 G 9 o t Y 4 z I k x x q + w W g m b K z t Z c 2 f X + 3 C o n r a Z n 9 c O t W v v V / H 5 Z d R 4 7 G 4 T x 3 N 2 n L b z 1 j l w j p 3 A
A e e T 8 9 n 5 U v l a + V 7 5 U f k 5 g 1 5 b m e 9 5 5 J S k 8 u s 3 w D X N u w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H I U 0 P J u 2 U 2 D 0 y 8
H E k 1 i k q H 7 z s M = " > A A A F O X i c h V R b b 9 M w F M 6 g w C i 3 D R s d Z W Q m h U S b U x 7 Q G p M I S Q A G 2 j u 0 l 1 N T n J S W P V T i L b v c n y L + A V f g 2 / h E f e E K / 8 A Z y 0 o 0 v R 4 E h O T s 7 f C 7 f c e x n j E r l u t 9 W r l 2 v 3 L h a / V 2 9 c 7 d e / c f r K 0 / P J H p U A R w H K Q s F W c + k c B o A s e K K g Z n m Q D C f Q a n / m A v 9 + O Q E i a J k d q m k G P k 3 C I x o Q Z U 2 H / v l a z W 2 6 h a C / F W + u 1 J y H J y v V z Z w m A Z D D o k K G J G y 6 7 m Z 6 m k i F A 0 Y m G o D D y V k J B i Q P n R l R h I q 4 7 2 i Q 9 s y U m y j A E n Q U / T J B s q S A J T v e T X k 6 K 9 s o 1 w K a f c N 6 j B i Y r l s i 8 3 X u W L 0 k T J c r S A + 4 L 2 Y 1 W y d o + 8 n s 7 B s 4 I E J D A O U s J E m o s e A g R G T J l d B a P y t F 8 X o 4 z y o c S b u Z P m Q y D w J C S x T r p 4 K q m L c s W b g D 6 t 3 4 I 2 R
A l H 6 d G h 3 a N Y 7 p j M j L q Z / i i H D K p v P 0 G s v o o h L U a K D 9 h E 0 R j Z C K A e X n A e U N o H R m u J g X o h K p F P m A J E k k k i B o V C 2 1 I A P C b J m s 3 J h U V o G k b x v Z H B M 5 t R M w d h 8 n A 1 s 2 A 6 V A V P G i V D x h E K m C W C J E O u 6 2 e t 2 e b l Q R w j B R 7 c K G d q 1 o z N K k X w Z H l L G 2 0 T X P r p Y x p b h X o G e h F 9
4 x x S f l k H m E 4 E X G x F 2 a k l R l R L 4 e 1 / w 9 h 0 T E N Q l I X Q t c c A H V U d t S U w S y J B O W n E / c F j u 1 L 1 3 G H j C A s / D W v b m a Q B V n 6 W d P D 7 z 9 k 9 r A s Q 4 t w B V q A C m K V 2 o H m 3 O f K A o n 5 A E T S b P E h f r l n d H P b 2 3 b x O H S N 1 Z / j W L l F S q P 3 j Q 6 M f m P 0 k d E d a z P 5 e O z x w A q k q n m 6 X l S 2 1 F / O M d 4 s S g 4 E z Z Q s M v 4 P 9 G 9 o t Y 4 z I k x x q + w W g m b K z t Z c 2 f X + 3 C o n r a Z n 9 c O t W v v V / H 5 Z d R 4 7 G 4 T x 3 N 2 n L b z 1 j l w j p 3 A
A e e T 8 9 n 5 U v l a + V 7 5 U f k 5 g 1 5 b m e 9 5 5 J S k 8 u s 3 w D X N u w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H I U 0 P J u 2 U 2 D 0 y 8
H E k 1 i k q H 7 z s M = " > A A A F O X i c h V R b b 9 M w F M 6 g w C i 3 D R s d Z W Q m h U S b U x 7 Q G p M I S Q A G 2 j u 0 l 1 N T n J S W P V T i L b v c n y L + A V f g 2 / h E f e E K / 8 A Z y 0 o 0 v R 4 E h O T s 7 f C 7 f c e x n j E r l u t 9 W r l 2 v 3 L h a / V 2 9 c 7 d e / c f r K 0 / P J H p U A R w H K Q s F W c + k c B o A s e K K g Z n m Q D C f Q a n / m A v 9 + O Q E i a J k d q m k G P k 3 C I x o Q Z U 2 H / v l a z W 2 6 h a C / F W + u 1 J y H J y v V z Z w m A Z D D o k K G J G y 6 7 m Z 6 m k i F A 0 Y m G o D D y V k J B i Q P n R l R h I q 4 7 2 i Q 9 s y U m y j A E n Q U / T J B s q S A J T v e T X k 6 K 9 s o 1 w K a f c N 6 j B i Y r l s i 8 3 X u W L 0 k T J c r S A + 4 L 2 Y 1 W y d o + 8 n s 7 B s 4 I E J D A O U s J E m o s e A g R G T J l d B a P y t F 8 X o 4 z y o c S b u Z P m Q y D w J C S x T r p 4 K q m L c s W b g D 6 t 3 4 I 2 R
A l H 6 d G h 3 a N Y 7 p j M j L q Z / i i H D K p v P 0 G s v o o h L U a K D 9 h E 0 R j Z C K A e X n A e U N o H R m u J g X o h K p F P m A J E k k k i B o V C 2 1 I A P C b J m s 3 J h U V o G k b x v Z H B M 5 t R M w d h 8 n A 1 s 2 A 6 V A V P G i V D x h E K m C W C J E O u 6 2 e t 2 e b l Q R w j B R 7 c K G d q 1 o z N K k X w Z H l L G 2 0 T X P r p Y x p b h X o G e h F 9
4 x x S f l k H m E 4 E X G x F 2 a k l R l R L 4 e 1 / w 9 h 0 T E N Q l I X Q t c c A H V U d t S U w S y J B O W n E / c F j u 1 L 1 3 G H j C A s / D W v b m a Q B V n 6 W d P D 7 z 9 k 9 r A s Q 4 t w B V q A C m K V 2 o H m 3 O f K A o n 5 A E T S b P E h f r l n d H P b 2 3 b x O H S N 1 Z / j W L l F S q P 3 j Q 6 M f m P 0 k d E d a z P 5 e O z x w A q k q n m 6 X l S 2 1 F / O M d 4 s S g 4 E z Z Q s M v 4 P 9 G 9 o t Y 4 z I k x x q + w W g m b K z t Z c 2 f X + 3 C o n r a Z n 9 c O t W v v V / H 5 Z d R 4 7 G 4 T x 3 N 2 n L b z 1 j l w j p 3 A
A e e T 8 9 n 5 U v l a + V 7 5 U f k 5 g 1 5 b m e 9 5 5 J S k 8 u s 3 w D X N u w = = < / l a t e x i t > sampling < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 l d S I
L q K Z a c / 6 0 Z i f c j X y Y Z w f z Y = " > A A A F S X i c h V T d b t M w F M 5 Y B 6 P 8 b I N L b q x 1 l R A a V V J t T L t A G h p C S I A 2 2 K 9 U V 5 O T n D R W b S e y 3 X W V 5 S f h F p 6 G J + A x u E N c 4 S Q d X Y Y G R 0 p 0 f M 7 n 8 / M d 2 2 H O q N K + / 3 3 u 1 n x j 4 f a d x b v N e / c f P F x a X n l 0 r L K R j O A o y l g m T 0 O i g F E B R 5 p q B q e 5 B M J D B i f h c L f w n 5 y D V D Q T h 3 q S Q 5 + T g a A J j Y h 2 p r P l J c z D 7 M I o w l 0 2 M b B n y y 2 / 4 5 e C / l a C q d L y p r J / t t J Y x X E W j T g I H T G i V C / w c 9 0 3 R G o a M b D N N h 4 p y E k 0 J A P o q Z w I q t K + C U k I 7 J q T 5 D k D T q K + o S I f a R C R b V 7 x m 4 u y 3 b q N c K U m P L S o z Y l O 1 X V f Y b z J l 2 R C q 3 q 0 i I e S D l J d s / Y O g 7 4 p w F V B E g S M o 4 x z I m K D J Y 8 h I S O m r c n T 8 3 q 0 k N f j n B d D i t e L v x I j H o K E 2 B H F B p m k O u V d R x Y + A P 1 u / A l y I N q 8 z q y J 3 T d O a U X k 1 d T P c E I 4 Z Z N p e o N V c l k J a r f R n m A T R B O k U 0 D F + U B F A y i r D J f z Q l Q h n a E Q k C J C I Q W S J s 1 a C y o i z J X J 6 o 0 p 7 R Q Q A 9 f I + p i o i Z u A d f s 4 G b q y G W g N s o l n p e I L B o k u i S V S Z u N e t 9 / r m 3 Y T I Q w X e q e 0 o W 0 n B r N M D O r g h D K 2 Y 0 0 r c F / X 2 l r c G 9 B V 6 N k a F 5 h y 6 R h k I Z F 4 t s F e l p 0 5 Y m S z F t 7 d I 8 Y m Y x q D p i y G n j s G + D C l 6 k B P G F R J F G h 3 f f y X O C 1 u 0 R o + I O c Q l / 5 W s G Y
r y I w s 8 7 w T 4 P c f c n d Y r k P L c C V a g o 5 S n b m B F t w X y g y J + R C k 6 H T 5 C L / a t a a z G W z 6 e B z 7 1 u k v c K r 9 M q U 1 e 9 Z E 1 r y x 5 t C a A 2 e z x X j c 8 c A a l G 4 F Z q 2 s 7 F p / B c d 4 v S w 5 k j T X q s z 4 P 9 C / o c 0 1 n B N Z v S r b p a B K 2 d q Y K t v B n 1 f l u N s J n P 5 x o 7 X j T 9 + X R e + J t + o 9 9 Q J v y 9 v x 3 n r 7 3 p E X e S P v s / f F + 9 r 4 1 v j R + N n 4 V U F v z U 3 3 P P Z q s j D / G w Z 3 0 2 o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 l d S I
L q K Z a c / 6 0 Z i f c j X y Y Z w f z Y = " > A A A F S X i c h V T d b t M w F M 5 Y B 6 P 8 b I N L b q x 1 l R A a V V J t T L t A G h p C S I A 2 2 K 9 U V 5 O T n D R W b S e y 3 X W V 5 S f h F p 6 G J + A x u E N c 4 S Q d X Y Y G R 0 p 0 f M 7 n 8 / M d 2 2 H O q N K + / 3 3 u 1 n x j 4 f a d x b v N e / c f P F x a X n l 0 r L K R j O A o y l g m T 0 O i g F E B R 5 p q B q e 5 B M J D B i f h c L f w n 5 y D V D Q T h 3 q S Q 5 + T g a A J j Y h 2 p r P l J c z D 7 M I o w l 0 2 M b B n y y 2 / 4 5 e C / l a C q d L y p r J / t t J Y x X E W j T g I H T G i V C / w c 9 0 3 R G o a M b D N N h 4 p y E k 0 J A P o q Z w I q t K + C U k I 7 J q T 5 D k D T q K + o S I f a R C R b V 7 x m 4 u y 3 b q N c K U m P L S o z Y l O 1 X V f Y b z J l 2 R C q 3 q 0 i I e S D l J d s / Y O g 7 4 p w F V B E g S M o 4 x z I m K D J Y 8 h I S O m r c n T 8 3 q 0 k N f j n B d D i t e L v x I j H o K E 2 B H F B p m k O u V d R x Y + A P 1 u / A l y I N q 8 z q y J 3 T d O a U X k 1 d T P c E I 4 Z Z N p e o N V c l k J a r f R n m A T R B O k U 0 D F + U B F A y i r D J f z Q l Q h n a E Q k C J C I Q W S J s 1 a C y o i z J X J 6 o 0 p 7 R Q Q A 9 f I + p i o i Z u A d f s 4 G b q y G W g N s o l n p e I L B o k u i S V S Z u N e t 9 / r m 3 Y T I Q w X e q e 0 o W 0 n B r N M D O r g h D K 2 Y 0 0 r c F / X 2 l r c G 9 B V 6 N k a F 5 h y 6 R h k I Z F 4 t s F e l p 0 5 Y m S z F t 7 d I 8 Y m Y x q D p i y G n j s G + D C l 6 k B P G F R J F G h 3 f f y X O C 1 u 0 R o + I O c Q l / 5 W s G Y
r y I w s 8 7 w T 4 P c f c n d Y r k P L c C V a g o 5 S n b m B F t w X y g y J + R C k 6 H T 5 C L / a t a a z G W z 6 e B z 7 1 u k v c K r 9 M q U 1 e 9 Z E 1 r y x 5 t C a A 2 e z x X j c 8 c A a l G 4 F Z q 2 s 7 F p / B c d 4 v S w 5 k j T X q s z 4 P 9 C / o c 0 1 n B N Z v S r b p a B K 2 d q Y K t v B n 1 f l u N s J n P 5 x o 7 X j T 9 + X R e + J t + o 9 9 Q J v y 9 v x 3 n r 7 3 p E X e S P v s / f F + 9 r 4 1 v j R + N n 4 V U F v z U 3 3 P P Z q s j D / G w Z 3 0 2 o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 l d S I
L q K Z a c / 6 0 Z i f c j X y Y Z w f z Y = " > A A A F S X i c h V T d b t M w F M 5 Y B 6 P 8 b I N L b q x 1 l R A a V V J t T L t A G h p C S I A 2 2 K 9 U V 5 O T n D R W b S e y 3 X W V 5 S f h F p 6 G J + A x u E N c 4 S Q d X Y Y G R 0 p 0 f M 7 n 8 / M d 2 2 H O q N K + / 3 3 u 1 n x j 4 f a d x b v N e / c f P F x a X n l 0 r L K R j O A o y l g m T 0 O i g F E B R 5 p q B q e 5 B M J D B i f h c L f w n 5 y D V D Q T h 3 q S Q 5 + T g a A J j Y h 2 p r P l J c z D 7 M I o w l 0 2 M b B n y y 2 / 4 5 e C / l a C q d L y p r J / t t J Y x X E W j T g I H T G i V C / w c 9 0 3 R G o a M b D N N h 4 p y E k 0 J A P o q Z w I q t K + C U k I 7 J q T 5 D k D T q K + o S I f a R C R b V 7 x m 4 u y 3 b q N c K U m P L S o z Y l O 1 X V f Y b z J l 2 R C q 3 q 0 i I e S D l J d s / Y O g 7 4 p w F V B E g S M o 4 x z I m K D J Y 8 h I S O m r c n T 8 3 q 0 k N f j n B d D i t e L v x I j H o K E 2 B H F B p m k O u V d R x Y + A P 1 u / A l y I N q 8 z q y J 3 T d O a U X k 1 d T P c E I 4 Z Z N p e o N V c l k J a r f R n m A T R B O k U 0 D F + U B F A y i r D J f z Q l Q h n a E Q k C J C I Q W S J s 1 a C y o i z J X J 6 o 0 p 7 R Q Q A 9 f I + p i o i Z u A d f s 4 G b q y G W g N s o l n p e I L B o k u i S V S Z u N e t 9 / r m 3 Y T I Q w X e q e 0 o W 0 n B r N M D O r g h D K 2 Y 0 0 r c F / X 2 l r c G 9 B V 6 N k a F 5 h y 6 R h k I Z F 4 t s F e l p 0 5 Y m S z F t 7 d I 8 Y m Y x q D p i y G n j s G + D C l 6 k B P G F R J F G h 3 f f y X O C 1 u 0 R o + I O c Q l / 5 W s G Y
r y I w s 8 7 w T 4 P c f c n d Y r k P L c C V a g o 5 S n b m B F t w X y g y J + R C k 6 H T 5 C L / a t a a z G W z 6 e B z 7 1 u k v c K r 9 M q U 1 e 9 Z E 1 r y x 5 t C a A 2 e z x X j c 8 c A a l G 4 F Z q 2 s 7 F p / B c d 4 v S w 5 k j T X q s z 4 P 9 C / o c 0 1 n B N Z v S r b p a B K 2 d q Y K t v B n 1 f l u N s J n P 5 x o 7 X j T 9 + X R e + J t + o 9 9 Q J v y 9 v x 3 n r 7 3 p E X e S P v s / f F + 9 r 4 1 v j R + N n 4 V U F v z U 3 3 P P Z q s j D / G w Z 3 0 2 o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 l d S I
L q K Z a c / 6 0 Z i f c j X y Y Z w f z Y = " > A A A F S X i c h V T d b t M w F M 5 Y B 6 P 8 b I N L b q x 1 l R A a V V J t T L t A G h p C S I A 2 2 K 9 U V 5 O T n D R W b S e y 3 X W V 5 S f h F p 6 G J + A x u E N c 4 S Q d X Y Y G R 0 p 0 f M 7 n 8 / M d 2 2 H O q N K + / 3 3 u 1 n x j 4 f a d x b v N e / c f P F x a X n l 0 r L K R j O A o y l g m T 0 O i g F E B R 5 p q B q e 5 B M J D B i f h c L f w n 5 y D V D Q T h 3 q S Q 5 + T g a A J j Y h 2 p r P l J c z D 7 M I o w l 0 2 M b B n y y 2 / 4 5 e C / l a C q d L y p r J / t t J Y x X E W j T g I H T G i V C / w c 9 0 3 R G o a M b D N N h 4 p y E k 0 J A P o q Z w I q t K + C U k I 7 J q T 5 D k D T q K + o S I f a R C R b V 7 x m 4 u y 3 b q N c K U m P L S o z Y l O 1 X V f Y b z J l 2 R C q 3 q 0 i I e S D l J d s / Y O g 7 4 p w F V B E g S M o 4 x z I m K D J Y 8 h I S O m r c n T 8 3 q 0 k N f j n B d D i t e L v x I j H o K E 2 B H F B p m k O u V d R x Y + A P 1 u / A l y I N q 8 z q y J 3 T d O a U X k 1 d T P c E I 4 Z Z N p e o N V c l k J a r f R n m A T R B O k U 0 D F + U B F A y i r D J f z Q l Q h n a E Q k C J C I Q W S J s 1 a C y o i z J X J 6 o 0 p 7 R Q Q A 9 f I + p i o i Z u A d f s 4 G b q y G W g N s o l n p e I L B o k u i S V S Z u N e t 9 / r m 3 Y T I Q w X e q e 0 o W 0 n B r N M D O r g h D K 2 Y 0 0 r c F / X 2 l r c G 9 B V 6 N k a F 5 h y 6 R h k I Z F 4 t s F e l p 0 5 Y m S z F t 7 d I 8 Y m Y x q D p i y G n j s G + D C l 6 k B P G F R J F G h 3 f f y X O C 1 u 0 R o + I O c Q l / 5 W s G Y
r y I w s 8 7 w T 4 P c f c n d Y r k P L c C V a g o 5 S n b m B F t w X y g y J + R C k 6 H T 5 C L / a t a a z G W z 6 e B z 7 1 u k v c K r 9 M q U 1 e 9 Z E 1 r y x 5 t C a A 2 e z x X j c 8 c A a l G 4 F Z q 2 s 7 F p / B c d 4 v S w 5 k j T X q s z 4 P 9 C / o c 0 1 n B N Z v S r b p a B K 2 d q Y K t v B n 1 f l u N s J n P 5 x o 7 X j T 9 + X R e + J t + o 9 9 Q J v y 9 v x 3 n r 7 3 p E X e S P v s / f F + 9 r 4 1 v j R + N n 4 V U F v z U 3 3 P P Z q s j D / G w Z 3 0 2 o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 l d S I
L q K Z a c / 6 0 Z i f c j X y Y Z w f z Y = " > A A A F S X i c h V T d b t M w F M 5 Y B 6 P 8 b I N L b q x 1 l R A a V V J t T L t A G h p C S I A 2 2 K 9 U V 5 O T n D R W b S e y 3 X W V 5 S f h F p 6 G J + A x u E N c 4 S Q d X Y Y G R 0 p 0 f M 7 n 8 / M d 2 2 H O q N K + / 3 3 u 1 n x j 4 f a d x b v N e / c f P F x a X n l 0 r L K R j O A o y l g m T 0 O i g F E B R 5 p q B q e 5 B M J D B i f h c L f w n 5 y D V D Q T h 3 q S Q 5 + T g a A J j Y h 2 p r P l J c z D 7 M I o w l 0 2 M b B n y y 2 / 4 5 e C / l a C q d L y p r J / t t J Y x X E W j T g I H T G i V C / w c 9 0 3 R G o a M b D N N h 4 p y E k 0 J A P o q Z w I q t K + C U k I 7 J q T 5 D k D T q K + o S I f a R C R b V 7 x m 4 u y 3 b q N c K U m P L S o z Y l O 1 X V f Y b z J l 2 R C q 3 q 0 i I e S D l J d s / Y O g 7 4 p w F V B E g S M o 4 x z I m K D J Y 8 h I S O m r c n T 8 3 q 0 k N f j n B d D i t e L v x I j H o K E 2 B H F B p m k O u V d R x Y + A P 1 u / A l y I N q 8 z q y J 3 T d O a U X k 1 d T P c E I 4 Z Z N p e o N V c l k J a r f R n m A T R B O k U 0 D F + U B F A y i r D J f z Q l Q h n a E Q k C J C I Q W S J s 1 a C y o i z J X J 6 o 0 p 7 R Q Q A 9 f I + p i o i Z u A d f s 4 G b q y G W g N s o l n p e I L B o k u i S V S Z u N e t 9 / r m 3 Y T I Q w X e q e 0 o W 0 n B r N M D O r g h D K 2 Y 0 0 r c F / X 2 l r c G 9 B V 6 N k a F 5 h y 6 R h k I Z F 4 t s F e l p 0 5 Y m S z F t 7 d I 8 Y m Y x q D p i y G n j s G + D C l 6 k B P G F R J F G h 3 f f y X O C 1 u 0 R o + I O c Q l / 5 W s G Y
r y I w s 8 7 w T 4 P c f c n d Y r k P L c C V a g o 5 S n b m B F t w X y g y J + R C k 6 H T 5 C L / a t a a z G W z 6 e B z 7 1 u k v c K r 9 M q U 1 e 9 Z E 1 r y x 5 t C a A 2 e z x X j c 8 c A a l G 4 F Z q 2 s 7 F p / B c d 4 v S w 5 k j T X q s z 4 P 9 C / o c 0 1 n B N Z v S r b p a B K 2 d q Y K t v B n 1 f l u N s J n P 5 x o 7 X j T 9 + X R e + J t + o 9 9 Q J v y 9 v x 3 n r 7 3 p E X e S P v s / f F + 9 r 4 1 v j R + N n 4 V U F v z U 3 3 P P Z q s j D / G w Z 3 0 2 o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 l d S I
L q K Z a c / 6 0 Z i f c j X y Y Z w f z Y = " > A A A F S X i c h V T d b t M w F M 5 Y B 6 P 8 b I N L b q x 1 l R A a V V J t T L t A G h p C S I A 2 2 K 9 U V 5 O T n D R W b S e y 3 X W V 5 S f h F p 6 G J + A x u E N c 4 S Q d X Y Y G R 0 p 0 f M 7 n 8 / M d 2 2 H O q N K + / 3 3 u 1 n x j 4 f a d x b v N e / c f P F x a X n l 0 r L K R j O A o y l g m T 0 O i g F E B R 5 p q B q e 5 B M J D B i f h c L f w n 5 y D V D Q T h 3 q S Q 5 + T g a A J j Y h 2 p r P l J c z D 7 M I o w l 0 2 M b B n y y 2 / 4 5 e C / l a C q d L y p r J / t t J Y x X E W j T g I H T G i V C / w c 9 0 3 R G o a M b D N N h 4 p y E k 0 J A P o q Z w I q t K + C U k I 7 J q T 5 D k D T q K + o S I f a R C R b V 7 x m 4 u y 3 b q N c K U m P L S o z Y l O 1 X V f Y b z J l 2 R C q 3 q 0 i I e S D l J d s / Y O g 7 4 p w F V B E g S M o 4 x z I m K D J Y 8 h I S O m r c n T 8 3 q 0 k N f j n B d D i t e L v x I j H o K E 2 B H F B p m k O u V d R x Y + A P 1 u / A l y I N q 8 z q y J 3 T d O a U X k 1 d T P c E I 4 Z Z N p e o N V c l k J a r f R n m A T R B O k U 0 D F + U B F A y i r D J f z Q l Q h n a E Q k C J C I Q W S J s 1 a C y o i z J X J 6 o 0 p 7 R Q Q A 9 f I + p i o i Z u A d f s 4 G b q y G W g N s o l n p e I L B o k u i S V S Z u N e t 9 / r m 3 Y T I Q w X e q e 0 o W 0 n B r N M D O r g h D K 2 Y 0 0 r c F / X 2 l r c G 9 B V 6 N k a F 5 h y 6 R h k I Z F 4 t s F e l p 0 5 Y m S z F t 7 d I 8 Y m Y x q D p i y G n j s G + D C l 6 k B P G F R J F G h 3 f f y X O C 1 u 0 R o + I O c Q l / 5 W s G Y
r y I w s 8 7 w T 4 P c f c n d Y r k P L c C V a g o 5 S n b m B F t w X y g y J + R C k 6 H T 5 C L / a t a a z G W z 6 e B z 7 1 u k v c K r 9 M q U 1 e 9 Z E 1 r y x 5 t C a A 2 e z x X j c 8 c A a l G 4 F Z q 2 s 7 F p / B c d 4 v S w 5 k j T X q s z 4 P 9 C / o c 0 1 n B N Z v S r b p a B K 2 d q Y K t v B n 1 f l u N s J n P 5 x o 7 X j T 9 + X R e + J t + o 9 9 Q J v y 9 v x 3 n r 7 3 p E X e S P v s / f F + 9 r 4 1 v j R + N n 4 V U F v z U 3 3 P P Z q s j D / G w Z 3 0 2 o = < / l a t e x i t > b (1) < l a t e x i t s h a 1 _ b a s e 6 4 = " d c c y H l A O 9 6 L 1 x V X V n E V B f z P 1 9 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 9 5 j c 7 r c c N t u I e h v x Z s q D W c q + 6 c r t V U c p s G I Q 6 I C R q T s e W 6 m + p o I R Q M G p t 7 C I w k Z C Y Z k A D 2 Z k Y T K u K 9 9 4 g O b c 5 I s Y 8 B J 0 N c 0 y U Y K k s D U L / n 1 e d F k 1 U a 4 l B P u G 9 T i R M V y 3 p c b r / J F a a J k N V r A f U E H s a p Y e w d e X + f g s i A B C Y y D l H O S h B o L H k J E R k w Z n c V n 1 W g + r 8 Y 5 y 0 c T r u d P m Y y 4 D w J C S x Q b p I K q m H c s W b g L 6 s 3 4 A 2 R A l H 6 Z G h 3 a N Y 5 p S e T l 1 E 9 w R D h l k 2 l 6 j W V 0 U Q l q t d B e w i a I R k j F g P J T g f I G U F o a L u a F q E Q q R T 4 g S R K J J A g a 1 S s t y I A w W y a r N i a
9 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j MV V S A Z 2 E b W x 0 R O 7 A S M 3 c f J 0 J b N Q C k Q d T w r F Z 8 z i F R B L B E i H f c 6 / V 5 f t + o I Y T h X O 4 U N b V v R m K X J o A q O K G M 7 R j c 8 u z r G V O J e g S 5 D z 7 5 x j i k + L Y P M J w L P N p i L s l N L j K h X w t u / h 7 H J m I a g K A u h Z 4 8 B P o i p 7 K o J g z K J B O W n 5 + 5 z H N u X b u I u O Y O w 8 D e 8 p i k h M 7 L 0 0 7 a H 3 7 7 L 7 G G Z h x b h C r Q A F c Q q t Q P N u c + V G R L z I Y i k 3 e E
j / G L X 6 P a m t + n i c e g a q z / D s X K L l E b v G R 0 Y / c r o A 6 O 7 1 m b y 8 d j j g R V I 1 f B 0 s 6 h s r r + c Y 7 x e l B w I m i l Z Z P w f 6 N / Q e h N n R J S 3 y n Y h q F S 2 N q b K t v f n V j n q t D 2 r v 9 9 o 7 D S n 9 8 u S 8 8 h Z d d Y c z 9 l y d p z X z r 5 z 6 A T O 0 P n k f H a + 1 L 7 W v t d + 1 H 6 W 0 M W F 6 Z 6 H T k V q v 3 4 D m O r P s Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d c c y H l A O 9 6 L 1 x V X V n E V B f z P 1 9 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 9 5 j c 7 r c c N t u I e h v x Z s q D W c q + 6 c r t V U c p s G I Q 6 I C R q T s e W 6 m + p o I R Q M G p t 7 C I w k Z C Y Z k A D 2 Z k Y T K u K 9 9 4 g O b c 5 I s Y 8 B J 0 N c 0 y U Y K k s D U L / n 1 e d F k 1 U a 4 l B P u G 9 T i R M V y 3 p c b r / J F a a J k N V r A f U E H s a p Y e w d e X + f g s i A B C Y y D l H O S h B o L H k J E R k w Z n c V n 1 W g + r 8 Y 5 y 0 c T r u d P m Y y 4 D w J C S x Q b p I K q m H c s W b g L 6 s 3 4 A 2 R A l H 6 Z G h 3 a N Y 5 p S e T l 1 E 9 w R D h l k 2 l 6 j W V 0 U Q l q t d B e w i a I R k j F g P J T g f I G U F o a L u a F q E Q q R T 4 g S R K J J A g a 1 S s t y I A w W y a r N i a
9 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j MV V S A Z 2 E b W x 0 R O 7 A S M 3 c f J 0 J b N Q C k Q d T w r F Z 8 z i F R B L B E i H f c 6 / V 5 f t + o I Y T h X O 4 U N b V v R m K X J o A q O K G M 7 R j c 8 u z r G V O J e g S 5 D z 7 5 x j i k + L Y P M J w L P N p i L s l N L j K h X w t u / h 7 H J m I a g K A u h Z 4 8 B P o i p 7 K o J g z K J B O W n 5 + 5 z H N u X b u I u O Y O w 8 D e 8 p i k h M 7 L 0 0 7 a H 3 7 7 L 7 G G Z h x b h C r Q A F c Q q t Q P N u c + V G R L z I Y i k 3 e E
j / G L X 6 P a m t + n i c e g a q z / D s X K L l E b v G R 0 Y / c r o A 6 O 7 1 m b y 8 d j j g R V I 1 f B 0 s 6 h s r r + c Y 7 x e l B w I m i l Z Z P w f 6 N / Q e h N n R J S 3 y n Y h q F S 2 N q b K t v f n V j n q t D 2 r v 9 9 o 7 D S n 9 8 u S 8 8 h Z d d Y c z 9 l y d p z X z r 5 z 6 A T O 0 P n k f H a + 1 L 7 W v t d + 1 H 6 W 0 M W F 6 Z 6 H T k V q v 3 4 D m O r P s Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d c c y H l A O 9 6 L 1 x V X V n E V B f z P 1 9 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 9 5 j c 7 r c c N t u I e h v x Z s q D W c q + 6 c r t V U c p s G I Q 6 I C R q T s e W 6 m + p o I R Q M G p t 7 C I w k Z C Y Z k A D 2 Z k Y T K u K 9 9 4 g O b c 5 I s Y 8 B J 0 N c 0 y U Y K k s D U L / n 1 e d F k 1 U a 4 l B P u G 9 T i R M V y 3 p c b r / J F a a J k N V r A f U E H s a p Y e w d e X + f g s i A B C Y y D l H O S h B o L H k J E R k w Z n c V n 1 W g + r 8 Y 5 y 0 c T r u d P m Y y 4 D w J C S x Q b p I K q m H c s W b g L 6 s 3 4 A 2 R A l H 6 Z G h 3 a N Y 5 p S e T l 1 E 9 w R D h l k 2 l 6 j W V 0 U Q l q t d B e w i a I R k j F g P J T g f I G U F o a L u a F q E Q q R T 4 g S R K J J A g a 1 S s t y I A w W y a r N i a
9 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j MV V S A Z 2 E b W x 0 R O 7 A S M 3 c f J 0 J b N Q C k Q d T w r F Z 8 z i F R B L B E i H f c 6 / V 5 f t + o I Y T h X O 4 U N b V v R m K X J o A q O K G M 7 R j c 8 u z r G V O J e g S 5 D z 7 5 x j i k + L Y P M J w L P N p i L s l N L j K h X w t u / h 7 H J m I a g K A u h Z 4 8 B P o i p 7 K o J g z K J B O W n 5 + 5 z H N u X b u I u O Y O w 8 D e 8 p i k h M 7 L 0 0 7 a H 3 7 7 L 7 G G Z h x b h C r Q A F c Q q t Q P N u c + V G R L z I Y i k 3 e E
j / G L X 6 P a m t + n i c e g a q z / D s X K L l E b v G R 0 Y / c r o A 6 O 7 1 m b y 8 d j j g R V I 1 f B 0 s 6 h s r r + c Y 7 x e l B w I m i l Z Z P w f 6 N / Q e h N n R J S 3 y n Y h q F S 2 N q b K t v f n V j n q t D 2 r v 9 9 o 7 D S n 9 8 u S 8 8 h Z d d Y c z 9 l y d p z X z r 5 z 6 A T O 0 P n k f H a + 1 L 7 W v t d + 1 H 6 W 0 M W F 6 Z 6 H T k V q v 3 4 D m O r P s Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d c c y H l A O 9 6 L 1 x V X V n E V B f z P 1 9 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 9 5 j c 7 r c c N t u I e h v x Z s q D W c q + 6 c r t V U c p s G I Q 6 I C R q T s e W 6 m + p o I R Q M G p t 7 C I w k Z C Y Z k A D 2 Z k Y T K u K 9 9 4 g O b c 5 I s Y 8 B J 0 N c 0 y U Y K k s D U L / n 1 e d F k 1 U a 4 l B P u G 9 T i R M V y 3 p c b r / J F a a J k N V r A f U E H s a p Y e w d e X + f g s i A B C Y y D l H O S h B o L H k J E R k w Z n c V n 1 W g + r 8 Y 5 y 0 c T r u d P m Y y 4 D w J C S x Q b p I K q m H c s W b g L 6 s 3 4 A 2 R A l H 6 Z G h 3 a N Y 5 p S e T l 1 E 9 w R D h l k 2 l 6 j W V 0 U Q l q t d B e w i a I R k j F g P J T g f I G U F o a L u a F q E Q q R T 4 g S R K J J A g a 1 S s t y I A w W y a r N i a
9 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j MV V S A Z 2 E b W x 0 R O 7 A S M 3 c f J 0 J b N Q C k Q d T w r F Z 8 z i F R B L B E i H f c 6 / V 5 f t + o I Y T h X O 4 U N b V v R m K X J o A q O K G M 7 R j c 8 u z r G V O J e g S 5 D z 7 5 x j i k + L Y P M J w L P N p i L s l N L j K h X w t u / h 7 H J m I a g K A u h Z 4 8 B P o i p 7 K o J g z K J B O W n 5 + 5 z H N u X b u I u O Y O w 8 D e 8 p i k h M 7 L 0 0 7 a H 3 7 7 L 7 G G Z h x b h C r Q A F c Q q t Q P N u c + V G R L z I Y i k 3 e E
j / G L X 6 P a m t + n i c e g a q z / D s X K L l E b v G R 0 Y / c r o A 6 O 7 1 m b y 8 d j j g R V I 1 f B 0 s 6 h s r r + c Y 7 x e l B w I m i l Z Z P w f 6 N / Q e h N n R J S 3 y n Y h q F S 2 N q b K t v f n V j n q t D 2 r v 9 9 o 7 D S n 9 8 u S 8 8 h Z d d Y c z 9 l y d p z X z r 5 z 6 A T O 0 P n k f H a + 1 L 7 W v t d + 1 H 6 W 0 M W F 6 Z 6 H T k V q v 3 4 D m O r P s Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d c c y H l A O 9 6 L 1 x V X V n E V B f z P 1 9 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 9 5 j c 7 r c c N t u I e h v x Z s q D W c q + 6 c r t V U c p s G I Q 6 I C R q T s e W 6 m + p o I R Q M G p t 7 C I w k Z C Y Z k A D 2 Z k Y T K u K 9 9 4 g O b c 5 I s Y 8 B J 0 N c 0 y U Y K k s D U L / n 1 e d F k 1 U a 4 l B P u G 9 T i R M V y 3 p c b r / J F a a J k N V r A f U E H s a p Y e w d e X + f g s i A B C Y y D l H O S h B o L H k J E R k w Z n c V n 1 W g + r 8 Y 5 y 0 c T r u d P m Y y 4 D w J C S x Q b p I K q m H c s W b g L 6 s 3 4 A 2 R A l H 6 Z G h 3 a N Y 5 p S e T l 1 E 9 w R D h l k 2 l 6 j W V 0 U Q l q t d B e w i a I R k j F g P J T g f I G U F o a L u a F q E Q q R T 4 g S R K J J A g a 1 S s t y I A w W y a r N i a
9 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j MV V S A Z 2 E b W x 0 R O 7 A S M 3 c f J 0 J b N Q C k Q d T w r F Z 8 z i F R B L B E i H f c 6 / V 5 f t + o I Y T h X O 4 U N b V v R m K X J o A q O K G M 7 R j c 8 u z r G V O J e g S 5 D z 7 5 x j i k + L Y P M J w L P N p i L s l N L j K h X w t u / h 7 H J m I a g K A u h Z 4 8 B P o i p 7 K o J g z K J B O W n 5 + 5 z H N u X b u I u O Y O w 8 D e 8 p i k h M 7 L 0 0 7 a H 3 7 7 L 7 G G Z h x b h C r Q A F c Q q t Q P N u c + V G R L z I Y i k 3 e E
j / G L X 6 P a m t + n i c e g a q z / D s X K L l E b v G R 0 Y / c r o A 6 O 7 1 m b y 8 d j j g R V I 1 f B 0 s 6 h s r r + c Y 7 x e l B w I m i l Z Z P w f 6 N / Q e h N n R J S 3 y n Y h q F S 2 N q b K t v f n V j n q t D 2 r v 9 9 o 7 D S n 9 8 u S 8 8 h Z d d Y c z 9 l y d p z X z r 5 z 6 A T O 0 P n k f H a + 1 L 7 W v t d + 1 H 6 W 0 M W F 6 Z 6 H T k V q v 3 4 D m O r P s Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d c c y H l A O 9 6 L 1 x V X V n E V B f z P 1 9 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 9 5 j c 7 r c c N t u I e h v x Z s q D W c q + 6 c r t V U c p s G I Q 6 I C R q T s e W 6 m + p o I R Q M G p t 7 C I w k Z C Y Z k A D 2 Z k Y T K u K 9 9 4 g O b c 5 I s Y 8 B J 0 N c 0 y U Y K k s D U L / n 1 e d F k 1 U a 4 l B P u G 9 T i R M V y 3 p c b r / J F a a J k N V r A f U E H s a p Y e w d e X + f g s i A B C Y y D l H O S h B o L H k J E R k w Z n c V n 1 W g + r 8 Y 5 y 0 c T r u d P m Y y 4 D w J C S x Q b p I K q m H c s W b g L 6 s 3 4 A 2 R A l H 6 Z G h 3 a N Y 5 p S e T l 1 E 9 w R D h l k 2 l 6 j W V 0 U Q l q t d B e w i a I R k j F g P J T g f I G U F o a L u a F q E Q q R T 4 g S R K J J A g a 1 S s t y I A w W y a r N i a
9 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j MV V S A Z 2 E b W x 0 R O 7 A S M 3 c f J 0 J b N Q C k Q d T w r F Z 8 z i F R B L B E i H f c 6 / V 5 f t + o I Y T h X O 4 U N b V v R m K X J o A q O K G M 7 R j c 8 u z r G V O J e g S 5 D z 7 5 x j i k + L Y P M J w L P N p i L s l N L j K h X w t u / h 7 H J m I a g K A u h Z 4 8 B P o i p 7 K o J g z K J B O W n 5 + 5 z H N u X b u I u O Y O w 8 D e 8 p i k h M 7 L 0 0 7 a H 3 7 7 L 7 G G Z h x b h C r Q A F c Q q t Q P N u c + V G R L z I Y i k 3 e E
j / G L X 6 P a m t + n i c e g a q z / D s X K L l E b v G R 0 Y / c r o A 6 O 7 1 m b y 8 d j j g R V I 1 f B 0 s 6 h s r r + c Y 7 x e l B w I m i l Z Z P w f 6 N / Q e h N n R J S 3 y n Y h q F S 2 N q b K t v f n V j n q t D 2 r v 9 9 o 7 D S n 9 8 u S 8 8 h Z d d Y c z 9 l y d p z X z r 5 z 6 A T O 0 P n k f H a + 1 L 7
W v t d + 1 H 6 W 0 M W F 6 Z 6 H T k V q v 3 4 D m O r P s Q = = < / l a t e x i t > b (2) < l a t e x i t s h a 1 _ b a s e 6 4 = " n V o 9 m O o 5 k m z f 7 G D F 8 k 0 1 A 8 o W b s w = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j M 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 5 3 H 5 n S 5 4 b b d Q t D f i j d V G s 5 U 9 k 9 X a q s 4 T I M R h 0 Q F j E j Z 8 9 x M 9 T U R i g Y M T L 2 F R x I y E g z J A H o y I w m V c V / 7 x A c 2 5 y R Z x o C T o K 9 p k o 0 U J I G p X / L r 8 6 L J q o 1 w K S f c N 6 j F i Y r l v C 8 3 X u W L 0 k T J a r S A + 4 I O Y l W x 9 g 6 8 v s 7 B Z U E C E h g H K e c k C T U W P I S I j J g y O o v P q t F 8 X o 1 z l o 8 m X M + f M h l x H w S E l i g 2 S A V V M e 9 Y s n A X 1 J v x B 8 i A K P 0 y N T q 0 a x z T k s j L q Z / g i H D K J t P 0 G s v o o h L U a q G 9 h E 0 Q j Z C K A e W n A u U N o L Q 0 X M w L U Y l U i n x A k i Q S S R A 0 q l d a k A F h t k x W b U w q q 0 A y s I 2 s j 4 m c 2 A k Y u 4 + T o S 2 b g V I g 6 n h W K j 5 n E K m C W C J E O u 5 1 + r 2 + b t U R w n C u d g o b 2 r a i M U u T Q R U c U c Z 2 j G 5 4 d n W M q c S 9 A l 2 G n n 3 j H F N 8 W g a Z T w S e b T A X Z a e W G F G v h L d / D 2 O T M Q 1 B U R Z C z x 4 D f B B T 2 V U T B m U S C c p P z 9 3 n O L Y v 3 c R d c g Z h 4 W 9 4 T V N C Z m T p p 2 0 P v 3 2 X 2 c M y D y 3 C F W g B K o h V a
g e a c 5 8 r M y T m Q x B J u 8 N H + M W u 0 e 1 N b 9 P F 4 9 A 1 V n + G Y + U W K Y 3 e M z o w + p X R B 0 Z 3 r c 3 k 4 7 H H A y u Q q u H p Z l H Z X H 8 5 x 3 i 9 K D k Q N F O y y P g / 0 L + h 9 S b O i C h v l e 1 C U K l s b U y V b e / P r X L U a X t W f 7 / R 2 G l O 7 5 c l 5 5 G z 6 q w 5 n r P l 7 D i v n X 3 n 0 A m c o f P J + e x 8 q X 2 t f a / 9 q P 0 s o Y s L 0 z 0 P n Y r U f v 0 G n c H P s g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6
4 = " n V o 9 m O o 5 k m z f 7 G D F 8 k 0 1 A 8 o W b s w = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j M 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 5 3 H 5 n S 5 4 b b d Q t D f i j d V G s 5 U 9 k 9 X a q s 4 T I M R h 0 Q F j E j Z 8 9 x M 9 T U R i g Y M T L 2 F R x I y E g z J A H o y I w m V c V / 7 x A c 2 5 y R Z x o C T o K 9 p k o 0 U J I G p X / L r 8 6 L J q o 1 w K S f c N 6 j F i Y r l v C 8 3 X u W L 0 k T J a r S A + 4 I O Y l W x 9 g 6 8 v s 7 B Z U E C E h g H K e c k C T U W P I S I j J g y O o v P q t F 8 X o 1 z l o 8 m X M + f M h l x H w S E l i g 2 S A V V M e 9 Y s n A X 1 J v x B 8 i A K P 0 y N T q 0 a x z T k s j L q Z / g i H D K J t P 0 G s v o o h L U a q G 9 h E 0 Q j Z C K A e W n A u U N o L Q 0 X M w L U Y l U i n x A k i Q S S R A 0 q l d a k A F h t k x W b U w q q 0 A y s I 2 s j 4 m c 2 A k Y u 4 + T o S 2 b g V I g 6 n h W K j 5 n E K m C W C J E O u 5 1 + r 2 + b t U R w n C u d g o b 2 r a i M U u T Q R U c U c Z 2 j G 5 4 d n W M q c S 9 A l 2 G n n 3 j H F N 8 W g a Z T w S e b T A X Z a e W G F G v h L d / D 2 O T M Q 1 B U R Z C z x 4 D f B B T 2 V U T B m U S C c p P z 9 3 n O L Y v 3 c R d c g Z h 4 W 9 4 T V N C Z m T p p 2 0 P v 3 2 X 2 c M y D y 3 C F W g B K o h V a
g e a c 5 8 r M y T m Q x B J u 8 N H + M W u 0 e 1 N b 9 P F 4 9 A 1 V n + G Y + U W K Y 3 e M z o w + p X R B 0 Z 3 r c 3 k 4 7 H H A y u Q q u H p Z l H Z X H 8 5 x 3 i 9 K D k Q N F O y y P g / 0 L + h 9 S b O i C h v l e 1 C U K l s b U y V b e / P r X L U a X t W f 7 / R 2 G l O 7 5 c l 5 5 G z 6 q w 5 n r P l 7 D i v n X 3 n 0 A m c o f P J + e x 8 q X 2 t f a / 9 q P 0 s o Y s L 0 z 0 P n Y r U f v 0 G n c H P s g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6
4 = " n V o 9 m O o 5 k m z f 7 G D F 8 k 0 1 A 8 o W b s w = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j M 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 5 3 H 5 n S 5 4 b b d Q t D f i j d V G s 5 U 9 k 9 X a q s 4 T I M R h 0 Q F j E j Z 8 9 x M 9 T U R i g Y M T L 2 F R x I y E g z J A H o y I w m V c V / 7 x A c 2 5 y R Z x o C T o K 9 p k o 0 U J I G p X / L r 8 6 L J q o 1 w K S f c N 6 j F i Y r l v C 8 3 X u W L 0 k T J a r S A + 4 I O Y l W x 9 g 6 8 v s 7 B Z U E C E h g H K e c k C T U W P I S I j J g y O o v P q t F 8 X o 1 z l o 8 m X M + f M h l x H w S E l i g 2 S A V V M e 9 Y s n A X 1 J v x B 8 i A K P 0 y N T q 0 a x z T k s j L q Z / g i H D K J t P 0 G s v o o h L U a q G 9 h E 0 Q j Z C K A e W n A u U N o L Q 0 X M w L U Y l U i n x A k i Q S S R A 0 q l d a k A F h t k x W b U w q q 0 A y s I 2 s j 4 m c 2 A k Y u 4 + T o S 2 b g V I g 6 n h W K j 5 n E K m C W C J E O u 5 1 + r 2 + b t U R w n C u d g o b 2 r a i M U u T Q R U c U c Z 2 j G 5 4 d n W M q c S 9 A l 2 G n n 3 j H F N 8 W g a Z T w S e b T A X Z a e W G F G v h L d / D 2 O T M Q 1 B U R Z C z x 4 D f B B T 2 V U T B m U S C c p P z 9 3 n O L Y v 3 c R d c g Z h 4 W 9 4 T V N C Z m T p p 2 0 P v 3 2 X 2 c M y D y 3 C F W g B K o h V a
g e a c 5 8 r M y T m Q x B J u 8 N H + M W u 0 e 1 N b 9 P F 4 9 A 1 V n + G Y + U W K Y 3 e M z o w + p X R B 0 Z 3 r c 3 k 4 7 H H A y u Q q u H p Z l H Z X H 8 5 x 3 i 9 K D k Q N F O y y P g / 0 L + h 9 S b O i C h v l e 1 C U K l s b U y V b e / P r X L U a X t W f 7 / R 2 G l O 7 5 c l 5 5 G z 6 q w 5 n r P l 7 D i v n X 3 n 0 A m c o f P J + e x 8 q X 2 t f a / 9 q P 0 s o Y s L 0 z 0 P n Y r U f v 0 G n c H P s g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6
4 = " n V o 9 m O o 5 k m z f 7 G D F 8 k 0 1 A 8 o W b s w = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j M 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 5 3 H 5 n S 5 4 b b d Q t D f i j d V G s 5 U 9 k 9 X a q s 4 T I M R h 0 Q F j E j Z 8 9 x M 9 T U R i g Y M T L 2 F R x I y E g z J A H o y I w m V c V / 7
x A c 2 5 y R Z x o C T o K 9 p k o 0 U J I G p X / L r 8 6 L J q o 1 w K S f c N 6 j F i Y r l v C 8 3 X u W L 0 k T J a r S A + 4 I O Y l W x 9 g 6 8 v s 7 B Z U E C E h g H K e c k C T U W P I S I j J g y O o v P q t F 8 X o 1 z l o 8 m X M + f M h l x H w S E l i g 2 S A V V M e 9 Y s n A X 1 J v x B 8 i A K P 0 y N T q 0 a x z T k s j L q Z / g i H D K J t P 0 G s v o o h L U a q G 9 h E 0 Q j Z C K A e W n A u U N o L Q 0 X M w L U Y l U i n x A k i Q S S R A 0 q l d a k A F h t k x W b U w q q 0 A y s I 2 s j 4 m c 2 A k Y u 4 + T o S 2 b g V I g 6 n h W K j 5 n E K m C W C J E O u 5 1 + r 2 + b t U R w n C u d g o b 2 r a i M U u T Q R U c U c Z 2 j G 5 4 d n W M q c S 9 A l 2 G n n 3 j H F N 8 W g a Z T w S e b T A X Z a e W G F G v h L d / D 2 O T M Q 1 B U R Z C z x 4 D f B B T 2 V U T B m U S C c p P z 9 3 n O L Y v 3 c R d c g Z h 4 W 9 4 T V N C Z m T p p 2 0 P v 3 2 X 2 c M y D y 3 C F W g B K o h V a g e a c 5 8 r M y T m Q x B J u 8 N H + M W u 0 e 1 N b 9 P F 4 9 A 1 V n + G Y + U W K Y 3 e M z o w + p X R B 0 Z 3 r c 3 k 4 7 H H A y u Q q u H p Z l H Z X H 8 5 x 3 i 9 K D k Q N F O y y P g / 0 L + h 9 S b O i C h v l e 1 C U K l s b U y V b e / P r X L U a X t W f 7 / R 2 G l O 7 5 c l 5 5 G z 6 q w 5 n r P l 7 D i v n X 3 n 0 A m c o f P J + e x 8 q X 2 t f a / 9 q P 0 s o Y s L 0 z 0 P n Y r U f v 0 G n c H P s g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " n V o 9 m O o 5 k m z f 7 G D F 8 k 0 1 A 8 o W b s w = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j M 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 5 3 H 5 n S 5 4 b b d Q t D f i j d V G s 5 U 9 k 9 X a q s 4 T I M R h 0 Q F j E j Z 8 9 x M 9 T U R i g Y M T L 2 F R x I y E g z J A H o y I w m V c V / 7 x A c 2 5 y R Z x o C T o K 9 p k o 0 U J I G p X / L r 8 6 L J q o 1 w K S f c N 6 j F i Y r l v C 8 3 X u W L 0 k T J a r S A + 4 I O Y l W x 9 g 6 8 v s 7 B Z U E C E h g H K e c k C T U W P I S I j J g y O o v P q t F 8 X o 1 z l o 8 m X M + f M h l x H w S E l i g 2 S A V V M e 9 Y s n A X 1 J v x B 8 i A K P 0 y N T q 0 a x z T k s j L q Z / g i H D K J t P 0 G s v o o h L U a q G 9 h E 0 Q j Z C K A e W n A u U N o L Q 0 X M w L U Y l U i n x A k i Q S S R A 0 q l d a k A F h t k x W b U w q q 0 A y s I 2 s j 4 m c 2 A k Y u 4 + T o S 2 b g V I g 6 n h W K j 5 n E K m C W C J E O u 5 1 + r 2 + b t U R w n C u d g o b 2 r a i M U u T Q R U c U c Z 2 j G 5 4 d n W M q c S 9 A l 2 G n n 3 j H F N 8 W g a Z T w S e b T A X Z a e W G F G v h L d / D 2 O T M Q 1 B U R Z C z x 4 D f B B T 2 V U T B m U S C c p P z 9 3 n O L Y v 3 c R d c g Z h 4 W 9 4 T V N C Z m T p p 2 0 P v 3 2 X 2 c M y D y 3 C F W g B K o h V a g e a c 5 8 r M y T m Q x B J u 8 N H + M W u 0 e 1 N b 9 P F 4 9 A 1 V n + G Y + U W K Y 3 e M z o w + p X R B 0 Z 3 r c 3 k 4 7 H H A y u Q q u H p Z l H Z X H 8 5 x 3 i 9 K D k Q N F O y y P g / 0 L + h 9 S b O i C h v l e 1 C U K l s b U y V b e / P r X L U a X t W f 7 / R 2 G l O 7 5 c l 5 5 G z 6 q w 5 n r P l 7 D i v n X 3 n 0 A m c o f P J + e x 8 q X 2 t f a / 9 q P 0 s o Y s L 0 z 0 P n Y r U f v 0 G n c H P s g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " n V o 9 m O o 5 k m z f 7 G D F 8 k 0 1 A 8 o W b s w = " > A A A F P 3 i c h V T d b t M w F M 5 G g V H + N r j k x l p b a a B R J d X G t A u k o S G E B G i D 7 k + q y + Q k J 4 1 V O 4 l s d 1 1 l + S G 4 h a f h M X g C 7 h C 3 3 O E k H V 2 K B k d y c n L O 5 / P z H c d + x q h U r v t t Y f F a 7 f q N m 0 u 3 6 r f v 3 L 1 3 f 3 n l w Z F M R y K A w y B l q T j x i Q R G E z h U V D E 4 y Q Q Q 7 j M 4 9 o e 7 u f / 4 D I S k a X K g J h n 0 O R k k N K I B U d Z 0 7 H / U a 5 3 H 5 n S 5 4 b b d Q t D f i j d V G s 5 U 9 k 9 X a q s 4 T I M R h 0 Q F j E j Z 8 9 x M 9 T U R i g Y M T L 2 F R x I y E g z J A H o y I w m V c V / 7 x A c 2 5 y R Z x o C T o K 9 p k o 0 U J I G p X / L r 8 6 L J q o 1 w K S f c N 6 j F i Y r l v C 8 3 X u W L 0 k T J a r S A + 4 I O Y l W x 9 g 6 8 v s 7 B Z U E C E h g H K e c k C T U W P I S I j J g y O o v P q t F 8 X o 1 z l o 8 m X M + f M h l x H w S E l i g 2 S A V V M e 9 Y s n A X 1 J v x B 8 i A K P 0 y N T q 0 a x z T k s j L q Z / g i H D K J t P 0 G s v o o h L U a q G 9 h E 0 Q j Z C K A e W n A u U N o L Q 0 X M w L U Y l U i n x A k i Q S S R A 0 q l d a k A F h t k x W b U w q q 0 A y s I 2 s j 4 m c 2 A k Y u 4 + T o S 2 b g V I g 6 n h W K j 5 n E K m C W C J E O u 5 1 + r 2 + b t U R w n C u d g o b 2 r a i M U u T Q R U c U c Z 2 j G 5 4 d n W M q c S 9 A l 2 G n n 3 j H F N 8 W g a Z T w S e b T A X Z a e W G F G v h L d / D 2 O T M Q 1 B U R Z C z x 4 D f B B T 2 V U T B m U S C c p P z 9 3 n O L Y v 3 c R d c g Z h 4 W 9 4 T V N C Z m T p p 2 0 P v 3 2 X 2 c M y D y 3 C F W g B K o h V a g e a c 5 8 r M y T m Q x B J u 8 N H + M W u 0 e 1 N b 9 P F 4 9 A 1 V n + G Y + U W K Y 3 e M z o w + p X R B 0 Z 3 r c 3 k 4 7 H H A y u Q q u H p Z l H Z X H 8 5 x 3 i 9 K D k Q N F O y y P g / 0 L + h 9 S b O i C h v l e 1 C U K l s b U y V b e / P r X L U a X t W f 7 / R 2 G l O 7 5 c l 5 5 G z 6 q w 5 n r P l 7 D i v n X 3 n 0 A m c o f P J + e x 8 q X 2 t f a / 9 q P 0 s o Y s L 0 z 0 P n Y r U f v 0 G n c H P s g = = < / l a t e x i t > b (t) < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 g Y x Y h 1 A s 1 X J R c u g + g Q Q J D o 3 i i f t R g s 6 Y s K V K Z q N a X Q K Z G P X y N q M 6 b m b g H X 7 J J u 4 s g U g g m r T Z a n 0 S E C C F b F M q X w 2 H I y G I 9 N r E 0 L h C L c q G 9 l 0 Y q j I s 3 E T n H A h t q z p B G 4 N r G 3 E / Q u 6 D r 3 8 p i W m + n Q M i p A p u t x g j 8 v O H T G q 3 Q j v / h 4 h 5 j M e A 3 I R w 9 A d A 7 q X c r 2 L c w F 1 E g 0 Y 5 k f + Q 5 q 6 l + n S X X Y I c e X v B F 1 b Q 5 Z k m X v 9 g L 5 4 W b j D c h J a h a v Q C j B K M X c D L b k v l S W S y g m o r D + Q U / r o s T X 9 j W D D p 7 P Y t 0 6 / T 1 P 0 q 5 T W b F s T W f P U m j 1 r d p 3 N l u N x x 4 M i a O w E p l t V d q K / k m O 6 V p U c K V 6 g r j L + D / R v a L t L C 6 b q W 2 W z E l I r D 9 Y X y m b w 6 1 Z 5 M + g H T n + 1 3 t n q L u 6 X F e + m d 8 u 7 7 Q X e A 2 / L e + b t e P t e 5 E 2 8 D 9 5 H 7 1 P r c + t r 6 1 v r e w 0 9 f W q x 5 4 b X k N a P n 9 0 + z / Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 g Y x Y h 1 A s 1 X J R c u g + g Q Q J D o 3 i i f t R g s 6 Y s K V K Z q N a X Q K Z G P X y N q M 6 b m b g H X 7 J J u 4 s g U g g m r T Z a n 0 S E C C F b F M q X w 2 H I y G I 9 N r E 0 L h C L c q G 9 l 0 Y q j I s 3 E T n H A h t q z p B G 4 N r G 3 E / Q u 6 D r 3 8 p i W m + n Q M i p A p u t x g j 8 v O H T G q 3 Q j v / h 4 h 5 j M e A 3 I R w 9 A d A 7 q X c r 2 L c w F 1 E g 0 Y 5 k f + Q 5 q 6 l + n S X X Y I c e X v B F 1 b Q 5 Z k m X v 9 g L 5 4 W b j D c h J a h a v Q C j B K M X c D L b k v l S W S y g m o r D + Q U / r o s T X 9 j W D D p 7 P Y t 0 6 / T 1 P 0 q 5 T W b F s T W f P U m j 1 r d p 3 N l u N x x 4 M i a O w E p l t V d q K / k m O 6 V p U c K V 6 g r j L + D / R v a L t L C 6 b q W 2 W z E l I r D 9 Y X y m b w 6 1 Z 5 M + g H T n + 1 3 t n q L u 6 X F e + m d 8 u 7 7 Q X e A 2 / L e + b t e P t e 5 E 2 8 D 9 5 H 7 1 P r c + t r 6 1 v r e w 0 9 f W q x 5 4 b X k N a P n 9 0 + z / Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 g Y x Y h 1 A s 1 X J R c u g + g Q Q J D o 3 i i f t R g s 6 Y s K V K Z q N a X Q K Z G P X y N q M 6 b m b g H X 7 J J u 4 s g U g g m r T Z a n 0 S E C C F b F M q X w 2 H I y G I 9 N r E 0 L h C L c q G 9 l 0 Y q j I s 3 E T n H A h t q z p B G 4 N r G 3 E / Q u 6 D r 3 8 p i W m + n Q M i p A p u t x g j 8 v O H T G q 3 Q j v / h 4 h 5 j M e A 3 I R w 9 A d A 7 q X c r 2 L c w F 1 E g 0 Y 5 k f + Q 5 q 6 l + n S X X Y I c e X v B F 1 b Q 5 Z k m X v 9 g L 5 4 W b j D c h J a h a v Q C j B K M X c D L b k v l S W S y g m o r D + Q U / r o s T X 9 j W D D p 7 P Y t 0 6 / T 1 P 0 q 5 T W b F s T W f P U m j 1 r d p 3 N l u N x x 4 M i a O w E p l t V d q K / k m O 6 V p U c K V 6 g r j L + D / R v a L t L C 6 b q W 2 W z E l I r D 9 Y X y m b w 6 1 Z 5 M + g H T n + 1 3 t n q L u 6 X F e + m d 8 u 7 7 Q X e A 2 / L e + b t e P t e 5 E 2 8 D 9 5 H 7 1 P r c + t r 6 1 v r e w 0 9 f W q x 5 4 b X k N a P n 9 0 + z / Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 g Y x Y h 1 A s 1 X J R c u g + g Q Q J D o 3 i i f t R g s 6 Y s K V K Z q N a X Q K Z G P X y N q M 6 b m b g H X 7 J J u 4 s g U g g m r T Z a n 0 S E C C F b F M q X w 2 H I y G I 9 N r E 0 L h C L c q G 9 l 0 Y q j I s 3 E T n H A h t q z p B G 4 N r G 3 E / Q u 6 D r 3 8 p i W m + n Q M i p A p u t x g j 8 v O H T G q 3 Q j v / h 4 h 5 j M e A 3 I R w 9 A d A 7 q X c r 2 L c w F 1 E g 0 Y 5 k f + Q 5 q 6 l + n S X X Y I c e X v B F 1 b Q 5 Z k m X v 9 g L 5 4 W b j D c h J a h a v Q C j B K M X c D L b k v l S W S y g m o r D + Q U / r o s T X 9 j W D D p 7 P Y t 0 6 / T 1 P 0 q 5 T W b F s T W f P U m j 1 r d p 3 N l u N x x 4 M i a O w E p l t V d q K / k m O 6 V p U c K V 6 g r j L + D / R v a L t L C 6 b q W 2 W z E l I r D 9 Y X y m b w 6 1 Z 5 M + g H T n + 1 3 t n q L u 6 X F e + m d 8 u 7 7 Q X e A 2 / L e + b t e P t e 5 E 2 8 D 9 5 H 7 1 P r c + t r 6 1 v r e w 0 9 f W q x 5 4 b X k N a P n 9 0 + z / Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 g Y x Y h 1 A s 1 X J R c u g + g Q Q J D o 3 i i f t R g s 6 Y s K V K Z q N a X Q K Z G P X y N q M 6 b m b g H X 7 J J u 4 s g U g g m r T Z a n 0 S E C C F b F M q X w 2 H I y G I 9 N r E 0 L h C L c q G 9 l 0 Y q j I s 3 E T n H A h t q z p B G 4 N r G 3 E / Q u 6 D r 3 8 p i W m + n Q M i p A p u t x g j 8 v O H T G q 3 Q j v / h 4 h 5 j M e A 3 I R w 9 A d A 7 q X c r 2 L c w F 1 E g 0 Y 5 k f + Q 5 q 6 l + n S X X Y I c e X v B F 1 b Q 5 Z k m X v 9 g L 5 4 W b j D c h J a h a v Q C j B K M X c D L b k v l S W S y g m o r D + Q U / r o s T X 9 j W D D p 7 P Y t 0 6 / T 1 P 0 q 5 T W b F s T W f P U m j 1 r d p 3 N l u N x x 4 M i a O w E p l t V d q K / k m O 6 V p U c K V 6 g r j L + D / R v a L t L C 6 b q W 2 W z E l I r D 9 Y X y m b w 6 1 Z 5 M + g H T n + 1 3 t n q L u 6 X F e + m d 8 u 7 7 Q X e A 2 / L e + b t e P t e 5 E 2 8 D 9 5 H 7 1 P r c + t r 6 1 v r e w 0 9 f W q x 5 4 b X k N a P n 9 0 + z / Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 g Y x Y h 1 A s 1 X J R c u g + g Q Q J D o 3 i i f t R g s 6 Y s K V K Z q N a X Q K Z G P X y N q M 6 b m b g H X 7 J J u 4 s g U g g m r T Z a n 0 S E C C F b F M q X w 2 H I y G I 9 N r E 0 L h C L c q G 9 l 0 Y q j I s 3 E T n H A h t q z p B G 4 N r G 3 E / Q u 6 D r 3 8 p i W m + n Q M i p A p u t x g j 8 v O H T G q 3 Q j v / h 4 h 5 j M e A 3 I R w 9 A d A 7 q X c r 2 L c w F 1 E g 0 Y 5 k f + Q 5 q 6 l + n S X X Y I c e X v B F 1 b Q 5 Z k m X v 9 g L 5 4 W b j D c h J a h a v Q C j B K M X c D L b k v l S W S y g m o r D + Q U / r o s T X 9 j W D D p 7 P Y t 0 6 / T 1 P 0 q 5 T W b F s T W f P U m j 1 r d p 3 N l u N x x 4 M i a O w E p l t V d q K / k m O 6 V p U c K V 6 g r j L + D / R v a L t L C 6 b q W 2 W z E l I r D 9 Y X y m b w 6 1 Z 5 M + g H T n + 1 3 t n q L u 6 X F e + m d 8 u 7 7 Q X e A 2 / L e + b t e P t e 5 E 2 8 D 9 5 H 7 1 P r c + t r 6 1 v r e w 0 9 f W q x 5 4 b X k N a P n 9 0 + z / Q = < / l a t e x i t > mini-batch < l a t e x i t s h a 1 _ b a s e 6 4 = " y F H j K 2 O P 4 1 b Q w K 0 5 T Z 5 e d c b N 8 U 8 = " > A A A F S 3 i c h V T d b t M w F M 5 G B 6 P 8 b X D J j b W u E k J b l V Q b 0 y 6 Q h o Y Q E q A N 9 i v V 1 e Q k J 4 1 V 2 4 l s d 1 1 l + V G 4 h a f h A X g O 7 h A X O E l H l 6 H B k R I d n / P 5 / H z H d p g z q r T v f 5 + b v 9 V Y u H 1 n 8 W 7 z 3 v 0 H D x 8 t L T 8 + V t l I R n A U Z S y T p y F R w K i A I 0 0 1 g 9 N c A u E h g 5 N w u F v 4 T 8 5 B K p q J Q z 3 J o c / J Q N C E R k Q 7 0 9 n S M u Z h d m E 4 F X Q 9 J D p K 7 d l S y + / 4 p a C / l W C q t L y p 7 J 8 t N 1 Z w n E U j D k J H j C j V C / x c 9 w 2 R m k Y M b L O N R w p y E g 3 J A H o q J 4 K q t G 9 C E g K 7 5 i R v x W s 2 g o y I 8 u s d w L 8 / k P u D s t 1 a B m u R E t w 9 1 B n b q A F 9 4 U y Q 2 I + B C k 6 X T 7 C r 3 a t 6 W w G m z 4 e x 7 5 1 + g u c a r 9 M a c 2 e N Z E 1 b 6 w 5 t O b A 2 W w x H n c 8 s A a l W 4 F Z L S u 7 1 l / B M V 4 r S 4 4 k z b U q M / 4 P 9 G 9 o c x X n R F a v y n Y p q F K 2 N q b K d v D n V T n u d g K n f 9 x o 7 f j T 9 2 X R e + q t e M + 8 w N v y d r y 3 3 r 5 3 5 E X e 2 P v s f f G + N r 4 1 f j R + N n 5 V 0 P m 5 6 Z 4 n X k 0 W F n 4 D 0 g P T / w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y F H j K 2 O P 4 1 b Q w K 0 5 T Z 5 e d c b N 8 U 8 = " > A A A F S 3 i c h V T d b t M w F M 5 G B 6 P 8 b X D J j b W u E k J b l V Q b 0 y 6 Q h o Y Q E q A N 9 i v V 1 e Q k J 4 1 V 2 4 l s d 1 1 l + V G 4 h a f h A X g O 7 h A X O E l H l 6 H B k R I d n / P 5 / H z H d p g z q r T v f 5 + b v 9 V Y u H 1 n 8 W 7 z 3 v 0 H D x 8 t L T 8 + V t l I R n A U Z S y T p y F R w K i A I 0 0 1 g 9 N c A u E h g 5 N w u F v 4 T 8 5 B K p q J Q z 3 J o c / J Q N C E R k Q 7 0 9 n S M u Z h d m E 4 F X Q 9 J D p K 7 d l S y + / 4 p a C / l W C q t L y p 7 J 8 t
n t A = " > A A A F P 3 i c h V T N b t N A E D Y Q o I T f w p H L i i Q S o B L Z U Q v q A a k I h J A
n t A = " > A A A F P 3 i c h V T N b t N A E D Y Q o I T f w p H L i i Q S o B L Z U Q v q A a k I h J A
n t A = " > A A A F P 3 i c h V T N b t N A E D Y Q o I T f w p H L i i Q S o B L Z U Q v q A a k I h J A
n t A = " > A A A F P 3 i c h V T N b t N A E D Y Q o I T f w p H L i i Q S o B L Z U Q v q A a k I h J A
n t A = " > A A A F P 3 i c h V T N b t N A E D Y Q o I T f w p H L i i Q S o B L Z U Q v q A a k I h J A
n t A = " > A A A F P 3 i c h V T N b t N A E D Y Q o I T f w p H L i i Q S o B L Z U Q v q A a k I h J A
N 1 Z w n E U j D k J H j C j V C / x c 9 w 2 R m k Y M b L O N R w p y E g 3 J A H o q J 4 K q t G 9 C E g K 7 5 i R 5 z o C T q G + o y E c a R G S b V / z m o m y 4 b i N c q Q k P L W p z o l N 1 3 V c Y b / I l m d C q H i 3 i o a S D V N e s v c O g b w p w V Z A E A e M o 4 5 y I 2 G D J Y 0 j I i G l r 8 v S 8 H i 3 k 9 T j n x Z j i t e K v x I i H I C F 2 R L F B J q l O e d e R h Q 9 A v x t / g h y I N q 8 z a 2 L 3 j V N a E X k 1 9 X O c E E 7 Z Z J r e Y J V c V o L a b b Q n 2 A T R B O k U U H F C U N E A y i r D 5 b w Q V U h n K A S k i F B I g a R J s 9 a C i g h z Z b J 6 Y 0 o 7 B c T A N b I 2 J m r i J m D d P k 6 G r m w G W o N s 4 l m p + I J B o k t i i Z T Z u N f t 9 / q m 3 U Q I w 4 X e K W 1 o 2 4 n B L B O D O j i h j O 1 Y 0 w r c 1 7 W 2 F v c G d B V 6 t s Y F p l w 6 B l l I J J 5 t s J d l Z 4 4 Y 2 a y F d z e J s c m Y x q A p i 6 H n j g E + T K k 6 0 B M G V R I F 2 l 0 g / y V O i 3 u 0 i g / I O c S l v x W s 2 g o y I 8 u s d w L 8 / k P u D s t 1 a B m u R E t w 9 1 B n b q A F 9 4 U y Q 2 I + B C k 6 X T 7 C r 3 a t 6 W w G m z 4 e x 7 5 1 + g u c a r 9 M a c 2 e N Z E 1 b 6 w 5 t O b A 2 W w x H n c 8 s A a l W 4 F Z L S u 7 1 l / B M V 4 r S 4 4 k z b U q M / 4 P 9 G 9 o c x X n R F a v y n Y p q F K 2 N q b K d v D n V T n u d g K n f 9 x o 7 f j T 9 2 X R e + q t e M + 8 w N v y d r y 3 3 r 5 3 5 E X e 2 P v s f f G + N r 4 1 f j R + N n 5 V 0 P m 5 6 Z 4 n X k 0 W F n 4 D 0 g P T / w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y F H j K 2 O P 4 1 b Q w K 0 5 T Z 5 e d c b N 8 U 8 = " > A A A F S 3 i c h V T d b t M w F M 5 G B 6 P 8 b X D J j b W u E k J b l V Q b 0 y 6 Q h o Y Q E q A N 9 i v V 1 e Q k J 4 1 V 2 4 l s d 1 1 l + V G 4 h a f h A X g O 7 h A X O E l H l 6 H B k R I d n / P 5 / H z H d p g z q r T v f 5 + b v 9 V Y u H 1 n 8 W 7 z 3 v 0 H D x 8 t L T 8 + V t l I R n A U Z S y T p y F R w K i A I 0 0 1 g 9 N c A u E h g 5 N w u F v 4 T 8 5 B K p q J Q z 3 J o c / J Q N C E R k Q 7 0 9 n S M u Z h d m E 4 F X Q 9 J D p K 7 d l S y + / 4 p a C / l W C q t L y p 7 J 8 t N 1 Z w n E U j D k J H j C j V C / x c 9 w 2 R m k Y M b L O N R w p y E g 3 J A H o q J 4 K q t G 9 C E g K 7 5 i R 5 z o C T q G + o y E c a R G S b V / z m o m y 4 b i N c q Q k P L W p z o l N 1 3 V c Y b / I l m d C q H i 3 i o a S D V N e s v c O g b w p w V Z A E A e M o 4 5 y I 2 G D J Y 0 j I i G l r 8 v S 8 H i 3 k 9 T j n x Z j i t e K v x I i H I C F 2 R L F B J q l O e d e R h Q 9 A v x t / g h y I N q 8 z a 2 L 3 j V N a E X k 1 9 X O c E E 7 Z Z J r e Y J V c V o L a b b Q n 2 A T R B O k U U H F C U N E A y i r D 5 b w Q V U h n K A S k i F B I g a R J s 9 a C i g h z Z b J 6 Y 0 o 7 B c T A N b I 2 J m r i J m D d P k 6 G r m w G W o N s 4 l m p + I J B o k t i i Z T Z u N f t 9 / q m 3 U Q I w 4 X e K W 1 o 2 4 n B L B O D O j i h j O 1 Y 0 w r c 1 7 W 2 F v c G d B V 6 t s Y F p l w 6 B l l I J J 5 t s J d l Z 4 4 Y 2 a y F d z e J s c m Y x q A p i 6 H n j g E + T K k 6 0 B M G V R I F 2 l 0 g / y V O i 3 u 0 i g / I O c S l v x W s 2 g o y I 8 u s d w L 8 / k P u D s t 1 a B m u R E t w 9 1 B n b q A F 9 4 U y Q 2 I + B C k 6 X T 7 C r 3 a t 6 W w G m z 4 e x 7 5 1 + g u c a r 9 M a c 2 e N Z E 1 b 6 w 5 t O b A 2 W w x H n c 8 s A a l W 4 F Z L S u 7 1 l / B M V 4 r S 4 4 k z b U q M / 4 P 9 G 9 o c x X n R F a v y n Y p q F K 2 N q b K d v D n V T n u d g K n f 9 x o 7 f j T 9 2 X R e + q t e M + 8 w N v y d r y 3 3 r 5 3 5 E X e 2 P v s f f G + N r 4 1 f j R + N n 5 V 0 P m 5 6 Z 4 n X k 0 W F n 4 D 0 g P T / w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y F H j K 2 O P 4 1 b Q w K 0 5 T Z 5 e d c b N 8 U 8 = " > A A A F S 3 i c h V T d b t M w F M 5 G B 6 P 8 b X D J j b W u E k J b l V Q b 0 y 6 Q h o Y Q E q A N 9 i v V 1 e Q k J 4 1 V 2 4 l s d 1 1 l + V G 4 h a f h A X g O 7 h A X O E l H l 6 H B k R I d n / P 5 / H z H d p g z q r T v f 5 + b v 9 V Y u H 1 n 8 W 7 z 3 v 0 H D x 8 t L T 8 + V t l I R n A U Z S y T p y F R w K i A I 0 0 1 g 9 N c A u E h g 5 N w u F v 4 T 8 5 B K p q J Q z 3 J o c / J Q N C E R k Q 7 0 9 n S M u Z h d m E 4 F X Q 9 J D p K 7 d l S y + / 4 p a C / l W C q t L y p 7 J 8 t N 1 Z w n E U j D k J H j C j V C / x c 9 w 2 R m k Y M b L O N R w p y E g 3 J A H o q J 4 K q t G 9 C E g K 7 5 i R 5 z o C T q G + o y E c a R G S b V / z m o m y 4 b i N c q Q k P L W p z o l N 1 3 V c Y b / I l m d C q H i 3 i o a S D V N e s v c O g b w p w V Z A E A e M o 4 5 y I 2 G D J Y 0 j I i G l r 8 v S 8 H i 3 k 9 T j n x Z j i t e K v x I i H I C F 2 R L F B J q l O e d e R h Q 9 A v x t / g h y I N q 8 z a 2 L 3 j V N a E X k 1 9 X O c E E 7 Z Z J r e Y J V c V o L a b b Q n 2 A T R B O k U U H F C U N E A y i r D 5 b w Q V U h n K A S k i F B I g a R J s 9 a C i g h z Z b J 6 Y 0 o 7 B c T A N b I 2 J m r i J m D d P k 6 G r m w G W o N s 4 l m p + I J B o k t i i Z T Z u N f t 9 / q m 3 U Q I w 4 X e K W 1 o 2 4 n B L B O D O j i h j O 1 Y 0 w r c 1 7 W 2 F v c G d B V 6 t s Y F p l w 6 B l l I J J 5 t s J d l Z 4 4 Y 2 a y F d z e J s c m Y x q A p i 6 H n j g E + T K k 6 0 B M G V R I F 2 l 0 g / y V O i 3 u 0 i g / I O c S l v x W s 2 g o y I 8 u s d w L 8 / k P u D s t 1 a B m u R E t w 9 1 B n b q A F 9 4 U y Q 2 I + B C k 6 X T 7 C r 3 a t 6 W w G m z 4 e x 7 5 1 + g u c a r 9 M a c 2 e N Z E 1 b 6 w 5 t O b A 2 W w x H n c 8 s A a l W 4 F Z L S u 7 1 l / B M V 4 r S 4 4 k z b U q M / 4 P 9 G 9 o c x X n R F a v y n Y p q F K 2 N q b K d v D n V T n u d g K n f 9 x o 7 f j T 9 2 X R e + q t e M + 8 w N v y d r y 3 3 r 5 3 5 E X e 2 P v s f f G + N r 4 1 f j R + N n 5 V 0 P m 5 6 Z 4 n X k 0 W F n 4 D 0 g P T / w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y F H j K 2 O P 4 1 b Q w K 0 5 T Z 5 e d c b N 8 U 8 = " > A A A F S 3 i c h V T d b t M w F M 5 G B 6 P 8 b X D J j b W u E k J b l V Q b 0 y 6 Q h o Y Q E q A N 9 i v V 1 e Q k J 4 1 V 2 4 l s d 1 1 l + V G 4 h a f h A X g O 7 h A X O E l H l 6 H B k R I d n / P 5 / H z H d p g z q r T v f 5 + b v 9 V Y u H 1 n 8 W 7 z 3 v 0 H D x 8 t L T 8 + V t l I R n A U Z S y T p y F R w K i A I 0 0 1 g 9 N c A u E h g 5 N w u F v 4 T 8 5 B K p q J Q z 3 J o c / J Q N C E R k Q 7 0 9 n S M u Z h d m E 4 F X Q 9 J D p K 7 d l S y + / 4 p a C / l W C q t L y p 7 J 8 t N 1 Z w n E U j D k J H j C j V C / x c 9 w 2 R m k Y M b L O N R w p y E g 3 J A H o q J 4 K q t G 9 C E g K 7 5 i R 5 z o C T q G + o y E c a R G S b V / z m o m y 4 b i N c q Q k P L W p z o l N 1 3 V c Y b / I l m d C q H i 3 i o a S D V N e s v c O g b w p w V Z A E A e M o 4 5 y I 2 G D J Y 0 j I i G l r 8 v S 8 H i 3 k 9 T j n x Z j i t e K v x I i H I C F 2 R L F B J q l O e d e R h Q 9 A v x t / g h y I N q 8 z a 2 L 3 j V N a E X k 1 9 X O c E E 7 Z Z J r e Y J V c V o L a b b Q n 2 A T R B O k U U H F C U N E A y i r D 5 b w Q V U h n K A S k i F B I g a R J s 9 a C i g h z Z b J 6 Y 0 o 7 B c T A N b I 2 J m r i J m D d P k 6 G r m w G W o N s 4 l m p + I J B o k t i i Z T Z u N f t 9 / q m 3 U Q I w 4 X e K W 1 o 2 4 n B L B O D O j i h j O 1 Y 0 w r c 1 7 W 2 F v c G d B V 6 t s Y F p l w 6 B l l I J J 5 t s J d l Z 4 4 Y 2 a y F d z e J s c m Y x q A p i 6 H n j g E + T K k 6 0 B M G V R I F 2 l 0 g / y V O i 3 u 0 i g / I O c S l v x W s 2 g o y I 8 u s d w L 8 / k P u D s t 1 a B m u R E t w 9 1 B n b q A F 9 4 U y Q 2 I + B C k 6 X T 7 C r 3 a t 6 W w G m z 4 e x 7 5 1 + g u c a r 9 M a c 2 e N Z E 1 b 6 w 5 t O b A 2 W w x H n c 8 s A a l W 4 F Z L S u 7 1 l / B M V 4 r S 4 4 k z b U q M / 4 P 9 G 9 o c x X n R F a v y n Y p q F K 2 N q b K d v D n V T n u d g K n f 9 x o 7 f j T 9 2 X R e + q t e M + 8 w N v y d r y 3 3 r 5 3 5 E X e 2 P v s f f G + N r 4 1 f j R + N n 5 V 0 P m 5 6 Z 4 n X k 0 W F n 4 D 0 g P T / w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y F H j K 2 O P 4 1 b Q w K 0 5 T Z 5 e d c b N 8 U 8 = " > A A A F S 3 i c h V T d b t M w F M 5 G B 6 P 8 b X D J j b W u E k J b l V Q b 0 y 6 Q h o Y Q E q A N 9 i v V 1 e Q k J 4 1 V 2 4 l s d 1 1 l + V G 4 h a f h A X g O 7 h A X O E l H l 6 H B k R I d n / P 5 / H z H d p g z q r T v f 5 + b v 9 V Y u H 1 n 8 W 7 z 3 v 0 H D x 8 t L T 8 + V t l I R n A U Z S y T p y F R w K i A I 0 0 1 g 9 N c A u E h g 5 N w u F v 4 T 8 5 B K p q J Q z 3 J o c / J Q N C E R k Q 7 0 9 n S M u Z h d m E 4 F X Q 9 J D p K 7 d l S y + / 4 p a C / l W C q t L y p 7 J 8 t N 1 Z w n E U j D k J H j C j V C / x c 9 w 2 R m k Y M b L O N R w p y E g 3 J A H o q J 4 K q t G 9 C E g K 7 5 i R 5 z o C T q G + o y E c a R G S b V / z m o m y 4 b i N c q Q k P L W p z o l N 1 3 V c Y b / I l m d C q H i 3 i o a S D V N e s v c O g b w p w V Z A E A e M o 4 5 y I 2 G D J Y 0 j I i G l r 8 v S 8 H i 3 k 9 T j n x Z j i t e K v x I i H I C F 2 R L F B J q l O e d e R h Q 9 A v x t / g h y I N q 8 z a 2 L 3 j V N a E X k 1 9 X O c E E 7 Z Z J r e Y J V c V o L a b b Q n 2 A T R B O k U U H F C U N E A y i r D 5 b w Q V U h n K A S k i F B I g a R J s 9 a C i g h z Z b J 6 Y 0 o 7 B c T A N b I 2 J m r i J m D d P k 6 G r m w G W o N s 4 l m p + I J B o k t i i Z T Z u N f t 9 / q m 3 U Q I w 4 X e K W 1 o 2 4 n B L B O D O j i h j O 1 Y 0 w r c 1 7 W 2 F v c G d B V 6 t s Y F p l w 6 B l l I J J 5 t s J d l Z 4 4 Y 2 a y F d z e J s c m Y x q A p i 6 H n j g E + T K k 6 0 B M G V R I F 2 l 0 g / y V O i 3 u 0 i g / I O c S l v x W s 2 g o y I 8 u s d w L 8 / k P u D s t 1 a B m u R E t w 9 1 B n b q A F 9 4 U y Q 2 I + B C k 6 X T 7 C r 3 a t 6 W w G m z 4 e x 7 5 1 + g u c a r 9 M a c 2 e N Z E 1 b 6 w 5 t O b A 2 W w x H n c 8 s A a l W 4 F Z L S u 7 1 l / B M V 4 r S 4 4 k z b U q M / 4 P 9 G 9 o c x X n R F a v y n Y p q F K 2 N q b K d v D n V T n u d g K n f 9 x o 7 f j T 9 2 X R e + q t e M + 8 w N v y d r y 3 3 r 5 3 5 E X e 2
P v s f f G + N r 4 1 f j R + N n 5 V 0 P m 5 6 Z 4 n X k 0 W F n 4 D 0 g P T / w = = < / l a t e x i t > L unsup ( , ✓) < l a t e x i t s h a 1 _ b a s e 6 4 = " k 1 i A A 4 f 6 p u q 1 8 W w D / U 7 S N i X j h M U = " > A A A F Z 3 i c h V T r i t Q w F K 6 6 4 2 W 8 7 K o g g n + i s w M q 6 9 A O X t g f w o o i g o q X 3 V V h M i x p e z o N m 6 Q l O d 3 Z I e Q p f B r / 6 l P 4 C L 6 F a W f W s S t q o O X k n C / n 8 p 2 T x K X g B s P w + 4 m T p 1 Y 6 p 8 + c P d c 9 f + H i p d W 1 y 1 c + m K L S C e w m h S j 0 p 5 g Z E F z B L n I U 8 K n U w G Q s 4 G O 8 / 7 S 2 f z w A b X i h d n B W w l i y i e I Z T x h 6 1 d 7 a P S o Z 5 g k T 9 p X b s 1 T G x a G l y N W M V M p U p X O 3 a Z n z D Y o 5 I L u z t 9 Y L B 2 G z y J 9 C t B B 6 w W K 9 3 b u 8 c o u m R V J J U J g I Z s w o C k s c W 6 a R J w J c t 0 8 r A y V L 9 t k E R q Z k i p t 8 b G M W g z h m Z G U p Q L J k b L k q K w S V u O 5 v d n v Y k N H W M W n M T M a O 9 O s q z X F b r f y b L S s U m r a 3 R M a a T 3 J s a U c 7 0 d j W 4 H l C G h R M k 0 J K p l J L t U w h Y 5 V A Z 8 v 8 o O 0 t l m 0 / B 3 U L 0 4 3 6 b 1 Q l Y 9 C Q e q L E p N A c c z n 0 Z N F t w J f T 9 1 A C Q / u s c D b 1 3 z T n c y J / D 3 2 X Z k x y M V u E t 9 R k R 5 m Q f p + 8 U W J G e E Z 8 V 0 k 9 P a Q u g B R z x V G / C D c E C x I D M U w Z Y k D z r N s q w f i 5 8 W m K d m E G v Q B q 4 g v Z m D I z 8 x 1 w / p x k + z 5 t A Y i g u 3 S Z K j 0 U k G F D L N O 6 m I 6 G 4 9 H Y 9 r u E U D j E r U Z H N v 2 y V B R q 0 g Z n X I g t Z 3 u R / 4 b O t f z + B T 1 3 v d z T G t N s P Y M i Z p o u D 7 i j t A t P j O 6 2 3 P t b J s R s y l N A L l I Y + T G g O z k 3 2 z g T M A 9 i A P 1 9 C h / T v L 5 W 6 3 S b H U D a 2 H v R u p t D l m T Z e 4 O I v n p d + m E 5 D m 3 c N W g N m O R Y + I b W 3 N f C E k n l P m g 1 G M q K P n n q 7 O B B 9 C C k 0 z R 0 X n 5 I c w y b k M 6 + c T Z x 9 r m z O 8 5 u e 5 2 r 2 + P H g y I Y 7 E V 2 v c n s W H 0 1 x 3 S j S T n R v E T T R P w f 6 N / Q 7 j o t m X b N q 7 L Z L D I X H t 1 f C J v R r 1 f l w 3 A Q e f n d / d 5 W u H h f z g Y 3 g l v B 7 S A K H g V b w Y v g b b A b J M H n 4 E v w N f i 2 8 q O z 2 r n W u T 6 H n j y x O H M 1 a K 3 O z Z 9 M M d 0 W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k 1 i A A 4 f 6 p u q 1 8 W w D / U 7 S N i X j h M U = " > A A A F Z 3 i c h V T r i t Q w F K 6 6 4 2 W 8 7 K o g g n + i s w M q 6 9 A O X t g f w o o i g o q X 3 V V h M i x p e z o N m 6 Q l O d 3 Z I e Q p f B r / 6 l P 4 C L 6 F a W f W s S t q o O X k n C / n 8 p 2 T x K X g B s P w + 4 m T p 1 Y 6 p 8 + c P d c 9 f + H i p d W 1 y 1 c + m K L S C e w m h S j 0 p 5 g Z E F z B L n I U 8 K n U w G Q s 4 G O 8 / 7 S 2 f z w A b X i h d n B W w l i y i e I Z T x h 6 1 d 7 a P S o Z 5 g k T 9 p X b s 1 T G x a G l y N W M V M p U p X O 3 a Z n z D Y o 5 I L u z t 9 Y L B 2 G z y J 9 C t B B 6 w W K 9 3 b u 8 c o u m R V J J U J g I Z s w o C k s c W 6 a R J w J c t 0 8 r A y V L 9 t k E R q Z k i p t 8 b G M W g z h m Z G U p Q L J k b L k q K w S V u O 5 v d n v Y k N H W M W n M T M a O 9 O s q z X F b r f y b L S s U m r a 3 R M a a T 3 J s a U c 7 0 d j W 4 H l C G h R M k 0 J K p l J L t U w h Y 5 V A Z 8 v 8 o O 0 t l m 0 / B 3 U L 0 4 3 6 b 1 Q l Y 9 C Q e q L E p N A c c z n 0 Z N F t w J f T 9 1 A C Q / u s c D b 1 3 z T n c y J / D 3 2 X Z k x y M V u E t 9 R k R 5 m Q f p + 8 U W J G e E Z 8 V 0 k 9 P a Q u g B R z x V G / C D c E C x I D M U w Z Y k D z r N s q w f i 5 8 W m K d m E G v Q B q 4 g v Z m D I z 8 x 1 w / p x k + z 5 t A Y i g u 3 S Z K j 0 U k G F D L N O 6 m I 6 G 4 9 H Y 9 r u E U D j E r U Z H N v 2 y V B R q 0 g Z n X I g t Z 3 u R / 4 b O t f z + B T 1 3 v d z T G t N s P Y M i Z p o u D 7 i j t A t P j O 6 2 3 P t b J s R s y l N A L l I Y + T G g O z k 3 2 z g T M A 9 i A P 1 9 C h / T v L 5 W 6 3 S b H U D a 2 H v R u p t D l m T Z e 4 O I v n p d + m E 5 D m 3 c N W g N m O R Y + I b W 3 N f C E k n l P m g 1 G M q K P n n q 7 O B B 9 C C k 0 z R 0 X n 5 I c w y b k M 6 + c T Z x 9 r m z O 8 5 u e 5 2 r 2 + P H g y I Y 7 E V 2 v c n s W H 0 1 x 3 S j S T n R v E T T R P w f 6 N / Q 7 j o t m X b N q 7 L Z L D I X H t 1 f C J v R r 1 f l w 3 A Q e f n d / d 5 W u H h f z g Y 3 g l v B 7 S A K H g V b w Y v g b b A b J M H n 4 E v w N f i 2 8 q O z 2 r n W u T 6 H n j y x O H M 1 a K 3 O z Z 9 M M d 0 W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k 1 i A A 4 f 6 p u q 1 8 W w D / U 7 S N i X j h M U = " > A A A F Z 3 i c h V T r i t Q w F K 6 6 4 2 W 8 7 K o g g n + i s w M q 6 9 A O X t g f w o o i g o q X 3 V V h M i x p e z o N m 6 Q l O d 3 Z I e Q p f B r / 6 l P 4 C L 6 F a W f W s S t q o O X k n C / n 8 p 2 T x K X g B s P w + 4 m T p 1 Y 6 p 8 + c P d c 9 f + H i p d W 1 y 1 c + m K L S C e w m h S j 0 p 5 g Z E F z B L n I U 8 K n U w G Q s 4 G O 8 / 7 S 2 f z w A b X i h d n B W w l i y i e I Z T x h 6 1 d 7 a P S o Z 5 g k T 9 p X b s 1 T G x a G l y N W M V M p U p X O 3 a Z n z D Y o 5 I L u z t 9 Y L B 2 G z y J 9 C t B B 6 w W K 9 3 b u 8 c o u m R V J J U J g I Z s w o C k s c W 6 a R J w J c t 0 8 r A y V L 9 t k E R q Z k i p t 8 b G M W g z h m Z G U p Q L J k b L k q K w S V u O 5 v d n v Y k N H W M W n M T M a O 9 O s q z X F b r f y b L S s U m r a 3 R M a a T 3 J s a U c 7 0 d j W 4 H l C G h R M k 0 J K p l J L t U w h Y 5 V A Z 8 v 8 o O 0 t l m 0 / B 3 U L 0 4 3 6 b 1 Q l Y 9 C Q e q L E p N A c c z n 0 Z N F t w J f T 9 1 A C Q / u s c D b 1 3 z T n c y J / D 3 2 X Z k x y M V u E t 9 R k R 5 m Q f p + 8 U W J G e E Z 8 V 0 k 9 P a Q u g B R z x V G / C D c E C x I D M U w Z Y k D z r N s q w f i 5 8 W m K d m E G v Q B q 4 g v Z m D I z 8 x 1 w / p x k + z 5 t A Y i g u 3 S Z K j 0 U k G F D L N O 6 m I 6 G 4 9 H Y 9 r u E U D j E r U Z H N v 2 y V B R q 0 g Z n X I g t Z 3 u R / 4 b O t f z + B T 1 3 v d z T G t N s P Y M i Z p o u D 7 i j t A t P j O 6 2 3 P t b J s R s y l N A L l I Y + T G g O z k 3 2 z g T M A 9 i A P 1 9 C h / T v L 5 W 6 3 S b H U D a 2 H v R u p t D l m T Z e 4 O I v n p d + m E 5 D m 3 c N W g N m O R Y + I b W 3 N f C E k n l P m g 1 G M q K P n n q 7 O B B 9 C C k 0 z R 0 X n 5 I c w y b k M 6 + c T Z x 9 r m z O 8 5 u e 5 2 r 2 + P H g y I Y 7 E V 2 v c n s W H 0 1 x 3 S j S T n R v E T T R P w f 6 N / Q 7 j o t m X b N q 7 L Z L D I X H t 1 f C J v R r 1 f l w 3 A Q e f n d / d 5 W u H h f z g Y 3 g l v B 7 S A K H g V b w Y v g b b A b J M H n 4 E v w N f i 2 8 q O z 2 r n W u T 6 H n j y x O H M 1 a K 3 O z Z 9 M M d 0 W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k 1 i A A 4 f 6 p u q 1 8 W w D / U 7 S N i X j h M U = " > A A A F Z 3 i c h V T r i t Q w F K 6 6 4 2 W 8 7 K o g g n + i s w M q 6 9 A O X t g f w o o i g o q X 3 V V h M i x p e z o N m 6 Q l O d 3 Z I e Q p f B r / 6 l P 4 C L 6 F a W f W s S t q o O X k n C / n 8 p 2 T x K X g B s P w + 4 m T p 1 Y 6 p 8 + c P d c 9 f + H i p d W 1 y 1 c + m K L S C e w m h S j 0 p 5 g Z E F z B L n I U 8 K n U w G Q s 4 G O 8 / 7 S 2 f z w A b X i h d n B W w l i y i e I Z T x h 6 1 d 7 a P S o Z 5 g k T 9 p X b s 1 T G x a G l y N W M V M p U p X O 3 a Z n z D Y o 5 I L u z t 9 Y L B 2 G z y J 9 C t B B 6 w W K 9 3 b u 8 c o u m R V J J U J g I Z s w o C k s c W 6 a R J w J c t 0 8 r A y V L 9 t k E R q Z k i p t 8 b G M W g z h m Z G U p Q L J k b L k q K w S V u O 5 v d n v Y k N H W M W n M T M a O 9 O s q z X F b r f y b L S s U m r a 3 R M a a T 3 J s a U c 7 0 d j W 4 H l C G h R M k 0 J K p l J L t U w h Y 5 V A Z 8 v 8 o O 0 t l m 0 / B 3 U L 0 4 3 6 b 1 Q l Y 9 C Q e q L E p N A c c z n 0 Z N F t w J f T 9 1 A C Q / u s c D b 1 3 z T n c y J / D 3 2 X Z k x y M V u E t 9 R k R 5 m Q f p + 8 U W J G e E Z 8 V 0 k 9 P a Q u g B R z x V G / C D c E C x I D M U w Z Y k D z r N s q w f i 5 8 W m K d m E G v Q B q 4 g v Z m D I z 8 x 1 w / p x k + z 5 t A Y i g u 3 S Z K j 0 U k G F D L N O 6 m I 6 G 4 9 H Y 9 r u E U D j E r U Z H N v 2 y V B R q 0 g Z n X I g t Z 3 u R / 4 b O t f z + B T 1 3 v d z T G t N s P Y M i Z p o u D 7 i j t A t P j O 6 2 3 P t b J s R s y l N A L l I Y + T G g O z k 3 2 z g T M A 9 i A P 1 9 C h / T v L 5 W 6 3 S b H U D a 2 H v R u p t D l m T Z e 4 O I v n p d + m E 5 D m 3 c N W g N m O R Y + I b W 3 N f C E k n l P m g 1 G M q K P n n q 7 O B B 9 C C k 0 z R 0 X n 5 I c w y b k M 6 + c T Z x 9 r m z O 8 5 u e 5 2 r 2 + P H g y I Y 7 E V 2 v c n s W H 0 1 x 3 S j S T n R v E T T R P w f 6 N / Q 7 j o t m X b N q 7 L Z L D I X H t 1 f C J v R r 1 f l w 3 A Q e f n d / d 5 W u H h f z g Y 3 g l v B 7 S A K H g V b w Y v g b b A b J M H n 4 E v w N f i 2 8 q O z 2 r n W u T 6 H n j y x O H M 1 a K 3 O z Z 9 M M d 0 W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k 1 i A A 4 f 6 p u q 1 8 W w D / U 7 S N i X j h M U = " > A A A F Z 3 i c h V T r i t Q w F K 6 6 4 2 W 8 7 K o g g n + i s w M q 6 9 A O X t g f w o o i g o q X 3 V V h M i x p e z o N m 6 Q l O d 3 Z I e Q p f B r / 6 l P 4 C L 6 F a W f W s S t q o O X k n C / n 8 p 2 T x K X g B s P w + 4 m T p 1 Y 6 p 8 + c P d c 9 f + H i p d W 1 y 1 c + m K L S C e w m h S j 0 p 5 g Z E F z B L n I U 8 K n U w G Q s 4 G O 8 / 7 S 2 f z w A b X i h d n B W w l i y i e I Z T x h 6 1 d 7 a P S o Z 5 g k T 9 p X b s 1 T G x a G l y N W M V M p U p X O 3 a Z n z D Y o 5 I L u z t 9 Y L B 2 G z y J 9 C t B B 6 w W K 9 3 b u 8 c o u m R V J J U J g I Z s w o C k s c W 6 a R J w J c t 0 8 r A y V L 9 t k E R q Z k i p t 8 b G M W g z h m Z G U p Q L J k b L k q K w S V u O 5 v d n v Y k N H W M W n M T M a O 9 O s q z X F b r f y b L S s U m r a 3 R M a a T 3 J s a U c 7 0 d j W 4 H l C G h R M k 0 J K p l J L t U w h Y 5 V A Z 8 v 8 o O 0 t l m 0 / B 3 U L 0 4 3 6 b 1 Q l Y 9 C Q e q L E p N A c c z n 0 Z N F t w J f T 9 1 A C Q / u s c D b 1 3 z T n c y J / D 3 2 X Z k x y M V u E t 9 R k R 5 m Q f p + 8 U W J G e E Z 8 V 0 k 9 P a Q u g B R z x V G / C D c E C x I D M U w Z Y k D z r N s q w f i 5 8 W m K d m E G v Q B q 4 g v Z m D I z 8 x 1 w / p x k + z 5 t A Y i g u 3 S Z K j 0 U k G F D L N O 6 m I 6 G 4 9 H Y 9 r u E U D j E r U Z H N v 2 y V B R q 0 g Z n X I g t Z 3 u R / 4 b O t f z + B T 1 3 v d z T G t N s P Y M i Z p o u D 7 i j t A t P j O 6 2 3 P t b J s R s y l N A L l I Y + T G g O z k 3 2 z g T M A 9 i A P 1 9 C h / T v L 5 W 6 3 S b H U D a 2 H v R u p t D l m T Z e 4 O I v n p d + m E 5 D m 3 c N W g N m O R Y + I b W 3 N f C E k n l P m g 1 G M q K P n n q 7 O B B 9 C C k 0 z R 0 X n 5 I c w y b k M 6 + c T Z x 9 r m z O 8 5 u e 5 2 r 2 + P H g y I Y 7 E V 2 v c n s W H 0 1 x 3 S j S T n R v E T T R P w f 6 N / Q 7 j o t m X b N q 7 L Z L D I X H t 1 f C J v R r 1 f l w 3 A Q e f n d / d 5 W u H h f z g Y 3 g l v B 7 S A K H g V b w Y v g b b A b J M H n 4 E v w N f i 2 8 q O z 2 r n W u T 6 H n j y x O H M 1 a K 3 O z Z 9 M M d 0 W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " k 1 i A A 4 f 6 p u q 1 8 W w D / U 7 S N i X j h M U = " > A A A F Z 3 i c h V T r i t Q w F K 6 6 4 2 W 8 7 K o g g n + i s w M q 6 9 A O X t g f w o o i g o q X 3 V V h M i x p e z o N m 6 Q l O d 3 Z I e Q p f B r / 6 l P 4 C L 6 F a W f W s S t q o O X k n C / n 8 p 2 T x K X g B s P w + 4 m T p 1 Y 6 p 8 + c P d c 9 f + H i p d W 1 y 1 c + m K L S C e w m h S j 0 p 5 g Z E F z B L n I U 8 K n U w G Q s 4 G O 8 / 7 S 2 f z w A b X i h d n B W w l i y i e I Z T x h 6 1 d 7 a P S o Z 5 g k T 9 p X b s 1 T G x a G l y N W M V M p U p X O 3 a Z n z D Y o 5 I L u z t 9 Y L B 2 G z y J 9 C t B B 6 w W K 9 3 b u 8 c o u m R V J J U J g I Z s w o C k s c W 6 a R J w J c t 0 8 r A y V L 9 t k E R q Z k i p t 8 b G M W g z h m Z G U p Q L J k b L k q K w S V u O 5 v d n v Y k N H W M W n M T M a O 9 O s q z X F b r f y b L S s U m r a 3 R M a a T 3 J s a U c 7 0 d j W 4 H l C G h R M k 0 J K p l J L t U w h Y 5 V A Z 8 v 8 o O 0 t l m 0 / B 3 U L 0 4 3 6 b 1 Q l Y 9 C Q e q L E p N A c c z n 0 Z N F t w J f T 9 1 A C Q / u s c D b 1 3 z T n c y J / D 3 2 X Z k x y M V u E t 9 R k R 5 m Q f p + 8 U W J G e E Z 8 V 0 k 9 P a Q u g B R z x V G / C D c E C x I D M U w Z Y k D z r N s q w f i 5 8 W m K d m E G v Q B q 4 g v Z m D I z 8 x 1 w / p x k + z 5 t A Y i g u 3 S Z K j 0 U k G F D L N O 6 m I 6 G 4 9 H Y 9 r u E U D j E r U Z H N v 2 y V B R q 0 g Z n X I g t Z 3 u R / 4 b O t f z + B T 1 3 v d z T G t N s P Y M i Z p o u D 7 i j t A t P j O 6 2 3 P t b J s R s y l N A L l I Y + T G g O z k 3 2 z g T M A 9 i A P 1 9 C h / T v L 5 W 6 3 S b H U D a 2 H v R u p t D l m T Z e 4 O I v n p d + m E 5 D m 3 c N W g N m O R Y + I b W 3 N f C E k n l P m g 1 G M q K P n n q 7 O B B 9 C C k 0 z R 0 X n 5 I c w y b k M 6 + c T Z x 9 r m z O 8 5 u e 5 2 r 2 + P H g y I Y 7 E V 2 v c n s W H 0 1 x 3 S j S T n R v E T T R P w f 6 N / Q 7 j o t m X b N q 7 L Z L D I X H t 1 f C J v R r 1 f l w 3 A Q e f n d / d 5 W u H h f z g Y 3 g l v B 7 S A K H g V b w Y v g b b A b J M H n 4 E v w N f i 2 8 q O z 2 r n W u T 6 H n j y x O H M 1 a K 3 O z Z 9 M M d 0 W < / l a t e x i t > L selfsup ( , ) < l a t e x i t s h a 1 _ b a s e 6 4 = " M g e C P P t K c M D P + K f 2 L G n c x Y U y u 5 k = " > A A A F Z 3 i c h V T r b h M 5 F B 6 g W S D s b g t I C I k / h j Q S i 0 o 0 E 3 F R f y A V g R A S r L i 0 B a Q 4 q j w z Z z J W b Y 9 l n z S N L D 8 F T 8 N f e A o e g b f A M 0 k 3 T F e A p R k d n / P 5 X L 5 z 7 F Q L b j G O v 5 4 5 e 2 6 t 8 8 f 5 C x e 7 l / 7 8 6 + / 1 j c t X 3 t l q a j L Y z y p R m Q 8 p s y C 4 g n 3 k K O C D N s B k K u B 9 e v i k t r 8 / A m N 5 p f Z w r m E s 2 U T x g m c M g + p g 4 y 6 V D M u M C f f S H z g q 0 + r Y U e R q T o L P w k 6 1 9 7 e p L v k W 1 Z b / c 7 D R i w d x s 8 j / h W Q p 9 K L l e n 1 w e e 0 W z a t s K k F h J p i 1 o y T W O H b M I M 8 E + G 6 f T i 1 o l h 2 y C Y y s Z o r b c u x S l o I 4 Z W R a C 5 A s G z u u 9 B R B Z b 7 7 g 9 0 d N 2 S 0 d U x a O 5 e p J / 2 6 S n v a V i t / Z i s q h b b t L Z O p 4 Z M S W 9 r R X j J 2 N X i R k A E F s 6 y S k q n c U S N z K N h U o H e 6 P G p 7 S 2 X b z 1 H d w n y r / l s 1 l S k Y y A N R Y l I Z j q U c B r L o L u C L 2 V v Q w N A 9 r b z L w z c r + Y L I H 0 P f o Q W T X M y X 4 R 2 1 x U k m p N 8 n r 5 S Y E 1 4 Q L I H U 0 0 P q A k i 1 U J z 0 i 3 B L s C I p E M u U D f N g e N F t l W D D 3 I Q 0 R b s w i 0 E A N Q m F b M 2 Y n Y c O + H B O s s O Q t g B E M F 2 6 S p U e C y i w I Z Y Z U 8 1 G w / F o 7 P p d Q i g c 4 0 6 j I 9 t h O S o q N W m D C y 7 E j n e 9 J H x D 7 1 t + f 4 J e u F 7 t a Y 1 p t o F B k T J D V w f 8 S d p V I M Z 0 W + 7 D L R N i P u M 5 I B c 5 j M I Y 0 L 2 S 2 1 2 c C 1 g E s Y D h P s W P a F l f q 0 2 6 y 4 4 g b + y 9 Z N M v I C u y 3 N 1 B Q l / + q 8 O w n I Y 2 7 h q 0 A c x K r E J D a + 5 r Y Y W k 8 h C M G g z l l D 5 + 4 t 3 g f n I / p r M 8 9 k F + Q E u M m 5 D e v f I u 8 + 6 Z d 3 v e 7 Q a d r 9 s T x o M i W O w l b r P J 7 F R 9 N c d 0 q 0 k 5 M 1 y j b S L + D v R r a H e T a m Z 8 8 6 p s N 4 s s h I f 3 l s J 2 8 t + r 8 m 4 4 S I L 8 5 l 5 v J 1 6 + L x e i G 9 G t 6 H a U R A + j n e h 5 9 D r a j 7 L o Y / Q p + h x 9 W f v W W e 9 c 6 1 x f Q M + e W Z 6 5 G r V W 5 + Z 3 P X v d E w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " M g e C P P t K c M D P + K f 2 L G n c x Y U y u 5 k = " > A A A F Z 3 i c h V T r b h M 5 F B 6 g W S D s b g t I C I k / h j Q S i 0 o 0 E 3 F R f y A V g R A S r L i 0 B a Q 4 q j w z Z z J W b Y 9 l n z S N L D 8 F T 8 N f e A o e g b f A M 0 k 3 T F e A p R k d n / P 5 X L 5 z 7 F Q L b j G O v 5 4 5 e 2 6 t 8 8 f 5 C x e 7 l / 7 8 6 + / 1 j c t X 3 t l q a j L Y z y p R m Q 8 p s y C 4 g n 3 k K O C D N s B k K u B 9 e v i k t r 8 / A m N 5 p f Z w r m E s 2 U T x g m c M g + p g 4 y 6 V D M u M C f f S H z g q 0 + r Y U e R q T o L P w k 6 1 9 7 e p L v k W 1 Z b / c 7 D R i w d x s 8 j / h W Q p 9 K L l e n 1 w e e 0 W z a t s K k F h J p i 1 o y T W O H b M I M 8 E + G 6 f T i 1 o l h 2 y C Y y s Z o r b c u x S l o I 4 Z W R a C 5 A s G z u u 9 B R B Z b 7 7 g 9 0 d N 2 S 0 d U x a O 5 e p J / 2 6 S n v a V i t / Z i s q h b b t L Z O p 4 Z M S W 9 r R X j J 2 N X i R k A E F s 6 y S k q n c U S N z K N h U o H e 6 P G p 7 S 2 X b z 1 H d w n y r / l s 1 l S k Y y A N R Y l I Z j q U c B r L o L u C L 2 V v Q w N A 9 r b z L w z c r + Y L I H 0 P f o Q W T X M y X 4 R 2 1 x U k m p N 8 n r 5 S Y E 1 4 Q L I H U 0 0 P q A k i 1 U J z 0 i 3 B L s C I p E M u U D f N g e N F t l W D D 3 I Q 0 R b s w i 0 E A N Q m F b M 2 Y n Y c O + H B O s s O Q t g B E M F 2 6 S p U e C y i w I Z Y Z U 8 1 G w / F o 7 P p d Q i g c 4 0 6 j I 9 t h O S o q N W m D C y 7 E j n e 9 J H x D 7 1 t + f 4 J e u F 7 t a Y 1 p t o F B k T J D V w f 8 S d p V I M Z 0 W + 7 D L R N i P u M 5 I B c 5 j M I Y 0 L 2 S 2 1 2 c C 1 g E s Y D h P s W P a F l f q 0 2 6 y 4 4 g b + y 9 Z N M v I C u y 3 N 1 B Q l / + q 8 O w n I Y 2 7 h q 0 A c x K r E J D a + 5 r Y Y W k 8 h C M G g z l l D 5 + 4 t 3 g f n I / p r M 8 9 k F + Q E u M m 5 D e v f I u 8 + 6 Z d 3 v e 7 Q a d r 9 s T x o M i W O w l b r P J 7 F R 9 N c d 0 q 0 k 5 M 1 y j b S L + D v R r a H e T a m Z 8 8 6 p s N 4 s s h I f 3 l s J 2 8 t + r 8 m 4 4 S I L 8 5 l 5 v J 1 6 + L x e i G 9 G t 6 H a U R A + j n e h 5 9 D r a j 7 L o Y / Q p + h x 9 W f v W W e 9 c 6 1 x f Q M + e W Z 6 5 G r V W 5 + Z 3 P X v d E w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " M g e C P P t K c M D P + K f 2 L G n c x Y U y u 5 k = " > A A A F Z 3 i c h V T r b h M 5 F B 6 g W S D s b g t I C I k / h j Q S i 0 o 0 E 3 F R f y A V g R A S r L i 0 B a Q 4 q j w z Z z J W b Y 9 l n z S N L D 8 F T 8 N f e A o e g b f A M 0 k 3 T F e A p R k d n / P 5 X L 5 z 7 F Q L b j G O v 5 4 5 e 2 6 t 8 8 f 5 C x e 7 l / 7 8 6 + / 1 j c t X 3 t l q a j L Y z y p R m Q 8 p s y C 4 g n 3 k K O C D N s B k K u B 9 e v i k t r 8 / A m N 5 p f Z w r m E s 2 U T x g m c M g + p g 4 y 6 V D M u M C f f S H z g q 0 + r Y U e R q T o L P w k 6 1 9 7 e p L v k W 1 Z b / c 7 D R i w d x s 8 j / h W Q p 9 K L l e n 1 w e e 0 W z a t s K k F h J p i 1 o y T W O H b M I M 8 E + G 6 f T i 1 o l h 2 y C Y y s Z o r b c u x S l o I 4 Z W R a C 5 A s G z u u 9 B R B Z b 7 7 g 9 0 d N 2 S 0 d U x a O 5 e p J / 2 6 S n v a V i t / Z i s q h b b t L Z O p 4 Z M S W 9 r R X j J 2 N X i R k A E F s 6 y S k q n c U S N z K N h U o H e 6 P G p 7 S 2 X b z 1 H d w n y r / l s 1 l S k Y y A N R Y l I Z j q U c B r L o L u C L 2 V v Q w N A 9 r b z L w z c r + Y L I H 0 P f o Q W T X M y X 4 R 2 1 x U k m p N 8 n r 5 S Y E 1 4 Q L I H U 0 0 P q A k i 1 U J z 0 i 3 B L s C I p E M u U D f N g e N F t l W D D 3 I Q 0 R b s w i 0 E A N Q m F b M 2 Y n Y c O + H B O s s O Q t g B E M F 2 6 S p U e C y i w I Z Y Z U 8 1 G w / F o 7 P p d Q i g c 4 0 6 j I 9 t h O S o q N W m D C y 7 E j n e 9 J H x D 7 1 t + f 4 J e u F 7 t a Y 1 p t o F B k T J D V w f 8 S d p V I M Z 0 W + 7 D L R N i P u M 5 I B c 5 j M I Y 0 L 2 S 2 1 2 c C 1 g E s Y D h P s W P a F l f q 0 2 6 y 4 4 g b + y 9 Z N M v I C u y 3 N 1 B Q l / + q 8 O w n I Y 2 7 h q 0 A c x K r E J D a + 5 r Y Y W k 8 h C M G g z l l D 5 + 4 t 3 g f n I / p r M 8 9 k F + Q E u M m 5 D e v f I u 8 + 6 Z d 3 v e 7 Q a d r 9 s T x o M i W O w l b r P J 7 F R 9 N c d 0 q 0 k 5 M 1 y j b S L + D v R r a H e T a m Z 8 8 6 p s N 4 s s h I f 3 l s J 2 8 t + r 8 m 4 4 S I L 8 5 l 5 v J 1 6 + L x e i G 9 G t 6 H a U R A + j n e h 5 9 D r a j 7 L o Y / Q p + h x 9 W f v W W e 9 c 6 1 x f Q M + e W Z 6 5 G r V W 5 + Z 3 P X v d E w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " M g e C P P t K c M D P + K f 2 L G n c x Y U y u 5 k = " > A A A F Z 3 i c h V T r b h M 5 F B 6 g W S D s b g t I C I k / h j Q S i 0 o 0 E 3 F R f y A V g R A S r L i 0 B a Q 4 q j w z Z z J W b Y 9 l n z S N L D 8 F T 8 N f e A o e g b f A M 0 k 3 T F e A p R k d n / P 5 X L 5 z 7 F Q L b j G O v 5 4 5 e 2 6 t 8 8 f 5 C x e 7 l / 7 8 6 + / 1 j c t X 3 t l q a j L Y z y p R m Q 8 p s y C 4 g n 3 k K O C D N s B k K u B 9 e v i k t r 8 / A m N 5 p f Z w r m E s 2 U T x g m c M g + p g 4 y 6 V D M u M C f f S H z g q 0 + r Y U e R q T o L P w k 6 1 9 7 e p L v k W 1 Z b / c 7 D R i w d x s 8 j / h W Q p 9 K L l e n 1 w e e 0 W z a t s K k F h J p i 1 o y T W O H b M I M 8 E + G 6 f T i 1 o l h 2 y C Y y s Z o r b c u x S l o I 4 Z W R a C 5 A s G z u u 9 B R B Z b 7 7 g 9 0 d N 2 S 0 d U x a O 5 e p J / 2 6 S n v a V i t / Z i s q h b b t L Z O p 4 Z M S W 9 r R X j J 2 N X i R k A E F s 6 y S k q n c U S N z K N h U o H e 6 P G p 7 S 2 X b z 1 H d w n y r / l s 1 l S k Y y A N R Y l I Z j q U c B r L o L u C L 2 V v Q w N A 9 r b z L w z c r + Y L I H 0 P f o Q W T X M y X 4 R 2 1 x U k m p N 8 n r 5 S Y E 1 4 Q L I H U 0 0 P q A k i 1 U J z 0 i 3 B L s C I p E M u U D f N g e N F t l W D D 3 I Q 0 R b s w i 0 E A N Q m F b M 2 Y n Y c O + H B O s s O Q t g B E M F 2 6 S p U e C y i w I Z Y Z U 8 1 G w / F o 7 P p d Q i g c 4 0 6 j I 9 t h O S o q N W m D C y 7 E j n e 9 J H x D 7 1 t + f 4 J e u F 7 t a Y 1 p t o F B k T J D V w f 8 S d p V I M Z 0 W + 7 D L R N i P u M 5 I B c 5 j M I Y 0 L 2 S 2 1 2 c C 1 g E s Y D h P s W P a F l f q 0 2 6 y 4 4 g b + y 9 Z N M v I C u y 3 N 1 B Q l / + q 8 O w n I Y 2 7 h q 0 A c x K r E J D a + 5 r Y Y W k 8 h C M G g z l l D 5 + 4 t 3 g f n I / p r M 8 9 k F + Q E u M m 5 D e v f I u 8 + 6 Z d 3 v e 7 Q a d r 9 s T x o M i W O w l b r P J 7 F R 9 N c d 0 q 0 k 5 M 1 y j b S L + D v R r a H e T a m Z 8 8 6 p s N 4 s s h I f 3 l s J 2 8 t + r 8 m 4 4 S I L 8 5 l 5 v J 1 6 + L x e i G 9 G t 6 H a U R A + j n e h 5 9 D r a j 7 L o Y / Q p + h x 9 W f v W W e 9 c 6 1 x f Q M + e W Z 6 5 G r V W 5 + Z 3 P X v d E w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " M g e C P P t K c M D P + K f 2 L G n c x Y U y u 5 k = " > A A A F Z 3 i c h V T r b h M 5 F B 6 g W S D s b g t I C I k / h j Q S i 0 o 0 E 3 F R f y A V g R A S r L i 0 B a Q 4 q j w z Z z J W b Y 9 l n z S N L D 8 F T 8 N f e A o e g b f A M 0 k 3 T F e A p R k d n / P 5 X L 5 z 7 F Q L b j G O v 5 4 5 e 2 6 t 8 8 f 5 C x e 7 l / 7 8 6 + / 1 j c t X 3 t l q a j L Y z y p R m Q 8 p s y C 4 g n 3 k K O C D N s B k K u B 9 e v i k t r 8 / A m N 5 p f Z w r m E s 2 U T x g m c M g + p g 4 y 6 V D M u M C f f S H z g q 0 + r Y U e R q T o L P w k 6 1 9 7 e p L v k W 1 Z b / c 7 D R i w d x s 8 j / h W Q p 9 K L l e n 1 w e e 0 W z a t s K k F h J p i 1 o y T W O H b M I M 8 E + G 6 f T i 1 o l h 2 y C Y y s Z o r b c u x S l o I 4 Z W R a C 5 A s G z u u 9 B R B Z b 7 7 g 9 0 d N 2 S 0 d U x a O 5 e p J / 2 6 S n v a V i t / Z i s q h b b t L Z O p 4 Z M S W 9 r R X j J 2 N X i R k A E F s 6 y S k q n c U S N z K N h U o H e 6 P G p 7 S 2 X b z 1 H d w n y r / l s 1 l S k Y y A N R Y l I Z j q U c B r L o L u C L 2 V v Q w N A 9 r b z L w z c r + Y L I H 0 P f o Q W T X M y X 4 R 2 1 x U k m p N 8 n r 5 S Y E 1 4 Q L I H U 0 0 P q A k i 1 U J z 0 i 3 B L s C I p E M u U D f N g e N F t l W D D 3 I Q 0 R b s w i 0 E A N Q m F b M 2 Y n Y c O + H B O s s O Q t g B E M F 2 6 S p U e C y i w I Z Y Z U 8 1 G w / F o 7 P p d Q i g c 4 0 6 j I 9 t h O S o q N W m D C y 7 E j n e 9 J H x D 7 1 t + f 4 J e u F 7 t a Y 1 p t o F B k T J D V w f 8 S d p V I M Z 0 W + 7 D L R N i P u M 5 I B c 5 j M I Y 0 L 2 S 2 1 2 c C 1 g E s Y D h P s W P a F l f q 0 2 6 y 4 4 g b + y 9 Z N M v I C u y 3 N 1 B Q l / + q 8 O w n I Y 2 7 h q 0 A c x K r E J D a + 5 r Y Y W k 8 h C M G g z l l D 5 + 4 t 3 g f n I / p r M 8 9 k F + Q E u M m 5 D e v f I u 8 + 6 Z d 3 v e 7 Q a d r 9 s T x o M i W O w l b r P J 7 F R 9 N c d 0 q 0 k 5 M 1 y j b S L + D v R r a H e T a m Z 8 8 6 p s N 4 s s h I f 3 l s J 2 8 t + r 8 m 4 4 S I L 8 5 l 5 v J 1 6 + L x e i G 9 G t 6 H a U R A + j n e h 5 9 D r a j 7 L o Y / Q p + h x 9 W f v W W e 9 c 6 1 x f Q M + e W Z 6 5 G r V W 5 + Z 3 P X v d E w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " M g e C P P t K c M D P + K f 2 L G n c x Y U y u 5 k = " > A A A F Z 3 i c h V T r b h M 5 F B 6 g W S D s b g t I C I k / h j Q S i 0 o 0 E 3 F R f y A V g R A S r L i 0 B a Q 4 q j w z Z z J W b Y 9 l n z S N L D 8 F T 8 N f e A o e g b f A M 0 k 3 T F e A p R k d n / P 5 X L 5 z 7 F Q L b j G O v 5 4 5 e 2 6 t 8 8 f 5 C x e 7 l / 7 8 6 + / 1 j c t X 3 t l q a j L Y z y p R m Q 8 p s y C 4 g n 3 k K O C D N s B k K u B 9 e v i k t r 8 / A m N 5 p f Z w r m E s 2 U T x g m c M g + p g 4 y 6 V D M u M C f f S H z g q 0 + r Y U e R q T o L P w k 6 1 9 7 e p L v k W 1 Z b / c 7 D R i w d x s 8 j / h W Q p 9 K L l e n 1 w e e 0 W z a t s K k F h J p i 1 o y T W O H b M I M 8 E + G 6 f T i 1 o l h 2 y C Y y s Z o r b c u x S l o I 4 Z W R a C 5 A s G z u u 9 B R B Z b 7 7 g 9 0 d N 2 S 0 d U x a O 5 e p J / 2 6 S n v a V i t / Z i s q h b b t L Z O p 4 Z M S W 9 r R X j J 2 N X i R k A E F s 6 y S k q n c U S N z K N h U o H e 6 P G p 7 S 2 X b z 1 H d w n y r / l s 1 l S k Y y A N R Y l I Z j q U c B r L o L u C L 2 V v Q w N A 9 r b z L w z c r + Y L I H 0 P f o Q W T X M y X 4 R 2 1 x U k m p N 8 n r 5 S Y E 1 4 Q L I H U 0 0 P q A k i 1 U J z 0 i 3 B L s C I p E M u U D f N g e N F t l W D D 3 I Q 0 R b s w i 0 E A N Q m F b M 2 Y n Y c O + H B O s s O Q t g B E M F 2 6 S p U e C y i w I Z Y Z U 8 1 G w / F o 7 P p d Q i g c 4 0 6 j I 9 t h O S o q N W m D C y 7 E j n e 9 J H x D 7 1 t + f 4 J e u F 7 t a Y 1 p t o F B k T J D V w f 8 S d p V I M Z 0 W + 7 D L R N i P u M 5 I B c 5 j M I Y 0 L 2 S 2 1 2 c C 1 g E s Y D h P s W P a F l f q 0 2 6 y 4 4 g b + y 9 Z N M v I C u y 3 N 1 B Q l / + q 8 O w n I Y 2 7 h q 0 A c x K r E J D a + 5 r Y Y W k 8 h C M G g z l l D 5 + 4 t 3 g f n I / p r M 8 9 k F + Q E u M m 5 D e v f I u 8 + 6 Z d 3 v e 7 Q a d r 9 s T x o M i W O w l b r P J 7 F R 9 N c d 0 q 0 k 5 M 1 y j b S L + D v R r a H e T a m Z 8 8 6 p s N 4 s s h I f 3 l s J 2 8 t + r 8 m 4 4 S I L 8 5 l 5 v J 1 6 + L x e i G 9 G t 6 H a U R A + j n e h 5 9 D r a j 7 L o Y / Q p + h x 9 W f v W W e 9 c 6 1 x f Q M + e W Z 6 5 G r V W 5 + Z 3 P X v d E w = = < / l a t e x i t >
f 2 i i h a F 0 = " > A A A E e n i c h V P b b h M x E H X b A C X c 2 v L I i 9 U k E p c Q 7 V Y t V R + Q i o o Q E q A W k l 6 k O K q 8 3 k n W q r 2 7 s p 2 k k e V f 4 R V + i X / h A e 8 m p U 0 Q M N K u j m a O Z 8 6 M x 1 E u u D Z B 8 G N p e a V y 6 / a d 1 b v V e / c f P H y 0 t r 5 x o r O h Y n D M M p G p s 4 h q E D y F Y 8 O N g L N c A Z W R g N P o 4 q C I n 4 5 A a Z 6 l H T P J o S f p I O V 9 z q j x r v O 1 D S K j 7 N I S z X A M L I t B u f O 1 W t A K S s N / g n A G a m h m R + f r K 0 c k z t h Q Q m q Y o F p 3 w y A 3 P U u V 4 U y A q z b I U E N O 2 Q U d Q F f n N O U 6 6 d m I R i A W g j T P B U j K e p a n + d B A y l z 1 R t x S q f V E R g 4 3 J D W J X o w V z r / F + l l q 9 H w 2 J i P F B 4 m Z 8 3 Y 7 Y c 8 W 5 L J 4 g y h I Y c w y K W k a W 6 J k D H 0 6 F M b Z P B n N p 4 v k f C I 1 F B A 3 R 8 X V x M 3 i r 9 O h j E B B 7 G c j B p n i J p F b f j 6 k D e b D + A v k Q I 1 9 m z k b + 2 + c 8 O n s b g p 4 T v p U c j G Z i f D 3 1 r / S g x s N f J i K C e Z 9 b B L A x V b g o g + c T R 1 X V 4 S 5 x i b D E W B N U 4 0 1 K N 6 v z j W i G R V e p p h v T x s P I B 3 4 R p p j q i f + I p w / d y 3 O K 6 V C T M Y 8 B s N F D F 0 / S N J J u G 6 b i Q D b q G J M N B i / b 8 F r k h R r V y d t O o K 4 j N f C u p t S r u v Y l 6 2 Q f P y U + 2 k v U s t 0 J V u B Y Y n J i h 3 2 s g t w z S T y A l T a 2 p J D 8 u b A 2 d Z O u B O Q c R w 4 j 1 + R x A R l S W c P n W X O v n O 2 4 2 z b + 1 z R m Z 8 s M a B N L b T 1 U t l C f 7 Y W O t I s J T P F c 6 P L i v 8 j / Z t a r Z O c z t 7 g X m l 4 C n a 3 Z 2 A v / P 0 G T 7 Z a o c e f t 2 v 7 w e w 1 r q I n a B M 9 R S H a R f v o P T p C x 4 i h S / Q V f U P f V 3 5 W N i v P K i + m 1 O W l 2 Z n H a M 4 q 2 7 8 A J a O B b Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / g N J F r X 0 W S f W Q S G F M 8 1 f 2 i i h a F 0 = " > A A A E e n i c h V P b b h M x E H X b A C X c 2 v L I i 9 U k E p c Q 7 V Y t V R + Q i o o Q E q A W k l 6 k O K q 8 3 k n W q r 2 7 s p 2 k k e V f 4 R V + i X / h A e 8 m p U 0 Q M N K u j m a O Z 8 6 M x 1 E u u D Z B 8 G N p e a V y 6 / a d 1 b v V e / c f P H y 0 t r 5 x o r O h Y n D M M p G p s 4 h q E D y F Y 8 O N g L N c A Z W R g N P o 4 q C I n 4 5 A a Z 6 l H T P J o S f p I O V 9 z q j x r v O 1 D S K j 7 N I S z X A M L I t B u f O 1 W t A K S s N / g n A G a m h m R + f r K 0 c k z t h Q Q m q Y o F p 3 w y A 3 P U u V 4 U y A q z b I U E N O 2 Q U d Q F f n N O U 6 6 d m I R i A W g j T P B U j K e p a n + d B A y l z 1 R t x S q f V E R g 4 3 J D W J X o w V z r / F + l l q 9 H w 2 J i P F B 4 m Z 8 3 Y 7 Y c 8 W 5 L J 4 g y h I Y c w y K W k a W 6 J k D H 0 6 F M b Z P B n N p 4 v k f C I 1 F B A 3 R 8 X V x M 3 i r 9 O h j E B B 7 G c j B p n i J p F b f j 6 k D e b D + A v k Q I 1 9 m z k b + 2 + c 8 O n s b g p 4 T v p U c j G Z i f D 3 1 r / S g x s N f J i K C e Z 9 b B L A x V b g o g + c T R 1 X V 4 S 5 x i b D E W B N U 4 0 1 K N 6 v z j W i G R V e p p h v T x s P I B 3 4 R p p j q i f + I p w / d y 3 O K 6 V C T M Y 8 B s N F D F 0 / S N J J u G 6 b i Q D b q G J M N B i / b 8 F r k h R r V y d t O o K 4 j N f C u p t S r u v Y l 6 2 Q f P y U + 2 k v U s t 0 J V u B Y Y n J i h 3 2 s g t w z S T y A l T a 2 p J D 8 u b A 2 d Z O u B O Q c R w 4 j 1 + R x A R l S W c P n W X O v n O 2 4 2 z b + 1 z R m Z 8 s M a B N L b T 1 U t l C f 7 Y W O t I s J T P F c 6 P L i v 8 j / Z t a r Z O c z t 7 g X m l 4 C n a 3 Z 2 A v / P 0 G T 7 Z a o c e f t 2 v 7 w e w 1 r q I n a B M 9 R S H a R f v o P T p C x 4 i h S / Q V f U P f V 3 5 W N i v P K i + m 1 O W l 2 Z n H a M 4 q 2 7 8 A J a O B b Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / g N J F r X 0 W S f W Q S G F M 8 1 f 2 i i h a F 0 = " > A A A E e n i c h V P b b h M x E H X b A C X c 2 v L I i 9 U k E p c Q 7 V Y t V R + Q i o o Q E q A W k l 6 k O K q 8 3 k n W q r 2 7 s p 2 k k e V f 4 R V + i X / h A e 8 m p U 0 Q M N K u j m a O Z 8 6 M x 1 E u u D Z B 8 G N p e a V y 6 / a d 1 b v V e / c f P H y 0 t r 5 x o r O h Y n D M M p G p s 4 h q E D y F Y 8 O N g L N c A Z W R g N P o 4 q C I n 4 5 A a Z 6 l H T P J o S f p I O V 9 z q j x r v O 1 D S K j 7 N I S z X A M L I t B u f O 1 W t A K S s N / g n A G a m h m R + f r K 0 c k z t h Q Q m q Y o F p 3 w y A 3 P U u V 4 U y A q z b I U E N O 2 Q U d Q F f n N O U 6 6 d m I R i A W g j T P B U j K e p a n + d B A y l z 1 R t x S q f V E R g 4 3 J D W J X o w V z r / F + l l q 9 H w 2 J i P F B 4 m Z 8 3 Y 7 Y c 8 W 5 L J 4 g y h I Y c w y K W k a W 6 J k D H 0 6 F M b Z P B n N p 4 v k f C I 1 F B A 3 R 8 X V x M 3 i r 9 O h j E B B 7 G c j B p n i J p F b f j 6 k D e b D + A v k Q I 1 9 m z k b + 2 + c 8 O n s b g p 4 T v p U c j G Z i f D 3 1 r / S g x s N f J i K C e Z 9 b B L A x V b g o g + c T R 1 X V 4 S 5 x i b D E W B N U 4 0 1 K N 6 v z j W i G R V e p p h v T x s P I B 3 4 R p p j q i f + I p w / d y 3 O K 6 V C T M Y 8 B s N F D F 0 / S N J J u G 6 b i Q D b q G J M N B i / b 8 F r k h R r V y d t O o K 4 j N f C u p t S r u v Y l 6 2 Q f P y U + 2 k v U s t 0 J V u B Y Y n J i h 3 2 s g t w z S T y A l T a 2 p J D 8 u b A 2 d Z O u B O Q c R w 4 j 1 + R x A R l S W c P n W X O v n O 2 4 2 z b + 1 z R m Z 8 s M a B N L b T 1 U t l C f 7 Y W O t I s J T P F c 6 P L i v 8 j / Z t a r Z O c z t 7 g X m l 4 C n a 3 Z 2 A v / P 0 G T 7 Z a o c e f t 2 v 7 w e w 1 r q I n a B M 9 R S H a R f v o P T p C x 4 i h S / Q V f U P f V 3 5 W N i v P K i + m 1 O W l 2 Z n H a M 4 q 2 7 8 A J a O B b Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / g N J F r X 0 W S f W Q S G F M 8 1 f 2 i i h a F 0 = " > A A A E e n i c h V P b b h M x E H X b A C X c 2 v L I i 9 U k E p c Q 7 V Y t V R + Q i o o Q E q A W k l 6 k O K q 8 3 k n W q r 2 7 s p 2 k k e V f 4 R V + i X / h A e 8 m p U 0 Q M N K u j m a O Z 8 6 M x 1 E u u D Z B 8 G N p e a V y 6 / a d 1 b v V e / c f P H y 0 t r 5 x o r O h Y n D M M p G p s 4 h q E D y F Y 8 O N g L N c A Z W R g N P o 4 q C I n 4 5 A a Z 6 l H T P J o S f p I O V 9 z q j x r v O 1 D S K j 7 N I S z X A M L I t B u f O 1 W t A K S s N / g n A G a m h m R + f r K 0 c k z t h Q Q m q Y o F p 3 w y A 3 P U u V 4 U y A q z b I U E N O 2 Q U d Q F f n N O U 6 6 d m I R i A W g j T P B U j K e p a n + d B A y l z 1 R t x S q f V E R g 4 3 J D W J X o w V z r / F + l l q 9 H w 2 J i P F B 4 m Z 8 3 Y 7 Y c 8 W 5 L J 4 g y h I Y c w y K W k a W 6 J k D H 0 6 F M b Z P B n N p 4 v k f C I 1 F B A 3 R 8 X V x M 3 i r 9 O h j E B B 7 G c j B p n i J p F b f j 6 k D e b D + A v k Q I 1 9 m z k b + 2 + c 8 O n s b g p 4 T v p U c j G Z i f D 3 1 r / S g x s N f J i K C e Z 9 b B L A x V b g o g + c T R 1 X V 4 S 5 x i b D E W B N U 4 0 1 K N 6 v z j W i G R V e p p h v T x s P I B 3 4 R p p j q i f + I p w / d y 3 O K 6 V C T M Y 8 B s N F D F 0 / S N J J u G 6 b i Q D b q G J M N B i / b 8 F r k h R r V y d t O o K 4 j N f C u p t S r u v Y l 6 2 Q f P y U + 2 k v U s t 0 J V u B Y Y n J i h 3 2 s g t w z S T y A l T a 2 p J D 8 u b A 2 d Z O u B O Q c R w 4 j 1 + R x A R l S W c P n W X O v n O 2 4 2 z b + 1 z R m Z 8 s M a B N L b T 1 U t l C f 7 Y W O t I s J T P F c 6 P L i v 8 j / Z t a r Z O c z t 7 g X m l 4 C n a 3 Z 2 A v / P 0 G T 7 Z a o c e f t 2 v 7 w e w 1 r q I n a B M 9 R S H a R f v o P T p C x 4 i h S / Q V f U P f V 3 5 W N i v P K i + m 1 O W l 2 Z n H a M 4 q 2 7 8 A J a O B b Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / g N J F r X 0 W S f W Q S G F M 8 1 f 2 i i h a F 0 = " > A A A E e n i c h V P b b h M x E H X b A C X c 2 v L I i 9 U k E p c Q 7 V Y t V R + Q i o o Q E q A W k l 6 k O K q 8 3 k n W q r 2 7 s p 2 k k e V f 4 R V + i X / h A e 8 m p U 0 Q M N K u j m a O Z 8 6 M x 1 E u u D Z B 8 G N p e a V y 6 / a d 1 b v V e / c f P H y 0 t r 5 x o r O h Y n D M M p G p s 4 h q E D y F Y 8 O N g L N c A Z W R g N P o 4 q C I n 4 5 A a Z 6 l H T P J o S f p I O V 9 z q j x r v O 1 D S K j 7 N I S z X A M L I t B u f O 1 W t A K S s N / g n A G a m h m R + f r K 0 c k z t h Q Q m q Y o F p 3 w y A 3 P U u V 4 U y A q z b I U E N O 2 Q U d Q F f n N O U 6 6 d m I R i A W g j T P B U j K e p a n + d B A y l z 1 R t x S q f V E R g 4 3 J D W J X o w V z r / F + l l q 9 H w 2 J i P F B 4 m Z 8 3 Y 7 Y c 8 W 5 L J 4 g y h I Y c w y K W k a W 6 J k D H 0 6 F M b Z P B n N p 4 v k f C I 1 F B A 3 R 8 X V x M 3 i r 9 O h j E B B 7 G c j B p n i J p F b f j 6 k D e b D + A v k Q I 1 9 m z k b + 2 + c 8 O n s b g p 4 T v p U c j G Z i f D 3 1 r / S g x s N f J i K C e Z 9 b B L A x V b g o g + c T R 1 X V 4 S 5 x i b D E W B N U 4 0 1 K N 6 v z j W i G R V e p p h v T x s P I B 3 4 R p p j q i f + I p w / d y 3 O K 6 V C T M Y 8 B s N F D F 0 / S N J J u G 6 b i Q D b q G J M N B i / b 8 F r k h R r V y d t O o K 4 j N f C u p t S r u v Y l 6 2 Q f P y U + 2 k v U s t 0 J V u B Y Y n J i h 3 2 s g t w z S T y A l T a 2 p J D 8 u b A 2 d Z O u B O Q c R w 4 j 1 + R x A R l S W c P n W X O v n O 2 4 2 z b + 1 z R m Z 8 s M a B N L b T 1 U t l C f 7 Y W O t I s J T P F c 6 P L i v 8 j / Z t a r Z O c z t 7 g X m l 4 C n a 3 Z 2 A v / P 0 G T 7 Z a o c e f t 2 v 7 w e w 1 r q I n a B M 9 R S H a R f v o P T p C x 4 i h S / Q V f U P f V 3 5 W N i v P K i + m 1 O W l 2 Z n H a M 4 q 2 7 8 A J a O B b Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / g N J F r X 0 W S f W Q S G F M 8 1 f 2 i i h a F 0 = " > A A A E e n i c h V P b b h M x E H X b A C X c 2 v L I i 9 U k E p c Q 7 V Y t V R + Q i o o Q E q A W k l 6 k O K q 8 3 k n W q r 2 7 s p 2 k k e V f 4 R V + i X / h A e 8 m p U 0 Q M N K u j m a O Z 8 6 M x 1 E u u D Z B 8 G N p e a V y 6 / a d 1 b v V e / c f P H y 0 t r 5 x o r O h Y n D M M p G p s 4 h q E D y F Y 8 O N g L N c A Z W R g N P o 4 q C I n 4 5 A a Z 6 l H T P J o S f p I O V 9 z q j x r v O 1 D S K j 7 N I S z X A M L I t B u f O 1 W t A K S s N / g n A G a m h m R + f r K 0 c k z t h Q Q m q Y o F p 3 w y A 3 P U u V 4 U y A q z b I U E N O 2 Q U d Q F f n N O U 6 6 d m I R i A W g j T P B U j K e p a n + d B A y l z 1 R t x S q f V E R g 4 3 J D W J X o w V z r / F + l l q 9 H w 2 J i P F B 4 m Z 8 3 Y 7 Y c 8 W 5 L J 4 g y h I Y c w y K W k a W 6 J k D H 0 6 F M b Z P B n N p 4 v k f C I 1 F B A 3 R 8 X V x M 3 i r 9 O h j E B B 7 G c j B p n i J p F b f j 6 k D e b D + A v k Q I 1 9 m z k b + 2 + c 8 O n s b g p 4 T v p U c j G Z i f D 3 1 r / S g x s N f J i K C e Z 9 b B L A x V b g o g + c T R 1 X V 4 S 5 x i b D E W B N U 4 0 1 K N 6 v z j W i G R V e p p h v T x s P I B 3 4 R p p j q i f + I p w / d y 3 O K 6 V C T M Y 8 B s N F D F 0 / S N J J u G 6 b i Q D b q G J M N B i / b 8 F r k h R r V y d t O o K 4 j N f C u p t S r u v Y l 6 2 Q f P y U + 2 k v U s t 0 J V u B Y Y n J i h 3 2 s g t w z S T y A l T a 2 p J D 8 u b A 2 d Z O u B O Q c R w 4 j 1 + R x A R l S W c P n W X O v n O 2 4 2 z b + 1 z R m Z 8 s M a B N L b T 1 U t l C f 7 Y W O t I s J T P F c 6 P L i v 8 j / Z t a r Z O c z t 7 g X m l 4 C n a 3 Z 2 A v / P 0 G T 7 Z a o c e f t 2 v 7 w e w 1 r q I n a B M 9 R S H a R f v o P T p C x 4 i h S / Q V f U P f V 3 5 W N i v P K i + m 1 O W l 2 Z n H a M 4 q 2 7 8 A J a O B b Q = = < / l a t e x i t >
L sup ( , ) < l a t e x i t s h a 1 _ b a s e 6 4 = " u E E 9 J W T v q Y U f U B e R y u G Z f 1 7 x U U k = " > A A A F Y 3 i c h V T r b h M 7 E F 6 g 4 R J u 5 f I P I V m k k Q C V a D f i o v 5 A K g K h I x 0 Q c N o C U h x V 3 t 3 Z r F X b a 9 m T p p H l Z + B p + A v P w Q P w H n g 3 K W G L 4 F j a 1 X j m 8 1 y + G T v V g l u M 4 2 + n T p 9 Z 6 5 w 9 d / 5 C 9 + K l y 1 e u r l + 7 / t 5 W U 5 P B X l a J y n x M m Q X B F e w h R w E f t Q E m U w E f 0 o P n t f 3 D I R j L K 7 W L c w 1 j y S a K F z x j G F T 7 6 / e o Z F h m T L h X f t 9 R m V Z H j i J X c 2 K n 2 v u 7 V J d 8 k 2 r L 7 + 2 v 9 + J B 3 C z y u 5 A s h V 6 0 X G / 3 r 6 3 d o X m V T S U o z A S z d p T E G s e O G e S Z A N / t 0 6 k F z b I D N o G R 1 U x x W 4 5 d y l I Q J 4 x M a w G S Z W P H l Z 4 i q M x 3 f 7 G 7 o 4 a I t o 5 J a + c y 9 a R f V 2 h P 2 m r l n 2 P x / 0 B / h 3 Y 3 q G b G N 6 / K V r P I Q n j y c C l s J T 9 f l f f D Q R L k d w 9 7 2 / H y f T k f 3 Y r u R H e j J H o S b U f / R G + j v S i L P k W f o y / R 1 7 X v n Y u d 6 5 2 b C + j p U 8 s z N 6 L W 6 t z + A U o z 2 0 E = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " u E E 9 J W T v q Y U f U B e R y u G Z f 1 7 x U U k = " > A A A F Y 3 i c h V T r b h M 7 E F 6 g 4 R J u 5 f I P I V m k k Q C V a D f i o v 5 A K g K h I x 0 Q c N o C U h x V 3 t 3 Z r F X b a 9 m T p p H l Z + B p + A v P w Q P w H n g 3 K W G L 4 F j a 1 X j m 8 1 y + G T v V g l u M 4 2 + n T p 9 Z 6 5 w 9 d / 5 C 9 + K l y 1 e u r l + 7 / t 5 W U 5 P B X l a J y n x M m Q X B F e w h R w E f t Q E m U w E f 0 o P n t f 3 D I R j L K 7 W L c w 1 j y S a K F z x j G F T 7 6 / e o Z F h m T L h X f t 9 R m V Z H j i J X c 2 K n 2 v u 7 V J d 8 k 2 r L 7 + 2 v 9 + J B 3 C z y u 5 A s h V 6 0 X G / 3 r 6 3 d o X m V T S U o z A S z d p T E G s e O G e S Z A N / t 0 6 k F z b I D N o G R 1 U x x W 4 5 d y l I Q J 4 x M a w G S Z W P H l Z 4 i q M x 3 f 7 G 7 o 4 a I t o 5 J a + c y 9 a R f V 2 h P 2 m r l n 2
x F p d C 2 v W U y N X x S Y k s 7 2 k 3 G r g Y v E j K g Y J Z V U j K V O 2 p k D g W b C v R Ox F p d C 2 v W U y N X x S Y k s 7 2 k 3 G r g Y v E j K g Y J Z V U j K V O 2 p k D g W b C v R OE E 9 J W T v q Y U f U B e R y u G Z f 1 7 x U U k = " > A A A F Y 3 i c h V T r b h M 7 E F 6 g 4 R J u 5 f I P I V m k k Q C V a D f i o v 5 A K g K h I x 0 Q c N o C U h x V 3 t 3 Z r F X b
a 9 m T p p H l Z + B p + A v P w Q P w H n g 3 K W G L 4 F j a 1 X j m 8 1 y + G T v V g l u M 4 2 + n T p 9 Z 6 5 w 9 d / 5 C 9 + K l y 1 e u r l + 7 / t 5 W U 5 P B X l a J y n x M m Q X B F e w h R w E f t Q E m U w E f 0 o P n t f 3 D I R j L K 7 W L c w 1 j y S a K F z x j G F T 7 6 / e o Z F h m T L h X f t 9 R m V Z H j i J X c 2 K n 2 v u 7 V J d 8 k 2 r L 7 + 2 v 9 + J B 3 C z y u 5 A s h V 6 0 X G / 3
x F p d C 2 v W U y N X x S Y k s 7 2 k 3 G r g Y v E j K g Y J Z V U j K V O 2 p k D g W b C v R OE E 9 J W T v q Y U f U B e R y u G Z f 1 7 x U U k = " > A A A F Y 3 i c h V T r b h M 7 E F 6 g 4 R J u 5 f I P I V m k k Q C V a D f i o v 5 A K g K h I x 0 Q c N o C U h x V 3 t 3 Z r F X b
a 9 m T p p H l Z + B p + A v P w Q P w H n g 3 K W G L 4 F j a 1 X j m 8 1 y + G T v V g l u M 4 2 + n T p 9 Z 6 5 w 9 d / 5 C 9 + K l y 1 e u r l + 7 / t 5 W U 5 P B X l a J y n x M m Q X B F e w h R w E f t Q E m U w E f 0 o P n t f 3 D I R j L K 7 W L c w 1 j y S a K F z x j G F T 7 6 / e o Z F h m T L h X f t 9 R m V Z H j i J X c 2 K n 2 v u 7 V J d 8 k 2 r L 7 + 2 v 9 + J B 3 C z y u 5 A s h V 6 0 X G / 3
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a 9 m T p p H l Z + B p + A v P w Q P w H n g 3 K W G L 4 F j a 1 X j m 8 1 y + G T v V g l u M 4 2 + n T p 9 Z 6 5 w 9 d / 5 C 9 + K l y 1 e u r l + 7 / t 5 W U 5 P B X l a J y n x M m Q X B F e w h R w E f t Q E m U w E f 0 o P n t f 3 D I R j L K 7 W L c w 1 j y S a K F z x j G F T 7 6 / e o Z F h m T L h X f t 9 R m V Z H j i J X c 2 K n 2 v u 7 V J d 8 k 2 r L 7 + 2 v 9 + J B 3 C z y u 5 A s h V 6 0 X G / 3
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a 9 m T p p H l Z + B p + A v P w Q P w H n g 3 K W G L 4 F j a 1 X j m 8 1 y + G T v V g l u M 4 2 + n T p 9 Z 6 5 w 9 d / 5 C 9 + K l y 1 e u r l + 7 / t 5 W U 5 P B X l a J y n x M m Q X B F e w h R w E f t Q E m U w E f 0 o P n t f 3 D I R j L K 7 W L c w 1 j y S a K F z x j G F T 7 6 / e o Z F h m T L h X f t 9 R m V Z H j i J X c 2 K n 2 v u 7 V J d 8 k 2 r L 7 + 2 v 9 + J B 3 C z y u 5 A s h V 6 0 X G / 3
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o S Z U 2 n O C F K T 8 z 5 a t N r e 6 W g P x V / p j S d m R y c r z X W c Z S F I w 6 p C h m R s u d 7 u e p r I h Q N G R i 3 h U c S c h I O y Q B 6 M i c p l U l f B y Q A t u A k e c 6 A k 7 C v a Z q P F K S h c a / 4 9 a R s s m 4 j X M o p D w x q c a I S u e g r j H / z x V m q Z D 1 a y A N B B 4 m q W X t H f l 8 X 4 K o g A S m M w 4 x z k k Y a C x 5 B T E Z M G Z 0 n F / V o A a / H u S h G E 2 0 W T 5 m O e A A C I k s U G 2 S C q o R 3 L F n 4 E N S b 8 Q f I w U 7 j Z W Z 0 Z N c 4 o R W R V 1 M / w T H h l E 1 n 6 T W W 8 W U l q N V C + y m b I h o j l Q A q T g U q G k B Z Z b i c F 6 I S q Q w F g C R J J Z I g a O z W W p A h Y b Z M V m 9 M K q t A O r C N b I 6 J n N o J G L u P k 6 E t m 4 F S I F w 8 L x V P G M S q J J Y I k Y 1 7 n X 6 v r 1 s u Q h g m q l v a 0 K 4 V j V m W D u r g m D L W N b r p 2 9 U x p h b 3 L + g q 9 P w b F 5 j y 0 z L I A i L w f I O 5 L D u z x A i 3 F t 7 + P Y x N x z Q C R V k E P X s M 8 F F C 5 a G a M q i S S F B B N v G e 4 8 S + 9 A Y + J B c Q l f 6 m v 2 E q y J w s / b T t 4 7 f v c n t Y F q F l u B I t Q I W J y u x A C + 4 L Z Y 7 E f A g i b X f 4 C L / Y M 7 q 9 7 W 9 7 e B x 5 x u r P c K K 8 M q X R + 0 a H R r 8 y + s j o Q 2 s z x X j s 8 c A K p G r 6 e q O s b K G / g m O 8 W Z Y c C p o r W W b 8 H + j f U H c D 5 0 R U t 8 p u K a h S d r Z m y q 7 / + 1 Y 5 6 b R 9 q 7 / f a n a 9 2 f 2 y 4 j x y 1 p 3 H j u / s O F 3 n t X P g H D u h M 3 Q + O p + c z 4 0 v j W + N 7 4 0 f F X R 5 a b b n o V O T x s 9 f h m r Q j A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H A r M o t u 6 G u s B 9 J e i 6 p G K 1 R 0 a 9 T 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g R H + N r j k x l p X C a F R J d X G t A u k o i G E B G i D / U p 1 N T n J S W P V T i L b X V t Z f g h u 4 W l 4 D J 6 A O 8 Q t d z h J R 5 c i 4 E h O T s 7 5 f H 6 + 4 z j I G Z X K 8 7 4 u L V 9 r X L 9 x c + W W e / v O 3 X v 3 V 9 c e n M h s J
E I 4 D j O W i b O A S G A 0 h W N F F Y O z X A D h A Y P T Y L h X + E 8 v Q E i a p U d q m k O f k 0 F K Y x
o S Z U 2 n O C F K T 8 z 5 a t N r e 6 W g P x V / p j S d m R y c r z X W c Z S F I w 6 p C h m R s u d 7 u e p r I h Q N G R i 3 h U c S c h I O y Q B 6 M i c p l U l f B y Q A t u A k e c 6 A k 7 C v a Z q P F K S h c a / 4 9 a R s s m 4 j X M o p D w x q c a I S u e g r j H / z x V m q Z D 1 a y A N B B 4 m q W X t H f l 8 X 4 K o g A S m M w 4 x z k k Y a C x 5 B T E Z M G Z 0 n F / V o A a / H u S h G E 2 0 W T 5 m O e A A C I k s U G 2 S C q o R 3 L F n 4 E N S b 8 Q f I w U 7 j Z W Z 0 Z N c 4 o R W R V 1 M / w T H h l E 1 n 6 T W W 8 W U l q N V C + y m b I h o j l Q A q T g U q G k B Z Z b i c F 6 I S q Q w F g C R J J Z I g a O z W W p A h Y b Z M V m 9 M K q t A O r C N b I 6 J n N o J G L u P k 6 E t m 4 F S I F w 8 L x V P G M S q J J Y I k Y 1 7 n X 6 v r 1 s u Q h g m q l v a 0 K 4 V j V m W D u r g m D L W N b r p 2 9 U x p h b 3 L + g q 9 P w b F 5 j y 0 z L I A i L w f I O 5 L D u z x A i 3 F t 7 + P Y x N x z Q C R V k E P X s M 8 F F C 5 a G a M q i S S F B B N v G e 4 8 S + 9 A Y + J B c Q l f 6 m v 2 E q y J w s / b T t 4 7 f v c n t Y F q F l u B I t Q I W J y u x A C + 4 L Z Y 7 E f A g i b X f 4 C L / Y M 7 q 9 7 W 9 7 e B x 5 x u r P c K K 8 M q X R + 0 a H R r 8 y + s j o Q 2 s z x X j s 8 c A K p G r 6 e q O s b K G / g m O 8 W Z Y c C p o r W W b 8 H + j f U H c D 5 0 R U t 8 p u K a h S d r Z m y q 7 / + 1 Y 5 6 b R 9 q 7 / f a n a 9 2 f 2 y 4 j x y 1 p 3 H j u / s O F 3 n t X P g H D u h M 3 Q + O p + c z 4 0 v j W + N 7 4 0 f F X R 5 a b b n o V O T x s 9 f h m r Q j A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H A r M o t u 6 G u s B 9 J e i 6 p G K 1 R 0 a 9 T 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g R H + N r j k x l p X C a F R J d X G t A u k o i G E B G i D / U p 1 N T n J S W P V T i L b X V t Z f g h u 4 W l 4 D J 6 A O 8 Q t d z h J R 5 c i 4 E h O T s 7 5 f H 6 + 4 z j I G Z X K 8 7 4 u L V 9 r X L 9 x c + W W e / v O 3 X v 3 V 9 c e n M h s J
E I 4 D j O W i b O A S G A 0 h W N F F Y O z X A D h A Y P T Y L h X + E 8 v Q E i a p U d q m k O f k 0 F K Y x
o S Z U 2 n O C F K T 8 z 5 a t N r e 6 W g P x V / p j S d m R y c r z X W c Z S F I w 6 p C h m R s u d 7 u e p r I h Q N G R i 3 h U c S c h I O y Q B 6 M i c p l U l f B y Q A t u A k e c 6 A k 7 C v a Z q P F K S h c a / 4 9 a R s s m 4 j X M o p D w x q c a I S u e g r j H / z x V m q Z D 1 a y A N B B 4 m q W X t H f l 8 X 4 K o g A S m M w 4 x z k k Y a C x 5 B T E Z M G Z 0 n F / V o A a / H u S h G E 2 0 W T 5 m O e A A C I k s U G 2 S C q o R 3 L F n 4 E N S b 8 Q f I w U 7 j Z W Z 0 Z N c 4 o R W R V 1 M / w T H h l E 1 n 6 T W W 8 W U l q N V C + y m b I h o j l Q A q T g U q G k B Z Z b i c F 6 I S q Q w F g C R J J Z I g a O z W W p A h Y b Z M V m 9 M K q t A O r C N b I 6 J n N o J G L u P k 6 E t m 4 F S I F w 8 L x V P G M S q J J Y I k Y 1 7 n X 6 v r 1 s u Q h g m q l v a 0 K 4 V j V m W D u r g m D L W N b r p 2 9 U x p h b 3 L + g q 9 P w b F 5 j y 0 z L I A i L w f I O 5 L D u z x A i 3 F t 7 + P Y x N x z Q C R V k E P X s M 8 F F C 5 a G a M q i S S F B B N v G e 4 8 S + 9 A Y + J B c Q l f 6 m v 2 E q y J w s / b T t 4 7 f v c n t Y F q F l u B I t Q I W J y u x A C + 4 L Z Y 7 E f A g i b X f 4 C L / Y M 7 q 9 7 W 9 7 e B x 5 x u r P c K K 8 M q X R + 0 a H R r 8 y + s j o Q 2 s z x X j s 8 c A K p G r 6 e q O s b K G / g m O 8 W Z Y c C p o r W W b 8 H + j f U H c D 5 0 R U t 8 p u K a h S d r Z m y q 7 / + 1 Y 5 6 b R 9 q 7 / f a n a 9 2 f 2 y 4 j x y 1 p 3 H j u / s O F 3 n t X P g H D u h M 3 Q + O p + c z 4 0 v j W + N 7 4 0 f F X R 5 a b b n o V O T x s 9 f h m r Q j A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H A r M o t u 6 G u s B 9 J e i 6 p G K 1 R 0 a 9 T 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g R H + N r j k x l p X C a F R J d X G t A u k o i G E B G i D / U p 1 N T n J S W P V T i L b X V t Z f g h u 4 W l 4 D J 6 A O 8 Q t d z h J R 5 c i 4 E h O T s 7 5 f H 6 + 4 z j I G Z X K 8 7 4 u L V 9 r X L 9 x c + W W e / v O 3 X v 3 V 9 c e n M h s J
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o S Z U 2 n O C F K T 8 z 5 a t N r e 6 W g P x V / p j S d m R y c r z X W c Z S F I w 6 p C h m R s u d 7 u e p r I h Q N G R i 3 h U c S c h I O y Q B 6 M i c p l U l f B y Q A t u A k e c 6 A k 7 C v a Z q P F K S h c a / 4 9 a R s s m 4 j X M o p D w x q c a I S u e g r j H / z x V m q Z D 1 a y A N B B 4 m q W X t H f l 8 X 4 K o g A S m M w 4 x z k k Y a C x 5 B T E Z M G Z 0 n F / V o A a / H u S h G E 2 0 W T 5 m O e A A C I k s U G 2 S C q o R 3 L F n 4 E N S b 8 Q f I w U 7 j Z W Z 0 Z N c 4 o R W R V 1 M / w T H h l E 1 n 6 T W W 8 W U l q N V C + y m b I h o j l Q A q T g U q G k B Z Z b i c F 6 I S q Q w F g C R J J Z I g a O z W W p A h Y b Z M V m 9 M K q t A O r C N b I 6 J n N o J G L u P k 6 E t m 4 F S I F w 8 L x V P G M S q J J Y I k Y 1 7 n X 6 v r 1 s u Q h g m q l v a 0 K 4 V j V m W D u r g m D L W N b r p 2 9 U x p h b 3 L + g q 9 P w b F 5 j y 0 z L I A i L w f I O 5 L D u z x A i 3 F t 7 + P Y x N x z Q C R V k E P X s M 8 F F C 5 a G a M q i S S F B B N v G e 4 8 S + 9 A Y + J B c Q l f 6 m v 2 E q y J w s / b T t 4 7 f v c n t Y F q F l u B I t Q I W J y u x A C + 4 L Z Y 7 E f A g i b X f 4 C L / Y M 7 q 9 7 W 9 7 e B x 5 x u r P c K K 8 M q X R + 0 a H R r 8 y + s j o Q 2 s z x X j s 8 c A K p G r 6 e q O s b K G / g m O 8 W Z Y c C p o r W W b 8 H + j f U H c D 5 0 R U t 8 p u K a h S d r Z m y q 7 / + 1 Y 5 6 b R 9 q 7 / f a n a 9 2 f 2 y 4 j x y 1 p 3 H j u / s O F 3 n t X P g H D u h M 3 Q + O p + c z 4 0 v j W + N 7 4 0 f F X R 5 a b b n o V O T x s 9 f h m r Q j A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H A r M o t u 6 G u s B 9 J e i 6 p G K 1 R 0 a 9 T 0 = " > A A A F P 3 i c h V T d b t M w F M 5 G g R H + N r j k x l p X C a F R J d X G t A u k o i G E B G i D / U p 1 N T n J S W P V T i L b X V t Z f g h u 4 W l 4 D J 6 A O 8 Q t d z h J R 5 c i 4 E h O T s 7 5 f H 6 + 4 z j I G Z X K 8 7 4 u L V 9 r X L 9 x c + W W e / v O 3 X v 3 V 9 c e n M h s J
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o S Z U 2 n O C F K T 8 z 5 a t N r e 6 W g P x V / p j S d m R y c r z X W c Z S F I w 6 p C h m R s u d 7 u e p r I h Q N G R i 3 h U c S c h I O y Q B 6 M i c p l U l f B y Q A t u A k e c 6 A k 7 C v a Z q P F K S h c a / 4 9 a R s s m 4 j X M o p D w x q c a I S u e g r j H / z x V m q Z D 1 a y A N B B 4 m q W X t H f l 8 X 4 K o g A S m M w 4 x z k k Y a C x 5 B T E Z M G Z 0 n F / V o A a / H u S h G E 2 0 W T 5 m O e A A C I k s U G 2 S C q o R 3 L F n 4 E N S b 8 Q f I w U 7 j Z W Z 0 Z N c 4 o R W R V 1 M / w T H h l E 1 n 6 T W W 8 W U l q N V C + y m b I h o j l Q A q T g U q G k B Z Z b i c F 6 I S q Q w F g C R J J Z I g a O z W W p A h Y b Z M V m 9 M K q t A O r C N b I 6 J n N o J G L u P k 6 E t m 4 F S I F w 8 L x V P G M S q J J Y I k Y 1 7 n X 6 v r 1 s u Q h g m q l v a 0 K 4 V j V m W D u r g m D L W N b r p 2 9 U x p h b 3 L + g q 9 P w b F 5 j y 0 z L I A i L w f I O 5 L D u z x A i 3 F t 7 + P Y x N x z Q C R V k E P X s M 8 F F C 5 a G a M q i S S F B B N v G e 4 8 S + 9 A Y + J B c Q l f 6 m v 2 E q y J w s / b T t 4 7 f v c n t Y F q F l u B I t Q I W J y u x A C + 4 L Z Y 7 E f A g i b X f 4 C L / Y M 7 q 9 7 W 9 7 e B x 5 x u r P c K K 8 M q X R + 0 a H R r 8 y + s j o Q 2 s z x X j s 8 c A K p G r 6 e q O s b K G / g m O 8 W Z Y c C p o r W W b 8 H + j f U H c D 5 0 R U t 8 p u K a h S d r Z m y q 7 / + 1 Y 5 6 b R 9 q 7 / f a n a 9 2 f 2 y 4 j x y 1 p 3 H j u / s O F 3 n t X P g H O 3 c u s 4 g N y C n H l b w W r d g q Z k 2 W e d g L 8 9 l 3 u D s t F a B W u Q k v Q U a q F G 2 j J f a n M k Z g P Q W a d L h / h F 7 v W d D a D T R + P Y 9 8 6 / R l O t V + l t G b P m s i a V 9 Y c W n P g b L Y c j z s e W I P S r c C s V p V d 6 K / k G K 9 X J U e S 5 l p V G f 8 H + j e 0 u Y p z I m 1 1 q 2 x X g q b K 1 s Z M 2 Q 7 + 3 C r H 3 U 7 g 9 P c b r R 1 / d r 8 s e g + 9 R 9 6 a F 3 h b 3 o 7 3 2 t v 3 j r z I M 9 4 X 7 6 v 3 r f G 9 8 b P x a 2 E G v X p l p t z 3 a r K w 9 B s + T t S Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l 6 r I g k p X z O 3 c u s 4 g N y C n H l b w W r d g q Z k 2 W e d g L 8 9 l 3 u D s t F a B W u Q k v Q U a q F G 2 j J f a n M k Z g P Q W a d L h / h F 7 v W d D a D T R + P Y 9 8 6 / R l O t V + l t G b P m s i a V 9 Y c W n P g b L Y c j z s e W I P S r c C s V p V d 6 K / k G K 9 X J U e S 5 l p V G f 8 H + j e 0 u Y p z I m 1 1 q 2 x X g q b K 1 s Z M 2 Q 7 + 3 C r H 3 U 7 g 9 P c b r R 1 / d r 8 s e g + 9 R 9 6 a F 3 h b 3 o 7 3 2 t v 3 j r z I M 9 4 X 7 6 v 3 r f G 9 8 b P x a 2 E G v X p l p t z 3 a r K w 9 B s + T t S Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l 6 r I g k p X z O 3 c u s 4 g N y C n H l b w W r d g q Z k 2 W e d g L 8 9 l 3 u D s t F a B W u Q k v Q U a q F G 2 j J f a n M k Z g P Q W a d L h / h F 7 v W d D a D T R + P Y 9 8 6 / R l O t V + l t G b P m s i a V 9 Y c W n P g b L Y c j z s e W I P S r c C s V p V d 6 K / k G K 9 X J U e S 5 l p V G f 8 H + j e 0 u Y p z I m 1 1 q 2 x X g q b K 1 s Z M 2 Q 7 + 3 C r H 3 U 7 g 9 P c b r R 1 / d r 8 s e g + 9 R 9 6 a F 3 h b 3 o 7 3 2 t v 3 j r z I M 9 4 X 7 6 v 3 r f G 9 8 b P x a 2 E G v X p l p t z 3 a r K w 9 B s + T t S Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l 6 r I g k p X z l M j z q Z / g h H D K i l l 6 g 1 V y V g l q t 9 F e x g p E E 6 R T Q O U p Q W U D S E w N Z / N C V C E t U A h I k U w h B Z I m z V o L K i L M l c n q j S n t F M g G r p H 1 M V G F m 4 B 1 + z g Z u r I Z a A 2 y i e e l 4 g m D R F f E E i n F u N f t 9 / q m 3 U Q I w 0 T v V D a 0 7 c R g J r J B H Z x Q x n a s a Q V u d a 2 t x b 0 E P Q 0 9 / 8 Y l p v p 0 D L K Q S D z f Y M / K F o 4 Y 2 a y F d 3 8 T Y 8 W Y x q A p i 6 H n j g E + T K k 6 0 A W D a R I F O h Q T / z l O 3 c u s 4 g N y C n H l b w W r d g q Z k 2 W e d g L 8 9 l 3 u D s t F a B W u Q k v Q U a q F G 2 j J f a n M k Z g P Q W a d L h / h F 7 v W d D a D T R + P Y 9 8 6 / R l O t V + l t G b P m s i a V 9 Y c W n P g b L Y c j z s e W I P S r c C s V p V d 6 K / k G K 9 X J U e S 5 l p V G f 8 H + j e 0 u Y p z I m 1 1 q 2 x X g q b K 1 s Z M 2 Q 7 + 3 C r H 3 U 7 g 9 P c b r R 1 / d r 8 s e g + 9 R 9 6 a F 3 h b 3 o 7 3 2 t v 3 j r z I M 9 4 X 7 6 v 3 r f G 9 8 b P x a 2 E G v X p l p t z 3 a r K w 9 B s + T t S Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l 6 r I g k p X z A C v 8 o M i h I h r C Z 7 L 8 e 4 = " > A A A F T 3 i c h V R b b 9 M w F A 7 Q w V Z u G 4 g n X i y 6 S g O N K q k 2 p j 0 g D Q 0 h J E A b 7 A J S X U 1 O c t J Y t e P I d t d G x j + G V / g 1 P P J L e E M 4 a U e X o c G R n J y c 8 / l c v u M 4 z B l V 2 v d / X L l 6 r b F w / c b i U v P m r d t 3 7 i 6 v 3 D t W Y i Q j O I o E E / J T S B Q w m s G R p p r B p 1 w C 4 S G D j + F w t / R / P A W p q M g O d Z F D n 5 N B R h M a E e 1 M J 8 s P c E q 0 K S w m e S 7 F B O V r x e f J 4 5 P l l t / x K 0 F / K 8 F M a X k z 2 T 9 Z a T z C s Y h G H D I d M a J U L / B z 3 T d E a h o x s M 0 2 H i n I S T Q k A + i p n G R U p X 0 T k h D Y B a c r h A E n U d / Q L B 9 p y C L b P O c 3 k 6 r p u o 1 w p Q o e W t T m R K f q o q 8 0 X u Z L R K Z V P V r E Q 0 k H q a 5 Z e 4 d B 3 5 T g a U E S M h h H g n O S x Q Z L H k N C R k x b k 6 e n 9 W g h r 8 c 5 L U c V r 5 d P l Y 1 4 C B J i R x Q b C E l 1 y r u O L H w A + s 3 4 A + T g Z v N S W B O 7 N U 7 p l M j z q Z / g h H D K i l l 6 g 1 V y V g l q t 9 F e x g p E E 6 R T Q O U p Q W U D S E w N Z / N C V C E t U A h I k U w h B Z I m z V o L K i L M l c n q j S n t F M g G r p H 1 M V G F m 4 B 1 + z g Z u r I Z a A 2 y i e e l 4 g m D R F f E E i n F u N f t 9 / q m 3 U Q I w 0 T v V D a 0 7 c R g J r J B H Z x Q x n a s a Q V u d a 2 t x b 0 E P Q 0 9 / 8 Y l p v p 0 D L K Q S D z f Y M / K F o 4 Y 2 a y F d 3 8 T Y 8 W Y x q A p i 6 H n j g E + T K k 6 0 A W D a R I F O h Q T / z l O 3 c u s 4 g N y C n H l b w W r d g q Z k 2 W e d g L 8 9 l 3 u D s t F a B W u Q k v Q U a q F G 2 j J f a n M k Z g P Q W a d L h / h F 7 v W d D a D T R + P Y 9 8 6 / R l O t V + l t G b P m s i a V 9 Y c W n P g b L Y c j z s e W I P S r c C s V p V d 6 K / k G K 9 X J U e S 5 l p V G f 8 H + j e 0 u Y p z I m 1 1 q 2 x X g q b K 1 s Z M 2 Q 7 + 3 C r H 3 U 7 g 9 P c b r R 1 / d r 8 s e g + 9 R 9 6 a F 3 h b 3 o 7 3 2 t v 3 j r z I M 9 4 X 7 6 v 3 r f G 9 8 b P x a 2 E G v X p l p t z 3 a r K w 9 B s + T t S Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l 6 r I g k p X z A C v 8 o M i h I h r C Z 7 L 8 e 4 = " > A A A F T 3 i c h V R b b 9 M w F A 7 Q w V Z u G 4 g n X i y 6 S g O N K q k 2 p j 0 g D Q 0 h J E A b 7 A J S X U 1 O c t J Y t e P I d t d G x j + G V / g 1 P P J L e E M 4 a U e X o c G R n J y c 8 / l c v u M 4 z B l V 2 v d / X L l 6 r b F w / c b i U v P m r d t 3 7 i 6 v 3 D t W Y i Q j O I o E E / J T S B Q w m s G R p p r B p 1 w C 4 S G D j + F w t / R / P A W p q M g O d Z F D n 5 N B R h M a E e 1 M J 8 s P c E q 0 K S w m e S 7 F B O V r x e f J 4 5 P l l t / x K 0 F / K 8 F M a X k z 2 T 9 Z a T z C s Y h G H D I d M a J U L / B z 3 T d E a h o x s M 0 2 H i n I S T Q k A + i p n G R U p X 0 T k h D Y B a c r h A E n U d / Q L B 9 p y C L b P O c 3 k 6 r p u o 1 w p Q o e W t T m R K f q o q 8 0 X u Z L R K Z V P V r E Q 0 k H q a 5 Z e 4 d B 3 5 T g a U E S M h h H g n O S x Q Z L H k N C R k x b k 6 e n 9 W g h r 8 c 5 L U c V r 5 d P l Y 1 4 C B J i R x Q b C E l 1 y r u O L H w A + s 3 4 A + T g Z v N S W B O 7 N U 7 p l M j z q Z / g h H D K i l l 6 g 1 V y V g l q t 9 F e x g p E E 6 R T Q O U p Q W U D S E w N Z / N C V C E t U A h I k U w h B Z I m z V o L K i L M l c n q j S n t F M g G r p H 1 M V G F m 4 B 1 + z g Z u r I Z a A 2 y i e e l 4 g m D R F f E E i n F u N f t 9 / q m 3 U Q I w 0 T v V D a 0 7 c R g J r J B H Z x Q x n a s a Q V u d a 2 t x b 0 E P Q 0 9 / 8 Y l p v p 0 D L K Q S D z f Y M / K F o 4 Y 2 a y F d 3 8 T Y 8 W Y x q A p i 6 H n j g E + T K k 6 0 A W D a R I F O h Q T / z l O 3 c u s 4 g N y C n H l b w W r d g q Z k 2 W e d g L 8 9 l 3 u D s t F a B W u Q k v Q U a q F G 2 j J f a n M k Z g P Q W a d L h / h F 7 v W d D a D T R + P Y 9 8 6 / R l O t V + l t G b P m s i a V 9 Y c W n P g b L Y c j z s e W I P S r c C s V p V d 6 K / k G K 9 X J U e S 5 l p V G f 8 H + j e 0 u Y p z I m 1 1 q 2 x X g q b K 1 s Z M 2 Q 7 + 3 C r H 3 U 7 g 9 P c b r R 1 / d r 8 s e g + 9 R 9 6 a F 3 h b 3 o 7 3 2 t v 3 j r z I M 9 4 X 7 6 v 3 r f G 9 8 b P x a 2 E G v X p l p t z 3 a r K w 9 B s + T t S Q < / l a t e x i t > Fig. 1. Sketch of the architectures studied in this paper.
D. Unsupervised Training
As illustrated in Fig.1, the composition of the encoder p θ (x|b) and the decoder q φ (b|x) leads to a stochastic autoencoder. This model can be trained without supervision using variational methods [10]. If S = {x (1) , x (2) , . . . , x (n) } denotes the set of training examples, the negative log-likelihood corresponding to a single data point x ( ) ∈ S, can be upper bounded by the following loss function [17]:
L ( ) unsup = E q φ (b|x ( ) ) − log p θ (x ( ) , b) + log q φ (b|x ( ) ) (1) = E q φ (b|x ( ) ) − log p θ (x ( ) |b) L1 +λ D KL q φ (b|x ( ) )||p θ (b) L2 .
The term L 1 measures the expected error in the reconstruction of x from the hash code b. For instance, if the decoder is Gaussian, L 1 is proportional to the squared loss
x − x 2 between the decoder's outputx and the original observation x. The term L 2 on the other hand, measures the Kullback-Leibler divergence between the distribution learnt by the encoder q φ (b|x) and some prior p θ (b). For hashing applications with Bernoulli autoencoders, this can be chosen as p θ (b i ) = Ber(0.5) ∀i ∈ [B], which expresses a preference for balanced hash tables. With this choice, L 2 can be computed analytically [17].
The main complexity of optimizing (1) using standard techniques such as backpropagation [7] is that the neural nets f (x; φ) and g(b; θ), parametrizing the autoencoder, get connected by sampling. The net f (x; φ) predicts the bit activation probabilities and then a hash b ∈ {0, 1} B is drawn to feed g(b; θ). Fortunately, one of the main achievements of the last years in deep learning has been adapting backpropagation to "pass" through stochastic layers like these [10]. [3] has shown that this method works well for hashing with Gaussian representations. In the case of discrete distributions the gradients can be estimated using the so-called Gumbel-Softmax reparametrization trick [9]. Experiments in [17] show that this method is stable and effective for hashing.
E. Semi-Supervised Training
If some examples in the training set S = {x ( ) } have been annotated with class labels describing its semantic content y ( ) ⊂ Y = {t 1 , t 2 , . . . , t K }, the unsupervised objective can be expanded to exploit this information. Hereafter we assume that the annotations have been one-hot encoded as probability distributions, i.e. y j = 1 if x ∈ S has been annotated with a label t j and y j = 0 otherwise. To accommodate the semisupervised scenario, we assume that only the first s n examples from S have been labelled. a) Pointwise Supervision: A simple way to guide the model towards a more discriminative latent representation for a pattern x is to train the model to learn not only p(x) but also p(y|x). More specifically, if the neural net implementing the encoder is f = f 1 • . . . f L−1 • f L , we burden the model with the task of inferring p(y|x) from the representation z = f 1 • . . . • f L−1 computed immediately before the bitactivation probabilities α(x). This approach is illustrated in Fig.1. Intuitively, if two patterns x (1) , x (2) have the same annotations, the model should learn that p(y|x (1) ) ≈ p(y|x (2) ). As p(y|x) is computed from z, we should have p(y|x (1) ) ≈ p(y|x (2) ) ⇒ z (1) ≈ z (2) . However, as also the codes are computed from z, we should have z (1) ≈ z (2) ⇒ b (1) ≈ b (2) . Therefore, the model should learn that patterns with the same annotations have to be allocated in nearby addresses of the hash table.
A similar type of supervision has been used in [3] and [4]. However, in these works, p(y|x) is approximated from the representation computed immediately after the decoder's output. Our choice is tailored to the self-supervision method introduced below.
As sketched in Fig.1, we can approximate the distribution p(y|x), augmenting the autoencoder with an extra fully connected layerŷ(z; ψ). The parameters ψ of this layer can be jointly trained with the rest of the architecture to minimize the cross-entropy loss between the predicted label distribution for a labelled example x ( ) and the ground-truth. In the simplest case (one-hot vectors representing mutually exclusive labels), the loss takes the form
L ( ) sup = −E y ( ) log p ψ (y|x ( ) ) = − k y ( ) k logŷ ( ) k . (2)
b) Pairwise Supervision: A more explicit way to enforce a consistency between the similarities in the code space and the similarities in the label space can be obtained by equipping the model with a pairwise loss function. If b ( ) and b ( ) denote the codes assigned to a pair of examples x ( ) , x ( ) sampled from the labelled dataset, and y ( ) , y ( ) denote their groundtruth labels, a loss that penalizes/rewards differences between the codes of similar/dissimilar pairs is
where D ± denote distance functions in the Hamming space. Often, D + is chosen to be the standard Hamming distance b ( ) − b ( ) H , but D − is shrunk as D − = −(ρ − b ( ) − b ( ) H ) + to avoid wasting efforts in separating dissimilar pairs beyond a margin ρ. This loss has been used in a plethora of hashing algorithms (see e.g. [16]). In the context of variational autoencoders, it has been proposed in [4]. c) Self-Supervision: Both the pointwise and the pairwise schemes of supervision suffer the lack of labelled examples. However, a hashing algorithm using the pairwise loss can deteriorate faster than a method using only pointwise supervision. Arguably, this happens because if the labelled subset is reduced to a fraction ρ of the training set, the fraction of pairs that can be generated reduces to ρ 2 . This issue makes the method more prone to over-fitting in semi-supervised scenarios with label scarcity. To address this problem, we propose a self-supervised learning mechanism in which the ground-truth labels required for Eqn.(3) and substituted bŷ y(z; ψ), the predictions of the pointwisely supervised layer of the autoencoder. To formally define the new loss function, we first write (3) in matrix form. Indeed, as y ( )T y ( ) = 0 if the points x ( ) , x ( ) have different labels and otherwise y ( )T y ( ) = 1, we have that the pairwise loss is equivalent to
L ( , ) pair = y ( )T y ( ) D + , − (1 − y ( )T y ( ) )D − , ,(4)
where we have used D ± , as a short-hand for D ± (b ( ) , b ( ) ). The self-supervised loss is thus defined as
L ( , ) selfsup =ŷ ( )Tŷ( ) D + , − (1 −ŷ ( )Tŷ( ) )D − , .(5)
Intuitively, minimizing the pointwise loss (2) requires less annotations than learning a label-consistent hash function. Thus, after some training rounds we will have thatŷ ( ) will approximate y ( ) for many unlabelled observations. Hereafter the self-supervised loss (5) approximates the more conventional pairwise loss (4). Note in addition that, as the label distribution in (5) is now trainable, the loss can be examined as a function of the labelsŷ assigned by the algorithm to the different points of the Hamming space. Reordering the terms,
L ( , ) selfsup = (D + , + D − , )ŷ ( )Tŷ( ) − D − , ,(6)
we see that the loss function penalizes correlations between the label distributions in a way proportional to D + , + D − , . The loss is 0 if and only if pairs ( , ) for which D ± > 0 get assigned orthogonal label distributions. As there is a finite number of (normalized) distributions on Y which are mutually orthogonal, the proposed loss is minimized by reserving a different labelling to distant regions of the Hamming space (D ± 0).
F. Efficient Implementation
The final objective function for training the autoencoder in semi-supervised scenarios is
L = n =1 L ( ) unsup + β s =1 L ( ) sup + α n , =1 L ( , ) selfsup(7)
where β, α > 0 are hyper-parameters. Note that only the supervised loss L sup is computed on labelled instances. The unsupervised loss and the self-supervised loss are computed using all the available observations. amount of supervision decreases down to 10%. These results suggest that, in scenarios of label scarcity, learning the label distribution, i.e. minimising the expected pointwise loss, may be easier than learning label consistent hash codes. As the lack of annotations has a quadratic effect on the number of groundtruth pairs that can be used, the pairwise approach is more fragile in semi-supervised scenarios. The results in TMC and 20News/16Bits, in which PSH-GS and VDSH-S deteriorate at more similar rates, may be explained by the fact that PSH-GS implements both types of supervision, pointwise and pairwise.
In Tab.I and Fig.2 we can see that the proposed method, which uses the ground-truth labels to learn the label distribution, but employs its own predictions to implement the pairwise loss, is much more robust to the lack of annotations. Its performance decreases more smoothly as the fraction of labelled instances reduces down, achieving noticeable improvements on PSH-GS for small amounts of supervision. This illustrates the interest of the approach for semi-supervised scenarios. For instance, with only 10% of training documents labeled, the proposed method gives a precision of 73.4% in 20News/32Bits, an (absolute) improvement of 14.5% compared to PSH-GS and 8.6% over VDSH-S. For the same level of supervision, it gets a precision of 81.6% in CIFAR/32Bits, a clear improvement over the 63.5% of the pairwise approach. In Snippets/16Bits, SSB-VAE provides a precision only 2% less than the precision achieved in the fully supervised case. In some cases (TMC), the pairwise approach is able to keep its advantage on VDSH-S almost uniformly as the supervision becomes lower. If this is the case, the proposed method is still competitive or better than the best baseline. Although the most significant improvements are obtained for smaller ρ, we also confirm that in scenarios of label abundance, using pairwise supervision based on the ground-truth distributions does not give a very significant advantage over the proposed approach. Indeed, in many cases (6/8) SSB-VAE achieves slightly better scores. To obtain an overall conclusion regarding the robustness of the proposed method, two types of statistical tests are conducted. We employ the Friedman test to assess whether there is enough statistical evidence to reject the hypothesis that the three methods are statistically equivalent (in terms of p@100), when considering different levels of supervision. In this design, the method (SSB-VAE, PSH, VDSH) serves as the group variable, and the level of supervision serves as the blocking variable. In addition, when rejecting the null hypothesis of Friedman's test, we compare the proposed method against PSH and VDSH using the Nemenyi post-hoc test, to check for statistically significant differences. The obtained p-values are presented in Tab.II. In all but two cases we obtain values below 5%.
V. CONCLUSIONS
We studied the performance of semi-supervised hashing algorithms based on variational autoencoders in scenarios of label scarcity. It was found that training the model to explicitly preserve pairwise similarities derived from the annotations, often yields better results than using pointwise supervision (only), confirming results of previous works. However, we also found that methods based on this type of supervision tend to deteriorate more sharply when the number of labelled observations decreases. To overcome this problem, we proposed a new type of supervision in which the model uses its own beliefs about the class distribution to enforce a consistency between the similarities in the code space and the similarities in the label space. Experiments in text and image retrieval tasks confirmed that this method degrades much more gracefully when the models are stressed with scarcely annotated data, and very often outperforms the baselines by a significant margin. As in scenarios of label abundance, the proposed method proved to be competitive or better than the best baseline, we can conclude that it is a robust approach to semi-supervised hashing. In future work we plan to equip the method with adaptive loss weights and extend the experiments to crossdomain information retrieval.
=
I(y ( ) = y ( ) )D + (b ( ) , b ( ) )
TABLE I P@100
IOF THE DIFFERENT METHODS FOR DIFFERENT LEVELS OF SUPERVISION ρ. A) 32 HASHING BITS AND B) 16 HASHING BITS. THE ALGORITHMS PHS-GS [4] AND VDHS-S [3] ARE ABBREVIATED PSH AND VDSH.A)
20-NEWS
CIFAR
SNIPPETS
TMC
ρ
PSH SSB-VAE VDSH
PSH SSB-VAE VDSH
PSH SSB-VAE
VDSH
PSH SSB-VAE VDSH
0.1 0.589
0.734 0.648
0.687
0.825 0.805
0.501
0.565
0.540 0.738
0.750 0.730
0.2 0.606
0.765 0.697
0.708
0.840 0.816
0.490
0.599
0.558 0.749
0.754 0.725
0.3 0.
TABLE II P
II-VALUES OF THE STATISTICAL TESTSNemenyi Test
Friedman Test
PSH
VDSH
20-NEWS
1.1 × 10 −4
6.3 × 10 −5
3.7 × 10 −2
32 Bits
SNIPPETS
7.4 × 10 −3
1.0 × 10 −2
8.9 × 10 −1
− I(y ( ) = y ( ) )D − (b ( ) , b ( ) ) ,
(1), . . . , x (n) } and semantic labels y ( ) for the first s. Output: Trained parameters φ, θ, ψ.1Initialize φ, θ, ψ; 2 while not converged do3Randomly split S into n/M batches of size M ;4foreach mini-batch B j do5Predictŷ ( ) for any x ( ) ∈ B j ;6Average the gradients of (1) and(5)Average the gradient of (2) w.r.t. ψ among the labelled examples in B j ;8Perform backpropagation updates for φ, θ, ψ; 9 end foreach 10 end whileWe optimize (7) using backpropagation. Indeed, thanks to the Gumbel-Softmax estimator[9], we can efficiently compute the gradients of L ( ) unsup and L ( , ) selfsup w.r.t. all the model's parameters φ, θ, ψ. Beingŷ ( ) independent on the stochastic layer, the gradients of L ( ) sup can be computed classically. However, the direct computation of the total gradient/loss has a quadratic computational complexity in the number of examples. As sketched in Alg.1, we circumvent this problem by forming the pairs required for L ( , ) selfsup at a mini-batch level. In this way, the computational cost of the algorithm is only O(nM ), where M is the mini-batch size, a small constant.IV. EXPERIMENTSWe conduct experiments to evaluate the robustness of semisupervised variational autoencoders for hashing in scenarios of label scarcity. The proposed approach is compared with previous methods on text and image retrieval tasks, widely used to assess this type of algorithms. Our code along with instructions to reproduce the results is made publicly available at: https://github.com/amacaluso/SSB-VAE. a) Data: The text retrieval tasks are defined on three annotated corpora: 20 Newsgroups, containing 18000 newsgroup posts on 20 different topics; TMC containing 28000 air traffic reports annotated using 22 tags; and Google Search Snippets, with 12000 short documents organized in 8 classes (domains). We define an image retrieval task using the dataset CIFAR-10, containing 60000 32 × 32 RGB images of 10 different classes[11]. To facilitate comparisons, we represent the text using TD-IDF features as in[3]and[17]. 20 Newsgroups (hereafter abbreviated 20News) and TMC are used with the train/validation/test split used in[3]. For Snippets, we follow[17]and randomly sample a test set of 1200 texts, a validation set of the same size, and leave the rest for training. Images are represented using deep VGG descriptors as in[5]. For CIFAR-10, we use the pre-defined test set[11]. A validation set of the same size is randomly sampled from the training set. b) Methods: We compare three semi-supervised methods based on variational autoencoders: (1) VDHS-S, a variational autoencoder proposed in[3]employing Gaussian latent variables, unsupervised learning and pointwise supervision; (ii) PHS-GS, a variational autoencoder proposed in[4]employing Bernoulli latent variables, unsupervised learning, pointwise supervision and pairwise supervision; and (iii) SSB-VAE, our proposed method based on Bernoulli latent variable, unsupervised learning, pointwise supervision and self-supervision. As you could note, other combinations of latent variables and types of supervision are possible. For sake of brevity we compare only with published methods. Note also that, as we use the same features and train/validation/test split that[3]for the text datasets, our results can be directly compared with the performance of many other deep learning methods assessed in this work, as in[4].c) Technical Details: To implement the neural nets corresponding to the encoder/decoder, we adopted the same architectures used in[3], for all the methods. We trained the models using 30 epochs, batch size M = 100, and the Adam learning rate scheduler[7]. The KL weight λ in Eqn.(1)was set to the values reported in[17]. The parameters β and α required for PHS-GS and SSB-VAE were selected on the validation set, using a logarithmic search grid in the range [10 −6 , 10 6 ]. For a fair comparison, we also allowed VDHS-S to select the weight of the supervised loss in the objective function. The scores reported in figures and tables were obtained as an average over 5 runs. All the codes were implemented in Python 3.7 with TensorFlow 2.1 and executed using a small GPU (GTX 1080Ti). d) Evaluation: To evaluate the effectiveness of the hash codes, each document/image in the test set is used as a query to search for similar items in the training set. Following previous works[4,5], a relevant search result is one which has the same ground-truth label (topic) as the query. To favour comparisons, the performance is measured using p@100, the precision within the first k = 100 retrieved documents/images, sorted according to the Hamming distances of their corresponding hash codes to that of the query. We also compute the mean average precision, the average of p@k varying k from 1 to the length of the retrieved list. This score penalizes missing relevant items among the first positions of the list.To assess the robustness of the algorithms to label scarcity, we train and evaluate the models at varying levels of supervision ρ = s/n, the ratio of labelled examples in the training set. Starting from ρ = 1, which represents a fully supervised setting, we stress the algorithms reducing ρ by steps of 0.1 until we get a 10% of supervision. e) Results & Discussion:Table Ishows the precision of the different methods on the four datasets used for evaluation. We present results for code lengths of B = 16 and B = 32 bits. It can be confirmed that when all the training instances are labelled (ρ = 1), the model based on pointwise supervision (PSH-GS) often achieves better results than the method based on pointwise supervision (VDSH-S). This tendency, previously reported in the literature[4], is no longer clear if we reduce the supervision level, specially below 50% (ρ ≤ 0.5). For instance, with only 20% of training images labeled, VDSH-S gives a precision of 81.6% in CIFAR, 10% over PSH-GS, in absolute terms. We see something similar in Snippets and 20News/32Bits. In these cases, the performance of PSH-GS suffers significantly more the lack of supervision, with a precision loss over 20% in CIFAR, 25% in 20News and 20% in Snippets. To better illustrate this point, we display inFig.2the mean average precision (MAP) of the algorithms as a function of ρ. In many cases (CIFAR, Snippets and 20News/32Bits) the performance of PSH-GS is clearly decreasing faster as the
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] | [] | Low-contrast all-optical Zeno switching has been demonstrated in a Si3N4 microdisk resonator coupled to a hot atomic vapor. The device is based on the suppression of the field build-up within a microcavity due to non-degenerate two-photon absorption. This experiment used one beam in a resonator and one in free-space due to limitations related to device physics. These results suggest that a similar scheme with both beams resonant in the cavity would correspond to input power levels near 20 nW. | 10.1103/physreva.87.023808 | [
"https://arxiv.org/pdf/1206.0930v1.pdf"
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All-Optical Switching Demonstration using Two-Photon Absorption and the Classical Zeno Effect
(Dated: May 1, 2014) 5 Jun 2012
S M Hendrickson
Applied Physics Laboratory
The Johns Hopkins University
20723LaurelMaryland
C N Weiler
Applied Physics Laboratory
The Johns Hopkins University
20723LaurelMaryland
R M Camacho
Sandia National Laboratories
87185AlbuquerqueNew Mexico
P T Rakich
Sandia National Laboratories
87185AlbuquerqueNew Mexico
A I Young
Sandia National Laboratories
87185AlbuquerqueNew Mexico
M J Shaw
Sandia National Laboratories
87185AlbuquerqueNew Mexico
T B Pittman
University of Maryland
Baltimore County
21250BaltimoreMaryland
J D Franson
University of Maryland
Baltimore County
21250BaltimoreMaryland
B C Jacobs
Applied Physics Laboratory
The Johns Hopkins University
20723LaurelMaryland
All-Optical Switching Demonstration using Two-Photon Absorption and the Classical Zeno Effect
(Dated: May 1, 2014) 5 Jun 2012
Low-contrast all-optical Zeno switching has been demonstrated in a Si3N4 microdisk resonator coupled to a hot atomic vapor. The device is based on the suppression of the field build-up within a microcavity due to non-degenerate two-photon absorption. This experiment used one beam in a resonator and one in free-space due to limitations related to device physics. These results suggest that a similar scheme with both beams resonant in the cavity would correspond to input power levels near 20 nW.
The quantum Zeno effect (QZE) can prevent a randomly occurring process by frequent measurement [1]. It has previously been shown [2][3][4] that this effect could be used to suppress errors in quantum logic gates using strong two-photon absorption (TPA). Recently, this work was extended to show that the QZE has a classical analog that could be used to create a low-loss all-optical switch [5] capable of operating at low powers.
Whereas the QZE prevents the buildup of a probability amplitude, the classical Zeno effect suppresses the coherent buildup of the electromagnetic field amplitude within a microresonator. To see how this can be used to create a switch, consider a system in which the resonator is strongly coupled to a two-photon absorbing medium such that two distinct frequencies are required for absorption to take place. With the resonator critically coupled to two waveguides, the presence of a resonant input at either of the two frequencies will result in the light coupling into the resonator and leaving the opposite waveguide. This is due to the destructive interference between the light remaining in the waveguide and the built-up field amplitude in the cavity that couples back to the waveguide. When both frequencies are present in the cavity the TPA prevents the coherent intra-cavity field buildup and the input beams pass by the resonator because there is now insufficient amplitude in the cavity to result in interference.
Based on these principles, groups have proposed alloptical Zeno switches employing other dissipative mechanisms, such as saturated absorption in a quantum dot coupled to a photonic crystal cavity [6], inverse Raman scattering (IRS) in a Silicon microdisk [7], IRS in an optical fiber [8] and sum and difference frequency generation in a χ (2) microdisk [9]. More generally, other techniques have recently been investigated to demonstrate all-optical switching with the intent of reducing operating power levels [10][11][12][13][14][15].
Here we present experimental progress towards a clas- small mode volume. This has recently been observed at nanowatt power levels in submicron-diameter tapered optical fibers [16,17]. Additional enhancement in the TPA rate has been achieved by using high quality factor (Q) resonators to increase the field strength and interaction time [14,18,19].
(a) P in −→ P drop ←− P through −→ (b) FIG
To demonstrate Zeno switching, we used a Si 3 N 4 microdisk with a diameter of 26 µm and a thickness of 250 nm. These devices were chosen over other high-Q microcavity designs because they have integrated waveguide coupling and a high index of refraction that supports mode compression. The resonators contain no cladding material, allowing the evanescent field to extend outside the device and overlap considerably with the atomic vapor. Each microdisk was critically coupled to two waveguides in an add-drop configuration, and then coupled to four single-mode fibers using a V-groove chip with compatible spacing. A chip-level schematic is shown in Figure 1(a) and an SEM image of the waveguide-coupled microdisk is shown in Figure 1
(b).
The setup used in this experiment is shown in Figure 2. Two frequency-stabilized external-cavity diode lasers were used, one at 780 nm and one at 1529 nm. Both of these beams passed through fiber splitters and into a reference cell as well as a vacuum system containing a microdisk. The two beams were focused through a Rb reference cell ( 87 Rb and 85 Rb) in a counter-propagating configuration to allow monitoring of the TPA condition while their frequencies were measured using a wavelength meter. The 780 nm beam passed through a low-noise tapered amplifier prior to entering the vacuum system through a view-port where it was focused onto the microdisk. The 1529 nm beam remained in fiber and was coupled to the resonator via on-chip waveguides. The 1529 nm through-and drop-ports were measured using femtowatt detectors after passing through 1500 nm longpass filters.
The microdisk was mounted in a vacuum system with the fibers entering through a Teflon feedthrough [20]. The vacuum system was lightly baked to 120 • C until pressures reached 10 −8 Torr and then cooled to about 80 • C. Rb vapor was injected into the system using an array of getters. The atomic density was estimated using a fit to the tails of the absorption spectrum of a probe beam. This method resulted in an average density between 10 11 and 10 12 cm −3 . The section containing the microdisk was surrounded by view-ports to allow optical access from the top, as will be discussed below.
Our scheme uses the two-photon transition 5S 1/2 → 5P 3/2 → 4D 3/2 that consists of the well-known D 2 line near 780 nm followed by a transition near 1529 nm. Because coupling gaps between the waveguide and the resonator were designed for wavelengths near 1529 nm, the 780 nm transition was excited using a focused free-space beam.
The matching of the cavity resonance (λ c ) to the 1529 atomic resonance (λ 1529 Rb ) was accomplished using a number of steps. Initially, the approximate diameter of the microdisk was chosen to optimize the mode volume according to [5] with λ c near λ 1529
Rb . The diameter was then varied incrementally for a series of disks laid-out across each substrate such that a device could be selected after fabrication, coupled to fibers and secured using vacuumcompatible, high-temperature UV-curable epoxy. Once the devices were inserted into the vacuum chamber they experienced a downward shift in λ c due to the change in index. It has also been observed that Si 3 N 4 devices undergo an upward shift in λ c when exposed to Rb in vacuum as a result of accumulation [21] although this change can be reversed by rinsing the device in distilled water. We have observed that this Rb shift in λ c tends to increase with the maximum vapor density achieved and have used it as a tool to shift the resonance by as much as 7 nm. The resonance can also be shifted down by etching with Hydrofluoric acid (HF) [21]. Ultimately, each of the aforementioned shifts were compensated for and then final tuning was accomplished thermally with a 25 mW 405 nm laser focused on the resonator.
While it has been predicted that classical Zeno switching could result in excellent signal contrast [5], the Q of this device (≈ 10 5 ) is currently below what is needed for optimal performance. With expectations of a somewhat modest signal we designed a data collection scheme to reduce noise through averaging. A calibration algorithm was used to vary the atomic intermediate state detuning (δ Rb ) by scanning the frequencies of the 780 nm and 1529 nm lasers by an equal and opposite amount such that their total energy remained fixed and resonant with the two-photon transition described previously. This allowed a full scan across the profile of the cavity resonance with the conditions for TPA satisfied, although the strength of the absorption varied as a function of intermediate state detuning. As a control, each resonant TPA scan was followed by a scan with the frequency of the 780 beam offset by 10 GHz such that the cavity resonance was measured in the absence of TPA. To compensate for any thermal drift when averaging data trials, each cavity profile was zeroed to a new frequency axis (∆) by fitting to a Lorentzian and then adding a frequency offset (f 0 ) to the atomic intermediate state detuning (δ Rb ),
∆ = δ Rb − f 0 .
(1) Figure 3 shows the strength of the TPA rate in the reference cell under conditions typical of each trial. As can be seen in the plot, the TPA condition is maintained throughout a scan of approximately 5 GHz. The shape of the graph corresponds to the well-known δ −2 Rb dependence predicted by perturbation theory [5], as shown by the fit.
An example of Zeno switching is shown in Figure 4. Approximately 100 trials were averaged for each condition. The blue lines in the upper plot show the cavity drop-port (peak) and through-port (dip) with the 780 nm laser detuned so that the conditions for TPA were not satisfied. The red lines in the upper plot show the cavity response in the presence of TPA. The difference in these two situations is highlighted in the lower plot, where it is emphasized that the effect of the additional absorption is to increase the transmission of the through-port and decrease the transmission of the drop-port, consistent with a low-contrast Zeno switching process. Figure 5 shows a control case with the only difference being the absence of Rb vapor. As can be seen in the data, there is no similar change in the transmission of the cavity to suggest a switching event. Although not shown here, additional controls were also performed to validate our data. For example, in one test the cavity was thermally detuned from two-photon resonance with the power levels, density and frequency of the 780 nm free-space beam unchanged. In another, the position of the 780 nm beam was moved from the resonator to the waveguides to determine whether absorption before or after the resonator was an issue. All of our controls supported the conclusion that switching was due to TPA in the evanescent field of the microdisk. Assuming symmetric fiber-coupling losses, the 1529 nm power in the input waveguide can be estimated to be roughly 14 nW for the switching data shown in Figure 4. The control data shown in Figure 5 corresponds to 56 nW. The power in the free-space 780 nm beam was 15 mW for both cases. The intensity of the 1529 beam in the resonator was 490 W/cm 2 assuming a mode volume of 1.9 ×10 −11 cm 3 (42 (λ/n) 3 ). The intensity of the freespace beam (470 W/cm 2 ) would correspond to 13 nW in the input waveguide.
The primary factor that affects switching contrast is the difference between the coupled cavity Q with and without TPA. One way to improve the contrast of the switch is to increase the intrinsic quality factor allowing the total system Q to be increased by decreasing the waveguide coupling strengths proportionally, leaving the coupling regime unchanged. Higher Qs in Si 3 N 4 devices have been demonstrated [21] using a variety of techniques and we are optimistic that we can improve the results presented here. We are also investigating the suitability of other types of micro-cavities for use in Zeno switching experiments, weighing factors such as intrinsic quality factor and mode volume with practical issues such as chip-scale integration and robustness.
The primary advantage of using an atomic vapor as compared to trapped atoms is the increased number of atoms in the mode and the practical advantages asso- ciated with not working at low temperature. The disadvantages of using atomic vapors are the effects of the atomic velocities and the buildup of the atoms on the surface of the device. The atomic velocities reduce the average time each atom spends in the mode, resulting in time-of-flight broadening [22]. In schemes such as the one demonstrated here, where the energies of the two atomic transitions are different, Doppler-free schemes are not effective and Doppler broadening dominates. A Doppler-free scheme could be performed using the 5S 1/2 → 5P 3/2 → 5D 5/2 transition in Rb, for example [5,23,24]. The atomic build-up on the surface of the device has been used as a tool in this experiment but it also results in slight degradations in the Q that may become more important with lower-loss resonators. This coating on the surface of the device may also decrease the fraction of the mode that overlaps with the atomic vapor thereby reducing the effect of the atoms. Investigations are underway to determine the extent of these effects on switching performance. To our knowledge, these results represent the first published experimental demonstration of Zeno optical switching based on TPA. Using estimates based on the power of the free-space beam, all-optical switching in the low nanowatt range seems attainable. Although the switching contrast is currently low, we believe the perfor-mance can be improved by addressing challenges related to device physics.
Funding was provided by the DARPA ZOE program (Contract No. W31P4Q-09-C-0566).
PACS numbers: 42.65.Pc, 42.65.-k, 42.82.Et
profile of the TPA signal in the reference cell as a function of intermediate state detuning of the 1529 nm beam. The frequencies of both lasers were scanned simultaneously such that their total energy remained fixed with the 85 Rb F=3 5S 1/2 → 5P 3/2 → 4D 3/2 transition. The typical cavity position during data collection (∆ = 0) was approximately equal to f0 = 6 GHz.
FIG. 4 :
4Switching data: The upper plot shows cavity transmission with the conditions for TPA satisfied in red and a control in blue. The lower plot shows the difference in the signals.
FIG. 5 :
5Control data demonstrating that no similar change is seen when the experiment is repeated with the frequency of the 1529 nm beam in a different range, as described in the text.
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| [] |
[
"Collapse and Fragmentation of Magnetized Cylindrical Clouds *",
"Collapse and Fragmentation of Magnetized Cylindrical Clouds *"
] | [
"Kohji Tomisaka \nFaculty of Education\nNiigata University\n8050 Ikarashi-2950-21NiigataJapan\n"
] | [
"Faculty of Education\nNiigata University\n8050 Ikarashi-2950-21NiigataJapan"
] | [] | Gravitational collapse of the cylindrical elongated cloud is studied by numerical magnetohydrodynamical simulations. In the infinitely long cloud in hydrostatic configuration, small perturbations grow by the gravitational instability. The most unstable mode indicated by a linear perturbation theory grows selectively even from a white noise. The growth rate agrees with that calculated by the linear theory. First, the density-enhanced region has an elongated shape, i.e., prolate spheroidal shape. As the collapse proceeds, the high-density fragment begins to contract mainly along the symmetry axis. Finally, a spherical core is formed in the non-magnetized cloud. In contrast, an oblate spheroidal dense disk is formed in a cloud in which the magnetic pressure is nearly equal to the thermal one. The radial size of the disk becomes proportional to the initial characteristic density scale-height in the r-direction. As the collapse proceeds, a slowly contracting dense part is formed ( < ∼ 10% in mass) inside of the fast contracting disk. And this is separated from other part of the disk whose inflow velocity is accelerated as reaching the center of the core. From arguments on the Jeans mass and the magnetic critical mass, it is concluded that the fragments formed in a cylindrical elongated cloud can not be supported against the self-gravity and it will eventually collapse. | 10.1086/175067 | [
"https://arxiv.org/pdf/astro-ph/9306028v1.pdf"
] | 15,358,414 | astro-ph/9306028 | 7f9eb9d35608de955a9925acba23eab42a6826a4 |
Collapse and Fragmentation of Magnetized Cylindrical Clouds *
29 Jun 1993
Kohji Tomisaka
Faculty of Education
Niigata University
8050 Ikarashi-2950-21NiigataJapan
Collapse and Fragmentation of Magnetized Cylindrical Clouds *
29 Jun 1993Received: April 16, 1993; accepted:arXiv:astro-ph/9306028v1Subject Headings: Interstellar: Matter -Interstellar: Magnetic Fields -Hydromagnetics - Stars: Formation
Gravitational collapse of the cylindrical elongated cloud is studied by numerical magnetohydrodynamical simulations. In the infinitely long cloud in hydrostatic configuration, small perturbations grow by the gravitational instability. The most unstable mode indicated by a linear perturbation theory grows selectively even from a white noise. The growth rate agrees with that calculated by the linear theory. First, the density-enhanced region has an elongated shape, i.e., prolate spheroidal shape. As the collapse proceeds, the high-density fragment begins to contract mainly along the symmetry axis. Finally, a spherical core is formed in the non-magnetized cloud. In contrast, an oblate spheroidal dense disk is formed in a cloud in which the magnetic pressure is nearly equal to the thermal one. The radial size of the disk becomes proportional to the initial characteristic density scale-height in the r-direction. As the collapse proceeds, a slowly contracting dense part is formed ( < ∼ 10% in mass) inside of the fast contracting disk. And this is separated from other part of the disk whose inflow velocity is accelerated as reaching the center of the core. From arguments on the Jeans mass and the magnetic critical mass, it is concluded that the fragments formed in a cylindrical elongated cloud can not be supported against the self-gravity and it will eventually collapse.
Introduction
The process of star formation begins in the interstellar clouds as a fragmentation of the clouds. Massive stars are formed only in giant molecular clouds, while less massive stars are born also in less massive dark clouds (for a review, see Larson 1991). This observational fact seems to suggest that the process of massive star formation and that of less massive stars are different. This seems to be related to the difference between the collapse of supercritical clouds and subcritical clouds (Shu, Adams, & Lizano 1987). If a mass of the cloud is larger than a critical mass (for a magnetic cloud, ∼ magnetic flux/G 1/2 ), there is no equilibrium for the thermal pressure, the Lorentz force, and the centrifugal force to counter-balance the self-gravity. The supercritical clouds begin dynamical collapse. However, the subcritical magnetohydrostatic cloud is thought to evolve slowly only by the plasma drift (ambipolar diffusion; Mouschovias 1977, Nakano 1979 and/or the magnetic braking (Mouschovias 1979). The structure of the cloud changes in a relatively long time scale of the plasma drift and the magnetic braking. Finally, if the cloud becomes supercritical due to the decrease of magnetic flux at the center of the cloud or the decrease of angular momentum to support the cloud, it begins to dynamical collapse, too. However, this is true only when the cloud in a static equilibrium is stable. When the time scale of some dynamical instabilities is shorter than the evolutionary time scale, the cloud may begin a dynamical contraction or fragmentation before it reaches the condition for the supercritical cloud. In a series papers, the fragmentation process in a subcritical, magnetohydrostatic cloud is studied by a non-linear magnetohydrodynamical simulation in two-dimension. This paper is devoted to the filamentary cloud and its gravitational fragmentation.
The process of fragmentation and thus gravitational collapse of the high-density portion of the cloud are studied by a linear and non-linear analyses. Linear analyses of the gravitational instability in an isothermal slab have been done many authors (see Spitzer 1978 for a review). The qualitative answer is as follows: the slab is unstable for a perturbation whose wavelength is longer than a critical length and the most unstable perturbation has the wavelength of λ max ≃ 20c s /(4πGρ c ) 1/2 and the e-growing time scale of τ max ∼ 2(4πGρ c ) −1/2 , where ρ c represents the density on the midplane of the disk (Elmegreen & Elmegreen 1978). The cylindrical cloud has a similar characteristic wavelength, λ max ≃ 20c s /(4πGρ c ) 1/2 and a longer growth time scale as τ max ∼ 3(4πGρ c ) −1/2 (Nagasawa 1987). It is concluded that if there is an inhomogeneity with an amplitude of ∆ρ/ρ ∼ 5%, in 3τ max < ∼ 10(4πGρ c ) −1/2 the dense part of the cloud grows as a fragment with non-linear density contrast ∆ρ/ρ ∼ 1. This time scale is comparable to that for the ambipolar diffusion (Nakano 1988). Therefore, the dynamical instability in the subcritical static cloud should be studied.
The isothermal non-magnetic cylindrical cloud has a critical mass per unit length beyond which no equilibrium is achieved as
λ c = ∞ 0 2πρrdr = 2 c 2 s G , (1.1)
where c s represents the isothermal sound speed in the cloud. The cylindrical cloud with a mass per unit length, λ = λ c , has the infinite density contrast between the center and the surface. As for the spherical isothermal cloud, the situation is different; the critical cloud has a finite density contrast of ρ c /ρ s = 14 (Bonner 1956;Ebert 1955 where p ext represents the external pressure on the surface of the cloud. Finally, in the slab geometry, no critical column density of the self-gravitating disk exists. These differences come from the geometry or the dimension of the system. Generally, the magnetic field and the centrifugal force in rotating clouds have an effect of increasing the critical mass. From studying magnetohydrostatic equilibrium, the critical mass of the magnetized cloud is obtained as
M c mag ≃ 1.4 1 − 0.17 dm/d(Φ B /G 1/2 ) 2 −3/2 c 4 s p 1/2 ext G 3/2 ,(1.3)
where dm/d(Φ B /G 1/2 ) means the mass-to-magnetic flux ratio at the center of the cloud (Tomisaka, Ikeuchi, &Nakamura 1988). Further the rotating cloud has a larger critical mass as
M c rot ≃ M 2 c mag + 4.8c s j G 2 1/2 ,(1.4)
where M c mag and j represent, respectively, the mass which can be supported without any rotation [Eq.(1.
3)] and a specific angular momentum of the cloud (Tomisaka, Ikeuchi, & Nakamura 1989).
Anyhow, the cloud with λ > λ c or M > M c eventually collapses, unless the excessive mass is eroded by any processes. Study of the dynamical evolution of the magnetized cloud has been restricted for supercritical clouds (Scott & Black 1980;Black & Scott 1982;Phillips 1986a, b;Dorfi 1982Dorfi , 1989. The supercritical cloud contracts as a whole and forms a dense contracting core inevitably. The authors reported that they did not observed any fragmentation in the process of contraction. This seems to correspond to the fact that free-fall time scale is shorter than the growing time scale of the gravitational instability (see above). The situation seems to be much different for subcritical clouds.
As for the non-magnetic cylindrical clouds, Bastien (1983), and Bastien et al. (1991) have studied the contraction of initially uniform cylindrical cloud with finite length. They achieved some conclusions on the fate of the above clouds: for example, for the elongated cloud with the ratio of length to diameter of the cylinder > ∼ 2, using the initial Jeans number, J 0 , which is defined as the ratio of gravitational to the thermal energies, the evolution is determined. When J 0 < ∼ J 2−f rag , two subcondensations are formed. For more large Jeans number as J 2−f rag < ∼ J 0 < ∼ J spindle , these two subcondensations collapses into one object. For the extreme case J 0 > ∼ J spindle , the cloud contracts onto a line and forms a spindle. However, their simulation has a restriction that the initial state is far from the hydrostatic equilibrium. Excess free energy is liberated in the process of the relaxation from the initial state to the equilibrium, which may act an important role. To understand the relatively slow evolution in the subcritical cloud, we should take a (magneto-)hydrostatic configuration as the initial state for the simulation.
Model and Numerical Method
Initial Condition
As described in the preceding section, we assume that the initial state is in a hydrostatic equilibrium. Using the cylindrical coordinate (z, r, φ), if the initial cylindrical isothermal cloud is homogeneous in the z−direction, the magnetohydrostatic equilibrium configuration is obtained by the equation of hydrostatic balance and the Poisson equation as
− ∂ψ ∂r − c 2 s ρ ∂ρ ∂r − 1 8πρ ∂B 2 z ∂r = 0, (2.1) 1 r ∂ ∂r r ∂ψ ∂r = 4πGρ, (2.2)
where ψ, ρ, B, c s , and G represent, respectively, the gravitational potential, density, magnetic field, isothermal sound speed, and the gravitational constant. To derive equation(2.1), it is assumed there is no helical magnetic field component (B φ = 0). The density distribution depends upon that of the magnetic field B z . In this paper, we restrict ourselves to two cases: (i) uniform magnetic field B z =constant, and (ii) the ratio of the magnetic pressure to the thermal one is constant. In the first model, the density distribution becomes identical to that of nonmagnetic isothermal cylinder as
ρ(r) = ρ c 1 + r 2 8 4πGρ c c 2 s 2 , (2.3)
where ρ c means the density at the center of the cylinder (r = 0). As easily seen, the density reaches zero only at the infinity (r = ∞). Thus the isothermal cloud should be bounded by the external pressure, p ext . The cloud has a boundary where ρ(r s ) is equal to p ext /c 2 s . The cloud radius is expressed as
r s = 2 3/2 ρ c ρ s 1/2 − 1 1/2 c s (4πGρ c ) 1/2 . (2.4)
where ρ s is the density on the cloud surface and equals to p ext /c 2 s . In the present paper, we use the normalization as c s = 4πG = p ext = 1. Thus the unit of the distance is chosen as H = c s /(4πGρ s ) 1/2 . The normalized density distribution, f (r) ≡ ρ(r)/ρ s , and the radius of the surface, ξ s ≡ r s /H, are expressed as
f (ξ) = F 1 + F 8 ξ 2 2 , (2.5) ξ s = 2 3 F 1/2 F 1/2 − 1 1/2 , (2.6)
where ξ ≡ r/H represents the normalized distance and F = ρ c /ρ s denotes the density contrast between the center and the surface. When the magnetic field plays a role in supporting the cloud, the density distribution is different from equation(2.3). Assuming that the magnetic pressure is proportional to the thermal pressure, i.e., B 2 z /8π ∝ c 2 s ρ, the distribution of the density becomes as
f (ξ) = F 1 + F 8 ξ 2 1 + α/2 2 , (2.7)
where the parameter α is defined as
α ≡ B 2 z /4π c 2 s ρ ≡ 2 β , (2.8)
in terms of the plasma β. This shows that the cloud becomes thick with increasing α, and the density scale-height changes in proportion to (1 + α/2) 1/2 . Typical density distributions are shown in Figure 1.
We assumed a hypothetical situation that the cloud is confined in a low-density ambient medium which has no importance as the source of gravity but has a finite pressure p ext . Thus, the initial solution of ρ(r) has a boundary where ρ(r) = p ext /c 2 s . Beyond this cloud boundary, a tenuous and thus hot medium is assumed to extend. However, as seen in the next subsection, since we assume the isothermal equation of state, it is difficult to calculate two kinds of gases, cold cloud component and warm intercloud component, separately. Therefore, we adopt a one-fluid approximation, where the gas has an identical temperature; but the ambient gas is defined as the portion with ρ < p ext /c 2 s ≡ ρ s , and this gas has no effect of the gravitational field made by the cloud. This gives a virtual distribution of ambient gas as
f (ξ) = ξ ξ s − 4(F 1/2 −1) F 1/2 , (2.9)
to counter-balance the gravity by the cloud as
g(ξ) = ∂ ln ρ ∂ξ = − 4 ξ F 1/2 − 1 F 1/2 .
(2.10)
Basic Equations
The cylindrical symmetry is assumed: ∂/∂φ = 0. For the dense gas found in interstellar clouds with ≃ 10K, the equation of state is well approximated with the isothermal one. We assume here that the gas obeys the isothermal equation of state. Thus the basic equations become the equations of magnetohydrodynamics for isothermal gases as
∂ρ ∂t + ∂ ∂z (ρv z ) + 1 r ∂ ∂r (rρv r ) = 0, (2.11) ∂ρv z ∂t + ∂ ∂z (ρv z v z ) + 1 r ∂ ∂r (rρv z v r ) = −c 2 s ∂ρ ∂z − ρ ∂ψ ∂z − 1 4π ∂B r ∂z − ∂B z ∂r B r , (2.12) ∂ρv r ∂t + ∂ ∂z (ρv r v z ) + 1 r ∂ ∂r (rρv r v r ) = −c 2 s ∂ρ ∂r − ρ ∂ψ ∂r + 1 4π ∂B r ∂z − ∂B z ∂r B z , (2.13) ∂B z ∂t = 1 r ∂ ∂r [r(v z B r − v r B z )], (2.14) ∂B r ∂t = − ∂ ∂z (v z B r − v r B z ), (2.15) ∂ 2 ψ ∂z 2 + 1 r ∂ ∂r r ∂ψ ∂r = 4πGρ, (2.16)
where the variables have their ordinary meanings.
To include only the gas density in the cloud as the source of the self-gravity, the right-hand side of the equation (2.16) is calculated as follows:
ρ(z, r) = ρ, if ρ > ρ s , 0, if ρ < ρ s . (2.17)
As long as the cloud keeps its hydrostatic configuration, the solution of the above equations coincides with the initial configuration, equations (2.5) and (2.9). However, if the ambient matter accretes onto the cloud by the gravity or the cloud material evaporates by the pressure force, the above procedure may give wrong results. We monitored the mass of the cloud in numerical runs, and check the validity of the ambient matter condition. We confirmed that the cloud mass only changed less than ∼ 3%. Nondimensional variables are used as follows: the density, ρ ′ ≡ ρ/ρ s , the velocity, v ′ ≡ v/c s , the gravitational potential, ψ ′ ≡ ψ/c 2 s , the magnetic fields, B ′ ≡ B/(4πc 2 s ρ s ) 1/2 , the time, t ′ ≡ t(4πGρ s ) 1/2 , and the linear size, r ′ ≡ r/H. Using this normalization, equations (2.12), (2.13), and (2.2) become
∂ρ ′ v ′ z ∂t ′ + ∂ ∂z ′ (ρ ′ v ′ z v ′ z ) + 1 r ′ ∂ ∂r ′ (r ′ ρv ′ z v ′ r ) = − ∂ρ ′ ∂z ′ − ρ ′ ∂ψ ′ ∂z ′ − ∂B ′ r ∂z ′ − ∂B ′ z ∂r ′ B ′ r , (2.18) ∂ρ ′ v ′ r ∂t ′ + ∂ ∂z ′ (ρ ′ v ′ r v ′ z ) + 1 r ′ ∂ ∂r ′ (r ′ ρ ′ v ′ r v ′ r ) = − ∂ρ ′ ∂r ′ − ρ ′ ∂ψ ′ ∂r ′ + ∂B ′ r ∂z ′ − ∂B ′ z ∂r ′ B ′ z , (2.19) ∂ 2 ψ ′ ∂z ′2 + 1 r ′ ∂ ∂r ′ r ′ ∂ψ ′ ∂r ′ = ρ ′ .
(2.20)
Other equations are identical even when these normalized variables are used. Hereafter, we will use these nondimensional variables and abbreviate the prime unless otherwise mentioned.
Numerical Method
The basic equations are solved by the finite difference method. The mesh spacing (∆z, ∆r) is constant spatially except for models CB and CB2 in table 1. Unequal spacing is used for model CB and CB2 to see the fragment more closely. Cell numbers in one dimension are chosen N = 200 − 800 (table 1). The "Monotonic Scheme" (van Leer 1977;Norman & Winkler 1986) is adopted to solve the hydrodynamical equations, i.e., equations (2.11)-(2.13), and the "Constrained Transport" method (Evans & Hawley 1988) is adopted to solve the induction equations of the magnetic field, i.e., equations (2.14) and (2.15). Although the equation of state is different, this scheme is essentially the same as that used by Tomisaka (1992), For numerical stability at the shock front, an artificial viscosity is included in equations (2.12) and (2.13). A tensor artificial viscosity term is used as shown in the Appendix of Tomisaka(1992). The program for adiabatic gas was checked by several test problems: Sedov solution, spherical stellar wind bubble solution (Weaver et al. 1977), one-dimensional magnetic Rieman problem (Brio & Wu 1988). Further, the contraction of a spherical isothermal cloud is compared with solutions of Larson (1969) and Penston (1969). The radial density distribution is well fitted by ∝ r −2 and derived velocity profiles agree with their results by one-dimensional calculation.
To solve the Poisson equation is reduced to find the solution of simultaneous linear equations with a dimension of N 2 . The linear equations are solved by the "Conjugate Gradient Squared method" (CGS method, Dongarra et al. 1991) preconditioned by the modified incomplete LU decomposition (MILUCGS) (Meijerink & van der Vorst 1977;Gustafsson 1978). A brief description of this method and its performance are described in Appendix.
On the upper (z = l z ) and lower (z = 0) boundary, the cyclic boundary condition is applied. On the outer boundary the fixed boundary condition is applied.
Result
A Typical Evolution
First, the numerical result without any initial perturbation is seen. This corresponds to the model with α = 1 and F ≡ c 2 s ρ c /p ext = ρ c /ρ s = 100 (model A). The size of the calculated region is taken as 1.93H, which coincides with the wavelength which has the maximum growth rate in the gravitational instability (Nakamura, Hanawa, & Nakano 1993). Figure 2 shows the evolutions of density (left panel) and magnetic fields (right panel). In t = 3.6, no prominent fragmentation appears. In the stage shown in Figure 2c, a high-density region elongating in the z-direction appears. The shape of this high-density region is prolate, i.e., if it fits to an ellipsoid the major axis coincides with the symmetric axis (the z-axis). This agrees with the result of linear analyses (Nakamura et al. 1993;Nagasawa 1987); The wavelength of the most unstable perturbation was predicted from the linear analysis as ≃ 2 × H. And the global shape of the high-density region in Figure 2c coincides with an expected shape from the eigen function of the linear analysis. From this figure, it is confirmed that the perturbation with a shorter wavelength than that of the most unstable mode does not appear at all, before the most unstable mode grows into the non-linear region. However, a perturbation with a longer wavelength than the numerical box can not be properly calculated. We have calculated in model A2 (not shown) the same problem with a larger spacing and thus with a numerical box twice as large as model A. In model A2, two high-density regions appear and each fragment shows the identical evolution as model A, which confirms the most unstable mode expected from the linear analysis grows first.
After the stage in Figure 2c, collapse along the symmetry axis proceeds as well as the radial direction. The final structure is a disk contracting towards the center seen in Figure 2d. Since the gas easily falls down along the magnetic field lines, this disk is perpendicular to the symmetric axis of the cylinder. The magnetic fields are squeezed by the effect of radial contraction near the disk (right panel). Since this configuration is unstable against the Parker instability (1979 and references therein), the gas is accumulated flowing along the magnetic fields. Contracting speed in the zdirection is ≃ twice as fast as the radially contracting speed. In the central region, a high-density core is formed, on to which accretion of extra mass continues. Finally, near the high-density core the infalling speed exceeds the isothermal sound speed at this stage. It is concluded that a typical product of the self-gravitational instability in the cylindrical cloud is oblate spheroidal disks separated by λ max .
The phase of the position where the disk is formed (z c ) is not determined a priori. This is related to the question that where the irregularity comes from even if the initial state is uniform along the z-axis. After several numerical tests, it is concluded that the main contribution of irregularity is coming from the Poisson solver; Even if the density distribution, ρ, is uniform in the z-direction, the numerical values of the potential, ψ, is not completely uniform. This is due to the CGS scheme. Although the typical relative error in the potential is small, ∆ψ/ψ ∼ 10 −7 , this irregularity grows by the effect of gravitational instability and finally it forms fragmentations.
To confirm the idea, another model B is calculated, whose parameters are identical with model A except for the existence of initial irregularity. In model B, a perturbation with relative contrast δ ini ≡ ∆ρ/ρ = 10 −2 is added as
ρ(z, r) = ρ 0 (r) 1 − δ ini cos 2πz l z , (3.1)
where ρ 0 and l z represent, respectively, the initial density determined by f (ξ) of equation(2.7) and the z-length of numerical box. The wavelength of this perturbation is again taken identical with that of the most unstable mode in the linear analysis. The structure at t = 1.26 is shown in Figure 3. This shows that the final structure of this model is identical with that of the model A (Fig.2d), although the time necessary for the fragment to grow is shortened much (from t = 5.6 to t = 1.26). It is clear that this quick evolution is due to a finite amplitude of the perturbation added to the initial state, compared with the intrinsic irregularity from the Poisson solver in model A. However, the evolution itself is very similar with each other. To confirm this, the time evolution of the amplitude of the density irregularity is plotted for both models A and B in Figure 4. Using the average < ρ(r = 0) > and the maximum ρ(r = 0)| max density on the z-axis, the relative density contrast
δ max ≡ ρ(r = 0)| max < ρ(r = 0) > − 1, (3.2)
is plotted against the time passed after the calculation begins. This shows that in model A the perturbation grows in accordance with ∝ exp [ωt] for 1 < ∼ t < ∼ 5, from 10 −8 to 0.1. Model B shows the similar growth but the region in which it shows the exponential growth is much restricted. However, if the origin of time for model B is moved, two curves for models A and B coincides. This indicates that the evolution is similar whether the most unstable mode is overlaid on to the initial density distribution (model B) or not (model A). This confirms the expectation that the most unstable mode grows selectively even from the irregularity of white noise. Thus, we add the perturbation of the expected most unstable mode to the initial density distribution as model B.
Non-Magnetic Cylinder
To see the effect of the magnetic field, we calculated the non-magnetic isothermal cylinder as model C.
From the linear analysis, the wavelength of the most unstable mode is equal to ≃ 2.21H (Nakamura et al. 1993). Since the magnetic field has an effect of supporting the cloud, the radial size of the cloud becomes more compact than that of models A and B. In Figure 5, the evolution of model C is shown. In t = 1.2, the fragment, whose density contrast δ reaches ∼ 3, has entered in non-linear region (Fig.5a). However, comparing with Figure 3 (these two models are assumed to have irregularities with the same initial amplitude), it is shown that this model evolves slower than model B. This difference is understood as follows: (1) since the magnetic fields of model B has a configuration unstable for the Parker instability, the system becomes more unstable including the magnetic fields, or (2) the mass per unit length of the cylinder and thus the effect of the self-gravity becomes larger when the magnetic fields are supporting the cloud. Figure 5b shows the structure at the age of t = 1.4. In this phase, the maximum density at the center of the core reaches 10 5 . The envelope with low-density ≃ 10 1.5 seems to have the shape of prolate spheroid as well as the magnetic models. In contrast, the high-density central region appears to be almost spherical symmetric. The difference between structures appearing in the magnetic cloud and the non-magnetic one is apparent. Since there is no lateral restoring force by the magnetic fields, the high density fragment shows spherical shape, while in the magnetic cloud the fragment is formed from the matter flowing along the magnetic fields as a disk perpendicular to the magnetic fields.
Central Core
To see the structure of the central core, the density profile is plotted against log |z − z c | and log r in Figure 6 for models B and C, where z c represents the position of the core center.
There exist two characteristic power law solutions. One is a singular isothermal sphere solution (Chandrasekhar 1939):
ρ sing = c 2 s 2πGr 2 ,(3.3)
or in a non-dimensional form: ρ sing ρ s = 2 (r/H) 2 .
(3.4)
The other is an asymptotic solution of equation(2.5) for ξ ≫ 1 or r ≫ H:
ρ out = ρ c r 4 8 2 (4πGρ c ) 2 c 2 s , (3.5)
or in a non-dimensional form:
ρ out ρ s = 1 F 8 ξ 2 2 .
(3.6)
As seen in Figure 6, the distribution in the r-direction is well fitted by equation (3.6) for the outer envelope. As collapse proceeds, the radial density distribution in the inner part of the cloud becomes similar to that of the singular isothermal sphere [equation (3.
3)].
As for the z-distribution, in the non-magnetic cloud (model C), the density distribution is very similar to that in the r-direction. However, in the magnetic cloud (model B), the r-and z-distributions are much different. This difference comes from the effect of magnetic fields, that is, in models A and B a disk is finally formed, although in model C an almost spherical core is formed. Since the gravities by a disk and a sphere are quit different, the resultant density distributions are different.
When the spherical cloud has the density distribution like equation (3.3), the total mass inside of the sphere R is written as
M (R) = R 0 ρ sing 4πr 2 dr = 2c 2 s G R, = 2 π 1/2 c 4 s [ρ sing (R)c 2 s ] 1/2 G 3/2 , (3.7)
using the density ρ(R) at r = R. This represents the Jeans mass for the spherical cloud with infinite center-to-surface density contrast, M J (R). Thus, when the density increases more rapid than equation (3.3) toward the center, the actual mass is larger than this Jeans mass and contraction never stops. In contrast, when the density distribution is flat, the actual mass M (r) never surpasses M J (r) in any region of r < R. Considering this, since the radial density distribution ρ > ∼ ρ sing , the spherical core formed in the center of the non-magnetic cloud seems to continue to contract. From the studies on magnetohydrostatic equilibrium (Tomisaka et al. 1988), the mass-to-flux ratio at the center of the cloud, M/(Φ/G 1/2 ), has a crucial role to divide the clouds into super-and subcritical clouds; that is, considering a cloud whose mass is larger than the non-magnetic critical mass ( > ∼ M cr of eq.[1.2]), when the mass-to-flux ratio is larger than ≃ 1/(2π) the cloud has no equilibrium solutions; while the cloud has at least one hydrostatic solution, when the ratio is smaller than this value. Since the mass-to-flux ratio remains constant along the contraction, the mass-to-flux ratio in the fragment can be written using quantities in the initial stage as
M Φ/G 1/2 = ρ c l z G 1/2 B c = ρ 1/2 c l z G 1/2 (4πα) 1/2 c s , (3.8)
where l z represents the z-length of the numerical box and is taken equal to the wavelength of the most unstable perturbation, ≃ 20c s /(4πGρ c ) 1/2 . This ratio is approximately equal to
M Φ/G 1/2 ≃ 1.59 α 1/2 ,(3.9)
Thus, for the cloud with the mass appreciably larger than that of equation (1.2), only when the magnetic field is extremely strong as α > ∼ 100, the magnetized fragment may be subcritical. The contraction of the core can not be stopped by a magnetic field, which is assumed ordinarily, α ≃ 1. This is valid until the wavelength of the most unstable perturbation is much longer than 20c s /(4πGρ c ) 1/2 in a cloud with low density contrast F . In conclusion, the cores formed in the fragment of the cylindrical cloud seems to continue to contract, until other effects not considered here may work in supporting the fragment, such as, the equation of state of the gas changes from the isothermal to more hard one.
Diffuse Cloud
Here, we compare the cloud with high central density and that with low density contrast calculated in models D. In Figure 7a, the evolution of model DB is plotted. Both models B and DB have the same parameters except for the center-to-surface density ratio, F . Difference between models B and DB are apparent: although a disk is formed for both models by the effect of the magnetic field, the disk perpendicular to the magnetic fields of model DB is much larger than that of model B. The disk has a radial size of 0.16 (model B) and 0.71 (model DB) (the size is measured as a radius of the largest contour line which has an oblate shape).
The radial extent of the finally formed disk seems to correspond to the size of the initial central cylindrical core, i.e., in equation (2.3) the initial density is almost uniform irrespective of the radial distance for r < ∼ r c = 2 1.5 (1 + α) 1/2 c s /(4πGρ c ) 1/2 . Since this gives
r c ≃ 0.4 F 100 −1/2 1 + α 2 1/2 c s (4πGρ s ) 1/2 , (3.10)
the above result shows that a disk threaded by magnetic fields whose radial extent is equal to ∼ 40 -55 % of the initial core size is formed. If this is true, does the cloud with a large α form a large disk? Yes. In model DD with α = 4 (Fig. 7b), the disk with a radial size of ∼ 1.2 is formed, while r c ≃ 2(F/10) 1/2 [(1 + α)/5] 1/2 . Thus, it can be concluded that in the magnetized cylindrical cloud a disk is formed perpendicularly threaded by the magnetic fields and its radial extent is related to r c .
The next characteristic point of the evolution of diffuse clouds is a low growth rate of fragmentation. A typical time scale for the perturbation to evolve into the nonlinear stage is t ∼ 4.72 for α = 0 (model DC), t ∼ 4.12 for α = 1 (model DB), and t ∼ 3.31 for α = 4 (model DD). This is much longer than that of models B and C with F = 100. This apparently comes from the fact that the growth time-scale is proportional to the free-fall time for the central density. Finally, comparing the non-magnetic clouds of models C and DC, Figures 4 (model C) and 7c (model DC) look very similar. These figures are scaled in proportional to l z , and l z is chosen as λ max ≃ 20c s /(4πGρ c ) 1/2 . This means that the actual size of the spherical core of model DC is just 10 1/2 times larger than that of model C. If we scale the size by c s /(4πGρ c ) 1/2 instead of H = c s /(4πGρ s ) 1/2 , the structures of these models becomes almost identical. Therefore, the structure of the fragment is mainly determined by the initial highest density at the center of the cylinder cloud for the non-magnetic cloud.
Further Evolution
To see the structure of the contacting central core in more detail, we have to execute larger calculation with finer resolution using a plenty of meshes. However, this approach will meet an inevitable limit of available memories on any computers. Therefore, we avoid this difficulty by varying mesh sizes, as a finer mesh is used around the expected position where the fragment is formed and a coarser mesh is used far from the cloud. We apply so-called log-mesh: the size of the grid is chosen like ∆z i = F z ∆z i−1 for i > i c , ∆z i = F z ∆z i+1 for i < i c , and ∆r j = F r ∆r j−1 (F z , F r > 1), where i and j represent the sequential numbers of the cell in the z-direction and in the r-direction, respectively, and subscript c means the the midpoint of the z-axis. Models CB and CB2 have the same physical parameters as model B but they have ≃ 2 -4 times closer spatial resolution than model B, respectively. Although these two models use different initial amplitude of perturbations (table 1), qualitative evolutions are similar with each other.
Due to a finer resolution, the evolution can be traced till the maximum density reaches ≃ 10 6 = 10 4 × initial central density. In Figure 8, we show the structure at t = 0.705 for model CB2 (due to a larger amplitude of δ ini = 10 −1 in model CB2, irregularities grow faster than models CB and B. This time scale is equivalent to t = 1.26 for models in which smaller initial perturbation is assumed as δ ini = 10 −2 ). As shown in Figure 8c and d, collapse speed becomes faster as reaching the fragment center. And the maximum falling speed increases as collapse proceeds. From Figure 8c, at that time, the maximum infall velocity in the z-direction agrees with that observed in the asymptotic solution for the spherical isothermal collapse by Larson(1969) and Penston(1969), 3.28c s . It is shown that high-density region with a thickness of ∼ 10 −2 , i.e., |z − z c | < ∼ 10 −2 H = 10 −1 c s /(4πGρ c init ) 1/2 , is formed, in which contraction speed is decreased as reaching z c . In the r-direction, in contrast, very smooth distributions of density, magnetic field strength, and velocity are seen. This numerical result qualitatively agrees with the calculation by Hanawa, Nakamura, & Nakano (private communication) using a different numerical scheme.
Thin dense object which is contracting slowly now seems to be separated from the outer inflow region seen in |z − z c | > ∼ 10 −2 . The fraction of mass which form this relatively static high-density disk is estimated as 6% -10 %. So, if stars are formed from the relatively static matter, the ratio of the mass of newly formed protostars to that of a parent cloud becomes as 6 % -10 %.
Therefore, the next evolution stage after this simulation ends seems to be a proto-stellar system consisting of a proto-star and an accreting matter onto the proto-star, whose luminosity comes from the liberated gravitational energy of falling material.
Discussion
A Model with Uniform Magnetic Fields
In the preceding section, we investigated the models that the cloud is supported by magnetic fields. In contrast, here, we study the model with uniform magnetic field and compare the structure of fragments. Instead of B 2 z /8π ∝ c 2 s ρ, we assume that the strength of B z is constant. Radial density distribution is given by equation (2.3). Relative strength of the magnetic fields is expressed by a parameter α c which is equivalent to the ratio of B 2 z /4π to c 2 s ρ at the center of the cloud. In Figure 9, the structure at t = 1.74 is shown. This figure shows several differences with Figures 1 and 2 (constant α model). (i) the cloud outer boundary: the distance from the z-axis increases near z ≃ l z /2, and decreases far from the fragment. This seems to come from the fact that the magnetic fields becomes more important far from the z-axis and the gas moves toward z ≃ l z /2 without any effects on the magnetic fields (Fig.9b). As a result, the density contours indicates a convex shape. Contrarily, in models with constant α, the gas falls down along the valley of magnetic fields. Thus, the cylindrical tube of the cloud shrinks near z ≃ l z /2 and inflates near z ≃ 0 and z ≃ l z . (ii) the magnetic field line: as seen in Figure 9b the field line breaks weaker than previous models. This seems to come from the fact that the configuration is not unstable against the Parker instability, that is, the magnetic valley does not necessarily promote the instability in this case. That is, there is no positive feedback. The second factor is: since the ratio of magnetic pressure to the thermal one, α/2, increases outwardly, the magnetic field becomes relatively important for the dynamics with increasing r, even if α c is taken ≃ 1. Thus the field line is almost locked in the initial configuration.
We studied two models for the magnetic field configuration, α =constant and B z =constant, which correspond two extremes which are realized in the real interstellar space.
Classification of the Cylindrical Cloud
How does a cylindrical cloud with a finite line density λ and a magnetic field strength evolve? In the preceding section, it is shown that even in a cloud in the hydrostatic equilibrium, fragments are formed and dense part of the fragment falls into a runaway collapse. From equation (2.7), the line density of a cloud with F = ρ c /ρ s , is written as
λ(F ) = 2c 2 s G F 1/2 − 1 F 1/2 1 + α 2 . (4.1)
This leads to the maximum line density which can be supported by the thermal and magnetic pressure as
λ max = 2c 2 s G 1 + α 2 , for F → ∞,(4.2)
which include the critical line density: equation(1.1). Therefore, if the cloud has a line density smaller than λ max , which is related to the isothermal sound speed, c s , and the Alfvén speed, α 1/2 c s , the cloud has a magnetohydrostatic equilibrium. In such a cloud, irregularity grows in several ×τ max depending on the amplitude of irregularities.
Adopting 3τ max as a typical growth time, this gives τ grow ≃ 2.4 × 10 6 yr F 100
−1/2 ρ s 2 × 10 −22 g cm −3 −1/2 . (4.3)
Final phase of the contraction, that is, the phase when the maximum density in the high-density core increases rapidly, continues only in a short duration ∼ 0.1 × τ grow . On the other hand, a cloud with super-critical line density λ > λ max does not have any magnetohydrostatic configuration. Thus, the cloud contracts as a whole, and forms a thin cloud like a string. Recently, hydrodynamical simulation has been done for such a massive contracting cylindrical clouds by Inutsuka and Miyama (private communication) for non-magnetic clouds. They found that in such clouds, the irregularity grows relatively slowly compared with the contraction time-scale ≃ free-fall time-scale and the cloud does not fragment but forms a string as long as the gas obeys the isothermal equation of state.
Comparing these two, it is expected that super-and subcritical cylindrical cloud show completely different evolutions. This may be related to the mass function of the new-born stars.
Summary
We studied the process of fragmentation in an isothermal cylindrical cloud with infinite length. By a numerical magnetohydrodynamical method, the evolution of the fragment was investigated from the linear stage to nonlinear stage throughout. It is shown that the fastest growing perturbation has the same wavelength as predicted with the linear theory. In a linear stage the fragment appears as a prolate spheroidal shape. However, in a nonlinear stage, the fragment threaded by the magnetic field forms a disk perpendicular to the field line. At the center of the fragment there forms a high-density core, which continues to collapse. Non-magnetized cloud only forms collapsing spherical cores which are separated by λ max . For magnetic clouds, it is shown that when the collapsing velocity reaches ∼ 3.5c s , relatively slowly contracting dense inner part is formed in the contracting disk, which also extends perpendicular to the magnetic fields. This disk is in almost static and is separated from the accretion flow which is accelerated as reaching the center of the core.
I would like to thank T. Hanawa (Nagoya University) for stimulating discussion. We compared the numerical results with different schemes and could confirm the reliability of the schemes and the physics. I also thank S. Inutsuka, S. M. Miyama (National Astronomical Observatory) and M. Y. Fujimoto (Niigata University) for useful discussions. Numerical calculations were carried out by supercomputers: Hitac S-820/80's (Hokkaido University and University of Tokyo), Facom VP200 (Institute of Space and Aeronautical Sciences). This work was supported in part by Grants-in-Aid for Science Research from the Ministry of Education, Science, and Culture (04233211 and 05217208). Tables TABLE 1: a In a series of models with character D, diffuse clouds are considered. b Spatially varying spacing grid is used. The physical parameters are taken as identical as model B. c Uniform magnetic fields are assumed. In this model, α represents the value at the center of the cloud (r = 0), that is α c .
A MILUCGS
Here, a brief description of MILUCGS and its performance are described. The finite difference scheme for the Poisson equation is written
ψ 1 ψ 2 ψ 3 . . . . . . . . . . . . ψ nm = ρ 1 ρ 2 ρ 3 . . . . . . . . . . . . ρ nm , (A.1)
where two-dimensional expression of ψ i,j and ρ i,j (1 ≤ i ≤ m and 1 ≤ j ≤ n) are converted to a one-dimensional expression ψ k and ρ k with k = (j − 1) × m + i. m and n represent, respectively, the cell numbers in the z-and r-direction. The coefficients db and ub correspond to the derivatives with respect to the z-direction, and the coefficients of dc and uc correspond to those of r-direction.
If the problem is a simple Neuman or Dirichret boundary condition, the matrix to be solved is like equation (A.1). However, since we apply here a cyclic boundary in the z-direction, we have to add two coefficients expressing this boundary condition. That is, the term of ψ m,j = ψ jm should connect with ψ m−1,j = ψ jm−1 and ψ 1,j = ψ jm−m+1 instead of ψ jm+1 . The equation for cyclic boundary becomes
ψ 1 ψ 2 . . . . . . ψ m+1 . . . . . . ψ nm = ρ 1 ρ 2 . . . . . . ρ m+1 . . . . . . ρ nm , (A.2)
where only dd j×m as well as ud (j−1)m+1 has non-zero components and the others have the value of zero. This linear simultaneous equation is solved by conjugate gradient squared method (Dongarra et al. 1991). If we write equation(A.2) as Aψ = ρ, the algorithm is as follows:
1. Prepare the initial guessψ 0 r 0 = ρ − Aψ 0 p 0 = e 0 =r 0 l = 0 2. while r l > ǫ ρ do (a) α l = (r 0 ,r l )/(r 0 , Ap l )
(b) h l+1 = e l − α l Ap l (c)r l+1 =r l − α l A(e l + h l+1 ) (d)ψ l+1 =x l + α l (e l + h l+1 ) (e) β l = (r 0 ,r l+1 )/(r 0 ,r l ) (f) e l+1 =r l+1 + β l h l+1 (g)p l+1 = e l+1 + β(h l+1 + β lpl ) (h) l = l + 1.
The convergence of this method is improved much by preconditioning. In the present paper, we apply the incomplete LU-decomposition. The incomplete LU-decomposition is a method that matrix A is decomposed as LDU − R, where L, D, U , and R are a lower triangle matrix, a diagonal matrix, an upper triangle matrix, and a residual matrix. Instead of Aψ = ρ, we apply the CGS method to the reduced equation of (LDU ) −1 Aψ = (LDU ) −1 ρ.
(A.3)
Since (LDU ) −1 A is nearly equal to the identity matrix I and the eigen values of this matrix gather around unity, the convergence of the CGS method is much accelerated. The way to decompose A = LDU − R is arbitrary. Here, we use the method by Meijerink & van der Vorst (1977) and Gustafsson (1978). Describing briefly, L, D, and U are chosen as
L = , (A.4) D = d 1d 2 0 . . . . . . . . . . . . 0d mn−1d mn , (A.5) U = . . . . (A.6)
If we choose the diagonal element of D as
d −1 = a k − db kdk−1 ub k−1 − dc kdk−m uc k−m − dd kdk−m+1 ud k−m+1 , (A.7)
the diagonal part of the matrix A becomes equal to that of LDU . However, when the matrix LDU is operated onψ, it generates extra terms (fill-in) such as
db kdk−1 uc k−1ψk+m−1 + dc kdk−m ub k−mψk−m+1 (A.8)
as well as the ordinary terms:
a kψk + ub kψk+1 + db kψk−1 + uc kψk+m + dc kψk−m + ud kψk+m−1 + dd kψk−m+1 . (A.9)
To suppress the effect of these extra terms, the diagonal termd is modified as
d −1 = a k − db kdk−1 ub k−1 − dc kdk−m uc k−m − dd kdk−m+1 ud k−m+1 − γ(db kdk−1 uc k−1 + dc kdk−m ub k−m )
, (A.10) where the factor γ is chosen as the way by Gustafsson (1978). Using the LDU-decomposition, in the CGS algorithm the product of Av is replaced with (LDU ) −1 Av. The operation multiplying (LDU ) −1 is done as follows: y = (LDU ) −1 x is equivalent to x = (LDU )y, i.e.,
Lz = x (A.11) DU y = z. (A.12)
The equation(A.11) is solved as: for k = 1, 2, · · · , mn
z k =d k (x k − db k z k−1 − dc k x k−m − dd k x k−m+1 ), (A.13)
and then equation (A.12) is solved as: for k = mn, mn − 1, · · · , 1 y k = z k −d k (ub k y k+1 + uc k y k+m + ud k y k+m−1 ). (A.14)
Necessary CPU time is proportional to the cell number, mn, and to loop count. A numerical experiment was done and we compared CGS with MILUCGS. The adopted test problem is as follows: the gravitational potential for a uniform cylinder is solved with square grids of m = n = N . The same problem was solved by a supercomputer Hitachi S-820/80, a general purpose main frame machine Hitachi M-682H, and a workstation Sun Sparc IPX. The allowable maximum relative error is chosen as ǫ = 10 −7 and calculations are done using 64 bits. As a result, about 80 × (N/200) cycles are necessary to achieve a solution for MILUCGS, while the simple CGS requires 410 × (N/200). Since the number of cycles necessary for convergence is proportional to N , CPU time is proportional to N 3 irrespective with the preconditioning or not.
In table A1, we summarize the CPU time necessary for solving the Poisson equation by respective machines. Comparing CPU time, the preconditioning reduces the CPU time about < ∼ 1/3 for scalar machines (M-682 and Sparc IPX). Since the amount of operations in MILUCGS is larger than that of CGS, the factor is not equal to ≃ 1/5 (the ratio of necessary loops) but to < ∼ 1/3. However, this clearly shows the advantage of MILUCGS over the simple CGS. The difference for the supercomputer is relatively small, for in order to vectorize the operation of multiplying (LDU ) −1 [equations (A.11, A.12)] a more complicated algorithm with list vectors is necessary. The initial radial density distributions. A solid line corresponds to models A and B, i.e., a magnetized cloud with α = 1. A dashed line shows the density of model C which has no magnetic fields α = 0. Both correspond to the models with the center-to-surface density ratio, F = 100. The dotted part of the curves represents the ambient medium f < 1 which is assumed to have no effect as a source of gravity. Fig.2: The time evolutions of density (left panel) and magnetic fields (right panel) for model A. The z-axis and r-axis are directed horizontally and vertically, respectively, in contrast to an ordinary fashion. The number attached to the density contour lines represent the logarithm of the density: log 10 ρ. The step of the contour lines is taken constant = 0.25. Four snapshots, i.e., t = 2.4 (a), t = 3.6 (b), t = 4.9 (c), and t = 5.6 (d) are shown. In t = 3.6, no prominent fragmentations appear. In the stage shown in (c), a high-density region elongating in the z-direction appears. After (c), contraction along the symmetry axis proceeds. The final structure is a disk contracting towards the center seen in (d). The disk is perpendicular to the symmetric axis of the cylinder. . It is shown that a fragment contracts and forms a disk perpendicular to the symmetric axis. Cross-cut views along the z-axis (c) and the r-axis (d) are plotted. In panel c, a logarithmic plot of ρ(z, 0) and B z (z, 0) and a linear plot of v z (z, 0)/10 are illustrated. Similarly, in panel d, a logarithmic plot of ρ(z c , r) and B z (z c , r) and a linear plot of v r (z c , r)/10 are shown, where z c represents the position of the center of the fragment = l z /2. This snapshot corresponds to the state of t = 1.26. Fig.4: Time evolutions of the relative density enhancement, δ max , for models A and B. This is defined as δ max ≡ ρ(r = 0)| max / < ρ(r = 0) > −1. The curve for model B is similar to the late phase (t > ∼ 4) of the model A. This shows that the structure and evolution of the fragment can be calculated from the initial state which has the most unstable perturbation with a finite amplitude. Fig.5: The time evolution of density for Model C. In this model, the fragmentation in non-magnetic cloud is studied. Since the magnetic field, which has the effect of supporting cloud laterally, is not included, the radial extent of the cloud is thinner than that of models A and B. Two snapshots of density distribution at t = 1.2 (a) and t = 1.4 (b) are plotted. Fig.6: The cross-cut views of the cloud of models B (a) and C (b). The left panel shows the variation of ρ(z, 0) against log |z − z c |, and the right panel shows that of ρ(z c , r) against log r. Two characteristic solutions [(3.4), and (3.6)] are also shown in right panels. Five snapshots at t = 0.3, 0.6, 0.9, 1.2, 1.4 are plotted. Fig.7: The density distributions of models DB (a), DD (b), and DC (c). Model DB (α = 1) has the same parameter as model B except for the density contrast between the center and the surface, F = 10. Model DD corresponds to the cloud with stronger magnetic fields (α = 4). Model DC (α = 0) has the same parameter as model C except for the density contrast between the center and the surface, F = 10. For panel c, comparing how the density contour lines are running with that for Fig.5, it can be seen that the density distributions are very similar with each other. Since the length of numerical box in the z-direction l z is taken as proportional to ρ −1/2 c , it is seen that the fragments of models C and F have an almost similar structure, if the size is scaled with the "scale-height" at the center, c s /(4πGρ c ) 1/2 instead of H. The number attached to the contour line, k, indicates that the value of the contour is equal to ρ = 10 k/4 . Fig.8: The structure of the fragment of model CB2 at t = 0.705. Physical parameters of this model are chosen identical with model B but this model has 4.28 times higher spatial resolution than model B. In this model a perturbation with a larger amplitude δ ini = 10 −1 is assumed initially to reduce a computing time. This time corresponds to 1.26 for model CB with δ ini = 10 −2 . The panel a shows the density and velocity fields and b shows the magnetic field lines. To see the structure near the fragment clearly, in panel a, only a region of 0 < z < l z /2 and r < l z /2 is plotted. Velocity vectors are plotted every 16 grids. Panels c and d are respectively, the cross-cut views along the z-axis and r-axis. A logarithmic plot of ρ(z, 0) and B z (z, 0) and a linear plot of v z (z, 0) are illustrated against log |z − z c | in panel c. Similarly, in panel d, a logarithmic plot of ρ(z c , r) and B z (z c , r) and a linear plot of v r (z c , r) are shown against log r. Fig.9: The structure of the fragment formed in a cylindrical cloud threaded by a uniform magnetic field (model UB). The initial radial distribution of the density is as the same as model C (nonmagnetic model), since the magnetic fields do not play a role in supporting the cloud. The density contour (a) and the magnetic fields (b) are illustrated. Panels c and d are, respectively, the crosscut views along the z-axis and the r-axis. That is, a logarithmic plot of ρ(z, 0) and B z (z, 0) and a linear plot of v z (z, 0) are illustrated in panel c. Similarly, in panel d, a logarithmic plot of ρ(z c , r) and B z (z c , r) and a linear plot of v r (z c , r) are shown. This snapshot corresponds to the state of t = 1.26.
Figure Captions
Fig. 1 :
1Fig.1: The initial radial density distributions. A solid line corresponds to models A and B, i.e., a magnetized cloud with α = 1. A dashed line shows the density of model C which has no magnetic fields α = 0. Both correspond to the models with the center-to-surface density ratio, F = 100. The dotted part of the curves represents the ambient medium f < 1 which is assumed to have no effect as a source of gravity. Fig.2: The time evolutions of density (left panel) and magnetic fields (right panel) for model A. The z-axis and r-axis are directed horizontally and vertically, respectively, in contrast to an ordinary fashion. The number attached to the density contour lines represent the logarithm of the density: log 10 ρ. The step of the contour lines is taken constant = 0.25. Four snapshots, i.e., t = 2.4 (a), t = 3.6 (b), t = 4.9 (c), and t = 5.6 (d) are shown. In t = 3.6, no prominent fragmentations appear. In the stage shown in (c), a high-density region elongating in the z-direction appears. After (c), contraction along the symmetry axis proceeds. The final structure is a disk contracting towards the center seen in (d). The disk is perpendicular to the symmetric axis of the cylinder. Fig.3: The structure of the cloud for model B. The upper-left panel shows the density distribution (a) and the upper-right does the magnetic field lines (b). It is shown that a fragment contracts and forms a disk perpendicular to the symmetric axis. Cross-cut views along the z-axis (c) and the r-axis (d) are plotted. In panel c, a logarithmic plot of ρ(z, 0) and B z (z, 0) and a linear plot of v z (z, 0)/10 are illustrated. Similarly, in panel d, a logarithmic plot of ρ(z c , r) and B z (z c , r) and a linear plot of v r (z c , r)/10 are shown, where z c represents the position of the center of the fragment = l z /2. This snapshot corresponds to the state of t = 1.26. Fig.4: Time evolutions of the relative density enhancement, δ max , for models A and B. This is defined as δ max ≡ ρ(r = 0)| max / < ρ(r = 0) > −1. The curve for model B is similar to the late phase (t > ∼ 4) of the model A. This shows that the structure and evolution of the fragment can be calculated from the initial state which has the most unstable perturbation with a finite amplitude. Fig.5: The time evolution of density for Model C. In this model, the fragmentation in non-magnetic cloud is studied. Since the magnetic field, which has the effect of supporting cloud laterally, is not included, the radial extent of the cloud is thinner than that of models A and B. Two snapshots of density distribution at t = 1.2 (a) and t = 1.4 (b) are plotted. Fig.6: The cross-cut views of the cloud of models B (a) and C (b). The left panel shows the variation of ρ(z, 0) against log |z − z c |, and the right panel shows that of ρ(z c , r) against log r. Two characteristic solutions [(3.4), and (3.6)] are also shown in right panels. Five snapshots at t = 0.3, 0.6, 0.9, 1.2, 1.4 are plotted. Fig.7: The density distributions of models DB (a), DD (b), and DC (c). Model DB (α = 1) has the same parameter as model B except for the density contrast between the center and the surface, F = 10. Model DD corresponds to the cloud with stronger magnetic fields (α = 4). Model DC (α = 0) has the same parameter as model C except for the density contrast between the center and the surface, F = 10. For panel c, comparing how the density contour lines are running with that for Fig.5, it can be seen that the density distributions are very similar with each other. Since the length of numerical box in the z-direction l z is taken as proportional to ρ
Fig. 3 :
3The structure of the cloud for model B. The upper-left panel shows the density distribution (a) and the upper-right does the magnetic field lines (b)
). The non-magnetic critical mass is equal toM c = 1.18
c 4
s
p
1/2
ext G 3/2
,
(1.2)
TABLE A1 :
A1Necessary CPU time to solve the Poisson equation once Workstation) . . . . . . . . . . . . . . . . . . . .Machine
CGS (sec) MILUCGS (sec) N
Sparc IPX (505
143
200
M-682H
(General Purpose Computer) . . . . . .
43.1
14.1
200
S-820/80
(Supercomputer) . . . . . . . . . . . . . . . . .
4.1
2.4
400
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| [] |
[
"A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue",
"A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue"
] | [
"M Menzel \nInstitute of Neuroscience and Medicine (INM-1)\n\n",
"K Michielsen \nJülich Supercomputing Centre\nForschungszentrum Jülich\nWilhelm-Johnen-Straße52425JülichGermany\n",
"H De Raedt \nDepartment of Applied Physics\nZernike Institute for Advanced Materials\nUniversity of Groningen\nNijenborgh 49747 AGGroningenThe Netherlands\n",
"J Reckfort \nInstitute of Neuroscience and Medicine (INM-1)\n\n",
"K Amunts \nInstitute of Neuroscience and Medicine (INM-1)\n\n\nCécile and Oskar Vogt Institute of Brain Research\nUniversity of Düsseldorf\n40204DüsseldorfGermany\n",
"M Axer \nInstitute of Neuroscience and Medicine (INM-1)\n\n"
] | [
"Institute of Neuroscience and Medicine (INM-1)\n",
"Jülich Supercomputing Centre\nForschungszentrum Jülich\nWilhelm-Johnen-Straße52425JülichGermany",
"Department of Applied Physics\nZernike Institute for Advanced Materials\nUniversity of Groningen\nNijenborgh 49747 AGGroningenThe Netherlands",
"Institute of Neuroscience and Medicine (INM-1)\n",
"Institute of Neuroscience and Medicine (INM-1)\n",
"Cécile and Oskar Vogt Institute of Brain Research\nUniversity of Düsseldorf\n40204DüsseldorfGermany",
"Institute of Neuroscience and Medicine (INM-1)\n"
] | [] | The neuroimaging technique three-dimensional polarized light imaging (3D-PLI) provides a high-resolution reconstruction of nerve fibres in human post-mortem brains. The orientations of the fibres are derived from birefringence measurements of histological brain sections assuming that the nerve fibres -consisting of an axon and a surrounding myelin sheath -are uniaxial birefringent and that the measured optic axis is oriented in direction of the nerve fibres (macroscopic model). Although experimental studies support this assumption, the molecular structure of the myelin sheath suggests that the birefringence of a nerve fibre can be described more precisely by multiple optic axes oriented radially around the fibre axis (microscopic model).In this paper, we compare the use of the macroscopic and the microscopic model for simulating 3D-PLI by means of the Jones matrix formalism. The simulations show that the macroscopic model ensures a reliable estimation of the fibre orientations as long as the polarimeter does not resolve structures smaller than the diameter of single fibres. In the case of fibre bundles, polarimeters with even higher resolutions can be used without losing reliability. When taking the myelin density into account, the derived fibre orientations are considerably improved. | 10.1098/rsif.2015.0734 | [
"https://arxiv.org/pdf/1506.02832v3.pdf"
] | 8,842,515 | 1506.02832 | b20dd13cdbd9e84b4b97f81200ea79539914d9aa |
A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue
M Menzel
Institute of Neuroscience and Medicine (INM-1)
K Michielsen
Jülich Supercomputing Centre
Forschungszentrum Jülich
Wilhelm-Johnen-Straße52425JülichGermany
H De Raedt
Department of Applied Physics
Zernike Institute for Advanced Materials
University of Groningen
Nijenborgh 49747 AGGroningenThe Netherlands
J Reckfort
Institute of Neuroscience and Medicine (INM-1)
K Amunts
Institute of Neuroscience and Medicine (INM-1)
Cécile and Oskar Vogt Institute of Brain Research
University of Düsseldorf
40204DüsseldorfGermany
M Axer
Institute of Neuroscience and Medicine (INM-1)
A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue
polarized light imagingnerve fibre architectureopticsbirefringenceJones matrix calculuscomputer simulation
The neuroimaging technique three-dimensional polarized light imaging (3D-PLI) provides a high-resolution reconstruction of nerve fibres in human post-mortem brains. The orientations of the fibres are derived from birefringence measurements of histological brain sections assuming that the nerve fibres -consisting of an axon and a surrounding myelin sheath -are uniaxial birefringent and that the measured optic axis is oriented in direction of the nerve fibres (macroscopic model). Although experimental studies support this assumption, the molecular structure of the myelin sheath suggests that the birefringence of a nerve fibre can be described more precisely by multiple optic axes oriented radially around the fibre axis (microscopic model).In this paper, we compare the use of the macroscopic and the microscopic model for simulating 3D-PLI by means of the Jones matrix formalism. The simulations show that the macroscopic model ensures a reliable estimation of the fibre orientations as long as the polarimeter does not resolve structures smaller than the diameter of single fibres. In the case of fibre bundles, polarimeters with even higher resolutions can be used without losing reliability. When taking the myelin density into account, the derived fibre orientations are considerably improved.
INTRODUCTION
Unravelling the architecture and connectivity of nerve fibres in the human brain is one of the greatest challenges in neuroscience. Over the past years, several methods have been developed to reconstruct the human connectome [1][2][3]. The neuroimaging technique three-dimensional polarized light imaging (3D-PLI) has been employed to reconstruct the three-dimensional architecture of nerve fibres in human post-mortem brains with a resolution of a few micrometres [4,5]. 3D-PLI enables the investigation of the pathways of long-range fibre bundles as well as single fibres and thus serves as a bridging technology between the macroscopic and the microscopic scale.
The spatial orientations of the nerve fibres are derived by transmitting polarized light through histological brain sections in a polarimeter and measuring their birefringence. To relate the measured signal to the fibre orientation, an effective model of birefringence is used which assumes that the fibre density is constant over the whole brain section [4] and that the measured optic axis indicates the predominant fibre orientation [5,6]. This assumption is based on various experimental studies on white matter which show that the average birefringence of parallel nerve fibres is negatively uniaxial and that the measured optic axis is oriented along the length of the fibres [7][8][9][10].
The majority of nerve fibres in the brain consist of an axon and a surrounding myelin sheath. The cytoplasm of the axon contains tubular polymers (microtubules) and neurofilaments running along the length of the axon [11,12]. The myelin sheath is formed by oligodendrocytes (glial cells) which are spirally wrapped around the axon. The cell membranes are bimolecular layers consisting of lipid molecules and membrane proteins. The membrane proteins are embedded in the bilayer or attached to the membrane surface [13][14][15], whereas the lipid molecules are oriented radially to the fibre axis [15][16][17]. The cell organelles of the axon and the protein framework of the myelin sheath lead to a weak positive birefringence with respect to the longitudinal fibre axis [7,8,10,[17][18][19][20]. The anisotropic structure of the lipid molecules causes a positive birefringence with respect to the radial fibre axis [7,8,15,18,21].
The effective model of uniaxial negative birefringence that is currently used in 3D-PLI seems reasonable for sufficiently low optical resolutions. However, it might no longer be valid if the anisotropic molecular structure of the nerve fibres is resolved. In this paper, we investigated the limitations of the effective model in terms of the optical resolution of the polarimeter using numerical simulations. The simulations were performed with a modified version of SimPLI [6], a simulation method that models the birefringence of the fibres with the Jones matrix calculus and allows data to be generated from synthetic fibre constellations that is comparable to experimental data. In order to study and understand the most dominant effects that generate the birefringence signals in 3D-PLI, the anisotropic molecular structure of the nerve fibres was described by a simplified birefringence model with radial optic axes (microscopic model) and the effective model of uniaxial negative birefringence by a birefringence model with axial optic axes (macroscopic model). To investigate the limitations of the effective model, the transition between the microscopic and the macroscopic model was investigated depending on the optical resolution of the imaging system.
THREE-DIMENSIONAL POLARIZED LIGHT IMAGING (3D-PLI)
The neuroimaging technique 3D-PLI determines the orientation of nerve fibres in post-mortem brains at the micrometre scale. The principles of 3D-PLI have been explained in detail by LARSEN et al. [22] and AXER et al. [4,5]. This section describes the measurement and data analysis procedures that are relevant for this study.
Measurement
To determine the orientation of the nerve fibres, a post-mortem brain -obtained from a body donor in accordance with ethical requirements -is fixed in buffered formaldehyde for several months, frozen and cut with a cryotome into histological sections of 70 µm, which are measured with a polarimeter. For the 3D-PLI measurement, two state-of-theart polarimeters with different optical resolutions and sensitivities are employed: The large-area polarimeter (LAP) has a pixel size of 64 µm and is mainly used for single-shot images of whole human brain sections. The polarizing microscope (PM) has a pixel size of 1.33 µm (i. e. down to small axonal diameters), which enables complex fibre constellations to be disentangled.
The LAP contains a pair of crossed linear polarizers and a quarter-wave retarder (with its fast axis adjusted at an angle of −45 • with respect to the transmission axis of the first linear polarizer), see Fig. 1a. The employed light source emits incoherent, non-polarized, diffusive light with a peak wavelength of 525 nm. During the measurement, the polarizers and the quarter-wave retarder are rotated simultaneously around the stationary tissue sample. For each rotation angle ρ = 0 • , 10 • , ..., 170 • , the transmitted light intensity is recorded by a CCD camera so that a series of 18 images is acquired.
The imaging principle works as follows: The quarter-wave retarder transforms the linearly polarized light from the first polarizer into circularly polarized light. The birefringent brain tissue induces an additional phase shift so that the outgoing light is elliptically polarized. The fraction of light that then passes the second linear polarizer depends on the local orientation of the optic axis of the birefringent tissue, which is assumed to coincide with the local fibre orientation.
The polarimetric set-up of the PM is slightly different to the set-up of the LAP (the order of the optical elements is reversed and only the first linear polarizer is rotatable). However, the imaging principle and the signal analysis are similar [4] so that the following considerations are only described for the LAP.
Signal analysis
The measured light intensity of an individual pixel describes a sinusoidal curve across the acquired image series, which depends on the orientation of the fibres within this pixel (see Fig. 1b). A physical description of the measured light intensity profile can be derived with the Jones matrix calculus [23,24], assuming that the light is coherent and completely polarized and that the optical elements are linear. For simplicity, the derivation is shown for a single pixel at a certain rotation angle ρ.
In the Jones matrix calculus, all optical elements in the polarimeter are represented by Jones matrices (cf. Fig. 1a). The Jones matrices of the crossed linear polarizers are given by [25]:
P x = 1 0 0 0 , P y = 0 0 0 1 . (2.1)
The Jones matrix of a wave retarder that is rotated by an angle ψ in counterclockwise direction and induces along the fast axis a phase shift δ between the two orthogonal components of the light wave is given by [25]:
M δ (ψ) = R(ψ) · M δ · R(−ψ) = cos ψ − sin ψ sin ψ cos ψ e i δ /2 0 0 e − i δ /2 cos ψ sin ψ − sin ψ cos ψ . (2.2)
In the experimental set-up, the fast axis of the quarter-wave retarder is rotated by −45 • with respect to the axis of the first linear polarizer. Thus, the quarter-wave retarder can be described by the Jones matrix of a rotated wave retarder as given in Eq. (2.2) with a rotation angle of ψ = −45 • and a phase shift of δ = 90 • :
M λ /4 ≡ M 90 • (−45 • ) = 1 √ 2 1 − i − i 1 . (2.3)
Under the assumption that the birefringence of the brain tissue can locally be described as negatively uniaxial with the optic axis indicating the predominant fibre direction (effective model), the brain tissue can locally be represented by a wave retarder that introduces a phase shift δ along the fast axis (fibre axis). During the measurement, the two polarizers and the quarter-wave retarder are rotated simultaneously around the specimen stage in counterclockwise direction by a rotation angle ρ. For simplicity, the equivalent case is considered in which the brain tissue is rotated by an angle (−ρ) in counterclockwise direction while the other optical elements are fixed. Thus, the brain tissue can be described by the Jones matrix of a rotated wave retarder as given in Eq. (2.2) with phase shift δ and rotation angle ψ = ϕ − ρ, where ϕ denotes the in-plane orientation of the optic axis:
M tissue ≡ M δ (ϕ − ρ) = cos(ϕ − ρ) − sin(ϕ − ρ) sin(ϕ − ρ) cos(ϕ − ρ) e i δ /2 0 0 e − i δ /2 cos(ϕ − ρ) sin(ϕ − ρ) − sin(ϕ − ρ) cos(ϕ − ρ) . (2.4)
When light with an electric field vector E 0 passes through the 3D-PLI set-up, the resulting output beam with electric field vector E T can be described by multiplication of the associated Jones matrices. As the Jones matrix calculus cannot be used to describe the non-polarized light emitted by the employed light source, the Jones vector E x = P x · E 0 is used to describe the horizontally polarized light after the first linear polarizer (cf. Fig. 1a):
E T = P y · M tissue · M λ /4 · E x .
(2.5)
Using I T ∼ | E T | 2 , the transmitted light intensity is calculated, yielding a sinusoidal intensity profile (see Fig. 1b):
I T (ρ) = I T,0 2 1 + sin 2(ρ − ϕ) sin δ ,(2.6)
where I T,0 ∼ | E x | 2 corresponds to the transmitted light intensity averaged over all rotation angles (here referred to as transmittance) and | sin δ | to the peak-to-peak amplitude of the normalized sinusoidal intensity profile (here referred to as retardation). The phase shift δ is given by (see Appx. A):
δ ≈ 2π λ t ∆n cos 2 α ,(2.7)
where λ is the wavelength of the incident light, t the thickness of the brain section, ∆n the local birefringence of the sample and α the local out-of-plane inclination angle of the fibre. Thus, the intensity profile in Eq. (2.6) is a direct measure of the spatial fibre orientation defined by the direction angle ϕ and the inclination angle α (see Fig. 1c). In order to compute transmittance, direction and retardation, the intensity profile is fitted by means of a discrete harmonic Fourier analysis [4,26]. The inclination angle α is calculated from the measured retardation |sin δ | by rearranging Eq. (2.7). The direction and inclination angles are combined to a unit vector indicating the local fibre orientation in three dimensions. Putting all unit vectors of several adjacent brain sections together, a three-dimensional volume of vectors is created and the fibre tracts are reconstructed with streamline algorithms.
SIMULATION OF 3D-PLI USING THE JONES MATRIX FORMALISM
Simulation model
3D-PLI derives the nerve fibre orientations based on the fact that the average birefringence of parallel fibres is negatively uniaxial [7][8][9][10] and assuming that the orientation of the measured optic axis corresponds to the local fibre orientation. To investigate the limitations of this effective birefringence model, a straight single fibre and a hexagonal bundle of straight parallel fibres were simulated and the birefringence of the fibres was modelled according to a microscopic and a macroscopic model for different optical resolutions of the simulated imaging system.
Microscopic model: The microscopic model of birefringence considers the anisotropic molecular structure of a single nerve fibre. To investigate and understand the predominant effects generating the birefringence signals in 3D-PLI, a simplified model of birefringence was chosen for the simulations. As stated in Sec. 1, the average birefringence of parallel nerve fibres is negative with respect to the longitudinal fibre axis. Therefore, the positive birefringence of the axon and the myelin proteins is weak as compared to the birefringence effects of the myelin lipids [8,9,18,20,21]. Since the exact contribution of the different birefringence effects to the overall birefringence is unknown, the birefringence effects of the nerve fibres were modelled by considering only the anisotropic radial structure of the myelin sheaths:
The fibres were simulated as hollow tubes (representing the myelin sheaths) with positive birefringence and radial optic axes (cf. lower Fig. 2b). The axons were considered to be non-birefringent.
Macroscopic model: To compare the simulation results of the microscopic model with the effective model of uniaxial negative birefringence, a macroscopic model of birefringence was defined. According to the assumptions made in the effective model, a single nerve fibre was simulated as negatively birefringent with axial optic axes oriented along the length of the fibre (cf. upper Fig. 2b). DOHMEN et al. [6] used this simulation model to investigate the effect of crossing fibre constellations. As this study concentrates on straight parallel fibres, the macroscopic model only serves as a reference for the effective model to verify the simulations of the microscopic model. To ensure a better comparison with the microscopic model, the fibres were simulated as hollow tubes (and not as solid cylinders as in [6]).
Simulation method
The basic idea of the simulation method is to model the birefringent myelin sheaths as series of linear optical retarder elements which are represented by Jones matrices. By defining the direction of the optic axes (radial/axial), both the microscopic and the macroscopic model can be simulated. The simulation approach is based on the simulation tool SimPLI developed by DOHMEN et al. [6]. For this study, the simulation tool was extended by the microscopic model and modified such that various fibre configurations with individual orientations, radii and myelin sheath thickness can be realized. The simulation tool is based on several assumptions and simplifications: First of all, the use of the Jones matrix calculus requires linear optical elements and perfect polarizers (i. e. the outgoing light is assumed to be completely polarized). Another assumption is that the incident light can be described by parallel rays of light with straight optical pathways, i. e. the light is assumed to be non-diffusive and refraction, diffraction and scattering are neglected. For this study, a parallel and straight beam of light seems a reasonable approximation for the LAP because the imaging system has a small numerical aperture (the acceptance angle of the objective lens is less than 1 • ) so that the camera only captures light rays that are almost parallel to each other.
The simulation consists of several steps:
1.) Generation of synthetic nerve fibres in a three-dimensional volume: The nerve fibres are modelled as hollow tubes representing the myelin sheaths (see Fig. 2a). In order to approximate the geometry of the fibres, the simulation volume is discretized into small cubic volume elements (called voxels), as indicated schematically by the grid in Fig. 2c.
2.) Generation of a three-dimensional vector field: For sufficiently small voxel sizes, the birefringence of the myelin sheaths can approximately be described by assigning each myelin voxel j a unit vector that indicates the direction of the optic axis (ϕ j , α j ) within the myelin sheath. In the macroscopic model, the vectors are oriented parallel to the fibre axis. In the microscopic model, the vectors are oriented radially to the fibre axis (see Fig. 2b).
3.) Generation of a synthetic 3D-PLI image series:
In order to model the birefringence effect of the myelin sheaths, each myelin voxel is represented by the Jones matrix of a rotated wave retarder. The retarder axis is aligned with the optic axis within the myelin voxel (see Fig. 2c). The synthetic 3D-PLI image series is calculated analogously to the derivation of the sinusoidal intensity profile as given in Eq. (2.5), with M tissue being replaced by the product of N matrices representing the myelin voxels along the optical path:
E T = P y · (M N · M N−1 · · · M 1 ) · M λ /4 · E x . (3.1) The matrix M j ≡ M δ j (ϕ j − ρ) is
Simulation parameters
The choice of the simulation parameters was inspired by real experimental conditions. According to typical dimensions of large nerve fibres in human white matter [16,[27][28][29], the diameter and the myelin sheath thickness of the simulated nerve fibres were chosen to be 15 µm and 2.5 µm, respectively (see Fig. 3a). The fibres were generated in a simulation volume with dimensions x × y × z = 64 × 64 × 70 µm 3 , corresponding to the pixel size of the LAP and the thickness of the brain section. The simulation volume was discretized into cubic voxels with a side length of ∆x sim . In a preliminary study (see later, Sec. 4.1), the optimal voxel size was determined to be ∆x sim = 0.1 µm, which was used for all following simulations. Note that the dimensions are given in micrometres to meet the experimental conditions. As only relative length scales matter for the qualitative simulation results, the units could be chosen arbitrarily.
Since measuring the birefringence of the micrometre-thick brain sections is impossible with the employed setups and literature values are not given for the currently used preparation technique, an upper limit for the birefringence of the myelin sheaths ∆n was estimated: Under the assumption that a brain section that is completely filled with a homogeneous birefringent material with in-plane optic axis (α = 0 • ) induces a maximum possible retardation (| sin δ | = 1 ⇔ δ = π/2), the upper limit of the birefringence was calculated by rearranging Eq. (2.7): ∆n = λ /(4t) = (525 nm)/(4 · 70 µm) ≈ 0.001875. Note that the choice of ∆n only changes the overall magnitude of the retardation and does not affect the simulation results qualitatively. In the macroscopic (microscopic) model, the myelin voxels were simulated with axial (radial) optic axes and negative (positive) birefringence with respect to the optic axes.
The wavelength of the incident light was chosen to correspond to the peak wavelength of the LAP (λ = 525 nm). To study only the birefringence effect of the nerve fibres, the fibres were simulated without any absorption.
Simulation of the optical resolution
To investigate the effect of different optical resolutions on the measured 3D-PLI signal, the synthetic 3D-PLI image series were downsampled using the open-source image processing programme Fiji [30]: To account for the limited optical resolution of the polarimeter, the image series were first convoluted with a two-dimensional Gaussian filter with a standard deviation σ . Then, the effect of the spatial discretisation of the CCD chip was modelled by resampling the resulting images with a sampling factor f s (average when downsizing). To determine realistic parameters for σ and f s , the imaging properties of the LAP were considered as a point of reference (see Appx. B).
Based on these considerations, the synthetic 3D-PLI image series were downsampled with different parameter sets (see Tab. 1), yielding images with different pixel sizes ∆x. The pixel size of the downsampled images was chosen such that a multiple of the pixel size corresponds to the side length of the simulation volume (∆x = 64 µm/n, with n = 4, 8, 16, 32). The standard deviation was calculated as a linear function of the pixel size (σ = 0.714 ∆x, see Appx. B) and the sampling factor was calculated by dividing the pixel size of the high-resolution image series by the pixel size of the downsampled image ( f s = ∆x sim /∆x = 0.1 µm /∆x). In the following, the optical resolution of the imaging system will be given in terms of the pixel size, which defines the set of downsampling parameters (σ and f s ) in Tab. 1. Note that the simulation results will not change qualitatively as long as the ratio between the fibre dimensions and the downsampling parameters remains the same.
Calculation of the retardation curve
The determination of the inclination angle α is challenging for 3D-PLI because the peak-to-peak amplitude of the measured intensity profile (| sin δ |) is highly sensitive to noise and -amongst others -influenced by the density of myelinated nerve fibres (see below).
In the standard 3D-PLI analysis, the inclination angle is calculated from the measured intensity profile assuming that the brain tissue can locally be described by the effective model of uniaxial negative birefringence. In order to investigate whether the effective model can be used to extract the correct fibre inclinations, the retardation computed from Eq. (2.7) was compared to the retardation values derived from simulations using the macroscopic and the microscopic model (see Sec. 3.1). For that purpose, the retardation images were calculated for different fibre inclinations and different optical resolutions, respectively. For a better comparison between the retardation values of the single fibre and the fibre bundle, only the pixel in the centre of each (downsampled) retardation image was considered for evaluation. If pixels at other locations had been chosen, the retardation values of the single fibre would have been influenced by boundary effects that do not exist for the fibre bundle or real brain tissue which are completely filled with fibres. The retardation values from the centre of each downsampled retardation image were plotted against the corresponding inclination angle, yielding a retardation curve for each downsampling step. The retardation curves were compared to the normalized retardation curve of the effective model (cf. Eq. (2.7)), in the following referred to as theoretical curve: | sin δ | = |sin (π/2) cos 2 α |.
To be able to compare different retardation curves, the retardation was normalized for each curve with the maximum retardation value, respectively:
sinδ = sin π 2 δ δ max . (3.2)
As only birefringent material (mainly myelin) is responsible for the phase shift in Eq. (2.7), t describes not the thickness of the whole brain section but rather the local myelin thickness t m , i. e. the combined thickness of myelin sheaths along the optical path. Due to the inhomogeneity of brain tissue, the local myelin density of a brain section is less than 100 %, i. e. the maximum possible retardation is | sin (δ α=0 • ,max ) | < 1. If the inclination is calculated under the assumption that the maximum possible retardation equals 1, the inclination angle will be overestimated. In order to obtain a more precise estimation of the inclination angle, a so-called myelin density correction was applied to the downsampled retardation images:
In the case of the macroscopic model, in which the optic axes within one fibre have the same orientations, δ scales linearly with t m . In the case of the microscopic model, in which the optic axes within one fibre have different inclination angles, the upper limit of δ scales linearly with t m as long as the optic axes of neighbouring myelin voxels have similar orientations (see Appx. C). Thus, the dependence on the myelin density can be eliminated to the greatest possible extent by multiplying the phase shift δ with a correction factor (t/t m ):
| sin (δ corr ) | = sin t t m δ . (3.3)
In order to apply the myelin density correction to the downsampled retardation images, t m was replaced by the combined thickness of myelin voxels along the optical path (after applying the Gaussian filter and resampling). The resulting retardation images were normalized according to Eq. (3.2), yielding | sin(δ corr )|.
SIMULATION RESULTS
Comparison of analytical and numerical solution
To estimate the accuracy of the simulation results for the microscopic model, a single fibre with radial optic axes and perpendicularly incident light (see Fig. 3b) was generated for different voxel sizes (∆x sim ) and the numerically computed phase difference between extraordinary and ordinary wave (∆Φ num ) was compared to the analytical solution (∆Φ ana ).
Assuming that reflection and refraction effects can be neglected so that associated extraordinary and ordinary wave follow the same pathway, BEAR and SCHMIDT derived an analytical expression for the phase difference [18]: where Γ is the optical path length difference between extraordinary and ordinary wave, r 1 the radius of the whole nerve fibre (outer cylinder), r 2 the radius of the non-birefringent axon (inner cylinder) and r 0 the distance at which the light is incident perpendicular to the fibre axis (see Fig. 3b).
∆Φ ana = 2π λ Γ ≈ 4π λ r 0 ∆n arccos r 0 r 1 − arccos r 0 r 2 ,(4.
In order to compute ∆Φ num , the propagation of ordinary and extraordinary wave were simulated separately: In the case of the ordinary wave, the light is polarized parallel to the longitudinal axis of the fibre. In the case of the extraordinary wave, the light is polarized perpendicular to the longitudinal axis of the fibre (see Fig. 3b). The phase for both the ordinary wave (Φ o ) and the extraordinary wave (Φ e ) was calculated from the corresponding electric field vector E T in Eq. (3.1):
Φ = arctan Im(| E T |) Re(| E T |) . (4.2)
The numerically computed phase difference ∆Φ num = Φ e − Φ o was evaluated at various distances 0 < r 0 < 5 µm away from the centre of the fibre and compared to the analytical solution given in Eq. (4.1), with r 1 = 7.5 µm, r 2 = 5 µm, λ = 525 nm and ∆n = 0.001875 (cf. Sec. 3.3). In order to study the impact of the spatial discretisation on the accuracy of the numerical solution, the simulation was performed for various voxel sizes 1.50 µm > ∆x sim > 0.06 µm. As a measure of consistency between the numerical and the analytical solution, the relative phase difference was calculated: (∆Φ ana − ∆Φ num )/∆Φ ana . Figures 4a and 4b show the relative phase difference plotted against r 0 for various voxel sizes ∆x sim . As can be seen, the numerical solution fluctuates around the analytical solution for voxel sizes of 0.5 µm and less. With smaller voxel sizes, the numerical solution approaches the analytical solution (indicated by the dashed black line). This behaviour is especially evident when considering the mean of the absolute relative phase difference for each voxel size (see Fig. 4c): For a voxel size of ∆x sim = 1.5 µm (corresponding to one tenth of the fibre diameter), the mean absolute relative phase difference is about 12 %. For ∆x sim = 0.5 µm, it is about 6 % and for ∆x sim = 0.06 µm, it is only 0.8 %. This demonstrates that the simulation tool produces correct results.
As a good compromise between computation time and accuracy, all following fibre simulations were performed with a voxel size of ∆x sim = 0.1 µm (corresponding to 1/150 of the fibre diameter). For this voxel size, the relative phase difference is no more than 4 % (see Fig. 4b) and the mean relative phase difference is about 1.3 % (see Fig. 4c).
Simulation of a single fibre
In a preliminary study, the limitations of the effective model of uniaxial negative birefringence were first studied for a straight single fibre. The fibre was simulated according to both the macroscopic and the microscopic model with An example of downsampled and corrected retardation images can be found in Appx. D. Figure 5 shows the dimensions of the simulated single fibre and the corresponding retardation curves (continuous lines) for both simulation models and different optical resolutions (according to Tab. 1). The theoretical retardation curve of the effective model is indicated by a dashed black line. In the case of the macroscopic model, the uncorrected retardation curves (see Fig. 5a) are already very similar to the theoretical curve for all investigated optical resolutions. After the myelin density correction (see Fig. 5c), all retardation curves match the theoretical curve exactly, independently of the optical resolution. In the case of the microscopic model (see Figs. 5b and 5d), the retardation curves for a pixel size much smaller than the fibre diameter (∆x < 2 µm) are inverted as compared to the theoretical curve for α < 90 • , i. e. the microscopic and the macroscopic model yield totally different results. For intermediate pixel sizes (2 µm ≤ ∆x ≤ 8 µm), the retardation curves are non-monotonic, i. e. the assignment of the inclination angle is ambiguous. Finally, for pixel sizes larger than the fibre diameter (∆x = 16 µm), the uncorrected retardation curve (see Fig. 5b) is similar to the theoretical curve. After the myelin density correction (see Fig. 5d), the retardation curve matches the theoretical curve almost exactly.
Simulation of a fibre bundle
In brain tissue, nerve fibres are usually organised in hexagonal close-packed fibre bundles [13]. In order to investigate the effect of fibre bundles on the 3D-PLI signal, a hexagonal bundle of straight parallel fibres with an inter-fibre spacing of 1 µm was simulated (see Fig. 6e). In order to obtain comparable results, the same dimensions and simulation parameters were chosen as for the single fibre. Figure 6 shows the normalized retardation curves for both simulation models and different optical resolutions (according to Tab. 1). The (downsampled) retardation images that were used to compute the corrected retardation curves of the microscopic model are shown in Appx. D. In the case of the macroscopic model, the uncorrected retardation curves (see Fig. 6a) are very similar to the theoretical retardation curve of the effective model (dashed black line) for all investigated optical resolutions. As compared to the retardation curves of the single fibre (see Fig. 5a), the retardation curves of the fibre bundle are closer to the theoretical curve. After the myelin density correction (see Fig. 6c), the curves are almost identical. In the case of the microscopic model, the uncorrected retardation curves (see Fig. 6b) are also closer to the theoretical curve as compared to the uncorrected retardation curves of the single fibre (see Fig. 5b). The myelin density correction (see Fig. 6d) makes only a small difference, especially for low optical resolutions. For the simulated fibre bundle, the transition between the microscopic and the macroscopic model already occurs for pixel sizes larger than the fibre radius (∆x ≥ 8 µm).
DISCUSSION
In 3D-PLI, the fibre orientations are derived under the assumption that the brain tissue can (locally) be described as a homogeneous and uniaxial birefringent material with the optic axis indicating the predominant fibre direction. Furthermore, the density of myelinated fibres is assumed to be the same for the whole brain section. In this paper, the limitations of this effective birefringence model have been studied for the first time. For that purpose, a single fibre and a hexagonal fibre bundle (with diameters d) were simulated based on the Jones matrix calculus, employing a microscopic and a macroscopic model of birefringence and different optical resolutions (defined by the pixel size ∆x as given in Tab. 1).
The transition between the two models is apparent when analysing the retardation curves: For high optical resolutions (∆x << d), the radial optic axes of the microscopic model are resolved. In this case, the optic axes are oriented perpendicular to the longitudinal fibre axis so that the retardation curves are inverted as compared to the macroscopic model and fibres with high inclination angles are interpreted as flat fibres. The zero retardation value for α = 90 • is an artifact arising from the fact that the retardation is evaluated at the centre of the retardation image which -in the case of vertical fibres -contains no myelin (cf. Fig. 8, upper right corner). For intermediate optical resolutions (∆x < d), there is a transition zone between the microscopic and the macroscopic model so that an unambiguous assignment between retardation and inclination is not possible. For sufficiently low optical resolutions (single fibre: ∆x > d; fibre bundle: ∆x > d/2), the microscopic and the macroscopic model yield similar results (see Figs. 5d and 6d) so that the effective model of uniaxial negative birefringence can be used to compute the fibre inclinations.
Thus, for the simulated fibre bundle (consisting of five fibre layers with d = 15 µm), the effective model can be used to interpret LAP measurements (∆x LAP = 64 µm > d/2), but not to interpret PM measurements (∆x PM = 1.33 µm < d/2). However, the diameters of the simulated fibres represent an upper estimate of typical fibre diameters in the human brain. The diameters of myelinated nerve fibres range from 0.3 to 15 µm [16,[27][28][29] and the majority of the fibres (e. g. 80 % in the corpus callosum [27]) have diameters of 1 µm or below so that the condition ∆x PM > d/2 is still fulfilled. In addition, fibre diameters much smaller than 15 µm implicate that the measured brain section (with thickness 70 µm) contains much more fibre layers than the simulated fibre bundle. A comparison between the simulated single fibre and the fibre bundle suggests that the more fibre layers are located along the optical path, the smaller is the minimum pixel size for which the effective model is still valid. To verify this hypothesis, the limitations of the effective model should also be studied in terms of the number of fibre layers along the optical path. However, a larger number of fibre layers also increases the probability that fibres with different spatial orientations are measured within the same volume, which poses a major challenge for 3D-PLI [6]. In future studies, the limitations of the effective model should therefore also be investigated for non-parallel fibre structures.
The simulations have shown that -in regions with parallel fibre structures -the effective model of uniaxial negative birefringence is valid for the employed optical set-ups. For imaging systems with very high optical resolutions, the effective model needs to be reconsidered. Even if the optical resolution is too high to extract the correct fibre inclinations, 3D-PLI remains a valuable neuroimaging technique as the image contrasts of transmittance and retardation still provide detailed structural information on the two-dimensional nerve fibre architecture in large histological brain sections.
The effective model that is currently used for the data analysis in 3D-PLI does not only assume parallel fibre structures, but also a uniform myelin density. The simulations have shown that the retardation signal is considerably influenced by the myelin density, which impairs the reconstructed fibre orientations. It could be demonstrated that the estimation of the fibre inclination is considerably improved by the myelin density correction which incorporates the local myelin thickness of the examined tissue into the calculation of the inclination angle. While the correction has a large effect on the retardation curves of the single fibre, the effect is smaller for the fibre bundle which is much more homogeneous than the single fibre. Thus, the myelin density correction is especially useful for regions with an inhomogeneous density of myelinated nerve fibres (e. g. for transition zones between white and grey matter). In the case of the microscopic model, the correction does not work as well as for the macroscopic model because the retardation also depends on the direction of the radially oriented optic axes in the myelin sheath, but it is still a considerable improvement. In order to incorporate the myelin density correction into the 3D-PLI signal analysis, the local myelin thickness t m of the sample needs to be determined. The intensity values of the transmittance image seem to be a good measure of the local myelin thickness in brain tissue [5].
The purpose of this study was to explore and understand the most dominant effects that generate the birefringence signals in 3D-PLI. To fully understand the physical processes behind 3D-PLI and to improve the interpretation of the reconstructed fibre orientations, a direct comparison between simulation and experiment is required. The long-term aim should be to develop a simulation tool of 3D-PLI that considers all relevant effects needed for reproducing the experimental results. To this end, the simulation model should be extended step by step and the relevant effects should be identified.
Although the simulations show that the simplified microscopic model can already be used to explain the effective negative birefringence of parallel nerve fibres, future studies should include the positive birefringence of the axon and investigate how this modification changes the transition between the microscopic and the macroscopic model.
So far, only straight and parallel fibres have been investigated. To provide more realistic fibre models, the fibres should be simulated with varying fibre diameters, myelin sheath thickness and spatial orientations. As fibres with different spatial orientations pose a major challenge for 3D-PLI [6], future studies should focus on investigating inhomogeneous, non-parallel fibre structures. To enable a direct comparison with the experiment, the simulated fibre configurations should be based on experimentally determined fibre structures.
In addition to a more realistic fibre model, the propagation of light should also be simulated more realistically. In this study, the incident light was described by a parallel beam of light. However, in the experiment, the employed light source emits diffusive light, i. e. the sample is illuminated by light with slightly different angles of incidence. As the measured birefringence signals depend on the angle between the light wave and the nerve fibres, a non-zero angle of incidence changes the retardation curves. For the LAP, which has a small numerical aperture, the effect can presumably be neglected. However, for systems with higher optical resolutions and higher numerical apertures, the effect of a Gaussian distribution of incident angles should be investigated further.
Moreover, the simulations were based on the Jones matrix calculus which is only applicable to completely polarized and coherent light. As the light source emits incoherent light and the polarizers are not perfect, the Jones matrices should be replaced by Müller matrices [31] which enable to study partially polarized and incoherent light.
Finally, the assumption of a linear optical pathway is a great simplification. The refractive index of the myelin sheath is higher than the refractive indices of the inner axon and the surrounding tissue [9,32,33] which will cause refraction/reflection at the interfaces and scattering of light. In future studies, the effects of refraction and scattering on the measured birefringence signal should be investigated in more detail. As the used simulation tool (SimPLI) is based on a matrix calculus, other simulation approaches will be required to investigate such non-linear pathways.
CONCLUSION
In this study, we laid a theoretical foundation for 3D-PLI. The effective model of uniaxial negative birefringence, which is currently used to compute the nerve fibre orientations from experimental data, has been validated for the first time. Using simulations based on the Jones matrix calculus, we have shown that the effective model can be used for the employed optical set-ups, i. e. as long as the polarimeter does not resolve structures smaller than the diameter of single nerve fibres. The developed Jones matrix formalism for simulating 3D-PLI has proven to be a powerful tool to gain a deeper theoretical understanding of the physical processes behind 3D-PLI and to better interpret the experimental data. The simulations enable not only to validate the computational model of the fibre reconstruction, but also to optimise the experimental set-up and the measurement method.
the interpretation of the simulated data, and helped draft the manuscript. H. De Raedt contributed to the interpretation of the simulated data, to the theoretical considerations and to the revision of the manuscript. J. Reckfort conducted experimental measurements, helped transfer the measurement results to the simulation and revised the manuscript. K. Amunts contributed to the anatomical content of the study and to the revision of the manuscript. M. Axer coordinated the study, participated in the conception and design, contributed to the analysis and interpretation of the simulated data, and helped draft the manuscript. All authors read the final manuscript and gave final approval for publication.
AKNOWLEDGEMENTS
The authors thank Melanie Dohmen for the introduction to the simulation software SimPLI.
A. DERIVATION OF THE PHASE SHIFT
When polarized light passes through the birefringent brain section, it is split into an ordinary and an extraordinary wave which both experience different refractive indices. The refractive index n e that the extraordinary wave experiences when passing through the birefringent tissue under an angle θ with respect to the optic axis, is given by [34]:
1 n e (θ ) 2 = 1 n 2 o cos 2 θ + 1 n 2 E sin 2 θ (A.1) ⇔ 1 n 2 o − 1 n e (θ ) 2 = 1 n 2 o − 1 n 2 E sin 2 θ , (A.2)
where n o is the ordinary refractive index and n E ≡ n e (θ = 90 • ) the principal extraordinary refractive index of the brain tissue.
The birefringence of biological tissue (∆n = n E − n o = 10 −3 ...10 −2 [35]) is small as compared to the values of the refractive indices n o and n E (n = 1.3-1.5 [33]). Therefore, a Taylor expansion can be applied to the function
f (∆n) ≡ 1 n 2 o − 1 n 2 E = 1 n 2 o − 1 (n o + ∆n) 2 (A.3)
in ∆n = 0:
f (∆n) = ∞ ∑ l=0 f (l) (0) l! (∆n) l = f (0) + f (0) ∆n + ... = 0 + 2 n 3 o ∆n + ... (A.4)
The same expansion can be done for 1/n 2 o − 1/n e (θ ) 2 in (∆n(θ ) = n e (θ ) − n o 1). With these Taylor expansions, Eq. (A.2) can be written as:
∆n(θ ) ≈ ∆n sin 2 θ . (A.5)
Choosing a coordinate system in which the light propagates in the z-direction and the brain tissue lies in the xyplane, the optic axis (oriented in the direction of the nerve fibres) makes an angle θ with the z-axis, i. e. the out-of-plane inclination angle of the fibre is α = 90 • − θ . With this definition follows: ∆n(θ ) ≈ ∆n cos 2 α.
Thus, when the light passes through a brain section of thickness t, the extraordinary wave experiences a phase shift with respect to the ordinary wave which depends on the inclination angle of the optic axis:
δ = 2π λ t ∆n(θ ) ≈ 2π λ t ∆n cos 2 α. (A.6)
This is the formula of the phase shift as given in Eq. (2.7).
B. DERIVATION OF THE DOWNSAMPLING PARAMETERS
In previous measurements, the optical resolution of the LAP was investigated by employing a USAF test chart which contains line pairs (lp) with different spacings [36]. From the measured line intensity profiles, the Michelson contrast C was computed:
C = I max − I min I max + I min , (B.1)
where I max corresponds to the mean intensity of the maxima and I min to the mean intensity of the (local) minima in the line intensity profile (cf. Fig. 7b). The largest number of line pairs per millimetre that can just be resolved (according to the Rayleigh criterion) was determined to be 5.66 lp/mm, which corresponds to a width per line pair of l LAP = 176.7 µm and a contrast of C LAP = 20.1 %. A width per line pair of 157.5 µm yields a Michelson contrast of 11.2 %.
According to these measurement results, a test image with three lines (pixel size: 0.1 µm) and a line width of l LAP /2 ≈ 88.4 µm was created, and the downsampling procedure (Gaussian filter and resampling) was applied to the test image (see Fig. 7a). The sampling factor was calculated by dividing the pixel size of the test image by the pixel size of the LAP: f s,LAP = 0.1 µm/64 µm. To reproduce the measured contrast of the line intensity profile (see Fig. 7b), a Gaussian filter with a standard deviation of σ PM = 45.7 µm was applied. To avoid boundary effects and ensure a symmetric line intensity profile, the dimensions of the image (1216 µm ×1216 µm) were chosen such that the downsampled image consists of an odd number of pixels (19 px ×19 px). Based on the determined parameters for the LAP (∆x LAP , σ LAP , f s,LAP ), downsampling parameters for imaging systems with other optical resolutions (see Tab. 1) were derived: The ratio between the pixel size of the LAP image and the determined standard deviation of the two-dimensional Gaussian filter is σ LAP /∆x LAP = 45.7 µm /64 µm ≈ 0.714. Analogous measurements of the PM yield a similar ratio between pixel size and standard deviation [36]. Assuming that this ratio is the same for all simulated imaging systems, the standard deviation of the two-dimensional Gaussian filter was calculated from the pixel size ∆x of the resulting downsampled image:
σ = 0.714 ∆x. (B.2)
After applying the Gaussian filter, the synthetic image series (with pixel size ∆x sim ) was resampled with a sampling factor of
f s = ∆x sim ∆x = 0.1 µm ∆x , (B.3)
yielding a downsampled image series with pixel size ∆x.
C. DEPENDENCE OF THE PHASE SHIFT ON THE LOCAL MYELIN THICKNESS
Each myelin voxel of a simulated nerve fibre is represented by the Jones matrix of a rotated wave retarder as defined in Eq. (2.2). Depending on what kind of model is used (macroscopic or microscopic), the retarder axis is either oriented parallel or radially to the fibre axis (see Fig. 2b). In the macroscopic model, the optic axes of the myelin voxels are all oriented in the fibre direction (ϕ, α) so that the voxels can be described by the same Jones matrix M δ (β ) with phase shift δ and β ≡ ϕ − ρ. When the light propagates through N voxels of myelin, the multiplication of the N corresponding Jones matrices yields (using Eq. In other words, the phase shift δ (and for small δ also the retardation | sin δ |) scales linearly with the combined thickness of myelin voxels (N ∆t), i. e. with the local myelin thickness t m . In the microscopic model, the optic axes of the myelin voxels along the optical path all have different orientations (ϕ j , α j ), see Fig. 2c. If the optic axes of neighbouring myelin voxels have a similar direction (ϕ 2 − ϕ 1 1 and α 2 − α 1 1), the multiplication of the N Jones matrices of the voxels can be simplified. For ϕ 2 − ϕ 1 1, one can define β 2 − β 1 ≡ η 21 1 and the multiplication of a pair of rotation matrices yields: R(−β 2 ) · R(β 1 ) = cos(β 2 ) sin(β 2 ) − sin(β 2 ) cos(β 2 ) cos(β 1 ) − sin(β 1 ) sin(β 1 ) cos(β 1 ) = cos(β 2 − β 1 ) sin(β 2 − β 1 ) − sin(β 2 − β 1 ) cos(β 2 − β 1 ) ≈ 1 η 21 −η 21 ≈ R(β 2 ) e i(δ 1 +δ 2 )/2 η 21 e i(δ 2 −δ 1 )/2 −η 21 e − i(δ 2 −δ 1 )/2 e − i(δ 1 +δ 2 )/2 R(−β 1 ).
(C.4)
The multiplication of four Jones matrices yields (ignoring terms in the order of η 2 ji ): If the number of myelin voxels (i. e. the number of matrices M j ) is small, the elements of the secondary diagonal in the resulting matrix can be neglected for η jk 1. If the number of myelin voxels is large, the arguments of the exponential functions in the secondary diagonal will take all possible values and cancel each other for η jk ≈ η lm ∀ j, k, l, m. In both cases, the multiplication of N Jones matrices yields: given that the optic axes of neighbouring myelin voxels have similar directions. The analytical considerations have shown that the phase shifts of individual voxels add together in both simulation models: In the macroscopic model, the phase shift scales linearly with the local myelin thickness t m . In the microscopic model, this is only true for the upper limit of the phase shift. The dependence on the local myelin thickness is taken into account in the myelin density correction (see Sec. 3.5).
M 4 · M 3 · M 2 · M 1 ≈ R(β 4 ) e i(M
D. RETARDATION IMAGES OF THE FIBRE
Figure 1 .
1(a) Measurement set-up of 3D-PLI (for the LAP): The brain tissue is placed between a pair of crossed linear polarizers and a quarter-wave retarder, which are rotated simultaneously by 18 discrete rotation angles ρ. The transmitted light intensity is calculated with the Jones calculus, in which each optical element is represented by a Jones matrix (bold symbols). (b) The normalized transmitted light intensity I T (ρ)/I T,0 describes a sinusoidal curve for each image pixel. The phase ϕ corresponds to the local fibre direction angle and the peak-to-peak amplitude |sin δ | to the local fibre inclination angle. (c) The three-dimensional orientation of a fibre is defined by the direction angle ϕ and the inclination angle α.
the Jones matrix of a rotated wave retarder as given in Eq. (2.2) and represents the j-th myelin voxel. The rotation angle depends on the in-plane direction angle ϕ j of the optic axis and the phase shift δ j on the out-of-plane inclination angle α j . The Jones matrices of the linear polarizers and the quarter-wave retarder are given by Eqs. (2.1) and (2.3). For each rotation angle of the polarimeter (ρ = 0 • , 10 • , ..., 170 • ), all Jones matrices along the optical path are multiplied (see Fig. 2c), yielding a series of 18 synthetic 3D-PLI images with a sinusoidal intensity profile for each image pixel.
Figure 2 .
2Simulation method: (a) Generation of synthetic nerve fibres in a three-dimensional volume. (b) Generation of a three-dimensional vector field according to the macroscopic model (axial optic axes) and the microscopic model (radial optic axes). (c) Generation of a synthetic 3D-PLI image series (illustrated for a large fibre in the microscopic model): The simulation volume is discretized into small volume elements (voxels). Each myelin voxel (grey) is represented by the Jones matrix of an optical retarder (M j ) whose axis is oriented in direction of the optic axes (arrows). The polarizing filters of the 3D-PLI set-up (seeFig. 1a) are also represented by Jones matrices. For each rotation angle of the polarimeter, all Jones matrices along the optical path (highlighted column) are multiplied.
Figure 3 .
3(a) Dimensions of the simulated fibre (cross-sectional view). (b) Simulation model for comparison with the analytical solution: A horizontal fibre is simulated with outer radius r 1 = 7.5 µm, inner radius r 2 = 5 µm, and radial optic axes. The light is incident perpendicular to the fibre axis at distance r 0 . The electric field vector of the ordinary wave ( E o ) is oriented parallel to the longitudinal axis of the fibre. The electric field vector of the extraordinary wave ( E e ) is oriented perpendicular to the fibre axis.
Figure 4 .
4(a,b) Relative difference between analytically and numerically calculated phase difference (∆Φ ana and ∆Φ num ) for various voxel sizes ∆x sim , evaluated at different distances r 0 away from the centre of the fibre. For reasons of clarity, the results are presented in two diagrams: (a) ∆x sim = 1.50-0.21 µm, (b) ∆x sim = 0.17-0.06 µm. The dashed black lines indicate the point at which the numerical values match the analytical solution. (c) Mean absolute relative phase difference plotted against the voxel size ∆x sim . The arrow indicates the voxel size (∆x sim = 0.1 µm) that is chosen for the fibre simulations. different inclination angles (α = 0 • , 10 • , . . . , 90 • ) and different optical resolutions. The dimensions of the fibre and the other simulation parameters were chosen as described in Sec. 3.3. The retardation curves were calculated from the downsampled retardation images (without/with myelin density correction) and normalized as described in Sec. 3.5.
Figure 5 .Figure 6 .
56(a-d) Normalized retardation curves of a straight single fibre simulated according to the macroscopic model (a,c) and the microscopic model (b,d) for different optical resolutions. Graphs (a,b) show the uncorrected retardation curves, graphs (c,d) show the retardation curves after the myelin density correction. For reasons of clarity, only selected graphs are shown. The legend indicates the pixel sizes of the retardation images from which the retardation curves have been calculated. The pixel size ∆x of the downsampled retardation images determines the parameters used for simulating the optical resolution (see Tab. 1). For better comparison, ∆x is also given in terms of the fibre diameter (d = 15 µm). (e) Dimensions of the simulated single fibre. (a-d) Normalized retardation curves of a hexagonal fibre bundle simulated according to the macroscopic model (a,c) and the microscopic model (b,d) for different optical resolutions. Graphs (a,b) show the uncorrected retardation curves, graphs (c,d) show the retardation curves after the myelin density correction. For reasons of clarity, only selected graphs are shown. The legend indicates the pixel sizes of the retardation images from which the retardation curves have been calculated. The pixel size ∆x of the downsampled retardation images determines the parameters used for simulating the optical resolution (see Tab. 1). For better comparison, ∆x is also given in terms of the fibre diameter (d = 15 µm). (e) Dimensions of the simulated fibre bundle.
was supported by the Helmholtz Association portfolio theme "Supercomputing and Modeling for the Human Brain" and by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 604102 (Human Brain Project). [36] J. Reckfort, H. Wiese, M. Dohmen, D. Grässel, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer. Extracting the inclination angle of nerve fibers within the human brain with 3D-PLI independent of system properties. In J. A. Shaw and D. A. LeMaster, editors, SPIE Proceedings, Polarization Science and Remote Sensing VI, volume 8873, San Diego, CA, 2013. SPIE -The International Society for Optical Engineering.
Figure 7 .
7(a) Downsampling of a test image (grey values: black = 0, white = 1): A Gaussian filter with standard deviation σ LAP and resampling with sampling factor f s,LAP are applied to the test image. (b) Line profile of the downsampled test image: The determined contrast C = (I max − I min )/(I max + I min ) matches approximately the contrast C LAP obtained from experimental measurements.
the N myelin voxels with thickness ∆t (along the optical path) and phase shift δ can be replaced by one myelin voxel with side length (N ∆t ≡ t m ) and phase shift:
first-order approximation in η 21 . The multiplication of two Jones matrices yields (with M j ≡ M δ j (β j )):
δ 1 +δ 2
2+δ 3 +δ 4 )/2 η η e − i(δ 1 +δ 2 +δ 3 +δ 4 )/2 R(−β 1 ), (C.5)where the elements of the secondary diagonal are given by:η = η 21 e i(δ 4 +δ 3 +δ 2 −δ 1 )/2 + η 32 e i(δ 4 +δ 3 −δ 2 −δ 1 )/2 + η 43 e i(δ 4 −δ 3 −δ 2 −δ 1 )/2 , (C.6) η = −η 21 e − i(δ 4 +δ 3 +δ 2 −δ 1 )/2 − η 32 e − i(δ 4 +δ 3 −δ 2 −δ 1 )/2 − η 43 e − i(δ 4 −δ 3 −δ 2 −δ 1 )/2 . (C.7)
N · M N−1 · · · M 1 ≈ R(β N ) e i(δ 1 +···+δ N )/2 0 0 e − i(δ 1 +···+δ N )/2 R(−β 1 ). (C.8)Thus, the N myelin voxels with thickness ∆t and phase shift δ j can be replaced by one myelin voxel with thickness (N∆t = t m ) and phase shift:
Figure 8 .
8Retardation images of the hexagonal fibre bundle for selected fibre inclination angles α, simulated according to the microscopic model. The retardation values have been computed from (downsampled) image series with different pixel sizes ∆x (according to Tab. 1) after the myelin density correction. The pixel sizes are indicated by a square in the left bottom corner of the retardation images. To enhance the image contrast, a different scale bar is used for each inclination angle (the minimum value of the high-resolution retardation image is encoded in black, the maximum value in white).
Table 1. Downsampling parameters (selected values): To obtain an image with pixel size ∆x, a two-dimensional Gaussian filter with standard deviation σ = 0.714 ∆x and resampling with sampling factor f s = 0.1 µm/∆x are applied to the image.∆x [µm] σ [µm]
f s
2.00
1.43
1/20
4.00
2.86
1/40
8.00
5.71
1/80
16.00
11.43
1/160
COMPETING INTERESTSThe authors declare that they have no competing interests.AUTHORS' CONTRIBUTIONSM. Menzel substantially contributed to the conception and design of the study as well as to the acquisition, analysis and interpretation of the simulated data. She carried out the simulations as well as the analytical calculations and drafted the manuscript. K. Michielsen participated in the design of the study, contributed to theoretical considerations and to
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| [] |
[
"THE ERDŐS-MOSER EQUATION 1 k + 2 k + · · · + (m − 1) k = m k REVISITED USING CONTINUED FRACTIONS",
"THE ERDŐS-MOSER EQUATION 1 k + 2 k + · · · + (m − 1) k = m k REVISITED USING CONTINUED FRACTIONS"
] | [
"Yves Gallot ",
"ANDPieter Moree ",
"Wadim Zudilin "
] | [] | [] | If the equation of the title has an integer solution with k ≥ 2, then m > 10 9.3·10 6 . This was the current best result and proved using a method due to L.Moser (1953). This approach cannot be improved to reach the benchmark m > 10 10 7 . Here we achieve m > 10 10 9 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N , with N an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.12 bis rue Perrey, | 10.1090/s0025-5718-2010-02439-1 | [
"https://arxiv.org/pdf/0907.1356v1.pdf"
] | 16,305,654 | 0907.1356 | bbedb7523a1ec54482475dc53166badff1db49cd |
THE ERDŐS-MOSER EQUATION 1 k + 2 k + · · · + (m − 1) k = m k REVISITED USING CONTINUED FRACTIONS
8 Jul 2009
Yves Gallot
ANDPieter Moree
Wadim Zudilin
THE ERDŐS-MOSER EQUATION 1 k + 2 k + · · · + (m − 1) k = m k REVISITED USING CONTINUED FRACTIONS
8 Jul 2009
If the equation of the title has an integer solution with k ≥ 2, then m > 10 9.3·10 6 . This was the current best result and proved using a method due to L.Moser (1953). This approach cannot be improved to reach the benchmark m > 10 10 7 . Here we achieve m > 10 10 9 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N , with N an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.12 bis rue Perrey,
Introduction
In this note we are interested in non-trivial integer solutions, that is, solutions with k ≥ 2, of the equation
1 k + 2 k + · · · + (m − 2) k + (m − 1) k = m k .(1)
Conjecturally such solutions do not exist. For k = 1 one has clearly the solution 1 + 2 = 3 (and no further ones). From now on we will assume that k ≥ 2. Moser [28] showed in 1953 that if (m, k) is a solution of (1), then m > 10 10 6 and k is even. His result has since then been improved on. Butske et al. [6] have shown by computing, rather than estimating, certain quantities in Moser's original proof that m > 1.485 · 10 9 321 155 . By proceeding along these lines this bound cannot be improved on substantially. Butske et al. [6, p. 411] expressed the hope that new insights will eventually make it possible to reach the more natural benchmark 10 10 7 . Using that Σ k (m) = 1 k + 2 k + · · · + m k ≤ m 1 t k dt and Σ k (m + 1) > m 0 t k dt we obtain that k + 1 < m < 2(k + 1). This shows that the ratio k/m is bounded. By a more elaborate reasoning along these lines Krzysztofek [20] obtained that k + 2 < m < 3 2 (k + 1). This implies that k ≥ 4 and hence k + 2 < m < 2k.
Dividing both sides of (1) by m k one sees that for every integer m ≥ 2, (1) has precisely one real solution k. It is known that lim m→∞ k/m = log 2 and we show here that in fact the behaviour of k as a function of m can be determined in a much more explicit way (Theorem 1 and Section 2). Moree et al. [27], using properties of the Bernoulli numbers and polynomials (an approach initiated in Urbanowicz [30]), showed that N 1 = lcm(1, 2, . . . , 200) | k.
Date: July 8, 2009July 8, . 2000 Mathematics Subject Classification. Primary 11D61, 11Y65; Secondary 11A55, 11B83, 11K50, 11Y60, 41A60. 1 Kellner [19] in 2002 showed that also all primes 200 < p < 1000 have to divide k. Actually, Moree et al. [27, p. 814] proved a slightly stronger result and on combining this with Kellner's, one obtains that N 2 | k with N 2 = 2 8 · 3 5 · 5 4 · 7 3 · 11 2 · 13 2 · 17 2 · 19 2 · 23≤p≤997 p > 5.7462 · 10 427 .
For some further references and info on the Erdős-Moser equation we refer to the book by Guy [14, D7].
In this note we attack (1) using the theory of continued fractions. This approach was first explored in 1976 by Best and te Riele [3] in their attempt to solve the related conjecture of Erdős [11] that there are infinitely many pairs (m, k) such that Σ k (m) ≥ m k and 2(m − 1) k < m k . In this context they also gave the following variant of one of their results (without proof), namely, (3) with O(m −2 ) replaced with o(m −1 ). The proof we give here uses the same circle of ideas as used by Best and te Riele. It seems that after their work continued fractions in the Erdős-Moser context have been completely ignored. We hope the present paper makes clear that this is unjustified.
Theorem 1. For integer m > 0 and real k > 0 satisfying equation (1), we have the asymptotic expansion
k = log 2 m − 3 2 − c 1 m + O 1 m 2 as m → ∞,(3)
with c 1 = 25 12 − 3 log 2 ≈ 0.00389 . . . . Moreover, if m > 10 9 then
k m = log 2 1 − 3 2m − C m m 2 , where 0 < C m < 0.004.(4)Corollary 1. If (m, k) is a solution of (1) with k ≥ 2, then 2k/(2m − 3)
is a convergent p j /q j of log 2 with j even.
Corollary 2. The number of solutions m ≤ x of (1), as x tends to infinity, is at most O(log x).
The equation (1) seems to be a sole example of an exponential Diophantine equation in just two unknowns for which even the finiteness of solutions is not yet established. The best result in this direction is given by Corollary 2, which is an immediate consequence of the exponential growth of p j as a function of j and Corollary 1.
Corollary 1 is not the only result which relates convergents to solutions of Diophantine equations. For example, if (x 0 , y 0 ) is a positive solution to Pell's equation x 2 − dy 2 = ±1, with d a positive square-free integer, then x 0 /y 0 is a convergent of the continued fraction expansion of √ d. On the other hand, in our situation the number in question, log 2, is transcendental and its continued fraction expansion is expected to be sufficiently 'generic' (unlike that of quadratic irrationals).
Corollary 1 naturally leads us to investigate common factors of k and 2m − 3. This can be done using the method of Moser, but is not in the literature, as before there was no special reason for considering 2m − 3.
A key role in this arithmetic study is played by the congruence
l−1 j=1 j r y = 0 (mod 1 2 ) if r > 1 is odd; − p|l, p−1|r 1 p (mod 1) otherwise.(5)
This identity can be proved using the Von Staudt-Clausen theorem; for alternative proofs see, e.g., Carlitz [7] or Moree [25]. Its relevance for the study of (1) was first pointed out by Moree [26].
Given N ≥ 1, put
P(N) = {p : p − 1 | N} ∪ {p : 3 is a primitive root modulo p}.
By a classical result of Hooley [16] it follows, assuming the Generalized Riemann Hypothesis (GRH), that P(N) has a natural density A, with A = 0.3739558136 . . . the Artin constant, in the set of primes. If 2k/(2m − 3) = p j /q j is a convergent of log 2 arising in Corollary 1, then it can be shown that (q j , 6) = 1 and, if p ∈ P(N 2 ) and p divides q j , then ν p (q j ) = ν p (3 p−1 − 1) + 1 ≥ 2, where we write ν p (n) = a if p a | n and p a+1 ∤ n. All primes p ≤ 2017 are in P(N 2 ). For p = 3 we have ν p (3 p−1 − 1) = 1 unless 3 p−1 ≡ 1 (mod p 2 ), that is, p is a Mirimanoff prime. (It is known that the only Mirimanoff primes p < 10 14 are 11 and 1006003.)
The main idea of this paper is, in essence, to make use of the fact that the convergents p j /q j of log 2 have no reason to also satisfy N 2 | p j . The first piece of information comes from asymptotic analysis and the latter piece from arithmetic. Analysis and arithmetic give rise to conditions on the solutions that 'do not feel each other' and this is exploited in our main result: Theorem 2. Let N ≥ 1 be an arbitrary integer. Let
log 2 2N
= [a 0 , a 1 , a 2 , . . . ] = a 0 + 1 a 1 + 1 a 2 + · · · be the (regular) continued fraction of (log 2)/(2N), with p i /q i = [a 0 , a 1 , . . . , a i ] its i-th partial convergent.
Suppose that the integer pair (m, k) with k ≥ 2 satisfies (1) with N | k. Let j = j(N) be the smallest integer such that:
(a) j is even; (b) a j+1 ≥ 180N − 2; (c) (q j , 6) = 1; and (d) ν p (q j ) = ν p (3 p−1 − 1) + ν p (N) + 1 for all primes p ∈ P(N) dividing q j .
Then m > q j /2.
Computing many partial quotients (that is, continued fraction digits) of log 2 is closely related to computing log 2 with many digits of accuracy. Indeed, it is a well-known result of Lochs that for a generic number knowing it accurately up to n decimal digits implies that we can compute about 0.97n (where 0.97 ≈ 6(log 2)(log 10)/π 2 ) continued fraction digits accurately. For example, knowing 1000 decimal digits of π allows one to compute 968 continued fraction digits.
It seems a hopeless problem to prove anything about E(log q j(N ) ), the expected value of log q j(N ) produced by the result. However, metric theory of continued fractions offers some hope of proving a non-trivial lower bound for E(log q j(N ) (ξ)), where we require conditions (a), (b), (c) and (d) to be satisfied but replace (log 2)/(2N) by a 'generic' ξ ∈ [0, 1] \ Q. In this context recall the result of Lévy [21] that, for such a ξ,
lim j→∞ log q j (ξ) j = π 2 12 log 2 ≈ 1.18.(6)
The Gauss-Kuz'min statistics asserts that, for a generic ξ, the probability that a given term in its continued fraction expansion is at least b, equals log 2 (1 + 1/b). This allows one to deal with the case where we only have condition (b). Likewise a result of Moeckel [23], reproved in a very different way a few years later by Jager and Liardet [17], allows one to deal with the case where we only focus on condition (c). Their result says that for a generic ξ
∈ [0, 1] \ Q we have lim n→∞ {1 ≤ m ≤ n : q m (ξ) ≡ a (mod d)} n = d J(d) ϕ((a, d)) (a, d) ,
where ϕ denotes Euler's totient function, J(m) = m 2 p|m (1−1/p 2 ) Jordan's totient and (a, m) the greatest common divisor of a and m. This result shows that (q j , 6) = 1 with probability 1/2 (note that a natural number is coprime to 6 with probability 1/3). P. Liardet communicated to us that methods of his paper [22] can be used to take into account both conditions (a) and (c); also the authors of [15] claim that this can be done. We expect that there is a positive constant c 1 such that for a generic ξ satisfying conditions (a), (b) and (c), we have E(log q j(N ) (ξ)) ∼ c 1 N as N tends to infinity. Furthermore, we expect that for a generic ξ satisfying conditions (a), (b), (c) and (d), E(log q j(N ) (ξ)) ∼ c 2 N log β N for some positive constants c 2 and β; condition (b) is responsible for N, condition (d) for log β N, while conditions (a) and (c) affect c 2 . We are definitely not experts in metric aspects of number theory, thus leave this problem to the interested reader acquainted with the subject. Indeed, we even expect that going beyond computing the expected value of log q j(N ) (ξ) is possible, and a probability distribution for log q j(N ) (ξ) can be obtained.
Using the above results from metric theory of continued fractions and some heuristics we are led to believe that roughly speaking we can get m > 10 257N from Theorem 2. Being able to compute the convergents of (log 2)/(2N) arbitrarily far, we would expect (taking N = N 2 ) to show that m > 10 10 400 . With the current computer technology computing sufficiently many convergents is the bottleneck. Taking this into consideration we would expect to get m > 10 0.515r , from Theorem 2, where r is the number of convergents we can compute accurately and 0.515 is the base 10 logarithm of Lévy's constant (6). Note that the fact that N 2 has many divisors gives us some flexibility and increases the likelihood of the heuristics to be applicable. Indeed, our numerical experimenting agrees well with our heuristic considerations (see Section 4). Early 2009, A. Yee and R. Chan [31] reached r > 31 · 10 9 for log 2. On the other hand, Y. Kanada and his team [18] computed π to over 1.24 trillion decimal digits already in 2002, using formulae of the same complexity as those used for the computation of log 2 (see [2,Chapter 3] for details). Thus, given the present computer (im)possibilities, one could hope to show (with a lot of effort!) that m > 10 10 12 .
Applying Theorem 2 with N = 2 8 · 3 5 · 5 3 or N = 2 8 · 3 5 · 5 4 , and invoking the result of Moree et al. [27] that N | k, we obtain the following As an application we can show that ω(m − 1) ≥ 33, this improves on the result of Brenton and Vasiliu [5], who have shown that ω(m − 1) ≥ 26, where ω denotes the number of distinct prime divisors; see Section 5.1 for further details.
The fact N 2 | k naively implies that k is of size 10 427 (at least), which is much smaller than Moser's 10 10 6 . However, in this paper we show that the fact actually yields that k > 10 10 9 (and likely even k > 10 10 400 ) -a modestly small number dividing k leads to a huge lower bound for k. Thus, on revisiting [27] after 16 years, its main result is seen to be far more powerful than the second author thought at that time.
In the three following sections we prove Theorems 1, 2 and 3, respectively. Our final Section 5 is devoted to discussing some problems related to the Erdős-Moser equation.
Asymptotic dependence of k in terms of m
Our proof of Theorem 1 makes use of the following lemma.
Lemma 1. For any real k > 0, we have (1−y) k = e −ky 1− k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + k(5k − 6) 30 y 5 +O(y 6 ) as y → 0. (7)
Moreover, for k > 8 and 0 < y < 1, the inequality
e −ky 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + k(5k − 6) 30 y 5 − k 3 6 y 6 < (1 − y) k < e −ky 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + k 2 2 y 5(8)
holds.
Proof. As for the asymptotic relation in (7), we simply develop the Taylor expansion of (1 − y) k e ky up to y 5 . Unfortunately, estimates coming from the classical forms for the remainder are not sufficient to derive a sharp dependence on k as in (8) for the last term. Therefore, we need more drastic methods to quantify the asymptotics in (7) when 0 < y < 1. First note that
(1 − y)e y = 1 − ∞ n=2 n − 1 n! y n = 1 − y 2 2 − y 3 3 − y 4 8 − y 5 30 − · · · , 0 < y < 1.
Since all coefficients, starting from n = 2, in this power series are negative and their sum is exactly −1, for these values of y we have the inequality
1 − y 2 2 − y 3 3 − y 4 8 − y 5 30 − y 6 120 < (1 − y)e y < 1 − y 2 2 − y 3 3 − y 4 8 .(9)
The quantities
x 1 = y 2 2 + y 3 3 + y 4 8 and x 2 = y 2 2 + y 3 3 + y 4 8 + y 5 30 + y 6 120 ,(10)
which appear in (9), lie between 0 and 1 for 0 < y < 1. Our next ingredient is Gerber's generalization of the Bernoulli inequality [12] (see also Alzer [1]). It states that the remainder after k terms of the (possibly divergent) binomial series for (1 + x) a (a, x real with −1 < x) has the same sign as the first neglected term. In particular we have for real k > 2 and 0 < x < 1,
(1 − x) k < 1 − kx + k(k − 1) 2 x 2 ,(11)
and for real k > 3 and 0 < x < 1,
(1 − x) k > 1 − kx + k(k − 1) 2 x 2 − k(k − 1)(k − 2) 6 x 3 .(12)
Using the right inequality in (9) and taking x = x 1 in (11) we obtain, for k > 2,
(1 − y) k e ky < 1 − k y 2 2 + y 3 3 + y 4 8 + k(k − 1) 2 y 2 2 + y 3 3 + y 4 8 2 = 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + k(k − 1)y 5 1 6 + 17 144 y + 1 24 y 2 + 1 128 y 3 < 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + 385 1152 k(k − 1)y 5(13)
implying the upper estimate in (8). In the same vein, the application of the left identity in (9) and of (12) with x = x 2 results, for k > 3, in
(1 − y) k e ky > 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + k(5k − 6) 30 y 5 − ky 6 12 n=0
(a n k 2 + b n k + c n )y n ,
where the polynomials p n (k) = a n k 2 + b n k + c n , n = 0, 1, . . . , 12, all have positive leading coefficients a n ; moreover, p n (k) > 0 for k > 3 and n = 2, 3, . . . , 12, p 1 (k) = 1 24 k 2 − 11 60 k + 17 120 > 0 for k > 4, and p 0 (k) = 1 48 k 2 − 13 72 k + 121 720 > 0 for k > 8. Using this positivity of the polynomials we can continue the inequality in (14) for k > 8 as follows:
(1 − y) k e ky > 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + k(5k − 6) 30 y 5 − ky 6 12 n=0
(a n k 2 + b n k + c n )
= 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + k(5k − 6) 30 y 5 − ky 6 1 6 k 2 − 17 24 k + 11 20 ,(15)
from which we deduce the left inequality in (8), and the lemma follows.
Proof of Theorem 1. The original equation (1) is equivalent to
1 = m−1 j=1 1 − j m k .(16)
Applying to each term on the right-hand side the inequality from (8) we obtain
S 0 − k 2m 2 S 2 − k 3m 3 S 3 + k(k − 2) 8m 4 S 4 + k(5k − 6) 30m 5 S 5 − k 3 6m 6 S 6 < m−1 j=1 1 − j m k < S 0 − k 2m 2 S 2 − k 3m 3 S 3 + k(k − 2) 8m 4 S 4 + k 2 2m 5 S 5 ,(17)
with the notation
S 0 = m−1 j=1 z j = z 1 − z − z m 1 − z ,
the second term as well as its z d dz -derivatives are bounded:
0 < z m 1 − z z=e −k/m < 3e −k and 0 < z d dz n z m 1 − z z=e −k/m < 3 n+1
m n e −k , for n = 1, 2, . . . . (17) as
Therefore, we can write the inequality in
S ′ 0 − k 2m 2 S ′ 2 − k 3m 3 S ′ 3 + k(k − 2) 8m 4 S ′ 4 + k(5k − 6) 30m 5 S ′ 5 − k 3 6m 6 S ′ 6 − 3 3 k 2 + 3 4 k 3 + 3 7 k 3 6 e −k < m−1 j=1 1 − j m k < S ′ 0 − k 2m 2 S ′ 2 − k 3m 3 S ′ 3 + k(k − 2) 8m 4 S ′ 4 + k 2 2m 5 S ′ 5 + 3 + 3 5 k(k − 2) 8 + 3 6 k 2 2 e −k implying S ′ 0 − k 2m 2 S ′ 2 − k 3m 3 S ′ 3 + k(k − 2) 8m 4 S ′ 4 + k(5k − 6) 30m 5 S ′ 5 − k 3 6m 6 S ′ 6 − 500k 3 e −k < m−1 j=1 1 − j m k < S ′ 0 − k 2m 2 S ′ 2 − k 3m 3 S ′ 3 + k(k − 2) 8m 4 S ′ 4 + k 2 2m 5 S ′ 5 + 500k 2 e −k ,(18)
where
S ′ n = ∞ j=1 j n z j z=e −k/m = z d dz n z 1 − z z=e −k/m = (−1) n z d dz n 1
z − 1 z=e k/m for n = 0, 1, . . . ;
in particular,
S ′ 0 = 1 z − 1 , S ′ 2 = z + z 2 (z − 1) 3 , S ′ 3 = z + 4z 2 + z 3 (z − 1) 4 , S ′ 4 = z + 11z 2 + 11z 3 + z 4 (z − 1) 5 , S ′ 5 = z + 26z 2 + 66z 3 + 26z 4 + z 5 (z − 1) 6 , S ′ 6 =
z + 57z 2 + 302z 3 + 302z 4 + 57z 5 + z 6 (z − 1) 7 with z = e k/m . Since 500k 3 e −k < (2k) −3 < m −3 for k > m/2 > 30, using our equation (16) we can write the estimates (18) as
k(5k − 6) 30m 5 S ′ 5 − k 3 6m 6 S ′ 6 − 1 m 3 < 1 − S ′ 0 + k 2m 2 S ′ 2 + k 3m 3 S ′ 3 − k(k − 2) 8m 4 S ′ 4 < k 2 2m 5 S ′ 5 + 1 m 3 .(19)
Noting that e 1/2 < z = e k/m < e, we find 0 < S ′ 5 < e + 26e 2 + 66e 3 + 26e 4 + e 5 (e 1/2 − 1) 6 < 41438, 0 < S ′ 6 < e + 57e 2 + 302e 3 + 302e 4 + 57e 5 + e 6 (e 1/2 − 1) 7 < 658544, we continue (19) as follows:
1 − 1 z − 1 + k 2m 2 z + z 2 (z − 1) 3 + k 3m 3 z + 4z 2 + z 3 (z − 1) 4 − k(k − 2) 8m 4 z + 11z 2 + 11z 3 + z 4 (z − 1) 5 < 110000 m 3 ,(20)
where z = e k/m . We already know that k/m is bounded as m → ∞; making the ansatz k/m = c + O(1/m), hence z = e k/m = e c + O(1/m), we find from (20) that
1 − 1 e c − 1 = O 1 m as m → ∞,
hence e c = 2 and c = log 2. Now we take
k m = log 2 + a m + b m 2 + O 1 m 3 as m → ∞, hence z = e k/m = 2 + 2a m + a 2 + 2b m 2 + O 1 m 3 as m → ∞.
Substituting these formulas into (20) results in
+ log 2 + O(m −1 ) 3m 2 26 + O(m −1 ) 1 + O(m −1 ) − log 2 2 + O(m −1 ) 8m 2 150 + O(m −1 ) 1 + O(m −1 ) + O 1 m 3 = 2a + 3 log 2 m − 3a 2 − 3a + 13a log 2 − 2b + 75 4 log 2 2 − 26 3 log 2 m 2 + O 1 m 3 ,
hence a = − 3 2 log 2, b = (3 log 2 − 25 12 ) log 2 and, finally, we get the asymptotic formula (3).
To quantify this asymptotic expansion, we introduce the function
f m (C) = 1 − 1 z − 1 + λ 2m z + z 2 (z − 1) 3 + λ 3m 2 z + 4z 2 + z 3 (z − 1) 4 − λ(λ − 2/m) 8m 2 z + 11z 2 + 11z 3 + z 4 (z − 1) 5 z=e λ , where λ = λ(C) = log 2 1 − 3 2m − C m 2
agrees with our k/m up to O(m −2 ). Direct computation then shows that
f m (0) > 0.005m −2 − 100m −3 and f m (0.004) < −0.00015m −2 + 100m −3
for m ≥ 100. Therefore, f m (0) > 110000/m 3 for m > 2202 · 10 4 and f m (0.004) < −110000/m 3 for m > 734 · 10 6 , so that |f m (C)| < 110000/m 3 is possible only if 0 < C < 0.004. Comparing this result with (20) we conclude that, for k and m > 10 9 satisfying (16), we necessarily have
k m = log 2 1 − 3 2m − C m m 2 with 0 < C m < 0.004.
Clearly, the strategy to deduce further terms in the expansion (3) remains the same, but in order to achieve precision O(m −n ) for an integer n ≥ 2 we have to use the Taylor expansion of (1 − y) k e ky up to y 2n+1 (each new term in (3) requires two extra terms in the expansion of (1 − y) k e ky ). In this way we get
k = cm − 3 2 c − 25 12 c − 3c 2 m −1 + − 73 8 c + 61 2 c 2 − 25c 3 m −2 + − 41299 720 c + 657 2 c 2 − 598c 3 + 1405 4 c 4 m −3 + O(m −4 ) ≈ 0.69314718m − 1.03972077 − 0.00269758m −1 + 0.00323260m −2 + 0.00217182m −3 + O(m −4 ),(21)
where c = log 2. However, we do not possess any clear general strategy to quantify such expansions. Already proving a sharp dependence on k for the remainder of the n-th truncation of the Taylor expansion of (1 − y) k e ky (like we do for n = 4 in Lemma 1) seems to be a difficult task. We discuss related problems in Section 5.
Proof of Corollary 1. Let (m, k) be a non-trivial integer solution of (1). By Moser's result we know that m > 10 9 . It follows from Theorem 1 that
0 < log 2 − 2k 2m − 3 < 0.0111 (2m − 3) 2 .(22)
By Legendre's theorem, | log 2 − p/q| < 1/(2q 2 ) implies that p/q is a convergent of log 2, while log 2 > p/q insures that the index of the convergent is even. Thus, 2k/(2m − 3) is a convergent p j /q j of the continued fraction of log 2 with j even.
The proof of the main theorem
In this section we prove Theorem 2. The restrictions on the prime factorization of q j in that result are established using an argument in the style of Moser given in the proof of the following lemma.
Lemma 2. Let (m, k) be a solution of (1) with k ≥ 2. Let p be a prime divisor of 2m − 3. If p − 1 | k, then
ν p (2m − 3) = ν p (3 p−1 − 1) + ν p (k) + 1 ≥ 2.
If 3 is a primitive root modulo p, then p − 1 | k.
Proof. Using that k must be even, we find that
2m−4 j=1 j k ≡ m−1 j=1 j k + m−3 j=1 (2m − 3 − j) k ≡ m−1 j=1 j k + m−3 j=1 j k (mod 2m − 3) ≡ m k + m k − (m − 1) k − (m − 2) k ≡ 2(3 k − 1)(m − 1) k (mod 2m − 3), where we used that m k ≡ (2m − 3 + m) k ≡ 3 k (m − 1) k (mod 2m − 3) and (m − 2) k ≡ (2m − 3 − m + 1) k ≡ (m − 1) k (mod 2m − 3)
. On applying (5) with l = 2m − 3 and r = k we then obtain that
2(3 k − 1)(m − 1) k 2m − 3 ≡ − p|2m−3 p−1|k 1 p (mod 1).(23)
If p | 2m − 3 and p − 1 | k, the p-order of the right-hand side is −1. The p-order of the left-hand side must also be −1, that is, we must have
ν p (2m − 3) = ν p (3 k − 1) + kν p (m − 1) + 1 = ν p (3 p−1 − 1) + ν p (k) + 1,
where we used that m − 1 and 2m − 3 are coprime. Now suppose that p | 2m − 3 and 3 is a primitive root modulo p (thus p | 3 k − 1 implies p − 1 | k). If p − 1 ∤ k, the p-order of the left-hand side is ≤ −1 and > −1 on the right-hand side. Thus, we infer that p − 1 | k.
This completes the required ingredients needed in order to prove the main result.
Proof of Theorem 2. Since by assumption N | k, we can write k = Nk 1 and thus rewrite (22) as
0 < log 2 2N − k 1 2m − 3 < 0.0111 2N(2m − 3) 2 .(24)
We infer that k 1 /(2m−3) = p j /q j is a convergent to (log 2)/(2N) with j even. Since p | m implies p − 1 ∤ k (see, e.g., Moree [26, Proposition 9]), we have (6, q j ) = 1. We rewrite (24) as
0 < log 2 2N − p j q j < 0.0111 2Nd 2 q 2 j ,
with d the greatest common divisor of k 1 and 2m − 3. On the other hand,
log 2 2N − p j q j > 1 (a j+1 + 2)q 2 j ,
hence (a j+1 + 2) −1 < 0.0111/(2Nd 2 ), from which the result follows on also noting that 2m − 3 ≥ q j and invoking Lemma 2 (note that if ν p (q j ) ≥ 1, then ν p (q j ) = ν p (2m − 3) − ν p (k 1 )).
To prove that p | m implies p −1 ∤ k one uses that k must be even and takes l = m in (5), showing that p|m, p−1|k 1 p must be an integer. Since a sum of reciprocals of distinct primes can never be an integer, the result follows.
Computation of the continued fractions
We make use of conditions (a), (b), (c) of Theorem 2. We recall that we expect E(log q j(N ) (ξ)) ∼ c 1 N for a generic ξ ∈ [0, 1] satisfying these conditions. Indeed, on the basis of theoretical results, heuristics and numerical experiments, we conjecture that c 1 = 60π 2 . The computation of (log 2)/(2N) is done in two steps. First, we generate d digits of log 2. For this we use the γ-cruncher [31]. With this program, A. Yee and R. Chan computed 31 billion decimal digits of log 2 in about 24 hours. Second, we set a rational approximation of (log 2)/(2N) with a relative error bounded by 10 −d . Then partial quotients of the continued fraction of (log 2)/(2N) are computed: about 0.97d of them can be evaluated, with safe error control [4] (cf. the result of Lochs mentioned in Section 1). We maintain a floating point approximation of numbers q j (rounded down) and residues of q j (mod 6) by the formula q i+1 = a i+1 q i + q i−1 for i ≥ 0, where q 0 = 1 and q −1 = 0. Table 1 was created with the 'basic method' of [4] for N ≤ 2 8 · 3 4 . It was fast enough to reach the benchmark m > 10 10 7 in four days with 50 · 10 6 digits of log 2. Bit-complexity of this algorithm (or of the indirect or direct methods [4]) is quadratic and reaching the m > 10 10 10 milestone would take centuries.
N j = j(N) a j+1 q j (rounded down) q j mod 6 p = p(q j )
Some subquadratic GCD algorithms were discovered that have asymptotic running time O(n(log n) 2 log log n) [24]. A faster version of the program was written: this time a recursive HGCD method is applied. It is adapted for computing a continued fraction by using Lemma 3 of [4] (which is similar to Algorithm 1.3.13 of [8]) for error control. With it the program leaps over 10 10 8 in just about one hour. Finally, the new benchmark m > 10 10 9 is established in no more than 10 hours with 3 · 10 9 digits of log 2, N = 1555200 and condition (d): the first found solution fits conditions (a)-(c), but not (d). With N = 7776000, m > 10 10 9 is achieved for the smallest j. See Table 1: in the last column, p is a prime such that p ∈ P(N) and ν p (q j ) = 1, that is, such that condition (d) of Theorem 2 is violated. Now, computation time is not a problem to achieve the m > 10 10 10 milestone, a few days will be sufficient on a computer with a large amount of memory. We remark that the complexity and hardware requirement for computation of the digits of log 2, respectively for computation of its continued fraction expansion, are similar.
Miscellaneous
5.1. The number of distinct prime factors of m − 1. There is a different application of Theorem 3 suggested by the work of Brenton and Vasiliu [5], to factorization properties of the number m − 1 coming from a non-trivial solution (m, k) of (1). A result of Moser [28] (which can also be deduced from the key identity (5), cf. the proof of Lemma 2 above) asserts that
p|m−1 1 p + 1 m − 1 ∈ Z;(25)
in particular, the number m − 1 is square-free. Since the sum of reciprocals of the first 58 primes is less than 2, we conclude that either ω(m − 1) ≥ 58 or the integer in (25) is equal to 1. In the latter case, we can apply Curtiss' bound [9] for positive integer solutions of Kellogg's equation
n i=1 1 x i = 1,
namely, max i {x i } ≤ A n −1, where the Sylvester sequence {A n } n≥1 = {2, 3, 7, 43, . . . } is defined by the recurrence A n = 1 + n−1 i=1 A i (for some further info, see e.g. Odoni [29]). From this result and the estimate A n < (1.066 · 10 13 ) 2 n−7 , we infer m < (1.066 · 10 13 ) 2 ω(m−1)−6 , which together with the lower bound on m from Theorem 3 yields ω(m − 1) ≥ 33. A similar estimate on the basis of another (25)-like identity of Moser implies that ω(m + 1) ≥ 32.
5.2.
Generalized EM equation. The method we use in Section 2 for deriving the asymptotics of k in terms of m works for the more general equation
1 k + 2 k + · · · + (m − 1) k = tm k ,(26)
with t ∈ N fixed, as well. Indeed, the coefficients in the Taylor series expansion
(1 − y) k e ky = 1 − k 2 y 2 − k 3 y 3 + k(k − 2) 8 y 4 + · · · = ∞ n=0 g n (k)y n(27)
are polynomials satisfying g 0 (k) = 1, g 1 (k) = 0, and deg k g n (k) = n 2 , g n (0) = 0 for n ≥ 2;
and so on. Note that c n (−(t + 1)) = (−1) n+1 c n (t) for n = 0, 1, 2, . . . ; this reflects the equivalence of equation (26) and 1 k + 2 k + · · · + (m − 1) k + m k = (t + 1)m k .
From this asymptotics we see that
2k 2m − t 1 = c + t 3 1 c 2 − 2t 2 1 c − t 1 c 2 + 4c/3 2(2m − t 1 ) 2 + O 1 (2m − t 1 ) 3 ,(31)
where t 1 = 2t+1 and c = log(1+1/t). It can be checked that for all positive integers t we have the inequality −0.22 < t 3 1 c 2 − 2t 2 1 c − t 1 c 2 + 4c 3 < 0, and hence 2k/(2m − 2t − 1) is a convergent (with even index) of this logarithm c = log(1 + 1/t) for m large enough.
Saddle-point method.
A different approach to treat the asymptotic behaviour of k in terms of m for k and m satisfying (1) (or, more generally, (26)) is based on the integral representation
1 k + 2 k + · · · + (m − 1) k = Γ(k) 2πi C+i∞ C−i∞ e mz (e z − 1)z k+1 dz,
where C is an arbitrary positive real number (cf. [10, p. 273]). On noting that e mz e z − 1 = e (m−1)z 1 − e −C 1 + 1 − e z−C e z − 1 one obtains, on taking C = (k + 1)/(m − 1) and after invoking some rather trivial estimates, that 1 k + 2 k + · · · + (m − 1) k = (m − 1) k 1 − e −(k+1)/(m−1) 1 + ρ k (m) ,
with |ρ k (m)| < 2(k + 1)C √ π(k − 1)(e C − 1) .
(This part of the argument is due to Delange; for more details see [10, pp. 273-274].) By (2), C is bounded and we infer that |ρ k (m)| = O(k −1/2 ) = O(m −1/2 ). On putting m k on the left-hand side of (32) and using (1 − 1/m) m = exp(−1 + O(m −1 )), we immediately conclude that, as m → ∞,
k m = log 2 + O 1 √ m ,
where the implied constant is absolute. A more elaborate analysis, using the saddlepoint method, will very likely allow one as many terms in the latter expansion as required.
Experimental asymptotics.
It is worth mentioning a fast experimental approach of doing asymptotics like (21). Given numerically a few hundred terms of a sequence s = {s n } n≥1 that one believes has an asymptotic expansion in inverse powers of n, one can try to apply the asymp k trick, a simple but often powerful method to numerically determine the coefficients in the ansatz s n ∼ c 0 + c 1 n + c 2 n 2 + · · · . As a second step one tries to identify the so-found coefficients with (linear combinations of) known constants. Thus, one arrives at a conjecture that hopefully can be turned into a proof. For more details and some 'victories' achieved by the asymp k method, see Grünberg and Moree [13].
D. Zagier has applied this trick to the sequence of k = k(m) obtained from (1) on letting m run through the first thousand values. Excellent agreement with our theoretical results was obtained in this way.
Theorem 3 .
3If an integer pair (m, k) with k ≥ 2 satisfies (1), then m > 2.7139 · 10 1 667 658 416 .
By (2) we have e −1 < z < e −1/ 2
2, where z = e −k/m , and hence 1/(1 − z) < 1/(1 − e −1/2 ) < 3, and in the closed-form expression of the sum
Acknowledgements. The second author is very indebted to Jerzy Urbanowicz for involving him in the early nineties in his EM research (with[27]as visible outcome). The second and third author would like to thank D. Zagier for verifying some of our results using the asymp k trick and for some informative discussions regarding the saddle-point method (reflected in the final section). H. te Riele provided us with the unpublished report[3], which became the 'initial spark' for the current project. C. Baxa pointed out the relevance of[15]to us. Further thanks are due to T. Agoh and I. Shparlinski.This research was carried out whilst the third author was visiting in the Max Planck Institute for Mathematics (MPIM) and the Hausdorff Center for Mathematics (HCM) financially supported by these institutions. He and the second author thank the MPIM and HCM for providing such a nice research environment.(28)the latter follows from raising the series (1 − y)e y = 1 − y 2 /2 − y 3 /3 − · · · to the power k. In these settings, equation(26)becomeshence in the notation λ = k/m and x = 1/m we haveSearching λ in the form λ = c 0 + c 1 x + c 2 x 2 + · · · , we find successively
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| [] |
[] | [] | [] | [] | par J.-L. Colliot-Thélène Résumé. -Dans http://arxiv.org/abs/1410.5671, Jahnel et Loughrań etablissent un principe local-global pour l'existence d'espaces linéaires de dimension r dans les intersections complètes lisses de deux quadriques dans un espace projectif de dimension 2r + 2. On donne une démonstration alternative d'un résultat un peu plus général.Abstract. -In http://arxiv.org/abs/1410.5671, Jahnel and Loughran prove the local principle for linear spaces of dimension r on smooth complete intersections of two quadrics in projective space of dimension 2r+2. We present an alternative proof of a slightly more general result. | null | [
"https://arxiv.org/pdf/1503.06444v1.pdf"
] | 118,106,196 | 1503.06444 | d5b94e4664367232303f03bfacb6cbf13d48dc48 |
22 Mar 2015
22 Mar 2015arXiv:1503.06444v1 [math.NT]
par J.-L. Colliot-Thélène Résumé. -Dans http://arxiv.org/abs/1410.5671, Jahnel et Loughrań etablissent un principe local-global pour l'existence d'espaces linéaires de dimension r dans les intersections complètes lisses de deux quadriques dans un espace projectif de dimension 2r + 2. On donne une démonstration alternative d'un résultat un peu plus général.Abstract. -In http://arxiv.org/abs/1410.5671, Jahnel and Loughran prove the local principle for linear spaces of dimension r on smooth complete intersections of two quadrics in projective space of dimension 2r+2. We present an alternative proof of a slightly more general result.
Le théorème
Théorème 1.1. -Soit k un corps de nombres. Soient n un entier naturel et f (x 0 , . . . , x n ) et g(x 0 , . . . , x n ) deux formes quadratiquesà coefficients dans k. Supposons qu'il existe une forme non singulière dans le pinceau de formes quadratiques λf + µg. Dans chacun des cas suivants :
(a) n = 2r + 2, (b) n = 2r + 1, la k-variété X définie par f = g = 0 dans P n k contient un sous-espace linéaire défini sur k de dimension r si et seulement si elle en possède un sur chaque complété k v de k, pour v parcourant l'ensemble Ω des places de k.
Démonstration. -Soit t une variable. La forme quadratique q(t) := f + tg sur le corps k(t) est par hypothèse non dégénérée. Soit δ(t) ∈ k(t) × son discriminant.
D'après le théorème d'Amer [1,5,4], sur tout corps F contenant k, la F -variété X × k F contient un F -sous-espace linéaire de dimension r si et seulement si la quadrique dans P n F (t) définie par la forme quadratique q(t) = f + tg sur le corps F (t) s'annule sur un espace linéaire de dimension r. Comme la forme quadratique q(t) est non dégénérée, cette dernière propriétééquivaut au fait que la forme quadratique q(t) contient en facteur direct la somme orthogonale de r + 1 facteurs hyperboliques H =< 1, −1 >.
Pour n = 2r + 2, ceciéquivaut au fait que la forme quadratique f + tg est isomorpheà la forme quadratique q 1 (t) := (r + 1).H ⊥< (−1) r+1 .δ(t) > .
Pour n = 2r + 1, ceci estéquivalent au fait que la forme quadratique f + tg est isomorpheà la forme quadratique q 2 (t) := (r + 1
W (k(t)) → v∈Ω W (k v (t))
est injective. Le théorème de simplification de Witt permet de déduire de ce résultat l'énoncé : Si deux formes quadratiques sur k(t) sont isomorphes sur chacun des k v (t) pour v ∈ Ω, alors elles sont isomorphes sur k(t).
Ceci achève la démonstration. générique dans le pinceau. Soit C ⊂ P 3 Q une courbe de genre 1 contre-exemple au principe de Hasse, donnée par un système de deux formes quadratiques
Remarques
q 1 (x 1 , x 2 , x 3 , x 4 ) = q 2 (x 1 , x 2 , x 3 , x 4 ) = 0.
Les deux formes en 7 variables Q 1 = q 1 (x 1 , . . . , x 4 ) + x 5 x 6 et Q 2 = q 2 (x 1 , . . . , x 4 ) + x 5 x 7 s'annulent simultanément sur un espace vectoriel de dimension 3 sur chaque complété de Q mais pas sur Q. On le voit en appliquant le théorème d'Amer-Brumer [1,4]à Q 1 + tQ 2 . Ceci correspond au cas r = 2 et n = 2r + 2 = 6 du théorème 1.1.
De même, les deux formes en 10 variables Q 1 = q 1 (x 1 , . . . , x 4 ) + x 5 x 6 + x 8 x 9 et Q 2 = q 2 (x 1 , . . . , x 4 ) + x 5 x 7 + x 8 x 10 s'annulent sur un espace vectoriel de dimension 5 sur chaque complété de Q, mais pas sur Q. Ceci correspond au cas r = 4 et n = 2r + 1 = 9 du théorème 1.1.
Remarque 2. 1 .
1-Dans [JL], Thm. 1.5, Jahnel et Loughran considèrent le cas où X est une intersection complète lisse de deux quadriques et où n = 2r + 2. Ils montrent dans ce cas le résultat plus précis : si pour presque toute place v de k, la k-variété X × k k v contient un k v -espace linéaire de dimension r, alors elle contient un k-espace linéaire de dimension r. Ceci tient au fait que dans leur cas le principe local-global qui est utilisé est : unélément d'un corps de nombres K est un carré si et seulement si il l'est sur presque tout complété de K.
Remarque 2.2. -Dans le cas n = 2r + 1, si X est une intersection complète lisse de deux quadriques, la dimension maximale d'un sous-espace linéaire contenu dans X est r − 1 ([6, Cor. 2.4]). Le théorème est donc vide dans ce cas. Il n'est intéressant que pour X singulière.Remarque 2.3. -Comme me le fait observer David Leep (20 février 2015), dans le théorème 1.1 on ne peut pas omettre l'hypothèse de non singularité
M Amer, Quadratische Formenüber Funktionenkörpern, Dissertation. MainzJohannes Gutenberg UniversitätM. Amer, Quadratische Formenüber Funktionenkörpern, Dissertation, Johannes Gutenberg Universität, Mainz 1976.
Descente et principe de Hasse pour certaines variétés rationnelles. J.-L Colliot-Thélène, D Coray, J.-J Sansuc, J. reine und angew. Math. 320J.-L. Colliot-Thélène, D. Coray et J.-J. Sansuc, Descente et principe de Hasse pour certaines variétés rationnelles, J. reine und angew. Math. 320 (1980) 150-191.
J Jahnel, D Loughran, The Hasse principle for lines on Del Pezzo surfaces. J. Jahnel et D. Loughran, The Hasse principle for lines on Del Pezzo surfaces, http://arxiv.org/abs/1410.5671
D Leep, The Amer-Brumer theorem over arbitrary fields. D. Leep, The Amer-Brumer theorem over arbitrary fields http://www.ms.uky.edu/~leep/Amer-Brumer_theorem.pdf
A Pfister, Quadratic Forms with Applications to Algebraic Geometry and Topology. Cambridge University PressA. Pfister, Quadratic Forms with Applications to Algebraic Geometry and Topo- logy, Cambridge University Press (1995).
The complete intersection of two or more quadrics. M Reid, Ph.D. thesis, CambridgeM. Reid, The complete intersection of two or more quadrics, Ph.D. thesis, Cam- bridge, 1972. 19 novembre 2014
. J.-L Colliot-Thélène, C N R S , Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France • E-mailUniversité Paris SudJ.-L. Colliot-Thélène, C.N.R.S., Université Paris Sud, Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France • E-mail : [email protected]
| [] |
[
"Non-standard interactions with high-energy atmospheric neutrinos at IceCube",
"Non-standard interactions with high-energy atmospheric neutrinos at IceCube"
] | [
"Jordi Salvado \nInstituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain\n",
"Olga Mena \nInstituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain\n",
"Sergio Palomares-Ruiz \nInstituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain\n",
"Nuria Rius \nInstituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain\n"
] | [
"Instituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain",
"Instituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain",
"Instituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain",
"Instituto de Física Corpuscular (IFIC)\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValenciaSpain"
] | [] | Non-standard interactions in the propagation of neutrinos in matter can lead to significant deviations from expectations within the standard neutrino oscillation framework and atmospheric neutrino detectors have been considered to set constraints. However, most previous works have focused on relatively low-energy atmospheric neutrino data. Here, we consider the one-year highenergy through-going muon data in IceCube, which has been already used to search for light sterile neutrinos, to constrain new interactions in the µτ -sector. In our analysis we include several systematic uncertainties on both, the atmospheric neutrino flux and on the detector properties, which are accounted for via nuisance parameters. After considering different primary cosmic-ray spectra and hadronic interaction models, we obtain the most stringent bound on the off-diagonal εµτ parameter to date, with the 90% credible interval given by −6.0 × 10 −3 < εµτ < 5.4 × 10 −3 . In addition, we also estimate the expected sensitivity after 10 years of collected data in IceCube and study the precision at which non-standard parameters could be determined for the case of εµτ near its current bound.PACS numbers: 95.85. Ry, 14.60.Pq, 95.55.Vj, 29.40.Ka | 10.1007/jhep01(2017)141 | [
"https://arxiv.org/pdf/1609.03450v2.pdf"
] | 53,999,380 | 1609.03450 | 8dc747f3b8bc247d929a5f6a5bb7d4976e5ad906 |
Non-standard interactions with high-energy atmospheric neutrinos at IceCube
Jordi Salvado
Instituto de Física Corpuscular (IFIC)
CSIC-Universitat de València
Apartado de Correos 22085E-46071ValenciaSpain
Olga Mena
Instituto de Física Corpuscular (IFIC)
CSIC-Universitat de València
Apartado de Correos 22085E-46071ValenciaSpain
Sergio Palomares-Ruiz
Instituto de Física Corpuscular (IFIC)
CSIC-Universitat de València
Apartado de Correos 22085E-46071ValenciaSpain
Nuria Rius
Instituto de Física Corpuscular (IFIC)
CSIC-Universitat de València
Apartado de Correos 22085E-46071ValenciaSpain
Non-standard interactions with high-energy atmospheric neutrinos at IceCube
IFIC/16-64
Non-standard interactions in the propagation of neutrinos in matter can lead to significant deviations from expectations within the standard neutrino oscillation framework and atmospheric neutrino detectors have been considered to set constraints. However, most previous works have focused on relatively low-energy atmospheric neutrino data. Here, we consider the one-year highenergy through-going muon data in IceCube, which has been already used to search for light sterile neutrinos, to constrain new interactions in the µτ -sector. In our analysis we include several systematic uncertainties on both, the atmospheric neutrino flux and on the detector properties, which are accounted for via nuisance parameters. After considering different primary cosmic-ray spectra and hadronic interaction models, we obtain the most stringent bound on the off-diagonal εµτ parameter to date, with the 90% credible interval given by −6.0 × 10 −3 < εµτ < 5.4 × 10 −3 . In addition, we also estimate the expected sensitivity after 10 years of collected data in IceCube and study the precision at which non-standard parameters could be determined for the case of εµτ near its current bound.PACS numbers: 95.85. Ry, 14.60.Pq, 95.55.Vj, 29.40.Ka
I. INTRODUCTION
Neutrino oscillations have been robustly established over the past decades and this has been deservedly awarded during the last years. From neutrino oscillation experiments we know neutrinos have mass, which implies the first departure from the Standard Model (SM) of particle physics. Oscillation data provide information on the mixing angles and on the mass squared differences, which are, in the minimal and largely successful three-neutrino scenario, the solar mass splitting (∆m 2 12 7.5 × 10 −5 eV 2 ) and the atmospheric mass splitting (|∆m 2 23 | 2.5 × 10 −3 eV 2 ) [1][2][3]. Despite the enormous observational success achieved in constraining the leptonic mixing sector, there are still some unknowns in the neutrino mixing picture. Namely, the sign of the largest mass splitting remains unknown, as well as the octant of the mixing angle θ 23 and the possible existence of leptonic CP violation. Neutrino physics has already entered into the high-precision measurements era and subleading effects due to exotic couplings, affecting neutrino production, propagation and/or detection processes, may also appear in the neutrino sector [4]. These, so-called, non-standard interactions (NSI) have been subject of extensive work in the past years (for recent reviews see, e.g., Refs. [5,6]), both from a pure theoretical perspective (see, e.g., Refs. [7][8][9]) and with more phenomenological approaches, constraining their relative size with different experimental setups (see, e.g., Refs. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). Although constructing (SU (2) L ×U (1) Y gauge-invariant) models with large neutrino NSI and consistent with all current experimental constraints, mainly from charged-lepton flavor-violating processes, requires a certain amount of fine-tuning [9], they cannot be completely excluded. Therefore, from the phenomenological point of view, it is worth to exploit all available data to constrain neutrino NSI.
The relative size of NSI with respect to standard neutrino oscillations depends on the neutrino energy. At very low (sub-GeV) energies, the NSI terms are sub-dominant with respect to standard (vacuum) neutrino oscillations. At intermediate energies, O(1 − 10) GeV, NSI can interfere with the standard matter potential and vacuum oscillation terms, modifying the neutrino propagation through the Earth. At higher energies, NSI effects may dominate. Notice, however, that such an energy dependence is different if the NSI are due to light mediators (see, for instance, Refs. [27,28]). In this case the effects depend on the high-energy, gauge-invariant, completion of each scenario. We will not consider this possibility, since our analysis is based on model-independent four-fermion effective operators, that we assume to be generated above the electroweak scale. Therefore, exploiting the NSI energy dependence over a large range of energies and baselines seems a promising way of constraining these new potential neutrino interactions. As the presence of the NSI affects neutrino propagation in a medium, having a large range of available neutrino baselines crossing the Earth would help enormously in disentangling standard oscillations from NSI. Thus, atmospheric neutrinos provide a unique and ideal tool to test and constrain the size of NSI effects , as their spectrum covers a huge energy range (∼ 0.1 − 10 5 GeV) and, depending on their arrival direction, they may travel distances across the Earth ranging from tens to several thousands kilometers. Most works in the literature have focused on the capabilities of past and current [31-39, 41, 44-46, 48, 49] or future detectors [40-43, 45, 47, 50] using atmospheric neutrino events in the O(10 GeV) energy range, where interference effects may take place. In particular, the Super-Kamiokande (SK) collaboration (exploiting the sub-GeV, multi-GeV, stopping and through-going muon samples) obtained the most stringent bounds on the diagonal and off-diagonal NSI parameters in the µτ -sector [39,46] and recently, the IceCube Collaboration has also presented a preliminary analysis [49] using the DeepCore three-year muon disappearance result [51], with slightly more restrictive limits than SK.
With the development of neutrino observatories, NSI searches via atmospheric neutrino fluxes benefit from larger detector sizes (and consequently, larger atmospheric neutrino event samples) and major improvements in energy reconstruction at neutrino energies above O(10 GeV), and up to O(10 PeV), although they have higher energy thresholds. This has been the main goal of Refs. [40,41,43,44,48,49], where the iceČerenkov IceCube neutrino observatory and/or its low energy extensions, the DeepCore or the future PINGU detectors, have been considered as the ideal targets where to test neutrino NSI, exploiting atmospheric neutrino fluxes. On the other hand, although at high-energies the standard neutrino oscillation phase, which is inversely proportional to the neutrino energy, is very small, in the presence of NSI, oscillations are not suppressed with energy and they only depend on the baseline. The data from the 79-string configuration in IceCube [52] was used to set constraints on NSI in Ref. [41].
Here we perform an analysis of the NSI effects on the propagation of high-energy atmospheric neutrinos by considering the publicly available IceCube one-year upgoing muon sample [53], referred to as IC86 (IceCube 86-string configuration), which contains 20145 muons detected over a live time of 343.7 days. We focus on the high-energy region of the atmospheric neutrino spectrum, and thus our results are complementary to those of previous analyses of IceCube data [41,48,49], some of them dealing exclusively with the low-energy atmospheric neutrino sample observed at the DeepCore detector [48,49]. In order to perform the analysis, we use the public IceCube Monte Carlo 1 that models the detector realistically and allows us to relate physical quantities, as the neutrino energy and direction, to observables, as the reconstructed muon energy and zenith angle. To account for some possible systematic uncertainties on the atmospheric neutrino flux, neutrino parameters and detector properties, we also include a number of nuisance parameters. We obtain the most stringent limits to date on the off-diagonal NSI parameter ε µτ . Finally, we also present a forecast of the sensitivity to NSI from future high-energy atmospheric neutrino data. We simulate 10 years of collected data in IceCube and assess how the bounds would improve and how well the presence of NSI could be determined, in case they exist.
The paper is organized as follows. In Sec. II we briefly review the NSI formalism relevant for the data we consider, i.e., for high-energy atmospheric neutrinos crossing the Earth, and describe the main features of the NSI effects. Then, in Sec. III we describe the data we use and explicitly show the potential effects of NSI on this type of observations. In Sec. IV, we first describe the likelihood and the different systematic uncertainties included in the analysis, presenting then the current bounds on NSI using the one-year through-going muon IceCube data. We finish that section by discussing the prospects for future limits with improved statistics (10 years of data) and we summarize our findings in Sec. V.
II. FORMALISM
We consider neutrino NSI that are generated by new physics above the electroweak scale, so that at low centerof-mass energies, E m W (or, equivalently, E m X , where m X is the mass of the heavy mediator), they can be described via model-independent four-fermion effective operators. These can be of neutral current (NC) type [4],
L NC NSI = −2 √ 2 G F ε f P αβ (ν α γ ρ Lν β )(f γ ρ P f ) ,(1)
where ε f P αβ are the NC NSI parameters (by hermiticity ε f P αβ = (ε f P βα ) * ), P = {L, R} (with L and R the left and right quirality projectors) and f is any SM fermion, as well as of charged-current (CC) type [4,54],
L CC NSI = − 2 √ 2 G F ε δσP αβ (ν α γ ρ Lν β )(¯ δ γ ρ P σ ) ,(2)L CCq NSI = − 2 √ 2 G F ε qq P αβ (ν α γ ρ L β )(qγ ρ P q ) + h.c. ,(3)
where ε δσP αβ and ε qq P αβ are the leptonic and hadronic CC NSI parameters (for the leptonic case, δ = σ corresponds to NC NSI), β is a charged lepton of flavor β, q is a down-type quark and q an up-type quark. In what follows, we neglect possible CP violation in the new interactions (this has been considered in different contexts [14,35,38,), so we take all NSI parameters ε f P αβ , ε δσP αβ and ε qq P αβ to be real. In the literature, NC NSI are frequently called matter NSI, since they modify neutrino propagation through matter, while the CC ones are referred to as production and detection NSI. Moreover, given that the neutrino flavor is always tagged through the flavor of the charged lepton associated with it, in the presence of CC NSI the neutrino flavor basis is not well-defined [4,54], since the neutrino detected or produced in association with a charged lepton does not necessarily share its flavor. In this case, flavor conversion is present at the interaction level, and the standard oscillation formulae become more cumbersome [54,56]. Model-independent bounds on both, NC and CC NSI have been derived in Ref. [20], where it was found that, in general, the limits on production and detection NSI are one order of magnitude more stringent than those on matter NSI. Model-dependent bounds in several new physics scenarios [4,54] also indicate that constraints on CC NSI are typically much more stringent. Therefore, we shall neglect CC NSI and concentrate only on NC NSI in the following.
The standard evolution Hamiltonian for neutrinos includes the coherent forward scattering on fermions of the type f , ν α + f → ν β + f , given by the matter interaction potential (defined in Eq. (5) below), which affects neutrino oscillations. However, neutrinos propagating through the Earth can also interact inelastically with matter, either via CC or NC processes. As the neutrino-nucleon cross section increases with energy, for energies above ∼TeV, the neutrino flux gets attenuated [82,83]. Whereas in the case of ν e 's and ν µ 's, the neutrino flux is absorbed via CC interactions and redistributed (degraded in energy) via NC interactions [84], in the case of ν τ 's, there is another effect. Unlike what happens for ν e and ν µ CC interactions, where charged leptons are quickly brought to rest and do not contribute to the high-energy neutrino flux, the tau leptons produced after ν τ CC interactions can decay before being stopped, so ν τ 's are not absorbed, but the flux gets regenerated (at lower energies) [85][86][87][88][89][90][91]. Thus, for each ν τ which is absorbed via CC interactions, another ν τ with lower energy is produced, and the Earth does not become opaque to high-energy ν τ 's. In addition, secondary ν e 's and ν µ 's are also produced after tau leptons decay into leptonic channels [92,93]. For high-energy neutrinos, oscillation, attenuation and regeneration effects occur simultaneously when they travel across the Earth, and the evolution equations should, in principle, include them. Notice that conventional neutrino oscillation analyses do not take into account attenuation and regeneration effects, which is a good approximation, provided the energy of the detected neutrinos is low enough. Nevertheless, this is not the case for the high-energy IceCube sample of atmospheric neutrinos we consider in this work, for which attenuation needs to be included. On the contrary, for atmospheric neutrinos, the effects of ν τ regeneration and production of secondary ν e and ν µ fluxes are very small. The explanation is two fold. On one hand, ν τ 's are very rarely produced after cosmic-ray interactions in the atmosphere, and therefore the atmospheric ν τ flux is negligible. On the other hand, these effects are only relevant for very hard spectra. Therefore, for the sake of computational time, we shall not include ν τ regeneration in this study, which nevertheless implies negligible corrections.
In what follows, we use the density matrix,
ρ(E ν , x) = ν(E ν , x) ⊗ ν(E ν , x) † , formalism,
where E ν is the neutrino energy and x the path variable. In the case of neglecting neutrino regeneration, the density matrix for neutrinos traversing the Earth obeys the evolution equation [94] dρ
(E ν , x) dx = −i[H(E ν , x), ρ(E ν , x)] − α 1 2 λ α (E ν , x) {Π α (E ν ), ρ(E ν , x)} + ∞ Eν ρ(E ν , x) 1 n N (x) dσ NC (E ν , E ν ) dE ν dE ν , (4) where Π α is the ν α projector, λ α (E ν , x) = 1/[n N (x) σ tot α (E ν )]
is the attenuation length of ν α , with n N (x) the nucleon number density in the Earth 2 and σ tot α (E ν ) the ν α total (CC+NC) cross section, and dσ NC /dE ν is the differential NC cross section. The first term on the right-hand side represents neutrino oscillations, the second term neutrino absorption and the third term the redistribution of the flux due to NC interactions.
In the presence of NSI, the effective Hamiltonian that controls neutrino propagation in matter can be written as
H(E ν , x) = 1 2E ν U M 2 U † + diag(V e , 0, 0) + f V f ε f V ,(5)
where U is the PMNS mixing matrix, M 2 = diag(0, ∆m 2 21 , ∆m 2 31 ), with ∆m 2 ij ≡ m 2 i − m 2 j the neutrino mass square differences and V e (x) = √ 2 G F n e (x) corresponds to the standard neutrino flavor potential in matter, with n e (x) the electron number density. The effect of NSI is encoded in the last term of Eq.
(5), where V f (x) = √ 2 G F n f (x)
, with n f (x) the number density of fermion f , and ε f V is the matrix in lepton flavor space that contains the vector combination of the NSI chiral parameters, ε f V αβ = ε f R αβ + ε f L αβ . As in the case of SM interactions, the matter term for antineutrinos changes sign and one has to make the substitution V f → −V f (and U → U * ). On the other hand, it is convenient to define effective NSI parameters for a given medium (from now on we omit for simplicity the x dependence of the number densities) by normalizing the fermion number density, n f , to the density of d-quarks, n d ,
ε αβ ≡ f n f n d ε f V αβ ,(6)
so that f V f ε f V ≡ V e r ε = V d ε, and r = n d /n e . For the Earth, n n ≈ n p and therefore, r ≈ 3. Given the current constraints on the electron neutrino NSI parameters ε eα , and, for energies above the resonance in the 13-sector (E ν 20 GeV), one of the mass eigenstates (mostly ν e orν e ) decouples from the other two states. Therefore, the ν e → ν µ transition does not affect the IceCube events as it is strongly suppressed and moreover, the initial atmospheric ν e andν e fluxes are much smaller than the ν µ andν µ fluxes. Thus, we can approximately describe the evolution of the system as that of a two-neutrino system, focusing on the 23-block of the evolution Hamiltonian, Eq. (5). Recall that neutrino oscillations are only sensitive to the difference in the diagonal effective parameters, i.e., ε = ε τ τ − ε µµ , which modifies the oscillation probability due a change of the effective matter density felt by neutrinos, while the off-diagonal term, ε µτ , shifts the effective mixing angle in the medium. The diagonal parameter ε characterizes the lack of universality of NC in the µτ -sector, and the off-diagonal ε µτ quantifies the strength of flavor changes in NC interactions.
Before discussing the main features of the transition probabilities at high energies, we would like to point out that the effects of NSI in high-energy atmospheric neutrinos in IceCube differ from the standard approach at lower energies in two ways: 1) Usually, only NSI of neutrinos with quarks and leptons of the first generation can be bounded, via the V f ε f V contributions to the matter Hamiltonian and via the ε udV contributions to CC interactions with pions and nucleons, in addition to the ε eµV contributions at production via muon decay. However, very energetic neutrinos (E ν TeV) can see the strange quark contribution inside nucleons, since for such high energies the strange quark parton distribution function is not negligible. As a consequence, there is an effective energy dependence of production and detection NSI terms (if NSI do not affect all quark flavors with the same strength) through the different contribution of the corresponding parton distribution at different energies. As mentioned above, here we do not consider CC NSI and, as done in the literature, we assume the NC NSI parameters to be equal for all quarks inside the nucleons. Relaxing these assumptions, IceCube data could also be used to bound strange quark CC NSI with neutrinos, by properly taking into account the energy dependence of the s quark contribution to the parton distribution functions.
2) Matter NSI could also modify the total inelastic scattering cross section, by altering the NC cross section, and thus, the absorption term in Eq. (4). Attenuation is negligible at low energies, but it is relevant for the highenergy IceCube neutrinos, so there could be some sensitivity to the presence of NSI. However, CC interactions are ∼ 2.4 times larger than the NC cross sections [82,83], so the latter dominate the absorption term and the effect of NC NSI can be safely neglected. Moreover, CC NSI could also be present, but for the values of the CC NSI parameters currently allowed, the NSI effects on attenuation would be very small, implying corrections to the results presented in this work below the percent level. On the other hand, by modifying the NC cross section, NSI would also alter the degradation in energy of the neutrino flux while crossing the Earth. Given the fact that for atmospheric neutrinos this effect is subdominant, we also neglect the NSI correction on the last term of Eq. (4).
In our calculations, we solve numerically the full three-neutrino evolution equation, using the values of the neutrino mixing parameters from Ref. [3] assuming normal hierarchy and including the effects mentioned above. To compute the neutrino propagation through the Earth, we use the publicly available libraries SQuIDS and ν-SQuIDS [95,96] in the Trunk version found in the repositories [97,98]. Nevertheless, in order to understand the effects of the diagonal (ε ) Comparison of the ratios of atmospheric νµ (solid lines) andνµ (dashed lines) fluxes at the detector (after propagation) with NSI to those without NSI, for εµτ = 0.003 (thin blue lines) and 0.006 (thick red lines). In both panels, the ratios are shown for cos θz = −1 and we have chosen ε = 0. For illustration, we show the gray area, which corresponds to the energy interval that produced 90% of the events in the entire sample considered here in the absence of NSI effects.
and off-diagonal (ε µτ ) NSI parameters in the energy range we consider, it is interesting to note that, for atmospheric neutrinos, the interplay of neutrino oscillations and attenuation in the Earth can be well described by an overall exponential suppression in the oscillated fluxes, i.e.,
φ α (E ν , θ z ) = φ 0 µ (E ν , θ z ) P (ν µ → ν α ; E ν , L(θ z )) exp{− L(θz) 0 dx/λ α (E ν , x)} ,(7)
where φ 0 µ is the atmospheric ν µ (orν µ ) flux before entering the Earth and L(θ z ) is the baseline across the Earth in a direction with zenith angle θ z . In this way, it is illustrative to study analytically the oscillation probabilities in the approximation of constant matter density (assuming constant NSI parameters), i.e., the solution of the evolution equation neglecting attenuation and energy degradation, regeneration and secondary production (see Ref. [41] for a detailed discussion) and for constant density. In this case, the two-neutrino oscillation probability after propagating over a distance L, P (ν µ → ν τ ) = 1 − Tr{Π µ ρ}, is given by [99]
P (ν µ → ν τ ) = sin 2 2θ mat sin 2 ∆m 2 31 L 4 E ν R ,(8)
where
R 2 = 1 + R 2 0 + 2 R 0 cos 2(θ 23 − ξ) ,(9)sin 2 2θ mat = (sin 2θ 23 + R 0 sin 2ξ) 2 R 2 ,(10)
with
R 0 = φ mat φ vac = V NSI L/2 ∆m 2 31 L/4E ν ,(11)V NSI = V d 4 ε 2 µτ + ε 2 ,(12)sin 2ξ = 2 ε µτ 4 ε 2 µτ + ε 2 .(13)
For the energies we consider in this work (E ν > 100 GeV), neutrino oscillations in vacuum are suppressed for baselines comparable to or smaller than the Earth diameter, φ vac ≡ ∆m 2 31 L/4E ν 1. Therefore, in the case in In both panels we choose the NSI off-diagonal parameter to be εµτ = 0.006 and the diagonal parameter to be ε = 0. We also show two gray lines, which bound the energy interval in which 90% of the events in the entire sample ( assuming no NSI) are produced.
which the vacuum and matter terms in the oscillation phase are of the same order of magnitude, i.e., R 0 = O(1), the transition probability approximately reads
P (ν µ → ν τ ) (sin 2θ 23 + R 0 sin 2ξ) 2 φ 2 vac = sin 2θ 23 ∆m 2 31 2 E ν + 2 V d ε µτ 2 L 2 2 .(14)
Considering normal hierarchy, for neutrinos (R 0 sin 2ξ > 0) this probability is enhanced and thus, the ν µ flux is suppressed with respect to the case without NSI (vacuum oscillations, R 0 sin 2ξ = 0), whereas for antineutrinos (R 0 sin 2ξ < 0) it is the other way around. It is interesting to note that inČerenkov detectors like IceCube, neutrinos cannot be distinguished from antineutrinos, so this effect tends to partially cancel out, as we will see below. These differences can be clearly seen for neutrino energies E ν = O(100 GeV) in both panels of Fig. 1, where we show the effect of neutrino propagation through the Earth with and without NSI. This regime corresponds to the low-energy part of the event sample we consider, as illustrated by the gray area in both panels, which represents the energy interval that produced 90% of the events in the entire sample assuming no NSI (5% upper cut and 5% lower cut). Notice that in this section we use true neutrino energies and zenith angles, whereas the IceCube events are described by reconstructed variables which differ from the true ones, mainly in the case of the reconstructed energy, which is always smaller than the true neutrino energy.
In the left panel of Fig. 1, we plot the ratio of the propagated to unpropagated atmospheric ν µ (solid lines) andν µ (dashed lines) fluxes with (thick red lines) and without (thin green lines) NSI, whereas in the right panel, we show the ratios of atmospheric ν µ andν µ fluxes at the detector (after propagation) with NSI to those without NSI. In the left panel we show the case of ε µτ = 0.006 (thick red lines) and no NSI, ε µτ = 0, (thin green lines) and in the right panel we depict the ratios for two representative values of the NSI off-diagonal parameter ε µτ = 0.003 (thin blue lines) and 0.006 (thick red lines). In both panels, we consider muon neutrinos and antineutrinos traversing the entire Earth, cos θ z = −1, and we have set ε = 0 (sin 2 2ξ = 1). In the left panel, in addition to the effect of oscillations at low energies (even without NSI), we can clearly see the effect of attenuation (and the subdominant degradation in energy via NC interactions) at higher energies. These curves approximately represent the product of the oscillation and attenuation terms in the right-hand side of Eq. (7). The differences between the neutrino and antineutrino results are two fold: at low energies the oscillation probabilities and, at all energies in the plot, the total cross sections, and thus, the attenuation factors, are different for neutrinos and antineutrinos. On the other hand, the effect of attenuation is factored out in the right panel, which approximately represents the ratio of the survival probabilities with and without NSI. Note that, at first order, the transition probability for energies E ν = O(100 GeV), Eq. (14), is independent of ε and thus, there is little sensitivity to the diagonal NSI parameter. In the limit of ε ε µτ (sin 2ξ 0), at high energies, vacuum mimicking is realized [100], but the oscillation phase is suppressed. Hence, there is significantly more sensitivity to ε for E ν < 100 GeV [22, 31-38, 40-45, 47, 50].
On the other hand, in the high-energy limit, the matter term dominates over vacuum oscillations, i.e., φ mat φ vac or R 0 1. In this regime, the two-neutrino oscillation probability, Eq. (8), is approximately given by
P (ν µ → ν τ ) sin 2 2ξ sin 2 φ mat ,(15)
where
φ mat = V d L 2 4 ε 2 µτ + ε 2 30 ρ 8 g/cm 3 L 2 R ⊕ 4 ε 2 µτ + ε 2 ,(16)
with R ⊕ the radius of the Earth. Then, for φ mat 1, the transition probability
P (ν µ → ν τ ) sin 2 2ξ φ 2 mat = (ε µτ V d L) 2(17)
is proportional to ε 2 µτ and becomes independent of ε [101], and the same result holds for antineutrinos. This is also clearly seen in the high-energy regime shown in the right panel of Fig. 1, where one can see that the neutrino and antineutrino ratios of (approximately) oscillation probabilities coincide. As a consequence, the high-energy IceCube atmospheric neutrino sample cannot significantly constrain the diagonal NSI parameter ε , so in our analysis we use information on ε based on the SK limits [39] (see below), obtained from its potential effects at lower energies using the zenith distribution.
Finally, in Fig. 2, in addition to the energy dependence, we also show the dependence on the zenith angle of the ratios depicted in Fig. 1. The effect of NSI in the neutrino propagation through the Earth is illustrated for ε µτ = 0.006 and ε = 0. In the left panel, analogously to the left panel of Fig. 1, we show the ratio between the initial atmospheric ν µ flux and the ν µ flux in the detector. In the right panel, we isolate the effect produced by the NSI on the oscillation probabilities, as in the right panel of Fig. 1, displaying the ratio between the final fluxes for ε µτ = 0.006 and ε µτ = 0., i.e., with and without NSI. Note that the left vertical axis (cos θ z = −1) in both panels corresponds to the red solid lines in both panels of Fig. 1. We clearly see the well-known effects of attenuation that shift to higher energies for more horizontal trajectories (left panel) and the NSI-induced oscillations of the atmospheric ν µ flux, which represent a flux suppression that, at high energies, only depends on the zenith angle (right panel).
III. DATA DESCRIPTION
The IceCube data we consider in this paper is the same sample used to search for light sterile neutrino signatures [53]. It contains 20145 events detected during 343.7 days of live data in the period 2011-2012 using the full IceCube 86-string configuration. These events correspond to upgoing neutrinos from the Northern hemisphere, which are dominantly produced by atmospheric ν µ andν µ CC interactions with nucleons of the material surrounding the detector, the so-called through-going muon tracks. The contamination from other sources is found to be below the 0.1% level [53]. The reconstructed muon energies in the detector of this sample lie in the range E rec µ (300 GeV − 20 TeV) and the neutrino energies mostly contributing to these events are indicated by the gray bands (lines) in Fig. 1 (Fig. 2).
Through-going track events are produced after ν µ andν µ CC interactions produce muons outside the instrumented volume, that traverse the detector while depositing energy along their trajectory (the track). At these energies, muons travel along (almost) the same direction of the parent neutrino, which is reconstructed with very good angular resolution (within one degree or better, i.e., σ cos θz 0.005−0.015 [53]). On the other hand, due to radiative losses, the fact that the position of the interaction vertex is unknown implies a large uncertainty in the estimation of the initial muon energy, which in turn is always smaller than the incoming neutrino energy. The muon energy when entering the detector is estimated based on the energy losses along the track [102] with a resolution of σ log(Eµ/GeV) ∼ 0.5 [53].
For our analysis, we use the high-statistics Monte Carlo released by the IceCube collaboration along with the data, which allows us to relate the true variables (neutrino energy and direction) to the reconstructed observables (deposited energy and track zenith angle) and to do a realistic treatment of the detector systematic uncertainties, which are described below.
In order to understand the different features in the neutrino propagation induced by the NSI effects, and discussed in detail in previous sections, we simulate 1000 realizations of mock data corresponding to one year of observation. In the left panel of Fig. 3, we show the expected difference in the number of events between the hypotheses without NSI (ε µτ = ε = 0) and that in which NSI are included, with ε µτ = 0.006 (and ε = 0), as a function of the reconstructed muon energy E rec µ and zenith angle θ rec z . It is clear from that panel that the largest differences in the expected number of events with and without NSI occur for neutrinos crossing the core of the Earth with energies ∼TeV.
Although Fig. 3 clearly illustrates the region in the parameter space which is sensitive to the NSI effects, we also quantify it statistically by defining a Poisson likelihood for each i-th bin (defined by an interval in E rec µ and θ rec z )
L i = e − N sim i N sim i N sim i N sim i ! ,(18)
where N sim i (N sim i ) refers to the average over all realizations of the number of simulated events (number of simulated events for an individual realization) in the i-th bin. On the right panel of Fig. 3, we show the expected average over all realizations of the log-likelihood difference between the null (assuming no NSI) and the NSI hypotheses (for ε µτ = 0.006 and ε = 0). Notice that the two panels of Fig. 3 look very similar, which indicates that the impact of NSI is what pulls the statistical significance, rather than the higher statistics around the horizon, with shows a negligible dependence on NSI effects. Therefore, as already anticipated, the most sensitive region in the reconstructed variables is that corresponding to neutrinos that travel through the core of the Earth, i.e., cos θ rec z −0.8, with reconstructed energies for which the data sample has the higher statistics, i.e., E rec µ ∼ O(TeV). This is expected, as the NSI effects turn out to be approximately energy independent.
Indeed, this energy independence can be clearly noticed from the results shown in the right panel of Fig. 4, where we depict the ratio of neutrino plus antineutrino events including NSI (ε µτ = 0.006 and ε = 0) to the number of events without NSI (i.e., with ε µτ = 0 and ε = 0). In analogy to the right panel of Fig. 1, in Fig. 4, we also show the ratios of neutrino (left panel) and antineutrino (middle panel) events including NSI (ε µτ = 0.006 and ε = 0) to either neutrino or antineutrino events without NSI (ε µτ = 0 and ε = 0). As expected, when NSI are at play, the ratio of events grows with energy for neutrinos and decreases with energy for antineutrinos in a very similar manner, and, consequently, once these two contributions are summed up (representing the measurable quantity in the IceCube neutrino telescope), their energy dependence approximately cancels out.
IV. RESULTS
In this section we describe the results arising from our analyses. Firstly, we describe the different ingredients that enter into the definition of the likelihood and then we show the results obtained with the current one-year through-going muon IceCube data [53]. Finally, we also perform forecast analyses with 10 years of simulated data considering two different hypotheses, with or without NSI.
A. Analysis methodology
Our analyses include several nuisance parameters that take into account systematic uncertainties in the atmospheric neutrino flux, in the neutrino parameters and in the detector properties. We include nuisance parameters for the normalization of the atmospheric neutrino flux, N , for the pion-to-kaon ratio in the atmospheric neutrino flux, π/K, and for the spectral index of the atmospheric neutrino spectrum, ∆γ. Furthermore, we include a nuisance parameter that accounts for uncertainties in the efficiency of the digital optical modules of the detector, DOM eff . As for the neutrino parameters, we also take into account the current uncertainties in ∆m 2 31 and θ 23 . In addition, other potentially important systematic errors come from uncertainties in the primary cosmic-ray flux and the hadronic interaction models. Our default choice for most of the results presented below is the combined Honda-Gaisser model and Gaisser-Hillas H3a correction (HG-GH-H3a) for the primary cosmic-ray flux [103] and the QGSJET-II-4 hadronic model [104], although we also consider the Zatsepin-Sokolskaya (ZS) flux [105] and the SIBYILL2.3 hadronic model [106].
The uncertainty on the flux normalization represents an overall normalization of the number of events which affects equally all bins in relative terms, and we allow it to vary freely within a factor of 2 (larger than current uncertainties [107,108]) of the central value. It is important to fit this parameter because it can be significantly different from one, mainly for the ZS primary cosmic-ray flux. The pion-to-kaon ratio affects the relative contribution to the neutrino flux from pion or kaon decays. A larger value π/K implies a softer spectrum, as the neutrino flux from kaon decays is harder. We use π/K normalized to one and consider a Gaussian prior of 10%. The uncertainty on the spectral index represents a tilt in the energy spectrum of the atmospheric neutrino flux with a pivot energy near the median of the neutrino energy distribution (so this correction is not very correlated with the normalization), and we apply a Gaussian prior with a 5% error. Finally, the uncertainty in the optical efficiency affects the determination of the reconstructed energy, so that a larger DOM eff implies a shift to larger energies. For this nuisance parameter we consider a flat prior, which in practice equals to allow it to float freely.
On the other hand, as we discussed above, the high-energy IceCube atmospheric neutrino sample cannot significantly constrain the diagonal NSI parameter ε , so we constrain this parameter by means of the SK limits, which were obtained using data at lower energies. The SK bound reads [39],
|ε | = |ε τ τ − ε µµ | < 0.049 , 90% confidence level (C.L.) ,(19)
and from Fig. 4 in Ref. [39], we set the 1σ C.L. prior on ε to σ ε = 0.040. To quantitatively assess the power of the high-energy atmospheric neutrino one-year IceCube data to constrain NSI in neutrino propagation in matter, we perform a likelihood analysis using all the events in the data sample and characterizing each event by its reconstructed muon energy and zenith angle. The full likelihood is defined as the bin product of the Poisson probability of measuring N data i for the expected value N th i times the product of Gaussian probabilities for the pulls of the nuisance parameters. The log-likelihood (up to a constant) is given by
ln L(ε µτ , ε ; η) = i∈bins N data i ln N th i (ε µτ , ε ; η) − N th i (ε µτ , ε ; η) − ε 2 2 σ 2 ε − j (η j − η 0 j ) 2 2 σ 2 j ,(20)
where the subindex i refers to a bin in E rec µ and cos θ rec z , N th i (ε µτ , ε ; η) is the expected number of evens for a given value of the NSI (ε µτ and ε ) and nuisance (η ≡ {N, π/K, ∆γ, DOM eff , ∆m 2 31 , θ 23 }) parameters in the i-th bin, and N data i is the number of data events in the same i-th bin. The index j corresponds to the nuisance parameters with Gaussian prior (π/K, ∆γ, ∆m 2 31 and θ 23 ) and σ j is the Gaussian error. To compute the likelihood for a given value of the parameters, we first propagate the neutrino fluxes from the atmosphere to the detector for both neutrinos and antineutrinos, then we weigh the events from the IceCube Monte Carlo with the propagated flux, which is a function of the true neutrino energy E ν and the zenith angle θ z , and we construct two-dimensional histograms as a function of the reconstructed variables: E rec µ and θ rec z (20 energy and 20 angular bins). With this likelihood, we perform a Bayesian analysis using the MultiNest nested sampling algorithm [109][110][111] in the NSI and nuisance parameter space. All the parameters, together with their range of variation and the type of prior considered, are summarized in Tab. I.
B. Current bounds
The results, using our default models for the primary cosmic-ray spectrum and hadronic interactions, are shown in Fig. 5, where we depict the 68% and 95% credible contours (posterior probabilities). Concerning the NSI parameter ε µτ , which is the main goal of this paper, its correlation with the continuous systematic parameters we consider is small. This is somehow expected, as in the O(TeV) energy range, the main signature of the presence of matter NSI is via the distortion of the angular distribution of the atmospheric neutrino events and all these systematics mostly affect the atmospheric neutrino energy spectrum, modifying very little its angular distribution. Notice indeed that most of the parameters are not much correlated among themselves, an exception being the pion-to-kaon ratio (π/K) and the flux normalization (N ), which show a clear anticorrelation.
From this analysis, using the high-energy atmospheric neutrino IceCube data, we obtain the most stringent bound on ε µτ to date,
− 6.0 × 10 −3 < ε µτ < 5.4 × 10 −3 , 90% credible interval (C.I.).(21)
The interval is rather symmetric with respect to zero, as the NSI effects depend mainly on ε 2 µτ . Our result improves over the SK limit [39,46],
|ε µτ | < 1.1 × 10 −2 90% C.L. (SK) ,(22)
over the result of a preliminary analysis of three-year DeepCore data [49],
− 6.7 × 10 −3 < ε µτ < 8.1 × 10 −3 90% C.L. (DeepCore) ,(23)
and it is very similar to that obtained in Ref. [41] using 79-string IceCube configuration and DeepCore data,
− 6.1 × 10 −3 < ε µτ < 5.6 × 10 −3 , 90% C.L. (IC79 + DeepCore) ,(24)
although note that we have included a number of nuisance parameters not considered in Ref. [41]. Posterior (68% and 95%) probability contours for the NSI parameters εµτ and ε along with several nuisance parameters, using the one-year through-going muon IceCube data [53]. On the right panels, we also depict the one-dimensional posterior probability distribution of the parameter corresponding to each column. In all the panels we also include the uncertainties on ∆m 2 31 and θ23.
To further assess the lack of correlation of ε µτ with the nuisance parameters and the stability of our results with respect to their variation, in the left panel of Fig. 6 we overimpose the contours obtained when fixing all nuisance parameters to their default values (see Tab. I) to those shown in Fig. 5, where they are varied as described above. It is clear that these systematics affect very little the final bound on ε µτ , which gets modified as
for the most optimistic case of not including systematic uncertainties in the analysis. This is a more fair comparison with the results of Ref. [41]. We also study the impact of using different primary cosmic-ray spectra and different hadronic interaction models on our results. As discussed above, neutrino NSI in matter may produce a suppression in the high-energy upgoing atmospheric muon data in IceCube, with a characteristic angular dependence (and little energy dependence). Different combinations of primary cosmic-ray spectrum and hadronic models imply slightly different angular distributions for the atmospheric neutrino flux and hence, potentially, they are an important source of systematic uncertainties on the NSI sensitivity reach of neutrino telescopes. In the right panel of Fig. 6 we show the results for different choices of cosmic-ray spectra and hadronic interaction models. We depict the posterior probabilities for ε µτ , marginalized with respect to the rest of the nuisance parameters and ε , for each of the four possible combinations. Indeed, the allowed range of ε µτ arising from our default combination of models (HG-GH-H3a + QGSJET-II-4) turns out to be very similar to the resulting ones from all possible combinations, whose bounds on ε µτ are:
− 5.5 × 10 −3 < ε µτ < 5.1 × 10 −3 , 90% C.I. (HG-GH-H3a + SIBYLL2.3) ,(26)−6.2 × 10 −3 < ε µτ < 5.8 × 10 −3 , 90% C.I. (ZS + SIBYLL2.3) .(27)
Finally, in Fig. 7, we show the event spectrum, integrated in the entire interval in reconstructed muon energy 3 , as a function of cos θ rec z . We show the histogram of the detected through-going atmospheric muon events after one year (black dots), together with their error bars and the expectation for the cases without NSI (red histogram) and when NSI are included with ε µτ = 0.006 and ε = 0 (blue histogram). We indicate the uncertainty due to the choice of the primary cosmic-ray spectrum and hadronic models by the width of the histograms, although the variation is very small. In all cases we consider the best fit values for the parameters. As discussed in previous sections, we see that the presence of NSI implies a suppression of the atmospheric neutrino flux, and hence the observed through-going muon spectra, for neutrinos crossing the core of the Earth, i.e., for cos θ z −0.8.
All our results are summarized in Tab. II. including NSI with εµτ = 0.006 and ε = 0 (blue solid histogram). The uncertainties due to the choice of the primary cosmic-ray spectrum and hadronic models are represented by the width of the histograms.
C. Forecast analyses
Finally, in order to assess the capabilities of the IceCube detector when using the high-energy atmospheric neutrino data to constrain matter NSI, we perform two forecast analyses for 10 years of simulated data. Therefore, we simulate 10 years of data and use the same priors on all parameters as described above, except from ∆m 2 31 and θ 23 which we fix to their best fit values and from ε which we take σ ε = 0.03, and our default combination of primary cosmic-ray and hadronic interaction models (HG-GH-H3a + QGSJET-II-4). The results are shown in Fig. 8 and in the left panel of Fig. 6.
On one hand, we simulate data assuming the case without NSI to be the true case realized in Nature (blue contours). As expected from current bounds (see Fig. 5), the systematics described by the nuisance parameters we consider do not play a significant role for future analyses, although some small correlations start to show up more clearly, which partially limit the expected reach. After 10 years of collecting data, we would expect ε µτ to be constrained in the interval − 3.3 × 10 −3 < ε µτ < 3.0 × 10 −3 90% C.I. (10-year forecast analysis).
Note that, with a factor of 10 more statistics, the improvement in the limits on NSI will be of about a factor of two. This can also be seen in the left panel of Fig. 6, where we show the 68% and 95% credible contours in the ε µτ − ε plane (black closed curves).
On the other hand, it is also interesting to consider the discovery potential in case of the presence of matter NSI. In order to do this, we simulate 10 years of data including NSI assuming that Nature has chosen ε µτ = 0.006 (which represents a value allowed by current data with about 90% probability) and ε = 0. As it is clear from Fig. 8, for this large value of ε µτ (red contours), future IceCube measurements would be able to detect the presence of matter NSI at a high significance, although the quadratic ε µτ -dependence of the effects would render impossible to determine the sign of ε µτ . FIG. 8. Posterior (68% and 95%) probability contours for two 10-year forecasts of high-energy atmospheric neutrino data in IceCube. We show the results assuming the data corresponds to the case without NSI (blue contours) and when the data includes NSI with εµτ = 0.006 and ε = 0 (red contours). On the right panels, we also show the one-dimensional probability distribution of the parameter corresponding to each column. The atmospheric neutrino parameters ∆m 2 31 and θ23 are fixed to their current best fit values.
V. SUMMARY AND CONCLUSIONS
The IceCube neutrino telescope, along with its low-energy extension DeepCore, is currently the leading experiment to detect high-energy neutrinos. After a few years of operation, statistics have been accumulated and a number of studies of atmospheric neutrinos have been performed. Atmospheric neutrino flux measurements have been carried out in a wide range of energies [112][113][114][115][116][117], low-energy atmospheric neutrino data have been considered to constrain neutrino oscillation parameters to levels comparable to other neutrino oscillation experiments [51,52], low-energy data have also been used to set constraints on matter NSI [49], and light sterile neutrinos, claimed as possible explanations of short baseline neutrino anomalies [118][119][120][121][122][123][124][125], have been searched for with atmospheric neutrinos with energies up to O(10 TeV) [53].
Although there are still unknowns within the standard picture of neutrino oscillations, new interactions in the neutrino sector, driven by dimension six (or higher) operators, could also give rise to sub-leading effects in neutrino production, propagation and detection. In this work we have considered the high-energy atmospheric neutrino data (previously used to search for sterile neutrinos in Ref. [53]), i.e., high-energy through-going muon events, to evaluate the impact of matter NSI in the neutrino propagation through the Earth. Our analysis represents the most stringent limit on the µτ -sector off-diagonal parameter ε µτ to date and it is complementary to other studies in the literature, which focus on atmospheric neutrino events at lower energies [34-39, 41, 44-46, 48, 49] and to the results obtained using the 79-string IceCube configuration [41].
We have first reviewed the formalism of matter NSI, which is relevant for high-energy atmospheric neutrinos (Sec. II). Although we have computed neutrino propagation in a full three-neutrino framework taking into account Earth attenuation and degradation in energy due to NC interactions, at the energies we consider, the description in terms of a two-neutrino system represents a very good approximation, as the (mostly) ν e state decouples. Therefore, we have described the main features within this approximation as two-neutrino oscillations in the µτ -sector. We have illustrated the effect of flavor transitions, with and without NSI, and attenuation by depicting ratios of propagated to unpropagated neutrino and antineutrino fluxes (left panels of Figs. 1 and 2) and have also isolated the effect of NSI by considering ratios of propagated fluxes with and without NSI (right panels of Figs. 1 and 2).
Then, we have briefly described the high-energy upgoing muon sample used to perform our analysis (Sec. III) and, to understand the features previously discussed at the level of fluxes, we have also studied their impact on the current observables, which cannot distinguish neutrinos from antineutrinos. In order to do so, we have simulated mock data and have shown the difference of expected number of events with and without NSI (left panel of Fig. 3) and the statistical pulls of NSI effects (right panel of Fig. 3) as a function of observables: reconstructed muon energy and zenith angle. As the energy dependence of the NSI effects for neutrinos and antineutrinos either tend to cancel out (at low energies in our sample) when both contributions are summed up, or are the same (at higher energies), the dominant distortion of the total event spectrum occurs in the angular distribution (Fig. 4).
In the likelihood we define to perform our statistical analysis (Sec. IV A), we have also included several nuisance parameters to describe systematic uncertainties on the atmospheric neutrino flux, on the neutrino parameters and on the detector properties, and have used a prior on the value of the diagonal NSI parameter ε from SK measurements (Tab. I). Our results (Sec. IV B) turn out not to be very correlated with any of the continuous parameters we consider (Fig. 5) and we obtain a limit on the off-diagonal term ε µτ (Tab. II). The bound, for the combination of primary cosmic-ray and hadronic models HG-GH-H3a + QGSJET-II-4, is − 6.0 × 10 −3 < ε µτ < 5.4 × 10 −3 90% C.I. ,
which represents the most stringent limit on this parameter to date.
This limit is very stable with respect to all the continuous nuisance parameters we consider, which can be safely fixed to their default values without affecting significantly the bound (left panel of Fig. 6). On the other hand, we find the main source of systematic uncertainties to lie in the choice of the combination of primary cosmic-ray and hadronic interaction models (right panel of Fig. 6). This is explained by the angular dependence of the event spectrum, which is slightly different for each of these combinations. We have also shown this uncertainty by depicting the event spectrum with and without NSI (Fig. 7) as a function of the zenith angle, accounting for the range of the four combinations we consider. Finally, we have also performed a forecast with simulated data for 10 years in IceCube (Sec. IV C) and noted that, although limits will not improve dramatically (within a factor of two), in case of the existence of large NSI, consistent with current limits, IceCube high-energy atmospheric neutrino data can establish its presence at high confidence (Fig. 8).
Unveiling the values of the parameters of the neutrino sector with high precision requires experimental setups that allow us to test different ranges of energies and baselines. In particular, it requires high-precision measurements, sensitive to non-canonical, sub-leading effects, as those caused by the potential existence of NSI. Neutrino telescopes as IceCube are sensitive to a wide range of energies and baselines and provide an excellent tool to explore possible new neutrino interactions beyond the standard neutrino oscillation paradigm by means of high-energy atmospheric neutrinos. Higher statistics will allow us to further test this scenario and in this regard, a future high-energy extension of the IceCube detector [126] and the planned KM3NeT telescope [127] will have a crucial role.
FIG. 1 .
1Left panel: Comparison of the ratios of propagated to unpropagated atmospheric νµ (solid lines) andνµ (dashed lines) fluxes for values of the NSI off-diagonal parameter εµτ = 0.006 (thick red lines) and εµτ = 0 (thin green lines). Right panel:
FIG. 2 .
2Left panel: Ratio of propagated to unpropagated atmospheric νµ fluxes as a function of the neutrino energy and the zenith angle. Right panel: Ratio of atmospheric νµ fluxes at the detector (after propagation) with NSI to those without NSI.
FIG. 3 .
3Left panel: Difference of the expected number of events (from neutrinos and antineutrinos) between the case of εµτ = 0.006 and εµτ = 0, as a function of reconstructed energy and zenith angle. Right panel: Statistical pulls as a function of reconstructed energy and zenith angle for the same set of parameters. In both panels we set ε = 0.
FIG. 4 .
4Left panel: Ratio of the number of events produced by neutrinos including NSI (εµτ = 0.006 and ε = 0) to that without NSI (εµτ = 0 and ε = 0), as a function of the reconstructed muon energy and zenith angle. Middle panel: Same but for antineutrinos. Right panel: Same but for the total number of events, i.e., neutrino plus antineutrino events.
FIG. 5. Posterior (68% and 95%) probability contours for the NSI parameters εµτ and ε along with several nuisance parameters, using the one-year through-going muon IceCube data [53]. On the right panels, we also depict the one-dimensional posterior probability distribution of the parameter corresponding to each column. In all the panels we also include the uncertainties on ∆m 2 31 and θ23.
− 5.1 × 10 −3 < ε µτ < 4.3 × 10 −3 ,90% C.I. (no systematic uncertainties) ,
FIG. 6 .
6Left panel: Comparison of the 68% and 95% credible contours in the εµτ − ε plane for our default analysis (filled blue regions) with those obtained when all nuisance parameters are fixed at their default values (red closed curves), see Tab. I. We also show the result expected in the case of no NSI after 10 years of data taking (black closed curves), see Sec. IV C. Right panel: Posterior probabilities of εµτ , after marginalizing with respect to the rest of parameters, for the four combinations of primary cosmic-ray spectrum and hadronic models: our default choice, HG-GH-H3a + QGSJET-II-4 (black solid curve); HG-GH-H3a + SIBYLL2.3 (red dashed curve); ZS + QGSJET-II-4 (blue dot-dashed curve) and ZS + SIBYLL2.3 (green dotted curve).
−6. 0 ×
010 −3 < ε µτ < 5.1 × 10 −3 , 90% C.I. (ZS + QGSJET-II-4) ,
FIG. 7 .
7Event spectrum: data points (black dots and error bars), expected results without NSI (red solid histogram) and
TABLE II. Mean value and standard deviation for the parameters and systematics of this analysis, for each of the four combinations of primary cosmic-ray flux and hadronic models.Parameter
Default value
Range
Prior
Description
εµτ
0.006
[−1, 1]
Flat
NSI flavor off-diagonal term
ε
0
[−1, 1]
Gaussian: σ = 0.04 NSI flavor diagonal term
N
1
[0.5, 2.0]
Flat
Normalization of the energy spectrum
π/K
1
[0.7, 1.5]
Gaussian: σ = 0.10 Pion-to-kaon ratio contribution
∆γ
0
[−0.2, 0.2] Gaussian: σ = 0.05 Tilt of the energy spectrum
DOM eff
0.99
[0.90, 1.19]
Flat
Optical efficiency
∆m 2
31 /10 −3 [eV 2 ]
2.484
[2.3, 2.7]
Gaussian: σ = 0.048 Atmospheric mass square difference
θ23 [ • ]
49.3
[43.0, 54.4]
Gaussian: σ = 1.7
Atmospheric mixing angle
TABLE I. Parameters, default values for plots (using the HG-GH-H3a primary cosmic-ray flux and the QGSJET-II-4 hadronic
model), their range of variation and priors (flat or Gaussian) for the different systematics considered in our statistical analysis.
HG-GH-H3a + QGSJET-II-4 HG-GH-H3a + SIBYLL2.3 ZS + QGSJET-II-4
ZS + SIBYLL2.3
Parameter
Mean
Std. dev.
Mean
Std. dev.
Mean Std. dev.
Mean Std. dev.
εµτ
-0.0004
0.0034
0.0001
0.0035
-0.0005
0.0036
-0.0002
0.0035
ε
0.000
0.047
-0.003
0.045
0.002
0.046
0.001
0.046
N
1.013
0.056
0.911
0.051
1.257
0.066
1.123
0.063
π/K
1.078
0.084
1.059
0.080
1.073
0.080
1.067
0.083
∆γ
-0.050
0.013
-0.092
0.013
0.066
0.012
0.102
0.012
DOM eff
0.9869
0.0064
0.9863
0.0061
0.9910
0.0061
0.9885
0.0058
∆m 2
31 /10 −3 [eV 2 ]
2.484
0.046
2.485
0.047
2.487
0.044
2.480
0.043
θ23 [ • ]
49.3
1.8
49.3
1.7
49.3
1.7
49.2
1.7
https://icecube.wisc.edu/science/data/IC86-sterile-neutrino
In general, one has to include all possible targets, as electrons, but at the energies of interest in this work, interactions with electrons have a negligible effect in the attenuation of the neutrino flux.
As the effect of NSI, at high energies, depends very mildly on the reconstructed muon energy, varying the energy range included inFig. 7does not change the relevant features.
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| [] |
[
"Entropy Minimizing Matrix Factorization",
"Entropy Minimizing Matrix Factorization"
] | [
"Mulin Chen ",
"Fellow, IEEEXuelong Li "
] | [] | [] | Nonnegative Matrix Factorization (NMF) is a widely-used data analysis technique, and has yielded impressive results in many real-world tasks. Generally, existing NMF methods represent each sample with several centroids, and find the optimal centroids by minimizing the sum of the approximation errors. However, the outliers deviating from the normal data distribution may have large residues, and then dominate the objective value seriously. In this study, an Entropy Minimizing Matrix Factorization framework (EMMF) is developed to tackle the above problem. Considering that the outliers are usually much less than the normal samples, a new entropy loss function is established for matrix factorization, which minimizes the entropy of the residue distribution and allows a few samples to have large approximation errors. In this way, the outliers do not affect the approximation of the normal samples. The multiplicative updating rules for EMMF are also designed, and the convergence is proved both theoretically and experimentally. In addition, a Graph regularized version of EMMF (G-EMMF) is also presented to deal with the complex data structure. Clustering results on various synthetic and real-world datasets demonstrate the reasonableness of the proposed models, and the effectiveness is also verified through the comparison with the state-of-the-arts. | 10.1109/tnnls.2022.3157148 | [
"https://arxiv.org/pdf/2103.13487v1.pdf"
] | 232,352,320 | 2103.13487 | b3e51834099db2f32b492762a2de910ea509bda6 |
Entropy Minimizing Matrix Factorization
Mulin Chen
Fellow, IEEEXuelong Li
Entropy Minimizing Matrix Factorization
Index Terms-Nonnegative Matrix factorizationentropy lossrobustnessclustering !
Nonnegative Matrix Factorization (NMF) is a widely-used data analysis technique, and has yielded impressive results in many real-world tasks. Generally, existing NMF methods represent each sample with several centroids, and find the optimal centroids by minimizing the sum of the approximation errors. However, the outliers deviating from the normal data distribution may have large residues, and then dominate the objective value seriously. In this study, an Entropy Minimizing Matrix Factorization framework (EMMF) is developed to tackle the above problem. Considering that the outliers are usually much less than the normal samples, a new entropy loss function is established for matrix factorization, which minimizes the entropy of the residue distribution and allows a few samples to have large approximation errors. In this way, the outliers do not affect the approximation of the normal samples. The multiplicative updating rules for EMMF are also designed, and the convergence is proved both theoretically and experimentally. In addition, a Graph regularized version of EMMF (G-EMMF) is also presented to deal with the complex data structure. Clustering results on various synthetic and real-world datasets demonstrate the reasonableness of the proposed models, and the effectiveness is also verified through the comparison with the state-of-the-arts.
INTRODUCTION
N ONNEGATIVE MATRIX FACTORIZATION (NMF) is a popular unsupervised machine learning technique for handling matrix data. Based on the matrix factorization theory [1], Lee and Seung [2] imposed the nonnegative constraint to learn the parts-of-whole interpretations. After that, NMF has atrracted sufficent attention due to its simplicity and interpretability, and shown encouraging performance in many real-world tasks, such as face recognition [3], document analysis [4], hyperspectral imagery [5] and recommendation systems [6].
Specifically, NMF approximates the nonnegative data matrix with the product of two nonnegative factor matrices. One is consist of the basis vectors, and another one is regarded as the coefficient matrix. By minimizing the approximation error, each sample is represented by the linear combination of the basis vectors, and the sample is associated with the basis which contributes the most to the representation. Therefore, the basis vectors act as the cluster centroids, and the coefficient matrix can be regarded as the cluster indicator. In addition, benefited from the nonnegative constraint, NMF allows only additive operation. Consequently, a parts-based representation is achieved, which is able to provide an interpretable understanding about the input data.
Over the past decades, NMF have been studied from a wide variety of perspectives. For example, researchers have proved the connection between NMF and some popular machine learning techniques [7], [8], [9], such as kmeans, spectral clustering and linear discriminant analysis. A number of techniques [10], [11], [12], [13], [14] have been proposed to perform NMF in the subspace. Some works improved NMF by exploiting the data geometry [3], [15], [16], [17], [18], while some others deal with the missing data [19], [20], [21], [22]. Recently, deep NMF [5], [6], [23], [24] has became an attractive research area. Despite the its salient properties and wide usage, NMF has some major drawbacks. In this paper, we devote to tackle the robustness problem. NMF is sensitive to the outliers because it employs the least square error function as the objective. For the outliers, which deviate from the normal distribution, their approximation errors are squared and may dominate the objective function. As a result, they affect the final results seriously. To improve the robustness, some variants of NMF have been presented. In stead of using Frobenius-norm, Ke and Kanade [25] proposed the 1 -norm NMF. The influence of the outliers is alleviated, since the approximation errors are summed up directly without taking the square. To maintain the feature rotation invariance, Kong et al. [26] utilized the 2,1 -norm formulation. Recently, Qi et al. [27] calculated the residue with the logarithmic loss function. Therefore, the objective value increases more slowly with the approximation error. The above methods weaken the effect of the outliers by utilizing different loss functions. However, the effects of outliers still exist. If the error is extremely large, the outliers will affect the results as well. Gao et al. [28] designed the capped norm matrix factorization model. They found the outliers directly by thresholding the approximation error, and set their residues as a constant. This strategy remove the outliers thoroughly, but it is unrealistic to find a suitable threshold for various real-world applications.
In this paper, an Entropy Minimizing Matrix Factorization framework (EMMF) is presented to improve the robustness. Different from the previous works, we do not approximate all the samples. A new entropy loss function is designed, which models the whole distribution of the residues and avoids the effect of outliers naturally. The proposed loss function could also be applied in other tasks involving matrix computation. In order to preserve the intrinsic geometry, the Graph regularized EMMF (G-EMMF) is also developed. The main contributions made in this study arXiv:2103.13487v1 [eess.IV] 24 Mar 2021 are summarized as follows.
1. We design a general entropy loss function for matrix factorization. By minimizing the entropy of the residue distribution, the proposed EMMF allows a few samples to be with relatively large errors, and focuses on approximating the most of the rest. Therefore, the outliers do not affect the updating of centroids. 2. We provide the efficient optimization algorithms for the proposed framework. The optimal solution can be obtained by the multiplicative updating rules with proved convergence. The computation costs of the optimization algorithm is almost the same as NMF, which guarantees the practicability for real-world tasks. 3. We conduct extensive experiments to validate the reasonableness and effectiveness of the proposed framework. As demonstrated by the results, the objective function is insensitive to the outliers with extremely large errors, and it works well for the data without outliers. The proposed G-EMMF also outperforms the existing graph-regularized NMF methods.
The paper is organized as follows. Section 2 reviews some existing NMF algorithms. Section 3 introduces the EMMF formulation, and provides the corresponding optimization algorithm. Section 4 presents the G-EMMF. Section 5 gives the experimental results of EMMF, and discusses its advantages. Section 6 shows the clustering performance of G-EMMF. Section 7 concludes this article Notations: in this paper, we write the matrices as uppercase and write the vectors as lowercase. For a matrix A, its (i, k)-th element is defined as A ik . Its i-th row, column are denoted as a i,: and a i respectively. The trace of A is defined as Tr(A). The transpose of A and a i are indicated by A T and a T i . I is the identity matrix. The ρ norm of a i is calculated as
||a i || ρ = d k=1 |A ki | ρ (ρ > 0).
PRELIMINARY
In this section, we revisited the formulation of NMF and some existing methods. Numerous algorithms have been proposed to improve NMF from different aspects, and we mainly focus on the robust variants.
Nonnegative Matrix Factorization
Supposing the data matrix is X = [x 1 , x 2 , · · · , x n ] ∈ R d×n and the desired centroid number is c, NMF aims to find the nonnegative matrices U ∈ R d×c and V ∈ R n×c which satisfy X ≈ UV T . The least square error objective function is formulated as
min U≥0,V≥0 ||X − UV T || 2 F = n i=1 ||x i − Uv T i,: || 2 2 ,(1)
where || · || F is the Frobenius norm. Lee and Seung derived the multiplicative updating rules for the above problem:
U ik ← U ik (XV) ik (UV T V) ik , V jk ← V jk (X T U) jk (VU T U) jk .
Instead of using Euclidean distance, another commonly used NMF uses the "divergence" as the loss function:
min U≥0,V≥0 DIV(X||UV T ),(2)
where
DIV(A||B) = d i=1 n k=1 (A ij log Aij Bij − A ij + B ij )
is called as the divergence measurement. Similarly, the updating rules are given as
U ik ← U ik n j=1 X ij V jk /(UV T ) ij n j=1 V jk , V jk ← V jk d i=1 X ij U ik /(UV T ) ij d i=1 U ik ,
By solving U and V iteratively, the local optimal solution of both problem (1) and (2) will be found. Besides, the problems can be decompose into the approximation of each sample, i.e. x i ≈ Uv T i,: . The largest element in v i,: indicates the closet centroid to x i . Therefore, V indicates the clustering results directly.
Since NMF minimizes the residues of all the samples, outliers with large approximation errors will affect the optimization inevitably. Furthermore, the square of the errors compounds the problem severely.
Robust NMF
To improve the robustness to outliers, some variants of NMF have been put forward. Ke and Kanade [25] replaced the Frobenius norm with the 1 norm, which yields the following problem
min U≥0,V≥0 ||X − UV T || 1 = n i=1 |x i − Uv T i,: |,(3)
and the model is solved by convex programming. Instead of squaring the approximation errors, problem (3) takes the 1 norm as the objective function. Therefore, the large residues are depressed.
Considering that 1 norm is sensitive to feature rotation, Kong et al. [26] proposed the 2,1 norm NMF with the following formulation min U≥0,V≥0
||X − UV T || 2,1 = n i=1 ||x i − Uv T i,: || 2 .(4)
Similar with the 1 norm NMF, problem (4) takes off the square operation. Besides, because ||A|| 2,1 equals to ||AR|| 2,1 for any rotation matrix R, 2,1 norm NMF is invariant to the feature rotation. Furthermore, it also achieves the structural sparsity. Considering the above advantages, 2,1 norm NMF has been extensively studied in the literature [16], [17], [29], [30], and became one of the most popular robust NMF. Based on 2,1 norm NMF, Huang et al. [8] removed the nonnegative constraint on U to handle the negative data, leading to
min V≥0,VV T =I ||X − UV T || 2,1 + λTr(V T LV).
The Laplacian graph L is utilized to preserve the local manifold structure. Cluster indicator V is constrained to be orthogonal to keep the uniqueness of solution. Ding et al. [31] further pointed out that the orthogonal constraint facilitates the interpretation of the clustering results. Du et al. [32] employed the correntropy induced metric to calculate the error, and proposed the following problem
max β,U ≥0,V ≥0 d i=1 n k=1 exp[−(X ik − c j=1 U ij V jk ) 2 /2β 2 ],
where β is the optimal Gaussian variance to be learned. The above model can handle the non-Gaussian outliers. To further weaken the large residues, Qi et al. [27] designed the logarithmic loss function:
min U≥0,V≥0 n i=1 log(1 + ||x i − Uv T i,: || 2 ).
There are also many other robust cost functions for NMF, such as the 1,2 norm loss [33], [34] and hypersurface loss [35]. However, all of them share the same drawback with the classical NMF that they minimize the errors of all the samples. Consequently, the outliers with extremely large errors may still affect the results. Gao et al. [28] proposed to remove the sample if its error exceeds a certain threshold, which may be inappropriate for various kinds of tasks.
ENTROPY MINIMIZING MATRIX FACTORIZATION
In this section, the Entropy Minimizing Matrix Factorization framework (EMMF) is introduced. The optimization strategy and the convergence analysis are also given.
Methodology
Before describing the formulation of EMMF, we first introduce the concept of entropy. Defining {p i } as the probability distribution of a random variable, the Shannon entropy is given by
H = − i p i logp i .
According to the information theory [36], the entropy is maximized when the distribution is uniform, i.e. all the probabilities are with the same value. Conversely, the less value of the entropy indicates the imbalance distribution. Defining M = X − UV T , we define p i as
p i = ||m i || 2 ||M|| 2,1 , i ∈ [1, n].
According to the aforementioned definitions, it is manifest that n i=1 p i = 1. Therefore, {p i } is exact the samples' residue distribution, and the entropy is computed as
H(M) = − n i=1 ||m i || 2 ||M|| 2,1 log ||m i || 2 ||M|| 2,1 .(5)
The value of H(M) is minimized when the residue distribution is extremely imbalance. However, the distribution can not reflect the exact value of the residues, i.e. H(M) equals to H(ρM) for any ρ > 0. To keep the uniqueness, the matrix residue ||M|| 2,1 should be also minimized. Since both the entropy and matrix residue are with positive values, we propose to minimize their product
min U,V H(M) × ||M|| 2,1 , s.t.M = X − UV T , U ≥ 0, V ≥ 0.
which yields the objective function of EMMF
min U,V − n i=1 ||m i || 2 log ||m i || 2 ||M|| 2,1 , s.t.M = X − UV T , U ≥ 0, V ≥ 0.(6)
If several samples are with large approximation errors, EMMF just let them be and moves the centroids towards the remaining ones. By searching an imbalance residue distribution, the effects of the outliers are avoided. One may doubt the correctness of the model for the data without outliers. In fact, since all the samples are with uniform distribution, the centroids will not change too much if a few samples are considered as outliers. Therefore, the mistaken outliers can still be connected with the correct centroid. This statement will be verified in Section 5.1.
Optimization
Due to the dependency between M, U and V, it is difficult to solve problem (6) directly. In this part, we transform the objective into the trace from.
Taking m i as the variable to be optimized, the first order derivative of the objective is − mi ||mi||2 log ||mi||2 ||M||2,1 ≥ 0, and the second order derivative is
1 ||M||2,1 − 1 ||mi||2 ≤ 0. Thus, the proposed function is monotonic increasing w.r.t. m i . The Lagrange function is L 1 (m i ) = − n i=1 ||m i || 2 log ||m i || 2 ||M|| 2,1 +G(m i , Λ 1 ),
where G(m i , Λ 1 ) represents the constraint on m i , Λ 1 is the Lagrange multiplier. Computing the derivative of L 1 (m i ) and setting it as zero, we get the optimal solution as
Q ii m i + ∂G(m i , Λ 1 ) ∂m i = 0,(7)
where Q ∈ R n×n is the diagonal matrix with
Q ii = − 1 ||m i || 2 log ||m i || 2 ||M|| 2,1 .(8)
In implementation, we add a small enough factor ε on ||m i || 2 to prevent it from being zero. When Q is set as sta-tionary, Eq. (7) is also the optimal solution to the following problem
min M Tr(MQM T ) = n i=1 Q ii ||m i || 2 2 , s.t.M = X − UV T , U ≥ 0, V ≥ 0.(9)
Then the optimal M of objective (6) can be obtained by solving problem (9). Accordingly, we search the optimal U and V by solving
min U≥0,V≥0 Tr[(X − UV T )Q(X − UV T ) T ].(10)
In each iteration, Q is updated with the current U, V according to Eq. (8). The updating rules of U and V as follows.
Updating U, problem (10) becomes
min U≥0 Tr(UV T QVU T ) − 2Tr(XQVU T ).(11)
The above sub-problem is convex, and the Lagrangian function is
L 2 (U) = Tr(UV T QVU T ) − 2Tr(XQVU T ) + Tr(Λ 2 U T ),(12)
where Λ 2 ∈ R d×c is the Lagrangian multiplier. Let ∂L2(U) ∂U to be zero, we have
2UV T QV − 2XQV + Λ 2 = 0.(13)
According to the KKT conditions (Λ 2 ) ik U ik = 0, we get
(UV T QV) ik U 2 ik − (XQV) ik U 2 ik = 0.(14)
Then the updating rule of U is
U ik ← U ik (XQV) ik (UV T QV) ik .(15)
At convergence, the equality holds for Eq. (15), so the condition in Eq. (13) is satisfied.
Updating V, the sub-problem is min V≥0 Tr(V T QVU T U) − 2Tr(V T QX T U).(16)
The Lagrangian function is
L 3 (V) = Tr(V T QVU T U) − 2Tr(V T QX T U) + Tr(Λ 3 V T ),
where Λ 3 ∈ R n×c is the Lagrangian multiplier. Similar with Eq. (14), we have
(QVU T U) ik V 2 ik − (QX T U) ik V 2 ik = 0, and the updating rule of V is V ik ← V ik (QX T U) ik (QVU T U) ik .(17)
At convergence, V satisfies the condition ∂L(V,Λ3) ∂V = 0. The details of the optimization for problem (10) is described in Algorithm 1. In each iteration, the computation of the diagonal matrix Q requires O(ndc) operations. The costs for updating U and V are also O(ndc). After t iterations, the overall cost of EMMF is O(tndc).
Algorithm 1 Optimization algorithm of EMMF
Convergence
The convergence analysis consists of two parts. First, we demonstrate that problem (9) converges to the optimal solution to objective (6). After that, since the updating rules of U and V are in similar form, we only prove the convergence of problem (11).
Convergence of problem (9): we introduce the following theorem.
Theorem 1. The optimization of problem (10) decreases the objective value of problem (6) monotonically.
Proof. Denote the value of M, Q at the t-th iteration are M (t) and Q (t) , and suppose the value of problem (9) decreases through the optimization, i.e.
n i=1 Q (t) ii ||m (t) i || 2 2 ≥ n i=1 Q (t+1) ii ||m (t+1) i || 2 2 .
With the definition of Q in Eq. (8), the above inequality is transformed into
n i=1 − ||m i (t) || 2 2 ||m i (t) || 2 log ||m i (t) || 2 ||M (t) || 2,1 ≥ n i=1 − ||m i (t+1) || 2 2 ||m i (t) || 2 log ||m i (t) || 2 ||M (t) || 2,1 , which leads to n i=1 (||m (t+1) i || 2 − ||m (t) i || 2 ) log ||m (t) i || 2 ||M (t) || 2,1 ≥ 0.(18)
According to the log sum inequality [37], we know that
n i=1 ||m (t+1) i || 2 (log ||m (t+1) i || 2 ||m (t) i || 2 − log ||M (t+1) || 2,1 ||M (t) || 2,1 ) ≥ 0.(19)
Summing up Eq. (18) and (19), the following inequality holds:
n i=1 −||m (t) i || 2 log ||m (t) i || 2 ||M (t) || 2,1 ≥ n i=1 −||m (t+1) i || 2 log ||m (t+1) i || 2 ||M (t+1) || 2,1 ,
which completes the proof.
Convergence of problem (11): to demonstrate that the updating rule (15) decreases the value of problem (11), the following definition [2] is introduced.
Definition 1. g(U,Ũ) is the auxiliary function for f (U) if for any U andŨ it satisfies g(U,Ũ) ≥ f (U), g(U, U) = f (U).(20)
As proved by Lee and Seung [2]], we have the following lemma:
Lemma 1. Given the auxiliary function g(U,Ũ), f (U (t+1) ) ≤ f (U (t) ) holds if U (t+1) is the solution to min U g(U, U (t) ).(21)
We propose the following theorem to demonstrate the convergence of problem (16).
Theorem 2.
Updating rule (15) decreases the Lagrangian function L 2 (U) in Eq. (12) monotonically.
Proof. As Ding et al. [38] pointed out, for any nonnegative
matrices A ∈ R d×d , B ∈ R c×c , C ∈ R d×c , G ∈ R d×c , if A
and B are symmetric, the following inequality holds:
Tr(C T ACB) ≤ d i=1 c k=1 (AGB) ik C 2 ik G ik .
Based on the above equation, the upper bound of the first term of f (U) is written as
Tr(UV T QVU T ) = Tr(U T UV T QV) ≤ d i=1 c k=1 (U (t) V T QV) ik U 2 ik U (t) ik .
For any scalar ρ ≥ 0, we have ρ ≥ 1 + log ρ, which leads to the following lower bound
T r(XQVU T ) = d i=1 c k=1 (XQV) ik U ik ≥ d i=1 c k=1 (XQV) ik U (t) ik (1 + log U ik U (t) ik
).
The last term of L 2 (U) equals to zero, so we do not consider it. Based on the bounds of the first two terms, the auxiliary function of L 2 (U) is
g(U, U (t) ) = d i=1 c k=1 (U (t) V T QV) ik U 2 ik U (t) ik − 2 d i=1 c k=1 (XQV) ik U (t) ik (1 + log U ik U (t) ik ),
which satisfies the conditions in Eq. (20). The first-order derivative of g(U, U (t) ) is
∂g(U, U (t) ) ∂U ik = 2(U (t) V T QV) ik U ik U (t) ik − 2(XQV) ik U (t) ik U ik ,
The Hessian matrix is
∂ 2 g(U, U (t) ) ∂U ik ∂U jl = 2δ ij δ kl ( (U (t) V T QV) ik U (t) ik + (XQV) ik U (t) ik U 2 ik ),
where δ ij is the defined as
δ ij = 1, if i = j 0, otherwise .
The Hessian matrix is a positive definite diagonal matrix, so g(U, U (t) is convex on U. The global optimal solution U (t+1) to min U g(U, U (t) ) is computed by setting the firstorder derivative to zero:
U (t+1) ik = U (t) ik (XQV) ik (U (t) V T QV) ik .
According to Lemma 1, L 2 (U) is non-increasing with the above updating rule.
GRAPH REGULARIZED EMMF
EMMF uses the global centroids to represent the samples, so it cannot the data with complex manifold structures. To explore the local data relationship, the graph-regularized EMMF (G-EMMF) is introduced. Since this research mainly focus on the robustness, we simply incorporate a graph regularization term into EMMF to improve the performance.
Methodology
Supposing S ∈ R n×n is the similarity graph of the data matrix X, a large value of S ij indicates the high similarity between x i and x j . Intuitively, if x i is similar to x j , their coefficient vectors should also be similar. Using the inner product to measure the distance between the vectors, the graph regularization term is given as
min V≥0 ||S − VV T || 2 2 .(22)
Ideally, we can obtain a block diagonal VV T Kuang et al. [39] proved that the above term is equivalent to spectral clustering if S is doubly stochastic and V is orthogonal.
Normalizing the graph S = D − 1 2 SD − 1 2 , where D is the degree matrix of S, problem (22) becomes min V≥0,V T V=I ||S − VV T || 2 2 .(23)
As mentioned in Section 2, the orthogonal constraint also facilitates the clustering interpretation. Combining Eq. (23) with the objective (6), the model of G-EMMF is
min U,V − n i=1 ||m i || 2 log ||m i || 2 ||M|| 2,1 + λ||S − VV T || 2 2 , s.t.M = X − UV T , U ≥ 0, V ≥ 0, V T V = I,(24)
where λ is the regularization parameter. With the above formulation, the coefficient matrix V preserves the local correlations between the samples.
Optimization
The objective of G-EMMF is equivalent to
min U,V Tr[(X−UV T )Q(X − UV)] + λ||S − VV T || 2 2 , s.t.U ≥ 0, V ≥ 0, V T V = I.
Removing the irrelevant terms, the above problem is simplified into
min U,V Tr[(X−UV T )Q(X − UV)] − 2λTr(V T SV), s.t.U ≥ 0, V ≥ 0, V T V = I.(25)
The updating rule of U is the same with EMMF, so we only give the updating rule of V.
Updating V, problem (25) is transformed into min V Tr(V T QVU T U) − 2Tr(V T QX T U) − 2λTr(V T SV), s.t.V ≥ 0, V T V = I.(26)
The Lagrangian function is
L 4 (V) =Tr(V T QVU T U) − 2Tr(V T QX T U) − 2λTr(V T SV) + Tr(Λ 4 V T ) + Tr[Λ 5 (V T V − I) T ](27)
where Λ 4 ∈ R n×c and Λ 5 ∈ R c×c are Lagrangian multipliers. Given
(Λ 4 ) ik V ik = 0, setting ∂L4(V) ∂V = 0 gives rise to (QVU T U − QX T U − 2λSV + VΛ 5 ) ik V 2 ik = 0, so the updating rule is V ik ← V ik (QX T U + 2λSV + VΛ − 5 ) ik (QVU T U + VΛ + 5 ) ik ,(28)
where Λ − 5 , Λ + 5 are the negative and positive parts of Λ 5 , i.e.
(Λ − 5 ) ik = |(Λ5) ik |−(Λ5) ik 2 and (Λ + 5 ) ik = |(Λ5) ik |+(Λ5) ik 2 .
Since we also have
(QVU T U − QX T U − 2λSV + VΛ 5 ) ik V ik = 0, Λ 5 is computed as Λ 5 = V T QX T U + 2λV T SV − V T QVU T U.
and Λ − 5 and Λ + 5 are computed as
Λ − 5 = V T QVU T U, Λ + 5 = V T QX T U + 2λV T SV.
Therefore, the updating rule in Eq. (28) is rewritten as
V ik ← V ik (QX T U + 2λSV + VV T QVU T U) ik (QVU T U + VV T QX T U + 2λVV T SV) ik .(29)
The optimization strategy for G-EMMF is described in Algorithm 2. G-EMMF takes O(n 2 d) additional complexity to construct and normalize the similarity graph, and the remaining costs are the same with EMMF, i.e. O(tndc). After t iterations, the overall computational cost of G-EMMF is O(n 2 d + tndc).
Convergence
Here we demonstrate the convergence of problem (26). We fist introduce the following theorem. Proof. According to the proof of Theorem 2, denoting the value of V at the t-th iteration as V (t) , the bounds of the non-zero terms in L 4 (V) are as follows.
Tr(V T QVU T U) ≤ n i=1 c k=1 (QV (t) U T U) ik V 2 ik V (t) ik , Tr(V T QX T U) ≥ n i=1 c k=1 (QX T U) ik V (t) ik (1 + log V ik V t ik ), Tr(V T SV) ≥ n i=1 c k=1 n l=1 S il V (t) ik V (t) lk (1 + log V ik V lk V (t) ik V (t) lk ), Tr(Λ − 5 V T V) ≥ n i=1 c k=1 c l=1 (Λ − 5 ) kl V (t) ik V (t) il (1 + log V ik V il V (t) ik V (t) il ), Tr(Λ + 5 V T V) ≤ n i=1 c k=1 (V (t) Λ + 5 ) ik V 2 ik V (t) ik .
Combining the bounds, the auxiliary function for L 4 (V) is
g(V, V (t) ) = n i=1 c k=1 (QV (t) U T U) ik V 2 ik V (t) ik − 2 n i=1 c k=1 (QX T U) ik V (t) ik (1 + log V ik V t ik ) − 2λ n i=1 c k=1 n l=1 S il V (t) ik V (t) lk (1 + log V ik V lk V (t) ik V (t) lk ) − n i=1 c k=1 c l=1 (Λ − 5 ) kl V (t) ik V (t) il (1 + log V ik V il V (t) ik V (t) il ) + n i=1 c k=1 (V (t) Λ + 5 ) ik V 2 ik V (t) ik .
The Hessian matrix is positive definite, so the optimal solution V (t+1) that minimizes g(V, V (t) ) is calculated by setting the first-order derivative to zero:
V (t+1) ik = V (t) ik (QX T U + λSV (t) + V (t) Λ − 5 ) ik (QV (t) U T U + V (t) Λ + 5 ) ik .
According to Lemma 1, the above updating rule decreases L 4 (V).
EVALUATION OF EMMF
In this section, the proposed EMMF is evaluated on several synthetic and real-world datasets.
Experiments on Synthetic Datasets
Synthetic datasets are constructed to validate the robustness of EMMF. Some static properties of EMMF are also discussed.
As shown in Fig. 1 (a), the first dataset consists of 13 two dimensional samples. The first ten samples are with normal distribution, and the last three samples are outliers. Fig. 1 (a) also visualizes the approximation results of NMF [2], 2,1 -NMF [26] and EMMF. The approximated samples of NMF deviates the normal distribution largely, which indicates the approximation is dominated by the outliers. As shown in Fig. 1 (b), the errors of some normal samples are larger than the outliers. 2,1 -NMF shows better performance because the outliers are depressed. However, the outliers still affect the approximation. EMMF just let the outliers to be with large errors, as shown in Fig. 1 (d), such that the normal samples are approximated correctly. Therefore, it shows more robustness. Fig. 2 shows the approximation results for the data without outliers. It can be seen that all the methods approximate the samples successfully. As visualized in Fig. 2 (c) and (d), both 2,1 -NMF and EMMF cannot achieve the zero error due to the factor ε added on Q. In Fig. 2 (d), some samples are with relative large errors because EMMF treat them as outliers. However, the errors of most samples are close to zero, so the overall error is still less than 2,1 -NMF. That is to say, a few mistaken outliers can not affect the approximation largely, and EMMF works well even when there is no outlier.
We also demonstrate that the objective of EMMF is unlikely to be dominant by outliers. Supposing x i is the outlier, its effects on the objectives of NMF, 2,1 -NMF and EMMF are computed as
φ NMF (x i ) = ||m i || 2 2 n i=1 ||m i || 2 2 , φ 2,1NMF (x i ) = ||m i || 2 n i=1 ||m i || 2 , φ EMMF (x i ) = ||m i || 2 log ||mi||2 ||M||2,1 n i=1 ||m i || 2 log ||mi||2 ||M||2,1 .
We randomly generate a matrix X ∈ R 50×50 , where 0 ≤ X ik ≤ 1. Adding a noise factor σ on X 11 , we investigate the effect of outlier x 1 on the objective value with varying σ. As shown in Fig. 3, the outlier effect on NMF increases dramatically. Compared with NMF, φ 2,1NMF (x 1 ) increases slower. But both of them reach to 1 when σ is very large. Meanwhile, φ EMMF (x 1 ) decreases when σ exceeds a certain value. Therefore, EMMF is insensitive to the outliers with extremely large errors.
Suppose there is only one outlier x i , and its residue ratio is p, i.e. ||mi||2 ||M||2,1 = p. The entropy of the distribution is minimized when all the remaining samples are with the same residue ratio, i.e. 1−p n−1 . In such situation, φ EMMF (x i ) is
φ EMMF (x i ) = p log(p) p log(p) + (1 − p)[log(1 − p) − log(n − 1)] .
(30) Given a certain n, the upper bound is calculated as the maximum value when varying p within the range (0, 1]. Taking the step length of p as 0.01 and increasing n, the upper bound curve is plotted in Fig. 4. We can see that the Samples 1440 7200 165 213 575 2000 1404 360 Dimension 1024 1024 256 676 644 240 320 90 Class 20 100 15 10 20 10 36 15 Fig. 3. Outlier effects of different methods with different value of σ on the randomly generated data X ∈ R 3×3 . The ratio of EMMF decreases when σ exceeds a certain value.
upper bound of φ EMMF (x i ) decreases monotonically with the value of n, which complies with the human perception that the effect of an outlier should be small when there are many normal samples.
Experiments on Real-world Datasets
The clustering performance of EMMF is evaluated on realworld datasets. Clustering accuracy (ACC) and Normalized Mutual Information (NMI) are used as measurements. Datasets: eight benchmarks for clustering are employed, including two object image datasets, i.e. COIL20 and COIL100 [15], three face image datasets, i.e. YALE [40], JAFFE [41] and UMIST [42], a multi-feature handwritten dataset, i.e. Multiple features (Mfeat) [43], a handwritten digit dataset, i.e. Binary Alphabet (BA) [44], and a hand movement dataset, i.e. Movement [43]. All the samples are normalized as the unit vectors in the experiments. The details of the datasets are exhibited in Table 1.
Competitors: six state-of-the-arts are taken for comparison, including • PNMF [45]: probabilistic NMF solved by variational Bayesian.
• GSNMF [46]: generalized separable NMF, which approximates both the rows and columns of the data matrix.
• 2,1 -NMF [26]: NMF with the 2,1 norm loss function.
• Hx-NMF [27]: NMF with the logarithmic loss function.
For PNMF and GSNMF, the best parameters are obtained by searching the grid {10 −3 , 10 −2 , · · · , 10 3 }, and they are initialized with the approaches suggested by the authors. For the other methods, including the proposed EMMF, we initialize U and V with k-means [47]. For PNMF, variational Bayesian gives the best results after 100 iterations. The maximum iteration number for all the other methods is set as 500. To alleviate the influence of initiation condition, all the methods are performed for twenty repetitions, and the averaged results are reported. Performance: the clustering results on different datasets are exhibited in Table 2. The proposed EMMF shows best performance in terms of ACC and NMI. On these datasets, most samples obey the normal distribution. The performance of PNMF and GSNMF are unsatisfying due to the lack of clustering interpretation. Specifically, their coefficient matrices do not contain the clear cluster structure. 2,1 -NMF and Hx-NMF show better results than NMF, which demonstrates the advantage of improving robustness. EMMF outperforms all the competitors because it moves the centroids towards the samples with small approximation errors. In this way, most of the samples are well represented by the centroids, and the clustering performance is improved. The convergence curves of EMMF are also given in Fig. 5. The optimization algorithm converges within 40 iterations on all the datasets, which ensures the efficiency.
To further prove the robustness of EMMF, we introduce large outliers into the real-world datasets. For each dataset, randomly generated vectors are added into the data matrix. The elements of the vectors are within the range of [0, 10×σ], where σ is the largest value in the original data matrix. We perform all the methods methods on the outlier datasets with different number of outliers ([0, n 3 ]), and show the clustering results of the original samples in Fig. 6. As the outlier number increases, the performance of NMF and NMF-DIV drops dramatically, especially on YALE and JAFFE. 2,1 -NMF and Hx-NMF show similar robustness by depressing the outliers. The ACC curves of EMMF are more stable than the competitors on all the datasets, because the outliers have less effect on the updating of centroids. Therefore, EMMF is able to handle the data with large outliers. [8]: robust manifold NMF, which integrates the spectral clustering term into the objective of 2,1 -NMF,
• NLCF [48]: nonnegative local coordinate factorization, which uses local coordinate learning to encode the data structure.
• LCF [10]: local coordinate concept factorization, which combines local coordinate learning and convex matrix factorization.
• LSNMF [49]: local-centroids structured NMF, which employs multiple local centroids to represent a cluster.
• SRMCF [11]: self-representative manifold concept factorization, which optimizes the data graph during performing convex matrix factorization.
• NMFAN [50]: NMF with adaptive neighbors, which learns the local data relationship adaptively.
The best parameters of the competitors are found by searching the grid {10 −5 , 10 −4 , · · · , 10 5 }. The graph regularization parameter λ in G-EMMF is found within the grid {10 1 , 10 3 , · · · , 10 9 }. The methods are initialized with kmeans, and the 0 − 1 weighting 5 nearest neighbor graph is constructed as the similarity graph. The stop criteria for all the methods is set as 500 maximum iterations. After repeating the methods twenty times, the average ACC and NMI are reported. Performance: the results of the graph regularized NMF methods are given in Table 3. RMNMF fixes the data graph during optimization. The outliers may affect the graph quality, and further influence the clustering results. Compared with RMNMF, the other competitors show better performance because they learn the data relationship iteratively, such that the graph quality is improved. Similar with RMNMF, the proposed G-EMMF also relies on the input graph. It still achieves the best performance because the noises in the graph has less effect on the learned U. In addition, compared with the results of EMMF in Table 2, G-EMMF improves the clustering performance on all the datasets, which demonstrates the necessity of exploiting data structure. The convergence curves are shown in Fig. 7, which verifies the effectiveness of the optimization algorithm. Besides, to investigate the effect of parameter λ on the clustering results, we plot the curves of ACC and NMI with varying λ, as shown in Fig. 8. The clustering performance is not very sensitive to λ within a wide range. When λ becomes very large, the performance tends to decreases because the matrix approximation error increases.
The outlier datasets used for the graph-regularized methods are different from those used in Section 5.2, because LSNMF is inapplicable when the outliers are directly added into the data matrix. Given the faces images from YALE, we randomly select three images from each class and add block noises on them, so these images become outliers, as shown in Fig. 9. The clustering results are given in Table 4. We can see RMNMF outperforms several competitors due to the robust formulation. LSNMF shows unsatisfying results since it aims to learn a graph with exact c connected components, which is unrealistic on the data with outliers. G-EMMF outperforms the competitors on different scale of block noises.
•
All the authors are with the Center for Optical Imagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China. E-mails: [email protected], [email protected]. X. Li is the corresponding author.
Theorem 3 .
3Updating rule (28) decreases Lagrangian function L 4 (V) in Eq. (27) monotonically.
Fig. 1 .
1(a) Results on the synthetic dataset with outliers. (b-d) Approximation error ||x i − Uv T i,: || 2 calculated by different methods.
Fig. 2 .
2(a) Results on the synthetic dataset without outliers. (b-d) Approximation error ||x i − Uv T i,: || 2 calculated by different methods.
Fig. 4 .
4Upper bound of φ EMMF (x i ) with different value of n. The upper bound decreases with n.
•
NMF[2]: NMF with the Frobenius norm formulation.• NMF-DIV[2]: NMF with divergence formulation.
Fig. 6 .
6Performance of different methods on datasets with increasing number of outliers.
Fig. 7 .
7Convergence curves of G-EMMF.
Fig. 8 .
8Performance of G-EMMF with different value of λ.
Input: Data matrix X, centroid number c. 1: Initialize U and V. Update V with Eq. (17). 6: until Converge Output: Optimal U, V.2: repeat
3:
Compute Q with Eq. (8).
4:
Update U with Eq. (15).
5:
Algorithm 2 Optimization algorithm of EMMF Input: Data matrix X, centroid number c, graph S. Update U with Eq. (15). Update V with Eq. (29). 7: until Converge Output: Optimal U, V.1: Initialize U and V.
2: Normalize S as D − 1
2 SD − 1
2 .
3: repeat
4:
Compute Q with Eq. (8).
5:
6:
TABLE 1
1Description on the real-world datasets.COIL20
COIL100
YALE
JAFFE
UMIST
Mfeat
BA
Movement
TABLE 2
2Performance of EMMF on real-world datasets. Best results are in bold face.ACC
COIL20
COIL100
YALE
JAFFE
UMIST
Mfeat
BA
Movement
NMF
0.4486
0.2532
0.3412
0.6390
0.3403
0.4233
0.1603
0.3297
NMF-DIV
0.5097
0.3954
0.4188
0.9136
0.3831
0.5945
0.2369
0.4372
PNMF
0.4882
0.2297
0.2909
0.3991
0.3443
0.5245
0.1538
0.3694
GSNMF
0.3431
0.2507
0.3152
0.2723
0.2226
0.4005
0.2486
0.3806
2,1-NMF
0.5826
0.4256
0.4182
0.9329
0.4026
0.5821
0.2203
0.4453
Hx-NMF
0.5808
0.4374
0.4097
0.9357
0.4090
0.5790
0.2211
0.4475
EMMF
0.5972
0.4542
0.4327
0.9423
0.4132
0.6195
0.2553
0.4542
NMI
COIL20
COIL100
YALE
JAFFE
UMIST
Mfeat
BA
Movement
NMF
0.5731
0.5273
0.4197
0.6702
0.4804
0.3878
0.2779
0.3882
NMF-DIV
0.6413
0.6534
0.4589
0.8909
0.5497
0.5553
0.3966
0.5554
PNMF
0.5682
0.4760
0.3276
0.4654
0.4342
0.4903
0.2589
0.4347
GSNMF
0.4459
0.5028
0.3047
0.2248
0.2921
0.3133
0.3549
0.4661
2,1-NMF
0.6937
0.6653
0.4541
0.9114
0.5665
0.5549
0.3698
0.5638
Hx-NMF
0.6909
0.6687
0.4485
0.9134
0.5658
0.5591
0.3695
0.5633
EMMF
0.7059
0.6889
0.4686
0.9238
0.5827
0.5771
0.4075
0.5813
0
100
200
300
400
500
600
700
800
900 1000
0
0.2
0.4
0.6
0.8
1
(x
1 )
NMF
l 2,1 -NMF
EMMF
TABLE 3
3Performance of G-EMMF on real-world datasets. Best results are in bold face.ACC
COIL20
COIL100
YALE
JAFFE
UMIST
Mfeat
BA
Movement
RMNMF
0.5792
0.4149
0.4448
0.9108
0.4191
0.5386
0.3154
0.4167
NLCF
0.6431
0.4153
0.4352
0.9315
0.3910
0.7472
0.3652
0.4444
LCF
0.6583
0.4596
0.4158
0.9268
0.3920
0.7538
0.3929
0.4667
LSNMF
0.6528
0.4651
0.2485
0.7502
0.3986
0.6166
0.0879
0.3278
SRMCF
0.6660
0.3990
0.4121
0.9615
0.4254
0.7036
0.4103
0.4583
NMFAN
0.6500
0.4558
0.4230
0.9305
0.3969
0.7064
0.3111
0.4417
G-EMMF
0.6882
0.4803
0.4558
0.9812
0.4588
0.7833
0.4795
0.4778
NMI
COIL20
COIL100
YALE
JAFFE
UMIST
MNIST
BA
Movement
RMNMF
0.6944
0.6605
0.4909
0.8894
0.5938
0.5206
0.4780
0.5217
NLCF
0.7364
0.6616
0.4867
0.9197
0.5883
0.7112
0.4810
0.6158
LCF
0.7483
0.7213
0.4809
0.9037
0.5901
0.7203
0.5288
0.6237
LSNMF
0.7699
0.6633
0.2510
0.8404
0.5310
0.6551
0.0827
0.4393
SRMCF
0.7699
0.6686
0.4745
0.9556
0.6292
0.6464
0.5453
0.5900
NMFAN
0.7492
0.7330
0.4682
0.9208
0.5963
0.6690
0.4433
0.6167
G-EMMF
0.8055
0.7547
0.5203
0.9731
0.6696
0.7888
0.6191
0.6362
0
20
40
60
80
100
120
140
160
180
200
Iteration
2.28
2.3
2.32
2.34
2.36
2.38
2.4
2.42
2.44
2.46
Objective value
10 7
(a) COIL20
0
20
40
60
80
100
120
140
160
180
200
Iteration
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
Objective value
10 8
(b) COIL100
0
20
40
60
80
100
120
140
160
180
200
Iteration
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
Objective value
10 6
(c) YALE
0
20
40
60
80
100
120
140
160
180
200
Iteration
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Objective value
10 6
(d) JAFFE
0
20
40
60
80
100
120
140
160
180
200
Iteration
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
9.4
Objective value
10 6
(e) UMIST
0
20
40
60
80
100
120
140
160
180
200
Iteration
2.62
2.63
2.64
2.65
2.66
2.67
2.68
2.69
Objective value
10 7
(f) Mfeat
0
20
40
60
80
100
120
140
160
180
200
Iteration
1.5
1.55
1.6
1.65
1.7
1.75
1.8
Objective value
10 7
(g) BA
0
20
40
60
80
100
120
140
160
180
200
Iteration
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Objective value
10 6
(h) Movement
(a) outliers with 4×4 block noises (b) outliers with 6×6 block noises (c) outliers with 8×8 block noises (d) outliers with 10×10 block noisesFig. 9. Illustration of outliers for graph regularized NMF methods.. Competitors: six graph-regularized NMF methods are used for comparison, including•
RMNMF
TABLE 4
4Performance of graph regularized NMF methods on YALE datasets with outliers. Best results are in bold face.ACC
4×4
6×6
8×8
10×10
RMNMF
0.4242
0.3818
0.3697
0.3212
NLCF
0.3697
0.3939
0.3212
0.3273
LCF
0.3576
0.3697
0.3333
0.3333
LSNMF
0.2727
0.1879
0.1636
0.1818
SRMCF
0.4242
0.3939
0.3455
0.3455
NMFAN
0.3636
0.3818
0.3333
0.3515
G-EMMF
0.4364
0.4121
0.4061
0.3939
NMI
4×4
6×6
8×8
10×10
RMNMF
0.4818
0.4298
0.4464
0.4041
NLCF
0.4439
0.4273
0.3623
0.3775
LCF
0.4506
0.4398
0.3749
0.3869
LSNMF
0.2613
0.1696
0.1178
0.1789
SRMCF
0.4703
0.4207
0.4005
0.3843
NMFAN
0.4333
0.4079
0.3720
0.3850
G-EMMF
0.5063
0.4708
0.4609
0.4519
EVALUATION OF G-EMMFThe effectiveness of G-EMMF is validated through experiments. The datasets used in this part are the same as in Section 5.2.
CONCLUSION AND FUTURE WORKThis paper proposes an Entropy Minimizing Matrix Factorization (EMMF) framework. A novel matrix factorization formulation with entropy loss is designed. Instead of approximating all the samples, the proposed model pursues an imbalance residue distribution, and the outliers with relative large errors are not taken into consideration. In this way, the outliers have less effect on the learned centroids. In addition, the graph regularized EMMF is introduced to handle the data with complex structures. The models can be solved by the suggested optimization algorithms efficiently. Experiments on various datasets demonstrate the robustness of the proposed methods, and show their applicability on data without outliers. Comparison with the state-of-thearts validates the superiorities of our methods.In the future work, we are desired to develop the deep model of EMMF, and apply it in large-scale clustering tasks. We also want to incorporate the graph information into EMMF without introducing any additional parameter.
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| [] |
[
"A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of τ",
"A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of τ"
] | [
"Miles Eli Jones \nDepartment of Mathematics\nDepartment of Mathematics\nUniversity of California\nSan Diego La Jolla92093-0112CAUSA\n",
"Jeffrey B Remmel [email protected] \nUniversity of California\nSan Diego La Jolla92093-0112CAUSA\n"
] | [
"Department of Mathematics\nDepartment of Mathematics\nUniversity of California\nSan Diego La Jolla92093-0112CAUSA",
"University of California\nSan Diego La Jolla92093-0112CAUSA"
] | [] | In this paper, we introduce a new method for computing generating functions with respect to the number of descents and left-to-right minima over the set of permutations which have no consecutive occurrence of τ where τ starts with 1. In particular, we study the generating function n≥0 | null | [
"https://arxiv.org/pdf/1201.0142v1.pdf"
] | 119,712,152 | 1201.0142 | 29e506600bbb838425a59bcf52c04a8780082330 |
A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of τ
30 Dec 2011
Miles Eli Jones
Department of Mathematics
Department of Mathematics
University of California
San Diego La Jolla92093-0112CAUSA
Jeffrey B Remmel [email protected]
University of California
San Diego La Jolla92093-0112CAUSA
A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of τ
30 Dec 2011Submitted: Date 1; Accepted: Date 2; Published: Date 3.arXiv:1201.0142v1 [math.CO]MR Subject Classifications: 05A1505E05 keywords: permutationpattern matchdescentleft to right minimumsymmetric polynomialexponential generating function
In this paper, we introduce a new method for computing generating functions with respect to the number of descents and left-to-right minima over the set of permutations which have no consecutive occurrence of τ where τ starts with 1. In particular, we study the generating function n≥0
Introduction
Given a sequence σ = σ 1 . . . σ n of distinct integers, let red(σ) be the permutation found by replacing the i th largest integer that appears in σ by i. For example, if σ = 2 7 5 4, then red(σ) = 1 4 3 2. Given a permutation τ = τ 1 . . . τ j in the symmetric group S j , we say a permutation σ = σ 1 . . . σ n ∈ S n has a τ -match starting at position i provided red(σ i . . . σ i+j−1 ) = τ . Let τ -mch(σ) be the number of τ -matches in the permutation σ. Given a permutation σ = σ 1 . . . σ n ∈ S n , we let des(σ) = |{i : σ i > σ i+1 }|. We say that σ j is a left-to-right minimum of σ if σ j < σ i for all i < j. We let LRmin(σ) denote the number of left-to-right minima of σ. Let N M n (τ ) denote the set of permutations in S n with no τ -matches and let N M τ,n (x, y) = σ∈N Mn(τ )
x LRmin(σ) y 1+des(σ) .
(1)
The main goal of this paper is to give a new method to compute generating functions of the form N M τ (t, x, y) = n≥0 t n n! N M τ,n (x, y) (2) where where τ is a permutation in S j which starts with 1. This method was first outlined in the [8] where the authors considered the generating function N M 1423 (t, x, y). In this paper, we shall develop a more general approach that applies to any τ that starts with 1 and has one descent. Our method does not compute N M τ (t, x, y) directly. Instead, we assume that N M τ (t, x, y) = 1 U τ (t, y)
x where U τ (t, y) = 1 + n≥1 U τ,n (y) t n n! .
Thus U τ (t, y) = 1 1 + n≥1 N M τ,n (1, y) t n n! (4) Our method gives a combinatorial interpretation the right-hand side of (4) and then uses that combinatorial interpretation to develop simple recursions on the coefficients U τ,n (y).
The first question to ask is why can we assume that N M τ (t, x, y) can be expressed in the form N M τ (t, x, y) = 1 U τ (t, y)
x when τ starts with 1. The fact that this is assumption is justified follows from previous work of the authors [7] where they introduced the general study of patterns in the cycle structure of permutations. That is, suppose that τ = τ 1 . . . τ j is a permutation in S j and σ is a permutation in S n with k cycles C 1 . . . C k . We shall always write cycles in the form C i = (c 0,i , . . . , c p i −1,i ) where c 0,i is the smallest element in C i and p i is the length of C i . Then we arrange the cycles by decreasing smallest elements. That is, we arrange the cycles of σ so that c 0,1 > · · · > c 0,k . Then we say that σ has a cycle-τ -match (c-τ -mch) if there is an i such that C i = (c 0,i , . . . , c p i −1,i ) where p i ≥ j and an r such that red(c r,i c r+1,i . . . c r+j−1,i ) = τ where we take indices of the form r + s modulo p i . Let c-τ -mch(σ) be the number of cycle-τ -matches in the permutation σ. For example, if τ = 2 1 3 and σ = (4, 7, 5, 8, 6)(2, 3)(1, 10,9), then 9 1 10 is a cycle-τ -match in the third cycle and 7 5 8 and 6 4 7 are cycle-τ -matches in the first cycle so that c-τ -mch(σ) = 3.
Given a cycle C = (c 0 , . . . , c p−1 ) where c 0 is the smallest element in the cycle, we let cdes(C) = 1 + des(c 0 . . . c p−1 ). Thus cdes(C) counts the number of descent pairs as we traverse once around the cycle because the extra 1 counts the descent pair c p−1 > c 0 . For example if C = (1, 5, 3, 7, 2), then cdes(C) = 3 which counts the descent pairs 53, 72, and 21 as we traverse once around C. By convention, if C = (c 0 ) is a one-cycle, we let cdes(C) = 1. If σ is a permutation in S n with k cycles C 1 . . . C k , then we define cdes(σ) = k i=1 cdes(C i ). We let cyc(σ) denote the number of cycles of σ.
In [7], Jones and Remmel studied generating functions of the form N CM τ (t, x, y) = 1 + n≥1 t n n! σ∈N CMn(τ )
x cyc(σ) y cdes (σ) where N CM n (τ ) is the set of permutations σ ∈ S n which have no cycle-τ -matches. The basic approach used in that paper was to use theory of exponential structures to reduce the problem of computing N CM τ (t, x, y) to the problem of computing similar generating functions for n-cycles.
That is, let N CM n,k (τ ) be the set of permutations σ of S n with k cycles such that σ has no cycle-τ -matches and L ncm m (τ ) denote the set of m-cycles γ in S m such γ has no cycle-τ -matches. The following theorem follows easily from the theory of exponential structures as is described in [15], for example. It turns out that if τ ∈ S j is a permutation that starts with 1, then we can reduce the problem of finding N CM τ (t, x, y) to the usual problem of finding the generating function of permutations that have no τ -matches. Letσ be the permutation that arises from C 1 · · · C k by erasing all the parentheses and commas. For example, if σ = (7,10,9,11) (4,8,6) (1,5,3,2), then σ = 7 10 9 11 4 8 6 1 5 3 2. It is easy to see that the minimal elements of the cycles correspond to left-to-right minima inσ. It is also easy to see that under our bijection σ →σ, cdes(σ) = des(σ) + 1 since every left-to-right minima other than the first element ofσ is part of a descent pair inσ. For example, if σ = (7,10,9,11) (4,8,6) (1, 5, 3, 2) so thatσ = 7 10 9 11 4 8 6 1 5 3 2, cdes((7, 10, 9, 11)) = 2, cdes((4, 8, 6)) = 2, and cdes((1, 5, 3, 2)) = 3 so that cdes(σ) = 2+2+3 = 7 while des(σ) = 6. This given, Jones and Remmel [7] proved that if τ ∈ S j and τ starts with 1, then for any σ ∈ S n , (1) σ has k cycles if and only ifσ has k left-to-right minima, (2) cdes(σ) = 1 + des(σ), and (3) σ has no cycle-τ -matches if and only ifσ has no τ -matches. Thus they proved the following theorem.
Theorem 2. Suppose that τ = τ 1 . . . τ j ∈ S j and τ 1 = 1. Then
N CM τ (t, x, y) = N M τ (t, x, y).(6)
It follows from Theorems 1 and 2 that if τ ∈ S j and τ starts with 1, then
N M τ (t, x, y) = F (t, y) x(7)
for some function F (t, y). Thus our assumption that
N M τ (t, x, y) = 1 U τ (t, y) x(8)
is fully justified in the case when τ starts with 1. We should note that if a permutation τ does not start with 1, then it may be that case that |N M n (τ )| = |N CM n (τ )|. For example, Jones and Remmel [7] computed that |N CM 7 (3142)| = 4236 and |N M 7 (3142)| = 4237.
Jones and Remmel [7] were able compute functions of the form N CM τ (t, x, y) when τ starts with 1 by combinatorially proving certain recursions for C∈L ncm m (τ ) y cdes(C) which led to certain sets of differential equations satisfied by N CM τ (t, x, y). For example, using such methods, they were able to prove the following two theorems.
Theorem 3. Let τ = τ 1 . . . τ j ∈ S j where j ≥ 3 and τ 1 = 1 and τ j = 2. Then
N CM τ (t, x, y) = N M τ (t, x, y) = 1 1 − t 0 e (y−1)s− y des(τ ) s j−1 (j−1)! ds x(9)
Theorem 4. Suppose that τ = 12 . . . (j − 1)(γ)j where γ is a permutation of j + 1, . . . , j + p and j ≥ 3. Then
N CM τ (t, x, y) = N M τ (t, x, y) = 1 (U τ (t, y)) x where U τ (t, y) = 1 + n≥1 U τ,n (y)
t n n! , U 1 (y) = −y, and for all n ≥ 2,
U τ,n (y) = (1 − y)U τ,n−1 (y) − y des(τ ) n − j p U τ,n−p−j+1 (y).(10)
The next step in our approach is to use the homomorphism method to give a combinatorial interpretation to
U τ (t, y) = 1 N M τ (t, 1, y) .(11)
That is, Remmel and various coauthors [1,9,10,11,12,13,14] developed a method called the homomorphism method to show that many generating functions involving permutation statistics can be derived by applying a homomorphism defined on the ring of symmetric functions Λ in infinitely many variables x 1 , x 2 , . . . to simple symmetric function identities such as
H(t) = 1/E(−t)(12)
where
H(t) = n≥0 h n t n = i≥1 1 1 − x i t and E(t) = n≥0 e n t n = i≥1 1 + x i t(13)
are the generating functions of the homogeneous symmetric functions h n and the elementary symmetric functions e n in infinitely many variables x 1 , x 2 , . . .. Now suppose that we define a homomorphism θ on Λ by setting θ(e n ) = (−1) n n! N M τ,n (1, y).
Then θ(E(−t)) = n≥0 N M τ,n (1, y) t n n! = 1 U τ (t, y) .
Thus θ(H(t)) should equal U τ (t, y). We shall then show how to use the combinatorial methods associated with the homomorphism method to develop recursions for the coefficient of U τ (t, y) similar to those in Theorem 4. For example, in this paper, we shall study the generating functions for permutations τ of the form τ = 1324 . . . p where p ≥ 4. That is, τ arises from the identity by interchanging 2 and 3. We shall show that U 1324,1 (y) = −y and for n ≥ 2,
U 1324,n (y) = (1 − y)U 1324,n−1 (y) + ⌊n/2⌋ k=2 (−y) k−1 C k−1 U 1324,n−2k+1 (y)(14)
where C k is the k-th Catalan number. For any p ≥ 5, we shall show that U 1324...p,n (y) = −y and for n ≥ 2, U 1324...p,n (y) = (1 − y)U 1324...p,n−1 (y) +
⌊ n−2 p−2 ⌋ k=1 (−y) k U 1324...p,n−(k(p−2)+1) (y).(15)
The outline of this paper is as follows. In Section 2, we shall briefly recall the background in the theory of symmetric functions that we will need for our proofs. Then in Section 3, we shall illustrate our method by proving (14) and (15). Then in Section 4, we shall show how the results of Section 3 allow us to compute explicit generating functions for the number of permutations of S n that have no consecutive occurrences of the pattern 1324 . . . p and exactly k descents for p ≥ 4 and k = 1, 2. Finally, in Section 5, we shall discuss our conclusions as well as some other results that follow by using the same method.
Symmetric functions.
In this section, we give the necessary background on symmetric functions that will be needed for our proofs.
Given a partition λ = (λ 1 , . . . λ ℓ ) where 0 < λ 1 ≤ · · · ≤ λ ℓ , we let ℓ(λ) be the number of nonzero integers in λ. If the sum of these integers is equal to n, then we say λ is a partition of n and write λ ⊢ n.
Let Λ denote the ring of symmetric functions in infinitely many variables x 1 , x 2 , . . .. The n th elementary symmetric function e n = e n (x 1 , x 2 , . . .) and n th homogeneous symmetric function h n = h n (x 1 , x 2 , . . .) are defined by the generating functions given in (13). For any partition λ = (λ 1 , . . . , λ ℓ ), let e λ = e λ 1 · · · e λ ℓ and h λ = h λ 1 · · · h λ ℓ . It is well known that {e λ : λ is a partition} is a basis for Λ. In particular, e 0 , e 1 , . . . is an algebraically independent set of generators for Λ and, hence, a ring homomorphism θ on Λ can be defined by simply specifying θ(e n ) for all n.
A key element of our proofs is the combinatorial description of the coefficients of the expansion of h n in terms of the elementary symmetric functions e λ given by Egecioglu and Remmel in [6]. They defined a λ-brick tabloid of shape (n) to be a rectangle of height 1 and length n which is covered by "bricks" of lengths found in the partition λ in such a way that no two bricks overlap. For example, Figure 1 shows one brick (2, 3, 7)-tabloid of shape (12). Let B λ,n denote the set of λ-brick tabloids of shape (n) and let B λ,n be the number of λ-brick tabloids of shape (n). If B ∈ B λ,n we will write B = (b 1 , . . . , b ℓ(λ) ) if the lengths of the bricks in B, reading from left to right, are b 1 , . . . , b ℓ(λ) . Through simple recursions, Egecioglu and Remmel [6] proved that
h n = λ⊢n (−1) n−ℓ(λ) B λ,n e λ .(16)
3 Computing U 1324...p,n (y).
The main goal of this section is to prove (14) and (15). We shall start by proving (15). Suppose that τ ∈ S j which starts with 1 and des(τ ) = 1. Our first step is to give a combinatorial interpretation to
U τ (t, y) = 1 N M τ (t, 1, y) = 1 1 + n≥1 t n n! N M τ,n (1, y) (17)
where N M τ,n (1, y) = σ∈N Mn(τ ) y 1+des(σ) .
To this end, we define a ring homomorphism θ τ on the ring of symmetric functions Λ by setting θ τ (e 0 ) = 1 and θ τ (e n ) = (−1) n n! N M τ,n (1, y) for n ≥ 1.
It follows that
θ τ (H(t)) = n≥0 θ τ (h n )t n = 1 θ τ (E(−t)) = 1 1 + n≥1 (−t) n θ τ (e n ) = 1 1 + n≥1 t n n! N M τ,n (1, y)
which is what we want to compute.
By (16), we have that
n!θ τ (h n ) = n! µ⊢n (−1) n−ℓ(µ) B µ,n θ τ (e µ ) = n! µ⊢n (−1) n−ℓ(µ) (b 1 ,...,b ℓ(µ) )∈Bµ,n ℓ(µ) i=1 (−1) b i b i ! N M τ,b i (1, y) = µ⊢n (−1) ℓ(µ) (b 1 ,...,b ℓ(µ) )∈Bµ,n n b 1 , . . . , b ℓ(µ) ℓ(µ) i=1 N M τ,b i (1, y).(19)
Our next goal is to give a combinatorial interpretation to the right-hand side of (19). If we are given a brick tabloid B = (b 1 , . . . , b ℓ(µ) ), then we can interpret the multinomial coefficient n b 1 ,...,bµ as all ways to assign sets S 1 , . . . , S ℓ(µ) to the bricks of B in such a way that |S i | = b i for i = 1, . . . , ℓ(µ) and the sets S 1 , . . . , S ℓ(µ) form a set partition of {1, . . . , n}. Next for each brick b i , we use the factor
N M τ,b i (1, y) = σ∈S b i ,τ -mch(σ)=0 y des(σ)+1
to pick a rearrangement σ (i) of S i which has no τ -matches to put in cells of b i and then we place a label of y on each cell that starts a descent in σ (i) plus a label of y on the last cell of b i . Finally, we use the term (−1) ℓ(µ) to turn each label y at the end of brick to a −y. We let
Next we define a weight-preserving sign-reversing involution I τ on O τ,n . Given an element O ∈ O τ,n , scan the cells of O from left to right looking for the first cell c such that either (i) c is labeled with a y or (ii) c is a cell at the end of a brick b i , the number in cell c is greater than the number in the first cell of the next brick b i+1 , and there is no τ -match in the cells of bricks b i and b i+1 .
In case (i), if c is a cell in brick b j , then we split b j in to two bricks b ′ j and b ′′ j where b ′ j contains all the cells of b j up to an including cell c and b ′′ j consists of the remaining cells of b j and we change the label on cell c from y to −y. In case (ii), we combine the two bricks b i and b i+1 into a single brick b and change the label on cell c from −y to y. For example, consider the element O ∈ O 13245,13 pictured in Figure 2. Note that even though the number in the last cell of the first brick is greater than the the number in the first cell of the second brick, we cannot combine these two bricks because the numbers 4 8 7 10 11 would be a 13245-match. Thus the first place that we can apply the involution is on cell 5 which is labeled with a y so that I τ (O) is the object pictured in Figure 3. We claim that I τ is an involution so that I 2 τ is the identity. To see this, consider case (i) where we split a brick b j at cell c which is labeled with a y. In that case, we let a be the number in cell c and a ′ be the number in cell c + 1 which must also be in brick b j . It must be the case that there is no cell labeled y before cell c since otherwise we would not use cell c to define the involution. However, we have to consider the possibility that when we spilt b j into b ′ j and b ′′ j that we might then be able to combine the brick b j−1 with b ′ j because the number in that last cell of b j−1 is greater than the number in the first cell of b ′ j and there is no τ -match in the cells of b j−1 and b ′ j . Since we always take an action on the left most cell possible when defining I τ (O), we know that we cannot combine b j−1 and b j so that there must be a τ -match in the cells of b j−1 and b j . Clearly, that match must have involved the number a ′ and the number in cell d which is the last cell in brick b j−1 . But that is impossible because then there would be two descents among the numbers between cell d and cell c + 1 which would violate our assumption that τ has only one descent. Thus whenever we apply case (i) to define I τ (O), the first action that we can take is to combine bricks b ′ j and b ′′ j so that I 2 τ (O) = O. If we are in case (ii), then again we can assume that there are no cells labeled y that occur before cell c. When we combine brick b i and b i+1 , then we will label cell c with a y. It is clear that combining the cells of b i and b i+1 cannot help us combine the resulting brick b with an earlier brick since it will be harder to have no τ -matches with the larger brick b. Thus the first place cell c where we can apply the involution will again be cell c which is now labeled with a y so that I 2
τ (O) = O if we are in case (ii).
It is clear from our definitions that if
I τ (O) = O, then sgn(O)W (O) = −sgn(I τ (O))W (I τ (O)). Hence it follows from (20) that n!θ τ (h n ) = O∈Oτ,n sgn(O)W (O) = O∈Oτ,n,Iτ (O)=O sgn(O)W (O).(21)
Thus we must examine the fixed points of I τ . So assume that O is a fixed point of I τ . First it is easy to see that there can be no cells which are labeled with y so that numbers in each brick of O must be increasing. Second we cannot combine two consecutive bricks b i and b i+1 in O which means either that there is an increase between the bricks b i and b i+1 or there is a decrease between the bricks b i and b i+1 , but there is a τ -match in the cells of the bricks b i and b i+1 . We claim that, in addition, the numbers in the first cells of the bricks must form an increasing sequence, reading from left to right. That is, suppose that b i and b i+1 are two consecutive bricks in a fixed point O of I τ and that a > a ′ where a is the number in the first cell of b i and a ′ is the number in the first cell of b i+1 . Then clearly the number in the last cell of b i must be greater than a ′ so that it must be the case that there is a τ -match in the cells of b i and b i+1 . However a ′ is the least number that resides in the cells of b i and b i+1 which means that the only way that a ′ could be part of a τ -match that occurs in the cells of b i and b i+1 is to have a ′ play the role of 1. But since we are assuming that τ starts with 1, this would mean that if a ′ is part of a τ -match, then that τ -match must be entirely contained in b i+1 which is impossible. Thus a ′ cannot be part of any τ -match that occurs in the cells of b i and b i+1 . However, this would mean that the τ -match that occurs in the cells of b i and b i+1 must either be contained entirely in the cells of b i or entirely in the cells of b i+1 which again is impossible. Hence it must be the case that a < a ′ . Thus we have proved the following.
Lemma 5. Suppose that τ ∈ S j , τ starts with 1, and des(τ ) = 1. Let θ τ : Λ → Q(y) be the ring homomorphism defined on Λ where Q(y) is the set of rational functions in the variable y over the rationals Q, θ τ (e 0 ) = 1 and θ τ (e n ) = (−1) n n! N M τ,n (1, y) for n ≥ 1. Then
n!θ τ (h n ) = O∈Oτ,n,Iτ (O)=O sgn(O)W (O) (22)
where O τ,n is the set of objects and I τ is the involution defined above. Moreover, every fixed point O of I τ has the following three properties. Now we specialize to the case of τ = 1324 . . . p where p ≥ 5. In this case, we can make a finer analysis of the fixed points of I τ . Let O be a fixed point of I τ . By Lemma 5, we know that 1 is in the first cell of O. We claim that 2 must be in the second or third cell of O. That is, suppose that 2 is in cell c where c > 3. Then since there are no descents within any brick, 2 must be the first cell of a brick. Moreover, since the minimal numbers in the bricks of O form an increasing sequence, reading from left to right, 2 must be in the first cell of the second brick. Thus if b 1 and b 2 are the first two bricks in O, then 1 is in the first cell of b 1 and 2 is in the first cell of b 2 . But then we claim that there is no τ -match in the cells of b 1 and b 2 . That is, since c > 3, b 1 has at least three cells so that O starts with an increasing sequence of length 3. But this means that 1 cannot be part of a τ -match. Similarly, no other cell of b 1 can be part of τ -match because the 2 in cell c is less than any of the remaining numbers of b 1 . Thus if there is a τ -match among the cells of b 1 and b 2 , it would have to be entirely contained in b 2 which is impossible. But this would mean that we could apply case (ii) of the definition of I τ to b 1 and b 2 which would violate our assumption that O is a fixed point of I τ . Thus, we have two cases.
Case 1. 2 is in cell 2 of O.
In this case there are two possibilities, namely, either (i) 1 and 2 are both in the first brick b 1 of O or (ii) brick b 1 is a single cell filled with 1 and 2 is in the first cell of the second brick b 2 of O. In either case, it is easy to see that 1 is not part of a τ -match in O and if we remove cell 1 from O and subtract 1 from the numbers in the remaining cells, we would end up with a fixed
point O ′ of I τ in O τ,n−1 . Now in case (i), it is easy to see that sgn(O)W (O) = sgn(O ′ )W (O ′ ) and in case (ii) since b 1 will have a label −y on the first cell, sgn(O)W (O) = (−y)sgn(O ′ )W (O ′ ).
It follows that fixed points in Case 1 will contribute (1 − y)U τ,n−1 (y) to U τ,n (y).
Case 2. 2 is in cell 3 of O.
Let O(i) denote the number in cell i of O and b 1 , b 2 , . . . be the bricks of O, reading from left to right. Since there are no descents within bricks in O and the first numbers of each brick are increasing, reading from left to right, it must be the case that 2 is in the first cell of b 2 . Thus b 1 has two cells. Note that b 2 must have at least p − 2 cells since otherwise, there could be no τ -match contained in the cells of b 1 and b 2 and we could combine bricks b 1 and b 2 which would mean that O is not a fixed point of I τ . But then the only reason that we cannot combine bricks b 1 and b 2 is that there is a τ -match in the cells of b 1 and b 2 which could only start at position 1.
Next we claim that O(p − 1) = p − 1. That is, since there is a τ -match starting at position 1 and p ≥ 5, we know that all the numbers in the first p − 2 cells of O are strictly less than Since O is a fixed point of I τ , this must mean that there is a τ -match in the cells of b 2 and b 3 . But since τ has only one descent, this τ -match can only start at the cell c which is the second
O(p − 1). Thus O(p − 1) ≥ p − 1. Now if O(p − 1) > p − 1,to the last cell of b 2 . Thus c could be p − 1 if b 2 has p − 2 cells or c > p − 1 if b 2 has more than p − 2 cells. In either case, p − 1 < O(p − 1) ≤ O(c) < O(c + 1) > O(c + 2) = i. But this is impossible since to have a τ -match starting at cell c, we must have O(c) < O(c + 2). Thus it must be the case that O(p − 1) = p − 1 and {O(1), . . . , O(p − 1)} = {1, . . . , p − 1}.
We now have two subcases.
Case 2.a. There is no τ -match in O starting at cell p − 1.
Then we claim that O(p) = p. That is, if O(p) = p, then p cannot be in b 2 so that p must be in the first cell of the brick b 3 . But then we claim that we could combine bricks b 2 and b 3 . That is, there will be a decrease between bricks b 2 and b 3 since p < O(p) and O(p) is in b 2 . Since there is no τ -match in O starting at cell p − 1, the only possible τ -match among the cells of b 2 and b 3 would have to start at a cell c = p − 1. But it can't be that c < p − 1 since then it would be the case that O(c) < O(c + 1) < O(c + 2). Similarly, it cannot be that c > p − 1 since then O(c) > p and p has to be part of the τ -match which is impossible since O(c) must play the role of 1 in the τ -match. Thus it must be the case that O(p) = p. It then follows that if we let O ′ be the result of removing the first p − 1 cells from O and subtracting p − 1 from the remaining numbers, then O ′ will be a fixed point of I τ in O τ,n−(p−1) . Note that if b 2 has p − 2 cells, then O ′ will start with a brick with one cell and if b 2 has more than p − 2 cells, then O ′ will start with a brick with at least two cells. Since there is −y coming from the brick b 1 , it is easy to see that the fixed points in Case 2.a will contribute −yU τ,n−(p−1) (y) to U τ,n (y).
Case 2.b. There is a τ -match starting cell p − 1 in O.
In this case, it must be that O(p − 1) < O(p) > O(p + 1) so that b 2 must have p − 2 cells and brick b 3 starts at cell p + 1. We claim that b 3 must have at least p − 2 cells. That is, if b 3 has less than p − 2 cells, then there could be no τ -match among the cells of b 2 and b 3 so then we could combine b 2 and b 3 violating the fact that O is a fixed point of I τ .
In the general case, assume that in O, the bricks b 2 , . . . , b k−1 all have (p − 2) cells. Then let r 1 = 1 and for j = 2, . . . , k − 1, let
r j = 1 + (j − 1)(p − 2). Thus r j is the position of the second to last cell of brick b j for 1 ≤ j ≤ k − 1. Furthermore, assume that there is a τ -match starting at cell r j for 1 ≤ j ≤ k − 1. It follows that O(r k−1 ) < O(r k−1 + 1) > O(r k−1 + 2)
so that brick b k must start at cell r k−1 + 2 and there is a decrease between bricks b k−1 and b k . But then it must be the case that b k has at least p − 2 cells since if b k has less than p − 2 cells, we could combine bricks b k−1 and b k violating the fact that O is a fixed point of I τ . Let r k = 1 + (k − 1)(p − 2) and assume that O does not have a τ -match starting at position r k . Thus we have the situation pictured below. (1), . . . , O(r j−1 )}, it follows that O(r j ) ≥ r j . Next suppose that O(r j ) > r j . Then let i be the least number that does not lie in the bricks b 1 , . . . , b j . Because the numbers in each brick increase and the minimal numbers in the bricks are increasing, it must be the case that i is in the first cell of the next brick b j+1 . Now it cannot be that j < k because then we have that i = O(r j + 2) ≤ r j < O(r j ) < O(r j+1 ) which would violate the fact that there is a τ -match in O starting at cell r j . If j = k, then it follows that there is a decrease between bricks b k and b k+1 since b k+1 starts with i ≤ r k < O(r k ). Since O is a fixed point of I τ , this must mean that there is a τ -match in the cells of b k and b k+1 . But since τ has only one descent, this τ -match can only start at the cell c which is the second to the last cell of b k . Thus c must be greater than r k because by hypothesis there cannot be a τ -match starting at cell r k . So b k+1 must have more than p − 2 cells. In this case, we have that This means that the sequence O(1), . . . , O(r k ) is completely determined. Next we claim that since there is no τ -match starting at position r k , it must be the case that O(r k + 1) = r k + 1. That is, if O(r k + 1) = r k + 1, then r k + 1 cannot be in brick b k so then r k + 1 must be in the first cell of the brick b k+1 . But then we claim that we could combine bricks b k and b k+1 . That is, there will be a decrease between bricks b k and b k+1 since r k + 1 < O(r k + 1) and O(r k + 1) is in b k . Since there is no τ -match starting in O at cell r k , the only possible τ -match among the cells of b k and b k+1 would have to start at a cell c = r k . Now it cannot be that c < r k since then O(c) < O(c + 1) < O(c + 2). But it cannot be that c > r k since then O(c) > r k + 1 and r k + 1 would have to be part of the τ -match which means that O(c) could not play the role of 1 in the τ -match. Thus it must be the case that O(r k + 1) = r k + 1. It then follows that if we let O ′ be the result of removing the first r k cells from O and subtracting r k from each number in the remaining cells, then O ′ will be a fixed point I τ in O τ,n−r k . Note that if b k has p − 2 cells, then the first brick of O ′ will have one cell and if b k has more than p − 2 cells, then the first brick of O ′ will have at least two cells. Since there is a factor −y coming from each of the bricks b 1 , . . . , b k−1 , it is easy to see that the fixed points in Case 2.b will contribute k≥3 (−y) k−1 U τ,n−((k−1)(p−2)+1) (y) to U τ,n (y). Thus we have proved the following theorem.
O(1) O(2) b 1 r 2 O(3) ··· O(r 2 ) O(r 2 +1) b 2 ... r k−1 ··· O(r k−1 ) O(r k−1 +1) b k−1 r k ··· O(r k ) O(r k +1) ··· b ki ≤ r k < O(r k ) ≤ O(c) < O(c + 1) > O(c + 2) = i.Theorem 6. Let τ = 1324 . . . p where p ≥ 5. Then N M τ (t, x, y) = 1 U τ (t, y) x where U τ (t, y) = 1 + n≥1 U τ,n (y)
t n n! , U τ,1 (y) = −y, and for n ≥ 2,
U τ,n (y) = (1 − y)U τ,n−1 (y) + ⌊ n−2 p−2 ⌋ k=1 (−y)U τ,n−(k(p−2)+1) (y).
For example, we have computed the following.
U 13245,1 (y) = −y, U 13245,2 (y) = −y + y 2 , U 13245,3 (y) = −y + 2y 2 − y 3 , U 13245,4 (y) = −y + 3y 2 − 3y 3 + y 4 , U 13245,5 (y) = −y + 5y 2 − 6y 3 + 4y 4 − y 5 , U 13245,6 (y) = −y + 7y 2 − 12y 3 + 10y 4 − 5y 5 + y 6 , U 13245,7 (y) = −y + 9y 2 − 21y 3 + 23y 4 − 15y 5 + 6y 6 − y 7 , U 13245,8 (y) = −y + 11y 2 − 34y 3 + 47y 4 − 39y 5 + 21y 6 − 7y 7 + y 8 , U 13245,9 (y) = −y + 13y 2 − 51y 3 + 88y 4 − 90y 5 + 61y 6 − 28y 7 + 8y 8 − y 9 , U 13245,10 (y) = −y + 15y 2 − 72y 3 + 153y 4 − 189y 5 + 156y 6 − 90y 7 + 36y 8 − 9y 9 + y 10 , U 13245,11 (y) = −y + 17y 2 − 97y 3 + 250y 4 − 368y 5 + 361y 6 − 252y 7 + 127y 8 − 45y 9 + 10y 10 − y 11 .
U 132456,1 (y) = −y, U 132456,2 (y) = −y + y 2 , U 132456,3 (y) = −y + 2y 2 − y 3 , U 132456,4 (y) = −y + 3y 2 − 3y 3 + y 4 , U 132456,5 (y) = −y + 4y 2 − 6y 3 + 4y 4 − y 5 , U 132456,6 (y) = −y + 6y 2 − 10y 3 + 10y 4 − 5y 5 + y 6 , U 132456,7 (y) = −y + 8y 2 − 17y 3 + 20y 4 − 15y 5 + 6y 6 − y 7 , U 132456,8 (y) = −y + 10y 2 − 27y 3 + 38y 4 − 35y 5 + 21y 6 − 7y 7 + y 8 , U 132456,9 (y) = −y + 12y 2 − 40y 3 + 68y 4 − 74y 5 + 56y 6 − 28y 7 + 8y 8 − y 9 , U 132456,10 (y) = −y + 14y 2 − 57y 3 + 114y 4 − 146y 5 + 131y 6 − 84y 7 + 36y 8 − 9y 9 + y 10 , U 132456,11 (y) = −y + 16y 2 − 78y 3 + 182y 4 − 270y 5 + 282y 6 − 216y 7 + 120y 8 − 45y 9 + 10y 10 − y 11 .
U 1324567,1 (y) = −y, U 1324567,2 (y) = −y + y 2 , U 1324567,3 (y) = −y + 2y 2 − y 3 , U 1324567,4 (y) = −y + 3y 2 − 3y 3 + y 4 , U 1324567,5 (y) = −y + 4y 2 − 6y 3 + 4y 4 − y 5 , U 1324567,6 (y) = −y + 5y 2 − 10y 3 + 10y 4 − 5y 5 + y 6 , U 1324567,7 (y) = −y + 7y 2 − 15y 3 + 20y 4 − 15y 5 + 6y 6 − y 7 , U 1324567,8 (y) = −y + 9y 2 − 23y 3 + 35y 4 − 35y 5 + 21y 6 − 7y 7 + y 8 , U 1324567,9 (y) = −y + 11y 2 − 34y 3 + 59y 4 − 70y 5 + 56y 6 − 28y 7 + 8y 8 − y 9 , U 1324567,10 (y) = −y + 13y 2 − 48y 3 + 96y 4 − 130y 5 + 126y 6 − 84y 7 + 36y 8 − 9y 9 + y 10 , U 1324567,11 (y) = −y + 15y 2 − 65y 3 + 150y 4 − 230y 5 + 257y 6 − 210y 7 + 120y 8 − 45y 9 + 10y 10 − y 11 .
Of course, one can use these initial values of the U 1324...p,n (y) to compute the initial values of N M 1324...p (t, x, y). For example, we have used Mathematica to compute the following initial terms of N M 13245 (t, x, y), N M 132456 (t, x, y), and N M 1324567 (t, x, y). N M 13245 (t, x, y) = 1 + xyt + 1 2
xy + x 2 y 2 t 2 + 1 6 xy + xy 2 + 3x 2 y 2 + x 3 y 3 t 3 + 1 24 xy + 4xy 2 + 7x 2 y 2 + xy 3 + 4x 2 y 3 + 6x 3 y 3 + x 4 y 4 t 4 + 1 120
xy + 10xy 2 + 15x 2 y 2 + 11xy 3 + 30x 2 y 3 + 25x 3 y 3 + xy 4 + 5x 2 y 4 + 10x 3 y 4 + 10x 4 y 4 + x 5 y 5 t 5 + 1 720
xy + 24xy 2 + 31x 2 y 2 + 62xy 3 + 140x 2 y 3 + 90x 3 y 3 + 26xy 4 + 91x 2 y 4 + 120x 3 y 4 + 65x 4 y 4 + xy 5 + 6x 2 y 5 + 15x 3 y 5 + 20x 4 y 5 + 15x 5 y 5 + x 6 y 6 t 6 + 1 5040
xy + 54xy 2 + 63x 2 y 2 + 273xy 3 + 553x 2 y 3 + 301x 3 y 3 + 292xy 4 + 840x 2 y 4 + 875x 3 y 4 + 350x 4 y 4 + 57xy 5 + 238x 2 y 5 + 406x 3 y 5 + 350x 4 y 5 + 140x 5 y 5 + xy 6 + 7x 2 y 6 + 21x 3 y 6 + 35x 4 y 6 + 35x 5 y 6 + 21x 6 y 6 + x 7 y 7 t 7 + 1 40320
xy + 116xy 2 + 127x 2 y 2 + 1068xy 3 + 2000x 2 y 3 + 966x 3 y 3 + 2228xy 4 + 5726x 2 y 4 + 5152x 3 y 4 + 1701x 4 y 4 + 1171xy 5 + 4016x 2 y 5 + 5474x 3 y 5 + 3640x 4 y 5 + 1050x 5 y 5 + 120xy 6 + 575x 2 y 6 + 1176x 3 y 6 + 1316x 4 y 6 + 840x 5 y 6 + 266x 6 y 6 + xy 7 + 8x 2 y 7 + 28x 3 y 7 + 56x 4 y 7 + 70x 5 y 7 + 56x 6 y 7 + 28x 7 y 7 + x 8 y 8 t 8 + · · · N M 132456 (t, x, y) = 1 + xyt + 1 2 xy + x 2 y 2 t 2 + 1 6 xy + xy 2 + 3x 2 y 2 + x 3 y 3 t 3 + 1 24 xy + 4xy 2 + 7x 2 y 2 + xy 3 + 4x 2 y 3 + 6x 3 y 3 + x 4 y 4 t 4 + 1 120 xy + 11xy 2 + 15x 2 y 2 + 11xy 3 + 30x 2 y 3 + 25x 3 y 3 + xy 4 + 5x 2 y 4 + 10x 3 y 4 + 10x 4 y 4 + x 5 y 5 t 5 + 1 720 xy + 25xy 2 + 31x 2 y 2 + 66xy 3 + 146x 2 y 3 + 90x 3 y 3 + 26xy 4 + 91x 2 y 4 + 120x 3 y 4 + 65x 4 y 4 + xy 5 + 6x 2 y 5 + 15x 3 y 5 + 20x 4 y 5 + 15x 5 y 5 + x 6 y 6 t 6 + 1 5040
xy + 55xy 2 + 63x 2 y 2 + 297xy 3 + 581x 2 y 3 + 301x 3 y 3 + 302xy 4 + 868x 2 y 4 + 896x 3 y 4 + 350x 4 y 4 + 57xy 5 + 238x 2 y 5 + 406x 3 y 5 + 350x 4 y 5 + 140x 5 y 5 + xy 6 + 7x 2 y 6 + 21x 3 y 6 + 35x 4 y 6 + 35x 5 y 6 + 21x 6 y 6 + x 7 y 7 t 7 + 1 40320
xy + 117xy 2 + 127x 2 y 2 + 1153xy 3 + 2092x 2 y 3 + 966x 3 y 3 + 2401xy 4 + 6086x 2 y 4 + 5348x 3 y 4 + 1701x 4 y 4 + 1191xy 5 + 4096x 2 y 5 + 5586x 3 y 5 + 3696x 4 y 5 + 1050x 5 y 5 + 120xy 6 + 575x 2 y 6 + 1176x 3 y 6 + 1316x 4 y 6 + 840x 5 y 6 + 266x 6 y 6 + xy 7 + 8x 2 y 7 + 28x 3 y 7 + 56x 4 y 7 + 70x 5 y 7 + 56x 6 y 7 + 28x 7 y 7 + x 8 y 8 t 8 + · · · N M 1324567 (t, x, y) = 1 + xyt + 1 2 xy + x 2 y 2 t 2 + 1 6 xy + xy 2 + 3x 2 y 2 + x 3 y 3 t 3 + 1 24 xy + 4xy 2 + 7x 2 y 2 + xy 3 + 4x 2 y 3 + 6x 3 y 3 + x 4 y 4 t 4 + 1 120 xy + 11xy 2 + 15x 2 y 2 + 11xy 3 + 30x 2 y 3 + 25x 3 y 3 + xy 4 + 5x 2 y 4 + 10x 3 y 4 + 10x 4 y 4 + x 5 y 5 t 5 + 1 720 xy + 26xy 2 + 31x 2 y 2 + 66xy 3 + 146x 2 y 3 + 90x 3 y 3 + 26xy 4 + 91x 2 y 4 + 120x 3 y 4 + 65x 4 y 4 + xy 5 + 6x 2 y 5 + 15x 3 y 5 + 20x 4 y 5 + 15x 5 y 5 + x 6 y 6 t 6 + 1 5040
xy + 56xy 2 + 63x 2 y 2 + 302xy 3 + 588x 2 y 3 + 301x 3 y 3 + 302xy 4 + 868x 2 y 4 + 896x 3 y 4 + 350x 4 y 4 + 57xy 5 + 238x 2 y 5 + 406x 3 y 5 + 350x 4 y 5 + 140x 5 y 5 + xy 6 + 7x 2 y 6 + 21x 3 y 6 + 35x 4 y 6 + 35x 5 y 6 + 21x 6 y 6 + x 7 y 7 t 7 + 1 40320
xy + 118xy 2 + 127x 2 y 2 + 1185xy 3 + 2128x 2 y 3 + 966x 3 y 3 + 2416xy 4 + 6126x 2 y 4 + 5376x 3 y 4 + 1701x 4 y 4 + 1191xy 5 + 4096x 2 y 5 + 5586x 3 y 5 + 3696x 4 y 5 + 1050x 5 y 5 + 120xy 6 + 575x 2 y 6 + 1176x 3 y 6 + 1316x 4 y 6 + 840x 5 y 6 + 266x 6 y 6 + xy 7 + 8x 2 y 7 + 28x 3 y 7 + 56x 4 y 7 + 70x 5 y 7 + 56x 6 y 7 + 28x 7 y 7 + x 8 y 8 t 8 + · · ·
We note that there are many terms in these expansions which are easily explained. For example, we claim that for any p ≥ 4, the coefficient of x k y k in N M 1324...p,n (x, y) is always the Stirling number S(n, k) which is the number of set partitions of {1, . . . , n} into k parts. That is, a permutation σ ∈ S n that contributes to the coefficient x k y k in N M 1324...p,n (x, y) must have k left-to-right minima and k −1 descents. Since each left-to-right minima of σ which is not the first element is always the second element of descent pair, it follows that if 1 = i 1 < i 2 < i 3 < · · · < i k are the positions of the left to right minima, then σ must be increasing in each of the intervals We claim that N M 1324...p,n (x, y)| xy 2 = 2 n−1 − n if n < p and 2 n−1 − n − (n − (p − 1)) = 2 n−1 − 2n + p − 1 if n ≥ p.
That is, suppose that σ ∈ S n contributes to N M 1324...p,n (x, y)| xy 2 . Then σ must have 1 left-toright minima and one descent. It follows that σ must start with 1 and have one descent. Now if A is any subset of {2, . . . , n} and B = {2, . . . , n} − A, then we let σ A be the permutation σ A = 1 A ↑ B ↑. The only choices of A that do not give rise to a permutation with one descent are ∅ and {2, . . . , i} for i = 2, . . . , n. It follows that there 2 n−1 −n permutations that start with 1 and have 1 descent. Next consider when such a σ A could have a 1234 . . . p-match. If the 1234 . . . pmatch starts at position i, then it must be the case that red(σ i σ i+1 σ i+2 σ i+3 ) = 1324. This means that the only descent is at position i + 1 and all the elements σ j for j ≥ i + 3 are greater than or equal to σ i+3 . But this means that all the elements between 1 and σ i+2 must appear in increasing order in σ 2 . . . σ i−1 . It follows that σ A is of the form 1 . . . (q − 2)q(q + 2)(q + 1)(q + 2) . . It then follows that if we let O ′ be the result of removing the first 3 cells from O and subtracting 3 from the remaining elements, then O ′ will be a fixed point I 1324 in O 1324,n−3 . Since there is −y coming from the brick b 1 , it is easy to see that the fixed points in Case II.a will contribute −yU 1324,n−3 (y) to U 1324,n (y).
Case II.b. There is a 1324-match starting a 3 in O.
In this case, it must be that O(3) < O(4) > O(5) so that b 2 must have two cells and brick b 3 starts at cell 5. We claim that b 3 must have at least two cells. That is, if b 3 has one cell, then there could be no 1324-match among the cells of b 2 and b 3 so that we could combine b 2 and b 3 violating the fact that O is a fixed point of I 1324 .
In the general case, assume that in O, the bricks b 2 , . . . , b k−1 all have two cells and there are 1324-matches starting at cells 1, 3, . . . , 2k − 3 but there is no 1324-match starting at cell 2k − 1 in O. Then we know that b k has least two cells. Let c i < d i be the numbers in the first two cells of brick b i for i = 1, . . . , k. Then we have that red(
c i d i c i+1 d i+1 ) = 1324 for 1 ≤ i ≤ k − 1. This means that c i < c i+1 < d i < d i+1 .
First we claim that it must be the case that {O (1) Since numbers in bricks are increasing, m must occupy the first cell of b k+1 . But then there is a descent between bricks b k and b k+1 so that m must be part of a 1324-match. But the only way this can happen is if in the 1324-match involving m, m plays the role of 2 and the numbers in the last two cells of brick b k play the role of 1 3. Since, we are assuming that a 1324-match does not start at cell 2k − 1 which is the cell that the number c k occupies, the numbers in the last two cells of brick b k must be greater than or equal to d k = M which is impossible since m < M . Thus it must be the case that {O(1), . . . , O(2k)} = {1, . . . , 2k} and that d k = 2k. It now follows that if we remove the first 2k − 1 cells from O and replace each remaining number i in O by i − (2k − 1), then we will end up with a fixed point in O ′ of I 1324 in O n−(2k−1) . Thus each such fixed point O will contribute (−y) k−1 U n−2k+1 (y) to U n (y).
The only thing left to do is to count the number of such fixed points O. That is, we must count the number of sequences
c 1 d 1 c 2 d 2 . . . c k d k such that (i) c 1 = 1, (ii) c 2 = 2, (iii) d k = 2k, (iv) {c 1 , d 1 , . . . , c k , d k } = {1, 2, . . . , 2k}, and (v) red(c i d i c i+1 d i+1 ) = 1324 for each 1 ≤ i ≤ k − 1.
We claim that there are C k−1 such sequences where C n = 1 n+1 2n n is the n-th Catlan number. It is well known that C k−1 counts the number of Dyck paths of length 2k − 2. A Dyck path of length 2k − 2 is a path that starts at (0, 0) and ends at (2k − 2, 0) and consists of either up-steps (1,1) or down-steps (1,-1) in such a way that the path never goes below the x-axis. Thus we will give a bijection φ between the set of Dyck paths of length 2k − 2 and the set of sequences c 1 , d 1 , . . . , c k , d k satisfying conditions (i)-(v). The map φ is quite simple. That is, suppose that we start with a Dyck path P = (p 1 , p 2 , . . . , p 2k−2 ) of length 2k − 2. First, label the segments p 1 , . . . , p 2k−2 with 2, . . . , 2k − 1, respectively. Then φ(P ) is the sequence c 1 d 1 . . . c k d k where c 1 = 1 and c 2 . . . c k are the labels of the up-steps of P , reading from left to right, d 1 . . . d k−1 are the labels of the down steps, reading from left to right, and d 2k = 2k. We have pictured an example in Figure 4 of the bijection φ in the case where k = 6. It is easy to see by construction that if P is a Dyck path of length 2k − 2 and φ(P ) = c 1 d 1 . . . c k d k , then c 1 < c 2 < · · · < c k and d 1 < d 2 < · · · < d k . Moreover, since each Dyck path must start with an up-step, we have that c 2 = 2. Clearly c 1 = 1, d k = 2k, and {c 1 , d 1 , . . . , c k , d k } = {1, . . . , 2k} by construction. Thus c 1 d 1 . . . c k d k satisfies conditions (i)-(iv). For condition (v), note that c 1 = 1 < d 1 > 2 = c 2 < d 2 so that red(c 1 d 1 c 2 d 2 ) = 1 3 2 4. If 2 ≤ i ≤ k − 1, then note that c i equals the label of the (i − 1)st up-step, c i+1 equals the label of the i-th up-step, and d i is the label of i-th down-step. Since in a Dyck path, the i-th down-step must occur after the i-th up-step, it follows that c i < c i+1 < d i < d i+1 so that red(c i d i c i+1 d i+1 ) = 1 3 2 4. Vice versa, if we start with a sequence c 1 d 1 . . . c k d k satisfying conditions (i)-(v) and create a path P = (p 1 , . . . , p 2k−2 ) with labels 2, . . . , 2k − 1 such that p j is an up-step if j + 1 ∈ {c 2 , . . . , c k } and p j is an down-step if j + 1 ∈ {d 1 , . . . , d k−1 }, then condition (iii) ensures P starts with an up-step and condition (v) ensures that the i-th up-step occurs before the i-th down step so that P will be a Dyck path. Thus φ is a bijection between the set of Dyck paths of length 2k − 2 and the set of sequence c 1 , d 1 , . . . , c k , d k satisfying conditions where C k is the k th Catalan number.
In this case, one can easily compute that U 1324,1 (y) = −y, U 1324,2 (y) = −y + y 2 , U 1324,3 (y) = −y + 2y 2 − y 3 , U 1324,4 (y) = −y + 4y 2 − 3y 3 + y 4 , U 1324,5 (y) = −y + 6y 2 − 8y 3 + 4y 4 − y 5 , U 1324,6 (y) = −y + 8y 2 − 18y 3 + 13y 4 − 5y 5 + y 6 , U 1324,7 (y) = −y + 10y 2 − 32y 3 + 36y 4 − 19y 5 + 6y 6 − y 7 , U 1324,8 (y) = −y + 12y 2 − 50y 3 + 85y 4 − 61y 5 + 26y 6 − 7y 7 + y 8 , U 1324,9 (y) = −y + 14y 2 − 72y 3 + 166y 4 − 170y 5 + 94y 6 − 34y 7 + 8y 8 − y 9 , U 1324,10 (y) = −y + 16y 2 − 98y 3 + 287y 4 − 412y 5 + 296y 6 − 136y 7 + 43y 8 − 9y 9 + y 10 , U 1324,11 (y) = −y + 18y 2 − 128y 3 + 456y 4 − 854y 5 + 824y 6 − 473y 7 + 188y 8 − 53y 9 + 10y 10 − y 11 .
This, in turn, allows us to compute the first few terms of the generating function N M 1324 (t, x, y).
That is, one can use Mathematica to compute that N M 1324 (t, x, y) = 1 + txy + 1 2 t 2 xy + x 2 y 2 + 1 6 t 3 xy + xy 2 + 3x 2 y 2 + x 3 y 3 + 1 24 t 4 xy + 3xy 2 + 7x 2 y 2 + xy 3 + 4x 2 y 3 + 6x 3 y 3 + x 4 y 4 + 1 120 t 5 xy + 9xy 2 + 15x 2 y 2 + 8xy 3 + 25x 2 y 3 + 25x 3 y 3 + xy 4 + 5x 2 y 4 + 10x 3 y 4 + 10x 4 y 4 + x 5 y 5 + 1 720 t 6 xy + 23xy 2 + 31x 2 y 2 + 47xy 3 + 119x 2 y 3 + 90x 3 y 3 + 20xy 4 + 73x 2 y 4 + 105x 3 y 4 + 65x 4 y 4 + xy 5 + 6x 2 y 5 + 15x 3 y 5 + 20x 4 y 5 + 15x 5 y 5 + x 6 y 6 + 1 5040 t 7 xy + 53xy 2 + 63x 2 y 2 + 221xy 3 + 490x 2 y 3 + 301x 3 y 3 + 202xy 4 + 637x 2 y 4 + 749x 3 y 4 + 350x 4 y 4 + 47xy 5 + 196x 2 y 5 + 343x 3 y 5 + 315x 4 y 5 + 140x 5 y 5 + xy 6 + 7x 2 y 6 + 21x 3 y 6 + 35x 4 y 6 + 35x 5 y 6 + 21x 6 y 6 + x 7 y 7 + 1 40320
xy + 115xy 2 + 127x 2 y 2 + 922xy 3 + 1838x 2 y 3 + 966x 3 y 3 + 1571xy 4 + 4421x 2 y 4 + 4466x 3 y 4 + 1701x 4 y 4 + 795xy 5 + 2890x 2 y 5 + 4270x 3 y 5 + 3164x 4 y 5 + 1050x 5 y 5 + 105xy 6 + 495x 2 y 6 + 1008x 3 y 6 + 1148x 4 y 6 + 770x 5 y 6 + 266x 6 y 6 + xy 7 + 8x 2 y 7 + 28x 3 y 7 + 56x 4 y 7 + 70x 5 y 7 + 56x 6 y 7 + 28x 7 y 7 + x 8 y 8 t 8 + · · ·
We note that there are other methods to compute N M 1324 (t, 1, 1). That is, Elizalde [4] developed recursive techniques to find the coefficients of the series N M 1324 (t, 1, 1).
4 Permutations with no 1324 . . . p-matches and one or two descents
In this section, we will show how we can use Theorem 6 and Theorem 7 to find the generating function for the number of permutations σ ∈ S n which have no 1324 . . . p-matches and have exactly k descents for k = 1, 2 and p ≥ 4. That is, fix p ≥ 4 and let d (i) n,p denote the number of σ ∈ S n such that 1324 . . . p-mch(σ) = 0 and des(σ) = i. Our goal is to compute
D (i) p (t) = n≥0 d (i) n,p t n n! = N M 1324...p (t, 1, y)| y i+1
for i = 1 and i = 2.
To this end, we first want to compute U 1324...p,n (y)| y , U 1324...p,n (y)| y 2 , and U 1324...p,n (y)| y 3 . That is, we want to compute the number of fixed points of I 1324...p that have either 1, 2, or 3 bricks. Clearly there is only one fixed point of I 1324...p of length n which has just one brick since in that case, the underlying permutation must be the identity. In such a situation, the last cell of the brick is labeled with −y so that for all n ≥ 1 and all p ≥ 4, U 1324...p,n (y)| y = −1. Hence
U 1324...p (t, y)| y = 1 − e t .(23)
Next we consider the fixed points of I 1324...p which are of length n and consists of two bricks, a brick B 1 of length b 1 followed by a brick B 2 of length b 2 . Note that in this case, the last cells of B 1 and B 2 are labeled with −y so that the weight of all such fixed points is y 2 . Suppose the underlying permutation is σ = σ 1 . . . σ n . Then there are two cases. Case 1. There is an increase between the two bricks, i.e. σ b 1 < σ b 1 +1 .
In this case, it easy to see that σ must be the identity permutation and, hence, there are n − 1 fixed points in case 1 since b 1 can range from 1 to n − 1.
Case 2.
There is an decrease between the two bricks, i.e. σ b 1 > σ b 1 +1 .
In this case, there must be a 1324 . . . p-match in the elements in the bricks of B 1 and B 2 which means that it must be the case that red(
σ b 1 −1 σ b 1 σ b 1 +1 . . . σ b 1 +p−2 ) = 1324 . . . p. Now suppose that σ b 1 −1 = x. Since σ b 1 +1
is the smallest element in brick B 2 and the elements in brick B 2 increase and σ b 1 > σ b 1 +1 , it must be the case that 1, . . . , x − 1 must lie in brick B 1 . It cannot be that σ b 1 = x + 1 since σ b 1 > σ b 1 +1 . Thus it must be the case that σ b 1 +1 = x + 1 and σ b 1 = x + 2 since σ b 1 < σ b 1 +2 . Thus brick B 1 consists of the elements 1, . . . , x, x + 2. Hence there are n − p + 1 possibilities in case 2 if n ≥ p and no possibilities in case 2 if n < p.
It follows that
U 1324...p,n | y 2 = 0 if n = 0, 1 n − 1 if 2 ≤ n ≤ p − 1 and 2n − p if n ≥ p.(24)
Note that
n≥p (2n − p) t n n! = 2t n≥p t n−1 (n − 1)! − p n≥p t n n! = 2t(e t − p−2 n=0 t n n! ) − p(e t − p−1 n=0 t n n! ) = (2t − p)e t + p + p−1 n=1 (p − 2n) t n n! .
Thus
U 1324...p (t, y)| y 2 = (2t − p)e t + p + p−1 n=1 (p − 2n) t n n! + p−1 n=2 (n − 1) t n n! = (2t − p)e t + p + p−1 n=1 (p − n − 1) t n n! .(25)
Next we consider a fixed point of I 1324...p which has 3 bricks, B 1 of size b 1 followed by B 2 of size b 2 followed by B 3 of size b 3 . Let σ = σ 1 . . . σ n be the underlying permutation. The weight of all such fixed points is −y 3 . We then have 4 cases.
Case a. There are increases between B 1 and B 2 and between B 2 and B 3 , i.e. σ b 1 < σ b 1 +1 and σ b 1 +b 2 < σ b 1 +b 2 +1 .
In this case, it is easy to see that σ must be the identity permutation so that there are n−1 2 possibilities in case 1 if n ≥ 3.
Case b.
There is an increase between B 1 and B 2 and a decrease between B 2 and B 3 , i.e. σ b 1 < σ b 1 +1 and σ b 1 +b 2 > σ b 1 +b 2 +1 .
In this case, it must be the case that σ 1 < · · · < σ b 1 +b 2 and
red(σ b 1 +b 2 −1 σ b 1 +b 2 σ b 1 +b 2 +1 . . . σ b 1 +b 2 +p−2 ) = 1324 . . . p.
Then we can argue exactly as in case 2 above that there must exist an x such that σ b 1 +b 2 −1 = x and 1, . . . , x − 1 must occur to the left of σ b 1 +b 2 −1 , σ b 1 +b 2 = x + 2 and σ b 1 +b 2 +1 = x + 1. Then for any fixed x ≥ 2, we have x − 1 choices for the length of B 1 so that we have Case c. There is a decrease between B 1 and B 2 and an increase between B 2 and B 3 , i.e.
σ b 1 > σ b 1 +1 and σ b 1 +b 2 < σ b 1 +b 2 +1 .
In this case, it must be the case that σ b 1 +1 < · · · < σ n and red(σ b 1 −1 σ b 1 σ b 1 +1 . . . σ b 1 +p−2 ) = 1324 . . . p. Again we can argue as in case 2 above that there must exist an x such that σ b 1 −1 = x and 1, . . . , x−1 must occur to the left of σ b 1 −1 , σ b 1 = x+2 and σ b 1 +1 = x+1. Then for any fixed x, we have n−1−(x+p−1) choices for the length of B 2 so that we have n−p x=1 n−x−p−1 = n−p+1 2 possibilities if n ≥ p + 1 and no possibilities if n ≤ p.
Case d. There are decreases between B 1 and B 2 and between B 2 and B 3 , i.e. σ b 1 > σ b 1 +1 and σ b 1 +b 2 > σ b 1 +b 2 +1 .
In this case, there are two subcases. Subcase d.1 p = 4. Now we must have red(σ b 1 −1 σ b 1 σ b 1 +1 σ b 1 +2 ) = 1324. First suppose that b 2 = 2. Then we also have that red(σ
b 1 +1 σ b 1 +2 σ b 1 +3 σ b 1 +4 ) = 1324. It follows that x = σ b 1 −1 < σ b 1 +1 < σ b 1 +3 .
Since σ b 1 +1 is the smallest element in brick B 2 and σ b 1 +3 is the smallest element in brick B 3 , it must be the case that 1, . . . x − 1 lie in brick B 1 and that σ b 1 +1 = x + 1. It also must be the case that σ b 1 < σ b 1 +2 < σ b 1 +4 so that σ b 1 , σ b 1 +2 ∈ {x + 2, x + 3} and σ b 1 +4 = x + 4. Thus there are two possibilities for each x. As x can vary between 1 and n − 5 in this case, we have 2(n − 5) possibilities if b 2 = 2 and n ≥ 6 and no possibilities if n < 6.
Next consider the case where b 2 ≥ 3. Again we must have red(σ b 1 −1 σ b 1 σ b 1 +1 σ b 1 +2 ) = 1324. Similarly we must have red(σ b 1 +b 2 −1 σ b 1 +b 2 σ b 1 +b 2 +1 σ b 1 +b 2 +2 ) = 1324, but this condition does not involve σ b 1 +1 . Nevertheless, these conditions still force that σ b 1 < σ b 1 +b 2 +1 so that if σ b 1 −1 = x, then x is less than the least elements in bricks B 2 and B 3 so that 1, . . . , x − 1 must be in brick B 1 and σ b 1 +1 = x + 1. However in this case, red(σ b 1 +b 2 −1 σ b 1 +b 2 σ b 1 +b 2 +1 σ b 1 +b 2 +2 ) = 1324 ensures that σ b 1 +2 is also less than the least element of B 3 and since σ b 1 < σ b 1 +2 , we must have σ b 1 = x + 2 and σ b 1 +2 = x + 3. If we then remove the first x + 2 cells which contain the numbers 1, . . . , x + 2, then we must be left with a fixed point which has two bricks on n − x − 2 cells. Then by our analysis of case 2, there are n − x − 2 − 3 possibilities for B 3 so that we have a total of n−6 x=1 n − x − 5 = n−5 2 possibilities. It follows that in subcase d.1 where τ = 1324, we have 2(n − 5) + n−5 2 possibilities if n ≥ 7 and no possibilities if n < 7.
Subcase d.2. p ≥ 5.
In this case, we must have a 1324 . . . p-match among the elements of bricks B 1 and B 2 which can only happen if red(σ b 1 −1 σ b 1 σ b 1 +1 . . . σ b 1 +p−2 ) = 1324 . . . p and b 2 ≥ p − 2. Similarly we must have have 1324 . . . p-match among the elements of bricks B 2 and B 3 which can only happen if red(σ b 1 +b 2 −1 σ b 1 +b 2 σ b 1 +b 2 +1 . . . σ b 1 +b 2 +p−2 ) = 1324 . . . p which is a condition that does not involve σ b 1 +1 . Nevertheless, these conditions still force that σ b 1 , σ b 1 +1 < σ b 1 +b 2 +1 so that if σ b 1 −1 = x, then x is less than the least elements in bricks B 2 and B 3 so that 1, . . . , x − 1 must be in brick B 1 and σ b 1 +1 = x + 1. We also have that σ b 1 < σ b 1 +2 and that σ b 2 +2 must be less than the least element in brick B 3 which is σ b 1 +b 2 +1 . It follows that it must be the case that σ b 1 = x + 2 and σ b 1 +2 = x + 3. If we then remove the first x + p − 3 cells which contain the numbers 1, . . . , x + p − 3, then we will be left with a fixed point which has two bricks on n − x − p + 3 cells. Then by our analysis of case 2, there are n − x − p + 3 − p + 1 possibilities for B 3 so that we have a total of
Now for any τ , we can write N M τ (t, 1, y) = 1 U τ (t, y) = 1 1 − (A τ (t)y − B τ (t)y 2 + C τ (t)y 3 + O(y 4 )) = 1 + n≥1 (A τ (t)y − B τ (t)y 2 + C τ (t)y 3 + O(y 4 )) n .
It then follows that N M τ (t, 1, y)| y = A τ (t),
N M τ (t, 1, y)| y 2 = (A τ (t)) 2 − B τ (t), and N M τ (t, 1, y)| y 3 = (A τ (t)) 2 − 2A τ (t)B τ (t) + C τ (t).
One can easily modify the direct argument that we used to prove d (1) n,4 = 2 n − 2n + 2 for n ≥ 4 to give a direct proof of this result.
One can also use Mathematica to compute that D (2(p − n − 1) t n n! + 2p−3 n=0 f (n, p) t n n!
where f (n, p) is defined as in (29). For example, one can compute that D
5 (t) = e 3t + (7 − 4t)e 2t + (25 − 17t −
2t 3 3! )e t − 33 − 21t − 6t 2 − t 3 − 3t 4 4! − t 5 5! , D(2)
6 (t) = e 3t + (9 − 4t)e 2t + (46 − 23t − t 2 −
4t 3 3! − 2t 4 4! )e t −56 − 40t − 27t 2 2 − 17t 3 3! − 3t 4 4! − 6t 5 5! − 3t 6 6! − t 7 7! , and D(2)7 (t) = e 3t + (11 − 4t)e 2t + (73 − 29t − 2t 2 − t 3 − 4t 4 4! − 2t 5 5! )e t −85 − 65t − 48t 2 2 − 34t 3 3! − 23t 4 4! − 15t 5 5! − 10t 6 6! − 6t 7 7! − 3t 8 8! − t 9 9! .
It then follows that for n ≥ 2p − 2, d (2) n,p = 3 n + (2p − 3 − 2n)2 n + 3p 2 − 12p + 10 + (13 − 6p)n + (5 − p)n(n − 1) − p−2 k=3 2(p − k − 1) k! n(n − 1) · · · (n − k + 1). (38)
For example, for n ≥ 8, d (2) n,5 = 3 n + (7 − 2n)2 n + 25 − 53n 3 + n 2 − n 3 3 .
For n ≥ 10, d
n,6 = 3 n + (9 − 2n)2 n + 46 − 136n 6 + n 2 12 − n 3 6 − n 4 12 .
For n ≥ 12, d
n,7 = 3 n + (11 − 2n)2 n + 73 − 142n 5 + 7n 3 12 − n 5 60 .
Conclusions
In this paper, we showed that if τ is a permutation which starts with 1, then N M τ (t, x, y) is always of the form 1 Uτ (t,y)
x where U τ (y) = 1+ n≥1 U τ,n (y) t n n! . In the special case where τ has
m
(τ ) y cdes(C) .(5)
Figure 1 :
1A (2, 3, 7)-brick tabloid of shape(12).
O τ,n denote the set of all objects created in this way. For each element O ∈ O τ,n , we define the weight of O, W (O), to be the product of y labels and the sign of O, sgn(O), to be (−1) ℓ(µ) . For example, if τ = 13245, then such an object O constructed from the brick tabloid B = (2, 8, 3) is pictured in Figure 2 where W (O) = y 7 and sgn(O) = (−1) 3 . It follows that n!θ τ (h n ) = O∈Oτ,n sgn(O)W (O).
Figure 2 :
2An element of O 13245,13 .
Figure 3 :
3I τ (O) for O inFigure 2.
1 .
1There are no cells labeled with y in O so that the numbers in each brick of O are increasing, 2. the first numbers in each brick of O form an increasing sequence, reading from left to right, and 3. if b i and b i+1 are two consecutive bricks in O, then either (a) there is increase between b i and b i+1 or (b) there is a decrease between b i and b i+1 , but there is τ -match in the cells of b i and b i+1 .
then let i be least number in the set {1, . . . , p − 1} that is not contained in bricks b 1 and b 2 . Since the numbers in each brick are increasing and the minimal numbers of the bricks are increasing, the only possible position for i is the first cell of brick b 3 . But then it follows that there is a decrease between bricks b 2 and b 3 .
First
we claim O(r j ) = r j and {1, . . . , r j } = {O(1), . . . , O(r j )} for j = 1, . . . , k. We have shown that O(1) = 1 and that O(r 2 ) = O(p−1) = p−1 and {O(1), . . . , O(p−1)} = {1, . . . , p−1}. Thus assume by induction, O(r j−1 ) = r j−1 and {1, . . . , r j−1 } = {O(1), . . . , O(r j−1 )}. Since there is a τ -match that starts at cell r j−1 and p ≥ 5, we know that all the numbers O(r j−1 ), O(r j−1 + 1), . . . , O(r j−1 + p − 3) are less than O(r j ) = O(r j−1 + p − 2). Since {1, . . . , r j−1 } = {O
But this cannot be since to have a τ -match starting at cell c, we must have O(c) < O(c + 2). Thus it must be the case that O(r j ) = r j . But then it must be the case that r j−1 = O(r j−1 ) < O(d) < O(r j ) = r j for r j−1 < d < r j so that {O(1), . . . , O(r j )} = {1, . . . , r j } as desired. Thus we have proved by induction that O(r j ) = r j and {1, . . . , r j } = {O(1), . . . , O(r j )} for j = 1, . . . , k.
[ 1 ,
1i 2 ), [i 2 , i 3 ), . . . , [i k−1 , i k ), [i k , n]. It is then easy to see that {σ 1 , . . . , σ i 2 −1 }, {σ i 2 , . . . , σ i 3 −1 }, . . . , {σ i k−1 , . . . , σ i k −1 }, {σ i k , . . . , σ n } is just a set partition of {1, . . . , n} ordered by decreasing minimal elements. Moreover, it is easy to see that no such permutation can have a 1324 . . . p-match for any p ≥ 4. Vice versa, if A 1 , . . . , A k is a set partition of {1, . . . , n} such that min(A 1 ) > · · · > min(A k ), then the permutation σ = A k ↑ A k−1 ↑ . . . A 1 ↑ is a permutation with k left-to-right minima and k − 1 descents where for any set A ⊆ {1, . . . , n}, A ↑ is the list of the element of A in increasing order. It follows that for any p ≥ 4, 1. N M 1324...p,n (x, y)| xy = S(n, 1) = 1, 2. N M 1324...p,n (x, y)| x 2 y 2 = S(n, 2) = 2 n−1 − 1, 3. N M 1324...p,n (x, y)| x n y n = S(n, n) = 1, and 4. N M 1324...p,n (x, y)| x n y n = S(n, n − 1) = n 2 .
. n. There are no such permutations if n ≤ p − 1 and there are n − (p − 1) such permutations if n ≥ p as q can range from 1 to n − (p − 1).We end this section by considering the special case where τ = 1324 where the analysis of the fixed points of I 1324 is a bit different. Let O be a fixed point of I 1324 . By Lemma 5, we know that 1 is in the first cell of O. Again, we claim that 2 must be in the second or third cell of O. That is, suppose that 2 is in cell c where c > 3. Then since there are no descents within any brick, 2 must be in the first cell of a brick. Moreover, since the minimal numbers in the bricks of O form an increasing sequence, reading from left to right, 2 must be in the first cell of the second brick. Thus if b 1 and b 2 are the first two bricks in O, then 1 is in the first cell of b 1 and 2 is in the first cell of b 2 . But then we claim that there is no 1324-match in the elements of b 1 and b 2 . That is, since c > 3, b 1 has at least three cells so that O starts with an increasing sequence of length 3. But this means that 1 can not be part of a 1324-match. Similarly, no other cell of b 1 can be part of 1324-match because the 2 in cell c is smaller than any of the remaining numbers of b 1 . But this would mean that we could apply case (ii) of the definition of I 1324 to b 1 and b 2 which would violate our assumption that O is a fixed point of I 1324 . Thus, we have two cases.Case I. 2 is in cell 2 of O. In this case there are two possibilities, namely, either (i) 1 and 2 lie in the first brick b 1 of O or (ii) brick b 1 has one cell and 2 is the first cell of the second brick b 2 of O. In either case, it is easy to see that 1 is not part of a 1324-match and if we remove cell 1 from O and subtract 1 from the elements in the remaining cells, we would end up with a fixed point O ′ of I 1324 in O 1324,n−1 . Now in case (i), it is easy to see that sgn(O)W (O) = sgn(O ′ )W (O ′ ) and in case (ii) since b 1 will have a label −y on the first cell, sgn(O)W (O) = (−y)sgn(O ′ )W (O ′ ). It follows that fixed points in Case 1 will contribute (1 − y)U 1324,n−1 (y) to U 1324,n (y). Case II. 2 in cell 3 of O. Let O(i) denote the element in i cell of O and b 1 , b 2 , . . . be the bricks of O, reading from left to right. Since there are no descents within bricks in O and the minimal elements in the bricks are increasing, we know that 2 is in the first cell of a brick b 2 . Thus b 1 has two cells. But then b 2 must have at least two cells since if b 2 has one cell, there could be no 1324-match contained in the cells of b 1 and b 2 and we could combine bricks b 1 and b 2 which would mean that O is not a fixed point of I 1324 . Thus b 1 has two cells and b 2 has at least two cells. But then the only reason that we could not combine bricks b 1 and b 2 is that there is a 1324-match in the cells of b 1 and b 2 which could only start at the first cell. We now have two subcases. Case II.a. There is no 1324-match in O starting at cell 3. Then we claim that {O(1), O(2), O(3), O(4)} = {1, 2, 3, 4}. That is, if {O(1), O(2), O(3), O(4)} = {1, 2, 3, 4}, then let i = min({1, 2, 3, 4} − {O(1), O(2), O(3), O(4)}). Since there is a 1324-match starting at position 1, it follows that O(4) > 4 since O(4) is the fourth largest element in {O(1), O(2), O(3), O(4)}. Since the minimal elements of the bricks of O are increasing, it must be that i is the first element in brick b 3 . But then we claim that we could combine bricks b 2 and b 3 . That is, there will be a decrease between bricks b 2 and b 3 since i < O(4) and O(4) is in b 2 . Since there is no 1324-match in O starting at cell 3, the only possible 1324-match among the elements in b 2 and b 3 would have start at a cell c > 3. But then O(c) > i, which is impossible since it would have to play the role of 1 in the 1324-match and i would have to play the role of 2 in the 1324-match since i occupies the first cell of b 3 . Thus it must be the case that O(1) = 1, O(2) = 3, O(3) = 2, and O(4) = 4.
, . . . , O(2k)} = {1, . . . , 2k}. If not there is a number greater than 2k that occupies one of the first 2k cells. Let M be the greatest such number. If M occupies one of the first 2k cells then there must be a number less than 2k that occupies one of the last n − 2k cells. Let m be the least such number. Since numbers in bricks are increasing, M must occupy the last cell in one of the first k − 1 bricks or occupy cell 2k. If M occupies the last cell in one of the first k − 1 bricks, then M is part of a τ -match . . . c i M c i+1 d i+1 . . . . But then red(c i M c i+1 d i+1 ) = 1 3 2 4 implies that M < d i+1 which contradicts our choice of M as the greatest number in the first 2k cells. Thus M cannot occupy the last cell in one of the first k − 1 bricks. This means that M must occupy cell 2k in O.
Figure 4 :
4The bijection φ.
(i)-(v). It follows that fixed points O of I 1324 where the bricks b 1 , b 2 , . . . , b k−1 are of size 2 and there are 1324-matches starting at positions 1, 3, . . . , 2k − 3 in O, but there is no 1324-match starting at position 2k − 1 in O contribute C k−1 (−y) k−1 U n−2k+1 (y) to U n (y). Thus we have proved the following theorem.
U(
1324,1 (y) = −y, and for n ≥ 2, U 1324,n (y) = (1 − y)U 1234,n−1 −y) k−1 C k−1 U 1324,n−2k+1 (y)
n ≥ p + 1 and no possibilities if n ≤ p.
2t 2 − 10t + 18)e t − 18 − 8t − t 2 + that for p ≥ 5, U 1324...p (t, y)| y 3 = (2t 2 + (5 − 4p)t + 3p 2 − 8p + 7)One can then use Mathematica to show thatU 1324...p (t, y)| y 3 = (2t 2 − (5 − 4p)t + 3p 2 − 8p + 7)e t + 2 + 8p − 7 if n = 0, −3p 2 + 12p − 12 if n = 1, −3p 2 + 16p − 21 if n = 2, − 3n 2 2 + (4p − 9 2 ) − 3p 2 + 8p − 6 if 3 ≤ n ≤ p, and − n 2 2 + (2p − 7 2 ) − 3p 2 + 7p − 6 if p + 1 ≤ n ≤ 2p − 3.
A 1324 (t) = e t − 1, B 1324 (t) = (2t − 4)e t + 4 + 2t + t 2 2 , and C 1324 (t) = (2t 2 − 10t + 18)e t − 18 − 8t − t 2 + t 3 3! + t 4 4! .One can then use Mathematica to compute that D (1)It follows that for n ≥ 4, dn,4 = 2 n − 2n + 2.This is easy to explain directly. That is, if σ ∈ N M 1324,n and has one descent, then σ has either one or two left-right-minima. Thus for p ≥ 4,One can also use Mathematica or Maple to compute thatIt then follows that for n ≥ 5, dn,4 = 3 n + (5 − 2n)2 n + n 2 − 11n + 5.We do not know of a simple direct proof of this result. We have shown that for p ≥ 5,One can then use Mathematica to compute thatIt follows that for n ≥ p − 1, d (1) n,p = 2 n − 2n − p + 2.one descent, we showed how to use the homomorphism method to give a simple combinatorial description of U τ,n (y) for any n ≥ 1. We then used this combinatorial description to show that the U τ,n (y)s satisfy simple recursions in the special case where τ = 1324 . . . p and p ≥ 4. The methods introduced in this paper can be use to prove several other similar results for other collections of patterns that start with 1 and have one descent. For example, suppose that τ = 1p2 . . . (p − 1) where p ≥ 4. Then we can prove thatt n n! , U τ,1 (y) = −y, and for n ≥ 2,If p ≥ 5 and τ = 134 . . . (p − 1)2p, then we can prove thatt n n! , U τ,1 (y) = −y, and for n ≥ 2, U τ,n (y) = (1 − y)U τ,n−1 (y) +These results will appear in subsequent papers.
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| [] |
[
"THREE-TORSION SUBGROUPS AND CONDUCTORS OF GENUS 3 HYPERELLIPTIC CURVES",
"THREE-TORSION SUBGROUPS AND CONDUCTORS OF GENUS 3 HYPERELLIPTIC CURVES"
] | [
"Elvira Lupoian "
] | [] | [] | We give a practical method for computing the 3-torsion subgroup of the Jacobian of a genus 3 hyperelliptic curve. We define a scheme for the 3-torsion points of the Jacobian and use complex approximations, homotopy continuation and lattice reduction to find precise expression for the 3-torsion. In the latter stages of the paper, we explain how the 3-torsion subgroup can be used to compute the wild part of the local exponent of the conductor at 2. | null | [
"https://export.arxiv.org/pdf/2210.02225v2.pdf"
] | 252,716,022 | 2210.02225 | e61e7cc6c9779923fe21b7113fceae70d4ab4f21 |
THREE-TORSION SUBGROUPS AND CONDUCTORS OF GENUS 3 HYPERELLIPTIC CURVES
8 Oct 2022
Elvira Lupoian
THREE-TORSION SUBGROUPS AND CONDUCTORS OF GENUS 3 HYPERELLIPTIC CURVES
8 Oct 2022
We give a practical method for computing the 3-torsion subgroup of the Jacobian of a genus 3 hyperelliptic curve. We define a scheme for the 3-torsion points of the Jacobian and use complex approximations, homotopy continuation and lattice reduction to find precise expression for the 3-torsion. In the latter stages of the paper, we explain how the 3-torsion subgroup can be used to compute the wild part of the local exponent of the conductor at 2.
Introduction
Let C be a smooth, projective, hyperelliptic curve of genus 3 defined over Q and let J be its Jacobian variety. Recall that J is a 3-dimensional abelian variety whose points can be identified with elements of the zero Picard group of C, Pic 0 (C). An affine model of such a curve is C ∶ y 2 = f (x) where f (x) ∈ Q[x] has degree 7 or 8, and no repeated roots. The Mordell-Weil theorem states that J (L) is a finitely generated group for any number field L; that is, J (L) ≅ J (L) tors ⊕ Z r where J (L) tors is the finite torsion subgroup and r is the rank. For a hyperelliptic curve we can compute a large part of the 2-torsion subgroup of J, J [2] = {P ∈ J ∶ 2P = 0}. For any two roots of f , x 1 and x 2 , the class of the divisor (x 1 , 0) − (x 2 , 0) − ∞ 1 − ∞ 2 is a non-zero element of J [2], where ∞ 1 , ∞ 2 are two marked points on the projective curve, and ∞ 1 = ∞ 2 when f has degree 7. Moreover, all points of order 2 are of this from when f has degree 7, see [1] or [5].
The problem of finding a point of order 3 is not as straightforward. In Section 2, we will show that all 3-torsion elements correspond to ways of expressing f , or a scalar multiple of f , as f (x) (x + α 1 ) 2 + α 7 x 3 + α 8 x 2 + α 9 x + α 10 3 = α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 2 when f has degree 7, and −x 6 − a 7 2
x 5 − − a 6 2 + a 2 7 8 x 4 + α 1 −x 5 − a 7 2 x 4 − α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 2 = α 7 x 3 + α 8 x 2 α 9 x + α 10 3 + x 2 + α 1 x + α 2 2 f (x)
when f has degree 8, for some α 1 , . . . , α 10 ∈ Q, where a 6 and a 7 are coefficients of f . The above correspondence can be used to define schemes parametrising the 3-torsion points of J. In Sections 3 we give a method of approximating the points of such schemes as complex numbers using homotopy continuation and the Newton-Raphson method. These numerical analysis techniques are used to efficiently compute approximations with a large precision, around 5000 decimal places, and in Section 4 we explain how such approximations are used to find algebraic expressions for the 3-torsion points of J, using lattice reduction. In Section 5, we compute the 3-torsion subgroups of the modular Jacobians J 0 (30) and J 0 (40). A similar method of complex approximations and lattice reduction was used in [2] to compute the 2-torsion subgroup of some non-hyperelliptic modular Jacobians. The second half of this paper will explain how the 3-torsion subgroup J [3] can be used to determine the local conductor exponent of C at 2. Recall that the conductor of a curve C Q is a representation theoretic constant, defined as a product N = ∏ p p np over the primes p where C has bad reduction. Thus the problem of computing the conductor of C reduces to computing the local exponents n p for all primes of bad reduction p. When C is an elliptic curve, the n p can be computed using Tate's algorithm (see [14,Chapter 4]). For hyperelliptic curves of arbitrary genus, there are formulae for n p for all p ≠ 2, see [4]. For curves of genus 2, Dokchitser and Dorris [3] give an algorithm for n 2 . In [3], the authors take C to be a non-singular projective curve of genus 2, defined over a finite extension K of Q 2 . Then, n 2 is the sum of the tame and wild parts, n 2 = n tame + n wild where n tame can be deduced from a regular model of the curve and n wild is the Swan conductor of the 3-adic Tate module of the Jacobian of C K, and it can be computed from the action of
Gal (K (J[3]) K) on J[3].
In the final two sections, we will assume C to be a smooth, projective and hyperelliptic curve of genus 3, defined over Q 2 , and following [3] we use the action of Gal Q Q on J [3] to compute n wild when C is hyperelliptic of genus 3. In Section 6, we give a brief theoretic overview of how the local conductor exponent at 2 is calculated using a regular model of the curve and the 3-torsion subgroup of its Jacobian. In Section 7, we compute the wild part of n 2 for the modular curves X 0 (30) and X 0 (40) using the 3-torsion subgroups computed in Section 5.
Acknowledgements. I would like to thank my supervisors Samir Siksek and Damiano Testa for their continued support, the many helpful conversations and their invaluable suggestions throughout this project. I would also like to thank Tim Dokchitser for the helpful conversation regarding the tame part of the conductor and for his computation of the regular model of X 0 (30).
Scheme of 3-torsion points
Let C be a smooth, projective, hyperelliptic curve of genus 3, defined over a number field K. By possible passing to a quadratic extension of K, C has an affine model of the form
y 2 = f (x) where f (x) ∈ K [x]
is monic, has degree 7 or 8 and has no repeated roots.
The projective closure of C in P 2 is defined by
Y 2 Z d−2 = Z d f (X Z) where d is the degree of f .
Remark 2.1. We refer to the points of C no appearing on the affine model as the points at infinity. These correspond to Z = 0, and we observe that there is a single such point, namely (0 ∶ 1 ∶ 0) when the degree of f is 7; and 2 points: (1 ∶ 1 ∶ 0) and (1 ∶ −1 ∶ 0) when the degree of f is 8.
Let J be the Jacobian variety of C. Recall that J is a 3 dimensional, abelian variety over K, whose points can be identified with points of Pic 0 (C), the zero Picard group of C. From now on we simply regard points on J as classes of divisors of degree 0 on C. See [1] or [5] for details on the arithmetic of hyperelliptic curves.
The 3-torsion subgroup of J consists of all elements [D] ∈ Pic 0 (C) such that 3D = div (h), where h is a rational function on C. To parametrise all such points, we treat the two degree cases separately. We begin with the following straightforward result, which is required throughout the remainder of the section. Lemma 1. Let C be a smooth, projective and hyperelliptic curve of genus g over a number field K and let K (C) be its function field. Let y 2 = f (x) be an affine model of the curve with f ∈ K [x]. Suppose g (x) is any polynomial in x, which is also an element of K (C) and its divisor of zeros is of the form 3D, where D is an effective divisor. Then g (x) is a cube as an element of K [x].
Proof. We can write g as
g (x) = α (x − β 1 ) r1 . . . (x − β s ) rs (x − γ 1 ) t1 . . . (x − γ n ) tn where f (β i ) = 0 for all i = 1 . . . s, f (γ j ) ≠ 0 for all j = 1 . . . n and α ∈ K × .The divisor of zero of g is s i=1 2r i (β i , 0) + n j=1 t j γ j , f (γ j ) + γ j , − f (γ j )
By assumption, this must equal 3D, and hence 3 divides 2r i and t j for all i = 1 . . . s and j = 1 . . . n, and the result follows.
Proposition 1. Let C be an odd degree hyperelliptic curve of genus 3, over a number field K, with an affine model y 2 = f (x) = x 7 + a 6 x 6 + a 5 x 5 + a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 where a i ∈ K and f has no repeated roots. Let J be the Jacobian of C. Then any non-zero 3-torsion point of J is the form
1 3 div (h) where h = y (x + α 1 ) + α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 with α 1 , . . . , α 6 ∈ K satisfying f (x) (x + α 1 ) 2 + α 7 x 3 + α 8 x 2 + α 9 x + α 10 3 = α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 2
for some α 7 , α 8 , α 9 , α 10 ∈K. Furthermore this correspondence preserves the action of G K = Gal K K .
Proof. Let ∞ be the unique point at infinity on this model and [D] ∈ J [3] ∖ 0. By Riemann-Roch there exists a unique, effective divisor D 0 = P 1 + P 2 + P 3 such that
D ∼ D 0 − 3∞ As 3D is principal, 3D 0 − 9∞ = div (h) ,
where h is a rational function on C. Thus h is in the Riemann-Roch space L (9∞) which has basis 1, x, x 2 , x 3 , x 4 , xy, y Then, replacing h by a scalar multiple if necessary, h is either a polynomial in x of degree at most 4, or
h = y + k (x) with k (x) ∈ K [x], deg (x) ≤ 4, or h = y (x + α 1 ) + k (x) with α 1 ∈ K, k (x) ∈ K [x] and deg (x) ≤ 4. Case 1. Suppose h ∈ K [x] and d = deg (h) ≤ 4. Let θ 1 , . . . , θ d be the roots of h. The divisor of zeros of h is 3D 0 = 3P 1 + 3P 2 + 3P 3 since div (h) = 3D 0 − 9∞.
We can also compute the divisor of zeros directly, and find it to be
d i=1 θ i , f (θ i ) + θ i , − f (θ i )
The above divisor has degree at most 8, whilst deg (3D 0 ) = 9, and hence they cannot be equal. Thus h cannot be a polynomial in x of degree at most 4.
Case 2. Suppose h = y +g (x) where g ∈ K [x]
and deg (g) ≤ 4, and leth = −y +g (x). As before, the divisor of zeros of h is 3D 0 , and the divisor of zeros ofh is
3ι (D 0 ) = 3ι (P 1 ) + 3ι (P 2 ) + 3ι (P 3 )
where ι ∶ C → C denotes the hyperelliptic involution on C. The divisor of zeros of hh is 3D 0 + 3ι (D 0 ), and hence hh = −f (x) + g (x)
2 is necessarily a cube as an
element of K [x] by Lemma 1. However, this is a contradiction since −f (x) + g (x) 2
has degree 7 or 8.
Case 3. This is the only remaining case.
Suppose h = y (x + α 1 ) + g (x) where α 1 ∈ K, g (x) ∈ K [x]
and g has degree at most 4, and leth = −y (x + α 1 ) + g (x). Arguing as before, the divisor of zeros of hh is 3D 0 + 3ι (D 0 ) and hence by Lemma
1, hh ∈ K [x]
is necessarily a cube. Hence
hh = (y (x + α 1 ) + g (x)) (−y (x + α 1 ) + g (x)) = −f (x) (x + α 1 ) 2 + g (x) 2 = α 7 x 3 + α 8 x 2 + α 9 x + α 10 3 for some α 7 , . . . , α 10 ∈ K, where g (x) = α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 for some α 2 , . . . , α 6 ∈ K.
Equating coefficients in this expression
f (x) (x + α 1 ) 2 + α 7 x 3 + α 8 x 2 + α 9 x + α 10 3 = α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 2
gives 10 equations in α 1 , . . . , α 10 , where (α 1 , . . . , α 6 ) define a 3-torsion point. We will refer to the scheme defined by these 10 equations as the scheme of 3-torsion points.
Proposition 2. Let C be an even degree hyperelliptic curve of genus 3, over a number field K, with an affine model
y 2 = f (x) = x 8 + a 7 x 7 + a 6 x 6 + a 5 x 5 + a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0
where a i ∈ K and f has no repeated roots. Let J be the Jacobian of C. Then any non-zero 3-torsion point of J is the form
1 3 div (h) where h = x 2 y − x 6 − a 7 2 x 5 + − a 6 2 + a 2 7 8 x 4 + α 1 xy − x 5 − a 7 2 x 4 + α 2 y − x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 for some α 1 , . . . , α 6 ∈ K satisfying −f (x) l (x) 2 + g (x) 2 = α 7 x 3 + α 8 x 2 + α 9 x + α 10 3 for some α 7 , . . . , α 10 ∈ K where l (x) = x 2 + α 1 x + α 2 g (x) = −x 6 + − a 7 2 − α 1 x 5 + − a 6 2 + a 2 7 8 − α 1 a 7 2 − α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6
Furthermore this correspondence preserves the action of G K = Gal K K .
Proof. Let ∞ + and ∞ − be the two points at infinity on this model and
[D] ∈ J [3]∖0.
By Riemann-Roch there exists a unique, effective divisor D 0 = P 1 + P 2 + P 3 such that
D ∼ D 0 − ∞ + − 2∞ − As 3D is principal, 3D 0 − 3∞ + − 6∞ − = div (h) , where h is a rational function on C. Thus h is in the Riemann-Roch space L (3∞ + + 6∞ − ) which has basis 1, x, x 2 , x 3 , y − x 4 , xy − x 5 − a7 2 x 4 , x 2 y − x 6 − a7 2 x 5 + − a6 2 + a 2 7 8
x 4 By possibly replacing h by a scalar multiple, h will necessarily fall in one of the following four cases.
Case 1. Suppose h is a polynomial in x of degree d ≤ 3. Let θ 1 , . . . , θ d be the roots of h. The divisor of zeros of h is 3D 0 since div (h) = 3D 0 − 9∞.
We can also compute the divisor of zeros directly, and find it to be
d i=1 θ i , f (θ i ) + θ i , − f (θ i )
The above divisor has degree at most 6, whilst deg (3D 0 ) = 9, and hence they cannot be equal. Thus h cannot be a polynomial in x of degree at most 3.
Case 2. Suppose h is of the form h = y − x 4 + α 1 x 3 + α 2 x 2 + α 3 x + α 4 = y + g (x) for some α 1 , . . . , α 4 ∈ K, where g (x) = −x 4 +α 1 x 3 +α 2 x 2 +α 3 x+α 4 . Leth = −y+g (x).
Arguing as in the proof of the previous proposition, the divisor of zeros of h is 3D 0 ; and the divisor of zeros ofh is 3ι
(D 0 ). The divisor of zeros of hh ∈ K [x] is 3D 0 + 3ι (D 0 ), and thus by Lemma 1, hh ∈ K [x]
is necessarily a cube. We find that
hh = (y + g (x)) (−y + g (x)) = −f (x) + g (x)
has degree at most 7, and hence it has degree 6 or 3 if it is indeed a cube. Suppose hh has degree 6, so hh = q 3 where q ∈ K [x] is a quadratic polynomial. Let θ 1 , θ 2 be the roots of q. Then the divisor of zeros of hh is
3 2 i=1 θ i , f (θ i ) + θ i , − f (θ i )
and by considering the degree of this divisor, it cannot equal 3D 0 . A very similar argument shows that the deg hh = 3 also leads to a contradiction. Thus h cannot be of the stated form.
Case 3. Suppose h is of the form h = xy − x 5 − a 7 2 x 4 + α 1 y − x 4 + α 2 x 3 + α 3 x 2 + α 4 x + α 5 = l (x) y + g (x) for some α 1 , . . . , α 5 ∈ K, where l (x) = x + α 1 , g (x) = −x 5 − a7 2 x 4 − α 1 x 4 + α 2 x 3 + α 3 x 2 + α 4 x + α 5 . Leth = −l (x) y + g (x). Arguing as before, hh ∈ K [x] is a cube. We find that hh = (l (x) y + g (x)) (−l (x) y + g (x)) = −l (x) 2 f (x) + g (x) 2
has degree at most 8, and hence it must have degree 3 or 6 if it is a cube. As in case 2, both possible degrees lead to a contradiction. Hence h cannot be of the stated form.
Case 4. Suppose h is of the form h = x 2 y − x 6 − a 7 2 x 5 + − a 6 2 + a 2 7 8 x 4 + α 1 xy − x 5 − a 7 2 x 4 + α 2 y − x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 = l (x) y + g (x)
for some α 1 , . . . ,
α 6 ∈ K, where l (x) = x 2 +α 1 x+α 2 , g (x) = −x 6 − a7 2 x 5 − − a6 2 + a 2 7 8 x 4 + α 1 −x 5 − a7 2 x 4 − α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 . Following previous arguments, set h = −l (x) y + g (x)
, then by considering the divisor of zeros of hh we find that hh ∈ K [x] must be a cube. In general,
hh = (l (x) y + g (x)) (−l (x) y + g (x)) = −l (x) 2 f (x) + g (x) 2
has degree 9, and so it must be the cube of a degree 3 polynomial; and so there exist α 7 , . . . , α 10 ∈ K such that
−l (x) 2 f (x) + g (x) 2 = α 7 x 3 + α 8 x 2 α 9 x + α 10 3
Thus such h define 3-torsion points on J.
Equating coefficients in the expression
−x 6 − a 7 2 x 5 − − a 6 2 + a 2 7 8 x 4 + α 1 −x 5 − a 7 2 x 4 − α 2 x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6 2 = α 7 x 3 + α 8 x 2 + α 9 x + α 10 3 + x 2 + α 1 x + α 2 2 f (x)
gives 10 equations in α 1 , . . . , α 10 , where (α 1 , . . . , α 6 ) define a 3-torsion point. We will refer to the scheme defined by these 10 equations as the scheme of 3-torsion points.
Complex Approximations and Homotopy Continuation
Let e 1 , . . . , e 10 be ten equations in α 1 , . . . , α 10 defining a scheme of 3-torsion points as in the previous section. The solutions of e 1 , . . . , e 10 can be approximated as complex points using the Newton-Raphson method. We give a brief overview of this, a detailed explanation can be found in [6,Page 298] Let E = (e 1 , . . . , e 10 ) and view it as a function C 10 → C 10 . Let dE be the Jacobian matrix of E. Suppose x 0 is an approximate solution to E with dE (x 0 ) invertible. For k ≥ 1, define
x k = x k−1 − dE (x k−1 ) −1 E (x k−1 )
Provided the initial approximation x 0 is a good enough approximation, the resulting sequence {x k } k≥0 converges to a root of E, with each iterate having increased precision. This method requires initial complex approximations to the solutions of E. These can be obtained using homotopy continuation and its implementation in Julia (see [7]).
3.1. Homotopy Continuation. Homotopy continuation is a method for numerically approximating the solutions of a system of polynomial equations by deforming the solutions of a similar system whose solutions are known. We give a brief sketch of the idea, but a more detailed explanation if this theory can be found in [7] or [8] .
The total degree of E is deg (E) = ∏ i deg (e i ), where deg (e i ) is the maximum of the total degrees of the monomials of e i .
Let F be a system of 10 polynomials in α 1 , . . . , α n , which has exactly deg (E) solutions and these solutions are known. The system F will be known as a start system. The standard homotopy of F and E is a function H (x, s N ), this is a system of 10 polynomials in α 1 , . . . , α n .
H ∶ C 10 × [0, 1] → C 10 H (x, t) = (1 − t) F (x) + tE (x) Fix N ∈ N, and for any s ∈ [0, N ] ∩ N define H s (x) =
For N large enough, the solutions of H s (x) are good approximations of the solutions of H s+1 (x), and using the Newton-Raphson method we can increase their precision. The solutions of H 0 (x) = F (x) are known, and they can be used to define solution paths to approximate solutions of H N (x) = E (x).
There are two important things to highlight.
1. Given any E, a start system ( and its solutions) can always be computed.
2.
A start system can be modified to ensure solutions paths are non-overlapping and converging to approximate solutions of E. Homotopy Continuation is implemented in the Julia package HomotopyContinuation.jl (see [7]).
Remark 3.1. The implementation of homotopy continuation in Julia gives approximates to solutions of E which are accurate to 16 decimal places. For our computations we used the approximate solutions and 1000 iterations of Newton-Raphson to obtain an accuracy of 5000 decimal places.
Algebraic Expressions
Suppose (α 1 , . . . , α 10 ) is a point on a scheme of 3-torsion points defined by E = (e 1 , . . . , e 10 ), which has a complex approximation (a 1 , . . . , a 10 ), accurate to k decimal places. We use the short vector algorithm to find the minimal polynomials of the α i and define the corresponding 3-torsion point.
Throughout this section we'll assume K = Q to simplify notation. At the end of the section, we explain how to deal with a general number field K.
Minimal Polynomials.
Fix i, 1 ≤ i ≤ 10 and let α = a i , θ = α i . As α is an algebraic number, there exists d ∈ N and c 0 , . . . , c d ∈ Z such that c d α d + . . . + c 1 α + c 0 = 0 Suppose θ ∈ R. Fix a constant C ∼ 10 k and let L k be the lattice generated by the columns of the (d + 1) × (d + 1) matrix
A k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 . . . 0 0 0 . . . 0 0 ⋮ ⋱ ⋮ ⋮ 0 . . . 1 0 Cθ d . . . [Cθ] [C] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = (v d , . . . , v 1 , v 0 )
where [a] denotes the integer part of a. As c 0 , . . . , c d ∈ Z
c k = ⎛ ⎜ ⎜ ⎜ ⎝ c d ⋮ c 1 a ⎞ ⎟ ⎟ ⎟ ⎠ = c d v n + . . . + c 0 v 0 ∈ L k where a = c d Cθ d + . . . + c 1 [Cθ] + c 0 [C]
. We can recover c ∞ = (c d , . . . , c 0 ) from c k by setting
c 0 = a − c d Cθ d + . . . + c 1 [Cθ] For any k ≥ 1 c k = c 2 d + . . . + c 2 1 + a 2 ≤ c 2 d + . . . + c 2 1 + c 2 d + . . . + c 2 1 + c 2 0 2 ∼ c ∞
so c k ∼ c ∞ and c k ∈ L k for all k ≥ 1. Thus for k large enough, we expect c k to be the shortest vector in the lattice L k . We use Hermite's theorem to determine when the shortest vector in L k is a suitable candidate for the vector of coefficients. There are bounds on these µ n given in [9, Page 66]. For a general lattice of full rank, we expect this bound to be close to the actual size of the shortest non-zero vector in the lattice.
Proof. See [9, Page 66]
Heuristic. For a full rank lattice L ⊂ R n , the length of the shortest vector in L is approximately d (L) 1 n Hermite's theorem suggests that the length of the shortest vector in L k is approximately d (L k ) 1 d+1 , so for k large enough, c k should be much smaller than this bound. This helps in identifying c k and the corresponding vector c ∞ . We use the short vector algorithm implemented in Magma (see [10]) to find short vectors in L k .
Remark 4.1. When the imaginary part of θ is not 0, the same method can be used but with L k being generated by the columns of
A k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 . . . 0 0 0 . . . 0 0 ⋮ ⋱ ⋮ ⋮ 0 . . . 1 0 CRe θ d . . . [CRe (θ)] [C] CIm θ d . . . [CIm (θ)] 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
where Re (θ) and Im (θ) denote the real and imaginary parts of θ.
To summarise, the strategy for finding the coefficients of the minimal polynomial of α is as follows.
1. Choose d.
2. Define the lattice L k . 3. In L k look for vectors which are shorter than, say 1 1000d (L k ) 1 d+1 . If such a vector doesn't exists, either increase the precision k and start again, or choose a different degree and start again. 4. If such a vector exists, verify that θ is an approximate solution of the corresponding polynomial. If this is not the case, choose a different degree and start again. Firstly, we can try to express α 2 , . . . , α 6 in terms of powers of α. Let K 1 = Q (α) be the number field defined by α. If f 2 has a root over K 1 , we can write it as
b d1 α 2 = b d1−1 α d1−1 + . . . + b 1 α + b 0 for some b 0 , . . . , b d1 ∈ Z.
Let a 1 , a 2 be complex approximations of α, α 2 correct to k decimal places. If a 1 , a 2 ∈ R, we search for b 0 , . . . , b d1 by looking for short vectors in the lattice generated by the columns of
A k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 . . . 0 0 0 0 . . . 0 0 0 ⋮ ⋱ ⋮ ⋮ ⋮ 0 . . . 0 1 0 Ca d1−1 1 . . . [Ca 1 ] [Ca 2 ] [C] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
where C is a constant of order 10 k . If a 1 , a 2 ∈ R, we instead search for short vectors in the lattice generated by the columns of
A k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 . . . 0 0 0 0 . . . 0 0 0 ⋮ ⋱ ⋮ ⋮ ⋮ 0 . . . 0 1 0 CRe a d1−1 1 . . . [CRe (a 1 )] [CRe (a 2 )] [C] CIm a d1−1 1 . . . [CIm (a 1 )] [CIm (a 2 )] 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Remark 4.2.
If no relations as above exist, we can use a similar lattice method to look for higher order relations, that could help to identify the corresponding the root of f 2 .
When the degrees d i are large, coefficient relations can be difficult to find. We can instead factorise the minimal polynomials. Suppose a i is a complex approximation of α i , and f i is the minimal polynomial of α i . Over C, f i can be factorised into linear factors.
f i = s 1 . . . s di
For k large enough, there is an n such that s n (a i ) is almost zero. Thus s n corresponds to the required root of f i .
Algebraic Expressions over Number Fields.
Let K be a number field of degree n. Then, by the Primitive Element theorem, K = Q (θ) for some θ ∈ K. Suppose we want to find the minimal polynomial of α ∈ K given a complex approximation a ∈ C of α, accurate to k decimal places. Suppose a ∈ R and θ ∈ R.
As α ∈ K, there exist d ∈ N and c 0 , . . . , c d ∈ K such that
c d α d + . . . + c 1 α + c 0 = 0 As c i ∈ K, there exist c i,j ∈ Q with c i = n−1 j=0 c i,j θ j
We can further assume c i,j ∈ Z by clearing denominators in the above expression. Therefore, there exist c i,j ∈ Z such that ∑ n−1 j=0 c d,j θ j α d + . . . + ∑ n−1 j=0 c 1,j θ j α + ∑ n−1 j=0 c 0,j θ j = 0 As before, to find c i,j ∈ Z, we take C to be a constant of order 10 k and look for short vectors in the lattice generated by the n (d + 1) × n (d + 1) matrix
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 . . . 0 0 . . . 0 . . . 0 0 0 . . . 0 0 . . . 0 . . . 0 0 ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ ⋮ 0 . . . 0 0 . . . 0 . . . 1 0 Cθ n−1 a d . . . Cθa d Ca d . . . Cθ n−1 . . . [Cθ] [C] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
where we view θ as a real number. When the imaginary part of a or θ (or both) is not 0, we generalise the above as in the Section 4.1. As before, we look for relations amongst the roots of each minimal polynomial to determine the 3-torsion points.
Remark 4.3. This method is extremely inefficient for a number field of large degree.
Examples
Using the method described in sections 2-4, we computed the 3-torsion subgroup of the modular Jacobians J 0 (30) and J 0 (40).
J 0 (30) [3].
We work with the model of the modular curve X 0 (30) given by Magma
y 2 + −x 4 − x 3 − x 2 y = 3x 7 + 19x 6 + 60x 5 + 110x 4 + 121x 3 + 79x 2 + 28x + 4
Completing the square gives a model of the form required by section 2,
y 2 = x 8 + 14x 7 + 79x 6 + 242x 5 + 441x 4 + 484x 3 + 316x 2 + 112x + 16
The scheme of 3-torsion points is defined by 10 equations − α 2 2 + α 2 6 − α 7 α 3 10 , − 2α 1 α 2 − 14α 2 2 + 2α 5 α 6 − 3α 7 α 9 α 2 10 , − α 2 1 − 28α 1 α 2 − 79α 2 2 − 2α 2 + 2α 4 α 6 + α 2 5 − 3α 7 α 8 α 2 10 − 3α 7 α 2 9 α 10 , − 14α 2 1 − 158α 1 α 2 − 2α 1 − 242α 2 2 − 28α 2 + 2α 3 α 6 + 2α 4 α 5 − 6α 7 α 8 α 9 α 10 − α 7 α 3 9 − 3α 7 α 2 10 , − 79α 2 1 − 484α 1 α 2 − 112α 1 α 6 − 28α 1 − 441α 2 2 − 2α 2 α 6 − 158α 2 + 2α 3 α 5 + α 2 4 + 2820α 6 − 3α 7 α 2 8 α 10 − 3α 7 α 8 α 2 9 − 6α 7 α 9 α 10 − 1,
− 242α 2 1 − 882α 1 α 2 − 112α 1 α 5 − 2α 1 α 6 − 158α 1 − 484α 2 2 − 2α 2 α 5 − 484α 2 + 2α 3 α 4 + 2820α 5 − 112α 6 − 3α 7 α 2 8 α 9 − 6α 7 α 8 α 10 − 3α 7 α 2 9 − 14, − 441α 2 1 − 968α 1 α 2 − 112α 1 α 4 − 2α 1 α 5 − 484α 1 − 316α 2 2 − 2α 2 α 4 − 882α 2 + α 2 3 + 2820α 4 − 112α 5 2α 6 − α 7 α 3 8 − 6α 7 α 8 α 9 − 3α 7 α 10 − 79, − 484α 2 1 − 632α 1 α 2 − 112α 1 α 3 − 2α 1 α 4 − 882α 1 − 112α 2 2 − 2α 2 α 3 − 968α 2 + 2820α 3 − 112α 4 − 2α 5 − 3α 7 α 2 8 − 3α 7 α 9 − 242, 2820α 2 1 − 112α 1 α 2 − 2α 1 α 3 − 158888α 1 − 3452α 2 − 112α 3 − 2α 4 − 3α 7 α 8 + 1987659, 2820α 1 − 112α 2 − 2α 3 − α 7 − 158404,
where the 3-torsion points are classes of divisors of the form 1
3 div (h) h = x 2 y − x 6 − 7x 5 − 15x 4 + α 1 xy − x 5 − 7x 4 + α 2 y − x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6
By approximating the solutions of the above system and then finding precise algebraic expressions for the 3-torsion points, we find that J 0 (30) [3] ≅ (Z 3Z) 6 can be generated using 3 Galois orbits, 2 consisting of 8 points each, and 1 consisting of 6 points. For each orbit, we give the minimal polynomial of α 1 and expressions for α 2 , . . . , α 6 in terms of α 1 .
u 6 − 21u 5 + 184u 4 − 861u 3 + 2296u 2 − 3381u + 2439 α 1 = u α 2 = u − 2 α 3 = (1 639) 4u 5 − 70u 4 + 704u 3 − 3962u 2 − 3192u − 10638 α 4 = (1 213) 4u 5 − 70u 4 + 704u 3 − 3962u 2 + 5541u − 7230 α 5 = (1 213) 4u 5 − 70u 4 + 704u 3 − 3962u 2 + 8310u − 8934) α 6 = (1 639) 4u 5 − 70u 4 + 704u 3 − 3962u 2 + 9588u − 10638 u 8 − 28u 7 + 343u 6 − 2401u 5 + 10414u 4 − 28147u 3 + 45290u 2 − 39200u + 13925 α 1 = u α 2 = 2u − 2 α 3 = (1 2169) 16u 7 − 392u 6 + 4116u 5 − 24010u 4 + 83312u 3 − 168882u 2 + 113309u − 54568 α 4 = (1 723) 32u 7 − 784u 6 + 8232u 5 − 48020u 4 + 166624u 3 − 337764u 2 + 326392u − 119258 α 5 = (1 723) 64u 7 − 1568u 6 + 16464u 5 − 96040u 4 + 333248u 3 − 675528u 2 + 699056u − 279004 α 6 = (1 2169) 128u 7 − 3136u 6 + 32928u 5 − 192080u 4 + 666496u 3 − 1351056u 2 + 1427032u − 592712 u 8 − 86u 7 + 2449u 6 − 33383u 5 + 252436u 4 − 1109723u 3 + 2786294u 2 − 3689116u + 2224811 α 1 = u α 2 = (1 1214905376480298255)(29876018790328u 7 − 2417413903833052u 6 + 60684703080638118u 5 − 674976608990629628u 4 + 3832952879194486442u 3 − 11087064205570838970u 2 + 16027124735004738752u − 11008190935547438114) α 3 = (−1 1214905376480298255)(226884728945872u 7 − 18363364083540328u 6 + 460287793516793082u 5 − 5069981080078429502u 4 + 28138121917331765018u 3 − 77651266046887373580u 2 + 119961145357139022083u − 45446963859192685796) α 4 = (−1 404968458826766085)(221239419854296u 7 − 17945665351388284u 6 + 451320054316335906u 5 − 4984082896579474376u 4 + 27907810187789236094u 3 − 78911274653216131110u 2 + 111314232875845983914u − 60056349012914418458) α 5 = (−1 1214905376480298255)(593735119981072u 7 − 48868278713945128u 6 + 1258524218508960012u 5 − 14170047695264405192u 4 + 81156089944126098548u 3 − 237313632893922545220u 2 + 339196464518479391108u − 198602211969557067116) α 6 = (−1 242981075296059651)(35094171383296u 7 − 3004896812056480u 6 + 81787675076005272u 5 − 944075033879118080u 4 + 5498949927080657672u 3 − 16418946803186159928u 2 + 23935267946866848320u − 14729053581484018328)
The field of definition of definition of all 3-torsion points defined by the above expressions is the degree 144 number field L defined as follows. Let K be the degree 48 number field defined by
x 48 − 9x 47 + 36x 46 − 75x 45 + 57x 44 + 45x 43 + 114x 42 − 1134x 41 + 2649x 40 − 2694x 39 − 9x 38 + 3708x 37 − 4208x 36 − 549x 35 − 477x 34 + 24297x 33 − 35388x 32 − 15957x 31 − 58908x 30 + 587655x 29 − 1095192x 28 + 147498x 27 + 2477835x 26 − 4287114x 25 + 2891076x 24 + 570960x 23 − 2932713x 22 + 2692353x 21 − 803187x 20 − 889560x 19 + 1287588x 18 − 729954x 17 + 58869x 16 + 358671x 15 − 388314x 14 + 194094x 13 − 21821x 12 − 50094x 11 +63396x 10 −45024x 9 +22035 * x 8 −8640x 7 +2955x 6 −684x 5 +111x 4 −24x 3 +1
then, L is the degree 6 extension of K defined by
x 6 − 21x 5 + 184x 4 − 861x 3 + 2296x 2 − 3381x + 2439 5.2. J 0 (40) [3].
We work with the model of the modular curve X 0 (30) given by Magma
y 2 + (−x 4 − 1)y = 2x 6 − x 4 + 2x 2
Completing the square gives a model of the form required by section 2,
y 2 = x 8 + 8x 6 − 2x 4 + 8x 2 + 1
The scheme of 3-torsion points is defined by 10 equations α 2 2 − α 2 6 − α 3 9 α 10 , 2α 1 α 2 − 2α 5 α 6 − 3α 8 α 2 9 α 10 , α 2 1 + 8α 2 2 + 2α 2 − 2α 4 α 6 − α 2 5 − 3α 7 α 2 9 α 10 − 3α 2 8 α 9 α 10 , 16α 1 α 2 + 2α 1 − 2α 3 α 6 − 2α 4 α 5 − 6α 7 α 8 α 9 α 10 − α 3 8 α 10 − 3α 2 9 α 10 , 8α 2 1 − 2α 2 2 + 2α 2 α 6 + 16α 2 − 2α 3 α 5 − α 2 4 + 8α 6 − 3α 2 7 α 9 α 10 − 3α 7 α 2 8 α 10 − 6α 8 α 9 α 10 + 1, − 4α 1 α 2 + 2α 1 α 6 + 16α 1 + 2α 2 α 5 − 2α 3 α 4 + 8α 5 − 3α 2 7 α 8 α 10 − 6α 7 α 9 α 10 − 3α 2 8 α 10 , − 2α 2 1 + 2α 1 α 5 + 8α 2 2 + 2α 2 α 4 − 4α 2 − α 2 3 + 8α 4 + 2α 6 − α 3 7 α 10 − 6α 7 α 8 α 10 − 3α 9 α 10 + 8, 16α 1 α 2 + 2α 1 α 4 − 4α 1 + 2α 2 α 3 + 8α 3 + 2α 5 − 3α 2 7 α 10 − 3α 8 α 10 , 8α 2 1 + 2α 1 α 3 + 8α 2 + 2α 4 − 3α 7 α 10 − 18,
8α 1 + 2α 3 − α 10 ,
where the 3-torsion points are classes of divisors of the form 1
3 div (h), h = x 2 y − x 6 − 4x 4 + α 1 xy − x 5 + α 2 y − x 4 + α 3 x 3 + α 4 x 2 + α 5 x + α 6
By approximating the solutions of the above system and then finding precise algebraic expressions for the 3-torsion points, we find that J 0 (40) [3] ≅ (Z 3Z) 6 can be generated using 3 Galois orbits, 2 consisting of 6 points each, and 1 consisting of 8 points.
For each orbit, we give the minimal polynomial of α 1 and expressions for α 2 , . . . , α 6 in terms of α 1 .
u 6 + 4u 4 − 8u 2 + 12 α 1 = u α 2 = u + 1 α 3 = (−1 9) (u 5 + u 3 + 16u + 18) α 4 = (−1 3) (u 5 + u 3 + 4u + 3) α 5 = (−1 3) (u 5 + u 3 + u − 6) α 6 = (−1 9) (u 5 + u 3 + 7u + 9) u 6 − 6u 5 + 4u 4 + 24u 3 + 256u 2 − 576u + 324 α 1 = u α 2 = (1 198) (−u 4 + 4u 3 + 58u 2 − 124u + 126) α 3 = (−1 99) (u 4 − 4u 3 − 58u 2 + 322u + 468) α 4 = (−1 99) (u 4 − 4u 3 − 58 * u 2 − 74u + 765) α 5 = (−1 99) (u 4 − 4u 3 − 58u 2 − 173u + 468) α 6 = (1 198) (u 4 − 4u 3 − 58u 2 + 520u − 522) u 8 − 126u 4 − 648u 2 − 1323 α 1 = u α 2 = −1 α 3 = (1 189) (u 7 − 63u 3 − 648u) α 4 = 3 α 5 = −u α 6 = 1
The field of definition of the 3-torsion subgroup is the degree 48 number field defined by
x 48 − 22x 47 + 220x 46 − 1298x 45 + 4840x 44 − 10758x 43 + 7848x 42 + 30564x 41 − 90644x 40 − 54378x 39 + 983934x 38 − 3228430x 37 + 6037118x 36 − 6706868x 35 + 3859158x 34 − 6290682x 33 + 41469355x 32 − 151827480x 31 + 375328308x 30 − 727099012x 29 + 1204881284x 28 − 1812362612x 27 + 2558319144x 26 − 3402905364x 25 + 4192192588x 24 −4669768140x 23 +4602283152x 22 −3939374364x 21 +2873125672x 20 − 1738390504x 19 + 830314684x 18 − 275496188x 17 + 30094447x 16 + 31178478x 15 − 22364652x 14 + 5362086x 13 + 2307708x 12 − 2995626x 11 + 1676724x 10 − 615660x 9 + 121728x 8 + 25686x 7 − 31194x 6 + 9162x 5 + 1458x 4 − 2088x 3 + 738x 2 − 126x + 9
6. Local Conductor Exponent at 2
Throughout this section, C K will denote a smooth, projective, hyperelliptic curve defined over K, a finite extension of Q 2 . Let J be the Jacobian variety associated to C, T = T l J the l-adic Tate module and V = V l J = T ⊗ Z l Q l the associated l-adic representation, where l is any prime different from 2. The conductor exponent of such a representations, as defined in [3] and [11] , is
n = ∞ −1 codimV G u K du
where G K = Gal K K is the absolute Galois group of K and {G u K } u≥−1 denote the ramification groups of G K in upper numbering. The tame and wild parts are defined as
n tame = 0 −1 codimV G u K du n wild = ∞ 0 codimV G u K du
Remark 6.1. The definition is independent of the choice of prime l, see [11].
Our approach is to take l = 3 and use the 3-torsion subgroup, computed as in Section 2 to 4. 6.1. Tame Conductor. From the above, the tame part of the conductor can be computed as n tame = 6 − dimV 3 J I where I ⩽ G K is the inertia subgroup.
Alternatively, we can also also deduce the tame part of the conductor from the regular model of C. From a regular model of C over Z 2 we can calculate
• the abelian part a, equal to the sum of the genera of all components of the model • the toric part t, equal to the number of loops in the dual graph of C Then, the tame part of the exponent is equal to 6 − 2a − t, see [12,Chapter 9] for details. Regular models can often be computed using the method described in [13], however, this is often a challenging problem. 6.2. Wild Conductor. Recall that we wish to compute
n wild = ∞ 0 codimV G u K du
For u ≥ 0, G u K is pro-p and codimV G u K = codimV G u K = codimJ [3]
G u K , see [11]. We may replace G K by G = Gal (K (J[3]) K), and thus
n wild = ∞ 0 codimJ[3] G u du
Alternatively, using the definition of G u and G u , the ramification groups in upper numbering and lower numbering respectively, we find
n wild = ∞ 0 codimJ[3] Gu [G 0 ∶ G u ] du = ∞ k=0 codimJ[3] G k [G 0 ∶ G k ]
Remark 6.2. Using our presentation of J [3], the ramification groups G u and their action on J [3] are completely explicit; and as a result n wild is a straightforward computation.
7. Examples 7.1. J 0 (30). A regular model of X 0 (30) at 2 is y 2 + −x 4 − x 3 − x 2 + 2x + 4 y = 3x 7 + 19x 6 + 61x 5 + 113x 4 + 124x 3 + 80x 2 + 24 and from the dual graph, we deduce that a = 2 and t = 1 in this case, thus n tame = 1. Recall that the 3-torsion subgroup J 0 (30) [3] is defined over the degree 144 number field defined by the polynomial f , stated in Section 5.1. Over Q 2 , f splits as the product of 6 degree 24 factors, and the splitting field of f , denoted by L, is a degree
Theorem 1 .
1(Hermite) Let L be an n dimensional lattice and M the length of the shortest non-zero vector in L. There exist constant µ n ∈ R ≥0 depending only on n such that M n ≤ µ n d (L)2 where d (L) is the discriminant of L.
4. 2 .
2Coefficient Relations. Suppose (α 1 , . . . , α 6 ) define a rational function h on C, which corresponds to a 3-torsion point as in Section 2. Let f i be the minimal polynomial of α i and set d i = deg (f i ). For a fixed root α = α 1 of f 1 , we want to determine the roots of f 2 , . . . , f 6 defining h, and thus the corresponding 3-torsion point.
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. CV4. 7Mathematics Institute, University of WarwickUnited Kingdom Email address: [email protected] Institute, University of Warwick, CV4 7AL, United Kingdom Email address: [email protected]
| [] |
[
"The Noisy Oscillator : Random Mass and Random Damping",
"The Noisy Oscillator : Random Mass and Random Damping"
] | [
"Stanislav Burov \nPhysics Department\nBar-Ilan University\nRamat Gan 52900Israel\n",
"Moshe Gitterman \nPhysics Department\nBar-Ilan University\nRamat Gan 52900Israel\n"
] | [
"Physics Department\nBar-Ilan University\nRamat Gan 52900Israel",
"Physics Department\nBar-Ilan University\nRamat Gan 52900Israel"
] | [] | The problem of a linear damped noisy oscillator is treated in the presence of two multiplicative sources of noise which imply a random mass and random damping. The additive noise and the noise in the damping are responsible for an influx of energy to the oscillator and its dissipation to the surrounding environment. A random mass implies that the surrounding molecules not only collide with the oscillator but may also adhere to it, thereby changing its mass. We present general formulas for the first two moments and address the question of mean and energetic stabilities. The phenomenon of stochastic resonance, i.e. the expansion due to the noise of a system response to an external periodic signal, is considered for separate and joint action of two sources of noise and their characteristics. * | 10.1103/physreve.94.052144 | [
"https://arxiv.org/pdf/1607.06289v2.pdf"
] | 23,064,303 | 1607.06289 | a267c0242694835624190c478f3d8b40dad6cfc8 |
The Noisy Oscillator : Random Mass and Random Damping
16 Nov 2016
Stanislav Burov
Physics Department
Bar-Ilan University
Ramat Gan 52900Israel
Moshe Gitterman
Physics Department
Bar-Ilan University
Ramat Gan 52900Israel
The Noisy Oscillator : Random Mass and Random Damping
16 Nov 20161
The problem of a linear damped noisy oscillator is treated in the presence of two multiplicative sources of noise which imply a random mass and random damping. The additive noise and the noise in the damping are responsible for an influx of energy to the oscillator and its dissipation to the surrounding environment. A random mass implies that the surrounding molecules not only collide with the oscillator but may also adhere to it, thereby changing its mass. We present general formulas for the first two moments and address the question of mean and energetic stabilities. The phenomenon of stochastic resonance, i.e. the expansion due to the noise of a system response to an external periodic signal, is considered for separate and joint action of two sources of noise and their characteristics. *
I. INTRODUCTION
One of the most general and most widely used models in physics is the damped linear harmonic oscillator, which is described by the following equation
m d 2 x dt 2 + γ dx dt + ω 2 x = 0(1)
This model has been applied in many fields, ranging from quarcks to cosmology. The ancient
Greeks already had a general idea of oscillations and used them in musical instruments.
Many applications have been found in the last 400 years [1]. The solution of Eq. (1) depends on the parameters γ/m and ω 2 /m. For a solution of the type x = exp (αt), one
obtains α = − γ 2m ± γ 2 4m 2 − ω 2 m .
For (γ/m) 2 ≥ 4 (ω 2 /m) , α is real and negative, i. e. for t → ∞, x monotonically goes to zero, as requiered for a stable system. However, for (γ/m) 2 < 4 (ω 2 /m) , α is complex, which means that approach of x to zero takes place with periodically decreasing amplitude. Equation (1) describes a pure mechanical system in the classical sense, i.e., zero temperature, while for quantum description the fluctuations persist even in the zero temperature limit. For non-zero temperature, the deterministic equation (1) has to be supplemented by thermal noise η(t),
m d 2 x dt 2 + γ dx dt + ω 2 x = η (t)(2)
where η (t) is a random variable with zero mean η (t) = 0 and a two-point correlation function η(t)η(t ) = 2Dδ(t − t ), which for thermal noise must satisfy the fluctuationdissipation theorem [2] η 2 (t) = 4γκT, where κ is the Boltzmann constant. For m = 0 and ω = 0, Eq. (2), describes an over damped Brownian particle, first introduced by Einstein more than 100 years ago.
Another generalization of Eq. (1) consists in adding external noise, which enters the equation of motion multiplicatively. For example, random damping yields
m d 2 x dt 2 + [1 + ξ (t)] γ dx dt + ω 2 x = η (t) .(3)
This equation was first used for the problem of water waves influenced by a turbulent wind field [3]. By replacing the coordinate x and time t by the order parameter and coordinate, respectively, Eq. (1) can be transformed into the stationary linearized Ginzburg-Landau equation with a convective term, which describes phase transitions in moving systems [4].
There are an increasing number of problems in which particles advected by the mean flow pass through the region under study. These include problems of phase transition under shear [5], open flows of liquids [6], Rayleigh-Benard and Taylor-Couette problems in fluid dynamics [7], dendritic growth [8], chemical waves [9], and the motion of vortices [10].
There is also a different type of Brownian motion, in which the surrounding molecules are capable not only of colliding with the Brownian particle, but also adhere to it for some random time, thereby changing its mass [11]. Such a process is described by the following stochastic equation
m [1 + ξ (t)] d 2 x dt 2 + γ dx dt + ω 2 x = η (t) .(4)
There are many situations in chemical and biological solutions in which the surrounding medium contains molecules which are capable of both colliding with the Brownian particle and also adhering to it for a random time. There are also some applications of a variable-mass oscillator [12]. Modern applications of such a model include a nano-mechanical resonator which randomly absorbs and desorbs molecules [13]. The diffusion of clusters with randomly growing masses has also been considered [14]. There are many other applications of an oscillator with a random mass [15], including ion-ion reactions [16]- [17], electrodeposition [18], granular flow [19], cosmology [20]- [21], film deposition [22], traffic jams [23]- [24], and the stock market [25]- [26].
In this paper we further generalize Eq. (1) to include the case of all three previously mentioned sources of noise, the additive part of Eq. (2) and the multiplicative parts of Eqs. (3)(4). Such an equation will describe a coarse-grained situation when a particle is affected by random kicks from its nearby environment (additive noise), adhesion of the molecules in the environment (random mass) and changes in the nearby environment (random friction).
While additive random noise is usually taken to be a Gaussian δ correlated (i.e. white) noise, this is not the case for multiplicative noise. It is natural to include correlations for the multiplicative part, since for example it can take some time for the attached molecule to return to the environment. Another complication is the value of the noise. While the random additive kick can be of any magnitude and sign (i.e. ±), the multiplicative noise does not have such luxury. Indeed, for the random mass case, a large negative value of the noise would imply a non-physical negative mass. Although friction can attain negative magnitude, it is much more common for friction to be strictly positive. To overcome such restrictions, we use exponentially correlated dichotomous noise for multiplicative noises [1].
A noise ξ(t) is called dichotomous when it randomly jumps between two states and its correlation function ξ(t )ξ(t ) decays exponentially. The advantage of such a choice for the noise is that it is not only correlated and bounded, it is also simple enough to serve as a test case for more complicated noise [27].
The paper is structured as follows. In Sec. II, we introduce the generalization of Eq. (1) for the case of random mass and random damping. The specific noise and the main mathematical tool (Shapiro-Loginov formula) are described. Section III is devoted to the calculation of the first and second moments of x. For each moment, two stability criteria are discussed, using the roots of an appropriate characteristic polynomial. The question of response to an external time-dependent periodic driving force is addressed in Sec. IV. We use examples of strictly random mass and strictly random friction to explain various types of observed stochastic resonances.
II. RANDOM MASS AND RANDOM DAMPING
We start with the generalization of the equation of a linear damped oscillator as previously described. In our generalization the noise perturbs both the mass of the oscillator and the friction m(1 + ξ 1 (t))
d 2 x dt 2 + γ(1 + ξ 2 (t)) dx dt + ω 2 x = η(t).(5)
The additive noise is taken to be zero average, δ correlated η(t 1 )η(t 2 ) = 2Dδ(t 1 − t 2 ) and it is uncorrelated with the multiplicative noise terms η(t 1 )ξ 1 (t 2 ) = η(t 1 )ξ 2 (t 2 ) = 0.
The multiplicative noise terms are both assumed to be symmetrical dichotomous noise with two-point correlation function
ξ 1 (t 1 )ξ 1 (t 2 ) = σ 2 1 exp(−λ 1 |t 1 − t 2 |), ξ 2 (t 1 )ξ 2 (t 2 ) = σ 2 2 exp(−λ 2 |t 1 − t 2 |).(6)
We further assume that the multiplicative noise terms are uncorrelated ξ 1 (t 1 )ξ 2 (t 2 ) = 0.
An advantage of treating the noise as symmetrical dichotomous noise is that it allows one to obtain results for the case of white noise. In the limit λ 1 → ∞ (with constant σ 2 1 /λ = D 1 ), the noise ξ 1 transforms to white (i.e. δ) correlated noise (a similar transformation holds of ξ 2 ). Before turning to the calculation of the moments of x, we mention the central tool we apply to obtain a solution. For an exponentially correlated stochastic process ξ (i.e. Eq. (6)) and some general function of the process g(ξ), the following relation holds
d dt + λ n ξg = ξ d n g dt n ,(7)
where n is a positive integer. Equation (7) is the Shapiro-Loginov formula [28] and its generalization for the case of two sources of noise is (d/dt + (λ 1 + λ 2 )) n ξ 1 ξ 2 g = ξ 1 ξ 2 d n g/dt n .
III. CALCULATION OF THE MOMENTS
A. Behavior of the Mean
We perform four operations upon Eq. (5) : (i) averaging with respect to the noise; (ii) multiplying by ξ 1 (t) and averaging; (iii) multiplying by ξ 2 (t) and averaging; (iv) multiplying by ξ 1 (t)ξ 2 (t) and averaging. By exploiting the property of dichotomous noise ξ 1 (t)ξ 1 (t) = σ 2 1 and ξ 2 (t)ξ 2 (t) = σ 2 2 and applying the Shapiro-Loginov formula (as given by Eq. (7)) we obtain
a d dt · ξ 1 x ξ 2 x ξ 1 ξ 2 x x = 0 (8) where a d dt = 0 b 2 2 + γ m b 2 + ω 2 m b 2 3 σ 2 2 γ m d dt b 2 1 + γ m b 1 + ω 2 m 0 γ m b 3 σ 2 1 d 2 dt 2 b 2 1 γ m b 2 0 d 2 dt 2 + γ m d dt + ω 2 m σ 2 2 γ m b 1 σ 2 1 b 2 2 b 2 3 + γ m b 3 + ω 2 m 0 .(9)
In Eq. (9) b 1 = (d/dt + λ 1 ), b 2 = (d/dt + λ 2 ) and b 3 = (d/dt + (λ 1 + λ 2 )). The well known Cramer's rule yields
a d dt x = 0.(10)
Substituting the expressions for a ij yields a differential equation of eighth order with constant be explored by studing the asymptotic behavior of x . The behavior will be stable if
coefficients i=8 i=0 c 8−i d i x dt i = 0.x(t) → 0 as t → ∞.
The general criteria for stability is the condition that for all α, the oscillations are diverging. Those results were obtained both by solution of Eq. (10) and while the symbols are obtained from numerical simulation of the process. Triangles (σ 2 = 1.45) are below the transition to instability, circles (σ 2 = 1.612..) at the transition and squares (σ 2 = 1.7) above the transition. The numerical data (symbols) was obtained by simulating 10 6 realizations of the process, each simulation performed by drawing the random times between switches of 1 ± σ 1 from an exponential distribution and similarly drawing random times between the switches of 1 ± σ 2 . During the instances when neither of the noises switched, the system was forwarded in time by exact integration. numerical simulation of the stochastic process.
○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ △ △ △ △ △△ △ △ △ △ △ △△ △ △ △ △ △△△ △ △ △ △ △△ △ △ △ △ △△△△ △ △ △△ △△ △ △ △△ △△△△ △△ △△△△△ △△△ △△△△△ △△△△△△△△ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □□ □ □ □ □ □ □□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ 0 5 10 15 20 25 30 35 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 t < x(t)>
B. Behavior of x 2
The stability criteria in the mean sense, as described in the previous section, can be rather unsatisfying. Indeed, the convergence of the mean to zero in the long run does not provide any certainty that the process x (as described by Eq. (5)) will be in the vicinity of zero. For example, the simple random walk starting from zero will on average be at zero, but the divergence of the second moment of a simple random walk produces very long excursions towards ±∞. It is thus preferable to obtain conditions for stability based on the behavior of the second moment x 2 . Generally the divergence of specific moment
x n (t) depends on the properties of the tail of the time dependent distribution of x, P (x, t).
The case when P (x, t) decays as |x| −1−z , with 1 < z < 2, produces stable solution for the mean but divergence of the second comment. The ability to compute the full distribution P (x, t) is beyond the scope of this study (or any other study to the best of our knowledge) and we therefor proceed to the exploration of the second comment. We note that in the literature [30,31] the instability based on the behavior of the second moment is addressed as an energetic instability. In order to obtain the various possible behaviors of x 2 , we now turn to Eq. (5) similarly to what was done for x .
We rewrite Eq. (5) in the following form dx dt =y
dy dt = − γ m 1 + ξ 2 1 + ξ 1 y − ω 2 m 1 1 + ξ 1 x − m 1 + ξ 1 η(t),(11)
and then obtain from Eq. (11) three equations after multiplying them by x and by y and summing up the mixed terms (i.e ydx/dt + xdy/dt)
dx 2 dt = 2xy dy 2 dt = − 2γ m 1 + ξ 2 1 + ξ 1 y 2 − 2ω 2 m 1 1 + ξ 1 xy + 2 m(1 + ξ 1 ) yη(t) dxy dt = − γ m 1 + ξ 2 1 + ξ 1 xy + y 2 − ω 2 m 1 1 + ξ 1 x 2 + 1 m(1 + ξ 1 ) xη(t) .(12)
First average Eq. (12) with respect to η. Since the multiplicative noise terms ξ 1 , ξ 2 are uncorrelated with η, we treat them as constants and only need to compute the correlators xη(t) η and yη(t) η . The symbol . . . η means average only with respect to η. Since η(t)
is a Gaussian δ correlated noise we can invoke Novikov Theorem [32] for the correlators.
The theorem states that for a vector u = (u 1 , u 2 , . . . , u n ) of dimension n and Gaussian δ correlated noise η(t) which satisfy the following relation
du dt = f (u) + g(u)η(t),(13)8 where f (u) = (f 1 (u), f 2 (u), .
. . , f n (u)) and g(u) = (g 1 (u)g 2 (u), . . . , g n (u)), the correlators satisfy
g i (u)η(t) η = D n j=1 ∂g i (u) ∂u j g j (u) η .(14)
From Eq. (12), we define u = (x 2 , y 2 , xy) and g(u) = (0, 2
m(1+ξ 2 ) y 2 , 1 m(1+ξ 2 ) √ x 2 ). Applying
Novikov Theorem yields
yη η = D m(1 + ξ 1 ) xη η =0 .(15)
Averaging Eq. (12) with respect to η and inserting Eq. (15) for the correlators, we obtain
d x 2 η dt − 2 xy η =0 (1 + ξ 1 ) 2 d y 2 η dt + 2γ m (1 + ξ 2 )(1 + ξ 1 ) y 2 η + 2ω 2 m (1 + ξ 1 ) xy η − 2D m 2 =0 (1 + ξ 1 ) d xy η dt + γ m (1 + ξ 2 ) xy η − (1 + ξ 1 ) y 2 η + ω 2 m x 2 η =0(16)
Equation (16) multiplying by ξ 1 (t) and averaging; (iii) multiplying by ξ 2 (t) and averaging; (iv) multiplying by ξ 1 (t)ξ 2 (t) and averaging. Since all sources of noise are uncorrelated we can switch the order of averaging. The outcome of the averaging order switching is that we may treat x 2 η , y 2 η , xy η as x 2 , y 2 , xy and after applying the Shapiro-Loginov procedure (Eq. (7)), only terms of the type ( x 2 , y 2 , xy , ξ 1 x 2 , . . . ) remain. The final result of the averaging is written in matrix form
M · X = X 0(17)
where M is given by
M d dt = d dt 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 b1 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 b1 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 b1 0 −2 0 (1+σ 2 1 ) d dt + 2γ m 2ω 2 m 0 2b 1 + 2γ m 2ω 2 m 0 2γ m 0 0 2γ m 0 0 2σ 2 1 d dt + 2γ m σ 2 1 2ω 2 m σ 2 1 0 (1+σ 2 1 )b1+ 2γ m 2ω 2 m 0 2γ m σ 2 1 0 0 2γ m 0 0 2γ m σ 2 2 0 0 2γ m σ 2 2 0 0 (1+σ 2 1 )b 2 + 2γ m 2ω 2 m 0 2b 3 + 2γ m 2ω 2 m 0 2γ m σ 2 1 σ 2 2 0 0 2γ m σ 2 2 0 0 2σ 2 1 b 2 + 2γ m σ 2 1 2ω 2 m σ 2 1 0 (1+σ 2 1 )b 3 + 2γ m 2ω 2 m ω 2 m −1 d dt + γ m 0 −1 b 1 0 0 γ m 0 0 0 0 −σ 2 1 σ 2 1 d dt ω 2 m −1 b 1 + γ m 0 0 0 0 0 γ m 0 0 γ m σ 2 2 0 0 0 ω 2 m −1 b 2 + γ m 0 −1 b 3 0 0 0 0 0 γ m σ 2 2 0 −σ 2 1 σ 2 1 b 2 ω 2 m −1 b 3 + γ m (18)
where X = ( x 2 , y 2 , xy , ξ 1 x 2 , ξ 1 y 2 , ξ 1 xy , ξ 2 x 2 , ξ 2 y 2 , ξ 2 xy , ξ 1 ξ 2 x 2 , ξ 1 ξ 2 y 2 , ξ 1 ξ 2 xy ) and X 0 = (0, 0, 0, 0, 2D/m 2 , 0, 0, 0, 0, 0, 0, 0). Cramer's rule implies
M d dt x 2 = M 1,5 d dt 2D m 2 ,(19)
where We would like to address the question of a response of a noisy oscillator with random mass and random damping to an external time-dependent driving term. The external driving term is taken to be a simple sinusoidal form A 0 cos (Ωt). Our general Equation (5) then becomes
m(1 + ξ 1 (t)) d 2 x dt 2 + γ(1 + ξ 2 (t)) dx dt + ω 2 x = η(t) + A 0 cos (Ωt) .(21)
Repeating the steps of Sec. III A and using the fact that A 0 cos (Ωt) and the multiplicative sources of noise are uncorrelated , i.e. ξ 1 (t) cos (Ωt) = ξ 2 (t) cos (Ωt) = ξ 1 (t)ξ 2 (t) cos (Ωt) = 0, we obtain
a d dt · ξ 1 x ξ 2 x ξ 1 ξ 2 x x = 0 0 A 0 cos (Ωt) 0 (22)
where a (d/dt) is defined by Eq. (9). The behavior of x is given by Cramer's rule
a d dt x = − a 4,3 d dt A 0 cos (Ωt) ,(23)
where |a 4,3 (d/dt)| is the {4, 3} minor of a (d/dt). In the limit t → ∞, when a stable solution for a d dt x = 0 exists and equals to 0, x is given by
x = A cos(Ωt + φ)(24)
with
A/A 0 = |a 4,3 (−iΩ)| |a 4,3 (iΩ)| |a (−iΩ)| |a (iΩ)|(25)
and
tan(φ) = |a 4,3 (−iΩ)| |a (iΩ)| + |a 4,3 (iΩ)| |a (−iΩ)| |a 4,3 (−iΩ)| |a (iΩ)| − |a 4,3 (iΩ)| |a (−iΩ)| i(26)
The response of x to the external driving term equals to A/A 0 (Eq. (25)) when a stable solution exists.
A. Various Aspects of Response
The expression for the response A/A 0 depends on seven parameters of the system and Ω. In order to obtain insight into the various possible types of behavior, we first treat the two simpler cases where only one source of multiplicative noises is present, i.e. (i) random damping (Eq. (4)) or (ii) random mass (Eq. (3)). The equation describing the case of a random mass and random damping, i.e. Eq. (5), reduces to case (i) by taking σ 2 and λ 2 to zero and to case (ii) by taking σ 1 and λ 1 to zero. Therefore, the response to an external periodic driving term for both simpler cases is provided by A/A 0 in Eq. (25) by setting the appropriate parameters to zero. We note that both of these simpler cases were previously treated [1]. In the following mainly the behavior of A/A 0 as a function of Ω is presented.
The behavior of A/A 0 as a function of σ 1 and σ 2 is presented in the Appendix.
Random Mass
The response for the case of a random mass is presented in Fig. 3, panels (a)-(c). In panel (a) a resonance is found for quite small values of noise strength (σ 2 1 = 0.01). Increasing the noise strength while keeping the correlation parameter λ 1 constant produces an additional maximum for A/A 0 , as shown in panel (b). This second resonance is due to the splitting of the first peak and decreasing its height. Such splitting occurs while the value of λ 1 is quite small, i.e. large correlation times of ξ 1 .
In order to understand the observed effect we notice the fact that random noise ξ 1 produces two mass values and creates two intrinsic states for the oscillator. In each of the states the oscillator behaves as a simple oscillator with additive noise. Existence of a resonance will depend on specific parameters of the state : m i , γ i and ω (subscript i runs over possible state indexes). The resonant frequency Ω R (if exisits) is provided by the well known formula [33]
Ω R = ω 2 m i − γ 2 i 2m 2 i .(27)
In the case of random mass m 1 = m 2 and γ 1 = γ 2 . If the oscillator can attain a resonance in both of the states, and the frequencies of those resonances are sufficiently distinct, we expect to observe two resonant frequencies as described in Fig. 3. Each of the resonant frequencies will correspond to an intrinsic regime/state of the oscillator and the splitting effect artificially resembles splitting of states in quantum system. Existence of two states for the oscillator is not sufficient for appearance of two resonant frequencies, the oscillator must also spend a sufficient amount of time (on average) in each of these states in order to attain a resonance. Since the oscillator constantly jumping from one state to the other, the time to build up a "proper" response to external field might be insufficient. The oscillator will jump to the other state where a different response will start to build up. It is thus important that the noise correlation time will be long enough. Indeed, this effect is shown in panel (c) of
Random Damping
The response for the case when only random damping exists is presented in panels ((d)- Fig. 4. Panel (d) shows a resonance for a small strength of the multiplicative noise ξ 2 , σ 2 = 0.01. In panel (e), the value of σ 2 2 was taken to be 0.9, yielding a threefold increase in the peak value of A/A 0 . The effect of resonant frequency splitting, similar to the random mass case, is not observed. The oscillator attains two intrinsic states with γ 1 = γ 2 and m 1 = m 2 .
(f )) of
The functional form of Eq. (27) allows two different resonant frequencies for two states with specific values of ω and damping. But in contrast to the random mass case the difference between two resonant frequencies is not sufficient (0 < σ 2 < 1). Random transitions between two states and the differences in response for each intrinsic state (i.e. decrease in response of one state while increase of the other) will smear presences of two maxima if the maxima frequencies are not sufficiently separated. It seems that for random damping the frequency separation is not sufficient and no splitting is observed. The increase in the resonance strength due to increase in the damping noise can be explained as a pronounced resonance in a state where the damping is very low (i.e. γ(1 − σ 2 )). This response increase is expected to disappear when the time the oscillator spends in a given state will decrease, as explained for the random mass case. Indeed when we decrease this time by increasing λ 2 the effect disappears. Panel (f ) of Fig. 4 shows the disappearance of the threefold increase of the peak value of the resonance after a significant decrease in the damping noise correlation time,
λ 2 → 10.
Random Mass and Damping
When both sources of noise (random mass and random damping) are present, we expect that a mixture of the previously discussed cases to take place. In panel (g) of the response to previously observed cases.
In the case of random damping, the presence of two states does not lead to appearance of resonant splitting. Interestingly enough, when both random damping and random mass present, an additional resonance splitting can occur. By keeping the temporal correlation of both sources of noise sufficiently long λ 1 = λ 2 = 0.1, we take the limit of very large strength of a random mass noise (σ 1 = 0.995) and large strength of random damping noise (σ 2 = 0.7). The result of additional resonance is presented in panel (l) of Fig. 5. Obviously, the simplistic approach that describes each resonant frequency as a frequency that correspond to a resonance for one of the states of the oscillator, fails here.
In order to study this effect further we present the behavior of the resonant frequency Ω R . In Fig. 6 panel (a) the behavior of the resonant frequency is presented for the case of random mass without random damping and compared to the predictions of Eq. (27). The second resonant frequency appears only when the frequencies of the two states are sufficiently distinct, and in general the behavior of the noisy case follows the predictions for the two different states. Even the non-monotonicity of Ω R for random mass is a consequence of the non-monotonicity of Ω R in Eq. (27). When also random damping is present the situation is quite similar while σ 1 is small enough. In panel (b) of Fig. 6 While for majority of the cases we managed to describe the response behavior in terms of response of the intrinsic states of the oscillator, there are exceptional situations. In those situations appearance of an additional resonance must be interpreted as an interference between various intrinsic states of the oscillator and not an attribute of a response in a single state. The noises in our oscillator model are not only capable of creating an intrinsic state that will attain a proper response. An effective coupling between transitions manages to create a preferable response to an external filed. Further study of such coupling is needed.
V. CONCLUSIONS
We considered an oscillator with two multiplicative random forces, which define the random damping and the random mass. The random mass means that the molecules of the surrounding medium, not only collide with an oscillator, but also adhere to it for a random time, thereby changing the oscillator mass. We calculated the first and the second moments of the oscillator coordinate by considering these two moments in the form of the damped exponential functions of time, exp(αt). The signs of α , which are obtained numerically, define the mean and energetic stability of the system. Stable solutions of the moments were represented by determinants of appropriate matrices. We brought references to many applications of such calculations to physics, chemistry and biology. Specifically we have shown that for the mean stable oscillations persist at the transition to instability.
The last section described the stochastic resonance phenomenon, that is the noise increased the applied periodic signal by helping the system to absorb more energy from the external force [35] . We presented the stochastic resonance as the function of the frequency Ω of the applied periodic signal, first separately for a random mass and random damping, and for the case of joint action of both these sources of noise. For most cases we managed to describe the observed phenomena in terms of simple intrinsic states of the oscillator and presence/non-presence of resonance for those states. Description by the means of underlying intrinsic states might become useful in experimental situations where the intrinsic states are explored by the means of response to external field, e.g., biomolecule folding/unfolding experiments [36,37] where distinct folded/unfolded sates are explored by external pulling . While the description by the means of response of the intrinsic states holds for majority of the cases, we found exceptions to this simple description. Specifically, we argue that appearance of additional resonant frequency at a regime where intrinsic resonance frequency dies out occurs due to transitions between states and not a presence of a single preferable response in an intrinsic state. It is the regime where the interference between states creates a preferable response.
VI. APPENDIX
In the main text we presented the response A/A 0 as a function of Ω. In this Appendix we present the response as a function of noise strength σ 1 and σ 2 . In general the dependence of A/A 0 on the noise strength, for specific value of Ω, is associated with the chosen Ω.
Non-monotonic behavior is expected in regions of Ω where the resonant frequency Ω R will be shifted when changing the noise strength (σ 1 or σ 2 ). If Ω R will coincide with the chosen Ω for some value 0 < σ 1 < 1 (or σ 2 ) a maxima of A/A 0 will appear for this specific value of σ 1 (or σ 2 ). When such crossover doesn't occur the behavior of A/A 0 is monotonic as displayed in Fig. 7 panels (a) and (c). When a crossover of Ω R occurs a modest maxima will be observed, as described in panels (b) and (d). Appearance of maxima as a function of and σ 2 are non-zero, where two maxima of A/A 0 appear (as a function of Ω). Existence of two (or even three) Ω R suggest that when those resonant frequencies are shifted one might observe also two maxima for A/A 0 as function of the noise strength. Due to the fact that the maxima of A/A 0 (as function of Ω) are well separated (in Ω) we were unable to find parameters where this phenomena might occur.
FIG. 1 :
1Seeking a solution of the form e αt , we obtain that α is a solution of |a (α)| = 0 The expressions for various α can only be found numerically. The stability of the system can Values of α s which satisfy |a (α)| = 0, plotted on the complex plane for two different sets of parameters (each dot represent different α) : (a) γ/m = 0.1, ω 2 /m = 0.1, λ 1 = 1, λ 2 = 0.1, σ 1 = 0.1, σ 2 = 0.9. (b) γ/m = 4, ω 2 /m = 1, λ 1 = 1, λ 2 = 1, σ 1 = 0.1, σ 2 = 1.5. The dashed line separates the complex plane into Re[α] < 0 and Re[α] > 0 .
which satisfy |a (α)| = 0, the value of α has a negative real part. The Routh-Hurowitz theorem[29] provides the condition for all the roots of polynomial to have a negative real part. The condition involves the calculation of the determinants of matrices up to 15 × 15 and is rather cumbersome. Instead, one can plot the various roots α on the complex plane and investigate their positions for various values of the parameters γ/m, ω 2 /m, λ 1 , λ 2 , σ 1 , σ 2 . InFig. 1, two examples are presented. In panel (a) the configuration of the roots is such that for all eight α, Re[α] < 0 and eventually x decays to zero. When there is at least one α for which Re[α] ≥ 0, i.e. panel (b), x does not converge to zero and the behavior is not stable in the mean sense. We note that the transition to instability can be achieved in various ways. There are various configurations of parameters for which exactly at the transition point, x will exhibit stable oscillations. Specifically, this occurs for γ/m = 1, ω 2 /m = 1, λ 1 = 1, λ 2 = 1, σ 1 = 1/10, σ 2 = 1.612443.... InFig. 2the behavior of x(t) is plotted as function of time for the mentioned parameters and three different values of σ 2 . Below the transition to instability (σ 2 = 1.45), decaying oscillations occur. At the instability (σ 2 = 1.612...) the oscillations are stable, and above the transition (σ 2 = 1.7)
FIG. 2 :
2Temporal behavior of x(t) for three different values of the random damping noise strength σ 2 while other parameters are kept constant γ/m = 1,ω 2 /m = 1,λ 1 = 1,λ 2 = 1,σ 1 = 1/10. The thick lines are the solutions of Eq. (10)
is then treated in the same fashion as Eq. (5) in Sec. III A. Four operations are performed upon each line in Eq. (16) : (i) averaging with respect to the noises; (ii)
From
|M 1,5 | is the {1, 5} minor of matrix M, i.e determinant of matrix M where the first column and fifth row were removed from the matrix. The determinants on both sides of Eq. (19) are differential operators and since |M 1,5 (d/dt)| operates on a constant it can be replaced by |M 1,5 (0)|. The stable solution is Eq. (20), it is clear that when |M (0)| = 0, the system is not stable and the second moment diverges. As was the case for x , we can write a more general condition. We search a solution of M d dt x 2 = 0 (i.e. the homogeneous part of Eq. (20)) in the form of exp(αt). This solution will be stable if ∀α (such that |M (α)| = 0) Re[α] < 0. Then this is the stability criterion and it includes the special case of α = 0 that zeros |M|. The search for the criteria of a negative real part of |M (α)| = 0 can be performed by plotting different values α on the complex plane and searching for situations where Re[α] ≥ 0. Specifically for the mentionedcase when x is stable (γ/m = 1, ω 2 /m = 1, λ 1 = 1, λ 2 = 1, σ 1 = 1/10, σ 2 = 1.45) the second moment x 2 will diverge.IV.RESPONSE TO EXTERNAL DRIVING TERM.
FIG. 3 :
3Response A/A 0 as a function of angular frequency (Ω) of the periodic external driving force as given in Eq. (25) for the case of only a random mass. Maximum of A/A 0 for specific parameters of the system describes a resonance between the behavior of x and the external driving force. (a) γ/m = 0.1, ω 2 /m = 1, λ 1 = 0.1, σ 1 = 0.1, σ 2 = 0. (b) The same parameters as in (a) except that σ 1 = 0.5. (c) The same parameters as in (b), except that λ 1 = 0.35.
Fig. 3 .
3While keeping the strength of the noise the same as in (b), λ 1 was increased and the collapse of the two resonances was obtained. The case of a random mass can thus contribute to the existence of a single stochastic resonance, but can also split a single resonance into two resonances (when the correlation time of the noise is sufficiently long). Appearance of multiple resonances was also observed in different noisy representations of the harmonic oscillator[31,34]
FIG. 4 :
4Response A/A 0 as a function of angular frequency (Ω) of the periodic external driving as given in Eq. (25) for the case of only random damping. Maximum of A/A 0 for specific parameters of the system describes a resonance between the behavior of x and the external driving force. (d) γ/m = 0.1, ω 2 /m = 1, λ 2 = 0.1, σ 1 = 0, σ 2 = 0.1. (e) The same parameters as in (d), except that σ 2 = 0.9. (f ) The same parameters as in (e), except that λ 2 = 10.
Fig. 5 ,FIG. 5 :FIG. 6 :
556A/A 0 exhibits a resonance for specific Ω, while the strengths of the sources of noise are quite small, σ 1 = 0.1 and σ 2 = 0.1. Increasing the strength of the random mass noise, while leaving the strength of the random damping noise constant, splits the resonance. Panel (h) of Fig. 5shows two maxima for A/A 0 and the effect is similar to the case of only a random mass, Response A/A 0 as a function of angular frequency (Ω) of the periodic external driving force as given in Eq.(25). Maximum of A/A 0 for specific parameters of the system describes a resonance between the behavior of x and the external drive. (g) γ/m = 0.2,ω 2 /m = 1, λ 1 = 0.1, λ 2 = 0.1, σ 1 = 0.1, σ 2 = 0.1. (h)The same parameters as in (g),except that σ 1 = 0.7. (i) The same parameters as in (h), except that σ 2 = 0.95. (j) Thesame parameters as in (i), except that λ 1 = 10. (k) The same parameters as in (i), except that λ 2 = 10. (l)The same parameters as in (i), except that σ 1 = 0.995 and σ 2 = 0.75. as described in panel (b) of Fig. 3. The presence of a small noise term for the damping does not qualitatively change the effect. But if in addition to increasing the strength of ξ 1 , one also increase the strength of ξ 2 ( i.e. random damping), a non-symmetric effect occurs. For the case of only random damping, an increase of noise strength expands the size of the resonance (panel (e) of Fig. 4). In panel (i) of Fig. 5, we see that as the strength of ξ 2 increases, it does not affect the values of the maxima in the same fashion. While the second maxima that appeared in panel (h) of Fig. 5 expanded significantly , the first maxima grew only slightly. This asymmetry arises due to the asymmetry of the resonant frequencies (as function of mass and damping) at each intrinsic state of the oscillator (i.e. Eq. (27)).m appears in the denominator and affects Ω R more violently than γ that appears in the numerator. Due to this fact a significant effect is expected for the state with smaller m and small γ. The temporal correlation must be long enough in order to observe the mentioned effect and indeed increasing either λ 1 (panel (j) ofFig. 5) or λ 2 (panel (k) ofFig. Resonantfrequencies Ω R as function of ξ 1 strength, for the case of random mass and random damping (thick lines) and for the specific states of the oscillator (dashed lines).
( a )
aOnly random mass noise is present γ/m = 0.2, ω 2 /m = 1, λ 1 = 0.1, λ 1 = 0.0,σ 2 = 0. The arrow points to an emergence of a second resonance. (b) Both noises are present. γ/m = 0.2, ω 2 /m = 1, λ 1 = 0.1, λ 1 = 0.1,σ 2 = 0.9. The left arrow shows the position of emergence of the second resonance while right arrow to the position of emergence of the third resonance. Four different dashed lines are presented, two bottom lines almost coincide for the whole range of σ 1 . The different color of the thick line is plotted for the part when three resonances occur. (c) Zoom into the range 0.9 ≤ σ 1 ≤ 1 of panel (b).
FIG. 7 :
7Response A/A 0 as a function of noise strenght (σ 1 and σ 2 ) of the periodic external driving force as given in Eq. (25) for spcific values of Ω. (a) γ/m = 0.2, ω 2 /m = 1, λ 1 = 0.5, λ 2 = 0.5, σ 2 = 0.5, Ω = 0.5. (b) The same parameters as in (a), except that Ω = 2. (c) γ/m = 0.2, ω 2 /m = 1, λ 1 = 0.5, λ 2 = 0.5, σ 1 = 0, Ω = 30. (d) The same parameters as in (c), except that σ 1 = 0.994. σ 2 occurs for non zero values of σ 1 . In the main text we described situations, when both σ 1
behavior similar to panel (a) is presented. The four different states appear as two states (the dashed lines are very close to one another) and generally there is almost no obvious effect of the additional noise. For large enough σ 1 Eq. (27) predicts disappearance of resonance for one of the states of the oscillator (one of the dashed lines drops to zero). Inside this region where only twostates with resonance exist suddenly appears additional resonance for the noisy case (lower red line). We cannot attribute this resonance to a resonance in an intrinsic state of the oscillator, since this intrinsic resonance does not exist for this range of parameters.
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| [] |
[
"MOORE-PENROSE INVERSE OF SOME LINEAR MAPS ON INFINITE-DIMENSIONAL VECTOR SPACES",
"MOORE-PENROSE INVERSE OF SOME LINEAR MAPS ON INFINITE-DIMENSIONAL VECTOR SPACES"
] | [
"Víctor Cabezas ",
"Sánchez ",
") Fernando ",
"Pablos Romo "
] | [] | [] | The aim of this work is to characterize linear maps of inner product infinite-dimensional vector spaces where the Moore-Penrose inverse exists. This MP inverse generalizes the well-known Moore-Penrose inverse of a matrix A ∈ Mat n×m (C). Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, we study the least norm solution of an infinite linear system from the Moore-Penrose inverse offered. | 10.13001/ela.2020.4979 | [
"https://arxiv.org/pdf/1810.04082v1.pdf"
] | 119,667,593 | 1810.04082 | 7421be440ac398a8d0dea5bc35054cd21f797d29 |
MOORE-PENROSE INVERSE OF SOME LINEAR MAPS ON INFINITE-DIMENSIONAL VECTOR SPACES
9 Oct 2018
Víctor Cabezas
Sánchez
) Fernando
Pablos Romo
MOORE-PENROSE INVERSE OF SOME LINEAR MAPS ON INFINITE-DIMENSIONAL VECTOR SPACES
9 Oct 2018
The aim of this work is to characterize linear maps of inner product infinite-dimensional vector spaces where the Moore-Penrose inverse exists. This MP inverse generalizes the well-known Moore-Penrose inverse of a matrix A ∈ Mat n×m (C). Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, we study the least norm solution of an infinite linear system from the Moore-Penrose inverse offered.
Introduction
Given again a matrix A ∈ Mat n×m (C), the Moore-Penrose inverse of A is the unique matrix A † ∈ Mat m×n (C) such that:
• A A † A = A; • A † A A † = A † ; • (A A † ) * = A A † ;
• (A † A) * = A † A; B * being the conjugate transpose of the matrix B. The Moore-Penrose inverse of A always exists, it is a reflexive generalized inverse of A, [A † ] † = A and, if A ∈ Mat n×n (C) is non-singular, then A † coincides with the inverse matrix A −1 .
Recently, generalized inverses of matrices A ∈ Mat n×m (C) have been extended to some linear maps on infinite-dimensional vector spaces. Indeed, the authors have computed explicit solutions of infinite systems of linear equations from reflexive generalized inverses of finite potent endomorphisms in [3] and, also, the second-named author has generalized the notion of Drazin inverse to finite potent endomorphisms in [4].
The aim of this work is to characterize linear maps of inner product infinitedimensional vector spaces where the Moore-Penrose (MP) inverse exists. This MP inverse generalizes the Moore-Penrose inverse A † of a matrix A ∈ Mat n×m (C). Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, we study the least norm solution of an infinite linear system from the Moore-Penrose inverse offered.
The paper is organized as follows. In section 2 we recall the basic definitions of this work: inner product vector spaces, finite potent endomorphisms, reflexive generalized inverse and Moore-Penrose inverse of a (n × m)-matrix. Also, in this section, we briefly describe the construction of Jordan bases for endomorphisms admitting an annihilator polynomial.
Section 3 contains the main results the this work: the definition of linear map admissible for the Moore-Penrose inverse (Definition 3.9), the proof of the existence and uniqueness of the MP inverse for these linear maps (Theorem 3.11) and the conditions for computing the MP inverse for some endomorphisms on infinitedimensional vector spaces from the MP inverses of (n×n)-matrices (Theorem 3.19).
Finally, Section 4 is devoted to study infinite systems of linear equations from the Moore-Penrose Inverse. Thus, Proposition 4.4 shows that if (V, g) and (W,ḡ) are two arbitrary inner product vector spaces over R of C, f : V → W is a linear map admissible for the Moore-Penrose inverse and f (x) = w is a linear system, then f † (w) is the unique minimal leastḡ-norm solution of this linear system.
Preliminaries
This section is added for the sake of completeness.
2.1. Inner Product Vector Spaces. Let k be the field of the real numbers or the field of the complex numbers, and let V be a k-vector space.
An inner product on V is a map g : V × V → k satisfying that:
• g is linear in its first argument:
g(λv 1 + µv 2 , v ′ ) = λg(v 1 , v ′ ) + µg(v 2 , v ′ ) for every v 1 , v 2 , v ′ ∈ V ; • g(v ′ , v) = g(v, v ′ ) for all v, v ′ ∈ V , where g(v, v ′ ) is the complex conjugate of g(v, v ′ ); • g is positive definite: g(v, v) ≥ 0 and g(v, v) = 0 ⇐⇒ v = 0 . Note that g(v, v) ∈ R for each v ∈ V , because g(v, v) = g(v, v). A pair (V, g) is named "inner product vector space". If (V, g)
is an inner product vector space over C, it is clear that g is antilinear in its second argument, that is:
g(v, λv ′ 1 + µv ′ 2 ) =λg(v, v ′ 1 ) +μg(v, v ′ 2 ) for all v, v ′ 1 , v ′ 2 ∈ V ,
andλ andμ being the conjugates of λ and µ respectively. Nevertheless, if (V , g) is an inner product vector space over R, then g is symmetric and bilinear.
The norm on an inner product vector space (V, g) is the real-valued function
· g : V −→ R v −→ + g(v, v) ,
and the distance is the map
d g : V × V −→ R (v, v ′ ) −→ v ′ − v g .
Simple examples of inner product vector spaces are Euclidean finite-dimensional real vector spaces and complex Hilbert spaces.
Let us now consider two inner product vector spaces: (V, g) and (W,ḡ). If f : V → W is a linear map, a linear operator f * : W → V is called the adjoint of f when
g(f * (w), v) =ḡ(w, f (v)) ,
for all v ∈ V and w ∈ W . If f ∈ End k (V ), we say that f is self-adjoint when f * = f . Moreover, if (V, g) and (W,ḡ) are finite-dimensional inner vector spaces over C, B = {v 1 , . . . , v m } and B ′ = {w 1 , . . . , w n } are orthonormal bases of V and W respectively, f : V → W is a linear map, A ∈ Mat n×m (C) and f ≡ A in these bases, then f * ≡ A * ∈ Mat m×n (C) in the same bases, where A * is the conjugate transpose of A.
2.2.
Jordan bases for endomorphisms admitting an annihilator polynomial. Let V be an arbitrary vector space over a ground field k, and let f ∈ End k (V ) be an endomorphism of V admitting an annihilator polynomial
a f (x) = p 1 (x) n1 · · · · · p r (x) nr , p i (x) being irreducible polynomials in k[x].
For each j ∈ {1, . . . , r}, we can consider
ν i (V, p j (f )) = dim Kj Ker p j (f ) i [Ker p j (f ) i−1 + p j (f ) Ker p(f ) i+1 ] , with K j = k[x] p j (x). Henceforth, S νi(V,pj(f )) will be a set such that #S νi(V,pj (f )) = ν i (V, p j (f )), with S νi(V,pj (f )) ∩ S ν h (V,pj (f )) = ∅ for i = h.
According to the statements of [5] there exist families of vectors
{v ij h } h∈S ν i (V,p j (f )) with v ij h ∈ Ker p j (f ) i v ij h / ∈ Ker p j (f ) i−1 + p j (f ) Ker p j (f ) i+1 ,
for all 1 ≤ j ≤ r and 1 ≤ i ≤ n j , such that if we set
H ij h =< v ij h > f = 0≤s≤i−1 {p j (f ) s [v ij h ], p j (f ) s [f (v ij h )], . . . , p j (f ) s [f dj−1 (v ij h )]} , then 1 ≤ j ≤ r 1 ≤ i ≤ n j h ∈ S νi(V,pj (f )) < v ij h > f
is a Jordan basis of V for f , and this basis determines a decomposition
(2.1) V = 1 ≤ j ≤ r 1 ≤ i ≤ n j h ∈ S νi(V,pj (f )) H ij h .
Example 1 (Jordan bases for a nilpotent endomorphism). Let V again be a vector space over an arbitrary field k and let f ∈ End k (V ) be a nilpotent endomorphism. If n is the nilpotency index of f , setting W f
i = Ker f i /[Ker f i−1 + f (ker f i+1 )] with i ∈ {1, 2, . . . , n}, µ i (V, f ) = dim k W f i and S µi(V,f ) a set such that #S µi(V,f ) = µ i (V, f ) with S µi(V,f ) ∩ S µj (V,f ) = ∅ for all i = j,
one has that there exists a family of vectors {v si } that determines a Jordan basis of V for f :
(2.2) B = s i ∈ S µi(V,f ) 1 ≤ i ≤ n {v si , f (v si ), . . . , f i−1 (v si )} . Moreover, if we write H f si = v si , f (v si ), . . . , f i−1 (v si ) , the basis B induces a de- composition (2.3) V = s i ∈ S µi(V,f ) 1 ≤ i ≤ n H f si .
Finite Potent Endomorphisms.
Let k be an arbitrary field, let V be a k-vector space and let ϕ ∈ End k (V ). We say that ϕ is "finite potent" if ϕ n V is finite dimensional for some n. This definition was introduced by J. Tate
if V admits a ϕ-invariant decompo- sition V = U ϕ ⊕ W ϕ such that ϕ |U ϕ is nilpotent, W ϕ is finite dimensional, and ϕ |W ϕ : W ϕ ∼ −→ W ϕ is an isomorphism.
2.4. Reflexive Generalized Inverses. Let C be the field of complex numbers. Given a matrix A ∈ Mat n×m (C), a reflexive generalized inverse of A is a matrix A + ∈ Mat m×n (C) such that:
• A A + A = A;
• A + A A + = A + . In general, the reflexive generalized inverse of a matrix A is not unique.
The notion of reflexive generalized inverse in arbitrary vector spaces is the following:
Definition 2.1. If V and W are k-vector spaces, given a morphism f : V → W , a linear map f + : W → V is a "reflexive generalized inverse" of f when:
•
f • f + • f = f ; • f + • f • f + = f + .
For every reflexive generalized inverse f + of f , if w ∈ W , it is known that
(2.4) w ∈ Im f ⇐⇒ (f • f + )(w) = w .
2.5. Moore-Penrose Inverse of an (n × m)-Matrix. Given again a matrix A ∈ Mat n×m (C), the Moore-Penrose inverse of A is a matrix A † ∈ Mat m×n (C) such that:
• A A † A = A; • A † A A † = A † ; • (A A † ) * = A A † ; • (A † A) * = A † A;
B * being the conjugate transpose of the matrix B.
The Moore-Penrose inverse of A always exists, it is unique, it is a reflexive generalized inverse of A, [A † ] † = A and, if A ∈ Mat n×n (C) is non-singular, then the Moore-Penrose inverse of A coincides with the inverse matrix A −1 .
For details, readers are referred to [2].
3. Moore-Penrose Inverse of Linear Maps on Infinite-Dimensional Vector Spaces
Henceforth, given a family of subspaces {H i } i∈I of an arbitrary vector space V over a field k, we shall write V = ⊕ i∈I H i to indicate that the natural morphism
⊕ i∈I H i −→ V (v i ) −→ i∈I v i , is an isomorphism.
This section is devoted to proving the existence of a Moore-Penrose inverse of some linear maps on infinite-dimensional vector spaces, such that it generalizes the notion and the properties of the Moore-Penrose inverse of an (n × m)-matrix with entries in C. Our generalization will be valid for linear maps on inner product vector spaces over k = R and k = C and we shall give conditions for computing the Moore-Penrose inverse of endomorphisms on infinite-dimensional vector spaces.
To do so, we shall first study some properties of the Moore-Penrose inverse of an endomorphism on a finite-dimensional inner product space.
3.1. Moore-Penrose inverse of an endomorphism on finite-dimensional inner product vector spaces. Let us now consider a finite dimensional inner product vector space (E, g) over k = R or k = C.
If f ∈ End k (E), one has that E = Ker f ⊕ [Ker f ] ⊥ = Im f ⊕ [Im f ] ⊥ , and there exists an isomorphism f : [Ker f ] ⊥ ∼ −→ Im f .
Thus, the Moore-Penrose of f is the unique linear map f † ∈ End k (E) such that
(3.1) f † (e) = (f | [Ker f ] ⊥ ) −1 (e) if e ∈ Im f 0 if e ∈ [Im f ] ⊥ .
It is known that f † is the unique linear map such that:
• f † is a reflexive generalized inverse of f ; • f † • f and f • f † are self-adjoint. Definition 3.1. If f ∈ End k (E) and H f = {H 1 , . . . , H n } is a family of subspaces of E invariants for f such that E = H 1 ⊕ · · · ⊕ H n , we define the endomorphism f + H f ∈ End k (E) as the unique linear map such that [f + H f ] |H i = [f |H i ] † for each i ∈ {1, . . . , n} .
If we denote f i = f |H i for each i ∈ {1, . . . , n}, it is clear that for every vector e ∈ E, such that e = h j1 + · · · + h jn with h ji ∈ H i , then
f + H f (e) = f † 1 (h j1 ) + · · · + f † n (h jn ) .
Moreover, it is immediately observed, from Definition 3.1 and from the properties of the Moore-Penrose inverse, that f + H f is a reflexive generalized inverse of f for every family H f .
Keeping the previous notation and given a subspace W ⊆ E, such that W ⊂ H i for a certain i ∈ {1, . . . , n}, we shall denote
W ⊥ i = {v i ∈ H i such that g(w, v i ) = 0 for all w ∈ W } . Lemma 3.2. If U ⊂ E is a subspace and {U 1 , . . . , U n } are subspaces of U such that U = U 1 ⊕ · · · ⊕ U n with U i ⊆ H i , then [U 1 ] ⊥ 1 ⊕ · · · ⊕ [U n ] ⊥ n ⊆ U ⊥ ⇐⇒ [U i ] ⊥ i ⊂ [ j =i U j ] ⊥ for all i ∈ {1, . . . , n} .
Proof.
If [U 1 ] ⊥ 1 ⊕ · · · ⊕ [U n ] ⊥ n ⊆ U ⊥ =⇒ [U i ] ⊥ i ⊆ U ⊥ for all i ∈ {1, . . . , n} =⇒ =⇒ [U i ] ⊥ i ⊆ [ j =i U j ] ⊥ for all i ∈ {1, . . . , n} . Conversely, if [U i ] ⊥ i ⊆ [ j =i U j ] ⊥ for all i ∈ {1, . . . , n} =⇒ g(v 1 + · · · + v n , u 1 + · · · + u n ) = 0 with v i ∈ [U i ] ⊥ i and u i ∈ U i for all i ∈ {1, . . . , n} =⇒ [U 1 ] ⊥ 1 ⊕ · · · ⊕ [U n ] ⊥ n ⊆ U ⊥ .
Lemma 3.3. Using the previous notation, we have that:
(1) Im f = Im f 1 ⊕ · · · ⊕ Im f n ;
(2) If H ⊥ f = [Im f 1 ] ⊥ 1 ⊕ · · · ⊕ [Im f n ] ⊥ n , then E = Im f ⊕ H ⊥ f ; (3) Ker f = Ker f 1 ⊕ · · · ⊕ Ker f n ; (4) If H ⊥ f = [Ker f 1 ] ⊥ 1 ⊕ · · · ⊕ [Ker f n ] ⊥ n , then E = Ker f ⊕ H ⊥ f ;
(5) f induces an isomorphism between H ⊥ f and Im f and
(3.2) [f + H f ](e) = (f | H ⊥ f ) −1 (e) if e ∈ Im f 0 if e ∈ H ⊥ f . Proof. (1) It is clear that Im f 1 ⊕ · · · ⊕ Im f n ⊆ Im f . Moreover, given e ∈ Im f , if e = f (e ′ ) with e ′ = e ′ j1 + · · · + e ′ jn and e ′ ji ∈ H ij for all i ∈ {1, . . . , n}, then e = f (e ′ j1 ) + · · · + f (e ′ jn ) ∈ Im f 1 ⊕ · · · ⊕ Im f n , because f (v ′ ji ) ∈ H i for every i ∈ {1, . . . , n}. (2) Since H i = Im f i ⊕ [Im f i ] ⊥ i for all i ∈ {1, . . . , n}, we have that E = H 1 ⊕ · · · ⊕ H n = (Im f 1 ⊕ [Im f 1 ] ⊥ 1 ) ⊕ · · · ⊕ (Im f n ⊕ [Im f n ] ⊥ n ) = = (Im f 1 ⊕ · · · ⊕ Im f n ) ⊕ ([Im f 1 ] ⊥ 1 ⊕ · · · ⊕ [Im f n ] ⊥ n ) = Im f ⊕ H ⊥ f . (3) It is immediate that Ker f 1 ⊕· · ·⊕Ker f n ⊆ Ker f . Furthermore, ifē ∈ Ker f and e =ē j1 + · · · +ē jn withē ji ∈ H i for every i ∈ {1, . . . , n} , since f (ē j1 ) + · · · + f (ē j k ) = 0 and H i ∩ [ r =i H r ] = 0 ,
we conclude that f (ē ji ) = 0 for all i ∈ {1, . . . , n}, from which it is deduced that Ker f ⊆ Ker f 1 ⊕ · · · ⊕ Ker f n . (4) Similar to (2). (5) It is a direct consequence of (2), (4) and the definition of f + H f .
Lemma 3.4. With the previous notation, in general H ⊥ f = [Ker f ] ⊥ and H ⊥ f = [Im f ] ⊥ .
Proof. The statement is deduced from the following counter-example.
Let
E = v 1 , v 2 , v 3 , v 4 be a inner product k-vector space, B = {v 1 , v 2 , v 3 , v 4
} being an orthonormal basis, and let us consider the endomorphism f : E → E defined as:
f (v i ) = v 1 if i = 1, 2 ; −2v 1 if i = 3 ; v 1 + v 2 + v 3 if i = 4 . If we set H 1 = v 1 , v 2 and H 2 = v 1 + v 2 + v 3 , v 4
, it is clear that H 1 and H 2 are f -invariant and, if we again denote f 1 = f |H 1 and f 2 = f |H 2 , we have that:
• Im f 1 = v 1 and [Im f 1 ] ⊥ 1 = v 2 ; • Im f 2 = v 1 + v 2 + v 3 and [Im f 2 ] ⊥ 2 = v 4 ; • Im f = v 1 , v 1 + v 2 + v 3 and [Im f ] ⊥ = v 2 − v 3 , v 4 ; • Ker f 1 = v 1 − v 2 and [Ker f 1 ] ⊥ 1 = v 1 + v 2 ; • Ker f 2 = v 1 + v 2 + v 3 and [Ker f 2 ] ⊥ 2 = v 4 ; • Ker f = v 1 − v 2 , v 1 + v 2 + v 3 and [Ker f ] ⊥ = v 1 + v 2 − 2v 3 , v 4 .
Accordingly,
H ⊥ f = [Ker f ] ⊥ and H f ⊥ = [Im f ] ⊥ in this case.
A direct consequence of Lemma 3.4 is:
Corollary 3.5. Given an endomorphism f ∈ End k (E), in general f + H f is not the Moore-Penrose inverse of f .f + H f (v i ) = 1 2 v 1 + 1 2 v 2 if i = 1 v 4 − 1 2 v 1 − 1 2 v 2 if i = 3 0 if i = 2, 4 and f † (v i ) = 1 6 v 1 + 1 6 v 2 − 1 3 v 3 if i = 1 − 1 12 v 1 − 1 12 v 2 + 1 6 v 3 + 1 2 v 4 if i = 2 − 1 12 v 1 − 1 12 v 2 + 1 6 v 3 + 1 2 v 4 if i = 3 0 if i = 4
.
Readers can easily check that
f + H f is a reflexive generalized inverse of f (Defini- tion 2.1), although it is clear that f + H f = f † .
Lemma 3.6. With the above notation, we have that
H ⊥ f = [Im f ] ⊥ if and only if [Im f i ] ⊥ i ⊆ [ j =i Im f j ] ⊥ for every i ∈ {1,
3.2.
Moore-Penrose inverse of linear maps on arbitrary inner product Spaces. We shall now generalize the notion of Moore-Penrose inverse to some endomorphisms of arbitrary vector spaces, in particular some infinite-dimensional vector spaces. Henceforth (V, g) and (W,ḡ) will be inner product vector spaces over k, with k = C or k = R.
V such that V = U ⊕ U ⊥ . In this case, if V = U ⊕ W , it is clear that the linear map f U ∈ End k (V ) defined as f U (v) = 0 if v ∈ U v if v ∈ W
is not admissible for the Moore-Penrose inverse. (1) f † is a reflexive generalized inverse of f ;
(2) f † • f and f • f † are self-adjoint, that is:
• g([f † • f ](v), v ′ ) = g(v, [f † • f ](v ′ ); •ḡ([f • f † ](w), w ′ ) =ḡ(w, [f • f † ](w ′ );
for all v, v ′ ∈ V and w, w ′ ∈ W . The operator f † is named the Moore-Penrose inverse of f .
Proof. If f is admissible of the Moore-Penrose inverse (Definition 3.9), then the restriction f | [Ker f ] ⊥ is an isomorphism between [Ker f ] ⊥ and Im f and there exists an unique linear map satisfying that
f † (w) = (f | [Ker f ] ⊥ ) −1 (w) if w ∈ Im f 0 if w ∈ [Im f ] ⊥ .
We shall now check that f † satisfies the conditions of the statement.
Firstly, since
(f • f † )(w) = w if w ∈ Im f 0 if w ∈ [Im f ] ⊥ and (f † • f )(v) = v 1 with v = v 1 + v 2 (v 1 ∈ [Ker f ] ⊥ and v 2 ∈ Ker f ), then it is clear that f † is a reflexive generalized inverse of f because: • (f • f † • f )(v) = f (v); • (f † • f • f † )(w) = f † (w).
Moreover,
g([f • f † ](w), w ′ ) =ḡ(w, [f • f † ](w ′ )) = ḡ (w, w ′ ) if w, w ′ ∈ Im f 0 if w ∈ [Im f ] ⊥ 0 if w ′ ∈ [Im f ] ⊥ And, if v, v ′ ∈ V with v = v 1 + v 2 , v ′ = v ′ 1 + v ′ 2 , v 1 , v ′ 1 ∈ [Ker f ] ⊥ and v 2 , v ′ 2 ∈ Ker f , one has that g([f † • f ](v), v ′ ) = g(v 1 , v ′ 1 ) = g(v, [f † • f ](v ′ )
. Hence, we conclude that f † satisfies the conditions of the Theorem.
For proving the uniqueness of the Moore-Penrose inverse of f , let us consider a linear map f :
W → V such that (1) f is a reflexive generalized inverse of f ; (2) g([ f • f ](v), v ′ ) = g(v, [ f • f ](v ′ ); (3)ḡ([f • f ](w), w ′ ) =ḡ(w, [f • f ](w ′ ); for all v, v ′ ∈ V and w, w ′ ∈ W . A direct consequence of (1) is that ( f • f ) 2 = f • f . Hence f • f is a projection and, since Im f = Im (f • f • f ) ⊆ Im (f • f ) ⊆ Im f ,
then Im (f • f ) = Im f . Accordingly, given w ∈ Imf , there existsw ∈ W such that (f • f )(w), and then
(f • f )(w) = (f • f ) 2 (w) = (f • f )(w) = w . Furthermore, if w ′ ∈ [Im f ] ⊥ , we have that 0 =ḡ([f • f ] 2 (w ′ ), w ′ ) =ḡ([f • f ](w ′ ), [f • f ](w ′ )) =⇒ [f • f ](w ′ ) = 0 .
Thus,
(f • f )(w) = w if w ∈ Im f 0 if w ∈ [Im f ] ⊥ and, in particular, f (w ′ ) = 0 when w ′ ∈ [Im f ] ⊥ .
In line with the above arguments, one has that (f
• f ) 2 = f • f and Im ( f • f ) = Im f . Now, if v ∈ Im f , v = [ f • f ](v) and v ′ ∈ Ker f , then g(v, v ′ ) = g([ f • f ](v), v ′ ) = g(v, 0) = 0 , and we deduce that Im f ⊆ [Ker f ] ⊥ . Finally, since f | [Ker f ] ⊥ : [Ker f ] ⊥ ∼ −→ Im f and (f • f ) | Im f = Id | Im f , then ( f ) | Im f = (f | [Ker f ] ⊥ ) −1 =⇒ f = f † .
Conversely, let us assume that there exists the Moore-Penrose inverse f † : W → V of a linear map f : V → W . Based on the same arguments as above one immediately has that:
•
f • f † and f † • f are projections; • Im (f • f † ) = Im f ; • [Im f ] ⊥ ⊆ Ker(f • f † ); • Im (f † • f ) ⊆ [Ker f ] ⊥ . Moreover, if w / ∈ [Im f ] ⊥ there existsw ∈ W such that 0 =ḡ([f • f † ] 2 (w), w) =ḡ([f • f † ](w), [f • f † ](w)) , from where we deduce that w / ∈ Ker(f • f † ) and [Im f ] ⊥ = Ker(f • f † ). On the other hand it is clear that Ker f ⊆ Ker(f † •f ) and, if v ∈ V with f (v) = 0 then v / ∈ Ker(f † • f ), because f (v) = (f • f † • f )(v) = 0 .
Hence, Ker f = Ker(f † • f ) and, bearing in mind that if g ∈ End k (V ) is a projection then V = Ker g ⊕ Im g, one concludes that
V = Ker(f † • f ) ⊕ Im (f † • f ) = Ker f ⊕ [Ker f ] ⊥ and W = Ker(f • f † ) ⊕ Im (f • f † ) = Im f ⊕ [Im f ] ⊥
. Accordingly, f is admissible for the Moore-Penrose inverse and the statement is deduced.
Since each isomorphism g : V → W is admissible for the Moore-Penrose inverse, a direct consequence of Theorem 3.11 is that g † = g −1 , where g −1 is the inverse map of g. Proof. This statement is deduced from Theorem 3.11 bearing in mind that:
• Im f † = [Ker f ] ⊥ ; • [Im f † ] ⊥ = Ker f ; • Ker f † = [Im f ] ⊥ ; • [Ker f † ] ⊥ = Im f .
Moreover, if f : V → W is a linear map admissible for the Moore-Penrose inverse and P [ker f ] ⊥ and P Im f are the projections induced by the decompositions V = Ker f ⊕ [Ker f ] ⊥ and W = Im f ⊕ [Im f ] ⊥ , respectively, we obtain from the arguments of the proof of Theorem 3.11 that Corollary 3.13. If (V, g) and (W,ḡ) are inner product spaces over k and f : V → W is a linear map admissible for the Moore-Penrose inverse, then:
•
f † • f = P [ker f ] ⊥ ; • f • f † = P Im f .
Computation of the Moore-Penrose inverse of Endomorphisms on
Arbitrary inner product Spaces. Similar to the finite-dimensional situation, given an inner product space (V, g) over k, f ∈ End k (V ) let us assume that there exists a family of f -invariant finite-dimensional subspaces,
H f = {H i } i∈I , such that V = i∈I H i .
Note that this assumption is always satisfied when f admits an annihilator polynomial.
To simplify, fixing a family H f , we shall denote
f i = f |H i .
Definition 3.14. We shall call reflexive generalized inverse of f associated with the family H f to the unique linear map f +
H f ∈ End k (V ) such that [f + H f ] |H i = f † i for every i ∈ I. For each vector v ∈ V , if v = v i1 + · · · + v is with v ij ∈ H ij , then f + H f (v) = f † i1 (v 1 ) + · · · + f † is (v s ) .
If f is admissible for the Moore-Penrose inverse, our purpose is to determine when f + H f = f † . To do this, the generalization onto infinite-dimensional vector spaces of Lemma 3.3 is: Lemma 3.15. We have that:
(1) Im f = i∈I Im f i ;
(2) If H ⊥ f = i∈I [Im f i ] ⊥ i , then V = Im f ⊕ H ⊥ f ; (3) Ker f = i∈I Ker f i ; (4) If H ⊥ f = i∈I [Ker f i ] ⊥ i , then V = Ker f ⊕ H ⊥ f ;
(5) f induces an isomorphism between H ⊥ f and Im f . Proof.
(1) It is clear that
i∈I Im f i ⊆ Im f . Moreover, given v ∈ Im f , if v = f (v ′ ) with v ′ = v ′ i1 + · · · + v ′ is and v ′ ij ∈ H ij for all j ∈ {1, . . . , s}, then v = f (v ′ i1 ) + · · · + f (v ′ is ) ∈ i∈I Im f i , because f (v ′ ij ) ∈ H ij for every j ∈ {1, . . . , s}. (2) Since H i = Im f i ⊕ [Im f i ] ⊥ i for all i ∈ I, we have that V = i∈I H i = i∈I Im f i ⊕ [Im f i ] ⊥ i = = i∈I Im f i ⊕ i∈I [Im f i ] ⊥ i = Im f ⊕ H ⊥ f . (3) It is immediate that i∈I Ker f i ⊆ Ker f . Furthermore, ifv ∈ Ker f and v =v i1 + · · · +v i k withv ij ∈ H ij for every j ∈ {1, . . . , k} , since f (v i1 ) + · · · + f (v i k ) = 0 and H ij ∩ [ r =j H ir ] = 0 ,
we conclude that f (v ij ) = 0 for all j ∈ {1, . . . , k}, from which it is deduced that Ker f ⊆ i∈I Ker f i .
(4) Similar to (2). (5) It is a direct consequence of (2) and (4).
It is now easy to prove that the generalization onto an arbitrary vector space V of the Lemma 3.2 is the following:
f = {U i } i∈I , if U ⊂ V is a subspace and {U i } i∈I is a family of subspaces of U such that U = ⊕ i∈I U i with U i ⊆ H i , then ⊕ i∈I [U i ] ⊥ i ⊆ U ⊥ ⇐⇒ [U i ] ⊥ i ⊂ [ j =i U j ] ⊥ for all i ∈ I .
Accordingly we have that:
Lemma 3.17. If (V, g) is an inner product vector space over k, f ∈ End k (V ), and
H f = {H i } i∈I with V = i∈I H i and each H i is f -invariant, then H ⊥ f = [Im f ] ⊥ if and only if [Im f i ] ⊥ i ⊆ [ j =i Im f j ] ⊥ for every i ∈ I .
Proof. This statement is the generalization of Lemma 3.6 to arbitrary vector spaces.
Moreover, similar to Lemma 3.7 one has that:
Lemma 3.18. If (V, g) is an inner product vector space over k, f ∈ End k (V )
,
and H f = {H i } i∈I with V = i∈I H i and each subspace H i is f -invariant, then H ⊥ f = [Ker f ] ⊥ if and only if [Ker f i ] ⊥ i ⊆ [ j =i
Ker f j ] ⊥ for every i ∈ I .
(1) [Im f i ] ⊥ i ⊆ [ j =i Im f j ] ⊥ for every i ∈ I; (2) [Ker f i ] ⊥ i ⊆ [ j =i Ker f j ] ⊥ for all i ∈ I. Proof.
The claim is a direct consequence of Lemma 3.17 and Lemma 3.18.
Corollary 3.20. If (V, g) is an inner product vector space over k, f ∈ End k (V ), Let ϕ ∈ End k (V ) the finite potent endomorphism defined as follows:
H f = {H i } i∈I with V = i∈I H i and each subspace H i is f -invariant, and H i ⊆ [ j =i H j ] ⊥ for every i ∈ I,ϕ(v i ) = v 2 + v 5 + v 7 if i = 1 v 1 + 3v 2 if i = 2 v 4 if i = 3 v 1 − v 3 if i = 4 −v 3 + 2v 5 + 2v 7 if i = 5 3v i+1 if i = 5h + 1 0 if i = 5h + 2 −v i−2 + 2v i+1 if i = 5h + 3 v i−2 + v i+1 if i = 5h + 4 −v i−4 + 5v i−3 if i = 5h + 5 for all h ≥ 1.
We have that the AST-decomposition V = U ϕ ⊕ W ϕ is determined by the subspaces
W ϕ = v 1 , v 2 , v 3 , v 4 , v 5 + v 7 and U ϕ = v j j≥6 , .
In this basis of W ϕ one has that
ϕ |W ϕ ≡ A Wϕ = 0 1 0 1 0 1 3 0 0 0 0 0 0 −1 −1 0 0 1 0 0 1 0 0 0 2 . Moreover, we can write U ϕ = i≥2 H i with H i = v 5i−4 , v 5i−3 , v 5i−2 , v 5i−1 , v 5i for
all i ≥ 2, and in the same bases we have that
ϕ |H i ≡ 0 0 −1 0 −1 3 0 0 1 5 0 0 0 0 0 0 0 2 0 0 0 0 0 1 0 for all i ≥ 2.
We can now consider the family of ϕ-invariant subspaces H ϕ = {H 1 = W ϕ , H i } i≥2 . Thus, bearing in mind that:
• Im
ϕ 1 = H 1 , Ker ϕ 1 = {0}, [Im ϕ 1 ] ⊥ 1 = {0} and [Ker ϕ 1 ] ⊥ 1 = H 1 ; • Im ϕ i = v 5i−4 , v 5i−3 , v 5i−1 , v 5i , ker ϕ i = v 5i−3 , [Im ϕ i ] ⊥ i = v 5i−2 and [Ker ϕ i ] ⊥ i = v 5i−4 , v 5i−2 , v 5i−1 , v 5i for all i ≥ 2; • Im ϕ = H 1 ⊕ [ ⊕ i≥2 v 5i−4 , v 5i−3 , v 5i−1 , v 5i ] and [Im ϕ] ⊥ = ⊕ i≥2 v 5i−2 ; • ker ϕ = ⊕ i≥2 v 5i−3 and [ker ϕ] ⊥ = H 1 ⊕ [ ⊕ i≥2 v 5i−4 , v 5i−2 , v 5i−1 , v 5i ]; we have that H ⊥ ϕ = [Ker ϕ] ⊥ and H ⊥ ϕ = [Im ϕ] ⊥ .
Hence, ϕ is admissible for the Moore-Penrose inverse and ϕ † = ϕ + Hϕ . Accordingly, a computation shows that
(ϕ † ) |W ϕ = A −1 Wϕ = 6 −2 6 0 3 −2 1 −2 0 −1 0 0 0 1 0 3 −1 2 0 1 −3 1 −3 0 −1 and (ϕ † ) |H i ≡ 5 3 1 3 0 5 6 − 1 3 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 1 −1 0 0 − 1 2 0 for all i ≥ 2, from where the endomorphism ϕ † is determined. Thus ϕ † (v i ) = 6v 1 − 2v 2 + 3v 4 − 3v 5 − 3v 7 if i = 1 −2v 1 + v 2 − v 4 + v 5 + v 7 if i = 2 6v 1 − 2v 2 + 2v 4 − 3v 5 − 3v 7 if i = 3 v 3 if i = 4 3v 1 − v 2 + v 4 − v 5 − v 7 if i = 5 5 3 v i − v i+4 if i = 5h + 1 1 3 v i−1 if i = 5h + 2 0 if i = 5h + 3 5 6 v i−3 + 1 2 v i−1 − 1 2 v i+1 if i = 5h + 4 − 1 3 v i−4 + v i−1 if i = 5h + 5 for all h ≥ 1.
Remark 3.21. In Example 3 it is clear that H ϕ satisfies the conditions of Theorem 3.19, but the hypothesis of Corollary 3.20 is not satisfied for this map, because v 5 + v 7 ∈ H 1 and v 5 + v 7 / ∈ H ⊥ 2 . Thus, this example allows us to deduce that the statement of Corollary 3.20 is a sufficient condition, but is not a necessary condition, for the computing of f † from a family of f -invariant subspaces H f . Moreover, this example shows that the Moore-Penrose inverse of a finite potent endomorphism is not, in general, a finite potent endomorphism.
Study of Infinite systems of linear equations from the Moore-Penrose Inverse
The aim of this final section is to study solutions of infinite system of linear equations from the Moore-Penrose inverse of linear maps characterized in the previous section.
Definition 4.1. If V and W are two arbitrary k-vector spaces and f : V → W is a linear map, a linear system is an expression
f (x) = w ,
where w ∈ W . This system is called "consistent" when w ∈ Im f .
If V and W are infinite-dimensional vector spaces, fixing bases of V and W , the linear system f (x) = w is equivalent to an infinite system of linear equations.
If a linear system f (x) = w is consistent and f (v 0 ) = w for a certain v 0 ∈ V , then the set of solutions of this system is v 0 + Ker f . The vector v 0 is named "particular solution" of the system.
Let us now consider two arbitrary inner product vector spaces (V, g) and (W,ḡ) over k, let f : V → W be a linear map admissible for the Moore-Penrose inverse and let f † be its Moore-Penrose inverse. If f (x) = w is a linear system , since f † is a reflexive generalized inverse of f , then f is consistent if and only if (f • f † )(w) = w and, in this case, the set of solutions of the linear system is
f † (w) + Ker f .
In finite-dimensional inner product vector spaces it is known that the Moore-Penrose inverse is useful for studying the least squares solutions of a linear system. To complete this work, we shall generalize this notion to arbitrary vector spaces.
Definition 4.2. If (V, g) and (W,ḡ) are two arbitrary inner product vector spaces over k, then v ′ ∈ V is called "leastḡ-norm solution" of a linear system f (
x) = w when f (v ′ ) − w ḡ ≤ f (v) − w ḡ for all v ∈ V .
Note that v ′ ∈ V is a leastḡ-norm solution of the linear system f (x) = w if and only if dḡ(w, f (v ′ )) = d g (w, Im f ). Definition 4.3. If (V, g) and (W,ḡ) are two arbitrary inner product vector spaces over k, thenṽ ∈ V is called "minimal leastḡ-norm solution" of a linear system
f (x) = w when ṽ g ≤ v ′ g for every leastḡ-norm solution v ′ ∈ V .(4.1) f (v) − w 2 g = [f (v) − f (f † (w))] + [f (f † (w)) − w] 2 g = = f (v) − f (f † (w)) 2 g + f (f † (w)) − w 2 g for all v ∈ V . Hence, f (f † (w)) − w ḡ ≤ f (v) − w ḡ
for all v ∈ V , and we deduce that f † (w) is a leastḡ-norm solution of f (x) = w. Moreover, it follows from (4.1) that v ′ ∈ V is a leastḡ-norm solution of this linear system if and only if f (v ′ ) − f (f † (w)) = 0, that is, v ′ is a solution of the consistent system f (x) − f (f † (w)) = 0 .
Thus, for each leastḡ-norm solution v ′ ∈ V , one has that
f † (w) = v ′ + h ,
with h ∈ Ker f and, bearing in mind that f † (w) ∈ [Ker f ] ⊥ , we conclude that f † (w) is the unique minimal leastḡ-norm solution of f (x) = w because
f † (w) g < v ′ g ,
for every v ′ = f † (w). to denote the well-defined vector
x = i∈N x i · v i ∈ V .
Let ϕ ∈ End k (V ) the finite potent endomorphism studied in Example 3. We can consider the system ϕ(x) = w, where w = (α i ) i∈N and whose explicit expression is:
(4.2) x 2 + x 4 = α 1 x 1 + 3x 2 = α 2 −x 4 − x 5 = α 3 x 3 = α 4 x 1 + 2x 5 = α 5 −x 8 − x 10 = α 6
x 1 + 2x 5 + 3x 6 + x 9 + 5x 10 = α 7 0 = α 5h+3 for all h ≥ 1 2x 5h+3 = α 5h+4 for all h ≥ 1
x 5h+4 = α 5h+5 for all h ≥ 1 −x 5h+3 − x 5h+5 = α 5h+1 for all h ≥ 2 3x 5h+1 + x 5h+4 + 5x 5h+5 = α 5h+2 for all h ≥ 2 . By Proposition 4.4, we have that the unique minimal least g-norm solution of this linear system is (β i ) i∈N = ϕ † (w) and, bearing in mind the expression of ϕ † obtained in Example 3, an easy computation shows that Moreover, one have that
β i = 6α 1 − 2α 2 + 6α 3 + 3α 5 if i = 1 −2α 1 + α 2 − 2α 3 − α 5 if i = 2 α 4 if i = 3 3α 1 − α 2 + 2α 3 + α 5 if i = 4 −3α 1 + α 2 − 3α 3 − α 5 if i = 5 −(ϕ • ϕ † )(v i ) = 0 if i = 5h + 3 v i otherwise .
Thus, according to (2.4), the system (4.2) is consistent if and only if α 5h+3 = 0 for all h ≥ 1, and, in this case, (β i ) i∈N is a particular solution of it.
Example 2 .
2If f and H f = {H 1 , H 2 } are as in the proof of Lemma 3.4, a computation shows that
Definition 3. 9 .
9Given a linear map f : V → W , we say that f is admissible for the Moore-Penrose inverse when V = Ker f ⊕ [Ker f ] ⊥ and W = Im f ⊕ [Im f ] ⊥ .
Remark 3 . 10 .
310It is known that there exist infinite-dimensional vector spaces V and vector subspaces U ⊂
Theorem 3. 11 (
11Existence and uniqueness of Moore-Penrose inverse). If (V, g) and (W,ḡ) are inner product spaces over k, then f : V → W is a linear map admissible for the Moore-Penrose inverse if and only if there exists a unique linear map f † : W → V such that:
Corollary 3 . 12 .
312If (V, g) and (W,ḡ) are inner product spaces over k and f : V → W is a linear map admissible for the Moore-Penrose inverse, then f † is also admissible for the Moore-Penrose inverse and (f † ) † = f .
Lemma 3 . 16 .
316With the previous assumptions on V and H
Theorem 3 . 19 .
319If (V, g) is an inner product vector space over k, f ∈ End k (V ) is admissible for the Moore-Penrose inverse, and H f = {H i } i∈I with V = i∈I H i and each subspace H i is f -invariant, then f + H f = f † if and only if the following conditions are satisfied:
then f is admissible for the Moore-Penrose inverse andf + H f = f † .Proof. With the hypothesis of this Corollary the conditions of Theorem 3.19 are satisfied.
Example 3 .
3Let (V, g) be an inner product vector space of countable dimension over k. Let {v 1 , v 2 , v 3 , . . . } be an orthonormal basis of V indexed by the natural numbers.
Proposition 4. 4 .
4If (V, g) and (W,ḡ) are two arbitrary inner product vector spaces over k, f : V → W is a linear map admissible for the Moore-Penrose inverse and f (x) = w is a linear system, then f † (w) is the unique minimal leastḡ-norm solution of this linear system.Proof. Firstly, since Im(f • f † − Id) ⊆ [Im f ] ⊥ , one has that
Example 4 .
4Let (V, g) be an inner product vector space of countable dimension over k. Let {v 1 , v 2 , v 3 , . . . } be an orthonormal basis of V indexed by the natural numbers. If (x i ) i∈N ∈ i∈N k, since x i = 0 for almost all i ∈ N, we shall write x = (x i )
in[7] as a basic tool for his elegant definition of Abstract Residues.In 2007 M. Argerami, F. Szechtman and R. Tifenbach showed in[1] that an endomorphism ϕ is finite potent if and only
. . . , n} . Proof. Considering that E = Im f ⊕ H ⊥ f and [Im f ] ⊥ ∩ Im f = {0}, the statement is deduced bearing in mind that this condition is equivalent to H ⊥ f ⊆ [Im f ] ⊥ (Lemma 3.2). Similarly, one can prove that Lemma 3.7. We have that H ⊥ f = [Ker f ] ⊥ if and only if Ker f j ] ⊥ for every i ∈ {1, . . . , n} . Proposition 3.8. If f ∈ End k (E) and H f = {H 1 , . . . , H n } is a family of subspaces of E invariants for f such that E = H 1 ⊕ · · · ⊕ H n and H i ⊆ [ j =i H j ] ⊥ for all i ∈ {1, . . . , n}, then f + H f = f † . Proof. If H i ⊆ [ j =i H j ] ⊥ for all i ∈ {1, . . . , n}, then the conditions of Lemma 3.6 and Lemma 3.7 hold. Hence, the claim is deduced.[Ker f i ] ⊥
i ⊆ [
j =i
i+4 if i = 5h + 1, for all h ≥ 21
3
α 5 +
5
3
α 6 +
1
3
α 7 +
5
6
α 9 −
1
3
α 10 if i = 6
0
if i = 5h + 2, for all h ≥ 1
1
2
α i+1
if i = 5h + 3, for all h ≥ 1
α i+1
if i = 5h + 4, for all h ≥ 1
−α i−4 −
1
2
α i−1
if i = 5h + 5, for all h ≥ 1
5
3
α i +
1
3
α i+1 +
5
6
α i+3 −
1
3
α .
On Tate's trace, Linear Multilinear Algebra. M Argerami, F Szechtman, R Tifenbach, 55Argerami, M.; Szechtman, F.; Tifenbach, R. On Tate's trace, Linear Multilinear Alge- bra 55(6), (2007) 515-520.
S L Campbell, C D MeyerJr, 978-0-486-66693-8Generalized Inverses of Linear Transformations. DoverCampbell, S. L.; Meyer, Jr., C. D.; Generalized Inverses of Linear Transformations, Dover, (1991). ISBN 978-0-486-66693-8.
Explicit solutions of infinite systems of linear equations from reflexive generalized inverses of finite potent endomorphisms. Cabezas Sánchez, V Pablos Romo, F , Linear Algebra Appl. 559Cabezas Sánchez, V.; Pablos Romo, F.; Explicit solutions of infinite systems of linear equations from reflexive generalized inverses of finite potent endomorphisms, Linear Algebra Appl. 559, (2018) 125-144.
On the Drazin Inverse of Finite Potent Endomorphisms. Pablos Romo, F , 10.1080/03081087.2018.1484421Linear and Multilinear Algebra. Pablos Romo, F. On the Drazin Inverse of Finite Potent Endomorphisms, Linear and Multilinear Algebra (2018), DOI: 10.1080/03081087.2018.1484421.
On The Classification of Endomorphisms on Infinite-Dimensional Vector Spaces. Pablos Romo, F , University of SalamancaPreprintPablos Romo, F. On The Classification of Endomorphisms on Infinite-Dimensional Vector Spaces, Preprint 2018, University of Salamanca.
C R Rao, S K Mitra, ISBN 0-471-70821-6Generalized Inverse of Matrices and its Applications. New YorkJohn Wiley -SonsRao, C. R.; Mitra, S. K.; Generalized Inverse of Matrices and its Applications, New York: John Wiley -Sons, (1971). ISBN 0-471-70821-6.
Residues of Differentials on Curves. J Tate, Ann. Scient.Éc. Norm. Sup. 14a sérieTate, J. Residues of Differentials on Curves, Ann. Scient.Éc. Norm. Sup. 1, 4a série, (1968) 149-159.
. Matemáticas Departamento De, Plaza de la Merced 1-4, 37008Universidad de SalamancaDepartamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008
España E-mail address: (*) [email protected] E-mail address: (**) [email protected]. Salamanca, Salamanca, España E-mail address: (*) [email protected] E-mail address: (**) [email protected]
| [] |
[
"REGULARITY OF MULTIFRACTAL SPECTRA OF CONFORMAL ITERATED FUNCTION SYSTEMS",
"REGULARITY OF MULTIFRACTAL SPECTRA OF CONFORMAL ITERATED FUNCTION SYSTEMS"
] | [
"Johannes Jaerisch ",
"Marc Kesseböhmer "
] | [] | [] | We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we introduce the notion of regular convergence for families of cIFS not necessarily sharing the same index set, which guarantees the convergence of the multifractal spectra on the interior of their domain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us to carry over results for finite to infinite systems, and in this way to establish a multifractal analysis without the usual regularity conditions. Finally, we discuss the connections to the λ -topology introduced by Roy and Urbański. | 10.1090/s0002-9947-2010-05326-7 | [
"https://arxiv.org/pdf/0902.2473v2.pdf"
] | 14,425,416 | 0902.2473 | 8adc8076e3d9a0c1379b8b722de87538fef72229 |
REGULARITY OF MULTIFRACTAL SPECTRA OF CONFORMAL ITERATED FUNCTION SYSTEMS
30 Mar 2010
Johannes Jaerisch
Marc Kesseböhmer
REGULARITY OF MULTIFRACTAL SPECTRA OF CONFORMAL ITERATED FUNCTION SYSTEMS
30 Mar 2010arXiv:0902.2473v2 [math.DS]
We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we introduce the notion of regular convergence for families of cIFS not necessarily sharing the same index set, which guarantees the convergence of the multifractal spectra on the interior of their domain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us to carry over results for finite to infinite systems, and in this way to establish a multifractal analysis without the usual regularity conditions. Finally, we discuss the connections to the λ -topology introduced by Roy and Urbański.
INTRODUCTION AND STATEMENT OF RESULTS
The theory of multifractals has its origin at the boarderline between statistical physics and mathematics -classical references are e. g. [FP85,Man74,Man88,HJK + 86]. In this paper we study multifractal spectra in the setting of infinite conformal iterated functions systems (cIFS). These systems are given by at most countable families Φ = (ϕ e : X → X) e∈I , I ⊂ N, of conformal contractions on a compact connected subset X of the euclidean space R D , · , D ≥ 1. The set of cIFS with fixed phase space X will be denoted by CIFS (X) (see Section 2 for definitions). For ω ∈ I N we let ω |k := ω 1 · · · ω k and ϕ ω |k := ϕ ω 1 • · · ·• ϕ ω k . Then for each ω ∈ I N the intersection ∞ k=1 ϕ ω |k (X) is always a singleton given rise to a canonical coding map π Φ : I N → X. Its image Λ Φ := π Φ I N will be called the limit set of Φ. Given a Hölder continuous function ψ : I N −→ R the multifractal analysis of the system Φ with respect to the potential ψ is in our context understood to be the analysis of the level sets F α := π Φ ω ∈ I N : lim k→∞ S k ψ (ω) log ϕ ′ ω |k X = α in terms of their Hausdorff dimension f (α) := dim H (F α ). In here, S k ψ := ∑ k−1 n=0 ψ • σ n denotes the Birkhoff sum of ψ with respect to the shift map σ : I N → I N on the symbolic space, and ϕ ′ ω |k X := sup x∈X ϕ ′ ω |k (x) with ϕ ′ ω |k (x) denoting the operator norm of the derivative. A good reference for this kind of multifractal analysis is provided e. g. in [Pes97].
Let us define the geometric potential function associated with Φ by ζ : I N → R − 0 , ζ (ω) := log ϕ ′ ω 1 (π (σ (ω))) . It is well known that in the case of finite cIFS, that is card (I) < ∞, f can be related to the Legendre transform of the free energy function t : R → R, which is defined implicitly by the pressure equation (cf. Definition 2.4) P (t (β ) ζ + β ψ) = 0, β ∈ R.
(1.1) More precisely, there exists a closed finite interval J ⊂ R such that for all α ∈ J we have
f (α) = −t * (−α) := − sup β {−β α − t (β )} = inf β {t (β ) + β α} , (1.2)
and for α / ∈ J we have F α = ∅ ([Pes97, Theorem 21.1], [Sch99]). If we consider infinite cIFS, i. e. card (I) = card (N), we have to take into account that the pressure function might behave irregularly and hence it is not always possible to find a solution of (1.1). For the special case in which (1.1) has a unique solution the multifractal analysis has been discussed in [MU03,Section 4.9]. Further interesting results on the spectrum of local dimension for Gibbs states can be found [RU09].
Our first task is to generalise this concept to the case when the free energy cannot be defined by the unique solution of (1.1). This leads to the following modified definition of the free energy function. Notice that our definition of the free energy function generalises the definition given for the multifractal analysis presented in [MU03,Section 4.9] or in [KU07], where the existence of a zero of the pressure function t −→ P(tζ + β ψ) is always required. Our definition is rather in the spirit of [MU03, Theorem 4.2.13], which gives a version of Bowen's formula, without assuming a zero of the pressure function to exist. More precisely, we have
dim H (Λ Φ ) = inf {t ∈ R : P (tζ ) ≤ 0} ,
which immediately implies that t (0) = dim H (Λ Φ ). In fact, Lemma 3.1 shows that Definition 1.1 gives rise to a proper convex function. This concept of the free energy function has been investigated further in [JKL10] as a special case of the induced topological pressure for arbitrary countable Markov shifts. We would like to point out that this new formalism gives rise to further interesting exhausting principles similar to Example 1.6 and Corollary 1.9 above.
To state our first main result we set
α − := inf −t − (x) : x ∈ Int (dom (t)) and α + := sup −t + (x) : x ∈ Int(dom (t)) ,
where t + , resp. t − , denotes the derivative of t from the right, resp. from the left, Int (A) denotes the interior of the set A, and dom(t) := {x ∈ R : t (x) < +∞} refers to the effective domain of t.
Theorem 1.2. For α ∈ R we have f (α) ≤ max {−t * (−α) , 0} and for α ∈ (α − , α + ) we have f (α) = −t * (−α).
This first main result is essentially a consequence of the multifractal regularity property of sequences of tuples (Φ n , ψ n ) n of iterated function systems and potentials, which is the second main concern of this paper.
We adapt the definition of pointwise convergence in CIFS (X) as used by Roy and Urbański in [RU05] to our setting, allowing us to investigate also families of cIFS with associated potentials not sharing the same index set N. To simplify notation let us write h Ω := sup ω∈Ω |h (ω)| for the supremum norm of the map h :
Ω −→ (V, | · |) from Ω to the normed space (V, | · |). For Φ 1 , Φ 2 ∈ CIFS (X) we define ρ Φ 1 , Φ 2 := ∑ i∈I 1 ∩I 2 2 −i ϕ 1 i − ϕ 2 i X + (ϕ 1 i ) ′ − (ϕ 2 i ) ′ X + ∑ i∈I 1 △I 2 2 −i , (1.4)
where A△B denotes the usual symmetric difference of the sets A and B. It will turn out that ρ defines a metric on CIFS (X). For ω ∈ N k and k ∈ N we let [ω] := τ ∈ N N : τ |k = ω denote the cylinder set of ω.
In order to set up a multifractal spectrum we restrict our analysis to families of Hölder continuous functions ψ n : I N n −→ R and ψ : I N −→ R with I n ⊂ I ⊂ N, n ∈ N. Definition 1.3. We say that (Φ n , ψ n ) n −→ (Φ, ψ) converges pointwise if,
(A) Φ n −→ Φ in the ρ-metric and (B) for all k ∈ I we have lim n→∞ ψ n − ψ [k]∩I N n = 0.
Notice, that the convergence in ρ-metric implies that ψ n − ψ [k]∩I N n in (B) is well defined for all sufficiently large n. For a further discussion of the above defined property see also the remark succeeding Lemma 2.6.
As discussed in [RU05] pointwise convergence topology leads to discontinuities of the Hausdorff dimension of the limit sets. By introducing a weaker topology called the λtopology in [RU05] the Hausdorff dimension of the limit set depends continuously on the system (see also [RSU09]). Convergence in λ -topology requires the additional condition (6.1) below. As a corollary we will also establish the continuity of the Hausdorff dimension under weaker assumptions.
We are going to employ similar assumptions on the convergence of the pairs (Φ n , ψ n ) n and (Φ, ψ) to obtain continuity of the multifractal spectra. This is the purpose of the following definition. For this let ζ n denote the geometric potential associated with Φ n . Definition 1.4. We say that(Φ n , ψ n ) n converges regularly to (Φ, ψ), if (Φ n , ψ n ) n −→ (Φ, ψ) converges pointwise, and if for t, β ∈ R with P(tζ + β ψ) < ∞ there exists k ∈ N and a constant C > 0 such that for all n ∈ N and all ω ∈ (I n ) k we have exp sup
τ∈I N n ∩[ω]
(S k (tζ n + β ψ n ) (τ)) ≤ C exp sup
ρ∈I N ∩[ω]
(S k (tζ + β ψ)(ρ)) .
The assumption in Definition 1.4 is similar to the corresponding inequality in the definition of the λ -topology in [RU05] but depends additionally on the potentials ψ n and ψ. For particular cases we will show that the convergence Φ n −→ Φ in the λ -topology immediately implies the conditions in Definition 1.4. This is demonstrated in the following example providing an analysis of the (inverse) Lyapunov spectrum. This example is covered by Proposition 6.4 (2) stated in Section 6.
Example 1.5 (λ λ λ -topology). Let Φ n = (ϕ n e ) e∈I n , Φ = (ϕ e ) e∈N be elements of CIFS (X) with Φ n → Φ converging in the λ -topology and let ψ n = ψ = 1. Then (Φ n , ψ n ) n −→ (Φ, ψ) converges regularly.
The second example -eventhough straightforward to verify -is not only interesting for itself but will be of systematic importance for the proof of Theorem 1.2. See also Remark 5.1 and Example 1.9 for further discussion of this example. If the multifractal regularity property is satisfied we are able to prove the regularity of the free energy functions.
Theorem 1.7. If (Φ n , ψ n ) n −→ (Φ, ψ) converges regularly then t n converges pointwise to t on R.
To state our second main result on the regularity of the multifractal spectra let
F n α := π Φ n ω ∈ I N n : lim k→∞ S k (ψ n ) (ω) log (ϕ n ω |k ) ′ X = α , f n (α) := dim H (F n α ) ,
and with t n denoting the free energy function of (Φ n , ψ n ) let α n − := inf −t − n (x) : x ∈ Int(dom(t n )) and α n + := sup −t + n (x) : x ∈ Int(dom (t n )) . Theorem 1.8. Let Φ n = (ϕ n e ) e∈I n , Φ = (ϕ e ) e∈I be elements of CIFS (X) and ψ n , ψ be Hölder potentials such that (Φ n , ψ n ) n −→ (Φ, ψ) converges regularly. Then for each α ∈ (α − , α + ) we have
• lim n→∞ −t * n (−α) = f (α) = −t * (−α), • f n (α) = −t * n (−α)
, for all n sufficiently large. In particular, we have lim sup n α n − ≤ α − ≤ α + ≤ lim inf n α n + . If additionally sup dom(t) = +∞ then lim inf n α n − ≥ α − , whereas, if inf dom (t) = −∞ then lim sup n α n + ≤ α + .
Combining the above theorems with Example 1.6 we obtain the following application of our analysis.
α ∈ (α − , α + ) we have lim n −t * n (−α) = f (α) = −t * (−α) and f n (α) = −t * n (−α)
, for all n sufficiently large. For the boundary points of the spectrum we have the following.
(1) If dom (t) = R then lim n→∞ α n ± = α ± , (2) if sup dom (t) < +∞ then lim sup n α n − = −∞ and for all α < α − we have lim sup
n→∞ −t * n (−α) ≤ f (α) ,
(3) if inf dom (t) > −∞ then lim inf n α n + = +∞ and for all α > α + we have lim sup
n→∞ −t * n (−α) ≤ f (α) .
In Example 1.13 below we demonstrate how the lower bound on f stated in Corollary 1.9
(3) can be applied.
Note that by virtue of Proposition 6.4 we have on the one hand that the convergence Φ n −→ Φ in the λ -topology implies that (Φ n , 0) n −→ (Φ, 0) converges regularly. On the other hand we have t n (0) = dim H (Λ Φ n ). Hence, the following corollary is straightforward and may be viewed as a generalisation of the continuity results in [RSU09,RU05] for the Hausdorff dimension of the limit sets.
Corollary 1.10 (Continuity of Hausdorff dimension). Let
Φ n = (ϕ n e ) e∈I n , Φ = (ϕ e ) e∈I be elements of CIFS (X) such that (Φ n , 0) n −→ (Φ, 0) converges regularly. Then lim n→∞ dim H (Λ Φ n ) = dim H (Λ Φ ) .
Finite-to-infinite phase transition.
To complete the discussion of the Exhausting Principle we would like to emphasise that the boundary values of the approximating spectra in general do not converge to the corresponding value of the limiting system, i. e. we may have
f n α n ± → f (α ± ) .
(1.5) We refer to the property of an infinite system having a discontinuity of this kind in one of the boundary points as a finite-to-infinite phase transition in α + , resp. α − . Let us illustrate this property with the following concrete example. Example 1.11 (Gauss system). Let Φ := {ϕ e : x → 1/ (x + e) : e ∈ N} denote the Gauss system and the potential ψ is given by ψ (ω) := −2 logω 1 , for ω ∈ N N . In [JK10] we have shown that the multifractal spectrum is unimodal, defined on [0, 1], and in the boundary points of the spectrum we have f (0) = 0 and f (1) = 1/2. Nevertheless, for the exhausting systems Φ n := (ϕ e ) 1≤e≤n and ψ n := ψ {1,...,n} N we have for their corresponding multifractal spectra f n α n + = 0 for all n ∈ N giving rise to a finite-to-infinite phase transition (see Fig 1.1). A proof of this will be postponed to the end of Section 5.
1 1 ½ 0 0 0 α f ( ) α α α f ( ) α n f ( ) α m 1 ½ 0 β t ( ) β FIGURE 1.1.
Sketch illustrating the finite-to-infinite phase transition for the Gauss system. The dashed graphs are associated to the approximating spectra f n : 0, α n + → R + and f m : 0, α m + → R + , m > n, of finite sub-systems to the multifractal spectrum f of the infinite system. Example 1.12 (Lüroth system). In the following example the effective domain of the free energy function is not equal to R, which leads to an interesting boundary behaviour. For this let us consider the Lüroth system Φ := {ϕ n : x → x/(n(n + 1)) + 1/ (n + 1) : n ∈ N} (essentially a linearised Gauss system) and the potential functions ψ given by ψ (ω) := −ω 1 , ω ∈ N N . Then in virtue of our theorems the spectrum is given by the Legendre transform of t on (2/ log (6) , +∞) via f (α) = −t * (−α). Similarly as for the Gauss system in the example above, one can show that f (2/ log(6)) = f n (2/ log (6)) = f n α n
+ = 0, n ∈ N. Since we have Lebesgue almost everywhere that lim k→∞ ∑ k i=1 a i / ∑ k i=1 log (a i (a i + 1)) = π −1 Φ (x) 1 dλ /´log π −1 Φ (x) 1 π −1 Φ (x) 1 + 1 dλ = +∞ we find f (+∞) = 1.
Hence as above, we have a finite-to-infinite phase transition -this time at infinity. Example 1.13 (Generalised Lüroth system). In the following example the effective domain of the free energy function is again not equal to R and additionally we have a second order phase transition. Let us consider the generalised Lüroth system Φ := {ϕ n : x → 4x/ (n(n + 1)(n + 2))+ 2/ ((n + 1)(n + 2)) : n ∈ N} and the potential functions ψ given by ψ (ω) := −ω 1 , ω ∈ N N . Then in virtue of our theorems the spectrum is given
by the Legendre transform of t on (3/ log (15), α + ) via f (α) = −t * (−α), where α + := 2 ∑ n≥1 log((n(n+1)(n+2))/4) (n(n+1)(n+2)) −1 .
Using the Corollary 1.9 (Exhausting Principle II) (3) we gather some extra information on the spectrum. Since we have lim n −t * n (−t ′ n (0)) = 1 and α n + = n/ log(n(n + 1)(n + 2)/4) → ∞, we deduce that 1 is a lower bound for f (α) for all α ≥ α + . Similarly as for the Gauss system in the example above, one can show that f (3/ log (15)) = f n (3/ log (15)) = f n α n + = 0, n ∈ N. Hence, f (α) = −t * (−α) for all α ≥ 3/ log(15). (cf. Fig. 1.2).
Generalising further the latter two examples our analysis has successfully been applied in [KMS10] to determine the Lyapunov spectrum of α-Farey-Lüroth and α-Lüroth systems.
1 0 + α α α α 1 0 β β t ( ) f ( ) n n n /log( ( +1)( +2)/4) n 3 15 /log( ) f ( ) n +∞ FIGURE 1.2.
Sketch of the free energy function t and the multifractal spectrum f for the generalised Lüroth system. The dashed graph is associated to the approximating spectra f n : [3/ log(15), n/ log(n(n + 1)(n + 2)/4)] → R + of finite sub-system to the multifractal spectrum f of the infinite system. Example 1.14 (Irregular cIFS). For this example we suppose that Φ is an irregular infinite cIFS, that is the range of the pressure function p : t → P (tζ ) consists of the negative reals and infinity (see [MU03] for explicit examples), and let ψ be constantly equal to −1. We suppose that p (δ ) = η < 0, where δ is the critical value as well as the Hausdorff dimension of the limit set. Then the free energy function t is given by t (β ) = p −1 (β ) for β < η and constantly equal to δ for β ≥ η. The corresponding spectrum will have a linear part in (0, α − ) if −p + (δ ) = 1/α − < ∞ and hence for α − > 0, we observe a second order phase transition (see Fig. 1
.3). 0 f ( ) α α δ 0 β t ( ) β δ α + α - η FIGURE 1.3.
Sketch of the free energy function t and the multifractal spectrum f for an irregular system with constant negative potential. Note that in this situation we have a second order phase transition in α − and the spectrum f is linear on (0, α − ).
The paper is organised as follows. In Section 2 we recall the basic notions relevant for cIFS. In Section 3 we show the regularity of the free energy function proving Theorem 1.7. Section 4 provides us with the necessary prerequisites from convex analysis allowing us to deduce the multifractal regularity in Section 5. In particular, we prove Theorem 1.2 and 1.8, and the finite-to-infinite phase transition for the Gauss system. The final section is devoted to the connection between our notion of regularity and the λ -topology.
PRELIMINARIES
Let us recall the definition of a conformal iterated function system (see [MU03] for further details). Let X be a compact metric space. For an alphabet I ⊂ N with card (I) ≥ 2 we call Φ = (ϕ e ) e∈I an iterated function system (IFS), where ϕ e : X → X are injective contractions, e ∈ I, with Lipschitz constants globally bounded away from 1.
Let I * := n∈N I n denote the set of all finite subwords of I N . We will consider the left shift map σ :
I N → I N defined by σ (ω i ) := (ω i+1 ) i≥1 .
For ω ∈ I * we let |ω| denote the length of the word ω, i. e. the unique n ∈ N such that ω ∈ I n .
The space I N is equipped with the metric d given by d(ω, τ) := exp (−|ω ∧ τ|) , where ω ∧ τ ∈ I * ∪ I N denotes the longest common initial block of the infinite words ω and τ.
We now describe the limit set of the iterated function system Φ. For each ω ∈ I * , say ω ∈ I n , we consider the map coded by ω, ϕ ω := ϕ ω 1 • · · · • ϕ ω n : X → X.
For ω ∈ I N , the sets ϕ ω| n (X) n≥1 form a descending sequence of non-empty compact sets and therefore n≥1 ϕ ω| n (X) = ∅. Since for every n ∈ N, diam ϕ ω| n (X) ≤ s n Φ diam (X), we conclude that the intersection ϕ ω| n (X) ∈ X is a singleton and we denote its only element by π Φ (ω). In this way we have defined the coding map π = π Φ : I N → X. The set Λ = Λ Φ = π I N will be called the limit set of Φ.
Definition 2.1. We call an iterated function system conformal (cIFS) if the following conditions are satisfied.
(a) The phase space X is a compact connected subset of a Euclidean space R D , D ≥ 1, such that X is equal to the closure of its interior, i. e. X = Int(X). (b) (Open set condition (OSC)) For all a, b ∈ I, a = b, ϕ a (Int(X)) ∩ ϕ b (Int(X)) = ∅.
(c) There exists an open connected set W ⊃ X such that for every e ∈ I the map ϕ e extends to a C 1 conformal diffeomorphism of W into W . (d) (Cone property) There exist γ, l > 0, γ < π/2, such that for every x ∈ X ⊂ R D there exists an open cone Con(x, γ, l) ⊂ Int(X) with vertex x, central angle of measure γ, and altitude l.
(e) There are two constants L = L Φ ≥ 1 and α = α Φ > 0 such that
|ϕ ′ e (y)| − |ϕ ′ e (x)| ≤ L Φ (ϕ ′ e ) −1 X y − x α
for every e ∈ I and every pair of points x, y ∈ X.
For the following let
s Φ := sup e∈I (ϕ e ) ′ X < 1. (2.1)
For a fixed phase space X satisfying (a) the set of conformal iterated function systems will be denoted CIFS (X) := Φ = (ϕ e : X −→ X) e∈I cIFS, I ⊂ N .
The following fact was proved in [MU03].
Proposition 2.2. For D ≥ 2, any family Φ = (ϕ e ) e∈I satisfying condition (a) and (c) also satisfies condition (e) with α = 1.
In [MU03] we also find the following straightforward consequence of (e).
Lemma 2.3.
If Φ = (ϕ e ) e∈I is a cIFS, then for all ω ∈ I * and all x, y ∈ W , we have
log |ϕ ′ ω (y)| − log|ϕ ′ ω (x)| ≤ L 1 − s α y − x α .
Another consequence of (e) is (f) (Bounded distortion property). There exists K Φ ≥ 1 such that for all ω ∈ I * and all x,
y ∈ X |ϕ ′ ω (y)| ≤ K Φ |ϕ ′ ω (x)|.
In [MU03, Lemma 2.3.1] it has been shown that for a Hölder continuous function g : J N → R, J ⊂ I, we have for all ω ∈ J * and all x, y ∈ [ω] that exp S |ω| g (x) ≤ K g exp S |ω| g (y) .
In here, the constant K g ≥ 1 only depends on the Hölder norm and the Hölder exponent of g, as well as the metric on J N . With
Z n (g) := ∑ ω∈J n exp sup τ∈[ω]∩J N (S n g (τ))
we will denote the n-th partition function of g.
Definition 2.4. The topological pressure P( f ) of a continuous function f : I N → R is defined by the following limit, which always exists (possibly equal to +∞),
P( f ) := lim n→∞ 1 n log Z n ( f ) = inf n 1 n log Z n ( f ) .
At the end of this section we would like to comment on the topology of pointwise convergence. ρ is well defined, since ϕ 1 i −ϕ 2 i X + (ϕ 1 i ) ′ −(ϕ 2 i ) ′ X is bounded by diam (X)+2. Using the fact that A△C ⊂ A△B ∪ B△C for arbitrary sets A, B,C, we readily observe that ρ as given in (1.4) actually defines a metric on CIFS (X). This metric induces the topology of pointwise convergence on CIFS (X). Let Φ n = (ϕ n i : X −→ X) i∈I n , n ∈ N, and Φ = (ϕ i : X −→ X) i∈I be elements of CIFS (X) with Φ n → Φ pointwise. Then for every k ∈ N we find an integer N k such that and all n ≥ N k we have I n △I ⊂ {k + 1, k + 2, . . .} . Similarly as in [RU05, Lemma 5.1] it follows that pointwise convergence in CIFS (X) is equivalent to the following condition. Proof. Using the above Condition 2.5 we find for ω ∈ I * and n sufficiently large that
max log (ϕ n ω ) ′ − log (ϕ ω ) ′ X , S |ω| ψ n − S |ω| ψ [ω]∩I N n ≤ 1.
Then we have with K Φ and K ψ denoting the bounded distortion constants as defined above
S |ω| ζ n (η) − S |ω| ζ (τ) = log (ϕ n ω ) ′ (π Φ n (σ n (η))) − log (ϕ ω ) ′ (π Φ (σ n (τ))) ≤ log (ϕ n ω ) ′ (π Φ n (σ n (η))) − log (ϕ ω ) ′ (π Φ n (σ n (η))) + log (ϕ ω ) ′ (π Φ n (σ n (η))) − log (ϕ ω ) ′ (π Φ (σ n (τ))) ≤ 1 + logK Φ as well as S |ω| ψ n (η) − S |ω| ψ (τ) ≤ S |ω| ψ n (η) − S |ω| ψ (η) + S |ω| ψ (η) − S |ω| ψ (τ) ≤ 1 + logK ψ .
Letting M := 3 · max K Φ , K ψ the lemma follows.
Remark 2.7. Note that we may replace condition (B) in Definition 1.3 by the slightly weaker conditions on ψ n and ψ stated in the above Lemma combined with the condition that ψ n converges uniformly to ψ on compact σ -invariant subsets of I N .
REGULARITY OF THE FREE ENERGY FUNCTION
In this section we give a proof of Theorem 1.7. Let ζ denote the geometric potential function associated with Φ as defined in the introduction. For Φ ∈ CIFS (X) and a Hölder continuous potential ψ : I N −→ R let t denote the free energy function of (Φ, ψ) as introduced in Definition 1.1, i. e. t (β ) := inf {t : P (tζ + β ψ) ≤ 0}. Clearly, if there exists a zero of t −→ P(tζ + β ψ) then t (β ) is the unique zero of this function (which in particular is the case for a finite alphabet I). Also, t (β ) = +∞ if and only if {t : P (tζ + β ψ) ≤ 0} = ∅.
Lemma 3.1. The free energy t of (Φ, ψ) is a proper (not necessarily closed) convex function on R.
Proof. Fix β 1 , β 2 ∈ R, λ ∈ (0, 1) and ε > 0. Using the convexity of the topological pressure we have
P ((λ t (β 1 ) + (1 − λ )t (β 2 ) + ε) ζ + (λ β 1 + (1 − λ )β 2 ) ψ) = P (λ ((t (β 1 ) + ε) ζ + β 1 ψ) + (1 − λ )((t (β 2 ) + ε) ζ + β 2 ψ)) ≤ λ P ((t (β 1 ) + ε)ζ + β 1 ψ) + (1 − λ )P ((t (β 2 ) + ε)ζ + β 2 ψ) ≤ 0.
Hence, by definition of t, this implies t (λ β 1 + (1 − λ )β 2 ) ≤ λ t (β 1 ) + (1 − λ )t (β 2 ) + ε. Since ε > 0 was arbitrary this shows the convexity.
To see that t is a proper convex function observe that −∞ < P (β ψ) and hence, for t < 0, we have
1 n log ∑ ω∈I n exp sup τ∈[ω] (S n tζ (τ) + β ψ) ≥ t log (s Φ ) + P (β ψ) → ∞ for t → −∞. Consequently, t (β ) > −∞ for all β ∈ R. Since also t (0) = dim H (Λ Φ ) < ∞ we have that t is proper.
Lemma 3.2. Let (Φ n , ψ n ) n −→ (Φ, ψ) converge regularly. Then for all t, β ∈ R and n tending to infinity we have P(tζ n + β ψ n ) −→ P(tζ + β ψ).
Proof. Fix t, β ∈ R with P(tζ + β ψ) < ∞ and ε > 0. With k ∈ N and C > 0 chosen according to Definition 1.4 choose m ∈ N large enough such that (mk) −1 log Z mk (tζ + β ψ) ≤ P (tζ + β ψ) + ε/2 and (mk) −1 log (K + 1) < ε/2, where K := M |β |+|t| and M is the constant defined in the proof of Lemma 2.6. We prove that for all n ∈ N sufficiently large we have
(mk) −1 log Z mk (tζ n + β ψ n ) ≤ (mk) −1 log Z mk (tζ + β ψ) + ε/2. (3.1)
This would imply
P(tζ n + β ψ n ) ≤ (mk) −1 log Z mk (tζ n + β ψ n ) ≤ (mk) −1 log Z mk (tζ + β ψ) + ε/2 ≤ P (tζ + β ψ) + ε
for sufficiently large n ∈ N. To prove (3.1) we first choose a finite set F ⊂ I such that
∑ ω∈I mk \F mk exp sup ρ∈[ω]∩I N (S mk (tζ + β ψ)(ρ)) < 1 (CK) m Z mk (tζ + β ψ) .(S mk (tζ n + β ψ n ) (ρ)) ≤ ∑ ω∈I k n \F k exp sup ρ∈I N n ∩[ω] (S k (tζ n + β ψ n ) (ρ)) m ≤ C m ∑ ω∈I k n \F k exp sup ρ∈I N ∩[ω] (S k (tζ + β ψ)(ρ)) m ≤ (CK) m ∑ ω∈I mk n \F mk exp sup ρ∈I N ∩[ω]
(S mk (tζ + β ψ)(ρ)) < Z mk (tζ + β ψ)
Hence, on the one hand, we have
Z mk (tζ n + β ψ n ) = ∑ ω∈I mk n ∩F mk exp sup ρ∈I N n ∩[ω]
(S mk (tζ n + β ψ n ) (ρ))
+ ∑ ω∈I mk n \F mk exp sup ρ∈I N n ∩[ω] (S mk (tζ n + β ψ n ) (ρ)) ≤ ∑ ω∈I mk n ∩F mk exp sup ρ∈I N n ∩[ω]
(S mk (tζ n + β ψ n ) (ρ)) + Z mk (tζ + β ψ).
To find an upper bound also for the finite sum in the latter inequality we note that by Lemma 2.6 we have for every t ∈ R and ω ∈ I mk n ∩ F mk and for sufficiently large n that exp sup
τ∈I N n ∩[ω] (S mk (tζ n + β ψ n ) (τ)) ≤ K exp sup ρ∈I N ∩[ω]
(S mk (tζ + β ψ)(ρ)) .
Since I mk n ∩ F mk is finite we have on the other hand for n sufficiently large that
∑ ω∈I mk n ∩F mk e sup ρ∈I N n ∩[ω] (S mk (tζ n +β ψ n )(ρ)) ≤ K ∑ ω∈I mk n ∩F mk e sup ρ∈I N ∩[ω] (S mk (tζ +β ψ)(ρ)) .
Combining both estimates we find for n sufficiently large Z mk (tζ n + β ψ n ) ≤ (K + 1)Z mk (tζ + β ψ).
Taking logarithm and dividing by mk proves (3.1).
To prove the reverse inequality lim inf n P(tζ n + β ψ n ) ≥ P(tζ + β ψ) let us fix ε > 0. Using [MU03, Theorem 2.15] we can choose a finite set F ⊂ I such that
P(tζ F N + β ψ F N ) ≥ P(tζ + β ψ) − ε.ζ n F N − ζ F N F N → 0 as well as ψ n F N − ψ F N F N → 0. Since f → P ( f ) is Lipschitz-continuous with respect to the · F N -norm (cf. [Wal82, Theorem 9.7]) we conclude lim inf n P(tζ n + β ψ n ) ≥ lim inf n P(tζ n F N + β ψ n F N ) = P(tζ F N + β ψ F N ) ≥ P(tζ + β ψ) − ε.
Proof of Theorem 1.7. Fix β ∈ R. To verify lim sup n t n (β ) ≤ t (β ) we may assume t (β ) < ∞. Since the map p β : t → P(tζ + β ψ) is strictly decreasing on dom p β we have that P ((t (β ) + δ ) ζ + β ψ) < 0 for every δ > 0. As a consequence of Lemma 3.2 we have P ((t (β ) + δ ) ζ n + β ψ n ) < 0 for all n sufficiently large. This implies t n (β ) ≤ t (β ) + δ for all δ > 0 and therefore lim sup n t n (β ) < t (β ).
To verify lim inf n t n (β ) ≥ t (β ) we first assume that t (β ) < ∞. By definition of t we have P ((t (β ) − δ ) ζ + β ψ) > 0 for every δ > 0. Then again by Lemma 3.2 we also have P ((t (β ) − δ ) ζ n + β ψ n ) > 0 for all n sufficiently large, which in turn implies t n (β ) ≥ t (β ) − δ for all n large enough. Finally, let t (β ) = ∞, i. e. P (tζ + β ψ) = ∞ for all t. By Lemma 3.2 we have for any t ∈ R fixed P (tζ n + β ψ n ) > 0 for n large enough and hence t n (β ) ≥ t for n large enough. Since t ∈ R was arbitrary it follows that t n (β ) tends to infinity as n increases.
CONVERGENCE AND CONJUGACY OF CONVEX FUNCTIONS
In this section we collect the necessary basic facts from convex analysis needed for the multifractal analysis in Section 5. We closely follow [SW77], and all details can be found either therein or in [Roc70].
The following proposition is a direct consequence of [SW77, Corollaries 2C and 3B] combined with the fact that Legendre conjugation is continuous with respect to the convergence of epigraphs in the classical sense as defined e. g. by Kuratowski in [Kur66].
Proposition 4.1. Let g n , g, n ∈ N, be closed convex functions on R such that Int(dom (g)) = ∅ and g n −→ g pointwise. Then pointwise on Int(dom(g * )), we have g * n −→ g * .
The following corollary allows us to apply Proposition 4.1 also in the case when the functions g n , g are not closed.
Corollary 4.2. Let g n , g, n ∈ N, be convex functions on R and α ∈ R such that there exist x 1 , x 2 ∈ Int(dom(g)) with x 1 < x 2 and g + (x 1 ) < α < g − (x 2 ). Furthermore, assume that there exists an open neighbourhood U ⊂ dom (g) containing x 1 , x 2 such that g n U −→ g U pointwise. Then we have g * n (α) −→ g * (α).
Proof. Without loss of generality we have g n U < ∞ for all n ∈ N. Let ½ A denote the indicator function on the set A and let g n , g denote the closed convex functions given by g n := g n ½ U + ∞½ R\U and g := g½ U + ∞½ R\U . Notice that these closed convex functions agree on U with the original functions. By Proposition 4.1 we conclude that g * n −→ g * pointwise on Int (dom g * ). Clearly by our assumptions, α ∈ Int(dom g * ) and α belongs to the subdifferential ∂ g (x) := {a ∈ R : ∀x ′ ∈ R g(x ′ ) − g(x) ≥ a (x ′ − x)} for some x ∈ U, and hence g * (α) = αx − g (x) = g * (α) by [Roc70,Theorem 23.5]. It remains to show that g * n (α) = g * n (α) for n sufficiently large. Since by [Roc70,Theorem 24.5] the subdifferentials converge, the assumption g + (x 1 ) < α < g − (x 2 ) implies that α ∈ ∂ g n (y n ) for some y n ∈ U and n large. Then again by [Roc70,Theorem 23.5] we have g * n (α) = αy n − g n (y n ) = g * n (α).
REGULARITY OF THE MULTIFRACTAL SPECTRUM
We proceed by proving the Theorems 1.2 and 1.8. Recall that throughout we use the generalised version of the free energy function t as stated in Definition 1.1.
Proof of Theorem 1.2. Using the definition of topological pressure and a standard covering argument (just cover F α with cylinder sets) we obtain f (α) ≤ max {−t * (−α) , 0} for every α ∈ R.
We will use the Exhausting Principle to prove the reverse inequality. Let Φ n = (ϕ e ) e∈I n with I n := I ∩ {1, . . . n} and ψ n := ψ I N n , n ∈ N. Clearly, (Φ n , ψ n ) n −→ (Φ, ψ) converges regularly (Example 1.6) and hence by Theorem 1.7, we conclude that t n −→ t pointwise on R. Note that for α ∈ (α − , α + ) we find x 1 , x 2 ∈ Int (dom (t)) with x 1 > x 2 and −t − (x 1 ) < α < −t + (x 2 ). Hence by Corollary 4.2, we conclude
lim n→∞ −t * n (−α) = −t * (−α) .
Since the functions t n are finite and differentiable on R we conclude by [Roc70,Theorem 24.5] that α ∈ −t ′ n (R) for all n large enough. Recall that in the finite alphabet case it is well-known that f n (α) = −t * n (−α). By construction we have F n α ⊂ F α and hence
−t * (−α) ≥ f (α) ≥ f n (α) = −t * n (−α) −→ −t * (−α) .
Proof of Theorem 1.8. By Theorem 1.7 the free energy functions t n converge pointwise to t on R. For α ∈ (α − , α + ) we have by Corollary 4.2 that lim n −t * n (−α) = −t * (−α). Furthermore by Theorem 1.2, we have −t * (−α) = f (α). We also have −t * n (−α) = f n (α) for large n by Theorem 1.2, since [Roc70, Theorem 24.5] implies α ∈ α n − , α n + for large n. This proves the first part of the theorem.
We now consider the case sup dom (t) = +∞ (the second case is proved along the same lines). Let us assume on the contrary that there exists ε > 0 with α n − < α − − ε for infinitely many n. For fixed K > 0 we find x ∈ dom (t) with x > (K + 1)/ε and such that t ′ (x) exists. By Theorem 1.7 we find n ∈ N such that |t n (x) − t (x)| < 1 and α n − < α − − ε. Furthermore, we can choose α ∈ α n − , α n + satisfying α < α − − ε. By Theorem 1.2 we have f n (α) = −t * n (−α). Since α + t ′ (x) ≤ α − α − < −ε we have by [Roc70,Theorem 23.5]
f n (α) + t * t ′ (x) = −t * n (−α) + t * t ′ (x) = inf β {t n (β ) + β α} − t (x) + xt ′ (x) ≤ (t n (x) + xα) − t (x) + xt ′ (x) ≤ 1 + x α + t ′ (x) < 1 − εx < −K.
Since t * (t ′ (x)) ≥ −t (0) = − dim H (Λ) and K can be chosen arbitrary large we get a contradiction to f n (α) ≥ 0.
Remark 5.1. Assume in the situation of Example 1.6 that (α − , α + ) is a bounded interval. If t * n (α ± ) −→ t * (α ± ) then f n (α ± ) −→ f (α ± ) and we have −t * (−α ± ) = f (α ± ). To see this notice that a → −t * (−a) is bounded on (α − , α + ) since it coincides with the Hausdorff dimension of certain sets. Furthermore, by [Roc70,Theorem 12.2] we have that t * is closed and hence −t * (−α ± ) < ∞. By our assumption we have −t * n (−α ± ) < ∞ for n large, hence α ± ∈ −t ′ n (R). Then the claim follows by observing that
−t * n (−α ± ) = f n (α ± ) ≤ f (α ± ) ≤ −t * (−α ± ) .
Finally, we will sketch the proof of (1.5) for the Gauss system as announced in the introduction. Let n := (n, n, n, . . .), n ≥ 2, and suppose that ω ∈ {1, . . . , n} k , k ∈ N, differs from n k in at least ℓ positions. Let q k (ω) denote the denominator of the k's approximant of the continued fraction expansion [ω 1 , . . . , ω k ] ∈ [0, 1]. Then, by the recursive definition of q k , we have
q k (ω) q k (n) > n − 1 n ℓ .
(5.1)
From this it follows that
α n + = ψ (n) ζ (n) = lim k→∞ k log (n) log q k (n) = − log (n) log −n/2 + n 2 /4 + 1 < 1 (cf. [JK10,
Fact 3]). Now we will argue similar as in [KS07]. To prove that f n α n + = 0 it is sufficient to verify that
µ ∈ M {1, . . . , n} N , σ :ˆψ dµ/ˆζ dµ = α n + = δ n ,
where M {1, . . . , n} N , σ denotes the set of shift invariant measures and δ x the Dirac measure centred on x. For the detailed argument see [KS07]. We prove this fact by way of contradiction. Assume there exists µ ∈ M {1, . . . , n} N , σ with µ = δ n . By convexity of the set of measures under consideration we may assume that µ is ergodic. Then by our assumption there exists ℓ < n with µ ([ℓ]) = η > 0. Then for all µ-typical points ω we have lim k S k ψ (ω) /S k ζ (ω) = lim k ∑ k i=1 log ω i / log q k (ω) = α n + and S k ½ [ℓ] (ω) ≥ ηk/2 for all k ∈ N sufficiently large. Hence, using (5.1), we obtain
k log (n) log q k (n) − ∑ k log ω i log q k (ω) = = k log (n) log q k (ω) − logq k (n) ∑ k log ω i log q k (ω) log q k (n) = ∑ k i=1 log (n) − log(ω i ) log q k (ω) + (log q k (ω) − logq k (n)) ∑ k log ω i log q k (ω) log q k (n) ≥ k (η/2) (log (n) − log(n − 1)) log q k (ω) + (log q k (ω) − logq k (n)) ∑ k i=1 log ω i log q k (ω) log q k (n) ≥ kη (log (n) − log (n − 1)) log q k (ω) − ∑ k i=1 log ω i 2 log q k (ω) log q k (n) →
ηα n + 2 log n (log (n) − log(n − 1)) 1 − α n + > 0.
Since lim k→∞ k log (n) / log q k (n) = α n + we obtain a contradiction.
THE EXTENDED λ -TOPOLOGY
In this section we compare the notion of regular convergence with the λ -topology introduced by Roy and Urbański. In particular, as a consequence of Proposition 6.4 we will verify Example 1.5.
For ease of notation we will always assume I = N. Let us first recall the definition of the λ -topology from [RU05] and then give a generalisation to adapt this concept to our purposes.
For Φ n = (ϕ n e ) e∈I , Φ = (ϕ e ) e∈I elements of CIFS (X) sharing the same alphabet I we say that Φ n converges to Φ in the λ -topology, if Φ n → Φ in the ρ-metric and there exists R > 1 such that for all sufficiently large n and all e ∈ I we have
R −1 ≤ (ϕ n e ) ′ X ϕ ′ e X ≤ R. (6.1)
We shall generalise this to the case where we have I n ⊂ I = N. We say that Φ n = (ϕ n e ) e∈I n converges to Φ = (ϕ e ) e∈I in the extended λ -topology of CIFS (X), if they converge in the ρ-metric and there exists D > 1 such that for all n sufficiently large and all e ∈ I n the assumption (6.1) holds.
Let us begin with the following basic lemma.
Lemma 6.1. Let Φ n = (ϕ n e ) e∈I n , Φ = (ϕ e ) e∈N be elements of CIFS (X) with Φ n → Φ converging in the extended λ -topology. Then with s Φ defined in (2.1), we have lim n s Φ n = s Φ < 1.
Proof.
By the open set condition (OSC) there exists e ∈ I with ϕ ′ e X = sup e ϕ ′ e X = s Φ as well as for every n ∈ N there exists e n ∈ I n satisfying (ϕ n e n ) ′ X = sup e (ϕ n e ) ′ X = s Φ n . Since lim n (ϕ n e ) ′ X = ϕ ′ e X we have lim inf n s Φ n ≥ s Φ . Next we conclude that {e n : n ∈ N} is contained in a finite set F ⊂ N. This follows by way of contradiction. Assume the set is infinite. Then there exists a subsequence n k such that on the one hand (ϕ ′ e n k ) X → 0 and on the other hand (ϕ n k e n k ) ′ X → lim inf n s Φ n ≥ s Φ > 0. This would contradict property (6.1) defining the extended λ -topology. Now by the definition of the ρ-metric we have for all ℓ ∈ F that lim n (ϕ n ℓ ) ′ X = (ϕ ℓ ) ′ X ≤ s Φ . This gives lim sup n s Φ n ≤ s Φ .
For the following let ψ n : I N n −→ R and ψ : I N −→ R be Hölder continuous functions, satisfying condition (B) in Definition 1.3. Assumption 6.2. Additionally, we assume that there exist k ∈ N and M ∈ N, such that for all n ∈ N and for all ω ∈ I k n , τ ∈ I N n ∩ [ω] and η ∈ I N ∩ [ω] we have M −1 ≤ exp (S k (ψ n ) (τ)) exp (S k (ψ) (η)) ≤ M.
Remark 6.3. Assumption 6.2 is for instance satisfied, if
(1) sup n ψ n I N n < ∞ and ψ I N < ∞, or (2) sup n ψ n − ψ I N n < ∞.
For the following proposition recall that K Φ denotes the bounded distortion constant for Φ as stated in condition (f) of the definition of a cIFS.
Proposition 6.4. Let Φ n = (ϕ n e ) e∈I n , Φ = (ϕ e ) e∈N be elements of CIFS (X) with Φ n → Φ converging in the extended λ -topology. Let ψ n : I N n −→ R and ψ : I N −→ R be Hölder continuous functions satisfying condition (B) in Definition 1.3 as well as Assumption 6.2. Then (Φ n , ψ n ) n → (Φ, ψ) converges regularly, if one of the following conditions is satisfied:
(1) sup n K Φ n < ∞.
(2) ψ I N < ∞.
(3) inf n α Φ n > 0 and sup n L Φ n < ∞.
(4) D ≥ 2 and the maps Φ n e extend to conformal diffeomorphisms on a common neighbourhood W ⊃ X into W for all e ∈ I n and n ∈ N.
Proof. Clearly (Φ n , ψ n ) n → (Φ, ψ) converges pointwise. Hence, we are left to verify the condition in Definition 1.4 under the assumption (1) as well as under the assumption (2), and then show that both (3) and (4) imply (1).
ad (1):
For t ≥ 0 we argue as follows. Let k ∈ N, ω ∈ I k n and n sufficiently large, such that (6.1) and Assumption 6.2 hold. Using this and the bounded distortion property of Φ from Lemma 2.3 with bounded distortion constant K = K Φ we obtain exp sup
τ∈I N n ∩[ω] (S k (ζ n ) (τ)) ≤ (ϕ n ω ) ′ X ≤ k ∏ i=1 (ϕ n ω i ) ′ X ≤ R k k ∏ i=1 (ϕ ω i ) ′ X ≤ R k K k (ϕ ω ) ′ X ≤ K k+1 R k exp sup ρ∈I N ∩[ω]
(S k (ζ ) (ρ)) .
Combining this with Assumption 6.2 we have with C := K t(k+1) R tk M |β | and t ≥ 0 exp sup
τ∈I N n ∩[ω]
(S k (tζ n + β ψ n ) (τ)) ≤ exp sup
τ∈I N n ∩[ω]
(S k (tζ n ) (τ)) exp sup
τ∈I N n ∩[ω] (S k (β ψ n ) (τ)) ≤ C exp sup ρ∈I N ∩[ω] (S k (tζ ) (ρ)) exp inf τ∈I N ∩[ω] (S k (β ψ) (τ)) ≤ C exp sup ρ∈I N ∩[ω]
(S k (tζ + β ψ)(ρ)) .
For t < 0 we have by the assumption (1) with K := sup K Φ n < ∞ that exp sup
τ∈I N n ∩[ω] (S k (ζ n ) (τ)) ≥ K −1 (ϕ n ω ) ′ X ≥ K −k−1 k ∏ i=1 (ϕ n ω i ) ′ X ≥ R −k K −k−1 k ∏ i=1 (ϕ ω i ) ′ X ≥ R −k K −k−1 (ϕ ω ) ′ X ≥ R −k K −k−1 exp sup ρ∈I N ∩[ω]
(S k (ζ ) (ρ)) .
As above we have with C := R −tk K −t(k+1) M |β | and t < 0 exp sup τ∈I N n ∩[ω]
(S k (tζ n + β ψ n ) (τ)) ≤ exp sup (S k (tζ + β ψ)(ρ)) .
ad (2):
Since the potential ψ is bounded and the topological entropy infinite we have P (β ψ) = ∞. Hence, to verify the condition in Definition 1.4 we only have to concider the case t > 0, since for t ≤ 0 we have P (tζ + β ψ) ≥ P (β ψ) = ∞. But this case has been treated in (1) without any additional assumption on Φ.
(3) =⇒ =⇒ =⇒ (1): Since by Lemma 6.1 the contraction ratios s Φ n of Φ n defined in (2.1) converge to s Φ < 1 we conclude by Lemma 2.3 that sup n K Φ n < ∞.
Definition 1. 1 .
1Let Φ ∈ CIFS (X) and ψ : I N −→ R be a potential function. Then the free energy function t : R −→ R ∪ {∞} for the pair (Φ, ψ) is given by t (β ) := inf {t ∈ R : P (tζ + β ψ) ≤ 0} .(1.3)
Example 1. 6 (
6Exhausting Principle I). Let Φ = (ϕ e ) e∈N be an element of CIFS (X) and ψ : I N −→ R be Hölder continuous. Define I n := I ∩ {1, . . . n}, n ∈ N and let Φ n = (ϕ e ) e∈I n and ψ n := ψ I N n . Then (Φ n , ψ n ) n −→ (Φ, ψ) converges regularly.
Corollary 1. 9 (
9Exhausting Principle II). Let Φ = (ϕ e ) e∈N be an element of CIFS (X), ψ : I N −→ R be Hölder continuous, and Φ n = (ϕ e ) e∈I n and ψ n := ψ I N n with I n := I ∩{1, . . . n}, n ∈ N. Then for each
Condition 2. 5 .
5We have (Φ n , ψ n ) → (Φ, ψ) pointwise if and only if for everyω ∈ I * lim n→∞ ϕ n ω − ϕ ω X + (ϕ n ω ) ′ − ϕ ′ ω X = 0 and lim n→∞ S |ω| ψ n − S |ω| ψ [ω]∩I N n = 0Using this condition we are able to prove a technical property which will be crucial in the proof of our main theorems.Lemma 2.6. Assume that (Φ n , ψ n ) → (Φ, ψ) converges pointwise. Then there exists M > 0 such that for every ω ∈ I * fixed and all sufficiently large n ∈ N (depending on ω) we have for all η ∈ [ω] ∩ I N n and τ ∈ [ω] ∩ I N e S |ω| ψ n (η) e S |ω| ψ(τ) , e S |ω| ζ n (η) e S |ω| ζ (τ) ∈ M −1 , M .
(
S k (tζ n ) (τ)) exp sup τ∈I N n ∩[ω] (S k (β ψ n ) (τ)) ≤ C exp sup ρ∈I N ∩[ω] (S k (tζ ) (ρ)) exp inf τ∈I N ∩[ω] (S k (β ψ) (τ)) ≤ C exp sup ρ∈I N ∩[ω]
( 4 )
4=⇒ =⇒ =⇒ (1): This implication is an immediate consequence of [MU03, Theorems 4.1.2 and 4.1.3]. See also Proof of Claim in the proof of Theorem 5.20 in [RSU09] for a similar argument.
Then by Definition 1.4 and the choice of F we have∑
ω∈I mk
n \F mk
exp sup
ρ∈I N
n ∩[ω]
Acknowledgement.We would like to thank Mario Roy, Hiroki Sumi and Mariusz Urbański for their helpful comments on an earlier draft of this paper.
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"\nSTABILITY OF UNIQUENESS AND COEXISTENCE OF EQUILIBRIUM STATES OF THE ISING MODEL UNDER LONG RANGE PERTURBATIONS SHUNSUKE USUKI\nKyoto University\nKitashirakawa Oiwake-cho, Sakyo-ku606-8502KyotoJapan\n"
] | [
"STABILITY OF UNIQUENESS AND COEXISTENCE OF EQUILIBRIUM STATES OF THE ISING MODEL UNDER LONG RANGE PERTURBATIONS SHUNSUKE USUKI\nKyoto University\nKitashirakawa Oiwake-cho, Sakyo-ku606-8502KyotoJapan"
] | [] | In this paper, we study perturbations of the d-dimensional Ising model for d ≥ 2, including long range ones to which the Pirogov-Sinai theory is not applicable. We show that the uniqueness of the equilibrium state of the Ising model at high temperature and the coexistence of equilibrium states at low temperature are preserved by spin-flip symmetric perturbations. | null | [
"https://arxiv.org/pdf/2110.15540v2.pdf"
] | 249,191,786 | 2110.15540 | 6d15a3ee5f17292106f3c8d765583d8b942c1286 |
May 2022
STABILITY OF UNIQUENESS AND COEXISTENCE OF EQUILIBRIUM STATES OF THE ISING MODEL UNDER LONG RANGE PERTURBATIONS SHUNSUKE USUKI
Kyoto University
Kitashirakawa Oiwake-cho, Sakyo-ku606-8502KyotoJapan
May 2022arXiv:2110.15540v2 [math-ph] 302020 Mathematics Subject Classification 82B2082B2682B05
In this paper, we study perturbations of the d-dimensional Ising model for d ≥ 2, including long range ones to which the Pirogov-Sinai theory is not applicable. We show that the uniqueness of the equilibrium state of the Ising model at high temperature and the coexistence of equilibrium states at low temperature are preserved by spin-flip symmetric perturbations.
Introduction
In this paper, we deal with Z d -lattice systems for d ≥ 2. We begin with the definition of them. Let F be a finite set and Ω = F Z d . Then Ω is a compact metrizable space with respect to the product topology. For Λ ⊂ Z d , we write Ω Λ = F Λ and ω = {ω(x) |x ∈ Λ } as an element of Ω Λ . For each a ∈ Z d , the translation by a is canonically defined, that is, for each Λ ⊂ Z d , we define τ a : Ω Λ → Ω Λ−a by (τ a ω)(x) = ω(x + a), x ∈ Λ − a for ω ∈ Ω Λ . When Λ = Z d , these translations define the Z d -action on Ω. In this paper, we write Λ ⋐ Z d when Λ ⊂ Z d and Λ is a finite set. We say Φ is an interaction if Φ is a family of functions Φ Λ : Ω Λ → R for each Λ ⋐ Z d and write Φ = {Φ Λ } Λ⋐Z d . We always assume in this paper that an interaction Φ is translation invariant, that is, for any a ∈ Z d and Λ ⋐ Z d , Φ Λ (ω) = Φ Λ−a (τ a ω). We also assume that Φ is absolutely summable, that is, the norm
Φ = 0∈Λ⋐Z d sup ξ∈Ω Λ |Φ Λ (ξ)|
is finite. Under such an interaction Φ, for each Λ ⋐ Z d and ω ∈ Ω, the Hamiltonian is defined by
H Φ,Λ (ω) = ∆∈F Λ Φ ∆ (ω),
where F Λ = ∆ ⋐ Z d |∆ ∩ Λ = ∅ and Φ ∆ (ω) = Φ ∆ ({ω(x)|x ∈ ∆}). By the translation invariance and the absolute summability of Φ, it is easily seen that the infinite sum converges.
Then Ω is interpreted as the configuration space and H Φ,Λ (Λ ⋐ Z d ) define the energy for each configuration ω ∈ Ω. Then we regard Ω with Φ as a Z d -lattice system. We say that an interaction Φ is finite range if r(Φ) = inf R > 0 Φ Λ = 0 for any Λ ⋐ Z d with diam(Λ) > R < ∞.
In the above, for Λ ⊂ Z d ,
diam(Λ) = sup { x − y ∞ | x, y ∈ Λ} ,
where x ∞ = max {|x 1 |, . . . , |x d |} for x = (x 1 , . . . , x d ) ∈ Z d . We emphasize that, in this paper, we deal with interactions which are long range, that is, r(Φ) = ∞ in general.
The Ising model is a famous and important example of a Z d -lattice system, which we mainly deal with in this paper. This is a simple mathematical model of ferromagnetism.
1 = d i=1 |x i | for x = (x 1 , . . . , x d ) ∈ Z d .
Here, β is the parameter representing the inverse of the temperature of the system.
For a Z d -lattice system, Borel probability measures on Ω which represent statistical states of the system stable under the interaction are determined and called Gibbs states. Translation invariant probability measures representing stable statistical states are called equilibrium states. We give the rigorous definition in Section 2.1. It is an important problem to determine whether there are more than one equilibrium state for the system or not. For the case of d = 1, it is known that there exists the only one equilibrium state for any interaction under a reasonable condition. For example, finite range interactions satisfy this condition (see [17,Chapter 5]). On the other hand, the system might have more than one equilibrium state when d ≥ 2. Concerning this problem, the Ising model defined in Definition 1.1 shows the remarkable property below (for example, see [10,Chapter 3] (1) If β < β c , then Φ β has a unique Gibbs state and, in particular, a unique equilibrium state.
(2) If β > β c , then Φ β has more than one equilibrium state.
This phenomenon is called phase transition on the Ising model and β c as above the critical inverse temperature. (i) For the Ising models with β > β c , the concrete structure of the set of equilibrium states is known (see [1] or [13] for d = 2 and [4] for d ≥ 3). We will mention this structure in detail later (Section 4). (ii) It seems difficult to study the Ising model at the critical temperature. However, it is shown in [2] that, at β = β c , Φ βc has a unique Gibbs state.
The central question of this paper is to determine for the Ising model whether the uniqueness or coexistence of equilibrium states is preserved if we perturb the interaction Φ β . Here a perturbation of Φ β means an interaction Φ β + Ψ defined by
(Φ β + Ψ) Λ = Φ β Λ + Ψ Λ for each Λ ⋐ Z d for some small perturbation interaction Ψ.
Perturbations of the interaction may correspond to some influence from the surroundings to the system or some noise in the interaction. Hence it is interesting to study what happens under perturbations of the interaction. We remark that the 'upper semicontinuity' of equilibrium states in perturbations of interactions holds in the following sense. Proposition 1.2. Let Φ 0 be an interaction, I Φ 0 be the set of equilibrium states for Φ 0 and I be the set of translation invariant Borel probability measures on Ω. Then, for any neighborhood U of I Φ 0 in I with respect to the weak*-topology, there exists 0 < δ < 1 such that, for any interaction Ψ with Ψ < δ, we have
I Φ 0 +Ψ ⊂ U.
We give the proof in the appendix. We notice that Proposition 1.2 does not ensure the stability of the uniqueness or coexistence of equilibrium states.
If the perturbations are finite range, then this perturbation problem can be studied by the known theory, called the Pirogov-Sinai theory and it is known that the coexistence of equilibrium states at low temperature is stable under finite range perturbations which preserve some symmetry of the Ising model, called the spin-flip symmetry.
Definition 1.2. An interaction Ψ on Ω = {1, −1} Z d is spin-flip symmetric if Ψ Λ (ω) = Ψ Λ (−ω) for Λ ⋐ Z d and ω ∈ Ω Λ .
Our main theorem says that the same statement holds under long range perturbations to which the Pirogov-Sinai theory is not applicable. We remark in detail the relation between our main theorem and the Pirogov-Sinai theory in Secrion 2.4. We consider perturbations by small parturbation interactions with respect to the norm |||·|||, which we call the d-th order decaying norm. The following is our main theorem. Theorem 1.3 (Main theorem). Let d ≥ 2 and, for β > 0, Φ β be the interaction of the d-dimensional Ising model at the inverse temperature β. We write β c for the critical inverse temperature. Then we have the following.
(1) There exist 0 < β h < β c and δ > 0 such that, for any 0 < β ≤ β h and any translation invariant interaction Ψ with |||Ψ||| < δ, Φ = Φ β + Ψ has a unique Gibbs state and, in particular, a unique equilibrium state. (2) There exist β c < β l < ∞ and δ > 0 such that, for any β ≥ β l and any translation invariant interaction Ψ with |||Ψ||| < δ which is spin-flip symmetric, Φ = Φ β + Ψ has more than one equilibrium state.
The definition of |||·||| is given in Section 2.2. We notice that this norm is stronger than · and weaker than the exponentially decaying norm.
Remark 1.2.
(i) Actually, Theorem 1.3 (1) is a direct collorary of Dobrushin's criterion, which gives a condition for an interaction to have a unique Gibbs state. We remark this in detail in Section 2.3. (ii) Theorem 1.3 (2) follows from stronger results, Theorems 3.1 and 3.2 below. (iii) In Theorem 1.3, the d-th order decaying condition is crucial. See Remark 3.1.
We notice that Theorem 1.3 (2) fails if we drop the spin-flip symmetry condition. To see this, we see the phase diagram of the Ising model with external fields. For β, h ∈ R, we define the interaction
{Φ β,h Λ } Λ⋐Z d by Φ β,h Λ (ω) = −βω(x)ω(y), Λ = {x, y} with x − y 1 = 1 −hω(x), Λ = {x} 0, otherwise,
for each Λ ⋐ Z d and ω ∈ Ω Λ . When h = 0 then Φ β,0 = Φ β and, by taking h = 0 arbitrarily small, we can think Φ β,h as a perturbation of Φ β . Then the following holds (see [10,Chapter 3] We notice that the perturbations Φ β,h (h = 0) are not spin-flip symmetric. Hence Proposition 1.4 says that, for sufficiently large β, non-spin-flip symmetric perturbations of Φ β can break the structure of the equilibrium states. Moreover, it is known that the uniqueness of the equilibrium state is a 'generic' property in the space of interactions (with appropriate norms). This fact is discussed in [6], [9], [18] and [14]. Our main theorem 1.3 says that spin-flip symmetric perturbations are 'not generic' and preserve the coexistence.
In Section 2.1, we give the rigorous definition of Gibbs states and equilibrium states. We give the definition of the d-th order decaying norm |||·||| is Section 2.2. In Section 2.3, we give Dobrushin's criterion and deduce Theorem 1.3 (1) from it. We remark the relation between our main theorem and the Pirogov-Sinai theory in Secrion 2.4. We prove the main theorem at low temperature in Section 3. We use in the proof some extension of Peierls' argument. We notice further questions in Section 4.
The author is grateful to Masayuki Asaoka and Mitsuhiro Shishikura for their helpful advice, and Takehiko Morita, Masaki Tsukamoto and Tom Meyerovitch for their useful comments.
Preliminaries
2.1. Gibbs states and equilibrium states. In this section, for a Z d -lattice system on Ω = F Z d with an interaction Φ, we give the rigorous definition of Gibbs states and equilibrium states. We first define Gibbs states by the thermodynamical condition, called the DLR condition * . We next define equilibrium states by the variational principle in the theory of dynamical systems and notice that these different definitions are equivalent under the translation invariant condition.
For Λ ⋐ Z d , η ∈ Ω Z d \Λ and ξ Λ ∈ Ω Λ , we write ξ Λ ∨ η for the element of Ω defined by ξ Λ ∨ η| Λ = ζ Λ and ξ Λ ∨ η| Z d \Λ = η. We define the finite Gibbs state µ η Φ,Λ in Λ with the boundary condition η as the Borel probability measure on Ω Λ defined by
(2.1) µ η Φ,Λ ({ω Λ }) = 1 Z η Φ,Λ exp (−H Φ,Λ (ω Λ ∨ η)) for each ω Λ ∈ Ω Λ , where Z η Φ,Λ = ξ Λ ∈Ω Λ exp (−H Φ,Λ (ξ Λ ∨ η)) .
We call Z η Φ,Λ the partition function in Λ with the boundary condition η. Let C(Ω) be the real Banach space of continuous real-valued functions on Ω with the supremum norm and M(Ω) be the set of Borel probability measures on Ω. M(Ω) can be viewed as a subset of the dual space of the Banach space C(Ω). Then M(Ω) is a compact, convex and metrizable space with respect to the weak*-topology.
Definition 2.1 (Gibbs State). σ ∈ M(Ω) is a Gibbs state for an interaction Φ if for each Λ ⋐ Z d , there exists a Borel probability measure σ Z d \Λ on Ω Z d \Λ such that for any ω Λ ∈ Ω Λ , σ({ξ ∈ Ω | ξ| Λ = ω Λ }) = Ω Z d \Λ µ η Φ,Λ ({ω Λ }) dσ Z d \Λ (η). Definition 2.1 is equivalent to the following: for each Λ ⋐ Z d and η ∈ Ω Z d \Λ , the conditional probability that ξ| Λ = ω Λ under ξ| Z d \Λ = η is µ η Φ,Λ ({ω Λ })
. This condition is called the DLR condition. Let K Φ be the set of Gibbs states for Φ. This is a nonempty, compact and convex subset of M(Ω) (see [17,Chapter 1]). We notice that K Φ+C = K Φ for any constant interaction C (that is, an interaction such that C Λ is constant for each Λ ⋐ Z d ).
Next, we state the variational principle in the theory of dynamical systems and give the definition of equilibrium states. Let I be the set of Borel probability measures µ on Ω which are translation invariant, that is, τ a * µ = µ for each a ∈ Z d . This is a nonempty, compact and convex subset of M(Ω). For a sequence {Λ n } ∞ n=1 of finite subsets of Z d , we write Λ n ր ∞ (limit in the sense of van Hove) when |Λ n | → ∞ and, for any a ∈ Z d , |(Λ n +a)\Λ n | |Λ n | → 0. An example of such a sequence is [17,Chapter 3]). h(µ) is called the measure-theoretic entropy of µ. Definition 2.3. For each interaction Φ and Λ ⋐ Z d , We define the partition function in Λ with free boundary condition by
{B(n)} ∞ n=1 , where B(n) = {−n, . . . , n} d . Definition 2.2. For each µ ∈ I, the limit h(µ) = lim n→∞ − 1 |Λ n | ω∈Ω Λn µ Λn ({ω}) log µ Λn ({ω}), where µ Λn ({ω}) = µ({ξ ∈ Ω | ξ| Λn = ω}), exists and is independent of the choice of sequence {Λ n } ∞ n=1 such that Λ n ր ∞ (seeZ Φ,Λ = ω∈Ω Λ exp − ∆⊂Λ Φ ∆ (ω) .
Then, the limit
P Φ = lim n→∞ 1 |Λ n | log Z Φ,Λn exists and is independent of the choice of sequence {Λ n } ∞ n=1 such that Λ n ր ∞ (see [17, Chapter 3]). P Φ is called the pressure of Φ.
For an interaction Φ, we define A Φ : Ω → R by
A Φ (ω) = − Λ Φ Λ (ω),
where the sum runs over those Λ ⋐ Z d such that Λ contains 0 in Z d and 0 is the middle element of Λ (that is, the ⌊(|Λ| + 1) /2⌋-th † element) with respect to the lexicographic order on Λ. Since Φ is absolutely summable, A Φ is continuous. −A Φ (ω) physically represents the contribution of 0 ∈ Z d to the energy in the configuration ω. The variational principle is the statement which connencts entropy, pressure and A Φ .
Proposition 2.1 (Variational Principle). For each interaction Φ, P Φ = sup µ∈I h(µ) + Ω A Φ dµ .
Moreover, there exists some µ ∈ I such that µ achieves the supremum (see [17,Theorem 3.12]).
Definition 2.4 (Equilibrium State
). An element µ ∈ I is an equilibrium state for an interaction Φ if µ achieves the supremum of Proposition 2.1.
Let I Φ be the set of the equilibrium states for Φ. This is a nonempty, compact and convex subset of M(Ω). The definituon of the equilibrium states turn out to be a charactarization of the translation invariant Gibbs states, that is, the following proposition holds (see [17,Theorem 4.2]).
Proposition 2.2. For each interaction Φ,
I Φ = K Φ ∩ I.Definition 2.5. An interaction Ψ is d-th order decaying if |||Ψ||| = 0∈Λ⋐Z d (diam(Λ) + 1) d sup ω∈Ω Λ |Ψ Λ (ω)| < ∞.
We call the norm |||·||| the d-th order decaying norm. † For a ∈ R, ⌊a⌋ denotes the largest integer that is not larger than a. (1) Finite range interactions are clearly d-th order decaying. (2) (Two body interactions.) Let Ψ be a translation invariant interaction on Ω which satisfies Ψ Λ = 0 unless Λ consists of distinct two points. We emphasize that these two body interactions are not finite range in general. We assume that Ψ is (2d + ε)-th order decaying, that is, there exists ε > 0 such that
Ψ ε = sup x − y 2d+ε ∞ Ψ {x,y} (ω) x, y ∈ Z d , x = y, ω ∈ Ω {x,y} < ∞. We have |||Ψ||| = x∈Z d \{0} ( x ∞ + 1) d sup ω∈Ω {0,x} Ψ {0,x} (ω) ≤ x∈Z d \{0} ( x ∞ + 1) d x −(2d+ε) ∞ Ψ ε =M ε Ψ ε , where M ε = ∞ n=1 ((2n + 1) d − (2n − 1) d )(n + 1) d n −(2d+ε) < ∞. Hence Ψ is d-th order decaying.
Here, we give some remark on two body interactions. If a two body interaction Ψ has a form Ψ {x,y} (ω) = K x,y ω(x)ω(y), K x,y ∈ R, a condition for Ψ to have more than one equilibrium state is given in [12]. From this condition, we can see that, if a perturbation interaction Ψ is a two body interaction which has a form as above, the coexistence of equilibrium states holds under weaker assumption than Theorem 1.3 (2).
2.3.
High temperature cases and Dobrushin's criterion. In this section, we give Dobrushin's criterion and see that Theorem 1.3 (1) follows from this. Let Φ be a translation invariant interaction. For any
x ∈ Z d \ {0}, we define ρ Φ (x) = sup µ η Φ,{0} ({1}) − µ ζ Φ,{0} ({1}) η, ζ ∈ Ω Z d \{0} , η| Z d \{0,x} = ζ| Z d \{0,x} . Proposition 2.3 (Dobrushin's Criterion). If x∈Z d \{0} ρ Φ (x) < 1,
then Φ has a unique Gibbs state.
The following is a convenient version of Dobrushis's criterion. We define
var Λ Φ = sup ω,ω ′ ∈Ω Λ |Φ Λ (ω) − Φ Λ (ω ′ )| for each Λ ⋐ Z d and Φ var = 0∈Λ⋐Z d (|Λ| − 1)var Λ Φ,
where |Λ| be the cardinality of Λ. It is obvious that
(2.2) Φ var ≤ 2|||Φ|||
for any d-th order decaying interaction Φ.
Proposition 2.4. If Φ var < 2,
then Φ satisfies Dobrushin's criterion and hence has a unique Gibbs state.
We refer [11,Chapter 8] for proofs of the above propositions. It is easy to see that Φ β var = 4dβ. From this, Inequality (2.2) and Proposition 2.4, we have Theorem 1.3 (1).
2.4.
Low temperature cases and Pirogov-Sinai theory. We notice the relation between Theorem 1.3 (2) and the Pirogov-Sinai theory. If β is sufficiently large and a perturbation interaction Ψ is finite range, then the equilibrium states of Φ = Φ β + Ψ can be studied by an application of the Pirogov-Sinai theory. It was introduced by S. A. Pirogov and Ya. G. Sinai in [16] and sophisticated in [19] to study phase diagrams of more general finite range interactions at low temperature. We refer [10,Chapter 7] for an intelligible introduction of this theory. The periodic ground states of Φ β , that is, the states which have the lowest energy under Φ β , are exactly the constant configurations η + and η − ∈ Ω, which is constant 1 and −1 on Z d , respectively, and it can be seen that sufficiently small and spin-flip symmetric perturbations of Φ β preserves η + and η − as periodic ground states. Hence, by the Pirogov-Sinai theory, if Ψ is sufficiently small, spin-flip symmetric and finite range interaction, then Φ = Φ β + Ψ has equilibrium states corresponding to two periodic ground states, η + and η − .
The assumption that interactions are finite range is crucial in the Pirogov-Sinai theory. Hence Theorem 1.3 (2), including results in the case of long range perturbations, does not follows from it. The Pirogov-Sinai theory was extended to some extent in [15], but this extention is possible only for sum of finite range interactions and two body interactions satisfying some exponentially decaying condition. It was also extended to quantum cases in [5], or a series of papers [7] and [8]. In these cases, perturbation interactions can be long range but must exponentially decay. Hence Theorem 1.3 (2), allowing d-th order decaying perturbations, also does not follow from these extended version. On the other hand, we can not study spin-flip symmetric perturbations (which are long range in general) in detail using the techniques in this paper as well as finite range perturbations using the Pirogov-Sinai theory. We will mention this later (Section 4).
3.
Perturbations of the Ising model at low temperature 3.1. The main theorem in low temperature cases and reformalization of perturbation interactions. In Section 3, we give a proof of Theorem 1.3 (2). As we mentioned in Remark 1.2, Theorem 1.3 (2) follows from stronger results: Theorems 3.1 and 3.2 below. We first state these results.
We introduce a norm |||·||| ′ , which is weaker than the d-th order decaying norm |||·|||. For a translation invariant interaction Ψ, we define
|||Ψ||| ′ = 0∈Λ⋐Z d |Λ| −1 (diam(Λ) + 1) d sup ω∈Ω Λ |Ψ Λ (ω)| .
It is clear that |||Φ||| ′ ≤ |||Φ|||. Then we have the following theorem.
Theorem 3.1. Let d ≥ 2.
Then there exists 0 < L < ∞ satisfying the following. If β, δ > 0 satisfy β − δ > L, then, for any translation invariant interaction Ψ with |||Ψ||| ′ < δ which is spin-flip symmetric, Φ = Φ β + Ψ has more than one equilibrium state.
Theorem 3.1 follows from Theorem 3.2 below. We say that Λ ⊂ Z d is · 1 -connected if for each x and y ∈ Λ there exist finite points x 0 , x 1 , . . . , x n−1 , x n ∈ Λ such that x 0 = x, x n = y, x i+1 − x i 1 = 1 for i = 0, . . . , n − 1. We call such a finite sequence x 0 , x 1 , . . . , x n−1 , x n a · 1 -path from x to y. For an interaction Ψ, we say that Ψ is zero on non-
· 1 -connected sets if Ψ Λ ≡ 0 whenever Λ ⋐ Z d is not · 1 -connected. Theorem 3.2. Let d ≥ 2.
Then there exists 0 < L < ∞ satisfying the following. If β, δ > 0 satisfy β − δ > L, then, for any translation invariant interaction Ψ with Ψ < δ which is spin-flip symmetric and zero on non-· 1 -connected set, Φ = Φ β + Ψ has more than one equilibrium state.
We see that how Theorem 3.1 reduces to Theorem 3.2. For this purpose, we reformalize a perturbation interaction. Let Ψ be a translation invariant interaction with |||Ψ||| ′ < ∞. For
R ⋐ Z d , we say R is a rectangle if R = [a 1 , b 1 ] × · · · × [a d , b d ] ∩ Z d for some a 1 ≤ b 1 , . . . , a d ≤ b d ∈ Z.
For each finite Λ ⋐ Z d , we write R(Λ) for the minimal rectangle which contains Λ with respect to the partial order given by inclusion. We notice that diam(Λ) = diam(R(Λ)). For each rectangle R, we define S (R) = Λ ⋐ Z d |R(Λ) = R . For Ψ as above, we define an interaction Ψ as
Ψ Λ ≡ 0 if Λ is not a rectangle and Ψ R (ω) = Λ∈S (R) Ψ Λ (ω| Λ )
for any rectangle R ⋐ Z d and ω ∈ Ω R . This interaction Ψ is made by putting interactions Ψ Λ on Λ ∈ S (R) together into the rectangle R. Obviously Ψ is a translation invariant interaction. We write S 0 for the set of Λ ⋐ Z d such that Λ contains 0 in Z d and 0 is the middle element of Λ (that is, the [(⌊Λ⌋ + 1) /2]-th element) with respect to the lexicographic order on Λ. Then, by the translation-invariance of Ψ, we have
Ψ = 0∈R⋐Z d ,R:rectangle sup ω∈Ω R | Ψ R (ω)| = 0∈R⋐Z d ,R:rectangle, R∈S 0 |R| sup ω∈Ω R | Ψ R (ω)| ≤ 0∈R⋐Z d ,R:rectangle, R∈S 0 |R| Λ∈S (R) sup ω∈Ω Λ |Ψ Λ (ω)| = Λ⋐Z d , R(Λ)∈S 0 |R(Λ)| sup ω∈Ω Λ |Ψ Λ (ω)| ( * ) = 0∈Λ⋐Z d |Λ| −1 |R(Λ)| sup ω∈Ω Λ |Ψ Λ (ω)| ≤ 0∈Λ⋐Z d |Λ| −1 (diam(Λ) + 1) d sup ω∈Ω Λ |Ψ Λ (ω)| = |||Ψ||| ′ . (3.1)
Here the equation ( * ) holds since Ψ is translation invariant and, for each Λ ⋐ Z d , there is exactly one translation of Λ that appears in the summation Λ⋐Z d ,R(Λ)∈S 0 . The following lemma shows that Ψ and Ψ induces the same statistical systems. In particular, by Proposition 2.2, Φ and Φ have the same equilibrium states.
Proof. We take arbitrary Λ ⋐ Z d and η ∈ Ω Z d \Λ . For any ω Λ ∈ Ω Λ , we have, by (2.1),
µ η Φ,Λ ({ω Λ }) = exp − ∆∈F Λ Φ ∆ (ω Λ ∨ η) + Ψ ∆ (ω Λ ∨ η) ξ Λ ∈Ω Λ exp − ∆∈F Λ Φ ∆ (ξ Λ ∨ η) + Ψ ∆ (ξ Λ ∨ η) = exp − ∆∈F Λ Φ ∆ (ω Λ ∨ η) − R∈R Λ S∈S (R) Ψ S (ω Λ ∨ η) ξ Λ ∈Ω Λ exp − ∆∈F Λ Φ ∆ (ξ Λ ∨ η) − R∈R Λ S∈S (R) Ψ S (ξ Λ ∨ η) ,
where R Λ = R ⋐ Z d R is a rectangle and R ∩ Λ = ∅ . Here, for any ξ Λ ∈ Ω Λ ,
R∈R Λ S∈S (R) Ψ S (ξ Λ ∨ η) = R∈R Λ S∈S (R)∩F Λ Ψ S (ξ Λ ∨ η) + R∈R Λ S∈S (R)\F Λ Ψ S (η) = ∆∈F Λ Ψ ∆ (ξ Λ ∨ η) + R∈R Λ S∈S (R)\F Λ Ψ S (η)
and the second term is independent of ξ Λ . Hence we have
µ η Φ,Λ ({ω Λ }) = exp − ∆∈F Λ Φ ∆ (ω Λ ∨ η) − R∈R Λ S∈S (R) Ψ S (ω Λ ∨ η) ξ Λ ∈Ω Λ exp − ∆∈F Λ Φ ∆ (ξ Λ ∨ η) − R∈R Λ S∈S (R) Ψ S (ξ Λ ∨ η) = exp − ∆∈F Λ (Φ ∆ (ω Λ ∨ η) + Ψ ∆ (ω Λ ∨ η)) ξ Λ ∈Ω Λ exp − ∆∈F Λ (Φ ∆ (ξ Λ ∨ η) + Ψ ∆ (ξ Λ ∨ η)) = µ η Φ,Λ ({ω Λ }).
This implies that each finite Gibbs state coinsides for Φ and Φ. Hence the statement holds.
Obviously Ψ is zero on non-· 1 -connected sets. Hence, by replacing Ψ with Ψ in Theorem 3.1 and using (3.1) and Lemma 3.3, we see that Theorem 3.1 follows from Theorem 3.2.
3.2.
Gibbs states with the +, −-boundary condition. We will give a proof of Theorem 3.2. Let Φ be a translation invariant interaction such that Φ < ∞. For Λ ⋐ Z d , we define the finite Gibbs state with the + and −-boundary condition: µ + Φ,Λ and µ − Φ,Λ as the Borel probability measures on Ω Λ defined by (2.1) with η = η + Z d \Λ and η − Z d \Λ : the constant 1 and −1 configurations on Z d \ Λ, respectively. That is,
µ + Φ,Λ ({ω}) = 1 Z + Φ,Λ exp − ∆∈F Λ Φ ∆ (ω ∨ η + Z d \Λ ) and µ − Φ,Λ ({ω}) = 1 Z − Φ,Λ exp − ∆∈F Λ Φ ∆ (ω ∨ η − Z d \Λ ) (3.2) for each ω ∈ Ω Λ , where Z + Φ,Λ = ω∈Ω Λ exp − ∆∈F Λ Φ ∆ (ω ∨ η + Z d \Λ ) and Z − Φ,Λ = ω∈Ω Λ exp − ∆∈F Λ Φ ∆ (ω ∨ η − Z d \Λ ) .
Here, we state the key proposition: Proposition 3.4 and see Theorem 3.2 follows from it. We say that Λ ⊂ Z d is · ∞ -connected if for each x and y ∈ Λ there exist finite points x 0 , x 1 , . . . , x n−1 , x n ∈ Λ such that x 0 = x, x n = y, x i+1 − x i ∞ = 1 for i = 0, . . . , n − 1. We call such a finite sequence x 0 , x 1 , . . . , x n−1 , x n a · ∞ -path from x to y. Moreover, we say that Λ is c-connected if Λ and Z d \ Λ are · ∞ -connected.
Proposition 3.4. Let d ≥ 2.
Then there exists sufficiently large 0 < L < ∞ and 0 < ε(L) < 1/2 with ε(L) → 0 as L → ∞ satisfying the following. Let β, δ > 0 and β − δ > L and Ψ be a translation invariant interaction with Ψ < δ which is spin-flip symmetric and zero on non-· 1 -connected sets. We write Φ = Φ β + Ψ. Then, for any c-connected Λ ⋐ Z d and x ∈ Λ, we have
µ + Φ,Λ ({ω ∈ Ω Λ |ω(x) = 1}) > 1 − ε(L) and µ − Φ,Λ ({ω ∈ Ω Λ |ω(x) = −1 }) > 1 − ε(L).
This proposition means that, with respect to µ + Φ,Λ and µ − Φ,Λ , the spin at x is magnetized for each x ∈ Λ. Let us see that Theorem 3.2 follows from Proposition 3.4. For a sequence {Λ n } ∞ n=1 of finite subsets of Z d , we weite Λ n ↑ Z d when Λ n ⊂ Λ n+1 for each n and ∞ n=1 Λ n = Z d . For example, Λ n = B(n) satisfies these conditions. For each Λ ⋐ Z d , we canonically regard µ + Φ,Λ and µ − Φ,Λ as Borel probability measures on Ω with µ + Φ,Λ (Ω + Λ ) = 1 and µ − Φ,Λ (Ω − Λ ) = 1, where Ω + Λ = ω ∈ Ω ω(x) = 1, x ∈ Z d \ Λ and Ω − Λ = ω ∈ Ω ω(x) = −1, x ∈ Z d \ Λ , respectively. It is known that if µ + Φ,Λn (resp. µ − Φ,Λn ) converges to some µ + (resp. µ − ) in M(Ω) as n → ∞, then µ + ∈ K Φ (resp. µ − ∈ K Φ ) (see [17,Chapter 1]
+ Φ and µ − Φ are in K Φ . By Proposition 3.4, µ + Φ satisfies µ + Φ ({ω ∈ Ω |ω(x) = 1 }) = lim k→∞ µ + Φ,Λn k ({ω ∈ Ω + Λn k |ω(x) = 1 }) ≥ 1 − ε(L) (3.3) for each x ∈ Z d . For each N ∈ N, we define ν + Φ,N ∈ M(Ω) by ν + Φ,N = 1 |B(N)| x∈B(N ) τ x * µ + Φ .
Since Φ is translation invariant and µ + Φ ∈ K Φ , τ x * µ + Φ ∈ K Φ for each x ∈ B(N). Hence, by the convexity of K Φ , ν + Φ,n ∈ K Φ . Moreover, from Inequality (3.3), we have
ν + Φ,N ({ω ∈ Ω |ω(0) = 1 }) = 1 |B(N)| x∈B(N ) µ + Φ ({ω ∈ Ω |ω(x) = 1 }) ≥ 1 − ε(L). (3.4)
We can take a divergent subsequence {N l } ∞ l=1 ⊂ N such that ν + Φ,N l converges to some ν + Φ in M(Ω). It is seen that ν + Φ ∈ I and, since K Φ ⊂ M(Ω) is closed, ν + Φ ∈ K Φ . Hence, by Proposition 2.2 we have ν + Φ ∈ I Φ . Moreover, from Inequality (3.4), we have
ν + Φ ({ω ∈ Ω |ω(0) = 1 }) = lim l→∞ ν + Φ,N l ({ω ∈ Ω |ω(0) = 1}) ≥ 1 − ε(L).
By doing the same argument to
µ − Φ , we have ν − Φ ∈ I Φ such that ν − Φ ({ω ∈ Ω |ω(0) = −1 }) ≥ 1 − ε(L).
Since ε(L) < 1/2, we have ν + Φ = ν − Φ and complete the proof.
3.3.
Contours for a configration and Peierls' argument. We will give a proof of Proposition 3.4. We notice that the following argument is based on Peierls' argument, which was used to show the coexistence of equilibrium states for nonperturbed Ising models at low temperature (see [10, Section 3.7.2, 5.7.4]). We introduce the notion of a contour on Z d . We follow the notion in [10,Chapter 5]. For each x ∈ Z d , we write I x for the closed unit cube in R d centered at x:
I x = x + − 1 2 , 1 2 d ,
and, for each Λ ⊂ Z d , define
M (Λ) = x∈Λ I x ⊂ R d . Definition 3.1. γ ⊂ R d is a contour if γ = ∂M (Λ)
for some c-connected Λ ⋐ Z d (where ∂ denotes the boundary with respect to the usual topology of R d ). It is seen that, for each a contour γ, Λ ⋐ Z d such that γ = ∂M (Λ) is uniquely determined. We call such Λ the interior of γ and write intγ.
It is seen that a contour γ is a connected sum of plaquettes, which are (d−1)-dimensional faces of I x , x ∈ Z d , such that γ devides Z d into two · ∞ -connected sets: the interior and exterior of γ (for example, see [10, Appendix B]). We write |γ| for the number of plaquettes which are contained in γ. We write E = {x, y} ⊂ Z d | x − y 1 = 1 and call an element of E an edge (considering the graph (Z d , E )). There is the one-to-one mapping between plaquettes and E , associating a plaquette with the unique edge crossing it. By this mapping, the plaquettes contained in γ correspond to elements of {x, y} ∈ E x ∈ intγ, y ∈ Z d \ intγ .
For each Λ ⋐ Z d , let Ω + Λ = ω ∈ Ω ω(x) = 1 on Z d \ Λ . We write Ω + = Λ⋐Z d Ω + Λ . For each ω ∈ Ω + , we define Λ − (ω) = x ∈ Z d |ω(x) = −1 ⋐ Z d and M (ω) = M (Λ − (ω)).
We decompose ∂M (ω) into connected components:
∂M (ω) = n i=1 γ i ,
then it is shown that each γ i (i = 1, . . . , n) is a contour and, by the mapping mentioned above, a plaquette contained in some γ i is associated with an edge {x, y} ∈ E such that ω(x) = 1 and ω(y) = −1 (see [10,Section 5.7.4]). We write Γ(ω) = {γ 1 , . . . , γ n }.
Let us consider the Ising model with inverse temparature β > 0. For Λ ⋐ Z d , we define
E Λ = {{x, y} ∈ E | {x, y} ∩ Λ = ∅}. Then for ω ∈ Ω + Λ , we have exp − ∆∈F Λ Φ β ∆ (ω) = exp β {x,y}∈E Λ ω(x)ω(y) = e β|E Λ | e −2β|{{x,y}∈E Λ |ω(x)=1,ω(y)=−1 }| = e β|E Λ | γ∈Γ(ω) e −2β|γ| .
Suppose Λ is c-connected. For ω ∈ Ω + Λ and γ ∈ Γ(ω), we define ω γ ∈ Ω by
ω γ (x) = −ω(x), x ∈ intγ ω(x), x ∈ Z d \ intγ.
Then, by the assumption that Λ is c-connected, ω γ ∈ Ω + Λ and
(3.5) Γ(ω γ ) = Γ(ω) \ {γ}.
Hence we have
exp − ∆∈F Λ Φ β ∆ (ω) = e −2β|γ| exp − ∆∈F Λ Φ β ∆ (ω γ ) .
This is a key equation in Peierls' argument. Let us consider a perturbation of Φ β by a spin-flip symmetric interaction Ψ such that Ψ < δ and it takes zero on non-· 1 -connected sets. Then, since ω γ = −ω on intγ and ω γ = ω on Z d \ intγ, by the spin-flip symmetry of Ψ, we have
∆∈F Λ Ψ ∆ (ω) = ∆∈F Λ , ∆⊂intγ or ∆∩intγ=∅ Ψ ∆ (ω) + ∆∈F Λ , ∆ ⊂intγ,∆∩intγ =∅ Ψ ∆ (ω) = ∆∈F Λ , ∆⊂intγ or ∆∩intγ=∅ Ψ ∆ (ω γ ) + ∆∈F Λ , ∆ ⊂intγ,∆∩intγ =∅ Ψ ∆ (ω).
If ∆ ∈ F Λ is · 1 -connected, ∆ ⊂ intγ and ∆ ∩ intγ = ∅, then ∆ contains some edge {x, y} ∈ E associated with some plaquette contained in γ. Hence we have
∆∈F Λ , ∆ ⊂intγ,∆∩intγ =∅ Ψ ∆ (ω) ≤ |γ| Ψ < |γ|δ
and the same holds for ω γ . From these, we have
exp − ∆∈F Λ Φ β + Ψ ∆ (ω) = e −2β|γ| exp − ∆∈F Λ Φ β ∆ (ω γ ) exp − ∆∈F Λ Ψ ∆ (ω) = e −2β|γ| exp − ∆∈F Λ Φ β + Ψ ∆ (ω γ ) exp − ∆∈F Λ , ∆ ⊂intγ,∆∩intγ =∅ (Ψ ∆ (ω) − Ψ ∆ (ω γ )) = e −2β|γ|+δ(Λ,ω,γ) exp − ∆∈F Λ Φ β + Ψ ∆ (ω γ ) (3.6) with (3.7)
|δ (Λ, ω, γ)| < 2|γ|δ.
3.4. Proof of Proposition 3.4. First, we prove the following lemma about the number of contours surrounding a fixed point. The result for d = 2 is proved and used in [11,Chapter 6]. We give a proof for the completeness. By elementary arguments of graph theory, it is seen that, for any finite connected subset S ⊂ V and any H ∈ S, there exists a path p in G such that p starts from and ends at H , passes through every vertex in S and does not pass through any other vertices and the length of p is bounded by 2|S|. We fix i = 0, . . . , n. Then, by regarding a contour as a finite connected subset in V , we can associate each γ ∈ Γ n,0 such that H i ⊂ γ with a path p in G starting form and ends at H i which satisfies the conditions for γ as above. Since the number of paths starting from H i the length of which is bounded by 2n is bounded by C 2n+1 d , we obtain |Γ n,0 | ≤ (n + 1)C 2n+1 d . The next lemma is a key to prove Proposition 3.4.
Lemma 3.6. Let β > 0, 0 < δ < 1 and Ψ be a tranlation invariant and spin-flip symmetric interaction which is zero on non-· 1 -connected sets and Ψ < δ. We write Φ = Φ β + Ψ. Let Λ ⋐ Z d be c-connected and γ 1 , . . . , γ n be contours such that intγ 1 , . . . , intγ n ⊂ Λ and they are pairwise disjoint. Then we have
µ + Φ,Λ ω ∈ Ω + Λ |γ 1 , . . . , γ n ∈ Γ(ω) ≤ exp (−2(β − δ)(|γ 1 | + · · · + |γ n |))
(where we canonically regard µ + Φ,Λ as a measure on Ω + Λ ).
Proof. For each i = 1, . . . , n, we define a map θ γ i : ω ∈ Ω + Λ |γ i ∈ Γ(ω) ∋ ω → ω γ i ∈ ω ∈ Ω + Λ |γ i / ∈ Γ(ω) . It is clear that θ γ i is injective. From Equation (3.5), we can define the composition θ = θ γn • · · · • θ γ 1 : ω ∈ Ω + Λ |γ 1 , . . . , γ n ∈ Γ(ω) → Ω + Λ and it is injective. Moreover, for each ω ∈ Ω + Λ such that γ 1 , . . . , γ n ∈ Γ(ω), from Equation (3.6) and Inequality (3.7), we have
exp − ∆∈F Λ Φ ∆ (ω) ≤ e −2(β−δ)|γ 1 | exp − ∆∈F Λ Φ ∆ (θ γ 1 (ω)) ≤ e −2(β−δ)|γ 1 | e −2(β−δ)|γ 2 | exp − ∆∈F Λ Φ ∆ (θ γ 2 (θ γ 1 (ω)))
. . . ≤ exp(−2(β − δ)(|γ 1 | + · · · + |γ n |)) exp − ∆∈F Λ Φ ∆ (θ(ω)) .
Hence we have
µ + Φ,Λ ω ∈ Ω + Λ |γ 1 , . . . , γ n ∈ Γ(ω) = 1 Z + Φ,Λ ω∈Ω + Λ , γ 1 ,...,γn∈Γ(ω) exp − ∆∈F Λ Φ ∆ (ω) ≤ exp(−2(β − δ)(|γ 1 | + · · · + |γ n |)) · 1 Z + Φ,Λ ω∈Ω + Λ , γ 1 ,...,γn∈Γ(ω) exp − ∆∈F Λ Φ ∆ (θ(ω)) ≤ exp(−2(β − δ)(|γ 1 | + · · · + |γ n |))
and obtain the statement.
We prove Proposition 3.4 using these lemmas.
Proof of Proposition 3.4. Let β, δ > 0 and Ψ be a tranlation invariant and spin-flip symmetric interaction which is zero on non-· 1 -connected sets and Ψ < δ. We write Φ = Φ β + Ψ. We take arbitrary Λ ⋐ Z d and x ∈ Λ and write Ω +,− Λ,x = ω ∈ Ω + Λ |ω(x) = −1 . Let Γ x = {γ : contour |x ∈ intγ }. For each ω ∈ Ω +,− Λ,x , it is seen that Γ(ω)∩Γ x = ∅. Hence, by Lemmas 3.6 and 3.5, we have
µ + Φ,Λ Ω +,− Λ,x ≤ γ∈Γx µ + Φ,Λ ({ω ∈ Ω + Λ |γ ∈ Γ(ω)}) ≤ γ∈Γx e −2(β−δ)|γ| = ∞ k=1 γ∈Γ k,0 e −2(β−δ)k ≤ ∞ k=1 (k + 1)C 2k+1 d e −2(β−δ)k .
uniformly in Λ and x. Hence, if we take log C d < L < ∞ sufficiently large and β − δ > L we have
µ + Φ,Λ Ω +,− Λ,x ≤ ∞ k=1 (k + 1)C 2k+1 d e −2(β−δ)k ≤ C d ∞ k=1 (k + 1)e −2(L−log C d )k
= ε(L) < 1 2 and ε(L) → 0 as L → ∞. We obtain the statement in the case of +. By the same argument in the case of −, we obtain Proposition 3.4. Remark 3.1. In the proof of Theorem 1.3 (2) above, we showed that there exist equilibrium states ν + Φ and ν − Φ for Φ = Φ β + Ψ such that the magnetization is positive and negative, respectively, that is,
Ω ω(0) dν + Φ (ω) > 0 and Ω ω(0) dν − Φ (ω) < 0.
It can be seen that some condition on order of decay of Ψ, like d-th order decaying condition, is necessary for Φ to have non-magnetization-vanishing equilibrium states. In [9], a translation invariant and spin-flip symmetric two body interaction Φ is constructed such that every equilibrium state µ for Φ = Φ β + Ψ vanishes magnetization. This interaction Ψ is antiferromagnetic type (that is, Ψ {x,y} (ω) = K x,y ω(x)ω(y), K x,y ≥ 0) and small with respect to · , but not d-th order decreasing. Moreover, it is shown in [3] that, if d < s ≤ d + 1, then, for the antiferromagnetic interaction Ψ of the form K x,y = δ/|x − y| s , δ > 0 (where | · | is the Euclidean distance), every equilibrium state for Φ = Φ β + Ψ vanishes magnetization. (It seems unknown whether the uniqueness of the equilibrium state holds for these perturbations.)
Further questions
Here we notice some further questions next to our results. We showed that there are more than one equilibrium state for Φ = Φ β + Ψ, where β > 0 is sufficiently large and Ψ is a translation invariant and spin-flip symmetric interaction such that |||Ψ||| is sufficiently small. When Ψ = 0, that is, Φ = Φ β , the nonperturbed Ising model at sufficiently low temperature, the structure of I Φ β is completely known. Remember µ + Φ β ,Λ and µ − Φ β ,Λ defined for each Λ ⋐ Z d in (3.2). converge to some µ + Φ β and µ − Φ β in M(Ω), respectively and they are independent of the choice of {Λ n } ∞ n=1 . µ + Φ β and µ − Φ β are translation invariant Gibbs states for Φ β and, then µ + Φ β , µ − Φ β ∈ I Φ β . Furthermore, µ + Φ β and µ − Φ β span I Φ β , that is,
I Φ β = pµ + Φ β + (1 − p)µ − Φ β |0 ≤ p ≤ 1 .
For a proof of the first half of the statement, see [10,Chapter 3] and for that of the second statement, see [1] or [13] for d = 2 and [4] for d ≥ 3. These results are due to the FKG inequality, the crucial property of ferromagnetic interactions. If β > 0 is sufficinetly large and a perturbation interaction Ψ is finite lange, then, as we saw in Section 2.4, we can apply the Pirogov-Sinai theory to Φ = Φ β + Ψ and the same statement holds (see [19]).
Problem 4.2.
For sufficiently large β > 0, determine whether the same statement as Proposition 4.1 holds or not in general when Ψ is a translation invariant and spin-flip symmetric interaction such that |||Ψ||| is sufficiently small.
We notice that these general cases include many cases to which the FKG inequality and existing extensions of the Pirogov-Sinai theory can not be applicable.
We notice another question. In Theorem 1.3, we saw the stability of uniquness of the equilibrium state for sufficiently small β and coexistence for sufficiently large β. Then what is about the case when β > 0 is intermediate? At β = β c , it is clear that the stability is broken. Hence we have a problrm as follows.
Problem 4.3.
(1) Determine for each 0 < β < β c whether the stability of uniqueness of the equilibrium state as Theorem 1.3 (1) holds or not.
(2) Determine for each β c < β < ∞ whether the stability of coexistence of equilibrium states as Theorem 1.3 (2) holds or not.
Appendix
Here we give a proof of Proposition 1.2.
Proof of Proposition 1.2. Let Φ 0 be an interaction.
We take an open set I Φ 0 ⊂ U ⊂ I. For an interaction Φ, we define the function F Φ : I → R as
F Φ (µ) = h(µ) + Ω A Φ dµ, µ ∈ I.
Then F Φ achieves P Φ = sup µ∈I F Φ (µ) exactly on I Φ . Moreover, since I ∋ µ → Ω A Φ dµ ∈ R is continuous and I ∋ µ → h(µ) ∈ R is upper semicontinuous, F Φ (µ) is upper semicontinuous in µ ∈ I. Hence, for Φ = Φ 0 , we have I \ U ⊂ µ ∈ I F Φ 0 (µ) < P Φ 0 and the left-hand side is compact and the right-hand side is open in I. Then, using the upper semicontinuity of F Φ 0 again, we have 0 < ε < 1 such that
I \ U ⊂ µ ∈ I F Φ 0 (µ) < P Φ 0 − ε .
We set δ = ε/4 and take an arbitrary interaction Ψ with Ψ < δ. We write Φ = Φ 0 + Ψ. Then, for any µ ∈ I, we have
|F Φ (µ) − F Φ 0 (µ)| = Ω A Φ dµ − Ω A Φ 0 dµ ≤ Ω |A Φ − A Φ 0 | dµ ≤ Ψ < δ and P Φ − P Φ 0 = sup µ∈I F Φ (µ) − sup µ∈I F Φ 0 (µ) ≤ δ.
If µ ∈ I \ U, then F Φ 0 (µ) < P Φ 0 − ε. Hence we have
F Φ (µ) ≤ F Φ 0 (µ) + δ < P Φ 0 − ε + δ ≤ P Φ − ε + 2δ = P Φ − ε 2
and µ does not achieve the supremum. Hence µ / ∈ I Φ .
Definition 1 . 1 (
11The Ising model). Let β > 0. The d-dimensional Ising model at the inverse temperature β is a Z d -lattice system on Ω = {1, −1} Z d with a finite range interaction Φ β defined by Φ β Λ (ω) = −βω(x)ω(y), if Λ = {x, y} with x − y 1 = 1 0, otherwise, for each Λ ⋐ Z d and ω ∈ Ω Λ = {1, −1} Λ , where x
2.2. d-th order decaying interactions. In this section, we introduce d-th order decaying interactions on Ω = {−1.1} Z d and give the definition of the d-th order decaying norm |||·||| in Theorem 1.3.
Lemma 3 . 3 .
33Let Ψ and Ψ be the interactions as above. For any interaction Φ 0 , interactions Φ = Φ 0 + Ψ and Φ = Φ 0 + Ψ have the same Gibbs states.
Lemma 3. 5 .
5For any n ∈ N, let Γ n,0 = {γ : contour ||γ| = n, 0 ∈ intγ } .Then we have |Γ n,0 | ≤ (n + 1)C 2n+1 d , where C d ∈ N is the number of (d − 1)-dimensional faces of I x , x ∈ Z d which are not {1/2} × [−1/2, 1/2] d−1 and intersect with it. Proof. Let γ ∈ Γ n,0 . Then it is easily seen that, if we write H i = {1/2 + i} × [−1/2, 1/2] d−1 : the (d − 1)-dimensional faces for i ∈ Z, then H i ⊂ γ for some i = 0, . . . , n. We define the graph G = (V, E), where the set of vertices V is the set of all (d − 1)-dimensional faces of I x , x ∈ Z d and the set of edges is the set of all pairs {H , H ′ } of elements of V such that H ∩ H ′ = ∅. For any H ∈ V , the number of edges which has H as an end point is C d .
Proposition 4 . 1 (
41Completeness of phase diagram). Let β > β c . We take a sequence {Λ n } ∞ n=1 of finite and c-connected subsets of Z d such that Λ n ↑ Z d . Then {µ + Φ β ,Λn } ∞ n=1 and {µ − Φ β ,Λn } ∞ n=1
) .
)Proposition 1.1 (Phase transition on the Ising model). Let d ≥ 2. Then there exists β c > 0 depending on d such that the followings hold.
).Proposition 1.4. For any β, h ∈ R with h = 0, Φ β,h has a unique Gibbs state.
It is obvious that Ψ ≤ |||Ψ|||for any d-th order decaying interaction Ψ on Ω. We see some examples of d-th order decaying interactions. Clearly, we have many d-th order decaying interactions other than these examples.Example 2.1.
) .
)Proof of Theorem 3.2. Assume that Proposition 3.4 holds. We take L, β, δ, Ψ and Φ as in Proposition 3.4. We take a sequence {Λ n } ∞ n=1 of finite and c-connected subsets of Z d such that Λ n ↑ Z d . then we have divergent subsequences {n k } ∞k=1 and {m k } ∞
k=1 of N such that
µ +
Φ,Λn k
and µ −
Φ,Λm k
converges to some µ +
Φ and µ −
Φ in M(Ω), respectively. Then, as we mentioned
above, µ
* This is named after R. L. Dobrushin, O. E. Lanford and D. Ruelle.
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Differentiability properties of the pressure in lattice systems. H A M Daniëls, A C D Van Enter, Comm. Math. Phys. 71H.A.M. Daniëls, A.C.D. van Enter: Differentiability properties of the pressure in lattice systems, Comm. Math. Phys. 71 (1980), 65-76.
Fröhlich: Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states. N Datta, R Fernández, J , J. Statist. Phys. 843-4N. Datta, R. Fernández, J. Fröhlich: Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states, J. Statist. Phys. 84 (3-4) (1996), 455-534.
Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy. N Datta, J Fröhlich, L Rey-Bellet, J. Helv. Phys. Acta. 69N. Datta, J. Fröhlich, L. Rey-Bellet: Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, J. Helv. Phys. Acta 69 (1996), 752-820.
A note on the stability of phase diagrams in lattice systems. A C D Van Enter, Comm. Math. Phys. 79A.C.D. van Enter: A note on the stability of phase diagrams in lattice systems, Comm. Math. Phys. 79 (1981), 25-32.
S Friedli, Y Velenik, Statistical mechanics of lattice systems: a concrete mathematical introduction. Cambridge University PressS. Friedli, Y. Velenik: Statistical mechanics of lattice systems: a concrete mathematical introduction, Cambridge University Press, 2017.
H Georgii, Gibbs measures and phase transitions. BerlinWalter De Gruytersecond editionH. Georgii: Gibbs measures and phase transitions, second edition, Walter De Gruyter, Berlin, 2011.
Ruelle: Condensation of lattice gases. J Ginibre, A Grossmann, D , Comm. Math. Phys. 3J. Ginibre, A. Grossmann, D. Ruelle: Condensation of lattice gases, Comm. Math. Phys. 3 (1966), 187-193.
On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model. Y Higuchi, Colloquia Mathematica Societatis János Bolyai. North-Holland, AmsterdamIRandom fieldsY. Higuchi: On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model, in Random fields, Vol. I, II (Esztergom, 1979), Colloquia Mathematica Societatis János Bolyai 27, 517-534, North-Holland, Amsterdam.
Generic triviality of phase diagrams in spaces of long-range interactions. R B Israel, Comm. Math. Phys. 106R.B. Israel: Generic triviality of phase diagrams in spaces of long-range interactions, Comm. Math. Phys. 106 (1986), 459-466.
187-218, and Extension of Pirogov-Sinai theory of phase transitions to infinite range interactions. II. Phase diagram. Y M Park, Comm. Math. Phys. 114Comm. Math. Phys.Y. M. Park: Extension of Pirogov-Sinai theory of phase transitions to infinite range interactions I. Cluster expansion, Comm. Math. Phys. 114 (1988), 187-218, and Extension of Pirogov-Sinai theory of phase transitions to infinite range interactions. II. Phase diagram, Comm. Math. Phys. 114 (1988), 219-241.
English translation: Phase diagrams of classical lattice systems. S A Pirogov, Ya G Sinai, and Teor. Mat. Fiz. 25Theor. Math. Phys.S.A. Pirogov, Ya.G. Sinai: Teor. Mat. Fiz. 25 (1975), 358-369, in Russian; English translation: Phase diagrams of classical lattice systems, Theor. Math. Phys. 25 (1975), 1185-1192, and Teor. Mat. Fiz. 26 (1976), 358-369, in Russian; English translation: Phase diagrams of classical lattice systems (Continua- tion), Theor. Math. Phys. 26 (1976), 39-49,
D Ruelle, Thermodynamic formalism: The mathematical structures of equilibrium statistical mechanics. CambridgeCambridge University Presssecond editionD. Ruelle: Thermodynamic formalism: The mathematical structures of equilibrium statistical mechanics, second edition, Cambridge University Press, Cambridge, 2004.
Sokal: More surprises in the general theory of lattice systems. A D , Comm. Math. Phys. 86A. D. Sokal: More surprises in the general theory of lattice systems, Comm. Math. Phys. 86 (1982), 327-336.
An alternate version of Pirogov-Sinai Theory. M Zahradník, Comm. Math. Phys. 93M. Zahradník: An alternate version of Pirogov-Sinai Theory, Comm. Math. Phys. 93 (1984), 559-581.
| [] |
[
"arXiv:1307.6193v1 [physics.plasm-ph] The Magneto-Rotational Decay Instability in Keplerian Disks",
"arXiv:1307.6193v1 [physics.plasm-ph] The Magneto-Rotational Decay Instability in Keplerian Disks"
] | [
"Yuri Shtemler \nDepartment of Mechanical Engineering\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael\n",
"Edward Liverts \nDepartment of Mechanical Engineering\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael\n",
"Michael Mond \nDepartment of Mechanical Engineering\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael\n"
] | [
"Department of Mechanical Engineering\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael",
"Department of Mechanical Engineering\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael",
"Department of Mechanical Engineering\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael"
] | [] | The saturation of the magnetorotational (MRI) instability in thin Keplerian disks through threewave resonant interactions is introduced and discussed. That mechanism is a natural generalization of the fundamental decay instability discovered five decades ago for infinite, homogeneous and immovable plasmas. The decay instability relies on the energy transfer from the MRI to stable slow Alfvén-Coriolis (AC) as well as magnetosonic (MS) waves. A second order forced Duffing amplitude equation for the initially unstable MRI as well as two first order equations for the other two waves are derived. The solutions of those equations exhibit bounded bursty nonlinear oscillations for the MRI as well as unbounded growth for the linearly stable slow AC and MS perturbations, thus giving rise to the magneto-rotational decay instability (MRDI). | 10.1103/physrevlett.111.231102 | [
"https://arxiv.org/pdf/1307.6193v1.pdf"
] | 20,561,290 | 1307.6193 | 1ed2f56fac128442b03a1cab0aac1148609c38cc |
arXiv:1307.6193v1 [physics.plasm-ph] The Magneto-Rotational Decay Instability in Keplerian Disks
23 Jul 2013 (Dated: May 11, 2014)
Yuri Shtemler
Department of Mechanical Engineering
Ben-Gurion University of the Negev
84105Beer-ShevaIsrael
Edward Liverts
Department of Mechanical Engineering
Ben-Gurion University of the Negev
84105Beer-ShevaIsrael
Michael Mond
Department of Mechanical Engineering
Ben-Gurion University of the Negev
84105Beer-ShevaIsrael
arXiv:1307.6193v1 [physics.plasm-ph] The Magneto-Rotational Decay Instability in Keplerian Disks
23 Jul 2013 (Dated: May 11, 2014)
The saturation of the magnetorotational (MRI) instability in thin Keplerian disks through threewave resonant interactions is introduced and discussed. That mechanism is a natural generalization of the fundamental decay instability discovered five decades ago for infinite, homogeneous and immovable plasmas. The decay instability relies on the energy transfer from the MRI to stable slow Alfvén-Coriolis (AC) as well as magnetosonic (MS) waves. A second order forced Duffing amplitude equation for the initially unstable MRI as well as two first order equations for the other two waves are derived. The solutions of those equations exhibit bounded bursty nonlinear oscillations for the MRI as well as unbounded growth for the linearly stable slow AC and MS perturbations, thus giving rise to the magneto-rotational decay instability (MRDI).
Introduction -
The magnetorotational instability (MRI) [1]- [2] is believed to play an important role in the dynamical evolution of thin astrophysical disks [3]- [4]. The analytical understanding of the processes that are responsible for the nonlinear evolution of the MRI is therefore crucial for assessing the true importance of that linear instability to such phenomena as turbulence generation in the disk and angular momentum transfer. First attempts to analyze the nonlinear evolution of the MRI focused on the dissipative saturation of the instability ( [5]- [6]) in environments that are characteristic of laboratory experiments. Recently however, a non dissipative mechanism has been proposed in the context of a thin disk geometry, according to which the MRI saturates to bounded bursty non linear oscillations by non resonantly driving a zero frequency magnetosonic (MS) wave ( [7]- [8]). The scope of the non dissipative mechanism of interacting waves is widened in the current work to include resonant interactions of three linear eigenoscillations of the system. Extending thus the weakly nonlinear analysis entails a surprising result. While the amplitude of the original MRI saturates via periodical nonlinear oscillations just as in the non-resonant case, it is shown in the current work that the amplitudes of the other two linearly stable modes that participate in the resonant triad may grow exponentially through the nonlinear magneto-rotational decay instability (MRDI) mechanism. This result provides a natural generalization of the decay instability mechanism discovered five decades ago for infinite, homogeneous, and immovable plasmas [9], to the geometry of thin, rotating, and axially stratified disks. The resonantly interacting triads of eigenmodes may serve therefore as building blocks of a turbulence model in thin magnetized disks. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
The Physical Model -The thin disk asymptotic expansion procedure [10]- [14] is applied to the magnetohydrodynamic (MHD) equations in order to study the weakly nonlinear evolution of the MRI in Keplerian disks that are subject to the action of an axial magnetic field. A detailed description of that procedure and its results for the steady-state as well as the linear problem is presented in [15]. The main results are hereby summarized: 1. Steady-State: Assuming axially isothermal steadystate the normalized mass density profiles are given by n(r, ζ) = N (r)Σ(η), where Σ(η) = e −η 2 /2 , N (r) is an arbitrary function of r, the radial coordinate, η = ζ/H(r), ζ = z/ǫ is the stretched axial coordinate, and H(r) is the semi thickness of the disk. The latter [or alternatively the temperature profile T (r)] is an arbitrary function of r. 2. Linear perturbations: Modifying the axial mass density profile toΣ(η) = sech 2 η enables the analytical solution of the linearized set of equations for small perturbations. The resulting eigenmodes are thus divided into two families. The first family, the Alfvén-Coriolis (AC) one, represents in-plain perturbations and includes two sets of axially discrete modes. The fast AC modes are stable while the slow AC modes may become unstable. The number of unstable slow AC modes is determined by the local plasma beta which is given by β(r) = β 0 N (r)C 2 s (r)/B 2 z (r) where β 0 is the beta value at some reference radius, and C s (r) and B z (r) are some arbitrary profiles of the sound velocity and the axial steady-state magnetic field, respectively. Thus, the threshold for exciting k unstable modes is given by β k cr = k(k + 1)/3, k = 1, 2, . . .. It is those unstable slow AC modes that constitute the MRI. The eigenfunctions of both sets of AC modes may be expressed in terms of the Legendre polynomials. Of particle importance is the fact that for β = β k cr the k-th eignevalue of the slow AC modes is zero with multiplicity two. The other family of eigen-oscillations in thin Keplerian disks includes the vertical magnetosonic (MS) modes. The latters are stable, possess a continuous spectrum, and their eigenfunctions are localized about the mid-plain and may be expressed in terms of some Hypergeometric functions. The two families of the linear eigenmodes, namely the AC and the MS modes, are the building blocks of the nonlinear analysis to be unfolded in the next sections.
Resonant interactions -
The scenario that is introduced in the current work is the following: a large amplitude MRI forms a triad of resonantly interacting modes with a stable fast or slow AC mode, and a stable MS wave. Such interaction is a direct result of the influence of the perturbed in-plain magnetic pressure gradients on the acoustic modes, and the simultaneous axial convection of the AC modes by the acoustic perturbations. Such mechanism underlies the well-known decay instability in plasmas that has been discovered five decades ago [9] and was shown to be of a fundamental nature. Thus, to illustrate the main idea, following [16] consider a parent Alfvén wave with amplitude a 1 (t), and two daughter waves, one of which is another Alfvén wave with amplitude a 2 (t), while the other one is a sound wave with amplitude a s (t), all co-exist in an infinite uniform, and immovable plasma. A resonant interaction between those three modes occurs if the following resonant conditions are satisfied: ω 2 = ω 1 + ω s , and k 2 = k 1 + k s . Thus, assuming that the amplitudes vary on a slower time scale than each of the inverse eigen-frequencies, the equations that describe the evolution of the interacting triad may be cast in the following way [16]:
da 1 dτ = iΓa 2 a * s (1) da 2 dτ = −iΓa s a 1 (2) da s dτ = iΓa * 1 a 2 ,(3)
where τ is a slow time variable. The solution of the above set under initial conditions that a 1 is much bigger than the other two amplitudes is characterized by cycles of exponential growth of a 2 and a s and decay of a 1 , followed by the saturation and decay of the formers and restitution of the latter. During those portions of the cycles that are marked by exponential growth of the daughter waves, a 2 and a s grow as e ντ where ν = Γa 1 (τ = 0).
Back to thin rotating disks and the MRI, the physics of resonantly interacting triads of eigenmodes is in principle similar to that described above. For simplicity it is assumed that the β value of the system is just above the first threshold for instability. Consequently, there is just one unstable MRI mode, characterized by axial wave number k = 1. The role of the large amplitude parent mode is played therefore by the k = 1 MRI, while the daughter waves are stable k = 2 slow AC and MS modes. Thus, contributions of the various modes to the pertur-bations may be expressed in the following way:
δB ⊥ (z, t) = f 1 (ζ, τ ) + f 2 (ζ, τ )e −iω2t (4) δρ(z, t) = g 1 (ζ, τ ) + g 2 (ζ, τ )e −iω2t(5)
Equations (4) and (5) describe the AC and MS modes, respectively. The first term on the right hand side of eq. (4) represents the parent MRI mode, whose real part of the frequency is zero (ω 1 = 0), while the second term describes the contribution of the k = 2 daughter slow AC mode that is characterized by the eigenvalue ω 2 . A main difference from the classical infinite plasma case is the presence of the first term on the right hand side of eq. 5 that represents the zero-frequency MS perturbations that are inevitably non resonantly driven by the parent MRI (see [8]). The second term describes the contribution of the MS eigenmode with frequency ω 2 so that the resonant condition on the frequencies is fulfilled due to the continuous nature of the MS spectrum. Time t is normalized with the local inverse rotation frequency of the disk Ω −1 (r), and the slow time is defined as τ = γt where γ << 1 is the growth rate of the parent MRI normalized with Ω(r). The amplitudes of the various modes in eqs. (4) and (5) may be postulated to be of the following form:
f 1 (ζ, τ ) = a 1 (τ )P 1 (ζ) + a * 2 (τ )a s (τ )ψ 2,s (ζ) + a 3 1 (τ )ψ 1,1 (ζ) (6) f 2 (ζ, τ ) = a 2 (τ )P 2 (ζ) + a 1 (τ )a s (τ )ψ 1,s (ζ)(7)
g 2 (ζ, τ ) = a s (τ )Q 2 (ζ) + a 1 (τ )a 2 (τ )ψ 1,2 (ζ)
(8) g 1 (ζ, τ ) = a 2 1 (τ )φ 1,1 (ζ)(9)
The first terms on the right hand sides of eqs. (6)-(8) represent the three linear modes that participate in the resonantly interacting triad where P i , i = 1, 2 are the eigenfunctions of the MRI and the stable slow AC mode (both expressed, as mentioned above, in terms of the Legendre polynomials), while Q 2 is the eigenfunction of the daughter MS mode (expressed in terms of hypergeometric functions). The second terms on the right hand side of (6)-(8) describe the nonlinear resonant interactions through the yet unknown coupling functions ψ i,j (ζ), i, j = 1, 2, s. Equation (9) describes the zero-frequency MS wave that is non resonantly forced by the parent MRI, while the last term on the right hand side of eq. (6) describe its back reaction on the MRI. It should finally be emphasized that unlike in the classical decay instability, since the parent MRI is of zero frequency, a 1 (τ ) be assumed to be real. The other two amplitudes are generally complex-valued.
During the linear stage all three modes are independent of each other so that a 1 (τ ) = a + 1 e τ + a − 1 e −τ (this form of a 1 echoes the multiplicity 2 of the corresponding eigenvalue for γ = 0), while a 2 and a s are constants. However, as a 1 grows, the nonlinear terms become progressively more important and the temporal behavior of the amplitudes change significantly. It is thus the main goal of the current work to derive the equations that govern the dynamical evolution of the three amplitudes a 1 (τ ), a 2 (τ ), and a s (τ ).
Guided by the equations of the classical decay instability [i.e. eqs. (1)-(3)], the equations for a 2 (t) and a s (t) are postulated to be of the following form:
da 2 dτ = −iΓ 2 a s a 1 (10) da s dτ = iΓ s a 1 a 2 .(11)
The equation for a 1 (τ ) however is different from its classical counterpart. First, a 1 is the amplitude of the MRI mode slightly above the instability threshold where, as indicated above, the eigenvalue is zero with multiplicity two. Hence, the equation for a 1 is expected to be of second order ( [8], [17], [18]). Furthermore, that equation has to include the influence of the driven zero-frequency magnetosonic perturbations. Taking all that into acount, and recalling the a 1 is real, the equation for a 1 is:
d 2 a 1 dτ 2 = a 1 + Ea 3 1 + Γ 1 (a 2 a * s + a * 2 a s ).(12)
The first term on the right hand side of the last equation describes the two linear modes (one exponentially growing, the MRI, and the other one evanescent) that coalesce at the threshold to a double zero eigenvalue. The second term describes the contribution of the driven zerofrequency MS perturbations, while the last two terms mark the resonant interaction with the other two modes of the triad. The calculation of the four real coupling coefficients in eqs. (10)-(12), namely Γ 1 , Γ 2 , Γ s and E, starts by realizing that those equations are written by tacitly assuming some ordering scheme among the various amplitudes. Thus, recalling that τ = γt, all terms in eqs. (10)- (12) are of the same order if the amplitude of the parent MRI is proportional to γ while the corresponding amplitudes of the daughter modes are proportional to γ 3/2 and γ 3/2 . Equations (10)- (12) are inserted now into the reduced thin-disk MHD equations [15] which are subsequently solved order by order in γ. Not surprising, the lowest order reproduces the linear results. The next order yields four non homogeneous ordinary differential equations for the coupling functions ψ 1,2 (ζ), ψ 1,s (ζ), ψ 2,s (ζ) and ψ 1,1 (ζ). The four solvability conditions for those equations (that provide a generalization of the resonant condition on the wave vectors in the classical case), result in four values for the coupling coefficients. As expected, E has the same value as in the non-resonant case, i.e., E = −27/35 [8]. The discussion concerning the values of the rest of the three coupling coefficients and their significance is deferred however until after the derivation of the solutions of eqs. (10)- (12).
Solution of the dynamical amplitude equations -Multiplying eqs. (10) and (11) by a * s and a * 2 , respectively, and summing the resulting equations yield:
d dτ [a 2 a * s + a * 2 a s ] = 0.(13)
Consequently, eq. (12) may be written as the following Duffing equation with a constant forcing term:
d 2 a 1 dτ 2 = a 1 + Ea 3 1 + Γ 10 ,(14)
where Γ 10 = Γ 1 (a 20 a * s0 + a * 20 a s0 ), and a j0 , j = 2, s are the initial values of the corresponding amplitudes. The value of Γ 10 varies within a wide range due to the arbitrariness of the initial data. The equation for a 1 may be solved now separately from those of the other two amplitudes. The value of Γ 10 determines the number of fixed points for a 1 , whether it is one (for |Γ 10 | > 2/ √ −27E) or three (for |Γ 10 | < 2/ √ −27E). However, regardless of the value of Γ 10 , the amplitude of the parent MRI, while initially growing exponentially, saturates and eventually oscillates nonlinearly in a bursty fashion with a constant amplitude, as is exemplified in Fig. (1). After solving for a 1 , the dynamical equations for the daughter modes are easily solved by defining the following new time variable:
τ ′ = |Γ 2 Γ s | τ 0 a 1 (ξ) dξ.(15)
The nature of the solution of eqs. (10) and (11) depends now on σ = sign(Γ 2 Γ s ), and is given by:
a 2 (τ ′ ) = a 20 cosh( √ στ ′ ) + iσa s0 α 2 sinh( √ στ ′ ) (16) a s (τ ′ ) = a s0 cosh( √ στ ′ ) + iσa 20 α s sinh( √ στ ′ ),(17)
where α j = Γ j / |Γ 2 Γ s |, j = 2, s. When the daughter AC mode is a k = 2 slow wave σ can be shown to be equal to 1. Equations (16) and (17) reveal therefore the following result: If Γ 2 Γ s > 0 the linearly stable AC and MS modes that participate in the resonant triad are nonlinearly destabilized by energy transfer from the linearly unstable MRI mode, which is consequently saturated. An effective growth rate of the MRDI of the daughter modes may thus be estimated as γ nl = | a 1 | |Γ 2 Γ s |, where a 1 = lim τ →∞ τ −1 τ 0 a 1 (ξ)dξ. Results - Figure (1 (2) and (3) describe the simultaneous exponential growth of the daughter waves for two different values of Γ 10 . It can be easily seen that the growth rate of the daughter waves does indeed depend on their initial conditions through the parameter Γ 10 . As the latter grows, so does | a 1 | and with it γ nl . In addition, as Γ 10 grows, the steady-state solution for a 1 changes its nature from a three fixed-points solution to a single fixedpoint one. This transition occurs for |Γ 10 | = 2/ √ −27E. Conclusions -The mechanism that is classically known as the decay instability is revisited and adapted to the geometry and physics of thin magnetized Keplerian disks. The resulting MDRI mechanism is conjectured to play an important role in the nonlinear evolution of the MRI. In the classical decay instability scenario developed for infinite homogeneous and immovable plasma, energy is transferred back and forth between a parent Alfvén wave and Alfvén and acoustic daughter waves through a three-wave resonant interaction. The thin disk version of the decay instability that has been introduced in the current work is shown to deviate significantly from its classical predecessor. Instead of the classical stable Alfvén wave, the role of the parent wave is currently played by an MRI mode that is slightly above the instability threshold. Hence, its amplitude is governed by a second order forced Duffing equation. The daughter waves are invariably AC and MS modes. In particular, it has been shown that for all possible initial conditions the parent MRI saturates in a bursty oscillatory manner. Furthermore, when the AC daughter wave is a slow AC mode, the linearly stable pair of daughter waves is nonlinearly destabilized and grow exponentially in time by tapping into the MRI energy. If, however, the role of the AC daughter wave is played by a stable fast mode, the amplitudes of all three modes remain bounded as they exchange energy periodically in a manner that resembles the classical decay instability. The picture of a resonantly interacting triad of modes may be easily generalized to a cluster of triads for a given parent MRI mode.
FIG. 2 :
2τ ′ as a function of τ [see eqs. (15)-(17)]. Same parameters as in Fig. 1.
FIG. 3: τ ′ as a function of τ [see eqs. (15)-(17)]. a1(0) = 0.05, da1/dτ (0) = 0.5, a2(0) = 1, as(0) = 0, Γ10 = 10/ √ −27E.
) demonstrate the saturation of the MRI while figures
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| [] |
[
"Statistics of relativistic electrons radiating in periodic fields",
"Statistics of relativistic electrons radiating in periodic fields",
"Statistics of relativistic electrons radiating in periodic fields",
"Statistics of relativistic electrons radiating in periodic fields"
] | [
"Eugene Bulyak \nNational Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine\n",
"Nikolay Shul'ga \nNational Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine\n",
"Eugene Bulyak \nNational Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine\n",
"Nikolay Shul'ga \nNational Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine\n"
] | [
"National Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine",
"National Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine",
"National Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine",
"National Science Center 'Kharkov Institute of Physics and Technology'\n1 Academichna strKharkivUkraine"
] | [] | We developed a general method for assessing the evolution of the energy spectrum of relativistic charged particles that have undergone small quantum losses, such as the ionization losses when the electrons pass through matter and the radiation losses in the periodic fields. These processes are characterized by a small magnitude of the recoil quantum as compared with the particle's initial energy. We convey the statistical consideration of the radiating electrons and demonstrate that at a small average number of the recoils, the electron's spectrum can be described as a composition of consecutive convolutions of the recoil spectrum with itself, taken with the Poisson mass. In this stage, the electron's spectrum reveals some individual characteristics of the recoil spectrum. Furthermore, the spectrum loses individuality and allows for proximate description in the terms of statistical parameters. This consideration reveals that the width of the electron's spectrum is increasing with the number of recoils according to the power law, with the power index being inverse to the stability parameter, which gradually increases with the number of recoils from one to two. Increase of the spectrum width limits the ability of the beam to generate coherent radiation in hard x-ray and gamma-ray region. | 10.1103/physrevaccelbeams.22.040705 | [
"https://arxiv.org/pdf/1812.07234v3.pdf"
] | 119,209,831 | 1812.07234 | 404a176ac8c264ba1401987b45aa1bcd69679520 |
Statistics of relativistic electrons radiating in periodic fields
18 Mar 2019
Eugene Bulyak
National Science Center 'Kharkov Institute of Physics and Technology'
1 Academichna strKharkivUkraine
Nikolay Shul'ga
National Science Center 'Kharkov Institute of Physics and Technology'
1 Academichna strKharkivUkraine
Statistics of relativistic electrons radiating in periodic fields
18 Mar 2019arXiv:1812.07234v3 [physics.acc-ph] (Dated: March 19, 2019)
We developed a general method for assessing the evolution of the energy spectrum of relativistic charged particles that have undergone small quantum losses, such as the ionization losses when the electrons pass through matter and the radiation losses in the periodic fields. These processes are characterized by a small magnitude of the recoil quantum as compared with the particle's initial energy. We convey the statistical consideration of the radiating electrons and demonstrate that at a small average number of the recoils, the electron's spectrum can be described as a composition of consecutive convolutions of the recoil spectrum with itself, taken with the Poisson mass. In this stage, the electron's spectrum reveals some individual characteristics of the recoil spectrum. Furthermore, the spectrum loses individuality and allows for proximate description in the terms of statistical parameters. This consideration reveals that the width of the electron's spectrum is increasing with the number of recoils according to the power law, with the power index being inverse to the stability parameter, which gradually increases with the number of recoils from one to two. Increase of the spectrum width limits the ability of the beam to generate coherent radiation in hard x-ray and gamma-ray region.
INTRODUCTION
In a number of processes involving beams of highenergy electrons, such as radiation in the periodic structures, ionization losses in matter, etc., the energy degradation of an incident electron is in the form of the small portions (recoil quanta), which spectrum is almost independent of the electron's energy. In our previous papers [1,2], we have considered the evolution of the spectrum of such electron beam. It was shown, that the spectrum is determined by the parameters of a single recoil and the average number of the recoils. This paper is concentrated on the dependence of the spectrum width on the number of recoils in the intermediate range, in between a small average number where the electron's spectrum can be described as a composition of consecutive convolutions of the recoil spectrum with itself taken with the Poisson mass, and the diffusion limit where the width increases as square root of the number of recoils. This paper is organized as follows: in the first section, we present a method of assessing the evolution of the straggling function that describes the distribution of the energy losses in the interim range of the number of recoils. In the second section, we validate the method by comparing it to the known theories at the limiting cases. The third section presents the results of the study of the kinetics of the radiating electrons in short undulators. The fourth section summarizes the results. * [email protected]; Also at V.N. Karazin National University, 4 Svodody sq., Kharkiv, Ukraine † Also at V.N. Karazin National University, 4 Svodody sq., Kharkiv, Ukraine
I. STATISTICS OF THE RADIATING ELECTRONS
A. Preliminaries
Distinguishing features of the considered system are: (i) Big number of the ensemble members (electrons in the radiating bunch) n ∼ 10 10 . (ii) Small average number of recoils (defined as ratio of the energy emitted by the electron to the mean energy of the spectrum of the radiation) x < 10 4 .
We adopt the assumption, that the spectrum of emitting quantum of the radiation inducing energy loss (recoil), w(ω) is 'physical': it has compact support, 0 ≤ ω min ≤ ω ≤ ω max < ∞ with ω being the energy of the recoil. The spectrum is normalized to unity, w(ω) dω = 1. (Here and below we drop out the infinite limits in integration.)
In this paper, we use the reduced energy units: ǫ for the energy in the straggling spectrum, ω for the energy of the spectrum of the recoil quantum, both are dimensionless, normalized to the energy unit, e.g., to the charged particle rest energy, [2]. We use a convention for the Fourier transform in the form of:
(F f )(s) =f (s) = e −2iπωs f (ω) dω
with s being the variable in the Fourier transform domain that complements to ω (or ǫ). For the inverse Fourier transform, (F −1 f )(s) =f (s), the (−) sign in the exponent of the integrand is replaced with (+) sign.
A sketch of the straggling electron's trajectory in the plane (x, ǫ) is presented in Fig. 1. The trajectory is composed of the free paths of random length, with the mean unit value and the (positive) random jumps having the same probability density distribution w(ω). Such a process belongs to the subclass of the Subordinate to the Compound Poisson Process, which in turn belongs to the α-stable (or Lévy) processes, see, e.g., [3].
The evolution of the electron's bunch spectrum is described by a transport equation, [4,5]:
∂f (x, γ) ∂x = ∞ −∞ [w(ω, γ + ω)f (x, γ + ω)− (1) w(ω, γ)f (x, γ)] dω ,(2)
with γ being the dimensionless particle energy (Lorentz factor). A solution to (1) in a form of the characteristic function (Fourier transform of the distribution density), [1,2] is:
f =f 0 exp[x(w − 1)] ,(3)
where w(ω) is the recoil's spectrum, and f 0 is the initial spectrum. The parameter x > 0 is the ensemble average number of the recoils undergone by an electron since entering the driving force, [1,2]. The equation (3) may be generalized and simplified due to the model's assumption of the independence of recoils on the electron's energy, as proposed in [4]. Instead of the beam spectrum, we consider the distribution density of losses: so called straggling function [2]. The straggling function, normalized to unity, presents the loss spectrum: only the particles that have undergone at least one recoil contribute to it.
The characteristic function for the straggling function S x and the Poisson-weighted expansion arê
S x =ŵe x(ŵ−1) , S x (ǫ) = ∞ n=0 e −x x n Γ(n + 1) F n (ǫ) ,(4)F n = F n−1 * w , F 0 = w .
where * stands for the convolution operation. The first three moments of the straggling function-mean, variance, and skewness-read:
ǫ = (1 + x) ω ; (5a) Var[ǫ] ≡ (ǫ − ǫ) 2 = (1 + x) ω 2 − ω 2 ; (5b) Sk[ǫ] ≡ (ǫ − ǫ) 3 = (1 + x) ω 3 − 3 ω 2 ω + 2 ω 3 . (5c)
Here ω, ω 2 , ω 3 are the raw moments of the recoil spectrum w(ω), ω n ≡ ω n w(ω) dω, the 'overline' sign indicates the ensemble average. A universal solution for the straggling function (4) allows for accurate evaluation at the beginning of the process, x 1 when the series may be limited to a few selfstates F n ; and in the opposite limit of the large number of recoils, x → ∞ when a few first moments (5) adequately represent the function. The first moment-mean energy loss-always holds, since it presents the energy conservation law.
B. Statistical properties for finite number of recoils
From a practical point of view,the most interesting for the physically realizable systems is a medium number of recoils, when a particle, after entering the system, lost a small fraction of its initial energy in the moderate number of the recoils, xω ≪ γ 0 with γ 0 being the initial electron energy.
To evaluate the functional dependency of the spectrum width against the average number of recoils, σ = σ(x), we compare the distribution (4) with the Lévy α-stable distributions, the only ones to which the sum of independent identically distributed variables is attracted, see [6].
It should be emphasized, that despite the relevance of the trajectory to the α-stable class, the bunch of such trajectories do not rigorously match the class, since individual trajectories come into stage at different x, as is depicted in Fig. 2. Nevertheless, at x ≫ 1 when almost all of the particles have been recoiled and the width of the distribution exceeds the width of the spectrum of recoils, the bunch of the trajectories is expected to obey a stable law.
The characteristic function of α-stable process, see e.g. [6], has a general form:
φ(s) = exp {−2πisµ − |2πσs| α [1 − iβ sign(s)Φ]} , (6) with Φ = tan πα 2 , α = 1, − 2 π log |s|, α = 1,
The parameters of the stable distribution are α ∈ (0, 2] the stability parameter, σ > 0 the scale parameter, β the skewness parameter, and µ the location parameter.
The model under consideration allows for a reduction of range of the parameters: the stability parameter should be in the range α ∈ (1, 2] due to a finite mean of the recoil spectrum; the location parameter is simply equal to the first moment of the straggling function (5a).
Comparison of the characteristic function of straggling (4) with that of the stable distribution (6) leads to two important consequences: (i) the scaling parameter σ is determined by the real part of the exponent, and (ii) the Fourier transform of a recoil spectrum in general may not be of the power form, ∝ |s| α with α = const. Because of aiming the study at evaluation the scale parameter of distribution and taking into account the similarity theorem-the width of the distribution is inversely proportional to the width of its Fourier transform-we suggest evaluating the stability parameter at s = s * where the real part of the exponent of the characteristic function equals to unity:
ℜ[x (1 −ŵ(s * ))] = 1 = |πσs * | α .(7)
Here in the square brackets of left-hand side of (7), we intentionally omit the small term ℜ[logŵ(s * )] ∝ −s 2 * . From this suggestion, an expression for the stability parameter is readily derived:
α = sD s ℜ[ŵ] 1 − ℜ[ŵ] s=s * ,(8)
where D s ≡ ∂ ∂s and s * = s * (x) > 0 is the root of (7).
Substituting the explicit expression for ℜ[ŵ],
|πσs * | α = x w(ω) [1 − cos(−2πs * ω)] dω = x w(ω) [πs * ω] α dω = xm α [w] [πs * ] α ,
we get a general dependence of the scale parameterthe width of the straggling distribution-on number of recoils:
σ(x) = [xm α [w]] 1/α ,(9)
where m α [w] is the raw generalized α-moment of the recoil spectrum:
m α [w] ≡ ω α w(ω) dω .
Thus, the width of the spectrum increases with the average number of the recoils as ∝ x 1/α(x) . The stability parameter α(x), in turn, increases with x from unity to two.
It should be noted, that the scale parameter is equal to half-width of the distribution at 1/e of the maximum. At α → 2 when the distribution approaches the normal (Gaussian) distribution, the scale parameter approaches the square root of Gaussian variance divided by two,
σ(x) → (ǫ − ǫ) 2 /2 1/2 .
II. VERIFICATION OF THE METHOD
The two known functional limits of the considered process, 1 ≤ α ≤ 2 with α = 1 being the Landau distribution and α = 2 the Fokker-Plank (diffusive limit, the Gaussian distribution), may be considered the benchmarks of the method, see, e.g., [7].
A. The diffusion limit
As stated in the Central Limit Theorem, the sum of independent identically distributed variables with the finite variance should approach the normal (Gaussian) distribution, which is a limiting case of the stable distributions with α = 2. Directly following from (4), at x → ∞, the real part of the exponent approaches Gaussian:
ℜ[ŵ − 1] ≈ 2π 2 s 2 ω 2 .
The same result, α = 2, directly stems from (8) since s * → 0 when x → ∞.
B. The Landau distribution
A particular case of the stable distributions, the Landau distribution function [4] (α = 1, β = 1), is of special importance since it has undergone extensive study and experimental validation, see [8,9]. The process of ionization losses described by the Landau distribution, agrees with the assumption of the small recoils, whose spectrum is independent of the energy of the particles. The problem is the employment of the idealized unbound recoil spectrum of ∝ ω −2 dependence on energy. This spectrum-the Rutherford cross section-can not be normalized (it has infinite moments).
To avoid the divergence, we consider a truncated recoil spectrum, 0 < a ≤ ω ≤ b < ∞, then take the limits a → 0, b → ∞, and keep the total energy losses finite. A 'physical' normalized Rutherford cross section (see [10,11]) reads:
w L (ω) = sign(ω − a) − sign(ω − b) 2ω 2 ab (b − a) .(10)
Its raw moments are finite:
ω = ab b − a log b a , ω 2 = ab .
Fourier transform for this cross-section is:
w L (s) = 1 (b − a) √ 2π × (11) {b cos(as) − a cos(bs) + sab [Si(as) − Si(bs)] − i [sab (Ci(as) − Ci(bs)) − b sin(as) + a sin(bs)]}
where Si(z) = z 0 sin(t)/t dt is the integral sinus, and Ci(z) = − ∞ z cos(t)/t dt is the integral cosine. Explicitly, we have for the model:
α L (s; a, b) = 1 + a cos(2πbs) − b cos(2πas) ab(b − a) + 2πsab (b − a) [Si(2bπs) − Si(2aπs)] ,(12)
with s = s * being the root of (7). The stability parameter (12) has two limits: (i) when a, b finite, x → ∞ (accordingly s * → 0) and (ii) x finite, a → 0, b → ∞ (Rutherford cross section):
lim s→0 α L (s; a, b) = 2 , 0 < a < b < ∞ ; (13a) lim a→0,b→∞ α L (s; a, b) = 1 , 0 < s .(13b)
Thus, the stability parameter (12) coincides with the Landau distribution at the Rutherford cross section and with the Gaussian distribution at x → ∞ and the finitemoments recoil spectra.
For the physically grounded cases of ionization losses, both the Landau and the Vavilov formulas are valid. The Landau distribution evolves into the Gaussian well beyond the physical region, as is illustrated in Fig. 3 for 50 MeV electrons traversing liquid hydrogen (in this case the ionization losses are dominant).
As it can be seen from Fig. 3, the Landau distribution adequately describes evolution of the straggling function. The width of the distribution linearly within the range of validity linearly increases with the mean losses. (Small oscillations in the stability parameter occur because of errors in the numerical computation of the root s * .) FIG. 3. The stability parameter (black curve) and the scale parameter (blue) against mean energy loss. The vertical green line indicates the limit of validity of the Landau distribution (10% loss, see [12]), the red line indicates the physical limit: all the energy radiated out.
III. RADIATION IN PERIODIC STRUCTURE
As an example of application of the method to a practical case, we consider evolution of the straggling function due to emission of the undulator radiation, see, e.g., [13]. The undulator parameter K, which is:
K = eBλ u 2πm e c ,
where B is the magnetic field strength, λ u is the spatial period of the magnetic field, e, m e are the electron charge and the rest mass, resp., c the speed of light. Evolution of the straggling function for a long undulator and K 1 approximates the diffusion process [14]. On the other hand, for the dipole radiation K ≪ 1, and a short undulator (or the entrance section of a long undulator), x 5 this function is asymmetric and non-Gaussian [1,15]. Figure 4 represents the straggling function profiles computed in accordance to (4) for a small average number of recoils. It shows, that the straggling function resembles the spectrum of the recoil at x ≪ 1, then it gradually spreads out and smoothes, approximating to some degree the Landau distribution.
The stability parameter against the number of recoils, computed based on (8) for different undulator parameters K is presented in Fig. 5. As it can be seen from the figure, the wider the recoil spectrum, the later the stability parameter approaches the diffusion limit of α = 2.
When the stability parameter approaches the 'diffusion' value of α = 2 (still remaining below it), the third centered moment (5c) stays positive and increases with x. We can derive practical information about the mode of the distribution. Making use of Pearson's skewness for a distribution close to normal, see, e.g. [16],
ǫ − ǫ mode σ = Sk[ǫ] 2σ 3 ,
where ǫ mode is the maximum of the distribution density, we get:
ǫ mode = ǫ − Sk[ǫ] 2σ 2 = (1 + x)ω − (1 + x)ω 3 − 3ω 2 ω + 2ω 3 2 (1 + x)ω 2 − ω 2 .(14)
For a big number of the recoils, x → ∞, the shift of the mode from the mean is almost independent of the number of recoils:
ǫ mode − ǫ ≈ − ω 3 2ω 2 .
The mode-position of the maximum-is shifted from the mean to smaller energy losses by the constant value, which is determined by the raw moments of the recoil spectrum.
IV. SUMMARY
A general dependence of the distribution of energy losses by the relativistic electrons due to radiation in periodic structures or ionization losses in matter was analyzed. The straggling function-distribution density of fluctuations-is determined solely by the ensembleaverage number of recoils having undergone by the particle since entering the field (or medium in the case of ionization losses), and the spectrum of the recoil.
The straggling function was compared to the Lévy stable process as the only attractor of such processes according to the Generalized Central Limit Theorem. The results of this consideration reveal that the width of the electron spectrum is increasing with the number of recoils according to the power law, with the power index being inverse to the stability parameter, i.e., linearly with the number of recoils at the beginning of the process, and in proportion to the square root from the number of recoils at the diffusion limit.
Increase of the spectrum width limits the ability of the beam to generate coherent radiation in hard x-ray and gamma-ray region.
Despite of the assumed independence of the recoil spectrum on the electron's energy, a 'negligible' (from the electron's point of view) change in this spectrum may play an important role in the reduction of the brightness of sources of hard x-rays and gamma-rays, which employed relativistic electrons.
It occurs due to the fact that only a small fraction of the spectrum is used: the pin-hole fraction of the spectrum has strong dependency upon the energy spread of
FIG. 1 .
1A sketch of straggling process. The distribution of recoil magnitudes resembles the dipole radiation spectrum.
FIG. 2 .
2Four simulated trajectories for the recoils from the dipole radiation emission (top). Density of the non-recoiled particles is indicated in grey on the bottom panel.
FIG. 4 .FIG. 5 .
45Straggling distribution function caused by recoils in helical undulator, K = 1, x = 0.01, 0.Stability parameter for K = 0.01 (blue), K = 0.3 (green), and K = 1 (red) vs average number of recoils. (The inset presents the corresponding recoil spectra.)
The attainable width of the pin-hole collimated radiation-upper limit of it-will exceed double of the electron bunch energy spread. 17electrons. The attainable width of the pin-hole col- limated radiation-upper limit of it-will exceed double of the electron bunch energy spread, [17].
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| [] |
[
"CP Asymmetry Measurements in D Decays from Belle",
"CP Asymmetry Measurements in D Decays from Belle"
] | [
"A J Schwartz \nPhysics Departement\nUniversity of Cincinnati\nP.O. Box 21001145221CincinnatiOhioUSA\n"
] | [
"Physics Departement\nUniversity of Cincinnati\nP.O. Box 21001145221CincinnatiOhioUSA"
] | [
"PRESENTED AT The 6 th International Workshop on Charm Physics (CHARM 2013)"
] | We present measurements of CP asymmetries in D decays performed by the Belle experiment running at the KEKB asymmetric-energy e + e − collider. | null | [
"https://arxiv.org/pdf/1312.5272v2.pdf"
] | 118,158,273 | 1312.5272 | 847f1514c9ddc50dc18be6e9524e1999f3a893b4 |
CP Asymmetry Measurements in D Decays from Belle
31 August -4 September, 2013
A J Schwartz
Physics Departement
University of Cincinnati
P.O. Box 21001145221CincinnatiOhioUSA
CP Asymmetry Measurements in D Decays from Belle
PRESENTED AT The 6 th International Workshop on Charm Physics (CHARM 2013)
Manchester, UK31 August -4 September, 2013
We present measurements of CP asymmetries in D decays performed by the Belle experiment running at the KEKB asymmetric-energy e + e − collider.
Introduction
The phenomenon of CP violation (CP V ) is well-established in the K 0 -K 0 and B 0 -B 0 systems [1,2]. The rate observed confirms the Cabibbo-Kobayashi-Maskawa (CKM) theory of quark flavor mixing [3]. The CKM theory predicts only tiny CP V in the D 0 -D 0 system [4], and to-date such CP V has not been observed. Both timeindependent and time-dependent measurements of partial widths can exhibit CP asymmetries. The former results mainly from interference between two decay amplitudes with different weak phases; this is called "direct" CP V . The latter results from either unequal rates of flavor mixing (called "indirect" CP V ) or interference between a mixed and an unmixed decay amplitude. In all cases new physics can increase the rate of CP V significantly above that predicted by the CKM theory [4]. Here we present results from searches for CP V in D decays from the Belle experiment. For a review of mixing and CP V formalism, see Ref. [5].
2 Time-dependent D 0 (t) → K + K − /π + π − The Belle experiment has measured the mixing parameter y CP and the CP -violating parameter A Γ using 977 fb −1 of data [6]. Both observables depend on mixing parameters x and y and CP V parameters |q/p| and φ. To lowest order the relations are y CP = (|q/p| + |p/q|) 2 y cos φ − (|q/p| − |p/q|) 2
x sin φ (1)
A Γ = (|q/p| − |p/q|) 2 y cos φ − (|q/p| + |p/q|) 2 x sin φ(2)
The parameters are determined by measuring lifetimes of D 0 and D 0 mesons to flavor-specific and CP -specific final states, e.g.,
y CP = τ (K − π + ) τ (K + K − ) − 1 (3) A Γ = τ (D 0 → K + K − ) − τ (D 0 → K + K − ) τ (D 0 → K + K − ) + τ (D 0 → K + K − ) .(4)
The latter measurement requires tagging the flavor of the decaying D meson, and this is done by reconstructing D * + → D 0 π + and D * − → D 0 π − decays, i.e., the charge of the accompanying π ± (which has low momentum and is often called the "slow pion") identifies the D flavor. Both K + K − and π + π − final states are used by Belle, and the fitted lifetime distributions are shown in Fig. 1. The precision of the measurement depends upon good understanding of the decay time resolution of the detector. For Belle, the resolution function depends upon θ * , the polar angle with respect to the e + beam of the D 0 in the e + e − center-of-mass (CM) frame. Thus the ratios in Eqs. (3) and (4) are measured in bins of cos θ * . The resolution function also depends on the detector configuration: for the first 153 fb −1 of data a 3-layer silicon vertex detector (SVD) was used, while for the remaining data a 4-layer SVD was used. These running periods ("SVD1" and "SVD2") are treated separately. The results for SVD2 data are plotted in Fig. 2. Fitting both SVD1 and SVD2 values to constants gives
y CP = (+1.11 ± 0.22 ± 0.11)% (5) A Γ = (−0.03 ± 0.20 ± 0.08)% .(6)
As a test of the resolution function, the absolute lifetime for D 0 → K − π + [7] decays is also measured. The result is 408.5 ± 0.5 fs, which is consistent with the world average [8]. 3
Time-integrated D 0 → K + K − /π + π −
The D → K + K − /π + π − samples used previously can be integrated over all decay times to measure the CP asymmetry A f CP , defined as
A f CP ≡ Γ(D 0 → f ) − Γ(D 0 → f ) Γ(D 0 → f ) + Γ(D 0 → f ) .(7)
This parameter is a difference in partial widths rather than a difference in lifetimes and thus depends strongly on the specific final state. In addition to the underlying CP asymmetry, there is a "forward-backward" asymmetry (A F B ) in D 0 /D 0 production due to γ-Z 0 electroweak interference and higher order QED effects in e + e − → cc; and there is an asymmetry A π ε in the reconstruction of π ± s from D * ± → Dπ ± s decays used to tag the D flavor. The reconstructed asymmetry one measures is the sum of all three:
A f recon = A f CP + A F B + A π ε .
Belle has measured A f recon for D 0 → K + K − /π + π − decays using 977 fb −1 of data [9] to determine A KK CP and A ππ CP . To correct for A π ε , Belle measures A f recon for flavor-tagged and untagged D 0 → K − π + decays. These decays have an additional asymmetry due to differences in the reconstruction efficiency of K − π + versus K + π − ; this difference is denoted A Kπ ε . Thus
A Kπ tagged = A Kπ CP + A F B + A Kπ ε + A π ε (8) A Kπ untagged = A Kπ CP + A F B + A Kπ ε ,(9)
and taking the difference A Kπ tagged − A Kπ untagged yields A π ε . In practice this is done by re-weighting events: A Kπ tagged is calculated by weighting D 0 decays by a factor 1 − A Kπ untagged (p D 0 , cos θ D 0 ) and weighting D 0 decays by a factor 1+A Kπ untagged (p D 0 , cos θ D 0 ), where p D and θ D are the momentum and polar angle with respect to the e + beam of the D. The resulting A Kπ tagged equals A π ε . The signal asymmetry A f recon is then calculated by weighting D 0 decays by a factor 1 − A π ε (p π , cos θ π ) and D 0 decays by a factor 1 + A π ε (p π , cos θ π ). The asymmetry A π ε is calculated in bins of p π and θ π to reduce systematic errors. The result equals A f CP + A F B . Since A F B is an odd function of cos θ * , where θ * is the polar angle of the D in the CM frame, and A f CP is nominally an even function of θ * , the individual asymmetries are extracted via
A f CP = A f,corr recon (cos θ * ) + A f,corr recon (− cos θ * ) 2 (10) A F B = A f,corr recon (cos θ * ) − A f,corr recon (− cos θ * ) 2 .(11)
The results of Eq. (10) for K + K − and π + π − final states are plotted in Fig. 3 for each bin of cos θ * . Fitting these values to constants yields
A KK CP = (−0.32 ± 0.21 ± 0.09)% (12) A ππ CP = (+0.55 ± 0.36 ± 0.09)% ,(13)
and
∆A CP ≡ A KK CP − A ππ CP = (−0.87 ± 0.41 ± 0.06)%. D K K D 4.058 / 3 P1 -0.3249E-02 0.2088E-02 |cos θ * | A CP a) 6.537 / 3 P1 0.5470E-02 0.3575E-02 |cos θ * | A CP b) -0.
Direct CP V in Neutral D 0 Decays
The above measurements of A Γ , A KK CP , and A ππ CP are sensitive to underlying parameters [4] a indir
CP = (|q/p| + |p/q|) 2 x sin φ − (|q/p| − |p/q|) 2 y cos φ (14) a dir CP = 2 A A sin φ sin(δ − δ) ,(15)
where
A ≡ A(D 0 → h + h − ), A ≡ A(D 0 → h + h − )
, and δ(δ) is the strong phase for amplitude A(A). Parameters a indir CP and a dir CP parameterize the amounts of indirect and direct CP violation in D decays, respectively. Since D decays are well-dominated by tree amplitudes, the phase φ ≡ Arg[(q/p)(A/A)] ≈ Arg(q/p) is "universal," i.e., common to all D 0 decay modes. From Eq. (14) this implies that a indir CP is also universal. On the other hand, a dir CP depends on the final state. The relations between the observables and parameters are, to subleading order [10],
A Γ ≈ −a indir CP − a dir CP y cos φ(16)A hh CP ≈ a dir CP − A Γ t τ ,(17)
where t is the mean decay time for D 0 → h + h − and τ is the D 0 lifetime. The second term in Eq. (16) is the subleading contribution -compare to Eq. (2). It is O(10 −4 ) or smaller and usually neglected; i.e., A Γ ≈ −a indir CP and is considered universal. Inserting Eq. (16) into Eq. (17) gives
A hh CP = a dir CP + a indir CP t τ + a dir CP y cos φ t τ .(18)
Experimentally, many systematic errors cancel when measuring the difference ∆A CP ≡ A KK CP − A ππ CP . Using Eq. (18) to calculate this difference, one obtains
∆A CP = ∆a dir CP + a indir CP ∆ t τ + a KK dir CP t KK τ − a ππ dir CP t ππ τ y cos φ (19) = ∆a dir CP 1 + y cos φ t τ + a indir CP + a dir CP y cos φ ∆ t τ (20) ≈ ∆a dir CP 1 + y cos φ t τ + a indir CP ∆ t τ ,(21)where ∆a dir CP ≡ a KK dir CP − a ππ dir CP , a dir CP ≡ a KK dir CP + a ππ dir CP /2, ∆ t ≡ t KK − t ππ ,
and t ≡ t KK + t ππ /2. Using Eq. (21) and A Γ = −a indir CP , one can fit the measured values of A Γ and ∆A CP for parameters a dir CP and a indir CP . A deviation from zero of either of these parameters would indicate CP V in D decays. Such an observation would hint at new physics. To perform this fit requires knowledge of t , ∆ t , and y cos φ.
The Heavy Flavor Averaging Group (HFAG) [11] performs this fit for all available data: Belle, BaBar, CDF, and LHCb measurements. They use values of t and ∆ t specific to each experiment, and y cos φ is calculated using world average values [12]. The resulting fit is shown in Fig. 4, which plots all relevant measurements in the two-dimensional ∆a dir CP -a indir CP plane. The most likely values and ±1σ errors are [13]
Direct
CP V in D + → K 0 S K +
The decay D + → K 0 S K + is self-tagging, there is no D * ± → Dπ ± decay, and thus there is no correction for A π ε . However, the final state K ± introduces a correction ( CP -a indir CP plane with constraints from individual experiments overlaid, from Ref. [13]. The cross denotes the fitted central value, and the ellipses denote 1σ, 2σ, and 3σ confidence regions. due to possible differences in K + and K − reconstruction efficiencies. In addition, as the neutral K 0 or K 0 is reconstructed via K 0 S → π + π − decay, there is an asymmetry (A K 0 CP ) due to the small difference in rates between K 0 → K 0 S and K 0 → K 0 S , or equivalently between K 0 → π + π − and K 0 → π + π − [14]. Thus:
A K + ε ) indA K S K + recon = A K 0 K + CP + A K 0 CP + A F B + A K + ε .
Belle has measured A K S K + recon using 977 fb −1 of data [15] to determine A K 0 K + CP . To determine A K + ε , Belle measures the asymmetries for untagged samples of D 0 → K − π + and D + s → φπ + decays. These asymmetries have the following components:
A K − π + recon = A F B + A π + ε − A K + ε(24)A φπ + recon = A F B + A π + ε .(25)
To isolate A K + ε , the weighting procedure performed for time-integrated D 0 → K + K − decays (see Section 3) is repeated here: D 0 → K − π + decays are weighted by a factor (1 − A φπ + recon ), and D 0 → K + π − decays are weighted by a factor (1 + A φπ + recon ). With this weighting the asymmetry A K − π + recon is calculated; the result equals −A K + ε . This procedure is then repeated for the signal sample: D + → K 0 S K + decays are weighted by a factor (1 − A K + ε ), and D − → K 0 S K − decays are weighted by a factor (1 + A K + ε ). The resulting A K S K + recon equals
A K 0 K + CP + A K 0 CP + A F B .
Since the sum A K 0 K + CP + A K 0 CP is an even function of cos θ * (θ * is the polar angle with respect to the e + beam of the D + in the e + e − CM frame), and A F B is an odd function, the two types of asymmetries are separated by taking sums and differences as done in Eqs. (10) and (11). The results are plotted in Fig. 5 in bins of cos θ * ; fitting these values to a constant yields
A K 0 K + CP + A K 0 CP = (−0.25 ± 0.28 ± 0.14)% .(26)
To extract
A K 0 K + CP from A K 0 K + CP + A K 0
CP , one corrects for A K 0 CP using the calculation of Ref. [14]. The result is
A K 0 K + CP = (+0.08 ± 0.28 ± 0.14)% ,(27)
which is consistent with zero. Figure 5:
A K S K + CP = A K 0 K + CP + A K 0 CP (top) and A F B (bottom) measured from D ± → K 0 S K ± decays [15]. 6 Direct CP V in D + → K 0 S π +
The decay D + → K 0 S π + is similar to D + → K 0 S K + in that it is also self-tagging and thus has no correction for D * ± → Dπ ± . However, the final state π + introduces a correction A π + ε due to possible differences between π + and π − reconstruction efficiencies. The asymmetry A K 0 CP is also present and must be corrected for as done for D + → K 0 S K + decays. Thus:
A K S π + recon = A K 0 π + CP + A K 0 CP + A F B + A π + ε .(28)
Belle has measured A K S π + recon using 977 fb −1 of data [16] to determine A K 0 π + CP .
To determine A π + ε , Belle measures the asymmetries for untagged samples of threebody D + → K − π + π + and D 0 → K − π + π 0 decays. These asymmetries have the following components:
A K − π + π + recon = A F B + A K − π + ε + A π + ε (29) A K − π + π 0 recon = A F B + A K − π + ε .(30)
To isolate A π + ε , the weighting procedure done for time-integrated D 0 → K + K − decays (Section 3) and D + → K 0 S K + decays (Section 5) is repeated again: D + → K − π + π + decays are weighted by a factor (1 − A K − π + π 0 recon ), and D − → K + π − π − decays are weighted by a factor (1 + A K − π + π 0 recon ). The resulting asymmetry A K − π + π + recon equals A π + ε . The weighting procedure is then repeated for the signal sample: D + → K 0 S π + decays are weighted by a factor (1 − A π + ε ), and D − → K 0 S π − decays are weighted by a factor (1 + A π + ε ). The resulting A K S π + recon equals A K 0 π + CP + A K 0 CP + A F B . Taking sums and differences in bins of cos θ * as done in Eqs. (10) and (11) isolates A K 0 π + CP + A K 0 CP . The results are plotted in Fig. 6; fitting these values to a constant yields
A K 0 π + CP + A K 0 CP = (−0.363 ± 0.094 ± 0.067)% .(31)
To extract A K 0 π + CP , one applies a correction for A K 0 CP [14]. The result is
A K 0 π + CP = (−0.024 ± 0.094 ± 0.067)% .(32)
The statistics of this measurement are high enough to observe the asymmetry due to A K 0 CP with a significance of 3.2σ. However, after correcting for A K 0 CP the result for A K 0 π + CP is consistent with zero.
Other Searches for Direct CP V
In addition to the searches described above, there have been numerous other searches for direct CP V in D decays at Belle. A complete listing of these searches and their results is given in Table 1. In all cases there is no evidence for direct CP V .
Figure 1 :
1Decay time distributions for D 0 → K + K − /π + π − , D 0 → K + K − /π + π − , and D 0 → K − π + . Fitting these distributions yields y CP and A Γ via Eqs.(3)and(4). The shaded histograms show background contributions; the lower plots show the fit residuals.
Figure 2 :
2Fitted values of y CP , A Γ , and τ D 0 for SVD2 data in 18 bins of cos θ * (see text). Fitting these points to constants yields the overall values listed.
Figure 3 :
3Asymmetries A KK CP (left) and A ππ CP (right) calculated in bins of cos θ * (see text). Fitting these values to constants yields the results listed in Eqs.(12) and(13).
CP = (−0.333 ± 0.120)% . (23) Whereas a indir CP is consistent with zero, ∆a dir CP indicates direct CP V . The p-value of the no-CP V point (a indir CP , ∆a dir CP ) = (0, 0) is 0.02.
Figure 4 :
4Two-dimensional ∆a dir
Figure 6 :
6A K S π + CP = A K 0 π + CP + A K 0 CP (top) and A F B (bottom) measured from D ± → K 0 S π ± decays[16].
→ K + π − π + π − 281 fb −1Table 1: Searches for direct CP V in D 0 , D + , and D + s decays at Belle.Decay mode
Data
A CP (%)
Reference
D 0 → π + π −
977 fb −1
(+0.55 ± 0.36 ± 0.09)%
[9]
D 0 → K + K −
977 fb −1
(−0.32 ± 0.21 ± 0.09)%
[9]
D 0 → K 0
S π 0
791 fb −1
(−0.28 ± 0.19 ± 0.10)%
[17]
D 0 → K 0
S η
791 fb −1
(0.54 ± 0.51 ± 0.16)%
[17]
D 0 → K 0
S η ′
791 fb −1
(0.98 ± 0.67 ± 0.14)%
[17]
D 0 → K + π − π 0
281 fb −1
(−0.6 ± 5.3)%
[18]
D 0 (−1.8 ± 4.4)%
[18]
D + → K 0
S K +
977 fb −1
(0.08 ± 0.28 ± 0.14)%
[15]
D + → K 0
S π +
977 fb −1 (−0.024 ± 0.094 ± 0.067)%
[16]
D + → φ π +
955 fb −1
(0.51 ± 0.28 ± 0.05)%
[19]
D + → π + η
791 fb −1
(1.74 ± 1.13 ± 0.19)%
[20]
D + → π + η ′
791 fb −1
(−0.12 ± 1.12 ± 0.17)%
[20]
D +
s → K 0
S π +
673 fb −1
(5.45 ± 2.50 ± 0.33)%
[21]
D +
s → K 0
S K +
673 fb −1
(0.12 ± 0.36 ± 0.22)%
[21]
ACKNOWLEDGEMENTSWe are thankful to the organizers of CHARM 2013 for a fruitful and enjoyable workshop.
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| [] |
[
"TRANSFINITE NORMAL AND COMPOSITION SERIES OF MODULES",
"TRANSFINITE NORMAL AND COMPOSITION SERIES OF MODULES"
] | [
"R A Sharipov "
] | [] | [] | Normal and composition series of modules enumerated by ordinal numbers are studied. The Jordan-Hölder theorem for them is discussed. | null | [
"https://arxiv.org/pdf/0909.2068v1.pdf"
] | 16,958,856 | 0909.2068 | 6236092992219fc69804424fecc1cef71fabf548 |
TRANSFINITE NORMAL AND COMPOSITION SERIES OF MODULES
10 Sep 2009
R A Sharipov
TRANSFINITE NORMAL AND COMPOSITION SERIES OF MODULES
10 Sep 2009arXiv:0909.2068v1 [math.RT]
Normal and composition series of modules enumerated by ordinal numbers are studied. The Jordan-Hölder theorem for them is discussed.
Introduction and some preliminaries.
Transfinite normal and composition series of groups were studied in [1]. They generalize classical normal and composition series which are finitely long. In this paper we reproduce the results of [1] in the case of left modules over associative rings and associative algebras. In both cases each element a ∈ A produces an operator ϕ(a) that can act upon any element v ∈ V . The result of applying ϕ(a) to v is denoted as
u = ϕ(a)(v).
(1.1)
In some cases the formula (1.1) is written as follows:
u = ϕ(a)v. (1.2)
2000 Mathematics Subject Classification. 16D70.
Typeset by A M S-T E X
The endomorphism ϕ(a) ∈ End(V ) in this formula is presented as a left multiplier for the element v ∈ V . In some cases the symbol ϕ is also omitted:
u = a v. (1.3)
The formula (1.3) is a purely algebraic version of the formulas (1.1) and (1.2). The action of ϕ(a) upon v here is written as a multiplication of v on the left by the element a ∈ A. This is the reason for which V is called a left A-module. Using (1.3), the definitions 1.1 and 1.2 can reformulated as follows.
Definition 1.3. A left module over an associative ring A is an additive Abelian group V equipped with an auxiliary operation of left multiplication by elements of the ring A such that the following identities are fulfilled:
1) a (v 1 + v 2 ) = a v 1 + a v 2 for all a ∈ A and for all v 1 , v 2 ∈ V ; 2) (a 1 + a 2 ) v = a 1 v + a 2 v for all a 1 , a 2 ∈ A and for all v ∈ V ; 3) (a 1 a 2 ) v = a 1 (a 2 v) for all a 1 , a 2 ∈ A and for all v ∈ V .
Definition 1.4. A left module over an associative K-algebra A is a linear vector space V over the field K equipped with an auxiliary operation of left multiplication by elements of the algebra A such that the following identities are fulfilled:
1) a (v 1 + v 2 ) = a v 1 + a v 2 for all a ∈ A and for all v 1 , v 2 ∈ V ; 2) (a 1 + a 2 ) v = a 1 v + a 2 v for all a 1 , a 2 ∈ A and for all v ∈ V ; 3) (a 1 a 2 ) v = a 1 (a 2 v) for all a 1 , a 2 ∈ A and for all v ∈ V ; 4) (k a) v = a (k v) = k (a v) for all k ∈ K, a ∈ A, and v ∈ V .
Right A-modules are similar to left ones. Here elements v ∈ V are multiplied by elements a ∈ A on the right. However, each right A-module can be treated as a left A • -module, where A • is the opposite ring (opposite algebra) for A (see Chapter 5 in [2]). For this reason below we consider left modules only.
Submodules and factormodules.
Definition 2.1. Let V be a left A-module, where A is a ring. A subset W ⊆ V is called a submodule of V if it is closed with respect to the inversion, with respect to the addition, and with respect to the multiplication by elements of the ring A:
1) v ∈ W implies −v ∈ W ; 2) v 1 ∈ W and v 2 ∈ W imply (v 1 + v 2 ) ∈ W ; 3) v ∈ W and a ∈ A imply (a v) ∈ W .
In other words, a submodule W is a subgroup of the additive group V invariant with respect to the endomorphisms ϕ(a) for all a ∈ A.
Definition 2.2. Let V be a left A-module, where A is some K-algebra. A subset W ⊆ V is called a submodule of V if
it is closed with respect to the addition and with respect to the multiplication by elements of the field K and the algebra A:
1) v ∈ W and k ∈ K imply (k v) ∈ W ; 2) v 1 ∈ W and v 2 ∈ W imply (v 1 + v 2 ) ∈ W ; 3) v ∈ W and a ∈ A imply (a v) ∈ W .
In other words, a submodule W is a subspace of the vector space V invariant with respect to the endomorphisms ϕ(a) for all a ∈ A.
In both cases a submodule W ⊆ V inherits the structure of a left A-module from V . Each submodule W is associated with the corresponding factorset V /W . The factorset V /W is composed by cosets
Cl W (v) = {u ∈ V : u = v + w for some w ∈ W }. (2.1) The element v in (2.1) is a representative of the coset Cl W (v). Any element v ∈ Cl W (v)
can be chosen as its representative. In both cases (where A is a ring or where A is an algebra) the factorset V /W inherits the structure of a left A-module from V . For this reason it is called a factormodule. Algebraic operations with cosets are given by the formulas
1) Cl W (v 1 ) + Cl W (v 2 ) = Cl W (v 1 + v 2 ) for any v 1 , v 2 ∈ V ; 2) a Cl W (v) = Cl W (a v)
for any a ∈ A and for any v ∈ V .
If A is a K-algebra, then
3) k Cl W (v) = Cl W (k v) for any k ∈ K and for any v ∈ V .
3. Transfinite normal and composition series.
Definition 3.1. Let V be a left A-module. A transfinite sequence of its submodules {0} = V 1 V 2 . . . V n = V (3.1) is called a transfinite normal series for V if V α = β<α V β for any limit ordinal α n.
The term "normal series" in the above definition comes from the group theory where each normal series {1} = G 1 G 2 . . . G n = G of a group G should be composed by its subgroups such that G i is a normal subgroup of G i+1 . Otherwise we would not be able to build the factorgroup G i+1 /G i . In the case of modules each submodule V i with i < n produces the factormodule V i+1 /V i . So the term "normal series" here is vestigial, it is used for ear comfort only. The concept of hypertranssimplicity is nontrivial in the case of groups. In the case of modules each nontrivial submodule W of V produces the nontrivial normal
series {0} = V 1 V 2 V 3 = V , where V 2 = W .
Intersections and sums of submodules.
The intersection of two or more submodules of a given module V is again a submodule of V . As for unions, the union of submodules in general case is not a submodule. Unions of submodules are used to define their sums.
Definition 4.1. Let U i with i ∈ I be submodules of some module V . The submodule U generated by the union of submodules U i is called their sum:
U = i∈I U i = i∈I U i . (4.1)
If the number of submodules in (4.1) is finite, one can use the following notations:
U = U 1 + . . . + U n = U 1 ∪ . . . ∪ U n . (4.2)
In both cases (4.1) and (4.2) each element u ∈ U is presented as a finite sum u = u i1 + . . . + u is , where u ir ∈ U ir and i r ∈ I for all r = 1, . . . , s. The lemma 4.1 is also known as the butterfly lemma. Typically the butterfly lemma is formulated for groups (see § 3 of Chapter I in [3]). However, its proof can be easily adapted for the case of modules. The rest is to prove that Ker ϕ = Ker ψ. Assume that a ∈ Ker ϕ. In this case a ∈Ũ ∩W and a ∈ M . The inclusion a ∈ M means that a = u + w, where u ∈ U and w ∈Ũ ∩ W . SinceŨ ∩ W ⊆Ũ ∩W , from u = a − w we derive u ∈Ũ ∩W . On the other hand u ∈ U . Hence u ∈ U ∩ (Ũ ∩W ), which means u ∈ U ∩W . Thus we have proved that each element a ∈ Ker ϕ is presented as a sum a = u + w, where u ∈W ∩ U and w ∈Ũ ∩ W.
(4.7)
Conversely, it is easy to see that the presentation (4.7) leads to a ∈W ∩Ũ and a ∈ M , i. e. a ∈ Ker ϕ. For this reason we have Ker ϕ = (W ∩ U ) + (Ũ ∩ W ). The equality Ker ψ = (W ∩ U ) + (Ũ ∩ W ) is proved similarly. Now the formula (4.4) is immediate from Ker ϕ = Ker ψ due to the surjectivity of the homomorphisms (4.5) and (4.6). The butterfly lemma 4.1 is proved.
5. The Jordan-Hölder theorem.
Definition 5.1. A transfinite normal series {0} =Ṽ 1 Ṽ 2 . . . Ṽ p = G is called a refinement for a transfinite normal series {0} = V 1 V 2 . . . V n = G if each submodule V i coincides with some submoduleṼ j . Definition 5.2. Two transfinite normal series {0} = V 1 V 2 . . . V n = V and {0} = W 1 W 2 . . . W m = V of a module V are called isomorphic if
there is a one-to-one mapping that associates each ordinal number i < n with some ordinal number j < m in such a way that V i+1 /V i ∼ = W j+1 /W j .
Theorem 5.1. Arbitrary two transfinite normal series of a left A-module V have isomorphic refinements.
Lemma 5.1. If {1} = G 1 . . . G n = G is a transfinite composition series of a group G, then it has no refinements different from itself.
Theorem 5.2 (Jordan-Hölder). Any two transfinite composition series of a left A-module V are isomorphic.
The theorem 5.2 is immediate from the theorem 5.1 and the lemma 5.1. As for the theorem 5.1 and the lemma 5.1, their proof is quite similar to the proof of the theorem 3.1 and the lemma 3.10 in [1]. The algebraic part of this proof is based on the Zassenhaus butterfly lemma. Its version for modules is given above (see lemma 4.1). The other part of the proof deals with indexing sets and ordinal numbers, not with algebraic structures. For this reason it does not differ in the case of groups and in the case of modules.
External direct sums.
Sums and direct sums introduced in the definitions 4.1 and 4.2 are internal ones. They are formed by submodules of a given module. External direct sums are formed by separate modules which are not necessarily enclosed in a given module.
v = v i1 + . . . + v is ,
where v ir ∈ V ir and i r ∈ I for all r = 1, . . . , s, (6.1)
constitute a left A-module V which is called the direct sum of the modules V i .
Once the external direct sum V is constructed, we find that it comprises submodules U i isomorphic to the initial modules V i . Indeed, we can set s = 1 in (6.1).
Formal sums (6.1) with exactly one summand v = v i , where v i ∈ V i , constitute a submodule U i of V isomorphic to the module V i . For this reason the constructions of internal and external direct sums are the same in essential.
According to the well-known Zermelo theorem (see Appendix 2 in [4]), every set I can be well ordered and then associated with some ordinal number n (see Proposition 3.8 in Appendix 3 of [4]). Therefore the external direct sum V in the above definition 6.1 can be written as
V = α<n V α . (6.2)
Relying on (6.2), for each i n we introduce the following submodule of V :
W i = α<i V α . (6.3)
The submodules (6.3) constitute a transfinite normal series of submodules for V :
{0} = W 1 W 2 . . . W n = V. (6.4)
The factormodules of the sequence (6.4) are isomorphic to the modules V i , namely we have W i+1 /W i ∼ = V i . If the modules V i are simple, then (6.4) is a transfinite composition series (see Definitions 3.4 and 3.5).
A remark. The formulas (6.2), (6.3), and (6.4) are equally applicable for internal and external direct sums. For this reason, applying the Jordan-Hölder theorem 5.2, we get the following result.
Theorem 6.1. If a module V is presented as a direct sum of its simple submodules V i , then these submodules are unique up to the isomorphism and some permutation of their order in the direct sum.
There is a special case of the external direct sum (6.2). Assume that U is some simple left A-module. Let's replicate this module into multiple copies and denote these copies through V α . Then V α ∼ = U . In this case the module V in (6.2) is denoted through N U , where N = |n| is the cardinality of the ordinal number n. Such a notation is motivated by the following theorem.
Theorem 6.2. Let U be a simple left A-module and let N U and M U be two external direct sums of the form (6.2) built by the copies of the module U :
N U = α<n U, M U = α<m U.
Then N U is isomorphic to M U if and only if N = M , i. e. if |n| = |m|.
The theorem 6.2 is easily derived from the theorem 6.1.
Concluding remarks.
The results of this paper are rather obvious and are known to the algebraists community. However, they are dispersed in various books as preliminaries to more special theories. Treated as obvious, these results are usually not equipped with explicit proofs and even with explicit statements. We gather them in this paper for referential purposes.
Definition 1 . 1 .
11Let A be an associative ring and let V be an additive Abelian group. The Abelian group V is called a left module over the ring A (or a left A-module) if some homomorphism of rings ϕ : A → End(V ) is given and fixed. Here End(V ) is the ring of endomorphisms of the additive Abelian group V . Definition 1.2. Let A be an associative algebra over some field K and let V be a linear vector space over the same field K. The linear vector space V is called a left module over the algebra A (left A-module) if some homomorphism of algebras ϕ : A → End(V ) is given and fixed. Here End(V ) is the algebra of endomorphisms of the linear vector space V .
Definition 3.2. A module V is called hypertranssimple if it has no normal series (neither finite nor transfinite) other than trivial one {0} = V 1 V 2 = V .
Definition 3.3. A transfinite normal series (3.1) of a module V is called a transfinite composition series of V if for each ordinal number i < n the factormodule V i+1 /V i is hypertranssimple.
For this reason the concept of hypertranssimplicity of modules reduces to the standard concept of simplicity. The definitions 3.2 and 3.3 then are reformulated as follows.
Definition 3.4. A module V is called simple if it has no submodules other than trivial ones V 1 = {0} and V 2 = V .
Definition 3.5. A transfinite normal series (3.1) of a module V is called a transfinite composition series of V if for each ordinal number i < n the corresponding factormodule V i+1 /V i is simple.
. 2 .
2The sum of submodules (4.1) is called a direct sum if for each element u ∈ U its presentation (4.3) is unique.
Lemma 4. 1 (
1Zassenhaus). LetŨ andW be submodules of some left A-module and let U and W be submodules ofŨ andW respectively. Then (U + (Ũ ∩W ))/(U + (Ũ ∩ W )) ∼ = (W + (W ∩Ũ))/(W + (W ∩ U )).(4.4)
Proof. Let's denote M = U + (Ũ ∩ W ) and N = W + (W ∩ U ). Elements of the factormodule in the left hand side of the formula (4.4) are cosets of the form Cl M (u + a), where u ∈ U and a ∈Ũ ∩W . Note that U ⊆ M . Therefore Cl M (u + a) = Cl M (a) which means that each element of the factormodule (U + (Ũ ∩W ))/M is represented by some element a ∈Ũ ∩W . Thus we have a surjective homomorphism of modules ϕ :Ũ ∩W −→ (U + (Ũ ∩W ))/M. (4.5) Repeating the above arguments for the factormodule in the right hand side of the formula (4.4), we get another surjective homomorphism of modules ψ :Ũ ∩W −→ (W + (W ∩Ũ ))/N. (4.6)
Definition 6. 1 .
1Let V i be left A-modules enumerated by elements i ∈ I of some indexing set I. Finite formal sums of the form
Transfinite normal and composition series of groups. R A Sharipov, arXiv:0908.2257Electronic Archive. e-printSharipov R. A., Transfinite normal and composition series of groups, e-print arXiv:0908.2257 in Electronic Archive http://arXiv.org.
Introduction to algebra. P J Cameron, Oxford University PressNew YorkCameron P. J., Introduction to algebra, Oxford University Press, New York, 2008.
. S Lang, Algebra, Springer-VerlagNew York, Berlin, HeidelbergLang S., Algebra, Springer-Verlag, New York, Berlin, Heidelberg, 2002.
5 Rabochaya street, 450003 Ufa, Russia Cell Phone: +7(917)476 93 48 E-mail address: r-sharipov@mail. P A Grillet, ru R [email protected] URL. Springer Science + Business MediaAbstract AlgebraGrillet P. A., Abstract Algebra, Springer Science + Business Media, New York, 2007. 5 Rabochaya street, 450003 Ufa, Russia Cell Phone: +7(917)476 93 48 E-mail address: [email protected] R [email protected] URL: http://www.freetextbooks.narod.ru http://sovlit2.narod.ru
| [] |
[
"Continuous Gaussian multifractional processes with random pointwise Hölder regularity (long version)",
"Continuous Gaussian multifractional processes with random pointwise Hölder regularity (long version)"
] | [
"Antoine Ayache \nUniversité Lille\n\n"
] | [
"Université Lille\n"
] | [] | Let {X(t)} t∈R be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Hölder regularity over any compact interval, that is the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon happens with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent?In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.Running Title: Gaussian multifractional processes with random exponents | 10.1007/s10959-012-0418-3 | [
"https://arxiv.org/pdf/1109.1617v2.pdf"
] | 117,817,095 | 1109.1617 | 002d9da90fbac4296854536bc75b55b184bb8d6a |
Continuous Gaussian multifractional processes with random pointwise Hölder regularity (long version)
18 Feb 2012 February 21, 2012
Antoine Ayache
Université Lille
Continuous Gaussian multifractional processes with random pointwise Hölder regularity (long version)
18 Feb 2012 February 21, 2012Hölder regularitypointwise Hölder exponentsmultifractional Brownian motionlevel setswavelet decompositions AMS Subject Classification: 60G1560G17
Let {X(t)} t∈R be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Hölder regularity over any compact interval, that is the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon happens with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent?In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.Running Title: Gaussian multifractional processes with random exponents
Introduction
Let {X(t)} t∈R be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions over the real line R. The global Hölder regularity of one of them, t → X(t, ω), over a non-degenerate 1 compact interval J ⊂ R, is measured through the uniform Hölder exponent β X (J, ω) defined as, β X (J, ω) = sup β ≥ 0 : sup
t ′ ,t ′′ ∈J |X(t ′ , ω) − X(t ′′ , ω)| |t ′ − t ′′ | β < ∞ . (1.1)
The local Hölder regularity of the trajectory t → X(t, ω) in a neighborhood of some fixed point s ∈ R, is measured through two different exponents: the local Hölder exponent α X (s, ω) and the pointwise Hölder exponent α X (s, ω). They are defined as: α X (s, ω) = sup β X [u, v], ω : u, v ∈ R and s ∈ (u, v) (1.2) and α X (s, ω) = sup α ≥ 0 : lim sup h→0 |X(s + h, ω) − X(s, ω)| |h| α = 0 .
(1.3)
Notice that one always has, α X (s, ω) ≤ α X (s, ω).
Moreover, the function s → α X (s, ω) is always lower semicontinuous over R [34], while the function s → α X (s, ω) does not necessarily satisfy such a nice property; in fact the latter function can be the liminf of any arbitrary sequence of continuous functions with values in [0, 1] (see [2,11,20,5,34]), therefore its behavior can be quite erratic. The notion of pointwise Hölder exponent is a fundamental concept in the area of the multifractal analysis of deterministic and random functions [22,23]. It provides a sharp estimation of the asymptotic behavior of the modulus of local continuity of the function t → X(t, ω) at any fixed point s ∈ R (see e.g. [28] page 214 for the defintion of this modulus of continuity); indeed, (1.3) implies that, α X (s, ω) = sup α ≥ 0 : lim sup ρ→0 + sup |X(t, ω) − X(s, ω)| ρ α : t ∈ R and |t − s| ≤ ρ = 0 .
In order to explain the main motivation behind our article, let us state the following theorem, whose proof which is given in Subsection 4.1, implicitly relies on zero-one law.
Theorem 1.1 One denotes by {X(t)} t∈R an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions over the real line R. The following two results hold.
(i) For each non-degenerate compact interval J ⊂ R, let b X (J) be the deterministic quantity defined as,
b X (J) = sup b ≥ 0 : sup t ′ ,t ′′ ∈J E|X(t ′ , ω) − X(t ′′ , ω)| 2 |t ′ − t ′′ | 2b < ∞ . (1.4)
Then, one has, P β X (J) = b X (J) = 1, (1.5) where P β X (J) = b X (J) denotes the probability that the uniform Hölder exponent β X (J) be equal to b X (J).
(ii) There exists Ω an event of probability 1, non depending on s, such that, the local Hölder exponent α X satisfies, α X (s, ω) = a X (s), for all (s, ω) ∈ R × Ω, (1.6) where a X (s) is the deterministic quantity defined as, a X (s) = sup b X [u, v] : u, v ∈ R and s ∈ (u, v) .
(1.7)
Notice that Theorem 1.1 Part (ii), has already been obtained in [19] (see Corollary 3.15 in this article) under the assumption that a X is continuous. This result means that, with probability 1, the function s → α X (s, ω) does not depend on the choice of ω, whatever the centered continuous nowhere differentiable Gaussian process {X(t)} t∈R might be. The main goal of our article is to show that, in some cases, a different phenomenon happens for the function s → α X (s, ω); namely we construct multifractional Gaussian processes {X(t)} t∈R with continuous nowhere differentiable trajectories, such that with a strictly positive probability the function s → α X (s, ω) depends on the choice of ω. To this end, we draw a close connection between the values of the latter function and the zero-level set s ∈ R : Y (s, ω) = 0 of a Gaussian process {Y (s)} s∈R closely related to X and very similar to it.
It is worth noticing that for each s ∈ R, there always exists a deterministic quantity a X (s) ∈ [0, 1] such that P α X (s) = a X (s) = 1.
(1.8)
Relation (1.8) corresponds to Lemma 3.5 in [4] and Proposition 6.2 in [5], it means that the deterministic function a X is a modification of the stochastic process α X . In view of (1.8), the fact that, with a strictly positive probability, the function s → α X (s, ω) depends on ω, implies that the deterministic function a X and the stochastic process α X are not indistinguishable;
not indistinguishable formally means that: for all event Ω of probability 1, there exists ω 0 ∈ Ω and s 0 (ω 0 ) ∈ R, such that, α X s 0 (ω 0 ), ω 0 = a X s 0 (ω 0 ) .
In order to show that with a strictly positive probability, the function s → α X (s, ω) depends on ω, we use the framework of multifractional Brownian motion (mBm). Let us now make a few recalls concerning this Gaussian process. We denote by {B(t, θ)} (t,θ)∈R×(0,1) the centered Gaussian field defined for all (t, θ) as the Wiener integral, (1.9) with the convention that for every (u, θ) ∈ R 2 , (u) Notice that in Theorem 2.1, we introduce a modification B of the field B defined as a random wavelet type series (see Subsection 4.2); let us stress that, for the sake of simplicity, in our article, B is often identified with B and the process X with its modification X, defined for all t ∈ R, as X(t) = B(t, H(t)).
B(t, θ) = R (t − x) θ−1/2 + − (−x) θ−1/2 + dW (x),
MBm is an extension of fractional Brownian motion (fBm), indeed, when the function H is a constant denoted by h, then X reduces to a fBm of Hurst parameter h; the latter Gaussian process has been widely studied since several decades, we refer to e.g. [33,13,1,24,14] for a presentation of its main properties. MBm was introduced, independently in [31] and [9], to overcome an important drawback of fBm due to the fact that its pointwise Hölder exponent remains constant all along its trajectory. Since several years there is an increasing interest in the study of multifractional processes (see for instance [3,4,5,6,15,16,17,18,29,35,36]).
It has been proved in [9,31] that when H is a Hölder function over a non-degenerate compact interval J and satisfies the condition max t∈J H(t) < β H (J), (1.11) where β H (J) denotes the uniform Hölder exponent of H over J, then for all s ∈J (note that one restricts toJ, the interior of J, in order to avoid the border effect), one has P α X (s) = H(s) = 1.
(1.12)
Later it has been shown in [5], that when the condition (1.11) is satisfied, then {H(s)} s∈J and {α X (s)} s∈J are indistinguishable; namely there exists Ω, an event of probability 1, non depending on s (and also non depending on J), such that, where α H (s) denotes the pointwise Hölder exponent of H at s. Observe that (1.14) has been derived in [18,19], for a definition of mBm slightly different from (1.10), yet the proof also works in the latter case.
α X (s, ω) = H(s), for all (s, ω) ∈ (J , Ω
In our article, we construct examples of Gaussian mBm's X with continuous nowhere differentiable trajectories which satisfy the following property: there exists an event D of strictly positive probability, such that for all ω ∈ D, one has,
α X s 0 (ω), ω = min H s 0 (ω) , α H s 0 (ω) ,
for some s 0 (ω) ∈ R. In view of (1.14), the latter property means that the pointwise Hölder regularity of X is random, in other words, it depends on the choice of a trajectory of X.
Note in passing, that there are many examples of non Gaussian processes whose regularity is random, as for instance, discontinuous Lévy processes [21], multifractional processes with random exponent [4,5], self-regulating processes [8], or pure jump Markov processes [7].
The remaining of this article is structured in the following way. In Section 2, we introduce a modification B of the field B, having some nice properties which are useful for the study of the pointwise Hölder regularity of the mBm X. Then, denoting by J ⊆ (0, +∞) an arbitrary open non-empty interval, under some condition on its parameter H, we show that the pointwise Hölder regularity of {X(t)} t∈J , is closely connected with the zeros of the process {Y (s)} s∈J = (∂ θ B)(s, H(s)) s∈J ; thus it turns out that this regularity is random, when the level set s ∈ J : Y (s) = 0 is non-empty with a (strictly) positive probability. In Section 3, we prove that this is indeed the case, namely with a strictly positive probability the latter level set, is rather large: it has a Hausdorff dimension bigger than 1 − η − inf s∈J H(s), where η is a fixed strictly positive and arbitrarily small real number. Finally, some technical proofs, mainly related to wavelet methods, are given in Section 4 (the Appendix).
Construction of the counterexamples
In order to construct continuous Gaussian multifractional Brownian motions with random pointwise Hölder regularity, first we need to show that the Gaussian random field {B(t, θ)} (t,θ)∈R×(0,1) defined in (1.9), has a modification { B(t, θ)} (t,θ)∈R×(0,1) satisfying some nice properties.
Namely, we need the following theorem.
Theorem 2.1 Let {B(t, θ)} (t,θ)∈R×(0,1) be the field defined in (1.9). There exists an event Ω * of probability 1 and there is a modification of {B(t, θ)} (t,θ)∈R×(0,1) denoted by { B(t, θ)} (t,θ)∈R×(0,1) , such that, for each ω ∈ Ω * , the following four results hold.
(i) The function (t, θ) → B(t, θ, ω) is continuous over R × (0, 1).
(ii) For every fixed arbitrarily small real number ε > 0 and (s, θ) ∈ R × (0, 1), one has
lim sup h→0 B(s + h, θ, ω) − B(s, θ, ω) |h| θ−ε = 0 (2.1) and lim sup h→0 B(s + h, θ, ω) − B(s, θ, ω) |h| θ+ε = +∞. (2.2)
(iii) For each fixed t ∈ R, the function θ → B(t, θ, ω) is C ∞ over (0, 1); its derivative, of any order n ∈ Z + , at θ, is denoted by ∂ n θ B (t, θ, ω).
(iv) For every fixed n ∈ Z + , arbitrarily small real number ε > 0 and real numbers M, a, b
satisfying M > 0 and 0 < a < b < 1, there exists a constant C(ω) > 0, only depending on ω, n, ε, M, a, b, such that the inequality,
∂ n θ B (t 1 , θ 1 , ω) − ∂ n θ B (t 2 , θ 2 , ω) ≤ C(ω) |t 1 − t 2 | max{θ 1 ,θ 2 }−ε + |θ 1 − θ 2 | , (2.3) holds, for all (t 1 , θ 1 ) ∈ [−M, M ] × [a, b] and (t 2 , θ 2 ) ∈ [−M, M ] × [a, b].
Remark 2.2
• The field B was introduced in [4] and Theorem 2.1 Parts (i) and (iii) were derived in the latter article (see [4], pages 463 to 470); notice that in [4], B was denoted by B.
• A less precise inequality than (2.3), was obtained in [4], in the particular case where It is worth noticing that a straightforward consequence of Part (ii) of Theorem 2.1, is the following:
Proposition 2.3 For all fixed θ ∈ (0, 1), we denote by B θ = {B θ (t)} t∈R the process { B(t, θ)} t∈R ;
observe that B θ is a fBm of Hurst parameter θ. There exists an event Ω * of probability 1, non depending on s and θ, such that one has, for each ω ∈ Ω * and for all (s, θ) ∈ R × (0, 1),
α B θ (s, ω) = θ,
where α B θ (s, ω) is the pointwise Hölder exponent at s of the function t → B θ (t, ω).
Observe that the fact that the pointwise Hölder exponent of the fBm B θ , is equal, almost surely for all s ∈ R, to the Hurst parameter θ, is a classical result (see for example [37,1,5]); the novelty in Proposition 2.3, is that this equality holds on an event Ω * of probability 1, which does not depend on the Hurst parameter θ (notice that the event Ω * also does not depend on s).
The proof of Theorem 2.1 mainly relies on wavelet techniques, rather similar to those used in [4,5]; it is not really the core of the article, this is why it is given in Subsection 4.2.
From now on, it is important that the reader keeps in his mind the following remark.
Remark 2.4
In the remaining of this section as well as in the next section,
• the Gaussian field {B(t, θ)} (t,θ)∈R×(0,1) defined in (1.9), will be always identified with its modification { B(t, θ)} (t,θ)∈R×(0,1) introduced in Theorem 2.1;
• the mBm {X(t)} t∈R of functional parameter H, defined in (1.10), will be always identified with its modification { X(t)} t∈R , defined for every real number t and all ω ∈ Ω (the underlying probability space), as X(t, ω) = B(t, H(t), ω).
Let us now state the main result of our article. (i) For all ω ∈ Ω * and s ∈ J , satisfying (∂ θ B)(s, H(s), ω) = 0, one has α X (s, ω) = α H (s).
(ii) For all ω ∈ Ω * and s ∈ J , satisfying (∂ θ B)(s, H(s), ω) = 0, one has α X (s, ω) = H(s).
(iii) There exists Ω * * ⊆ Ω * an event of probability 1, such that for all ω ∈ Ω * * ,
dim H s ∈ J : (∂ θ B)(s, H(s), ω) = 0 = 1,
where dim H (·) denotes the Hausdorff dimension; in other words, dim H s ∈ J : α X (s, ω) = α H (s) = 1.
(iv) For each arbitrarily small η > 0, there exists D ⊆ Ω * * , an event of (strictly) positive probability, which a priori depends on η, such that for all ω ∈ D,
dim H s ∈ J : (∂ θ B)(s, H(s), ω) = 0 ≥ 1 − η − inf s∈J H(s) > 0;
in other words,
dim H s ∈ J : α X (s, ω) = H(s) ≥ 1 − η − inf s∈J H(s) > 0.H(s) = 1 3 + 1 − 2 −1/4 15 +∞ j=0 2 j −1 k=0 2 −jζ(k2 −j ) T (2 j s − k),
where ζ : [0, 1] → [1/4, 7/24] is an arbitrary C 1 function satisfying ζ(0) = ζ(1) and, where
T : R → [0, 1] is the function defined, for each x ∈ R, as, T (x) = 1 − |2x − 1| if x ∈ [0, 1]
and T (x) = 0 else; by slightly adapting the proof of Proposition 5 in [11], one can show that it is convenient to introduce the following concise notation: let f be a real-valued function defined on neighborhood of 0, and let τ ∈ [0, +∞) be fixed, we assume that the notation:
|f (h)| ≍ |h| τ ,
means that for all arbitrarily small ε > 0, one has:
lim sup h→0 |f (h)| |h| τ −ε = 0 and lim sup h→0 |f (h)| |h| τ +ε = +∞.
Heuristic proof of Theorem 2.5 Parts (i) and (ii): For all fixed ω ∈ Ω * and s ∈ J , the increment X(s + h, ω) − X(s, ω), of the mBm {X(t)} t∈R = {B(t, H(t))} t∈R , can be expressed as:
X(s + h, ω) − X(s, ω) = ∆B H(s) (s, h, ω) + R(s, h, ω),(2.H(s + h) − H(s) ≍ |h| α H (s) . (2.10)
Also observe that, in view of (2.9) and the fact that (t, θ) → (∂ θ B)(t, θ, ω) is a continuous function, one has that, Then putting together, (2.4), (2.7), (2.12) and the inequality α H (s) < H(s), one obtains that
(∂ θ B) s + h, θ(s, h, ω), ω − −− → h→0 (∂ θ B) s, H(s), ω .X(s + h, ω) − X(s, ω) ≍ |h| α H (s) ,
which proves that Part (i) of the theorem holds. In the case where (∂ θ B) s, H(s), ω = 0;
(2.3), (2.9), (2.10) and the inequality α H (s) < H(s), entail that, for all arbitrarily small
ε > 0, (∂ θ B) s + h, θ(s, h, ω), ω = O |h| α H (s)−ε . (2.13)
Then (2.8), (2.10) and (2.13) imply that for all arbitrarily small ε > 0,
R(s, h, ω) = O |h| 2α H (s)−ε .
(2.14)
Finally, it follows from (2.4), (2.7), (2.14) and the inequality H(s) < 2α H (s), that
X(s + h, ω) − X(s, ω) ≍ |h| H(s) ,
which proves that Part (ii) of the theorem holds.
In order to give a rigorous proof of Parts (i) and (ii) of Theorem 2.5, we need some preliminary results. In the remaining of this section, for the sake of simplicity, we assume that J = (0, 1).
Let us first give two lemmas which, generally speaking (even in the case where condition (A) fails to be satisfied), provide upper and lower bounds, for absolute increments of a typical trajectory of the mBm X, in a neighborhood of an arbitrary fixed point s ∈ [0, 1].
Lemma 2.7
For all fixed arbitrarily small ε > 0, ω ∈ Ω * (the event of probability 1 introduced in Theorem 2.1) and s ∈ [0, 1], there is a constant C(ω) > 0, such that the following inequality,
X(s + h, ω) − X(s, ω) ≤ C(ω) |h| min{H(s),2α H (s)}−ε + ∂ θ B (s, H(s), ω) × |h| α H (s)−ε ,
holds, for every real number h satisfying s + h ∈ [0, 1].
Lemma 2.8
For all fixed arbitrarily small ε > 0, ω ∈ Ω * and s ∈ [0, 1], there exist two constants C(ω) > 0 and C ′ (ω) > 0 , such that, the inequalities,
X(s + h, ω) − X(s, ω) (2.15) ≥ ∂ θ B (s, H(s), ω) × H(s + h) − H(s) − C(ω)|h| min{H(s),2α H (s)}−ε and X(s + h, ω) − X(s, ω) ≥ B(s + h, H(s), ω) − B(s, H(s), ω) (2.16) −C ′ (ω) ∂ θ B (s, H(s), ω) × |h| α H (s)−ε + |h| min{H(s)+α H (s),2α H (s)}−ε .
hold, for every real number h, satisfying s + h ∈ [0, 1].
Remark 2.9
Let ω ∈ Ω * and s ∈ (0, 1) be fixed. Recall that the pointwise Hölder exponent at s, of the function t → X(t, ω), is denoted by α X (s, ω). It follows from the previous lemma that:
(i) When ∂ θ B (s, H(s), ω) = 0, one has α X (s, ω) ≥ min{H(s), α H (s)}. (ii) When ∂ θ B (s, H(s), ω) = 0, one has α X (s, ω) ≥ min{H(s), 2α H (s)}.
Let us now give the proofs of these two lemmas.
Proof of Lemma 2.7: First observe that using the first equality in (1.10) as well as the triangle inequality, one has that
X(s + h, ω) − X(s, ω) (2.17) ≤ B(s + h, H(s + h), ω) − B(s + h, H(s), ω) + B(s + h, H(s), ω) − B(s, H(s), ω) .
The function θ → B(s + h, θ, ω) being continuously differentiable over (0, 1) (see Part (iii) of Theorem 2.1), it follows from the Mean Value Theorem that there is
θ(s, h, ω) ∈ min{H(s + h), H(s)}, max{H(s + h), H(s)} ,
such that
B(s + h, H(s + h), ω) − B(s + h, H(s), ω) = ∂ θ B s + h, θ(s, h, ω), ω × H(s + h) − H(s) . (2.18)
Moreover, the triangle inequality implies that
∂ θ B s+h, θ(s, h, ω), ω ≤ ∂ θ B s, H(s), ω + ∂ θ B s+h, θ(s, h, ω), ω − ∂ θ B s, H(s), ω . (2.19)
Part (iv) of Theorem 2.1 (in which ε is replaced by ε/2 and one takes n = 1, M = 1,
a = min x∈[0,1] H(x) and b = max x∈[0,1] H(x)), entails that ∂ θ B s + h, θ(s, h, ω), ω − ∂ θ B s, H(s), ω ≤ C 1 (ω) |h| max{H(s), θ(s,h,ω)}−ε/2 + θ(s, h, ω) − H(s) ≤ C 1 (ω) |h| H(s)−ε/2 + H(s + h) − H(s) ,(2.20)
where C 1 (ω) is a constant non depending on s and h. Putting together (2.18), (2.19), (2.20) and the inequality |H(s + h) − H(s)| ≤ c|h| α H (s)−ε/2 (c being a constant), one gets that
B(s + h, H(s + h), ω) − B(s + h, H(s), ω) (2.21) ≤ C 2 (ω) ∂ θ B (s, H(s), ω) × |h| α H (s)−ε/2 + |h| min{H(s)+α H (s),2α H (s)}−ε ,
where C 2 (ω) > 0 is a constant only depending on ω, s and ε. On the other hand,
(2.1) implies that B(s + h, H(s), ω) − B(s, H(s), ω) ≤ C 3 (ω)|h| H(s)−ε , (2.22) where C 3 (ω) > 0∂ θ B s+h, θ(s, h, ω), ω ≥ ∂ θ B s, H(s), ω − ∂ θ B s+h, θ(s, h, ω), ω − ∂ θ B s, H(s), ω ,(2.B(s + h, H(s + h), ω) − B(s + h, H(s), ω) ≥ ∂ θ B s, H(s), ω − ∂ θ B s + h, θ(s, h, ω), ω − ∂ θ B s, H(s), ω × H(s + h) − H(s) ≥ ∂ θ B (s, H(s), ω) × H(s + h) − H(s) −C 1 (ω) |h| H(s)−ε/2 + H(s + h) − H(s) × H(s + h) − H(s) ≥ ∂ θ B (s, H(s), ω) × H(s + h) − H(s) − C 2 (ω) |h| H(s)−ε/2 + H(s + h) − H(s) 2 , (2.25)
where C 1 (ω) > 0 and C 2 (ω) > 0 are two constants only depending on ω, s and ε. Putting 3 Hausdorff dimension of the zero-level set of the process
{(∂ θ B)(s, H(s))} s∈J
The goal of this section is to show that Theorem 2.5 Parts (iii) and (iv) hold. Notice that in all the sequel, we do not necessarily impose on the continuous functional parameter H of mBm to satisfy condition (A) (see Theorem 2.5). Also, in all the sequel, we denote by
{Y (s)} s∈R the centered Gaussian process {(∂ θ B)(s, H(s))} s∈R , where the centered Gaussian field { ∂ θ B (t, θ)} (t,θ)∈R×(0,1) = { ∂ θ B (t, θ)} (t,θ)∈R×(0,1) has been introduced in Theorem 2.1 Part (iii). Let us first give stochastic integral representations (modifications) of { ∂ θ B (t, θ)} (t,θ)∈R×(0,1)
and {Y (s)} s∈R .
Proposition 3.1 One has for all (t, θ) ∈ R × (0, 1), almost surely, ∂ θ B (t, θ) = R (t − x) θ−1/2 + log (t − x) + − (−x) θ−1/2 + log (−x) + dW (x),
with the convention that (y) θ−1/2 + log (y) + = 0 for every real numbers θ ∈ (0, 1) and y ≤ 0.
As a straightforward consequence, one has for all s ∈ R, almost surely,
Y (s) = R (s − x) H(s)−1/2 + log (s − x) + − (−x) H(s)−1/2 + log (−x) + dW (x). (3.1)
The proof of Proposition 3.1 is given in Subsection 4.3, since it relies on wavelet techniques similar to those used in Subsection 4.2, in order to show that Theorem 2.1 holds.
From now on, we assume that I = [δ 1 , δ 2 ] ⊂ J is a compact interval such that 0 < δ 2 − δ 1 < 1 and
0 < b = max s∈I H(s) ≤ η + inf s∈J H(s) < 1, (3.2)
where η > 0 is an arbitrarily small fixed real number; the open interval J has been introduced in Theorem 2.5 and one has that δ 1 > 0 since J ⊆ (0, +∞). Observe that such an interval I exists, since we assume that H is a continuous function over R. The following lemma shows that Var Y (s) , s ∈ I, is bounded away from zero.
Lemma 3.2
There is a constant c > 0, only depending on δ 1 , such that for all s ∈ I, one has
Var Y (s) ≥ c.
v = s − x, that Var Y (s) = R (s − x) H(s)−1/2 + log (s − x) + − (−x) H(s)−1/2 + log (−x) + 2 dx ≥ s 0 (s − x) 2H(s)−1 log 2 (s − x)] dx = s 0 v 2H(s)−1 log 2 (v) dv ≥ min(δ 1 ,1) 0 v log 2 (v) dv > 0.
Now we are in position to prove Theorem 2.5 Part (iii).
Proof of Theorem 2.5 Part (iii): Let us fix s 0 ∈ I ⊂ J . Observe that the centered Gaussian random variable Y (s 0 ) = (∂ θ B)(s 0 , H(s 0 )) has a non-zero standard deviation (see Lemma 3.2) and thus its probability density function exists; the fact that the latter function is strictly positive on the whole real line, implies that (∂ θ B)(s 0 , H(s 0 )) = 0 almost surely.
Then, noticing that the event Ω * (see Theorem 2.1) is of probability 1, it follows that the probability of the event
Ω * * = ω ∈ Ω * : (∂ θ B)(s 0 , H(s 0 ), ω) = 0 ,s ∈ J : (∂ θ B)(s, H(s), ω) = 0 = J ∩ (∂ θ B)(·, H(·), ω) −1 R \ {0} ,
is a non-empty open subset of R, which implies that its Hausdorff dimension is equal to 1.
The proof of Theorem 2.5 Part (iv), is in the same spirit as that of Theorem 8.4.2 in [1] and Relation (5.9) in [29]; basically, it relies on the following proposition which shows that the process {Y (s)} s∈I satisfies the so-called property of one sided strong local nondeterminism.
A detailed presentation of the important concept of local nondeterminism and related topics can be found in [10] and in [38], for instance.
s 1 < . . . < s n , (3.4) one has, Var Y (s n )|Y (s 1 ), . . . , Y (s n−1 ) ≥ 2 −1 s n − s n−1 2H(s n ) log 2 (s n − s n−1 ) , (3.5) where Var Y (s n )|Y (s 1 ), . . . , Y (s n−1 ) is the conditional variance of Y (s n ) given Y (s 1 ), . . . , Y (s n−1 ).
Proof of Proposition 3.3: In view of the definition of Var Y (s n )|Y (s 1 ), . . . , Y (s n−1 ) and the Gaussianity of the process {Y (s)} s∈I , it is sufficient to show that, for every integer n ≥ 2, for all real numbers a 1 , . . . , a n−1 and for each s 1 , . . . , s n ∈ I satisfying (3.4), one has
E Y (s n ) − n−1 l=1 a l Y (s l ) 2 ≥ 2 −1 s n − s n−1 2H(s n ) log 2 (s n − s n−1 ) . (3.6)
It follows from (3.1) and the isometry property of Wiener integral, that
E Y (s n ) − n−1 l=1 a l Y (s l ) 2 = R (s n − x) H(s n )−1/2 + log (s n − x) + − (−x) H(s n )−1/2 + log (−x) + (3.7) − n−1 l=1 a l (s l − x) H(s l )−1/2 + log (s l − x) + − (−x) H(s l )−1/2 + log (−x) + 2 dx.
Next, observe that for every x ∈ [s n−1 , s n ], one has −x < 0 and, as a consequence, for all
E Y (s n ) − n−1 l=1 a l Y (s l ) 2 ≥ s n s n−1 (s n − x) 2H(s n )−1 log 2 (s n − x) dx = s n −s n−1 0 (s n − s n−1 − x) 2H(s n )−1 log 2 (s n − s n−1 − x) dx. (3.10)
Next, setting in the last integral, v = x/(s n − s n−1 ) and using the fact that 0 < s n − s n−1 ≤ δ 2 − δ 1 < 1, one obtains that
s n −s n−1 0 (s n − s n−1 − x) 2H(s n )−1 log 2 (s n − s n−1 − x) dx = (s n − s n−1 ) 1 0 (s n − s n−1 ) − (s n − s n−1 )v 2H(s n )−1 log 2 (s n − s n−1 ) − (s n − s n−1 )v dv = (s n − s n−1 ) 2H(s n ) 1 0 (1 − v) 2H(s n )−1 log (1 − v) −1 + log (s n − s n−1 ) −1 2 dv ≥ (s n − s n−1 ) 2H(s n ) log 2 (s n − s n−1 ) 1 0 (1 − v) dv = 2 −1 (s n − s n−1 ) 2H(s n ) log 2 (s n − s n−1 ) . (3.11)
Finally combining (3.10) with (3.11), it follows that (3.6) holds.
Now it is convenient to make a few recalls concerning the so-called (deterministic) measures of finite γ energy, more information about them can be found in [25]. In the sequel, we always assume that γ ∈ (0, 1). A measure µ defined on the Borel sets of R is said to be of finite γ energy, if the integral
I γ (µ) = R R |s − t| −γ dµ(s)dµ(t),(3.12)
which is usually called the γ energy of µ, exists and is finite. M γ , the class of these measures, forms a Hilbert space equipped with the inner product
(µ, ν) γ = R R |s − t| −γ dµ(s)dν(t);
the corresponding norm is denoted by · γ . Moreover, M + γ , the subset of the positive measures of M γ , is a complete metric space, for the metric
µ − ν γ = R R |s − t| −γ d(µ − ν)(s)d(µ − ν)(t) = I γ (µ − ν).
(3.13)
One of the main interests of the positive measures of finite γ energy comes from the following lemma which is a straightforward consequence of the Frostman Theorem, the latter theorem is presented in e.g. [24] pages 132 and 133 (see also [14]).
Lemma 3.4 Let K be a compact subset of R. If K carries a positive non-vanishing measure of finite γ energy (i.e. if there is a positive non-vanishing measure of finite γ energy whose support is contained in K), then the Hausdorff dimension of K is greater than or equal to γ.
We are now in position to prove Theorem 2.5 Part (iv).
Proof of Theorem 2.5 Part (iv): For all ω ∈ Ω * (the event of probability 1 introduced in Theorem 2.1), we set L Y (ω) = s ∈ I : (∂ θ B)(s, H(s), ω) = 0 .
Recall that I = [δ 1 , δ 2 ] ⊂ J is a compact interval such that 0 < δ 2 − δ 1 < 1 and (3.2) holds.
Also, recall that the process {(∂ θ B)(s, H(s))} s∈I is denoted by {Y (s)} s∈I .
We will show that for all γ < 1 − b there is D γ ⊆ Ω * an event, which a priori depends on γ but whose probability is bigger than a strictly positive constant non depending on γ, such that for all ω ∈ D γ , the set L Y (ω) carries a positive non-vanishing deterministic measure µ(·, ω), whose γ energy is finite. Roughly speaking, the idea for obtaining µ(·, ω) is somehow similar to the one which consists in constructing a Dirac measure as a limit of Gaussian measures; namely, µ(·, ω) will be the limit, in the sense of the norm · γ , of some of the positive measures µ n (·, ω), n ∈ N, defined for each Borel subset A of R, as,
µ n (A, ω) = A∩I Φ n (t, ω) dt,(3.14)
where for all t ∈ I,
Φ n (t, ω) = 2πn 1/2 exp − nY (t, ω) 2 2 . (3.15)
Notice that (3.15) and the Fourier inversion formula, imply that for all t ∈ I and ω ∈ Ω * , Moreover, µ n (·, ω) is of finite γ energy for any γ ∈ (0, 1). Indeed, in view of (3.12) and (3.14), I γ µ n (·, ω) can be expressed as,
Φ n (t, ω) = R exp − ξ 2 2n + iξY (t, ω) dξ.I γ µ n (·, ω) = I I |s − t| −γ Φ n (s, ω)Φ n (t, ω) ds dt; (3.18)
then, (3.15) implies that
I γ µ n (·, ω) ≤ 2πn I I |s − t| −γ ds dt < ∞.
Observe that one can more generally, show in the same way, that for all integers n ≥ 1 and m ≥ 1,
I I |s − t| −γ Φ n (s, ω)Φ m (t, ω) ds dt < ∞.
Let us now construct a subsequence l → n l satisfying the following property: for all γ < 1− b, there isΩ γ ⊆ Ω * an event of probability 1, which a priori depends on γ, such that for each ω ∈Ω γ , µ n l (·, ω) l is a Cauchy sequence, in the sense of the norm · γ . To this end, one needs to give, for all integers n ≥ 1 and p ≥ 0, a convenient upper bound of the quantity E µ n+p − µ n 2 γ . By using (3.13), (3.18), Fubini Theorem and (3.16), one gets that,
E µ n+p − µ n 2 γ = I I |s − t| −γ E Φ n+p (s) − Φ n (s) Φ n+p (t) − Φ n (t) ds dt (3.19) = I I R R |s − t| −γ exp − ξ 2 2(n + p) − exp − ξ 2 2n exp − η 2 2(n + p) − exp − η 2 2n ×E exp i ξY (s) + ηY (t) dξ dη ds dt.
Moreover, in view of the fact that (ξ, η) → E exp i ξY (s) + ηY (t) is the characteristic function of the centered Gaussian vector Y (s)
Y (t) , one has that E exp i ξY (s) + ηY (t) = exp − 1 2 ξ η t Γ Y (s, t) ξ η , (3.20) where ξ η t is the transpose of ξ η and where Γ Y (s, t) = E Y (s) 2 E Y (s)Y (t) E Y (s)Y (t) E Y (t) 2 is the covariance matrix of Y (s) Y (t)
. Also, observe that for all integers n ≥ 1, p ≥ 0 and real number ξ,
0 ≤ exp − ξ 2 2(n + p) − exp − ξ 2 2n ≤ exp − ξ 2 2(n + p) 1 − exp − p 2n(n + p) ξ 2 ≤ 1 − exp − 1 2n ξ 2 . (3.21)
Putting together, (3.19), (3.20) and (3.21), one obtains that for all integers n ≥ 1, p ≥ 0
0 ≤ E µ n+p − µ n 2 γ ≤ U n ,(3.22)
where
U n = I I R R |s − t| −γ 1 − exp − 1 2n ξ 2 1 − exp − 1 2n η 2 (3.23) × exp − 1 2 ξ η t Γ Y (s, t) ξ η dξ dη ds dt.
Let us now show that lim n→+∞ U n = 0.
(3.24)
The equality (3.24) results from the dominated convergence Theorem. Indeed, for almost all (s, t, ξ, η) ∈ I 2 × R 2 , one has
lim n→+∞ |s − t| −γ 1 − exp − 1 2n ξ 2 1 − exp − 1 2n η 2 (3.25) × exp − 1 2 ξ η t Γ Y (s, t) ξ η = 0.
Moreover, using the equalities
R R exp − 1 2 ξ η t Γ Y (s, t) ξ η dξ dη = 2π det Γ Y (s, t) −1/2 ,(3.I I R R |s − t| −γ exp − 1 2 ξ η t Γ Y (s, t) ξ η dξ dη ds dt = 4π δ 2 δ 1 t δ 1 (t − s) −γ det Γ Y (s, t) −1/2 ds dt = 4π δ 2 δ 1 t δ 1 (t − s) −γ Var Y (s) × Var Y (t)|Y (s) −1/2 ds dt ≤ c 1 δ 2 δ 1 t δ 1 (t − s) −γ−b log(t − s) −1 ds dt < ∞,(3.28)
where c 1 is a finite constant only depending on δ 1 ; observe that the last integral is finite since
γ < 1 − b.
Relations (3.25) and (3.28), allow us to use the dominated convergence Theorem and to obtain Relation (3.24). Next, it follows from (3.24), that there is an increasing subsequence l → n l such that for all l one has U n l ≤ 2 −l .
Then setting in (3.22), n = n l and p = n l+1 − n l , and using Cauchy-Schwarz inequality, one obtains that E µ n l+1 − µ n l γ ≤ 2 −l/2 and consequently that
E +∞ l=0 µ n l+1 − µ n l γ < ∞.
This implies that there exists an eventΩ γ ⊆ Ω * of probability 1 such that for all ω ∈Ω γ , +∞ l=0 µ n l+1 (·, ω) − µ n l (·, ω) γ < ∞.
Therefore µ n l (·, ω) l is a Cauchy sequence for the norm · γ and, as a consequence, it converges to a positive measure µ(·, ω) of finite γ energy. In view of (3.17), one clearly has that Supp µ(·, ω) ⊆ I, let us show that one even has,
Supp µ(·, ω) ⊆ L Y (ω). (3.29)
Let g be a bounded continuous function on the real line which vanishes on a neighborhood of L Y (ω) and let K be the compact set defined as K = I ∩ (Supp g). Observe that, in view of the definition of L Y (ω) and of the continuity of the function t → Y (t, ω) 2 , there is a constant C 2 (ω) > 0 such that for all t ∈ K, Y (t, ω) 2 ≥ C 2 (ω). Therefore, using (3.14) and (3.15), one has for all l,
R g(t) dµ n l (t, ω) = (2πn l ) 1/2 K g(t)e −n l Y (t,ω) 2 /2 dt ≤ (2πn l ) 1/2 e −n l C 2 (ω)/2 max t∈K |g(t)|,
and consequently that To this end, we will use the following lemma whose proof is elementary (see e.g. [24] page 8).
Lemma 3.5 Let X be a real-valued nonnegative random variable with a finite non-vanishing second moment. Then one has for all λ ∈ (0, 1),
P X ≥ λE(X) ≥ (1 − λ) 2 E 2 (X) E(X 2 ) . (3.32)
It follows from (3.14), (3.16), Fubini Theorem and the fact that ξ → E[e iξY (t) ] is the characteristic function of the centered Gaussian random variable Y (t), that
E µ n l (I) = I R e −ξ 2 /2n l E e iξY (t) dξ dt = I R e −ξ 2 /2n l e −σ Y (t) 2 ξ 2 /2 dξ dt,
where σ Y (t) is the standard deviation of Y (t). Then, using the dominated convergence Theorem and Lemma 3.2, one has that
lim l→+∞ E µ n l (I) = √ 2π I σ Y (t) −1 dt = c 5 > 0.0 < c 6 = lim l→+∞ E µ n l (I) 2 = 2π I I det Γ Y (s, t) −1/2 ds dt (3.34) ≤ c 1 δ 2 δ 1 t δ 1 (t − s) −b log(t − s) −1 ds dt < ∞,
where c 1 is the finite constant already introduced in (3.28) and where the last integral is finite since b < 1. Let λ 0 ∈ (0, 1) be such that P µ I = λ 0 c 5 = 0. By using (3.33) as well as the fact that one has, almost surely (more precisely, on the eventΩ γ ),
µ I = lim l→+∞ µ n l (I),
one obtains that P µ I ≥ λ 0 c 5 = lim l→+∞ P µ n l I ≥ λ 0 E µ n l (I) .
Then
that for all ω ∈ D m , dim H L Y (ω) ≥ 1 − b − m −1 . Therefore taking D = +∞ m>(1−b) −1 D m ∩ Ω * * ,
where Ω * * is the event of probability 1 introduced in Theorem 2.5 Part (iii), one obtains Theorem 2.5 Part (iv).
|X(t ′ , ω) − X(t ′′ , ω)| |t ′ − t ′′ | λ < ∞, almost surely. (4.2)
Next (4.2) and the Gaussianity of the process {X(t)} t∈J , entail (see [26]) that
sup t ′ ,t ′′ ∈J E|X(t ′ , ω) − X(t ′′ , ω)| 2 |t ′ − t ′′ | 2λ ≤ E sup t ′ ,t ′′ ∈J |X(t ′ , ω) − X(t ′′ , ω)| 2 |t ′ − t ′′ | 2λ < ∞
and, as a consequence (see (1.4)), that λ ≤ b X (J). One gets, from the latter inequality, that b X (J) ≤ b X (J), since λ ∈ (0, b X (J)) is arbitrary.
Proof of Theorem 1.1 Part (ii): First observe that, assuming that J 1 and J 2 are two arbitrary non-degenerate compact intervals satisfying J 1 ⊆ J 2 , one has, in view of (1.4) and (1.1), that,
b X (J 1 ) ≥ b X (J 2 ) and β X (J 1 , ω) ≥ β X (J 2 , ω) for all ω.
Therefore (1.7) and (1.2), imply that,
a X (s) = sup b X [u, v] : u, v ∈ Q and s ∈ (u, v) (4.3)
and, for each ω,
α X (s, ω) = sup β X [u, v], ω : u, v ∈ Q and s ∈ (u, v) , (4.4)
where Q denotes the set of the rational numbers. On the other hand, (1.5) and the fact that Q is a countable set, entail that,
P β X [u, v] = b X [u, v] : for all u, v ∈ Q such that u < v = 1.
Proof of Theorem 2.1
As we have already mentioned (see Remark 2.2), Parts (i) and (iii) of Theorem 2.1 have been obtained in [4], however, we need several ingredients of their proofs, in order to derive the other parts of the theorem. This is the reason why these proofs will be recalled in the sequel.
The modification { B(t, θ)} (t,θ)∈R×(0,1) of the field {B(t, θ)} (t,θ)∈R×(0,1) (see (1.9)), will be defined as a random wavelet type series. Let us first introduce some notations related to wavelets.
• We denote by ψ a Lemarié-Meyer real-valued mother wavelet [27,30,12]. Recall that it satisfies the following three nice properties:
(a) ψ belongs to the Schwartz class S(R); which means that ψ is a C ∞ function and decreases at infinity, as well as all its derivatives of any order, faster than any polynomial.
(b) The support of ψ, the Fourier transform of ψ, is contained in the ring ξ ∈ R : 2π 3 ≤ |ξ| ≤ 8π 3 }; throughout this subsection, the Fourier transform of a function f ∈ L 1 (R) is definied for every real number ξ, as f (ξ) = R e −iξx f (x) dx.
Sometime, we denote f by F(f ).
(c) The collection of the functions:
ML = 2 j/2 ψ(2 j · −k) : (j, k) ∈ Z 2 ,
forms an orthonormal basis of the Hilbert space L 2 (R).
Observe that (a) and (b) imply that ψ belongs to Lizorkin space (see e.g. page 148 in [32] for a definition of this space).
• We denote by Ψ the real-valued function defined for every (y, θ) ∈ R × (0, 1) as,
Ψ(y, θ) = R (y − x) θ−1/2 + ψ(x) dx. (4.6)
Observe that for all fixed θ ∈ (0, 1), the Fourier transform F Ψ(·, θ) of the function t → Ψ(t, θ), satisfies, for all real number ξ = 0,
F Ψ(·, θ) (ξ) = Γ(θ + 1/2)e −i sgn(ξ)(θ+1/2) π 2 ψ(ξ) |ξ| θ+1/2 , (4.7)
where Γ is the usual Gamma function, defined for every real number z > 0, as,
Γ(z) = +∞ 0 x z−1 e −x dx.
Relation (4.7) comes from the fact that for each fixed θ ∈ (0, 1), the function Ψ(·,θ) Γ(θ+1/2) , is the left-sided fractional primitive of ψ of order θ + 1/2, we refer to Chapter 2 of [32] for its proof. It is worth noticing that by using (4.7) and a method quite similar to that which allowed to obtain Lemma 2.1 in [4] and Lemma 2.4 in [6], one can show that Ψ is C ∞ over R × (0, 1) and also, that it is, as well as its partial derivatives of any order, well-localized in the variable y ∈ R, uniformly in the variable θ ∈ (0, 1); in other words, for every nonnegative integers m and n, one has sup 2 + |y| 2 (∂ m y ∂ n θ Ψ)(y, θ) : (y, θ) ∈ R × (0, 1) < ∞. (4.8)
• We denote by ε j,k : (j, k) ∈ Z 2 the sequence of the real-valued independent N (0, 1)
Gaussian random variables defined, for all (j, k) ∈ Z 2 , as,
ε j,k = 2 j/2 R ψ(2 j x − k) dW (x). (4.9)
Roughly speaking, the field { B(t, θ)} (t,θ)∈R×(0,1) is defined as,
B(t, θ) = (j,k)∈Z 2 2 −jθ ε j,k Ψ(2 j t − k, θ) − Ψ(−k, θ) .
In order to precisely explain its definition, one needs some preliminary results. The following lemma allows to almost surely control the increase of the sequence ε j,k : (j, k) ∈ Z 2 .
Lemma 4.1 There is Ω * an event of probability 1, such that every ω ∈ Ω * satisfies the following two properties.
(i) There exists C a positive random variable, non depending on (j, k) and of finite moment of any order, such that for all (j, k) ∈ Z 2 , one has |ε j,k (ω)| ≤ C(ω) log(2 + |j|) log(2 + |k|).
(4.10)
(ii) For each fixed s ∈ R and j ∈ N, let τ j (s) be the random variable defined as, τ j (s) = max |ε j,k | : k ∈ Z and |s − 2 −j k| ≤ j2 1−j ; (4.11)
then one has,
lim inf j→+∞ τ j (s, ω) ≥ 1/4. (4.12)
The proof of Lemma 4.1 has been omitted, since Part (i) can be obtained similarly to Lemma 4 in [3] and Part (ii) similarly to Lemma 4.1 in [5]. The precise definition of the field { B(t, θ)} (t,θ)∈R×(0,1) is provided by the following proposition.
Proposition 4.2 Let Ψ be the function introduced in (4.6) and let ε j,k : (j, k) ∈ Z 2 be the sequence of the real-valued independent N (0, 1) Gaussian random variables defined in (4.9).
For all fixed ω ∈ Ω * and (t, θ) ∈ R × (0, 1), one has
(j,k)∈Z 2 2 −jθ ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) < ∞. (4.13)
Therefore, the series of real numbers
(j,k)∈Z 2 2 −jθ ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) ,
converges to a finite limit which does not depend on the way the terms of the series are ordered; this limit is denoted by B(t, θ, ω). Moreover for each ω / ∈ Ω * and every (t, θ) ∈ R × (0, 1), one sets B(t, θ, ω) = 0.
Proof of Proposition 4.2:
Let ω ∈ Ω * and (t, θ) ∈ R × (0, 1) be arbitrary and fixed. By using the triangle inequality, (4.8) in which one takes m = n = 0, and (4.10), it follows that for all arbitrary fixed j ∈ N,
k∈Z ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) ≤ C 1 (ω) log(2 + j) k∈Z log(2 + |k|) 2 + |2 j t − k| 2 + log(2 + |k|) 2 + |k| 2 ≤ C 2 (ω) log(2 + j) log(2 + 2 j |t|) k∈Z log(2 + |k|) 2 + |2 j t − [2 j t] − k| 2 + log(2 + |k|) 2 + |k| 2 , (4.14)
where [2 j t] denotes the integer part of 2 j t and where C 1 (ω) and C 2 (ω) are two finite constants non depending on k, j and t. Then, noticing that,
sup y∈[0,1] k∈Z log(2 + |k|) 2 + |y − k| 2 < ∞,(4.15)
it follows from (4.14) that j∈N k∈Z
2 −jθ ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) < ∞. (4.16)
Let us now prove that,
j∈Z − k∈Z 2 −jθ ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) < ∞. (4.17)
Assume that j ∈ Z − is arbitrary and fixed. Applying, for any fixed k ∈ Z, the Mean Value
Theorem to the function y → Ψ(y −k) on the interval min{2 j t, 0}, max{2 j t, 0} ⊆ −|t|, |t| , one has that there exists u ∈ − |t|, |t| , such that,
Ψ(2 j t − k, θ) − Ψ(−k, θ) = 2 j t(∂ y Ψ)(u − k, θ). (4.18)
Next, denote by K(t) and K c (t) the sets K(t) = {k ∈ Z : |k| ≤ |t|} and K c (t) = Z \ K(t);
observe that the cardinality of K(t) is bounded from above by 2|t| + 1. Putting together (4.18), (4.8) in which one takes (m, n) = (1, 0), and (4.10), one obtains that,
k∈K(t) ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) ≤ C 3 (ω) sup y∈R (∂ y Ψ)(y, θ) 2 j |t| log(2 + |j|) k∈K(t) log(2 + |k|) ≤ C 3 (ω)|t| 2|t| + 1 log(2 + |t|) sup y∈R (∂ y Ψ)(y, θ) 2 j log(2 + |j|) (4.19)
and k∈K c (t)
ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) ≤ C 3 (ω)2 j |t| log(2 + |j|) k∈K c (t) sup y∈[−|t|,|t|] (∂ y Ψ)(y − k, θ) log(2 + |k|) ≤ C 4 (ω)2 j |t| log(2 + |j|) +∞ k=[|t|]+1 log(2 + k) 1 + k − [|t|] 2 ≤ C 5 (ω)|t| log(2 + |t|) +∞ k=0 log(2 + k) 2 + k 2 2 j log(2 + |j|),(4.20)
where C 3 (ω) is the positive finite constant C(ω) in (4.10), and where C 4 (ω) and C 5 (ω) are two positive finite constants non depending on k, j and t. Next combining (4.19) and (4.20), with the fact that θ ∈ (0, 1), one gets (4.17). Finally (4.16) and (4.17) show that (4.13) holds.
Let us now explain the reason why the random field { B(t, θ)} (t,θ)∈R×(0,1) can be identified with the random field {B(t, θ)} (t,θ)∈R×(0,1) defined in (1.9). Proof of Proposition 4.3: The proof is quite classical in the area of multifractional processes (see e.g. [9,4,5,6]). First one expands for all fixed (t, θ) ∈ R × (0, 1) the function
x → (t − x) θ−1/2 + − (−x) θ−1/2 +
in the basis ML, then one makes, in the deterministic integrals corresponding to the coefficients, the change of variable u = 2 j x−k, finally using the isometry property of the Wiener integral in (1.9), one can show that the series
(j,k)∈Z 2 2 −jθ ε j,k Ψ(2 j t − k, θ) − Ψ(−k, θ) ,
converges in L 2 (Ω) (Ω is the underlying probability space) to the random variable B(t, θ).
Let us now notice that for all ω ∈ Ω * and (t, θ) ∈ R × (0, 1), B(t, θ, ω) can be expressed as,
B(t, θ, ω) = B 1 (t, θ, ω) + B 2 (t, θ, ω) − R(θ, ω),(4.21)
where
B 1 (t, θ, ω) = 0 j=−∞ 2 −jθ k∈Z ε j,k (ω) Ψ(2 j t − k, θ) − Ψ(−k, θ) , (4.22) B 2 (t, θ, ω) = +∞ j=1 2 −jθ k∈Z ε j,k (ω)Ψ(2 j t − k, θ),(4.23)
and
R(θ, ω) = +∞ j=1 2 −jθ k∈Z ε j,k (ω)Ψ(−k, θ). (4.24)
In the sequel, we show that the functions (t, θ) → B 1 (t, θ, ω) and (t, θ) → R(θ, ω) are C ∞ over R×(0, 1); thus it turns out that for proving Theorem 2.1, it is sufficient to show that it is true when the field { B(t, θ)} (t,θ)∈R×(0,1) , is replaced by the more simple field { B 2 (t, θ)} (t,θ)∈R×(0,1) .
The following technical lemma will play a crucial rôle in the sequel.
Lemma 4.4
For all ω ∈ Ω * and j ∈ Z, we denote by S j (·, ·, ω) the real-valued function defined for every (t, θ) ∈ R × (0, 1) , as,
S j (y, θ, ω) = k∈Z ε j,k (ω)Ψ(y − k, θ). (4.25)
Then, the following two results hold, for each ω ∈ Ω * .
(i) For all j ∈ Z, the function S j (·, ·, ω) is C ∞ over R × (0, 1) and one has for every nonnegative integers m, n and each (y, θ) ∈ R × (0, 1), ≤ C(ω) log(2 + |z|) log(2 + |j|). Proof of Lemma 4.4: First observe that by using (4.10) and (4.8), one can prove that for each fixed (j, y, θ, ω) ∈ Z × R × (0, 1) × Ω * , the series in (4.25) is absolutely convergent, which implies that the function S j (·, ·, ω) is well-defined. Let us now show that Part (i) holds, to this end, it is sufficient to prove that for all nonnegative integers m, n, the series arbitrary real numbers satisfying M > 0 and 0 < a < b < 1. In order to derive the latter result, we will show that, k∈Z |ε j,k (ω)|s k < ∞, (4.28) where for each k ∈ Z,
(∂ m y ∂ n θ S j )(y, θ, ω) = k∈Z ε j,k (ω)(∂ m y ∂ n θ Ψ)(y − k, θ).k∈Z ε j,k (ω)(∂ m y ∂ n θ Ψ)(y − k, θ),s k = sup (∂ m y ∂ n θ Ψ)(y − k, θ) : (y, θ) ∈ [−M, M ] × [a, b] .
Observe that, in view of (4.8), one has, for some constant c 1 > 0 and all k ∈ Z, satisfying |k| > M ,
s k ≤ c 1 2 + |k| − M −2 .
Therefore using (4.10), one gets (4.28). Let us now prove that Part (ii) holds. It follows from
(∂ m y ∂ n θ S j )(y, θ, ω) ≤ k∈Z |ε j,k (ω)| (∂ m y ∂ n θ Ψ)(y − k, θ) ≤ C 2 (ω) log(2 + |j|) k∈Z log(2 + |k|) 2 + |y − k| 2 ≤ C 3 (ω) log(2 + |j|) log(2 + z) k∈Z log(2 + |k|) 2 + |y − [y] − k| 2 ,(4.29)
where [y] is the integer part of y and where C 2 (ω) and C 3 (ω) are two constants non depending on j, y, z and θ. Then, combining (4.29) with (4.15), one obtains (4.27).
The following lemma corresponds to Proposition 2.1 in [4], yet we prefer to give a short proof of it, for the sake of clarity.
Lemma 4.5 For all ω ∈ Ω * the function (t, θ) → B 1 (t, θ, ω) (see (4.22)) is C ∞ over R × (0, 1).
Proof of Lemma 4.5: In view of (4.22) and (4.25), one has that
B 1 (t, θ, ω) = 0 j=−∞ Q j (t, θ, ω),
where for each j ∈ Z − , Q j (·, ·, ω) is the C ∞ function over R × (0, 1), defined as,
Q j (t, θ, ω) = 2 −jθ S j (2 j t, θ, ω) − S j (0, θ, ω) . (4.30)
Therefore, in order to show the lemma, it is sufficient to prove that for each nonnegative integers p, q, and real numbers M, a, b satisfying M > 0 and 0 < a < b < 1, one has,
0 j=−∞ sup (∂ p t ∂ q θ Q j )(t, θ, ω) : (t, θ) ∈ [−M, M ] × [a, b] < ∞. (4.31)
Let us first study the case where p = 0 and q is an arbitrary nonnegative integer. Using (4.30) and the Leibniz formula, it follows that
(∂ q θ Q j )(t, θ, ω) = q l=0 q l (−j log 2) l 2 −jθ (∂ q−l θ S j )(2 j t, θ, ω) − (∂ q−l θ S j )(0, θ, ω) ,(4.32)
where q l is the binomial coefficient q! l!×(q−l)! . Next applying the Mean Value Theorem, one gets that
sup (∂ q−l θ S j )(2 j t, θ, ω) − (∂ q−l θ S j )(0, θ, ω) : (t, θ) ∈ [−M, M ] × [a, b] ≤ 2 j M sup (∂ y ∂ q−l θ S j )(y, θ) : (y, θ) ∈ [−M, M ] × [a, b] . (4.33)
Next putting together (4.32), (4.33) and (4.27), it follows that (4.31) holds in the case where p = 0. Let us now study the case where p ≥ 1 and q is an arbitrary nonnegative integer. In view of (4.30), one has that,
(∂ p t Q j )(t, θ, ω) = 2 j(p−θ) (∂ p y S j )(2 j t, θ, ω).
Therefore, using the Leibniz formula, one obtains that,
(∂ p t ∂ q θ Q j )(t, θ, ω) = q l=0 q l (−j log 2) l 2 j(p−θ) (∂ p y ∂ q−l θ S j )(2 j t, θ, ω). (4.34)
Finally, combining (4.34) with (4.27), one gets (4.31).
The proof of the following lemma has been omitted since it is rather similar to that of Lemma 4.5.
Lemma 4.6 For all ω ∈ Ω * the function (t, θ) → R(θ, ω) (see (4.24)) is C ∞ over R × (0, 1).
Let us now give some results concerning the global regularity of the function (t, θ) → B 2 (t, θ, ω) (see (4.23)).
Lemma 4.7
For all ω ∈ Ω * , the following three results hold.
(i) The function (t, θ) → B 2 (t, θ, ω) is continuous over R × (0, 1).
(ii) For each fixed t ∈ R, the function θ → B 2 (t, θ, ω) is C ∞ over (0, 1).
(iii) For every fixed nonnegative integer n, the function (t, θ) → (∂ n θ B 2 )(t, θ, ω) is continuous over R × (0, 1).
Proof of Lemma 4.7: In view of (4.23) and (4.25), one has that
B 2 (t, θ, ω) = +∞ j=1 V j (t, θ, ω),(4.35)
where for each j ∈ N, V j (·, ·, ω) is the C ∞ function over R × (0, 1), defined as,
V j (t, θ, ω) = 2 −jθ S j (2 j t, θ, ω). (4.36)
Therefore, in order to show the lemma, it is sufficient to prove that for each nonnegative integer q, and real numbers M, a, b satisfying M > 0 and 0 < a < b < 1, one has,
+∞ j=1 sup (∂ q θ V j )(t, θ, ω) : (t, θ) ∈ [−M, M ] × [a, b] < ∞. (4.37)
By using (4.36) and the Leibniz formula, it follows that, Let us now turn to the proof of Theorem 2.1 Part (iv). We need the following two lemmas.
(∂ q θ V j )(t, θ, ω) = q l=0 q l (−j log 2) l 2 −jθ (∂ q−l θ S j )(2 j t, θ, ω).(∂ n θ B 2 )(t, θ 1 , ω) − (∂ n θ B 2 )(t, θ 2 , ω) ≤ C(ω)|θ 1 − θ 2 |, (4.39)
where the finite constant C(ω), only depends on ω, n, M, a, b.
Observe that in the case where n = 0, Lemma 4.8 can be related to Part (c) of Proposition 2.2 in [4].
Proof of Lemma 4.8: The lemma easily results from the Mean Value Theorem and Lemma 4.7.
Lemma 4.9 For all ω ∈ Ω * , for each nonnegative integer n, for every arbitrarily small real number ε > 0 and for every real numbers M, a, b satisfying M > 0 and 0 < a < b < 1, one has, for each θ ∈ [a, b] and t 1 ,
t 2 ∈ [−M, M ], (∂ n θ B 2 )(t 1 , θ, ω) − (∂ n θ B 2 )(t 2 , θ, ω) ≤ C 1 (ω)|t 1 − t 2 | θ−ε . (4.40)
where the finite constant C(ω), only depends on ω, n, ε, M, a, b.
Proof of Lemma 4.9: The proof is inspired by that of Proposition 4.2 in [5]. First observe that, in view of (4.35) and (4.38), in order to derive (4.40), it is sufficient to show that for all l ∈ {0, . . . , n}, one has,
+∞ j=1 j l 2 −jθ (∂ n−l θ S j )(2 j t 1 , θ, ω) − (∂ n−l θ S j )(2 j t 2 , θ, ω) ≤ C 1 (ω)|t 1 − t 2 | θ−ε ,(4.41)
where C 1 (ω) is a finite constant only depending on ω, n, ε, M, a, b. Also observe that there exists a constant c 2 > 0, only depending, on ε, M and n such that, for all integer j ≥ 1,
j l 2 −jθ log(2 + M 2 j ) log(2 + |j|) ≤ c 2 2 −j(θ−ε) . (4.42)
The inequality (4.41) is clearly satisfied when t 1 = t 2 , so from now on we assume that |t 1 − t 2 | > 0. Let then j 0 ≥ 1 be the unique integer such that
M 2 −j 0 +1 < |t 1 − t 2 | ≤ M 2 −j 0 +2 . (4.43)
By using (4.27), (4.42) and (4.43), one has that,
+∞ j=j 0 +1 j l 2 −jθ sup (∂ n−l θ S j )(2 j t, θ, ω) : (t, θ) ∈ [−M, M ] × [a, b] (4.44) ≤ C 3 (ω) +∞ j=j 0 +1 2 −j(θ−ε) ≤ C 3 (ω) 1 − 2 −(a−ε) −1 2 −(j 0 +1)(θ−ε) ≤ C 4 (ω)|t 1 − t 2 | θ−ε ,
where C 3 (ω) and C 4 (ω) are two finite constants only depending on ω, n, ε, M, a, b. On the other hand, it follows from the Mean Value Theorem, that, for all j ∈ N,
(∂ n−l θ S j )(2 j t 1 , θ, ω) − (∂ n−l θ S j )(2 j t 2 , θ, ω) (4.45) ≤ 2 j |t 1 − t 2 | sup (∂ y ∂ n−l θ S j )(y, θ, ω) : (y, θ) ∈ [−2 j M, 2 j M ] × [a, b] .
Then, putting together, (4.45), (4.27), (4.42) and (4.43), one obtains that,
j 0 j=1 j l 2 −jθ (∂ n−l θ S j )(2 j t 1 , θ, ω) − (∂ n−l θ S j )(2 j t 2 , θ, ω) ≤ C 5 (ω)|t 1 − t 2 | j 0 j=1 2 (1−θ+ε)j ≤ C 5 (ω)|t 1 − t 2 | 2 (1−b+ε) − 1 −1 2 (1−θ+ε)(j 0 +1) ≤ C 6 (ω)|t 1 − t 2 | θ−ε ,(4.∂ n θ B 2 . Let (t 1 , θ 1 ) ∈ [−M, M ] × [a, b] and (t 2 , θ 2 ) ∈ [−M, M ] × [a, b]
, there is no restriction to assume that θ 1 = max{θ 1 , θ 2 }. Using the triangle inequality, one has that,
(∂ n θ B 2 )(t 1 , θ 1 , ω) − (∂ n θ B 2 )(t 2 , θ 2 , ω) ≤ (∂ n θ B 2 )(t 1 , θ 1 , ω) − (∂ n θ B 2 )(t 2 , θ 1 , ω) + (∂ n θ B 2 )(t 2 , θ 1 , ω) − (∂ n θ B 2 )(t 2 , θ 2 , ω) .
Next, combining the latter inequality with Lemmas 4.8 and 4.9, we can finish our proof.
Let us now turn to the proof of Part (ii) of Theorem 2.1. Notice that (2.1) is a straightforward consequence of (2.3) in which one takes n = 0 and M, a, b such that (s, θ) ∈
(−M, M ) × [a, b].
In order to show that (2.2) holds, one needs to introduce the real-valued function Ψ, defined for every (y, θ) ∈ R × (0, 1) as, Ψ(y, θ) = 1 2πΓ(θ + 1/2) R e iyξ e −i sgn(ξ)(θ+1/2) π 2 |ξ| θ+1/2 ψ(ξ) dξ, (4.47) where ψ is the Fourier transform of the Lemarié-Meyer mother wavelet ψ introduced at the very beginning of this subsection. Let us now give some useful properties of Ψ. The proof of the following lemma, has been omitted since it is very similar to those of Lemmas 2.1 and
in [4].
Lemma 4.10 (i) Ψ is C ∞ over R × (0, 1) and its partial derivatives of any order are welllocalized in the variable y ∈ R, uniformly in the variable θ ∈ (0, 1); in other words, one has, for every nonnegative integers m and n, sup 2 + |y| 2 (∂ m y ∂ n θ Ψ)(y, θ) : (y, θ) ∈ R × (0, 1) < ∞. (ii) For all θ ∈ (0, 1), the first moment of the function Ψ(·, θ) vanishes, that is R Ψ(y, θ) dy = 0. (4.49)
(iii) For all θ ∈ (0, 1), the system of functions 2 j ′ /2 Ψ(2 j ′ · −k ′ , θ) : j ′ ∈ Z, k ′ ∈ Z (recall that Ψ has been introduced in (4.6)) and 2 j/2 Ψ(2 j · −k, θ) : j ∈ Z, k ∈ Z is biorthogonal. This means that for all j, k, j ′ , k ′ ∈ Z, one has 2 (j+j ′ )/2 R Ψ(2 j ′ t − k ′ , θ) Ψ(2 j t − k, θ) dt = δ(j, k; j ′ , k ′ ), (4.50)
where δ(j, k; j ′ , k ′ ) = 1 if (j, k) = (j ′ , k ′ ) and 0 otherwise.
The following lemma, which can be related to Part (c) of Proposition 3.3 in [5], allows one to understand the motivation behind the introduction of Ψ.
Lemma 4.11
For all ω ∈ Ω * , let B 2 (·, ·, ω) be the function introduced in (4.23). Then for each fixed (j, k, θ, ω) ∈ N × Z × (0, 1) × Ω * , the integral, W j,k (θ, ω) = 2 j(1+θ) R B 2 (s, θ, ω) Ψ(2 j s − k, θ) ds, (4.51)
is well-defined. Moreover, one has, W j,k (θ, ω) = ε j,k (ω), (4.52)
where ε j,k is the N (0, 1) Gaussian random variable introduced in (4.9).
Proof of Lemma 4.11: First observe that by using the second inequality in (4.27), in the case where m = n = 0, z = 1 + |s| and θ ∈ [a, b], one obtains, in view of (4.23), that Proof of Part (ii) of Theorem 2.1: As we have mentioned before, Relation (2.1) is a straightforward consequence of Part (iv) of the theorem, which has already been proved. So, it remains to show that (2.2) holds; the proof, we are going to give, is inspired by that of Proposition 4.1 in [5]. It follows from (4.21), Lemma 4.5 and Lemma 4.6, that it is sufficient to show that (2.2) holds, when B is replaced by B 2 . Suppose ad absurdum that the latter relation is not satisfied for some ω 0 ∈ Ω * , (s 0 , θ 0 ) ∈ R × (0, 1) and ε 0 > 0 (notice that there is no restriction to assume that θ 0 + ε 0 < 1), then there exists a finite constant C 1 (ω 0 ) > 0 such that for some finite constant η 0 > 0 and for all real number s satisfying |s − s 0 | ≤ η 0 , one has B 2 (s, θ 0 , ω 0 ) − B 2 (s 0 , θ 0 , ω 0 ) ≤ C 1 (ω 0 )|s − s 0 | θ 0 +ε 0 . (4.54)
B 2 (s, θ, ω) ≤ +∞ j ′ =1 2 −j ′ θ k ′ ∈Z |ε j ′ ,k ′ (ω) Ψ(2 j ′ s − k ′ , θ) ≤ C 1 (ω) log(3 + |s|),(4.
On the other hand, by using the fact that s → B 2 (s, θ 0 , ω 0 ) − B 2 (s 0 , θ 0 , ω 0 ) is a continuous function over R as well as (4.53), one obtains that, sup B 2 (s, θ 0 , ω 0 ) − B 2 (s 0 , θ 0 , ω 0 ) |s − s 0 | θ 0 +ε 0 : s ∈ R and |s − s 0 | ≥ η 0 < ∞. R |s − s 0 | θ 0 +ε 0 Ψ(2 j s − k, θ 0 ) ds ≤ C 2 (ω 0 )2 jθ 0 R 2 −j t + 2 −j k − s 0 θ 0 +ε 0 | Ψ(t, θ 0 )| dt ≤ C 3 (ω 0 )2 −jε 0 1 + 2 j s 0 − k θ 0 +ε 0 , (4.57)
where C 3 (ω 0 ) is a finite constant non depending on j and k. Observe that to derive the last inequality in (4.57), we have used the fact that, for some finite constant c 4 and for every real numbers u, v, one has, |u| + |v| θ 0 +ε 0 ≤ c 4 |u| θ 0 +ε 0 + |v| θ 0 +ε 0 , and we have also used (4.48).
Finally, (4.57) and (4.11) entail that τ j (s 0 , ω 0 ) = O(2 −jε 0 /2 ), which contradicts (4.12).
Proof of Proposition 3.1
Let us denote by {Z(t, θ)} (t,θ)∈R×(0,1) the field defined for each (t, θ) as the Wiener integral,
Z(t, θ) = R (t − x) θ−1/2 + log (t − x) + − (−x) θ−1/2 + log (−x) + dW (x).
(4.58)
By expanding for every fixed (t, θ) ∈ R × (0, 1) the function
x → (t − x) θ−1/2 + log (t − x) + − (−x) θ−1/2 + log (−x) + ,
in the Lemarié-Meyer orthonormal wavelet basis ML (see the beginning of Subsection 4.2), and by using the isometry property of the Wiener integral, it follows that Z(t, θ) = j∈Z k∈Z ε j,k d j,k (t, θ) − d j,k (0, θ) , (4.59) where the N (0, 1) Gaussian random variables ε j,k have been introduced in (4.9) and where d j,k (t, θ) = 2 j/2
R (t − x) θ−1/2 + log (t − x) + ψ(2 j x − k) dx. (4.60)
Observe that the series in (4.59) is convergent, for every fixed (t, θ), in L 2 (Ω), where Ω is the underlying probability space. Setting in (4.60) s = 2 j x−k, and using Lemma 4.4, one obtains after a sequence of standard computations, that for every ω ∈ Ω * (the event of probability 1 introduced in Theorem 2.1) and for each (t, θ) ∈ R × (0, 1), one has when j ≤ 0, (∂ θ Q j )(t, θ, ω) = k∈Z ε j,k (ω) d j,k (t, θ) − d j,k (0, θ) . and, one has when j > 0, (∂ θ V j )(t, θ, ω) − (∂ θ V j )(0, θ, ω) = k∈Z ε j,k (ω) d j,k (t, θ) − d j,k (0, θ) ; (4.62)
recall that the C ∞ functions Q j (·, ·, ω) and V j (·, ·, ω) have been introduced in (4.30) and in (4.36). Moreover using Proposition 4.2, (4.31) and (4.37), it follows that for all ω ∈ Ω * and (t, θ) ∈ R × (0, 1),
(∂ θ B)(t, θ, ω) = 0 j=−∞ (∂ θ Q j )(t, θ, ω) + +∞ j=1
(∂ θ V j )(t, θ, ω) − (∂ θ V j )(0, θ, ω) .
Let H be an arbitrary fixed continuous function defined on R and with values in the open interval (0, 1). The mBm of functional parameter H, is the centered Gaussian process {X(t)} t∈R defined for all t as,
n
= 0 and [−M, M ] is replaced by [0, 1] (see in [4], Theorem 2.1 and Proposition 2.2 Part (b)).
Theorem 2. 5
5Let H : R → (0, 1) be a continuous function, which is nowhere differentiable on some open non-empty interval J ⊆ (0, +∞) and satisfies on it, the condition:α H (s) < H(s) < 2α H (s), for each s ∈ J ,(A)where α H (s) is the pointwise Hölder exponent of H at s. We denote by {α X (s)} s∈R , the pointwise Hölder exponent of {X(t)} t∈R = {B(t, H(t))} t∈R , the mBm of functional parameter H, and we assume that Ω * is the event of probability 1, introduced in Theorem 2.1. Then, the following four results hold.
α
H (s) = ζ(s) ∈ [1/4, 7/24] for any real number s. The proofs of Parts (iii) and (iv) of Theorem 2.5 are postponed to the next section, since they require a specific treatment. Parts (i) and (ii) will result from Theorem 2.1, let us present the main ideas of their proof, before giving the technical details of it. To this end,
study two cases: (∂ θ B) s, H(s), ω = 0 and (∂ θ B) s, H(s), ω = 0. In the case where (∂ θ B) s, H(s), ω = 0; (2.8), (2.10) and (2.11), imply that R(s, h, ω) ≍ |h| α H (s) .(2.12)
together ( 2 .
222), (2.25), (2.23) and the inequality |H(s + h) − H(s)| ≤ c|h| α H (s)−ε/2 (c being a constant), one obtains (2.15). Let us now show that (2.16) holds. It follows from the first equality in (1.10) as well as the triangle inequality that X(s+h, ω)−X(s, ω) ≥ B(s+h, H(s), ω)−B(s, H(s), ω) − B(s+h, H(s+h), ω)−B(s+h, H(s), ω) . Then combining the latter inequality with (2.21), one gets (2.16). Rigorous proof of Theorem 2.5 Parts (i) and (ii): The proof can easily be done by making use of condition (A), Remark 2.9, Lemma 2.8, (2.2) in which one takes θ = H(s),
is equal to 1 .
1Next, in view of the fact that, for all fixed ω ∈ Ω * * ⊆ Ω * , (∂ θ B)(s 0 , H(s 0 ), ω) = 0 and s → (∂ θ B)(s, H(s), ω) is a continuous function over the real line (this easily results from Theorem 2.1 Part (iv) as well as from the continuity of H), one has that
Proposition 3. 3
3For all integer n ≥ 2 and for each s 1 , . . . , s n ∈ I satisfying
observe that for every x ∈ [s n−1 , s n ] and l ∈ {1, . . . , n − 1}, one has, in view of (3.4), that s l − x ≤ 0, therefore, (s l − x)
µ n (·, ω) = I.(3.17)
Y (t, s) = Var Y (s) × Var Y (t)|Y (s) ,(3.27) and using the fact that I = [δ 1 , δ 2 ], Lemma 3.2, Proposition 3.3 and (3.2), one has
t) dµ n l (t, ω) = 0; thus one obtains(3.29). Let us now show that there are two constants c 3 > 0 and c 4 > 0, which do not depend on γ, and that there exists an event D γ ⊆Ω γ , a priori depending on γ, which satisfies P(D γ ) ≥ c 3(3.30) and for all ω ∈ D γ , µ(L Y (ω), ω) = µ(I, ω) ≥ c 4 .(3.31)
Then using the dominated convergence Theorem, (3.26), (3.27), the equality I = [δ 1 , δ 2 ], Lemma 3.2, Proposition 3.3 and (3.2), one obtains that
, c 4 = λ 0 c 5 and D γ = {ω ∈Ω γ : µ(I, ω) ≥ c 4 }, one gets(3.30) and(3.31). Next it follows from Lemma 3.4 that, for all real number γ, satisfyingγ < 1 − b and for all ω ∈ D γ , one has dim H L Y (ω) ≥ γ.(3.35) For every integer m > (1 − b) −1 , let D m be the event defined as D m = +∞ n=m D 1−b−n −1 . It is clear that for all m, D m+1 ⊆ D m . Moreover, (3.30) implies that P(D m ) ≥ c 3 and (3.35)
Proof of Theorem 1.1 Part (i): First observe that by using the same method as in the proofs of Lemma 3.5 in[4] and Proposition 3.6 in[5], one can show that there exists a deterministic quantity b X (J) ∈ [0, 1], such that,P β X (J) = b X (J) = 1.(4.1) Let us prove that b X (J) = b X (J). It follows from (1.4), (1.1), (4.1) and Lemma 2.3 in [4], that b X (J) ≥ b X (J). In view of the latter inequality and the fact that b X (J) ≥ 0, it is clear that one has b X (J) = b X (J) when b X (J) = 0. So from now on, we assume that b X (J) ∈ (0, 1]. Let then λ be an arbitrary deterministic real number belonging to the open interval (0, b X (J)). Relations (4.1) and (1.1), imply that sup t ′ ,t ′′ ∈J
Proposition 4. 3
3The field { B(t, θ)} (t,θ)∈R×(0,1) introduced in Proposition 4.2, is a modification of the field {B(t, θ)} (t,θ)∈R×(0,1) defined in (1.9).
) For all real numbers 0 < a < b < 1 and for every nonnegative integers m, n, there exists a finite constant C(ω) only depending on a, b, m, n, ω, such that for all j ∈ Z and positive real number z, one has, sup (∂ m y ∂ n θ S j )(y, θ, ω) : (y, θ) ∈ [−z, z] × [a, b] ≤ sup k∈Z |ε j,k (ω)| (∂ m y ∂ n θ Ψ (y − k, θ) : (y, θ) ∈ [−z, z] × [a, b]
is uniformly convergent, on each compact set of the form [−M, M ] × [a, b], where M, a, b are
( 4 .
426),(4.8) and(4.10), that, for all j ∈ Z and (y,θ) ∈ [−z, z] × [a, b],
combining (4.38) with (4.27), one gets (4.37). Now we are in position to prove Parts (i) and (iii) of Theorem 2.1.Proof of Parts (i) and (iii) of Theorem 2.1: These two parts are straightforward consequences of (4.21), and Lemmas 4.5, 4.6 and 4.7.
Lemma 4. 8
8For all ω ∈ Ω * , for each nonnegative integer n, and for every real numbers M, a, b satisfying M > 0 and 0 < a < b < 1, one has for all θ 1 , θ 2 ∈ [a, b],sup t∈[−M,M ]
combining (4.54) with (4.55), it follows that, there is a finite constant C 2 (ω 0 ) > 0, such that for all s ∈ R,B 2 (s, θ 0 , ω 0 ) − B 2 (s 0 , θ 0 , ω 0 ) ≤ C 2 (ω 0 )|s − s 0 | θ 0 +ε 0 . (4.56)Then using (4.51), (4.52), (4.49), (4.56) and the change of variable t = 2 j s − k, one gets that for all integer j ≥ 1 and k ∈ Z,|ε j,k (ω 0 )| = 2 j(1+θ 0 ) R B 2 (s, θ 0 , ω 0 ) Ψ(2 j s − k, θ 0 ) ds = 2 j(1+θ 0 ) R B 2 (s, θ 0 , ω 0 ) − B 2 (s 0 , θ 0 , ω 0 ) Ψ(2 j s − k, θ 0 ) ds ≤ C 2 (ω 0 )2 j(1+θ 0 )
together (4.59), (4.61), (4.62) and (4.63), one obtains the proposition.
46 )
46where C 5 (ω) and C 6 (ω) are two finite constants only depending on ω, n, ε, M, a, b. Finally Now we are in position to prove Part (iv) of Theorem 2.1. of Part (iv) of Theorem 2.1: it follows from (4.21), Lemma 4.5 and Lemma 4.6, that it is sufficient to show that (2.3) holds, when ∂ n θ B is replaced bycombining (4.44) with (4.46), one gets (4.41).
Proof
53 )
53where C 1 (ω) is a constant only depending on a, b, ω. Thus (4.53) and (4.48) imply that the integral in (4.51) is well-defined. Moreover (4.23), the dominated convergence Theorem and (4.50), entail that (4.52) holds. Now, we are in position to prove Part (ii) of Theorem 2.1.
Acknowledgement.The author thanks the anonymous referees and Professor Davar Khoshnevisan, for their valuable comments, which have led to improvements of the manuscript.Some parts of it have been written, while the author was invited Professor at the department of Mathematics of the Wuhan University in China; he is very grateful to this department for its financial support, also he thanks all of its members for their kindness and in particular Professor Yijun Hu.
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. Bâtiment. Antoine Ayache: U.M.R. CNRS 8524, Laboratoire Paul Painlevé[email protected] Ayache: U.M.R. CNRS 8524, Laboratoire Paul Painlevé, Bâtiment M2, Uni- versité Lille 1, 59655 Villeneuve d'Ascq Cedex, France. E-mail: [email protected]
| [] |
[
"Lagrangian stochastic models with specular boundary condition",
"Lagrangian stochastic models with specular boundary condition"
] | [
"Mireille Bossy ",
"Jean-François Jabir "
] | [] | [] | In this paper, we prove the well-posedness of a conditional McKean Lagrangian stochastic model, endowed with the specular boundary condition, and further the mean no-permeability condition, in a smooth bounded confinement domain D. This result extends our previous work[5], where the confinement domain was the upper-half plane and where the specular boundary condition has been constructed owing to some well known results on the law of the passage times at zero of the Brownian primitive. The extension of the construction to more general confinement domain exhibits difficulties that we handle by combining stochastic calculus and the analysis of kinetic equations. As a prerequisite for the study of the nonlinear case, we construct a Langevin process confined in D and satisfying the specular boundary condition. We then use PDE techniques to construct the time-marginal densities of the nonlinear process from which we are able to exhibit the conditional McKean Lagrangian stochastic model. | 10.1016/j.jfa.2014.11.016. | [
"https://arxiv.org/pdf/1304.6050v4.pdf"
] | 119,655,869 | 1304.6050 | 102206b27707e11a3b6d01c3ae79ba7dbd490b95 |
Lagrangian stochastic models with specular boundary condition
9 Dec 2014 December 10, 2014
Mireille Bossy
Jean-François Jabir
Lagrangian stochastic models with specular boundary condition
9 Dec 2014 December 10, 2014Lagrangian stochastic modelmean no-permeabilitytrace problemMcKean-Vlasov- Fokker-Planck equation AMS 2010 Subject classification: 58J65, 34B15, 35Q83, 35Q84
In this paper, we prove the well-posedness of a conditional McKean Lagrangian stochastic model, endowed with the specular boundary condition, and further the mean no-permeability condition, in a smooth bounded confinement domain D. This result extends our previous work[5], where the confinement domain was the upper-half plane and where the specular boundary condition has been constructed owing to some well known results on the law of the passage times at zero of the Brownian primitive. The extension of the construction to more general confinement domain exhibits difficulties that we handle by combining stochastic calculus and the analysis of kinetic equations. As a prerequisite for the study of the nonlinear case, we construct a Langevin process confined in D and satisfying the specular boundary condition. We then use PDE techniques to construct the time-marginal densities of the nonlinear process from which we are able to exhibit the conditional McKean Lagrangian stochastic model.
Introduction
We are interested in the well-posedness of the stochastic process ((X t , U t ); 0 ≤ t ≤ T ), for any arbitrary finite time T > 0, whose time-evolution is given by
X t = X 0 + t 0 U s ds, U t = U 0 + t 0 B[X s ; ρ(s)]ds + σW t + K t , K t = − 0<s≤t 2 (U s − · n D (X s )) n D (X s )½ {Xs∈∂D} , ρ(t)
is the probability density of (X t , U t ) for all t ∈ (0, T ],
(1.1)
where (W t , t ≥ 0) is a standard R d -Brownian motion, the diffusion σ is a positive constant, D is an open bounded domain of R d , and n D is the outward normal unit vector of ∂D. Eq. (1.1) provides a Lagrangian model describing, at each time t, the position X t and the velocity U t of a particle confined within D.
The drift coefficient B is the mapping from D × L 1 (D × R d ) to R d defined by
B[x; ψ] = R d b(v)ψ(t, x, v)dv R d ψ(t, x, v)dvU t = U 0 + t 0 E[b(U s )|X s ]ds + σW t + K t .
In the dynamics of U , the càdlàg process (K t ) confines the component X in D by reflecting the velocity of the outgoing particle. This particular confinement is linked with the specular boundary condition:
γ(ρ)(t, x, u) = γ(ρ)(t, x, u − 2(u · n D (x))n D (x)), dt ⊗ dσ ∂D ⊗ du-a.e. on (0, T ) × ∂D × R d , (1.3) where σ ∂D denotes the surface measure of ∂D and where γ(ρ) stands for the trace of the probability density ρ on (0, T ) × ∂D × R d . As already noticed in [5,Corollary 2.4], under integrability and positiveness properties on γ(ρ), the specular condition (1.3) implies the mean no-permeability boundary condition:
R d (u · n D (x))γ(ρ)(t, x, u) du R d γ(ρ)(t, x, u) du = 0, for dt ⊗ dσ ∂D -a.e. (t, x) ∈ (0, T ) × ∂D. (1.4)
The function (t, x) → R d (u · n D (x))γ(ρ)(t, x, u) du R d γ(ρ)(t, x, u) du , on (0, T ) × ∂D, serves here as a formal representation of the normal component of the bulk velocity at the boundary, so that (1.4) can be seen as E[(U t · n D (X t ))|X t = x] = 0, for dt ⊗ dσ ∂D -a.e. (t, x) ∈ (0, T ) × ∂D.
In view of (1.4), an appropriate notion of the trace of ρ is given with the following Σt (u · n D (x)) γ(ρ)(s, x, u)f (s, x, u) ds dσ ∂D (x) du
= − D×R d f (t, x, u)ρ t (x, u) dx du + D×R d f (0, x, u)ρ 0 (x, u) dx du + Qt ∂ s f + u · ∇ x f + B[·; ρ · ] · ∇ u f + σ 2 2
△ u f (s, x, u)ρ s (x, u) ds dx du (1.5) and, for dt ⊗ dσ ∂D a.e. (t, x) in (0, T ) × ∂D,
R d |(v · n D (x))|γ(ρ)(t, x, v) dv < +∞, (1.6a) R d γ(ρ)(t, x, v) dv > 0. (1.6b)
In addition to the well-posedness of (1.1), we prove that the solution admits a trace in a sense of Definition 1.1 and thus satisfies the specular condition and the mean no-permeability boundary condition (1.3)- (1.4).
Our interest in the model (1.1) and its connection with (1.4) arises with the modeling of boundary conditions of the Lagrangian stochastic models for turbulent flows. These models are developed in the context of Computational Fluid Dynamics (CFD) and feature a class of stochastic differential equations with singular coefficients (we refer to Bernardin et al. [2], Bossy et al. [6] for an account of the various theoretical and computational issues related to these models). The design of boundary conditions for the Lagrangian stochastic models according to some Dirichlet condition or some physical wall law, the analysis of their effects on the nonlinear dynamics and their momenta are among the current challenging issues raised by the use of Lagrangian stochastic models in CFD.
In the kinetic theory of gases, the specular boundary condition belongs to the family of the Maxwell boundary conditions which model the interaction (reflection, diffusion and absorption phenomena) between gas particles and solid surface (see Cercignani [10]). Specifically, the specular boundary condition models the reflection of the particles at the boundary of a totally elastic wall (no loss of mass nor energy).
The intrinsic difficulty to the well-posedness of (1.1) lies in the study of the hitting times {τ n , n ≥ 0} of the particle position (X t ) on the boundary ∂D, defined by τ n = inf{τ n−1 < t ≤ T ; X t ∈ ∂D}, for n ≥ 1, τ 0 = 0, which must tend to infinity to ensure that (K t ) is well defined (with the convention that inf{∅} = +∞). By Girsanov Theorem, it is not difficult to see that the sequence {τ n ; n ≥ 0} is related to the attaining times of the primitive of the Brownian motion on a smooth surface.
In the previous work [5], we established the well-posedness of (1.1) in the case where D is the upper half-plane R d−1 × (0, +∞). In this situation, only one component of the process is confined in [0, +∞), and our construction of the confined process mainly relies on the explicit distribution of the zero-sets of the primitive of one dimensional Brownian motion given in McKean [25] and Lachal [21]. To the best of our knowledge, similar results on these attaining times have only been extended in the case of bounded interval. Note also that in the case treated in [5], the existence problem of trace functions in the sense of Definition 1.1 is solved thanks to the explicit construction of the confined linear Langevin process.
Here some new difficulties are enhanced by the boundary reflection generalized to any smooth bounded domain D. Those difficulties appear first in the construction of the confined linear Langevin process (see Eq. (2.1) that corresponds to (1.1) with b = 0), next in the treatment of the McKean nonlinearity in (1.1) and in the verification of the mean-no-permeability condition.
The approach that we propose in this paper strongly mixes stochastic analysis with PDE analysis.
Main result
From now on, we implicitly assume that σ is positive and that D is an open bounded domain in R d . In addition, the set of hypotheses for the main theorem below is denoted by (H). In this set we distinguish (H Langevin ), the hypotheses for the construction of the linear Langevin process, and (H MVFP ) the hypotheses for the well-posedness of the nonlinear Vlasov-Fokker-Planck equation related to (1.1) (see Eq. (3.1)), as follows:
(H Langevin )-(i) The initial condition (X 0 , U 0 ) is assumed to be distributed according to a given initial law µ 0 having its support in D × R d and such that D×R d |x| 2 + |u| 2 µ 0 (dx, du) < +∞.
(H Langevin )-(ii) The boundary ∂D is a compact C 3 submanifold of R d .
(H MVFP )-(i) b : R d → R d is a bounded measurable function.
(H MVFP )-(ii) The initial law µ 0 has a density ρ 0 in the weighted space L 2 (ω, D × R d ) with ω(u) := (1 + |u| 2 ) α 2 for some α > d ∨ 2 (see the Notation subsection below for a precise definition).
(H MVFP )-(iii) There exist two measurable functions P 0 , P 0 : R + −→ R + such that 0 < P 0 (|u|) ≤ ρ 0 (x, u) ≤ P 0 (|u|), a.e. on D × R d ;
and R d (1 + |u|)ω(u)P We further introduce the set Π ω := Q, probability measure on T s.t., for all t ∈ [0, T ], Q • (x(t), u(t)) −1 ∈ L 2 (ω; D × R d ) . (1.1) in Π ω . Moreover the set of time-marginal densities (ρ(t, ·), t ∈ [0, T ]) is in V 1 (ω, Q T ) and admits a trace γ(ρ) in the sense of Definition 1.1 which satisfies the no-permeability boundary condition (1.4).
Theorem 1.2. Under (H), there exists a unique solution in law to
The precise definition of the weighted Sobolev space V 1 (ω, Q T ) is given in the Notation subsection below.
The rest of the paper is devoted to the proof of Theorem 1.2. In Section 2, we set the linear basis of our approach: we construct the solution to the confined linear Langevin process (solution to (2.1)) and we study the property of its semi-group. This latter will rely on a Feynman-Kac interpretation of the semi-group and the analysis of the boundary value problem
∂ t f (t, x, u) − (u · ∇ x f (t, x, u)) − σ 2 2 △ u f (t, x, u) = 0, ∀ (t, x, u) ∈ (0, T ] × D × R d , lim t→0 + f (t, x, u) = f 0 (x, u), ∀ (x, u) ∈ D × R d ,
f (t, x, u) = q(t, x, u), ∀ (t, x, u) ∈ Σ + T ,
(1.7)
for which we prove the existence of a smooth solution, continuous at the boundary (see Theorem 2.5). In Section 3, using a PDE approach, we construct a set of density functions (ρ(t); t ∈ [0, T ]) that features the time-marginal densities of a solution in law to (1.1). More precisely, we construct a weak solution to the following nonlinear McKean-Vlasov-Fokker-Planck equation with specular boundary condition:
∂ t ρ + (u · ∇ x ρ) + (B[· ; ρ] · ∇ u ρ) − σ 2 2 △ u ρ = 0, on (0, T ] × D × R d , ρ(0, x, u) = ρ 0 (x, u), on D × R d , γ(ρ)(t, x, u) = γ(ρ)(t, x, u − 2(u · n D (x))n D (x)), on (0, T ) × ∂D × R d ,
where γ(ρ) stands for the trace of ρ in the sense of Definition 1.1 (see Theorem 3.3 for the existence result). In particular, the verification of the properties (1.6a) and (1.6b) is obtained thank to the construction of Maxwellian bounds for the solution to the nonlinear PDE and its trace at the boundary. Starting from this solution, we set a drift B(t, x) = B[x; ρ(t)] from (1.2) and we construct a process candidate to be a solution of (1.1) using a change of probability measure from the confined Langevin law constructed in Section 2. We achieve the proof of Theorem 1.2 in Section 4, by proving that the resulting set of time-marginal densities coincides with the solution to the McKean-Vlasov-Fokker-Planck equation considered in Section 3. We also prove the uniqueness in law for the solution of (1.1) in Π ω .
Notation
For all t ∈ (0, T ], we introduce the time-phase space
Q t := (0, t) × D × R d ,
and the boundary sets:
Σ + := (x, u) ∈ ∂D × R d s.t. (u · n D (x)) > 0 , Σ + t := (0, t) × Σ + , Σ − := (x, u) ∈ ∂D × R d s.t. (u · n D (x)) < 0 , Σ − t := (0, t) × Σ − , Σ 0 := (x, u) ∈ ∂D × R d s.t. (u · n D (x)) = 0 , Σ 0 t := (0, t) × Σ 0 , and further Σ T := Σ + T ∪ Σ 0 T ∪ Σ − T = (0, T ) × ∂D × R d .
Denoting by dσ ∂D the surface measure on ∂D, we introduce the product measure on Σ T :
dλ Σ T := dt ⊗ dσ ∂D (x) ⊗ du.
We set the Sobolev space
H(Q t ) = L 2 ((0, t) × D; H 1 (R d ))
equipped with the norm H(Qt) defined by
φ 2 H(Qt) = φ 2 L 2 (Qt) + ∇ u φ 2 L 2 (Qt) .
We denote by H ′ (Q t ), the dual space of H(Q t ), and by ( , ) H ′ (Qt),H(Qt) , the inner product between H ′ (Q t ) and H(Q t ).
We define the weighted Lebesgue space
L 2 (ω, Q t ) := ψ : Q t → R s.t √ ωψ ∈ L 2 (Q t ) ,
with the weight function u → ω(u) on the velocity variable
ω(u) := (1 + |u| 2 ) α 2 , for α > d ∨ 2. (1.8) We endow L 2 (ω, Q t ) with the norm L 2 (ω,Qt) defined by φ 2 L 2 (ω,Qt) = √ ωφ L 2 (Qt) .
We introduce the weighted Sobolev space
H(ω, Q t ) := ψ ∈ L 2 (ω, Q t ) s.t |∇ u ψ| ∈ L 2 (ω, Q t )
with the norm H(ω,Qt) defined by
φ 2 H(ω,Qt) = φ 2 L 2 (ω,Qt) + ∇ u φ 2 L 2 (ω,Qt) .
Finally, we define the set
V 1 (ω, Q T ) = C [0, T ]; L 2 (ω, D × R d ) ∩ H(ω, Q T ),
equipped with the norm
φ 2 V 1 (ω,Q T ) = max t∈[0,T ] D×R d ω(u) |φ(t, x, u)| 2 dx du + Q T ω(u)|∇ u φ(t, x, u)| 2 dt dx du.
We further introduce the spaces
L 2 (Σ ± T ) = ψ : Σ ± T → R s.t Σ ± T |(u · n D (x))| |ψ(t, x, u)| 2 dλ Σ T (t, x, u) < +∞ , L 2 (ω, Σ ± T ) = ψ : Σ ± T → R s.t Σ ± T ω(u)|(u · n D (x))| |ψ(t, x, u)| 2 dλ Σ T (t, x, u) < +∞ ,
equipped with their respective norms
ψ L 2 (Σ ± T ) = Σ ± T |(u · n D (x))| |ψ(t, x, u)| 2 dλ Σ T (t, x, u), ψ L 2 (ω,Σ ± T ) = Σ ± T ω(u)|(u · n D (x))| |ψ(t, x, u)| 2 dλ Σ T (t,
x, u).
Preliminaries: the confined Langevin process
In this section, we prove the well-posedness of the confined linear Langevin equation, namely there exists a unique solution, defined on a filtered probability space (Ω, F,
(F t ; t ∈ [0, T ]), P) endowed with an R d -Brownian motion W , to X t = x 0 + t 0 U s ds, U t = u 0 + σW t + K t , K t = −2 0≤s≤t (U s − · n D (X s )) n D (X s )½ {X s ∈ ∂D} , ∀t ∈ [0, T ],(2.1)
for any (x 0 , u 0 ) ∈ (D ×R d )∪(Σ\Σ 0 ). We further investigate some properties of its semigroup (notably the L p -stability).
Well-posedness of (2.1)
We focus our well-posedness result to the case where (x 0 , u 0 ) ∈ (D × R d ) ∪ Σ − , which is the situation where either the particle starts inside D or starts at boundary with an ingoing velocity. We naturally extend the solution to the case where (x 0 , u 0 ) ∈ Σ + (namely the situation of an initial outgoing velocity) by defining (with the flow notation) the solution of (2.1) starting from (x 0 , u 0 ) ∈ Σ + by
(X t , U t ) 0,x 0 ,u 0 ; t ∈ [0, T ] = (X t , U t ) 0,x 0 ,u 0 −2(u 0 ·n D (x 0 ))n D (x 0 ) ; t ∈ [0, T ] .
The construction presented hereafter takes advantage of the regularity of ∂D to locally straighten the boundary, in the same manner than the construction of the diffracted process across a submanifold in [4]. This allows us to adapt the one dimensional construction proposed in [5] and based on the explicit law of the sequence of passage times at zero of the 1D-Brownian motion primitive (see [25], [21]). Our main result is the following:
Theorem 2.1. Under (H Langevin )-(ii), for any (x 0 , u 0 ) ∈ (D × R d ) ∪ Σ − ,
there exists a weak solution to (2.1). Moreover, the sequence of hitting times τ n = inf{τ n−1 < t ≤ T ; X t ∈ ∂D}, for n ≥ 1, τ 0 = 0, is well defined and grows to infinity. The pathwise uniqueness holds for the solution of (2.1).
For the sake of completeness, we recall some results related to the local straightening of the boundary ∂D as given in [4]. Since ∂D is C 3 , one can construct a C 2 b -mapping π from a neighborhood N of ∂D to ∂D such that
|x − π(x)| = d(x, ∂D), ∀x ∈ N ,
where d(x, ∂D) denotes the distance between x and the set ∂D. Note that, reducing N if necessary, we can always assume that π is C 2 b (N ). For all x ∈ N , we set
ς(x) := (x − π(x)) · n D (π(x)) ,(2.2)
so that ς(x) is the signed distance to ∂D (positive in R d \ D, negative in D) and is of class C 2 b (N ). We still denote by ς a C 2 (R d )-extension of this function to the whole Euclidean space. It is well-known (see e.g. [19, p. 355
]) that ∇ς(x) = n D (π(x)), ∀x ∈ N .(2.{U 1 , . . . , U M −1 } such that ∂D ⊂ ∪ M −1 i=1 U i , and a family of R d -valued functions {ψ 1 , . . . , ψ M −1 } such that, for all 1 ≤ i ≤ M − 1, ψ i = (ψ (1) i , . . . , ψ (d) i ) is a C 2 b diffeomorphism from U i to ψ i (U i ), admitting a C 2 b extension on U i and satisfying for all x ∈ U i ψ (d) i (x) = ς(x), ∇ψ (k) i (x) · n D (π(x)) = 0, ∀k ∈ {1, 2, . . . , d − 1}, ∂ψ −1 i ∂x d (ψ i (x)) = n D (π(x)
).
(2.4)
Note that by (2.4), ψ i (U i ∩ ∂D) ⊂ R d−1 × {0}, which justifies the term "local straightening" of the boundary.
Let
U M be an open subset of R d such that ∂D ∩ U M = ∅ and ∪ M i=1 U i = R d , and set ψ M (x) := x on U M .
Proof of Theorem 2.1. For any (x, u) ∈ (D×R d )∪Σ − , we consider the flow of processes ((x t , u t ) s,x,u ; s ≤ t ≤ T ) in R 2d , defined by
x s,x,u t = x + t s u s,x,u r dr, u s,x,u t = u + σ(W t − W s ).
(2.5)
For notation convenience, we set (x t , u t ) := (x t , u t ) 0,x 0 ,u 0 for (x 0 , u 0 ) ∈ (D × R d ) ∪ Σ − ,
and assume that the process is constructed on the canonical filtered probability space (Ω, F, (F t ; t ∈ [0, T ]), P x 0 ,u 0 ). We introduce the index i 1 that corresponds to the smallest index of the open subsets for which x 0 is the most "deeply" contained:
i 1 = min i ∈ {1, . . . , M }; d(x 0 , R d \ U i ) = max 1≤m≤M d(x 0 , R d \ U m ) .
We consider also the exit time
ζ 1 = inf {t ≥ 0 ; x t / ∈ U i 1 } . If i 1 = M , we set for all t ≤ ζ 1 , (X t , U t ) = (x t , u t ).
Else, as the diffeomorphism ψ i 1 is C 2 b on U i 1 and satisfies (2.4), we can apply the Itô formula between 0 and t ≤ ζ 1 to the vector process
(Y t , V t ) = ((Y (k) t , V (k) t ); k = 1, ..., d) given by (Y (k) t , V (k) t ) := ψ (k) i 1 (x t ), (∇ψ (k) i 1 (x t ) · u t ) = ψ (k) i 1 (x t ), d l=1 ∂ x l ψ (k) i 1 (x t )u (l) t .
For all 0 ≤ t ≤ ζ 1 , we obtain that (Y, V ) is a solution to the following system of SDE: for k = 1, . . . , d,
Y (k) t =ψ i 1 (x 0 ) + t 0 V (k) s ds, V (k) t = ∇ψ (k) i 1 (x 0 ) · u 0 + σ t 0 d l=1 ½ {Ys∈ψ i 1 (U i 1 )} ∂ x l ψ (k) i 1 (ψ −1 i 1 (Y s )) + 1 √ d ½ {Ys / ∈ψ i 1 (U i 1 )} dW (l) s , + t 0 ½ {Ys∈ψ i 1 (U i 1 )} 1≤l,n≤d ∂ 2 x l ,xn ψ (k) i 1 (ψ −1 i 1 (Y s ))(∇ψ −1 i 1 (Y s )V s ) (l) (∇ψ −1 i 1 (Y s )V s ) (n) ds.
(2.6) The SDE above has a non-homogeneous diffusion coefficient and a drift coefficient with quadratic growth. Nevertheless, since max t∈[0,T ] |u t | 2 is finite P-a.s., the same holds true for max t∈[0,T ] |V t∧ζ 1 | 2 so that the solution does not explode at finite time and is pathwise unique.
Note that, from (2.4) and (2.3), |∇ψ
(d) i 1 (x)| = 1 on U i 1 , so that the stochastic integral in V (d) t
is a local martingale with a quadratic variation given by
d l=1 ½ {Ys∈ψ i 1 (U i 1 )} ∂ x l ψ (d) i 1 (ψ −1 i 1 (Y s )) + 1 √ d ½ {Ys / ∈ψ i 1 (U i 1 )} dW (l) s t = t.
By Levy characterization, it follows that
W (d) t := t 0 d l=1 ½ {Ys∈ψ i 1 (U i 1 )} ∂ x l ψ (d) i 1 (ψ −1 i 1 (Y s )) + 1 √ d ½ {Ys / ∈ψ i 1 (U i 1 )} dW (l)
s is a standard Brownian motion in R. We also set
W (k) t := t 0 d l=1 (½ {Ys∈ψ i 1 (U i 1 )} ∂ x l ψ (k) i 1 (ψ −1 i 1 (Y s )) + 1 √ d ½ {Ys / ∈ψ i 1 (U i 1 )} )dW (l) s , for k = 1, . . . , d − 1. Now, from the identity ψ −1 (ψ(x)) = x, we easily derive that d i=1 ∂ x i ψ −1 (k) (ψ(x))∂ x l ψ (i) (x)= δ kl ,
where δ kl is the Kronecker delta, and
ψ −1 i 1 (Y t ), ∇ψ −1 i 1 (Y t )V t = (x t , u t ).
Then for any component k, the drift term of V
(k) t writes t 0 ½ {Ys∈ψ i 1 (U i 1 )} 1≤l,n≤d ∂ 2 x l ,xn ψ (k) i 1 (x s )u (l) s u (n) s ds
and is such that
E t 0 ½ {Ys∈ψ i 1 (U i 1 )} 1≤l,n≤d ∂ 2 x l ,xn ψ (k) i 1 (x s )u (l) s u (n) s 2 ds ≤ sup 1≤l,n≤d ∂ 2 x l ,xn ψ (k) i 1 L ∞ (U i 1 ) E t 0 u (l) s u (n) s 2 ds < +∞.
Consequently (see Lipster-Shiryaev [23,Theorem 7.4]), the law of (
(Y t , V t ); t ∈ [0, T ]) is absolutely continuous w.r.t. the law of ((Y t , V t ); t ∈ [0, T ]), solution to Y (k) t = ψ (k) i 1 (x 0 ) + t 0 V (k) s ds, V (k) t = d l=1 ∂ x l ψ (k) i 1 (x 0 )u (l) 0 + σ W (k) t .
(2.7)
In particular, Y (d) is the primitive of the Brownian motion W (d) and McKean [25] has shown that if
(Y d 0 , V d 0 ) = (0, 0) then, P-almost surely, the path t → (Y (d) t , V (d)
t ) never crosses (0, 0). Thus the sequence of passage times at zero of (Y (d) t ) tends to infinity, as (x 0 , u 0 ) ∈ (D × R d ) ∪ Σ − , and the same holds true for the sequence of passage times at zero (β 1 n , n ≥ 1,
β 1 0 = 0) of (Y (d)
t ) as well as for the sequence of hitting times of the boundary ∂D by (x t ). In particular
β 1 1 = inf{t > 0; Y (d) t = 0} = inf{t > 0; Y (d) t = 0} = inf{t > 0; x t ∈ ∂D}. Now we set (X t , U t ) = (x t , u t ), for all 0 ≤ t < β 1 1 ∧ ζ 1 .
Suppose that β 1 1 < ζ 1 . At time β 1 1 , as x β 1 1 ∈ ∂D with (u β 1 1 · n D (x β 1 1 )) > 0 P-a.s., we reflect the velocity as follows:
(X β 1 1 , U β 1 1 ) = x β 1 1 , u β 1 1 − − 2(u β 1 1 − · n D (x β 1 1 ))n D (x β 1 1 )
. We resume the first step of our construction: we set θ 0 = ζ 0 = β 0 1 = 0 and we have defined
i 1 = min i ∈ {1, . . . , M }; d(X θ 0 , R d \ U i ) = max 1≤m≤M d(X θ 0 , R d \ U m ) , ζ 1 = inf{t > θ 0 ; x t / ∈ U i 1 }, β 1 1 = inf{t > θ 0 ; x t ∈ ∂D}.
We set
θ 1 = β 1 1 ∧ ζ 1 and (X t , U t ) = (x t , u t ), for all θ 0 ≤ t < θ 1 , and (X θ 1 , U θ 1 ) = x θ 1 , u θ − 1 − 2(u θ − 1 · n D (x θ 1 ))n D (x θ 1 )½ {X θ 1 ∈∂D} .
We iterate the construction as follows: assume that we have constructed the process (X t , U t ) on [0, θ n ∧ T ]. We define
i n+1 = min i ∈ {1, . . . , M }; d(X θn , R d \ U i ) = max 1≤m≤M d(X θn , R d \ U m ) , (2.8) ζ n+1 = inf{t > θ n ; x t / ∈ U i n+1 }, (2.9) β n+1 1 = inf{t > θ n ; x t ∈ ∂D},(2.10)
where now (x t , u t ) denotes the solution of (2.5) starting at (θ n , X θn , U θn ). We set
θ n+1 = β n+1 1 ∧ ζ n+1 , (2.11)
(X t , U t ) = (x t , u t ) θn,X θn ,U θn , for all θ n ≤ t < θ n+1 , (2.12) and
(X θ n+1 , U θ n+1 ) = x θ n+1 , u θ − n+1 − 2(u θ − n+1 · n D (x θ n+1 ))n D (x θ n+1 )½ {X θ n+1 ∈∂D} . (2.13)
Note that the index n above is always greater or equal to the number of times (X t , t ≤ θ n ) hits the boundary. By construction (X t , U t ) satisfies (2.1) and the sequence of hitting times {τ m ; m ≥ 0} is well defined as each τ m corresponds to some β n 1 . We conclude on the existence of a solution to (2.1) with Lemma 2.3 proved below.
The pathwise uniqueness result is consequence of the well-posedness of the hitting time sequence {τ m , m ≥ 0}: consider (X, U ) and ( X, U ), two solutions to (2.1) defined on the same probability space, endowed with the same Brownian motion. Let us denote by τ 1 the first hitting time of X, we observe that τ 1 ∧ T = τ 1 ∧ T due to the continuity of X and X. It follows that U τ 1 ∧T = U τ 1 ∧T , so that (X, U ) and ( X, U ) are equal up to τ 1 ∧ T . By induction, one checks that this assertion holds true up to τ n ∧ T for all n ∈ N. As τ n tends to +∞ P x 0 ,u 0 -a.s., (X, U ) and ( X, U ) are equal on [0, T ].
Lemma 2.3. For any
(x 0 , u 0 ) ∈ (D × R d ) ∪ Σ − , P x 0 ,u 0 -a.
s., the sequence {θ n ; n ∈ N} given in (2.11) grows to infinity as n tends to +∞.
We also emphasize that, starting from (x 0 , u 0 ) ∈ (D × R d ) ∪ Σ − , P x 0 ,u 0 -almost surely any path of the free process (x t , u t ) solution of (2.5) never crosses Σ 0 . Indeed, the probability for the solution of (2.5) to cross Σ 0 is dominated by the probability that a piece of straightened path (starting in
D × R d ) crosses (R d−1 × {0}) 2 which is nil by the McKean result.
Proof of Lemma 2.3. We already know that the sequence {β n 1 ; n ∈ N} is well defined and grows to infinity. So we only have to prove that {ζ n ; n ∈ N} grows to infinity. We simplify the presentation of the proof by considering the sequence of stopping times {ζ n ; n ∈ N} omitting the intercalation with {β n 1 ; n ∈ N}:
i n+1 = min i ∈ {1, . . . , M }; d(x ζn , R d \ U i ) = max 1≤m≤M d(x ζn , R d \ U m ) , (2.14) ζ n+1 = inf{t ≥ ζ n ; x t / ∈ U i n+1 }, ζ 0 = 0 (2.15)
where ((x t , u t ); t ∈ [0, +∞)) is the solution of (2.5) starting at (0, x 0 , u 0 ). For any i ∈ {1, . . . , M } and any
x ∈ ∂U i , max j =i d(x, R d \ U j ) is strictly positive and continuous with respect to x ∈ ∂U i . Since U i is bounded for 1 ≤ i ≤ M − 1 and ∂U M ⊂ M −1 i=1 U i , the set ∂U i is compact for any i ∈ {1, . . . , M }.
Hence, we can define the strictly positive constant γ 0 as the minimal distance that allows the process (x t ) to go from one U i to another U j ,
γ 0 := min 1≤i≤M inf x∈∂U i max j =i d(x, R d \ U j ) > 0.
The idea is to prove that, almost surely, ζ k − ζ k−1 ≥ T infinitely often w.r.t. k ≥ 1 for T small enough. We fix a constant T > 0 and we consider
A k = {sup 0≤t≤T |x ζ k−1 +t − x ζ k−1 | ≤ γ 0 }, such that P(A c k | F ζ k−1 ) ≤ T γ 0 E sup 0≤t≤T |u t | .
Since A k ∈ F ζ k for all k ≥ 1, for all m < n,
P m≤k≤n A c k = E ½ { m≤k≤n−1 A c k } P(A c n | F ζ n−1 ) ≤ P m≤k≤n−1 A c k T γ 0 E sup 0≤t≤T |u t | ≤ . . . ≤ T γ 0 E sup 0≤t≤T |u t | n−m+1 . Therefore, choosing T such that T γ 0 E[sup 0≤t≤T |u 0 + σW t |] < 1, it follows that lim n→+∞ P m≤k≤n A c k = 0, ∀m ≥ 1.
This entails that the events A k occur infinitely often P-a.s. We thus found T > 0 such that the events {ζ k − ζ k−1 ≥ T } occur a.s. infinitely often.
The following lemma is used in Section 4 to achieve the construction of the nonlinear process.
(x 0 , u 0 ) ∈ (D × R d )∪(Σ \ Σ 0 ), for all integer K > 1, the measure K n=0 P x 0 ,u 0 (τ n ∈ dt, X τn ∈ dx, U τn ∈ du) is absolutely continuous w.r.t the measure λ Σ T (dt, dx, du) = dt ⊗ dσ ∂D (x) ⊗ du.
Proof. Owing to the strong Markov property of (X, U ), it is sufficient to prove that for all (x 0 , u 0 ) ∈ Σ − , the probability P x 0 ,u 0 • (τ 1 , X τ 1 , U τ 1 ) −1 is absolutely continuous w.r.t. λ Σ T . Using the same notation than in the proof of Theorem 2.1, for (x 0 , u 0 ) corresponding to some (x ζn , u ζn ) in the iterative construction ((2.8)-(2.13)), it is sufficient to prove that P
x 0 ,u 0 • (β 1 1 , x β 1 1 , u β 1 1 ) −1 and λ Σ T are equivalent, where ((x t , u t ); t ∈ [0, T ])
is the solution of (2.5). Let i 1 be the index of the subset U i 1 such that (2.6)). Then for any measurable test function f ,
x x 0 ,u 0 β 1 1 ∈ U i 1 ∩ ∂D, ψ i 1 (x t ) = Y t , (∇ψ i 1 (x t ) · u t ) = V t and β 1 1 = inf{t > 0; Y (d) t = 0} (with ((Y t , V t ); t ∈ [0, T ]) solution toE Px 0 ,u 0 f (β 1 1 , x β 1 1 , u β 1 1 ) = E Py 0 ,v 0 f (β 1 1 , ψ −1 i 1 (Y β 1 1 ), (∇ψ −1 i 1 (Y β 1 1 ) · V β 1 1 )) ,
where (y 0 , v 0 ) := (ψ −1 i 1 (x 0 ), (∇ψ −1 i 1 · u 0 )). At this point, and owing to the equivalence between the laws of ((Y t , V t ); t ∈ [0, T ]) and ((Y t , V t ); t ∈ [0, T ]), solution to (2.7), we are reduce to prove that the law of (β 1
1 , ψ −1 i (Y β 1 1 ), (∇ψ −1 i 1 (Y β 1 1 ) · V β 1 1 )) is absolutely continuous w.r.t. λ Σ T .
Let us first recall that the joint law of (β 1 1 , Y
(d) β 1 1 , V (d) β 1 1
) is explicitly known (see [21,Theorem 1]) and is absolutely continuous
w.r.t. the product measure dt ⊗ du (d) . Furthermore, β 1 1 is independent of the (d − 1)-first components of (Y t , V t , t ∈ [0, T ])
. Hence we remark that the law of (β 1
1 , Y β 1 1 , V β 1 1 ) is absolutely continuous w.r.t. dt ⊗ dy ′ ⊗ du, where dy ′ is the surface measure of ∂(R d−1 × R + ).
Let us next recall the characterization of the surface measure σ ∂D (see e.g [1, Chapter 5]). As ∂D is C 3 , it can be (locally) represented as the graph of a C 3 function: for all x ∈ ∂D, there exists an open neighborhood U
x ⊂ R d of x and a C 3 b (R d−1 ) function φ x , such that for all y ∈ ∂D ∩ U x , y (d) = φ x (y (1) , . . . , y (d−1) ) and D ∩ U x = {y = (y ′ , y (d) ) ∈ U x , s.t y (d) < φ x (y ′ )}. Hence, for all C ∞ -function g with compact support in ∂D ∩ U x , we have that ∂D g(x)dσ ∂D (x) = R d−1 g((y ′ , φ x (y ′ ))) 1 + |∇φ x (y ′ )| 2 dy ′ .
Owing to this characterization and the preceding remark on the law of (β 1 1 , Y β 1 1 , V β 1 1 ), we conclude on the result.
On the semigroup of the confined Langevin process
In this section we investigate some estimates related to the semigroup associated to the solution of the SDE (2.1); namely, for a nonnegative test function
ψ ∈ C ∞ c (D × R d ), for all (x, u) ∈ (D × R d )∪(Σ \ Σ 0 ), we define Γ ψ (t, x, u) := E P [ψ(X x,u t , U x,u t )] , (2.16) where ((X x,u t , U x,u t ); t ∈ [0, T ]) is the solution of (2.1) starting from (0, x, u) and ((X s,x,u t , U s,x,u t ); t ∈ [0, T ])
is the solution of (2.1) starting from (s, x, u).
As in [5], this semigroup is of particular interest for the construction of the solution to the nonlinear equation (1.1). Due to the pathwise uniqueness of the confined Langevin process, one has that, for all
0 ≤ s ≤ t ≤ T , Γ ψ (t − s, x, u) = E P [ψ(X s,x,u t , U s,x,u t )] ,(2.17)
so that the estimates hereafter can be extended to the semigroup transitions of the process. We consider also the semigroup related to the stopped process:
Γ ψ n (t, x, u) = E P ψ(X x,u t∧τ x,u n , U x,u t∧τ x,u n ) ,
where {τ x,u n ; n ∈ N} is the sequence of hitting times defined in Theorem 2.1 and Γ ψ 0 (t, x, u) = ψ(x, u). The estimates on {Γ ψ n ; n ≥ 1} and Γ ψ rely on the following PDE result, the proof of which is postponed in the next Subsection 2.3.
Theorem 2.5. Assume (H Langevin ). Given two nonnegative functions
f 0 ∈ L 2 (D × R d ) ∩ C b (D × R d ) and q ∈ L 2 (Σ + T ) ∩ C b (Σ + T ), there exists a unique nonnegative function f ∈ C 1,1,2 b (Q T ) ∩ C((0, T ] × (D × R d \ Σ 0 )) ∩ L 2 ((0, T ) × D; H 1 (R d )) which is a solution to ∂ t f (t, x, u) − (u · ∇ x f (t, x, u)) − σ 2 2 △ u f (t, x, u) = 0, for all (t, x, u) ∈ Q T , f (0, x, u) = f 0 (x, u), for all (x, u) ∈ D × R d , f (t, x, u) = q(t, x, u), for all (t, x, u) ∈ Σ + T . (2.18) In addition, for (x x,u t , u x,u t ; t ∈ [0, T ]) solution to (2.5) starting from (x, u) ∈ D × R d at t = 0 and β x,u := inf{t > 0 ; x x,u t ∈ ∂D}, we have f (t, x, u) = E P f 0 (x x,u t , u x,u t )½ {t ≤ β x,u } + E P q(t − β x,u , x x,u β x,u , u x,u β x,u )½ {t > β x,u } . (2.19)
Furthermore, for all t ∈ (0, T ), f satisfies the energy equality: 20) and the L p -estimate: for 1 ≤ p < +∞,
f (t) 2 L 2 (D×R d ) + f 2 L 2 (Σ − t ) + σ 2 ∇ u f 2 L 2 (Qt) = f 0 2 L 2 (D×R d ) + q 2 L 2 (Σ + t ) ,(2.f (t) p L p (D×R d ) + f p L p (Σ − t ) ≤ f 0 p L p (D×R d ) + q p L p (Σ + t ) .
(2.21)
From Theorem 2.5, we deduce the following result for {Γ ψ n , n ≥ 1}:
Corollary 2.6. Assume (H Langevin ). Then, for all nonnegative ψ ∈ C c (D × R d ) and all n ∈ N * , Γ ψ n is a nonnegative function in C 1,1,2 b (Q T ) ∩ C(Q T \ Σ 0 ) and satisfies the PDE ∂ t Γ ψ n (t, x, u) − (u · ∇ x Γ ψ n (t, x, u)) − σ 2 2 △ u Γ ψ n (t, x, u) = 0, for all (t, x, u) ∈ Q T , Γ ψ n (0, x, u) = ψ(x, u), for all (x, u) ∈ D × R d , Γ ψ n (t, x, u) = Γ ψ n−1 (t, x, u − 2(u · n D (x))n D (x))
, for all (t, x, u) ∈ Σ + T .
(2.22)
In addition, the set {Γ ψ n , n ≥ 1, Γ ψ 0 = ψ} belongs to L 2 ((0, T ) × D; H 1 (R d )) and satisfies the energy equality
Γ ψ n (t) 2 L 2 (D×R d ) + σ 2 ∇ u Γ ψ n 2 L 2 (Qt) + Γ ψ n 2 L 2 (Σ − t ) = ψ 2 L 2 (D×R d ) + Γ ψ n−1 2 L 2 (Σ − t ) . (2.23) Proof. For n > 1, let us assume that Γ ψ n−1 ∈ C(Q T \ Σ 0 ) with Γ ψ n−1 | Σ − T ∈ L 2 (Σ − T ). Since x → n D (x) is continuous on ∂D, the function (t, x, u) → Γ ψ n−1 (t, x, u − 2(u · n D (x))n D (x)) is in C b (Σ + T ). Furthermore, Γ ψ n−1 (t, x, u−2(u·n D (x))n D (x))| Σ + T is in L 2 (Σ + T ) since, by using the change of variables u → u := u − 2(u · n D (x))n D (x) for fixed x ∈ ∂D, we have Σ + T |(u · n D (x))| Γ ψ n−1 (t, x, u − 2(u · n D (x))n D (x)) 2 dλ Σ T (t, x, u) = Σ − T |(u · n D (x))| Γ ψ n−1 (t, x, u) 2 dλ Σ T (t, x, u) = Γ ψ n−1 2 L 2 (Σ − T ) < +∞.
(2.24)
From the strong Markov property of the solution of (2.1), we get that for all (
x, u) ∈ (D × R d )∪ Σ + , E[ψ(X x,u t∧τ x,u n , U x,u t∧τ x,u n )½ {τ x,u 1 <t} ] = E[Γ ψ n−1 (t − τ x,u 1 , X x,u τ x,u 1 , U x,u τ x,u 1 )½ {τ x,u 1 <t} ]. (2.25) Considering a sequence (x m , u m , m ∈ N) in D × R d converging to (x, u) ∈ Σ + , and t > 0, we have, for m large enough (τ xm,um 1 , X xm,um t , U xm,um t , t < τ xm,um 1 ) = (β xm,um , x xm,um t , u xm,um t , t < β xm,um ) (X xm,um τ xm,um 1 , U xm,um τ xm,um 1 ) = (x xm,um β xm,um , u xm,um β xm,um ).
Hence, from the continuity of (y, v) → (β y,v , x y,v t , u y,v t ) proved with Proposition 2.13,
lim m→+∞ (τ xm,um 1 , X xm,um t∧τ xm,um 1 , U xm,um t∧τ xm,um 1 ) = (0, x, u − 2(u · n D (x))n D (x)).
The right-hand side of (2.25) is then continuous on (D × R d ) ∪ Σ + , as well as
E[ψ(X x,u t∧τ x,u n , U x,u t∧τ x,u n )½ {τ x,u 1 ≥t} ] = E[ψ(X x,u t∧τ x,u 1 , U x,u t∧τ x,u 1 )½ {τ x,u 1 ≥t} ]. Moreover, for (t, x, u) ∈ Σ + T , Γ ψ n (t, x, u) = lim m→+∞ E[ψ(X xm,um t∧τ xm ,um n , U xm,um t∧τ xm,um n )½ {τ xm,um 1 <t} ] + E[ψ(X xm,um t∧τ xm ,um n , U xm,um t∧τ xm,um n )½ {τ xm,um 1 ≥t} ] = Γ ψ n−1 (t, x, u − 2(u · n D (x))n D (x)).
Now Theorem 2.5 ensures that there exists a classical solution
f n to (2.18) for f 0 = ψ and q(t, x, u) = Γ ψ n−1 (t, x, u − 2(u · n D (x))n D (x)) on Σ + T . According to (2.19), we have, for (t, x, u) ∈ Q T f n (t, x, u) = E P ψ(x x,u t , u x,u t )½ {t ≤ β x,u } + E P Γ ψ n−1 t − β x,u , x x,u β x,u , u x,u β x,u − 2(u x,u β x,u · n D (x x,u β x,u ))n D (x x,u β x,u ) ½ {t > β x,u } .
One can observe that
E P ψ(x x,u t , u x,u t )½ {t ≤ β x,u } = E P ψ(X x,u t , U x,u t )½ {t ≤ τ x,u 1 } = E P ψ(X x,u t∧τ x,u n , U x,u t∧τ x,u n )½ {t ≤ τ x,u 1 }
and that
E P Γ ψ n−1 t − β x,u , x x,u β x,u , u x,u β x,u − 2(u x,u β x,u · n D (x x,u β x,u ))n D (x x,u β x,u ) ½ {t > β x,u } = E P Γ ψ n−1 (t − τ x,u 1 , X x,u τ x,u 1 , U x,u τ x,u 1 )½ {t > τ x,u 1 } = E P ψ(X x,u t∧τ x,u n , U x,u t∧τ x,u n )½ {t > τ x,u 1 } ,
where the second equality follows from the strong Markov property of (X x,u t , U x,u t ). Therefore
f n (t, x, u) = E P ψ(X x,u t∧τ x,u n , U x,u t∧τ x,u n ) = Γ ψ n (t, x, u), from which we deduce that Γ ψ n ∈ C 1,1,2 b (Q T ) ∩ C(Q T \ Σ 0 T ) ∩ L 2 ((0, T ) × D; H 1 (R d )) is a solution to (2.22) with Γ ψ n | Σ − T ∈ L 2 (Σ − T ). Moreover, according to (2.24), for all t ∈ (0, T ), Γ ψ n (t) 2 L 2 (D×R d ) + Γ ψ n 2 L 2 (Σ − t ) + σ 2 ∇ u Γ ψ n 2 L 2 (Qt) = ψ 2 L 2 (D×R d ) + q 2 L 2 (Σ + t ) = ψ 2 L 2 (D×R d ) + Γ ψ n−1 2 L 2 (Σ − t ) .
For n = 1, setting f 0 = ψ and q = ψ| Σ +
T = 0 (since ψ has its support in the interior of D × R d ), one can check that Γ ψ 1 ∈ C 1,1,2 b (Q T ) ∩ C(Q T \ Σ 0 T ) ∩ L 2 ((0, T ) × D; H 1 (R d )
) satisfies (2.22) and (2.23). By induction, we end the proof.
Corollary 2.7. Assume (H Langevin ). For all nonnegative ψ ∈ C c (D × R d ), Γ ψ is a nonnegative function that belongs to L 2 ((0, T ) × D; H 1 (R d ))
and satisfies the energy equality:
Γ ψ (t) 2 L 2 (D×R d ) + σ 2 ∇ u Γ ψ 2 L 2 (Qt) = ψ 2 L 2 (D×R d ) , ∀ t ∈ (0, T ). (2.26) Furthermore, Γ ψ (t) is solution in the sense of distributions of ∂ t Γ ψ − (u · ∇ x Γ ψ ) − σ 2 2 △ u Γ ψ = 0, on Q T , Γ ψ (0, x, u) = ψ(x, u), on D × R d , Γ ψ (t, x, u) = Γ ψ (t, x, u − 2(u · n D (x))n D (x)), on Σ + T . (2.27)
Proof. We first observe that since ψ| ∂D×R d = 0,
Γ ψ n (t, x, u) = E P ψ(X x,u t∧τ x,u n , U x,u t∧τ x,u n ) = E P ψ(X x,u t , U x,u t )½ {τ x,u n ≥t} .
Next, there exists a nonnegative function β ∈ L 2 (R) such that β(|u|) = 1 on the support of ψ and
ψ ≤ Cβ(|u|), with C := sup (x,u)∈D×R d ψ(x, u). Then 0 ≤ Γ ψ n (t, x, u) ≤ CE P β(|U x,u t |)½ {τ x,u n ≥t} . As E P [β(|U x,u t |)] is equal to the convolution product (G(σt) * β)(|u|), where G denotes the heat kernel on R d , we obtain 0 ≤ Γ ψ n (t, x, u) ≤ C(G(σt) * β)(|u|), on Q T . (2.28)
Owing to the continuity of Γ ψ n , from the interior of Q T to its boundary, (2.28) still holds true along Σ ± T .
Let us now recall that, for a.e. (x, u) ∈ (D × R d )∪(Σ \ Σ 0 ), P (x,u) -a.s. τ n grows to ∞ as n increases, and then
lim n→+∞ Γ ψ n (t, x, u) = Γ ψ (t, x, u), for a.e. (t, x, u) ∈ Q T , λ Σ T -a.e. (t, x, u) ∈ Σ T \ Σ 0 T . (2.29) Indeed, Γ ψ n (t, x, u) − Γ ψ (t, x, u) = E P ψ(X x,u t , U x,u t )½ {τ x,u n ≤t} ≤ ψ ∞ P(τ x,u n ≤ t).
In particular, (2.28) is also true for Γ ψ . We conclude by the Lebesgue Dominated Convergence Theorem that Γ ψ n converges to Γ ψ in L 2 (D × R d ). Next we deduce that the norms involving Γ ψ n in the left-hand side of (2.23) are finite for all t, uniformly in n (as the right-hand side of (2.23) is bounded uniformly in n by the Maxwellian bound (2.28)). Therefore, the estimate (2.23) is also true for Γ ψ (see e.g. [8]), and Γ ψ n converges to Γ ψ in
L 2 ((0, T ) × D; H 1 (R d ))
.
Corollary 2.8 (L p -estimates). Given ψ ∈ C c (D × R d ) nonnegative, the kernels {Γ ψ n , n ≥ 1} and Γ ψ satisfy for all p ∈ [1, +∞) Γ ψ n (t) p L p (D×R d ) + Γ ψ n p L p (Σ − t ) ≤ ψ p L p (D×R d ) + Γ ψ n−1 p L p (Σ − t ) , ∀ t ∈ (0, T ), (2.30) Γ ψ (t) p L p (D×R d ) ≤ ψ p L p (D×R d ) , ∀ t ∈ (0, T ). (2.31)
Proof. Applying estimate (2.21) to the solution of (2.22), it follows that for all t ∈ (0, T ),
Γ ψ n (t) p L p (D×R d ) + Γ ψ n p L p (Σ − t ) ≤ ψ p L p (D×R d ) + Γ ψ n p L p (Σ + t ) . Since Γ ψ n p L p (Σ + t ) = Γ ψ n−1 p L p (Σ − t )
we deduce (2.30). Using the convergence of Γ ψ n to Γ ψ and the uniform bounds (2.28) on Σ − T , we also deduce (2.31).
On the boundary value problem (2.18)
In this section, we prove Theorem 2.5. We consider the inputs (f 0 , q) and assume the following
(H f 0 ,q ): f 0 ∈ L 2 (D × R d ) ∩ C b (D × R d ) and q ∈ L 2 (Σ + T ) ∩ C b (Σ + T ) are nonnegative functions.
The main difficulty in the well-posedness of the boundary value problem (2.18) lies in the degeneracy of the diffusion operator and in the fact that we want to obtain the continuity of f up to and along Σ T \Σ 0 T . Such a problem has been addressed in Fichera [15] for second order differential operators of the form
L(f )(z) = Trace(a(z)∇ 2 f (z)) + (b(z) · ∇f (z)) + c(z)f (z) − h(z), z ∈ R N
where ϑ is some smooth bounded open domain of R N and, for all z ∈ ϑ, a(z) is only assumed to be a positive semi-definite matrix, that is (ξ · a(z)ξ) ≥ 0, for all ξ ∈ R N . Consider PDE of the form
L(f ) = 0, on ϑ (2.32)
submitted to some Dirichlet boundary condition. Denoting by ν(z) the unit outward normal vector to ∂ϑ, the boundary ∂ϑ may be split into four parts: the so-called non-characteristic part Σ 3 := {z ∈ ∂ϑ; (ν(z) · a(z)ν(z)) > 0}, the relevant part Σ 2 := {z ∈ ∂ϑ/Σ 3 ; (b(z) · ν(z)) + Trace(a(z)∇ν(z)) > 0}, the irrelevant part Σ 1 := {z ∈ ∂ϑ/Σ 3 ; (b(z) · ν(z)) + Trace(a(z)∇ν(z)) < 0} and the sticking part
Σ 0 := {z ∈ ∂ϑ/Σ 3 ; (b(z) · ν(z)) + Trace(a(z)∇ν(z)) = 0}.
The term relevant refers to the boundary part where the boundary condition has to be specified:
f = g on Σ 2,3 = Σ 2 ∪ Σ 3 . (2.33)
The existence of solutions f in C(ϑ ∪ Σ 2,3 ) to (2.32)-(2.33) has been studied by several authors, among them Kohn and Nirenberg [20], Oleȋnik [28], Bony [3], and also Manfredini [24] in the context of ultra-parabolic equations with Dirichlet boundary condition along the position×velocity domain (the velocity space is assumed to be bounded). Stochastic interpretation of (2.32)-(2.33) has been studied in Stroock and Varadhan [31], Freȋdlin [16], and Friedman [18]. However to the best of our knowledge, the regularity of f along Σ 1 has not been considered outside a few works. We shall mention the works of Oleȋnik and Radkevič [29] and Taira [32] who have shown the well-posedness of analytic solutions (on ϑ) to the elliptic equation (2.32) with the homogeneous boundary condition
f = 0, on Σ 2,3 ,
under the particular assumption that the sets (Σ i , i = 0, 1, 2, 3) are closed and that Σ 2 ∪ Σ 3 and Σ 0 ∪ Σ 1 are disjoint. Note that such assumption does not hold in the situation of kinetic equations. In that situation, existence of weak solution is well known (see, e.g., Degond [12], Carrillo [9]). In particular, Carrillo [9] considers the situation where (2.33) is the specular boundary condition (1.3) and, establishes the existence of trace functions and a Green identity related to the transport operator T = ∂ t + (u · ∇ x ). As a preliminary for the proof of Theorem 2.5, let us recall a well-known existence result for equation (2.18).
f 0 ∈ L 2 (D × R d ) and q ∈ L 2 (Σ + T ), there exists a unique nonnegative function f in C([0, T ]; L 2 (D × R d )) ∩ H(Q T ) admitting a nonnegative trace γ(f ) ∈ L 2 (Σ T ) along the boundary Σ T , satisfying equation (2.18) in the sense that ∂ t f − (u · ∇ x f ) − σ 2 2 △ u f = 0, in H ′ (Q T ), f (t = 0, x, u) = f 0 (x, u), on D × R d , γ(f )(t, x, u) = q(t, x, u), on Σ + T .
(2.34)
In particular, for all t ∈ (0, T ),
f (t) 2 L 2 (D×R d ) + σ 2 ∇ u f 2 L 2 (Qt) + γ(f ) 2 L 2 (Σ − t ) = f 0 2 L 2 (D×R d ) + q 2 L 2 (Σ + t ) . (2.35) If, in addition f 0 ∈ L p (D × R d ), q ∈ L p (Σ + T ) for p ∈ [1, +∞), then for all t ∈ (0, T ), f (t) p L p (D×R d ) + f p L p (Σ − t ) + σ 2 p(p − 1) Qt |∇ u f | 2 f p−2 ≤ f 0 p L p (D×R d ) + q p L p (Σ + tt ∈ [0, T ], for all ψ ∈ C ∞ b (Q t ), we have Qt f (s, x, u) ∂ s ψ − (u · ∇ x ψ) − σ 2 2 △ u ψ (s, x, u) ds dx du = D×R d ψ(s, x, u)f (s, x, u) s=t s=0 dx du − Σ − t (u · n D (x))γ(f )(s, x, u)ψ(s, x, u) dλ Σ T (s, x, u) − Σ + t (u · n D (x))q(s, x, u)ψ(s, x, u) dλ Σ T (s, x, u),(2.
36)
which expresses that f is a solution to (2.18) in the sense of distributions. Let us further notice that the trace function γ(f ) in L 2 (Σ T ) is characterized by the Green formula related to the transport operator ∂ t + (u · ∇ x ) (we refer to Subsection 3.1 for more details).
Considering the solution f in C([0, T ]; L 2 (D × R d )) ∩ H(Q T ) of (2.34), given by Proposition 2.9, we show its interior regularity and its continuity up to and along Σ T \ Σ 0 T . The proof of the following proposition is postponed to Appendix A.2.
Proposition 2.11 (Interior regularity).
Under (H f 0 ,q ), the unique solution f of (2.34) belongs to C 1,1,2 (Q T ).
Proposition 2.12 (Continuity up to Σ + T ). Assume (H Langevin )-(ii) and (H f 0 ,q ). Let f ∈ C 1,1,2 (Q T ) ∩ C([0, T ]; L 2 (D × R d )) ∩ H(Q T ) be the solution to (2.34) with inputs (f 0 , q). Then f is continuous up to Σ + T .
Proof. To show the continuity up to the boundary Σ + T , we follow the classical method of local barrier functions (see e.g. [19]).
Let (t 0 , x 0 , u 0 ) ∈ Σ + T . Since q is continuous in Σ + T , we can assume that for any ǫ > 0, there exists a neighborhood O t 0 ,x 0 ,u 0 of (t 0 , x 0 , u 0 ) such that q(t 0 , x 0 , u 0 ) − ǫ ≤ q(t, x, u) ≤ q(t 0 , x 0 , u 0 ) + ǫ, ∀(t, x, u) ∈ O t 0 ,x 0 ,u 0 ∩ Σ + T . In addition, since (u 0 · n D (x 0 )) > 0, by reducing O t 0 ,x 0 ,u 0 , we can assume that ς (the signed distance to ∂D given in (2.2)) is in C 2 (O t 0 ,x 0 ,u 0 ) and that (u · ∇ς(x)) > η for all (t, x, u) ∈ O t 0 ,x 0 ,u 0 , for some positive η 1 . Consequently, by setting ̺(x) = −ς(x) and L := ∂ t − (u · ∇ x ) − σ 2 2 △ u , we observe that, for all (t, x, u) ∈ O t 0 ,x 0 ,u 0 , L(̺)(t, x, u) = −(u · ∇̺(x)) = (u · ∇ς(x)) > η > 0. (2.37) Reducing again O t 0 ,x 0 ,u 0 , we can assume that O t 0 ,x 0 ,u 0 has the form (t 0 − δ, t 0 + δ) × B x 0 (δ ′ ) × B u 0 (δ ′ ) (where B x 0 (δ ′ ) [resp. B x 0 (δ ′ )] is the ball centered in x 0 [resp. u 0 ] of radius δ ′ ) for some positive constants δ, δ ′ > 0 such that 0 ≤ t 0 − δ < t 0 + δ ≤ T .
We can construct the barrier functions related to
(t 0 , x 0 , u 0 ) ∈ Σ + T with ω ǫ (t, x, u) = q(t 0 , x 0 , u 0 ) + ǫ + k ǫ ψ x 0 (x) + K ǫ ̺(x), ω ǫ (t, x, u) = q(t 0 , x 0 , u 0 ) − ǫ − k ǫ ψ x 0 (x) − K ǫ ̺(x). (2.38)
where ψ x 0 (x) = (x − x 0 ) 2 and where the parameters k ǫ , K ǫ ∈ R + are chosen large enough so that, for
M + [resp. M − ] an upper-bound [resp. lower-bound] of f on ∂O t 0 ,x 0 ,u 0 ∩ Q T , we have k ǫ inf Ot 0 ,x 0 ,u 0 ∩Q T L(ψ x 0 ) + K ǫ inf Ot 0 ,x 0 ,u 0 ∩Q T L(̺) ≥ 0, k ǫ inf ∂Ot 0 ,x 0 ,u 0 ∩Q T ψ x 0 + K ǫ inf ∂Ot 0 ,x 0 ,u 0 ∩Q T ̺ ≥ M + − (q(t 0 , x 0 , u 0 ) + ǫ) ∨ (q(t 0 , x 0 , u 0 ) − ǫ) .
For example, setting η :
= inf Ot 0 ,x 0 ,u 0 ∩Q T (u · ∇ς(x)), as inf ∂Ot 0 ,x 0 ,u 0 ∩Q T ψ x 0 = δ ′2 , one can choose k ǫ and K ǫ such that −(δ ′ ) 2 k ǫ + K ǫ η = 0, k ǫ (δ ′ ) 2 = M + − q(t 0 , x 0 , u 0 ) ∨ q(t 0 , x 0 , u 0 ).
Thus, ω ǫ and ω ǫ satisfy the properties
(P)- (a) ω ǫ (t, x, u) ≥ q(t, x, u) ≥ ω ǫ (t, x, u) for all (t, x, u) ∈ O t 0 ,x 0 ,u 0 ∩ (0, T ) × ∂D × R d , (b) L(ω ǫ ) ≥ 0 ≥ L(ω ǫ ) for all (t, x, u) ∈ O t 0 ,x 0 ,u 0 ∩ Q T , (c) ω ǫ (t, x, u) ≥ M + , and ω ǫ (t, x, u) ≤ M − , for all (t, x, u) ∈ ∂O t 0 ,x 0 ,u 0 ∩ Q T , (d) lim ǫ→0 + ω ǫ (t 0 , x 0 , u 0 ) = lim ǫ→0 + ω ǫ (t 0 , x 0 , u 0 ) = q(t 0 , x 0 , u 0 ).
Now we shall prove that, for f the solution to (2.18),
ω ǫ ≤ f ≤ ω ǫ on O t 0 ,x 0 ,u 0 ∩ Q T . Owing to the property (P)-(d), this allows to conclude that f (t, x, u) tends to q(t 0 , x 0 , u 0 ) as (t, x, u) tends to (t 0 , x 0 , u 0 ), for all (t 0 , x 0 , u 0 ) of Σ + T .
For the local comparison between ω ǫ and f , let us consider the positive part (f − ω ǫ ) + of f − ω ǫ . Let η 0 denote some nonnegative cut-off function defined in a neighborhood of (t 0 , x 0 , u 0 ) such that η 0 (t, x, u) = 0 for all (t, x, u) ∈ ∂O t 0 ,x 0 ,u 0 , and let β be a real parameter that we will specify later. The function △ u |(f − ω ǫ ) + | 2 is well defined a.e. on Q T since, using Theorem A.2 (see e.g Tartar [33]
), one can check that △ u |(f − ω ǫ ) + | 2 = 2∇ u · ((f − ω ǫ ) + ∇ u (f − ω ǫ )) = 2((f − ω ǫ ) + △ u (f − ω ǫ )) + 2 |∇ u (f − ω ǫ )| 2 ½ {f >ωǫ} . We shall observe that L(η 0 exp {βt} (f − ω ǫ ) + 2 ) = (f − ω ǫ ) + 2 L(η 0 exp {βt}) + η 0 exp {βt}L( (f − ω ǫ ) + 2 ) − σ 2 exp {βt} ∇ u η 0 · ∇ u (f − ω ǫ ) + 2 . The property (P)-(b) ensures that L( (f − ω ǫ ) + 2 ) = (f −ω ǫ ) + L(f −ω ǫ )− σ 2 2 |∇ u (f − ω ǫ )| 2 ½ {f >ωǫ} ≤ − σ 2 2 |∇ u (f − ω ǫ )| 2 ½ {f >ωǫ} ≤ 0,
and thus
L(η 0 exp {βt} (f − ω ǫ ) + 2 ) ≤ exp {βt} (f − ω ǫ ) + 2 {L(η 0 ) + βη 0 } − σ 2 exp {βt} ∇ u η 0 · ∇ u (f − ω ǫ ) + 2 .
We integrate the two sides above over O t 0 ,x 0 ,u 0 ∩ Q T . Since η 0 = 0 on ∂O t 0 ,x 0 ,u 0 , an integration by parts yields
Ot 0 ,x 0 ,u 0 ∩Q T L(η 0 exp {βt} (f − ω ǫ ) + 2 )(t, x, u) = − Ot 0 ,x 0 ,u 0 ∩Σ T (u · n D (x))η 0 (t, x, u) exp {βt}|(γ(f ) − ω ǫ ) + (t, x, u)| 2 dλ Σ T (t, x, u).
Hence, we have obtained that
− Ot 0 ,x 0 ,u 0 ∩Σ T (u · n D (x))η 0 (t, x, u) exp {βt}|(γ(f ) − ω ǫ ) + (t, x, u)| 2 dλ Σ T (t, x, u) ≤ Ot 0 ,x 0 ,u 0 ∩Q T L(η 0 ) + βη 0 + σ 2 △ u η 0 exp {βt}|(f − ω ǫ ) + | 2 , or equivalently, since η 0 = 0 on Σ − T , (as (x 0 , u 0 ) ∈ Σ + ) − Ot 0 ,x 0 ,u 0 ∩Σ + T (u · n D (x))η 0 (t, x, u) exp {βt}|(q − ω ǫ ) + (t, x, u)| 2 dλ Σ T (t, x, u) ≤ Ot 0 ,x 0 ,u 0 ∩Q T L(η 0 ) + βη 0 + σ 2 △ u η 0 exp {βt}|(f − ω ǫ ) + | 2
From (P)-(a) and (P)-(c), the integral along O t 0 ,x 0 ,u 0 ∩ Σ + T is nonnegative. By choosing η 0 and β ∈ R such that L(η 0 ) + βη 0 + σ 2 △ u η 0 < 0,
we conclude that f ≤ ω ǫ on O t 0 ,x 0 ,u 0 . Similar arguments entail that ω ǫ ≤ f .
Feynman-Kac representation and continuity up to and along
Σ − T .
We prove the Feynman-Kac representation (2.19) by replicating the arguments of Friedman [17, Chapter 5, Theorem 5.2]: for
(y, v) ∈ D × R d fixed, let ((x y,v t , u y,v t ); t ∈ [0, T ]) satisfy x y,v t = y + t 0 u y,v s ds, u y,v t = v + σW t , where (W t ; t ≥ 0) is an R d -f (t − s, x y,v s∧β y,v δ , u y,v s∧β y,v δ ), for s ∈ [0, t], yields f (t, y, v) = E P f 0 (x y,v t , u y,v t )½ {t ≤ β y,v δ } + E P f (t − β y,v δ , x y,v β y,v δ , u y,v β y,v δ )½ {t > β y,v δ } .
Since P-a.s., β y,v δ tends to β y,v = inf{t > 0 ; d(x y,v t , ∂D) = 0}, as δ tends to 0, and thanks to Proposition 2.12, one obtains (2.19).
Proposition 2.13. Assume (H f 0 ,q ). Let f ∈ C 1,1,2 (Q T ) ∩ C(Q T ∪ Σ + T )
be the solution to (2.18). Then f is continuous along and up to Σ − T .
Proof. According to (2.19) and since f 0 and q are continuous, the continuity of f up to Σ − T will follow from the continuity of (y, v) → (β y,v , x y,v t , u y,v t ). P-almost surely, for all t ≥ 0, the flow (y, v)
→ (x y,v t , u y,v t ) is continuous on R d × R d . As (y, v) / ∈ Σ 0 ∪ Σ + , we have β y,v = τ y,v := inf{t > 0; x y,v t / ∈ D}.
To prove that (y, v) → τ y,v is continuous up to Σ − , we follow the general proof of the continuity of exit time related to a flow of continuous processes given in Proposition 6.3 in Darling and Pardoux [11]. First, replicating the argument of the authors, one can show that, for all (y m , v m ) ∈ D × R d such that lim m→+∞ (y m , v m ) = (y, v) ∈ Σ − , lim sup m→+∞ τ ym,vm ≤ τ y,v .
Next, it is sufficient to check that τ y,v ≤ lim inf m→+∞ τ ym,vm .
Following [11], we may observe that, as in the proof of Theorem 2.1 , for a.e.
(y, v) ∈ D × R d ∪ Σ − , the path t → (x y,v t , u y,v t ) never hits Σ 0 ∪Σ − , and, since P-a.s. (t, y, v) → (x y,v t , u y,v t ) is continuous on [0, +∞) × D × R d , one can check that {(x ym,vm τ ym,vm , u ym,vm τ ym,vm ); m ∈ N} ⊂ Σ + ,
and that (x y,v lim inf m→+∞ τ ym,vm , u y,v lim inf m→+∞ τ ym,vm ) ∈ Σ + . Since τ y,v = inf{t > 0; (x y,v t , u y,v t ) ∈ Σ + }, we deduce that τ y,v ∈ [0, lim inf m→+∞ τ ym,vm ].
This ends the proof of Theorem 2.5.
On the conditional McKean-Vlasov-Fokker-Planck equation
In this section, we construct a probability density function satisfying (in the sense of distribution):
∂ t ρ + (u · ∇ x ρ) + (B[· ; ρ] · ∇ u ρ) − σ 2 2 △ u ρ = 0, on (0, T ) × D × R d , (3.1a) ρ(0, x, u) = ρ 0 (x, u), on D × R d , (3.1b) γ(ρ)(t, x, u) = γ(ρ)(t, x, u − 2(u · n D (x))n D (x)), on (0, T ) × ∂D × R d , (3.1c)
where B is defined as in (1.2). Clearly (3.1) is the equation of the time marginal law of ((X t , U t ); t ∈ [0, T ]) solution to (1.1). In particular (3.1c) takes into account the specular reflection resulting from the confinement component
(K t ; t ∈ [0, T ]) in (1.1).
Throughout this section, we refer to equation (3.1) as the conditional McKean-Vlasov-Fokker-Planck equation. Furthermore, for notation convenience, we denote by T the transport operator in (3.1a), namely for all test function ψ on Q T ,
T (ψ) = ∂ t ψ + (u · ∇ x ψ). (3.2)
As mentioned in Subsection 2.3, the well-posedness of the linear Vlasov-Fokker-Planck equation and the related trace problem has been well studied in the literature of kinetic equation (we particularly refer to Degond [12], Degond and Mas-Gallic [13], Carrillo [9] and Mischler [26]). More recently, in his study of geometric Kramers-Fokker-Planck operators with boundary conditions [27], Nier showed how to associate a boundary condition operator with the linear Vlasov-Fokker-Planck equation, corresponding to a Langevin stochastic dynamics with jump process at the boundary. This methodology may offer some other perspectives to extend the construction of the nonlinear Langevin dynamics with a general class of boundary conditions.
For the study of Eq. Definition 3.1. For given a ∈ R, µ > 0, P 0 ∈ L 1 (R d ), such that P 0 ≥ 0 on R d , a Maxwellian distribution with parameters (a, µ, P 0 ) is a function P :
R + × R d → R + such that P (t, u) = exp{at} [m(t, u)] µ ,(3.
3)
where m :
R + ×R d → R + is defined by m(t, u) = (G(σ 2 t) * P 1 µ 0 )(u), with G(t, u) = 1 2πt d 2 exp{ −|u| 2 2t }.
Remark 3.2.
Let p be a Maxwellian distribution with parameters (a, µ, p 0 ). If p 0 (u) = p 0 (|u|) then, the Maxwellian distribution is invariant for specular reflection. More precisely, for all vector n ∈ R d such that n = 1,
p(t, u − 2(u · n) n) = p(t, u), for a.e. (t, u) ∈ (0, +∞) × R d .
This section is now devoted to the proof of the following existence result.
ψ ∈ C ∞ c (Q t ), Qt ρT (ψ) + ψ (B[· ; ρ] · ∇ u ρ) + σ 2 2 (∇ u ψ · ∇ u ρ) (s, x, u) ds dx du = D×R d ρ(t, x, u)ψ(t, x, u) dx du − D×R d ρ 0 (x, u)ψ(0, x, u) dx du + Σ + t (u · n D (x))γ + (ρ)(s, x, u)ψ(s, x, u)dλ Σ T (s, x, u) + Σ − t (u · n D (x))γ + (ρ)(s, x, u − 2(u · n D (x))n D (x))ψ(s, x, u)dλ Σ T (s, x, u). (3.4)
In addition, there exist a couple of Maxwellian distributions P , P such that
P ≤ ρ ≤ P , a.e. on Q T , P ≤ γ ± (ρ) ≤ P , λ Σ T -a.e. on Σ ± T ,(3.
5)
P and P satisfy the specular boundary condition (3.1c), and for all t ∈ (0, T ], Let us also emphasize that the weight ω(u) defined in (1.8) is useful here to preserve the probabilistic interpretation of (3.1) while working in L 2 -space. Later it also allows a fixed point argument. Let us remark the following properties:
sup t∈(0,T ) R d (1 + |u|)ω(u) P (t, u) 2 du < +∞, (3.6a) inf t∈(0,T ) R d P (t, u) du > 0. (3.6b) Remark 3.4. As D is a bounded domain, the lower-bound in (3.5) is well defined in L 2 (ω, Q T ) (since P L 2 (ω,Q T ) = |D| P L 2 (ω,(0,T )×R d ) ),Lemma 3.5. For the weight function u → ω(u) = (1 + |u| 2 ) α/2 , α > d ∨ 2, for all u and u ′ in R d , (i) ω(u + u ′ ) ≤ 2 α 2 (ω(u) + ω(u ′ )) ,(ii)
(u · ∇ω(u)) ≥ 0, and |∇ω(u)| ≤ αω(u),
|(u · ω(u))| ≤ α 4 ω(u), |∇ ω(u)| ≤ α 2 ω(u), △ω(u) ≤ α(α − 2 + d)ω(u), (iii) R d du ω(u) < +∞.
Proof. The assertions (i) and (ii) are directly deduced from the calculations:
ω(u + u ′ ) = 1 + u + u ′ 2 α 2 ≤ 1 + 2|u| 2 + 2|u ′ | 2 α 2 , (u · ∇ω(u)) = α 2 |u| 2 (1 + |u| 2 ) α 2 −1 , |∇ω| (u) = α |u| 1 + |u| 2 α 2 −1 ≤ αω(u), ∇ ω(u) = α 2 |u| 1 + |u| 2 α 4 −1 , △ω(u) = αd ω(u) (1 + |u| 2 ) + 2α α 2 − 1 ω(u) |u| 2 (1 + |u| 2 ) 2 .
For (iii), by a change of variable in the polar coordinates, we have
R d du ω(u) = |S d−1 | R + 1 + r 2 − α 2 r d−1 dr ≤ |S d−1 | R + 1 + |r| 2 − α+d−1 2 dr
where |S d−1 | is the Lebesgue measure of the unit sphere of R d . Since the right member is finite for α > d ∨ 2, (iii) follows.
On the linear Vlasov-Fokker-Planck equation
In this section, we set up the framework for the proof of Theorem 3.3, based on the existence result of the linear Vlasov-Fokker-Planck equation and the associated spaces. First we give some properties of the operator T defined in (3.2), that were initially stated in [9], inspired from ideas in [13] and [12]. For all t ∈ (0, T ], we consider the space
Y(Q t ) = φ ∈ H(Q t ) s.t. T (φ) ∈ H ′ (Q t ) , equipped with the norm φ 2 Y(Qt) = φ 2 H(Qt) + T (φ) 2 H ′ (Qt) .
We consider also the subset Y(Q t ) of all elements of C 1,1,2 c (Q t ), vanishing at the neighborhood of the boundaries {0} × ∂D × R d , {t} × ∂D × R d and Σ 0 t . We recall the following. Lemma 3.6 (Carrillo [9], Lemma 2.3 and its proof). For all ψ ∈ Y(Q t ), there exists a sequence {ψ n ; n ∈ N} of Y(Q t ) such that ψ n tends to ψ for the norm Y(Qt) when n tends to +∞. Moreover, ψ has trace values γ
+ (ψ) ∈ L 2 (Σ + T ) (resp. γ − (ψ) ∈ L 2 (Σ − T )) on Σ + T (resp. on Σ − T ) given by γ ± (ψ) = lim n→+∞ ψ n , in L 2 (Σ ± t )
and, for all t ∈ (0, T ], ψ(t, ·) belongs to L 2 (D × R d ), with
ψ(t, ·) = lim n→+∞ ψ n (t, ·), in L 2 (D × R d ).
With the help of Lemma 3.6, we adapt the Green formula stated in [9, Lemma 2.3] to the weighted spaces considered in this section. Lemma 3.7. Let ψ ∈ H(ω, Q T ) be such that T ( √ ωψ) ∈ H ′ (Q T ). Then ψ has traces γ ± (ψ) ∈ L 2 (ω, Σ ± T ), and ψ(t, ·) ∈ L 2 (ω,
D × R d ). Moreover, for all φ in Y(Q T ), t ∈ [0, T ], (T (ψ), φ) H ′ (Qt),H(Qt) + (T (φ), ψ) H ′ (Qt),H(Qt) = D×R d ψ(t, x, u)φ(t, x, u) dx du − D×R d ψ(0, x, u)φ(0, x, u) dx du + Σ + t (u · n D (x))γ + (ψ)(s, x, u)γ + (φ)(s, x, u)dλ Σt (s, x, u) + Σ − t (u · n D (x))γ − (ψ)(s, x, u)γ − (φ)(s, x, u)dλ Σt (s, x, u). (3.7)
Proof. In [9, Lemma 2.3], the existence of the trace values (in time and space) and the Green formula (3.7) are obtained for all ψ ∈ H(Q T ) such that T (ψ) ∈ H ′ (Q T ). Here, we have that T ( √ ωψ) = √ ωT (ψ) and, for all φ ∈ H(Q T ),
| (T (ψ), φ) H ′ (Q T ),H(Q T ) | = |(T ( √ ωψ), φ √ ω ) H ′ (Q T ),H(Q T ) | ≤ T ( √ ωψ) H ′ (Q T ) φ √ ω H(Q T ) ≤ α 2 T ( √ ωψ) H ′ (Q T ) φ H(Q T ) .
We deduce that T (ψ) is in H ′ (Q T ), and hence we deduce the Green formula.
For given q, B and g, let us consider the linear Vlasov-Fokker-Planck equation:
T (f ) = σ 2 2 △ u f − (∇ u · Bf ) + g, in H ′ (Q T ), (3.8a) f (0, x, u) = ρ 0 (x, u), on D × R d , (3.8b) γ − (f )(t, x, u) = q(t, x, u), on Σ − T .f (t) 2 L 2 (ω,D×R d ) + σ 2 ∇ u f 2 L 2 (ω,Qt) + γ + (f ) 2 L 2 (ω,Σ + t ) = ρ 0 2 L 2 (ω,D×R d ) + q 2 L 2 (ω,Σ − t ) + Qt σ 2 2 △ω + (∇ u ω · B) |f | 2 + 2 Qt ωgf. (3.9)
When g = 0, if ρ 0 and q are nonnegative, then f and γ + (f ) are nonnegative.
Proof. In our situation of weighted spaces, it is easy to deduce from the original proof of Carrillo [9], that there exists a unique solution f ∈ H(ω, Q T ) to (3.8) and that √ ωf ∈ Y(Q T ) (for the sake of completeness, the proof of this well-posedness result is given in the Appendix A.3).
Then, using Lemma 3.7, one can take φ = ψ = √ ωf in the Green formula (3.7) and combined with (3.8a), we obtain that:
2(T ( √ ωf ), √ ωf ) H ′ (Qt),H(Qt) = f (t) 2 L 2 (ω,D×R d ) − f (0) 2 L 2 (ω,D×R d ) + γ + (f ) 2 L 2 (ω,Σ + T ) − γ − (f ) 2 L 2 (ω,Σ − T ) = −σ 2 ∇ u f 2 L 2 (ω,Qt) + Qt ( σ 2 2 △ω + (∇ u ω · B))|f | 2 + 2 Qt ωgf.
Using (3.8b) and (3.8c), we deduce (3.9). In order to conclude that f ∈ V 1 (ω, Q T ), it remains to show the continuity of the mapping t → f (t) in L 2 (ω, D × R d ). Let us start by establishing the right continuity. For 0 < h < T , we define f h (t, x, u) := f (t + h, x, u) on Q T −h . Observe that, for any χ : t ∈ [0, +∞) → χ(t) ∈ [0, +∞) in C 1 b ([0, +∞)), by (3.8a), (3.8b), and (3.8c), one can check that
χ(f h −f ) ∈ H(ω, Q t−h ), T (χ(f h −f )) ∈ H ′ (Q T −h ) and T (χ(f h − f )) = σ 2 2 △ u χ(f h − f ) − (∇ u · Bχ(f h − f )) + χ(g h − g) + χ ′ (f h − f ), in H ′ (Q T −h ), (χ(f h − f ))(0, x, u) = χ(0)(f (h, x, u) − f 0 (x, u)), on D × R d , γ − (χ(f h − f ))(t, x, u) = χ(t)(q h − q)(t, x, u), on Σ − T −h .
Since χ(f h − f ) ∈ H(ω, Q T −h ), using (3.9), one obtains that, for all t ∈ (0, T )
χ(t)(f h − f )(t) 2 L 2 (ω,D×R d ) − χ(0)(f (h) − f 0 ) 2 L 2 (ω,D×R d ) + σ 2 ∇ u χ(f h − f ) 2 L 2 (ω,Q t+h ) + χ(γ + (f h ) − γ + (f )) 2 L 2 (ω,Σ + t+h ) = χ(q h − q) 2 L 2 (ω,Σ − t−h ) + Q t+h ( σ 2 2 △ω + (∇ u ω · B))|χ(f h − f )| 2 + Q t+h χ ′ χω|(f h − f )| 2 + 2 Q t+h ωχ 2 (g h − g)(f h − f ).
(3.10)
Since χ and χ ′ are bounded, by using Corollary A.1 and the estimations on ω and its derivatives in Lemma 3.5, all the terms above with an integral in time tend to 0 when h goes to 0. Hence we have, for a fixed t ∈ (0, T ], lim h→0 +
χ(t)(f (t + h) − f (t)) L 2 (ω,D×R d ) − χ(0)(f (h) − f 0 ) L 2 (ω,D×R d ) = 0.
We get the right continuity at time t, by choosing χ(0) = 0 and χ(t) = 1. The continuity at time t = 0 is given by taking χ(0) = 1 and χ(t) = 0. The left continuity is proved in an analogous way.
The Maxwellian bounds for the linear Vlasov-Fokker-Planck equation
We state the existence of lower and upper-bounds for the solution in V 1 (ω, Q T ) to the linear problem (a1) µ > 1, and µ ∈ ( 1 2 , 1).
T (S) + (B · ∇ u S) − σ 2 2 △ u S = 0, in H ′ (Q T ), S(0, x, u) = ρ 0 (x, u), on D × R d , γ − (S)(t, x, u) = q(t, x, u), on Σ − T .(a2) a ≤ −µ 2σ 2 (µ − 1) B 2 L ∞ ((0,T )×D;R d ) , and a ≥ µ 2σ 2 (1 − µ) B 2 L ∞ ((0,T )×D;R d ) .
Then the following properties hold: (d2) Let S be the unique weak solution of (3.11) with inputs ρ 0 , q and B . If p ≤ q ≤ p, λ Σ T -a.e. on Σ − T , then p ≤ S ≤ p, a.e. on Q T , and p ≤ γ + (S) ≤ p, λ Σ T -a.e. on Σ + T .
For the proof of Proposition 3.9, we will use the notions of super-solution and sub-solution of Maxwellian type related to the operator
L B (ψ) = T (ψ) + (B · ∇ u ψ) − σ 2 2 △ u ψ.
P is a sub-solution of Maxwellian type for
L B if −∞ < L B (P ) ≤ 0, a.e. on Q T .
The proof of Proposition 3.9 proceeds as follows. In Step 1, we exhibit a class of Maxwellian distributions satisfying (d1) and some regularity properties (see Lemma 3.11). In Step 2, we establish a comparison principle between super-solutions and sub-solutions of Maxwellian type and the weak solution to (3.11) (see Lemma 3.12). We thus deduce a particular class of Maxwellian distributions satisfying the properties (d1) and (d2). In Step 3, we identify the class of super-solutions and subsolutions of Maxwellian type for the operator L B for all B fixed in L ∞ ((0, T ) × D; R d ) (see Lemma 3.13). In Step 4, combining these results, we conclude on Proposition 3.9.
Step 1. We start by emphasizing some technical properties of the Maxwellian distributions. Then the following properties hold:
(i1) sup t∈[0,T ] R d (1 + |u|)ω(u) |p(t, u)| 2 du < +∞. (i2) inf t∈[0,T ] R d p(t, u)du > 0, if µ > 1.
(i3) There exists a sequence of positive reals {ǫ k ; k ∈ N} such that lim k→+∞ ǫ k = 0 and lim k→+∞ p(ǫ k , ·) = p 0 (·), in L 2 (R d ).
(i4) For all δ > 0, ∂ t p belongs to L 2 ((δ, T ) × R d ).
(i5) p ∈ H(Q T ).
The proof of this lemma relies on some well-known properties of Gaussian distributions and is postponed in the Appendix A.4. Lemma 3.11 enables us to identify the class of Maxwellian distributions satisfying (d1) in Proposition 3.9. The properties ((i3), (i4), (i5)) emphasize regularities that we will need in the sequel.
Step 2. Comparison principle and Maxwellian bounds.
Lemma 3.12. Let B ∈ L ∞ ((0, T ) × D; R d ) be fixed and let p, p be two Maxwellian distributions, sub-solution and super-solution for L B respectively with parameters a, µ, p 0 and (a, µ, p 0 ) such that 2µ ∧ 2µ > 1 and p 0 , p 0 satisfying (3.13). If p 0 ≤ ρ 0 ≤ p 0 , a.e. on D × R d , and p ≤ q ≤ p, λ Σ T -a.e. on Σ − T , then we have p ≤ S ≤ p, a.e. on Q T , (3.14)
p ≤ γ + (S) ≤ p, λ Σ T -a.e. on Σ + T , (3.15)
for S the weak solution in V 1 (ω, Q T ) to (3.11) with inputs (ρ 0 , q, B).
Proof. Let us first prove the implication for the upperbounds
ρ 0 ≤ p 0 , a.e. on (0, T ) × D, q ≤ p, λ Σ T -a.e. on Σ − T . =⇒ S ≤ p, a.e. on Q T , γ + (S) ≤ p, λ Σ T -a.e. on Σ + T .
(3.16)
Defining F , γ + (F ) with F (t, x, u) = (p(t, u) − S(t, x, u)) , γ + (F )(t, x, u) = p(t, u) − γ + (S)(t, x, u) , (3.16) is equivalent to the inequality: ∀ t ∈ (0, T ], (F (t)) − 2 L 2 (D×R d ) + γ + (F ) − 2 L 2 (Σ + t ) ≤ (p 0 − ρ 0 ) − 2 L 2 (D×R d ) + (p − q) − 2 L 2 (Σ − t ) .
(3.17) In order to obtain (3.17), we establish a Green identity on a smooth approximation of (F ) − (where (F ) − refers to the negative part of F ). For fixed t in (0, T ], by Lemma 3.6, there exists a sequence of C ∞ c (Q t )-functions {f n ; n ∈ N}, such that
lim n→+∞ f n = S, in H(Q t ), lim n→+∞ T (f n ) = T (S), in H ′ (Q t ), lim n→+∞ D×R d |f n (s, x, u) − S(s, x, u)| 2 dx du = 0, ∀ s ∈ [0, t], lim n→+∞ Σ ± t |(u · n D (x))| f n (s, x, u) − γ ± (S)(s, x, u) 2 dλ Σ T (s, x, u) = 0. (3.18)
In addition, according to Lemma 3.11, p satisfies (i1) to (i5). Let us define the sequence of
C 2 b (Q t )- functions {F n ; n ∈ N} by F n (s, x, u) = p(s, x, u) − f n (s, x, u).
Then by definition of L B in (3.12), for a.e. (s, x, u) in Q t ,
T (F n )(s, x, u) = L B (p − f n )(s, x, u) − (B (s, x) · ∇ u F n (s, x, u)) + σ 2 2 △ u F n (s, x, u) ≥ −L B (f n )(s, x, u) − (B (s, x) · ∇ u F n (s, x, u)) + σ 2 2 △ u F n (s, x, u) (3.19) since L B (p − f n ) ≥ −L B (f n )
as p is a super-solution for L B . Using the sequence {ǫ k ; k ∈ N} given by (i3) and by taking k such that 0 < ǫ k ≤ t, (i4) and (i5) ensure that T (F n ) ∈ L 2 (Q ǫ k ,t ) and F n ∈ H (Q ǫ k ,t ) for Q ǫ k ,t := (ǫ k , t) × D × R d . These properties are also true for (F n ) − (see Theorem A.2). Multiplying both sides of (3.19) by (F n ) − , and integrating the resulting expression on Q ǫ k ,t , we obtain
Qǫ k ,t T (F n ) (F n ) − ≥ − Qǫ k ,t (B · ∇ u F n ) (F n ) − + σ 2 2 Qǫ k ,t (△ u F n ) (F n ) − − Qǫ k ,t L B (f n ) (F n ) − .
(3.20)
For the l.h.s. in (3.20), an integration by parts yields
Qǫ k ,t T (F n ) (F n ) − = − Qǫ k ,t T ((F n ) − ) (F n ) − = − Qǫ k ,t ∂ t (F n ) − + u · ∇ x (F n ) − (F n ) − = 1 2 (F n (ǫ k )) − 2 L 2 (D×R d ) − 1 2 (F n (t)) − 2 L 2 (D×R d ) − 1 2 (F n ) − 2 L 2 (Σ + ǫ k ,t ) + 1 2 (F n ) − 2 L 2 (Σ − ǫ k ,t ) ,
with Σ ± ǫ k ,t := (s, x, u) ∈ Σ ± t ; s ∈ (ǫ k , t) . In the r.h.s. in (3.20), as B depends only on x, we get
− Qǫ k ,t (B · ∇ u F n ) (F n ) − + σ 2 2 Qǫ k ,t (△ u F n ) (F n ) − = σ 2 2 Qǫ k ,t ∇ u (F n ) − 2 ≥ 0.
Coming back to (3.20), it follows that
(F n (ǫ k )) − 2 L 2 (D×R d ) + (F n ) − 2 L 2 (Σ − ǫ k ,t ) ≥ (F n (t)) − 2 L 2 (D×R d ) + (F n ) − 2 L 2 (Σ + ǫ k ,t ) − 2 Qǫ k ,t L B (f n ) (F n ) − .
Taking the limit k → +∞, (i3) implies that lim
k→+∞ (F n (ǫ k )) − = (p 0 − f n (0)) − . Thus (p 0 − f n (0)) − 2 L 2 (D×R d ) + (F n ) − 2 L 2 (Σ − t ) ≥ (F n (t)) − 2 L 2 (D×R d ) + (F n ) − 2 L 2 (Σ + t ) − 2 Qt L B (f n ) (F n ) − (3.21)
It remains to study the limit w.r.t. n. By (3.18),
lim n→+∞ (p 0 − f n (0)) − 2 L 2 (D×R d ) + (F n ) − 2 L 2 (Σ − t ) = (p 0 − ρ 0 ) − 2 L 2 (D×R d ) + (p − g) − 2 L 2 (Σ − t ) , lim n→+∞ (F n (t)) − 2 L 2 (D×R d ) + (F n ) − 2 L 2 (Σ + t ) = (F (t)) − 2 L 2 (D×R d ) + γ + (F ) − 2 L 2 (Σ + t ) .
For the last term in (3.21), an integration by parts yields
Qt L B (f n ) (F n ) − = T (f n ), (p − f n ) − H ′ (Qt),H(Qt) + Qt (B · ∇ u f n ) (p − f n ) − + σ 2 2 Qt ∇ u f n · ∇ u (p − f n ) − . As lim n→+∞ (p − f n ) − = (p − S) − in H(Q T ) and lim n→+∞ T (f n ) = T (S) in H ′ (Q T ), we get lim n→+∞ Qt L B (f n ) (F n ) − = T (S), (p − S) − H ′ (Qt),H(Qt) + Qt (B · ∇ u S) (p − S) − + σ 2 2 Qt ∇ u S · ∇ u (p − S) − = 0 since T (S) + (B · ∇ u S) − σ 2 2 △ u S = 0 in H ′ (Q T )
. Coming back to (3.21), we take the limit n → +∞ to get (3.17). The implication for the lower-bounds
p 0 ≤ ρ 0 , a.e. on (0, T ) × D, p ≤ q, λ Σ T -a.e. on Σ − T .
=⇒ p ≤ S, a.e. on Q T , p ≤ γ + (S), λ Σ T -a.e. on Σ + T .
(3.22)
is proved in the same way. By defining
J(t, x, u) := S(t, x, u) − p(t, u) , γ + (J)(t, x, u) := γ + (S)(t, x, u) − p(t, u) ,
we can then establish that with {f n ; n ∈ N} satisfying (3.18), and using the fact that p is a sub-solution of Maxwellian type, we obtain: for all t ∈ (0, T ] fixed, for a.e. (s, x, u) in Q t , it holds
(ρ 0 − p 0 ) − 2 L 2 (D×R d ) + (q − p ) − 2 L 2 (Σ − t ) ≥ (J(t)) − 2 L 2 (D×R d ) + γ + (J) − 2 L 2 (Σ + t ) ,(3.T (J n )(s, x, u) ≥ L B (f n )(s, x, u) − (B (s, x) · ∇ u J n (s, x, u)) + σ 2 2 △ u J n (s, x, u).
Replicating the arguments in (3.16), we get
(f n (0) − p 0 ) − 2 L 2 (D×R d ) + (J n ) − 2 L 2 (Σ − t ) ≥ (J n (t)) − 2 L 2 (D×R d ) + (J n ) − 2 L 2 (Σ + t ) + 2 Qǫ k ,t L B (f n ) (J n ) − .
We then obtain (3.23) by taking the limit n → +∞.
Step 3. Existence of sub-and super-solutions of Maxwellian type. Lemma 3.13. Let p be a Maxwellian distribution with parameters (a, µ, p 0 ). For B ∈ L ∞ ((0, T ) × D; R d ), let L B be the operator defined in (3.12). Then the following properties hold.
(i) If µ ∈ (0, 1) and a ≥ µ
2σ 2 (1 − µ) B 2 L ∞ ((0,T )×D;R d ) , then p is a super-solution for L B . (ii) If µ > 1 and a ≤ −µ 2σ 2 (µ − 1) B 2 L ∞ ((0,T )×D;R d ) , then p is a sub-solution for L B .
Proof. By using the explicit form (3.3) of the considered Maxwellian distribution, we have
L B (p)(t, x, u) = a exp {at} m µ (t, u) + µ exp {at} ∂ t m(t, u) − σ 2 2 △ u m(t, u) m µ−1 (t, u) + µ exp {at} (B (t, x) · ∇ u m(t, u)) m µ−1 (t, u) − σ 2 2 µ(µ − 1) exp {at} |∇ u m(t, u)| 2 m µ−2 (t, u).
Since m is a classical solution of the heat equation, the previous equality reduces to
L B (p)(t, x, u) = exp {at} m µ−2 (t, u) a m 2 (t, u) − σ 2 2 µ(µ − 1)|∇ u m(t, u)| 2 + µ m(t, u) (B (t, x) · ∇ u m(t, u)) .
The sign of L B (p) is thus determined by the function:
J(t, x, u) := a m 2 (t, u) − σ 2 µ(µ − 1) 2 |∇ u m(t, u)| 2 + µm(t, u) (B (t, x) · ∇ u m(t, u)) . (3.24)
• When a and µ satisfy (i), using the identity (u 1 · u 2 ) = 1
2 ǫu 1 + u 2 ǫ 2 − ǫ 2 |u 1 | 2 2 − |u 2 | 2 2ǫ 2 , for u 1 , u 2 ∈ R d , ǫ > 0, we have µ m(t, u) (B (t, x) · ∇ u m(t, u)) = 1 2 ǫ∇ u m(t, u) + 1 ǫ m(t, u)B (t, x) 2 − 1 ǫ 2 m 2 (t, u)|B (t, x)| 2 − ǫ 2 |∇ u m(t, u)| 2 .
Inserting this equality into (3.24)
, with ǫ = σ √ 1 − µ √ µ (> 0 since 0 < µ < 1), it follows that J(t, x, u) = a − µ 2σ 2 (1 − µ) |B (t, x)| 2 |m(t, u)| 2 + 1 2 σ µ(1 − µ)∇ u m(t, u) + √ µ σ √ 1 − µ m(t, u)B (t, x) 2 ,
where, under (i),
a − µ 2σ 2 (1 − µ) |B (t, x)| 2 ≥ a − µ 2σ 2 (1 − µ) B 2 L ∞ ((0,T )×D;R d ) ≥ 0.
We thus deduce that J (and consequently L B (p)) is nonnegative in the situation (i).
• When a and µ satisfy (ii), using the identity (u 1 · u 2 ) = − 1
2 ǫu 1 − u 2 ǫ 2 + ǫ 2 |u 1 | 2 2 + |u 2 | 2 2ǫ 2 , for u 1 , u 2 ∈ R d , ǫ > 0, we get µ m(t, u) (B (t, x) · ∇ u m(t, u)) = 1 2ǫ 2 |B (t, x)| 2 |m(t, u)| 2 + ǫ 2 2 µ 2 |∇ u m(t, u)| 2 − 1 2 ǫm(t, u)B (t, x) − µ ǫ ∇ u m(t, u) 2 . Taking ǫ = σ √ µ−1 √ µ it follows that J(t, x, u) = a + µ 2σ 2 (µ − 1) |B (t, x)| 2 m 2 (t, u) − 1 2 σ √ µ − 1 √ µ m(t, u)B (t, x) − µ √ µ σ √ µ − 1 ∇ u m(t, u) 2 . As (ii) ensures that a + µ 2σ 2 (1 − µ) |B (t, x)| 2 ≤ a + µ 2σ 2 (1 − µ) B 2 L ∞ ((0,T )×D;R d ) ≤ 0,
we conclude that L B (p) is non-positive.
Step 4. Proof of Proposition 3.9. Let p, p be a couple of Maxwellian distributions of parameters (a, µ, P 0 ) and (a, µ, P 0 ) such that (P 0 , P 0 ) satisfy (H MVFP )-(iii) and such that (a, µ) and (a, µ) satisfy the properties (a1) and (a2) in Proposition 3.9. Applying Lemma 3.13, p, p are respectively a sub-solution and a super-solution of Maxwellian type for the linear operator. Moreover, recalling that P 0 , P 0 ∈ L 2 (ω, Q T ) are positives, these Maxwellian distributions satisfy the conditions of Lemma 3.11 and Lemma 3.12, and then p, p satisfy the properties (d1), (d2) of Proposition 3.9.
Construction of a weak solution to the conditional McKean-Vlasov-Fokker-Planck equation
Introduction of the specular boundary condition
We consider now a linear problem endowing a given convection term B ∈ L ∞ ((0, T ) × D; R d ) and submitted to the specular boundary condition: Let P , P be a couple Maxwellian distributions with parameters a, µ, P 0 and a, µ, P 0 respectively, satisfying the hypotheses of Proposition 3.9. Then there exists a unique weak solution S in V 1 (ω, Q T ) of (3.25) such that
T (S) + (B · ∇ u S) − σ 2 2 △ u S = 0, in H ′ (Q T ), S(0, x, u) = ρ 0 (x, u), on D × R d , γ − (S)(t, x, u) = γ + (S)(t, x, u − 2(u · n D (x))n D (x)), on Σ − T .P (t, u) ≤ S (t, x, u) ≤ P (t, u), for a.e. (t, x, u) ∈ Q T , P (t, u) ≤ γ ± (S )(t, x, u) ≤ P (t, u), for λ Σ T -a.e. (t, x, u) ∈ Σ ± T . Proof.
For the existence claim, let us introduce the functional space E = ψ ∈ V 1 (ω, Q T ) ; ψ admits trace functions γ ± (ψ) on Σ ± T belonging to L 2 (ω,Σ ± T ) , equipped with the norm
ψ 2 E = ψ 2 V 1 (ω,Q T ) + γ + (ψ) 2 L 2 (ω,Σ + T ) + γ − (ψ) 2 L 2 (ω,Σ − T )
. For all f ∈ E, we denote by S (f ) the unique weak solution (in the sense of Lemma 3.8) to the linear Vlasov-Fokker-Planck equation
T (S (f )) + (B · ∇ u S (f )) − σ 2 2 △ u S (f ) = 0, in H ′ (Q T ), S (f )(0, x, u) = ρ 0 (x, u), on D × R d , γ − (S(f ))(t, x, u) = γ + (f )(t, x, u − 2 (u · n D (x)) n D (x)), on Σ − T .
(3.26) Lemma 3.8 ensures that S (f ) ∈ V 1 (ω, Q T ), and that the trace functions γ ± (S (f )) belong to L 2 (ω, Σ ± T ). Therefore, we can define the mapping
S : f ∈ E −→ S (f ) ∈ E.
If S admits a fixed point S, then it naturally satisfies the specular boundary condition γ − (S)(t, x, u) = γ + (S)(t, x, u − 2 (u · n D (x)) n D (x)), on Σ − T , implying that S is a weak solution to (3.25). In order to establish the existence of this fixed point, let us observe the following properties of S (the proof of which are postponed at the end on this section):
(p1) S is Lipschitz-continuous on E;
(p2) S is an increasing mapping on E w.r.t. the following order relation:
f 1 ≤ f 2 on E ⇐⇒ f 1 (t, x, u) ≤ f 2 (t, x, u), for a.e. (t, x, u) ∈ Q T , γ(f 1 )(t, x, u) ≤ γ(f 2 )(t, x, u), for λ Σ T −a.e. (t, x, u) ∈ Σ T ; (p3) If P ≤ γ + (f ) ≤ P , λ Σ T -a.e. on Σ + T , then P ≤ S(f ) ≤ P on E.
The fixed point of S arises from the convergence of the sequence {S n ; n ∈ N} defined iteratively by S 0 = P and S n+1 = S (S n ).
Indeed, the monotone property (p2) of S implies that {S n ; n ∈ N} is increasing in E. In addition, since P = S 0 ≤ P , Proposition 3.9 ensures that P ≤ S 1 ≤ P and P ≤ γ + (S 1 ) ≤ P . Using repeatedly (p3), it holds that ∀ n ∈ N, P ≤ S n ≤ P , on E.
(3.27)
The sequence {S n ; n ∈ N} being increasing and uniformly bounded on E, we deduce that {S n ; n ∈ N} and {γ ± (S n ); n ∈ N} converge. Set
S(t, x, u) := lim n→+∞ S n (t, x, u), for (t, x, u) ∈ Q T , and γ ± (S)(t, x, u) = lim n→+∞ γ ± (S n )(t, x, u), for (t, x, u) ∈ Σ ± T .
According to (3.27), P ≤ S ≤ P on E. Since P ∈ L 2 (ω, Q T ) and P ∈ L 2 (ω, Σ T ), by dominated convergence, we obtain that We furthermore observe that, owing to (3.27), (3.9) ensures that
lim n→+∞ S − S n L 2 (ω,Q T ) = 0, lim n→+∞ γ ± (S) − γ ± (S n ) L 2 (ω,Σ ± T ) = 0.sup n ∇ u S n 2 L 2 (ω,Qt) ≤ C ρ 0 2 L 2 (ω,D×R d ) + sup t∈(0,T ) P (t) 2 L 2 (ω,R d )
where C is some constant depending only on d, T, σ, b L ∞ (R d ;R d ) , α and the Lebesgue measure of D.
It follows that S ∈ H(ω, Q T ) with ∇ u S = lim n→+∞ ∇ u S n in L 2 (ω, Q T ). We thus conclude that S has a fixed point S in E.
For the uniqueness claim, consider two weak solutions S 1 , S 2 to (3.25). Set
R(t, x, u) := S 1 (t, x, u) − S 2 (t, x, u) for (t, x, u) ∈ Q T , γ ± (R)(t, x, u) := γ ± (S 1 ) − γ ± (S 2 ) (t, x, u) for (t, x, u) ∈ Σ ± T .
Then, R and γ ± (R) satisfy
T (R) + (B · ∇ u R) − σ 2 2 △ u R = 0, in H ′ (Q T ), R(0, x, u) = S 1 (0) − S 2 (0) = 0, on D × R d , γ − (R)(t, x, u) = γ + (S 1 ) − γ + (S 2 ) (t, x, u − 2(u · n D (x))n D (x)), on Σ − T .
Using Lemma 3.8, one has for all t ∈ (0, T ],
R(t) 2 L 2 (ω,D×R d ) + σ 2 ∇ u R 2 L 2 (ω,Q T ) + γ + (R) 2 L 2 (ω,Σ + t ) = γ − (R) 2 L 2 (ω,Σ − t ) + Qt σ 2 2 △ω + (B · ∇ u ω) |R| 2 .
Since S 1 and S 2 satisfy the specular boundary condition,
Σ − t (u · n D (x))ω(u) γ − (R)(s, x, u) 2 dλ Σ T (s, x, u) = − Σ + t (u · n D (x))ω(u) γ + (R)(s, x, u) 2 dλ Σ T (s, x, u),
so that, using Lemma 3.5, the previous inequality is reduced to
min(1, σ 2 ) R 2 V 1 (ω,Qt) ≤ σ 2 2 α(α − 2 + d) + α B L ∞ ((0,T )×D;R d ) t 0 R 2 V 1 (ω,Qs) ds
which ensures the uniqueness result by applying Gronwall's Lemma.
Proof of (p1). For f 1 , f 2 ∈ E, we set
R(t, x, u) = S (f 1 )(t, x, u) − S (f 2 )(t, x, u), for (t, x, u) ∈ Q T , γ ± (R)(t, x, u) = γ ± (S(f 1 )) − γ ± (S(f 2 )) (t, x, u), for (t, x, u) ∈ Σ ± T .
Then, R and γ ± (R) satisfy
T (R) + (B · ∇ u R) − σ 2 2 △ u R = 0, in H ′ (Q T ), R(0, x, u) = S(f 1 )(0) − S(f 2 )(0) = 0, on D × R d , γ + (R)(t, x, u) = (γ + (f 1 ) − γ + (f 2 )) (t, x, u − 2(u · n D (x))n D (x)), on Σ − T .
Replicating the proof of the uniqueness result for (3.25), one has
min(1, σ 2 ) R 2 V 1 (ω,Qt) + γ + (R) 2 L 2 (ω,Σ + t ) ≤ γ − (R) 2 L 2 (ω,Σ − t ) + C t 0 R 2 V 1 (ω,Qs) ds for C := σ 2 2 α(α − 2 + d) + α B L ∞ ((0,T )×D;R d )
. Applying Gronwall's Lemma, it follows that
S (f 2 ) − S (f 1 ) 2 E ≤ C γ − (S (f 2 )) − γ − (S (f 2 )) 2 L 2 (ω,Σ − T ) = C γ + (f 2 ) − γ + (f 1 ) 2 L 2 (ω,Σ + T ) , which enables us to deduce S (f 2 ) − S (f 1 ) 2 E ≤ C f 2 − f 1 2 E
, and thus that S is Lipschitz-continuous.
Proof of (p3). Assume that f ∈ E is such that
P (t, u) ≤ γ + (f )(t, x, u) ≤ P (t, u), for λ Σ T -a.e. (t, x, u) ∈ Σ + T . (3.28)
Remark 3.2 implies that P (t, u) = P (t, u − 2(u · n D (x))n D (x)), P (t, u) = P (t, u − 2(u · n D (x))n D (x)).
Hence (3.28) is equivalent to
P (t, u) ≤ γ + (f )(t, x, u − 2(u · n D (x))n D (x)) = γ − (S)(t, x, u) ≤ P (t, u), λ Σ T -a.e. on Σ − T . (3.29)
Applying Proposition 3.9, it follows that P ≤ S(f ) ≤ P , a.e. on Q T , and P ≤ γ + (S(f )) ≤ P , λ Σ T -a.e. on Σ + T .
Proof of (p2). Let f 1 , f 2 be such that
f 1 ≤ f 2 on E. The difference S (f 2 ) − S (f 1 )
is then a weak solution to the linear Vlasov-Fokker-Planck equation (3.11) for ρ 0 = 0 and
g(t, x, u) = γ + (f 2 ) − γ + (f 1 ) (t, x, u − 2(u · n D (x))n D (x)); namely T (S (f 2 ) − S (f 1 )) + (B · ∇ u (S (f 2 ) − S (f 1 ))) − σ 2 2 △ u (S (f 2 ) − S (f 1 )) = 0, in H ′ (Q T ), S (f 2 )(0, x, u) − S (f 1 )(0, x, u) = 0, on D × R d , γ − (S (f 2 ) − S (f 1 )) (t, x, u) = γ + (f 2 ) − γ + (f 1 ) (t, x, u − 2 (u · n D (x)) n D (x)), on Σ − T .
Therefore, applying Lemma 3.8, we obtain that
S (f 2 ) − S (f 1 ) = S (f 2 − f 1 ) ≥ 0 and that γ + (S (f 2 )) − γ + (S (f 1 )) = γ + (S (f 2 ) − S (f 1 )) = γ + (S (f 2 − f 1 )) ≥ 0. We conclude that S is nondecreasing on E since γ − (S (f 2 ))(t, x, u) − γ − (S (f 1 ))(t, x, u) = (γ + (f 2 ) − γ + (f 1 )) (t, x, u − 2(u · n D (x))n D (x)) ≥ 0.
Introduction of the nonlinear drift (end of the proof of Theorem 3.3)
Hereafter we end the proof of Theorem 3.3 by introducing the nonlinear coefficient B[· ; ·] in the equation (3.25). To this aim, we consider (P , P ), a couple of Maxwellian distributions with parameters (a, µ, P 0 ) and (a, µ, P 0 ) such that
2µ > 1, 2µ > 1, a ≤ −µ 2σ 2 (µ − 1) b 2 L ∞ (R d ;R d ) , a ≥ µ 2σ 2 (1 − µ) b 2 L ∞ (R d ;R d ) .
(3.30)
We also consider the sequence {ρ (n) ; n ∈ N} in V 1 (ω, Q T ), defined by
• ρ (0) = ρ 0 on Q T ;
• For n ≥ 1 and ρ (n−1) given, we define B x; ρ (n−1) (t) as in (1.2).
Under (H MVFP )-(i), we have |B| ≤ b L ∞ (R d ;R d ) . We define ρ (n) as the unique solution in V 1 (ω, Q T ) of T (ρ (n) ) + B[·; ρ (n−1) ] · ∇ u ρ (n) − σ 2 2 △ u ρ (n) = 0, in H ′ (Q T ), γ − (ρ (n) )(t, x, u) = γ + (ρ (n) )(t, x, u − 2(u · n D (x))n D (x)), on Σ − T , ρ (n) (0, x, u) = ρ 0 (x, u), on D × R d .
(3.31) According to Proposition 3.14, for all n ≥ 1, P ≤ ρ (n) ≤ P , a.e. on Q T , P ≤ γ ± (ρ (n) ) ≤ P , λ Σ T -a.e. on Σ ± T , (3.32) so that ρ (n) is positive on Q T and
B[x; ρ (n) (t)] = R d b(v)ρ (n) (t, x, v) dv R d ρ (n) (t, x, v) dv , for a.e. (t, x) ∈ (0, T ) × D.
Proposition 3.15. Assume (H MVFP ). Let (P , P ) be a couple of Maxwellian distributions with respective parameters (a, µ, P 0 ) and (a, µ, P 0 ) satisfying (3.30). Then the sequence {ρ (n) ; n ∈ N} converges in V 1 (ω, Q T ) to a weak solution ρ to the non-linear equations (3.1a)-(3.1c). Moreover this solution satisfies P ≤ ρ ≤ P , a.e. on Q T , P ≤ γ ± (ρ) ≤ P , λ Σ T -a.e. on Σ ± T . Proof. First, we establish the following uniform estimation for {ρ (n) ; n ∈ N}:
sup n≥1 ρ (n) 2 V 1 (ω,Q T ) ≤ K ρ 0 2 L 2 (ω,D×R d ) + P 2 L 2 (ω,(0,T )×R d ) ,(3.33)
for some constant K > 0. Indeed, using the energy estimate (3.9), we obtain that, for all t ∈ (0, T ],
min 1, σ 2 ρ (n) 2 V 1 (ω,Qt) ≤ ρ 0 2 L 2 (ω,D×R d ) + σ 2 2 α (α − 2 + d) + α b L ∞ (R d ;R d ) t 0 ρ (n) (s) 2 L 2 (ω,D×R d ) ds.
Using the Maxwellian upper-bound (3.32), we obtain (3.33).
Since (V 1 (ω, Q T ), V 1 (ω,Q T ) ) is a Banach space, it is sufficient to establish that {ρ (n) ; n ∈ N} is a Cauchy sequence in V 1 (ω, Q T ). For n, m > 1, the functions R (n,n+m) := ρ (n+m) − ρ (n) , on Q T , γ ± (R (n,n+m) ) := γ ± (ρ (n+m) ) − γ ± (ρ (n+m) ), on Σ ± T , satisfy T (R (n,n+m) ) − σ 2 2 △ u R (n,n+m) = ∇ u · B[·; ρ (n−1) ]ρ (n) − B[·; ρ (n+m−1) ]ρ (n+m) , in H ′ (Q T ), R (n,n+m) (0, x, u) = 0, on D × R d ,
γ − (R (n,n+m) )(t, x, u) = γ + (R (n,n+m) )(t, x, u − 2(u · n D (x))n D (x)), on Σ − T . Then, according to (3.9), it follows that for all t ∈ (0, T ],
D×R d ω R (n,n+m) (t, x, u) 2 dx du + σ 2 Qt ω ∇ u R (n,n+m) 2 ≤ σ 2 2 Qt △ω R (n,n+m) 2 + 2 Qt ωR (n,n+m) ∇ u · B[· ; ρ (n−1) ]ρ (n) − B[· ; ρ (n+m−1) ]ρ (n+m) . But 2 Qt ωR (n,n+m) ∇ u · B[· ; ρ (n−1) ]ρ (n) − B[· ; ρ (n+m−1) ]ρ (n+m) ≤ 2 Qt ω α|R (n,n+m) | + |∇ u R (n,n+m) | B[· ; ρ (n−1) ]ρ (n) − B[· ; ρ (n+m−1) ]ρ (n+m) ≤ Qt α 2 ω(R (n,n+m) ) 2 + Qt σ 2 2 ω|∇ u R (n,n+m) | 2 + (1 + 2 σ 2 ) Qt ω B[· ; ρ (n−1) ]ρ (n) − B[· ; ρ (n+m−1) ]ρ (n+m) 2
and hence, using Lemma 3.5,
D×R d ω R (n,n+m) (t, x, u) 2 dx du + σ 2 2 Qt ω ∇ u R (n,n+m) 2 ≤ σ 2 2 α(α − 2 + d) + α 2 t 0 R (n,n+m) 2 V 1 (ω,Qs) ds + (1 + 2 σ 2 ) Qt ω B[· ; ρ (n−1) ]ρ (n) − B[· ; ρ (n+m−1) ]ρ (n+m) 2 .
Let us now observe that
Qt ω B[· ; ρ (n−1) ]ρ (n) − B[· ; ρ (n+m−1) ]ρ (n+m) 2 ≤ 1 2 Qt ω ρ (n+m) − ρ (n) 2 B[· ; ρ (n+m−1) ] 2 + 1 2 Qt ω ρ (n) 2 B[· ; ρ (n+m−1) ] − B[· ; ρ (n−1) ] 2 .
For the first term, we have
Qt ω ρ (n+m) − ρ (n) 2 B[· ; ρ (n+m−1) ] 2 ≤ b 2 L ∞ (R d ;R d ) Qt ω ρ (n+m) − ρ (n) 2 . (3.34)
For the second term, let us set
M := inf t∈(0,T ) R d P (t, u)du, M := sup t∈(0,T ) R d ω(u) P (t, u) 2 du.
According to the definition of B[·; ·] in (1.2),
B[x; ρ (n+m−1) (s)] − B[x; ρ (n−1) (s)] 2 = R d b(v)ρ (n+m−1) (s, x, v)dv R d ρ (n+m−1) (s, x, v)dv − R d b(v)ρ (n−1) (s, x, v)dv R d ρ (n−1) (s, x, v)dv 2 ≤ 1 2 R d b(v) ρ (n+m−1) − ρ (n−1) (s, x, v)dv 2 R d ρ (n+m−1) (s, x, v)dv 2 + 1 2 R d ρ (n+m−1) − ρ (n−1) (s, x, v)dv 2 R d b(v)ρ (n−1) (s, x, v)dv 2 R d ρ (n+m−1) (s, x, v)dv R d ρ (n−1) (s, x, v)dv 2 ≤ b 2 L ∞ (R d ;R d ) R d ρ (n+m−1) − ρ (n−1) (s, x, v)dv 2 (M ) 2 ≤ w b 2 L ∞ (R d ;R d ) (M ) 2 R d ω(v) ρ (n+m−1) − ρ (n−1) (s, x, v) 2 dv, since w := R d 1 ω(v)
dv < +∞ (see Lemma 3.5), from which it follows that
Qt ω|ρ (n) | 2 B[· ; ρ (n+m−1) ] − B[· ; ρ (n−1) ] 2 ≤ b 2 L ∞ (R d ;R d ) w M 2 M 2
Qt ω ρ (n+m−1) − ρ (n−1) 2 .
Combining this inequality with (3.34), we obtain that
Qt ω ρ (n+m) B[· ; ρ (n+m−1) ] − ρ (n) B[· ; ρ (n−1) ] 2 ≤ b 2 L ∞ (R d ;R d ) Qt ω R (n,n+m) 2 + b 2 L ∞ (R d ;R d ) w M 2 M 2 Qt ω R (n−1,n+m−1) 2 .
Then for all t in (0, T ],
R (n,n+m) 2 V 1 (ω,Qt) ≤ C 1 t 0 R (n,n+m) 2 V 1 (ω,Qs) ds + C 2 t 0 R (n+m−1,n−1) 2 V 1 (ω,Qs) ds, with C 1 = σ 2 2 α(α − 2 + d) + α 2 + b 2 L ∞ (R d ;R d ) (1 + 2 σ 2 ) min 1, σ 2 2 , C 2 = (1 + 2 σ 2 ) b 2 L ∞ (R d ;R d ) w M 2 M 2 min 1, σ 2 2 .
Therefore, applying Gronwall's Lemma, we deduce that
R (n,n+m) 2 V 1 (ω,Qt) ≤ C 2 t 0 (1 + exp {C 1 (t − s)}) R (n+m−1,n−1) 2 V 1 (ω,Qs) ds.
Iterating n − 1 times this inequality, we obtain
R (n,n+m) 2 V 1 (ω,Q T ) ≤ (C 2 ) n−1 T 0 (1 + exp{C 1 (T − t n )}) t n−1 0 · · · · · · t 2 0 (1 + exp{C 1 (T − t 1 )}) ρ (m+1) − ρ (1) 2 V 1 (ω,Qt 1 ) dt 1 · · · dt n−1 dt n ≤ (C T ) n−1 (n − 1)! ρ (m+1) − ρ (1) 2 V 1 (ω,Q T ) with C T := C 2 T 0 (1 + exp{C 1 (T − t)})dt.
Using the estimation (3.33), it follows that
sup m∈N ρ (n+m) − ρ (n) 2 V 1 (ω,Q T ) ≤ (C T ) n n! 2K ρ 0 2 L 2 (ω,D×R d ) + M .
Since n∈N (C T ) n n! = exp {C T } < +∞, (C T ) n n! tends to 0 as n tends to infinity. Therefore ρ (n) ; n ∈ N is a Cauchy sequence in V 1 (ω, Q T ). Let us denote by ρ, the limit of ρ (n) ; n ≥ 1 in V 1 (ω, Q T ). According to (3.32), we have P ≤ ρ ≤ P , a.e. on Q T . Now we check that ρ is a weak solution to (3.1a)-(3.1c). Since {ρ n ; n ≥ 1} tends to ρ in V 1 (ω, Q T ), and so in H(ω, Q T ), we can consider a subsequence still denoted by {ρ (n) ; n ≥ 1} such that lim n→+∞ ρ (n) (t, x, u) = ρ(t, x, u), for a.e. (t, x, u) ∈ Q T , lim n→+∞ ∇ u ρ (n) (t, x, u) = ∇ u ρ(t, x, u), for a.e. (t, x, u) ∈ Q T , and sup n≥1 ρ (n) (t, x, u) + ∇ u ρ (n) (t, x, u) ∈ L 2 (ω, Q T ).
Therefore, by dominated convergence, we deduce the convergence of the coefficients
lim n→+∞ B[x; ρ (n) (t)] = B[x; ρ(t)]
, a.e. on (0, T ) × D.
We further observe that for ψ ∈ C ∞ c (Q T ),
|(T ( √ ωρ), ψ) H ′ (Q T ),H(Q T ) | = Q T T (ψ) √ ωρ ≤ lim sup n→+∞ Q T T (ψ) √ ωρ (n) ≤ lim sup n→+∞ Q T √ ωψ B · ; ρ (n−1) · ∇ u ρ (n) − σ 2 2 ∇ u ( √ ωψ) · ∇ u ρ (n) ≤ b L ∞ (R d ;R d ) + σ 2 2 1 + α 2 ψ H(Q T ) lim sup n→+∞ ρ (n) V 1 (ω,Q T ) .
Owing to (3.33), the right-hand side is uniformly bounded, so that T ( √ ωρ) ∈ H ′ (Q T ) and, from (3.31),
T (ρ) + (B[· ; ρ] · ∇ u ρ) − σ 2 2 △ u ρ = 0 , in H ′ (Q T ).
According to Lemma 3.7, ρ admits traces functions along the frontier Σ ± T , which belongs to L 2 (ω, Σ ± T ). It remains to check the specular boundary condition and the Maxwellian bounds for ρ:
γ − (ρ)(s, x, u) = γ + (ρ)(s, x, u − 2(u · n D (x))n D (x)), λ Σ T -a.e. on Σ − T , P ≤ γ ± (ρ) ≤ P , λ Σ T -a.e. on Σ ± T . (3.35) For all ψ ∈ C ∞ c (Q T ), we have Q T T (ψ) ρ − ρ (n) + Q T ψ (B[· ; ρ] · ∇ u ρ) − B · ; ρ (n−1) · ∇ u ρ (n) + σ 2 2 Q T ∇ u ψ · ∇ u ρ − ρ (n) = − D×R d ψ(T, x, u) ρ(T, x, u) − ρ (n) (T, x, u) dx du − Σ + T (u · n D (x))ψ(s, x, u) γ + (ρ) − γ + (ρ (n) ) (s, x, u)dλ Σ T (s, x, u) − Σ − T (u · n D (x))ψ(s, x, u) γ − (ρ)(s, x, u) − γ − (ρ (n) )(s, x, u) dλ Σ T (s, x, u). Hence, for all ψ ∈ C ∞ c (Q T ) vanishing on {T } × D × R d and Σ + T , we have lim n→+∞ Σ − T (u · n D (x))ψ(s, x, u) γ − (ρ)(s, x, u) − γ − (ρ (n) )(s, x, u) dλ Σ T (s, x, u) = 0.
It follows that lim
n→+∞ γ − (ρ) − γ − (ρ (n) ) L 2 (Σ − T ) = 0.
Since, for all n ≥ 1, ρ (n) satisfies the specular boundary condition and the Maxwellian bounds (3.32), we deduce (3.35).
Well-posedness for the nonlinear Lagrangian stochastic model with specular boundary condition 4.1 Construction of a solution
Under the hypotheses (H), we construct a solution to the Lagrangian stochastic model with specular boundary condition (1.1) that satisfies the mean no permeability condition (1.4). Let us consider a probability P given by Theorem 2.1, on the sample space T , such that under P the canonical process ((
x(t), u(t)); t ∈ [0, T ]) of T satisfies x(t) = x(0) + t 0 u(s) ds, u(t) = u(0) + σ w(t) − 0<s≤t 2 u(s − ) · n D (x(s)) n D (x(s))½ {x(s) ∈ ∂D} ,
where ( w(t); t ∈ [0, T ]) is an R d -Brownian motion and P • (x(0), u(0)) −1 = µ 0 . Next, we consider a solution ρ FP ∈ V 1 (ω, Q T ) to the conditional McKean-Vlasov-Fokker-Planck equation (3.1) that we have constructed in Theorem 3.3. We also denote by γ ± (ρ FP ) ∈ L 2 (ω, Q T ) its trace functions. Due to the Maxwellian bounds (3.5), one can check that the function
γ(ρ FP )(t, x, u) := γ + (ρ FP )(t, x, u) on Σ + T , γ − (ρ FP )(t, x, u) on Σ − T ,
is a trace of ρ FP in the sense of Definition 1.1. In particular, the integrability and positivity requirement (1.6b) is an immediate consequence of the Maxwellian bounds (3.5), and the initial bounds in (H MVFP )-(iii). For (1.6a), using the estimate (3.6a), and since 1
ω L 1 (R d ) < +∞, Σ T |(u · n D (x))|γ(ρ FP )(t, x, u) dλ Σ T (t, x, u) = 2 Σ + T |(u · n D (x))|γ + (ρ FP )(t, x, u) dλ Σ T (t, x, u) ≤ C (0,T )×R d |u|P (t, u)du dt ≤ C (0,T )×R d |u|ω(u)P 2 (t, u)dt du 1 2 < +∞,
where C is a constant depending only on T , ∂D, and 1 ω L 1 (R d ) . The mean no-permeability condition (1.4) is then satisfied, since ρ FP satisfies the specular boundary condition.
We now introduce the probability measure Q defined by is an R d -valued Q-Brownian motion, and Q(x(0) ∈ dx, u(0) ∈ du) = ρ 0 (x, u)dx du. To prove that Q is a solution in law to (1.1), we check that the time-marginals Q • (x(t), u(t)) −1 satisfy a mild equation. More precisely, we show that Q • (x(t), u(t)) −1 admits a density function equal to ρ FP (t), so that B[x(t); ρ FP (t)] is equal to E Q [b(u(t))|x(t)]. For this purpose, we introduce the following linear mild equation: for all t ∈ (0, T ], for all ψ ∈ C c (D × R d ),
dQ dP = exp 1 σ T 0 B[x(t); ρ FP (t)] d w(t) − 1 2σ 2 T 0 B[x(t); ρ FP (t)] 2 dt .ψ, ρ(t) = Γ ψ (t), ρ 0 + t 0 ∇ u Γ ψ (t − s), B[·; ρ FP (s)]ρ(s) ds,(4.2)
where ·, · stands for the inner product in L 2 (D × R d ) and Γ ψ (t, x, u) is defined as in (2.16). According to Corollary 2.7, for all ψ ∈ C c (D × R d ), Γ ψ belongs to L 2 ((0, T ) × D; H 1 (R d )) hence (4.2) is well defined. Furthermore, we have Proof. Let ρ 1 , ρ 2 ∈ C([0, T ]; L 2 (D × R d )) be two mild solutions to (4.2). Then, for all t ∈ [0, T ],
ρ 1 (t) − ρ 2 (t) 2 L 2 (D×R d ) = sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 ( ψ, (ρ 1 − ρ 2 )(t) ) 2 = sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 Qt ∇ u Γ ψ (t − s, x, u), B[x; ρ FP (s)] (ρ 1 − ρ 2 )(s, x, u) ds dx du 2 ≤ b 2 L ∞ (R d ;R d ) sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 ∇ u Γ ψ 2 L 2 (Qt) t 0 ρ 1 (s) − ρ 2 (s) 2 L 2 (D×R d ) ds.
By using the estimate (2.26) in Corollary 2.7 on the decomposition (ψ) + and (ψ) − , we obtain that ∇ u Γ ψ L 2 (ω,Q T ) ≤ 1/σ 2 . It follows that
ρ 1 (t) − ρ 2 (t) 2 L 2 (D×R d ) ≤ b 2 L ∞ (R d ;R d ) σ 2 t 0 ρ 1 (s) − ρ 2 (s) 2 L 2 (D×R d ) ds.
By Gronwall's Lemma, we conclude on the uniqueness of solutions to (4.2). Combining Proposition 4.2 and Proposition 4.1, we conclude on the equality Q • (x(t), u(t)) −1 = ρ FP (t, x, u)dx, du. We also conclude that Q is a solution in law to (1.1) in Π ω . The set of time marginal densities of Q naturally inherits the trace functions of ρ FP so that the specular and the mean no-permeability boundary conditions are both satisfied.
Proof of Proposition 4.2. We first prove (i). According to Theorem 3.3,
ρ FP ∈ V 1 (ω, Q T ) ⊂ C([0, T ]; L 2 (D× R d )) satisfies: for all t ∈ [0, T ], Ψ ∈ C ∞ c (Q t ), Qt ρ FP (s, x, u) ∂ s Ψ + (u · ∇ x Ψ) + σ 2 2 △ u Ψ (s, x, u) + B[x; ρ FP (s)] · ∇ u Ψ(s, x, u) ρ FP (s, x, u) ds dx du = − D×R d ρ FP (t, x, u)Ψ(t, x, u) dx du + D×R d ρ 0 (x, u)Ψ(0, x, u) dx du − Σ + t (u · n D (x))γ + (ρ FP )(s, x, u) (Ψ(s, x, u) − Ψ(s, x, u − 2(u · n D (x))n D (x))) dλ Σ T (s, x, u). (4.3)
Using convolution approximation on the test function and since ρ FP , ∇ u ρ FP and γ + (ρ FP ) are squareintegrable, one can extend the preceding formula for all
Ψ ∈ C b (Q t ) ∩ C 1,1,2 b (Q t ). Using Corollary 2.6, we know that (s, x, u) ∈ Q t → Γ ψ n (t − s, x, u) is a smooth function that satisfies ∂ s Γ ψ n (t − s) + (u · ∇ x Γ ψ n (t − s)) + σ 2 2 △ u Γ ψ n (t − s) = 0, on Q t , lim s→t Γ ψ n (t − s, x, u) = ψ(x, u), on D × R d , Γ ψ n (t − s, x, u) = Γ ψ n−1 (t − s, x, u − 2(u · n D (x))n D (x)), on Σ + t . (4.4)
Hence, in the case Ψ(s,
x, u) = Γ ψ n (t − s, x, u), (4.3) reduces to D×R d ρ FP (t, x, u)ψ(x, u) dx du = D×R d ρ 0 (x, u)Γ ψ n (t, x, u) dx du + Qt B[x; ρ FP (s)] · ∇ u Γ ψ n (t − s, x, u) ρ FP (s, x, u) ds dx du − Σ − t (u · n D (x))γ − (ρ FP )(s, x, u) Γ ψ n (t − s, x, u) − Γ ψ n−1 (t − s, x, u) dλ Σ T (s, x, u).
(4.5) Owing to (2.29), we obtain (4.2) by taking the limit n → +∞. Now we prove (ii). Let Q be defined as in (4.1). Let us also introduce the time-marginal probability measures (µ m (t) := Q • (x(t ∧ τ m ), u(t ∧ τ m )) −1 ; m ≥ 1) where τ m is the mth-time x(t) hits ∂D. Since (t, x) → B[x; ρ FP (t)] is uniformly bounded, owing to Girsanov transform and Theorem 2.1, one can easily check that Q is absolutely continuous w.r.t. P so that the sequence (τ m ; m ≥ 1) is well-defined and grows to ∞ under Q. As a first step, we show the existence of an L ∞ ((0, T ); L 2 (D × R d ))-density of µ m . Using a Riesz representation argument, it is sufficient to show that there exists K > 0, possibly depending on m, such that
∀t ∈ (0, T ), ∀ψ ∈ C ∞ c (D × R d ), nonnegative, D×R d ψ(x, u)µ m (t, dx, du) ≤ K ψ L 2 (D×R d ) .
(4.6) To prove (4.6), let us observe that for all nonnegative ψ ∈ C ∞ c (D × R d ), and for all α ∈ (1, +∞), using Girsanov's change of probability and the boundedness of B[·; ρ FP (·)],
D×R d ψ(x, u)µ m (t, dx, du) = E Q [ψ(x(t ∧ τ m ), u(t ∧ τ m ))] ≤ exp b L ∞ (R d ;R d ) T 2(α + 1) (E P [ψ α (x(t ∧ τ m ), u(t ∧ τ m ))]) 1 α .
(4.7)
We will specify later the appropriate α. Now observe that
E P [ψ α (x(t ∧ τ m ), u(t ∧ τ m ))] = Γ (ψ) α m (t), ρ 0 , for Γ (ψ) α m
given as in (2.16). Let us observe that, according to Corollary 2.8, for all β ∈ (1, 2),
Γ (ψ) α m (t) β L β (D×R d ) + Γ (ψ) α m β L β (Σ − t ) ≤ ψ α β L β (D×R d ) + Γ (ψ α ) m−1 β L β (Σ − t ) .
Iterating this inequality m times and since Γ (ψ α ) 0 = ψ = 0 on Σ T , one gets
Γ (ψ) α m (t) β L β (D×R d ) ≤ m ψ α β L β (D×R d ) .
It thus follows that, for β ′ conjugate to β,
Γ (ψ α ) m (t), ρ 0 ≤ Γ (ψ α ) m (t) L β (D×R d ) ρ 0 L β ′ (D×R d ) ≤ m 1 β ψ α β L β (D×R d ) ρ 0 β ′ L β ′ (D×R d )
.
Coming back to (4.7), we deduce that
D×R d ψ(x, u)µ m (t, dx, du) ≤ m 1 βα exp b L ∞ T 2(α + 1) ψ L βα (D×R d ) ρ 0 L β ′ (D×R d ) 1 α (4.8)
For the special case where α and β are such that βα = 2 and owing to (H MVFP )-(iii), we get (4.6) for a constant K depending on m, T , ρ 0 L 2 (D×R d ) and b L ∞ (R d ;R d ) .
Now applying the Itô formula to (s, x, u) → Γ ψ n (t − s, x, u) and using Eq. (4.4), it follows that, Q-a.s.,
ψ(x(t ∧ τ m ), u(t ∧ τ m )) = Γ ψ n (t, x(0 ∧ τ m ), u(0 ∧ τ m )) + t∧τm 0 ∇ u Γ ψ n (t − s, x(s), u(s)) dw(s) + B[x(s); ρ FP (s)] ds + m k=0 Γ ψ n (t − τ k , x(τ k ), u(τ k )) − Γ ψ n (t − τ k , x(τ k ), u(τ − k )) ½ {τm≤t} .
Since Γ ψ n (t − τ k , x(τ k ), u(τ − k )) = Γ ψ n−1 (t − τ k , x(τ k ), u(τ k )), taking the expectation on both sides of the equality yields
ψ, µ m (t) = Γ ψ n (t), ρ 0 + E Q t∧τm 0 ∇ u Γ ψ n (t − s, x(s), u(s))B[x(s); ρ FP (s)] ds + E Q m k=0 Γ ψ n (t − τ k , x(τ k ), u(τ k )) − Γ ψ n−1 (t − τ k , x(τ k ), u(τ k )) ½ {τ k ≤t} .
(4.9)
We take the limit n → +∞ in (4.9). According to Lemma 2.4, m k=0 P • (τ k , x(τ k ), u(τ k )) −1 is a finite measure on Σ − T and is absolutely continuous w.r.t. λ Σ T . By Girsanov Theorem, the same holds true for m k=0 Q • (τ k , x(τ k ), u(τ k )) −1 . Hence using the λ Σ T -a.e. convergence given in (2.29) we get that
lim n→+∞ E Q m k=0 Γ ψ n (t − τ k , x(τ k ), u(τ k )) − Γ ψ n−1 (t − τ k , x(τ k ), u(τ k )) = 0.
Since Γ ψ n and ∇ u Γ ψ n converge to Γ ψ and ∇ u Γ ψ in L 2 (Q T ) and since µ m ∈ L ∞ ((0, T ); L 2 (D × R d )) we get ψ, µ m (t) = Γ ψ (t), ρ 0 + E Q t∧τm 0 ∇ u Γ ψ (t − s, x(s), u(s)), B[x(s); ρ FP (s)] ds Next, one can observe that, for all t ∈ (0, T ),
ψ, µ m (t) = D×R d |Γ ψ (t)|(x, u)|ρ 0 |(x, u) dx du + t 0 ∇ u Γ ψ (t, s), B[x; ρ FP (s)] µ m (s, x, u) ds dx du (4.10) ≤ Γ ψ (t) L 2 (D×R d ) ρ 0 L 2 (D×R d ) + b L ∞ (R d ;R d ) ∇ u Γ ψ L 2 (Q T ) µ m L 2 (Qt) .
By Gronwall's Lemma, this shows that µ m is bounded in L 2 (Q T ) uniformly in m. In addition, for all t ∈ (0, T ], µ m (t) converges weakly toward Q • (x(t), u(t)) −1 . Thus, using again a Riesz representation argument, we deduce that Q admits a set of time marginal densities (ρ(t); t ∈ [0, T ]) in L 2 (Q T ).
Taking the limit m → +∞ in (4.10), we further conclude that (ρ(t); t ∈ [0, T ]) in L 2 (Q T ) is solution to (4.2).
Uniqueness
Under (H), let us observe that any solution in law Q ∈ Π ω to (1.1) has time marginal densities ρ(t) ∈ L 2 (ω, D × R d ), solution of the following weighted nonlinear mild equation: for all t ∈ (0, T ], for all
ψ ∈ L 2 (D × R d ), √ ωψ, ρ(t) = Γ ψ (t), √ ωρ 0 + t 0 ∇ u Γ ψ (t − s), √ ωB[·; ρ(s)]ρ(s) ds + t 0 Γ ψ (t − s), (∇ u log( √ ω) · √ ωB[·; ρ(s)])ρ(s) ds 5 + σ 2 2 t 0 (△ u √ ω)Γ ψ (t − s), ρ(s) ds.
(4.11)
We prove (4.11) by replicating some proof steps of Proposition 4.2-(ii): for fixed nonnegative ψ ∈ C ∞ c (D × R d ), t ∈ (0, T ], and m, n ∈ N * , using Corollary 2.6, Itô's formula applied to (s, x, u) ∈ Q t → ω(u)Γ ψ n (t − s, x, u) yields
E Q ω(u(t))ψ(x(t ∧ τ m ), u(t ∧ τ m )) = E Q ω(u(0))Γ ψ n (t, x(0), u(0)) + E Q t∧τm 0 ω(u(s)) ∇ u Γ ψ n (t − s, x(s), u(s)) · B[x(s); ρ(s)] ds + E Q t∧τm 0 Γ ψ n (t − s, x(s), u(s)) ∇ u ω(u(s)) · B[x(s); ρ(s)] + σ 2 2 △ √ ω(u(s)) ds + E Q m k=0 ω(u(τ k )) Γ ψ n (t − τ k , x(τ k ), u(τ k )) − Γ ψ n−1 (t − τ k , x(τ k ), u(τ k )) ½ {τ k ≤t} .
Then taking the limit n → +∞, the boundary term vanishes. Next as m → +∞, we prove (4.11) for any nonnegative test function ψ ∈ C c (D × R d ). This result extends to any test function ψ ∈ C c (D × R d ), using the linearity of ψ → Γ ψ and the decomposition ψ = (ψ) + − (ψ) − . Since the drift coefficient in (1.1) is bounded, the fact that two solutions in law of (1.1) coincide is equivalent with the equality between the time marginal densities of these two solutions. We thus conclude the proof of the uniqueness part of Theorem 1.2 with the following. Proof. Using (4.11), for all t ∈ [0, T ], we have
ρ FP (t) − ρ(t) 2 L 2 (ω,D×R d ) = √ ω(ρ FP (t) − ρ(t)) 2 L 2 (D×R d ) = sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 √ ωψ, ρ FP (t) − ρ(t) 2 ≤ 2 sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 t 0 ∇ u Γ ψ (t − s) + Γ(t − s)∇ u log( √ ω), √ ω B[·; ρ FP (s)]ρ FP (s) − B[·; ρ(s)]ρ(s) ds 2 + σ 2 sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 t 0 Γ ψ (t − s)△ u √ ω, ρ FP (s) − ρ(s) ds 2 .
Using Cauchy-Schwarz's inequality and Lemma 3.5, it follows that
ρ FP (t) − ρ(t) 2 L 2 (ω,D×R d ) ≤ 2 sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 ∇ u Γ ψ 2 L 2 (Qt) + α 4 Γ ψ 2 L 2 (Qt)+ σ 2 2α( α 2 − 1) + αd sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 Γ ψ 2 L 2 (Qt) t 0 ρ FP (s) − ρ(s) 2 L 2 (ω,D×R d ) ds.
Using Corollary 2.7, one can deduce that ρ FP (t) − ρ(t) 2 L 2 (ω,D×R d ) ≤
+ R d b(v)ρ FP (s, x, v) dv R d ρ FP (s, x, v) − ρ(s, x, v) dv R d ρ FP (s, x, v) dv R d ρ(s, x, u) dv .
Using the Maxwellian bounds P and P of ρ FP given in Theorem 3.3, one has
R d ω(u) ρ FP (s, x, u) 2 du R d b(v) ρ FP (s, x, v) − ρ(s, x, v) dv R d ρ FP (s, x, v) dv 2 ≤ R d ω(u) P (s, u) 2 du R d P (s, u) du 2 R d b(v) ρ FP (s, x, v) − ρ(s, x, v) dv 2 ≤ 2 b L ∞ (R d ;R d ) sup t∈[0,T ] R d ω(u) P (t, u) 2 du inf t∈[0,T ] R d P (t, u) du 2 R d ω −1 (v) dv R d ω(v) ρ FP (s, x, v) − ρ(s, x, v) 2 dv, and R d ω(u) ρ FP (s, x, u) 2 du R d b(v)ρ FP (s, x, v) dv R d ρ FP (s, x, v) − ρ(s, x, v) dv R d ρ FP (s, x, v) dv R d ρ(s, x, u) dv 2 ≤ b L ∞ (R d ;R d ) R d ω(u) ρ FP (s, x, u) 2 du R d ρ FP (s, x, v) dv 2 R d b(v) ρ FP (s, x, v) − ρ(s, x, v) dv 2 ≤ b L ∞ (R d ;R d ) sup t∈(0,T ) R d ω(u) P (t, u) 2 du inf t∈(0,T ) R d P (t, u) du 2 R d ω(v) ρ FP (s, x, v) − ρ(s, x, v) 2 dv,
Therefore, for some constant C > 0, where C depends on ρ 0 , d, σ and α. Using Gronwall's Lemma, this ends the proof.
Acknowledgment. The authors would like to thank Pr. Pierre-Louis Lions for having pointed to us the PDE approach developed in Section 3 and for his helpful remarks.
A Appendix
A.1 Some recalls Corollary A.1 (Rana [30]). If φ ∈ L p (R d ) for p ∈ [1, +∞) then lim |δ|→0 + |φ(z + δ) − φ(z)| p dz = 0.
Theorem A.2 (Tartar [33], Chapter 4). Let V be an open subset of R d and ψ ∈ L 2 (V) such that ∇ v ψ ∈ L 2 (V). Then
∇ v (ψ) + , ∇ v (ψ) − ∈ L 2 (V) with ∂ v i (ψ) + = ∂ v i ψ½ {ψ≥0} and ∂ v i (ψ) − = −∂ v i ψ½ {ψ≤0} .
Theorem A.3 (Lions and Magenes [22]). Let E be a Hilbert space with the inner product ( , ) E . Let F ⊂ E equipped with the norm | | F such that the canonical injection of F into E is continuous. Assume that A : E × F → R is a bilinear application satisfying:
We obtain the regularity w.r.t. u by applying the differential operator ∂ u i to Eq. (A.2). Hence ∂ u i f r 1 satisfies
∂ t ∂ u i f r 1 − (u · ∇ x ∂ u i f r 1 ) − σ 2 2 △ u ∂ u i f r 1 = ∂ u i g r 1 + ∂ x i f r 1 , in (C ∞ c (R × R d × R d )) ′ , (A.5) where ∂ u i g r 1 = (∂ u i Γ r 1 )f + Γ r 1 ∂ u i f + (Ψ r 1 · ∇ u ∂ u i f ) + (∂ u i Ψ r 1 · ∇ u f ) .
Theorem A.4-(a) ensures that △ u f r 0 L 2 (R×R d ×R d ) < +∞. As f r 0 has a compact support, standard arguments give that 1≤i,j≤d
∂ 2 u i ,u j f r 0 2 L 2 (R×R d ×R d ) = △ u f r 0 2 L 2 (R×R d ×R d ) < +∞
and thus 1≤i,j≤d ∂ 2 u i ,u j f 2 L 2 (Bz 0 (r 1 )) = 1≤i,j≤d ∂ 2 u i ,u j f r 0 L 2 (Bz 0 (r 1 )) < +∞.
Now we set h = ∂ u i g r 1 +∂ x i f r 1 with h L 2 (R×R d ×R d ) ≤ ∇ u g r 1 L 2 (R×R d ×R d ) + ∇ x f r 1 L 2 (R×R d ×R d ) < +∞, since ∇ u g r 1 L 2 (R×R d ×R d ) ≤ C f L 2 (Bz 0 (r 1 )) + ∇ u f L 2 (Bz 0 (r 1 )) + 1≤i,j≤d ∂ 2 u i ,u j f 2 L 2 (Bz 0 (r 1 )) < +∞.
applying Theorem A.4-(a), we deduce as before that △ u D η u f ∈ L 2 (B z 0 ( R N 2 )), which ensures that ∇ u D η u f | ∈ L 2 (B z 0 ( R N 2 )). By applying Theorem A.4-(b) three times, we obtain that |∇ x D η u f | ∈ L 2 (B z 0 ( R N 2 3 )). This ends the proof of the induction N + 1. We iterate m times this induction and conclude that, for r := r 0 2 3m ,
η∈N d ;|η|≤m 1 µ 0 (u) µ du.
In addition, since µ > 1, Hölder's inequality yields
R d G(ν, u) G(σ 2 t) * p 1 µ 0 (u) du µ ≤ G(σ 2 ν) µ L µ ′ (R d ) R d G(σ 2 t) * p 1 µ 0 (u) µ du,
where µ ′ is the conjugate of µ, ν is a positive constant and G(ν) µ
L µ ′ (R d ) = (2πσ 2 ν) −d 2 (µ ′ ) −d 2(µ ′ −1) > 0.
Setting C µ,ν := exp{|a|T } G(σ 2 ν) µ L µ ′ (R d ) , we then observe that Since p 0 is assumed to be not identically zero on R d , we conclude (i2).
Proof of (i3). For all µ > 0, it is enough to show that for some real sequence {ǫ k ; k ∈ N} decreasing to 0,
lim k→+∞ R d p 1 µ (ǫ k , u) − p 1 µ 0 (u) 2µ du = 0. (A.8)
Indeed, in the case µ ≤ 1, recalling that:
||c| − |b|| q ≤ ||c| q − |b| q | , for c, b ∈ R, q ≥ 1, it holds, for all k,
|p(ǫ k , u) − p 0 (u)| 2 = |p(ǫ k , u) − p 0 (u)| 1 µ 2µ ≤ |p 1 µ (ǫ k , u) − p 1 µ 0 (u)| 2µ .
Then (A.8) will imply (i3) for all {ǫ k ; k ∈ N} considered above.
In the case µ > 1, (A.8) yields the convergence p(ǫ k ) → p 0 (·) in L 2µ (R d ). Applying [Theorem 4.9, Brezis [8]], we deduce the existence of a subsequence of {p(ǫ k ); k ∈ N} such that lim k→+∞ p(ǫ k , u) = p 0 (u) a.e. u ∈ R d , and sup k∈N |p(ǫ k , u)| 1 µ ∈ L 2µ (R d ).
Since sup k∈N |p(ǫ k , u)| 2 ≤ (sup k∈N |p(ǫ k , u)| 1 µ ) 2µ , (i3) follows from the Lebesgue Dominated Convergence Theorem.
We now check that (A.8) holds true: By definition R d |p According to assumption (3.13), R d p 2 0 (u) du is finite so the first term in the expression above tends to 0 when k goes to infinity. For the second term, a change of variables and Hölder's inequality give
R d |m(ǫ k , u) − p 1 µ 0 (u)| 2µ du = R d R d G(σ 2 , u 0 )p 1 µ 0 (u − √ ǫ k u 0 ) du 0 − p 1 µ 0 (u) 2µ du ≤ R d G(σ 2 , u 0 ) R d |p 1 µ 0 (u − √ ǫ k u 0 ) − p 1 µ 0 (u)| 2µ du du 0 .
Since p We check also that R G(σ 2 , u 0 ) sup
k R d |p 1 µ 0 (u − √ ǫ k u 0 ) − p 1 µ 0 (u)| 2µ du du 0 ≤ 2 R d p 2 0 (u) du < +∞,
in order to conclude that lim t→0 + R d |m(t, u) − p 1 µ 0 (u)| 2µ du = 0 by dominated convergence.
By integrating by parts the right member of (A.10) and using the heat equation △ u m = 2 σ 2 ∂ t m, we get (ǫ k ,T )×R d exp{2at} |∇ u m(t, u)| 2 m 2µ−2 (t, u) dt du
= −1 2µ − 1 (ǫ k ,T )×R d exp{2at}△ u m(t, u)m 2µ−1 (t, u) dt du = −2 σ 2 (2µ − 1) (ǫ k ,T )×R d exp{2at}∂ t m(t, u)m 2µ−1 (t, u) dt du = −1 σ 2 µ(2µ − 1) (ǫ k ,T )×R d exp{2at}∂ t m 2µ (t, u) dt du
Using again an integration by parts enables us to obtain the equality (ǫ k ,T )×R d exp{2at} |∇ u m(t, u)| 2 m 2µ−2 (t, u) dt du
= −1 σ 2 µ(2µ − 1) R d p 2 (T, u) du − R d p 2 (ǫ k , u) du + 2a σ 2 µ(2µ − 1) (ǫ k ,T )×R d p 2 (t, u) dt du.
Coming back to (A.10) and letting k increase to +∞, it follows that
(0,T )×R d |∇ u p(t, u)| 2 dt du = −µ σ 2 (2µ − 1) R d p 2 (T, u) du − R d p 2 0 (u) du + 2aµ σ 2 (2µ − 1) (0,T )×R d p 2 (t, u) dt du.
Thanks to the assumption (3.13), p 0 ∈ L 2 (R d ) and (i1), the right member is finite. We conclude (i5).
: R d → R d is a given measurable function. Formally the function (t, x) → B[x; ρ(t)] in (1.1) corresponds to the conditional expectation (t, x) → E[b(U t )|X t = x]and the velocity equation in (1.1) rewrites
Definition 1. 1 .
1Let (ρ(t); t ∈ [0, T ]) be the time-marginal densities of a solution to (1.1). We say thatγ(ρ) : (0, T ) × ∂D × R d → R is the trace of (ρ(t); t ∈ [0, T ]) along (0, T ) × ∂D × R d if it is a nonnegative function γ(ρ) satisfying, for all t in (0, T ], f in C ∞ c ([0, T ] × D × R d ):
2 0
2(|u|)du < +∞.Let us precise the notion of solution that we consider for (1.1). A probability measure Q, in the samplespace T := C([0, T ]; D) × D([0, T ]; R d ) with canonical process (x(t), u(t); t ∈ [0, T ]), is a solution in law to (1.1) if for all t ∈ [0, T ], Q • (x(t), u(t)) −1 admits a density function ρ(t) with ρ(0) = ρ 0 and there exists an R d -Brownian motion (w(t); t ≥ 0) under Q, such that Q-a.s. x(s); ρ(s)] ds + σw(t) − 0<s≤t 2 u(s − ) · n D (x(s)) n D (x(s))½ {x(s) ∈ ∂D} .
Proposition 2. 9 (
9Carrillo [9], Theorem 2.2, Proposition 2.4 and Lemma 3.4). Given two nonnegative functions
valued Brownian motion defined on some probability space (Ω, F, P). Set β y,v δ := inf{t > 0 ; d(x y,v t , ∂D) ≤ δ}. Since f is smooth in the interior of Q T and satisfies (2.18), applying Itô's formula to
(3.1), the two main difficulties are in the fractional form of B[· ; ρ], and in the verification of the properties (1.6a) and (1.6b) of the trace function γ(ρ) (see Definition 1.1). For this purpose, starting from the assumption (H MVFP )-(iii), we prove the existence of a solution to (3.1), as well as the existence of related Maxwellian upper and lower-bounds. These Maxwellian bounds are of the following form.
Theorem 3. 3 .
3Under (H MVFP ), there exists a function ρ ∈ V 1 (ω, Q T ), and there exist γ + (ρ), γ − (ρ) defined on Σ + T and Σ − T respectively, with γ ± (ρ) ∈ L 2 (ω, Σ ± T ), such that, for all t ∈ (0, T ], for all
and does not contradict the fact that ρ ∈ V 1 (ω, Q T ). In the case of an unbounded domain D, the Maxwellian bounds should involve some Maxwellian distributions in the space variable also. The main steps of the proof of Theorem 3.3 are the following. First, in the next Subsection 3.1, we consider a linear version of equation (3.1) where a Dirichlet condition is imposed on Σ − T , and where the drift coefficient is given in L ∞ ((0, T ) × D; R d ). Under (H MVFP )-(ii) and (H MVFP )-(iii), the problem is well-posed in V 1 (ω, Q T ) (see Lemma 3.8).As a preliminary step, we also highlight some meaningful properties on the transport operator T and the Green identity related to the Vlasov-Fokker-Planck equation in the weighted spaces V 1 (ω, Q T ). Then, in Subsection 3.2, we show the existence of Maxwellian bounds satisfying the requirements (3.6) (see Proposition 3.9) and which are identified as super-solution and sub-solution for the linear problem. Next in Subsection 3.3, by means of fixed point methods, we successively construct a solution to the equation with the specular boundary condition (see Proposition 3.14) and with the nonlinear term B[· ; ρ] (see Proposition 3.15).
. 8 .
8Assume (H MVFP )-(ii). Then, given B ∈ L ∞ ((0, T ) × D; R d ), q ∈ L 2 (ω, Σ − T ), and g ∈ L 2 (ω, Q T ), there exists a unique solution f in V 1 (ω, Q T ) to(3.8). In addition, this solution admits trace functions γ ± (f ) in L 2 (ω, Σ ± T ) and, for all t ∈ (0, T ],
. 9 .
9Assume (H MVFP )-(ii) and (H MVFP )-(iii). For B ∈ L ∞ ((0, T )×D; R d ), for (P 0 , P 0 ) as in (H MVFP )-(iii), let p, p be a couple of Maxwellian distributions with parameters a, µ, P 0 and a, µ, P 0 satisfying:
( 3 . 12 )
312Definition 3.10. Let P be a Maxwellian distribution with parameters (a, µ, P 0 ). For B ∈ L ∞ ((0, T ) × D; R d ), we say that: 1. P is a super-solution of Maxwellian type for L B if 0 ≤ L B (P ) < +∞, a.e. on Q T .
Lemma 3 . 11 .
311Let p be a Maxwellian distribution equipped with the parameters (a, µ, p 0 ) such that 2µ > 1, p 0 is not identically equal to zero, and R d (1 + |u|)ω(u)p 2 0 (u) du < +∞.(3.13)
23) from which we conclude on(3.22). The inequality (3.23) is proved by using the sequence J n (s, x, u) := f n (s, x, u) − p(s, u),
. 14 .
14Assume (H MVFP )-(ii), (H MVFP )-(iii), and B ∈ L ∞ ((0, T ) × D; R d ).
Owing to the continuity of S given in (p1), we deduce that S = lim n→+∞ S n+1 = lim n→+∞ S (S n ) = S (S) , γ ± (S) = lim n→+∞ γ ± (S n+1 ) = lim n→+∞ γ ± (S (S n )) = γ ± (S (S)) .
B
according to Girsanov Theorem, ((x(t), u(t)); t ∈ [0, T ]) satisfies the confined Langevin equation with the additional drift (t, x) → B[x; ρ FP (t)]; namely, Q-a.s., [x(s); ρ FP (s)] ds + σw(t) − 0<s≤t 2 u(s − ) · n D (x(s)) n D (x(s))½ {x(s) ∈ ∂D} , FP (s)] ds; t ∈ [0, T ])
Proposition 4. 1 .
1There exists at most one solution in C([0, T ]; L 2 (D × R d )) to the linear mild equation (4.2).
Proposition 4. 2 .
2(i) The solution (ρ FP (t); t ∈ [0, T ]) ∈ V 1 (ω, Q T ) of(3.1), constructed in Theorem 3.3 is solution to the mild equation (4.2).
(
ii) For Q defined in (4.1), for all t ∈ [0, T ], the time marginal Q • (x(t), u(t)) −1 admits a density ρ(t) ∈ L 2 (ω, D × R d ) which is solution to the mild equation(4.2).
Lemma 4. 3 .
3Under (H), any solution ρ(t) ∈ L 2 (ω; D × R d ) to the non-linear mild equation (4.11) is equal to ρ FP (t) for all t ∈ [0, T ].
t 0 ρ
0FP (s)B[·; ρ FP (s)] − ρ(s)B[·; ρ(s)] 2 L 2 (ω,D×R d ) ds
ρρ
FP (s)B[·; ρ FP (s)] − ρ(s)B[·; ρ(s)] 2 L 2 (ω,D×R d ) FP (s) − ρ(s) 2 L 2 (ω,D×R d ) ds.(4.12) Now observe thatQ T ω(u) ρ FP (s, x, u)B[x; ρ FP (s)] − ρ(s, x, u)B[x; ρ(s)] 2 ds dx du ≤ b 2 L ∞ (R d ;R d ) ρ FP − ρ 2 L 2 (ω,Q T ) u) ρ FP (s, x, u) 2 du B[x; ρ FP (s)] − B[x; ρ(s)] 2 ds dx and that B[x; ρ FP (s)] − B[x; ρ(s)] = R d b(v) ρ FP (s, x, v) − ρ(s, x, v) dv R d ρ FP (s, x, v) dv
u) ρ FP (s, x, u) 2 du B[x; ρ FP (s)] − B[x; ρ(s)] 2 ds dx ≤ 2C ρ FP − ρ 2 L 2 (ω,Qt)Coming back to (4.12), we deduceρ FP (t) − ρ(t) 2 L 2 (ω,D×R d ) ≤ C sup ψ∈Cc(D×R d ); ψ L 2 (D×R d ) =1 t 0 ∇ u Γ ψ (t − s) 2 L 2 (ω,D×R d ) ds t 0 ρ FP (s) − ρ(s) 2 L 2 (ω,D×R d ) ds.
) .
)Remark 2.10. Eq. (2.34) provides a variational formulation of the abstract Cauchy problem (2.18) in the sense that if f and γ(f ) satisfy Eq. (2.34) then, for all
For instance, set D µ := {x ∈ D; −ς(x) ≤ µ}. Assuming that ς ∈ C 2 for all x ∈ D µ for some µ > 0, by setting C0 := (u0 · nD(x0)) > 0, one can choose δ ′ ∈ (0, µ) so that δ ′ sup x∈D µ |∇ς(x)| + (δ ′ + |u0|) sup x∈D µ |∇ 2 ς(x)| < C0/2. Therefore, we have (u · nD(x)) ≥ (u0 · nD(x0)) − |(u0 · nD(x0)) − (u · ∇ς(x))| ≥ C0 − δ ′ sup D µ |∇ς| + (δ ′ + |u0|) sup D µ |∇ 2 ς| > C0/2 := η.
For ⌊x⌋ the nearest integer lower than x ∈ R + .
This can be shown by applying a Cauchy-Schwarz inequality in the alternative definition of the fractional derivative in L 2 via Fourier transform, see e.g.[14].
A.2 Proof of Proposition 2.11To prove this proposition, it is sufficient to show that, for all z 0 := (t 0 , x 0 , u 0 ) in Q T , there exists r > 0 such that f belongs to C 1,1,2 (B z 0 (r)) where B z 0 (r) ⊂ Q T is the open ball centered at z 0 of radius r. To this end, we use the Sobolev embeddings (see e.g.[8], Corollary 9.15): for m = ⌊d/2⌋ + 2 − ⌊1 − d/2 − ⌊d/2⌋⌋, we have 2 W 2,2 ((0, T )) ⊂ C 1 ([0, T ]), W m,2 (B x 0 (r)) ⊂ C 1 (B x 0 (r)), W m+1,2 (B u 0 (r)) ⊂ C 2 (B u 0 (r)).We thus have to prove that for some r > 0,D κ u f L 2 (Bz 0 (r)) < +∞, (A.1)where D η x and D κ u refer to the differential operators given byThe proof of (A.1), is based on a bootstrap argument that uses the regularity results (in fractional Sobolev spaces) obtained in Bouchut[7]for the solution to kinetic equation (see Theorem A.4).Step 1. Let us start with the regularity along the (x, u)-variables. We proceed by induction on a truncated version of f .For any r 0 > 0 such that B z 0 (r 0 ) Q T , we denote by β r 0 : Q T → [0, 1], a C ∞ c (Q T )-cutoff function such thatWe further assume that there exists a constant C depending on r 0 such thatStarting from f ∈ L 2 ((0, T ) × D; H 1 (R d )) given in Proposition 2.9, the truncated function f r 0 := β r 0 f satisfies, in the sense of distributions,where g r 0 := Γ r 0 f + (Ψ r 0 · ∇ u f ). Let us now recall Theorem 1.5 (and its proof) in[7]: for α ∈ (0, 1), we further denote by D α x the fractional derivative w.r.t. x-variables, defined as the fractional Laplace operator of order αThen there exists a positive constant C(d) depending on the dimension such that:Since2 ), this particularly ensures thatBy setting r 1 := r 0 2 and f r 1 := β r 1 f , it follows that D 1/3Applying the differential operator D 1/3). We sum up the estimations we have obtained asWe extend (A.6) to higher order differentials through the following induction argument: we have proved that for N = 1,Starting from the induction assumption that D ηApplying three times Theorem A.4-(b), we deduce as before thatx, u), and η is a positive real parameter that we explicit later. Setting ψ E := (ψ, ψ) E for the norm of E, we observe that ψ E ≤ |ψ| F for ψ ∈ F . The continuity of the injection J : F → E obviously holds true, as well as the continuity of the application A(·, ψ) : E → R for ψ ∈ F fixed. For the coercivity of A, we check that, for all ψ ∈ F ,Since, by Lemma 3.5, for all (t, x, u) ∈ Q T ,the coercivity of A on F is established by choosing η large enough. Theorem A.3 then ensures the existence of f ∈ E such that, for all φ in F ,In the case ψ = φ √ ω for φ ∈ C ∞ c (Q T ), the preceding expression writes asfrom which we deduce thatAccording to Lemma 3.7, f admits traces γ ± ( f ) on Σ ± t satisfying the Green formula (3.7). In particular,From this expression, replicating the arguments of[9], we establish that f (0, ·) = ρ 0 on D × R d andx, u), and observing thatwe deduce that f ∈ H(ω, Q T ) is a weak solution to(3.8).A.4 Proof of Lemma 3.11Proof of (i1). Since 2µ > 1, by Jensen's inequality, m 2µ (t, u) ≤ (G(σ 2 t) * p 2 0 )(u). By setting f (u) := (1 + |u|)ω(u), we thus have, by Lemma 3.5-(i),Since the Gaussian density has all its moments finite, R d f (u)G(σ 2 t, u) du < +∞ and (i1) follows from the assumption (3.13).Proof of (i2). Let us remark thatfrom which we get, for all t > 0,By (i1), R d p 2 (t, u) du is finite. For the second term, observe thatUsing the inequalityUsing the upper-bound m(t, u) ≤ 2Since p 0 ∈ L 2 (R d ), we deduce (i4).Proof of (i5). To prove that ∇ u p ∈ L 2 ((0, T ) × R d ), we use the sequence {ǫ k ; k ∈ N} given in (i3). For all k, we have
A is coercive on F that is there exists a constant c > 0 such that A(ψ, ψ) ≥ c|ψ| 2 F , ∀ ψ ∈ F. A is coercive on F that is there exists a constant c > 0 such that A(ψ, ψ) ≥ c|ψ| 2 F , ∀ ψ ∈ F .
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Second Order Equations with Nonnegative Characteristic Form. O A Oleȋnik, E V Radkevič, Paul C. FifePlenum PressNew YorkO. A. Oleȋnik and E. V. Radkevič. Second Order Equations with Nonnegative Characteristic Form. Plenum Press, New York, 1973. Translated from Russian by Paul C. Fife.
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| [] |
[
"A Gini approach to spatial CO 2 emissions",
"A Gini approach to spatial CO 2 emissions"
] | [
"Bin Zhouid \nPotsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany\n\nJacob Blaustein Institutes for Desert Research\nBen-Gurion University of the Negev\nBeershebaIsrael\n",
"Stephan Thies \nPotsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany\n",
"Ramana Gudipudiid \nPotsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany\n",
"Matthias K B Lü Deke \nPotsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany\n",
"Jü Rgen ",
"P Kropp \nPotsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany\n\nDepartment of Geo-and Environmental Sciences\nUniversity of Potsdam\nPotsdam, Germany\n",
"Diego Rybski \nPotsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany\n\nDepartment of Environmental Science Policy and Management\nUniversity of California Berkeley\nBerkeleyCAUnited States of America\n"
] | [
"Potsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany",
"Jacob Blaustein Institutes for Desert Research\nBen-Gurion University of the Negev\nBeershebaIsrael",
"Potsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany",
"Potsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany",
"Potsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany",
"Potsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany",
"Department of Geo-and Environmental Sciences\nUniversity of Potsdam\nPotsdam, Germany",
"Potsdam Institute for Climate Impact Research\nMember of the Leibniz Association\nPotsdam, Germany",
"Department of Environmental Science Policy and Management\nUniversity of California Berkeley\nBerkeleyCAUnited States of America"
] | [] | Combining global gridded population and fossil fuel based CO 2 emission data at 1 km scale, we investigate the spatial origin of CO 2 emissions in relation to the population distribution within countries. We depict the correlations between these two datasets by a quasi-Lorenz curve which enables us to discern the individual contributions of densely and sparsely populated regions to the national CO 2 emissions. We observe pronounced country-specific characteristics and quantify them using an indicator resembling the Gini-index. As demonstrated by a robustness test, the Gini-index for each country arise from a compound distribution between the population and emissions which differs among countries. Relating these indices with the degree of socio-economic development measured by per capita Gross Domestic Product (GDP) at purchase power parity, we find a strong negative correlation between the two quantities with a Pearson correlation coefficient of -0.71. More specifically, this implies that in developing countries locations with large population tend to emit relatively more CO 2 , and in developed countries the opposite tends to be the case. Based on the relation to urban scaling, we discuss the implications for CO 2 emissions from cities. Our results show that general statements with regard to the (in)efficiency of large cities should be avoided as it is subject to the socio-economic development of respective countries. Concerning the political relevance, our results suggest a differentiated spatial prioritization in deploying climate change mitigation measures in cities for developed and developing countries.A Gini approach to spatial CO 2 emissions PLOS ONE | https://doi.org/10. | 10.1371/journal.pone.0242479 | null | 53,318,430 | 1810.01133 | cf6a93d5240e72e733a41df2411f3d84d01a0287 |
A Gini approach to spatial CO 2 emissions
Bin Zhouid
Potsdam Institute for Climate Impact Research
Member of the Leibniz Association
Potsdam, Germany
Jacob Blaustein Institutes for Desert Research
Ben-Gurion University of the Negev
BeershebaIsrael
Stephan Thies
Potsdam Institute for Climate Impact Research
Member of the Leibniz Association
Potsdam, Germany
Ramana Gudipudiid
Potsdam Institute for Climate Impact Research
Member of the Leibniz Association
Potsdam, Germany
Matthias K B Lü Deke
Potsdam Institute for Climate Impact Research
Member of the Leibniz Association
Potsdam, Germany
Jü Rgen
P Kropp
Potsdam Institute for Climate Impact Research
Member of the Leibniz Association
Potsdam, Germany
Department of Geo-and Environmental Sciences
University of Potsdam
Potsdam, Germany
Diego Rybski
Potsdam Institute for Climate Impact Research
Member of the Leibniz Association
Potsdam, Germany
Department of Environmental Science Policy and Management
University of California Berkeley
BerkeleyCAUnited States of America
A Gini approach to spatial CO 2 emissions
RESEARCH ARTICLE
Combining global gridded population and fossil fuel based CO 2 emission data at 1 km scale, we investigate the spatial origin of CO 2 emissions in relation to the population distribution within countries. We depict the correlations between these two datasets by a quasi-Lorenz curve which enables us to discern the individual contributions of densely and sparsely populated regions to the national CO 2 emissions. We observe pronounced country-specific characteristics and quantify them using an indicator resembling the Gini-index. As demonstrated by a robustness test, the Gini-index for each country arise from a compound distribution between the population and emissions which differs among countries. Relating these indices with the degree of socio-economic development measured by per capita Gross Domestic Product (GDP) at purchase power parity, we find a strong negative correlation between the two quantities with a Pearson correlation coefficient of -0.71. More specifically, this implies that in developing countries locations with large population tend to emit relatively more CO 2 , and in developed countries the opposite tends to be the case. Based on the relation to urban scaling, we discuss the implications for CO 2 emissions from cities. Our results show that general statements with regard to the (in)efficiency of large cities should be avoided as it is subject to the socio-economic development of respective countries. Concerning the political relevance, our results suggest a differentiated spatial prioritization in deploying climate change mitigation measures in cities for developed and developing countries.A Gini approach to spatial CO 2 emissions PLOS ONE | https://doi.org/10.
Introduction
Urbanization is an ongoing process in many parts on the globe. It is projected that due to rural-urban migration much of the future urbanization is going to take place in developing and transition countries. This leads to ever more mega-cities [1,2]. In parallel, humanity is facing another challenge, namely climate change. To date, cities, despite occupying less than 1% of the global land area, account for more than 70% of the anthropogenic green house gas (GHG) emissions [3]. Therefore, cities are often identified as the key focal areas for global mitigation actions. While a large contribution of the global CO 2 emissions is commonly attributed to cities [4], the CO 2 reduction role of further urbanization is also discussed with the argument of efficiency gains associated with the high densities in cities [5]. Moreover, cities are known to perform more efficiently in addressing the basic needs of human beings [5]. Hence, a diversified view on cities is needed and in view of climate change mitigation, a better understanding of the interplay between urbanization, origin of CO 2 emissions, and socio-economic development is of great interest. Globally, cities are characterized by higher population densities compared to rural areas. Recent literature has identified the crucial role played by population density in either increasing or decreasing the emission efficiency in cities [6][7][8][9][10]. The impact of population density on reducing/increasing CO 2 emissions in these studies is either calculated based on specific assumptions made to calculate the city specific CO 2 emissions or through the construction of city clusters using a clustering algorithm, see [11,12]. However, most of these studies are limited to a specific country or a region. Therefore, there is a gap in the existing literature about the sub-national origin of CO 2 emissions at a global scale. Bridging this gap would provide better insights as to whether population density is a crucial factor in improving/decreasing emission efficiency and would identify other factors that influence CO 2 emissions at a country scale.
Here, we investigate how the spatial origin of CO 2 emissions relates to the spatial distribution of population. We address the questions, to which extent locations of large population also emit the most CO 2 and if there is any dependence on human development. In order to avoid discussions about the proper city definition, the correlations are analyzed on the level of grid cells-keeping in mind that locations of high population are likely to belong to cities. Thus, we analyze population and CO 2 emissions by employing a quasi-Lorenz curve that relates the cumulative population and cumulative emissions for entire countries on a grid-cell level (the Lorenz curve was originally used to describe unequal income distribution).
The shape of these curves explains whether the emissions are concentrated in locations of high or low population. Inspired by the apparent similarity, we extend the well-known Giniindex. Based on the data employed, we find that within many countries, locations with high or low population exhibit different relative emissions. We thus compare the extended "Giniindex" with the economic strength of the considered countries (as captured by the GDP per capita) which can be to some extent interpreted as a measure for the stage of development. We further hypothesize that the development stage of respective countries plays an important role in explaining this relationship.
Earlier studies attempted to address the emission efficiency of densely populated regions by means of urban scaling, where an urban indicator is plotted against the city size in terms of population [13]. The exponent, estimated as the slope of a linear regression in the log-log representation, quantifies efficiency gains of large or small cities. However, in case of urban CO 2 emissions, published results from urban scaling leave an inconclusive picture (for an overview we refer to [14,15]). In the present work we address this issue by combining high resolution, global population and CO 2 emission data sets in order to quantify whether locations with high or low population emit more or less CO 2 . We further discuss an analytic link between our approach and urban scaling.
Materials and methods
Population data
We used the Gridded Population of the World, version 4 (GPWv4) population count data for the year 2010 [16]. GPWv4 data allocate the population counts of census units collected globally from various institutions into standard 1 × 1 km 2 grid cells by means of an arealweighting interpolation [17]. Fig 1(a) illustrates the GPWv4 data in the year 2010 for the contiguous US. The distribution of population in the US exhibits an inhomogeneity. The metropolitan urban agglomerations accommodate a large share of population in the US, whereas the states in the Mountain West are generally sparsely populated.
CO 2 emissions data
Fossil fuel based CO 2 emission estimates are obtained from the Open source Data Inventory of Anthropogenic CO 2 (ODIAC) emissions of version ODIAC2015a available globally at 1 × 1 km 2 grid for the year 2010 [18]. In the ODIAC dataset, point sources, i.e. power plant emissions obtained from the database CARMA (Carbon Monitoring and Action) are directly assigned to the grids, while non-point sources (e.g. emissions from transport, industrial, residential, and commercial sectors) are disaggregated based on global and national emission estimates made by the Carbon Dioxide Information Analysis Center (CDIAC) [19], using remotely sensed nightlight data as a proxy. An exception is the emissions from cement production which have point source origins but are spatially disaggregated as non-point sources. Non-land emissions, such as those from international bunkers (international aviation and maritime shipping), are assigned to the non-point emissions.
Compared with conventional population-based approaches, the nightlight data can trace the human activities more appropriately [20,21]. Worthy of special mention is that the gridded emission data of ODIAC used in this study is not disaggregated using population density as a proxy. Therefore, the two datasets depict distinguishing zonal patterns, as shown in the example for the New York metropolitan region in Fig 1(b) and 1(d). Without relying on the time-consuming update of census data, emissions allocated using nightlights can be updated more frequently and may be of particular importance for developing countries where conducting census is still a challenge. Fig 1(c) shows the gridded total anthropogenic CO 2 emissions (in tons) for the year 2010 for the contiguous US, analogous to the population data shown in Fig 1(a). As observed, the emissions also exhibit pronounced inhomogeneities.
In order to check the consistency of the results obtained, we further compare our results obtained from the ODIAC data with other CO 2 emission datasets, namely the Fossil Fuel Data
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Assimilation System (FFDAS) version 2.0 [22] and the Emission Database for Global Atmospheric Research (EDGAR) version 4.3.2 [23]. Both data are for the year 2010. For the subnational analysis we also analyze the Vulcan data, which has been analyzed before [12]. However, we focus on ODIAC, since it has the highest resolution, and we discuss the results of other datasets in comparison to the ODIAC results.
The fundamentals of creating the four gridded CO 2 emission inventories used in this study have been compared and discussed in detail in [24]. In general, they differentiate themselves in terms of 1) the energy statistics used which determines the sectors included in calculating the total national CO 2 emissions, and 2) the approach to disaggregating and allocating the CO 2 emissions to a regular grid.
Dissimilarities among the inventories may be dominated by the disaggreation method. FFDAS applies the Kaya identity to balance CO 2 emissions across regions, relying on population and nightlight data [22,25] (see also [15] for further information on the Kaya identity). Viewed as the most accurate emissions inventory, a bottom-up method has been used for the Vulcan data allocate large point sources, road-specific emissions, and non-point emissions to census tracts, and further resampled to a 10-km grid [26]. However, since the subnational emissions data are not always available, the Vulcan data is restricted to the USA at the moment.
Inhomogeneity index of CO 2 emissions G e
In order to characterize the relation between country-wise population and CO 2 emissions, we plotted the cumulative quantities against each other. We sorted the grid cells of a country by population in ascending order and calculated the cumulative share of population and CO 2 emissions arising therefrom. Then we interpreted the cumulative emissions as a function of the corresponding cumulative population.
The plotted curves resemble the so-called concentration curves used to describe socio-economic inequalities. The most popular concentration curve is the Lorenz curve usually employed to visualize income inequalities. Other applications of concentration curves include for example the analysis of socio-economic inequalities in the health sector (e.g. [27]). Since the curves we compute here, do not agree exactly with the classical definition of a concentration curve we will refer to them as quasi-Lorenz curves. We justify the choice of this method by its simplicity-it only requires sorting-and the fact that it does not require any parameters or assumptions on functional forms.
To quantify the curves, we break the shape of each curve down to a single number. As it is well known and used in this context, we generalize the Gini coefficient, which originally has been introduced to quantify income inequality [28]. As illustrated in Fig 2, we distinguish between curves above or below the dashed line with a slope of 45˚-the line of equality. For the blue quasi-Lorenz curve, we defined an inhomogeneity index as the ratio of the area between the curve and the line of equality (marked as A) to the total area below the line of equality (A + B). Analogously, the inhomogeneity index of the green curve is −A 0 /(A 0 + B 0 ). We arbitrarily assign the inhomogeneity index for the curves above the line of equality negative, and below positive. In Germany, per capita CO 2 emissions of large cities are smaller than those of small ones, but the difference seems to be minor [29]. In contrast, per capita CO 2 in the UK emissions remarkably diverge between large and small cities, ranging from 25.6 tonnes per capita in Middlesbrough to 5.4 tonnes per capita in London in 2012, reflecting the impact of industrial base [30,31].
Results
Inhomogeneity of emissions across countries
Interestingly, in Fig 3 developed countries seem to belong to the group where the curves extend to the upper left corner and less developed countries seem to belong to the group where the curves extend to the opposite corner.
G e versus GDP per capita at trans-national level
In order to verify whether there is a systematic relationship between the curve type and the level of countries' economic development, we plotted the values of the inhomogeneity index G e for a large number of countries against the logarithm of GDP Purchasing Power
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Parity (PPP) per capita obtained from the World Bank, an important indicator for economic development.
As observed in Fig 4, the two quantities correlate (with a Pearson correlation coefficient ρ = −0.71, p � 0.01). In general, for developed countries G e tends to have smaller values, and for developing ones it tends to have larger values. Thus, we generalize that in economically developing countries, high population densities are more emission intense and the opposite is the case in economically developed countries.
We repeated the analysis for the FFDAS and the EDGAR data. For reasons of consistency, we also analyze the ODIAC data aggregated to 10 × 10 km 2 resolution. For the three datasets, the resulting G e -values are plotted against the GPD per capita, analogous to Fig 5(a)]. In contrast, for the EDGAR data [ Fig 5(c)] the development dependence vanishes and is even slightly inverted (ρ = 0.29, p � 0.01). Differences between the G e -values of the EDGAR and ODIAC or FFDAS data are most pronounced for developing countries. The difference in these results could be attributed to the poor quality of population census, high demographic dynamics, and insufficient geo-spatial data in developing countries. However, further investigation is needed in order to understand which of these methodological differences factor more with regard to the pronounced dissimilarities in the results. EDGAR relies
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on road networks, population density and agriculture land use data to downscale the national emissions, which renders it more sensible to errors embedded in the proxy datasets. In comparison, FFDAS uses, besides population density, nightlight data to disaggregate emissions.
Moreover, since ODIAC and FFDAS are at least partly based on nightlight data for the subnational disaggregation [25,33,34], one may argue that the development dependence in Fig 5 (a) and 5(b) are simply due to such an effect in the nightlight data. Thus, we also analyzed the nightlight data from the Visible Infrared Imaging Radiometer Suite (VIIRS) Day/Night Band (DNB) data [32] in an analogous way as the emissions data and the results are displayed in Fig 5(d). As can be seen, for nightlight data, we do not see any correlations between G e -values and the GDP per capita. Accordingly, we conclude that the development dependence found in the ODIAC and FFDAS data is not stemming from the nightlight data. Overall, G e -values tend to be negative for nightlights, indicating that locations of low population have a relatively strong contribution.
G e versus GDP per capita at sub-national level
Next we analyzed whether the correlations between G e (for ODIAC) and GDP per capita among countries also appear within a country. Taking China as an example, we disaggerate the national data into provinces. Analogously as for the countries, we calculate cumulative emissions vs. cumulative population and determine the inhomogeneity index at the province level. In Fig 6(a) the G e -values are plotted vs. the corresponding GDP per capita values, as in We performed the corresponding sub-national analysis for the USA on the state level. However, we could not find significant correlations (see S1 Fig in S1 File). Despite this lack of correlations, we find a spatial pattern in the USA. States at the west coast and in the Northeast tend to have larger G e -values. This is also the case for other states at the east coast and in the Midwest. States in the south as well as Montana, North Dakota, South Dakota tend to have more extreme G e -values. Repeating the analysis for the Vulcan data, which might be considered the most detailed data, still no correlations between G e and GDP per capita within the USA are found (S2 Fig in S1 File). However, the analysis does show weak correlations between the G e -values of Vulcan and ODIAC data. This may imply that, albeit based on a relatively simple disaggregation scheme, the ODIAC datasets are able to describe the spatial inhomogeneity of CO2 emissions at a large scale comparably well as a more complex bottom-up based CO2 emission data, particularly in countries where an accurate CO2 inventory is available.
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Robustness of G e
Lastly we checked the robustness of the G e coefficient. We explored different forms of sampling and randomization. In order to check the influence of outliers, we create random subsamples of the ODIAC data. We constructed a set with 50% of the original size by randomly selecting pairs of population and emissions values from the original set without replacement for 1000 iterations. We calculated the cumulative quantities as before and determined the inhomogeneity index. Repeating the procedure we can assess the statistical spread. As observed in Fig 7, the resampling has minor influence on the shape of the curve and the resulting G e -values. For Germany, Fig 7(a), 95% of the realizations lead to G e -values in the range of -0.131 to 0.063, with a median and mean of -0.032, which is very close to the measured value -0.033. The sub-sampled robustness check for the UK led to analogous findings [Fig 7(b)].
Another way to randomize is to shuffle. Since in the analysis we have already sorted the data, we now shuffle only the emissions data and destroy the correlations between emissions and population. Then we perform the whole analysis and obtain cumulative emissions and population curves as well as G e -values. Repeating the procedure we can assess the statistical spreading. The results are also displayed in Fig 7, and we find that the curves for the shuffled data are very different from the original curves which shows that the actual shapes in Fig 3 are due to the correlations between emissions and population. The shape of the curves for the shuffled data differs between Germany and the UK , Fig 7(a) and 7(b). Since shuffling destroys any correlations, the actual form of the curves can be attributed to a combination of the probability distributions of the population and emissions which differ among the countries.
Relation to urban scaling
The analysis of CO 2 efficiency that is carried out here using quasi-Lorenz curves can be related to the urban scaling approach as advocated in [13]. The urban scaling approach aims to
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establish a parametric relationship between the urban population P u of a city and the respective emissions E u . In our analysis we do not analyze urban population and urban emissions explicitly but examine gridded population P g and emission data E g within countries. Since urban areas are usually characterized by high population densities (depending on the pixel size), one could transfer the idea of urban scaling to our setting and assume the scaling relationship E g � P b
g . The case of β < 1 indicates CO 2 efficiency gains with increasing population (density) while β > 1 is associated with efficiency losses. Here it is of interest how the non-parametric quasi-Gini coefficient G e is related to the parametric scaling exponent β.
Generally there is no simple association between β and G e . Empirically, the β coefficient is usually estimated as the slope of a linear regression of the logarithmic quantities. Hence, it depends on the correlations among the logarithmic quantities cor{log P g , log E g } and on the variance of log P g and log E g only. By contrast, G e as a non-parametric estimator depends on the exact form of the joint distribution of P g and E g . However, it is possible to determine a specific expression for the relationship between G e and β under certain conditions. The coefficients are related via [35]
G e ¼ b À 1 2l À b À 1 ;ð1Þ
if P g is Pareto distributed with shape parameter λ > 1 and a scaling relation of the form E g � P b g with β < λ holds exactly. For a detailed derivation see Sec.2 in SI. The formula shows that a scaling coefficient β > 1 is associated with G e > 0. Equivalently, β < 1 implies G e < 0. If E g � P b g holds only approximatively, Eq (1) should still give a reasonable approximation.
Under this scenario, our finding of development dependent G e -values implies a corresponding development dependence of the scaling exponent β. Accordingly, in developing countries large cities are typically less emission efficient and vice versa in developed countries.
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Discussion and conclusions
In summary, we have analyzed the correlations between the spatial distribution of population with CO 2 emissions using high resolution datasets. In order to understand these correlations we employed the quasi-Lorenz curve. The shape of the curve indicates to which extent locations of high or low population emit relatively more or less CO 2 . We characterized the inhomogeneity by a generalized Gini coefficient. For the ODIAC and FFDAS data it depends on the socio-economic development of the considered country (developing countries exhibit relatively more emissions in locations of high population). For the EDGAR data there is no development dependence (overall relatively more emissions in locations of low population). Within China, the development dependence persists for the ODIAC data, but within the USA it vanishes for the ODIAC and Vulcan data. Sub-sampling and shuffling supports the robustness of our analysis.
There is a well-known association between urbanization, economic development, and carbon emissions. However, the quantitative relations behind this association are less understood.
Here we show that also the location of emissions is influenced by the economic development. We conclude that during the course of development a spatial separation of emission source and population happens, based on the results for the ODIAC and FFDAS data. This means to some extent high-emitting sources relocate away from locations of large population. A possible explanation could be an increasing environmental consciousness and adoption of cleaner technologies-a trend similar to the environmental Kuznets curve (EKC). Another possibility could be altering composition of economic sectors from agriculture over emission intensive industry to service [36]. While a majority of national mitigation strategies target specific sectors, our results suggest a complementary spatial perspective to prioritize mitigation actions. Depending on the considered scope of emissions, these would be sparsely populated regions in developed countries and densely populated regions in developing and transition countries. Particular attentions should be paid to the latter, as these countries are projected to become more urbanized in the upcoming decades, which entails further rural-urban migration.
The difficulty in explaining the observed phenomenon of country-specific inhomogeneity indices may be attributable to a complex interplay of human activity on local, country, and international scale which entails more evaluation. Concentration or dispersion of human activities is strongly linked to the extent of urban sprawl. Such structural properties certainly affect both the population and the emissions. Moreover, as mentioned earlier, the proxies used to downscale national level CO 2 emissions and the sectors included while calculating the national level emission data will also impact the spatial inhomogeneity of the origin of CO 2 emissions. In addition, the location of point sources is an important aspect that can hardly be generalized on the national or even international scale. Maybe, a starting point could be a better understanding of the spatial characteristics of CO 2 efficiency. Explaining the presented phenomenon-i.e. development dependent concentration of emissions in locations of high or low population-remains a challenge for future research.
Our results for the ODIAC and FFDAS data are consistent with previously reported findings [14], according to which in developing countries large cities are comparably less efficient in terms of CO 2 emissions, and in developed ones small cities are less efficient. On the one hand, the present study provides stronger empirical evidence, e.g. because it is based on more data and the signatures are more pronounced. On the other hand, the methodology of the present study does not rely on any city definition [37,38] or any assumption about the functional form of the correlations between population and emissions [39].
We argue that the affirmation "large cities are less green" [11] needs to be revised. According to our results only in developing countries large cities are less green. In developed countries, including the USA, the opposite is the case, relatively more emissions stem from small cities. Anyways, we find it misleading to speak about "green cities" in the context of urban CO 2 emissions [7], since greenness usually refers to urban vegetation or metaphorically to pollution (while CO 2 is a colorless gas which as a GHG contributes to global warming).
Certainly, our analysis also has some potential caveats which we want to discuss briefly. The analysis stands and falls with the employed input data, so we cannot exclude to obtain other results if we use other population or emissions data as inputs. Why the EDGAR data leads to different results compared to ODIAC and FFDAS is an interesting problem requiring further research. Moreover, our curves, such as in Fig 3, can have (multiple) crossings with the diagonal, and the index G e cannot capture to a full extent more complex shapes of the curves.
Another aspect that could be addressed in future studies is the role of the population density [6,12,14,40,41]. Here we avoid any discussion about city definitions by simply taking gridded data. Since the grid cells are approximately of equal area, the population count and the density are approximately identical. In order to investigate the influence of the density, a suitable city definition-joining grid-cells-will be necessary.
Supporting information S1 File. (PDF)
Fig 1 .
1Population and CO 2 emissions for the contiguous USA. (a) Gridded Population of the World, version 4, GPWv4 [17] at 1 × 1 km 2 spatial resolution in 2010 and (c) total anthropogenic CO 2 emission from ODIAC data [18] for the same region, year, and resolution. (b) and (d) depict magnified views of population and CO 2 emission for the New York metropolitan region, respectively. Visually, large agglomerations of population coincide with large amounts of emissions. To which extent they relate proportionally is the subject of this paper. https://doi.org/10.1371/journal.pone.0242479.g001
Fig 3
3shows the quasi-Lorenz curves for a few countries. The slopes of the curves depend on per capita emissions. In the case of constant emissions per capita, the cumulative share of population and CO 2 emissions are proportional to each other and follow the diagonal. This is approximately the case for Germany,Fig 3(e). In the case of the USA,Fig 3(a), and the UK,Fig 3(b), the curves are bent to the upper left corner, i.e. the grid cells with small population already include a large amount of emissions (large slope). On the contrary, the curves, e.g., for Uganda,Fig 3(h), and Kenya,Fig 3(g), are bent to the lower right corner. Many grid cells with small population are necessary to include a fair amount of emissions (small slope). Accordingly, curves bent to upper left indicate high per capita emissions in sparsely populated cells and comparably lower per capita emissions in densely populated cells, and vice versa.
Fig 2 .
2Illustration of the inhomogeneity index G e . Quasi-Lorenz curves (solid lines) and the calculation of G e , which is inspired by the Gini coefficient: G e+ = A/(A + B) and G e− = −A 0 /(A 0 + B 0 ). https://doi.org/10.1371/journal.pone.0242479.g002
Fig 4.
Fig 5(b)reveals a very similar development dependence when FFDAS data are applied as for ODIAC [
Fig 3 .
3Quasi-Lorenz curves and corresponding inhomogeneity index G e for selected countries. The country-specific curves are drawn by plotting the accumulated population (in ascending order) on the horizontal axis against the accumulated share of CO 2 emissions of the corresponding grid cells. The panels show the curves for (a) USA, (b) UK, (c) Brazil, (d) France, (e) Germany, (f) China, (g) Kenya, and (h) Uganda. If the curves follow the diagonal, then low and high densities have the same emissions per capita. If the curves are bent to the lower right corner, then cells of small density exhibit relatively low emissions and high population cells exhibit relatively high emissions. Curves in the upper left corner indicate the opposite behavior. The inhomogeneity index G e is positive or negative. It can be seen, that various countries exhibit non-proportional relations between population and emissions. The inhomogeneity index G e seems to be related to the development of the country. https://doi.org/10.1371/journal.pone.0242479.g003
Fig 4 .
4Development dependence of CO 2 -population-inhomgeneity. The inhomogeneity index G e is plotted vs. Gross Domestic Product (GDP) per capita (PPP) for 94 countries on a semi-logarithmic scale. For better readability only the symbols of a sub-set of countries are labeled. As can be seen, the G e correlate with the economic development. The Pearson correlation coefficient between G e and GDP on a logarithmic scale is ρ = −0.71 (p � 0.01). In more developed countries high population densities have lower emissions as low densities. The GDP data were obtained from the World Bank (http://data.worldbank.org), measured in USD of the year 2010. https://doi.org/10.1371/journal.pone.0242479.g004 PLOS ONE Fig 4 but now for provinces. Similar to the country analysis and even more pronounced, we find statistically significant correlations (ρ = −0.87, p-value: <0.01).
Fig 5 .
5Comparison of CO 2 -population-inhomgeneity for different CO 2 datasets and nightlights. (a) ODIAC, (b) FFDAS, (c) EDGAR, (d) nightlights [32]. Each panel is analogous to Fig 4, but for consistency of spatial resolution the underlying ODIAC data in (a) has been aggregated to 10 km resolution. https://doi.org/10.1371/journal.pone.0242479.g005
Fig 6 .
6Sub-national inhomogeneity index G e . We calculated the G e on the province level for China. In (a) the G e -values are plotted against the corresponding province GDP per capita values on a logarithmic scale, analogous to Fig 4. The dashed line indicates the country-level mean G e . Panel (b) shows a map of China where the provinces are color-coded according to the inhomogeneity index G e . It can be seen that the development dependence as found in Fig 4 does also hold on the sub-national scale-at least in China. Provinces with lowest and highest G e -values are Hong Kong and Tibet, respectively. Note, however, that for the USA we do not find sub-national correlations (S1 Fig in S1 File). (Data source of China level-1 administrative boundaries: https://www.naturalearthdata.com/downloads/ 10m-cultural-vectors/10m-admin-1-states-provinces/). https://doi.org/10.1371/journal.pone.0242479.g006
Fig 7 .
7Robustness of G e . In order to illustrate the robustness of the curves in Fig 3 we compare them with curves when the data is subsampled or shuffled. The panels for (a) Germany and (b) UK include the curves for the full samples (Fs), median and envelop for the subsampled data (Sb, green), and the median and envelop for the shuffled data (Sf, orange). The insets show histograms of the corresponding inhomogeneity indices. It can be seen that sub-samples of the data lead to similar results as for the full sample so that the results are not due to individual pixels. The G e values from the shuffling approach −1, as the correlations between population and CO 2 are destroyed. https://doi.org/10.1371/journal.pone.0242479.g007
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AcknowledgmentsWe thank M. Barthelemy for useful discussions.Author ContributionsConceptualization: Bin Zhou, Diego Rybski. Kropp, Diego Rybski.
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| [] |
[
"Effective Actions of IIB Matrix Model on S 3",
"Effective Actions of IIB Matrix Model on S 3"
] | [
"Hiromichi Kaneko [email protected]†e-mailaddress:[email protected]‡e-mailaddress:[email protected] \nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan\n\nIntroduction\n\n",
"Yoshihisa Kitazawa \nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan\n\nDepartment of Particle and Nuclear Physics\nThe Graduate University for Advanced Studies (SOKENDAI)\n305-0801TsukubaIbarakiJapan\n\nIntroduction\n\n",
"Koichiro Matsumoto \nDepartment of Particle and Nuclear Physics\nThe Graduate University for Advanced Studies (SOKENDAI)\n305-0801TsukubaIbarakiJapan\n"
] | [
"High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan",
"Introduction\n",
"High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan",
"Department of Particle and Nuclear Physics\nThe Graduate University for Advanced Studies (SOKENDAI)\n305-0801TsukubaIbarakiJapan",
"Introduction\n",
"Department of Particle and Nuclear Physics\nThe Graduate University for Advanced Studies (SOKENDAI)\n305-0801TsukubaIbarakiJapan"
] | [] | S 3 is a simple principle bundle which is locally S 2 × S 1 . It has been shown that such a space can be constructed in terms of matrix models. It has been also shown that such a space can be realized by a generalized compactification procedure in the S 1 direction. We investigate the effective action of supersymmetric gauge theory on S 3 with an angular momentum cutoff and that of a matrix model compactification. The both cases can be realized in a deformed IIB matrix model with a Myers term. We find that the highly divergent contributions at the tree and 1-loop level are sensitive to the UV cutoff. However the 2-loop level contributions are universal since they are only logarithmically divergent. We expect that the higher loop contributions are insensitive to the UV cutoff since 3-dimensional gauge theory is super renormalizable. *(2.4) J 1i and J 2i obey the following commutation relations:where J 1± = J 11 ± iJ 12 and J 2± = J 21 ± iJ 22 . We can express these operators in terms of the coordinates system (2.2). For example,where we set that p/2 =M 1 , q/2 =M 2 and (−p − q)/2 =M 3 , and omit a zero mode of J 1 , J 2 and J 3 . We have the cutoff scale 2Λ on h, and divide the summation over J 1 , J 2 and J 3 into two sections at Λ as following: | 10.1103/physrevd.76.084024 | [
"https://arxiv.org/pdf/0706.1708v2.pdf"
] | 17,729,538 | 0706.1708 | 7727d5b4b2f4119e6fa554d3c8b4b815c876ce89 |
Effective Actions of IIB Matrix Model on S 3
4 Jul 2007 June. 2007
Hiromichi Kaneko [email protected]†e-mailaddress:[email protected]‡e-mailaddress:[email protected]
High Energy Accelerator Research Organization (KEK)
305-0801TsukubaIbarakiJapan
Introduction
Yoshihisa Kitazawa
High Energy Accelerator Research Organization (KEK)
305-0801TsukubaIbarakiJapan
Department of Particle and Nuclear Physics
The Graduate University for Advanced Studies (SOKENDAI)
305-0801TsukubaIbarakiJapan
Introduction
Koichiro Matsumoto
Department of Particle and Nuclear Physics
The Graduate University for Advanced Studies (SOKENDAI)
305-0801TsukubaIbarakiJapan
Effective Actions of IIB Matrix Model on S 3
4 Jul 2007 June. 2007
S 3 is a simple principle bundle which is locally S 2 × S 1 . It has been shown that such a space can be constructed in terms of matrix models. It has been also shown that such a space can be realized by a generalized compactification procedure in the S 1 direction. We investigate the effective action of supersymmetric gauge theory on S 3 with an angular momentum cutoff and that of a matrix model compactification. The both cases can be realized in a deformed IIB matrix model with a Myers term. We find that the highly divergent contributions at the tree and 1-loop level are sensitive to the UV cutoff. However the 2-loop level contributions are universal since they are only logarithmically divergent. We expect that the higher loop contributions are insensitive to the UV cutoff since 3-dimensional gauge theory is super renormalizable. *(2.4) J 1i and J 2i obey the following commutation relations:where J 1± = J 11 ± iJ 12 and J 2± = J 21 ± iJ 22 . We can express these operators in terms of the coordinates system (2.2). For example,where we set that p/2 =M 1 , q/2 =M 2 and (−p − q)/2 =M 3 , and omit a zero mode of J 1 , J 2 and J 3 . We have the cutoff scale 2Λ on h, and divide the summation over J 1 , J 2 and J 3 into two sections at Λ as following:
Introduction
In our universe, there are many mysteries that remain to be understood such as the selection mechanism of spacetime dimension, gauge groups and matter contents. It is important that we make progress to resolve these problems because we are increasing well informed how our universe is formed. Especially, we focus on the question why the 4-dimensionality of spacetime is selected in our universe.
It is considered that superstring theory provides an effective tool to explain the 4-dimensionality of spacetime. Since superstring theory is a unified theory including the gravity, we may hope to derive all physical predictions from the first principle. Unfortunately, superstring theory suggests 10-dimensional spacetime on the perturbative analysis. We believe that the nonperturbative analysis of superstring theory is needed to explain the 4-dimensionality of our universe. This question may be addressed in the matrix models which are proposed for nonperturbative formulations of superstring theory [1,2]. IIB matrix model is a candidate for the nonperturbative formulation of IIB superstring theory [2,3]. It is defined by the following action:
S IIB = −Tr 1 4 [A µ , A ν ] 2 + 1 2ψ Γ µ [A µ , ψ] ,(1.1)
where A µ is a 10-dimensional vector and ψ is a 10-dimensional Majorana-Weyl spinor field respectively, and both fields are N × N Hermitian matrices. There are considerable amount of investigations toward understanding the 4-dimensionality of spacetime by using IIB matrix model. For example, we may list the following studies: branched polymer picture [4], complex phase effects [5,6] and mean-field approximations [7][8][9]. These studies seem to suggest that IIB matrix model predicts 4-dimensionality of spacetime.
But it is difficult to analyze dynamics of IIB matrix model in a generic spacetime. So we would like to understand general mechanisms to single out the 4-dimensionality of spacetime through the studies of concrete examples. We have successfully constructed fuzzy homogeneous spaces using IIB matrix model [10]. We construct the homogeneous spaces as G/H where G is a Lie group and H is a closed subgroup of G. When we give a background field to A µ , we can examine the stability of this matrix configurations by investigating the behavior of the effective action under the change of some parameters of the background. We have investigated the stabilities of fuzzy S 2 [11], fuzzy S 2 × S 2 [12], fuzzy CP 2 [14] and fuzzy S 2 × S 2 × S 2 [15] in the past. We have found that IIB matrix model favors the configurations of 4-dimensionality and more symmetric manifolds.
In this paper, we investigate the 3-dimensional sphere S 3 configuration. We calculate the effective action of a deformed IIB matrix model by introducing a Myers term up to the 2-loop level on S 3 . S 3 is a simple principle bundle which is locally S 2 × S 1 . It has been shown that such a space can be constructed in terms of matrix models [20]. It has been also shown that such a space can be realized by a generalized compactification procedure in the S 1 direction [21]. We investigate the effective action of supersymmetric gauge theory on S 3 with an angular momentum cutoff and that of a matrix model compactification. The both cases can be realized in a deformed IIB matrix model with a Myers term. We find that the highly divergent contributions at the tree and 1-loop level are sensitive to the UV cutoff. However the 2-loop level contributions are universal since they are only logarithmically divergent. We expect that the higher loop contributions are insensitive to the UV cutoff since 3-dimensional gauge theory is super renormalizable.
The organization of this paper is as follows: In section 2, we review the properties of the S 3 . In section 3, we investigate the effective action of supersymmetric gauge theory on S 3 with an angular momentum cutoff up to the 2-loop level. In section 4, we investigate the corresponding effective action of a matrix model compactification. In section 5,we conclude with discussions.
Derivatives on S 3
In this section, we construct the derivatives on a 3-dimensional sphere: S 3 . First of all, we review basic properties of an S 3 [16][17][18][19]. S 3 is defined by the following condition involving four Cartesian coordinates x n : It is more convenient to introduce the following parametrization:
u = x 1 + ix 2 = cos θ e iφ , v = x 3 + ix 4 = sin θ e iφ ,(2.2)
where 0 ≤ θ ≤ π/2, 0 ≤ φ < 2π and 0 ≤φ < 2π. The S 3 has an SO(4) symmetry while the S 2 has an SO(3) symmetry. It is a well-known fact that the SO(4) is isomorphic to SU(2)×SU(2). Let J 1i and J 2i denote the generators of each SU(2) subgroup respectively, where i = 1, 2, 3.
We define the generators of the SO(4) rotation group by the following linear combinations of the SU(2) generators:
M i = J 1i + J 2i = −iǫ ijk x j ∂ ∂x k , N i = J 1i − J 2i = i x 4 ∂ ∂x i − x i ∂ ∂x 4 . (2.3)
In fact, the operators M i represent the rotations around the x i directions and the operators N i represent the rotations in the x i -x 4 planes. They obey the following commutation relations:
∂ ∂φ + ∂ ∂φ , J 23 = − i 2 ∂ ∂φ − ∂ ∂φ . (2.6)
The raising and lowering operators are
J 1± = 1 2 e ±iφ e ±iφ ± ∂ ∂θ + i tan θ ∂ ∂φ − i cot θ ∂ ∂φ , J 2± = 1 2 e ±iφ e ∓iφ ∓ ∂ ∂θ + i tan θ ∂ ∂φ + i cot θ ∂ ∂φ . (2.7)
The S 3 is isomorphic to the SU(2) group manifold as an element of SU(2) is represented by the Pauli matrices σ i :
U = x 4 1 + i 3 i=1 x i σ i .
When we impose the condition of the unitarity: U † U = UU † = 1 and speciality: detU = 1, we obtain the equation (2.1).
This space is a homogeneous space as we recall the following relation:
SO(4)/SU(2) = SU(2) ∼ S 3 . (2.8)
The homogeneous space is represented as G/H, where G is a Lie group and H is a closed subgroup of G. We can construct the S 3 by using the subgroup SU(2) of the SO(4). Let us choose the basis vectors as follows:
e (1) 1 = (x 4 , x 3 , −x 2 , −x 1 ), e (1) 2 = (−x 3 , x 4 , x 1 , −x 2 ), (2.9) e (1) 3 = (x 2 , −x 1 , x 4 , −x 3 ),
where (a, b, c, d) denotes the Cartesian components of a vector. We find that the components of a vector V is denoted by V
(1) i = V · e (1)
i , and then the derivatives ∂ (1) i along these axes are denoted by
∂ (1) i = −2iJ 1i , ∂ (1) i , ∂ (1) j = 2ǫ ijk ∂ (1)k . (2.10)
The derivatives on the S 3 is constructed by a Lie algebra of the SU(2). On the other hand, we can choose another different basis vectors as follows:
e (2) 1 = (x 4 , −x 3 , x 2 , −x 1 ), e (2) 2 = (x 3 , x 4 , −x 1 , −x 2 ), (2.11) e (2) 3 = (−x 2 , x 1 , x 4 , −x 3 ).
In the same way, we can construct another set of derivatives on S 3
∂ (2) i = 2iJ 2i , ∂ (2) i , ∂ (2) j = −2ǫ ijk ∂ (2)k . (2.12)
In the parameterization (2.2) of the S 3 , we find the Laplacian on the S 3 is as follows:
△ 3 = 1 sin θ cos θ ∂ ∂θ sin θ cos θ ∂ ∂θ + 1 cos 2 θ ∂ 2 ∂φ 2 + 1 sin 2 θ ∂ 2 ∂φ 2 . (2.13)
It is easy to recognize that the Casimir operators act as the Laplacian on the S 3 :
J 2 1 = J 2 2 = − 1 4 △ 3 .
(2.14)
We may consider an eigenvalue equation of the Laplacian on the S 3 as follows:
△ 3 Y (θ, φ,φ) = −λY (θ, φ,φ), (2.15)
where Y (θ, φ,φ) represents the solutions of this equation. We can solve this equation by separating the variables completely: Y (θ, φ,φ) = Θ(θ) exp(imφ + imφ). We find the following differential equation:
1 − z 2 d 2 P (z) dz 2 − 2z dP (z) dz + 1 4 λ − m 2 +m 2 2 1 1 − z 2 + m 2 −m 2 2 z 1 − z 2 P (z) = 0, (2.16)
where z = cos 2θ and P (z) = Θ(θ). The m andm take integers because exp(imφ + imφ) has a periodicity of 2π. If we consider the case m =m ≡ k, we obtain the Legendre's differential equation:
1 − z 2 d 2 P (z) dz 2 − 2z dP (z) dz + 1 4 λ − k 2 1 − z 2 P (z) = 0. (2.17)
We can solve this equation by a series expansion. We obtain Legendre polynomials as the solutions with the eigenvalues: λ = n(n + 2) where n is a positive integer. We can subsequently operate the raising and lowering operators J 1± and J 2± to the Legendre polynomials and obtain the complete solutions of (2.16) for m =m. In this way we find that the solutions of the differential equation (2.16) are the spherical harmonics on the S 3 :
Y n mm (θ, φ,φ) = √ n + 1 e imφ+imφ d n 2 1 2 (m+m), 1 2 (m−m) (2θ) ,(2.18)
where the d functions are given in terms of the Jacobi polynomials [22,23], −n/2 ≤ (m+m)/2 ≤ n/2 and −n/2 ≤ (m −m)/2 ≤ n/2. When the J 13 and J 23 operate on the spherical harmonics Y n mm (Ω) on the S 3 , we obtain eigenvalues (m +m) /2 and (m −m) /2 respectively. They satisfy the following orthogonality condition:
dΩ Y n 1 m 1m1 * (Ω) Y n 2 m 2m2 (Ω) = δ n 1 n 2 δ m 1 m 2 δm 1m2 . (2.19)
We define the volume element dΩ as follows:
dΩ = 1 2π 2 sin θ cos θ dθ dφ dφ. (2.20)
3 Effective action on S 3
In this section, we investigate the effective action of a deformed IIB matrix model on the S 3 background. In order to obtain S 3 as classical solutions of IIB matrix model, we deform the action of IIB matrix model by adding a Myers term as follows:
S = −Tr 1 4 [A µ , A ν ] 2 + 1 2ψ Γ µ [A µ , ψ] − i 3 f µνρ [A µ , A ν ] A ρ ,(3.1)
where f µνρ is the structure constant of SU(2). When we assume ψ = 0, we obtain the equation of motion for A µ :
A µ , [A µ , A ν ] − if µνρ [A µ , A ρ ] = 0. (3.2)
The nontrivial classical solutions of this equation of motion are
A a = t a , other A µ = 0, (3.3)
where t a 's satisfy the Lie algebra. To calculate the effective action of the deformed IIB matrix model, we decompose the matrices A µ and ψ into the backgrounds and the quantum fluctuations as follows:
A µ = p µ + a µ , ψ = χ + ϕ, (3.4)
where p µ and χ are the backgrounds, and a µ and ϕ are the quantum fluctuations. When we expand the action (3.1) around the quantum fluctuations, we also add the gauge fixing term and the Faddeev-Popov ghost term as follows:
S g.f. = − 1 2 Tr [p µ , a µ ] 2 , S F.P. = Tr b p µ , [p µ + a µ , c] ,(3.5)
where c and b are ghosts and anti-ghosts, respectively. In this way we find that the following action:
S = S + S g.f. + S F.P. = −Tr 1 4 [p µ , p ν ] 2 + 1 2χ Γ µ [p µ , χ] − i 3 f µνρ [p µ , p ν ] p ρ − a ν p µ , [p µ , p ν ] + 1 2 {χΓ ν , χ} +χΓ µ [p µ , ϕ] − if µνρ [p µ , p ν ] a ρ + 1 2 [p µ , a ν ] 2 + [p µ , p ν ][a µ , a ν ] − b p µ , [p µ , c] + 1 2φ Γ µ [p µ , ϕ] +χΓ µ [a µ , ϕ] + if µνρ a µ [p ρ , a ν ] + [p µ , a ν ][a µ , a ν ] − b p µ , [a µ , c] + 1 2φ Γ µ [a µ , ϕ] − i 3 f µνρ [a µ , a ν ] a ρ + 1 4 [a µ , a ν ] 2 .
(3.6)
Since we investigate the effective action on the S 3 background, we may substitute the derivatives i ∂ (1) i on the S 3 for the backgrounds as follows:
p i = i ∂ (1) i = αJ 1i , other p µ = 0, χ = 0, (3.7)
where α is a scale factor and i = 1, 2, 3. In [20], the authors have found that the bosonic part A µ of IIB matrix model can be interpreted as differential operators on the principle bundles. One of the goals of our investigations is to obtain deeper understanding of such an interpretation through a concrete example: S 3 . When the backgrounds is a flat space, we expand the quantum fluctuations by a plane wave. Since we consider the S 3 background, it is natural to expand the quantum fluctuations by a spherical harmonics on the S 3 :
a µ = nmm a n µmm Y n mm (Ω) , ϕ = nmm ϕ n mm Y n mm (Ω) ,(3.8)
where a n µmm and ϕ n mm are expansion coefficients. In the same way, ghosts and anti-ghosts fields are expanded by the spherical harmonics on the S 3 :
c = nmm c n mm Y n mm (Ω) , b = nmm b n mm Y n mm (Ω) ,(3.9)
The structure constants f µνρ are given as follows:
f ijk = αǫ ijk , other f µνρ = 0. (3.10)
The gauge fixed action around the backgrounds (3.7) take the following form:
S = −Tr 1 4 [p i , p j ] 2 − i 3 f ijk [p i , p j ]p k + 1 2 [p i , a µ ] 2 − b p i , [p i , c] + 1 2φ Γ i [p i , ϕ] + [p i , a µ ][a i , a µ ] − b p i , [a i , c] + 1 2φ Γ µ [a µ , ϕ] − i 3 f ijk [a i , a j ]a k + 1 4 [a µ , a ν ] 2 .
(3.11)
Using the above actionS, we can evaluate the effective action W of the deformed IIB matrix model on the S 3 background as follows in a background gauge method:
W = − log da dϕ dc db e −S . (3.12)
Firstly, we evaluate the effective action at the tree level
W tree = −Tr 1 4 [p i , p j ] 2 − i 3 f ijk [p i , p j ]p k = − α 4 6 Tr (J 1i ) 2 . (3.13)
In the last step, we substitute the derivatives: i ∂ (1) i = αJ 1i on the S 3 for the backgrounds p i of the deformed IIB matrix model. We evaluate the effective action of the deformed IIB matrix model by taking the continuous limit as follows:
TrX −→ dΩ Ω| X |Ω .
(3.14)
We may consider that this limit corresponds to a semi-classical limit. We evaluate the trace of the Casimir operator as: In the second line, we have used the complete set of the eigenstates for the Laplacian on the S 3 :
Tr (J 1i ) 2 −→ − 1 4 dΩ Ω| △ 3 |Ω
nmm |n, m,m n, m,m| = 1. In the last line, we have used the fact that Y n mm (Ω) = Ω|n, m,m and the degeneracy factors coming from m andm are (n + 1) 2 . Therefore,
W tree −→ − α 4 24 n n(n + 2)(n + 1) 2 . (3.16)
Since n takes a positive integer, the summation over n is formally divergent. It is necessary to impose a cutoff at a some large but finite n. When impose a cutoff at n = l, the tree level effective action is evaluated as follows:
W tree −→ − α 4 24 l n=1 n(n + 2)(n + 1) 2 ∼ − α 4 24 l 5 . (3.17)
Secondly, we evaluate the effective action at the 1-loop level as follows:
W 1−loop = 1 2 Tr log P 2 i δ µν − Tr log P 2 i − 1 4 Tr log P 2 i + i 2 F ij Γ ij 1 + Γ 11 2 , (3.18) where [p i , X] = P i X, [f ij , X] = F ij X, if ij = [p i , p j ] . (3.19)
The first and the second terms are the bosonic contributions while the third term is the fermionic contribution. We have to include a projection operator (1+Γ 11 )/2 in the fermionic part because we consider a 10-dimensional Majorana-Weyl spinor field. We expand the third term of the equation (3.18) into the power series of P i and F ij . In this way, we obtain the leading term of the 1-loop level effective action as follows:
W 1−loop ∼ −Tr 1 P 2 i 2 F ij F ji = Tr 2α 2 P 2 i . (3.20)
When we impose the same cutoff procedure: n ≤ l, we obtain the effective action at the 1-loop level as follows:
W 1−loop −→ 8 l n=1 (n + 1) 2 n(n + 2) ∼ 8 l. (3.21)
Finally, we evaluate the effective action at the 2-loop level due to planar diagrams. We describe the detailed calculations of the 2-loop effective action in appendix B. The effective action at the 2-loop level is:
W 2−loop = 2304 α 4 n 1 n 2 n 3 (n 1 + 1)(n 2 + 1)(n 3 + 1) n 1 (n 1 + 2)n 2 (n 2 + 2)n 3 (n 3 + 2)
.
(3.22)
We impose the same the cutoff procedure of the summations over n 1 , n 2 and n 3 :
l n 1 =1 l n 2 =1 l n 3 =1 (n 1 + 1)(n 2 + 1)(n 3 + 1) n 1 (n 1 + 2)n 2 (n 2 + 2)n 3 (n 3 + 2) ≡ f (l) . (3.23)
where we recall the following selection rules that |n 1 − n 2 | ≤ n 3 ≤ n 1 + n 2 and n 1 + n 2 + n 3 must be even numbers. To the leading logarithmic order, we can analytically evaluate it: We illustrate the comparison between the numerical evaluation and the analytic expression (3.24) in Fig. 1. We find that the analytic expression is valid to the leading logarithmic order as the slope of the two lines are identical. We thus conclude that the 2-loop level effective action is
f (l) ∼ π 2 4 log l,(3.W 2−loop ∼ 576π 2 α 4 log l. (3.25)
In this way we can summarize the effective action with an angular momentum cutoff l up to the 2-loop level:
W ∼ − α 4 24 l 5 + 8 l + 576π 2 α 4 log l. (3.26)
While the tree and 1-loop contributions are highly divergent, the 2-loop contribution is only logarithmically divergent.This result is consistent with the fact that 3-dimensional gauge theory is super renormalizable and we expect that the higher loop contributions are finite. Since 1/α 4 acts as the loop expansion parameter, we need to assume α ∼ O (1). We then conclude that the effective action of the deformed IIB matrix model on the S 3 is stable against the quantum corrections as it is dominated by the tree level contribution.
IIB matrix model compactification on S 3
There is an interesting construction of an S 3 background in matrix models recently. In [21], the authors have concluded that the S 3 is realized by three matrices. They have proved the following two relations between the vacua of different gauge theories:
(i) The theory around each vacuum of super Yang-Mills theory on S 2 (SYM S 2 ) is equivalent to the theory around a certain vacuum of a matrix model.
(ii) The theory around each vacuum of the super Yang-Mills theory on S 3 /Z k (SYM S 3 /Z k ) is equivalent to the theory around a certain vacuum of SYM S 2 with periodic identifications.
They selected the following nontrivial vacua of SYM S 2 :
Φ = µ 2 N 1 α 1 · · · α 1 N 2 α 2 · · · α 2 · · · N T α T · · · α T , (4.1) A θ = 0, A φ = 1 µ (1 − cos θ) Φ for 0 ≤ θ < π 2 + ǫ − 1 µ (1 + cos θ) Φ for π 2 − ǫ < θ ≤ π , (4.2)
where α s 's (s = 1, · · · , T ) parameterize monopole charges:
q st = 1 2 (α s − α t ) ,(4.3)
and all α s 's are different. Additionally we have the following relation
N 1 + · · · + N T =Ñ . (4.4)
The radius of the S 2 is fixed to be 1/µ. The fields in SYM S 2 are split into the blocks of N s × N t rectangular matrices around the above nontrivial vacua. On the other hand, a vacuum of a matrix model is represented as follows:
Y i = −µL i ,(4.5)
where
L i = N 1 L [j 1 ] i · · · L [j 1 ] i N 2 L [j 2 ] i · · · L [j 2 ] i · · · N T L [j T ] i · · · L [j T ] i .
(4.6)
The L i is a reducibleN-dimensional representation of SU (2), and obeys the following commutation relation: (2), and obeys the commutation relation:
[L i , L j ] = iǫ ijk L k , (4.7) where (2j 1 + 1)N 1 + (2j 2 + 1)N 2 + · · · + (2j T + 1)N T =N. (4.8) L [js] i (s = 1, · · · , T ) is the (2j s + 1) × (2j s + 1) spin j s representation of SUL [js] i , L [js] j = iǫ ijk L [js]k .
(4.9)
The Casimir operator of L
[js] i is that
L [js] i L [js]i = j s (j s + 1) 1 2js+1 . (4.10)
This vacuum (4.6) can be interpreted as a set of coincident N s fuzzy spheres with the radii µ j s (j s + 1), where all the fuzzy spheres are concentric. In order to prove the above two relations, they expand the theories around various vacua using appropriate spherical harmonics respectively. The spherical harmonics
Y JMM on an S 3 is relevant for SYM S 3 /Z k (J = n/2, M = (m +m)/2 andM = (m −m)/2 in our convention) where J = 0, 1/2, 1, · · · , M = −J, −J + 1, · · · , J − 1, J andM = −J, −J + 1, · · · , J − 1, J.
The monopole harmonic functionỸ JM qst is used to expand around the background of SYM S 2 , where J = |q st | , |q st | + 1, |q st | + 2, · · · , and M = −J, −J + 1, · · · , J − 1, J. For a matrix model, fuzzy sphere harmonicsŶ (jsjt) JM : the harmonic function on a set of fuzzy spheres with different radii, is used where J = |j s − j t | , |j s − j t | + 1, · · · , j s + j t and M = −J, −J + 1, · · · , J − 1, J. The equivalence (i) is proved when the following conditions are imposed on the parameters of the vacua of SYM S 2 and the vacua of a matrix model:
j s − j t = 1 2 (α s − α t ) = q st , j s , j t −→ ∞. (4.11)
The equivalence (ii) is proved under the following conditions
α s = sk, N s = N, s = 1, · · · , ∞. (4.12)
Additionally, they identify the fuzzy spheres by imposing the periodicity on the (s, t) blocks and by factoring out the overall factor. Combining the equivalences (i) and (ii), they have concluded that the theory around the trivial vacua of SYM S 3 /Z k is equivalent to the theory around the vacua of a matrix model. In order to draw the above conclusion, the following conditions are necessary:
j s − j t = k 2 (s − t) = q st , N s = N, j s , j t −→ ∞, s, t = 1, · · · , ∞. (4.13)
The condition that j s − j t = k(s − t)/2 can be also written as 2j s + 1 = N 0 + ks, where N 0 is a positive integer. The condition that j s → ∞ corresponds to N 0 → ∞. We make use of the work [21] to make a connection between super Yang-Mills theory on an S 3 background and IIB matrix model on an S 3 background. We expand the deformed IIB matrix model action (3.1) around the backgrounds in an analogous way in section 3. We introduce the matrix Y i as the backgrounds of the deformed IIB matrix model:
p i = Y i = −µL i = βL i , other p µ = 0, χ = 0. (4.14)
where β is a scale factor and i = 1, 2, 3. We make a mode expansion of the quantum fluctuations using the fuzzy sphere harmonics:
a (s,t) µ = js+jt J=|js−jt| J M =−J a (s,t) µJM ⊗Ŷ (jsjt) JM , ϕ (s,t) = js+jt J=|js−jt| J M =−J ϕ (s,t) JM ⊗Ŷ (jsjt) JM , (4.15)
where the suffix (s, t) represents the (s, t) block in anN ×N matrix and s, t = 1, · · · , T . The coefficients of the mode expansion a JM are N s × N t matrices and the fuzzy sphere harmonicsŶ When we expand the deformed IIB matrix model up to the forth order of the quantum fluctuations, we obtain the following action:
(jsjt) JM is a (2j s + 1) × (2j t + 1) matrix. Therefore, a (s,t) µ and ϕ (s,t) are N s (2j s + 1) × N t (2j t + 1)S = −Tr stu 1 4 [p i , p j ] 2 − i 3 f ijk [p i , p j ]p k + 1 2 p i , a (s,t) µ 2 − b (s,t) p i , [p i , c (t,s) ] + 1 2φ (s,t) Γ i p i , ϕ (t,s) + p i , a (s,t) µ a (t,u)i , a (u,s)µ − b (s,t) p i , [a (t,u)i , c (u,s) ] + 1 2φ (s,t) Γ µ a (t,u) µ , ϕ (u,s) − i 3 f ijk a (s,t)i , a (t,u)j a (u,s)k + 1 4 a (s,t) µ , a (t,u) ν 2 .
(4.18)
Because of the j st = j s − j t depends only on s − t, we can impose the following condition on the quantum fluctuations and ghosts and anti-ghosts fields:
a (s+1,t+1) µ = a (s,t) µ , ϕ (s+1,t+1) = ϕ (s,t) , c (s+1,t+1) = c (s,t) , b (s+1,t+1) = b (s,t) .
(4.19)
From the above condition, we obtain the condition for the mode expansion coefficients:
a (s+1,t+1) µJM = a (s,t) µJM , ϕ (s+1,t+1) JM = ϕ (s,t) JM , c (s+1,t+1) JM = c (s,t) JM , b (s+1,t+1) JM = b (s,t) JM .
(4.20)
We can rewrite this condition as follows:
a (s,t) µJM = a µJM jst , ϕ (s,t) JM = ϕ JM jst , c (s,t) JM = c JM jst , b (s,t) JM = b JM jst .
(4.21)
We rewrite the action (4.18) by using (4.21). For example, we consider the following term:
− Tr where we use the property of fuzzy spherical harmonics in the appendix C. We set that s−t = p and s = h, so p and h are integers, to make the connection between super Yang-Mills theory on the S 3 and IIB matrix model on the S 3 . In this way, we obtain that
st 1 2 p i , a (s,t) µ 2 = β 2 2 Tr st a (s,t) µ L i • L i • δ µν a (t,s) ν = β 2 2 Tr st J 1 M 1 J 2 M 2 a (s,t) µJ 1 M 1 ⊗Ŷ (jsjt) J 1 M 1 J 2 (J 2 + 1) δ µν a (t,s) νJ 2 M 2 ⊗Ŷ (jtjs) J 2 M 2 = N 0 β 2 2N 0 β 2 2 h ∞ J=0 J M =−J J M =−J a µJMM (−1) −M −M J (J + 1) δ µν a νJMM ,(4.23)
where we set that p/2 =M and a (s,t) µJM = a µJMM . In a similar way, we can evaluate the other terms in the action (4.18). Then, the overall factor h appears in front of the action. After factoring out the overall factor h , we find that the mode expansion of the deformed IIB matrix model around the background Y i is identical to the action of super Yang-Mills on the S 3 .
In what follows, we investigate the relations between the gauge theory on the S 3 background and the matrix models on S 3 . Although they are classically equivalent, the relation is more subtle at the quantum level since the matrix model compactification assumes a definite cutoff procedure. We have evaluated the effective action of the gauge theory on S 3 using the background field method in section 3. In this section, we evaluate an effective actionŴ of the deformed IIB matrix model on the S 3 background Y i :
W = − log da dϕ dc db e −Ŝ ,(4.24)
whereŜ is the action (4.18). Firstly, we evaluate the effective action at the tree level:
W tree = −Tr 1 4 [p i , p j ] 2 − i 3 f ijk [p i , p j ]p k = − β 4 6 Tr T s=1 N s L [js] i 2 = − β 4 6 T s=1 N s j s (j s + 1) (2j s + 1) . (4.25)
We impose the conditions 2j s + 1 = N 0 + s in order to connect super Yang-Mills theory on the S 3 and IIB matrix model on the S 3 . We obtain
W tree = − β 4 24 ∞ h=1 (N 0 + h) 3 − (N 0 + h) ,(4.26)
where we set that s = h, and N s = N = 1 for simplicity. Since h takes integer values, the sum over h is formally divergent. We have a cutoff scale on h at a number 2Λ = T which is equal to the number of (s, t) blocks. When we take a large N 0 limit in such a way that N 0 ≫ Λ, we obtain the following effective action at the tree level:
W tree −→ N 0 →∞ N 0 ≫Λ 2Λ h=1 − β 4 24 N 3 0 . (4.27)
Secondly, we calculate the effective action at the 1-loop level as follows:
W 1−loop ∼ −Tr st 1 P 2 i 2 F ij F ji = Tr st 2β 2 P 2 i = st js+jt J=|js−jt| 2 J (J + 1) (2J + 1) . (4.28)
We obtain thatŴ
1−loop ∼ ∞ h=1 ∞ J=0 J M =−J 2 J (J + 1) (2J + 1) . (4.29)
We have a cutoff such that h < 2Λ, so that the maximal value of J andM are N 0 and Λ, respectively. We divide the summation over J into two parts at the value Λ as the following:
W 1−loop ∼ 2Λ h=1 Λ J=1/2 2 J (J + 1) (2J + 1) 2 + 2Λ h=1 N 0 J=Λ+1/2 2 J (J + 1) (2J + 1) (2Λ + 1) , (4.30)
where we omit a zero mode of J. When we take a large N 0 limit with N 0 ≫ Λ, we obtain the effective action at the 1-loop level:
W 1−loop −→ N 0 →∞ N 0 ≫Λ 2Λ h=1 (8Λ + 8Λ log N 0 ) . (4.31)
Finally, we calculate the effective action at the 2-loop level due to planar diagrams. We describe the detailed calculations of the 2-loop effective action in appendix D. The result iŝ
W 2−loop ∼ 36 β 4 N 0 stu × (2Λ + 1) (2J 1 + 1) (2J 2 + 1) (2J 3 + 1) J 1 (J 1 + 1) J 2 (J 2 + 1) J 3 (J 3 + 1) J 1 J 2 J 3 ΛM 2M3 2 + 3 · 36 β 4 N 0 2Λ h=1 N 0 J 1 =Λ+1/2 N 0 J 2 =Λ+1/2 Λ J 3 =1/2 J 3 M 3 =−J 3 × (2Λ + 1) 2 (2J 1 + 1) (2J 2 + 1) (2J 3 + 1) J 1 (J 1 + 1) J 2 (J 2 + 1) J 3 (J 3 + 1) J 1 J 2 J 3 Λ ΛM 3 2 + 36 β 4 N 0 2Λ h=1 N 0 J 1 =Λ+1/2 N 0 J 2 =Λ+1/2 N 0 J 3 =Λ+1/2 × (2Λ + 1) 2 (2J 1 + 1) (2J 2 + 1) (2J 3 + 1) J 1 (J 1 + 1) J 2 (J 2 + 1) J 3 (J 3 + 1) J 1 J 2 J 3 Λ Λ Λ 2 . (4.34)
When we take the large N 0 limit with N 0 ≫ Λ, we find that the first term of (4.34) is logarithmically infinite while the others are finite. We describe the detailed calculations of (4.34) in appendix D. 9. In this way, we obtain the effective action at the 2-loop level:
W 2−loop −→ N 0 →∞ N 0 ≫Λ 2Λ h=1 576π 2 β 4 1 N 0 log Λ ,(4.35)
We summarize the effective action of the deformed IIB matrix model on S 3 in a matrix model compactification procedure up to 2-loop level:
w ≡Ŵ / h −→ N 0 →∞ N 0 ≫Λ − β 4 24 N 3 0 + 8Λ + 8Λ log N 0 + 576π 2 β 4 1 N 0 log Λ,(4.36)
where we have factored out the overall factor h in front of the effective action. We make the comparison between the effective action (3.26) of the gauge theory on S 3 and the effective action (4.36) of a matrix model compactified on S 3 . Tr over matrices corresponds to the integration over the volume as follows:
1 N 0 Tr −→ dΩ. (4.37)
Therefore, we obtain the following relation by comparing the tree level action:
α 4 = β 4 N 0 . (4.38)
as they are classically equivalent. We note here that the tree level and the 1-loop contributions are highly divergent just like the gauge theory on S 3 case in the preceding section. In fact they are very sensitive to the cutoff procedure. However we find the identical 2-loop contribution as it is only logarithmically divergent. We also expect that the higher loop contributions are finite since the 3-dimensional gauge theory is super renormalizable. With our identification of the inverse coupling constant α 4 = β 4 N 0 , we conclude again that the effective action of the deformed IIB matrix model on the S 3 is stable against the quantum corrections as it is dominated by the tree level contribution.
Conclusions and discussions
In this paper, we have shown that the S 3 background is one of nontrivial solutions of a deformed IIB matrix model. We have evaluated the effective actions of a deformed IIB matrix model on the S 3 background up to the 2-loop level in a two different cutoff procedure. We have found that the effective action of the deformed IIB matrix model on the S 3 background is stable on the condition that the coupling constant is O(1). Since we have evaluated only planar diagrams, our investigation is valid in the large N limit of U(N) gauge theory on S 3 . In section 3, we have used the three derivatives i ∂ (1) i on the S 3 to evaluate the effective action on the S 3 background. In [20], the authors pointed out that the bosonic parts A µ of IIB matrix model can be interpreted as derivatives on a curved space. They claim that IIB matrix model represents generic curved spaces in their interpretation. We believe that their claim is clarified by our concrete investigations on S 3 especially at the quantum level.
In [21], the S 3 is realized by three matrices Y i which are the vacuum configuration of a matrix model. We have also investigated the effective actions on the S 3 background Y i in this generalized matrix model compactification. In both cases, we find that the highly divergent contributions at the tree and 1-loop level are sensitive to the UV cutoff. However the 2-loop level contributions are universal since they are only logarithmically divergent. We expect that the higher loop contributions are insensitive to the UV cutoff since 3-dimensional gauge theory is super renormalizable. We can thus conclude that the effective action of the deformed IIB matrix model on the S 3 is stable against the quantum corrections as it is dominated by the tree level contribution.
We recall here that we have obtained the identical conclusions for the S 2 case. We thus believe that the 2-and 3-dimensional spheres are classical objects in IIB matrix model since the tree level effective action dominates. We can in turn conclude that they are not the solutions of the IIB matrix model without a Myers term.
We have investigated the IIB matrix model in a matrix model compactification procedure by imposing the periodicity on the blocks of the matrices. We believe it is also interesting to investigate the same background without such a condition. Such a case appears to correspond to the quenched matrix model in the flat background case. It is also interesting to investigate higher dimensional spaces such as S 4 or S 3 ×R as they are physically and quantum mechanically more interesting.
Acknowledgments
This work is supported in part by the Grant-in-Aide for Scientific Research from the Ministry of Education, Science and Culture of Japan. We would like to thank S. Shimasaki for discussions during the 1st Asian Winter School on "String Theory, Geometry, Holography and Black Holes".
A Spherical harmonics on S 3
In this appendix, we summarize the spherical harmonics on the S 3 . We adopt the following representation of the spherical harmonics on the S 3 [17]: Y n mm satisfies the following equations:
Y n mm (Ω) = √ n + 1 e imφ+imφ dJ 2 1 Y n mm (Ω) = n 2 n 2 + 1 Y n mm (Ω) , J 2 2 Y n mm (Ω) = n 2 n 2 + 1 Y n mm (Ω) , J 13 Y n mm (Ω) = 1 2 (m +m) Y n mm (Ω) , J 23 Y n mm (Ω) = 1 2 (m −m) Y n mm (Ω) , (A.4)
where J 1i and J 2i are the generators of SU(2) algebra. Y n mm are normalized as follows:
dΩ Y n 1 m 1m1 (Ω) Y n 2 m 2m2 (Ω) = (−1) −m 1 δ n 1 n 2 δ m 1 −m 2 δm 1 −m 2 , (A.5)
The complex conjugate of Y n mm is that
Y n * mm (Ω) = (−1)mY n −m−m (Ω) . (A.6)
The integrals of the product of three spherical harmonics [18,19] can be evaluated as
dΩ Y n 1 m 1m1 (Ω) Y n 2 m 2m2 (Ω) Y n 3 m 3m3 (Ω) = (−1) (n 1 +n 2 +n 3 )/2 (n 1 + 1)(n 2 + 1)(n 3 + 1) × n 1 2 n 2 2 n 3 2 M 1 M 2 M 3 n 1 2 n 2 2 n 3 2 M 1M2M3 , (A.7)
where (· · · ) represents the 3-j symbol of Wigner [22,23].
B Two-loop effective action on background i ∂
(1) i
In this appendix, we evaluate the effective action at the 2-loop level on the background: i ∂
i . We can evaluate the effective action W of the deformed IIB matrix model on the S 3 as follows:
W = − log da dϕ dc db e −S = W tree + W 1−loop + W 2−loop , (B.1) where W 2−loop = exp Tr [p i , a µ ][a i , a µ ] − b p i , [a i , c] + 1 2φ Γ µ [a µ , ϕ] − i 3 f ijk [a i , a j ]a k + 1 4 [a µ , a ν ] 2 1PI , (B.2)
and · · · 1PI represents that we sum only the 1PI (1-Particle-Irreducible) diagrams. There are five 1PI diagrams to evaluate which are illustrated in Fig. 2
B.1 Bosonic propagators
From the quadratic terms for the gauge fields a µ , we can read out the propagators of gauge boson modes a n µmm .
Tr 1 2 a µ P 2 i δ µν a ν −→ dΩ 1 2 n 1 m 1m1 n 2 m 2m2 a n 1 µm 1m1 Y n 1 m 1m1 (Ω) α 2 4 n 2 (n 2 + 2) δ µν a n 2 νm 2m2 Y n 2 m 2m2 (Ω) = 1 2 n 1 m 1m1 n 2 m 2m2 a n 1 µm 1m1 α 2 4 n 2 (n 2 + 2) δ µν (−1) −m 1 δ n 1 n 2 δ m 1 −m 2 δm 1 −m 2 a n 2 νm 2m2 . (B.3)
In the first step, we have taken the semi-classical limit and substituted the quantum fluctuations which are expanded by the spherical harmonics on the S 3 :
a µ = nmm a n µmm Y n mm (Ω) .
(B.4)
Therefore, a n 1 µm 1m1 a n 2 νm 2m2 = 4 α 2 (−1)m 1 n 1 (n 1 + 2) δ µν δ n 1 n 2 δ m 1 −m 2 δm 1 −m 2 .
(B.5)
The propagators of gauge boson fields become
a µ a ν = 4 α 2 nmm (−1)m n(n + 2) δ µν Y n mm (Ω 1 ) Y n −m−m (Ω 2 ) . (B.6)
In the same way, we can read off the propagators of ghost fields.
c n 1 m 1m1 b n 2 m 2m2 = 4 α 2 (−1)m 1 n 1 (n 1 + 2) δ n 1 n 2 δ m 1 −m 2 δm 1 −m 2 , c b = 4 α 2 nmm (−1)m n(n + 2) Y n mm (Ω 1 ) Y n −m−m (Ω 2 ) . (B.7)
We have introduced the quantum fluctuations which are expanded by the spherical harmonics on the S 3 as follows:
c = nmm c n mm Y n mm (Ω) , b = nmm b n mm Y n mm (Ω) . (B.8)
B.2 Contribution from four-point gauge boson vertex (a)
We evaluate the 1PI diagram involving a 4-point gauge boson vertex.
V 4 = 1 4
Tr [a µ , a ν ] 2 = 1 2 Tr (a µ a ν a µ a ν − a µ a µ a ν a ν ) .
(B.9)
We can calculate V 4 using the Wick contraction.
V 4 −→ 1 2 (10 + 10) − 1 2 10 2 + 10 dΩ 1 4 α 2 2 n 1 m 1m1 n 2 m 2m2
(−1)m 1 +m 2 n 1 (n 1 + 2)n 2 (n 2 + 2)
×Y n 1 m 1m1 (Ω 1 ) Y n 1 −m 1 −m 1 (Ω 1 ) Y n 2 m 2m2 (Ω 1 ) Y n 2 −m 2 −m 2 (Ω 1 ) = −45 dΩ 1 dΩ 2 4 α 2 2 n 1 m 1m1 n 2 m 2m2
(−1)m 1 +m 2 n 1 (n 1 + 2)n 2 (n 2 + 2)
×Y n 1 m 1m1 (Ω 1 ) Y n 1 −m 1 −m 1 (Ω 2 ) Y n 2 m 2m2 (Ω 1 ) Y n 2 −m 2 −m 2 (Ω 2 ) δ (Ω 1 − Ω 2 ) . (B.10)
Here we can insert the complete set as follows:
nmm (−1)m Y n mm (Ω 1 ) Y n −m−m (Ω 2 ) = δ (Ω 1 − Ω 2 ) . (B.11)
Therefore, we can get
V 4 −→ −45 123 Ψ * 123 (Ω 2 ) 1 P 2 Q 2 Ψ 123 (Ω 1 ) , (B.12)
where P , Q and R are defined as follows:
P i Y n 1 m 1m1 (Ω) ≡ p i Y n 1 m 1m1 (Ω), Q i Y n 2 m 2m2 (Ω) ≡ p i Y n 2 m 2m2 (Ω), R i Y n 3 m 3m3 (Ω) ≡ p i Y n 3 m 3m3 (Ω). (B.13)
We have introduced the wave functions such that
Ψ 123 (Ω) ≡ dΩ Y n 1 m 1m1 (Ω) Y n 2 m 2m2 (Ω) Y n 3 m 3m3 (Ω) .
(B.14)
123 denotes n 1 m 1m1 n 2 m 2m2 n 3 m 3m3 .
B.3 Contribution from three-point gauge boson vertex (b)
We evaluate the 1PI diagram involving 3-point gauge boson vertices. We can express the contribution corresponding the diagram (b) as follows:
V 3 = 1 2 Tr[p i , a µ ][a i , a µ ] 2 . (B.15)
We can express the result as a compact form:
V 3 −→ 9 2 123 Ψ * 123 (Ω 2 ) 2P 2 − P · Q − P · R P 2 Q 2 R 2 Ψ 123 (Ω 1 ) . (B.16)
We use the following relation: 17) and the momentum conserved relation: P + Q + R = 0. Therefore, we can simplify the result as
P · Q Ψ 123 (Ω) = R 2 − P 2 − Q 2 2 Ψ 123 (Ω) ,(B.V 3 −→ 27 3 123 Ψ * 123 (Ω 2 ) 1 P 2 Q 2 Ψ 123 (Ω 1 ) . (B.18)
We note the relation that
mm (−1)mY n mm (Ω) P i Y n −m−m (Ω) = − mm (−1)m (P i Y n mm (Ω)) Y n −m−m (Ω) . (B.19)
B.4 Contribution from the Myers type interaction (c)
The diagram involving the Myers type interactions is represented as follows:
V M = − 1 18 Tr f ijk [a i , a j ]a k 2 . (B.20)
We get the following result
V M −→ 4α 2 123 Ψ * 123 (Ω 2 ) 1 P 2 Q 2 R 2 Ψ 123 (Ω 1 ) , (B.21)
where we have used the relation: f ijk f ijk = 6α 2 .
B.5 Contribution from the ghost interaction (d)
We evaluate the contribution from the ghost interactions.
V gh = 1 2 Tr [p i , b][a i , c] 2 . (B.22)
The result is
V gh −→ − 1 2 123 Ψ * 123 (Ω 2 ) 1 P 2 Q 2 Ψ 123 (Ω 1 ) .
(B.23)
B.6 Fermion propagator
We can read off the fermion propagator from the quadratic terms in the action (3.11):
Tr − 1 2φ Γ i P i ϕ −→ dΩ − 1 2 n 1 m 1m1 n 2 m 2m2φ n 1 m 1m1 Y n 1 * m 1m1 (Ω) Γ i P i ϕ n 2 m 2m2 Y n 2 m 2m2 (Ω) = − 1 2 n 1 m 1m1 n 2 m 2m2φ n 1 m 1m1 Γ i P i δ n 1 n 2 δ m 1 m 2 δm 1m2 ϕ n 2 m 2m2 , (B.24)
where we have expanded the quantum fluctuations by the spherical harmonics on the S 3 as follows:
ϕ = nmm ϕ n mm Y n mm (Ω) . (B.25) Therefore, ϕ n 1 m 1m1φ n 2 m 2m2 = − 1 Γ i P i δ n 1 n 2 δ m 1 m 2 δm 1m2 . (B.26)
The fermion propagator is
ϕφ = nmm − 1 Γ i P i (−1)mY n mm (Ω 1 ) Y n −m−m (Ω 2 ) . (B.27)
We can further expand the fermion propagator in powers of 1/P 2 as follows:
− 1 Γ i P i = − 1 P 2 + i 2 F ij Γ ij Γ k P k = − 1 P 2 Γ i P i + i 2 1 P 2 2 F ij Γ i Γ j Γ k P k + 1 4 1 P 2 3 F ij Γ ij F kl Γ kl Γ a P a + · · · .
(B.28)
The second term of (B.28) is that
i 2 1 P 2 2 F ij Γ i Γ j Γ k P k = i 2 1 P 2 2 f ijk Γ ijl P k P l − 2i Γ · P + · · · = i 2 1 P 2 2 f ijk Γ ijl P k P l + 1 P 2 2 Γ · P + · · · . (B.29)
Here we have used the formula as follows:
Γ i Γ j Γ k = Γ ijk + δ ij Γ k − δ ik Γ j + δ jk Γ i . (B.30)
While the third term of (B.28) is
1 4 1 P 2 3 F ij Γ ij F kl Γ kl Γ a P a = 1 4 1 P 2 3 −4Γ · P P 2 + · · · = − 1 P 2 2 Γ · P + · · · . (B.31)
Here we have used the formula as follows:
Γ ij Γ kl Γ a = Γ ijkla − δ ik Γ jla + δ il Γ jka − δ ia Γ jkl + δ jk Γ ila − δ jl Γ ika + δ ja Γ ikl −δ ka Γ ijl + δ la Γ ijk − δ ik δ jl Γ a + δ ik δ ja Γ l + δ il δ jl Γ k − δ il δ ja Γ k +δ ia δ jk Γ l − δ ia δ jl Γ k + δ ka δ il Γ j − δ ka δ jl Γ i − δ la δ ik Γ j + δ la δ jk Γ i . (B.32)
In this way, we obtain
− 1 Γ i P i = − 1 P 2 Γ i P i + i 2 1 P 2 2 f ijk Γ ijl P k P l + O 1 P 2 3 .
(B.33)
B.7 Contribution from the fermion interaction (e)
Finally, we evaluate the diagram involving fermion interactions.
V F = 1 8 (TrφΓ µ [a µ , ϕ]) 2 = 1 2 (TrφΓ µ a µ ϕ) 2 . (B.34)
We perform the Wick contractions and evaluate it in the semi-classical limit:
V F −→ 1 2 dΩ 1 dΩ 2 ×tr n 1 m 1m1 1 P 2 Γ i P i − i 2 1 P 2 2 f ijk Γ ijl P k P l + · · · 1 + Γ 11 2 Γ µ ×(−1)m 1 Y n 1 m 1m1 (Ω 1 ) Y n 1 −m 1 −m 1 (Ω 2 ) × n 2 m 2m2 4 α 2 (−1)m 2 n 2 (n 2 + 2) δ µν Y n 2 m 2m2 (Ω 1 ) Y n 2 −m 2 −m 2 (Ω 2 ) × n 3 m 3m3 − 1 P 2 Γ a P a + i 2 1 P 2 2 f abc Γ abd P c P d + · · · 1 + Γ 11 2 Γ ν ×(−1)m 3 Y n 3 m 3m3 (Ω 1 ) Y n 3 −m 3 −m 3 (Ω 2 ) , (B.35)
where tr represents the trace over gamma matrices. Firstly, we evaluate the leading term in the 1/P 2 expansion. The trace of products of gamma matrices in the leading term is as evaluated follows:
trΓ i Γ µ Γ a Γ µ 1 + Γ 11 2 = −8trΓ i Γ a 1 + Γ 11 2 = −128 δ ia . (B.36)
The trace of products of gamma matrices in the next leading term is evaluated as follows:
trΓ i Γ µ Γ abd Γ µ 1 + Γ 11 2 = −8trΓ i Γ abd 1 + Γ 11 2 = 0. (B.37)
The trace of products of gamma matrices in the next next leading term is evaluated as follows:
trΓ ijl Γ µ Γ abd Γ µ 1 + Γ 11 2 = 4trΓ ijl Γ adb 1 + Γ 11 2 = −64 δ ia δ jd δ lb − δ ia δ jb δ ld − δ ja δ id δ lb +δ ja δ ib δ ld + δ la δ id δ jb − δ la δ ib δ jd . (B.38)
In this way, we obtain the following result
V F ∼ 123 Ψ * 123 (Ω 2 ) −64 P · Q P 2 Q 2 R 2 + 32α 2 P 2 Q 2 (P 2 ) 2 (Q 2 ) 2 R 2 Ψ 123 (Ω 1 ) .
(B.39)
B.8 Two-loop effective action
We summarize the 2-loop effective action on the S 3 as follows:
W 2−loop ∼ V 4 + V 3 + V M + V gh + V F = 36α 2 123 Ψ * 123 (Ω 2 ) 1 P 2 Q 2 R 2 Ψ 123 (Ω 1 ) = 2304 α 4
n 1 n 2 n 3 (n 1 + 1)(n 2 + 1)(n 3 + 1) n 1 (n 1 + 2)n 2 (n 2 + 2)n 3 (n 3 + 2)
.
(B.40)
Here we have used the formula (A.7), and evaluated the summations over m 1 , m 2 , m 3 ,m 1 ,m 2 andm 3 .
C Fuzzy sphere harmonics
In this appendix, we summarize the fuzzy sphere harmonics based on the work [21]. The fuzzy sphere harmonics is the eigen function on a set of fuzzy S 2 with different radii. Let us consider a set of linear maps M jj ′ from a (2j + 1)-dimensional complex vector space V j to a (2j ′ + 1)dimensional complex vector space V j ′ , where j and j ′ are non-negative half-integers. M jj ′ is [(2j + 1) × (2j ′ + 1)]-dimensional complex vector space. They constructed a basis of M jj ′ to use a basis of the spin j and j ′ representations of SU(2) as a basis of V j and V j ′ :
|jr j ′ r ′ | , (C.1)
where r = −j, −j + 1, · · · , j − 1, j and r ′ = −j ′ , −j ′ + 1, · · · , j ′ − 1, j ′ . Then, an arbitrary element of M jj ′ : M is represented by
M = rr ′ M rr ′ |jr j ′ r ′ | . (C.2)
They defined linear maps from M jj ′ to M jj ′ by its operation on the basis as follows:
L i • |jr j ′ r ′ | = L i |jr j ′ r ′ | − |jr j ′ r ′ | L i , (C.3)
where L i is a representation matrix (4.6) for aN -dimensional representation of SU (2). The matrix element M rr ′ is transformed under linear maps from M jj ′ to M jj ′ :
(L i • M) rr ′ = L [j] i rp M pr ′ − M rp ′ L [j ′ ] i p ′ r ′ , (C.4) where L [j]
i is the (2j + 1) × (2j + 1) representation matrix of for the spin j representation of SU (2). Additionally, the following identity holds:
(L i • L j • −L j • L i •) |jr j ′ r ′ | = iǫ ijk L k • |jr j ′ r ′ | . (C.5)
The fuzzy sphere harmonics is defined as a basis of M jj ′ :
Y (jj ′ ) JM = rr ′ (−1) 3j−r ′ +2J+M −(ζ+ζ ′ )/2 N 0 (2J + 1) j J j ′ −r m r ′ |jr j ′ r ′ | , (C.6)
where J = |j − j ′ | , |j − j ′ | + 1, · · · , j + j ′ − 1, j + j ′ and M = −J, −J + 1, · · · , J − 1, J. Furthermore, they have introduced a positive integer N 0 as the following parameter:
2j + 1 = N 0 + ζ, 2j ′ + 1 = N 0 + ζ ′ . (C.7)
Therefore, J = |ζ − ζ ′ | /2, |ζ − ζ ′ | /2 + 1, · · · , (ζ + ζ ′ ) /2 + N 0 − 2, (ζ + ζ ′ ) /2 + N 0 − 1. N 0 plays a role of an ultraviolet cutoff scale for the angular momentum. SinceŶ
(jj ′ )
JM is the basis of the spin J irreducible representation of SU(2), the following equations hold
L i • L i •Ŷ (jj ′ ) JM = J (J + 1)Ŷ (jj ′ ) JM , L 3 •Ŷ (jj ′ ) JM = MŶ (jj ′ ) JM . (C.8) Y (jj ′ )
JM is normalized as follows:
1 N 0 TrŶ (jj ′ ) J 1 M 1Ŷ (j ′ j) J 2 M 2 = (−1) M 1 −(j−j ′ ) δ J 1 J 2 δ M 1 −M 2 , (C.9)
where Tr stands for a trace over (2j + 1) × (2j + 1) matrices. The hermitian conjugate ofŶ
(jj ′ ) JM is defined asŶ (jj ′ ) † JM = (−1) M −(j−j ′ )Ŷ (j ′ j) J−M . (C.10)
The product of three fuzzy spherical harmonics can be evaluated as
1 N 0 TrŶ (jj ′ ) J 1 M 1Ŷ (j ′ j ′′ ) J 2 M 2Ŷ (j ′′ j) J 3 M 3 = (−1) −j+j ′ +J 1 +J 2 +J 3 −ζ/2−3ζ ′ /2−ζ ′′ N 0 (2J 1 + 1) (2J 2 + 1) (2J 3 + 1) × J 1 J 2 J 3 M 1 M 2 M 3 J 1 J 2 J 3 j ′′ j j ′ , (C.11)
where (· · · ) and {· · · } represent the 3-j and 6-j symbol of Wigner, respectively [22,23]. In the large N 0 limit, we can obtain that
1 N 0 TrŶ D[p i , a (s,t) µ ][a (t,u)i , a (u,s)µ ] − b (s,t) p i , [a (t,u)i , c (u,s) ] + 1 2φ (s,t) Γ µ [a (t,u) µ , ϕ (u,s) ] − i 3 f ijk [a (s,t)i , a (t,u)j ]a (u,s)k + 1 4 a (s,t) µ , a (t,u) ν 2 1PI , (D.2)
and · · · 1PI implies that we sum only the 1PI diagrams. The evaluation procedure parallels to that of appendix A. There are precisely identical five 1PI diagrams to evaluate which are illustrated in Fig. 2.
D.1 Bosonic propagators
From the quadratic terms for the gauge fields a
J 1 M 1 J 2 M 2 a (s,t) µJ 1 M 1 ⊗Ŷ (jsjt) J 1 M 1 β 2 J 2 (J 2 + 1) δ µν a (t,s) νJ 2 M 2 ⊗Ŷ (jtjs) J 2 M 2 = 1 2 st J 1 M 1 J 2 M 2 a (s,t) µJ 1 M 1 β 2 J 2 (J 2 + 1) δ µν (−1) −M 1 +(js−jt) N 0 δ J 1 J 2 δ M 1 −M 2 a (t,s) νJ 2 M 2 . (D.3)
We expand the quantum fluctuations by the fuzzy spherical harmonics as follows:
a (s,t) µ = JM a (s,t) µJM ⊗Ŷ (jsjt) JM . (D.4)
Therefore, a (s,t) We have introduced the following wave function:
µJ 1 M 1 a (t,s) νJ 2 M 2 = 1 β 2 N 0 (−1) M 1 −(js−jt) J 1 (J 1 + 1) δ µν δ J 1 J 2 δ M 1 −M 2 .Ψ 123 ≡ 1 N 0 Tr stuŶ (jsjt) J 1 M 1Ŷ (jtju) J 2 M 2Ŷ (jujs) J 3 M 3 .
(D.14)
123 denotes J 1 M 1 J 2 M 2 J 3 M 3 .
D.3 Contribution from three-point gauge boson vertex (b)
We evaluate the 1PI diagram involving 3-point gauge boson vertices. We can express the contribution corresponding to the diagram (b) as follows:
V 3 = 1 2 Tr stu p i , a (s,t) µ a (t,u)i , a (u,s)µ 2 . (D.15)
We can express the result as a following compact form:
V 3 = 9 2N 0ˆ 123Ψ † 123
D.4 Contribution from the Myers type interaction (c)
The diagram involving the Myers type interactions is represented as follows:
V M = −
D.5 Contribution from the ghost interaction (d)
We evaluate the contribution from the ghost interactions.
D.6 Fermion propagator
We can read off the fermion propagator from the quadratic term of ϕ in the action (4.18): We can further expand the fermion propagator in powers of 1/P 2 as in appendix A.6.
Tr st − 1 2φ (s,t) Γ i P i ϕ (t,s) = Tr st − 1 2 J 1 M 1 J 2 M 2φ (s,t) J 1 M 1 ⊗Ŷ (jsjt) † J 1 M 1 Γ i P i ϕ (t,s) J 2 M 2 ⊗Ŷ (jtjs) J 2 M 2 = − 1 2 st J 1 M 1 J 2 M 2φ (s,t) J 1 M 1 Γ i P i N 0 δ J 1 J 2 δ M 1 M 2 ϕ (t,− 1 Γ i P i = − 1 P 2 Γ i P i + i 2 1 P 2 2 f ijk Γ ijl P k P l + O 1 P 2 3 .
(D.28)
D.7 Contribution from the fermion interaction (e)
Finally, we evaluate the diagram involving fermion interactions. We can perform the Wick contractions:
V F = 1 2 Tr stu Tr pqr × J 1 M 1 1 N 0 1 P 2 Γ i P i − i 2 1 P 2 2 f ijk Γ ijl P k P l + · · · 1 + Γ 11 2 Γ µ ×(−1) M 1 −(jp−jr)Ŷ (jsjt) J 1 M 1Ŷ (jrjp) J 1 −M 1 × J 2 M 2 1 β 2 N 0 (−1) M 2 −(jr−jq) J 2 (J 2 + 1) δ µνŶ (jtju) J 2 M 2Ŷ (jqjr) J 2 −M 2 × J 3 M 3 1 N 0 − 1 P 2 Γ a P a + i 2 1 P 2 2 f abc Γ abd P c P d + · · · 1 + Γ 11 2 Γ ν ×(−1) M 3 −(jq−jp)Ŷ (jujs) J 3 M 3Ŷ (jpjq) J 3 −M 3 . (D.30)
We can evaluate the traces of products of gamma matrices as in appendix A.7. We obtain the following resultV
F ∼ˆ 123 1 N 0Ψ † 123 −64P ·Q P 2Q2R2 + 32β 2P 2Q2 (P 2 ) 2 (Q 2 ) 2R2 Ψ 123 .
(D.31)
D.8 Two-loop effective action
We summarize the 2-loop effective action on the background Y i as follows:
W 2−loop ∼V 4 +V 3 +V M +V gh +V F = 36β 2 N 0ˆ 123Ψ † 123 1 P 2Q2R2Ψ 123 −→ 36 β 4 N 0 stu J 1 J 2 J 3 (2J 1 + 1) (2J 2 + 1) (2J 3 + 1) J 1 (J 1 + 1) J 2 (J 2 + 1) J 3 (J 3 + 1) J 1 J 2 J 3 M 1M2M3 2 .
(D.32)
Here we have used the formula (C.12), and performed the summations over M 1 , M 2 and M 3 . D.9 Two-loop effective action at large N 0 limit We evaluate the 2-loop effective action in such a large N 0 limit that N 0 ≫ Λ. We impose cutoff scale 2Λ on h, and separate the summation over J 1 , J 2 and J 3 into two parts at a cutoff scale Λ as (4.34). The first term of (4.34) is calculated by using the result of (3.24): where we have set that J 1 = n 1 /2, J 2 = n 2 /2 and J 3 = n 3 /2. Therefore, we can estimate it aŝ
W (4) 2−loop −→ N 0 →∞ N 0 ≫Λ h (const) , (D.50)
We conclude that the 2-loop effective action is given as follows in such a large N 0 limit that N 0 ≫ Λ:Ŵ
2−loop −→ N 0 →∞ N 0 ≫Λ h 576π 2 β 4 N 0 log Λ . (D.51)
Figure 1 :
1The numerical and analytic calculations of the 2-loop corrections.
rectangular matrices. Similarly, ghosts and anti-ghosts fields are expanded by the fuzzy sphere harmonics: JM are the expansion coefficients. As for structure constant, we adopt that f ijk = βǫ ijk , other f µνρ = 0. (4.17)
×
(cos θ) M +M (sin θ) M −M P (M −M, M +M ) · · · , M = (m +m) /2 = −J, −J + 1, · · · , J − 1, J andM = (m −m) /2 = −J, −J + 1, · · · , J − 1, J. Additionally, we make use of the Rodrigues formulas for the Jacobi polynomial: P (M −M, M +M) J−M (cos 2θ) = (−1) J−M 2 J−M (J − M)! (1 − cos 2θ) −M +M (1 + cos 2θ) −M −M × d J−M d cos 2θ J−M (1 − cos 2θ) J−M (1 + cos 2θ) J+M . (A.3)
. The diagrams (a), (b) and (c) represent the contributions from gauge fields, and (c) involves the Myers type interaction. The diagrams (d) and (e) represent the contributions from ghost and fermion fields respectively.
Figure 2 :
2Feynman diagrams of 2-loop corrections to the effective action[12].
of gauge boson fields become a (s,t) µ a (t,s) ν = 1 β 2 N 0 JM (−1) M −(js−jt) J (J + 1) δ µνŶ (jsjt) JMŶ (jtjs) J−M . (D.6)
N
0ˆ 123Ψ † 123 1 P 2Q2R2Ψ 123 , (D.21)where we have used the relation: f ijk f ijk = 6β 2 .
expanded the quantum fluctuations by the fuzzy spherical harmonics as follows:
(s,t) Γ µ a (t,u) ,t) Γ µ a (t,u) µ ϕ (u,s) 2 .(D.29)
Two-loop effective action on background Y i In this appendix, we calculate the effective action at the 2-loop level on the background Y i . We can investigate the effective actionŴ of the deformed IIB matrix model on the background Y i as follows:Ŵ = − log da dϕ dc db e −Ŝ=Ŵ tree +Ŵ 1−loop +Ŵ 2−loop ,
(D.1)
wherê
W 2−loop =
exp Tr
stu
P 2Q2Ψ 123 , (D.12) whereP ,Q andR are defined as follows:P iŶ (jsjt) J 1 M 1 ≡ p i ,Ŷ(jsjt)J 1 M 1 , Q iŶ (jsjt) J 2 M 2 ≡ p i ,Ŷ (jsjt) J 2 M 2 , R iŶ (jsjt) J 3 M 3 ≡ p i ,Ŷ (jsjt) J 3 M 3 .(D.13)
c (s,t)We expand the ghost fields by the fuzzy spherical harmonics as follows:D.2 Contribution from four-point gauge boson vertex (a)We evaluate the 1PI diagram involving a 4-point gauge boson vertex:We can calculateV 4 by performing the Wick contraction.(D.10)Here we have inserted the complete set as follows:
In the same way, we can read off the propagators of ghost fields: References. In the same way, we can read off the propagators of ghost fields: References
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| [] |
[
"Robust Combination of Local Controllers",
"Robust Combination of Local Controllers"
] | [
"Carlos Guestrin [email protected] \nDepartment of Computer Science\nDepartment of Computer Science\nStanford University Stanford\nStanford University Stanford\n94305-9010, 94305-9010CA, CA\n"
] | [
"Department of Computer Science\nDepartment of Computer Science\nStanford University Stanford\nStanford University Stanford\n94305-9010, 94305-9010CA, CA"
] | [] | Finding solutions to high dimensional Markov Decision Processes (MDPs) is a difficult prob lem, especially in the presence of uncertainty or if the actions and time measurements are contin uous. Frequently this difficulty can be alleviated by the availability of problem-specific knowledge.For example, it may be relatively easy to design controllers that are good locally, though having no global guarantees. We propose a non paramet ric method to combine these local controllers to obtain globally good solutions. We apply this formulation to two types of problems: motion planning (stochastic shortest path problems) and discounted-cost MDPs. For motion planning, we argue that only considering the expected cost of a path may be overly simplistic in the presence of uncertainty. We propose an alternative: find ing the minimum cost path, subject to the con straint that the robot must reach the goal with high probability. For this problem, we prove that a polynomial number of samples is sufficient to obtain a high probability path. For discounted MDPs, we consider various problem formulations that explicitly deal with model uncertainty. We provide empirical evidence of the usefulness of these approaches using the control of a robot arm. | null | [
"https://arxiv.org/pdf/1301.2273v1.pdf"
] | 15,527,123 | 1301.2273 | 0921d77df800c167a557a4cb0c41571f3b998ca4 |
Robust Combination of Local Controllers
Carlos Guestrin [email protected]
Department of Computer Science
Department of Computer Science
Stanford University Stanford
Stanford University Stanford
94305-9010, 94305-9010CA, CA
Robust Combination of Local Controllers
178 GUESTRIN & ORMONEIT UA12001
Finding solutions to high dimensional Markov Decision Processes (MDPs) is a difficult prob lem, especially in the presence of uncertainty or if the actions and time measurements are contin uous. Frequently this difficulty can be alleviated by the availability of problem-specific knowledge.For example, it may be relatively easy to design controllers that are good locally, though having no global guarantees. We propose a non paramet ric method to combine these local controllers to obtain globally good solutions. We apply this formulation to two types of problems: motion planning (stochastic shortest path problems) and discounted-cost MDPs. For motion planning, we argue that only considering the expected cost of a path may be overly simplistic in the presence of uncertainty. We propose an alternative: find ing the minimum cost path, subject to the con straint that the robot must reach the goal with high probability. For this problem, we prove that a polynomial number of samples is sufficient to obtain a high probability path. For discounted MDPs, we consider various problem formulations that explicitly deal with model uncertainty. We provide empirical evidence of the usefulness of these approaches using the control of a robot arm.
Introduction
Planning is a central problem in artificial intelligence.
In robot motion planning, for example, one is faced with the task of finding a collision-free path between an initial and a final configuration of a robot. For ex ample, in a chemical plant, we may need to control the flow of chemicals to maintain certain characteristic re actions balanced. Markov Decision Processes (MDPs) have been used extensively as a framework for tack ling such planning problems. Unfortunately, planning problems are frequently intractable due to the high di mensionality of the underlying system1. Many of the 1 Motion planning, for example, is known to be PSPACE-hard [9). On the other hand, problem specific characteristics can sometimes be exploited to tackle high dimensionality and continuous spaces. For instance, it is often possi ble to design simple controllers that work well locally, but have no global performance guarantees. In motion planning, one can use potential fields to design a good controller that can locally avoid obstacles well even in the presence of uncertainty, but cannot overcome local minima, preventing the robot from reaching a global goal [7]. Other examples of local controllers in clude PID-controllers, subsumption architectures and other user-designed local heuristics. A straightforward approach to simplify the solution of difficult planning problems is to combine several of these local controllers in a globally optimal fashion.
Of course, the general idea of combining local con trollers is far from novel. For example, Hauskrecht et al. [4] as well as Parr [8] consider the optimal combination of local controllers in discrete MDPs. In this paper, we propose an approach that is suitable for continuous state spaces. We associate controllers with regions of the state space around anchor points, called milestones. The domain of the local controllers is the Voronoi partition implied by the milestones. We call this procedure nonparametric combination of lo cal controllers, in analogy to nonparametric estima tion. We describe algorithms to compute a global strategy from this type of policy representation in two types of problems: motion planning (stochastic short est path problems) and discounted-cost MDPs. Our algorithm for motion planning is related to the proba bilistic roadmap algorithm proposed for deterministic problems [6,5]. Other algorithms for motion planning under uncertainty, e.g., Preimage Backchaining [7], have mainly focused on two dimensional problems and would be difficult to extend to higher dimensions. The MDP community has also studied this problem as the stochastic shortest path problem [1], however, most al gorithms focus on discrete cases.
A particular focus is on the "robustness" of the solu tion: if the transition probabilities of the MDP must be estimated from data, there is model uncertainty in addition to the uncertainty inherent to the MDP. In the motion planning case, we suggest algorithms that account for model uncertainty by minimizing the path cost subject to a constraint on the minimum allowable probability of success. Experimental results illustrate the qualitative and quantitative characteristics of this type of optimization. Also, for the motion planning case, it is possible to prove that in expansive spaces, a concept used previously in motion planning [5], it is possible to obtain a high probability path with a polynomial number of samples. For discounted-cost MDPs, we suggest that it is also possible to nonpara metrically combine local controllers. We propose a formal framework to account for model uncertainty in this case, and review algorithms to deal with this type of uncertainty.
2
Motion planning using local controllers In this section, we present an application of the con cept of non parametric combinations of local controllers to motion planning. There are several sources of un certainty in this case, e.g., the control of the robot may not be precise, the position of the obstacles may not be known exactly, there may be moving obstacles for which the motion is uncertain, etc. In the presence of uncertainty, it is hence crucial to find a path from a starting to a goal configuration that has a high proba bility of success; that is, a high probability of reaching the goal without collisions. We present an algorithm to compute the path that is effective in this sense in obstacles, self collisions, etc. As we mentioned above, the objective is to find a collision free path between a starting configuration x5 E C and a goal configuration xG E C which minimizes the cost c of the path.
We modify this definition of the deterministic motion planning problem to account for uncertainty as follows:
It may not be possible to obtain a path that is guaran teed to be collision free in the presence of uncertainty.
Instead we content ourselves with a path that has a high probability of success. This is where the notion of local controllers becomes important. We can de sign local controllers that will take the robot from one configuration (milestone) to another nearby configura tion (milestone) with high probability. Thus, we rede fine the objective as one of finding a set of milestones X= (xo, ... ,xn)', x; E C, such that xo = x5, Xn = xG and the local controller can take the robot from x; to Xi+I with high probability. Below we present an al gorithm to address this problem, after defining a local controller.
Local controllers: A local controller a has as pa rameters a starting configuration x; and a goal config uration x3. The objective is to steer from x; towards the local goal Xj· Here the controllers may be either smart and try to avoid obstacles (or even model un certainty locally) or they may be simple and, say, try to maintain a straight line path to the goal.
The local controller evaluates whether it can {locally) connect two configurations x; and Xj using a generative model. That is, it repeatedly starts at x; and simulates the local control as it tries to reach xi. Each simulated transition t has associated with it a cost cl , j, e.g., the distance traveled by the robot along the path from x; to X j · There are two termination conditions for each simulation: First, the controller may hit an obstacle or the search may exceed a time limit which corresponds to a failure of the local connection. Second, it may reach an "endgame region" around the goal x i in which case the controller stops successfully and returns the accumulated cost ctJ. In unsuccessful trials cl , j = 0.
Error probabilities: We carry out a fixed number T of these trials and let Tsuccess be the number of successful trials. Let Ci,j := 1 /Tsuccess "L,'{=I c�,j and Pi,j = Tsuccess/T denote the average costs and the em pirical frequency of success of this experiment, respec tively. As we mentioned above, Pi,j may deviate from the true transition probability Pi,j due to the random sampling, and it is important to account for this model uncertainty when searching robust paths. To express the model uncertainty formally, note that Pi,j can be viewed as a random variable, resulting from averaging independent Bernoulli variables (simulation outcomes) with probability of success equal to the (unknown) true Pi,J. Hence Pi,J is -except for the normalizationbinomially distributed with unknown parameters. We can approximate this binomial distribution with the normal distribution N (Pi,J, Pi,j -PT,J) asymptotically. The straightforward way to define a lower bounds on
Pi, j with reliability of at least say "( is to consider the corresponding tail probability of the normal distribu tion, q.-l b), rescaled by the root of the empirical vari ance iJf, j = T_:l [Tsuccess(l-2 p;,j ) + Tfiz,j]: Ther e sulting (lower) bound for P i,j is:
Ei,j = max{Pi,J-6-; , jcl>-1 ("(), 0}.
We consider the lower bound because we are interested in guaranteeing good performance in the worst-case scenario. Naturally, this perspective may not lead to any feasible solution in the case where we have only very few trials for each connection.
2.1
Finding the path with the least probability of error
We can now generate a graph {V, E), or roadmap, that succinctly represents how the local controller can act globally. The vertices of this graph are the milestones. There is an edge between two milestones x; and X j if Pi,j =j: 0 and each edge is augmented with the values Pi,J and c;,j. The algorithm proceeds as follows:
1. Sample n -1 milestones x1, . . . , Xn-l uniformly from F by taking samples uniformly from C and rejecting those samples that result in collisions with certainty. 2. Try to connect each milestone x; to its k nearest neighbors using the local controller as described above. This gives the cost c;,j and the probability of failure Pi,J. 3. For each new query for a path from a starting position x8 to a goal position xG, include x8 and xG into the set of milestones using the indices 0 and n, repsectively. That is, X = (xo, ... , xn)' with x0 = x8 and Xn = xG as described above.
Also, use the local controller to detmine P o,j and Pi,n· 4. Determine the shortest path between x0 and Xn in the graph induced by X using -log( P i,j ) as the edge weights.
5.
Apply local controller along the shortest path.
Thus, the algorithm we present finds the set of mile stones that gives the highest probability path in the graph from start to goal. As we will prove in Sec. 2.3, it is possible to obtain (with high probability) a path that has high success probability, even with a polyno mial number of milestones. Another issue is the cost of the resulting path. In the next section we present an approptiate tradeoff between the costs and the proba bility of success.
I � --. . ,( a )
2.2
The shortest path subject to robustness constraints In the previous section, we considered the problem of finding a viable connection between two locations in the state space with a low probability of failure. In most practical problems an additional requirement is that the path followed should be "cheap" in a suitable sense, e.g., it is typically desirable to move a robot arm to its goal position without unnecessary detours.
More formally, we need to take into account the cost Ci, J associated with the motion between locations i and j in addition to the probability of failure Pi , i in troduced above. This extension immediately leads to a difficulty with our problem definition: clearly, the tasks of finding a path with a low probability of error and that of finding a path with low costs may be con tradictory. The shortest path must usually run very close to obstacles, e.g., when turning a corner the robot must almost touch the corner to make the shortest path. However, paths closer to obstacles usually im ply higher probability of colliding with them. This is illustrated in Fig. 1, where we are interested in plan ning a path for two degrees of freedom ( dof) holonomic circular robot. However, the positions of the obstacles are not perfectly known, e.g., they could have been measured by a noisy sensor. The noise in this exam ple is gaussian and the shaded areas are a graphical representation of this distribution. In Fig. l(a), we try to find a path with very high probability of suc cess. This is a conservative path that chooses long but safe route. On the other hand, in Fig. l(b), lowering the constraint on the success probability, we obtain a much shorter path taking a "risky" passage between obstacles.
There are several ways to tradeoff costs and proba bility of success mathematically. From a conservative viewpoint, it is more important to reach the goal at all than to reach it cheaply. Hence we cast our algorithm as a constrained optimization problem, where the ob jective is to minimize the path cost and the constraint is that the path chosen must lead to an overall prob ability of success of at least some pre-specified value Pmin· This problem formulation is natural in many practical cases. For example, in motion planning it is very important that the robot arm does not crash into any obstacle and reaches the goal whereas it is less relevant whether or not the path chosen is slightly suboptimal in terms of its length.
Note that the optimization we propose is different from the usual objective in MDP algorithms, which is to minimize the expected cost. In the usual MDP formu lation, one would have to define, in addition to the cost function associated with the problem, a cost for hitting obstacles and another (reward) for reaching the goal. The tradeoff becomes implicit and difficult to control, forcing the user to tune the cost function, balancing the relative weights of these three quantities.
Our new constrained optimization problem can be cast as a restricted shortest-path problem. We modify the fourth step of the algorithm of Sec. 2.1 to incorpo rate this new optimality criteria. Theoretically it is well-known that restricted shortest-path problems are NP-hard, but fully polynomial approximation schemes (FPAS) have been developed for their approximate so lution [3]. In this section, we describe an algorithm that is not FPAS but that is easy to implement and worked very well in our experiments.
We discretize the range [pmin, 1] into S + 1 values us ing q(s) := (Pmin)s/S for s = 0, . . . , S. Intuitively we think of the value V (s, i) as the minimum cost to reach Xi from x0 with a probability of success of at least q(s). For simplicity of exposition, we will assume that all success probabilities are strictly smaller than one (Pi,j < 1). 2 The algorithm computes the value function at each vertex of the graph by a simple dy namic programming algorithm: probability of success. If this is at an acceptable level, then we can set Pmin to a value less than or equal to this maximal probability and run the constrained opti mization algorithm. If the probability of success is not at an acceptable level, we add more milestones to the graph or use a better local controller. The remaining question is how many milestones are needed to obtain high success probability paths. In the next section, we provide a formal answer to this question, which implies that a high probability path can be obtained with a polynomial number of samples.
2.3
Theoretical analysis
In this section, we analyze the algorithm presented in Sec. 2.1 and show that it is possible to obtain a path with high probability of success using a polynomial number of samples. There are two key issues to be considered: first, how many milestones are needed to make it sufficiently likely that the graph contains a path between x5 and xG; second, what is the success probability of the resulting path. As we will show, these two issues can be addressed theoretically by ex tending the analysis of Hsu, Latombe, and Motwani [5] for the deterministic motion planning case.
The analysis of these authors shows that if the con figuration space is "expansive", it is possible to find a path from the starting point to the goal using a poly nomial number of samples [ 5, Hsu et al.]. In the rest of this section, we extend their formal framework to problems with uncertainty.
First, we extend some definitions. Let a p..good ex pansive space ( E, a:, /3, p) under some local controller a be defined as follows: By analogy to the deterministic case, we let R.p(x) denote the region reachable by the local controller starting at x, i.e., the set of points that can be reached from the configuration x using the lo cal controller a with probability of success of at least p. Let J.t(A) denote the volume of the set A. Now , we need to extend the concept of £-goodness, which im plies that the controller can reach at least some pro portion of the free space with high probability: Next, we need to define the lookout of a set S:
Definition 2.1 Let E E (0, 1] beLOOKOUT$(S) = {x E SIJ.-L(R.p(x) \ S) � {3.J-L(F' \ S)}.
Intuitively, this quantity measures the proportion of S that can reach many points outsideS. Finally, we can extend the concept of (o:, /3,p)-expansiveness: independent uniformly sampled milestones from the free space :F; then, with probability at least 1 -'Y, the roadma.p will contain a path between any two mile stones in the same connected component with proba bility of success of at least jJ3fl3+ l. Proof sketch: the arguments are similar to those for the deterministic motion planning case of Hsu el al. [5].
Our extension of the definition of expansiveness to the stocha.'ltic case allows us, using similar arguments, to obtain results equivalent to Lemmas 1 and 2 in [5]. Our final result, which is analogous to the determin istic case addressed by Hsu et al. in Theorem 1, uses a similar linking sequences argument [5]. As a result of this argument, any two milestones q0 and qm+l in the same connected component will, with probability 1 -'Y, be connected by a path that contains at most m � 3/.8 milestones { q0, q1, ... , qm, qm+l}. Further more, for each i, qi+l E n;.(qi), i.e., the local controller must be able to navigate from q; to q;+1 with success probability of at least jJ. Implying that the complete path has success probability of at least jJ3ff'+l. I Note that we do not impose any explicit smoothness constraints on the transition density in this work. This is by contrast to an algorithm by Rust [10] for the ap proximate solution of continuous state, discrete actions and discrete time discounted MDPs, which depends crucially on the Lipschitz continuity of the transition density. Rust's algorithm is more effective when the Lipschitz constant is small, i.e., there is a lot of ran domness in the system. On the other hand, our al gorithm becomes more efficient as the controllers be come more effective (deterministic) locally. We believe that, in most practical robotics applications, local con trollers are close to deterministic, thus, making our algorithm more suitable.
In summary, in the context of expansive spaces, it is possible to obtain, with high probability, a path be tween two milestones using a number of samples that is polynomial in (1/t:,1/a,1/,8,ln1fr). In particu lar, the number of samples needed depends only on the number of "critical" dimensions of the problem, as was argued by Hsu et al. [5]. This number is fre quently relatively small in practice, allowing for a good performance of the algorithm even in reatively high dimensional spaces.
Discounted MDPs
In the previous section, we suggested a robust algo rithm for motion planning using local controllers that minimizes the cost of a path subject to a reliability constraint. The idea of using local controllers can be very useful not only for motion planning, but also for discounted MDPs.
As an example, consider the task of riding a bicycle.
Here we may have different controllers at our disposal that show different degrees of reliability in different re gions of the state-space. For instance, one controller may be better suited for "normal" riding scenarios and another controller may work well in "emergency situ ations". A globally optimal strategy would identify these scenarios and assign the local controllers accord ingly to achieve an overall good performance.
3.1
Evaluating combinations of local controllers For discounted MDPs, we assume we have several dif ferent local controllers, indicated by the variable a. The task is then to choose one controller at each state in a way that minimizes the overall total expected dis counted cost. In this problem formulation, future re wards are discounted using a discount factor a: E [0, 1).
The use of local controllers (macro-actions) has been explored in discrete MDPs, e.g. [4,8]. Our approach circumvents this problem by implicitly defining the boundaries using a set of milestones: We assume that a set of local policies is given, and we divide the state space into local regions using a set of milestones { x0, ... , Xn}, as in Section 2. These regions are defined in terms of a Voronoi partition of the state space; that is, each location is assigned to the nearest milestone. Forthermore, each region can employ one of several local controllers and the objective is to find the optimal assignment of controllers to regions.
We explore the connectivity between the milestones by simulating each controller a starting from uniformly generated starting points in :F. Each point is associ ated with its nearest milestone, xi, and the simulation is terminated as soon as the trajectory is closer to an other milestone Xj than to Xi· We let I(i,j) be the indicator that a simulation starting in the vicinity of Xi terminates in the vicinity of Xj, and we let r de note the corresponding "stopping time" . Our goal is to approximate the discounted transition probability Note that these estimates are heuristic, because we stop the simulation at the boundary of the neighboring cell, as described above. As a consequence, there arises a bias because we ignore information regarding the motion within the neighboring cell. 3
In our second step, we use the estimates Fa and Ca to determine the optimal assignment of controllers to regions. A straightforward approach to compute the solution of the discrete-state, discrete-action MDP de fined by the estimates Pa and Ca for this purpose. How ever, like in the motion planning case it is very impor tant to account for the random variation in Fa(i,j) due to the random simulation. In the next section, we suggest approaches to deal with this type of model uncertainty in the context of discounted MDPs.
3.2
Robust solution of the approximate MDP
We emphasized above that a drawback of any simula tion based algorithm for planning under uncertainty, e.g. the algorithms of Sections 2 and 3.1, is the model uncertainty that may produce misleading results. It is important to account for this uncertainty as it was shown in Sections 2.1 and 2.2 for motion planning. The case of discounted MDPs is mathematically more involved, and we give an overview of useful algorithms in this section.
Algorithms for discounted MDPs with model uncer tainty have been discussed in some detail in the op erations research and computer science literatures. A comprehensive mathematical treatment of MDPs with "imprecise transition probabilities" can be found in a paper by White and Eldeib [11]. These authors are concerned with the case where we have a number of linear constraints on the ith transition probability vec tor Fa(i, ·). Givan et al. [2] specialize this method to express model uncertainty in the form of elementwise bounds on the parameters ca(i) and Fa(i,j). They also derive modifications of the policy iteration and value 3 iteration algorithms for this generalized class of MDPs and highlight their relationship to stochastic games.
Applied to the current context, their algorithm amounts to defining elementwise bounds Pa(i,j) and P a (i,j) by analogy to Section 2, as well as correspond ing bounds on the cost function, Q, ( i) and Ca ( i). . J
This algorithm converges to a unique fixed point that characterizes the optimal solution, defined using a suit able notion of robustness [2].
In practice, a potential difficulty with the algorithm by Givan et al. is that, depending on the bounds on Pa and ca, the range of possible value functions that are consistent with V(i) and V(i) may be too big and, as a consequence, these bounds may not be sufficiently in formative to discriminate between good and bad poli cies. Hence it is important to define the sets Pa and Ca in a way that is as restrictive as possible in practice.
In particular, the elementwise bounds introduced above may be too loose in many applications, and there are many ways to arrive at tighter constraints at the cost of additional mathematical sophistication and computational effort. From a statistical perspec tive, a natural way to define those constraints is by formulating likelihood functions that involve Ca and Pa as parameters, and then to use the sublevel sets of the likelihood. While the constraints derived in this manner are frequently convexand hence amenable to numerical optimization -they can still be diffi cult to deal with computationally. A reasonable com promise is to derive spherical bounds using a Laplace approximation of the likelihood: We consider the sub level sets of a second-order Taylor expansion of the log-likelihood, which can be described as spheres with respect to the inverse Fisher information matrix n. In the case of the transition matrix Pa, the resulting con straints on the ith row can be written in the form: where llzlln = � is the D-weighted Eucledian norm.
Pa(i, · ) E Pa,i = { Fa(i, ·) + 8w I llwlln � 1, w'n = 0},
A robust algorithm is to find a policy that is optimal with respect to the most adverse transition probabil ity in this class, by analogy to the interval value it eration algorithm described above. For this purpose, note that the equation above amounts to a set of linear equality and quadratic inequality constraints. Evaluating maxP.EP. {ca(i) + L j Pa(i,j)V(j) } in the up date equation for v' (i) in the interval value iteration algorithm thus requires solving a linear program with (convex) quadratic constraints. The iterated solution of this program gives a robust estimate of the value function. It can be used to derive an approximately optimal strategy in a straightforward manner.
To summarize, our overall algorithm consists of fi rst sampling a set of milestones, then estimating the tran sition probabilities between each neighboring region using simulation, and finally solving a "robust" MDP to derive a suitable global strategy. There are several ways to define the robust MDP that vary in their al gorithmic complexity and the expressiveness of their solution. A useful heuristic is to start from relatively simple (elementwise) bounds in practice, and to resort to more sophisticated methods only as additional pre cision is needed.
Experiments
In this section, we present an implementation of the motion planning algorithm described in Section 2. The test cases involve uncertainty in the position of obstacles. This type of uncertainty could result from noisy sensor data, for example. The noise in these examples is gaussian and the shaded areas are a graphical representation of the uncertainty.
Since it is difficult to represent paths in a paper, we strongly encourage the reader to see animated paths and other examples on the project website:
robotics.stanford.edu/-guestrin/Research/RobustLocalControl/.
We have not tried to do any local optimization or smoothing of the paths in these experiments. This would make paths somewhat shorter and more natural, but might hide certain characteristics of the algorithm. To navigate between nearby milestones, x; to Xj, we used a simple local controller that attempts to traverse a straight line path from Xi to x J.
The first example, shown in Fig. 2, deals with the centralized control of two circular holonomic robots. The goal is to minimize the total costs while avoid ing the obstacles at the same time. Accordingly, the paths in Fig. 2 have qualitatively different behaviors as we relax the constraints on the minimum allow able success probability. For the highest success prob ability ( Fig. 2(a)), both robots must take a long path around the obstacles to avoid regions of likely colli sions. As we relax the constraint on the success prob ability ( Fig. 2(b)), the planner is able to route one robot through a risky (high uncertainty in the position of obstacles) short cut, making the overall path length shorter. Finally, as we relax the constraints further, both robots take the risky short cut. The quantitative differences are:
Example Path length Prob. success Fig. 2(a) 3.53 0.99 Fig. 2(b) 2.79 0.54 Fig. 2(c) 1.53 0.13
The second example concerns the control of a 5 de grees of freedom (do£) robot arm. The most probable path (Fig. 3) still has to pass through an area of high uncertainty; this is a narrow passage that cannot be avoided. However, it is able to avoid other uncertain areas. As we relax the constraint on the failure prob abilities, we can obtain shorter paths (Fig. 4) at the cost of entering areas of high risk. Quantitatively:
Example Path length Prob. success Fig. 3 10.07 0.95 Fig. 4 (a) 9.23 0.81 Fig. 4 (b) 7.81 0.60
To illustrate the degree of uncertainty present in the problem, the histogram in Fig. 5 shows the distribu tion in values of Pi,i determined during the simulation. Note that 70% of the edges have probability of suc cess less than 1, demonstrating the relevance of dealing with uncertainty explicitly.
Discussion
We presented new algorithms for planning under un certainty in continuous state and action spaces, which are based on the combination local controllers. For motion planning, we argue that expected cost -the usual objective function for planning using MOPs may be inappropriate in situations where the cost of a path is dominated by the need to eliminate the prob ability of error. We propose an alternative, minimiz ing path cost subject to a constraint on the minimum acceptable probability of reaching the goal. This con cepts leads to an algorithm that is guaranteed to yield a robust (high success probability) solution path. Ex periments show our problem formulation successfully trades off risk and reward in two planning scenarios.
Also for discounted MDPs model uncertainty is an im portant issue, and we suggest a similar approach to combine local controllers robustly in this context. In detail, we suggest a way to quantify the model uncer tainty mathematically and review techniques to solve the resulting robust MDP, varying in mathematical sophistication and expressiveness of the solution.
of planning algorithms are fur ther complicated by the presence of uncertainty. The control of the robot may be imprecise or the sensors measuring chemical levels may be noisy. Furthermore, the state and the action spaces of MDPs are frequently continuous; few algorithms can deal with this situa tion.
Figure 1 :
1Single robot example: (a) setting success proba bility Pmin = 0.99 yields a path of length 1.75; (b) loweringPmin to 0.51 the path is 1.08 long.
1. V(s, 0 )
0:= 0 for s = 0, ... , S. 2. V(O,i) := oo fori= 1, ... ,n. 3. For s := 1, . . . , S; For j := 1, ... , n: V(s,j) := min { �i�k�p:, r :�ts(j) { V (l s -Sk,jJ, k) + Ck,j } } 'where Sk,j := S logpk,i/ logPmin · In practice, we suggest applying first the algorithm described in the previous section to find the maximal 2Probability one transitions can be dealt with by adding an extra loop to the algorithm.
a constant. A free space F is E:-p-good if for every connected component :F' � F, for every x E :F', J.-L(R..p(x)) � E.J.-L(F').
Definition 2. 2
2Let a: E (0, 1] and /3 E (0, 1] be con stants, a free space :F is (o:,j3,p)-expansive if for ev ery connected component :F' � :F, for every S � P, J.t(LOOKOUT$ (S)) � o:.J-L( S) .These extensions of the original definitions of expan sive spaces allow us to prove the following theorem: Theorem 2.3 For any r E (0, 1], let a roadmap be generated from a set M of 2f8ln(8/wr)/m+3/.Bl +2
P
;:(i,j) = Er[O:r-II(i,j)], where the term ar-l ac counts for the different stopping times. For this pur-pose, suppose we have T; trial simulations that start in the vicinity of each state x; for each controller a.Let ri�J be the stopping time of the kth simulation starting from x; that ended at Xj. Furthermore, let the discounted cost associated with the path generated during the kth trial be denoted as c� . Then straight forward estimates of the transition costs and of the discounted transition probabilities are defined accord ing to:T; ca(i) :::: 1 /T; I>�; and Fa(i, j) :::: 1/T; L a.,-'�i-1• k=l .,-k
.
For mally, these bounds translate into the sets of feasible transition matrices and cost vectors 'Pa = {Pa)E a � Pa � Pa} and Ca = {cak a � Ca � ca}, where the'�' is to be interpreted elementwise. In addition, they define interval value functions in terms of elementwise upper and lower bounds V(i) and V(i), respectively. These value functions are updated recursively in a variant of the value iteration algorithm as follows: -min max {ca( i) + L Pa(i,j)V(j) }; {fa( i) + L P a(i,j)V(j) } . a P.EP.
Figure 2 :
2Two robots, roadmap with 2000 milestones, considering 10 nearest neighbors: (a) most conservative path (pmin = 0.99); (b) as more risk is allowed CPmin = 0.54) one robot enters uncertain narrow passage; (c) both robots use narrow passage in riskiest situation (Pmin = 0.13).
Figure 3 :Figure 4 :
34Five dof arm, roadmap with 1000 milestones, considering 10 nearest neighbors: highest probability path (Pmin = 0.95) requires traversal risky passage (3rd image) and detour (4rd image). (a) shorter path (Pmin = 0.81) requires entering risky area two more times; (b) much shorter path is obtained by entering more areas of high collision probabilities (pmin = 0.60).
Figure 5 :
5Distribution of edge probabilities Pi,j.
Motion planning problem: A deterministic mo tion planning problem is defined by a compact config uration space C, that is the set of all possible configu rations x of the robot, and by an open subset :F � C that is free in the sense that the configurations in :FSection 2.1. A conflicting goal is to minimize the costs
of the path, e.g. the distance traveled, time and energy
spent on the way from the initial to the goal configu
ration. A conservative mindset suggests placing more
emphasis on the probability of error than on the cost
of a path. Hence a sensible approach is to minimize
the costs using the probability of error as a robustness
constraint (Section 2.2). Section 2.3 gives a theoretical
result regarding the probability of finding a path with
low probability of error. Prior to discussing these re
sults, we first formalize the basic building blocks for a
motion planning problem under uncertainty with local
controllers:
are physically possible, thus excluding collisions with
The algorithm by Hauskrecht et al. assumes some partitioning of the state space into regions, it is then able to generate lo cal policies within regions and combines them into aglobal policy. Unfortunately, extending this concept
to continuous problems is difficult. The boundaries
between regions can be complicated to describe. Fur
thermore, in ad-dimensional continuous state space,
the boundaries define a (d-1) -dimensional man
ifold which is difficult to deal with computationally.
GUESTRIN & ORMONEIT UA12001
AcknowledgmentsWe would like to thank Daphne Koller for help and sup port. This work was supported by the ONR under the MURI program "Decision Making Under Uncertainty", by the ARO under the MURI program "Integrated Approach to Intelligent Systems", and by the Sloan Foundation.
Neuro-Dynamic Pro gramming. D Bertsekas, J Tsitsiklis, Athena Scientific, Mass. D. Bertsekas and J. Tsitsiklis. Neuro-Dynamic Pro gramming. Athena Scientific, Mass., 1996.
Bounded-parameter Markov decision processes. R Givan, S Leach, T Dean, Artif. Intelligence. R. Givan, S. Leach, and T. Dean. Bounded-parameter Markov decision processes. Artif. Intelligence, 2000.
Approximation schemes for the re stricted shortest path problem. Refael Hassin, Mathematics of Oper ations Research. 171Refael Hassin. Approximation schemes for the re stricted shortest path problem. Mathematics of Oper ations Research, 17(1):36 -42, Feb. 1992.
Hierarchical solution of markov de cision processes using macro-actions. M Hauskrecht, N Meuleau, L Kaelbling, T Dean, C Boutilier, UAI-98. M. Hauskrecht, N. Meuleau, L. Kaelbling, T. Dean, and C. Boutilier. Hierarchical solution of markov de cision processes using macro-actions. In UAI-98, 1998.
Path plan ning in expansive configuration spaces. D Hsu, J.-C Latombe, R Motwani, Int. JourD. Hsu, J.-C. Latombe, and R. Motwani. Path plan ning in expansive configuration spaces. Int. Jour.
. Comput. Geometry 8 App. 9Comput. Geometry 8 App., 9(4-5):495-512, 1999.
Probabilistic roadmaps for path planning in high-dimensional configuration spaces. L E Kavraki, P Svestka, J C Latombe, M Over, IEEE Trans. on Robotics and Automation. 124L.E. Kavraki, P. Svestka, J.C. Latombe, and M. Over mars. Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. on Robotics and Automation, 12(4):566-580, 1996.
Robot Motion Planning. Jean-Claude Latombe, Jean-Claude Latombe. Robot Motion Planning.
. Kluwer, Dordrecht, The NetherlandsKluwer, Dordrecht, The Netherlands, 1991.
Flexible decomposition algorithms for weakly coupled markov decision problems. R Parr, UAI-98. R. Parr. Flexible decomposition algorithms for weakly coupled markov decision problems. In UAI-98, 1998.
Complexity of the mover's problem and generalizations. J H Reif, 20th IEEE Symp. on Foundations of Computer Sc ience. Puerto RicoJ. H. Reif. Complexity of the mover's problem and generalizations. In 20th IEEE Symp. on Foundations of Computer Sc ience, Puerto Rico, Oct. 1979.
Using randomization to break the curse of dimensionality. John Rust, Econometrica. 653John Rust. Using randomization to break the curse of dimensionality. Econometrica, 65{3):487 -516, 1997.
Markov decision pro cesses with imprecise transition probabilities. C C White, H Eldeib, Opera tions Research. 424C. C. White and H. K Eldeib. Markov decision pro cesses with imprecise transition probabilities. Opera tions Research, 42(4):739-749, 1994.
| [] |
[
"Contemporary continuum QCD approaches to excited hadrons",
"Contemporary continuum QCD approaches to excited hadrons"
] | [
"Bruno El-Bennich \nLaboratório de Física Teórica e Computacional\nUniversidade Cruzeiro do Sul\nRua Galvão Bueno 86801506-000São PauloSPBrazil\n\nInstituto de Física Teórica\nUniversidade Estadual Paulista\nRua Bento Teobaldo Ferraz 27101140-070São PauloSPBrazil\n",
"Eduardo Rojas \nInstituto de Física\nUniversidad de Antioquia\nCalle 70, no. 52-21MedellínColombia\n"
] | [
"Laboratório de Física Teórica e Computacional\nUniversidade Cruzeiro do Sul\nRua Galvão Bueno 86801506-000São PauloSPBrazil",
"Instituto de Física Teórica\nUniversidade Estadual Paulista\nRua Bento Teobaldo Ferraz 27101140-070São PauloSPBrazil",
"Instituto de Física\nUniversidad de Antioquia\nCalle 70, no. 52-21MedellínColombia"
] | [] | Amongst the bound states produced by the strong interaction, radially excited meson and nucleon states offer an important phenomenological window into the longrange behavior of the coupling constant in Quantum Chromodynamics. We here report on some technical details related to the computation of the bound state's eigenvalue spectrum in the framework of Bethe-Salpeter and Faddeev equations.Introduction: excited states as an eigenvalue problemA great deal of research activity in hadron physics is concerned with hadron structure and revolves around two fundamental questions: what constituents are the hadrons made of and how does Quantum Chromodynamics (QCD), the strong interaction component of the Standard Model, produce them? These are simple questions which, however, may not entail simple answers. To understand the measurable content of QCD, spectroscopy is a valuable and time-honored tool -suffice it to mention the inestimable progress made in the computation of atomic or molecular spectra and subsequent comparison with experiments that lead to a deeper understanding of Quantum Electrodynamics.The same is true for QCD: if confinement is related to the analytic properties of QCD's Schwinger functions, then light-quark confinement ought to be understood by mapping out the infrared behavior of the theory's universal β function. Obviously, this cannot be possibly achieved in perturbation theory. A nonperturbative continuum approach to QCD is provided by Dyson-Schwinger equations (DSE) which relate the theory's β function to experimental observables[1]. Therefore, comparison between DSE predictions embedded in bound-state calculations and observations of the hadron mass spectrum as well as of elastic and transition form factors can be used to study the long-range behavior of QCD's interaction. As the properties of excited hadron states are considerably more sensitive to the long-range behavior of the strong interaction than those of ground states[2][3][4], excited mesons and nucleons[5]are an important source of information and complement our understanding of the strong interaction in light mesons [6-9] and in heavy-light mesons with disparate energy scales[10][11][12][13].Of course, the properties of these hadrons can only be understood, and the functional behavior of the β function be inferred therefrom, by studying the quark's DSE in conjunction with quark-antiquark a | 10.1051/epjconf/201611305003 | [
"https://www.epj-conferences.org/articles/epjconf/pdf/2016/08/epjconf_fb2016_05003.pdf"
] | 119,227,069 | 1509.02919 | 95dffecf723b353e42c13dac9b06428a98e90b1d |
Contemporary continuum QCD approaches to excited hadrons
Bruno El-Bennich
Laboratório de Física Teórica e Computacional
Universidade Cruzeiro do Sul
Rua Galvão Bueno 86801506-000São PauloSPBrazil
Instituto de Física Teórica
Universidade Estadual Paulista
Rua Bento Teobaldo Ferraz 27101140-070São PauloSPBrazil
Eduardo Rojas
Instituto de Física
Universidad de Antioquia
Calle 70, no. 52-21MedellínColombia
Contemporary continuum QCD approaches to excited hadrons
Amongst the bound states produced by the strong interaction, radially excited meson and nucleon states offer an important phenomenological window into the longrange behavior of the coupling constant in Quantum Chromodynamics. We here report on some technical details related to the computation of the bound state's eigenvalue spectrum in the framework of Bethe-Salpeter and Faddeev equations.Introduction: excited states as an eigenvalue problemA great deal of research activity in hadron physics is concerned with hadron structure and revolves around two fundamental questions: what constituents are the hadrons made of and how does Quantum Chromodynamics (QCD), the strong interaction component of the Standard Model, produce them? These are simple questions which, however, may not entail simple answers. To understand the measurable content of QCD, spectroscopy is a valuable and time-honored tool -suffice it to mention the inestimable progress made in the computation of atomic or molecular spectra and subsequent comparison with experiments that lead to a deeper understanding of Quantum Electrodynamics.The same is true for QCD: if confinement is related to the analytic properties of QCD's Schwinger functions, then light-quark confinement ought to be understood by mapping out the infrared behavior of the theory's universal β function. Obviously, this cannot be possibly achieved in perturbation theory. A nonperturbative continuum approach to QCD is provided by Dyson-Schwinger equations (DSE) which relate the theory's β function to experimental observables[1]. Therefore, comparison between DSE predictions embedded in bound-state calculations and observations of the hadron mass spectrum as well as of elastic and transition form factors can be used to study the long-range behavior of QCD's interaction. As the properties of excited hadron states are considerably more sensitive to the long-range behavior of the strong interaction than those of ground states[2][3][4], excited mesons and nucleons[5]are an important source of information and complement our understanding of the strong interaction in light mesons [6-9] and in heavy-light mesons with disparate energy scales[10][11][12][13].Of course, the properties of these hadrons can only be understood, and the functional behavior of the β function be inferred therefrom, by studying the quark's DSE in conjunction with quark-antiquark a
or three-quark bound-state equations, the Bethe-Salpeter and Faddeev equations, respectively. Both are treated as eigenvalue problems which we here exemplify with the relativistic bound-state equation of pseudoscalar J P = 0 − mesons. The homogeneous Bethe-Salpeter equation for this qq bound state with relative momentum p and total momentum P can be generally written as,
Γ 0 − (p, P) = Λ k K(p, k, P) S (k + η + P) Γ 0 − (k, P) S (k − η − P) ,(1)
where S (k ± η ± P) are dressed quark propagators with η + + η − = 1 (NB: in a Poincaré invariant calculation numerical results are independent of the momentum partition parameter η ± ). For the sake of simplicity, we omit flavor and Dirac indices, as the following discussion is independent of them. In the same spirit, we restrict ourselves to the rainbow-ladder truncation of the interaction kernel,
K(p, k, P) = − Z 2 2 G(q 2 ) q 2 λ a 2 γ μ T μν (q) λ a 2 γ ν ,(2)
with the transverse projection operator T μν (q) := g μν − q μ q ν /q 2 , q = p − k, Z 2 is the wave-function renormalization constant and λ a are the SU(3) color matrices in the fundamental representation. Various model ansätze [14][15][16][17] have been proposed for the effective interaction, G(q 2 ), which emulates the combined effect of the gluon and quark-gluon vertex dressing functions and most recent efforts extend these models to include important transverse components of this vertex beyond the leading truncation [18]. Most importantly, Eq. (2) satisfies the axialvector Ward-Takahashi identity [19] and therefore ensures a massless pion in the chiral limit. As we shall see below, Eqs. (1) and (2) define an eigenvalue problem with physical solutions at the mass-shell points, P 2 = −m 2 , where m is the bound-state mass. The Bethe-Salpeter equation's Poincaré-invariant solutions can be cast in the form,
Γ 0 − (p, P) = γ 5 i I D E 0 − (p, P) + / PF 0 − (p, P) + / p(p · P) G 0 − (p, P) + σ μν p μ P ν H 0 − (p, P) ,(3)
Note that this Euclidean-metric basis, A α (p, P) = γ 5 i I D , / P, / p(p · P), σ μν p μ P ν , is nonorthogonal with respect to the Dirac trace. The functions F α 0 − (p, P) = E 0 − (p, P), F 0 − (p, P), G 0 − (p, P), H 0 − (p, P) are Lorentz-invariant scalar amplitudes and are extracted from the Bethe-Salpeter amplitude (3) with appropriate projectors, P a (p, P), 1
1 4 P αβ (p, P) Tr D A β (p, P)A γ (p, P) = δ αγ ; P α (p, P) = P αβ (p, P) A β (p, P) ,(4)
where α, β = 1, ..., 4 and the projection is given by:
F α 0 − (p, P) = 1 4 Tr D P α (p, P) Γ 0 − (p, P) .(5)
Using the Bethe-Salpeter equation, this leads to the eigenvalue problem,
λ(P 2 ) F α 0 − (p, P) = Λ k K αβ (p, k, P) F β 0 − (k, P) ,(6)
where K αβ (p, k, P) stems from the projection of Eq. (1) using Eqs. (2) and (12):
K αβ (p, k, P) = − Z 2 2 4 G(q 2 ) q 2 T μν (q) tr CD P α (p, P) γ μ λ a S (k + ) A β (k, P) S (k − ) γ ν λ a .(7)
1 see, e.g., Ref. [20] for the derivation of the change of basis coefficients P αβ (p, P).
EPJ Web of Conferences
05003-p.2
In Eq. (6), λ(P 2 ) is a scalar function and the eigenvalue equation has a solution for every value of P 2 . Typically, iterative eigenvalue algorithms are employed as only under simplifying assumptions an inversion of the Bethe-Salpeter problem is possible [21][22][23]. To elucidate the iterative procedure, we simplify Eq. (6) (see also Ref. [24]), so that:
λ(P 2 ) |Φ = K(P 2 ) |Φ .(8)
The kernel, K(P 2 ), has a complete set of real eigenvectors φ i with eigenvalues λ i (P 2 ) which are ordered as λ 0 (P 2 ) > λ 1 (P 2 ) > λ 2 (P 2 ) > .... > λ i (P 2 ). Thus, any solution can be written as a linear superposition,
|Φ = ∞ i=1 a i | φ i ,(9)
where a i are real constants and the vector must be normalized. To begin the iterative process, one may "guess" as solution, such as in Eq. (9), for a given value of P 2 and n successive actions of the kernel lead to,
| φ n := K n (P 2 ) |Φ = ∞ i=1 λ n i a i | φ i = λ n 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ a 0 | φ 0 + ∞ i=1 λ i λ 0 n a i | φ i ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .(10)
Since λ 0 > λ i , the coefficients of | φ i converge to zero for sufficiently large values of n and therefore the amplitude | φ n converges to the ground state amplitude | φ 0 :
| φ n n→∞ = λ n 0 a 0 | φ 0 λ 0 K n−1 (P 2 ) |Φ .(11)
This is the most basic method of computing the largest eigenvalue λ 0 (P 2 ) and its associated eigenvector to any required accuracy and is referred to as power or von Mises iteration. The trajectory of this eigenvalue function for a range of P 2 values yields the ground-state meson mass, that is one finds λ 0 (P 2 ) = 1 when P 2 = −m 2 0 .
Gram-Schmidt orthogonalization
A common procedure to extract the wave function of excited states from the spectrum of the interaction kernel is based on the Gram-Schmidt orthogonalization process. As mentioned in Section 1, the homogeneous Bethe-Salpeter equation is an eigenvalue problem, where the largest eigenvalue the interaction kernel produces corresponds to the ground state. However, the eigenvalue spectrum is not limited to the ground state and excited states with smaller eigenvalues can be determined with the same iterative methods discussed above. As sketched, e.g. in Ref. [24], one chooses a mass that is larger than the ground state mass, P 2 = −m 2 < −m 2 0 , and applies the iterative procedure outlined in Section 1:
• find the largest eigenvalue, λ 0 , and the associated eigenvector for m 2 > m 2 0 and λ 0 (m 2 ) > λ 0 (m 2 0 ). This is the unphysical "ground state" at the mass scale P 2 = −m 2 .
• make again a guess for the Bethe-Salpeter amplitude now projecting out the eigenvector that pertains to the largest eigenvalue λ 0 (m 2 ) in a first iteration.
• use the eigenvector obtained in the previous step as input for the Bethe-Salpeter amplitude and project out again the eigenvector associated with λ 0 (m 2 ) in a second iteration; the resulting eigenvector must be projected as before in a third iteration and so on.
• the iteration converges after the nth projection which yields an eigenvalue, λ 1 (m 2 ), with an eigenvector orthogonal to the one associated with the "ground state" eigenvalue λ 0 (m 2 ) > λ 1 (m 2 ). • by varying P 2 one obtains the mass evolution of the second largest eigenvalue λ 1 (P 2 ), which is exemplified in Figure 1. The solution for λ 1 (P 2 ) = 1 yields the mass of the first excited state, P 2 = −m 2 1 where m 1 > m 0 , at which the eigenvector is the state's Bethe-Salpeter amplitude. The normalized projection within this Gram-Schmidt procedure is effected by,
|Φ = |Φ − φ 0 | Φ φ 0 | φ 0 |φ 0 ,(12)
where |Φ is the initial guess for the Bethe-Salpeter amplitude in Eq. (3) and | φ 0 is the vector of the ground-state Bethe-Salpeter amplitude. In projecting out this ground-state, one must define a norm in Euclidean space via an inner product, as is evident from Eq. (12). Within Quantum Field Theory in rainbow-ladder approximation, this product is defined as:
Ψ | Φ := tr CD Λ k Ψ(k, −P) S (k + η + P) Φ(k, p) S (k − η − P) .(13)
Orthogonality is thus defined by the condition, Ψ|Φ = 0, which expresses the vanishing of the overlap amplitude at P 2 . Note that Eq. (13) is not valid for computations beyond the leading approximation, i.e. the rainbow-ladder truncation.
Krylov subspace and Arnoldi iteration
After momentum discretization and Chebyshev expansion of Eq. (6), the numerical kernel of the Bethe-Salpeter equation (or the Faddeev equation for the three-body problem) is a non-symmetric matrix of large dimensions and with some of its eigenvalues close to zero. In general, non-symmetric matrices have eigenvalues and eigenvectors which are very sensitive to small changes in the matrix elements due to the lack of symmetries on which traditional methods rely to ensure numerical stability [27]. This is particularly true for radial excitations, where angular dependence encoded in higher Chebyshev moments contributes in a nontrivial manner. Therefore, we make use of the numerical ARPACK library [28] which is designed to compute a few eigenvalues and corresponding eigenvectors of a general n × n matrix by means of the implicitly restarted Arnoldi method (IRAM). The strength of the Arnoldi method [29] lies in the application of the stabilized orthogonalization algorithm in the Krylov space of a given matrix K to find its eigenvectors, as will be explained shortly. The most important feature of this algorithm is its ability to decouple the calculation of the eigenvectors corresponding to the eigenvalues with the largest absolute value from the eigenvectors whose eigenvalues have an absolute value close to zero. This is highly desirable as the calculation of the latter are plagued by numerical instabilities.
In algebra, the order-r Krylov subspace is generated by an operator whose representation is given by an n × n matrix K = K i j , and a vector Φ of dimension n. It corresponds to the linear subspace spanned by actions of K on Φ:
S r := Φ, KΦ, K 2 Φ, K 3 Φ, ...., K r−1 Φ .(14)
The power iteration of K yields the sequence in Eq. (14) and converges to the eigenvector associated with the largest eigenvalue λ 0 . However, hereby one only makes very limited use of the stored information, as only the final vector, K r−1 Φ, is kept. On the other hand, the basis for the Krylov subspace is derived from the Cayley-Hamilton theorem which implies that the inverse of a matrix can be expressed as a linear combination of its powers. Thus, Krylov subspaces play an important role in contemporary iterative methods to obtain one or few eigenvalues of large sparse matrices or to solve large systems of linear equations. Instead of heavy matrix operations, these methods rely on successive multiplications of vectors by the matrix, thus forming a Krylov subspace, and then employ the resulting vectors. The vectors of the Krylov space are initially not ortogonal and usually become almost linearly dependent due to the properties of the matrix power iteration. Nonetheless, an orthogonal matrix can be constructed from the basis vectors by means of the Gram-Schmidt process described in Section 2. This method proves to be unstable but its shortcoming can be overcome with the Arnoldi iteration [29] which uses the stabilized Gram-Schmidt process and can be applied to general, possibly non-Hermitian matrices. The Arnoldi method generalizes the Gram-Schmidt process by computing the eigenvalues of the orthogonal projection of K onto the Krylov subspace, where the projection is represented by the upper Hessenberg matrix H r [28]. For Hermitian (symmetric) matrices, the Arnoldi iteration is analogous to the Lanczos iteration.
A note on orthogonality
In the context of Bethe-Salpeter and Faddeev equations, orthogonality is defined by Eq. (13). In vectorial form, using the shorthand x = k 2 and y = p 2 , Eq. (6) can be written as,
λ(y) F R (y) = dx K(x, y) · F R (x) ,(15)
which in a numerical treatment is evaluated as the sum,
λ(x j ) F R (x j ) = i w i K(x j , x i ) · F R (x i ) ,(16)
and similarly,
λ(x j ) F L (x j ) = i w i F L (x i ) · K(x j , x i ) ,(17)
where x i are the nodes of a given quadrature with weights w i . Since the integral in Eq. (13) is over a charge-conjugate Bethe-Salpeter amplitude,Ψ(k, −P) := C Ψ T (−k, −P)C T , we introduce in Eqs. (16) and (17) "left" and "right" eigenvectors, F L and F R , respectively. Multiplying Eq. (16) from the left by j w j F L (x j ) we obtain,
λ j w j F L (x j ) · F R (x j ) = i, j w i w j F L (x j ) · K(x j , x i ) · F R (x i ) = λ i w i F L (x i ) · F R (x i ) ,(18)
where F L (x j ) is the eigenvector associated with the eigenvalue λ . For λ λ this relation implies,
dx F L (x) · F R (x) = 0 ,(19)
The left eigenvector is given by the Bethe-Salpeter wave function, S (k − η − P) Φ λ (k, −P)S (k + η + P), and the right eigenvector is the Bethe-Salpeter amplitude [30]. We thus deduce from Eq. (19),
tr CD k S (k − η − P) Φ λ (k, −P) S (k + η + P) Φ λ (k, P) = 0, for λ λ ,(20)
and because the trace is cyclic this orthogonality condition is equivalent to that described in the paragraph below Eq. (13). Moreover, we verify that the Bethe-Salpeter equation spectrum of Eq. (16) is equal to that in Eq. (17). We stress that Eq. (19) is generally valid and no assumption was made about the kernel structure or about the eigenvectors.
Examples in hadron physics
The meson and nucleon resonance structure has been the object of a long history of studies and we here limit ourselves to the approaches based on the combined Dyson-Schwinger and Bethe-Salpeter (or Faddeev) equations, in particular their application to excited mesons and nucleons to compute their masses, weak decay constants and/or electromagnetic form factors making use of the techniques described in Sections 1-4. Seminal studies on the ground state spectrum of light pseudoscalar mesons established that the π(1300) can be described as the first radially excited state of the Goldstone boson and furthermore that the decay constant of excited states, P n , vanishes identically in the chiral limit [31],
fm =0 P n (μ) ≡ 0 , n ≥ 1 ,(21)
wherem is the renormalization-group invariant current-quark mass. Electromagnetic properties of ground and excited state pseudoscalar mesons, making use of the the Gram-Schmidt process illustrated in Section 2, were studied in Ref. [32]. A first approach to computing the eigenvalues of unflavored light and heavy mesons and their corresponding Bethe-Salpeter amplitudes with the implicitly restarted Arnoldi factorization, as implemented in the ARPACK [28] library, is expounded in Refs. [30,33] and has successively been applied to the light [35] and heavy [34,35] quarkonia spectrum for pseudoscalar, vector and tensor states. A recent detailed analysis of the Maris-Tandy interaction [16] parameter space for ground and radially excited states is presented in Ref. [35], where the authors conclude that the preferred inverse effective range of the Maris-Tandy interaction in rainbow-ladder truncation, ω, is shorter for heavy quarkonia than for the light quarkonium spectrum: ω QQ = 0.7 GeV vs. ω qq = 0.5 GeV.
The same library has then been employed to study radial excitations of flavor-singlet and flavored pseudoscalar mesons within the framework of the rainbow-ladder truncation [4,36,37]. A summary of the theoretical values for the pseudoscalar's ground-and excited-state masses and weak decay constants for flavor-singlet and nonsinglet J P = 0 − mesons is reproduced in Table 1 for two parameter [17]. Namely, Model 1 and Model 2 correspond to the interaction parameters, ω = 0.4 GeV, ωD = (0.8 GeV) 3 and ω = 0.6 GeV, ωD = (1.1 GeV) 3 ; see Ref. [4] for details and Ref. [37] for a discussion of the effective interaction. Note that we explored a range of parameter combinations for ωD const., yet could not find a unique set which produces theoretical observables that compare well with all data in Table 1. We therefore observe that we are unable to consistently and simultaneously describe both ground and excited state observables with the given interaction [2] in the rainbow-ladder truncation -whereas one parameter set successfully reproduces experimental ground state masses and decay constants but not the experimental data on excited states, the second parameter set only provides a reasonable description of the excited pseudoscalar mass spectrum. As it becomes clear from the left graph of Fig. 2, for p 2 1 GeV 2 the amplitude's lowest Chebyshev projection, 0 E P 1 (p 2 ) associated with the leading covariant iγ 5 of the pseudoscalar's Bethe-Salpeter amplitude, becomes negative definite in case of the first radial excitations, π(1300) and (ss) n=1 , whereas it remains positive for the ground states. It is remarkable that the behavior of the Bethe-Salpeter amplitudes parallels the pattern of wave functions in quantum mechanics, namely, the number of zeros can be associated with a principal quantum number n. As just mentioned, the ground state amplitude has no zeros and can thus be associated with n = 0. The amplitude of the next highest mass mesons possesses one zero and one assigns the quantum number n = 1 and so on. An analogous behavior is found for the nucleon's first excited state Faddeev amplitude for which we plot the first three Chebyshev moments of its leading S -wave component. Indeed, very recently a range of properties of the proton's radial excitation was predicted which strongly suggests that the nucleon's first radial excitation is the Roper resonance [26]. In particular, in Fig. 3 of Ref. [26] one can appreciate that the overlap amplitude between the nucleon and its first excited state described by the nucleon-Roper transition form factor F * 1 (Q 2 ) vanishes at Q 2 = 0 and therefore satisfies orthogonality.
Figure 1 .
1First two eigenvalue trajectories of the quark-diquark kernel used in Refs.[25,26] as a function of √ −P 2 . Note that the eigenvector solutions are here differently normalized so that the eigenvalues are λ i = 1/g 2 s instead of λ i = 1 for P 2 = −m 2 i ; see Ref.[25] for details. The intersections of the solid horizontal line with the trajectories locates the mass-pole position of the nucleon and its first excited state identified with the Roper.
Figure 2 .
2Left panel: Lowest Chebyshev moment, 0 E P 1 (p 2 ), associated with the leading Dirac structure E P 1 (p 2 ) of the pseudoscalar's Bethe-Salpeter amplitude for the first radial excitations P 1 = π(1300), (ss) n=1 and η c (2S ). Right panel: First three normalized Chebyshev moments, i S 1 (p 2 ), i = 0, 1, 2, of the leading S -wave component in the nucleon's first excited-state Faddeev amplitude.
Table 1 .
1Masses and decay constants for flavor singlet and nonsinglet J P = 0 − mesons; seeSection 5. sets of the interaction introduced by Qin et al.Model 1 [GeV] Model 2 [GeV] Reference
m π
0.138
0.153
0.139 [38]
f π
0.139
0.189
0.1304 [38]
m π(1300)
0.990
1.414
1.30 ± 0.10 [38]
f π(1300)
−1.1 × 10 −3
−8.3 × 10 −4
m K
0.493
0.541
0.493 [38]
f K
0.164
0.214
0.156 [38]
m K(1460)
1.158
1.580
1.460 [38]
f K(1460)
−0.018
−0.017
ms s
1.287
1.702
fs s
−0.0214
−0.0216
m η c (1S )
3.065
3.210
2.984 [38]
f η c (1S )
0.389
0.464
0.395 [39]
m η c (2S )
3.402
3.784
3.639 [38]
f η c (2S )
0.089
0.105
-p.5
-p.7
-p.9
AcknowledgementsWe thank the organizers of the 21 International Conference on Few-Body Problems in Physics in Chicago for the opportunity to present our work at this successful event. The work of B. E. is supported in parts by a State of São Paulo Research Foundation (FAPESP) grant, an MCTI/CNPq/Universal grant and by the Brazilian agency CNPq with a visiting fellowship at the Institute for Theoretical Physics, State University of São Paulo (IFT-Unesp). The work of E. R. is supported by "Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación, Francisco José de Caldas" and by "Sostenibilidad-UDEA 2014-2015".
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| [] |
[
"CHROMATIC PICARD GROUPS AT LARGE PRIMES",
"CHROMATIC PICARD GROUPS AT LARGE PRIMES"
] | [
"Piotr Pstrągowski "
] | [] | [] | We show that the Hopkins' Picard group of the K(n)-local category coincides with its algebraic approximation when 2p − 2 > n 2 + n. | null | [
"https://arxiv.org/pdf/1811.05415v2.pdf"
] | 119,327,223 | 1811.05415 | a102a979446842f3731656b91ff8aa46b4dc5788 |
CHROMATIC PICARD GROUPS AT LARGE PRIMES
28 Jan 2022
Piotr Pstrągowski
CHROMATIC PICARD GROUPS AT LARGE PRIMES
28 Jan 2022
We show that the Hopkins' Picard group of the K(n)-local category coincides with its algebraic approximation when 2p − 2 > n 2 + n.
Introduction
If C is a symmetric monoidal ∞-category, then we can consider equivalence classes of invertible objects X, that is, those such that there exists a Y satisfying X ⊗ Y ≃ ½. This is often a set, rather than a proper class, and it inherits a group multiplication induced from the tensor product. We call the resulting group the Picard group and denote it by Pic(C).
Following ideas of Hopkins, the study of Picard groups was brought into chromatic homotopy theory [HMS94], [Str92]. In this context, C is usually taken to be the ∞-category of E(n)-or K(n)-local spectra at a fixed prime.
As a general rule, one expects the answers to be algebraic when the prime is large compared to the height. To explain what we mean, let us focus on the E(n)-local case first. In this context, taking rational homology defines a homomorphism HQ * : Pic(Sp E(n) ) → Pic(Q),
where by the latter we denote the Picard group of graded rational vector spaces, which is isomorphic to Z. This homomorphism is in fact a split surjection, with splitting k → S k E(n) . Then, it is a result of Hovey and Sadofsky that when 2p − 2 > n 2 + n, the algebraic comparison map is an isomorphism, so that we have Pic(Sp E(n) ) ≃ Z [HS99a]. This is in stark contrast with what happens at small primes; for example, we have Pic(Sp E(1) ) ≃ Z ⊕ Z/2 at p = 2, and Pic(Sp E(2) ) ≃ Z ⊕ Z/3 ⊕ Z/3 at p = 3 [HS99a], [GHMR14].
To study the K(n)-local case, one needs a more subtle algebraic invariant. More precisely, we define the completed E-homology as E ∨ * X := π * L K(n) (E ∧ X), where E is the Morava E-theory spectrum of height n. When it's finitely generated, E ∨ * X has a canonical structure of an L-complete comodule over
E ∨ * E [Bak09] [BH16][1.22].
The latter Hopf algebroid can be described explicitly as E ∨ * E ≃ map c (G n , E * ), the space of continuous functions on the Morava stabilizer group, with structure maps induced from the action of G n [DH04].
If X is K(n)-locally invertible, then E ∨ * X is an invertible E ∨ * E-comodule, which gives a homomorphism Pic(Sp K(n) ) → Pic(E ∨ * E) into the algebraic Picard group, given by isomorphisms classes of invertible comodules.
The algebraic Picard group can be expressed in terms of cohomology of the Morava stabilizer group; to do so, one observes that an invertible E ∨ * E-comodule is the same as an invertible E *module equipped with a compatible continuous action of G n . Since E 0 is a regular local ring, any such module is free of rank one, and so we have a short exact sequence
0 → Pic 0 (E ∨ * E) → Pic(E ∨ * E) → Z/2 → 0, where Pic 0 (E ∨ * E)
is the subgroup of those invertible modules which are concentrated in even degrees. Since E * is 2-periodic, any such module is determined by its degree zero part, which yields an isomorphism Pic 0 (E ∨ * E) ≃ H 1 c (G n , E × 0 ) by standard considerations [GHMR14].
Due to a classical argument using the sparsity of the Adams-Novikov spectral sequence, one knows that Pic(Sp K(n) ) → Pic(E ∨ * E) is injective when 2p − 2 > n 2 and (p − 1) ∤ n [HMS94][7.5]. On the other hand, surjectivity was not known except at low heights, where both sides can be computed explicitly.
Our main result gives a range in which the comparison map is in fact an isomorphism.
Theorem 1.1 (2.5). When 2p − 2 > n 2 + n, Pic(Sp K(n) ) → Pic(E ∨ * E) is an isomorphism. The proof of Theorem 1.1 rests on the recent chromatic algebraicity result of the author which states that at large primes, there exists an equivalence hSp E ≃ hD(E * E) between the homotopy categories of E-local spectra and differential E * E-comodules [Pst21].
In fact, to prove the isomorphism between Picard groups, we do not need the equivalence of homotopy categories, but only the weaker statement that any E * E-comodule can be canonically realized as a homology of a certain E-local spectrum. Thus, Theorem 1.1 holds in a slightly larger range of primes than chromatic algebraicity.
The arguments we use are quite general, and we believe could be applied in many other profinite contexts. In particular, a generalization to the case of K(n)-locally dualizable spectra appears in the work of Barthel, Heard and Naumann [BHN20].
As was pointed to us by Paul Goerss, a more natural proof of Theorem 1.1 would use the descent spectral sequence for S 0 K(n) → E, which is K(n)-local pro-Galois extension with Galois group G n [Rog08]. No such spectral sequence was known at the time this article first appeared, but it has been since then constructed by Heard [Hea21,§6C].
Since it is of potential interest, we present this alternative approach in Remark 2.6. The proof given in the main body of the article is independent from this argument.
Acknowledgements
I would like to thank my supervisor Paul Goerss for his support and guidance, as well as for helpful comments on the structure of this paper.
Chromatic Picard groups at large primes
We let p denote the prime and n the height, both of which are fixed. By E we denote the Morava E-theory spectrum, this is an even periodic Landweber exact spectrum associated to the Lubin-Tate ring E 0 ≃ W (F p n )[[u 1 , . . . , u n−1 ]] of the Honda formal group law over F p n , see for example [GHMR05,§1] for more on the setup. In particular, E 0 is a complete regular local ring of dimension n, with maximal ideal m = (p, u 1 , . . . , u n−1 ).
The completion functor M → lim ← − M/m k M on E * -modules is neither right or left exact, but it has a right exact left derived functor which we denote by
L 0 [HS99b, Appendix A]. We say a module M is L-complete if the natural map M → L 0 M is an isomorphism. If M, N are modules, then we denote their L-complete tensor product by M ⊗ E * N := L 0 (M ⊗ E * N ).
We let K denote the associated Morava K-theory spectrum; this is the unique up to equivalence E-module which admits an E-algebra structure whose unit induces an isomorphism K * ≃ E * /m [LH]. The spectrum K is Bousfield equivalent to the classical Morava K-theory spectrum K(n)
satisfying K(n) * ≃ F p [v ±1
n ]. Following [HMS94], [Str92], for a spectrum X we define its completed E-homology as E ∨ * X := π * L K (E ∧ X). One can show that E ∨ * X is an L-complete E * -module, and if it is finitely generated, then it has a structure of a comodule over
E ∨ * E [HS99b, 8.5], [Bak09], [BH16, 1.22]. If X is finite, or more generally if E * X is L-complete, then E ∨ * X ≃ E * X [Hov04, 3.2]
. We would like to understand the Picard group Pic(Sp K ) of equivalence classes of invertible Klocal spectra. We begin by recalling the following fundamental result.
Theorem 2.1 ( [HMS94](1.3)). A spectrum X is K-locally invertible if and only if E ∨ * X is free of rank one over E * ; equivalently, is an invertible E ∨ * E-comodule. As a consequence of Theorem 2.1, we obtain a homomorphism E ∨ * : Pic(Sp K ) → P ic(E ∨ * E) from the K-local Picard group into the Picard group of E ∨ * E, given by isomorphisms classes of E ∨ * E-comodules which are invertible under the tensor product. One can describe E ∨ * E and the associated Picard group in terms of the Morava stabilizer group, which we now recall.
Since E is even periodic, it is complex orientable and the associated formal group is the universal deformation of the Honda formal group law Γ of height n over F p n . This endows E 0 with an action of the Morava stabilizer group G n := Aut(F p n , Γ) ≃ Aut(Γ) ⋊ Gal(F p n /F p ), which by the Goerss-Hopkins-Miller theorem lifts to an action on E by maps of commutative ring spectra [GH], [PV21].
The action of G n on E induces an isomorphism E ∨ * E ≃ map c (G n , E * ), where the latter is the space of continuous functions on the Morava stabilizer group [DH04]. If M is an E ∨ * E-comodule, then this identification endows it with an action of G n , and if M is finitely generated over E * , then this action is continuous in the m-adic topology and any such continuous action determines a comodule structure [BH16,5.4].
We deduce that the data of an invertible E ∨ * E-comodule is the same as that of a an invertible E *module equipped with a compatible continuous action of G n , this allows one to give a homological description of Pic(E ∨ * E), as we recalled in the introduction. Our goal is to prove that the homomorphism Pic(Sp K ) → Pic(E ∨ * E) is an isomorphism at large primes. We start with injectivity, which is classical, but since the proof is enlightening, and not particularly difficult, we briefly recall the argument.
Proposition 2.2 ([HMS94](7.5)). If 2p − 2 ≥ n 2 and (p − 1) ∤ n, then the comparison map
E ∨ * : Pic(Sp K ) → Pic(E ∨ * E) is injective. Proof.
Suppose that X ∈ Pic(Sp K ); since E ∨ * X is free of rank one, in particular finitely generated, we have the K-local E-based Adams spectral sequence of the form Ext s,t E ∨ * E (E * , E ∨ * X) ⇒ π t−s X and an isomorphism Ext s,t
E ∨ * E (E * , E ∨ * X) ≃ H s c (G n , E ∨ t X)
between the E 2 -term and the continuous cohomology of the Morava stabilizer group [BH16, 3.1, 4.1].
The E 2 -term is concentrated in internal degrees divisible by 2p − 2 and if (p − 1) ∤ n, then it has a horizontal vanishing line at n 2 , the homological dimension of the Morava stabilizer group [Hea15, 4.2.1]. It follows that under the given assumptions the spectral sequence collapses for degree reasons. Now, suppose that X is in the kernel of E ∨ * : Pic(Sp K ) → Pic(E ∨ * E), so that we have an isomorphism E ∨ * X ≃ E ∨ * S 0 K ≃ E * . As observed above, the E-based Adams spectral sequence collapses, and it follows that the chosen isomorphism is necessarily an infinite cycle and so descends to an equivalence X ≃ S 0 K . This ends the argument.
We move on to the surjectivity of the comparison map; this is the heart of the problem. We start with two short, technical lemmas.
Lemma 2.3. We have lim − → Ext E * (E * /m k , K * ) ≃ K * , concentrated in homological degree zero.
Proof. Since E * is 2-periodic and the above modules are even graded, it is enough to prove that lim − → Ext E0 (E 0 /m k , K 0 ) ≃ K 0 , concentrated in homological degree zero. Since E 0 is a regular local ring, local duality implies that lim − → Ext i E0 (E 0 /m k , K 0 ) ≃ Ext n−i E0 (K 0 , E 0 ) ∨ , where by (−) ∨ we denote the Matlis dual [BH98]. Because K 0 ≃ E 0 /m is the unique simple E 0 -module, it is Matlis self-dual and we deduce that it is enough to show that Ext E0 (K 0 , E 0 ) ≃ K 0 , concentrated in homological degree n.
More generally, we claim that Ext E0 (E 0 /I k , E 0 ) ≃ E 0 /I k , concentrated in homological degree k, where I k = (p, u 1 , . . . , u k−1 ) for any 0 ≤ k ≤ n. This is clear for k = 0 and the general case follows by induction from the long exact sequence of Ext-groups associated to
0 → E 0 /I k−1 → E 0 /I k−1 → E 0 /I k → 0,
which ends the proof.
Lemma 2.4. Let X ≃ lim ← − X i be a limit diagram of K-local spectra such that X i and X are K-locally dualizable. Then, for any
K-local spectrum Y we have L K (Y ∧ X) ≃ lim ← − L K (Y ∧ X i ).
Proof. Consider the collection of all K-local spectra Y such that the needed condition holds. Since X and X i are dualizable, smashing with them preserves K-local limits and we deduce that this collection is closed under limits. Since it also contains S 0 K by assumption, we deduce that it is necessarily all of Sp K by [HS99b,7.5].
The following is the main result of this note.
Theorem 2.5. Let 2p − 2 > n 2 + n. Then, E ∨ * : Pic(Sp K ) → Pic(E ∨ * E) is an isomorphism.
Proof. If 2p − 2 > n 2 + n, then 2p − 2 ≥ n 2 and (p − 1) ∤ n and we've seen in Proposition 2.2 that under these conditions the homomorphism between Picard groups is injective.
To verify surjectivity, we have to prove that if M ∈ Pic(E ∨ * E), there exists a K-locally invertible spectrum X with E ∨ * X ≃ M . Observe that as an E * -module, M is necessarily free of rank one [HS99b, A.9] and, without loss of generality, we can assume that it is even graded. Then, for each k ≥ 1 we have M/m k M ≃ E * /m k as an E * -module and so
E ∨ * E ⊗ E * M/m k M ≃ E * E ⊗ E * M/m k M ≃ E * E ⊗ E * M/m k M ,
where the first isomorphism is [HS99b, A.7] and the second follows from the fact that the last term is an E * /m k -module and so is already L-complete. Thus, we deduce that M/m k M is an E * E-comodule in the usual, non-complete sense. Under the assumption 2p−2 > n 2 +n, in [Pst21, 2.14] we construct the Bousfield splitting functor β : Comod E * E → hSp E valued in the homotopy category of E-local spectra with the property that E * βM ≃ M for any M ∈ Comod E * E .
By construction, we have E * β(M/m k M ) ≃ M/m k M and since the latter is L-complete, we deduce from [Hov04, 3.2] that
E ∨ * β(M/m k M ) ≃ E * β(M/m k M ) ≃ M/m k M. Note that we have a universal coefficient spectral sequence of signature (1) Ext E * (M/m k M, K * ) ⇒ K * β(M/m k M ).
As E 0 is a regular local ring of dimension n, this has a horizontal vanishing line at s = n and in particular is strongly convergent; this will be important below.
We let X k := L K β(M/m k M ); by the above, this is a K-local spectrum with E ∨ * X k ≃ M/m k M . In particular, E ∨ * X k is degreewise finite and so X k is a finite K-local spectrum of type n by [HS99b,8.5].
As β is a functor, we have maps X k → X k−1 induced from the projections M/m k → M/m k−1 , well-defined up to homotopy, and we let X := lim ← − X k denote the corresponding homotopy limit. Here, by the latter we mean that we pick a lift of the tower of X k to the ∞-category Sp K and we compute the limit there. It is classical that up to equivalence the homotopy limit does not depend on the choice of that lift, since it can be defined using the triangulated structure alone.
We first show that X is invertible. Since X k are dualizable, we have X ≃ D(lim − → DX k ), where D := F (−, S 0 K ) is the K-local Spanier-Whitehead dual and the colimit is the K-local one. Thus, it is enough to show that lim − → DX k is invertible; which we will verify by showing that
K * (lim − → DX k ) ≃ lim − → K * (DX k ) ≃ K * .
As the universal coefficient spectral sequences of (1) have all the same horizontal vanishing line, by taking filtered colimits we obtain a strongly convergent spectral sequence of signature
lim − → Ext E * (M/m k M, K * ) ⇒ K * (lim − → DX k )
Since M is a free E * -module of rank one, the needed statement follows from Lemma 2.3, and we deduce that lim − → DX k , hence X, is invertible. Since X k and X are K-locally dualizable,
L K (E ∧ X) ≃ lim ← − L K (E ∧ X k )
by Lemma 2.4. After passing to homotopy groups, we obtain the Milnor exact sequence
0 → lim ← − 1 (M/m k M )[−1] → E ∨ * X → lim ← − M/m k M → 0
and since M is free of rank one, the lim ← − 1 -term vanishes. We deduce that the second map must be an isomorphism, which ends the proof since M ≃ lim ← − M/m k M .
Remark 2.6. The following alternative argument, based on the descent spectral sequence, was pointed to us by Paul Goerss. The needed spectral sequence was not known at the time this article first appeared, but it has been since then constructed in the work of Heard [Hea21,§6C], giving an alternative proof of Theorem 2.5. If C is a presentably symmetric monoidal ∞-category, then the Picard group can be lifted to the Picard space, which we will denote by Pic(C) [MS16]. The latter is the ∞-groupoid of invertible objects in C; it is an E ∞ -space with multiplication induced from the tensor product.
The Picard group itself can be recovered through the relation Pic(C) = π 0 Pic(C). The higher homotopy groups of the Picard space are easy to describe, as we have π t Pic(C) ≃ π t−1 aut C (½, ½) for t > 0, where ½ is the monoidal unit and aut C denotes the space of self-equivalences.
By the work of Devinatz and Hopkins, the map S 0 K → E of commutative ring spectra is a K(n)-local pro-Galois extension in the sense of Rognes with Galois group G n [DH04], [Rog08]. In [Hea21], Heard proves that associated to this extension we have a spectral sequence of signature 1 H s c (G n , π t Pic(Mod E )) ⇒ π t−s Pic(Sp K ),
with differentials d r of degree (r, r − 1) and where the action of G n on Mod E is induced from that on E. This extends the previous work in the case of the finite Galois group [MS16], [GL21].
To get hold on the E 2 -term, we need to understand the homotopy of the Picard space of Mod E , but this is not difficult. Since E is even periodic and E 0 is regular local, any invertible E-module is free and so π 0 Pic(Mod E ) ≃ Z/2 [BR05]. Moreover, because E is the monoidal unit of Mod E , we have π 1 Pic(Mod E ) ≃ E × 0 and π t Pic(Mod E ) ≃ E t−1 for t ≥ 2. If (p − 1) ∤ n, then the Morava stabilizer group is of finite homological dimension n 2 and the E 2term has a horizontal vanishing line. Furthermore, by standard considerations H s (G n , E t ) vanishes unless t is divisible by 2p − 2 [Hea15, 4.2.1].
It follows that if 2p − 2 ≥ n 2 and (p − 1) ∤ n, then if drawn using the Adams grading, the −1 ≤ t − s ≤ 1 region of the above spectral sequence looks like
H 1 c (G n , Z/2) 0 H 1 c (G n , E × 0 ) H 0 c (G n , Z/2) 0 H 0 c (G n , E × 0 ) t − s ,
with only zeroes above. We deduce that in this range this spectral sequence collapses and yields a short exact sequence 0 → H 1 c (G n , E × 0 ) → π 0 Pic(Sp K ) → Z/2 → 0. This means that the topological Picard group Pic(Sp K ) ≃ π 0 Pic(Sp K ) fits into a short exact sequence of the same form as the algebraic one, as explained in the introduction. One can then verify that Pic(Sp K ) → Pic(E ∨ * E) fits into a map of short exact sequences which is then an isomorphism by the five-lemma, giving a different proof of Theorem 2.5.
In fact, the bound obtained in this way is slightly sharper, as one only needs 2p − 2 ≥ n 2 , rather than 2p − 2 > n 2 + n. This comes from the fact that this argument avoids the use of the E-local category, since the homological dimension of E * E is n 2 + n, while the homological dimension of G n is just n 2 .
To be more precise, Heard does not identify all of the E 2 -term with continuous group cohomology, but he identifies it in the range large enough ((s, t) = (0, 0), (1, 1) or s ≥ 2) for our argument to go through.
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| [] |
[
"On the functional equations for polylogarithms in one variable",
"On the functional equations for polylogarithms in one variable"
] | [
"Daniil Rudenko "
] | [] | [] | We develop a new approach to the study of the functional equations satisfied by classical polylogarithms, inspired by Goncharov's conjectures. We prove a sharpened version of Zagier's criterion for such an equation and explain, how our approach leads to a very simple description of the equations in one variable, satisfied by dilogarithm and trilogarithm.Our main result is the complete description of the functional equations for weight four polylogarithm in one variable. discussions and suggestions. My special gratitude goes to H. Gangl for showing me his computations for polylogarithms in weight four. | null | [
"https://arxiv.org/pdf/1511.09110v1.pdf"
] | 73,680,987 | 1511.09110 | d0ec153c725a7159f9489fc0f34b39dac6ce4743 |
On the functional equations for polylogarithms in one variable
2 Nov 2015 December 1, 2015
Daniil Rudenko
On the functional equations for polylogarithms in one variable
2 Nov 2015 December 1, 2015
We develop a new approach to the study of the functional equations satisfied by classical polylogarithms, inspired by Goncharov's conjectures. We prove a sharpened version of Zagier's criterion for such an equation and explain, how our approach leads to a very simple description of the equations in one variable, satisfied by dilogarithm and trilogarithm.Our main result is the complete description of the functional equations for weight four polylogarithm in one variable. discussions and suggestions. My special gratitude goes to H. Gangl for showing me his computations for polylogarithms in weight four.
for odd n > 1
Li n (z) = ℜ n−1 k=0 β k log k |z| · Li n−k (z) , for even positive n Li n (z) = ℜ n−1 k=0 β k log k |z| · Li n−k (z) ,
where 2x e 2x − 1 = ∞ k=0 β k x k .
Zagier polylogarithm is known to be single valued and continuous on CP 1 and real analytic on CP 1 \{0, 1, ∞}. Next, we define an abelian group, generated by single-variable functional equations, satisfied by Li n (z). For arbitrary set S denote by Q[S] a vector space with a basis, indexed by elements of S. For n i ∈ Q and ϕ i ∈ C(t) the sum k i=1 n i [ϕ i (t)] lies in A n if and only if k i=1 n i Li n (ϕ i (t)) = const. We will denote the inclusion map from A n to Q[C(t)] by I n .
In this work we will describe groups A n for weight less than or equal to four. The first result is a complete criterion for an element of Q[C(t)] to lie in A n . This is a sharpened version of Zagier's proposition 1 in [3]. To state it we need to introduce several axillary definitions.
Denote by A a Q−vector space of zero degree divisors on CP 1 . For a function ϕ(t) denote by (ϕ) it's divisor. We will use inhomogeneous notation, where [∞] = 0 in A and A is a free vector space with the basis on all points of A 1 C . Next, for each p ∈ C denote by ψ p an element of A * dual to p ∈ A. On a function ϕ(t) it equals the order of zero (pole) of ϕ in p.
Next, for vector space V denote by Sym i V it's i−th symmetric power and by i V it's i−th wedge power.
Let S n−1,1 * V be the following functor: S n−1,1 * V ∼ = Sym n−1 V ⊗ V Sym n V . For n = 1 we put S 0,1 * V ∼ = V. We used this symbol, because S n−1,1
The statement of this result can be rephrased as follows: k i=1 n i Li n (ϕ i (t)) = const if and only if k i=1 (ϕ i (t)) n−1 ⊗ (1 − ϕ i (t)) = 0 in S n−1,1 * A. Using the duality between S n−1,1 * and the corresponding Schur functor one can give yet another formulation of our result: k i=1 n i Li n (ϕ i (t)) = const if and only if for arbitrary p 1 , ..., p n ∈ C k i=1 n i ψ p1 (ϕ i ) · ... · ψ pn−2 (ϕ i ) · ψ pn−1 (ϕ i )ψ pn (1 − ϕ i ) − ψ pn−1 (1 − ϕ i )ψ pn (ϕ i ) = 0.
Our next result completely describes groups A i for i ≤ 3.
Theorem 1.5. The maps J 1 , J 2 and J 3 are surjective. Equivalently,
Q[C(t)] 1 A 1 ∼ = A, Q[C(t)] 2 A 2 ∼ = A ∧ A, Q[C(t)] 3 A 3 ∼ = Sym 2 A ⊗ A Sym 3 A .
From the proof of this result we will derive two corollaries. The first one is so-called Roger's identity, claiming that for arbitrary rational function f (t)
Li 2 (f (t)) = a,b∈C ψ a (f )ψ b (1 − f )Li 2 t − a b − a + const.
For linear fractional f (t) this identity reduces to the five-term relation.
The second identity, which can be found in works of Wojtkowiak, claims that for arbitrary rational function f (t) such that both functions f (t) and 1 − f (t) are finite and invertible at t = ∞,
Li 3 (f (t)) = − 1 2 a,b,c∈C ψ a (f )ψ b (f )ψ c (1 − f )Li 3 (t − a)(c − b) (t − b)(c − a) + const. For f (t) = (t − a) 2 (t − b)(t − c)
this reduces to the classical Kummer's equation and for
f (t) = (t − a)(t − b) (t − c)(t − d) this gives
Goncharov's equation.
Our last and the most novel result describes the image of the map J 4 . To give its formulation we need to give several more definitions. For four points a, b, c, d ∈ CP 1 let us denote their cross-ratio by r(a, b, c, d)
= (a − b)(c − d) (c − b)(a − d)
.
Then, B 2 (C) is a vector space Q[C]/R 2 (C), where R 2 (C) is generated by symbols 0, 1, ∞ and linear combinations 4 i=0 (−1) i r(x 0 , ...,x i , ..
., x 4 ). Note that every element of R 2 (C) gives rise to a relation between values of Li 2 . This follows from the so-called five-term relation, found by Abel:
4 i=0 (−1) i Li 2 (r(x 0 , ...,x i , ..., x 4 )) = 0. Element of B 2 (C), coming from [x] ∈ Q[C] will be denoted by {x} 2 .
Theorem 1.6. The following sequence is exact:
0 −→ A 4 I4 −→ Q[C(t)] 4 J4 −→ S 3,1 * A K4 −→ (A ∧ A) ⊗ B 2 (C) −→ 0, where K 4 (x 1 x 2 x 3 ⊗ y) = σ∈S3 x σ(1) ∧ x σ(2) ⊗ {r(x σ(1) , y, x σ(3) , ∞)} 2 .
By quadratic rational function we mean a fraction of two polynomials in variable t of degree 2 with complex coefficients. We call a quadratic rational function q(t) special if q(∞) ∈ {0, 1, ∞} or q(t) has a double root. The next corollary from the proof of theorem 1.6 gives general functional equations for classical polylogarithm of weight four. Corollary 1.7. For arbitrary rational function f (t) ∈ C(t) there exists an integer k, rational numbers a i and special quadratic rational functions q i (t) such that
Li 4 (f (t)) = k i=1 a i Li 4 (q i (t)) + const.
2 Examples and motivation.
We start this section with the proof of theorem 1.5 and its corollaries. These results are very simple and serve as an illustration of our method of dealing with functional equations. Next, we explain how general conjectures of Goncharov about mixed Tate motives lead to theorems 1.4 and 1.6.
Proof. First, we need to prove that J 1 is surjective. This is true, since for arbitrary a ∈ C, obviously, J 1 (t− x) = [x] ∈ A. Since elements of this type form a basis of A, J 1 is surjective. We remember that in our inhomogenious
notation [∞] = 0 Similarly, J 2 is surjective, since J 2 ( t − a b − a ) = [a] ∧ [b] ∈ A ∧ A = S 2 * (A).
The last claim is surjectivity of J 3 . Consider the following filtration on S 2,1 * V :
S 2,1 * V 1 ⊆ S 2,1 * V 2 ⊆ S 2,1 * V 3 , where S 2,1 * V i is generated by tensors [x 1 ][x 2 ] ⊗ [x 3 ] ∈ S 2,1 * V such that between x j there are at most i different.
Let gr i S 2,1 * V be associated graded factors S 2,1
* V i / S 2,1 * V i−1 . First note that S 2,1 * V 1 vanishes, since [a] 2 ⊗ [a] ∈ Sym 3 (V ). Next, J 3 is surjective on gr 2 S 2,1 * V, since J 3 t − a b − a = [a] 2 ⊗ [b] and [a][b] ⊗ [a] = − 1 2 [a] 2 ⊗ [b]
. Finally, we need to show that J 3 is surjective on gr 3 S 2,1 * V.
This is true, since J 3 (t − a)(c − b) (t − b)(c − a) = ([a] − [b]) 2 ⊗ ([c] − [b]) = −2[a][b] ⊗ [c].
Next, we are going to prove Rogers identity and that of Wojtkowiak. To show that
Li 2 (f (t)) = a,b∈C ψ a (f )ψ b (1 − f )Li 2 t − a b − a + const,
we need to prove that
[f (t)] − a,b∈C ψ a (f )ψ b (1 − f ) t − a b −J 2 (f (t)) = (f ) ∧ (1 − f ) = a,b∈C ψ a (f )ψ b (1 − f )a ∧ b = a,b∈C ψ a (f )ψ b (1 − f )J 2 t − a b − a .
The trilogarithmic identity of Wojtkowiak claims that for f (t) ∈ C(t) such that both functions f (t) and 1 − f (t) are invertible at t = ∞,
Li 3 (f (t)) = − 1 2 a,b,c∈C ψ a (f )ψ b (f )ψ c (1 − f )Li 3 (t − a)(c − b) (t − b)(c − a) + const.
This is equivalent to the fact that
[f (t)] + 1 2 a,b,c∈C ψ a (f )ψ b (f )ψ c (1 − f ) (t − a)(c − b) (t − b)(c − a)
lies in A 3 . By theorem 1.4 it is enough to check that J 3 vanishes on this element. First note that since f and 1 − f are invertible in ∞, a∈C ψ a (f ) = 0, b∈C ψ b (f ) = 0 and c∈C ψ c (1 − f ) = 0. From this we can prove the claim:
J 3 (f ) = a,b,c∈C ψ a (f )ψ b (f )ψ c (1 − f )[a][b] ⊗ [c] = = − 1 2 a,b,c∈C ψ a (f )ψ b (f )ψ c (1 − f )([a] − [b]) 2 ⊗ ([c] − [b]) = = − 1 2 a,b,c∈C ψ a (f )ψ b (f )ψ c (1 − f )J 3 (t − a)(c − b) (t − b)(c − a) .
Our next goal is to show, how theorems 1.4 and 1.6 follow from general conjectures of Goncharov, explained in [2]. These conjectures explain the structure of the category M T (F ) of mixed Tate motives over field F . Note that even the existence of such a category with desirable properties is proved only for some types of fields. Category M T (F ) should be Tannakian, generated by tensor powers of simple object Q(1). By general properties of Tannakian categories there should exist a graded Lie coalgebra L MT (F ) such that M T (F ) is equivalent to the category of representations of L MT (F ). For simplicity we will denote it just by L(F ). Beilinson and Deligne showed in [1] that for arbitrary a ∈ F there should exist an element in L(F ), denoted by Li M n (a), called motivic polylogarithm. Their main property is that for F = C their Hodge realization gives classical polylogarithms, so all linear relations between motivic polylogarithms give rise to relations for numbers Li n (a).
The Lie coalgebra L(F ) is positively graded by integers i > 0, and one can show that L 1 (F ) should be isomorphic to F * , generated by motivic logarithms log M (a). Let I(F ) be a coideal of elements in L(F ) of degree greater than one. It gives rise to an exact sequence of Lie coalgebras
0 −→ F * −→ L(F ) −→ I(F ) −→ 0.
The Freeness Conjecture of Alexander Goncharov claims that I(F ) is cofree Lie coalgebra. Furthermore, its first homology group in degree d is generated by elements Li M d (a), for a ∈ F. Following Goncharov, we will denote this group by B d (F ).
It is reasonable to assume that L(C) and L(C(t)) are related by the following exact sequence:
0 −→ L(C) −→ L(C(t)) −→ L A −→ 0,
where L A is a cofree Lie coalgebra, generated by A. It is an object, graded dual to the graded free Lie algebra, generated by A * . The exact sequence above is an instance of Van Kampen theorem: fundamental group of a Riemann sphere with all points removed is freely generated by infinitesimal loops around all points. This exact sequence gives rise to the corresponding exact sequence of coideals
0 −→ I(C) −→ I(C(t)) −→ I A −→ 0.
Since all the three coideals are cofree Lie coalgebras, according to Goncharov's Conjecture, only their zero and first homology are nontrivial. Application of Hochshild-Serre spectral sequence leads to the following exact sequence:
0 −→ H 1 (I(C)) −→ H 1 (I(C(t))) −→ H 1 (I A ) −→ H 1 (I(C), H 1 (I A )) −→ 0.
Another part of Freeness Conjecture claims that in degree d H 1 (I(C)) ∼ = B d (C) and H 1 (I(C(t))) ∼ = B d (C(t)). Since B d (C(t)) is a group, generated by motivic polylogarithms of rational functions, it is natural to assume that
B d (C(t))/B d (C) ∼ = Q[C(t)] d /A d .
A simple application of Koszul resolution leads to the computation of H 1 (I A ) which is isomorphic to S d−1,1 *
A. So, we come to an exact sequence
0 −→ A d −→ Q[C(t)] d −→ S d−1,1 * A −→ H 1 (I(C), H 1 (I A )) −→ 0.
The first three terms of this sequence lead to the theorem 1.4.
Due to degree reasons, H 1 (I(C), H 1 (I A )) vanishes in degree less than 4. This leads to theorem 1.5.
In degree four H 1 (I(C), H 1 (I A )) ∼ = A ∧ A ⊗ B 2 (C)
. This leads to theorem 1.6.
Proof of theorem 1.4.
We need to show that the following sequence is exact:
0 −→ A n In −→ Q[C(t)] d Jn −→ S n−1,1 * A.
We will prove it by induction on n. For arbitrary p ∈ C we introduce formal analogs of partial derivatives D p .
Define D p : Sym n A −→ Sym n−1 A by formula D p ([x 1 ][x 2 ]...[x n ]) = n i=1 ψ p (x i ) · [x 1 ][x 2 ]...[x i ]...[x n ].
For us will be important that D p X n = nψ p (X) · X n−1 .
Lemma 3.1. The map ⊕D p : Sym n A −→ p∈C Sym n−1 A is injective. Proof. For arbitrary points p i ∈ C the composition D p1 • D p2 • ... • D pn is a functional on Sym n A.
It is easy to see that such elements generate the dual vector space to Sym n A. So, if for some element Y ∈ Sym n A D p Y = 0 for arbitrary p, then all linear functionals on Sym n A vanish on Y and we conclude that Y = 0.
Next, we let the operators D p act on all members of Koszul resolution Sym n−k A ⊗ k A as D p ⊗ id. It can be shown that D p form a chain map from Koszul resolution of weight n to Koszul resolution of weight n − 1. So, the map D p :
S n−1,1 * A −→ S n−2,1 * A is correctly defined. Corollary 3.2. The map ⊕D p : S n−1,1 * A −→ p∈C S n−2,1 * A is injective.
Proof. X p form a chain map of Koszul resolutions so the following diagram is commutative.
S n−1,1 * A ⊕Dp − −−− → p∈C S n−2,1 * A Kn−1 ⊕Kn−2 Sym n−2 A ⊗ A ∧ A ⊕Dp − −−− → p∈C Sym n−3 A ⊗ A ∧ A
Since both vertical and the lower horizontal maps are injective, the upper horizontal map is injective as well.
Now we introduce similarly denoted maps D p from Q[C(t)] n to Q[C(t)] n−1 . For n ≥ 3 we put D p [ϕ] n = ψ p (ϕ) · [ϕ] n−1 and for n = 2 we put D p [ϕ] 2 = ψ p (ϕ) · [1 − ϕ] 1 − ψ p (1 − ϕ) · [ϕ] 1 .
It is easy to check that D p • J n = J n−1 • D p . Now theorem 1.4 is a simple consequence of the following lemma:
Lemma 3.3. Element F ∈ Q[C(t)] n lies in A n if and only D p F lies in A n−1 for every p.
Proof. The "only if" part is the direct corollary of theorem 1.17 in [?]. The proof is based on the analysis of singularities of the function dLi n (F ) dt .
The "if" part is slightly more tricky. For arbitrary abelian group G denote by G Q a Q−vector space G ⊗ Z Q. We first recall the well-known criteria of Zagier for polylogarithmic functional equations. It claims that if for some n i ∈ Z and ϕ i ∈ C(t) equality
n i [ϕ i ] n−2 ⊗ [ϕ i ] ∧ [1 − ϕ i ] = const holds in Sym n−2 C(t) * Q ⊗ C(t) * Q ∧ C(t) * Q , then n i Li n (ϕ i (t)) = const, so n i [ϕ i ] ∈ A n . Here by const in Sym n−2 C(t) * Q ⊗ C(t) * Q ∧ C(t) * Q we mean an element in Sym n−2 C * Q ⊗ C * Q ∧ C * Q
Let us suppose that for some X = n i [ϕ i ] for arbitrary p ∈ C holds D p (X) = 0. We will deduce the fact that X ∈ A n from Zagier criterion. Denote the element
n i [ϕ i ] n−2 ⊗ [ϕ i ] ∧ [1 − ϕ i ] by Z(X).
We will consider two sets of functionals α : C(t) * Q −→ Q, which we will call "constantial" and "valuational." To construct constantial functionals, consider some(uncountable) basis of C * Q /Q and take the dual basis. Then each element of this dual basis can be extended to whole C(t) * Q by letting it vanish on all monomials (t − λ). In this way we get constantial functionals. Valuational functionals are just ψ p for p ∈ C. If each constantial and each valuational functional annulates an element of C(t) * Q , then this element vanishes. Similarly, to show that Z(X) = 0 it is enough to prove that all functionals ψ 1 · .... · ψ n−2 ⊗ ψ n−1 ∧ ψ n vanish on Z(X), where each ψ i is either constantial or valuational. We prove this statement by induction on the number C of constantial functionals.
Applying the "only if" part of the lemma to each D p X n − 1 times, we deduce that if all ψ i are valuational, than any functional ψ 1 · .... · ψ n−2 ⊗ ψ n−1 ∧ ψ n vanishes on Z(X). This proves the base of induction C = 0. Now, let us suppose that for s = k −1 the statement is proved. The value of the functional ψ 1 ·....·ψ n−2 ⊗ψ n−1 ∧ψ n on Z(X) does not change after a permutation of the first n − 2 functionals and change sign after the transposition of the last two. Moreover, the following sum vanishes on Z(X) :
ψ 1 · .... · ψ n−3 · ψ n−2 ⊗ ψ n−1 ∧ ψ n + ψ 1 · .... · ψ n−3 · ψ n ⊗ ψ n−2 ∧ ψ n−1 + ψ 1 · .... · ψ n−3 · ψ n−1 ⊗ ψ n ∧ ψ n−2 .
From this it follows that it is enough to show that under the induction hypothesis functionals ψ 1 · .... · ψ n−2 ⊗ ψ n−1 ∧ ψ n vanishes on Z(X) with ψ n−1 constantial and ψ n -valuational.
The last claim follows from the following identity: for arbitrary constantial functional α :
C * Q −→ Q, p ∈ C and ϕ(t) ∈ C(t) α(ϕ)ψ p (1 − ϕ) − α(1 − ϕ)ψ p (ϕ) = q∈C α(p − q)(ψ p (ϕ)ψ q (1 − ϕ) − ψ p (1 − ϕ)ψ q (ϕ)).
Indeed, from this identity we deduce that ψ 1 · .... · ψ n−2 ⊗ α ∧ ψ p equals to q∈C α(p − q)ψ 1 · .... · ψ n−2 ⊗ ψ p ∧ ψ q on Z(X). The latter number vanishes by the induction hypothesis.
To complete the proof of the lemma it remains only to prove the identity above. For this let
ϕ(t) = A ϕ q∈C (t − q) ψq(ϕ) , 1 − ϕ(t) = A 1−ϕ q∈C (t − q) ψq(1−ϕ) .
Since α is constantial, it vanishes on all monomials (t − λ), so α(ϕ) = α(A ϕ ) and
α(1 − ϕ) = α(A 1−ϕ ). The function ϕ(t) ψp(1−ϕ) (1 − ϕ(t)) ψp(ϕ) equals to 1 in p, so q∈C (p − q) ψp(ϕ)ψq(1−ϕ)−ψp(1−ϕ)ψq(ϕ) = A ψp(1−ϕ) ϕ A ψp(ϕ) 1−ϕ .
The identity follows by taking the value of functional α of both parts. Now we derive theorem 1.4 from lemma 3.3.
Proof. We prove it by induction on n. For n = 1 we need to show that the sequence
0 −→ A 1 I1 −→ Q[C(t)] 1 J1 −→ A
is exact. An element F ∈ A 1 is a linear combination n i [ϕ i ], such that n i log|ϕ i (t)| = const. We may assume that n i are integers. From the previous identity it follows that log | ϕ ni i | = const. By Liouville's theorem we deduce that ϕ ni i = const, so n i (ϕ i ) = 0 in A. This means that J 1 • I 1 = 0. Similarly one can show that Ker(J 1 ) = Im(I 1 ).
Next suppose that theorem 1.4 is proved for n = k − 1. Let us show that J k • I k = 0. For arbitrary F ∈ Q[C(t)] k lying in the image of A k and for any p ∈ C by lemma 3.3 D p F lies in A k−1 , so by inductive hypothesis D p • J k F = J k−1 • D p F = 0. By corollary 3.2 ⊕D p is injective, so J k F = 0. To check that Ker(J k ) = Im(I k ) take F ∈ Ker(J k ). Then D p F ∈ Ker(J k−1 ), so D p F ∈ A k−1 . By lemma 3.3 we deduce that F ∈ A k .
Proof of theorem 1.6
We need to show that the following sequence is exact:
0 −→ A 4 I4 −→ Q[C(t)] 4 J4 −→ S 3,1 * A K4 −→ (A ∧ A) ⊗ B 2 (C) −→ 0, where K 4 (x 1 x 2 x 3 ⊗ y) = σ∈S3 x σ(1) ∧ x σ(2) ⊗ {r(x σ(1) , y, x σ(3) , ∞)} 2 .
We will introduce a filtration on S 3,1 * V similar to the one, we used in the proof of theorem 1.5:
S 3,1 * V 1 ⊆ S 3,1 * V 2 ⊆ S 3,1 * V 3 ⊆ S 3,1 * V 4 , where S 3,1 * V i is generated by tensors [x 1 ][x 2 ][x 3 ] ⊗ [x 4 ] ∈ S 3,1 * V such that between x j there is at most i different. Let gr i S 3,1 * V be associated graded factors S 3,1 * V i / S 3,1 * V i−1 .
For simplicity, we denote gr i S 3,1 * A just by gr i . Let gr i denote the factor of gr i by the image of morphism J 4 .
By theorem 1.4, the sequence
0 −→ A 4 I4 −→ Q[C(t)] 4 J4 −→ S 3,1 * A is exact. Next, K 4 is surjective, since K 4 1 2 [a] 2 [b] ⊗ a − bz 1 − z = a ∧ b ⊗ {z} 2 .
Lemma 4.1. The composition K 4 • J 4 vanishes.
Proof. To show that K 4 • J 4 = 0 let us recall the Rogers identity: for arbitrary f (t) ∈ C(t) the following equality holds in B 2 (C) :
{f (t)} 2 − {f (∞)} 2 = a,b∈C ψ a (f )ψ b (1 − f ){r(b, t, a, ∞)} 2 .
From this equality it follows that
K 4 • J 4 ([f ]) = K 4 ([f ] 3 ⊗ [1 − f ]) = x1,x2,x3,y∈C ψ x1 (f )ψ x2 (f )ψ x3 (f )ψ y (1 − f )K 4 (x 1 x 2 x 3 ⊗ y) = σ∈S3 x1,x2,x3,y∈C ψ x1 (f )ψ x2 (f )ψ x3 (f )ψ y (1 − f )x σ(1) ∧ x σ(2) ⊗ {r(x σ(1) , y, x σ(3) , ∞)} 2 = 3 · x1,x2,x3,y∈C ψ x1 (f )ψ x2 (f )ψ x3 (f )ψ y (1 − f )x 1 ∧ x 2 ⊗ ({r(x 1 , y, x 3 , ∞)} 2 − {r(x 2 , y, x 3 , ∞)} 2 ) = 3 · x1,x2∈C ψ x1 (f )ψ x2 (f )x 1 ∧ x 2 ⊗ ({f (x 1 )} 2 − {f (x 2 )} 2 ).
The last sum vanishes, since if ψ x1 = 0, then {x 1 } 2 = 0 and if ψ x2 = 0, then {x 2 } 2 = 0.
It remains only to show that ImJ 4 = KerK 4 .
Lemma 4.2. gr 1 = 0, gr 2 = 0, gr 4 = 0.
Proof. S 3,1 * V 1 = 0, since all elements aaa⊗ a ∈ Sym 4 A, so gr 1 = 0. Next,
J 4 t − a b − a = [a] 3 ⊗ [b], J 4 t − a t − b = −[a − b] 3 ⊗ [b], so [a] 3 ⊗ [b] = 0 and [a] 2 [b] ⊗ [b] = [a][b] 2 ⊗ [b] ∈ gr 2 . Since in S 3,1 * A holds equality 3[a] 2 [b] ⊗ [a] + [a] 3 ⊗ [b] = 0, we deduce that gr 2 = 0.
Let a 1 , a 2 , a 3 , b 1 , b 2 , b 3 be six points on CP 1 such that there exists a projective involution I which translates
a i to b i . Let's denote by a 1 a 2 a 3 b 1 b 2 b 3 the rational function (t − a 1 )(t − b 1 )(a 3 − a 2 )(a 3 − b 2 ) (t − a 2 )(t − b 2 )(b 3 − a 1 )(b 3 − b 1 )
. This is exactly the unique quadratic function q such that q • I = q and q(a 1 ) = 0, q(a 2 ) = ∞, q(a 3 ) = 1.
It is easy to see that in gr 4 the following holds:
J 4 a 1 a 2 a 3 b 1 ∞ b 3 = ([a 1 ]+[b 1 ]−[a 2 ]) 3 ⊗([a 3 ]+[b 3 ]−[a 2 ]) = −6[a 1 ][b 1 ][a 2 ] ⊗ ([a 3 ] + [b 3 ])
. For four points a, b, c, d ∈ CP 1 consider three involutions: I a which maps a to ∞ and b to c, I b which maps b to ∞ and a to c, and I c which maps c to ∞ and b to a. Then, obviously, these involutions are commuting and I a • I b • I c = id. From this it follows that involution I a maps I b (d) to I c (d).
Consider the following element of S 3,1 * A :
X = a b d c ∞ I b (d) + a c d b ∞ I c (d) − b a I b (d) c ∞ I c (d) .
Then
J 4 (X) = −6[a][b][c]⊗([d]+[I b (d)])−6[a][b][c]⊗([d]+[I c (d)])+6[a][b][c]⊗([I c (d)]+[I b (d)]) = −12[a][b][c]⊗[d].
From this it follows that gr 4 = 0.
In view of lemma 4.2 we need only to show that K 4 :
gr 3 −→ (A ∧ A) ⊗ B 2 (C)
is an isomorphism. For this we will construct a map L 4 in reverse direction. We define it on A ⊗ A ⊗ Z[C] by formula
L 4 ([a] ⊗ [b] ⊗ {z} 2 ) = 1 2 [a] 2 [b] ⊗ a − bz 1 − z .
Obviously, K 4 •L 4 = id, so theorem 1.6 will be proven if we show that L 4 is correctly defined on (A∧A)⊗B 2 (C). For this we need to check four identities in gr 3 : three more simple and one much harder. The last one is the following "five-term" identity:
L 4 [a] ∧ [b] ⊗ 4 i=0 (−1) i r(x 0 , ...,x i , ..., x 4 ) = 0.
The following lemma contains three more simple identities:
Lemma 4.3.
In gr 3 theimage of the following three elements under the map L 4 vanish:
[a] ⊗ [b] ⊗ {z} 2 + [b] ⊗ [a] ⊗ {z} 2 , [a] ⊗ [b] ⊗ {z} 2 + [a] ⊗ [b] ⊗ 1 z 2 , [a] ⊗ [b] ⊗ {z} 2 + [a] ⊗ [b] ⊗ {1 − z} 2 .
Proof. The space gr 3 is generated by elements
[a] 2 [b] ⊗ [c], because 2[a][b][c] ⊗ [a] = −[a] 2 [b] ⊗ [c] − [a] 2 [c] ⊗ [b]. Since J 4 (t − a)(c − b) (t − b)(c − a) = 3[a][b] 2 ⊗ [c] − 3[a] 2 [b] ⊗ [c], in gr 3 we have [a][b] 2 ⊗ [c] = [a] 2 [b] ⊗ [c]. It is easy to see that J 4 (t − a)(t − b) (d − a)(d − b) = ([a] + [b]) 3 ⊗ ([d] + [a + b − d]) = 6[a] 2 [b] ⊗ ([d] + [a + b − d])
. From this the equality L 4 ((a ⊗ b + b ⊗ a) ⊗ {z} 2 ) = 0 follows. The second identity follows from the fact that
L 4 (a ∧ b ⊗ {z}) = L 4 (b ∧ a ⊗ { 1 z }). Finally, J 4 (t − b) 2 z(1 − z) (t − a)(a − b) = (2[b] − [a]) 3 ⊗ a − bz 1 − z + a − b + bz z − [a] = = −6[b][a] 2 ⊗ a − bz 1 − z + a − b + bz z
. This is equivalent to the third identity.
It remains to show that the "five term" identity holds. For this we need to do some preparation. Consider arbitrary distinct five points x 1 , ..., x 5 on P 1 C . They determine fifteen other points on P 1 C in the following way: for each choice of two distinct pairs out of these points one can construct an image of the fifth point under the involution, interchanging points in pairs. For instance, if we pair x 1 with x 2 and x 3 with x 4 we construct a point y such that there exist an involution sending x 1 to x 2 , x 3 to x 4 and x 5 to y. This point will be denoted by symbol [x 1 x 2 |x 3 x 4 ]. Note that this symbol makes sense only when all the five initial points are given. It is not changed after interchanging of the first pair of elements, second pair of elements and two pairs with each other. The fifteen points constructed all together form an interesting configuration on P 1 C . The following lemma summarizes its main properties.
Lemma 4.4. An involution, interchanging x 1 with x 2 and x 3 with x 4 also interchanges [
x 1 x 2 |x 3 x 4 ] with x 5 . The same involution interchanges [x 1 x 3 |x 2 x 4 ] with [x 1 x 4 |x 2 x 3 ].
An involution, which fixes x 1 and interchanges
x 2 with x 3 , also interchanges [x 1 x 4 |x 2 x 3 ] with [x 1 x 5 |x 2 x 3 ].
Proof. The first statement of the lemma was checked in the proof of lemma 4.2 via the trick with three commuting involutions. Let us check the second statement. For this denote point [x 1 x 4 |x 2 x 3 ] by A and [x 1 x 5 |x 2 x 3 ] by B. Cross ratio is preserved by a projective involution, so r(
x 1 , x 2 , x 3 , A) = r(x 4 , x 3 , x 2 , x 5 ) = r(B, x 2 , x 3 , x 1 ) = r(x 1 , x 3 , x 2 , B)
. From this the claim of the lemma follows.
Using this lemma we will construct three types of quadratic rational functions, which we will use to construct an element in Q[C(t)] 4 whose image under J 4 will coincide with an image under the map L 4 of
[a] ∧ [b] ⊗ 4 i=0 (−1) i r(x 0 , ...,x i , ..., x 4 ) .
Suppose that distinct five points x 1 , ..., x 5 ∈ P 1 C are given. Let q[x 1 x 2 |x 3 x 4 ] be a quadratic function with zeros The next lemma is the central part of the proof of the theorem 1.6. To simplify its formulation and proof let us make some conventions. Suppose that five points x 1 , x 2 , x 3 , x 4 and y are given. Quadratic functions, like q[x 1 x 3 |x 2 y], should be understood with respect to these five points. By we will mean summation over 24 elements of the group S 4 of all permutations of symbols x i in the summand. For instance,
q[x 1 x 2 , x 3 , y] means σ∈S4 q[x σ(1) x σ(2) , x σ(3) , y].
Lemma 4.5. The element
− 1 2 [x 1 ][x 2 ] 2 + 1 3 [x 1 ][x 2 ][x 3 ] ⊗ ([x 1 x 2 |x 3 x 4 ] + [x 1 x 3 |x 2 x 4 ] + [x 1 x 4 |x 2 x 3 ]) coincides with − 1 2 [x 1 ][x 2 ] 2 + 1 3 [x 1 ][x 2 ][x 3 ] ⊗ ([x 1 ] + [x 2 ] + [x 3 ] + [x 4 ] − [y]) modulo the image of J 4 .
Proof. Denote these elements by X and Y . They lie in Sym 3 A ⊗ A Sym 4 A . Throughout the proof we will work in gr 3 ,
so will suppose that [a] 3 ⊗ [b], [a] 2 [b] ⊗ [a], [a] 2 [b] ⊗ [b], and [a] 2 [b] ⊗ [c] − [a][b] 2 ⊗ [c] vanish.
First, notice that the difference of X and Y is homogeneous, that is is not changed if some [λ] in A is added to all variables in X − Y . To see it notice that
[x 1 x 2 |x 3 x 4 ] + [x 1 x 3 |x 2 x 4 ] + [x 1 x 4 |x 2 x 3 ] − [x 1 ] − [x 2 ] − [x 3 ] − [x 4 ] + [y]
is of degree zero and
− 1 2 [x 1 ][x 2 ] 2 + 1 3 [x 1 ][x 2 ][x 3 ] = 1 36 (2[x 1 ] − [x 2 ] − [x 3 ]) 3 .
From the homogeneity of X − Y and the fact that all elements in the image of J 4 are homogenious as well it follows that it is enough to prove the statement of the lemma, supposing that y = ∞ and neglecting it.
I claim that 6(X − Y ) coincides with the image under J 4 of the following element G:
q[x 1 x 2 |x 3 ∞] + 1 2 q[x 2 1 |x 2 x 3 ] + 1 4 q[∞ 2 |x 1 x 2 ] − 1 2 q[x 2 1 |x 2 ∞].
To check it we need to compute J 4 of each term. I claim that
1 6 J 4 q[x 1 x 2 |x 3 ∞] = 1 6 ([x 1 ]+[x 2 ]−[x 3 ]) 3 ⊗([x 1 x 2 |x 3 x 4 ]+[x 4 ]−[x 3 ]) = ([x 1 ][x 2 ] 2 −[x 1 ][x 2 ][x 3 ])⊗([x 1 x 2 |x 3 ∞]+ [x 4 ] − [x 3 ]). 1 6 J 4 q[x 2 1 |x 2 x 3 ] = −([x 1 ][x 2 ] 2 + [x 2 ][x 3 ] 2 + [x 3 ][x 1 ] 2 − 2[x 1 ][x 2 ][x 3 ]) ⊗ ([x 2 x 3 |x 1 x 4 ] + [x 2 x 3 |x 1 ∞] − [x 3 ] − [x 2 ]). 1 6 J 4 q[∞ 2 |x 1 x 2 ] = −[x 1 ][x 2 ] 2 ⊗ ([x 1 x 2 |x 3 ∞] + [x 1 x 2 |x 4 ∞]) 1 6 J 4 q[x 2 1 |x 2 ∞] = −[x 1 ][x 2 ] 2 ⊗ ([x 1 x 3 |x 2 ∞] + [x 1 x 4 |x 2 ∞]) For instance, 1 6 J 4 q[x 2 1 |x 2 ∞] = 1 6 (2[x 1 ] − [x 2 ] − [∞]) 3 ⊗ ([x 1 x 3 |x 2 ∞] + [x 1 x 4 |x 2 ∞] − [x 2 ] − [∞]) = 1 6 (2[x 1 ] − [x 2 ]) 3 ⊗([x 1 x 3 |x 2 ∞]+[x 1 x 4 |x 2 ∞]−[x 2 ]) = 1 6 (8[x 1 ] 3 −12[x 1 ] 2 [x 2 ]+6[x 1 ][x 2 ] 2 −[x 2 ] 3 )⊗([x 1 x 3 |x 2 ∞]+[x 1 x 4 |x 2 ∞]− [x 2 ]) = −[x 1 ][x 2 ] 2 ⊗ ([x 1 x 3 |x 2 ∞] + [x 1 x 4 |x 2 ∞]).
The first thing to check is that in 6J 4 G all terms of the type * ⊗[x 1 , x 2 |x 3 , ∞] vanish. For this let's combine all the terms of this type in J 4 G. Contribution from q[x 1 x 2 |x 3 ∞] will be 2([
x 1 ][x 2 ] 2 −[x 1 ][x 2 ][x 3 ])⊗[x 1 , x 2 |x 3 , ∞]. Term 1 2 q[x 2 1 |x 2 x 3 ] will give −([x 1 ][x 2 ] 2 + [x 2 ][x 3 ] 2 + [x 3 ][x 1 ] 2 − 2[x 1 ][x 2 ][x 3 ]) ⊗ [x 1 x 2 |x 3 ∞]. The contribution from 1 4 q[∞ 2 |x 1 x 2 ] will be −[x 1 ][x 2 ] 2 ⊗ [x 1 x 2 |x 3 ∞] and − 1 2 q[x 2 1 |x 2 ∞] will give ([x 2 ][x 3 ] 2 + [x 1 ][x 3 ] 2 ) ⊗ [x 1 x 2 |x 3 ∞]. Since 2([x 1 ][x 2 ] 2 − [x 1 ][x 2 ][x 3 ]) − ([x 1 ][x 2 ] 2 + [x 2 ][x 3 ] 2 + [x 3 ][x 1 ] 2 − 2[x 1 ][x 2 ][x 3 ]) − [x 1 ][x 2 ] 2 + [x 2 ][x 3 ] 2 + [x 1 ][x 3 ] 2 vanishes, the statement follows.
The remaining check of the fact that (X − Y ) coincides 1 6 J 4 G is straightforward. Now we are ready to prove the "five-term" identity. Summation of these four elements shows that P lies in the image of J 4 .
From lemmas 4.3 and 4.6 we conclude that L 4 is correctly defined map from (A ∧ A) ⊗ B 2 (C), to S 31 A ImJ 4 , such that L 4 • K 4 vanishes on S 31 A ImJ 4 . From this it follows that Ker(K 4 ) = Im(J 4 ) and the theorem 1.6 is proved.
Definition 1 . 3 .
13The group of single-variable functional equations for Li n (z) is denoted by A n . It is a sub-vector space of Q[C(t)] and characterized by the following property:
x 1 and x 2 , poles x 3 and x 4 and equal to 1 in x 5 and [x 1 x 2 |x 3 x 4 ]. Next, q[x 1 x 2 ||x 3 x 4 ] will be a quadratic function with zeros x 1 and x 2 , poles x 3 and x 4 and equal to 1 in [x 1 x 3 |x 2 x 4 ] and [x 1 x 4 |x 2 x 3 ]. Finally, q[x 2 1 |x 2 x 3 ] will be a quadratic function double zero x 1 , poles x 2 and x 3 and equal to 1 in [x 1 x 4 |x 2 x 3 ] and [x 1 x 5 |x 2 x 3 ].
Lemma 4. 6 .
6The image under L 4 of [a] ∧ [b] ⊗ 4 i=0 (−1) i {r(x 0 , ...,x i , ..., x 4 )} 2 lies in the image of J 4 .Proof. Without loss of generality we may suppose that x 1 = a,x 2 = b, x 3 = c, x 4 = d and x 5 = ∞. By definition of L the image of the element a ∧ b ⊗ (a, x, b, ∞) equals 1 2 [a] 2 [b] ⊗ [x] and of the element a ∧ b ⊗ (b, x, a, ∞) equals − 1 2 [a] 2 [b] ⊗ [x]. By lemma 4.3, the following equalities of cross-ratios hold:r(b, c, d, ∞) = r(∞, a, [ca|b∞], b) = r(b, [ca|b∞], a, ∞), r(a, c, d, ∞) = r(∞, b, [cb|a∞], a) = r(a, [cb|a∞], b, d) = 1 − r(a, c, b, ∞) = 1 − r(b, [ab|d∞], a, ∞). So, 4 i=0 (−1) i {r(x 0 , ...,x i , ..., x 4 )} 2 = {r(b, c, d, ∞)} 2 − {r(a, c, d, ∞)} 2 + {r(a, b, d, ∞)} 2 − {r(a, b, c, ∞)} 2 + {r(a, b, c, d)} 2 . By lemma 4.3, L 4 (a ∧ b ⊗ {r(b, c, d, ∞)} 2 ) = −L 4 (a ∧ b ⊗ {r(a, [ca|b∞], b, ∞)} 2 ) = − 1 2 [a] 2 [b] ⊗ [ca|b∞], L 4 (a ∧ b ⊗ {r(a, c, d, ∞)} 2 ) = L 4 (a ∧ b ⊗ {r(a, [cb|a∞], b, ∞)} 2 ) = 1 2 [a] 2 [b] ⊗ [cb|a∞], L 4 (a ∧ b ⊗ {r(a, b, d, ∞)} 2 ) = −L 4 (a ∧ b ⊗ {r(a, d, b, ∞)} 2 ) = − 1 2 [a] 2 [b] ⊗ [d], L 4 (a ∧ b ⊗ {r(a, b, c, ∞)} 2 ) = −L 4 (a ∧ b ⊗ {r(a, c, b, ∞)} 2 ) = − 1 2 [a] 2 [b] ⊗ [c], L 4 (a ∧ b ⊗ {r(a, b, c, d)} 2 ) = L 4 (a ∧ b ⊗ {r(b, a, ∞, [ab, c∞])} 2 ) = − 1 2 [a] 2 [b] ⊗ [ab|d∞]. So, we need to show that an element P = [a] 2 [b] ⊗ ([ca|b∞] + [cb|a∞] + [ab|c∞] + [d] − [c])lies in the image on J 4 . For this let us apply lemma 4.5 to five elements x 1 = a, x 2 = b, x 3 = c, x 4 = ∞, y = d. we will have the following relation in gr 3 :
(−[a] 2 [b] − [b] 2 [c] − [c] 2 [a] + [a][b][c]) ⊗ ([ca|b∞] + [cb|a∞] + [ab|c∞]) = (−[a] 2 [b] − [b] 2 [c] − [c] 2 [a] + [a][b][c]) ⊗ ([a] + [b] + [c] − [d]). It is easy to see that image under J 4 of the element [ba||c∞] + [ab|c∞] equals ([a] 2 [b] − [a][b][c]) ⊗ ([ca|b∞] + [cb|a∞] + [ab|c∞] + [d] − 2[c]), so in gr 3 we have an equality (−[a] 2 [b] + [a][b][c]) ⊗ ([ca|b∞] + [cb|a∞] + [ab|c∞]) = (−[a] 2 [b] + [a][b][c]) ⊗ (2[c] − [d]). [c] − [a][b][c]) ⊗ ([ca|b∞] + [cb|a∞] + [ab|c∞]) = ([a] 2 [c] − [a][b][c]) ⊗ (2[b] − [d]), ([b] 2 [c] − [a][b][c]) ⊗ ([ca|b∞] + [cb|a∞] + [ab|c∞]) = ([b] 2 [c] − [a][b][c]) ⊗ (2[a] − [d]).Similarly we have
([a] 2
Acknowledgments. I would like to thank S. Bloch, A. Suslin, A. Beilinson and A. Goncharov for invaluable
Interpretation motivique de la conjecture de Zagier. A A Beilinson, P Deligne, Symp. in Pure Math., v. 552Beilinson A.A., Deligne P, Interpretation motivique de la conjecture de Zagier. Symp. in Pure Math., v. 55, part 2, (1994).
Polylogarithms and motivic Galois group. A B Goncharov, Proc. AMS Conference on Motives. AMS Conference on MotivesGoncharov, A. B. Polylogarithms and motivic Galois group. Proc. AMS Conference on Motives.
Dedekind zeta functions and the algebraic K-theory of fields. D Zagier, Polylogarithms, Proceedings of the Texel conference on Arithmeti-cal Algebraic Geometry. Zagier D., Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields. Proceedings of the Texel conference on Arithmeti-cal Algebraic Geometry, (1990).
| [] |
[
"High-energy emission from pulsars in polar-cap models with CR-induced cascades",
"High-energy emission from pulsars in polar-cap models with CR-induced cascades"
] | [
"B Rudak \nN. Copernicus Astronomical Center\nRabiańska 887-100ToruńPoland\n",
"J Dyks \nN. Copernicus Astronomical Center\nRabiańska 887-100ToruńPoland\n"
] | [
"N. Copernicus Astronomical Center\nRabiańska 887-100ToruńPoland",
"N. Copernicus Astronomical Center\nRabiańska 887-100ToruńPoland"
] | [] | For a subclass of polar-cap models based on electromagnetic cascades induced by curvature radiation (CR) we calculate broad-band high-energy spectra of pulsed emission expected for classical and millisecond pulsars. The spectra are a combination of curvature and synchrotron components. The spectrum of curvature component breaks at 150 MeV, and neither its slope nor level below this energy are compatible with phase-averaged spectra of pulsed X-ray emission inferred from observations. Spectral properties in the combined energy range of ROSAT and ASCA (0.1 -10 keV) depend upon the location of cyclotron turnover energy ǫ ct =h eB mec / sin ψ in the synchrotron component. Unlike in outer-gap models, the available range of pitch angles ψ is rather narrow and confined to low values. For classical pulsars, a gradual turnover begins already at ∼ 1 MeV, and the level of the synchrotron spectrum decreases. At ∼ 10 keV the curvature component eventually takes over, but with photon index α = 2/3, in disagreement with observations. For millisecond pulsars, the X-ray spectra are dominated by synchrotron component with α ≃ 1.5, and a sharp turnover into α ≃ −1 at ǫ ct ∼ 100 eV. Relations of pulsed luminosity L X to spin-down luminosity L sd are presented for classical and millisecond pulsars. We conclude that spectral properties and fluxes of pulsed non-thermal X-ray emission of some objects, like the Crab or the millisecond pulsar B1821-24, pose a challenge to the subclass of polar-cap models based on curvature and synchrotron radiation alone. | 10.1046/j.1365-8711.1999.02329.x | [
"https://arxiv.org/pdf/astro-ph/9812212v1.pdf"
] | 119,352,634 | astro-ph/9812212 | f520f442ae60d5053ddcbdea720e981769e6f06e |
High-energy emission from pulsars in polar-cap models with CR-induced cascades
Printed 13 June 2021
B Rudak
N. Copernicus Astronomical Center
Rabiańska 887-100ToruńPoland
J Dyks
N. Copernicus Astronomical Center
Rabiańska 887-100ToruńPoland
High-energy emission from pulsars in polar-cap models with CR-induced cascades
Printed 13 June 2021Accepted 1998 November 30. Received 1998 June 2; in original form 1997 March 4arXiv:astro-ph/9812212v1 10 Dec 1998 Mon. Not. R. Astron. Soc. 000, 000-000 (0000) (MN L A T E X style file v1.4)pulsars: general -X-rays: observations -X-rays: theory
For a subclass of polar-cap models based on electromagnetic cascades induced by curvature radiation (CR) we calculate broad-band high-energy spectra of pulsed emission expected for classical and millisecond pulsars. The spectra are a combination of curvature and synchrotron components. The spectrum of curvature component breaks at 150 MeV, and neither its slope nor level below this energy are compatible with phase-averaged spectra of pulsed X-ray emission inferred from observations. Spectral properties in the combined energy range of ROSAT and ASCA (0.1 -10 keV) depend upon the location of cyclotron turnover energy ǫ ct =h eB mec / sin ψ in the synchrotron component. Unlike in outer-gap models, the available range of pitch angles ψ is rather narrow and confined to low values. For classical pulsars, a gradual turnover begins already at ∼ 1 MeV, and the level of the synchrotron spectrum decreases. At ∼ 10 keV the curvature component eventually takes over, but with photon index α = 2/3, in disagreement with observations. For millisecond pulsars, the X-ray spectra are dominated by synchrotron component with α ≃ 1.5, and a sharp turnover into α ≃ −1 at ǫ ct ∼ 100 eV. Relations of pulsed luminosity L X to spin-down luminosity L sd are presented for classical and millisecond pulsars. We conclude that spectral properties and fluxes of pulsed non-thermal X-ray emission of some objects, like the Crab or the millisecond pulsar B1821-24, pose a challenge to the subclass of polar-cap models based on curvature and synchrotron radiation alone.
INTRODUCTION
According to recent results from ROSAT and ASCA (Becker &Trümper 1997 andSaito 1998, respectively) luminosities of pulsed and unpulsed components in the X-ray emission from pulsars are related to spin-down luminosities L sd , suggesting thus a rotation-powered origin of X-rays. The unpulsed emission is usually interpreted as synchrotron radiation from an unresolved nebula surrounding the pulsar. A magnetospheric wind of ultrarelativistic particles (with Lorentz factors about 10 7 ) will lead to X-ray emission provided the nebula contains magnetic field of the order of ∼ 10 −4 G (e.g. Manning & Willmore (1994) estimate the magnetic field strength of 0.17 × 10 −4 G for the hypothetical nebula around B0950+08).
Pulsed components are of particular interest, since they are the direct signatures of non-thermal processes within a magnetosphere. Very likely, pulsed non-thermal X-ray emission is nothing but a low-energy tail of gamma-ray emission, which in most models is a superposition of curvature (CR) and synchrotron (SR) emission. In a subclass of polar cap models developed by Dermer & Sturner (1994) gamma-ray photons have different origin -they are thermal UV/X-ray photons subject to Lorentz-boost by inverse Compton scattering (IC) with beam particles.
In polar-cap scenarios pulses in gamma-rays should be accompanied by pulses in X-rays, with similar shapes, and in phase. With such criteria the Crab pulsar would be a model example. However, one has to show first that its contributions to X-rays and gamma-rays are in right proportions, i.e. in (at least qualitative) accord with model predictions.
Phase-averaged broad-band (Xγ) spectra of six, out of seven, gamma-ray pulsars are relatively steep, with photon index αXγ decreasing with pulsar's characteristic age (see Thompson et al. 1997 for recent review): For the Crab αXγ ≈ 2, and therefore, luminosities per logarithmic energy bandwidth in X-rays and in gamma-rays are comparable. For B1055-52, which is the oldest gamma-ray pulsar, αXγ ≈ 1.6, and its pulsed X-ray luminosity LX is roughly one per cent of its Lγ .
[Note: To make this comparison we followed the convention used in gamma-rays (Ωγ = 1 sr is the solid angle of gammaray emission) taking ΩX = 1 sr for the solid angle of X-ray emission.]
These observed spectral properties have been quite successfully accounted for in a newly proposed model of 'thick outer gap' by Zhang & Cheng (1997). Unlike polar-cap models, outer-gap models [see also Wang et al. (1997)] may take advantage of the full range of pitch angles, which in turn makes it possible to trigger electromagnetic cascades even for relatively soft gamma-ray photons ( > ∼ 1 MeV) on magnetic field lines (also on closed ones). The energy distribution function for e ± pairs produced in the cascades may easily become steep enough (N (γ±) ∝ γ −3 ± ) over a wide range of energy, for subsequent synchrotron spectrum (αXγ = 2) to account for the observed soft gamma-rays and X-rays. In particular, Cheng et al. (1998) managed to reconstruct simple empirical relation between LX and L sd , which reads (assuming ΩX = 4π sr) LX ≃ 10 −3 L sd (Becker & Trümper 1997). Moreover, cascades propagating starward are subject to substantial lateral spread, leading thus to rather weak X-ray modulations.
The aim of this paper is to present general features of broad-band Xγ spectra expected for the subclass of polar cap models, with Xγ emission due to CR and SR (e.g. Daugherty & Harding 1982, Daugherty & Harding 1996 and to show that luminosity of pulsed X-rays observed for some pulsars is too high compared to Lγ (or to L sd if there is no information about gamma-rays) to be understood within polar-cap models unless some unorthodox assumptions are accepted. Some of these features are only weakly modeldependent and may, therefore, play a decisive role in asessing validity of polar-cap models. We were motivated by the fact, that neither proponents nor oponents of the polar-cap models seem to acknowledge a real challenge posed to these models by observed properties of non-thermal, pulsed X-ray emission. It is this emission which is more promising as a potential discriminator between rivalring classes (polar-cap, outer-gap) of models than gamma-ray emission itself. In Section 2 we present model spectra of CR and SR emission per parent (beam) particle, expected in pulsars with high (∼ 10 12 G) and low (∼ 10 9 G) magnetic fields. Section 3 compares LX expected within energy band 0.1 keV−10 keV with observations for a wide range of spin-down luminosities L sd . Section 4 contains discussion and conclusions.
SPECTRAL PROPERTIES OF CURVATURE AND SYNCHROTRON RADIATION
The energy distribution Ne(γ) of particles is governed by their injection rate Qe(γ) (also called a source function), cooling rateγ, and a characteristic time scale of their escape tesc, via steady-state kinetic equation
∂ ∂γ (|γ|Ne) = Ne/tesc − Qe,(1)
where γ is particle energy in units of mec 2 . The solution of eq.1 yields two asymptotic relations
Ne(γ) = 1 |γ| γmax γ Qe(γ)dγ, |γ|/γ ≫ t −1 esc ; Qe · tesc, |γ|/γ ≪ t −1 esc .(2)
If the solution to eq.1 is a power-law function with index p, Ne ∝ γ −p , the subsequent photon spectrum of SR or CR will have a form
Nν (ǫ) ∝ ǫ − p+1 q ,(3)
(but the value of p must not be too low) where q = 2 for SR, or q = 3 for CR.
[Note: Throughout this section we use a short notation for electron and photon distribution functions: Ne(γ) = dNe/dγ and Nν (ǫ) = dNν /dǫ, respectively.]
Curvature Radiation
First, let us consider spectral component due to CR. If the source function Qe(γ) of beam particles is monoenergetic
Qe(γ) = C0 δ(γ0 − γ),(4)
and redistribution over the energy space is governed by the CR cooling (|γ| = |γcr| ∝ γ 4 ), then from the first relation of eq.2 it follows that Ne ∝ γ −4 (i.e. p = 4) for γ < γ0. The lower limit (let us denote it as γ break ) to this distribution is determined by the condition, that a cooling time-scale due to CR, tcr ≡ γ/|γcr|, is shorter than tesc. According to eq.3, the photon spectrum of CR becomes Ncr(ǫ) ∝ ǫ − 5 3 . It remains unchanged down to photon energy ǫ break , which may be approximated with a characteristic energy of CR
ǫcrit = 3 2 ch γ 3 ρcr ,(5)
(where ρcr is a local radius of curvature) taken for γ break . Below ǫ break , the photon spectrum follows the low-energy tail of CR, i.e. Ncr(ǫ) ∝ ǫ − 2 3 . In order to find ǫ break we should first estimate the time scale tesc. We assume that tesc ≈ ρcr/c. The CR cooling rate for a single particle of energy γ iṡ
γcr = − 2 3 e 2 mc γ 4 ρ 2 cr .(6)
The condition tcr = tesc becomes then
γ 3 break ρcr = 3 2 mc 2 e 2 ,(7)
and from eq.5 it follows that
ǫ break = 9 4h c r0 ≈ 150 MeV,(8)
where r0 is the classical electron radius. Note, that the photon energy ǫ break at which the spectral break occurs does not depend on any pulsar parameters, as long as our estimation of tesc is accurate. In particular, it does not depend on magnetic field structure -should it be pure dipole with ρcr > ∼ 10 8 √ P cm, or dominated by high-order multipolar components with ρcr ∼ 10 6 cm. Fig.1 shows numerically calculated shapes of the CR spectrum produced by beam particles injected at the outer rim of a canonical polar cap with a dipolar magnetic field. The magnetic strength Bpc at the polar cap is 10 9 G (upper panel), and 10 12 G (lower panel). The period of rotation P is 0.003 s and 0.06 s, respectively. For the initial energy E0 we choose 1.1 × 10 7 MeV and 6.7 × 10 6 MeV, respectively (see section 3 for explanation). The dotted line corresponds to the unabsorbed spectrum, and the solid line is the spectrum with magnetic absorption. The spectral break (where the power-law spectrum changes its slope by one power of ǫ) is visible in both panels, and its location at ∼ 150 MeV agrees quite well with the analytical estimate (eq.8). A word of comment on high-energy cutoffs seems appropriate here, though it is not linked directly to the subject of this paper: The high-energy cutoff for the case of 10 12 G does not exceed 10 GeV, whereas the cutoff for the low-B case reaches ∼ 0.1 TeV (the energy range accessible with ground-based Cherenkov techniques). The explanation is twofold: (1) for low values of B the magnetic absorption is weak; (2) very short periods P infer small dipolar curvature radii, and a characteristic photon energy (eq.5) taken for γ0 = E0/mec 2 increases.
With the spectral break at ǫ break ≈ 150 MeV, the curvature radiation becomes energetically unimportant below this energy. Such objects would be dim in X-rays, which apparently is not the case. Moreover, the slope −2/3 in the X-ray energy range is in clear conflict with spectral analysis of X-ray data (Becker & Trümper 1997).
A way out may be offered by including synchrotron radiation (next subsection) and/or by relaxing the assumption about the monoenergetic source function of beam particles (eq.4). Suppose, for example, that we want the slope of −5/3 to extend down to energy ǫx in the X-ray range: Nν (ǫ) ∝ ǫ − 5 3 for ǫ > ∼ ǫx. According to eq.2 (lower relation) we now have the energy distribution Ne(γ) = Qe · tesc. This, in turn, requires that the source function of beam particles is
Qe ∝ γ −4 , for γ > ∼ γx,(9)
a very steep function of γ (for whatever the reason), where γx is linked to ǫx via eq.5. To find γx we solve the equation ǫx = ǫcrit(γx) for γx: For dipolar magnetic fields, the smallest curvature radii ρcr are those at outer rim of the polar cap. They may be approximated with ρcr ≈ √ r RL, where r is the radial coordinate of the particle, and RL = c P/2π is the pulsar's light cylinder. Within the region of r ≃ 2RNS we have therefore tesc ≈ 10 −2.5 √ P s. From eq.5 we obtain
ǫx ≈ 0.3 (γx/10 5 ) 3 P −0.5 keV.(10)
For a pulsar with P = 0.1 s, the source function Qe(γ) ∝ γ −4 would have to extend down to γx ∼ 10 5 for ǫx to reach ∼ 1 keV. This value of γx is about 55 times lower than the value of γ break (eq.7) corresponding to ǫ break (eq.8).
Synchrotron Radiation
Since CR is not a promising candidate for hard X-rays, the standard explanation involves SR, produced by e ± pairs created in the process of magnetic absorption of high-energy CR photons. Total contribution of the SR component to the overall (SR+CR) energy of radiation per beam particle depends on number of created pairs n±. One may expect, therefore, that at least for pulsars with strong magnetic fields (∼ 10 12 G), the energy contained in SR should be comparable or exceed the energy contained in CR. However, spectral properties of the SR component depend not only on local values of B, but also on the available range of pitch angles ψ between magnetic field lines and the direction of propagation of created pairs. The most popular assumption -about an isotropic distribution -may be fully justified in the case of outer-gap models (Zhang & Cheng 1997), but is not obvious by any means for polar-cap scenarios. Below, we'll present numerical calculations of synchrotron spectra due to e ± -pairs which are assumed (after Daugherty & Harding 1982) to be directed along the direction of propagation of their parent CR-photons at the moment of the creation. This assumption, along with narrow opening angles of magnetic field lines available for beam particles, results in a confinement of pitch angles to a narrow range of low values. For a pulsar with a dipolar magnetic field and rotating with period P , we expect then sin ψ ≤ 0.01/ √ P as a first approximation. The character of the source function Q± of e ± -pairs depends primarily on the reachness of the cascades. In the case of low magnetic fields (∼ 10 9 G) only one generation of e ± -pairs is produced, and their formation region is narrow. For these reasons Q± is almost monoenergetic (see Fig.1, upper panel), and basic spectral properties of SR calculated for a canonical millisecond pulsar are easily understood with well known analytical considerations:
Let us denote by γ the particle Lorentz factor along the local magnetic field line. We will ignore possible curvature of the line, and assume that γ remains constant. The particle energy (in units of mec 2 ) to be emitted in bursts of SR corresponds then to Lorentz factor γ ⊥ of gyration. The total particle energy is γ = γ ⊥ γ . As long as γ ≫ 1, the pitch angle of the particle is determined by sin ψ ≈ γ −1 .
The rate of SR cooling iṡ
γsr = − 2 3 r 2 0 mec B 2 γ 2 ⊥ = − 2 3 r 2 0 mec B 2 sin 2 ψ γ 2 .(11)
Unlike in the case of CR, the SR cooling rate is enormous and affects the energy distribution of pairs until γ ⊥ ∼ 1 where the synchrotron approximation breaks (O'Dell & Sartori 1970). Therefore, from eqs.2, 11 and 3, we'll get photon spectrum of SR with well known single power-law shape Nsr(ǫ) ∝ ǫ − 3 2 . Main contribution to the spectrum at energy ǫ comes from particles with γ for which ǫ = ǫsr(γ), where
ǫsr = 3 2 ch eB mec γ 2 sin ψ,(12)
is a critical photon energy in SR. The spectrum spreads between a high-energy cutoff ǫsr(γmax) and a low-energy turnover ǫct determined by the condition γ ⊥ ∼ 1 (O'Dell & Sartori 1970):
ǫct ≡ ǫsr(γ = γ ) = 3 2 ch eB mec 1 sin ψ .
Below ǫct, the spectrum flattens, and may be described asymptotically as Nsr(ǫ) ∝ ǫ +1 . It is built up by contributions from low-energy tails emitted by particles with γ ⊥ ≫ 1, and each low-energy tail is assumed to cut off at local gyrofrequency, which in the reference frame comoving with the center of gyration is ωB = eB mec γ ⊥ . The numerical example presented in the upper panel of Fig.1 reveals all these 'classical textbook' features so clearly, because (for reasons mentioned before) the source function of pairs in the case of a dipolar field with Bpc = 10 9 G is practically monoenergetic. The turnover energy is located Figure 1. The radiation energy spectrum per logarithmic energy bandwidth. The spectrum is normalized to the energy of the parent particle, E 0 . It consists of two components: curvature and synchrotron. Dotted line is the curvature spectrum before correction for the magnetic absorption effects. Solid line connecting filled dots, is the curvature spectrum with the magnetic absorption taken into account. Solid line without any symbols overlayed is the synchrotron component. Dot-dashed line indicates superposition of the two. In addition to the electromagnetic spectra, we show the spectra of e ± pairs: the line connecting filled squares is for the energy parallel to local magnetic field lines (labeled with E ); the line connecting open squares is the initial distribution of energy perpendicular to local magnetic lines (labeled with E ⊥ ). The upper panel is for Bpc = 10 9 G, and E 0 = 1.08×10 7 MeV (see eq.14) which corresponds to P = 3.1×10 −3 s (i.e. L sd = 10 35 erg s −1 ). The lower panel is for Bpc = 10 12 G, and E 0 = 6.68 × 10 6 MeV which corresponds to P = 5.6 × 10 −2 s (i.e. L sd = 10 36 erg s −1 ) around 0.1 keV, i.e. in the soft range of X-rays. More importantly, the SR component dominates over the CR component in the entire energy range of X-rays. The lower panel of Fig.1 shows analogous results obtained for Bpc = 10 12 G. The created pairs belong now to two generations, and they are rich (∼ 400 times more numerous than for 10 9 G). Therefore, the SR component is now energetically comparable to CR. There is no single, well defined turnover energy ǫct anymore.
The spectrum of the source function Q± is spread over two decades in energy, and in consequence the SR spectrum reveals a gradual turnover, which starts already at ∼ 1 MeV, due to high values of γ as well as strong local B (see eq.13).
Our approximate treatment of SR spectral shapes at low-energy limit requires a word of comment: As electrons go to the ground Landau level, the spectrum reveals its harmonic structure and the analytical formula that we used, should be treated with caution. Comparison of Monte Carlo spectra (calculated by summing-up the rates of the quantum transitions), with analytical formulae was carried out by Harding & Preece (1987). They found that the analytical formula for the low-energy tail of SR with a cutoff at cyclotron frequency generally overestimates (but not dramatically) the actual level of photon spectrum. Therefore, our results for SR in the context of X-rays should be treated rather as upper limits, but this does not change our conclusions.
RELATIONS BETWEEN X-RAYS AND GAMMA-RAYS
We calculated combined spectra of CR and SR (as shown in Fig.1), emitted by a beam particle with initial energy E0, for two canonical values of Bpc : 10 9 G and 10 12 G, and for a range of periods P in order to cover a full possible range of spin-down luminosities L sd . Formula for the energy E0 was adopted from Rudak & Dyks (1998) [RD98]. This energy is only a few times higher than threshold energy Emin required for magnetic pair creation in a dipolar magnetic field:
E0 = 2.5 × Emin,(14)
with Emin = 1.2 × 10 7 B 10 12 G −1/3
P 1/3 MeV,(15)
but at the same time it must never exceed any of the following limits -Emax and Ew (the first restriction is due to curvature cooling, the second one is due to potential drop across the polar cap). The original motivation for this formula was to reproduce Lγ for the seven gamma-ray pulsars. For 10 9 G, E0 ≈ 10 7 MeV, while L sd covers the range between 10 34 erg s −1 and 10 37 erg s −1 . In practice, the values of E0 do not change dramatically over the allowed ranges of L sd : For 10 12 G, E0 starts with 3 × 10 6 MeV, increases to 9.8×10 6 MeV, and goes down to 4×10 6 MeV, for L sd changing from 10 31 erg s −1 to 10 39 erg s −1 . What became clear already in the previous section is that the range of X-rays is never energetically important. Although for 10 9 G the spectrum of SR extends well into soft X-ray band and dominates there, its fractional contribution to the total radiation energy output is low. It varies between 0.003 (for L sd = 10 34 erg s −1 ) and 0.22 (for 10 37 erg s −1 ). For the case of 10 12 G, the energy content of SR becomes significant for L sd > 10 33 erg s −1 , but the spectrum of SR is now confined to gamma-rays, and it turns over at < ∼ 1 MeV. In either case, the bulk of particle energy converted into radiation is concentrated within gamma-rays.
Gamma-ray luminosity Lγ in RD98 was identified with the power of outflowing particles
L particles = η E±n±ṄGJ,(16)
where E± (characteristic energy attained by a fraction of secondary particles, η n±, due to possible acceleration) was assumed to be of the order of E0, andṄGJ was the Goldreich-Julian rate of outflow of beam particles. The parameter η = 0.004 was used to match the model with the joint EGRET and COMPTEL luminosity inferred for B1951+32 (see RD98 for detalis). Here we made more accurate calculations for Lγ , by replacing the particle energy E0 with its radiation energy yield Eγ for photon energy Figure 2. Evolution of high-energy, non-thermal luminosity across the spin-down luminosity space. The long upper curve connecting filled dots is the track of gamma-ray luminosity Lγ (> 100 keV), and the long lower curve connecting filled squares is the track of X-ray luminosity L X (0.1 − 10 keV), both calculated according to eq.17, for a pulsar with Bpc = 10 12 G. [The dotted line is the track of L particles (eq.16), which in RD98 was identified with Lγ .] The short upper curve with open circles is for gamma-ray luminosity Lγ (> 100 keV), and the short lower curve with open squares is for X-ray luminosity L X (0.1 − 10 keV), calculated according to eq.17, for Bpc = 10 9 G. The short-dashed line marks L = L sd . The long-dashed line, marking the empirical relation of Becker & Trümper (1997) rewritten for Ω X = 1 sr, has been added for reference. The fluctuations of L X (0.1 − 10 keV) (for 10 12 G only) between 10 36 erg s −1 and 10 37 erg s −1 in L sd , are of numerical origin: in this range of L sd the low-energy tail of the synchrotron component crosses the curvature component just around 10 keV, and any fluctuations due to Monte Carlo treatment of cutting off synchrotron spectra influence the level of the total spectrum.
ǫ ≥ 100 keV. Similarly, we calculate expected X-ray luminosity LX within (0.1 keV − 10 keV). Accordingly, we take
Lγ = L particles × Eγ E0 , and LX = L particles × EX E0 ,(17)
where Eγ E 0 and E X E 0 are fractional radiation energy yields per particle, calculated by integrating the differential (per logarithmic energy bandwidth) spectra [which are simply
1 E0 ǫ dNν d ln ǫ(18)
-see Fig.1 for two examples], over the range ǫ ≥ 100 keV and 0.1 keV ≤ ǫ ≤ 10 keV, respectively.
Extremely low values of relative energy output in Xray photons and gamma-ray photons per particle imply equally low values of LX /Lγ (between 10 −6 and 10 −4 ). Fig.2 presents Lγ and LX calculated according to eqs.17, 18, and 16 as a function of L sd . It shows unambiguously, that some pulsars from the list of X-ray and gamma-ray source do not fit the picture: The Crab pulsar, for which the observed ratio of LX /Lγ is about 0.47, remains certainly a challenge. Also among millisecond pulsars, there is a 'Crab-like' example, which does not fit the model. B1821-24 observed with ASCA (Saito et al. 1997) reveals clear, double-peaked pulsations resembling those of the Crab. Apart of the disagreement between the observed spectral slope (∼ 1.9) and the predicted one (∼ 1.5), the predicted ratio of LX /Lγ ∼ 2 × 10 −4 (we took LX for 1sr) would make Lγ to exceed L sd . Or, alternatively, using the upper limit for the gamma-ray luminosity, Lγ < 0.028 L sd (Nel et al. 1996), the "observed" ratio LX /Lγ is > 2.5 × 10 −3 , in disagreement with predictions.
However, among pulsars there are also strong sources of steady X-ray emission, with no trace of pulsations. B1706-44, a strong gamma-ray pulsar detected with EGRET, with Lγ = 2.5 × 10 34 erg s −1 (Thompson et al. 1996) shows no pulsations in X-rays. Its unpulsed, non-thermal X-ray emission has the luminosity LX = 1.3 × 10 33 erg s −1 (Becker et al. 1993). Apparently, pulsed X-rays of B1706-44, which we predict to reach the luminosity of ∼ 3 × 10 29 erg s −1 , are dwarfed by nearby nebular steady emission.
CONCLUSIONS
Recent results from ROSAT and ASCA suggest that the Xray spectra of pulsars may be a superposition of thermal and non-thermal components (see e.g. Zavlin & Pavlov 1997 for discussion). The existence of these two components is in general expected in both classes of rival models (polar-cap and outer-gap models; e.g. Arons 1981 andCheng, Ho &Ruderman 1986, respectively) and the thermal component results from polar cap heating by inflowing particles. However, clear empirical relations between thermal and non-thermal components are yet to be determined. The latest review on the status of pulsar' X-ray properties by Becker & Trümper (1997) favoures pure non-thermal spectral models (with four exceptions of additional thermal components due to initial cooling of the star itself).
Within the framework of polar-cap models with magnetospheric activity induced by curvature radiation of beam particles we have calculated broad-band photon spectra of non-thermal origin, for both classical and millisecond pulsars. This non-thermal emission is a superposition of curvature and synchrotron radiation. It is expected to be pulsed, like its high-energy extension -the gamma-ray emission. Our objective was to estimate spectral properties and the level of this emission in the energy domain of X-rays.
Our calculations show, that for beam particles with monoenergetic source function, the properties of photon spectra in the combined energy range for ROSAT and ASCA (0.1 -10 keV) depend primarily on the magnetic field strength at the polar cap. Millisecond pulsars, with dipolar magnetic fields of the order of 10 9 G, have X-ray spectra dominated by synchrotron emission. On the other hand, pulsars with B of the order of 10 12 G have X-ray spectra almost exclusively due to low-energy tail of curvature emission. However, the spectrum of the curvature component breaks already at 150 MeV regardless the pulsar parameters, and neither its slope nor level in the X-ray domain is compatible with phaseaveraged spectra of pulsed X-ray emission inferred from observations.
Detailed properties of the broad-band spectrum, depend upon the location(s) of cyclotron turnover energy ǫct =h eB mec / sin ψ of the synchrotron emission. Unlike in outer-gap models, the available range of pitch angles ψ is rather narrow and confined to low values (sin ψ < 0.01/ √ P ). We find that for classical pulsars (10 12 G) the upper limit for turnover energy ǫct occurs already at ∼ 1 MeV. Below this energy, the level of synchrotron spectrum gradually decreases, and around ∼ 10 keV the curvature component takes over. Unless the spectrum of primary beam particles is very steep (N b ∝ γ −4 b ) and extends down to Lorentz factor γ b ≈ 10 5 , the X-ray spectrum below ∼ 10 keV is a powerlaw with photon index α ph = 2/3. Therefore, the model is not able to explain the non-thermal X-ray spectra with α ph ≃ 1.5÷2 inferred from observations (Becker & Trümper 1997). The expected luminosity LX of pulsed component is a negligible fraction of gamma-ray luminosity Lγ , and for L sd ≥ 10 33 erg s −1 an approximate relation log LX ≈ −1.5 + 0.83 log L sd
follows from Fig.2. The situation is qualitatively different for millisecond pulsars. The X-ray spectra are dominated now by synchrotron components, with a photon index α ph ≃ 3/2, which extends down to ǫct ∼ 100 eV before breaking sharply into α ph ≃ −1. The expected LX is still low with respect to Lγ , though not as low as for classical pulsars. For L sd ≥ 10 34 erg s −1 we find log LX ≈ −40.3 + 1.90 log L sd .
We conclude that spectral properties and fluxes of pulsed non-thermal X-ray emission of some objects, like the Crab or the millisecond pulsar B1821-24, pose a real challenge to the subclass of polar-cap models based on curvature and synchrotron radiation alone.
c 0000 RAS, MNRAS 000, 000-000
ACKNOWLEDGEMENTSThis work has been financed by the KBN grant 2P03D-00911. BR acknowledges discussions with K.S. Cheng, J. Gil, and W. Kluźniak. We thank the anonymous referee for useful suggestions which helped to clarify the paper.
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| [] |
[
"ELECTROMAGNETIC WAVE PROPAGATION IN GLHUA INVISIBLE SPHERE BY GL NO SCATERING FULL WAVE MODELING AND INVERSION",
"ELECTROMAGNETIC WAVE PROPAGATION IN GLHUA INVISIBLE SPHERE BY GL NO SCATERING FULL WAVE MODELING AND INVERSION"
] | [
"Jianhua Li \nChinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China\n",
"Feng Xie \nChinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China\n",
"Lee Xie \nChinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China\n",
"Ganquan Xie \nChinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China\n",
"Ganquan Xie \nChinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China\n"
] | [
"Chinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China",
"Chinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China",
"Chinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China",
"Chinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China",
"Chinese Dayuling Supercomputational Sciences Center\nGL Geophysical Laboratory\nUSA, China"
] | [] | Using GL no scattering full wave modeling and inversion, we create a GLHUA pre cloak electromagnetic (EM) material in the virtual sphere that makes the sphere is invisible. The invisible sphere is called GLHUA sphere. In GLHUA sphere, the Pre cloak relative parameter is not less than 1; the parameters and their derivative are continuous across the boundary r=R2 and the parameters are going to infinity at origin r=0. The phase velocity of EM wave in the sphere is less than light speed and going to zero at origin. The EM wave field excited in the outside of the sphere can not be disturbed by GLHUA sphere. By GL full wave method, we rigorously proved the incident EM wave field excited in outside of GLHUA sphere and propagation through the sphere without any scattering by the sphere, the total EM field in outside of the sphere equal to the incident wave field. Moreover, we prove that in GLHUA sphere with the pre cloak material, when r is going to origin, EM wave field propagation in GLHUA sphere is going to zero. All copyright and patent of the GLHUA EM cloaks,GLHUA sphere and GL modeling and inversion methods are reserved by authors in GL Geophysical Laboratory. | null | [
"https://arxiv.org/pdf/1701.02583v2.pdf"
] | 119,383,095 | 1701.02583 | bb0dac8041450b6a1253158caa3433d417b7e799 |
ELECTROMAGNETIC WAVE PROPAGATION IN GLHUA INVISIBLE SPHERE BY GL NO SCATERING FULL WAVE MODELING AND INVERSION
26 Dec 2016 (Dated: December 26,2016)
Jianhua Li
Chinese Dayuling Supercomputational Sciences Center
GL Geophysical Laboratory
USA, China
Feng Xie
Chinese Dayuling Supercomputational Sciences Center
GL Geophysical Laboratory
USA, China
Lee Xie
Chinese Dayuling Supercomputational Sciences Center
GL Geophysical Laboratory
USA, China
Ganquan Xie
Chinese Dayuling Supercomputational Sciences Center
GL Geophysical Laboratory
USA, China
Ganquan Xie
Chinese Dayuling Supercomputational Sciences Center
GL Geophysical Laboratory
USA, China
ELECTROMAGNETIC WAVE PROPAGATION IN GLHUA INVISIBLE SPHERE BY GL NO SCATERING FULL WAVE MODELING AND INVERSION
26 Dec 2016 (Dated: December 26,2016)arXiv:1701.02583v1 [physics.class-ph]
Using GL no scattering full wave modeling and inversion, we create a GLHUA pre cloak electromagnetic (EM) material in the virtual sphere that makes the sphere is invisible. The invisible sphere is called GLHUA sphere. In GLHUA sphere, the Pre cloak relative parameter is not less than 1; the parameters and their derivative are continuous across the boundary r=R2 and the parameters are going to infinity at origin r=0. The phase velocity of EM wave in the sphere is less than light speed and going to zero at origin. The EM wave field excited in the outside of the sphere can not be disturbed by GLHUA sphere. By GL full wave method, we rigorously proved the incident EM wave field excited in outside of GLHUA sphere and propagation through the sphere without any scattering by the sphere, the total EM field in outside of the sphere equal to the incident wave field. Moreover, we prove that in GLHUA sphere with the pre cloak material, when r is going to origin, EM wave field propagation in GLHUA sphere is going to zero. All copyright and patent of the GLHUA EM cloaks,GLHUA sphere and GL modeling and inversion methods are reserved by authors in GL Geophysical Laboratory.
I. INTRODUCTION
Using GL no scattering modeling and inversion and many GL method simulation [1] [2] [3], we find a class nonzero relative anisotropic parameter solution of EM zero scattering inversion in the sphere r ≤ R 2 We create a novel material in GLHUA sphere with relative EM parameter not less than 1 that makes the sphere is invisible. The parameters and their derivative are continuous across the boundary r = R 2 and the parameters are going to infinity at origin r = 0. The phase velocity of EM wave in the sphere is less than light speed and going to zero at origin. We discovered and proved an essential property that in the local sphere r ≤ R O without EM source, including R 2 < r < R O annular layer in free space and r ≤ R 2 in GLHUA sphere, on any spherical surface with radius r < R O , the spherical surface integral of E r sin θ and H r sin θ is zero, (E r is radial electric wave field, H r is radial magnetic wave field). The essential property of EM wave is a key difference from the acoustic wave and seismic wave. Based on the essential property, by GL full wave method, we rigorously proved the incident EM wave field excited in the outside of GLHUA sphere and propagation through the sphere without any scattering from GLHUA sphere, the total EM field in the outside of GLHUA sphere equal to the incident wave field. EM wave field excited in the outside of the sphere can not * Also at GL Geophysical Laboratory, USA, [email protected] be disturbed by GLHUA sphere. GLHUA sphere is complete invisible. Moreover, based on the above essential property of EM wave, we prove that in GLHUA sphere with the pre cloak material, when r is going to origin, EM wave field propagation in GLHUA sphere is going to zero. We propose a special GL transform to map GL-HUA outer annular layer cloak to GLHUA sphere. The anisotropic relative parameters of EM material in physical GLHUA outer annular layer cloak, ε p,r = µ p,r = 1,
ε p,θ = 1 2 r−R1 R2−R1 + R2−R1 r−R1
, ε p,θ = ε p,φ = µ p,θ = µ p,φ are mapping to anisotropic relative parameter material in GLHUA invisible sphere that satisfy GLHUA pre invisible cloak material conditions. From the theoretical proof in GLHUA invisible sphere, we can rigorously prove GLHUA outer annular layer cloak is invisible cloak with concealment. The major new ingredients in this paper are proposed in 7 sections.The introduction is presented in the section 1. Second order Maxwell electromagnetic equation in anisotropic material in spherical coordinate, and the basic fundamental electromagnetic wave in free space is presented in the section 2. The Global and Local (GL) method for radial electromagnetic equation and the essential property of radial EM wave are presented in the section 3. In the section 4, we propose GL EM wave Greens equation and GL EM Greens function. GL EM Integral equation and its theoretical proof are proposed in the section 5. In the section 6, we propose GLHUA EM invisible sphere, in this section, we rigorously proved that EM wave field propagation is going to zero at origin in GLHUA sphere. In section 7, the discussion and conclusion is presented.
∇ × E = 1 r 2 sin θ r r θ r sin θ φ ∂ ∂r ∂ ∂θ ∂ ∂φ E r rE θ r sin θE φ = −iω (H r , H θ , H φ ) µ r µ θ µ φ µ 0 (1) ∇ × H = 1 r 2 sin θ r r θ r sin θ φ ∂ ∂r ∂ ∂θ ∂ ∂φ H r rH θ r sin θH φ = iω (E r , E θ , E φ ) ε r ε θ ε φ ε 0 + δ( r − r s ) e s (2) ∇ · B = 1 r 2 ∂ ∂r r 2 µ r H r + 1 sin θr ∂ ∂θ sin θµ θ H θ + 1 r sin θ ∂ ∂φ µ φ H φ = 0,(3)∇ · D = 1 r 2 ∂ ∂r r 2 ε r E r + 1 sin θr ∂ ∂θ sin θε θ E θ + 1 r sin θ ∂ ∂φ ε φ E φ = 4πρ,(4)
where ρ is the electric charge, ε 0 is the basic constant electric permittivity, µ 0 is the basic constant magnetic permeability, ε r , relative radial electric permittivity, ε θ and ε φ , relative angular electric permittivity, µ r , relative radial magnetic permeability, µ θ and µ φ , relative angular magnetic permeability, E r is the radial electric field, E θ is the electric field in angular θ direction, E φ is the electric field in angular φ direction, E = (E r , E θ , E φ ) is the electric field vector, D is the electric displacement vector, B is the magnetic flux vector, H r is the radial magnetic field, H θ is the magnetic field in angular θ direction, H φ is the magnetic field in angular φ direction, H = (H r , H θ , H φ ) is the magnetic field vector, ω = 2πf is the angular frequency, f is the frequency Hz, r = (r, θ, φ), the vector in spherical coordinate.
∂φ 2 + k 2 µ θ ε r r 2 E r = − 1 sin θ ∂ ∂θ sin θ ∂ ∂r 1 iωε θ rJ θ − 1 sin θ ∂ ∂φ ∂ ∂r 1 iωε θ rJ φ − ∂ ∂r 1 iωε θ ∂ ∂r r 2 J r − (−iωµ θ )r 2 J r ,(5)∂φ 2 + k 2 ε θ µ r r 2 H r = 1 sin θ ∂ ∂θ sin θrJ φ − 1 sin θ ∂ ∂φ rJ θ ,(6)
D. Second Order Maxwell Electromagnetic Equation On Angular Electric Wave Field In Spherical Coordinate
After solving equation (5) and (6) and obtaining the radial electric wave field E r (r, θ, φ) and radial magnetic wave field H r (r, θ, φ), the following second order Maxwell electromagnetic equation govern angular electric wave field E θ (r, θ, φ) and E φ (r, θ, φ) ∂ ∂r After solving equation (5) and (6) and obtaining the radial electric wave field E r (r, θ, φ) and radial magnetic wave field H r (r, θ, φ) , the following second order Maxwell electromagnetic equation govern angular magnetic wave field H θ (r, θ, φ) and H φ (r, θ, φ)
1 (µ θ ) ∂rE θ ∂r + k 2 ε θ rE θ = −iωµ 0 1 sin θ ∂Hr ∂φ + ∂ ∂r 1 µ θ ∂Er ∂θ −iωµ 0 rJ θ (7) ∂ ∂r 1 (µ θ ) ∂rE φ ∂r + k 2 ε θ rE φ = ∂ ∂r 1 (µ θ ) 1 sin θ ∂Er ∂φ + iωµ 0 ∂Hr ∂θ −iωµ 0 rJ φ (8) E. Second∂ ∂r 1 ε θ ∂ ∂r rH θ + k 2 µ θ rH θ = iωε 0 1 sin θ ∂Er ∂φ + ∂ ∂r 1 ε θ ∂Hr ∂θ + ∂ ∂r 1 ε θ rJ φ , (9) ∂ ∂r 1 ε θ ∂ ∂r rH φ + k 2 µ θ rH φ = 1 sin θ ∂ ∂r 1 ε θ ∂Hr ∂φ − iωε 0 ∂Er ∂θ − ∂ ∂r 1 ε θ rJ θ ,(10)
F. Fundamental acoustic wave field Let g( r, r s ) is the fundamental acoustic wave field, which is solution of following acoustic equation
∂ ∂r r 2 ∂g ∂r + 1 sin θ ∂ ∂θ sin θ ∂g ∂θ + 1 sin 2 θ ∂ 2 g ∂φ 2 +k 2 r 2 g = δ(r − r s ) δ(θ−θs) sin θ δ(φ − φ s ),(11)g( r, r s ) = − 1 4π e −ik| r− rs| | r − r s | (12) | r − r s | = | r − r s | 2 , | r − r s | 2 = r 2 + r 2 s −2rr s sin θ sin θ s cos(φ − φ s ) −2rr s cos θ cos θ s ,(13)
where r s = (r s , θ s , φ s ) is point source location, r = (r, θ, φ) is variable spherical coordinate, i.e. observation point.
G. Basic fundamental electromagnetic wave field in free space
The basic fundamental electromagnetic wave field in free space can be excited by current source J = δ( r − r s ) e j , or magnetic moment source M = δ( r − r s ) e j .We consider the current source in this paper, similarly theorem and proof are suitable for the magnetic moment source. Let ε r = ε θ = ε φ = 1 and µ r = µ θ = µ φ = 1, electric current source J = δ( r − r s ) e j ,The basic fundamental electromagnetic wave field in free space are solutions of equations (5)-(6) in free space,
E b j = 1 iωε 0 ∇∇ · (g e j ) + k 2 g e j (14) E b j,r = 1 iωε0 e jr ∂ 2 g ∂r 2 + k 2 g + e jθ ∂ ∂r 1 r ∂g ∂θ +e jφ 1 iωε0 ∂ ∂r 1 r sin θ ∂g ∂φ (15) H b j = ∇ × (g e j ), H b j,r = e jφ 1 r ∂g ∂θ − e jθ 1 r sin θ ∂g ∂φ ,(16)
where the above equation are in the spherical coordinate system, g( r, r s ) is denoted in (11)-(13), ∇· is diverge operator E b j is basic fundamental electric wave, , H b j is basic fundamental magnetic wave, which is excited by source J = δ( r − r s ) e j , the up script b means basic fundamental electric wave, the lower script j means source with excited vector e j . E b j,r is radial basic fundamental electric wave, i.e. r component of E b j , H b j,r is radial basic fundamental magnetic wave, i.e. r component of H b j , e j , j=1,2,3, is a unite vector,
e 1 = e x = sin θ cos φ cos θ cos φ − sin φ , e 2 = e y = sin θ sin φ cos θ sin φ cos φ , e 3 = e z = cos θ − sin θ 0 ,(17)
III. GLOBAL AND LOCAL (GL) METHOD FOR RADIAL ELECTROMAGNETIC EQUATION AND THE ESSENTIAL PROPERTY OF THE RADIAL ELECTROMAGNETIC WAVE
In the section 1, we proposed second order Maxwell electromagnetic equation in anisotropic material in spherical coordinate. The fundamental electromagnetic wave field in free space in sphere coordinate is presented in section 2. These basic equations are used in this and next sections.
A. Global and Local GL radial electric second order differential equation
For R 2 > 0, we consider electromagnetic equation (5)-(10) in the sphere r ≤ R 2 with anisotropic media, in the outside of the sphere, r > R 2 with basic isotropic electric permittivity ε 0 and magnetic permeability µ 0 in free space. In our paper, we suppose that the electromagnetic source set is bounded, the bounded source set Ω s is in outside of the large sphere with radius R O , R O > R 2 ,for example, the point source located in outside of the sphere, r s > R O > R 2 .
Using Global and Local (GL) field method, we propose radial GL electromagnetic field and GL electromagnetic equation. and study the following GL radial electromagnetic equation in the spherer ≤ R 2 .
Definition of GL radial electromagnetic wave field
E( r) = ε r r 2 E r ( r), H( r) = µ r r 2 H r ( r),(18)
From the radial electric equation (5) , we propose GL radial electric second order differential equation
∂ ∂r 1 ε θ ∂ ∂r E + 1 εrr 2 1 sin θ ∂ ∂θ sin θ ∂E ∂θ + 1 εr r 2 1 sin 2 θ ∂ 2 E ∂φ 2 + k 2 µ θ E = J s ,(19)where k = 2πf √ ε 0 µ 0 , J s = δ(r − r s ) 1 sin θ δ(θ − θ s )δ(φ − φ s )e r electric point source r s > R O . The incident GL electric wave field in free space, E b j ( r) = r 2 E b j,r = 1 iωε0 r 2 e jr ∂ 2 g ∂r 2 + k 2 g + 1 iωε0 r 2 e jθ ∂ ∂r 1 r ∂g ∂θ + e jφ ∂ ∂r 1 r sin θ ∂g ∂φ ,(20)
where g = g(r, r s ) is the fundamental solution of the acoustic wave equation, (11-13).
Let E b ( r) to denote one of E b j ( r) = r 2 E b j,r , j = 1, 2, 3, it obvious that lim r→0 E b ( r) = 0,(21)lim r→0 ∂ ∂r E b ( r) = 0,(22)
B. Global and Local GL radial magnetic field second order differential equation
By definition of GL magnetic wave H( r) = µ r r 2 H r ( r) in (18), from the radial magnetic equation (6), we propose GL radial magnetic field second order differential equation
∂ ∂r 1 µ θ ∂ ∂r H + 1 µr r 2 1 sin θ ∂ ∂θ sin θ ∂H ∂θ + 1 µr r 2 1 sin 2 θ ∂ 2 H ∂φ 2 + k 2 ε θ H = M s ,(23)
M s is magnetic point source, incident GL magnetic wave in free space is,
H b j = r 2 H b j,r = r e jφ ∂ ∂θ g − e jθ 1 sin θ ∂ ∂φ g ,(24)Let H b ( r) to denote one of the H b j ( r) = r 2 H b j,r , j = 1, 2, 3, it obvious lim r→0 H b ( r) = 0,(25)lim r→0 ∂ ∂r H b ( r) = 0,(26)
C. Spherical surface integral of incident GL electromagnetic wave is vanished
Define spherical surface integral of incident GL electric wave in free space as
E b 0 (r) = 1 4π π 0 2π 0 E b ( r) sin θdθdφ,(27)
Define spherical surface integral of incident GL magnetic wave in free space as
H b 0 (r) = 1 4π π 0 2π 0 H b ( r) sin θdθdφ,(28)
D. Essential property of GL radial electromagnetic wave T heorem 3.1 :, Suppose that the electromagnetic source set is bounded, the bounded source set is in outside of the sphere with large radius R O , r s > R O . In the no source domain, with weight sin θ spherical surface integral of incident radial GL electromagnetic wave is zero.
E b 0 (r) = 0, H b 0 (r) = 0,(29)
The spherical surface integral of radial GL electromagnetic wave is zero,
E 0 (r) = 0, H 0 (r) = 0,(30)
P roof :, By equation (19), the GL radial electric second order differential equation in free space is ∂ 2
∂r 2 E b + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂E b ∂θ + 1 r 2 1 sin 2 θ ∂ 2 E b ∂φ 2 + k 2 E b = S E(31)
The electromagnetic source S E is denoted by (5), (6), because the bounded source set is in outside of the sphere, with large radius R O ,r s > R O , there exist the no source domain r < R O , in the no source domain or for plane electromagnetic wave without source,S E = 0. Use sin θ times both sides of (31) and take spherical surface integral and by integral by parts, we get Linville ordinary equation
d 2 dr 2 E b 0 + k 2 E b 0 = 0,(32)
From (21) and (22), the initial condition is
lim r→0 E b 0 (r) = 0 (33) lim r→0 ∂ ∂r E b 0 (r) = 0,(34)
The equation system (32)-(34) has only zero solution. We have proved that E b 0 (r) is complete vanished
E b 0 (r) = 0,
Similarly, we have proved that H b 0 (r) is complete vanished,
H b 0 (r) = 0.
The first part (29) of the theorem 3.1 is proved. Next, we prove the second part of the theorem. Suppose that the electromagnetic source set is bounded, the bounded source set is in outside of the sphere with radius R O , in the GLHUA sphere, r ≤ R 2 < R O , the source term is zero in the right hand of GL electric equation (19). Use sin θ times both sides of (19) and take sphere surface integral and by integral by parts, we have ordinary equation
∂ ∂r 1 ε θ ∂ ∂r E 0 (r) + k 2 µ θ E 0 (r) = 0, r ≤ R 2 ,(35)
By above proof, we have initial condition E b 0 (R 2 ) = 0 and ∂ ∂r E b 0 (R 2 ) = 0. Because in the GLHUA sphere, electromagnetic material parameters and their derivative are continuous across the outer boundary r = R 2 , the radial GL electromagnetic wave and their derivative are continuous across the boundary r = R 2 .
E 0 (R 2 ) = E b 0 (R 2 ) = 0,(36)∂ ∂r E 0 (R 2 ) = ∂ ∂r E b 0 (R 2 ) = 0,(37)
The solution of ordinary differential equation (35) with zero initial boundary condition (36) and (37) must be zero. Therefore,E 0 (r) = 0 similar H 0 (r) = 0, we proved theorem 3.1 that spherical surface integral of GL electromagnetic wave is vanished.
T heorem 3.2 : Spherical surface integral of incident radial electromagnetic wave is zero. Spherical surface integral of radial electromagnetic wave is zero. P roof :, For EM point source and r s > R O , and r < R O ,also from (15) and (16), by direct integral, we can calculate
E b 0 r (r) = 1 4πr 2 π 0 2π 0 E b ( r) sin θdθdφ = 0,(38)H b 0 r (r) = = 1 4πr 2 π 0 2π 0 H b ( r) sin θdθdφ = 0,(39)E b 0 jr (r) = = 1 4π π 0 2π 0 E b jr ( r) sin θdθdφ = 0,(42)
and
H b 0 jr (r) = = 1 4π π 0 2π 0 H b jr ( r) sin θdθdφ = 0,(43)
Here, we direct integral of (43), because (16)
H b jr = 1 r 2 sin θ ∂ ∂θ (r sin θe jφ g) − 1 r 2 sin θ ∂ ∂φ (re jθ g), j = 1, 2, 3,
e j is an unit vector of the source, j=1,2,3, in (17),
H b 0jr (r) = 1 4π π 0 2π 0 H b jr ( r) sin θdθdφ = 1 4π 1 r 2 2π 0 π 0 ∂ ∂θ (r sin θe jφ g)dθdφ − 1 4π 1 r 2 π 0 2π 0 ∂ ∂φ (re jθ g)dφdθ = 0,(43)
is already proved by direct integral. Similarly, by direct integral, we can prove (42).
From theorem 3.1,E 0 (r) = 0, by definition of GL electromagnetic wave field E( r) = ε r r 2 E r ( r), H( r) = µ r r 2 H r ( r),
we have E 0,r ( r) = 1 ε r r 2 E 0 ( r) = 0 and H 0.r ( r) = 1 µ r r 2 H 0 ( r) = 0 . The theorem 3.2 is proved.
The spherical surface integral of incident radial electromagnetic wave is zero that is essential property. The key property is essential different between electromagnetic wave and acoustic wave. Note that GL electromagnetic wave (18) is not Maxwell electromagnetic field wave, also is not flux nor displace current. In the GL method, Global and Local virtual wave E and H in (18) is convenient under any coordinate transform. Global and Local virtual wave (18) and GL second order differential equation (19) (23) are important for study GL-HUA sphere and GLHUA cloak. It is shown that the GL electromagnetic differential equation and their incident wave (18)-(21) and (22)-(25) have sane equation form and theoretical properties. For simply, we study GL electric differential equation and its incident wave (18)-(21) in detail.
IV. GL ELECTROMAGNETIC GREENS EQUATION
A. We propose GL electromagnetic Greens equation for GL electric equation (18) and GL magnetic equation (23)
∂ ∂r ∂ ∂r G( r, r ′ ) + 1
r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ G( r, r ′ ) + 1 r 2 1 sin 2 θ ∂ 2 ∂φ 2 G( r, r ′ ) +k 2 G( r, r ′ ) = δ( r − r ′ ),(44)
B. GL electromagnetic Greens function
We find and propose GL Greens function G( r, r ′ ) which is the solution of above GL electromagnetic Greens equation (44),
G( r, r ′ ) = rr ′ g( r, r ′ ),(45)g( r, r ′ ) = − 1 4π e −ik| r− r ′ | | r − r ′ | ,(46)
Where
| r − r ′ | = | r − r ′ | 2 , | r − r ′ | 2 = r 2 − 2rr ′ sin θ sin θ ′ cos(φ − φ ′ ) −2rr ′ cos θ cos θ ′ + r ′2 ,
The GL Greens equation (44) and GL Greens function (45) are suitable for all global free space which is as background of local cloak.
C. The spherical surface integral of the GL electromagnetic Greens equation
We take
G 0 (r, r ′ ) = 1 4π π 0 2π 0 G( r, r ′ ) sin θdθdφ,(47)
Take sphere surface integral of GL electromagnetic Greens equation (44), then sphere surface integral of Greens function, G 0 (r, r ′ ), in (44) satisfy
∂ ∂r ∂ ∂r G 0 (r, r ′ ) +k 2 G 0 (r, r ′ ) = δ(r − r ′ ),(48)
We find spherical surface integral of GL Greens function G 0 (r, r ′ ) which is the solution of above equation (48),
G 0 (r, r ′ ) = rr ′ ikj 0 (kr) (j 0 (kr ′ ) − iy 0 (kr ′ )) , r ≤ r ′ , G 0 (r, r ′ ) = rr ′ ikj 0 (kr ′ ) (j 0 (kr) − iy 0 (kr)) , r ′ ≤ r,(49)
j 0 (kr) = sin kr kr , y 0 (kr) = − cos kr kr ,
V. GL ELECTROMAGNETIC INTEGRAL EQUATION
We propose the Global and Local GL integral equation on GL electric wave field E( r) in (18) in the sphere body
r ≤ R 2 E( r ′ ) = E b ( r ′ )− S(r≤R2) 1 − 1 ε θ ∂ ∂r G ∂ ∂r EdV + S(r≤R2) 1 r 2 1 − 1 εr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 (1 − µ θ ) GEdV ,(51)
The equivalent between GL integral equation (51) and GL electric wave differential equation (19) is proved next theorem 5.1. In integral equation (51), we change E to H, ε r to µ r , ε θ to µ θ , we obtain GL magnetic integral equation.
H( r ′ ) = H b ( r ′ ) − S(r≤R2) 1 − 1 µ θ ∂ ∂r G ∂ ∂r HdV + S(r≤R2) 1 r 2 1 − 1 µr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 HdV + S(r≤R2) k 2 (1 − ε θ ) GHdV ,(52)
A.
The equivalent between GL integral equation (51) and GL electric wave differential equation (19) T heorem 5.1 : Suppose that the radial electric wave E( r) is solution of the GL radial electric wave differential equation (19) with incident wave (20)-(22), Greens function G( r, r ′ ) in (45) satisfy the GL electromagnetic Greens differential equation (44), then E( r) satisfy the GL integral equation (51) P roof : By using Greens function G( r, r ′ ) in (45) to time the GL electric differential equation (19) and after doing some calculation and integral by part, we have
S(r≤R2) ∂ ∂r 1 ε θ ∂ ∂r E G dV − S(r≤R2) 1 ε θ ∂ ∂r E ∂ ∂r GdV + S(r≤R2) 1 εr r 2 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 µ θ GEdV = E b ( r)(53)
By using unknown wave function E( r, r s ) to time the GL Greens equation (44) and after do some calculation and integral by part, we have
S(r≤R2) ∂ ∂r ∂ ∂r G E dV − S(r≤R 2 ) ∂ ∂r G ∂ ∂r EdV + S(r≤R2) 1 r 2 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + R 3 k 2 GEdV = E( r),(54)
To subtract (53) from (54)
E( r ′ ) = E b ( r ′ ) + e −ikr ′ 4π π 0 2π 0 E(0, θ, φ) sin θdθdφ − S(r≤R2) 1 − 1 ε θ ∂ ∂r G ∂ ∂r EdV + S(r≤R2) 1 r 2 1 − 1 εr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 (1 − µ θ ) GEdV .(55)
Substitute (27) for sphere surface integral of electric wave into the (55), because the anisotropic inhomogeneous electromagnetic relative material parameter are variable in the sphere,r ≤ R 2 The outside of sphere, r > R 2 is free space with relative electric permittivity diag(1, 1, 1) and magnetic permeability diag(1, 1, 1) , because (30)
e −ikr ′E( r ′ ) = E b ( r ′ ) − S(r≤R2) 1 − 1 ε θ ∂ ∂r G ∂ ∂r EdV + S(r≤R2) 1 r 2 1 − 1 εr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 (1 − µ θ ) GEdV . (51)
The theorem 5.1 is proved.
VI. THEORY OF GLHUA ELECTROMAGNETIC INVISIBLE SPHERE AND BEHAVIOR OF THE ELECTROMAGNETIC WAVE FIELD PROPAGATION
A. GLHUA pre cloak material conditions on relative electric permittivity and magnetic permeability for invisible sphere
In this section, we propose GLHUA pre cloak material conditions of anisotropic relative electric permittivity and magnetic permeability for invisible virtual sphere r ≤ R 2 , which are of the following properties for invisible sphere r ≤ R 2 . The electric permittivity is the product of relative electric permittivity and ε 0 , the magnetic permeability is the product of relative magnetic permeability and µ 0 .ε 0 Is the basic constant electric permittivity, µ 0 is basic constant magnetic permeability. GLHUA pre cloak material conditions in invisible virtual sphere r ≤ R 2 are as follows:
(6.1)µ r (r) = ε r (r), µ θ (r) = µ φ (r) = ε θ (r) = ε φ (r) , in the sphere r ≤ R 2 are continuous differentiable function of r, (56) (6.2)these parameter functions and their derivative functions are continuous across boundary
r = R 2 ,(57)(6.3) lim r→0 r 2 µ r (r) = lim r→0 r 2 ε r (r) = ∞,(58)
(6.4)µ θ (r) = µ φ (r) = ε θ (r) = ε φ (r) = f (r) 1 r 2 , and their derivative are continuous across boundary
r = R 2 , lim r→0 f (r) = R 2 2 2 , lim r→0 1 r f ′ (r) = 0,(59)
B. GL electromagnetic field and are approaching to zero at r=0 in invisible sphere T heorem 6.1 : Suppose that the anisotropic relative electric permittivityε r (r),ε θ (r) ε φ (r) and magnetic permeability, µ r (r),µ θ (r) µ φ (r) , satisfy the above GL-HUA pre cloak material conditions in invisible sphere (6.1) to (6.4), also we suppose that
S(r≤R2) |E( r)| 2 + |H( r)| 2 dV isf inite,(60)
then GL radial electromagnetic wave field is vanished at origin,r = 0,
lim r→0 E( r) = 0,(61)lim r→0 H( r) = 0,(62)P roof lim r→0 r 2 E( r) = 0, lim r→0 r 2 H( r) = 0,(63)
is derived from (60). the additional condition (60) is reasonable finite energy condition. From GL radial electric integral equation (51)), we have
E( r ′ ) = E b ( r ′ )− − S(r≤R2) ∂ ∂r 1 − 1 ε θ ∂ ∂r G E dV + S(r≤R2) ∂ ∂r 1 − 1 ε θ ∂ ∂r G EdV + S(r≤R2) 1 r 2 1 − 1 εr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 (1 − µ θ ) GEdV,(64)
The integral equation (51) is translated to
1 ε θ E( r ′ ) = E b ( r ′ )− − S(r≤R2) ∂ ∂r 1 ε θ ∂G ∂r EdV + S(r≤R2) 1 − 1 ε θ ∂ 2 ∂r 2 G EdV + S(r≤R2) 1 r 2 1 − 1 εr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 (1 − µ θ )GEdV,(65)
The above equation (65) becomes
1 ε θ E( r ′ ) = E b ( r ′ )− − S(r≤R2) ∂ ∂r 1 ε θ ∂G ∂r EdV − S(r≤R2) 1 r 2 1 − 1 ε θ 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV − S(r≤R2) k 2 1 − 1 ε θ GEdV + S(r≤R2) 1 r 2 1 − 1 εr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 (1 − µ θ )GEdV.(66)
The GL electric integral equation (51) becomes to
1 ε θ E( r ′ ) = E b ( r ′ )− − S(r≤R2) ∂ ∂r 1 ε θ ∂G ∂r EdV + S(r≤R2) 1 r 2 1 ε θ − 1 εr 1 sin θ ∂ ∂θ sin θ ∂G ∂θ + 1 sin 2 θ ∂ 2 G ∂φ 2 EdV + S(r≤R2) k 2 1 ε θ − µ θ GEdV ,(67)
Let
G L (r, r ′ ) = G( r, r ′ ) − G 0 (r, r ′ )(68)
to substitute (68) for G( r, r ′ ) into (67) , and by using E 0 (r) = 0 in (30), the integral equation (67) becomes
1 ε θ E( r ′ ) = E b ( r ′ )− − S(r≤R2) ∂ ∂r 1 ε θ ∂GL ∂r EdV + S(r≤R2) 1 r 2 1 ε θ − 1 εr 1 sin θ ∂ ∂θ sin θ ∂GL ∂θ + 1 sin 2 θ ∂ 2 GL ∂φ 2 EdV + S(r≤R2) k 2 1 ε θ − µ θ G L EdV ,(69)
where
G L (r, r ′ ) = G( r, r ′ ) − G 0 (r, r ′ ) = rr ′ ∞ l=1 g l (r, r ′ ) l m=−l Y m * l (θ, φ)Y m l (θ ′ , φ ′ ),(70)
g l (r, r ′ ) = ikj l (kr)(j l (kr ′ ) − iy l (kr ′ )), r ≤ r ′ (71) g l (r, r ′ ) = ik(j l (kr) − iy l (kr))j l (kr ′ ), r ≥ r ′
Substitute the G L (r, r ′ ) in (70-72) into the integral equation (69), the equation (69) becomes to the following integral equation,
1 ε θ E( r ′ ) = E b ( r ′ ) −r ′ ∞ l=1 S(r≤R2) ∂ ∂r 1 ε θ ∂ ∂r rg l (r, r ′ ) l m=−l Y m * l (θ, φ)EdV Y m l (θ ′ , φ ′ ) +r ′ ∞ l=1 S(r≤R2) 1 r 2 1 ε θ − 1 εr l(l + 1)rg l (r, r ′ ) l m=−l Y m * l (θ, φ)EdV Y m l (θ ′ , φ ′ ) +r ′ ∞ l=1 S(r≤R2) k 2 1 ε θ − µ θ rg l (r, r ′ ) l m=−l Y m * l (θ, φ)EdV Y m l (θ ′ , φ ′ ),(73)lim r ′ →0 1 ε θ E( r ′ ) = lim r ′ →0 E b ( r ′ ) − lim r ′ →0 r ′ ∞ l=1 S(r≤R2) ∂ ∂r 1 ε θ ∂ ∂r rg l (r, r ′ ) l m=−l Y m * l (θ, φ)EdV Y m l (θ ′ , φ ′ ) + lim r ′ →0 r ′ ∞ l=1 S(r≤R2)
From the equation (60) lim
r ′ →0 1 ε θ E( r ′ ) = 0(75)
From the equation (21) lim r ′ →0
I = lim r ′ →0 E b ( r ′ ) = 0,(76)
In next, we will prove
P roof :, Using similar proof process, the theorem 6.2 can be proved.
T heorem 6.3, Suppose that the anisotropic relative electric permittivity ε r (r),ε θ (r) ε φ (r) and magnetic permeability, µ r (r),µ θ (r) µ φ (r) , satisfy the above GL-HUA pre cloak material conditions in invisible sphere (6.1) to (6.4), also angular electromagnetic wave satisfy following finite energy condition (63)
S(r≤R2) |rE θ ( r)| 2 + |rH θ ( r)| 2 + |rE φ ( r)| 2 + |rH φ ( r)| 2 dV, is f inite,(107)
E
(0, θ, φ) sin θdθdφ = 0 the integral equation (55) will become
:, Because the source is outside sphere,r s > R O > R 2 , In the Sphere 0 < r ≤ R 2 , from (4)sin θrE φ = −iωµ 0 H,
B. Second Order Maxwell Electromagnetic Equation On Radial Electric Wave In SphericalCoordinateWe use Electric-Magnetic-Electric EME translate pro-
cess to translate Maxwell Equation in Spherical Coor-
dinate (S1) and (S2) with anisotropic material into the
Second order Maxwell Electromagnetic Equation on Ra-
dial Electric Wave Field In Spherical Coordinate
∂
∂r
1
ε θ
∂
∂r ε r r 2 E r + 1
sin θ
∂
∂θ sin θ ∂Er
∂θ +
1
sin 2 θ
∂ 2 Er
AcknowledgmentsWe wish to acknowledge the support of the GL Geophysical Laboratory and thank the GLGEO Laboratory to approve the paper publication.Substitute(70)and(71)into the (77) and using LHO-PITAL ROLE, we prove the limitation equation (77) in detail,Because the finite energy condition (60) and GLHUA pre cloak material condition (6.1) to (6.4), the integral in above equation(83)is finite integrative,Substitute(71)and(72)for g l (r, r ′ ) into the (88),Because for incident wave with electric current pointLetTo use ℜ(θ, φ) in (97) to make action to both side of the limitation equation (96), we haveFor every (θ ′ , φ ′ ) and k ≥ k 0 > 0Other proof approach is as follows: By using above similar with proof process of (98), for every term l and k, we can prove that, Because E 0 (r) = 0 in (30),l ≥ 1 and for k ≥ k 0 > 0The limitation equation(61)is proved. Similarly, we can prove limitation equation(62)limSimilarly, for incident plane electromagnetic wave , we also can prove limitation equation (61) and (62). The theorem 6.1 is proved.T heorem 6.2, Suppose that the anisotropic relative electric permittivity ε r (r),ε θ (r) ε φ (r) and magnetic permeability, µ r (r),µ θ (r) µ φ (r) , satisfy the above GL-HUA pre cloak material conditions in invisible sphere (6.1) to (6.4), and finite energy condition (60), thenTo use product of GLHUA Greens function matrix by sin θ, G(θ, θ ′ , φ, φ ′ ) sin θ, to multiply the matrix equation (113) and take sphere surface integral of resulted equation on [0, π; 0, 2π] π 0 2π 0To subtract equation(116)from(117), we haveBased on theorem 6.1 and theorem 6.2, we haveWe have proved the first part of (108) of theorem 6.3, similarly, we can prove second part of (109). The theorem 6.3 is proved.T heorem 6.4, Suppose that the anisotropic relative electric permittivity ε r (r),ε θ (r) ε φ (r) and magnetic permeability, µ r (r),µ θ (r) µ φ (r) , satisfy the above GL-HUA pre cloak material conditions in invisible sphere (6.1) to (6.4), also angular electromagnetic wave satisfy following finite energy condition (63), then lim r→0 1 ε θ ∂ ∂r rE θ ( r) = 0, limP roof :, By using the similar proof process on the 8.3, we can prove the theorem 6.4.VII. DISCUSSION AND CONCLUSIONThe pre cloak material conditions (6.1) to (6.4) in GL-HUA sphere is from GL zero scattering inversion and GL no scattering modeling. Many GL no scattering modeling simulations show that under the conditions
These condition is not unique and can be relaxed. We publish this paper to arXiv for support our paper arXiv.org/abs/1612.02857. Our GLHUA cloak and GL-HUA sphere publication in arXiv are for open review. Please colleague give comments to me by my email or give open comments in arXiv. All copyright and patent of the GLHUA EM cloaks,GLHUA sphere and GL modeling and inversion methods are reserved by authors in GL Geophysical Laboratory. E(r) = 0, and lim r→0 H(r) = 0, are verified. If some colleague cite our paper in his work paper, please cite our paper as reference in his paper.E(r) = 0,, and lim r→0 H(r) = 0, are ver- ified. These condition is not unique and can be relaxed. We publish this paper to arXiv for support our paper arXiv.org/abs/1612.02857. Our GLHUA cloak and GL- HUA sphere publication in arXiv are for open review. Please colleague give comments to me by my email or give open comments in arXiv. All copyright and patent of the GLHUA EM cloaks,GLHUA sphere and GL mod- eling and inversion methods are reserved by authors in GL Geophysical Laboratory.If some colleague cite our pa- per in his work paper, please cite our paper as reference in his paper..
3-D electromagnetic modeling and nonlinear inversion. G Xie, J H Li, E Majer, D Zuo, M Oristaglio, Geophysics. 653Xie, G., J.H. Li, E. Majer, D. Zuo, M. Oristaglio "3-D electromagnetic modeling and nonlinear inversion," Geo- physics, Vol. 65, No. 3, 804-822, 2000.
New GL method and its advantages for resolving historical diffculties. G Xie, F Xie, L Xie, J Li, Progress In Electromagnetics Research, PIER. 63Xie, G., F. Xie, L. Xie, and J. Li, "New GL method and its advantages for resolving historical diffculties," Progress In Electromagnetics Research, PIER 63, 141-152, 2006.
GL metro carlo EM inversion. G Xie, J Li, L Xie, F Xie, Journal of Electromagnetic Waves and Applications. 2014Xie, G., J. Li, L. Xie, and F. Xie, "GL metro carlo EM inversion," Journal of Electromagnetic Waves and Appli- cations, Vol. 20, No. 14, 1991-2000, 2006.
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[
"CLOSURE METHOD FOR SPATIALLY AVERAGED DYNAMICS OF PARTICLE CHAINS",
"CLOSURE METHOD FOR SPATIALLY AVERAGED DYNAMICS OF PARTICLE CHAINS"
] | [
"Alexander Panchenko ",
"ANDLyudmyla L Barannyk ",
"Robert P Gilbert "
] | [] | [] | We study the closure problem for continuum balance equations that model mesoscale dynamics of large ODE systems. The underlying microscale model consists of classical Newton equations of particle dynamics. As a mesoscale model we use the balance equations for spatial averages obtained earlier by a number of authors: Murdoch and Bedeaux, Hardy, Noll and others. The momentum balance equation contains a flux (stress), which is given by an exact function of particle positions and velocities. We propose a method for approximating this function by a sequence of operators applied to average density and momentum. The resulting approximate mesoscopic models are systems in closed form. The closed from property allows one to work directly with the mesoscale equaitons without the need to calculate underlying particle trajectories, which is useful for modeling and simulation of large particle systems. The proposed closure method utilizes the theory of ill-posed problems, in particular iterative regularization methods for solving first order linear integral equations. The closed from approximations are obtained in two steps. First, we use Landweber regularization to (approximately) reconstruct the interpolants of relevant microscale quantitites from the average density and momentum. Second, these reconstructions are substituted into the exact formulas for stress. The developed general theory is then applied to non-linear oscillator chains. We conduct a detailed study of the simplest zero-order approximation, and show numerically that it works well as long as fluctuations of velocity are nearly constant. | 10.1016/j.nonrwa.2010.10.021 | [
"https://arxiv.org/pdf/1010.4832v1.pdf"
] | 9,618,121 | 1010.4832 | 1404111af645f176b13b7f674c7d91d7b027b1de |
CLOSURE METHOD FOR SPATIALLY AVERAGED DYNAMICS OF PARTICLE CHAINS
Alexander Panchenko
ANDLyudmyla L Barannyk
Robert P Gilbert
CLOSURE METHOD FOR SPATIALLY AVERAGED DYNAMICS OF PARTICLE CHAINS
FPU chainparticle chainoscillator chainupscalingmodel reductiondimension reductionclosure problem
We study the closure problem for continuum balance equations that model mesoscale dynamics of large ODE systems. The underlying microscale model consists of classical Newton equations of particle dynamics. As a mesoscale model we use the balance equations for spatial averages obtained earlier by a number of authors: Murdoch and Bedeaux, Hardy, Noll and others. The momentum balance equation contains a flux (stress), which is given by an exact function of particle positions and velocities. We propose a method for approximating this function by a sequence of operators applied to average density and momentum. The resulting approximate mesoscopic models are systems in closed form. The closed from property allows one to work directly with the mesoscale equaitons without the need to calculate underlying particle trajectories, which is useful for modeling and simulation of large particle systems. The proposed closure method utilizes the theory of ill-posed problems, in particular iterative regularization methods for solving first order linear integral equations. The closed from approximations are obtained in two steps. First, we use Landweber regularization to (approximately) reconstruct the interpolants of relevant microscale quantitites from the average density and momentum. Second, these reconstructions are substituted into the exact formulas for stress. The developed general theory is then applied to non-linear oscillator chains. We conduct a detailed study of the simplest zero-order approximation, and show numerically that it works well as long as fluctuations of velocity are nearly constant.
Introduction
In a series of papers, [1], [2], [3], [4], Murdoch and Bedeaux studied continuum mechanical balance equations for mesoscopic space time averages of discrete systems. Earlier work of Irving and Kirkwood [14], Noll [16], and Hardy [13] on closely related topics should be also mentioned here. The fluxes in balance equations (e. g. stress) are given by exact formulas as functions of particle positions and velocities. This is useful for linking microscale dynamics with mesoscale phenomena. However, using these formulas requires a complete knowledge of underlying particle dynamics. Since many particle systems of interest have enormous size, direct simulation of particle trajectories may be intractable. Consequently, it makes sense to look for closed form approximations of fluxes in terms of other mesoscale quantities (e.g., average density and velocity), rather than microscopic variables.
In this paper we address the above closure problem for spatially averaged mesoscale dynamics of large size classical particle chains. The design of the method was influenced by the following considerations.
(1) The quantities of interest are space-time continuum averages, such as density, linear momentum, stress, energy and others. This choice of averages is natural because these quantities are experimentally measurable, and also because of their importance in coupled multiscale simulations involving both continuum and discrete models. In addition, by working directly with space-time averages instead of ensemble averages one can bypass a difficult problem of relating probabilistic and space-time averages.
(2) It is desirable to be able to predict behavior of averages on arbitrary time intervals, no matter how short. This perspective comes from PDE problems, where observation time is often arbitrary and long time behavior is not of interest. When one tracks an ODE systems on an arbitrary time interval, transients may be all that is observed. Therefore, we do not use qualitative theory of ODEs, primarily concerned with describing long time features of dynamics. This significantly decreases the range of available tools. However, the closure problem for mesoscopic PDEs turned out to be a question that can be still answered in a satisfactory way. The methods developed in this fashion can be helpful in situations where long time features are not of interest: modeling transient and shortlived phenomena, working with metastable systems, and dealing with problems for which relaxation times can be hard to estimate. (3) We consider particle systems with initial conditions that either known precisely, or ar least such that the possible initial positions and velocities are strongly restricted by available a priori information. This is in contrast to statistical mechanics, where uncertainty of initial conditions is a major problem.
In this regard we note that our approach makes sense for discrete models of solid-fluid continuum systems, where the smallest relevant length scale is still much larger than a typical intermolecular distance. For other particle systems, our method can be used to run deterministic simulations repeatedly, in order to accumulate statistical information about the underlying probability density. (4) Because of widespread use of computers in physical and engineering sciences, it is useful to develop theories tailored for computation, rather than "paper and pencil" modeling. As far as the closure problem is concerned, traditional phenomenological approach to formulating constitutive equations can be subsumed by a more general problem of finding a computational closure method. In particular, a closure method can be realized as an iterative procedure where one inputs the values of the primary variables (e.g. density and velocity) computed at the previous moment of time, and the algorithm generates the flux (e.g. stress) at the next moment. Then primary variables are updated using mesoscopic balance equations, and the process is repeated. In addition, focusing on computing one can obtain unconventional but useful continuum mechanical models. By replacing a simple, but possibly crude, Taylor series truncation with an algorithm we make it harder to obtain exact solutions. Since such solutions are rarely available even for simple classical systems, (e.g. Navier-Stokes equations), this is not a serious drawback. On the positive side, computational closure generally contains an explicit (explicitly computable) link between micro-and mesoscale properties. (5) An important potential application of closure is development of fast numerical methods for simulating meso-scopic dynamics of particle systems. Mesoscale solvers usually employ coarse meshes with mesh size much larger than a typical interparticle distance. Then the averages would be usually given by their coarse mesh values, while interpolants of microscale quantities are discretized on a fine scale mesh. Consequently, a closure method might consist of two generic blocks: (i) reconstruction on mesoscale mesh thereby a coarse approximations of fine scale quantities are obtained from averages; and (ii) interpolation of the obtained coarse scale discretizations to fine scale.
The closure algorithm developed in the paper is based on iterative regularization methods for solving first kind integral equations. We observe that primary mesoscale averages are related to the interpolants of microscale variables via a linear convolution operator. The kernel of this operator is the "window function" used in [1] to generate averages. Such integral operators are usually compact. A compact operator may be invertible, but the inverse operator is not continuous. Therefore, the problem of reconstructing microscale quantities from given averages is ill-posed. Such problems are well studied in the literature [12,6,9,15,18]. A particular method used in the paper for inverting convolutions is Landweber iteration [5], [10]. It is known that if the error in the data tends to zero, the Landweber method produces successive approximations converging to the exact solution. For the merely bounded data error, convergence is replaced by a stopping criterion. This criterion provides the optimal number of iterations needed to approximate the solution with the accuracy proportional to the error in the data. As a consequence, our method has desirable feature: one can improve the approximation quality at the price of increasing the algorithm complexity. This means that predictive capability of the method can be regulated depending on available computing power.
The paper is organized as follows. In Section 2 we describe a general multi-dimensional microscopic model. The equations of motion are classical Newton equations. We limit ourselves to the case of short range interaction forces that may be either conservative or dissipative. The scaling of particle masses and forces reflects a continuum mechanical perspective, that is a family of particle systems of increasing size should represent a hypothetical continuum material. As N → ∞, the total mass of the system should remain fixed, and the total particle energy should be either fixed, or at least bounded independent of N . Next, we recall the main points of averaging theory of Murdoch-Bedeaux and provide mesoscopic balance equations and exact formulas for the stress from [4]. In Section 3 we develop integral approximations of averages, and describe the use of Landweber iterative regularization for approximate reconstruction. Section 4 contains the formulation of the scaled ODE equations of the so-called Fermi-Pasta-Ulam (FPU) chains. In Section 5 we derive closed form mesoscopic continuum equations of chain dynamics. The complexity of these continuum models increases with the order n of the iterative deconvolution approximation. Section 6 is devoted to the detailed study of the simplest closed model with n = 0, which we call zero-order closure. Essentially, zero-order closure means that the microscopic quantities are replaced by their averages. Such an approximation can work well only for systems with small fluctuations. To quantify fluctuation size we introduce upscaling temperature and the related notion of quasi-isothermal dynamics. For such dynamics, we show how to interpolate averages given by mesoscopic mesh values, in order to initialize approximate particle positions and velocities. The interpolation procedure is problem-specific: it conserves microscopic energy and preserves quasi-isothermal nature of the dynamics. Section 7 contains the results of computational tests. Here we apply our zero-closure algorithm to a Hamiltonian chain with the finite range repulsive potential U , decreasing as a power of distance. The results show good agreement of zero-order approximations with the exact stress produced by direct simulations with 10000-80000 particles, provided the initial conditions have small fluctuations. In our example, the initial conditions are such that the upscaling temperature is nearly zero during the observation time. We also demonstrate that increasing fluctuations of initial velocities leads to a considerable increase in the approximation error, indicating that higher order closure algorithms should be used instead of zero-order closure. Applicability of the zero-order closure is further discussed in Section 8. Finally, conclusions are provided in Section 9.
2. Microscale equations and mesoscale spatial averages 2.1. Scaled ODE problems. The starting point is the microscale ODE problem. In this paper we shall work with classical Newton equations of point particle dynamics. The same equations may arise as discretization of the momentum balance equation for continuum systems. Consider a system containing N 1 identical particles, denoted by P i . The mass of each particle is M N , where M is the total mass of the system. Suppose that during the observation time T , P i remain inside a bounded domain Ω in R d , where d is the physical space dimension, usually 1, 2 or 3. The positions q i (t) and velocities v i (t) of particles satisfy a system of ODEsq
i = v i , (2.1) M Nv i = f i + f (ext) i , (2.2)
subject to the initial conditions
(2.3) q i (0) = x i , v i (0) = v 0 i . Here f (ext) i
denotes external forces, such as gravity and confining forces. The interparticle forces f i = j f ij , where f ij are pair interaction forces which depend on the relative positions and velocities of the respective particles.
We are interested in investigating asymptotic behavior of the system as N → ∞. Thus it is convenient to introduce a small parameter
(2.4) ε = N −1/d ,
characterzing a typical distance between neighboring particles. As ε approaches zero, the number of particles goes to infinity, and the distances between neighbors shrink. Consequently, the forces in (2.2) should be properly scaled. The guiding principle for scaling is to make the energy of the system bounded independent of N , as N → ∞. In addition, the energy of the initial conditions should be bounded uniformly in N .
As an example of scaling, consider forces generated by a finite range potential U and assume that each particle interacts with no more than a fixed number of neighbors (this is the case, e.g., for particle chains with nearest neighbor interaction, where a particle always interacts with two neighbors). The fixed number of interacting neighbors implies that there are about N interacting pairs. Assuming also that the system is sufficiently dense, and variations of particle concentrations are not large, we can suppose that a typical distance between interacting particles is on the order N −1/d L = εL. The resulting scaling
(2.5) f ij = − 1 εN ∇ x U q j − q k ε
makes the potential energy of an isolated system bounded independent of N . Kinetic energy will be under control provided the total energy of the initial conditions is bounded independent of N . If exterior forces are present, they should be scaled as well.
Remark. Superficially, the system (2.1), (2.2) looks similar to the parameter-dependent ODE systems studied in numerous works on ODE time homogenization (see e g. [17] and references therein). In the problem under study, ε depends on the system dimension N , while in the works on time-homogenization and ODE perturbation theory, the system size is usually fixed as ε → 0.
2.2. Length scales. We introduce the following length scales: -macroscopic length scale L = diam(Ω); -microscopic length scale εL; -mesoscopic length scale ηL, where 0 < η < 1 is a parameter that characterizes spatial mesoscale resolution. This parameter is chosen based on the desired accuracy, the computational cost requirements, available information about initial conditions and behavior of ODE trajectories etc. The computational domain Ω is subdivided into mesoscopic cells C β , β = 1, 2, . . . , B, with the side length on the order of ηL. The centers x β of C β are the nodes of the meso-mesh. The number of unknowns in the mesoscopic system will be on the order of B. For computational efficiency, one should have B N . This does not mean that η is close to one. In fact, it makes sense to keep η as small as possible in order to have an additional asymptotic control over the system behavior. Decreasing η will in general make computations more expensive.
Averages and their evolution.
To define averages we first select a fast decreasing window function ψ satisfying ψ(x)dx = 1. There are many possible choices of the window function. In the paper we assume, unless otherwise indicated, that ψ is a compactly supported, differentiable on the interior of its support, and non-negative. Next, define
ψ η (x) = η −d ψ x η .
Once the window function is chosen, we can evaluate the averages of various continuum mechanical variables, following [1], [4]. The mesoscopic average density and momentum are given by
(2.6) ρ η (t, x) = M N N i=1 ψ η (x − q i (t)), (2.7) ρ η v η (t, x) = M N v i (t)ψ η (x − q i (t)).
The meaning of the above definitions becomes clear if one considers ψ = (c d ) −1 χ(x), where χ is a characteristic function of the unit ball in R d , and c d is the volume of the unit ball. Then
ρ η = 1 c d η d M N χ x − q i (t) η .
The sum in the right hand side gives the number of particles located within distance η of x at time t. Multiplying by M/N we get the total mass of these particles, and dividing by c d η d (the volume of η-ball) gives the usual particle density. Differentiating (2.6), (2.7) in t, and using the ODEs (2.1), (2.2) one can obtain [1] exact mesoscopic balance equations for all primary variables. For example, for an isolated system with (f (ext) i = 0), mass conservation and momentum balance equations take the form:
(2.8) ∂ t ρ η + div(ρ η v η ) = 0, (2.9) ∂ t (ρ η v η ) + div (ρ η v η ⊗ v η ) − divT η = 0. The stress T η = T η (c) + T η (int) [4], where (2.10) T η (c) (t, x) = − m i (v i − v η (t, x, )) ⊗ (v i − v η (x, t))ψ(x − q i )
is the convective stress, and
(2.11) T η (t, x) (int) = (i,j) f ij ⊗ (q j − q i ) 1 0 ψ s(x − q j ) + (1 − s)(x − q i ) ds
is the interaction stress. The summation in (2.11) is over all pairs of particles (i, j) that interact with each other. Discretizing balance equations on the mesoscopic mesh yields a discrete system of equations, called the meso-system, written for mesh values of ρ η β , (ρ η v η ) β and T η β . The dimension of the meso-system is much smaller than the dimension of the original ODE problem. However, at this stage we still have no computational savings, since the meso-system is not closed. This means that mesoscopic fluxes such as (2.10), (2.11) are expressed as functions of the microscopic positions and velocities. To find these positions and velocities, one has to solve the original microscale system (2.1), (2.2). To achieve computational savings we need to replace exact fluxes with approximations that involve only mesoscale quantities. We refer to the procedure of generating such approximations as a closure method. This closure-based approach has much in common with continuum mechanics. The important difference is that the focus is on computing, rather than continuum mechanical style modeling of constitutive equations.
Closure via regularized deconvolutions
3.1. Outline. Our approach is based on a simple idea: the integral approximations of primary averages (such as density and velocity) are related to the corresponding microscopic quantities via convolution with the kernel ψ η . Therefore, given primary variables we can (approximately) recover the microscopic positions and velocities by numerically inverting convolution operators. The results are inserted into equations for secondary averages (or fluxes), such as stress in the momentum balance. This yields closed form balance equations that can be simulated efficiently on the mesoscopic mesh.
3.2.
Integral approximation of discrete averages. To exploit the special structure of primary averages, it is convenient to approximate sums such as
(3.1) g η = 1 N N j=1 g(v j , q j )ψ η (x − q j )
by integrals. Since particle positions q j are not periodically spaced, (3.1) is not in general a Riemann sum for gψ η (x − ·). To interpret the sum correctly, we introduce interpolantsq(t, X),ṽ(t,q) of positions and velocities, associated with the microscopic ODE system (2.1), (2.2). At t = 0 these interpolants satisfỹ
q(0, X j ) = q 0 j ,ṽ(0,q(0, X j )) = v 0 j , where X j , j = 1, 2, .
. . , N are points of ε-periodic rectangular lattice in Ω. At other times,
q(t, X j ) = q j (t),ṽ(t,q(t, X j )) = v j (t).
Then we can rewrite (3.1) as
(3.2) g η = 1 |Ω| N j=1 |Ω| N g (ṽ (t,q(t, X j )) ,q(t, X j ) ψ η (x −q(t, X j )),
where |Ω| denotes the volume (Lebesgue measure) of Ω. Eq. (3.2) is a Riemann sum generated by partitioning Ω into N cells of volume |Ω|/N centered at X j . This yields
(3.3) g η = 1 |Ω| Ω g (ṽ(t,q(t, X)),q(t, X)) ψ η (x −q(t, X))dX,
up to discretization error. Now suppose that the mapq(·, X) is invertible for each t, that is X =q −1 (t,q). Changing the variables in the integral y =q(t, X) we obtain a generic integral approximation
(3.4) g η = 1 |Ω| Ω g (ṽ(t, y), y) ψ η (x − y)J(t, y) dy, where (3.5) J = | det ∇q −1 |,
up to discretization error.
Regularized deconvolutions.
Define an operator R η by
R η [f ](x) = ψ η (x − y)f (y)dy.
To simplify exposition, suppose that R η is injective. For example, a Gaussian ψ η produces an injective operator, which is not difficult to check using Fourier transform and uniqueness of analytic continuation. If R η is injective, then there exists the single-valued inverse operator R −1 η , that we call the deconvolution operator. Unfortunately, this operator is unbounded, since R η is compact in L 2 (Ω). This is the underlying reason for the popular belief that averaging destroys the high-frequency information contained in the microscopic quantities. In fact, this information is still there (the inverse operator exists), but it is difficult to recover in a stable manner, because of unboundedness. This does not make the situation hopeless, as has been recognized for some time. Reconstructing f from the knowledge of R η [f ]) is a classical example of an unstable ill-posed problem (small perturbations of the right hand side may produce large perturbations of the solution). The exact nature of ill-posedness and methods of regularizing the problem are well investigated both analytically and numerically (see, e. g. [6,9,15,18,11,12]). Accordingly, we interpret notation R −1 η as a suitable regularized approximation of the exact operator. Many regularizing techniques are currently available: Tikhonov regularization, iterative methods, reproducing kernel methods, the maximum entropy method, the dynamical system approach and others. It is very fortunate that this vast array of knowledge can be used for the ODE model reduction. On the conceptual level, our approach makes it clear that instability associated with ill-posedness is a fundamental difficulty in the process of closing the continuum mechanical equations.
A family of Landweber iterative deconvolution methods [5], [10] seems to be particularly convenient in the present context. In the simplest version, approximations g n to the solution of the operator equation
(3.6) R η [g] = g η
are generated by the formula
(3.7) g n = n k=0 (I − R η ) n g η , g 0 = g η .
The number n of iterations plays the role of regularization parameter. In (3.7), I denotes the identity operator. The first three low-order approximations are g 0 = g η n = 0, (3.8)
g 1 = g η + (I − R η )[g η ] n = 1, (3.9) g 2 = g η + (I − R η )[g η ] + (I − R η ) 2 [g η ] n = 2.
(3.10)
Microscale particle chain equations
In this section, the general method outlined above is detailed in the case of one-dimensional Hamiltonian chain of oscillators that consists of N identical particles with nearest neighbor interaction. The domain Ω is an interval (0, L). Particle positions, denoted by q j = q j (t), j = 1, . . . , N , satisfy 0 < q 1 < q 2 < . . . < q N < L at all times, i.e. the particles cannot occupy the same position or jump over each other. Next, define a small parameter
ε = 1 N ,
and microscale step size
(4.1) h = L N .
The interparticle forces
(4.2) f jk = q j − q k |q j − q k | U |q j − q k | ε
are defined by a finite range potential U . We suppose that U (ξ) ≥ 0 for all ξ within the range. Note that k in (4.2) can take only two values: j − 1 or j + 1. Also, observe f jk = −f kj , as it should be by the third law of Newton, and also that the sign of f jk is the same as sign of q j − q k . This means that the force exerted on P j by say, P j+1 is repulsive. The total interaction force acting on the particle P j is
f j = f j,j−1 + f j,j+1 ,
for j = 2, 3, . . . , N − 1.
Each particle has mass m = M/N = M ε, where M is the total mass of the system. Particles have velocities denoted by v j , j = 1, . . . , N . Writing the second Newton's law as a system of first order equations yields the scaled microscale ODE system
(4.3)q j = v j , εMv j = f j , j = 1, . . . , N subject to the initial conditions (4.4) q j (0) = q 0 j , v j (0) = v 0 j .
Integral approximation of stresses for particle chains. Mesoscopic continuum equations
In the one-dimensional case stress is a scalar quantity, and (2.10), (2.11) reduce to, respectively,
(5.1) T η (c) (t, x) = − N j=1 M N (v j − v η (t, x)) 2 ψ η (x − q j ), and (5.2) T η (int) (t, x) = N −1 j=1 f j,j+1 (q j+1 − q j ) 1 0 ψ η (x − sq j+1 − (1 − s)q j )ds.
The sum in (5.2) is simplified compared to the general expression, since we have exactly N − 1 interacting pairs of particles.
To obtain integral approximations of stresses, we define interpolantsq,ṽ, as in Sect. 3.2. Assuming as before thatq is invertible and repeating the calculations we get
(5.3) T η (c) (t, x) = − M L L 0 (ṽ(t, y) − v η (t, x)) 2 ψ η (x − y)J(t, y)dy.
Remark. Many equalities in the paper, including (5.3) hold up to a discretization error. To simplify presentation, we do not mention this in the sequel when discrete sums are approximated by integrals. The interaction stress can be rewritten as
(5.4) T η (int) (t, x) = − N − 1 N N −1 j=1 L N − 1 U q j+1 − q j h L q j+1 − q j h 1 0 ψ η (x − sq j+1 − (1 − s)q j )ds.
Next we approximate (q j+1 −q j )/h byq (t, X j ). This approximation is in fact exact, provided the interpolant is chosen to be piecewise linear. Note also that
q (t, X) = 1 (q −1 ) (t,q(t, X)) = 1 J(t,q(t, X))
.
Inserting this into (5.4), replacing Riemann sum with an integral and changing variable of integration as in Sect. 3.2, we obtain the integral approximation of the interaction stress:
(5.5) T η (int) (t, x) = − L 0 U L J(t, y) 1 0 ψ η x − y −
sh J(t, y) ds dy.
Equations (5.3), (5.5) contain two microscale quantities: J andṽ. Approximating sums in the definitions of the primary averages (2.6), (2.7) by integrals we see that ρ η and v η are obtained by applying the convolution operator R η to, respectively J and Jṽ:
(5.6) ρ η = M L R η [J], ρ η v η = M L R η [Jṽ].
The discretization error in (5.6) can be made small by imposing suitable requirements on the microscopic interpolants. Fortunately, the theory of ill-posed problems allows for errors in the right hand side of integral equations. The size of the error determines the choice of regularization parameter. In the present case, the error determines the number of iterations needed for the optimal reconstruction, according to so-called stopping criteria. These criteria are available in the literature on ill-posed porblems (see e.g. [9]). Detailed investigation of these questions is left to future work. Denote by R −1 η,n the iterative Landweber regularizing operators
R −1 η,n = n k=0 (I − R η ) k .
Applying R −1 η,n in (5.6) yields a sequence of approximations
(5.7) J n = L M R −1 η,n [ρ η ],ṽ n = R −1 η,n [ρ η v η ] R −1 η,n [ρ η ]
, and a corresponding sequence of closed form mesoscopic continuum equations (written here for an isolated system with zero exterior forces)
∂ t ρ η + ∂ x (ρ η v η ) = 0, (5.8) ∂ t (ρ η v η ) + ∂ x ρ η (v η ) 2 − ∂ x (T η (c),n + T η (int),n ) = 0, (5.9)
where T η (c),n , T η (int),n are given by
(5.10) T η (c),n = − M L L 0 (ṽ n (t, y) − v η (t, x)) 2 ψ η (x − y)J n (t, y)dy, (5.11) T η (int),n = − L 0 U L J n (t, y) 1 0 ψ η x − y − sh J n (t, y) ds dy,
with J n ,ṽ n given by (5.7).
6. Zero-order closure for particle chains
Let us consider zero-order approximations in detail. The mesoscopic mesh consists of points
x β = β − 1 2 ηL, β = 1, 2, . . . , B,
where B = 1/η, presumed to be an integer satisfying B N . Meso-cells are intervals I β of length
L η = L/B = ηL, centered at x β .
Suppose that the only primary variables of interest are density ρ η and linear momentum ρ η v η . These variables will be computed by the mesoscale solver. For simplicity, suppose that the meso-solver is explicit in time. Then the average density and average velocity will be available at the previous moment of time. Our task is to design an update step for computing density and velocity at the next time moment. To construct a closed form update step, we need to approximate stress T η in (2.9) in terms of ρ η , ρ η v η . From the knowledge of ρ η , ρ η v η we can approximately recover J and Jṽ. The zero-order approximation (3.8) corresponds to
J(t, x) ≈ L M ρ η (t, x), (6.1) J(t, x)ṽ(t, x) ≈ L M ρ η v η (t, x). (6.2)
In other words, the microscale quantities are approximated by their averages. The corresponding closed form approximations for stress are obtained by inserting (6.1), (6.2) into (5.10), (5.11):
(6.3) T η (c),0 (t, x) = − L 0 (v η (t, y) − v η (t, x)) 2 ψ η (x − y)ρ η (t, y)dy, (6.4) T η (int),0 = − L 0 U M ρ η (t, y) 1 0 ψ η x − y − shM Lρ η (t, y) ds dy.
For computation, a numerical quadrature should be used. In this regard, note that all average quantities are computed on the mesoscale mesh, while the formulas (2.10), (2.11) are fine scale discretizations. Therefore, one might wonder if a straightforward mesoscale quadrature of (6.3), (6.4) is too crude. A better approach is to interpolate ρ η , v η by prescribing approximate particle positionsq j and velocitiesv j , j = 1, 2, . . . , N , compatible with the given ρ η , v η . Once this is done, (6.3), (6.4) can be discretized on a fine scale mesh with mesh nodesq j . Interpolants cannot be unique. For zero-order closure, we are choosing positions and velocities that produce the given average density and average velocity. Clearly, there are many different position-velocity configurations with the same averages. The choice made in the paper is motivated by the practical requirement of achieving low operation count, as well as by certain expectations about the nature of dynamics. From the continuum mechanical point of view, if a system can be adequately modeled by balance equations of mass and momentum, then it must have have a trivial energy balance. Most often this means that the deformation is nearly isothermal. To mimic such isothermal dynamics we suppose that at each time step, there exists a positive number κ 2 (it can be called "upscaling temperature") such that
(6.5) j∈J β (v β j − v η (t, x β )) 2 ψ η (x β −q j ) = κ 2 .
Here the summation is over all particles located in a meso-cell I β . The temperature κ 2 is the same for all β = 1, 2, . . . , B. We emphasize that the actual value of κ 2 is not as important as the fact that its value is the same for all meso-cells. This is because (6.5) would yield constant mesoscale mesh node values of T η (c),0 As a result, the finite difference approximation of ∂ x T η (c),0 on the mesoscale mesh is identically zero. We interpret this by saying that convective stress does not contribute to the mesoscopic dynamics in the isothermal case. Another observation is that κ need not be the same at different moments of time, so our assumption is somewhat more flexible than the standard isothermal deformation approximation. Also, we note that validity (6.5) depends on the choice of η. For bigger η, it is more likely that (6.5) holds for the same underlying microscopic dynamics. Details on this are provided below in Section 6.2. Additionally, other features of the microscopic dynamics should be taken into account. Most importantly, interpolated velocitiesv j = v η (q j ) must be such that the collectionq j ,v j , j = 1, 2, . . . , N conserves microscopic energy E:
(6.6) E = 1 2 M N N j=1 (v j ) 2 + U( Q)
where U( Q) is the microscale potential energy corresponding to the positionsq j .
6.1. Prescribing particle positions. The objective of this section is to assign approximate particle positionsq j . We start by interpolating J. The simplest interpolant is piecewise-constant:
J(t, x) ≈ B β=1 J(t, x β )χ β (x) = B β=1 L M ρ η (t, x β )χ β (x)
where χ β is the characteristic function of the meso-cell I β . A simple choice of the position map compatible with this interpolant is a piecewise linear map having the prescribed constant value of J in each meso-cell. In practical terms, this means that in each meso-cell, particles are spaced at equal intervals from each other. The local interparticle spacing (6.7) ∆ β = M ρ η (t, x β )N is determined by the mesh value of the average density. To explain (6.7), note that the total mass of particles contained in the meso-cell I β can be approximated by ρ η (t, x β )L η . Dividing by the mass M/N of one particle, we obtain an approximate number of particles inside I β :
n β = ρ η (t, x β )L η N M ,
and thus ∆ β = L η /n β . We emphasize thatq j are chosen based only on the known mesh values of the density ρ η , and thatq j will be different from the actual particle positions q j . Now we approximate the integral in (6.4) by its Riemann sum generated by the partition {q j , j = 1, 2, . . . , N }:
T η (int),0 ≈ − N −1 j=1 U (N (q j+1 −q j )) (q j+1 −q j ) 1 0 ψ η (x − sq j+1 − (1 − s)q j ) ds. (6.8)
6.2. Prescribing particle velocities. In order to approximate the convective stress in the fine scale discretization of (6.3), we need to choose approximationsv j of the true particle velocities v j . The choice ofv j must satisfy (6.6) and be compatible with the available average velocity at the mesoscale mesh nodes.
For eachq j ∈ I β , we setv
j = v η β + δv β j ,
where v η β is the local average velocity, and δv β j is a perturbation to be defined. Next, we show that the energy-conserving collection of δv j velocity always exists, provided its upscaling temperature is suitably prescribed. This prescription will be based only on the available mesoscale information. For definitiveness, in the rest of this section we suppose that ψ η satisfies the following condition:
ψ η (x β −q j ) > 0, ifq j ∈ I β . (6.9)
To make the algebra simpler, we make another assumption: for each β = 1, 2, . . . , B,
(6.10) N j=1 f (v j )ψ η (x β −q j ) ≈ j∈J β f (v j )ψ η (x β −q j ),
where f is eitherv j or (v j ) 2 . The second summation is over all j such thatq j ∈ I β . Assumption (6.10) holds when ψ η (x β − y) is small outside of I β .
Averaging ofv j should produce the known average velocity v η β . This yields an equation for δv j :
(6.11) M N j∈J βv j ψ η (x β −q j ) = ρ η β v η β . Since M N j∈J βv j ψ η (x β −q j ) = v η β M N j∈J β ψ η (x β −q j ) + M N j∈J β δv β j ψ η (x β −q j ) = ρ η β v η β + M N j∈J β δv β j ψ η (x β −q j ),(6.
11) holds provided
(6.12) M N j∈J β δv β j ψ η (x β −q j ) = 0.
Now we look for perturbations in the form
(6.13) δv β j = a β j ψ η (x β −q j ) ,
where the a β j are to be determined. Next, narrow down the choice of a β j by setting (6.14) a β j = tã β j , whereã β j is either one or negative one. To satisfy (6.12) we need n β to be even (one more point can be easily inserted if the actual n β is odd); in addition, the number of positive and negativeã β j must be the same.
To simplify further calculations, we write conservation of energy (6.6) in the form
(6.15) j∈J β (δv β j ) 2 = K β ,
where K β are any numbers satisfying
(6.16) K β > 0, B β=1 K β = 2N M E − U( Q) − 1 2 M N B β=1 (v η β ) 2 n β .
Our goal now is to show that there is a choice of K β ,κ 2 and t such that δv β j defined by (6.13), (6.14) satisfy equations (6.5), (6.15), and (6.16). Inserting (6.13) into (6.5) and (6.15) yields, respectively,
t 2 j∈J β 1 ψ η (x β −q j ) =κ 2 , (6.17) t 2 j∈J β 1 (ψ η (x β −q j )) 2 = K β . (6.18)
Combining these equations we get
(6.19) t 2 =κ 2 j∈J β 1 ψ η (x β −q j ) −1 , (6.20)κ 2 j∈J β 1 (ψ η (x β −q j )) 2 j∈J β 1 ψ η (x β −q j ) −1 = K β .
Substituting into (6.16) yields the choice ofκ:
(6.21)κ 2 = 2N M E − U( Q) − 1 2 M N B β=1 (v η β ) 2 n β B β=1 j∈J β 1 (ψη(x β −qj )) 2 j∈J β 1 ψη(x β −qj ) −1 .
The choice of all constants now should be made as follows: 1) Given E, v η β ,q j , findκ by (6.21); 2) Determine K β from (6.20); 3) Determine t from (6.19); 4) Choose δv β j by (6.13), (6.14). Note that step 4 introduces non-uniqueness, but we are concerned only with existence of suitable velocity perturbations. The actual choice of δv β j will not change the mesoscopic discretization of momentum balance equation. Indeed, onceq j ,v j are chosen, we can approximate the integral in (6.3) (for x at the mesoscale mesh nodes) by a Riemann sum corresponding to the partitionq j of (0, L):
T η (c),0 (t, x α ) = − B β=1 j∈J β L η n β (δv β j ) 2 ψ η (x α −q j )ρ β (6.22) = − j∈Jα (δv α j ) 2 ψ η (x α −q j ) = −κ 2 , α = 1, 2, . . . , B.
Therefore, the mesoscale mesh values of T η (c),0 are all equal. This implies that a finite difference approximation of ∂ x T η (c),0 on the mesoscale mesh vanishes. The conclusion is that for isothermal dynamics, any suitable choice of a velocity perturbation produces a convective stress that has zero divergence on the mesoscale. 6.3. Zero-order isothermal continuum model. Combining the approximation ∂ x T η (c),0 = 0 with (6.3), (6.4) we obtain an isothermal zero-order continuum model
∂ t ρ η + ∂ x (ρ η v η ) = 0, (6.23) ∂ t (ρ η v η ) + ∂ x ρ η (v η ) 2 − ∂ x T η (int),0 = 0, (6.24)
where T η (int),0 is given by an integral expression (6.4) (or by a discretization (6.8)). We can interpret interaction stress as pressure. Then (6.4) provides dependence of pressure on density, which is non-local in space and non-linear. For small h (large N ), it can be approximated by
T η (int),0 ≈ − L 0 U M ρ η (t, y) ψ η (x − y) dy,
which is still non-local. In the limiting case η → 0, observing that convolution with ψ η is an approximate identity, this equation can be reduced a local equation of state
T η (int),0 ≈ −U M ρ η (t, x) .
If ρ η is nearly constant, this equation can be linearized to produce a classical gas dynamics linear equation of state. This shows that zero-order closure (6.4) generalizes several classical phenomenological equations of state. The connection between micro-and mesoscales is made explicit in (6.4). Using higher order closure approximations, one can obtain other non-classical continuum models worth further investigation.
Computational results
In this section, the method developed in the previous sections is tested for a chain of N = 10, 000 to N = 80, 000 particles interacting with a non-linear finite rage potential
(7.1) U (ξ) = C r 1 1−p ξ 1−p x − ξx 1−p + p p−1 x 2−p , if ξ ∈ (0, x ] 0, if ξ > x
where p > 1, x = αL, α ≈ 1 and C r is material stiffness. This potential mimics a Hertz potential used in modeling of granular media. Particles in this model are centers of lightly touching spherical granules arranged in a chain, and the ODEs model acoustic wave propagation in this chain. The microscale equations are (4.3), (4.4) with initial conditions given below. The forces include the pair interaction forces defined by U , and the exterior confining forces acting on the first and last particles. The parameters of U are chosen so that all particles stay within the interval [0, L] for the duration of a simulation. To ensure that particles do not leave the interval [0, L], its endpoints are modeled as stationary particles that interact with the moving particles with forces generated by the same potential U . If needed, stiffness of walls can be increased by using a different value of C r . Let x β be the centers of the mesocells I β , β = 1, 2, . . . , B, as defined in Section 6. Next, let a window function ψ(x) be the characteristic function of the interval [− 1 2 ηL, 1 2 ηL). The average density and momentum are defined by (2.6) and (2.7), respectively, and the average velocity is
v β (t) = N j=1 v j (t)ψ(x β − q j (t)) N j=1 ψ(x β − q j (t))
.
The average density ρ η evaluated at the center x β of a mesocell I β can be written as
(7.2) ρ η β (t) = ρ η (t, x β ) = M N N j=1 1 ηL ψ x β − q j (t) η = B N M L N j=1 ψ x β − q j (t)0 j = γ, if 0 ≤ q 0 j ≤ L 5 , γ − 5 L q 0 j + 2 , if L 5 ≤ q 0 j ≤ 2L 5 , 0,
if 2L 5 ≤ q 0 j ≤ L using the Velocity Stormer-Verlet method. We use L = 1, p = 2, α = 1, γ = 0.3 and C r = 100. This velocity profile initiates an acoustic wave that propagates to the right. The stiffness constant C r can be used to ensure that particles have only small displacements from their equilibrium positions. Using initial velocity with higher γ would require a higher value of C r to enforce smallness of typical particle displacements.
With fixed N = 40, 000, B = 50, we integrate microscopic equations (4.3), (4.4) until the acoustic wave reaches the right wall, interacts with it and is about of being reflected to the left, which corresponds to t = 0.07. To capture the most interesting dynamics, we present snapshots of results at times t = 0, 0.01, 0.03, 0.05, 0.06 and 0.07. To test our closure method, we compute microscopic positions q j and velocities v j , j = 1, . . . , N , at every time step and use them to evaluate primary mesoscopic variables: average density ρ η β and average velocity v η β , at mesocell centers x β , β = 1, . . . , B. These mesoscopic quantities are defined in (2.6), (2.7) (see also (5.6)). They are then employed in computing of the zero-order approximation T η (int),0 (t, x β ) defined in (6.4). We compare this mesoscopic approximation with the "exact" microscopic interaction stress T η (int) (t, x β ) defined in (5.2), and also test other approximations given by (6.1), (6.2). Comparing v j , j = 1, . . . , N and v η β , β = 1, 2, . . . , B (not shown here) we find that micro-and mesoscale velocities are essentially indistinguishable during the simulation time. In Fig. 1, we analyze microscopic Jacobian J(t, x β ) together with its zero-order mesoscopic approximation L M ρ η (t, x β ) obtained according to (6.1). In this and other figures, we plot "exact" microscopic quantities using a dashed line while mesoscopic quantities are depicted with a solid line. Results shown in Fig. 1 indicate that L M ρ η (t, x β ) exhibits some oscillations whose amplitude is about 10 −3 as compared to Jacobian J(t, x β ). The oscillations are likely caused by the choice of a window function ψ. For computational testing, we chose ψ to be a characteristic function. The main reason was to try "the worst case scenario" concerning smoothness of ψ. We expected that this window function would produce more oscillations than a smoother ψ. A good agreement between our approximation and the direct simulation results strongly suggest that the proposed method is viable. We believe that it should perform better with a smoother choice of ψ. The oscillations present in L M ρ η (t, x β ) are amplified in the approximated stress T η (int),0 (t, x β ), due to the rather high stiffness constant C r = 100, as shown in Fig. 2. We also compare microscopic J(t, x β )ṽ(t, x β ) with its zero-order approximation L M ρ η (t, x β )v η (t, x β ) according to (6.2). Graphs are not shown here but we find that these quantities agree very well similar to micro-and mesoscale velocities. This is expected since the average density is approximately identity with small oscillations. Finally, we verify that the dynamics is quasi-isothermal by plotting the convective stress T η (c) (t, x β ) defined in (5.11) in Fig. 3. As can be seen, fluctuations in the convective stress do not exceed 10 −4 throughout computational time, therefore, the kinetic energy of velocity fluctuations is small.
We next tested the effect of the scale separation on the quality of the zero-order approximation. With fixed B = 50, we allowed N vary from 10, 000 to 80, 000 and followed the evolution of mesoscale quantities of interest: L M ρ η (t, x β ) and T η Fig. 1).
In the above example, fluctuations of microscopic velocities about their average values were very small and the zero-order approximation worked well. Next we show that if microscopic velocities have high fluctuations then the zero-order approximation is not capable of captioning an appropriate dynamics.
We demonstrate this by imposing high frequency k oscillations with relatively large amplitude a on the nonzero portion of the initial velocity used in the previous experiments. The initial velocity is
v 0 j = γ + a sin( 5kπ L q 0 j ), if 0 ≤ q 0 j ≤ L 5 , γ − 5 L q 0 j + 2 + a sin( 5kπ L q 0 j ), if L 5 ≤ q 0 j ≤ 2L 5 , 0,
if 2L 5 ≤ q 0 j ≤ L. and it is plotted in the left panel of Fig. 6. We use a = 5 and k = 20 that gives one period of imposed oscillations per mesocell. This microscopic initial velocity has a property that the average velocity at time t = 0 is exactly the same as in the previous example. Simulations were done with N = 10, 000 until the same t = 0.07. The right panel of Fig. 6 shows a typical microscopic velocity profile together with its average velocity (taken at t = 0.01): to the left from the wave front, the microscopic velocity has large frequency oscillations (due to dispersion?) with an amplitude sometimes exceeding the initial amplitude by a factor of 1.5 and to the right from the wave front, the microscopic velocity is zero. Clearly, the average velocity is very different from the microscopic velocity.
Analysis of microscopic Jacobian J(t, x β ) and mesoscopic L M ρ η (t, x β ) reveals that these functions have qualitatively the same dynamics as micro-and mesoscale velocities, respectively, shown in Fig. 6. We plot the former in Fig. 7 where the left panel has graphs at t = 0 while the right panel shows typical structure with data taken at t = 0.01. It is interesting to note that the zero-order approximation T η (int),0 (t, x β ) to the interaction stress T η (int) (t, x β ) plotted in Fig. 8 does not agree in those areas that were affected by large magnitude oscillations in microscopic velocities while agrees well in those areas to which oscillations have not come yet. This finding suggests that indeed the zero-order approximation should not be used for large frequency oscillations in microscopic velocities and a higher order approximation is needed.
Finally, in Fig. 9 we plot the convective stress T η (c) (t, x β ) whose large values confirm that oscillations in microscopic velocities are much bigger during the computational time than those in the first example. When the initial velocity has fluctuations with frequency higher than k = 20, discrepancy between microand mesoscale quantities is even more pronounced.
Zero-order closure: applicability
Zero-order closure is very similar to the use of the Cauchy-Born rule in quasi-continuum simulations of solids. Here, the nodes of the mesoscale mesh can be thought of as "representative particles". These particles are moved with the average velocity, while the velocities of other particles are assigned by interpolation. A construction of an interpolant should take into account the physics of the microscopic model such as energy conservation. In the computational example of Section 7, zero-order approximation turns out to be quite accurate, when non-oscillatory initial conditions are imposed. In this case, we found that approximate and exact stresses agree rather well, and this agreement becomes better with increasing scale separation. This does not mean that zero-order closure always works well. Our numerical simulations suggest that applicability of zero-order closure is determined by initial conditions, exterior forces and interaction potential (arranged in order or importance).
Approximating functions by their averages we neglect fluctuations. Therefore, the initial velocities should have small fluctuations. Initial positions should be chosen so that the number of particles in a meso-cell varies slightly from one cell to another. The initial velocity fluctuations in our first example are small, and convective stress at later times is by three orders of magnitude smaller than interaction stress. This remains true on the time interval sufficient for the traveling wave to reach the opposite end of the chain.
For one-dimensional problems, convective stress is proportional to the kinetic energy of velocity fluctuations. This kinetic energy can be naturally associated with upscaling temperature. Relative smallness of the convective stress mens that upscaling temperature is nearly zero. Therefore, the corresponding dynamics can be termed cold. We also note that cold dynamics is a special case of isothermal dynamics, considered in Section 6. As has been remarked earlier, isothermal dynamics implies that divergence of the convective stress is nearly zero on mesoscale, and thus can be neglected compared with the divergence of the interaction stress.
Another consideration is related to inhomogeneity in actual particle distribution. In our example, deviations of about 4% in relative particle positions produced visible oscillations in the approximation of the interaction stress. This amplification of small perturbations is due to the stiffness of the interaction potential. However, the same stiffness prevents particle aggregation, keeping the interparticle distances bounded from below. Bounds from above are difficult to enforce with the chosen potential because it does not have a potential well. The isolated particle system with this potential would just fall apart. This phenomenon is common place for granular materials. The particles remain confined to the domain (container) only because they are repelled by the walls. Walls have very little direct influence on the interparticle distances in the systems' interior. Therefore, applicability of zero-order closure also depends on the stiffness of the problem, and more generally on how well the potential enforces uniform particle distribution. In that sense, zeroorder closure makes a reasonable approximation for lattice systems modeling small deformation of solids at constant temperature.
To further understand limitations of zero-order closure, consider the effect of increasing the order n of the Landweber approximations (3.7). The Fourier transform the kernel of I − R η is equal to 1 − e −η 2 π 2 ξ·ξ .
It is very small for ξ close to zero, and then increases to one as |ξ| goes to infinity. Therefore, I − R η acts as a filter damping low frequencies and thus emphasizing higher frequency content of the signal. Higher order approximations amount to applying convolutions n k=1 (I − R η ) k to mesoscale averages. As n increases, high frequency content of the reconstruction will be increasingly amplified. This suggests that systems capable of producing large fluctuations should be handled with higher order approximations.
A related comment is that averages of fluctuations can become additional state variables in a mesoscale continuum model. A familiar example is the use of the averaged energy balance equation (see [1] for derivation), in addition to the mass and momentum balance. The energy balance equation describes evolution of the density of kinetic energy of velocity fluctuations. An intriguing question here is how the model with just two equations of balance but high order closure approximation compares with a zero-order closure model containing all three balance equations. In classical physics, additional balance equations are often introduced as a means of compensating for errors introduced by replacing state variables with their averages. Use of higher order closure could offer an alternative to this approach. Indeed, suppose that one is interested only in tracking density and velocity on mesoscale. The corresponding two balance equations contain only two microscale quantities: velocity fieldṽ and the Jacobian J of the inverse position mapq −1 . Ifṽ and J can be accurately reconstructed from their averages, we do not need to deal with the energy balance equation. This observation offers a new way of reducing computational cost. Higher order approximation are more expensive than zero-order, but using more balance equations also increases computational cost. We also note that increasing the order of closure approximations involves repeated convolutions with the window function ψ η . On the other hand, simulating an energy balance involves numerical integration of an additional non-linear integral-differential equation, a much more difficult task.
Conclusions
We propose a closure method that gives closed form approximations for mesoscale continuum mechanical fluxes (such as stress) in terms of primary mesoscopic variables (such as average density and velocity). Our closure construction is based on iterative regularization methods for solving first kind integral equations. Such integral equations are relevant because mesoscopic density and velocity are related to the corresponding microscopic quantities via a linear convolution operator. The problem of inverting convolution operators is unstable (ill-posed) and requires regularization. Use of the well known Landweber iterative regularization yields successive approximations, of orders zero, one, two and so forth, to interpolants of particle positions and velocities in terms of available averages. Closure is achieved by inserting any of these approximations into the equations for fluxes instead of the actual particle positions and velocities. Low order approximations are simpler to implement, while higher order approximations can be used to more accurately reproduce the high frequency content of the microscopic quantities.
The above general strategy is applied in the paper to spatially averaged dynamics of classical particle chains. We focus on the simplest zero-order approximation and show numerically that it works reasonably well as long as initial conditions have small velocity fluctuations. The case of large fluctuations in velocities should be handled by higher order approximations.
Figure 1 .Figure 2 .
12η that shows that average density is proportional to the scale separation B/N . We solve microscopic equations (4.3), (4.4) subject to the initial positions q 0 j = j − 1 2 h, j = 1, 2, . . . , N, h = N = 40, 000, B = 50. Dashed line: Jacobian J(t, x β ); solid line: its mesoscale approximation L M ρ η (t, x β ), β = 1, 2, . . . , B according to (6.1). Blowup of results at t = 0.01 shows discrepancy between J(t, x β ) and L M ρ η (t, x β ). N = 40, 000, B = 50. Dashed line: exact interaction stress T η (int) (t, x β ); solid line: its approximation T η (int),0 (t, x β ), β = 1, 2, . . . , B, defined in (6.4). Blowup of results at t = 0.001 shows difference between exact stress and its approximation. and the initial velocities
Figure 3 .
3N = 40, 000, B = 50. Convective stress T η (c) (t, x β ), β = 1, 2, . . . , B, defined in (6.3).
(int),0 (t, x β ). Snapshots of these functions at the same representative time t = 0.01 are plotted in Figs. 4 and 5, respectively, with N = 10, 000, N = 20, 000 and N = 80, 000. The results with N = 40, 000 at the same time are given the middle top panels in Figs. 1, 2 for comparison. It is clear that as scale separation increases, oscillations in both L M ρ η (t, x β ) and T η (int),0 (t, x β ) diminish and when N = 80, 000, the exact microscopic quantities and their mesoscale approximations are almost indistinguishable.
Figure 4 .Figure 5 .
45Effect of the scale separation on L M ρ η . B = 50 is fixed, N varies, data is taken at the same t = 0.01. Dashed line: Jacobian J(t, x β ); solid line: its mesoscale approximation L M ρ η (t, x β ), β = 1, 2, . . . , B. Effect of the scale separation on T η (int),0 . B = 50 is fixed, N varies, data is taken at the same t = 0.01. Dashed line: exact interaction stress T η (int) (t, x β ); solid line: its mesoscale approximation T η (int),0 (t, x β ), β = 1, 2, . . . , B.
Figure 6 .Figure 7 .
67Example with imposed high frequency oscillations. N = 10, 000, B = 50. Dashed line: exact velocity v j , j = 1, 2, . . . , N ; solid line: average velocity v β , β = 1, 2, . . . , B. Example with imposed high frequency oscillations. N = 10, 000, B = 50. Dashed line: Jacobian J(t, x β ); solid line: its mesoscale approximation L M ρ η (t, x β ), β = 1, 2, . . . , B (compare with
Figure 8 .
8Example with imposed high frequency oscillations. N = 10, 000, B = 50. Dashed line: exact interaction stress T η (int) (t, x β ); solid line: its mesoscale approximation T η (int),0 (t, x β ) (compare withFig. 2).
Figure 9 .
9Example with imposed high frequency oscillations. N = 10, 000, B = 50. Convective stress T η (c) (t, x β ), β = 1, 2, . . . , B (compare with Fig. 3).
AcknowledgmentsWork of Alexander Panchenko was supported in part by DOE grant DE-FG02-05ER25709 and by NSF grant OISE-0438765. Work of R. P. Gilbert was supported in part by NSF grants OISE-0438765 and DMS-0920850, and by the Alexander v. Humboldt Senior Scientist Award at the Ruhr Universitat Bochum.
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| [] |
[
"Gaussian information matrix for Wiener model identification *",
"Gaussian information matrix for Wiener model identification *"
] | [
"Kaushik Mahata [email protected]‡email:[email protected] \nDepartment of Electrical Engineering\nUniversity of Newcastle\nNSW-2308CallaghanAustralia\n",
"Johan Schoukens \nDepartment ELEC\nVrije Universiteit\nBuilding K, Pleinlaan 21050Brussel, BrusselsBelgium\n"
] | [
"Department of Electrical Engineering\nUniversity of Newcastle\nNSW-2308CallaghanAustralia",
"Department ELEC\nVrije Universiteit\nBuilding K, Pleinlaan 21050Brussel, BrusselsBelgium"
] | [] | We present a closed form expression for the information matrix associated with the Wiener model identification problem under the assumption that the input signal is a stationary Gaussian process. This expression holds under quite generic assumptions. We allow the linear sub-system to have a rational transfer function of arbitrary order, and the static nonlinearity to be a polynomial of arbitrary degree. We also present a simple expression for the determinant of the information matrix. The expressions presented herein has been used for optimal experiment design for Wiener model identification. | null | [
"https://arxiv.org/pdf/1510.03013v1.pdf"
] | 15,981,568 | 1510.03013 | 85df03514e7849cc7c63ca816606cec47285d1da |
Gaussian information matrix for Wiener model identification *
11 Oct 2015 October 13, 2015
Kaushik Mahata [email protected]‡email:[email protected]
Department of Electrical Engineering
University of Newcastle
NSW-2308CallaghanAustralia
Johan Schoukens
Department ELEC
Vrije Universiteit
Building K, Pleinlaan 21050Brussel, BrusselsBelgium
Gaussian information matrix for Wiener model identification *
11 Oct 2015 October 13, 20151Wiener model identificationInformation matrixGaussian inputDeterminant *
We present a closed form expression for the information matrix associated with the Wiener model identification problem under the assumption that the input signal is a stationary Gaussian process. This expression holds under quite generic assumptions. We allow the linear sub-system to have a rational transfer function of arbitrary order, and the static nonlinearity to be a polynomial of arbitrary degree. We also present a simple expression for the determinant of the information matrix. The expressions presented herein has been used for optimal experiment design for Wiener model identification.
Introduction
In this paper we present an expression of the information matrix associated with the Wiener model identification problem under a generic setting. Our analysis allows a rational model the linear subsystem, and a polynomial nonlinearity. The rational transfer function can be of arbitrary oder and a nonlinearity with arbitrary polynomial order can be allowed. The analysis assumes Gaussian stationary input process.
This work is mainly motivated by the experiment design problem. The standard approach in experiment design is to choose the input excitation in order to optimize some monotonic function of the information matrix [1-3, 8, 9, 11]. When the linear subsystem G of the underlying Wiener system has a generic rational transfer function, it has an infinite impulse response. Consequently, the information matrix J becomes a function of the of higher order joint moments of the input process u(t), u(t − 1), u(t − 2), . . .. It is very challenging to optimize any criterion of J with respect to the all higher order moments of potentially infinitely many consecutive samples of the input u(t). In fact, it is quite difficult to just compute J. Firstly, the available formulae for calculating higher order moments are quite challenging to program. More importantly, the complexity of the resulting algorithm typically grows exponentially with the length of the impulse response of G [12]. In fact, to the best of our knowledge no previous authors have considered handling this issue when G is not a finite impulse response system. Even when a finite impulse response system is considered in the literature, the order of the system have been restricted to 4 or less. In fact, when compared with the experiment design literature for linear system identification [1-3, 8, 9, 11], the the line of research in the nonlinear input design has undergone a significant paradigm shift. Most of the preliminary studies reported so far [6,7,10,13,16], have considered a deterministic setting. Among these the multi-level excitation approach [4][5][6]13] appears to be popular lately. These deterministic methods do have their limitations. The multi-level approach is often not tractable when we increase the number of levels. The majority of these methods are unable to handle IIR-type non-linear systems.
We show when the input process is Gaussian there is a simple algorithm to compute J. This analysis also reveals some interesting mathematical structures, that allows us to parameterize the set of all admissible information matrices with a finite number of parameters. Hence the experiment designer needs to solve only a finite dimensional problem. See [14] for the details of how the expressions presented herein can be used in experiment design.
Information matrix and its determinant
In this section we present our main findings about the information matrix J and its determinant. We start in Section 2.1 with the basic notation and introduce the formal problem setting. In particular, we introduce a generalized framework for setting up the constraint to ensure unique identifiability of the Wiener model. Next in Section 2.2 we list the main results. In particular we use a state space representations of underlying transfer functions. We believe this approach simplifies the analysis, and illuminates the underlying mathematical structures to a significant extent.
Model parameterization and identifiability
A Wiener system is a cascade of a linear time invariant system followed by a static nonlinearity. We assume that the linear sub-system has a rational transfer function G(z, θ) = g 0 + g 1 z −1 + · · · + g n z −n 1 + a 1 z −1 + · · · + a n z −n ,
parameterized by the parameter vector θ defined as θ = [ a 1 · · · a n g 1 · · · g n g 0 ] ⊺ .
The output of the linear model is denoted by w:
w(t, θ) = G(z, θ)u(t).(3)
The static nonlinearity is modeled by a polynomial ℘ of order m:
℘(x,ᾱ) = α 0 + α 1 x + · · · + α m x m ,
parameterized by the vector of polynomial coefficients
α = [ α 0 α 1 · · · α m ] ⊺ .
Therefore, the Wiener model equation takes the form
M(ϑ, u t ) = ℘{G(z, θ)u(t),ᾱ}.(4)
It is tempting to choose ϑ = [ᾱ ⊺ θ ⊺ ] ⊺ . But this parameterization fails to ensure unique identifiability. We cannot allow all the components of θ andᾱ to vary freely while remaining independent of each other. The reason is straightforward. The transfer operator between u and y does not change by dividing G by a scalar ̺ = 0, and multiplying α k by ̺ k for all k = 1, 2, . . . , m.
For this reason we must impose some additional constraint on the parameters. In this paper we allow varying the static gain of G freely, and impose a normalization constraint onᾱ.
Assumption 1 There is a known vector
υ = [ υ 0 υ 1 · · · υ m ] ⊺(5)
such that
α 0 υ 0 + α 1 υ 1 + · · · + α m υ m = 1,(6)
where υ ℓ = 0 for some known ℓ ∈ {1, 2, . . . , m}.
The choice of ℓ is often governed by the prior knowledge on the type of nonlinearity. Typically ℓ = 0, because it is often the case that α 0 = 0. For an odd (even) nonlinearity ℓ must be an odd (even) number. In our experience, the choice of ℓ does not influence the asymptotic large sample accuracy of the estimated model. Example 1 It is common to take υ = (0, 1, . . . , 0) or υ = (0, . . . , 0, 1). Another possibility would be to take υ = (1, . . . , 1) implying ℘(1) = 1. Note that the choice υ = (1, 0, · · · , 0) is forbidden. It leads to a model that is not identifiable.
Since υ ℓ = 0 under Assumption 1, we can rewrite (6) as
α ℓ = 1 υ ℓ 1 − m k = 0 k = ℓ υ k α k .(7)
Equation (7) can be built into the identification algorithm. We do not identify α ℓ separately, but express it using (7). We define the parameter vector
α := [ α i 1 · · · α im ] ⊺ ,(8)
where the indices i k ∈ {0, 1, . . . , m} are chosen such that i k = ℓ for all k, and i k = i j whenever k = j. Note that mapping k → i k is quite flexible, and we need not impose any further restriction on this mapping. The identification algorithm estimates
ϑ = [ α ⊺ θ ⊺ ] ⊺ from the data. Defining L = 1 0 · · · 0 −υ i 1 /υ ℓ 0 1 · · · 0 −υ i 2 /υ ℓ . . . . . . . . . . . . 0 0 · · · 1 −υ im /υ ℓ , P = [ e i 1 · · · e im e ℓ ] ⊺ ,(9)
with e k denoting the k + 1 th column of (m + 1) × (m + 1) identity matrix, it can be verified from
(7) that [ α ⊺ α ℓ ] ⊺ = Pᾱ = L ⊺ α.(10)
Main theoretical results
Let a = [ a 1 · · · a n ] ⊺ , and g = [ g 1 · · · g n ] ⊺ . Then we can write (1) as
G(z, θ) = g 0 + (g − ag 0 ) ⊺ (zI − A 1 ) −1 b 1 ,(11)where (A 1 , b 1 ) is in controllable canonical form, i.e. A 1 = −a 1 · · · −a n−1 −a n 1 · · · 0 0 . . . . . . . . . . . . 0 · · · 1 0 , b 1 = 1 0 . . . 0 .(12)
Note that we can impose the structure (11) and (12) without any loss of generality. We make the following assumption throughout the paper, whereθ denotes the true value of θ.
Assumption 2 G(z,θ) is asymptotically stable. Consequently, all the eigenvalues ofÅ 1 (which denotes the true value of A 1 ) are located inside the unit disc in the complex plane. In addition, the state space realization (11) is minimal.
Lemma 1 Define the matrices A, b, c andC as
C = I −g 0 I 0 0 I 0 0 −å ⊺ 1 , b = 0 n×1 0 n×1 1 , c = 0 n×1 g g 0 , A =C Å 1 −b 1 (g −åg 0 ) ⊺ 0 n×1 0 n×nÅ1 b 1 0 1×n 0 1×n 0 C −1 ,(13)
whereÅ 1 ,g 0 , etc are obtained by setting θ =θ in A 1 , g 0 , etc. Consider the stochastic process x which is given in state space form as
x(t) = Ax(t − 1) + bu(t).(14)
Then
w(t,θ) = c ⊺ x(t), and v t = LPz(t,θ) x(t)α ⊺ 2 z(t,θ) ,(15)
where we define
z(t, θ) := [ 1 w(t, θ) {w(t, θ)} 2 · · · {w(t, θ)} m ] ⊺ ,(16)α 2 = [α 1 2α 2 · · · mα m 0 ] ⊺ ,(17)
withα k being the true value of α k .
Proof: See Section A.
Remark 1 Lemma 1 does cover the case when G is of finite impulse response type, i.e., G(z, θ) = g 0 + g 1 z −1 + · · · + g n z −n .
In this case θ = [ g ⊺ g 0 ] ⊺ , and a = 0. The expressions (11) and (12) still hold with a = 0. While finding a realization of G 1 we do not need to consider the derivatives with respect to a. As a result we get
A = Å 1 b 1 0 1×n 0 , b = 0 n×1 1 , C = I and c = θ.
The consequence of Lemma 1 is that J = E{v t v ⊺ t } is a function of the moments of the state vector x. For the purpose of setting up an input design problem we can parameterize J in terms of the moments of the random vector x. In particular, when u(t) is Gaussian, then so is x(t). Hence for a Gaussian input J is a function of the mean and the covariance matrix of x(t). As the next Theorem reveals, we can obtain a closed form expression for J. Assumption 3 The input process u(t) is stationary Gaussian with mean η u .
Under Assumption 3, x is a Gaussian random vector with mean
η := E{x(t)} = (I − A) −1 bη u . (18) Let us define Σ = E{[x(t) − η][x(t) − η] ⊺ }.(19)
Consequently, c ⊺ x(t) is a Gaussian random variable such that
γ := E{c ⊺ x(t)} = c ⊺ η. (20a) σ := E{c ⊺ x(t) − γ} 2 = c ⊺ Σc. (20b)
In the rest of the paper we denote
Λ := E{z(t,θ)[z(t,θ)] ⊺ }.
Remark 2 It is possible to express Σ as well in terms of A, b, and the power spectral density Φ of u. However, we postpone that for a while. We first express J in terms of Σ and η, and later connect Φ with Σ. This approach suits the purpose of input design, where it is simpler to work with a parameterization of Σ than to work with Φ directly.
Remark 3
The correlation matrix Λ can be given entirely as a function of the mean γ and variance σ of c ⊺ x(t). Many different ways are used in the literature to express the moments of the scalar valued normal density. There are some explicit formulae for smaller orders. We find it convenient to use a recursive formula in the implementation. Let us denote µ k (γ, σ) := E{(c ⊺ x) k }. So µ k is a function of σ and µ. Then µ k (γ, σ) satisfies the recursion [15,Chapter 5]:
µ k (γ, σ) = γ k + k(k − 1) 2 σ 0 µ k−2 (τ, σ) dτ.(21)
Note that the recursion (21) needs to be carried out separately for even and odd values of k. For even valued k one can initialize the recursion with µ 0 (γ, σ) = 1, and for the odd values of k we initialize with µ 1 (γ, σ) = γ. This allows us to form
Λ = µ 0 (γ, σ) µ 1 (γ, σ) · · · µ m (γ, σ) µ 1 (γ, σ) µ 2 (γ, σ) · · · µ m+1 (γ, σ) . . . . . . . . . µ m (γ, σ) µ m+1 (γ, σ) · · · µ 2m+1 (γ, σ) .
Since x is Gaussian, all the moments of x can be expressed as functions of η and Σ. This allows us to derive manageable expressions of J as a function of η and Σ. This is shown next.
Theorem 1 Define
α 1 = [ 0α 1 , 2α 2 · · · mα m ] ⊺ , β = α ⊺ 2 Λα 2 ,(22)Q = 1 σ − γ σ 0 1 , F := [ Σc η ], H = βσ 0 0 0 . Partition J as J = J 11 J ⊺ 21 J 21 J 22 ,
where J 11 is of size m × m, while J 22 is of size (2n + 1) × (2n + 1). Then
J 11 = L 1 ΛL ⊺ 1 ,(23)J 21 = FQL 2 ΛL ⊺ 1 J 22 = FQ(L 2 ΛL ⊺ 2 − H)Q ⊺ F ⊺ + βΣ, where L 1 := LP,(24)L 2 := α ⊺ 1 α ⊺ 2 = 0α 1 2α 2 · · · mα m α 1 2α 2 · · · mα m 0 . Proof: See Appendix B.
Remark 4 Expressions given by Theorem 1 allow us to compute J in a simple way. To the best of our knowledge there is no similar expressions in the literature allowing this computational advantage.
The matrices Q, H, Λ, L 1 , L 2 and β share an interesting property. They depend only on the true parameter vectorθ and the second order statistics (consisting of γ and σ) of the stochastic process w(t,θ) = c ⊺ x(t). These quantities remain constant so long γ and σ remain constant, even though the input power spectral density (and thus Σ) may vary. This is due to the fact that the estimation accuracy of the static nonlinearity depends only on the amplitude distribution of w(t), regardless of Σ (or equivalently, Φ). This observation plays a key role in the sequel, and is formalized via the following definition.
Definition 1 A quantity is called w-dependent if it is a function ofθ, σ and γ only.
The expressions given in Theorem 1 may not seem appealing from the point of view of setting up an optimization problem for input design that can be solved in a tractable manner. The next result is more attractive in that regard. is w-dependent, and remains constant when the statistics of w(t, θ 0 ) remain invariant. On the other hand it is well-known from the literature on the input design for linear systems that we can parameterize det(Σ) in a convex manner using a finite number of parameters. When the mean η u of the input is kept fixed, then the above facts let us solve the D-optimal design problem for Wiener models via an one dimensional search in σ. To emphasize the w-dependence of f we write it as f (γ, σ). When we consider a situation where γ is fixed and known, then we write it as f (σ).
Remark 6
Note that J is singular when r 1 = 0. This means that the normalization of the form described in Assumption 1 ensures identifiability (and thus a non-singular information matrix) only when By assumption, (α 0 ,α 1 , · · · ,α m ) ∈ H. The model is identifiable when H intersects with the manifold M = {(α 0 , ̺α 1 , · · · , ̺ mα m ) : ̺ = 0} only at the point (α 0 ,α 1 , · · · ,α m ), which corresponds to ̺ = 1. We have local identifiability at (α 0 ,α 1 , · · · ,α m ) only if M is not oriented along H at (α 0 ,α 1 , · · · ,α m ), i.e., ̺ = 1. In other words, we do not want the directional derivative (0, 2α 1 , · · · , mα m ) =: α 1 of M at ̺ = 1 to be perpendicular to υ, which is identical to (27).
0 = α ⊺ 1 υ = υ 1α1 + 2υ 2α2 + · · · + mυ mαm ,(27)
Conclusions
We have presented several new results on the analysis of Wiener model identification using Gaussian input processes. One of the main results in this paper is Theorem 1, which gives a closed form expression of the associated information matrix J. This expression holds under very generic assumptions on the model structure. In addition, unlike other similar formulae available in the literature, our expression for J is easy to compute. This aspect makes it attractive in input design problems. Theorem 2 gives a simple expression for the determinant of J. These expressions are particularly useful in experiment design, see [14] for details.
A Proof of Lemma 1
By definition of P in (9) we have PP ⊺ = I. Using this in (10) gives
α = P ⊺ L ⊺ α.(28)
Using (28) and the definition of z(t, θ) in (16) in (4) we have
M(ϑ, u t ) =ᾱ ⊺ z(t, θ) = α ⊺ LPz(t, θ).(29)
Hence
∂M(θ, u t ) ∂α = LPz(t,θ).(30)
Also using the definition of z(t, θ) in (16) and differentiating M(t, ϑ) in (29) with respect to θ we get
∂M(θ, u t ) ∂θ = ∂w(t,θ) ∂θα ⊺ LP 0 1 2w(t,θ)
. . .
m{w(t,θ)} m−1 = ∂w(t,θ) ∂θ α ⊺ 2 z(t,θ),(31)
where the last equality follows from the definition of α 2 in (17) and the definition of z(t, θ) in (16). The proof for the expression of v t in (15) will be complete if we can show
∂w(t,θ) ∂θ = ∂G(z,θ) ∂θ u(t) = x(t).(32)
This is done next. By differentiating G with respect to a, g and g 0 we get
∂G(z, θ) ∂a = −(zI − A 1 ) −1 b 1 g 0 − (zI − A 1 ) −1 b 1 (g − ag 0 ) ⊺ (zI − A 1 ) −1 b 1 , (33a) ∂G(z, θ) ∂g = (zI − A 1 ) −1 b 1 ,(33b)∂G(z, θ) ∂g 0 = 1 − a ⊺ (zI − A 1 ) −1 b 1 .(33c)
Using (33) and (13) it can be verified by direct calculations that
∂G(z,θ) ∂θ = (I − Az −1 ) −1 b,(34)
implying (32).
To show w(t,θ) = c ⊺ x(t) verify from (11) and (33) that
G(z,θ) = [ 0 ⊺g⊺g 0 ] ∂G(z,θ) ∂θ = c ⊺ (I − Az −1 ) −1 b.
B Proof of Theorem 1
Since Σ is positive definite, Σc = 0. Hence there exists a full column rank (2n + 1) × (2n) matrix C such that the column space of C is the orthogonal complement of Σc, i.e., C ⊺ Σc = 0. Hence
c ⊺ C ⊺ Σ [ c C ] = σ 0 0 Σ 1 ,(35)
The block diagonal structure of the matrix in the right hand side of (35) ensures that by premultiplying x by [ c C ] ⊺ we get two mutually uncorrelated components c ⊺ x and
x 1 := C ⊺ x, with γ := E{x 1 } = C ⊺ η, Σ 1 := E{[x 1 − γ][x 1 − γ] ⊺ } = C ⊺ ΣC.(36)
Because x is a Gaussian random vector, we conclude that [ c ⊺ x x ⊺ 1 ] ⊺ too is a jointly Gaussian random vector. Since uncorrelated Gaussian variables are independent, c ⊺ x and x 1 are mutually independent.
Define the (m + 2n + 1) × (m + 2n + 1) matrix
T = I 0 0 c ⊺ 0 C ⊺ ,(37)
where the identity matrix appearing in (37) in the north-west corner is of size m × m. Premultiplying v t in (16) by T we note that
Tv t = L 1 z(t,θ) c ⊺ x(t)α ⊺ 2 z(t,θ) C ⊺ x(t)α ⊺ 2 z(t,θ) .(38)
From Lemma 1 recall that c ⊺ x(t) = w(t,θ). Then from the definition of z(t, θ) in (16), the definitions α 1 and α 2 in (22) and (17) we have
c ⊺ x(t)α ⊺ 2 z(t,θ) = w(t,θ)α ⊺ 1 z(t,θ). In addition, C ⊺ x(t) = x 1 (t). Hence Tv t = L 1 z(t,θ) α ⊺ 1 z(t,θ) x 1 (t)α ⊺ 2 z(t,θ) .(39)
Since x 1 (t) is independent of w(t,θ) = c ⊺ x(t), it is also independent of z(t,θ), see (16). Using this and (39) we get
TJT ⊺ = E {[Tv t ] [Tv t ] ⊺ } = L 1 ΛL ⊺ 1 L 1 Λα 1 L 1 Λα 2 γ ⊺ α ⊺ 1 ΛL ⊺ 1 α ⊺ 1 Λα 1 α ⊺ 1 Λα 2 γ ⊺ γα ⊺ 2 ΛL ⊺ 1 γα ⊺ 2 Λα 1 α ⊺ 2 Λα 2 (γγ ⊺ + Σ 1 ) .(40)
Define the vector d and the (2n + 1) × (2n) matrix D by partitioning the inverse
c ⊺ C ⊺ −1 = [ d D ].(41)
Then (37) and (40) imply
J = I 0 0 0 d D (TJT ⊺ ) I 0 0 d ⊺ 0 D ⊺ .
Using expression of TJT ⊺ in (40) we get
J 11 = L 1 ΛL ⊺ 1 ,(42a)J 21 = [ d Dγ ] L 2 ΛL ⊺ 1 (42b) J 22 = [ d Dγ ] L 2 ΛL ⊺ 2 [ d Dγ ] ⊺ + βDΣ 1 D ⊺ .(42c)
Now from (35) and (41) we obtain
Σ = [ d D ] σ 0 0 Σ 1 d ⊺ D ⊺ = dσd ⊺ + DΣ 1 D ⊺ .(43)
By definition of d and D in (41) we know
c ⊺ C ⊺ [ d D ] = 1 0 0 I ,
and this implies C ⊺ d = 0, ⇒ d = kΣc. In addition, 1 = c ⊺ d = kc ⊺ Σc = kσ. Consequently,
d = 1 σ Σc.(44)
On the other hand
I = [ d D ] c ⊺ C ⊺ = dc ⊺ + DC ⊺ = 1 σ Σcc ⊺ + DC ⊺ ,(45)
Now multiply both sides of (45) by η to get
η − γ σ Σc = Dγ(46)
From (44)
J 21 = FQL 2 ΛL ⊺ 1 J 22 = FQL 2 ΛL ⊺ 2 Q ⊺ F ⊺ + β(Σ − Σcc ⊺ Σ/σ) = βΣ + FQL 2 ΛL ⊺ 2 Q ⊺ F ⊺ − FQHQ ⊺ F ⊺ = FQ(L 2 ΛL ⊺ 2 − H)Q ⊺ F ⊺ + βΣ.J 22 − J 21 J −1 11 J ⊺ 21 = βΣ + FQ[L 2 υ(υ ⊺ Λ −1 υ) −1 υ ⊺ L ⊺ 2 − H]Q ⊺ F ⊺ .
Proof: In this proof we let Γ be the Cholesky factor of Λ, i.e., Λ = ΓΓ ⊺ . From the expressions of J 11 , J 21 and J 22 in Theorem 1 it follows that
J 22 − J 21 J −1 11 J ⊺ 21 = βΣ + FQ[L 2 ΠL ⊺ 2 − H]Q ⊺ F ⊺ ,(47)
where we define
Π = Λ − ΛL ⊺ 1 (L 1 ΛL ⊺ 1 ) −1 L 1 Λ = Γ[I − Γ ⊺ L ⊺ 1 (L 1 ΓΓ ⊺ L ⊺ 1 ) −1 L 1 Γ]Γ ⊺ .(48)
However, the matrixΠ :
= I − Γ ⊺ L ⊺ 1 (L 1 ΓΓ ⊺ L ⊺ 1 ) −1 L 1 Γ is the orthogonal projection operator onto the nullspace of L 1 Γ.
From (5), (9) and the definition of L 1 in (24) verify that that L 1 υ = LPυ = 0. This means
L 1 ΓΓ −1 υ = 0,
i.e. the vector Γ −1 υ spans the one dimensional nullspace of L 1 Γ. HenceΠ is also the orthogonal projection operator onto the span of Γ −1 υ. Hencē
Π = Γ −1 υ(υ ⊺ Λ −1 υ) −1 υ ⊺ Γ −⊺ .
Substituting this expression in (48) gives
Π = υ(υ ⊺ Λ −1 υ) −1 υ ⊺ ,
which upon substitution in (47) yields the desired result.
Define
r i := α ⊺ i υ(υ ⊺ Λ −1 υ) −1/2 , i = 1, 2.(49)
Note that
L 2 υ(υ ⊺ Λ −1 υ) −1 υ ⊺ L ⊺ 2 = r 1 r 2 r 1 r 2 ⊺ ,(50)
see the definition of L 2 in Theorem 1. When r 2 = 0 the matrix L 2 υ(υ ⊺ Λ −1 υ) −1 υ ⊺ L ⊺ 2 − H is of rank 1. Then the calculations turn out to be quite different from the case where r 2 = 0.
Proof: When r 2 = 0 then using (49), definition of Q in Theorem 1 and the expressions given by Lemma 2 we get
J 22 − J 21 J −1 11 J ⊺ 21 = βΣ + FQ r 2 1 − βσ 0 0 0 Q ⊺ F ⊺ = βΣ + F 1 σ − γ σ 0 1 r 2 1 − βσ 0 0 0 Q ⊺ F ⊺ = βΣ + F r 2 1 /σ − β 0 0 0 1 σ 0 − γ σ 1 F ⊺ = βΣ + F r 2 1 /σ 2 − β/σ 0 0 0 F ⊺ = βΣ + (r 2 1 /σ 2 − β/σ)Σcc ⊺ Σ.(52)
In this proof we write q = r 2 1 /σ 2 − β/σ compactly. From (52)
The result of Theorem 2 is immediate from (62) once we use the expression for det(J 22 −J 21 J −1 11 J ⊺ 21 ) given by (53).
Theorem 2
2The determinant of J is given bydet(J) = β 2n r 2 1 σ det(J 11 ) det(Σ).(25)where r 1 = α ⊺ 1 υ(υ ⊺ Λ −1 υ) −1/2 .
see the definition of r 1 in the statement of Theorem 2. We can easily construct a case where (27) fails to hold. That is υ = (1, 0, . . . , 0). It is straightforward to see why this choice leads to lack of identifiability: it still allows us to simultaneously vary the gain of the linear subsystem and the factors {α k } m k=1 , while the constraint (6) is satisfied. By imposing the constraint (6) we restrict the search space to the hyperplane H = (α 0 , α 1 , · · · α m ) : m k=0 α k υ k = 1
we have det(J 22 − J 21 J −1 11 J ⊺ 21 ) = det (βΣ + qΣcc ⊺ Σ) Substituting the value of q we get (51). Lemma 4 Suppose that r 2 = 0. Then det(J 22 − J 21 J cη ⊺ − I [βΣ] −1 r 2 r 1 cη ⊺ − I of the formula for det(J) Using Schur's determinant formula we know det(J) = det(J 11 ) det(J 22 − J 21 J −1 11 J ⊺ 21 ).
and (46) it follows that [ d Dγ ] = FQ. to eliminate d and D from the expressions of J 12 and J 22 . We haveNow we use (43), (44), and (46) in (42)
C Proof of Theorem 2 C.1 Some Schur complement expressions Lemma 2 The Schur complement J 22 − J 21 J −1 11 J ⊺ 21 admits an expression
Proof: DefineRecall that ζ = r 1 /r 2 . Hence from (50) we getHence by definition of Q, see Theorem 1, we get(56)Taking determinant we haveOn the other hand, recall that F = [ Σc η ]. Hence using (20) we getCombining(56)and(58)we getNow using Lemma 2 and (55) we know
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| [] |
[
"Modelling jet quenching",
"Modelling jet quenching"
] | [
"Thorsten Renk \nDepartment of Physics\nUniversity of Jyväskylä\nP.O. Box 35FI-40014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 64FI-00014Finland\n"
] | [
"Department of Physics\nUniversity of Jyväskylä\nP.O. Box 35FI-40014Finland",
"Helsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 64FI-00014Finland"
] | [
"Nuclear Physics A"
] | High P T measurements of hard hadrons or jets at RHIC and LHC appear contradictory and in some cases counterintuitive, but upon closer investigation they represent a coherent picture of jet-medium interaction physics which can be established with systematic comparisons of models against a large body of data. This picture is consistent with a perturbative QCD mechanism and does not require exotic assumptions. This overview outlines how several key measurements each partially constrain shower-medium interaction physics and how from the sum of those the outlines of the mechanism of jet quenching can be deduced. Most current jet results from LHC can be naturally understood in this picture. A short summary of what can be established about the nature of parton-medium interaction with current data is given in the end. | 10.1016/j.nuclphysa.2012.12.081 | [
"https://arxiv.org/pdf/1207.4885v1.pdf"
] | 119,109,493 | 1207.4885 | d14a30c8e2f22dd32d645fcadafd5611d2b48273 |
Modelling jet quenching
2014
Thorsten Renk
Department of Physics
University of Jyväskylä
P.O. Box 35FI-40014Finland
Helsinki Institute of Physics
University of Helsinki
P.O. Box 64FI-00014Finland
Modelling jet quenching
Nuclear Physics A
002014jet quenching
High P T measurements of hard hadrons or jets at RHIC and LHC appear contradictory and in some cases counterintuitive, but upon closer investigation they represent a coherent picture of jet-medium interaction physics which can be established with systematic comparisons of models against a large body of data. This picture is consistent with a perturbative QCD mechanism and does not require exotic assumptions. This overview outlines how several key measurements each partially constrain shower-medium interaction physics and how from the sum of those the outlines of the mechanism of jet quenching can be deduced. Most current jet results from LHC can be naturally understood in this picture. A short summary of what can be established about the nature of parton-medium interaction with current data is given in the end.
Introduction
High transverse momentum (P T ) processes in Quantum Chromodynamics (QCD) have been suggested as a probe to investigate the physics of ultrarelativistic heavy-ion (A-A) collisions more than a decade ago and have become a cornerstone of the experimental heavy-ion program both at RHIC and LHC. Yet, compared with for instance the modelling of the bulk medium evolution using hydrodynamics where an era of precision fits to differential data has begun (see e.g. [1]), theoretical progress in the physics of hard probes is slow.
Partially, this can be attributed to the structure of the problem: Hard probes are at the same time sensitive to the microscopical degrees of freedom of the medium and the macroscopical medium density distribution and evolution. Thus high P T observables have the capability to potentially do both tomography and transport coefficient measurements in the medium, i.e. to constrain its microscopical and macroscopical dynamics. While the microscopical physics of the medium and hence the basic structure of the parton-medium interaction is currently in detail unknown, the density evolution (in terms of fluid-dynamical modelling) is constrained by bulk data, but not unambiguously determined. As a result, the description of any single observable in terms of a parton-medium interaction scenario and a medium evolution model is usually far from unique and does not allow a firm conclusion with regard to either question.
However, systematic multi-observable studies of a matrix of parton-medium interaction and medium evolution models can reduce the allowed model space significantly, resulting in a class of models which shows good consistency across RHIC and LHC observables.
The vacuum baseline
In vacuum, at the core of the computation of hard processes in perturbative QCD (pQCD) is the factorization of the hard, short-distance interaction part and the long-distance physics of initial state parton distributions and final state parton showers. In a medium, QCD factorization is not strictly proven, but remains a plausible working assumption. A sufficiently large separation of hard and thermal momentum scale then ensures that the hard process itself receives no medium corrections and can serve as a 'standard-candle', producing high p T partons at a known and calculable rate.
The partons emerging from the hard process are typically characterized by a large virtuality scale Q. If one takes the uncertainty relation to estimate the timescale at which intermediate shower states at lower virtuality can be formed, one finds τ ∼ E/Q 2 with E the virtual parton energy. Comparing this scale with typical medium lifetimes O(10) fm, one finds that for typical RHIC and LHC kinematics a significant part of the partonic shower evolution (i.e. with Q > Λ QCD , m h ) takes place in the medium, with the shower being gradually boosted out of the medium at top LHC energies and on the other hand hadrons with large mass m h being produced in the medium. However, over a large kinematical range and for sufficiently light hadrons it can reasonably be assumed that the medium modification largely concerns the evolution of a parton shower with subsequent hadronization as in vacuum.
As a baseline example for in-medium shower modelling, consider a virtuality-ordered parton shower such as implemented in the PYSHOW algorithm [2]. This is treated as an interated series of splittings of a parent into two daughter partons a → bc where E b = zE a and E c = (1 − z)E a and where the virtuality of partons in terms of t = ln Q 2 /Λ 2 QCD decreases in each branching. The splitting probability is given by
dP a = b,c α s (t) 2π P a→bc (z)dtdz .
where P a→bc (z) are perturbatively calculable objects, the so-called splitting kernels. The splittings cease when the parton virtuality reaches a non-perturbative lower scale O(1) GeV at which point a hadronization model needs to be used. In this way, the original high virtuality of the shower initiator is converted into the transverse momentum distribution of the final shower partons (and the initial virtuality sets a kinematic bound for observables like jet shapes).
An important thing to note is that the splitting kernels P a→bc (z) are scale invariant, i.e. they do not depend on an absolute momentum scale. As a result, the fragmentation functions generated by a series of splittings is self-similar (the momentum distribution of hadrons in a subjet looks almost the same as the distribution in the whole jet when plotted as a function of z) and up to logarithmic corrections has no strong energy dependence.
Estimating the timescale associated with the initial, hard branchings which largely determine jet shape and subjet structure, one finds τ ∼ 0.01 fm, i.e. well before any medium can be formed. This gives rise to the expectation that subject structure and jet shape at high P T should not be very sensitive to medium modifications.
In-medium showers
There are two main processes by which this pattern can be modified in a medium: 1) direct loss of energy (by elastic collisions, drag, . . . ) into the medium as parametrized by a transport coefficientê and 2) enhanced soft gluon radiation by medium-induced virtuality as parametrized by a transport coefficientq. The first mechanism is in general incoherent and the energy is lost into the medium, the second mechanism involves a de-coherence time for the radiated gluon and transfers energy from the leading parton into an increased number of subleading shower partons. Note that this distinction can be made on average on the basis of transport coefficients, but not for a single interaction graph with the medium which may involve both direct and radiative energy loss. A recently suggested additional mechanism is the modification of color flow by the medium, leading to a modification of the hadronic shower evolution [3].
Models may now be classified based on what part of the dynamics they include, how they treat the medium and how they solve the shower evolution equations. The most complete approach are models which simulate the whole in-medium shower evolution such as JEWEL [4], YaJEM [5,6], Q-PYTHIA [7] or resummed higher twist (HT) [8]. However, if one is only interested in observables which probe the leading shower fragments, such as single inclusive high P T hadron suppression, which probe dominantly showers in which a large fraction of the energy flows through a single parton, then leading parton energy loss modelling in which the medium evolution is cast into an energy loss probability P(∆E) before vacuum fragmentation is applied is sufficient. Models of this type include the ASW [9], AMY [10,11] or WHDG [12] frameworks. Finally, in hybrid approaches a vacuum shower is computed down to some scale, then the produced partons are put on-shell and each propagated through the medium using a leadingparton energy loss model. An example for hybrid modelling is MARTINI [13].
Of considerable interest is how the medium is included into the modelling process. A number of models come with an explicit treatment of medium partons and their interaction with shower partons, typically as a thermal gas of quark and gluon quasiparticles [4,10,11,13]. Such a description requires that the medium is, at least to some degree, perturbatively tractable, which is not in line with the basic assumption underlying almost ideal hydrodynamical modelling which assumes that the medium is strongly coupled and the mean free path essentially vanishes. Usually an explicit treatment of interactions with medium partons breaks the scale invariance of the fragmentation function around the momentum scale of the medium partons and below, while it largely remains intact above. In addition, the medium provides additional kinematic phase space for transverse shower broadening.
A different approximation is to cast all medium effects into a modification of the splitting probability, in essence replacing P a→bc (z) → P a→bc (z) with the detailed form of P motivated by a specific physics scenario as done e.g. in Q-PYTHIA [7]. Such a prescription explicitly conserves energy and momentum inside the shower, i.e. there is no additional phase space for transverse broadening. It also perserves scale invariance of the fragmentation function, but it alters its shape to a different functional form.
Finally, the medium can be (without any explicit reference to its degrees of freedom) appear via transport coefficients such asq,ê and alter the kinematics of propagating partons as implemented in YaJEM [5,6]. This again breaks scale invariance of the fragmentation function around the medium momentum scale and below and leads to additional phase space for transverse broadening.
There are two main strategies to solve the shower evolution equations -analytical and Mote-Carlo (MC) techniques. In general, analytical techniques get exact treatment of quantum interference effects while they often rely on kinematic approximations such as eikonal propagation or infinite parent parton energy which violate energymomentum conservation. In contrast, MC frameworks usually have exact kinematics, but have to resort to phenomenological prescriptions for interference effects. These different approaches to solving the in-medium shower evolution equations are summarized for several wellknown models in table 1. Model results are also quantitatively sensitive to implementation details such as cutoffs or the choice of light-cone vs. energy splitting. However, these implementation specifics do not seem to lead to qualitative changes of the model results. A very instructive review of such effects can be found in [17].
Physics assumptions in jet quenching models
In addition to implementing different approximations and solution strategies, models also show genuine differences with regard to assumptions about the physics underlying parton-medium interaction, and this leads to various testable consequences. In particular, the pathlength dependence of the mean energy loss ∆E of a parton traversing a constant medium is parametrically different dependent on what one assumes to be the relevant physics (note that ∆E is not as such a very well-defined concept for an in-medium shower, but it can be extracted by tagging a quark flavour as done e.g. in [6]).
For any incoherent mechanism, the mean energy loss largely tracks the number of scatterings, which in a constant medium are proportional to the length, i.e. ∆E ∼ L. For radiative processes, there is a decoherence time involved and hence generic arguments can be made (e.g. [18]) that ∆E ∼ L 2 . It is however known that this holds only for sufficiently large parent parton energy and finite energy corrections quickly change this to a linear dependence [19,20,21]. In strong coupling scenarios motivated by AdS/CFT approaches to the medium dynamics, a longitudinal drag rather than transverse random momentum transfer lead to induced radiation, which changes the above argument dimensionally, leading to ∆E ∼ L 3 [22], to which finite energy correction have so far not been obtained, but would be expected to weaken the dependence.
This discussion is somewhat correlated to the way the medium is implemented in models as discussed in the previous section, as models which include explicit interactions with perturbative medium quasiparticles typically find 50% elastic (incoherent) energy loss for reasonable values of the strong coupling α s ≈ 0.3 [12,23]. In contrast, many older leading parton radiative energy loss (e.g. [9,16]) start from the assumption of static scattering centers in the medium and hence do not have any incoherent component.
Yet a different source of pathlength dependence is the idea that the minimum virtuality scale down to which a shower can evolve in-medium should be constrained by uncertainty arguments to Q 0 = √ E/L [8]. This leads to a non-linear pathlength as well as additional energy dependence of the mean energy loss [20].
It is important to note that in a real fluid-dynamical background the actual pathlength dependence is drastically changed by the spacetime evolution of the medium density. In this sense, expressions like 'quadratic pathlength dependence' are to be understood as labels what a model would do if applied to a constant medium, not as descriptions of what is experimentally measurable. In the following, we aim to establish experimental tests sensitive to these different physics assumptions and implementation differences such that the data can be used to discriminate between models.
Single hadron suppression
In general, the P T dependence of the single inclusive hadron suppression factor R AA is more driven by generic pQCD effects such as the functional form of the produced parton momentum spectrum than by jet quenching model specific effects (see [28] for a discussion). As a result, within the uncertainty associated with the choice of the bulk medium evolution model, models tuned to RHIC data tend to extrapolate reasonably well to LHC conditions (see Fig. 1). [29] and ALICE [30] in various models [24,25,26,27].
The important exception is the strong coupling scenario denoted as AdS which overquenches significantly. This can be understood as follows: Due to dimensional reasons, the L 3 pathlength dependence requires a scaling of the quenching effect with the medium temperature ∼ T 4 whereas all other scenarios scale with the approximate medium density ∼ T 3 . Such a strong increase of the quenching power is however not indicated by the data, essentially ruling out this particular application of AdS/CFT ideas to jet quenching. R AA is, however, not expected to be a very sensitive probe [28] -due to the need to average over the full medium geometry and to fold with the pQCD parton spectra, only a narrow region of the medium-modified fragmentation function (MMFF) is probed by the observable, and thus models predicting a very different MMFF can result in nearly the same R AA . One way to constain models better is to study the dependence of R AA on the angle of the hadron with the reaction plane φ. In this way, the pathlength dependence of the parton-medium interaction model in combination with a particular medium evolution model is tested. Such investigations have established two crucial insights [28,31]:
• There is about a factor two uncertainty associated with the choice of the medium model provided that the medium model is a fluid-dynamical model constrained by bulk observables. One implication is that R AA (φ) has the capability to discriminate between fluid-dynamical models, i.e. it is a true tomographical measurement. However, this result also indicates that any results obtained with a schematical medium evolution model not constrained by bulk data (such as the Bjorken cylinder) should be disregarded, as they have easily a factor 10 systematic uncertainty.
• Within the factor two uncertainty, most scenarios describe the data reasonably well. The one exception which fails by a huge margin (factor 6) is a linear pathlength dependence [32]. Based on comparison with the data, any component with linear pathlength dependence must be smaller than about 10% [28,32,33]. This does not only disfavour scenarios with a 50% elastic energy loss component, but also any radiative scenario with finite energy corrections. The only realistic pathlength-dependence generating option viable with the combined data is hence the prescription for the minimum in-medium virtuality Q 0 = √ E/L as suggested in [8] (both radiative energy loss from an infinite energy parton and an AdS L 3 dependence would work with the data, but the first is not a sufficiently realistic scenario and the second alternative does not agree well with the extrapolation from RHIC to LHC as discussed above).
Hard dihadron correlations
The normalized dihadron away side correlation strength I AA is an interesting observable, since the requirement of a hard near side trigger adds a series of biases on parton type and kinematics while the away side fragmentation pattern has no additional bias [34]. As a result, the MMFF is probed in more detail in such measurements.
This allows to study the redistribution of longitudinal momentum inside a shower in quite some detail. The two main physics scenarios outlined above expect rather different behaviour: In a direct, elastic energy transfer into the medium, longitudinal momentum lost from the jet is carried by the medium at thermal momentum scales (i.e. not in hard modes). In contrast, in a radiative energy loss scenario, longitudinal momentum is carried by additional soft gluon radiation, part of which is still 'hard' as compared to a thermal momentum scale. Dihadron correlation are able to discriminate between these patterns. Figure 2: Away side normalized dihadron correlation strength I AA as compared with leading parton energy loss models and in-medium shower models [34,35]. This is illustrated in Fig. 2 where the away side correlation strength as a function of z T = E hadron /E trigger for central 200 AGeV Au-Au collisions and a 8-15 GeV trigger is shown. In leading parton energy loss calculations, energy is implicitly assumed to be transferred into the medium and to reappear at thermal momentum scales (below the z T range of the measurement). As a result, the curves bend downward at low z T which is not in agreement with the trend seen in the data. In contrast, in in-medium shower pictures where the enhanced production of subleading jet fragments is explicitly treated, the curves reflect the enhanced soft gluon production by bending upward.
However, under the assumption that all energy re-appears in soft gluon production, the computation overshoots the data. The tend seen in the data can only be reproduced under the assumption that about 10% of the energy is directly tranferred into the soft medium [35]. This lower limit for the direct elastic energy transfer agrees nicely with the upper limit of the same order as obtained by a study of pathlength dependent observables. Thus, this observable clearly shows the limits of leading parton energy loss modelling and also disfavours all scenarios in which there is no possibility of direct energy transfer into the medium.
Clustering and suppression of jets
The clustering of hadron showers into jets by means of a clustering algorithm is designed to suppress the dependence of observables to soft physics such as hadronization and to provide a good comparison point between hard pQCD calculations and measurements without the need to model physics close to Λ QCD . As a result, clustering suppresses many effects of a medium modifiaction since these take place at a scale T ∼ Λ QCD . To provide a concrete example, while the medium-induced emission of an almost collinear gluon leads to a modification of the leading hadron spectrum since energy has been taken from the leading parton, it does not lead to a modification of the jet spectrum since the emitted gluon is clustered back into the jet. Thus, unless distributions of hadrons inside the clustered jets are considered, jets are significantly less sensitive to medium modifications than single hadrons [36].
There is, however, a two stage mechanism which is able to suppress jets: While hard partons with p T T are kinematically very robust against interactions with the medium and cannot easily be scattered out of the jet, the interaction can nevertheless induce the emission of additional soft gluons at a thermal energy scale. Such soft gluons however are not kinematically robust and can be scattered easily out of the jet cone, leading to jet energy loss. Since gluons at a thermal scale are indistinguishable from medium gluons, the energy radiated into these modes is quickly thermalized and flows in hydrodynamical excitations to large angles. This is thus a very generic mechanism independent of the specific in-medium shower physics. The characteristics of the parton-medium interaction are then only apparent from semi-soft gluons above the thermal scale. Such a scenario is well in line with jet quenching as characterized by RHIC observables discussed before.
This mechanism of jet suppression has recently been denoted as frequency collimation [37] but has been implemented into MC and observed earlier in [38]. Since it relies on extra available phase space for rescattered soft gluons, it can not be observed in purely probabilistic approximations of the jet-medium interaction.
Jet observables
The first LHC observable involving reconstructed jets has been the dijet energy imbalance A J = E T 1 −E T 2 E T 1 +E T 2 or simply the ratio E T 2 /E T 1 of reconstructed jet energies E T 1 on the near side and E T 2 on the away side [39,40]. For the reasons mentioned above, clustering largely removes the sensitivity to specific characteristics of the jet quenching model, and as a result the data can be described in various models making somewhat different assumptions about the dynamics of parton-medium interaction and medium geometry [36,41,42,43]. The trigger energy dependence of the imbalance is somewhat more constraining, it probes the medium-induced broadening of the jet [44] (see Fig. 3).
A greater challenge to models is posed by the CMS observation that the longitudinal momentum distribution of reconstructed jets, when plotted as a fraction of the reconstructed jet energy, appears unchanged from vacuum even for highly imbalanced events [45]. This is a highly unexpected finding for any model which implements the interaction with the medium probabilistically and hence uses a modified set of splitting kernels P a→b,c (z) to build the fragmentation function, as in such a model the MMFF would be self-similar, but not show the same functional form as in vacuum. It is however a natural outcome for models which break scale invariance only below a fixed momentum scale O(T ), since in such frameworks the splitting kernels generate a self-similar fragmentation function with the same Figure 3: Dijet asymmetry as measured by ATLAS [39] and CMS [40] as compared with medium induced radiative energy loss scenarios [41,44] functional form as in vacuum above the breaking scale. The flattening of I AA (z T ) for z T > 0.5 in Fig. 2 is a manifestation of the same dynamics. From the RHIC data, significant deviations from the vacuum dynamics are expected to occur around 2-3 GeV, below the range of the CMS measurement. Thus, the CMS finding of an apparently unmodified fragmentation pattern strongly disfavours an probabilistic implementation of medium effects in the splitting kernel, but is not unexpected for other models, indeed almost unmodified jet properties above a momentum cut of 4 GeV with significant modifications below had been predicted in [38].
Summary
Our knowledge of the viability of various models, given the available data and aking use of the full uncertainty range given by the choice of the hydrodynamical evolution model can be summarized as in Table 2: Table 2: Viability of different parton-medium interaction models tuned to the P T dependence of R AA in 200 AGeV Au-Au collisions given various data sets under the assumption that the best possible hydrodynamical evolution scenario is chosen. The various labels refer to: elastic [33], elMC [32], ASW [18], AdS [22], YaJEM [5,6], YaJEM-D [20], YaJEM-DE [35].
R AA (φ)@RHIC R AA @LHC (P T ) I AA @RHIC I AA @LHC A J E T 2 /E T 1 (E trig )
It is immediately evident that the combination of all observables cuts the available model space down much more than any single observable. Most of the constraints are provided by the combination of the P T dependence of R AA at LHC which probes the dynamics of leading partons in combination with a dihadron correlation I AA which is a probe of subleading fragment dynamics. Current jet measurements do not have strong constraining power beyond this, however they do highlight the need to go beyond a purely probabilistic implementation of medium modification and also provide constraints for transverse dynamics of showers. A number of conclusions with regard to the nature of jet-medium interaction can be drawn from this:
• The combined data is compatible with a fairly standard pQCD picture of medium-induced radiation and a subleading component of elastic interactions leading to energy transfer into the medium • There is no evidence for exotic scenarios, the data neither indicate an L 3 dependence of energy loss as suggested by some variants of AdS/CFT inspired models nor a modification of the hadronization stage.
• Data from RHIC and LHC consistently indicate that some moderate amount of energy is transferred from the hard parton directly into the non-perturbative medium rather than into subleading jet fragments. However, the amount of direct energy transfer is constrained by pathlength-dependent observables and found to be significantly smaller than if the medium can be described as a near-ideal quark-gluon gas. This may imply that the observed degrees of freedom in the medium are massive or correlated quasiparticles.
To make these conclusions more quantitative, further high-statistics multi-differential measurements are needed from the experimental side, followed by systematic multi-observable studies from theory. In particular, the systematics of h-h, jet-h and γ-h correlations (with decreasing geometry bias) as a function of the reaction plane angle for various system centralities is expected to be a sensitive probe of both the parton-medium interaction mechanism and the medium density evolution. Once these questions are reliably addressed, the precise nature of the mechanism by which the energy deposited from hard partons is carried by the medium presents itself as the next goal.
Figure 1 :
1Pre-and postdictions for the nuclear suppression factor R AA in 2.76 ATeV central Pb-Pb collisions comparison with data from CMS
Table 1 :
1A summary table of a rough classification of several well-known jet quenching models.
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| [] |
[
"Network reconstruction from random phase-resetting",
"Network reconstruction from random phase-resetting"
] | [
"Zoran Levnajić \nDepartment of Physics and Astronomy\nUniversity of Potsdam\n14476PotsdamGermany\n",
"Arkady Pikovsky \nDepartment of Physics and Astronomy\nUniversity of Potsdam\n14476PotsdamGermany\n"
] | [
"Department of Physics and Astronomy\nUniversity of Potsdam\n14476PotsdamGermany",
"Department of Physics and Astronomy\nUniversity of Potsdam\n14476PotsdamGermany"
] | [] | We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase-resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony.Complex networks of many interacting units found at all scales in nature are the subject of intense research in many scientific areas [1]. Among the central issues in this field are the exploration and development of methods for determining the architecture of a network based on the observable data. Knowing the network structure helps in understanding its collective behavior, and indicates ways to engineer networks with the desired properties. For instance, it has been realized that inferring the topology of gene regulatory networks is crucial for completing our knowledge about the inner workings of cells[2]. Many real networks display modular and community structure that is essential for their functioning [3] and can be extracted using a variety of methods [4], as done for yeast metabolic network[5]. Reconstruction techniques often rely on examining the time-series of network dynamics that can reveal its interaction functions[6]. The network topology can be detected by studying the interchanges among its collective behaviors or investigating its response dynamics[7]. Structural properties can be determined from various time-scales in the emergence of synchronization [8], or by employing specific control theory methods[9]. Recently proposed techniques involve noisy dynamical correlations between the nodes [10], and even tackle models with non-equilibrium dynamics[11]. | 10.1103/physrevlett.107.034101 | [
"https://arxiv.org/pdf/1012.3624v3.pdf"
] | 18,011,173 | 1012.3624 | 4f89dd2b62abdfb41352784e050c5b3c57826e2d |
Network reconstruction from random phase-resetting
6 May 2011
Zoran Levnajić
Department of Physics and Astronomy
University of Potsdam
14476PotsdamGermany
Arkady Pikovsky
Department of Physics and Astronomy
University of Potsdam
14476PotsdamGermany
Network reconstruction from random phase-resetting
6 May 2011
We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase-resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony.Complex networks of many interacting units found at all scales in nature are the subject of intense research in many scientific areas [1]. Among the central issues in this field are the exploration and development of methods for determining the architecture of a network based on the observable data. Knowing the network structure helps in understanding its collective behavior, and indicates ways to engineer networks with the desired properties. For instance, it has been realized that inferring the topology of gene regulatory networks is crucial for completing our knowledge about the inner workings of cells[2]. Many real networks display modular and community structure that is essential for their functioning [3] and can be extracted using a variety of methods [4], as done for yeast metabolic network[5]. Reconstruction techniques often rely on examining the time-series of network dynamics that can reveal its interaction functions[6]. The network topology can be detected by studying the interchanges among its collective behaviors or investigating its response dynamics[7]. Structural properties can be determined from various time-scales in the emergence of synchronization [8], or by employing specific control theory methods[9]. Recently proposed techniques involve noisy dynamical correlations between the nodes [10], and even tackle models with non-equilibrium dynamics[11].
However, existing reconstruction methods, that often use network models with single-node dynamics represented by different types of oscillators [12], typically require long time-series of dynamical data, or a certain level of complexity in the emergent dynamics [6,7]. Since synchronization destroys the initial node-related information, detecting network topology in such cases is extremely difficult. Some methods are applicable only to sparse or non-directed networks, often providing results with only a limited precision [10].
In this Letter, we propose a novel method of reconstructing the topology and interactions of a general oscillator network. Our idea relies on repeatedly reinitializing the network dynamics (e.g. by performing random phase-resets), in order to produce an ensemble of the initial dynamical data. We design the quantities obtained by averaging this ensemble, whose values reveal the desired details of the network structure. Our method is applicable to any directed and weighted network, with general interaction functions and oscillator frequencies, and with arbitrary emergent dynamics, while avoiding the need for long time-series.
In the context of phase-resets, one is typically interested in the phase-resetting curves, which specify the system's response to weak external perturbations [13]. They have been investigated both experimentally [14] and theoretically [15,16], and been shown to contain properties relevant for determining network details such as clustering [17]. An algorithm for the estimation of neuron interaction and its stability based on phase-resets has been proposed [14]. We here employ phase-resetting somewhat differently, since our interest lies in the internal network interactions, rather than its response to stimuli. Contrary to [14], we use phase-resets only as a natural way to re-initialize the dynamics of an oscillator network, without measuring the phase shifts occurring due to resetting.
Our model consists of N oscillators (nodes), characterized by their phases ϕ i ∈ [0, 2π) and natural frequencies ω i . They are coupled pair-wise, via general 2π-periodic interaction functions f ij with zero mean:
ϕ i = ω i + N j=1 f ji (ϕ j − ϕ i ) .(1)
Models of this type include the famous Kuramoto model and its generalizations, widely used in theoretical studies, as well as for describing specific experimental situations [8,12,14]. The functions f ij (φ) are generally nonsymmetric with respect to exchange of indices, and thus fully define the dynamical network (order of indices determines the direction of interaction). Network adjacency matrix given as A ij = sgn |f ij | specifies its topology. Dynamics starts from a set of initial phases (i.p.) which we denote as ϕ = (ϕ 1 , . . . ϕ N )(t = 0), chosen from a distribution ρ(ϕ) > 0 normalized to (2π) N . The method is based on two assumptions: (i) we are able to arbitrarily re-initialize the network dynamics I times, by independently resetting the phases of all nodes to a new state ϕ; (ii) we are able to measure all the values ϕ l , and all initial instantaneous frequenciesφ l , each time the dynamics is re-initialized (for l = 1, . . . I). As we show in what follows, the ensemble of data for I ≫ 1 created under these assumptions yields the entire network structure. Introducing a 2π-periodic test-function g = g(ϕ i − ϕ j ) with zero mean, our aim is to compute the reconstruction index S ij defined as:
S ij [g] = (2π) −N [0,2π] N dϕ g(ϕ i − ϕ j )φ j (ϕ) .(2)
Taking the functions f ij in Eq.(1) to be generally given by the Fourier series f ij (φ) = n a (n) ij sin nφ+b (n)
ij cos nφ, we obtain the following expression for S ij :
S ij = (2π) −N N k=1 ∞ n=1 [0,2π] N dϕ g(ϕ i − ϕ j ) × a (n) kj sin(nϕ k − nϕ j ) + b (n) kj cos(nϕ k − nϕ j ) ,
which is independent of the frequencies ω i . The integral over ϕ i vanishes unless i = k. This implies that if A ij = 0, the corresponding S ij = 0, independently of the choice of g. The non-zero entries of S ij directly reveal the presence of network links. In addition, matrix S ij detects the desired properties of the interaction functions for appropriately selected test-function g. In particular, using g(φ) = 2e inφ we obtain the Fourier harmonics of f ij , which are the interaction parameters a (n)
ij and b (n) ij : S ij [2e inφ ] = b (n) ij + ia (n) ij = 1 π 2π 0 f ij (φ)e inφ dφ . (3)
Computation of S ij for adequate g amounts for reconstruction of any dynamical network described by Eq.(1). Depending on the properties of f ij that are to be examined, other choices of g are also possible. When dealing with the empirical interaction functions involving an unknown number of Fourier harmonics, a specifically designed g based on the experimental assumptions about f ij might be useful. This result is largely independent of the frequencies ω i , the network's directedness, and the distribution ρ. In particular, it is also independent of the network's final dynamical state, whether dependent on ρ or not. However, a constant component in case of f ij with non-zero mean cannot be detected, since its presence is indistinguishable from the natural frequency ω.
To practically implement our method, we need to convert the integral from Eq.(2) into an average involving discrete non-uniformly distributed empirical data {ϕ l } I l=1 and {φ l } I l=1 . To that end, we represent the functionφ j (ϕ) using the kernel smoother Q(ϕ − ϕ l ) [18] as:
ϕ j (ϕ) = I l=1 Q(ϕ − ϕ l )φ j (ϕ l ) I l=1 Q(ϕ − ϕ l ) .
The denominator is just the empirical density ρ(ϕ) = l Q(ϕ − ϕ l ) obtained via kernel distribution estimate [18]. Since the integration over ϕ already provides smoothing, we take Q(ϕ − ϕ l ) → δ(ϕ − ϕ l ), and replace the Eq.(2) with a practical formula for S ij :
S ij [g] = φ j g(ϕ i − ϕ j ) ρ(ϕ) = 1 I I l=1φ j (ϕ l )g(ϕ i − ϕ j ) ρ(ϕ l ) ,(4)
which is the average of empiricalφ j g weighted by 1 ρ . The most trivial way to obtain the ensemble {ϕ l } I l=1 would be to pick the values from a fixed distribution ρ(ϕ). Instead, we seek to mimic an experimentally feasible situation by performing I random phase-resets of the network dynamics, separated by the time interval τ . Mathematically, this amounts to adding the term [16]. For each reset l and each oscillator i, we independently pick the kicking strength K i,l from a zero mean Gaussian distribution with standard deviation K = 1, and the phase-shift α i,l uniformly from [0, 2π). The ensemble is constructed by storing the phase values immediately after resets. The resulting artificially created ensemble has little in common with the natural distribution of phases, and can be considered as approximately independent. This is expressed by separability of ρ(ϕ) into a product of N one-dimensional distributions ρ i (ϕ i ):
I l=1 K i,l sin(ϕ i + α i,l )δ(t− lτ ) to the RHS of Eq.(1)ρ(ϕ) = N i=1 ρ i (ϕ i ) ,(5)
each of which we determine from generated data using the kernel estimation method [19]. After each reset, the ensemble ofφ is computed using a small time interval. The phase value prior to reset is of no importance, since our interest is not in the phase-resetting curves, but in modeling a realistic way to create the ensemble ϕ. The described procedure is quite similar to the recent experimental implementation of the randomized phaseresetting of epileptic neurons aimed at their transient desynchronization [20]. In these experiments, however, the problem of simultaneous read-out of phases and frequencies remains a challenge. We now illustrate our theoretical findings through numerical simulations on simple network examples, computing the reconstruction index S ij as described above. Consider a simple network with N = 4 oscillators shown in Fig.1. We pick the natural frequencies at random from
ij , b (1) ij ∈ [−1, 1], while taking a (n) ij = b (n) ij = 0 for n ≥ 2.
Since such a network typically does not synchronize, our approximation of independent i.p. after resetting is appropriate. We take g = 2e iφ and compute S ij from an ensemble of I = 10 4 i.p. to obtain the numerical approximations of a (1) ij and b (1) ij via Eq.(4). In Fig.2 we compare the numeri-cal a node pairs (different from zero) and non-linked node pairs (zero). We have not only revealed the adjacency matrix A ij , but also found the interaction parameters a (1) ij and b (1) ij , thus reconstructing the entire dynamical network.
Below we discuss the limitations of our method. If the available data ensemble I is too small, the statistics is poor and the obtained network characteristics have large uncertainties, which typically decrease as ∼ I − 1 2 . To illustrate this, in Fig.3a we present the numerical values of parameter a (1) ij , computed for network in Fig.1 using the ensemble of i.p. ϕ of size I. While the distinction between links and non-links can already be seen for I ∼ 10 3 , for good approximation one needs I 10 4 (as done in Fig.2). For higher Fourier harmonics, the convergence is gradually slower, but maintains the same properties.
Another limitation is related to the validity of our independence assumption for the ensemble of i.p. which is expressed by the separability of distribution ρ(ϕ) Eq.(5). This heavily depends on the network's dynamical regime and the resetting strength. For a full synchrony and weak kicking, the reset state is expected to be strongly correlated, whereas for chaotic dynamics and strong resets, the independence assumption is essentially correct. To study this, we consider again the network from Fig.1, but now we fix all frequencies to ω i = 1, and take all interactions to be attractive a
(1) ij = 1, b(1)
ij = 0 (Kuramoto-type model with identical oscillators). We apply random kicking as described above after allowing the network to synchronize (τ ≫ t synch ), but this time with a variable standard deviation of kicking strength 0 < K < 10. For each value of K we create an ensemble of I = 10 4 i.p., and use it to compute a (1) ij as done previously. In Fig.3b we show the reconstructed values of a (1) ij for links and non-links in relation to K. Sufficiently strong kicking (K 5) succeeds in destroying the network's synchrony and generating the independent i.p., from which a good approximation of a (1) ij is computed. Moderate kicking K ∼ 1 applied previously are now insufficient. This furthermore depends on the relation between τ and t synch : if τ t synch (frequent resets) the separability of ρ is easier to achieve.
Too strong kicking can also induce correlations in ϕ, regardless of dynamical regime and τ . However, note that ρ can be estimated using the techniques more elaborate than simple one-dimensional kernels [18], which can in principle yield a good estimate even in the non-separable case. On the other hand, phase-resetting is potentially not the only mechanism of obtaining the ensemble ϕ; recall that our theory with a known ρ(ϕ) works equally well for any case, including full synchrony and inseparability. Adding noise terms to RHS of Eq.(1) does not formally change the derivation of our main result, rendering our theory valid in the presence of noise. However, in the light of discussion above, noise will have an effect on the performance of method: additional uncertainty due to larger fluctuations of the estimatedφ require larger ensembles to achieve the desired precision. On the other hand, noise may play a constructive role by destroying the undesired correlations within ϕ, and thus facilitating the separability of ρ.
While the experimental techniques for measuring ϕ are already in use [14], in a potential realistic applica-tion of our method a problem may arise in relation to the measurement ofφ. The entire cycle of a real oscillator is often not accessible; instead, one can observe only a single event per period (e.g. a spike produced by a neuron). In such cases, one is forced to estimate the instantaneous frequencies relying solely on the time intervals between the spikes. To illustrate this, we consider again the system studied in Fig.3b, but now we replace a (1) ij with εa (1) ij . The parameter ε (coupling strength) controls the ratio between the oscillation time-scale (period) and the interaction time-scale (synchronization). Rather than computing instantaneousφ after each reset, we observe only the event of an oscillator passing through the phase value ϕ = 0 (spike), and estimate both ϕ andφ from the first two spikes observed after resetting. We then reconstruct the values of a (1) ij using the ensemble of I = 10 5 i.p. as done before (strong resetting is applied immediately after the spikes are recorded). The results shown in Fig.3c have a clear physical interpretation: for too small coupling ε 0.03 the links can not be revealed since the interaction is too weak. For too large ε 0.4 the two time-scales are too close, and the detection is again impossible since the distribution of phases changes significantly over a period. However, between these extremes, there is a range of coupling around ε ∼ 0.1 where the two time-scales are well separated allowing a reliable reconstruction. This shows that with an adequately big ensemble our method works even if the entire oscillator cycle is not accessible: errors in the estimation of ϕ anḋ ϕ play a role similar to the noise. The method fails in the case of too strong coupling, similarly to the case of too weak resetting after synchronization (cf. Fig.3b).
In conclusion, we proposed a method of reconstructing oscillator networks by repeating random phase-resets, applicable to a general network irrespectively of the dynamical regimes (the feasibility of such resetting has been recently demonstrated for neural tissue [20]). Our theory emphasizes the importance of the transient dynamics in the context of network reconstruction, thus complementing the available techniques that rely on timeseries recorded in final stationary state. Our theoretical model can be straightforwardly generalized to other models beyond Eq.(1). If the couplings depend on two phases in a more general way, or depend on more than two phases, one should use more elaborate test-functions (e.g. in a form of general complex exponentials); however, even a theoretical description of such networks is already a challenge. For high-dimensional oscillators only a single scalar might be observable: our method can still be applied through the appropriate transformation to phases [6]. Another generalization regards the reconstruction of sub-networks, in the case that only information on some nodes is accessible. The problem here is to infer the distribution of i.p. for the non-accessible nodes. Finally, a real experimental situation may involve a network whose dynamics cannot be reset for all nodes simultaneously, which renders the independence assumption invalid. This is a much more challenging, although very realistic case that requires additional study. Support from DFG via project FOR868 is acknowledged. Thanks to A. Díaz-Guilera for useful discussions.
FIG. 1
1: 4-node network used for illustrating our method.
ω i ∈ [−1, 1].The interaction functions f ij are defined for linked node pairs by randomly choosing a
FIG. 2 :
2Reconstruction of the network from Fig.1. Circles: actual parameter values, crosses: numerically obtained values for I = 10 4 . Left: a (1) ij , right: b
, for each node pair i → j.
network in Fig.1, for links (cyan/gray) and non-links (black). (a) computed from ensemble of I i.p. (cf. Fig.2). (b) computed from I = 10 4 for network with attractive interactions, and with resetting done at synchronous state using kicking strength K. (c) computed from I = 10 5 for network with attractive interactions where only spikes (ϕ = 0) are observable, in relation to coupling strength ε (see text for details).
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. Z Levnajić, A Pikovsky, Phys. Rev. E. 8256202Z. Levnajić and A. Pikovsky, Phys. Rev. E 82, 056202 (2010).
. S Achuthan, C C Canavier, J. Neuroscience. 295218S. Achuthan and C. C. Canavier, J. Neuroscience 29, 16, 5218 (2009).
Kernel smoothing (Chapman & Hall. M P Wand, M C Jones, LondonM. P. Wand and M. C. Jones, Kernel smoothing (Chap- man & Hall, London, 1995);
Multivariate density estimation. D W Scott, J. Wiley & SonsNew YorkD. W. Scott, Multivariate density estimation (J. Wiley & Sons, New York, 1992).
This value defines 1-D distribution estimate ρi(ϕi) = l Q(ϕi − ϕ l ) = l e σ i (ϕ i −ϕ l ) for this node. Using the kernel Q = e σ(ϕ−ϕ l ). 18we find for each node i the optimal value σi by cross-validation via maximal likelihood method. based on the empirical reset ensemble {ϕ l } I l=1Using the kernel Q = e σ(ϕ−ϕ l ) , we find for each node i the optimal value σi by cross-validation via maximal like- lihood method [18]. This value defines 1-D distribution estimate ρi(ϕi) = l Q(ϕi − ϕ l ) = l e σ i (ϕ i −ϕ l ) for this node, based on the empirical reset ensemble {ϕ l } I l=1 .
. C Hauptmann, J. Neural Eng. 666003C. Hauptmann et al., J. Neural Eng. 6, 066003 (2009);
. P A Tass, Phys. Rev. E. 8011902P. A. Tass et al., Phys. Rev. E 80, 011902 (2009);
. C Hauptmann, P A Tass, J. Neural Eng. 756008C. Hauptmann and P. A. Tass, J. Neural Eng. 7, 056008 (2010).
| [] |
[
"Quantum Faraday Effect in Double-Dot Aharonov-Bohm Ring",
"Quantum Faraday Effect in Double-Dot Aharonov-Bohm Ring"
] | [
"Kicheon Kang \nDepartment of Physics\nInstitute of Experimental and Applied Physics\nChonnam National University\n500-757GwangjuRepublic of Korea\n\nUniversity of Regensburg\n93040RegensburgGermany\n"
] | [
"Department of Physics\nInstitute of Experimental and Applied Physics\nChonnam National University\n500-757GwangjuRepublic of Korea",
"University of Regensburg\n93040RegensburgGermany"
] | [] | We investigate Faraday's law of induction manifested in the quantum state of Aharonov-Bohm loops. In particular, we propose a flux-switching experiment for a double-dot AB ring to verify the phase shift induced by Faraday's law. We show that the induced Faraday phase is geometric and nontopological. Our study demonstrates that the relation between the local phases of a ring at different fluxes is not arbitrary but is instead determined by Faraday's inductive law, which is in strong contrast to the arbitrary local phase of an Aharonov-Bohm ring for a given flux. | 10.1209/0295-5075/99/17005 | [
"https://arxiv.org/pdf/1102.5261v1.pdf"
] | 117,802,981 | 1102.5261 | 2d4d71c258cbc38c5408e67728080e31cc07ab12 |
Quantum Faraday Effect in Double-Dot Aharonov-Bohm Ring
25 Feb 2011
Kicheon Kang
Department of Physics
Institute of Experimental and Applied Physics
Chonnam National University
500-757GwangjuRepublic of Korea
University of Regensburg
93040RegensburgGermany
Quantum Faraday Effect in Double-Dot Aharonov-Bohm Ring
25 Feb 2011(Dated: January 13, 2013)
We investigate Faraday's law of induction manifested in the quantum state of Aharonov-Bohm loops. In particular, we propose a flux-switching experiment for a double-dot AB ring to verify the phase shift induced by Faraday's law. We show that the induced Faraday phase is geometric and nontopological. Our study demonstrates that the relation between the local phases of a ring at different fluxes is not arbitrary but is instead determined by Faraday's inductive law, which is in strong contrast to the arbitrary local phase of an Aharonov-Bohm ring for a given flux.
PACS numbers: 03.65.Vf,03.65.Wj We begin by pointing out an apparent paradox between the two well known facts in quantum theory: (1) the local phase factor of the wave function in an Aharonov-Bohm (AB) loop is arbitrary [1]; (2) the wave function (more generally, the density matrix) of a quantum system can be reconstructed by a technique of the quantum state tomography (QST) on repeated preparation of the system [2,3]. Aharonov-Bohm (AB) effect [4] is one of the most striking phenomena discovered in quantum theory. In the case of an arbitrary AB loop at equilibrium, any physical quantity is periodic in Φ with a period Φ 0 = e/hc, the flux quantum ("Byers-Yang's theorem") [5]. AB effect is gauge-invariant and appears as a manifestation of the gauge-invariant phase factor. The choice of a particular gauge is arbitrary, and therefore the local phase factor of the wave function is also arbitrary.
It is interesting to note that this arbitrary local phase factor of an AB loop is inconsistent with the fact that the wave function can be reconstructed by the technique of QST. The gap between the two facts needs to be clarified. In this Letter, we show that the law of Faraday's induction plays a central role when we try to reconstruct the wave function of an AB loop. For a full reconstruction of the wave function, change of the AB flux is inevitable. The change of the AB flux results in Faraday's induction, and gives rise to an additional phase shift. We find that this Faraday-induced phase shift is geometric and nontopological. It is geometric in the sense that it depends only on the net change of the magnetic field. On the other hand, it is nontopological since it depends not only on the flux change but also on the specific geometry of the system.
We analyze the characteristics of the quantum Faraday effect for a tunnel-coupled double-dot AB loop with a localized magnetic flux Φ penetrating the hole ( Fig. 1(a)). This is the simplest two-state problem involving an AB flux. The Hamiltonian of a double-dot loop is given by
H = α=1,2 ε α c † α c α − (te iφa c † 1 c 2 + te iφ b c † 2 c 1 ) − h.c.,(1a)
where c α (c † α ) annihilates(creates) an electron at QD-α (α = 1, 2) with its energy level ε α . For simplicity, tunneling amplitude t is assumed to be identical for both the upper (path a) and the lower (path b) paths. The AB phase φ (= 2πΦ/Φ 0 ) is given by φ = φ a + φ b . The "local" phases φ a and φ b are not gauge-invariant. Only φ is a gauge-invariant phase. This Hamiltonian can be simplified as
H = α=1,2 ε α c † α c α + t φ c † 1 c 2 + t * φ c † 2 c 1 ,(1b)
with the effective tunneling amplitude
t φ = −2te i(φa−φ b ) cos (φ/2) .(1c)
Note that t φ depends on the choice of gauge due to the phase factor e i(φa−φ b ) . It is useful to rewrite the Hamiltonian (Eq. 1) in a Bloch-sphere (pseudospin) representation with | ↑ = |1 and | ↓ = |2 :
H = − 1 2 σ · B,(2a)
where
B = (−2Re(t φ ), 2Im(t φ ), ∆ε) (2b)
is a pseudo-magnetic field, and σ = (σ x , σ y , σ z ). ∆ε = ε 2 − ε 1 is the energy level detuning of the two dots. We imposed the condition ε 1 + ε 2 = 0 without loss of generality.
It is straightforward to obtain the eigenstate energies and the corresponding eigenvectors. Although a multi-valued wave function cannot be ruled out a priori [5], we do not consider this possibility because a QST with multi-valued wave function is meaningless [6]. The two eigenstate energies are E ± (B) = ∓ 1 2 |B| = ∓ 1 2 (∆ε) 2 + 4|t φ | 2 , with the corresponding eigenvectors being given by
| ± (B) = α ± | ↑ + β ± | ↓ ,(3a)
where
α ± = t φ (ε 1 − E ± ) 2 + |t φ | 2 , β ± = − ε 1 − E ± (ε 1 − E ± ) 2 + |t φ | 2 .
(3b) Although the energy eigenvalue E ± is periodic in the AB phase φ with a period of 2π (as indicated by Byers-Yang's theorem [5]), the wave function does not show such periodicity. Instead, the wave function depends on the arbitrary choice of the phases φ a and φ b . This is evident from the gauge-dependent factor t φ in Eq. (3b) (see also Eq. (1c)). Now, let us discuss the following flux-switching and pseudospin precession experiment (see also the illustration of Fig. 1(b)). This kind of time-domain experiment is an essential part of a QST with a system involving an AB flux [7]. Initially, the AB flux is prepared to have an arbitrary, fixed value φ, and the system is in the ground state | + (B) of Eq. (3). It is assumed that thermal fluctuations are small enough that the system is in the ground state, that is, k B T ≪ |B|. Then, the external magnetic flux is suddenly dropped to zero, and the pseudo-magnetic field B changes immediately to the value given by Eq. (2b) with φ = 0 and φ b = 0. (φ b might be chosen to be nonzero in general, but it does not make any change to our findings here.) This corresponds to
B → B 0 = (4t, 0, ∆ε).(4)
Because of the sudden change in B, the state of the electron, denoted by |ψ φ→0 (t) , will precess according to the relation (shown as a dashed line in Fig. 1
(b)) |ψ φ→0 (t) = e i 2 σ·nB0t | + (B)
, wheren is the unit vector parallel to B 0 , and B 0 is its magnitude. We obtain
|ψ φ→0 (t) = α 0 (t)| ↑ + β 0 (t)| ↓ ,(5a)
where
α 0 (t) = α + cos B 0 t 2 + i (α + n z + β + n x ) sin B 0 t 2 ,(5b) β 0 (t) = β + cos B 0 t 2 − i (β + n z − α + n x ) sin B 0 t 2 . (5c)
Here n x (n z ) is the x(z) component of the vectorn. The time evolution of the state, |ψ φ→0 (t) , depends on the gauge, simply because B depends on the gauge. Furthermore, this gauge dependence is shown in physical quantities, for instance, in the time evolution of the electron number at QD-1 (or at QD-2):
n 1 (t) = ψ φ→0 (t)|c † 1 c 1 |ψ φ→0 (t) = |α 0 (t)| 2 .(6)
The time evolution of n 1 is displayed for two different choices of gauge: (i) symmetric gauge (φ a = φ b = φ/2) (Fig. 2), and (ii) "φ b = 0" gauge ( Fig. 3). (In fact, an infinite number of gauge choices exists which satisfy the condition φ a + φ b = φ). For the symmetric gauge, the effective tunneling amplitude is t φ = −2t cos (φ/2). This gauge leads to a 4π-periodicity of the Hamiltonian and its eigenstates (Eqs. (1,3)). On the other hand, for the "φ b = 0" gauge, t φ = −t(e iφ + 1), which provides a 2πperiodicity of the Hamiltonian and its eigenstates. Obviously, a different choice of gauge leads to a different result.
Of course, the gauge dependence of the results shown in Fig. 2 and Fig. 3 should not exist in reality. The result should be unique in spite of the choice of gauge. In fact, the contradiction is resolved if we take into account Faraday's law induction in the flux-switching procedure. When the localized magnetic field changes in time, the electric field
E = −∇V − 1 c ∂A ∂t (7)
is induced, where V and A are the scalar and the vector potentials, respectively. The choice of the gauges for V and A should provide the correct value of E, which was not taken into account in obtaining the results displayed Figs. 2 and 3. Still there is a freedom to choose the gauge, and we choose V to be time-independent, and then the inductive component of the field, E t , is given by E t = − 1 c ∂A ∂t . For the system under consideration, the following relations are derived from Faraday's law:
γ E t · dR = − 1 c ∂ ∂t γ A · dr = −h e ∂φ γ ∂t ,(8)
where γ represents an integral over the path γ (a or b). The question at this point is how to choose a gauge giving the correct inductive field. Actually, the induced electric field depends not only on the time-dependent flux Φ(t) but also on the specific geometry of the system and the distribution of the localized magnetic flux. Let us consider, for example, a highly symmetric double-dot ring: with circular symmetric AB flux and identical paths for the upper (path a) and the lower (path b) parts of the ring. Then, the symmetry of the system leads to the relation a E t ·dR = b E t ·dR, or ∂φa ∂t = ∂φ b ∂t . We find that this condition is fulfilled by choosing the symmetric gauge for the time-dependent phases: φ a (t) = φ b (t) = φ(t)/2. Therefore, the results shown in Fig. 2 are correct for the symmetric system because they give the correct value of E t . In general, the gauge should be selected to give the correct value of E t , which depends on the specific geometry of the system. An important implication of the above discussion on Faraday's induction is that it gives an additional local phase in the wave function of the system. In the following, we show that Faraday's induction gives a geometrical (but nontopological) phase shift. Here we discuss it for a specific double-dot ring system, but it can be applied equally to any AB loops. The initial (local) phases at t = t i are represented by φ i a and φ i b for paths a and b, respectively. These phases evolve as the magnetic field changes in time, and the final values (at t = t f ) are given by φ i a + ∆φ a and φ i b + ∆φ b , respectively. During the change in the magnetic field, the inductive field induces a momentum kick ∆p(r) which depends on the position r as
∆p(r) = e t f ti E t dt = − e c ∆A(r),(9)
where ∆A(r) is the change in the vector potential. This momentum kick induces a local phase shift
φ F (r) = 1 h ∆p(r) · r,(10)
in the wave function. From this relation, one can find that Faraday's induction gives the relative phases of the two quantum dots
φ F a = 1 h a ∆p(r) · dr = −∆φ a ,
for path a. Similarly, it gives φ F b = −∆φ b for path b. It is interesting to note that the Faraday phase for one loop is equivalent to the negative of the change in the AB phase: φ F one loop = 1 h ∆p(r) · dr = −∆φ . In contrast to the AB effect, not only the phase for one loop but also the local phase φ F (r) is physically meaningful because the latter is directly related to the inductive field, a physical quantity. Note that, the local phase φ F (r) is geometric in the sense that it depends only on the net change of the vector potential ∆A(r). However, this phase is nontopological because it depends not only on the topology of the ring, but also on the specific geometry. The 4π-periodicity of a symmetric double-dot ring can be understood from the 4π-periodicity of the phases φ F a and φ F b , since it satisfies
φ F a = φ F b = −∆φ/2.(11)
The effect of the Faraday phase can be observed even in an adiabatic change of the magnetic flux. Two conditions are necessary for this purpose. First, a nonstationary initial state should be prepared. Otherwise, the adiabatic evolution of the magnetic field gives just an adiabatic evolution of the ground state, and the Faraday phase is not observable. Second, the characteristic time scale of the flux change, denoted by ∆t, should meet the conditionh/|B| ≪ ∆t ≪ t deph , where t deph is the dephasing time of the initial nonstationary state. The flux should change at a much slower rate than the precession frequency of the pseudospin (adiabaticity), but should be switched faster than the dephasing time, in order to observe the evolution of the nonstationary state.
The procedure for a possible experiment is as follows: (i) a ground state is prepared with ∆ε = 0 and B = B i . (ii) A nonstationary state is initialized by a sudden switching of the level detuning from zero to a finite value ∆ε. (iii) The magnetic flux is adiabatically switched so that the pseudo-magnetic field B changes accordingly. (iv) Finally, the time evolution of the electron number is measured in one of the QDs.
The adiabatic process of (iii) is described by the Hamiltonian H{B(t)} = − σ · B(t)/2 with an adiabatic change of B. The nonstationary initial state immediately after process (ii) (for a symmetric ring with the initial AB phase in the range of π < φ < 3π),
|ψ 0 = 1 √ 2 (| ↑ − | ↓ ) ,(12a)
evolves upon the adiabatic change in B(t). It satisfies the adiabatic evolution of the eigenstates
|ψ ± (B(t)) ≃ e iγ±(B) e −i t 0 E±(t ′ )dt ′ | ± (B(t) ,(12b)
where γ ± denotes the geometric phase acquired during the adiabatic change of B(t) [8]. We find that γ ± = 0 for the symmetric ring (imposed by the relation φ a (t) = φ b (t)). Therefore, the time evolution of the state upon adiabatic change is given by
|ψ(t) = c + e iφ + dy (t) | + (B(t)) + c − e iφ − dy (t) | − (B(t)) , (12c) where c ± = ±(B i )|ψ 0 , and φ ± dy (t) ≡ − t E ± (B(t ′ ))dt ′ = ± 1 2 t B(t ′ )dt ′ (12d)
is a dynamical phase. It is also useful to define the difference between the two dynamical phases, ∆φ dy (t) ≡ φ + dy (t) − φ − dy (t). The time evolution of the electron number at QD-1 (state | ↑ ) is
n 1 (t) = | ↑ |ψ(t) | 2 .(13)
We find from Eq. (12) that ↑ |ψ(t) = u(t) cos (∆φ dy (t)/2) + iv(t) sin (∆φ dy (t)/2), (14a) where
u(t) = 1 √ 2 cos θ i − θ(t) 2 − sin θ i − θ(t) 2 , (14b) v(t) = 1 √ 2 cos θ i + θ(t) 2 − sin θ i + θ(t) 2 . (14c)
θ(θ i ) is the angle between the z-axis and B(B i ) in the Bloch sphere. Fig. 4 displays the time evolution of the occupation number, n 1 , as a function of ∆φ dy (t), for an adiabatic change in the AB phase of the form φ(t) = φi of change. As discussed above, we have another condition φ a (t) = φ b (t) = φ(t)/2 for a symmetric ring, which takes Faraday's induction into account. The time evolution of n 1 for φ i = 2π with η = 0.04 is plotted in Fig. 4 (solid line). This is compared to the case of the static AB phase, φ(t) = 2π (dashed line). The result shows that the adiabatic change of the AB phase from 2π to zero indeed leads to out-of-phase oscillation of n 1 . This shift of the phase in n 1 results from the Faraday phase, φ F a = φ F b = −∆φ/2 = π, as shown in Eq. (11). Note that the AB effect does not play any role when φ changes by 2π.
The double-dot AB ring system is equivalent to a single Cooper pair box (SCB) composed of two Josephson junctions with an AB flux [9,10], if the two QD states are replaced by the two charge states in the SCB. The experiments described above can be applied equally to a SCB. It could be more easily realized with a SCB, considering the recent progress made in controlling superconducting qubits [10].
At this stage, we are able to address the question raised at the very beginning of this Letter: the inconsistency between the arbitrary local phase in an AB loop and the possibility of its measurement with a QST. Of course, a measurement of the local phase of a static AB ring is meaningless since it is arbitrary. However, for a complete QST, one should also change the localized AB flux. What is measured during a QST (which inevitably involves a change of the flux) is not an arbitrary local phase but the Faraday phase induced by the change in the flux itself.
In conclusion, we have shown that the relative local phase at different strengths of flux in an AB loop is not arbitrary but is instead determined by Faraday's law of induction. This is in strong contrast to the arbitrary local phase factor of an AB loop which depends on the choice of gauge in the vector potential. Faraday's induction provides a geometric and nontopological contributions to the local phase of a ring. Flux-switching experiments for double-dot rings have been proposed to verify the effect of the Faraday phase. Measurement of the Faraday phase in our setup of a double-dot ring is just one example of the very general nature of the problem. It should be observable in various different types of AB loops, which calls for further study.
This work was supported by National Research Foundation of Korea under Grant No. 2009-0072595 andNo. 2009-0084606, andby LG Yeonam Foundation.
[1] See e.g., S. Gasiorowicz, Quantum Physics, 3rd ed. (John Wiley & Sons, New Jersey, 2003
0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 2 Π 3 0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 4 Π 3 0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 2 Π 0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 8 Π 3 0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 10 Π 3 0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 4 Π
FIG. 2: (Color online) Time dependence of the electron number at QD-1 (n1(t)) for the symmetric gauge, upon a sudden drop in the AB phase from φ to zero. Six different input values of φ are used, and ∆ε = 4t in all graphs.
[4] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959);ibid. 123, 1511(1961
0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 2 Π 3 0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 4 Π 3 0 Π 2 Π 3 Π 4 Π B0 t 0.5 1 n1 t Φ 2 Π
FIG. 3: (Color online) Time dependence of the electron number at QD-1 (n1(t)) for "φ b = 0" gauge, upon a sudden drop in the AB phase from φ to zero. Three different input values of φ are used, and ∆ε = 4t for all graphs.
online) (a) Schematic of an Aharonov-Bohm ring composed of two quantum dots. (b) Illustration of the state evolution |ψ φ→0 (t) in the Bloch-sphere representation after a sudden drop in the AB flux.
). [2] K. Vogel and H. Risken, Phys. Rev. A 40, R2847 (1989); U. Leonhardt, Phys. Rev. Lett. 74, 4101 (1995). [3] For a review, see e.g., Quantum State Estimation, ed., by M. Paris and J. Rehácek (Springer, Berlin, 2004).
).[5] N.Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961). [6] For a useful discussion on the single valuedness of the wave function, see e.g., E. Merzbacher, Am. J. Phys. 30, 237 (1962). [7] Y.-x. Liu, L. F. Wei, and F. Nori, Phys. Rev. B 72, 014547 (2005). [8] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984). [9] Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Nature 398, 786 (1999). [10] For a review, see e.g., Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001); J. Clarke and F. K. Wilhelm, Nature 453, 1031 (2008).
[1 − tanh (η ∆φ dy (t))]. The initial phase φ = φ i at ∆φ dy (t) ≪ −η −1 changes adiabatically to φ = 0 at ∆φ dy (t) ≫ η −1 . The parameter η determines the rate
Color online) Time dependence of n1 upon an adiabatic change of the flux for a nonstationary initial state of Eq. (12a) (solid line). The parameters used here are ∆ε = 0 → 4t, φi = 2π, and η = 0.04. For comparison, the time evolution of n1 for a static AB phase (φ(t) = 2π) is also plotted. 4dashed lineFIG. 4: (Color online) Time dependence of n1 upon an adi- abatic change of the flux for a nonstationary initial state of Eq. (12a) (solid line). The parameters used here are ∆ε = 0 → 4t, φi = 2π, and η = 0.04. For comparison, the time evolution of n1 for a static AB phase (φ(t) = 2π) is also plotted (dashed line).
| [] |
[
"JOINT DYNAMIC MODELS AND STATISTICAL INFERENCE FOR RECURRENT COMPETING RISKS, LONGITUDINAL MARKER, AND HEALTH STATUS",
"JOINT DYNAMIC MODELS AND STATISTICAL INFERENCE FOR RECURRENT COMPETING RISKS, LONGITUDINAL MARKER, AND HEALTH STATUS"
] | [
"Lili Tong [email protected] \nDepartment of Biostatistics\nDepartment of Mathematical Sciences\nDepartment of Statistics\nUniversity of Nebraska Medical Center Omaha\nBentley University Waltham\nUniversity of South Carolina Columbia\n68198, 02452, 29208NE, MA, SC\n",
"Piaomu Liu [email protected] \nDepartment of Biostatistics\nDepartment of Mathematical Sciences\nDepartment of Statistics\nUniversity of Nebraska Medical Center Omaha\nBentley University Waltham\nUniversity of South Carolina Columbia\n68198, 02452, 29208NE, MA, SC\n",
"Edsel A Peña [email protected] \nDepartment of Biostatistics\nDepartment of Mathematical Sciences\nDepartment of Statistics\nUniversity of Nebraska Medical Center Omaha\nBentley University Waltham\nUniversity of South Carolina Columbia\n68198, 02452, 29208NE, MA, SC\n"
] | [
"Department of Biostatistics\nDepartment of Mathematical Sciences\nDepartment of Statistics\nUniversity of Nebraska Medical Center Omaha\nBentley University Waltham\nUniversity of South Carolina Columbia\n68198, 02452, 29208NE, MA, SC",
"Department of Biostatistics\nDepartment of Mathematical Sciences\nDepartment of Statistics\nUniversity of Nebraska Medical Center Omaha\nBentley University Waltham\nUniversity of South Carolina Columbia\n68198, 02452, 29208NE, MA, SC",
"Department of Biostatistics\nDepartment of Mathematical Sciences\nDepartment of Statistics\nUniversity of Nebraska Medical Center Omaha\nBentley University Waltham\nUniversity of South Carolina Columbia\n68198, 02452, 29208NE, MA, SC"
] | [] | Consider a subject or unit in a longitudinal biomedical, public health, engineering, economic or social science study which is being monitored over a possibly random duration. Over time this unit experiences recurrent events of several types and a longitudinal marker transitions over a discrete state-space. In addition, its "health" status also transitions over a discrete state-space with at least one absorbing state. A vector of covariates will also be associated with this unit. Of major interest for this unit is the time-to-absorption of its health status process, which could be viewed as the unit's lifetime. Aside from being affected by its covariate vector, there could be associations among the recurrent competing risks processes, the longitudinal marker process, and the health status process in the sense that the time-evolution of each process is associated with the other processes. To obtain more realistic models and enhance inferential performance, a joint dynamic stochastic model for these components is proposed and statistical inference methods are developed. This joint model, formulated via counting processes and continuous-time Markov chains, has the potential of facilitating 'personalized' interventions. This could enhance, for example, the implementation and adoption of precision medicine in medical settings. Semi-parametric and likelihood-based inferential methods for the model parameters are developed when a sample of these units is available. Finite-sample and asymptotic properties of estimators of model parameters, both finite-and infinite-dimensional, are obtained analytically or through simulation studies. The developed procedures are illustrated using a real data set. | null | [
"https://arxiv.org/pdf/2103.12903v2.pdf"
] | 246,430,578 | 2103.12903 | 9e2bea552c98e8d62ad3b3a650a9c6ad84e8d879 |
JOINT DYNAMIC MODELS AND STATISTICAL INFERENCE FOR RECURRENT COMPETING RISKS, LONGITUDINAL MARKER, AND HEALTH STATUS
February 1, 2022
Lili Tong [email protected]
Department of Biostatistics
Department of Mathematical Sciences
Department of Statistics
University of Nebraska Medical Center Omaha
Bentley University Waltham
University of South Carolina Columbia
68198, 02452, 29208NE, MA, SC
Piaomu Liu [email protected]
Department of Biostatistics
Department of Mathematical Sciences
Department of Statistics
University of Nebraska Medical Center Omaha
Bentley University Waltham
University of South Carolina Columbia
68198, 02452, 29208NE, MA, SC
Edsel A Peña [email protected]
Department of Biostatistics
Department of Mathematical Sciences
Department of Statistics
University of Nebraska Medical Center Omaha
Bentley University Waltham
University of South Carolina Columbia
68198, 02452, 29208NE, MA, SC
JOINT DYNAMIC MODELS AND STATISTICAL INFERENCE FOR RECURRENT COMPETING RISKS, LONGITUDINAL MARKER, AND HEALTH STATUS
February 1, 2022Continuous-time Markov chain · Counting process · Dynamic models · Intensity-based model · Personalized medicine · Conditional independence · Parametric and semi-parametric estimation MSC2020 SUBJECT CLASSIFICATIONS: Primary: 62N0160J27; Secondary: 62G0562M02
Consider a subject or unit in a longitudinal biomedical, public health, engineering, economic or social science study which is being monitored over a possibly random duration. Over time this unit experiences recurrent events of several types and a longitudinal marker transitions over a discrete state-space. In addition, its "health" status also transitions over a discrete state-space with at least one absorbing state. A vector of covariates will also be associated with this unit. Of major interest for this unit is the time-to-absorption of its health status process, which could be viewed as the unit's lifetime. Aside from being affected by its covariate vector, there could be associations among the recurrent competing risks processes, the longitudinal marker process, and the health status process in the sense that the time-evolution of each process is associated with the other processes. To obtain more realistic models and enhance inferential performance, a joint dynamic stochastic model for these components is proposed and statistical inference methods are developed. This joint model, formulated via counting processes and continuous-time Markov chains, has the potential of facilitating 'personalized' interventions. This could enhance, for example, the implementation and adoption of precision medicine in medical settings. Semi-parametric and likelihood-based inferential methods for the model parameters are developed when a sample of these units is available. Finite-sample and asymptotic properties of estimators of model parameters, both finite-and infinite-dimensional, are obtained analytically or through simulation studies. The developed procedures are illustrated using a real data set.
Introduction and Motivation
Consider a unit -for example, a human subject in a medical or a social science study, an experimental animal in a biological experiment, a machine in an engineering or reliability setting, or a company in an economic or business situation -in a longitudinal study monitored over a period [0, τ ], where τ is possibly random. Associated with the unit is a covariate row vector X = (X 1 , X 2 , . . . , X p ). Over time, the unit will experience occurrences of Q competing types of recurrent events, its recurrent competing risks (RCR) component; transitions of a longitudinal marker (LM) process W (t) over a discrete state space W; and transitions of a 'health' status (HS) process V (t) over a discrete state space V = V 0 V 1 , with V 0 = ∅ being absorbing states. If the health status process transitions into an absorbing state prior to τ , then monitoring of the unit ceases, so time-to-absorption serves as the lifetime of the unit. To demonstrate pictorially, the two panels in Figure 1 : Realized data observables from two distinct study units. The first plot panel is for a unit which was rightcensored prior to reaching an absorbing state, while the second plot panel is for a unit which reached an absorbing state prior to getting right-censored.
V = {v 0 , v 1 , v 2 }
with v 0 an absorbing state. In panel 1, the unit did not transition to an absorbing state prior to reaching τ ; whereas in panel 2, the unit reached an absorbing state prior to τ . Two major questions arise: (a) how do we specify a dynamic stochastic model that could be a generative model for such a data, and (b) how do we make statistical inferences for the model parameters and predictions of a unit's lifetime from a sample of such data?
To address these questions, the major goals of this paper are (i) to propose a joint stochastic model for the random observables consisting of the RCR, LM, and HS components for such units, and (ii) to develop appropriate statistical inference methods for the proposed joint model when a sample of units are observed. Achieving these two goals will enable statistical prediction of the (remaining) lifetime of a, possibly new, unit; allow for the examination of the synergistic association among the RCR, LM, and HS components; and provide a vehicle to compare different groups of units and/or study the effects of concomitant variables or factors. More importantly, a joint stochastic model endowed with proper statistical inference methods could potentially enable unit-based interventions which are performed after a recurrent event occurrence or a transition in either the LM or HS processes. As such it could enhance the implementation of precision or personalized decision-making; for instance, precision medicine in a medical setting.
A specific situation where such a data accrual occurs is in a medical study. For example, a subject may experience different types of recurring cancer, with the longitudinal marker being the prostate-specific antigen (PSA) level categorized into a finite ordinal set, while the health status is categorized into either a healthy, diseased, or dead state, with the last state absorbing. A variety of situations in a biomedical, engineering, public health, sociology, and economics settings, where such data structure arise, are further described in Section 2. Several previous works dealing with modeling have either focused in the marginal modeling of each of the three data components, or in the joint modeling of two of the three data components. In this paper we tackle the problem of simultaneously modeling all three data components: RCR, LM, and HS, in order to account for the associations among these components, which would not be possible using either the marginal modeling approach or the joint modeling of pairwise combinations of these three components. A joint full model could also subsume these previous marginal or joint models -in fact, our proposed class of models subsumes as special cases models that have been considered in the literature. In contrast, only by imposing restrictive assumptions, such as the independence of the three model components, could one obtain a joint full model from marginal or pairwise joint models. As such, a joint full model will be less likely to be mis-specified, thereby reducing the potential biases that could accrue from mis-specified models among estimators of model parameters or when predicting residual lifetime.
A joint modeling approach has been extensively employed in previous works. For instance, joint models for an LM process and a survival or time-to-event (TE) process have been proposed in [29], [32], [27], [16], [30], and [22]. Also, the joint modeling of an LM process and a recurrent event process has also been discussed in [15] and [11], while the joint modeling of a recurrent event and a lifetime has also been done such as in [19]. An important and critical theoretical aspect that could not be ignored in these settings is that when an event occurrence is terminal (e.g., death) or when there is a finite monitoring period, informative censoring naturally occurs in the RCR, LM, or HS components, since when a terminal event or when the end of monitoring is reached, then the next recurrent event occurrence, the next LM transition, or the next HS transition will not be observed.
Another aspect that needs to be taken into account in a dynamic modeling approach is that of performed interventions, usually upon the occurrence of a recurrent event. For instance, in engineering or reliability systems, when a component in the system fails, this component will either be minimally or partially repaired, or altogether replaced with a new component (called a perfect repair); while, with human subjects, when a cancer relapses, a hospital infection transpires, a gout flare-up, or alcoholism recurs, some form of intervention will be performed or instituted. Such interventions will impact the time to the next occurrence of the event, hence it is critical that such intervention effects be reflected in the model; see, for instance, [14] and [15]. In addition, models should take into consideration the impacts of the covariates and the effects of accumulating event occurrences on the unit. Models that take into account these considerations have been studied in [23] and [25]. Appropriate statistical inference procedures for these dynamic models of recurrent events and competing risks have been developed in [23] and [28]. Extensions of these joint dynamic models for both RCR and TE can be found in [20]. Some other recent works in joint modeling included the modeling of the three processes: LM, RCR (mostly, a single recurrent event), and TE simultaneously in [17], [7], [18], [21] and [5]. The joint model that will be proposed in this paper will take into consideration these important aspects.
We now outline the remainder of this paper. Prior to describing formally the joint model in Section 3, we first present in Section 2 some concrete situations in science, medicine, engineering, social, and economic disciplines where the data accrual described above could arise and where the joint model will be relevant. Section 3 formally describes the joint model using counting processes and continuous-time Markov chains (CTMCs), and provide interpretations of model parameters. In subsection 3.3 we discuss in some detail a special case of this joint model obtained using independent Poisson processes and homogeneous CTMCs. Section 4 deals with the estimation of the parameters. In subsection 4.1 we demonstrate the estimation of the model parameters in the afore-mentioned special case to pinpoint some intricacies of joint modeling and its inferential aspects. The general joint model contains nonparametric (infinite-dimensional) parameters, so in subsection 4.2 we will describe a semi-parametric estimation procedure for this general model. Section 5 will present asymptotic properties of the estimators, though we will not present in this paper the rigorous proofs of these analytical results but defer them to a more theoretically-focused paper. Section 6 will then demonstrate the semi-parametric estimation approach through the use of a synthetic or simulated data using an R [26] program we developed. In Section 7, we perform simulation studies to investigate the finite-sample properties of the estimators arising from semi-parametric estimation procedure and compare these finite-sample results to the theoretical asymptotic results. An illustration of the semi-parametric inference procedure using a real medical data set is presented in Section 8. Section 9 contains concluding remarks describing some open research problems.
Concrete Situations of Relevance
To demonstrate potential applicability of the proposed joint model, we describe in this section some concrete situations arising in biomedical, reliability, engineering, and socio-economic settings where the data accrual described in Section 1 could arise.
• A Medical Example: Gout is a form of arthritis characterized by sudden and severe attacks of pain, swelling, redness and tenderness in one of more joints in the toes, ankles, knees, elbows, wrists, and fingers (see, for instance, Mayo Clinic publications about gout). When a gout flare occurs, it renders the person incapacitated (personally attested by the senior author) and the debilitating condition may last for several days. Since the location of the gout flare could vary, we may consider gout as competing recurrent events -competing with respect to the location of the flare, and recurrent since it could keep coming back. Gout occurs when urate crystals accumulate in the joints, which in turn is associated with high levels of uric acid in the blood. Other covariates, such as gender, blood pressure, weight, etc., could also impact the occurrence of gout flares, uric acid level, and CKD. When a gout flare occurs, lifestyle interventions could be performed such as (i) consuming skim milk powder enriched with the two dairy products glycomacropeptide (GMP) and G600 milk fat extract; or (ii) consuming standard milk or lactose powder. The purpose of such interventions is to lessen gout flare recurrences. Of major interest is to jointly model the competing gout recurrences, the categorized SUR process, and the CKD process. A study consisting of n subjects could be performed with each subject monitored over some period, with the time of gout flare recurrences of each type, SUR levels, and CKD states recorded over time, aside from relevant covariates. Based on such a data, it will be of interest to estimate the model parameters and to develop a prediction procedure for time-to-absorption to End Stage CKD for a person with gout recurrences.
• A Reliability Example: Observe n independent cars over each of their own monitoring period until the car is declared inoperable or the monitoring period ends. Cars are complex systems in that they are constituted by different components, which could be subsystems or modules, configured according to some coherent structure function [3]. For each car, the states of Q components (such as its engine subsystem; transmission subsystem; brake subsystem; electrical subsystem; etc.) are monitored. Furthermore, its covariates such as weight; initial mileage; current mileage; years of operation; and other characteristics (for example, climate in which car is mostly being driven) are observed. Also, its 'health status', which is either functioning, functioning with some problems, or total inoperability (an absorbing state), is tracked over the monitoring period. Meanwhile, a longitudinal marker such as its oil quality indicator (which is either excellent; good; or poor) and the occurrences of failures of any of the Q components are also recorded over the monitoring period. When a component failure occurs, a repair or replacement of the component is undertaken. Given the data for these n cars, an important goal is to predict the time to inoperability of another car. Note that this type of application could occur in more complex systems such as space satellites, nuclear power plants, medical equipments, etc.
• A Social Science Example: Observe n independent newly-married couples over a period of years (say, 20 years). Over this follow-up period, the marriage could end in separation or divorce, remain intact, or end due to the death of at least one of them. Each couple will have certain characteristics: their ages when they got married; working status of each; income level of each; education level of each; number of children (this is a time-dependent covariate); net worth of couple; etc. Possibly through a regularly administered questionnaire, the couple provides information from which their "marriage status" could be ascertained (either very satisfied; satisfied; poor; separated or divorced). Competing recurrent events for the couple could be changes in job status for either; addition in the family; educational changes of children; and major disagreements. A longitudinal marker could be the financial health status of the couple reflected by their categorized FICO scores. A goal is to infer about the parameters of the joint model based on the observations from these n couples, and to predict if separation or divorce will occur for a married couple, who are not in the original sample, and if so, to obtain a prediction interval of the time of such an occurrence.
• A Financial Example: Track n independent publicly-listed companies over their monitoring periods. At the end of its monitoring period, a company could be bankrupt, still in business, or could have been acquired by another company. Each company has its own characteristics, such as total assets, number of employees, number of branches, etc. Note that these are all time-dependent characteristics. The "health status" of a company is rated according to four categories (A: Exceptional; B: Less than Exceptional; C: Distressed; D: Bankrupt). The bankrupt status is the absorbing state. The company's liability relative to its asset, categorized into High, Medium, Low, Non-Existent could be an important longitudinal marker. Recurrent competing risks will be the occurrence of an increase (decrease) of at least 5% during a trading day in its stock share price. Based on data from a sample of these companies, it could be of interest to predict the time to bankruptcy of another company that is not in the sample.
• COVID- Let τ , the end of monitoring period, have a distribution function G(·), which may be degenerate. The covariate vector will be X = (X 1 , . . . , X p ), assumed to be time-independent, though the extension to time-dependent covariates are possible with additional assumptions. For the RCR component, let
N R = {N R q (s) ≡ (N R (s; q), q ∈ I Q ) : s ≥ 0}, with index set I Q = {1, .
. . , Q}, be a Q-dimensional multivariate counting (column) vector process such that, for q ∈ I Q , N R q (s) is the number of observed occurrences of the recurrent event of type q over [0, s], with N R q (0) = 0. Thus, the sample path s → N R q (s) takes values in Z 0,+ = {0, 1, 2, . . .}, is a non-decreasing step-function, and is right-continuous with left-hand limits. We denote by
dN R q (s) = N R q (s) − N R q (s−),= {(w 1 , w 2 ) : w 1 , w 2 ∈ W, w 1 = w 2 }, we can convert W into a (column) 2 |W| 2 -dimensional multivariate counting process {N W ≡ (N W (s; w 1 , w 2 ), (w 1 , w 2 ) ∈ I W ) : s ≥ 0}, where N W (s; w 1 , w 2 ) is the number of observed transitions of W from state w 1 into w 2 over the period [0, s], that is, for (w 1 , w 2 ) ∈ I W , N W (s; w 1 , w 2 ) = t≤s I{W (t−) = w 1 , W (t) = w 2 },
with I(·) denoting indicator function. Thus, for each (w 1 , w 2 ) ∈ I W , the sample path s → N W (s; w 1 , w 2 ) takes values in Z 0,+ , is a nondecreasing step-function, right-continuous with left-hand limits, and with N W (s; w 1 , w 2 ) = 0. Furthermore, (w1,w2)∈I W dN W (s; w 1 , w 2 ) ∈ {0, 1} for every s ≥ 0.
I V = {(v 1 , v) : v 1 ∈ V 1 , v ∈ V; v 1 = v}, whose cardinality is |I V | = |V 1 ||V| − |V 1 |. We convert V into a (column) |I V |-dimensional multivariate counting process {N V ≡ (N V (s; v 1 , v), (v 1 , v) ∈ I V ) : s ≥ 0}, where N V (s; v 1 , v) is the number of observed transitions of V from state v 1 into state v over the period [0, s], that is, for (v 1 , v) ∈ I V , N V (s; v 1 , v) = t≤s I{V (t−) = v 1 , V (t) = v}.
For each (v 1 , v) ∈ I V , the sample path s → N V (s; v 1 , v) takes values in Z 0,+ , and is a nondecreasing step-function, right-continuous with left-hand limits, and with N W (0; v 1 , v) = 0. In addition, (v1,v)∈I V dN V (s; v 1 , v) ∈ {0, 1} for every s ≥ 0. Next, we combine the multivariate counting processes N R , N W , and N V into one (column) multivariate counting process N = {N (s) : s ≥ 0} of dimension Q + |I W | + |I V |, where, with T denoting vector/matrix transpose,
N (s) = (N R (s)) T , (N W (s)) T , (N V (s)) T T .
An important point needs to be stated regarding the observables in the study, which will have an impact in the interpretation of the parameters of the joint model. This pertains to the "competing risks" nature of all the possible events at each time point s. The possible Q recurrent event types, as well as the potential transitions in the LM and HS processes, are all competing with each other. Thus, suppose that at time s 0 , the event that occurred is a recurrent event of type q 0 , that is, dN R (s 0 ; q 0 ) = 1. This means that this event has occurred in the presence of the potential recurrent events from the other Q − 1 risks, and the potential transitions from either the LM and HS processes. This will entail the use of crude hazards, instead of net hazards, in the joint modeling, and this observation will play a critical role in the dynamic joint model since each of the competing event occurrences at a given time point s from all the possible event sources (RCR, LM, and HS) will be affected by the history of all these processes just before time s. This is the aspect that exemplifies the synergistic association among the three components.
Another observable process for our joint model is a vector of effective (or virtual) age processes E = {(E 1 (s), . . . , E Q (s)) : s ≥ 0}, whose components are F-predictable processes with sample paths that are nonnegative, left-continuous, piecewise nondecreasing and differentiable. These effective age processes will carry the impact of interventions performed after each recurrent event occurrence or a transition in either the LM process or the HS process. For recent articles dealing with virtual ages, see the philosophically-oriented article [12] and the very recent review article [4].
Finally, we define the time-to-absorption of the unit to be τ A = inf{s ≥ 0 : V (s) ∈ V 0 } with the convention that inf ∅ = ∞. Using this τ A and τ , we define the unit's at-risk process to be Y = {Y (s) : s ≥ 0}, with Y (s) = I{min(τ, τ A ) ≥ s}. In addition, we define LM-specific and HS-specific at-risk processes as follows: For w ∈ W, define Y W (s; w) = I{W (s−) = w}; and, for v ∈ V 1 , define Y V (s; v) = I{V (s−) = v}. For a unit, we could then concisely summarize the random observables in terms of stochastic processes as:
D = (X, N, E, Y, Y W , Y V ) ≡ {X, (N (s), E(s), Y (s), Y W (s), Y V (s) : s ≥ 0)}.(1)
Note that the processes are undefined for s > min(τ A , τ ) ≡ inf{s ≥ 0 : Y (s+) = 0} since monitoring of the unit had by then ceased.
Joint Model Specification for One Unit
The joint model specification will be through the specification of the compensator process vector and the predictable quadratic variation (PQV) process matrix of the multivariate counting process N . The predictable process vector
A = {A(s) : s ≥ 0} is of dimension Q + |I W | + |I V |A = (A R ) T , (A W ) T , (A V ) T T and M = (M R ) T , (M W ) T , (M V ) T , where, with q ∈ I Q , (w 1 , w 2 ) ∈ I W , (v 1 , v) ∈ I V , and s ≥ 0, A R = {(A R (s; q))} and M R = {(M R (s; q))}; A W = {(A W (s; (w 1 , w 2 ))} and M W = {(M W (s; (w 1 , w 2 ))}; A V = {(A V (s; (v 1 , v))} and M V = {(M V (s; (v 1 , v))},
with A R and M R of dimensions Q; A W and M W of dimensions |I W |; and A V and M V of dimensions |I V |. The matrix M, M could then be partitioned similarly to reflect these block components.
We can now proceed with the specification of the compensator process vector and the PQV process matrix. For conciseness, we introduce the generic mapping ι defined as follows: For a set A with m elements, A = {a 1 , . . . , a m }, let ι A (a) = (I(a 2 = a), . . . , I(a m = a)), a row vector of m − 1 elements. Here ι A (a) is the indicator vector of a excluding the first element of A, so that ι A (a 1 ) = (0). Excluding a 1 in the ι mapping is for purposes of model identifiability. One may think of the mapping ι as a converter to dummy variables. We will also need the following quantities or functions.
• For each q ∈ I Q there is a baseline (crude) hazard rate function λ 0q (·) with associated baseline (crude) cumulative hazard function Λ 0q (·). We also denote byF 0q (·) = · v=0 [1 − Λ 0q (dv)] the associated baseline (crude) survivor function, where is the product-integral symbol.
• For each q ∈ I Q there is a mapping ρ q (·; ·) : Z Q 0,+ × dq → 0,+ , where the d q 's are known positive integers. There is an associated vector α q ∈ dq .
• There is a collection of non-negative real numbers η = {η(w 1 , w 2 ) : (w 1 , w 2 ) ∈ I W }, and we define for every w 1 ∈ W, η(w 1 , w 1 ) = − w∈W;w =w1 η(w 1 , w).
• There is a collection of non-negative real numbers
ξ = {ξ(v 1 , v) : (v 1 , v) ∈ I V }, and we define for every v 1 ∈ V, ξ(v 1 , v 1 ) = − v∈V;v =v1 ξ(v 1 , v), and with ξ(v 1 , v 2 ) = 0 for every v 1 ∈ V 0 and v 2 ∈ V.
We then define the observables and associated finite-dimensional parameters, respectively, for each of the three model components according to
B R (s) = [X, ι V (V (s)), ι W (W (s))] and θ R = [(β R ) T , (γ R ) T , (κ R ) T ] T ; B W (s) = [X, ι V (V (s)), N R (s)] and θ W = [(β W ) T , (γ W ) T , (ν W ) T ] T ; B V (s) = [X, ι W (W (s)), N R (s)] and θ V = [(β V ) T , (κ V ) T , (ν V ) T ] T .
In addition to the λ 0q 's, α q 's, η, ξ, the θ R , θ W , and θ V will constitute all of the model parameters. For the proposed model, the compensator process components are given by, for q ∈ I Q , (w 1 , w 2 ) ∈ I W , and (v 1 , v) ∈ I V :
A R (s; q) = s 0 Y (t)λ 0q [E q (t)]ρ q [N R (t−); α q ] exp{B R (t−)θ R }dt; A W (s; w 1 , w 2 ) = s 0 Y (t)Y W (t; w 1 )η(w 1 , w 2 ) exp{B W (t−)θ W }dt; A V (s; v 1 , v) = s 0 Y (t)Y V (t; v 1 )ξ(v 1 , v) exp{B V (t−)θ V }dt.
In the left-hand side of the equations above, we have suppressed writing the dependence on the model parameters. With Dg(a) denoting the diagonal matrix formed from vector a, the PQV process is specified to be
M, M (s) = Dg[(A R (s)) T , (A W (s)) T , (A V (s)) T ].
Observe the dynamic nature of this model in that an event occurrence at an infinitesimal time interval [s, s + ds) depends on the history of the processes before time s. According to the theory of counting processes, we have the the following probabilistic interpretations (statements are almost surely):
E{dN R (s; q)|F s− } = dA R (s; q); E{dN W (s; w 1 , w 2 )|F s− } = dA W (s; w 1 , w 2 ); E{dN V (s; v 1 , v)|F s− } = dA V (s; v 1 , v), and V ar{dN R (s; q)|F s− } = dA R (s; q); V ar{dN W (s; w 1 , w 2 )|F s− } = dA W (s; w 1 , w 2 ); V ar{dN V (s; v 1 , v)|F s− } = dA V (s; v 1 , v),
together with the conditional covariance, given F s− , between any pair of elements of dN (s) being equal to zero, e.g., Cov{dN R (s), dN W (s; w 1 , w 2 )|F s− } = 0. However, note that we are not assuming that the components of N R , N W , and N V are independent, nor even conditionally independent. A way to view this model is that, given F s− , the history just before time s, on the infinitesimal interval [s,
s + ds), dN (s) = (dN R (s) T , dN W (s) T , dN V (s) T ) T has a multinomial distribution with parameters 1 and dA(s) = (dA R (s) T , dA W (s) T , dA V (s) T ) T .
As such, for every s ≥ 0, the following constraint holds:
dN • (s) = dN R • (s) + dN W • (s) + dN V • (s) ∈ {0
, 1}, with the notation that a subscript of • means the sum over all the appropriate index set, e.g., dN R
• (s) = q∈I Q dN R (s; q) and dA • (s) = dA R • (s) + dA W • (s) + dA V • (s).
The multinomial distribution above could actually be approximated by independent Bernoulli distributions. To see this, if we have real numbers p k , k = 1, . . . , K, with 0 < p k ≈ 0 for each k = 1, . . . , K, and with K k=1 p k ≈ 0, then we have the approximation
1 − K k=1 p k ≈ K k=1 (1 − p k ).
Consequently, in the equation below, the multinomial probability on the left-hand side is approximately the product of (independent) Bernoulli probabilities in the right-hand side.
q∈I Q [dA q (s)] dN R q (s) (w1,w2)∈I W [dA W (s; w 1 , w 2 )] dN W (s;w1,w2) × (v1,v)∈I V [dA V (s; v 1 , v)] dN V (s;v1,v) [1 − dA • (s)] 1−dN•(s) ≈ q∈I Q [dA q (s)] dN R q (s) [1 − dA R q (s)] 1−dN R q (s) (w1,w2)∈I W [dA W (s; w 1 , w 2 )] dN W (s;w1,w2) [1 − dA W (s; w 1 , w 2 )] 1−dN W (s;w1,w2) (v1,v)∈I V [dA V (s; v 1 , v)] dN V (s;v1,v) [1 − dA V (s; v 1 , v)] 1−dN V (s;v1,v) .
This approximate equivalence informs the manner in which we generate data from the model later in the sections dealing with an illustrative data set (Section 6) and the simulation studies (Section 7) where we used this product of Bernoulli approach.
Consider a unit that is still at risk just before time s whose LM process is at state w 1 and HS process at state v 1 / ∈ V 0 . Two questions of interest are:
(a) What is the distribution of the next event occurrence? (b) Given in addition that an event occurred at time s + t, what are the conditional probabilities of each of the possible events?
Denote by T the time to the next event occurrence starting from time s. Then,
P{T > t|F s− } = s+t u=s 1 − Q q=1 λ 0q [E q (u)]ρ q [N R (u−); α q ] exp{B R (u−)θ R }− η(w 1 , w 1 ) exp{B W (u−)θ W } − ξ(v 1 , v 1 ) exp{B V (u−)θ V } du = exp − s+t s Q q=1 λ 0q [E q (u)]ρ q [N R (u−); α q ] exp{B R (u−)θ R }− η(w 1 , w 1 ) exp{B W (u−)θ W } − ξ(v 1 , v 1 ) exp{B V (u−)θ V } du = exp − exp{B R (s−)θ R } Q q=1 ρ q [N R (s−); α q ] s+t s λ 0q [E q (u)]du+ η(w 1 , w 1 ) exp{B W (s−)θ W }t + ξ(v 1 , v 1 ) exp{B V (s−)θ V }t ,
with the second equality obtained by invoking the product-integral identity
s∈I [1 − dA(s)] 1−dN (s) = exp − s∈I dA(s) and since no events in [s, s+t) means dN R • (u)+dN W • (u)+dN V • (u) = 0 for u ∈ [s, s+t)
, and the last equality arising since, prior to the next event, there will be no changes in the values of N R , B R , B W , and B V from their respective values just before time s. In particular, if the λ 0q s are constants, corresponding to the hazard rates of an exponential distribution, then the distribution of the time to the next event occurrence is exponential. Given that the event occurred at time s + t, then the conditional probability that it was an RCR-type q event is
e {B R (s−)θ R } ρ q [N R (s−); α q ]λ 0q [E q (s + t)]/C(s, t), with C(s, t) = e {B R (s−)θ R } Q q =1 ρ q [N R (s−); α q ]λ 0q [E q (s + t)]− η(w 1 , w 1 )e {B W (s−)θ W } − ξ(v 1 , v 1 )e {B V (s−)θ V }
Similarly, the conditional probability that it was a transition to state w 2 for the LM process is
η(w 1 , w 2 )e {B W (s−)θ W } /C(s, t)
, and the conditional probability that is was a transition to state v, possibly to an absorbing state, for the HS process is
ξ(v 1 , v)e {B V (s−)θ V } /C(s, t)
. These probabilities demonstrate the competing risks nature of the different possible events. They also provide a computational approach to iteratively generate data from the joint model for use in simulation studies, with the basic idea being to first generate a time to any type of event, then to mark the type of event or update each of the counting processes by using the above conditional probabilities.
Denoting by Θ the set of all parameters of the model, the likelihood function arising from observing D, with p (W,V ) (·, ·) the initial joint probability mass function of (W (0), V (0)), is given by
L(Θ|D) = p (W,V ) (W (0), V (0))× (2) ∞ s=0 q∈I Q [dA R (s; q)] dN R (s;q) (w1,w2)∈I W [dA W (s; w 1 , w 2 )] dN W (s;w1,w2) × (v1,v)∈I V [dA V (s; v 1 , v)] dN V (s;v1,v) [1 − dA • (s)] 1−dN•(s)
The likelihood in (2) could be rewritten in the form
L(Θ|D) = p (W,V ) (W (0), V (0))× (3) ∞ s=0 q∈I Q [dA R (s; q)] dN R (s;q) (w1,w2)∈I W [dA W (s; w 1 , w 2 )] dN W (s;w1,w2) (v1,v)∈I V [dA V (s; v 1 , v)] dN V (s;v1,v) exp{−A • (∞)}.
Let us examine the meaning of
A • (∞) ≡ A R • (∞) + A W • (∞) + A V • (∞). Simplifying, we see that this equals A • (∞) = ∞ 0 Y (s)T (s)ds = τ ∧τ A 0 T (s)ds, where T (s) = q∈I Q λ 0q [E q (s)]ρ q [N R (s−); α q ] exp{B R (s−)θ R } − w1∈W Y W (s; w 1 )η(w 1 , w 1 ) exp{B W (s−)θ W } − v1∈V1 Y V (s; v 1 )ξ(v 1 , v 1 ) exp{B V (s−)θ V }.
Recall that η(w 1 , w 1 ) and ξ(v 1 , v 1 ) are non-positive real numbers. Thus, T (s)ds could be interpreted as the total risk of an event, either a recurrent event in the RCR component or a transition in the LM or HS components, occurring from all possible sources (RCR, LM, HS) that the unit is exposed to at the infinitesimal time interval [s, s + ds), given the history F s− just before time s.
We provide further explanations of the elements of the joint model. First, there is a tacit assumption that no more than one event of any type could occur at any given time s. Second, the event rate at any time point s for any type of event is in the presence of the other possible risk events. Thus, consider a specific q 0 ∈ I Q and assume that the unit is still at-risk at time s 0 . Then,
P{dN R q0 (s 0 ) = 1|F s0− } ≈ λ 0q0 [E q0 (s 0 )]ρ q0 [N R (s 0 −); α q0 ] exp{B R (s 0 −)θ R }ds 0(4)
is the conditional probability, given F s0− , that an event occurred at [s 0 , s 0 + ds 0 ) and is of RCR type q 0 and all other event types did not occur, which is the essence of what is called a crude hazard rate, instead of a net hazard rate. Third, the effective (or virtual) age functions E q (·)s, which are assumed to be dynamically determined and not dependent on any unknown parameters, encodes the impact of performed interventions that are performed after each event occurrence. Several possible choices of these functions are:
• E q (s) = s for all s ≥ 0 and q ∈ I Q . This is usually referred to as a minimal repair intervention, corresponding to the situation where an intervention simply puts back the system at the age just before the event occurrence.
• E q (s) = s − S N•(s−) where 0 = S 0 < S 1 < S 2 < . .
. are the successive event times. This corresponds to a perfect intervention, which has the effect of re-setting the time to zero for each of the RCRs after each event occurrence. In a reliability setting, this means that all Q components (unless having exponential lifetimes) are replaced by corresponding identical, but new, components at each event occurrence.
• E q (s) = s − S R N R • (s−) where 0 = S R 0 < S R 1 < S R 2 < . .
. are the successive event times of the occurrences of the RCR events.
• E q (s) = s − S R qN R (s−;q) where 0 = S R q0 < S R q1 < S R q2 < . .
. are the successive event times of the occurrences of RCR events of type q. • Other general forms are possible, such as those in [9] and [14], the latter employing ideas of Kijima. See also the discussion on the 'reality' of virtual or effective ages in the paper by [12], as well as the recent review paper by [4].
Fourth, the impact of accumulating RCR event occurrences, which could be adverse, but could also be beneficial as in software engineering applications, is incorporated in the model through the ρ q (·; ·) functions. One possible choice is an exponential function, such as ρ q (N R (s−); α q ) = exp{N R (s−) T α q }, but other choices could be made as well. Finally, the modulating exponential link function in the model is for the impact of the covariates as well as the values of the RCR, LM, and HS processes just before the time of interest, with the vector of coefficients quantifying the effects of the covariates. The use of N (s−) in the model could be viewed as using them as internal covariates, or that the dynamic model is a self-exciting model.
Similar interpretations hold for the parameters
{η(w 1 , w 2 ) : (w 1 , w 2 ) ∈ I W } and {ξ(v 1 , v) : (v 1 , v) ∈ I V }. Thus, if just before time s 0 , W (s 0 −) = w 1 and V (s 0 −) = v 1 , indicated by F s0− (w 1 , v 1 ), then P{W (s 0 + ds 0 ) = w 2 |F s0 (w 1 , v 1 )} ≈ (5) η(w 1 , w 2 ) exp{Xβ W + γ W j(v1) + N R (s 0 −)ν W }ds 0 ; P{V (s 0 + ds 0 ) = v|F s0 (w 1 , v 1 )} ≈ (6) ξ(v 1 , v) exp{Xβ V + κ V j(w1) + N R (s 0 −)ν V }ds 0 , where j(v 1 ) is the index associated with v 1 in V 1 and j(w 1 ) is the index associated with w 1 in W.
Special Case: Independent Poisson Processes and CTMCs for One Unit
There is a special case arising from this general joint model obtained when we set λ 0q (s) = λ 0q , q ∈ I Q ; ρ q = 1; θ R = 0; θ W = 0; and θ V = 0. In this situation, we have
dA R (s; q) = λ 0q ds, q ∈ I Q ; dA W (s; w 1 , w 2 ) = η(w 1 , w 2 )Y (s)Y W (s; w 1 )ds, (w 1 , w 2 ) ∈ I W ; dA V (s; v 1 , v) = ξ(v 1 , v)Y (s)Y W (s, v 1 )ds, (v 1 , v) ∈ I V .
It is easy to see that this particular model coincides with the model where we have the following situations:
(i) N R (·; q), q ∈ I Q , are independent homogeneous Poisson processes with respective rates λ 0q , q ∈ I Q ;
(ii) W (·) is a continuous-time Markov chain (CTMC) with infinitesimal generator matrix (IGM) consisting of {η(w 1 , w 2 )}; (iii) V (·) is a CTMC with IGM consisting of {ξ(v 1 , v 2 )};
(iv) N R , W , and V are independent; and
(v) Processes are observed over [0, min(τ, τ A )], where τ is the end of monitoring period, while τ A is the absorption time of V into V 0 .
In this special situation, the λ 0q 's are both crude and net hazard rates. Also, due to the memoryless property of the exponential distribution, interventions performed after each event occurrence will have no impact in the succeeding event occurrences. This specific joint model further allows us to provide an operational interpretation of the model parameters. Thus, suppose that at time s, the LM process is at state w 1 and the HS process is at state v 1 / ∈ V 0 . Then, the distribution of the time to the next event occurrence of any type (the holding or sojourn time at the current state configuration) has an exponential distribution with parameter C = λ 0• − η(w 1 , w 1 ) − ξ(v 1 , v 1 ). When an event occurs, then the (conditional) probability that it is (i) an RCR event of type q is λ 0q /C; (ii) a transition to LM state w 2 = w 1 is η(w 1 , w 2 )/C; or (iii) a transition to an HS state v = v 1 is ξ(v 1 , v)/C. This is the essence of the competing risks nature of all the possible event types: an RCR event, an LM transition, and an HS transition. As such, the more general model could be viewed as an extension of this basic model with independent Poisson processes for the RCR component and CTMCs for the LM and HS components. For this special case, the likelihood function in (3) simplifies to the expression
L(Θ|D) = p (W,V ) (W (0), V (0)) q∈I Q λ N R (τ ∧τ A ;q) 0q × (7) (w1,w2)∈I W η(w 1 , w 2 ) N W (τ ∧τ A ;w1,w2) (v1,v)∈I W ξ(v 1 , v) N V (τ ∧τ A ;v1,v) × exp − τ ∧τ A 0 T (s)ds .
where
T (s) = λ 0• − w1∈W η(w 1 , w 1 )Y W (s; w 1 ) − v1∈V1 ξ(v 1 , v 1 )Y V (s; v 1 )
. Note that T (s) = T (s; λ 0 , η, ξ), that is, it is a quantity instead of a statistic. Here X 1 ∼ BER(0.5), X 2 ∼ N (0, 1). X 1 and X 2 are generated independently of each other. Panel 2: Recurrent competing risks occurrences with three types of competing risks. Each unit is either censored ("+") or reaches the absorbing status ("×"). Panel 3: Marker processes. Panel 4: Health status processes, with state "1" absorbing.
Estimation of Model Parameters
Parametric Estimation
Having introduced the joint model, we now address in this section the problem of making inferences about the model parameters. We assume that we are able to observe n independent units, with the ith unit having data (1). The full sample data will then be represented by
D i = (X i , N i , E i , Y i , Y W i , Y V i ) as inD = (D 1 , D 2 , . . . , D n ),(8)
while the model parameters will be represented by, with the convention that q ∈ I Q , (w 1 , w 2 ) ∈ I W , and
(v 1 , v) ∈ I V , Θ ≡ {λ 0q (·), α q }, {η(w 1 , w 2 )}, {ξ(v 1 , v)}, θ R , θ W , θ V .
The λ 0q s could be parametrically-specified, hence will have finite-dimensional parameters, so Θ will also then be finite-dimensional. Except for the special case mentioned above, our main focus will be the case where the λ 0q s are nonparametric. The distributions, G i s, of the end of monitoring periods, τ i s, also have model parameters, but they are not of main interest. To visualize the type of sample data set that accrues, Figure 2 provides a picture of a simulated sample data with n = 50 units.
The full likelihood function, given D, is (3). If the λ 0q (·)s are parametrically-specified, then estimators of the finite-dimensional model parameters could be obtained as the maximizers of this full likelihood function, and their finite and asymptotic properties will follow from the general results for maximum likelihood estimators based on counting processes; see, for instance, [6] and [1].
L(Θ|D) = n i=1 L(Θ|D i ), where the L(Θ|D i ) is of the form in
We illustrate this situation for the special case of the model given in subsection 3.3, so that the parameter is simply
Θ = [{λ 0q }, {η(w 1 , w 2 )}, {ξ(v 1 , v)}].
In this situation, from (7), the full likelihood reduces to, with τ * i = τ i ∧ τ iA ,
L(Θ|D) = n i=1 p (W,V ) (W i (0), V i (0))× q∈I Q λ n i=1 N R i (τ * i ;q) 0q (w1,w2)∈I W η(w 1 , w 2 ) n i=1 N W i (τ * i ;w1,w2) × (v1,v)∈I V ξ(v 1 , v) n i=1 N V i (τ * i ;v1,v) exp − n i=1 τ * i 0 T i (s; λ 0 , η, ξ)ds , where T i (s; λ 0 , η, ξ) = λ 0• − w1∈W η(w 1 , w 1 )Y W i (s; w 1 ) − v1∈V ξ(v 1 , v 1 )Y V i (s; v 1 ). The score function U (Θ|D) = ∇ Θ log L(Θ|D) has elements U R (Θ; q) = n i=1 N R i (τ * i ; q) λ 0q − n i=1 τ * i , q ∈ I Q ; U W (Θ; w 1 , w 2 ) = n i=1 N W i (τ * i ; w 1 , w 2 ) η(w 1 , w 2 ) − n i=1 τ * i 0 Y W i (s; w 1 )ds, (w 1 , w 2 ) ∈ I W ; U V (Θ; v 1 , v) = n i=1 N V i (τ * i ; v 1 , v) ξ(v 1 , v) − n i=1 τ * i 0 Y V i (s; v 1 )ds, (v 1 , v) ∈ I V .
Equating these equations to zeros yield the ML estimators of the parameters, which are given below and possess the interpretation of being the observed "occurrence-exposure" rates.
λ 0q = n i=1 N R i (τ * i ; q) n i=1 τ * i = n i=1 ∞ 0 dN R i (s; q) n i=1 ∞ 0 Y i (s)ds , q ∈ I Q ; η(w 1 , w 2 ) = n i=1 N W i (τ * i ; w 1 , w 2 ) n i=1 τ * i 0 Y W i (s; w 1 )ds = ∞ 0 dN W i (s; w 1 , w 2 ) n i=1 ∞ 0 Y i (s)Y W i (s; w 1 )ds , (w 1 , w 2 ) ∈ I W ; ξ(v 1 , v) = n i=1 N V i (τ * i ; v 1 , v) n i=1 τ * i 0 Y V i (s; v 1 )ds = ∞ 0 dN V i (s; v 1 , v) n i=1 ∞ 0 Y i (s)Y V i (s; v 1 )ds , (v 1 , v) ∈ I V .
In these estimators, the numerators are total event counts, e.g.,
n i=1 N R i (τ * i ; q) is the total number of observed RCR type q events; n i=1 N W i (τ * i ; w 1 , w 2 )
is the total number of observed transitions in the LM process from state w 1 into w 2 ; and
n i=1 N V i (τ * i ; v 1 , v)
is the total number of observed transitions in the HS process from state v 1 into v. On the other hand, the denominators are total observed exposure times, e.g.,
n i=1 ∞ 0 Y i (s)ds is the total time at-risk for all the units; n i=1 ∞ 0 Y i (s)Y W i (s; w 1 )
dw is the total observed time of all units that they were at-risk for a transition in the LM process from state w 1 ; and
n i=1 ∞ 0 Y i (s)Y V i (s; v 1 )
ds is the total observed time of all units that they were at-risk for a transition in the HS process from state v 1 . An important and crucial point to emphasize here is that in these exposure times, they all take into account the time after the last observed events in each component process until the end of monitoring, whether it is a censoring (reaching τ i ) or an absorption (reaching τ iA ). If one ignores these right-censored times, then the estimators could be severely biased. This is a critical aspect we mentioned in the introductory section and re-iterate at this point that this should not be glossed over when dealing with recurrent event models.
The elements of the observed information matrix, I(Θ; D) = −∇ Θ T U (Θ|D), which is a diagonal matrix, have diagonal elements given by:
I R (Θ; q) = n i=1 N R i (τ * i ; q) λ 2 0q , q ∈ I Q ; I W (Θ; w 1 , w 2 ) = n i=1 N W i (τ * i ; w 1 , w 2 ) η(w 1 , w 2 ) 2 , (w 1 , w 2 ) ∈ I W ; I V (Θ; v 1 , v) = n i=1 N V i (τ * i ; v 1 , v) ξ(v 1 , v) 2 , (v 1 , v) ∈ I V .
Abbreviating the estimators intoΘ = (λ 0 ,η,ξ), we obtain the asymptotic result, that as n → ∞,
Θ ∼ AsyMVN(Θ, I(Θ; D) −1 ),(9)
with AsyMVN meaning asymptotically multivariate normal. Thus, this result seems to indicate that the RCR, LM, and HS components or the estimators of their respective parameters do not have anything to do with each other, which appears intuitive since the RCR, LM, and HS processes were assumed to be independent processes to begin with. But, let us examine this issue further. The result in (9) is an approximation to the theoretical result that
Θ ∼ AsyMVN Θ, 1 n I(Θ) −1 ,(10)
where 1 n I(Θ; D) pr → I(Θ). Evidently, I is a diagonal matrix, so let us examine its diagonal elements. Let q ∈ I Q . Then we have, with 'pr-lim' denoting in-probability limit,
λ 2 0q I R (Θ; q) = pr-lim n→∞ 1 n n i=1 ∞ 0 dN R i (s; q) = pr-lim n→∞ 1 n n i=1 ∞ 0 dM R i (s; q) + λ 0q pr-lim n→∞ 1 n n i=1 ∞ 0 Y i (s)ds .
The first term on the right-hand side (RHS) converges in probability to zero by the weak law of large numbers and the zero-mean martingale property. The second term in the RHS converges in probability to its expectation, hence
I R (Θ; q) = 1 λ 0q ∞ 0 lim n→∞ 1 n n i E[Y i (s)] ds .
But, now,
lim n→∞ 1 n n i E[Y i (s)] = lim n→∞ 1 n n i P{τ i ≥ s}P{τ A i ≥ s} = lim n→∞ 1 n n iḠ i (s−)P{V i (u) / ∈ V 0 , u ≤ s}. withḠ i = 1 − G i .
The last probability term above will depend on the generators
{ξ(v 1 , v) : (v 1 , v) ∈ I V } of the CTMC {V (s) : s ≥ 0},
so that the theoretical Fisher information or the asymptotic variance associated with the estimatorλ 0q depends after all on the HS process, as well as on the G i s, contrary to the seemingly intuitive expectation that it should not depend on the LM and HS processes. This result is a subtle one which arise because of the structure of the observation processes. If G i = G, i = 1, . . . , n, then we have that
I R (Θ; q) = 1 λ 0q ∞ 0Ḡ (s−)P{V (u) / ∈ V 0 , u ≤ s}ds, since P{V i (u) / ∈ V 0 , u ≤ s} = P{V (u) / ∈ V 0 , u ≤ s}, i = 1, .
. . , n. If we denote by Γ the generator matrix of {V (s) : s ≥ 0} and let Γ 1 be the sub-matrix associated with the V 1 states, then if p V 0 ≡ (p V (v 1 ), v 1 ∈ V 1 ) T is the initial probability mass function of V (0), we have
P{V (u) / ∈ V 0 , u ≤ s} = P{V (s) ∈ V 1 } = (p V 0 ) T e sΓ11 1 |V1|
where the matrix exponential is e sΓ11 ≡ ∞ k=0 s k Γ k 11 k! and 1 K is a column vector of 1s of dimension K. Thus, we obtain
I R (Θ; q) = 1 λ 0q (p V 0 ) T ∞ k=0 ∞ 0Ḡ (s−) s k k! ds Γ k 11 1 |V1| .
For example, ifḠ(s) = exp(−νs), that is, τ i 's are exponentially-distributed with mean 1/ν, then the above expression simplifies to
I R (Θ; q) = 1 λ 0q 1 ν (p V 0 ) T ∞ k=0 Γ 11 ν k 1 |V1| .
For computational purposes, one may use an eigenvalue decomposition of Γ 11 : Γ 11 = U Dg(d)U −1 where d consists of the eigenvalues of Γ 11 and U is the matrix of eigenvectors associated with the eigenvalues d. The main point of this example though is the demonstration that estimators of the parameters associated with the RCR, LM, or HS process will depend on features of the other processes, even when one starts with independent processes.
We remark that the estimatorsλ 0q s,η(w 1 , w 2 )s, andξ(v 1 , v)s could also be derived as method-of-moments estimators using the martingale structure. The inverse of the observed Fisher information matrix coincides with an estimator using the optional variation (OV) matrix process, while the inverse of the Fisher information matrix coincides with the limit-in-probability of the predictable quadratic variation matrix. To demonstrate for λ 0q , we have that
n i=1 M R i (s; q) = n i=1 N R i (s; q) − s 0 Y i (t)λ 0q dt : s ≥ 0
is a zero-mean square-integrable martingale. Letting s → ∞, setting n i=1 M R i (∞; q) = 0, and solving for λ 0q yieldŝ λ 0q . Next, we haveλ
0q = n i=1 ∞ 0 dN R i (s; q) n i=1 ∞ 0 Y i (s)ds = λ 0q + n i=1 ∞ 0 dM R i (s; q) n i=1 ∞ 0 Y i (s)ds so that √ n[λ 0q − λ 0q ] = ∞ 0 1 n n i=1 Y i (s)ds −1 1 √ n n I=1 ∞ 0 dM R i (s; q).
We have already seen where ∞ 0 1 n n i=1 Y i (s)ds converges in probability, whereas by the Martingale Central Limit Theorem, we have that
1 √ n n i=1 ∞ 0 dM R i (s; q) d → N (0, σ 2 R (q)) with σ 2 R (q) = ∞ 0 pr-lim n→∞ 1 n n i=1 d M R i (·; q), M R i (·; q) (s) = ∞ 0 pr-lim n→∞ 1 n n i=1 Y i (s)λ 0q ds. Therefore, we have √ n[λ 0q − λ 0q ] d → N 0, λ 0q ∞ 0 pr-lim n→∞ 1 n n i=1 Y i (s) ds = λ 0q ∞ 0 lim n→∞ 1 n n i=1 E[Y i (s)] ds ,
which is the same result stated above using ML theory. Analogous asymptotic derivations can be done forη(w 1 , w 2 ) andξ(v 1 , v), though the resulting limiting variances will involve expected occupation times for their respective states of the W i -processes coming from the Y W i (·; w 1 ) terms and the V i -processes from the Y V i (·; v 1 ) terms. Note that, asymptotically, these estimators are independent, but their limiting variances depend on the parameters from the other processes.
Semi-Parametric Estimation
In this section we consider the estimation of model parameters when the hazard rate functions λ 0q (·)s are specified nonparametrically. We shall denote by Λ 0q (t) = t 0 λ 0q (u)du, q ∈ I Q , the associated cumulative hazard functions. To simplify notation, we let
ψ R (s; θ R ) = exp{B R (s)θ R }; ψ W (s; θ W ) = exp{B W (s)θ W }; ψ V (s; θ V ) = exp{B V (s)θ V }.
Using these functions, for q ∈ I Q , (w 1 , w 2 ) ∈ I W , and (v 1 , v) ∈ I V , we then have
dA R (s; q) = Y (s)λ 0q [E q (s)]ρ q (N R (s−); α q )ψ R (s−; θ R )ds; dA W (s; w 1 , w 2 ) = Y (s)Y W (s; w 1 )η(w 1 , w 2 )ψ W (s−; θ W )ds; dA V (s; v 1 , v) = Y (s)Y V (s; v 1 )ξ(v 1 , v)ψ V (s−; θ V )ds.
We also abbreviate the vector of model parameters into Θ ≡ (Λ 0 , α, η, ξ, θ R , θ W , θ V ). Our goal is to obtain estimators for these parameters based on the sample data D = (D 1 , D 2 , . . . , D n ). In a nutshell, the basic approach to obtaining our estimators is to first assume that (α, θ R , θ W , θ V ) are known, then obtain 'estimators' of (Λ 0 , η, ξ). Having obtained these 'estimators', in quotes since they are not yet estimators when (α, θ R , θ W , θ V ) are unknown, we plug them into the likelihood function to obtain a profile likelihood function. From the resulting profile likelihood function, which depends on (α, θ R , θ W , θ V ), we obtain its maximizers with respect to these finite-dimensional parameters to obtain their estimators. These estimators are then plugged into the 'estimators' of (Λ 0 , η, ξ) to obtain their estimators.
The full likelihood function based on the sample data D = (D 1 , . . . , D n ) could be written as a product of three "major" likelihood functions corresponding to the three model components:
L(Θ|D) = q∈I Q L R (Λ 0q , α, θ R ; q|D) × (11) (w1,w2)∈I W L W (η, θ V ; w 1 , w 2 |D) × (v1,v)∈I V L V (ξ, θ W ; v 1 , v|D) ,
where, suppressing writing of the parameters in the functions,
L R (Λ 0q , α, θ R ; q|D) = n i=1 ∞ s=0 dA R i (s; q) dN R i (s;q) × exp − n i=1 ∞ 0 dA R i (s; q) ; L W (η, θ V ; w 1 , w 2 |D) = n i=1 ∞ s=0 dA W i (s; w 1 , w 2 ) dN W i (s;w1,w2) × exp − n i=1 ∞ 0 dA W i (s; w 1 , w 2 ) ; L V (η, θ V ; v 1 , v|D) = n i=1 ∞ s=0 dA V i (s; v 1 , v) dN V i (s;v1,v) × exp − n i=1 ∞ 0 dA V i (s; v 1 , v) .
Let 0 = S 0 < S 1 < S 2 < . . . < S K < S K+1 = ∞ be the ordered distinct times of any type of event occurrence for all the n sample units. Also, let 0 = T 0 < T 1 < T 2 < . . . < T L < T L+1 = ∞ be the ordered distinct values of the set {E iq (S j ) : i = 1, . . . , n; q ∈ I Q ; j = 0, 1, . . . , S K }. Recall that τ * i = τ i ∧ τ iA . Observe that both {S k : k = 0, 1, . . . , K, K + 1} and {T l : l = 0, 1, . . . , L, L + 1} partition [0, ∞). For each i = 1, . . . , n, and q ∈ I Q , E iq (·) is observed, hence defined, only on [0, τ * i ). However, for notational convenience, we define E iq (s) = 0 for s > τ * i . In addition, on each non-empty interval (S j−1 ∧ τ * i , S j ∧ τ * i ], E iq (·) has an inverse which will be denoted by E −1 iqj (·). Henceforth, for brevity of notation, we adopt the mathematically imprecise convention that 0/0 = 0.
Proposition 4.1. For q ∈ I Q , if (α q , θ R ) is known, then the nonparametric maximum likelihood estimator (NPMLE) of Λ 0q (·) is given byΛ
0q (t; α q , θ R ) = l:T l ≤t n i=1 K j=1 I{E iq (S j ) = T l }dN R i (S j ; q) S 0R q (T l |α q , θ R ) (12) where S 0R q (u|α q , θ R ) = n i=1 K j=1 ρ q [N R i (E −1 iqj (u)−); α q ]ψ R i (E −1 iqj (u)−; θ R ) E iq [E −1 iqj (u)] ×(13)I{E iq (S j−1 ∧ τ * i ) < u ≤ E iq (S j ∧ τ * i )}} .
Proof. The likelihood L R q (Λ 0q , α q , θ R |D) could be written as follows:
L R q = n i=1 K j=1 [Y i (S j )λ 0q [E iq (S j )ρ q [N R i (S j −); α q ]ψ i (S j −; θ R )] dN R i (Sj ;q) × exp − n i=1 K+1 j=1 Sj Sj−1 Y i (s)λ 0q [E iq (s)]ρ q [N R i (s−); α q ]ψ i (s−; θ R )ds .
Focusing on the nonparametric parameter Λ 0q (·), the first term of L R q could be written as
n i=1 K j=1 [λ 0q [E iq (S j )] dN R i (Sj ;q) = L l=1 [λ 0q (T l )] dN R • (∞,T l ;q) = L l=1 [dΛ 0q (T l )] N R • (∞,T l ;q) where dN R • (∞, T l ; q) = n i=1 K j=1 I{E iq (S j ) = T l }dN R i (S j ; q).
The exponent in the second term of L R q could be written as follows, the second equality obtained after an obvious change-of-variable and using the definition of S 0R q (·; ·, ·) in the proposition:
n i=1 K+1 j=1 Sj Sj−1 Y i (s)λ 0q [E iq (s)]ρ q [N R i (s−); α q ]ψ i (s−; θ R )ds = n i=1 K+1 j=1 Sj ∧τ * i Sj−1∧τ * i λ 0q [E iq (s)]ρ q [N R i (s−); α q ]ψ i (s−; θ R )ds = ∞ 0 S 0R q (u|α q , θ R )dΛ 0q (u) = L+1 l=1 T l T l−1 S 0R q (u; α q , θ R )dΛ 0q (u) = L l=1 S 0R q (T l |α q , θ R )dΛ 0q (T l ) + L+1 l=1 u∈(T l−1 ,T l ) S 0R q (u|α q , θ R )dΛ 0q (u)
Therefore, L R q , when viewed solely in terms of the parameter Λ 0q equals
L R q = L l= [dΛ 0q (T l )] dN R • (∞,T l ;q) × exp − L l=1 S 0R q (T l |α q , θ R )dΛ 0q (T l ) + L+1 l=1 u∈(T l−1 ,T l ) S 0R q (u|α q , θ R )dΛ 0q (u)
Since L+1 l=1 u∈(T l−1 ,T l ) S 0R q (u|α q , θ R )dΛ 0q (u) ≥ 0, then we obtain the upper bound for L R q by setting this term to be equal to zero:
L R q ≤ L l=1 [dΛ 0q (T l )] dN R • (∞,T l ;q) exp − L l=1 S 0R q (T l |α q , θ R )dΛ 0q (T l ) .
The upper bound is maximized by setting
dΛ 0q (T l |α q , θ R ) = dN R • (∞, T l ; q) S 0R q (T l |α q , θ R )
, l = 1, 2, . . . , L.
For u ∈ (T l−1 , T l ), we then takeΛ 0q (u|α q , θ R ) =Λ 0q (T l−1 |α q , θ R ) which will satisfy the condition u∈(T l−1 ,T l ) S 0R q (u|α q , θ R )dΛ 0q (u) = 0 for all l = 1, 2, . . . , L + 1. Thus,
Λ 0q (t|α q , θ R ) = l:T l ≤t dΛ 0q (T l ; α q , θ R ) = l:T l ≤t dN R • (∞, T l ; q) S 0R q (T l |α q , θ R ) ,
which is a step-function with jumps only on T l s with dN R • (∞, T l ; q) > 0, maximizes Λ 0q (·) → L R q (Λ 0q (·)|α q , θ R |D) for given (α q , θ R ), completing the proof of the proposition.
A more elegant representation of the Aalen-Breslow-Nelson type estimatorΛ 0q (·; α q , θ R ) in Proposition 4.1, which shows that the estimator is also moment-based, aside from being useful in obtaining finite and asymptotic properties, is through the use doubly-indexed processes as in [25] for a setting with only one recurrent event type and without LM and HS processes. Define the doubly-indexed processes
{(N R i (s, t; q), A R i (s, t; q|Λ 0q , α 1 , θ R ) : (s, t) ∈ 2 + } where N R i (s, t; q) = s 0 I{E iq (v) ≤ t}dN R i (v; q); A R i (s, t; q|Λ 0q , α q , θ R ) = s 0 I{E iq (v) ≤ t}dA R i (v; q|Λ 0q , α q , θ R ). Also, let N R • (s, t; q) = n i=1 N R i (s, t; q) and A R • (s, t; q|Λ 0q , α q , θ R ) = n i=1 A R i (s, t; q|Λ 0q , α q , θ R ).
Then, for fixed t,
{M R • (s, t; q|Λ 0q , α q , θ R ) = N R • (s, t; q) − A R • (s, t; q|Λ 0q , α q , θ R ) : s ≥ 0} is a zero-mean square- integrable martingale. Thus, E[N R • (s, t; q)] = E[A R • (s, t; q|Λ 0q , α q , θ R )]. Proposition 4.2. For q ∈ I Q , A R • (s, t; q|Λ 0q , α q , θ R ) = t 0 S 0R q (s, u|α q , θ R )dΛ 0q (u), where S 0R q (s, u|α q , θ R ) = n i=1 K j=1 ρ q [N R i (E −1 iqj (u)−); α q ]ψ R i (E −1 iqj (u)−; θ R ) E iq [E −1 iqj (u)] × I{E iq (S j−1 ∧ τ * i ∧ s) < u ≤ E iq (S j ∧ τ * i ∧ s)}} ; and dN R • (s, T l ; q) = n i=1 K j=1 I{S j ≤ s; E iq (S j ) = T l }dN R i (S j ; q).
Proof. Similar to steps in the proof of Proposition 4.1.
By first assuming that (α q , θ R ) is known, then from the identities in Proposition 4.2, a method-of-moments type estimator of Λ 0q (t) is given bŷ
Λ 0q (s, t|α q , θ R ) = l:T l ≤t dN R • (s, T l ; q) S 0R q (s, T l |α q , θ R ) = t 0 N R • (s, du; q) S 0R q (s, u|α q , θ R ) .(14)
When s → ∞, thisΛ 0q (s, t|α q , θ R ) converges to the estimatorΛ 0q (t|α q , θ R ) in Proposition 4.1. Next, we obtain estimators of the η(w 1 , w 2 )s and ξ(v 1 , v)s, again by first assuming first that θ W and θ V are known. Proposition 4.3. If (θ W , θ V ) are known, the ML estimators of the η(w 1 , w 2 )s and ξ(v 1 , v)s are the "occurrenceexposure" ratesη
(w 1 , w 2 |θ W ) = n i=1 K j=1 dN W i (S j ; w 1 , w 2 ) S 0W (w 1 ; θ W ) , ∀(w 1 , w 2 ) ∈ I W ;(15)ξ(v 1 , v|θ V ) = n i=1 K j=1 dN V i (S j ; v 1 , v) S 0V (v 1 ; θ V ) , ∀(v 1 , v) ∈ I V ,(16)
where
S 0W (w 1 ; θ W ) = ∞ 0 n i=1 Y i (s)Y W i (s; w 1 )ψ W i (s−; θ W )ds; S 0V (v 1 ; θ V ) = ∞ 0 n i=1 Y i (s)Y V i (s; v 1 )ψ V i (s−; θ V )ds.
Proof. Follows immediately by maximizing the likelihood functions L W and L V with respect to the η(w 1 , w 2 )s and ξ(v 1 , v)s, respectively.
We can now form the profile likelihoods for the parameters ({α q , q ∈ I Q }, θ R , θ W , θ V ). These are the likelihoods that are obtained after plugging-in the 'estimators'Λ 0q (·; α q , θ R )s,η(w 1 , w 2 )s, andξ(v 1 , v)s in the full likelihoods. The resulting profile likelihoods are reminiscent of the partial likelihood function in Cox's proportional hazards model [8,2]. Proposition 4.4. The three profile likelihood functions L R pl , L W pl and L V pl are given by
L R pl (α q , q ∈ I Q , θ R |D) = q∈I Q n i=1 K j=1 L l=1 ρ q [N R i (S j −); α q ]ψ R i (S j −; θ R ) S 0R q (T l |α q , θ R ) I{Eiq(Sj )=T l }dN R i (Sj ;q) ; L W pl (θ W |D) = w1∈W n i=1 K j=1 ψ W i (S j −; θ W ) S 0W (w 1 ; θ W ) dN W i (Sj ;w1,•) ; L V pl (θ V |D) = v1∈V1 n i=1 K j=1 ψ V i (S j −; θ V ) S 0V (v 1 ; θ V ) dN V i (Sj ;v1,•)
.
with dN W i (S j ; w 1 , •) = w2∈W; w2 =w1 dN W i (S j ; w 1 , w 2 ), the number of transitions from state w 1 at time S j for unit i, and dN V i (S j ; v 1 , •) = v∈V; v =v1 dN V i (S j ; v 1 , v), the number of transitions from state v 1 at time S j for unit i.
Proof. These follow immediately by plugging-in the 'estimators' into the three main likelihoods in (11) and then simplifying.
From these three profile likelihoods, we could obtain estimators of the parameters α q s, θ R , θ W , and θ V as follows:
(α q , q ∈ I Q ,θ R ) = arg max (αq,θ R ) L R pl (α q , q ∈ I Q , θ R |D);
θ W = arg max θ W L W pl (θ W |D) andθ V = arg max θ V L V pl (θ V |D)
. Equivalently, these estimators are maximizers of the logarithm of the profile likelihoods. These log-profile likelihoods are more conveniently expressed in terms of stochastic integrals as follows:
l R pl = q∈I Q n i=1 ∞ 0 ∞ 0 log ρ q [N R i (s−); α q ] + log ψ R i (s−; θ R )− log S 0R q (t; α q , θ R ) N R i (ds, dt; q); l W pl = w1∈W n i=1 ∞ 0 log ψ W i (s−; θ W ) − log S 0W (s; w 1 |θ W ) N W i (ds; w 1 , •); l V pl = v1∈V1 n i=1 ∞ 0 log ψ V i (s−; θ V ) − log S 0V (s; v 1 |θ V ) N V i (ds; v 1 , •).
Associated with each of these log-profile likelihood functions are their profile score vector (the gradient or vector of partial derivatives) and profile observed information matrix (negative of the matrix of second partial derivatives):
U R (α, θ R ); U W (θ W ) and U V (θ V ) for the score vectors, and I R (α, θ R ), I W (θ W ) and I V (θ V ) for the observed information matrices. The estimators could then be obtained as the solutions of the set of equations U R pl (α,θ R ) = 0; U W pl (θ W ) = 0; U V pl (θ V ) = 0. A possible computational approach to obtaining the estimates is via the Newton-Raphson iteration procedure:
(α,θ R ) ← (α,θ R ) + I R (α,θ R ) −1 U R (α,θ R ); θ W ←θ W + I W (θ W ) −1 U W (θ W );θ V ←θ V + I V (θ V ) −1 U V (θ V ).
Having obtained the estimates of α, θ W and θ V , we plug them intoΛ 0q (t|α q , θ R )s, η(w 1 , w 2 |θ W )s and ξ(v 1 , v|θ V )s to obtain the estimatorsΛ 0q (t)s,η(w 1 , w 2 )s andξ(v 1 , v)s:
Λ 0q (t) =Λ 0q (t|α q ,θ R );η(w 1 , w 2 ) =η(w 1 , w 2 |θ W );ξ(v 1 , v) =ξ(v 1 , v|θ V ).(17)
Asymptotic Properties of Estimators
In this section we provide some asymptotic properties of the estimators, though we do not present the rigorous proofs of the results due to space constraints and instead defer them to a separate paper. To make our exposition more concise and compact, we introduce additional notation. Consider a real-valued function h defined on I W . Then, h(w 1 , w 2 ) will represent the value at (w 1 , w 2 ), but when we simply write h it means the |I W | × 1 vector consisting of h(w 1 , w 2 ), (w 1 , w 2 ) ∈ I W . Thus, η is an |I W | × 1 (column) vector with elements η(w 1 , w 2 )s; N W i (s) is an |I W | × 1 vector consisting of N W i (s; w 1 , w 2 )s; S 0W (s|θ W ) is an |I W | × 1 vector with elements S 0W (s; w 1 |θ W ), (w 1 , w 2 ) ∈ I W ; and ψ W i (s|θ W ) is an |I W | × 1 vector consisting of the same elements. Similarly for those associated with the HS process where functions are defined over I V ; e.g.,
η = (η(v 1 , v), (v 1 , v) ∈ I V ), N V i (s) = (N V i (s; v 1 , v) : (v 1 , v) ∈ I V ), etc.
Let us first consider the profile likelihood function L W pl and the estimatorsθ W andη. Using the above notation, the associated profile log-likelihood function, score function, and observed information matrix could be written as follows:
l W pl (θ W ) = n i=1 ∞ 0 log ψ W i (s − |θ W ) − log S 0W (s|θ W ) T dN W i (s); U W pl (θ W ) = n i=1 ∞ 0 H W 1i (s|θ W ) T dN W i (s); I W pl (θ W ) = n i=1 ∞ 0 V W 1 (s|θ W ) T dN W i (s), where H W 1i (s|θ W ) = · ψ W i (s − |θ W ) ψ W i (s − |θ W ) − · S 0W (s|θ W ) S 0W (s|θ W ) ≡ · ψ W i (s − |θ W ) ψ W i (s − |θ W ) − · S 0W (s; w 1 |θ W ) S 0W (s; w 1 |θ W ) : (w 1 , w 2 ) ∈ I W ; and V W 1 (s|θ W ) = ·· S 0W (s|θ W ) S 0W (s|θ W ) − · S 0W (s|θ W ) S 0W (s|θ W ) ⊗2 ≡ ·· S 0W (s; w 1 |θ W ) S 0W (s; w 1 |θ W ) − · S 0W (s; w 1 |θ W ) S 0W (s; w 1 |θ W ) ⊗2 : (w 1 , w 2 ) ∈ I W .
A '·' over a function means gradient with respect to the parameter vector, while a '··' over a function means the matrix of second-partial derivatives with respect to the element of the parameter vector. In obtaining I W pl , we also used the fact that ψ W i is an exponential function. We now present some results aboutθ W . We also let
I W pl = pr-lim 1 n I W pl (θ W ) .
This will be a function of all the model parameters, since the limiting behavior of the Y i s and Y W i s will depend on all the model parameters, owing to the interplay among the RCR, LM, and HS processes, as could be seen in the special case of Poisson processes and CTMCs.
(i) (Consistency)θ W p → θ W ; (ii) (Asymptotic Representation) √ n[θ W − θ W ] = 1 n I W pl (θ W ) −1 1 √ n n i=1 ∞ 0 H W 1i (s|θ W ) T dM W i (s|η, θ W ) + o p (1); (iii) (Asymptotic Normality) √ n[θ W − θ W ] d → N 0, I W pl −1 .
We point out an important result needed for the proof of Proposition 5.1(iii), which is that
1 n I W pl (θ W ) = (w1,w2)∈I W 1 n n i=1 ∞ 0 H W 1i (s; w 1 |θ W ) ⊗2 Y i (s)Y W i (s; w 1 )η(w 1 , w 2 )ψ W i (s − |θ W )ds +o p (1) p → I W pl
. This asymptotic equivalence indicates where the involvement of the at-risk processes come into play in the limiting profile information matrix, hence the dependence on all the model parameters. This also shows that a natural consistent estimator of I W pl is I W pl (θ W )/n.
Next, we are now in position to present asymptotic results aboutη. First, let s 0W and · s 0W be deterministic functions satisfying
1 n ∞ 0 S 0W (z|θ W )dz − ∞ 0 s 0W (z)dz → 0; 1 n ∞ 0 · S 0W (z|θ W )dz − ∞ 0 · s 0W (z)dz → 0.
Proposition 5.2. Under certain regularity conditions, we have as n → ∞,
(i) (Consistency)η p → η; (ii) (Asymptotic Normality) √ n(η − η) d → N (0, Σ), where Σ = Dg ∞ 0 s 0W (z)dz −1 Dg(η) + Dg ∞ 0 s 0W (z)dz −1 Dg(η) ∞ 0 · s 0W (z)dz T (I W pl ) −1/2 ⊗2 .
(iii) (Joint Asymptotic Normality) √ n(η − η) and √ n(θ W − θ W ) are jointly asymptotically normal with means zeros and asymptotic covariance matrix
Acov( √ n(θ W − θ W ), √ n(η − η)) = −(I W pl ) −1 ∞ 0 · s 0W (z)dz Dg(η)Dg ∞ 0 s 0W (z)dz −1 .
We remark that in result (ii) for the asymptotic covariance matrix Σ in Proposition 5.2, the additional variance term in the right-hand side is the effect of plugging-in the estimatorθ W for θ W in the 'estimators'η(w 1 , w 2 |θ W )s to obtain η(w , w 2 )s. Without having to write them down explicitly, similar results are obtainable for the estimatorsθ V andξ.
Next, we present results concerning the asymptotic properties of the estimators of α q s, θ R , and Λ 0q (·)s. Define the restricted profile likelihood for (α q , θ R ) based on data D observed over [0, s * ] via
L R pl (α, θ R ) ≡ L R pl (α, θ R |s * , t * ) ≡ L R pl (α, θ R |D(s * , t * )) = Q q=1 n i=1 s * s=0 ρ q [N R i (s−); α q ]ψ R i (s−; θ R ) S 0R q (s * , E iq (s)|α q , θ R ) I{Eiq(s)≤t * }dN R i (s;q)
with S 0R q (·, ·|·, ·) as defined in Proposition 4.2. This is restricted in the sense that we only consider events that happened when the effective age are no more than t * and happened over [0, s * ]. Note that as we let t * → ∞ and s * → ∞, we obtain the profile likelihood in Proposition 4.4, The log-profile likelihood function is
l R pl (α, θ R |s * , t * ) ≡ log L R pl (α, θ R |s * , t * ) = Q q=1 n i=1 s * 0 log ρ q [N R i (s−); α q ] + log ψ R i (s − |θ R ) − log S 0R q (s * , E iq (s)|α q , θ R ) × I{E iq (s) ≤ t * }dN R i (ds; q)
. The associated profile score function is
U R pl (α, θ R |s * , t * ) = U R pl (α, θ R ) = n i=1 Q q=1 s * 0 H R iq (s|s * , α, θ R )N R i (ds, t * ; q)
where, for q ∈ I Q ,
H R iq (s|s * , α, θ R ) T = 0 T , . . . , 0 T , H i1q (s|s * , α, θ R ) T , 0 T , . . . , 0 T , H i2q (s|s * , α, θ R ) T , a (Q + 1) × 1 block matrix and with H R i1q (s|s * , α q , θ R ) = ∇ αq ρ q [N R i (s−); α q ) ρ q [N R i (s−); α q ) − ∇ αq S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) ; H R i2q (s|s * , α q , θ R ) = ∇ θ R ψ R i (s − |θ R ) ψ R i (s − |θ R ) − ∇ θ R S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R )
.
Note the dimensions of these block vectors: the qth block is of dimension
dim(α q ) × 1, while the (Q + 1)th block is of dimension dim(θ R ) × 1. An important martingale representation of U R pl (α, θ R |s * , t * ), which is straight-forward to establish, is U R pl (α, θ R |s * , t * ) = n i=1 Q q=1 s * 0 H R iq (s|s * , α, θ R )M R i (ds, t * ; q).
The predictable variation associated with this score function is
U R pl (α, θ R |·, t * ) (s * ) = n i=1 n q=1 s * 0 [H R iq (s|s * , α, θ R )] ⊗2 A R i (ds, t * ; q) with A R i (ds, t; q) = I{E iq (s) ≤ t * }dA R i (s; q)
. The estimatorsα q (s * , t * ), q = 1, . . . , Q, andθ R (s * , t * ) satisfy the equation U R pl (α,θ R |s * , t * ) = 0. The observed profile information matrix I R pl (α, θ R |s * , t * ) is a (Q + 1) × (Q + 1) symmetric block matrix with the following block elements, for q, q = 1, . . . , Q:
I R pl,qq (α, θ R |s * , t * ) = n i=1 s * 0 V R i11qq (s|s * , α, θ R )N R i (ds, t * ; q) for q = q 0 for q = q ; I R pl,(Q+1)q (α, θ R |s * , t * ) = n i=1 s * 0 V R i21q (s|s * , α, θ R )N R i (ds, t * ; q); I R pl,(Q+1)(Q+1) (α, θ R |s * , t * ) = n i=1 Q q=1 s * 0 V R i22q (s|s * , α, θ R )N R i (ds, t * ; q), where V R i11qq is of dimension dim(α q ) × dim(α q ) for q, q = 1, 2, . . . , Q; V R i21q and (V R i12q ) T have dimensions dim(θ R ) × dim(α q ); and V R i22q has dimension dim(θ R ) × dim(θ R )
. These are given by the following expressions, for q, q = 1, . . . , Q:
V R i11qq (s|s * , α, θ R ) = − ∇ αqα T q ρ q [N i (s−); α q ] ρ q [N i (s−); α q ] − ∇ αq ρ q [N i (s−); α q ] ρ q [N i (s−); α q ] ⊗2 + ∇ αqα T q S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) − ∇ αq S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) ⊗2 ; V R i11qq (s|s * , α, θ R ) = 0, q = q ; V R i21q (s|s * , α, θ R ) = − ∇ αq(θ R ) T S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) − ∇ αq S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) ∇ (θ R ) T S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) ; V R i22q (s|s * , α, θ R ) = − ∇ θ R (θ R ) T S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) − ∇ θ R S 0R q (s * , E iq (s)|α q , θ R ) S 0R q (s * , E iq (s)|α q , θ R ) ⊗2 ,
where for the last expression we used the fact that ψ i (z) = exp(z).
A condition needed for the asymptotic results is that there is an invertible matrix function I R pl (α, θ R |s * , t * ) which equals the in-probability limit of 1 n I R pl (α, θ R |s * , t * ) as n → ∞. Under this condition, we have the following consistency and asymptotic normality results for (α,θ R ): Proposition 5.3. Under regularity conditions and as n → ∞,
(i) (α(s * , t * ),θ R (s * , t * )) p → (α, θ R ); (ii) √ n α(s * , t * ) − α θ R (s * , t * ) − θ R d → N 0, [I R pl (α, θ R |s * , t * )] −1 .
Important equivalences to note are, for q, q = 1, . . . , Q:
1 n I R pl,qq (α, θ R |s * , t * ) = 1 n n i=1 s * 0 V R i11qq (s|s * , α, θ R )I{E iq (s) ≤ t * } × Y i (s)ρ q [N R i (s−); α q ]ψ R i (s−; θ R )λ 0q [E iq (s)]ds + o p (1); 1 n I R pl,(Q+1)q (α, θ R |s * , t * ) = 1 n n i=1 s * 0 V R i21q (s|s * , α, θ R )I{E iq (s) ≤ t * } × Y i (s)ρ q [N R i (s−); α q ]ψ R i (s−; θ R )λ 0q [E iq (s)]ds + o p (1); 1 n I R pl,(Q+1)(Q+1) (α, θ R |s * , t * ) = 1 n n i=1 Q q=1 s * 0 V R i22q (s|s * , α, θ R )I{E iq (s) ≤ t * } × Y i (s)ρ q [N R i (s−); α q ]ψ R i (s−; θ R )λ 0q [E iq (s)]ds + o p (1)
. In addition, the regularity conditions must imply that, we have
1 n I R pl,qq (α, θ R |s * , t * ) − 1 n n i=1 s * 0 [H R i1q (s|s * , α, θ R )] ⊗2 A R i (ds, t * ; q) p → 0; 1 n I R pl,(Q+1)(Q+1) (α, θ R |s * , t * ) − 1 n n i=1 s * 0 [H R i2q (s|s * , α, θ R )] ⊗2 A R i (ds, t * ; q) p → 0; 1 n I R pl,(Q+1)q (α, θ R |s * , t * ) − 1 n n i=1 s * 0 H R i1q (s|s * , α, θ R )[H R i2q (s|s * , α, θ R )] T A R i (ds, t * ; q) p → 0.
These conditions imply that 1 n U R pl − 1 n I R pl p → 0 as n → ∞. These are the analogous results to those in the usual asymptotic theory of maximum likelihood estimators, where the Fisher information is equal to the expected value of the squared partial derivative with respect to the parameter of the log-likelihood function and also the negative of the expected value of the second partial derivative of the log-likelihood function. They are usually satisfied by imposing a set of conditions that allows for the interchange of the order of integration with respect to s and the partial differentiation with respect to the parameters; see, for instance, [6], [2], and [24] in similar but simpler settings.
The proof of Proposition 5.3 then relies on the asymptotic representation
√ n α(s * , t * ) − α θ R (s * , t * ) − θ R = 1 n I R pl (α, θ R |s * , t * ) −1 1 √ n U R pl (α, θ R |s * , t * ) + o p (1).
In fact, this representation is also crucial for finding the asymptotic properties of the estimators of the Λ 0q (·)s, which we will now present. By first-order Taylor expansion and under regularity conditions, we have the representations, for each q = 1, . . . , Q, given bŷ
Λ 0q (s * , t) = t 0 I{S 0R q (s * , w|α(s * , t * ),θ R (s * , t * )) > 0} S 0R q (s * , w|α(s * , t * ),θ R (s * , t * ) n i=1 N R i (s * , dw; q) = t 0 I{S 0R q (s * , w|α, θ R ) > 0} S 0R q (s * , w|α, θ R ) n i=1 N R i (s * , dw; q) + B 1q (s * , t|α, θ R ), B 2q (s * , t|α, θ R ) α(s * , t * ) − α θ R (s * , t * ) − θ R + o p (1/ √ n) where B 1q (s * , t|α, θ R ) = − t 0 I{S 0R q (s * , w|α, θ R ) > 0}× ∇ α S 0R q (s * , w|α, θ R ) [S 0R q (s * , w|α, θ R )] 2 n i=1 N R i (s * , dw; q); B 2q (s * , t|α, θ R ) = − t 0 I{S 0R q (s * , w|α, θ R ) > 0}× ∇ R θ S 0R q (s * , w|α, θ R ) [S 0R q (s * , w|α, θ R )] 2 n i=1 N R i (s * , dw; q).
For q = 1, . . . , Q, let
Λ * 0q (s * , t) = t 0 I{S 0R q (s * , w|α, θ R ) > 0}λ 0q (w)dw. Observe that t 0 I{S 0R q (s * , w|α, θ R ) > 0} S 0R q (s * , w|α, θ R ) n i=1 A R i (s * , dw|α, θ R ) = Λ * 0q (s * , t), implying that t 0 I{S 0R q (s * , w|α, θ R ) > 0} S 0R q (s * , w|α, θ R ) n i=1 N R i (s * , dw; q) = t 0 I{S 0R q (s * , w|α, θ R ) > 0} S 0R q (s * , w|α, θ R ) n i=1
M R i (s * , dw; q) + Λ * 0q (s * , t).
Let us also defineẐ
R q (s * , t) = [Λ 0q (s * , t) − Λ * 0q (s * , t)]− B 1q (s * , t|α, θ R ), B 2q (s * , t|α, θ R ) α(s * , t * ) − α θ R (s * , t * ) − θ R ; H R iq (w|s * ) = j:Sj ≤s * H R iq [E −1 iqj (w)|s * ]I{E iq (S j−1 ) < w ≤ E iq (S j )};
and form the vectors of functionsẐ = (Ẑ q , q = 1, . . . , Q) andH R i = (H R iq , q = 1, . . . , Q). Then, the main asymptotic representation leading to the asymptotic results for the RCR component parameters is given by, for t ∈ [0, t * ],
√ n α (s * , t * ) − α θ R (s * , t * ) − θ R Z(s * , t) = 1 n I R pl (s * , t * ) −1 0 0 I × 1 √ n t 0 H R i (w|s * ) Dg I{S 0R (s * ,w)/n>0} S 0R (s * ,w)/n M R i (s * , dw) + o p (1).(18)
Using this representation, we obtain the following asymptotic properties, though not stated in the most general form. Proposition 5.4. Under certain regularity conditions, we have
(i) √ n α(s * , t * ) − α θ R (s * , t * ) − θ R and √
nẐ(s * , t) are asymptotically independent;
(ii) For each q = 1, . . . , Q,Λ 0q (s * , t) converges uniformly in probability to Λ 0q (t) for t ∈ [0, t * ];
(iii) For each q = 1, . . . , Q, and for t ∈ [0, t * ], √ n[Λ 0q (s * , t) − Λ 0q (t)] converges in distribution to a normal distribution with mean zero and variance
Γ q (s * , t) = t 0 dΛ 0q (w) s 0R q (s * , w) + (b 1q (s * , t), b 2q (s * , t))[I R pl (s * , t * )] −1 (b 1q (s * , t), b 2q (s * , t)) T ,
where (b 1q (s * , t), b 2q (s * , t)) = pr-lim 1 n (B 1q (s * , t), B 2q (s * , t)) .
(iv) More, generally, for q = 1, . . . , Q, the stochastic process
{ √ n[Λ 0q (s * , t) − Λ 0q (t)] : t ∈ [0, t * ]} converges weakly in Skorokhod's space (D[0, t * ], D[0, t * ]
) to a zero-mean Gaussian process with covariance function
Γ q (s * , t 1 , t 2 ) = min(t1,t2) 0 dΛ 0q (w) s 0R q (s * , w) + (b 1q (s * , t 1 ), b 2q (s * , t 1 ))[I R pl (s * , t * )] −1 (b 1q (s * , t 2 ), b 2q (s * , t 2 )) T .
This covariance function is consistently estimated by replacing the unknown functions by their empirical counterparts and replacing the finite-dimensional parameters by their estimators.
We point out that theΛ 0q , q = 1, . . . , Q, are asymptotically dependent, and are also not independent of (α(s * , t * ),θ R (s * , t * )). The results stated above generalize those in [2] and [24] to a more complex and general situation.
Illustration of Estimation Approach on Simulated Data
In this section we provide a numerical illustration of the estimation procedure when given a sample data. The illustrative sample data set with n = 50 units is depicted in Figure 2. This is generated from the proposed model with the following characteristics. For the ith sample unit, the covariate values are generated according to X i1 ∼ BER(0.5), X i2 ∼ N (0, 1) with X i1 and X i2 independent, where BER(p) is the Bernoulli distribution with success probability of p. The end of monitoring time is τ i ∼ EXP(5), where EXP(λ) is the exponential distribution with mean λ. For the RCR component with Q = 3, the baseline (crude) hazard rate function for risk q ∈ {1, 2, 3}, is a two-parameter Weibull given by
λ 0q (t|κ * q , θ * q ) = κ * q θ * q t θ * q κ * q −1 , t ≥ 0,
with κ * q ∈ {2, 2, 3} and θ * q ∈ {0.9, 1.1, 1}. The associated (crude) survivor function for risk q is Table 1: The second column contains the true values of the parameters in the first column of the model that generated the illustrative sample data in Figure 2 and also used in the simulation study. The third column are the estimates of these parameters arising from the illustrative sample data, while the fourth column contains the information-based estimates of the standard errors of the estimators. The RCR component model parameters are the αs and those with superscript R. The LM component parameters are those with superscript of W . Finally, the HS component parameters are those with superscript V .
F 0q (t|κ * q , θ * q ) = κ * q θ * q t θ * q κ * q −1 exp − t θ * q κ * q , t ≥ 0.
For each replication, the realized data for the ith unit among the n units are generated in the following manner. At time s = 0, we first randomly assign an initial LM state (either 1, 2, or 3) and HS state (either 2 or 3) uniformly among the allowable states and independently for the LM and HS processes. We specify a fixed length of the intervals partitioning [0, ∞), which in the simulation runs was set to ds = 0.001, so we have intervals I k = [s k , s k+1 ) = [k(ds), (k + 1)(ds)), k = 0, 1, 2, . . .. A smaller value of ds will make the data generation coincide more closely to the model, but at the same time will also lead to higher computational costs, especially in a simulation study with many replications. The data generation proceeds sequentially over k = 0, 1, 2, . . .. Suppose that we have reached interval I k = [s k , s k+1 ) with W (s k ) = w 1 and V (s k ) = v 1 . For q ∈ I Q , generate a realization e R q of E R q according to a BER(p R q ) with p R q given by (4). Also, for w 2 = w 1 , generate a realization e W w2 of E W w2 according to a BER(p W w2 ) with p W w2 given by (5). Finally, for v = v 1 , generate a realization e V v of E V v according to a BER(p V v ) with p V v given by (6).
• If all the realizations e R q , q ∈ I Q , e W w2 , w 2 ∈ I W , w 2 = w 1 , and e V v , v ∈ I V , v = v 1 are zeros, which means no events occurred in the interval I k , then we proceed to the next interval I k+1 , provided that τ / ∈ I k , otherwise we stop.
• If exactly one of the realizations e R q , q ∈ I Q , e W w2 , w 2 ∈ I W , w 2 = w 1 , and e V v , v ∈ I V , v = v 1 equals one, so that an event occurred, then we update the values of N R , N W , N V and proceed to the next interval I k+1 , unless e V v = 1 for a v ∈ V 0 (i.e., there is a transition to an absorbing state) or τ ∈ I k , in which case we stop.
In our implementation, since 0 < ds ≈ 0, the success probabilities in the Bernoulli distributions above will all be close to zeros, hence the probability of more than one of the e R q , q ∈ I Q , e W w2 , w 2 ∈ I W , w 2 = w 1 , and e V v , v ∈ I V , v = v 1 taking values of one is very small. But, in case there were at least two of them with values of one, then we randomly choose one to take the value of one and the others are set to zeros. Thus, we always have
q∈I Q e R q + w2∈I W ; w2 =w1 e W w2 + v∈I V ; v =v1 e V v ∈ {0, 1},
which means there is at most one event in any interval. Note that whether we reach an absorbing state or not, there will always be right-censored event times or sojourn times.
We remark that the event time generation could also have been implemented by first generating a sojourn time and then deciding which event occurred according to a multinomial distribution at the realized sojourn time as indicated in subsection 3.2. In addition, depending on the form of the baseline hazard rate functions λ 0q (·)s (e.g., Weibulls) and the effective age processes E iq (·)s (e.g., backward recurrence times), a more direct and efficient manner of generating the event times using a variation of the Probability Integral Transformation is possible, without having to do the partitioning of the time axis as done above. However, the method above is more general, though approximate, and applicable even if the λ 0q 's or the E iq 's are of more complex forms.
Graphical plots associated with the generated illustrative sample data are provided in Figure 2 of Section 4.1. For this sample data set there were 36 units that reached the absorbing state, with a mean time to absorption of about t = 1; while 14 units did not reach the absorbing state before hitting their respective end of their monitoring periods, with the mean monitoring time at about t = 2. Recall that τ i 's were distributed as EXP(5) so the mean of τ i is 5. One may be curious why the mean monitoring time for those that did not get absorbed is about 2 and not close to 5. The reason for this is because of an induced selection bias. Those units who got absorbed will tend to have longer monitoring times, hence those that were not absorbed will tend to have shorter monitoring times, explaining a reduction in the mean monitoring times for the subset of units that were not absorbed.
We have developed programs in R [26] for implementing the semi-parametric estimation procedure for the joint model described above. We used these programs on this illustrative sample data set to obtain estimates and their estimated standard errors of the model parameters. The third and fourth columns of Table 1 contain the parameter estimates and estimates of their standard errors, respectively, of the finite-dimensional parameters in the first column, whose true values are given in the second column. Figure 3 depicts the true baseline survivor functions, which are Weibull survivor functions, together with their semi-parametric estimates for each risk of the three risks in the RCR component. The estimates of the baseline survivor functions are obtained using the product-integral representation of cumulative hazard functions. For this generated sample data, the estimates obtained and the function plots demonstrate that there is reasonable agreement between the true parameter values and true functions and their associated estimates, indicating that the semi-parametric estimation procedure described earlier appears to be viable.
Finite-Sample Properties via Simulation Studies
Simulation Design
We have provided asymptotic results of the estimators in Section 5. In this section we present the results of simulation studies to assess the finite-sample properties of the estimators of model parameters. This will provide some evidence whether the semi-parametric estimation procedure, which appears to perform satisfactorily for the single illustrative sample data set in Section 6, performs satisfactorily over many sample data sets. These simulation studies were implemented using R programs we developed, in particular, the programs utilized in estimating parameters in the preceding section. In these simulation studies, as in the preceding section, when we analyze each of the sample data, the baseline hazard rate functions are estimated semi-parametrically, even though in the generation of each of the sample data sets, two-parameter Weibull models were used in the RCR components. Aside from the set of model parameters described in Section 6, the simulation study have the additional inputs which are the sample size n and the number of simulation replications Mreps, the latter set to 1000. The sample sizes used in the two simulation studies are n ∈ {50, 100}. For fixed n, for each of the Mreps replications, the sample data generation is as described in Section 6. Table 2 presents some summary results pertaining to the three processes based on the Mreps replications. The first three rows indicate the means and standard deviations of the number of event occurrences per unit for each risk, and the mean time for the first event occurrence of each risk. For example, risk 1 occurs about 2.6 times per unit with a standard deviation 3.57. The mean time for risk 1 to occur for the first time is about 0.48. We notice that occurrence frequencies for three risks are ordered according to Risk 1 Risk 2 Risk 3, and consequently risk 1 tends to have the shortest mean time to the first event. Also note that the mean number of event occurrences per unit for each risk is around 2, which implies that there are not too few RCR events or too many RCR events (see the property of "explosion" as discussed in [13]) per unit. This indicates that the choice of the effective age function E iq (·) and the accumulating event function ρ q (·), together with the parameter values we chose for the data generation, were reasonable.
The fourth to ninth rows show the mean and standard deviation of the number of transitions to specific states per unit, the mean and standard deviation of occupation times per unit for specific states, the mean and standard deviation of sojourn times for specific states. For example, column 4 tells us that (i) the mean number of transitions to state 2 of the HS process (HS 2 for short) per unit is 2.34; (ii) a unit would stay in HS 2 for an approximate time of 0.8 on average; (iii) the mean sojourn time for HS 2 is about 0.34. We do not include information for HS 1 since it is an absorbing state. Comparing the V = 2 and V = 3 columns, we find that units tend to transit to HS state 2 more often than to HS state 3. The mean occupation time for state 2 per unit is longer compared to state 3. For the last three columns, there are more transitions to state 2 than to other states. Thus, a unit tends to stay in LM state 1 more often than in the other two LM states. Table 2: Summary statistics for three processes for Mreps replications of data set. The first three rows are for RCR events. The first three rows indicate the mean/standard deviation of the number of recurrent event occurrences per unit for each risk, the mean time for one recurrent event for each risk. The fourth to ninth rows are for HS and LM events. They indicate the mean/standard deviation of the number of transitions to the specific state per unit, the mean/standard deviation of the occupation time for the specific state per unit, the mean/standard deviation of the sojourn time for the specific state. State 1 of the HS process V is absorbing, hence not included in the table.
Also, to obtain some insights into the model-induced dependencies among the components, we also obtained the correlations among RCR, LM, and HS processes over time from the simulated data. We first constructed a vector of six variables over a finite number of time points S ⊂ [0, τ ] given by
{Z(s) ≡ [I(V i (s) = 2), I(W i (s) = 2), I(W i (s) = 3), N R i1 (s), N R i2 (s), N R i3 (s)] T : s ∈ S}.
For each s ∈ S, we then obtained the sample correlation matrix C(s) from {Z i (s), i = 1, 2, . . . , n}. Each of the Mreps replications then yielded a C(s), so we took the mean, element-wise, of these Mreps correlation matrices. The matrix of scatterplots in Figure 4 provides the plots of these mean correlation coefficients over time points s ∈ S. The point we are making here is that the joint model does induce non-trivial patterns of dependencies over time among the three model components.
Finite-Sample Properties of Estimators
The set of estimates obtained for one sample data set in the last section is insufficient to assess the performance of the semi-parametric estimation procedure. To get a sense of its performance we performed simulation studies with Mreps = 1000 replications and sample sizes n ∈ {50, 100}. For each replication, we generated a sample data set according to the same joint model, then obtained the set of estimates, via the semi-parametric procedure, for this data set. Summary statistics, such as the means, standard deviations, asymptotic standard errors, percentiles, and boxplots for all Mreps estimates were then obtained or constructed. Table 3 shows these summary statistics of the Mreps estimates for each parameter. The asymptotic standard errors reported in Table 3 are the means of the asymptotic standard errors over the Mreps replications. Also included are the percentile 95% confidence intervals for each of the unknown parameters based on the Mreps replications stratified according to n = {50, 100}. Since the Λ 0q s are functional nonparametric parameters, we only provide the estimates at four selected time points. Due to space limitation, we also only provide the results for a small subset of the η and ξ parameters in Table 3. From these simulation results, we observe that the estimates are close to the true values of the model parameters, and the sample standard deviations are close to the asymptotic standard errors, providing some validation to the semi-parametric estimation procedure for the joint model and empirically lending support to the asymptotic results. By comparing the results for n = 50 and n = 100 in Table 3, we find that when the sample size n increases, the performance of the estimators of the parameters improves with biases and standard errors decreasing.
The graphical summary of these centered estimates for Mreps replications of n = 50 units is given in Figure 5. Centering for each estimator is done by subtracting the true value of the parameter being estimated. We observe that the medians of all these centered estimates are close to 0. We also observe some outliers, but most of the centered Mreps estimates are close to 0. In Figure 6, we show three types of plots for the baseline survivor function for each risk in the RCR component. The true Weibull type baseline survivor function is plotted in red color, the overlaid plots of a random selection of ten estimates of the baseline survivor functions are in green color, and the mean baseline survivor function based on the Mreps estimates is shown in blue color. We observe that there is close agreement between the true (red) and mean (blue) curves. Based on these simulation studies, the semi-parametric estimation procedure for the
W i (s) = 3), N R i1 (s), N R i2 (s), N R i3 (s)].
The sample correlation matrix is computed based on the n = 50 sample units. The element-wise means of the Mreps correlation matrices were then computed. The plots depict these mean sample correlations for the pairs of variables in C(s) over time s. joint model appears to provide reasonable estimates of the true finite-and infinite-dimensional model parameters, at least for the choices of parameter values for these particular simulations. Further simulation and analytical studies are still needed to substantively assess the performance of the semi-parametric estimation procedure for the proposed joint model.
Illustration on a Real Data Set
To illustrate our estimation procedure on a real data set, we apply the joint model and the semi-parametric estimation procedure to a medical data set with n = 150 patients diagnosed with metastatic colorectal cancer which cannot be controlled by curative surgeries. This data set was gathered in France from 2002-2007 and was used in [10]. It consists of two data sets which are deposited in the frailtypack package in the R Library [26]: data set colorectal.Longi and data set colorectal. The data set colorectal.Longi includes the follow-up period, in years, of the patient's tumor size measurements. The times of first measurements of tumor size vary from patient to patient, so to have all of them start at time 'zero', our artificial time origin, we consider these first measurements as their initial states. Subsequent times of measuring tumor size are then in terms of the lengths of time from their time origin. There were a total of 906 tumor size measurements for all the patients. In order to conform to our discrete-valued LM model, we classify (arbitrarily) the tumor size into three categories (states): 1, 2, and 3, if the tumor size belongs in the intervals [3.4, 6.6], [2, 3.4], and (0, 2), respectively. Since the tumor size is only measured at discrete times, instead of continuously, we assumed that the tumor size state is constant between tumor size measurement times, and consequently the tumor size process could only transition at the times in which tumor size is measured. This assumption is most likely unrealistic, but we do so for the purpose of illustration. The data set colorectal contains some information about the patient's ID number, covariates X 1 and X 2 , with X 1 = 1(0) if patient received treatment C (S); X 2 consists of two dummy variables, with X 2 = (0, 0), (1, 0), (0, 1) if the initial WHO performance status is 0, 1, 2, respectively; the time (in years) of each occurrence of a new lesion since baseline measurement time; and the final right-censored or death time. There were 289 occurrences of new lesions and 121 patients died during the study.
Clearly, this data set is a special case of the type of data appropriate for our proposed joint model, having only one type of recurrent event, one absorbing health status state (dead), and one transient health status state (alive). We assume the effective age E i (s) = s − S R iN R i (s−) after each occurrence of a new lesion, and use ρ(k); α) = α log(1+k) . The unknown model parameters in the RCR (here, just a recurrent event) component of the model includes α in the ρ(·) function, We fitted the joint model to this data set. The resulting model parameter estimates along with the information-based standard error estimates are given in Table 4. The standard errors are obtained by taking square roots of the diagonal elements of the observed inverse of the profile likelihood information matrix. Based on these estimates, we could also perform hypothesis tests. Thus, for instance, the p-values associated with the two-tailed hypothesis tests are also given in the table. The null hypothesis being tested for α is that H 0 : α = 1, while the null hypotheses for the other model parameters are that their true parameter values are zeros. We test α = 1 instead of α = 0 because α = 1 means that the accumulating number of recurrent event occurrences does not have an impact in subsequent recurrent event occurrences. From the values in Table 4, the estimate of α is less than one, which may indicate that each occurrence of new lesion decreases the risk of future occurrences of new lesions, though from the result of the statistical test we cannot conclude statistically that α < 1. Based on the set of p-values, we find that the initial WHO performance state of 1 and the tumor size state of 2 are associated with decreased risk of new lesion occurrences, while an initial WHO performance state of 2 is associated with an increased risk of new lesion occurrences. An initial WHO performance state of 2 and the number of occurrences of new lesions are associated with an increased risk of death in the health status.
Finally, we want to emphasize the importance of the effective age process. An inappropriate effective age may lead to misleading estimates. Parameter model estimates under a mis-specified effective age could lead to biases and potentially misleading conclusions. This is one aspect where domain specialists and statisticians need to consider when assessing the impact of interventions since the specification of the effective or virtual age have important and consequential implications. For more discussions about effective or virtual ages, see the recent papers [12,4], the last one also touching on the situation where the virtual age function depends on unknown parameters. 0.14 0.00 Table 4: Parameter estimates, information-based standard errors, and p-values for the RCR, LM, and HS processes based on the real data set. The p-value is based on the two-tailed hypothesis test that the model parameter is zero (except for the parameter α where H 0 : α = 1). The top block includes the model parameters in the RCR component, the middle block includes the model parameters in the LM component, and the bottom block includes the model parameters in the HS component.
Concluding Remarks
For the general class of joint models for recurrent competing risks, longitudinal marker, and health status proposed in this paper, which encompasses many existing models considered previously, there are still numerous aspects that need to be addressed in future studies. Foremost among these aspects is a more refined analytical study of the finite-sample and asymptotic properties of the estimators of model parameters, together with other inferential and prediction procedures. The finite-sample and asymptotic results could be exploited to enable performing tests of hypothesis and construction of confidence regions for model parameters. There is also the interesting aspect of computationally estimating the standard errors of the estimators. How would a bootstrapping approach be implemented in this situation? Another important problem that needs to be addressed is how to perform goodness-of-fit and model validation for this joint model. Though the class of models is very general, there are still possibilities of model mis-specifications, such as, for example, in determining the effective age processes, or in the specification of the ρ q (·)-functions. What are the impacts of such model mis-specifications? Do they lead to serious biases that could potentially result in misleading conclusions? These are some of the problems whose solutions await further studies.
A potential promise of this joint class of models is in precision medicine. Because all three components (RCR, LM, HS) are taken into account simultaneously, in contrast to a marginal modeling approach, the synergy that this joint model allows may improve decision-making -for example, in determining interventions to be performed for individual units. In this context, it is of utmost importance to be able to predict in the future the trajectories of the HS process given information at a given point in time about all three processes. Thus, an important problem to be dealt with in future work is the problem of forecasting using this joint model. How should such forecasting be implemented? This further leads to other important questions. One is determining the relative importance of each of the components in this prediction problem. Could one ignore other components and still do as well relative to a joint model-based prediction approach? If there are many covariates, how should the important covariates among these numerous covariates be chosen in order to improve prediction of, say, the time-to-absorption?
Finally, though our class of joint models is a natural extension of earlier models dealing with either recurrent events, competing recurrent events, longitudinal marker, and terminal events, one may impugn it as not realistic, but instead view it as more of a futuristic class of models, since existing data sets were not gathered in the manner for which these joint models apply. For instance, in the example pertaining to gout in Section 2, the SUR level and CKD status are not continuously monitored. However, with the advent of smart devices, such as smart wrist watches, embedded sensors, black boxes, etc., made possible by miniaturized technology, high-speed computing, almost limitless cloud-based memory capacity, and availability of rich cloud-based databases, the era is, in our opinion, fast approaching when continuous monitoring of longitudinal markers, health status, occurrences of different types of recurrent events, be it on a human being, an experimental animal or plant, a machine such as an airplane or car, an engineering system, a business entity, etc. will become more of a standard rather than an exception. By developing the models and methods of analysis for such future complex and advanced data sets, even before they become available and real, will hasten and prepare us for their eventual arrival.
depict the time-evolution for two distinct units, where Q = 3, W = {w 1 , w 2 , w 3 }, arXiv:2103.12903v2 [stat.ME] 29 Jan 2022
Figure 1
1Figure 1: Realized data observables from two distinct study units. The first plot panel is for a unit which was rightcensored prior to reaching an absorbing state, while the second plot panel is for a unit which reached an absorbing state prior to getting right-censored.
For the HS process, let V = {V (s) : s ≥ 0}, where V (s) takes values in the finite state space V = V 0 V 1 , where states in V 0 are absorbing states, and with |V 0 | > 0. V (s) is the state occupied by the HS process at time s, so that if V (s) ∈ V 0 then V (s ) = V (s) for all s > s. Similar to the LM process, let
and is such that the vector process M = N − A = {M (s) = N (s) − A(s) : s ≥ 0} is a zero-mean square-integrable martingale process with PQV matrix process M, M . The vectors A and M are actually partitioned into three vector components reflecting the RCR, LM, and HS components, according to
Figure 2 :
2Simulated sample data with n = 50 units. Panel 1: Covariate values of the n = 50 units. Values of X 2 are indicated by rectangles if X 1 = 0, while values of X 1 are indicated by circles if X 1 = 1.
For.
risk q, the effective age process function is E iq (s) = s − S R iqN R iq (s−) , the backward recurrence time for this risk. For the effects of the accumulating event occurrences,ρ q (N R i (s−); α q ) = (α q ) log(1+N R iq (s−)). For the HS component, V = {1, 2, 3} with state '1' an absorbing state, so γ is a scalar. For the LM component, W = {1 = High, 2 = Normal, 3 = Low}, so κ is a two-dimensional vector. The infinitesimal generator matrices η for the LM process and ξ for the HS process are, respectively,The values in the first row for the ξ-matrix are all zeros because state 1 in HS is absorbing. The true values of the remaining model parameters are given in the second column of
Figure 3 :
3The true (red, dashed) and estimated (blue, solid) baseline (crude) survivor functions for each of the three risks based on the simulated illustrative sample data set with n = 50 units depicted inFigure 2.
Figure 4 :
4Plots of sample (Pearson) correlations over a finite set of time points in [0, 3]. The random vector, at each time s and for each sample unit, is C(s) = [I(V i (s) = 2), I(W i (s) = 2), I(
Figure 5 :
5Boxplots of the centered parameter estimates from Mreps replications for simulated data sets each with n = 50 units. Centering is done by subtracting the true parameter value from each of the Mreps estimates.
Figure 6 :
6Overlaid plots of the true baseline survivor function (in red), ten simulated estimates of the baseline survivor function (in green), and the mean baseline survivor function based on Mreps simulations (in blue) for each of the three risks in the RCR component.
mg/dL ≤ SUR ≤ 6.0 mg/dL for females). The SUR level could be considered a longitudinal marker. Kidneys are associated with the excretion of uric acid in the body. Thus, chronic kidney disease (CKD) impacts the level of uric acid in the body, hence the occurrence of gout. The ordinal stages of CKD, based on the value of the Glomerular Filtration Rate (GFR), are as follows: Stage 1 (Normal) if GFR ≥ 90 mL/min; Stage 2 (Mild CKD) if 60 mL/min ≤ GFR ≤ 89 mL/min; Stage 3A (Moderate CKD) if 45 mL/min ≤ GFR ≤ 59 mL/min; Stage 3B (Moderate CKD) if 30 mL/min ≤ GFR ≤ 44; Stage 4 (Severe CKD) if 15 mL/min ≤ GFR ≤ 29 mL/min; and Stage 5 (End Stage CKD) if GRF ≤ 14 mL/min. The state of Stage 5 (End Stage CKD) can be viewed as an absorbing state. The CKD status could be viewed as the "health status" of the person.The
level of uric acid is measured by the Serum Urate Level (SUR), which can be categorized as Hyperuricemia (if
SUR > 7.2 mg/dL for males; if SUR > 6.0 mg/dL for females), or Normal (if 3.5 mg/dL ≤ SUR ≤ 7.2 mg/dL
for males; if 2.6
19 Example: Consider a vaccine trial where n subjects are randomized into different vaccine groups, including a no vaccine group. Group membership could be coded using dummy covariates. Each subject is monitored over an observation period, until loss to follow-up, or until death. Aside from the group membership, other covariates (e.g., age or age-group, gender, BMI, race, pre-existing conditions such whether immunocompromised or not, political affiliation, religious affiliation, etc.) for each subject are also observed. A longitudinal marker for each subject could be the viral load, categorized into none, low, medium, or high. See[31] for other examples of longitudinal medical markers observed in Covid-19 studies. The health status for each subject could be classified into free of Covid-19, moderately infected, severely infected, or dead. Competing recurrent events could be the occurrence of abdominal problems, severe coughing, body temperature reaching 103 degrees Fahrenheit. Possible goals of the study are to compare the different vaccine groups in terms of preventing Covid-19 infection; with respect to the mean or median time to absorption; or with respect to mean or median time to transitioning out of infection state given Covid-19 infection.Denote by (Ω, F, P) the basic filtered probability space with filtration F = {F s : s ≥ 0} where all random entities under consideration are defined. We begin by describing the joint model for the data observable components for one unit.3 Joint Model of RCR, LM, and HS Processes
3.1 Data Observables for One Unit
the jump at time s of N R q . For the LM process, let W = {W (s) : s ≥ 0}, where W (s) takes values in a finite state space W with cardinality |W|. W (s) represents the state of the longitudinal marker at time s. The sample path s → W (s) is a step-function which is right-continuous with left-hand limits. By introducing the index set I W
Proposition 5.1. Under certain regularity conditions, we have, as n → ∞,
Table 1 .
1Parameter True Estimate Est. Standard Error
α 1
1.50
1.58
0.13
α 2
1.20
1.16
0.15
α 3
2.00
2.10
0.21
β R
1
1.00
1.17
0.09
β R
2
-1.00
-0.94
0.10
γ R
1
1.00
1.17
0.09
κ R
1
1.00
1.10
0.09
κ R
2
-1.00
-0.68
0.10
β W
1
1.00
1.02
0.28
β W
2
-1.00
-0.78
0.28
γ W
1
1.00
0.97
0.27
ν W
1
1.00
0.82
0.06
ν W
2
1.00
1.11
0.09
ν W
3
-2.00
-2.01
0.09
β V
1
1.00
0.93
0.17
β V
2
-1.00
-1.00
0.16
κ V
1
1.00
1.00
0.20
κ V
2
-1.00
-1.59
0.40
ν V
1
1.00
1.24
0.04
ν V
2
1.00
1.04
0.06
ν V
3
-2.00
-2.30
0.06
Table 3 :
3Summary statistics of the parameter estimates for the Mreps replications in the simulation runs for sample sizes n = {50, 100}. The columns are the true values of the model parameters, the sample mean of the estimates, the sample standard deviations, the asymptotic standard errors, the 2.5% percentile, and the 97.5% percentile. The sample standard deviations are estimates of the standard errors of the estimators. The asymptotic standard errors are the means of the asymptotic standard errors over the Mreps replications. The RCR component includes model parameters Λs, αs and those parameters with superscript R. The LM component includes model parameters ηs and those parameters with superscript W . The HS component includes model parameters ξs and those parameters with superscript V . ] for the covariates, and [κ R 1 , κ R 2 ] for the LM state. The unknown model parameters in the LM component of the model are β W for the covariates and ν W 1 for the recurrent event process. The unknown parameters in the HS component of the model includes β V for the covariates, [κ V 1 , κ V 2 ] for the LM state, and ν V 1 for the recurrent event counting process.β R = [β R
1 , β R
2 , β R
3
Parameter Estimate Est. Standard Error p-valueα
0.77
0.22
0.30
β R
1
-0.16
0.20
0.43
β R
2
-0.42
0.21
0.05
β R
3
0.88
0.41
0.03
κ R
1
-0.52
0.27
0.05
κ R
2
-0.25
0.25
0.31
β W
1
0.08
0.25
0.75
β W
2
-0.00
0.25
0.99
β W
3
-0.51
0.73
0.48
ν W
1
-0.23
0.18
0.19
β V
1
0.49
0.30
0.10
β V
2
-0.11
0.30
0.71
β V
3
1.20
0.41
0.00
κ V
1
-0.43
0.31
0.16
κ V
2
-0.18
0.33
0.59
ν V
1
0.71
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| [] |
[
"An original unified approach for the description of phase transformations in steel during cooling: first application to binary Fe-C",
"An original unified approach for the description of phase transformations in steel during cooling: first application to binary Fe-C"
] | [
"Olivier Bouaziz \nLaboratoire d'Étude des Microstructures et de Mécanique des Matériaux (LEM3)\nCNRS\nUniversité de Lorraine\nArts et Métier Paris TechF 57000MetzFrance\n\nIntroduction\n\n"
] | [
"Laboratoire d'Étude des Microstructures et de Mécanique des Matériaux (LEM3)\nCNRS\nUniversité de Lorraine\nArts et Métier Paris TechF 57000MetzFrance",
"Introduction\n"
] | [] | Exploiting Landau's theory of phase transformations, defining an original order parameter and using the phenomenological transformation temperatures, it is reported that it is possible to describe in a global approach the conditions for the formation of each constituent (ferrite, bainite, martensite) from austenite during cooling in steel. It allowed to propose a new rigorous classification of the different thermodynamic conditions controlling each phase transformation. In a second step, the approach predicts naturally the effect of cooling rate on the bainite start temperature. Finally, perspectives are assessed to extend the approach in order to take into account the effect of an external field such as applied stress. | 10.1051/metal/2019037 | [
"https://www.metallurgical-research.org/articles/metal/pdf/2019/06/metal180248.pdf"
] | 118,510,448 | 1202.6194 | f9b2b426c1d753bf0e79d4cd60e35ec8d8513ee1 |
An original unified approach for the description of phase transformations in steel during cooling: first application to binary Fe-C
Olivier Bouaziz
Laboratoire d'Étude des Microstructures et de Mécanique des Matériaux (LEM3)
CNRS
Université de Lorraine
Arts et Métier Paris TechF 57000MetzFrance
Introduction
An original unified approach for the description of phase transformations in steel during cooling: first application to binary Fe-C
10.1051/metal/2019037Received: 7 September 2018 / Accepted: 9 August 2019REGULAR ARTICLEphase transformation
Exploiting Landau's theory of phase transformations, defining an original order parameter and using the phenomenological transformation temperatures, it is reported that it is possible to describe in a global approach the conditions for the formation of each constituent (ferrite, bainite, martensite) from austenite during cooling in steel. It allowed to propose a new rigorous classification of the different thermodynamic conditions controlling each phase transformation. In a second step, the approach predicts naturally the effect of cooling rate on the bainite start temperature. Finally, perspectives are assessed to extend the approach in order to take into account the effect of an external field such as applied stress.
Introduction
In the field of solid-state phase transformation in metallic alloys [1], the transformation of austenite in steel on cooling can occur by a variety of mechanisms including the formation of ferrite, bainite and martensite. The bainitic transformation occurs in a range between purely diffusional transformation to ferrite or pearlite and low temperature transformation to martensite by a displacive mechanism. Thus, the bainite transformation exhibits features of both diffusional and displacive transformations and has given rise to a large amount of research activity (see [2,3] for reviews). A major part of the research has concerned modelling of the kinetics of the transformations by detailed descriptions of the thermodynamic conditions operating at the interface [4][5][6][7]. However, the crucial understanding of the physically based conditions of the start of each phase transformation is less understood, especially for bainite.
Usually the Ar 3 temperature is defined as the maximum temperature for any phase transformation of austenite to ferrite during cooling [8]:
Ar 3 o C À Á ¼ 910 À 230C À 21Mn À 15Ni þ 45Si þ 32Mo:ð1Þ
For temperature lower than Ar 3 , the basical tools of physical metallurgy are the definition of the martensite start temperature M s and the bainite start temperature B s expressed phenomenologically as functions of chemical composition as:
M s o C À Á ¼ 539 À 423C À 30:4Mn À 17:7Ni À 12:1:Cr À 11Si;ð2Þ
where alloying element are expressed in weight(%) [9], and [10]:
B s o C À Á ¼ 870 À 270C À 90Mn À 37Ni À 70Cr À 83Mo:ð3Þ
In addition, another characteristic temperature is defined as the temperature where austenite and ferrite has the same thermo-chemical free energy determined as [2,11]:
T 0 o C À Á ¼ 835 À 198:C À 91Mn À 36Ni À 73Cr À 15Si À 87Mo:ð4Þ
In this publication, it is showed that it is possible to describe in a global approach the conditions for each phase transformation exploiting completely Landau's theory [12][13][14][15] of phase transformations including an original order parameter and to propose a new classification of the different phase transformation in steel during cooling and to predict naturally the effect of cooling rate on bainite start temperature.
The proposed approach
In the framework of Landau's theory [12][13][14], for a second order phase transition, the free energy is expressed as:
F x;T Þ ¼ F 0 T ð Þ þ A T ð Þ:x 2 þ C:x 4 ; Àð5Þ
where T is the temperature, x the order parameter, F 0 (T) the thermo-chemical free energy, A(T) a function of temperature and C a positive constant. The simplest expression for A(T) is:
A T ð Þ ¼ A 0 T À T c ð Þ;ð6Þ
with A 0 a positive constant and T c a critical temperature where A(T) changes of sign. Usually in phase transformation of steels, F 0 (T) is known [2,3,7]. So, the identification of the total free energy F (x,T) requires the determination of three parameters: A 0 , T c , C.
For T > T c , F (x,T) exhibits one minimum as a function of x:
∂F x; T ð Þ ∂x ¼ 0; ð7Þ for x = 0
For T T c , F (x,T) exhibits two minima as a function of x:
∂F x; T ð Þ ∂x ¼ 0; ð8Þ for x ¼ ± ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 0 T c À T ð Þ 2:C r :ð9Þ
In order to exploit this approach to a classification of phase transformations in steel, it is now assumed that:
T c ¼ B s ;ð10Þ
where B s is the bainite start temperature. By convention, it is chosen to have an order parameter for martensite:
x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 0 B s À M s ð Þ 2:C r ¼ 1:ð11Þ
Giving a first relationship:
A 0 2:C ¼ 1 B s À M s :ð12Þ
Therefore, the order parameter is:
x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B s À T B s À M s r :ð13Þ
As B s to M s temperatures depends on the carbon content, this one controls the order parameter.
In order to illustrate quantitatively this law, the evolution of the order parameter from B s to M s temperatures for two different Fe-C compositions has been plotted in Figure 1.
It is now proposed to define clearly what could be the parameter of order for phase transformation in steels. If Cf is the carbon content of the phase appearing, it is reasonable to propose that:
x ¼ C À C a;eq C g À C a;eq ;ð14Þ
Where C a;eq is the solubility of carbon in ferrite at equilibrium and C g is the carbon in austenite. So, the order parameter can be a sursaturation in carbon in the binary Fe-C system. In addition, chemical free energy at M s has been determined as [14]:
F 0 M s ; T ð Þ¼F 0 M s ð ÞþS 0 : T À M s ð Þ ;ð15Þ
with F 0 (M s ) = 1250 J/mol and S 0 = À6.8 J.mol À1 .K À1 , which are independent of the composition. The energy value F 0 (M s ) should correspond to the maximum at M s for x = 0. As for any T < B s this maximum exists for the same value of x = 0, it is written for M s T < B s :
F 0; T ð Þ ¼ F 0 T À M s ð ÞþF 0 M s ð Þ;ð16Þ
with when T is near M s but it can be completely different for higher temperature.
F 0 T À M s ð Þ¼S 0 : T À M s ð Þ ;ð17Þ
In order to determine A 0 and C, as the total free energy at M s for x = 1 has to be zero, it comes:
F 0 M s ð Þþ A 0 M s À B s ð ÞþC ¼ 0; ð18Þ or C ¼ 2C À F 0 M s ð Þ;ð19Þ
and
C ¼ F 0 M s ð Þ:ð20Þ
Consequently:
A 0 ¼ 2:F 0 M s ð Þ B s À M s :ð21Þ
Finally it is possible to express completely the free energy:
F x; T ð Þ ¼ F 0 T À M s ð ÞþF 0 M s ð Þ þ 2:F 0 M s ð Þ B s À M s T À B s ð Þ:x 2 þ F 0 M s ð Þ:x 4 ;ð22Þ
or
F x; T ð Þ ¼ F 0 T À M s ð ÞþF 0 M s ð Þ þ F 0 M s ð Þ:x 2 2: T À B s ð Þ B s À M s þ x 2 :ð23Þ
In order to highlight the key role of the order parameter, it has been drawn in Figure 2. The evolution of the rightend side term of the expression:
F x; T ð ÞÀF 0 T À M s ð Þ¼F 0 M s ð Þ þ F 0 M s ð Þ:x 2 2: T À B s ð Þ B s À M s þ x 2 :ð24Þ
Finally, the quantitative developed approach can be used in order to summarized the thermodynamic conditions for the formation of each phase, as it is summarized in Table 1, providing more rigorous occurrence criterion especially to distinguish ferrite, bainite or martensite formation.
Extended approach including cooling rate
It is well known that B s decreases if the cooling rate is increased. In order to capture this effect using the previous approach, it is assumed that a kinetic equation for the order parameter can be written for T < B s as:
∂x ∂t ¼ Àh: ∂F x; T ð Þ ∂T ;ð25Þ
where h is independent of x and of T, giving for T < B s :
∂x ∂t ¼ h:A 0 :x 2 :ð26Þ
Using equation (13), it is also possible to express:
∂x ∂T ¼ À ffiffiffiffiffiffiffiffiffiffiffi 1 B s ÀM s q 2: ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi B s À T p ¼ À 1 2: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B s À T ð Þ : B s À M s ð Þ p :ð28Þ
Therefore: It comes using equations (26) and (28):
dT dt ¼ ∂x ∂t . ∂x ∂T :ð27ÞFerrite ∂F x; T ð Þ ∂x ¼ 0 x ¼ 0 Bainite ∂F x; B s ð Þ dx ¼ 0 ∂ 2 F x; B s ð Þ ∂x 2 ¼ 0 T <B s ;x ¼ ffiffiffiffiffiffiffiffiffiffi ffi B s ÀT B s ÀM s q Martensite F x;M s Þ¼0 ð ∂F x; M s ð Þ ∂x ¼ 0 x ¼ 1 OdT dt ¼ À2:h:A 0 : B s À T ð Þ 3 = 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B s À M s p :ð30Þ
Combining with equation (21), we have:
dT dt ¼ À2:h:F 0 M s ð Þ: B s À T B s À M s 3 = 2 :ð32Þ
Thus, the temperature T respecting equation (32) is:
T ¼ B s À B s À M s ð Þ : À1 2:h:F 0 M s ð Þ dT dt 2 = 3 ;ð33Þ
with dT dt 0 during cooling. This temperature is the bainite start temperature affected by the cooling rate.
As T ≥ M s , equation (33) gives a critical cooling rate for T = M s :
dT dt ¼ À2:h:F 0 M s ð Þ:ð34Þ
In addition, a formula has been determined for the critical cooling rate in order to form 1% of martensite from austenite [16]:
Log ÀC r ð Þ¼4:5 À 2:7C À 0:95Mn À 0:18Si À 0:38Cr À 1:29ðC:CrÞ; ð35Þ
where C r is expressed in°C/s. Moreover, using equation (34) and (35), it is consistent to impose that:
2:h:F 0 M s Þ ¼ C r ; ð ð 36Þ and h ¼ C r 2:F 0 M s Þ: ðð37Þ
Finally, the law predicting the evolution of the bainite start temperature as a function of cooling rate is:
T ¼B s ÀðB s ÀM s Þ: À1 C r : dT dt 2 3 :ð38Þ
In order to illustrate quantitatively this law, the evolution of the bainite start temperature as a function of cooling rate for two different Fe-C compositions has been plotted in Figure 3.
Conclusions
Exploiting Landau's theory of phase transformations, defining an original order parameter and using the phenomenological transformation temperatures, it is reported that it is possible to describe in a global approach the conditions for the formation of each phase (ferrite, bainite, martensite) from austenite during cooling in steel. It allowed to propose a new rigorous classification of the different thermodynamic conditions controlling each phase transformation. In a second step, the approach predicts naturally the effect of cooling rate on the bainite start temperature. In perspectives, the approach can be extended to take into account external fields by adding a linear term in free energy linearly proportional to the order of parameter and proportional to the potential energy of the external field [12][13][14]. For instance, in the case of an uniaxial applied stress s, the contribution to the free energy is expressed as:
F x;sÞ ð Þ¼ ± s 2 2:E :x;ð39Þ
where E is the elastic modulus and the sign depends on the compressive or tensile stress. This term breaks the symmetry of the total free energy as a function of the parameter of order and it can change the occurrence conditions of each phase. A lot of experimental data are available is the literature in order to validate this last point which will be investigated in a next future.
Fig. 1 .
1Evolution of the order parameter from B s to M s temperatures for two different Fe-C compositions.
Fig. 2 .
2Evolution of the F x;T ÞÀF 0 T ÀM s Þ ð ðas a function of the order parameter at B s and at M s temperatures.
Table 1 .
1Classification of phase transformation conditions from austenite during cooling.Phase
Free energy
Order parameter
O. Bouaziz: Metall. Res.Technol. 116, 615 (2019)
Acknowledgement. The author thanks Dr. F. Levy for stimulating discussions.
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Alloy phase diagrams. M Hillert, Paraequilibrium and other restricted equilibria. L.H. Bennet, T.B. Massalski, B.C. Giessen19295M. Hillert, Paraequilibrium and other restricted equilibria, in: L.H. Bennet, T.B. Massalski, B.C. Giessen (Eds), Alloy phase diagrams, Mater. Res. Soc. Symp. Proc. 19, 295 (1983).
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Cite this article as: Olivier Bouaziz, An original unified approach for the description of phase transformations in steel during cooling: first application to binary Fe-C. Metall. Res. Technol. 116615Cite this article as: Olivier Bouaziz, An original unified approach for the description of phase transformations in steel during cooling: first application to binary Fe-C, Metall. Res. Technol. 116, 615 (2019)
| [] |
[
"SOME SUPPORT PROPERTIES FOR A CLASS OF Λ-FLEMING-VIOT PROCESSES",
"SOME SUPPORT PROPERTIES FOR A CLASS OF Λ-FLEMING-VIOT PROCESSES"
] | [
"Huili Liu ",
"Xiaowen Zhou "
] | [] | [] | For a class of Λ-Fleming-Viot processes with underlying Brownian motion whose associated Λ-coalescents come down from infinity, we prove a one-sided modulus of continuity result for their ancestry processes recovered from the lookdown construction of Donnelly and Kurtz. As applications, we first show that such a Λ-Fleming-Viot support process has one-sided modulus of continuity (with modulus function C t log (1/t)) at any fixed time. We also show that the support is compact simultaneously at all positive times, and given the initial compactness, its range is uniformly compact over any finite time interval. In addition, under a mild condition on the Λ-coalescence rates, we find a uniform upper bound on Hausdorff dimension of the support and an upper bound on Hausdorff dimension of the range. | 10.1214/13-aihp598 | [
"https://arxiv.org/pdf/1307.3990v2.pdf"
] | 119,504,523 | 1307.3990 | 297335d2db56b55c345615644b6c2f5ca567a21a |
SOME SUPPORT PROPERTIES FOR A CLASS OF Λ-FLEMING-VIOT PROCESSES
20 Sep 2013
Huili Liu
Xiaowen Zhou
SOME SUPPORT PROPERTIES FOR A CLASS OF Λ-FLEMING-VIOT PROCESSES
20 Sep 2013arXiv:1307.3990v2 [math.PR]
For a class of Λ-Fleming-Viot processes with underlying Brownian motion whose associated Λ-coalescents come down from infinity, we prove a one-sided modulus of continuity result for their ancestry processes recovered from the lookdown construction of Donnelly and Kurtz. As applications, we first show that such a Λ-Fleming-Viot support process has one-sided modulus of continuity (with modulus function C t log (1/t)) at any fixed time. We also show that the support is compact simultaneously at all positive times, and given the initial compactness, its range is uniformly compact over any finite time interval. In addition, under a mild condition on the Λ-coalescence rates, we find a uniform upper bound on Hausdorff dimension of the support and an upper bound on Hausdorff dimension of the range.
Introduction
Fleming-Viot process arises as a probability-measure-valued stochastic process on the distribution of allelic frequencies in a selectively neutral population with mutation. We refer to Ethier and Kurtz [17] and Etheridge [18] for surveys on the Fleming-Viot process and related mathematical models from population genetics.
Moments of the classical Fleming-Viot process can be expressed in terms of a dual process involving Kingman's coalescent and semigroup for the mutation operator. The Λ-Fleming-Viot process generalizes the classical Fleming-Viot process by replacing Kingman's coalescent with the Λ-coalescent allowing multiple collisions. Formally, the Λ-Fleming-Viot process is a Fleming-Viot process with general reproduction mechanism so that the total number of children from a parent can be comparable to the size of population. We refer to Birkner et al. [5] for a connection between mutationless Λ-Fleming-Viot processes and continuous state branching processes. In this paper we only consider the Fleming-Viot process with Brownian mutation that can also be interpreted as underlying spatial Brownian motion.
The support properties are interesting in the study of measure-valued processes. For the Dawson-Watanabe superBrownian motion arising as high density limit of empirical measures for near critical branching Brownian motions, the modulus of continuity and the carrying dimensions have been studied systematically for its support process. We refer to Chapter 7 of Dawson [7], Chapter 9 of Dawson [8] and Chapter III of Perkins [21] and references therein for a collection of these results. The proofs involve the historically cluster representation, the Palm distribution for the canonical measure and estimates obtained from PDE associated with the Laplace functional. For a superBrownian motion with a general branching mechanism, Delmas [13] obtained the Hausdorff dimension for its range using Brownian snake representation with subordination.
However, the approaches for Dawson-Watanabe superBrownian motions do not always apply to Fleming-Viot processes which are not infinitely divisible. Consequently, there are only a few results available for Fleming-Viot support processes so far. The earliest work on the compact support property for classical Fleming-Viot processes is due to Dawson and Hochberg [9]. It was shown in [9] that at any fixed time T > 0 the classical Fleming-Viot process with underlying Brownian motion has a compact support with Hausdorff dimension not greater than two. Using non-standard techniques Reimers [23] improved the above result by showing that the carrying dimension of the support is at most two simultaneously for all positive times. Applying a generalized Perkins disintegration theorem, the support dimension was found in Ruscher [24] for a Fleming-Viot-like process obtained from mass normalization and time change of superBrownian motion with stable branching. The Λ-Fleming-Viot process does not have a compact support if the associated Λ-coalescent does not come down from infinity. Liu and Zhou [20] recently extended the results in [9] to a class of Λ-Fleming-Viot processes whose associated Λ-coalescents come down from infinity. We are not aware of any results on the modulus of continuity for Fleming-Viot support processes although the modulus of continuity for superBrownian motion support had been first recovered by Dawson et al. [11] more than twenty years ago and further studied in Dawson and Vinogradov [12] and in Dawson el al. [10].
The lookdown construction of Donnelly and Kurtz [14,15,16] is a powerful technique in the study of the Fleming-Viot processes. Loosely speaking, the idea of lookdown construction is a discrete representation that leads to a nice version of the corresponding measure-valued process. The lookdown construction naturally results in a genealogy process describing the genealogical structure of the particles involved. In a sense it plays the role of historical processes for Dawson-Watanabe superprocesses.
Donnelly and Kurtz [14] first proposed the lookdown construction of a system of countable particles embedded into the classical Fleming-Viot process. They showed the duality between classical Fleming-Viot process and Kingman's coalescent and recovered some previous results on the classical Fleming-Viot process using this explicit representation. This representation was later extended in Donnelly and Kurtz [16] via a modified lookdown construction to a larger class of measure-valued processes including the Λ-Fleming-Viot processes and the Dawson-Watanabe superprocesses. Donnelly and Kurtz [15] also found a discrete representation for the classical Fleming-Viot models with selection and recombination.
Birkner and Blath [4] further discussed the modified lookdown construction in [16] for the Λ-Fleming-Viot process with jump type mutation operator. They also described how to recover the Λ-coalescent from the modified lookdown construction.
For the Ξ-coalescent allowing simultaneous multiple collisions, a Poisson point process construction of the Ξ-lookdown model can be found in Birkner et al. [6] by extending the modified lookdown construction of Donnelly and Kurtz [16]. It was proved in [6] that the empirical measure of the exchangeable particles converges almost surely in the Skorohod space of measure-valued paths to the so called Ξ-Fleming-Viot process with jump type mutation.
Using the modified lookdown construction of Donnelly and Kurtz, Liu and Zhou [20] proved that a class of Λ-Fleming-Viot processes with underlying Brownian motion have compact supports at any fixed time T > 0 provided the associated Λ-coalescents come down from infinity fast enough. Further, both lower and upper bounds were found in [20] on Hausdorff dimension for support of the Λ-Fleming-Viot process at the time T , where the exact Hausdorff dimension was shown to be two whenever the associated Λcoalescent has a nontrivial Kingman component. These results generalize the previous results of Dawson and Hochberg [9] on the classical Fleming-Viot processes.
In this paper, for the class of Λ-Fleming-Viot processes in [20], we refine the arguments in [20] to further study their support properties. Our first result is a one-sided modulus of continuity type result for the ancestry process defined via the lookdown construction. The second result is the one-sided modulus of continuity for the Λ-Fleming-Viot support process at any fixed time. The third is on the uniform compactness of the Λ-Fleming-Viot support and the associated range. Under an additional mild condition on the coalescence rates of the corresponding Λ-coalescent, we also obtain two results on support dimensions. One result is an uniform upper bound on Hausdorff dimension for the support at all positive times. The other is an upper bound on Hausdorff dimension for the range of the Λ-Fleming-Viot support process. Again, the lookdown construction plays a key role throughout our arguments.
The paper is arranged as follows. In Section 2 we introduce the Λ-coalescent and the corresponding coming down from infinity property. In Section 3 we briefly discuss the lookdown construction for Λ-Fleming-Viot process with underlying Brownian motion and the associated ancestry process recovered from the lookdown construction. In Section 4 we present the main results of this paper together with corollaries and propositions. Proofs of the main results are deferred to Section 5.
2. The Λ-coalescent 2.1. The Λ-coalescent. We first introduce some notation. Put [n] ≡ {1, . . . , n} and [∞] ≡ {1, 2, . . .}. An ordered partition of D ⊂ [∞] is a countable collection π = {π i , i = 1, 2, . . .} of disjoint blocks such that ∪ i π i = D and min π i < min π j for i < j. Then blocks in π are ordered by their least elements.
Denote by P n the set of ordered partitions of and π ∈ P ∞ , let R n (π) ∈ P n be the restriction of π to [n].
Kingman's coalescent is a P ∞ -valued time homogeneous Markov process such that all different pairs of blocks independently merge at the same rate. Pitman [22] and Sagitov [25] generalized the Kingman's coalescent to the Λ-coalescent which allows multiple collisions, i.e., more than two blocks may merge at a time. The Λ-coalescent is defined as a P ∞ -valued Markov process Π ≡ (Π(t)) t≥0 such that for each n ∈ [∞], its restriction to [n], Π n ≡ (Π n (t)) t≥0 is a P n -valued Markov process whose transition rates are described as follows: if there are currently b blocks in the partition, then each k-tuple of blocks (2 ≤ k ≤ b) independently merges to form a single block at rate
λ b,k = [0,1] x k−2 (1 − x) b−k Λ(dx),(1)
where Λ is a finite measure on [0, 1]. It is easy to check that the rates (λ b,k ) are consistent so that for all 2 ≤ k ≤ b,
λ b,k = λ b+1,k + λ b+1,k+1 .(2)
Consequently, for any 1 ≤ m < n ≤ ∞, the coalescent process R m (Π n (t)) given Π n (0) = π n has the same distribution as Π m (t) given Π m (0) = R m (π n ).
With the transition rates determined by (1), there exists a one to one correspondence between Λ-coalescents and finite measures Λ on [0, 1].
For n = 2, 3, . . ., denote by
(3) λ n = n k=2
n k λ n,k the total coalescence rate starting with n blocks. It is clear that (λ n ) n≥2 is an increasing sequence, i.e., λ n ≤ λ n+1 for any n ≥ 2. In addition, denote by
γ n = n k=2 (k − 1) n k λ n,k
the rate at which the number of blocks decreases.
Coming down from infinity. Given any
Λ-coalescent Π ≡ (Π(t)) t≥0 with Π(0) = 0 [∞]
, let #Π(t) be the number of blocks in the partition Π(t). The Λ-coalescent Π comes down from infinity if P (#Π(t) < ∞) = 1 for all t > 0 and it stays infinite if P (#Π(t) = ∞) = 1 for all t > 0. Suppose that the measure Λ has no atom at 1. It is shown by Schweinsberg [26] that
• the Λ-coalescent comes down from infinity if and only if ∞ n=2 γ −1 n < ∞; • the Λ-coalescent stays infinite if and only if ∞ n=2 γ −1 n = ∞. It is pointed out in Bertoin and Le Gall [3] that for
ψ(q) = [0,1] (e −qx − 1 + qx)x −2 Λ(dx), ∞ n=2 γ −1 n < ∞ if and only if ∞ a 1 ψ(q) dq < ∞,
where the integral is finite for some (and then for all) a > 0.
Example 2.1. In case of Λ = δ 0 , the corresponding coalescent is Kingman's coalescent and comes down from infinity.
Example 2.2. If β ∈ (0, 2) and
Λ(dx) = Γ(2) Γ(2 − β)Γ(β) x 1−β (1 − x) β−1 dx,
the corresponding coalescent is Beta(2 − β, β)-coalescent.
• In case of β ∈ (0, 1], it stays infinite.
• In case of β ∈ (1, 2), it comes down from infinity.
3. The Λ-Fleming-Viot process and its lookdown construction
In this section, we first discuss the lookdown construction of Λ-Fleming-Viot process with underlying Brownian motion. Then we explain how to recover the Λ-coalescent from the lookdown construction. Finally, we introduce the ancestry process for the Λ-Fleming-Viot process from the lookdown construction.
3.1. Lookdown construction of Λ-Fleming-Viot process with underlying Brownian motion. Donnelly and Kurtz [16] introduced a modified lookdown construction with the empirical measure process converging to measure-valued stochastic process. A key advantage of the lookdown construction is its projective property. Intuitively, in the lookdown model each particle is attached a "level" from the set {1, 2, . . .}. The evolution of a particle at level n only depends on the evolution of the finite particles at lower levels. This property allows us to construct approximating particle systems, and their limit as n → ∞ in the same probability space.
Following Birkner and Blath [4], we now give a brief introduction on the modified lookdown construction of the Λ-Fleming-Viot process with underlying Brownian motion.
Let (X 1 (t), X 2 (t), X 3 (t), . . .) be an (R d ) ∞ -valued random variable, where for any i ∈ [∞]
, X i (t) represents the spatial location of the particle at level i. We require the initial values {X i (0), i ∈ [∞]} to be exchangeable random variables so that the limiting empirical measure
lim n→∞ 1 n n i=1 δ X i (0)
exists almost surely by de Finetti's theorem.
Let Λ be the finite measure associated to the Λ-coalescent. The reproduction in the particle system consists of two kinds of birth events: the events of single birth determined by measure Λ({0})δ 0 and the events of multiple births determined by measure Λ restricted to (0, 1] that is denoted by Λ 0 .
To describe the evolution of the system during events of single birth, let {N ij (t) : 1 ≤ i < j < ∞} be independent Poisson processes with common rate Λ({0}). At a jump time t of N ij , the particle at level j looks down at the particle at level i and assumes its location (therefore, particle at level i gives birth to a new particle). Values of particles at levels above j are shifted accordingly, i.e., for ∆N ij (t) = 1, we have
X k (t) = X k (t−), if k < j, X i (t−), if k = j, X k−1 (t−), if k > j.(4)
For those events of multiple births we can construct an independent Poisson point processÑ on
R + × (0, 1] with intensity measure dt ⊗ x −2 Λ 0 (dx). Let {U ij , i, j ∈ [∞]} be i.i.d. uniform [0, 1] random variables. Jump points {(t i , x i )} forÑ correspond to the multiple birth events. For t ≥ 0 and J ⊂ [n] with |J| ≥ 2, define N n J (t) ≡ i:t i ≤t j∈J 1 {U ij ≤x i } j∈[n]\J 1 {U ij >x i } .(5)
Then N n J (t) counts the number of birth events among the particles from levels {1, 2, . . . , n} such that exactly those at levels in J are involved up to time t. Intuitively, at a jump time t i , a uniform coin is tossed independently for each level. All the particles at levels j with U ij ≤ x i participate in the lookdown event. More precisely, those particles involved jump to the location of the particle at the lowest level involved. The spatial locations of particles on the other levels, keeping their original order, are shifted upwards accordingly, i.e., if t = t i is the jump time and j is the lowest level involved, then
X k (t) = X k (t−), for k ≤ j, X j (t−), for k > j with U ik ≤ x i , X k−J k t (t−), otherwise, where J k t i ≡ #{m < k, U im ≤ x i } − 1.
Between jump times of the Poisson processes, particles at different levels move independently according to Brownian motions in R d .
We assume that the above-mentioned lookdown construction is carried out in a probability space (Ω, F, P) .
For each t > 0, X 1 (t), X 2 (t), . . . are known to be exchangeable random variables so that
X(t) ≡ lim n→∞ X (n) (t) ≡ lim n→∞ 1 n n i=1 δ X i (t)
exists almost surely by de Finetti's theorem and follows the probability law of the Λ-Fleming-Viot process with underlying Brownian motion. Further, we have that X (n) converges to X in the path space D M 1( R d ) ([0, ∞)) equipped with the Skorohod topology, where M 1 R d denotes the space of probability measures on R d equipped with the topology of weak convergence. See Theorem 3.2 of [16].
In the sequel we always write X for such a Λ-Fleming-Viot process. Write supp µ for the closed support of measure µ. Lemma 3.1. For any t ≥ 0, P-a.s. the spatial locations of the countably many particles in the lookdown construction satisfy
{X 1 (t), X 2 (t), X 3 (t), . . .} ⊆ supp X(t).
Proof. In the lookdown construction, (X n (t)) n≥1 are exchangeable at any time t ≥ 0. By de Finetti's theorem (cf. Aldous [1]) such a system is a mixture of i.i.d. sequence, i.e., given the empirical measure
X(t) = lim n→∞ 1 n n i=1 δ X i (t) ,
the random variables {X i (t), i = 1, 2, . . .} are jointly distributed as i.i.d. samples from the directing measure X(t). Therefore, X n (t) ∈ supp X(t) for any n ∈ [∞].
3.2.
The Λ-coalescent in the lookdown construction. The birth events induce a family structure to the particle system so we can present the genealogy process first introduced in Donnelly and Kurtz [16]. For any 0 ≤ t ≤ s and n ∈ [∞], denote by L s n (t) the ancestor's level at time t for the particle with level n at time s. Given s and n, L s n (t) is nondecreasing and left continuous in t. Moreover, the genealogy processes (L s n ) s≥0 , n = 1, 2, . . . satisfy the equations
L s n (t) = n − 1≤i<j<n s t− 1 {L s n (u)>j} dN ij (u) − 1≤i<j≤n s t− (j − i) 1 {L s n (u)=j} dN ij (u) − J⊂[n] s t− (L s n (u) − min J) 1 {L s n (u)∈J} dN n J (u) − J⊂[n] s t− (|J ∩ {1, . . . , L s n (u)}| − 1) × 1 {L s n (u)>min J,L s n (u) ∈J} dN n J (u)
.
Given T > 0, for any 0 ≤ t ≤ T and i ∈ [∞], L T i (T − t) represents the ancestor's level at time T − t of the particle with level i at time T and X L T i (T −t) ((T − t)−) represents that ancestor's location.
Write Π T (t) 0≤t≤T for the P ∞ -valued process such that i and j belong to the same
block of Π T (t) if and only if L T i (T − t) = L T j (T − t),
i.e., i and j belong to the same block if and only if the two particles with levels i and j, respectively, at time T share a common ancestor at time T − t. The process Π T (t) 0≤t≤T turns out to have the same law as the Λ-coalescent running up to time T . See Donnelly and Kurtz [16] and Birkner and Blath [4].
The next property of the genealogy process can be found in Lemma 3.1 of [20].
Lemma 3.2. For any fixed T > 0, let Π T (t) 0≤t≤T be the Λ-coalescent recovered from the lookdown construction. Then given t ∈ [0, T ] and the ordered random partition Π T (t) = π l (t) : l = 1, . . . , #Π T (t) , we have L T j (T − t) = l for any j ∈ π l (t). 3.3. Ancestry process. For any T > 0, denote by (X 1,s , X 2,s , X 3,s , . . .) 0≤s≤T the ancestry process with X i,s defined by For any s ≥ 0, we can recover the Λ-coalescent (Π s (t)) 0≤t≤s from the lookdown construction. For any 0 ≤ r < s, set
(6) X i,s (t) ≡ X L s i (t) (t−) for 0 ≤ t ≤ s. Intuitively X i,N r,s ≡ #Π s (s − r)
and
Π s (s − r) ≡ {π l : 1 ≤ l ≤ N r,s },
where π l ≡ π l (r, s), 1 ≤ l ≤ N r,s are all the disjoint blocks of Π s (s − r) ordered by their least elements. Let H(r, s) be the maximal dislocation between the countably many particles at time s and their respective ancestors at time r. Applying Lemma 3.2, we have
H (r, s) ≡ max 1≤l≤N r,s max j∈π l X j (s) − X L s j (r) (r−) = max 1≤l≤N r,s max j∈π l |X j (s) − X l (r−)| .
4. Some properties of the Λ-Fleming-Viot process 4.1. Main results. For any T > 0, let (Π T (t)) 0≤t≤T be the Λ-coalescent recovered from the lookdown construction with Π T (0) = 0 [∞] . Write Π ≡ (Π(t)) t≥0 for the unique (in law) Λ-coalescent such that (Π(t)) 0≤t≤T has the same distribution as (Π T (t)) 0≤t≤T . We call Π the Λ-coalescent associated to the Λ-Fleming-Viot process X.
For any positive integer m, set
(7) T m ≡ inf t ≥ 0 : #Π(t) ≤ m with the convention inf ∅ = ∞. Given η > 0, for any Borel set A ⊂ R d , let B (A, η) be its closed η-neighborhood such that B (A, η) ≡ x∈A B (x, η),
where B (x, η) denotes the closed ball centered at x with radius η.
We now recall the definition of Hausdorff dimension. Given A ⊂ R d and β > 0, η > 0, let
H β η (A) ≡ inf {S l }∈ϕη l d (S l ) β ,
where d (S l ) denotes the diameter of ball S l in R d and ϕ η denotes the collection of ηcovers of set A by balls with diameters at most η. The Hausdorff β-measure of A is defined by
H β (A) = lim η→0 H β η (A) . The Hausdorff dimension of A is defined by dim A ≡ inf β > 0 : H β (A) = 0 = sup β > 0 : H β (A) = ∞ .
Recall that X is the Λ-Fleming-Viot process with underlying Brownian motion. For
any subset I ⊂ R ∩ [0, ∞), let R(I) ≡ ∪ t∈I supp X(t)
be the range of supp X on the time interval I.
Throughout the paper, we always write C or C with subscript for a positive constant and write C(x) for a constant depending on x whose values might vary from place to place. The main results of this paper are the following theorems. We defer the proofs to Section 5. Assumption I: There exists a constant α > 0 such that the associated Λ-coalescent Π satisfies lim sup m→∞ m α ET m < ∞.
[0, T ] satisfying 0 < s − r ≤ θ, we have H (r, s) ≤C (s − r) log (1/ (s − r)).(8)
Theorem 4.2. Under Assumption I and given any fixed t ≥ 0, there exist a positive random variable θ ≡ θ (t, d, α) < 1 and a constant C ≡ C(d, α) such that for any ∆t with 0 < ∆t ≤ θ we have P-a.s.
supp X (t + ∆t) ⊆ B supp X(t), C ∆t log (1/∆t) . (9) Theorem 4.3. Under Assumption I, supp X(t) is compact for all t > 0 P-a.s.. Further, if supp X(0) is compact, then R([0, t)) is compact for all t > 0 P-a.s.. Condition A: There exists a constant α > 0 such that the associated Λ-coalescent Π satisfies lim sup m→∞ m α ∞ b=m+1 λ b −1 < ∞. Remark 4.4. The Kingman's coalescent satisfies Condition A with α = 1. In case of β ∈ (1, 2), the Beta(2 − β, β)-coalescent satisfies Condition A with α = β − 1.
A sufficient condition.
Recall the Markov chain introduced in [20]. For any n, (Π n (t)) t≥0 is the Λ-coalescent Π restricted to [n] with Π n (0) = 0 [n] . For any n > m, the block counting process (#Π n (t) ∨ m) t≥0 is a Markov chain with initial value n and absorbing state m. For any n ≥ b > m, let (µ b,k ) m≤k≤b−1 be its transition rates such that
µ b,b−1 = b 2 λ b,2 , µ b,b−2 = b 3 λ b,3 , · · · · · · µ b,m+1 = b b−m λ b,b−m , µ b,m = b k=b−m+1 b k λ b,k .(10)
The total transition rate is
µ b = b−1 k=m µ b,k = b k=2 b k λ b,k = λ b .
For b > m, let γ b,m be the total rate at which the block counting Markov chain starting at b is decreasing, i.e.,
γ b,m = b−m k=2 (k − 1) b k λ b,k + b k=b−m+1 (b − m) b k λ b,k , if b ≥ m + 2, b k=2 b k λ b,k , if b = m + 1.(11)
Condition B: There exists a constant α > 0 such that
lim sup m→∞ m α ∞ b=m+1 γ b,m −1 < ∞.
Remark 4.7. It follows from the proof of Lemma 4.4 in [20] that
ET m ≤ ∞ b=m+1 γ b,m −1 .
Recalling the definitions of γ b,m by (11) and λ b by (3), we have λ b ≤ γ b,m for any b > m.
Then for any α > 0, we have
m α ET m ≤ m α ∞ b=m+1 γ b,m −1 ≤ m α ∞ b=m+1 λ b −1 .
Therefore, Condition A implies Condition B which is sufficient for Assumption I. Condition A is not a strong requirement since for the Beta coalescents Condition A is sufficient and necessary for coming down from infinity.
The speed of coming down from infinity for Λ-coalescent is discussed in Berestycki et al [2]. It is shown that there exists a deterministic function ν : (0, ∞)→(0, ∞) such that #Π(t)/ν(t)→1 as t→0 both almost surely and in L p for p ≥ 1. For our purpose, it is possible to replace Assumption I with an assumption on the behavior of ν(t) for t close to 0.
Some Corollaries and Propositions.
For t > 0, let
r(t) ≡ inf {R ≥ 0 : supp X (t) ⊆ B (0, R)} .
The next result is similar to Theorem 2.1 of Tribe [27] on the support process of superBrownian motion; also see Theorem 9.3.2.3 of Dawson [8]. It follows immediately from Theorem 4.2.
P δ 0 lim sup t↓0 sup 0≤u≤t r(u) t log (1/t) ≤ C = 1,
where P δ 0 denotes the law of X with X (0) = δ 0 .
Corollary 4.9. Suppose that Condition A holds. For any T > 0, we have
P δ 0 (dim R ([0, T )) ≤ 2 + 2/α) = 1.
We defer the proof of Corollary 4.9 to Section 5. The next result follows from the proof of Theorem 4.5 and a standard result of Hausdorff measure; see Lemma 6.3 of Falconer [19]. For any 0 < t < 1, let (12) h(t) ≡ t log (1/t). Proof. Since Λ({0}) > 0, the Λ-coalescent has a nontrivial Kingman component. Then
λ b ≥ 1 2 Λ({0})b(b − 1) and ∞ b=m+1 1 λ b ≤ ∞ b=m+1 2 Λ({0})b(b − 1) = 2 Λ({0})m ,
i.e., Condition A holds with α = 1. Therefore, the results follow from Remark 4.7 and Theorems 4.2-4.6.
Remark 4.12. The uniform upper bound on the Hausdorff dimension of classical Fleming-Viot support process was first proved by Reimers [23], where a non-standard construction of the classical Fleming-Viot process is used to establish this result.
Recall the (c, ǫ, γ)-property introduced in [20]. We say that a Λ-coalescent has the (c, ǫ, γ)-property, if there exist constants c > 0 and ǫ, γ ∈ (0, 1) such that the measure Λ restricted to [0, ǫ] is absolutely continuous with respect to Lebesgue measure and
Λ(dx) ≥ cx −γ dx for all x ∈ [0, ǫ].
The Λ-coalescents with the (c, ǫ, γ)-property come down from infinity. Proposition 4.13. Let X be any Λ-Fleming-Viot process with underlying Brownian motion in R d for d ≥ 2. If the associated Λ-coalescent has the (c, ǫ, γ)-property, then given any fixed t ≥ 0, with probability one the process supp X(t) has the one-sided modulus of continuity with respect to Ch, where C ≡ C(d, γ) is the constant determined in Theorem 4.2. Further, with probability one supp X(t) is compact for all t > 0 and if supp X(0) is compact, then R ([0, t)) is also compact for all t > 0. In addition, with probability one dim supp X(t) ≤ 2/γ for all t > 0. Finally, given any 0 < δ < T , with probability one dim R ([δ, T )) ≤ 2 + 2/γ.
Proof. It has been proved by Lemma 4.13 of [20] that for any n ≥ 2, there exists a positive constant C(c, ǫ, γ) such that the total coalescence rate of the Λ-coalescent with the (c, ǫ, γ)-property satisfies λ n ≥ C(c, ǫ, γ)n 1+γ .
Then ∞ b=m+1 1 λ b ≤ 1 C(c, ǫ, γ) ∞ m 1 x 1+γ dx ≤ 1 γC(c, ǫ, γ)m γ ,
i.e., Condition A holds with α = γ. Consequently, the results follow from Remark 4.7 and Theorems 4.2-4.6. Now we discuss the support properties for Beta(2 − β, β)-Fleming-Viot process with underlying Brownian motion. It is known that the Beta(2−β, β)-coalescent stays infinite if β ∈ (0, 1] and comes down from infinity if β ∈ (1, 2). For β ∈ (1, 2), given any ǫ ∈ (0, 1), the Beta(2 − β, β)-coalescent has the (c, ǫ, β − 1)-property. Therefore, the conclusions of Proposition 4.13 hold with γ = β − 1.
For t ≥ 0 put S t ≡ ∩ ∞ n=1 R([t, t + 1/n)). Proposition 4.14. Under Assumption I and for any T > 0, there exist a positive random variable θ ≡ θ (T, d, α) < 1 and a constant C ≡ C(d, α) such that P-a.s.
suppX(t + ∆t) ⊆ B(S t , Ch(∆t))
for all 0 ≤ t < t + ∆t ≤ T and 0 < ∆t ≤ θ.
We also defer the proof of Proposition 4.14 to Section 5.
5.
Proofs of Theorems 4.1-4.6, Corollary 4.9 and Proposition 4.14 5.1. Modulus of continuity for the ancestry process. In this subsection we first obtain some estimates on the Λ-coalescent and on the maximal dislocation of the particles from their respective ancestors.
Denote by ⌊x⌋ the integer part of x for any x ∈ R. Given T > 0 and ∆ > 0, we can divide the interval [0, T ] into subintervals as follows: Set ∆ ≡ ∆ n = 2 −n . Let S T n be the collection of the endpoints of the first ⌊2 n T ⌋ subintervals, i.e.,
S T n ≡ k2 −n : 0 ≤ k ≤ 2 n T . Put S T ≡ n≥1 S T n = n≥1 k2 −n : 0 ≤ k ≤ 2 n T .
Clearly, given any T > 0, S T is the collection of all the dyadic rationals in [0, T ]. So S T is a dense subset of [0, T ]. For any n ∈ [∞], let {A n,k : 1 ≤ k ≤ 2 n T } be the collection of the first ⌊2 n T ⌋ subintervals in the partition so that A n,k ≡ (k − 1)2 −n , k2 −n .
For simplicity, we denote N n,k ≡ N (k−1)2 −n ,k2 −n . Also denote by H n,k the maximal dislocation over interval A n,k of all the Brownian motions followed by the countably many particles alive at time k2 −n and their respective ancestors at time (k − 1) 2 −n , i.e.,
H n,k ≡ H (k − 1) 2 −n , k2 −n .
For any positive integer m, let T n,k m ≡ inf t ∈ [0, 2 −n ] : #Π k2 −n (t) ≤ m with the convention inf ∅ = 2 −n . Notice that for any fixed n ∈ [∞] and m, the random times {T n,k m : 1 ≤ k ≤ 2 n T } follow the same distribution. Write T n,k
x ≡ T n,k ⌊x⌋ for any x > 0.
We need a standard estimate on Brownian motion.
Lemma 5.1. Given any x > 0 and d-dimensional standard Brownian motion (B (s)) s≥0 , we have Proof. Given any n and 1 ≤ k ≤ 2 n T , we first divide each interval A n,k into countably many subintervals as follows: For l = 0, 1, 2, . . ., let D n,k l be the maximal dislocation of the ancestors (for those countably many particles alive at time k2 −n ) at time b n,k l+1 from their respective ancestors at time b n,k l , i.e.,
P sup 0≤s≤t |B(s)| > x ≤ 8d 3 t π 1 x exp − x 2 2dt .J n,k 0 ≡ (k − 1)2 −n ,D n,k l ≡ max 1≤i≤N b n,k l ,k2 −n max j∈π i X L k2 −n j b n,k l+1 b n,k l+1 − − X i b n,k l − ,(13)
where π i : 1 ≤ i ≤ N b n,k l ,k2 −n denotes the collection of all the disjoint blocks of partition Π k2 −n k2 −n − b n,k l ordered by their least elements.
In the case of b n,k l+1 = b n,k l , i.e., J n,k l = 0, which corresponds to the situation of either T n,k 2 (3n+3l+3)/α = 2 −n or T n,k 2 (3n+3l+3)/α = T n,k 2 (3n+3l)/α , it follows from Lemma 3.2 that For dimension d and constant α in Assumption I, let C 1 (d, α) be a positive constant satisfying C 1 (d, α) > 2d (3/α + 1).
L k2 −n j b n,k l+1 = L k2 −n j b n,k l = i for any j ∈ π i with 1 ≤ i ≤ N b n,
Now we estimate the total maximal dislocation D n,k as follows. Let
I n ≡P max 1≤k≤2 n T D n,k > ∞ l=0 C 1 (d, α) h 2 −(n+2l) .
Since D n,k = ∞ l=0 D n,k l , we have
D n,k > ∞ l=0 C 1 (d, α) h 2 −(n+2l) ⊆ ∞ l=0 D n,k l > C 1 (d, α) h 2 −(n+2l) .
Therefore,
I n ≤ 2 n T k=1 ∞ l=0 P D n,k l > C 1 (d, α) h 2 −(n+2l) .
Under Assumption I, there exists a positive constant C such that for N large enough and for all n > N, ET 8 n/α ≤ C8 −n . For all those n > N, since D n,k l = 0 for those l with interval length J n,k l = 0, we only need to consider the case of J n,k l > 0. Observe that for l = 0, 1, 2, . . ., the total number of Brownian motion paths connecting the ancestors (of the countably many particles alive at k2 −n ) at time b n,k l+1 to their respective ancestors at earlier time b n,k l is at most 8 (n+l+1)/α . Since |J n,k
0 | = b n,k 1 − b n,k 0 ≤ 2 −n , we have P D n,k 0 > C 1 (d, α) h 2 −n ≤ 8 n+1 α P sup 0≤s≤2 −n |B (s) | > C 1 (d, α) h 2 −n .
For l = 1, 2, . . . , we have
P D n,k l > C 1 (d, α) h 2 −(n+2l) ≤P J n,k l > 2 −(n+2l) + P D n,k l > C 1 (d, α) h 2 −(n+2l) , 0 < J n,k l ≤ 2 −(n+2l) .
Since |J n,k l | ≤ T n,k 2 (3n+3l)/α , for any n > N the length of interval J n,k l satisfies P J n,k
l > 2 −(n+2l) ≤P T n,k 2 (3n+3l)/α > 2 −(n+2l) ≤2 n+2l ET n,k 2 (3n+3l)/α ≤ C2 −(2n+l) . We further have P D n,k l > C 1 (d, α) h 2 −(n+2l) ≤C2 −(2n+l) + 8 n+l+1 α P sup 0≤s≤2 −(n+2l) |B (s) | > C 1 (d, α) h 2 −(n+2l) .
Therefore,
I n ≤2 n T 8 n+1 α P sup 0≤s≤2 −n |B (s) | > C 1 (d, α) h 2 −n + 2 n T ∞ l=1 C2 −(2n+l) + 8 n+l+1 α P sup 0≤s≤2 −(n+2l) |B (s) | > C 1 (d, α) h 2 −(n+2l) = ∞ l=1 CT 2 −(n+l) + 2 n T ∞ l=0 8 n+l+1 α P sup 0≤s≤2 −(n+2l) |B (s) | > C 1 (d, α) h 2 −(n+2l) .
It follows from Lemma 5.1 that
P sup 0≤s≤2 −(n+2l) |B (s) | > C 1 (d, α) h 2 −(n+2l) ≤ 1 C 1 (d, α) 8d 3 π(n + 2l) log 2 exp − C 2 1 (d, α) (n + 2l) log 2 2d ≤ 1 C 1 (d, α) 8d 3 π log 2 2 − C 2 1 (d,α)(n+2l) 2d ≡C 2 (d, α)2 − C 2 1 (d,α)(n+2l) 2d
. Therefore, for any n > N we have
I n ≤ CT 2 −n + 2 n T ∞ l=0 8 n+l+1 α C 2 (d, α)2 − C 2 1 (d,α)(n+2l) 2d ≤ CT 2 −n + ∞ l=0 T C 2 (d, α)2 − C 2 1 (d,α) 2d − 3 α −1 n− C 2 1 (d,α) d − 3 α l+ 3 α .
Since C 1 (d, α) > 2d (3/α + 1), it follows that
I n ≤CT 2 −n + T C 3 (d, α)2 − C 2 1 (d,α) 2d − 3 α −1 n ,(14)
where
C 3 (d, α) ≡ ∞ l=0 C 2 (d, α)2 − C 2 1 (d,α) d − 3 α l+ 3 α .
Both terms on the right hand side of (14) are summable with respect to n. Thus, n I n < ∞, and it follows from the Borel-Cantelli lemma that P-a.s.
max 1≤k≤2 n T D n,k ≤ ∞ l=0 C 1 (d, α) h 2 −(n+2l) ≤C 1 (d, α) 2 −n n log 2 1 + ∞ l=1 √ 2 −2l+1 l ≡C 4 (d, α) 2 −n n log 2
for n large enough. By the lookdown construction and the arguments in Lemmas 4.6-4.7 of [20] we have H n,k ≤ D n,k . Thus, P-a.s. We are ready to prove the one-sided modulus of continuity for the ancestry process.
Proof of Theorem 4.1. We first show that P-a.s. for all r, s ∈ S T satisfying 0 < s − r ≤ θ,
H (r, s) ≤Ch (s − r) .
The following argument is similar to that in Section III.1 of Perkins [21]. By Lemma 5.2, given T > 0, there exist an event Ω T,d,α of probability one, and an integer-valued random variable N(T, d, α) big enough such that 2 −N(T,d,α) ≤ e −1 and max 1≤k≤2 n T H n,k ≤ C 4 (d, α)h 2 −n , n > N(ω, T, d, α), ω ∈ Ω T,d,α .
Let θ ≡ θ (ω, T, d, α) = 2 −N(ω,T,d,α) . For any r, s ∈ S T with 0 < s − r ≤ 2 −N(ω,T,d,α) = θ, there exists an n ≥ N(ω, T, d, α) such that 2 −(n+1) < s − r ≤ 2 −n . Recall that S T k = l2 −k : 0 ≤ l ≤ 2 k T and S T = ∪ k≥1 S T k = [0, T ]. For any k > n, choose s k ∈ S T k such that s k ≤ s and s k is the largest such value. Then s k ↑ s, s k+1 = s k + j k+1 2 −(k+1) with j k+1 ∈ {0, 1}.
Since s ∈ S T , then (s k ) k>n is a sequence with at most finite terms that are not equal to s. Applying (15), we have (16) H (s k , s k+1 ) ≤ C 4 (d, α)j k+1 h 2 −(k+1) .
By Lemma 5.3,
H (s n+1 , s) ≤ ∞ k=n+1 H (s k , s k+1 ) ≤ ∞ k=n+1 C 4 (d, α)j k+1 h 2 −(k+1) ≤C 4 (d, α) ∞ k=n+1 2 −(k+1) (k + 1) log 2 ≤C 4 (d, α) 2 −(n+1) (n + 1) log 2 ∞ k=1 √ 2 −k+1 k ≡C 5 (d, α) 2 −(n+1) (n + 1) log 2,(17)
where observe that only finitely many terms are nonzero in the summation on the right hand side of the first inequality. Similarly, for any k > n, choose r k ∈ S T k such that r k ≥ r and r k is the smallest such value. Then
r k ↓ r, r k+1 = r k − j ′ k+1 2 −(k+1) with j ′ k+1 ∈ {0, 1}.
Applying (15), we have
H (r k+1 , r k ) ≤ C 4 (d, α)j ′ k+1 h 2 −(k+1) .
Similar to (17), by Lemma 5.3 we have
H (r, r n+1 ) ≤ ∞ k=n+1 H (r k+1 , r k ) ≤ ∞ k=n+1 C 4 (d, α)j ′ k+1 h 2 −(k+1) ≤C 5 (d, α) 2 −(n+1) (n + 1) log 2.(18)
Since 2 −(n+1) < s − r ≤ 2 −n , we have 0 ≤ s n+1 − r n+1 ≤ i n+1 2 −(n+1) with i n+1 ∈ {0, 1, 2}. It comes from (16) and Lemma 5.3 that
H (r n+1 , s n+1 ) ≤2C 4 (d, α)h 2 −(n+1) =2C 4 (d, α) 2 −(n+1) (n + 1) log 2.(19)
Combining (17), (18) and (19), we have P-a.s. for all r, s ∈ S T with 0 < s − r ≤ θ H (r, s) ≤H (r, r n+1 ) + H (r n+1 , s n+1 ) + H (s n+1 , s)
≤2C 4 (d, α) 2 −(n+1) (n + 1) log 2 + 2C 5 (d, α) 2 −(n+1) (n + 1) log 2 ≤C(d, α) 2 −(n+1) (n + 1) log 2, where C(d, α) ≡ 2C 4 (d, α) + 2C 5 (d, α).
Function h is increasing on (0, e −1 ]. Since
2 −(n+1) < s − r ≤ θ ≤ e −1 ,
we have
H (r, s) ≤C(d, α)h 2 −(n+1) ≤ C(d, α)h (s − r)(20)
for all r, s ∈ S T satisfying 0 < s − r ≤ θ.
Finally, for any 0 < r < s < T with s − r < θ/2, find sequences (r m ) ⊆ S T and (s n ) ⊆ S T with r m ↑ r and s n ↓ s. By the lookdown construction, for any j ∈ [∞],
|X j (s) − X L s j (r) (r−)| ≤|X j (s) − X j (s n )| + |X j (s n ) − X L sn j (rm) (r m −)| + |X L sn j (rm) (r m −) − X L sn j (r) (r−)| + |X L sn j (r) (r−) − X L s j (r) (r−)|.(21)
Let both n and m be big enough such that 0 < s n − r m ≤ θ. It follows from (20) that the second term on the right hand side of (21) is bounded from above by C (d, α) h (s n − r m ). First fix n and let m → ∞. The third term tends to 0 because X L sn j (·) (·−) is continuous for any j ∈ [∞]. Then letting n → ∞, the first term tends to 0 because X j (·) is right continuous for any j ∈ [∞]. The last term is equal to 0 for large n since s n is then so close to s that there is no lookdown event involving levels {1, 2, . . . , j} during time interval (s, s n ]. Consequently,
|X j (s) − X L s j (r) (r−)| ≤ lim n→∞ |X j (s) − X j (s n )| + lim n→∞ lim m→∞ C (d, α) h (s n − r m ) + lim n→∞ lim m→∞ |X L sn j (rm) (r m −) − X L sn j (r) (r−)| + lim n→∞ |X L sn j (r) (r−) − X L s j (r) (r−)| =C (d, α) h (s − r) .
Then (8) follows.
Remark 5.4. It follows from estimate (14) that there exist positive constants C 6 ≡ C 6 (T, d, α) and C 7 ≡ C 7 (d, α) such that for ǫ > 0 small enough
P(θ ≤ ǫ) ≤ C 6 ǫ C 7 .
5.2.
Modulus of continuity for the Λ-Fleming-Viot support process and uniform compactness for the support and range. We will need the following observation on weak convergence. Lemma 5.5. If (ν n ) n≥1 , ν ⊆ M 1 R d and ν n weakly converges to ν, then we have supp ν ⊆ ∩ m≥1 ∪ n≥m supp ν n .
Proof. Suppose that there exists an x ∈ R d such that
x ∈ supp ν ∩ ∪ n≥m supp ν n c for some m. Since ∪ n≥m supp ν n c is an open set, there exists a positive value δ such that {y : |y − x| < δ} ⊆ ∪ n≥m supp ν n c . We can define a nonnegative and continuous function g satisfying g > 0 on {y : |y − x| < δ/2} and g = 0 on {y : |y − x| ≥ δ}. Then ν n , g = 0 for any n ≥ m but ν, g > 0. Consequently, ν n , g → ν, g , which contradicts the fact that ν n weakly converges to ν.
Proof of Theorem 4.2. Applying Theorem 4.1, there exist a positive random variable θ ≡ θ (T, d, α) and a constant C ≡ C (d, α) such that given any fixed t ∈ [0, T ), P-a.s. for all r ∈ S T ∩ (t, t + θ], we have
H (t, r) ≤ Ch (r − t) ,
which gives the upper bound for the maximal dislocation between the countably many particles at time r and their corresponding ancestors at time t. By Lemma 3.2, the ancestors at time t are exactly {X 1 (t−) , X 2 (t−) , . . . , X N t,r (t−)}, so we have P a.s.
{X 1 (r) , X 2 (r) , . . .} ⊆ 1≤i≤N t,r B (X i (t−) , Ch (r − t)) .
For the given t ∈ [0, T ), P a.s.
X i (t) = X i (t−) for any i ∈ [∞],
where X i (0−) ≡ X i (0), so for any r ∈ S T ∩ (t, t + θ], we have P a.s.
{X 1 (r) , X 2 (r) , . . .} ⊆ 1≤i≤N t,r B (X i (t) , Ch (r − t)) .(22)
Apply Lemma 3.1, for the given t ∈ [0, T ), P a.s.
{X 1 (t) , X 2 (t) , . . . , X N t,r (t)} ⊆ supp X (t) .
It follows from (22) that
{X 1 (r) , X 2 (r) , . . .} ⊆ B (supp X (t) , Ch (r − t)) .
For all r ∈ S T ∩ (t, t + θ], we have P-a.s.
X (n) (r) ≡ 1 n n i=1 δ X i (r) → X (r) .
Clearly,
supp X (n) (r) ⊆ {X 1 (r) , X 2 (r) , . . .} ⊆ B (supp X (t) , Ch (r − t))
for all n, which implies (23) supp X (r) ⊆ B (supp X (t) , Ch (r − t)) .
Then for any s satisfying t < s ≤ (t + θ/2) ∧ T , we can choose a sequence (s l ) l≥1 ⊆ S T ∩ (t, t + θ] such that s l ↓ s. It follows from the right continuity of X and Lemma 5.5 that supp X (s) ⊆ m≥1 l≥m supp X (s l ).
By (23), we have supp X (s l ) ⊆ B (supp X(t), Ch (s l − t)) for all l. Consequently, for any t < s
≤ (t + θ/2) ∧ T , supp X (s) ⊆ m≥1 l≥m B (supp X(t), Ch (s l − t)) = m≥1 B (supp X(t), Ch (s m − t)) = B (supp X(t), Ch (s − t)) .
Therefore, given any fixed t ≥ 0, there exist a positive random variable θ ≡ θ (t, d, α) and a constant C ≡ C(d, α) such that for any ∆t with 0 < ∆t ≤ θ, P-a.s. supp X (t + ∆t) ⊆ B (supp X(t), Ch (∆t)) = B supp X(t), C ∆t log (1/∆t) .
C (d, α) = 2C 4 (d, α) + 2C 5 (d, α) = 2C 4 (d, α) + 2C 4 (d, α) ∞ k=1 √ 2 −k+1 k = 2C 1 (d, α) 1 + ∞ l=1 √ 2 −2l+1 l 1 + ∞ k=1 √ 2 −k+1 k ,
where C 1 (d, α) is any constant satisfying C 1 (d, α) > 2d (3/α + 1). Proof. Under Assumption I, there exists a positive constant C such that (24) ET m ≤ Cm −α for m large enough. Given n, T n,k 4 n/α n 2/α , 1 ≤ k ≤ 2 n T are i.i.d. random variables following the same distribution as T 4 n/α n 2/α ∧ 2 −n . Consequently, N n,k , 1 ≤ k ≤ 2 n T are also i.i.d. random variables. Choosing 4 n/α n 2/α large enough, by (24) we have P max = 2 n T P T 4 n/α n 2/α ≥ 2 −n ≤ 2 n T ET n,k 4 n/α n 2/α /2 −n ≤ CT n −2 , which is summable with respect to n. Applying Borel-Cantelli lemma, we then have P-a.s. max
for n large enough. Given any positive constants σ and T with 0 < σ < T , we first show that R([σ, T )) is a.s. compact. Applying Theorem 4.1, there exist a positive random variable θ ≡ θ (T, d, α) > 0 and a constant C ≡ C(d, α) such that P-a.s. for all r, s ∈ S T satisfying 0 < s − r ≤ θ, H (r, s) ≤Ch (s − r) .
For the given σ, choose n big enough so that 2 −n ≤ θ ∧ σ and (25) holds. For any 1 ≤ k ≤ 2 n T and t ∈ S T ∩ [k2 −n , (k + 1)2 −n ∧ T ), we have
H (k − 1) 2 −n , t ≤ H (k − 1) 2 −n , k2 −n + H k2 −n , t ≤ 2Ch 2 −n .
It follows from the lookdown construction and Lemma 3.2 that
supp X (t) ⊆ 1≤i≤N (k−1)2 −n ,t B X i (k − 1) 2 −n − , 2Ch 2 −n .
By (25) we have N (k−1)2 −n ,t ≤ N (k−1)2 −n ,k2 −n = N n,k < 4 n/α n 2/α . Consequently, (26) supp
X (t) ⊆ 1≤i<4 n/α n 2/α B X i (k − 1) 2 −n − , 2Ch 2 −n .
For general t ∈ [k2 −n , (k + 1)2 −n ∧ T ). We can select a decreasing sequence
t n,k l l≥1 ⊆ S T ∩ [k2 −n , (k + 1)2 −n ∧ T ) satisfying t n,k l ↓ t as l → ∞.
Since the Λ-Fleming-Viot process X is right continuous, it follows from Lemma 5.
B X i (k − 1) 2 −n − , 2Ch 2 −n .
Therefore, for any t ∈ [k2 −n , (k + 1)2 −n ∧ T ), we also have
(27) supp X (t) ⊆ 1≤i<4 n/α n 2/α B X i (k − 1) 2 −n − , 2Ch 2 −n ,
i.e., R ([k2 −n , (k + 1)2 −n ∧ T )) is contained in at most ⌊4 n/α n 2/α ⌋ closed balls each of which has radius bounded from above by 2Ch (2 −n ). Then
R ([σ, T )) ⊆ R 2 −n , T ⊆ 1≤k≤2 n T R k2 −n , (k + 1) 2 −n ∧ T ⊆ 1≤k≤2 n T 1≤i<4 n/α n 2/α B X i (k − 1) 2 −n − , 2Ch 2 −n ,(28)
where the right hand side is the union of at most ⌊2 n T ⌋× ⌊4 n/α n 2/α ⌋ closed and bounded balls. So R ([σ, T )) is compact. Consequently, the random measure X(t) has compact support for all times t ∈ [σ, T ) simultaneously. Let σ = 1/T and T → ∞. Then the random measure X (t) has compact support for all times t ∈ (0, ∞) simultaneously.
Further, given that supp X(0) is compact, we can adapt the above-mentioned strategy to find a finite cover for R([0, T )). Applying Theorem 4.2, for n large enough, we have
R [0, 2 −n ) = t∈[0,2 −n ) supp X(t) ⊆ B supp X (0) , Ch 2 −n . Then R ([0, T )) ⊆ 0≤k≤2 n T R k2 −n , (k + 1) 2 −n ∧ T ⊆ B supp X (0) , Ch 2 −n 1≤k≤2 n T 1≤i<4 n/α n 2/α B X i (k − 1) 2 −n − , 2Ch 2 −n ,
where the right hand side is compact given the compactness of supp X (0). So, R ([0, T )) is compact.
Note that R ([0, T )) is increasing with respect to T . Let T → ∞. It is clear that R ([0, t)) is compact for all t > 0 P-a.s..
Upper bounds on
Hausdorff dimensions for the support and range. Given any Λ-coalescent (Π(t)) t≥0 with Π(0) = 0 [∞] , recall that For any x > 0, write T n x ≡ T n ⌊x⌋ and T x ≡ T ⌊x⌋ . Let (T n ) n≥2 be independent random variables such thatT n has the same distribution as T n n−1 .
T m ≡ inf t
Lemma 5.8. For any n > m, T n m is stochastically less than n i=m+1T i , i.e., for any t > 0,
(29) P (T n m ≥ t) ≤ P n i=m+1T i ≥ t .
Proof. We use a coupling argument by defining an auxiliary [n] × [n]-valued continuous time Markov chain (Y 1 , Y 2 ) describing the following urn model. Intuitively, there are balls in an urn of color either white or black. Let Y 1 (t) and Y 2 (t) represent the number of white and black balls at time t, respectively. After each independent exponential sampling time a random number of balls are taken out of the urn and then immediately replaced with certain white or black colored balls so that the total number of balls in the urn decreases exactly by one overall afterwards. More precisely, given that there are w white balls and b black balls in the urn, at rate λ w+b,k each group of k balls with k ≤ w + b is independently removed. Suppose that w ′ white balls and k − w ′ black balls have been chosen and removed at time t, we then immediately return k − 1 balls to the urn so that among the returned balls, either one is white and all the others are black if w ′ > 0 or all of them are black if w ′ = 0. At such a sampling time t we define
Y 1 (t) = w − w ′ + 1 and Y 2 (t) = b + w ′ − 2 = w + b − 1 − Y 1 (t), if w ′ > 0; Y 1 (t) = w and Y 2 (t) = b − 1, if w ′ = 0,
and the value of (Y 1 , Y 2 ) keeps unchanged between the sampling times. The abovementioned procedure continues until there is one white ball left in the urn. Suppose that there are n white balls and no black balls in the urn initially, i.e., (Y 1 (0), Y 2 (0)) = (n, 0).
Observe that Y 1 follows the law of the Λ-coalescent starting with n-blocks and (T i ) i≤n has the same distribution as the inter-decreasing times for process Y 1 + Y 2 . Plainly, inf{t : Y 1 (t) ≤ m} ≤ inf{t : Y 1 (t) + Y 2 (t) ≤ m}.
Inequality (29) thus follows.
The estimate in Lemma 5.7 is not enough for the proofs of Theorems 4.5-4.6. A sharper estimate is obtained in the following result under a stronger condition. Lemma 5.9. Suppose that Condition A holds. We have P-a.s. Proof. Under Condition A, there exists a positive constant C such that for n large enough and for any b > 2 n/α n 2/α , (31) λ b ≥ (C⌊2 n/α n 2/α ⌋ −α ) −1 > 2 n+1 n.
Letting n → ∞ in (29), for any t > 0 and m ∈ [∞] we have whereT i follows an exponential distribution with parameter λ i . It follows from (31) that when n is large enough, λ i > 2 n n for any i > 2 n/α n 2/α , which guarantees the existence of moment generating function forT i . As a result, P max for n large enough.
Proof of Theorem 4.5. Given any 0 < σ < T , we first consider the uniform upper bound on Hausdorff dimensions for supp X(t) at all times t ∈ [σ, T ). We adapt the same idea as the proof of Theorem 4.3 to find a cover for the support at any time t ∈ [σ, T ). Since we have a sharper estimate for N n,k under Condition A, for n large enough, (27) in the proof of Theorem 4.3 can be replaced by supp X (t) ⊆ 1≤i<2 n/α n 2/α B X i (k − 1) 2 −n − , 2Ch 2 −n for any t ∈ [k2 −n , (k + 1)2 −n ∧ T ) and 1 ≤ k ≤ 2 n T , i.e., for any t ∈ [σ, T ) ⊆ [2 −n , T ), supp X(t) is contained in at most ⌊2 n/α n 2/α ⌋ closed balls each of which has a radius bounded from above by 2Ch (2 −n ). which implies H 2+ǫ α (supp X(t)) = 0. Since ǫ is arbitrary, the Hausdorff dimensions for supp X(t) at all times t ∈ [σ, T ) are uniformly bounded from above by 2/α. Finally, let σ ≡ 1/T and T → ∞. The Hausdorff dimension for supp X(t) has uniform upper bound 2/α at all positive times simultaneously.
Proof of Theorem 4.6. Given any 0 < δ < T , we also follow the proof of Theorem 4.3 to find a finite cover for R ([δ, T )). Choose n large enough such that 2 −n ≤ θ ∧ δ and (30) holds. Similarly as (28) in the proof of Theorem 4.3, we have R ([δ, T )) ⊆ R 2 −n , T ⊆ 1≤k≤2 n T R k2 −n , (k + 1) 2 −n ∧ T ⊆ 1≤k≤2 n T 1≤i<2 n/α n 2/α B X i (k − 1) 2 −n − , 2Ch 2 −n , which implies that R ([δ, T )) is contained in at most ⌊2 n T ⌋ × ⌊2 n/α n 2/α ⌋ closed balls, each of which has radius bounded from above by 2Ch(2 −n ).
For any ǫ > 0, it follows that lim n→∞ ⌊2 n T ⌋ × 2 Since ǫ is arbitrary, the Hausdorff dimension for the range R ([δ, T )) is bounded from above by 2/α + 2.
[n] and by P ∞ the set of ordered partitions of [∞]. Write 0 [n] ≡ {{1}, . . . , {n}} for the partition of [n] consisting of singletons and 0 [∞] for the partition of [∞] consisting of singletons. Given n ∈ [∞]
s keeps track of locations for all the ancestors of the particle with level i at time s.
Theorem 4. 5 .
5Suppose that Condition A holds. Then dim supp X(t) ≤ 2/α for all t > 0 P-a.s..
Theorem 4. 6 .
6Suppose that Condition A holds. Then for any 0 < δ < T , dim R([δ, T )) ≤ 2 + 2/α P-a.s..
Corollary 4. 8 .
8Under Assumption I, there exists a constant C > 0 such that
Proposition 4 . 10 .
410Suppose that Condition A holds. Then P-a.s. for all t > 0 and ǫ > 0 we have lim sup r→0+ X(t)(B(x, r)) r 2/α+ǫ > 0 for X(t) almost all x.
Proposition 4. 11 .
11Let X be any Λ-Fleming-Viot process with Λ({0}) > 0 and underlying Brownian motion in R d for d ≥ 2. Then given any fixed t ≥ 0, with probability one the process supp X(t) has the one-sided modulus of continuity with respect to Ch, where C ≡ C(d) is the constant determined in Theorem 4.2. Further, with probability one supp X(t) is compact for all t > 0 and if supp X(0) is compact, then R ([0, t)) is also compact for all t > 0. In addition, with probability one dim supp X(t) ≤ 2 for all t > 0. Finally, given any 0 < δ < T , with probability one dim R ([δ, T )) ≤ 4.
[ 0 ,
0∆], [∆, 2∆], . . . , [⌊T /∆ − 1⌋∆, ⌊T /∆⌋∆] , [⌊T /∆⌋∆, T ].
Lemma 5. 2 .
2Under Assumption I and for any T > 0, there exists a positive constant C 4 (d, α) such that P-a.s. max 1≤k≤2 n T H n,k ≤ C 4 (d, α)h 2 −n for n large enough, where h is defined by(12).
≡ 0 ≡
0k2 −n − T n,k 8 (n+l)/α , k2 −n − T n,k 8 (n+l+1)/α for l = 1, 2, 3, . . .. Consequently, the lengths of these countably many subintervals satisfy that J n,k 0 ≤ 2 −n and J n,k l ≤ T n,k 8 (n+l)/α = T n,k 2 (3n+3l)/α for l = 1, 2, 3, . . . .The right endpoints of these subintervals b sequence of random times converging increasingly to k2 −n . Set b n,k (k − 1) 2 −n for convenience.
lookdown construction and the coming down from infinity property, there exists a finite number of ancestors at each time b n,k l , l = 0, 1, 2, . . . for those countably many particles alive at time k2 −n , i.∈ [∞] < ∞. So both maximums in (13) are in fact taken over finite sets. Put
,k ≤ C 4 (d, α)h 2 −n for n large enough. Lemma 5.3 follows from the lookdown construction.
Lemma 5 . 3 .
53For any r, t, s with 0 ≤ r ≤ t ≤ s we have H (r, s) ≤ H (r, t) + H (t, s) with the convention H (r, r) = H (s, s) ≡ 0.
Remark 5 . 6 .
56The constants C ≡ C (d, α) in Theorems 4.1 and 4.2 are the same. From the proofs of Lemma 5.2 and Theorems 4.1-4.2, it is clear that
Lemma 5. 7 .
7Under Assumption I, we have P-a.s.
≥ 0 :
0#Π(t) ≤ m with the convention inf ∅ = ∞. (Π n (t)) t≥0 is its restriction to [n] with Π n (0) = 0 [n]. For any n ≥ m, let
inf t ≥ 0 : #Π n (t) ≤ m with the convention inf ∅ = ∞.
( 32 )E
32P (T m ≥ t) ≤ P i>mT i ≥ t .With estimate (32) we can find a sharper uniform upper bound for the maximal number of ancestors as follows: exp 2 n nT i ,
Theorem 4.1. Under Assumption I and for any T > 0, there exist a positive random variable θ ≡ θ (T, d, α) < 1 and a constant C ≡ C(d, α) such that P-a.s. for all r, s ∈
Proof of Corollary 4.9. With initial value δ 0 , applying Theorem 4.2, it is clear that almost surely R [0, 2 −n ) ⊆ B 0, Ch(2 −n ) for n large enough. From the proof of Theorem 4.6, we havefor n large enough. Therefore, R ([0, T )) is contained in at most ⌊2 n T ⌋ × ⌊2 n/α n 2/α ⌋ + 1 closed balls, each of which has radius bounded from above by 2Ch(2 −n ).For any ǫ > 0, we haveSince ǫ is arbitrary, the Hausdorff dimension for the range R ([0, T )) is bounded from above by 2/α + 2.Proof of Proposition 4.14. Let {t i } be any dense subset of [0, T ]. Combining the proofs for Theorem 4.1 and Theorem 4.2, there exist θ ≡ θ(T, d, α) < e −1 and C ≡ C(d, α) such that P-a.s.for all i and 0 < ∆t ≤ θ ∧ (T − t i ). Then for any t ∈ [0, T ), there exists a subsequence (t i j ) with t i j ↓ t such that given any n > 0,⊆ B(R ([t, t + 1/n)) , Ch(∆t)) for 0 < ∆t ≤ θ ∧ (T − t) and j large enough. So, supp X(t + ∆t) ⊆ B(S t , Ch(∆t)) since n is arbitrary.
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| [] |
[
"Why planetary and exoplanetary protection differ: The case of long duration Genesis missions to habitable but sterile M-dwarf oxygen planets",
"Why planetary and exoplanetary protection differ: The case of long duration Genesis missions to habitable but sterile M-dwarf oxygen planets"
] | [
"Claudius Gros \nInstitute for Theoretical Physics\nGoethe University\nFrankfurtGermany\n"
] | [
"Institute for Theoretical Physics\nGoethe University\nFrankfurtGermany"
] | [] | Time is arguably the key limiting factor for interstellar exploration. At high speeds, flyby missions to nearby stars by laser propelled wafersats taking 50-100 years would be feasible. Directed energy launch systems could accelerate on the other side also crafts weighing several tons to cruising speeds of the order of 1000 km/s (c/300). At these speeds, superconducting magnetic sails would be able to decelerate the craft by transferring kinetic energy to the protons of the interstellar medium. A tantalizing perspective, which would allow interstellar probes to stop whenever time is not a limiting factor. Prime candidates are in this respect Genesis probes, that is missions aiming to offer terrestrial life new evolutionary pathways on potentially habitable but hitherto barren exoplanets.Genesis missions raise important ethical issues, in particular with regard to planetary protection. Here we argue that exoplanetary and planetary protection differ qualitatively as a result of the vastly different cruising times for payload delivering probes, which are of the order of millennia for interstellar probes, but only of years for solar system bodies. Furthermore we point out that our galaxy may harbor a large number of habitable exoplanets, M-dwarf planets, which could be sterile due to the presence of massive primordial oxygen atmospheres. We believe that the prospect terrestrial life has in our galaxy would shift on a fundamental level in case that the existence of this type of habitable but sterile oxygen planets will be corroborated by future research. It may also explain why our sun is not a M dwarf, the most common star type, but a medium-sized G-class star. | 10.1016/j.actaastro.2019.01.005 | [
"https://arxiv.org/pdf/1901.02286v1.pdf"
] | 57,721,174 | 1901.02286 | 419035fa681c6386fcae8d355a91c782acb2ae6d |
Why planetary and exoplanetary protection differ: The case of long duration Genesis missions to habitable but sterile M-dwarf oxygen planets
8 Jan 2019 January 9, 2019
Claudius Gros
Institute for Theoretical Physics
Goethe University
FrankfurtGermany
Why planetary and exoplanetary protection differ: The case of long duration Genesis missions to habitable but sterile M-dwarf oxygen planets
8 Jan 2019 January 9, 2019interstellar missionsdirected energy launch systemsmagnetic sailstransiently habitable exoplanetsoxygen planetGenesis project
Time is arguably the key limiting factor for interstellar exploration. At high speeds, flyby missions to nearby stars by laser propelled wafersats taking 50-100 years would be feasible. Directed energy launch systems could accelerate on the other side also crafts weighing several tons to cruising speeds of the order of 1000 km/s (c/300). At these speeds, superconducting magnetic sails would be able to decelerate the craft by transferring kinetic energy to the protons of the interstellar medium. A tantalizing perspective, which would allow interstellar probes to stop whenever time is not a limiting factor. Prime candidates are in this respect Genesis probes, that is missions aiming to offer terrestrial life new evolutionary pathways on potentially habitable but hitherto barren exoplanets.Genesis missions raise important ethical issues, in particular with regard to planetary protection. Here we argue that exoplanetary and planetary protection differ qualitatively as a result of the vastly different cruising times for payload delivering probes, which are of the order of millennia for interstellar probes, but only of years for solar system bodies. Furthermore we point out that our galaxy may harbor a large number of habitable exoplanets, M-dwarf planets, which could be sterile due to the presence of massive primordial oxygen atmospheres. We believe that the prospect terrestrial life has in our galaxy would shift on a fundamental level in case that the existence of this type of habitable but sterile oxygen planets will be corroborated by future research. It may also explain why our sun is not a M dwarf, the most common star type, but a medium-sized G-class star.
Introduction
Space exploration is confronted inherently with the extended travel times needed to traverse the voids of space extending between earth and the destination. In addition to travel times, key limiting factors are cost considerations and the time devoted to mission development and design. The latter is in particular the case for further away destinations, such as missions to the outer solar system. Typically examples of past and future missions are here the Voyager crafts [1], the Europa Clipper [2] and missions searching for life in the subglacial waters of ice moons [3]. Is there however a maximal planning horizon societies would be willing to support? It is not uncommon for projects involving large-scale scientific or technology development tasks to span decades. Long-term collaborative efforts, like the ITER fusion reactor [4], are nevertheless more often than not accompanied by continuous controversies concerning the ultimate effort to utility ratio, with a central reason being the rational to discount future rewards [5]. It is hence unlikely that explorative space missions taking centuries or even millennia to complete would ever survive the initial cost-to-benefit evaluation. The situation may however change for endeavors not designed for their usefulness in terms of science data or other return values. This will be the case, as we argue here, for Genesis missions aiming to establish an ecosphere of unicellular lifeforms on potentially habitable but hitherto barren exoplanets.
2 The utility vs. realizability dilemma of deep space exploration Solar system traveling times are long, but somewhat manageable. The recent surge of interest in directed energy launching systems [6] has presented us on the other side with the prospect that interstellar space missions may become realizable within several decades. The development of mission scenarios for interstellar probes has hence left the realm of fusion-size spaceships [7,8], becoming instead a question of pros and cons. We argue here that two types of interstellar probes may be considered, fast data return probes and comparatively slow Genesis crafts.
Data gathering by fast flyby probes
Directed energy launch systems may accelerate wafersat spacecrafts weighing a few gram up to 20% of the speed of light [9], at least modulo a substantial technology development effort. Wafer-sized interstellar probes capable of reaching the nearest stars within several decades may hence be employed, as envisioned by the Breakthrough Starshot Initiative [10], for science data return missions. The technical challenges, ranging from material issues [11], the stability at launch [12], to the interaction of a relativistic spacecraft with electromagnetic forces [13], and with the interstellar medium [14], are immense, but not insurmountable. Interstellar science probes need to be fast, as data return missions taking centuries would not be considered worth the investment. Flyby missions are consequently the only viable option. Decelerating with the aim to enter solar or planetary orbits involves on a practical level the transfer of the kinetic energy to either the photons of the target star or to the ionized particles of the interstellar medium. Solar sail deceleration, the first method, is however not possible when the craft travels at relativistic speeds; the probe would have bypassed the target star long before coming to a stop. For the second method, magnetic sails weighing of the order of several hundreds tons would be necessary [15]. To accelerate a craft of that size to close to the speed of light is however beyond launch infrastructures potentially realizable within the next generations. The same holds for active deceleration of massive spaceships by TW-sized solar-system based laser beams [16]. Interstellar science probes are consequently realizable only if they are fast, viz when limited to flyby investigations.
Payload delivery by slow interstellar crafts
Fast and slow are highly relative terms in the realm of space travel. The Voyager spaceprobes cruise at speeds of the order of 20 km/s, which is high with respect to everyday's velocities, but slow when it comes to interstellar distances. Here we consider an interstellar spacecraft to be 'slow' when cruising roughly 50 times faster than the Voyager probes, at 1000 km/s, which corresponds to 1/300 of the speed of light. The millennia needed to reach the nearest stars at such a velocity exceed typical human planing horizons by far, which implies that it is not possible to assess the potential benefits of slow interstellar missions with standard utility criteria. A slow craft has on the other hand enough time at its disposal to decelerate via magnetic braking, that is by transferring momenta to the interstellar protons [17]. It has been estimated in this regard, that the magnetic field created by a 1.5 ton superconducting loop with a radius of 50 km would be able of doing the job [15]. Magnetic sails are moreover self-deploying, as wires with opposite currents repel each other. Slow interstellar spacecrafts are hence suited for delivering ton-sized payloads to far away destinations.
Slow interstellar ton-sized crafts may be launched, importantly, by the same directed energy launch systems envisioned for fast flyby missions, with the reduced velocity trading off the increased weight [18]. Comparatively slow, that is non-relativistic interstellar crafts, could be accelerated alternatively by the type of advanced ion engines that are being developed within NASA's evolutionary Xenon thruster (NEXT) effort [19,20]. Laser arrays of the order of 100 MW would be used in this case not to propel a reflecting lightsail, but to power the solar cells of the craft [21]. A conversion rate of 70%, potentially achievable when the sail performance is tuned to the laser frequency, would then be enough to power 70 MW lithium-fueled gridded ion thrusters [22].
Overall we are confident that interstellar spaceprobes entering solar or planetary orbit on arrival are potentially realizable, albeit at the cost of prolonged mission times. Less likely seem in contrast the perspective that fast probes could decelerate, independently of the technique envisioned for the braking maneuver.
Interstellar deceleration
The lack of an in-place infrastructure implies, on arrival, that it is substantially more challenging to decelerate an interstellar craft than to speed it up in first place. This is in especially true when the craft is fast and when mission durations should be kept within human planing horizons. Given enough time, and a magnetic sail of substantial size [15,23,24], braking from the protons of the interstellar medium is however feasible, as discussed in the previous sections.
Regarding solar sails, it has been proposed that graphene may be the optimal candidate material [25], both for launching an interstellar probe and for braking from the photons of the target star [26]. Physically, a graphene monolayer is characterized by an ultralow areal mass density of 7.4 · 10 −7 kg/m 2 , a negligible reflectivity and a flat absorption coefficient A ω = πα ≈ 0.023, as resulting from the Dirac cone, where α = e 2 /(hc) ≈ 1/137 is the fine structure constant [27]. Assuming that the properties of a graphene monolayer could be improved by about a factor 100 without a corresponding weight increase, namely to reflectivity values of 99.99% -99.999%, a fast interstellar craft could decelerate at α-Centauri using stellar photon pressure [26], at least as a matter of principle. How to realize the required performance boost is presently however unclear.
The field of a magnetic sail is produced by a large superconducting loop. Alternatively one may consider an electric sail [28], which consists of electrically charged structures of similar extensions. In this case it is the electric field of the charged craft that reflects the protons of the interstellar medium [17]. It would however be a challenge for an interstellar craft to power the electron gun needed to maintain the required potential difference between the craft and the surrounding rarefied medium. Electric sails may however be advantageous for solar system application, as their performance decays only as 1/r, as a function of the distance to the sun, and not as 1/r 2 , as for solar sails [29].
The slow path from prokaryotes to eukaryotes
Payload delivering interstellar crafts come with cruising times of a few millennia, at least, that is with timescales that may seem extraordinary long to human planning horizons. A handful of millenia are on the other hand irrelevant from the perspective of evolutionary processes. On earth it took about one billion years, that is until the end of the archean genetic expansion [30], to develop modern prokaryotes, viz bacteria, and another billion years for the basis of all complex life, eukaryotic cells, to emerge [31]. It is not a coincidence, that higher life forms are made of eukaryotic and not of prokaryotic cells, but a consequence of the energy barrier that prevents prokaryotic cells to support genomes of eukaryotic size [32]. The massive genomes necessary for the coding of complex eukaryotic morphologies are typically four to six orders of magnitude larger than the genetic information encoding prokaryotic life [33].
The emergence of eukaryotic cells has been on earth the key bottleneck along the route from uni-cellular to multi-cellular and morphological complex life. Taking the timescale of terrestrial evolutionary processes as a reference, we may hence postulate that exobiological lifeforms could need similar time spans, if at all, to evolve to complexity levels comparable to terrestrial eukaryotic life [34]. There are moreover arguments for the possibility, as discussed further below, that a relatively high percentage of potentially habitable planets may harbor either no life at all, or only lifeforms equivalent in complexity below that of modern terrestrial bacteria. A payload of single-cell eukaryotes, either as germs or in terms of codings for an onboard gen laboratory, would hence be a valuable payload for slow cruising Genesis probes destined to habitable but otherwise barren exoplanets. Operationally, instead of landing, the Genesis probe would carry out the seeding process from orbit via the retrograde expulsion of micro-sized drop capsules. The goal would be in the end to lay the foundations for a self-developing ecosphere of initially unicellular organisms [35].
The case for habitable but sterile oxygen planets
A Genesis probe should comply with planetary protection considerations and target only certain types of potentially habitable planets [36]. One possibility is that the candidate planet is only transiently habitable, that is for time spans that are too short for complex single-or multi-cellular life to develop [35]. Examples for causes for limited habitability are orbital instabilities of the hosting planetary system and geological disruptions due to the absence of plate tectonics [35]. Of interest in this respect are furthermore planets orbiting brown dwarfs [37], that is failed stars having 13-75 times the mass of Jupiter. The mass of brown dwarfs is too low for hydrogen fusion, the energy source of main-sequence stars, with the consequence that the star cools progressively by radiative dissipation of its initial reservoir of thermal energy. Depending on the mass of the star, on the orbital distances of the planets and on other parameters, like the impact of gravitational and atmospheric tides, a given brown-dwarf planet could remain habitable for periods ranging from a few hundred million years to a few billion years [37]. Brown dwarf planets are hence interesting Genesis candidate planets.
Abiotic oxygen buildup in the runaway greenhouse state of young M-dwarf planets
Stars with a mass greater than about 0.075 the mass of the sun are heavy enough to produce energy via hydrogen burning. A well known example is the Trappist-1 system [38], a system composed of seven earth-sized planets orbiting a M-dwarf star at distances that are either within or close to the nominal habitable zone. The mass of the central star is in this case about 0.08 the mass of the sun, which is not a coincidence. Estimates show [39], that a majority of rocky habitable zone planets is expected to orbit M dwarfs, that is low-mass stars like Trappist-1. M dwarfs are characterized by an extended Kelvin Helmholtz contraction time, which is the time it takes for a protostar to reach the main sequence by shedding its initial reservoir of gravitational energy radiatively. The Kelvin Helmholtz timescale extends from about 10 million years for sun-like stars to several hundred million years for late M dwarfs [40]. Planets orbiting low mass stars at a distance corresponding to the mainsequence habitable zone will hence experience an extended initial runaway greenhouse state induced by the increased irradiation from the initially substantially larger host star. With the ending of the Kelvin Helmholtz contraction of the central star the atmosphere of the planet cools correspondingly.
In the initial greenhouse state the stratosphere is wet. The UV radiation of the host star leads in this stage to the photodissociation of water, and with it to the loss of hydrogen to space, with the oxygen staying mostly behind [41]. Depending on the initial reservoir, several earth oceans worth of water may be lost altogether [42]. For the habitable-zone planets of the Trappist-1 system the resulting buildup of abiotic oxygen has been estimated to reach partial pressures of 350-490 bars [43]. It is presently not clear to which extent the buildup of abiotic oxygen during the Greenhouse state is countered by redox reactions resulting from the interaction of the atmosphere with a magma ocean [44]. It is likely that the final oxygen content of the atmosphere is reduced, but still substantial. Primordial oxygen partial pressures of several bars and more may hence be a common feature of rocky M-dwarf planets.
Are oxygen planets sterile?
The chemical environments of oxygen planets, that is of planets disposing of a substantial amount of primordial atmospheric oxygen, are expected to differ substantially from the one of archean earth. The origins of life on earth are yet not understood [45], it is however clear that abiogenesis may occur only in microstructured chemo-physical reaction environments [46] that are driven by a sustained energy source [47], as realized within the alkaline hydrothermal vent scenario [48]. Potential birthing places of life such as submarine alkaline vents are conjectured furthermore to be characterized by steep electrochemical concentration gradients [49], as a necessary precondition for the emergence of prebiotic vectorial reaction pathways. Primordial oxygen, when present, could disrupt however the formation of these electronchemical disequilibria [50]. An important point in this context is a well-known relationship between oxygen and cellular energy 1 , namely that the synthesis of the chemical constituents of cells, like amino acids, bases and lipids, from glucose and ammonium, demands about 13 times more energy per cell in the presence of O 2 than in the absence of oxygen [51,52]. It is hence conceivable that the emergence of life could be preempted on otherwise habitable M-dwarf planets by the presence of primordial oxygen. A substantial amount of future research effort is clearly warranted in order to corroborate, or to disprove this presumption. In case, we would live in a galaxy where habitable but sterile planets abound. Oxygen planets would then be prime candidates for Genesis missions.
Planetary vs. exoplanetary protection
An endeavor aiming to endow other planets with life raises a series of ethical issues. From a utilitarian perspective it may be considered in fact unethical to allocate a substantial amount of resources to projects not contributing to the overall welfare of humanity [53]. We will not pursue this argument further, focusing instead on two key aspects of planetary protection.
Planetary protection for human benefit
Planetary protection had been formulated historically with the exploration of the solar system in mind [54]. Back contamination needs to be avoided, clearly, such that "earth is protected from the potential hazard posed by extraterrestrial matter carried by a spacecraft returning from another planet". Space exploration should be carried out, furthermore, in a manner that does not jeopardize "the conduct of scientific investigations of possible extraterrestrial life forms, precursors and remnants". Human benefit considerations have hence been, as these formulations of the planetary protection policy of the International Committee on Space Research (COSPAR) show, the core original rational for avoiding not only backward, but also forward contamination [55].
The very existence of extraterrestrial life is a subject of debate. Remote sensing attempts, like the detection of extrasolar life via a direct or indirect spectral analysis of exoplanetary atmospheres [56], will be carried out in the next years. In situ investigations of extrasolar life are in contrast unlikely to be ever undertaken. On one side because the delivery of the required landing modules by slow-cruising interstellar probes would take millennia. The second point is that we may expect science to progress within the intervening centuries to a point that would allow for a near to full understanding of the possible routes to abiogenesis and of the spectra of possible lifeforms. Another aspect is that computer experiments can be anticipated to advance to a point that would allow, eventually, to retrace the geophysical evolution of a given non-solar planetary system in detail, possibly when supplemented by flyby observations. Relatively little could be added in this case by additional in situ investigations. Protecting the rudimentary biosphere of an exoplanet for science purposes is hence not as relevant as it is for solar system bodies.
Ethically grounded planetary protection
Common ethical imperatives are ambiguous when human activities impact higher but nonhuman life forms, in particular with regard of the relative relevance of anthropocentric and non-anthropocentric values [57]. There is however a deeply rooted common-sense notion that humanity should protect life forms of a certain level of complexity, at least whenever possible. This notion withstands the Darwinian nihilist viewpoint [58], attributing instead value to life per se [59].
Taking the evolution of terrestrial biota as a reference [30,35], we may classify nonsolar ecosystems into four categories: primitive-prokaryotic, prokaryotic, unicellular eukaryotic and multi-cellular eukaryotic, viz complex life. For terrestrial life it is custom to attribute value nearly exclusively to complex life, viz to animals and plants. Killing a few billion bacteria while brushing teeth does not cause, to give an example, moral headaches. The situation changes however when it comes to extrasolar life, for which we may attribute value also to future evolutionary pathways. This is a delicate situation. Is it admissible to bring eukaryotes to a planet in a prokaryotic state, superseding such indigenous life with lifeforms having the potential to develop into complex ecologies? Our prevalence to attribute value predominately to complex lifeforms would suggest that this would be ethically correct [58,59], in particular if we could expect our galaxy to harbor large numbers of planets in prokaryotic states. Endowing a selected number of exoplanets with the possibility to evolve higher life forms would in this case not interfere with the evolution of yet simple life forms on potentially billions of other planets.
Genesis missions would comply with the common-sense norm to attribute value to complex lifeforms, the very rational to undertake them in first place, and abort whenever the target planet harbors life that can be detected from orbit. Considering the case of Mars, it is however clear that it will be hard to rule out unambiguously the existence of ecospheres of exceedingly low bioproductivity. Protocols regulating the necessary level of confidence are hence needed. It would be meaningful to embargo the entire extrasolar system in case that complex life would be detected by flyby probes on one of its planets.
Discussion & outlook
The recent advent of directed energy launch concepts demonstrates that interstellar space probes may become realizable within the foreseeable future [60]. The technical challenges involved are daunting. An example is the development of self-healing electronics [61], that is of circuits that would be capable to withstand decades to millennia of cosmic bombardment [62]. It is hence important to assess and to classify the range of possible interstellar missions. The first option is a high speed flyby mission by gram-sized wafersats that have been accelerated to a sizable fraction of the speed of light [18], say 20%. Here we have pointed out that the directed energy launch systems envisioned for fast flyby missions would be suited to launch in addition payload delivering probes cruising at reduced velocities of typical 1000 km/s. These probes would weigh of the order of several tons, in particular due to the weight demands of the magnetic sail that would needed for braking off the interstellar medium [15]. The long arrival times of a minimum of several thousand years require however an in depth analysis of the rational for carrying out this kind of comparatively slow-cruising interstellar missions. One possibility would be the Genesis project [35], which proposes to initiate the development of precambrian ecospheres of unicellular organisms on transiently habitable exoplanets.
We have pointed out, in addition, that the existence of habitable but sterile oxygen planets would alter radically our view of our cosmic neighborhood, in particular from the perspective of interstellar mission planing. The number of potentially habitable Mdwarf planets has been estimated to be substantial [63], with the consequence that it is not implausible that a rich biosphere might be detected eventually on a nearby M-dwarf planet via remote sensing. Biosphere compatibility considerations suggest in this case that we should not consider in-situ investigations of exoplanets teeming with life [35], with the reason being that such an endeavor could be catastrophic for the indigenous biosphere.
The situation changes, in contrast, if the target habitable planet contains a substantial amount of primordial atmospheric oxygen and if primordial oxygen preempts the emergence of life. Habitable oxygen planets would then be sterile. Oxygen, which is otherwise a preconditions for multi-cellular and hence complex life to thrive, is expected to be generated in vast amounts during the the initial runaway greenhouse state occurring during the extended Kevin Helmholtz contraction phase of nominally habitable late M dwarfs planets [42]. It is presently not known if the resulting primordial oxygen atmosphere, which may differ drastically from planet to planet in volume [44], would inhibit life to originate in first place. The existence of habitable but sterile oxygen planets, that is of worlds that would offer terrestrial life nearly unlimited grounds for the pursue of new evolutionary pathways, would revolutionize in any case our view of our galactic neighborhood.
The initial Kevin Helmholtz contractions phase of yellow G-class stars like our sun is relatively short, typically of the order of several million years. Potentially habitable planets orbiting not a M dwarf, but G stars, are hence not forced to go through an extended initial Greenhouse state, even though they can enter one, like Venus, as a consequence of the final orbital parameters. One may speculate whether this circumstance is the reason why earth is not orbiting a red M dwarf, the most frequent star type of the galaxy, but a star type which is substantially less common, a yellow G star.
Regarding the difference between the protection of solar system bodies and exoplanets we have pointed out that the extended time scales necessary for an in-situ exploration of exoplanets changes the rational. Financing a deep-space mission taking several millenia cannot be justified along the lines of solar system exploration, viz for the advancement of science. It is interesting in this context to connect to the ongoing controversy [64], whether the hypothetical counterfactual of planetary protection, "You protect what you want to study, but you cannot study what you protect.", does impede the search for life on Mars [65]. Protecting life on exoplanets for the sake of science is in analogy not a valid rational, as it could be studied in any case only on time scales far exceeding standard human planning horizons.
Acknowledgments
The author thanks Andrés de la Escosura Navazo, Phil Holliger, Kepa Ruiz Mirazo, Steven Beckwith, William F. Martin and Michael J. Russell for exchanges regarding the likelihood of life emerging in the presence of oxygen.
W. F. Martin, private communication
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| [] |
[
"Regularity theory for nonlinear systems of SPDEs",
"Regularity theory for nonlinear systems of SPDEs"
] | [
"Dominic Breit \nLMU Munich\nMathematical Institute\nTheresienstraße 3980333MunichGermany\n"
] | [
"LMU Munich\nMathematical Institute\nTheresienstraße 3980333MunichGermany"
] | [] | We consider systems of stochastic evolutionary equations of the typewhere S is a non-linear operator, for instance the p-Laplacianwith p ∈ (1, ∞) and Φ grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity:where F(ξ) = (1 + |ξ|) p−2 2 ξ. If we have Uhlenbeck-structure then E ∇u q q is finite for all q < ∞ if the same is true for the initial data. | 10.1007/s00229-014-0704-8 | [
"https://arxiv.org/pdf/1311.2001v4.pdf"
] | 119,156,141 | 1311.2001 | 4bb8093dae4d0a9f709323fed82161a13411d576 |
Regularity theory for nonlinear systems of SPDEs
Aug 2014
Dominic Breit
LMU Munich
Mathematical Institute
Theresienstraße 3980333MunichGermany
Regularity theory for nonlinear systems of SPDEs
Aug 2014Parabolic stochastic PDE'sNon-linear Laplacian-type systemsExistence of weak solutionsRegularity of solutions2010 MSC: 35R60, 35D30, 60H15, 35K55, 35B65
We consider systems of stochastic evolutionary equations of the typewhere S is a non-linear operator, for instance the p-Laplacianwith p ∈ (1, ∞) and Φ grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity:where F(ξ) = (1 + |ξ|) p−2 2 ξ. If we have Uhlenbeck-structure then E ∇u q q is finite for all q < ∞ if the same is true for the initial data.
Introduction
We will study existence and regularity of solutions u : Q → R D , Q := (0, T ) × G with T > 0 and G ⊂ R d bounded, to systems of stochastic PDE's of the type du = div S(∇u) dt + Φ(u)dW t u(0) = u 0 .
(1.1)
Here S : R d×D → R d×D is a non-linear operator and u 0 some in general random initial datum. The most famous example is the p-Laplacian operator S(ξ) = (1 + |ξ|) p−2 ξ, ξ ∈ R d×D , (1.2) with p ∈ (1, ∞). Equation (1.1) is an abbreviation for u(t) = u 0 + t 0 div S(∇u) dσ + t 0 Φ(u) dW σ (1.3) P ⊗ L 1 -a.e. We assume that W is a Brownian motion with values in a Hilbert space (see section 2 for details) . We suppose linear growth of Φ -roughly speaking |Φ(u)| ≤ c(1 + |u|) and |Φ ′ (u)| ≤ c (for a precise formulation see (2.9) in section 2). The motivation for this is an interaction between the solution and the random perturbation caused by the Brownian motion. For large values of |u| we expect a larger perturbation than for small values.
Since the p-Laplace equation is a basic problem in non-linear PDE's it can be understood as a model-problem to a large class of equations. In view of applications we especially mention the flow of Non-Newtonian fluids (see for instance [DRW], [BrDS], [TeYo] and [MNRR]) which might be the topic of some future projects. The deterministic equivalent to the equation mentioned above is already well understood, we refer to [Gi], [Giu] and [Uh] for the stationary case and to [LaUrSo] [DiFr], [Wi] for the evolutionary situation. We also refer to the survey papers [Mi] and [DuMiSt] giving a nice overview.
Regarding the stochastic problem there is a lot of literature regarding the existence of solutions to nonlinear evolutionary equations. The popular variational approach by Pardoux [Pa] for SPDEs provides an existence theory for a quite general class of equations. It requires a Banach space V which is continuously embedded into the Hilbert space H on which the equation is considered. The main part of the equation is to be understood in the dual V * . In the situation (1.1)-(1.2) we have V =W 1,p (G) and H = L 2 (G). Although this does not include the case p ≤ 2d d+2 (this bound arises from Sobolev's Theorem) system (1.1) can still be treated by slightly modified arguments. For recent developments we refer to [LiRo] and [PrRo]. However, there is not much literature about the regularity for nonlinear stochastic problems like (1.1). Certain regularity results about nonlinear stochastic PDEs are known:
• In [Ho1] and [Ho2] semilinear stochastic PDEs are considered, were also regularity statements are shown. Anyway, the elliptic part of the equations studied there is still linear.
• Zhang [Zh] observes non-linear stochastic PDEs but only in space-dimension one.
• Very recently the regularity of certain nonlinear parabolic systems with stochastic perturbation was investigated in [BeFl]. The results are C α -estimates for the solution under a quadratic growth assumption.
The literature dedicated to the regularity theory for linear SPDEs is quite extensive, we refer to [Kr], [KrRo1], [KrRo2], [Fl1] and the references therein. The situation in the nonlinear case is different, as explained above and to our best knowledge there is nothing about regularity for the stochastic p-Laplacian system. Hence this is the aim of the present paper. We will prove the following statements:
• The weak solution u is a strong solution to (1.1) and it holds E sup Theorem 4 and Theorem and 5).
t∈(0,T ) G ′ |∇u(t)| 2 dx + T 0 G ′ |∇F(∇u)| 2 dx dt < ∞ (1.4) for all G ′ ⋐ G, where F(ξ) = (1 + |ξ|) p−2 2 ξ (see
• Let S(ξ) = ν(|ξ|)ξ for ν : [0, ∞) → [0, ∞) 1 and p > 2 − 4 d . Then the strong solution u satisfies
E T 0 G ′ |∇u| q dx dt < ∞ (1.5)
for all G ′ ⋐ G and all q < ∞ (see Theorem 7).
Remark 1. (a)
The estimate in (1.4) is the natural extension of the results for non-linear PDE's in the deterministic situation to the stochastic setting, see [DuMiSt] (chapter 5). In the deterministic case, it is also quite standard to get regularity results in time: testing with ∂ 2 t u gives ∂ t u ∈ L ∞ (L 2 ) and ∂ t F(∇u) ∈ L 2 (Q). Due to the appearance of the Brownian motion such a results cannot be true for the stochastic problem.
(b) We consider only the non-degenerated case, see (2.8). However, the regularity estimates are independent of Λ which means it is possible to obtain results for the degenerate case via approximation.
(c) It is only a technical matter to assume general Dirichlet boundary conditions. In order to keep the proofs easier, we assume them to be zero.
(d) A lot of other statements which are known in the deterministic situation are still open for the stochastic problem. For instance partial regularity for the parabolic problems with p-structure which is shown in [DuMi] via the A-caloric approximation method.
(e) The proof of (1.5) is based on Moser iteration (see for instance [GT], ch. 8.5, for a nice presentation in the stationary deterministic case). Moser iteration in the stochastic setting also appears in [DMS1]- [DMS3]. The authors study estimates and maximum princples for the solution to SPDEs with a linear operator in the main part. This paper is concerned with gradient estimates for nonlinear systems of SPDEs.
Our procedure is as follows: In section 3 we study the case p ≥ 2. We apply the difference quotient method to gain higher differentiability and the corresponding estimates in the superquadratic case (section 4). Since this does not work immediately if p < 2 we approximate by a quadratic problem and show uniform estimates. We have to combine the techniques from non-linear PDE's with stochastic calculus for martingales. Note that it is not possible to work directly with test functions. Instead we apply Itô's formula to certain functions of u. Finally we prove arbitrarily high integrability of ∇u under special structure assumptions. This is done by Moser iteration.
Probability framework
Let (Ω, F, P) be a probability space equipped with a filtration {F t , 0 ≤ t ≤ T }, which is a nondecreasing family of sub-σ-fields of F, i.e. F s ⊂ F t for 0 ≤ s ≤ t ≤ T . We further assume that {F t , 0 ≤ t ≤ T } is right-continuous and F 0 contains all the P-negligible events in F. For a Banach space (X, · X ) we denote for 1 ≤ p < ∞ by L p (Ω, F, P; X) the Banach space of all F-measurable functions v : Ω → X such that
E v p X < ∞,
where the expectation is taken w.r.t. (Ω, F, P). Let U, H be two separable Hilbert spaces and let (e k ) k∈N be an orthonormal basis of U . We denote by L 2 (U, H ) the set of Hilbert-Schmidt operators from U to H . Throughout the paper we consider a cylindrical Wiener process W = (W t ) t∈[0,T ] which has the form
W(σ) = k∈N β k (σ)e k (2.6)
with a sequence (β k ) of independent real valued Brownian motions on (Ω, F, P). Now t 0 ψ(σ)dW σ , ψ ∈ L 2 (Ω, F, P; L 2 (0, T ; L 2 (U, H ))), with ψ progressively (F t )-measurable, defines a P-almost surely continuous L 2 (Ω) D valued F t -martingale. 2 Moreover, we can multiply with test-functions since
t 0 ψ(σ)dW σ , ϕ H = ∞ k=1 t 0 ψ(σ)(e k ), ϕ H dβ k (σ), ϕ ∈ H ,
is well-defined (the series converges in L 2 (Ω, F, P; C[0, T ])).
Our actual aim is the study of the system (1.1), where H = L 2 (G), V =W 1,p (G):
du = div S(∇u) dt + Φ(u) dW t u(0) = u 0 , (2.7)
where S : R d×D → R d×D is C 1 and fulfils
λ(1 + |ξ|) p−2 |ζ| 2 ≤ DS(ξ)(ζ, ζ) ≤ Λ(1 + |ξ|) p−2 |ζ| 2 (2.8)
for all ξ, ζ ∈ R d×D with some positive constants λ, Λ and p ∈ (1, ∞). Suppose that Φ satisfies (2.9).
Definition 2 (weak solution). Let W be a Brownian motion as in (2.6) on a probability space (Ω, F, P) with filtration
(F t ). A function u ∈ L 2 (Ω, F, P; L ∞ (0, T ; L 2 (G))) ∩ L p (Ω, F, P; L p (0, T ;W 1,p (G))) which is progressively (F t )-measureable is called a weak solutions to (1.1) if for every ϕ ∈ C ∞ 0 (G) it holds for a.e. t G u(t) · ϕ dx + G t 0 S(∇u(σ)) : ∇ϕ dx dσ = G u 0 · ϕ dx + G t 0 Φ(u) dW σ · ϕ dx P-almost surely.
In order to show regularity of solutions we suppose the following linear growth assumptions on Φ (following [Ho1]): For each z ∈ L 2 (G) there is a mapping Φ(z) : U → L 2 (G) D defined by Φ(z)e k = g k (·, z(·)). In particular, we suppose that g k ∈ C 1 (G × R D ) and the following conditions
k∈N |g k (x, ξ)| 2 ≤ c(1 + |ξ| 2 ), k∈N |∇ ξ g k (x, ξ)| 2 ≤ c, ξ ∈ R D , k∈N |∇ x g k (x, ξ)| 2 ≤ c(1 + |ξ| 2 ).
(2.9)
Definition 3 (strong solution).
A weak solution is called a strong solutions to (1.1) if div S(∇u) ∈ L 1 (Ω, F, P; L 1 loc (Q)) and
u(t) = u 0 + t 0 div S(∇u) dσ + t 0 Φ(u) dW σ holds P ⊗ L d+1 -a.e.
2 for stochastic calculus in infinite dimensions we refer to [DaZa] 3. Regularity for p ≥ 2
Throughout this section we study problems of the type (1.1) with (2.8) for p ≥ 2. In the following section we consider subquadratic problems regularized by quadratic ones.
Theorem 4 (Regularity). Assume u 0 ∈ L 2 (Ω, F 0 , P,W 1,2 (G)), (2.8) with p ≥ 2 and (2.9). Then the unique weak solution u to (1.1) is a strong solution and satisfies E sup
t∈(0,T ) G ′ |∇u(t)| 2 dx + T 0 G ′ |∇F(∇u)| 2 dx dt < ∞ for all G ′ ⋐ G.
Proof. Since u 0 ∈ L 2 (Ω, F 0 , P,W 1,2 (G)) and p ≥ 2 the existence of a unique weak solution (in the sense of defintion 2) follows by the common variational approach (see for instance [PrRo]) and satisfies
• u ∈ L 2 (Ω, F, P; L ∞ (0, T ; L 2 (G))); • u ∈ L p (Ω, F, P; L p (0, T ;W 1,p (G))).
We consider a cut-off function η ∈ C ∞ 0 (G) and the difference quotient
∆ γ h in direction γ ∈ {1, ..., d} with |h| < 1 2 dist(supp η, ∂Ω). We apply Itô's formula to the function f (v) = 1 2 η∆ γ h v 2 L 2 (G)
. In appropriate version it is shown in [DHoV], Prop. A.1. Although only the L 2 -case is considered there it is straightforward to extend it to the L p -setting.This shows
1 2 η∆ γ h u(t) 2 L 2 (G) = 1 2 η∆ γ h u 0 2 L 2 (G) + t 0 f ′ (u)du σ + 1 2 t 0 f ′′ (u)d u σ = 1 2 η∆ γ h u 0 2 L 2 (G) + G t 0 η 2 ∆ γ h u σ dx + 1 2 G t 0 η 2 d · 0 ∆ γ h Φ(u) dW σ dx =: (I) + (II) + (III).
We consider the three integrals separately. For the second one we get
(II) = −(II) 1 − (II) 2 + (II) 3 , (II) 1 := t 0 G η 2 ∆ γ h S(∇u) : ∆ γ h ∇u dx dσ, (II) 2 := t 0 G ∆ γ h S(∇u) : ∇η 2 ⊗ ∆ γ h u dx dσ, (II) 3 := t 0 G η 2 ∆ γ h u · ∆ γ h Φ(u) dW σ dx.
Using the assumptions for S in (2.8) we get
(II) 1 = t 0 G η 2 1 0 DS(∇u + sh∆ γ h ∇u) ds ∆ γ h ∇u, ∆ γ h ∇u dx dσ ≥ λ t 0 G η 2 1 0 (1 + |∇u + sh∆ γ h ∇u|) p−2 ds|∆ γ h ∇u| 2 dx dσ ≥ c t 0 G η 2 (1 + |∇u| + |h∆ γ h ∇u|) p−2 |∆ γ h ∇u| 2 dx dσ ≥ c t 0 G η 2 |∆ γ h F(∇u)| 2 dx dσ.
In the second last step we used [AF], Lemma 2.1. For the second term we obtain by similar arguments
(II) 2 ≤ c t 0 G η 1 0 (1 + |∇u + sh∆ γ h ∇u|) p−2 ds|∆ γ h ∇u||∇η||∆ γ h u| dx dσ ≤ δ t 0 G η 2 1 0 (1 + |∇u + sh∆ γ h ∇u|) p−2 ds|∆ γ h ∇u| 2 dx dσ + c(δ) t 0 G 1 0 (1 + |∇u + sh∆ γ h ∇u|) p−2 ds|∇η| 2 |∆ γ h u| dx dσ ≤ c(δ) t 0 supp η (1 + |∇u| + |h∆ γ h ∇u|) p−2 |∆ γ h u| 2 dx dσ + δ t 0 G η 2 |∆ γ h F(∇u)| 2 dx dσ.
Here we used Young's inequality for an arbitrary δ > 0. Moreover, we have by (2.9)
(III) = 1 2 G t 0 η 2 d · 0 ∆ γ h Φ(u) dW σ dx = 1 2 k G t 0 η 2 d · 0 ∆ γ h Φ(u)e k dβ k σ dx ≤ 1 2 k G t 0 η 2 1 0 ∇ ξ g k (·, u + sh∆ γ h u) ds ∆ γ h u 2 dσ dx + 1 2 k G t 0 η 2 1 0 ∂ γ g k (x + she γ , u) ds 2 dσ dx ≤ c G t 0 η 2 |∆ γ h u| 2 dσ dx + c G t 0 η 2 |u| 2 dσ dx.
Plugging all together and using E[(II) 3 ] = 0 we see
E G η 2 |∆ γ h u(t)| 2 dx + Q η 2 |∆ γ h F(∇u)| 2 dx dt ≤ c E G |∇u 0 | 2 dx + c t 0 E G η 2 |∆ γ h u| 2 + |u| 2 dx dσ + c E t 0 supp η η 2 (1 + |∇u| + |h∆ γ h ∇u|) p−2 |∆ γ h u| 2 dx dσ .
By Gronwall's Lemma and since u 0 ∈ L 2 (Ω, F 0 , P;W 1,2 (G)) we end up with
E G η 2 |∆ γ h u(t)| 2 dx + Q η 2 |∆ γ h F(∇u)| 2 dx dt ≤ c(η) 1 + E t 0 supp η (1 + |∇u| + |h∆ γ h ∇u|) p−2 |∆ γ h u| 2 dx dσ .
Here we also took into account u ∈ L 2 (Ω × Q). If p > 2 we gain by Young's inequality for the exponents p 2 and p
p−2 3 (RHS) ≤ c(η) 1 + t 0 G |∇u| p dx dσ + t 0 supp η |h∆ γ h ∇u| p dx dσ ≤ c(η) 1 + t 0 G |∇u| p dx dσ
which is bounded as well (independent of h). This means we have shown
E G η 2 |∆ γ h u(t)| 2 dx + Q η 2 |∆ γ h F(∇u)| 2 dx dt ≤ c(η).
Now we want to interchange supremum and expectation value. Applying similar arguments as before we obtain
E sup (0,T ) G η 2 |∆ γ h u(t)| 2 dx + E Q η 2 |∆ γ h F(∇u)| 2 dx dt ≤ c(η) + c E sup (0,T ) |(II) 3 | .
(3.10)
Using the assumptions on W (see (2.9)) we see
(II) 3 = G t 0 η 2 ∆ γ h u · ∆ γ h Φ(u)e k dβ k (σ) dx = k G t 0 η 2 ∆ γ h u · ∆ γ h g k (·, u) dβ k (σ) dx = k G t 0 η 2 1 0 ∇ ξ g k (·, u + sh∆ γ h u) ds (∆ γ h u, ∆ γ h u) dβ k (σ) dx + k G t 0 η 2 ∆ γ h u · 1 0 ∂ γ g k (x + she γ , u) ds dβ k (σ) dx = G t 0 η 2 G ξ (∆ γ h u, ∆ γ h u) dβ k (σ) dx + G t 0 η 2 G x (u) · ∆ γ h u dβ k (σ) dx =: (II) 1 3 + (II) 2 3
where we abbreviated
G ξ := k G ξ k := k 1 0 ∇ ξ g k (·, u + sh∆ γ h u) ds , G x (u) := k G x k (u) := k 1 0 ∂ γ g k (x + she γ , u) ds.
3 This step is trivial if p = 2.
On account of assumption (2.9) Burkholder-Davis-Gundy inequality and Young's inequality imply for arbitrary δ > 0 E sup
t∈(0,T ) |(II) 1 3 | ≤ E sup t∈(0,T ) t 0 k G η 2 G ξ k (∆ γ h u, ∆ γ h u) dx dβ k (σ) ≤ c E T 0 G η 2 G ξ (∆ γ h u, ∆ γ h u) dx 2 dt 1 2 ≤ c E T 0 G η 2 |∆ γ h u| 2 dx 2 dt 1 2 ≤ δ E sup (0,T ) G η 2 |∆ γ h u| 2 dx + c(δ) E Q η 2 |∆ γ h u| 2 dx dt .
By similar arguments we gain
E sup t∈(0,T ) |(II) 2 3 | ≤ c E T 0 G η 2 G x (u) · ∆ γ h u dx 2 dt 1 2 ≤ c E T 0 G η 2 |∆ γ h u||u| dx 2 dt 1 2 ≤ c E sup (0,T ) G η 2 |u| 2 dx + c E Q η 2 |∆ γ h u| 2 dx dt .
Combining this with (3.10), using u ∈ L 2 (Ω, F, P; L ∞ (0, T ; L 2 (G))) and choosing δ sufficiently small shows
E sup (0,T ) G η 2 |∆ γ h u(t)| 2 dx + E Q η 2 |∆ γ h F(∇u)| 2 dx dt ≤ c(η). (3.11)
This finally proves the claim (see [BeFl], section 3.2, for difference quotients and differentiability in the stochastic setting).
The subquadratic case: p < 2
Throughout this section we study problems of the type (1.1) with (2.8) and p ≤ 2. We add the Laplacian to the main part in order to get a problem with quadratic growth. Let u ε be the solution to
du ε = div S(∇u ε ) dt + ε∆udt + Φ(u ε )dW t , u(0) = u 0 .
(4.12)
From Theorem 4 we know that the solution has the following properties
• u ε ∈ L 2 (Ω, F, P; L ∞ (0, T ; L 2 (G)));
• ∇u ε ∈ L 2 (Ω, F, P; L 2 (0, T ; W 1,2 loc (G))). We will prove the following a priori estimates which are uniform in ε:
E sup t∈(0,T ) G |u ε (t)| 2 dx + Q |∇u ε | p dx dt + ε Q |∇u ε | 2 dx dt ≤ c 1 + E G |u 0 | 2 dx .
(4.13)
We apply Itô's formula to the function f (v) = 1 2 v 2 L 2 (G) which shows
1 2 u ε (t) 2 L 2 (G) = 1 2 u 0 2 L 2 (G) + t 0 f ′ (u ε )du ε σ + 1 2 t 0 f ′′ (u ε ) d u ε σ = 1 2 u 0 2 L 2 (G) − ε G t 0 |∇u ε | 2 dx dσ − G t 0 S(∇u ε ) : ∇u ε dx dσ + G t 0 u ε · Φ(u ε ) dW σ dx + G t 0 d · 0 Φ(u ε ) dW σ dx.
(4.14)
Now we can follow, building expectations and using (2.8), that
E G |u ε (t)| 2 dx + ε t 0 G |∇u ε | 2 dx dσ + t 0 G |∇u ε | p dx dσ ≤ c E 1 + u 0 2 L 2 (G) + E J 1 (t) + E J 2 (t) .
Here we abbreviated
J 1 (t) = G t 0 u ε · Φ(u ε ) dW σ dx, J 2 (t) = G t 0 d · 0 Φ(u ε ) dW σ dx.
Using (2.9) we gain
E[J 2 ] = E t 0 ∞ i=1 G |Φ(u ε )e i | 2 dx dσ = E t 0 ∞ i=1 G |g i (·, u ε )| 2 dx dσ ≤ c E 1 + t 0 G |u ε | 2 dx dσ .
Clearly, we have E[J 1 ] = 0. So interchanging the time-integral and the expectation value and applying Gronwall's Lemma leads to
sup t∈(0,T ) E G |u ε (t)| 2 dx + εE Q |∇u ε | p dx dt + E Q |∇u ε | p dx dt ≤ c E 1 + G |u 0 | 2 dx . (4.15) A similar observation shows E sup t∈(0,T ) G |u ε (t)| 2 dx ≤ c E 1 + G |u 0 | 2 dx + T 0 G |u ε | 2 dx dσ + c E sup t∈(0,T ) |J 1 (t)| .
(4.16)
On account of the Burkholder-Davis-Gundy inequality, (2.9) and Young's inequality we obtain for arbitrary κ > 0 E sup
t∈(0,T ) |J 1 (t)| = E sup t∈(0,T ) t 0 G u ε Φ(u ε ) dx dW σ = E sup t∈(0,T ) t 0 i G u ε · g i (·, u ε ) dx dβ i (σ) ≤ c E T 0 i G |u ε |g i (·, u ε ) dx 2 dt 1 2 ≤ c E 1 + T 0 G |u ε | 2 dx 2 dσ 1 2 ≤ κE sup t∈(0,T ) G |u ε | 2 dx + c(κ)E 1 + T 0 G |u ε | 2 dx dσ
Inserting this in (4.16), choosing κ small enough and using (4.15) proves (4.13). After passing to a (not relabeled) subsequence we have for a certain function u u ε ⇁ u in L p (Ω, F, P; L p (Q)), u ε ⇁ u in L 2 (Ω, F, P; L r (0, T ; L 2 (G))) ∀r < ∞, ∇u ε ⇁ ∇u in L p (Ω, F, P; L p (Q)), ε∇u ε → 0 in L 2 (Ω, F, P; L 2 (Q)).
(4.17)
Theorem 5 (Regularity). Assume (2.8) with p ≤ 2, (2.9) and u 0 ∈ L 2 (Ω, F 0 , P,W 1,2 (G)). Then there is a unique weak solution u to (1.1) which is a strong solution and satisfies E sup
t∈(0,T ) G ′ |∇u(t)| 2 dx + T 0 G ′ |∇F(∇u)| 2 dx dt < ∞ for all G ′ ⋐ G, where F(ξ) = (1 + |ξ|) p−2 2 ξ.
Remark 6. In cthe ase 1 < p < 2d d+2 even the existence of a weak solution is not contained in literature. In this case no Gelfand triple is available and hence the general results for evolutionary SPDEs based on the variational approach (see for instance [PrRo,Thm. 4
.2.4])
do not hold. The uniqueness is again classical and follows from the monotonicity of the coefficients.
Proof. From the proof of Theorem 4 we can quote (recall (4.13))
E G η 2 |∆ γ h u ε (t)| 2 dx + Q η 2 |∆ γ h F(∇u ε )| 2 dx dt ≤ c(η) 1 + E t 0 supp η (1 + |∇u ε | + |h∆ γ h ∇u ε |) p−2 |∆ γ h u ε | 2 dx dσ + c(η)ε E t 0 supp η |∆ γ h u ε | 2 dx dσ
since the arguments up to this step also work for p ≤ 2. All involved quantities have weak derivatives so we can go to the limit h → 0. This shows by (4.13)
E G η 2 |∇u ε (t)| 2 dx + Q η 2 |∇F(∇u ε )| 2 dx dt ≤ c(η) 1 + E t 0 G (1 + |∇u ε |) p−2 |∇u ε | 2 dx dσ + c(η)ε E t 0 G |∇u ε | 2 dx dσ . ≤ c(η) 1 + E t 0 G (1 + |∇u ε |) p−2 |∇u ε | 2 dx dσ ≤ c(η) 1 + E t 0 G |∇u ε | p dx dσ .
Using similar arguments as in the last section we can interchange supremum and integral and conclude E sup
t∈(0,T ) G η 2 |∇u ε (t)| 2 dx + Q η 2 |∇F(∇u ε )| 2 dx dt ≤ c(η) 1 + E Q |∇u ε | p dx dσ ≤ c(η).
(4.18)
Now we have to go to the limit in the equation. We get
S(∇u ε ) ⇁:S in L p ′ (Ω, F, P; L p ′ (Q)), Φ(u ε ) ⇁:Φ in L 2 (Ω, F, P; L 2 (0, T ; L 2 (U, L 2 (G) D ))). (4.19)
One can now pass to the limit in the equation to obtain the corresponding equation for u with S andΦ instead of S(∇u) and Φ(u), respectively. The passage to the limit in the stochastic integral is justified since the mapping L 2 (Ω, F, P; L 2 (0, T ; L 2 (U ; L 2 (G)))) → L 2 (Ω, F, P; L 2 (0, T ; L 2 (G))),
ϕ → t 0 ϕ dW σ ,
is continuous hence weakly continuous. We have to show thatS = S(∇u) andΦ = Φ(u) hold. Subtracting the formula for u ε 2 L 2 (G) and u 2 L 2 (G) (see (4.14)) shows
1 2 E G |u ε (T ) − u(T )| 2 dx +E G T 0 S(∇u ε ) − S(∇u) : ∇ u ε − u dx dσ + ε E T 0 G |∇u ε | 2 dx dσ = E − G u ε (T ) · u(T ) dx + E G T 0 S − S(∇u ε ) : ∇u dx dσ − G T 0 S(∇u) : ∇ u ε − u dx dσ + E G T 0 u ε · Φ(u ε )dW σ − u ·ΦdW σ dx + E G T 0 d · 0 Φ(u ε )dW σ − · 0Φ dW σ dx .
By (4.17) u ε (T ) is bounded in L 2 (Ω×G, P⊗L d ). Which gives u ε (T ) ⇁ u(T ) in the same space at least for a subsequence (note that both are weakly continuous in L 2 (Ω × G, P ⊗ L d ) with respect to t which can be shown by the equations). Letting ε → ∞ shows for a subsequence using (4.17) and (4.19)
lim ε E G |u ε (T ) − u(T )| 2 dx + G T 0 S(∇u ε ) − S(∇u) : ∇ u ε − u dx dσ ≤ lim ε E G T 0 d · 0 Φ(u ε )dW σ − · 0Φ dW σ dx .
Following essential ideas of [ChCh] (section 6) the last integralT can be written as
T = i E G T 0 |Φ(u ε )e i | 2 dx dσ − i E G T 0 |Φe i | 2 dx dσ = E T 0 Φ(u ε ) 2 L 2 (U,L 2 (G)) dt − E T 0 Φ 2 L 2 (U,L 2 (G)) dt = E T 0 Φ(u ε ) −Φ 2 L 2 (U,L 2 (G)) dt + 2 E T 0 Φ(u ε ),Φ L 2 (U,L 2 (G)) dt − 2 E T 0 Φ 2 L 2 (U,L 2 (G)) dt .
On account of (4.19) for ε → 0 we only have to consider the first term which can be written as
E T 0 Φ(u ε ) −Φ 2 L 2 (U,L 2 (G)) dt = E T 0 Φ(u ε ) − Φ(u) 2 L 2 (U,L 2 (G)) dt − E T 0 Φ(u) −Φ 2 L 2 (U,L 2 (G)) dt + 2 E T 0 Φ(u ε ) −Φ, Φ(u) −Φ L 2 (U,L 2 (G)) dt
Using again (4.19) and also (2.9) implies
lim εT ≤ lim ε E T 0 Φ(u ε ) − Φ(u) 2 L 2 (U,L 2 (G)) dt ≤ c lim ε E T 0 G |u ε − u| 2 dx dt .
We finally gain on account of Grownwall's lemma after interchanging expectation and integral
E G T 0 S(∇u ε ) − S(∇u) : ∇ u ε − u dx dσ = 0.
From this we deduce, by monotonicity of S that
∇u ε −→ ∇u P ⊗ L d+1 − a.e.
This means we have shownS = S(∇u). Now we combine the uniform L p -estimates for ∇u ε with Vitali's Theorem to get ∇u ε −→ ∇u in L q (Ω × (0, T ) × G; P ⊗ L d+1 ) for all q < p.
(4.20)
Of course this also means compactness of u ε in the same space (we have zero traces). Therefore, we gainΦ = Φ(u). Now we can pass to the limit in the approximated equation and finish the proof of Theorem 5.
Uhlenbeck-structure
In order to get better results we assume Uhlenbeck structure for the non-linear tensor S. If D ≥ 2 we suppose
S(ξ) = ν(|ξ|)ξ (5.21) for a C 1 -function ν : [0, ∞) → [0, ∞).
Theorem 7 (Higher integrability). Assume (2.8), (5.21), (2.9) and u 0 ∈ L q (Ω, F 0 , P,W 1,q (G)) for all q < ∞. If p > 2 − 4 d then the solution u to (1.1) satisfies
E T 0 G ′ |∇u| q dx dt < ∞
for all G ′ ⋐ G and all q < ∞.
Since we now assume higher moments for the initial data we gain higher moments for the solution as well.
Lemma 8.
Under the assumptions of Theorem 7 we have E sup
t∈(0,T ) G |u(t)| 2 dx + T 0 G |∇u| p dx dt q < ∞ for all q < ∞.
Proof. Due to the regularity results from Theorem 4 and 5 we have a strong solution and Itô's formula can be directly applied to the funtion f (v) = 1 2 v 2 L 2 (G) . Using the growth condition on S from (2.8), taking the supremum and the q-th power of both sides of the equation and applying expectations shows
E sup (0,T ) G |u(t)| 2 dx + T 0 G |∇u| p dx dσ q ≤ c E 1 + G |u 0 | 2q dx + E sup (0,T ) |J 1 (t)| q + E sup (0,T ) |J 2 (t)| q , J 1 (t) = G t 0 u · Φ(u) dW σ dx, J 2 (t) = G t 0 d · 0 Φ(u) dW σ dx.
Using (2.9) we gain E sup
t∈(0,T ) |J 2 (t)| q = E sup t∈(0,T ) t 0 ∞ i=1 G |Φ(u)e i | 2 dx dσ q ≤ E T 0 ∞ i=1 G |g i (·, u)| 2 dx dσ q ≤ c E 1 + T 0 G |u| 2 dx dσ q .
On account of the Burkholder-Davis-Gundi inequality, (2.9) and Young's inequality we obtain for arbitrary ε > 0 E sup
t∈(0,T ) |J 1 (t)| q = E sup t∈(0,T ) i t 0 G u · g i (·, u) dx dβ i (σ) q ≤ c E T 0 i G u · g i (·, u) dx 2 dt q 2 ≤ c E 1 + T 0 G |u| 2 dx 2 dσ q 2 ≤ εE sup t∈(0,T ) G |u| 2 dx q + c(ε)E T 0 G |u| 2 dx dσ q .
Choosing ε small enough and using Gronwall's lemma proves the claim. Since the calculations above are not well-defined a priori one can work with a quadratic approximation for the function z → z q .
Before we begin with the proof of Theorem 7 which is based on the Moser iteration (see [GT] for a nice presentation in the easier elliptic case) we need some preparations. The basic idea is estimating higher powers of |∇u| by lower powers and iterate this. Therefore, we define h(s) := s 0 (1 + θ) α θ dθ, α ≥ 0, which behaves like s α+2 for large s. Unfortunately we cannot work directly with h, we need an approximation h L which grows quadratically and converges to h. We follow the approach in [BiFu] and define for L ≫ 1
h L (s) := s 0 τ g L (τ ) dτ, g L (τ ) := g(0) + τ 0 ψ(θ)g ′ (θ) dθ, g(θ) := h ′ (θ) θ .
(5.22)
Here ψ ∈ C 1 ([0, ∞)) denotes a cut-off function with the properties 0 ≤ ψ ≤ 1, ψ ′ ≤ 0, |ψ ′ | ≤ c/L, ψ ≡ 1 on [0, 3L/2] and ψ ≡ 0 on [2L, ∞). For the function h L we obtain the following properties (see [Br], Lemma 2.1, and [Br2], section 2)
Lemma 9. For the sequence (h L ) we have:
(a) h L ∈ C 2 [0, ∞), h L (s) = h(s) for all t ≤ 3L/2 and lim L→∞ h L (s) = h(s) for all s ≥ 0; (b) h L ≤ h, g L ≤ g and h ′′ L ≤ c(L) on [0, ∞); (c) It holds h ′ L (s) s ≤ h ′′ L (s) ≤ c(α + 1) h ′ L (s) s and h ′ L (s)s ≤ ch L (s) uniformly in L. (d) We have for all s, t ≥ 0 uniformly in L h ′ L (s) s t 2 ≤ c 1 + h L (s) + h L (t)t 2 .
With this preparations the following calculations are well-defined by Theorem 4 and Theorem 5.
Lemma 10.
Under the assumptions of Theorem 7 we have E sup
t∈(0,T ) G |u(t)| q dx < ∞ for all q < ∞.
Proof. We apply Itô's formula to the function
f L (v) := G H L (v) dx := G h L (|v|) dx,
where h L is defined in (5.22) and set α = q − 2. We obtain
G h L (|u|) dx = G η 2 h L (|u 0 |) dx + t 0 f ′ L (u)du σ + 1 2 t 0 f ′′ L (u) d u σ σ = G η 2 h L (|u 0 |) dx + G t 0 DH L (u) · du σ dx + G t 0 D 2 H L (u) d · 0 Φ(u) dW σ dx =: (I) q + (II) q + (III) q .
We consider the three integrals separately and decompose the second one into
(II) q = −(II) 1 q − (II) 2 α + (II) 3 q , (II) 1 q := t 0 G h ′ L (|u|) |u| S(∇u) : ∇u dx, (II) 2 q := t 0 G S(∇u) : ∇ h ′ L (|u|) |u| ⊗ u dx dσ, (II) 3 q := t 0 G h ′ L (|u|) |u| u · Φ(u) dW σ dx.
Using the Uhlenbeck structure (5.21) and Lemma 9 c) we gain
(II) 1 q = t 0 G h ′ L (|u|) |∇u| ν(|∇u|)|∇u| 2 dx ≥ 0, (II) 2 q = t 0 G ν(|∇u|)∇u : h ′′ L (|u|)|u|−h ′ L (|u|) |u| 2 ∇|u| ⊗ u dx dσ = 1 4 t 0 G ν(|∇u|) h ′′ L (|u|)|u|−h ′ L (|u|) |u| 3 ∇|∇u| 2 2 dx dσ ≥ 0.
This and the assumptions on u 0 imply E sup
t∈(0,T ) G h L (|u|) dx ≤ c E 1 + sup t∈(0,T ) |(II) 3 q | + sup t∈(0,T ) |(III) q |
We have by (2.9) and Lemma 9
sup
t∈(0,T ) |(III) q | = 1 2 k G T 0 D 2 H L (u) d · 0 g k (·, u)dβ k σ dx ≤ 1 2 k G T 0 |D 2 H L (u)||g k (·, u)| 2 dσ dx ≤ c(q) k G T 0 h ′ L (|u|) |u| |g k (·, u)| 2 dσ dx ≤ c(q) G T 0 h ′ L (|u|)|u| 2 dσ dx ≤ c(q) G T 0 h L (|u|) dσ dx.
Similar to the proof of Theorem 4 we gain using
h ′ L (s) s + h ′′ L (s) s 2 ≤ c(α)h L (s) uniformly in L (recall Lemma 9 c)) E sup t∈(0,T ) |(II) 3 q | ≤ c E T 0 G h L (|u|) dx 2 dt 1 2 ≤ δ E sup (0,T ) G h L (|u|) dx + c(δ) E Q h L (|u|) dx dt .
Finally we have shown
E sup t∈(0,T ) G h L (|u|) dx ≤ c E 1 + T 0 G h L (|u|) dx dt and by Gronwall's Lemma E sup t∈(0,T ) G h L (|u|) dx ≤ c
uniformly in L. Passing to the limit L → ∞ yields the claim.
Proof. (of Theorem 7) We apply Itô's formula to the function
f L (v) := G η 2 H L (∇v) dx := G η 2 h L (|∇v|) dx,
where η ∈ C ∞ 0 (G) is a cut-off function and h L is defined in (5.22). We obtain
G η 2 h L (|∇u|) dx = G η 2 h L (|∇u 0 |) dx + t 0 f ′ L (u)du σ + 1 2 t 0 f ′′ L (u) d u σ σ = G η 2 h L (|∇u 0 |) dx + G t 0 η 2 DH L (∇u) : d∇u σ dx + G t 0 η 2 D 2 H L (∇u) d · 0 ∇ Φ(u) dW σ dx =: (I) α + (II) α + (III) α .
We consider the three integrals separately and decompose the second one into
(II) α = −(II) 1 α − (II) 2 α − (II) 3 α + (II) 4 α , (II) 1 α := t 0 G η 2 h ′ L (|∇u|) |∇u| DS(∇u) ∂ γ ∇u, ∂ γ ∇u , (II) 2 α := t 0 G h ′ L (|∇u|) |∇u| DS(∇u) ∂ γ ∇u, ∇η 2 ⊗ ∂ γ u dx dσ, (II) 3 α := t 0 G η 2 DS(∇u) ∂ γ ∇u, ∇ h ′ L (|∇u|) |∇u| ⊗ ∂ γ u dx dσ, (II) 4 α := t 0 G η 2 h ′ L (|∇u|) |∇u| ∇u : ∇ Φ(u) dW σ dx.
Using the assumptions on S, see (2.8), we obtain
(II) 1 α ≥ c t 0 G η 2 h ′ L (|∇u|) |∇u| (1 + |∇u|) p−2 |∇ 2 u| 2 dx dσ.
For the second term we gain for every δ > 0 using Young's inequality and Lemma 9
(II) 2 α ≤ δ(II) 1 α + c(δ) t 0 supp η h ′ L (|∇u|) |∇u| DS(∇u) ∇η 2 ⊗ ∂ γ u, ∇η 2 ⊗ ∂ γ u dx dσ ≤ δ(II) 1 α + c(δ) t 0 supp η (1 + |∇u|) p−2 h L (|∇u|) dx dσ.
Thanks to assumption (5.21) and Lemma 9 it holds 4
(II) 3 α = t 0 G η 2 DS ∂ γ ∇u, ∇ h ′ L (|∇u|) |∇u| ⊗ ∂ γ u dx dσ = 1 2 t 0 G η 2 DS e γ ⊗ ∇ h ′ L (|∇u|) |∇u| , e γ ⊗ ∇|∇u| 2 dx dσ = 1 2 t 0 G η 2 h ′′ L (|∇u|)|∇u|−h ′ L (|∇u|) |∇u| 3 DS e γ ⊗ ∇|∇u| 2 , e γ ⊗ ∇|∇u| 2 dx dσ ≥ 0.
Moreover, we have by (2.9) and Lemma 9
(III) α = 1 2 k G t 0 η 2 D 2 H L (∇u) d · 0 ∇ g k (·, u) dβ k σ dx ≤ 1 2 k G t 0 η 2 |D 2 H L (∇u)||∇ ·, g k (u) | 2 dσ dx ≤ k G t 0 η 2 h ′′ L (|∇u|) + h ′ L (|∇u|) |∇u| |∇ g k (·, u) 2 dσ dx ≤ c(α + 1) k G t 0 η 2 h ′ L (|∇u|) |∇u| |∇ ξ g k (·, u)∇u| 2 + |∇ x g k (·, u)| 2 dσ dx ≤ c(α + 1) G t 0 η 2 h ′ L (|∇u|) |∇u| |∇u| 2 + |u| 2 dσ dx ≤ c(α + 1) G t 0 η 2 1 + h L (|∇u|) + h L (|u|)|u| 2 dσ dx.
In the last step we applied Lemma 9 c) and d).Thus we obtain taking the supremum, the q-th power and applying expectations E sup If we choose δ small enough we can remove the term involving (II) 4 α from the right-hand-side of (5.23). By Gronwall's Lemma, the assumptions on u 0 and Lemma 10 we end up with
(0,T ) G η 2 h L (|∇u|) dx + T 0 G η 2 h ′ L (|∇u|) |∇u| (1 + |∇u|) p−2 |∇ 2 u| 2 dx dσ q ≤ c(η) E G h L (|∇u 0 |) dx + T 0 supp η (1 + |∇u|) p−2 h L (|∇u|) dx dσ q + c(α + 1) T 0 E G η 2 h L (|∇u|) + h L (|u|) dx q dσ + c E supE G η 2 h L (|∇u(t)|) dx + Q η 2 h ′ L (|∇u|) |∇u| (1 + |∇u|) p−2 |∇ 2 u| 2 dx dt q (5.24) ≤ c(η, α)E 1 + t 0 supp η (1 + |∇u|) p−2 h L (|∇u|) dx dσ q .
Assume for a moment that ∇u ∈ L q (Ω, F, P; L p+α ((0, T ) × G ′ )) ∀G ′ ⋐ G, ∀q < ∞, (5.25) u ∈ L q (Ω, F, P; L ∞ (0, T ; L α+2 (G ′ ))) ∀G ′ ⋐ G, ∀q < ∞, (5.26)
Then we are allowed to go to the limit L → ∞ on the r.h.s. of (5.24). By Fatou's Theorem we are now allowed to do this on the l.h.s. as well. We obtain E sup ∈ L q (Ω, F, P; L 2 (0, T ; W 1,2 loc (G))) ∩ L q (Ω, F, P; L ∞ (0, T ; L 2 α+2 α+p loc (G))) ∀q < ∞.
A parabolic interpolation (see for instance [Am], Thm. 3.1) shows on account of p > 2 − 4 d ∇u ∈ L q (Ω, F, P; L ω(α) (0, T ) × G ′ )) ∀G ′ ⋐ G, ∀q < ∞, (5.27) ω(α) := (p + α) 1 + 2 d α + 2 α + p .
(5.28)
Since (5.25) is true for α = 0 (by Lemma 8) we start an iteration procedure by α 0 := 0, α k+1 := ω(α k ) − p, k ∈ N.
On account of α k → ∞ the claim is proven.
Remark 11. As already observed in [DiFr], Remark 2.1., for the deterministic problem, it is not possible to obtain L ∞ -bounds for ∇u except of the case p = 2via Moser iteration. So it is an open question if one can gain Lipschitz regularity for the stochastic problem. In the deterministic case this is shown using the DeGiorgi method (see [DiFr], Lemma 2.3).
However it is not clear if similar arguments will work for stochastic problems.
( 0 ,
0T ) G η 2 h(|∇u(t)|) dx + Q η 2 |∇(1 + |∇u|)
the so-called Uhlenbeck-structure, see[Uh]
for a detailed explanation of this step we refer to[Bi], (32) on p. 62.
Acknowledgement• The work of the author was supported by Leopoldina (German National Academy of Science).• The author wishes to thank the referee for the careful reading of the manuscript and many helpful advises.
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| [] |
[
"PSI-PR-16-15 Complementarity in lepton-flavour violating muon decay experiments",
"PSI-PR-16-15 Complementarity in lepton-flavour violating muon decay experiments"
] | [
"A Crivellin \nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n",
"S Davidson \nIPNL\nCNRS/IN2P3\n4 rue E. Fermi69622Villeurbanne cedexFrance\n\nUniversité Lyon 1\nVilleurbanne\n\nUniversité de Lyon\nF-69622LyonFrance\n",
"G M Pruna \nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n",
"A Signer \nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n\nPhysik-Institut\nUniversität Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland\n"
] | [
"Paul Scherrer Institut\nCH-5232Villigen PSISwitzerland",
"IPNL\nCNRS/IN2P3\n4 rue E. Fermi69622Villeurbanne cedexFrance",
"Université Lyon 1\nVilleurbanne",
"Université de Lyon\nF-69622LyonFrance",
"Paul Scherrer Institut\nCH-5232Villigen PSISwitzerland",
"Paul Scherrer Institut\nCH-5232Villigen PSISwitzerland",
"Physik-Institut\nUniversität Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland"
] | [] | This note presents an analysis of lepton-flavour-violating muon decays within the framework of a low-energy effective field theory that contains higher-dimensional operators allowed by QED and QCD symmetries. The decay modes µ → eγ and µ → 3e are investigated below the electroweak symmetry-breaking scale, down to energies at which such processes occur, i.e. the muon mass scale. The complete class of dimension-5 and dimension-6 operators is studied systematically at the tree level, and one-loop contributions to the renormalisation group equations are fully taken into account. Current experimental limits are used to extract bounds on the Wilson coefficients of some of the operators and, ultimately, on the effective couplings at any energy level below the electroweak symmetry-breaking scale. Correlations between two couplings relevant to both processes illustrate the complementarity of searches planned for the MEG II and Mu3e experiments.a Corresponding author: [email protected]. | null | [
"https://arxiv.org/pdf/1611.03409v1.pdf"
] | 119,099,464 | 1611.03409 | e0269a69c5b2a9715942fce22975644295c2895f |
PSI-PR-16-15 Complementarity in lepton-flavour violating muon decay experiments
A Crivellin
Paul Scherrer Institut
CH-5232Villigen PSISwitzerland
S Davidson
IPNL
CNRS/IN2P3
4 rue E. Fermi69622Villeurbanne cedexFrance
Université Lyon 1
Villeurbanne
Université de Lyon
F-69622LyonFrance
G M Pruna
Paul Scherrer Institut
CH-5232Villigen PSISwitzerland
A Signer
Paul Scherrer Institut
CH-5232Villigen PSISwitzerland
Physik-Institut
Universität Zürich
Winterthurerstrasse 190CH-8057ZürichSwitzerland
PSI-PR-16-15 Complementarity in lepton-flavour violating muon decay experiments
This note presents an analysis of lepton-flavour-violating muon decays within the framework of a low-energy effective field theory that contains higher-dimensional operators allowed by QED and QCD symmetries. The decay modes µ → eγ and µ → 3e are investigated below the electroweak symmetry-breaking scale, down to energies at which such processes occur, i.e. the muon mass scale. The complete class of dimension-5 and dimension-6 operators is studied systematically at the tree level, and one-loop contributions to the renormalisation group equations are fully taken into account. Current experimental limits are used to extract bounds on the Wilson coefficients of some of the operators and, ultimately, on the effective couplings at any energy level below the electroweak symmetry-breaking scale. Correlations between two couplings relevant to both processes illustrate the complementarity of searches planned for the MEG II and Mu3e experiments.a Corresponding author: [email protected].
Introduction
This note presents a specific example of a correlation that occurs in lepton-flavour-violating (LFV) muonic decays in the context of effective field theories (EFTs).
Whilst in the neutrino sector evidence for LFV is now established beyond doubt 1,2,3 , the absence of experimental hints of LFV in the charged lepton sector, together with the smallness of the neutrino mass scale, indicate that a very incisive flavour conservation mechanism is at work. Although allowed in the Standard Model (SM) with right-handed neutrinos, the branching ratios (BRs) of such transitions are suppressed by (m ν /M W ) 4 , making them too small to be observable in any conceivable experiment. Consequently, any LFV production channel or decay mode offers a promising benchmark against which to search for physics beyond the SM.
Among charged LFV processes, muonic transition occurs in a relatively clean experimental environment, to the point that the MEG experiment has recently set a stringent limit 4 on BR (µ → eγ). This represents the strongest existing bound on 'forbidden' decays, while the SINDRUM result 5 obtained almost three decades ago is still very competitive with regard to the current experimental status of other sectors. The well-known outcomes of these experiments are:
Br (µ → eγ) ≤ 4.2 × 10 −13 ,(1)
Br (µ → 3e) ≤ 1.0 × 10 −12 .
Furthermore, there are good prospects for future MEG II and Mu3e experiments. The former is expected 6 to reach a limit of 4 × 10 −14 , while the latter might even achieve a four-orders-ofmagnitude improvement 7,8 on the existing limit. All the aforementioned experiments are being carried out at the Paul Scherrer Institut's experimental facilities. The present analysis does not consider LFV transitions in a nuclear environment (coherent and incoherent muon conversion in nuclei). See Refs 9 and 10 for extensive treatments of this topic.
From a theoretical perspective, LFV processes have been studied in many specific extensions of the SM. In some cases the matching of such extensions to a low-energy effective theory has also been considered 11,12 . However, this analysis follows a bottom-up approach in which effective interactions are included in a low-energy Lagrangian 13 that respects the SU (3) c and U (1) EM gauge symmetries. In exploiting the Appelquist-Carazzone theorem 14 , it is possible to extend the QCD and QED Lagrangian b with higher-dimensional operators
L eff = L QED+QCD + 1 Λ k C (5) k Q (5) k + 1 Λ 2 k C (6) k Q (6) k + O 1 Λ 3 .(3)
Here, Λ is the ultraviolet (UV) completion energy scale, which in this context is required not to exceed the electroweak symmetry-breaking (EWSB) scale, where the SM dynamic degrees of freedom and symmetries must be adequately restored 15,16 and matched with those of the low-energy theory.
Having established the theoretical background, the main focus is on the interpretation of correlations between operators in the BRs of both µ → eγ and µ → 3e at the muon mass energy scale and beyond. Experimental limits are then applied to the parameter space in a search for allowed regions.
The popular parametrisation of dipole and four-fermion LFV operators 17
L = m µ (k + 1) Λ 2 (μ R σ µν e L ) F µν + k (k + 1) Λ 2 (μ L γ µ e L ) f γ µ f ,(4)
where k is an ad hoc parameter to be interpreted strictly at the muon mass energy scale, allows to switch from a pure dipole interaction (k ∼ 0) to a pure four-fermion interaction (k ∼ ∞). Although this approach ensures a descriptive phenomenological understanding of the contributions of different operators to different observables, a more consistent theoretical approach can be achieved without losing interpretive power. The advantage of a systematic effective QFT approach lies in the fact that it can be used to link phenomenological observables at different energy scales unambiguously through the renormalisation-group evolution (RGE) of the Wilson coefficients. In this regard, the RGE between the muon mass energy scale and the EWSB scale is calculated at the leading order (up to the one-loop level) in QED and QCD for any operator contributing to LFV muon decays. This encompasses possible mixing effects among operators, which in this study are taken into account in a similar way to recent theoretical works 18 . From this analysis, it is possible to extract limits both on the Wilson coefficients defined at the phenomenological energy scale and on the coefficients defined at the UV matching scale.
This paper is organised as follows. Section 2 introduces the LFV effective Lagrangian, and in Section 3 the observables connected with the µ → eγ and µ → 3e searches are briefly discussed. Section 4 provides a brief phenomenological analysis, and in Section 5 conclusions are drawn. Formulae relevant to the RGE of the Wilson coefficients are provided in the appendix.
b Without the top quark field.
LFV effective Lagrangian at the muon energy scale
The Appelquist-Carazzone theorem 14 is exploited to construct an effective Lagrangians valid below the EWSB scale, with higher-dimensional operators that respect the QCD SU (3) c and QED U (1) EM symmetries. This allows for an interpretation of BSM effects at high energy scales in terms of new, non-renormalisable interactions at the low energy scale.
In this respect, all possible QCD and QED invariant operators relevant to µ → e transitions are considered up to dimension 6. These can be arranged in the following effective Lagrangian with dimensionless Wilson coefficients C and the decoupling energy scale M W ≥ Λ m b :
L eff = L QED+QCD + 1 Λ 2 C D L O D L + f =q, C V LL f f O V LL f f + C V LR f f O V LR f f + C S LL f f O S LL f f + f =q,τ C T LL f f O T LL f f + C S LR f f O S LR f f + L ↔ R + H.c.,(5)
where q and l specify that sums run over the quark and lepton flavours, respectively. The explicit structure of the operators is given by
O D L = e m µ (ēσ µν P L µ) F µν ,(6)
O
V LL f f = (ēγ µ P L µ) f γ µ P L f ,(7)
O
V LR f f = (ēγ µ P L µ) f γ µ P R f ,(8)O S LL f f = (ēP L µ) f P L f ,(9)O S LR f f = (ēP L µ) f P R f ,(10)
O
T LL f f = (ēσ µν P L µ) f σ µν P L f ,(11)
and an analogous notation is assumed for cases in which the L ↔ R exchange is applied. In the previous equations, the convention P L/R = 1 ∓ γ 5 /2 is understood. Apart from being multiplied by the QED coupling e, the operator in Eq.6 is also rearranged into a dimension-6 operator with an appropriate normalisation factor m µ . The reason is that this operator is directly related to a dimension-6 operator in the SMEFT 19,20 . Direct comparison of Eq. 5 and Eq. 4 reveals that the latter assumes a tree-level correlation between independent operators. This assumption is manifestly inconsistent when quantum fluctuations are considered. Notably, an analysis of LFV transitions in nuclei calls for a further dimension-7 operator relating to the leading-order muon-electron-gluon interaction, which is generated by threshold corrections induced by the heavy quark operators (see Ref. 21 for details).
Lepton-flavour-violating muonic observables
This section describes two of the most relevant LFV muon decay processes, µ + → e + γ and µ + → e + e − e + . Since the following analysis does not include a study of angular distributions (as in Ref. 22 for the case of polarised τ -lepton decays), the charges of the external states need not be specified. The following partial widths should be divided by the total muon decay width, i.e. Γ µ G 2 F m 5 µ / 192π 3 , in order to obtain the corresponding BRs.
µ → eγ
The simplest and most investigated LFV muonic process is µ → eγ. On the one hand, the serious experimental bounds 4 on this kinematically simple transition clearly indicate that there is an indisputable conservation mechanism at work. On the other hand, any observation of a non-zero µ → eγ in current or future experiments would indicate the existence of BSM physics. The Lagrangian in Eq. 5 results in a branching ratio
Γ (µ → eγ) = e 2 m 5 µ 4πΛ 4 C D L 2 + C D R 2 ,(12)
from which it is clear that, with the Wilson coefficients defined at the muon energy scale, the associated BR is related only to the dipole operators C D L/R . According to the RGEs presented in Eq. 17, these operators will receive contributions from scalar (C S ll with l = e, µ) and tensor (C T τ τ and C T qq with q = u, c, d, s, b) operators, with non-vanishing coefficients at higher scales.
µ → eee
The second representative channel for muonic LFV decays is µ → eee. Prospects for future experimental developments in this rare muon process are very promising: the current experimental limit 5 is expected to be improved considerably by the Mu3e experiment. Again, any signal of such a rare decay would be a clear signal for BSM physics.
The partial width reads
Γ (µ → 3e) = = α 2 m 5 µ 12Λ 4 π C D L 2 + C D R 2 8 log m µ m e − 11 + m 5 µ 3Λ 4 (16π) 3 C S LL ee 2 + C S RR ee 2 + 8 2 C V LL ee 2 + C V LR ee 2 + C V RL ee 2 + 2 C V RR ee 2 − αm 5 µ 3Λ 4 (4π) 2 ( [C D L C V RL ee + 2C V RR ee * ] + [C D R 2C V LL ee + C V LR ee * ]),(13)
where a more complicated interplay between operators occurs. The next section provides an explicit example of a correlation between the coefficients in Eqs. 12 and 13 with respect to the two experimental bounds on LFV transitions.
Limits on Wilson coefficients and correlations
In this section, the present experimental limits together with anticipated updates are applied to the observables of Eqs. 12 and 13 defined at a UV-completion energy scale. Closer examination of Eqs. 12 and 13 together with the RGE equations in the appendix reveals that only two classes of operators -the dipole (O D ) and the scalar (O S ee ) -are manifestly correlated at the one-loop level in two self-consistent systems (separate by chirality) of ordinary differential equations (ODE). In principle, more complicated relations occur if non-zero tensorial quark or τ -lepton operators are considered. In addition, at the two-loop level, even the vectorial operators mix with the dipole. However, a complete quantitative treatment of all possible correlations is beyond the scope of this analysis.
For illustrative purposes, in the following discussion, we consider a scenario where an underlying UV-complete theory produces non-vanishing SMEFT coefficients. We assume that matching this SMEFT to the low-energy Lagrangian of Eq. 5, only two categories of non-vanishing coefficients are produced, namely C D and C S ee . According to the RGE described by Eqs. 17 and 18, if the RGE effects are neglected for the EM coupling and fermion masses c , then the running of these two operators can be described by a relatively simple system of two ODE. The solutions are
C D L/R (µ) µ m Z 4 α C D L/R (m Z ) − m e 16απm µ µ m Z 3 α m α Z − µ α m α Z C S LL/RR ee (m Z ) , (14) C S LL/RR ee (µ) µ m Z 3 α C S LL/RR ee (m Z ) ,(15)
where µ is the phenomenological energy scale at which the coefficients should be evaluated, and α = α/π is the normalised EM coupling. By combining these results with the BRs of Section 3 and applying the experimental limits, at the muon mass scale µ = m µ , we obtain the constraints on the coefficients C D (M Z ) and C S ee (M Z ) shown in Figure 1 (right-chirality ones give the same result). Note that the evolution of the EM coupling and fermion masses is taken into account in these numerical results.
First, it must be appreciated that the limits originating from the non-observation of LFV muon decays in different experiments are manifestly complementary. In particular, for µ → eγ there is a region of the parameter space in which an explicit cancellation occurs between the contributions of the two operators. This effect is due to the relative sign in the evolution equation, which implies that
C D L/R (m µ ) is small if C D L/R (m Z ) m e 16απm µ m α Z − m α µ m α µ C S LL/RR ee (m Z ) .(16)
Thus for MEG there is a blind direction in parameter space. In contrast, the µ → 3e decay mode is not subject to any cancellation among effective couplings, meaning that only the future Mu3e experiment will be able to explore this corner of the parameter space, as the SINDRUM experiment did in the past. A second important aspect is that the last stage of the Mu3e experiment will cover a wider region of the parameter space than the MEG II experiment (in the absence of other correlations between operators), producing better limits for both the dipole and four-fermion effective couplings.
A much more involved scenario might arise if other operators are taken into account. For example, if C T bb is generated at the EWSB energy scale, the evolution of the dipole operator changes dramatically. However, salient aspects of the complementarity of the two experimental searches will remain qualitatively unaltered.
Conclusion
In this note, LFV muon decays have been analysed within the framework of an effective field theory with higher-dimensional operators at low energy scales.
The processes µ → eγ and µ → 3e have been investigated below the EWSB energy scale, down to the natural energy regime at which such processes occur, i.e. the muon mass scale. The complete class of contributing dimension-5 and dimension-6 operators allowed by QED and QCD have been systematically studied at the tree level, and one-loop contributions to the RGE have been taken into account.
The current experimental limits from the MEG and SINDRUM experiments have been used to extract bounds on some of the Wilson coefficients of the effective theory and, ultimately, on the Wilson coefficients at any energy level below the EWSB scale. This note has also presented an explicit example of a correlation between dipole and fourfermion scalar effective couplings, under the assumption that they are the only two non-vanishing couplings generated at the EWSB energy scale by an underlying BSM theory, illustrating the complementarity of the searches planned for the MEG II and Mu3e experiments. In particular, it has been shown that the µ → 3e channel allows for exploration of a region of the parameter space which µ → eγ experiments are unable to investigate. Furthermore, in the absence of any other correlation it was shown that the last experimental phase of Mu3e will provide the best bound on the parameter space for both considered operators. However, this assertion might be invalid in the presence of other operators that mix in some way with the tree-level Wilson coefficients.
This appendix presents the anomalous dimensions of the operators exploited in the phenomenological analysis of Section 4. The corresponding equations for the chirality-flipped operators are obtained by the label interchange R ←→ L.
The dipole operator runs according to
where N c is the number of colours, and Q l , Q u and Q d are the charges associated with leptons, u-type and d-type quarks, respectively. The running of the leptonic scalar and tensorial operators is summarised by the following equations:
16π 2 ∂C S RR ee/µµ ∂ (log µ) = 12e 2 Q 2 l C S RR ee/µµ ,(18)16π 2 ∂C S RR τ τ ∂ (log µ) = −12e 2 Q 2 l C S RR τ τ + 8C T RR τ τ ,(19)16π 2 ∂C S RL τ τ ∂ (log µ) = −12e 2 Q 2 l C S RL τ τ ,(20)16π 2 ∂C T RR τ τ ∂ (log µ) = −2e 2 Q 2 l C S RR τ τ − 2C T RR τ τ .(21)
The running of the scalar and tensorial quark operators is given by
16π 2 ∂C S RR qq ∂ (log µ) = −6 Q 2 l + Q 2 q e 2 + 1 − N 2 c g 2 S C S RR qq − 96e 2 Q l Q q C T RR qq ,(22)
and
16π 2 ∂C T RR qq ∂ (log µ) = −2e 2 Q l Q q C S RR qq + 2 Q 2 l + Q 2 q e 2 + N 2 c − 1 N c g 2 S C T RR qq .(23)
The running of vector operators is decoupled from the dipole operator C D at the oneloop level. Nevertheless, it is well known that a non-vanishing mixing occurs at the two-loop level 23,24 . However, inclusion of these effects is beyond the scope of the present analysis and will be provided in a future publication 25 .
Figure 1 -
1Allowed parameter space for the two coefficients C D L and C S LL ee (non-vanishing at the Λ = mZ energy scale) by the µ → eγ (red regions) and µ → 3e experiments (yellow/green regions). Present (solid lines) and anticipated bounds (dashed/dashed-dotted lines) are plotted on a linear (upper frame) and pseudo-logarithmic scale (lower frame). In evaluating of the RGE, the running of the gauge coupling and fermion masses are included.
µ − m q ).
c If the running of the electromagnetic (EM) coupling and the fermion masses is taken into account, then the evolution of the couplings is more involved, but at the same time the qualitative conclusion of this note will remain unchanged.
AcknowledgementsAC's work is supported by an Ambizione grant from the Swiss National Science Foundation (SNF). GMP's work is supported by SNF under contract 200021 160156. GMP thanks Angela Papa and Ann-Kathrin Perrevoort for their insight and expertise, which greatly improved the manuscript.Appendix -Anomalous dimensions
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| [] |
[
"EPSILON-STRONGLY GRADED RINGS, SEPARABILITY AND SEMISIMPLICITY",
"EPSILON-STRONGLY GRADED RINGS, SEPARABILITY AND SEMISIMPLICITY",
"EPSILON-STRONGLY GRADED RINGS, SEPARABILITY AND SEMISIMPLICITY",
"EPSILON-STRONGLY GRADED RINGS, SEPARABILITY AND SEMISIMPLICITY"
] | [
"Patrik Nystedt ",
"Héctor Pinedo ",
"\nDepartment of Engineering Science\nJOHANÖINERT Department of Mathematics and Natural Sciences\nUniversity West\nSE-46186TrollhättanSweden\n",
"\nEscuela de Matemáticas\nBlekinge Institute of Technology\nSE-37179KarlskronaSweden\n",
"\nUniversidad Industrial de Santander\nCarrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678BucaramangaColombia\n",
"Patrik Nystedt ",
"Héctor Pinedo ",
"\nDepartment of Engineering Science\nJOHANÖINERT Department of Mathematics and Natural Sciences\nUniversity West\nSE-46186TrollhättanSweden\n",
"\nEscuela de Matemáticas\nBlekinge Institute of Technology\nSE-37179KarlskronaSweden\n",
"\nUniversidad Industrial de Santander\nCarrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678BucaramangaColombia\n"
] | [
"Department of Engineering Science\nJOHANÖINERT Department of Mathematics and Natural Sciences\nUniversity West\nSE-46186TrollhättanSweden",
"Escuela de Matemáticas\nBlekinge Institute of Technology\nSE-37179KarlskronaSweden",
"Universidad Industrial de Santander\nCarrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678BucaramangaColombia",
"Department of Engineering Science\nJOHANÖINERT Department of Mathematics and Natural Sciences\nUniversity West\nSE-46186TrollhättanSweden",
"Escuela de Matemáticas\nBlekinge Institute of Technology\nSE-37179KarlskronaSweden",
"Universidad Industrial de Santander\nCarrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678BucaramangaColombia"
] | [] | We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nǎstǎsescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings. Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Simón concerning when graded rings can be presented as partial crossed products. We also provide some interesting classes of examples of separable epsilon-strongly graded rings, with finite as well as infinite grading groups. In particular, we obtain an answer to a question raised by Le Bruyn, Van den Bergh and Van Oystaeyen in 1988. | 10.1016/j.jalgebra.2018.08.002 | [
"https://arxiv.org/pdf/1606.07592v3.pdf"
] | 119,602,888 | 1606.07592 | c0bfcbd26618662385cbea569c215f9e4407e88a |
EPSILON-STRONGLY GRADED RINGS, SEPARABILITY AND SEMISIMPLICITY
12 Feb 2018
Patrik Nystedt
Héctor Pinedo
Department of Engineering Science
JOHANÖINERT Department of Mathematics and Natural Sciences
University West
SE-46186TrollhättanSweden
Escuela de Matemáticas
Blekinge Institute of Technology
SE-37179KarlskronaSweden
Universidad Industrial de Santander
Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678BucaramangaColombia
EPSILON-STRONGLY GRADED RINGS, SEPARABILITY AND SEMISIMPLICITY
12 Feb 2018
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nǎstǎsescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings. Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Simón concerning when graded rings can be presented as partial crossed products. We also provide some interesting classes of examples of separable epsilon-strongly graded rings, with finite as well as infinite grading groups. In particular, we obtain an answer to a question raised by Le Bruyn, Van den Bergh and Van Oystaeyen in 1988.
Introduction
Let S be an associative ring equipped with a non-zero multiplicative identity element 1. Let S/R be a ring extension. By this we mean that R is a subring of S containing 1. Recall that S/R is called separable if the multiplication map m : S ⊗ R S → S is a splitting epimorphism of R-bimodules. Equivalently, this can be formulated by saying that there is x ∈ S ⊗ R S satisfying m(x) = 1 and that, for every s ∈ S, the relation sx = xs holds. In that case, x is called a separability element of S ⊗ R S. Separable ring extensions are a natural generalization of the classical separability condition for algebras over fields which in turn is a generalization of separable field extensions (see e.g. [7]). Nǎstǎsescu, Van den Bergh and Van Oystaeyen [21] have generalized this even further by introducing the notion of a separable functor. They show that a ring extension is separable precisely when the associated restriction functor is separable. A lot of work has been devoted to the question of when ring extensions are separable (see e.g. [1], [14], [2], [3], [7], [9], [10], [15], [16], [19], [21], [23] and [24]). One reason for this intense interest is that some properties of the ground ring R automatically are inherited by S, such as semisimplicity and hereditarity (see e.g. [21]).
In the context of group graded rings, necessary and sufficient criteria for separability has been obtained in two different cases (see Theorem 1 and Theorem 2 below). Indeed, let G be a group with identity element e. Let S be graded by G. Recall that this means that, for all g, h ∈ G, there is an additive subgroup S g of S such that S = ⊕ g∈G S g and S g S h ⊆ S gh . The subring R = S e is called the principal component of S.
In the first case, S is strongly graded. Recall that this means that S g S h = S gh , for all g, h ∈ G. This makes each S g , for g ∈ G, an invertible R-bimodule which implies that there is a unique ring automorphism β g : Z(R) → Z(R) such that β g (r)s = sr, for r ∈ Z(R) and s ∈ S g (see [19] or e.g. Definition 9 and Proposition 12). If G is finite, then the trace function tr β : Z(R) → Z(R) is defined by tr β (r) = g∈G β g (r), for r ∈ Z(R).
Theorem 1 (Nǎstǎsescu, Van den Bergh and Van Oystaeyen [21]). If S is strongly graded by G, then S/R is separable if and only if G is finite and 1 ∈ tr β (Z(R)).
In the second case, S is a unital partial crossed product of G over R. Recall that a unital twisted partial action of G on R is a triple α = ({D g } g∈G , {α g } g∈G , {w g,h } (g,h)∈G×G ) where for each g ∈ G, D g is a unital ideal of R having an (not necessarily non-zero) identity element 1 g which is central in R, α g : D g −1 → D g is an isomorphism of rings, and for each (g, h) ∈ G × G, w g,h is an invertible element from D g D gh , satisfying the following assertions for all g, h, l ∈ G:
(P1) α e = id R ; (P2) α g (D g −1 D h ) = D g D gh ; (P3) if r ∈ D h −1 D (gh) −1 , then α g (α h (r)) = w g,h α gh (r)w −1 g,h ; (P4) w e,g = w g,e = 1 g ; (P5) if r ∈ D g −1 D h D hl , then α g (rw h,l )w g,hl = α g (r)w g,h w gh,l .
Given a unital twisted partial action of G on R, the unital partial crossed product R ⋆ w α G is the direct sum ⊕ g∈G D g δ g , in which the δ g 's are formal symbols, and the multiplication is defined by the biadditive extension of the relations (P6) (rδ g )(r ′ δ h ) = rα g (r ′ 1 g −1 )w g,h δ gh , for g, h ∈ G, r ∈ D g and r ′ ∈ D h . If G is finite, then the trace function tr α : Z(R) → Z(R) is defined by tr α (r) = g∈G α g (r1 g −1 ), for r ∈ Z(R).
Theorem 2 (Bagio, Lazzarin and Paques [1]). If S is a unital partial crossed product of a finite group G over R, then S/R is separable if and only if 1 ∈ tr α (Z(R)).
In this article, we wish to unify Theorem 1 and Theorem 2 (see Theorem 3) for a class of rings which properly contains both the class of strongly graded rings and the class of partial crossed products. We call this class of rings epsilon-strongly graded. The term "epsilonstrongly" is supposed to be suggestive of the fact that the grading is "an epsilon away" from being strong. Let us briefly describe the idea behind this class of rings. Suppose that S is a ring graded by a group G and take g, h ∈ G. Instead of postulating that S g S h = S gh , as in the strongly graded case, we relax this condition by saying that S g S g −1 and S h −1 S h are unital ideals of R such that the equalities S g S h = S g S g −1 S gh = S gh S h −1 S h hold. The multiplicative identity element in S g S g −1 is denoted by ǫ g . Here is an outline of the article.
In Section 2, we introduce epsilon-strongly graded rings (see Definition 4) and we give several equivalent characterizations of them (see Proposition 7).
In Section 3, we show that if S is epsilon-strongly graded by G, then we can define a trace function tr γ : Z(R) fin → Z(R) (see Definition 14) which generalizes the trace functions from both the strongly graded case and the partial crossed product situation. Here Z(R) fin is the set of r ∈ Z(R) with the property that for all but finitely many g ∈ G, the relation rǫ g = 0 holds. At the end of Section 3, we show the following simultaneous generalization of Theorem 1 and Theorem 2. Notice that our result holds for any, possibly infinite, group G.
Theorem 3. If S is epsilon-strongly graded by G, then S/R is separable if and only if 1 ∈ tr γ (Z(R) fin ).
In Section 4, we use Theorem 3 to find criteria for when epsilon-strongly graded rings are semisimple, hereditary or Frobenius (see Theorem 23 and Theorem 24).
In Section 5, we show that a result concerning simplicity for strongly graded rings from [17,Theorem 6.6] can be generalized to epsilon-strongly graded rings (see Proposition 29).
In Section 6, we introduce epsilon-crossed products (see Definition 32). We show that the class of epsilon-crossed products coincides with the class of unital partial crossed products (see Theorem 33). This is an epsilon-analogue of how the classical crossed products appear in the class of strongly graded rings (see e.g. [20]). Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Simón [8] concerning when graded rings can be presented as partial crossed products. At the end of the section, we use Theorem 3 to reformulate Theorem 2 so that it holds for any, possibly infinite, group G (see Theorem 35).
In Section 7, we provide a class of examples of separable epsilon-strongly Z 2 -graded rings, neither of which are strongly graded, nor partial crossed products, in any natural way (see Proposition 38 and Proposition 39). Thereby, we provide the first known non-trivial example of a ring, graded by a finite group, which is separable over its principal component but yet not strongly graded (see Remark 40 and [14, Remark II.5.1.6]).
In Section 8, we consider Morita rings which are in a natural way Z-graded. We show that, under weak assumptions, they are in fact epsilon-strongly graded and separable over their principal components (see Proposition 42).
Some Characterizations of Epsilon-strongly Graded Rings
In this section, we introduce epsilon-strongly graded rings (see Definition 4) and we give several equivalent characterizations of them (see Proposition 7). Throughout the rest of this article, unless otherwise stated, let G be an arbitrary group with identity element e. In this section, let S be an arbitrary unital ring which is graded by G and put R = S e . Definition 4. Let S be a ring which is graded by G. We say that S is epsilon-strongly graded by G if for each g ∈ G, S g S g −1 is a unital ideal of R such that for all g, h ∈ G the equalities S g S h = S g S g −1 S gh = S gh S h −1 S h hold. In that case, for each g ∈ G, we let ǫ g denote the multiplicative identity element of S g S g −1 .
Proposition 5. If S is epsilon-strongly graded by G, then, for every g ∈ G, ǫ g ∈ Z(R).
Proof. Take g ∈ G and r ∈ R. Since S g S g −1 is an R-ideal it follows that ǫ g r, rǫ g ∈ S g S g −1 .
Using that ǫ g is a multiplicative identity element of S g S g −1 , we therefore get that ǫ g r = (ǫ g r)ǫ g = ǫ g (rǫ g ) = rǫ g . Definition 6. Following [4,Definition 4.5] we say that S is symmetrically graded if for every g ∈ G, the equality S g S g −1 S g = S g holds. Proposition 7. The following assertions are equivalent:
(i) S is epsilon-strongly graded by G;
(ii) S is symmetrically graded by G, and for every g ∈ G the R-ideal S g S g −1 is unital;
(iii) For every g ∈ G there is an element ǫ g ∈ S g S g −1 such that for all s ∈ S g the relations ǫ g s = s = sǫ g −1 hold; (iv) For every g ∈ G the left R-module S g is finitely generated and projective, and the map n g : (S g ) R → Hom R ( R S g −1 , R) R , defined by n g (s)(t) = ts, for s ∈ S g and t ∈ S g −1 , is an isomorphism of right R-modules.
Proof. (i)⇒(ii): This follows immediately from Definition 4 by putting h = e.
(ii)⇒(iii): Take g ∈ G and s ∈ S g . Let ǫ g denote the multiplicative identity element of S g S g −1 . Using that S is symmetrically graded, we may write s = n i=1 a i b i c i where a 1 , . . . , a n , c 1 , . . . , c n ∈ S g and b 1 , . . . , b n ∈ S g −1 . This yields
ǫ g s = n i=1 ǫ g a i b i ∈SgS g −1 c i = n i=1 a i b i c i = s and similarly sǫ g −1 = n i=1 a i b i c i ∈S g −1 Sg ǫ g −1 = n i=1 a i b i c i = s.
(iii)⇒(i): Take g, h ∈ G. Then it follows that
S g S h = ǫ g S g S h ⊆ S g S g −1 S g S h ⊆ S g S g −1 S gh ⊆ S g S h and S g S h = S g S h ǫ h −1 ⊆ S g S h S h −1 S h ⊆ S gh S h −1 S h = S g S h .
It is clear that ǫ g is a multiplicative identity element of S g S g −1 .
(iii)⇒(iv): Suppose that (iii) holds. From the relation ǫ g −1 ∈ S g −1 S g it follows that there is n ∈ N and u i ∈ S g −1 and v i ∈ S g , for i ∈ {1, . . . , n}, such that n i=1 u i v i = ǫ g −1 . For each i ∈ {1, . . . , n}, define the left R-linear map f i : S g → R by the relations f i (s) = su i , for s ∈ S g . Take s ∈ S g . Then s = sǫ
g −1 = n i=1 su i v i = n i=1 f i (s)v i . Therefore {v i } n i=1 and {f i } n i=1
form "dual bases" and S g is therefore finitely generated and projective as a left Rmodule. Next we show that n g is a monomorphism. Suppose that s ∈ S g satisfies n g (s) = 0. Then s = ǫ g s ∈ S g S g −1 s = S g n g (s)(S g −1 ) = {0}. Therefore s = 0. Now we show that n g is surjective. There is n ∈ N and a i ∈ S g and v i ∈ S g −1 , for i ∈ {1, . . . , n}, such that n i=1 a i b i = ǫ g . For each i ∈ {1, . . . , n}, define the left R-linear map g i : S g −1 → R by the relations g i (s) = sa i , for s ∈ S g −1 . Take t ∈ S g −1 . Then
t = tǫ g = n i=1 ta i b i = n i=1 g i (t)b i . Take f ∈ Hom R ( R S g −1 , R) R . Then f (t) = n i=1 g i (t)f (b i ) = t n i=1 a i f (b i ) = n g ( n i=1 a i f (b i ))(t). Therefore, f = n g ( n i=1 a i f (b i ))
. Hence, n g is surjective. (iv)⇒(iii): Take a ∈ S g and b ∈ S g −1 . Suppose that the left R-module S g is finitely generated and projective, and that the map n g is an isomorphism of right R-modules. The dual basis lemma shows that there are b 1 , . . . , b n ∈ S g −1 and f 1 , . . . , f n ∈ Hom R (S g −1 , R) such
that b = n i=1 f i (b)b i . For every i ∈ {1, . . . , n}, there is a i ∈ S g such that n g (a i ) = f i . Hence b = n i=1 n g (a i )(b)b i = n i=1 ba i b i = bǫ g , where ǫ g = n i=1 a i b i ∈ S g S g −1 . This shows that S g −1 (1 − ǫ g ) = {0}. Therefore S g −1 (1 − ǫ g )a = {0} and thus n g ((1 − ǫ g )a)(S g −1 ) = {0}.
This implies that n g ((1 − ǫ g )a) = 0. But since n g is injective, we finally get that (1 − ǫ g )a = 0 and hence a = ǫ g a. Therefore S is epsilon-strongly graded by G.
Proposition 8. If S is epsilon-strongly graded by G, then S is strongly graded by G if and only if for every g ∈ G the equality ǫ g = 1 holds.
Proof. Suppose that S is strongly graded by G. Take g ∈ G. Since S g S g −1 = R, we get that ǫ g = 1. Now suppose that S is epsilon-strongly graded with ǫ g = 1, for all g ∈ G. Since S g S g −1 is a unital ideal of R with 1 as a multiplicative identity, it follows that R = R1 ⊆ RS g S g −1 ⊆ R. Therefore S g S g −1 = R.
Separability
In this section, we shall assume that S is an arbitrary unital ring which is epsilon-strongly graded by G. We will, for each g ∈ G, introduce an additive function γ g : S → S (see Definition 9). These functions will in turn be used to define a trace function tr γ : Z(R) fin → Z(R) (see Definition 14). At the end of this section, we prove Theorem 3. Let N denote the set of positive integers. Definition 9. Let g ∈ G be arbitrary. From the relation ǫ g ∈ S g S g −1 it follows that there is n g ∈ N, and u
(i) g ∈ S g and v (i) g −1 ∈ S g −1 , for i ∈ {1, . . . , n g }, such that ng i=1 u (i) g v (i) g −1 = ǫ g .
Unless otherwise stated, the elements u (i) g and v (i) g −1 are fixed. We also assume that n e = 1 and u
(1) e = v (1) e = 1. Define the additive function γ g : S → S by γ g (s) = ng i=1 u (i) g sv (i) g −1 , for s ∈ S.
Proposition 10. For any g ∈ G and r ∈ Z(R), the definition of γ g (r) does not depend on the choice of the elements u
(i) g and v (i) g −1 . Proof. Take m g ∈ N, s (j) g ∈ S g and t (j) g −1 ∈ S g −1 , for j ∈ {1, . . . , m g }, such that mg j=1 s (j) g t (j) g −1 = ǫ g . Then γ g (r) = ng i=1 u (i) g rv (i) g −1 = ng i=1 ǫ g u (i) g rv (i) g −1 = ng i=1 mg j=1 s (j) g t (j) g −1 u (i) g rv (i) g −1 .
Since t (j)
g −1 u (i) g ∈ R and r ∈ Z(R), the last sum equals ng i=1 mg j=1 s (j) g rt (j) g −1 u (i) g v (i) g −1 = mg j=1 s (j) g rt (j) g −1 ǫ g = mg j=1 s (j) g rt (j) g −1 .
Proposition 11. For any g, h ∈ G and r ∈ Z(R), γ g (γ h (r)) = γ gh (r)ǫ g holds.
Proof. From the definitions of γ g and γ h , it follows that
γ g (γ h (r)) = n h i=1 γ g (u (i) h rv (i) h −1 ) = n h i=1 ng j=1 u (j) g u (i) h rv (i) h −1 v (j) g −1 .
Since u
(j) g u (i) h ∈ S gh , the last sum equals n h i=1 ng j=1 ǫ gh u (j) g u (i) h rv (i) h −1 v (j) g −1 = n h i=1 ng j=1 n gh k=1 u (k) gh v (k) h −1 g −1 u (j) g u (i) h rv (i) h −1 v (j) g −1 .
Using that r ∈ Z(R) and v
(k) h −1 g −1 u (j) g u (i) h ∈ R, the last sum equals n h i=1 ng j=1 n gh k=1 u (k) gh rv (k) h −1 g −1 u (j) g u (i) h v (i) h −1 v (j) g −1 = ng j=1 n gh k=1 u (k) gh rv (k) h −1 g −1 u (j) g ǫ h v (j) g −1 .
Since
v (k) h −1 g −1 u (j) g ∈ S h −1 , the last sum equals ng j=1 n gh k=1 u (k) gh rv (k) h −1 g −1 u (j) g v (j) g −1 = n gh k=1 u (k) gh rv (k) h −1 g −1 ǫ g = γ gh (r)ǫ g .
Proposition 12. Let g ∈ G be arbitrary. The additive function γ g : S → S restricts to a surjective ring homomorphism Z(R) → Z(R)ǫ g which satisfies the relation γ g (r)s g = s g r, for all r ∈ Z(R), s g ∈ S g . This function, in turn, restricts to a ring isomorphism Z(R)ǫ g −1 → Z(R)ǫ g .
Proof. First we show that γ g (Z(R)) ⊆ Z(R)ǫ g . From the definition of γ g it follows that γ g (S) ⊆ Sǫ g . Thus, since ǫ g is idempotent, we only need to show that γ g (Z(R)) ⊆ Z(R). To this end, take r ∈ Z(R) and r ′ ∈ R. Then, since v
(i) g −1 r ′ ∈ S g −1 , we get that γ g (r)r ′ = ng i=1 u (i) g rv (i) g −1 r ′ = ng i=1 u (i) g rv (i) g −1 r ′ ǫ g = ng i=1 ng j=1 u (i) g rv (i) g −1 r ′ u (j) g v (j) g −1 .
Using that r ∈ Z(R), v
(i) g −1 r ′ u (j) g ∈ R and r ′ u (j) g ∈ S g , the last sum equals ng i=1 ng j=1 u (i) g v (i) g −1 r ′ u (j) g rv (j) g −1 = ng j=1 ǫ g r ′ u (j) g rv (j) g −1 = ng j=1 r ′ u (j) g rv (j) g −1 = r ′ γ g (r).
This shows that γ g (r) ∈ Z(R). Now we show that the restriction of γ g to Z(R) respects multiplication. Take r, r ′ ∈ Z(R). Then
γ g (rr ′ ) = ng i=1 u (i) g rr ′ v (i) g −1 = ng i=1 ǫ g u (i) g rr ′ v (i) g −1 = ng i=1 ng j=1 u (j) g v (j) g −1 u (i) g rr ′ v (i) g −1 .
Since r ∈ Z(R) and v (j)
g −1 u (i) g ∈ R, the last sum equals ng i=1 ng j=1 u (j) g rv (j) g −1 u (i) g r ′ v (i) g −1 = ng j=1 u (j) g rv (j) g −1 ng i=1 u (i) g r ′ v (i) g −1 = γ g (r)γ g (r ′ ).
Next, we show that the restriction Z(R) → Z(R)ǫ g is surjective. Take r ∈ Z(R). From Proposition 11, we get that γ g (γ g −1 (rǫ g )) = γ e (rǫ g )ǫ g = rǫ 2 g = rǫ g . For any s g ∈ S g , using that v
(i) g −1 s g ∈ R, we conclude that γ g (r)s g = ng i=1 u (i) g rv (i) g −1 s g = ng i=1 u (i) g v (i) g −1 s g r = ǫ g s g r = s g r.
Finally, we need to show that the restriction Z
(R)ǫ g −1 → Z(R)ǫ g is injective. Suppose that r ∈ Z(R) is chosen so that γ g (rǫ g −1 ) = 0. From Proposition 11, we get that rǫ g −1 = rǫ 2 g −1 = γ e (rǫ g −1 )ǫ g −1 = γ g −1 (γ g (rǫ g −1 )) = 0.
Remark 13. Notice that, by Proposition 11 and Proposition 12, the collection of (restriction) maps γ g : Z(R)ǫ g −1 → Z(R)ǫ g , for g ∈ G, yields a partial action of G on Z(R).
Definition 14.
Let Z(R) fin denote the set of r ∈ Z(R) such that for all but finitely many g ∈ G, the relation rǫ g = 0 holds. Define the trace function tr γ : Z(R) fin → Z(R) by tr γ (r) = g∈G γ g (r), for r ∈ R.
Remark 15. Recall that if we for every g ∈ G put
(S ⊗ R S) g = (g ′ ,g ′′ )∈G×G, g ′ g ′′ =g S g ′ ⊗ R S g ′′ ,
then this defines a graded R-bimodule structure on S ⊗ R S. We will refer to this as the G-grading of S ⊗ R S. There is another type of grading on S ⊗ R S that we will also use. If we, for every (g, h) ∈ G × G, put
(S ⊗ R S) (g,h) = S g ⊗ R S h ,
then this defines a graded additive structure on S ⊗ R S. We will refer to this as the (G × G)grading on S ⊗ R S. For more details concerning these gradings, see [20].
Proof of Theorem 3. First we show the "if" statement. Suppose that there is c ∈ Z(R) fin such that tr γ (c) = 1. We wish to show that S/R is separable. To this end, put
x = g∈G ng i=1 u (i) g c ⊗ v (i) g −1 .
From the definition of Z(R) fin it follows that x is well-defined, since for all but finitely many g ∈ G, we get that u
(i) g c = u (i) g ǫ g −1 c = u (i) g 0 = 0. Now m(x) = g∈G ng i=1 u (i) g cv (i) g −1 = g∈G γ g (c) = tr γ (c) = 1.
Next we show that x commutes with all elements of S. To this end, take h ∈ G and s ∈ S h . Using that su
(i) g ∈ S hg , we get that sx = g∈G ng i=1 su (i) g c ⊗ v (i) g −1 = g∈G ng i=1 ǫ hg su (i) g c ⊗ v (i) g −1 = g∈G ng i=1 n hg j=1 u (j) hg v (j) g −1 h −1 su (i) g c ⊗ v (i) g −1 .
Since c ∈ Z(R) and v (j)
g −1 h −1 su (i) g ∈ R, the last sum equals g∈G ng i=1 n hg j=1 u (j) hg c ⊗ v (j) g −1 h −1 su (i) g v (i) g −1 = g∈G n hg j=1 u (j) hg c ⊗ v (j) g −1 h −1 sǫ g . Using that v (j) g −1 h −1 s ∈ S g −1 , the last sum equals g∈G n hg j=1 u (j) hg c ⊗ v (j) g −1 h −1 s = xs.
Therefore, x is a separability element for S/R. Now we show the "only if" statement. Suppose that x ∈ S ⊗ R S is a separability element for S/R. Then x satisfies m(x) = 1 and, for each s ∈ S, the relation xs = sx holds. From the G-grading on S ⊗ R S it follows that there are unique y ∈ (S ⊗ R S) e and z ∈ ⊕ g∈G\{e} (S ⊗ R S) g such that x = y + z. Then we get that 1 = m(x) = m(y) + m(z). But since 1 ∈ R and m(z) ∈ ⊕ g∈G\{e} S g , we get that m(z) = 0 and m(y) = 1. We claim that for each s ∈ S, the relation ys = sy holds. To this end, take h ∈ G and s ∈ S h . Then 0 = sx − xs = sy − ys + sz − zs.
Since sy − ys ∈ (S ⊗ R S) h and sz − zs ∈ g∈G\{e} ((S ⊗ R S) hg + (S ⊗ R S) gh ), we get that sy − ys = 0 and sz − zs = 0. In particular, y is also a separability element for S/R. Now, for each g ∈ G, there is d g ∈ S g ⊗ R S g −1 such that for all but finitely many g ∈ G, d g = 0, and y = g∈G d g . Furthermore, for each g ∈ G, there is l g ∈ N, and a (i)
g ∈ S g and b (i) g −1 ∈ S g −1 , for i ∈ {1, . . . , l g }, such that d g = lg i=1 a (i) g ⊗ b (i) g −1 .
For each g ∈ G, put c g = m(d g ). Take h ∈ G. From the (G × G)-grading on S ⊗ R S it follows that d h commutes with every r ∈ R. Therefore, it follows that c h ∈ Z(R). Take s ∈ S h . From the (G × G)-grading on S ⊗ R S it also follows that sd e = d h s. Applying m gives us that sc e = c h s. In particular, we get that
γ h (c e ) = n h i=1 u (i) h c e v (i) h −1 = n h i=1 c h u (i) h v (i) h −1 = c h ǫ h = ǫ h c h = ǫ h m(d h ) = m(d h ) = c h .
Using that d h = 0, for all but finitely many h ∈ G, the same holds for c h and hence also for γ h (c e ). By summing over h ∈ G, we get that
tr γ (c e ) = h∈G γ h (c e ) = h∈G c h = h∈G m(d h ) = m(y) = 1.
Moreover, by Proposition 11 it follows that for all but finitely many h ∈ G, the relation c e ǫ h −1 = γ h −1 (γ h (c e )) = 0 holds. Thus, c e ∈ Z(R) fin .
Definition 16. Let Z(R) γ fin denote the set of r ∈ Z(R) fin with the property that for every g ∈ G, the relation γ g (r) = rǫ g holds.
Lemma 17. With the above notation, the following assertions hold:
(a) The set Z(R) fin is a (possibly non-unital) subring of Z(R). 1 ∈ Z(R) fin if and only if for all but finitely many g ∈ G, ǫ g = 0.
(b) The set Z(R) γ fin is a (possibly non-unital) subring of Z(R) fin . 1 ∈ Z(R) γ fin if and only if for all but finitely many g ∈ G, ǫ g = 0.
(c) tr γ (Z(R) fin ) ⊆ Z(R) γ fin and the function tr γ : Z(R) fin → Z(R) γ fin is a Z(R) γ fin -bimodule homomorphism.
(d) Suppose that ǫ g = 0, for all but finitely many g ∈ G. If r ∈ Z(R) γ fin and r is invertible in Z(R), then r −1 ∈ Z(R) γ fin . Proof. (a): Take r, r ′ ∈ Z(R) fin . Let X(r) denote the finite set {g ∈ G | rǫ g = 0}. Then X(rr ′ ) ⊆ X(r ′ ) which is a finite set. Also X(r + r ′ ) ⊆ X(r) ∪ X(r ′ ) which is a finite set. Therefore Z(R) fin is a subring of Z(R). Also 1 ∈ Z(R) fin precisely when ǫ g = 1ǫ g = 0 for all but finitely many g ∈ G.
(b): Take r, r ′ ∈ Z(R) γ fin . From (a) we know that rr ′ , r + r ′ ∈ Z(R) fin . Take g ∈ G. From Proposition 12, it follows that γ g (rr ′ ) = γ g (r)γ g (r ′ ) = rǫ g r ′ ǫ g = rr ′ ǫ g . Also γ g (r + r ′ ) = γ g (r) + γ g (r ′ ) = rǫ g + r ′ ǫ g = (r + r ′ )ǫ g . Therefore rr ′ , r + r ′ ∈ Z(R) γ fin . Since γ g (1) = ǫ g , the last part of (b) follows from (a).
(c): Take r ∈ Z(R) fin and g ∈ G. By Proposition 11, we get that γ g (tr γ (r)) = h∈G γ g (γ h (r)) = h∈G γ gh (r)ǫ g = tr γ (r)ǫ g . Therefore, tr γ (r) ∈ Z(R) γ fin . To prove the last statement, take r ∈ Z(R) fin and r ′ , r ′′ ∈ Z(R) γ fin . Then Proposition 12 implies that
tr γ (r ′ rr ′′ ) = g∈G γ g (r ′ rr ′′ ) = g∈G γ g (r ′ )γ g (r)γ g (r ′′ ) = g∈G r ′ ǫ g γ g (r)r ′′ ǫ g = g∈G r ′ ǫ g γ g (r)ǫ g r ′′ = g∈G r ′ γ g (r)r ′′ = r ′ tr γ (r)r ′′ .
(d): From the relation rr −1 = 1, we get that γ g (r)γ g (r −1 ) = ǫ g . Since r ∈ Z(R) γ fin , we get that rǫ g γ g (r −1 ) = ǫ g . Thus, r −1 rγ g (r −1 ) = r −1 ǫ g . Hence, γ g (r −1 ) = r −1 ǫ g .
Corollary 18. Suppose that ǫ g = 0, for all but finitely many g ∈ G. If tr γ (1) is invertible in R, then S/R is separable.
Proof. From Lemma 17(d), we get that tr γ (1) −1 ∈ Z(R) γ fin . Thus, by Lemma 17(c), it follows that tr γ (tr γ (1) −1 1) = tr γ (1) −1 tr γ (1) = 1. Hence, S/R is separable due to Theorem 3.
Remark 19. The sufficient condition concerning invertibility of tr γ (1) in Corollary 18 is not necessary for separability (see Proposition 39).
Semisimplicity, Hereditarity and Frobenius Properties
In this section, we use Theorem 3 to find criteria for when epsilon-strongly graded rings are semisimple, hereditary and Frobenius (see Theorem 23 and Theorem 24). For the rest of this section, S/R denotes a ring extension. Let res denote the restriction functor S-mod → Rmod. The following two results are quite well-known, but, for the convenience of the reader, we have chosen to include the proofs. Proof. We only show the "left" part of the proof. The "right" part is shown in an analogous way and is therefore omitted. Take n ∈ N and s j , t j ∈ S, for j ∈ {1, . . . , n}, such that x = n j=1 s j ⊗ t j is a separability element of S ⊗ R S. Since res(M ) is projective, M has a dual Rbasis {m i } i∈I and {f i } i∈I . For each i ∈ I, define F i : M → S by F i (m) = n j=1 s j f i (t j m), for m ∈ M . We wish to show that {F i } i∈I and {m i } i∈I is a dual S-basis for M . First of all, clearly, each F i is additive. Next, for each m ∈ M , i∈I F i (m)m i = n j=1 s j i∈I f i (t j m)m i = n j=1 s j t j m = 1m = m. Finally, take i ∈ I, s ∈ S and m ∈ M . We need to show that F i (sm) = sF i (m), or, in other words, that n j=1 s j f i (t j sm) = n j=1 ss j f (t j m). To see this, first notice that xs = sx implies that n j=1 s j ⊗ t j s = n j=1 ss j ⊗ t j . From the left S ⊗ R S-module structure on S ⊗ R M it follows that n j=1 s j ⊗ t j sm = n j=1 ss j ⊗ t j m. By applying the function S ⊗ R M ∋ a ⊗ b → a ⊗ f i (b) ∈ S ⊗ R M to the last equality, we get that n j=1 s j ⊗ f i (t j sm) = n j=1 ss j ⊗ f i (t j m). Finally, by applying the function S ⊗ R M ∋ a ⊗ b → ab ∈ M to the last equality, we get that n j=1 s j f i (t j sm) = n j=1 ss j f i (t j m).
Lemma 21. S is projective as a left R-module, if and only if, the functor res preserves projectives.
Proof. Suppose that M is a projective left S-module. We wish to show that M , considered as a left R-module, is projective. Take a dual S-basis {m i , f i } i∈I for M and a dual R-basis {s j , g j } j∈J for S. Then {s j m i , g j •f i } (i,j)∈I×J is a dual basis of M as a left R-module. Indeed, for m ∈ M we have that
(i,j)∈I×J (g j • f i )(m)(s j m i ) = i j g j (f i (m))s j m i = i f i (m)m i = m.
The converse is clear.
Corollary 22. Let S/R be separable. If R is semisimple (left/right hereditary and S is projective as a left/right R-module), then S is semisimple (left/right hereditary).
Proof. This follows from Proposition 20, Lemma 21 and the fact that a ring is semisimple (hereditary) if and only if every module (or submodule of a projective module) over the ring is projective (see [13,Theorem (2.8)
]).
It is easy to see that if S is a ring graded by a group G and we put R = S e , then semisimplicity (left/right hereditarity) of R is always necessary for S to be semisimple (left/right hereditary). In fact, this is true even in the more general setting of rings graded by categories (see [16,Proposition 3]). Now we determine sufficient conditions for semisimplicity (left/right hereditarity).
Theorem 23. Let S be epsilon-strongly graded by a group G and put R = S e . Suppose that R is semisimple (hereditary). If 1 ∈ tr γ (Z(R) fin ) (and every S g , for g ∈ G, is projective as a left/right R-module), then S is semisimple (left/right hereditary). In particular, if ǫ g = 0 for all but finitely many g ∈ G and tr γ (1) is invertible in R (and every S g , for g ∈ G, is projective as a left/right R-module), then S is semisimple (left/right hereditary).
Proof. This follows from Theorem 3, Corollary 18, Corollary 22 and the fact that a direct sum of projective modules is projective.
Recall that S/R is called a Frobenius extension if there is a finite set J, x j , y j ∈ S, for j ∈ J, and an R-bimodule map E : S → R such that, for every s ∈ S, the equalities s = j∈J x j E(y j s) = n j∈J E(sx j )y j hold. In that case, (E, x j , y j ) is called a Frobenius system. Theorem 24. If S is epsilon-strongly graded by a finite group G and we put R = S e , then S/R is a Frobenius extension.
Proof. Put J = {(g, i) | g ∈ G, 1 ≤ i ≤ n g }, where n g is given by Definition 9. Since G is finite, J is finite. For each j = (g, i) ∈ J, define x j = u
x j E(y j s) = g∈G ng i=1 u (i) g E(v (i) g −1 s) = g∈G ng i=1 u (i) g v (i) g −1 s g = g∈G ǫ g s g = s and n j∈J E(sx j )y j = g∈G ng i=1 E(su (i) g )v (i) g −1 = g∈G ng i=1 s g −1 u (i) g v (i) g −1 = g∈G s g −1 ǫ g = s.
Remark 25. The conclusion of Theorem 24 follows from Proposition 7(iv). Indeed, since G is finite, it follows that S is finitely generated and projective as a left R-module. Using the notation used in Proposition 7, put n = ⊕ g∈G n g . Then n is an isomorphism of R-modules S R → Hom R ( R S, R) R . Hence S/R is a Frobenius extension, according to [12,Theorem 1.2].
Recall that if T is a non-empty subset of S, then C S (T ) denotes the set of s ∈ S such that for every t ∈ T , the relation st = ts holds.
Proposition 26. Let S/R be a Frobenius extension with Frobenius system (E, x j , y j ). Then S/R is separable if and only if there is d ∈ C S (R) such that j∈J x j dy j = 1.
Proof. See [12, Corollary 2.17].
Remark 27. Suppose that S is epsilon-strongly graded by a finite group G and put R = S e . Using Theorem 24 and Proposition 26, we can, in this case, prove Theorem 3 in a different way. Indeed, using the above results, we can conclude that S/R is separable if and only if
there is d ∈ C S (R) such that g∈G γ g (d) = g∈G ng i=1 u (i) g dv (i)
g −1 = 1. Since 1 ∈ R it follows from the grading that S/R is separable if and only if there is c = d e ∈ C R (R) = Z(R) such that tr γ (c) = 1.
Simplicity
In this short section, we show that a result concerning simplicity for strongly graded rings from [17,Theorem 6.6] can be generalized to epsilon-strongly graded rings (see Proposition 29). Throughout this section, S denotes an arbitrary unital ring which is epsilon-strongly graded by G and we put R = S e . Recall that R is called a maximal commutative subring of S if C S (R) = R.
Lemma 28. If I is a non-zero ideal of S, then I ∩ C S (Z(R)) = {0}.
Proof. We claim that S is right non-degenerate in the sense of [18,Definition 2]. If we assume that the claim holds, then the desired result follows from [18,Theorem 3]. Now we show the claim. Take g ∈ G and any non-zero s ∈ S g . Seeking a contradiction, suppose that sS g −1 = {0}. Then sS g −1 S g = {0}. But since ǫ g −1 ∈ S g −1 S g we get that s = sǫ g −1 = 0, which is a contradiction.
Recall that an ideal I of S is said to be graded if I = ⊕ g∈G (I ∩ S g ) holds. If {0} and S are the only graded ideals of S, then S is said to be graded simple.
Proposition 29. If R is a maximal commutative subring of S, then S is simple if and only if S is graded simple.
Proof. The "only if" statement is clear. Now we show the "if" statement. Let I be a non-zero ideal of S. By the assumption we have C S (Z(R)) = C S (R) = R. Hence, by Lemma 28 the set J = I ∩ C S (Z(R)) is a non-zero ideal of R. The set SJS is a non-zero graded ideal of S and thus, by graded simplicity of S, we get that S = SJS = J. This shows that S is a simple ring.
Partial Crossed Products
In this section, we introduce epsilon-crossed products (see Definition 32). We show that the class of epsilon-crossed products coincides with the class of unital partial crossed products (see Theorem 33). Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a more general result by Dokuchaev, Exel and Simón [8, Theorem 6.1] concerning when graded rings can be presented as partial crossed products. At the end of this section, we use Theorem 3 to reformulate Theorem 2 so that it holds for any, possibly infinite, group G (see Theorem 35).
Definition 30. Let S be a ring which is epsilon-strongly graded by G. Take g ∈ G and s ∈ S g . Then s is called epsilon-invertible if there is t ∈ S g −1 such that st = ǫ g and ts = ǫ g −1 .
We will refer to t as the epsilon-inverse of s.
The usage of the term "the epsilon-inverse" is justified by the next result.
Proposition 31. Epsilon-inverses are unique.
Proof. Suppose that g ∈ G, s ∈ S g and r, t ∈ S g −1 satisfy the equalities st = sr = ǫ g and ts = rs = ǫ g −1 . Then r = rǫ g = rst = ǫ g −1 t = t.
Definition 32. Let S be a ring which is epsilon-strongly graded by G. We say that S is an epsilon-crossed product by G if for each g ∈ G, there is an epsilon-invertible element in S g .
Theorem 33. Let S be a ring which is epsilon-strongly graded by G. Then S is an epsiloncrossed product if and only if S is a unital partial crossed product.
Proof. First we show the "only if" statement. Suppose that S is an epsilon-crossed product. We will present S as a unital partial crossed product. Take g, h ∈ G. Fix an epsilon-invertible element s g ∈ S g with epsilon-inverse t g −1 ∈ S g −1 . We may assume that s e = t e = 1. Put D g = S g S g −1 = Rǫ g , 1 g = ǫ g and δ g = s g . Furthermore, define α g : D g −1 → D g by α g (rǫ g −1 ) = s g rt g −1 , for r ∈ R. Then α g is well-defined. Indeed, if r, r ′ ∈ R satisfy rǫ g −1 = r ′ ǫ g −1 , then α g (rǫ g −1 ) = s g rt g −1 = s g rǫ g −1 t g −1 = s g r ′ ǫ g −1 t g −1 = s g r ′ t g −1 = α g (r ′ ǫ g −1 ). The function α g is bijective with inverse given by α −1 g (rǫ g ) = t g −1 rs g . Indeed, take r ∈ R. Then α −1 g (α g (rǫ g −1 )) = t g −1 s g rt g −1 s g = ǫ g −1 rǫ g −1 = rǫ g −1 and α g (α −1 g (rǫ g )) = s g t g −1 rs g t g −1 = ǫ g rǫ g = rǫ g . The function α g is clearly additive. Also α g (ǫ g −1 ) = s g ǫ g −1 t g −1 = s g t g −1 = ǫ g . Now we show that α g is multiplicative. Take r, r ′ ∈ R. Then α g (rr ′ ǫ g ) = s g rr ′ t g −1 = s g rǫ g −1 r ′ t g−1 = s g rt g −1 s g r ′ t g −1 = α g (rǫ g −1 )α g (r ′ ǫ g −1 ).
Next put w g,h = s g s h t (gh) −1 . Since w g,h ∈ R, ǫ g s g = s g and t (gh) −1 ǫ gh = t (gh) −1 , it follows that w g,h ∈ D g D gh . Now we show that w g,h is a unit in D g D gh . To this end, first notice that α g (ǫ g −1 ǫ h ) = ǫ g ǫ gh . In fact, from (P2) (see below), we get that there is r ∈ D g −1 D h such that α g (r) = ǫ g ǫ gh . Since ǫ g ǫ gh is the identity of D g D gh , we get that
α g (ǫ g −1 ǫ h ) = α g (ǫ g −1 ǫ h )ǫ g ǫ gh = α g (ǫ g −1 ǫ h )α g (r) = α g (ǫ g −1 ǫ h r) = α g (r) = ǫ g ǫ gh . Put v g,h = s gh t h −1 t g −1 ǫ g ǫ gh .
Then v g,h ∈ D g D gh and
w g,h v g,h = s g s h t (gh) −1 s gh t h −1 t g −1 = s g s h ǫ (gh) −1 t h −1 t g −1 = s g s h t h −1 t g −1 = s g ǫ h t g −1 = s g ǫ h ǫ g −1 t g −1 = α g (ǫ h ǫ g −1 ) = ǫ g ǫ gh and v g,h w g,h = s gh t h −1 t g −1 s g s h t (gh) −1 = s gh t h −1 ǫ g −1 s h t (gh) −1 = s gh t h −1 s h t (gh) −1 = s gh ǫ h −1 t (gh) −1 = s gh ǫ h −1 ǫ (gh) −1 t (gh) −1 = α gh (ǫ (gh) −1 ǫ h −1 ) = ǫ gh ǫ g ǫ gh = ǫ gh ǫ g .
Now we check conditions (P1)-(P6) from the introduction.
(P1): Using that ǫ e = 1, we get that D e = R. Since γ e = id R , we get that α e = id R .
(P2): First notice that α g (D g −1 D h ) = s g D g −1 D h t g −1 = s g S g −1 S g S h S h −1 t g −1 .
Since s g ∈ S g , and thereby s g S g −1 ∈ R, we can conclude that
α g (D g −1 D h ) = ǫ g s g S g −1 ǫ gh S g S h S h −1 t g −1 = ǫ g ǫ gh (s g S g −1 S g S h S h −1 t g −1 ) ⊆ D g D gh R = D g D gh .
Now we show the reversed inclusion. Take r ∈ R. Put r ′ = t g −1 rǫ g ǫ gh s g ∈ R. Then ǫ g −1 r ′ = r ′ . Also, since ǫ gh s g ∈ S gh S (gh) −1 s g ⊆ S gh S h −1 , it follows that r ′ ǫ h = r ′ . Thus, r ′ ∈ D g −1 D h . Now, α g (r ′ ) = s g t g −1 rǫ g ǫ gh s g t g −1 = ǫ g rǫ g ǫ gh ǫ g = rǫ g ǫ gh .
(P3): Take r ∈ D h −1 D (gh) −1 . Then
α g (α h (r))w g,h = s g s h rt h −1 t g −1 s g s h t (gh) −1 = s g s h rt h −1 ǫ g −1 s h t (gh) −1 ,
and the last expression equals
s g s h rt h −1 s h t (gh) −1 = s g (s h r)ǫ h −1 t (gh) −1 = s g s h rt (gh) −1 = s g s h rǫ (gh) −1 t (gh) −1 = (s g s h t (gh) −1 )s gh rt (gh) −1 = w g,h s gh rt (gh) −1 = w g,h α gh (r).
(P4): Using that s e = 1, we get that w g,e = s g s e t g −1 = s g t g −1 = ǫ g and w e,g = s e s g t g −1 = s g t g −1 = ǫ g .
(P6): Notice first that S g = D g δ g . In fact, since s g ∈ S g and D g ⊆ R it follows that S g ⊇ D g s g . On the other hand, take s ′ g ∈ S g . Then s ′ g = s ′ g ǫ g −1 = s ′ g t g −1 s g = s ′ g t g −1 ǫ g s g ∈ D g s g . Thus S g ⊆ D g s g . So we get that S = ⊕ g∈G D g s g . Also, if s = g∈G r g s g , for r g ∈ D g , then the r g 's are unique. Indeed, suppose that r g s g = r ′ g s g for some r g , r ′ g ∈ D g . Using that ǫ g is the multiplicative identity element of D g = S g S g −1 , we get that r g = r g ǫ g = r g s g t g −1 = r ′ g s g t g −1 = r ′ g ǫ g = r ′ g . Take r ∈ D g and r ′ ∈ D h . Then
(rs g )(r ′ s h ) = rs g ǫ g −1 r ′ ǫ g −1 s h = rs g ǫ g −1 r ′ t g −1 s g s h = rα g (ǫ g −1 r ′ )s g s h = rα g (ǫ g −1 r ′ )s g s h ǫ (gh) −1 = rα g (ǫ g −1 r ′ )s g s h t (gh) −1 s gh = rα g (ǫ g −1 r ′ )w g,h s gh .
(P5): Take r ∈ D g −1 D h D hl . Then (s g rs h )s l = (α g (r)s g s h )s l = (α g (r)w g,h s gh )s l = α g (r)w g,h w gh,l s ghl and s g (rs h s l ) = s g (rw h,l s hl ) = α g (rw h,l )w g,hl s ghl .
The claim now follows from the proof of (P6) and associativity. Now we show the "if" statement. Suppose that S = ⊕ g∈G D g δ g is a unital partial crossed product. Take g ∈ G. Since S g S g −1 = D g δ e , we can put ǫ g = 1 g δ e . What remains to show is associativity of S. This has already been shown in a more general context (see [8,Theorem 2.4]). Here we provide a short direct proof for unital twisted partial actions. To this end, take g, h, l ∈ G, a ∈ D g , b ∈ D h and c ∈ D l . Then
(aδ g bδ h )cδ l = (aα g (1 g −1 b)w g,h δ gh )cδ l = aα g (1 g −1 b)w g,h α gh (1 (gh) −1 c)w gh,l δ ghl .
By (P2), the last expression equals
aα g (1 g −1 b)w g,h α gh (1 h −1 1 (gh) −1 c)w gh,l δ ghl ,
which, in turn, by (P3), equals
aα g (1 g −1 b)α g (α h (1 h −1 1 (gh) −1 c))w g,h w gh,l δ ghl = aα g (1 g −1 bα h (1 h −1 1 (gh) −1 c))w g,h w gh,l δ ghl = aα g (1 g −1 bα h (1 h −1 c))w g,h w gh,l δ ghl .
By (P5), this equals
aα g (1 g −1 bα h (1 h −1 )w h,l )w g,hl δ ghl = aδ g (bα h (1 h −1 c)w h,l δ hl ) = aδ g (bδ h cδ l ).
Definition 34. Let S = R ⋆ w α G be a unital partial crossed product, and let Z(R) α,fin denote the set of r ∈ Z(R) with the property that for all but finitely many g ∈ G, the relation r1 g = 0 holds. Define the trace map t α : Z(R) α,fin → Z(R) by t α (r) = g∈G α g (r1 g −1 ), for r ∈ Z(R) α,fin .
Theorem 35. If S is a unital partial crossed product of a group G over R, then S/R is separable if and only if 1 ∈ tr α (Z(R) fin ).
Proof. This follows from Theorem 3 and Theorem 33.
Lemma 36. Let S be a unital partial crossed product of a group G over R and take g ∈ G.
If D g is projective as a left (right) R-module, then S g = D g δ g is projective as a left (right) R-module.
Proof. The "left" part is trivial since the left action of R on S g is defined by the left action of R on D g . Now we show the "right" part. Suppose that D g is projective as a right R-module.
Let {d i , f i } i∈I be a dual basis for D g as a right R-module. For each i ∈ I, define f i : D g → R by the relations f i (d) = α −1 g (1 g f i (d)), for d ∈ D g . Then {d i δ g , f i } i∈I is a dual basis for S g = D g δ g as a right R-module. In fact, if d ∈ D g , then
i∈I (d i δ g )(f i (d)δ e ) = i∈I d i α g (f i (d))w g,e δ g = i∈I d i 1 g f i (d)δ g = dδ g .
Theorem 37. Let S be a unital partial crossed product of a group G over R, and let R is semisimple (left/right hereditary). If 1 ∈ tr γ (Z(R) fin ), then S is semisimple (left/right hereditary). In particular, if ǫ g = 0 for all but finitely many g ∈ G and tr γ (1) is invertible in R, then S is semisimple (left/right hereditary).
Proof. The "semisimple" part follows from Theorem 23. The "hereditary" part follows from Theorem 23, Lemma 36 and the fact that all the ideals D g , for g ∈ G, of the hereditary ring R, are left/right projective.
Examples: A Dade-Like Construction
In this section, we provide a class of examples of separable epsilon-strongly graded rings, neither of which are strongly graded, nor partial crossed products, in any natural way. Our inspiration comes from the first known example (due to E. Dade, according to [6, Example 2.9]) of a strongly graded ring which is not a crossed product. Namely, suppose that A is a commutative unital ring with a non-zero multiplicative identity 1 A . Put
S = M 3 (A), R = A A 0 A A 0 0 0 A and T = 0 0 A 0 0 A A A 0 .
Then S is strongly Z 2 -graded with S 0 = R and S 1 = T , but S is not a crossed product of Z 2 over R since T does not contain any element which is invertible in S. Our idea is to postulate another unital commutative ring B with a non-zero multiplicative identity 1 B such that B is an ideal of A with B A. Now we modify Dade's example by putting
S = A A B A A B B B A , R = A A 0 A A 0 0 0 A and T = 0 0 B 0 0 B B B 0 .
Proposition 38. The ring S is epsilon-strongly Z 2 -graded with S 0 = R and S 1 = T . With this grading, S is neither strongly graded, nor a partial crossed product. Moreover, the ring extension S/R is separable.
Proof. If we put
ǫ 0 = 1 A 0 0 0 1 A 0 0 0 1 A and ǫ 1 = 1 B 0 0 0 1 B 0 0 0 1 B ,
then it is clear that RR = Rǫ 0 and T T = ǫ 1 R. Hence S is epsilon-strongly Z 2 -graded. The equality T T = ǫ 1 R also shows that S is not strongly graded, since 1 A / ∈ B. Seeking a contradiction, suppose that this grading presents S as a partial crossed product of Z 2 over R. Then there is δ 1 ∈ T and a non-zero unital ideal D of R such that T = Dδ 1 . Take unital ideals I and J of A, of which at least one is non-zero, such that
D = I I 0 I I 0 0 0 J , take b 1 , b 2 , b 3 , b 4 ∈ B such that δ 1 = 0 0 b 3 0 0 b 4 b 1 b 2 0
and take w 11 , w 12 , w 21 , w 22 ∈ I and w 33 ∈ J such that
w −1 1,1 = w 11 w 12 0 w 21 w 22 0 0 0 w 33 . From (P6) it follows that (1) (w −1 1,1 δ 1 )(1 D δ 1 ) = 1 D δ 0 .
By a straightforward calculation, (1) can be rewritten as
(2) 1 J b 1 (w 11 b 3 + w 12 b 4 ) 1 J b 2 (w 11 b 3 + w 12 b 4 ) 0 1 J b 1 (w 21 b 3 + w 22 b 4 ) 1 J b 2 (w 21 b 3 + w 22 b 4 ) 0 0 0 1 I w 33 (b 1 b 3 + b 2 b 4 ) = 1 I 0 0 0 1 I 0 0 0 1 J .
From (2) it follows in particular that
1 J b 1 (w 11 b 3 + w 12 b 4 ) = 1 I and 1 I w 33 (b 1 b 3 + b 2 b 4 ) = 1 J .
Thus, 1 I ∈ J and 1 J ∈ I. Hence I = J and so we get that 1 I = 1 J = 0. By a straightforward calculation the determinant of the left hand side of (2) is zero. This contradicts the fact that the determinant of the right hand side of (2) equals 1 I = 0. Now we show that S/R is separable. First of all, it is easy to show that
Z(R) = a 0 0 0 a 0 0 0 a ′ a, a ′ ∈ A .
We know that γ 0 : Z(R) → Z(R) is the identity map on Z(R). Now we determine γ 1 : Z(R) → Z(R). To this end, let e ij denote the 3×3 matrix over A with 1 A in the ijth position, and zeros elsewhere. Since By Theorem 3, we can deduce that S/R is separable if we can find
r = a 0 0 0 a 0 0 0 a ′ ∈ Z(R) such that tr γ (r) = 1 A 0 0 0 1 A 0 0 0 1 A .
By a straightforward calculation, the last relation is equivalent to the set of equations a + 1 B a ′ = 1 A and a ′ + 1 B a = 1 A . It is easy to see that this set of equations is satisfied if we e.g. put a = 1 A and a ′ = 1 A − 1 B . Therefore, S/R is separable.
It is easy to give concrete examples of rings A and B which fit into the above construction. In fact, from now on in this section, suppose that F is a field, A = F × F and B = F × {0}. In that case, the sufficient condition for separability in Corollary 18 is not necessary.
Proposition 39. With the above notation, tr γ (1 R ) is invertible in R if and only if char(F) = 2.
Proof. Since
1 R = 1 A 0 0 0 1 A 0 0 0 1 A ,
we get, from the proof of Proposition 38, that
tr γ (1 R ) = 1 A + 1 B 0 0 0 1 A + 1 B 0 0 0 1 A + 1 B .
Since 1 A + 1 B = (1, 1) + (1, 0) = (2, 1), we get that tr γ (1 R ) is invertible in R if and only if char(F) = 2.
Remark 40. In [14, Remark II.5.1.6] the authors write that "If S is an arbitrary graded ring by a finite group G we do not know whether separability of S over S e implies that S is strongly graded. This seems very likely however." Proposition 38 is a counterexample to this assumption. Moreover, it is the first known example for which all the homogeneous components of the grading are non-zero. Using the same method as in [1, Remark 3.2], one may construct a counterexample with a trivial grading, i.e. with S = S e . The associated Morita ring is the set
S = A M N B
equipped with the natural addition and with a multiplication defined by a 1 m 1 n 1 b 1 * a 2 m 2 n 2 b 2 = a 1 a 2 + ϕ(m 1 ⊗ n 2 ) a 1 m 2 + m 1 b 2 n 1 a 2 + b 1 n 2 φ(n 1 ⊗ m 2 ) + b 1 b 2 for a 1 , a 2 ∈ A, b 1 , b 2 ∈ B, m 1 , m 2 ∈ M and n 1 , n 2 ∈ N . Let G be an infinite cyclic group, generated by g. We can define a G-grading on S by putting R = S e = A 0 0 B , S g = 0 M 0 0 , S g −1 = 0 0 N 0 and S h = {( 0 0 0 0 )} for every h ∈ G \ {e, g, g −1 }. It is easy to see that S g S g −1 = Im(ϕ) 0 0 0 = A 0 0 0 and S g −1 S g = 0 0 0 Im(φ) = 0 0 0 B and thus S is obviously not strongly graded. However, S is epsilon-strongly graded. Indeed, if we put ǫ g = 1 A 0 0 0 , ǫ g −1 = 0 0 0 1 B and ǫ e = ǫ g + ǫ g −1 then it is easy to verify that this yields an epsilon-strong G-grading on S. From the fact that Supp(S) = {g ∈ G | S g = {0}} is finite, we immediately see that
Z(R) fin = Z(R) = Z(A) 0 0 Z(B) .
Remark 41. With this grading one can find examples in which the Morita ring S is not a partial crossed product of G over R = S e . Indeed, let P be a progenerator in the category mod − R, of right R-modules. It follows by [11,Theorem 3.20] that (EndP R , R, P, P * = hom(P R , R), ϕ, φ), where ϕ : P * ⊗ EndP R P ∋ f ⊗ p → f (p) ∈ R and ϕ : P ⊗ R P * ∋ p ⊗ f → f p ∈ EndP R , and f p (r) = pf (r), for all r ∈ R, is a strict Morita context. Consider the associated Morita ring S, and put D g = S g S g −1 .
Since, in general, as left EndP R -modules, EndP R is not isomorphic to P , it follows by [8, Theorem 6.5] that S is not a partial crossed product of G over S e . A concrete example is obtained by taking a commutative unital ring R and P = R n , for some n > 1. Proof. We know that γ 0 : Z(R) → Z(R) is the identity map on Z(R). Now we determine γ g and γ g −1 . Let i m i ⊗n i ∈ M ⊗ B N be such that i ϕ(m i ⊗n i ) = 1 A and j n j ⊗m j ∈ N ⊗ A M with j φ(n j ⊗ m j ) = 1 B . Then for ae 11 + be 22 ∈ Z(R) one has that From this it follows that tr(1 A e 11 ) = 1 A e 11 + 1 B e 22 , and hence S/R is separable due to Theorem 3.
Example 43. Let T = ⊕ g∈G T g be a ring which is strongly graded by a group G. Fix g ∈ G and consider the strict Morita context (T e , T e , T g , T g −1 , ϕ, φ) where ϕ : T g ⊗ Te T g −1 → T e and φ : T g −1 ⊗ Te T g → T e are the canonical T e -bimodule isomorphisms (see [22,Corollary 3.1.2] and [5]). The corresponding Morita ring S = Te Tg T g −1 Te is epsilon-strongly graded by an infinite cyclic group G, generated by g, as described above. By Proposition 42, S is separable over R = Te 0 0 Te . Remark 44. If S is a ring which is strongly graded by G, then G = Supp(S) = {g ∈ G | S g = {0}} necessarily holds. However, if S is only epsilon-strongly graded by G, then Supp(S) need not even be a subgroup of G. Indeed, consider Example 43 and notice that g belongs to Supp(S) but that S g 2 = {( 0 0 0 0 )}. Hence, in this case Supp(S) is not closed under group multiplication.
acknowledgement
The authors are grateful to Ruy Exel for having pointed out the equivalence between (ii) and (iii) in Proposition 7.
Proposition 20 .
20Let S/R be separable and let M be a left (right) S-module. If res(M ) is left (right) projective, then M is left (right) projective.
1 . Define E : S → R by E(s) = s e , for s ∈ S. Then, clearly, E is an R-bimodule map. Take s ∈ S. Then
8 .
8Examples: Morita rings Let (A, B, A M B , B N A , ϕ, φ) be a strict Morita context. It consists of unital rings A and B, an A−B-bimodule M , an B −A-bimodule N , an A−A-bimodule epimorphism ϕ : M ⊗ B N → A and an B − B-bimodule epimorphism φ : N ⊗ A M → B.
Proposition 42 .
42Let (A, B, A M B , B N A , ϕ, φ) be a strict Morita context, and let S be the associated Morita ring. Then the extension S/R is separable.
γ g (ae 11 + be 22 ) = i ϕ(m i b ⊗ n i )e 11 and γ g −1 (ae 11 + be 22 ) = j φ(n j a ⊗ m j )e 22 .Then the trace map tr γ : Z(R) → Z(R) is given byae 11 + be 22 → ae 11 + be 22 + i ϕ(m i b ⊗ n i )e 11 + j φ(n j a ⊗ m j )e 22 .
Recall that R is called left (right) semisimple if all left (right) R-modules are semisimple. Notice that since R is left semisimple if and only if R is right semisimple, (see [13, Corollary (3.7)]) the left/right distinction is therefore unnecessary. Recall that R is called left (right) hereditary if all submodules of left (right) projective modules over R are again projective.
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| [] |
[
"Optimal evolutionary control for artificial selection on molecular phenotypes",
"Optimal evolutionary control for artificial selection on molecular phenotypes"
] | [
"Armita Nourmohammad ",
"Ceyhun Eksin ",
"\nDepartment of Physics\nMax Planck Institute for Dynamics and Self-organization\nDepartment of Industrial and Systems Engineering\nUniversity of Washington\n3910 15th Ave Northeast, am Faßberg 1798195, 37077Seattle, GöttingenWAGermany\n",
"\nTexas A&M University\n77845College StationTX\n"
] | [
"Department of Physics\nMax Planck Institute for Dynamics and Self-organization\nDepartment of Industrial and Systems Engineering\nUniversity of Washington\n3910 15th Ave Northeast, am Faßberg 1798195, 37077Seattle, GöttingenWAGermany",
"Texas A&M University\n77845College StationTX"
] | [] | Controlling an evolving population is an important task in modern molecular genetics, including in directed evolution to improve the activity of molecules and enzymes, breeding experiments in animals and in plants, and in devising public health strategies to suppress evolving pathogens. An optimal intervention to direct evolution should be designed by considering its impact over an entire stochastic evolutionary trajectory that follows. As a result, a seemingly suboptimal intervention at a given time can be globally optimal as it can open opportunities for desirable actions in the future. Here, we propose a feedback control formalism to devise globally optimal artificial selection protocol to direct the evolution of molecular phenotypes. We show that artificial selection should be designed to counter evolutionary tradeoffs among multi-variate phenotypes to avoid undesirable outcomes in one phenotype by imposing selection on another. Control by artificial selection is challenged by our ability to predict molecular evolution. We develop an information theoretical framework and show that molecular time-scales for evolution under natural selection can inform how to monitor a population to acquire sufficient predictive information for an effective intervention with artificial selection. Our formalism opens a new avenue for devising optimal artificial selection for directed evolution of molecular functions. | 10.1101/2019.12.27.889592 | [
"https://arxiv.org/pdf/1912.13433v1.pdf"
] | 209,515,711 | 1912.13433 | 9578e18019cb6070aebe8cd976569e492953dfd1 |
Optimal evolutionary control for artificial selection on molecular phenotypes
Armita Nourmohammad
Ceyhun Eksin
Department of Physics
Max Planck Institute for Dynamics and Self-organization
Department of Industrial and Systems Engineering
University of Washington
3910 15th Ave Northeast, am Faßberg 1798195, 37077Seattle, GöttingenWAGermany
Texas A&M University
77845College StationTX
Optimal evolutionary control for artificial selection on molecular phenotypes
Controlling an evolving population is an important task in modern molecular genetics, including in directed evolution to improve the activity of molecules and enzymes, breeding experiments in animals and in plants, and in devising public health strategies to suppress evolving pathogens. An optimal intervention to direct evolution should be designed by considering its impact over an entire stochastic evolutionary trajectory that follows. As a result, a seemingly suboptimal intervention at a given time can be globally optimal as it can open opportunities for desirable actions in the future. Here, we propose a feedback control formalism to devise globally optimal artificial selection protocol to direct the evolution of molecular phenotypes. We show that artificial selection should be designed to counter evolutionary tradeoffs among multi-variate phenotypes to avoid undesirable outcomes in one phenotype by imposing selection on another. Control by artificial selection is challenged by our ability to predict molecular evolution. We develop an information theoretical framework and show that molecular time-scales for evolution under natural selection can inform how to monitor a population to acquire sufficient predictive information for an effective intervention with artificial selection. Our formalism opens a new avenue for devising optimal artificial selection for directed evolution of molecular functions.
Controlling an evolving population is an important task in modern molecular genetics, including in directed evolution to improve the activity of molecules and enzymes, breeding experiments in animals and in plants, and in devising public health strategies to suppress evolving pathogens. An optimal intervention to direct evolution should be designed by considering its impact over an entire stochastic evolutionary trajectory that follows. As a result, a seemingly suboptimal intervention at a given time can be globally optimal as it can open opportunities for desirable actions in the future. Here, we propose a feedback control formalism to devise globally optimal artificial selection protocol to direct the evolution of molecular phenotypes. We show that artificial selection should be designed to counter evolutionary tradeoffs among multi-variate phenotypes to avoid undesirable outcomes in one phenotype by imposing selection on another. Control by artificial selection is challenged by our ability to predict molecular evolution. We develop an information theoretical framework and show that molecular time-scales for evolution under natural selection can inform how to monitor a population to acquire sufficient predictive information for an effective intervention with artificial selection. Our formalism opens a new avenue for devising optimal artificial selection for directed evolution of molecular functions.
I. INTRODUCTION
The concept of feedback control in molecular evolution was first advocated by A. Wallace as a way of describing natural selection [1]. Wallace hypothesized that similar to the centrifugal governor of the steam engine, the action of natural selection is like a controller that balances organismic traits, such that weak feet are often accompanied with powerful wings [1]. Such evolutionary tradeoffs are ubiquitous in natural fitness landscapes. For example, experiments on a protein transport system has shown that the fitness landscape for the underlying biochemical network is tuned to exploit optimal control with feedback throughout evolution [2]. However, it remains to be determined whether these structures are solely reflective of biochemical constraints or have emerged as incidences of fitness landscapes that could accommodate for long-term evolutionary survival.
Evolution as a feedback control is also reflected in the inheritance strategies and phenotypic response of populations to time-varying environments. A prominent example of such adaptation is observed in bacteria where cells use phenotypic switches to produce slowly replicating bacteria with tolerance and persistence against antibiotics. Populations use this Lamarckian-type phenotypic response [3] to hedge their bets against fluctuating environments [4,5]-an optimal response that can be * correspondence should be addressed to Armita Nourmohammad: [email protected] viewed as an evolutionary feedback control [6].
Another approach to evolutionary control is through external interventions with artificial selection to direct populations to acquire a desired trait. Fig. 1 demonstrates artificial selection with a feedback control to breed "pink cows", which are otherwise not favored by natural selection. Such selective breeding has long been used to domesticate animals or to improve agricultural yield in crops and became the basis for Mendelian genetics [7].
Another important avenue for artificial selection is to characterize intervention protocols against rapidly evolving pathogens, for example to counter emergence of drug resistance in bacteria, escape of viral populations from immune challenge, or progression of evolving cancer tumors [8,9]. Artificial selection also plays a significant role in improving molecular functions. Importantly, directed evolution in the lab is currently being employed to improve the activity and selectivity of molecules and enzymes [10][11][12], often desirable in industry or for pharmaceutical purposes.
Designing any artificial selection protocol is limited by our ability to predict the outcome of evolution, which is often challenging due to a multitude of stochastic forces at play, such as mutations, reproductive stochasticity (genetic drift) and environmental fluctuations [13,14]. In contrast to strongly divergent evolution at the genetic level, there is growing experimental evidence for convergent predictable evolution at the phenotypic level [15][16][17], including for complex molecular phenotypes like RNA polymerase function [18]. We will exploit this evolutionary predictability and focus on designing artificial selection for molecular phenotypes, which are key links be- Artificial selection is an external intervention to select for a desired trait (i.e., pinkness of cows) in a population, which is otherwise not favored by natural selection. Artificial selection should be viewed as an optimal feedback control, whereby monitoring a stochastically evolving population informs the intervention protocol to optimally bias breeding and reproduction over generations. (B) The graph shows different paths with indicated costs for a system to evolve from a start to a target state. Bellman's principle of optimality states that at each step an optimal decision is made, assuming that the following steps are also determined optimally. Although the first step (full line) of the blue path is more costly compared to the others (dotted lines), its cumulative cost is minimum, and hence, it should be chosen as the optimal path. This decision can be made best recursively, known algorithmically as dynamic programming.
tween genotypic information, organismic functions, and evolutionary fitness [13].
Fitness and large-scale organismic traits are often encoded by a number of co-varying molecular phenotypes, linked through genetic interactions; pigmentation patterns on the wings or body of fruit flies are among such multi-dimensional traits, impacted by the expression level of many interacting genes.
A central issue in designing artificial selection for multi-variate phenotypes is to avoid the undesirable (side)effects of selection, which can arise due to evolutionary tradeoffs, e.g. between function and stability of a protein. Evolutionary interventions on multi-variate phenotypes should be designed by assuming their impact over an entire evolutionary trajectory that follows. As a result, a locally optimal action at a given time point may be sub-optimal once considering all the actions that are required to follow in order to direct the correlated evolution of the phenotypes towards their targets; see Finding a globally optimal protocol to direct a stochastic evolution is a topic of control theory, known for its impact in engineering, economics and other fields [19]. Here, we introduce a formalism based on optimal control to devise a population-level artificial selection strategy and drive the stochastic evolution of multi-variate molecular phenotypes towards a desired target. We will show how uncertainty and lack of evolutionary predictability can limit the efficacy of such artificial selection. Finally, we will discuss how to best monitor a population and acquire a sufficient predictive information in order to optimally intervene with its evolution.
II. RESULTS
A. Model of multi-variate phenotypic evolution Molecular phenotypes are often polymorphic due to genetic variation in their encoding sequence within a population. Here, we primarily focus on phenotypes that are encoded by a relatively large number of genetic loci and hence, are approximately normally distributed within a population-this Gaussian approximation however, can be relaxed as discussed in ref. [20]. In the case of normally distributed k-dimensional phenotypes, we characterize the population's phenotype statistics by the average x = [x 1 , x 2 , . . . , x k ] and a symmetric covariance matrix K, where the diagonal elements K ii (x) indicate the variance of the i th phenotype and the off-diagonal entries K ij (x) indicate the covariance between different phenotypes.
The primary evolutionary forces that shape the composition of phenotypes within a population are selection, mutations and genetic drift at the constitutive genetic sites. Molecular phenotypes are often encoded in confined genomic regions of about a few 100 bps, and hence, are not strongly impacted by recombination, even in sexually reproducing populations. The impact of the evolutionary forces on phenotypes can be directly projected from the evolutionary dynamics in the high-dimensional space of the encoding genotypes [21,22]. The effect of selection on the mean phenotype is proportional to the covariance between fitness and phenotype within a population [23]. For Gaussian distributed phenotypes, the change in mean phenotype dx over a short time interval dt simplifies to a stochastic process [24],
dx = (K · ∇F + ∇M )dt + 1 N Σ · dW(1)
where, F and M are fitness and mutation potentials, re-spectively. The gradient functions (denoted by ∇F and ∇M ) determine the forces acting on the phenotypes by selection and mutation, respectively. dW is a differential that the reflects the stochastic effect of genetic drift by a multi-dimensional Wiener noise process [25]. The amplitude of the noise is proportional to Σ, which is the square root of the covariance matrix (i.e., Σ Σ ≡ K), scaled by the effective population size N that adjusts the overall strength of the noise. In other words, the fluctuations of the mean phenotype across realizations of an evolutionary process is proportional to the intra-population variance K and inversely scales with the effective population size (i.e., the sample size) N . Most of our analyses are applicable to general fitness and mutation landscapes. However, we characterize in detail the features of artificial selection to direct evolution on quadratic fitness and mutation landscapes, where phenotypes evolve by natural selection towards an evolutionary optimum [26]. In this case, the impacts of selection and mutation follow linear functions in the high-dimensional phenotypic space, ∇F = −2C 0 · x, ∇M = −2L · x, where x denotes the shifted phenotype vector centered around its stationary state and C 0 and L are selection and mutation strengths, respectively. We can formulate the evolution of mean phenotypes by,
dx = −2KC x dt + Σ dW(2)
where C ≡ N (C 0 + K −1 L) is the adaptive pressure, scaled by the population size, which quantifies the potential of a phenotype to accumulate mutations under selection. In the rest of the manuscript, we will use F as a short hand for the adaptive landscape under natural selection, whose gradient characterizes the adaptive pressure, ∇F = −2Cx in eq. 2. We have also rescaled time with the effective population size (i.e., t → N t), which is the coalescence time in neutrality [27]. Similar to the mean, the covariance matrix K is a timedependent variable, impacted by evolution. However, fluctuations of covariance are much faster compared to the mean phenotype, and therefore, covariance can be approximated by its stationary ensemble-averaged estimate [20,21]. Moreover, even in the presence of moderately strong selection pressure, the phenotypic covariance depends only weakly on the strength of selection and is primarily determined by the supply of mutations in a population [21,22]. Therefore, we also assume that the phenotypic covariance matrix remains approximately constant over time, throughout evolution. With these approximations, evolution of the mean phenotype can be described as a stochastic process with a constant adaptive pressure that approaches its stationary state over a characteristic equilibration time ∼ (2KC) −1 .
The stochastic evolution of the mean phenotype in eq. 2 defines an ensemble of evolutionary trajectories. We can characterize the statistics of these evolutionary paths by the dynamics of the underlying conditional probability density P (x , t |x, t) for a population to have a mean phenotype x at time t , given its state x at an earlier time t < t . The dynamics of this probability density follows a high-dimensional Fokker-Planck equation [21],
∂ ∂t P (x , t |x, t) = 1 2N TrK∇xx − ∇(K · ∇F ) P (x , t |x, t)(3)
where we introduced the compact notation,
TrK∇ xx ≡ ij K ij ∂ ∂xi ∂ ∂xj .
As a result, the conditional distribution of phenotypes follows an Ornstein-Uhlenbeck process, described by a time-dependent multi-variate Gaussian distribution; see Appendix A.
B. Artificial selection to optimally direct evolution Natural selection in eqs. 2,3 drives populations towards an optimum, which in general is a function of the organism's environmental and physiological constraints. Artificial selection aims to relax or tighten some of the natural constraints to drive evolution towards an alternative desired state x * . In general, we can formulate evolution subject to artificial selection as,
dx = K · ∇F + u(x, t) A(x,t) dt + ΣdW(4)
where u(x, t) is a time-and phenotype-dependent vector, which denotes the impact of artificial selection and A(x, t) is the total force incurred by natural and artificial selection on the phenotypes. Our goal is to find an optimal protocol for artificial selection u(x, t) in order to reach the target x * by a desired time t f , while minimizing the cost function,
Ω(x, u, t) = V (x, t) + 1 2 u Bu(5)
over an entire evolutionary trajectory. Here, V (x, t) ≡ V (|x t − x * |) is the cost for deviation of the phenotype state x t at time t from the desired target x * , and B is a matrix that characterizes the cost for imposing artificial selection u ≡ u(x, t) and intervening with natural evolution of each phenotype. Finding an optimal strategy u * (x, t) to steer a stochastic process is a topic of control theory, known for its impact in engineering, economics and other fields [19]. To solve the optimal control problem (i.e., to characterize an optimal artificial selection strategy), we define the costto-go function,
J(x, t) = min u Q(x, t f ) + t f t ds V (xs) + 1 2 u s Bus evol.(6)
where the angular brackets · indicate expectation over stochastic evolutionary histories from time t until the tar-
get time t f . Here, Q(x, t f ) ≡ Q(|x t f − x * |)
characterizes the cost of deviation from the target at the end of the evolutionary process t f , which could be chosen to be different from the path cost V (x).
An optimal artificial selection protocol should be designed by considering its impact over an entire stochastic evolutionary trajectory that follows. As a result, a seemingly suboptimal intervention at a given time can be globally optimal as it can open opportunities for more desirable actions in the future; see schematic Fig. 1B. To characterize a globally optimal artificial selection protocol at each time point u * (x, t), we will assume that the following selection strategies will be also determined optimally. This criteria is known as Bellman's "principle of optimality" [28], and would allow us to express the optimal control problem in a recursive form (Appendix A), known as dynamic programming in computer science [28]. As a result, we can formulate a dynamical equation for the cost-to-go function, known as Hamilton-Jacobi-Bellman (HJB) equation [29],
− ∂J(x, t) ∂t = min u Ω(xt, ut) + A(xt) · ∇J + 1 2 TrK∇xxJ (7)
with the boundary condition J(x, t f ) = Q(x, t f ) at the end of the process; see Appendix A. Importantly, the HJB equation (7) indicates that the cost-to-go J(x, t) is a potential function based on which the optimal artificial selection can be evaluated,
u * (x, t) = −B −1 · ∇J(x, t).(8)
In other words, the cost-to-go function characterizes a time-and phenotype-dependent artificial fitness landscape that determines the strength of artificial selection u * (x, t).
C. Artificial selection by path integral control
The solution to the HJB equation (7) for the costto-go function J(x, t) and the artificial selection u * (x, t) can be complex time-and state-dependent functions, described by non-linear evolutionary operators (Appendix A). Here, we consider a class of control problems, known as "path integral control" [30][31][32], where the cost matrix B for artificially intervening with evolution is inversely proportional to the phenotypic covariance K, i.e., B = λK −1 , where λ is a constant that determines the overall cost of artificial selection. In other words, we assume that imposing artificial selection on highly conserved phenotypes is more costly than on variable phenotypes. This is intuitive as conserved phenotypes are often essential for viability of an organism and it is best to design a control cost function that limits the access to such phenotypes through artificial selection.
Importantly, the path-integral control assumption results in a significant mathematical simplification for the dynamics of the of the cost-to-go function J(x, t) and makes the inference of optimal artificial selection more tractable; see Appendices A, B. We can characterize the evolution of the conditional distribution P u (x , t |x, t) for a population under optimal artificial selection u * (x, t) to be in the phenotypic state x at time t , given its state x at time t by,
∂ ∂t Pu(x , t |x, t) = 1 2N TrK∇xx − ∇(K∇F ) − 1 λ V (x, t) Pu(x , t |x, t)(9)
with the initial condition P u (x , t|x, t) = δ(x − x ) (see Appendix A). The cost-to-go function J(x, t), and consequently the optimal control u * (eq. 8), can be directly evaluated as a marginalized form of the conditional probability density in eq. 9 (Appendix A),
J(x, t) = −λ log dx Pu(x , t f |x, t) exp[−Q(x , t f )/λ](10)
where Q(x , t f ) is the end point cost (eq. 6). Evolution under artificial selection (eq. 9) resembles the natural evolutionary dynamics (eq. 3 with u = 0) with an extra annihilation term V (x, t)/λ [31]. Therefore, artificial selection acts as an importance sampling protocol over each selection cycle (e.g. each generation) that removes (annihilates) individuals from the population with a rate proportional to their distance from the evolutionary target ∼ V (|x t −x * |)/λ; see Fig. 2. Specifically, at each time point, this protocol generates a phenotypic distribution consistent with the evolutionary process under optimal artificial selection in eq. 4, without an explicit knowledge of the selection protocol u * (x, t); see Appendix A.
Moreover, in the path integral control regime (i.e., B = λK −1 ), the cost-to-go function scaled by the overall control factor, J(x, t)/λ, determines a time-and phenotype-dependent fitness landscape associated with artificial selection F art. (x, t); see eq. 8). Throughout an artificial selection process, populations experience an effective landscapeF (x, t) = F (x) + F art. (x, t), as a combination of the natural fitness landscape F (x) and the artificial fitness landscape, F art. (x, t). The overall cost of control λ determines the impact of artificial selection on evolution relative to natural selection, and when the control cost is small (i.e., λ 1), artificial selection can dominate the course of evolution.
D. Artificial selection for phenotypes under stabilizing selection
Here, we focus on the specific case of natural evolution in a high dimensional quadratic fitness landscape (eq. 2). In addition, we assume a quadratic form for the cost function throughout the evolutionary process,
V (x, t) = 1 2 (x t − x * ) G(x t − x * ) and also at the end point Q(x, t f ) = 1 2 (x t f − x * ) G (x t f − x * ).
Characterizing an artificial selection protocol under such quadratic constraints falls within the class of standard stochastic control problems, known as linear-quadratic-Gaussian (LQG) control [19]. However, we will present our analyses based on the path integral control approach in eq. 9, which is generalizable beyond LQG and can be applied to arbitrary cost functions and fitness landscapes (see Appendix B).
Let us imagine that our criteria is to drive evolution towards the optimum x * by time t f , which implies that the path cost is zero G = 0 but the end-point cost is non-zeroG > 0; see the above section and Appendix B for the general scenario including the case with G > 0. In this case, we infer that the strength of the optimal artificial selection should strongly increase as we approach the end point,
u * (x, t) = − K λ e −2CKτG I − Kτ λ I + Kτ λG −1G (e −2KCτ x − x * )(11)
where τ = t f − t is the remaining time to the end point and K τ = x(t), x(t f ) is the time-dependent covariance matrix for the conditional probability density of natural evolutionary process; see Appendix B for the case of G > 0 and Appendix C for detailed derivation. At the end point, the optimal artificial selection keeps the population close to the target with a strength,
u * (τ → 0) = − 1 λ KG(x − x * )(12)
which resembles the breeder's equation [33] for artificial selection with a heritability factor, h 2 = KG/λ. When the goal is to drive the population towards a target by an end point t f , the effective fitnessF (x, t) remains close to the natural landscape for an extended period. As time approaches the end point, populations transition from evolving in their natural landscape F (x) to the artificially induced fitness landscape F art. (x, t f ); see Figs. 3 and 4 for evolution in one and two dimensions, respectively. Moreover, towards the end point, the fitness peak and the strength of selection approach their final values, determined by the target and the cost functions in eq. 5, in an exponentially rapid manner; Fig. 3B,C.
E. Artificial selection for multi-variate phenotypes
One of the main issues in designing breeding experiments in plants and animals is the undesirable (side)effects of artificial selection on covarying phenotypes, primarily due to evolutionary tradeoffs [34] e.g. between sturdiness and flavor of vegetables like tomatoes [35]. Similarly, tradeoffs among covarying molecular phenotypes (e.g. function vs. stability of a protein) could lead to undesirable outcomes for artificial selection at the molecular level.
To demonstrate the consequences of phenotypic covariation, let us consider a simple example for artificial selection on two covarying phenotype (x, y). We aim to drive the phenotype x towards the target x * > 0 by artificial selection while keeping the phenotype y at its stationary state value y * = 0. An optimal artificial selection protocol (eq. 11) defines an effective two-dimensional quadratic fitness landscape that biases the evolutionary trajectories towards the target state (Fig. 4A). As a result, the phenotype distributions at the end of the process become significantly distinct from the expectation under natural selection, and remain strongly restricted around their targets values; Fig. 4B.
The peak of this fitness landscape (i.e., the effective optimum) changes from the natural state (0, 0) to the target state (x * , y * ) by the end of the selection process; Fig. 4C and Fig. S1. The fitness peak moves monotonically along the x-phenotype from the natural optimum 0 towards the target x * , irrespective of the correlation ρ x,y between the two phenotypes; Fig. 4C. However, the dynamics of the fitness peak along the y-phenotype is generally non-monotonic and strongly dependent on the phenotypic correlation ρ x,y . An increase in x drives the positively (negatively) correlated y towards higher (lower) values. Therefore, in the beginning of the process, the optimal artificial selection protocol sets the fitness peak for the y-phenotype at an opposite direction to counter-balance the effect of evolutionary forces due to phenotypic covariation. As the end-point approaches, artificial selection becomes significantly strong with an effective fitness optima set at the target for each phenotype x * and y * (eq. 12). Therefore, the optimum y-value should return to its target state (y * = 0), resulting a non-monotonic behavior in the dynamics of the fitness peak along the y-phenotype; see Fig. 4C. Moreover, the strength of selection also transitions over time and becomes stronger towards the target phenotypes at the end point (heatmaps in Fig. 4C and Fig. S2).
The optimal artificial selection protocol in Fig. 4 is strongly sensitive to the structure of the phenotypic covariance matrix K (eq. 11), and hence, any protocol should be carefully designed to achieve the intended phenotypic desirability.
F. Artificial selection with intermittent monitoring
Imposing artificial selection based on continuous feedback from the state of the population (Fig. 1) requires complete monitoring and the possibility of continuous evolutionary intervention-a criteria that is often not met in real conditions. In general, discrete selection protocols based on a limited information can be inefficient for devising evolutionary feedback control [8,36]. Here, we characterize the limits of discrete optimal interventions based on the evolutionary response of the population to artificial selection. We consider a simple scenario where in a stationary state we aim to keep a population at the target phenotype x * , using discrete monitoring and interventions at time points (i = 1, . . . , M ) with a time separation τ ≡ t i+1 − t i . We define a stationary cost-to-go function, J(x, tm; τ ) = min (13) where the devision by the total time M τ assures that the cost-to-go remains finite. To further simplify, we only consider one dimensional phenotype x with intrapopulation variance k, the cost of deviation V (x) = g(x − x * ) 2 /2 from target x * , and the cost of intervention βu 2 /2 with artificial selection u. However, our analyses can be easily generalized to multi-variate phenotypes.
u lim M →∞ 1 (M − m)τ M i=m u i Bui + t M t i V (xt)dt evol.
In the stationary state and in the regime of small perturbations (gk/λ 1), the optimal artificial selection protocol u * should be a variant of the case with full information with a strength of selection α τ dependent on the time window τ , u * τ = −kα τ (x − x * ); see Appendix D. We can characterize the optimal strength of artificial selection α τ by minimizing the cost-to-go function in eq. 13,
α τ = α 0 1/(4c)(1 − e −2τ ) + 2(x * ) 2 (1 − e −τ ) τ /(2c)(1 + 4c(x * ) 2 )(14)
where α 0 = g/λ is the optimal selection strength under continuous monitoring. Here, time τ is measured in units of the characteristic time for evolution under natural selection, i.e., (2kc) −1 . The partially informed artificial selection α τ depends on most of the evolutionary parameters similar to selection with complete monitoring α 0 . Importantly, the ratio α τ /α 0 , depends only weakly on the strength of natural selection c (Fig. 5A; top) and the target for artificial selection x * (Fig. 5A; bottom) and it is insensitive to the phenotypic diversity k and the parameter λ (eq. 14).
However, the optimal artificial selection α τ strongly depends on the time interval τ and it decays as the time interval τ broadens (Fig. 5A). This decay is linear and relatively slow up to the characteristic time for evolution under natural selection (2kc) −1 . This is the time scale over which an intermittent artificial selection can still contain the population around the desired target x * . If interventions are separated further in time (i.e., τ 1), the optimal selection strength decays rapidly as ∼ τ −1 . Imposing a strong artificial selection in this regime is highly ineffective as populations can easily drift away from the target and towards their natural state within each time interval, and any artificial selection would only contribute to the intervention cost ∼ u 2 without offering any benefits.
G. Information cost for artificial selection
Artificial selection is an evolutionary drive that shifts the equilibrium state of the population under natural selection to a new state around the target. As a result, the phenotypic histories x t0,...t f over the period of (t 0 , . . . , t f ) are statistically distinct for evolution under natural and artificial selection (Fig. 4A). This deviation can be quantified by the Kullback-Leibler distance D KL (P u (x)||P(x)) between the distribution of histories under artificial selection P u (x) ≡ P u (x t0,...t f ) and un-der natural selection P(x). In the stationary state, the Kullback-Leibler distance quantifies the impact of artificial selection on evolution and can be interpreted as the amount of work W t f t0 (u) done by external forces [37] (i.e., the artificial selection) to shift the population from its natural equilibrium to the artificial equilibrium,
W t f t0 (u) = D KL (P u (x)||P(x)) = dx t f t0 P u (x) log P u (x) P(x)(15)
The cumulative work is related to the cost of artificial selection, and for the case of path integral control, it is equivalent to the cumulative cost of control
W t f t 0 (u) = 1 2 u K −1 u dt = 1 2λ
u Bu dt , where the angular brackets · denote expectation over the ensemble of evolutionary histories under artificial selection; see refs. [38,39] and Appendix E. The power (i.e., work per unit time), associated with intermittent artificial selection, can be expressed as the amount of work per time interval τ ,
Power(τ ) = lim M →∞ 1 M τ M i=1 W (ti) = 1 2τ u τ K −1 uτ (16)
The expected work, and hence the power, depend on the time interval τ through various factors. Work depends quadratically on the strength of artificial selection α τ and on the expected population's deviation from the target (x τ − x * ) 2 . On the one hand, the optimal strength of artificial selection α τ decays with increasing the time interval; see Fig. 5A and eq. 14. On the other hand, as the time interval broadens, populations deviate from the target towards their natural state, resulting in an increase in the expected work by artificial selection. Moreover, since interventions are done once per cycle, the power has an overall scaling with the inverse of the cycle interval ∼ τ −1 . These factors together result in a reduction of the expected power associated with artificial selection as the time interval widens; see Fig. 5B.
Power depends strongly on the parameters of natural evolution including the strength of natural selection (c) and the phenotypic diversity within a population (k); Fig. 5B. This is due to the fact that steering evolution under strong natural selection (i.e., with large k and c) is more difficult and would require a larger power by artificial selection. However, the dependence of power on the evolutionary parameters (k, c) remain approximately separated from its dependence on the time interval τ . Thus, power rescaled by its expectation at the characteristic time τ = 1 shows a universal timedecaying behavior, independent of the evolutionary parameters (Fig. 5B).
H. Predictive information as a limit for efficient artificial selection
Artificial selection can only be effective to the extent that an intervention is predictive of the state of the population in the future. The mutual information between artificial selection and the future state of the population quantifies the amount of predictive information [40] by artificial selection, or alternatively, the memory of the population from the selection intervention. We characterize the predictive information I τ as a time-averaged mutual information I(x t , x 0 ) between an intervention (at time t = 0) and the state of the population at a later time t, (0 < t < τ ), during each intervention cycle in the stationary state,
I τ = 1 τ τ 0 dt I(x t , x 0 ) = 1 τ dt dx 0 dx t P (x t , x 0 ) log P (x t |x 0 ) P (x t )(17)
The predictive mutual information monotonically decreases as the time interval τ increases and the population evolves away from the selection target; see Fig. 5C. Predictive information in eq. 17 quantifies the impact of artificial selection on the future of a population. The information theoretical measure of power in eq. 16 on the other hand, quantifies how the optimal artificial selection protocol distinguishes a population's past evolutionary history from the expectation under natural selection. The efficacy of any intervention (i.e., power) is tightly bounded by the impact it may have on the state of the population in the future (i.e., predictive information); see Fig. 5D. Any non-optimal artificial selection effort should lie below this bound and within the accessible region of the information-power plane (Fig. 5D).
Phenotypic diversity k characterizes the rate at which a population evolves away from the target and towards its natural state during an intervention interval (eq. 2). As a result, the information bound for artificial selection is tighter in more diverse populations, which can rapidly evolve away from the target and towards their natural state during each time interval τ .
As interventions become more frequent, predictive mutual information increases but more slowly than the amount of power necessary to induce an effective artificial selection (Fig. 5D). Importantly, the gain in predictive information becomes much less efficient for time intervals shorter than the characteristic time of natural selection (τ 1). Trading power with information provides a guideline for scheduling selection interventions. The characteristic time for evolution under natural selection is a powerful gauge for scheduling the interventions. Importantly, setting the time interval within the range of the characteristic evolutionary time τ ∼ 1 could provide a sufficient power-to-information ratio for an optimal artificial se-lection protocol. However, as information becomes less predictive or the inferred selection protocol becomes less optimal, it would be necessary to monitor and intervene more frequently.
III. DISCUSSION
An optimal intervention should be designed by considering its impact over an entire evolutionary trajectory that follows. Here, we infer an artificial selection strategy as an optimal control with feedback to drive multivariate molecular phenotypes towards a desired target. This selection protocol is optimal over the defined time course and may seem sub-optimal on short time-scales as it drives one phenotype away from its target while driving another towards the target to overcome tradeoffs (Fig. 4C). Monitoring and feedback from the state of a population are key for imposing an effective artificial selection strategy. We show that the schedule for monitoring should be informed by the molecular timescales of evolution under natural selection, which set the rate at which a population loses memory of artificial interventions by evolving towards its favorable state under natural selection.
Being able to control evolution could have significant applications in designing novel molecular functions or in suppressing the emergence of undesirable resistant strains of pathogens or cancers. Interventions that select for desired phenotypes have become possible in molecular engineering [41,42] and in targeted immune-based therapies against evolving pathogens [43]. However, the efficacy of these actions are limited by our ability to monitor and predict the evolutionary dynamics in response to interventions.
Evolution is shaped by a multitude of stochastic effects, including the stochasticity in the rise of novel beneficial mutations and fluctuations in the environment, which at best, raise skepticism about predicting evolution [13,14]. However, evolutionary predictability is not an absolute concept and it depends strongly on the time window and the molecular features that we are interested in. For example, despite a rapid evolutionary turnover in the influenza virus, a number of studies have successfully forecasted the dominant circulating strains for a one year period [44,45]. Similarly, phenotypic convergence across parallel evolving populations has been reported as an evidence for phenotypic predictability, despite a wide genotypic divergence [15][16][17][18]. Therefore, to exploit the evolutionary predictability for the purpose of control, it is essential to identify the relevant degrees of freedom (e.g., phenotypes vs. genotypes) that confer predictive information and to characterize the evolutionary time scales over which our observations from a population can inform our interventions to drive the future evolution.
We focus on modeling control of molecular phenotypes. Phenotypic diversity within a population provides standing variation that selection can act upon. To allow for a continuous impact of artificial selection over many generations, we have limited our analyses to a regime of moderate time-and phenotype-dependent artificial selection to sustain the phenotypic variability in a population. However, it would be interesting to allow for stronger artificial selection to significantly impact the standing variation and the structure of the phenotypic covariance within a population over time. Indeed, allowing a population to rapidly collapse as it approaches a desired target is a common strategy in evolutionary optimization algorithms [46]-a strategy that could accelerate the design of new functions with directed molecular evolution.
In this work, we assume a stochastic model for evolution of multi-variate molecular phenotypes, which has been powerful in describing a range biological systems, including the evolution of gene expression levels [47]. Indeed, optimal control protocols are often designed by assuming a specific model for the underlying dynamics. However, in most biological systems, we lack a knowledge of the details and the relevant parameters of the underlying evolutionary process. The ultimate goal is to simultaneously infer an effective evolutionary model based on the accumulating observations and to design an optimal intervention to control the future evolution-an involved optimization problem known as dual adaptive control [48].
t f t+δt Ω(x s , u s )ds = lim δt→0 min ut→t f J(x t+δt , t + δt) + t+δt t Ω(x s , u s )ds = J(x t , t) + min ut→t f Ω(x s , u s )δt + ∂ ∂t J(x t , t) + (A(x t ) + u) (∇J) + 1 2 ij K ij ∂ ∂x i ∂ ∂x j J δt (A5)
where we used Ito calculus to expand the cost-to-go function, J(x t+δt , t + δt); see e.g. ref. [2]. By reordering the terms in eq. A5, we arrive at the Hamilton-Jacobi-Bellman (HJB) equation, The functional form for the optimal artificial selection u * follows by minimizing the right hand side of eq. A6 with respect to u,
− ∂ ∂t J(x, t) = min u Ω(x t , u t ) + A(x t ) · ∇J + 1 2 TrK∇ xx J = min u 1 2 u Bu + u · ∇J + V (x) + (A(x t ) + u) · ∇J + 1 2 TrK∇ xx J(u * = −B −1 ∇J. (A7)
Therefore, the time-and phenotype-dependent solution for the cost-to-go function J(x, t) determines the optimal protocol for artificial selection u * (x, t). By substituting the form of the optimal control u * in eq. A6, we arrive at a non-linear partial differential equation for the cost-to-go function,
− ∂ ∂t J(x, t) = − 1 2 (∇J) B −1 ∇J + V (x) + A(x t ) · ∇J + 1 2 TrK∇ xx J (A8)
which should be solved with a boundary condition J(x, t f ) = Q(x, t f ) at the end point. We introduce a new variable Ψ = exp[−J/λ] as the exponential of the cost-to-go function. The dynamics of Ψ follows,
λ Ψ ∂ ∂t Ψ = − λ 2 2Ψ 2 (∇Ψ) B −1 ∇Ψ + V (x) − λ Ψ A(x t ) · (∇Ψ) − λ 2 K −1 Ψ 2 (∇Ψ) · ∇Ψ + 1 Ψ ∇ xx Ψ (A9)
The dynamics of Ψ linearizes if and only if there exists a scalar λ that relates the control cost to the covariance matrix such that B = λK −1 . This criteria is known as the path-integral control condition [3,4] by which we can map a generally non-linear control problem onto a linear stochastic process. The path-integral control condition implies that the cost of artificial selection on each phenotype should be inversely proportional to the phenotype's fluctuations. In other words, artificially tweaking with highly conserved phenotypes should be more costly than with variable phenotypes. In this case, the HJB equation for the transformed cost-to-go function Ψ follows,
∂ ∂t Ψ = −A(x) · ∇Ψ − 1 2 TrK∇ xx Ψ + 1 λ V (x)Ψ ≡ −L † Ψ (A10)
where L † is a linear operator acting on the function Ψ. Equation A10 has the form of a backward Fokker-Planck equation with the boundary condition at the end point Ψ(
x, t f ) = exp[−J(x, t f )/λ] = exp[Q(x, t f )/λ].
We can define a conjugate function P u that evolves forward in time according to the Hermitian conjugate of the operator L † . This conjugate operator L can be characterized by evaluating the inner product of the two functions,
LP u |Ψ = P u |L † Ψ = dx P u (x, t)L † Ψ(x, t) = dx P u (x, t) A(x) · ∇Ψ + 1 2 TrK∇ xx Ψ − 1 λ V (x)Ψ Ψ(x, t) = dx − 1 λ V (x)P u (x, t) − ∇A(x)P u + 1 2 TrK∇ xx P u Ψ(x, t) (A11)
where we performed integration by part and assumed that the function P u vanishes at the boundaries. This formulation suggests a forward evolution by the operator L † for the function P u (x , t |x, t) ,
∂ ∂t P u (x , t |x, t) = LP u (x , t |x, t) = 1 2 TrK∇ xx − ∇A(x) L0 − 1 λ V (x) P u (x , t |x, t) (A12)
with a boundary condition at the initial time point P (x , t|x, t) = δ(x − x ). Importantly, the linear operator L resembles the forward Fokker Planck operator L 0 for evolution under natural selection (i.e., the dynamics in eq. A2 with u = 0) with an extra annihilation term V (x)/λ. The evolutionary dynamics with the L 0 operator under natural selection conserves the probability density. The annihilation term on the other hand, eliminates the evolutionary trajectories with a rate proportional to their cost V (x, t)/λ at each time point.
Since Ψ evolves backward in time according to L † and P u evolves forward in time according to L, their inner product P u |Ψ = dx P u (x , t |x, t)Ψ(x , t ) remains time-invariant 1 . Therefore, the inner product of the two functions at the initial and the final time points are equal, which follows,
P u (t)|Ψ(t) = P u (t f )|Ψ(t f ) → dx P u (x , t|x, t)Ψ(x , t) = dx P u (x , t f |x, t)Ψ(x , t f ) → Ψ(x, t) = dx P u (x , t f |x, t) exp[−Q(x , t f )/λ] (A13)
where we substituted the boundary condition for P u at the initial time t and for Ψ at the finial time t f . Thus, the cost-to-go function follows,
J(x, t) = −λ log Ψ(x, t) = −λ log dx P u (x , t f |x, t) exp[−Q(x , t f )/λ] (A14)
Appendix B: Path integral solution to stochastic adaptive control Given the structure of the linear forward operator L, we can either exactly compute the conditional function P u (x , t f |x, t) or to use approximation methods common for path integrals (e.g. the semi-classical method) to evaluate cost-to-go function in eq. A14. To formulate a path integral for P u (x , t f |x, t), we discretize the time window [t : t f ] into n small time slices of length , (t 0 , t 1 , . . . , t n ), with n = t f − t. The conditional probability P u (x , t f |x, t) can be written as an integral over all trajectories that start at the phenotypic state x at time t 0 ≡ t and end at x at time t n ≡ t f ,
P u (x , t f |x, t) ∼ n i=1 dx i P u (x i , t i |x i−1 , t i−1 ) δ(x n − x ) (B1)
The short-time propagator P u (x i , t i |x i−1 , t i−1 ) follows a simple Gaussian form [1],
P u (x i , t i |x i−1 , t i−1 ) ∼ exp − 1 λ x i − x i−1 − A(x i ) λK −1 2 x i − x i−1 − A(x i ) + V (x i ) = exp − λ x i − x i−1 − A(x i ) B 2 x i − x i−1 − A(x i ) + V (x i ) (B2)
where we used, K = λB −1 . We can express the cost-to-go function (eq. A14) as a path integral,
e −J(x,t)/λ = dx P u (x , t f |x, t) exp[−Q(x , t f )/λ] ∼ n i=1 dx i exp − λ x i − x i−1 − A(x i ) B 2 x i − x i−1 − A(x i ) + V (x i ) + Q(x n ) ∼ D(x) exp − 1 λ Q(x(t f )) + t f t dt dx(t) dt − A(x(t), t) B 2 dx(t) dt − A(x(t), t) + V (x, t) ≡ D(x) exp − 1 λ S path (x(t → t f )) (B3) 1
The inner product of the two conjugate functions Pu|Ψ is time invariant:
Pu(t )|Ψ(t ) ≡ dx Pu(x , t |x, t)Ψ(x , t ) = e L(t −t) Pu(t)|e −L † (t −t) Ψ(t) = Pu(t)|e L † (t −t) e −L † (t −t) Ψ(t) ≡ Pu(t)|Ψ(t)
cost of deviation from the optimum throughout each interval. The stationary cost-to-go function follows,
J(x, tm; τ ) = min u lim M →∞ 1 (M − m)τ M i=m u i Bui + t M t i V (xt)dt evol. (D1)
where we have normalized the path cost by the interval τ to assure that the cost-to-go remains finite.
To further simplify, we only consider one dimensional phenotype x with intra-population variance k, the cost of deviation V (x) = g(x − x * ) 2 /2 from target x * , and the cost of intervention βu 2 /2 with artificial selection u. In the stationary state and in the regime of small interventions (gk/λ < 1), we assume that the optimal artificial selection protocol u * should be a variant of the case with full information with a strength of selection α τ dependent on the time window τ , u * τ = −kα τ (x − x * ). Our goal is to characterize the strength of optimal artificial selection α τ . The total cost over an interval τ in the stationary state follows,
Ω τ (x) = β 2 u 2 + 1 τ ti+τ t=ti V (x t )dt = β 2 k 2 α 2 τ (x τ − x * ) 2 + 1 2τ γ τ t=0 (x t − x * ) 2 dt (D2)
We are interested in the regime of moderate to weak interventions (gk/λ < 1), for which the linear response theory can characterize the change in the state of the system after each intervention. In this regime, evolution under artificial selection can be approximated as a perturbation from the dynamics under natural selection. The evolutionary dynamics of the phenotype distribution is governed by a time-dependent Fokker Planck operator, L(x, t),
∂ ∂t P u;τ (x, t) = [L 0 (x) + L u (x)Y (t)]P u;τ (x, t) (D3)
where P u;τ (x, t) is the full probability density under intermittent artificial selection, which can be split into the stationary solution under natural selection and the time-dependent deviation due to artificial selection: P u;τ (x, t) = P 0 (x) + P u (x, t; τ ). L 0 (x) is the Fokker Planck operator associated with evolution under natural selection (i.e., the dynamics in eq. A2 with u = 0), L u (x) = ∂ x kα τ (x − x * ) is the state-dependent operator associated with artificial selection and Y (t) = lim M →∞ M i=1 δ(t − t i ) characterizes a time-dependent component due to artificial selection interventions at the end of each time interval.
In the regime of linear response, where the impact of artificial selection is small, the deviation ∆z of an expected value of an arbitrary function z(x) from it stationary state (i.e., under natural selection) follows (cf. [1]),
∆z(t) = z(x)P u (x, t)dx ≡ ∞ −∞ R z,Lu (t − t )Y (t )dt (D4)
where R z,Lu (t) is the response function to artificial selection L u ,
R z,Lu (t) = z(x) e L0(x) ·t L u (x)P 0 (x)dx for t ≥ 0 0 for t < 0 (D5)
At end of each time interval, artificial selection imposes a shock-type perturbation to the evolutionary process. The immediate response of the population to this selection pressure can be characterized by the instantaneous response function (i.e., with Y (t ) = δ(t − t )), resulting in the change in a given phenotypic statistics z (cf. [1]),
∆z(t) = z(x)L u (x)P 0 (x)dx = 1 Z dx z(x) ∂ ∂x kα τ (x − x * ) exp − x 2 2var st0 = kα τ z(x) st0 − 1 var st0 z(x) x 2 st0 − x * z(x) x st0(D6)
where P 0 (x) is the Gaussian stationary distribution for the mean phenotype under natural selection, var st0 = 1/4c is the stationary ensemble variance for the mean phenotype under natural selection, Z is the normalization factor for the underlying stationary distribution, and · st0 indicates expectation values under the stationary distribution. At the beginning of each interval t = 0 the deviation of the mean phenotype from its expectation under natural selection x st0 = 0 follows, ∆x = x(t = 0) = kα τ x * (D7)
Similarly, the deviation in the second moment of the phenotypic distribution from the expectation under natural selection x 2 st0 = var st0 follows, ∆x 2 = x 2 (t = 0) − var st0 = α τ var st0 − 1 var st0
x 4 st0 = −2kα τ var st0 = −kα τ /2c (D8)
Thus, the phenotypic variance at the beginning of each interval follows,
var u (t = 0) = x 2 (t = 0) − x(t = 0) 2 = var st0 [1 − 2kα τ ] − [kα τ x * ] 2(D9)
Following an intervention at time t = 0, populations evolve according to natural selection until the next intervention. Therefore, the phenotype statistics during each time interval (0 < t < τ ) follow,
x(t) = α τ x * e −2kct (D10) var(t) = var st0 (1 − 4kc) − (kα τ x * ) 2 e −4kct + var st0 (1 − e −4kct ) = var st0 (1 − 2kα τ e −4kct ) − (kα τ x * ) 2 e −4kct(D11)
We can now evaluate the cumulative cost function (eq. D2)
Ω τ (x) = β 2 u 2 + 1 2τ γ τ t=0 (x t − x * ) 2 dt = β 2 k 2 α 2 τ (x τ − x * ) 2 + 1 2τ γ τ t=0 (x t − x * ) 2 dt = β 2 k 2 α 2 τ ( x τ − x * ) 2 + var(τ ) + 1 2τ γ τ t=0 ( x t − x * ) 2 + σ 2 (t) dt = β 2 k 2 α 2 τ var st0 (1 − 2kα τ e −4kcτ ) + (x * ) 2 (1 − 2kα τ e −2kcτ ) + 1 2τ γ −α τ 2c
(1 − e −4kcτ )var st0 + 2(1 − e −2kcτ )(x * ) 2 + (var st0 + (x * ) 2 )τ (D12)
where we have used the time-dependent expectation and variance in eqs. D10 and D11. The optimal strength of artificial selection α * τ for intermittent interventions can be characterized by minimizing the cost function (eq. D12) with respect to α τ ,
α * τ = γ λ (1 − e −τ )(1 + 8c(x * ) 2 + e −τ ) 2τ (1 + 4c(x * ) 2 ) + O (kγ/λ) 2 (D13)
which in the limit of small separation time (τ → 0) approaches the expectation under continuous monitoring in the stationary state (eq. C6), α * (τ → 0) = γ/λ.
FIG. S1: Effective fitness optimum for 2D covarying phenotypes under artificial selection. The dynamics of the effective fitness peak is shown over time (colors) for 2D covarying phenotypes with correlations ρxy indicated by the shape of the markers, and for increasing end-point cost of deviation from target along the x-phenotype, gx = 1, 2, 3 from left to right, with gy = 2 and for increasing strength of natural selection on the x-phenotype, cx = 3, 5, 7 from top to bottom with cy = 5.
Other parameters: x * = 3, y * = 0; cxy = 0; kx = ky = 0.02; λ = 0.1.
FIG. S2: Effective strength of selection for 2D covarying phenotypes under artificial selection . The dynamics of the effective strength of selection is shown over time (colors) for 2D covarying phenotypes with correlations ρxy indicated by the shape of the markers. The parameters in each panels are the same as in Fig. S1.
FIG. 1 :
1Artificial selection as an optimal stochastic adaptive control strategy. (A)
Fig. 1B.
FIG. 2 :
2Artificial selection with stochastic optimal annihilation. Phenotypic trajectories starting from three distinct initial states (open circles) are shown for evolution under natural selection in a 1D quadratic landscape of eq. 2. The trajectories are annihilated ( † and low opacity lines) with a rate proportional to the cost of their deviation from target (dotted line) V (|x − x * |)/λ (eq. 9). At each time point, the survived trajectories characterize the ensemble of evolutionary states for evolution under optimal artificial selection to reach the target at x * = 1. Time is measured in units of the characteristic time for natural evolution (1/2kc). Parameters: k = 0.4; c = 1; λ = 0.01; g = 2.
FIG. 3 :
3to end point, t / t f time relative to end point, t / t f Optimal artificial selection for a 1D phenotype. The impact of artificial selection intensifies as time approaches the end point of the process. (A) The interplay between artificial and natural selection defines an effective time-dependent (colors) fitness landscapeF with an optimumx(t) that approaches the phenotypic target for artificial selection (x * = 3) and an effective selection pressureĉ that intensifies as time approaches end point t/t f → 1. Other parameters: λ = 0.1; c = 2; g = 2. (B) and (C) show the effective fitness peak relative to the targetx/x * and the relative selection pressure of the effective fitness landscapeĉ(t)/ĉ(t f ) as a function of time, for a range of relative artificial to natural selection pressures g/λc (colors).
FIG. 4 :
4Artificial selection on covarying phenotypes. (A) Trajectories for evolution under natural (orange) and artificial (blue) selection are shown for a 2D phenotype (x, y), in a quadratic landscape. Parameter: cx = 2, cy = 4, cxy = 0; x * = 1.2, y * = 0; kx = 0.02; ky = 0.05; kxy = 0; gx = gy = 2; λ = 0.01. (B) The distribution of phenotypes at the end point of an artificial selection protocol (blue) is compared to the phenotypic distribution under natural selection (orange). Evolutionary parameters are the same as in (A). (C) The dynamics of the effective fitness peak is shown over time (colors) for 2D covarying phenotypes with correlations ρxy indicated by the shape of the markers. From left to right, panels show increasing end-point cost of deviation from the target along the x-phenotype, gx = 1, 2, 3 , with gy = 2. Heatmaps show the effective fitness landscapes associated with a specific fitness peak (indicated by the dotted arrows) for anti-correlated phenotypes at three different time points. The direction and length of the red arrows in each heatmap indicate the direction and the strength of selection pressure towards the effective optimum. Parameters: x * = 3, y * = 0; cx = cy = 5, cxy = 0; kx = ky = 0.02; λ = 0.1.
FIG. 5 :
5Artificial selection with limited information. (A) Relative strength of artificial selection ατ /α0 (eq. 14) is shown as a function of the time interval for monitoring and intervention τ , measured in units of the characteristic time for evolution under natural selection (1/2kc). The selection ratio is shown for various strengths of natural selection c (top; with x * = 1) and for various targets of artificial selection x * (bottom; with c = 1). (B) Power (eq. 16) is shown as a function of the time interval τ for a range of parameters for the phenotypic diversity k (full line) and the strength of natural selection c (dotted line). The insert shows a collapse plot for power scaled with the expectation at the characteristic time for natural selection Power(τ )/Power(τ = 1). (C) Predictive mutual information I(τ ) (eq. 17) is shown to decay with the time interval τ for a wide range of parameters (k, c). Insert zooms into a narrower range for the time interval τ < 1. (D) Predictive information (eq. 17)is shown as a function of the scaled power for optimized artificial selection (eq. 16). Each curve sets an information bound for artificial selection for a given set of evolutionary parameters (k, c). A non-optimal selection intervention should lie below the information curve, shown by the gray shaded area as the accessible region associated with the dark blue curve. Color code in (C,D) is similar to (B). Other parameters: λ = 0.6; x * = 3; g = 2.
t f t0 ) and F u = log P u (x t f t0 ) (cf.[5]). The
AcknowledgementsThis work has been supported by the DFG grant (SFB1310) for Predictability in Evolution (AN, CE) and the MPRG funding through the Max Planck Society (AN).Appendix A: Hamilton-Jacobi-Bellman equation for optimal controlWe define a general stochastic evolutionary process for a population of size N with an evolutionary drive due to natural selection and mutations A(x, t) and an external artificial selection u(x, t), dx = (A(x) + u(x, t))dt + Σ(x)dW (A1)Here, time t is measured in units of the coalescence time N (i.e., the effective population size). dW is a differential random walk due to an underlying Wiener process with an amplitude Σ, which is the square root of the phenotypic covariance matrix K: Σ Σ ≡ K. The stochastic evolution in eq. A1 defines an ensemble of phenotypic trajectories, the statistics of which can be characterized by a conditional probability density P (x, t|x , t ) for a population to have a phenotype x at time t, given its state x at a previous time t < t. For a given artificial selection protocol u(x, t), the conditional probability density evolves according to a Fokker-Planck equation[1],where we have used the short hand notation,∂xi∂xj O, as operators that act on the function O in front of them.The purpose of artificial selection is to minimize a cost function,is the cost for deviating from the desired target x * during evolution and B is the cost for intervening in natural evolution and applying artificial selection u ≡ u(x, t).We define the cost-to-go function J(x, t) as the expected value for the cumulative cost from time t to end of the process t f , subject to the evolutionary dynamics and under an optimal control u * t→t f ,is the cost of deviation from the target at the end point t f , which in general can be distinct from the path cost V (x t ). We can formulate a recursive relation for the cost-to-go function J(x, t),In the case that the path cost is zero V (x) = 0, the artificially and naturally selected trajectories become distinct only due to the contributions from the end-point cost at t = t f . For the choice of a linear evolutionary force A(x) = −2KCx and a quadratic end-point cost,, evolution follows an Ornstein-Uhlenbeck (OU) process and the solution to eq. (A12) takes a Gaussian form (see e.g. ref.[2]),with a conditional time-dependent mean,and a covariance matrix,In this case, the cost-to-go in eq. (A14) can be evaluated by a Gaussian integral to marginalize over the end stateresulting in an optimal artificial selection protocol,As the time approaches the end point (t → t f or τ → 0), optimal artificial selection acts as a linear attractive force (i.e., a stabilizing selection)to maintain the population close to the phenotypic target, with an effective strength of artificial stabilizing selectioñ G/λ.Appendix D: Artificial selection with intermittent monitoringHere, we assume that artificial selection is imposed in discrete steps with time interval τ . Similar to the continuous control, the cost function has two components: the cost of control at the end of each intervention and the cumulative ). In analogy to thermodynamics, we can associate a free energy to these distributions, as F 0 = log P 0 (x expected difference between the free energy of the two phenotypic configurations can be interpreted as the amount of work done by artificial selection, which corresponds to the Kullback-Leibler distance between the two distributions,where Dx is the integral measure over all trajectories. The estimate of work in eq. E1 however should not be interpreted as a physical work, rather as an information theoretical measure of discrimination between the two phenotype distributions due to artificial selection. The evolution of the distribution for phenotype trajectories P u (x t f t0 ) under a given artificial selection protocol, u t f t0is Markovian (eqs. A1,A2). To characterize this path probability density, we will follow the path integral formulation in eq. B1 and discretize the time window [t 0 : t f ] into n small time slices of length , (t 0 , t 1 , . . . , t n ), with n = t f − t.The probability of a given trajectory P u (x t f t0 ) can be written as a product of short-term propagators (i.e., conditional probabilities); cf.[1],The Kullback-Leibler distance between the two distributions follows,where we have used dx t − A(x t )dt = u(x t , t)dt + dW t , with dW t as the stochastic differential measure for a multivariate Wiener process (cf.[2]). Importantly, with the criteria of path integral control (i.e., K −1 = B/λ), the Kullback-Leibler distance between the artificially and naturally selected phenotype distributions is equivalent to the cumulative cost of intervention, divided by the overall cost of artificial selection λ,which can intuitively be interpreted as the amount of work done by artificial selection.
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| [] |
[
"STELLAR METALLICITY GRADIENTS IN SDSS GALAXIES",
"STELLAR METALLICITY GRADIENTS IN SDSS GALAXIES"
] | [
"Benjamin Roig ",
"Michael R Blanton ",
"Renbin Yan "
] | [] | [] | We infer stellar metallicity and abundance ratio gradients for a sample of red galaxies in the Sloan Digital Sky Survey (SDSS) Main galaxy sample. Because this sample does not have multiple spectra at various radii in a single galaxy, we measure these gradients statistically. We separate galaxies into stellar mass bins, stack their spectra in redshift bins, and calculate the measured absorption line indices in projected annuli by differencing spectra in neighboring redshift bins. After determining the line indices, we use stellar population modeling from the EZ Ages software to calculate ages, metallicities, and abundance ratios within each annulus. Our data covers the central regions of these galaxies, out to slightly higher than 1R e . We find detectable gradients in metallicity and relatively shallow gradients in abundance ratios, similar to results found for direct measurements of individual galaxies. The gradients are only weakly dependent on stellar mass, and this dependence is well-correlated with the change of R e with mass. Based on this data, we report mean equivalent widths, metallicities, and abundance ratios as a function of mass and velocity dispersion for SDSS early-type galaxies, for fixed apertures of 2.5 kpc and of 0.5 R | 10.1088/0004-637x/808/1/26 | [
"https://arxiv.org/pdf/1506.05013v1.pdf"
] | 53,605,046 | 1506.05013 | 707017ad8a326bf841432fe687a12610a9cecefc |
STELLAR METALLICITY GRADIENTS IN SDSS GALAXIES
16 Jun 2015 Draft version June 17, 2015 June 17, 2015
Benjamin Roig
Michael R Blanton
Renbin Yan
STELLAR METALLICITY GRADIENTS IN SDSS GALAXIES
16 Jun 2015 Draft version June 17, 2015 June 17, 2015Preprint typeset using L A T E X style emulateapj v. 5/2/11 Draft version
We infer stellar metallicity and abundance ratio gradients for a sample of red galaxies in the Sloan Digital Sky Survey (SDSS) Main galaxy sample. Because this sample does not have multiple spectra at various radii in a single galaxy, we measure these gradients statistically. We separate galaxies into stellar mass bins, stack their spectra in redshift bins, and calculate the measured absorption line indices in projected annuli by differencing spectra in neighboring redshift bins. After determining the line indices, we use stellar population modeling from the EZ Ages software to calculate ages, metallicities, and abundance ratios within each annulus. Our data covers the central regions of these galaxies, out to slightly higher than 1R e . We find detectable gradients in metallicity and relatively shallow gradients in abundance ratios, similar to results found for direct measurements of individual galaxies. The gradients are only weakly dependent on stellar mass, and this dependence is well-correlated with the change of R e with mass. Based on this data, we report mean equivalent widths, metallicities, and abundance ratios as a function of mass and velocity dispersion for SDSS early-type galaxies, for fixed apertures of 2.5 kpc and of 0.5 R
INTRODUCTION
The study of stellar population and metallicity gradients has been an ongoing topic because it places relatively strong constraints on the evolutionary history of galaxies. This has been a much-studied field for the past 50-60 years, with a wealth of information available that allows the interpretation of galaxy spectra in terms of stellar populations (Faber et al. 1985 for example). Early studies that correlated observable metal absorption lines with stellar population properties only could examine small numbers of galaxies (e.g. Couture & Hardy 1988;Munn 1992;Gonzalez & Gorgas 1995), but led to the acceptance that galaxies typically had lower metallicities at larger radii. Later studies have increased the sample size, measured additional metal absorption indices, and added detail to that basic picture (Carollo et al. 1993;Koleva et al. 2011).
Theoretical studies have also, of course, addressed the question of what causes this common trend in elliptical galaxies (Martinelli et al. 1998;Ogando et al. 2006 are several of many). Mergers, star formation histories, gas flow, and other mechanisms help determine the metallicity gradients. Mergers can drastically change the distribution of stars in the final galaxies, affecting the observed metallicity gradients dramatically as well (Di Matteo et al. 2009). Conversely, a lack of mergers can lead to the evolution of a metallicity gradient after the infall of cooling gas (Pipino et al. 2010). Predicting the observed stellar metallicity gradients correctly requires a model of the star formation and formation history (mergers or otherwise) of galaxies. This suggests that a large-scale study of metallicity gradients in elliptical galaxies may help our understanding of the average path an elliptical galaxy takes in formation -how many mergers, mass ratios of the mergers, when star formation 1 Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 Washington Place, New York, NY 10003 2 University of Kentucky bursts occur, and how long these formation bursts last, among other properties. Equally, observing metallicity gradients in a large sample of galaxies has been a challenge, as it requires spectra of multiple regions of the same galaxy. Usually, then, observational studies that attempt to constrain theoretical models are restricted to a relatively small number of nearby galaxies (Greene et al. 2013;Spolaor et al. 2010;Kuntschner et al. 2010;Rawle et al. 2010;Pastorello et al. 2014), which makes them sensitive to the specific choice of galaxies to observe. Newer projects (including González Delgado et al. (2014) with the CALIFA survey and Bundy et al. (2015) with MaNGA) study these gradients using integral field spectroscopy (IFS) to attempt to improve our understanding. Here, we study a large number of galaxies observed by the Sloan Digital Sky Survey (SDSS) and instead of finding metallicity gradients in individual galaxies, we average galaxies in redshift bins and calculate statistical gradients between annuli found by subtracting galaxy fluxes at different redshifts from each other. This process yields the population-averaged metallicity and abundance profiles of early type galaxies.
We begin by selecting a sample of non-starforming galaxies in SDSS, as detailed in Section 2.
We measure the standard Lick indices used to calculate age and metallicity, and make use of EZ AGES (Graves & Schiavon 2008) to obtain those parameters. With our sample complete, we compare our calculated gradients to several other studies in Section 6 to verify the accuracy of our approach, focusing on studies that also are able to find gradients in the inner 1R e of galaxies rather than ones that examine the full outer regions as well. We then examine several papers that discuss theoretical models for the formation of these galaxies to see which our observations support and what, if any, conclusions we can draw about likely evolutionary histories of our galaxies in Section 7.
Our sample is composed of a subset of the NYU Value-Added Galaxy Catalog (NYU-VAGC; Blanton et al. 2005), based on the SDSS Data Release 7 (DR7; Abazajian et al. 2009). We use 686,356 Main sample galaxies (Strauss et al. 2002) with observed spectra and several extracted values from the observations (redshift, a half-light radius, magnitudes in several bands, and an estimate of the stellar mass). The stellar mass we use is estimated from the K-corrected mass-to-light ratios in the ugriz and JHK bands (see Blanton & Roweis 2007 for details) . We wish to select a uniform sample of non-starforming galaxies so that we can have confidence that the properties remain relatively the same across the entire redshift range of our study and avoid any emission line contamination of the features we seek to measure. To do this, we perform a few simple cuts involving [OII] and Hα equivalent widths (EWs) as previously done in Yan et al. (2006) that yield cuts in spectroscopic properties. First, objects that have [OII] emission but no Hα are kept in the sample. Second, any object without either [OII] or Hα emission is kept. Finally, for objects with both [OII] and Hα, we keep ones with a high ratio of [OII]/Hα and reject those with a low ratio. We define this ratio as Yan et al. (2006) does (EW([OII]) > 5EW(Hα)−7) and we keep all objects that pass this ratio of those with both lines detected. These cuts may potentially exclude some red, non-starforming galaxies as the cost of being sure the number of blue, starforming galaxies remaining in the sample is very low (approximately 3% per Yan et al. 2006). We do not, however, make any morphology cuts; this suggests about 40% of the sample will be pure elliptical galaxies, with the rest a mix of S0 or Sa morphologies (Blanton & Moustakas 2009). The dependence of gradients on a mix of E, S0, and Sa morphologies has been found to have limited effect (see González Delgado et al. 2014, Fig. 5), so this may not dramatically impact our results, but should be remembered as a caveat.
This leaves us with 266,195 non-starforming galaxies to work with. The next set of cuts is to ensure that the data remaining has correctly measured results for all the important parameters that we will need. We remove all objects with masses less than 10 7 and all objects that do not have measured EWs for the metal indices (Mg b, Fe5270, Fe5335, Ca4227, and C4668) that we will be using, as well as Hβ, which tends to happen only in the rare case of a badly fit absorption line. This is a relatively minor adjustment and only reduces our sample by approximately 4%.
Finally, metallicities are dependent on the stellar masses of the galaxies, and so we break up the data into mass bins, each individually volume limited, the details of which are given in Table 1. We have also alternatively divided the galaxies into velocity dispersion bins, with the name number of bins and volume limiting cuts in redshift and magnitude.
As a note, the mass bins are large enough to still leave around a factor of two difference in the brightest versus dimmest galaxies in each bin; however, in tests to see if this luminosity range overly weighted the brightest galaxies in each bin in measuring line fluxes we found less than a 5% steepening of the indicator gradients when normalizing luminosities versus leaving them unchanged. For the results presented in this paper, we do not normalize Note. -4 stellar mass bins are chosen for the galaxes in this paper to reduce the effects of the mass dependence of metallicity. Each bin has a redshift range and magnitude limit selected to maximize the number of objects in the sample while ensuring that the sample remains volume limited. each galaxy in a mass bin to the same luminosity, but rather leave the luminosities as they are for each individual galaxy. Figure 1 shows the resulting samples with the cuts in red lines.
In preparation for Lick index measurement, we must make two adjustments to our spectra. The first is a resolution correction -as Zhu et al. (2010) notes when performing the same corrections, SDSS spectra are not at the same resolution at which the originally defined Lick indices were observed. Schiavon (2007) contains several tables (Tables 43-46, depending on galaxy age) in which corrections for a given velocity dispersion to the Lick IDS resolution are defined, but to establish as common a baseline as we can between mass bins, we smooth all galaxies in our sample to the same velocity dispersion of 325 km s −1 . Any errors in the resolution correction from Schiavon (2007) will thus at least be the same for every galaxy, hopefully ensuring that any gradient trends we report will be independent of these errors. We use the table of corrections for a 7.9 Gyr population, noting that while some of our galaxies will be older and some younger, the differences are only a few percent for corrections of a different stellar population age. The second correction, also defined by Schiavon (2007), is a fluxing correction to bring the SDSS spectra in line with the Lick IDS system. It too is empirical, with potentially large error, but we apply this correction after each Lick index is measured to fully bring our results in line with the Lick IDS system, noting unfortunately that errors from these corrections cannot be included in our error budget as well.
Each object that is kept in the sample then has its Lick indices for Mg b, Fe 5335, Fe 5270, Ca 4227, C 4668, and Hβ measured. In addition to the Lick index, we also measure the absorption line flux and continuum values directly over the same wavelength range. We also measure the [OIII] line flux to correct for Hβ line infill due to emission (see Zhu et al. 2010;Trager et al. 2000;Mehlert et al. 2000 for some discussion on the process and difficulties behind this correction). We use the standard correction factor of ∆EW Hβ = 0.6EW([OIII]) that these prior works settle on with the understanding that it is empirical and has relatively large scatter, instead of attempting a correction where we would directly adjust the flux (instead of the equivalent width) of the Hβ absorption based on the measured flux of the [OIII] emission. In this [OIII] line fitting, we ignore any lines that are weak detections (defined as indistinguishable from zero line flux or EW at the one sigma level), and simply do not correct Hβ for those objects. We do have data for other commonly used indicators such as D4000 or CN1 and CN2, but the population modeling code we use does not use these as inputs to determine metallicity or age, so we do not present them here.
We convert several other of the above measured Lick indices to commonly used combinations in calculations of metallicity as well. First we compute [MgFe] ′ = Mgb × (0.72Fe5270 + 0.28Fe5335), and secondly we compute Fe = (Fe5270 + Fe5335)/2. Both these combinations are chosen because they more directly correlate with metallicity than any single Lick index . This gives us 5 metallicity indicators ([MgFe] ′ , Fe , Mg b, Ca 4227, C 4668) and one age-related indicator (corrected Hβ) for each of our galaxies that we will analyze.
MOCK CATALOG COMPARISON AND CORRECTION
To confirm that we can accurately measure gradients using this annulus method, we test the procedure on mock galaxy spectra generated with known metallicity gradients. To create this catalog, we use the NASA-Sloan Atlas (described in Blanton et al. 2011) as our reference for the properties of similar red and old galaxies. We apply the same selection criteria to the NASA-Sloan Atlas as in Section 2 to ensure we only have red galaxies, and then randomly select galaxies from the remaining catalog. We assign the Sersic index n, the Sersic halflight radius (in kpc), the absolute magnitude in the ugriz bands, the stellar mass, and the axis ratio b/a for each randomly chosen galaxy; we then assign a random redshift in our sample (0.02 to 0.24) to the galaxy. Based on this new redshift, we rescale the half-light radii into angular units and the luminosities into fluxes for each galaxy. We then perform the identical volume limiting cuts based on redshift and V-band magnitude for each galaxy, creating a sample of red galaxies that would pass all our real samples cuts as well.
We next create mock spectra using Flexible Stellar Population Synthesis code (FSPS); (Conroy et al. 2009;Conroy & Gunn 2010), run via the Python-FSPS modules written by Daniel Foreman-Mackey, for a range of metallicities from 0.10 to −1.00, spaced at 0.01 dex, all with identical 10 Gyear ages. To properly create the mock spectra we would observe for a galaxy with a metallicity gradient, we create a pixel grid (resolution of 0.1 arcsec/pixel) where each pixels distance from the center determines its metallicity and therefore which FSPSgenerated spectrum it is assigned. We generate the profile of our mock galaxies given their Sersic indices and radii and the axis ratio b/a, which lets us know the fraction of the luminosity coming from each pixel. We then assign a metallicity to each pixel based on its radial distance from the center by using a simple linear fit with constant and slope parameters chosen by us. The metallicity profile follows the light profile of the galaxysteeper along the minor axis as defined by the b/a ratio of the galaxy.
Finally, we must generate a model for the aperture these galaxies are observed with. We convolve an image of the SDSS-I and -II fiber aperture (3 arcsecond diameter), placed at the exact center of each galaxy, with a double-Gaussian PSF. The Gaussians are both wavelength-dependent and variable across a range of possible seeings based on the actual BOSS seeing. For the core Gaussian, we define a mean FWHM of 1.5 arcseconds at 6000 angstroms, with variability around the mean of 0.3 arcseconds. The second Gaussian has a mean FWHM of 5.0 arcseconds at 6000 angstroms, also with variability around the mean of 0.3 arcseconds. For the wavelength dependence, we assume a λ −1/5 dependence for both Gaussians. The second Gaussian integral is weighted with a factor of 0.1 relative to the first.
With our aperture and galaxy image created, we then multiply the two together to model how much light at each pixel we would observe, and then weight the FSPSgenerated spectrum of the metallicity of the pixel by that factor, finally summing all the pixels to generate a single mock spectrum that represents what we would observe for a galaxy with the metallicity gradient we have assigned.
We do not include noise in our procedure. Although noise in the spectra will cause noise in the result, it will not change the expectation value of the result; i.e. it will not change the expected slope of the measured profile.
Following this process, we create a mock catalog of 22,000 galaxies with known metallicity gradients to analyze identically to our real sample. We run it through the same procedures as for the real data to extract the metallicity indicators in each annulus for each mass bin and compare to the metallicity indicators we input at each radial point. The results of this are shown in Fig. 2. The black lines are the gradients in each indicator that were input, and the blue line shows what our annulus measuring code returns as outputs. We test a range of input metallicity gradients of similar magnitudes to our measured results to ensure that our method works for both steep and shallow gradients. We find that our method leads to a slight constant offset from our inputs and a minor slope steepening (around 15 − 25%, depending on mass bin and indicator). However, the changes are not so dramatic as to invalidate the method. This steepening of the gradient likely occurs for two reasons: first, with an axis ratio not equal to one, our circular annuli are overlayed on elliptical constant-metallicity contours (based on the galaxy b/a axis ratio). This will cause some data from lower metallicities to be included. This lowers the indicator values. Secondly, the PSF we use smears light from the entire galaxy image into the aperture, albeit at a very low weighting on the low-metallicity edges. This, too, will lower the indicator values, but will have a greater impact at high redshift (larger effective aperture) due to the differences in how fast the PSF falls off away from the center (a Gaussian) and how fast the metallicity declines away from the center (log-linearly), resulting in a steepening gradient as well as a constant offset.
To examine these differences, we compare the measured Lick index values that we input to our recorded output values after the analysis, as shown in Fig. 2.
We find that our results are sensitive to the axis ratio distribution and the number of objects in each bin -a large percentage of objects in a bin with extreme axis ratios can cause large (and not real) scatter in the two annulus points calculated from that bin. We find it that this effect is minimized when we work with data sets larger than 20,000 galaxies for our mock catalogs, and as our real sample is approximately 6 times larger than that, we are confident that the scatter introduced by our methodology due to this effect is minor.
As a result of these tests, we can estimate the correction factor in the slope needed to make our mock catalog output the same metallicity gradient that we input in each indicator. For our presented results below, we will apply this small correction factor, which should account for some of these effects in the real sample.
DATA ANALYSIS AND PROCEDURES
The SDSS Main sample of galaxies does not have many spectra of multiple parts of individual galaxies to study how the metallicity varies as a function of radius in a single object. Instead, this sample was entirely observed with single, central fibers with 3 arcsec diameters. This constant angular aperture means that observing a galaxy at a low redshift will measure the metallicity in a smaller physical aperture around the center of the galaxy than a galaxy observed at higher redshift. This approach follows that of Yan & Blanton (2012), who measured emission lines. Here, we are attempting to localize the absorption lines by stacking galaxies and computing the values in question in annuli of the galaxies.
Each stellar mass bin is handled separately. Within each stellar mass bin, the objects are divided up into redshift bins with the goal of having the maximal number of bins without having any span too large a redshift range or contain too few objects for statistical power. We need to balance the concern of flattening the true gradient by having too few bins (see discussion in Yuan et al. 2013) with the concern that too many bins will cause differentials between bins to be too small to be observed with our uncertainty being on the same order of magnitude as the actual changes. We tested many binning choices and find that changes to the binning choices have minimal impact in the determination of the gradients, indicating that our results are relatively robust.
Once the stellar mass bins are subdivided into redshift bins, we compute the average value for the line and continuum luminosity of each metallicity indicator measured above for all the galaxies falling into that bin. For details of this sample's binning, see Table 2.
We then can compute the Lick index for an annulus by calculating line luminosity zbin1 − line luminosity zbin2 continuum luminosity zbin1 − continuum luminosity zbin2
(1) for adjacent redshift bins "zbin 1" and "zbin 2." Knowing the redshift of each object in a redshift bin and the SDSS aperture size (3 arcseconds diameter), we can then compute the average redshift of each galaxy in a bin and then convert that to a physical radius assuming standard cosmology (H 0 = 70 km s −1 Mpc −1 , Ω M = 0.3, Ω Λ = 0.7). A radius is then assigned to each annulus by taking the midpoint of the two redshift bins' physical radii that the annulus is computed from. Thus, we are left with the pairs of points (Lick index, physical radius) for each stellar mass bin, which are plotted in Figure 3 to measure a gradient. The one exception to this is the innermost point -this one is not an annulus but rather just the innermost redshift bin, and it is assigned a radius of half its maximum extent. One important detail that results from treating the innermost bin this way and binning in equal-redshift spacings is that the innermost redshift bin in each mass bin has a very large amount of impact on the gradient calculations -it not only is used twice (once in the difference between bins 1 and 2 and once by itself as the innermost point), but it covers the largest range of radius and has the smallest number of objects of all the redshift bins in that stellar mass bin. Finally, because our annuli are no longer fully independent of each other, there should be nonzero off-diagonal values in the covariance matrix. We neglect these terms and only consider the diagonal variance in plotting error bars for each point and in fitting for gradients. This analysis is repeated identically, but replacing stellar mass with velocity dispersion in the initial step. We perform this step mainly for confirmation that our derived masses are accurate, but also so that we can more directly compare the results we find to papers that only report trends with velocity dispersion. Below we will present the results for both binning schemes, but as will be seen there are only a small number of differences.
In addition, we also calculate an average galaxy radius for each stellar mass bin using the average elliptical massradius relation discussed in several papers (Chiosi et al. 2012;Shen et al. 2003, among others). For high-mass ellipticals such as our sample here, the relation is approximately log R 1/2 = 0.54 log M/M * − 5.25. For the stellar mass value, we use the average stellar mass in each stellar mass bin. As mentioned in those papers, this half-mass radius is not strictly identical to the half-light, or effective radius R e , but is usually quite close. In this paper we will use R 1/2 as a proxy for R e in comparison to other works and refer to it as R e only from now on. This will allow for better comparison to theoretical works later in -Differences between the gradient input to galaxies and the gradient we measure using our annulus method for each indicator. Only one mass bin (10.3 < log(M ) < 10.7) is shown for simplicity, but all mass bins were very similar. The input data is shown with black diamonds, and the output measured is shown with blue stars. There is a consistent steepening of the true gradient, as well as a slight constant offset, which we will take into account with a correction factor in our analysis and future plots. this paper.
With the values computed for the various metallicity indicators in each annulus, we chose to use EZ Ages (Graves & Schiavon 2008) to compute the actual metallicities as well as several other parameters. EZ Ages uses iterative fitting of the stellar population models described in Schiavon (2007) to determine the best-fit metallicity and age from the Lick indices we have measured. We applied EZ Ages to the stacked annuli, not the individual objects. We use solar isochrones and a Salpeter initial mass function with exponent 1.35 as inputs for all galaxies. Changing to α-enhanced isochrones was tested with minimal impact on the overall results, although a small number of points no longer fit on the age-metallicity grid. The results of the EZ Ages fitting is shown in Figure 4. With metallicities computed, we perform linear regression to compute the gradient for each stellar mass and velocity dispersion bin separately.
One caveat of the use of EZ Ages is that it uses SSP models. As we are averaging over a range of galaxy types and potential evolution paths, the already-approximate SSP approach will have more issues. We thus suggest that care be taken in interpreting the age and metallicity results too strongly.
RESULTS
Our results are shown in Figures 3-10 below. The first four (Figures 3, 4, 5, and 6) show the results as a function of mass. Figure 3 displays the Lick indices as a function of physical annulus radius. Figure 4 shows the resulting EZ Ages model parameters as a function of physical radius. Figures 5 and 6 show the same data, but versus R/R e , the radius scaled to the galaxy effective radius. The second four (Figures 7-10) show the same results as a function of velocity dispersion.
The lines plotted in all eight are the linear fitting results shown in Table 3 and Table 4 (for stellar mass binning) and Table 5 and Table 6 (for velocity dispersion binning) where the exact values are listed for more precise comparisons. The units are dex kpc −1 in radius. In addition, we list the line intercept at R = 2.5 kpc and R/R e = 0.5, effectively the mean aperture-corrected Lick indices and parameters of early-type SDSS galaxies as a function of mass and velocity dispersion.
Finally, we calculate gradients versus log r for [Fe/H], [Mg/Fe], and [C/Fe] for each stellar mass and velocity dispersion bin. These logarithmic gradients are shown in Figures 11 and 12. The values are given in Tables 7 and 8. We find in general very little dependence of the gradients on either mass or velocity dispersion. The only exceptions are a very small trend towards flattening gradients in [Fe/H] as a function of velocity dispersion or mass, with the exception of the lowest velocity dispersion or mass bin gradient of [Fe/H] which is much steeper than the others.
Note that in the above results we found a systematic mis-measurement for one annulus of the H β line in the -Details of how galaxies are binned by stellar mass in this paper. A bin is first divided on the basis of the stellar mass of the galaxies (units are solar masses). Following that, the galaxies are divided into bins of equal size in redshift. All the limits are set such that the lower limit is inclusive and the upper is exclusive to avoid overlap. 10.7 < log M < 11.0 mass bin, due to part of the redshift range it covers. The H β line for one bin is redshifted on top of the OI 5577Å sky line; errors in the sky subtraction at that location lead to an abnormally low flux measured in the line. This leads to Hβ absorption that appears too large, and an age that is too young, as well as a slight increase in [Fe/H] and [Mg/Fe] for that one point. We retain this data point on all plots, but do not use it in the fits; it is tinted a slightly lighter red in color to indicate the presence of bad data. This issue is not as clearly present in the velocity dispersion binning, likely due to its wider redshift coverage per bin, which dilutes the effect of the error; no masking is used in this case.
There are a wide range of important conclusions to draw from these data. Before moving to discuss them in the context of previous observations and theory, it's worth simply listing some of the notable facts. 6. Most of the dependence of the gradient with physical radius on mass is accounted for by the change in R e with mass.
7. There are no dramatic differences between the results from the stellar mass binning and the velocity dispersion binning.
We can determine metallicities rather well, but due to the uncertainty in our Hβ measurements and corrections, our age determinations have a large amount of scatter. Some of the Hβ uncertainty is driven by the inherent scatter in the [OIII] correction to Hβ line infill, some results from the extra noise in measuring two lines (instead of one), and some results from the expected slight differences amongst these galaxies that will add some scatter to our points as well. In addition to this, the redshift range that a single data point covers will include some real age differences of the galaxies due to the universe age at that point, which is not accurate if we wish to infer age gradients in an actual galaxy. This effect is at most 2 Gyears across a single bin, though. Due to the lack of trend we find in the age points and the likelihood of these gradients not measuring what we wish to measure, we do not fit to these data points -they are simply shown with error bars at each radius.
We can also see that our stellar mass bins and velocity dispersion bins show similar trends, as we would expect. If we consider the R/R e plots (Figures 5,6,9, 10; values given in Table 4 and 6), we also find that stellar mass or velocity dispersion only influences the fitted gradients slightly -the [Fe/H] plot shows a steadily increasingly negative central metallicity as stellar mass increases, but the gradient stays mostly constant (contrary to expectations, although similar to recent findings in Pastorello et al. 2014). This trend in central metallicities is loosely reflected in the [MgFe] ′ and Fe plots as well, as would be expected. Other metals such as [C/Fe] and the indicators C 4668 and Mg b show no clear dependencies on stellar mass in gradient or central values, however.
Below, we will use both the mass and the velocity dispersion results for comparison to prior works, depending on what data the paper in question offers, because our qualitative conclusions are the same regardless. Some of the papers we will examine do not use the same stellar population models -they instead use the slightly different models from Thomas et al. (2003) (TMB). These models differ slightly in the outputs they which we will use to compare our observations to those papers that use the TMB models. For the α-elements, the TMB models assume that all α-elements track Mg, so we can compare our [Mg/Fe] to the [α/Fe] values. [C/Fe] will have no analog in those cases.
Gradients
Two large studies integral field studies of the metallicity gradients in inner regions of galaxies cover well the same radial range we cover: Rawle et al. (2010) and Kuntschner et al. (2010) , which indicates a slight trend unlike our data, but isn't outside of our error bar range. Mehlert et al. (2003) study the metallicity gradients of early-type galaxies in the Coma Cluster out to about R e , also similar to our measurements, using the data from Mehlert et al. (2000). Rather than using EZ Ages to model the metallicity and age, they use the models of Thomas et al. (2003), with the differences discussed above. The results in Mehlert et al. (2003) show declining [Z/H] and constant age and [α/Fe] gradient. We can calcuate similar logarithmic index gradients (per Equation 5 in Mehlert et al. 2003, given in Spolaor et al. (2010) cover a slightly larger range of radii, out to between 1 and 3 R e . They analyzed 14 low-mass ellipticals in the Fornax and Virgo clusters and compared them to several higher mass ellipticals from previous studies. They interpret the stellar populations using the models of Thomas et al. (2003). They find a declining [Z/H] (−0.22 ± 0.14 averaged across all mass bins) and roughly constant [α/Fe] gradient, in agreement with our results. Greene et al. (2013) study gradients in 33 nearby galaxies, also out to a much larger radius of the galaxies than we do, about 14 kpc. They measure similar metallicity indicators to our survey, also using EZ Ages to convert the data into physical parameters. They find a [Fe/H] gradient of around −0.3 to −0.5 dex kpc −1 , significantly steeper than ours (Note that here they report gradients versus radius rather than versus logarithmic radius). This discrepancy may indicate that the metallicity gradients steepen in the outer parts of galaxies. cases aside from that of Rawle et al. (2010), it is reasonable that our [Fe/H] results and the [Z/H] results from these studies are in agreement.
Trend in central values with mass
We next turn to trends with mass in the stellar population parameters at the centers of galaxies. Our centermost bins are R ∼ 0.2R e or ∼ 0.5 kpc. We find only weak (at best) trends with mass. Central [Fe/H] shows a dependence that is not monotonic and is < 0.1 dex across our whole sample. Central [Mg/Fe] shows an small increase with velocity dispersion that is ∼ 0.1 dex across our whole range of masses or velocity dispersion, but the same trend does not exist for mass binning Kuntschner et al. (2010) reports their central values at R e /8, slightly more internal than ours but comparable. They too find no monotonic dependence of the central metallicities on velocity disperson or mass, although their results have more scatter (ranging from 0.2 to around −0.4) after we apply the [Z/H] to [Fe/H] conversion. Again similarly, they find that the central [α/Fe] does increase with mass or velocity dispersion, although there is a constant offset of about 0.2 dex higher in their central values than we find, part of which can be explained by the more central location. Their trend is much clearer than ours is as well.
Turning to Rawle et al. (2010), we find central values recorded at R e /3, again comparable to our central annulus radius. As with Kuntschner et al. (2010) an increase as velocity dispersion increases, but as with Kuntschner et al. (2010), they also find systematically higher values for the central [α/Fe] than we record and a clearer indication of the trend than we see. Kuntschner et al. (2010) and Rawle et al. (2010).
Using data with similar radial coverage, Mehlert et al. (2003) found similar dependences. These results indicate stronger mass dependence than we find by about 0.1 dex over this range for [α/Fe]. Assuming the conversion about between [Z/H] and [Fe/H] it also implies a stronger mass dependence in [Fe/H] than we find, also by about 0.1 dex.
The galaxies in Greene et al. (2013) also show an increase in central metallicity with mass at a fixed radius, but their centermost bin is around 2-3 kpc. Our galaxies show a similar trend at that physical radius. The trends with mass agree with those in Greene et al. (2013) in the [C/Fe] and [Mg/Fe] ratio, with the former showing slight increase at all our radii with stellar mass, and the latter being roughly constant. This trend with mass persists across all radii in our survey (with the exception of the lowest mass or velocity dispersion bin). Figures 11 and 12 show the logarithmic gradients as a function of stellar mass and velocity dispersion. Our metallicity gradients are close to constant or perhaps slightly steepening within the errors as stellar mass increases with the only exception being the lowest stellar mass and velocity dispersion bins. In the higher mass bins, the [Fe/H] gradient anti-correlates with both the stellar mass and the velocity dispersion; however, both trends are very low significance. For [Mg/Fe] and [C/Fe], we find no significant trends in the gradients as a function of mass or velocity dispersion.
Trend in gradients with mass
Both Rawle et al. (2008) and its followup paper, Rawle et al. (2010), as well as Kuntschner et al. (2010) find a large amount of scatter in gradients. Kuntschner et al. (2010) note a slight trend that is partially reflected in our data.
At low masses, Kuntschner et al. (2010) reports an increasingly negative metallicity gradient (up to about 3.5 × 10 10 solar masses), followed by a flattening metallicity gradient as mass increases further. We have far fewer points to use to determine trends with mass, but our lowest-mass point is indeed the steepest gradient, with all higher-mass ones showing some evidence of flattening out (see Figures 11 and 12, and Tables 7 and 8). However the granularity of our data prevents us from checking in detail at which mass we find this turnover to be at to further compare to the observations in Kuntschner et al. (2010). Spolaor et al. (2010) also covers enough of a stellar mass range to check for any mass dependency of these gradients, and does find a slight flattening of the metallicity gradient as stellar mass increases for high-mass ellipticals, although the scatter increases significantly as well. The bins that overlap most completely (covering a range of velocity dispersions 2.1 < log σ < 2.6) have very similar measured gradients as a function of R/R e (−0.2 to −0.4 in Spolaor et al. (2010) and −0.15 to −0.25 in our sample), although the trend is reversed.
The work of Ogando et al. (2006) provides another reference on metallicity gradients for us to compare to. Ogando et al. (2006) measure Mg 2 and Hβ only to supply to the TMB stellar models, with a forced [α/Fe] of 0.3, slightly higher than our model fits would predict, but within most of the error bars. With that, they find [Fe/H] logarithmic gradients (converted from the [Z/H] gradients in the paper) ranging from −0.1 to −1.2, but with the bulk of the galaxies found to have gradients between −0.3 and −0.8. They record these as functions of galaxy stellar velocity dispersion and mass, and find a slight trend to a steepening [Fe/H] gradient as mass or velocity dispersion increases, matching our 3 highest mass bins; however, the scatter is large. This is in agreement with the results in Kuntschner et al. (2010) and our results, although again, the observed steepening with increased mass is paired with an increase in scatter as well.
Finally, at the larger radii that Greene et al. (2013) measure, they find qualitatively similar trends, with roughly constant gradients in all indicators as a function of mass.
As will be discussed more in Section 7, the scatter observed in the gradients could be due to the wide variety of merger histories available to high-mass ellipticals (Di Matteo et al. 2009), leading to a wide variety of potential final, observed gradients for individual galaxies.
Realistically, the large error bars present in the determination of the metallicity gradients prevent us from concluding anything too firm; we mostly can say that it appears that the gradient very slightly flattens with increasing stellar mass or velocity dispersion, and isclose to constant or slightly declining above a certain stellar mass or velocity dispersion cutoff around 3×10 10 solar masses, which is not in disagreement with any of the studies we found.
General conclusions
There is good agreement with finding a negative [Fe/H] gradient, a constant [Mg/Fe], and a very slightly negative [C/Fe] gradient, even if the numerical results presented have a fairly large amount of scatter. For trends in central values as mass (or velocity dispersion) increases, we find general agreement with an increasing central [Mg/Fe] (although our dependence is less than other compared studies by about 0.1 dex) and constant central [Fe/H], and increasing central [C/Fe] (where central in this data set means within ∼ 0.5 kpc).
Finally, we find that in general we agree with the most of the reported trends in gradients of [Fe/H] with mass as well -with the most negative gradient in the lowest mass objects, and then a flatter although still negative gradient for higher mass galaxies, with the gradient leveling out to be roughly constant with mass or perhaps very slightly steepening.
[Mg/Fe] shows agreement here as well, with no real trends reported with mass in the gradients.
Because it is a statistical average, our data set is not sensitive to the increase in scatter with mass found by Kuntschner et al. (2010) and others in the [Fe/H] gradient.
COMPARISON TO THEORY
According to a number of theoretical investigations, mergers tend to flatten gradients and monolithic collapse models tend to steepen them (Pipino et al. 2010;Di Matteo et al. 2009). Because the detailed history of each galaxy's growth involves some features of both monolithic collapse and hierarchical merging models, the predicted results lie along a continuum of flat to steep gradients, with substantial scatter amongst the results due to differing degrees of these two effects for each individual galaxy (Pipino et al. 2010). Thus we do not expect perfect agreement with any individual models but rather that our data lie somewhere in between results reported by merger-focused simulations and monolithic collapse-focused simulations.
When comparing our gradient values to those in Hopkins et al. (2009), who simulated merger models, we find general agreement. Most of their metallicity gradients are between −0.1 and −0.6, while our results (shown in Table 7) are around −0.1 to −0.4 dex. Similar conclusions are drawn by Kobayashi (2003) who finds gradients in the range of −0.2 to −0.8. Kobayashi (2003) simulates both merging and monolithically collapsing galaxies, and it is important to note that within the spread of the results, our data agrees with both sets. Because there is likely to be natural variation among galaxies in our sample, we do not rule out that some galaxies have the steep gradients predicted by models. Di Matteo et al. (2009) handles mergers in more detail than most simulations, measuring gradients that result from various mass ratios and various initial metallicity gradients of merging galaxies. The results again show metallicity gradients of about −0.1 to −0.4, but are dependent on the type of mergers that have occurred to form the final galaxies as expected. For reasonable initial metallicity gradients and merger histories, our results appear in agreement with the simulations of Di Matteo et al. (2009).
Another interesting property to investigate is the stellar mass trend. Our results (see Figure 11) suggest that all three gradients are roughly constant with stellar mass, with perhaps a slightly flattening [Fe/H] depending on how much the lowest stellar mass bin is to be believed. Simulations give varied results depending on the merger history of the galaxies, with Di Matteo et al. (2009) noting processes that can give rise to both flatter and steeper gradients depending on the stellar masses of merging galaxies and their initial gradients, and thus proposing that little trend should exist overall but scatter should increase. Other models predict no clear trend and increasing scatter as well (see Pipino et al. 2010;Kobayashi 2003). Some models do predict a correlation (Kawata & Gibson 2003), but only take into account monolithic collapse and not the interplay of mergers as well. In general, though, the conclusions are broadly suggestive that metallicity gradients should slightly steepen with increasing stellar mass, albeit with increased scatter as well (Ogando et al. 2005) due to higher stellar mass galaxies being allowed a larger variety of possible evolution paths that will change how their gradients develop. We find no clear trend in [Fe/H] gradient with stellar mass or velocity dispersion, with only a weakly detected flattening. The error bars in our results are largely driven by statistical considerations based on the number of objects in each stellar mass bin, so we are unable to detect any potential intrinsic scatter in the data that may exist due to our method.
The potential for many types of mergers and the differing gradients that result from the different stellar mass ratios and initial metallicity gradients of the progenitors as discussed in detail in Di Matteo et al. (2009) make it difficult to conclude definitively if our observations agree with theory as simply adjusting the progenitor properties within reasonable values can change the model predictions dramatically. To make a more detailed comparison of our results to theory would require modeling expected merger rates and stellar mass ratios. However, for now, we can conclude that our results do not indicate any clear disagreements -the gradients we find are within the ranges predicted by models that incorporate both monolithic collapse and mergers of many types into the evolution of elliptical galaxies, and our gradients follow a generally observed trend with stellar mass.
CONCLUSIONS
The approach used here has drawbacks and challenges but also significant advantages relative to previous studies. We can determine only mean metallicity and abundance gradients, with little power to constrain how the gradients are distributed about the mean. However, by being able to work with single-spectrum galaxies, we potentially can examine a much larger sample of galaxies than previously possible.
Relative to previous studies, our measured gradients in [Fe/H] are similar but on the whole slightly shallower, while our [Mg/Fe] gradient matches all the compared studies by being flat. In terms of stellar mass dependence, we see a flattening of metallicity gradients as mass increases in line with the compared studies. We find fairly similar central values for [α/Fe] (although slightly smaller), and also observe an increase in central [α/Fe] with stellar mass as the other studies do. Our metallicity also matches previous studies both in values and trends with mass.
In sum, we find that using this new technique to find the metallicity inside an annulus of averaged galaxies roughly agrees with both observations of individual galaxies and simulated predictions of galaxies that have formed from mergers and/or monolithic collapse, a conclusion which is supported by our analysis of mock data to ensure this method is valid. This technique may be of further use. With this same data set, one could extend the analysis to include the broader wavelength range accessible to the SDSS spectrograph than to most integral field observations. In addition, this technique could be used for some higher redshift surveys to measure a similar mean gradient for galaxies at higher redshift, for example in the AGN and Galaxy Evolution Survey (AGES), GAMA, or the planned DESI Bright Galaxy Survey.
We thank the referee for many helpful comments, especially in directing us to consider simulations to confirm our method's validity.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, Univer- 11.0 < log(M ) < 11.5 -0.2851 0.2300
Note. -The calculated logarithmic gradients for the metallicity and [α/Fe] ratios for all four stellar mass bins.
Fig. 1 .
1-The cuts made to create a volume-limited sample for each stellar mass bin.
Fig. 2.-Differences between the gradient input to galaxies and the gradient we measure using our annulus method for each indicator. Only one mass bin (10.3 < log(M ) < 10.7) is shown for simplicity, but all mass bins were very similar. The input data is shown with black diamonds, and the output measured is shown with blue stars. There is a consistent steepening of the true gradient, as well as a slight constant offset, which we will take into account with a correction factor in our analysis and future plots.
6 .
6COMPARISON TO PREVIOUS OBSERVATIONAL STUDIES We have three main results: gradients for the EZ AGES outputs ([Fe/H], [C/Fe], and age), the trend in their central values with mass, and the trend in their gradients with mass. Each of these have been studied in other papers and we compare with those results here.
Fig. 3 .Fig. 4 .
34-The gradients found for each metallicity indicator. Each stellar mass bin is represented with a different color: 10.0 < log M < 10.3 is blue, 10.3 < log M < 10.7 is green, 10.7 < log M < 11.0 is red, and 11.0 < log M < 11.-The gradients found for the metallicity, age, and several [α/Fe] ratios computed using EZ Ages. Colors are the same asFigure 3. Binning is done based on stellar mass. produce, with the TMB models giving [Z/H] and [α/Fe] instead of the EZ AGES output of [Fe/H], [Mg/Fe], and [C/Fe]. Thomas et al. (2003) proposes a conversion between the two measures of [Fe/H]=[Z/H]−0.94[α/Fe],
(2010) finds an average gradient of −0.05 ± 0.05 dex −1 and Kuntschner et al. (2010) finds an average gradient of −0.25 ± 0.11 dex −1 , with values ranging from −0.1 to −0.5 much like our results. Also in agreement with our observations, Kuntschner et al. (2010) report an [α/Fe] gradient consistent with zero. Rawle et al. (2010) finds a very slightly negative [α/Fe] gradient of −0.06 ± 0.03 dex −1
Fig. 5 .Fig. 6 .
56-Similar toFigure 3, but with all radii scaled to the effective radius. Colors are the same asFigure 3. Binning is done based on stellar mass. -Similar toFigure 4, but with all radii scaled to the effective radius. Colors are the same asFigure 3. Binning is done based on stellar mass.
Fig. 7 .Fig
7-The gradients found for each metallicity indicator, but with the initial binning done by velocity dispersion and not stellar mass. Each velocity dispersion bin is a different color: 30 km s −1 to 125 km s −1 blue, 125 km s −1 to 185 km s −1 green, 185 km s −1 to 230 km s −1 red, 230 km s −1 to 325 km s −1 brown. . 8.-The gradients found for the metallicity, age, and several [α/Fe] ratios computed using EZ Ages. Colors are the same asFigure 7. and binning is done on the basis of velocity dispersion.
Fig. 9 .Fig
9, after converting their reported [Z/H] values to [Fe/H], there is no clear trend with central velocity dispersion and a larger scatter than we report. Their [α/Fe] central trend with central velocity dispersion also matches ours with -Similar to Figure 7, but with all radii scaled to the effective radius. Colors are the same as Figure 7. Binning is done based on velocity dispersion. . 10.-Similar to Figure 8, but with all radii scaled to the effective radius. Colors are the same as Figure 7. Binning is done based on velocity dispersion.
Fig. 11 .
11-The stellar mass dependence of the [Fe/H], [Mg/Fe], and [C/Fe] logarithmic gradients as calculated in
TABLE 1 Volume
1Limited Mass Binning of Galaxies in this Paperlog (M ) Range
z Range
Max V Mag Objects
10.0 < log M < 10.3 0.02 < z < 0.09
-19.35
23324
10.3 < log M < 10.7 0.02 < z < 0.12
-20.05
57381
10.7 < log M < 11.0 0.03 < z < 0.18
-21.05
50880
11.0 < log M < 11.5 0.07 < z < 0.24
-21.80
22029
TABLE
1 .
1All the metal-related Lick indices show a signficantly negative gradient when plotted both versus physical radius and R/R e .2. Gradients for stellar population parameters [Fe/H]
and [C/Fe] are generally declining, while the gra-
dient for [Mg/Fe] is generally flat.
3. The stellar population ages have very large scatter,
with ages ranging from 3 to 12 billion years.
4. There is a slight increase in the central value of
[C/Fe] with mass, while the central [Fe/H] and
[Mg/Fe] values are nearly constant (where "cen-
tral" corresponds to the inner 0.5 kpc).
5. The logarithmic gradients have little detectable de-
pendence on mass or velocity dispersion, with the
greatest dependence being a slight flattening of the
[Fe/H] gradient as those parameters increase.
. Both these studies measure [Z/H] values out to 1R e . Both only calculate a [α/Fe] ratio (which tracks Mg) and not a [C/Fe] ratio. When applying the [Z/H] to [Fe/H] conversion to match our results, both studies find a slightly negative logarithmic metallicity gradient (however, note that the [Fe/H gradient inRawle et al. (2010) is consistent with zero), matching our observations -Rawle et al.
Table 7 )
7to compare the values of the metallicity indicators in our stacked annuli to their results. Their [Z/H] gradient matches our [Fe/H] gradient well, with a gradient of around −0.1 to −0.2.
Their findings for [Mg/Fe] and [C/Fe] are similar to ours, with a constant [Mg/Fe] gradient in both velocity dispersion bins like all four of ours, and a [C/Fe] gradient of around −0.1 to −0.2 dex kpc −1 as our data shows as well.Because the [α/Fe] gradients are relatively flat in all
0
1
2
3
4
5
6
Center radius of annulus (kpc)
2.6
2.8
3.0
3.2
3.4
3.6
[MgFe]' (Angstroms)
0
1
2
3
4
5
6
Center radius of annulus (kpc)
2.2
2.4
2.6
2.8
3.0
<Fe> (Angstroms)
0
1
2
3
4
5
6
Center radius of annulus (kpc)
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
Mg b (Angstroms)
0
1
2
3
4
5
6
Center radius of annulus (kpc)
4
5
6
7
C 4668 (Angstroms)
0
1
2
3
4
5
6
Center radius of annulus (kpc)
1.2
1.4
1.6
1.8
Ca 4227 (Angstroms)
0
1
2
3
4
5
6
Center radius of annulus (kpc)
1.6
1.8
2.0
2.2
2.4
EQW Corrected Hβ (Angstroms)
Table 7 .
7Broadly speaking we find a constant [Fe/H] (with the exception of the lowest stellar mass bin which shows a much steeper gradient), [Mg/Fe] and [C/Fe] gradient as a function of stellar mass. et al. (2010) detect an increase with mass of central [Z/H] and [α/Fe] at radii similar to the smallest radii we probe (R e /8); between 100 and 300 km s −1 Fig. 12.-The velocity dispersion dependence of the [Fe/H], [Mg/Fe], and [C/Fe] logarithmic gradients as calculated in Table 8. Broadly speaking we find a very slightly flattening [Fe/H] gradient, and a constant [Mg/Fe] and [C/Fe] gradient as a function of velocity dispersion. they detect an increase in [Z/H] of 0.3 dex and in [α/Fe] of 0.2 dex. When applying the conversion from [Z/H] to [Fe/H], this is close to our results, although a slight central [F e/H] increase as a function of velocity still remains that we do not detect and the dependence of [α/Fe] is stronger than what we find, matching more closelySpolaor
TABLE 3
3Gradients in Metallicity Indicators Versus Physical Radius Note. -The calculated gradients for each measured metallicity indicator and the EZ Ages results binned by stellar mass. The units are dex kpc −1 .Measure
log M Range
Gradient Uncertainty in Gradient Value at 2.5 kpc
[M gF e] ′
10.0 < log(M ) < 10.3
-0.3229
0.0165
2.4379
[M gF e] ′
10.3 < log(M ) < 10.7
-0.2035
0.0071
2.8636
[M gF e] ′
10.7 < log(M ) < 11.0
-0.1526
0.0050
3.1023
[M gF e] ′
11.0 < log(M ) < 11.5
-0.1385
0.0059
3.2685
< F e >
10.0 < log(M ) < 10.3
-0.2493
0.0274
2.3472
< F e >
10.3 < log(M ) < 10.7
-0.1262
0.0119
2.7090
< F e >
10.7 < log(M ) < 11.0
-0.0974
0.0081
2.8277
< F e >
11.0 < log(M ) < 11.5
-0.1032
0.0093
2.8801
Mg b
10.0 < log(M ) < 10.3
-0.4068
0.0182
2.4345
Mg b
10.3 < log(M ) < 10.7
-0.2961
0.0081
2.9186
Mg b
10.7 < log(M ) < 11.0
-0.2078
0.0059
3.3138
Mg b
11.0 < log(M ) < 11.5
-0.1658
0.0071
3.6103
C 4668
10.0 < log(M ) < 10.3
-1.1205
0.0282
3.7319
C 4668
10.3 < log(M ) < 10.7
-0.7542
0.0123
5.1096
C 4668
10.7 < log(M ) < 11.0
-0.4804
0.0092
6.0734
C 4668
11.0 < log(M ) < 11.5
-0.3499
0.0133
6.6346
Ca 4227
10.0 < log(M ) < 10.3
-0.1652
0.0247
1.3840
Ca 4227
10.3 < log(M ) < 10.7
-0.1241
0.0104
1.5666
Ca 4227
10.7 < log(M ) < 11.0
-0.0910
0.0077
1.6885
Ca 4227
11.0 < log(M ) < 11.5
-0.0806
0.0100
1.7647
Corrected Hβ 10.0 < log(M ) < 10.3
0.1185
0.0210
2.1687
Corrected Hβ 10.3 < log(M ) < 10.7
0.1049
0.0095
2.0620
Corrected Hβ 10.7 < log(M ) < 11.0
0.0590
0.0084
1.8736
Corrected Hβ 11.0 < log(M ) < 11.5
0.1025
0.0099
1.8275
[F e/H]
10.0 < log(M ) < 10.3
-0.1171
0.0135
-0.2301
[F e/H]
10.3 < log(M ) < 10.7
-0.0301
0.0058
-0.0079
[F e/H]
10.7 < log(M ) < 11.0
-0.0114
0.0039
0.0264
[F e/H]
11.0 < log(M ) < 11.5
-0.0166
0.0043
0.0092
[M g/F e]
10.0 < log(M ) < 10.3
0.0407
0.0309
0.0061
[M g/F e]
10.3 < log(M ) < 10.7
0.0046
0.0179
-0.0430
[M g/F e]
10.7 < log(M ) < 11.0
-0.0146
0.0117
-0.0315
[M g/F e]
11.0 < log(M ) < 11.5
0.0165
0.0089
0.0198
[C/F e]
10.0 < log(M ) < 10.3
-0.0602
0.0252
-0.0543
[C/F e]
10.3 < log(M ) < 10.7
-0.0611
0.0218
-0.0136
[C/F e]
10.7 < log(M ) < 11.0
-0.0242
0.0136
0.0951
[C/F e]
11.0 < log(M ) < 11.5
-0.0091
0.0108
0.1771
TABLE 4
4Gradients in Metallicity Indicators Versus Scaled Radius Note. -The calculated gradients for each measured metallicity indicator and the EZ Ages results binned by stellar mass. The units are dex.Measure
log M Range
Gradient Uncertainty in Gradient Value at 0.5 Re
[M gF e] ′
10.0 < log(M ) < 10.3
-0.5658
0.0281
2.9623
[M gF e] ′
10.3 < log(M ) < 10.7
-0.5865
0.0187
3.0790
[M gF e] ′
10.7 < log(M ) < 11.0
-0.6961
0.0202
3.1358
[M gF e] ′
11.0 < log(M ) < 11.5
-1.0237
0.0394
3.1031
< F e >
10.0 < log(M ) < 10.3
-0.4494
0.0466
2.7456
< F e >
10.3 < log(M ) < 10.7
-0.3929
0.0312
2.8281
< F e >
10.7 < log(M ) < 11.0
-0.4828
0.0327
2.8298
< F e >
11.0 < log(M ) < 11.5
-0.7990
0.0623
2.7386
Mg b
10.0 < log(M ) < 10.3
-0.7023
0.0310
3.1003
Mg b
10.3 < log(M ) < 10.7
-0.8199
0.0213
3.2489
Mg b
10.7 < log(M ) < 11.0
-0.9070
0.0240
3.3797
Mg b
11.0 < log(M ) < 11.5
-1.1906
0.0475
3.4294
C 4668
10.0 < log(M ) < 10.3
-1.9689
0.0480
5.5489
C 4668
10.3 < log(M ) < 10.7
-2.1291
0.0322
5.9306
C 4668
10.7 < log(M ) < 11.0
-2.1628
0.0374
6.1929
C 4668
11.0 < log(M ) < 11.5
-2.6084
0.0891
6.2052
Ca 4227
10.0 < log(M ) < 10.3
-0.2767
0.0420
1.6587
Ca 4227
10.3 < log(M ) < 10.7
-0.3531
0.0274
1.7002
Ca 4227
10.7 < log(M ) < 11.0
-0.4164
0.0314
1.7078
Ca 4227
11.0 < log(M ) < 11.5
-0.6056
0.0671
1.6634
Corrected Hβ 10.0 < log(M ) < 10.3
0.2036
0.0358
1.9744
Corrected Hβ 10.3 < log(M ) < 10.7
0.2800
0.0249
1.9396
Corrected Hβ 10.7 < log(M ) < 11.0
0.2458
0.0342
1.8490
Corrected Hβ 11.0 < log(M ) < 11.5
0.6925
0.0663
1.9176
[F e/H]
10.0 < log(M ) < 10.3
-0.2127
0.0228
-0.0437
[F e/H]
10.3 < log(M ) < 10.7
-0.0808
0.0132
0.0176
[F e/H]
10.7 < log(M ) < 11.0
-0.1652
0.0154
-0.0189
[F e/H]
11.0 < log(M ) < 11.5
-0.1966
0.0294
-0.0446
[M g/F e]
10.0 < log(M ) < 10.3
0.1090
0.0483
-0.0537
[M g/F e]
10.3 < log(M ) < 10.7
-0.0067
0.0369
-0.0473
[M g/F e]
10.7 < log(M ) < 11.0
-0.0689
0.0558
-0.0326
[M g/F e]
11.0 < log(M ) < 11.5
0.0902
0.0653
0.0295
[C/F e]
10.0 < log(M ) < 10.3
-0.1611
0.0469
0.0352
[C/F e]
10.3 < log(M ) < 10.7
-0.1965
0.0410
0.0648
[C/F e]
10.7 < log(M ) < 11.0
-0.0777
0.0601
0.1057
[C/F e]
11.0 < log(M ) < 11.5
-0.3059
0.0717
0.1476
TABLE 5
5Gradients in Metallicity Indicators Versus Physical Radius Measure σ Range (km s −1 ) Gradient Uncertainty in Gradient Value at 2.5 kpc [M gF e] ′ Note. -The calculated gradients for each measured metallicity indicator and the EZ Ages results for the data binned by velocity dispersion. The units are dex kpc −1 .30 < σ < 125
-0.1152
0.0204
2.5996
[M gF e] ′
125 < σ < 185
-0.1336
0.0082
2.8832
[M gF e] ′
185 < σ < 230
-0.1307
0.0066
3.0963
[M gF e] ′
230 < σ < 325
-0.1185
0.0069
3.3224
< F e >
30 < σ < 125
-0.1281
0.0341
2.4848
< F e >
125 < σ < 185
-0.1060
0.0137
2.7160
< F e >
185 < σ < 230
-0.1042
0.0108
2.8100
< F e >
230 < σ < 325
-0.0833
0.0109
2.9161
Mg b
30 < σ < 125
-0.0934
0.0230
2.6544
Mg b
125 < σ < 185
-0.1679
0.0094
2.9668
Mg b
185 < σ < 230
-0.1547
0.0079
3.3209
Mg b
230 < σ < 325
-0.1443
0.0083
3.6861
C 4668
30 < σ < 125
-0.4376
0.0354
4.4164
C 4668
125 < σ < 185
-0.5219
0.0142
5.2554
C 4668
185 < σ < 230
-0.3636
0.0124
6.1250
C 4668
230 < σ < 325
-0.3249
0.0153
6.7582
Ca 4227
30 < σ < 125
-0.0772
0.0315
1.4471
Ca 4227
125 < σ < 185
-0.0834
0.0121
1.5851
Ca 4227
185 < σ < 230
-0.0760
0.0104
1.6896
Ca 4227
230 < σ < 325
-0.0658
0.0116
1.7962
Corrected Hβ
30 < σ < 125
-0.0255
0.0260
2.1447
Corrected Hβ
125 < σ < 185
0.0525
0.0109
2.0780
Corrected Hβ
185 < σ < 230
0.0730
0.0102
1.9444
Corrected Hβ
230 < σ < 325
0.0908
0.0115
1.7897
[F e/H]
30 < σ < 125
-0.0791
0.0163
-0.1343
[F e/H]
125 < σ < 185
-0.0545
0.0066
-0.0342
[F e/H]
185 < σ < 230
-0.0193
0.0044
0.0252
[F e/H]
230 < σ < 325
-0.0069
0.0042
0.0232
[M g/F e]
30 < σ < 125
0.0694
0.0268
0.0359
[M g/F e]
125 < σ < 185
-0.0051
0.0124
-0.0248
[M g/F e]
185 < σ < 230
0.0071
0.0095
-0.0125
[M g/F e]
230 < σ < 325
0.0154
0.0103
0.0143
[C/F e]
30 < σ < 125
-0.0107
0.0266
-0.0537
[C/F e]
125 < σ < 185
-0.0606
0.0156
-0.0056
[C/F e]
185 < σ < 230
0.0056
0.0109
0.1232
[C/F e]
230 < σ < 325
-0.0153
0.0079
0.2063
TABLE 6
6Gradients in Metallicity Indicators Versus Scaled Radius Measure σ Range (km s −1 ) Gradient Uncertainty in Gradient Value at 0.5 Re [M gF e] Note. -The calculated gradients for each measured metallicity indicator and the EZ Ages results binned by velocity dispersion. The units are dex.30 < σ < 125
-0.2322
0.0377
2.7716
[M gF e]
125 < σ < 185
-0.4138
0.0221
3.0104
[M gF e]
185 < σ < 230
-0.6042
0.0269
3.1210
[M gF e]
230 < σ < 325
-0.7705
0.0399
3.2333
< F e >
30 < σ < 125
-0.2667
0.0629
2.6716
< F e >
125 < σ < 185
-0.3492
0.0370
2.8064
< F e >
185 < σ < 230
-0.5079
0.0435
2.8166
< F e >
230 < σ < 325
-0.5760
0.0630
2.8363
Mg b
30 < σ < 125
-0.1845
0.0425
2.7957
Mg b
125 < σ < 185
-0.4954
0.0253
3.1388
Mg b
185 < σ < 230
-0.6881
0.0319
3.3636
Mg b
230 < σ < 325
-0.9081
0.0480
3.5929
C 4668
30 < σ < 125
-0.8821
0.0654
5.0693
C 4668
125 < σ < 185
-1.5585
0.0384
5.7810
C 4668
185 < σ < 230
-1.6801
0.0500
6.1940
C 4668
230 < σ < 325
-2.1148
0.0888
6.5130
Ca 4227
30 < σ < 125
-0.1371
0.0582
1.5715
Ca 4227
125 < σ < 185
-0.2527
0.0327
1.6673
Ca 4227
185 < σ < 230
-0.3539
0.0421
1.7027
Ca 4227
230 < σ < 325
-0.4386
0.0675
1.7414
Corrected Hβ
30 < σ < 125
-0.0446
0.0481
2.1860
Corrected Hβ
125 < σ < 185
0.1457
0.0294
2.0196
Corrected Hβ
185 < σ < 230
0.3013
0.0414
1.9126
Corrected Hβ
230 < σ < 325
0.5337
0.0667
1.8296
[F e/H]
30 < σ < 125
-0.2311
0.0303
-0.0513
[F e/H]
125 < σ < 185
-0.0709
0.0151
0.0185
[F e/H]
185 < σ < 230
-0.1157
0.0176
-0.0040
[F e/H]
230 < σ < 325
-0.1740
0.0290
-0.0325
[M g/F e]
30 < σ < 125
0.1363
0.0551
-0.0522
[M g/F e]
125 < σ < 185
-0.0068
0.0377
-0.0474
[M g/F e]
185 < σ < 230
0.1155
0.0411
-0.0038
[M g/F e]
230 < σ < 325
0.0784
0.0567
0.0236
[C/F e]
30 < σ < 125
-0.1861
0.0517
0.0284
[C/F e]
125 < σ < 185
-0.2005
0.0419
0.0616
[C/F e]
185 < σ < 230
-0.0758
0.0575
0.1060
[C/F e]
230 < σ < 325
-0.1251
0.0710
0.1785
TABLE 7 Logarithmic
7Gradients: Stellar Mass Binning M g/F e] 10.0 < log(M ) < 10.3 0.1552 0.2066 [M g/F e] 10.3 < log(M ) < 10.7 -0.0075 0.1302 [M g/F e] 10.7 < log(M ) < 11.0 -0.0712 0.1519 [M g/F e] 11.0 < log(M ) < 11.5Measure
log M
d(Measure)/d log r
Error
[F e/H]
10.0 < log(M ) < 10.3
-0.3028
0.0657
[F e/H]
10.3 < log(M ) < 10.7
-0.0906
0.0296
[F e/H]
10.7 < log(M ) < 11.0
-0.1709
0.0343
[F e/H]
11.0 < log(M ) < 11.5
-0.1832
0.0702
[0.0840
0.2212
[C/F e]
10.0 < log(M ) < 10.3
-0.2294
0.1886
[C/F e]
10.3 < log(M ) < 10.7
-0.2204
0.1292
[C/F e]
10.7 < log(M ) < 11.0
-0.0803
0.1836
[C/F e]
TABLE 8 Logarithmic
8Gradients: Velocity Dispersion Binning Note. -The calculated logarithmic gradients for the metallicity and [α/Fe] ratios for all four velocity dispersion bins.Measure
σ
d(Measure)/d log r
Error
[F e/H]
30 < σ < 125
-0.3069
0.0853
[F e/H]
125 < σ < 185
-0.0708
0.0343
[F e/H]
185 < σ < 230
-0.1203
0.0403
[F e/H]
230 < σ < 325
-0.1866
0.0776
[M g/F e]
30 < σ < 125
0.1810
0.2223
[M g/F e] 125 < σ < 185
-0.0068
0.1312
[M g/F e] 185 < σ < 230
0.1201
0.1250
[M g/F e] 230 < σ < 325
0.0840
0.2212
[C/F e]
30 < σ < 125
-0.2472
0.1957
[C/F e]
125 < σ < 185
-0.2005
0.1293
[C/F e]
185 < σ < 230
-0.0788
0.1791
[C/F e]
230 < σ < 325
-0.1341
0.2528
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| [] |
[
"All-optical pump-and-probe detection of two-time correlations in a Fermi gas",
"All-optical pump-and-probe detection of two-time correlations in a Fermi gas"
] | [
"T.-L Dao \nCentre de Physique Théorique\nÉcole Polytechnique\nCNRS\n91128PalaiseauFrance\n\nLaboratoire Charles Fabry de l'Institut d'Optique\nCNRS and Univ. Paris-Sud\nCampus Polytechnique, RD 128F-91127Palaiseau cedexFrance\n",
"C Kollath \nCentre de Physique Théorique\nÉcole Polytechnique\nCNRS\n91128PalaiseauFrance\n",
"I Carusotto \nDipartimento di Fisica\nCNR-INFM BEC Center\nUniversità di Trento\n38050PovoItaly\n",
"M Köhl \nCavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AvenueCB3 0HECambridgeUnited Kingdom\n"
] | [
"Centre de Physique Théorique\nÉcole Polytechnique\nCNRS\n91128PalaiseauFrance",
"Laboratoire Charles Fabry de l'Institut d'Optique\nCNRS and Univ. Paris-Sud\nCampus Polytechnique, RD 128F-91127Palaiseau cedexFrance",
"Centre de Physique Théorique\nÉcole Polytechnique\nCNRS\n91128PalaiseauFrance",
"Dipartimento di Fisica\nCNR-INFM BEC Center\nUniversità di Trento\n38050PovoItaly",
"Cavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AvenueCB3 0HECambridgeUnited Kingdom"
] | [] | We propose an all-optical scheme to probe the dynamical correlations of a strongly-interacting gas of ultracold atoms in an optical lattice potential. The proposed technique is based on a pump-and-probe scheme: a coherent light pulse is initially converted into an atomic coherence and later retrieved after a variable storage time. The efficiency of the proposed method to measure the two-time one-particle Green function of the gas is validated by numerical and analytical calculations of the expected signal for the two cases of a normal Fermi gas and a BCS superfluid state. Protocols to extract the superfluid gap and the full quasi-particle dispersions are discussed. | 10.1103/physreva.81.043626 | [
"https://arxiv.org/pdf/0905.2826v2.pdf"
] | 119,198,194 | 0905.2826 | ee608354dd41e0534cab28e29ad44106cb265c93 |
All-optical pump-and-probe detection of two-time correlations in a Fermi gas
26 Mar 2010 (Dated: March 29, 2010)
T.-L Dao
Centre de Physique Théorique
École Polytechnique
CNRS
91128PalaiseauFrance
Laboratoire Charles Fabry de l'Institut d'Optique
CNRS and Univ. Paris-Sud
Campus Polytechnique, RD 128F-91127Palaiseau cedexFrance
C Kollath
Centre de Physique Théorique
École Polytechnique
CNRS
91128PalaiseauFrance
I Carusotto
Dipartimento di Fisica
CNR-INFM BEC Center
Università di Trento
38050PovoItaly
M Köhl
Cavendish Laboratory
University of Cambridge
JJ Thomson AvenueCB3 0HECambridgeUnited Kingdom
All-optical pump-and-probe detection of two-time correlations in a Fermi gas
26 Mar 2010 (Dated: March 29, 2010)arXiv:0905.2826v2 [cond-mat.quant-gas]numbers: 0375Ss4250Gy7847jc7110Fd
We propose an all-optical scheme to probe the dynamical correlations of a strongly-interacting gas of ultracold atoms in an optical lattice potential. The proposed technique is based on a pump-and-probe scheme: a coherent light pulse is initially converted into an atomic coherence and later retrieved after a variable storage time. The efficiency of the proposed method to measure the two-time one-particle Green function of the gas is validated by numerical and analytical calculations of the expected signal for the two cases of a normal Fermi gas and a BCS superfluid state. Protocols to extract the superfluid gap and the full quasi-particle dispersions are discussed.
I. INTRODUCTION
Many-body quantum systems exhibit truly remarkable features such as high-temperature superconductivity and the fractional quantum Hall effect. Traditionally, these phenomena are studied in the solid state. However, in recent years dilute, yet strongly interacting, atomic gases have started providing a novel class of systems to investigate this fascinating physics. Their outstanding cleanliness, control, and precise microscopic understanding will push forward the fundamental understanding of quantum many-body physics [1].
Strongly interacting atomic quantum gases are generally prepared by trapping atoms in vacuum in a magnetic or optical potential. This offers two remarkable opportunities: Firstly, a superb isolation from the environment opens the door to fascinating experiments out of equilibrium to investigate genuine quantum dynamics. Secondly, a variety of coherent optical processes are available to selectively probe the quantum system without being disturbed by a surrounding bulk medium. In particular, these optical detection techniques can provide repetitive and almost non-destructive in-situ measurements [2,3]. The combination of these two features makes ultracold quantum gases ideal systems to study the non-equilibrium and dynamic properties of isolated quantum many body systems. However, the experimental study of these properties requires the development of novel detection schemes that are sensitive to a wider variety of observables of the quantum gas, e.g. its multi-time correlation functions.
The prime example of an atomic quantum system mimicking the physics of the solid state are interacting fermionic atoms in artificial lattices structures, the so called optical lattices [4]. The preparation of strongly correlated states in an optical lattice will allow for an analog simulation of complex quantum many body Hamiltonians. Recently, evidence for the stabilization of a Mott-insulating phase has been obtained by looking at density related quantities of the gas [5][6][7][8]. The identification and characterization of more complex quantum phases requires, however, the measurement of time-resolved single-particle correlation functions, also called Green func-tions, of the form ψ † σ,r (t)ψ σ ′ ,r ′ (t ′ ) . Here ψ ( †) σ,r (t) is the annihilation (creation) operator for the internal atomic state σ at position r and time t. The single-particle two-time correlation function reveals profound information about the macroscopic coherence and decoherence of the systems and keeps track of the subtle properties of quantum phases which are not densityordered, e.g. the existence of quasi-particles in a strongly correlated Fermi liquid. This same correlation function plays a even more crucial role in the case non-equilibrium situations: as the most celebrated example, the particular relaxation behaviour of glasses is almost invisible in one-time correlations, while it can be followed in full detail by measuring the twotime ones [9].
Up to now, the single particle equal time correlation function out of equilibrium was investigated for bosonic atoms [10]. For fermions, only the energy resolved correlation function of an equilibrium state has so far been probed by momentum-integrated [11][12][13][14][15][16] and momentum resolved [17,18] radio-frequency or two-photon spectroscopy. A more elaborated scheme for the detection of the two-time correlation function based on the immersion of a ion into quantum gases has been proposed in [19].
Here, we propose an all-optical pump-and-probe scheme to extract quantitative information on the microscopic physics of a Fermi gas and in particular on its two-time correlation functions. A pump sequence firstly brings the system into a quantum superposition of its initial state and an excited state. The response of the system to a second probe pulse sequence is then measured after a variable time delay. In this way, information on the time evolution of the atomic two-time correlations is converted into easily detectable observables, such as the intensity and the phase of the outgoing light.
From an alternative point of view, our scheme can be seen as an application of light storage techniques [20][21][22] to the diagnostic of many-body systems: a coherent pulse of light is stored in a quantum gas and retrieved at a later time after a variable interval. Information on the system is extracted from the properties of the retrieved light. Differently from standard light storage experiments where it is a purely detrimental effect, decoherence of the stored pulse as a function of storage time is in our scheme the crucial tool to obtain information on the many-body dynamics of the underlying quantum gas.
The structure of the paper is the following. In Sec.II the measurement schemes are introduced and analytical expression relating the observed signal to the many-body observables are given. In Sec.III an application to a fermionic system is discussed in detail and experimental protocols to extract the superfluid gap and the full quasi-particle dispersions of a BCS superfluid are outlined. Conclusions and future perspectives are given in Sec.IV. Even though the application scope of our measurement procedure is much wider, we focus our attention here onto the case of fermionic atoms in a two-dimensional lattice geometry. This geometry lies at the heart of quantum simulations with the aim of exploring the mechanisms underlying high-temperature superconductivity [23]. A tight optical confinement potential freezes the atomic motion into a single xy plane. Additionally, a periodic optical lattice potential is applied along the x and y directions to generate a twodimensional lattice structure [24].
II. THE MEASUREMENT PROCEDURE
The gas consists of a mixture of atoms in two hyperfine ground states g and g ′ that feel an identical confinement potential. Our all-optical probing scheme involves three atomic levels (g, e, and m) arranged in a Λ scheme as schematically shown in Fig. 1. The m state is a long-lived electronic ground state whereas the e state is an electronically excited state. With a suitable choice of polarization and frequency, the g ′ atoms experience a negligible coupling to the pump and probe light fields.
The diagnostic scheme (Fig. 2) starts with the creation of a coherent excitation by adiabatically switching on a laser of (spatially uniform) Rabi frequency Ω in 2 and then a weaker collinear laser of (spatially dependent) Rabi frequency Ω 1 (r) [25]. The two beams are then suddenly and simultaneously switched-off. The frequency ω 1,2 of the beams are chosen to be resonant with the g → e and m → e transitions, respectively. To ensure adiabaticity of the preparation stage, the switch-on of the two lasers has to be performed on a timescale long as compared to the internal atomic dynamics and to the Rabi frequencies Ω 1 , Ω in 2 . On the other hand, the switchoff has to be much faster than all these frequencies.
Provided the whole excitation stage takes place on a fast time scale as compared to the many-body dynamics, the atomic position can be considered as fixed and the optical process described by a single atom picture where the atomic |g state is adiabatically transformed into a dark state [26][27][28][29] |dark = e iθ
|Ω 1 | 2 + |Ω in 2 | 2 Ω in 2 |g − Ω 1 |m (1)
which is fully decoupled from the excitation lasers. All other bright eigenstates are energetically separated and do not get mixed with the dark state provided the switch-on phase is performed in a slow enough manner. Assuming that the phase of the Ω 1,2 Rabi frequencies (once the carriers at ω 1,2 are eliminated) is constant during the whole sequence, the global Berry phase
θ = i dt dark(t)| d dt |dark(t)(2)
acquired by the atom during the adiabatic evolution is easily shown to vanish. In the |Ω 1 | ≪ |Ω in 2 | limit under investigation here, the effect of the excitation stage on the initial many-body state |φ 0 in the lattice representation can be expressed as the transformation:
|φ d ≃ 1 − i Ω 1 (r i ) Ω in 2ψ † m,riψ g,ri |φ 0 .(3)
Here, theψ σ,ri (ψ † σ,ri ) lattice operators destroy (create) a fermionic atom in the σ = g, m state at the lattice site r i , respectively. Ω 1,2 are the Rabi frequencies in the lattice representation. Initially, only the g and g ′ states are assumed to be occupied.
The use of a locally focused laser of amplitude Ω 1 (r) on the g → e transition allows one to selectively address a well defined region of the sample. The space selection is useful to eliminate inhomogeneous broadening effects due to the spatially varying density in e.g. trapped systems [30,31]. If no spatial selection is performed, the final signal would include the contributions of different regions of the system.
Once the two beams are switched off, the system evolves according to its many-body Hamiltonian for a storage time t s from the state |φ d to the new many-body state
|φ(t s ) = U mb (t s ) |φ d = e −i (H0+Hm+Hint) ts/ |φ d (4)
where U mb (t s ) is the many body time-evolution operator and the system Hamiltonian involves three contributions: H 0 is the Hamiltonian acting on the states g and g ′ , H m the Hamiltonian of the atoms in the m state, and H int contains the interaction processes between the g, g ′ and m states. This timeevolution will change the coherence between g and m present in the prepared dark state. The g, m coherence remaining at the end of the storage time is finally probed.
The detection of the remaining coherence can be achieved by different schemes: Either a fast π pulse is applied to coherently transfer all atoms from the m to the e state and then coherent photons emitted on the e → g transition are detected. Or the excitation is slowly released by means of a weak field of frequency ω 2 and Rabi frequency in the continuum Ω out 2 ≪ γ e that transfers the atoms adiabatically from the m state into a coherent superposition of m and e. In both cases, the electric dipole that is responsible for the emission at frequency ω 1 on the e → g transition is proportional to the coherence between the g and m atomic states,
d(r) = Dψ † g,rψ m,r .(5)
The constant D depends on the details of the process and determines the duration in time τ r of the released pulse τ −1 r = γ e |d eg /D| 2 , where γ e is the radiative decay rate of e state atoms. In the π pulse case, the constant D is equal to the electric dipole matrix element between the state g and e, D = d ge .
In the case of a slow release, D is approximately given by D = 2iΩ out 2 d ge /γ e . In order for the many-body dynamics not to interfere with the release process, the time duration τ r of this latter has to be shorter than the characteristic time scales of the many-body dynamics.
The near field pattern of the emitted light amplitude is determined by the expectation value of the local dipole operator on the final state |φ(t s )
d(r, t s ) = φ(t s )|d(r)|φ(t s )(6)
We switch now to the lattice representation by relating the field operators in the continuum to the lattice operators via the Wannier functions w σ (r) as followingψ σ,r = i w σ (r − r i )ψ σ,ri . We checked numerically that for tight atomic Wannier functions on deep lattices, we can keep only the terms with Wannier factors taken at the same sites and neglect all other contributions. This leads to the following expression for the dipole operator
d(r, t s ) ≃ D i W ri (r) φ(t s )|ψ † g,riψ m,ri |φ(t s ) ,(7)
with W ri (r) = w * g (r−r i )w m (r−r i ). Inserting the expression (4) of the final state and switching to the Heisenberg representation for the operatorsψ x,ri (t s ) = U † mb (t s )ψ x,ri U mb (t s ), this has the form
d(r, t s ) = D i W ri (r) × φ d |U † mb (t s )ψ † g,riψm,ri U mb (t s )|φ d = D i W ri (r) φ d |ψ † g,ri (t s )ψ m,ri (t s )|φ d .(8)
Inserting into (8) the explicit expression (3) for the dark state and taking into account that no atoms were initially present in the m state, this expression can be written in the compact form
d(r, t s ) = − D Ω in 2 i,j Ω 1 (r j )W ri (r) × φ 0 |ψ † g,ri (t s )ψ m,ri (t s )ψ †
m,rj (0)ψ g,rj (0)|φ 0 (9) that only involves a time-dependent correlation function taken on the initial many-body state |φ 0 .
As no atoms are initially present in the m state, the initial many-body state |φ 0 exactly factorizes in a complex manybody state for the g, g ′ subspace and vacuum for the m one.
Assuming that the few atoms that are transferred into the m state during the preparation stage do not significantly interact with the majority of atoms left in the g, g ′ states [18,32,33] allows us to neglect the m − g, g ′ interaction term of the Hamiltonian H int in (4) and write the time-evolution operator U mb in the factorized form U mb (t s ) = e −iH0ts/ e −iHmts/ . As a direct consequence, the Heisenberg evolution of thê ψ m,ri (t) operator is only determined by the H m part of the evolution operator, while theψ g,ri (t) andψ g ′ ,ri (t) operators evolve with the many-body H 0 Hamiltonian in the g, g ′ space.
These simple facts allow one to rewrite the dipole expectation value in the final form:
d(r, t s ) = − D Ω in 2 i,j Ω 1 (r j ))W ri (r) × vac|ψ m,ri (t s )ψ † m,rj (0)|vac × φ 0 |ψ † g,ri (t s )ψ g,rj (0)|φ 0 .(10)
where all the expectation values are to be evaluated on the initial many-body state before the preparation stage, with no occupation in the m state. In particular, the m state propagator describes the free-particle evolution in the lattice potential. The far-field pattern in a directionθ is proportional to the spatial Fourier transform of d(r, t s ) evaluated at a wavevector k equal to the projection of the emission wavevectorθ ω 1 /c along the xy plane. Here, c is the velocity of light. This leads to the following expression for the far-field emission amplitude at a distance R = Rθ:
E out θ (t s ) = C k N q e −iωm(q+k)ts φ 0 |ψ † g,q (t s )ψ g,q (0)|φ 0(11)
where ω m (q) is the free-particle dispersion of m state atoms in the lattice potential and the coefficient C k is defined as
C k = D ω 2 1 Ω 1 (k) W (k) 4πǫ 0 R c 2 Ω in 2 .(12)
Invariance under translations along the plane guarantees that the coherent emission amplitude in theθ direction, i.e. with an in-plane wavevector k, only depends on the incident probe amplitude Ω 1 (k) at the same k. Here we have set Ω 1 (k) = j Ω 1 (r)e −ik.rj , we have definedψ g,q = (N ) −1/2 jψ g,rj e −iq.rj , and we have used a lattice with N sites neglecting boundary effects. The factor W (k) = d 2 rw * g (r)w m (r)e −ik.r is a slowly varying envelop stemming from the tight atomic Wannier functions.
Expression (11) relates the coherent amplitude E out θ (t s ) of the released light to the time-dependent one-body Green function of a generic many-body gas. It is one key result of the present paper. In the limiting case ω m (q) = ω o m where the m atoms do not appreciably move during the time t s , the far-field amplitude (11) can be further simplified into the form E out θ (t s ) = C k e −iω o m ts φ 0 |ψ † g,ri (t s )ψ g,ri (0)|φ 0 , (13) which only involves the local value of the Green function of g atoms.
Experimentally, the coherent E out θ amplitude can be measured by homodyne detection of the emission with a stronger reference beam at ω 1 . The intensity and phase of E out θ is inferred from the amplitude and phase of the oscillations in the interference signal as a function of the mixing phase. This procedure requires coherence at the g → m frequency which can be easily achieved if all Ω 1 , Ω in,out 2 fields are obtained from a single laser source.
Another quantity of interest is the total (i.e. coherent and incoherent) intensity pattern in either the far-or the near-field. Differently from the coherent amplitude (11), these involve higher order correlations of the many-body gas. For instance, the near-field dipole pattern I(r) = d † (r)d(r) reads:
I(r, t s ) = |D| 2 |Ω in 2 | 2 i,j |Ω 1 (r j )W ri (r)| 2 × vac|ψ m,rj (0)ψ † m,ri (t s )ψ m,ri (t s )ψ † m,rj (0)|vac × φ 0 |ψ † g,rj (0)ψ g,ri (t s )ψ † g,ri (t s )ψ g,rj (0)|φ 0 .
For a localized beam Ω 1 (r j ), the I(r) signal is proportional to a fixed envelope determined by the motion of atoms in the m state times a two-body Green function of g atoms. The correlation function of the g state can be understood as measuring the density at time t s at site r i if one has removed an atom at time 0 at site r j . This scheme looks promising e.g. to follow the dynamics of holes in anti-ferromagnetic states [35,36].
III. APPLICATION TO BCS SUPERFLUID
In order to demonstrate the efficiency of the proposed detection technique, we now calculate the signal that is expected for a weakly attractive, unpolarized two-component Fermi gas in an optical lattice at half filling. In particular we show how the proposed method is able to identify a superfluid state and its quasiparticles from the measured two-time correlation function.
In the normal state, the dispersion relation of quasiparticles is given by the free-particle dispersion in the lattice. Here we take the tight-binding form ω g,g ′ ,m (q) = ω o g,g ′ ,m − 2J g,g ′ ,m [cos(q x a) + cos(q y a)]. While the g, g ′ atoms feel the same potential, J g = J g ′ , the hopping J m for the m state atoms can be different. In the following we set ω o g ′ = ω o g = 0 and focus on the case of half-filling.
In the superfluid state, the quasiparticle dispersion predicted by BCS theory consists of two branches E ± q = ± [ ω g (q)] 2 + ∆ 2 separated by a gap of amplitude 2∆. The one-body Green function G g (q, t) = ψ † g,q (t)ψ g,q (0) for the BCS phase reads [34]
G g (q, t) = u 2 q f (E + q )e i(ω mf +E + q / )t + v 2 q f (E − q )e i(ω mf +E − q / )t .
The Bogoliubov coefficients are defined as u 2 q , v 2 q = 1 2 1 + ω g (q)/E ± q and the Fermi distribution as f (E) = (1 + e E/kB T ) −1 . ω mf is the mean-field shift [39]. In what follows, we shall focus our attention on low temperature T for which the upper branch E + q is almost empty and can be neglected. Under such an assumption the emission amplitude (11) becomes
E out θ (t s ) = C k N q e −i(ωm(q+k)−ω mf −E − q / )ts v 2 q f (E − q ).
Its Fourier transform with respect to the storage time t s has the form
E out θ (ω s ) = C k N q v 2 q f (E − q ) δ(ω s −ω m (q+k)+ω mf +E − q / ).
For each value ω s of the frequency, the signal comes from the wavevectors q which fulfill
ω s = ω o m − ω mf + r ω g (q + k) − E − q / .(14)
In the following we will neglect the contributions by ω o m −ω mf since these can be eliminated in the homodyne detection [40]. Several regimes can be identified depending on the value of the hopping ratio r = J m /J g . Experimentally, the hopping amplitudes can be varied within some range by tuning the frequency and polarization of the lattice beams, or, if necessary, by using more complex multi-photon transitions instead than the simple Raman scheme discussed so far.
A. Small hopping ratio r ≪ 1
The physics is the simplest in the r ≪ 1 case where the atoms in the m state do not move during the experiment and This physics is easily understood looking at the corresponding frequency spectra plotted in Fig. 4. In the limiting case r → 0, the spectrum recovers the density of states for quasiparticles. In the normal state, the spectrum has a broad shape extending up to ω max = 4J g (1 − r) and showing a singularity at ω s = 0 as a consequence of the perfect nesting of the square Fermi surface at half-filling. In the superfluid state, the dominant feature is the peak at ω s ≃ ∆ that limits the spectrum from below and from which the BCS gap is immediately extracted. In this state the upper limit of the signal is shifted to ω max = −4rJ g + 16J 2 g + ∆ 2 . The coherent emission spectra in the case of equal hopping amplitude r = 1 show a rich structure that strongly depends on the wavevector k (Fig. 5). Even if the physics is somehow more involved than in the r ≪ 1 case considered in the previous subsection, still the observed signal can be used to obtain useful information on the many-body system, e.g. its superfluid gap.
Let us first focus on the coherent emission in the k = 0 direction. At the lower boundary a large signal is found at ω min ≈ (∆ 2 /4J g )/2 for ∆ ≪ 4J g (see the inset) which originates from quasiparticles at q = 0. The long tail that appears at high frequencies past ω s = ∆ is a direct consequence of the smearing out of the Fermi surface on an energy scale ∆ in the BCS state.
The emission spectrum in the direction along the diagonal of the Brillouin zone k = (k, k) with k = π/2a (a is the lattice constant) is characterized by two peaks and a broad background with quite sharp edges: most visible is the strong peak at ω s = ∆ that originates from the divergence of the density of states at the Fermi level in a BCS state. This peak persists for different values of k (cf. Fig. 5 k = π/4a) and its position can be used to experimentally measure the amplitude ∆ of the gap.
C. High hopping ratio r ≫ 1
We conclude our study with a brief account of the case of a high hopping ratio r ≫ 1. Examples of spectra for r = 3 are plotted in Fig. 7): in particular, they show a clear peak at ω s = ∆ independent of the direction of the light. This distinctive feature allows for a direct measurement of the gap amplitude ∆.
Furthermore, the full dispersion of the BCS quasiparticles E − q can be extracted from the position of the lower edge of the spectrum. For r ≫ 1 the r-dependent term in Eq. (14) dominates and determines the q values that correspond to the spectral edges: the contribution of quasi-particles with momentum q = −k determines the sharp lower edge at ω min = −4rJ g − E − −k . The dependence of the lower spectral edge on the emission direction (k = k x = k y ) is shown in Fig. 6 for different values of r and compared to the quasi-particle dispersion. While the agreement is limited to the special points k = 0, π/2a for r = 1, it quickly improves for larger r; a reasonably accurate image of the quasi-particle dispersion around k = 0, π/2a is already recovered for r 3.
IV. CONCLUSIONS AND PERSPECTIVES
In summary, we have proposed a novel all-optical, spatially selective and almost non-destructive technique to probe in situ the microscopic many-body dynamics of a gas of interacting ultracold atoms. The technique is inspired to recent light storage experiments and is based on the creation of an atomic coherence by coherent absorption of a pump laser pulse and its later retrieval after a variable storage time: information on the many-body dynamics of the quantum gas is extracted from the amplitude and coherence properties of the retrieved light. Differently from most previous measurement schemes, the use of a spatially localized pump spot will allow to individually address the different coexisting quantum phases that can appear in a trapped system.
The efficiency of the proposed measurement scheme is tested on the specific, analytically tractable example of a twodimensional BCS superfluid. Protocols to extract the superfluid gap and the quasi-particle dispersion are presented, which take into account some most significant difficulties that arise from the internal structure of the atoms.
As our scheme consists of the measurement of two-times correlation functions, it is expected to be of great utility in the study of the non-equilibrium dynamics of a quantum system: on one hand, its almost non-destructive nature suggests that a series of many measurements can be performed at a high repetition rate without significantly perturbing the system dynamics. On the other hand, the observed quantities play a crucial role in the characterization of relaxation dynamics [9,38]: for instance,they may serve to identify the glassiness of a system in the presence of disorder [9].
Future work will investigate the extension of the method to more complex, three-dimensional geometries: differently from the two-dimensional geometry considered so far, this requires a careful treatment of light propagation across a bulk sample in both the excitation and the retrieval stages. Preliminary work in this direction has appeared as [37].
FIG. 1 :
1(Color online) Diagram of the internal atomic levels involved in the proposed detection scheme.
FIG. 2 :
2(Color online) Snapshots of the measurement procedure. From left to right: adiabatic storage of coherence from incident beams, free many-body evolution, light re-emission and detection.
online) Time-dependence of the emission amplitude in the normal (k = 0) direction for a normal state N (blue solid line) and a BCS superfluid with gap ∆/4Jg = 0.2 (red dashed line). Hopping ratio r = 0.1. Temperature kBT = Jg/50. online) Frequency-dependence of the emission amplitude in the normal (k = 0) direction for a normal state N (blue solid line) and a BCS superfluid with gap ∆/4Jg = 0.2 (red dashed line). Hopping ratio r = 0.1. Temperature kBT = Jg/50.the emission amplitude is determined by the local Green function(13). As one can see inFig. 3, the emission amplitude for a superfluid state as a function of storage time t s shows a slowly decaying oscillation at a low frequency determined by the BCS gap ∆. On top of this slow oscillation, faster and quickly decaying oscillations are visible at frequencies on the order of the Fermi energy (i.e. the band width J g ). The long lasting, slow oscillations are a signature of the superfluid state. They disappear in a normal state where one is left with fast and quickly decaying oscillations.
online) Frequency dependence of the emission amplitude from a BCS superfluid with ∆/4Jg = 0.2 at different angles, k = kx = ky = 0, π/4a, π/2a, with hopping ratio r = 1. Inset: magnified view of the k = 0 curve. The arrows indicate the corresponding spectral minimum ωmin. Temperature kBT = Jg/50.
online) k-dependence of the lower edge of the spectrum compared to the quasiparticle dispersion E − k of the BCS superfluid with ∆/4Jg = 0.2. From top to bottom, hopping ratio r = 1, 1.5, 3. Curves for different r are offset by 0.5 for better visionline) Frequency dependence of the emission amplitude from a BCS superfluid with ∆/4Jg = 0.2 at different angles, k = kx = ky = 0, π/4a, π/2a. Hopping ratio r = 3. The arrows indicate the corresponding spectral minimum ωmin. Temperature kBT = Jg/50.
can be detuned by the mean-field shift from the resonance m → e to eliminate oscillations of frequency ω o m − ω mf .
AcknowledgmentsWe are grateful to F. Bariani, M. Capone, M. Inguscio and the Quantum Optics group of ETH Zürich for stimulating discussions. We acknowledge support from the 'Triangle de la Physique', ANR ('FABIOLA' and 'FAMOUS'), the DARPA-OLE program, and EPSRC (EP/G029547/1). CK would like to acknowledge the IPAM for its hospitality.
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For the slow outcoupling procedure, the out-coupling laser Ω out. For the slow outcoupling procedure, the out-coupling laser Ω out
| [] |
[
"Radio spectra and polarisation properties of radio-loud Broad Absorption Line Quasars",
"Radio spectra and polarisation properties of radio-loud Broad Absorption Line Quasars"
] | [
"F M Montenegro-Montes \nIstituto di Radioastronomia\nINAF\nVia Gobetti 101I-40129BolognaItaly\n\nDepartamento de Astrofísica\nUniversidad de La Laguna\nAvda. Astrofísico Fco. Sánchez s/n\nE-38200La LagunaSpain\n",
"K.-H Mack \nIstituto di Radioastronomia\nINAF\nVia Gobetti 101I-40129BolognaItaly\n",
"M Vigotti \nIstituto di Radioastronomia\nINAF\nVia Gobetti 101I-40129BolognaItaly\n",
"C R Benn \nIsaac Newton Group\nApartado 321E-38700Santa Cruz de La PalmaSpain\n",
"R Carballo \nDpto. de Matemática Aplicada y Ciencias de la Computación. Univ. de Cantabria\nETS Ingenieros de Caminos Canales y Puertos. Avda. de los Castros s/n\nE-39005SantanderSpain\n",
"J I González-Serrano \nInstituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nE-39005SantanderSpain\n",
"J Holt \nLeiden Observatory. Leiden University\nP O Box 9513NL-2300 RALeidenThe Netherlands\n",
"F Jiménez-Luján \nInstituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nE-39005SantanderSpain\n\nDpto. de Física Moderna\nUniv. de Cantabria\nAvda de los Castros s/n, Spain JuneE-39005, 2008SantanderDraft\n"
] | [
"Istituto di Radioastronomia\nINAF\nVia Gobetti 101I-40129BolognaItaly",
"Departamento de Astrofísica\nUniversidad de La Laguna\nAvda. Astrofísico Fco. Sánchez s/n\nE-38200La LagunaSpain",
"Istituto di Radioastronomia\nINAF\nVia Gobetti 101I-40129BolognaItaly",
"Istituto di Radioastronomia\nINAF\nVia Gobetti 101I-40129BolognaItaly",
"Isaac Newton Group\nApartado 321E-38700Santa Cruz de La PalmaSpain",
"Dpto. de Matemática Aplicada y Ciencias de la Computación. Univ. de Cantabria\nETS Ingenieros de Caminos Canales y Puertos. Avda. de los Castros s/n\nE-39005SantanderSpain",
"Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nE-39005SantanderSpain",
"Leiden Observatory. Leiden University\nP O Box 9513NL-2300 RALeidenThe Netherlands",
"Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nE-39005SantanderSpain",
"Dpto. de Física Moderna\nUniv. de Cantabria\nAvda de los Castros s/n, Spain JuneE-39005, 2008SantanderDraft"
] | [
"Mon. Not. R. Astron. Soc"
] | We present multi-frequency observations of a sample of 15 radio-emitting Broad Absorption Line Quasars (BAL QSOs), covering a spectral range between 74 MHz and 43 GHz. They display mostly convex radio spectra which typically peak at about 1-5 GHz (in the observer's rest-frame), flatten at MHz frequencies, probably due to synchrotron self-absorption, and become steeper at high frequencies, i.e., ν 20 GHz. VLA 22-GHz maps (HPBW ∼ 80 mas) show unresolved or very compact sources, with linear projected sizes of 1 kpc. About 2/3 of the sample look unpolarised or weakly polarised at 8.4 GHz, frequency in which reasonable upper limits could be obtained for polarised intensity. Statistical comparisons have been made between the spectral index distributions of samples of BAL and non-BAL QSOs, both in the observed and the rest-frame, finding steeper spectra among non-BAL QSOs. However constraining this comparison to compact sources results in no significant differences between both distributions. This comparison is consistent with BAL QSOs not being oriented along a particular line of sight. In addition, our analysis of the spectral shape, variability and polarisation properties shows that radio BAL QSOs share several properties common to young radio sources like Compact Steep Spectrum (CSS) or Gigahertz-Peaked Spectrum (GPS) sources. | 10.1111/j.1365-2966.2008.13520.x | [
"https://arxiv.org/pdf/0805.4746v1.pdf"
] | 13,981,235 | 0805.4746 | d8b35c118cea75d08c115770f4cd3ed7f9291271 |
Radio spectra and polarisation properties of radio-loud Broad Absorption Line Quasars
2008
F M Montenegro-Montes
Istituto di Radioastronomia
INAF
Via Gobetti 101I-40129BolognaItaly
Departamento de Astrofísica
Universidad de La Laguna
Avda. Astrofísico Fco. Sánchez s/n
E-38200La LagunaSpain
K.-H Mack
Istituto di Radioastronomia
INAF
Via Gobetti 101I-40129BolognaItaly
M Vigotti
Istituto di Radioastronomia
INAF
Via Gobetti 101I-40129BolognaItaly
C R Benn
Isaac Newton Group
Apartado 321E-38700Santa Cruz de La PalmaSpain
R Carballo
Dpto. de Matemática Aplicada y Ciencias de la Computación. Univ. de Cantabria
ETS Ingenieros de Caminos Canales y Puertos. Avda. de los Castros s/n
E-39005SantanderSpain
J I González-Serrano
Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)
Avda. de los Castros s/nE-39005SantanderSpain
J Holt
Leiden Observatory. Leiden University
P O Box 9513NL-2300 RALeidenThe Netherlands
F Jiménez-Luján
Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)
Avda. de los Castros s/nE-39005SantanderSpain
Dpto. de Física Moderna
Univ. de Cantabria
Avda de los Castros s/n, Spain JuneE-39005, 2008SantanderDraft
Radio spectra and polarisation properties of radio-loud Broad Absorption Line Quasars
Mon. Not. R. Astron. Soc
0002008(MN L A T E X style file v2.2)quasars: absorption lines − radio continuum: galaxies − Polarisation
We present multi-frequency observations of a sample of 15 radio-emitting Broad Absorption Line Quasars (BAL QSOs), covering a spectral range between 74 MHz and 43 GHz. They display mostly convex radio spectra which typically peak at about 1-5 GHz (in the observer's rest-frame), flatten at MHz frequencies, probably due to synchrotron self-absorption, and become steeper at high frequencies, i.e., ν 20 GHz. VLA 22-GHz maps (HPBW ∼ 80 mas) show unresolved or very compact sources, with linear projected sizes of 1 kpc. About 2/3 of the sample look unpolarised or weakly polarised at 8.4 GHz, frequency in which reasonable upper limits could be obtained for polarised intensity. Statistical comparisons have been made between the spectral index distributions of samples of BAL and non-BAL QSOs, both in the observed and the rest-frame, finding steeper spectra among non-BAL QSOs. However constraining this comparison to compact sources results in no significant differences between both distributions. This comparison is consistent with BAL QSOs not being oriented along a particular line of sight. In addition, our analysis of the spectral shape, variability and polarisation properties shows that radio BAL QSOs share several properties common to young radio sources like Compact Steep Spectrum (CSS) or Gigahertz-Peaked Spectrum (GPS) sources.
INTRODUCTION
About 15 per cent of optically-selected quasars exhibit Broad Absorption Lines (BALs) in the blue wings of the UV resonance lines, due to gas with outflow velocities up to 0.2 c (Hewett & Foltz 2003). A classification of BALs has been done according to their association with highly ionised species like C iv, N v, Si iv (HiBALs) or, in addition, with low-ionisation species like Mg ii or Al iii (LoBALs). These broad troughs are the most evident manifestations of quasar outflows but many other intrinsic absorbers such as Associated Absorption Lines (AALs), mini-BALs or high velocity ⋆ E-mail:[email protected] Narrow Absorption Lines (NALs) are also related to them (Hamann & Sabra 2004;Ganguly & Brotherton 2008).
The study of BAL quasars (BAL QSOs) has become increasingly important in the last years. Outflows are not only useful as probes of the physics in the AGN environment, but also because they might be an important piece in many other puzzles like enrichment of the intergalactic medium, galaxy formation, evolution of the AGN and the host galaxy, cluster cooling flows, magnetisation of cluster and galactic gas and the luminosity function of quasars.
At the moment it is not clear how many quasars host outflows and why these can only be seen in a fraction of the quasar population. The most popular hypotheses proposed to explain this mainly differ in the role given to orientation. The main premise in the 'orientation scenario' is that the c 2008 RAS BAL phenomenon might be present in all quasars but intercepted by only ∼15 per cent of the lines of sight to the quasar (Weymann et al. 1991), e.g. within the walls of a bifunnel centred on the nucleus (Elvis 2000). This scenario is supported because in most respects, apart from the BALs themselves, BAL QSOs look similar to normal quasars. The small differences from non-BAL QSOs, e.g. slightly redder continua (Reichard et al. 2003b) and higher optical polarisation, could be a consequence of a preferred orientation of viewing angle.
The alternative hypothesis, the so-called 'unification by time', assumes that BALs appear during one or maybe more short periods in the lifetime of quasars. The observed rate of BAL QSOs would then be a function of the percentage of the quasar lifetime in which these outflows show up. This evolutionary scenario has been discussed by Becker et al. (2000) on the basis of the radio properties of their Faint Images Radio Sky at Twenty-centimeters (FIRST, Becker et al. 1995) BAL QSO sample. They note that 80 per cent of their BAL QSOs are unresolved at a 0.2-arcsec scale, with both flat and steep spectra. This latter group resembles the Compact Steep-Spectrum Sources (CSS) and their Gigahertz Peaked-Spectrum (GPS) sub-class. As a consequence, they suggest the picture in which BAL QSOs represent an early stage in the development of quasars.
This last scenario finds motivation on the collection of theoretical and phenomenological works that in recent years have contributed to build a first consistent picture of early AGN evolution. As an example, recent simulations (Hopkins et al. 2005) describe how quasar evolution can be understood as the result of merger events, where the supermassive black hole grows via accretion and, when it finally becomes an active 'protoquasar', expels out the surrounding material deposited by the merger through powerful winds. The fingerprints of these winds have been manifested not only as BAL troughs in UV spectra, but also as emission line outflows and sometimes associated to H I absorptions (see e.g., Holt et al. 2008). To that respect, even if the fraction of optically-selected quasars hosting BAL systems is estimated to be about 10-20 per cent of the quasar population, it has been argued that these samples might be substantially biased against BAL QSOs, as suggested by larger fractions of BAL QSOs found among infrared-selected quasars (Dai et al. 2008). In addition, Ganguly & Brotherton (2008) have recently estimated that ∼60 per cent of AGNs could host outflows when the different kinds of intrinsic absorptions are considered.
Radio emission has become, in fact, an important additional diagnostic tool when studying the orientation and evolutionary status of BAL QSOs. For a long time radio-loud BAL QSOs were believed to be extremely rare among the population of luminous quasars (Stocke et al. 1992). With the advent of large comprehensive radio surveys it has however become clear that radio-loud BAL QSOs are common. However the fraction of BAL QSOs seem to vary inversely with the radio-loudness parameter, with the radio-brightest quasars being 4 times less likely to exhibit BALs (Becker et al. 2001;Gregg et al. 2006). The evolutionary picture is, in fact, supported by some recent works coming from radio observations. For instance, EVN observations at 1.6 GHz of a few BAL QSOs from Becker's list (Jiang & Wang 2003) show that they present a still compact structure at these resolutions, with subkpc projected sizes, and a variety of orientations according to their radio geometry. In addition, the radio powerful CSS BAL QSO 1045+352 has also been observed with MERLIN and VLBA, and its complex radio structure has been interpreted as evidence of radio-intermittent activity (Kunert-Bajraszewska & Marecki 2007).
Some other works address the question of the orientation of BAL QSOs through radio variability (Zhou et al. 2006;Ghosh & Punsly 2007). They compare the 1.4-GHz flux densities of all quasars in the Sloan Digital Sky Survey (SDSS) quasar catalogues (Schneider et al. 2005;Schneider et al. 2007) with detections in both the FIRST and NRAO VLA Sky Survey (NVSS, Condon et al. 1998) surveys. The most strongly variable quasars have probably their jets closely aligned with the line of sight, being the relativistic beaming responsible for such high flux density variations. Some of those variable quasars present in addition the BAL phenomenon which is considered as a proof for the existence of at least a sub-population of polar BAL QSOs. At the same time, a few examples have been found of BAL QSOs associated to radio sources with an extended FR II morphology (Gregg et al. 2006) which suggest for those the opposite edge-on orientation.
This paper reports the first results of a study in which a systematic approach is taken to characterise the radio-loud BAL QSO population. The final goal is to locate these objects in the framework of an adequate orientation and/or evolutionary status scenario. In order to do this, the radio spectra and polarisation properties of a small sample of radio-loud BAL QSOs will be presented.
In Section 2 some samples of radio emitting QSOs existing in the literature are summarised, and among them a small sample of BAL QSOs is selected for multi-frequency continuum and polarisation observations. In Section 3 the observations, data reduction and the treatment of the errors are described. In Section 4 the results of the observations are presented. The radio morphologies of BAL QSOs, their polarisation properties and the variability of some of these objects with 2-epoch observations are analysed. In addition, the shape of the radio spectra is described and the radio spectral index distributions of radio quasars with and without associated BALs are compared. All these results are reviewed together in Section 5 and compared with other findings trying to put emphasis on those aspects related to the orientation vs. evolutionary dilemma. Some of the limitations of the analysis are noted and solutions to some of these caveats are proposed. Finally Section 6 summarises the main conclusions.
The cosmology adopted within the paper assumes a flat universe and the following parameters: H0=70 km s −1 Mpc −1 , ΩΛ=0.7, ΩM =0.3. The adopted convention for the spectral index definition is Sν ∝ ν α , except where otherwise specified.
SAMPLES OF BAL AND NON-BAL QSOS USED
The first significant sample of radio BAL QSOs was presented by Becker et al. (2000), which included 29 BAL QSOs. These were optical identifications of FIRST radio Columns are: (1) source ID; (2,3) Optical coordinates in J2000.0; (4) angular distance between the optical and FIRST radio positions; (5) optical spectroscopic redshift; (6) FIRST 1.4-GHz flux density; (7) APS E magnitude corrected from galactic extinction; (8) 5-GHz rest-frame luminosity, computed using the radio spectra presented in this paper; (9) reference where the BAL QSO appears in the literature and (10) sources in the FIRST Bright Quasar Survey (FBQS) showing BAL features in their optical spectra. Becker et al. (2000) performed radio observations of this list at two frequencies, 3.6 and 21 cm, in order to determine the radio spectral index of these sources. One of the aims of our work is to present a representative sample of radio-loud BAL QSOs with radio spectra covering a wide range in frequency (i.e. between 74 MHz and 43 GHz). The selection was based on the available samples of radio-loud BAL QSOs in the literature when the project was started. The main selection criterion was a cut in flux density of S1.4 GHz > 15 mJy in order to facilitate total power detections at high frequencies and also the detection of polarised emission from those sources with a higher degree of linear polarisation.
We have defined a sample of radio-loud BAL QSOs [RBQ sample] composed of 15 sources. Seven of them were present in the list of Becker et al. (2000) or its extension in the southern galactic cap (Becker et al. 2001). The remaining eight objects are SDSS quasars classified as BAL QSOs by visual inspection and associated to FIRST sources within a 2 ′′ radius. Two of these were presented by Menou et al. (2001), three more by Reichard et al. (2003a) and the unusual BAL QSO 1624+37 in the sample was discovered in a survey for z∼4 quasars (Holt et al. 2004). These 13 objects are the only BAL QSOs present in the mentioned samples with S1.4 GHz > 15 mJy. Finally, the other two BAL QSOs satisfying this flux density criterion were identified by one of us directly from the third data release of the SDSS (DR3). At a later stage, Trump et al. (2006) presented a catalogue with 4784 SDSS BAL QSOs selected using a more . Distribution of known radio-loud BAL QSOs in log S and z before going on-sky. The sample to be studied in this paper was selected by taking all those with S 1.4GHz > 15 mJy.
homogeneous selection criteria. They used an automated algorithm to fit the continuum and measure the AI (Absorption Index, Hall et al. 2002) in all quasars from the SDSS-DR3 quasar catalogue (Schneider et al. 2005). In fact, all SDSS quasars in the RBQ sample belong to the list of Trump et al. (2006), except 0039-00 which is not included in the SDSS-DR3 quasar catalogue, but appears in the more recent SDSS-DR5 quasar catalogue (Schneider et al. 2007). However, some care has to be taken when using the list of Trump et al. (2006). According to the more subjective classification by Ganguly et al. (2007), the absorption present in about 15 per cent of the quasars classified as BAL QSOs Becker et al. (2000) RBQ sample Vigotti et al. (1997) Figure 2. Same plot as Figure 1 in which the sample of normal radio-loud quasars from Vigotti et al. (1997) is shown, together with the samples of radio-fainter BAL QSOs.
z H G 4 . 1 ) y J m ( S g o l
in Trump's list might actually be due to intervening systems instead of having an intrinsic origin. Some examples of these false positives are shown in Figure 2 of Ganguly et al. (2007). In this work the radio spectra of BAL QSOs will be described, and in particular, the spectral index distribution will be compared with that of a non-BAL QSO sample. This is an interesting exercise because the spectral index is known to be an statistital indicator of the orientation of quasars, and this comparison can give some information about the orientation of BAL QSOs. This comparison sample of non-BAL QSOs has been extracted from the complete B3-VLA quasar catalogue (Vigotti et al. 1997). The B3-VLA survey (Vigotti et al. 1989) consists of 1049 sources selected from the B3 survey (Ficarra et al. 1985) in five flux density bins between declination 37 and 47 deg. It was designed to be complete down to flux densities of 100 mJy at 408 MHz. In addition to the original flux densities at 408 MHz the entire classical radio frequency range has been covered for most of the sources in the sample, both from already existent surveys (6C at 151 MHz, WENSS at 325 MHz, NVSS at 1400 MHz, GB6 at 4850 MHz) and from dedicated observations, mainly at higher frequencies (74 MHz, Mack et al. 2005;2.7 GHz, Klein et al. 2003;4.85 GHz, Vigotti et al. 1999 and10.5 GHz, Gregorini et al. 1998). Based on this radio survey, Vigotti et al. (1997) presented a complete sample of 125 1 quasars and this will be used as a comparison sample of radio-loud non-BAL quasars.
In Figure 2 the flux density is plotted against the redshift but now the sample of Vigotti et al. (1997) is presented together with that of Becker et al. (2000) and the one studied in this paper. The redshift range covered by the BAL QSOs goes from 0.656 to 3.124 (Becker's list) and to 3.416 (RBQ sample). The sources in Vigotti et al. (1997) list are, on average at smaller redshift (zmean=1.25), but they cover 45, 14.5, 22.5, 43.5 the redshift range between 0.096 and 2.757. Figure 2 also illustrates how the RBQ sample represent intermediate flux densities, between the flux density coverage spanned by the relatively faint sample of (Becker et al. 2000) and that covered by the list of brighter quasars from Vigotti et al. (1997).
OBSERVATIONS AND DATA REDUCTION
Radio-continuum flux densities with full polarisation information were collected for the RBQ sample at different frequencies in several runs with the Effelsberg 100-m radiotelescope and the Very Large Array (VLA). A summary of the different observing runs can be found in Table 2, and the list of frequencies used with their characteristics (e.g., angular resolutions) are displayed in Table 3. Since the sources in our sample are too faint to have a high signal-to-noise ratio per channel, only the wide-band channel centred at 2.65 GHz has been used. The receivers at 4.8 and 10.5 GHz have multi-feed capabilities with 2 and 4 horns, respectively, allowing real-time sky subtraction in every subscan measurement. The observational strategy consisted of cross-scanning the source position in azimuth and elevation. These perpendicular scans were added to gain a factor √ 2 in sensitivity since the higher resolution FIRST maps (∼ 5 ′′ ) showed that none of these sources has structure more extended than 1 arcmin, the higher resolution in our observations (see Table 3). The cross-scan length was chosen to be about four times the beam size at each frequency for a correct subtraction of linear baselines. The scanning speed ranged from 8 ′′ to 32 ′′ /s depending on the frequency, stacking typically 8 subscans at 2.65 GHz, and some 16 subscans in the rest of the frequencies (32 for the faintest sources at 10.5 GHz). This translates into on-source integration times between 15-75 sec per source and per frequency. Before combining, individual subscans were checked to remove those affected by radio frequency interference, bad weather, or detector instabilities.
Effelsberg 100-m telescope
The calibration sources 3C 286 and 3C 295 were regularly observed to correct for both time-dependent gain instabilities and elevation-dependent sensitivity of the antenna and to bring our measurements to an absolute flux density scale (Baars et al. 1977). The quasar 3C 286 was also used as a polarisation calibrator obtaining the polarisation degree m and the polarisation angle χ in agreement with the values in the literature (Tabara & Inoue 1980) within errors <1 per cent and <5 deg, respectively. The unpolarised planetary nebula NGC 7027 was also observed to have an estimation of the instrumental polarisation at each frequency.
The measurement of flux densities from the single-dish cross-scans was done by fitting Gaussians to the signal from all the polarimeter outputs (Stokes I, Q and U) and identifying the Gaussian amplitudes with the flux densities SI , SQ and SU . For all sources with significant SQ and SU contributions, the polarised flux density SP , the degree of linear polarisation m and the polarisation angle χ were computed, using the expressions given in Klein et al. (2003).
Very Large Array
VLA data were taken during February/March 2006 in the most extended A configuration using the receivers corresponding to the X, U, K and Q bands, i.e., at 8.4, 15, 22 and 43 GHz, respectively. Integration times were chosen on the basis of the predicted flux densities at these frequencies, after extrapolating from the Effelsberg frequencies with the adequate spectral indices. Those sources with indications of measurable polarised emission were observed more deeply and those expected to be too faint for this purpose were observed to have at least a good signal to noise ratio in SI . Integration times range from ∼1 minute at 8.4 GHz up to ∼1 hour at 22 or 43 GHz for some sources.
At the two highest frequencies the fast-switch method was used, quickly switching between target and calibrator in order to have phase stability, taking care of the rapid atmospheric fluctuations. The duration of the cycles targetcalibrator was of ∼ 150 to 200 seconds in the two highest frequencies.
3C 286 was the primary flux density calibrator and 3C 48 was used to test the goodness of the flux scale solution. Several phase calibrators chosen from the VLA calibrator manual 2 were observed during the run. Most were selected at about 2-5 deg from their target sources, and in all cases within 10 deg. This is especially important to avoid the loss of coherence at the highest frequencies. Again the flux density scale is the one of Baars et al. (1977) except at 22 and 43 GHz where the Baars expressions are not valid. At these frequencies the adopted scale is based on emission models and observations of planet Mars, and is supposed to be accurate at a level of 5-10 per cent.
The 31DEC06 version of the AIPS package was used to flag, calibrate, clean and image the VLA data. The standard recipe was followed in all frequencies. However, at 22 and 43 GHz opacity and gain curve corrections were applied as a previous step to the standard calibration, and the calibration table was gridded in intervals of 3 seconds to get rid of 2 Available at http://www.vla.nrao.edu/astro/calib/manual rapid atmospheric fluctuations. The appropriate clean component models were used for 3C 286 and 3C 48 at these two frequencies.
The peak flux densities were measured by fitting 2-D Gaussians to the source profiles. The integrated flux densities were computed using the BLSUM task, adding the signal in small boxes containing each source. The same boxes were used to measure the Stokes parameters SQ and SU . The local noise of the maps was estimated from a circular region around each individual source. Peak and integrated flux densities have essentially the same values because most sources are unresolved, as will be discussed below.
The polarisation calibration of the VLA data was determined using the AIPS tasks PCAL and RLDIF. The first solves for the feed parameters in each antenna, using repeated observations of a strong unresolved source (in our case several snapshots of 3C 286 were taken) covering a range of different parallactic angles. RLDIF determines and subtracts any phase difference between the right and left polarisation systems. The residual instrumental polarisation at the VLA frequencies was estimated measuring the linear polarisation of 3C 84 after having applied the polarisation calibration. Aller et al. (2003) report that this source shows a percentage of linear polarisation of 0.12 ± 0.01 at 14.5 GHz, and for our purposes it can be considered unpolarised.
Error determination
We consider three main contributions to the flux density error as in Klein et al. (2003). These are (i) the calibration error ∆S cal (in percentage) which is estimated by the dispersion of the different observations of the flux density calibrators; (ii) the error introduced by noise, ∆Sn,i, which is estimated from the local noise around the source; and (iii) the confusion error ∆S conf,i due to background sources within the beam area. This last term can be neglected in interferometric data where the synthesised beam has small dimensions.
Thus the expressions for the total uncertainty of Stokes parameters are Equations 1 and 2 for Effelsberg and the VLA, respectively:
∆Si = q (Si · ∆S cal ) 2 + ∆S 2 n,i + ∆S 2 conf,i i = I, Q, U (1) ∆Si = r (Si · ∆S cal ) 2 + S 2 n,i · Asrc A beam i = I, Q, U(2)
where Asrc is the area of the box used to measure the source flux density, and A beam is the beam area. The ex- pressions for the uncertainties of SP , m and χ can be taken from Klein et al. (2003). Table 4 shows the coefficients used for the calculation of errors ∆Si. The confusion limits at 2.7, 4.8 and 10.5 GHz have been extracted from Klein et al. (2003) and that at 8.35 GHz (not available in Klein et al. 2003) was obtained from our own observations inspecting the behaviour of the signal rms at long integration times. This value is consistent with the interpolation over frequency from the two adjacent bands (at 4.85 and 10.5 GHz). The last column in Table 4 lists the instrumental polarisation for the different frequencies and telescopes.
RESULTS
Radio flux densities
Tables 5 and 6 show the measured flux densities and errors (in mJy) between 74 MHz and 43 GHz for the RBQ sample. At frequencies below 2.6 GHz there are available flux densities from the literature and these are listed in Table 5. The flux densities at 325 MHz were obtained from the Westerbork Northern Sky Survey (WENSS; Rengelink et al. 1997). In addition, 3-σ upper limits and a detection from the Texas survey (Douglas et al. 1996) at 365 MHz are given for those sources not covered by the WENSS survey. At 74 MHz none of the sources is detected in the VLA Low-Frequency Sky Survey (VLSS; Cohen et al. 2007), but 3-σ upper limits have been computed measuring the local noise around the source position in the available images. Flux densities at 1.4 GHz from FIRST and NVSS (Condon et al. 1998) can be also found in Table 5.
The BAL QSO 1624+37 was observed in this campaign at 2.65 GHz with Effelsberg and at 15 and 43 GHz with the VLA. All the flux densities in the remaining frequencies were presented by Benn et al. (2005).
Morphologies
The FIRST maps at 1.4 GHz show a compact morphology for all 15 BAL QSOs in the RBQ sample, at a resolution of ∼5 ′′ . To quantify this we have computed the morphological parameter Θ, based on the FIRST integrated and peak flux densities, Θ = log(Sint/S peak ), and for all 15 sources Θ <0.03. All sources, with the exception of 1053−00, appear essentially point-like at this resolution. The FIRST map of 1053−00 is shown in Figure 3 where a point-like core and two faint components (5-and 8-σ detections) can be seen. The faint components seem to be real and this source has been included in the catalogue of double-lobed radio quasars from SDSS (de . This is thus an additional example of the rare class of BAL QSOs showing double-lobed morphology, and it is not included in the list of 8 SDSS BAL QSOs of this type compiled by Gregg et al. (2006). Our VLA maps at higher frequencies confirm the compactness of the 15 BAL QSOs, having all of them point-like structure at 8.4 and 15.0 GHz. The faint components of 1053−00 cannot be seen at these frequencies, probably due to a steep spectral index in these two regions.
The combination of the 43-GHz band with the A configuration provides the highest resolution available with the VLA (θHP BW ∼ 50 mas). However, as can be seen in Table 6 most sources are very faint at 43 GHz, i.e., close to the 1 mJy level. For those the self-calibration process is not appropriate and the way to recover the flux densities was to assume an initial point-source model before the first mapping. This procedure is to some extent well justified since at lower frequencies the morphology is still point-like, although the precise astrometric and morphological information about the sources at 43 GHz is lost.
Self-calibration at high frequency was only possible for the brightest sources: 0957+23, 1159+01 and 1312+23. The first two sources appear point-like in the 43-GHz maps. Figure 4 shows the naturally weighted maps of 1312+23 at 43 and 22 GHz, revealing an extension towards the north coming from the central unresolved component. Although the feature, possibly a jet, is detected at both frequencies, its significance is higher at 43 GHz.
The small feature in the 43-GHz map at ∼120 mas south from the central source is probably not real because it is located in a region affected by a stripe. It must be noticed that at this high frequency this flux density level is probably close to the limit where signal coherence allows accurate mapping. For the remaining 12 sources the mapping at 43 GHz was not Table 6. RBQ sample of 15 radio-loud BAL QSOs: Flux densities (in mJy) from 2.6 to 43 GHz obtained in this work. Superscripts on the errors indicate the run number of that observation, according to the key in Table 2. The exception is 1624+37 (with superscript 0), for which several flux densities were presented in Benn et al. (2005)
ID S 2.6 S 4.8 S 8.3 S 8.4 S 10.5 S 15 S 22 S 43 0039−00 − 19.3 ± 0.6 1 12.0 ± 0.4 1 12.3 ± 0.1 5 9.7 ± 0.4 1 6.2 ± 1.1 5 2.2 ± 1.2 5 1.1 ± 0.1 5 0135−02 − 27.2 ± 0.6 1 31.9 ± 0.8 1 30.8 ± 0.4 5 29.5 ± 0.7 3 22.7 ± 0.8 5 14.7 ± 0.2 5 2.3 ± 0.2 5 0256−01 − 12.0 ± 0.5 1 7.2 ± 0.3 1 7.2 ± 0.4 5 5.9 ± 0.4 1 4.3 ± 1.5 5 2.2 ± 0.3 5 1.4 ± 0.1 5 0728+40 6.7 ± 1.5 4 6.4 ± 0.7 1 3.1 ± 0.2 1 2.5 ± 0.3 5 1.6 ± 0.3 1 1.9 ± 0.3 5 0.83 ± 0.06 5 − 0837+36 50.4 ± 1.6 4 30.2 ± 2.3 2 12.1 ± 0.8 1 12.5 ± 0.1 6 13.0 ± 0.4 2 3.3 ± 1.3 6 2.2 ± 0.5 6 1.3 ± 0.2 6 0957+23
93.7 ± 2.4 4 78.7 ± 1.1 1 48.5 ± 1.2 1 53.2 ± 0.5 5 39.8 ± 1.2 1 31.4 ± 0.7 5 22.4 ± 0.3 5 9.8 ± 0.9 5 1053−00 13.6 ± 1.6 4 15.7 ± 0.6 1 11.8 ± 0.4 1 13.1 ± 0.1 5 12.3 ± 0.6 1 8.2 ± 1.6 5 5.9 ± 0.7 5 2.3 ± 0.7 5 1159+01
133.9 ± 2.0 4 137.8 ± 1.7 1 158.0 ± 2.0 2 160.8 ± 1.2 5 150.6 ± 3.7 2 123.8 ± 1.6 5 105.5 ± 1.1 5 74.6 ± 2.1 5 1213+01
23.0 ± 1.7 4 15.2 ± 0.5 1 9.7 ± 0.3 1 11.6 ± 0.1 5 8.2 ± 0.5 1 5.6 ± 1.4 5 3.6 ± 0.5 5 1.4 ± 0.2 5 1228−01
19.4 ± 1.6 4 18.6 ± 0.5 1 15.5 ± 0.4 1 16.9 ± 0.2 5 13.5 ± 0.7 1 11.5 ± 1.6 5 8.1 ± 0.2 5 2.5 ± 0.7 5 1312+23
28.9 ± 1.6 4 25.7 ± 0.6 1 19.
8 ± 0.5 1 19.4 ± 0.2 5 16.1 ± 0.6 1 10.5 ± 0.3 5 7.7 ± 0.1 6 5.4 ± 0.4 6 1413+42 7.6 ± 2.9 4 8.8 ± 0.7 2 13.4 ± 0.3 2 13.4 ± 0.1 6 12.7 ± 0.4 2 9.5 ± 1.5 6 7.0 ± 0.2 6 2.2 ± 0.4 6 1603+30 22.8 ± 1.7 4 26.1 ± 0.7 1 19.1 ± 0.5 1 22.1 ± 0.3 5 16.8 ± 0.6 1 12.5 ± 1.7 5 8.3 ± 0.1 5 1.9 ± 0.5 5 1624+37 33.5 ± 1.6 4 23.3 ± 1.1 0 − 15.0 ± 0.1 0 10.5 ± 0.8 0 9.6 ± 0.6 5 5.41 ± 0.02 0 2.1 ± 0.3 5 1625+48
17.6 ± 1.5 4 9.4 ± 0.7 2 8.0 ± 0.2 2 7.0 ± 0.2 5 6.3 ± 0.2 2 5.8 ± 1.3 5 1.9 ± 0.3 5 1.7 ± 0. possible and an upper limit on their sizes was obtained using the observations at 22 GHz (θHP BW ∼ 80 mas), where they are bright enough for mapping. For those, the task IMFIT was used to fit an elliptical Gaussian component and a zero level within a small box containing the radio source, yielding the major and minor axes of the fitted Gaussian, position angle and the formal beam-deconvolved dimensions. The uncertainty in the dimensions was estimated following Equation 1 of White et al. (1997). Eight of the twelve sources are clearly more extended than the beam size and their fitted parameters are given in Table 7. Also in this table are shown the dimensions of 1312+23 and 1624+37, as measured from the map in Figure 4 and from VLBA observations to be published in a subsequent paper, respectively. The remaining ± 20 † Not deconvolved size, but measured from the extended structure in Figure 4. † † Measured from VLBA map at 4.8 GHz to be presented in Montenegro-Montes et al. (in prep.) three sources (0039−00, 0256−01, 1053−00) plus 0957+23 and 1159+01 can be considered strictly unresolved at high frequencies.
VLBI observations of 0957+23 and 1312+23 have been presented by Jiang & Wang (2003) using the European VLBI Network at 1.6 GHz. With a restoring beam of only 18.5 × 6.86 mas, 0957+23 appears as a single point-like component, while 1312+23 is resolved. This last source shows a symmetric morphology with a central core and two components in the northern and southern directions with a total extension of ∼150 mas (i.e., about 1 kpc). Although an extended morphology is also present in our 22-GHz and 43-GHz maps (Fig. 4), only the northern part is detected.
Summarising, our VLA maps show very compact morphologies for all 15 sources at all frequencies, at a maximum resolution of 80 mas. The exceptions are 1312+23 which shows some elongation at 22 and 43 GHz, and 1053−00 which is extended at 1.4 GHz showing two faint lobes not detected at higher frequencies.
Variability
We have checked for variability at 1.4 GHz comparing the flux densities from FIRST and NVSS epochs in those 14 sources with NVSS measurement. The same has been done at 8.4 GHz for those 5 sources in common with the list of Becker et al. (2000). They observed these sources with the VLA in A and D configuration (resolutions of 0.8 and 9 arcsec respectively) and all 5 sources presented point-like structure at both resolutions. As mentioned before this is again confirmed by our observations at 8.4 GHz in A configuration. This means that flux densities of both measurements can be directly compared. The same is true at 1.4 GHz because all sources except 1053−00 are point-like at the resolution of FIRST (B configuration) and the resolu- tion of NVSS is poorer (VLA D and CnB configurations).
Although 1053−00 is resolved, it has a compact nucleus and the two extensions are not expected to contribute much to the total flux density (Section 4.2). We measure the flux density variations with the parameter V ar∆S, as defined e.g. by Torniainen et al. (2005):
V ar∆S = Smax − Smin Smin(3)
As an estimate of the significance of the source variability the σvar parameter defined by Zhou et al. (2006) is used, adopting the integrated flux densities S1 and S2 at the two epochs:
σV ar = |S2 − S1| p σ 2 2 + σ 2 1(4)
where σ2 and σ1 are the uncertainties in the integrated flux densities. Table 8 shows flux densities, V ar∆S and σV ar at 1.4 and 8.4 GHz, for those sources exhibiting σV ar > 3. For the flux densities of Becker et al. (2000), whose errors are not given in their work, a 5 per cent error is assumed. A low percentage of the sample (only 2 out of 14 sources) shows variability at 1.4 GHz, both sources varying about 20 per cent and with a significance just above 3 standard deviations. At 8.4 GHz, 3 out of 5 sources are variable at this level and 1312+23 constitutes the most extreme case, showing variations of about 50 per cent with significance above 10.
Flux density variations have recently been used to constrain the orientation of several BAL QSOs. Zhou et al. (2006) and later Ghosh & Punsly (2007) found a few examples of BAL QSOs with variability as high as 40 per cent at 1.4 GHz between the NVSS and FIRST, i.e., in periods of about 1-5 years. These variations were high enough to make the brightness temperature TB exceed the 10 12 K limit associated to the Compton catastrophe. This was interpreted as a sign of beaming and the jet was supposed to be oriented at a few degrees from the line of sight.
The same procedure can be applied to 0256−01 and 1213+01, accepting the 3-σ significance as real. Following the formalism in Ghosh & Punsly (2007) and knowing that the time differences between the NVSS and FIRST observations are 2.1 and 3.4 years, respectively, the observed variations imply brightness temperatures of T b =3.1·10 13 K for 0256−01 and 1.3·10 13 K for 1213+01. These translate into upper limits for the jet orientation angle with respect to the line of sight of ∼20 and 25 deg, respectively. One concern about this approach is that it is based on variability at a single frequency and from only two measurements. Moreover, the flux densities we obtained for 0256-01 at 8.4 GHz, with Effelsberg and almost one year later with the VLA, show a good match, i.e., 7.2±0.3 and 7.2±0.4 mJy respectively, in contradiction with the expected stronger variability at high frequencies.
4.4 Shape of the radio spectra 4.4.1 Sample of radio BAL QSOs Figure 5 show the spectra of the 15 objects. Many of them have a relatively steep spectrum at high frequency. About 75 per cent of the sample show flattening of the spectrum at low frequencies (typically below 1-5 GHz) which could indicate synchrotron self-absorption. However, in some cases some variability cannot be excluded, since the 1.4-GHz measurements were taken some years before the others. About 1/3 of the sample show, in addition, enhanced emission at MHz frequencies which could be indications of a second component emitting mostly in this range. Again, variability cannot be ruled out, but any possible time variability is expected to be less pronounced at lower frequencies. The source 1159+01 is the one which best fits in this category with both, a depression of the spectrum towards low frequency and a contribution from a second component in its spectrum.
Following the approach by Dallacasa et al. (2000), we have fitted each spectrum with the analytical function:
logSν = a − p b 2 + (c log(ν) − d) 2(5)
where a,b,c,d are numerical constants. This function, presented in Dallacasa et al. (2000) was used by these authors to obtain the frequency turnover, ν peak , of sources with convex spectra peaking above a few GHz, referred to in that paper as High Frequency Peakers. The value of ν peak approximately marks the frequency region where the source starts to be optically thick to synchrotron radiation. These fits have been done excluding some points in order to obtain a reduced χ 2 below 0.05. This was possible for most of the sources which present a smooth radio spectrum, specially at high frequency, but not for 0728+40 (χ 2 =0.08) which presents a large scatter. At low frequencies, as we noted before, there are sources for which a possible second component arises. Again in these cases we decided whether to include or not in the fit these (low frequency) points trying to maintain the reduced χ 2 within the mentioned limit. An exception was done with 0957+23 because even if an acceptable χ 2 was obtained when including the 1.4 GHz flux densities, the resulting fit was not compatible with the 365 MHz upper limit.
In our sample we find 9 sources for which the spectrum peaks at a frequency higher than 1 GHz in the observer's frame and there are 4 displaying a peak in the range 300-700 MHz. For the remaining three objects (0256−01, 0957+23 and 1312+23) the spectrum is slightly convex but not showing an obvious peak. However, power-law fits would not be compatible with the Texas upper limits and then extrapolated peaks compatible with both the spectrum curvature and these upper limits have been determined. The peak frequencies have to be multiplied by (1+z) to obtain intrinsic peak-frequencies.
In order to understand the global trends on the spectral shape of BAL QSOs, we compare in Figure 6 all the spec-tra bringing them first to the rest-frame in order to avoid the effect of the redshift. The rest-frame normalisation frequency was chosen to be 25 GHz. This is a compromise to choose a region representative of pure synchrotron emission, where neither synchrotron self-absorption nor synchrotron losses might be present in the majority of sources. This frequency also makes the dispersion of the normalised spectra relatively small compared to other normalisation frequencies.
We see in Figure 6 a variety of spectral slopes around 25 GHz. There is one object in the sample which differ more strongly from the main trend, This is 1413+42 which has a spectrum with two components separated at 4 GHz (see Figure 5), and for this reason it shows in Figure 6 an inverted spectrum around the normalisation frequency. Apart from this object, and assuming that variability effects are not very important, Figure 6 reflects the rest-frame spectrum of a typical BAL QSO in the RBQ sample. It is in general convex, quite flat below 10 GHz, and becoming steeper around 25 GHz. About half of the objects become even steeper at frequencies higher than 50 GHz.
The spectral shape can be quantified by means of the spectral indices. Table 9 provides the turnover frequencies and representative spectral indices at different frequencies, both in the observer's and in the rest-frame. The spectral indices α 1.4 0.365 , α 15 4.8 and α 43 8.4 describe the low-, medium-and high-frequency regions. Moreover, the index α 8.4
1.4 is shown for comparison with Becker et al. (2000). It is worth noting that the spectral indices α 8.4
1.4 shown in Table 9 are not based on simultaneous observations, but as we previously have shown, variability might not be very important at this frequency, and in any case it only affects a small number of sources. In the rest-frame the indices α 25 6.0 , α 25 12 , α 50 25 and α 100 50 are also given in Table 9.
Looking at the rest-frame spectral indices of BAL QSOs and Figure 6, we see on average a flattening below 10 GHz, which is probably due to synchrotron self-absorption, and an overall convex shape. The low-frequency flattening is reflected in the median α 25 6 =−0.32, while the median α 25 12 decreases to −0.57. The steepening at high frequency can be seen from the median α 50 25 dropping to −0.92 and finally becoming steeper at higher frequencies with a median α 100 50 =−1.24.
The radio spectral index is known to be a statistical indicator of the orientation of radio sources (Orr & Browne 1982). Becker et al. (2000) found respectively two thirds and one third of the radio-loud BAL QSOs in their list with steep and flat spectral index. However, it is worth to note that they define the limit between flat and steep to be α=−0.5, and about one third of the sample has spectal index close to this value −0.6 α −0.4. They suggest that this result is not consistent with BAL QSOs being oriented along a particular direction with respect to the line of sight to them. We find in our sample, which overlaps with that of Becker et al. (2000) in 5 objects, 60 per cent (9/15) of flat and 40 per cent of steep (6/15) spectrum sources, with 27 per cent of objects (4/15) being in the range −0.6 α −0.4. These percentages are roughly consistent within the errors, which are large because the samples are relatively small. 9. Turnover frequencies (in GHz) and spectral indices for the RBQ sample in the observer's frame (columns 2-6) and the rest frame (columns 7-11). The spectral indices in the observer's frame have been computed from each pair of observed flux densities while the rest-frame spectral indices are based on interpolation over the whole spectrum. Median values of the spectral indices are given in the last row.
Comparison with non-BAL QSOs
To further investigate the question of the orientation of BAL QSOs we want to compare in this paper the spectral index distribution of quasars in a pure BAL QSO sample with that of a sample of non-BAL QSOs with the idea of interpreting possible differences with different orientations of both samples. In this comparison we will use the spectral range 1.4-8.4 GHz in the observer's frame and 6.0-25 GHz in the rest frame. The list of non-BAL QSOs will be that of Vigotti et al. (1997) presented in Section 2, but in all the following discussion we will restrict it to objects with z>0.5. This restriction only reduces the total number of objects from 124 to 119. We have computed the spectral indices in the same way as in our sample, considering pairs of observed flux densities in the observer's frame and interpolating over the whole spectra in the rest frame. The spectral range covered by the non-BAL QSO sample goes from 74 MHz up to 10.5 GHz, and only those sources with z>1.5 will reach 25 GHz in the rest frame.
As a BAL QSO sample we will add to our RBQ sample the list of (Becker et al. 2000). They only provide flux densities in two frequencies, i.e., 1.4 and 8.4 GHz. Thus, combining these we end up with a combined sample of 38 BAL QSOs. These spectral indices can be found in Table 9 for objects in the RBQ sample, and in Table 2 of (Becker et al. 2000) for the remaining objects.
The Vigotti et al. sample could in principle contain both BAL QSOs and non-BAL QSOs. However, a low contamination of BAL QSOs is expected. Becker et al. (2001) showed that in the radio-powerful regime the fraction of BAL QSOs is small. They used the definition of radio-loudness parameter R * given by Stocke et al. (1992), i.e., the ratio of radio power at 5 GHz to optical luminosity at 2500Å in the rest frame. In the regime of QSOs with log(R * ) >2 they estimated a fraction of 4 per cent (<1.5 per cent) of HiBAL We have calculated log(R * ) for all quasars in Vigotti's list. For this purpose, the radio luminosity at 5 GHz has been computed on the basis of the observed radio spectra, applying the appropriate K-correction interpolating between each pair of observed frequencies with the adequate spectral index. The optical luminosities at 2500Å were computed using the POSS-I APM E magnitudes published in Vigotti et al. (1997), but corrected from Galactic extinction following the maps of Schlegel et al. (1998) and assuming an optical spectral index of αopt = −1 (Sν ∝ ν α ), which is the value adopted by Becker et al. (2001). Figure 7. Histogram of radio-loudness parameter R * as defined by Stocke et al. (1992) for the sample of Vigotti et al. (1997).
In Figure 7 the histogram shows how log(R * ) ranges from 1.9 to 4.6. To estimate the fraction of BAL QSOs following the mentioned proportions, we split this sample in two groups, one with 0.5<z 1.5 and the other with z>1.5, comprising 80 and 44 quasars respectively. About 1-2 Hi-BAL QSOs are expected in the high-z subsample and about 1 BAL QSO in the low-z subsample.
We made use of the NASA Extragalactic Database (NED) and the SDSS-DR6 database to look for the optical spectra of QSOs from Vigotti et al. (1997) and check whether they are BAL QSOs. From 119 objects with z<0.5, 55 were found in the SDSS-DR6 database. These were inspected visually to look for the presence of BAL features. All of them are non-BAL QSOs and only one of them, (SDSS J080016.09+402955.6, associated to the radio source B3 0756+406) shows some broad absorption component bluewards of the C iv emission line, also present in Si iv. However, this absorption has a width of ∼ 1000 km s −1 and it is located within 3000 km s −1 from the line emission. It is therefore not consistent with the definition of BAL QSO but it could be better classified as a mini-BAL or a Narrow Absorption Line (NAL) system (see Figure 8).
We searched for the remaining 64 spectra in the NED and found 14 of them in different publications, most of them in Vigotti et al. (1997) and Lahulla et al. (1991). From these 14, only 4 could be identified as normal QSOs without BALs: B3 0034+393 with z= 1.94 (Osmer et al. 1994); B3 0219+443 with z=0.85 (Djorgovski et al. 1990); B3 0249+383 with z=1.12 (Henstock et al. 1997) and B3 2351+456 with z=2.00 (Stickel & Kühr 1993). The other 10 are difficult to classify because they are either faint, with low signal-to-noise ratio in the continuum (just good enough for a correct redshift identification), or the wavelength coverage does not include the region where the absorbing troughs could be present. Summing up, 59 out of 119 QSOs (roughly half of the sample) were inspected, with no one clearly showing the BAL phenomenon. These results justify the assumption that the list of (Vigotti et al. 1997) can be used as a representative sample of non-BAL quasars. Table 10 summarises the number of elements and the basic statistics on the spectral index for the samples un- der comparison. Since BAL QSOs are compact objects as Becker et al. (2000) discovered and we confirm in this work, we will also consider in our comparison a subsample of non-BAL QSOs extracted from Vigotti et al. (1997) which includes only those compact sources with angular sizes <0.5 arcsec at 1.4 GHz. The angular sizes at 1.4 GHz in Vigotti's sample were measured from maps taken with the VLA in C and D configurations (Vigotti et al. 1989) and for the most compact sources from observations in A configuration (priv. communication).
The first four rows in Table 10 show that non-BAL QSOs have a median α 8.4
1.4 lower than BAL QSOs (−0.92 and −0.53, respectively). The conclusion could be that BAL QSOs tend to have flatter spectra and are, on average, oriented at a lower angle with respect to the line of sight than normal QSOs. On the other hand, when the non-BAL QSOs sample is restricted to compact objects only, the median spectral index becomes −0.49, very similar to that of BAL QSOs. When the full sample of non-BAL QSOs is considered, those extended objects have a higher contribution of emission produced in the lobes, which translates into steeper spectral indices for non-BAL QSOs. A similar behaviour can be found when analysing the rest-frame spectral indices α 25 12 and also α 25 6 although in this last case the median values for BAL QSOs and compact non-BAL QSOs are more distant, i.e. −0.32 and −0.47 respectively.
A Kolmogorov-Smirnov (K-S) test has been done in order to compare the α 8.4
1.4 spectral index distributions of these samples. The test yields a probability P<0.001 that both distributions are statistically equal, confirming that they are indeed different populations. A similar K-S test restricted to only compact sources from Vigotti et al. (1997) yields P=0.204 which is consistent with both samples having a similar distribution. The percentile plot in the upper panel of Figure 9 illustrates the shape of the spectral index distribution of these three samples. It can be seen that the important contribution of extended sources from Vigotti et al. (1997) makes the percentile curve rise more rapidly than in the other samples. This result essentially means that the differences in spectral indices between BAL and non-BAL Table 10. Statistics on spectral indices. B00: Becker et al. (2000); V97: Vigotti et al. (1997); V97c: sources from Vigotti et al. (1997) QSOs arise only because BAL QSOs are compact sources, while normal QSOs include both compact and extended. We will also compare the rest-frame spectral indices indices to look for possible intrinsic variations between BAL and non-BAL QSOs, once the redshift effect has been corrected for. The most reasonable comparison is using α 25 12 where the effect of synchrotron self-absorption seem less pronounced. Two K-S tests have been done comparing these samples. The first compares Vigotti's sources and RBQ yielding P=0.048, while the second compares only the compact sources in Vigotti et al. (1997) and RBQ, yielding P=0.629. Again we can talk about statistically different populations when all QSOs are included in the comparison, but not when only spectral indices of compact sources are compared. The percentile plot is shown in the lower panel of Figure 9.
Even if the previous statistical test are consistent with similar spectral index distributions for BAL QSOs and non-BAL compact sources, it is interesting to note in Figure 9 that there might be a small deficit of steep sources among BAL QSOs with spectral index ∼ −0.7, compared to the compact sources in Vigotti et al. (1997). This effect is more marked in the upper panel, but also insinuated in the lower panel where, nevertheless, the number of objects is significantly smaller.
Polarisation
The RBQ sample was searched in the NVSS database looking for polarised emission and only one BAL QSO, 1159+01, shows a high amount of linear polarisation at 1.4 GHz. For the rest of the sources only upper limits could be established typically below 1.5 per cent at this frequency.
At higher frequencies and up to 43 GHz we have measured the SQ and SU parameters as explained in Section 3, in order to obtain the degree of linear polarisation, m, and the polarisation angle χ. Table 11 shows the fractional polarised intensity, mostly upper limits, for the BAL QSOs in the RBQ sample. The only sources which are clearly polarised showing significant m at various frequencies are 1159+01 and 1624+37. These sources are presented in different tables (Table 12 in this paper and Table 2 from Benn et al. 2005). These two are the only bright sources displaying relatively high (i.e., >10 percent) polarised intensity at some frequency, having 1159+01 m1.4=15 per cent and 1624+37 with m10.5=11 per cent. Apart from these, also the weak source 1625+48 with only 1.7 mJy at 43 GHz shows m43 ∼18 per cent but this measurement is based on a 3-σ detection in SU and the uncertainty in this measurement is high. From the remaining sources in Table 11, 1312+23 shows significant polarisation at 8.4 and 22 GHz, and there are 5 more sources with significant detections at only one frequency, in all cases below 4 per cent.
It should be noted from Table 11 that the most stringent upper limits are given by the VLA observations at 8.4 GHz. At this frequency the sources are bright enough to have reliable detections and the relatively low noise achieved makes possible the determination of reasonable upper limits for the non-detections. At higher frequencies this is more difficult because the total power spectra drops down to fainter levels.
At 8.4 GHz, one third of the sample (i.e, 5 out of 15) show at least a 3-σ detection in SQ, SU or both. From these Table 11. Degree of polarisation, mν (in percentage), at several frequencies, ν (in GHz), for the RBQ sample. Most of the values are 3-σ upper limits and those showing a measured value can have a relatively large error but come from an at least 3-sigma detection in Stokes Q or Stokes U. The BAL QSOs 1159+01 and 1624+37 presented separately (see Table 12 in this paper and Table 2 from Benn et al. 2005) are the only two sources with significantly detected polarisation. (Condon et al. 1998) 5 BAL QSOs only 1624+37 is strongly polarised showing 6.5 per cent of linearly polarised flux density (Benn et al. 2005).
The fractional polarised intensity in the other four objects (1053−00, 1159+01, 1312+23 and 1603+30) is below 3 per cent. The upper limits for most of the 10 undetected sources are below 1 per cent indicating strong depolarisation. For 1159+01 and 1624+37 it is possible the determination of the Rotation Measure, RM, (χ = χ0 + RM · λ 2 ) fitting a slope to the polarisation angles χi as a function of the square of the observed wavelenghts. Benn et al. (2005) already reported for 1624+37 a Rotation Measure of −990 ± 30 rad m −2 , which in the rest-frame translates into the extremely high intrinsic Rotation Measure, RM = -18350 ± 570 rad m −2 . For 1159+01 we have fitted the angles χi in Table 12. Our observations give a poor upper-limit at 2.65 GHz, but an additional measurement at this frequency has been obtained by Simard-Normandin et al. (1981). We get a best fit with a Rotation Measure RM=(−72.1 ± 1.4) rad m −2 and an intrinsic polarisation angle χ0 = −24 ± 3 deg. This observed value should be corrected by the RM introduced by our own Galaxy, which is quite difficult to estimate. To have a rough estimate we have inspected the allsky map presented by Wielebinski & Krause (1993) where the RM of 976 extragalactic sources are plotted. The neighbouring sources around the position of 1159+01 have small (i.e., |RM| <30 rad m −2 ) Rotation Measures, which suggests that the Galactic contribution might be smaller than the determined value. Assuming no Galactic correction, the fitted Rotation Measure brought to the rest-frame would be higher by a factor (1+z) 2 becoming RM=(644 ± 12) rad m −2 .
DISCUSSION
The morphologies and dimensions of the radio sources in BAL QSOs seem to be the keys to understand the orientation and evolutionary status of these sources. There is evidence that BAL QSOs are associated with compact radio sources, which are supposed to constitute a high fraction of the young population of radio sources, like e.g., CSS or GPS sources. Becker et al. (2000) noted that 90 per cent of their sample of radio-loud BAL QSOs (extracted from the FBQS survey) present point-like structure at the resolution of 5 ′′ in contrast to the diversity of both point-like and extended sources they found within the whole population of FBQS quasars. In fact, only a few BAL QSOs are known to have an extended FR II radio structure (Gregg et al. 2006). Our 22-GHz observations of the RBQ sample confirm this tendency, because most sources appear unresolved at all frequencies constraining their apparent dimensions up to 0.1 arcsec. This translates into projected linear sizes (LS) 1 kpc which are typical of GPS/CSS sources. It could be argued that an extended component with a steep spectrum might be present, observable only at lower frequencies, and thus being the source sizes larger than those found at high frequencies. This hypothesis is a priori no supported by the fact that most sources are still point-like at 1.4 GHz as can be seen in the FIRST maps, with the exception of 1053−00.
The radio spectra in Figure 5 also display the typical convex shape of GPS-like sources, in most cases with a peak at ∼ 0.5−10 GHz suggested by at least one or two points, or by upper limits at MHz frequencies. One concern about this result is that FIRST or WENSS flux densities were observed at different epochs and variability might be an issue, although noticeable flux density variations are likely to be stronger in the optically thin region of the spectrum. However, relatively deep integrations at MHz frequencies would be desirable to better constrain the shape of the spectra at low frequency. In addition, a project with VLBI multi-frequency observations and the analysis on the pcscale properties of several BAL QSOs of the RBQ sample has been started by us, and results will be presented in a subsequent paper.
If radio sources associated to BAL QSOs are actually CSS/GPS sources, they should follow the anticorrelation between intrinsic turnover frequency and projected linear size found by Fanti et al. (1990) for CSS sources. This was, for instance, quantified by O' Dea & Baum (1997) in a combined sample of CSS and GPS sources compiled by Fanti et al. (1990) and Stanghellini (1992), respectively. This relationship was found to be valid for both galaxies and quasars. The anticorrelation suggests that the mechanism producing the turnover simply depends on the source size. Figure 11 shows the relationship between intrinsic ν peak and projected LS for the samples of CSS and CPS sources mentioned before, and we have also plotted for comparison our sample of radio-loud BAL QSOs. The turnover frequencies for the BAL QSOs appear in Table 9 while projected linear sizes have been extracted from the deconvolved sizes in Table 7.
A group of 5 BAL QSOs match quite well the distribution of CSS/GPS sources, and they all show simple convex spectra, i.e., 0728+40, 1213+01, 1312+23, 1624+37 and 1625+48. The three additional BAL QSOs displaying simple convex spectrum, 0039−00, 0256−01 and 0957+23 are unresolved in our observations and their position can only be constrained by upper limits. Nevertheless these 8 sources with simple complex spectrum are consistent with the correlation.
From the remaining 7 objects, 5 are resolved in our observations (0135−02, 0837+36, 1228−01, 1413+42 and 1603+30) and they are located above the cloud of CSS/GPS sources. All of them have in common a complex spectrum with indications of a double component. In these sources the turnover has been determined using the high frequency component but it is possible in the spectra of 1413+42 and 1603+30 to fit a second peak at lower frequencies (at 0.7 and 0.8 GHz respectively). A second peak at low frequency could also be fitted to 1159+01 but we will not do that because being this source unresolved we cannot give the exact location in the diagram. When considering the low frequency peaks of 1413+42 and 1603+30 in the diagram, these two sources also agree very well with the anticorrelation as shown in Figure 11. This better agreement could mean that the high-frequency part of these spectra could be dominated by the emission coming from a compact and active (possibly beamed) region smaller than the entire source, like a hotspot. Unfortunately we do not have enough data to check whether this could be the case also for 0135-02, 0837+36 and 1228-01. Low-frequency data will be important to sample this part of the spectra to search for a possible second peak. Again, VLBI observations are crucial to further investigate this interpretation.
In fact, it has been shown that some GPS sources associated to optical QSOs can be just normal flat-spectrum quasars for which a jet/knot/hot-spot dominates the spectrum, which then adopts the characteristic convex shape of a GPS source, as discussed e.g. by Snellen et al. (1999). Recently Stanghellini et al. (2005) and Orienti et al. (2006) have studied VLBI samples of GPS sources and High Frequency Peakers (HFP, 5 GHz < ν pk < 10 GHz) confirming this scenario. In fact, these "contaminant" quasars seem to be present, even in high percentages, in many well-defined samples of GPS sources (Torniainen et al. 2005). Two characteristics of "genuine" young GPS radio sources are low polarisation and low variability (O'Dea 1998), while flatspectrum quasars show higher polarisation and variability. Torniainen et al. (2005), who have used an extensive database of sources with multifrequency data from several long-term variability programs, define as "bona fide" GPS sources those with a maximum variability V ar∆S <3. From our comparison in Section 4 which is, of course, based on only two epochs we see that many sources do not show significant variations at 1.4 GHz, and 3 out of 5 inspected at 8.4 GHz show significant but not very strong variations. We cannot thus exclude these sources from the candidates to young sources. As also shown in Torniainen et al. (2005) the maximum variations are found at high frequencies, so multifrequency multi-epoch observations of the RBQ sample will be very interesting for a more precise variability study.
Compact GPS sources are supposed to be completely embedded in their narrow line region. On this basis, (Cotton et al. 2003) proposed the existence of a typical frequency-dependent scale, below which the GPS source becomes completely depolarised, the so-called "Cotton effect". Fanti et al. (2004) and later Rossetti et al. (2008) have confirmed this behaviour analysing the polarisation properties of the B3-VLA CSS sample (Fanti et al. 2001), and they propose models to describe the depolarising screen. The BAL QSOs in the RBQ sample have probably physical sizes of 1 kpc and if Faraday effects are important they should be completely depolarised even up to relatively high frequencies. This is the case for all sources except 1159+01 at 1. GHz. At these two frequencies the upper limits obtained for the degree of linear polarisation are reasonably small. From Table 11 only 1312+23 seems to be weakly polarised at 8.4 and 22 GHz, but no polarisation was found at 15 GHz.
The two exceptions are 1159+01 and 1624+37. The first source is weakly polarised at 8.4 GHz but it is worth to note that the degree of linear polarisation increases as frequency decreases, becoming strongly polarised at 1.4 GHz. The opposite is expected when a depolarising plasma curtain is present. The reason for this anomalous behaviour might be that the emission at low frequencies arises from a different component, as suggested by the complex radio spectrum (see Figure 5). An hypothetical extended structure with a relatively ordered magnetic field would contribute with substantial linear polarisation, and this may only be detected at low frequencies if a steep spectral index is assumed for the emission of this extended component.
The moderate observed Rotation Measure of (−72.1 ± 1.4) rad m −2 in 1159+01 is consistent with the fact that it is strongly polarised in the NVSS. Should the Rotation Measure be much higher, the large bandwidth used for the NVSS observations would produce a strong bandwidth depolarisation effect, probably hiding any measurable polarised intensity. The extreme Rotation Measure of J1624+37, RM=−18350 rad m −2 (Benn et al. 2005) which is the second highest value known among QSOs and the highest among BAL QSOs, seems to be an exception and not representative of the BAL QSOs family.
The median m8.4 of the 5 polarised BAL QSOs in the RBQ sample is ∼1.3 per cent, but the remaining 10 BAL QSOs show m8.4 1. As a comparison Saikia et al. (1987) found a median polarisation degree at 6 GHz of about 2 per cent in a subsample of quasars extracted from the sample of ∼400 compact sources of Perley (1982). Saikia et al. (1987) found no significant differences between the median value of flat-spectrum cores and CSS quasars whereas that radio sources associated to galaxies or to empty fields (no optical identifications) were found to be less polarised at 6 GHz, with a median value of ∼0.5 per cent. More recently, Stanghellini (2003) found a mean fractional polarisation of 1.2 and 1.8 per cent for GPS and flat-spectrum quasars, respectively, and m < 0.3 per cent for galaxies.
The sources showing some polarisation in the RBQ sample have probably polarised intensities consistent with flatspectrum quasars. It is also likely that some of them (e.g., 1159+01 or 1603+30) can show polarised emission due to Doppler boosted knots in jets, which is again compatible with their multi-component radio spectra peaking at high frequencies. However, our whole sample seem to be in better agreement with the GPS class than with the class of flat-spectrum quasars, because of the high percentage of unpolarised or weakly polarised sources. Of course, to firmly test this hypothesis it will be particularly interesting to do polarimetric multi-frequency observations of a radio brighter sample, which is now available after the recent releases of SDSS.
As far as the spectral index distribution is concerned, the different statistical tests suggest that the spectral index distributions of BAL and non-BAL QSOs are different. This difference is however due to the fact that all BAL QSOs in the sample have compact structures, while non-BAL QSOs have both extended and compact morphologies. When comparing the samples of BAL QSOs and compact non-BAL QSOs, no significant differences are found in the spectral index distributions. Thus, from our comparison no preferred orientation can be attributed to BAL QSOs. When larger samples become available, the comparison of spectral indices of BAL and non-BAL QSOs as a function of radio power and UV luminosity will be possible, allowing us to look for particular orientations in different bins of luminosity, as has been suggested by Elvis (2000).
CONCLUSIONS
• A sample of 15 BAL QSOs associated with FIRST sources brighter than 15 mJy has been built. Radio continuum observations for these sources have been collected from 2.6 up to 43 GHz in full polarisation. Flux densities covering this radio range have been presented complemented by archive data at lower frequencies from 74 MHz up to 1.4 GHz.
• VLA maps in the most extended configuration show very compact morphologies for most sources at all frequencies being unresolved or slightly resolved at 22 GHz with a resolution of 80 mas. These translate into projected linear sizes 1 kpc, which are the typical sizes of CSS/GPS sources.
• The spectra of these sources are typically convex, i.e., they seem to become flat or inverted at MHz frequencies probably due to synchrotron self-absorption, while at frequencies higher than 15 GHz they are definitely steeper. The spectra typically peak between 1 and 5 GHz in the observer's frame and about 1/3 of the sample shows complex spectra suggesting different components.
• All BAL QSOs in our sample presenting simple convex spectra follow the anticorrelation between projected linear size and turnover frequency found for CSS and GPS sources. Two BAL QSOs with complex radio spectra are also in agreement with this relationship if we consider the lower frequency peak of their spectra. The higher frequency peak might be in this two sources due to emission from a hot-spot or a knot in a jet.
• A high percentage of the sample does not show significant variability when comparing the flux densities in 2 different epochs. Some BAL QSOs show significant variability at 8.4 GHz but not strong enough to exclude them from the candidates to young radio sources.
• Most BAL QSOs in our list are not strongly polarized either at 1.4 GHz or at 8.4 GHz, and sensible upper limits are given at these frequencies. Only two sources show significant polarisation at several frequencies. From these two, only the unusual BAL 1624+37 has a high Rotation Measures suggesting strong depolarization. The median fractional polarisation of the sample is in better agreement to mean values found in CSS/GPS sources, than those found for flatspectrum quasars.
• A series of statistical tests have been done to compare the spectral index distribution of BAL and non-BAL QSOs finding that these distributions are different. However, no significant differences were found when comparing to compact non-BAL QSOs only. This is consistent with BAL QSOs spanning the same range of orientations as normal quasars with respect to our line of sight to them.
BAL classification as given in that reference. References key: 1-Menou et al. (2001); 2-Becker et al. (2000); 3-Becker et al. (2001); 4-Reichard et al. (2003a); 5-Holt et al. (2004); 6-Identified directly from SDSS-DR3 database.
Figure 1
1Figure 1. Distribution of known radio-loud BAL QSOs in log S and z before going on-sky. The sample to be studied in this paper was selected by taking all those with S 1.4GHz > 15 mJy.
The single-dish observations at 4.8, 8.4 and 10.5 GHz took place in January 2005. Some sources were re-observed in July and October 2005 in order to confirm doubtful values or improve the signal-to-noise ratio. Observations at 2.65 GHz were carried out in June 2006. All receivers used in these observations are mounted on the secondary focus of the 100-m antenna. The recently installed 2.7-GHz hybrid feed can observe simultaneously in 9 channels, 8 of them 10-MHz wide and the ninth covering the full 80-MHz range.
2 5 CONTFigure 3 .
53: J105352-1053.LBAND.1 Cont peak flux = 2.3693E-02 JY/BEAM Levs = 1.600E-04 * (Map of BAL QSO 1053−00 from FIRST at 1.4 GHz. The synthesized beam-size is shown in the lower-left corner.
Figure 4 .
4VLA map of BAL QSO 1312+23 at 22 GHz (top) and at 43 GHz (bottom). The synthesized beam-size is shown in the lower-left corner of each map.
Figure 5 .
5Radio spectra of the 15 BAL QSOs in the RBQ sample. Errors are shown if larger than the symbol size. Triangles mean 3-σ upper limits. Solid lines are fits to the analytical function give in Section 4.4.1. The peak frequencies given at the top-right of each pannel were obtained from the fit. Some sources like 1159+01, 1413+42 and 1603+30 seem to have a second component at low frequencies.
Figure 6 .
6Rest-frame spectra of 15 BAL QSOs in the RBQ sample. They have been normalised to νrest=25.0 GHz (log(νrest)=1.398) (LoBAL) QSOs from the FBQS, in the interval of redshift [1.5,4.0] ([0.5,1.5]) where the C iv (Mg ii) absorbing features would be covered in optical spectra.
Figure 8 .
8Spectrum of SDSS J080016.09+402955.6, showing a relatively narrow absorption in C IV and Si IV.
Figure 9 .
9Percentile plot of the samples involved in the K-S tests, illustrating the shape of the spectral index distributions.
Figure 10 .
10Determination of the RM for 1159+01. c 2008 RAS, MNRAS 000, 1-18
Figure 11 .
114 GHz, and also for about 70 per cent of the sample at 8.4 Rest-frame turnover frequency versus projected linear size relation for samples of CSS and GPS sources (adapted from O'Dea & Baum 1997). Galaxies are represented by the symbol '×', quasars by the simbol '+' and the RBQ sample is plotted as solid circles. Upper limits for those BAL QSOs unresolved at 22 or 43 GHz are marked by triangles. The dashed line represent the anticorrelation found by O'Dea & Baum (1997): log ν peak = −0.21−0.65 log LS. In those two resolved sources with two peaks in the spectrum (1413+43 and 1603+30) the grey points indicate the new location in the plot when the lower frequency peak is considered.
Table 1 .
1Sample of 15 radio-loud BAL QSOs studied in this paper.ID
RA
DEC
rro
z
S peak
1.4 GHz
E
log(L 5 GHz ) Ref
Type
(J2000)
(J2000)
(arcsec)
(mJy/beam)
(W Hz −1 )
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
0039−00
00:39:23.18
−00:14:52.6
0.22
2.233
21.2
19.50
26.33
1
HiBAL
0135−02
01:35:15.22
−02:13:49.0
0.31
1.820
22.4
16.79
26.22
3
LoBAL
0256−01
02:56:25.56
−01:19:12.1
0.66
2.491
27.6
18.57
26.54
3
HiBAL
0728+40
07:28:31.64
+40:26:16.0
0.33
0.656
17.0
15.27
24.97
2
LoBAL
0837+36
08:37:49.59
+36:41:45.4
0.20
3.416
25.5
19.16
26.66
6
LoBAL
0957+23
09:57:07.37
+23:56:25.2
0.12
1.995
136.1
17.64
27.05
2
HiBAL
1053−00
10:53:52.86
−00:58:52.7
0.14
1.550
24.7
18.03
26.02
4
LoBAL
1159+01
11:59:44.82
+01:12:06.9
0.07
1.989
268.4
17.30
27.31
1
HiBAL
1213+01
12:13:23.94
+01:04:14.7
0.13
2.836
21.5
19.69
26.65
4
HiBAL
1228−01
12:28:48.21
−01:04:14.5
0.27
2.653
29.4
17.74
26.62
4
HiBAL
1312+23
13:12:13.57
+23:19:58.6
0.10
1.508
43.3
17.13
26.32
2
HiBAL
1413+42
14:13:34.38
+42:12:01.7
0.10
2.810
17.8
17.63
26.42
2
HiBAL
1603+30
16:03:54.15
+30:02:08.6
0.28
2.028
53.7
17.61
26.61
2
HiBAL
1624+37
16:24:53.47
+37:58:06.6
0.05
3.377
56.1
17.62
27.06
5
HiBAL
1625+48
16:25:59.90
+48:58:17.5
0.02
2.724
25.3
17.41
26.59
6
HiBAL
Table 1
1shows the main properties of the RBQ sample. Figure 1 shows flux density at 1.4 GHz versus redshift for all BAL QSOs belonging to the mentioned samples. Sources above the horizontal line of 15 mJy define the RBQ sample.
Table 2 .
2Summary of the observations.Run Date
Telescope
Frequencies (GHz)
1
26−30 Jan 05
Effelsberg
4.85, 8.35, 10.5
2
15−16 Jul 05
Effelsberg
4.85, 8.35, 10.5
3
27 Oct 05
Effelsberg
4.85, 8.35, 10.5
4
23−26 Jun 06
Effelsberg
2.65
5
13 Feb 06
VLA(A)
8.45, 14.5, 22.5, 43.5
6
7 Mar 06
VLA(A)
8.
Table 3 .
3Observing frequencies and beam sizes.Telescope
Frequency
Bandwidth
θ HPBW
(GHz)
(MHz) (arcsec)
Effelsberg
2.65
350
265
Effelsberg
4.85
500
145
Effelsberg
8.35
1200
80
Effelsberg
10.5
300
65
VLA(A)
8.45
700
0.24
VLA(A)
14.5
700
0.14
VLA(A)
22.5
2000
0.08
VLA(A)
43.5
10000
0.05
Table 4 .
4Values used to calculate the errors of the Stokes parameters and obtained instrumental polarisations.Telescope
Frequency
∆S conf
∆S cal
P instr
(GHz)
(I)
(Q,U)
(per cent)
Effelsberg
2.65
1.5
0.5
0.8
0.7
Effelsberg
4.85
0.45
0.15
0.9
0.7
Effelsberg
8.35
0.23
−
1.0
0.6
Effelsberg
10.5
0.08
−
1.9
0.8
VLA
8.45
−
−
1.9
<0.1
VLA
15.0
−
−
2.9
0.6
VLA
22.5
−
−
1.8
0.7
VLA
43.5
−
−
5.3
2.1
Table 5 .
5RBQ sample of 15 radio-loud BAL QSOs: Flux densities (in mJy), errors or 3-σ upper limits from 74 MHz to 1.4 GHz obtained from the literature.The symbol † indicate data from WENSS at 325 MHz, the rest are from the Texas Survey at 365 MHz.ID
S VLSS
0.074
S †
0.365
S FIRST
1.4
S NVSS
1.4
0039−00
<475
<80
21.2
19.5 ± 0.7
0135−02
<350
<80
22.4
22.8 ± 1.1
0256−01
<385
<80
27.6
22.3 ± 0.8
0728+40
<300
15 ± 3.3 †
17.0
17.2 ± 0.6
0837+36
<290
184 ± 3.4 †
25.5
−
0957+23
<260
<80
136.1
136.9 ± 4.1
1053−00
<500
<80
24.7
22.7 ± 1.1
1159+01
<320
887 ± 30
268.5
275.6 ± 8.3
1213+01
<340
<80
21.5
27.5 ± 7.9
1228−01
<340
<80
29.4
29.1 ± 1.0
1312+23
<300
<80
43.3
46.5 ± 1.4
1413+42
<330
16 ± 3.2 †
17.8
16.8 ± 0.6
1603+30
<225
33 ± 4.4 †
53.7
54.1 ± 1.7
1624+37
<355
59 ± 4.0 †
56.1
55.6 ± 1.7
1625+48
<290
26 ± 3.7 †
25.3
26.0 ± 0.9
Table 7 .
7Angular dimensions for the extended sources.Source
Freq.
θ maj
θ min
PA θ
(dec)
maj
(GHz)
(mas) (mas)
(deg)
(mas)
0135−02
22
112.4
78.7
−12
11 ± 4
0728+40
22
85.5
73.0
26
26 ± 6
0837+36
22
94.0
85.9
34
19 ± 10
1213+01
22
111.7
75.5
−19
27 ± 11
1228−01
22
95.3
76.3
−30
19 ± 4
1312+23
43
54.5
41.8
64
69 ± 12 †
1413+42
22
92.0
69.5
−60
9 ± 3
1603+30
22
95.8
71.0
−71
15 ± 3
1624+37
4.8
−
−
− 36 ± 5 † †
1625+48
22
157.0
71.8
76
63
CONT: 13122+23 IPOL 22460.100 MHZ 1312K.PBCOR.1Cont peak flux = 7.1157E-03 JY/BEAM
Levs = 6.000E-05 * (-3, 3, 4, 5, 8, 16, 32, 64)
DECLINATION (J2000)
RIGHT ASCENSION (J2000)
13 12 13.60
13.59
13.58
13.57
13.56
13.55
23 19 59.0
58.9
58.8
58.7
58.6
58.5
58.4
58.3
58.2
CONT: 13122+23 IPOL 43339.900 MHZ 1312Q5.ICL001.10
Cont peak flux = 2.9973E-03 JY/BEAM
Levs = 1.250E-04 * (-3, 3, 4, 5, 8, 16, 32, 64)
DECLINATION (J2000)
RIGHT ASCENSION (J2000)
13 12 13.595 13.590 13.585 13.580 13.575 13.570 13.565 13.560 13.555
23 19 58.85
58.80
58.75
58.70
58.65
58.60
58.55
58.50
58.45
58.40
58.35
1312+23 Total Intensity
43340 MHz
Levs = 6.000E−03 (−3, 3, 4, 5, 8, 16, 32, 64)
mJy/beam
Levs = 1.250E−01 (−3, 3, 4, 5, 8, 16, 32, 64) mJy/beam
1312+23 Total Intensity
22460 MHz
Table 8 .
8Sources showing significant flux density variability. At
1.4 GHz, S 1 comes from NVSS and S 2 from FIRST. At 8.4 GHz,
S 1 comes from Becker et al. (2000), S 2 from this work (Table 6)
Source
Freq
S 1
S 2
V ar ∆S
σ V ar
(GHz)
(mJy)
(mJy)
0256−01
1.4
22.3±0.8 D
27.5 B
0.23
3.2
1213+01
1.4
27.5±0.9 D
22.9 B
−0.20
3.2
1312+23
8.4
12.6 A
19.4±0.2 A
0.54
10.3
1413+42
8.4
11.3 D
13.4±0.1 A
0.19
3.7
1603+30
8.4
18.1 A
22.1±0.3 A
0.22
4.2
Table
with sizes <5 arcsec at 1.4 GHzSample
N
Min
Max
Mean
Median
Std
Observer's frame: α [1.4 − 8.4] GHz
RBQ+B00
38
−1.50
0.70
−0.50
−0.53
0.43
RBQ
15
−1.19
0.20
−0.52
−0.44
0.34
V97
119
−1.47
0.86
−0.80
−0.92
0.43
V97c
50
−1.41
0.86
−0.53
−0.49
0.49
Rest frame: α [6.0 − 25] GHz
RBQ
15
−0.87
0.17
−0.38
−0.32
0.30
V97
51
−1.27
0.17
−0.71
−0.82
0.38
V97c
27
−1.21
0.17
−0.53
−0.47
0.40
Rest frame: α [12 − 25] GHz
RBQ
15
−1.00
0.45
−0.40
−0.58
0.43
V97
51
−1.39
0.39
−0.71
−0.76
0.42
V97c
27
−1.31
0.39
−0.54
−0.62
0.44
Table 12 .
12Polarisation properties of BAL QSO 1159+01. The second measurement at 2.65 GHz comes fromSimard-Normandin et al. (1981) Frequency
Telescope
m
χ
(GHz)
(per cent)
(deg)
1.40
NVSS
15.03 ± 0.44
−31.1 ± 0.6
2.65
Effelsberg
<16
−
2.65
−
3.1 ± 1.9
111 ± 17
4.85
Effelsberg
2.1 ± 0.2
−42 ± 6
8.35
Effelsberg
0.95 ± 0.05
−29 ± 5
8.45
VLA
0.9 ± 0.4
−29 ± 7
10.5
Effelsberg
<1.2
−
15.0
VLA
<1.2
−
22.5
VLA
<0.4
−
43.5
VLA
<1.2
−
RM= -72.1 1.4
±
2
−
m
d
a
r
0
χ = -24
3
±
g
e
d
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
c 2008 RAS, MNRAS 000, 1-18
In practise 124 because the redshift of quasar B3 2329+398 is not known.
ACKNOWLEDGEMENTSWe are grateful to F. Mantovani and M. Orienti for helping us during the observations at the 100-m Effelsberg telescope, and to A. Kraus for support and help with the calibration of the Effelsberg data. The authors acknowledge financial support from the Spanish Ministerio de Educación y Ciencia under project PNAYA2005-00055. This work has benefited from research funding from the European Community's sixth Framework Programme under RadioNet R113CT 2003 5058187. This work has been partially based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research has made use of the NASA/IPAC Infrared Science Archive and NASA/IPAC Extragalactic Database (NED) which are both operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Use has been made of the Sloan Digital Sky Survey (SDSS) Archive. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the participating institutions: The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The John Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.
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| [] |
[
"Rise of azimuthal anisotropies as a signature of the Quark-Gluon-Plasma in relativistic heavy-ion collisions",
"Rise of azimuthal anisotropies as a signature of the Quark-Gluon-Plasma in relativistic heavy-ion collisions"
] | [
"V P Konchakovski \nInstitute for Theoretical Physics\nUniversity of Giessen\nGiessenGermany\n\nBogolyubov Institute for Theoretical Physics\nKievUkraine\n",
"E L Bratkovskaya \nInstitute for Theoretical Physics\nUniversity of Frankfurt\nFrankfurtGermany\n\nFrankfurt Institute for Advanced Studies\nFrankfurtGermany\n",
"W Cassing \nInstitute for Theoretical Physics\nUniversity of Giessen\nGiessenGermany\n",
"V D Toneev \nFrankfurt Institute for Advanced Studies\nFrankfurtGermany\n\nJoint Institute for Nuclear Research\nDubnaRussia\n",
"V Voronyuk \nBogolyubov Institute for Theoretical Physics\nKievUkraine\n\nFrankfurt Institute for Advanced Studies\nFrankfurtGermany\n\nJoint Institute for Nuclear Research\nDubnaRussia\n"
] | [
"Institute for Theoretical Physics\nUniversity of Giessen\nGiessenGermany",
"Bogolyubov Institute for Theoretical Physics\nKievUkraine",
"Institute for Theoretical Physics\nUniversity of Frankfurt\nFrankfurtGermany",
"Frankfurt Institute for Advanced Studies\nFrankfurtGermany",
"Institute for Theoretical Physics\nUniversity of Giessen\nGiessenGermany",
"Frankfurt Institute for Advanced Studies\nFrankfurtGermany",
"Joint Institute for Nuclear Research\nDubnaRussia",
"Bogolyubov Institute for Theoretical Physics\nKievUkraine",
"Frankfurt Institute for Advanced Studies\nFrankfurtGermany",
"Joint Institute for Nuclear Research\nDubnaRussia"
] | [] | The azimuthal anisotropies of the collective transverse flow of hadrons are investigated in a large range of heavy-ion collision energy within the Parton-Hadron-String Dynamics (PHSD) microscopic transport approach which incorporates explicit partonic degrees of freedom in terms of strongly interacting quasiparticles (quarks and gluons) in line with an equation-of-state from lattice QCD as well as dynamical hadronization and hadronic dynamics in the final reaction phase. The experimentally observed increase of the elliptic flow v2 with bombarding energy is successfully described in terms of the PHSD approach in contrast to a variety of other kinetic models based on hadronic interactions. The analysis of higher-order harmonics v3 and v4 shows a similar tendency of growing deviations between partonic and purely hadronic models with increasing bombarding energy. This signals that the excitation functions of azimuthal anisotropies provide a sensitive probe for the underling degrees of freedom excited in heavy-ion collisions. PACS numbers: 25.75.-q, 25.75.Ag | 10.1103/physrevc.85.011902 | [
"https://arxiv.org/pdf/1109.3039v2.pdf"
] | 119,225,365 | 1109.3039 | e8b1c127d857e20958ae1797bb3c9fc600b51655 |
Rise of azimuthal anisotropies as a signature of the Quark-Gluon-Plasma in relativistic heavy-ion collisions
11 Jan 2012
V P Konchakovski
Institute for Theoretical Physics
University of Giessen
GiessenGermany
Bogolyubov Institute for Theoretical Physics
KievUkraine
E L Bratkovskaya
Institute for Theoretical Physics
University of Frankfurt
FrankfurtGermany
Frankfurt Institute for Advanced Studies
FrankfurtGermany
W Cassing
Institute for Theoretical Physics
University of Giessen
GiessenGermany
V D Toneev
Frankfurt Institute for Advanced Studies
FrankfurtGermany
Joint Institute for Nuclear Research
DubnaRussia
V Voronyuk
Bogolyubov Institute for Theoretical Physics
KievUkraine
Frankfurt Institute for Advanced Studies
FrankfurtGermany
Joint Institute for Nuclear Research
DubnaRussia
Rise of azimuthal anisotropies as a signature of the Quark-Gluon-Plasma in relativistic heavy-ion collisions
11 Jan 2012
The azimuthal anisotropies of the collective transverse flow of hadrons are investigated in a large range of heavy-ion collision energy within the Parton-Hadron-String Dynamics (PHSD) microscopic transport approach which incorporates explicit partonic degrees of freedom in terms of strongly interacting quasiparticles (quarks and gluons) in line with an equation-of-state from lattice QCD as well as dynamical hadronization and hadronic dynamics in the final reaction phase. The experimentally observed increase of the elliptic flow v2 with bombarding energy is successfully described in terms of the PHSD approach in contrast to a variety of other kinetic models based on hadronic interactions. The analysis of higher-order harmonics v3 and v4 shows a similar tendency of growing deviations between partonic and purely hadronic models with increasing bombarding energy. This signals that the excitation functions of azimuthal anisotropies provide a sensitive probe for the underling degrees of freedom excited in heavy-ion collisions. PACS numbers: 25.75.-q, 25.75.Ag
Introduction. A few decades of experimental studies at the Schwerionen-Synchrotron (SIS), the Alternating Gradient Synchroton (AGS) and the Super Proton Synchroton (SPS) have shown that the physics of nuclear collisions at moderate relativistic energies is dominated by the nonequilibrium dynamics of hadronic resonance matter, i.e. the confined phase of QCD. The body of data extends and builds up the knowledge gained about dense hadronic matter, in particular, at the SPS/CERN. The SPS heavy-ion data have shown several signatures that hinted at the onset of a quark-gluon plasma (QGP) formation [1,2]. With the Relativistic Heavy Ion Collider (RHIC) the center-of-mass energy could be increased by a factor of 10 relative to the SPS, and the experiments at RHIC assured that a new form of matter -well above the deconfinement transition point -was created in the laboratory.
Indeed, the discovery of a large azimuthal anisotropic flow of hadrons at RHIC provides a conclusive evidence for the creation of dense partonic matter in ultrarelativistic nucleus-nucleus collisions. The strongly interacting medium in the collision zone can be expected to achieve a local equilibrium and exhibit an approximately hydrodynamic flow [3][4][5]. The momentum anisotropy is generated due to pressure gradients in a collective expansion of an initial geometry of an "almond-shaped" collision zone produced in noncentral collisions [3,4]. The pressure gradients translate early stage coordinate space asymmetry to final-state momentum space anisotropy [6]. The picture thus emerges that the medium created in ultra-relativistic collisions for a couple of fm/c interacts more strongly than hadron resonance matter and exhibits collective properties that resemble those of a liquid of a very low shear viscosity η to the entropy density s ratio, η/s, close to a nearly perfect fluid [7][8][9].
An experimental manifestation of this collective flow is the anisotropic emission of particles in the plane transverse to the beam direction. A quark number scaling of the elliptic flow proposed in Ref. [10] was observed at RHIC for a broad range of particle species, collision centralities, and transverse kinetic energy, presumably interpreted as due to the development of substantial collectivity in the early partonic phase [11].
It was shown that higher-order anisotropy harmonics, in particular the hexadecupole moment v 4 , can provide a more sensitive constraint on the magnitude of η/s and the freeze-out dynamics, and the ratio v 4 /(v 2 ) 2 might indicate whether a full local equilibrium is achieved in the QGP [12]. Recently, the importance of the triangular flow v 3 , which originates from fluctuations in the initial collision geometry, has been pointed out [13,14]. The participant triangularity characterizes the triangular anisotropy of the initial nuclear overlap geometry and arises from event-by-event fluctuations in the participantnucleon collision points and corresponds to a large third Fourier component in two-particle azimuthal correlations at large pseudo-rapidity separation ∆η. This fact suggests a significant contribution of the triangular flow to the ridge phenomenon and broad away-side structures observed in the RHIC data [13].
A large number of anisotropic flow measurements have been performed by many experimental groups at SIS, AGS, SPS and RHIC energies over the past 20 years. Very recently the azimuthal asymmetry has also been measured at the Large Hadron Collider (LHC) at CERN [15]. The Beam Energy Scan (BES) program proposed at RHIC [16] covers the energy interval from √ s N N = 200 GeV, where partonic degrees of freedom (DOF) play a decisive role, down to the AGS energy √ s N N ≈ 5 GeV, where most experimental data can be described successfully in terms of hadronic DOF Lowering the collision energy and studying the energy dependence of an anisotropic flow allows one to search for the onset of the transition to a phase with partonic DOF at an early stage of the collision as well as possibly identify the location of the expected critical end-point that terminates the first order phase transition at high quarkchemical potential [11,17]. This work aims to study excitation functions for different harmonics of the charged particle anisotropy in momentum space in a wide collision energy range, i.e. from the AGS to the top RHIC energy regime. We want to clarify how the interplay of quark and hadron DOF is changed with increasing bombarding energy. In this study we investigate the excitation function of different flow coefficients. Our analysis of the STAR/PHENIX RHIC data -based on recent results of the BES program -will be performed within the PHSD transport approach [18] that includes explicit partonic DOF as well as a dynamic hadronization scheme for the transition from partonic to hadronic DOF and vice versa.
The PHSD approach. The dynamics of partons, hadrons and strings in relativistic nucleus-nucleus collisions is analyzed here within the Parton-Hadron-String Dynamics approach [18]. In this transport approach the partonic dynamics is based on Kadanoff-Baym equations for Green functions with self-energies from the Dynamical QuasiParticle Model (DQPM) [19,20] which describes QCD properties in terms of "resummed" single-particle Green functions. In Ref. [21], the actual three DQPM parameters for the temperature-dependent effective coupling were fitted to the recent lattice QCD results of Ref. [22]. The latter lead to a critical temperature T c ≈ 160 MeV which corresponds to a critical energy density of ǫ c ≈ 0.5 GeV/fm 3 . In PHSD the parton spectral functions ρ j (j = q,q, g) are no longer δ-functions in the invariant mass squared as in conventional cascade or transport models but depend on the parton mass and width parameters which were fixed by fitting the lattice QCD results from Ref. [22]. We recall that the DQPM allows one to extract a potential energy density V p from the space-like part of the energy-momentum tensor as a function of the scalar parton density ρ s . Derivatives of V p with respect to ρ s then define a scalar mean-field potential U s (ρ s ) which enters into the equation of motion for the dynamic partonic quasiparticles. Furthermore, a two-body interaction strength can be extracted from the DQPM as well from the quasiparticle width in line with Ref. [9]. The transition from partonic to hadronic DOF (and vice versa) is described by covariant transition rates for the fusion of quark-antiquark pairs or three quarks (antiquarks), respectively, obeying flavor current-conservation, color neutrality as well as energymomentum conservation [18,21]. Since the dynamical quarks and antiquarks become very massive close to the phase transition, the formed resonant "prehadronic" color-dipole states (qq or qqq) are of high invariant mass, too, and sequentially decay to the ground-state meson and baryon octets increasing the total entropy.
On the hadronic side PHSD includes explicitly the baryon octet and decouplet, the 0 − -and 1 − -meson nonets as well as selected higher resonances as in the Hadron-String-Dynamics (HSD) approach [23,24]. Hadrons of higher masses (> 1.5 GeV in case of baryons and > 1.3 GeV for mesons) are treated as "strings" (color dipoles) that decay to the known (low-mass) hadrons, according to the JETSET algorithm [25]. Note that PHSD and HSD merge at low energy density, in particular below the critical energy density ǫ c ≈ 0.5 GeV/fm 3 .
The PHSD approach was applied to nucleus-nucleus collisions from √ s N N ∼ 5 to 200 GeV in Refs. [18,21] in order to explore the space-time regions of "partonic matter". It was found that even central collisions at the top-SPS energy of √ s N N = 17.3 GeV show a large fraction of nonpartonic, i.e., hadronic or stringlike matter, which can be viewed as a hadronic corona. This finding implies that neither hadronic nor only partonic "models" can be employed to extract physical conclusions in comparing model results with data. All these previous findings provide promising perspectives to use PHSD in the whole range from about √ s N N = 5 to 200 GeV for a systematic study of azimuthal asymmetries of hadrons produced in relativistic nucleus-nucleus collisions.
Calculational results and comparison to data. The anisotropy in the azimuthal angle ψ is usually characterized by the even order Fourier coefficients v n = exp( ı n(ψ − Ψ RP )) , n = 2, 4, ..., since for a smooth angular profile the odd harmonics become equal to zero. As noted above, Ψ RP is the azimuth of the reaction plane and the brackets denote averaging over particles and events. In particular, for the widely used secondorder coefficient, denoted as an elliptic flow, we have
v 2 = cos(2ψ − 2Ψ RP ) = p 2 x − p 2 y p 2 x + p 2 y ,(1)
where p x and p y are the x and y components of the particle momenta. This coefficient can be considered as a function of centrality, pseudorapidity η, and/or transverse momentum p T . We note that the reaction plane in PHSD is given by the (x − z) plane with the z axis in the beam direction. Integrated v 3 and v 4 coefficients are calculated by the two-particle correlation method in line with Ref. [26]:
cos(nψ 1 − nψ 2 )) = v 2 n + δ n ,(2)
which is based on the assumption that nonflow contributions δ n are small and account for event-by-event fluctuations of the event plane. In Fig. 1 the experimental v 2 excitation function in the transient energy range is compared to the results from the PHSD calculations; HSD model results are given as well for reference. We note that the centrality selection and acceptance are the same for the data and models. We recall that the HSD model has been very successful in describing heavy-ion spectra and rapidity distributions from SIS to SPS energies. A detailed comparison The v2 STAR data compilation for minimal bias collisions are taken from Ref. [27] (stars) and the preliminary PHENIX data [28] are plotted by solid circles.
of HSD results with respect to a large experimental data set was reported in Ref. [29] for central Au+Au (Pb+Pb) collisions from AGS to top SPS energies. Indeed, as shown in Fig. 1 (dashed lines), HSD is in good agreement with experiment for both data sets at the lower edge ( √ s N N ∼ 10 GeV) but predicts an approximately energy-independent flow v 2 at larger energies and, therefore, does not match the experimental observations. This behavior is in quite close agreement with another independent hadronic model, the UrQMD (Ultra relativistic Quantum Molecular Dynamics) [30] (cf. with [27]).
From the above comparison one may conclude that the rise of v 2 with bombarding energy is not due to hadronic interactions and models with partonic DOF have to be addressed. Indeed, the PHSD approach incorporates the parton medium effects in line with a lQCD equation of state, as discussed above, and also includes a dynamic hadronization scheme based on covariant transition rates. It is seen from Fig. 1 that PHSD performs better: The elliptic flow v 2 from PHSD (solid curve) is fairly in line with the data from the STAR and PHENIX collaborations and clearly shows the growth of v 2 with the bombarding energy.
Since partonic DOF come into play, it is interesting to compare with another parton-hadron model, i.e. the AMPT (A Multi Phase Transport) model [31]. This model is based on a perturbative QCD description of partonic interactions, including the production of multiple minijet partons according to the number of binary initial collisions. As shown in Ref. [27], the AMPT model predicts an approximately constant v 2 with √ s N N similar to the hadronic models HSD and UrQMD; however, the v 2 values match the experimental data at the top RHIC energy. This discrepancy is due to a pQCD description of the partonic phase in AMPT where the minijet partons are treated as massless and their potentials are disregarded when they undergo scattering. Note that PHSD and AMPT (with the additional strong hadron melting assumption in AMPT) practically give the same elliptic flow at the top RHIC energy of √ s N N = 200 GeV. We note that the PHSD model includes more realistic properties of dynamical quasiparticles especially in the vicinity of the critical energy density. Furthermore, the quark-gluon transport in PHSD naturally passes on to the (hadronic) HSD model at lower √ s N N . In Fig. 2 we display the PHSD and HSD results for the anisotropic flows v 3 and v 4 of charged particles at midpseudorapidity for Au + Au collisions from √ s N N = 5 to 200 GeV. The triangular flow increases with √ s N N having negative values for √ s N N < ∼ 10 GeV. The pure hadronic model HSD gives v 3 ≈ 0 for √ s N N > ∼ (20-30) GeV. Accordingly, the results from PHSD (solid red lines) are systematically larger than those from HSD (dashed blue lines). Unfortunately, our statistics is not high enough to allow for more precise conclusions. The hexadecupole flow v 4 stays almost constant in the considered energy range; here PHSD gives slightly higher values than HSD.
The v 2 increase is clarified in Fig. 3 where the partonic fraction of the energy density at mid-pseudorapidity with respect to the total energy density in the same pseudorapidity interval is shown. We recall that the repulsive scalar mean field potential U s (ρ s ) for partons in the PHSD model leads to an increase of the flow v 2 as compared to that for HSD or PHSD calculations without partonic mean fields [21]. As follows from Fig. 3, the energy fraction of the partons substantially grows with increasing bombarding energy while the duration of the partonic phase is roughly the same. Accordingly, the increasing influence of the repulsive partonic mean-field U s (ρ s ) leads to an increase of the flow v 2 with bombarding energy. We point out that the increase of v 2 in PHSD relative to HSD is also partly due to the higher interaction rates in the partonic medium because of a lower ratio of η/s for partonic degrees of freedom at energy densities above the critical energy density than for hadronic media below the critical energy density [32,33]. The relative increase in v 3 and v 4 in PHSD essentially is due to the higher partonic interaction rate and, thus, to a lower ratio η/s in the partonic medium, which is mandatory to convert initial spacial anisotropies to final anisotropies in momentum space [34].
Conclusions. The anisotropic flows -elliptic v 2 , tri-angular v 3 , and hexadecupole v 4 -are reasonably described within the PHSD model in the whole transient energy range naturally connecting the hadronic processes at moderate bombarding energies with ultrarelativistic collisions at RHIC energies where the quark-gluon DOF become dominant due to a growing number of partons. The smooth growth of the elliptic flow with collision energy demonstrates the increasing importance of partonic DOF This feature is reproduced by neither explicit hadronic kinetic models like HSD or UrQMD nor the AMPT model treating the partonic phase on the basis of pQCD with massless partons and a noninteracting equation-of-state for the partons. Further signatures of the transverse collective flow, the higher-order harmonics of the transverse anisotropy v 3 and v 4 change only weakly from √ s N N ≈ 7 GeV to the top RHIC energy √ s N N = 200 GeV, roughly in agreement with preliminary experimental data. Certainly, new measurements within the BES program at RHIC, especially for higher-order harmonics, will further constrain the partonic dynamics.
FIG. 1 :
1Average elliptic flow v2 of charged particles at midrapidity for two centrality selections calculated within the PHSD (solid curves) and HSD (dashed curves) approaches.
FIG. 2 :
2Average anisotropic flows v3 and v4 of charged particles at mid-pseudorapidity for minimum bias collisions of Au + Au calculated within the PHSD (solid lines) and HSD (dashed lines) models.
FIG. 3 :
3The evolution of the parton fraction of the total energy density at the mid-pseudorapidity for different collision energies.
Acknowledgements. We are thankful to S. Voloshin for constructive remarks and O. Linnyk for useful discussions. This work has been supported in part by the DFG Grant WA 431/8-1, the DFG Grant CA 124/7-1, the RFFI Grants 08-02-01003-a, the Ukrainian-RFFI Grant 09-02-90423-ukr-f-a, and the LOEWE center HIC for FAIR.
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"Hu Xu \nDepartment of Computer Science\nUniversity of Illinois at Chicago\n\n\nUber AI\n\n",
"Zhaleh Feizollahi [email protected] \nUber AI\n\n",
"Bing Liu [email protected] \nDepartment of Computer Science\nUniversity of Illinois at Chicago\n\n",
"Piero Molino [email protected] \nStanford University and ML Collective\n\n"
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"Department of Computer Science\nUniversity of Illinois at Chicago\n",
"Uber AI\n",
"Uber AI\n",
"Department of Computer Science\nUniversity of Illinois at Chicago\n",
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"Department of Computer Science\nUniversity of Illinois at Chicago\n",
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"Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: Findings"
] | This work introduces Focused-Variation Network (FVN), a novel model to control language generation. The main problems in previous controlled language generation models range from the difficulty of generating text according to the given attributes, to the lack of diversity of the generated texts. FVN addresses these issues by learning disjoint discrete latent spaces for each attribute inside codebooks, which allows for both controllability and diversity, while at the same time generating fluent text. We evaluate FVN on two text generation datasets with annotated content and style, and show state-of-the-art performance as assessed by automatic and human evaluations. | 10.18653/v1/2020.findings-emnlp.339 | [
"https://www.aclweb.org/anthology/2020.findings-emnlp.339.pdf"
] | 221,949,069 | 2009.12046 | 782daebab8aedd5bde1d935f4c20691594ba4d6d |
Controllable Text Generation with Focused Variation
Association for Computational LinguisticsCopyright Association for Computational LinguisticsNovember 16 -20, 2020. 2020
Lei Shu
Department of Computer Science
University of Illinois at Chicago
Alexandros Papangelis [email protected]
Uber AI
Yi-Chia Wang [email protected]
Uber AI
Hu Xu
Department of Computer Science
University of Illinois at Chicago
Uber AI
Zhaleh Feizollahi [email protected]
Uber AI
Bing Liu [email protected]
Department of Computer Science
University of Illinois at Chicago
Piero Molino [email protected]
Stanford University and ML Collective
Controllable Text Generation with Focused Variation
Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: Findings
the 2020 Conference on Empirical Methods in Natural Language Processing: FindingsAssociation for Computational LinguisticsNovember 16 -20, 2020. 20203805
This work introduces Focused-Variation Network (FVN), a novel model to control language generation. The main problems in previous controlled language generation models range from the difficulty of generating text according to the given attributes, to the lack of diversity of the generated texts. FVN addresses these issues by learning disjoint discrete latent spaces for each attribute inside codebooks, which allows for both controllability and diversity, while at the same time generating fluent text. We evaluate FVN on two text generation datasets with annotated content and style, and show state-of-the-art performance as assessed by automatic and human evaluations.
Introduction
Recent developments in language modeling Dai et al., 2019;Radford et al., 2018;Holtzman et al., 2020;Khandelwal et al., 2020) make it possible to generate fluent and mostly coherent text. Despite the quality of the samples, regular language models cannot be conditioned to generate language depending on attributes. Conditional language models have been developed to solve this problem, with methods that either train models given predetermined attributes (Shirish Keskar et al., 2019), use conditional generative models (Kikuchi et al., 2014;Ficler and Goldberg, 2017), fine-tune models using reinforcement learning (Ziegler et al., 2019), or modify the text on the fly during generation (Dathathri et al., 2020).
As many researchers noted, injecting style into natural language generation can increase the naturalness and human-likeness of text by including pragmatic markers, characteristic of oral language (Biber, 1991;Paiva and Evans, 2004;Mairesse and Walker, 2007). Text generation with * Work done while at Uber AI Labs. style-variation has been explored as a special case of conditional language generation that aims to map attributes such as the informational content (usually structured data representing meaning like frames with keys and values) and the style (such as personality and politeness) into one of many natural language realisations that conveys them (Novikova et al., 2016(Novikova et al., , 2017Wang et al., 2018). As the examples in Table 1 show, for one given content frame there can be multiple realisations.When a style (a personality trait in this case) is injected, the text is adapted to that style (words in red) while conveying the correct informational content (words in blue). A key challenge is to generate text that respects the specified attributes while at the same time generating diverse outputs, as most existing methods fail to correctly generate text according to given attributes or exhibit a lack of diversity among different samples, leading to dull and repetitive expressions.
Conditional VAEs (CVAE) (Sohn et al., 2015) and their variants have been adopted for the task and are able to generate diverse texts, but they suffer from posterior collapse and do not strictly follow the given attributes because their latent space is pushed towards being a Gaussian distribution irrespective of the different disjoint attributes, conflating the given content and style.
An ideal model would learn a separate latent space that focuses on each attribute independently. For this purpose, we introduce a novel natural language generator called Focused-Variation Network (FVN) 1 . FVN extends the Vector-Quantised VAE (VQ-VAE) (van den Oord et al., 2017), which is non-conditional, to allow conditioning on attributes (content and style). Specifically, FVN:
(1) models two disjoint codebooks for content and style respectively that memorize input text vari- ations;
(2) further controls the conveyance of attributes by using content and style specific encoders and decoders; (3) computes disjoint latent space distributions that are conditional on the content and style respectively, which allows to sample latent representations in a focused way at prediction time. This choice ultimately helps both attribute conveyance and variability. As a result, FVN can preserve the diversity found in training examples as opposed to previous methods that tend to cancel out diverse examples. FVN's disjoint modeling of content and style increases the conveyance of the generated text, while at the same time generating more natural and fluent text. We tested FVN on two datasets, PersonageNLG (Oraby et al., 2018) andE2E (Dušek et al., 2020) that consist of content-utterance pairs with personality labels in the first case, and the experimental results show that it outperforms previous state-ofthe-art methods. A human evaluation further confirms that the naturalness and conveyance of FVN generated text is comparable to ground truth data.
Related Work
Our work is related to CVAE based text generation (Bowman et al., 2016;Shen et al., 2018;, where the goal is to control a given attribute of the output text (for example, style) by providing it as additional input to a regular VAE. For instance, the controlled text generation method proposed by Hu et al. (2017) extends VAE and focuses on controlling attributes of the generated text like sentiment and style. Differently from ours, this method does not focus on generating text from content meaning representation (CMR) or on diversity of the generated text. (Song et al., 2019) use a memory augmented CVAE to control for persona, but with no control over the content.
The works of (Oraby et al., 2018;Harrison et al., 2019;Oraby et al., 2019) on style-variation generators adopt sequence-to-sequence based models and use human-engineered features (Juraska and Walker, 2018) (e.g. personality parameters or syntax features) as extra inputs alongside the content and style to control the generation and enhance text variation. However, using human-engineered features is labor-intensive and, as it is not possible to consider all possible feature combinations, performance can be sub-optimal. In our work we instead rely on codebooks to memorize textual variations.
There is a variety of works that address the problem of incorporating knowledge or structured data into the generated text (for example, entities retrieved from a knowledge base) (Ye et al., 2020), or that try generate text that is in line with some given story (Rashkin et al., 2020). None of these works focuses specifically on generating text that conveys content while at the same time controlling style. Last, there are works such as (Rashkin et al., 2018) that focus on generating text consistent with an emotion (aiming to create an empathetic agent) without, however, directly controlling the content.
Methodology
Our proposed FVN architecture (Figure 1) has the goal to generate diverse texts that respect every attribute provided as controlling factor. We describe a specific instantiation where the attributes Figure 1: Focused-Variation Network (FVN) has four encoders (text-to-content encoder, text-to-style encoder, content encoder and style encoder), two codebooks (for content and style), and one text decoder. The training data contains ground-truth text with associated content and style. The text decoder uses v C and v S , latent vectors of content and style, as well as the latent vectors e C k and e S n from codebooks (the nearest to the z C and z S vectors produced by the text-to-content and text-to-style encoders) to generate text back. To further control the content and style of the generated text, we feed the o L output vectors of the generated text t to text encoders (content and style). o L are aligned to a word embedding codebook.
are content (a frame CMR containing slots keys and values) and style (personality traits). However, the same architecture can be used with additional attributes and / or with different types of content attributes (structured data tables and knowledge graphs for instance) and style attributes (linguistic register, readability, and many others). To encourage conveyance of the generated texts, FVN learns disjoint discrete content-and style-focused representation codebooks inspired by VQ-VAE as extra information along with the representations of intended content and style, which avoids the posterior collapse problem of VAEs.
During training, FVN receives as input an intended content c and style s as well as a reference text t. The reference text is passed through two encoders (text-to-content and text-to-style), while content and style are encoded with a content encoder and a style encoder. The text-to-content encoder maps input text t into a content latent vector z C and the text-to-style encoder maps the input text t into a latent style vector z S . The closest vectors to z C and z S from the content codebook e C and style codebook e S , e C k and e S n , are selected. The content encoder encodes the intended content frame into a latent vector v C and the style encoder encodes the intended style into a latent vector v S . A text decoder then receives e C k , e S n , v C and v S and generates the output text t . The generated text is subsequently fed to a content and a style decoder that predict the intended content and style.
At prediction time (Figure 2), only content c and style s are given, and in order to obtain e C k and e S n without an input text, we A) collect a distribution over the codebook indices by counting, for each training datapoint containing a specific value for c and s, the amount of times a specific index is used, and B) sample e C k and e S n from these frequency distributions. These disjoint distributions allow the model to focus on specific content and style by using them for conditioning and the sampling allows for variation, hence the name of focused variation. v C and v S obtained from the content and style encoders and the sampled e C k and e S n are provided to the text generator that generates t .
The rest of this section will detail each component and the training and prediction processes.
Encoding and Codebooks
As shown in Figure 1, FVN uses four encoders and one decoder during training: the text-to-content encoder Enc TC (·), the text-to-style encoder Enc TS (·), the content encoder Enc C (·), the style encoder Enc S (·), and the text decoder Dec(·).
Text-to-* encoders The text-to-content encoder Enc TC (·) encodes a text t to a dense representation z C ∈ R D while the text-to-style encoder Enc TS (·) encodes a text t to a dense representation z S ∈ R D : z C = Enc TC (t) and z S = Enc TS (t).
In order to learn disjoint latent spaces for the different attributes we want to model, we train two codebooks, one for content e C ∈ R K×D and one for style e S ∈ R N ×D . They are shown as [e C 1 , . . . , e C K ] and [e S 1 , ..., e S N ] in Figure 1. These two codebooks are used to memorize the latent vectors for text-to-content variation and textto-style variation learned during training. Instead of using the z C and z S vectors as inputs to the de-coder, we find their nearest latent vectors in the codebooks e C k and e S n and use those nearest latent vectors for decoding the text instead of the original encoded dense representation. Formally, k = argmin i z C −e C i 2 and n = argmin j z S −e S j 2 . Like in VQ-VAE, we use the l2-norm error to move the latent vectors in the codebooks e towards the same space of the encoder outputs z:
L C VQ = sg(z C ) − e C k 2 2 + β C z C − sg(e C k ) 2 2 ,(1)L S VQ = sg(z S ) − e S n 2 2 + β S z S − sg(e S n ) 2 2 ,(2)
where sg(·) stands for the stop gradient operator.
Style and content encoders The content encoder encodes a CMR c treating it as a sequence of tokens and producing a matrix V C ∈ R L ×D , where L is the length of c, from which the last element v C ∈ R D is returned. The style encoder encodes a style s and obtains a dense representation v S ∈ R D selecting the last element of the matrix V S ∈ R L ×D . Ultimately, v C = Enc M (m) and v S = Enc S (s).
Both sets of vectors, e and v are needed as the former learn to memorize the encoded inputs z, while the latter learn regularities in the attributes.
Text Decoder
The decoder takes the e C k , e S n , v C and v S , which encode content and style, as input and decodes text t . We use an LSTM network to model our decoder and provide the initial hidden state h 0 and initial cell state c 0 . The initial hidden state is the concatenation of e C k and e S n , while the initial cell state is the concatenation of v C and v S :
c 0 = v C • v S and h 0 = e C k • e S n .
When we decode the l-th word, we encode the previous word t l−1 and pay attention to the encoded sequence of content v C and style v S using the last hidden state as a query. Since both content and style are sequences of words, the attention mechanism can help figure out which part of them is important for decoding the current word. We concatenate the embedded previous output word and the attention output as the input for LSTM x l . The LSTM updates the hidden state, cell state and produces an output vector g l ∈ R 2D . Since we want to feed the generated text back to text encoders for additional control, we reduce g l to a word embedding dimension vector o l by a linear transformation. Finally, we map o l to the size of the vocabulary and apply softmax to obtain a probability distribution over the vocabulary.
x l = Emb(t l−1 ) • Attn(h l−1 , V C • V S ), (3) g l , (h l , c l ) = LSTM x l , (h l−1 , c l−1 ) ,(4)o l = W emb · g l + b emb ,(5)P (t l ) = softmax(WV · o l + bV ).(6)
The loss for text decoding is the sum of cross entropy loss of each word os L Dec = − l log P (t l ).
Content and Style Decoders
To ensure the generated text t conveys the correct content and style, we feed them to content and style decoders to perform backward prediction tasks that better control the generator. The decoders contain two components: we first reuse the text-to-content and text-to-style encoders to encode the embedded predicted text o L and obtain latent representations z C and z S , and then we classify them to predict content c and style s , as shown in the right side of Figure 1: z C = Enc TC (o L ) and z S = Enc TS (o L ). Enc TC (·) and Enc TS (·) denote the same text-to-content and text-to-style encoders we defined previously. This design is inspired by work on text style transfer (dos Santos et al., 2018).
Both z vectors and e vectors are used by two classification heads F C (multi-label) and F S (multi-class) for predicting content and style respectively in order to force those vectors to encode attribute information. We use g to denote the g-th element in the set of possible key-value pairs in the CMR and m(·) to represent an indicator function that returns whether the g-th element is in the ground-truth CMR.
P y C z (g) = m(g) = F C (z C ),(7)P (y S z = s) = F S (z S ),(8)P y C e (g) = m(g) = F C (e C k ),(9)P (y S e = s) = F S (e S n ).(10)
The loss for training the two prediction heads is:
LCTRL = − g log P y C e (g) = m(g) − log P (y S e = s) − g log P y C z (g) = m(g) − log P (y S z = s).(11)
Finally, we also adopt vector quantization by mapping each generated word's representation o l to the word embedding e V ∈ R |V |×D to map the output of the decoder and the input of text encoders in the same space. This is needed because the textto-* encoders expect as input text embedded using Figure 2: At prediction time we encode c and d with encoders to obtain v C and v S and we select e C k by sampling k ∼ P (K|C = c) and e S n by sampling n ∼ P (N |S = s). Those four vectors are provided as input to the text decoder to generate text. word embeddings, but in this case we are providing o L as input, and without this vector quantization loss, o L will not be in the same space of the embeddings. As a result, there is another VQ loss:
L V VQ = sg(o l ) − e V v 2 2 + β V o l − sg(e V v ) 2 2 .
The total loss minimized during training is the sum of the losses for decoding the text, predicting the content and style, the VQ-loss from two codebooks, and the VQ-loss for word embedding:
L = L Dec + L CTRL + L C VQ + L S VQ + L V VQ .
Prediction
The whole prediction process is depicted in Figure 2. The trained text decoder expect four inputs: v C , v S , e C k , and e S n . At prediction time, only content c and style s are given. We can obtain v C , v S by providing c and s to their respective encoder, but we also need to obtain e C k and e S n without input text. At the end of the training phase, we map each content c ∈ C and style s ∈ S to the indices in the e C and e S codebooks by first obtaining z S and z C vectors from the training data associated with c and s, we find the index of the closest codebooks vectors by argmin k e C k − z C 2 and argmin n e S n − z S 2 and count how many times each index k ∈ K was the closes to each c ∈ C and likewise for indices n ∈ N for each s ∈ S. By normalizing the counts, we obtain a distribution P (K|C) for content and a distribution P (N |S) for style. The construction of the two distributions is performed only once at the end of the training phase.
To obtain e C k at prediction time, we select the k vector of the codebook by sampling k ∼ P (K|C = c) and likewise to obtain e S n with n ∼ P (N |S = s). Sampling from those distri- butions allows to both focus on specific content and style disjointly by conditioning on them, while at the same time allowing variability because of the sampling (we refer to this procedure as focused variation). v C , v S , e C k , and e S n are finally provided as inputs to the decoder to generate the text t . Content and style decoders mentioned in the training section are not needed for prediction.
Experiments
To test the capability of FVN to generate diverse texts that convey the content while adopting a certain style, we use the PersonageNLG text generation dataset for dialogue systems that contains CMR and style annotations. To test if FVN can convey the content (both slots and values) correctly on an open vocabulary, with complex syntactic structures and diverse discourse phenomena, we use the End-2-End Challenge dataset (E2E), a text generation dataset for dialogue systems that is annotated with CMR. (Dušek et al., 2018), we delexicalized only 'name' and 'near' keeping the remaining slots' values. Since the E2E dataset does not have style annotations but has lexicalized texts, we model the CMR in the same way we did for PersonageNLG, but we replace the style codebook with a slot-value codebook that help the text decoder generating the slot values in the CMR. We build the focused variation distribution for every slot-value independently over the codebook indices, e.g. P (N |s = P riceRange[high])P (N |s = F oodT ype[F rench]).... During prediction we sample codes for each slot value in the CMR and use their average to condition text decoding. This is particularly useful when the surface forms in the output text are not the slot values themselves, e.g. when "PriceRange[high]" should be generated as "expensive" rather than "high".
Datasets and Baselines
We use NLTK (Bird et al., 2009) to tokenize each sentence and de-lexicalize the text as described in (Dušek and Jurcicek, 2016a). We use 300-dimensional GloVe embeddings (Pennington et al., 2014) trained on 840B words. Words not in GloVe are initialized as the averaged embeddings of all other embeddings plus a small amount of random noise to make them different from each other. The details of each module in the FVN are listed in Table 2. We set D = 300, K = 512, N = |V |. The encoders are three-layer stacked Bi-LSTM and the text decoder is one-layer LSTM. The style/slotvalue codebook is initialized as pre-trained word embedding. The content codebook is uniformed initialized in the range of [−1/K, 1/K]. We use the Adam optimizer (Kingma and Ba, 2015) with a learning rate of 0.001 for minimizing the total loss. More dataset details are shown in Appendix A Table 13 and Table 14.
We compare our proposed model against the best performing models in both datasets. All of them are sequence-to-sequence based models. For Per-sonageNLG, TOKEN, CONTEXT (from (Oraby et al., 2018)) are variants of the TGEN (Novikova et al., 2017) The results of the baselines (Oraby et al., 2018;Harrison et al., 2019) are taken from their original papers, but it's unclear if they were evaluated using a single or multiple references (for this reason they are marked with †), but since these models are not dependent on sampling from a latent space, we would not expect that to change performance.
We also compare to conditional VAEs: CVAE implements the conditional VAE (Sohn et al., 2015) framework. Controlled CVAE implements the controlled text generation (Hu et al., 2017) framework. The architecture and hyper-parameters of CVAE and controlled CVAE are the same as FVN.
The FVN ablations used in our evaluation are: (1) FVN-ED does not use the codebooks, only uses the content and style encoders and decoders, and is equivalent to an attention-augmented sequenceto-sequence model; (2) FVN-VQ does not use the content and style encoders and decoders, it directly uses the sampled latent vector for text decoding (3.3); (3) FVN-EVQ does not use content and style decoders; (4) FVN is the full network. Refer to Table 2 for architecture details. All VAEs and FVN variants are evaluated using multiple references because the sampling from latent space may lead to generate a valid and fluent text that n-gram overlap metrics would not score high when evaluated against a single reference.
Automatic Evaluation
We evaluate the quality and diversity of the generated text on both dataset. PersonageNLG is styleannotated and delexicalized, so we also report style and content correctness for it.
To evaluate quality in the generated text, we use the automatic evaluation from the E2E generation challenge, which reports BLEU (n-gram precision) (Papineni et al., 2002), NIST (weighted n-gram pre- , 2005), and ROUGE (n-gram recall) (Lin, 2004) scores using up to 9-grams. To evaluate content correctness, we report micro precision, recall, and F 1 score of slot special tokens in the generated text, with respect to the slots in the given CMR c. To evaluate diversity, we report the distinct n-grams of groundtruth and baselines' examples. For style evaluation, we separately train a personality classifier (with GloVe embeddings, 3 bi-directional LSTM layers, 2 feed-forward linear layers) on the PersonageNLG training data. The macro precision, recall, and F 1 score of the personality classifier on the test set is 0.996. We use this classifier to evaluate the style of the generated text and report our results in Table 5.
PersonageNLG Human Evaluation
In addition to automatic evaluation, we conducted a crowdsourced evaluation to compare our model against the ground truth on the entire test set. We did not compare our model with baselines since a pilot evaluation on a random sample of 100 data points from the test set suggested that baselines did not produce fluent enough text to compare with FVN. We considered the ground truth to be a performance upper bound and compared against it to find how close FVN is to it. Crowdworkers were pre- sented with a personality and two sentences (one is ground truth and the other one was generated by FVN) in random order, and were asked evaluate A) the fluency of the sentences in a scale from 1 to 3 and B) which of the two sentences was most likely to be uttered by a person with a given personality (more details in Appendix C). This evaluation was conducted on the entire test set consisting of 1,390 data points, 278 per personality, and each data point was judged by three different crowdworkers. We report the result of Question A in Table 9. For each sentence, we averaged the scores across three judges. The overall performance of FVN is very close to the ground truth (2.81 vs. 2.9), which suggests that FVN can generate text of comparable fluency with respect to ground truth texts.
We evaluated Question B using a majority vote of the three crowdworkers. Considering the overall performance, 50.29% of times human evaluators considered FVN generated text equal or better at conveying personality than the ground truth. This suggests that FVN can generate text with comparable conveyance with respect to ground truth.
More details and a full breakdown on the human evaluation are available in Appendix C.
Results and Analysis
Tables 3, 4 and 5 show the results on text quality, content correctness, and style. As shown in Table 3, FVN significantly outperforms the state-ofthe-art methods (context-m), especially on BLEU and NIST, which evaluate the precision of generated text, with the caveat regarding single or multiple references explained above. We believe this is due to the fact that FVN explicitly models CMR and style, while context-m depends on human-engineered features. Comparing FVN with CVAE and controlled CVAE, which are similar methods that also sample from the latent space, FVN performs better on all the metrics. Human evaluation results in Section 4.3 show that FVN is close to the ground truth in fluency and style.
Regarding the content correctness evaluation in Table 4, FVN overall performs much better than other baselines, especially on the recall score. Methods with explicit control decoders (controlled CVAE and FVN) perform better than CVAE and FVN-EVQ, which suggests that the controlling module is useful to enhance the content conveyance. Regarding the style evaluation in Table 5, all methods have good performance. Style is likely easy to convey in the text (the markers are pretty specific) and easy to identify for the separately trained personality classifier. Nevertheless, FVN is the best performing model. The text diversity comparison in Table 8 shows how FVN and its ablations have a diversity of generated texts with respect to the ground truth texts, but so do VAE-based methods. The combination of these findings suggests that FVN can produce text with comparable or better diversity than VAEs and ground truth, while conveying content and style more accurately.
Comparing with the ablations, the full FVN always performs better than FVN-ED and FVN-VQ, especially on the recall of slot tokens. FVN-VQ is able to precisely generate slot tokens from the CMR, but it cannot generate all required slot tokens, while FVN can generate them with high precision and substantially higher recall. An explanation is that the latent vectors in the content codebook only memorize the representations of texts without generalizing properly to new CMRs: since FVN is able to generate text containing most of the required slots, that text is usually longer than FVN-VQ's, which also explains why FVN performs better than FVN-VQ on METEOR and ROUGE-L that evaluate the recall of n-grams, and suggests that all encoders and codebooks are indeed needed for obtaining high performance.
The comparison between FVN and FVN-EVQ shows how in some cases FVN-EVQ has higher quality, but FVN obtains better scores on correctness and style, suggesting the additional decoder improves conveyance sacrificing some fluency.
In Table 7, we compare our proposed model and variants against the best performing models in the E2E challenge: TGEN (Novikova et al., 2017), SLUG (Juraska et al., 2018), and Thomson Reuters NLG (Davoodi et al., 2018;Smiley et al., 2018). extravert Name EatType Food PriceRange CustomerRating Area FamilyFriendly Near same e C k let 's see what we can find on Name SLOT . yeah, it is FamilyFriendly SLOT with a CustomerRating SLOT rating , it is a EatType SLOT , it is a Food SLOT place in Area SLOT , it is pricerange SLOT near Near SLOT . different e S n i do n't know . Name SLOT is a EatType SLOT with a CustomerRating SLOT rating , also it is a FamilyFriendly SLOT , Area SLOT , and it is a Food SLOT place near Near SLOT , also it has a price range of pricerange SLOT . Name SLOT is a EatType SLOT , it is a FamilyFriendly SLOT , it 's a Food SLOT place , it is near Near SLOT , it has a CustomerRating SLOT rating , you know pal! it is in Area SLOT and has a price range of pricerange SLOT . different e C k Name SLOT is a EatType SLOT with a CustomerRating SLOT rating , also it is a Food SLOT place , you know ! and it is Area SLOT , also it is FamilyFriendly SLOT near Near SLOT , also it has a price range of pricerange SLOT . same e S n Name SLOT is a EatType SLOT , it is a FamilyFriendly SLOT , it 's a Food SLOT place , it is near Near SLOT , it has a CustomerRating SLOT rating , you know and it is in Area SLOT and pricerange SLOT . Name SLOT is a EatType SLOT , it is a Food SLOT place , it is FamilyFriendly SLOT , it 's in Area SLOT , it is near Near SLOT , it has a CustomerRating SLOT rating and a price range of pricerange SLOT, you know! . Table 11: Diversity in FVN-generated PersonageNLG examples. Given the CMR and style the the generated text varies depending on the vector sampled from the codebook.
agreeable "let 's see what we can find on" "well , i see" "did you say ?" "i suppose" "right" "okay ?" " you see ?" "it is somewhat" disagreeable "oh god i mean , everybody knows" "oh god" "i do n't know ." "i am not sure ." conscientious "let 's see what we can find on" "well , i see" "did you say " " sort of " "you see ?" "let 's see, " "..." unconscientious "oh god i , i do n't know ." "darn" " i mean ." "i ... i , i do n't know ." "i mean , i am not sure ." "damn" "!" "it has like a " extravert "oh god i am not sure ." "let 's see ," "..." "alright ?" "yeah" "i do n't know" "did you say ?" "you know !" "you know and" "pal" "! " We can see from the results that FVN performs better than all these state-of-the-art models. The reason of the low performance of CVAE-based methods on the E2E dataset is that the CMR are disjoint in the train and test sets (while in PersonageNLG they are overlapping) and CVAEs struggle to handle unseen CMRs. FVNs performs well because it builds focused variations for each attribute independently instead of the entire CMR. Table 11 shows texts generated by FVN under the same CMR (given 8 attributes, rare in training data) and extravert style. The first three samples have the same CMR latent vector, but different sampled style latent vectors. The remaining three examples have different sampled CMR latent vectors, but the same style latent vector. In the first three examples, the generated texts and the words representing the extravert style are different ("let 's see what we can", "I don't know", "you know"). In the latter three examples, the words representing style are similar ("you know"), but the aggregation of attributes is different. These examples suggest that the two codebooks learn disjoint information and that the sampling mechanism introduces the desired variation in the generated texts. Table 11 shows that FVN learns disjoint content and style codebooks and that the vectors in the codebook can be explicitly interpreted by sampling multiple texts and observing the generated patterns. This is useful because, beyond sampling correct style vectors, we can select the realization of a style we prefer (Table 12 shows linguistic patterns associated with the top codes of each style). These patterns are automatically learnt and suggest that there is no need to encode them with manual features. Conditional VAEs do not provide this capability.
Samples obtained providing the same CMR and style to different models and examples of the linguistic patterns learned by FVN's style codebook are provided in Appendix D. Diverse samples obtained from FVN by sampling different latent codes are shown in Table 15.
Conclusion
In this paper, we studied the task of controlling language generation, with a specific emphasis on content conveyance and style variation. We introduced FVN, a novel model that overcomes the limitations of previous models, namely lack of conveyance and lack of diversity of the generated text, by adopting disjoint discrete latent spaces for each of the desired attributes. Our experimental results show that FVN achieves state-of-the-art performance on Per-sonageNLG and E2E datasets and generated texts are comparable to ground truth ones according to human evaluators. https://github.com/ neural-dialogue-metrics Table 13 shows details of the PersonageNLG dataset, while Table 14 shows details of the E2E dataset.
B Baselines details
The first three baselines are taken from (Oraby et al., 2018) and adopt the TGen architecture (Dušek and Jurcicek, 2016b), an encoder-decoder network, with different kinds of input.
TOKEN adds a token of additional supervision to encode personality. Unlike other works that use a single token to control the generator's output (Hu et al., 2017), the personality token encodes a several different parameters that define style.
CONTEXT introduces a context vector that explicitly encodes a set of 36 manually-defined style parameters encoded as a vector of binary values. We then apply these style encoding approaches to three state of the art models taken from (Harrison et al., 2019), which extend (Oraby et al., 2018) by changing the basic encoder-decoder network to OpenNMT-py (Klein et al., 2017) in the following ways.
m1 inserts style information into the sequence of tokens that constitute the content c; m2 incorporates style information in the content encoding process by concatenating style representation with content representation before passing it to the content encoder;
m3 incorporates style information into the generation process by additional inputs to the decoder. At each decoding step, style representation is concatenated with each word's embedding and passed as input to the decoder.
token-m means that style (personality here) is encoded with a single token;
context-m means that style is encoded via the 36 parameters.
TGEN (Novikova et al., 2017) adopts a seq2seq model with attention (Bahdanau et al., 2015) with added beam search and a reranker penalizing outputs that stray away from the input CMR.
SLUG (Juraska et al., 2018) adopts seq2seqbased ensemble which uses LSTM/CNN as the encoders and LSTM as the decoder); heuristic slot aligner reranking and data augmentation. Both TGEN and SLUG use partial ('name' and 'near' slot) de-lexicalized texts .
Thomson
C Human evaluation details
Crowdworkers were presented with a personality and two sentences (one is ground truth and the other one was generated by our model) in random order, and were asked to answer the following two questions:
• Question A: On a scale of 1-3, how grammatical or natural is this sentence? (please answer for both sentences).
• Question B: Which of these two sentences do you think would be more likely to be said by a(n) person? (Fill in with the personality given, e.g. agreeable) Answers: Sentence 1, 2, equally Question A asked the crowdworkers to assess the degree of grammaticality / naturalness of a sentence while Question B was designed to evaluate which of the two sentences exhibits a specific personality. -use the top frequent code of each value-The Phoenix is a french pub near Crowne Plaza Hotel in the city centre . It is not children friendly and has a price range of more than £30 and has a customer rating of 5 out of 5 .
-use same area[city centre] code, other values' codes are sampled-The Phoenix is a pub in the city centre . It is a french food . It is located in the city centre . The Phoenix is a pub in the city centre . It is a french food . It is a high price range and is not child friendly .
-use same customer rating[5 out of 5] code, other values' codes are sampled-The Phoenix is a french pub located in the city centre . It is a high customer rating and is not children friendly . The Phoenix is a pub in the city centre near Crowne Plaza Hotel . It is a high customer rating and is not children friendly .
-use same eattype [pub] code, other values' codes are sampled-The Phoenix is a french pub near Crowne Plaza Hotel in the city centre . It is not children friendly and has a price range of more than £30 . The Phoenix is a french pub in the city centre near Crowne Plaza Hotel . It is not child friendly and has a high price range and a customer rating of 5 out of 5 .
-use same familyFriendly[no] code, other values' codes are sampled-The Phoenix is a french pub located in the city centre near Crowne Plaza Hotel . It is not family-friendly and has a customer rating of 5 out of 5 . The Phoenix is a french pub located in the city centre . It is not family-friendly and has a customer rating of 5 out of 5 .
-use same food [French] code, other values' codes are sampled-The Phoenix is a french pub in the city centre . It is a high customer rating and is not children friendly . The Phoenix is a french pub located in the city centre . It is not family-friendly .
-use same priceRange[more than £30] code, other values' codes are sampled-The Phoenix is a french pub in the city centre . It is not children friendly and has a price range of more than £30 . The Phoenix is a french pub near Crowne Plaza Hotel in the city centre . It is not children friendly and has a price range of more than £30 . We report the result of Question A in Table 9. For each sentence, we averaged the scores across three judges, and conducted a paired t-test between the ground truth and our model for each personality. The result shows that the FVN sentences were considered significantly more grammatical / natural on conscientiousness and disagreeableness, the ground truth sentences were better on agreeable and unconscientiousness, and no difference was found for extravert. The overall performance of FVN is very close to the ground truth (2.81 vs. 2.9), which suggests that FVN can generate text of comparable fluency with respect to ground truth texts.
We evaluated Question B using a majority vote of the three crowdworkers. Table 10 shows the percentage frequency distribution for each personality and the entire test set. We found that our FVN model performs better than the ground truth on agreeable and conscientiousness, while the ground truth is better for the rest of the three personalities. Specifically, 53% and 67% of the time, the crowdworkers judge the agreeable and conscientious sentences generated by our model to be better than the ground truth sentences. This finding is surprising, since we consider the ground truth be an upper bound in this task, and our model outperforms it two out of five personalities. One possible explanation about why FVN only performs better on agreeable and conscientiousness is that the language patterns of agreeableness and conscientiousness are more systematic and thus easier to learn by the model. In Table 10 we also report a column that shows the percentage frequency of text where the judgment was equal or in favor of FVN. Underlined rows show when the number of equal judgments or judgments favorable to FVN exceeds the judgments that preferred the ground truth text. Considering the overall performance, 50.29% of times human evaluators considered FVN generated text equal or better at conveying personality than the ground truth. This finding suggests that FVN can generate text with comparable conveyance with respect to ground truth texts. Table 15 shows generated examples from FVN trained on E2E. Given a CMR, we sample a code for each slot value. The first part shows the generated text using the most frequent code for each slot value. We can see that the text is fluent and conveys the CMR precisely. In the remaining part, we keep one slot-value's code fixed and the remaining slot codes are sampled. The fixed slot-value is present, but some of the other slot-values are missing in the generated text. One explanation is that in the training data the text associated with a CMR can also contain missing values and therefore the codebook memorizes this behavior.
D Generated Samples and Linguistic Patterns
Emb(|V |, D), LSTM(D,2D), Attn(2D, 2D) Dense(2D, D), Dense(D,|V |)
architecture, while token-m* and context-m* are from (Harrison et al., 2019) (which adoptOpenNMT-py (Klein et al., 2017) as the basic encoder-decoder architecture). token-* base-lines use a special style token to provide style information while context-* baselines use 36 human defined pragmatic and aggregation-based features to provide style information '-m*' indicates variants of how the style information is injected into the encoder and the decoder. For the E2E challenge dataset, TGEN(Novikova et al., 2017),SLUG (Juraska et al., 2018), and Thomson Reuters NLG (Davoodi et al., 2018;Smiley et al., 2018) are the best performing models. They have different architectures, re-rankers, beam search and data augmentation strategies. More details are provided in Appendix B.
Reuters NLG (Davoodi et al., 2018; Smiley et al., 2018) use fully de-lexicalized text and a seq2seq model with hyperparameter tunning.
Fitzbillies is a pub with a decent rating. It is a moderately priced Italian restaurant in riverside near The Sorrento. It is not familyfriendly. Delex. Text 1 Name SLOT is a EatType SLOT with a CustomerRating SLOT rating. It is a PriceRange SLOT priced Food SLOT restaurant in Area SLOT near Near SLOT. It is FamilyFriendly SLOT. Agreeable Let's see what we can find on Name SLOT. I see, well, it is an EatType SLOT with a CustomerRating SLOT rating, also it is a PriceRange SLOT priced Food SLOT restaurant in Area SLOT and near Near SLOT, also it is FamilyFriendly SLOT, you see? Disagreeable I mean, everybody knows that Name SLOT is an EatType SLOT with a CustomerRating SLOT rating. It is a PriceRange SLOT priced Food SLOT restaurant in Area SLOT near Near SLOT. It is FamilyFriendly SLOT. Name SLOT is a Food SLOT place near Near SLOT in Area SLOT and PriceRange SLOT priced. It has a CustomerRating SLOT rating. It is an EatType SLOT and FamilyFriendly SLOT kid friendly. Conscientious Let's see what we can find in Name SLOT. Emm ... it is a Food SLOT place near Near SLOT in Area SLOT and PriceRange SLOT priced. It has a CustomerRating SLOT rating. It is an EatType SLOT and FamilyFriendly SLOT. Unconscientious Oh god yeah, I don't know. Name SLOT is a Food SLOT place near Near SLOT in Area SLOT and PriceRange SLOT priced. It has a CustomerRating SLOT rating. It is an EatType SLOT and FamilyFriendly SLOT kid friendly.CMR
Name[Fitzbillies], EatType[pub], Food[Italian], CustomerRating[decent], Area[Riverside], FamilyFriendly[No], Near[The Sorrento],
PriceRange[Moderate]
Text 1
Delex. Text 2
Table 1 :
1Text generation with focused variations (underlined red denotes personality, italics blue denotes content).The styles are personality traits (dis/agreeable, un/conscientious, extrovert). The content meaning representation
and neutral text (Text 1 and 2) are shown at the top. When given a style, the generated text strictly follows it. Delex
denotes delexicalised text.
Table 2 :
2Details of Modules in FVN: D = 300, K = 512, N = |V |
PersonageNLG contains 88,855 training and 1,390 test examples. We reserve 10% of the train set for validation. There are 8 slots in the CMR and 5 kinds of style: agreeable, disagreeable conscientious, unconscientious, and extravert personality traits. The styles are evenly distributed in both train and test sets. All slots' values are delexicalized. We model the focused variation distribution of the content by jointly modeling the presence of slot names in the CMR, e.g. P (K|P riceRange ∈ c and F oodT ype ∈ c), because there are no slot values. Style is modeled as a single categorical variable, e.g. P (N |s = Agreable). E2E contains 42,061 training examples (4,862 CMRs), 4,672 development examples (547 CMRs) and 4,693 test examples (630 CMRs). Like in the PersonageNLG dataset, there are 8 slots in the CMR. Each CMR has up to 5 realisations (references) in natural language. Differently from Per-sonageNLG, the CMRs in the different splits are disjoint and the texts are lexicalized. Following the challenge guidelines
Table 3 :
3Quality Evaluation for PersonageNLG.Model Precision Recall F 1 score
CVAE
0.961
0.942
0.952
Controlled CVAE
0.961
0.969
0.965
FVN-ED
0.997
0.748
0.855
FVN-VQ
0.87
0.799
0.833
FVN-EVQ
0.963
0.989
0.976
FVN
0.987
1.0
0.994
Table 4 :
4Content Correctness Evaluation for Peron-ageNLG. Micro precision, recall and F 1 score for "* SLOT" tokens.Model Precision Recall F 1 score
CVAE
0.973
0.973
0.973
Controlled CVAE
0.981
0.981
0.981
FVN-ED
0.996
0.996
0.996
FVN-VQ
1.0
1.0
1.0
FVN-EVQ
1.0
1.0
1.0
FVN
1.0
1.0
1.0
Table 5 :
5Style Evaluation on PersonageNLG. Macro precision, recall and F 1 score for the style of generated text based on a separately trained style classifier.Model 1-gram 2-gram 3-gram 4-gram
ground truth
0.74
0.902
0.924
0.905
CVAE 0.738
0.896
0.919
0.902
Controlled CVAE 0.715
0.869
0.902
0.899
FVN-ED 0.508
0.618
0.668
0.71
FVN-VQ
0.68
0.849
0.896
0.894
FVN-EVQ 0.738
0.883
0.907
0.901
FVN 0.720
0.870
0.906
0.904
Table 6 :
6Diversity Evaluation on PersonageNLG. Distinct n-grams between generated texts and ground truth.Model BLEU NIST METEOR ROUGE-L
TGEN 0.659 8.609
0.448
0.685
SLUG 0.662 8.613
0.445
0.677
Thomson Reuters NLG 0.681 8.778
0.446
0.693
Thomson Reuters NLG 0.674 8.659
0.450
0.698
CVAE 0.377 6.624
0.336
0.525
Controlled CVAE 0.404 6.852
0.346
0.544
FVN-ED 0.665 8.359
0.428
0.699
FVN-VQ 0.681 8.864
0.422
0.698
FVN-EVQ 0.711 9.066
0.453
0.721
FVN 0.714 9.004
0.451
0.719
Table 7 :
7Quality Evaluation on E2E.Model 1-gram 2-gram 3-gram 4-gram
ground truth 0.878
0.949
0.915
0.876
CVAE 0.841
0.931
0.900
0.859
Controlled CVAE 0.834
0.927
0.900
0.859
FVN-ED 0.839
0.924
0.898
0.858
FVN-VQ 0.855
0.943
0.91
0.869
FVN-EVQ 0.855
0.943
0.914
0.876
FVN 0.841
0.935
0.913
0.878
Table 8 :
8Diversity Evaluation on E2E. Distinct n-grams between generated texts and ground truth.Personality
GT
FVN
p
agreeable 2.8309 2.4412 ***
conscientiousness 2.9808 2.9976 **
disagreeable 2.8345 2.9388 ***
extravert 2.9221 2.8933
unconscientiousness 2.9365 2.7962 ***
overall 2.9001 2.8134 ***
*:p < 0.05, **:p < 0.01, ***:p < 0.001
Table 9 :
9The analysis result of Question A -grammaticality / naturalness. cision) (Doddington, 2002), METEOR (n-grams with synonym recall) (Banerjee and Lavie
Table 10 :
10The analysis result of Question B -person-ality. The percentage frequency distribution (%) over
three possible answers (GT, equal, FVN) for each per-
sonality is reported, with an additional column report-
ing the sum of equal and FVN. In this column, under-
lined values are those that exceed the ones reported in
the GT column.
Table 12 :
12Top codes' linguistic pattern of each style
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Dataset
Pairs
Number of Slots in CMR
CMRs
3
4
5
6
7
8
Train 88,855
600
0.13 0.30 0.29 0.22 0.06 0.01
Test
1,390
35
0.02 0.04 0.06 0.15 0.35 0.37
Table 13 :
13Distribution of slots in the CMR in both training and test splits of PersonageNLG. Pairs refer to content-utterance pairs.Dataset
Pairs
CMRs
Number of Slots in CMR
CMRs
3
4
5
6
7
8
Train 42061
4862
0.05 0.18 0.32 0.28 0.14 0.03
Dev
4672
547
0.09 0.11 0.05 0.35 0.30 0.10
Test
4693
630
0.01 0.03 0.08 0.17 0.34 0.37
Table 14 :
14Distribution of slots in the CMR in both training, development and test splits of E2E. Pairs refer to content-utterance pairs.These are the links to the adopted datasets and the code for computing the metrics.PersonageNLG text generation dataset: https://nlds.soe.ucsc.edu/ stylistic-variation-nlgEnd-2-End Challenge dataset (E2E): http://www.macs.hw.ac.uk/ InteractionLab/E2E/ Automatic evaluation metrics code from the E2E generation challenge: https://github.com/ tuetschek/e2e-metricsA Experiments details
Distinct
n-grams
metric
code:
area[city centre] customer rating[5 out of 5] eatType[pub] familyFriendly[no] food[French] name[The Phoenix] near[Crowne Plaza Hotel] priceRange[more than £30]
Table 15 :
15Diversity in FVN-generated E2E examples.
The code is available at https://leishu02. github.io/
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| [] |
[
"The need for a local nuclear physics feature in the neutron-rich rare-earths to explain solar r-process abundances",
"The need for a local nuclear physics feature in the neutron-rich rare-earths to explain solar r-process abundances"
] | [
"Nicole Vassh \nTRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBritish ColumbiaCanada\n\nDepartment of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA\n",
"Gail C Mclaughlin \nDepartment of Physics\nNorth Carolina State University\n27695RaleighNorth CarolinaUSA\n",
"Matthew R Mumpower \nTheoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n\nCenter for Theoretical Astrophysics\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n",
"Rebecca Surman \nDepartment of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA\n"
] | [
"TRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBritish ColumbiaCanada",
"Department of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA",
"Department of Physics\nNorth Carolina State University\n27695RaleighNorth CarolinaUSA",
"Theoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA",
"Center for Theoretical Astrophysics\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA",
"Department of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA"
] | [] | We apply Markov Chain Monte Carlo to predict the masses required to form the observed solar rprocess rare-earth abundance peak. Given highly distinct astrophysical outflows and nuclear inputs, we find that results are most sensitive to the r-process dynamics (i.e. overall competition between reactions and decays), with similar mass trends predicted given similar dynamics. We show that regardless of whether fission deposits into the rare-earths or not, our algorithm consistently predicts the need for a local nuclear physics feature of enhanced stability in the neutron-rich lanthanides.For more than 60 years the solar abundances have been providing clues to the astrophysical origins of heavy elements[1]. Although the era of multi-messenger astronomy presents new paths to understanding single events[2][3][4][5][6][7][8], the solar abundances still serve as the key informant of the contributions of a given site to the enrichment of the Solar System. To model the dominant astrophysical source of rapid neutron capture (r-process) elements in a modern way, statistical methods offer a fresh and innovative approach. With such methods, observational and experimental data can be used to trace back to more fundamental nuclear physics properties[9][10][11][12][13][14].The rare-earth abundance peak seen in the r-process residuals at A ∼ 164 is ideal for first applications of such statistical methods to the solar abundances since many nuclear species of importance have yet to be probed experimentally but some relevant nuclear physics information is available to guide the calculation. Unlike the second (A ∼ 130) and third (A ∼ 195) peaks linked to the neutron shell closures at N = 82 and N = 126 respectively, the mechanism by which the rare-earth peak forms is presently uncertain. Since the abundances in this region are highly sensitive to the astrophysical environment in which heavy element synthesis occurs, the answer to rare-earth peak origins can provide hints to the source of r-process elements in our galaxy. Rare-earth abundances are also highly sensitive to the nuclear properties of lanthanides thereby permitting small changes to inputs introduced via statistical methods to greatly influence the predicted outcome. Such an investigation is timely both due to the importance of lanthanide abundances for kilonova signals as well as the significant advancements at current and upcoming nuclear physics facilities.It is possible to use an MCMC procedure to derive mass adjustments to the Duflo-Zuker (DZ) mass model * [email protected], [email protected][15]which, when used in the r-process, produce consistency with the rare earth solar data[16][17][18][19]. Using MCMC with astrophysical outflows typical of simulations of accretion disk wind ejecta[20][21][22][23], predicts two different peak formation mechanisms that are associated with two broad classes of reaction dynamics (i.e overall competition between reactions and decays)[19]. In outflows with hot dynamics, (n,γ) (γ,n) equilibrium persists for long timescales and shapes the r-process path (location of most abundance species at a given Z). In contrast, in cold outflows photodissociation falls out of equilibrium early with the competition between neutron capture and β-decay largely determining how the r-process proceeds. When compared to independent, precision mass measurements, MCMC mass surface predictions in the hot case are more consistent than those in the cold case[19]. The primary impactful feature in the mass surface for the hot case is at N = 104 and produces a persistent 'pile-up' at this neutron number with the path having its highest abundances at N = 104 for many proton numbers[18,19]. This feature is just outside the latest measurements at N = 104.These conclusions were reached for moderately neutron-rich conditions, leaving a process which could play a key role in shaping rare-earth abundances remained unexplored, that is, fission. One might expect fission product deposition into the rare earth region to impact peak formation. In this case the rare-earth abundances could be influenced by the properties of actinide species lying far away in the nuclear chart rather than being purely shaped by the local nuclear features of lanthanide isotopes. In fact there has been speculation that late-time fission deposition could be the exclusive origin of the rare-earth peak[28]. Here we consider the sensitivity of the predictions of local nuclear physics feature to fission deposition by using yield predictions from the 2016 and 2018 versions of the GEF code [29] (GEF16 and GEF18), as well as a symmetric split of all fissioning nuclei in half (50/50) and compare to results obtained in arXiv:2202.09437v1 [nucl-th] | null | [
"https://arxiv.org/pdf/2202.09437v1.pdf"
] | 247,011,368 | 2202.09437 | ef870fb2298b67366d7a53e512a0de5138a2c28a |
The need for a local nuclear physics feature in the neutron-rich rare-earths to explain solar r-process abundances
18 Feb 2022
Nicole Vassh
TRIUMF
4004 Wesbrook MallV6T 2A3VancouverBritish ColumbiaCanada
Department of Physics
University of Notre Dame
Notre Dame
46556IndianaUSA
Gail C Mclaughlin
Department of Physics
North Carolina State University
27695RaleighNorth CarolinaUSA
Matthew R Mumpower
Theoretical Division
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
Center for Theoretical Astrophysics
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
Rebecca Surman
Department of Physics
University of Notre Dame
Notre Dame
46556IndianaUSA
The need for a local nuclear physics feature in the neutron-rich rare-earths to explain solar r-process abundances
18 Feb 2022(Dated: February 22, 2022)
We apply Markov Chain Monte Carlo to predict the masses required to form the observed solar rprocess rare-earth abundance peak. Given highly distinct astrophysical outflows and nuclear inputs, we find that results are most sensitive to the r-process dynamics (i.e. overall competition between reactions and decays), with similar mass trends predicted given similar dynamics. We show that regardless of whether fission deposits into the rare-earths or not, our algorithm consistently predicts the need for a local nuclear physics feature of enhanced stability in the neutron-rich lanthanides.For more than 60 years the solar abundances have been providing clues to the astrophysical origins of heavy elements[1]. Although the era of multi-messenger astronomy presents new paths to understanding single events[2][3][4][5][6][7][8], the solar abundances still serve as the key informant of the contributions of a given site to the enrichment of the Solar System. To model the dominant astrophysical source of rapid neutron capture (r-process) elements in a modern way, statistical methods offer a fresh and innovative approach. With such methods, observational and experimental data can be used to trace back to more fundamental nuclear physics properties[9][10][11][12][13][14].The rare-earth abundance peak seen in the r-process residuals at A ∼ 164 is ideal for first applications of such statistical methods to the solar abundances since many nuclear species of importance have yet to be probed experimentally but some relevant nuclear physics information is available to guide the calculation. Unlike the second (A ∼ 130) and third (A ∼ 195) peaks linked to the neutron shell closures at N = 82 and N = 126 respectively, the mechanism by which the rare-earth peak forms is presently uncertain. Since the abundances in this region are highly sensitive to the astrophysical environment in which heavy element synthesis occurs, the answer to rare-earth peak origins can provide hints to the source of r-process elements in our galaxy. Rare-earth abundances are also highly sensitive to the nuclear properties of lanthanides thereby permitting small changes to inputs introduced via statistical methods to greatly influence the predicted outcome. Such an investigation is timely both due to the importance of lanthanide abundances for kilonova signals as well as the significant advancements at current and upcoming nuclear physics facilities.It is possible to use an MCMC procedure to derive mass adjustments to the Duflo-Zuker (DZ) mass model * [email protected], [email protected][15]which, when used in the r-process, produce consistency with the rare earth solar data[16][17][18][19]. Using MCMC with astrophysical outflows typical of simulations of accretion disk wind ejecta[20][21][22][23], predicts two different peak formation mechanisms that are associated with two broad classes of reaction dynamics (i.e overall competition between reactions and decays)[19]. In outflows with hot dynamics, (n,γ) (γ,n) equilibrium persists for long timescales and shapes the r-process path (location of most abundance species at a given Z). In contrast, in cold outflows photodissociation falls out of equilibrium early with the competition between neutron capture and β-decay largely determining how the r-process proceeds. When compared to independent, precision mass measurements, MCMC mass surface predictions in the hot case are more consistent than those in the cold case[19]. The primary impactful feature in the mass surface for the hot case is at N = 104 and produces a persistent 'pile-up' at this neutron number with the path having its highest abundances at N = 104 for many proton numbers[18,19]. This feature is just outside the latest measurements at N = 104.These conclusions were reached for moderately neutron-rich conditions, leaving a process which could play a key role in shaping rare-earth abundances remained unexplored, that is, fission. One might expect fission product deposition into the rare earth region to impact peak formation. In this case the rare-earth abundances could be influenced by the properties of actinide species lying far away in the nuclear chart rather than being purely shaped by the local nuclear features of lanthanide isotopes. In fact there has been speculation that late-time fission deposition could be the exclusive origin of the rare-earth peak[28]. Here we consider the sensitivity of the predictions of local nuclear physics feature to fission deposition by using yield predictions from the 2016 and 2018 versions of the GEF code [29] (GEF16 and GEF18), as well as a symmetric split of all fissioning nuclei in half (50/50) and compare to results obtained in arXiv:2202.09437v1 [nucl-th]
We apply Markov Chain Monte Carlo to predict the masses required to form the observed solar rprocess rare-earth abundance peak. Given highly distinct astrophysical outflows and nuclear inputs, we find that results are most sensitive to the r-process dynamics (i.e. overall competition between reactions and decays), with similar mass trends predicted given similar dynamics. We show that regardless of whether fission deposits into the rare-earths or not, our algorithm consistently predicts the need for a local nuclear physics feature of enhanced stability in the neutron-rich lanthanides.
For more than 60 years the solar abundances have been providing clues to the astrophysical origins of heavy elements [1]. Although the era of multi-messenger astronomy presents new paths to understanding single events [2][3][4][5][6][7][8], the solar abundances still serve as the key informant of the contributions of a given site to the enrichment of the Solar System. To model the dominant astrophysical source of rapid neutron capture (r-process) elements in a modern way, statistical methods offer a fresh and innovative approach. With such methods, observational and experimental data can be used to trace back to more fundamental nuclear physics properties [9][10][11][12][13][14].
The rare-earth abundance peak seen in the r-process residuals at A ∼ 164 is ideal for first applications of such statistical methods to the solar abundances since many nuclear species of importance have yet to be probed experimentally but some relevant nuclear physics information is available to guide the calculation. Unlike the second (A ∼ 130) and third (A ∼ 195) peaks linked to the neutron shell closures at N = 82 and N = 126 respectively, the mechanism by which the rare-earth peak forms is presently uncertain. Since the abundances in this region are highly sensitive to the astrophysical environment in which heavy element synthesis occurs, the answer to rare-earth peak origins can provide hints to the source of r-process elements in our galaxy. Rare-earth abundances are also highly sensitive to the nuclear properties of lanthanides thereby permitting small changes to inputs introduced via statistical methods to greatly influence the predicted outcome. Such an investigation is timely both due to the importance of lanthanide abundances for kilonova signals as well as the significant advancements at current and upcoming nuclear physics facilities.
It is possible to use an MCMC procedure to derive mass adjustments to the Duflo-Zuker (DZ) mass model * [email protected], [email protected] [15] which, when used in the r-process, produce consistency with the rare earth solar data [16][17][18][19]. Using MCMC with astrophysical outflows typical of simulations of accretion disk wind ejecta [20][21][22][23], predicts two different peak formation mechanisms that are associated with two broad classes of reaction dynamics (i.e overall competition between reactions and decays) [19]. In outflows with hot dynamics, (n,γ) (γ,n) equilibrium persists for long timescales and shapes the r-process path (location of most abundance species at a given Z). In contrast, in cold outflows photodissociation falls out of equilibrium early with the competition between neutron capture and β-decay largely determining how the r-process proceeds. When compared to independent, precision mass measurements, MCMC mass surface predictions in the hot case are more consistent than those in the cold case [19]. The primary impactful feature in the mass surface for the hot case is at N = 104 and produces a persistent 'pile-up' at this neutron number with the path having its highest abundances at N = 104 for many proton numbers [18,19]. This feature is just outside the latest measurements at N = 104.
These conclusions were reached for moderately neutron-rich conditions, leaving a process which could play a key role in shaping rare-earth abundances remained unexplored, that is, fission. One might expect fission product deposition into the rare earth region to impact peak formation. In this case the rare-earth abundances could be influenced by the properties of actinide species lying far away in the nuclear chart rather than being purely shaped by the local nuclear features of lanthanide isotopes. In fact there has been speculation that late-time fission deposition could be the exclusive origin of the rare-earth peak [28]. Here we consider the sensitivity of the predictions of local nuclear physics feature to fission deposition by using yield predictions from the 2016 and 2018 versions of the GEF code [29] (GEF16 and GEF18), as well as a symmetric split of all fissioning nuclei in half (50/50) and compare to results obtained in moderately neutron rich conditions. To this end, we consider very neutron-rich outflows (Y e ∼ 0.01) that significantly populate fissioning nuclei. Such outflows have been consistently predicted to dominate the dynamical ejecta from a neutron star merger [26,27,[30][31][32]. In addition to the participation of fission, further distinctions between very neutron-rich conditions as compared to moderately neutron-rich cases exist. First, the r-process path tends to reach isotopes further from stability. Second, the higher neutron richness implies a greater potential for late-time neutron capture to play a role in shaping the rare-earth peak. Thus, considering ejecta which reaches fissioning nuclei applies our method to cases which have the maximal possible deviation from the moderately neutron-rich cases considered in [19] due to both the distinct evolution of their hydrodynamic properties as well as the impact of fission deposition on abundances. Therefore this work reports on the masses predicted to be needed for rare-earth peak formation given the largest set of astrophysical and theoretical nuclear physics inputs considered to date.
As was done with the moderately neutron-rich cases, we consider several very neutron-rich conditions with a range of hydrodynamic properties whose initial conditions are outlined in Table I. In addition to diverse initial conditions, the cases considered also have distinct temperature/density evolutions. For instance, 0.1 sec af-ter the 8 GK initial conditions, the densities and temperatures of the moderately neutron-rich cases hot wind and cold wind have dropped to ∼ 10 5 , 10 3 g/cm 3 and 3.2, 0.7 GK respectively while the densities and temperatures of the very neutron-rich dynamical ejecta outflows hot dyn sf , cold dyn sf , and cold dyn 5050 have fallen to ∼ 10 6 , 10 4 , 10 4 g/cm 3 and 0.5, 1.0, 0.4 GK respectively (with the low temperature, low density condition being that of tidal tail ejecta and the highest density condition being that of shock-heated ejecta).
In very neutron-rich ejecta, the nuclear reheating effects on the temperature become more pronounced than in moderately neutron-rich conditions. We therefore consider the impact of reheating on the dynamics of the outflow by using a cold condition with a heating efficiency of zero (cold dyn sf ) as well as outflows for which the reheating efficiency is non-zero. We note that even with reheating included, there remain outflow conditions under which the dynamics remain of the cold type (cold dyn 5050, cold dyn GEF 16, cold dyn GEF 18). Considering a range of outflow evolutions is an important part of analyzing the robustness of our previous findings that cases with distinct dynamics (hot vs. cold) require distinct lanthanide masses in order to form the rare-earth peak. Thus for the present investigation of very neutron-rich ejecta, we again separately consider peak formation in hot and cold scenarios. On the left abundances are the unscaled network output with mass fractions summing to one whereas on the right results are scaled to the solar abundances between A = 150 − 180. The solar abundances and uncertainties were derived from those in [33,34] as described in [19].
We show abundance patterns for all the astrophysical outflows and nuclear inputs outlined in Table I in Figure 1. In all cases, our baseline abundances are flat on average with no peak structure or show a rough peak which is off center from that seen the solar data. Here the influence of the fission prescription on the final abundances is well demonstrated since prior to applying our MCMC procedure, cases with 50/50 yields (hot dyn sf , hot dyn 5050, cold dyn sf , and cold dyn 5050) concentrate deposition near the second peak and leave the rare-earth abundances to be structured by local effects whereas in the case of GEF16 yields (hot dyn GEF 16 and cold dyn GEF 16) deposition occurs between the second peak and rare-earth peak. A stronger influence from fission deposition with GEF18 yields (hot dyn GEF 18 and cold dyn GEF 18) is evident from the enhanced abundances near A ∼ 160. When considering the χ 2 fit in the A = 150 − 180 region, this enhancement in abundances from fission deposition to the left of the rare-earth peak produces a higher χ 2 value when the abundances scaled to the peak are compared to the solar data.
In the right panel we show the results of the application of the MCMC method. We start with a mass parameter-ization:
M (Z, N ) = M DZ (Z, N ) + a N e −(Z−C) 2 /2f(1)
where M DZ (Z, N ) is the DZ mass of nuclear species (Z, N ), a N are the mass adjustments determined by the MCMC procedure, and the exponential acts to center the adjustments in the neutron-rich region where masses are largely unmeasured. At each MCMC step, neutron capture rates and beta decay properties are adjusted according to the test masses, and a nucleosynthesis calculation is performed and compared to solar data to set the evolution of the Markov chain. Our general procedure was first introduced in [16,17] and revised and refined in [18,19]. Even though here we consider cases with higher initial χ 2 values than were considered in [19], in all cases we are able to obtain substantially lower χ 2 values after applying our MCMC procedure, as can be seen in the right panels of Figure 1. For instance, for all the hot, very neutron-rich dynamical ejecta considered here, our MCMC procedure is able to find solutions with χ 2 ≤ 40, with most being around 35, which is comparable to what the MCMC was able to achieve when faced with moderately neutron-rich conditions. Therefore, despite a wide variety of astrophysical outflow types and nuclear inputs, our MCMC algorithm is able to find a solution for the masses of neutron-rich rare-earths which gives abundances that are significantly more consistent with the solar data. We next more explicitly demonstrate the influence of fission deposition on rare-earth peak abundances using the hot dynamical ejecta cases. In Figure 2 we show a snapshot of the summed neutron-induced and β-delayed fission flow of fissioning nuclei multiplied by their fission yield for the three distinct fission yield treatments considered in this work. The r-process path is also shown to demonstrate that fission deposition is occurring during a time at which the nuclei which will go on to form the rare-earth peak are undergoing pile-up. Given a symmetric 50/50 split for all nuclei, deposition remains isolated near the N = 82 shell closure. The yields predicted by the GEF16 model transition from asymmetric to symmetric across the nuclear chart (see [35]), which populates neutron-rich isotopes that will decay back to set abundances on the left side of the rare-earth peak. When compared to GEF16, the yields predicted by GEF18 have enhanced asymmetries for several nuclei, such as those near N = 184, thereby having greater amounts of deposition into regions set to populate central species in the rare-earth peak. Note that in an astrophysical outflow with fission cycling such as those considered here, nuclei in the rare-earths go through a wave or multiple waves of first being maximally populated to then be depleted as they undergo capture to heavier species, with fission eventually repopulating these nuclei so that the process can then repeat. Since this all occurs as the ejecta is expanding and cooling, there is a time sensitive connection between fission product deposition and the ultimate population of lanthanides versus actinides. Despite this complex interplay between the lanthanides and actinides, any isotopes that remain in the rare-earth region after the conditions can no longer support capture up to the actinides must decay back to stability and be subject to local nuclear structure influences.
The masses derived using our MCMC method in all hot and cold dynamical ejecta cases are shown in Fig. 3, with an explicit comparison to our previous findings for moderately neutron-rich ejecta in the top panels. For the very neutron-rich outflow results in the top panels, the hot dynamical ejecta case hot dyn sf uses an outflow tracer from [25] which accounts for nuclear reheating. The cold dynamical ejecta case cold dyn sf uses an outflow tracer from [26] and exemplifies an extreme of cold conditions since the efficiency of nuclear heating is assumed to be zero. For the cases featured in the top panels, we consider a simplified fission treatment in which all nuclei with A ≥ 250 spontaneously fission at a very fast rate to then deposit near the second r-process peak and not in the rare-earth peak by applying symmetric 50/50 fission yields. Such a simplification was first considered for two main reasons. Firstly it allows to build predictions with comparable statistics to the moderately neutron- [19] (red band) compared to results with the very neutron-rich condition hot dyn sf (yellow band), (top right) the moderately neutron-rich condition cold wind explored in [19] (dark blue band) compared to results for the very neutron-rich condition cold dyn sf (light blue band). (Bottom left) Results for the very neutron-rich conditions of hot dyn 5050 (dark pink band) compared to cases hot dyn GEF 16 (orange band) and hot dyn GEF 18 (light pink band) which apply yields that give some late time deposition into the rare-earth region. (Bottom right) Results given the very neutron-rich conditions of cold dyn 5050 (teal band) as compared to the cold dyn GEF 16 (purple band) and cold dyn GEF 16 (medium blue band) cases. The AME2012 data [36] used to guide the calculation is also shown, along with AME2016 [37] and CPT at CARIBU [18] data of which the calculation was not informed.
rich cases (both derived from 50 MCMC runs) since running the nucleosynthesis network at each MCMC step is much less costly than is the case when more proper fission rates and yields are considered. Secondly our simplified fission treatment removes the effects of late time fission deposition on rare-earth abundances (by contributing exclusively to the abundances near A ∼ 130) thereby permitting us to isolate how our predictions change due solely to the distinctions in the outflow properties of very neutron-rich cases as compared to moderately neutronrich cases. We note that as in [19], we center our mass adjustments at C = 60 in the hot cases and C = 58 in the cold cases (the r-process paths in cold cases tend to lie further from stability, therefore in test runs mass adjustments near C = 58 were favored). As can be seen from the top panels of Fig. 3, although some differences in the masses predictions exist, the MCMC results for the very neutron-rich hot dyn sf case have key similarities to the results when the moderately neutron-rich case hot wind is considered. Both show a rise in the mass surface near N = 102 followed by the dip at N = 104 to be the features primarily responsible for the peak structure near A = 164. Key similarities are also observed in results for the cold outflows cold wind and cold dyn sf except that an influence from a nuclear physics feature at N = 110 emerges in the dynamical ejecta case considered here, whereas in the moderately neutron-rich cold wind case N = 103 coupled with N = 108 features were sufficient to form the peak. Therefore although the neutronrichness does influence the details of peak formation, similar peak formation mechanisms are needed for conditions with similar r-process dynamics. Particularly, in the case of hot outflows, the mechanism which we previously found to be responsible for peak formation, that is the pile-up at N = 104, is robust. We now discuss whether the fission treatment significantly modifies the expected local nuclear masses of the lanthanides needed in order to form the peak. To do so, we move away from the simplified treatment previously described and apply a more proper treatment of fission rates by utilizing predictions from CoH and BeoH Hauser-Feshbach codes [38] when DZ masses and FRLDM barriers [39] are assumed (similar to the treatment described in [35]). We also advance the fission yield treatment by implementing the three models previously described. The bottom panels of Fig. 3 show the MCMC predictions when these nuclear data inputs are considered in both hot dynamical ejecta (hot dyn 5050, hot dyn GEF 16, and hot dyn GEF 18) and cold dynamical ejecta (cold dyn 5050, cold dyn GEF 16, and cold dyn GEF 18). Additionally, we note that here for the cold case we consider a merger tidal tail ejecta tracer from [27] where we have included the effect of nuclear reheating on the trajectory but this cases nevertheless retains its cold dynamics. In all scenarios considered in the bottom panel, running the nucleosynthesis network on each MCMC step is costly due to the need to follow hundreds of fission products. Therefore to reduce the computational cost, we apply the method outlined in [35] whereby only the fissioning nuclei which most influence the abundances are treated with GEF16 and GEF18 yields and a 50/50 split is applied for all other species. Although this reduces the cost of running the network significantly, there is nevertheless a greater computational expense than in the simplified fission case considered in the top panels of Fig. 3. Therefore since here our primary aim is to test the robustness of the similarities we see in the predictions for hot versus cold cases as well as the robustness of the nuclear physics feature we repeatedly see at N = 104 in hot cases, our MCMC predictions for calculations which apply a more proper fission treatment are sufficiently explored through 25 MCMC runs in the hot dynamical ejecta cases and 15 MCMC runs in the cold dynamical ejecta cases.
The influence of the fission yield treatment is evident in the bottom panels of Fig. 3 for both the hot and cold cases. For hot dynamical ejecta, the build-up in the abundances to the left of the rare-earth peak due to the deposition from GEF16 and GEF18 fission yields (as can be seen in Fig. 1) must be suppressed. This is achieved by introducing a dip feature in the mass surface at N = 99 which was not needed for the hot dyn 5050 and hot dyn sf cases for which fission deposition plays no role in setting rare-earth abundances. In the case of hot dyn GEF 18, the GEF18 yield prescription also requires a mass surface feature at N = 108 in order for the late-time deposition seen with this model to stay contained in the rare-earth region, although this works to mostly fill in the right side of the peak. Strikingly, and most importantly, in the hot case all yield models considered point to the now familiar need for a nuclear physics feature of enhanced stability at N = 104 to produce the pile-up which ultimately creates the rare-earth peak at A = 164. For the cold dynamical ejecta MCMC runs, it is primarily a feature at N = 101 which is responsible for peak formation which is in tension with the latest precision mass measurements. Therefore despite the distinct ways in which nuclei are populated in the rare-earth region, our MCMC algorithm finds that all fission yield models and all astrophysical outflows considered require local assistance to form the peak via a nuclear physics feature emerging in neutron-rich lanthanides.
In this work, we have presented the MCMC mass predictions which are capable of forming the r-process rareearth abundance peak given highly distinct astrophysical outflows and nuclear data treatments. While our study of moderately neutron-rich nuclei in [19] was able to show that: (1) outflows with distinct reaction dynamics (hot vs. cold) require distinct mass surfaces in order to form the rare-earth peak and (2) the mass predictions in hot outflows are most consistent with the latest measurements up to N = 102, the investigation presented in this work is able to significantly broaden what can be concluded from our MCMC procedure. The detailed exploration of peak formation in very neutron-rich ejecta presented here shows that such conditions require mass trends which are similar to those found given moderately neutron-rich outflows. This highlights that it is the rprocess dynamics (e.g. hot vs. cold), rather than the exact details of the hydrodynamic properties, which is most influential on peak formation. This work therefore could be used alongside simulation advancements to infer the astrophysical site at which solar lanthanides where dominantly formed by considering the degree of participation of photodissociation in simulation outflows. Additionally the calculations presented in this work show that although the fission treatment does produce some noteworthy differences in the masses predictions, the algorithm nevertheless consistently predicts the need for a local nuclear physics feature in the neutron-rich lanthanides in order to form the peak, despite the differences in fission deposition and outflow properties considered here. In all results given astrophysical outflows with cold dynamics, our algorithm predicts features which are in tension with the latest experimental data, although here features at neutron numbers which are several neutrons higher than what has been probed are important. In the case of hot outflows, our MCMC mass predictions persistently allude to the presence of a nuclear physics feature causing enhanced stability in the lanthanides at N = 104, just a few neutron numbers outside current measurements. Therefore, since our algorithm finds that fission deposition must be aided by local lanthanide nuclear features to form the peak, near future theoretical and experimental campaigns to map out neutron-rich lanthanide properties, as could be possible at ARIEL at TRIUMF, FRIB, and N=126 Factory at ANL, are well poised to explicitly address a long-standing mystery of heavy element production, that is, the origin the rare-earth peak.
FIG. 1 .
1(Top left) Baseline abundances for all hot astrophysical outflows and fission treatments outlined in Table I (with χ 2 values of 200, 121, 236, 212, and 211 respectively). (Top right) Final abundances for the hot conditions given the mass solutions found for each case using our MCMC procedure (with χ 2 values of 23, 37, 40, 36, and 35). (Bottom left) Same as top left but for all cold astrophysical outflows considered (with χ 2 values of 286, 361, 211, 499, and 488). (Bottom right) Same as top right but for the case of cold conditions (with χ 2 values of 22, 106, 68, 185, and 148).
FIG. 2 .FIG. 3 .
23The summed fission flow (rate×abundance) of neutron-induced and β-delayed fission for each fissioning species multiplied by its fission yield to demonstrate deposition with (top) 50/50 splits, (middle) GEF16 yields, and (bottom) GEF18 yields (cases hot dyn 5050, hot dyn GEF 16 and hot dyn GEF 18, respectively) at a given instance in time (1.4 seconds with temperature 1.1 GK and density 3.4×10 3 g/cm 3 ). For reference, the dark pink (top), orange (middle) and light pink (bottom) show the r-process path at this time and the grey shows the DZ dripline. Purple boxes highlight the stable nuclei with A = 150 − 180, whose abundances following the β-decay of r-process species determines the structure of the rare-earth peak, and black boxes denote all other stable species. The MCMC predicted masses for neodymium (Z = 60), relative to the DZ mass model, given (top left) the moderately neutron-rich hot wind outflow explored in
TABLE I .
IDescription of all astrophysical outflows and nuclear inputs considered in this work. Note the initial conditions (Ye, entropy, and density) are all reported at 8 GK with the entropies given assuming the SFHo equation of state[24].Label
Description
Type Ye
Initial
Entropy
(s/kB)
Initial
Density
(g/cm 3 )
Fission
Yields
Fission Rates
hot wind
parameterized, low entropy outflow
as from an accretion disk
hot 0.2
30
8.4×10 6 N/A
N/A
cold wind
parameterized, low entropy outflow
as from an accretion disk
cold 0.2
10
6.1×10 7 N/A
N/A
hot dyn sf
NSM dynamical ejecta simulation
[25] with reheating
hot 0.01 9
1.1×10 9 50/50
instantaneous spontaneous
fission for A ≥ 250
hot dyn 5050
NSM dynamical ejecta simulation
[25] with reheating
hot 0.01 9
1.1×10 9 50/50
neutron-induced,
β-delayed, and spontaneous
fission from FRLDM
barriers + DZ masses
hot dyn GEF 16
NSM dynamical ejecta simulation
[25] with reheating
hot 0.01 9
1.1×10 9 GEF16 same as hot dyn 5050
hot dyn GEF 18
NSM dynamical ejecta simulation
[25] with reheating
hot 0.01 9
1.1×10 9 GEF18 same as hot dyn 5050
cold dyn sf
NSM dynamical ejecta simulation
[26], no reheating
cold 0.01 22
2.8×10 7 50/50
instantaneous spontaneous
fission for A ≥ 250
cold dyn 5050
NSM dynamical ejecta simulation
[27] with reheating
cold 0.02 4
7.0×10 10 50/50
neutron-induced,
β-delayed, and spontaneous
fission from FRLDM
barriers + DZ masses
cold dyn GEF 16
NSM dynamical ejecta simulation
[27] with reheating
cold 0.02 4
7.0×10 10 GEF16 same as cold dyn 5050
cold dyn GEF 18
NSM dynamical ejecta simulation
[27] with reheating
cold 0.02 4
7.0×10 10 GEF18 same as cold dyn 5050
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| [] |
[
"Reverse Engineering and Symbolic Knowledge Extraction on Lukasiewicz Fuzzy Logics using Linear Neural Networks",
"Reverse Engineering and Symbolic Knowledge Extraction on Lukasiewicz Fuzzy Logics using Linear Neural Networks"
] | [
"Carlos Leandro [[email protected]] \nDepartamento de Matemática\nInstituto Superior de Engenharia de Lisboa\nPortugal\n"
] | [
"Departamento de Matemática\nInstituto Superior de Engenharia de Lisboa\nPortugal"
] | [] | This work describes a methodology to combine logic-based systems and connectionist systems. Our approach uses finite truth valued Lukasiewicz logic, where we take advantage of fact, presented by Castro in[6], what in this type of logics every connective can be define by a neuron in an artificial network having by activation function the identity truncated to zero and one. This allowed the injection of first-order formulas in a network architecture, and also simplified symbolic rule extraction.Our method trains a neural network using Levenderg-Marquardt algorithm, where we restrict the knowledge dissemination in the network structure. We show how this reduces neural networks plasticity without damage drastically the learning performance. Making the descriptive power of produced neural networks similar to the descriptive power of Lukasiewicz logic language, simplifying the translation between symbolic and connectionist structures.This method is used in the reverse engineering problem of finding the formula used on generation of a truth table for a multi-valued Lukasiewicz logic. For real data sets the method is particulary useful for attribute selection, on binary classification problems defined using nominal attribute. After attribute selection and possible data set completion in the resulting connectionist model: neurons are directly representable using a disjunctive or conjunctive formulas, in the Lukasiewicz logic, or neurons are interpretations which can be approximated by symbolic rules. This fact is exemplified, extracting symbolic knowledge from connectionist models generated for the data set Mushroom from UCI Machine Learning Repository. | null | [
"https://arxiv.org/pdf/1604.02774v1.pdf"
] | 3,030,573 | 1604.02774 | 474e8093e5e6b04df2851b86070b327253651113 |
Reverse Engineering and Symbolic Knowledge Extraction on Lukasiewicz Fuzzy Logics using Linear Neural Networks
11 Apr 2016 April 12, 2016
Carlos Leandro [[email protected]]
Departamento de Matemática
Instituto Superior de Engenharia de Lisboa
Portugal
Reverse Engineering and Symbolic Knowledge Extraction on Lukasiewicz Fuzzy Logics using Linear Neural Networks
11 Apr 2016 April 12, 2016
This work describes a methodology to combine logic-based systems and connectionist systems. Our approach uses finite truth valued Lukasiewicz logic, where we take advantage of fact, presented by Castro in[6], what in this type of logics every connective can be define by a neuron in an artificial network having by activation function the identity truncated to zero and one. This allowed the injection of first-order formulas in a network architecture, and also simplified symbolic rule extraction.Our method trains a neural network using Levenderg-Marquardt algorithm, where we restrict the knowledge dissemination in the network structure. We show how this reduces neural networks plasticity without damage drastically the learning performance. Making the descriptive power of produced neural networks similar to the descriptive power of Lukasiewicz logic language, simplifying the translation between symbolic and connectionist structures.This method is used in the reverse engineering problem of finding the formula used on generation of a truth table for a multi-valued Lukasiewicz logic. For real data sets the method is particulary useful for attribute selection, on binary classification problems defined using nominal attribute. After attribute selection and possible data set completion in the resulting connectionist model: neurons are directly representable using a disjunctive or conjunctive formulas, in the Lukasiewicz logic, or neurons are interpretations which can be approximated by symbolic rules. This fact is exemplified, extracting symbolic knowledge from connectionist models generated for the data set Mushroom from UCI Machine Learning Repository.
Introduction
There are essentially two representation paradigms, namely, connectionist representations and symbolic-based representations, usually taken as very different. On one hand, symbolic based descriptions is specified through a grammar having a fairly clear semantics, can codify structured objects, in some cases support various forms of automated reasoning and can be transparent to users. On the other hand the usual way to see information presented using connectionist description, is its codification on a neural network. Artificial neural networks in principle combine, among other things, the ability to learn (and be trained) with massive parallelism and robustness or insensitivity to perturbations of input data. But neural networks are usually taken as black boxes providing little insight into how the information is codified. They have no explicit, declarative knowledge structure that allows the representation and generation of explanation structures. Thus, knowledge captured by neural networks is not transparente to users and cannot be verified by domain experts. To solve this problem, researchers have been interested in developing a humanly understandable representation for neural networks.
It is natural to seek a synergy integrating the white-box character of symbolic base representation and the learning power of artificial neuro networks. Such neuro-symbolic model are currently a very active area of research. One particular aspect of this problem which been considered in a number of papers, see [5] [17] [18] [19] [20], is the extraction of logic programs from trained networks.
Our approach to neuro-symbolic models and knowledge extraction is based on trying to find a comprehensive language for humans representable directly in a neural network topology. This has been done for some types of neuro networks like Knowledge-based networks [10] [27]. These constitute a special class of artificial neural network that consider crude symbolic domain knowledge to generate the initial network architecture, which is later refined in the presence of training data. In the other direction there has been widespread activity aimed at translating neural language in the form of symbolic relations [11] [12] [26]. This processes served to identify the most significant determinants of decision or classification. However this is a hard problem since often an artificial neural network with good generalization does not necessarily imply involvement of hidden units with distinct meaning. Hence any individual unit cannot essentially be associated with a single concept of feature of the problem domain. This the archetype of connectionist approaches, where all information is stored in a distributed manner among the processing units and their associated connectivity. In this work we searched for a language, based on fuzzy logic, where the formulas are simple to inject in a multilayer feedforward network, but free from the need of given interpretation to hidden units in the problem domain.
For that we selected the language associated to a many-valued logic, the Lukasiewicz logic. We inject and extract knowledge from a neural network using it. This type of logic have a very useful property motivated by the "linearity" of logic connectives. Every logic connective can be define by a neuron in an artificial network having by activation function the identity truncated to zero and one [6]. Allowing the direct codification of formulas in the network architecture, and simplifying the extraction of rules. This type of back-propagation neural network can be trained efficiently using the Levenderg-Marquardt algorithm, when the configuration of each neuron is conditioned to converge to predefined patters associated or having directed representation in Lukasiewicz logic.
This strategy presented good performance when applied to the reconstruction of formulas from truth tables. If the truth table is generated using a formula from the language Lukasiewicz first order logic the optimum solution is defined using only units directly translated in formulas. In this type of reverse engineering problem we presuppose no noise. However the process is stable for the introduction of Gaussian noise on the input data. This motivate the application of this methodology to extract comprehensible symbolic rules from real data. However this is a hard problem since often an artificial neural network with good generalization does not necessarily imply that neural units can be translated in a symbolic formula. We describe, in this work, a simple rule to generate symbolic approximation to these unrepresentable configurations.
The presented process, for reverse engineering, can be applied to data sets characterizing a property of an entity by the truth value for a set of propositional features. And, it proved to be an excelente procedure for attribute selection. Allowing the data set simplification, by removing irrelevant attributes. The process when applied to real data generates potencial unrepresentable models. We used the relevant inputs attributes on this models as relevante attributes to the knowledge extraction problem, deleting others. This reduces the problem dimension allowing the potencial convergence to a less complex neuronal network topology.
Overview of the paper: After present the basic notions about may valued logic and how can Lukasiewicz formulas be injected in a neural network. We describe the methodology for training a neural network having dynamic topology and having by activation function the identity truncated to zero and one. This methodology uses the Levenderg-Marquardt algorithm, with a special procedure called smooth crystallization to restrict the knowledge dissemination in the network structure. The resulting configuration is pruned used a crystallization process, where only links with values near 1 or -1 survive. The complexity of the generate network is reduced by applying the "Optimal Brain Surgeon" algorithm proposed by B. Hassibi, D. G. Stork and G.J. Stork. If the simplified network doesn't satisfies the stoping criteria, the methodology is repeated in a new network, possibly with a new topology. If the data used on the network train was generated by a formula, and have sufficient cases, the process converge for a prefect connectionist presentation and every neural unit in the neural network can be reconverted to a formula. We finish this work by presenting how the describe methodology could be used to extract symbolic knowledge from real data, and how the generated model could be used as a attribute selection procedure.
Preliminaries
We begin by presenting the basic notions we need from the subjects of many valued logics, and how formulas in its language can be injected and extracted from a back-propagations neural network.
Many valued logic
Classical propositional logic is one of the earliest formal systems of logic. The algebraic semantics of this logic is given by Boolean algebra. Both, the logic and the algebraic semantics have been generalized in many directions. The generalization of Boolean algebra can be based in the relationship between conjunction and implication given by
x ∧ y ≤ z ⇔ x ≤ y → z ⇔ y ≤ x → z.
These equivalences, called residuation equivalences, imply the properties of logic operators in a Boolean algebras. They can be used to present implication as a generalize inverse for the conjunction.
In application of fuzzy logic the properties of Boolean conjunction are too rigid, hence it is extended a new binary connective ⊗, usually called fusion. Extending the commutativity to the fusion operation, the residuation equivalences define an implication denoted in this work by ⇒ :
x ⊗ y ≤ z ⇔ x ≤ y ⇒ z ⇔ y ≤ x ⇒ z.
This two operators are supposed defined in a partially ordered set of truth values (P, ≤), extending the two valued set of an Boolean algebra. This defines a residuated poset (P, ⊗, ⇒, ≤), where we interprete P as a set of truth values. This structure have been used on the definition of many types of logics. If P have more than two values the associated logics are called many-valued logics. An infinite-valued logic is a many valued logic with P infinite.
We focused our attention on many-valued logics having [0, 1] as set of truth values. In this logics the fusion operator ⊗ is known as a t -norm. In [13] it is defined as a binary operator defined in [0, 1] commutative and associative, non-decreasing in both arguments and 1 ⊗ x = x and 0 ⊗ x = 0.
The following are example of t-norms. All are continuous t-norms
1. Lukasiewicz t-norm: x ⊗ y = max(0, x + y − 1).
2. Product t-norm: x ⊗ y = xy usual product between real numbers.
3. Gödel t-norm: x ⊗ y = min(x, y).
In [9] all continuous t-norms are characterized as ordinal sums of Lukasiewicz, Gödel and product t-norms. Many-valued logics can be conceived as a set of formal representation languages that proven to be useful for both real world and computer science applications. And when they are defined by continuous t-norms they are known as fuzzy logics.
Processing units
As mention in [1] there is a lack of a deep investigation of the relationships between logics and neural networks. In this work we present a methodology using neural networks to learn formulas from data. And where neural networks are trate as circuital counterparts of (functions represented by) formulas. They are either easy to implement and high parallel objects.
In [6] it is shown what, by taking as activation function ψ the identity truncated to zero and one, also named saturating linear transfer function ψ(x) = min(1, max(x, 0)) it is possible to represent the corresponding neural network as combination of propositions of Lukasiewicz calculus and viceversa [1].
Usually Lukasiewicz logic sentences are built, as in first-order logic languages, from a (countable) set of propositional variables, a conjunction ⊗ (the fusion operator), an implication ⇒ and the truth constant 0. Further connectives are defined as follows:
1. ϕ 1 ∧ ϕ 2 is ϕ 1 ⊗ (ϕ 1 ⇒ ϕ 2 ), 2. ϕ 1 ∨ ϕ 2 is ((ϕ 1 ⇒ ϕ 2 ) ⇒ ϕ 2 ) ∧ ((ϕ 2 ⇒ ϕ 1 ) ⇒ ϕ 1 ) 3. ¬ϕ 1 is ϕ 1 ⇒ 0 4. ϕ 1 ⇔ ϕ 2 is (ϕ 1 ⇒ ϕ 2 ) ⊗ (ϕ 2 ⇒ ϕ 1 ) 5. 1 is 0 ⇒ 0
The usual interpretation for a well formed formula ϕ is defined recursive defining by the assignment of truth values to each proposicional variable. However the application of neural network to learn Lukasiewicz sentences seems more promisor using a non recursive approach to proposition evaluation. We can do this by defining the first order language as a graphic language. In this language, words are generate using the atomic componentes presented on figure 2, they are networks defined linking this sort of neurons. This is made gluing atomic componentes, satisfy the neuron signature, i.e. it is an unit having several inputs and one output. This task of construct complex structures based on simplest ones can be formalized using generalized programming [8].
In other words Lukasiewicz logic language is defined by the set of all neural networks, where its neurons assume one of the configuration presented in figure 2. A networks of this type can be interpreted as a function, see figure 3, generically denoted by ψ b (w 1 x 1 , w 2 x 2 ). In this context a network is the functional interpretation for a sentence when its interpretation is the sentence truth table. The fact of networks and interpretation have a similar structure preserved by For instance, the semantic for sentence ϕ = (x ⊗ y ⇒ z) ⊕ (z ⇒ w) can be described using the bellow network or codified by the bellow set of matrixes. We must note, in the example, what the partial interpretation of each unit is a simple exercise of pattern checking, relating the weights signs and the neuron bias.
x 1 ❆ ❆ ❆ ❆ −1 ?>=< 89:; ⊗ y 1 ⑦ ⑦ ⑦ ⑦ x −1 ❈ ❈ ❈ ❈ 1 ?>=< 89:; ⇒ y 1 ④ ④ ④ ④ x 1 ❆ ❆ ❆ ❆ 0 ?>=< 89:; ⊕ y 1 ⑦ ⑦ ⑦ ⑦ x 0 ❅ ❅ ❅ ❅ 1 7654 0123 1 y 0 ⑦ ⑦ ⑦ ⑦ x 0 ❅ ❅ ❅ ❅ 0 7654 0123 0 y 0 ⑦ ⑦ ⑦ ⑦ 1 x −1 7654 0123 ¬x 1 ❈ ❈ ❈ ❈ −1 ?>=< 89:; ⊗ −1 ❈ ❈ ❈ ❈ ❈ 1 y 1 ⑧ ⑧ ⑧ ⑧ ⑧ 7654 0123 = 1 ?>=< 89:; ⇒ 1 ❇ ❇ ❇ ❇ ❇ 0 z −1 ❄ ❄ ❄ ❄ ❄ 1 ⑧ ⑧ ⑧ ⑧ ⑧ 1 0 0 ?>=< 89:; ⊕ ?>=< 89:; ⇒ 1 7654 0123 = 1 ⑤ ⑤ ⑤ ⑤ ⑤ w 1 ③ ③ ③ ③ x y z w b's partial interpretation i 1 i 2 i 3 1 1 0 0 0 0 1 0 0 0 −1 1 −1 0 1 x ⊗ y z z ⇒ w i 1 i 2 i 3 j 1 j 2 −1 1 0 0 0 1 1 0 i 1 ⇒ i 2 i 3 j 1 j 2 1 1 0 j 1 ⊕ j 2 INTERPRETATION: j 1 ⊕ j 2 = (i 1 ⇒ i 2 ) ⊕ (i 3 ) = = ((x ⊗ y) ⇒ z) ⊕ (z ⇒ w)
In this sense this neural network can be seen as an interpretation for sentence ϕ, it codifies f ϕ , the proposition truth table. And it can be presented in string base notation by writing:
f ϕ (x, y, z, w) = ψ 0 (ψ 0 (ψ 1 (−z, w)), ψ 1 (ψ 0 (z), −ψ −1 (x, y)))
However f ϕ is a continuo structure, for our propose, it must be discretized using a finite structure but having suficiente information to describe the original formula.
Similarity between a configuration and a formula
We called Castro neural network to a neural network having as activation function ψ the identity truncated to zero and one and where its weights are -1, 0 or 1 and having by bias an integer. And a Carlos neural network is called representable if it is codified as a binary neural network i.e. a Castro neural network where each neuron don't have more than two inputs. A network is called unrepresentable if it can't be codified using a binary Castro neural network. In figure 4, we present an example of an unrepresentable network configuration, as we will see in the following.
Note what binary Castro neural network can be translated directory in the Lukasiewicz first order language, and in this sense are called them Lukasiewicz neural network. Bellow we presented examples of the functional interpretation for formulas with two propositional variables. They can be organized in two class:
x −1 ❆ ❆ ❆ ❆ 0 y 1 ?>=< 89:; ψ w ⇔ w = ψ 0 (−x, y, z) z 1 ⑦ ⑦ ⑦ ⑦ Figure 4: An unrepresentable neural network Disjunctive interpretations Conjunctive interpretations ψ 0 (x 1 , x 2 ) = f x 1 ⊕x 2 ψ −1 (x 1 , x 2 ) = f x 1 ⊗x 2 ψ 1 (x 1 , −x 2 ) = f x 1 ⊕¬x 2 ψ 0 (x 1 , −x 2 ) = f x 1 ⊗¬x 2 ψ 1 (−x 1 , x 2 ) = f ¬x 1 ⊕x 2 ψ 0 (−x 1 , x 2 ) = f ¬x 1 ⊗x 2 ψ 2 (−x 1 , −x 2 ) = f ¬x 1 ⊕¬x 2 ψ 1 (−x 1 , −x 2 ) = f ¬x 1 ⊗¬x 2
And correspond to all possible configurations of neurons with two inputs. The other possible configurations are constant and can also be seen as repre-
sentable configurations. For instance ψ b (x 1 , x 2 ) = 0, if b < −1, and ψ b (−x 1 , −x 2 ) = 1, if b > 1.
In this sense every representable network can be codified by a neural network where the neural units satisfy the above patterns. Bellow we present examples of representable configurations with three inputs and how they can be codified using representable neural networks having units with two inputs.
Disjunctive configurations
ψ −2 (x 1 , x 2 , x 3 ) = ψ −1 (x 1 , ψ −1 (x 2 , x 3 )) = f x 1 ⊗x 2 ⊗x 3 ψ −1 (x 1 , x 2 , −x 3 ) = ψ −1 (x 1 , ψ 0 (x 2 , −x 3 )) = f x 1 ⊗x 2 ⊗¬x 3 ψ 0 (x 1 , −x 2 , −x 3 ) = ψ −1 (x 1 , ψ 1 (−x 2 , −x 3 )) = f x 1 ⊗¬x 2 ⊗¬x 3 ψ 1 (−x 1 , −x 2 , −x 3 ) = ψ 0 (−x 1 , ψ 1 (−x 2 , −x 3 )) = f ¬x 1 ⊗¬x 2 ⊗¬x 3 Conjunctive interpretations ψ 0 (x 1 , x 2 , x 3 ) = ψ 0 (x 1 , ψ 0 (x 2 , x 3 )) = f x 1 ⊕x 2 ⊕x 3 ψ 1 (x 1 , x 2 , −x 3 ) = ψ 0 (x 1 , ψ 1 (x 2 , −x 3 )) = f x 1 ⊕x 2 ⊕¬x 3 ψ 2 (x 1 , −x 2 , −x 3 ) = ψ 0 (x 1 , ψ 2 (−x 2 , −x 3 )) = f x 1 ⊕¬x 2 ⊕¬x 3 ψ 3 (−x 1 , −x 2 , −x 3 ) = ψ 1 (−x 1 , ψ 2 (−x 2 , −x 3 )) = f ¬x 1 ⊕¬x 2 ⊕¬x 3 Constant configurations ψ b (x 1 , x 2 , x 3 ) = 0, if b < −2, and ψ b (−x 1 , −x 2 , −x 3 ) = 1, if b > 3
, are also representable. However there are example an unrepresentable network with three inputs in fig. 4. Naturally, a neuron configuration when representable can by codified by different structures using Lukasiewicz neural network. Particularly we have:
Proposition 2 If the neuron configuration α = ψ b (x 1 , x 2 , . . . , x n−1 , x n ) is rep- resentable,
but not constant, it can be codified in a Lukasiewicz neural network with structure:
β = ψ b1 (x 1 , ψ b2 (x 2 , . . . , ψ bn−1 (x n−1 , x n ) . . .)).
And since the n-nary operator ψ b is comutativa in function β variables could interchange its position without change operator output. By this we mean what, in the string based representation, variable permutation generate equivalent formulas. From this we can concluded what:
Proposition 3 If α = ψ b (x 1 , x 2 , .
. . , x n−1 , x n ) is representable, but not constant, it is the interpretation of a disjunctive formula or of a conjunctive formula.
Recall that disjunctive formulas are written using only disjunctions and negations, and conjunctive formulas are written using only conjunctions and negations. This live us with the task of classify a neuron configuration according with its representation. For that, we established a relation using the configuration bias and the number of negative and positive inputs.
Proposition 4 Given the neuron configuration
α = ψ b (−x 1 , −x 2 , . . . , −x n , x n+1 , . . . , x m )
with m = n + p inputs and where n and p are, respectively, the number of negative weights and the number of positive, on the neuron configuration.
1. If b = −(m − 1) + n (i.e. b = −p + 1) the neuron is called a conjunction
and it is a interpretation for
¬x 1 ⊗ . . . ⊗ ¬x n ⊗ x n+1 ⊗ . . . ⊗ x m 2. When b = n the neuron is called a disjunction and it is a interpretation of ¬x 1 ⊕ . . . ⊕ ¬x n ⊕ x n+1 ⊕ . . . ⊕ x m
From this we proposed the following estrutural characterization for representable neurons.
Proposition 5 Every conjunctive or disjunctive configuration α = ψ b (x 1 , x 2 , . . . , x n−1 , x n ), can be codified by a Lukasiewicz neural network
β = ψ b1 (x 1 , ψ b2 (x 2 , . . . , ψ bn−1 (x n−1 , x n ) . . .)), where b = b 1 + b 2 + · · · + b n−1 and b 1 ≤ b 2 ≤ · · · ≤ b n−1 .
This can be translated in the following neuron rewriting rule,
w 1 ❂ ❂ ❂ ❂ ❂ ❂ b . . . ?>=< 89:; ψ R / / wn ✁ ✁ ✁ ✁ ✁ ✁ w 1 ❂ ❂ ❂ ❂ ❂ ❂ b 0 . . . ?>=< 89:; ψ 1 ❁ ❁ ❁ ❁ ❁ ❁ ❁ b 1 w n−1 ✝ ✝ ✝ ✝ ✝ ✝ ✝ ?>=< 89:; ψ wn ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠
linking networks, where values b 0 and b 1 satisfy b = b 0 + b 1 and b 1 ≤ b 0 , and such that neither involved neurons have constant output. This rewriting rule can be used to like equivalent configurations like:
x −1 ❇ ❇ ❇ ❇ 2 y 1 7654 0123 ϕ R / / z −1 ⑤ ⑤ ⑤ ⑤ ⑤ w 1 ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ x −1 ❇ ❇ ❇ ❇ 2 y 1 7654 0123 ϕ 1 ❆ ❆ ❆ ❆ ❆ 0 R / / z −1 ✁ ✁ ✁ ✁ ✁ 7654 0123 ϕ w 1 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ x −1 ❇ ❇ ❇ ❇ 2 z −1 7654 0123 ϕ 1 ❆ ❆ ❆ ❆ ❆ 0 y 1 7654 0123 ϕ 1 ❆ ❆ ❆ ❆ ❆ 0 w 1 7654 0123 ϕ
Note what a representable Castro neural network can been transformed by the application of rule R in a set of equivalente Lukasiewicz neural network having less complex neurons. Then we have:
Proposition 6 Unrepresentable neuron configurations are those transformed by rule R in, at least, two not equivalent neural networks.
For instance unrepresentable configuration ψ 0 (−x 1 , x 2 , x 3 ) is transform by rule R in three not equivalent configurations:
1. ψ 0 (x 3 , ψ 0 (−x 1 , x 2 )) = f x 3 ⊕(¬x 1 ⊗x 2 ) , 2. ψ −1 (x 3 , ψ 1 (−x, x 2 )) = f x 3 ⊗(¬x 1 ⊗x 2 ) , or 3. ψ 0 (−x 1 , ψ 0 (x 2 , x 3 )) = f ¬x 1 ⊗(x 2 ⊕x 3 ) .
The representable configuration ψ 2 (−x 1 , −x 2 , x 3 ) is transform by rule R on only two distinct but equivalent configurations: From this we concluded that Castro neural networks have more expressive power than Lukasiewicz logic language. There are structures defined using Castro neural networks but not codified in the Lukasiewicz logic language.
We also want meed reverse the knowledge injection process. We want extracted knowledge from trained neural networks. For it we need translate neuron configuration in propositional connectives or formulas. However, we as just said, not all neuron configurations can be translated in formulas, but they can be approximate by formulas. To quantify the approximation quality we defined the notion of interpretations and formulas λ-similar.
Two neuron configurations α = ψ b (x 1 , x 2 , . . . , x n ) and β = ψ b ′ (y 1 , y 2 , . . . , y n ) are called λ-similar in a (m + 1)-valued Lukasiewicz logic if λ is the mean absolute error by taken the truth subtable given by α as an approximation to the truth subtable given by β. When this is the case we write α ∼ λ β.
If α is unrepresentable and β is representable, the second configuration is called a representable approximation to the first.
We have for instance, on the 2-valued Lukasiewicz logic (the Boolean logic case), the unrepresentable configuration α = ψ 0 (−x 1 , x 2 , x 3 ) satisfies: ψ 1 (−x 1 , x 2 )), and 1 , ψ 0 (x 2 , x 3 )).
1. ψ 0 (−x 1 , x 2 , x 3 ) ∼ 0.125 ψ 0 (x 3 , ψ 0 (−x 1 , x 2 )), 2. ψ 0 (−x 1 , x 2 , x 3 ) ∼ 0.125 ψ −1 (x 3 ,3. ψ 0 (−x 1 , x 2 , x 3 ) ∼ 0.125 ψ 0 (−x
And in this case, the truth subtables of, formulas
α 1 = x 3 ⊕ (¬x 1 ⊗ x 2 ), α 1 = x 3 ⊗ (¬x 1 ⊗ x 2 ) and α 1 = ¬x 1 ⊗ (x 2 ⊕ x 3 ) are both λ-similar to ψ 0 (−x 1 , x 2 , x 3 ),
where λ = 0.125 since they differ in one position on 8 possible positions. This mean that both formulas are 12.3% accurate. The quality of this approximations was checked by presenting values of similarity levels λ on other finite Lukasiewicz logics. For every selected logic both formulas α 1 , α 2 and α 3 have the some similarity level when compared to α: Lets see a more complex configuration α = ψ 0 (−x 1 , x 2 , −x 3 , x 4 , −x 5 ). From it we can derive, through rule R, configurations:
1. β 1 = ψ 0 (−x 5 , ψ 0 (x 4 , ψ 0 (−x 3 , ψ 0 (x 2 , −x 1 )))) 2. β 2 = ψ −1 (x 4 , ψ −1 (x 2 , ψ 0 (−x 5 , ψ 0 (−x 3 , −x 1 ))))
3. β 3 = ψ −1 (x 4 , ψ 0 (−x 5 , ψ 0 (x 2 , ψ 1 (−x 3 , −x 1 )))) 4. β 4 = ψ −1 (x 4 , ψ 0 (x 2 , ψ 0 (−x 5 , ψ 1 (−x 3 , −x 1 )))) since this configurations are not equivalents we concluded that α is unrepresentable. When we compute the similarity level between α and each β i using different finite logics we have: From this we may concluded that β 2 is a good approximation to α and its quality improve when we increase the number of truth values. The error increase at a low rate that the number of cases.
In this sense we will also use rule R in the case of unrepresentable configurations. From an unrepresentable configuration α we can generate the finite set S(α), with representable networks similar to α, using rule R. Note what from S(α) we may select as approximation to α the formula having the interpretation more similar to α, denoted by s(α). This identification of unrepresentable configuration by representable approximations can be used to transform network with unrepresentable neurons into representable neural networks. The stress associated to this transformation caracterizes the translation accuracy.
A neural network crystallization
Weights in Castro neural networks assume the values -1 or 1. However the usual learning algorithms process neural networks weights presupposing the continuity of weights domain. Naturally, every neural network with weighs in [−1, 1] can be seen as an approximation to a Castro neural networks. The process of identify a neural network with weighs in [−1, 1] with a Lukasiewicz neural networks was called crystallization. And essentially consists in rounding each neural weight w i to the nearest integer less than or equal to w i , denoted by ⌊w i ⌋.
In this sense the crystallization process can be seen as a pruning on the network structure, where links between neurons with weights near 0 are removed and weights near -1 or 1 are consolidated. However this process is very crispy. We need a smooth procedure to crystallize a network in each learning iteration to avoid the drastic reduction on learning performance. In each iteration we want restrict the neural network representation bias, making the network representation bias converge to a structure similar to a Castro neural networks. For that, we defined by representation error for a network N with weights w 1 , . . . , w n , as
∆(N ) = n i=1 (w i − ⌊w i ⌋).
When N is a Castro neural networks we have ∆(N ) = 0. And we defined a smooth crystallization process by iterating the function: Υ n (w) = sign(w).((cos(1 − abs(w) − ⌊abs(w)⌋).
π 2 ) n + ⌊abs(w)⌋)
where sign(w) is the sign of w and abs(w) its absolute value. We denote by Υ n (N ) the function having by input and output a neural network defined extending Υ(w i ) to all network weights and neurons bias. Since, for every network N and n > 0, ∆(N ) ≥ ∆(Υ n (N )), we have: The convergence speed dependes on parameter n. Increasing n speedup crystallization but reduces the network plasticity to the training data. For our applications, we selected n = 2 based on the learning efficiency on a set of test formulas. For grater values for n imposes stronger restritivos to learning. This induces a quick convergence to an admissible configuration of Castro neural network.
Learning propositions
We began the study of Castro neural network generation trying to do reverse engineering on a truth table. By this we mean what given a truth table from a (n + 1)-valued Lukasiewicz logic, generated by a formula in the Lukasiewicz logic language, we will try to find its interpretation in the form of a Lukasiewicz neural network. And from it rediscover the original formula.
For that we trained a Backpropagation neural networks using the truth table. Our methodology trains networks having progressively more complex topologies, until a crystalized network with good performance have been found. Note that this methodology convergence dependes on the selected training algorithm.
The bellow Algorithm 1 described our process for truth table reverse engineering:
Algorithm 1 Reverse Engineering algorithm Try a new network. Go to 3 12: end if 13: Refine the crystalized network Given a part of a truth table we try to find a Lukasiewicz neural network what codifies the data. For that we generated neural networks with a fixed number of hidden layers, on our implementation we used three. When the process detects bad learning performances, it aborts the training, and generates a new network with random heights. After a fixed number of tries the network topology is change. This number of tries dependes of the network inputs number. After trying configure a set of networks with a given complexity and bad learning performance, the system tries to apply the selected Backpropagation algorithm to a more complex set of networks. In the following we presented a short description for the selected learning algorithm.
If the continuous optimization process converges, i.e. if the system finds a network codifying the data, the network is crystalized. If the error associated to this process increase the original network error the crystalized network is throwaway, and the system returns to the learning fase trying configure a new network.
When the process converges and the resulting network can be codified as a crisp Lukasiewicz neural network the system prunes the network. The goal of this fase is the network simplification. For that we selected the Optimal Brain Surgeon algorithm proposed by G.J. Wolf, B. Hassibi and D.G. Stork in [16].
The figure 5 presents an example of the Reverse Engineering algoritmo input data set (a truth table) and output neural network structure.
x 1
x 2 x 1 xorx 2 [3]. The basic EBP algorithm adjusts the weights in the steepest descent direction. This is the direction in which the performance function is decreasing most rapidly. In the EBP algorithm, the performance index F (w) to be minimized is defined as the sum of squared erros between the target output and the network's simulated outputs, namely:
1 1 0 1 0 1 0 1 1 0 0 0 ⇒Reverse Engineering⇒ 1 −1 −1 1 0 0 x 1 ⊗ ¬x 2 ¬x 1 ⊗ x 2 1 1 0 i 1 ⊕ i 3 1 0
F (w k ) = e T k e k where the vector w k = [w 1 , w 2 , . . . , w n ] consists of all the current weights of the network, e k is the current error vector comprising the error for all the training examples.
When training with the EBP method, an iteration of the algorithm define the change of weights and have the form
w k+1 = w k − αG k
where G k is the gradient of F on w k , and α is the learning rate.
Note that, the basic step of the Newton's method can be derived dom Taylor formula and is as: Since Newton's method implicitly uses quadratic assumptions (arising from the neglect of higher order terms in a Taylor series), The Hessian need not to be evaluated exactly. Rather an approximation can be used like
w k+1 = w k − H −1 k G k where H k isH k ≈ J T k J k
where J k is the Jacobian matrix that contains first derivatives of the network errors with respect to the weights w k . The Jacobian matrix J k can be computed through a standard back propagation technique [22] that is much less complex than computing the Hessian matrix. The current gradient take the form G k = J T k e k , where e k is a vector of current network errors. Note what H k = J T k J k in linear case. The main advantage of this technique is rapid convergence. However, the rate of convergence is sensitive to the starting location, or more precisely, the linearity around the starting location.
It can be seen that simple gradient descent and Newton iteration are complementary in advantages they provide. Levenberg proposed an algorithm based on this observation, whose update rule is blend mentioned algorithms and is given as
w k+1 = w k − [J T k J k + µI] −1 J T k e k
where J k is the Jacobian matrix evaluated at w k and µ is the learning rate. This update rule is used as follow. If the error goes down following an update, it implies that our quadratic assumption on the function is working and we reduce µ (usually by a factor of 10) to reduce the influence of gradient descent. In this way, the performance function is always reduced at each iteration of the algorithm [14]. On the other hand, if the error up, we would like to follow the gradient more and so µ is increased by the same factor. The Levenberg algorithm is thus 1. Do an update as directed by the rule above.
2. Evaluated the error at the new weight vector.
3. If error has increased as the result the update reset the weights to their previous values and increase µ by a factor β. Then try an update again 4. If error has decreased as a result of the update, then accept the set and decrease µ by a factor β.
The above algorithm has the disadvantage that if the value of µ is large, the approximation to Hessian matrix is not used at all. We can derive some advantage out of the second derivative even in such cases by scaling each component of the gradient according to the curvature. This should result in larger movement along the direction where the gradient is smaller so the classic "error valley" problem does not occur any more. This crucial insight was provided by Marquardt. He replaced the identity matrix in the Levenberg update rule with the diagonal of Hessian matrix approximation resulting in the Levenberg-Marquardt update rule.
w k+1 = w k − [J T k J k + µ.diag(J T k J k )] −1 J T k e k
Since the Hessian is proportional to the curvature this rule implies a larger step in the direction with low curvatures and big step in the direction with high curvature. The standard LM training algorithm can be illustrated in the following pseudo-codes:
It is to be notes while LM method is no way optimal but is just a heuristic, it works extremely well for learn Lukasiewicz neural network. The only flaw is its need for matrix inversion as part of the update. Even thought the inverse is usually implemented using pseudo-inverse methods such as singular value decomposition, the cost of update become prohibitive after the model size increases to a few thousand weights.
The application of a soft cristalizador step in each iteration accelerates the convergence to a Castro neural network.
Applying reverse engineering on truth tables
Given a Lukasiewicz neural network it can be translated in the form of a string base formula if every neuron is representable. Proposition 4 defines a way to translate from the connectionist representation to a symbolic representation. And it is remarkable the fact that, when the truth table used in the learning is generate by a formula in a adequate n-valued Lukasiewicz logic the Reverse Engineering algorithm converges to a representable Lukasiewicz neural network and it is equivalent to the original formula.
When we generate a truth table in the 4-valued Lukasiewicz logic using formula
(x 4 ⊗ x 5 ⇒ x 6 ) ⊗ (x 1 ⊗ x 5 ⇒ x 2 ) ⊗ (x 1 ⊗ x 2 ⇒ x 3 ) ⊗ (x 6 ⇒ x 4 )
it have 4096 cases, the result of applying the algorithm is the 100% accurate neural network.
0 0 0 −1 0 1 0 0 0 1 1 −1 1 1 −1 0 0 0 −1 1 0 0 −1 0 0 −1 −1 2 ¬x 4 ⊗ x 6 x 4 ⊗ x 5 ⊗ ¬x 6 x 1 ⊗ x 2 ⊗ ¬x 3 ¬x 1 ⊕ x 2 ⊕ ¬x 5 −1 −1 −1 1 0 ¬i 1 ⊗ ¬i 2 ⊗ ¬i 3 ⊗ i 4 1 0 j 1
From it we may reconstructed the formula:
j 1 = ¬i 1 ⊗ ¬i 2 ⊗ ¬i 3 ⊗ i 4 = ¬(¬x 4 ⊗ x 6 ) ⊗ ¬(x 4 ⊗ x 5 ⊗ ¬x 6 ) ⊗ ¬(x 1 ⊗ x 2 ⊗ ¬x 3 ) ⊗ (¬x 1 ⊕ x 2 ⊕ ¬x 5 ) = = (x 4 ⊕ ¬x 6 ) ⊗ (¬x 4 ⊕ ¬x 5 ⊕ x 6 ) ⊗ (¬x 1 ⊕ ¬x 2 ⊕ x 3 ) ⊗ (¬x 1 ⊕ x 2 ⊕ ¬x 5 ) = = (x 6 ⇒ x 4 ) ⊗ (x 4 ⊗ x 5 ⇒ x 6 ) ⊗ (x 1 ⊗ x 2 ⇒ x 3 ) ⊗ (x 1 ⊗ x 5 ⇒ x2)
Note however the restriction imposed, in our implementation, to three hidden layers having the least hidden layer only one neuron, impose restriction to the complexity of reconstructed formula. For instance
((x 4 ⊗ x 5 ⇒ x 6 ) ⊕ (x 1 ⊗ x 5 ⇒ x 2 )) ⊗ (x 1 ⊗ x 2 ⇒ x 3 ) ⊗ (x 6 ⇒ x 4 )
to be codified in a three hidden layer network the last layer needs two neurons one to codify the disjunction and the other to codify the conjunctions. When the algorithm was applied to the truth table generated in the 4-valued Lukasiewicz logic having by stoping criterium a mean square error less than 0.0007 it produced the representable network:
0 0 0 1 0 −1 1 −1 0 1 1 −1 1 1 −1 0 0 0 1 −2 −1 x 4 ⊕ ¬x 6 x 1 ⊗ ¬x 2 ⊗ x 4 ⊗ x 5 ⊗ ¬x 6 x 1 ⊗ x 2 ⊗ ¬x 3 1 −1 −1 0 i 1 ⊗ ¬i 2 ⊗ ¬i 3 1 0 j 1
By this we may conclude what original formula can be approximate, or is λsimilar with λ = 0.002 to:
j 1 = i 1 ⊗ ¬i 2 ⊗ ¬i 3 = (x 4 ⊕ ¬x 6 ) ⊗ ¬(x 1 ⊗ ¬x 2 ⊗ x 4 ⊗ x 5 ⊗ ¬x 6 ) ⊗ ¬(x 1 ⊗ x 2 ⊗ ¬x 3 ) = = (x 4 ⊕ ¬x 6 ) ⊗ (¬x 1 ⊕ x 2 ⊕ ¬x 4 ⊕ ¬x 5 ⊕ x 6 ) ⊗ (¬x 1 ⊕ ¬x 2 ⊕ x 3 ) = = (x 6 ⇒ x 4 ) ⊗ ((x 1 ⊗ x 4 ⊗ x 5 ) ⇒ (x 2 ⊕ x 6 )) ⊗ (x 1 ⊗ x 2 ⇒ x 3 )
Note that j 1 is 0.002-similar to the original formula in the 4-valued Lukasiewicz logic but it is equivalente to the original in the 2-valued Lukasiewicz logic, i.e. in Boolean logic. The fixed number of layer also impose restrictions to reconstruction of formula. A table generated by:
(((i 1 ⊗ i 2 ) ⊕ (i 2 ⊗ i 3 )) ⊗ ((i 3 ⊗ i 4 ) ⊕ (i 4 ⊗ i 5 ))) ⊕ (i 5 ⊗ i 6 )
requires at least 4 hidden layers, to be reconstructed, this is the number os levels required by the associated parsing tree.
Bellow we can see all the fixed points found by the process, when applied on the 5-valued truth table for
x ∧ y := min(x, y).
These reversed formulas are equivalent in the 5-valued Lukasiewicz logic, and where find for different executions.
0 −1 y −1 7654 0123 ϕ −1 7654 0123 ϕ x 1 ✁ ✁ ✁ ✁ ✁−1 7654 0123 ϕ −1 ⑥ ⑥ ⑥ ⑥ ⑥ 1 0 0 y −1 7654 0123 ϕ −1 7654 0123 ϕ x 1 ✁ ✁ ✁ ✁ ✁ 1 7654 0123 ϕ 1 ⑥ ⑥ ⑥ ⑥ ⑥ 0 0 0 y 1 7654 0123 ϕ −1 7654 0123 ϕ x −1 ✁ ✁ ✁ ✁ ✁ 1 7654 0123 ϕ 1 ⑥ ⑥ ⑥ ⑥ ⑥ 0 1 0 y 1 7654 0123 ϕ 1 7654 0123 ϕ x −1 ✁ ✁ ✁ ✁ ✁−1 7654 0123 ϕ −1 ⑥ ⑥ ⑥ ⑥ ⑥ 1 ¬(¬¬(x ⇒ y) ⇒ ¬x) ¬(x ⇒ ¬(x ⇒ y)) ¬(¬(y ⇒ x) ⇒ x) ¬((x ⇒ y) ⇒ ¬x) 0 0 x −1 7654 0123 ϕ −1 7654 0123 ϕ y 1 ✁ ✁ ✁ ✁ ✁ 1 7654 0123 ϕ 1 ⑥ ⑥ ⑥ ⑥ ⑥ 0 0 −1 x 1 7654 0123 ϕ 1 7654 0123 ϕ y 1 ✁ ✁ ✁ ✁ ✁−1 7654 0123 ϕ 1 ⑥ ⑥ ⑥ ⑥ ⑥ 1 1 1 x −1 7654 0123 ϕ −1 7654 0123 ϕ y −1 ✁ ✁ ✁ ✁ ✁ 1 7654 0123 ϕ −1 ⑥ ⑥ ⑥ ⑥ ⑥ 0 1 0 x 1 7654 0123 ϕ 1 7654 0123 ϕ y −1 ✁ ✁ ✁ ✁ ✁−1 7654 0123 ϕ −1 ⑥ ⑥ ⑥ ⑥ ⑥ 1 ¬(y ⇒ ¬(y ⇒ x)) (y ⇒ x) ⊗ y ¬(¬(y ⇒ x) ⇒ ¬y) ¬((y ⇒ x) ⇒ ¬y)
The bellow table presents mean times need to find a configuration with a mean square error less than 0.002. Then mean time is computed using a 6 tries for some formulas on the 5-valued truth Lukasiewicz logic. We implementation the algorithm using the MatLab neural network package and run it in a AMD Athalon 64 X2 Dual-Core Processor TK-53 at 1.70 GH on a Windows Vista system with 1G of memory. 4 Applying the process on real data
The extraction of a rule from a data set is very different from the task of reverse engineering the rule used on the generation of a data set. In sense what, in the reverse engineering task we know the existence of a prefect description for the information, we know the adequate logic language to describe it and we have lack of noise. The extraction of a rule from a data set is made establishing a stopping criterium base on a language fixed by the extraction process. The expressive power of this language caracterize the learning algorithm plasticity. However very expressive languages produce good fitness to the trained data, but with bad generalization, and its sentences are usually difficult to understand. With the application of our process to real data we try to catch information in the data similar to the information described using sentences in Lukasiewicz logic language. This naturally means what, in this case, we will try to search for simple and understandable models for the data. And for this make sense strategy followed of train of progressively more complex models and subjected to a strong criteria of pruning. When the mean squared error stopping criteria is satisfied it has big probability of be the simplest one. However some of its neuron configurations may be unrepresentable and must be approximated by a formula without damage drastically the model performance.
Note however the fact what the use of the presented process can be prohibitive to train complex models having a grate number of attributes, i.e. learn formulas with many connectives and propositional variables. In this sense our process use must be preceded by a fase of attribute selection.
Mushrooms
Mushroom is a data set available in UCI Machine Learning Repository. Its records drawn from The Audubon Society Filed Guide to North American Mushrooms (1981) G. H. Lincoff (Pres.), New York, was donate by Jeff Schlimmer. This data set includes descriptions of hypothetical samples corresponding to 23 species of gilled mushrooms in the Agaricus and Lepiota Family. Each species is identified as definitely edible, definitely poisonous, or of unknown edibility and not recommended. This latter class was combined with the poisonous one. The Guide clearly states that there is no simple rule for determining the edibility of a mushroom. However we will try to find a one using the data set as a truth With the values assumed by this attributes we produce a new data set. After some tries the simples model generated was the following:
A1 − bruises? = t 1 ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ A2 − odor ∈ {a, l, n} 1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ 1 A3 − odor = c −1 ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ A4 − ring.type = e −1 7654 0123 ϕ A5 − spore.print.color = r −1 ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ A6 − population = c −1 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ A7 − habitat =② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ②
This model have an accuracy of 100%. From it, and since attribute values in A2 and A3, and in A7 and A8 are auto exclusive, we used propositions A1, A2, A3, A4, A5, A6, A7 to define a new data set. This new data set was enriched with new negative cases by introduction for each original case a new one where the truth value of each attribute was multiplied by 0.5. For instance the "eatable" mushroom case: This resulted in a convergence speed increase and reduced the occurrence of no representable configurations.
When we applied our "reverse engineering" algoritmo to the enriched data set, having by stoping criteria the mean square error less than mse. For mse = 0.003 the system produced the model:
0 1 0 0 −1 0 1 0 1 0 1 0 0 −1 −1 −1 A2 ⊗ ¬A5 ⊗ A7 A2 ⊗ A4 ⊗ ¬A7 1 1 0 i 1 ⊕ i 2 1 0
This model codifies the proposition
(A2 ⊗ ¬A5 ⊗ A7) ⊕ (A2 ⊗ A4 ⊗ ¬A8)
and misses the classification of 48 cases. It have 98.9% accuracy. More precise model can be produced, by restring the stoping criteria. However this, in general, produce more complex propositions and more dificulte to understand. For instance with mse = 0.002 the systems generated the bellow model. It misses 32 cases, having an accuracy of 99.2%, and easy to convert in a proposition.
0 0 0 −1 0 0 1 1 1 0 −1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 −1 −1 1 1 −1 0 −1 ¬A4 ⊕ A7 A1 ⊗ A2 ⊗ ¬A4 A7 A2 ⊗ ¬A5 ⊗ ¬A6 ⊗ A7 −1 0 1 0 1 −1 0 −1 1 0 ¬i 1 ⊕ i 3 i 1 ⊗ ¬i 2 ⊗ ¬i 4 1 −1 0 j 1 ⊗ ¬j 2
This neural network codifies Some times the algorithm converged to unrepresentable configurations like the one presented bellow, having however 100% accuracy. The frequency of this type of configurations increases with the increase of required accuracy.
−1 1 −1 1 0 −1 0 0 0 0 1 1 0 −1 1 1 0 0 0 0 −1 0 1 0 i 1 unrepresentable A4 ⊗ A5 ⊗ ¬A6 i 3 unrepresentable 1 −1 1 0 j 1 unrepresentable 1 0
Since, for the similarity evaluation on data set, we have:
1. i 1 ∼ 0.0729 ((¬A1 ⊗ A4) ⊕ A2) ⊗ ¬A3 ⊗ ¬A6 2. i 3 ∼ 0.0 (A1 ⊕ ¬A7) ⊗ A2
3. j 1 ∼ 0.0049 (i 1 ⊗ ¬i 2 ) ⊕ i 3
The formula is λ-similar, with λ = 0.0049 to the original neural network. Formula α misses the classification for 40 cases. Note what the symbolic model is stable to the bad performance of i 1 representation. Other example of unrepresentable is given bellow. This network structure can be simplified during the symbolic translation.
1 1 −1 1 0 0 1 0 0 1 −1 0 0 0 0 −1 0 1 1 0 −1 −1 0 2 i 1 unrepresentable A3 ⊗ ¬A4 ¬A2 ⊗ A4 ⊗ A5 ⊗ ¬A7 −1 0 1 0 1 1 0 1 ¬i 1 ⊗ i 3 1 −1 1 1 ¬j 1 ⊗ j 2 Since i 1 ∼ 0.0668 (A1 ⊗ A2 ⊗ A7) ⊕ ¬A3 ⊕ A4
the neural network is similar to,
α = ¬j 1 ⊗ j 2 = ¬(¬i 1 ⊗ i 3 ) ⊗ 1 = ((A1 ⊗ A2 ⊗ A7) ⊕ ¬A3 ⊕ A4) ⊕ ¬(¬A2 ⊗ A4 ⊗ A5 ⊗ ¬A7)
and the degree of similarity is λ = 0, i.e. the neural network interpretation is equivalent to formula α in the Mushrooms data set, in the sense what both produce equal classifications.
Conclusions
This methodology to codify and extract symbolic knowledge from a neuro network is very simple and efficient for the extraction of simple rules from medium sized data sets. From our experience the described algorithm is a very good tool for attribute selection, particulary when we have low noise and classification problems depending from few nominal attributes to be selected from a huge set of possible attributes.
In the theoretic point of view it is particularly interesting the fact what restricting the values assumed by neurons weights restrict the information propagation in the network. Allowing the emergence of patterns in the neuronal network structure. For the case of linear neuronal networks these structures are characterized by the occurrence of patterns in neuron configuration with a direct symbolic presentation in a Lukasiewicz logic.
Figure 1 :
1Saturating linear transfer function.
Figure 2 :
2Neural networks codifying formulas x ⊗ y, x ⇒ y, x ⊕ y, T rue, F alse, ¬x and x.
Figure 3 :
3functional interpretation for a neural network a graph homomorphism, called the translation morphism, simplifies the transformation between string base representations and the network representation, allowing to write: Proposition 1 Every well formed formula in Lukasiewicz logic language can be codified using a neural network.
A truth table f ϕ for a formula ϕ is a map f ϕ : [0, 1] m → [0, 1], where m is the number of propositional variables used in ϕ. For each integer n > 0, let S n be the set {0, 1 n , . . . , n−1 n , 1}. Each n > 0, defines a subtable for f ϕ defined by f (n) ϕ : (S n ) m → S n , and given by f
ϕ
(v) = f ϕ (v), and called the ϕ (n+1)-valued truth subtable.
1. ψ 0 (x 3 , ψ 2 (−x 1 , −x 2 )) = f x 3 ⊕¬(x 1 ⊗x 2 ) , or 2. ψ 1 (−x 2 , ψ 1 (−x 1 , x 3 )) = f ¬x 2 ⊕(¬x 1 ⊕x 3 )
• 3 -
3valued logic, λ = 0.1302, • 4-valued logic, λ = 0.1300, • 5-valued logic, λ = 0.1296, • 10-valued logic, λ = 0.1281, • 20-valued logic, λ = 0.1268, • 30-valued logic, λ = 0.1263, • 50-valued logic, λ = 0.1258.
• 2 -
2valued logic α ∼ 0.156 β 1 , α ∼ 0.094 β 2 , α ∼ 0.656 β 3 and α ∼ 0.531 β 4 , • 3-valued logic α ∼ 0.134 β 1 , α ∼ 0.082 β 2 , α ∼ 0.728 β 3 and α ∼ 0.601 β 4 , • 4-valued logic α ∼ 0.121 β 1 , α ∼ 0.076 β 2 , α ∼ 0.762 β 3 and α ∼ 0.635 β 4 , • 5-valued logic α ∼ 0.112 β 1 , α ∼ 0.071 β 2 , α ∼ 0.781 β 3 and α ∼ 0.655 β 4 , • 10-valued logic α ∼ 0.096 β 1 , α ∼ 0.062 β 2 , α ∼ 0.817 β 3 and α ∼ 0.695 β 4 ,
Proposition 7
7Given a neural networks N with weights in the interval [0, 1]. For every n > 0 the function Υ n (N ) have by fixed points Castro neural networks N ′ .
1 :
1Given a (n+1)-valued truth subtable for a Lukasiewicz logic proposition 2: Define an inicial network complexity 3: Generate an inicial neural network 4: Apply the Backpropagation algorithm using the data set5: if the generated network have bad performance then Do neural network crystallization using the crisp process.10: if crystalized network have bad performance then 11:
the Hessian matrix of the performance index at the current values of the weights.
Algorithm 2
2Levenberg-Marquardt algorithm with soft crystallization 1: Initialize the weights w and parameters µ = .01 and β = .1 2: Compute e the sum of the squared error over all inputs F (w) 3: Compute J the Jacobian of F in w 4: Compute the increment of weight ∆w = −[J T J + µdiag(J T k J k )] −1 J T e 5: Let w * be the result of applying to w + ∆w the soft crystallization process Υ 2 . 6: if F (w * ) < F (w
⊗ i 5 ⇒ i 6 ) ⊕ (i 1 ⊗ i 5 ⇒ i 2 )) ⊗ (i 1 ⊗ i 3 ⇒ i 2 ) 224.74 36475.47 6 ((i 4 ⊗ i 5 ⇒ i 6 ) ⊕ (i 1 ⊗ i 5 ⇒ i 2 )) ⊗ (i 1 ⊗ i 3 ⇒ i 2 ) ⊗ (i 6 ⇒ i 4 )368.32 55468.66
(
A1=0, A2=1, A3=0, A4=0, A5=0, A6=0, A7=0,A8=1,A9=0) was used on the definition of a new "poison" case (A1=0, A2=0.5, A3=0, A4=0, A5=0, A6=0, A7=0,A8=0.5,A9=0)
j 1
1⊗ ¬j 2 = (¬i 1 ⊕ i 3 ) ⊗ ¬(i 1 ⊗ ¬i 2 ⊗ ¬i 4 ) = = (¬(¬A4 ⊕ A7) ⊕ A7) ⊗ ¬((¬A4 ⊕ A7) ⊗ ¬(A1 ⊗ A2 ⊗ ¬A4) ⊗ ¬(A2 ⊗ ¬A5 ⊗ ¬A6 ⊗ A7)) = = ((A4 ⊗ ¬A7) ⊕ A7) ⊗ ((A4 ⊗ ¬A7) ⊕ (A1 ⊗ A2 ⊗ ¬A4) ⊕ (A2 ⊗ ¬A5 ⊗ ¬A6 ⊗ A7))
α
= (((((¬A1 ⊗ A4) ⊕ A2) ⊗ ¬A3 ⊗ ¬A6) ⊗ ¬(A4 ⊗ A5 ⊗ ¬A6)) ⊕ ((A1 ⊕ ¬A7) ⊗ A2)
EBP algorithm dependent methods. It gives a good exchange between the speed of Newton algorithm and the stability of the steepest descent methodFigure 5: Input and Output structures
2.1 Training the neural network
Standard Error Backpropagation algorithm (EBP) is a gradient descent algo-
rithm, in which the network weights are moved along the negative of the gradient
of the performance function. EBP algorithm has been a significant improve-
ment in neural network research, but it has a weak convergence rate. Many
efforts have been made to speed up EBP algorithm [4] [24] [25] [23] [21]. The
Levenderg-Marquardt algorithm (LM) [15] [2] [3] [7] ensued from development
of
table .
.The data set have 8124 instances defined using 22 nominally valued attributes presented in the table bellow. It has missing attribute values, 2480, all for attribute #11. 4208 instances (51.8%) are classified as editable and 3916 (48.2%) has classified poisonous. duced a model, having an architecture (2,1,1), a quite complex rule with 100% accuracy depending on 23 binary attributes defined by values of {odor,gill.size,stalk.surface.above.ring, ring.type, spore.print.color}
A8 − habitat ∈ {g, m, u, d, p, l} −1w
1
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
We used a unsupervised filter converting all nominal attributes into binary numeric attributes. An attribute with k values is transformed into k binary attributes if the class is nominal. This produces a data set with 111 binary attributes.After the binarization we used the presented algorithm to selected relevante attributes for mushrooms classification. After 4231.8 seconds the system pro-
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| [] |
[
"Hybrid Symplectic Integrators for Planetary Dynamics",
"Hybrid Symplectic Integrators for Planetary Dynamics"
] | [
"Hanno Rein \nDepartment of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada\n\nDepartment of Astronomy and Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada\n",
"David M Hernandez \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St., MS 5102138CambridgeMAUSA\n\nMIT Kavli Institute for Astrophysics and Space Research\n77 Massachusetts Ave02139CambridgeMAUnited States\n",
"Daniel Tamayo \nDepartment of Astrophysical Sciences\nPrinceton University\n08544PrincetonNew JerseyUnited States\n",
"† ",
"Garett Brown \nDepartment of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada\n\nDepartment of Physics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada\n",
"Emily Eckels \nDepartment of Mathematics\nEmory University\n201 Dowman Drive30322AtlantaGAUSA\n",
"Emma Holmes \nDepartment of Mathematics and Statistics\nMcMaster University\n1280 Main St. WL8S 4K1HamiltonOntarioCanada\n",
"Michelle Lau \nDepartment of Physics\nImperial College London\nSW7 2AZLondonUnited Kingdom\n",
"Réjean Leblanc \nDepartment of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada\n\nDepartment of Physics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada\n",
"Ari Silburt \nDepartment of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada\n\nDepartment of Astronomy and Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada\n"
] | [
"Department of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada",
"Department of Astronomy and Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden St., MS 5102138CambridgeMAUSA",
"MIT Kavli Institute for Astrophysics and Space Research\n77 Massachusetts Ave02139CambridgeMAUnited States",
"Department of Astrophysical Sciences\nPrinceton University\n08544PrincetonNew JerseyUnited States",
"Department of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada",
"Department of Physics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada",
"Department of Mathematics\nEmory University\n201 Dowman Drive30322AtlantaGAUSA",
"Department of Mathematics and Statistics\nMcMaster University\n1280 Main St. WL8S 4K1HamiltonOntarioCanada",
"Department of Physics\nImperial College London\nSW7 2AZLondonUnited Kingdom",
"Department of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada",
"Department of Physics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada",
"Department of Physical and Environmental Sciences\nUniversity of Toronto at Scarborough\nM1C 1A4TorontoOntarioCanada",
"Department of Astronomy and Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoOntarioCanada"
] | [
"Mon. Not. R. Astron. Soc"
] | Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations. | 10.1093/mnras/stz769 | [
"https://arxiv.org/pdf/1903.04972v1.pdf"
] | 119,269,113 | 1903.04972 | cebd849f20de354505ce03083fe34e4127d01b5c |
Hybrid Symplectic Integrators for Planetary Dynamics
Hanno Rein
Department of Physical and Environmental Sciences
University of Toronto at Scarborough
M1C 1A4TorontoOntarioCanada
Department of Astronomy and Astrophysics
University of Toronto
M5S 3H4TorontoOntarioCanada
David M Hernandez
Harvard-Smithsonian Center for Astrophysics
60 Garden St., MS 5102138CambridgeMAUSA
MIT Kavli Institute for Astrophysics and Space Research
77 Massachusetts Ave02139CambridgeMAUnited States
Daniel Tamayo
Department of Astrophysical Sciences
Princeton University
08544PrincetonNew JerseyUnited States
†
Garett Brown
Department of Physical and Environmental Sciences
University of Toronto at Scarborough
M1C 1A4TorontoOntarioCanada
Department of Physics
University of Toronto
M5S 3H4TorontoOntarioCanada
Emily Eckels
Department of Mathematics
Emory University
201 Dowman Drive30322AtlantaGAUSA
Emma Holmes
Department of Mathematics and Statistics
McMaster University
1280 Main St. WL8S 4K1HamiltonOntarioCanada
Michelle Lau
Department of Physics
Imperial College London
SW7 2AZLondonUnited Kingdom
Réjean Leblanc
Department of Physical and Environmental Sciences
University of Toronto at Scarborough
M1C 1A4TorontoOntarioCanada
Department of Physics
University of Toronto
M5S 3H4TorontoOntarioCanada
Ari Silburt
Department of Physical and Environmental Sciences
University of Toronto at Scarborough
M1C 1A4TorontoOntarioCanada
Department of Astronomy and Astrophysics
University of Toronto
M5S 3H4TorontoOntarioCanada
Hybrid Symplectic Integrators for Planetary Dynamics
Mon. Not. R. Astron. Soc
0000000Accepted for publication by MNRAS on 12th March 2019. Submitted on 19th January 2019.Printed 13 March 2019 (MN L A T E X style file v2.2)methods: numerical -gravitation -planets and satellites: dynamical evolution and stability
Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.
INTRODUCTION
Since ancient times astronomers have predicted the locations of planets in the night sky. But only the advent of computers has made it possible to calculate the orbital evolution of planetary systems over millions and even billions of years (Laskar & Gastineau 2009). One major breakthrough was the development of mixed variable symplectic maps for the N -body problem which we will refer to as Wisdom-Holman integrators (Wisdom & Holman 1992). The idea behind the Wisdom-Holman integrator is to split the motion of a planet into two steps: a dominant Keplerian motion around the star (the Kepler step), and the motion due to small interactions between the planets (the interaction step). E-mail: [email protected] † NHFP Sagan Fellow The Wisdom-Holman scheme works well as long as the interactions between planets can be considered perturbations to their predominant Keplerian motion around the star. However, if two planets come close to each other, the planet-planet interactions can become the dominant part of the motion. In that case the Wisdom-Holman integrator becomes inaccurate and might require excessively small timesteps to achieve an acceptable solution. Duncan et al. (1998) and Chambers (1999) provide a solution to this problem in the form of hybrid symplectic integrators. The idea is to move terms from the interaction part to the Keplerian part during close encounters, to ensure that one part always remains a perturbation. Although solving the Keplerian part during a close encounter now becomes more difficult than simply solving Kepler's equation, it can be done relatively efficiently with a high order integrator such as a Burlish-Stoer or Gauß-Radau scheme. As long as close encounters happen infrequently, the high order integrator is rarely used and has a negligible effect on the runtime. The scheme is therefore almost as fast as a Wisdom-Holman integrator. Furthermore, the scheme is also symplectic, ensuring good long term conservation of energy and angular momentum.
In this paper we look specifically at the MERCURY integrator. MERCURY is freely available online and has become a widely used tool in running many otherwise impossible types of simulations. We discuss inconsistencies in the original paper by Chambers (1999) which describes the MERCURY algorithm 1 . We note that Wisdom (2017) also looked at these inconsistencies and came independently of us to the same conclusion. Despite these inconsistencies, the resulting scheme has very good characteristics. We provide mathematical and physical explanations for them in this paper. With this new understanding, it is easy to construct new hybrid integrators. We describe one new integrator that is much simpler than MERCURY, but achieves the same accuracy.
HYBRID SYMPLECTIC INTEGRATORS
We begin by introducing the notation and coordinate system we use. We have one star and an additional N planetary bodies with Cartesian coordinates qi and canonical momenta pi in some inertial frame. We will identify the stellar object with i = 0 and the planets with i > 0. The N -body Hamiltonian in these coordinates is
H = N i=0 p 2 i 2mi − G N i=0 N j=i+1 mimj |qi − qj| .(1)
To evolve this system forward in time we could calculate the equations of motion for this Hamiltonian. However, in general the resulting differential equations cannot be solved analytically and are usually stiff and thus hard to solve numerically.
The hybrid integration schemes discussed in this paper make use of democratic heliocentric coordinates Qi and corresponding canonical momenta Pi (see Appendix A). The advantage of these coordinates is that the Hamiltonian can be written as a sum of four terms: where
H = H0 + HK + HI + HJ,(2)H0 = P 2 0 2M (3) HK = i=1 P 2 i 2mi − Gm0mi Qi −G i=1 j=i+1 mimj Qij (1 − K(Qij))(4)HI = −G i=1 j=i+1 mimj Qij K(Qij) (5) HJ = 1 2m0 i=1 Pi 2 .(6)
In the above splitting, we've introduced the arbitrary scalar function K(r) which depends only on the distance between pairs of particles. Note that the terms involving K cancel out when adding HK and HI and thus do not change the evolution of the system. In this paper, we will use the letter K to label switching functions that appear in the Hamiltonian and refer to them as Hamiltonian switching functions. Later, we will also encounter force switching functions that first appear in the equations of motions. To avoid any confusion, we will use the letter L to label force switching functions. There is a relation between K and L which we derive in Sec. 3.
Each of the above terms has a physical interpretation. H0 describes the motion of the centre of mass. The term HK describes the Keplerian motion of the planets, ignoring planet-planet interactions (while K = 1). This term is separable and can be solved for each planet independently. The term HI corresponds to the planet-planet interactions. The term HJ is related to the barycentric motion of the star and is referred to as the jump term (it changes the positions of particles instantaneously but keeps the momenta constant).
We can calculate the equations of motion for each of these Hamiltonians. The resulting differential equations for H0, HI , and HJ are trivial to solve. Only the equations for HK require more work, i.e. solving Kepler's equation (while K = 1). Evolving the system under the influence of a Hamiltonian can be thought of as an operator acting on a state corresponding to the initial conditions. The idea of an operator splitting integration method is to approximate the evolution of the full system by consecutively evolving the system under the partial Hamiltonians. We can construct arbitrarily high order integrators by applying these operators in the right order for different amounts of time (Yoshida 1990). In particular, a second-order integrator can be constructed from the above splitting by using the following chain of operators
HW H (dt) ≡ HI (dt/2) • HJ (dt/2) • H0(dt) • HK (dt) • HJ (dt/2) • HI (dt/2),(7)
where an operator HA(dt) corresponds to evolving the system under the Hamiltonian HA for a time dt. Note that we recover the standard Wisdom-Holman integrator in democratic heliocentric coordinates if we set K = 1. For any finite timestep dt, HW H is only approximately equal to H. We can analyze the error of this splitting scheme with the help of the Baker-Campbell-Hausdorff (BCH) for-mula. The splitting scheme error arises because some pairs of operators do not commute. The error consists of an infinite series of commutators between operators. In our case, the action of an operator corresponds to solving the differential equations resulting from its respective Hamiltonian. As a result, the splitting scheme error can be expressed as a series of nested Poisson brackets (see e.g. Hernandez & Dehnen 2017). One caveat to keep in mind when working with the BCH formula is that it is only a formal asymptotic series and does not necessarily converge everywhere in phase space. We come back to this issue later.
The commutators that appear in the BCH formula can be grouped together by powers of the timestep dt. The specific splitting scheme in Eq. (7) is time symmetric and therefore only even orders of dt appear (Yoshida 1990
In this paper, we do not attempt to calculate approximations to these expressions analytically using Poisson brackets, but do so numerically. This allows us to easily extend the calculations to arbitrarily high order (7th order in our case). There are two way to approach this. Consider a commutator of the form [ A, B]. Starting from a state z, we can calculate δz ≡ ( A B) z − ( B A) z, and similarly for higher order commutators. If δz = 0, then the commutators commute. To quantify the statement, we can use any norm on δz. The measure we are typically interested in is the energy, or more specifically the energy error. Note that calculating the energy of δz does not make sense. However, we can calculate the energy error by comparing the energy of the state z to that of the state z + δz. Alternatively, we can calculate the state z ≡ ( B −1 A −1 B A) z. Here, z is a valid state and we can directly compare the energy of the states z and z .
In this paper we use the latter approach. Note that it is a simple chain of operators and no additions or subtractions of state vectors are needed. This approach avoids some issues with finite floating point precision compared to the standard approach where one needs to subtract nearly equal numbers multiple times when calculating the energy difference of the states z and z + δz. As a specific example, consider the first (third order) commutator that appears in the BCH formula of the hybrid symplectic intergators: [[ HJ , HK ], HK ]. The effect of this commutator can be calculated by the following chain of operators:
HK (−dt) • HJ (−dt) • HK (dt) • HJ (dt) =[ H J , H K ] −1 (−dt) • HK (−dt) • HJ (−dt) • HK (−dt) • HJ (dt) • HK (dt) =[ H J , H K ](dt)
• HK (dt).
The two approaches are not exactly equal, only to within order dt. Our approach is only an approximation of the commutator appearing in the BCH formula and is accu-rate to within O(dt) only. We could apply the Zassenhaus formula (the dual of the BCH formula) to calculate higher order corrections, which are again chains of the same operators. However, the approximations are good enough for our purpose: they will allow us to demonstrate which are the dominant contributions to the error terms in the BCH formula and understand their origins, especially for higher order commutators which are otherwise not easily tractable.
THE MERCURY CODE
The Wisdom-Holman map (setting K = 1 in the above notation) performs poorly during close encounters because some of the commutators in the error term become large. To construct an integrator that can more accurately resolve close encounters, Chambers (1999) suggests to use a non-constant Hamiltonian splitting function K as a basis for the MERCURY scheme.
The idea, as outlined in Chambers (1999), is that the switching function keeps the interaction Hamiltonian small even during close encounters between planets. As a result the switching scheme errors from the BCH formula should remain small as well.
Let us use Hamilton's equations and derive the equations of motion for the Hamiltonian HK :
Qi = Vi (9) Vi = − Gm0 Q 3 i Qi −G j=1 j =i mj Q 3 ij Qij 1 − K(Qij) + QijK (Qij) ,(10)
where we have introduced the democratic heliocentric velocities
Vi ≡ Pi mi and thereforeVi =Ṗ i mi .(11)
We also used the fact that the gradient of K satisfies
∂K(Qij) ∂Qij = Qij Qij ∂K(r) ∂r r=Q ij = Qij Qij K (Qij).(12)
The equations of motion above are not those of Keplerian motion but now also include interaction terms. Nevertheless they can be solved with a high order integrator such as a Bulirsch-Stoer algorithm or a high order Gauß-Radau integrator (Everhart 1985;Rein & Spiegel 2015). Similarly, the equations of motion for HI can be derived aṡ
Qi = 0 (13) Vi = −G j=1 j =i mj Q 3 ij Qij K(Qij) − QijK (Qij)(14)
These equations remain trivial to solve even in the case of K = 1. Somewhat surprisingly, the equations of motion derived above are not the equations of motion implemented in the MERCURY code. Specifically, the terms involving K (Qij) are not present in the publicly available version of MERCURY.
Therefore, the MERCURY code does also not correspond to the Hamiltonian splitting in Eq. (2).
This then begs the question 2 : What is MERCURY solving? To find the answer to this question, we define a new switching function
L(r) ≡ K(r) − rK (r).(15)
With this definition, we can rewrite the equations of motion in terms of L. For example, Eq. (10) becomeṡ
Vi = − Gm0 Q 3 i Qi − G j=1 j =i mj Q 3 ij Qij (1 − L(Qij)) .(16)
Note that no derivatives of L appear in the equations of motion. We can think of L, which is changing between 0 and 1, as a weight in the inter-particle forces. We therefor refer to it as the force switching function. If we set L to the piecewise polynomial function
Lmerc(r) = 0 y 0 10y 3 − 15y 4 + 6y 5 0 < y < 1 1 y 1,(17)with y = r − 0.1rcrit 0.9rcrit(18)
then we recover exactly the integration scheme as implemented in the MERCURY code 3 . In the above equation, rcrit is some critical switching distance. The value depends on the specific application but it typically a small multiple of the mutual Hill radius of the particles.
To find out what the corresponding Hamiltonians are that this scheme is integrating, we need to solve the differential equation in Eq. (15) to find the Hamiltonian switching function K, given the force switching function L. The solution is a one parameter family due to a gauge symmetry. We choose the solution that is continuous by matching Kmerc(rcrit) = 1. We plot the functions Kmerc and Lmerc in the top panel of Fig. 1. Note that the function Kmerc does not reach exactly 0 at r = 0.1rcrit. This means that there is a finite contribution to the interaction Hamiltonian HI for any choice of r > 0 (see function Kmerc(r)/r in the bottom panel of Fig. 1). This is inconsistent with the discussion of Chambers (1999) which argues that the switching function keeps the interaction Hamiltonian small. That argument would only be correct if the force switching function Lmerc were used as a Hamiltonian switching function (in the MERCURY code Lmerc is used as a force switching function). As an illustration, we also plot the function Lmerc(r)/r in the bottom panel of Fig. 1. This is a potential which now reaches 0 at 2 This inconsistency was discussed at the Toronto Meeting on Numerical Integration Methods in Planetary Science in 2017 and an explanation was first published by Wisdom (2017).
We independently came to the same conclusion. 3 Note that this is not the function described in the paper which has roughly the same shape but is slightly different from the one implemented in the publicly available version of MERCURY. We here use the function implemented in the MERCURY code. r = 0.1rcrit. However, whereas using Lmerc as the Hamiltonian switching function would lead to a valid symplectic switching integrator, it will not perform as well as an integrator using Kmerc. Looking at the bottom panel of Fig. 1, this is because Lmerc(r)/r has a local minima around 0.8rcrit leading to an unphysical repulsive force between particles in some intermediate regime. One consequence of the force changing sign is that the differential equations corresponding to Lmerc(r)/r are stiffer during the close encounter. Note that in both cases, particles will feel no force from the potentials in the interaction Hamiltonian while r < 0.1rcrit. It is worth pointing out that for small r the shape of the effective potential Kmerc(r)/r is similar to that of potentials used in other N -body systems where one simply ignores the close range interactions by having a finite smoothing length (similar to a kernel in smooth particle hydrodynamics).
We would like to stress that despite this inconsistency in the derivation of the equations of motion, the MERCURY algorithm is solving a Hamiltonian system that converges to the original system in the limit of dt → 0.
For the remainder of this paper, we will refer to the force switching function L(r) simply as the switching function. The corresponding Hamiltonian switching function K(r) can always be calculated by solving Eq. (15).
We now explore different force switching functions and their effects on the accuracy of the integration. Our motivation comes from the fact that the function used by MERCURY has been determined heuristically and, as we have shown above, the original justification is not correct. It is thus not clear if one can improve the integration scheme by simply choosing a better switching function. In this paper, we will present and discuss results for the following four switching functions. These are extreme cases which will allow us to explain the general ideas behind choosing switching functions, which can then be applied to any arbitrary switching function.
(i) We set L(r) = 1. The integrator simply becomes the (non-switching) Wisdom-Holman integrator in democratic heliocentric coordinates.
(ii) We set L(r) to the discontinuous Heaviside function with a jump at r = rcrit.
(iii) Same as (ii) but the value of L is kept constant for the duration of the entire timestep. To do that we approximate the trajectories along straight lines for the duration of the timestep and set L = 0 if the trajectories come closer than rcrit, and L = 1 otherwise. There are cases where this breaks time reversibility. We could avoid this by doing this procedure iteratively. However, in all tests that we have performed this does not seem to be an issue. Further, note that this procedure formally makes the integrator phase space dependent in a similar way than adaptive timesteps do. We comment on whether this has any real world consequences in Sec. 7.
(iv) We use the switching function Lmerc(r) given in Eq.(17). Thus the integrator becomes the MERCURY integrator.
(v) Finally, we consider an infinitely differentiable switching function given by
L inf (r) = = f (y) f (y) + f (1 − y)(19)
where y is given by Eq. (18) and f (y) is
f (y) = 0 y 0 e −1/x y > 0 .(20)
The shape of the function L inf is very similar to that of Lmerc (shown in Fig. 1). However the function L inf (r) and all its derivatives are smooth and continuous for all r, even at the boundaries. Note however, that just as Lmerc(r), L inf (r) is not analytic, i.e. a Taylor series expansion will not converge globally.
TEST SETUP AND MERCURIUS
We now look at one representative test case in which a close encounter occurs between two giant planets. We work in units where G = 1 and the stellar mass is m0 = 1. Both planets have a mass similar to that of Jupiter, m1 = m2 = 10 −3 . The inner planet is on an orbit with semi-major axis a1 = 1 and eccentricity e1 = 0.1. The outer planet is on a nearby orbit with a2 = 1.1 and e = 0.2. We use a timestep corresponding to 0.5% of the inner planet's period. We adjust the phases such that a close encounter occurs at t ≈ 0 with a close approach distance of approximately 5 · 10 −5 .
To run these simulations we use MERCURIUS, our own implementation of the MERCURY integrator 4 . We here only give a short overview of the implementation as we plan to publish a detailed code description paper. The MERCURIUS integrator is freely available as part of the REBOUND integrator package. Internally it uses the Kepler solver of the WHFast integrator (Rein & Tamayo 2015) and the IAS15 integrator (Rein & Spiegel 2015) to evolve HK. Using MERCURIUS allows us to easily experiment with different switching functions. We can also call the individual operators manually which allows us to calculate the commutators numerically as described above. We also ran tests with the original MERCURY code and implemented wrapper functions to make the fortran code callable from python. These functions allow us to call MERCURY's time-stepping functions directly from python, avoiding any potential issues that may arise due to coordinate transformations, unit conversions, time-stepping logic, or input/output files. This might have been an issue in an earlier study looking at the symplecticity of MERCURY (see Appendix B). Aside from small differences at the floating point precision limit, if we use the same switching function as implemented in MERCURY, we find perfect agreement between MERCURY and MERCURIUS in all our results and therefore only show the results using MERCURIUS.
RESULTS
Figure 2 summarizes our results for the test case and switching functions introduced above. The columns correspond to the different switching functions. The horizontal axis of every panel corresponds to time, with the close encounter occurring at t ≈ 0. The particles enter the critical distance rcrit at t ≈ −1 and exit at it again at t ≈ 1. The top row shows the value of the switching function L as a function of time. The second row shows the absolute value of the gravitational force between the planets as calculated during the interaction step. We also show the first four derivatives of the force. The third, fourth, and fifth rows show approximations of the third, fifth, and seventh order commutators, respectively. We colour commutators that only include HK and HI (i.e. not HJ ) red, and all other commutators orange. We do not show commutators that are always zero (i.e. those only involving HI and HJ ). As a measure of the size of the commutators, we use the energy error as described above. Note that the energy error does not account for phase errors. Thus an energy error of zero does not imply a perfect solution. The plots for the commutators also show the actual energy error occurred over one timestep step, as well as the integrated energy error, i.e. the energy error measured relative to the beginning of the simulation.
The first column shows the evolution for the standard WH integrator. The forces, its derivatives, and therefore all commutators get very large during the close encounter. As a result, the integrated energy error goes off the chart. As expected, this integrator cannot accurately resolve the close encounter. Note that the higher order commutators become comparable to the lower order commutators during the close encounter. The physical interpretation for this is that timestep is too large to resolve the characteristic timescale (periastron passage of the close encounter). Also note that the dominant commutators are those involving only HK and HI .
The second column shows the evolution of using the Heaviside function as the switching function. Note that the commutators which only involve HK and HI vanish when L = 0. This happens because HI becomes the identity operator, i.e. not having any effect, and it thus commutes with any other operator. Commutators involving HK and HJ do not vanish when L = 0 and become the dominant contribution to the energy error during that time. However, these are several orders of magnitude smaller and do not diverge during the close encounter unless the planets are very eccentric and close to periastron. Note that there is a spike in most commutators exactly when L transitions at rcrit. This can be understood by imagining a scenario where the particles start out just outside of rcrit. To calculate the commutator [ HK , HI ], let us initially apply HI (dt). When calculating the effect of this operator the value of L will be 1. If we then apply HK (dt) to the result, it might take the particles inside of rcrit. Thus, if we then apply HI (−dt), the value of L will be 0. Let us finally apply HK (−dt) again which might take the particles outside of rcrit. This sequence of operators corresponds to the commutator [ HK , HI ] which is one of the nested commutators that appears at all orders in the BCH formula. The above argument makes it clear that the commutator will be particularly large when the switching function changes a lot during a timestep. This is the case for the Heaviside function, but it is important to note that for any other switching function that changes significantly during a timestep this is also the case, no matter if the switching function is smooth or has a discontinuity. The consequence is that the actual energy error (both the single step error and the integrated error) jump up when the particles transition rcrit. Despite this undesirable jump, the integrator is at least somewhat better at resolving the close encounter than the standard Wisdom-Holman integrator. A physical argument can be given in terms of timescales again: if the switching function changes a lot during one timestep, we're effectively introducing a very short timescale that the integrator is unable to resolve.
We can avoid large changes of the switching function during the timestep, by simply calculating the value of L once, at the beginning of the timestep, and then keeping it fixed. The results of this modification to the Heaviside switching function are shown in the third column. Several features are identical to the standard way of using the Heaviside switching function (second column). In particular, all commutators that only involve HK and HI still vanish when L = 0. However, we see that we have successfully removed the spikes in the commutators when L changes. The commutators involving HK and HI simply go to zero. As a consequence the energy error does not increase when particles pass over the rcrit transition, despite the sudden change in the value of the switching function L. This can be explained following the same scenario described in the last paragraph. This is a remarkable result: This integration scheme, effectively using a binary switching function, is significantly simpler to implement than those using a continuous switching function. However, as one can see in the figure, it provides equal accuracy for one close encounter, at reduced computational complexity. We further comment on this case below.
The fourth column shows the evolution using the standard MERCURY algorithm and its polynomial switching function. Once again all commutators that only involve HK and HI vanish when L = 0. However, in contrast to the integrators using the Heaviside function, this only happens for r < 0.1rcrit. Although these commutators are finite for r > 0.1rcrit they are significantly smaller than in the case of the Wisdom-Holman integrator. Note that the third order commutators are smooth. This is because the specific choice of switching function ensures that the forces and its first two derivatives are continuous. These derivatives appear in the Poisson bracket and therefore in the commutator. However looking at the higher order commutators, one can see the effect of non-continuous higher derivates due to the finite differentiable switching function Lmerc. Fortunately, these commutators are significantly smaller than the third order commutators and therefore do not contribute much to the total energy error. Remember that the plots only show approximations of the commutators. This becomes apparent in the 7th order commutators where the spikes near the discontinuities of the force derivatives at rcrit are somewhat smeared out.
The fifth column shows the evolution using an infinitely differentiable switching function. Here, all derivatives of the force, and therefore all Poisson brackets, are continuous and smooth. Because of the smoothness requirement, the higher order derivatives of the force become highly oscillatory. This is a generic feature of any infinitely differentiable switching function and not specific to the function we chose. The commutators look similar to those of the MERCURY algorithm but one significant difference is that the higher order commutators become larger and more oscillatory. As in the previous case this is because these commutators are directly related to the higher order derivatives of the forces. For moderate timesteps like the one we've chosen in this example, the energy error of the simulation is still dominated by the third order commutators. However, for larger timesteps, there can be cases where the higher order commutators will dominate the error. We can think of this in physical terms once again. The highly oscillatory derivatives of the force cor-responds to very short timescales. Because we are using a finite and fixed timestep, these timescales can at some point be no longer accurately resolved, leading to larger errors. The appearance of these small timescales is a purely numerical artefact (somewhat related to timestep resonances discussed by Rauch & Holman 1999). Mathematically speaking, the differential equations corresponding to the higher order commutators become very stiff. Note that in the standard Wisdom-Holman integrator (left panel), the forces get large, but do not oscillate.
CONCLUSIONS
We revisited the motivation behind hybrid symplectic integrators and the choice of switching functions. We found that the derivation of equations of motion by Chambers (1999) omits derivative terms. As we showed, it turns out that despite this inconsistency the MERCURY integrator is a switching integrator. However, it effectively uses a very different switching function than the one described by Chambers (1999).
Motivated by the somewhat arbitrary choice of switching function in MERCURY, we explored a wide range of different switching functions to see if it is possible to improve the accuracy of this kind of integrators any further. In particular, we presented results of two extreme cases: one infinitely differentiable function, and the Heaviside function. In summary, we found that the smoothness alone is not a good criterion for choosing a switching function.
Our results show that an infinitely differentiable switching function does not perform much better over one close encounter than the polynomial function used by MERCURY. We attribute this to two reasons. First, the beneficial effects of a smoother function only show up in higher order commutators which are several orders of magnitude smaller than the dominant third order commutators. Second, the higher order derivates of an infinitely differentiable switching function necessarily become highly oscillatory. The integrator cannot resolve the associated small timescales and thus behaves no better than an integrator involving finite differentiable functions.
We show that using a Heaviside switching function can have surprisingly good properties if the value of the switching function does not change during the timestep. The resulting scheme is significantly simpler than the MERCURY scheme but achieves the same accuracy in the test case of a single close encounter presented. What still needs to be studied in greater detail is the long term evolution in systems with repeated close encounters. We ran additional tests of 1000 highly collisional planetary systems and found that the integrator using the Heaviside function performs on average as well as MERCURY. We will discuss the implementation of this new integrator, as well as the implementation of the MERCURIUS integrator, in an upcoming code description paper.
In practice one is mostly interested in how accurate an integrator can resolve a close encounter. In any numerical simulation, one can calculate a precise answer to this question using any arbitrary metric one thinks is approriate.
A mathematical (or maybe even somehwat philosophical) question is whether these integrators we discussed in this paper are symplectic or not. The authors of this paper were not able to agree on the answer to this question. We will therefore leave the final answer up to the reader, but comment on a few aspects of this question.
First, it is important to keep in mind that we very quickly reach the limits of finite floating point precision in systems with close encounters 5 . One could argue that we should not talk about concepts of differential geometry such as symplecticity at all if we work with floating point numbers because derivatives formally do not even exist. We run simulations on a discrete phase space where only finite differences exist.
Second, note that all switching functions we used are non-analytic. That is not a coincidence but a requirement if we want the integrators to fall back to the Wisdom-Holman integrator whenever particles are far away from each other. As a consequence, the Taylor series of the switching function, and therefore the Poisson Brackets in the BCH formula will not converge to the correct solution globally.
Third, when choosing a switching function, differentiability is not the right criteria to consider. If the mathematical concept of differtiability were an important requirement for a switching function, we could simply come up with a new infinitely differentiable function that is arbitrarily close to the original non-smooth function. We have effectivly done that for Lmerc by introducing L inf , which did not lead to improvements in any of our tests. What matters is how much the switching function changes during a (finite) timestep. If it changes a lot, then the corresponding differential equations become stiff and the errors large. What also matters is that the switching function is continous to ensure the existance and uniquness of a solution 6 (Hernandez 2019). Note that integrator with the Heaviside function completely changes the nature of the algorithm and can among other things break time reversibility. This might be important for some long term integrations, but not for simulations where encounters are rare (for example for planetary system which go unstable), or if physical collisions break time reversibility anyway.
Fourth, all integrators will converge to the correct solution in the limit of the timestep going to zero. We thus argue that the simplest and most reliable way to test that a given integrator gives reliable answers is to change the timestep and watch for changes in the results, i.e. a classical convergence test. 5 Imagine a scenario where two planets on orbits with a ≈ 1 have a close encounter with a minimal encounter distance of one Earth radius, ≈ 10 −5 . Using double floating point precision, we only have ≈ 11 decimal digits to work with when representing the planets' positions in heliocentric coordinates. When calculating any derivative, which is required for checking the symplecticity, we further loose roughly half of the significant digits. 6 This only matters when calculating the solution of the H K step. It does not matter for the solution of the H I step because the switching function is constant.
APPENDIX A: DEMOCRATIC HELIOCENTRIC COORDINATES
The hybrid integration schemes discussed in this paper make use of democratic heliocentric coordinates (Duncan et al. 1998). Let us assume we have one star and an additional N planetary bodies with Cartesian coordinates qi and canonical momenta pi in some inertial frame. Then, the democratic heliocentric coordinates are defines as:
Qi = qi − q0 i = 0 1 M N −1 j=0 mjqj i = 0.(A1)
Here, M = N −1 j=0 mj is the total mass of the system. The corresponding canonical momenta are
Pi = pi − m i M N −1 j=0 pj i = 0 N −1 j=0 pj i = 0.(A2)
APPENDIX B: SYMPLECTICITY OF MERCURY
While working with MERCURY and MERCURIUS we also looked at measuring the symplecticity error. To do that we follow Hernandez (2016) in calculating the Jacobian using a finite difference approach. This is non-trivial because one quickly runs into issues of finite floating point precision. The results of Hernandez (2016) suggest that the MERCURY algorithm might be non-symplectic for their binary planet test case.
To investigate this issue, we implemented wrapper functions to directly call the MERCURY subroutines which evolve the state by one timestep, but ignore all the input, output and other bookkeeping login in the MERCURY package. Our results with this method differ from those obtained by Hernandez (2016) and indicate that MERCURY is symplectic (whether we should really call the scheme symplectic or not, becomes a somewhat philosophical rather than practical question, see discussion). We attribute this discrepancy to some non-symplectic operations that the MERCURY package applies to the state vector before and after the actual integration (e.g. unit conversions and coordinate transformation), and a loss of precision when using ASCII files as input and output files. We also ran tests with our own implementation of the algorithm, MERCURIUS, and confirm the results obtained with MERCURY.
Figure 1 .
1Top panel: the Hamiltonian switching function Kmerc and the corresponding force switching function Lmerc. Note thatChambers (1999) discusses Lmerc in the context of a Hamiltonian switching function, but the actual implementation in MERCURY uses Lmerc, a force switching function. Bottom panel: potentials in the interaction Hamiltonian if both Kmerc and Lmerc were used as Hamiltonian switching functions. Kmerc/r is the potential corresponding to the actual implementation of MERCURY.
Figure 2 .
2Comparison of different switching functions used in a hybrid symplectic integrator during a close encounter of two giant planets. Each column corresponds to a different switching function. The top row shows the value of the force switching function during the integration. The second row shows the absolute value of the gravitational force between the two planets in the interaction step. The bottom three rows show approximations of the third, fifth, and seventh order commutators that appear in the integrators' error terms. We work in a system of units where G = 1. See text for more details.
). Further note that HJ commutes with HI . Thus, the lowest order non-zero error terms that appear are:1
12
[[ HJ , HK ], HK ],
1
24
[[ HJ , HK ], HJ ],
1
12
[[ HI , HK ], HK ],
1
24
[[ HI , HK ], HI ],
and
1
12
[[ HJ , HK ], HI ].
c 0000 RAS, MNRAS 000, 000-000
In reference to and appreciation of MERCURY, we call our implementation MERCURIUS (the Latin word for Mercury).c 0000 RAS, MNRAS 000, 000-000
ACKNOWLEDGMENTSWe thank the referee John Chambers for providing us with helpful comments which allowed us to improve and clarify the manuscript. We thank Scott Tremaine for helpful discussions at various stages of this project. This research has been supported by the NSERC Discovery Grant RGPIN-2014-04553, the 2018 Fields Undergraduate Summer Research Program at the Fields Institute for Research in Mathematical Sciences in Toronto, and the Centre for Planetary Sciences at the University of Toronto Scarborough. Support for this work was also provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51423.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. This research was made possible by the open-source projects Jupyter(Kluyver et al. 2016), iPython(Pérez & Granger 2007), and matplotlib(Hunter 2007;Droettboom et al. 2016).
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| [] |
[
"Sub-Poissonian multiplicity distributions in jets produced in hadron collisions",
"Sub-Poissonian multiplicity distributions in jets produced in hadron collisions"
] | [
"Han Wei Ang \nDepartment of Physics\nNational University of Singapore\n117551Singapore\n",
"Maciej Rybczyński \nInstitute of Physics\nJan Kochanowski University\n25-406KielcePoland\n",
"Grzegorz Wilk \nNational Centre for Nuclear Research\n00-681WarsawPoland\n",
"Zbigniew W Lodarczyk \nInstitute of Physics\nJan Kochanowski University\n25-406KielcePoland\n"
] | [
"Department of Physics\nNational University of Singapore\n117551Singapore",
"Institute of Physics\nJan Kochanowski University\n25-406KielcePoland",
"National Centre for Nuclear Research\n00-681WarsawPoland",
"Institute of Physics\nJan Kochanowski University\n25-406KielcePoland"
] | [] | In this work we show that the proper analysis and interpretation of the experimental data on the multiplicity distributions of charged particles produced in jets measured in the ATLAS experiment at the LHC indicates their sub-Poissonian nature. We also show how, by using the recurrent relations and combinants of these distributions, one can obtain new information contained in them and otherwise unavailable, which may broaden our knowledge of the particle production mechanism. | 10.1103/physrevd.105.054003 | [
"https://arxiv.org/pdf/2111.07934v2.pdf"
] | 244,117,733 | 2111.07934 | 88050aec5d3de932b7bd996fd9d016dbba5707b4 |
Sub-Poissonian multiplicity distributions in jets produced in hadron collisions
Han Wei Ang
Department of Physics
National University of Singapore
117551Singapore
Maciej Rybczyński
Institute of Physics
Jan Kochanowski University
25-406KielcePoland
Grzegorz Wilk
National Centre for Nuclear Research
00-681WarsawPoland
Zbigniew W Lodarczyk
Institute of Physics
Jan Kochanowski University
25-406KielcePoland
Sub-Poissonian multiplicity distributions in jets produced in hadron collisions
numbers: 1385Hd2575Gz0250Ey
In this work we show that the proper analysis and interpretation of the experimental data on the multiplicity distributions of charged particles produced in jets measured in the ATLAS experiment at the LHC indicates their sub-Poissonian nature. We also show how, by using the recurrent relations and combinants of these distributions, one can obtain new information contained in them and otherwise unavailable, which may broaden our knowledge of the particle production mechanism.
I. INTRODUCTION
The experimentally measured multiplicity distributions P (N ) of the produced particles are the main source of information about the dynamics of their production processes [1,2]. In the theoretical description they are characterized by the generating functions,
G(z) = ∞ N =0 P (N )z N ,(1)
such that
P (N ) = 1 N ! d N G(z) dz N z=0 .(2)
Preliminary information about P (N ) is provided by the moments of this distribution,
m k = ∞ N =0 (N − c) k P (N ).(3)
In many cases, it is sufficient to analyze only the two lowest moments, namely the mean value N = m 1 being the first raw moment (c = 0), and variance V ar(N ) = m 2 , being the second central moment (c = N ). Another way to characterize P (N ) is through a recursive formula, (N + 1)P (N + 1) = g(N )P (N ),
that connects adjacent values of P (N ) for the production of N and (N + 1) particles. It is assumed here that every P (N ) is determined only by the next lower P (N − 1) value. In other words, a relationship with others P (N −j) for j > 1 is indirect. The final algebraic form of P (N ) is determined by the function g(N ). In its simplest form, * [email protected] † [email protected] ‡ [email protected] § [email protected] g(N ) is assumed to be a linear function of N given by g(N ) = α+βN . This form is enough to define commonly known and widely used distributions like Poisson Distribution (PD), for which β = 0), Binomial Distribution (BD) (for which β < 0) or Negative Binomial Distribution (for which β > 0). In general, by selecting the appropriate model, the form g(N ) can be chosen in such a way that the corresponding P (N ) describes the experimental data (for example, by introducing higher order terms [3] or by using its more involved forms [4,5]). The more promising approach is to use g(N ) which contains information about the interrelationship between the multiplicity N and all smaller multiplicities recursively, (N + 1)P (N + 1) = N N j=0 C j P (N − j).
The memory of that relationship is encoded in coefficients C j called modified combinants. They were introduced and intensively discussed in [6][7][8][9][10][11][12][13]. By inverting the recursion (5) we obtain an equation that allows us to determine C j from the measured P (N ),
N C j = (j+1) P (j + 1) P (0) − N j−1 i=0 C i P (j − i) P (0)(6)
(provided we have sufficient statistics). Modified combinants are closely related to combinants C j ,
C j = j + 1 N C j+1 ,(7)
introduced in [14][15][16] by means of the generating functions G(z) as
N C j = 1 j! d j+1 ln G(z) dz j+1 z=0 ,(8)
which have been discussed and used in many publications [1,2,[17][18][19][20][21]. Modified combinants are complementary to the commonly used factorial moments, F q and cumulant factorial arXiv:2111.07934v2 [hep-ph] 11 Feb 2022 moments, K q (see Appendix A for details). They differ in that while C j 's depend only on multiplicities smaller than their rank, K q 's require the knowledge of all P (N )'s and are therefore very sensitive to possible limitations of the available phase space [1,2]. However, both C j and K q share the property of additivity. It turns out that most of the measured multiplicity distributions P (N ) give oscillatory combinants C j with increasing index j which arise due to the presence of a BD component in the measured P (N ) [7][8][9][10][11][12].
Combinants are believed to be best suited for the study of sparsely populated areas of phase space (while cumulants are better suited for the study of densely populated areas) [1,2]. This feature makes them a potentially important tool for studying P (N ) in jets where the number of produced particles is small (on the order of ∼ 10). However, the ATLAS data [22] do not include P (0), which is crucial for their determination. This means that in our analysis, we must extrapolate the recurrent relation in Eq. (4) to evaluate P (0) and use the combinants only for additional verification of our conclusions.
In the next Section we provide details concerning AT-LAS data with particular attention to the fact that the measured multiplicity distributions in the jets are clearly sub-Poissonian in character (with details depending on the phase-space covered). This observation will be our main point for further discussion and calculations described in Sections III and IV. Section V summarizes and concludes our work.
II. MULTIPLICITY DISTRIBUTIONS OF PARTICLES IN ATLAS JETS
ATLAS data [22] of jets measured in proton-proton collisions at an energy √ s = 7 TeV (using a minimum bias trigger) were taken (data used here come from [23]). Jets were reconstructed using the anti-k t algorithm applied to charged particles produced in very narrow cones defined by radius parameter R = ∆η 2 + ∆φ 2 (where ∆φ and ∆η are the azimuthal angle and the pseudorapidity of the hadrons relative to that of the jet respectively. Here, η = − ln tan θ, with θ being the polar angle), with R = 0.4 and again at R = 0.6. Only data with high statistics within acceptable regions of phase space are selected. They are obtained over 5 different transverse momentum ranges across 4 rapidity ranges, giving a total of 20 possible combinations for each of the two radius parameters (with radius parameters R = 0.4 and R = 0.6, respectively). So far, the ATLAS data have been carefully analyzed in terms of possible self-similarity between p T distribution of jets and p T distributions of particles in these jets [24]. Indeed, their detailed analysis clearly indicates the self-similarity of the particle distributions in jets and the distributions of the jets themselves, indicative of the existence of a common mechanism behind all these processes. FIG. 1. Multiplicity distribution of charged particles per jet with 4 GeV < p jet T < 6 GeV , over the full measured rapidity range |y| < 1.9, with radius parameter R = 0.4. Points show data from ATLAS experiment [22]. P (N = 0) comes from extrapolation of experimental recurrent relation g (N ) to N = 0. The curve fitting this data comes from Eq. (13) with parameters α = 14.1 and δ = 1.07.
Taking advantage of the fact that ATLAS also publishes data for multiplicity distributions of particles produced in observed jets, we extend this analysis to study the nature of these distributions. The first observation is that the multiplicity distributions of the particles in the jets observed in the ATLAS experiment are sub-Poissonian, cf. Fig. 1 where V ar(N ) < N . Though BD is a possible sub-Poissonian distribution for g(N ), the plot in Fig. 2 derived from the recursive relationship in Eq. (4) is non-linear.
It means therefore that one should switch to scenario inspired by the possible non-linear form of recursion g(N ) such as
g(N ) = (N + 1) P (N + 1) P (N ) = α (N + 1) δ .(9)
As shown in Fig. 2, such form with δ = 1.07 fits data very well. Therefore, we note that for
(N + 1) P (N + 1) P (N ) = α(10)
we have the PD with,
P (N ) = c α N N ! where α = N , c = exp(−α). (11)
If we change this recursive relationship to a nonlinear one given by
(N + 1) 1+δ P (N + 1) P (N ) = α,(12)
we get sub-Poissonian distribution,
P (N ) = c α N (N !) 1+δ ,(13)
where c = P (0) is a normalization factor. In Fig. 1 we present a comparison of this multiplicity distribution with the experimental data from ATLAS. Once we know P (0) from extrapolation of experimental recurrent relation g (N ) to N = 0, we are able to determine the corresponding modified combinants from the measured multiplicity distributions P (N ) using Eq. (6) (which is important because in ATLAS, the P (0) data points are not directly measured). The C j 's derived in this way are plotted in Fig. 3 as red circles. They can now be compared with the combinants obtained in the same way but from the theoretical sub-Poissonian distribution P (N ) given by Eq. (13). The corresponding results are plotted in Fig. 3 as black squares. We observe characteristic oscillatory behaviour of modified combinants with only rough amplitude agreement. Despite nice agreement of multiplicity distributions, shown in Fig. 1, the mentioned combinants indicate difference between fit and experimental data.
In the event where δ = 1 Eq. (13) can be written in a closed form given by,
P (N ) = 1 I 0 (2 √ α) α N (N !) 2 ,(14)
where
α = N 2 and c = P (0) = 1 I 0 (2 √ α) ,(15)
with I 0 being the Bessel function of the first kind. In this case (δ = 1) the theoretical C j 's are given by the recurrent relation,
N C j = (j + 1)α j+1 [(j + 1)!] 2 − N j−i i=0 C i α j−i [(j − i)!] 2 ,(16)
and are of the form where the numbers β j are rational:
N C j = (−1) j β j α j+1(17)β 0 = 1, β 1 = 1 2 , β 2 = 1 3 , β 3 = 11 48 , β 4 = 19 120 , β 5 = 473 4320 , .
. . . They were first calculated by Euler in relation to the positive zeros γ l of the Bessel function J 0 (z) as [25]
β j+1 = ∞ l=1 2 γ l 2(j+l) , j = 0, 1, 2, . . . .(18)
For j ≥ 1 coefficients β j can be approximated by
β j ∼ = exp − j + 1 e .(19)
At this point, it is worth noting that C j can help in the search for the correct P (N ). Based on our experience thus far, let us assume, that strongly oscillating C j (especially with period, as seen in Fig. 3) indicate the presence of a single-component BD in some form. On closer inspection, it is clear that this cannot be the case since the amplitude of oscillations of C j in Fig. 3 grows approximately as 9.3 j with j. Should these C j originate from a single-component BD with amplitudes given by [p/(1 − p)] j , it would mean that p > 0.8. In addition, to reproduce N = Kp ∼ 3.3 as observed in data, one would require K < 5. Taken together, these would limit us to multiplicities of N < 5. This is in contradiction with the measured P (N ) where the observed multiplicities N = 5. By continuing to stick to BD, our previous experiences [7,[9][10][11] tell us that a potential solution might be to use the sum of two BD's instead of one. However, as we will show in the Appendix B, it is not possible to describe both P (N ) and its corresponding C j with this approach.
Continuing the approach based on g(N ), it turns out that with the increase in p T of jets (corresponding to an increase of N in our case), we observe a deviation from the form of g(N ) given in Eq. (9). The modified g(N ) can be made to describe data if expressed as a recursive relation given by (see Fig. 4)
g(N ) = (N + 1) P (N + 1) P (N ) = α (N + 1) δ + α 0 ,(20)
leading to the multiplicity distribution (see Fig. 5)
P (N ) = c N ! N i=1 α i δ + α 0(21)
with c = P (0). The corresponding C j are shown in Fig. 6.
Note that for integer parameter δ , Eq. (21)) has closed analytical form, namely for δ = 1
P (N ) = P (0) α N 0 (N !) 2 Γ 1 + α α0 + N Γ 1 + α α0 ,(22)
while for δ = 2
P (N ) = P (0) α N 0 (N !) 3 · · Γ 1 − −α α0 + N Γ 1 − −α α0 Γ 1 + −α α0 + N Γ 1 + −α α0 ,(23)
and for higher value of δ we have product of δ Pochhammer symbols,
(x l ) N = Γ (x i + N ) Γ (x i ) ,(24)
where x l = 1+(−1) l (−1) l/δ (α/α 0 ) 1/δ for l = 0, 1, . . . , δ− 1.
III. POSSIBLE EXPLANATION: MULTIPLICITY DEPENDENT BIRTH AND DEATH RATES
To interpret the results shown in the previous section, note that Eq. (13) actually represents the so-called COM-Poisson distribution introduced by Conway and Maxwell [26] as a model for steady state queuing systems with state-dependent arrival or service rates (in other words, birth-death process with Poisson arrival rate and exponential service rate). It was rediscovered in [27] where the term Conway-Maxwell-Poisson was proposed and a detailed study of its properties and applications was performed. More recent studies can be found in [28,29]. To our best knowledge, this distribution has not been used in the analysis of multiplicity distributions of particles produced in multiparticle production processes.
We will now show that the form of the COM-Poissonian distribution can be obtained from a stochastic Markov process with multiplicity-dependent birth and death rates denoted by λ N and µ N , respectively [30]. Let P (N, t) be the probability of having N particles at time t and let us consider a very general birth-death process given by the following equations: N + 1, t). (26) If we assume the forms
P (0, t) = −λ 0 P (0, t) + µ 1 P (1, t),(25)P (N, t) = − (λ N + µ N ) P (N, t) + +λ N −1 P (N − 1, r) + µ N +1 P (λ N = λ (N + 1) a and µ N = N b µ,(27)
we get
− λ (N + 1) a + N b µ P (N ) + + λ N a P (N − 1) + (N + 1) b µP (N + 1) = 0 (28)
for the steady state, where P (N, t) = 0. If we denote
λ µ = α(29)
we can re-write Eq. (28) as
− α (N + 1) a P (N ) − N b P (N ) + + α N a P (N − 1) + (N + 1) b P (N + 1) = 0 (30)
which leads to the recurrent relation
(N + 1) b P (N + 1) = α (N + 1) a P (N ).(31)
Further simplifications can be made by writing
a + b = ν(32)
which gives us
P (N + 1) P (N ) = α (N + 1) ν .(33)
For δ = ν − 1, this is just the recurrent form of COM-Poisson distribution defined by Eq. (12). The condition in Eq. (32) has allowed us to successfully re-parameterize the recurrent relation of Eq. (33) using α and ν = a + b. This corresponds to the entire class of Markov processes previously characterized by parameters a and b with the birth and death rates given in Eq. (27). However, to describe the distribution defined by equations (20) and (21) (which is no longer COM-Poisson distribution) still using the birth-death process we need to add an additional term λ to the birth rate λ N in the form given by
λ N = λ (N + 1) a + λ (N + 1) (b−1) .(34)
If we substitute (34) into (26) and denote
λ µ = α 0 (35)
we get the recurrent relation
P (N + 1) P (N ) = α (N + 1) ν + α 0 N + 1 (36)
corresponding to Eq. (20).
IV. SUMMARY OF RESULTS
The ATLAS data suitable for our purposes cover 5 different ranges of transverse momentum, each further divided into 4 different rapidity ranges totalling 20 different fragments of the phase space. In previous sec-
N |C j | = AB j .(37)
With increasing values of transverse momenta of jets, p jet T , the mean multiplicity in jet grows and affects parameters given in the tables. In Table I the parameters depend on N in the following ways: Table II shows results for broader jets with R = 0.6. Dependence of the parameters on N in this case are as follows:
δ = 0.94 + N 4.9 4.4 ,(38)α 0 = −4.94 + 1.7 N ,(39)α ∼ = A ∼ = B ∼ = 10 + N 3.05 6 ,(40)P (0) = 0.5 exp − N 0.7 .(41)δ = 0.66 + N 6.38 5.85 ,(42)α 0 = −9.56 + 2.17 N ,(43)α ∼ = A ∼ = B ∼ = 15.5 + N 5.9 17.5 ,(44)
Dependence of P (0) on N remains the same as in Eq.
(41). The value of N increases by a factor of 1.85 for R = 0.6 in comparison to the case of R = 0.4 as seen from Tables I and II. Other dependencies on N are similar to those from the above equations. However, in the interval of 24 GeV< p jet T < 40 GeV, we have δ < 0 and hence, this p jet T interval was omitted in determining the parameter dependence on N .
With regards to the relation between V ar(N ) and N , it is observed that for R = 0.4, we have dispersion σ = V ar(N ) = 0.33 N + 0.23 and for R = 0.6 we have V ar(N ) = 0.33 N + 0.10 1 . Note that mean multiplicities N within the various rapidity intervals do not depend significantly on the rapidity interval |y|, as shown in Table III, and are similar to N for the maximal interval 0 < |y| < 1.9. Similarly, the other parameters do not differ significantly from the corresponding parameters in the maximum rapidity range (no systematic changes with the width of the rapidity range).
V. CONCLUSIONS
Recurrent relation g(N ) = (N + 1)P (N + 1)/P (N ) leads to multiplicity distributions of the form
P (N ) = P (0) N ! N −1 i=0 g(i).(45)
For g(N ) given by Eq. (20) with α 0 = 0, we have PD for δ = 0, a sub-Poissonian distribution for δ > 0 also known as the Conway-Maxwell-Poisson distribution (COM-PD) [26][27][28][29][30] and a super-Poissonian distribution for δ < 0. It turns out, however, that the multiplicity distributions in jets prefer the recurrent relation with α 0 = 0 leading to multiplicity distributions of the form given by Eq. (21). For small N we observe sub-Poissonian distributions with modified combinants oscillating as
N C j ∝ (−1) j α j+1 .(46)
Parameters of multiplicity distributions (α, α 0 and δ) depend on N . For large N , distributions are super-Poissonian when −1 < δ < 0 2 . A sub-Poissonian distribution with V ar(N ) < N has not been very popular in the majority of discussions about multiplicity distributions in high-energy physics so far (mainly due the fact that this phenomenon is only observed when N is not too large). However, it is quite an important distribution [17,34]. The existence of a sub-Poissonian distribution implies at least one of the following scenarios: (1) the underlying elementary processes are not totally random (partially deterministic);
(2) the classical Markov processes describing them require further generalizations. Either conclusion forces us to modify the successful stochastic approach. In our case, we show that the sub-Poisonian multiplicity distributions describing the experimental data can be naturally interpreted as stochastic Markov processes in which the birth and death rates are both multiplicity-dependent.
Usually information contained in P (N ) is obtained by examining their corresponding factorial moments, F q , and cumulant factorial moments, K q , (or their ratios) (cf., [1,2]),
K q = F q − q−1 i=1 q − 1 i − 1 K q−i F i ,(A1)
where
F q = ∞ N =q N (N − 1)(N − 2) . . . (N − q + 1)P (N ), (A2)
are the factorial moments. As shown in [9,11] the K q can be expressed as an infinite series of the C j ,
K q = ∞ j=q (j − 1)! (j − q)! N C j−1 ,(A3)
and, conversely, the C j can be expressed in terms of the K q [1, 2],
C j = 1 N 1 (j − 1)! ∞ p=0 (−1) p p! K p+j .(A4)
Note that C j depends only on multiplicities smaller than their rank [14,15] while moments K q require the knowledge of all P (N ) and therefore are very sensitive to possible limitations of the available phase space [1,2]. On the other hand, calculations of combinants require the knowledge of P (0) which may not always be available. Both C j and K q exhibit the property of additivity. FIG. 7. Points: P (N ) from ATLAS data for jets with 4 GeV < p jet T < 6 GeV and radius parameter R = 0.4, over the full measured rapidity range |y| < 1.9. The curve fitting these data comes from generating function given by Eq. (B2).
Appendix B: Multiplicities from two Binomial Distributions
If we have two sources producing N 1 and N 2 particles respectively and each distributed according to BD defined by parameters (K 1 , p 1 ) and (K 2 , p 2 ), then the distribution of N = N 1 + N 2 particles,
P (N ) = min(N,K1,K2) i=0 P 1 (i)P 2 (N − i),(B1)
is described by a generating function comprising the product of generating functions for both sources, i.e. by
G(z) = (1 − p 1 + p 1 z) K1 · (1 − p 2 + p 2 z) K2 (B2)
In this case, the first two moments of the distribution P (N ) are given by
N = dG(z) dz z=1 = K 1 p 1 + K 2 p 2 , (B3) V ar(N ) = d 2 G(z) dz 2 z=1 + N − N 2 = = K 1 p 1 (1 − p 1 ) + K 2 p 2 (1 − p 2 ). (B4)
Denoting the modified combinant of the first and second BD component as C(1) j and C(2) j respectively, the overall modified combinant N C j can be written as
N C j = N 1 C(1) j + N 2 C(2) j = (B5) = (−1) j K 1 p 1 (1 − p 1 ) j+1 +K 2 p 2 (1 − p 2 ) j+1 .
The value of one of the p parameters must be carefully chosen to reflect the observed increase in the amplitude of C j . Choosing (indicative) K 1 = 2, p 1 = 0.9 and K 2 = 10, p 2 = 0.17 we have P (N ) and C j as shown in Figs. 7 and 8. For this set of parameters used to fit data, we have N = 3.5 and V ar(N ) = 1.6.
While it is possible to reasonably describe either P (N ) or C j with a suitable choice of parameters, it is not yet possible to describe both simultaneously. Figs. 7 and 8 show the extent of deviation from data of such an approach.
It turns out that while by suitable choice of parameters, we can describe (more or less reasonably) separately P (N ) or C j , but not simultaneously both observables. Figs. 7 and 8 demonstrate to what extent is it possible to get closer to this goal in such an approach.
FIG. 2 .
2Recurrent relations. Points show g(N ) for experimental P (N ) from Fig. 1. Curve fit using g(N ) from Eq. (9) with α = 14.1 and δ = 1.07.
FIG. 3 .
3Comparison of Cj for multiplicity distributions from Fig. 1. Red circles: Cj from data on P (N ) from Fig. 1. Black squares: from theoretical P (N ) defined by Eq.(13).
FIG. 4 .
4Recurrent relations g(N ) for multiplicity distributions P (N ) over the full measured rapidity range |y| < 1.9, in jets with radius parameter R = 0.4 and transverse momentum range 10 GeV < p jet T < 15 GeV . Points: g(N ) from experimental data[22]. Curve: fit using g(N ) from Eq.(20) with parameters: α = 29.4, δ = 2.20 and α0 = 4.37.
FIG. 5 .
5Points: P (N ) from ATLAS data for jets with 10 GeV < p jet T < 15 GeV and radius parameter R = 04, over the full measured rapidity range |y| < 1.9. The curve fitting these data comes from Eq.(21) with parameters: c = 3.2·10 −4 , α = 29.4, α0 = 4.37 and δ = 2.20.
FIG. 6 .
6Comparison of Cj for multiplicity distribution in jets. Red circles: Cj from data on P (N ), black squares: from theoretical P (N ) defined by Eq.(21). Parameters the same as inFig. 5.
tions we have plotted examples of multiplicity distributions and their corresponding combinants. Figs. 1 and 3 show the plots for 4 GeV < p jet T < 6 GeV at |y| < 1.9 with R = 0.4 while Figs. 5 and 6 show that for 10 GeV < p jet T < 15 GeV. Instead of showing all figures for P (N ) and C j which we have calculated for each possible g(N ) given by Eq. (20) and P (N ) using Eq. (21), the presented values of the parameters in are described by the formula
. Dependence on rapidity interval |y| for R = 0.4 and 10 < p jet T < 15 GeV. 5 − 1.9 5.061 3.854 30.7 31.0 32.632 2.370 4.310 0.000320779 the comparison of
FIG. 8 .
8Comparison of Cj for multiplicity distributions from Fig. 7. Red circles: Cj from data on P (N ) , black squares: Cj from Eq. (B4 obtained from theoretical P (N ) defined by the generating function given by Eq. (B2).
TABLE I .
IDependence on p jet T for R = 0.4 and 0 < |y| < 1.9.
It is worth recalling at this point that this linear relationship, V ar(N ) = a N + b, known as Wróblewski's law[31], is satisfied for a wide range of multiparticle production processes like pp collisions (here a = −b = 0.585), both π + p and π − p collisions (with a = 0.44 for π ± and b = −0.22 for π + and b = −0.9 for π − )[32] and for e + e − (where a = 0.25 and b = 0.7))[33].2 For δ = −1 we have well known NBD, for α 0 > α > 0 (where parameter k = α 0 /α + 1, if additionally α 0 = 0 we would have geometrical distribution, the most wide one) or BD for α 0 > 0 and α < 0 (where parameter K = −α 0 /α − 1).
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| [] |
[
"Nonextensive statistical field theory",
"Nonextensive statistical field theory"
] | [
"P R S Carvalho \nDepartamento de Física\nUniversidade Federal do Piauí\n64049-550TeresinaPIBrazil\n"
] | [
"Departamento de Física\nUniversidade Federal do Piauí\n64049-550TeresinaPIBrazil"
] | [] | We introduce a field-theoretic approach for describing the critical behavior of nonextensive systems, systems displaying global correlations among their degrees of freedom, encoded by the nonextensive parameter q. As some applications, we report, to our knowledge, the first analytical computation of both universal static and dynamic q-dependent nonextensive critical exponents for O(N ) vector models, valid for all loop orders and |q − 1| < 1. Then emerges the new nonextensive O(N )q universality class. We employ six independent methods which furnish identical results. Particularly, the results for nonextensive 2d Ising systems, exact within the referred approximation, agree with that obtained from computer simulations, within the margin of error, as better as q is closer to 1. We argue that the present approach can be applied to all models described by extensive statistical field theory as well. The results show an interplay between global correlations and fluctuations. | 10.1016/j.physletb.2022.137147 | [
"https://arxiv.org/pdf/2112.00678v2.pdf"
] | 244,773,033 | 2112.00678 | 2029f40ac64fa0cbf98624a4451d2f3c47c8e343 |
Nonextensive statistical field theory
6 Dec 2021
P R S Carvalho
Departamento de Física
Universidade Federal do Piauí
64049-550TeresinaPIBrazil
Nonextensive statistical field theory
6 Dec 2021
We introduce a field-theoretic approach for describing the critical behavior of nonextensive systems, systems displaying global correlations among their degrees of freedom, encoded by the nonextensive parameter q. As some applications, we report, to our knowledge, the first analytical computation of both universal static and dynamic q-dependent nonextensive critical exponents for O(N ) vector models, valid for all loop orders and |q − 1| < 1. Then emerges the new nonextensive O(N )q universality class. We employ six independent methods which furnish identical results. Particularly, the results for nonextensive 2d Ising systems, exact within the referred approximation, agree with that obtained from computer simulations, within the margin of error, as better as q is closer to 1. We argue that the present approach can be applied to all models described by extensive statistical field theory as well. The results show an interplay between global correlations and fluctuations.
Since a nonextensive generalization of both thermodynamics [1] and statistical mechanics [2] was proposed [3], many of their aspects were examined. In one of these investigations, it was shown that the nonextensive version of thermodynamics can be mapped into its extensive counterpart through a transformation of variables [4]. As thermodynamics permit us to explore physics just at large length scales, signatures of nonextensivity do not manifest itself at such large scales. So nonextensive effects can emerge only when all length scales are considered, whose appropriate scenario of investigation is that of statistical mechanics, particularly when we study the critical behavior of systems undergoing phase transitions [5,6]. So far, some nonextensive behaviors were observed experimentally [7][8][9]. Many other possible applications of nonextensivity were proposed, with theoretical, experimental and computational evidences [10,11]. From the generalization process, a parameter q ∈ R arises naturally, characterizing the nonextensivity of the theory, namely superextensivity (q < 1) and subextensivity (q > 1). The parameter q is interpreted as one encoding global correlations among the many degrees of freedom of the system [12]. The limit q → 1 recovers the extensive theory. Among the applications, that involving the critical behavior of nonextensive systems undergoing continuous phase transitions have attracted great attention in recent years [13][14][15][16][17][18][19]. It was discovered some qualitative agreement between the critical behavior of real systems as the so-called manganites, e.g. La 0.7 Sr 0.3 MnO 3 and La 0.60 Y 0.07 Ca 0.33 MnO 3 [20][21][22][23], and the nonexetnsive two-dimensional Ising model [13,14] (through computational methods). Despite this great attention, among the many theoretical methods employed for studying nonextensive systems [10,11], up to date, to our knowledge, there is not a field-theoretic formulation for attaining the displayed results available in the current literature. The aim of this Letter is to * [email protected] fill this gap by introducing such a formulation capable of furnishing results valid in the nonextensive domain. It is based on momentum-space field-theoretic renormalization group and ǫ-expansion techniques, namely the nonextensive statistical field theory (NSFT). Its corresponding extensive counterpart, i. e., statistical field theory (SFT) [24], has provided very accurate results, for example to the extensive O(N ) vector model [24], whose nonextensive version we have to study in this Letter. When distinct systems as a fluid and a ferromagnet are characterized by the same set of critical exponents, we say that they belong to the same universality class. Universal critical exponents satisfy to some scaling relations and do not depend on the microscopic details of the system or equivalently on its nonuniversal properties as the critical temperature, form of the lattice, etc.. They depend only on universal parameters as the dimension d of the system, N and symmetry of some N -component order parameter and if the interactions among their degrees of freedom are of short-or longrange type [25]. In SFT, the critical indices, for O(N ) vector models, are obtained through the fluctuation properties of some self-interacting fluctuating N -component scalar quantum field φ [5,6]. That properties are its number N of components, the properties of its internal space [25] (the symmetry of that space for example) and the dimension d of the spacetime where the field is embedded but not on the nature of the spacetime itself, as if it is curved [26] or a Lorentz-violating one [27] for example. Aspects like short-and long-range interactions are contained in the form of the free propagators, depending on momentum through quadratic powers [5,6] and other functional forms distinct from quadratic ones [25], respectively. Some scaling relations depend on the effective dimension d ef f of the system [5,6], which is given by d ef f = d and a function d ef f ≡ d ef f (ς i ) for systems with short-and long-range interactions, respectively. The parameters ς i characterize the long-range interactions [28,29]. Also, d ef f depends on the momentum mass dimension of the momenta present in the free propagator. Its functional dependence on momenta can be quadratic (d ef f = d) or distinct from quadratic one (d ef f ≡ d ef f (ς i )) and on the volume element in momentum space of the Feynman integrals [28,29]. A nonextensive scalar quantum field φ q , for arbitrary values of q, was defined [30,31]. Moreover, the resulting nonextensive quantum field theory is nonlinear. This does not permit us to apply the superposition principle to express a general solution for φ q in terms of creation and annihilation operators and then to compute the corresponding free propagator. Recently, a linearized version of that theory, for |q − 1| < 1, was proposed [32] for obtaining the referred propagator, where φ q turns out to be its linearized extensive counterpart, namely φ. Although that free propagator depends on q through its effective nonextensive mass m 2 q = qm 2 , the critical exponents obtained do not depend on q, even for all loop orders. Then, the critical indices are extensive, at least in that version. Now we have to introduce the NSFT, which yields nonextensive q-dependent critical exponents values. As aforementioned [32], the nonextensive quantum scalar field, its mass, equation of motion, Lagrangian and free propagator turn out to be their respective extensive counterparts for |q − 1| < 1. We expect a better result as q is closer to 1, as an effect of the linearization procedure for obtaining the free propagator.
The NSFT to be introduced in this Letter is obtained as an improvement of the earlier one of Ref. [32] by now considering the nonextensive form of the statistical weight (not considered in Ref. [32]). Then we apply the generating functional for the interacting Euclidean quantum field theory
Z[J] = N −1 exp q − d d xL int δ δJ(x) q × exp 1 2 d d xd d x ′ J(x)G 0 (x − x ′ )J(x ′ ) ,(1)
now without any approximation on q, where exp q (−x) =
[1 − (1 − q)x] 1/(1−q) +
is the q-exponential function [3]([y] + ≡ yθ(y) and θ(y) is the Heaviside step function), rather than the conventional one e −x [5,6]. The constant N is determined through the normalization condition Z[J = 0] = 1, G −1 0 (k) = k 2 + m 2 is the free propagator of the extensive theory (it is quadratic in momentum space, so d ef f = d and thus the nonextensive scaling relations are the same as that of the extensive theory) and J(x) is some external source to generate the correlation functions and later to be vanished. In Eq. (1), we have applied the consistent form of the q-distribution [33] (some q-dependent factors present in this form can be discarded because they do not alter the final result of the nonextensive critical exponents, once they represent only an overall factor in the Lagrangian of the system) and its second term is extensive, since it is related to the free propagator, which can be defined just in the extensive realm.
As some illustrative applications, for O(N )-symmetric vector models, we consider N -component λφ 4 scalar field theories. In the first application, we compute the static nonextensive critical exponents by applying six distinct and independent renormalization group methods and ǫ-expansion techniques in d = 4 − ǫ dimensions in dimensional regularization (we follow the notation of Ref. [32]). Motivated by the just mentioned qualitative agreement between the critical behavior of manganites and nonexetnsive two-dimensional Ising model, for which a continuous transition along the ferro-paramagnetic frontier was identified for 0.5 < q < 1 [21], we make q → 2 − q in the q-exponential function of Eq. (1) (additive duality [34]), but not on its power q. This procedure yields a physical, normalizable probability q-distribution for q < 1, for which S q (AB) ≤ S q (A) + S q (B) (this is a reasonable result, since the global correlations turns out the system more organized as its extensive counterpart). We attain the last result by applying the additive duality
for S q (AB) = S q (A) + S q (B) + (1 − q)S q (A)S q (B) and obtain S q (AB) = S q (A) + S q (B) + (q − 1)S q (A)S q (B). Moreover, it implies that 1 − (q − 1)x > 0 for x > 0, which is our case where x ∝ to the quartic interaction. Furthermore, for |q − 1| < 1 [3], [exp 2−q (−a)] q = e −a 1 − a(a + 2)z/2 + a 2 (3a + 4)z 2 /24 − · · · ,
where z = q − 1 and a = H/k B T , and thus we expect slight q-dependent contributions. Also, we expect nonextensive indices values being smaller than their extensive counterparts, since correlated systems are less susceptible to temperature fluctuations than uncorrelated ones. Then, the nonextensive response functions are less divergent than their extensive counterparts. For heat capacity the effect is the opposite, once for observing a unit change of temperature in a correlated system, we have to supply a larger amount of heat than for an uncorrelated one. Corresponding ideas apply to the equation of state, whose index is δ q . By applying the methods, we obtain that the corresponding theories are renormalizable at leading loop order (LO) for general values of q and we can compute q-dependent critical exponents at that order. However, at next-to-leading loop order (NLO) and beyond, the theories are nonrenormalizable for any value of q but they are renormalizable only for q = 1. Then the nonextensive indices for NLO and higher ones are that of the extensive theory. A field theory to be renormalizable at LO for general values of some parameter and nonrenormalizable for NLO and beyond for any of the values of this parameter, but otherwise to be renormalizable only for a specific value of that parameter for NLO and higher ones is a known feature in literature. This is the case, for example, of quantum field theories in curved spacetime [35,36]. The nonextensive indices values are then composed of three-level, NLO and higher ones extensive contributions (resulting from considering all length scales) and a LO q-dependent term (as an effect from small length scales). The latter ones represent slight contributions as expected, according to the discussion aforementioned. We have obtained identical results for the static nonextensive critical indices through the six distinct methods, showing the arbitrariness of the renormalization schemes employed and the importance of checking the results as provided by so many methods. All the methods furnish two independent identical critical indices valid for all loop orders, namely η q and ν q . They are given by
η q = η − (1 − q) (N + 2)ǫ 2 2(N + 8) 2 , ν q = ν − (1 − q) (N + 2)ǫ 4(N + 8) ,(2)
where ν and η are their corresponding all-loop extensive critical exponents values. The four remaining ones α q , β q , γ q and δ q are obtained through four scaling relations among them [5,6]. Their Landau values, which can also be obtained through thermodynamics (they are a result of considering only large length scales), do not depend on q. This fact is in accord with the general result of Ref. [4]. We have that η q < η, ν q < ν, etc. (α q > α and δ q > δ), as expected. As a check, in Table I, we compare the exponents numerical outcomes for some values of q, to 2d (ǫ = 2) nonextensive (q = 1) Ising (N = 1) systems, obtained from NSFT and Ref. [13] (in this Ref., β q and ν q , whose q-dependence is slight, as expected, are available and we have obtained the remaining ones from scaling relations as well as their respective relative errors from elementary relative error theory). The extensive indices values used are that from Onsager's exact solution [37]. In this sense our results for 2d Ising systems are exact, within the approximation of the present Letter. The agreement is satisfactory, within the margin of error (as better as q is closer to 1, as a result of the linearization process for obtaining the free propagator), although we are leading with the expansion parameter ǫ = 2. The nonextensive indices evaluated from NSFT satisfy to the scaling relations, as opposed to the ones with strong dependence on q obtained from Ref. [14] satisfying modified scaling relations with some effective dimension d+n q (n q = (q 2 + q − 2)/4). Such modified scaling relations are not expected, since the free propagator of the theory is quadratic. Also, in Ref. [14], ν q is independent of q and violates the modified α q = 2 − (d + n q )ν q scaling relation. In that Ref., the exponents were obtained through computer simulations for finite-size systems. Therefore, a finite-size scaling computation from NSFT furnishes the same bulk nonextensive exponents as that obtained in this Letter, since they do not depend on the size of the system. Their q-dependence gives rise to the new nonextensive O(N ) q universality class and implies that they satisfy to the universality hypothesis, since the q parameter encodes global correlations or equivalently represents effective interactions among the degrees of freedom of the system [11]. According to the universality hypothesis, this leads necessarily to q-dependent critical indices. Alternatively, we see that the nonextensive indices must depend on q, once the nonextensive statistical weight of Eq. (1) promotes a modification of the internal properties of the fluctuating quantum field φ thus modifying how it interacts (depending on q) with itself. This implies a relation between fluctuations and global correlations. This mechanism occurs in the internal space of φ and not in the spacetime where it is embedded. This leads, necessarily, to the emergence of q-dependent exponents. As another application, we compute the nonextensive dynamic critical exponent z q . Its extensive counterpart value is that evaluated up to five-loop order in Ref. [38]. We compare z q evaluated from NSFT with the respective numerical values obtained from computer simulations for 2d (ǫ = 2) nonextensive (q = 1) Ising (N = 1) systems in Ref. [13] for some values of q. Following similar steps as for the static case, we obtain [5]
z q = z − (1 − q) [6 ln(4/3) − 1](N + 2) 2(N + 8) 2 ǫ 2 .(3)
In Table II, once again, we observe that the agreement is satisfactory, although for an ǫ-expansion that takes into account the ǫ = 2 value as the expansion parameter.
TABLE II. Results for the dynamic nonextensive critical exponent zq, to 2d (ǫ = 2) nonextensive (q = 1) Ising (N = 1) systems, obtained from NSFT and Ref. [13]. q zq 1 [38] 2.140 1 [13] 2.240±0.007 0. 9 2.135 0.9 [13] 2.151±0.006 0.8 2.129 0.8 [13] 2.153±0.005 0.7 2.124 0.7 [13] 2.141±0.006 0. 6 2.118 0.6 [13] 2.151±0.007
Now we make some predictions on the values of both static and dynamic nonextensive critical exponents for the referred systems through Tables III -IX: TABLE III. Results for the static nonextensive critical exponents for some values of q, to 2d (ǫ = 2) nonextensive (q = 1) Self-avoiding random walk (N = 0) systems, obtained from NSFT. q αq βq γq 1Exact [5] For the 3d systems, both static and dynamic nonex-tensive critical exponents depend slightly on q, as in the 2d situation.
In summary, we have introduced a field-theoretic approach for describing the critical behavior of nonextensive systems. We have computed analytically both universal static and dynamic nonextensive critical exponents for O(N ) models, valid for all loop orders and |q − 1| < 1. Then, a new universality class emerged, namely the nonextensive O(N ) q one. Their values are composed of both three-level, NLO and higher ones extensive contributions (resulting from considering all length scales) and a LO q-dependent term (as an effect from small length scales), where the latter one is slight. Both 2d nonextensive static and dynamic critical indices are in agreement with those obtained from computer simulation for 2d nonextensive Ising model [13] (Tables I and II), within the margin of error, as better as q is clorer to 1, as a consequence of the linearization process for defining the free propagator. Our results for 2d Ising systems are exact, within the approximation of the present Letter. We have showed that the nonextensive critical indices are universal, satisfy to the scaling relations and reduce to their extensive values when q → 1. Furthermore, we have predicted results to the values of both static and dynamic nonextensive critical exponents for some other systems through Tables III -IX. These outcomes can be compared to their counterparts obtained though computer simulations as well as from experimental measurements [21][22][23] in a near future. The NSFT opens a new route for studying critical properties of nonextensive systems undergoing continuous phase transitions. Besides, it represents a new methodology and fills the gap in the literature for such a method like its extensive counterpart, so successful for describing extensive systems [24]. Then, it can be applied for many other extensive models as well [5].
TABLE I .
IResults for the static nonextensive critical expo-
nents for some values of q, to 2d (ǫ = 2) nonextensive (q = 1)
Ising (N = 1) systems, obtained from NSFT and Ref. [13].
q
αq
βq
γq
1 Exact
0.000
0.125
1.750
0.9
0.034
0.121
1.724
0.9[13]
0.020±0.014
0.120±0.000 1.740±0.014
0.8
0.066
0.117
1.699
0.8[13]
0.054±0.012
0.119±0.000 1.701±0.012
0.7
0.100
0.114
1.673
0.7[13]
0.082±0.014
0.116±0.000 1.686±0.014
0.6
0.134
0.110
1.647
0.6[13]
0.090±0.016
0.117±0.000 1.676±0.016
q
δq
ηq
νq
1 Exact
15.000
0.250
1.000
0.9
15.260
0.246
0.983
0.9[13]
15.529±0.008
0.242±0.002 0.990±0.007
0.8
15.461
0.243
0.967
0.8[13]
15.327±0.008
0.245±0.002 0.973±0.006
0.7
15.736
0.239
0.950
0.7[13]
15.529±0.008
0.242±0.002 0.959±0.007
0.6
16.021
0.235
0.933
0.6[13]
15.327±0.008
0.245±0.002 0.955±0.008
TABLE VI .
VITABLE VIII. Results for the dynamic nonextensive critical exponent zq for some values of q, to 3d (ǫ = 1) nonextensive (q = 1) Ising (N = 1) systems, obtained from NSFT. TABLE IX. Results for the dynamic nonextensive critical exponent zq for some values of q, to 3d (ǫ = 1) nonextensive (q = 1) Heisenberg (N = 3) systems, obtained from NSFT.Results for the static nonextensive critical expo-
nents for some values of q, to 3d (ǫ = 1) nonextensive (q = 1)
XY (N = 2) systems, obtained from NSFT.
q
αq
βq
γq
1 [5]
-0.011±0.004
0.3470±0.0016 1.3169±0.0020
0.9
0.019±0.004
0.3412±0.0016 1.2985±0.0046
0.8
0.049±0.004
0.3354±0.0016 1.2801±0.0046
0.7
0.079±0.004
0.3296±0.0016 1.26184±0.0046
0.6
0.109±0.004
0.3238±0.0016 1.2433±0.0046
q
δq
ηq
νq
1 [5]
4.7949±0.0140 0.0354±0.0025 0.6703±0.0015
0.9
4.8061±0.0140 0.0334±0.0025 0.6603±0.0015
0.8
4.8173±0.0141 0.0314±0.0025 0.6503±0.0015
0.7
4.8286±0.0141 0.0294±0.0025 0.6403±0.0015
0.6
4.8400±0.0141 0.0274±0.0025 0.6303±0.0015
TABLE VII. Results for the static nonextensive critical expo-
nents for some values of q, to 3d (ǫ = 1) nonextensive (q = 1)
Heisenberg (N = 3) systems, obtained from NSFT.
q
αq
βq
γq
1 [5]
-0.122±0.010
0.3662±0.0025 1.3895±0.0050
0.9
-0.088±0.008
0.3607±0.0027 1.3686±0.0086
0.8
-0.054±0.008
0.3596±0.0027 1.3477±0.0086
0.7
-0.020±0.008
0.3465±0.0027 1.3267±0.0086
0.6
-0.015±0.008
0.3399±0.0027 1.3056±0.0086
q
δq
ηq
νq
1 [5]
4.7943±0.0140 0.0355±0.0025 0.7073±0.0035
0.9
4.8061±0.0140 0.0334±0.0025 0.6959±0.0035
0.8
4.8173±0.0141 0.0314±0.0025 0.6846±0.0035
0.7
4.8292±0.0141 0.0293±0.0025 0.6732±0.0035
0.6
4.8411±0.0141 0.0272±0.0025 0.6618±0.0035
q
zq
1 [38]
2.0235
0.9
2.0222
0.8
2.0208
0.7
2.0195
0.6
2.0181
q
zq
1 [39]
2.0330
0.9
2.0315
0.8
2.0300
0.7
2.0285
0.6
2.0270
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| [] |
[
"2D van der Waals Nanoplatelets with Robust Ferromagnetism",
"2D van der Waals Nanoplatelets with Robust Ferromagnetism"
] | [
"Michael C De Siena \nDepartment of Chemistry\nUniversity of Washington\n98195SeattleWAUSA\n",
"Sidney E Creutz \nDepartment of Chemistry\nUniversity of Washington\n98195SeattleWAUSA\n",
"‡ ",
"Annie Regan \nDepartment of Chemistry\nUniversity of Washington\n98195SeattleWAUSA\n",
"§ Paul Malinowski \nDepartment of Physics\nUniversity of Washington\n98195SeattleWAUSA\n",
"Qianni Jiang \nDepartment of Physics\nUniversity of Washington\n98195SeattleWAUSA\n",
"Kyle T Kluherz \nDepartment of Chemistry\nUniversity of Washington\n98195SeattleWAUSA\n\nPhysical Sciences Division\nPacific Northwest National Laboratory\n99352RichlandWAUSA\n",
"Guomin Zhu \nDepartment of Materials Science and Engineering\nUniversity of Washington\n98195SeattleWAUSA\n\nPhysical Sciences Division\nPacific Northwest National Laboratory\n99352RichlandWAUSA\n",
"Zhong Lin \nDepartment of Physics\nUniversity of Washington\n98195SeattleWAUSA\n",
"James J De Yoreo \nDepartment of Chemistry\nUniversity of Washington\n98195SeattleWAUSA\n\nDepartment of Materials Science and Engineering\nUniversity of Washington\n98195SeattleWAUSA\n\nPhysical Sciences Division\nPacific Northwest National Laboratory\n99352RichlandWAUSA\n",
"Xiaodong Xu \nDepartment of Physics\nUniversity of Washington\n98195SeattleWAUSA\n\nDepartment of Materials Science and Engineering\nUniversity of Washington\n98195SeattleWAUSA\n",
"Jiun-Haw Chu \nDepartment of Physics\nUniversity of Washington\n98195SeattleWAUSA\n",
"Daniel R Gamelin \nDepartment of Chemistry\nUniversity of Washington\n98195SeattleWAUSA\n"
] | [
"Department of Chemistry\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Chemistry\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Chemistry\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Physics\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Physics\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Chemistry\nUniversity of Washington\n98195SeattleWAUSA",
"Physical Sciences Division\nPacific Northwest National Laboratory\n99352RichlandWAUSA",
"Department of Materials Science and Engineering\nUniversity of Washington\n98195SeattleWAUSA",
"Physical Sciences Division\nPacific Northwest National Laboratory\n99352RichlandWAUSA",
"Department of Physics\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Chemistry\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Materials Science and Engineering\nUniversity of Washington\n98195SeattleWAUSA",
"Physical Sciences Division\nPacific Northwest National Laboratory\n99352RichlandWAUSA",
"Department of Physics\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Materials Science and Engineering\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Physics\nUniversity of Washington\n98195SeattleWAUSA",
"Department of Chemistry\nUniversity of Washington\n98195SeattleWAUSA"
] | [] | We have synthesized unique colloidal nanoplatelets of the ferromagnetic twodimensional (2D) van der Waals material CrI 3 and have characterized these nanoplatelets structurally, magnetically, and by magnetic circular dichroism spectroscopy. The isolated CrI 3 nanoplatelets have lateral dimensions of ~25 nm and ensemble thicknesses of only ~4 nm, corresponding to just a few CrI 3 monolayers. Magnetic and magneto-optical measurements demonstrate robust 2D ferromagnetic ordering in these nanoplatelets with Curie temperatures similar to those observed in bulk CrI 3 , despite the strong spatial confinement. These data also show magnetization steps akin to those observed in micron-sized few-layer 2D sheets and associated with concerted spin-reversal of individual CrI 3 layers within few-layer van der Waals stacks. Similar data have also been obtained for CrBr 3 and anion-alloyed Cr(I 1-x Br x ) 3 nanoplatelets. These results represent the first example of laterally confined 2D van der Waals ferromagnets of any composition. The demonstration of robust ferromagnetism at nanometer lateral dimensions opens new doors for miniaturization in spintronics devices based on van der Waals ferromagnets. | 10.1021/acs.nanolett.0c00102 | [
"https://arxiv.org/pdf/2001.04594v1.pdf"
] | 210,473,134 | 2001.04594 | 9e1ff2eeb9325ae60e762b777741b4f5e55af2a9 |
2D van der Waals Nanoplatelets with Robust Ferromagnetism
Michael C De Siena
Department of Chemistry
University of Washington
98195SeattleWAUSA
Sidney E Creutz
Department of Chemistry
University of Washington
98195SeattleWAUSA
‡
Annie Regan
Department of Chemistry
University of Washington
98195SeattleWAUSA
§ Paul Malinowski
Department of Physics
University of Washington
98195SeattleWAUSA
Qianni Jiang
Department of Physics
University of Washington
98195SeattleWAUSA
Kyle T Kluherz
Department of Chemistry
University of Washington
98195SeattleWAUSA
Physical Sciences Division
Pacific Northwest National Laboratory
99352RichlandWAUSA
Guomin Zhu
Department of Materials Science and Engineering
University of Washington
98195SeattleWAUSA
Physical Sciences Division
Pacific Northwest National Laboratory
99352RichlandWAUSA
Zhong Lin
Department of Physics
University of Washington
98195SeattleWAUSA
James J De Yoreo
Department of Chemistry
University of Washington
98195SeattleWAUSA
Department of Materials Science and Engineering
University of Washington
98195SeattleWAUSA
Physical Sciences Division
Pacific Northwest National Laboratory
99352RichlandWAUSA
Xiaodong Xu
Department of Physics
University of Washington
98195SeattleWAUSA
Department of Materials Science and Engineering
University of Washington
98195SeattleWAUSA
Jiun-Haw Chu
Department of Physics
University of Washington
98195SeattleWAUSA
Daniel R Gamelin
Department of Chemistry
University of Washington
98195SeattleWAUSA
2D van der Waals Nanoplatelets with Robust Ferromagnetism
12D materialsnanocrystalsferromagnetismvan der Waalscolloidal
We have synthesized unique colloidal nanoplatelets of the ferromagnetic twodimensional (2D) van der Waals material CrI 3 and have characterized these nanoplatelets structurally, magnetically, and by magnetic circular dichroism spectroscopy. The isolated CrI 3 nanoplatelets have lateral dimensions of ~25 nm and ensemble thicknesses of only ~4 nm, corresponding to just a few CrI 3 monolayers. Magnetic and magneto-optical measurements demonstrate robust 2D ferromagnetic ordering in these nanoplatelets with Curie temperatures similar to those observed in bulk CrI 3 , despite the strong spatial confinement. These data also show magnetization steps akin to those observed in micron-sized few-layer 2D sheets and associated with concerted spin-reversal of individual CrI 3 layers within few-layer van der Waals stacks. Similar data have also been obtained for CrBr 3 and anion-alloyed Cr(I 1-x Br x ) 3 nanoplatelets. These results represent the first example of laterally confined 2D van der Waals ferromagnets of any composition. The demonstration of robust ferromagnetism at nanometer lateral dimensions opens new doors for miniaturization in spintronics devices based on van der Waals ferromagnets.
Long-range magnetic order has recently been demonstrated in micron-sized twodimensional (2D) magnetic semiconductors, 1-2 opening opportunities for atomically thin spintronics. [3][4][5] For high-density devices, however, a fundamental limitation is the instability of ferromagnetism as lateral sizes approach the nanoscale. [6][7] It is thus of both fundamental and technological importance to explore the magnetic properties of 2D layered magnets with strong lateral confinement. Here, we report successful synthesis of unique colloidal nanoplatelets of 2D magnets CrX 3 (X = I, Br) and their alloys. The miniaturization and associated solution compatibility demonstrated here suggest new opportunities for studying 2D spin effects at the nanoscale and exploiting nanostructuring in high-density spintronics.
An interesting aspect of the ferromagnetism in CrI 3 is that the ground-state ( 4 A 2 ) singleion anisotropy (D) of its pseudo-octahedral Cr 3+ constituents is negligible, 8 yet the Curie temperature is still relatively high (T C ~ 45 K in monolayers). The magnetic anisotropy underlying T C comes entirely from the 2D morphology of the individual layers, and specifically from the presence of in-plane (anisotropic) ferromagnetic superexchange interactions that pin the spin orientation of any given Cr 3+ and generate a gap in the spin-wave density of states. [8][9] As the lateral dimensions of the CrI 3 plane are decreased, this morphological stabilization should diminish due to broken symmetry at the edges, eventually causing the ferromagnetism to be surmountable by thermal fluctuations and the CrI 3 to become superparamagnetic. For example, ferromagnetic hcp cobalt metal has a very high T C of >1000 K in bulk but becomes superparamagnetic at room temperature in nanocrystals. 10 Theory predicts that the electronic structures of CrI 3 nanoribbons should be strongly dependent on their edge structures, displaying quantum size effects in the energy gap and new edge states around the Fermi level. 11 To date, however, there have been no experimental examples of laterally confined 2D magnets of any type. It remains an open question whether 2D ferromagnetism can survive lateral truncation of the 2D structural periodicity.
Here we report the bottom-up synthesis and physical characterization of colloidal fewlayer CrX 3 (X = I, Br) nanoplatelets with average lateral dimensions of only ~25 nm, focusing on CrI 3 . Common solution routes to colloidal metal-halide nanocrystals 12 frequently involve hightemperature reaction of halide sources (e.g., alkylammonium halides, benzoyl halides, trimethylsilyl halides) with simple metal salts (e.g., metal acetates, carbonates) solubilized by surfactants (e.g., long-chain carboxylates, amines). In our hands, attempts to synthesize CrX 3 via similar approaches using simple Cr(III) precursors and common surfactants did not work (see Supporting Information). Octahedral Cr(III) is canonically substitutionally inert, 13 making these simple Cr(III) precursors difficult to solubilize and relatively unreactive. We therefore sought more reactive precursors that would be highly soluble in nonpolar, noncoordinating solvents such as alkanes or toluene, and that would also react in a way that produces only innocuous and readily separable byproducts. We found that Cr(OR) 3 (R = 1,1-di-t-butylethoxide) meets all of these requirements. Similarly, trimethylsilyl iodide (TMSI) was identified as a reactive and alkane-soluble iodide precursor. 14 When an anhydrous toluene solution of Cr(OR) 3 with excess TMSI is immersed with vigorous stirring into an oil bath preheated to 135 o C, abrupt formation of a black precipitate is observed after ~5 min. The rapid onset of nucleation under these conditions is well-suited to the formation of high-quality nanocrystals. 15 Because no solubilizing ligands are present, the resulting nanocrystals precipitate from solution rather than forming a colloidal dispersion. They can be separated from the clear supernatant by centrifugation and washing with hexanes. The nanocrystals can then be resuspended in dichloromethane through vigorous ultrasonication to generate a clear, dark-green solution (Fig. 1a). 18 The arrow indicates the direction of the crystallographic c-axis. (C) Powder X-ray diffraction data for CrI 3 nanocrystals (blue) measured with Cu K α radiation, compared to a reference pattern for CrI 3 (C2/m, ICSD Coll. Code 251654). 18 Prominent reflections from planes parallel to the c-axis are highlighted in red and labeled with the Miller indices of the corresponding planes.
Interlayer magnetic coupling in CrI 3 is tightly connected to the layer stacking arrangement. [16][17] Figure 1b illustrates the lattice structure of bulk CrI 3 , which adopts a monoclinic C2/m structure at room temperature and transitions to a rhombohedral R-3 phase below 220 K. 18 Figure 1c shows powder X-ray diffraction (PXRD) data collected for the isolated nanocrystals. Although high sensitivity to air complicated the PXRD measurements (see Methods), these data confirm the identity, structure, and morphology of the isolated material as CrI 3 , best matching the C2/m monoclinic phase. In this phase, stacked CrI 3 monolayers are displaced along the a-axis with the nearly ABC arrangement 18 associated with antiferromagnetic interlayer coupling. [19][20] Notably, substantial Scherrer broadening is observed in Fig. 1c for reflections from planes parallel to the ab-plane. Only reflections from planes lying mostly or entirely parallel to the c-axis are reasonably narrow. This result suggests substantial crystalline shape anisotropy. As described below, these nanocrystals are in fact CrI 3 nanoplatelets with lateral dimensions ~5 times greater than their few-layer thicknesses. The (060) reflection at 2θ = 46 o is particularly narrow, and its width can be used to estimate a mean lateral nanoplatelet size of ~20 nm for the sample of Fig. 1. Notably, these PXRD peak widths and the colloidal stability of these nanocrystals preclude the existence of large bulk-like crystals within this ensemble.
Transmission electron microscopy (TEM) was used to further characterize the CrI 3 nanocrystals and the results are summarized in Fig. 2. Figure 2a shows an overview TEM image of an ensemble of CrI 3 nanocrystals. The absence of surface ligands causes the nanocrystals to aggregate when cast onto the TEM grids, making it difficult to identify many well-isolated nanocrystals. Figure 2b shows a selected-area electron diffraction (SAED) image of the same aggregated nanocrystals. Its integrated profile is shown in Fig. 2c compared to the calculated reference lines for CrI 3 (C2/m). These data confirm the identity of these nanocrystals as monoclinic CrI 3 . nanoplatelet thicknesses from a survey of over 100 nanoplatelets. The average nanoplatelet thickness is 3.7 ± 0.7 nm, corresponding to 6 ± 1 CrI 3 monolayers. All data were collected at room temperature.
The inset to Fig. 2a shows a low-resolution image of several individual nanocrystals sideby-side. These nanocrystals appear disk-like and have low contrast, consistent with thin nanoplatelets. Figure 2d shows a high-resolution image of a single nanoplatelet. Although faint, faceting consistent with hexagonal CrI 3 is observable. Careful inspection further reveals lattice fringes in this image. Figure 2e plots the Fourier transform of this image, revealing distinct peaks characteristic of single-crystal CrI 3 (C2/m). The sizes of over 100 individual nanoplatelets were measured, yielding an ensemble width of 26 ± 11 nm (Fig. 2f). The smallest nanoplatelets identified were only 10 nm across, and the largest nanoplatelet reached 66 nm across.
Several nanocrystals can be seen lying on their sides, resulting in thin dark stripes in Fig. 2a. Figure 2g shows a high-resolution TEM image of one isolated edge-on structure, in which six parallel thinner lines are observed. Figure 2h plots the Fourier transform of this image, revealing a series of spots that can be indexed to (00n) peaks. These data show an interlayer spacing of 0.69 ± 0.07 nm, consistent with the value of 0.66 nm expected from the CrI 3 (C2/m) crystal structure. Over 100 edge-on nanoplatelets were surveyed, from which an average thickness of 3.7 ± 0.7 nm is obtained (Fig. 2i), corresponding to only 6 ± 1 CrI 3 monolayers. Similar results were obtained for CrBr 3 nanoplatelets prepared analogously (see Supporting Information).
Beyond CrI 3 and CrBr 3 , this synthetic approach also allows formation of anion-alloyed Cr(I 1-
x Br x ) 3 nanoplatelets, providing a mechanism for continuous tuning of the nanoplatelet optical and magnetic properties (see Supporting Information).
The electronic structures and magnetism of CrBr 3 and CrCl 3 single crystals have previously been investigated by transmission and reflection magnetic circular dichroism (MCD) spectroscopies, [21][22] including down to a single monolayer. 2,23-24 Figure 3 summarizes the 5 K absorption (extinction) and transmission MCD spectra of representative CrBr 3 and CrI 3 nanoplatelet ensembles. MCD spectra are plotted as the differential absorbance of left and right circularly polarized light (∆A) normalized to the absorbance at the first peak maximum (A max ( 4 T 2 )) after accounting for scattering. Figure 3. Transmission magnetic circular dichroism (MCD) spectra of (A) CrI 3 and (B) CrBr 3 nanoplatelets measured at 5 K and at a magnetic field of 5 T, compared with the 5 K zero-field electronic absorption (extinction) spectra. Absorption axes are labeled by optical density (O.D.). MCD spectra are plotted as ∆A/A max ( 4 T 2 ), i.e., the differential absorbance of left and right circularly polarized light (∆A) normalized to the absorbance at the first peak maximum (A max ( 4 T 2 )) after accounting for the scattering baseline. The assignments of select optical transitions are indicated.
The CrI 3 nanoplatelet spectra (Fig 3a) are better understood by first examining the spectra of CrBr 3 nanoplatelets (Fig. 3b). As in bulk CrBr 3 , 21-22 the CrBr 3 nanoplatelet spectra show two weak absorption features at low energies (1.6, 2.2 eV) attributable to the 4 A 2 → 4 T 2 and 4 T 1 ligand-field transitions of pseudo-octahedral [CrBr 6 ] 3-. Each band is associated with an MCD feature in which one polarization dominates. To higher energy, a pair of more intense π-type ligand-to-metal charge-transfer (LMCT) transitions appears, centered at ~2.75 eV and with opposite MCD polarizations. [25][26] The absorption spectrum of the CrI 3 nanoplatelets (Fig. 3a) is very similar to the differential reflection spectrum of bulk CrI 3 , but it differs slightly from that reported for monolayer CrI 3 , 27 possibly due to interface effects in the latter. In Fig. 3a, the 4 T 2 band remains the lowest-energy transition, but the LMCT transitions have shifted down to ~2.1 eV. A redshift of ~1.1 eV is predicted from the electronegativity difference between Brand I -, 28 in good agreement with the data. To higher energies in the CrI 3 spectra, several additional absorption and MCD features are observed, but overlap between LMCT and ligand-field transitions complicates band assignments. Both CrI 3 and CrBr 3 thus have their absorption gaps determined by the same highly localized 4 A 2 → 4 T 2 Cr 3+ ligand-field transition. Notably, the MCD rotational strength of the 4 T 2 excitation is substantially (~5x) greater in the CrI 3 nanoplatelets than in the CrBr 3 nanoplatelets, likely reflecting enhanced configuration interaction with the strongly optically active LMCT states in CrI 3 due to their lower energies.
Variable-temperature and variable-field MCD measurements were performed to assess the magnetism of these CrI 3 and CrBr 3 nanoplatelets. Figure 4a plots CrI 3 MCD spectra collected at several temperatures between 5 and 200 K (see Supporting Information for CrBr 3 data). The inset to Fig. 4a plots the MCD amplitudes extracted from these data, from which T C = 54 K is determined. This value is smaller than in bulk (T C = 61 K), 18 but the trend is consistent with the decrease to T C = 45 K reported for monolayer CrI 3 sheets. 2 Figure 4b sheets. To illustrate, the shaded grey bars in Fig. 5 mark the fields at which abrupt magnetization steps occur in four-layer CrI 3 sheets with micron lateral dimensions when magnetized along their easy axis. 23 The steps in the nanoplatelet data appear less abrupt due to orientation averaging because of more gradual magnetization when the external field (B) is not along the nanoplatelet easy axis. These steps have been assigned to magnetization reversal of individual CrI 3 monolayers within multilayer stacks. 23 We thus attribute the inflections observed in Fig. 5a,b to magnetization reversal of individual monolayers within multilayer CrI 3 nanoplatelets, as depicted schematically in Fig. 5d. black and opaque, indicating the formation of the CrI 3 nanoplatelets as a precipitate. Heating was continued for an additional 10 min, and then the reaction was removed from the heating bath and allowed to cool to room temperature. The precipitate was separated from the colorless supernatant by centrifugation and washed three times with hexanes (5 mL). The precipitate was finally collected as a suspension in hexanes.
Synthesis of CrI 3 single crystals.
Single crystals of CrI 3 were grown by chemical vapor transport using iodine as a self-transport agent following a procedure adapted from the literature with modification. 18 Cr(0) pieces and solid crystalline I 2 were loaded into a quartz tube and sealed under an evacuated argon atmosphere. The quartz tube was 10 cm long, inner diameter 14 mm, and outer diameter 16 mm, and the amount of loaded iodine was determined by ensuring that the pressure inside of the tube reached a value near atmospheric pressure upon reaching the highest growth temperature. The transport of material was achieved using the natural temperature gradient of an open-ended horizontal furnace. The source end of the tube is placed in the hot end of a 650/550˚C temperature gradient and allowed to dwell for 7 days, and then allowed to slowly cool to room temperature. Crystals grew at the cold end of the tube as large, shiny black plates.
Powder X-ray diffraction (XRD) measurements. Samples were prepared for powder XRD by drop-casting suspensions of nanoplatelets from hexanes onto silicon wafers, and protecting them from air by sealing under Kapton film. Data were collected using a Bruker D8 Discover diffractometer.
Magnetic circular dichroism (MCD) and absorption (extinction) measurements.
Samples for transmission MCD measurements were prepared by mixing dried nanoplatelets in polydimethylsiloxane (PDMS, viscosity 1,000 cSt). This mixture was then sandwiched between two quartz discs to make a mull suspension. Low-temperature magnetic circular dichroism (MCD) spectra were conducted with the samples placed in a superconducting magneto-optical cryostat (Cryo-Industries SMC-1659 OVT) oriented in the Faraday configuration. Samples were loaded into the cryostat under helium gas to minimize air exposure and sample decomposition.
At liquid helium temperatures, the sample was screened for depolarization by matching the CD spectra of a chiral molecule placed along the optical path before and after the sample.
Depolarization by the samples was less than 5% in each case. Transmission MCD spectra were collected using an Aviv 40DS spectropolarimeter. UV-Vis absorption (extinction) measurements were performed on similar mulls using an Agilent Cary 5000 spectrophotometer, and sample cooling was achieved using a flow cryostat with a variable-temperature sample compartment.
Vibrating sample magnetometry (VSM) measurements. A Quantum Design PPMS
DynaCool was used for VSM measurements. Powders were loaded into plastic VSM powder sample holders and the single crystal was affixed to the end of a quartz paddle with varnish (VGE 7031). The paddle was then snapped into the VSM brass sample holder with another quartz paddle placed symmetrically above the sample to minimize the background coming from the quartz.
Transmission electron microscopy (TEM) measurements. TEM samples were prepared by drop casting suspensions of nanocrystals onto 400 mesh carbon-coated copper grids purchased from TED Pella, Inc. and dried under an inert atmosphere. Nanocrystal suspensions were prepared by ultrasonication of the materials in dichloromethane at a sonicator frequency of 20 kHz under an inert atmosphere. In a glovebox, TEM grids were loaded into a vacuum transfer holder to prevent sample exposure to air. TEM images were obtained on an FEI Titan microscope operated at 300 kV or on an FEI Tecnai F20 microscope operated at 200 kV. FFT images were generated using ImageJ, and brightness was adjusted to aid visualization. 31 Energy-dispersive X-ray spectroscopy (EDS) measurements. For EDS analysis of nanoplatelet compositions, samples were drop cast onto silicon substrates and coated with a ~200 nm thick layer of carbon; spectra were acquired in an FEI Sirion Scanning Electron Microscope operating at 30 kV using an Oxford EDS spectrometer. Standardless quantification was used.
Acknowledgments
The development, structural and analytical characterization, and magnetic/magneto-optical characterization of these 2D ferromagnets were all supported as part of Programmable Quantum Materials By contrast, CrX 3 nanocrystals could be successfully prepared using Cr(OR) 3 (R = 1,1-di-tbutylethoxide) as the Cr(III) precursor. Cr(OR) 3 is a three-coordinate, highly air-sensitive complex of the bulky alkoxide "ditox" (2,2-di-t-butylethoxy) ligand, and it is highly soluble in alkane and arene solvents including pentane, toluene, and octadecene. 3 The anion source was trimethylsilyl halide (TMSX, X = I, Br). Cr(OR) 3
S-2
Scheme S1. Synthesis of CrX 3 (X = I, Br) nanoplatelets Figure S1 shows data from TEM measurements on CrBr 3 nanoplatelets. These data are similar to those presented for CrI 3 nanoplatelets in the main text. Anion alloying has dramatic effects on the nanoplatelet optical spectra and, in particular, on the stability of the ferromagnetic phase. Figure S3b plots 5 K, 5 T MCD spectra of a series of Cr(I 1-x Br x ) 3 nanoplatelets with x ranging from 0 to 1. The energies of both the ligand-field and LMCT transitions redshift with decreasing x, as expected from the endpoint data in Fig. 3 of the main text. Figure S3c plots magnetic hysteresis data measured by MCD for the same samples.
Increasing x reduces the coercivity, narrowing the hysteresis. Whereas spin-flip inflections are S-5 clearly visible in the hysteresis data for the CrI 3 (x = 0.00) and lightly bromide-doped (x = 0.06) nanoplatelets, such features become less evident at higher x, where B sat is diminished.
Nevertheless, even in the CrBr 3 nanoplatelets a small foot in the magnetization curve is seen at ~0.3 T that appears to stem from the same phenomenon. Figure S3d plots the integrated MCD intensity (absolute) as a function of temperature for the various Cr(I 1-x Br x ) 3 nanoplatelets. The spectra used for this analysis are presented in Fig. S4. T C values determined from these data are summarized in the inset of Fig. S3d, which shows that T C decreases as x increases, reaching T C = 33 K at x = 1 (CrBr 3 ). This value of T C is slightly smaller than the value (37 K) reported for bulk CrBr 3 , 4 suggesting confinement effects in CrBr 3 similar to those found in the CrI 3 nanoplatelets. Figure S4: Variable-temperature magnetic circular dichroism spectra of Cr(I 1x Br x ) 3 nanoplatelets. Temperature dependence (5 to 150 K) of the MCD spectra of Cr(I 1-x Br x ) 3 nanoplatelets, measured at 5 T. These spectra were used to determine the data points in Fig. S3d.
S-6
S4: Estimation of the energy barrier to magnetization reversal in CrI 3 nanoplatelets.
The Néel-Arrhenius equation (eq S1) describes the time constant for aligned spins to reverse their orientation:
! ! = ! ! ! !" ! ! ! (S1)
where τ 0 is the attempt time, K is the magnetocrystalline anisotropy constant, and V is the particle volume. The product KV represents the energy barrier to spin reversal of the singledomain particle. Idealizing the CrI 3 nanoplatelets as hexagons, we can express this energy barrier as:
! !"#$ !"#"!$%& = ! • ! ! ! ! ! ℎ (S2)
where d is the platelet diagonal and h is its height. K has been measured for bulk CrI 3 where it is found to be temperature dependent. 5 Taking the values of K = 50 kJ/m 3 (0.31 meV/nm 3 ) measured at ~T C and 300 kJ/m 3 (1.86 meV/nm 3 ) measured at low temperature, and using the interlayer spacing of 0.65 nm for the hexagon height, Fig. S5 plots KV (meV) as a function of nanoplatelet size (d). For comparison, k B T C is also plotted. This analysis shows that with these assumptions, KV > k B T C for CrI 3 dimensions larger than d ~ 6.5 nm.
Figure 1 .
1(A) Photograph of a solution of CrI 3 nanocrystals in DCM. (B) Illustration of the room-temperature crystal structure of bulk CrI 3 , showing a section of three monolayers with the nearly ABC stacking arrangement of the monoclinic C2/m structure.
Figure 2 .
2(A) TEM image of an aggregate of CrI 3 nanoplatelets. Inset: Zoomed in view of a string of isolated nanoplatelets. (B) Selected area electron diffraction (SAED) image of aggregated CrI 3 nanoplatelets. The rings are indexed to their corresponding diffraction planes. (C) Integrated profile of the SAED diffraction rings, compared to the calculated reference lines for CrI 3 (C2/m). (D) High-resolution TEM image of a single isolated nanoplatelet. (E) Fourier transform of the image from panel D, showing peaks consistent with the nanoplatelet being a single crystalline domain with the monoclinic C2/m structure along the [001] zone axis. (F) Distribution of CrI 3 nanoplatelet lateral sizes from a survey of over 100 nanoplatelets, yielding an ensemble width of 26 ± 11 nm. (G) Edge-on HRTEM image of a single stack of nanoplatelets. (H) Fourier transform of the image from panel G, showing peaks consistent with van der Waals stacked CrI 3 nanoplatelets with the C2/m structure. (I) Distribution of CrI 3
plots variable-field MCD spectra of the CrI 3 nanoplatelets collected at 5 K. All of the spectral features show the same field dependence, confirming their common origin. The inset to Fig. 4b plots the CrI 3 nanoplatelet MCD intensity as a function of applied field, revealing a sizable hysteresis that confirms retention of ferromagnetism in these nanoplatelets. Multiple distinct inflections are observed at intermediate fields during the field sweep (see Supporting Information).
Figure 4 .
4(A) Variable-temperature (5 to 200 K) MCD spectra of CrI 3 nanoplatelets measured at 5 T. Inset: Plot of integrated absolute 5 T MCD intensity as a function of temperature, yielding a Curie temperature of ~54 K. The solid curve in the inset is a guide to the eye. (B) Variable-field (0 -5T) MCD spectra of CrI 3 nanoplatelets measured at 5 K. Inset: Magnetization vs magnetic field plot for CrI 3 nanoplatelets at 5 K, as probed using the MCD signal at 455 nm. The sample was first magnetized at 5 T then swept to -5 T and back to 5 T during data collection.Figures 5a,b compare the MCD hysteresis curve from Fig. 4c with magnetic hysteresis data for the same nanoplatelets collected by vibrating sample magnetometry (VSM) on a slightly expanded x-axis scale. Although less pronounced, several inflections are also observed in the VSM data. Interestingly, the fields at which these inflections occur coincide almost exactly with the fields at which magnetization steps are observed in mechanically exfoliated multilayer CrI 3
Figure 5 .
5Magnetization vs field plots measured by (A) MCD spectroscopy and (B) vibrating sample magnetometry (VSM) on the same sample of randomly oriented CrI 3 nanoplatelets. (C) VSM data from a CrI 3 single crystal aligned both parallel and perpendicular to the applied field are also plotted on the same field axis. The vertical gray bars represent the fields at which the spins of individual layers of a four-layer CrI 3 sheet have been reported to flip. 23 (D) Schematic of magnetic structures of a representative four-layer CrI 3 nanoplatelet. Purple spheres represent Cr 3+ ions and Iions are omitted. The fields at which transitions between these magnetic arrangements occur are determined by the weak antiferromagnetic interlayer coupling.The monolayer spin-flip events observed here are signatures of few-layer CrI 3 . In these nanoplatelets, however, the fraction of edge Cr 3+ ions is vastly greater than in exfoliated CrI 3 sheets: Idealizing the nanoplatelets as hexagons, the average CrI 3 nanoplatelet here contains ~2360 Cr 3+ ions, of which ~10% occupy edge sites. Because of broken translational symmetry, these edge spins are not pinned with the same energies as the core spins, leading to spin canting or lower-temperature spin fluctuations. Lateral confinement should thus reduce the barrier to magnetization reversal, but the data here do not show such a reduction. From the magnetocrystalline anisotropy of bulk CrI 3 ,29 we estimate that the barrier to monolayer magnetization reversal will not drop below the bulk value of k B T C until lateral dimensions reach < ~6.5 nm (~230 Cr 3+ ions, ~34% edge sites, see Supporting Information), consistent with the bulk-like T C observed inFig. 4a.The nanoplatelets reported here represent a fundamentally new morphology for this emergent class of materials. Few-layer van der Waals ferromagnets like CrI3 have previously only been prepared using mechanical exfoliation of bulk crystals. Remarkably, despite their extremely small (~25 nm) lateral dimensions, these CrI 3 nanoplatelets still show the same robust intra-layer ferromagnetism, interlayer antiferromagnetic coupling, and discrete layer-by-layer spin-flip magnetization as observed in mechanically exfoliated CrI 3 sheets with much larger dimensions. Moreover, T C appears undiminished by their very small volumes. This miniaturization, in conjunction with the unique solution processability of this form of CrI 3 , opens interesting avenues for constructing fundamentally new types of programmable 2D magnetic quantum materials, for example, through their integration as localized "spin bubbles" within otherwise non-magnetic 2D heterostructures, through self-assembly into spin-bubble superlattices, through nucleation of lateral heterostructures, or even through combination with other solution-phase materials to construct free-standing nanoscale 2D van der Waals heterostructures with magnetic functionality. More generally, the demonstration that robust ferromagnetism persists in nanoscale CrI 3 indicates the possibility of high-density spintronic devices based on nanopatterned 2D ferromagnetic semiconductors. Methods General considerations. Unless otherwise stated, all measurements and synthetic manipulations were performed using standard Schlenk techniques under a dinitrogen atmosphere, or in a glovebox under an atmosphere of purified dinitrogen. Anhydrous tetrahydrofuran (THF), dichloromethane (DCM), ethyl ether, pentane, and toluene were purified through an alumina column pressurized with argon. Hexanes was further dried over sodium benzophenone and distilled before use. Xylenes was dried by refluxing over calcium hydride and distilled before use. All solvents were stored over 4Å sieves. Chemicals. Unless otherwise stated, all chemicals were used as received without further purification. Zinc metal, chromium metal (99.995%), anhydrous CrCl 3 , and I 2 (99.99%) were purchased from Alfa Aesar. Hexamethylacetone, methyllithium (1.6 M in ether), trimethylsilyl bromide (97%), and trimethylsilyl iodide (97%) were purchased from Sigma Aldrich. Cr(OCMe t Bu 2 ) 3 was synthesized as previously reported. 30 Synthesis of CrX 3 nanoplatelets. The chromium precursor, Cr(OCMe t Bu 2 ) 3 (20 mg, 0.04 mmol) is dissolved in 2 mL of toluene (X = I) or xylenes (X = Br) in a 25 mL Schlenk tube equipped with a stir bar, giving a light blue solution. A total of 0.70 mmol of trimethylsilyl halide (bromide, iodide, or a mixture of both) is added to the reaction mixture. No observable change occurs. The Schlenk tube is sealed and immersed into a pre-heated oil bath at 135˚C (180˚C for CrBr 3 ) with rapid stirring. After approximately 5 min, the solutions abruptly turned
(Pro-QM), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award DE-SC0019443 (D.R.G., J.-H.C., and X.X.). Some of the electron microscopy data were collected in the William R. Wiley Environmental Molecular Sciences Laboratory (EMSL), a national scientific user facility sponsored by DOE's Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. Some of the nanocrystal spectroscopy work was supported by the Northwest Institute for Materials Physics, Chemistry, and Technology (NW IMPACT). This research was additionally partially supported by an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at UW (S.E.C.), administered by Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and the Office of the Director of National Intelligence. K.T.K. gratefully acknowledges support from the U.S. Department of Energy Office of Science Graduate Student Research (SCGSR) program for some of the TEM characterization. The SCGSR program is administered by the Oak Ridge Institute for Science and Education (ORISE) for the DOE. ORISE is managed by ORAU under contract number DE-SC0014664. Part of this work was conducted at the Molecular Analysis Facility, a National Nanotechnology Coordinated Infrastructure site at UW that is supported in part by the NSF (ECC-1542101), UW, the Molecular Engineering & Sciences Institute, the Clean Energy Institute, and the National Institutes of Health. Author Information ‡ These authors contributed equally to this work. ∥ Present address: Department of Chemistry, Mississippi State University, Box 9573, Mississippi State, Mississippi 39762, USA § Present address: School of Chemistry, Trinity College Dublin, The University of Dublin, College Green, Dublin 2, Ireland Contributions M.C.D., S.E.C., and D.R.G. conceived the experiments. S.E.C. and A.R. synthesized the nanocrystals and performed the analytical nanocrystal characterization. P.M., Q.J., and Z.L.
and TMSI do not undergo any observable reaction at room temperature or upon mild (<100 o C) heating, likely due to the considerable steric bulk of the reagents. Heating an anhydrous toluene solution of Cr(OR) 3 with excess Me 3 SiX to 135 o C initiates nucleation of CrX 3 nanoplatelets. Reaction of the alkane-soluble iodide precursor TMSX with the alkoxide ligands of Cr(OR) 3 putatively releases the halide concomitant with formation of a silyl ether byproduct (Me 3 SiOR). This silyl ether byproduct likely undergoes further decomposition in situ. The overall synthesis is summarized in Scheme S1.
Figure S1 :
S1Characterization of CrBr 3 nanoplatelets. (A) TEM image of an aggregate of CrBr 3 nanoplatelets. (B) HR-TEM image of a CrBr 3 nanoplatelet lying on its short side. Lattice fringes are clearly seen. The nanocrystal in this image shows 5 individual CrBr 3 monolayers within the van der Waals nanoplatelet structure. (C) Selected area electron diffraction (SAED) image of aggregated CrBr 3 nanoplatelets. (D) Distribution of CrBr 3 nanoplatelet thicknesses, determined by measuring the sizes of >40 individual nanocrystals. S2: Inflections in magnetization data of CrI 3 nanoplatelets. The data in the main text show multiple distinct inflections at intermediate fields during the MCD field sweep CrI 3 nanoplatelets. These inflections are more easily seen in the derivative of the field sweep data, as shown in Fig. S2.
Figure S2 :
S2Inflections in the MCD magnetization vs magnetic field data for CrI 3 nanoplatelets. First derivative of the MCD magnetization field sweep data presented in the inset to Fig. 4b of the main text. The derivative highlights the inflections in the hysteresis loop. These inflections are assigned to spin flip events of individual CrI 3 layers. S3: Anion alloying and its effects on spectroscopy and T C . Cr(I 1-x Br x ) 3 nanoplatelets were prepared by mixing TMSI and TMSBr anion precursors during the synthesis. Varying the ratio of TMSI to TMSBr changes the ratio of Brto Iincorporated into the nanoplatelets, and this ratio can be finely tuned across the entire range of 0 ≤ x ≤ 1, but the nanoplatelet stoichiometry differs from the nominal (precursor) stoichiometry of the reaction. To illustrate, Fig. S3a plots the analytical Icontents measured in a series of Cr(I 1-x Br x ) 3 nanoplatelets as a function of the amount of TMSI added during synthesis. Halide compositions were determined by analysis of EDX data and by Vegard's law analysis of the (060) XRD reflection (inset of Fig. S3a); both approaches yield the same results. The elevated Iincorporation into the nanoplatelets is attributed to the greater reactivity of TMSI compared to TMSBr. These data demonstrate facile chemical control over the anion compositions in these colloidal CrX 3 nanoplatelets.
Figure S3 :
S3Structural and magnetic characterization of alloyed Cr(I 1-x Br x ) 3 nanoplatelets. (A)Measured mole percentage of iodide in the product for a given nominal iodide percentage in the reactants (TMSI and TMSBr). Compositions were determined by both XRD (Vegard's law, red crosses) and SEM/EDX (purple circles) measurements, which agree well. Inset: powder XRD data for the different alloy compositions, ranging from CrI 3 (red) to CrBr 3 (purple). The (060) reflection, typically the best resolved in the PXRD data, is shown. The dashed curve is a guide to the eye. (B) MCD spectra measured at 5 K and 5 T for different alloy compositions (given compositions are as determined by PXRD). (C) Field vs. magnetization sweeps for four different compositions of randomly oriented Cr(I 1-x Br x ) 3 nanoplatelets. (D) Integrated absolute MCD signal as a function of temperature for four different compositions of Cr(I 1-x Br x ) 3 . Inset: Curie temperatures determined from the data in panel C and plotted as a function of iodide content for Cr(I 1-x Br x ) 3 nanoplatelets. The dashed curves are guides to the eye.
Figure S5 :
S5Size dependence of the barrier to magnetization reversal in CrI 3 nanoplatelets. Plots of estimated barriers to magnetization reversal (KV) for CrI 3 as a function of nanoplatelet size. Energy barriers were estimated from eq S2 for two different experimental (ref. 5) values of the volumetric anisotropy constant, K. K = 50 kJ/m 3 (0.31 meV/nm 3 ) was measured at ~T C , and K = 300 kJ/m 3 (1.86 meV/nm 3 ) was measured in the low-temperature limit. For comparison, the value of k B T at T C is also included in the plot.
Table of Contents Graphic
ofS-1
Supporting Information for
2D van der Waals Nanoplatelets
with Robust Ferromagnetism
*Corresponding author's e-mail: [email protected]
S1: Additional description of the synthesis of CrI 3 nanoplatelets. Attempts to synthesize
CrX 3 using, e.g., Cr(III) acetylacetonate, Cr(III) nitrate, or "Cr(III) acetate" 1 in carboxylate and
amine surfactants either gave no readily isolable product, formed unidentified amorphous
precipitates, or formed chromium oxide. 2 In most cases, we observed no reaction or even
dissolution of these simple precursors until very high temperatures (>330 o C) were reached and
sustained for extended periods.
Competing InterestsThe authors declare no competing financial interests.Supporting InformationAdditional synthesis discussion, TEM characterization of CrBr 3 nanoplatelets, derivative MCD magnetization vs magnetic field data, variable-temperature MCD spectra for CrBr 3 and Cr(I 1-x Br x ) 3 alloy nanoplatelets and discussion, estimation of CrI 3 nanoplatelet blocking temperatures.Author Information
. P M , Q J , performed the VSM magnetometrythe single-crystal sample, and P.M. and Q.J. performed the VSM magnetometry.
performed and analyzed the electron microscopy measurements. M C D , K T K , G , M.C.D., K.T.K., and G.Z. performed and analyzed the electron microscopy measurements.
C. performed the optical and magneto-optical measurements and analyzed the data. All authors contributed to analysis of the results and preparation of the manuscript. M C D , S E , ReferencesM.C.D. and S.E.C. performed the optical and magneto-optical measurements and analyzed the data. All authors contributed to analysis of the results and preparation of the manuscript. References
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| [] |
[
"Layer specific observation of slow thermal equilibration in ultrathin metallic nanostructures by femtosecond X-ray diffraction",
"Layer specific observation of slow thermal equilibration in ultrathin metallic nanostructures by femtosecond X-ray diffraction"
] | [
"J Pudell \nInstitut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany\n",
"A A Maznev \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n",
"M Herzog \nInstitut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany\n",
"M Kronseder \nPhysics Department\nTechnical University Munich\n85748GarchingGermany\n",
"C H Back \nPhysics Department\nTechnical University Munich\n85748GarchingGermany\n\nInstitut für Experimentelle und Angewandte Physik\nUniversität Regensburg\n93040RegensburgGermany\n",
"G Malinowski \nInstitut Jean Lamour (UMR CNRS 7198)\nUniversité Lorraine\n54506Vandoeuvre-lès-NancyFrance\n",
"A Von Reppert \nInstitut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany\n",
"& M Bargheer \nInstitut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany\n\nHelmholtz-Zentrum Berlin for Materials and Energy GmbH\nWilhelm-Conrad-Röntgen Campus\nBESSY II\nAlbert-Einstein-Str. 1512489BerlinGermany\n"
] | [
"Institut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany",
"Department of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMAUSA",
"Institut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany",
"Physics Department\nTechnical University Munich\n85748GarchingGermany",
"Physics Department\nTechnical University Munich\n85748GarchingGermany",
"Institut für Experimentelle und Angewandte Physik\nUniversität Regensburg\n93040RegensburgGermany",
"Institut Jean Lamour (UMR CNRS 7198)\nUniversité Lorraine\n54506Vandoeuvre-lès-NancyFrance",
"Institut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany",
"Institut für Physik & Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-2514476PotsdamGermany",
"Helmholtz-Zentrum Berlin for Materials and Energy GmbH\nWilhelm-Conrad-Röntgen Campus\nBESSY II\nAlbert-Einstein-Str. 1512489BerlinGermany"
] | [] | Ultrafast heat transport in nanoscale metal multilayers is of great interest in the context of optically induced demagnetization, remagnetization and switching. If the penetration depth of light exceeds the bilayer thickness, layer-specific information is unavailable from optical probes. Femtosecond diffraction experiments provide unique experimental access to heat transport over single digit nanometer distances. Here, we investigate the structural response and the energy flow in the ultrathin double-layer system: gold on ferromagnetic nickel. Even though the excitation pulse is incident from the Au side, we observe a very rapid heating of the Ni lattice, whereas the Au lattice initially remains cold. The subsequent heat transfer from Ni to the Au lattice is found to be two orders of magnitude slower than predicted by the conventional heat equation and much slower than electron-phonon coupling times in Au. We present a simplified model calculation highlighting the relevant thermophysical quantities. | 10.1038/s41467-018-05693-5 | null | 52,049,623 | 1803.04034 | ad15fc59cd5daa5808cbbddfeef01d1ed8ed6208 |
Layer specific observation of slow thermal equilibration in ultrathin metallic nanostructures by femtosecond X-ray diffraction
J Pudell
Institut für Physik & Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24-2514476PotsdamGermany
A A Maznev
Department of Chemistry
Massachusetts Institute of Technology
02139CambridgeMAUSA
M Herzog
Institut für Physik & Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24-2514476PotsdamGermany
M Kronseder
Physics Department
Technical University Munich
85748GarchingGermany
C H Back
Physics Department
Technical University Munich
85748GarchingGermany
Institut für Experimentelle und Angewandte Physik
Universität Regensburg
93040RegensburgGermany
G Malinowski
Institut Jean Lamour (UMR CNRS 7198)
Université Lorraine
54506Vandoeuvre-lès-NancyFrance
A Von Reppert
Institut für Physik & Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24-2514476PotsdamGermany
& M Bargheer
Institut für Physik & Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24-2514476PotsdamGermany
Helmholtz-Zentrum Berlin for Materials and Energy GmbH
Wilhelm-Conrad-Röntgen Campus
BESSY II
Albert-Einstein-Str. 1512489BerlinGermany
Layer specific observation of slow thermal equilibration in ultrathin metallic nanostructures by femtosecond X-ray diffraction
10.1038/s41467-018-05693-5ARTICLE OPEN. Correspondence and requests for materials should be addressed to A.v.R. ( 1
Ultrafast heat transport in nanoscale metal multilayers is of great interest in the context of optically induced demagnetization, remagnetization and switching. If the penetration depth of light exceeds the bilayer thickness, layer-specific information is unavailable from optical probes. Femtosecond diffraction experiments provide unique experimental access to heat transport over single digit nanometer distances. Here, we investigate the structural response and the energy flow in the ultrathin double-layer system: gold on ferromagnetic nickel. Even though the excitation pulse is incident from the Au side, we observe a very rapid heating of the Ni lattice, whereas the Au lattice initially remains cold. The subsequent heat transfer from Ni to the Au lattice is found to be two orders of magnitude slower than predicted by the conventional heat equation and much slower than electron-phonon coupling times in Au. We present a simplified model calculation highlighting the relevant thermophysical quantities.
U ltrafast heating and cooling of thin metal films has been studied extensively to elucidate the fundamentals of electron-phonon interactions [1][2][3][4][5][6][7] and heat transport at the nanoscale [8][9][10][11][12][13] . The energy flow in metal multilayers following optical excitation attracted particular attention in the context of heat-assisted magnetic recording 14,15 and all-optical magnetic switching [16][17][18] . The role of temperature in optically induced femtosecond demagnetization is intensely discussed, particularly with regard to multipulse switching scenarios 19 . Two-or threetemperature models (TTMs) are often used to fit the experimental observations 20 . The microscopic three-temperature model (M3TM) 20 , which uses Elliot-Yafet spin-flip scattering as the main mechanism for ultrafast demagnetization is often contrasted against superdiffusive spin transport 21 . Such electron transport is closely related to ultrafast spin-Seebeck effects 22,23 , which require a description with independent majority and minority spin temperatures. The heat flow involving electrons, phonons, and spins has been found to play a profound role in ultrafast magnetization dynamics 24,25 . The description of the observed dynamics in TTMs or the M3TM are challenged by ab initio theory which explicitly holds the nonequilibrium distribution responsible for the very fast photoinduced demagnetization 26,27 . The presence of multiple subsystems (lattice, electrons, and spins), e.g., in ferromagnetic metals 5,28 , poses a formidable challenge for experimental studies of their coupling and thermal transport on ultrafast time scales when these subsystems are generally not in equilibrium with each other 26,27,29 . Temperature dynamics in metal films are typically monitored using optical probe pulses via time-domain thermoreflectance (TDTR) 12 . This technique has been a workhorse of nanoscale thermal transport studies, but experiences significant limitations when applied to ultrathin multilayers with individual layer thicknesses falling below the optical skin depth, which are in the focus of ultrafast magnetism research [22][23][24][30][31][32] . Optical probes are generally sensitive to electronic and lattice temperatures, although in some cases the lattice temperature 13 or the spin temperature 24 may be deduced. In order to understand the thermal energy flow, it is highly desirable to directly access the temperature of the lattice, which provides the largest contribution to the specific heat. Ultrafast X-ray diffraction is selectively sensitive to the crystal lattice, and material-specific Bragg angles enable measurements of multiple layers even when they are thinner than the optical skin depth and/or buried below opaque capping layers [33][34][35] . The lattice constant variations of each layer can be measured with high absolute accuracy, making it possible to determine the amount of deposited heat in metal bilayers that was debated recently [30][31][32] . The great promise of ultrafast X-ray diffraction (UXRD) for nanoscale thermal transport measurements and ultrafast lattice dynamics has already been demonstrated in experiments with synchrotron-based sources [33][34][35][36] . However, limited temporal resolution of these experiments (~100 ps) only allowed to study heat transport on a relatively slow (nanosecond) time scale and over distances >100 nm. Ultrafast nanoscale thermal transport research will greatly benefit from femtosecond X-ray sources. While free electron laser facilities are in very high demand, an alternative is offered by laser-based plasma sources of femtosecond X-rays 37,38 , which lack the coherence and high flux of a free electron laser but are fully adequate for UXRD measurements 6,39,40 . As an example, a recent experiment on 6 nm thick Au nanotriangles 39 confirmed the τ 0 Au = 5 ps electron-phonon equilibration time generally accepted for high fluence excitation of Au 3,6,41,42 . For similar fluences ultrafast electron diffraction reported τ 0 Ni = 0.75 to 1 ps for Ni thin films between room temperature and Curie temperature T C 7,43 .
In this report, we demonstrate that the use of a femtosecond X-ray probe enables thermal transport measurements over a distance as small as~5 nm in a Au/Ni bilayer with thickness d Au = 5.6 nm and d Ni = 12.4 nm grown on MgO. By monitoring the dynamics of the lattice constants of Au and Ni, we find that the Ni lattice fully expands within about 2 ps, while the Au lattice initially remains cold even if a significant fraction of the excitation light is absorbed by the electronic subsystem in Au. The Au lattice then heats up slowly, reaching the maximum temperature about 80 ps after the optical excitation. The observed thermal relaxation of the bilayer structure is two orders of magnitude slower than 1 ps predicted by the heat equation and also much slower than the usual electron-phonon equilibration time τ 0 Au = 1-5 ps (see Table 1) 3,41,42 . We explain this surprising result in a model (see Fig. 1) based on the keen insight into the physics of the thermal transport in Au-Pt bilayers offered in recent studies 11,13 , which showed that nonequilibrium between electrons and lattice in Au persists for a much longer time in a bilayer than in a single Au film. We find, furthermore, that on the spatial scale of our experiment thermal transport by phonons in metals can no longer be neglected. Our results underscore challenges for thermal transport modeling on the nanometer scale. On the other hand, they demonstrate the great potential of the UXRD for monitoring thermal transport under experimental conditions typical for studies of ultrafast magnetism 20,44 .
Results
Experiment.
We use femtosecond laser pulses at 400 and 800 nm to excite the electron system of Au and Ni through the Au top layer. The sample structure and the calculated absorption profiles are shown in Fig. 1. We note that for 400 nm pulses the absorbed energy density ρ Q Au;Ni in Au and Ni is similar, whereas for 800 nm almost no light is absorbed in Au. The much higher absorption of 400 nm light in Au is a result of the larger real part of the refractive index 31,32 . For our 5.6 nm thick Au film, the destructive interference of light reflected at the interfaces additionally contributes to the suppressed absorption at 800 nm.
The strains ε Au,Ni determined via Bragg's law from UXRD data (Fig. 2b, Thermal conductivity, κ (Wm −1 K −1 ) 318 56 90 56 Thermal conductivity (lattice), κ ph (Wm −1 K −1 ) 5 56 9.6 56 Expansion coefficient with Poisson correction, α uf (10 −5 K −1 )
3.16 57
57
Literature values for material parameters relevant for modeling the heat transfer after laser excitation. For C ph we use the parameters at room temperature. The e-ph coupling time ranges are calculated for 1000 K to show that for an equilibrated electron system, the e-ph coupling time in Ni is much shorter than in Au using effective out-of-plane expansion coefficients α uf Au;Ni and specific heats C Au,Ni , which are generally temperature dependent. For our experimental conditions temperature-independent coefficients are good approximations. The effective expansion coefficients α uf Au;Ni take into account the crystalline orientation of the films and the fact that on ultrafast (uf) timescales the film can exclusively expand out-of plane, since the uniform heating of a large pump-spot region leads to a one-dimensional situation, as in-plane forces on the atoms by the thermal stresses vanish. For details about α uf Au;Ni and a description how heat in electrons and phonons drive the transient stress via macroscopic Grüneisen coefficients see the Methods section.
We now discuss the information that can be directly inferred from the measured transient strains (Fig. 3) in the laser-excited metallic bilayer without any advanced modeling. For convenience, we added two right vertical axes to Fig. 3a, b showing the layer-specific temperature and energy density according to Eqs. (1) and (2). Initially Ni expands, while the Au layer gets compressed by the expansion of the Ni film. Around 3 ps Au shows a pronounced expansion, when the compression wave turns into an expansion wave upon reflection at the surface. Less pronounced signatures of the strain wave are observed in Ni as well. A surprisingly long time of about 80 ps is required to reach the maximum expansion of Au by transport of heat from the adjacent Ni until T Au ≈ T Ni . For times t > 100 ps, cooling by heat transfer to the substrate dominates the signal. In Fig. 3c we show the heat energy ΔQ MgO flowing through a unit area A into the
Δρ Q Ni;Au ðtÞ = ρ Q Ni;Au ðtÞ À ρ Q Ni;Au ð0Þ are the changes of the energy densities ρ Q Ni and ρ Q Au with respect to the initially deposited energy densities. Even when the temperatures are equilibrated at t > 100 ps, ρ Q Ni and ρ Q Au differ strongly because of the different specific heat of Au and Ni. Figure 3c confirms that within the first 20 ps the heat energy ΔQ Au = d Au Δρ Q Au flowing from Ni into Au is similar to the amount ΔQ MgO transported into the substrate. At about 150 ps half of the energy deposited in the film has been transported into the substrate. However, leaking a fraction of the thermal energy to the insulating substrate does not explain why the ultrathin Au layer is not much more rapidly heated via electronic heat transport typical of metals.
Modeling. Inspired by the recent studies using TDTR 11,13 we set up a modified two-temperature model graphically represented in Fig. 1b to rationalize the slow Au heating observed in Fig. 3a. We first justify this simplified modeling. The high electron conductivity-potentially including ballistic and superdiffusive electrons-rapidly equilibrates the electron systems of Ni and Au. The fact that the Au layer is equally compressed in the first 2 ps irrespective of the excitation wavelengths is an experimental proof of the rapid equilibration of electron temperatures. Otherwise the high electron pressure in Au after 400 nm excitation (cf. Fig. 1c) would counterbalance the compression caused by the Ni expansion 6 . As Ni has a much larger Sommerfeld constant (Table 1) the electronic specific heat C e = γ S T is dominated by Ni and the ratio of energy densities ρ Q Ni =ρ Q Au ≈ 10 is large at 1 ps. A significant electronic interface resistance 45 that would prevent a rapid equilibration of electron temperatures in Au and Ni is clearly incompatible with our measurements at 400 nm. If the electrons did not equilibrate much faster than 1 ps and effectively remove the heat deposited in the electron system of Au, we would not observe the same strong compression of the Au lattice, since electronic pressure would instantaneously force the Au to expand 6,7,40,43 . In the diffuse-mismatch model, the electronic interface conductance of metals increases linearly with the temperature and can be calculated from the Sommerfeld constant and the Fermi velocity 45 . Immediately after excitation, the electron temperature reaches several thousand Kelvin, which leads to a subpicosecond thermalization of the electrons in simulations, including the interface resistance.
The electron-phonon coupling constant in Ni is much larger than in Au (Table 1). Consequently, nearly all photon energy initially absorbed in the electronic system is funneled into the Ni lattice, even when one third of the absorbed energy is initially deposited in the electronic system of Au with 400 nm excitation. In contrast, the electron-phonon coupling times τ 0 Au;Ni = C e Au;Ni =g Au;Ni for Au and Ni are not very different if the films are not in contact, because the large electronic specific heat C e Ni of Ni cancels its large electron-phonon coupling constant g Ni (see Table 1). However, in the bilayer, the electrons in Au and Ni rapidly form an equilibrated heat bath with C e tot % C e Ni . Now only the electron-phonon coupling constant determines the coupling time: τ Ni = C e tot =g Ni ( C e tot =g Au = τ Au . We start the numerical modeling when a quasi-equilibrium temperature in the combined system C com = C e Au þ C e Ni þ C ph Ni % C e Ni þ C ph Ni ≈ C Ni is established after electron-phonon equilibration in Ni around τ Ni = C e tot =g Ni % C e Ni =g Ni ≈ 1 ps. Since C ph Ni ) C e Ni ) C e Au and d Ni > d Au , we refer to the combined system as C Ni in the equations. Since the energy stored in each layer is proportional to their thickness and the energy transfer rate from electrons to phonons in Au is proportional to the Au volume V Au ∝ d Au , the differential equations describing this special TTM represented in Fig. 1b read
d Au C ph Au ∂T ph Au ∂t ¼ d Au g Au T Ni À T ph Au ð4Þ d Ni C Ni ∂T Ni ∂t ¼ d Au g Au T ph Au À T Ni :ð5Þ
Note that the two temperatures in this model are the temperature of the Au lattice, T ph Au and the temperature of the combined system, which is denoted as T Ni , keeping in mind that this Ni temperature equals the Au electron temperature. For small temperature changes over which the specific heats are approximately constant, the solution to this system of equations is an exponential decay of T Ni~e −t/τ and a concomitant rise of the Au lattice temperature T Au~( 1 − e −t/τ ) on the characteristic timescale
τ ¼ 1 g Au 1 C Au þ d Au d Ni 1 C Ni :ð6Þ
Due to the small film thickness and the rapid electronic heat diffusion, we do not assume any gradient in the temperatures of each film. At about 1 ps after excitation we define the initial conditions as T Ni ð1psÞ ¼ T i Ni and T i Au % 0. The final temperature after equilibrating the temperatures of the two thin films, neglecting heat transport to the substrate is
T f ¼ T i d Ni C Ni d Au C ph Au þ d Ni C Ni :ð7Þ
This very simple model (dashed lines of Fig. 4a) for the transient quasi-equilibrium temperatures agrees very well with the data. In particular, the exponential rise of T Au and the exponential decay of T Ni converge around 80 ps. Deviations at longer times originate mainly from heat transport into the MgO substrate, which is not included in the model (dashed lines).
The only fitting parameters of our model are the initial temperature T i and the electron-phonon coupling constant of Au. With our simple model we get the best fit using g Au = 6.5 × 10 16 W m −3 K −1 , which is somewhat larger than the range from 1 to 4 × 10 16 W m −3 K −1 reported in the literature 2,3 . If-as an example-we reduce the electron-phonon coupling constant to the value of 4 × 10 16 W m −3 K −1 , the calculated equilibration of T Au and T Ni is much too slow. Including electronic interface resistance would make it even slower. The missing energy transfer rate, however, can be easily rationalized by phonon heat conductivity κ ph . If we fully disregarded electronic heat conduction in Au, the literature value for κ ph Au given in Table 1 would lead to an equilibration of Au and Ni temperature exclusively via phonons three times faster than we observe. The phonon heat transport is probably much less efficient than this prediction because of additional interface resistances for phonon heat transport and because the mean free path of phonons is on the order of the layer thickness 8,10 . However, we do not attempt to quantify κ ph and g Au here. We only note qualitatively that to conform to the expected values of electron-phonon coupling in Au, the phonon heat conduction must become important in nanoscale multilayers, even though normally the heat conduction in metals is dominated by electrons (κ ) κ ph see Table 1). Phonon heat transport is not included in our numerical calculations, because in fact the heat diffusion equation is not valid at such small length scales below the phonon mean free path. Similarly, a complex theoretical modeling would be required to simulate the heat transport to the substrate, e.g., by heat transfer from Ni electrons to MgO phonons at the interface 46 . Figure 3c provides a benchmark of the experimentally determined phonon heat transport into the substrate.
Discussion
In summary, the modified TTM model (Eqs. (4) and (5)) captures the essence of heat transport between ultrathin metal films: the electrons in Au and Ni are rapidly equilibrated. This is evidenced by the fact that 400 and 800 nm excitation both initially only heat Ni, regardless of the energy absorbed in Au. For 400 nm excitation we showed an intricate process of shuttling heat energy back and forth between the layers: the electrons first rapidly transport energy from Au into Ni (e-e equilibration ( 1 ps) before they transport some of the heat back from the Ni phonons to the Au phonons. Finally, the heat flows back through Ni toward the substrate. Heat transport by phonons can account for a fraction of the Au heating. The energy transported from the Ni phonons via Ni and Au electrons into the Au lattice is throttled by the weak electron-phonon coupling in Au. We believe that our results will have an important impact on ultrafast studies of the spin-Seebeck effect, superdiffusive electron transport as well as optical demagnetization and remagnetization. Precise measurements of the total heat in the system after few picoseconds will help to determine the actually required laser fluence in ultrafast demagnetization studies, which currently diverge by an order of magnitude in the literature 44,47 . The lattice is not only discussed as the sink of angular momentum in the ultrafast demagnetization: with its dominant heat capacity the lattice constitutes the heat bath which controls the speed of reordering of the spin systems at high fluence 20,44 . Our detailed account of heat flow in Ni after photo-excitation must influence the interpretation of MOKE data, which were fitted in previous studies 20,48 by using a value for the specific heat of the Ni phonon system which is a factor of two below the Dulong-Petit value.
We have demonstrated the power of UXRD in probing nanoscale heat transport in an ultrathin metallic bilayer system which is relevant to current magnetic recording developments such as heat-assisted magnetic recording. To understand the alloptical-15 and helicity-dependent 49 switching in ferrimagnets and two different timescales observed in the demagnetization of transition metals 20,44 or rare earths 50,51 , precise calibration of the lattice temperature is crucial. We are convinced that the direct access to the lattice, the layer-specific information for layers thinner than the optical skin depth, the conceptual simplicity of the arguments and the experimental geometry make the paper particularly useful for comparisons to previous 20,[30][31][32]44 and future work on optical manipulation of spins.
Methods
Sample growth and UXRD. Ni/Au stacks with different Ni and Au thicknesses were grown by molecular beam epitaxy onto a MgO(001) substrate at 100°C. The MgO(001) substrates were degassed at 350°C for 10 min. The pressure during growth never exceeded 6 −10 mbar. We measured the layer thicknesses d Au = 5.6 nm and d Ni = 12.4 nm of the investigated sample by X-ray reflectivity. The 24 lattice planes of Au yield a symmetric (111) Bragg reflection (Fig. 2a) at ϑ = 19.29°, well separated from the symmetric (200) Ni peak at 25.92°originating from 70 lattice planes. The lattice strains ε Ni,Au (t) = −cot(ϑ(t))Δϑ(t) perpendicular to the sample surface are directly retrieved from the time-resolved Bragg-peak positions ϑ (t) (Fig. 2b, c) 5839,40 . These UXRD data were recorded at our laser-driven plasma X-ray source at the University of Potsdam, that emits 200 fs X-ray pulses with a photon energy of 8 keV. The sample was excited by p-polarized 400 and 800 nm laser pulses of about 100 fs duration with a pulse energy of 0.3 mJ and a diameter of 1.5 mm (FWHM). Since the angle between the pump pulse and the Bragg-reflecting X-ray probe pulse is fixed in the setup, we take into accout the modified angle of incidence of the optical pulse of 44°(51°) with respect to the surface normal for the Ni (Au) reflection to calculate the incident fluence of 9 (8) mJ/cm 2 and an absorbed fluence of 3 (2.9) mJ/cm 2 for our bilayer system using a matrix formalism, which also yields the absorption profiles at 400 and 800 nm excitation shown in Fig. 1c 52 .
The above values are for 800 nm excitation, and the 400 nm data in Fig. 3 are scaled up for better comparison of the two different excitation conditions.
Correction of the thermal expansion coefficient. The effective expansion coefficient α uf Au;Ni valid for heating a thin epitaxial layer is based on the lattice constants and strains predicted from equilibrium thermal expansion coefficients, corrected according to the Poisson effect 53 . In cubic materials with (100) surface orientation the ratio of the observed ultrafast (uf) strain and the strain ε eq = α eq (T)ΔT along the (100) direction calculated from equilibrium value (eq) is ε/ε eq = α uf (T)/α eq (T) = 1 + 2C 12 /C 11 = 2.2 for Ni and would be 2.6 for Au. For the Au (111) cubic crystal surface, the above equation is still valid if the elastic constants are calculated in the rotated coordinate system, in which the x-axis is [111]. We find that the newly obtained C 11 and C 12 coincidentally yield the same correction factor of 2.2 for Au (111) as for Ni (100).
Strain waves prove ultrafast electron-equilibration. The pronounced compression and expansion of the Au layer (see Fig. 4a) clearly originates from the laser-induced stress generated in Ni. In order to show that our modified TTM predicting negligible energy density in Au immediately after the excitation can quantitatively explain the signal oscillations, we have used the transient temperatures T Ni,Au (t) from our TTM as input parameters for a full thermo-elastic simulation using the udkm1Dsim toolbox, which are represented as solid lines in Fig. 4a 59 . For convenience, Fig. 4b shows the spatio-temporal strain map from which the solid lines in Fig. 4a are calculated by spatial averaging over the layer for each time delay. Multiple reflections of strain waves at the interfaces are strongly damped by transmission to the substrate.
Macroscopic Grüneisen coefficients. Several recent ultrafast X-ray diffraction and electron diffraction experiments on thin metal films have highlighted two contributions of electrons and phonons to the transient stress σ, which drives the observed strain waves. A very useful concept uses the macroscopic Grüneisen coefficient Γ e and Γ ph , which relate the energy densities ρ Q to the stress σ = Γρ. While in Au the electronic Grüneisen constant Γ e Au = 1.5 is about half of its phonon counterpart Γ stress by 15%. In Au the electron pressure is negligible in our bilayer system, since due to the large electronic specific heat of Ni and the subpicosecond equilibration among the electrons, all the energy is accumulated in Ni. The ab initio modeling discussed in connection to the recent UXRD study on Fe points out that both electron-phonon coupling parameters and phonon Grüneisen coefficients depend on the phonon mode 5,29 . While in that study the scattering of X-rays from individual phonon modes selected by the scattering geometry may require a modespecific analysis, we believe that measuring the lattice expansion via a Bragg-peak shift looks at an average response of the lattice to all phonon modes, and hence a mode-averaged analysis is reasonable if there is no selective excitation of modes with extraordinarily different Grüneisen coefficients.
Data availability. The data that support the findings of this study are available from the corresponding authors on reasonable request.
Received: 22 November 2017 Accepted: 18 May 2018
Fig. 1 Fig. 2
12Schematic of heat reservoirs in the sample structure. a Layer stacking of the metallic heterostructure: Au on Ni deposited on an MgO substrate. Each layer has a phonon heat reservoir. The metal layers additionally have an electronic heat reservoir. The heat contained in the Ni spin system is included in the electron system. The electron-phonon coupling constants g Au and g Ni parametrize the local energy flow among electrons and phonons within each layer, whereas the thermal conductivity κ indicates spatial heat transport. b Calculated optical absorption profiles in the metallic bilayer Experimental data. a X-ray diffraction pattern of the sample (see inset) evidencing the crystalline orientation of the Au and Ni nanolayers.Colored lines visualize transient shifts of the Bragg peaks at selected times. Their full time evolution is shown in panels (b) for Au and c for Ni along with the respective peak center positions (black line). The white dashed line indicates the axis break from linear to logarithmic time scale t (ps)
Fig. 4
4ph Au = 3.0, in Ni Γ e Ni = 1.5 is only slightly different from Γ ph Ni = 1.76,7 . For our analysis the distinction of the origin of pressure in Ni is not very relevant, since the redistribution of energy from electrons to phonons only increases the Comparison of models with the experimental data. a Dots indicate the measured strain ε. The dashed lines represent the strain calculated from the average heating of the layers according to the model visualized inFig. 1b. Solid lines are simulations, which are based on this model and additionally include the strain waves triggered by the impulsive excitation (see Methods section). Heat transport to the substrate is not included. b Color-coded strain ε as a function of sample depth and time t, which is simulated assuming a spatially homogeneous transient thermal stress in each layer which is proportional to the dashed lines in (a). Spatial averaging of the strain ε(t) in each layer yields the solid lines in panel (a)
c) can be converted to lattice temperature changes ΔT Au,Ni and energy density changes ρ Q Au;Ni viaε Au;Ni ¼ α uf
Au;Ni ΔT Au;Ni
ð1Þ
ε Au;Ni ¼
α uf
Au;Ni
C Au;Ni
ρ Q
Au;Ni
ð2Þ
Table 1 Thermophysical parameters of Au and Ni
Parameter
Gold
Nickel
Lattice specific heat, C ph (10 6 Jm −3 K −1 )
2.5 54
3.8 55
Sommerfeld constant, γ S (Jm −3 K −2 )
67.5 2
1074 2
Electron-phonon coupling constant,
g (10 16 Wm −3 K −1 )
1-4 2
36-105 2
e-ph coupling time isolated layers
@1000 K, τ 0 (ps)
1.7-6.7
1-3
e-ph coupling time equilibrated
electrons @1000 K, τ (ps)
26-107
1-3
Fig. 3Transient energy densities and temperatures. Transient lattice strain ε in the Au film (a) and the Ni film (b) as measured by UXRD after excitation with 400 nm (blue) and 800 nm (red) light pulses. The right axis label the temperature change ΔT and the energy density ρ Q calculated from ε. c Red and blue dots show the energy per unit area ΔQ/A obtained from (a, b) by multiplication with d Au,Ni . The red and blue lines show thermal dynamics with acoustic oscillations removed, yielding the true energy per unit area ΔQ/A. The black dashed line shows the sum of these energies. The gray line is the thermal energy that has been transported into the substrate ΔQ MgO ðtÞ=A ¼ Àd Au Δρ Q Au ðtÞ À d Ni Δρ Q Ni ðtÞ:ΔQ Au
ΔQ Ni
ΔQ layers
ΔQ MgO
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05693-5
ARTICLE
NATURE COMMUNICATIONS | (2018) 9:3335 | DOI: 10.1038/s41467-018-05693-5 | www.nature.com/naturecommunications
substrate, which we can directly calculate from the measured
energy densities via
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05693-5
NATURE COMMUNICATIONS | (2018) 9:3335 | DOI: 10.1038/s41467-018-05693-5 | www.nature.com/naturecommunications
© The Author(s) 2018
AcknowledgmentsWe acknowledge the BMBF for the financial support via 05K16IPA and the DFG via BA 2281/8-1. The contribution by A.A.M. was supported by the US Department of Energy Grant no. DE-FG02-00ER15087. We thank Daniel Schick for help regarding the calculations using the optical matrix formalism.Additional informationCompeting interests: The authors declare no competing interests.Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.
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| [] |
[
"Functional renormalization with fermions and tetrads",
"Functional renormalization with fermions and tetrads"
] | [
"P Donà \nInternational School for Advanced Studies\nItaly and INFN\nSezione di Trieste\nvia Bonomea 26534136TriesteItaly\n",
"R Percacci \nInternational School for Advanced Studies\nItaly and INFN\nSezione di Trieste\nvia Bonomea 26534136TriesteItaly\n"
] | [
"International School for Advanced Studies\nItaly and INFN\nSezione di Trieste\nvia Bonomea 26534136TriesteItaly",
"International School for Advanced Studies\nItaly and INFN\nSezione di Trieste\nvia Bonomea 26534136TriesteItaly"
] | [] | We investigate some aspects of the renormalization group flow of gravity in the presence of fermions, which have remained somewhat puzzling so far. The first is the sign of the fermionic contribution to the running of Newton's constant, which depends on details of the cutoff. We argue that only one of the previously used schemes correctly implements the cutoff on eigenvalues of the Dirac operator, and it acts in the sense of screening Newton's constant. We also show that Kähler fermions give the same contribution to the running of the cosmological and Newton constant as four Dirac spinors. We then calculate the graviton contributions to the beta functions by imposing the cutoffs on the irreducible spin components of the tetrad. In this way we can probe the gauge dependence of the off-shell flow. The results resemble closely those of the metric formalism, except for an increased scheme-and (off shell) gauge-dependence.I. CUTOFF SCHEMESThe functional renormalization group equation (FRGE) is a convenient method to calculate the quantum effective action that is not restricted to renormalizable field theories[1,2]. Instead of computing directly the effective action, one computes its derivative with respect to an external mass scale k called "the cutoff". This "beta functional" is automatically free of UV and IR divergences.The advantage of this procedure is that it provides a definition of the beta functions of a theory that does not refer to any UV regularization. Then, taking some (generally local) action as starting point at an UV scale Λ, one can obtain the effective action by integrating the FRGE from Λ to zero.The divergences of the theory reappear when one tries to shift Λ to infinity and can be analysed by integrating the FRGE in the direction of increasing k.For our purposes the most useful implementation of this idea is the Wetterich equation [2].It gives the k-dependence of a functional called the Effective Average Action (EAA), which is a modified effective action where the propagation of the low momentum modes is suppressed. Here "low momentum" generally means "momenta below a given scale k". The suppression is described, in flat space, by some function R k (q 2 ), where q 2 is the squared momentum. To be admissible, the | 10.1103/physrevd.87.045002 | [
"https://arxiv.org/pdf/1209.3649v2.pdf"
] | 119,197,964 | 1209.3649 | 35a46e512a24165e4367f4d575cb90ff9d2161dd |
Functional renormalization with fermions and tetrads
8 Oct 2012
P Donà
International School for Advanced Studies
Italy and INFN
Sezione di Trieste
via Bonomea 26534136TriesteItaly
R Percacci
International School for Advanced Studies
Italy and INFN
Sezione di Trieste
via Bonomea 26534136TriesteItaly
Functional renormalization with fermions and tetrads
8 Oct 2012
We investigate some aspects of the renormalization group flow of gravity in the presence of fermions, which have remained somewhat puzzling so far. The first is the sign of the fermionic contribution to the running of Newton's constant, which depends on details of the cutoff. We argue that only one of the previously used schemes correctly implements the cutoff on eigenvalues of the Dirac operator, and it acts in the sense of screening Newton's constant. We also show that Kähler fermions give the same contribution to the running of the cosmological and Newton constant as four Dirac spinors. We then calculate the graviton contributions to the beta functions by imposing the cutoffs on the irreducible spin components of the tetrad. In this way we can probe the gauge dependence of the off-shell flow. The results resemble closely those of the metric formalism, except for an increased scheme-and (off shell) gauge-dependence.I. CUTOFF SCHEMESThe functional renormalization group equation (FRGE) is a convenient method to calculate the quantum effective action that is not restricted to renormalizable field theories[1,2]. Instead of computing directly the effective action, one computes its derivative with respect to an external mass scale k called "the cutoff". This "beta functional" is automatically free of UV and IR divergences.The advantage of this procedure is that it provides a definition of the beta functions of a theory that does not refer to any UV regularization. Then, taking some (generally local) action as starting point at an UV scale Λ, one can obtain the effective action by integrating the FRGE from Λ to zero.The divergences of the theory reappear when one tries to shift Λ to infinity and can be analysed by integrating the FRGE in the direction of increasing k.For our purposes the most useful implementation of this idea is the Wetterich equation [2].It gives the k-dependence of a functional called the Effective Average Action (EAA), which is a modified effective action where the propagation of the low momentum modes is suppressed. Here "low momentum" generally means "momenta below a given scale k". The suppression is described, in flat space, by some function R k (q 2 ), where q 2 is the squared momentum. To be admissible, the
cutoff function R k must satisfy the following basic requirements: (i) it must be continuous and monotonically decreasing in q 2 and k, (ii) it must go rapidly to zero for q 2 > k 2 , (iii) it must tend to a positive (possibly infinite) value for q 2 → 0 and (iv) it must tend to zero for k → 0.
It is unavoidable that details of the flow depend on the shape of the cutoff function, even though the effective action, which is obtained by letting k → 0, is not. This is referred to as "scheme dependence" and is akin to the renormalization scheme dependence in perturbation theory.
When one considers anything more than a single scalar field, in addition to the shape of the function R k (z), one encounters further choices. We will be concerned mainly with gravity, which offers a particularly complex scenario. The application of the FRGE to gravity has been pioneered in [3], followed by [4]. Already in these early works the implementation of the cutoff followed different methods. In [3] the cutoff was imposed on the tracefree and trace parts of the metric fluctuation (a decomposition that is purely algebraic), whereas in [4] the cutoff was imposed separately on each irreducible spin component (a finer decomposition that involves differential conditions). Following [5], we will call the former a cutoff of type "a" and the latter of type "b". When gravity is involved, the kinetic operators governing the propagation of quantum fields typically have the structure −∇ 2 +aR+bE, where ∇ is the Riemannian covariant derivative, R stands here for some combination of curvature terms (Riemann, Ricci and scalar curvatures, with indices arranged in various ways) and E a term that may involve other background fields and possibly also couplings. Since Fourier analysis is not available in general curved spaces, in defining the cutoff one has to specify an operator whose eigenvalues form a basis in field space and in general play the role that plane waves play in flat space. One takes R k to be a function of this operator (or rather, its eigenvalues). In [5] the following terminology has been introduced: a cutoff is of type I if R k is a function of −∇ 2 , of type II if it is a function of −∇ 2 + aR and of type III if it is a function of −∇ 2 + aR + bE. It is clear that there are infinitely many more possible cutoff variants and that this classification is very incomplete, but it covers the most commonly used cutoffs.
The RG flow in the so-called Einstein-Hilbert truncation, where one retains in the action terms at most linear in curvature, has been studied in [6][7][8][9][10], with en extended ghost sector in [11][12][13] and using the Vilkoviski-de Witt formalism in [14]. Higher derivative terms have been analysed in [5,[15][16][17][18][19][20]. The contribution of matter fields has been taken into account in [4,[21][22][23][24] using a Type I cutoff and in [25] using a type II cutoff. Conversely, the effect of gravity on fermionic interactions has been studied in [26][27][28][29].
Physical observables, which are related to the full effective action at k = 0, will be independent of the cutoff choice. Furthermore, some terms in the beta functional, namely those that refer to dimensionless couplings, can actually be shown to be scheme-independent. It is also wellunderstood that if the system has a nontrivial fixed point, its position is not universal. In general, the scheme-dependence of the flow should therefore not be a cause of major concern [30].
In the literature on the renormalization group for gravity, there is however one result where the scheme dependence seems to be particularly nasty: it is the contribution of fermion fields to the beta function of Newton's constant. This contribution has been computed for example in section III of [5]. If there are n D Dirac fields it is
− n D 1 2 1 (4π) 2ˆd 4 x √ g 1 6 Q 1 ∂ t R k P k − 1 4 Q 2 ∂ t R k P 2 k 4 R(1)
if one uses a type I cutoff and
− n D 1 2 1 (4π) 2ˆd 4 x √ g 1 6 − 1 4 Q 1 ∂ t R k P k 4 R = n D 96π 2 Q 1 ∂ t R k P k ˆd 4 x √ g R(2)
if one uses a type II cutoff. Here Q n (W ) = 1 Γ(n)´∞ 0 dz z n−1 W (z). If we use the optimized cutoff [9] R k (z) = (k 2 − z)θ(k 2 − z) ,
we get Q 1 ∂tR k P k = 2k 2 , Q 2 ∂tR k P 2 k = k 2 and therefore the contribution of fermions to the relevant term of the beta functional is − n D 96π 2ˆd 4 x √ g R for the type I cutoff (4) n D 48π 2ˆd 4 x √ g R for the type II cutoff.
The latter agrees with an earlier independent calculation of the renormalization of G in [31]; the former differs not just in the value of the coefficient but even in the sign. 1 To check that this is not just a quirk of the optimized cutoff, consider an exponential cutoff
R k (z) = ze −a(z/k 2 ) b 1 − e −a(z/k 2 ) b .(6)
In order to guarantee that condition (iii) is satisfied, one has to assume b ≥ 1. For example if b = 1, Q 1 ∂tR k P k = π 2 k 2 /3a and Q 2 ∂tR k P 2 k = 2k 2 /a and therefore we encounter the same problem, independently of the parameter a. One can see that the same is true also for b > 1.
This sign ambiguity is puzzling. Since the beta function of G vanishes for G = 0, the sign of G can never change in the course of the RG flow (we disregard here as pathological the case when the inverse of G passes through zero). Therefore, if the physical RG trajectory approaches a fixed point in the UV, the sign of Newton's constant at low energy will be the same as that of Newton's constant at the fixed point. In a model where gravity is induced by minimally coupled fermions, the sign of Newton's constant would depend on the scheme.
One has to be careful in drawing physical conclusions from these calculations: the relation between the coupling G appearing in Γ k and the physical Newton's constant that is measured in the lab may not be as simple as it seems. The functional that obeys an exact RG equation is not a simple functional of a single metric but rather a functional of a background metric and (the expectation value of) the fluctuation of the metric. This functional is not invariant under shifts of the background and fluctuation that keep the sum constant, and therefore there may be several couplings that one could legitimately call "Newton's constant" (for a discussion see [33]). The one that we are discussing here is the one that multiplies the Hilbert action constructed from the background metric only, and it is not obvious that the other ones would behave in the same way. In order to make a completely well-defined statement one should really calculate a physical observable and in such a calculation all ambiguities should disappear. It is therefore possible, in principle, that even the sign difference between the two calculations discussed above may be resolved when one considers physical observables. In this paper we will argue for a different and simpler solution of the issue, namely that only the type II cutoff gives a result with the physically correct sign.
As an aside, we will also calculate the contribution to the beta functions of the cosmological constant and Newton constant due to Euclidean Kähler fermions in four dimensions. Kähler fermions are a way of describing fermion fields in terms of Grassmann algebra elements, instead of spinors and has the merit that it does not require the use of tetrad fields. We find that one Kähler fermion gives exactly the same contribution as four Dirac spinors, and that the same sign issue is present.
When gravity is coupled to fermions an additional question arises regarding the field carrying the gravitational degrees of freedom. In [4,21,22], where the contribution of graviton loops was added to that of matter fields, as well as in [26][27][28] where scalars and fermions were interacting with gravity, the carrier of the gravitational degrees of freedom was the metric. It has been pointed out in [34] that in the presence of fermions it may be more natural to use the tetrad. Even if one chooses a Lorentz gauge where the antisymmetric part of the tetrad fluctuation is suppressed, the two calculations are not the same, because one has to work off shell and the Hessian in the tetrad formalism contains some additional terms proportional to the equations of motion. The calculations in [34], which used a type Ia cutoff in the α = 1 gauge, show that a fixed point is still present, but is less stable than in the metric formulation. We will compute the gravitational contributions in the tetrad formalism, using type b cutoffs (both Ib and IIb); this allows us to analyse the off-shell α-dependence. The results will be found to be somewhat closer to those of the metric formalism than those found in [34] (which we have independently verified), but they still present a stronger dependence on the scheme and on the gauge parameter.
II. KÄHLER FERMIONS
Before discussing in detail the issue of cutoff scheme with ordinary spinor fermions, we would like to point out that precisely the same problem arises also in another representation of fermionic matter. This section is not strictly necessary for the following discussion so readers that are mainly interested in the solution of the puzzle presented above may go directly to section III.
As is well-known, in any dimension the Grassmann and the Clifford algebras are isomorphic as vector spaces. This is the basis for a representation of fermion fields as inhomogeneous differential forms [35][36][37]. Such fields are called Kähler fermions. In this representation the analogue of the Dirac operator is the first order operator d + δ, where d is the exterior derivative and δ is its adjoint. Note that the definition does not require the use of a tetrad. We would like to compare the contribution of Kähler fermions to the gravitational beta functions to the one of ordinary spinor fermions. In particular we would like to see whether the same sign issue arises when different cutoff types are used. Since the details of the calculation are strongly dimension-dependent, we shall restrict our attention to the case d = 4.
An inhomogeneous complex differential form Φ can be expanded as
Φ = ϕ(x)+ϕ µ (x)dx µ + 1 2! ϕ µν (x)dx µ ∧dx ν + 1 3! ϕ µνρ (x)dx µ ∧dx ν ∧dx ρ + 1 3! ϕ µνρσ (x)dx µ ∧dx ν ∧dx ρ ∧dx σ
We can map the 3-and 4-form via Hodge duality into a 1-and 0-form respectively. The field Φ thus describes a scalar, a pseudoscalar, a vector, a pseudovector and an antisymmetric tensor, for a total of 16 complex components. This is an early sign of the fact that one Kähler field is equivalent to four Dirac fields.
The square of the Kähler operator is the Laplacian on forms: ∆ = (d + δ) 2 = dδ + δd. On forms of degree 0, 1 and 2 it is given explicitly by
∆ (0) = −∇ 2(7)∆ (1)µ ν = −∇ 2 δ µ ν + R µ ν (8) ∆ (2) αβ γσ = −∇ 2 1 αβ γσ + R γ α δ σ β − R γ β δ σ α − 2R γ σ α β(9)
In order to read off the beta functions of the cosmological constant and Newton's constant it enough to consider a spherical (Euclidean de Sitter) background, with curvature tensor
R µνρσ = 1 d (d − 1) (g µρ g νσ − g µσ g νρ )R ; R µν = 1 d g µν R(10)
Then the operators defined above reduce to
∆ (1)µ ν = −∇ 2 + 1 4 R δ µ ν (11) ∆ (2) αβ γσ = −∇ 2 + 1 3 R 1 αβ γσ(12)
The ERGE for a Kähler fermion with a type II cutoff is
dΓ k dt = − 2 1 2 Tr (0) ∂ t R k (∆ (0) ) P k (∆ (0) ) − 2 1 2 Tr (1) ∂ t R k (∆ (1) ) P k (∆ (1) ) − 1 2 Tr (2) ∂ t R k (∆ (2) ) P k (∆ (2) ) =4 · 1 2 1 (4π) 2ˆd 4 x √ g −4 Q 2 ∂ t R k P k + 1 3 R Q 1 ∂ t R k P k .(13)
For a type I cutoff we find instead:
dΓ k dt = − 2 · 1 2 Tr (0) ∂ t R k (−∇ 2 ) P k (−∇ 2 ) − 2 · 1 2 Tr (1) ∂ t R k (−∇ 2 ) P k (−∇ 2 ) + R 4 − 1 2 Tr (2) ∂ t R k (−∇ 2 ) P k (−∇ 2 ) + R 3 = 4 · 1 2 1 (4π) 2ˆd 4 x √ g −4 Q 2 ∂ t R k P k − R 2 3 Q 1 ∂ t R k P k − Q 2 ∂ t R k P 2 k .(14)
Evaluating the Q-functionals for the cutoff (3) we get:
Type II: dΓ k dt = 4 · 1 2 1 (4π) 2ˆd 4 x √ g −4 k 4 + 2 3 R k 2(15)
Type I:
dΓ k dt = 4 · 1 2 1 (4π) 2ˆd 4 x √ g −4 k 4 − 1 3 R k 2(16)
In both cases the effect of one Kähler fermion is seen to match exactly the result of four spinors (n D = 4). This should not induce one to believe that spinors and Kähler fermions are equivalent:
in fact, their contributions to the curvature squared terms are quite different. Nevertheless, the puzzling sign issue of the R-term that afflicts spinor fermions is present in this case too.
III. CUTOFF CHOICE FOR FERMIONS
We now return to ordinary Dirac spinor fields and we reexamine in more detail their contribution to the gravitational effective action and beta functions. For the sake of generality we will now work in arbitrary dimension d. The standard way of defining the effective action for a fermion field that is minimally coupled to gravity is to exploit the properties of the logarithm and write
Γ = −Tr log(| / D|) = − 1 2 Tr log( / D 2 ) = − 1 2 Tr log −∇ 2 + R 4 .(17)
The corresponding EAA can then be defined as
Γ k = − 1 2 Tr log −∇ 2 + R 4 + R k .(18)
In the definition of this functional one encounters the same ambiguities that we have mentioned earlier for bosonic systems. In addition to the shape of the cutoff function R k , one seems to also have the freedom of choosing the argument of this function to be either −∇ 2 (type I cutoff) or −∇ 2 + R 4 (type II cutoff). The former choice has been made in [4,[21][22][23][24], the latter in [25]. Taking the t-derivative (where t = log(k/k 0 )) and defining P k (z) = z + R k (z), one has
dΓ k dt = − 1 2 Tr ∂ t R k (−∇ 2 ) P k (−∇ 2 ) + R 4
for a type I cutoff (19)
dΓ k dt = − 1 2 Tr ∂ t R k (−∇ 2 + R 4 ) P k (−∇ 2 + R 4 )
for a type II cutoff.
The first few terms in the curvature expansion of these traces can be evaluated, for any background, using heat kernel methods. However, for a spherical background, the spectrum of the Dirac operator is known explicitly and the same traces can also be computed directly as spectral sums. Comparison of these calculations indicates that only the type II cutoff correctly reflects the cutoff on eigenvalues of the Dirac operator.
A. Heat kernel evaluation
With a type I cutoff, the trace (19) giving contribution of n D Dirac spinors to the FRGE is
dΓ k dt = − n D 2 2 [d/2] (4π) d/2ˆd d x √ g Q d 2 ∂ t R k P k + 1 6 Q d 2 −1 ∂ t R k P k − 1 4 Q d 2 ∂ t R k P 2 k R+. . . .(21)
Here
2 [d/2] (where [x]
is the integer part of x) is the dimension of the representation. The first term proportional to R is proportional to the heat kernel coefficient b 2 (−∇ 2 ), and the second comes from the expansion of the denominator in (19). Evaluating the Q-functionals with the cutoff (3) we
obtain dΓ k dt = − 1 Γ d 2 + 1 2 [d/2] (4π) d/2 n Dˆd d x √ g k d + d − 3 12 k d−2 R .(22)
Evaluating (20) with the same techniques yields
dΓ k dt = − n D 2 2 [d/2] (4π) d/2ˆd d x √ g Q d 2 ∂ t R k P k − 1 12 R Q d 2 −1 ∂ t R k P k + . . . ,(23)
where the term proportional to R comes entirely from the heat kernel coefficient b 2 −∇ 2 + R 4 . Evaluating with the cutoff (3) we obtain
dΓ k dt = − 1 Γ d 2 + 1 2 [d/2] (4π) d/2 n Dˆd d x √ g k d − d 24 k d−2 R .(24)
In d = 4 this yields the results quoted in section I. We see that the sign issue is present in any dimension d > 3.
B. Spectral sums on S d
The heat kernel calculation of the preceding subsection can be done in an arbitrary background.
On the other hand, in the case of the spherical background we know explicitly the spectrum of the Dirac operator: the eigenvalues and multiplicities are
λ ± n = ± R d(d − 1) d 2 + n , m n = 2 [ d 2 ] n + d − 1 n , n = 0, 1, . . . .(25)
With this information one can compute the trace of any function of the Dirac operator as
Tr f ( / D) = ∞ n=0 m n f (λ n ).
We will now evaluate the r.h.s. of the FRGE by imposing a cutoff on the eigenvalues of the Dirac operator. The EAA can be defined directly in terms of the Dirac operator as
Γ k = −tr log | / D| + R D k (| / D|) ,(26)
where the cutoff R D k has to be a function of the modulus of the Dirac operator, since we want to suppress the modes depending on the wavelength of the corresponding eigenfunctions. This is also needed for reasons of convergence. Since the operator is first order, the conditions that R D k has to satisfy are similar to (i)-(iv) of section I, except for the replacement of q 2 and k 2 by λ n and k respectively. For the explicit evaluation, we will use the optimized profile
R D k (z) = (k − z)θ(k − z) , (z > 0) .(27)
Then we have
Tr ∂ t R D k (| / D|) P D k (| / D|) = n m n ∂ t R D k (|λ n |) P D k (|λ n |) = ± n m n θ(k − |λ n |) .(28)
The sum can be computed using the Euler-Maclaurin formula. Details are given in Appendix I.
The result is
dΓ k dt = −Tr ∂ t R D k P D k = − 1 Γ d 2 + 1 2 [ d 2 ] (4π) d 2 V (d) k d − d 24 k d−2 R + O R 2 ,(29)
where V (d) is the volume of the d-sphere. This agrees exactly with (24), which was obtained with the type II cutoff. (We have checked that the agreement extends also to the next order in the curvature expansion.)
C. Discussion
Note that computing the r.h.s. of the FRGE with a spectral sum is a much more direct procedure, since it avoids going through the square root of the square of the Dirac operator, and also avoids having to use the Laplace transform and the heat kernel. It is therefore also a more reliable procedure when there are ambiguities. The agreement of the spectral sum with the type II-heat kernel calculation is a useful consistency check and suggests that the latter gives the correct result whereas the type I cutoff does not.
If so, there remain to understand why the type I cutoff should not be admissible in this case.
One can get some hint by thinking of what this cutoff does in terms of eigenvalues of the Dirac operator. We begin by noting that (26) can be rewritten as follows: 2
Γ k = − 1 2 tr log | / D| + R D k (| / D|) 2 = − 1 2 tr log −∇ 2 + R 4 + 2| / D|R D k (| / D|) + R D k (| / D|) 2 .(30)
One can compare this with (18). Note that the cutoff R k in that formula could be a function of different operators which, on a sphere, differ by a constant shift. For the present purposes it is convenient to think of it as a function of / D 2 . CallingR k this function and calling z = | / D|, we havē
R k (z 2 ) = 2zR D k (z) + R D k (z) 2 .(31)
We can solve this relation to get
R D k (z) = −z + z 2 +R k (z 2 ) ,(32)
so for any cutoff imposed at the level of (18) one can reverse-engineer an effective cutoff to be imposed at the level of (26) that will give the same result.
In general, this cutoff may fail to satisfy the required conditions, in particular condition (iv). For
a type II cutoff, R k in (18) is a function of z 2 , soR k (z 2 ) = R k (z 2 )
. This implies thatR 0 (z 2 ) = 0
and thus also R D 0 (z) = 0 for all z > 0. For a type I cutoff, on the other hand, this may not be the case, as we will show in the following examples.
Consider first the optimized cutoffs. In the type II case one hasR k (z 2 ) = (k 2 − z 2 )θ(k 2 − z 2 ) and one finds that in this case the corresponding cutoff R D k (z) given by (32) is also optimized, and precisely of the form (27). This is a way of understanding why the two calculations give the same result. In the case of a type I cutoff, we have insteadR k (
z 2 ) = (k 2 − z 2 + R/4)θ(k 2 − z 2 + R/4), whence one derives R D k (z) = k 2 + R 4 − z θ k 2 + R 4 − z .(33)
This does not tend uniformly to zero when k → 0. In the case of an exponential type II cutoff with R k = R k given by (6), we have
R D k (z) = −z + z 1 − e −a(z 2 /k 2 ) b ,(34)
which has all the desired properties. On the other hand for an exponential type I cutoff withR k given by (6),
R D k (z) = −z + z 2 + z 2 − R 4 e −a(z 2 − R 4 ) b /k 2b 1 − e −a(z 2 − R 4 ) b /k 2b .(35)
For b odd, and in particular for the most natural case b = 1, this function does not tend uniformly to zero when k → 0 and therefore condition (iv) is not satisfied. 3
These arguments lend support to the view that only the type II cutoff gives the physically correct result. Of course not all results obtained from the type I cutoff have to be wrong, for example the leading term (renormalizing the cosmological constant) and, in d = 4, the curvature squared terms, are the same using the two cutoffs. These however are just the "universal" quantities that do not depend on the choice of the cutoff. We believe that for the generic dimensionful, non-universal quantities, the results obtained via a type I cutoff should not be trusted.
IV. TETRAD GRAVITY
We will now compute the graviton contribution to the running of Newton's constant and cosmological constant, in d-dimensions, when the tetrad is used as a field variable. This has been discussed recently in [34] using a type Ia cutoff, i.e. a cutoff that depends on −∇ 2 that is added to the full gravitational Hessian. In order to have a manageable, minimal Laplacian-type operator, this requires that the gravitational gauge-fixing parameter be fixed to α = 1. We will use instead cutoffs of type b, meaning that the graviton is first decomposed into irreducible components of spin 2, 1 and 0, and the cutoff is imposed separately one each component. This permits the discussion of general diffeomorphism gauges. We will use both type Ib and type IIb cutoffs.
A. Hessian and gauge fixing
The ansatz we make for the effective average action is the standard Einstein-Hilbert truncation
Γ k [e,ē] = Γ EH k [e,ē] + Γ GF k [e,ē] = − 1 16πG kˆd d x det e R(g(e)) − 2Λ k + Γ GF k [e,ē](36)
where we have indicated the k-dependence of the couplings and Γ GF k is the gauge-fixing term, to be specified below. In tetrad formulation the metric is represented in terms of vielbeins e a µ as g µν = e a µ e b ν η ab . If we decompose g µν ≡ḡ µν + h µν and e a µ ≡ē a µ + ε a µ we have the relation
h µν = 2ε (µν) + ε (µ ρ ε ν)ρ(37)
where Latin indices on ε have been transformed to Greek ones by contraction withē. Now substituting this formula in the Taylor expansion of the metric in terms of metric fluctuations we find:
Γ EH (e) = Γ EH (ē) +ˆδ Γ EH δg µν 2ε µν + ζˆδ Γ EH δg µν ε µ ρ ε νρ + 1 2ˆˆδ Γ EH δg µν δg ρσ 4 ε µν ε ρσ + . . .(38)
In the third term we have introduced by hand a factor 0 ≤ ζ ≤ 1 that interpolates continuously between the pure metric formalism (ζ = 0, h µν = 2ε (µν) ) and the tetrad formalism (ζ = 1). We see that the part of the action quadratic in ε differs from the one in the metric formalism by terms proportional to the equations of motion. Since in the derivation of the beta functions it is essential to work off shell, we cannot ignore these terms.
The gauge fixing terms for diffeomorphisms and local Lorentz transformations are
Γ GF k [e,ē] = 1 2αˆd d x √ḡḡ µν F µ F ν + 1 2α Lˆd d x √ḡ G ab G ab(39)
For diffeomorphisms we choose the condition
F µ = 1 √ 16πG ∇ νh µν − β 2∇ µh ,(40)
while for the internal O(d) transformation we choose a symmetric vielbein G ab = ε [ab] . We will choose α L = 0 in order to simplify the computation. This correspond to a sharp implementation of the Lorentz gauge fixing, where one can simply set ε [µν] = 0 and suppress the corresponding rows and columns in the Hessian.
Next we perform the TT decomposition on the symmetric part of the vielbein fluctuation
ε (µν) = h T T µν + ∇ µ ξ ν + ∇ ν ξ µ + ∇ µ ∇ ν σ − 1 d g µν ∇ 2 σ + 1 d g µν h 2 ,(41)
and the associated field redefinitions
ξ µ → −∇ 2 − R d ξ µ and σ → −∇ 2 − R (d−1) (−∇ 2 ) σ.
With these definitions, and dropping bars from the background quantities for notational simplicity, the quadratic part of the action (36) is
Γ (2) h T h T = 1 2ˆ√ g h T µν −∇ 2 + d (d − 3) + 4 d (d − 1) − ζ d − 2 2d R − (2 − ζ)Λ h T µν (42) Γ (2) ξξ = 1 αˆ√ g ξ ν −∇ 2 + α (d − 2) − 1 d − ζα d − 2 2d R − α (2 − ζ) Λ ξ ν (43) Γ (2) σσ = (d − 1) 2d 2 (d − 1) − α (d − 2) αd × (44) √ g σ −∇ 2 + (d − 2)(2 − ζ)α − 4 2 (d − 1) − α (d − 2) R − αd (2 − ζ) 2 (d − 1) − α (d − 2) Λ σ Γ (2) hh = d − 2 4d 2 d 2 − 3d + 2 α − (dβ − 2) 2 dα(d − 2) × (45) √ g h − ∇ 2 + α(d − 2) (d − 4 + ζ) 2 (d 2 − 3d + 2) α − (dβ − 2) 2 R − 2 dα (d − 2 + ζ) 2 (d 2 − 3d + 2) α − (dβ − 2) 2 Λ h Γ (2) hσ = −(d − 2)α + dβ − 2 dα (d − 1) dˆ√ g h −∇ 2 − R (d − 1) ∇ 2 σ(46)
We notice that for β = 1 d (α(d − 2) + 2) we can get rid of the mixed term. In the rest of the paper we will work in this "diagonal" gauge. In this case the trace part reduces to
Γ (2) hh = − 1 2 d − 2 2dˆ√ g h −∇ 2 + d − 4 + ζ 2(d − 1) − α(d − 2) R − 2d 2(d − 1) − α(d − 2) 1 + ζ d − 2 Λ h
After decomposing the diffeomorphism ghost in its transverse and longitudinal parts, and absorbing √ −∇ 2 in the latter, the ghost action is the sum of
Γ (2) c T ν c T µ =ˆ√gc T ν ∇ 2 + R d c T µ Γ (2) cc =ˆ√gc ∇ 2 + 2 R d c(47)
The Lorentz ghosts do not propagate and following standard perturbative procedure one could neglect them entirely, but we will follow [34] and introduce a cutoff for them too. The corresponding contribution to the FRGE is computed, together with other traces, in Appendix II.
B. Beta functions
The FRGE can now be calculated by introducing a cutoff separately in each spin sector. (This is known as a "type b" cutoff.) Using the same heat kernel methods that we have employed in section III A, the expression for ∂ t Γ k can be expanded to linear order in R. Comparing the terms of order zero and one in R in the FRGE yields:
∂ t 2Λ 16πG = k d 16π (A 1 + ηA 2 ) (48) −∂ t 1 16πG = k d−2 16π (B 1 + ηB 2 )(49)
where η = −∂ t G/G and A i , B i are in general polynomials inΛ = k −2 Λ. From here one can find the beta functions of the dimensionless parametersG = k d−2 G andΛ:
∂ tG =(d − 2)G + B 1G 2 1 +GB 2 (50) ∂ tΛ = − 2Λ +G A 1 + 2B 1Λ +G (A 1 B 2 − A 2 B 1 ) 2 1 + B 2G(51)
In the following two sections we will give explicit results using specific cutoffs.
For numerical results in d = 4 we will always use the optimized cutoff (3). For a discussion of the dependence on the shape of the function R k (z) we refer to [34]. We will instead concentrate on the differences between cutoffs of type I vs. II and type a vs b. For the type Ia case we refer again to the extensive discussion in [34], whose results we have checked independently. We will report in detail the results for the cases Ib and IIb, and highlight the differences with the case Ia.
C. Type Ib cutoff
First we choose as reference operator, in each spin sector, the "bare" Laplacian −∇ 2 . The cutoff is a function R k (−∇ 2 ). This is called a cutoff of type Ib. The calculation of the coefficients A 1 , A 2 , B 1 and B 2 for arbitrary dimension and cutoff shape is described in Appendix A. Here we just report the result in d = 4 and for the cutoff profile (3):
A 1 = 1 2π 5 1 − (2 − ζ)Λ + 3 1 − α(2 − ζ)Λ + 1 1 − 2α(2−ζ) 3−αΛ + 1 1 − 2(2+ζ) 3−αΛ − 8 + A L (μ)(52)A 2 = 1 12π 5 1 − (2 − ζ)Λ + 3 1 − α(2 − ζ)Λ + 1 1 − 2α(2−ζ) 3−αΛ + 1 1 − 2(2+ζ) 3−αΛ(53)B 1 = 1 24π − 20 1 − (2 − ζ)Λ − 5 (8 − 3ζ) (1 − (2 − ζ)Λ) 2 + 6 1 − α(2 − ζ)Λ + 9(1 − α(2 − ζ)) 4(1 − α(2 − ζ)Λ) 2 + 4 1 − 2α(2−ζ)Λ 3−α + 12 + 6α(ζ − 2) (3 − α) 1 − 2α(2−ζ)Λ 3−α 2 + 4 1 − 2(ζ+2)Λ 3−α − 6ζ (3 − α) 1 − 2(ζ+2)Λ 3−α 2 − 50 + B L (μ)(54)B 2 = 1 144π − 30 1 − (2 − ζ)Λ − 5 (8 − 3ζ) (1 − (2 − ζ)Λ) 2 + 9 1 − α(2 − ζ)Λ + 9(1 − α(2 − ζ) (1 − α(2 − ζ)Λ) 2 (55) + 6 1 − 2α(2−ζ)Λ 3−α + 12 + 6α(ζ − 2) (3 − α) 1 − 2α(2−ζ)Λ 3−α 2 + 6 1 − 2(ζ+2)Λ 3−α − 6ζ (3 − α) 1 − 2(ζ+2)Λ 3−α 2 .
The result is still quite general: it depends on the parameter ζ, which allows us to interpolate continuously between the purely metric formulation (ζ = 0) and the purely tetrad formulation I agrees with the third row in table II in [5].
Whereas with a type Ia cutoff the fixed point becomes UV-repulsive, and a limit cycle develops, forμ sufficiently large, with the type Ib cutoff it remains UV-attractive for arbitrarily largeμ.
This is a nice feature of this scheme, because it means that the fixed point can also be found if one adopts the perturbative prescription of neglecting the Lorentz ghosts entirely. However, the results in the tetrad formalism match most closely those of the metric formalism whenμ is chosen to be a bit larger than one. As with the type Ia cutoff, forμ smaller than a critical valueμ c , the critical exponents become real. We findμ c ≈ 0.705 for α = 0 andμ c ≈ 0.715 for α = 1.
The dependence of the universal quantities on the gauge parameters is illustrated in Fig. 1. The slow decrease of the real part of the critical exponent for 0 < α < 2 is in agreement with earlier calculations in the metric formalism (see e.g. fig. 9 in [7]). The results of different schemes seem to converge for α → 0 which, we recall, is believed to give the physically most reliable picture. On the other hand when α is greater than some value of order 2 the fixed point becomes repulsive, reproducing the behavior that had been observed in [34] for largeμ. It is tempting to conjecture that also in the cutoff scheme Ia used in [34] the fixed point would have the usual properties, even for largeμ, if one could choose α closer to zero. The strongμ-dependence that had been observed there may be due to a particularly strong α-dependence. Altogether it appears that with a type Ib cutoff, the tetrad formalism leads to results that are qualitatively similar to those of the metric formalism, and that the correspondence is best when 0 < α < 1 and the Lorentz ghosts are turned on, with a parameterμ that is a little larger than one.
D. Type IIb cutoff
We call type IIb a cutoff imposed separately on each spin-component of the graviton and taking as reference operator the Laplace-type operator that appears in the corresponding Hessian, including the curvature terms, but not the term proportional to the cosmological constant. The rationale for excluding the cosmological constant term is that the cosmological constant is a running coupling and if one included it in the reference operator, it would not remain fixed in the course of the flow.
Here we choose a reference operator that remains fixed along the flow. 4 As we have already seen in the case of the fermions, the type II cutoffs tend to give somewhat simpler final formulae than the corresponding type I cutoffs, because to leading order one always finds traces of the function ∂ t R k /P k and it is not necessary to expand the denominators.
The coefficients A 1 , A 2 are the same as with a type Ib cutoff and are given in (52,53). The coefficients B 1 and B 2 for arbitrary dimension and cutoff shape are given in Appendix III. In d = 4
and for the cutoff profile (3), they become found in [34] with the type Ia cutoff scheme (which we have independently verified). In particular we find that the FP has properties close to the standard ones of the metric formulation also when the Lorentz ghosts are neglected. Figure 2 gives the gauge-dependence of the universal quantities ΛG and ϑ. Note that the real part of the scaling exponent ϑ is particularly stable in this scheme, for 0 < α < 1.
B 1 = 1 12π − 10(10 − 3ζ) 1 − (2 − ζ)Λ + 6(4 − 3α(2 − ζ)) 1 − α(2 − ζ)Λ + 2 − 6ζ 3−α 1 − 2 2+ζ 3−αΛ + 14 − 6 4−αζ 3−α 1 − 2α 2−ζ 3−αΛ − 40 + B L(56)B 2 = 1 48π − 5(10 − 3ζ) 1 − (2 − ζ)Λ − 3(4 − 3α(2 − ζ) 1 − α(2 − ζ)Λ − 2 − 6ζ 3−α ) 1 − 2 2+ζ 3−αΛ − 14 − 6 4−αζ 3−α 1 − 2α 2−ζ 3−αΛ(57)
E. Type IIa cutoff
For completeness we mention here also the results for the cutoff of type IIa, which had been discussed first in section IVC of [5]. In this scheme only the gauge α = 1 is easily computable.
In this gauge it is enough to split the metric fluctuation into its trace and tracefree parts to write the Hessian of the Einstein-Hilbert action as two minimal Laplace-type operators. The cutoff is then defined as a function of these operators, including the curvature terms but excluding the cosmological constant term. This prescription leads to particularly simple expressions.
The coefficients A 1 , A 2 are the same as with a other cutoff types considered here and are given in (52,53). The coefficients B 1 and B 2 in d = 4 and for the cutoff profile (3) are simply
B 1 = 1 12π 2 − 3ζ 1 − (2 + ζ)Λ − 27(2 − ζ) 1 − (2 − ζ)Λ − 40 + B L (58) B 2 = 1 48π 2 − 3ζ 1 − (2 + ζ)Λ − 27(2 − ζ) 1 − (2 − ζ)Λ(59)
These expressions coincide with (56,57) when one puts α = 1 there. As a consequence, all properties of the flow are the same and we will not discuss this case further.
It is nevertheless interesting to understand the reason for this coincidence, which is not restricted to d = 4 and is also independent of the shape of the function R k . 5 For the sake of simplicity we shall discuss here only the case ζ = 0, but the result is general. Since in all cases the trace field h is treated separately, and its contribution is the same for type a and b cutoffs, it is enough to consider the tracefree part of the graviton, namely the components h T T µν , ξ µ and σ. In the gauge α = 1, the Hessian in the tracefree subsector is a minimal second order operator of the form
− ∇ 2 + C T R − 2Λ (60) where C T = d(d−3)+4 d(d−1)
. When one uses a cutoff of type IIa (no further decomposition) the contribution of this sector to the r.h.s. of the FRGE is
n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) − 2Λ (61)
where λ n are the eigenvalues of the operator O = −∇ 2 + C T R. One can divide these eigenvalues into three classes, depending on the spin of the corresponding eigenfunction. Upon using the TT decomposition (41) one finds that the eigenvalues of O on fields of type h T T µν , ∇ µ ξ ν − ∇ ν ξ µ and ∇ µ ∇ ν σ − 1 d ∇ 2 σ are equal to the eigenvalues of the operators in square brackets in (42,43,44), stripped of the Λ terms. We denote these operators
O T T = −∇ 2 + C T R, O ξ = −∇ 2 + C ξ R and
O σ = −∇ 2 + C σ R and the corresponding eigenvalues λ T T n , λ ξ n and λ σ n . So, the trace (63) is equal to
n ∂ t R(λ T T n ) − ηR k (λ T T n ) P k (λ T T n ) − 2Λ + n ∂ t R(λ ξ n ) − ηR k (λ ξ n ) P k (λ ξ n ) − 2Λ + n ∂ t R(λ σ n ) − ηR k (λ σ n ) P k (λ σ n ) − 2Λ(62)
Since for α = 1 the coefficients of Λ in (42,43,44) are all the same and equal to −2, this is recognized as the contribution of the fields h T T µν , ξ µ and σ to the r.h.s. of the FRGE when one uses a cutoff of type IIb. By a similar reasoning one also concludes that the ghost contribution is the same in the IIa and IIb schemes.
Things do not work in the same way for type I cutoffs, i.e. when the cutoff is a function of −∇ 2 .
For a type Ia cutoff the contribution of the tracefree sector to the r.h.s. of the FRGE is
n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) + C T R − 2Λ ,(63)
where λ n now denote the eigenvalues of −∇ 2 . This can be expanded as
T T n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) + C T R − 2Λ + ξ n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) + C T R − 2Λ + σ n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) + C T R − 2Λ ,(64)
where T T , ξ and σ denote the sum over eigenvalues of −∇ 2 on h T T µν , ∇ µ ξ ν − ∇ ν ξ µ and ∇ µ ∇ ν σ − 1 d ∇ 2 σ respectively. On the other hand for a type Ib cutoff the same contribution is
T T n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) + C T R − 2Λ + ξ n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) + C ξ R − 2Λ + n ∂ t R(λ n ) − ηR k (λ n ) P k (λ n ) + C σ R − 2Λ(65)
One clearly sees that the two traces are different.
V. DISCUSSION
The implementation of the FRGE in the presence of fermions and gravity presents some subtleties that had not been fully appreciated until recently. The sign ambiguity of the fermionic contribution to the running of Newton's constant had been known for a while, but it was regarded as just another aspect of the scheme dependence that is intrinsic to applications of the FRGE, albeit a particularly worrying one. Although a completely satisfactory understanding can only come from a treatment of physical observables, we have argued here that the correct treatment of fermion fields, when the Dirac operator is squared, is to use a cutoff that depends on −∇ 2 + R 4 (type II cutoff). There also follows from our discussion that using a cutoff that depends on −∇ 2 (type I cutoff) may give physically incorrect results. Unfortunately, several earlier studies (in particular [21,22]) have used this scheme, so some of those results may have to be revised. We plan to return to this point in a future publication.
Another issue is the use of tetrad vs. metric degrees of freedom. We have extended the analysis of tetrad gravity initiated in [34] by using a different cutoff (type Ib and IIb vs. type Ia) which allowed us to keep the diffeomorphism gauge parameter α arbitrary. We have found that the results for the running couplings using the tetrad formalism are qualitatively similar to those of the metric formalism, with some quirks. The following points should be noted.
(i) For ζ = 0 one recovers the standard metric formalism. In the case of the type Ib cutoff the results agree with the ones obtained earlier in the literature [4,5,7]. The type IIb cutoff with generic α had never been used before and the results obtained here are new. We have shown that for α = 1 they coincide with the ones found in [5] for the type IIa cutoff. For other values of α they differ only marginally from those obtained with other cutoff types and confirm the stability of the fixed point in the metric formalism. Type II cutoffs have the attractive feature that they lead to somewhat more compact expressions for the beta functions.
ii) The case ζ = 1 corresponds to the tetrad formalism. In this case a new ambiguity appears in the definition of the ghost sector: it can be parameterized by a mass µ that appears in the mixing between diffeomorphism and Lorentz ghosts, or by the corresponding dimensionless parameter µ = µ/k. This parameter is a priori arbitrary, but in order not to introduce additional mass scales into the problem it is natural to assume that it is of order one. On the other hand we recall that in perturbation theory and in the chosen gauge the Lorentz ghosts are neglected, since they do not propagate. This corresponds to takingμ = ∞.
iii) If one uses a type Ia cutoff and completely neglects the Lorentz ghosts, there is no attractive FP for positive G [34]. Instead, one finds a UV-repulsive fixed point surrounded by a UV-attractive limit cycle 6 . This is not the case when one imposes the cutoff separately on each spin component, as we did here. We find that both with type Ib and IIb cutoffs an attractive FP with complex critical exponents is present also when Lorentz ghosts are neglected, both for α = 1 and α = 0. This is reassuring because in the metric formalism the fixed point can be found even using the perturbative one loop beta functions. iv) If the contribution of Lorentz ghosts is added, as advocated in [34], its effect is weighted by the parameterμ: it is strong for smallμ and weak for largeμ. Since the ghosts are fermions, the fixed point is shifted towards negative Λ for decreasingμ. In addition, they have a systematic effect on the critical exponents: the modulus of the imaginary part decreases with decreasingμ and there is a critical valueμ c under which the critical exponents become real. In the gauge α = 1,μ c = 0.715 for a type Ib cutoff,μ c = 0.748 for a type IIb cutoff, andμ c ≈ 0.8 for a type Ia cutoff. Similar behaviour is observed also for α = 0. On the other hand, the real part of the critical exponent decreases with increasingμ. With the type Ia cutoff this effect is most dramatic: the real part becomes negative forμ ≈ 1.4, and this marks the appearance of the limit cycle. With the type II cutoffs discussed here the effect is much weaker and the fixed point becomes only slightly less attractive even in the limitμ → ∞, both for α = 1 and α = 0.
v) The closest match between the tetrad and metric results is typically obtained if one chooses some value ofμ that is not too far from one. This is always the case for the product ΛG, and in most cases also for the critical exponent. For the Ia cutoff this value was found to be approximately 1.2. For the Ib and IIb cutoffs it is somewhat larger, depending on the quantities one is comparing.
An exception occurs for the critical exponent in the case of a cutoff of type IIb in the gauge α = 0, for which the best match occurs forμ → ∞.
vi) Using type b cutoffs (i.e. decomposing the fields into irreducible components) has the advantage that one can keep the diffeomorphism gauge parameter α arbitrary. The gauge dependence of the critical exponents is similar to what had been observed previously in the metric formalism, as long as α is not too much greater than one. In particular the real part of the critical exponent tends to decrease as α increases, for small values of α. In the limit α → 0 theμ-dependence becomes very weak and the critical exponents nicely converges towards a common value. On the other hand for α somewhat larger than one the fixed point either becomes complex or repulsive. This is the behaviour that had been observed in [34] with the type Ia cutoff. This suggests that if we were able to compute the beta functions for this cutoff type with α = 1 we would find that also with largeμ the fixed point is present and has the standard properties for α sufficiently close to zero.
vii) Altogether the results are very similar to those found in the metric formalism, except for the dependence on the new parameterμ, which is particularly strong for the type Ia cutoff. As argued in [34], one can probably attribute the increased sensitivity of the results to the fact that in the tetrad formalism one has to deal with more unphysical degrees of freedom. The type Ia cutoff seems to be particularly sensitive to the off-shell, unphysical features of the flow. Type b cutoffs, where each spin component is treated separately, are less sensitive.
viii) All of the preceding discussion is in the context of a "single metric truncation", i.e. one assumes that the VEV of the fluctuation field is zero. As discussed in [33], the application of the FRGE to gravity requires that the effective average action be considered in general a function of two variables. We plan to consider these more general truncations in a future paper.
In conclusion, let us comment on the use of tetrad vs. metric variables. Since fermions exists in nature, it may seem in principle inevitable that gravity has to be described by tetrads. This would complicate the theory significantly. Every diffeomorphism-invariant functional of the metric can be viewed as a diffeomorphism and local Lorentz-invariant functional of the tetrad, but in the bimetric formalism there are many functionals of two tetrads that cannot be viewed as functionals of two metrics. Therefore, as already noted in [34], the tetrad theory space is much bigger than the metric theory space.
The necessity of using tetrads should, however, not be taken as a foregone conclusion. First of all, it is possible that the fermions occurring in nature are Kähler fermions. This would completely remove the argument for the use of tetrads, even in principle. Whether this is the case or not is a difficult issue that should be answered experimentally. For the time being one might just consider the use of Kähler fermions as a computational trick. Second, even if we stick to spinor matter, by squaring the Dirac operator and using a type II cutoff one can calculate many quantum effects due to fermions without ever having to use tetrad fields. The additional complications due to the Lorentz gauge fixing and the increased sensitivity to gauge and scheme choice advise against the use of the tetrad formalism, as a matter of practical convenience.
Acknowledgements R.P. thanks U. Harst, M. Reuter and F. Saueressig for discussions and hospitality at the University of Mainz during the preparation of this paper.
VI. APPENDIX I: DIRAC SPECTRAL SUMS
To compute the sum (28) we can use the Euler-Maclaurin formula
n i=0 F (i) =ˆn 0 F (x) dx−B 1 ·(F (n)+F (0))+ p k=1 B 2k (2k)! F (2k−1) (n) − F (2k−1) (0) +remainder (66)
where B i are the Bernoulli numbers. After collecting a volume contribution, the only terms we need to compute are the 0-th and 1-st power of R. Note that only the integral depends on R, and therefore, in dimensions d > 2 for the terms that we are interested in it is enough to compute the integral.
Since the volume of the d−sphere is
V (d) = 2 d! Γ d 2 + 1 (4π) d/2 (d−1)d R d/2
we only have to isolate the terms in the integral proportional to R −d/2 and R 1−d/2
2 2 [ d 2 ]ˆk d(d−1) R − d 2 0 dn n + d − 1 n = 2 2 [ d 2 ] (d − 1)!ˆk d(d−1) R − d 2 0 dn (n + d − 1) · · · (n + 1) (67) changing variables n → n ′ − d/2 2 2 [ d 2 ] (d − 1)!ˆk d(d−1) R d 2 dn ′ n ′ + d 2 − 1 · · · n ′ − d 2 − 1(68)
the terms we are interested in come from the integral of the two highest order power of n ′
n ′ + d 2 − 1 · · · n ′ − d 2 − 1 = n ′d−1 − n ′d−3 [ d−1 2 ] k=1 d 2 − k 2 + · · ·(69)
we can rewrite the sum
[ d−1 2 ] k=1 d 2 − k 2 = 1 24 d (d − 1) (d − 2)
, and perform the integral
Tr ∂ t R k P k = 2 2 [ d 2 ] (d − 1)! 1 d k d(d − 1) R d −2 2 [ d 2 ] (d − 1)! 1 d − 2 k d(d − 1) R d−2 1 24 d (d − 1) (d−2)+· · ·(70)
Collecting the volume of S d we obtain
dΓ k dt = −Tr ∂ t R k P k = − 1 Γ d 2 + 1 2 [ d 2 ] (4π) d 2 V (d) k d − d 24 k d−2 R + O R 2(71)
VII. APPENDIX II: TYPE IB CALCULATION
We report here the detailed computation of the A and B coefficients of (48,49) for a Type Ib cutoff. The FRGE is the sum of traces over the irreducible components of the metric fluctuation defined in (41). They give:
1 2 Tr (2) ∂ t R k + ηR k P k + d(d−3)+4 d(d−1) − ζ d−2 2d R − (2 − ζ)Λ = (72) 1 2 1 (4π) d/2ˆd x √ g (d − 2)(d + 1) 2 Q d 2 ∂ t R k + ηR k P k − (2 − ζ)Λ + (d − 5)(d + 1)(d + 2) 12(d − 1) Q d 2 −1 ∂ t R k + ηR k P k − (2 − ζ)Λ R − d (d − 3) + 4 d (d − 1) − ζ d − 2 2d (d − 2)(d + 1) 2 Q d 2 ∂ t R k + ηR k (P k − (2 − ζ)Λ) 2 R 1 2 Tr ′ (1) ∂ t R k + ηR k P k + α(d−2)−1 d − ζα d−2 2d R − α (2 − ζ) Λ = (73) 1 2 1 (4π) d/2ˆd x √ g (d − 1)Q d 2 ∂ t R k + ηR k P k − α (2 − ζ) Λ + (d − 3)(d + 2) 6d Q d 2 −1 ∂ t R k + ηR k P k − α (2 − ζ) Λ R − α (d − 2) − 1 d − ζα d − 2 2d (d − 1)Q d 2 ∂ t R k + ηR k (P k − α (2 − ζ) Λ) 2 R 1 2 Tr ′′ (0) ∂ t R k + ηR k P k + α(d−2)(2−ζ)−4 2(d−1)−α(d−2) R − αd(2−ζ) 2(d−1)−α(d−2) Λ = (74) 1 2 1 (4π) d/2ˆd x √ g Q d 2 ∂ t R k + ηR k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ + 1 6 Q d 2 −1 ∂ t R k + ηR k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ R − α(d − 2)(2 − ζ) − 4 2(d − 1) − α(d − 2) Q d 2 ∂ t R k + ηR k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ 2 R 1 2 Tr (0) ∂ t R k + ηR k P k + d−4+ζ 2(d−1)−α(d−2) R − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ = (75) 1 2 1 (4π) d/2ˆd x √ g Q d 2 ∂ t R k + ηR k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ + 1 6 Q d 2 −1 ∂ t R k + ηR k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ R − d − 4 + ζ 2(d − 1) − α(d − 2) Q d 2 ∂ t R k + ηR k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ 2 R
Here a prime or a double prime indicate that the first or the first and the second eigenvalues have to be omitted from the trace (because ξ µ and σ obey to some differential constraints, for more details see for example [5]). The contribution of the transverse and longitudinal parts of the diffeomorphism ghosts are −Tr (1) ∂
t R k P k − R d = − 1 (4π) d/2ˆd x √ g (d − 1)Q d 2 ∂ t R k P k − d − 1 d Q d 2 ∂ t R k P 2 k R + (d + 2) (d − 3) 6d Q d 2 −1 ∂ t R k P k R ; (76) −Tr (0) ∂ t R k P k − 2R d = − 1 (4π) d/2ˆd x √ g Q d 2 ∂ t R k P k − 2 d Q d 2 ∂ t R k P 2 k R + 1 6 Q d 2 −1 ∂ t R k P k R .(77)
Collecting the coefficients of´√g and −´√gR we extract the A and B coefficients:
A 1 = 1 2 16π (4π) d/2 (d − 2)(d + 1) 2Q d 2 ∂ t R k P k − (2 − ζ)Λ + (d − 1)Q d 2 ∂ t R k P k − α(2 − ζ)Λ +Q d 2 ∂ t R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ +Q d 2 ∂ t R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ − 2dQ d 2 ∂ t R k P k (78) A 2 = 1 2 16π (4π) d/2 (d − 2)(d + 1) 2Q d 2 R k P k − (2 − ζ)Λ + (d − 1)Q d 2 R k P k − α(2 − ζ)Λ +Q d 2 R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ +Q d 2 R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ (79) B 1 = 1 2 16π (4π) d/2 (d − 5)(d + 1)(d + 2) 12(d − 1)Q d 2 −1 ∂ t R k P k − (2 − ζ)Λ − d (d − 3) + 4 d (d − 1) − ζ d − 2 2d (d − 2)(d + 1) 2Q d 2 ∂ t R k (P k − (2 − ζ)Λ) 2 + (d − 3)(d + 2) 6dQ d 2 −1 ∂ t R k P k − α(2 − ζ)Λ − α (d − 2) − 1 d − ζα d − 2 2d (d − 1)Q d 2 ∂ t R k (P k − α(2 − ζ)Λ) 2 + 1 6Q d 2 −1 ∂ t R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ − α(d − 2)(2 − ζ) − 4 2(d − 1) − α(d − 2)Q d 2 ∂ t R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ 2 + 1 6Q d 2 −1 ∂ t R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ − d − 4 + ζ 2(d − 1) − α(d − 2)Q d 2 ∂ t R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ 2 − d 2 − 6 3dQ d 2 −1 ∂ t R k P k − 2 d + 1 dQ d 2 ∂ t R k P 2 k (80) B 2 = 1 2 16π (4π) d/2 (d − 5)(d + 1)(d + 2) 12(d − 1)Q d 2 −1 R k P k − (2 − ζ)Λ − d (d − 3) + 4 d (d − 1) − ζ d − 2 2d (d − 2)(d + 1) 2Q d 2 R k (P k − (2 − ζ)Λ) 2 + (d − 3)(d + 2) 6dQ d 2 −1 R k P k − α(2 − ζ)Λ − α (d − 2) − 1 d − ζα d − 2 2d (d − 1)Q d 2 R k (P k − α(2 − ζ)Λ) 2 + 1 6Q d 2 −1 R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ − α(d − 2)(2 − ζ) − 4 2(d − 1) − α(d − 2)Q d 2 R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ 2 + 1 6Q d 2 −1 R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ − d − 4 + ζ 2(d − 1) − α(d − 2)Q d 2 R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ 2 (81)
Here we have defined the dimensionless versions of the Q-functionals:
Q d 2 = k −d Q d 2 andQ d 2 −1 = k 2−d Q d 2 −1 .∂ t R k R k + 2µ 2 √ ζ = − 1 (4π) d/2 d(d − 1) 2ˆd x √ g Q d 2 ∂ t R k R k + 2µ 2 √ ζ + 1 6 Q d 2 −1 ∂ t R k R k + 2µ 2 √ ζ R (82)
Here we have introduced the arbitrary mass parameter µ (denotedμ in [34]). In particular note that in the limit µ → ∞ the ghost contribution vanishes and one recovers the standard perturbative result where the Lorentz ghosts are neglected. Let A L and B L be the contribution of the Lorentz ghosts to the coefficients A 1 and B 1 , defined in (48,49). From the above we read off
A L = − 16π (4π) d/2 d(d − 1) 2Q d 2 ∂ t R k R k + 2µ 2 √ ζ ; B L = − 16π (4π) d/2 d(d − 1) 12Q d 2 −1 ∂ t R k R k + 2µ 2 √ ζ .(83)
Note the appearance of R k instead of P k in the denominators. In general the Q functionals Q n ∂tR k R k +2µ 2 / √ ζ can be computed explicitly, with cutoff (3), in terms of hypergeometric functions.
For the calculations in four dimensions we only need the following
Q 1 ∂ t R k R k + 2µ 2 √ ζ = Log 1 + √ ζ 2μ 2 ;
(84)
Q 2 ∂ t R k R k + 2µ 2 √ ζ = −1 + 1 + 2μ 2 √ ζ Log 1 + √ ζ 2μ 2 .
(85)
VIII. APPENDIX III: TYPE IIB CALCULATION
We report here the A and B coefficients of (48,49) for a Type IIb cutoff. The contributions of the irreducible components of the metric fluctuation to the r.h.s. of the FRGE are
1 2 Tr (2) ∂ t R k + ηR k P k − (2 − ζ)Λ = 1 2 1 (4π) d/2ˆd x √ g (d − 2)(d + 1) 2 Q d 2 ∂ t R k + ηR k P k − (2 − ζ)Λ (86) − (d + 1) 12d 5d 2 − 22d + 48 − 3ζ (d − 2) 2 Q d 2 −1 ∂ t R k + ηR k P k − (2 − ζ)Λ R 1 2 Tr ′ (1) ∂ t R k + ηR k P k − α (2 − ζ) Λ = 1 2 1 (4π) d/2ˆd x √ g (d − 1)Q d 2 ∂ t R k + ηR k P k − α (2 − ζ) Λ (87) − d 2 + 5d − 12 + 3α(ζ − 2) (d − 1) (d − 2) 6d Q d 2 −1 ∂ t R k + ηR k P k − α (2 − ζ) Λ R 1 2 Tr ′′ (0) ∂ t R k + ηR k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ = 1 2 1 (4π) d/2ˆd x √ g Q d 2 ∂ t R k + ηR k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ (88)+ 14 + (2 − 7α − 3ζα)(d − 2) 6(2(d − 1) − α(d − 2)) Q d 2 −1 ∂ t R k + ηR k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ R 1 2 Tr (0) ∂ t R k + ηR k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ = 1 2 1 (4π) d/2ˆd x √ g Q d 2 ∂ t R k + ηR k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ (89)+ 22 − 4d + (2 − d)α − 6ζ 6(2(d − 1) − α(d − 2)) Q d 2 −1 ∂ t R k + ηR k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ R
The contribution of the diffeomorphism ghosts is −Tr (1) ∂ t R k P k = − 1
(4π) d/2ˆd x √ g (d − 1)Q d 2 ∂ t R k P k + d 2 + 5d − 12 6d R Q d 2 −1 ∂ t R k P k (90) −Tr (0) ∂ t R k P k = − 1 (4π) d/2ˆd x √ g Q d 2 ∂ t R k P k + d + 12 6d R Q d 2 −1 ∂ t R k P k .(91)
The contribution of Lorentz ghosts is the same as in the Type Ib case. The coefficients A 1 and A 2 are the same as in equations (78, 79), whereas
B 1 = 1 2 16π (4π) d/2 − (d + 1) 12d 5d 2 − 22d + 48 − 3ζ (d − 2) 2 Q d 2 −1 ∂ t R k P k − (2 − ζ)Λ (92) + d 2 + 5d − 12 + 3α(ζ − 2)(d − 1)(d − 2) 6dQ d 2 −1 ∂ t R k P k − α (2 − ζ) Λ + 22 − 4d + (2 − d)α − 6ζ 6(2(d − 1) − α(d − 2))Q d 2 −1 ∂ t R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ + 14 + (2 − 7α − 3ζα)(d − 2) 6(2(d − 1) − α(d − 2))Q d 2 −1 ∂ t R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ − d + 6 3Q d 2 −1 ∂ t R k P k B 2 = 1 2 16π (4π) d/2 − (d + 1) 12d 5d 2 − 22d + 48 − 3ζ (d − 2) 2 Q d 2 −1 R k P k − (2 − ζ)Λ + d 2 + 5d − 12 + 3α(ζ − 2)(d − 1)(d − 2) 6dQ d 2 −1 R k P k − α (2 − ζ) Λ + 22 − 4d + (2 − d)α − 6ζ 6(2(d − 1) − α(d − 2))Q d 2 −1 R k P k − 2d(1+ ζ d−2 ) 2(d−1)−α(d−2) Λ + 14 + (2 − 7α − 3ζα)(d − 2) 6(2(d − 1) − α(d − 2))Q d 2 −1 R k P k − αd(2−ζ) 2(d−1)−α(d−2) Λ (93)
(ζ = 1 )
1, on the arbitrary gauge parameter α, which allows us to test the gauge dependence of the results, and on the parameter µ that allows us to weigh differently the contribution of the Lorentz ghosts.Let us now describe the main features of these flows. Both in the metric and in the tetrad formulations, a UV-attractive fixed point is found for all values of µ and for α not too large. Its location and the corresponding critical exponents ϑ (which are defined as minus the eigenvalues of the stability matrix) are given in table I in the metric (ζ = 0) or tetrad (ζ = 1) formalism, in the gauges α = 0, α = 1 and with two different values of the dimensionless Lorentz ghost parameter µ = µ/k:μ = ∞, andμ = 1.2. The former corresponds to neglecting the Lorentz ghosts entirely and the latter is chosen to ease comparison with[34], who found that this value gives results that are closest to the metric formalism in the gauge and scheme they use. Note that the case ζ = 0, α = 1 correspond to the purely metric flow with type Ib cutoff, which had already been discussed previously in the literature. Indeed the second row in table
FIG. 1 :
1The non trivial fixed point in the type Ib cutoff in metric (ζ = 0) and tetrad (ζ = 1) formalism, in the gauges α = 0 and α = 1 and with different weights of the Lorentz ghosts. Recall Re(ϑ)>0 implies that the fixed point is UV attractive. Plot of universal quantities as functions of the gauge parameter α for type Ib cutoff. In the left panel the product Λ * G * , in the right panel the real part of the critical exponent.
TABLE
II: The non trivial fixed point in the type IIb cutoff in metric (ζ = 0) and tetrad (ζ = 1) formalism, in the gauges α = 0 and α = 1 and with different weights of the Lorentz ghosts.
FIG. 2 :
2II gives the UV-attractive fixed point and the corresponding critical exponents in the metric (ζ = 0) or tetrad (ζ = 1) formalism, in the gauges α = 0, 1, and with two different values of the Lorentz ghost parameter,μ = ∞, andμ = 1.2. The results are qualitatively similar to the ones obtained with the type Ib cutoff. This is in line with all the results obtained previously in the Einstein-Hilbert truncation. The non trivial FP exists and has complex critical exponents for all values ofμ greater than a critical valueμ c , which is approximately equal to 0.766 for α = 0 and 0.748 for α = 1. For smallμ the FP moves towards negative values ofΛ. For largeμ the fixed point remains UV attractive, in contrast to the result Plot of universal quantities as functions of the gauge parameter α for type IIb cutoff. In the left panel the product Λ * G * , in the right panel the real part of the critical exponent.
Finally let us consider the contribution of Lorentz ghosts. They do not propagate and therefore are usually neglected in the evaluation of the effective action in perturbation theory. Nevertheless if, following [34], we impose a cutoff on their determinant they contribute to the r.h.s. of the FRGE an amount − Tr
Table
This was noted a while ago by Calmet et al. while they were working on[32]. R.P. wishes to thank D. Reeb for correspondence on this point.
This is a formal relation because the functional Γ k is ill-defined, but one can write a corresponding relation for ∂tΓ k , with the same result.
Note that in the limit k → 0 the function R D k is non zero only for z < R/4. Since the smallest eigenvalue of the Dirac operator is R/3, it remains true that lim k→0 Γ k = Γ.
Cutoffs that depend on the full Hessian, including the terms proportional to the cosmological constant, were called of type III in[5], where they have been applied to the Einstein-Hilbert truncation.
The agreement between cutoffs IIa and IIb for α = 1 had been noticed before in[38].
This is different from the behaviour of the limit cycle discussed in[39].
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| [] |
[
"Slowly rotating super-compact Schwarzschild stars",
"Slowly rotating super-compact Schwarzschild stars"
] | [
"Camilo Posada \nDepartment of Physics and Astronomy\nUniversity of South Carolina\n712 Main Street29208ColumbiaSCUSA\n"
] | [
"Department of Physics and Astronomy\nUniversity of South Carolina\n712 Main Street29208ColumbiaSCUSA"
] | [
"MNRAS"
] | The Schwarzschild interior solution, or 'Schwarzschild star', which describes a spherically symmetric homogeneous mass with constant energy density, shows a divergence in pressure when the radius of the star reaches the Schwarzschild-Buchdahl bound. Recently Mazur and Mottola showed that this divergence is integrable through the Komar formula, inducing non-isotropic transverse stresses on a surface of some radius R 0 . When this radius approaches the Schwarzschild radius R s = 2M, the interior solution becomes one of negative pressure evoking a de Sitter spacetime. This gravitational condensate star, or gravastar, is an alternative solution to the idea of a black hole as the ultimate state of gravitational collapse. Using Hartle's model to calculate equilibrium configurations of slowly rotating masses, we report results of surface and integral properties for a Schwarzschild star in the very little studied region R s < R < (9/8)R s . We found that in the gravastar limit, the angular velocity of the fluid relative to the local inertial frame tends to zero, indicating rigid rotation. Remarkably, the normalized moment of inertia I/MR 2 and the mass quadrupole moment Q approach to the corresponding values for the Kerr metric to second order in Ω. These results provide a solution to the problem of the source of a slowly rotating Kerr black hole. | 10.1093/mnras/stx523 | [
"https://arxiv.org/pdf/1612.05290v3.pdf"
] | 119,413,139 | 1612.05290 | 1b76457af7d9311f0a2eaef67d1a8d88f0c9da99 |
Slowly rotating super-compact Schwarzschild stars
2016
Camilo Posada
Department of Physics and Astronomy
University of South Carolina
712 Main Street29208ColumbiaSCUSA
Slowly rotating super-compact Schwarzschild stars
MNRAS
0002016Preprint 20 February 2017 Compiled using MNRAS L A T E X style file v3.0Rotating GravastarInterior Schwarzschild solutionHartle's equations
The Schwarzschild interior solution, or 'Schwarzschild star', which describes a spherically symmetric homogeneous mass with constant energy density, shows a divergence in pressure when the radius of the star reaches the Schwarzschild-Buchdahl bound. Recently Mazur and Mottola showed that this divergence is integrable through the Komar formula, inducing non-isotropic transverse stresses on a surface of some radius R 0 . When this radius approaches the Schwarzschild radius R s = 2M, the interior solution becomes one of negative pressure evoking a de Sitter spacetime. This gravitational condensate star, or gravastar, is an alternative solution to the idea of a black hole as the ultimate state of gravitational collapse. Using Hartle's model to calculate equilibrium configurations of slowly rotating masses, we report results of surface and integral properties for a Schwarzschild star in the very little studied region R s < R < (9/8)R s . We found that in the gravastar limit, the angular velocity of the fluid relative to the local inertial frame tends to zero, indicating rigid rotation. Remarkably, the normalized moment of inertia I/MR 2 and the mass quadrupole moment Q approach to the corresponding values for the Kerr metric to second order in Ω. These results provide a solution to the problem of the source of a slowly rotating Kerr black hole.
INTRODUCTION
In classical General Relativity it is commonly accepted that the final state of complete gravitational collapse is a singular state called a 'black hole' (Misner et al. 1973). This object is characterized by a central space-time singularity at r = 0 surrounded by an event horizon, a null hypersurface located at the Schwarzschild radius R s = 2M which separates points connected to infinity by a timelike curve from those that are not. These features are a consequence of the exact solution to Einstein's field equations in the vacuum found a century ago by Schwarzschild (1916a) which describes the exterior space-time geometry of a spherically symmetric mass.
Despite the vast amount of literature [see e.g. (Wald 2001;Page 2005)], the physical reality of black holes has not only generated some skepticism (Abramowicz et al. 2002;Frolov 2014;Hawking 2014), but also has raised some paradoxical issues which have not been consistently solved. A pivotal one is the nonconservation of information by quantum matter falling into a black hole (Hawking 1976). Additionally in the original Hawking (1975) radiation derivation, the backward-in-time propagated mode seems to experience a large blueshift with energies larger than the Planck energy. It is expected that these highly 'blue-shifted' photons would leave a non-negligible 'imprint' on the spacetime geometry, making the approximation of fixed classical geometry background untenable (Mazur & Mottola 2001;Mottola 2011). Moreover, the arbi-Contact e-mail: [email protected] trarily large values of entropy at T H → 0 associated with the black hole as predicted by the Bekenstein (1974) formula in the classical limit → 0 produces serious challenges to the foundations of quantum mechanics. It is believed that the resolution of these issues will be achieved in the framework of a consistent theory of quantum gravity. We still do not posses such a theory, therefore it is valuable to investigate alternative solutions to the aforementioned problems.
Alternatives have been introduced to alleviate some of the black hole paradoxes (Stephens et al. 1994;Chapline 2001;Berezin 2003). In particular, we concentrate in the gravastar (vacuum condensate gravitational star) model proposed by Mazur & Mottola (2001, 2004a. A gravastar is basically the aftermath of the gravitational collapse of a star to the Schwarzschild radius R s , leaving a final state characterized by a modified de Sitter interior region with negative pressure and a finite surface tension. The exterior spacetime remains the standard spherically symmetric Schwarzschild exterior solution. In connection with the gravastar, Mazur & Mottola (2015) considered the constant density Schwarzschild interior solution, or 'Schwarzschild star'. It is well known that this interior solution shows a divergence in pressure when the radius of the star R = (9/4)M (Schwarzschild 1916b;Buchdahl 1959). The existence of this limit in addition to the homogeneous mass approximation, considered 'unrealistic', have been assumed to be sufficient reasons to exclude the Schwarzschild star from further investigation (Wald 1984). This complete disregard of the interior solution has left the interesting region R s < R < (9/8)R s unexplored.
In a bold approach Mazur & Mottola (2015) analyzed this 'forbidden' region and found that the divergence in the central pressure is integrable through the Komar formula (Komar 1959), producing a δ-function of transverse stresses implying a relaxation of the isotropic fluid condition on a surface of some radius R 0 . In the limit when R → R + s from above and R 0 → R − s from below, the interior region suffers a phase transition (starting at the centre) becoming one of negative pressure evoking a de Sitter spacetime. This nonsingular 'bubble' of dark energy which is matched to an external vacuum Schwarzschild spacetime, has zero entropy and temperature, so providing a consistent picture of a gravitational Einstein-Bose condensate, or gravastar, as the final state of complete gravitational collapse.
The relevance of gravastars follows from the fact that their physical properties and behaviour are governed by classical general relativity. Gravastars are being recognized as a very challenging alternative to black holes. Moreover, calculations of observational consequences of a merger of either two black holes or two gravastars in the context of gravitational waves, e.g. ringdowns (Chirenti & Rezzolla 2016) and afterglows (Abramowicz et al. 2016), may provide methods to discriminate between black holes and gravastars.
Some authors (Lobo 2006) have investigated possible sources for the interior of the gravastar, and the electrically charged case was considered by Horvat et al. (2009). The issue of stability against axial perturbations was studied by Chirenti & Rezzolla (2007). They found that gravastars are stable under axial perturbations, moreover, the quasi-normal modes of rotating gravastars deviate from those associated with a black hole. They concluded that this might help to distinguish observationally between a gravastar and a black hole. Radial and axial gravitational perturbations on thin-shell gravastars were studied by Pani et al. (2009Pani et al. ( , 2010.
Perturbation theory can also be applied to the study of equilibrium configurations of slowly rotating compact objects. In a seminal paper Hartle (1967) provided the relativistic structure equations to determine the equilibrium configurations of slowly rotating stars to second order in the angular velocity. In Hartle's model the interior of the star is composed of a fluid characterized by a general one-parameter equation of state (EOS). This configuration is matched to a stationary and axially symmetric exterior region across a timelike hypersurface. Chandrasekhar & Miller (1974) studied slowly rotating homogeneous masses characterized by a constant energy density, using Hartle's framework. For this configuration they solved numerically the structure equations for several values of the parameter R/R s where R is the radius of the star and R s is the Schwarzschild radius. Using these solutions Chandrasekhar & Miller calculated integral and surface equilibrium properties such as moment of inertia and mass quadrupole moment up to the Buchdahl bound. They found that the ellipticity of the star, considering constant mass and angular momentum, manifests a prominent maximum at the radius R/R s ∼ 2.4. One result of particular interest is that for a star with the 'minimum possible' radius R = (9/8)R s , the quadrupole mass moment is very close to the value associated with the Kerr metric to second order in the angular velocity.
Motivated by the aforementioned works, in this paper we report results of surface and integral properties of a slowly rotating Schwarzschild star in the unstudied region R s < R < (9/8)R s . These results extend those presented by Chandrasekhar & Miller (1974) which where considered up to the Buchdahl radius. We show that for a Schwarzschild star in the gravastar limit when R → R s , surface properties like moment of inertia, angular veloc-ity and mass quadrupole moment approach the corresponding Kerr metric values. These remarkable results provide a long sought solution to the problem of the source of rotation of a slowly rotating Kerr black hole. Throughout the paper, we use geometrized units where c = G = 1.
SCHWARZSCHILD STAR AND GRAVASTAR LIMIT
The Schwarzschild interior solution corresponding to a spherical configuration with constant energy density, is discussed in standard general relativity textbooks (see e.g. Wald 1984;Plebański & Krasiński 2006). The starting point is a spherically symmetric spacetime in Schwarzschild coordinates ds 2 = −e 2ν(r) dt 2 + e 2λ(r) dr 2 + r 2 dθ 2 + sin 2 θdφ 2 .
(1)
The stress-energy tensor for a spherically symmetric fluid is given by
T µ ν = − 0 0 0 0 p 0 0 0 0 p ⊥ 0 0 0 0 p ⊥ (2)
where , p and p ⊥ correspond to the energy density, radial pressure and tangential pressure respectively, which are functions of r only. The energy density and the pressure p are related through a given one-parameter EOS. The relevant components of the Einstein
equation G µ ν = 8πT µ ν are e −2λ 2r dλ dr − 1 + e 2λ = 8π r 2 ,(3)e −2λ 2r dν dr + 1 − e 2λ = 8πpr 2 ,(4)
jointly with the energy-momentum conservation relation
∇ µ T µ r = d p dr + ( + p) dν dr + 2 r (p − p ⊥ ) = 0,(5)
which corresponds to the relativistic generalization of the hydrostatic equilibrium equation or Tolman-Oppenheimer-Volkoff (TOV) equation. It is conventional to introduce
h(r) ≡ e −2λ(r) = 1 − 2m(r) r ,(6)
where the function m(r) is associated with the mass within a radius r and is given by the Misner-Sharp relation (Misner et al. 1973)
m(r) = r 0 dr4πr 2 .(7)
In terms of (7), (4) becomes
dν dr = m(r) + 4πpr 3 r [r − 2m(r)] ,(8)
which in the non-relativistic limit reduces to Poisson's equation dν/dr = m(r)/r 2 , where ν(r) is associated to the Newtonian gravitational potential. The interior solution, or Schwarzschild star, is matched at the boundary r = R to the asymptotically flat vacuum exterior Schwarzschild solution
e 2ν(r)ext = h ext (r) = 1 − 2M r , r R(9)
where M is the total mass and R is the radius of the star. The functions p(r) and m(r) satisfy the boundary conditions p(R) = 0, m(R) = M. The interior region is modeled as an incompressible and isotropic fluid p = p ⊥ with
=¯ = 3M 4πR 3 = const.(10)
It is useful to define (Mazur & Mottola 2015)
≡ 3H 2 8π , H 2 = R s R 3 ,(11)
where R s = 2M is the Schwarzschild radius. In terms of (11), equations (6) and (7) can be solved to obtain
m(r) = 4π 3¯ r 3 = M r R 3 , h(r) = 1 − H 2 r 2 , 0 r R.(12)
From (5) the pressure takes the form
p(r) =¯ √ 1 − H 2 r 2 − √ 1 − H 2 R 2 3 √ 1 − H 2 R 2 − √ 1 − H 2 r 2 .(13)
The metric function e 2ν(r) for r < R can be computed to give
f (r) ≡ e 2ν(r) = 1 4 3 √ 1 − H 2 R 2 − √ 1 − H 2 r 2 2 0.(14)
Across the boundary of the configuration r = R, this function must match the exterior metric (9). The continuity of f (r) guarantees that an observer crossing the boundary will not notice any discontinuity of time measurements. Notice that (14) is regular except at some radius R 0 where the denominator in (13)
D ≡ 3 √ 1 − H 2 R 2 − √ 1 − H 2 r 2 ,(15)
vanishes in the range 0 < r < R. Remarkably, it can be seen from (13) and (14) that the pressure goes to infinity at the same point where f (r) = 0. This singular radius can be found directly from (15) to be
R 0 = 3R 1 − 8 9 R R s ,(16)
which is imaginary for R/R s > 9/8. In this regime, p(r) and f (r) are positive. Moreover, when R → (9/8)R + s from above, (16) shows that R 0 approaches the real axis at R 0 = 0 and a divergence of the pressure appears jointly with f (r) → 0. This limit value R B = (9/8)R s , or Schwarzschild-Buchdahl bound (Schwarzschild 1916b;Buchdahl 1959), fixes the maximum possible mass for a star with given radius R. At this radius R B general relativity predicts that the star cannot remain in static equilibrium. Furthermore, once the star reaches this critical point, its gravitational collapse is inevitable.
Due to the manifestation of this divergence in pressure at the Buchdahl bound, in addition to the incompressible fluid approximation being considered artificial (Narlikar 2010), the Schwarzschild star solution below the Buchdahl bound has been ignored in the literature. Mazur & Mottola (2015) analyzed the region R s < R < (9/8)R s and they found that the zero of D given by (16) moves outwards from the origin to finite values 0 < R 0 < R (see Fig.1). Then there emerges a region where p(r) < 0, f (r) > 0 and D < 0, covering the range 0 r < R 0 . As the radius of the star keeps approaching the Schwarzschild radius from above R → R + s , R 0 → R − s from below, where R 0 is given by (16) which corresponds to the radius of the sphere where the pressure is divergent and f (R 0 ) = 0 (see Fig. 3). Analysis of (13) shows that the new interior region becomes one of constant negative pressure p = − for r < R = R 0 = R s (see Fig.2). In this limit, the interior metric function (14) becomes which is a patch of modified de Sitter spacetime. The exterior region r > R s remains the vacuum spherically symmetric Schwarzschild geometry (9), with an infinitesimal thin shell discontinuity at R s = 2M where there is a jump in pressure and the zeroes f = h = 0 of the interior modified de Sitter and exterior vacuum Schwarzschild spacetimes match. Although there is no event horizon, R = R s is a null hypersurface. However in contrast to the black hole, the gravastar does not require the interior region r < R s to be trapped. Moreover, the gravastar solution with interior p = − , has no entropy and zero temperature, validating its condensate state nature. Mazur & Mottola (2015) showed that the divergence in pressure at R 0 can be integrated through the Komar formula. However this integration demands that p ⊥ p, therefore breaking the isotropic fluid condition. Below the Buchdahl bound R < 9 8 R s the relation for pressure is
f (r) = 1 4 1 − H 2 r 2 = 1 4 h(r) = 1 4 1 − r 2 R 2 s , H = 1 R s(17)p/ R/R s =1.124 R/R s =1.10 R/R s =1.01 R/R s =1.0018π f h r 2 (p ⊥ − p) = 8π 3 R 3 0 δ(r − R 0 )(18)
indicating an anisotropy in pressure at r = R 0 . It is this δ-function integrable through the Komar formula, together with the relaxation Figure 3a, shows that R 0 emerges at the center of the star where the fluid suffers a phase transition. The region 0 r < R 0 with negative pressure starts approaching R s from below, meanwhile the radius of the star R approaches R s from above (see Fig. 3b). In the gravastar limit when R → R + s and R 0 → R − s , the whole interior region is one of constant negative pressure given by a static patch of modified de Sitter spacetime with a finite surface tension (see Fig. 3c). The exterior spacetime is described by the standard vacuum Schwarzschild metric. Instead of an event horizon, an infinitely thin shell forms at the Schwarzschild radius R s where there is a jump in pressure and the zeroes f = h = 0 of the interior modified de Sitter and exterior Schwarzschild solutions match. of the isotropic perfect fluid condition at r = R 0 that provide a physical interpretation of the Schwarzschild star. The surface energy is found to be
(a) R/R s = 1.124 (b) R/R s = 1.10 (c) R/R s = 1.0125E s = 8π 3 R 3 0 = 2M R 0 R 3 ,(19)
together with the discontinuity on the surface gravities
δκ ≡ κ + − κ − = R s R 0 R 3(20)
provides a surface tension at r = R 0 given by
τ s = MR 0 4πR 3 = ∆κ 8πG .(21)
In contrast to a black hole, (21) corresponds to a physical surface tension (localized in an infinitesimal thin shell at r = R s ) provided by a surface energy and positive transverse pressure as determined by the Komar formula. Notice that the Schwarzschild star solution provides an instructive limiting case of a stellar model in general relativity. Furthermore in the limit when R → R + s and R 0 → R − s , the Schwarzschild star turns out to be the non-singular gravitational condensate star or gravastar, with a surface tension at R s , proposed by Mazur & Mottola (2001, 2004a as an alternative to black holes as the final state of gravitational collapse.
In the next section we will review the perturbative method developed by Hartle (1967), to study equilibrium configurations of slowly rotating relativistic stars. We will apply these methods to the slowly rotating Schwarzschild star in the region R s < R < (9/8)R s .
HARTLE'S STRUCTURE EQUATIONS
In this section the equations of structure for slowly rotating masses derived by Hartle (1967) are summarized. The Hartle model is based on the consideration of an initially static configuration set in slow rotation. In this approximation, fractional changes in pressure, energy density and gravitational field are much less than unity. This condition implies that RΩ << 1 where R is the radius of the star and Ω its angular velocity. The appropriate line element for this situation is 1
ds 2 = −e 2ν 0 [1 + 2h 0 (r) + 2h 2 (r)P 2 (cos θ)] dt 2 + e 2λ 0 1 + e 2λ 0 r [2m 0 (r) + 2m 2 (r)P 2 (cos θ)] dr 2 + r 2 [1 + 2k 2 (r)P 2 (cos θ)] dθ 2 + [dφ − ω(r)dt] 2 sin 2 θ ,(22)
where P 2 (cos θ) is the Legendre polynomial of order 2; (h 0 , h 2 , m 0 , m 2 , k 2 ) are quantities of order Ω 2 ; and ω, which is proportional to the angular velocity of the star Ω, is a function of r that describes the dragging of the inertial frames. One can introduce a local Zero-Angular-Momentum-Observer (ZAMO), then the function ω(r) corresponds to the angular velocity of the local ZAMO relative to a distant observer. In the non-rotating case the metric (22) reduces to (1). In the coordinate system of (22), the fluid inside the configuration rotates uniformly with four-velocity components (Hartle & Thorne 1968)
u t = (−g tt − 2Ωg tφ − g φφ Ω 2 ) −1/2 = e −ν 0 1 + 1 2 r 2 sin 2 θ(Ω − ω) 2 e −ν 0 /2 − h 0 − h 2 P 2 (cos θ) , u φ = Ωu t , u r = u θ = 0.(23)
It is conventional to define
≡ Ω − ω,(24)
to be the angular velocity of the fluid as measured by the local ZAMO. The magnitude of the centrifugal force is determined by this quantity which, to first order in Ω, satisfies the equation
d dr r 4 j(r) d dr + 4r 3 d j dr = 0,(25)with j(r) ≡ e −(λ 0 +ν 0 ) .(26)
In the exterior empty region r > R, j(r) = 1 and (25) can be easily integrated to give
(r) = Ω − 2J r 3 ,(27)
where the constant J corresponds to the angular momentum of the star (Hartle 1967). Equation (25) will be integrated outward from the origin with the boundary conditions (0) = c = const., and d /dr = 0. The value of c is chosen arbitrarily. Once the solution on the surface is found, one can determine the angular momentum J and the angular velocity Ω through the formulas
J = 1 6 R 4 d dr r=R , Ω = (R) + 2J R 3 .(28)
The angular momentum is related linearly to Ω through the relation J = IΩ, where I is the relativistic moment of inertia. Additionally, due to the rotation, the star will deform carrying with it changes in pressure and energy density given by (Hartle & Thorne 1968)
p + ( + p) δp 0 + δp 2 P 2 (cos θ) ≡ p + ∆P (29) + ( + p)(d /d p) δp 0 + δp 2 P 2 (cos θ) ≡ + ∆(30)
where δp 0 and δp 2 are functions of r, proportional to Ω 2 , which correspond to perturbations in pressure and energy density. The spherical deformations can be studied from the l = 0 equations with the condition that the central energy density is the same as in the static configuration. The relevant expressions are
dm 0 dr = 4πr 2 ( + p) d d p δp 0 + 1 12 r 4 j 2 d dr 2 − 1 3 r 3 2 d j 2 dr ,(31)dh 0 dr = − d dr δp 0 + 1 3 d dr r 2 e −2ν 0 2 .(32)
These equations will be integrated outward from the origin, where the boundary conditions h 0 (0) = m 0 (0) = 0 must be satisfied. In this approximation, the slowly rotating configuration will have the same central pressure as in the static case. In the exterior region, (31) and (32) can be integrated explicitly to give
m 0 = δM − J 2 r 3 ,(33)h 0 = − δM r − 2M 0 + J 2 r 3 (r − 2M 0 ) ,(34)
where M 0 corresponds to the total mass of the star and δM is an integration constant which is associated to the change in mass due to the rotation. This constant δM can be found by matching the interior and exterior solutions for h 0 at the boundary r = R.
Recently Reina & Vera (2015) revisited Hartle's framework within the context of the modern theory of perturbed matchings. They found that the perturbative functions at first and second order are continuous across the boundary of the configuration except when the energy density is discontinuous there. In this particular case, the discontinuity in the radial function m 0 at the boundary is proportional to the energy density there. Furthermore, Reina and Vera showed that the manifestation of this jump in the perturbative function m 0 induces a modification to the original change of mass (33), which is given by (Reina 2016)
δM = δM H + δM C = m 0 (R) + J 2 R 3 + 4π R 3 M 0 (R − 2M 0 ) (R)δp 0 (R).(35)
where δM H corresponds to the original change of mass (33) and δM C is the correction term. Reina (2016) points out that this correction is relevant in configurations where the energy density does not vanish at the boundary, for instance for homogeneous masses. We shall consider the corrected expression (35) in our computations.
The quadrupole deformations of the star are computed from the integrals of the l = 2 equations which give
dv 2 dr = −2 dν 0 dr h 2 + 1 r + dν 0 dr 1 6 r 4 j 2 d dr 2 − 1 3 r 3 2 d j 2 dr , (36) dh 2 dr = − 2v 2 r [r − 2m(r)] (dν 0 /dr) + −2 dν 0 dr + r 2 [r − 2m(r)] (dν 0 /dr) 8π( + p) − 4m(r) r 3 h 2 + 1 6 r dν 0 dr − 1 2 [r − 2m(r)] (dν 0 /dr) r 3 j 2 d dr 2 − 1 3 r dν 0 dr + 1 2 [r − 2m(r)] (dν 0 /dr) r 2 2 d j 2 dr ,(37)m 2 = [r − 2m(r)] −h 2 − 1 3 r 3 d j 2 dr 2 + 1 6 r 4 j 2 d dr 2 (38)
where v 2 = h 2 + k 2 . These equations will be integrated outward from the center, where h 2 = v 2 = 0. Outside the star, (36) and (37) are integrated analytically
h 2 (r) = J 2 1 M 0 r 3 + 1 r 4 + KQ 2 2 r M 0 − 1 ,(39)v 2 (r) = − J 2 r 4 + K 2M 0 [r(r − 2M 0 )] 1/2 Q 1 2 r M 0 − 1 ,(40)
where K is an integration constant which can be found from the continuity of the functions h 2 , v 2 at the boundary, and Q m n are the associated Legendre functions of the second kind. The constant K in (39) and (40) is related to the mass quadrupole moment of the star, as measured at infinity, through the relation (Hartle & Thorne 1968
) Q = J 2 M 0 + 8 5 K M 3 0 .(41)
Due to the rotation, the surface of the Schwarzschild star will be deformed from the spherical shape it has in the static case, preserving the same central density. The modified radius of the slowly rotating isobaric surface is given by r(θ) = r 0 + ξ 0 (r 0 ) + ξ 2 (r 0 )P 2 (cos θ),
where r 0 corresponds to the radius of the spherical surface in the non-rotating case, and the deformations ξ 0 and ξ 2 satisfy
δp 0 = − 1 + p d p dr 0 ξ 0 (r 0 ), δp 2 = − 1 + p d p dr 0 ξ 2 (r 0 ).(43)
The ellipticity of the isobaric surfaces can be computed from (Thorne 1971;Miller 1977)
ε(r) = − 3 2r ξ 2 (r) + r(v 2 − h 2 ) ,(44)
which is correct to order Ω 2 .
STRUCTURE EQUATIONS FOR THE SCHWARZSCHILD STAR
In a seminal paper Chandrasekhar & Miller (1974) studied slowly rotating homogeneous masses using Hartle's framework. In that paper, the structure equations were integrated numerically and surface properties were computed for several values of the parameter R/R s , where R is the radius of the star and R s is the Schwarzschild radius. This procedure represented a quasi-stationary contraction of the star (Miller 1977). We will refer to the relevant equations of that paper prefixed by the letters CM, for example (CM.1) indicates equation (1) of (Chandrasekhar & Miller 1974).
The geometry of the Schwarzschild star was discussed in 2. In order to facilitate the numerical integrations, it is useful to introduce the coordinates (CM.35)
r = (1 − y 2 ) 1/2 , y 2 1 = 1 − R 2 α 2 = 1 − H 2 R 2 .(45)
Here r is being measured in the unit α = 1/H, where H is given by (11). In terms of these variables the Schwarzschild star solution (6), (12), (13) and (14) takes the form
e λ 0 = 1 y , e ν 0 = 1 2 |3y 1 − y|, p = y − y 1 3y 1 − y ,(46)j = 2y |3y 1 − y| , 2m(r) r = 1 − y 2 .(47)
Notice that, in contrast to (CM.34), here we consider the modulus of (3y 1 − y) in (46) in harmony with the fact that the metric element e 2ν 0 in (14) is a perfect square making it always a positive quantity. Notice that the function e ν 0 is negative when 3y 1 < y which occurs in the region below the Buchdahl bound. Therefore, in order to investigate the region R s < R < (9/8)R s it is important to specify the modulus condition. This specification was taken into account in the code to compute the numerical solutions. To facilitate computations, we define the quantity
k = |3y 1 − 1|,(48)
which is always positive as it is required for the analysis of the region R s < R < (9/8)R s . It is also advantageous to introduce the coordinate (CM.39)
x ≡ 1 − y = 1 − 1 − r α 2 1/2 ,(49)
where x covers the range (0, 1 − y 1 ]. In terms of (49), equation (25) reads
x 2k + (2 − k)x − x 2 d 2 dx 2 + 5k + (3 − 5k)x − 4x 2 d dx − 4(k + 1) = 0,(50)
Near the origin (x ≈ 0) satisfies
= 1 + 4(k + 1) 5k x c(51)
where is measured in the unit c , its value at the centre, which is arbitrary. The field equations (31), (32) take the forms (53) which, near the origin, satisfy
dm 0 dx = α 3 (1 − x) [x(2 − x)] 3/2 (k + x) 2 1 3 x(2 − x) d dx 2 + 8(k + 1) 3(k + x) 2 , (52) d dx δP 0 = − (k + 1) (1 − x)(k + x) δP 0 − 2 + (k + 1)(1 − x) − 3(1 − x) 2 (k + x)(1 − x) 2 [x(2 − x)] 3/2 α −1 m 0 + 8x(2 − x) 3(k + x) 2 d dx + [x(2 − x)] 2 3(1 − x)(k + x) 2 d dx 2 − 8 3 1 − (k + 1)(1 − x) (k + x) 3 2 ,m 0 = 32 √ 2(k + 1) 15k 3 x 5/2 α 3 2 c ,(54)δP 0 = 8x 3k 2 α 2 2 c .(55)
The equations for h 2 , v 2 as functions of x now take the form
dv 2 dx = − 2h 2 k + x + α 2 2 [x(2 − x)] 2 3(k + x) 3 1 + (k + 1)(1 − x) − 2(1 − x) 2 × d dx 2 + 4(k + 1) x(2 − x)(k + x) 2 , (56) dh 2 dx = (1 − x) 2 + (k + 1)(1 − x) − 2 x(2 − x)(k + x) h 2 − 2(k + x) [x(2 − x)] 2 v 2 + α 2 3 2[x(2 − x)] 2 − (k + x) 2 x(2 − x) (k + x) 3 d dx 2 + 4α 2 3 (k + 1) 2x 2 (2 − x) 2 + (k + x) 2 2 (k + x) 4 . (57)
The functions δP 0 , δP 2 , h 2 , k 2 and v 2 are measured in the unit α 2 ω 2 c and m 0 is measured in the unit α 3 ω 2 c . Solutions to (56) and (57) can be expressed as the superposition of a particular and a complementary solution (CM.61)
h 2 = h (p) 2 + βh (c) 2 , v 2 = v (p) 2 + βv (c) 2 ,(58)
with β being an integration constant. The complementary functions here satisfy the homogeneous forms of equations (56) and (57) dv (c)
2 dx = − 2h c 2 k + x ,(59)dh (c) 2 dx = (1 − x) 2 + (k + 1)(1 − x) − 2 x(2 − x)(k + x) h (c) 2 − 2(k + x) [x(2 − x)] 2 v (c) 2 ,(60)
which have the following behaviours near the origin (CM.64)
h (p) 2 = cx, v (p) 2 = ax 2 ,(61)h (c) 2 = −kBx, v (c) 2 = Bx 2 (62)
where 3k 2 (c − ka) = 8(k + 1) and B is an arbitrary constant. On the other hand the exterior solutions (33),(34),(39) and (40) take the forms 2
m 0 = δM − J 2 r 3 , h 0 = − m 0 r − (1 − y 2 1 ) 3/2 ,(63)h 2 = 2 (1 − y 2 1 ) 3/2 + 1 r J 2 r 3 + KQ 2 2 ,(64)v 2 = − J 2 r 4 + K (1 − y 2 1 ) 3/2 r r − (1 − y 2 1 ) 3/2 1/2 Q 1 2 .(65)
At the boundary of the configuration, the interior equations (50)-(57) must match the exterior solutions (63)-(65). Finally, from (44), the ellipticity of the embedded spheroid at the bounding surface takes the form
ε = 3(1 − x)(k + x) 2x(2 − x) h 2 + 4x(2 − x) 3(k + x) 2 (R) 2 − 3 2 (v 2 − h 2 ),(66)
where ε is being measured in the unit α 2 2 c . It can be observed that the structure equations preserve the same form as considered in Chandrasekhar & Miller (1974). The only significant change corresponds to the modulus condition (48) which, we emphasize, is crucial for the strict analysis of the regime R s < R < (9/8)R s .
RESULTS
In this section we present the results of integrations of the Hartle structure equations and derived surface properties for a slowly rotating Schwarzschild star with negative pressure, in the unstudied regime R s < R < (9/8)R s . We followed similar methods to those used by Chandrasekhar & Miller (1974), in particular, units were chosen such that derived quantities are dimensionless. Similarly we constructed several configurations under quasi-stationary contraction, by varying the radius of the star through the parameter R/R s . For convenience we introduce the 'Schwarzschild deviation parameter' ζ ≡ (R − R s )/R s .
The integrations were performed numerically using the Runge-Kutta-Fehlberg (RKF) adaptive method in Python 3.4 (Kiusalaas 2010;Newman 2012). It is well known that adaptive methods are usually more convenient, than the standard fourthorder Runge-Kutta, when the function to be integrated changes rapidly near some point. In such situations, setting a constant step of integration h on the whole integration range might not be appropriate and we are forced to adjust the step to maintain the truncation error within prescribed limits. In our particular case, we found that the (RKF) method provided a fast, reliable and stable technique to integrate the structure equations near and below the Buchdahl radius.
In our routine the condition (48) was specified which is key to analyze the region below the Buchdahl radius. We have checked our code by reproducing the computations found in (Chandrasekhar & Miller 1974) for R 1.125R s . We found agreement up to the fourth decimal place in some cases (see Table 1). In contrast to the CM paper, we are measuring the mass quadrupole moment Q in units
∆Q Q ≡ Q − Q kerr Q kerr(67)
which corresponds to the relative deviation of the quadrupole moment from the Kerr metric value (Q kerr = J 2 /M 0 ). One important observation by Chandrasekhar & Miller (1974) is that the structure equations can actually be integrated at the Buchdahl radius, i.e., y 1 = 1/3 and k = 0, by considering the expansions (CM.69)-(CM.71). We reproduced these results (see Table 1) with very good agreement, except for the mass quadrupole moment where we found Q = 2.02311 (in units of J 2 /R s ). The fact that the structure equations are integrable at R = (9/8)R s might be seen as an early indication that the region below this limit is potentially interesting. This hypothesis has been investigated in this paper inspired by the results of Mazur & Mottola (2015). In the following, we present some plots of our results and further analysis.
In Fig. 4 we plot the surface value of the fluid angular velocity relative to the local ZAMO (R), versus the 'compactness parameter' R/R s , above the Buchdahl bound. Notice that (R) reaches a maximum near to R = 1.4R s and then approaches zero in the Newtonian limit R → ∞. These results are in very good agreement with (Chandrasekhar & Miller 1974). Figure 5 shows the angular velocity (R) relative to the local ZAMO as a function of R/R s in the regime R s < R < (9/8)R s . It is observed that in the limit when the radius of the star R approaches the Schwarzschild radius R s , tends to zero. In connection with this, Fig. 7 shows the angular velocity Ω, relative to a distant observer, as a function of R/R s in the region R s < R < (9/8)R s . Notice the increase in the angular velocity reaching a maximum at R/R s ≈ 1.03, and the subsequent decrease tending to the value 2 (in units of J/R 3 s ) when R → R + s . In this limit, → 0 and the angular velocity Ω = ω is a constant indicating a rigidly rotating body with no differential surface rotation (Marsh 2014).
It can be shown that the value Ω = ω = 2 (in units of J/R 3 s ) for the angular velocity of the super-compact Schwarzschild star in the gravastar limit (ζ ∼ 10 −14 ) is consistent with that of the Kerr black hole limit. It is well known that in the Kerr spacetime, a radially falling test particle with zero angular momentum acquires an Table 1. Integral and surface properties of a slowly rotating 'Schwarzschild star' for several values of the deviation parameter ζ ≡ R−Rs Rs , where R is the radius of the star and R s = 2M 0 is the Schwarzschild radius. We use geometrized units (c = G = 1). The angular velocity relative to the local ZAMO (R) = (Ω − ω)| r=R is given in units of J/R 3 s . The moment of inertia I is in the unit R 3 s . The ratio δM H /M denotes the original Hartle's fractional change in mass, as given by (33), measured in units of J 2 /R 4 s . The ratio δM/M corresponds to the amended fractional change of mass as given by (35). The ratio ∆Q/Q defined in (67) corresponds to the relative deviation of the mass quadrupole moment from that of the Kerr metric. We measure the quadrupole moment Q in units of J 2 /M 0 so the Kerr factorq = QM 0 /J 2 corresponds to the unity. The ellipticity ε is measured in units of J 2 /M 4 . All the quantities are computed at the surface of the configuration. The digit in parenthesis following each entry corresponds to the power of ten by which the entry is multiplied. Ω/(J/R 3 s ) Figure 7. The angular velocity Ω (in units of J/R 3 s ) plotted as a function of the compactness parameter R/R s in the region R s < R < 1.125R s . angular velocity when it approaches the spinning black hole. The angular velocity as measured by a distant ZAMO is given by
ω = dφ dt = 2aM 0 r (r 2 + a 2 ) 2 − ∆(r)a 2 sin 2 θ ,(68)
where a ≡ J/M 0 and ∆(r) ≡ r 2 − 2M 0 r + a 2 . Notice that positive a implies positive ω, therefore the particle will rotate in the spinning direction of the black hole. This is the so-called dragging effect in Kerr geometry. At the 'event horizon' (68) satisfies
ω bh = a 2M 0 r + ,(69)
where r + = M 0 + (M 2 0 − a 2 ) 1/2 . Equation (69) corresponds to the angular velocity of the Kerr black hole. In the slowly rotating approximation (ξ ≡ a/M 0 << 1) a straightforward calculation from (69) shows that
Ω = ω bh ≈ a 4M 2 0 + O(ξ 2 ) = 2 J R 3 s + O(ξ 2 )(70)
which is consistent with our numerical results for Ω in the gravastar limit ζ ∼ 10 −14 (see Table 1). From Fig. 8 it can be observed that the normalized moment of inertia approaches the value 0.8 at the Buchdahl bound. For large values of R we notice that I N tends to the value 0.4 which is the well known moment of inertia of a sphere in consistency with the Newtonian limit. Figure 9 shows the normalized moment of inertia versus the factor R/R s in the region R s < R < (9/8)R s . Notice how I N approaches 1 systematically when R → R + s . This is in remarkable agreement with the Kerr value in the slowly rotating approximation which is given by
I = J ω bh ≈ 4M 3 0 + O(ξ 2 ).(71)
In Fig. 10 the original (33) and amended (35) change of mass are plotted as a function of the compactness parameter R/R s , for R (9/8)R s . Notice that δM/M 0 reaches a maximum at R/R s ∼ 2.81 and then decreases in the Newtonian limit. Notice that this decrease is slower than the one obtained from the original (33). These results are in very good agreement with (Reina 2016). Figure 11 shows the original and amended fractional change in mass, as a function of the parameter R/R s , for R s < R < (9/8)R s . Figure 9. The normalized moment of inertia I/M 0 R 2 plotted as a function of the compactness parameter R/R s in the region R s < R < 1.125R s . Notice the approach to 1 of the moment of inertia (normalized) in the gravastar limit R → R + s .
I/M 0 R 2
Notice the systematic decrease of δM H /M 0 and δM/M 0 for R below the Buchdahl bound, and their subsequent approach to the value 2 in the gravastar limit R → R + s . Notice that in this limit the additional term for the change of mass in (35) is negligible. Figures 12 and 13 show the ellipticity ε(R) of the bounding surface, as defined in (66), plotted as a function of the compactness parameter R/R s above the Buchdahl bound, for a Schwarzschild star with fixed total mass and angular momentum. Notice the nonmonotonic behaviour reaching a maximum at R/R s ∼ 2.4. In principle, under adiabatic contraction, the star would be flattened as expected due to the rotation. However, a peculiar behaviour occurs below the maximum R/R s ∼ 2.4 where ε(R) decreases indicating that the configuration becomes more spherical. These results are in very good agreement with (Chandrasekhar & Miller 1974;Miller 1977;Abramowicz & Miller 1990).
The reversal of ellipticity for a slowly rotating relativistic star, has been the subject of lively discussion in the literature (Chandrasekhar & Miller 1974;Miller 1977;Abramowicz & Miller 1990;Abramowicz 1993;Chakrabarti & Khanna 1992). In connection with this, in Fig. 14 we plot the ellipticity as a function of R/R s in the region 1.10 < R/R s < 2. Notice the continuous decrease of the ellipticity as the compactness increases. In the limit when R → R + s the eccentricity tends to the value 0.375 (in units of J 2 /M 4 ). Notice that in the relativistic case, in contrast to the Newtonian approximation, the eccentricity is a much more complicated quantity which depends not only of the centrifugal effects (as determined by ) but also of the behaviour of the functions h 2 (y 1 ) and v 2 (y 1 ). Finally in Fig. 15 the Kerr factorq = QM 0 /J 2 (Thorne 1971;Miller 1977) is plotted as a function of the parameter R/R s . Notice the approach to the Kerr valueq = 1 when R → R + s . A remarkable and unprecedented result is that relative deviations of the mass quadrupole moment as given by (67) are of the order of 10 −15 in the gravastar limit R → R + s with ζ ∼ 10 −14 . Therefore, we conclude that the exterior metric to a slowly rotating super-compact Schwarzschild star (with negative pressure) in the gravastar limit R → R + s , agrees to an accuracy of 1 part in 10 15 with the Kerr metric. MNRAS L A T E X guide for authors 11 Figure 15. The Kerr factorq = QM 0 /J 2 plotted as a function of the compactness parameter R/R s in the regime R s < R < (9/8)R s . Notice the approach to the Kerr valueq = 1 in the gravastar limit R → R + s . In Table 1 it is shown that the relative deviation ∆Q/Q is of the order of 10 −15 for ζ ∼ 10 −14 .
CONCLUDING REMARKS
Motivated by recent investigations of Mazur & Mottola (2015) and the methods introduced by Hartle (1967) and Chandrasekhar & Miller (1974) in the study of slowly rotating relativistic masses, we have presented in this paper results for integral and surface properties of a slowly rotating super-compact Schwarzschild star in the unstudied regime R s < R < (9/8)R s . We found that the angular velocity relative to the local ZAMO tends to zero in the gravastar limit R → R + s . This result indicates that the supercompact Schwarzschild star rotates rigidly with no differential surface rotation. Furthermore the angular velocity Ω of the supercompact Schwarzschild star, in the gravastar limit, is constant and approaches the corresponding Kerr value in the slowly rotating approximation.
Additionally, we found that the normalized moment of inertia I/M 0 R 2 approaches 1 systematically when R → R + s . This result is in agreement with the value corresponding to the slowly rotating Kerr metric. The most remarkable result concerns the mass quadrupole moment Q. We found that for a slowly rotating supercompact Schwarzschild star, in the gravastar limit, the relative deviation factor is ∆Q/Q ∼ 10 −15 . These aforementioned results indicate that the external metric of a slowly rotating super-compact Schwarzschild star in the gravastar limit, agrees with the Kerr metric to the requisite order to one part in 10 15 . These results provide the long-sought solution to the problem of the source of rotation of the slowly rotating Kerr metric.
Figure 1 .
1R 0 as a function of R (in units of R s ).
Figure 2 .
2Pressure (in units of ) as a function of r (in units of the stellar radius R) of the interior Schwarzschild solution for various values of the ratio R/R s below the Buchdahl bound. Notice the approach of the negative interior pressure p → − as R → R + s from above and R 0 → R + s from below.
Figure 3 .
3Pictorial diagram of the Schwarzschild star in the regime R s < R < (9/8)R s , showing the approach of the surface of the star R (Cyan) to the Schwarzschild surface R s (Red). The radius of the star is measured in units of the Schwarzschild radius R s . The surface R 0 (Black) where the pressure diverges (and f = h = 0) is shown at different stages.
Figure 4 .
4The angular velocity = (Ω−ω)| r=R (in units of J/R 3 s ) relative to the local ZAMO, plotted as a function of the compactness parameter R/R s above the Buchdahl-Bondi bound R > 1.125R s . of J 2 /M 0 . It is useful to introduce the quantity(Bradley & Fodor 2009)
Figure 5 .Figure 6 .
56The angular velocity = (Ω−ω)| r=R (in units of J/R 3 s ) relative to the local ZAMO, plotted as a function of the compactness parameter R/R s in the region R s < R < 1.125R s . The angular velocity Ω (in units of J/R 3 s ) relative to an observer at infinity, plotted as a function of the compactness parameter R/R s above the Buchdahl bound.
Figure 8 .
8The normalized moment of inertia I N plotted as a function of the compactness parameter R/R s above the Buchdahl bound.
Figure 10 .
10The original δM H /M 0 and amended δM/M 0 fractional change of mass against the compactness parameter R/R s above the Buchdahl bound.
Figure 11 .
11The original δM H /M 0 and amended δM/M 0 fractional change of mass as a function of the compactness parameter R/R s in the region R s < R < 1.125R s .
Figure 12 .Figure 13 .Figure 14 .
121314The ellipticity of the bounding surface (in units of J 2 /M 4 ) as a function of the compactness parameter R/R s above the Buchdahl bound. The ellipticity of the bounding surface (in units of J 2 /M 4 ) as a function of the compactness parameter R/R s above the Buchdahl limit. The horizontal axis has been plotted with higher resolution to show more detail. The ellipticity of the surface (in units of J 2 /M 4 ) plotted against R/R s in the regime 1.10 < R/R s < 2.
MNRAS 000, 1-11(2016)
The subscript (0) in the metric functions denotes quantities in the static configuration, except for the functions h 0 and m 0 which correspond to the l = 0 term in the harmonic expansion.
There is a misprint in equation (CM.53), equation(65)here. The numerator of the second term to the right should be (1 − y 2 1 ) 3/2 .
This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-11(2016)
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| [] |
[
"The production of unknown neutron-rich isotopes in 238 U+ 238 U collisions at near-barrier energy",
"The production of unknown neutron-rich isotopes in 238 U+ 238 U collisions at near-barrier energy"
] | [
"Kai Zhao \nDepartment of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China\n",
"Zhuxia Li \nDepartment of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China\n",
"Yingxun Zhang \nDepartment of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China\n",
"Ning Wang \nDepartment of Physics\nGuangxi Normal University\n541004GuilinPeople's Republic of China\n",
"Qingfeng Li \nSchool of Science\nHuzhou University\n313000HuzhouPeople's Republic of China\n",
"Caiwan Shen \nSchool of Science\nHuzhou University\n313000HuzhouPeople's Republic of China\n",
"Yongjia Wang \nSchool of Science\nHuzhou University\n313000HuzhouPeople's Republic of China\n",
"Xizhen Wu \nDepartment of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China\n"
] | [
"Department of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China",
"Department of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China",
"Department of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China",
"Department of Physics\nGuangxi Normal University\n541004GuilinPeople's Republic of China",
"School of Science\nHuzhou University\n313000HuzhouPeople's Republic of China",
"School of Science\nHuzhou University\n313000HuzhouPeople's Republic of China",
"School of Science\nHuzhou University\n313000HuzhouPeople's Republic of China",
"Department of Nuclear Physics\nInstitute of Atomic Energy\nP.O. Box 275(10)102413BeijingChina, People's Republic of China"
] | [] | The production cross sections for primary and residual fragments with charge number from Z=70 to 120 produced in the collision of 238 U+ 238 U at 7.0 MeV/nucleon are calculated by the improved quantum molecular dynamics (ImQMD) model incorporated with the statistical evaporation model (HIVAP code). The calculation results predict that about sixty unknown neutron-rich isotopes from element Ra (Z=88) to Db (Z=105) can be produced with the production cross sections above the lower bound of 10 −8 mb in this reaction. And almost all of unknown neutron-rich isotopes are emitted at the laboratory angles θ lab ≤ 60 • . Two cases, i.e. the production of the unknown uranium isotopes with A ≥ 244 and that of rutherfordium with A ≥ 269 are investigated for understanding the production mechanism of unknown neutron-rich isotopes. It is found that for the former case the collision time between two uranium nuclei is shorter and the primary fragments producing the residues have smaller excitation energies of ≤ 30 MeV and the outgoing angles of those residues cover a range of 30 • -60 • . For the later case, the longer collision time is needed for a large number of nucleons being transferred and thus it results in the higher excitation energies and smaller outgoing angles of primary fragments, and eventually results in a very small production cross section for the residues of Rf with A ≥ 269 which have a small interval of outgoing angles of θ lab =40 • -50 • . | 10.1103/physrevc.94.024601 | [
"https://arxiv.org/pdf/1605.07393v1.pdf"
] | 119,300,915 | 1605.07393 | 26ed3e073a1db1700d2952d0260ab2196cb011a9 |
The production of unknown neutron-rich isotopes in 238 U+ 238 U collisions at near-barrier energy
24 May 2016
Kai Zhao
Department of Nuclear Physics
Institute of Atomic Energy
P.O. Box 275(10)102413BeijingChina, People's Republic of China
Zhuxia Li
Department of Nuclear Physics
Institute of Atomic Energy
P.O. Box 275(10)102413BeijingChina, People's Republic of China
Yingxun Zhang
Department of Nuclear Physics
Institute of Atomic Energy
P.O. Box 275(10)102413BeijingChina, People's Republic of China
Ning Wang
Department of Physics
Guangxi Normal University
541004GuilinPeople's Republic of China
Qingfeng Li
School of Science
Huzhou University
313000HuzhouPeople's Republic of China
Caiwan Shen
School of Science
Huzhou University
313000HuzhouPeople's Republic of China
Yongjia Wang
School of Science
Huzhou University
313000HuzhouPeople's Republic of China
Xizhen Wu
Department of Nuclear Physics
Institute of Atomic Energy
P.O. Box 275(10)102413BeijingChina, People's Republic of China
The production of unknown neutron-rich isotopes in 238 U+ 238 U collisions at near-barrier energy
24 May 2016(Dated: May 25, 2016) 1numbers: 2570Hi2570Lm2570-z2790+b * Electronic address: zhaokai@ciaeaccn † Electronic address: lizwux@ciaeaccn 2
The production cross sections for primary and residual fragments with charge number from Z=70 to 120 produced in the collision of 238 U+ 238 U at 7.0 MeV/nucleon are calculated by the improved quantum molecular dynamics (ImQMD) model incorporated with the statistical evaporation model (HIVAP code). The calculation results predict that about sixty unknown neutron-rich isotopes from element Ra (Z=88) to Db (Z=105) can be produced with the production cross sections above the lower bound of 10 −8 mb in this reaction. And almost all of unknown neutron-rich isotopes are emitted at the laboratory angles θ lab ≤ 60 • . Two cases, i.e. the production of the unknown uranium isotopes with A ≥ 244 and that of rutherfordium with A ≥ 269 are investigated for understanding the production mechanism of unknown neutron-rich isotopes. It is found that for the former case the collision time between two uranium nuclei is shorter and the primary fragments producing the residues have smaller excitation energies of ≤ 30 MeV and the outgoing angles of those residues cover a range of 30 • -60 • . For the later case, the longer collision time is needed for a large number of nucleons being transferred and thus it results in the higher excitation energies and smaller outgoing angles of primary fragments, and eventually results in a very small production cross section for the residues of Rf with A ≥ 269 which have a small interval of outgoing angles of θ lab =40 • -50 • .
I. INTRODUCTION
The production of unknown neutron-rich nuclei, especially for unexplored superheavy nuclei and isotopes near r-process, in fusion, fission, fragmentation process and multinucleon transfer reactions has been of experimental and theoretical interest. A lot of more neutronrich isotopes below Ra in fragmentation process were produced in recent years [1]. But for the new nuclei in the 'northeast' area of the nuclear map, it is difficult to be reached in the fission reactions and fragmentation processes widely used nowadays. Due to the 'curvature' of the stability line, it is also difficult for reaching these new more neutron-rich nuclei in fusion reactions with stable projectiles because of the lack of neutron number. The revival interest of multinucleon transfer between actinide nuclei at low-energy collisions, such as two 238 U, has arisen. This type of reaction provides us with an alternative way to produce more neutron-rich actinide and transactinide isotopes through multinucleon transfer.
During later 1970s and early 1980s, the uranium beam available at GSI was used to investigate the gross features of the products of reactions 238 U+ 238 U and 238 U+ 248 Cm associated with the charge distribution and cross sections for heavy actinide isotopes [2][3][4][5][6][7]. The experimental data of the actinide and complementary products in the reaction 238 U+ 238 U were reexamined in 2013 [8]. At GANIL, the experiment on the collision of 238 U+ 238 U at energies between 6.09 MeV/nucleon and 7.35 MeV/nucleon was performed, and the dependence of production yield of products on the beam energy and on the angle of detection were measured [9]. However, new neutron-rich nuclei have not been experimentally reported in the reaction of 238 U+ 238 U up to now.
The production of unknown neutron-rich isotopes was predicted in low-energy dissipative collisions of 238 U+ 248 Cm through multinucleon transfer based on multidimensional Langevin equations [10,11]. The semiclassical model GRAZING with considering the competition between neutron emission and fission showed the production of a few unknown neutron-rich isotopes with Z=92 to 94 in 238 U+ 238 U at entering energy E lab =2059 MeV [12]. Due to a large number of degrees of freedom, such as deformations of two nuclei, neck formation, nucleon transfer, nucleon emission and different types of separation of the transient composite system being involved in the reaction, it is more suitable to apply a microscopic dynamical model to investigate the reaction mechanism and the production of unknown isotopes. TDHF approach was used to analyze the role of nuclear deformation on collision time and on nucleon transfer in central collisions of 238 U+ 238 U, but the production of residual fragments was not calculated yet [13,14]. Microscopic transport model such as QMD type models were also applied to study the low energy reactions of heavy nuclei systems such as 197 Au+ 197 Au, 238 U+ 238 U and 232 Th+ 250 Cf [15][16][17][18][19][20][21][22]. By ImQMD model incorporated with the statistical evaporation model (HIVAP code) [23,24], the mass distribution of products in 238 U+ 238 U at 7.0MeV/nucleon was calculated and generally in consistence with the experiment measurement of GANIL [22]. The calculated isotope distributions of the residual fragments and the most probable mass number of fragments were generally in agreement with experimental data of GSI, and the production mechanism of neutron-rich residual fragments was studied [25]. In this work, we will further investigate the production of the primary and residual fragments with charge number from Z=70 to 120 produced in reaction of 238 U+ 238 U at 7.0 MeV/nucleon. The structure of this paper is as follows. In Sec. II, the framework of the ImQMD model is briefly introduced. In Sec. III, the production cross sections for primary and residual fragments are calculated, and the production of unknown neutron-rich isotopes in the reactions of 238 U+ 238 U will be discussed. Further, the microscopic mechanism of producing these isotopes is carefully analyzed. Finally, a brief summary is given in Sec. IV.
II. THEORETICAL MODEL
As in the original QMD model [26][27][28], each nucleon is represented by a coherent state of a Gaussian wave packet in the ImQMD model. The time evolution of the coordinate and momentum for each nucleon in the mean field part is determined by Hamiltonian equations.
The Hamiltonian includes kinetic energy, nuclear potential energy and the Coulomb energy.
The nuclear potential energy is an integration of the Skyrme type potential energy density functional, which reads
V loc = α 2 ρ 2 ρ 0 + β γ + 1 ρ γ+1 ρ γ 0 + g 0 2ρ 0 (∇ρ) 2 + c s 2ρ 0 [ρ 2 − κ s (∇ρ) 2 ]δ 2 + g τ ρ η+1 ρ η 0 ,(1)
where ρ = ρ n +ρ p is the nucleon density and δ = (ρ n −ρ p )/(ρ n +ρ p ) is the isospin asymmetry.
ρ n , ρ p are neutron and proton density, respectively.
The Coulomb energy is written as a sum of the direct and the exchange contribution:
U Coul = 1 2 ρ p (r) e 2 |r − r ′ | ρ p (r ′ )drdr ′ − e 2 3 4 3 π 1/3 ρ 4/3 p dR.(2)
In the collision part, the phase space occupation constraint for single particle proposed by Papa et al. [29] is applied in each time evolution step. The isospin-dependent in-medium nucleon-nucleon scattering cross sections are applied. The Pauli-blocking effect is treated as the same as in reference [30], which is obtained according to the the Uehling-Uhlenbeck factor. The model parameters as those used in Ref. [20] are listed in TABLE I. More detailed description of ImQMD model and its applications can be found in Refs. [19,20,22,31,32].
α(MeV) β(MeV) γ g 0 (MeVf m 2 ) g τ (MeV) η c s (MeV) κ s (f m 2 ) ρ 0 (f m −3 )
-356 303 7/6 7.0 12.5 2/3 32 0.08 0.165
In this work, the binding energy per nucleon and deformation of 238 U are taken as E gs = 7.37 MeV, β 2 =0.215 and β 4 =0.093 given by Ref. [33]. For low-energy collision of 238 U+ 238 U, the initial condition of reaction, such as the properties of projectile and target nuclei, is of vital importance for the microscopic transport model. We check the binding energy, the root-mean-square radius and the deformation of the initial nuclei, as well as their time evolution carefully. Only those initially selected nuclei with no spurious particle emission and their properties, such as the binding energy, root-mean-square radius and deformation being stable within 1000fm/c are adopted. The orientations of the initial uranium nuclei in all events are sampled randomly with an equal probability. In the ImQMD model, the time evolution of the reaction for each event at different impact parameters can be tracked.
Both the formation time and the reseparation time of the transient composite system of 238 U+ 238 U can be recognized in the simulations [25]. The charge number Z, mass number A and the excitation energy E * of each fragment formed in each event can also be determined.
The cross section for producing the primary fragment with Z, A and E * is then calculated
by σ(Z, A, E * ) = bmax 0 2πbdb N f rag (Z, A, b, E * ) N tot (b) = bmax b=0 2πb∆b N f rag (Z, A, b, E * ) N tot (b) .(3)
Here b is the impact parameter, N f rag (Z, A, b, E * ) is the number of events in which a fragment (Z, A, E * ) is formed at a given impact parameter b. The excitation energy E * of the fragment with charge number Z and mass number A is obtained by subtracting the corresponding ground-state energy [33] from the total energy of the excited fragment in its rest frame.
N tot (b) is the total event number at a given impact parameter b. The outgoing angle of each primary fragment can be obtained from its momentum. In this work, the maximum impact parameter is taken to be b max =15 fm, and the impact parameter step is ∆b=0.15 fm. The initial distance between the centers of mass of projectile and target is taken to be 40 fm.
100,000 events for each impact parameter are simulated in this work.
At 1000fm/c after the re-separation of the composite system, the ImQMD simulation is terminated and the primary fragments are recognized at this time as that did in Ref. [25].
Then the de-excitation process, including the evaporation of γ, n, p and α particle and fission, for each excited primary fragment is performed by using the statistical evaporation model (HIVAP code) [23,24]. In the HIVAP code, the survival probability of the fragment with charge number Z, mass number A and excitation energy E * are calculated by branching ratios expressed by relative partial decay widths for all possible decay modes,
Γ i (Z, A, E * )/Γ tot (Z, A, E * ), where Γ tot (Z, A, E * ) = i Γ i (Z, A, E * )
, and i=γ, n, p, α, and fission.
III. RESULTS AND DISCUSSION
The production cross sections for primary fragments produced in reaction 238 U+ 238 U at 7.0 MeV/nucleon are calculated by using ImQMD model. In Fig.1 The production cross sections for residual fragments are obtained through de-excitation of primary fragments by using HIVAP code and shown in Fig.1 by colored rectangles. Here we set the lower bound cross section to be 10 −8 mb for the production of residual fragments in the figure. We find that the production cross sections for most of transactinide nuclei are smaller than 10 −8 mb because it is difficult for those primary fragments to survive against fission due to very low fission barrier. For comparison, the area of known nuclei taken from Ref. [34] are presented by the magenta thick line in the figure. Comparing the predicted produced residual fragments with the known nuclei area, one can find that quite a few unknown neutron-rich isotopes at the 'northeast' area of nuclear map can be produced through multinucleon transfer between two 238 U. Some of those residues are difficult to be produced by fusion reactions. And most of these unknown isotopes are located in the region of actinide elements, and are about three to six neutrons richer than the known most neutron-rich nuclei. For the predicted produced light uranium-like elements with Z < 92, we find that they can reach the border of the proton-rich side of known nuclei in the nuclear map. Because of the high fission barrier, the light uranium-like primary fragments can survive against fission more easily and de-excite through neutron evaporation leading to the production of proton-rich nuclei.
It is very useful to investigate the outgoing angles of primary and residual fragments for experimental measurement and also for understanding the reaction mechanism. Here we present the calculated results of production cross sections for primary fragments at angle Fig.2 In order to further investigate the contribution to the production of residual fragments from different outgoing angles shown in Fig.3 and the production mechanism of unknown neutron-rich nuclei, we take the unknown uranium isotopes with A ≥ 244 as an example. the angular distribution mainly comes from the reactions at impact parameters b=4-8 fm.
regions of θ lab =0 • -10 • ,...,70 • -80 • in
In order to track the origin of the two-hump behavior shown in Fig.4, we further study the dependence of average excitation energies < E * > of primary fragments of uranium with A ≥ 244 and the average lifetime < T lif e > of composite system formed in the corresponding events on the outgoing angles and impact parameters. In Fig.5, the < E * > and the < T lif e > are taken for the primary fragments in a small interval of outgoing angles ∆θ lab =1 • and from the reactions within an impact parameter interval ∆b=0.15 fm. In Fig.5 (a) and (b), one can find that the primary fragments of uranium with A ≥ 244 are separated into two branches in both panels: the upper one consists of the projectile-like primary fragments and the lower one higher. Eventually, the production cross section for the residues of Rf decreases quickly and the outgoing angle of residues becomes narrower. It is because the outgoing angles of residual fragments from projectile-like primary fragments decrease due to the rotation of the composite system.
IV. Summary
In this work we apply the improved quantum molecular dynamics (ImQMD) model incorporated with the statistical evaporation model (the HIVAP code) to study the reaction 238 U+ 238 U at 7.0 MeV/nucleon. The calculation results of the production cross sections for the primary and residual fragments with charge number from Z=70 to 120 are presented.
About sixty unknown neutron-rich isotopes from element Ra (Z=88) to Db (Z=105) with the production cross sections above the lower bound of 10 −8 mb among the residual fragments produced in the reaction are predicted. The outgoing angles of primary and residual fragments are also investigated. We find that for most of the unknown neutron-rich isotopes around uranium, the outgoing angles are in a wider range of θ lab =30 • -60 • , while for those of heavier transactinide isotopes of Rf the outgoing angles are in a narrower range of 40 • -50 • .
In order to understand the production mechanism of unknown neutron-rich isotopes, we study the impact parameter dependence of the excitation energies of primary fragments of uranium isotopes with A ≥ 244 and that of rutherfordium isotopes with A ≥ 269 and the lifetimes of their corresponding composite systems. We find that for the former case the collision time between two uranium nuclei is shorter and the primary fragments producing
FIG. 1 :
1(Color online) The landscape of the cross sections for primary and residual fragments produced in 238 U+ 238 U at 7.0 MeV/nucleon (logarithmic scale, the black contour lines for primary fragments and colored rectangles for residual fragments). The area of known nuclei are denoted by magenta thick line. most of reaction events have reached the isospin equilibrium at that time. The superheavy primary fragment (114,184) (the isospin asymmetry is 0.235) at the center of the first 'island of stability' denoted by cross symbol in red color is not far from this line.
. Clearly, the production cross sections for primary fragments vary with their emitting angles and most of transactinide primary fragments are emitted within angles θ lab ≤ 50 • . In the figure, red cross symbols denote the center of 'island of stability' (Z=114, N=184). It shows that the outgoing angles of primary fragments around (114,184) are within θ lab ≤ 40 • . The calculated production cross sections for the residual fragments emitted at angle regions θ lab =0 • -10 • ,...,70 • -80 • are shown in Fig.3. The dotted lines denote the position of the heaviest actinide element Lr (Z=103). After the de-excitation process, the production cross sections for superheavy nuclei around the 'island of stability' are smaller than 10 −8 mb. It is noted from the figure that the outgoing angles of unknown actinide and transactinide isotopes are around θ lab =0 • -60 • and 40 • -50 • , respectively.
FIGFIG
The calculated angular distribution of residual fragments of uranium with A ≥ 244 is shown in Fig.4. Here the star symbols denote the total angular distribution of residual fragments of uranium with mass number A ≥ 244 and the contributions from the reactions within different impact parameter intervals are shown with the lines with different color and different . 2: (Color online) The landscape of the production cross sections for primary fragments emitted within different laboratory angle range in 238 U+ 238 U at 7.0 MeV/nucleon. The red cross symbols denote the center of 'island of stability' (Z=114, N =184). symbols. For the reactions at the impact parameters b ≤ 6 fm, a double-hump distribution is observed, the left hump with smaller outgoing angles corresponds to the residues produced from the target-like primary fragments and the right ones with larger outgoing angles come from the projectile-like primary fragments. With the increasing of impact parameters, the width of the angular distribution for each hump decreases, and the peak of the left hump shifts from an angle of less than 10 • to about 32.5 • , while that of the right hump shifts from about 52.5 • to 47.5 • and the shift is much smaller compared with that of the left one. The total angular distribution of the residues by adding up the contributions from all impact parameters becomes flat with a wide hump around 30 • -55 • , which is the superposition of the contributions from projectile-like and target-like primary fragments. The hump part of . 3: (Color online) The landscape of the cross sections for residual fragments emitted within different laboratory angle range in 238 U+ 238 U at 7.0 MeV/nucleon. The area of known nuclei are denoted by magenta thick line. The dotted lines denote the position of the heaviest actinide element Lr (Z=103).
FIG. 4 :FIG. 5 :
45(Color online) The angular distribution of unknown isotopes of uranium (A ≥ 244) produced in reactions 238 U+ 238 U at different impact parameters. consists of target-like primary fragments. And there also exists a correspondence between the excitation energy of primary fragments and the life time of corresponding composite system, i.e. the longer the lifetime of the composite system is, the higher the excitation energy of the primary fragments produced from the composite system is. Further, the low excitation energy area of ≤ 30 MeV (the blue area) in panel (a) coincides with the area with short lifetime area of 200-400fm/c (the light blue area) in panel (b). Relating Fig.5 (a) and (b) to Fig.4, we can obtain a scene that most of the residual fragments of uranium with A ≥ 244 are produced among the reaction events, in which two uranium nuclei bombarding with impact parameters b=4-8 fm contact with each other for about 200-400 fm/c and then the composite system re-separate. Another example is the production of unknown isotopes of rutherfordium (Z=104) with A ≥ 269 for investigating the production mechanism of unknown neutron-rich transactinide nuclei. Fig.5 (c) and (d) show the average excitation energies of primary fragments with Z = 104 and A ≥ 269 and the average lifetimes of their corresponding transient composite systems as the function of impact parameters and outgoing angles. The same as in panels (a) and (b), (Color online)(a) The average excitation energies of primary fragments of unknown isotopes of uranium and (b) the average lifetime of composite system from which those primary fragments are produced. (c) and (d) are the same as (a) and (b) but for the unknown isotopes of rutherfordium. in Fig.5 (c) and (d) the primary fragments are also separated into two branches corresponding to projectile-like and target-like primary fragments, respectively. But if comparing these two panels with (a) and (b) more carefully, we can find large difference between the production of unknown rutherfordium isotopes (A ≥ 269) and that of unknown uranium isotopes (A ≥ 244). For the Rf case, because of a large number of nucleons (12 protons and over 19 neutrons) being transferred, the collision time between projectile and target become much longer as is seen from panel (d), where the shortest collision time is 400fm/c. Thus the average excitation energies of primary fragments are all larger than 30 MeV. Moreover the reactions at larger impact parameter have much less contribution compared with the case of the production of unknown uranium isotopes. Eventually, it leads to a very small cross section for the residual fragments of Rf with A ≥ 269. The outgoing angles of rutherfordium primary fragments are smaller compared with those of uranium primary fragments with A ≥ 244 due to the longer collision time of composite system (because of the rotation of the composite system). From Fig.3, one sees that the production cross sections for the residual fragments of Rf with A ≥ 269 are lower than 10 −6 mb and the outgoing angles are in a narrow interval of 40 • -50 • . From Fig.3, Fig.5(c) and Fig.5(d), we can deduce that those residues of Rf come from such kind of reaction events, in which the projectile-like fragments capturing a large number of nucleons from target bring a relatively larger collective kinetic energy (i.e. relatively lower excitation energies) and exit with laboratory angles around θ lab =40 • -50 • . From these two examples, we can learn that for the unknown neutron-rich uranium residues, both projectile-like and target-like primary fragments with low excitation energies of ≤ 30 MeV provide the comparable contributions and thus the residual fragments have a wider angular distribution of θ lab =30 • -60 • . As the number of transferred protons and neutrons increases, the collision time between projectile and target needed increases for the corresponding reaction events and the excitation energies of primary fragments become
those residual fragments have low excitation energies of ≤ 30 MeV and their outgoing angles covers a wider range of 30 • -60 • . And for the later case the longer collision time is needed for the transfer of a large number of nucleons and thus it results in the higher excitation energies and smaller outgoing angles of primary fragments and eventually results in a very small production cross sections and a narrower outgoing angle range of 40 • -50 • for the residual fragments of Rf with A ≥ 269. This study should be useful for us to select the suitable projectile and target to produce the unknown heavy neutron-rich isotopes.
TABLE I :
Ithe model parameters
, the cross sections are plotted by black contour lines. It shows that a large amount of primary fragments are produced via proton and neutron transfer between projectile and target. And the most probable isotopes of primary fragments are located near the line with the isospin asymmetry close to that of 238 U (the isospin asymmetry is 0.227) on the nuclear map. It indicates that1.0E-03
1.0E-04
1.0E-03
1.0E-04
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+03
100
120
140
160
180
200
70
80
90
100
110
120
Neutron number
Proton number
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
238
U+
238
U: 7.0 MeV/nucleon
Primary
fragments
Residual
fragments
Known nuclei
(114,184)
Cross section (mb)
2013CB834404. We acknowledge support by the computing server C3S2 in Huzhou University.
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| [] |
[
"ON PROOFS OF CERTAIN COMBINATORIAL IDENTITIES",
"ON PROOFS OF CERTAIN COMBINATORIAL IDENTITIES"
] | [
"George Grossman ",
"Akalu Tefera ",
"Aklilu Zeleke "
] | [] | [] | In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated proofs using the Wilf-Zeilberger(WZ) method. | 10.2140/involve.2021.14.697 | [
"https://arxiv.org/pdf/0709.1978v1.pdf"
] | 17,629,364 | 0709.1978 | ea6fe3dced783903e099411b30fda93fa678abac |
ON PROOFS OF CERTAIN COMBINATORIAL IDENTITIES
13 Sep 2007
George Grossman
Akalu Tefera
Aklilu Zeleke
ON PROOFS OF CERTAIN COMBINATORIAL IDENTITIES
13 Sep 2007
In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated proofs using the Wilf-Zeilberger(WZ) method.
Introduction
It is known that every integer can be written as a sum of integral powers of 2. A somewhat related problem is to find for every positive integer n a positive integer k depending on n with k(n) < k(n + 1) and integer coefficients a i , i = 0, 1, . . . , k − 1 such that
(1) n = k−1 i=0 a i 2 2 i .
The background and motivation for this problem lies in studying the zeros of the j−th order polynomial of the generalized Fibonacci sequence given by
(2) F j (x) = x j − x j−1 − · · · − x − 1 .
For studies related to the positive zeros of (2) we refer the reader to the papers by Dubeau ([D89]) and Flores ([F67]). It can be shown (see [GN99]) that for j even (3) F j (x) = (x − 2 + ǫ j )(x + 1 − δ j )(x j−2 + a j−3 x j−3 + ... + a 1 x + a 0 ) , where −1 + δ j and 2 − ǫ j are the negative and positive zeros of (2). Here {δ j } and {ǫ j } are positive, decreasing sequences. In a recent paper, Grossman and Zeleke ([GZ03]) have found an explicit form for the a i 's in terms of ǫ j and δ j for j ≥ 4. The explicit expressions for a i as well as special cases led to some interesting identities. In this paper we present different proofs of three such identities that are hypergeometric. The paper is organized as follows. In section 2, we formulate the main results. In section 3 we provide combinatorial proofs. This requires first finding combinatorial interpretations by counting words of certain properties and defining an appropriate sign reversing involution which we call "involution". Gessel and Stanley discuss the mathematical theory related to such proofs in ( [GS95]).
In section 4, we present computer generated proofs of the main results. It is to be noted that there are philosophical arguments over computer-based proofs to mathematical proofs in general. It will be clear from sections 3 and 4 that the WZ method gives a unified and structured approach to proving identities of this type. Introductions to the WZ method can be found among others in the book A = B ( [PWZ96]) or the website http://mathworld.wolfram.com/Wilf-ZeilbergerPair.html. Throughout this paper we denote the set {k, k + 1, k + 2, . . . } for k ∈ Z by N k .
Main Results
Theorem 1. [GZ03] n k=0 (−1) n+k n + k + 1 2 k + 1 2 2 k = n + 1, n ∈ N 0 .
Remark:
The following theorems will show that the coefficients a i 's in the expansion of positive integers are not unique.
Theorem 2. n+1 k=0 (−1) n+k+1 n + k + 1 2 k 2 2 k = 2 n + 3, n ∈ N −1 . Theorem 3. [GZ03] n−1 k=0 n+1+k m=2 k+2 (−1) m+k n + k + 1 m 2 m−1 = n (n + 1), n ∈ N 1 .
Proofs of the Main Results
Combinatorial Proofs.
For a fixed n ∈ N k , k ∈ Z, consider the set S of words in the alphabet {a, b, c} such that 2 (the number of a's) + 1 (the number of b's) + 1 (the number of c's) = p(n),
where p(n) is some polynomial of n. For w ∈ S, define the weight W t(w) of w by W t(w) = (−1) (the number of a's) .
Proof of Theorem 1. For n ∈ N 0 , take p(n) = 2 n + 1. Then
w∈S W t(w) = n k=0 (the number of w's with (n − k) a ′ s) W t(w) = n k=0 n + k + 1 2 k + 1 2 2 k+1 (−1) n+k = 2 n k=0 (−1) n+k n + k + 1 2 k + 1 2 2 k . Consider now S = T ∪ (S − T ),
where T is the set of all words in S of the form c m b 2 n+1−m for m = 0, 1, . . . , 2 n + 1. Here by, for x ∈ {a, b, c} the notation x 0 is used to denote the empty word. Define an "involution" as follows:
For w ∈ S read w left to right until you either get an a, or b c. If it is an a, make it a b c. If it is a b c, make it an a. This changes the sign of W t(w) and is an involution. Note that T has 2 n + 2 elements each of weight (−1) 0 = 1. From the involution it is clear that the sum of the weights of the elements of S − T is 0. Thus w∈S W t(w) = 2 n + 2. Hence the theorem follows.
Proof of Theorem 2. Let n ∈ N −1 and p(n) = 2 n + 2. Then
w∈S W t(w) = n+1 k=0 (the number of w's with (n + 1 − k) a ′ s) W t(w) = n k=0 n + k + 1 2 k 2 2 k (−1) n+k+1 = n k=0 (−1) n+k+1 n + k + 1 2 k 2 2 k .
Partition S as in the proof of Theorem 1 with T the set of all words in S of the form c m b 2 n+2−m for m = 0, 1, . . . , 2 n + 2. T has 2 n + 3 elements each of weight (−1) 0 = 1.
Using the same "involution" as in the proof of Theorem 1 the sum of the weights of the elements of S − T would be 0 and hence w∈S W t(w) = 2 n + 3.
Proof of Theorem 3.
n + k + 1 m 2 m−1 (−1) m+k .
Read a word w ∈ S from left to right. Count the number b and c until the sum is 3. Thus w has the form a l x a n y a m z * where l, m, n ∈ N 0 and x, y, z ∈ {b, c}. For such words, define a mapping σ as follows:
σ(w) =
a l x a n+1 y a m z * : if n, m have same parity and n = 1 , a l x a n−1 y a m z * : if n, m have same parity and n = 1 , a l x a n−1 y a m z * : if n, m have opposite parity and n = 0 , a l x a n+1 y a m z * : if n, m have opposite parity and n = 0 .
Clearly σ is an "involution". This involution is not defined for elements of S of length n + 1 and the number of b's and c's exactly 2. There are 4 n + 1 2 = 2 n(n + 1) such words each of weight (−1) n+1 and hence the theorem follows.
The WZ Method Proofs. Proof of Theorem 1.
Let F (n, k) = n + k + 1 2 k + 1 2 2 k (−1) k+n+1 n + 1 and let S(n) = n k=0 F (n, k). We want to show that S(n) = 1 for all n ∈ N 0 . F satisfies the recurrence equation: 1
F (n + 1, k) + F (n, k) = G(n, k + 1) − G(n, k),(4)
where G(n, k) = R(n, k) F (n, k) and R(n, k) = − k (2 k + 1) (n + 1 − k) (n + 2) .
By summing both sides of equation (4) with respect to k we get S(n + 1) − S(n) = 0. Moreover, S(0) = 1 and hence S(n) = 1 for all n ∈ N 0 .
Proof of Theorem 2.
Let F (n, k) = n + k + 1 2 k 2 2 k (−1) k+n+1 2 n + 3 and let S(n) = n+1 k=0 F (n, k). We want to show that S(n) = 1 for all n ∈ N −1 . F satisfies the recurrence equation: 1
F (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k),(5)
where G(n, k) = R(n, k) F (n, k) and R(n, k) = 2 k (2 k − 1) (n + 2 − k) (2 n + 5) .
By summing both sides of equation (5) with respect to k we get S(n + 1) − S(n) = 0. Moreover, S(−1) = 1 and hence S(n) = 1 for all n ∈ N −1 .
Proof of Theorem 3.
Reversing the order of summation, the identity can be rewritten as 2 n m=2 ⌊ m−2 2 ⌋ k=0 n + k + 1 m 2 m−1 (−1) m+k+n+1 = n (n + 1) .
Let us denote the left side of (6) by S(n) and its summand by F (n, k, m), i.e. F (n, k, m) = n + k + 1 m 2 m−1 (−1) m+k+n+1 .
Then F satisfies the recurrence equation: 2
F (n + 1, k, m) − F (n, k, m) = F (n, k + 1, m) − F (n, k, m) .(7)
Summing both sides of (7) with respect to k and with respect to m, we get
S(n + 1) − S(n) = 2 n m=2 n + ⌊m/2⌋ + 1 m 2 m−1 (−1) m+⌊m/2⌋+n+1 − 2 n m=2 n + 1 m 2 m−1 (−1) m+n+1 .(8n + 1 m 2 m−1 (−1) m+n+1 = (−1) n+1 2 n+1 m=0 n + 1 m (−2) m − (n + 1) (−1) n + (−1) n 2 = (−1) n+1 2 (1 − 2) n+1 − (n + 1) (−1) n + (−1) n 2 = 1 + (−1) n 2 − (n + 1) (−1) n ,(9)2 n m=2 n + ⌊m/2⌋ + 1 m 2 m−1 (−1) m+⌊m/2⌋+n+1 = n m=1 n + m + 1 2 m 2 2 m−1 (−1) m+n+1 + n−1 m=1 n + m + 1 2 m + 1 2 2 m (−1) m+n = n+1 m=0 n + m + 1 2 m 2 2 m−1 (−1) m+n+1 − 2 2 n+1 + (−1) n 2 + n m=0 n + m + 1 2 m + 1 2 2 m (−1) m+n − (n + 1) (−1) n − 2 2 n = n + 3 2 − 2 2 n+1 + (−1) n 2 + (n + 1) − (n + 1) (−1) n − 2 2 n .(10)
From equations (8), (9) and (10), we get S(n + 1) − S(n) = 2 (n + 1). Since S(1) = 2, and n (n + 1) satisfies the same recurrence relation, therefore S(n) = n (n + 1) for all n ∈ N 1 .
Some Corollaries.
For completeness, we state the following results from ( [GZ03]) and prove using theorems 1-3. (−1) m+k+n+1 n + k + 1 m 2 m−1 = (n + 1) 2 .
Proof: The result follows by adding theorems 1 and 3.
Corollary 2. (−1) m+k 2 n + k + 2 m 2 m−1 = (2 n + 1)(2 n + 2) .
Proof: Add theorem 3 and 1 and multiply the result by 2.
Corollary 3. (−1) m+k 2 n + k + 2 m 2 m−1 = (2 n + 2) 2 .
Proof: Replace n by 2 n + 1 in theorem 1. (−1) m+k+1 2 n + k + 1 m 2 m−1 = (2 n + 1) 2 .
Proof: Replace n by 2 n in theorem 1.
Corollary 5.
2 l k=0 2 k+2 m=1 (−1) m 2 k + m + 1 2 m − 1 4 m−1 = (2 l + 1)(2 l + 2) .
Proof: Replace n by 2 n + 1 in theorem 1 and sum k from 0 to 2 l.
For n ∈ N 1 , consider the set S of words in the alphabet {a, b, c} such that 1 (the number of a's) + 1 (the number of b's) + 1 (the number of c's) = n + 1 + k, for some k ∈ {0, . . . , n − 1} and (the number of b's) + (the number of c's) is at least 2 k + 2. For w ∈ S, define the weight W t(w) of w by W t(w) = (−1) (the number of a's) .
) The recurrence equation is automatically generated by a MAPLE package EKHAD which is available from htpp//www.math.rutgers.edu/~zeilberg/ 2 The recurrence equation is automatically generated by MultiSum, a Mathematica package which is available from1 htpp//www.risc.uni-linz.ac.at/research/combinat/risc/software/
But
2 n
m=2
Acknowledgement: The authors would like to thank Doron Zeilberger and the referee for their helpful suggestions on the combinatorial proofs of the main results using the involution approach.
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| [] |
[
"PROBABILISTIC DIMENSIONALITY REDUCTION VIA STRUCTURE LEARNING",
"PROBABILISTIC DIMENSIONALITY REDUCTION VIA STRUCTURE LEARNING"
] | [
"L I Wang "
] | [] | [] | We propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a smooth skeleton of embedding points in a low-dimensional space from high-dimensional noisy data. The formulation of the new model can be equivalently interpreted as two coupled learning problem, i.e., structure learning and the learning of projection matrix. This interpretation motivates the learning of the embedding points that can directly form an explicit graph structure. We develop a new method to learn the embedding points that form a spanning tree, which is further extended to obtain a discriminative and compact feature representation for clustering problems. Unlike traditional clustering methods, we assume that centers of clusters should be close to each other if they are connected in a learned graph, and other cluster centers should be distant. This can greatly facilitate data visualization and scientific discovery in downstream analysis. Extensive experiments are performed that demonstrate that the proposed framework is able to obtain discriminative feature representations, and correctly recover the intrinsic structures of various real-world datasets.L. Wang is with | 10.1109/tpami.2017.2785402 | [
"https://arxiv.org/pdf/1610.04929v1.pdf"
] | 15,233,044 | 1610.04929 | 96da654db0d9d37de373034c1f2393877addc5d9 |
PROBABILISTIC DIMENSIONALITY REDUCTION VIA STRUCTURE LEARNING
L I Wang
PROBABILISTIC DIMENSIONALITY REDUCTION VIA STRUCTURE LEARNING
We propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a smooth skeleton of embedding points in a low-dimensional space from high-dimensional noisy data. The formulation of the new model can be equivalently interpreted as two coupled learning problem, i.e., structure learning and the learning of projection matrix. This interpretation motivates the learning of the embedding points that can directly form an explicit graph structure. We develop a new method to learn the embedding points that form a spanning tree, which is further extended to obtain a discriminative and compact feature representation for clustering problems. Unlike traditional clustering methods, we assume that centers of clusters should be close to each other if they are connected in a learned graph, and other cluster centers should be distant. This can greatly facilitate data visualization and scientific discovery in downstream analysis. Extensive experiments are performed that demonstrate that the proposed framework is able to obtain discriminative feature representations, and correctly recover the intrinsic structures of various real-world datasets.L. Wang is with
Introduction
Contemporary simulation and experimental data acquisition technologies enable scientists and engineers to generate progressively large and inherently highdimensional data sampled from sources with unknown multivariate probability distributions. Data expressed with many degrees of freedom imposes serious problems for data analysis. It is often difficult to directly analyze these datasets in the highdimensional space, and is desirable to reduce the data dimensionality in order to overcome the curse of dimensionality and associate data with intrinsic structures for data visualization and subsequent scientific discovery.
Dimensionality reduction is a learning paradigm that transforms high-dimensional data into a low-dimensional representation. Ideally, the reduced representation should correspond to the intrinsic dimensionality of the original data, and it can often be of advantage in practical applications to analyze the intrinsic structure of data. Examples include clustering of gene expression data and text documents [14], and high-dimensional data visualization of image datasets [36,43]. Accordingly, many dimensionality reduction methods have been proposed with the aim to preserve certain information of data. Principal component analysis (PCA) [24] is a classic method for this purpose. It learns a subspace linearly spanned over some orthonormal bases by minimizing the reconstruction error [8]. However, a complex 2 L. WANG structure of the data could be misrepresented by a linear manifold constructed by PCA.
Complex structures have been studied to overcome the misrepresentation issue of PCA. Kernel PCA [38] first maps the original space to a reproducing kernel Hilbert space (RKHS) by a kernel function and then performs PCA in the RKHS space. Hence, KPCA is a nonlinear generalization of traditional PCA. Another approach is manifold learning [9], which aims to find a manifold close to the intrinsic structure of data. By projecting data onto a manifold, the low-dimensional representation of data can be obtained by unfolding the manifold. Isometric feature mapping (Isomap) [43] first estimates geodesic distances between data points using shortest-path distances on a K-nearest neighbor graph, and then use multidimensional scaling to find points in a low-dimensional Euclidean space where the distance between any two points match the corresponding geodesic distance. Local linear embedding (LLE) [37] finds a mapping that preserves local geometry where local patches based on K-nearest neighbors are nearly linear and overlap with one another to form a manifold. Laplacian eigenmap (LE) [2] is proposed based on spectral graph theory, and a K-nearest neighbor graph is used to construct a Laplacian matrix. Other methods related to neighborhood graphs are referred to survey papers [9,48], including maximum variance unfolding (MVU) [50], diffusion maps (DM) [26], Hession LLE [15], and local tangent space analysis (LTSA) [53].
Probabilistic models have also been studied for dimensionality reduction. Probabilistic PCA (PPCA) [45] generalizes PCA by applying the latent variable model to the representation of linear relationship between data and its embeddings. Gaussian process latent variable model (GPLVM) [28] takes an alternative approach to marginalize the linear projection matrix, and then parametrizes covariance matrix using a kernel function. GPLVM with a linear kernel is the dual interpretation of PPCA, while its nonlinear generalization is related to KPCA. Bayesian GPLVM [46] maximizes the likelihood of data by marginalizing out both projection matrix and embeddings using variational inference. Maximum entropy unfolding (MEU) [29] is proposed to directly model the density of observed data by minimizing Kullback-Leibler (KL) divergence under a set of constraints, and embedding points of data are obtained by maximizing the likelihood of the learned density. t-distributed stochastic neighbor embedding (tSNE) [47] employs a heavy-tailed distribution in the low-dimensional space to alleviate both the crowding problem and the optimization problem of SNE [21], which converts the high-dimensional Euclidean distances between data points into conditional probabilities that represent similarities.
Although the above two classes of methods work well under certain conditions, they lack a unified probabilistic framework to learn robust embeddings from noisy data by applying neighborhood graph to capture the locality information. Probabilistic models such as PPCA and GPLVM can deal with noisy data, but they are difficult to incorporate the neighborhood manifold, which has been proved to be effective for nonlinear dimensionality reduction. On the other hand, methods based on neighborhood manifold, such as MVU, LE and LLE, either are hard to learn the manifold structure of a smooth skeleton, or cannot be interpreted as probabilistic models for model selection and noise tolerance.
In addition, neighborhood graphs used in the above methods are generally constructed from data resided in a high-dimensional space if they are unknown. Graph structures that are commonly used in graph based clustering and semi-supervised learning are the K-nearest neighbor graph and the -neighborhood graph [3]. Dramatic influences of these two graphs on clustering techniques have been studied in [30]. Since the -neighborhood graph could result in disconnected components or subgraphs in the dataset or even isolated singleton vertices, the b-matching method is applied to learn a better b-nearest neighbor graph via loopy belief propagation [23]. However, it is improper to use a fixed neighborhood size since the curvature of manifold and the density of data points may be different in different regions of the manifold [17]. Moreover, most distance-based manifold learning methods suffer from the curse of dimensionality, i.e., there is little difference in the distances between different pairs of data points [4]. Hubs are closely related to the nearest neighbors [35], that is, points that appear in many more K-nearest neighbors lists than other points, effectively making them "popular" nearest neighbors. As alluded by [35], hubs can have a significant effect on dimensionality reduction and clustering, so we should take hubs into account in a way equivalent to the existence of outliers. Furthermore, if the data is noisy, a precomputed neighborhood graph to approximate the manifold of data is not reliable any more. As a result, it is less reliable to directly construct K-nearest neighbor graphs in a high-dimensional space.
To overcome the issues of constructing graphs, structure learning has had a great success in automatically learning explicit structures from data. A sparse manifold clustering and embedding (SMCE) [17] is proposed using 1 norm over the edge weights and 2 norm over the errors that measure the linear representation of every data point by using its neighborhood information. Similarly, 1 graph is learned for image analysis using 1 norm over both the edge weights and the errors for enhancing the robustness of the learned graph [10]. Instead of learning directed graphs by using the above two methods, an integrated model for learning an undirected graph by imposing the sparsity on a symmetric similarity matrix and a positive semidefinite constraint on the Laplacian matrix is proposed [27]. These are discriminative models, so they lack the ability to model noise of data. In addition to learning a general graph, a simple principal tree learning algorithm (SimplePPT) [32] aims to learn a spanning tree from data by minimizing data quantization error and the length of the tree. However, SimplePPT generates principal tree in the original space of the input data, so it is not applicable for dimensionality reduction.
To overcome the above issues, we propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. This framework is formulated in terms of empirical Bayesian inference where the likelihood function models the data generation process using noise model and the loss function penalizes the violations of expected pairwise distances between two original data points and their corresponding embedded points in a neighborhood. This framework generalizes the dimensionality reduction approach proposed in our preliminary work [33], which learns a mapping function that transforms data points in a high-dimensional space to latent points in a low-dimensional space such that these latent points directly forms a graph. Our previous work has proved that the proposed method is able to correctly discover the intrinsic structures of various real-world datasets including curves, hierarchies and a cancer progression path. Extensive experiments are conducted to validate our proposed methods for the visualization of learned embeddings for general structures and classification performance. The main contributions of this paper are summarized as follows:
1) We propose a novel probabilistic framework for dimensionality reduction, which not only takes the noise of data into account, but also utilizes the neighborhood graph as the locality information. To the best of our knowledge, there is no prior work that can model data generation error and pairwise distance constraints in a unified framework for dimensionality reduction.
2) We present a new model under the proposed framework using 2 loss function over the expected distances. Given a neighborhood graph, this model is able to learn a smooth skeleton structure of embedded points and retain the inherent structure from noisy data by imposing the shrinkage of the pairwise distances between data points.
3) To learn an explicit graph structure, we break down the optimization problem of the new model into two components: structure learning and projection matrix learning. By replacing the component of structure learning with the problem of learning an explicit graph, the new model reduces to our preliminary work [33].
4) The connections between the proposed models and various existing methods are discussed including reversed graph embedding, MEU, and the various structure learning methods mentioned above.
The rest of the paper is organized as follows. We first briefly introduce several existing methods from deterministic and probabilistic points of view in Section 2. In Section 3, we propose a unified probabilistic framework for dimensionality reduction and a new model for learning a smooth skeleton from noisy, high-dimensional data. We further generalize this framework for learning an explicit graph structure in Section 4 and discuss the connections to various existing works in Section 5. Extensive experiments are conducted in Section 6. We conclude this work in Section 7. Moreover, proofs and more experimental results are given in the supplementary materials.
Related Work
Let Y = {y i } N i=1 be a set of N data points where y i ∈ R D .
The goal of dimensionality reduction is to find a set of embedded data points
X = {x i } N i=1
, where x i ∈ R d and d < D, satisfying certain assumptions. Next, we briefly introduce several existing dimensionality reduction methods from both deterministic and probabilistic perspectives.
2.1. Deterministic Methods. The classic deterministic method for dimensionality reduction is MVU [50]. Its objective is to maximize the variance of the embedded points subject to constraints such that distances between nearby inputs are preserved. MVU consists of three steps. The first step is to compute the K-nearest neighbors N i of data point y i , ∀i. The second step is to solve the following optimization problem
max X N i=1 ||xi|| 2 (1) s.t. ||xi − xj|| 2 = ||yi − yj|| 2 , ∀i, j ∈ Ni,(2)N i=1 xi = 0,(3)
where constraints (2) preserve distances between K-nearest neighbors and constraint (3) eliminates the translational degree of freedom on the embedded data points by constraining them to be centered at the origin. Instead of optimizing over X, MVU reformulates (1) as a semidefinite programming by learning a kernel matrix K with the (i, j)th element denoted by κ i,j = x i , x j H with a semidefinite constraint K 0 for a valid kernel [39] where the corresponding mapping function lies in a RKHS H. Define φ i,j = ||y i −y j || 2 and ζ i,
j = ||x i −x j || 2 = κ i,i +κ j,j −2κ i,j .
The resulting semidefinite programming is
max K Tr(K) : s.t. i,j κi,j = 0, K 0, ζi,j = φi,j, ∀i, j ∈ Ni, where N i=1 x i , N j=1 x j H = i,j κ i,j =
0 is a relaxation of (3) for ease of kernelization. The last step is to obtain the embedding X by applying KPCA on the optimal K. The distance/similarity information on a neighborhood graph is widely used in the manifold-based dimensionality reduction methods such as locally linear embedding (LLE) and its variants [37], and Laplacian Eigenmap (LE) [2].
A duality view of MVU problem has been studied in [51]. Define E i,j as an N × N matrix consisting of only four nonzero elements:
E i,j [i, i] = E i,j [j, j] = 1, E i,j [i, j] = E i,j [j, i] = −1.
The preserving constraints can be rewritten as Tr(KE i,j ) = φ i,j , ∀i, j ∈ N i . Thus, the dual problem of the above semidefinite programming is given by
min {w i,j } i,j∈N i wi,jφi,j : s.t. λN−1(L) ≥ 1, L = i,j∈N i wi,jE i,j ,(4)
where w i,j is the dual variable subject to the preserving constraint associated to edge (i, j), and λ N −1 denotes the second smallest eigenvalue of a symmetric matrix [51]..
2.2.
Probabilistic Models. Probabilistic models are able to take the noise model of data generation into consideration. The observed data Y and the embedding X are treated as random variables. For dimensionality reduction, we associate a set of latent variables X ∈ R N ×d to a set of observed variables Y ∈ R N ×D . Latent variable models for dimensionality reduction generally assume the linear relationship between x i and y i with noise given by
yi = Wxi + i, ∀i,(5)
where W ∈ R D×d is a linear projection matrix, and i ∈ R D is the vector of noise values. Noise is independently sampled from a spherical Gaussian distribution with mean zero and covariance γ −1 I D where γ > 0 and I D is a D × D identity matrix. Thus, the likelihood of data point x i is
p(yi|xi, W, γ) = N (yi|Wxi, γ −1 ID),(6)
6 L. WANG and the likelihood of the whole data is p(Y|X, W, γ) = N i=1 p(y i |x i , W, γ) due to the independently and identically distributed (i.i.d.) assumption of data.
PPCA [45] further assumes that the latent variables {x i } N i=1 follow a unit covariance zero mean Gaussian distribution π(xi) = N (xi|0 d , I d ). (7) The projection matrix W is obtained by maximizing the likelihood of the given data max W p(Y|W, γ) (8) where p(Y|W, γ) = N i=1 p(y i |W, γ) and the marginal likelihood of each data is obtained by marginalizing out the latent variable X given by
p(yi|W, γ) = p(yi|xi, W, γ)π(xi)dxi = N (yi|0, WW T + γ −1 ID).(9)
Tipping and Bishop [45] showed that the principal subspace of the data is the optimal solution of problem (8) if γ approaches to infinity. Therefore, this model is viewed as a probabilistic version of PCA.
GPLVM [28] takes an alternative way to obtain marginal likelihood of data by marginalizing out W and optimizing with respect to X. Assume the prior distribution of W as
π(W) = D j=1 N (wj|0 d , I d )
. (10) where w j is the jth row of W. The marginal likelihood of the data is obtained by marginalizing out W given by (11) where MN N,D is the matrix normal distribution with zero mean, sample-based covariance matrix K = XX T + γ −1 I N and feature-based covariance matrix I D [20]. GPLVM obtains X by maximizing the marginal likelihood of the data max X log p(Y|X, γ).
p(Y|X, γ) = N i=1 p(yi|xi, W, γ)π(W)dW = MN N,D (0, K, ID)
Lawrence [28] showed that the optimal solution is equivalent to that obtained by PCA. The merit of this model is that different covariance functions can be incorporated for nonlinear representations since K is of the form of inner product matrix. Thus, the linear model is called the dual interpretation of PPCA, and the nonlinear model is related to KPCA [38].
Structured Dimensionality Reduction
We propose a novel dimensionality reduction framework based on regularized empirical Bayesian inference [54], where the unknown embedded data is not only decided by the observed data, but also regulated by manifold structures. Next, we first present the regularized empirical Bayesian inference, and expectation constraints for capturing manifold structure are then presented. With these two ingredients, we formulate the proposed framework for dimensionality reduction.
3.1. Regularized Empirical Bayesian Inference. Regularized empirical Bayesian inference [54] is an optimization formulation of a richer type of posterior inference, by replacing the standard normality constraint with a wide spectrum of knowledgedriven and/or data-driven constraints or regularization. Following the notation in [54], let M denote the space of feasible models, and M ∈ M represents an atom in this space. We assume that Π is absolutely continuous with respect to some background measure µ, so that there exists a density π such that dΠ = πdµ. Given a model, let D be a collection of observed data points, which are assumed to be i.i.d.. Define KL(q(M)|π(M)) = M q(M) log(q(M)/π(M))dµ(M) as the Kullback-Leibler (KL) divergence from q(·) to π(·).
In the presence of unknown parameters, e.g., hyper-parameters, empirical Bayesian inference is necessary where an estimation procedure such as maximum likelihood estimation is needed. Here, we focus on the expectation constraints, of which each one is a function of q(M) through an expectation operator. For example, let ψ = {ψ 1 , . . . , ψ T } be a set of feature functions, each of which is ψ t (M; D) defined on M and possibly data dependent. With unknown parameter Θ, regularized empirical Bayesian inference is formulated by solving the following optimization problem
+ U ({E q(M) [ψt(M; D)]} T t=1 ) (13) s.t. q(M) ∈ Ppost, Θ ∈ Θ, where E q(M) [ψ t ] is the expectation of ψ t over q(M), U is a function of {E q(M) [ψ t ]} T t=1
, Θ is the feasible set of the unknown parameter Θ, and P post is a subspace of distributions. Note that minimizing the first two terms of (13) with respect to q(M) and Θ leads to an optimal solution Θ * , which is equivalent to maximum likelihood estimation, Θ * = arg max Θ∈Θ log p(D|Θ) (14) where q(M) = p(M|D, Θ). Hence, problem (13) is called regularized empirical Bayesian inference where the regularization term is useful to capture domain knowledge or structure information of data.
3.2. Expectation Constraints over Pairwise Distances. Expectation constraints are widely employed for classification problems in the generalized maximum entropy model [16] and regularized Bayesian inference model [54] . Here, we are particularly interested in defining expectation constraints for dimensionality reduction.
Given a probabilistic density function q(M), the definition of feature function over data is one of the necessary ingredients to form an expectation constraint. For classification, the feature-label pair of one instance is naturally treated as ψ t . However, this is not suitable for dimensionality reduction, where features from a single instance are not enough to determine the embedding of the whole data. As discussed before, most discriminant methods take pairwise distance as the key information provided by the data. For example, MVU [50] takes pairwise distances over a neighborhood graph, and LE [2] transforms pairwise distances over a neighborhood graph to similarity. Thus, pairwise distance can be reasonably considered as the factor of the feature function for dimensionality reduction. Specifically, in 8 L. WANG this paper, the feature function ψ i,j represents the difference between pairwise distance of embedding points x i and x j and their corresponding distance φ i,j , i.e.,
ψ i,j (X; Y) = ||x i − x j || 2 − φ i,j .
Another necessary ingredient is to determine function U , which has significant influence on density function q(X). By incorporating specific prior information of data, we have various choices. One choice is to strictly preserve the pairwise distances over a given neighborhood graph, which corresponds to equality constraints (2) in MVU. To achieve this, we define U (q(X)) = i,j∈Ni I(E q(X) [ψ i,j (X; Y)] = 0), where I(a) is an indicator function that equals to 0 if the a = 0 is satisfied; otherwise ∞. As a result, the optimal q(X) must satisfy
E q(X) [||x i − x j || 2 ] = φ i,j , ∀i, j ∈ N i .
As discussed in [16], we can formulate different expectation constraints over pairwise distances. In this paper, we are more interested in the shrinkage effect of pairwise distances of data to form a smooth skeleton structure in the embedding space. This can be achieved by defining function
U (ξ) = i,j∈Ni ξ 2 i,j where the closeness tolerance ξ i,j is constrained by E q(X) [||x i − x j || 2 ] − φ i,j ≤ ξ i,j , ∀i, j ∈ N i . If ξ i,j ≤ 0, we have E q(X) [||x i − x j || 2 ] ≤ φ i,j + ξ i,j ≤ φ i,j so that E q(X) [||x i − x j || 2 ] cannot be bigger than φ i,j . On the other hand, if ξ i,j > 0, we have probably E q(X) [||x i − x j || 2 ] ≥ φ i,j but E q(X) [||x i − x j || 2 ] cannot be bigger than φ i,j + ξ i,j .
Thus, the above function U (ξ) = i,j∈Ni ξ 2 i,j prefers to shrink the pairwise distance of two original points as the pairwise distance of the corresponding two embedding points, but these constraints allow the violation of expansion of pairwise distance no more than ξ i,j if ξ i,j > 0. These constraints are quite different from the expectation constraints used in [16,54]. Moreover, they allow us to use efficient optimization tools, which will be illustrated in Section 3.3.
Structured Projection Learning.
We propose a new model by incorporating the newly defined expectation constraints into the regularized empirical Bayesian inference framework. Following PPCA, we treat X as a random variable and W as an unknown parameter. Given a neighborhood graph with a set N i as the neighbors of the ith vertex of the graph. According to the regularized empirical Bayesian inference, we formulate the following optimization problem
min W min q(X),ξ KL(q(X)||π(X)) − log p(Y|X, W, γ)q(X)dX + C||ξ|| 2 F (15) s.t. E q(X) ||xi − xj|| 2 − φi,j ≤ ξi,j, ∀i, j ∈ Ni q(X) ∈ P prob W T W = I d ,
where C > 0 is a regularization parameter, the orthogonal constraint is added for preventing arbitrarily scaling of the variable W, and P prob represents the feasible set of all density functions over X.
The following proposition shows that problem (15) has interesting property in terms of its partial dual problem. The proof is given in the supplementary materials. Proposition 1. Problem (15) has an analytic solution given by (16) and ξ i,j = si,j 2C where S and W can be obtained by solving the following optimization problem
q(X) ∝π(X)p(Y|X, W, γ) exp − i,j∈N i si,j||xi − xj|| 2max S min W d 2 log det((γ + 1)IN + 4L) − i,j∈N i si,jφi,j− 1 4C ||S|| 2 F − γ 2 2 Tr(W T Y T ((γ + 1)IN + 4L) −1 YW) (17) s.t. si,j = 0, ∀i, j ∈ Ni si,j ≥ 0, si,j = sj,i, ∀i, j, W T W = I d ,
and L = diag(S1) − S is the Laplacian matrix with 1 as the column vector of all ones.
According to the property of Laplacian matrix, we have L = diag(S1) − S 0 if S ≥ 0. In other words, L is guaranteed to be positive semidefinite for any non-
negative S. Define Q = diag(S1) − S + (γ+1) 4 I N and W = {W ∈ R D×d |W T W = I d }.
Due to the inversion of Q and nonconvexity of the objective function, it is challenging to solve problem (17) globally. In order to reach a stationary point, we take the projected subgradient ascend method [7] to solve problem (17). First, we denote a function with respect to S as
g(S) = max W∈W h(S, W). (18) where h(S, W) = γ 2 8 Tr(W T Y T Q −1 YW).
It is worth noting that h(S, W) is convex over W given any S since Q is positive definite. Thus, we can obtain the subgradient of g(S) as the convex hull of union of subdifferentials of active functions at S given by
∂g(S)=Co {∂h(S, W)|h(S, W)=h(S, W * ), ∀W ∈ W},(19)
where W * = arg max W∈W h(S, W). In this paper, we take ∂h(S, W * ) as the ascend direction. Given an S, maximizing h(S, W) with respect to W is equivalent to the problem of classical PCA where the covariance matrix is Y T Q −1 Y. Hence, W * consists of the eigenvectors corresponding to the d largest eigenvalues of the covariance matrix. Let
f (S) = d 2 log det(Q)− i,j∈N i si,jφi,j − 1 4C ||S|| 2 F − max W∈W h(S, W)
Thus, we can compute subgradient ∂f (S) based on the optimal W * . Considering the symmetric property of S, the derivative of the log det term with respect to s i,j and a pair of indexes (i, j)
∈ I N = {j < i ∧ j ∈ N i , ∀i} is obtained by ∂ log det(Q) ∂si,j = Tr ∂ log det(Q) ∂Q T ∂[diag(S1) − S + (γ+1) 4 IN ] ∂si,j = Tr(Q −1 Ai,j)
where the matrix A i,j can be represented by
[Ai,j](m, n) = 1, m = n = i or m = n = j −1, m = i ∧ n = j or m = j ∧ n = i 0, otherwise.
As a result, the subgradient of the objective function f (S) can be computed as,
∀(i, j) ∈ I N , L. WANG
Algorithm 1 Structured Projection Learning (SPL)
1: Input: Data Y, neighbors Ni, ∀i, reduced dimension d parameters γ and C 2: S = 0 3: repeat 4:
Q = diag(S1) − S + (γ+1) 4 IN 5: Perform eigendecomposition Y T Q −1 Y = UΓU T with diag(Γ) sorted in a descend- ing order 6: W = U(:, 1 : d) 7: Z = γ+1 4 Q −1 YW 8:
Compute subgradient using (20) 9:
s (t+1) = Π s≥0 s (t) + 1 t ∂sf (s (t) ) 10: until Convergence 11: Output: Embedding X = γ 4 Q −1 YW ∂s i,j f (s)= 1 2 Tr(Q −T (dIN + γ 2 4 P)Ai,j)− 1 C si,j −2||yi −yj|| 2 (20) where P = YW * W * T Y T Q −1 .
Finally, we can solve (17) by using the projected subgradient ascend method as
s (t+1) = Π s≥0 s (t) + αt∂sf (s (t) ) ,
where Π is the projection on the non-negative set, and α t is the step size in the tth iteration. In order to guarantee the convergence, the step size should have the properties:
∞ t=1 α 2 t < ∞ and ∞ t=1 α t = ∞.
In the experiments, we take a typical example as α t = 1/t [7].
After learning the posterior distribution (16) with optimal S and W, we obtain the embedded points X by maximizing the logarithm of (16) as the maximum a posteriori estimation, i.e., max X log q(X), which can be rewritten as
min X 1 2
Tr(X T (4L + (γ + 1)IN )X) + γTr(X T YW). (21) Problem (21) is a quadratic optimization problem since matrix 4L + (γ + 1)I N is positive definite. Thus, we can obtain an analytic solution by setting its derivative with respect to X to zero, given by
X = γ(4L + (γ + 1)IN ) −1 YW = γ 4 Q −1 YW.(22)
As observed, there is a trivial solution for above objective, i.e., X = 0 if γ = 0. To overcome this issues, we have the following observation: the posterior distribution is the matrix normal distribution [20] given by
q(X) ∼ MN N,d (0, Σ, I d )(23)
where Σ = (4L + I N ) −1 is the sample-based covariance matrix and can also be interpreted as a regularized Laplacian kernel with regularization parameter λ > 0 [40]. As a result, we can apply KPCA on Σ to achieve the embedded data points if γ = 0 to avoid trivial solution. The pseudo-code of our proposed structured projection learning (SPL) is described in Algorithm 1. Our method can also leverage the fast eigendecomposition methods for finding a small number of large eigenvalues and eigenvectors. Thus, the computational complexity of Algorithm 1 is same as that in most of spectral based methods, but is much faster than semidefinite programming used in MVU. The theoretical convergence analysis of Algorithm 1 follows the projected subgradient method [7].
Algorithm 1 takes two parameters into account, C and γ, except the lowdimensional parameter d, which is a common parameter for dimensionality reduction and will not be discussed in this paper. Parameter C regulates the error tolerance of pairwise distances between original points and embedding points. The larger the C is, the smaller the error tolerance is imposed. In the case of C = ∞, the model does not allow the error. Parameter γ controls the noise of data in the generative model (5). More interestingly, this parameter plays an important role on balancing two distinct models: deterministic model (1) and generative model (5). The role becomes clear by investigating the proposed unified model (17). Specifically, problem (17) only learns the similarity matrix S using pairwise distances of original data as input if γ = 0. On the other hand, if γ > 0, the random noise of original points is simultaneously incorporated by regulating similarity matrix learning of deterministic model and data reconstruction of generative model. The merit of non-zero γ leads to an easy embedding process and meanwhile maintaining the intrinsic structure of data in low-dimensional space, which will be investigated in Section 6.1.
The proposed embedding framework provides a novel way to automatically learn a sparse positive similarity matrix W from a set of pairwise distances, and the sparse positive similarity matrix is purposely designed for learning the embedded points. This also provides a probabilistic interpretation why MVU takes KPCA as the embedding method after learning a kernel matrix.
Learning Explicit Graph Structure
For certain applications, we know the explicit representation of latent graph structure that generates the observed data, but both the embedded points and the correspondence between vertexes of the graph and the observed data are unknown. In this section, we adapt structured projection learning to learn an explicit graph structure, so that the learned embedded points reside on the optimal graph inside a set of feasible graphs with the given graph representation in the latent space.
2(γ + 1) ||Y − ZW T || 2 F + 2 (γ + 1) 2 Tr(Z T LZ) − 1 2(γ + 1) ||Y|| 2 F ,(24)12 L. WANG where Z = γ+1 4 Q −1 YW.
According to the property of Laplacian matrix and the above proposition, we reformulate the problem (17) as
max S min W∈W − d 2 log det(Q) + S, Φ Y + 1 4C ||S|| 2 F + γ 2 2(γ + 1) ||Y − ZW T || 2 F + 2γ 2 (γ + 1) 2 Tr(Z T LZ),(25)
where Z is a matrix according to Proposition 2, Φ Y is a distance matrix with the (i, j)th element as ||y i − y j || 2 . From (25), given S, we have
min W,Z 1 2γ ||Y − ZW T || 2 F + 2 γ 2 Tr(Z T LZ) (26) s.t. W T W = I d ,
by removing the following three terms that regulate S, i.e., d 2 log det(γI N + 4L), S, Φ Y , and 1 4C ||S|| 2 F . Thus, we can view model (17) as an approach to learn S from data by simultaneously preserving expected distances and optimizing (26) as the learning criterion for dimensionality reduction based on a graph. Similarly, we can also learn an explicit graph structure by incorporating known constraints of certain graph structures and minimizing criterion (26).
Following the above annotations, we can define a general graph representation. Let G = (V, E) be an undirected graph, where V = {V 1 , . . . , V N } is a set of vertices and E is a set of edges. Suppose that every vertex V i corresponds to a point z i ∈ Z ⊂ R d , which lies in an intrinsic space of dimension d. Denote the weight of edge (V i , V j ) as s i,j , which represents the similarity (or connection indicator) between z i and z j in the intrinsic space Z. We assume that matrix S ∈ S with the (i, j)th element as s i,j can be used to define the representation of a latent graph, where S is a set of feasible graphs with the given graph representation.
By combining the above ingredients, we formulate the following optimization problem, given by
min S,W,Z 1 2γ ||Y − ZW T || 2 F + 2 γ 2 Tr(Z T LZ) (27) s.t. W T W = I d , S ∈ S.
Problem (27) is more flexible than problem (26) since the graph structure represented by S in (27) can be directly controlled according to S, but (26) cannot, even though they share the same objective function. Thus, formulation (27) is a general framework for dimensionality reduction by learning an intrinsic graph structure in a low-dimensional space. In order to instantiate a new method, we have to specify the feasible set S of graphs.
4.2.
Dimensionality Reduction via Learning a Tree. We investigate a family of tree structures, which can be used to deal with various real world problems.
Given a connected undirected graph G = (V, E) with a cost c i,j associated with edge (V i , V j ) ∈ E, ∀i, ∀j, let T = (V, E T ) be a tree with the minimum total cost and E T be the edges forming the tree. In order to represent and learn a tree, we consider |A| − 1, ∀A ⊆ V}. The first constraint of S enforce the symmetric connection of undirected graph, e.g. s i,j = s j,i . The second constraint states that the spanning tree only contains |V| − 1 edges. The third constraint imposes the acyclicity and connectivity properties of a tree. It is difficult to solve an integer programming problem optimally. Instead, we resort to a relaxed problem by letting s i,j ≥ 0, that is,
{s i,j } as binary variables where s i,j = 1 if (V i , V j ) ∈ E T ,min S∈S T i,j si,jci,j,(28)
where the set of linear constraints over convex domain is given by S T = {S ≥ 0}∩S . Problem (28) can be solved by Kruskal's algorithm [12].
Let λ = 8 γ . We can equivalently rewrite (27) as the following optimization problem
min W,Z,S N i=1 ||yi − Wzi|| 2 + λ 2 i,j si,j||zi − zj|| 2 (29) s.t. W T W = I d , S ∈ S, where W = [w 1 , . . . , w d ] ∈ R D×d is an orthogonal set of d linear basis vectors w l ∈ R D , ∀l, Z = [z 1 , . . . , z N ] T ∈ R N ×d is represented by the projected data points of D in the low-dimensional space R d , S = [s i,j ] ∈ R N ×N is an adjacent matrix of a tree T = (V, E T ) where E T = {(i, j) : s i,j = 0}. If λ = 0, problem (29) is equivalent to the optimization problem of PCA, oth- erwise the data points Y are mapped into a low-dimensional space where {z i } N i=1
form a tree. Therefore, PCA is a special case of problem (29). Another important observation is the distances between any two latent points are computed in a low-dimensional space, i.e., ||z i − z j || 2 since latent points {z i } N i=1 are in R d . As a result, problem (29) can effectively mitigate the curse of dimensionality. For ease of reference, we name the problem (29) with λ > 0 as dimensionality reduction tree (DRTree). 4.3. Discriminative DRTree. DRTree projects data points in a high-dimensional space to latent points that directly form a tree structure in the low-dimensional space. However, the tree structure achieved might be at the risk of losing clustering information. In other words, some data points are supposed to form a cluster, but they are scattered to different branches of the tree, and distances between them on the intrinsic structure become large.
To incorporate the discriminative information, we introduce another set of latent points
{c k } K k=1 as the centers of {z i } N i=1
where c k ∈ R d so as to minimize the trade-off between the objective functions of K-means and DRtree. As a result, we formulate the following optimization problem
min W,Z,S,C,P N i=1 ||yi − Wzi|| 2 + λ 2 k,k s k,k ||c k − c k || 2 + γ K k=1 j∈P k ||zj − c k || 2 (30) s.t. W T W = I d , S ∈ ST ,
where the third term of the objective function is same as the objective function of K-means, P = {P 1 , . . . , P K } is a partition of {1, . . . , N }, C = [c 1 , . . . , c K ] T ∈ R K×d and γ ≥ 0 is a trade-off parameter between the objective function of DRTree and empirical quantization error of latent points {z i } N i=1 and {c k } K k=1 . Unlike problem (29), problem (30) is now regularized on centers (29) is a special case of problem (30) if K = N and γ → ∞ since the third term can be removed without changing the optimal solution of (30) due to z i = y i , ∀i at optimum. In this case, problems (29) and (30) are equivalent. Except for the special case, problem (30) is able to achieve discriminative and compact feature representation for dimensionality reduction since clustering objective and DRTree are optimized in a unified framework.
{y k } K k=1 instead of {z i } N i=1 . However, problem
The hard partition imposed by K-means, however, has several drawbacks. First, parameter K is data-dependent, so it is hard to set properly. Second, it is sensitive to noise, outliers, or some data points that cannot be thought of as belonging to a single cluster [18]. Soft partition methods such as Gaussian mixture modeling have also been used in modeling principal curves [5,44]. However, the likelihood of a Gaussian mixture model tends to be infinite when a singleton is formed [44].
To alleviate the problems from which the aforementioned methods suffer, we propose to replace the hard partition K-means with a relaxed regularized empirical quantization error given by
min W,Z,S,C,R N i=1 ||yi − Wzi|| 2 + λ 2 k,k s k,k ||c k − c k || 2 + γ K k=1 N i=1 r i,k ||zi − c k || 2 + σΩ(R) (31) s.t. W T W = I d , S ∈ ST , K k=1 r i,k = 1, r i,k ≥ 0, ∀i, ∀k,
where R ∈ R N ×K with the (i, k)th entry as r i,k , Ω(R) = N i=1 K k=1 r i,k log r i,k is the negative entropy regularization, and σ > 0 is the regularization parameter. The negative entropy regularization transforms hard assignment used in K-means to soft assignment used in Gaussian mixture models [32], and is also used in other tasks [31].
The following proposition shows that problem (31) with respect to {C, R} by fixing the remaining variables is equivalent to the mean shift clustering method [11], which is able to determine the number of clusters automatically and initialize centers {c k } K k=1 by latent points
{z i } N i=1 , ∀i if K = N .
The following lemma further shows that the optimal solution R has an analytical expression if C is given. Lemma 1. Given {W, Z, S, C}, Problem (31) has the optimal solution R given by the following analytical form, ∀k, ∀i Compute R with each element as (32) 9:
r i,k = exp −||zi − c k || 2 /σ K k=1 exp −||zi − c k || 2 /σ .Γ = diag(1 T R) 10: Q = 1 1+γ I + R 1+γ γ λ γ L + Γ − R T R −1 R T
11:
Perform eigendecomposition Y T QY = UΛU T and diag(Λ) is sorted in a descending order.
C = λ γ L + Γ −1 R T Z 15: until Convergence
The key difference between problem (31) and the traditional mean shift is that the latent points {z i } N i=1 in our model are variables and can be affected by dimensionality reduction and tree structure learning. We also build a connection between problem (31) and problem (30) as shown in the following proposition. The above properties of problem (31) facilitates the setting of parameters in different contexts of applications. In the case of dimensionality reduction, discriminative information might be important for some applications such as clustering problems, and Proposition 3 provides a natural way to form a cluster without predefining the number of clusters. In the case of finding K clusters, we prefer problem (30) to (31) since (30) is formulated in terms of K clusters directly. According to Proposition 4, the purpose of clustering can also be achieved by solving problem (31) with a small σ.
Alternating structure optimization [1] is used to solve problem (31). We first partition variables into two groups {W, Z, C} and {S, R}, and then solve each subproblem iteratively until the convergence is achieved.
Given {S, R}, we can obtain an analytical solution by solving problem (31) with respect to {W, Z, C}, which is discussed in Proposition 5. Before presenting Proposition 5, we first state a necessary condition of the proposition in Lemma 2 by proving the existence of the inverse matrix of 1+γ
γ ( λ γ L + Γ) − R T R. Lemma 2. The inverse of matrix 1+γ γ ( λ γ L + Γ) − R T R exists if N i=1 r i,k > 0, ∀k, where Γ = diag(1 T R)
and Laplacian matrix over a tree encoded in S is L = diag(S1) − S.
The conditions
W = U(:, 1 : d), Z = QYW, C = λ γ L + Γ −1 R T Z (33) where Q = 1 1+γ I N + R 1+γ γ λ γ L + Γ − R T R −1
R T , U and diag(Λ) are the eigenvectors and eigenvalues of matrix Y T QY with diag(Λ) sorted in a descending order, respectively, Γ = diag(1 T R) and the Laplacian matrix over a tree encoded in S is defined as L = diag(S1) − S. By fixing {W, Z, Y}, problem (31) with respect to {S, R} is jointly convex optimization problem with respect to S and R. Importantly, the subproblems with respective to S and R can be solved independently. According to Lemma 1, the optimum R is given by equation (32). To obtain the optimum S, the optimization problem with respect to S, i.e., min S∈S T k,k s i,j ||c k − c k || 2 , can be solved by Kruskal's method.
As discussed in Section 4.2, PCA is a special case of DRTree, so variable Z can be naturally initialized by PCA. By Proposition 3, we can set K = N and initialize Y = Z. The pseudo-code of Discriminative DRTree is given in Algorithm 2, briefly named as DDRTree. The implementations of DRTree and DDRTree in both MATLAB and R can be freely available 1 .
Connections to Existing Methods
We have developed several methods based on regularized empirical Bayesian inference so that both the global assumption of generative model and the local assumption of manifold learning are naturally incorporated into a unified model. In addition to the relationships of our method to MVU and various probabilistic models such as PPCA and GPLVM discussed in the related work, we further present a detailed discussion of other existing methods that are closely related to our proposed model. 5.1. Connection to Reversed Graph Embedding. Reversed graph embedding [32] was proposed to learn a set of principal points in the original space. Given a dataset D = {x i } N i=1 , it formulates the following optimization problem to learn a set of latent variables {z 1 , . . . , z N } in low-dimensional space with z i ∈ Z given by
min G∈ G b min f G ∈F min {z 1 ,...,z N } N i=1 ||xi − fG(zi)|| 2 (34) + λ 2 (V i ,V j )∈E bi,j||fG(zi) − fG(zj)|| 2 ,
where λ ≥ 0 is a parameter that controls the trade-off between the data reconstruction error and the objective function of reverse graph embedding, and G b is a feasible set of graphs with the set V of vertices and a set E of edges specified by a set {b i,j } of edge weights. We consider learning a function f G ∈ F and f G : Z → X over G = (V, E) that maps the intrinsic space Z to the input space X . For simplicity, the work [32] consider learning f G (z i ) as one single variable for variables f G and z i . In contrast, we in this paper aim to learn a set of points and the projection matrix as two separate variables, so that we can control the reduced dimensionality of the intrinsic space where the graph structure may reside. Moreover, we provide a general similarity matrix learning framework (17) for principal graph learning and special tree structure learning formulations (27) and (30).
5.2.
Connection to Maximum Entropy Unfolding. MEU [29] was proposed to directly model the density of observed data Y = [y 1 , . . . , y N ] by minimizing the KL divergence between a base density m(Y) and the density p(Y) given by min p(Y) p(Y) log(p(Y)/m(Y)), under the constraints on the expected squared inter-point distances φ i,j of any two samples, y i and y j . Let m(Y) be a very broad, spherical Gaussian density with covariance λ −1 I. The density function is then constructed as
p(Y) ∝ exp − 1 2 Tr(λYY T ) exp − 1 2 i j∈N i wi,jφi,j ,
even though the explicit form of these constraints is not given. The Laplacian matrix L defined over similarity w i,j is achieved by maximizing the logarithmic function of p(Y). Finally, the embedding is obtained by applying KPCA on the kernel matrix K = (L + λI N ) −1 .
One of the key differences is that our framework directly models the posterior distribution p(X|Y) of latent data, while MEU models the density of observed data. As a result, MEU has to assume that the data features are i.i.d. given the model parameters. However, this assumption is hardly satisfied if feature correlation exists. In contrast, our model assumes that the reduced features in the latent space are i.i.d, which is more reasonable than that used in MEU since the latent space is generally assumed to be formed by a set of orthogonal bases, such as PCA and KPCA.
5.3.
Connection to Structure Learning. SMCE [17] was proposed using 2 norm over the errors that measure the linear representation of every data point by using its neighborhood information. Similarly, 1 graph was learned for image analysis using 1 norm over the errors for enhancing the robustness of the learned graph [10]. These two methods [10,17] learn a directed graph from data so that they might yield suboptimal results by heuristically transforming a directed graph to an undirected graph for clustering and dimensionality reduction.
Instead of learning directed graphs by using the above two methods, an integrated model for learning an undirected graph by imposing a sparsity penalty (i.e., 1 prior) on a symmetric similarity matrix and a positive semidefinite constraint on the Laplacian matrix was proposed [27], given by,
max Q,S,σ 2 log det(Q) − 1 D Tr(QYY T ) − β D ||S||1 (35) s.t. Q = diag(S1) − S + IN /σ 2 si,i = 0, sij ≥ 0, i = 1, . . . , N σ 2 > 0
where β > 0 is a regularization parameter. Another approach dimensionality reduction through regularization of the inverse covariance in the loglikelihood (DRILL) [29] was proposed by applying an 1 prior to the elements of an inverse covariance, given by, obtain embedding points with less noises (smooth S-shape skeleton); (ii) SPL with = 0 changes the spatial intrinsic structure from S-shape to V-shape structure, while the true structures can be retained by introducing the generative model using a small ; (iii) a large C is preferred since it does not allow a large violation of pairwise distances from original data to the embedding data; (iv) SPL can achieve relatively good embeddings in a large interval of and C. All these observations are consistent with algorithmic analysis in Section 3.4.
Synthetic Data
We conduct experiments for data visualization on six synthetic datasets by comparing the proposed SPL method with various dimensionality reduction methods, including KPCA [6], MVU [11], MEU [18], LE [9], LLE [8], GPLVM [16], tSNE [19], SMCE [24]. Two datasets, Spiral and Zigzag 2 , are used, which are also used in the principal curve learning [45] and the underlying structures are spiral and zigzag curves, respectively. Dataset 2moons 3 is used in manifold-based semisupervised learning [21] for distinguishing two classes, so the true skeleton structure consists of two smooth curves. Following the work [10], we generate a 3-D datasets, helix by using drtoolbox, where the data point y i is computed by y i = [(2 + cos(8p i )) cos(p i ), (2 + cos(8p i )) sin(p i ), sin(8p i )] where p i is a random number that is sampled from a uniform distribution with support [0, 1]. Two datasets, Tree and Three-clusters are from [46], where the underlying structures are tree and three clusters with weak connections, respectively. Each data is projected to a 2-D space for data visualization by the above mentioned methods. Gaussian kernel is used for kernel-based methods or methods with prefixed similarity measurement. We set C = 10 3 and = 10 3 for SPL for all synthetic datasets. The same neighborhood size parameter K = 10 is used to construct neighborhood graph for neighborhood based methods. Fig. 2 shows the visualization results of embedding points in 2-D space obtained by six methods on six synthetic datasets. More comparing results are presented in the supplementary materials due to the space limit. We made the following observations. Some methods, such as KPCA, MVU, LLE and GPLVM, cannot tolerate noise of data so that the learned embeddings do not show a smooth skeleton structure. Some methods like MEU, LE, tSNE and SMCE can learn a smooth skeleton structure from noise data, but most of them are not consistent with the underlying structures. curves from two-moons data, but the curves learned by SPL are much smoother and more similar to moon-shaped structures than tSNE. On helix and zigzag, most methods obtain the embeddings that are very different from the true skeleton structures. Only SPL can obtain smooth skeleton structures by maintaining the underlying structures on all six datasets. These observations imply that SPL is able to learn a smooth skeleton structure from noisy data, and simultaneously retain the underlying structure of data in 2-D space.
Visualization of Two Real Datasets
A collection of 400 teapot images from [47] are used 4 for the purpose of visualization. These images are taken successively as a teapot was rotated 360 . Our goal is to unveil the circular structure in 3-D space that organizes the 400 images. Each image consists of 76 ⇥ 101 RGB pixels and is represented as a vector of size 23, 028. Thus, this data is high-dimensional. We run our algorithm with parameters 2 {0, 10 3 } and C = 10 3 . The visualization results are reported in Fig. 3. A two-dimensional representation of the same set of teapot images is given in [47], where MVU can also successfully arrange these images in a circle (see Figure 3 in [47]). This is the same as our result with = 0. However, when = 10 3 , the visualization of embeddings is slightly different, although the similar circular structure can be achieved. 4. http://www.cc.gatech.edu/⇠lsong/data/teapotdata.zip Figure 1. The visualization results for parameter sensitivity analysis of SPL on DistortedSShape by either varying γ ∈ {0, 10 −3 , 1} with prefixed C = ∞, or varying C ∈ {10 −2 , 10 3 , ∞} with γ = 10 −3 using the neighborhood size as 10 in 2-D space.
max Λ D 2 log det(Λ + λIN ) − 1 2 Tr((Λ + λI)YY T ) − ||Λ||1,(36)
and an implied covariance matrix is K = (Λ + λI N ) −1 .
By comparing (17) with (35) and (36), we can see that (17) is more similar than (35) instead of (36) since the sparsity is imposed on S, not on Λ, which is analogous to the Laplacian matrix L = diag(S1) − S. In fact, the sparisties of S and L are the same except the diagonal, but the properties of two matrices are very different. Our model demonstrates two key differences from the two methods. First, (17) has additional term for modeling the data generation process. If γ = 0 and the absolute difference is used (see Section 3.2 ), the primal problem of (17) is equivalent to (35). γ > 0 is useful to retain the structure of data after dimensionality reduction. Second, our structure learning model DRTree and DDRTree takes the spanning trees as the candidate structure, which is very different from 1 regularization over S so that our structure learning methods can transform original points into embedded points that form spanning trees in the latent space.
Experiments
We perform extensive experiments to verify our proposed models, SPL and DDRTree, separately, by comparing them to various existing dimensionality reduction methods on a variety of synthetic and real world datasets. 6.1. Structured Projection Learning.
6.1.1. Parameter Sensitivity Analysis. We investigate the parameter sensitivity of the proposed SPL method by varying γ and C on DistortedSShape, a synthetic data of 100 data points, which has been used in [25]. For simplicity, we study the influence of parameters by varying one and fixing the other. The neighborhood size is set to 10 and the reduced dimension is 2. We vary γ ∈ {0, 10 −3 , 1} and C ∈ {10 −2 , 10 3 , ∞}. Fig. 1 shows the original data and the resulting embeddings using SPL by varying γ and C. We made the following observations from Fig. 1: (i) SPL with a small γ can obtain embedding points with less noises (smooth S-shape skeleton); (ii) SPL with γ = 0 changes the spatial intrinsic structure from S-shape to V-shape structure, while the true structures can be retained by introducing the generative model using a small γ; (iii) a large C is preferred since it does not allow a large violation of pairwise distances from original data to the embedding data; (iv) SPL can achieve relatively good embeddings in a large interval of γ and C. All these observations are consistent with algorithmic analysis in Section 3.4. 6.1.2. Synthetic Data. We conduct experiments for data visualization on six synthetic datasets by comparing the proposed SPL method with various dimensionality reduction methods, including KPCA [38], MVU [50], MEU [29], LE [2], LLE [37], GPLVM [28], tSNE [47], SMCE [17]. Two datasets, Spiral and Zigzag 2 , are used, which are also used in the principal curve learning [25] and the underlying structures are spiral and zigzag curves, respectively. Dataset 2moons 3 is used in manifold-based semi-supervised learning [3] for distinguishing two classes, so the true skeleton structure consists of two smooth curves. Following the work [48], we generate a 3-D datasets, helix by using drtoolbox, where the data point y i is computed by y i = [(2 + cos(8p i )) cos(p i ), (2 + cos(8p i )) sin(p i ), sin(8p i )] where p i is a random number that is sampled from a uniform distribution with support [0, 1]. Two datasets, Tree and Three-clusters are from [52], where the underlying structures are tree and three clusters with weak connections, respectively. Each data is projected to a 2-D space for data visualization by the above mentioned methods. Gaussian kernel is used for kernel-based methods or methods with prefixed similarity measurement. We set C = 10 3 and γ = 10 −3 for SPL for all synthetic datasets. The same neighborhood size parameter K = 10 is used to construct neighborhood graph for neighborhood based methods. Fig. 2 shows the visualization results of embedding points in 2-D space obtained by nine methods on six synthetic datasets. We made the following observations. Some methods, such as KPCA, MVU, LLE and GPLVM, cannot tolerate noise of data so that the learned embeddings do not show a smooth skeleton structure. Some methods like MEU, LE, tSNE and SMCE can learn a smooth skeleton structure from noise data, but most of them are not consistent with the underlying structures. For example, only tSNE and SPL can obtain two separate curves from two-moons data, but the curves learned by SPL are much smoother and more similar to moon-shaped structures than tSNE. On helix and zigzag, most methods obtain the embeddings that are very different from the true skeleton structures. Only SPL can obtain smooth skeleton structures by maintaining the underlying structures on all six datasets. These observations imply that SPL is able to learn a smooth skeleton structure from noisy data, and simultaneously retain the underlying structure of data in 2-D space.
Visualization of Two Real Datasets.
A collection of 400 teapot images from [49] are used 4 for the purpose of visualization. These images are taken successively as a teapot was rotated 360 • . Our goal is to unveil the circular structure in 3-D space that organizes the 400 images. Each image consists of 76×101 RGB pixels and is represented as a vector of size 23, 028. Thus, this data is high-dimensional. We run our algorithm with parameters γ ∈ {0, 10 −3 } and C = 10 3 . The visualization results are reported in Fig. 3. A two-dimensional representation of the same set of teapot images is given in [49], where MVU can also successfully arrange these images in a circle (see Figure 3 in [49]). This is the same as our result with γ = 0. es (smooth S-shape the spatial intrinsic ture, while the true ing the generative s preferred since it ise distances from v) SPL can achieve interval of and t with algorithmic lization on six synd SPL method with s, including KPCA GPLVM [16], tSNE Zigzag 2 , are used, e learning [45] and zigzag curves, renifold-based semiing two classes, so wo smooth curves. 3-D datasets, helix t y i is computed by i )) sin(p i ), sin(8p i )] s sampled from a . Two datasets, Tree e underlying strucweak connections, 2-D space for data methods. Gaussian s or methods with set C = 10 3 and atasets. The same s used to construct ased methods. lts of embedding ethods on six synre presented in the ce limit. We made ds, such as KPCA, e noise of data so curves from two-moons data, but the curves learned by SPL are much smoother and more similar to moon-shaped structures than tSNE. On helix and zigzag, most methods obtain the embeddings that are very different from the true skeleton structures. Only SPL can obtain smooth skeleton structures by maintaining the underlying structures on all six datasets. These observations imply that SPL is able to learn a smooth skeleton structure from noisy data, and simultaneously retain the underlying structure of data in 2-D space.
Visualization of Two Real Datasets
A collection of 400 teapot images from [47] are used 4 for the purpose of visualization. These images are taken successively as a teapot was rotated 360 . Our goal is to unveil the circular structure in 3-D space that organizes the 400 images. Each image consists of 76 ⇥ 101 RGB pixels and is represented as a vector of size 23, 028. Thus, this data is high-dimensional. We run our algorithm with parameters 2 {0, 10 3 } and C = 10 3 . The visualization results are However, when γ = 10 −3 , the visualization of embeddings is slightly different, although the similar circular structure can be achieved. Another data is USPS handwritten digits 5 , which contains handwritten digits from 0 to 9 with different written styles. Each one is a gray image of size 16 × 16. The vectorization of each image with label "1" is used to study the embeddings in 2-D space since these images demonstrate very clear written styles. There are 1, 100 images in total. Following the same setting as did for teapot data, the visualization results of USPS data are shown in Fig. 3, where each image is shown in the position located by its associated embedded point in 2-D space. We observe that the images of "1"s are sorted in an order such that the angular degree of image "1"s is changed continuously, and the perpendicular ones are shown in the middle of the learned skeleton structure. The visualization difference between γ = 0 and γ = 10 −3 is that the latter demonstrates a smoother change than the former since the structure of the latter is an arc while the structure of the former is a right angle.
6.1.4. Classification Performance of Embeddings. Table 1. The leave-one-out cross validation accuracy of one-nearest neighbor classifier over ten datasets. N is the number of data points. c is the true number of clusters. D is the original dimensionality and d is the reduced dimensionality. The best results are in bold. As shown in Table 1, ten datasets taken from the UCI and Statlib repositories are used to evaluate classification performance of embedded points learned by baseline methods same as those used in the experiments on synthetic data. The reduced dimensionality of data is shown in Table 1 by preserving 95% of energy of data. Following [50], we use the leave-one-out cross validation accuracy as the criterion for evaluating one-nearest neighbor classifier on the embeddings learned by these baseline methods. For methods that require K nearest neighbor graph as the input, we tune K ∈ [5,10,15,20,30,50]. We tune the parameter λ ∈ [0.01, 0.1, 1, 10] for SMCE. Other parameters are set as the default values in the drtoolbox 6 . In addition, we tune C = [10, 10 3 ] and γ ∈ [0, 10 −3 ]. The best results are reported for every baseline methods by tuning their own parameters. Table 1 shows the leave-one-out cross validation accuracy of one-nearest neighbor classifier over the embeddings learned by nine methods on ten benchmark datasets. It is clear to see that SPL is competitive to tSNE in terms of classification accuracy, and demonstrates much better than the others. As shown in [47], tSNE helps to achieve good classification performance by learning a new embedding of original data. The learning criterion of tSNE is better suitable for clustering/classification, but it is not appropriate for learning skeleton structures in a latent space as observed in Section 6.1.2 and Section 6.1.3. These results imply that SPL is not only suitable for learning skeleton structures in latent spaces from high-dimensional data, but also it can achieve competitive or better classification performance on the learned embedding points by comparing with various existing methods.
Iris
Learning Tree Structures From Real Datasets.
We investigate the ability of our proposed DDRTree method to automatically discover tree structures from three real-world datasets. The latent tree structures of these three datasets include principal curves, hierarchical tree structures, and a cancer progression path. 6.2.1. Principal Curve. The same teapot data in Section 6.1.3 is used to justify our tree structure learning methods. Similar to [41], the data in each dimension is normalized to have zero mean and unit standard deviation. And, a kernel matrix Y is generated where Y (i, j) = exp(−||y i − y j || 2 /D).
We run our proposed DDRTree method using the kernel matrix as the input. We set λ = 0.1 × N and d = 36 that keeps 95% of total energy. The experimental results of DDRTree are shown in Figure 4. The principal curve (Figure 4(a)) is shown in terms of the first 3 columns of the learned projection matrix W as the coordinates where each dot represents one image. The sampled images at intervals of 30 are plotted for the purpose of visualization. Figure 4(b) shows the linear chain dependency among teapot images following the consecutive rotation process. We can see that the curve generated by our method is consistent with the rotating process of the 400 consecutive teapot images.
A similar result is also recovered by CLUHSIC, which assumes that the label kernel matrix is a ring structure [41]. However, there are three main differences. First, we learn a projection space where images are arranged in the form of a principal curve, while CLUHSIC applies KPCA to transform original data to an orthogonal space where clustering is performed. Second, the principal curve generated by our DDRTree method is much smoother than that obtained by CLUHSIC (see Figure 4 in [41]). Third, our method learns the adjacency matrix from the given dataset, but CLUHSIC requires a label matrix as a prior. We attempted to run MVU by keeping 95% energy, i.e., d = 36. However, storage allocation fails due to the large memory requirement of solving a semidefinite programming problem in MVU. Hence, MVU fails to learn a relatively large intrinsic dimensionality. However, our method does not have this issue.
6.2.2. Hierarchical Tree. Facial expression data 7 is used for hierarchical clustering, which takes into account both the identities of individuals and the emotion being expressed [41]. This data contains 185 face images (308 × 217 RGB pixels) with three types of facial expressions (NE: neutral, HA: happy, SO: shock) taken from three subjects (CH, AR, LE) in an alternating order, with around 20 repetitions each. Eyes of these facial images have been aligned, and the average pixel intensities have been adjusted. As with the teapot data, each image is represented as a vector, and is normalized in each dimension to have zero mean and unit standard deviation. A kernel matrix is used as the input to DDRTree. λ = 0.1 × N and d = 185. The experimental results are shown in Figure 5. We can clearly see that three subjects are connected through different branches of a tree. If we take the black circle in Figure 5(a) as the root of a hierarchy, the tree forms a two-level hierarchical structure. As shown in Figure 5(b), all three facial expressions from three subjects are also clearly separated. A similar two-level hierarchy is also recovered by CLUHSIC (Figure 3(b) in [41]). However, the advantages of using DDRTree discussed above for teapot images are also applied here. In addition, we can observe more detailed information from the tree structure. For example, LE@SO is the junction to other two subjects, i.e., AR@SO and CH@SO, which can be observed from the 9th row of the adjacency matrix 5(b). This observation suggests that the shock is the most similar facial expression among three subjects. However, CLUHSIC is not able to obtain this information. We are particularly interested in studying human cancer, a dynamic disease that develops over an extended time period. Once initiated from a normal cell, the advance to malignancy can to some extent be considered a Darwinian process -a multistep evolutionary process -that responds to selective pressure [19]. The disease progresses through a series of clonal expansions that result in tumor persistence and growth, and ultimately the ability to invade surrounding tissues and metastasize to distant organs. As shown in Figure 6(a), the evolution trajectories inherent to cancer progression are complex and branching [19]. Due to the obvious necessity for timely treatment, it is not typically feasible to collect time series data to study human cancer progression [28]. However, as massive molecular profile data from excised tumor tissues (static samples) accumulates, it becomes possible to design integrative computation analyses that can approximate disease progression and provide insights into the molecular mechanisms of cancer. We have previously shown that it is indeed possible to derive evolutionary trajectories from static molecular data, and that breast cancer progression can be represented by a high-dimensional manifold with multiple branches [42].
We interrogate a large-scale, publicly available breast cancer dataset [13] for cancer progression modeling. The dataset contains the expression levels of over 25, 000 gene transcripts obtained from 144 normal breast tissue samples and 1, 989 tumor tissue samples. By using a nonlinear regression method, a total of 359 genes were identified that may play a role in cancer development [42]. In the analysis, we set λ = 5 × N and d = 80 that retains 90% of energy. Figure 6(b) shows the learned latent structures and latent points in a reduced dimensional space. Each tumor sample is colored with its corresponding PAM50 subtype label, a molecular approximation that uses a 50-gene signature to group breast tumors into five subtypes including normal-like, luminal A, luminal B, HER2+ and basal [34]. Basal and HER2+ subtypes are known to be the most aggressive breast tumor types. The learned graph structure in the low-dimensional space suggests a linear bifurcating progression path, starting from the normal tissue samples, and diverging to either luminal A or basal subtypes. The linear trajectory through luminal A continues to luminal B and to the HER2+ subtype. Significant sidebranches are evident for both luminal A and luminal B subtypes, suggesting that these subtypes can be further delineated. The revealed data structure is consistent with the proposed branching architecture of cancer progression shown in Figure 6.
Conclusion
In this paper, we proposed a general probabilistic framework for dimensionality reduction, which not only takes the noise of data into account, but also utilizes the neighborhood graph as the locality information. Based on this framework, we presented a model that can learn a smooth skeleton of embedding points from highdimensional, noisy data. In order to learn an explicit graph structure, we developed another new dimensionality reduction method that learns a latent tree structure and low-dimensional feature representation simultaneously. We extended the proposed method for clustering problems by imposing the constraint that data points belonging to the same cluster are likely to be close along the learned tree structure. The experimental results demonstrated the effectiveness of the proposed methods for recovering intrinsic structures from real-world datasets. Dimensionality reduction via learning a graph is formulated from a general graph, so the development of new dimensionality reduction methods for various specific structure is also possible.
Appendix A. Proofs
A.1. Proof of Proposition 1. In order to prove the results, we first derive the explicit expression of q(X) by taking the Lagrangian duality and obtaining the partial dual problem.
By fixing W and introducing dual variables S ∈ R N ×N with the (i, j)th element denoted as s i,j ≥ 0, the Lagrangian function L(q(X), W, S, ξ) can be formulated as KL(q(X)||π(X)) − log p(Y|X, W, γ)q(X)dX + C||ξ|| 2 F + i,j∈N i si,j E q(X) ||xi − xj|| 2 − φi,j − ξi,j .
According to the duality theorem, we have the following KKT conditions, 1 + log(q(X)/π(X)) − log p(Y|X, W, γ) + si,j||xi − xj|| 2 = 0 si,j E q(X) ||xi − xj|| 2 − φi,j − ξi,j = 0, ∀i, j ∈ Ni 2Cξi,j − si,j = 0, ∀i, j ∈ Ni.
The first equation leads to the following analytic form of posterior distribution q(X) ∝ π(X)p(Y|X, W, γ) exp − i,j∈N i si,j||xi − xj|| 2 .
The second KKT condition states that the distance is preserved if s i,j = 0 and ξ i,j = 0. Let S ∈ R N ×N with the (i, j)th element as s i,j if j ∈ N i and 0 otherwise. In other words, the non-zero elements of S stands for the distance preserving equalities. The third KKT condition leads to ξ i,j = si,j 2C . Its dual problem is then obtained by substituting ξ and q(X) back to Lagrangian function as where the third equality holds due to the orthogonal constraint, x q is the qth column of X, and w q is the qth column of W. After substituting it back to log Z(W, S), we have the optimization problem.
A.2. Proof of Proposition 2. Taking the first derivative of the objective function with respect to Z and setting it to zeros, we have − 1 (γ + 1) (Y − ZW T )W + 4 (γ + 1) 2 LZ = 0 and the optimal solution Z = (I N + 4 γ+1 L) −1 YW. By substituting Z back to the objective function, we obtain Since Y is constant and independent of W and Z, the proof is completed.
A.3. Proof of Lemma 1. Each row r i of R in Problem (31) can be solved independently. By introducing dual variables α, we have the Lagrangian function defined for each subproblem with respect to r i as L(r i , α) = K k=1 r i,k ||z i − c k || 2 + σ log r i,k + α( K k=1 r i,k − 1). The KKT conditions can be obtained as ||z i − c k || 2 + σ(1 + log r i,k ) + α = 0, ∀k and K k=1 r i,k = 1, r i,k ≥ 0, ∀k, which lead to the following analytic solution r i,k = exp(−||z i − c k || 2 /σ − (1 + α/σ)). According to the KKT condition K k=1 r i,k = 1, we have exp(1 + α/σ) = K k=1 exp(−||z i − c k || 2 /σ). This completes the proof.
A.4. Proof of Proposition 3. According to Lemma 1, we have an analytical solution of R shown in (32). By substituting (32) back into the objective function of (31), we have following derivations
K k=1 N i=1 r i,k ||zi − c k || 2 + σ N i=1 K k=1 r i,k log r i,k = −σ N i=1 log K k=1 exp −||zi − c k || 2 /σ .
The equality is obtained by the optimal solution (32) and the simplex constraint of (31). The optimization problem now becomes unconstrained optimization problem
min C −σ N i=1 log K k=1 exp −||zi − c k || 2 /σ .
According to first order optimal condition [6], we can obtain the optimal solution by letting the first derivative be zero, i.e.,
∇c k = −σ N i=1 exp −||zi − c k || 2 /σ (−2(c k − zi)/σ) K k=1 exp (−||zi − c k || 2 /σ) = 2 N i=1 (c k − zi) · exp −||zi − c k || 2 /σ K k=1 exp (−||zi − c k || 2 /σ) = 2 N i=1
(c k − zi) · r i,k = 0.
By solving the optimal condition problem, we have the optimal solution of c k as
c k = N i=1 r i,k zi N i=1 r i,k ,
which is equivalent to the update rule used in mean shift [11] if we consider the kernel function as a Gaussian distribution with bandwidth σ.
A.5. Proof of Proposition 4. The optimal solution (32) is a softmin function with respect to distance ||z i −c k || 2 . If σ → 0, r i,k = 1 if k = min k=1,...,K ||z i −c k || 2 , and otherwise r i,k = 0. In the case of σ → 0, we have
lim σ→0 K k=1 N i=1 r i,k ||zi − c k || 2 + σ N i=1 K k=1 r i,k log r i,k = N i=1 min k=1,...,K ||zi − c k || 2 = K k=1 i∈P k ||zi − c k || 2 ,
where the negative entropy is equal to zero. This completes the proof.
A.6. Proof of Lemma 2. To prove the existence of the inverse of matrix 1+γ γ ( λ γ L+ Γ) − R T R, we prove that this matrix is positive definite. Given any non-zero vector v ∈ R K , we have the following derivations
v T 1 + γ γ ( λ γ L + Γ) − R T R v ≥ 1 + γ γ v T Γv − v T R T Rv = 1 + γ γ v T Γv + v T (diag(R T R1) − R T R)v − v T diag(R T R1)v ≥ 1 γ v T Γv + v T Γv − v T diag(R T 1)v = 1 γ v T Γv,
where the first and second inequalities follow the fact that the Laplacian matrix is positive semi-definite. If where Γ = diag(1 T R) and the Laplacian matrix over a tree represented by S is L = diag(S1 K ) − S. Let h(W, Z, C) be the objective function of the above optimization problem. By setting the partial derivative of h(W, Z, C) with respect to C to zero ∂h(W, Z, C)/∂C = 2λLC−2γR T Z+2γΓC = 0, we have an analytical solution of C given by
C Z = λ γ L + Γ −1 R T Z.
By substituting C Z into h(W, Z, C), we yields
h(W, Z, C Z ) = ||Y − ZW T || 2 F + γTr(ZZ T ) − γTr Z T R λ γ L + Γ −1 R T Z .
Similarly, by setting the partial derivative of h(W, Z, C Z ) with respect to Z to zero, we obtain
Z W = (1 + γ)IN − γR λ γ L + Γ −1 R T −1
YW.
L. WANG
According to the Woodbury formula [22], the optimal solution of Z W can be further reformulated as Z W = QYW where
Q = (1 + γ)IN − γR λ γ L + Γ −1 R T −1 = 1 1 + γ IN − 1 (1 + γ) 2 R − 1 γ λ γ L + Γ + 1 1 + γ R T R −1 R T = 1 1 + γ IN + R 1 + γ γ λ γ L + Γ − R T R −1 R T .
According to Lemma 2, the inverse matrix exists. The objective function can thus be further written as a function with respect only to W, which is given by h(W, Z W , C Z W ) = Tr(Y T Y) − Tr(W T Y T QYW). The optimization problem of h(W, Z W , C Z W ) with respect to W is equivalent to the following maximization optimization problem,
max W trace(W T XQX T W), s.t. W T W = I d ,
which is similar to the problem of PCA and can be solved optimally by eigendecomposition on Y T QY.
p(D|M, Θ)q(M)dµ(M)
3. 4 .
4Algorithm Analysis. The computational complexity of Algorithm 1 can be estimated as follows: solving problem (17) takes approximately O(N 2.37 ) for computing logdet and inversion of matrix Q at each iteration; computing subgradient and function value of f (S) takes O(N 3 ) due to the eigendecomposition. The time complexity of Algorithm 1 takes the order of O(N 3 ).
4. 1 .
1Explicit Graph Structure Learning. Before presenting the model for explicitly learning a graph structure, we first introduce an important result.
Proposition 2 .
2Given an S, min W −h(S, W) is equivalent to the following optimization problem min W∈W,Z
and 0 otherwise. Denote S = [s i,j ] ∈ {0, 1} N ×N . The integer linear programming formulation of minimum spanning tree (MST) can be written as: min S∈S0 i,j s i,j c i,j , where S 0 = {S ∈ {0, 1} N ×N }∩S and S = {S = S T }∩{ 1 2 i,j s i,j = |V|−1}∩{ 1 2 Vi∈A,Vj ∈A s i,j ≤ PROBABILISTIC DIMENSIONALITY REDUCTION VIA STRUCTURE LEARNING 13
( 32 )
32Proposition 3. Given {W, Z, S}, λ = 0 and assuming K = N , problem (31) with respect to {C, R} can be solved by a mean shift clustering method by initializing C = Z. Algorithm 2 Discriminative DRTree (DDRTree) 1: Input: Data matrix Y, parameters λ, σ and γ 2: Initialize Z by PCA 3: K = N , C = Z 4: repeat 5:c k.k = ||c k − c k || 2 ,
Proposition 4 .
4If σ → 0, (31) is equivalent to(30).
N i=1 r i,k > 0, ∀k, always hold in the case of the soft-assignment obtained by Proposition 3.L. WANG Proposition 5. By fixing {S, R}, problem (31) with respect to {W, Z, C} has the following analytical solution:
Fig. 1 .
1The visualization results for parameter sensitivity analysis of SPL on DistortedSShape by either varying 2 {0, 10 3 , 1} with prefixed C = 1, or varying C 2 {10 2 , 10 3 , 1} with = 10 3 using the neighborhood size as 10 in 2-D space.
For
Fig. 3 .
3Visualization results of our proposed method with parameters 2 {0, 10 3 } and C = 10 3 on Teapot and USPS with tag 1.
Fig. 1 .Figure 2 .
12The embedding results of nine methods on six synthetic data. The visualization results of embedding points using six methods on six synthetic data. Due to the space limit, the visualization results of the remaining methods are reported in the supplementary materials. of SPL on DistortedSShape by either varying 2 {0, 10 3 , 1} with prefixed 10 3 using the neighborhood size as 10 in 2-D space.
Fig. 3 .
3Visualization results of our proposed method with parameters 2 {0, 10 3 } and C = 10 3 on Teapot and USPS with tag 1.
Figure 3 .
3Visualization results of our proposed method with parameters γ ∈ {0, 10 −3 } and C = 10 3 on Teapot and USPS with tag 1.
Figure 4 .
4Experimental results of DDRTree applied to Teapot images. (a) principal curve generated by DRTree. Each dot represents one teapot image. Images following the principal curve are plotted at intervals of 30 for visualization. (b) The adjacency matrix of the curve follows the ordering of the 400 consecutive teapot images with 360 • rotation.
Figure 5 .
5Experimental results of our DDRTree method performed on facial expression images. (a) A hierarchical tree generated by DRTree. Each dot represents one face image. Images of three types of facial expressions from three subjects are plotted for visualization. The black circle is the root of the hierarchical structure; (b) The adjacency matrix of the tree on nine blocks indicates that each block corresponds to one facial expression of one subject. 6.2.3. Cancer Progression Path.
Figure 6 .
6Graph structure learned by DDRTree on breast cancer dataset with d = 80 and visualized in three-dimensional space spanned by the first three components of the learned projection matrix.
N
partition function Z(W, S) is further simplified by marginalizing out X asZ(W, S) = π(X)p(Y|X, W, γ) exp − (xi|0, I d )N (yi|Wxi, γ −1 ID) exp(−2Tr(X T LX)q ) T ( (γ + 1)IN + 4L)x q + γ(x q ) T Yw q )dx q = 1 (2π) N D γ −N D exp(− γ 2 ||Y|| 2 ) det((γ + 1)IN + 4L) −d/2 · exp( γ 2 2Tr(W T Y T ((γ + 1)IN + 4L) −1 YW))
i,k > 0, ∀k, the matrix 1+γ γ ( λ γ L + Γ) − R T R is positive definite. A.7. Proof of Proposition 5. With simple matrix manipulation, problem (31) with respect to {W, Z, Y} by fixing {S, R} can be written as min W,Z,C γ Tr(ZZ T ) − 2Tr(R T ZC T ) + Tr(C T ΓC) + ||Y − ZW T || 2 F + λTr(C T LC) : W T W = I d ,
example, only tSNE and SPL can obtain two separate 2. https://www.lri.fr/⇠kegl/researchUdeM/research/pcurves /implementations/Samples/ 3. manifold.cs.uchicago.edu/manifold regularization/2moons.matTeapot
USPS-1
= 0
= 10 3
http://liwang8.people.uic.edu/
https://www.lri.fr/∼kegl/researchUdeM/research/pcurves /implementations/Samples/ 3 manifold.cs.uchicago.edu/manifold regularization/2moons.mat 4 http://www.cc.gatech.edu/∼lsong/data/teapotdata.zip
http://www.cs.nyu.edu/∼roweis/data/usps all.mat
https://lvdmaaten.github.io/drtoolbox/
http://www.cc.gatech.edu/∼lsong/data/facedata.zip
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| [] |
[
"Irreversibility and the Arrow of Time",
"Irreversibility and the Arrow of Time"
] | [
"Jürg Fröhlich "
] | [] | [] | Within the general formalism of quantum theory irreversibility and the arrow of time in the evolution of various physical systems are studied. Irreversible behavior often manifests itself in the guise of "entropy production." This motivates me to begin this paper with a brief review of quantum-mechanical entropy, a subject that Elliott Lieb has made outstanding contributions to, followed by an enumeration of examples of irreversible behavior and of an arrow of time analyzed in later sections. Subsequently, a derivation of the laws of thermodynamics from (quantum) statistical mechanics, and, in particular, of the Second Law of thermodynamics, in the forms given to it by Clausius and Carnot, is presented. In a third part, results on diffusive (Brownian) motion of a quantum particle interacting with a quasi-free quantum-mechanical heat bath are reviewed. This is followed by an outline of a theory of friction by emission of Cherenkov radiation of sound waves in a system consisting of a particle moving through a Bose-Einstein condensate and interacting with it. In what may be the most important section of this paper, the fundamental arrow of time inherent in Quantum Mechanics is discussed. | null | [
"https://arxiv.org/pdf/2202.04619v1.pdf"
] | 246,679,980 | 2202.04619 | 9b568fe92648c4c3eaf67eeab429f58a13fc9613 |
Irreversibility and the Arrow of Time
February 10, 2022
Jürg Fröhlich
Irreversibility and the Arrow of Time
February 10, 2022
Within the general formalism of quantum theory irreversibility and the arrow of time in the evolution of various physical systems are studied. Irreversible behavior often manifests itself in the guise of "entropy production." This motivates me to begin this paper with a brief review of quantum-mechanical entropy, a subject that Elliott Lieb has made outstanding contributions to, followed by an enumeration of examples of irreversible behavior and of an arrow of time analyzed in later sections. Subsequently, a derivation of the laws of thermodynamics from (quantum) statistical mechanics, and, in particular, of the Second Law of thermodynamics, in the forms given to it by Clausius and Carnot, is presented. In a third part, results on diffusive (Brownian) motion of a quantum particle interacting with a quasi-free quantum-mechanical heat bath are reviewed. This is followed by an outline of a theory of friction by emission of Cherenkov radiation of sound waves in a system consisting of a particle moving through a Bose-Einstein condensate and interacting with it. In what may be the most important section of this paper, the fundamental arrow of time inherent in Quantum Mechanics is discussed.
Introductory Remarks
As Elliott Lieb likes to say, it is often at least as rewarding to discover and / or explore new and sometimes unexpected consequences of known Laws of Nature as it is to discover new such Laws. In this paper, some ideas and results on irreversibility and the arrow of time in the behavior of physical systems, derived from mostly well established theory, are reviewed.
The arrow of time is a short hand for the basic dichotomy of past and future, the past consisting of facts or actualities, and the future consisting of uncertainties and potentialities. In a timereversal invariant, deterministic world inhabited by an infinitely clever mind who is able to compute all processes, past and future -definitely not our world! -there would be no arrow of time.
In this paper, a physical system is said to exhibit an arrow of time if complete knowledge of its past states (its actualities) does not suffice to predict its future states with certainty; and / or if knowledge of its present state does not contain enough information to enable one to reconstruct past states of the system, in case one failed to observe them, i.e., if information about the pastif not recorded -is dissipating. 1 For example, knowing the state of a Brownian walker at time t does not enable one to determine its states at times < t (in case one did not record them), nor its states at times > t; see Sect. 4. In quantum mechanics, knowing the exact state at time t of a system featuring events does usually not enable one to reconstruct its states at times < t, let alone to predict future states, with certainty; see Sect. 6. 2 By irreversibility I mean the following: A dynamical process happening in a physical system is called irreversible if, for all practical purposes, it is impossible to run the process backwards. For example, motion with friction, studied in Sect. 5, if run backwards, would become continuously accelerated motion, which one does not observe; or then, a steady flow of heat energy from a colder reservoir to a hotter reservoir is never observed; see Sect. 3. This paper is based on a lecture about irreversibility and the arrow of time given by the author at various institutions, during the past seven years. 3 It addresses a theme that has been discussed by numerous authors (see [1] and references given there), and one might expect that not much can be said that is really new. The results described on the following pages have grown out of collaborations of the author with quite a number of colleagues and friends that have extended over at least two decades; and I think that some of our results have clarified some important aspects of the general topic alluded to in the title and abstract.
In this paper, I focus on a quantum-mechanical description of the physical systems to be considered; but I am confident that many of the results discussed in Sections 3 -5 can also be derived -mutatis mutandis -within classical physics. My survey is intended for physicists; mathematical precision is not its main priority. However, where I state results in the form of theorems the reader can be confident that there are mathematically precise statements and proofs thereof in the literature referred to in the text.
It is with great pleasure that I dedicate this paper to an admired colleague, mentor and friend of many years, Elliott H. Lieb, on the occasion of his 90 th birthday -with all best wishes!
Relative Entropy
In this section we recapitulate the definition of quantum-mechanical entropy [2] and some of its properties used afterwards. Of particular importance for the purposes of this paper is relative entropy.
Let Ω be a density matrix acting on a Hilbert space H; (i.e., Ω is a quantum-mechanical "state"). The von Neumann entropy of Ω is defined by S(Ω) := −Tr Ω · lnΩ (1) It has the following properties:
1. S(Ω) ≥ 0, ∀Ω, with "=" only if Ω is pure, i.e., a rank-1 orthogonal projection.
2. S(·) is strictly concave.
3. S(·) is subadditive and strongly subadditive; see [2] Statement 1 is obvious. Statement 2 and subadditivity of entropy follow from Klein's Inequality: Let f be a real-valued, strictly convex function on the real line, and let A and B be bounded, self-adjoint, trace-class operators on H. Then
with "=" only if A = B. To prove concavity and subadditivity of S(·) one chooses f (x) = x · lnx.
Proof of Inequality (2): This proof is simple enough to be presented here. Let {ψ j } ∞ j=0 be a complete orthonormal system (CONS) of eigenvectors of B corresponding to eigenvalues β j , j = 0, 1, 2, . . . Let ψ be an arbitrary unit vector in H, and set c j := ψ j , ψ , ∀j. Then convexity of f implies the following two inequalities:
ψ, f (B)ψ = j |c j | 2 f (β j ) ≥ f j |c j | 2 β j = f ψ, Bψ (3) f ψ, Bψ ≥ f ( ψ, Aψ + f ψ, Aψ · ψ, (B − A)ψ .(4)
If ψ is an eigenvector of A then the right side of (4) is given by
= ψ, f (A) + f (A) · (B − A) ψ(5)
Eq. (2) follows by summing inequalities (3) and (5) over a CONS of eigenvectors of A. Let Σ and Ω be two arbitrary density matrices on H. The relative entropy of Σ with respect to Ω is defined by S(Σ Ω) := T r(Σ (lnΣ − lnΩ)), (6) assuming that ker(Ω) ⊆ kerΣ. Crucial properties of S(Σ Ω) are:
• Positivity: S(Σ Ω) ≥ 0, as follows from inequality (2)
• Convexity: S(Σ Ω) is jointly convex in Σ and Ω.
Joint convexity can be derived from the following inequality: Let T be a trace-preserving, completely positive map on the convex set of density matrices on H. Then S(Σ Ω) ≥ S(T (Σ) T (Ω)) .
Exercise: Show that this inequality, due to Lindblad and Uhlmann [3,4], implies Strong Subadditivity, first established by Lieb and Ruskai [2]:
S(Ω 12 ) + S(Ω 23 ) − S(Ω 123 ) − S(Ω 2 ) ≥ 0 ,
where Ω 123 is a density matrix on H 1 ⊗ H 2 ⊗ H 3 , and the density matrices Ω 12 , ... are obtained by taking partial traces. A well known application of strong subadditivity is to proving existence of the thermodynamic limit of specific entropy. For the purposes of this paper, inequality (7) is particularly important. It will be used to derive the 2 nd Law of Thermodynamics from quantum statistical mechanics; see Sect. 3.
I have learned the neat proof of Klein's Inequality and the right way of introducing the 2 nd Law of thermodynamics from my teacher Res Jost (see [5]). Incidentally, he also warned some of us that, at a party, one should never start a conversation about • irreversibility and the arrow of time;
• the foundations and the "interpretation" of Quantum Mechanics;
• religious faith.
For, most people mistakenly believe that they have some solid understanding of these topics, and they get surprisingly emotional when one tells them that they might actually be wrong. (I have made the experiment myself.) There appears to be much confusion -even among grown-up theoretical physicists -about irreversibility and the origin of time's arrow. For example, most people mistakenly believe that all fundamental laws of Nature treat past and future in a symmetric way (i.e., do not distinguish between past and future). And, in the author's modest opinion, there is quite an enormous confusion about the foundations of Quan-tum Mechanics and its deeper meaning! In this paper, I attempt to uncover causes of the arrow of time and of irreversibility in physics, in particular in quantum physics (touching on its foundations).
The following examples of physical systems exhibiting irreversible behavior and / or an arow of time will be considered.
• Irreversibility: A macroscopic physical system prepared in an unlikely initial state (e.g., a small subsystem coupled to some macroscopic reservoirs at different temperatures and / or chemical potentials; the Universe) -see Sects. 3 and 7.
• Arrow of time: A small subsystem, e.g., a quantum-mechanical particle, evolving under the influence of noise from its environment, such as a heat bath -this is the subject matter of Sect. 4, where a theory of Quantum Brownian Motion is sketched.
• Irreversibility: A physical system, such as a particle coupled to the electromagnetic field or to a Bose gas exhibting Bose-Einstein condensation, leaking energy and "information" into massless modes that propagate to ∞ and, hence, are not accessible to observation, anymorethis forms the contents of Sect. 5, where the phenomenon of friction by emission of Cherenkov radiation is analyzed.
• Arrow of time: Isolated physical systems featuring "events," as described by Quantum Mechanics -this fundamental example is treated in Sect. 6.
The Second Law of Thermodynamics -Clausius and Carnot
Elliott Lieb has had a successful and widely noted collaboration with J. Yngvason on the foundations of thermodynamics [6]. It may therefore be appropriate to review some recent and not so recent results concerning a derivation of the laws of thermodynamics from (quantum) statistical mechanics and their implications concerning the irreversibility of certain natural processes.
I. Preliminaries on thermodynamics from the point of view of quantum statistical mechanics
We begin this section by recalling some facts in thermodynamics that can be derived from quantum statistical mechanics.
Let R be a macroscopically large thermal reservoir (or heat bath) whose state space is a separable Hilbert space, denoted by H R , and whose Hamiltonian is denoted by H R . In order for our arguments to be meaningful mathematically, we will initiallly suppose that reservoirs are finitely extended in space, i.e., are confined to a compact region (e.g., a rectangle), Λ, in physical space, E 3 . Subsequently the thermodynamic limit, Λ E 3 , will most often be taken. But we will not always make it explicit whether the reservoirs are assumed to be finitely extended or to fill all of physical space, and we will not discuss conditions that guarantee the existence of the thermodynamic limit. These matters have been widely discussed in the literature; see, e.g., [7]. We suppose that every family of thermal reservoirs (indexed by spatial regions Λ they are confined to) considered in the following can be described by a net, A(O) O⊂E 3 , of local algebras of operators representing physical quantities of the reservoirs, where O is an arbitrary bounded open set in physical space, and [8]. We may then define
A(O 1 ) A(O 2 ) whenever O 1 ⊂ O 2 ; seeA := O⊂E 3 A(O),(8)
(or A = closure of the right side in the operator norm). We assume that, in the thermodynamic limit, a thermal reservoir R obeys the Zero th Law of thermodynamics: Initial states of R within a "large class of states" -when restricted to a local algebra A(O) -approach states that are indistinguishable from thermal equilibrium states, restricted to the algeba A(O), as time t tends to ∞, for an arbitrary bounded region O. (A somewhat more precise formulation of the Zero th Law can be found at the end of this section.) Let C be a "small system" coupled to R. The state space of C is denoted by H C ; the one of the total system, S = R ∨ C, is then chosen to be H S := H R ⊗ H C . The Hamiltonian of S is given by
H(t) := H R | H R ⊗ 1 H C + H C (t),
where H C (t) is a possibly time-dependent bounded quasi-local operator on H S . For the purposes of this paper quasi-local operators are defined to be operators contained in the algebra A ⊗ B(H C ), with B(H C ) the algebra of bounded operators on H C , and A as in (8). We assume that H(t) → H ∞ , as t → ∞, in a sense made more precise in Eq. (14) below. Let U (t, s) | t, s in R be the unitary propagator generated by the time-dependent Hamiltonians H(t), i.e.,
∂ ∂t U (t, s) = −iH(t) U (t, s) , ∀t, s,(9)
U (t, u) · U (u, s) = U (t, s) , U (t, t) = 1, ∀ t, u, s .
(We will always use units such that = 1.) See, e.g., [9] for results concerning (9). General states of S are described by density matrices, Ω, Σ, . . . , on H S . Let
Ω t = U (t, s)ΩU (s, t),
be the true state of S at time t, where Ω is the initial state of S prepared at time s. The equation of motion of Ω t is given by
∂ ∂t Ω t = −i[H(t), Ω t ] ≡ −iL(t) Ω t ,(10)
where L(t) is the so-called "Liouvillian" of the system; (the Liouvillians L(t), t ∈ R, can be viewed as self-adjoint operators on H S ⊗ H S ). We also define the instantaneous thermal equilibrium state at inverse temperature β = 1/k B T associated with the Hamiltonian H(t) to be given by
Ω β t := Z t (β) −1 exp[−β H(t)], with Z t (β) = Tr exp[−β H(t)]
, (assuming that the region Λ the system is confined to is bounded). It is convenient to use the notation
ω t (A) := Tr Ω t · A , ω β t (A) := Tr Ω β t · A ,(11)
where A ∈ A ⊗ B(H C ) is an arbitrary quasi-local operator. We will usually assume that, for every quasi-local A, the thermodynamic limit, Λ E 3 , of the expectation values ω t (A) exists (at least in the sense of convergent subsequences), uniformly in time t in bounded intervals of the time axis. (We will not indicate the dependence of these expectation values on Λ.)
Definition of some thermodynamic quantities:
• Internal energy: U (t) := ω t H(t)
and U C (t) := ω t H C (t) .
• Reservoir power:
P(t) ≡ P R (t) ≡Q(t) := d dt ω t H R = −iω t [H R , H C (t)] , where [H R , H C (t)
] is a quasi-local operator; (so that the right side may have a well defined thermodynamic limit).
We then have thatU
(t) =Q(t) +U C (t) = ω t Ḣ C (t) .(12)
For a time-dependent function f (t), t ∈ R, we define
∆f ≡ ∆ t s f := f (t) − f (s) = t sḟ (τ )dτ.
Integrating Eq. (12) in time, we find that
∆U C = ∆Q + ∆W,(13)
where ∆Q = −∆Q is the heat energy absorbed by the subsystem C, and
∆W ≡ ∆ t s W := t s ω τ Ḣ C (τ ) dτ
is the work done on C between time s and time t. Eq. (13) is the First Law of thermodynamics. Next, we review an important example of irreversible behavior, namely the phenomenon of return to equilibrium (see [10,11,12,13]), which expresses a remarkable stability property of thermal equilibrium against local perturbations. We assume that the Liouvillians of S (see Eq. (10)) have the following properties:
(i) The Liouvillians L t , t ∈ R, converge strongly to an operator L ∞ , as t → ∞, and there exists a finite constant C such that, uniformly in the region Λ ⊂ E 3 S is confined to,
∞ L τ − L ∞ L ∞ + i −1 dτ ≤ C .(14)
(ii) In the thermodynamic limit (Λ E 3 ), the spectrum of L(t) is absolutely continuous, except for a simple eigenvalue 0, for all times t.
Remark: When attempting to prove property (ii) for concrete models one usually has to assume that the Hamiltonians H C (t) satisfy a "Fermi Golden Rule"-type condition, for all times t ∈ R, 4 and that their norms, H C (t) , are sufficiently small, uniformly in t. Furthermore, for the time being, one has to assume that the reservoir R has good dispersive properties; e.g., that it is described by an ideal quantum gas.
The following result provides a first example of irreversible behavior.
Return to Equilibrium. We suppose that the thermodynamic limit of the system S has been constructed, and that properties (i) and (ii) stated above hold. Let the initial state, ω, of S at some time t 0 be given by
ω(A) = ω β t 0 M * A M /ω β t 0 (M * M ) , A ∈ A ⊗ B(H S ) ,
where A is the algebra defined in (8), ω β t 0 is the instantaneous equilibrium state at some inverse temperature β < ∞ (see Eq. (11)), and M is an arbitrary quasi-local operator. Let ω t be the true state of S at time t, with initial condition ω t 0 = ω. Then lim
t→∞ ω t (A) = ω β ∞ (A) , ∀ quasi-local operators A ,(15)
where ω β ∞ is the equilibrium state at inverse temperature β corresponding to the limiting Hamiltonian H ∞ .
Remarks: (1) Mathematical study of this phenomenon was initiated in [10].
(2) Given properties (i) and (ii), above, Return to Equilibrium is a simple consequence of the KMS condition [8] (exercise). The hard problem is to verify property (ii) for concrete models.
(3) 'Return to Equilibrium' can be extended in various ways. For example, one can prove a result on thermal ionization of atoms immersed in a heat bath (see [14]); and, assuming that H C (t) is periodic in time t with period τ , one can derive a result concerning the approach to time-τ -periodic states after having taken the thermodynamic limit (see [15]).
(4) An irreversible phenomenon more subtle, mathematically, than 'Return to Equilibrium' is the phenomenon of Relaxation of isolated systems (such as atoms or molecules) coupled to the quantized electromagnetic field to their Groundstates. In some models of atoms or molecules prepared in states with an energy below the ionization threshold, various rigorous results on Rayleigh scattering of light, the Bohr frequency condition, and relaxation to the groundstate have been established; see [16,17,18,19]. In these models, the atomic nuclei are assumed to be static, the electrons have non-relativistic kinematics, pair-creation is neglected, and the interactions of electrons with the quantized electromagnetic field are cut off at high energies (i.e., are regularized in the ultraviolet). The phenomenon of relaxation to the groundstate is a consequence of spontaneous emission of photons by electrons in excited states of the atom (as originally predicted by Einstein in 1916). Although this process has been understood at some heuristic level for a long time, the job to understand it mathematically rigorously has turned out to be quite involved. 5 In thermodynamics, quasi-static processes, and in particular quasi-static isother-mal processes (which are reversible), play an important role. Such processes can be described very nicely within quantum statistical mechanics. In [20] the following variant of the adiabatic theorem of quantum mechanics has been proven.
Isothermal Theorem. Consider a system S in the thermodynamic limit (Λ E 3 ), as above. We choose the Hamiltonian of C to be given by
H C (t) ≡ H (τ ) C (t) := K C (τ −1 · t) , τ > 0 , where K C (t) is τ -independent.
We assume that the corresponding Liouvillian, L t ≡ L (τ ) t := K t/τ , of S (see (10)) satisfies a number of conditions specified in [20], including self-adjointness, differentiability properties in time and spectral properties (in particular, that 0 is a simple eigenvalue of K t , for all times t). Let us suppose that the true state of S at some initial time t 0 is given by an instantaneous equilibrium state at some inverse temperature β < ∞, i.e., ω = ω t 0 = ω β t 0 . Then, for t in an arbitrary finite interval of length ∝ τ of the time axis R, the true state ω t of S follows the instantaneous equilibrium state ω β t corresponding to the Hamiltonian H (τ ) (t) = H (τ ) C (t) + H R , up to an error functional whose norm converges to 0, as the adiabatic time scale τ of the process tends to ∞.
This result (whose precise statement and proof can be found in [20]) has an interesting application in thermodynamics: Let t j = τ ·s j , j = 1, 2, with s 1 < s 2 , and let H C (t) := H
C (t) = K C (τ −1 t). Then ∆ t 2 t 1 U C + ∆ t 2 t 1 Q = ∆ t 2 t 1 W (13) = t 2 t 1 ω t Ḣ (τ ) C (t) dt = t 2 t 1 ω t d dt [K C (τ −1 t)] dt
Assuming that the initial state is an instantaneous equilibrium state, ω t 1 = ω β t 1 , and applying the Isothermal Theorem, we conclude that
∆ t 2 t 1 W = t 2 t 1 ω β t d dt [K C (τ −1 t)] dt + o(1) =∆ t 2 t 1 F C + o(1) ,(16)
i.e.,
∆W = ∆F C + o(1), where F C (t) := β −1 lnZ t (β) − ln Tr exp[−βH R ]
is the free energy of the small system C (coupled to the thermal reservoir). In words: In a quasistatic isothermal (hence reversible) process, the amount of work done on the subsystem C equals the change of its free energy. We conclude this subsection with some remarks about entropy. When a "small" system C is in contact with a macroscopically large environment R the entropy of C should be defined to be the negative relative entropy of the true state, Ω t , of the total system S = C ∨R, given a reference state, Ω ref , with the property that C and R are decoupled from one another. If R is a thermal reservoir then it is assumed to be in thermal equilibrium at some inverse temperature β before being coupled to C (Zero th Law ). Since we have chosen the state space of C to be finite-dimensional, we may choose the normalized trace, tr, as the reference state of C and set
Ω ref := Z(β) −1 exp[−βH R ] ⊗ 1| H C , with Z(β) = Tr exp[−βH R ] ⊗ 1| H C .
We define the entropy of C at time t to be given by
S C (t) := −k B lim Λ E 3
Tr Ω t lnΩ t − lnΩ ref
(6) ≤ 0 .(17)
We note that Tr Ω t lnΩ t and Tr Ω t · Z(β)1 are independent of time t. Hencė
S C (t) = − 1 T d dt ω t H R = − 1 TQ (t) , or ∆S C = − 1 T ∆Q = 1 T ∆Q ,(18)
as expected, where T = (k B β) −1 is the absolute temperature. It is instructive to specialize this equation to entropy changes observed in quasi-static isothermal processes. In such processes, ω t = ω β t , up to small error terms, where ω β t is the (thermodynamic limit of the) instantaneous equilibrium state introduced in Eq. (11). Specializing the definition of S C in (17) to this choice of states, one finds that
TṠ C (t) = d dt ω β t H C (t) − ω β t Ḣ C (t) ,
which, with (13) and after intergrating over time, yields ∆U C = T ∆S C + ∆W (18) = ∆Q + ∆W , or, with (16) ,
∆F C = ∆U C − T ∆S C .(19)
To conclude this subsection, we emphasize that, from the point of view of mathe-matical analysis, the hard problems are (1) to analyze the spectrum of the Liouvillians L t t∈R in the thermodynamic limit of the reservoirs and to establish property (ii) after Eq. (14), and (2) to prove the Isothermal Theorem.
II. The Second Law of thermodynamics in the formulation due to Clausius
We consider a physical system, S, consisting of two macroscopic thermal reservoirs, R 1 and R 2 , at temperatures T 1 > T 2 , respectively, coupled to one another by a "thermal contact", C, i.e., a small system separated from R 1 and R 2 by diathermal walls:
S = R 1 ∨ C ∨ R 2 .
We describe S quantum-mechanically, supposing, for example, that the reservoirs are filled with ideal quantum gases confined to some large disjoint regions, Λ 1 and Λ 2 , respectively, in physical space (which, later on, will approach infinitely extended half spaces in E 3 ). The state spaces of the reservoirs are denoted by H 1 and H 2 , the state space of C by H C , which, for simplicity, we assume to be finite-dimensional. The Hamilton operators describing the time evolution of R 1 and R 2 are denoted by H 1 and H 2 , respectively; the Hamiltonian of C, which includes interaction terms between C and R 1 ∨R 2 , by H C . As in I, above, H C is assumed to be a bounded quasi-local operator acting on
H S := H 1 ⊗ H C ⊗ H 2 .
The total Hamiltonian of S is given by the operator
H := H 1 + H 2 + H C on H S (with H 1 identified with H 1 | H 1 ⊗ 1| H C ⊗ 1| H 2 ,
and similarly for H 2 ). The state of S at a time t ∈ R is given by a density matrix, Ω t , see I. Before C is "opened", this state is assumed to be given by a tensor product of Gibbs states (as suggested by the Zero th Law of thermodynamics): The heat power, P i , absorbed by R i , i = 1, 2, is defined by
Ω ref := Z −1 exp(−β 1 H 1 ) ⊗ 1 C ⊗ exp(−β 2 H 2 ), β i := 1/k B T i .(20)P i (t) := d dt Tr(Ω t H i ) = −iTr Ω t [H i , H C ] .(21)
(Since the operators [H i , H C ] are quasi-local, the thermodynamic limit of right side of (21) may exist.) If R 1 and R 2 have "good dispersive properties", e.g., are filled with an ideal quantum gas or with black-body radiation, one proves that, in the thermodynamic limit,
ω t → t→∞ ω ∞ ,
where ω ∞ is a so-called non-equilibrium stationary state (NESS). This is among the somewhat hard results proven in this context; see [22,23] and followers. 6 As in Sect. 1.2 and I, we consider the relative entropy
S(Ω t Ω ref ) = Tr Ω t [lnΩ t − lnΩ ref ] ,
with Ω ref as in (20). It is easy to see that its time derivative is given bẏ
S(Ω t Ω ref ) = β 1 P 1 (t) + β 2 P 2 (t) .(22)
Thus, if ω t → ω ∞ , as t → ∞, with ω ∞ time-translation-invariant, then the following statements hold in the thermodynamic limit:
1.Ṡ(Ω t Ω ref ) has a limit, denoted by σ ∞ , as t → ∞, and it follows from the non-negativity of relative entropy, see (7), that
σ ∞ ≥ 0 (positivity of entropy production)(23)
and 2. P 1 (t) has a limit, denoted by −P ∞ , as t → ∞, and
P 1 (t) + P 2 (t) → 0 , as t → ∞.(24)
From Eqs. (22), (23) and (24) we derive the Second Law of thermodynamics in the formulation of Clausius:
P ∞ ( 1 T 2 − 1 T 1 ) >0 ≥ 0 ⇒ P ∞ ≥ 0,(25)
i.e., at large times when a stationary state has been reached, heat flows from the warmer reservoir, R 1 , to the colder one, R 2 . This is the Second Law of thermodynamics in the formulation of Clausius. So far, we have worked in the canonical ensemble: The subsystem C is connected to thermal reservoirs only through diathermal walls; matter is not exchanged between C and the reservoirs. Actually, it is easy to extend our analysis and the results reviewed so far to the grand-canonical ensemble, with subsystems connected to reservoirs through channels transmitting not only energy but also matter (atoms or molecules); see, e.g., [22].
Further results:
• For certain simple models one can show that, at large times, the entropy production σ ∞ is strictly positive, and
• Ohm's Law holds; see [22].
• The Onsager relations hold; see [24,25].
III. The Second Law in the formulation of Carnot
We may replace the thermal contact C between two thermal reservoirs R 1 and R 2 by a heat engine, e.g., a "locomotive", E, that extracts energy from a heater R 1 , releases part of it into a "cooler" R 2 , with T 1 > T 2 , and performs work. Thus, the system we consider is
S = R 1 ∨ E ∨ R 2 .
The engine E is driven periodically in time, with some period τ > 0. This means that the Hamiltonian, H E (t), depends periodically on time t, with period τ . Assuming that R 1 and R 2 have good dispersive properties, e.g., are filled with ideal quantum gases or black-body radiation, and developing a Floquet theory for Liouvillians (see [15]), one proves that, in the thermodynamic limit, the true state, ω t , of S approaches a time-periodic state, ω asy t , with the same period, τ , as the Hamiltonian of E.
Using this fact, we are able to derive Carnot's bound on the degree of efficiency, η, of heat engines. The existence of time-periodic asymptotic states ω asy t and the non-negativity of relative entropies, i.e., S(Ω t Ω ref t ) ≥ 0, with Ω ref as in Eq. (20) (with C replaced by E) imply that, asymptotically as time t → ∞, the entropy production per cycle (of length τ ), ∆S ∞ ≡ ∆S ∞ (τ ), is non-negative. Let ∆Q 1 be the amount of heat energy released per cycle by the heater R 1 into E, and let ∆Q 2 be the heat energy lost per cycle by E into the cooler R 2 , after the state of S has approached ω asy t . Using the identity (22)) we conclude that Carnot's bound on the degree of efficiency holds.
0 ≤ ∆S ∞ = − ∆Q 1 T 1 + ∆Q 2 T 2 (see
Theorem. (Carnot's bound)
The degree of efficiency, η, of the heat engine E satisifies the familiar bound η := ∆W
∆Q 1 = ∆Q 1 − ∆Q 2 ∆Q 1 ≤ T 1 − T 2 T 1 ≡ η Carnot , and η = η Carnot iff ∆S ∞ (τ ) = 0 .(26)
This is Carnot's formulation of the 2 nd Law of thermodynamics. (Note that, since, asymptotically, the state of S approaches one that is periodic in time, the change of the internal energy of E per cycle, ∆U E , very nearly vanishes at very large times. This explains why ∆U E does not appear in the formula for η.)
Remark: Defining the entropy of a small system, E, coupled to an unobserved environment, R (such as some thermal reservoirs), as (-1 ×) the relative entropy of the state of the total system, S = E ∨ R, with respect to a reference state in which E is decoupled from R, we find that, in stationary or time-periodic states, which are approached asymptotically, the entropy production is always non-negative (and strictly positive in certain simple model systems). This fact implies that in irreversible adiabatic processes of the small system E (defined appropriately) its entropy can never decrease (actually increases in certain simple model systems) -as expected on the basis of standard formulations of the Second Law.
IV. The Zero th Law of thermodynamics We conclude this section with a comment on the 0 th Law of thermodynamics (which is probably the deepest law of thermodynamics). It can be formulated as follows: Assume that the particleand energy densities of the initial state of a macroscopic system, R, are uniformly bounded in space, and that the dynamics of R is translation-invariant and time-independent. Then, asymptotically as time t tends to ∞ and as the thermodynamic limit is approached, the state of R approaches a state locally indistinguishable from an equilibrium state whose particle-and energy densities are given by the spatial averages of the particle-and energy densities of the initial state.
A fundamental problem is to find the right hypotheses on the properties of R enabling one to establish the validity of the 0 th Law in a form involving realistic time scales, and hence to show that heat baths or thermal reservoirs really exist. (This can also be phrased as the problem of "eth" vs. many-body localization). The 0 th Law represents a key problem encountered in trying to derive the laws of thermodynamics from statistical mechanics that is not understood very well, yet. Preliminary results can be found in [26].
Quantum Brownian Motion
In this section we describe results, formulated and established within quantum mechanics, concerning an example of a physical system exhibiting an arrow of time (in the sense specified at the beginning of this paper), namely the example of Brownian motion of a particle coupled to a thermal reservoir.
As is well known, the theoretical study of Brownian motion was initiated by Einstein and Smoluchowski in 1905, and their work played a significant role in finally establishing the atomistic nature of matter in experiments by Perrin and others. But it took a century before one succeeded in establishing the diffusive nature of motion of a quantum-mechanical particle interacting with a translation-invariant, infinitely extended quantum-mechanical thermal reservoir (disregarding from results on simple exactly solved model systems; see, e.g., [27], and references given there). Here we describe some fairly recent results on diffusive motion that have appeared in [28,29].
We consider a quantum-mechanical tracer particle hopping on a lattice Z 3 that has two internal states, a groundstate and an excited state. Its Hilbert space of pure state vectors is given by H P := 2 (Z 3 ) ⊗ C 2 , and its Hamiltonian is given by
H P := − ∆ X 2M ⊗ 1 + 1 ⊗ σ z ,(27)
where M is the mass of the tracer particle, ∆ X is the discrete Laplacian, X is the position operator of the particle, with specX = Z 3 , and σ z = 1 0 0 −1 ; we work in units where = 1.
The tracer particle is immersed in a Bose gas. 7 The atoms constituting the Bose gas are free, non-relativistic particles with Bose-Einstein statistics and a mass m = 1 2 (m M ) propagating freely in E 3 . The dynamics of the Bose gas is described by the usual Hamiltonian, denoted by H BG , which is self-adjoint and non-negative on the bosonic Fock space F.
The Hamiltonian of the coupled system, S = tracer particle ∨ Bose gas, is given by
H := H P + H BG + ν R 3 dx W (X − x){b * (x) + b(x)},(14)
acting on the Hilbert space H P ⊗ F, where W describes interactions of the tracer particle with the atoms of the Bose gas; it is given by some 2 × 2 matrix-valued pair potential that is chosen in such a way that it can cause transitions between the groundstate and the excited state of the particle (Fermi's Golden Rule!); the coupling constant ν is given by ν := ρ 0 /2, with ρ 0 a dimensionless quantity proportional to the density of the Bose gas, and b * (x) and b(x) are creation-and annihilation operators of phonons (= quanta of sound waves) in the Bose gas satisfying the usual canonical commutation relations.
We distinguish two regimes:
(A) ν small, M = ν −2 M 0 , where M 0 is a constant (kinetic regime); and (B) ν large, with M = ν 2 M 0 , ν −2 ↔ (mean-field regime).
We first study regime (A), assuming that the Bose gas is in thermal equilibrium at some finite positive temperature T = (k B β) −1 . The problem of interest to us is to understand the nature of the motion of the tracer particle. For this purpose we will study the diffusion constant, D, which characterizes its large-time behavior. We prepare the system in an initial state of the form ρ T := σ 0 ⊗ ω β (β = 1/k B T ), where σ 0 is given by a density matrix, Σ 0 , on H P localized near X = 0 (i.e., Tr Σ 0 X 2 is finite). We study this system in the Heisenberg picture: The state ρ T is taken to be time-independent, but the operators representing physical quantities of the system evolve in time according to Heisenberg's equations. Thus, X(t) is the operator representing the position of the tracer particle at time t. By (·) T we denote expectations with respect to the state ρ T .
One attempts to prove and, after much hard work, succeeds in proving [28,29] that
[X(t) − X(0)] 2 T ∼ D · t, as t → ∞,(28)
for a diffusion constant D proportional to ν 2 , as can be guessed on the basis of simple dimensional analysis:
D ≈ (v ν × t ν ) 2 /t ν ∝ ν 2 < ∞ ,
where v ν ∝ ν 2 is the average speed of the particle (recall that its mass is proportional to ν −2 ), and t ν ∝ ν −2 is the average time elapsing between two subsequent collisions of the particle with an atom in the Bose gas (recall that the strength of interaction between the particle and sound waves in the Bose gas is O(ν)).
Rough idea of the proof of (28): Evidently
[X(t) − X(0)] 2 T = t 0 dτ t 0 dσ Ẋ (τ ) ·Ẋ(σ) T .(29)
Assuming that Ẋ (τ ) ·Ẋ(σ) T decays integrably fast in |τ − σ|, the right side of (29) grows linearly in t, as t → ∞, which implies that D < ∞.
Integrable decay, with a decay time of O(ν 2 ), is due to fast de-correlation (in time) of the direction of motion of the particle as a result of collisions with atoms in the Bose gas. This can be established, mathematically, with the help of a rather intricate "cluster expansion in time," which yields precise information on quantities such as Ẋ (τ ) ·Ẋ(σ) T .
Some key ideas underlying this expansion are as follows: Let Z ν t denote the completely positive map acting on the algebra, B(H P ), of bounded operators on H P that takes an operator A ≡ A(0) representing a physical quantity, A, characteristic of the tracer particle at time 0 to the operator A(t) representing A at time t; assuming that the degrees of freedom of the heat bath (the Bose gas) have been "traced out." If the dynamics of the particle were ballistic then
A(t) = Z ν t (A) ≈ exp[itH ren P ] A exp[−itH ren P ], and [X(t) − X(0)] 2 T ∝ ν 4 · t 2 ,
for an effective (renormalized) Hamiltonian H ren P ∝ −ν 2 ∆ X . Instead, thanks to the interactions of the tracer particle with the atoms in the Bose gas, which are in a thermal state at temperature T , one expects that
A(t) = Z ν t (A) ≈ exp t i ad σ z + ν 2 M T (A), A = A(0) ∈ B(H P ),(30)
where M T is the generator of a semi-group of completely positive maps on B(H P ) to be chosen selfconsistently; (it is related to a linear Boltzmann equation for the Wigner distribution of the state σ t of the tracer particle in the Schrödinger picture). If the maps exp t i ad σ z + ν 2 M T , t ≥ 0, gave rise to the exact time evolution of operators related to the tracer particle then Eq. (28) would hold, for some positive constant D = O(ν 2 ). The idea is then to expand the effective dynamics, Z ν t (·), of the tracer particle around the right side of (30). Such an expansion has a chance to converge uniformly in time t, whereas an expansion of the effective dynamics around the free (ballistic) dynamics of the tracer particle cannot converge uniformly in t, because the latter has the wrong behavior for very large times.
The reason why we assume that the tracer particle has an internal degree of freedom is that, in this study, we would like to profit from the ultraviolet regularization of the problem provided by the lattice. We would therefore not like to have to re-scale space and time (in such a way that the continuum limit is approached). An internal degree of freedom makes it easy to satisfy momentumand energy conservation in collisions of the tracer particle with atoms in the Bose gas; (see [28] for a more detailed explanation).
We note that, in this quantum-mechanical study of diffusive motion,
v ν ∼ Ẋ (t) T = O(ν 2 ),
i.e., the expected speed of the tracer particle is finite (and small, for small ν). 8 This is in marked contrast to ordinary Brownian motion for whichẊ(t) does not exist. Furthermore, a variant of the Equipartition Theorem holds for Ẋ (t) 2 .
These results appear to be the first and only results on a derivation of diffusive motion from fundamental quantum dynamics in a model that cannot be solved exactly (see [28]; and [29] for stronger results and better methods of proof).
It may be of interest to note that if one couples a particle moving in a disordered (random) potential to a thermal reservoir then (Anderson) localization breaks down, and the particle exhibits diffusive motion, no matter how weak the coupling of the particle to the thermal reservoir is. Suppose the disorder in the random potential is large enough for the particle to be localized before it is coupled to the reservoir, no matter what its initial energy is. Then the diffusion constant tends to 0, as the strength of the coupling of the particle to the reservoir tends to 0; not because the mass of the particle diverges (as in the kinetic regime (A)), but because of Anderson localization in the absence of thermal noise. Results like this have been proven in [31] for simplified models with Markovian thermal reservoirs.
Hamiltonian Friction
"A moving body will come to rest as soon as the force pushing it no longer acts on it in the manner necessary for its propulsion." (Aristotle)
In this section we consider an archetypal example of an irreversible phenomenon, namely friction, in a model of a tracer particle propagating in an ideal Bose gas at zero temperature exhibiting Bose-Einstein condensation; (see [32,33,34] for various results on friction in mechanical models). This is a problem in "tribology," a branch of physics first explored by Aristotle, Leonardo da Vinci and Guillaume Amontons. In our model friction arises because the particle emits Cherenkov radiation of sound waves into the condensate of the Bose gas. 9 To be able to come up with mathematically precise results, we study the model in the mean-field regime introduced in Sect. 4, regime (B), where the mass, M , of the particle behaves like M = ν 2 M 0 , M 0 = O(1), ν −2 ∼ ρ −1 0 ↔ , with ν −2 → 0. We consider a particle without internal degrees of freedom moving in physical space E 3 , rather than hopping on Z 3 . We suppose that the particle interacts with the atoms of the Bose gas through a smooth pair potential, W (X − x), of short range, where X ∈ E 3 denotes the position of the tracer particle, and x ∈ E 3 is the position of one of the atoms in the Bose gas. The gas is in a state of high density ρ 0 and very low temperature T ≈ 0, exhibiting a Bose-Einstein condensate (i.e., the gas is in its groundstate).
In the mean-field limit described above, the dynamics of this system approaches the classical Hamiltonian dynamics of a particle, whose state is described by its position X ∈ E 3 and momentum P ∈ R 3 , coupled to a complex-valued (c-number) field, β(x), x ∈ E 3 , describing sound waves in the Bose gas, where β is an element of the Sobolev space H 1 (E 3 ). The phase space, Γ, of the system is given by
Γ := R 6 × H 1 (E 3 ) .
Its symplectic structure is encoded in the Poisson brackets
{β(x), β(y)} = iδ(x − y), {X i , P j } = −δ i j ,(31)
and
{β # (x), β # (y)} = {X i , X j } = {P i , P j } = 0 , ∀ x, y, i and j.
The Hamilton functional of the system is given by
H cl (X,P ; β, β) := P 2 2M 0 − F · X+ +2 E 3 dx W (X − x) Reβ(x) + E 3 dx (∇β)(x) · (∇β)(x) ,(32)
where F is a constant external force gently pushing the particle, and W is a smooth function on E 3 of short range. The equations of motion of the particle and of the sound-wave field β of the Bose gas are given byẊ
t = M −1 0 P t ,Ṗ t = F − 2 R 3 dx ∇W (X t − x) Reβ t (x),(33)
and
iβ t (x) = −∆β t (x) + W (X t − x)(34)
Remark: Canonical quantization of the Hamiltonian system (31) -(32) (with −1 ∼ ν 2 ∝ mass of particle ∝ density of Bose gas) reproduces the quantum system that we have started from. One can prove a Egorov-type theorem (see [36]): Quantization and time-evolution commute, up to errors of O(ν −2 ). This means that insights into the dynamics of the classical system reveal features of the quantummechanical dynamics over a finite period of time in a regime of large values of ν.
We first study "stationary" solutions of Eqs. (33) and (34), settinġ
P t = 0 and β t (x) = γ v (x − vt − X 0 ), with X t = X 0 + vt,
where v is the velocity of the particle (here assumed to be constant), and γ v is a function determined by plugging the above ansatz into (34). Eq. (33) then tells us that the external force F must be cancelled by the second term on the right side, which describes a friction force arising from the particle's emission of Cherenkov radiation of sound waves into the Bose gas.
Result: If W is smooth and of short range then there is a positive constant F max < ∞ such that:
• If |F | < F max then there are two stationary solutions of the equations of motion describing a particle propagating with a constant velocity v F , with |v| either given by v − (F ) (a stable solution) or by v + (F ) > v − (F ) (a "run-away" solution), accompanied by a splash in the condensate whose shape is given by γ v ± . The behavior of a particle perpared in an initial state with a speed v close to v + (F ) may be perceived as paradoxical: When the external force F pushing the particle decreases it becomes faster! (See Figure 2 shown below.)
• If |F | > F max then stationary solutions do not exist; the particle accelerates for ever! Figure 2: External force as a function of paticle speed It would be of interest to study the (rate of) approach of a solution of the equations of motion to a stationary solution and to test all these predictions in experiments.
Next, we study what happens to the particle when the external force F vanishes. As long as the speed of the particle is larger than the minimal speed of sound, v * , in the Bose gas, it keeps loosing energy into sound waves, which, thanks to the dispersive properties of the gas, propagate to spatial infinity. In an ideal Bose gas, v * = 0. In this case, the particle will come to rest, as time t tends to ∞, as imagined by Aristotle. Here is a theorem proven in [34] (see also [35]).
Theorem. (Friction by emission of Cherenkov radiation)
We consider a particle moving in an ideal Bose gas at zero temperature and interacting with the gas atoms through a two-body potential, W , that is smooth and of short range and such that
E 3 W (x) d 3 x = 0.
Then there exists a constant δ * ≈ 0.66 such that, given an arbitrary δ ∈ (0, δ * ), there exists an ε = ε(δ) > 0 with the property that, for initial conditions with
(1 + |x| 2 ) 3 2 β 0 (x) < ε, |P 0 | < ε,
the solution of the equations of motion (33) and (34) behave as follows:
|P t | ≤ O(t − 1 2 −δ ), β t − ∆ −1 W (X t − ·) ∞ → 0, as t → ∞.(35)
Choosing ε so small that δ > 1 2 , one has that X t → X ∞ , as t → ∞, with |X ∞ | < ∞, i.e., the particle comes to rest at a finite position in physical space, as t tends to ∞.
Remark: An analogous result for the model studied in the kinetic limit can be found in [37].
The proof of this theorem is quite technical, and I won't go into any details. But here is an idea. Given a particle trajectory
X := X t = X 0 + M −1 0 t 0 P s ds 0 ≤ t < ∞ ,
where P := P t 0 ≤ t ≤ ∞ is an element of a suitably chosen Banach space, we can solve Eq. (34) for β t (x) = β t (x|X). We then plug the result into the right side of Eq. (33), with F = 0. This yields an integro-differential equation with memory of the forṁ
P t = F (P t , t) + L(P t , t), with P t := P s 0 ≤ s ≤ t ,(36)
where F (P t , t) is a sum of three terms depending on P t and on W , one of which depends (linearly) on the initial condition, β 0 , of the Bose gas, and where L(P t , t) is a sum of two terms depending linearly on P t and quadratically on W , but not on β 0 . In order to show local well-posedness of Eq. (36), it is convenient to convert it into an integral equation for P . Let B δ be the Banach space of momentum trajectories, P , with the property that P δ := sup 0≤t<∞ (1 + t) 1 2 +δ |P t | < ∞, where δ is the parameter appearing in the statement of the theorem. One then shows that, for δ ∈ (0, δ * ), ε = ε(δ) can be chosen small enough such that if
(1 + |x| 2 ) 3 2 β 0 (x) < ε, |P 0 | < ε,
then the integral equation for P has a unique solution in B δ , with P δ ≤ O(ε). This is proven by applying a suitable fixed-point theorem on the Banach space B δ . For all further details, see [34].
Results extending those stated in the theorem above have been established for interacting Bose gases in the Bogoliubov limit, which have the property that the minimal speed of sound, v * , is strictly positive. In this situation a particle prepared in a state with speed below v * moves inertially for ever [38], while a particle prepared in a state with speed larger than v * decelerates until its speed approaches v * , at which point it starts to move inertially [39]. (The proof of this last result is considerably more intricate than the proof of the theorem stated above.) Somewhat related, interesting results have been published in [40] and references given there.
It would be good to test the theoretical predictions discussed above in experiments.
L'Insoutenable Flèche du Temps dans l'Évolution Quantique
"Alle Naturwissenschaft ist auf die Voraussetzung der vollständigen kausalen Verknüpfung jeglichen Geschehens begründet." (Albert Einstein, Zurich physical society, 1910)
Well, is it?
Analyzing this question within Quantum Mechanics (QM) is the goal of this section, which describes some fairly recent insights and ideas described in [41,42,43]. Quite to my surpise, these ideas appear to give rise to somewhat violent controversies, which is why I have chosen a somewhat shimmering title for this section. 10 In Quantum Mechanics, the basic difference between Past and Future is mirrored in the fundamental dichotomy 11 Past = a history of events / actualities -Future = a tree of potentialities or, put differently, in the "unbearable" arrow of time in the evolution of states of isolated open physical systems.
In non-relativistic Quantum Mechanics, an isolated physical system, S, is specified by the following data:
1. Physical quantities of S are represented by abstract self-adjoint operators
X = X * ∈ O S ,
where O S is a family of operators whose only properties are that it contains the identity 1 and that if X ∈ O S and F is a real-valued bounded continuous function on the real line then F ( X) ∈ O S . It is assumed that, at every time t, there is a representation of O S on a separable Hilbert space H: O S X → X(t), where X(t) is a self-adjoint bounded operator on H (i.e., a self-adjoint element of B(H)).
2.
We specify the quantum dynamics of an isolated system in the Heisenberg picture: Operators are time-dependent, their time-dependence is given by the Heisenberg equations
X(t ) = e i(t −t)H/ X(t)e −i(t −t)H/ , for t, t in R ,(37)
where H is the Hamiltonian of the system; (H is assumed to be a self-adjoint operator on H). Interactions between S and its complement (the rest of the Universe) are assumed to be negligibly small.
Because an operator X on H representing a physical quantity in O S is obtained by integrating some "density" (given by an operator-valued distribution depending on (space and) time, such as a charge-, spin-or energy density) over a compact interval of the time axis, every such X is localized in some interval, I = I X , of the time axis. (Bounded functions of) all operators localized in an interval I generate a * -algebra, E I , and we have that E I E I if I ⊂ I. This allows us to introduce the algebras
E ≥t := I⊂[t,∞) E I , t ∈ R ,(38)
where the closure is taken in the weak topology of B(H), 12 with E ≥t E ≥t , for t > t.
Definition of Events:
A potential event that might happen at time t or later is an element of a partition of unity, F := π ξ ∈ E ≥t | ξ ∈ X , by disjoint orthogonal projections acting on H, where X (= " spectrum of F ") is a finite or countably infinite set. The projections π ξ ∈ F have the properties π ξ = π * ξ , π ξ · π η = δ ξη π ξ , ∀ ξ, η ∈ X,
ξ∈X π ξ = 1 .(39)
3. A state of S at time t is given by a quantum probability measure on the lattice of orthogonal projections in E ≥t , i.e., it is a functional, ω t , with the properties
• ω t assigns to every orthogonal projection π ∈ E ≥t a non-negative number ω t (π) ∈ [0, 1], with ω t (1) = 1,
• ω t is additive, in the sense that
π∈F ω t (π) = 1, ∀ partitions of unity F ⊂ E ≥t .(40)
Remark: Gleason's theorem [44] (generalized by Maeda) says that states, ω t , of S at time t, as defined above, are positive, normal, normalized linear functionals on E ≥t , i.e., states on E ≥t in the usual sense of this notion. By definition,
B(H) ⊇ E ≥t ⊇ E ≥t , ∀t > t.
Thus, apparently, an isolated physical system is characterized by a co-filtration
E ≥t | t ∈ R
of algebras, with E ≥t generated by projections representing potential events possibly happening at time t or later. These data encode all potential events possibly happening in S, along with the deterministic Heisenberg-picture dynamics of S. They do not contain any grain of uncertainty or chance, yet! This observation compels us to ask where the probabilistic nature of Quantum Mechanics, expected to be fundamental, is hiding. It turns out that it is the quantum-mechanical time evolution of states of physical systems that is stochastic. This claim is evident when one considers measurements and observations within the so-called "Copenhagen Interpretation" of Quantum Me-chanics: Every measurement is thought to provoke a non-linear stochastic change of state. An interesting example is the observation of tracks of α-particles (helium nuclei) emitted from a perfectly round ball of radioactive material in a bubble chamber filled with a saturated gas: One may wonder how the perfect spherical symmetry of the initial state of this system gets broken in such a way that tracks of particles pointed in definite directions appear. Physicists tend to agree with the claim that Quantum Mechanics can at best predict the probability distribution on the possible directions of such tracks. 13 But does Quantum Mechanics, when formulated correctly, explain that tracks (rather than waves) will occur; and how do probabilities enter the theory?
In this section, we summarize a novel approach, dubbed ET H-Approach (see [41,42,43]), towards understanding the quantum-mechanical stochastic time evolution of states of isolated open systems featuring events. Definition 1. An Isolated Open Physical System is an isolated system, S, characterized by a co-filtration, E ≥t | t ∈ R , satisfying the Principle of Diminishing Potentialities (P DP ): In an Isolated Open Physical System featuring events the following strict inclusions hold
E ≥t E ≥t , for arbitrary t > t .(41)
Note: The "initial state" of S may be pure; but, since E ≥t B(H) , ∀t < ∞, assuming that (P DP ) holds, its state at time t will in general be a mixed state on E ≥t (entanglement! ) This observation opens the door towards a natural notion of Actual Events, or "actualities", in our formalism and to a theory of direct / projective mea-surements and observations.
In accordance with the "Copenhagen interpretation" of Quantum Mechanics, one may pretend that some potential event in a partition of unity F = {π ξ |ξ ∈ X} ⊂ E ≥t actually happens in the interval [t, ∞) of times, i.e., becomes an actual event setting in at time t, iff
ω t (A) = ξ∈X ω t (π ξ A π ξ ), ∀A ∈ E ≥t .(42)
No off-diagonal elements appear on the right side of (42), which describes an incoherent superposition of states in the images of disjoint projections. But what, in Quantum Mechanics, determines whether, at some time t, a partition of unity F exists such that Eq. (42) holds for the state of the system at time t? This is the question to be answered next. To answer it one would clearly not want to invoke anything like the "free will" of observers! In order to describe what I consider to be a satisfactory answer, I have to introduce some simple notions from algebra. 14 Let M be a * -algebra, and let ω be a state on M. We define the centralizer of ω by
Let S be an isolated open physical system (see Definition 1,above). In (43) we set M := E ≥t , and ω := ω t , where ω t is a state on E ≥t . Definition 2. Given that ω t is some state on E ≥t , we say that an Actual Event is setting in at time t iff Z ωt (E ≥t ) contains at least two non-zero orthogonal projections, π (1) , π (2) , which are disjoint, i.e., π (1) · π (2) = 0, and have non-vanishing "Born probabilities", i.e., 0 < ω t (π (i) ) < 1 , for i = 1, 2 .
Let us suppose for simplicity that Z ωt (E ≥t ) is generated by a partition of unity F t = {π ξ | ξ ∈ X ωt } of orthogonal projections, where X ωt , the spectrum of the abelian algebra Z ωt (E ≥t ), is assumed to be a countable set. Then Eq. (42) holds true for X = X ωt .
The Law describing the stochastic time evolution of states in Quantum Mechanics is derived from the following State Reduction-, or Collapse Postulate (which makes precise mathematical sense if time is discrete).
Let ω t be the state of S at time t. Let dt denote a time step; (dt is strictly positive if time is discrete; otherwise we will attempt to let dt tend to 0 at the end of our constructions). We define a state ω t+dt on the algebra E ≥t+dt (⊂ E ≥t ) by restriction of ω t , i.e.,
ω t+dt := ω t E ≥t+dt .
Axiom CP: Let F t+dt := π ξ | ξ ∈ X ω t+dt be the partition of unity generating the center, Z ω t+dt (E ≥t+dt ), of the centralizer of the state ω t+dt on E ≥t+dt . Then 'Nature' replaces the state ω t+dt by a state ω t+dt (·) ≡ ω t+dt,ξ (·) := ω t+dt (π ξ ) −1 · ω t+dt (π ξ (·)π ξ ) , for some ξ ∈ X ω t+dt , with ω t+dt (π ξ ) = 0. The probability, prob t+dt (ξ), for the state ω t+dt,ξ to be selected by 'Nature' as the state of S at time t + dt is given by
prob t+dt (ξ) = ω t+dt (π ξ ) (Born's Rule)(44)
The ET H-Approach to Quantum Mechanics sketched above leads to the following picture of dynamics in Quantum Mechanics: The evolution of states of an isolated open system S featuring events, in the sense of Definition 2, is determined by a stochastic branching process, whose state space is referred to as the non-commutative spectrum, Z S , of S (see [43]). Assuming that all the algebras E ≥t are isomorphic to one specific (universal) von Neumann algebra, denoted by N , 15 the non-commutative spectrum of S is defined by
Z S := ω ω , Z ω (N ) ,(45)
where the union over ω is a disjoint union, and ω ranges over all states of S of physical interest. 16 If time is taken to be discrete then Definition 2 of actual events and Born's Rule (44) completely specify the branching probabilities of this process. The mathematical theory obtained when the time step, dt, tends to 0 is, however, not developed rigorously, yet. I expect that this is will become a challenging topic for mathematicians.
Remarks:
• Here is an explanation of the meaning of the name "ET H-Approach": "E" stands for "events", "T " for "trees" -referring to the tree-like structure of the space of all actualities an isolated physical system could in principle encounter in the course of its evolution -and "H" stands for "histories," referring to the actual trajectory of states occupied by the system in the course of its evolution.
• Axiom CP (the Collapse Postulate), in combination with Eq. (42) and Born's Rule (44), is reminiscent of the collapse postulate of the Copenhagen interpretation of Quantum Mechanics. But the Principle of Diminishing Potentialities (P DP ) imparts on it a transparent, logically coherent status, without assigning any role to something like the free will of "observers."
• One might argue that (P DP ) and the Collapse Postulate just represent a precise version of the Many-Worlds Interpretation of Quantum Mechanics [47]. However, the ET H-Approach provides a precise rule for "branching," namely Axiom CP along with Born's Rule -which the Many-Worlds interpretation does not. And in the ET H-Approach there is no reason to imagine that, besides the world we actually perceive, other worlds exist.
• In [43] it is argued in which way the occurrence of an actual event may correspond to the measurement of a physical quantity. (Of course, in general, there are plenty of actualities happening that cannot be related to the measurement of a well-defined physical quantity.) Let ε 1 be a positive number. We define Z t,ε to be the abelian algebra generated by the projections in a family, π 1 , . . . , π n ⊂ Z ωt (E ≥t ), of n disjoint orthogonal projections with the property that
ω t (1 − n i=1 π i ) ≤ ε
Let X = X * be a self-adjoint operator in the algebra E ≥t representing some physical quantity X ∈ O S , and let Π 1 , . . . , Π k be k disjoint spectral projections of X with the property that
ω t (1 − k i=1 Π i ) ≤ ε.
If every projection Π i , i = 1, . . . , k, can be approximated in norm by operators in Z t,ε with a precision of O(ε) then one can argue that the occurrence of an event in Z ωt (E ≥t ) corresponds to the measurement of an approximate value of the physical quantity X at a time ≈ t, with a precision of O(ε).
A more compelling discussion of the relationship between actual events and measurements of physical quantities can be found, in the context of an analysis of simple models, in Sect. 5.4 of [42] and will be pursued in forthcoming work.
• For theoretical physicists, the most interesting problem connected with the ET H-Approach to Quantum Mechanics is to explore physical mechanisms leading to the Principle of Diminishing Potentialities (P DP ). Some simple models satisfying this principle are discussed in Sect. 5 of [42]. These models have the feature that if time is continuous their total Hamiltonian, H, is unbounded above and below. I have argued that (P DP ) is compatible with the usual spectrum condition, namely that H ≥ 0, only in models of local relativistic quantum theory -a rather tantalizing conclusion. In such models, a mechanism leading to (P DP ) is known:
It is related to Huygens' Principle valid in quantum electrodynamics and other theories with massless particles in four-dimensional space-time; see [46].
We expect that there are plenty of other mechanisms leading to (P DP ); but a systematic study has not been made, yet.
A variant of the ET H-Approach formulated within local relativistic quantum theory is discussed in [43].
Conclusions and Acknowledgements
It is a widespread conviction that the fundamental Laws of Nature treat past and future in a symmetric way and that, for this reason, they cannot only be used to predict the future, but they can also be applied to reconstructing the past from knowledge of the present. Irreversible behavior of physical systems and an arrow of time are thus often perceived as mysterious. In Sections 3 through 5, I have examined examples of physical systems exhibiting an arrow of time or irreversible behavior in spite of the fact that the underlying dynamics of these systems is time-reversal invariant.
In this context, results of Elliott Lieb and coworkers concerning properties of quantum-mechanical entropy furnish some of the elements of an explanation. In Sect. 6, I have argued, I hope quite convincingly, that the quantum-mechanical evolution of states of physical systems featuring events that can in principle be obser-ved is fundamentally stochastic and exhibits an arrow of time. So far, this view has not been widely accepted, yet. 17 However, I am confident that it has its merits and that it is firmly rooted in known properties of quantum field theories with massless modes, such as quantum electrodynamics. Moreover, simple models implement and illustrate it in a natural and convincing way (see [42]).
One might argue that the most fascinating example of a physical system exhibiting an arrow of time is the Universe. The ultimate theory to be used to describe the Universe is expected to be some generalization of quantum mechanics that people tend to call "quantum gravity." This theory, whose precise formulation remains to be discovered, ought to describe the big bang and the ensuing evolution of the Universe. Fundamental problems in a search for "quantum gravity" are, among others, to introduce a sharp and useful concept of cosmological "event" and to formulate a precise stochastic law of evolution of "space-time-matter." I expect that the ideas sketched in Sect. 6 and the results described in [43] might become relevant in this connection. A future theory should then elucidate why and how structure with classical features emerged from an initial highly isotropic and homogeneous quantum state of the Universe (during an epoch when there weren't any "observers" to perform measurements).
Of course, I am not competent to do justice to this example. Nevertheless I would like to sketch some basic facts related to the fundamentally irreversible evolution of the state of the Universe that remain highly enigmatic and wait to be unravelled.
• On very large distance scales, the Universe appears to be isotropic and homogeneous, although different patches could not have been causally connected at very early times. It has been proposed that Inflation, an epoch of glaringly irreversible, explosive evolution of the Universe, could explain these surprising features (besides solving several other puzzles).
• The Universe expands (rather than contracts), and its energy density, ρ, appears to be very nearly equal to the critical density for an open universe. Inflation appears to provide an explanation of this.
• The Universe exhibits a basic asymmetry between Matter and Antimatter.
• Galactic rotation curves and plenty of other phenomena (including, e.g., gravitational lensing) suggest that there exists Dark Matter in the Universe with an equation of state implying that 0 < p ρ, where p denotes pressure.
• The observed accelerated expansion of the Universe suggests that there exists Dark Energy with an equation of state p ≈ −ρ.
• At the present stage of evolution of the Universe, the contributions of visible matter, Dark Matter and Dark Energy to the energy density of the Universe have the same order of magnitude.
• There exist tiny, highly homogeneous intergalactic magnetic fields in the Universe stretched out over huge distances. ...
A daring conjecture: The facts just described must have something to do with one another and ought to have interrelated theoretical explanations! One may safely expect that these explanations will be based, in part, on "physics beyond the standard model" (e.g., they are likely to involve quantum fields of geometrical nature not present in the standard model of particle physics that will give rise to Dark Matter and Dark Energy). They may thus lead theorists to discover genuinely new physics. 18 To conclude, one may hope that revealing origins of time's arrow and of irreversibility in different physical theories remains a challenging endeavor in theoretical physics that is likely to give rise to surprising future discoveries.
Tr f (B) ≥ Tr f (A) + Tr f (A)(B − A) ,
Figure 1 :
1Reservoirs coupled by a thermal contact
Figure 3 :
3Tracks of α-particles in a bbubble chamber
C ω (M) := X ∈ M ω([A, X]) = 0, ∀A ∈ M Note that C ω (M) is a subalgebra of M and that ω is a normalized trace on C ω (M). The center, Z ω (M), of C ω (M) is defined by Z ω (M) := X ∈ C ω (M) [X, A] = 0, ∀A ∈ C ω (M)
Aristotle put it as follows: "Indeed, it is evident that the mere passage of time itself is destructive rather than generative ..., because change is primarily a 'passing away'." 2 "Every experiment destroys some of the knowledge of the system which was obtained by previous experiments."(Werner Heisenberg)
It was first presented, on invitation kindly extended to me by A. Alekseev, as a Mirimanoff lecture at the University of Geneva, in 2015.
I am grateful to Israel Michael Sigal for having taught me how to use Fermi-Golden-Rule conditions in the analysis of quantum-mechanical resonances and spectral problems.
An important role in this analysis is played by the fact that, in Rayleigh scattering, a genuine infrared problem does not arise -the number of photons created in such processes has a finite expectation value (see[18]), a conjecture originally proposed by the author.
An early analysis of such states appeared in[21]. A mathematically precise result of this type was established in the diploma thesis of S. Dirren, written under the supervision of G. M. Graf and the author (ETH Zurich, winter 1998/99), following ideas and methods in[10].
See[30] for comprehensive information on the mathematical theory of Bose gases -one of Elliott Lieb's many interests!
Of course, for a particle hopping on the lattice Z 3 to have a finite speed is not surprizing. The smallness of D and vν , as ν → 0, is due to the large mass, M = O(ν −2 ), of the particle.
Cherenkov radiation is observed in nuclear reactors: Electrons moving through water at a speed larger than the speed of light in water emit blue light. This causes them to decelerate until their speed has dropped to below the speed of light in water.
It is inspired by a novel of Milan Kundera, entitled: L'insoutenable légèrté de l'être.11 According to Edmund Husserl, actuality means existence in space and time, as opposed to possibility or potentiality, which refers to the capacity, power, ability, or chance for something to happen or to occur.
It is convenient to take weak closures, i.e., consider von Neumann algebras, because they have the property that, with every self-adjoint bounded operator, they also contain all its spectral projections.
For an explanation of the phenomenon of paticle tracks in detectors see[45], and references given there.14 The following remarks may look overly abstract; but this cannot be avoided, because the algebras E ≥t are usually not of type I.
This is the case in relativistic Quantum Electrodynmics[46] 16 "States of physical interest" are normal states a concrete system can actually be prepared in. Here we may leave this notion a little vague.
To quote Max Planck: "Eine neue wissenschaftliche Wahrheit pflegt sich nicht in der Weise durchzusetzen, daß ihre Gegner überzeugt werden und sich als belehrt erklären, sondern vielmehr dadurch, daß ihre Gegner allmählich aussterben und daß die heranwachsende Generation von vornherein mit der Wahrheit vertraut gemacht ist."
The reader may find a sketch of my views concerning these matters in[43,48] and references given there. Not being closely familiar with this subject I must refrain from quoting any literature.
I am grateful to these as well as to many further colleagues and friends, including D. Brydges, D. Buchholz, the late D. Dürr, S. Goldstein, G.M. Graf, E. Seiler and T. Spencer, for countless useful discussions on some of the ideas described in this paper. I thank T. Spencer for hospitality at the School of Mathematics of the Institute for Advanced Study, during a period when I was working on the material sketched in Section 6, and Ph. Blanchard and H. Siedentop for hospitality at their respective institutions.I am deeply thankful to Elliott Lieb for all he has taught me and for his friendship, and to my PhD advisor and colleague Klaus Hepp for -among many other things -having introduced me to Elliott and for sharing him with me as a common friend.
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| [] |
[
"First moments of the nucleon generalized parton distributions from lattice QCD",
"First moments of the nucleon generalized parton distributions from lattice QCD"
] | [
"A Sternbeck [email protected] \nInstitut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany\n",
"M Göckeler \nInstitut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany\n",
"Ph Hägler \nInstitut für Kernphysik\nJohannes Gutenberg-Universität Mainz\n55128MainzGermany\n",
"R Horsley \nSchool of Physics and Astronomy\nUniversity of Edinburgh\nEH9 3JZEdinburghUK\n",
"Y Nakamura \nRIKEN Advanced Institute for Computational Science\n650-0047KobeHyogoJapan\n",
"A Nobile \nJülich Research Centre\nJSC\n52425JülichGermany\n",
"D Pleiter \nInstitut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany\n\nJülich Research Centre\nJSC\n52425JülichGermany\n",
"P E L Rakow \nTheoretical Physics Division\nDepartment of Mathematical Sciences\nUniversity of Liverpool\nL69 3BXLiverpoolUK\n",
"A Schäfer \nInstitut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany\n",
"G Schierholz \nDeutsches Elektronen-Synchrotron DESY\n22603HamburgGermany\n",
"J Zanotti \nSchool of Chemistry and Physics\nUniversity of Adelaide\n5005SAAustralia\n"
] | [
"Institut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany",
"Institut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany",
"Institut für Kernphysik\nJohannes Gutenberg-Universität Mainz\n55128MainzGermany",
"School of Physics and Astronomy\nUniversity of Edinburgh\nEH9 3JZEdinburghUK",
"RIKEN Advanced Institute for Computational Science\n650-0047KobeHyogoJapan",
"Jülich Research Centre\nJSC\n52425JülichGermany",
"Institut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany",
"Jülich Research Centre\nJSC\n52425JülichGermany",
"Theoretical Physics Division\nDepartment of Mathematical Sciences\nUniversity of Liverpool\nL69 3BXLiverpoolUK",
"Institut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany",
"Deutsches Elektronen-Synchrotron DESY\n22603HamburgGermany",
"School of Chemistry and Physics\nUniversity of Adelaide\n5005SAAustralia"
] | [
"The XXIX International Symposium on Lattice Field Theory -Lattice"
] | We report on our lattice calculations of the nucleon's generalized parton distributions (GPDs), concentrating on their first moments for the case of N f = 2. Due to recent progress on the numerical side we are able to present results for the generalized form factors at pion masses as low as 260 MeV. We perform a fit to one-loop covariant baryon chiral perturbation theory with encouraging results. | 10.22323/1.139.0177 | [
"https://arxiv.org/pdf/1203.6579v1.pdf"
] | 73,583,202 | 1203.6579 | 25e4d1065aa48ddf6b4a0688da6a41c0f3cdcb60 |
First moments of the nucleon generalized parton distributions from lattice QCD
2011 July 10-16, 2011
A Sternbeck [email protected]
Institut für Theoretische Physik
Universität Regensburg
93040RegensburgGermany
M Göckeler
Institut für Theoretische Physik
Universität Regensburg
93040RegensburgGermany
Ph Hägler
Institut für Kernphysik
Johannes Gutenberg-Universität Mainz
55128MainzGermany
R Horsley
School of Physics and Astronomy
University of Edinburgh
EH9 3JZEdinburghUK
Y Nakamura
RIKEN Advanced Institute for Computational Science
650-0047KobeHyogoJapan
A Nobile
Jülich Research Centre
JSC
52425JülichGermany
D Pleiter
Institut für Theoretische Physik
Universität Regensburg
93040RegensburgGermany
Jülich Research Centre
JSC
52425JülichGermany
P E L Rakow
Theoretical Physics Division
Department of Mathematical Sciences
University of Liverpool
L69 3BXLiverpoolUK
A Schäfer
Institut für Theoretische Physik
Universität Regensburg
93040RegensburgGermany
G Schierholz
Deutsches Elektronen-Synchrotron DESY
22603HamburgGermany
J Zanotti
School of Chemistry and Physics
University of Adelaide
5005SAAustralia
First moments of the nucleon generalized parton distributions from lattice QCD
The XXIX International Symposium on Lattice Field Theory -Lattice
2011 July 10-16, 2011DESY 12-038, Edinburgh 2012/03, LTH 941 Squaw Valley, Lake Tahoe, California * Speaker. † Supported by the FP7-People Programme of the European Commission. First moments of the nucleon generalized parton distributions from lattice QCD A. Sternbeck
We report on our lattice calculations of the nucleon's generalized parton distributions (GPDs), concentrating on their first moments for the case of N f = 2. Due to recent progress on the numerical side we are able to present results for the generalized form factors at pion masses as low as 260 MeV. We perform a fit to one-loop covariant baryon chiral perturbation theory with encouraging results.
Introduction
The study of the internal structure of hadrons still presents an exciting challenge. Among the different types of studies, the computation of Generalized Parton Distributions (GPDs) is especially challenging, but also attractive, because of their potential for hadron physics.
GPDs were introduced in the late 90s. For a given hadron, they provide detailed information on the partonic structure with respect to spatial, momentum and spin degrees of freedom. GPDs combine the information of the traditional form factors and parton distribution functions (containing them as limiting cases) into a single set of functions and hence contain information also on the correlation between the momentum, spin and spatial degrees of freedom. For the nucleon, one hopes GPDs will provide one day a three-dimensional spatial picture, a better understanding of its spin structure and a value for the quark orbital angular momentum [1].
Beside the renormalization scale 1 , GPDs depend on three kinematic variables: the longitudinal parton momentum fraction x, the skewness parameter ξ and the virtuality t. The quark structure of a nucleon, for example, is governed by eight GPDs. Among these, the unpolarized GPDs H and E parametrize the off-diagonal matrix element
N(P )|O µ V (x)|N(P) = U(P ) γ µ H(x, ξ ,t) + iσ µν ∆ ν 2m N E(x, ξ ,t) U(P) + higher twist . (1.1)
Here P and P denote the incoming and outgoing nucleon momenta (and so ∆ = P − P, P = (P + P)/2, t = ∆ 2 and ξ = −n · ∆/2) and
O µ V (x) is the light-cone bilocal operator O µ V (x) = ∞ −∞ dλ 2π e iλ xq − λ 2 n γ µ Pe −ig −λ /2 λ /2 dα nA(αn) q λ 2 n ,(1.2)
which often arises in hard scattering processes (see, e.g., [3]). n (with P · n = 1) denotes a light cone vector in Eq.(1.2) and P the correct path-ordering of the gluon fields A. The polarized nucleon GPDs,H andẼ, are defined in a similar manner, replacing γ µ in Eq.(1.2) by γ µ γ 5 .
GPDs and the lattice
GPDs can be accessed experimentally, for instance, via deeply virtual Compton scattering. The analysis, however, is rather demanding and requires also a partial modeling of the combined x-, ξ -and t-dependence. Cross-checks to other methods are thus inevitable.
A promising method is given by lattice QCD computations. Although a direct determination of GPDs on the lattice is not possible, their (Mellin) moments
1 −1 dx x n−1 H(x, ξ ,t), 1 −1 dx x n−1 E(x, ξ ,t), . . . (2.1)
are accessible. For a nucleon, for example, these moments can be calculated via matrix elements N(P )|O|N(P) of local operators O. For H and E these operators read
O µν 1 ·ν n−1 V (z) = q(z) γ {µ iD ↔ ν 1 · · · iD ↔ ν n−1 } q(z) − traces (2.2)
where q refers to a quark field,
D ↔ ≡ D → − D ←
to the covariant derivative and {· · · } to a symmetrization of the Lorentz indices. For a definition and further details on the operators needed for the remaining nucleon GPDs, the reader may refer to [4].
Admittedly, the computation of such matrix elements is quite demanding already for n ≥ 2, and we are not yet in the stage to provide precision results close to the physical point. Nonetheless, such calculations have become more and more feasible in recent years, and hence have attracted interest from within the lattice community [5,6,7,8,9,10,11].
In what follows, we will restrict ourselves to the two nucleon GPDs H and E. Their moments are polynomials in ξ ,
1 −1 dx x n−1 H(x, ξ ,t) E(x, ξ ,t) = [(n−1)/2] ∑ k=0 (2ξ ) 2k A n,2k (t) B n,2k (t) ± δ n,even (2ξ ) n C n (t) . (2.
3)
The expansion coefficients A, B and C are real functions of the momentum transfer t (and the renormalization scale µ) and are known as the Generalized Form Factors (GFFs) of the nucleon. In this notation, for instance, A 10 and B 10 correspond to the electromagnetic form factors [12], and A 20 , B 20 and C 2 parametrize the matrix elements of the energy-momentum tensor O
µν V N(P )|O µν V |N(P) = U(P ) γ {µ P ν} A 20 (t) − i∆ ρ σ ρ{µ 2m N P ν} B 20 (t) + ∆ {µ ∆ ν} m N C 2 (t) U(P) . (2.4)
Below we present results for A 20 , B 20 and C 2 . They can be extracted from ratios of two-and three-point correlation functions
R(t, τ, p , p) = C 3 (t, τ, p , p) C 2 (t, p ) × C 2 (τ, p )C 2 (t, p )C 2 (t − τ, p) C 2 (τ, p)C 2 (t, p)C 2 (t − τ, p ) 1/2 , (2.5)
which are proportional to N(P )|O µν V |N(P) and constant in the limit 0 τ t T /2 (T is the temporal lattice extension). C 2 (t, p) is the nucleon two-point function with a source at time 0 and sink at time t, and C 3 (t, τ, p , p) is the three-point function with an operator insertion at time τ. The latter we calculate employing the sequential source technique. Our data for the GFFs is for gauge configurations thermalized using the standard Wilson gauge action and two flavors of clover-improved Wilson fermions. The gauge couplings are β = 5.25, 5.29 and 5.40; and the κ values are such that pion masses from 1 GeV down to 260 MeV are simulated, where we primarily work with the data in the mass range 260 MeV ≤ m π ≤ 490 MeV. The scale is fixed through setting r 0 = 0.5 fm. This is about the value we obtain from chiral extrapolations of our nucleon mass data [13] for the same set of configurations. The lattice sizes are 24 3 × 48, 32 3 × 64, 40 3 × 64 and 48 3 × 64. In particular the latter two provide us with a good signal-to-noise ratio. See, for example, Fig. 1, where data for A 20 (t) in the isovector channel is shown for the lattice sizes 24 3 × 48, 32 3 × 64 and 40 3 × 64 at β = 5.29 and κ = 0.13632 (m π = 287 MeV). The number of measurements is 2755, 3013 and 1478, respectively. The advantage of volume averaging is clearly evident as with about half the statistics, the data for the 40 3 × 64 lattice comes with much less statistical noise than that for the 24 3 × 48 lattice. Moreover, Fig. 1 indicates that finite size effects are small, at least at our level of precision.
Results
0.1 0.2 0.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 A u−d 20 (t) −t [GeV 2 ]
A selection of all of our GFF data is shown as a function of the momentum transfer −t in Fig. 2. There, the panels from top to bottom display the respective data for A 20 (t), B 20 (t) and C 2 (t). Left panels are for the isovector case, right panels for the isoscalar (without disconnected contributions). For simplicity, only data for five ensembles is shown, which fall into two groups of approximately equal pion mass. For the larger pion mass (430-490 MeV), we have results for three lattice spacings (a = 0.06, 0.07 and 0.08 fm), for the smaller one (260-287 MeV) we can show data for two sets (a = 0.06, 0.07 fm). For this (admittedly small) range of lattice spacings we observe, however, no systematic dependence on a. Apparently, there is a slight vertical shift in the data for A 20 (t < 0) [and in the opposite direction for B 20 (t < 0)] for the lighter sets, but we do not see these shifts for the heavier sets (including those not shown), at least with the available precision. It will be interesting to see how well our forthcoming results at β = 5.25, κ = 0.13620 (i.e., a = 0.084 fm, 260 MeV pion mass) fit to these findings. Similarly, we observe a trend for A 20 (t) if the pion mass is changed: The low-t dependence of A 20 (t) gains slope if m π is reduced from 430-490 MeV to 260-287 MeV. This effect, however, is small and we see no such effect in the data from 500 MeV to 1 GeV pion mass. It thus remains to be seen if this effect at lower m π stays or disappears with higher statistics. As above, a further check should become possible as soon as our results at β = 5.25 and κ = 0.13620 are available.
We can confirm though the (notorious) weak m π dependence of A u−d 20 at t = 0, i.e., of x u−d (see upper left panel in Fig. 2). From phenomenology one expects x u−d ≈ 0.16 at the physical point. So far, however, all available (world) lattice data for x u−d for pion masses above 200 MeV gives values for x u−d well above 0.16, and moreover, almost no signal for a downwards trend towards the physical point is seen (see, e.g., [4] and references therein). From baryon chiral perturbation theory (BChPT), for example, such a trend is expected, but it has not yet been demonstrated (convincingly) on the lattice.
It is however interesting that our data for |t| < 0.4GeV 2 indicates an almost linear t-dependence for A u−d 20 (t) and a flattening of the slope for B u−d 20 (t). This would be consistent with expectations from covariant BChPT at leading-one-loop order [14].
In Fig. 2, and in particular in Fig. 3, we show a first attempt of fitting our data to the BChPT expressions for A 20 , B 20 and C 2 as worked out in [14] (see the dashed lines at lower t). Note that such a fit has to be a combined fit to the data for all three GFFs simultaneously, because the parameter a 20 enters all of them. The dashed lines in Figs. 2 and 3 refer to such a fit which incorporates only the lighter data sets (full symbols) and points for |t| < 0.44 GeV 2 . Five parameters (a 20 , b 20 c 20 , c r 8 and c 12 ) were left free, while the phenomenological value ∆x phen u−d = 0.21 was used to constrain the coupling ∆a v 20 . The m π -dependence of the nucleon mass (entering the BChPT expression for B 20 and C 2 ) and the parameter M 0 were taken from our nucleon mass fits [13].
It turns out that for the isovector case the fit quality is quite good: A reduced χ 2 -value of about one is reached and the low-t dependence of A 20 and B 20 is roughly reproduced; actually, also for C 2 , as only the data point at the smallest |t| falls somewhat below the fitting curve. For the isoscalar case, the same fit works less satisfactorily. The reason might be that BChPT does not work at the pion masses under consideration, or that disconnected contributions are not negligible. The latter are certainly worth to be calculated.
Even though disconnected contributions are still missing, it is interesting to look at the total quark angular momentum
J u,d = 1 2 A u,d 20 (0) + B u,d 20 (0) ,(3.1)
to check if it is in the ballpark of expected values.
In Fig. 4 we show this data for our lighter sets, that is, for pion masses 261 and 287 MeV. Note that for A u±d 20 we have data directly at t = 0, but not for B u±d 20 (0). However, looking at Fig. 2 one easily sees that the main contribution to J u+d comes from A u+d 20 (0) and the t-dependence of B u±d 20 is comparably weak. We therefore approximate B u±d 20 (0) by our data for the smallest |t|. This should be perfectly fine for our purposes, given all the other uncertainties and the lack of disconnected contributions. Note that in Fig. 4 we have also included data for the quark spin
s q = 1 2 1 −1 dxH(x, ξ , 0) = 1 2Ã q 10 (t = 0) ,(3.2)
which we obtain from data 2 for the axial nucleon GFFsà u−d 10 andà u+d 10 . If we compare our data in Fig. 4 with that of the LHPC collaboration [15], we find good agreement (albeit their data is for N f = 2 + 1). We also see the same ordering for the total and orbital (L q = J q − s q ) angular momentum and the quark spin:
|J d | |J u |, |J d | |s d |, |L u+d | |L u |, |L d | .
It will be interesting to see how this figure changes when data at smaller pion masses becomes available and/or disconnected contributions are included.
Conclusions
We have presented an update on our efforts to calculate the generalized form factors for the nucleon. We have restricted ourselves here to the case of N f = 2 and reported only on results for the GFFs of the energy-momentum tensor (n = 2). Due to recent progress on the numerical side we are able to provide data for these GFFs for pion masses down to 260 MeV. In particular our lighter sets provide an improvement of the available data for these form factors: Large lattice volumes have allowed us to obtain a very good signal-to-noise ratio, and at low |t| our data starts to fulfill expectations from one-loop BChPT. When comparing our data to that of the LHPC collaboration presented at this conference [15] we see a small vertical offset for the GFF data, but overall agreement for angular momentum and spin. It remains to be seen if this offset is due to the different renormalization procedures of the lattice operators or due to the different N f .
Figure 1 :
1A 20 (t) for the isovector case for β = 5.29 and κ = 0.13632 and for different volumes.
Figure 2 :
2The generalized form factors A 20 , B 20 and C 2 (from top to bottom) vs. momentum transfer −t; left for the isovector channel, right for the isosinglet channel. The data is for three lattice spacings and two groups of approximately equal pion masses. If applicable, solid (dashed-dotted) lines represent dipole fits to the data. Dashed lines at low t result from a simultaneous fit of the low-pion-mass data (full symbols) to covariant chiral perturbation theory (see text and alsoFig. 3for more details).
Figure 3 :
3A u−d 20 (top) and B u−d 20 (bottom) vs. −t at the pion masses 261 and 288 MeV. Dashed lines result from a simultaneous fit of the data (including that for C u−d 2 ) to expectations from BChPT[14].
J
u : β = 5.29 β = 5.40 J d : β = 5.29 β = 5.40 s u : β = 5.29 s d : β = 5.29
Figure 4 :
4The total quark angular momentum J and spin s q vs. pion mass.
For simplicity, we drop the explicit reference to the renormalization scale µ in what follows. It is always implicitly understood. Our lattice data below has been nonperturbatively renormalized[2] and is for the MS scheme at µ = 2 GeV.
Unfortunately, there is no data forà u±d 10 at β = 5.4 for these small pion masses, but it will become available soon.
The numerical calculations have been performed on the APEmille, apeNEXT systems and PAX cluster at NIC / DESY (Zeuthen, Germany), the IBM BlueGene/L at EPCC (Edinburgh, UK), the BlueGene/P (JuGene) and the Nehalem Cluster (JuRoPa) at NIC (Jülich, Germany), and the SGI Altix and ICE 8200 systems at LRZ (Munich, Germany) and HLRN (Berlin-Hannover, Germany). We have made use of the Chroma software suite[16]. The BlueGene codes were optimised with Bagel[17]. This work has been supported in part by the DFG(
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| [] |
[
"Characterization and Modeling of 28-nm FDSOI CMOS Technology down to Cryogenic Temperatures",
"Characterization and Modeling of 28-nm FDSOI CMOS Technology down to Cryogenic Temperatures"
] | [
"Arnout Beckers \nIntegrated Circuits Laboratory (ICLAB)\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nSwitzerland\n",
"Farzan Jazaeri \nIntegrated Circuits Laboratory (ICLAB)\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nSwitzerland\n\nCEA-Léti\nGrenobleFrance\n",
"Heorhii Bohuslavskyi \nCEA-Léti\nGrenobleFrance\n",
"Louis Hutin \nCEA-Léti\nGrenobleFrance\n",
"Silvano De Franceschi \nIntegrated Circuits Laboratory (ICLAB)\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nSwitzerland\n",
"Christian Enz "
] | [
"Integrated Circuits Laboratory (ICLAB)\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nSwitzerland",
"Integrated Circuits Laboratory (ICLAB)\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nSwitzerland",
"CEA-Léti\nGrenobleFrance",
"CEA-Léti\nGrenobleFrance",
"CEA-Léti\nGrenobleFrance",
"Integrated Circuits Laboratory (ICLAB)\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nSwitzerland"
] | [] | This paper presents an extensive characterization and modeling of a commercial 28-nm FDSOI CMOS process operating down to cryogenic temperatures. The important cryogenic phenomena influencing this technology are discussed. The low-temperature transfer characteristics including body-biasing are modeled over a wide temperature range (room temperature down to 4.2 K) using the design-oriented simplified-EKV model. The trends of the free-carrier mobilities versus temperature in long and short-narrow devices are extracted from dc measurements down to 1.4 K and 4.2 K respectively, using a recently-proposed method based on the output conductance. A cryogenic-temperature-induced mobility degradation is observed on long pMOS, leading to a maximum hole mobility around 77 K. This work sets the stage for preparing industrial design kits with physicsbased cryogenic compact models, a prerequisite for the successful co-integration of FDSOI CMOS circuits with silicon qubits operating at deep-cryogenic temperatures. | 10.1016/j.sse.2019.03.033 | [
"https://arxiv.org/pdf/1809.09013v1.pdf"
] | 108,775,699 | 1809.09013 | 20d9fb5b211f65900ca6220a87ff327e85659a72 |
Characterization and Modeling of 28-nm FDSOI CMOS Technology down to Cryogenic Temperatures
Arnout Beckers
Integrated Circuits Laboratory (ICLAB)
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Switzerland
Farzan Jazaeri
Integrated Circuits Laboratory (ICLAB)
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Switzerland
CEA-Léti
GrenobleFrance
Heorhii Bohuslavskyi
CEA-Léti
GrenobleFrance
Louis Hutin
CEA-Léti
GrenobleFrance
Silvano De Franceschi
Integrated Circuits Laboratory (ICLAB)
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Switzerland
Christian Enz
Characterization and Modeling of 28-nm FDSOI CMOS Technology down to Cryogenic Temperatures
28 nm FDSOIcharacterizationcryogenic CMOScryogenic MOSFETdouble-gatelow temperaturemobilitymodeling42 K
This paper presents an extensive characterization and modeling of a commercial 28-nm FDSOI CMOS process operating down to cryogenic temperatures. The important cryogenic phenomena influencing this technology are discussed. The low-temperature transfer characteristics including body-biasing are modeled over a wide temperature range (room temperature down to 4.2 K) using the design-oriented simplified-EKV model. The trends of the free-carrier mobilities versus temperature in long and short-narrow devices are extracted from dc measurements down to 1.4 K and 4.2 K respectively, using a recently-proposed method based on the output conductance. A cryogenic-temperature-induced mobility degradation is observed on long pMOS, leading to a maximum hole mobility around 77 K. This work sets the stage for preparing industrial design kits with physicsbased cryogenic compact models, a prerequisite for the successful co-integration of FDSOI CMOS circuits with silicon qubits operating at deep-cryogenic temperatures.
Introduction
The birth of CMOS-compatible qubits in silicon [1,2,3,4] has rebooted the interest in cryogenic CMOS electronics for computing applications. Since the 1970s, MOSFET devices have been under investigation at cryogenic temperatures for use in custom applications, such as low-noise scientific equipment, spacecraft, power conversion etc. [5,6,7,8,9]. However, despite its many benefits for reaching high-performance and low-power computing [10], cryogenic cooling did not stay into practice for computing, abandoning the trend set by the ETA-10 liquid-nitrogen-cooled supercomputer [11].
Nowadays, co-integrating qubits and CMOS circuits on the same substrate can greatly aid the development of scalable quantum computers featuring massive parallelism and error correction [12,13,14]. In this context, a silicon-on-insulator (SOI) platform is particularly attractive since the back gate provides additional control over the electron-spin qubit, trapped under the front gate of a SOI (nanowire) MOSFET [12,15,16]. To integrate the control circuits with quantum devices working at deep-cryogenic temperatures, regular SOI MOSFETs need to demonstrate reliable digital, analog and RF functionalities at such low temperatures. Using SOI cryogenic control electronics, the back gate can prove a useful tool to control the threshold voltage and hence the power consumption in circuits integrated close to the qubits [17], benefiting qubit coherence time by lowering generated noise. The main focus is on advanced ultra-thin body fully-depleted SOI (FDSOI) technology, e.g., the 28-nm node, to enable ultimate scalability of the resulting hybrid quantum-classical system [16,18].
The 28-nm node, presently considered the ideal node for analog and RF applications at room temperature [19], has recently been tested for digital and analog functionality down to liquid-helium temperature (4.2 K) [17], and millikelvin temperature (20 mK) [20]. The improvement in RF characteristics has been verified down to liquid-nitrogen temperature (77 K) [21]. In addition to device characterization, it is mandatory that industry-standard compact models [22,23,24] become compatible with cryogenic temperatures, to achieve optimal cryogenic CMOS designs controlling a large number of qubits. To date, important temperature-related phenomena have been included only by fitting the characteristics using the existing temperature-scaling laws available in industry-standard compact MOS transistor models dedicated to room-temperature operation, i.e., for bulk [25] and double-gate MOSFET [26]. However, this approach cannot provide a physically-sound basis to further develop compact models targeting reliable CMOS designs at cryogenic temperatures. Recently, an analytical model for bulk cryogenic MOSFET operation [27] has been proposed, which has been developed starting from the Poisson equation at cryogenic temperatures, validating the Boltzmann statistics and taking into account the temperature dependencies of dopant freeze-out, bandgap widening and the Fermi-Dirac occupation of interface charge traps. This model provides the necessary analytical physics-based expressions for compact modeling purposes at cryogenic temperatures. Furthermore, a body-partitioning technique has been developed which can be used to calculate the extension of the ionized layer of dopants under the gate by field-assisted ionization when the substrate is initially frozen-out at cryogenic temperatures [28]. From this method, it can be expected that the low-doped thin film of silicon in a FDSOI MOSFET can be completely ionized at cryogenic temperature depending on the relative bias points of the front and back gates. In this work, as an initial investigation prior to further physical and compact modeling, we perform a cryogenic characterization and semi-empirical modeling of a commercial ultra-thin-body 28-nm FDSOI CMOS technology at temperatures down to 4.2 K, similar to earlier work on a commercial 28-nm bulk CMOS technology [29]. The low-temperature dc measurements (transfer and output characteristics) and the characterization down to 4.2 K are presented in Sec. 2 and 3, respectively. In Sec. 2.2, we qualitatively explain the influence of cryogenic temperatures on the electrical behavior of this technology, with a main focus on incomplete ionization (freeze-out) and interface charge traps. The free-carrier mobility trends versus temperature are obtained from the recently-proposed g ds -function method [30], which is derived here for FDSOI technologies (Sec. 3.2). This method allows to extract the mobility from dc transfer or output measurements. In Sec. 4, we illustrate how the drastic temperature reduction, the incomplete ionization, and the interface charge traps can be accounted for in circuit device-models, taking as an example the simplified-EKV model, a simple design-oriented model using only four parameters for short-channel devices.
Low-Temperature Measurements
Transfer characteristics in linear (|V DS | = 50 mV) and saturation (|V DS | = 0.9 V) were measured on various devices of a 28-nm FDSOI CMOS process from room temperature down to 4.2 K, including changes in the body bias [17]. Intermediate temperature steps were taken at 10, 36, 77, 110, 160, and 210 K, and the back-gate voltage (V back ) was ramped from −0.9 V to 0.9 V. Output characteristics were measured at zero V back using the same temperature steps, and additionally at 1.4 K for some devices. Figures (1), (2), and (3) show these measurements for large ( Fig. 1), narrow (Fig. 2), and small ( Fig. 3) nMOS and pMOS devices.
Discussion
A clear improvement in the subthreshold swing and transconductance is evident for the wide-long nMOS and pMOS devices in Fig. 1. However, it should be noted that the improvement is minimal between 10 K and 4.2 K with curves lying almost on top of each other. The on-state current increases with decreasing temperature in long nMOS, but decreases in long pMOS, as highlighted by the opposite temperature trends in the output characteristics in Figs.1-c and 1-f. This will be explained by a cryogenic-temperature-induced mobility degradation in pMOS in Sec. 3.2. The fact that the initial slope of the output characteristics in the linear regime (small V DS ) changes with temperature can be used to extract the mobility trend versus temperature down to 1.4 K, according to the g ds -function method described in Sec.3.2. Figure 2 shows similar temperaturedependent dc characteristics for narrow-short and narrow-long devices. However, conductance oscillations are observed on a narrow-short pMOS ( Fig. 2-d) in the deep-cryogenic range starting from 36 K. These oscillations have been attributed to the presence of dopants diffused from source and drain into the channel [17]. It can be noted that the oscillations becomes less pronounced with increasing temperature, gradually disappearing at 77 K and 110 K (green curves). In Fig. 2 (linear scale) at high gate voltages an impact of access resistance or mobility degradation due to the vertical field is noticeable on narrow-short nMOS in the linear regime. In Fig.3, for wideshort pMOS, conductance oscillations are also observed. In the output characteristics ( Fig. 3-c and 3-f) a draininduced-barrier-lowering is present at all temperatures, which is roughly temperature dependent. No kink effect is observed in this advanced fully-depleted technology. Since the output conductance in saturation is almost constant with temperature, the intrinsic gain versus temperature will follow the increase in transconductance with decreasing temperature.
Low-Temperature Phenomena
Important cryogenic phenomena influencing double-gate MOSFET performance have been extensively reviewed by Balestra and Ghibaudo [31,32,7], and Claeys and Simoen [8]. These phenomena, also present at room temperature but to a lower degree, include interface traps, dopant incomplete ionization, field-assisted ionization, mobility temperature-trend, bandgap temperature-trend, exponential temperature dependency of the intrinsic carrier concentration, and quantum effects. It should be noted that the kink effect in the output characteristics, prominently present in older technologies at cryogenic temperatures [33], has not been observed in this fully-depleted technology below the used supply voltage. Below follows a brief description of the phenomena which can impact a fully-depleted FDSOI technology, and how to model them:
• Incomplete ionization or substrate freeze-out In a MOSFET in thermal equilibrium (no voltage applied) dopant atoms will become deionized at a sufficiently low (cryogenic) temperature depending on the doping concentration in the range 10 12 to 10 18 cm −3 [34,35,36,37]. An overview of the freeze-out critical temperatures for each doping concentration in this range in silicon can be found in [28]. At higher doping concentrations, e.g., in the source and drain contacts, no freeze-out happens due to the formation of impurity bands which overlap with the conduction or valence band edges [38]. Therefore, freeze-out does not influence the access resistance improvement at cryogenic temperatures [39]. For a p-type silicon body, an acceptor dopant atom will be ionized from a theoretical viewpoint when the acceptor energy E A is occupied by an electron. Therefore, the ionized dopant concentration, N − A , is given by
a) b) d) e) c) f) T(K) T(K) T(K) T(K) T(K)N − A = N A f (E A ) = N A 1 + g A e E A −E F,n kT = N A 1 + g A e ψ A −(ψ−V ch ) U T ,(1)
where f (E A ) is the ionization probability given by a Fermi-Dirac distribution function, and the electron quasi-Fermi-level is E F,n = E F − qV ch with V ch the channel voltage. The RHS of (1) is convenient for direct inclusion in the Poisson-Boltzmann equation [27].
In FDSOI, the doping concentration is rather low (≈ 10 15 cm −3 ) compared to the inversion charge density. However, as illustrated in Fig. 4-a, in the flatband condition and at 4.2 K, approximately all dopants will be frozen-out, independent of the doping concentration in the range (10 12 − 10 18 cm −3 ). The calculated E F -position at 4.2 K lies under E A , leading to freeze-out or f (E A ) 1 ( Fig. 4-b). Nonetheless, the front-gate voltage will ionize the impurities under the surface of the front-gate, when E A bends under E F near the surface of the front-gate. In the subthreshold region, when E F ≈ E c − 3U T , complete ionization can be assumed under the front gate. This transition from freeze-out to complete ionization due to the applied field can lead to a kink in early depletion [27]. Note that depending on the band bending at the front and back gates in a certain mode of operation, it is possible that the dopants under the front gate are completely ionized but frozen-out under the back gate or vice versa. A complete comprehension of the field-assisted ionization effect on the mobile charge density in FDSOI below inversion and including body bias, would require a more in-depth physical analysis.
• Temperature-dependent occupation of interface charge traps [41,42] Interface traps need to be included for both the front and the back gate, as illustrated in Fig. 4-b. This adds two additional Fermi-Dirac temperature dependencies f (E t,j ) (with E t,j a trap energy-level at position j in the bandgap), apart from the ionization probability f (E A ). The interface traps can be modeled as a discrete summation of traps, as explained in [43,44,45]. The temperature-dependent occupation of interface traps is important for a correct derivation of the subthreshold-swing formula, leading to hyperbolic temperature dependency of the slope factor (ignoring coupling effects between front and back gates), which will be discussed in more detail in Sec. 3.
• Bandgap widening The total change of the silicon bandgap from room temperature down to 4.2 K is approximately 1.12 to 1.16 eV, widening with decreasing temperatures [40]. The temperature dependence in the cryogenic regime (< 100 K) is almost constant, as illustrated in Fig. 4-a. a) • Exponential temperature dependency of the intrinsic carrier concentration The intrinsic carrier concentration is given by
E c E D E A E v f(E A ) E i E t Buried oxide f(E) 4.2 K to substrate E F (undoped) E F (N A =10 15 cm −3 ) 0 E t 1 b) 4.2 Kn i = √ N c N v exp [−E g /(2kT )]
, which at 4.2 K leads to extremely small values lying outside IEEE double-precision arithmetic, in the order of 10 −650 cm −3 [27]. This is physically accurate since the overlap of a Fermi-Dirac function at 4.2 K, lying at the intrinsic level ( Fig. 4-b), and the density-of-states in the conduction band becomes very small.
• Quantum confinement and quantum transport Quantum effects become more pronounced in FDSOI MOS-FETs at cryogenic temperatures, since they are less obscured by thermal fluctuations when the quantized energy is similar to the thermal energy [46].
Characterization
In this section, the following technological parameters are extracted from the cryogenic measurements: the subthreshold swing (SS), slope factor (n), threshold voltage (V th ), transconductance in linear and saturation (G m,lin , G m,sat ), the on-state current (I on ), and the effective free-carrier (electron and hole) mobilities (µ ef f ).
Subthreshold swing, Threshold voltage, Transconductance, and On-state current
As illustrated in Fig. 5-a, for temperatures below ≈ 160 K, the extracted average SS-values show an increasing offset, ∆SS, from the thermal limit, U T ln 10, with U T kT /q the thermal voltage. ∆SS reaches around 10 mV/dec at 4.2 K for long nMOS, since U T ln 10 predicts ≈ 0.8 mV/dec. The slope factors required to reach such high SS-values are shown in Fig. 5-b (n = SS/(U T ln 10)). From this figure a hyperbolic temperaturedependency of n is evident, which is not strongly dependent on geometry at cryogenic temperatures. The values below 77 K in Fig. 5-a cannot be explained anymore by n 0 U T ln 10, where the slope-factor n 0 is limited by two, according to n 0 = 1 + C dep /C ox . Here the depletion capacitance C dep is smaller than the oxide capacitance C ox . Furthermore, including the interface-trap capacitance, C it = qN it , i.e. n 0 = 1 + (C dep + C it )/C ox with N it the density-of-interface-traps per unit area, would lead to very high extracted values for N it in the order of 10 13 cm −2 at 4.2 K [9,42,41], and 10 17 cm −2 at 20 mK [20]. The latter is higher than the density of surface-states in silicon (10 15 cm −2 ). However, it should be emphasized that in this n 0 -formula the temperature-dependent occupation of interface-traps is not taken into account (see Sec. 2.2). Reliable extraction of the interface-trap-density at deep-cryogenic and millikelvin temperatures using the standard SS-formula is therefore questionable. Inclusion of the interface-trap temperature dependency into the subthreshold swing theory of bulk MOSFET has been shown to yield lower extracted N it -values [27]. Similarly to the derivation for bulk MOSFET presented in [27], by including f (E t ) the temperature dependency of n ∝ 1/U T can be derived for the front-gate in FDSOI as well, ignoring the coupling effects between front and back gates. This gives SS = n(T )U T ln 10 = n 0 U T ln 10 + ∆SS, where n 0 is the slope factor without interface traps, and ∆SS the subthreshold-swing offset as observed in Fig. 5-a. ∆SS is given by (qN it /C ox ) ln 10 g t /(1 + g t ) 2 with N it the density-of-interface-traps and g t the trap degeneracy factor. Note that in this model, N it does not become multiplied with U T , resulting in reasonable extracted values for N it at cryogenic temperatures lower than found in [41,42,20]. The ∆SS-offset starts to increase below ≈ 160 K since the subthreshold region happens when E F lies closer to E c , where N it is observed to be higher already at 300 K (see also Fig. 4-b) [47]. The shift in threshold voltage at 4.2 K with respect to room temperature increases in the order of 0.1−0.3 V (Fig. 5-c). Note that the largest V th -increase is observed for pMOS, similarly to a 28-nm bulk process [29]. Furthermore, the maximum G m,sat and G m,lin (Figs.5d-e) improve down to 4.2 K, e.g. respectively × 3.4 (linear) and × 1.8 (saturation) for nMOS W/L = 1 µm / 1 µm. In Fig.5-f, I on is extracted at |V GS | = 1 V. Note that the actual trend of I on with temperature is strongly dependent on the bias and the device-type. At a standard supply voltage of 1 V, the on-state current increases with decreasing temperature for long nMOS ( Fig. 1-a-c), while it decreases for pMOS ( Fig. 1-d-f). However, a cryogenic-temperature-induced mobility degradation has not been extracted from measured CV characteristics on this device [17]. Therefore, in the next section, we take a second look at the characterization of the free-carrier mobility in this technology at cryogenic temperatures.
Free-Carrier Mobility
In doped bulk silicon, the free-carrier mobility is expected to drop when transitioning below a certain cryogenic temperature and Coulombic impurity scattering becomes dominant over phonon scattering, leading to a typical bell-shaped mobility trend with respect to temperature [38,5] This behavior can be different in MOSFET devices when the channel is ballistic.
The free-carrier mobility in MOSFETs is usually extracted from dc measurements using the Y -function approach [48,49,50], or from a combination of dc and capacitance measurements using the split-CV method. For advanced CMOS technologies, the split-CV method can yield unreliable results due to the dominance of extrinsic capacitances, and the Y -function has two effective slopes. Both methods are also used to characterize advanced devices down to deep-cryogenic (< 10 K) temperatures [51,52,32,17]. Dopant freeze-out and fieldassisted ionization may change the dependency of the scattering mechanisms on the gate voltage. Since in strong inversion field-assisted ionization is complete, the underlying assumption of the Y -function approach, i.e., the homographic gate-voltage dependent mobility law, µ = µ 0 / [1 + θ(V GS − V t )] [48], can still provide an adequate description of the mobility down to cryogenic temperatures. On the other hand, the split-CV method has to deal with the unknown thermal behavior of the extrinsic capacitances, and requires deep-cryogenic cooling for two types of measurements. Using CV measurements, a mobility degradation at cryogenic temperatures has not been observed for long pMOS in this technology [17]. In the next subsection, we obtain the free-carrier mobilities from dc measurements according to an approach recently developed by Jazaeri et al [30].
Free-Carrier Mobility Extraction using g ds -function
The Jazaeri mobility-extraction method (g ds -function) [30] does not assume any gate-voltage dependent mobility law a priori. The method only assumes drift-diffusion transport in advanced field-effect transistors as a a) b) c) Figure 6: Free-carrier mobilities in a 28-nm FDSOI CMOS process extracted using the g ds -function [30] and normalized to the room-temperature mobility, a) nMOS 1 µm/1 µm down to 1.4 K, b) pMOS 1 µm/1 µm down to 1.4 K, and c) nMOS 1 µm/28 nm down to 4.2 K. A free-carrier mobility degradation induced by cryogenic temperatures is extracted on long pMOS starting from around 77 K, and around 10 K in long nMOS. For the considered 28-nm process [17], the EOT is 1.55 nm for nMOS, and 1.7 nm for pMOS with thin front-gate oxide [17].
starting point of the derivation [30]. Drift-diffusion transport has been shown to give an accurate representation of the current down to 4.2 K, and the validity of the Boltzmann statistics has been demonstrated down to millikelvin temperatures [28,27]. Therefore, here we can extend this method to deep-cryogenic temperature operation. In what follows, the method will be briefly derived for SOI technology.
Following the approach in [30], for two different operating points in the linear regime, drift-diffusion gives
I D1 = −(W/L G )µQ m1 V DS1 , and I D2 = −(W/L G )µQ m2 V DS2 ,
where Q m is the mean value of the local mobile charge densities at source and drain, i.e., Q m (Q mS + Q mD )/2, and it is assumed that µ is not as a function of V DS for small V DS . Hence we can derive that
1 Q m2 ∂Q m ∂V DS ≈ I D1 − I D2 V DS1 − V DS2 1 I D2 − I D1 I D2 1 V DS1 (2)
in a first approximation. The RHS of (2) can be obtained from dc measurements. Once ∂Q m /∂V DS is known, the mobility can be extracted by merging (2) with the drift-diffusion expression for I D2 and eliminating the mobility [30]. In strong inversion, Q m can be estimated as
Q m = −C GG1 (V GS1 − V T 1 ) − C GG2 (V GS2 − V T 2 )
, with C GG the intrinsic gate capacitance per unit area. This expression is still valid down to deep-cryogenic temperatures since (i) the Maxwell-Boltzmann approximation has been verified down to millikelvin temperatures [27], (ii) in the inversion layer all the dopants are ionized due to field-assisted ionization [53,28], and (iii) interface traps only affect the DC current significantly in the subthreshold region, not in the inversion region [27,28]. Therefore, following [30] ∂Q m
∂V DS = C GG1 + C GG2 2 .(3)
With (2), we obtain
Q m2 ≈ (C GG1 + C GG2 )I D2 2 I D1 −I D2 VDS1−VDS2 − I D1 VDS1(4)
Thus the mobility is given by
µ ≈ − 2L G W (C GG1 + C GG2 )V DS2 × I D1 − I D2 V DS1 − V DS2 − I D1 V DS1 = − 2L G W (C GG1 + C GG2 ) × 1 V DS1 − V DS2 I D1 V DS1 − I D2 V DS2(5)
Therefore, for FDSOI the formula remains the same but only the two gate capacitances have to be added. To be very accurate, one would need to consider C GG (T ) as a function of temperature, but this would again require CV measurements. However, in the CV characteristics of this technology down to 4.2 K only a change in the threshold voltage has been observed, and not much change in the shape of C gg [17,54]. Therefore, in strong inversion, we can determine an effective mobility, µ ef f , setting C GG1 = ε SiO2 /EOT = C ox and C GG2 = C BOX = ε SiO2 /t BOX. . For this reason, in this work we will only investigate mobility-ratios with respect to room temperature, and not the exact values of the mobility. To extract the mobility at a constant back-gate voltage, it is allowed to take into account only the capacitance on the front-gate. This gives
µ ef f ≈ − 2L G W C front gate ∂g ds ∂V DS ,(6)
which is valid at small V DS (linear regime) and high V GS (strong inversion) [30]. Expression (6) is referred to as the g ds -function. According to this expression, the mobility is proportional to the curvature of the output characteristics at small V DS at a given gate voltage and temperature. Note the minus sign, leading to a positive mobility-value since ∂g ds /∂V DS is negative. The derivative in (6) can be calculated from the measurements as the difference in initial slopes using a back-difference method (g ds,1 − g ds,0 )/(V DS1 − V DS0 ) (at small V DS ). The method is versatile since the free-carrier mobility can be obtained either from measured output characteristics using (6), or from two linear transfer characteristics (5), depending on which low-temperature data is available.
Here we extract the mobility from the output characteristics which are available down to 1.4 K for the long devices, shown in Fig.1-c and 1-f. Figure 6 plots the ratio of the maximum effective mobility versus the room-temperature mobility for nMOS 1 µm/1 µm (Fig. 6-a), pMOS 1 µm/1 µm (Fig. 6-b), and nMOS 80 nm/46 nm (Fig. 6-c). A mobility degradation is observed for the long devices, where the temperature with maximum mobility is shifted between nMOS (≈10 K) and pMOS (≈77 K). No mobility degradation is observed on short nMOS down to 4.2 K.
Modeling
In this section, we model the low-temperature measurements (Sec. 2) using the design-oriented simplified EKV model, focusing on the measurements that do not show any oscillations.
A detailed overview of this model is presented in [55,56]. Its suitability for FDSOI processes has been assessed at room temperature, including body-biasing [57]. The model is valid in saturation, expressing the measured drain current in saturation in terms of an inversion coefficient, IC, given by IC I D,sat /I spec , where the specific current, I spec , is defined as I spec I spec (W/L), and the 'specific-current-per-square', I spec , is a parameter independent of dimensions given by 2nµC ox U 2 T . Many analog figure-of-merits can be expressed in terms of this inversion coefficient, which separates the different regions of inversion as follows:
• IC < 0.1: weak inversion The long-channel model is given by the following expression [55,56]:
v p − v s = ln( √ 4IC + 1 − 1) + √ 4IC + 1 − (1 + ln 2),(7)
and the short-channel model by
IC = 4(q 2 s + q s ) 2 + λ c + 4(1 + λ c ) + λ 2 c (1 + 2q s ) 2 , v p − v s = ln q s + 2q s ,(8)
where v p V P /U T is the normalized pinch-off voltage, q s Q s /Q spec the normalized inversion charge at the source (with Q spec −2nU T C ox ), and v s V S /U T the normalized source voltage. The velocity saturation parameter, λ c = L sat /L, is the ratio of the channel in full velocity saturation (near the drain) over the total length of the channel.
Starting from the measured drain current in saturation, the inversion coefficient is evaluated for each I D,sat at a given gate voltage, using a specific model parameter I spec . Depending on the length of the channel, we proceed as follows:
Procedure long-channel
For each IC, the normalized pinch-off voltage v p is obtained from 7. At a given temperature, the gate voltage follows from V g = nU T v p + V T0 , given specific n and V T0 model parameters. Initial guesses for these model parameters can be obtained from the extracted threshold voltage and slope factors in Fig. 5.
Procedure short-channel
For each IC, the first expression in (8) is numerically solved for q s , given a specific L sat . The v p -values are derived from all q s using the second expression in (8), Similar to the long-channel model, the gate voltage then follows from V g = nU T v p + V T0 , given specific n and V T0 model parameters.
Note that the long-channel model uses three model parameters (n, V T0 , I spec ), while the short-channel model uses four (n, V T0 , I spec , L sat ). By plotting I D,sat versus the obtained V g , the model curves can be validated with the measurements at each temperature, as will be illustrated in the next section.
Comparison with measurements
Using the model over a wide temperature range (from 300 down to 4.2 K), the transfer characteristics, backgate sensitivity, and transconductance efficiency can be accurately modeled in long and short FDSOI devices. Figures 7 to 9 show the modeled transfer characteristics in saturation at all considered temperatures down to 4.2 K at zero back-gate voltage for long nMOS (Fig. 7), long pMOS (Fig. 8), and short nMOS (Fig. 9). The model parameters are shown in the tables below the figures. The strong increase in the n model-parameter at deep-cryogenic temperatures corresponds to the interface-trapping process, as explained in Sec. 3. The V T0 model-parameter captures the change in the threshold voltage due to Fermi-Dirac scaling and incomplete ionization, increasing in the order of 0.1 V. Note that the used values for n and V T0 correspond to the extracted values in Fig. 5-b and 5-c. The I spec model-parameter decreases over one order of magnitude from 300 down to 4.2 K. For the short device, the L sat -parameter decreases from 11 to 5 nm due to a reduction in the phonon scattering, leading to a shorter part of the channel near the drain in velocity saturation. The lower impact of velocity saturation at lower temperatures becomes clear also by plotting the normalized transconductance efficiency, G m nU T /I D , versus the inversion coefficient at 300, 77, and 4.2 K, shown in Fig. 10. Using the same parameters for n, I spec , and L sat as in Fig. 9, good agreement is obtained between the modeled and measured transconductance efficiency at 300, 77, and 4.2 K. Fig. 10 verifies that the G m /I D design-methodology [58] remains valid for a 28 nm FDSOI technology down to 4.2 K, extending therefore its universality to advanced bulk and FDSOI CMOS operating at extremely-low temperatures. Furthermore, as illustrated in Figures 11 and 12, changing the V T0 model parameter allows to capture the effect of the back-gate at 4.2 K for long (11) and short devices. The n model parameter tends to increase with increasing absolute values of the back-gate voltage in both long and short devices, accounting for a change in SS induced by the back gate. The L sat model parameter maintains the same value (5 nm at 4.2 K) for different back-gate voltages, showing that the velocity saturation is not influenced by the back gate.
Conclusion
A 28-nm Fully-Depleted SOI CMOS process is characterized and modeled from room temperature down to liquid-helium temperature (4.2 K). Output characteristics and free-carrier mobilities are presented down to 1.4 K. The design-oriented simplified EKV model can accurately predict the impact of the temperature reduction on the transfer characteristics, back-gate sensitivity, and transconductance efficiency of 28-nm devices using four parameters: the slope factor n, threshold voltage V T0 , specific current I spec , and saturation length L sat . A new method is proposed to extract the free-carrier mobility-trends versus temperature in SOI technology from dc measurements. This method does not require CV measurements and can hence be used to extract the mobilitytrend also on short-narrow advanced CMOS devices where parasitic capacitances can dominate. Using this method, a degradation in the free-carrier mobility is observed at cryogenic temperatures in long nMOS and pMOS, and an increase in a short 46-nm nMOS. Table 3. Table 4. Table 5.
Figure 1 :
1Transfer and output characteristics measured in wide-long nMOS and pMOS devices (W/L = 1µm/1µm) of a
Figure 2
2: a)-b) Transfer characteristics measured in a narrow-short (W/L =210 nm/28 nm) nMOS device down to 4.2 K at zero back-gate voltage, c) Output characteristics measured in a narrow-long (W/L =210 nm/1 µm) nMOS device down to 1.4 K, d)e) Transfer characteristics measured on narrow-short pMOS (W/L =300 nm/28 nm) down to 4.2 K at zero back-gate voltage, f) Output characteristics measured in a narrow-long pMOS (W/L =300 nm/1 µm) down to 1.4 K, The color scheme for the intermediate temperatures is the same as inFig. 1.
Figure 3 :
3a)-b) Transfer characteristics measured in a wide-short (W/L =1 µm/28 nm) nMOS device down to 4.2 K at zero backgate voltage, c) Output characteristics measured in a small nMOS device with W/L =80 nm/46 nm down to 1.4 K at two different gate voltages, d)-e) Transfer characteristics measured in a wide-short pMOS (W/L =1 µm/28 nm) down to 4.2 K at zero back-gate voltage, f) Output characteristics measured in a small pMOS (W/L =80 nm/46 nm) down to 1.4 K at zero back-gate voltage. The color scheme for the intermediate temperatures is the same as in Fig. 1.
Figure 4 :
4a) Simulated position of the Fermi-level (red) in n-and p-type doped silicon as a function of temperature and doping concentration. At 4.2 K, the silicon is frozen-out for all doping concentrations in the range (10 12 -10 18 cm −3 ) since E F > E D or E F < E A , and f (E D ) and f (E A ) are close to step functions. The temperature dependency of the bandgap from Varshni[40] is used, b) Illustration of freeze-out and interface charge traps in the thin silicon film of a FDSOI MOSFET with a p-type body (N A = 10 15 cm −3 ). The two phenomena can be described by Fermi-Dirac statistics. The position of the Fermi level is shown in red. The probabilities of dopant ionization and interface-trap occupation depend on the position of the Fermi-level in the silicon film with respect to E A and Et, respectively. In case E A bends under E F near the surface of one of the gates, an ionized layer of dopants forms under the gate. Field-assisted ionization makes f (E A ) ≈ 1 before inversion is reached. In the figure it is assumed that the front and back gates are biased such that a flatband situation is created.
Figure 5 :
5Characterization of a 28-nm FDSOI CMOS technology down to liquid-helium temperature (4.2 K), a) Subthreshold swing SS, b) Slope factor (n = SS/(U T ln 10), c) Threshold voltage shift with respect to room temperature ∆V th = V th − V th,300 , d) Transconductance in saturation, e) Transconductance in linear, f) Ratio of low-temperature over room-temperature on-state current.
•
IC > 10: strong inversion.
Figure 9 :
9Modeling short 28-nm FDSOI nMOS down to 4.2 K. Model parameters are given in
Figure 10 :
10Modeling the normalized transconductance efficiency at 300, 77, and 4.2 K in a short 28-nm FDSOI nMOS in saturation. Model parameters are given in the figure.
Figure 11 :
11Modeling the body bias effect at 4.2 K in a long FDSOI nMOS. Model parameters are given in
Figure 12 :
12Modeling the body bias effect at 4.2 K in a short 28-nm FDSOI nMOS. Model parameters are given in
[ 17 ]
17H. Bohuslavskyi, S. Barraud, M. Cass, V. Barrai, B. Bertrand, L. Hutin, F. Arnaud, P. Galy, M. Sanquer, S. D. Franceschi, M. Vinet, 28nm Fully-depleted SOI technology: Cryogenic control electronics for quantum computing, in: 2017 Silicon Nanoelectronics Workshop (SNW), 2017, pp. 143-144. doi:10.23919/SNW.
At V GS = 1 V, the saturation current decreases with decreasing temperature for pMOS, while it increases for nMOS. This hints on a cryogenic-temperature-induced mobility degradation in long pMOS devices, as will be demonstrated in Sec. 3.2. The temperature dependencies of the free-carrier mobilities can be extracted from c) and f) knowing that the mobility is proportional to the derivative of the output conductance at small V DS (see Sec. 3.2).28-nm
FDSOI CMOS technology at zero back-gate voltage. The linear and saturation transfer characteristics are presented down to 4.2 K
(red: 300 K, brown: 210 K, orange: 160 K, light green: 110 K, dark green: 77 K, purple: 36 K, light blue: 10 K, and dark blue:
4.2 K). In subfigures a, b, and e the curves at 10 K and 4.2 K lie almost on top of each other. The output characteristics (c and f)
are presented down to 1.4 K (black: 1.4 K).
Table 1: Model parameters for nMOS W/L =1 µm/1 µm at V back =0 V and increasing temperatures, corresponding toFig. 7-
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Figure 7: Modeling long FDSOI nMOS down to 4.2 K. Model parameters are given in Table 1.
Temperature [K]
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Figure 8: Modeling long FDSOI pMOS down to 4.2 K. Model parameters are given inTable 2.1
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0
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2
V
S
G
[
V
]
W
/
L
=
1
µ
m
/
1
µ
m
p
M
O
S
V
S
D
=
0
.
9
V
M
o
d
e
l
M
e
a
s
u
r
e
m
e
n
t
s
V
b
a
c
k
=
0
V
T
[
K
]
=
4
.
2
,
3
6
,
7
7
,
1
1
0
,
1
6
0
,
2
1
0
,
3
0
0
Temperature [K]
n
V T0 [V] I spec [nA]
4.2
23
0.84
42
36
3.07
0.825
65
77
1.82
0.76
75
110
1.46
0.73
105
160
1.25
0.695
125
210
1.11
0.65
135
300
1.1
0.6
235
Table 2 :
2Model parameters for pMOS W/L =1 µm/1 µm at V back =0 V and increasing temperatures, corresponding toFig. 8.
Temperature [K] n V T0 [V] I spec [nA] L sat [nm]4.2
22
0.47
75
5
77
1.7
0.46
175
8
110
1.47
0.45
195
8.5
160
1.38
0.43
335
9
210
1.34
0.41
505
10
300
1.3
0.37
835
11
Table 3 :
3Model parameters for nMOS W/L =1 µm/28 nm at V back =0 V and increasing temperatures, corresponding toFig. 9.
Table 4 :
4Model parameters for nMOS W/L =1 µm/1 µm at 4.2 K and ramping V back , corresponding toFig. 11.Back-gate Voltage [V]
n
V T0 [V] I spec [nA] L sat [nm]
-0.9
22.8
0.522
77
5
-0.6
22.7
0.506
76
5
-0.3
22.6
0.49
75
5
0
22.7
0.473
70
5
0.3
23.9
0.455
80
5
0.6
25
0.435
85
5
0.9
29.3
0.42
115
5
Table 5 :
5Model parameters for nMOS W/L =1 µm/28 nm at 4.2 K and ramping V back , corresponding toFig. 12.
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Design-oriented modeling of 28 nm FDSOI CMOS technology down to 4.2 K for quantum computing. A Beckers, F Jazaeri, H Bohuslavskyi, L Hutin, S D Franceschi, C Enz, 10.1109/ULIS.2018.83547422018 Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EUROSOI-ULIS). A. Beckers, F. Jazaeri, H. Bohuslavskyi, L. Hutin, S. D. Franceschi, C. Enz, Design-oriented modeling of 28 nm FDSOI CMOS technology down to 4.2 K for quantum computing, in: 2018 Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EUROSOI-ULIS), 2018, pp. 1-4. doi:10.1109/ULIS.2018.8354742.
Effects of cryogenic temperatures on small-signal MOSFET capacitances. A Akturk, N Goldsman, Z Dilli, M Peckerar, 10.1109/ISDRS.2007.4422237International Semiconductor Device Research SymposiumA. Akturk, N. Goldsman, Z. Dilli, M. Peckerar, Effects of cryogenic temperatures on small-signal MOSFET capacitances, in: 2007 International Semiconductor Device Research Symposium, 2007, pp. 1-2. doi: 10.1109/ISDRS.2007.4422237.
Nanoscale MOSFET Modeling: Part 1: The Simplified EKV Model for the Design of Low-Power Analog Circuits. C Enz, F Chicco, A Pezzotta, 10.1109/MSSC.2017.2712318IEEE Solid-State Circuits Magazine. 93C. Enz, F. Chicco, A. Pezzotta, Nanoscale MOSFET Modeling: Part 1: The Simplified EKV Model for the Design of Low-Power Analog Circuits, IEEE Solid-State Circuits Magazine 9 (3) (2017) 26-35. doi:10.1109/MSSC.2017.2712318.
Using the Inversion Coefficient as the Primary Design Parameter. C Enz, F Chicco, A Pezzotta, Mosfet Nanoscale, Modeling, 10.1109/MSSC.2017.2745838doi:10.1109/ MSSC.2017.2745838IEEE Solid-State Circuits Magazine. 24C. Enz, F. Chicco, A. Pezzotta, Nanoscale MOSFET Modeling: Part 2: Using the Inversion Coefficient as the Primary Design Parameter, IEEE Solid-State Circuits Magazine 9 (4) (2017) 73-81. doi:10.1109/ MSSC.2017.2745838.
A design-oriented charge-based simplified model for FDSOI MOSFETs. A Pezzotta, F Jazaeri, H Bohuslavskyi, L Hutin, C Enz, 10.1109/ULIS.2018.83547642018 Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EUROSOI-ULIS). A. Pezzotta, F. Jazaeri, H. Bohuslavskyi, L. Hutin, C. Enz, A design-oriented charge-based simplified model for FDSOI MOSFETs, in: 2018 Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EUROSOI-ULIS), 2018, pp. 1-4. doi:10.1109/ULIS.2018.8354764.
A Gm/ID-based methodology for the design of CMOS analog circuits and its application to the synthesis of a silicon-on-insulator micropower OTA. F Silveira, D Flandre, P G A Jespers, 10.1109/4.535416IEEE Journal of Solid-State Circuits. 319F. Silveira, D. Flandre, P. G. A. Jespers, A Gm/ID-based methodology for the design of CMOS analog circuits and its application to the synthesis of a silicon-on-insulator micropower OTA, IEEE Journal of Solid-State Circuits 31 (9) (1996) 1314-1319. doi:10.1109/4.535416.
| [] |
[
"ABSENCE OF POINT SPECTRUM FOR THE SELF-DUAL EXTENDED HARPER'S MODEL",
"ABSENCE OF POINT SPECTRUM FOR THE SELF-DUAL EXTENDED HARPER'S MODEL"
] | [
"Rui Han "
] | [] | [] | We give a simple proof of absence of point spectrum for the self-dual extended Harper's model. We get a sharp result which improves that of[1]in the isotropic self-dual regime. | 10.1093/imrn/rnw279 | [
"https://arxiv.org/pdf/1909.03995v1.pdf"
] | 125,334,016 | 1909.03995 | a03d459fc35ea0e104fc85bf09e55b0a0e249db6 |
ABSENCE OF POINT SPECTRUM FOR THE SELF-DUAL EXTENDED HARPER'S MODEL
9 Sep 2019
Rui Han
ABSENCE OF POINT SPECTRUM FOR THE SELF-DUAL EXTENDED HARPER'S MODEL
9 Sep 2019
We give a simple proof of absence of point spectrum for the self-dual extended Harper's model. We get a sharp result which improves that of[1]in the isotropic self-dual regime.
Introduction
We study the extended Harper's model on l 2 (Z):
(H λ,α,θ u) n = c λ (θ + nα)u n+1 +c λ (θ + (n − 1)α)u n−1 + v(θ + nα)u n , (1.1) where c λ (θ) = λ 1 e −2πi(θ+ α 2 ) + λ 2 + λ 3 e 2πi(θ+ α 2 ) and v(θ) = 2 cos 2πθ.c λ (θ) = c λ (θ) for θ ∈ T and its analytic extension when θ / ∈ T. We refer to λ = (λ 1 , λ 2 , λ 3 ) as coupling constants, θ ∈ T = [0, 1] as the phase and α as the frequency.
In [2] the authors partitioned the parameter space into the following three regions. Region I: 0 ≤ λ 1 + λ 3 ≤ 1, 0 < λ 2 ≤ 1, Region II: 0 ≤ λ 1 + λ 3 ≤ λ 2 , 1 ≤ λ 2 , Region III: max{1, λ 2 } ≤ λ 1 + λ 3 , λ 2 > 0. According to the action of the duality transformation σ : λ = (λ 1 , λ 2 , λ 3 ) →λ = ( λ3 λ2 , 1 λ2 , λ1 λ2 ), we have the following observation [2]:
Observation 1.1. σ is a bijective map on 0 ≤ λ 1 + λ 3 , 0 < λ 2 . (i) σ(I • ) = II • , σ(III • ) = σ(III • ) 1
(ii) Letting L I := {λ 1 + λ 3 = 1, 0 < λ 2 ≤ 1}, L II := {0 ≤ λ 1 + λ 3 ≤ 1, λ 2 = 1}, and L III := {1 ≤ λ 1 + λ 3 = λ 2 }, σ(L I ) = L III and σ(L II ) = L II .
As σ bijectively maps III ∪ L II onto itself, the literature refers to III ∪ L II as the self-dual regime. We further divide III into III λ1=λ3 (isotropic self-dual regime) and III λ1 =λ3 (anisotropic self-dual regime).
A complete understanding of the spectral properties of the extended Harper's model for a.e. θ has been established:
Theorem 1.2. [1]
The following Lebesgue decomposition of the spectrum of H λ,α,θ holds for a.e.θ.
• For all Diophantine α, for Region I, H λ,α,θ has pure point spectrum.
• For all irrational α, for Regions II, III λ1 =λ3 , H λ,α,θ has purely absolutely continuous spectrum. • For all irrational α, for Region III λ1=λ3 , H λ,α,θ has purely singular continuous spectrum.
As pointed out in [1], the main missing link between [2,3] and Theorem 1.2 is the following theorem, excluding eigenvalues in the self-dual regime. We say θ is α-rational if 2θ ∈ Zα + Z, otherwise we say θ is α-irrational.
Theorem 1.3. [1] For all irrational α,
• for λ ∈ III λ1 =λ3 ∪ L II , H λ,α,θ has empty point spectrum for all α-irrational θ.
• for λ ∈ III λ1=λ3 , H λ,α,θ has empty point spectrum for a.e. θ.
In [1] the authors had to exclude more phases than α-rational θ in the isotropic self-dual regime.
In this paper we give a simple proof of the following theorem.
Theorem 1.4. For all irrational α, for λ ∈ III, H λ,α,θ has empty point spectrum for all α-irrational θ.
Remark 1.1. Our result for the isotropic self-dual regime III λ1=λ3 is sharp. Indeed, according to Proposition 5.1 in [1], for α-rational θ, H λ,α,θ has point spectrum.
We organize this paper in the following way: in Section 2 we include some preliminaries, in Section 3 we present two lemmas that will be used in Section 5, then we deal with III λ1=λ3 and III λ1 =λ3 ∩ {λ 1 + λ 3 = 1} in Section 4 and III λ1 =λ3 ∩ {λ 1 + λ 3 > 1} in Section 5.
Preliminaries
2.1. Rational approximation. Let { pm qm } be the continued fraction approximants of α, then 1 2q m+1 ≤ q m α T ≤ 1 q m+1 . (2.1)
The exponent β(α) is defined as follows
β(α) = lim sup m→∞ ln q m+1 q m . (2.2)
It describes how well is α approximated by rationals.
2.2. Self-dual extended Harper's model. Let |c| λ (θ) = c λ (θ)c λ (θ) be the analytic function that coincides with |c λ (θ)| when θ ∈ T.
The presence of singularities of c λ (θ) is explicit:
Observation 2.1. (e.g. [1])
The function c λ (θ) has at most two zeros. Necessary conditions for real roots are λ ∈ III λ1=λ3 or λ ∈ III λ1 =λ3 ∩ {λ 1 + λ 3 = λ 2 }. Moreover,
• for λ ∈ III λ1=λ3 , c λ (θ) has real roots determined by
2λ 3 cos 2π(θ + α 2 ) = −λ 2 , (2.3)
and giving rise to a double root at
θ = 1 2 − α 2 if λ ∈ III λ1=λ3 ∩ {λ 1 + λ 3 = λ 2 }. • for λ ∈ III λ1 =λ3 ∩ {λ 1 + λ 3 = λ 2 }, c λ (θ)
has only one simple real root at θ = 1 2 − α 2 . Remark 2.1. By the definition of the duality transformation σ, Observation 2.1 implies that cλ(θ) has singular point if and only if λ ∈ III λ1=λ3 or λ ∈ III λ1 =λ3 ∩ {λ 1 + λ 3 = 1}.
It will be clear in Section 4 that presence of singularities of cλ(θ) indeed simplifies the proof of empty point spectrum of H λ,α,θ .
Lemmas
Lemma 3.1. For λ ∈ III λ1 =λ3 ∩ {λ 1 + λ 3 > 1}, when λ 3 > λ 1 , we have cλ(θ) |c|λ(θ) = e −2πi(θ+ α 2 )+if (θ) andcλ (θ) |c|λ(θ) = e 2πi(θ+ α 2 )−if (θ) ,
for a real analytic function f (θ) on T with T f (θ)dθ = 0.
Proof. By the definition of cλ(θ) we have
cλ(θ) = λ 3 λ 2 e −2πi(θ+ α 2 ) + 1 λ 2 + λ 1 λ 2 e 2πi(θ+ α 2 ) (3.1) = λ 1 λ 2 e −2πi(θ+ α 2 ) (e 2πi(θ+ α 2 ) − y + )(e 2πi(θ+ α 2 ) − y − ), (3.2) where y ± = −1± √ 1−4λ1λ3 2λ1
. Note that
y + = y − with |y + | = |y − | = λ 3 λ 1 > 1, when 1 ≤ 2 λ 1 λ 3 , (3.3) y + , y − ∈ R with |y + | > |y − | = 2λ 3 λ 1 + √ 1 − 4λ 1 λ 3 > 1, when λ 1 + λ 3 > 1 > 2 λ 1 λ 3 . (3.4) Note that cλ(θ) |c|λ(θ) = cλ(θ) cλ(θ) = e −2πi(θ+ α 2 ) (e 2πi(θ+ α 2 ) − y + )(e 2πi(θ+ α 2 ) − y − ) (e −2πi(θ+ α 2 ) − y + )(e −2πi(θ+ α 2 ) − y − ) . (3.5) By (3.3), we have T arg (e 2πi(θ+ α 2 ) − y + )(e 2πi(θ+ α 2 ) − y − ) (e −2πi(θ+ α 2 ) − y + )(e −2πi(θ+ α 2 ) − y − ) dθ = 0, (3.6) and | (e 2πi(θ+ α 2 ) − y + )(e 2πi(θ+ α 2 ) − y − ) (e −2πi(θ+ α 2 ) − y + )(e −2πi(θ+ α 2 ) − y − ) | ≡ 1. (3.7)
Thus there exists a real analytic function g(θ) on T such that
(e 2πi(θ+ α 2 ) − y + )(e 2πi(θ+ α 2 ) − y − ) (e −2πi(θ+ α 2 ) − y + )(e −2πi(θ+ α 2 ) − y − ) = e ig(θ) ,f (x) + f (x + α) + · · · + f (x + q m l α − α) = 0 uniformly in x ∈ T.
Proof. Suppose f is analytic on |Imθ| ≤ δ 0 , then |f (n)| ≤ ce −2πδ0|n| for some constant c > 0. Case 1. If β(α) = 0, then by solving the coholomogical equation we get
f (x) = h(x + α) − h(x) for some analytic h(x). Then lim m→∞ (f (x) + f (x + α) + · · · + f (x + q m α − α)) = lim m→∞ (h(x + q m α) − h(x)) = 0 uniformly in x. Case 2. If β(α) > 0, choose a sequence m l such that q m l +1 ≥ e β 2 qm l . Then |f (x) + f (x + α) + · · · + f (x + q m l α − α)| =| |n|≥1f (n)(1 + e 2πinα + · · · + e 2πin(qm l −1)α )e 2πinx | =| |n|≥1f (n) 1 − e 2πinqm l α 1 − e 2πinα e 2πinx | ≤ 1≤|n|≤qm l −1 c 1 − e 2πinqm l α 1 − e 2πinα + |n|≥qm l ce −2πδ0|n| q m l ≤c q 3 m l q m l +1 + cq m l e −2πδ0qm l → 0 as l → ∞
uniformly in x.
Consequence of point spectrum
This part follows from [1]. We present the material here for completeness and readers' convenience. Suppose {u n } is an l 2 (Z) solution to H λ,α,θ u = Eu, where λ = (λ 1 , λ 2 , λ 3 ). This means (4.1) c λ (θ + nα)u n+1 +c λ (θ + (n − 1)α)u n−1 + 2 cos(2π(θ + nα))u n = Eu n .
Let u(x) = n∈Z u n e 2πinx ∈ L 2 (T). Multiplying (4.1) by e 2πinx and then summing over n, we get
(4.2) e 2πiθ cλ(x)u(x + α) + e −2πiθcλ (x − α)u(x − α) + 2 cos 2πx u(x) = E λ 2 u(x), whereλ = ( λ3 λ2 , 1, λ1 λ2
). If we multiply (4.1) by e −2πinx and sum over n, we get
(4.3) e −2πiθ cλ(x)u(−x − α) + e 2πiθcλ (x − α)u(−x + α) + 2 cos 2πx u(−x) = E λ 2 u(−x).
Thus writing (4.2), (4.3) in terms of matrices, we get
1 cλ(x) E λ2 − 2 cos 2πx −cλ(x − α) cλ(x) 0 u(x) u(−x) e −2πiθ u(x − α) e 2πiθ u(−(x − α)) = u(x + α) u(−(x + α)) e −2πiθ u(x) e 2πiθ u(−x) e 2πiθ 0 0 e −2πiθ (4.4) Let M θ (x) ∈ L 2 (T) be defined by M θ (x) = u(x) u(−x) e −2πiθ u(x − α) e 2πiθ u(−(x − α)) . Let Aλ ,E/λ2 (x) = 1 cλ(x) E λ2 − 2 cos 2πx −cλ(x − α)
cλ(x) 0 be the transfer matrix associated to Hλ ,α,θ and
R θ = e 2πiθ 0 0 e −2πiθ
be the constant rotation matrix. Then (4.4) becomes
Aλ ,E (x)M θ (x) = M θ (x + α)R θ . (4.5)
Taking determinant, we have the following proposition.
Proposition 4.1. [1] If θ is α-irrational, then | det M θ (x)| = b |c|λ(x − α) (4.6)
for some constant b > 0 and a.e.x ∈ T.
5.
Regions III λ1=λ3 and III λ1 =λ3 ∩ {λ 1 + λ 3 = 1}
We will show the following lemma.
Lemma 5.1. If θ is α-irrational, then for λ ∈ III λ1=λ3 or λ ∈ III λ1 =λ3 ∩ {λ 1 + λ 3 = 1}, H λ,α,θ has no point spectrum.
Proof.
According to Remark 2.1, we have cλ(x 0 ) = 0 for some x 0 ∈ T. Note that presence of singularity implies 1 cλ(x) / ∈ L 1 (T). Thus by (4.6), det M θ (x) / ∈ L 1 (T). This contradicts with M θ (x) ∈ L 2 (T).
6. Regions III λ1 =λ3 ∩ {λ 1 + λ 3 > 1} Without loss of generality, we assume λ 3 > λ 1 . Fix θ. Denote det M θ (x) = g(x) for simplicity.
Lemma 6.1. If θ is α-irrational, then H λ,α,θ has no point spectrum in the anisotropic self-dual region.
Proof. Taking determinant in (4.5), we get:
cλ(x − α)
cλ(x) g(x) = g(x + α).
This implies
g(x + kα) =cλ (x + kα − 2α) · · ·cλ(x)cλ(x − α) cλ(x + kα − α) · · · cλ(x + α)cλ(x) g(x). (6.1)
Taking k = q m l , as in Lemma 3.2, on one hand, since g(x) is an L 1 function, as the determinant of an L 2 matrix, and lim l→∞ q m l α T = 0, we have lim l→∞ g(x + q m l α) − g(x) L 1 = 0.
(3.8) with T g(θ)dθ = 0. Taking f (θ) = g(θ)/2 yields the desired the result.
the continued fraction approximants of α so that for any analytic function f on T with T f (θ)dθ = 0, we have lim l→∞
AcknowledgementThis research was partially supported by the NSF DMS-1401204. I would like to thank Svetlana Jitomirskaya for useful discussions.By (6.1), this impliesOn the other hand, by Lemma 3.1Combining the fact q m l α T → 0 with Lemma 3.2, we get pointwise convergence,Then by dominated convergence theorem, we get lim l→∞ I 2 = 0. Then (6.3) implies that for any small constant δ > 0,where |{x : 2q m l x + q 2 m l α ≥ δ}| |F m l ,δ | = 1 − 2δ. Thus.|c|λ(x−α) for some constant b > 0, thus g L 1 , g L ∞ are positive finite numbers, so one can choose δ ∼ 0 such that 4δ g L 1 − 8δ 2 g L ∞ is strictly positive. This contradicts with (6.2).
Spectral theory of extended Harper's model and a question by Erdős and Szekeres. A Avila, S Jitomirskaya, C A Marx, arXiv:1602.05111arXiv preprintAvila, A., Jitomirskaya, S. and Marx, C.A., 2016. Spectral theory of extended Harper's model and a question by Erdős and Szekeres. arXiv preprint arXiv:1602.05111.
Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model. S Jitomirskaya, C A Marx, Communications in mathematical physics. 3161Jitomirskaya, S. and Marx, C.A., 2012. Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model. Communications in mathematical physics, 316(1), pp.237-267.
Erratum to: Analytic quasi-perodic cocycles with singularities and the Lyapunov Exponent of Extended Harper's Model. Communications in mathematical physics. S Jitomirskaya, C A Marx, 31792717Department of Mathematics, University of California, Irvine CAJitomirskaya, S. and Marx, C.A., 2013. Erratum to: Analytic quasi-perodic cocycles with singularities and the Lyapunov Exponent of Extended Harper's Model. Communications in mathematical physics, 317, pp.269-271. Department of Mathematics, University of California, Irvine CA, 92717
E-mail address: rhan2@uci. E-mail address: [email protected]
| [] |
[
"Inhomogeneous superfluids",
"Inhomogeneous superfluids"
] | [
"Ignacio Salazar Landea \nInstituto de Física La Plata (IFLP)\nDepartamento de Física\nUniversidad Nacional de La Plata\nCC 671900La PlataArgentina\n"
] | [
"Instituto de Física La Plata (IFLP)\nDepartamento de Física\nUniversidad Nacional de La Plata\nCC 671900La PlataArgentina"
] | [] | We show examples of a striped superfluid in a simple λϕ 4 model at finite velocity and chemical potential with a global U (1) or U (2) symmetry. Whenever the chemical potential is large enough we find flowing homogeneous solutions and static inhomogeneous solutions at any arbitrary small velocity. For the U (1) model the inhomogeneous solutions found are energetically favorable for large enough superfluid velocity and the homogeneous and inhomogeneous phases are connected via a first order phase transitions. On the other hand, the U (2) model becomes striped as soon as we turn on the velocity through a second order phase transition. In both models increasing the velocity leads to a second order phase transition into a phase with no condensate. | null | [
"https://arxiv.org/pdf/1410.7865v2.pdf"
] | 119,225,650 | 1410.7865 | 95d14c71d3c8396035b0144e5ae45028c9588909 |
Inhomogeneous superfluids
Ignacio Salazar Landea
Instituto de Física La Plata (IFLP)
Departamento de Física
Universidad Nacional de La Plata
CC 671900La PlataArgentina
Inhomogeneous superfluids
We show examples of a striped superfluid in a simple λϕ 4 model at finite velocity and chemical potential with a global U (1) or U (2) symmetry. Whenever the chemical potential is large enough we find flowing homogeneous solutions and static inhomogeneous solutions at any arbitrary small velocity. For the U (1) model the inhomogeneous solutions found are energetically favorable for large enough superfluid velocity and the homogeneous and inhomogeneous phases are connected via a first order phase transitions. On the other hand, the U (2) model becomes striped as soon as we turn on the velocity through a second order phase transition. In both models increasing the velocity leads to a second order phase transition into a phase with no condensate.
Introduction
In this paper we will address the existence of inhomogeneous or striped superfluids. We have a two folded motivation to do so.
Firstly, superfluidity is an important phenomenon in both condensed matter and high energy physics. While superfluidity was first measured and studied in cold helium, it's also believed to play a role in high density states of nuclear matter.
On the other hand, gravitational duals of superfluids have been proposed recently in [3] and a recipe for making them flow was proposed in [4]. A linear analysis of the stability was made in [5], where a striped instability was found, with similar features to those found in the weak coupling limit [1,2]. Furthermore, evidence of the existence of such an inhomogeneous phase was also found in the T = 0 limit of a similar model in [6]. Similar hints were also found in brane models in [7].
Back to the real world, superfluid states of matter are likely to exist in the interior of compact stars. Neutrons living in the interior of a neutron star as well as quarks inside a hybrid star may become superfluid trough Cooper pairing. In this context, the works [1,2] have recently shown that Landau's λϕ 4 model has instabilities apparently towards an inhomogeneous phase when the superfluid flows fast enough.
Also a U (2) Landau's λϕ 4 model was introduced in the context of Kaon condensation [8]. When the condensation is induced by a strangeness chemical potential, Goldstone modes with a quadratic in momentum dispersion relation appear. The existence of such modes suggest that the theory will not be able to accommodate a superflow.
In this paper we will study the zero temperature Landau's λϕ 4 theory in the presence of superfluid velocity for both the U (1) and U (2) models. We will define the superfluid four-velocity as [9]
ξ µ = ig −1 ∂ µ g + A µ ,(1)
where g is an element of the symmetry group and A µ an external gauge field. With this definition the superfluid velocity will be basically the derivative of the Goldstone boson in the broken phase. For the U (1) model we find a first order phase transition at a critical velocity v c ≈ 0.365 at which the homogeneous solution is no longer preferred. On the other hand, for the U (2) model we find that the inhomogeneous solution has a smaller energy for arbitrary small values of the velocity. In both cases the inhomogeneous solution is static.
2 Inhomogeneous U (1) superfluid 2.1 The model
Lets consider a global U (1) Landau's λϕ 4 just like in [1].
L = ∂ µ ϕ∂ µ ϕ * − m 2 |ϕ| 2 − λ|ϕ| 4 ,(2)
where ϕ is a complex scalar field, m ≥ 0 its mass and the coupling constant λ > 0. The lagrangian is invariant under U (1) rotations ϕ → e iα ϕ which implies a conserved current. Notice that the condition of having a real mass implies that the quadratic term of the potential will be positive and that the spontaneous symmetry breaking occurs only if we introduce a chemical potential associated to the conserved current. If the chemical potential is greater then the mass m we get a Bose-Einstein condensate. In this formalism we will introduce a chemical potential trough a temporal dependence in the phase of the order parameter. We will choose the following ansatz for the Bose-Einstein condensate
ϕ(x) = e iψ( x) √ 2 ρ( x) .(3)
Here, ρ( x) is the modulus and ψ( x) the phase of the condensate. Introducing this ansatz in the condensate we get the following tree level lagrangian
L = 1 2 ∂ µ ρ∂ µ ρ + ρ 2 2 ∂ µ ψ∂ µ ψ − m 2 − λ 4 ρ 4 .(4)
The classical equations of motion read
ρ = ρ σ 2 − m 2 − λρ 2 , (5) ∂ µ ρ 2 ∂ µ ψ = 0 ,(6)
where we have called
σ 2 ≡ ∂ µ ψ∂ µ ψ .(7)
A simple classical solution to the equations of motion is
ψ = p µ x µ ,(8)ρ = p 2 − m 2 λ .(9)
This ansatz corresponds to an infinite superfluid flowing uniformly. The density and flow are determined by the components of p µ , that are simply numbers, they do not depend on x and they are not determined by the equations of motion. The value of p µ is determined by the boundary conditions, that specify the topology of the field configuration, i.e. the winding of the phase of the order parameter as we cross the space-time region in which the superfluid lives.
Notice that in this case p µ is nothing but the superfluid four-velocity defined by equation (1). We will consider, without any loss of generality, our four-velocity to have nontrivial components only in the t and x directions resulting on
p µ = (µ, v, 0, . . . ) .(10)
Here µ is the chemical and v the superfluid velocity. From now on we will consider the m = 0 case where we can scale away λ. Furthermore we can measure the velocity v in terms of the chemical potential, or equivalently we will set µ = 1.
Linear analysis
In this section we shall review the arguments of [1,2] about the existence of a instability towards an inhomogeneous phase. We will always work in the zero temperature limit.
Lets consider the following ansatz
ϕ = (ρ + δρ)e iψ+iδα ,(11)
where ρ and ψ are given by the classical homogeneous solutions (8)(9) and δα and δρ are small fluctuations around that solution. The linearized equations of motion read
δρ − 2(1 − v 2 )δρ + 2 √ 1 − v 2 (∂ t δα − v∂ x δα) = 0 ,(12)
δα
+ 2 √ 1 − v 2 (∂ t δρ − v∂ x δρ) = 0 ,(13)
and we will suppose an harmonic dependence ≈ e −iωt+ikxx+ik ⊥ y . We wish to look now for the velocity of the Goldstone modes around this background. The same can be computed taking simultaneously the limit of ω, k x and k ⊥ to zero in the mass matrix
2(v 2 − 1) + O(k 2 ) −2i √ 1 − v 2 (k x v + ω) + O(k 3 ) 2i √ 1 − v 2 (k x v + ω) + O(k 3 ) (v 2 − 1) (k 2 x + k 2 ⊥ − ω 2 + O(k 3 )) ,(14)
with k 2 = k 2 x + k 2 ⊥ and tan θ = k x /k ⊥ . The dispersion relations are obtained asking for ω such that the determinant of the mass matrix is zero. Then the sound velocity is just the linear in k coefficient of the dispersion relation ω ≈ v s · k + . . . . Measured at an angle θ with respect to the superfluid velocity it reads
v s = 2v cos θ − √ 1 − v 2 √ 3 − 2v 2 − v 2 cos 2θ v 2 − 3 .(15)
On the left hand side of Figure 1 we show a parametric plot of the sound velocity (15). As we can see, there is a critical superfluid velocity v * = 1/ √ 3 at which the excitations have zero velocity. According to the Landau criterion this might signal an instability which shall kill the superflow.
Going beyond the small momentum limit might give some insight on the fate of the superfluid after the instability is triggered. Indeed, as showed on the right hand side of Figure 1, for superflow velocities greater than v * the energy has a minimum at finite momentum, which signals an instability towards an inhomogeneous phase.
We have showed evidence for the existence of a phase transition to a non homogeneous phase at large enough superfluid velocities. In the next section we will address the issue of the construction of such a phase. The construction itself will show that the second order phase transition argued in this section is actually over-seeded by a first order phase transition at a smaller velocity.
v = 0 (blue), v = 0.4 (red), v = 0.6 (black), v = 1/ √ 3 (green).
(Right) Dispersion relation beyond the hydrodynamic limit for v = 0.5 (blue) and v = 0.6 (purple) and momentum in the direction of the superflow. The energy of the purple curve shows a minimum at finite momentum.
Construction of the inhomogeneous phase
In this section we shall construct the inhomogeneous phase whose existence was hinted in the previous section. In order to do so we will add a spatial dependence to the ansatz for the fields that does not change the boundary conditions, so that the winding of the phase across the superfluid does not change. A natural way to do so is to add a periodic x dependence.
To begin with, let us consider the following ansatz
ρ ≡ ρ(x) , (16) ψ ≡ −µ t + α(x) .(17)
Let us expand ρ(x) and α (x) alla Fourier
ρ(x) = nmax n=0 ρ n cos(nkx) ,(18)α (x) = nmax n=0 α n cos(nkx) ,(19)
where n max will be a numerical cutoff and the zeroth coefficient gives the velocity α 0 ≡ v.
We can now use Mathematica's FindRoot command to numerically integrate the equations of motion (5-6). The procedure consists in solving the equations for each Fourier mode up to a maximum n max , chosen large enough so that we can trust that the solution won't change significatively if further modes are taken into account. Two kind of solutions were found numerically. A first one in which all modes are null but the zero modes, and corresponds to the homogeneous phase. The second one is spatially modulated and the phase looks like a step function in the large n max limit. An example of both solutions can be observed in Figure 2 The picture would be the following. For certain critical velocity v c there is a first order phase transition. For low velocities v < v c the modulus of the condensate is constant and its phase is linear in x. As we increase the the velocity the phase at which the condensate is inhomogeneous dominates. In these solutions the phase of the order parameter is no longer linear with x, but a stairway of v sized steps. Since there is no continuous connection between homogeneous and inhomogeneous solutions at finite v, the phase transition must be first order. Being that the case, the linear analysis of the previous section will not shed light on this phase transition. In Figure 3 we plot the energy density of the homogeneous and inhomogeneous solutions. As we can see from the plot, the phase transition is indeed first order since the energy of the system (the energy of the solution with the lowest energy) has a discontinuity in its derivative, resulting from the fact that the inhomogeneous solution does not arise continuously from the homogeneous one. The critical velocity, where the energy of the different solutions is the same, is v c ≈ 0.365 < 1/ √ 3. This first order phase transition overseeds the lineal instability of the homogeneous background, and the stability analysis should be redone considering fluctuations around this new background.
Another interesting fact that can be observed in Figure 3 is that while the homogeneous solutions with ρ = 0 only exist for v < 1, inhomogeneous solutions exist for v <ṽ ≈ 1.92. The complete phase diagram would read as follows. For velocities larger thanṽ the system lives in the trivial solution, with no condensate. As we lower the velocity, a second order phase transition into the inhomogeneous solution occurs atṽ. If we keep on lowering the velocity a first order phase transition occur at v c into the homogenous phase. This phase dominates down to null velocity.
The existence of solutions with velocity larger than v = 1 is very counterintuitive and may even seem unphysical but are certainly needed to smoothly connect with the normal phase. This should not be an issue since the phase uses the places in space to jump up to the next step precisely where the modulus ρ is zero.Then we must analyze carefully the superfluid current
j µ = ρ(∂ µ α + A µ ) ,(20)
which is a physical observable. When doing so we reach to the conclusion that even though the average velocity is grater than one, one can see that the supercurrent is time-like everywhere. Furthermore, x component related to the flow is exactly zero everywhere since the phase is constant everywhere but in the places where the ρ = 0. We reach then to the conclusion that for high enough superfluid velocities the system goes trough a first order phase transition into a static striped phase.
3 Inhomogeneous U (2) superfluid
The model
Inspired by Kaon condensation in the color-flavor locked phase of QCD the authors of [11,12] studied QCD at finite strangeness chemical potential. It was shown that at a critical chemical potential equal to the mass of the Kaons, Kaon condensation occurs through a continuous phase transition. Furthermore, a Goldstone boson with a non relativistic dispersion relation ω ∼ p 2 appears in the condensed phase spectrum. To illustrate such a fact the authors consider the following model:
L = −(∂ 0 − iµ)φ † (∂ 0 + iµ)φ + ∂ i φ † ∂ i φ + M 2 φ † φ + λ 2 (φ † φ) 2 ,(21)
where φ is a complex scalar doublet,
φ = φ 1 φ 2 .(22)
Here we have introduced the chemical potential µ through an external gauge field, minimally coupled to the scalar doublet, following [11,12]. While µ < M the masses of the four excitations of the theory are the roots ω of
(ω ± µ) 2 = M 2 .(23)
All of them are doubly degenerated. It is immediate to check that as soon as µ = M the U (2) symmetry is broken and a new vacuum must be chosen:
φ = 0 ρ 0 , with ρ 2 0 = µ 2 − M 2 λ .(24)
If we study the fluctuations of φ around this background one finds two positive massive modes and two non massive modes with dispersion relations:
ω 2 1 = µ 2 − M 2 3µ 2 − M 2 k 2 + O(k 4 ) ,(25)ω 2 2 = 6µ 2 − 2M 2 + O(k 2 ) , (26) ω 2 3 = k 2 − 2µω 3 , (27) ω 2 4 = k 2 + 2µω 4 .(28)
If we focus in the positive roots we see that ω 1 is a normal Goldstone mode with a linear dispersion relation. The positive root of (27) is
ω 3 = k 2 2µ + O(k 4 ) .(29)
This is by definition a type II Goldstone mode: it has a nonlinear dispersion relation proportional to an even power of momentum. Since the theory has Lorentz symmetry we also have a negative mode with quadratic dispersion. This comes from the negative root of ω 3 .
Finally ω 2 and ω 4 are massive modes with
ω 4 = 2µ + O(k 2 ) .(30)
Since the symmetry breaking pattern is U (2) → U (1) we have three spontaneously broken generators but only two massless modes in the spectrum. This is due to the quadratic dispersion relation, and satisfies Chadha-Nielsen counting rules [13]. The role of ω 4 is special since it is the mode that couples with the type II Goldstone mode in (27) and (28). There is evidence that its energy at zero momentum is protected under quantum corrections [18,19,20]. Some recent related papers are [14,15,16,17]. As it has been pointed out in [10], the existence of this type II Goldstone modes should make the system unstable when an arbitrarily small velocity is turned on. We will address this issue more deeply.
Adding velocity naively
Let us naively add a velocity in the x direction by turning on an external A x = v gauge field. Immediately we can see that this contributes to the condensate as a positive mass term, so the homogeneous classical solution will read
φ = 0 ρ 0 , with ρ 2 0 = µ 2 − M 2 − v 2 λ .(31)
We shall consider again perturbations around this background. We can see that the lower sector is just that of the U (1) sector studied previously, and will obviously show the same instabilities. Now let us see what happens to the lower sector. We can see that the positive branch of the type II Goldstone now acquires a negative velocity
ω 3 = − vk x µ + −(A x k x ) 2 + µ 2 (k 2 x + k 2 ⊥ ) 2µ 3 + O(k 4 ) .(32)
This signals the fact that the energy minimum of the perturbation won't be at zero momentum and that the system will rather be in a striped phase.
In order to make a more direct connection with the previous section we will go to the conformal limit where M = 0, and we can choose λ = 1. We will work at fixed chemical potential µ = 1, or equivalently we can say that we measure the velocity in terms of the chemical potential.
In figure 4 we can see the dispersion relation for the ex type II Goldstone mode ω 3 plotted beyond the hydrodynamic limit, i.e. considering the expression (32) at all order in k. As we can see, as we increase the velocity the minimum in energy occurs at larger momentum. This hints a second order phase transition to an inhomogeneous phase as soon as we turn on a velocity. We will repeat the mechanism of the previous section in order to construct inhomogeneous solutions corresponding to the U (2) superfluid at finite velocity.
Constructing the inhomogeneous phase
In order to construct the inhomogeneous phase let us consider the following ansatz
φ = ρ u (x)e iαu(x) ρ d (x)e iα d (x) .(33)
The classical equations of motions for the fields read
ρ u = (v + α u ) 2 ρ u + (ρ 2 d + ρ 2 u − 1)ρ u ,(34)0 = ρ u α u + 2 (v + α u ) ρ u ,(35)ρ d = (v + α d ) 2 ρ d + (ρ 2 d + ρ 2 u − 1)ρ d ,(36)0 = ρ d α d + 2 (v + α d ) ρ d .(37)
Again we will do a Fourier decomposition of the fields
ρ u (x) = nmax n=0 ρ (n) u cos(nkx) ,(38)α u (x) = nmax n=1 α (n) u cos(nkx) ,(39)ρ d (x) = nmax n=0 ρ (n) d cos(nkx) ,(40)α d (x) = nmax n=1 α (n) d cos(nkx) ,(41)
where we have removed the zero modes of the phase since they will be taken into account in the spatial component of the external gauge field A x = v.
We will now solve the equations for each Fourier mode numerically. Considering ρ u = 0 we recover the same solutions that in the previous section. When we allow a non-trivial profile for ρ u we find a further numerical solution. Its Fourier coefficients satisfy
ρ (n) u = (−1) n ρ (n) d , α (n) u = (−1) n α (n) d ,(42)
and correspond to a solution where both condensates are modulated, with a half period relative phase in their oscillatory space dependence. Their phases α u,d are again step functions, and also have a half period relative phase. An example of these new solutions can be observed in Figure 5. In Figure 6 we show the energy density of this new configuration in contrast to the energy density of the solutions that also exist in the U (1) model, i.e. the homogeneous solution and the solution with condensate in only one component. We can see that as soon as we turn on a velocity, a second order phase transition occurs into an inhomogeneous phase with two spatially modulated condensates. This is in agreement with the linear analysis done in Αu v x Figure 6: Energy density of the homogeneous (blue) and inhomogenous solutions with condensate only in the lower component (black) and with condensate in both components (red). We have used n max = 40 to generate this plot.
Section 3.2. The order of the phase transition comes from the fact that the energy has the same slope for both solutions with respect to the velocity, which might be a bitt counter intuitive, since one solution does not emerge continuously from the other. We can see from Figure 6 that the solution with two condensates always have the smallest free energy until it no longer exist atṽ ≈ 1.965. At such critical velocity the superfluid solution is connected with the trivial solution.
Once again we can numerically check that the supercurrent is zero everywhere, so the system does not flow even though we turn on a superfluid velocity, but it develops stripes.
Conclusions
We have shown an example of a striped superfluid in a simple λϕ 4 model at finite velocity and chemical potential. We have studied two models one with global U (1) gauge symmetry and the other one with U (2).
For the U (1) model the inhomogeneous solutions found are energetically favorable for large enough superfluid velocity. The homogeneous and inhomogeneous phases are connected via a first order phase transition. Increasing the velocity leads to a second order phase transition into a phase with no condensate. This work somehow completes the picture shown in [1,2], about this very same model.
For the U (2) model on the other hand, as soon as we turn on the velocity we end in a striped phase. This is in agreement with Landau criterion for superfluidity, since this model has zero velocity excitations. Increasing the velocity leads to a second order phase transition into a phase with no condensate.
As a possible continuation of this work, we would like to compute a similar computation in the context of AdS/CFT, following [5]. There, it is shown that the holographic U (2) superfluid constructed in [21,22] is unstable at all the range of velocities numerically reachable, while the U (1) model of [4] is only unstable for large enough velocities, in agreement with the field theoretical predictions. The explicit construction of the holographic phases should be an interesting challenge.
Another interesting problem would be to analyze fluctuations around this inhomogeneous background, in order to address the problem of stability.
It would also be interesting to make a connection with fluids physics, looking at the hydrodynamic limit of this theory, following the steps of [1].
Figure 1 :
1(Left) Sound velocity for
Figure 2 :
2Modulus (left) and phase (right) for the homogeneous (blue) and inhomogeneous (black) solutions. This solutions correspond to v = 0.4 and n max = 60.
Figure 3 :
3Energy density of the homogeneous (blue) and inhomogenous (black) solutions. We have used n max = 60 to generate this plot.
Figure 4 :
4Dispersion relation of the type II Goldstone mode for v = 0 (blue), v = 0.2 (purple), v = 0.4 (yellow) and v = 0.6 (green).
Figure 5 :
5Homogeneous (blue) and inhomogeneous (red) solutions for v = 0.7 and n max = 60.
AcknowledgementsWe would like to thank Irene Amado, Daniel Areán, Raul Arias, God, Carlos Hoyos, Amadeo Jiménez, Karl Landsteiner, Luis Melgar, Andreas Schmitt and MVC for correspondence, inspiration and advise. Also we would like to special thank to Daniel Areán for "carefully" reading this manuscript.
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