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[
"Electrostatic Bender Fields, Optics, Aberrations, with Application to the Proton EDM Ring",
"Electrostatic Bender Fields, Optics, Aberrations, with Application to the Proton EDM Ring"
] |
[
"TRIUMFR Baartman "
] |
[] |
[] |
Electrostatic bender optics are derived up to second order (third order in fields and the Hamiltonian) and applied to the proposed EDM proton ring. The results for linear optics agree with those already presented by V. Lebedev (Nov. 18, 2013). Second order optics is not sensitive to the shape of the fringe fields and formulas are given. It is shown that the proposed electrode shape that linearizes the vertical electric field is no advantage to this order.
| null |
[
"https://arxiv.org/pdf/1508.00157v1.pdf"
] | 119,278,906 |
1508.00157
|
1ab30efe60a31c533d052d52be6f20aea77d2dc5
|
Electrostatic Bender Fields, Optics, Aberrations, with Application to the Proton EDM Ring
1 Aug 2015 Dec. 2013
TRIUMFR Baartman
Electrostatic Bender Fields, Optics, Aberrations, with Application to the Proton EDM Ring
1 Aug 2015 Dec. 2013
Electrostatic bender optics are derived up to second order (third order in fields and the Hamiltonian) and applied to the proposed EDM proton ring. The results for linear optics agree with those already presented by V. Lebedev (Nov. 18, 2013). Second order optics is not sensitive to the shape of the fringe fields and formulas are given. It is shown that the proposed electrode shape that linearizes the vertical electric field is no advantage to this order.
Introduction
The general electrostatic bender has a given curvature in the median plane (arbitrarily assumed here to be horizontal), and, to control the motion in the vertical direction, also has curvature in that direction. It is assumed that the two curvatures are constant radius, but different, so the equipotentials are sections of a torus. Lebedev [1] has mentioned that the electrodes can be shaped in such a way that the vertical field is linear, i.e., and so has no xy 2 coupling term. This shape will also be considered.
Though the electrodes would be continuous in the vertical plane, they are necessarily truncated in the horizontal plane to allow the particles to enter and exit the field. Thus the radius of curvature along the reference trajectory changes continuously from infinity outside the field region to a finite value inside. The electric potential of such a truncated torus is derived to third order in deviations from the reference trajectory; the third order term contains the second derivative of the curvature function thus enabling analysis of lowest order fringe field effects. The relativistically correct Hamiltonian is derived also to third order. A canonical transformation is found which simplifies the Hamiltonian by eliminating derivatives of the curvature function. This allows an analysis of the linear optics and simple formulas for second order aberrations arising both from the curved geometry and the fringe fields. The linear optics agree with known; the second order aberration formulas were presented previously for the non-relativistic limit [2].
The Potential
In a general curvilinear system the Laplacian of a potential V (q 1 , q 2 , q 3 ) is given by
∇ 2 V = 1 h 1 h 2 h 3 3 i=1 ∂ ∂q i h 1 h 2 h 3 h 2 i ∂V ∂q i ,(1)
where the h i are the Lamé coefficients and in general are functions of the q i ..
Toroidal coordinates
We start with a curvilinear coordinate system that conforms to the shape of the equipotentials: the coordinate along the intended reference orbit we call q 3 = s, the radially outward coordinate is q 1 = x, and the vertical is q 2 =ỹ. The horizontal curvature (reciprocal of radius) is h(s), and the vertical curvature is k, a constant. In such a system, the Lamé coefficients are h 1 = 1, h 2 = 1 + kx, h 3 = 1 + h(s)x. But in this system, the potential has no dependence onỹ, the direction orthogonal to x and s. We can therefore expand to third order as 1 V (x, s) = a 10 (s)x + a 20 (s)
x 2 2 + a 30 (s) x 3 6(2)
We know the electric field on the reference orbit imposes the curvature h, so a 10 (s) = h(s). We can find the other coefficients from Laplace's equation and eqn. 1:
V (x, s) = hx − 1 2 h(k + h)x 2 + 1 6 2h k 2 + kh + h 2 − h ′′ x 3 (3)
The Potential in Frenet-Serret Coordinates
The Frenet-Serret coordinate system is the usual one of accelerator physics. It has no curvature in the y-direction, so the factor h 2 = 1 in eqn. 1. The potential in the median plane is the same in both systems, we need only add the following two more terms allowed by the symmetry: a 02 (s)y 2 /2, a 12 (s)xy 2 /2.
V (x, y, s) =Ṽ (x, s) + 1 2 a 02 (s)y 2 + 1 2 a 12 (s)xy 2 .(4)
We again use the curvilinear Laplacian (1), but this time with h 2 = 1, set it to zero and solve term-wise in the expansion to find a 02 and a 12 :
V (x, y, s) = hx− h 2 (k+h)x 2 + hk 2 y 2 − kh 2 (2k+h)xy 2 + 1 6 [2h(k 2 +hk+h 2 )−h ′′ ]x 3
(5) Lebedev's [1] linear vertical electric field (VL) potential also can be augmented to include lowest order fringe field effects and still satisfy Laplace to third order. The result is slightly different from the one above:
V (x, y, s) = hx − h 2 (k + h)x 2 + hk 2 y 2 + 1 6 [h 2 (k + 2h) − h ′′ ]x 3(6)
The reason for the difference is that potential 5 is derived for electrodes whose vertical sections are exact circular arcs while the potential 6 was derived with the specific intention to zero the xy 2 term. We shall see how the dynamics of the two compare.
Dynamics
General Hamiltonian for Electrostatics
First, we must decide on the dynamical coordinates, especially the longitudinal (third) generalized momentum, which has been the source of considerable confusion. A common approach, used especially by those accustomed to only magnetic elements, is to let the third momentum be ∆p/p. This is not the simplest approach, since when electric fields are included, it is not conserved. This means that when a particle enters the electrostatic element off-axis, it must receive a "kick" to get into the potential field. This kicks p, but leaves E unchanged. For an electrostatic bend of radius A, this kick is ∆p/p = ±x/A; the upper sign is for entry, the lower for exit.
A better approach is therefore to let the third momentum be E. Since the fields are static, E is conserved; no kicks are required.
For independent variable s, on a reference trajectory curving in the xsplane with curvature h(s) = 1/R(s), the Hamiltonian H = −p s is wellknown, and I will not derive it here:
H = −(1 + hx) E − qΦ c 2 − m 2 c 2 − p 2 x − p 2 y(7)
We write E = E 0 + ∆E, and note that the large quantity under the square root sign is
p 2 0 = E 2 0 /c 2 − m 2 c 2 = (γ 2 − 1)m 2 c 2 = (βγmc) 2 ,(8)
the square of the reference momentum.
H = −(1 + hx)p 0 1 + 2E 0 (∆E − qΦ) p 2 0 c 2 + ∆E − qΦ p 0 c 2 − p 2 x p 2 0 − p 2 y p 2 0(9)
Let us transform so that the third coordinate is not time, but a relative distance deviation w.r.t. the reference particle: i.e. from (t, −∆E) to (τ, p τ ) where τ ≡ s − βct, p τ = ∆E/(βc). The generating function is
F (t, p τ ) = (s − βct)p τ(10)
then the new Hamiltonian is
K = H + ∂F/∂s = H + p τ K = p τ − (1 + hx)p 0 1 + 2 p τ p 0 − qΦ βcp 0 + β 2 p τ p 0 − qΦ βcp 0 2 − p 2 x p 2 0 − p 2 y p 2 0(11)
This cleans up considerably if we scale all momenta and the Hamiltonian by p 0 : P x = p x /p 0 , P y = p y /p 0 , P τ = p τ /p 0 ,K = K/p 0 , and introduce the scaled potential V = qΦ βcp 0 :
K = P τ − (1 + hx) 1 + 2 (P τ − V ) + β 2 (P τ − V ) 2 − P 2 x − P 2 y(12)
This also makes the momenta accord with the more usual definitions, since, P x = x ′ , P y = y ′ , and outside the electric field, P τ = ∆p/p. This is the Hamiltonian of the electrostatic bend. It is exact.
Linear Optics
Linear optics arise from second order Hamiltonian terms. Expanding to second order we get for both the potentials eqns. 5, 6:
V = hx − h(h + k) x 2 2 + hk y 2 2 ,(13)
where h = 1/R 0 and k = 1/R y is the curvature in the non-bend direction. The first term, needed to ensure that the reference trajectory is x = 0, yields the required electric field on the reference orbit:
E = − ∂Φ ∂x = − βcp 0 q ∂V ∂x x=0 = β 2 R 0 E 0 q .(14)
In the non-relativistic limit, the electric field is twice the beam kinetic energy divided by charge and bend radius: qE = mv 2 /R 0 . The first order terms in the resulting Hamiltonian all cancel as they should or the reference orbit would not be an orbit, so when expanded to second order it isK
1 = P 2 x 2 + P 2 y 2 + P 2 τ 2γ 2 − 2 − β 2 R 0 xP τ + ξ 2 2R 2 0 x 2 + η 2 2R 2 0 y 2(15)
The parameters ξ and η are introduced as they parametrize the x and y focusing strengths: ξ 2 + η 2 = 2 − β 2 = 1 + γ −2 , η 2 = k/h = R 0 /R y , R y being the curvature radius in the non-bend direction. (η 2 is given the symbol m in proton EDM group convention.) In the non-relativistic limit, for a cylindrical bend, ξ = √ 2, η = 0; for a spherical bend, ξ = η = 1.
The transfer matrix is easily derived from this HamiltonianK 1 :
cos ξθ R 0 ξ sin ξθ 0 0 0 2−β 2 ξ 2 R 0 (1 − cos ξθ) − ξ R 0 sin ξθ cos ξθ 0 0 0 2−β 2 ξ sin ξθ 0 0 cos ηθ R 0 η sin ηθ 0 0 0 0 − η R 0 sin ηθ cos ηθ 0 0 − 2−β 2 ξ sin ξθ − 2−β 2 ξ 2 R 0 (1 − cos ξθ) 0 0 1 R 0 θ 1 γ 2 − 2−β 2 ξ 2 1 − sin ξθ ξθ 0 0 0 0 0 1
(16) This matrix agrees precisely with that given by Lebedev [1], except that the 56 element explicitly includes the velocity dependent part 1/γ 2 , allowing to read directly the phase slip factor for a solid electrostatic ring: 1
γ 2 − 2−β 2 ξ 2 .
Moreover, for such a ring, the horizontal tune is Q
x = ξ = √ 2 − β 2 − m, vertical tune Q y = η = √ m, dispersion = 2−β 2 2−β 2 −m R 0 .
I have used this matrix to find the first order optics of a ring with 14 benders with η 2 = m = 0.199 and separated by 0.834 metre drifts: tunes, β-functions, dispersion and phase slip factor agree precisely with the results presented by Valeri Lebedev [1].
Second Order Aberrations
Expanding to third order, the Hamiltonian for the toroidal electrodes becomes
K 2 =K 1 − P τ 2 P 2 x + P 2 y + P 2 τ γ 2 − P τ x 2 kh 2γ 2 + 2 h 2 γ 2 − P τ y 2 kh 2γ 2 + x P 2 x + P 2 y + 2 P 2 τ γ 2 h +xy 2 −k 2 h + kh 2 2γ 2 + x 3 k 2 h 3 − kh 2 6 1 + 3 γ 2 − h 3 6 1 − 3 γ 2 − h ′′ 6(17)
For the potential 6, it is only slightly different:
K 2 =K 1 − P τ 2 P 2 x + P 2 y + P 2 τ γ 2 − P τ x 2 kh 2γ 2 + 2 h 2 γ 2 − P τ y 2 kh 2γ 2 + x P 2 x + P 2 y + 2 P 2 τ γ 2 h +xy 2 (1 + γ 2 )kh 2 2γ 2 + x 3 k 2 h 3 − kh 2 6 2 + 3 γ 2 − h 3 6 1 − 3 γ 2 − h ′′ 6(18)
Thus even though the latter potential was designed to eliminate the xy 2 term, it re-appears in the Hamiltonian. The reason can be traced to the factor 1 + hx in front of the square root; coupled with the focusing term ∝ khy 2 this results in a term ∝ kh 2 xy 2 which is in fact larger than the term that was eliminated. Thus it is not clear that special non-circular-section electrode shapes are warranted.
Fringe Field Shape Effect
The presence of the second derivative of the curvature in the Hamiltonian seems to suggest that the effect in this order of the fringe field can be minimized by shaping it. For example, if h(s) is linear, its contribution is zero. However, this technique proves to be futile: there necessarily will be regions of large h ′′ at either end of such a linear fringe field, and these will have the same integrated effect as a fringe field with no linear part. To be sure, third (often called octupole) and higher orders do depend on the fringe field shape; in general, the smoother and more extended the fringe field, the better. In fact, the second (often called sextupole) order effects of the fringe field are insensitive to the fringe field shape. This follows from the fact that the second derivative can be transformed away with a canonical transformation to a new horizontal transverse coordinate.
We transform (x, P x ) → (X, P X ) by means of the following generating function:
G(x, P X , s) = xP X + h ′ x 3 6 − h x 2 P X 2 (19) X = x − 1 2 x 2 h and P x = P X − xP X h + 1 2 x 2 h ′(20)
The new Hamiltonian isK(X, P X , y, P y , z, P z ; s) =K + ∂G ∂s . The same transformation applies to the (VL) potential that is designed to eliminate the xy 2 term. In that case only K Q is different. The result is:
K = P 2 X 2 + P 2 y 2 + P 2 τ 2γ 2 + X 2 − kh 2 + h 2 2 + h 2 2γ 2 + y 2 kh 2 − P τ X h + h γ 2 + (21) P 2 τ X 2h γ 2 + P τ − P 2 X 2 − P 2 y 2 − ky 2 h 2γ 2 + X 2 kh 2γ 2 − h 2 2 − 5h 2 2γ 2 − P 3 τ 2γ 2 + P 2 y Xh + (22) Xy 2 −k 2 h + kh 2 2γ 2 + X 3 1 3 k 2 h − 2 3 kh 2 − kh 2 2γ 2 +K Q = X 3 1 3 − 5m 6 + 1 γ 2 −
m 2γ 2 + Xy 2 m 2 + m 2γ 2 = 0.746X 3 + 0.163Xy 2 (25) As stated, the non-toroidal electrodes that make the vertical field linear appear to be of no advantage.
With no derivatives of h appearing, we can now to a good approximation assume h = 1/R 0 , a constant, and k = m/R 0 . Further, scale lengths to be in units of R 0 . Then K Q = X 3 (m − 1) values are for the proton EDM ring (γ = 1.2481, m = 0.199).
This potential is dimensionless. Multiply by βc(Bρ) = βcp 0 /q = β 2 γmc 2 /q to get the potential.
differ from each other only in higher order, the linear optics is unchanged. The Hamiltonian can be written as a sum of linear part K 1 (21), momentum-dependent nonlinear part K P (22). (x As (x, P , and spatial nonlinear part K Q (23As (x, P x ) and (X, P X ) differ from each other only in higher order, the linear optics is unchanged. The Hamiltonian can be written as a sum of linear part K 1 (21), momentum-dependent nonlinear part K P (22), and spatial nonlinear part K Q (23).
V Lebedev, Proton EDM Lattice and Experimental Parameters. EDM collab. teleconferenceV. Lebedev: Proton EDM Lattice and Experimental Parameters, November 18, 2013 EDM collab. teleconference.
End Effects of Beam Transport Elements. R Baartman, Talk at Snowmass. R. Baartman: End Effects of Beam Transport Elements, Talk at Snowmass 2001, http://lin12.triumf.ca/text/Talks/2001Snowmass/FF.pdf
|
[] |
[
"On-shell representations of two-body transition amplitudes: Single external current",
"On-shell representations of two-body transition amplitudes: Single external current"
] |
[
"R A Briceño ",
"A W Jackura ",
"F G Ortega-Gama ",
"K H Sherman ",
"\nOld Dominion University Old Dominion University ODU Digital Commons ODU Digital Commons Physics Faculty Publications Physics\nOld Dominion University\n\n"
] |
[
"Old Dominion University Old Dominion University ODU Digital Commons ODU Digital Commons Physics Faculty Publications Physics\nOld Dominion University\n"
] |
[
"Physical Review D"
] |
This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, 2 þ J → 2. Such amplitudes can provide structural information about the excited QCD spectrum. We derive an exact analytic representation for these reactions. From these amplitudes, we show how to rigorously define resonance and bound-state form factors. Furthermore, we explore the consequences of the narrow-width limit of the amplitudes as well as the role of the Ward-Takahashi identity for conserved vector currents. These results hold for any number of two-body channels with no intrinsic spin, and a current with arbitrary Lorentz structure and quantum numbers. This work and the existing finite-volume formalism provide a complete framework for determining this class of amplitudes from lattice QCD.
|
10.1103/physrevd.103.114512
| null | 229,371,240 |
2012.13338
|
158b078ff9a06a8f9b05b581020c13321016f91d
|
On-shell representations of two-body transition amplitudes: Single external current
2021
R A Briceño
A W Jackura
F G Ortega-Gama
K H Sherman
Old Dominion University Old Dominion University ODU Digital Commons ODU Digital Commons Physics Faculty Publications Physics
Old Dominion University
On-shell representations of two-body transition amplitudes: Single external current
Physical Review D
10311202110.1103/PhysRevD.103.114512(Received 12 January 2021; accepted 30 March 2021; published 22 June 2021)Follow this and additional works at: https://digitalcommons.odu.edu/physics_fac_pubs This Article is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected].
This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, 2 þ J → 2. Such amplitudes can provide structural information about the excited QCD spectrum. We derive an exact analytic representation for these reactions. From these amplitudes, we show how to rigorously define resonance and bound-state form factors. Furthermore, we explore the consequences of the narrow-width limit of the amplitudes as well as the role of the Ward-Takahashi identity for conserved vector currents. These results hold for any number of two-body channels with no intrinsic spin, and a current with arbitrary Lorentz structure and quantum numbers. This work and the existing finite-volume formalism provide a complete framework for determining this class of amplitudes from lattice QCD.
I. INTRODUCTION
Resolving the hadronic spectrum has proven to be a significant challenge due to the nonperturbative nature of quantum chromodynamics (QCD). In the case of the lowest-lying spinless hadrons, the pseudoscalar pions can be readily identified as the pseudo-Goldstone bosons of chiral symmetry; however, the scalar hadrons are notoriously difficult to characterize. This is not surprising given the multitude of Fock states allowed to participate in this channel, i.e., quark-antiquark pairs, mesonic molecules, tetraquarks, glueball states, etc. 1 A satisfactory interpretation of these states demands for a more comprehensive understanding of the dynamics of QCD.
For example, the determination of the mass and width of the f 0 ð500Þ=σ, the lightest QCD resonance, had been disputed since its discovery and only recently has reached consensus [2]. The difficulty to study this state arises in part due to its large decay width and the atypical shape of the cross section of its decay products, i.e., ππ. However, the nature of this state is not resolved from its mass and width alone, motivating the attention to other physical properties, like the charge radius or distribution functions, which naturally arise in transition amplitudes.
With this in mind, Ref. [3] calculated the ππ þ H → ππ transition with unitarized chiral perturbation theory (χPT), where H represents a scalar current. The scalar radius of the σ was in turn estimated by analytically continuing the amplitude to the resonance position. The value found for this parameter supports an interpretation of this resonance as a compact state for pions at their physical mass, and a molecular ππ description if the quark masses are modified such that the pion mass is greater than 400 MeV. 2 This demonstrates that transition amplitudes can play a role in the description of resonances. The question that arises is how to determine these amplitudes directly from the dynamics of QCD, and the best current answer is lattice QCD.
Lattice QCD is a numerical implementation of the path integral in a finite volume and can be used to calculate observables directly from QCD. In the past decades the scope of the field has increased substantially, moving past the studies of stable ground states into the more interesting region of resonances and excited states. Studies of excited and multiparticle states are challenging because of, among other things, the need for a formal connection between finite-and infinite-volume states, and matrix elements.
In the case of scattering amplitudes, the Lüscher formalism and its extensions [6][7][8][9][10][11][12][13][14][15][16][17][18][19] have been tested and applied successfully in numerous processes; see the recent review [20] and references therein. This includes determinations of the σ mass and width [4,5,21] all the way to resonances that involve multiple coupled channels and partial waves [22][23][24][25][26][27], a remarkable example is the recent study of the 1 −þ hybrid resonance [28].
Furthermore, the technology to compute transition matrix elements involving excited states from the lattice has already been implemented and employed [29]. In addition to this, the seminal work in Ref. [30] by Lellouch and Lüscher laid the foundation to develop a general technique to match finite-volume matrix elements to 1 þ J → 2 transition processes [31,32], where J is some external local current, 1 refers to a state of just a QCD-stable hadron and 2 is an asymptotic state of two hadrons. An application of this formalism was used to calculate the pion photoproduction in the π þ γ ⋆ → ππ process, from which the π þ γ ⋆ → ρ transition form factor was determined for heavier-than-physical pions by two distinct groups [33,34]. Carrying on this effort, some of the authors developed a framework that addresses the finitevolume effects of amplitudes with two hadrons in the initial and final states, i.e., 2 þ J → 2 [35,36]. It is precisely through these amplitudes that elastic form factors of resonant or shallow bound states can be determined.
The purpose of this work is to complement this technique, relevant when translating finite-volume matrix elements into infinite-volume amplitudes, by deriving the universal analytic structure that the 2 þ J → 2 amplitudes receive from Lorentz symmetry and unitarity in an infinitevolume. This is especially important when evaluating the amplitude in the complex energy plane where resonances and bound poles reside. This framework is also applicable for the case of nonresonant amplitudes. In the latter case, understanding the analytic structure is critical in order to prevent the incorrect identification of kinematic singularities as dynamical poles.
We begin by considering processes with only one open two-hadron channel, in an arbitrary partial-wave l. Then, we show how the generalization to an arbitrary number of two-hadron channels is straightforward. Whenever possible we will ignore subtleties associated with the spin of the hadrons in the initial and final states, letting the total angular momentum of the two-particle states be equal to l. We will keep the masses of the hadrons to be distinct throughout. For the sake of generality, we leave the Lorentz structure, e.g., scalar, vector, etc., of the current as generic whenever possible.
The formalism we exploit relies on generic properties of a quantum field theory based on self-consistent integral equations for the off-shell 2 → 2, 1 þ J → 2, and 2 þ J → 2 amplitudes. Our main results are presented in Sec. II, where we summarize the on-shell representation of each amplitude. After that, in Sec. III we investigate the implication of our results for resonances. We use our formalism and the Ward-Takahashi Identity to show that the charge of a resonance is protected to be the sum of the charge of its decay products. We also investigate the narrow-width limit of the resonance as a consistency cross-check.
This formalism is general and will have an impact in the study of elastic and transition form factors of a broad class of resonant states. In Sec. III B we discuss the timely example of the ρ resonance. This is the most extensively studied resonance via lattice QCD, and one of its transition form factors has already been constrained [33,34]. In this subsection we explain how the presented formalism provides the missing piece to be able to determine the ρ electromagnetic form factors from lattice QCD.
The derivation of our results is presented in Sec. IV. First, in Sec. IVA we recover the well-known analytic structure of the two-body scattering amplitude, which is also a direct consequence of unitarity. We use the fact that we are interested in a limited range of kinematics where only two-particle states can go on shell. It is by separating the singularities that appear at each order in the two-particle loops that we can express the amplitudes to all-orders in terms of kinematic functions that contain all the nonanalytic behavior, and real functions that encode the short-distance dynamics. Finally, we project the resulting equation on shell and partial-wave expand to yield amplitudes of definite angular momentum. After that, in Sec. IV B we use this technique to recover the analytic form of the 1 þ J → 2 amplitudes with any number of two-hadron coupled channels.
The derivation of the main result of this work is presented in Sec. IV C, where we apply the aforementioned formalism to the 2 þ J → 2 amplitude. Closely related techniques were used in Refs. [31,35] to study the finitevolume analogues of these reactions. Nontrivial checks of this formalism have been carried out [37,38]. We dedicate Sec. IV C 6 to highlighting the novelty of our result with respect to what has been done in past work. Finally, we summarize our results in Sec. V.
II. ANALYTIC REPRESENTATION OF AMPLITUDES
The remainder of this work proceeds to present exact forms of two-body hadronic amplitudes involving a single current insertion. For the sake of completeness, we consider all amplitudes of the form n þ J → m with n and m less or equal to two. We use all-orders perturbation theory to treat the hadronic contributions nonperturbatively within a generic effective field theory (EFT). Furthermore, since we focus on the on-shell behavior of amplitudes, our procedure is independent of the specifics about couplings or renormalization scheme, which are encoded into unknown short-distance functions. In the absence of insertions of external currents, all-orders perturbation theory provides results that are consistent with unitarity constraints. 3 In the presence of external currents, this provides a systematic procedure to asses the singularity structure of the resultant amplitudes.
Here we present the final results and leave the derivation for Sec. IV. In arriving at these results, we make only two assumptions throughout the work. First, that the asymptotic particles considered, which will be referred to as hadrons, 4 carry no intrinsic spin. In other words, they can be either scalars or pseudoscalars. Second, we assume the energies are above the lowest-lying two-particle threshold and below the first unaccounted inelastic threshold, e.g., the threeparticle threshold. This means that the results hold for generic external currents and that the kinematics can be such that any number of two-particle states may go on shell.
The number of classes of singularities and consequently the complexity of the amplitudes grows with the number of external current insertions and particles. However, the majority of these singularities are common across these amplitudes. As a result, the singularity structure of more complicated amplitudes can be written in terms of simpler ones representing subprocesses.
With this in mind, it is convenient to categorize the amplitudes according to the number of currents that are considered. We begin with the two-body scattering amplitude with no external currents, which we label as M. First, we show the case of a single channel system and then discuss the extensions to multiple scattering channels. In Sec. IVA, we prove the well-known result that the on-shell partial-wave scattering amplitude can be written in the form,
iMðsÞ ¼ iKðsÞ 1 1 − iρKðsÞ ;ð1Þ
which is a matrix equation that has elements M l 0 m l 0 ;lm l , where l is the angular momentum between the two particles defined in their center-of-momentum (CM) frame, and m l is its projection onto some fixed axis. 5 Figure 1(a) shows a diagrammatic representation of the amplitude.
Here s ¼ P 2 is the usual Mandelstam invariant with P being the four-momentum of the system, and K is the twobody K matrix, which is a real function in our kinematic region of interest. In general, this function contains branch points associated with crossed-channel processes and multiparticle thresholds, but since these are far from our kinematic region they can be described by smooth contributions. For the sake of brevity we will denote these as smooth throughout the text. Finally, ρ is the phase space which is defined in the standard way for a single channel,
ρ l 0 m l 0 ;lm l ¼ δ l 0 l δ m l 0 m l ξq ⋆ 8π ffiffi ffi s p ≡ δ l 0 l δ m l 0 m l ρ 0 ;ð2Þ
where ξ is a symmetry factor defined to be ξ ¼ 1=2 if the two scattering particles are identical and ξ ¼ 1 otherwise, and q ⋆ is the relative momentum between the two particles in their CM frame,
q ⋆ ¼ 1 2 ffiffi ffi s p λ 1=2 ðs; m 2 1 ; m 2 2 Þ;ð3Þ
where λða; b; cÞ ¼ a 2 þ b 2 þ c 2 − 2ðab þ bc þ caÞ is the Källén triangle function, and m 1 and m 2 are the two masses in the channel considered. Rotational invariance implies that the amplitude is diagonal in l and independent of m l , which reduces FIG. 1. Diagrammatic representation of the amplitudes considered in this work. Shown are the (a) 2 → 2, (b) 1 þ J → 1, (c) 1 þ J → 2, and (d) 2 þ J → 2 amplitudes along with momentum assignments. 3 Although evident for two-body systems, this was proven for three-particle systems in Refs. [39,40], where it was shown that previous results describing three-body amplitudes obtained using all-orders perturbation theory [41] and unitarity constraints [42] were consistent. 4 Even though our motivation is to understand reactions within QCD, we make no reference to the underlying theory. 5 Semicolons in matrix elements separate initial and final state indices. A P,-f-P,
(c) i1f.A = P1 -~ -P, (d) iWA
Eq. (1) to a single algebraic relation for each partial-wave, i.e., M l 0 m l 0 ;lm l ¼ δ l 0 l δ m l 0 m l M l and similarly for K. The on-shell behavior of the scattering amplitude is dictated by S matrix unitarity, which fixes the nonanalytic behavior of the amplitude, originating from direct channel pair production, as indicated by Eq. (2). However, kinematic singularities may remain due to the projection into the angular momentum basis, as discussed in Sec. IVA, which requires that the amplitude M l possesses a barrier suppression of the form q ⋆2l near threshold. This implies that the K matrix has the same threshold behavior. For kinematics where multiple two-body channels are open, in Sec. IVA 1 we show that these objects can easily be upgraded into matrices over the channel index. First, the masses of the particles would acquire an additional index to identify the individual particles in a given channel. We label the two particles in channel "a" to have masses m a1 and m a2 . The K matrix and phase space factor would both be matrices in channel space with components K l 0 m l 0 ;lm l → K al 0 m l 0 ;blm l and ρ l 0 m l 0 ;lm l → ρ al 0 m l 0 ;blm l , respectively. The phase space matrix ρ would be diagonal in this space with elements defined as
ρ al 0 m l 0 ;blm l ¼ δ l 0 l δ m l 0 m l δ ab ξ a q ⋆ a 8π ffiffi ffi s p ; ð4Þ where q ⋆ a ¼ λ 1=2 ðs; m 2 a1 ; m 2 a2 Þ=2
ffiffi ffi s p , and ξ a is the symmetry factor for each channel a. Therefore, Eq. (1) becomes an enlarged matrix within this channel space, which has all the properties of the amplitudes as before, except the barrier suppression is now channel-dependent, M al 0 m l 0 ;blm l ∼ q ⋆l 0 a q ⋆l b , where the dominant singular behavior is the lightest threshold.
A. Amplitudes with a single current insertion
Having shown the well-known result for the two-body hadronic amplitude, we proceed with the description of transitions induced by an external local current insertion. As stated above, we make no assumptions about the quantum numbers of the current, which we denote as J .
In particular, the current can have an arbitrary Lorentz structure with indices μ 1 Á Á Á μ N . Furthermore, other indices can include quantum numbers associated with, for example, flavor-changing processes. For simplicity, we will adopt a notation similar to that in Ref. [43] where all of the quantum numbers of the current, including the Lorentz indices, are absorbed into a single index A.
The simplest of these amplitudes is the one where an external current couples to a one-particle state of mass m, i.e., 1 þ J → 1, which we denote as w on and show diagrammatically in Fig. 1(b). This amplitude is given by
w A on ðk f ; k i Þ ¼ X j K A j ðk f ; k i Þf j ðQ 2 Þ;ð5Þ
where k i =k f are the initial/final four-momentum of the single particle. We write w on in terms of kinematic functions, K, dictated by the Lorentz structure of the current, and Lorentz invariant form factors, f, which depend on
Q 2 ¼ −ðk f − k i Þ 2 .
For a given current, there will always be a finite number of these form factors, for which we will label the jth form factor as f j . 6 The subscript "on" indicates that the form factors are on shell, while the kinematic tensor does not have to be. The arbitrariness of the current allows for the particle species to change; however, we focus only on kinematic regions where the form factors are analytic functions of Q 2 , i.e., above any pole of a state coupling to the current or particle production thresholds. When
k 2 i ¼ k 2 f ¼ m 2 , then Eq. (5) gives the single-particle matrix element hk f jJ A ðx ¼ 0Þjk i i.
Moving up in complexity, the next amplitude we consider is one where the current generates a transition between a one-and two-particle state, i.e., 1 þ J → 2. This amplitude, which we label as H and show in Fig. 1(c), has been previously studied in Ref. [31] for any number of channels. In Sec. IV B we reproduce the finding that this amplitude has an on-shell form given by
iH A ðP f ; P i Þ ¼ iMðs f ÞA A 21 ðP f ; P i Þ;ð6Þ
where P i and P f are the four-momenta of the initial singleparticle state and the final two-particle system, respectively, with s f ¼ P 2 f . The function A 21 is real and smooth 7 in s f , with the same caveats described earlier for the K-matrix, and characterizes the short-distance dynamics. Additionally, it contains the same type of singularities in Q 2 that appear in f j , these again can be described by smooth functions within our kinematic domain. The indices on A 21 refer to the number of hadrons coupling to this short-distance function. Unlike the 2 → 2 amplitude, both H and A 21 are Lorentz tensors which can be written in terms of kinematic prefactors and energy-dependent form factors, similar to the construction in Eq. (5), that depend on the final state angular momentum l as well as its projection m l .
This expression makes explicit that H inherits the analytic structure of M. For kinematics where a single channel is open, this is nothing more than the manifestation of Watson's final state theorem [44]. As with M, near the threshold the angular momentum decomposition of H requires that H lm l ∼ q ⋆l f , where q ⋆ f is as in Eq. (3) with s f . In conjunction with M l ∼ q ⋆2l f , this implies that any parametrization of the A 21 amplitudes necessitates a barrier enhancement of the form A 21;lm l ∼ q ⋆−l f at threshold. For 6 Since they do not possess Lorentz structure we drop the index A, but leave implicit their possible dependence on the current internal quantum numbers. 7 up to barrier factors associated with partial-wave projections. multiple open channels, Eq. (6) naturally extends such that H and A 21 become vectors in channel space. For further discussion on these aspects of H see Sec. IV B.
The main original result presented here is the 2 þ J → 2 amplitude. These amplitudes were introduced in Refs. [35,36] with the goal of trying to obtain them using lattice QCD. These studies were interested in finding a nonperturbative relation between the desired amplitudes and the finite-volume matrix elements that can be accessed via lattice QCD. In Sec. IV C we derive the exact analytic form that these amplitudes must take. We define the 2 þ J → 2 amplitude, which we label as W, via the matrix element,
W A ðP f ;p 0⋆ f ; P i ;p ⋆ i Þ ≡ hP f ;p 0⋆ f ; outjJ A ð0ÞjP i ;p ⋆ i ; ini conn ;ð7Þ
where the initial asymptotic two-particle state depends on the total four-momentum P i , as well as the orientationp ⋆ i of the relative momentum between the two particles in their CM frame, and similarly for the final state defined in its own CM frame. The subscript "conn" highlights that we only consider connected diagrams, i.e., topologies where the hadrons do not interact with each other or with the current are not included. Figure 1(d) shows a diagrammatic representation of the W amplitude, while Fig. 2(a) shows the momentum flow where we adopt the convention that the first and second particle have momenta p and P i − p for the initial state, respectively, and p 0 and P f − p 0 for the first and second particle in the final state, respectively. As mentioned before, we are focused on the kinematic region below three or more particle thresholds for both the initial and final two-particle states. Additionally, we restrict the momentum transfer Q 2 ¼ −ðP f − P i Þ 2 of the current, such that we do not probe any multiparticle production threshold in the Q 2 channel.
In Sec. IV C we derive that the exact analytic form can be separated into two types of terms depending on whether or not they contain single-particle poles associated with the current probing an external leg,
iW A ðP f ;p 0⋆ f ; P i ;p ⋆ i Þ ¼ X fiw A on iDiMg þ iW A df ðP f ;p 0⋆ f ; P i ;p ⋆ i Þ:ð8Þ
Starting with the first term, which represents the case in which the current probes an external leg, w on is as defined in Eq. (5) and D is the pole piece of the fully dressed singleparticle propagator which, for a particle with mass m α with α ¼ 1, 2, can be written as
iD α ðkÞ ¼ i k 2 − m 2 α þ iϵ :ð9Þ
The M was introduced in Refs. [35,36] and is the full 2 → 2 scattering amplitude which has additional barrier factors in its partial wave expansion to cancel out singularities of the spherical harmonics at threshold. For the lowest partial wave, l ¼ 0, M and M are identical. We give the exact definition of this in Sec. IV C 2 in Eq. (84). Finally, the symbol P reminds one to sum over all allowed insertions of the current over the external legs of the amplitude, illustrated in Fig. 2(b). For example, in the case where the current only couples to particle 2, only the first two diagrams of Fig. 2
(b) contribute, written explicitly as X fiw A on iDiMg ¼ iw A on;2 ðp 0 f ; p 0 i ÞiD 2 ðp 0 i ÞiMðp 0⋆ i ;p ⋆ i Þ þ iMðp 0⋆ f ; p ⋆ f ÞiD 2 ðp f Þiw A on;2 ðp f ; p i Þ;ð10Þ
where we use the notation p ð0Þ i=f ≡ P i=f − p ð0Þ , and the threevectors p 0⋆
i and p ⋆ f are the spatial part of the four-vectors p 0μ
= L {X } + iW.j'} p' p
and p μ , respectively, when evaluated in the CM frame indicated by the subscript. The quantity w on;2 is the elastic matrix element of particle 2. Equation (10) diverges whenever the four-vector in either D goes on shell, for example when P f ¼ P i . So far we have discussed the single-pole contribution to W, which can diverge for physical kinematics and as we have already seen, these singularities are completely described by simpler amplitudes. The more phenomenologically interesting component of W is the second term in Eq. (8), which is appropriately labeled with a subscript "df," meaning divergence free. In Sec. IV C we prove that it can be written in an on-shell partial-wave projected form as
iW A df ðP f ; P i Þ ¼ Mðs f Þ iA A 22 ðP f ; P i Þ þ X j;α if j;α ðQ 2 ÞG A j;α ðP f ; P i Þ × Mðs i Þ;ð11Þ
where A 22 is a real and smooth function in both s i and s f , up to barrier factors with the same caveats as K and A 21 . The symbol f j;α is the jth form factor of the αth particle as defined in Eq. (5), and G j;α is a kinematic function to be described shortly. Unlike the scattering amplitude Eq. (1), W df is in general a dense matrix in (l,m l )-space since the current can inject angular momentum. Similar to A 21 , defined in Eq. (6), A 22 contains barrier enhancements near threshold, in this case being of the form A 22;l 0 m l 0 ;lm l ∼ ðq ⋆ f Þ −l 0 ðq ⋆ i Þ −l . This function is unknown and can be parametrized with energy-dependent form factors which can be determined, e.g., via lattice QCD calculations using the formalism presented in Refs. [35,36].
The only quantity not yet defined is the triangle function G j;α , diagrammatically shown in Fig. 3, which occurs when the current probes either particle 1 or particle 2 in the intermediate state. For example with α ¼ 2, it has matrix elements given by
G A j;2;l 0 m l 0 ;lm l ðP f ; P i Þ ≡ Z d 4 k ð2πÞ 4 Y Ã l 0 m l 0 ðk ⋆ f ÞiK A j;2 ðk f ; k i ÞY lm l ðk ⋆ i Þ ðk 2 − m 2 1 þ iϵÞðk 2 f − m 2 2 þ iϵÞðk 2 i − m 2 2 þ iϵÞ ;ð12Þ
where k i=f ≡ P i=f − k, k ⋆ i=f is the spatial part of the fourvector k μ in the initial/final CM frame, and K are the kinematic functions defined in Eq. (5). The symbol Y l;m l contains a spherical harmonic multiplied by the necessary barrier factor to cancel its singular behavior as k ⋆ ≡ jk ⋆ j → 0 [35,36],
Y lm l ðk ⋆ Þ ¼ ffiffiffiffiffi ffi 4π p Y lm l ðk ⋆ Þ k ⋆ q ⋆ l :ð13Þ
The triangle function when α ¼ 1 is written as is shown in Eq. (12), but with labels 1 and 2 switched. In general, Eq. (12) is UV divergent and requires some regularization procedure. For a given scheme, A 22 will compensate this choice such that W df remains finite and scheme independent. Note that if the particles are identical, then there is no sum over the particle index α in Eq. (11). In addition to having threshold singularities, this kinematic function also has a new class of singularities, known as the triangle singularities [45]. The triangle singularities have a logarithmic behavior which we summarize here and give a full description in Appendix A 2. For example, in the case where α ¼ 2, both the initial and final states are in S wave, and for a scalar current with K j ¼ 1, the triangle function is given by
G 00;00 ðP f ; P i Þ ¼ i 32π ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðP f · P i Þ 2 − P 2 i P 2 f q log 1 þ z ⋆ f þ iϵ 1 − ðz ⋆ f þ iϵÞ þ log 1 þ z ⋆ i þ iϵ 1 − ðz ⋆ i þ iϵÞ þ …;ð14Þ
where the ellipsis represents nonsingular terms, z ⋆ f is defined in Eq. (A30) as function of P i , P f and the masses of the external particles, and z ⋆ i is the same function but with the labels f and i switched. The logarithmic singularities can be seen as divergences in the imaginary part, while the real part exhibits a discontinuity at the same energy. This energy is the threshold for which all three particles in the triangle are able to go on shell simultaneously.
If there are multiple open scattering channels, then Eqs. (8) and (11) generalize to matrices in channel space, e.g., W l 0 m l 0 ;lm l → W al 0 m l 0 ;blm l where a and b are channel indices. In general, the two-hadron form factor A 22 is a dense matrix in channel space, while P j f j G j is a dense 13), which have the angular momenta associated with the initial and final two-particle states.
ifj(Q 2 ) Qf(P1, Pi) = j P1 -~..r~, -k Pt -~.
.5pi ~ k matrix if the current allows for species transmutation, for example flavor changing processes (π þ W ⋆ → K) or radiative transitions (η þ γ → η 0 ). If the current does not change species, then P j f j G j is diagonal in channel space, similar to the two-body phase space factor in Eq. (4), since the off diagonal elements of f j;a;b are zero. For example, assuming l 0 ¼ l ¼ 0 and α ¼ 2, then Eq. (12) is modified as
G A j;2;a;b ðP f ; P i Þ ≡ δ a1;b1 Z d 4 k ð2πÞ 4 iK A j;2 ðk f ; k i Þ ðk 2 − m 2 a1 þ iϵÞðk 2 f − m 2 a2 þ iϵÞðk 2 i − m 2 b2 þ iϵÞ ;ð15Þ
where δ a1;b1 ensures that particle 1 appears in both channels. It is worth emphasizing the significance of being able to write W in the form shown in Eqs. (8) and (11). This demonstrates that even if the amplitude W is complex and features singularities, if one has previously determined the w on and M amplitudes, only one class of functions remains to be constrained, A 22 , which are purely real and smooth. In practice, given a finite amount of partial waves, these functions can be defined in terms of energy-dependent form factors associated with the desired angular momentum states. In turn, in the following section, we discuss the determination of resonant form factors in terms of the twohadron form factors A 22 .
III. IMPLICATION FOR RESONANCE FORM FACTORS
The analytic structure presented in the previous section is independent of the dynamics underlying the amplitudes. In this section we consider the implication of the expressions presented for systems that feature a hadronic resonance. For simplicity, we allow for the resonance to couple to a single channel. In particular, we review how one can obtain the mass, decay width, and transition form factors for a scalar resonance coupling to a scalar external current. Additionally, we explain how elastic form factors of this resonance can be obtained from W df . Although we consider the simplest possible systems, we stress that the following procedure is valid for higher partial waves, multiple open channels, and currents with any Lorentz structure.
We begin by providing the standard definition of resonances as complex-valued poles ðs R Þ in the analytic continuation of the scattering amplitude onto the second Riemann sheet,
lim s→s R M II ðsÞ ¼ − c 2 s − s R ;ð16Þ
where c is the coupling to the asymptotic states, and the II superscript has been introduced to emphasize that this is a pole in the second sheet of the scattering amplitude. The pole location can be related to the resonance mass (m R ) and its decay width (Γ R ) via ffiffiffiffiffi
s R p ¼ m R − iΓ R =2.
The amplitude on the second Riemann sheet is found by analytically continuing through the branch cut, which is demonstrated in Appendix B. Using the on-shell form Eq. (1), we find that the second sheet amplitude is expressed as
M II ðsÞ ¼ 1 K −1 ðsÞ þ iρ 0 ;ð17Þ
where the sign flip on the phase space factor arises from continuing through the branch cut.
If there is a resonance present in M, then both H and W df will inherit the singular behavior, as is evident from Eqs. (6) and (11), respectively. In the limit that one approaches the resonance pole, the relationship between the residues of the poles in these amplitudes and the desired transition and elastic form factors can be obtained using the Lehmann-Symanzik-Zimmermann (LSZ) reduction procedure. This is illustrated diagrammatically in Fig. 4. In our example of a scalar resonance and a scalar current, the transition amplitude H can be related to the transition form factor, f 1→R ðQ 2 Þ, via
lim s→s R H II ðs; Q 2 Þ ¼ − cf 1→R ðQ 2 Þ s − s R ;ð18Þ
where again this is defined on the second sheet. In Eq. (6), we wrote H in terms of A 21 . Because A 21 has been found to be nonsingular, continuing H to the second sheet amounts to a continuation in M from Eq. (6), giving
H II ðs; Q 2 Þ ¼ M II ðsÞA 21 ðs; Q 2 Þ:ð19Þ
Note that for scalar currents, since H and A 21 are Lorentz invariants, we can write s and Q 2 for their arguments, rather than P i and P f . In the case considered, A 21 can be understood as an energy-dependent form factor. We can rewrite the transition form factor in terms of A 21 ,
f 1→R ðQ 2 Þ ¼ lim s→s R ðs R − sÞ c M II ðsÞA 21 ðs; Q 2 Þ; ¼ lim s→s R ðs R − sÞ c − c 2 s − s R A 21 ðs; Q 2 Þ ; ¼ cA 21 ðs R ; Q 2 Þ:ð20Þ
Given that A 21 ðs; Q 2 Þ is real and has no nearby singularities, it is a convenient function to parametrize. This in part illustrates that if you know A 21 ðs; Q 2 Þ as a function of energy, it is straightforward to determine the transition form factor of the resonance of interest. This insight was used in, for example, recent exploratory determinations of the ρ → π electromagnetic form factor from lattice QCD [33,34,46]. Similarly, one can determine the elastic form factor of the resonance, f R→R ðQ 2 Þ, from the residue of W via the LSZ reduction as lim s i ;s f →s R W II;II ðs f ; Q 2 ; s i Þ ¼ lim
s i ;s f →s R W II;II df ðs f ; Q 2 ; s i Þ; ð21Þ ¼ −c s f − s R f R→R ðQ 2 Þ −c s i − s R ; ð22Þ where f R→R ðQ 2 Þ is defined at fixed s i ¼ s f ¼ s R .
The appearance of two II superscripts emphasizes that one must continue the amplitude in both s i and s f planes in order to evaluate the amplitude at the resonance pole. In the first equality, we have used the fact that the difference between W and W df , given in Eq. (8), only couples to either the initial or final resonance pole but not both. As a result, this is suppressed relative to the double pole. 8 From the on-shell representation of Eq. (11), we find that the W df amplitude on the second Riemann sheet in both variables takes the form,
W II;II df ðs f ;Q 2 ;s i Þ ¼ M II ðs f Þ½A 22 ðs f ;Q 2 ;s i Þ þ fðQ 2 ÞG II;II ðs f ;Q 2 ;s i ÞM II ðs i Þ;ð23Þ
where G II;II is the analytically continued triangle function which is
G II;II ðs f ; Q 2 ; s i Þ ¼ Gðs f ; Q 2 ; s i Þ − 2iImGðs f ; Q 2 ; s i Þ:ð24Þ
In arriving at Eq. (23), it was necessary to use the fact that A 22 is nonsingular in the kinematic region considered. A detailed proof of Eq. (23) is provided in Appendix B.
Using the all-order expression for W df in terms of the triangle loop and the energy-dependent form factor given in Eq. (11), one finds
f R→R ðQ 2 Þ ≡ lim s i ;s f →s R s i − s R −c s f − s R −c M II ðs f Þ½A 22 ðs f ; Q 2 ; s i Þ þ fðQ 2 ÞG II;II ðs f ; Q 2 ; s i ÞM II ðs i Þ; ¼ c 2 ½A 22 ðs R ; Q 2 ; s R Þ þ fðQ 2 ÞG II;II ðs R ; Q 2 ; s R Þ:ð25Þ
As previously stated, although the arguments presented here were for scalar currents for S wave systems, the relations easily generalize to arbitrary currents, partial waves, and channels. In the case of currents with nontrivial Lorentz structure, form factors are accompanied by kinematic Lorentz tensors, which do not alter the analytic structure. For multiple scattering channels, one must take care of which sheet the amplitude is continued to, following the same methodology as presented in, for example, Ref. [49].
In the special case of conserved vector currents, it was shown in Ref. [37] that current conservation via the Ward-Takahashi identity constrains the forward limit of the 2 þ J → 2 amplitude. For example, the amplitude for a two hadron system consisting of one neutral and one charged particle must follow the relation,
lim P i →P f W μ df ðP f ; P i Þ ¼ 2P μ Q 0 ∂ ∂s MðsÞ;ð26Þ
where Q 0 is the charge of the particle. This identity imposes further constraints on A 22 , namely that in the forward limit,
A μ 22 ðP; PÞ ¼ −2Q 0 P μ ∂ ∂s K −1 ðsÞ − Q 0 ReG μ ðP; PÞ;ð27Þ
which follows directly from Eq. (11), and noting that the imaginary part of G μ is proportional to ∂ρ=∂s, ensuring that A 22 is a real function.
If there is a resonance in this system, then at the resonance pole Eq. (26) further imposes that the form factor of the resonance at Q 2 ¼ 0 is the charge of the resonant state. We define the form factor for a scalar resonance with a vector current in an analogous way to Eq. (21) as
ðP f þ P i Þ μ f R→R ðQ 2 Þ ≡ lim s i ;s f →s R s f − s R c W II;II df;μ ðP f ; P i Þ s i − s R c :ð28Þ
Taking the P i → P f limit of Eq. (28), we then use Eq. (26) to find
2P μ f R→R ð0Þ ¼ 2P μ Q 0 lim s→s R ðs − s R Þ 2 c 2 ∂ ∂s M II ðsÞ; ð29Þ ¼ 2P μ Q 0 :ð30Þ
Therefore, we conclude that the resonance form factor for a conserved vector current yields its charge at Q 2 ¼ 0 as one may expect. The use of the Ward-Takahashi identity to impose additional constraints on two-hadron resonances has been explored, e.g., in Ref. [50] for the Roper and in Ref. [51] for the Δ.
In practical lattice QCD calculations, renormalizing a conserved current by demanding that the form factor at Q 2 ¼ 0 of one of the stable hadrons is equal to its physical charge, ensures that the charge of the rest of the stable hadrons is recovered. Since the normalization of form factors of resonances and bound states is fixed by this same charge, according to both the analytic expression presented here as well as the finite volume framework in Refs. [35,36], they do not require any additional renormalization.
A. Recovering the narrow-width approximation
For processes where resonances may appear as intermediate states, it is advantageous to perform an expansion about their decay width in the limit where it becomes infinitesimally small. See Ref. [52] for a recent example of this. In this limit, the scattering amplitude acquires a pole at physical energy values, thus violating unitarity. This can be remedied by calculating corrections to the nonzero width. In order to gain further insight into W df we explore its behavior in the presence of an infinitesimally narrow resonance. Given Eq. (21), we expect this to be dominated by a double pole structure.
For simplicity, we consider the case where all amplitudes are saturated by the l ¼ 0 partial wave and are dominated by a narrow resonance, this motivates the use of the S wave Breit-Wigner parametrization,
tan δðsÞ ¼ ffiffi ffi s p ΓðsÞ m 2 0 − s ;ð31ÞΓðsÞ ¼ g 2 6π m 2 0 s q ⋆ ¼ 4g 2 3ξ m 2 0 ffiffi ffi s p ρ 0 ;ð32Þ
where δðsÞ is the scattering phase shift, ΓðsÞ is the energydependent width, and m 0 and g are constants of the parametrization that in general do not possess direct physical interpretation. In the last equality, we used the phase-space definition given in Eq. (2). Finally, for a single partial wave the K matrix relates to the scattering phase shift via K −1 ¼ ρ 0 cot δðsÞ. Having this parametrization in place, we can analytically continue the corresponding amplitude to the resonant pole as discussed in Sec. III and relate these parameters to the pole location and residue. Given the definition of Γ above, it is clear that the narrow-width limit can be considered by expanding the Breit-Wigner amplitude about g ≈ 0. In this case, ffiffiffiffiffi
s R p ¼ m 0 þ Oðg 2 Þ.
In what follows, we will expand the amplitudes to leading order in g. In addition to verifying the expected behavior of the amplitudes, it informs us how the generalized form factors A 21 and A 22 behave as a function of g at leading order. At leading order in g, the purely hadronic amplitude is equal to
MðsÞ ¼ 4g 2 m 2 0 3ξ 1 m 2 0 − s þ Oðg 4 Þ:ð33Þ
This has the same form as Eq. (16), allowing us to identify the residue at the pole in terms of the parameters of the model. At leading order in g we get
c ¼ 2gm 0 ffiffiffiffiffi 3ξ p :ð34Þ
Given this, we can explore the narrow-width limit for the remaining amplitudes by performing an expansion in g and keeping the leading order term. To do this, it is useful to categorize the various building blocks of the aforementioned amplitudes in their leading behavior in g. For the purely hadronic ones it is evident that
MðsÞ ¼ Oðg 2 Þ:ð35Þ
For the 1 þ J → 2 transition amplitude it is slightly less obvious how it should scale. Focusing on scalar currents, we begin by anchoring their scaling in terms of the transition form factor, given in Eq. (20). Keeping only the leading order behavior in g, we find
f 1→R ðQ 2 Þ ¼ 2m 0 ffiffiffiffiffi 3ξ p gA 21 ðm 2 0 ; Q 2 Þ;ð36Þ
where we used the relationship between c and g, given in Eq. (34). Given that the form factor must in general be nonzero in the g → 0 limit, we find how A 21 ðs; Q 2 Þ must scale with g. This, in combination with Eq. (6), tells us that the scaling of the other 1 þ J → 2 transition building blocks is
A 21 ðs; Q 2 Þ ¼ Oðg −1 Þ; Hðs; Q 2 Þ ¼ OðgÞ:ð37Þ
We proceed to apply the same logic to the 2 þ J → 2 amplitude. In other words, we first look at the elastic resonant form factor, Eq. (25) and use the relationship between g and c to find
f R→R ðQ 2 Þ ¼ 4g 2 m 2 0 3ξ ½A 22 ðm 2 0 ; Q 2 ; m 2 0 Þ þ fðQ 2 ÞGðm 2 0 ; Q 2 ; m 2 0 Þ;ð38Þ
where we neglected higher-order corrections in g. Again, this is in general nonzero in the g → 0 limit, so we deduce that the term in brackets must scale as 1=g 2 . This scaling, of course, cannot come from either f or G. The former is the single particle form factor, which in general has no knowledge of the resonance being considered. The latter is a purely kinematic function which contains no information of dynamical quantities, like g. Therefore neither can have any information of g, leaving us to conclude that
A 22 ðm 2 0 ; Q 2 ; m 2 0 Þ ¼ Oðg −2 Þ:ð39Þ
As a result at leading order in g the form factor satisfies
f R→R ðQ 2 Þ ¼ 4g 2 m 2 0 3ξ A 22 ðm 2 0 ; Q 2 ; m 2 0 Þ:ð40Þ
Thus, from Eq. (11) we find the leading order behavior in g of the double pole contribution to W df to be
W df ðP f ; Q 2 ; P i Þ ¼ Oðg 2 Þ:ð41Þ
In conclusion, this suggests that for a narrow resonance within the Breit-Wigner parametrization, one should introduce
A 21 ðs; Q 2 Þ ¼ e A 21 ðs; Q 2 Þ g ; A 22 ðs f ; Q 2 ; s i Þ ¼ e A 22 ðs f ; Q 2 ; s i Þ g 2 ;ð42Þ
where e A 21 and e A 22 do not scale with g. It is worth commenting further as to why this is a sensible conclusion within an EFT point of view. Within an EFT, one can always introduce an auxiliary field for a narrow resonance that couples to asymptotic states. Defining the coupling to the scattering states to be proportional to g, one immediately finds that any loop would be Oðg 2 Þ. In other words, the s-channel loops appearing in M and H are Oðg 2 Þ suppressed, and these loops are the source of the resonance nonzero width. Similarly, in W df the schannel loops, including the triangle one in Fig. 3 are Oðg 2 Þ suppressed. This is consistent with the fact that A 22 is enhanced relative to the triangle function by 1=g 2 .
B. Example: ρ resonance form factors from lattice QCD
To illustrate the applicability of this framework, we overview the procedure for determining the electromagnetic form factors of the ρ resonance from lattice QCD. A charged ρ decays primarily to a ππ isovector state, e.g., ρ þ → π þ π 0 , assuming exact isospin symmetry. For heavier than physical quark masses such that m ρ < 2m π , the ρ becomes stable, and one may utilize the fact that matrix elements receive exponentially suppressed finite-volume corrections to directly extract the form factors [29]. If the light quark mass is such that m ρ > 2m π , then the decay channel opens, and one must resort to using the framework presented in Refs. [35,36], as well as the formalism presented in this work, to determine the resonant ρ form factors.
More explicitly, let us assume that both the I ¼ 1 ππ scattering amplitude and the π þ electromagnetic form factor have been previously determined, as in Refs. [29,53] respectively. Reference [36] gives an overview on how one uses the finite-volume formalism together with these measured quantities to determine W df . Given that in the elastic kinematic regime for l 0 ¼ l ¼ 1, W df has a decomposition given in Eq. (11), the finite-volume formalism ultimately allows us to fix the unknown A 22 much in the same way that K is fixed from the Lüscher framework. Once A 22 is determined, we may then follow the procedure outlined in the beginning of this section to determine the form factors of the ρ. Such a calculation would yield a first-principles determination of the resonant ρ form factors and gives a pathway for understanding the structure of excited hadrons.
IV. DERIVATION OF ON-SHELL REPRESENTATIONS
In this section, we derive the on-shell representations presented in Sec. II. Our main tool relies on summing diagrams to all-orders, separating singularities induced by particle production in the physical region, from shortdistance contributions. All short-distance physics is absorbed into a set of unknown functions which can be determined from lattice QCD. We first review the on-shell projection for the 2 → 2 hadronic amplitude in Sec. IVA, recovering the well-known K matrix representation. Following this, we turn to the transition amplitudes, first reviewing the known result for 1 þ J → 2 processes in Sec. IV B. We then present the derivation of the main original result in this article, the 2 þ J → 2 transition amplitude, in Sec. IV C.
A. Review of the 2 → 2 scattering amplitude
We begin with the hadronic 2 → 2 scattering amplitude in the kinematic region where only one channel, composed of two scalar particles with masses m 1 and m 2 , is open. In Sec. IVA 1, we lift this assumption to accommodate any number of intermediate two-particle states. It is convenient to consider the off-shell extension of this amplitude, Mðp 0 ; pÞ, where the initial state carries momenta p and P − p for particles 1 and 2, respectively, and the final state carries momenta p 0 and P − p 0 , for particles 1 and 2, respectively. Note, the momenta of the initial/final state appear in the rightmost/leftmost part of the arguments of Mðp 0 ; pÞ. We will follow this convention throughout. We leave the dependence on the total conserved momentum P in the amplitude M implicit for notational convenience.
where ⋆ denotes CM coordinates, andp ⋆ andp 0⋆ are the orientations of particle 1 in the initial and final state in this frame, respectively. These are not fixed when placing the particles on their mass shell.
It can be shown, e.g., summing to all-orders in perturbation theory, that Mðp 0 ; pÞ satisfies the self-consistent integral equation,
iMðp 0 ; pÞ ¼ iK 0 ðp 0 ; pÞ þ ξ Z d 4 kð2πÞ4 iMðp 0 ; kÞiΔ 1 ðkÞ × iΔ 2 ðP − kÞiK 0 ðk; pÞ;ð44Þ
where iK 0 is the Bethe-Salpeter kernel, which contains all s-channel two-particle irreducible diagrams, ξ is the symmetry factor defined in the previous section, iΔ 1 and iΔ 2 are the fully dressed propagators for particles 1 and 2, respectively, and where the integral runs over the fourmomentum of the intermediate state particles. This equation is depicted pictorially in Fig. 5. Note that Eq. (44) can be written such that M and K 0 in the second term are interchanged. In the following manipulations, we work with Eq. (44) as presented but remark that the same procedure holds for the alternative where M and K 0 are interchanged in the second term. For each particle α ¼ 1, 2, we choose the propagators to have unit residue at the pole mass,
iΔ α ðkÞ ¼ i k 2 − m 2 α þ iϵ þ iS α ðkÞ;ð45Þ≡iD α ðkÞ þ iS α ðkÞ;ð46Þ
where S α are nonsingular at the pole. For the kinematic region of interest, iK 0 is nonsingular and can be thought of as a smooth function. We proceed to now separate the on-shell components from Eq. (44), exploiting the fact that in the elastic kinematic domain the only singularity that the amplitude has is the two-particle intermediate state threshold. In other words, we will make explicit the on-shell singularities required by S matrix unitarity, and all off-shell contributions will be absorbed into some short-distance function to be determined, e.g., from lattice QCD calculations. We first iterate Eq.
We now take the second term, as well as the second loop in the third term, and separate out the on-shell contribution, which amounts to identifying all components which can give an imaginary part. In Appendix A 1 we describe in detail the procedure we follow, which closely resembles that presented in Ref. [8]. Following the operations outlined in the aforementioned Appendix, we find
ξ Z d 4 kð2πÞ4 iK 0 ðp 0 ; kÞiΔ 1 ðkÞiΔ 2 ðP − kÞiK 0 ðk; pÞ ¼ iK 1 ðp 0 ; pÞ þ Z dk ⋆ 4π iK 0 ðp 0 ;k ⋆ Þρ 0 iK 0 ðk ⋆ ; pÞ; ¼ iK 1 ðp 0 ; pÞ þ X l;m l iK 0;lm l ðp 0 Þρ 0 iK 0;lm l ðpÞ;ð48Þ
where K 1 is a smooth function, and ρ 0 is the two-particle phase space defined in Eq. (2)
where in the second term we recovered M using Eq. (44), and its intermediate state is projected on shell and into partial waves similar to Eq. (50). We see that the third and fourth terms form the same structure as our starting point Eq. (44), with a new kernel K 1 . Therefore, we can insert Eq. (44) into the fourth term, and repeat the on-shell separation as before, yielding new terms that go like Eq. (48) with the rightmost K 0 replaced by K 1 , and K 1 replaced by K 2 . This pattern continues for every loop with the new kernel defined by the previous separation. We therefore define the iterated loop identity,
ξ Z d 4 kð2πÞ
where in the last line we defined the K matrix as the sum of each iterated kernel. Since the intermediate state is now on shell, we project the initial and final states on their mass shell, and project the initial and final states via the partialwave expansion, (1) as a matrix in angular momentum space. The partial wave expansion induces kinematic singularities at threshold since the spherical harmonics become singular as q ⋆ → 0. Therefore, M l 0 m l 0 ;lm l ∼ q ⋆l 0 þl in order to compensate for this. As introduced in Sec. II, rotational invariance allows us to write the amplitude as M l 0 m l 0 ;lm l ¼ δ l 0 l δ m l 0 m l M l . As a final remark, the on-shell form Eq. (1) explicitly shows the singularities required by unitarity of the S matrix, which for partial waves states that the discontinuity of the amplitude across the real s axis must satisfy
Mðp 0⋆ ;p ⋆ Þ ¼ 4π X l 0 ;m l 0 X l;m l Y l 0 m l 0 ðp 0⋆ ÞM l 0 m l 0 ;lm l ðsÞY Ã lm l ðp ⋆ Þ;ð55ÞDiscM l ¼ 2iImM l ¼ 2iρ 0 jM l j 2 ;ð56Þ
for energies greater than the production threshold, s ≥ s th ≡ ðm 1 þ m 2 Þ 2 . The equivalence between the discontinuity and the imaginary part holds from the Schwartz reflection principle, i.e., Mðs Ã Þ ¼ M à ðsÞ.
Arbitrary number of channels
If one considers an arbitrary number of strongly interacting channels, then the previous discussions are extended such that the amplitudes are matrices in channel space. Let a, b, and c be the channel index, which ranges from 1 to N ch , where N ch indicates the number of channels. The selfconsistent integral equation for the scattering amplitude is then iM a;b ðp 0 ; pÞ ¼ iK 0;a;b ðp 0 ; pÞ þ where the 1c and 2c indices on the propagators indicate particle 1 and 2 in channel c, respectively. Following the same steps to project the system to its on-shell representation, with the simple extension that each kernel is a matrix in channel space, we arrive at the expression, where angular momentum indices are left implicit while we explicitly show the indices for channel space.
X N ch c¼1 ξ c Z d 4 kð2πÞ
B. The 1 + J → 2 transition amplitude
In performing the on-shell projection of the 2 → 2 amplitude in the previous section, we effectively separated the short and long distance contributions of this amplitude. A similar separation can be made for the 1 þ J → 2 transition amplitude, H. This can be done while placing no restrictions on the current, except that it is local. As a result, we consider an arbitrary external and local current. The final two-particle state has momenta p 0 and P f − p 0 for particles 1 and 2, respectively, while the initial state has only a single particle with momentum P i , and associated mass M. The external current carries a momentum transfer squared Q 2 ¼ −ðP f − P i Þ 2 . We again consider the off-shell extension Hðp 0 ; P f ; P i Þ. As explained in Sec. II A, we will label the current and the subsequent amplitudes with a single superscript A, which encodes all identifiers of the current.
The amplitude satisfies the following self-consistent integral equation [31]:
iH A ðp 0 ; P f ; P i Þ ¼ iH A 0 ðp 0 ; P f ; P i Þ þ Z d 4 k ð2πÞ 4 iMðp 0 ; kÞiΔ 1 ðkÞ × iΔ 2 ðP f − kÞiH A 0 ðk; P f ; P i Þ:ð59Þ
Here H 0 is a nonsingular and smooth function of s f in the kinematic region of interest, similar to K 0 , while Δ and M are as before. Diagrammatically, this is shown in Fig. 7. We now substitute the self-consistent relation for the M amplitude, given in Eq. (44), into Eq. (59). We then proceed as before separating out the on-shell behavior. For this situation, we cast the iterated loop identity as where H j is the jth iterated kernel which absorbs all real contributions from the loop and the previous iterated kernel, and ρ 0 depends on s f . The subscript j enumerates the number of K 0 kernels inserted in the projection. Applying the same procedure as with M we arrive at
ξ Z d 4 k ð2πÞ 4 iK 0 ðp 0 ;kÞiΔ 1 ðkÞiΔ 2 ðP f − kÞiH A j ðk;P f ;P i Þ ¼ iH A jþ1 ðp 0 ;P f ; P i Þ þ X l;m l iK 0;lm l ðp 0 Þρ 0 iH A j;lm l ðP f ;P i Þ;ð60ÞiH A lm l ðP f ; P i Þ ¼ iH A lm l ðP f ; P i Þ þ iM l ðs f Þρ 0 iH A lm l ðP f ; P i Þ;ð61Þ
where H is the infinite sum of all iterated kernels defined by Eq. (60) and is a real function in the kinematic domain of interest. The on-shell projection is illustrated diagrammatically in Fig. 8. Note that the partial-wave expansion for H is only on the final state,
H A ðp 0 ; P f ; P i Þ ¼ ffiffiffiffiffi ffi 4π p X l;m l Y lm l ðp 0⋆ f ÞH A lm l ðP f ; P i Þ;ð62Þ
where the subscript f onp 0⋆ f indicates that these angles are evaluated in the CM frame of the final state.
Using Eq. (1) for M, we can write the on-shell representation for H as
iH A lm l ðP f ; P i Þ ¼ 1 1 − K l ðs f Þiρ 0 iH A lm l ðP f ; P i Þ:ð63Þ
At this stage, we make note that H is an analytic function in the complex s f plane except for the branch cut associated with the pair production of the intermediate state and potential bound state poles. The K matrix could in principle have poles in s f for physical energies, which do not appear in the scattering amplitude as poles on the real axis. One can show, using all orders perturbation theory, that H must have these same poles in s f . If this were not the case, the unphysical poles would correspond to zeros of the H. This motivates us to introduce a parametrization,
iH A lm l ðP f ; P i Þ ¼ iK l ðs f ÞA A 21;lm l ðP f ; P i Þ;ð64Þ
where A 21 is a smooth function in the allowed kinematic domain, except for barrier factors near threshold. Combining this with Eq. (63), we arrive at Eq. (6). As mentioned in Sec. II, H and A 21 are Lorentz tensors which can be expanded in energy-dependent form factors.
The form of the on-shell amplitude satisfies Watson's final state theorem [44], meaning that the phase of the amplitude H is equal to that of M. This is a consequence of the unitarity condition for H, DiscH lm l ¼ 2iρ 0 M Ã l H lm l , from which one immediately identifies Eq. (6) as a solution.
Our results can be generalized to accommodate any number of two-body scattering channels. Using the results for the coupled-channel scattering amplitude and extending Eq. (59) for N ch channels, we find
iH A a ðp 0 ; P f ; P i Þ ¼ iH A 0;a ðp 0 ; P f ; P i Þ þ X N ch b¼1 ξ b Z d 4 k ð2πÞ 4 iM a;b ðp 0 ; kÞiΔ b1 ðkÞ × iΔ b2 ðP f − kÞiH A 0;b ðk; P f ; P i Þ:ð65Þ
The preceding arguments can be made to show that its onshell representation has the same analytic structure,
H A a ðP f ; P i Þ ¼ X N ch b¼1 M a;b ðs f ÞA A 21;b ðP f ; P i Þ;ð66Þ
which agrees with Eq. (6) and the expressions presented in Ref. [31]. Here, H a and A 21;a are elements of a vector in channel space for some given initial state, and M a;b is defined in Eq. (58). All arguments above were for the case when the current interacts with an initial single-particle state, 1 þ J → 2, and the same manipulations apply if one considers the case where the current is ejected in the final state, 2 → 1 þ J . Moreover, the previous derivation can be adapted to the case with no initial hadrons, i.e., J → 2, by replacing the kernel H 0 with one encoding the short-distance dynamics of pair creation. Therefore the on-shell amplitude of both processes has the same analytic structure. This is because the J → 2 reaction is analogous to a 1 þ J → 2 transition where the initial state happens to have vanishing momentum.
C. The 2 + J → 2 transition amplitude
Here we present an all orders calculation of the 2þJ →2 amplitude, W, where an external current with arbitrary Lorentz structure injects momentum into the initial two particle system. We consider the case where the species of the initial and final state particles are the same. In this case, the initial state particles have momenta p and P i − p for particles 1 and 2, respectively, while the final state has p 0 and P f − p 0 for particles 1 and 2, respectively. Therefore, the external current carries a momentum transfer squared
Q 2 ¼ −ðP f − P i Þ 2 .
The W amplitude is defined via the matrix element as in Eq. (7), and is the on-shell limit of the off-shell extension,
W A ðP f ;p 0⋆ f ; P i ;p ⋆ i Þ ¼ W A ðP f ; p 0 ; P i ; pÞ p 2 ¼p 02 ¼m 2 1 ;ðP i −pÞ 2 ¼ðP f −p 0 Þ 2 ¼m 2 2 :ð67Þ
Generally, the current can also be flavor changing, as in the 1 þ J → 2 case. We first focus on the case where the current is not flavor changing, so that the initial, intermediate, and final states are the same species. Furthermore, for the following derivation we consider the case where particle 1 is neutral with respect to the external current, i.e., the current only interacts with particle 2. At the end of this section, we comment on the extension in which both particles interact with the current, including the case where the particles are identical. Additionally, we show the result for an arbitrary number of channels to which the current can couple. We proceed as before, only now we have to separate out the on-shell behavior from both the initial and final state two-particle scatterings. However, we encounter a new feature which is not present in the previous case, namely the triangle diagram topology in which the current interacts with a single particle only. This introduces new kinematic singularities in addition to those already present in the twoparticle loop. Considering the elastic region for both the initial and final states, using all-orders perturbation theory the amplitude can be separated into two topologically distinct classes of amplitude,
iW A ¼ iW A 1B þ iW A 1B ;ð68Þ
where the subscript 1B stands for topologies where the current couples to a single hadron (one-body), cf. Fig. 9. Each amplitude contains a distinct kernel, which represents all short-range physics which cannot go on shell in the kinematic domain of our interest. First, the kernel contained in the amplitude W 1B we denote as W 0j0 , and it represents a short-range 2 þ J → 2 transition amplitude, which is twoparticle irreducible in both the s i and s f channels, where s i ¼ P 2
i and s f ¼ P 2 f . Like the kernel H 0 , the subscript "0" denotes the absence of two-particle dressings from K 0 , with the vertical line representing a distinction between initial and final states. These kernels are smooth, nonsingular functions in the elastic region of both the initial and final states. Dressing this kernel with all two-particle scatterings to all orders in the strong interactions, one can show that W 1B obeys the following equation: where we remind the reader of the shorthand notation k i ≡ P i − k, and similarly for the final state and k 0 momenta. Equation (69) is represented diagrammatically in Fig. 9.
iW A 1B ðP f ; p 0 ; P i ; pÞ ¼ iW A 0j0 ðP f ; p 0 ; P i ; pÞ þ ξ Z d 4 kð2πÞ
The second type of kernel we consider is the off-shell extension of the single hadron transition amplitude w, which can be expressed in terms of the off-shell extended form factors f j and kinematic functions K j as in Ref. [36],
w A ðk f ; k i Þ ¼ X j K A j ðk f ; k i Þf j ðQ 2 ; k 2 f ; k 2 i Þ;ð70Þ
where k i is the momentum flow from the initial state, and k f the momentum flow from the final state, giving the same invariant momentum transfer squared as before. Since we consider here the case where the external current couples only to particle 2, all quantities in Eq. (70) refer to this particle. This restriction can be trivially lifted by considering two sets of kernels w 1 and w 2 , each coupling to particles 1 and 2, respectively, which we discuss later. We consider an illustrative example of the explicit decomposition of Eq. (70) for the following two cases. If the current is a scalar and the initial and final particles are identical scalars, then the sum only includes a single form factor with no kinematic prefactor. If instead the current is a conserved vector current, the sum again only has one term and the prefactor is
K μ ðk f ; k i Þ ¼ ðk f þ k i Þ μ .
Physical single-hadron form factors are defined by the on-shell limit of this kernel, i.e., by continuing Eq. (70) to
k 2 i ¼ k 2 f ¼ m 2 2
, which is equivalent to the single hadron matrix element,
w A ðk f ; k i Þ k 2 f ¼m 2 2 ;k 2 i ¼m 2 2 ¼ w A on ðk f ; k i Þ;ð71Þ
where in this case we define the on-shell limit as affecting the form factors only, leaving the kinematic tensors unaffected. The on-shell form factor is denoted with a single argument Q 2 , differing from the off-shell extension where it depends on both the initial and final invariant mass,
f j ðQ 2 Þ ¼ lim k 2 i →m 2 2 k 2 f →m 2 2 f j ðQ 2 ; k 2 f ; k 2 i Þ:ð72Þ
There are three sets of diagrams that include the kernel w which contribute to the all-orders relations for W 1B . Two of them involve the case where the current probes either the initial or final state, and the third set of diagrams include the case where the current probes one of the propagators in an intermediate state. Summing to all orders, one finds the relation,
iW A 1B ðP f ; p 0 ; P i ; pÞ ¼ iw A ðp 0 f ; p 0 i ÞiΔ 2 ðp 0 i ÞiMðp 0 ; pÞ þ iMðp 0 ; pÞiΔ 2 ðp i Þiw A ðp f ; p i Þ þ Z d 4 k ð2πÞ 4 iMðp 0 ; kÞiΔ 1 ðkÞiΔ 2 ðk f Þiw A ðk f ; k i Þ × iΔ 2 ðk i ÞiMðk; pÞ:ð73Þ
Combining Eqs. (69), (73), and (68) leads to a complete all-orders description of the 2 þ J → 2 amplitude in terms of general one-and two-particle irreducible kernels. Diagrammatically, W 1B and W 1B are shown in Fig. 9.
On-shell projecting W 1B
We now project Eq. (69) such that intermediate states are on their mass-shell. This case follows the same manipulations as in H, except here there are two-body dressings on both the initial and final states. We split the on-shell projection into two steps. Initially, consider the first and second terms of Eq. (69), where the final state is dressed with two-particle scattering processes. The form of these two terms is similar to H, and we can project these two terms to their on-shell form, where the new kernel is defined as
iW A 0j0 ðP f ; p 0 ; P i ; pÞ þ ξ Z d 4 kð2πÞW A ∞j0 ¼ X ∞ j¼0 W A jj0 ;ð75Þ
and W jj0 is the jth iterated kernel for the final state, defined in the same way as Eq. (60),
ξ Z d 4 k ð2πÞ 4 iK 0 ðp 0 ; kÞiΔ 1 ðkÞiΔ 2 ðk f ÞiW A jj0 ðk; P f ; P i Þ ¼ iW A jþ1j0 ðp 0 ; P f ; P i Þ þ X l;m l iK 0;lm l ðp 0 Þρ 0 iW A jj0;lm l ðP f ; P i Þ:ð76Þ
Again, the subscript j indicates the number of absorbed kernels K 0 from the on-shell projection. In the second term, the intermediate-state particles are on shell, and we expanded both the amplitude and the kernel into partial waves as in Eq. (49). Recall that the external states remain off shell, which means that this same decomposition holds for the third and fourth terms of Eq. (69), in which the kernels in Eq. (74) are dressed with two-body scatterings on the initial state. Next, the third and fourth terms represent a similar form to before, except we trade the final for initial state two-body dressings. Repeating the same projection with this new kernel, now on the initial state, we arrive at the fully on-shell projected W 1B amplitude,
iW A 1B;l 0 m l 0 ;lm l ðP f ;P i Þ ¼ ½1 þ iM l 0 ðs f Þρ 0 iW A 1B;l 0 m l 0 ;lm l ðP f ;P i Þ½1 þ ρ 0 iM l ðs i Þ;ð77Þ
where we introduce a new short-distance kernel W 1B which absorbs all the off-shell contributions from the intermediate state on-shell projections, defined by
W A 1B ≡ W A ∞j∞ ¼ X ∞ j¼0 W A ∞jj ;ð78Þ
where W ∞jj is defined via an iterate loop identity which we forgo writing since it is structurally identical to Eq. (76) except for the swapping of the kernels. The phase space factors depend on the same total momentum as their respective adjacent factors of M. Additionally, we have placed the external states on their mass shell and expanded them into their respective partial-wave projections,
W A 1B ðP f ;k ⋆ f ; P i ;k ⋆ i Þ ¼ 4π X l 0 ;m l 0 X l;m l Y l 0 m l 0 ðk ⋆ f ÞW A 1B;l 0 m l 0 ;lm l ðP f ; P i ÞY Ã lm l ðk ⋆ i Þ;ð79Þ
which holds for both W 1B and W 1B . Unlike the 2 → 2 amplitude, the angular momentum between the hadrons is not conserved due to the insertion of the current, and thus is a dense matrix in this space. Equation (77) shows that the on-shell kernel is dressed by initial and final state rescatterings, similar to how the 1 þ J → 2 amplitude is dressed in the final state, cf. Eq. (61).
On-shell projecting W 1B
Moving on to W 1B we need to consider contributions due to the single hadron transition amplitude. This leads to a new on-shell function, namely the triangle diagram. We start with the first term of Eq. (73) by using Eq. (44) to obtain expressions where w is attached to the kernel K 0 ,
iw A ðp 0 f ; p 0 i ÞiΔ 2 ðp 0 i ÞiMðp 0 ; pÞ ¼ iw A ðp 0 f ; p 0 i ÞiΔ 2 ðp 0 i ÞiK 0 ðp 0 ; pÞ þ iw A ðp 0 f ; p 0 i ÞiΔ 2 ðp 0 i Þξ Z d 4 kð2πÞ
Following similar steps outlined in detail in Ref. [36], we use Eq. (45) to isolate the pole piece of the propagators. We expand the kernels on either side of the pole term about the pole. Next, we project the kinematics of the single-particle form factor(s), appearing in the definition of w, adjacent to the pole on shell; however we leave the kinematic tensors K j off shell. We keep the on-shell kernels multiplying the pole term and absorb all remaining contributions into a new smooth kernel which we define as W Lj0 , where L denotes that the current absorbed was from the left. These operations are summarized by the on-shell rule, diagrammatically shown in Fig. 10,
iw A ðp 0 f ;p 0 i ÞiΔ 2 ðp 0 i ÞiK 0 ðp 0 ;pÞ ¼ iw A on ðp 0 f ;p 0 i ÞiD 2 ðp 0 i Þ X lm ffiffiffiffiffi ffi 4π p Y lm ðp 0⋆ i Þ p 0⋆ i q ⋆ i l iK 0;lm ðpÞ þiW A Lj0 ðP f ;p 0 ;P i ;pÞ;ð81Þ
where we note that the final state for the kernel K 0 has an additional barrier factor in its partial wave expansion as compared to Eq. (49). This factor ensures that no spurious threshold singularities arise since the partial-wave kernel behaves like K 0;lm ðpÞ ∼ q ⋆l i , the relative momentum defined in Eq. (3) evaluated at s i , while the spherical harmonics have a ðp 0⋆ i Þ −l behavior. In order to elucidate the validity of this expression, we note that the partial-wave decomposition of the off-shell kernel can be written as
i Þ ffiffiffiffiffi ffi 4π p Y Ã lm l ðp ⋆ i Þ; ¼ X l 0 m l 0 ;lm l Y l 0 m l 0 ðp 0⋆ i ÞiK 0;l 0 m l 0 ;lm l ðs i Þ ffiffiffiffiffi ffi 4π p Y Ã lm l ðp ⋆ i Þ þ X l 0 m l 0 ;lm l Y l 0 m l 0 ðp 0⋆ i Þ q ⋆ i p 0⋆ i l 0 iK 0;l 0 m l 0 ;lm l ðp 02 i ; s i Þ − iK 0;l 0 m l 0 ;lm l ðs i Þ ffiffiffiffiffi ffi 4π p Y Ã lm l ðp ⋆ i Þ;ð82Þ
where in the first equality we have made the dependence on p 02 i explicit for the off-shell kernel. In the last equality, we have added and subtracted the value of K 0;lm;l 0 m 0 when all particles are placed on shell. The last term vanishes in the on-shell limit and does not have threshold singularities.
Substituting this into Eq. (80) and simplifying using Eq. (44), we get
iw A ðp 0 f ; p 0 i ÞiΔ 2 ðp 0 i ÞiMðp 0 ; pÞ ¼ iw A on ðp 0 f ; p 0 i ÞiD 2 ðp 0 i ÞiMðp 0⋆ i ; pÞ þ iW A Lj0 ðP f ; p 0 ; P i ; pÞ þ ξ Z d 4 k ð2πÞ 4 iW A Lj0 ðP f ; p 0 ; P i ; kÞiΔ 1 ðkÞiΔ 2 ðk i ÞiMðk; pÞ:ð83Þ
Here, M indicates that the partial wave expansion of the 2 → 2 amplitude contains a barrier factor [35,36]. In this case, the partial-wave projection looks like
Mðp 0⋆ i ;p i Þ ¼ ffiffiffiffiffi ffi 4π p X l 0 ;m l 0 X l;m l Y l 0 m l 0 ðp 0⋆ i ÞM l 0 m l 0 ;lm l ðs i ÞY Ã lm l ðp ⋆ i Þ:ð84Þ
From here on, all amplitudes which contain the overline follow this convention. Following similar manipulations for the second term in iW 1B , we obtain
iMðp 0 ; pÞiΔ 2 ðp f Þiw A ðp f ; p i Þ ¼ iMðp 0 ; p ⋆ f ÞiD 2 ðp f Þiw A on ðp f ; p i Þ þ iW A 0jR ðP f ; p 0 ; P i ; pÞ þ ξ Z iMðp 0 ; kÞiΔ 1 ðkÞiΔ 2 ðk f ÞiW A 0jR ðP f ; k; P i ; pÞ;ð85Þ
where W 0jR is a new nonsingular kernel arising from the on-shell projection with the current on the right.
For the final term in Eq. (73), we first use Eq. (44) and look at the triangle diagram with K 0 kernels on each side. In Appendix A 2, we consider this loop in detail. We show that this can be written in terms of three singular pieces and a new nonsingular contribution, which we label W 0jCj0 . Using the expression derived in Appendix A 2, Eq. (A19), as well as Eq. (81) and the equivalent for the iK 0 iΔ 2 iw case, we find
Z d 4 kð2πÞ
Equation (86) defines a new short-distance kernel W 0jCj0 which absorbs all off-shell contribution. We also see two contributions from a two particle cut, and associated kernels W Lj0 and W 0jR , which are the same kernels we found in Eqs. (83) and (85). This decomposition is illustrated in Fig. 11. The final term of Eq. (86) involves G, the triangle function, which is a purely kinematic function given in Eq. (12). In Appendix A 2, we present two different ways of defining this function. The first follows from Ref. [36], and the second follows from the Cutkosky rules [54]. The difference between these two is a smooth function and can be absorbed into the definition of W 0jCj0 .
Summing back the rescattering contributions using Eq. (44), and combining the result with Eqs. (83) and (85), we on-shell project the remaining rescattering effects on the kernels W Lj0 , W 0jR , and W 0jCj0 in a similar manner to the derivations of the preceding section. We arrive at the on-shell form for W 1B ,
iW A 1B ðP f ; p 0 ; P i ; pÞ ¼ iw A on ðp 0 f ; p 0 i ÞiD 2 ðp 0 i ÞiMðp 0⋆ i ; pÞ þ iMðp 0 ; p ⋆ f ÞiD 2 ðp f Þiw A on ðp f ; p i Þ þ X l 0 ;m l 0 X l;m l M l 0 m l 0 ðp 0 Þ X j if j ðQ 2 ÞG A j;l 0 m l 0 ;lm l ðP f ; P i ÞM lm l ðpÞ þ 4π X l 0 ;m l 0 X l;m l Y l 0 m l 0 ðp 0⋆ f Þ½1 þ iM l 0 ðs f Þρ 0 iW A 1B;l 0 m l 0 ;lm l ðP f ; P i Þ½1 þ ρ 0 iM l ðs i ÞY Ã lm l ðp ⋆ i Þ;ð87Þ
where W 1B is a new kernel which is defined as
W A 1B ¼ W A ∞jCj∞ þ W A Lj∞ þ W A ∞jR ;ð88Þ
in which each of these kernels are defined to be fully dressed by the two-body kernels K 0 which resulted from the on-shell projection, e.g., as in Eq. (75). Both scattering amplitudes in the third term have on-shell kinematics for the intermediate states, and the last term has an identical structure as in Eq. (77), which we will exploit in the next section.
Full on-shell result for W
Finally, we can combine Eqs. (77) and (87) into our on-shell expression for W,
iW A 1B ðP f ; p 0 ; P i ; pÞ ¼ iw A on ðp 0 f ; p 0 i ÞiD 2 ðp 0 i ÞiMðp 0⋆ i ; pÞ þ iMðp 0 ; p ⋆ f ÞiD 2 ðp f Þiw A on ðp f ; p i Þ þ X l 0 ;m l 0 X l;m l M l 0 m l 0 ðp 0 Þ X j ½if j ðQ 2 ÞG A j;l 0 m l 0 ;lm l ðP f ; P i ÞM lm l ðpÞ þ 4π X l 0 ;m l 0 X l;m l Y l 0 m l 0 ðp 0⋆ f Þ½1 þ iM l 0 ðs f Þρ 0 iW A l 0 m l 0 ;lm l ðP f ; P i Þ½1 þ ρ 0 iM l ðs i ÞY Ã lm l ðp ⋆ i Þ;ð89Þ
where W ¼ W 1B þ W 1B . The first two terms represent the case where the current probes an external particle, which yields a kinematic divergence from the pole term of the propagator. As these terms do not yield physics involving short-range twobody dynamics, we define a divergence-free amplitude, W df , which removes these two pole contributions,
iW A df ðP f ; p 0 ; P i ; pÞ ≡ iW A ðP f ; p 0 ; P i ; pÞ − iw A on ðp 0 f ; p 0 i ÞiD 2 ðp 0 i ÞiMðp 0⋆ i ; pÞ − iMðp 0 ; p ⋆ f ÞiD 2 ðp f Þiw A on ðp f ; p i Þ:ð90Þ
It is worth remarking that W df is the amplitude which naturally appears in the formalism for finite-volume two-body matrix elements of local currents [35,36]. Of course, one can use the identity (90) to obtain the full W amplitude. Removing external leg contributions, which solely depend on previously known dynamical functions, we arrive at the expression, FIG. 11. Decomposition of triangle diagram into on-shell pieces involving new kernels W 0jCj0 , W Lj0 , W Rj0 . iW A df;l 0 m l 0 ;lm l ðP f ;P i Þ
¼ 1 1 − K l 0 ðs f Þiρ 0 iW A l 0 m l 0 ;lm l ðP f ;P i Þ 1 1 − iρ 0 K l ðs i Þ þ M l 0 ðs f Þ X j if j ðQ 2 ÞG A j;l 0 m l 0 ;lm l ðP f ;P i ÞM l ðs i Þ:ð91Þ
In arriving at this expression, we have partial-wave projected W df using the expansion Eq. (79), and we used Eq. (1) for the initial and final state scattering amplitudes. As was the case for H in Sec. IV B, W contains potential K matrix poles. Because W depends on the energy of the initial and final states, it can have K matrix poles associated with both of these states. To make this explicit, we introduce one final parametrization, which closely mirrors Eq. (64),
W A l 0 m l 0 ;lm l ðP f ; P i Þ ¼ K l 0 ðs f ÞA 22;l 0 m l 0 ;lm l ðP f ; P i ÞK l ðs i Þ:ð92Þ
Combining this with the previous equation, we arrive at our final result,
iW A df;l 0 m l 0 ;lm l ðP f ; P i Þ ¼ M l 0 ðs f Þ iA A 22;l 0 m l 0 ;lm l ðP f ; P i Þ þ X j if j ðQ 2 ÞG A j;l 0 m l 0 ;lm l ðP f ; P i Þ M l ðs i Þ;ð93Þ
where we have made explicit all partial wave indices. This agrees with Eq. (11) when considering that only particle 2 couples to the current.
Generalizations for two charged species and identical particles
We can easily generalize the above relations for the case where both particles are charged under interaction of the external current. First, consider the case where the particles are distinguishable. Each particle has an associated form factor f j;α where α ¼ 1 or 2, and an associated kernel w α given by Eq. (70). Our starting all-orders equation Eq. (73) is augmented with the addition of two more terms where the external leg of particle 1 is probed by the current and an additional triangle diagram. These additional terms carry through the same on-shell projections as before, and we have the amplitude, iW A ðP f ; p 0 ; P i ; pÞ
¼ X fiw on iDiMg þ iW A df ðP f ; p 0 ; P i ; pÞ;ð94Þ
where W df now has the form, as a matrix in angular momentum space,
iW A df ðP f ; P i Þ ¼ Mðs f Þ iA 22 ðP f ; P i Þ þ X j if j;1 ðQ 2 ÞG A j;1 ðP f ; P i Þ þ X j if j;2 ðQ 2 ÞG A j;2 ðP f ; P i Þ Mðs i Þ;ð95Þ
where G j;2 is the kinematic triangle function associated with particle 2, as defined in Eq. (12), and G j;1 is the triangle function where particle 1 is switched with particle 2.
In the case of identical particles, there are still four terms in Eq. (94) associated with the current probing the external leg. In Eq. (95), however, there is only one triangle function contribution.
Arbitrary number of channels
For multiple two-particle scattering channels, W df , M, A 22 , and P j f j G j become matrices in channel space, where the latter is a dense matrix if the current allows for species transmutation. Suppressing the angular momentum indices, one finds that the coupled-channel W df takes the form, The definition of the other labels, like j and A, carry through from the previous expressions. In practice, for a specified current and in/out states, these expressions can be simplified further. This is the most general form we obtain for Eq. (11).
iW A df;a;b ðP f ; P i Þ ¼ X N ch c;c 0 ¼1 M a;c 0 ðs f Þ iA A 22;c 0 ;c ðP f ; P i Þ þ X 2 α¼1 X j if j;
Comparison with existing formalism
Having completed the derivation of our main result, Eq. (11), it is worth remarking on the comparison of this equation with existing formalism for 2 þ J → 2 amplitudes. In short, previous work, which includes Refs. [3,47,48,55], 9 considered the constructions of these amplitudes for a specific channel using a well motivated EFT. Furthermore, in all of these, the calculation was 9 For a study in 1 þ 1D of the elastic form factors of resonance, we point the reader to Ref. [56]. naturally performed to a finite order in the EFT expansion. This is to say, the goals and results of these studies were quite different than the one presented here.
Nevertheless, what is true is that because our result is the most general to date, one can cast it in the form of previous existing formalism, while the converse is not true. As an illustrative example, we consider the result presented in Ref. [3]. This study used next-to-leading (NLO) order χPT to determine the ππ þ J → ππ scattering amplitude in the presence of an external scalar current. In order to assure that these amplitudes satisfy unitarity nonperturbatively, the authors follow a unitarization procedure where the s-channel diagrams are effectively upgraded in the power counting and summed to all orders. Special attention was placed on the scalar/isoscalar channel, where the σ=f 0 ð500Þ resonance resides. By analytically continuing to the σ=f 0 ð500Þ pole, they were able to constrain the scalar radius of this state to be hr 2 i ¼ ð0.19 AE 0.02Þ − ið0.06 AE 0.02Þ fm 2 , suggesting it is a compact state.
The most general form of their result is in Eq. (71) of Ref. [3]. The notation used in that reference is slightly different than the one adopted here. Therefore to make a comparison we first provide a relationship between the notations. The amplitude the authors considered, which they label T S , does not include the single-particle poles. As a result, this can be interpreted as the analogue to W df . The denominator of this has the same denominator as Mðs f ÞMðs i Þ. More specifically the denominator of their expression can be written as V NLO ðs f ÞV NLO ðs i Þ=½Mðs f ÞMðs i Þ, where V NLO is the Bethe-Salpeter kernel obtained at NLO in χPT.
Within this scheme, the authors define the numerator at NLO in χPT. We label the numerator of Eq. (71) of Ref. [3] as N NLO , instead of W to avoid confusion. Given this, we can equate our main result Eq. (11) to Eq. (71) of Ref. [3], to find
ðiA 22 þ if · GÞ ≈ V −1 NLO · N NLO · V −1 NLO :ð97Þ
Here we have dropped the kinematic arguments and the spherical harmonic indices of the functions. This is an approximate relation due to the fact that the right-hand side is only defined perturbatively. Nevertheless, as illustrated in Ref. [3] going to NLO in the chiral expansion assures that N NLO will encode the triangle singularity. In other words, the right-hand side of Eq. (97) has the same singularities as the left-hand side. This example illustrates that given one EFT calculated to a sufficiently high order, one can recover the key features of our main result. Of course, the nonsingular contributions and the exact prefactor of the singularities are in general nonperturbative dynamical functions. Presently the only tool available for obtaining these is lattice QCD following the formalism and procedure outlined in Refs. [35,36].
V. CONCLUSION
We have presented a model independent on-shell decomposition for transition amplitudes of two hadrons interacting with an external current. Building off the known result for transitions involving 1 þ J → 2 processes, we sum to all orders in the strong interaction, while working to leading order in the current insertion, to find an on-shell representation for 2 þ J → 2 scattering. The result is valid for any number of channels involving two spinless hadrons which couple to an arbitrary partial wave, as well as any structure for the external current. Comparing to standard 2 → 2 or 1 þ J → 2 processes, the 2 þ J → 2 transition amplitude contains, as well as the two-particle branch cut, an additional singular structure in the form of the triangle function. The triangle function induces additional singularities stemming from on-shell intermediate states where one of the particles interacts with the external current.
We showed, given the on-shell 2 þ J → 2 transition amplitude, we can rigorously define resonance form factors by analytically continuing the amplitude to the unphysical Riemann sheet in both the initial and final state two-particle energies. The transition amplitudes presented connect to the previously studied finite-volume formalism [35,36] which links to matrix elements calculated with lattice QCD. This allows us to ascertain structural information of resonant states, such as charge radii, in an EFT independent way.
We close by remarking that this work provides a key step towards understanding the analytic decomposition of more complicated transition amplitudes. In particular, one class of amplitudes that are pressing are those involving twocurrent insertions, "in" þ J A → "out" þ J B . These play an important role in our understanding of phenomena ranging from the nature of low-lying QCD states to extensions of the Standard Model. For instance, there is a growing demand to have reliable estimates of nuclear matrix elements pertinent for neutrino-less double beta decay [57]. 10 As one would expect, the analytic structure of "in" þ J A → "out" þ J B amplitudes will, in general, depend on the amplitudes for the allowed subprocesses. Depending on the nature of the "in"/ "out" states, these subprocesses will include those described by M, H, and W df . This explains the claim made that understanding the analytic structure of W df , among other things, is a necessary step towards constraining the aforementioned processes.
ACKNOWLEDGMENTS
The authors would like to thank J. Dudek for useful comments on the manuscript, as well as M. Albaladejo, G. Blume, R. Edwards, M. Hansen, L. Leskovec, 10 For recent proposals and a detailed discussion for studying such amplitudes via lattice QCD, see Refs. [43,[58][59][60][61].
APPENDIX A: LOOP DIAGRAMS
The derivations considered in this work rely on the correct identification of the analytic structure of the s-channel intermediate loop diagrams that contribute to the nonanalytic part of the amplitudes. In this Appendix we review the origin of these contributions.
Bubble loop
We begin with the loop with no current insertions of Eq. (48). This is the diagram that visually resembles a bubble on the left-hand side of Fig. 6. It is well known that this loop leads to square-root singularities at the twoparticle thresholds. We provide a derivation of this result for two reasons. The first is completeness. The second is the fact that the techniques used for the bubble diagram will also be used for the more complicated triangle diagram below.
We will use generic end cap functions LðP; kÞ and RðP; kÞ, that can represent any of the different kernels used in our derivations. These are smooth for real energies and kinematics where two-particle states can go on shell. Away from this limited region, they can have singularities, in particular associated with other thresholds. Here, P denotes the total momentum of the system and k is the momentum of one of the internal particles, this is illustrated in Fig. 6. Furthermore, in general these will depend on the momenta of external states, which can be off shell. Because the momenta of the external legs play no role in the expressions that follow, they will be left implicit.
Using these end caps, the loop in full notation is written as
I 0 ðPÞ ¼ ξ Z d 4 kð2πÞ
where the propagators iΔ α ðkÞ and the symmetry factor ξ are described in the main text. Our goal is not to provide an exact form of this integral, but rather to isolate its singular piece. To be able to evaluate the integral exactly would require having a closed form for the end caps, which is not the case. Instead we recognize that the singularities of these diagrams emerge from intermediate particles going on shell. As a result, the singular contribution will only depend on the end caps evaluated on shell. With this in mind, we define partialwave projected on-shell end caps that contain barrier factors, so as to avoid spurious singularities from the spherical harmonics,
iLðP; k ⋆ Þ ≡ X l;m l iL lm l ðPÞY Ã lm l ðk ⋆ Þ; iRðP; k ⋆ Þ ≡ X l;m l Y lm l ðk ⋆ ÞiR lm l ðPÞ;ðA2Þ
where the function Y lm l is defined in Eq. (13). For values of k ⋆ putting the particles inside the loop on shell, these quantities are equal to the on-shell L and R respectively. The decomposition prescription of the off-shell kernels that we adopt in this case is
iLðP; kÞ ¼ iLðP; k ⋆ Þ þ ½iLðP; kÞδ; iRðP; kÞ ¼ iRðP; k ⋆ Þ þ δ½iRðP; kÞ;ðA3Þ
where in the first term on the rhs of each equation we have the kernel with the legs next to the loop evaluated on shell, i.e., k 2 ¼ m 2 1 and ðP − kÞ 2 ¼ m 2 2 . The terms with a δ operator next to them are inspired by the notation introduced in Ref. [35], in this case they will vanish when both internal legs of the loop are on shell and therefore will be proportional to at least one factor of ðk 2 − m 2 1 ÞððP − kÞ 2 − m 2 2 Þ. This means that a term with a δ operator times the propagators will be smooth. Furthermore, in order to obtain the singular contribution of the integral, we just need to consider the term with the propagators replaced by their singular pieces,
I 0 ðPÞ ¼ X l;m l ;l 0 ;m 0 l iL lm l ðPÞξ Z d 4 k ð2πÞ 4 ðY Ã lm l ðk ⋆ Þ × iD 1 ðkÞiD 2 ðP − kÞY l 0 m 0 l ðk ⋆ ÞÞiR l 0 m 0 l ðPÞ þ δI 0 ðPÞ;ðA4Þ
where δI 0 is purely smooth in the kinematic region of interest.
The singular contribution can be isolated from the first term above in a few different ways. First, one could evaluate the k 0 integral using Cauchy's theorem by closing the contour from below. The singular contribution would come from the pole at ω k1 ≡ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k 2 þ m 2 1 p in iD 1 ðkÞ.
The remaining three-dimensional integral can be written in spherical coordinates. The singular piece arises from the pole of the remaining propagator. By picking this contribution, one can evaluate the remaining spherical integral using the orthogonality relations of the spherical harmonics to find
I 0 ðPÞ ¼ X l;m l iL lm l ðPÞ ξq ⋆ 8π ffiffi ffi s p iR lm l ðPÞ þ δI 0 0 ðPÞ; ¼ X l;m l iL lm l ðPÞρ 0 iR lm l ðPÞ þ δI 0 0 ðPÞ;ðA5Þ
where δI 0 0 contains all the nonsingular terms and ρ 0 is the two-body phase space factor defined in Eq. (2). The application of this result is written diagrammatically in Fig. 6.
Alternatively, one can obtain the discontinuity using Cutkosky rules [54], which amount to replacing the propagators with Dirac delta functions,
iD 1 ðkÞ → 2πδðk 2 − m 2 1 Þθðk 0 Þ; iD 2 ðP − kÞ → 2πδððP − kÞ 2 − m 2 2 ÞθðP 0 − k 0 Þ:ðA6Þ
After doing the substitution, carrying out two integrals with the help of the Dirac deltas, and performing the remaining angular integration we obtain the well-known discontinuity of the bubble loop, We close by remarking that although we have in fact isolated the singular piece of the bubble diagram, which was our goal, there is a freedom as to how ρ is defined. For example, one can could shift it by an overall real function, while simultaneously redefining δI 0 0 , such that I 0 is unchanged. One example of an alternative definition of ρ includes the frequently used Chew-Mandelstam phase space [62]. In the following section we will make this freedom explicit when defining the triangle singularity.
Triangle loop
Let us move on to the loop integral of Eq. (86) that features a current insertion in one of the internal legs, the so-called triangle diagram,
I 1 ðP f ; P i ; Q 2 Þ ¼ Z d 4 kð2πÞ
with Q 2 ¼ −ðP f − P i Þ 2 , and where we are using the same shorthand notations k i ≡ P i − k, k f ≡ P f − k as in the main text. Again, we have left out the total energy dependence of the end caps to simplify the notation, but it should be remembered that the current can insert momentum and the initial (final) four-momentum is P i (P f ).
Although it is not absolutely necessary, we have simplified matters by assuming that the presence of the current only leads to elastic processes. In other words, the states with four-momenta k i and k f have the same mass. In general, this does not have to be the case, even when considering the insertion of an electromagnetic current, but this can be readily generalized.
To extract the nonanalytic behavior we will implement the second strategy that we used for the bubble loop. We perform the same separation of the on-shell part of the end caps as in Eq. (A2) with the only modification that these now depend on different total momenta,
iLðP f ; k ⋆ f Þ ≡ X l;m l iL lm l ðP f ÞY Ã lm l ðk ⋆ f Þ;ðA9ÞiRðP i ; k ⋆ i Þ ≡ X l;m l Y lm l ðk ⋆ i ÞiR lm l ðP i Þ;ðA10Þ
where the subscript of the vectors indicates the frame where it is evaluated, since the initial and final CM frames can be different for a nonzero value of the current momentum insertion. The on-shell decomposition prescription follows from Eq. (A3) with the additional subscripts for the initial and final states,
iLðP f ; kÞ ¼ iLðP f ; k ⋆ f Þ þ ½iLðP f ; kÞδ; iRðP i ; kÞ ¼ iRðP i ; k ⋆ i Þ þ δ½iRðP i ; kÞ:ðA11Þ
In Ref. [36] it was realized that it is more convenient for the δ operator to not act on the whole current insertion w, but only on the scalar form factors that contain the dynamics, such that we will need the Lorentz decomposition,
iwðk f ; k i Þ ¼ X j K j ðk f ; k i Þif j ðQ 2 ; k 2 f ; k 2 i Þ;ðA12Þ
where K j are kinematic known functions of P f , P i and k that depend on the Lorentz structure of the current insertion. The on-shell expansion of these form factors is
if j ðQ 2 ;k 2 f ;k 2 i Þ ¼ if j ðQ 2 Þ þ δ½if j ðk 2 f ;Q 2 Þ þ ½if j ðQ 2 ;k 2 i Þδ þ δ½if j ðk 2 f ;k 2 i Þδ;ðA13Þ
where the first term is the on-shell one-particle form factor, i.e., the form factor evaluated at k 2
f ¼ k 2 i ¼ m 2 2 .
The rest are terms that contain the off-shell behavior but vanish for values of k when the initial or final state on-shell conditions are met, depending whether the term is acted by a δ operator from the left or the right, or both. The quantities, divergences in the imaginary part, while a step shows up in the real part at the same energy. The imaginary part coincides exactly to what is found by evaluating explicitly G, while the real part only differs by smooth functions. This can be verified, for instance, by comparing it to Fig. 7 of Ref. [36] where the loop integral was calculated with Feynman parameters and a final one-dimensional numerical integration. The case s i ¼ s f ¼ s is found as the limit of Eq. (A44) to be SingG 00;00 ðsÞ ¼
i 32π ffiffi ffi s p q ⋆ s − m 2 2 þ m 2 1 s :ðA45Þ
The other case of interest is the vector current insertion with K j ¼ k μ . When thinking of the initial and final states to be in an S wave, one needs B μ;00;fðiÞ 00;00
¼ ½Λ −βi 32π ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðP f · P i Þ 2 − P 2 i P 2 f q log 1 þ z ⋆ f þ iϵ 1 − ðz ⋆ f þ iϵÞ ½Λ −β f μ 0 ω ⋆ q1;f þ q ⋆ f ½Λ −β f μ k ½P ⋆ i;f k z ⋆ f þ log 1 þ z ⋆ i þ iϵ 1 − ðz ⋆ i þ iϵÞ ½Λ −β i μ 0 ω ⋆ q1;i þ q ⋆ i ½Λ −β i μ k ½P ⋆ f;i k z ⋆ i − 2q ⋆ f ½Λ −β f μ k ½P ⋆ i;f k − 2q ⋆ i ½Λ −β i μ k ½P ⋆ f;i k :ðA48Þ
In the limit of P f ¼ P i ¼ P the above function simplifies to SingG μ 00;00 ðPÞ ¼ iP μ 16πs 3=2 q ⋆ ð2q ⋆2 þ m 2 1 Þ: ðA49Þ
APPENDIX B: ANALYTIC CONTINUATIONS TO UNPHYSICAL SHEETS
In this Appendix, we review the analytic continuation of a two particle scattering amplitude to the unphysical sheet and illustrate how the procedure extends to the transition amplitudes. Physical scattering amplitudes on the real s axis are boundary values of an analytic function, which has a discontinuity across the branch cut given by the unitarity relation Eq. (56). Therefore, we can formally define the second sheet amplitude by continuing through the branch cut, cf. Eq. (56), using the boundary condition,
M II ðs AE Þ ¼ Mðs ∓ Þ;ðB1Þ
where we have defined the short-hand notation s AE ¼ s AE iϵ and assume that ϵ → 0 þ . Using this short-hand, the unitarity relation can be expressed as
Mðs þ Þ − Mðs − Þ ¼ 2iρðs þ ÞMðs − ÞMðs þ Þ;ðB2Þ
where we have used the Schwartz reflection principle M à ðsÞ ¼ Mðs à Þ. For technical convenience, we choose to continue the amplitude to the second sheet in the upperhalf s plane, i.e., M II ðs þ Þ ¼ Mðs − Þ. We then use the Schwartz reflection principle to extend the result to the lower-half s plane, which is nearest to the physical region assuming the usual þiϵ prescription. The result is identical if one chooses to continue to the lower-half plane directly; however, we find this approach convenient to simplify the later derivation for the 2 þ J → 2 amplitude. Assuming a Ej/m continuation to the upper-half plane, we now insert Eq. (B1) into (B2), and solve for M II to find
M II ðsÞ ¼ 1 1 þ 2iρðsÞMðsÞ MðsÞ;ðB3Þ
where we have used Cauchy's theorem to extend the domain from near the real axis to the upper-half complex plane. One can make similar arguments for the lower-half plane, with an additional boundary condition for the phase space factor ρðs þ Þ ¼ −ρðs − Þ, finding the same form as Eq. (B3). Alternatively, we may analytically continue the on-shell form Eq. (1) directly, recognizing that the nonanalyticity arises solely from the phase space factor. Continuing this to the second sheet via the relation ρ II ¼ −ρ, which is due to the square root branch cut, then we recover directly Eq. (17) which agrees with Eq. (B3) when Eq. (1) is substituted.
The analytic continuation for the 1 þ J → 2 amplitude follows the same procedure as for the hadronic amplitude. Focusing on the case for scalar currents with the hadrons in S wave, the on-shell form Eq. (6) has a branch cut due to the hadronic scattering amplitude. To obtain the resonance form factors, we fix Q 2 to be real and analytically continue only in the s-plane. The transition amplitude has the same boundary condition as Eq. (B1), i.e., H II ðs þ ; Q 2 Þ ¼ Hðs − ; Q 2 Þ. From Eq. (6), we find that the analytic continuation to the unphysical sheet simply requires continuing M, which gives Eq. (19). As with the 2 → 2 amplitude, this can also be seen by using the unitarity relation, ImHðs; Q 2 Þ ¼ M Ã ðsÞρðsÞHðs; Q 2 Þ; ðB4Þ followed by writing the imaginary part as the discontinuity and imposing the boundary condition. We find that the second sheet 1 þ J → 2 amplitude takes the form, H II ðs; Q 2 Þ ¼ Hðs; Q 2 Þ − 2iM II ðsÞρðsÞHðs; Q 2 Þ; ðB5Þ
which after substitution of Eq. (B3) and using the on-shell form Eq. (6), we recover Eq. (19).
In the case of the 2 þ J → 2 amplitude W, we have to analytically continue both the initial and final state invariant mass squares s i and s f , respectively. It is sufficient to consider W df since this is the only contribution which can have both initial and final state resonance poles. Since both variables are continued, we impose the boundary condition, which shows there is an additional singular term arising from the triangle function. This additional term implies that we cannot just continue both the external M functions in Eq. (11), but that we also need to continue G. We first write the imaginary part as the difference, We now impose the boundary conditions Eqs. (B1) and (B6), again continuing to the upper-half planes, the 2 þ J → 2 amplitude on the second sheets is given by where the term in brackets is precisely the triangle function on the unphysical sheets as presented in Eq. (24), using similar arguments as above. As claimed in Sec. III, this gives the analytic continuation of W df on to the second Riemann sheets in both variables, Eq. (23). We comment that a similar procedure holds for arbitrary currents with the two hadrons in an arbitrary partial wave, noting that the Lorentz structure does not introduce any physical singularities in the s i =s f planes.
FIG. 2 .
2(a) The relation between the full 2 þ J → 2 amplitude and W df as defined in Eq.(8). (b) Depicted are all allowed current insertions over the external legs. The symbol "Σ" is used as a shorthand to express this sum. The dotted lines represent the pole piece D of the propagator, the other objects are defined inFig. 1.
FIG. 3 .
3Triangle-function contribution to W df , Eq. (11), written in terms of the single-hadron form factors (f j ) and the triangle loops (G j ) defined in Eq.(12). The gray circle and dashed lines were defined inFigs. 1 and 2,respectively. The open semicircles represent the modified spherical harmonics, defined in Eq. (
FIG. 4 .
4(a) 2 → 2, (b) 1 þ J → 2, and (c) 2 þ J → 2 amplitudes near the resonance pole, s R . The residues at the pole define the coupling, c, transition form factor, f 1→R , and elastic resonance form factor, f R→R .
The on-shell amplitude is recovered by placing the external legs on shell, Mðp 0⋆ ;p ⋆ Þ ¼ Mðp 0
×
(44) once by substituting the relation into itself, givingFIG. 5. (a) Self-consistent integral equation for the hadronic scattering amplitude M as given in Eq. (44). (b) Examples of diagrams contributing to the Bethe-Salpeter kernel, K 0 , which contains all s-channel two-particle irreducible diagrams. = P-p'xP-:_p = ):( = ):( => X , -:;ec_ iMðp 0 ; pÞ ¼ iK 0 ðp 0 iΔ 2 ðP − k 0 ÞiK 0 ðk 0 ; kÞiΔ 1 ðkÞ × iΔ 2 ðP − kÞiK 0 ðk; pÞ:
which is evaluated at s. The quantities K 0 ðp 0 ;k ⋆ Þ and K 0 ðk ⋆ ; pÞ are the kernels where the intermediate state is projected on shell. The intermediate states are further decomposed into partial waves K 0;lm l ðp 0 Þ and K 0;lm l ðpÞ defined asK 0 ðk ⋆ ; pÞ ¼ ffiffiffiffiffi ffi 4π p X l;m l Y l;m l ðk ⋆ ÞK 0;lm l ðpÞ; ð49Þ K 0 ðp 0 ;k ⋆ Þ ¼ ffiffiffiffiffi ffi 4π p X l;m l K 0;lm l ðp 0 ÞY Ã l;m l ðk ⋆ Þ;ð50Þwhere l is the angular momentum between the two particles defined in their CM frame, and m l is its projection onto some fixed axis. Note that we only projected the intermediate states of each kernel on shell, leaving the external kinematics off their mass-shell until after we iterated over all loops in the integral equation.Applying this loop identity, and combining terms using Eq.(44), M becomes iMðp 0 ; pÞ ¼ iK 0 ðp 0 ; pÞ þ X l;m l iM lm l ðp 0 Þρ 0 iK 0;lm l ðpÞ þ iK 1 ðp 0 ; pÞ þ ξ Z d 4 k ð2πÞ 4 iMðp 0 ; kÞiΔ 1 ðkÞ × iΔ 2 ðP − kÞiK 1 ðk; pÞ;
4 iK 0 ðp 0 ; kÞiΔ 1 ðkÞiΔ 2 ðP − kÞiK j ðk; pÞ ¼ iK jþ1 ðp 0 ; pÞ þ X l;m l iK 0;lm l ðp 0 Þρ 0 iK j;lm l ðpÞ; ð52Þwhich is structurally identical to that of Eq.(48), except for the involvement of the jth and (j þ 1)th kernels. The iterated loop identity is shown diagrammatically inFig. 6. Repeated use of Eqs.(44) and(52)to all orders allows us to write M as iMðp 0 ; pÞ ¼ lm l ðp 0 Þρ 0 iK lm l ðpÞ;
FIG. 6 .
6Iterated loop identity as given in Eq.(52), where the open circle is the Bethe-Salpeter kernel K 0 , the box with label j is the jth kernel, and the dashed line indicates the two-particle phase space cut which places intermediate state particles on shell.
4 ×
4iM a;c ðp 0 ; kÞiΔ 1c ðkÞiΔ 2c ðP − kÞ × iK 0;c;b ðk; pÞ;ð57Þ
;c ðsÞρ c;d iK d;b ðsÞ; ð58Þ
FIG. 7 .i
7Diagrammatic representation of the 1 þ J → 2 all-orders equation defined in Eq. (59), where the open circle representing H 0 contains all two-particle irreducible diagrams in the s f ¼ P 2 f channel. 'I-/.A= Pi<-P; = jr-+ )()--p'
FIG. 8 .
8On-shell projection of H, where the left-hand side is Eq. (59) and the right hand side is Eq. (61). The white box indicates the real function H.
4 iMðp 0 ; kÞiΔ 1 ðkÞiΔ 2 ðk f ÞiW A 0j0 ðP f ; k; P i ; pÞ ¼ iW A ∞j0 ðP f ; p 0 ; P i ; pÞ þ X l;m l iM l ðp 0 Þρ 0 iW A ∞j0;lm l ðP f ; P i ; pÞ;ð74Þ
4 iK 0 ðp 0 ; kÞiΔ 1 ðkÞ × iΔ 2 ðk i ÞiMðk; pÞ:
FIG. 10 .
10Diagrammatic representation of the expansion shown in Eq. (81). The dotted line in the first term represents the pole piece of the propagator, D 2 , and the second term is a new smooth kernel we define as W Lj0 .
4 iK 0 ðp 0 lm l ðP f ; p 0 ; P i Þρ 0 iK 0
000; kÞiΔ 1 ðkÞiΔ 2 ðk f Þiw A ðk f ; k i ÞiΔ 2 ðk i ÞiK 0 ðk; pÞ ¼ iW A 0jCj0 ðP f ; p 0 ; P i ; pÞ þ X l 0 ;m l 0 iK 0;l 0 m l 0 ðp 0 Þρ 0 iW A Lj0;l 0 m l 0 ðP f ; ðQ 2 ÞG A j;l 0 m l 0 ;lm l ðP f ; P i ÞK 0;lm l ðpÞ:
ðc 0 αÞ;ðcαÞ ðQ 2 ÞG A j;ðc 0 αÞ;ðcαÞ ðP f ; P i Þ M c;b ðs i Þ; ð96Þ where a=b denote the external channel space indices, and the c=c 0 indices are summing over the N ch intermediate channels. For each intermediate channel, the current can couple to two legs in the triangle diagram. This is encoded in the α index which can take on either 1 or 2.
lm l ðPÞ ξq ⋆ 8π ffiffi ffi s p θðs − s th ÞiR lm l ðPÞ; ðA7Þ which of course agrees with the result of Eq. (A5).
FIG. 13 .
13Singularities of the scalar and S wave case of the G loop integral as reproduced by the function SingG 00;00 . The value of the spatial momentum jP ⋆ f;i j ¼ ð2π=6Þm.
2iImW df ðs f ; Q 2 ; s i Þ ¼ W df ðs f;þ ; Q 2 ; s i;þ Þ − W Ã df ðs f;þ ; Q 2 ; s i;þ Þ; ¼ W df ðs f;þ ; Q 2 ; s i;þ Þ − W df ðs f;− ; Q 2 ; s i;− Þ;ðB8Þwhere in the second line we used the extension of the Schwartz reflection principle for multivariate functions, the edge-of-the-wedge theorem, to write the conjugated amplitude as a function of variables evaluated on the lowerhalf plane, i.e., W Ã df ðs f;þ ; Q 2 ; s i;þ Þ ¼ W df ðs f;− ; Q 2 ; s i;− Þ. Using the Schwartz reflection principle for the scattering amplitudes, Eqs. (B7) and (B8) give us the relation,W df ðs f;þ ; Q 2 ; s i;þ Þ − W df ðs f;− ; Q 2 ; s i;− Þ ¼ 2iMðs f;− Þρðs f;þ ÞW df ðs f;þ ; Q 2 ; s i;þ Þþ 2iW df ðs f;− ; Q 2 ; s i;− Þρðs i;þ ÞMðs i;þ Þ þ 2iMðs f;− ÞfðQ 2 ÞImGðs f;þ ; Q 2 ; s i;þ ÞMðs i;þ Þ: ðB9Þ
; P i ; kÞiΔ 1 ðkÞiΔ 2 ðk i ÞiMðk; pÞ; ð69Þ FIG. 9. Diagrammatic representations of the all-orders equations for W 1B given in Eq. (69) and W 1B in Eq. (73), where the open circle is the kernel iW 0j0 .4 iMðp 0 ; kÞiΔ 1 ðkÞiΔ 2 ðk f ÞiW A
0j0 ðP f ; k; P i ; pÞ
þ ξ
Z
d 4 k
ð2πÞ 4 iW A
0j0 ðP f ; p 0 ; P i ; kÞiΔ 1 ðkÞiΔ 2 ðk i ÞiMðk; pÞ
þ ξ
Z
d 4 k 0
ð2πÞ 4 ξ
Z
d 4 k
ð2πÞ 4 iMðp 0 ; k 0 ÞiΔ 1 ðk 0 ÞiΔ 2 ðk 0
f ÞiW A
0j0 ðP f ; k 0
and the rest of the Hadron Spectrum Collaboration for useful discussions. RAB and A. W. J. acknowledges support from U.S. Department of Energy Contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. RAB and K. H. S. acknowledge support of the US DOE Early Career Award, Contract No. DE-SC0019229. F. G. O. acknowledges support from the U.S. Department of Energy Contract No. DE-SC0018416 at William & Mary and the JSA/JLab Graduate Fellowship Program. K. H. S. acknowledges support by the U.S. Department of Energy, Office of Science Graduate Student Research (SCGSR) program. 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 No. DE-SC0014664. All opinions expressed in this paper are the authors and do not necessarily reflect the policies and views of DOE, ORAU, or ORISE.
4 iLðP; kÞiΔ 1 ðkÞiΔ 2 ðP − kÞiRðP; kÞ;
4 iLðP f ; kÞiΔ 1 ðkÞiΔ 2 ðk f Þ × iwðk f ; k i ÞiΔ 2 ðk i ÞiRðP i ; kÞ;
. ....,1
2.15
"' 0
s
-
2.10
B
0
-1
-1
E
Ef/m
-2
1.9
2.6
1.95
-3
fðiÞ Rðθ iðfÞ ; ϕ iðfÞ Þ μ ¼ ½Λ −β fðiÞ Rðθ iðfÞ ; ϕ iðfÞ Þ μ and the singularities of G μ 00;00 are captured by the function, SingG μ 00;00 ðP f ; P i Þ ¼0 ω ⋆
q1;fðiÞ ;
B
μ;10;fðiÞ
00;00
k
½ẑ k
ffiffi ffi
3
p ;
ðA46Þ
B
μ;00;fðiÞ
00;00
¼ ½Λ −β fðiÞ μ
0 ω ⋆
q1;fðiÞ ;
B
μ;10;fðiÞ
00;00
¼ ½Λ −β fðiÞ μ
k
½P ⋆
i;fðf;iÞ k
ffiffi ffi
3
p
;
ðA47Þ
W II ;
IIII df ðs f;AE ; Q 2 ; s i;AE Þ ¼ W df ðs f;∓ ; Q 2 ; s i;∓ Þ; ðB6Þ which the double superscript indicates both variables are continued to their respective second sheets. The on-shell representation Eq. (11) ensures that the imaginary part of W df takes the form, ImW df ðs f ; Q 2 ; s i Þ ¼ M Ã ðs f Þρðs f ÞW df ðs f ; Q 2 ; s i Þ þ W Ã df ðs f ; Q 2 ; s i Þρðs i ÞMðs i Þ þ M Ã ðs f ÞfðQ 2 ÞImGðs f ; Q 2 ; s i ÞMðs i Þ; ðB7Þ
W II ;
IIII df ðs f ;Q 2 ;s i Þ ¼ 1 1þ2iMðs f Þρðs f Þ W df ðs f ;Q 2 ;s i Þ½1−2iρðs i ÞM II ðs i Þ þM II ðs f ÞfðQ 2 Þ½Gðs f ;Q 2 ;s i Þ−2iImGðs f ;Q 2 ;s i ÞM II ðs i Þ; ðB10Þ where we have extended the domain from near the real axis to the entire upper-half complex planes via Cauchy's theorem, as before for the 2 → 2 amplitude. We now use the on-shell form Eq. (11), as well as (B3) to construct an on-shell form, W II;II df ðs f ; Q 2 ; s i Þ ¼ M II ðs f ÞfA 22 ðs f ; Q 2 ; s i Þ þ fðQ 2 Þ½Gðs f ; Q 2 ; s i Þ − 2iImGðs f ; Q 2 ; s i ÞgM II ðs i Þ; ðB11Þ
This procedure is followed in Ref.[3] for studying the σ as well as in Refs.[37,47,48] for theories with bound states.
Phase 3 Ej/m -zj=l -zj=-1 1 2 2.20
if j ðk 2 f ; Q 2 Þ ≡ if j ðQ 2 Þ þ δ½if j ðk 2 f ; Q 2 Þ; ðA14Þ if j ðQ 2 ; k 2 i Þ ≡ if j ðQ 2 Þ þ ½if j ðQ 2 ; k 2 i Þδ; ðA15Þwill also simplify the clutter of the derivation. The explicit dependence of these quantities makes clear that they contain on-shell and off-shell information of the initial or final particle leg respectively. Finally, as done in Ref.[35]we introduce the following shorthand notation for end caps with a current insertion in one of its external legs and the divergent piece originating from the intermediate propagator subtracted:½if j iΔ 2 iR df ðQ 2 ; P i ; kÞ ¼ if j ðQ 2 ; k 2 i ÞiΔ 2 ðk i ÞiRðP i ; kÞ − if j ðQ 2 ÞiD 2 ðk i ÞiRðP i ; k ⋆ i Þ: ðA17ÞWhen the end caps are K 0 , and each term is multiplied by K j and summed over j, these quantities correspond to the symbols W Lj0 and W 0jR introduced in Eq. (81) of the main text. Given all these definitions, we can separate the integral I 1 into terms with different analytic behavior, where terms that do not appear explicitly are smooth analytic functions of the external momenta within the kinematic region of interest. At this point there is not a unique choice about how to distill the singularities of these terms. Different choices would mean different functional forms of the analytic part of the amplitude, but all choices have to agree on the position and characteristics of any singularity in the amplitude.This freedom was emphasized at the end of the previous section. In here, we will write down two options, and the different smooth behaviors will be collected in the implicit definitions of smooth functions introduced below. Both of these prescriptions agree that this integral can be diagrammatically represented byFig. 11. The first, most conservative option, requires only the partial wave expansion of the end cap functions as we did before,where δI 1 is a residual, smooth function. The subscript in the phase space ρ indicates whether ρ 0 is evaluated in Eq. (2) with s i or s f . We have also introduced the partial-wave decomposition of the divergence-free kernels. This decomposition is similar to the ones from the simple kernels, except that we first multiply by the kinematic functions K j , and sum over all j,an analogous relation holds for ½iwiΔ 2 iR df;lm l ðP i ; Q 2 Þ. This prescription for the isolation of the triangle singularity requires the calculation of the following matrix in angular momentum space:which in general needs to be regularized for ultraviolet divergences, the needed counterterms would come from δI 1 since they have to be analytic functions. For more details about this matrix, and how to efficiently evaluate and regularize it, the interested reader is referred to Ref.[36].The second option to obtain the singularities from Eq. (A18) is to use the Cutkosky rules to extract discontinuities of the loops and verify with the Landau conditions[45]that all the singularities associated with the diagram are being described. In the case of the triangle loop, the Cutkosky rules require at least three cuts, one for each of the vertices. Since we are interested in the kinematic region where the current insertion energy is below the particle creation threshold, only two cuts contribute to the discontinuities associated with the triangle loop. These contain the branch-cuts associated with the initial/final two-particle states.As detailed discussion of Cutkosky rules can be found in[63], instead of jumping straight to the result, we provide some of the key steps that lead one to the final form. The two key conceptual points are the following. First, one must recognize that the discontinuity may only be obtained after replacing D 1 ðkÞ by its imaginary piece. Second, having made this replacement, the discontinuity can be obtained by evaluating the difference across the s i and s f branch cuts. This can be made clear by introducing explicitly the dependence of the propagators on ϵ, which will be fixed to be positive. For example, we intermediately replace D 2 ðk f Þ → D 2 ðk f ; ϵÞ and take the limit as ϵ → 0 afterwards.More explicitly, the discontinuity of the triangle loop iswhere we have used the following identity in the last step,The three-vector integral is easiest to carry out in the CM frame where the second delta Dirac of each cut imposes the on-shell condition, i.e.,After that, we are left only with the angular integral,where the dots indicate the second term, which is identical to the first one by exchanging the initial and final labels.In this integral the pole of the remaining propagator follows the principal value integration prescription. The angular dependence of the numerator can be expressed in terms of a single spherical harmonic and a set of B coefficients,where a similar decomposition but for the initial CM frame define the B JM;i coefficients. The B JM;f coefficients will include a relation from the spherical harmonics in the initial frame to 4-vectors, Lorentz boosts to the final frame, the relation from 4-vectors in the final frame to spherical harmonics, and the recombination of all the spherical harmonics into a single one. Explicit calculations of these coefficients are done in Appendix B1 of Ref.[36], they will depend only on the value of P f and P i .Note that there is a final spatial rotation Rðθ; ϕÞ ambiguity of the Lorentz vectors in the final CM frame, we will choose the Lorentz transformation that leaves the spatial part of P i pointing in the z-direction. For instance, in the case of a vector current insertion, we would use the following relationship:where the direction ðθ i ; ϕ i Þ refers to that of the spatial part of the four vector P μ i when acted by the pure boost ½Λ β f . This boost is such that when it acts on P μSimilarly, we need to make explicit the angular dependence of the remaining propagator in the respective CM frame. Since we have defined the transformation into the final CM frame such that P μ⋆ i;f ¼ ðP 0⋆ i;f ; jP ⋆ i;f jẑÞ, the propagator in this frame is equal towhere z ⋆ f , in terms of Lorentz scalars, is given byIn arriving at this result, we have two identities to rewrite P ⋆ i;f and other quantities above in a Lorentz invariant way,The quantityffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiThe azimuthal angle integral is proportional to δ M;0 since the only azimuthal dependence is contained in the spherical harmonics, while the polar angle integral is proportional to the Legendre functions of the second kind,Special care has to be taken when evaluating Q J at threshold, or when the initial and final CM frame are the same, i.e., when jP ⋆ i;f j ¼ 0.Since z ⋆ f ∝ 1=ðq ⋆ f jP ⋆ i;f jÞ, in either of these cases, the argument of Q J diverges. However it can be found, either by using the series expansion of the Legendre functions of the second type, or the expansion of the fraction in Eq. (A32) before performing the integration, thatso that even at x ¼ 0 a finite value can be assigned to the discontinuity. Putting all the terms together, we find the discontinuity of the triangle diagram to be DiscG j;lm l ;l 0 m 0 l ðP f ;In the case when the CM frame of the initial and final state coincide, i.e., β i ¼ β f , the discontinuity only depends on s i and s f , and simplifies toðA35Þ s ¼ P 2 and is equal toThe step function is able to move out of the derivative operator because the threshold behavior of B 00 j;lm l ;l 0 m 0 l is proportional to at least l þ l 0 powers of q ⋆ ; see Eq. (A26).The matrix DiscG in angular momentum space is equal to twice the imaginary part of G, modulo the phases inherited by the spherical harmonics encoded in the B coefficients. Singularities in this matrix arise from the Q J functions, which generically feature a regular polynomial part Q r J and a nonanalytic piece proportional to a Legendre polynomial of the same J,A careful inspection of the behavior of the z ⋆ variables as a function of the external kinematics, and the matrix DiscG for the general case, yields that only when z ⋆ f ¼ z ⋆ i ¼ 1 there is a logarithmic singularity, and that DiscG generically features square-root-type singularities at threshold. This is in agreement with the analysis of the Landau conditions of this diagram; see Ref.[65]for a detailed review of this procedure. What is more, the imaginary part of the loop yields all the information about its singularities, i.e., its nature and their coefficients. Once we know the coefficient of each singularity, we can make a continuation of the DiscG function to also reproduce its real part. However, given the behavior of the z ⋆ variables as a function of the external kinematics, only a specific continuation around the branch points of the Q J function is consistent. To illustrate this we will describe the behavior of the z ⋆ f variable while moving in the trajectory shown inFig. 12(a)Six points labeled A through F have been chosen on this plane. Correspondingly, the value of z ⋆ f at each of these points has been placed on the complex z ⋆ f plane onFig. 12(b). The color background on the latter figure represents the phase of the function logðð1 þ z ⋆ f Þ=ð1 − z ⋆ f ÞÞ, which generates the Riemann sheet structure to all Q J functions. In this figure we have chosen to push the branch cut of Q J , which conventionally runs from −1 to 1, to run from −1 to infinity in the negative imaginary semiplane and then come back to 1 through the positive imaginary semiplane. The branch cut is indicated by a gray dashed line.11This choice, as shown shortly, will allow the variable z ⋆ f to remain on the same sheet for the values of E ⋆ i and E ⋆ f within our kinematic region of interest. To describe the behavior of z ⋆ f let us begin in the kinematic region where E ⋆ f is below threshold. By extending the domain of q ⋆ f below threshold within the physical Riemann sheet in the s f complex plane, one sees from Eq. (A38) that z ⋆ f becomes purely imaginary and takes positive or negative imaginary values depending on the value of E ⋆ i . As a result, when moving from point F to point A, z ⋆ f will pass through zero, motivating the choice to not have a branch cut there. When moving from point A to point B, one must cross the threshold of the final twoparticle state, at which points z ⋆ f diverges; see Eq. (A38). Given that there is no branch point at infinity in the z ⋆ f plane, one should remain in the same sheet when making this move from A to B. This motivates having the negative imaginary infinity and the positive real infinity on the same side of the cut. In the trajectory BCDE there are a priori four options to go around the branch points, but only by going below the branch point at 1 and above the branch point at −1, as shown in the figure, the points E and F will be connected to form a closed trajectory. This choice around the branch points can be encoded as an addition of iϵ to the argument of Q J . As a short hand we will introduce Q c J as our choice of the analytic continuation of this function,where no absolute value of the argument of the logarithm is taken, and its range is extended into the complex plane. With this information we can describe all the singular behavior of G analytically with SingG j;lm l ;l 0 m 0 l ðP f ;in the case of arbitrary external kinematics, within our region of interest. Note the relative factor of 2 compared with the discontinuity, given in Eq. (A34). This is because, as previously mentioned, the discontinuity is equal to twice the imaginary piece of G. For the case where the CM frame of final and initial state coincide it simplifies further toSingG j;lm l ;l 0 m 0 l ðsÞ ¼ where we capture the smooth contributions in δI 0 1 . This is the main result from this section.Finally, let us calculate two explicit examples of the SingG j;lm l ;l 0 m 0 l loop. First, the case of a purely scalar kinematic function K j ¼ 1, and the S wave for both the initial and final partial waves. In that case, the integral Eq. (A22) converges and can be calculated analytically up to a single integration of a Feynman parameter. In this case the B coefficients are simply equal to B J0;f 00;00 ¼ B J0;i 00;00 ¼ δ J;0 , and the singular behavior of the triangle diagram is described by SingG 00;00 ðP f ;A plot of SingG 00;00 as a function of E ⋆ f is shown inFig. 13for multiple values of E ⋆ i and spatial momenta jP ⋆ f;i j ¼ ð2π=6Þ m. The logarithmic singularities can be seen as
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[] |
[
"Loglinear models for first-order probabilistic reasoning",
"Loglinear models for first-order probabilistic reasoning"
] |
[
"James Cussens [email protected] \nDepartment of Computer Science\nUniversity of York Heslington\nYOlO 5DDYorkUK\n"
] |
[
"Department of Computer Science\nUniversity of York Heslington\nYOlO 5DDYorkUK"
] |
[] |
Recent work on loglinear models in probabilistic constraint logic programming is applied to first order probabilistic reasoning. Probabilities are defined directly on the proofs of atomic formu lae, and by marginalisation on the atomic formu lae themselves. We use Stochastic Logic Pro grams (SLPs) composed of labelled and unla belled definite clauses to define the proof prob abilities. We have a conservative extension of first-order reasoning, so that, for example, there is a one-one mapping between logical and ran dom variables. We show how, in this framework, Inductive Logic Programming (ILP) can be used to induce the features of a loglinear model from data. We also compare the presented framework with other approaches to first-order probabilistic reasoning.
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[
"https://arxiv.org/pdf/1301.6687v1.pdf"
] | 1,816,018 |
1301.6687
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842e5e38137c3a3bf99d583137951002ec5793f4
|
Loglinear models for first-order probabilistic reasoning
James Cussens [email protected]
Department of Computer Science
University of York Heslington
YOlO 5DDYorkUK
Loglinear models for first-order probabilistic reasoning
126loglinear modelsconstraint logic program minginductive logic programming
Recent work on loglinear models in probabilistic constraint logic programming is applied to first order probabilistic reasoning. Probabilities are defined directly on the proofs of atomic formu lae, and by marginalisation on the atomic formu lae themselves. We use Stochastic Logic Pro grams (SLPs) composed of labelled and unla belled definite clauses to define the proof prob abilities. We have a conservative extension of first-order reasoning, so that, for example, there is a one-one mapping between logical and ran dom variables. We show how, in this framework, Inductive Logic Programming (ILP) can be used to induce the features of a loglinear model from data. We also compare the presented framework with other approaches to first-order probabilistic reasoning.
Introduction
A framework which merges first-order logical and prob abilistic inference in a theoretically sound and applicable manner promises many benefits. We can benefit from the compact knowledge representation of logic, and still rep resent and reason about the uncertainty found in most ap plications. Here we propose a conservative extension to the logic programming framework by defining probabilities di rectly on proofs and hence indirectly on atomic formulae. Our conservatism allows us to tie probabilistic and logical concepts very closely. Table I lists the linkages which the proposed approach establishes.
This paper is laid out as follows. We begin in Section 2, with a brief overview of logic programming concepts. Sec tion 3 forms the core of the paper where we introduce the loglinear model and Stochastic Logic Programs. Section 4 Logic Probability logical variable random variable instantiation instantiation relations joint distributions queries queries ground definitions probability tables disjunctive definitions mixture models defining relations in defining distributions In terms of other relations terms of other distributions Table I: Linking logic and probability then presents SLPs which represent Markov nets and then more complex models. Section 5 discusses the role of ILP in learning the structure of the loglinear model, focusing on the work of Dehaspe. We discuss related work in Section 6 and briefly mention future work in Section 7.
2 Logic programming essentials
We give a very brief overview of logic programming. For more details, the reader can consult any standard textbook on logic programming, e.g. (Lloyd, 1987). In this pa per we will consider only definite logic programs. Defi nite (logic) programs consist of a set of definite clauses, where each definite clause is a disjunctive first-order for mula such as p(X, Y) V --,q(X, Z) V ..,r(Z) ¢} p(X, Y) + q(X, Z) 1\ r(Z). All variables are implicitly universally quantified (we will denote variables by names starting with upper-case letters). A literal is an atomic formula (briefly atom) or the negation of an atom. Definite clauses consist of exactly one positive literal (p(X, Y)) in our example) and zero or more negative literals (such as q(X, Z) and r(Z)). The positive literal is the head of the clause and the negative literals are the body.
A goal or query is of the form +-Atom1 1\ Atom2 1\ · · · 1\ Atomn. A substitution, such as(} = {X fa, Y/Z} is a mapping from variables to first-order terms. If a substi tution maps variables to terms which do not include vari-abies, we will call it an instantiation. A substitution (} uni fies two atoms Atom1, Atom2 if Atom1(} ((} applied to Atom1) is identical to Atom29. Resolution is an infer ence rule that takes an atom Atom selected from a goal +-Atomtll· ·· /\Atom /\·· ·IIAtomn. unifies Atom with the head H of a clause H +-B using a substitution (} and returns ( +-Atom1 II · · · II B II · · · II Atomn )(} as a new goal. Note that B may be empty. With Prolog the selected atom is always the leftmost atom. An SW-refutation of a goal G is a sequence of resolution steps which produce the empty goal. The SW-tree for a goal G is a tree of goals, with G as root node, and such that the children of any node/goal G' are goals produced by one resolution step using G' (the empty goal has no children). Branches of the SLD-tree ending in the empty goal are success branches corresponding to successful refutations. The success set for a definite program is the set of all ground atoms Atom such that +-Atom has an SLD-refutation. The success set for an n-a rity predicate pjn, denoted SS(pjn), is all those atoms in the program's success set that have pin as their predicate symbol. The most general goal for a predicate pjn is of the form+-p(X1,X2, ••• ,Xn) where the X;
are distinct variables. The computed answer for a goal is a substitution for the variables in G produced by an SLD refutation of G. We will sometimes use Pro log notation, where p(X, Y) +-q(X, Z) II r(Z) is represented thus:
p(X,Y ) :-q(X, Z), r(Z)., and+-q(X, a) is represented thus:
-q(X, a).
3 Loglinear models for first-order probabilistic reasoning A loglinear probability distribution on a set fl is of the fol lowing form. For w E fl:
p(w) = z-i exp ( � Ad;(w)) (I)
where the /; are the features of the distribution, the A; are the model parameters and Z is a normalising constant.
Probabilistic Constraint Logic Programming
(Riezler, 1997) develops (Abney, 1997) by defining a log linear model on the proofs of formulae with some con straint logic program. This requires defining features on these proofs (the/;) and defining the model parameters (the A;).
The essentials of this approach can be given by using the logic programming framework. This is a special case of constraint logic programming where the only constraints allowed are equational constraints between terms. We will stay with the standard logic programming framework for simplicity. Consider the logic program LPl given in Fig I. s(X) :-p(X,Y), q(Y).
p(a, b). p(a,a). p(a, c). p(d,b). q(b). q(c). We can now define a loglinear distribution on refutations of +-s(X). Firstly, we define the features of the distri bution. Consider two features of refutations, /1 and /2.
For any refutation R, ft (R) = n if the goal+-q(b) ap pears n times in the R, and h(R) = m if+-q(c) ap pears m times in R. Let A1 = 0.2 and A2 = 0.4, then the leftmost proof of p(a) has probability z-l exp(0.2 X 1 + 0.4 x 0) and the one further to the right has probabil ity z-i exp(0.2 X 0 + 0.4 X 1). The probability of the single proof of p(d) is z-l exp(0.2 X 1 + 0.4 X 0). Z is simply exp(0.2 x 1 + 0.4 x 0) + exp(0.2 x 0 + 0.4 x 1) + exp(0.2 x 1 + 0.4 x 0) = 2e0·2 + e0·4 :::: : 3.9, so the three probabilities are 0.31, 0.38, 0.31 respectively. Having defined probabilities p on the proofs of these atomic formu lae, it is now trivial to define a distribution p' on the formu lae themselves: p'(Atom) = LR is a proof of AtomP(R).
We have p'(s(a)) = 0.69 and p'(s(b)) = 0.31, which is a distribution on S S ( s I 1), the success set for s I 1.
This approach applies very naturally to natural language processing (NLP). In NLP, a proof that a sentence belongs to a language amounts to a parse of that sentence, and the loglinear model can be used to find the most likely parse of any particular sentence. Riezler extends the improved iterative scaling algorithm of (Pietra et al., 1997) to induce features and parameters for a loglinear model from incom plete data. Incomplete data here consists of just atoms, rather than the proofs of those atoms. In an NLP context this means having a corpus of sentences rather than sen tences annotated with their correct parses, the former being a considerably cheaper resource.
Stochastic Logic Programs
Riezler's framework allows arbitrary features of SLD-trees, and recent experiments have used features "indicating the number of argument-nodes or adjunct-nodes in the tree, and features indicating complexity, parallelism or branching behaviour" (Stefan Riezler, personal communication).
Here we concentrate on a special case of Riezler' s frame work, where the clauses used in a proof are the features defining the probability of that proof, with clause labels de noting the parameters. (Eisele, 1994) examined this ap proach from an NLP perspective. (Muggleton, 1995) intro duced Stochastic Logic Programs, approaching the issue from a general logic programming perspective, with a view to applications in Inductive Logic Programming.
In these cases, Stochastic Context-Free Grammars (SCFGs) were "lifted" to stochastic feature grammars (SFGs) and stochastic logic programs (SLPs) respectively.
SCFGs are CFGs where each production is labelled, such that the labels for a particular non-terminal sum to one. The probability of a parse is then simply the product of the la bels of all production rules used in that parse. Sentence probabilities are given by the sum of all parses of a sen tence. The distributions so defined are special cases of log linear models where the grammar rules define the features f; and their labels are the parameters A;. Z is guaranteed to be one. This is because the labels for each non-terminal sum to one and because the context-freeness ensures that we never fail, and hence never have to backtrack, when generating a sentence from a SCFG-a production rule can always be applied to a nonterminal. Because of this a num ber of techniques (such as the inside-outside algorithm for parameter estimation (Lari and Young, 1990)) can be ap plied to SCFGs, but cannot be lifted to SFGs or SLPs. (See (Abney, 1997) for a demonstration of this.)
We define a stochastic logic program (SLP) as follows. An SLP is a logic program where some of the clauses are la belled with a non-negative number, and which satisfies the following constraints:
Constraint 1 If there is a refutation of the most general goal for a predicate that uses a labelled clause, then the predicate is distribution-defining. It is required that the computed answer substitutions for any unit goal where the predicate is distribution-defining be ground.
Constraint 2 The potential'ljJ(R) of any refutation R is the product of all the clause labels of the clauses used in R. If none of the clauses used in R have labels, then 1/!(R) is undefined. The potential 'I/!( G) of a goal is
L,R E ref ( G) 1/!(R), where re f ( G) is the set of all refu tations R of G such that 1/J(R) is defined. If 1/J(R) is undefined for all refutations R of a goal G, then 1/J (G)
is also undefined. We require that all goal potentials be finite.
Constraint 3 For every distribution-defining predicate, the potential of its most general goal must be positive.
Constraint 1 can be met by requiring SLPs to be range restricted: every variable appearing in the head of a clause must also appear in the body. The second condition is triv ially met by any SLP where there is a bound on the depth of any refutation, e.g. non-recursive SLPs, and can also be met by requiring the clause labels for the clauses defining any given predicate to sum to at most one. Our definition generalises that found in (Muggleton, 1995) Z, is simply the appropriate normalising constant, which can be found by simply summing the potentials of all refu tations of+-r(X1, X2, X3). By definition, this sum is the potential of the goal +-r(Xt, X2, X3). We have that
p,(R) = z; 1 exp ( � J o g (A; ) /(R, i)) (2) z-! II A J( R ,i ) r • (3) where A; isZ , = 1/!( +-r(A, B , C )) = L,AtomESS(r/3) 1/!( +-At om ).
This last equation holds because Constraint 1 ensures that every refutation of+-r(Xt, X2, X3) finds a member of SS(r/3), and all elements of SS(r/3) can be found this way. Constraints 2 and 3 ensure that 0 < Z, < oo, so Z; 1 is always defined.
We get a marginal distribution over SS(r/3): any ground atom A has probability p�(A) = L,R E r ef(<-A) Pr (R). Now consider the X; in+-r(X1, X2, X3). Each atom in SS(r /3) defines a joint instantiation of the X; and there fore the distribution on atoms defines a three-dimensional joint probability distribution over (Dt, D2, D3) , where D; is the domain of X; which is both a logical and random variable. D; is just the set of values found for X; in SS(r/3). In a standard logic program the D; will be fi nite or countably infinite.
We have used an example predicate r /3 to concreteness, but all of the above applies to predicates of any arity. We can use the logical structure of SLPs to define complex multi-dimensional joint distributions. The next section de scribes presents some SLPs, beginning with the simplest SLPs which represent Markov nets.
SLPmodels
SLPs and Markov nets
Markov nets (or undirected Bayes nets) are representations of graphical models, a special case of loglinear models. Figure 6 which represents a simple linear Markov net. We have that A is independent of C given B (A .L CIB). This conditional independence phe nomenon is central to probabilistic graphical models such as Markov nets. But note that A is independent of C given any value of B. Sometimes we may not be justified in mak ing such a strong assumption. It may be that A is only independent of C given particular values of B. This con ditional conditional independence1 or context-specific in dependence between A and C crops up often in applica tions and has been investigated by a number of workers, e.g. (Boutilier et a!., 1996).
To represent context-sensitive independence, we need to be able to differentiate between these two sorts of values of B. Let us assume we have two predicates, strong I 1 and weak/ 1 defined to be mutually exclusive which achieves Context-sensitivity occurs whenever backtracking (due to unification failure) is a possibility in the search for refu tations, and is ubiquitous in (real) natural language gram mars. Fig 8 shows an SLP defining a distribution over the non-context-free language { anbncn : n � 0}. Note that we can defi ne distributions using structured terms, not just constants, and that the domain of this distribution is count ably infi nite. 2\ + is ISO Pro log notation for not.
Inference in SLP models
Markov nets, mixtures of Markov nets and context sensitive stochastic grammars are all models that have been investigated in previous work, as have corresponding algo rithms for inference and learning. Our aim here is to use SLPs as a common framework which can bring out useful connections and contrasts between these different models and algorithms.
A basic probabilistic inference problem in SLPs is to take a query, e.g. f.-t(X 1 ,a,X3) and returnPrt(XI,X31Xz = a), where Prt is the distribution assocated with the predi cate t/3. The simple naive approach to inference in SLPs is to look for all refutations of f.-t(X 1 , a, X3), record their potentials and find Prt (X1, X3IX 2 = a) by marginalising.
Although this could be used where we know that goals will have few refutations, in general it will be very inefficient and will not even terminate for goals with infi nitely many refutations.
We do not have efficient general purpose algorithms for SLPs, so here we just sketch an approach. For a given query, find all clauses which have heads which unify with the goal, then apply the unifying substitution to the clause body, and then attempt to refute the subgoal composed of all the literals in the body that are not distribution-defi ning.
For each clause body, and for each successful refutation, we have a remaining subgoal involving only distribution defining predicates. Some of the variables in this remain ing subgoal may be instantiated, so the subgoal represents a partially instantiated Markov net, but one where the func tions defined on the cliques may not be represented by ta bles. When they are, we can use standard Markov net in ference algorithms. When they are not, one possibility is to call our sketch algorithm recursively, if the SLP is so de fined to guarantee termination. Note that we will only be interested in the distribution over the variables that appear in the head of the clause. These distributions can then be mixed according to the relevant clause labels to produce the final distribution.
Using ILP for feature construction
Since we use clauses to defi ne the structural features of our distribution, it is natural to look to ILP for techniques which induce such structural features from data. (Dehaspe, 1997) does just this using the MACCENT algorithm which con structs a log-linear model using boolean clausal constraints as features. Dehaspe uses the "learning from interpreta tions" ILP setting where each example is represented as a Prolog database. Dehaspe applies MACCENT to classi fication, using a simple animal classification task to illus trate his approach. To bring out the connections between Dehaspe's approach and that presented here, we can re write Dehaspe's clausal constraints as labelled clauses as in Fig 9. Dehaspe uses negation which is safe here since it is assumed that all queries are ground.
There are techniques for learning the structure of Bayes nets which start from an unconnected net and incremen tally add arcs. Such techniques are strongly related to ILP Ll : p(X,fish) :searches (likeDehaspe's) where we start from a maximally \+ has_legs (X), habitat (X, water).
general clause e.g. p(X, Y, Z) +-and re fine it by adding L2 : p (X, reptile) literals to the body until a 'best' (however defined) clause \ + has_ covering (X , hair) , \ + has _l egs (X) . is found. p(X, Y, Z) +-corresponds to an totally uncon- and hence, with suitable normalisation, conditional distri butions Pr(Classll). We have a bijection between proofs ofp/2 atoms and p/2 clauses, since each proof uses exactly one p/2 clause. This allows Dehaspe to treat each proof as a featu re, where the parameter associated with each fea ture (=proof) is the label of the p/2 clause used in that proof. These features are then used to define a probabil ity on atoms directly. This contrasts with the SLP and PFG approach where each proof has features (e.g. the set of labelled clauses used in the proof), and these are used to define a probability ove r proofs. To get an (unnormalised) probability on an atom with SLPs we have to sum up the probabilities of the proofs of that atom.
Dehaspe's approach allows a more direct defi nition of a distribution over atoms, but relies on each proof passing through exactly one labelled clause. SLPs are not so re stricted. Also with SLPs, the probability of an atom always increases with the number of proofs of that atom, which seems desirable. Following Dehaspe's approach this may not always be the case.
Dehaspe exploits the lattice structure of clauses and ap plies ILP techniques to guide the search for suitable con straints, searching for clauses with a general-to-specific beam search using the DLAB declarative bias language for malism. Dehaspe, like Riezler, keeps all the old parameters fixed when searching for the next constraint (= clause).
nected Markov net with three nodes. Refining this, to, say, p(X, Y, Z) +-q(X, Y) corresponds adding an arc between the X and Y nodes. Further exploration of this connection may well yield valuable cross-fertilisation between ILP and Bayes net structure learning.
Related work
We do not give anything like a comprehensive survey of the work on connecting logic and probability that can be found in the UAI, philosophical, statistical and logical literature. Instead we will contrast the approach presented here with a few examples of particularly closely related work.
This translation of the clique functions of a Markov net to a generalised relational database is essentially the same as that of (Wong et a!., 1995). Wong eta/ translate many of the graphical operations used with Markov nets to database op erations: product distributions are constructed using joins, conditional distributions by projection, and marginals by database operations which mimic the standard approach in the Markov net literature. Wong et a/'s argument is that since the operations required for effective use of Markov nets are defined on tables-for example, tables defining marginal probability distributions-one should use opti mised methods developed by the database community for manipulating tables. The current work seeks to extend that of Wong et a/ by moving from a relational database setting to the logic programming setting.
In Knowledge-based model construction (KBMC) (Ngo and Haddaway, 1997;Had daway, 1999) first-order rules with associated probabili ties are used to generate Bayesian networks for particular queries. As in SLD-resolution queries are matched to the heads of rules, but in KBMC this results in nodes repre senting ground facts being added to a growing (directed) Bayesian network. A context is defined using normal fi rst order rules, perhaps explicitly as a logic program (Ngo and Haddaway, 1997), which specifies logical conditions for labelled rules to be used. The ground facts are seen as boolean variables (either true or false). Once the Bayesian network is built it is then used to compute the probability that the query is true.
In KBMC, as in much of the work connecting logic and probability, parameterised first-order rules a : c(X) + a(X) are connected to conditional probability statements such as p(c(b)ia(b)) =a . Also the objective is to compute the probability that an atom is true. In the current paper, we focus on undirected representations, so that>. : p(X, Y) +q(X, Y), r(Y) forms part of the definition of a binary dis tribution associated with p/2 defined in terms of distribu tions associated with q/2 and r fl. We make no attempt to model causality.
Secondly, we do not use a labelled rule >. : p( X, Y) +q(X, Y), r(Y) to define the probability that some ground atom p( a, b) is true as in KBMC, or to provide bounds on the probability that p(a, b) is true as in (Shapiro, 1983;Ng and Subrahmanian, 1992). Instead, we have a binary distri bution associated with p(X, Y) which defines the probabil ity of instantiations such as {X fa, Yfb }. In order to reason about the probability of the truth of atoms, we simply aug ment atoms by introducing an extra logical-random vari able to represent the truth value of unaugmented atoms, and then treat this logical-random variable exactly as any other. This is in keeping with our conservative approach-if we are interested in the truth value of an atom as it varies across different "possible worlds" -then we model this variation in the standard way: with a random variable. an example from , where the "rule says that when a person's parent has a gene, the per son will inherit it with probability 0.5''. We would en code such "degree of belief" probability in an SLP with a boolean truth-value variable as in Fig 10. genotype (P,G,T) :-parent (P,Q), genotype (Q,G,l), half (T). To find the probability that genotype( bob, big..ears)
is true we are required to use the SLP to com pute the probabilities of genotype(bob, big ..ears, 1) and genotype( bob, big ..ears, 0). (In fact, all we need are un normalised potentials for these, which is simpler.) This amounts to demanding arguments (=proofs) for the truth of genotype(bob, big ..ears) and for its falsity. We then effec tively balance the strength of these proofs when deciding on the probability of truth.
Despite these differences in approach there are clear simi larities between KBMC query-specific Bayes net construc tion and the query-specific exploration of an SLD-tree by Pro log which deserve further investigation.
Another approach to relational probabilistic reasoning are the relational Bayesian networks of (Jaeger, 1997). Here whole interpretations are the nodes of a Bayesian net. It is conceivable that such networks could be implemented as an SL P, using some suitable object-level representation of an interpretation, but it is likely that they would be unwieldy in practice. Since, at the end of the day, we are interested in the truth-values of atoms, it seems easier to deal with these directly, perhaps resorting to quite complex SLPs to model complex interactions between degrees of belief.
Finally, SLPs are very closely related to the stochastic (functional) programs of . Stochastic execution of the functional program defines a distribution over outputs of the program. As we have done here, Koller et al show how Bayesian nets and SCFGs can be repre sented in their richer formalism. They base their repre sentation of directed Bayes nets on "the observation that each node in a Bayes net is a stochastic function of its parent's values." They also show how their formalism can exploit context-sensitive independence. Unlike the present paper, they also provide details of an efficient algorithm for probabilistic inference in their formalism, which mim ics standard efficient algorithms for Bayesian networks. , does not discuss methods for inducing stochastic functional programs, but it seems highly likely that ILP techniques could be applied.
Open questions and future work
We have shown how various properties of SLPs (shared variables, multi-clause programs, unification failure and existential variables) correspond to various existing mod els (graphical models, mixture models, context-sensitive models and marginalisation) and argued that existing algo rithms for these models can hence be used for inference and learning in SLPs. This work remains to be done. It is likely that suitable algorithms will mimic algorithms those used in Koller eta/'s stochastic programs. Work on the imple mentation of randomised algorithms in logic programming is likely to be relevant too (Angelopoulos et al., 1998). We also expect techniques from logic programming and com putational linguistics, such as Earley deduction and pro gram transformation to be useful. For example, when learn ing the parameters of SLPs, Riezler's approach of storing proofs in a chart using Earley deduction makes a lot more sense than continually re-refuting goals.
Probabilistic inference and learning by Markov Chain
Monte Carlo is also attractive for SLPs. For example, in a Gibbs sampling approach, all except one argument of a goal would be ground on each iteration. Such constrained goals generally have few refutations which might lead to an efficient method.
Finally, we hope that the current framework will stimu late further research into statistical ILP, and that such re search will benefit from and contribute to related work on inducing models from data in computational linguistics and Bayesian networks.
Figure I :Figure 2 :
I2LP1, a simple logic program Fig 2 shows the SLD-tree generated by the query+-s(X)(with empty goals omitted). This shows three refutations of+-s(X) which amount to two proofs of s(a) and one proof of s(d). SLD-tree for+-s(X) and LP1
the label of clause C; and f ( R, i) is the number of times the clause C; is used in R. So we have a loglinear model where the labelled clauses define features and the logs of the clause labels are the model parameters.
Fig 3
3shows the "Asia" Markov net used as a running ex ample in(Lauritzen and Spiegelhalter, 1988).
Figure 3 :
3"Asia" Markov net In general, let V c be the set of cliques of a Markov net G. A potential representation consists of evidence poten tials 1/J A . defined on the cliques. Potentials are real-valued non-negative functions depending only on the states of the variables in each clique. The evidence potentials define a joint distribution on the nodes V of the net as follows:p (V) = z-l II 1/normalising constant. If Z = 0 then p is undefined. (Z will always be finite.) Consider the Markov net inFig 3,which has six cliques. Each of the random variables in this net is binary, taking values t or f .Table 2gives a potential function defined on the clique {A, T}.
Figure 4
4Figure 4 gives an SLP representation of the clique poten tial defined in Ta ble 2. 0. 0005 0. 0095 0.0099 0.9801
Figure 4 :
4Ground SLP representation of clique potential on {A,T} We can then use a single unlabelled clause to represent the structure of a Markov net. The net in Fig 3 is represented by the unlabelled clause shown in Fig 5. Let us call SLPs that represent Markov nets in this fashion Markov net SLPs.Each ground goal has one refutation, so the probability of any ground atom is a normalised product; it is clear that the SLP and the Markov net represent the same loglinear dis tribution. Since they represent Markov nets, probabilistic inference based on such SLPs can use any of the standard algorithms for Markov nets.
Figure 5 :
5Clausal representation of the "Asia" Markov net 4.2 SLP mixture models for context-specifi c independence Consider the SLP in
Figure 6 :
61 conditional on a variable and conditional on values of Linear Markov net SLP this. We can then define the SLP inFig 7 thatdefi nes an appropriate mixture model. A neater alternative might be to use negation to differentiate, using strong (B) and \ + strong (B) 2 , but the use of negation in SLPs has yet to be properly investigated, hence our current restriction to definite clauses. Mixture models for context-specific inde pendence are investigated in(Thiessen et al., 1997), where learning of such models is considered. (One can view ta bles defining discrete distributions as inFig 4,as mixtures of degenerate distributions, but we will not do so.)
Figure 7 :
7Mixture model SLP defining context-specifi c in dependence
build( [a i AJ-Ap, [b i BJ-Bp, [c i CJ-Cp)build(A-Ap, B-Bp, C-Cp).
Figure 8 :
8Stochastic non context-free grammar defined with an SLP
Figure 9 :
9Dehaspe's clausal constraints as labelled clauses Dehaspe associates (modulo our rewrite) boolean features with each labelled clause, defined on (I, Class) pairs, where I denotes an, as yet, unclassified instance. /j(I, Class) = { � if B,Ci f-p(I,Class) otherwise B is background knowledge represented by a logic pro gram. This defines a distribution over pairs (I, Class), Pr(I,Class) = z-1 exp ( �.Xifi(I,Class))
Figure 10 :
10Representing degree of belief with an extra vari able
, where Mug gleton requires SLPs to be range-restricted and with labels for the same predicate summing exactly to one. Also, Mug gleton does not use SLPs to define a loglinear model as we do here.An SLP defines a distribution for every distribution defining predicate in the SLP. Suppose r /3 were a distribution-defining predicate, then we have a loglinear distribution over refutations R of the most general goal for this predicate+-r(Xt, X2, X3), as follows:
Table 2 :
2An evidence potential on the clique {A, T} Markov nets consist of a structure with associated param eters. Both can be represented easily using SLPs. Clique potentials are represented as tables of SLP ground facts.
AcknowledgementsMany thanks to Sara-Jayne Fanner for weeding out vari ous errors and omissions. Thanks to Stephen Muggleton for useful discussions on the role of normalisation in SLPs and to Stefan Riezler for clarifying his method. Thanks also Gillian, Jane and Robert Higgins for putting up with me. Finally thanks to Luc. de Raedt for encouraging me to investigate first-order Bayesian nets.
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"Determination of chirality and density control of Néel-type skyrmions with in-plane magnetic field",
"Determination of chirality and density control of Néel-type skyrmions with in-plane magnetic field"
] |
[
"Senfu Zhang \nPhysical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia\n",
"Junwei Zhang ",
"Yan Wen \nPhysical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia\n",
"Eugene M Chudnovsky \nPhysical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia\n\nPhysics Department\nLehman College and Graduate School\nThe City University of New York\n250 Bedford Park Boulevard West10468-1589BronxNew YorkUSA\n",
"Xixiang Zhang \nPhysical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia\n"
] |
[
"Physical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia",
"Physical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia",
"Physical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia",
"Physics Department\nLehman College and Graduate School\nThe City University of New York\n250 Bedford Park Boulevard West10468-1589BronxNew YorkUSA",
"Physical Science and Engineering Division (PSE)\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalSaudi Arabia"
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Magnetic skyrmions are topologically protected nanoscale spin textures exhibiting fascinating physical behaviors. Recent observations of room temperature Néel-type skyrmions in magnetic multilayer films are an important step towards their use in ultra-low power devices. We have investigated the magnetization reversal in a [Pt/Co/Ta] 11 multilayer sample under a tilted magnetic field using the in-situ Lorentz tunneling electron microscopy. On decreasing the magnetic field individual skyrmions appear that subsequently evolve into snake-like structures growing in the direction opposite to the in-plane magnetic field. We show that this unusual relation between the velocity vector and the magnetic field is dominated by the chirality of the Néel-type skyrmions. It allows one to extract the sign of the Dzyaloshinskii-Moriya constant. We also demonstrate that high concentration of skyrmions can be achieved on increasing the in-plane component of the field. Our micromagnetic simulations agree with our experimental results.
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10.1038/s42005-018-0040-5
|
[
"https://arxiv.org/pdf/1805.03875v1.pdf"
] | 119,459,749 |
1805.03875
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da7a1d841867cdf110e28ad0748b05501e211c2e
|
Determination of chirality and density control of Néel-type skyrmions with in-plane magnetic field
Senfu Zhang
Physical Science and Engineering Division (PSE)
King Abdullah University of Science and Technology (KAUST)
23955-6900ThuwalSaudi Arabia
Junwei Zhang
Yan Wen
Physical Science and Engineering Division (PSE)
King Abdullah University of Science and Technology (KAUST)
23955-6900ThuwalSaudi Arabia
Eugene M Chudnovsky
Physical Science and Engineering Division (PSE)
King Abdullah University of Science and Technology (KAUST)
23955-6900ThuwalSaudi Arabia
Physics Department
Lehman College and Graduate School
The City University of New York
250 Bedford Park Boulevard West10468-1589BronxNew YorkUSA
Xixiang Zhang
Physical Science and Engineering Division (PSE)
King Abdullah University of Science and Technology (KAUST)
23955-6900ThuwalSaudi Arabia
Determination of chirality and density control of Néel-type skyrmions with in-plane magnetic field
1 # Equally contributed to this work. * E-mail address: [email protected] (Xixiang Zhang), [email protected] (Eugene M Chudnovsky)
Magnetic skyrmions are topologically protected nanoscale spin textures exhibiting fascinating physical behaviors. Recent observations of room temperature Néel-type skyrmions in magnetic multilayer films are an important step towards their use in ultra-low power devices. We have investigated the magnetization reversal in a [Pt/Co/Ta] 11 multilayer sample under a tilted magnetic field using the in-situ Lorentz tunneling electron microscopy. On decreasing the magnetic field individual skyrmions appear that subsequently evolve into snake-like structures growing in the direction opposite to the in-plane magnetic field. We show that this unusual relation between the velocity vector and the magnetic field is dominated by the chirality of the Néel-type skyrmions. It allows one to extract the sign of the Dzyaloshinskii-Moriya constant. We also demonstrate that high concentration of skyrmions can be achieved on increasing the in-plane component of the field. Our micromagnetic simulations agree with our experimental results.
Magnetic skyrmions are nanoscale, relatively stable chiral spin textures that are protected topologically 1-4 . Owning to their topological property, small size, and high mobility, skyrmions can be manipulated by much lower current densities than current densities needed to manipulate magnetic domains 3,[5][6][7][8][9] . Skyrmions have been therefore proposed as promising candidates for the next generation, low power spintronic devices, such as non-volatile information storage 3,10,11 , spin transfer nano-oscillators 12,13 and logic devices 14,15 .
Skyrmions were first observed in B20 materials in which the Dzyaloshinskii-Moriya interaction (DMI) originates from a non-centrosymmetric crystalline structure and leads to the formation of the intriguing spin texture 5,[16][17][18][19][20][21][22][23][24][25][26] . In these B20 materials, the skyrmions are Bloch-type and can only exist at relatively low temperatures 4,27,28 . To create room temperature skyrmions suitable for industrial applications, hetero-structured magnetic thin-films were developed, in which the DMI is produced by breaking of the inversion symmetry at the interfaces of ferromagnetic layers and heavy metal layers with large spin-orbit coupling 27,[29][30][31][32][33][34][35][36][37] . This interfacial DMI can result in the formation of the Néel-type skyrmions, which has been widely observed in various thin-film stacks at room temperature 27,[29][30][31][32][33][34][35][36][37] . Furthermore, the formation and dynamics of the Néel-type skyrmions driven by the current have been investigated, which manifested an important step towards application of skyrmions in devices 9,11,33,35,38 . Usually, the out-of-plane magnetic field is very important for stabilization of skyrmions, thus, the study of the out-of-plane magnetic field on the skyrmions became one of the foci in this area 4,28 . However, the magnetization reversal and its dynamics are also governed by the in-plane magnetic field 9,36,39,40 , especially for the films with very weak perpendicular magnetic anisotropy (PMA). Consequently, the study of the effects induced by the in-plane magnetic field has fundamental importance.
Lorentz transmission electron microscopy (L-TEM) is one of the most direct methods with high spatial resolution to observe the magnetic domain structures, domain walls, and skyrmions, especially for objects with the Bloch-type spin rotation. Moreover, the chirality of the Bloch-type skyrmions can also be determined by the L-TEM images 4,22,23,25,26,41 , while observation of the Néel-type domain walls and skyrmions in the materials with PMA requires tilting of the sample and does not permit identification of the chirality of skyrmions from the L-TEM images alone 36,42 . In this work, we report the determination of the chirality of Néel-type skyrmions from the magnetization reversal behavior in [Pt/Co/Ta] 11 multilayers with weak PMA by introducing the in-plane magnetic field in the L-TEM imaging. We find that the in-plane magnetic field contributes to the creation of skyrmions and that high concentration of skyrmions can be achieved by increasing the in-plane field.
Our most remarkable finding is that the vector of the speed V with which Néel skyrmion snakes are growing in a certain direction (or the vector indicating that direction), is determined by the vector of the in-plane magnetic field H in , implying the relation V = p H in between the directions of the two vectors, with p being a scalar function. For most physical systems such a relation is prohibited by symmetry because V changes sign under spatial reflection, r -r, while H in does not. One must have a pseudo-scalar in the system, that is, a scalar function p that changes sign under spatial reflection, in order to have the direction of the velocity determined by the direction of the magnetic field. In our system that pseudo-scalar originates from the chirality of the Characterization of the samples. Figure 1(a) shows the high resolution, high-angle annular dark-field scanning transmission electron microscopy (HADDF-STEM) image of the cross-section of the multilayer, which exhibits a clear stack structure. Electron energy loss spectroscopy (EELS) mapping spectrum ( Fig. 1(b)) further reveals a periodic tri-layered structure showing that each tri-layer is composed of a Co layer sandwiched by Pt and Ta layers.
Magnetic properties of the multilayers were characterized using a SQUID-VSM magnetometer at room temperature. Figure 1(c) shows the normalized out-of-plane and in-plane hysteresis loops, which indicate that the sample possesses a weak PMA with an in-plane saturation field μ 0 H k of only 0.206 T. This observation suggests that the sample should be very sensitive to the in-plane field. Moreover, both loops show almost zero remanence. To explore the difference in the magnetic structures in the remanent states after the sample was saturated in the presence of outof-plane and in-plane fields, the magnetic force microscopy (MFM) measurements were performed at zero fields after each hysteresis loop measurement. The corresponding MFM images are shown in Fig. 1(d) and (e). As expected for the films with both PMA and DMI, a typical labyrinth domain structure is clearly seen after saturation by a perpendicular field.
Interestingly, aligned and stripe-like domains, sharply different from the labyrinth domain structure, were observed in the remanent state after the in-plane saturation, although the magnetization within the domains was still perpendicular to the film plane.
To understand the effect of the in-plane field on the magnetization reversal in the multilayer sample, in situ L-TEM measurements were performed. In L-TEM mode, the magnetic field parallel to the electron beam can be easily applied through the objective lens of the TEM. By tilting the sample, an in-plane field can also be applied. The ratio of the in-plane and out-of-plane field can be tuned by varying the tilt angle (θ) 36,42 . For the Bloch-type skyrmions, it is not necessary to tilt the sample to create the contrast in the L-TEM images. Depending on the sign of D, the core (shell) shows bright (dark) or dark (bright) contrast 25 . However, for the Néel-type skyrmions, the sample tilting is essential 36,42 . Here, we introduce briefly the physics of the L-TEM imaging for Néel-type skyrmions. Relationship between in-plane field direction and the preferred extension direction. The observation of the preferred extension direction governed by the in-plane field may become a useful and reliable approach to determine the skyrmion chirality, which is of great importance for manipulating skyrmions as well as for establishing the sign of the DMI constant 4 .
To further understand this phenomenon and also determine the chirality of skyrmions, micromagnetic simulations were performed with the mumax 3 software 43 . As an example, we first studied a system with an interfacial DMI of positive D. Following the experimental protocol, we created some skyrmions randomly and stabilized them with a magnetic field applied in the -z direction. Figure 3(a) shows the evolution of the magnetic structure when decreasing the magnetic field with the tilt angles of θ = 0 • and θ > 0 • . In the case of θ = 0 • , when no in-plane field is applied, the skyrmions extended isotropically, leading to a typical labyrinth domain structure, as we observed experimentally ( Fig. 1(d)). However, at θ > 0 • , when a non-zero inplane field component was applied along the +x axis, an anisotropic growth of the skyrmions was clearly observed. An important finding is that the snake-like structures grow parallel to the inplane magnetic field. We also found that the growing speed in the -x direction is much greater than in the +x direction. Thus, we concluded that the preferred extension direction is opposite to the direction of the in-plane magnetic field, when the magnetization of the skyrmion core points up for an interfacial DMI with positive D.
More simulations were performed for both Bloch-type skyrmions and Néel-type skyrmions with different sign of D. The results are summarized in Fig. 3(b-f). Fig. 3(b) shows the spin texture of the four types of skyrmions whose magnetization at the skyrmion core is pointing up. images for the Bloch-type skyrmions and their chirality could be distinguished from the contrast (bright or dark) under the skyrmion core (Fig. 3(c)), while for the Néel-type skyrmions, the sample tilting is essential for the L-TEM imaging and same contrasts (Fig. 3(c)) will be observed regardless of the sign of D. Therefore, their chirality cannot be identified by the normal L-TEM imaging. However, we could solve this problem by applying an in-plane magnetic field. The inplane field (+x direction here) breaks the symmetry of the magnetization distribution of a skyrmion as is shown in Fig. 3(d), The final domain structure at zero field is thus an orientated structure in the y directions as is shown in Fig. 3(f). This behavior has been observed experimentally 44 . For the interfacial DMI, the preferred extension direction is collinear with the in-plane field. By comparing theoretical prediction with experimental data on the relationship between the preferred extension direction and the in-plane field direction, we were able to identify the chirality of the skyrmions (or the sign of D). It is evident that D > 0 in our [Pt/Co/Ta] 11 multilayers.
Skyrmion creation assisted by the in-plane field. In Fig. 2, we can see that the achieved skyrmion densities are different for different tilting angles (or in-plane field). To understand this phenomenon, we increased the tilt angle θ from 3 o to 45 o , and studied the magnetization reversal using the in-situ L-TEM. As has been discussed in Fig. 2(a), the skyrmions appeared twice when the magnetic field was swept from negative saturation field to positive saturation filed. The first emergence of skyrmions corresponded to the nucleation from the ferromagnetic state at negative field and the second emergence corresponded to the creation of skyrmions by breaking the labyrinth domains by the positive field. Figure 4 shows the maximum densities d max1 and d max2 as functions of the tilt angle and it also shows the corresponding L-TEM images. One can see that when the tilt angle is very small (i.e., 3 o , consequently, the in-plane field is very small), d max1 is almost zero and the snake-like structures are created directly. However, many more skyrmions were created at the breaking process, when the field was increased from zero to positive saturation. The reason why d max2 is much larger than d max1 over the whole range of tilt angles is that the breaking process is energetically more favorable than the nucleation process. Interestingly, with increasing the tilt angle (increasing the in-plane field), we found that both d max1 and d max2 increase, which indicates that the in-plane field favors the creation of skyrmions. This can be understood in the following way. It has been shown that skyrmion density increased with increasing the critical material parameter In summary, we deposited [Pt/Co/Ta] 11 multilayer sample with weak PMA and investigated the magnetization reversal behavior for different tilted angles of the field, to have both magnetic field components parallel and perpendicular to the film plane, using in-situ L-TEM. We found that the chirality of skyrmions can be identified by their preferred extension direction under an in-plane magnetic field, which was confirmed by micromagnetic simulations. Furthermore, we found that the in-plane field facilitates the creation of skyrmions and increases their concentration.
Methods
L-TEM measurements:
In-situ Lorentz transmission electron microscopy (L-TEM) imaging was carried out by using a FEI Titan Cs Image TEM in Lorentz mode (the Fresnel imaging mode) at 300 kV. A double holder was used for the measurements. The magnetic field parallel to the electron beam can be easily applied and tuned by varying the current passing though the microscope's objective lens. By tilting the sample, an in-plane field can also be applied. The ratio of the in-plane and out-of-plane field can be tuned by varying the tilt angle (θ).
MFM measurements:
The magnetic force microscopy (MFM) experiments were performed using an Agilent 5500 Scanning Probe Microscope in tapping/lift mode.
Micromagnetic Simulations: mumax 3 software package, including the extension module of the DMI, was used for the micromagnetic simulations. A 1×1 μm 2 square system with 2D periodic boundary conditions was used as a material system. The mesh size is set to 2×2 nm 2 and the following parameters were chosen: M s = 7.0×10 5 A/m, K u = 3.
skyrmion that is determined by the sign of the Dzyaloshinskii-Moriya constant D in a noncentrosymmetric crystal. Thus, observation in which direction the skyrmion snakes are growing allows one to extract the skyrmions type and the sign of D.
Figure 1 (
1f) shows the top view schematic diagram of the L-TEM imaging. The sample is tilted from the xy plane to the x'y' plane. Blue, white and red contrasts represent the outside, the boundary and the inside of a skyrmion with negative, zero and positive magnetization along the z' axis. The corresponding xy plane projection of the magnetization is indicated with the brown arrows. The green arrows indicate the corresponding Lorenz force when electrons pass through the skyrmion. It is clear that the skyrmion edge does not show any intensity contrast in the L-TEM image because the intensity change caused by the Lorenz force at any point in this zone is compensated or cancelled. The intensity contrast is formed in the L-TEM images due to the contributions from both the outside and the core of the skyrmion. Therefore, the resulted image for the skyrmion should be of dark-bright contrast with dark at the top side and bright at the bottom side just under the Néel-skyrmion edge, as is shown in the right side of Fig. 1(f). The distance between the intensity minimum and the density maximum represents the skyrmion size. Magnetization reversal under a tilted magnetic field. We start with the tilting angles of α = 23 o and β = -20 o for the L-TEM imaging. Here, α is the tilting angle around the y axis and β around the x axis. The z axis is chosen to be parallel to the TEM cylinder. Thus, Before imaging, the sample was saturated in the down direction (-z) while the direction of the in-plane (x'y' plane) component of the magnetic field pointed to the top left as indicated by the arrow in up-right corner of Fig. 2(a). The evolution of the domain structure was imaged at the defocus of 7.62 mm by varying the magnetic field from the negative saturation field to the positive saturation field as is shown in Fig. 2 (a) (See the detailed reversal process in Supplementary Movie 1). When the magnetic field was changed to about -1648 Oe (the out-of-plane component H out =1427 Oe and in-plane component H in = 824 Oe), an isolated skyrmion emerged as marked by the green circle with the contrast of bright at upper right and dark at lower left. By continually varying the field, more skyrmions appeared (such as at -1600 Oe) and began to evolve into snake-like structures as is shown in the image obtained at -1464 Oe. The maximum skyrmion density before the formation of the snake-like structures is denoted as d max1 . By changing the field further (from -1464 Oe to -1320 Oe), the length of the snake-like structures increases. Meanwhile, some new skyrmions appeared (as indicated by yellow arrows in the image obtained at -1456 Oe) and rapidly extended to the snake-like structures. We compared the two images obtained at -1468 Oe and -1456 Oe and marked the structure differences with green dashed ellipses. Interestingly, we found that the snake-like structures mainly extended along the directions that are opposite to the direction of the in-plane field. Finally, the density of the snake-like structures achieved maximum over the entire film at about -680 Oe, forming an almost aligned arrangement. When the field approached zero, the in-plane field became zero too, the oriented stripe domains became less aligned. With increasing the field in the positive direction the in-plane field increases in the opposite direction. The stripe domains become better oriented again as seen in the image obtained at 1080 Oe. When increasing the field to 1400 Oe, the orientated domain structures fractured and a mixed state of skyrmions and snake-like structures appeared. Eventually, all snake-like domains broke into skyrmions at 1720 Oe. The maximum skyrmion density obtained in this image was defined as d max2 . With increasing the field further, the skyrmions annihilated gradually (1840 Oe) and the film was finally saturated to a ferromagnetic state at 1960 Oe. In-plane field induced anisotropic growth of the snake-like structures. To explore the correlation between the in-plane field direction and the preferred extension direction of the snake-like domains, more experiments were performed with different tilting angles and directions, and same phenomena were observed. Note that the extension of the snake-like structures is driven by the out-of-plane component of the field, but their anisotropic formation is induced by the in-plane field. Figures 2(b) and 2(c) show L-TEM images of the extending snakelike structures taken during changing the magnetic field from negative saturation field to zero with the tilting angles of α = 0 o ; β = -20 o and α = -28 o ; β = -20 o respectively. The correspondingdirections of in-plane fields were indicated by black arrows. Similarly, we indicated the changes in the images with green dashed ellipses. It is clear that for bothFig. 2(b) and 2(c), the snake-like structures preferred to grow along the directions opposite with that of the in-plane fields and finally formed an almost aligned arrangement.
type skyrmion, the in-plane spin components rotate clockwise for positive D and counterclockwise for negative D, while for the Néel-type skyrmions, the in-plane component spins point inside (positive D) or outside (negative D). Based on the principle of the contrast formation in the L-TEM imaging, it is not necessary to tilt the sample to create the contrast in the L-TEM
because the region with spins parallel to the in-plane component of the field grows while the region with the opposite spins shrinks. Though the inplane component of the fields is in the +x direction for all the four cases, it is found that the preferred extension directions are all different as marked in Fig. 3(e). For samples with bulk DMI, the preferred extension directions are in the +y or -y direction depending on the sign of D.
37 , where A is the exchange stiffness, K eff is the effective perpendicular anisotropy and D is the DMI constant. Application of the in-plane field will actually weaken the role of the perpendicular anisotropy, while keeping D and A unchanged, which leads to the increase of κ.
deposited simultaneously both on thermally oxidized Si substrates and carbon membranes by DC magnetron sputtering at room temperature. The pressure of the Ar gas was 0.4 Pa. The base pressure was lower than ~2×10 -5 Pa. The films grown on thermally oxidized Si substrates were used for the SQUID-VSM and MFM measurements. The films on carbon membranes were used for the in-situ L-TEM measurements. Transmission electron microscopy (TEM): TEM sample was prepared using focused ion beam (FIB) from the sample deposited on thermally oxidized Si substrates. The High angle annular dark field scanning TEM (HADDF-STEM) images and Electron energy loss spectroscopy (EELS) mapping spectrum were employed to analyze the structure of sample in a FEI TEM (Titan 60-300).
5×10 5 J/m 3 , A = 1.0×10 -11 J/m and T = 0 K. For comparison, both bulk DMI and interfacial DMI were considered and its magnitude was set to D = 1.3×10 -3 J/m 2 .
Figure 1
1Figure 1
Figure 1
1Structure analysis and magnetic properties. (a) HADDF-STEM image and (b) EELS mapping spectrum of the cross-section of the Ta(5 nm)/[Pt(4 nm)/Co(1.4 nm)/Ta(1.8 nm)] 11 multilayer sample. (c) Normalized out-of-plane and in-plane hysteresis loops of the sample. MFM images at zero fields after the (d) out-of-plane and (e) in-plane saturation fields. (f) Schematic diagram of a Néel-skyrmion on a tilting sample for L-TEM imaging. The blue/red contrast represents areas with negative/positive magnetization along z'. The corresponding xy plane projection is indicated with the brown arrows. The green arrows indicate the Lorenz force when electrons pass through the skyrmion. The expected Fresnel contrast is shown at the right side and the distance between the intensity minimum and density maximum represents the skyrmion size.
Figure 2
2Figure 2
Figure 2 .
2In-situ L-TEM observation of reversal behaviors. (a) In-situ L-TEM observation of the [Pt/Co/Ta] 11 multilayer's reversal behaviors. The images were taken at a tilt angle of α = 23 o and β = -20 o and the direction of in-plane component field is marked by black arrow. L-TEM images of the snake-like structures extension taken during changing the magnetic field from negative saturation field to zero with the tilting angles of (b) α = 0 o ; β = -20 o and (c) α = -28 o ; β = -20 o . The scale bar corresponds to 1 μm.
Figure 3
3Figure 3
Figure 3 .
3Micromagnetic simulations. (a) Magnetic structure evolution when increasing magnetic field from negative saturation to zero with a tilt angle of θ = 0 and θ > 0. (b) Four kinds of skyrmions and (c) the expected L-TEM contrast for Bloch-type skyrmion without tilting the sample and for Néel-type skyrmion on a tilt sample. (d) Magnetic structure when apply an inplane field. (e) The preferred direction when the in-plane field is in the +x direction. (f) The final orientated domain structure at zero fields.
Figure 4
4Figure 4
Figure 4 .
4Relationship between skyrmion densities and tilt angles. The maximum skyrmion densities d max1 and d max2 as functions of the tilt angles. The top shows the corresponding L-TEM images. The scale bar corresponds to 1μm.
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The design and verification of MuMax3. A Vansteenkiste, AIP Adv. 4107133Vansteenkiste, A. et al. The design and verification of MuMax3. AIP Adv. 4, 107133 (2014).
Enhanced Stability of the Magnetic Skyrmion Lattice Phase under a Tilted Magnetic Field in a Two-Dimensional Chiral Magnet. C Wang, Nano Lett. 17Wang, C. et al. Enhanced Stability of the Magnetic Skyrmion Lattice Phase under a Tilted Magnetic Field in a Two-Dimensional Chiral Magnet. Nano Lett. 17, 2921-2927 (2017).
|
[] |
[
"Double Self-weighted Multi-view Clustering via Adaptive View Fusion",
"Double Self-weighted Multi-view Clustering via Adaptive View Fusion"
] |
[
"Xiang Fang ",
"Member, IEEEYuchong Hu "
] |
[] |
[] |
Multi-view clustering has been applied in many realworld applications where original data often contain noises. Some graph-based multi-view clustering methods have been proposed to try to reduce the negative influence of noises. However, previous graph-based multi-view clustering methods treat all features equally even if there are redundant features or noises, which is obviously unreasonable. In this paper, we propose a novel multi-view clustering framework Double Self-weighted Multi-view Clustering (DSMC) to overcome the aforementioned deficiency. DSMC performs double self-weighted operations to remove redundant features and noises from each graph, thereby obtaining robust graphs. For the first self-weighted operation, it assigns different weights to different features by introducing an adaptive weight matrix, which can reinforce the role of the important features in the joint representation and make each graph robust. For the second self-weighting operation, it weights different graphs by imposing an adaptive weight factor, which can assign larger weights to more robust graphs. Furthermore, by designing an adaptive multiple graphs fusion, we can fuse the features in the different graphs to integrate these graphs for clustering. Experiments on six real-world datasets demonstrate its advantages over other state-of-the-art multi-view clustering methods.
| null |
[
"https://arxiv.org/pdf/2011.10396v2.pdf"
] | 227,119,065 |
2011.10396
|
3ece687decbad69d8ebc01b2c1f1b4d6d2259cc0
|
Double Self-weighted Multi-view Clustering via Adaptive View Fusion
Xiang Fang
Member, IEEEYuchong Hu
Double Self-weighted Multi-view Clustering via Adaptive View Fusion
1Index Terms-ClusteringData miningMachine learning
Multi-view clustering has been applied in many realworld applications where original data often contain noises. Some graph-based multi-view clustering methods have been proposed to try to reduce the negative influence of noises. However, previous graph-based multi-view clustering methods treat all features equally even if there are redundant features or noises, which is obviously unreasonable. In this paper, we propose a novel multi-view clustering framework Double Self-weighted Multi-view Clustering (DSMC) to overcome the aforementioned deficiency. DSMC performs double self-weighted operations to remove redundant features and noises from each graph, thereby obtaining robust graphs. For the first self-weighted operation, it assigns different weights to different features by introducing an adaptive weight matrix, which can reinforce the role of the important features in the joint representation and make each graph robust. For the second self-weighting operation, it weights different graphs by imposing an adaptive weight factor, which can assign larger weights to more robust graphs. Furthermore, by designing an adaptive multiple graphs fusion, we can fuse the features in the different graphs to integrate these graphs for clustering. Experiments on six real-world datasets demonstrate its advantages over other state-of-the-art multi-view clustering methods.
I. INTRODUCTION
Spectral clustering can be regarded as the graph-based clustering since its performance is directly determined by the obtained graph. Spectral clustering makes use of the spectral-graph structure of an affinity matrix to partition data into disjoint meaningful groups. In many real-world applications, we often collect multi-view data [1]- [3]. Therefore, multi-view clustering methods are needed to cluster these multi-view data [4]- [6]. Benefiting from its efficiency and good performance, multi-view spectral clustering has gained much attention in many fields of machine learning and data mining [7]- [9]. Up to present, several multi-view spectral clustering methods have been proposed [10], [11], but most of them perform clustering only based on the diversity between different views in multi-view data [12]- [14].
To learn optimal graphs, several state-of-the-art multi-view spectral clustering methods are proposed. [10] propose MLAN to learn the local structure of multi-view data by constructing a constrained graph Laplacian. [15] propose MMSC to learn a shared graph Laplacian matrix by integrating different image features. [16] propose RMSC to learn the standard Markov chain by constructing a graph with fewer noises for each view and a low-rank transition probability matrix shared by all the views. [17] propose SwMC to learn an optimal weight for each graph by constructing a Laplacian rank constrained graph. [18] propose AWP to learn optimal Procrustes [19] for all the views adaptively.
However, these multi-view spectral clustering methods mainly have the following two pivotal drawbacks which greatly limit their applications: i) they can not learn the intrinsic structure of data because they neglect of the local structure of data; ii) these learned graphs are not the optimal graphs for clustering because they ignore the connection between features in different views. These two drawbacks will cause these methods to fail in many multi-view clustering tasks with a large number of features because these methods are unable to extract favorable features for clustering from high-dimensional features with noises. For convenience, we call these features favorable for clustering "favorable features" and these features unfavorable for clustering "unfavorable features". For the sake of description, we do not distinguish between views and graphs in this paper. Therefore, to tackle these issues, we try to learn optimal graphs for multi-view clustering by introducing weight matrix to each graph in order to extract favorable features, and these graphs have the following characteristics: 1) each graph corresponds the optimal representation of a view (in the view, these features favorable for clustering are considered and these features unfavorable for clustering are ignored), 2) for all the graphs, these views favorable for clustering are preserved and these views unfavorable for clustering are ignored.
In this paper, we propose a Double Self-weighted Multiview Clustering (DSMC) scheme to obtain these optimal graphs. DSMC performs two self-weighted operations: a) for the first self-weighted operations, it first creates multiple initialized graphs, then weights the different features of these graphs by introducing weight matrices; b) for the second selfweighted operations, it weights different graphs by learning a weight coefficient. Furthermore, we design an adaptive multiple graphs fusion method to integrate these graphs by fusing favorable features, which can reduce the influence of noises.
In summary, we highlight the main contributions of our proposed DSMC as follows:
• DSMC is an innovative method to construct optimal graphs. It can weigh these graphs twice and fuses these graphs adaptively to learn optimal graphs for clustering, thus improving clustering results. • Double self-weighted operations are performed to integrate optimal graphs. For the first self-weighted operation, DSMC can remove unfavorable features and extract favorable features by introducing a weight matrix. For the second self-weighted operation, DSMC can integrate these features by learning a suitable weight for each graph. • It fuses these graphs adaptively to simplify the computation of our self-weighted framework and improve clustering performance by redefining the weight of each graph. Experimental results show that it achieves better performance than state-of-the-art multi-view spectral clustering approaches. Empirical comparisons also show the promising efficiencies of DSMC. The rest of the paper is organized as follows. Section II overviews related work, proposes our DSMC method and analyzes it. Section III leverages an iteration procedure to solve our DSMC. Section IV shows the experimental results and analysis. Section V concludes the paper.
Notation: For convenience, we introduce some notation through the paper. All the matrices are written as uppercase. For a matrix A, its ij-th element and i-th column are denoted by a ij and a i separately; the trace of A is denoted by Tr(A); the Frobenius norm of A is denoted by ||A|| F ; E is a matrix which all elements are 1. For the data matrix of one view, it is denoted by X ∈ R p×d , where p is the number of instances and d is the dimension of features. The weighted graph in spectral clustering is denoted by W and w ij = exp(−
||xi−xj || 2 2 2σ 2 ),
where σ is the bandwidth parameter. D ∈ R p×p denotes the diagonal matrix and d ii = p j=1 w ij . L denotes the normalized graph Laplacian matrix, and L = I − D −1/2 W D −1/2 , where I is an identity matrix. F denotes the clustering indicator matrix. c denotes the number of clusters. For a matrix A, A 1/2 is the element-wise square root of A, i.e., each element of A 1/2 is (a ij ) 1/2 . denotes the element-wise multiplication. k denotes the number of clusters. 1 is a column vector and each element of the vector is 1. For a matrix X, X ∈ Def denotes X ∈ {X ∈ {0, 1} p×k |X1 = 1}.
II. METHODOLOGY
In this section, we first revisit some classic methods of multi-view spectral clustering.
A. Multi-view Spectral Clustering Revisit
Multi-view spectral clustering is popular in many multiview clustering tasks for its simplicity and effectiveness. Given an input data matrix X v ∈ R p×dv , and each column of X v is an instance vector, where d v is the dimension of features in the v-th view. To group these instances into c clusters, the classic framework based on multi-view spectral clustering is as follows:
min n v=1 T r(F L (v) F ) s.t. F T F = I(1)
For the framework, a direct way to use each graph obtained from Eq. (1) is to stack up to a new normalized Laplacian matrix and put it into the standard spectral analysis model. But the simple way neglects the differences of different graphs and may add an unreliable graph to Eq. (1), which will load to unsatisfactory clustering results. To improve the clustering results, [18] propose AWP to learn optimal Procrustes for all the views adaptively, and the framework of AWP is as follows:
min n v=1 w v ||Y − F (v) R (v) || 2 F s.t. Y ∈ Def, (R (v) ) T R (v) = I.(2)
B. Dual Self-weighted Multi-view Clustering
Eq.
(2) treats all features (in each view) equally important for clustering. This treatment is harmful to cluster largescale multi-view data, which always has a large number of unimportant features that are useless for clustering (called terrible features), especially in image clustering tasks and text clustering tasks. However, a robust self-weighted multiview clustering methods should not only assign optimal weight to each view, but also extract important features to improve clustering results. Motivated by this, we propose Dual Selfweighted Multi-view Clustering (DSMC) as follows:
min n v=1 w v ||(M (v) ) 1/2 (Y − F (v) R (v) )|| 2 F + µ 2 ||M (v) || 2 F s.t. Y ∈ Def, (R (v) ) T R (v) = I, M (v) > 0, (M (v) ) T E = E,(3)
where M (v) is the weighted matrix, in the v-th view, to assign different weights to different features. (M (v) ) 1/2 is the element-wise square root of M (v) . denotes the element-wise multiplication. µ is a hyper-parameter to controls the trade-off of corresponding terms, and E is a matrix which all elements are 1. By introducing weighted matrix M (v) to Eq. (2), we can reinforce the effect of important features by assigning large weights to them, thus learning robust graphs. Besides, we add the constraint term (M (v) ) T E = E to Eq. (2) to treat each instance equally.
Similar to [18], we can obtain w v as follows:
w v = 1 2(||(M (v) ) 1/2 (Y − F (v) R (v) )|| F ) .(4)
From Eq. (3) and Eq. (4), our DSMC can be formulated as:
min v w v ||(M (v) ) 1/2 (Y − F (v) R (v) )|| 2 F + µ 2 ||M (v) || 2 F w v = 1 2(||(M (v) ) 1/2 (Y − F (v) R (v) )|| F ) s.t. Y ∈ Def, (R (v) ) T R (v) = I, S > 0, (M (v) ) T E = E(5)
Note that w v in Eq. (5) is dependent on the target variable Y , so it is not directly available. But we can first set w v stationary, and update it after obtaining Y [18].
C. Adaptive View Fusion
To further improve the clustering performance, we design an adaptive graph fusion method to adaptively fuse these graphs in Eq. (5) by integrating important features from different graphs. To simplify the calculation of Eq. (5), borrowing the idea of alternating direction method of multipliers (ADMM) [20], we can update Eq. (5) as follows:
min v w v ||(M (v) ) 1/2 U (v) || 2 F + µ 2 ||M (v) || 2 F w v = 1 2(||(M (v) ) 1/2 U (v) || F ) s.t. Y ∈ Def, (R (v) ) T R (v) = I, S > 0, (M (v) ) T E = E, U (v) = Y − F (v) R (v) .(6)
In Eq. (6), the difference between (Y − F (v) R (v) ) and U (v) will affect the update of w v , thus decrease clustering accuracy.
To improve clustering results, we redefine the w v as follows:
w v = 1 2(||(M (v) ) 1/2 (Y − F (v) R (v) )|| F )(7)
Our redefinition of w v is not only simple in form, but also has two advantages as follows:
1) It saves the computational cost. We set the update of Y and the update of R (v) as the first and second steps in optimization respectively (see Section III in more detail). Therefore, after Y and R (v) are updated, we can update w v immediately with the updated Y and R (v) (the update of w v can be viewed parallel with the update of other variables (i.e., U (v) , M (v) , C (v) )). Moreover, the time complexity of updating w v is much smaller than the sum of time complexity of updating other variables. Therefore, for the entire optimization, the calculation of updating w v does not affect the total time complexity. 2) It improves the accuracy of the calculation. It reduces the impact of difference between (Y − F (v) R (v) ) and U (v) . As a result, our final adaptive graph fusion method as follows:
min v w v ||(M (v) ) 1/2 U (v) || 2 F + µ 2 ||M (v) || 2 F w v = 1 2(||(M (v) ) 1/2 (Y − F (v) R (v) )|| F ) s.t. Y ∈ Def, (R (v) ) T R (v) = I, S > 0, (M (v) ) T E = E, U (v) = Y − F (v) R (v) .(8)
III. OPTIMIZATION Since Eq. (8) is not convex for all the variables simultaneously, inspired by ADMM [20], we leverages an iteration procedure to update these variables.
Firstly, we form the augmented Lagrangian function of Eq. (8) as follows: where C (v) is the Lagrange multiplier of the v-th view.
J = v (w v ||(M (v) ) 1/2 U (v) || 2 F + µ 2 ||M (v) || 2 F + µ 2 ||Y − F (v) R (v) − U (v) + C (v) µ || 2 F ),(9)
Then, we design a six-step iteration procedure to solve Eq. (9).
Step 1. Updating Y . Fix the other variables, and the problem to solve variable Y is degraded to minimize the following problem:
J(Y ) = ||Y − F (v) R (v) − U (v) + C (v) µ || 2 F .(10)
We can obtain Y by setting the derivative of J(Y ) w.r.t. Y to zero as follows:
∂J(Y ) ∂Y = Y − F (v) R (v) − U (v) + C (v) µ = 0 ⇔Y = F (v) R (v) + U (v) − C (v) µ .(11)
Step 2. Updating R (v) . Fix the other variables, and the problem to solve variable R (v) is degraded to minimize the following problem:
J(R (v) ) = ||Y − F (v) R (v) − U (v) + C (v) µ || 2 F .(12)
Note that the objective function Eq. (9) w.r.t. R (v) is additive and the constraints w.r.t. R (v) is separable. We can update R (v) individually, which is equivalent to the Orthogonal Procrustes Problem. We have following closed-form solution for it. Thus, for R (v) in Eq. (1), it is updated according to
Lemma 1. For problem min
R T R=I ||M − N R|| 2 F , there is a closed-form solution of R, that is R * = U V T , where U , VR (v) = U (v) V (v) T ,(13)
where
F (v) T (Y − U (v) + C (v) µ ) = U (v) Σ (v) V (v) T .
Step 3. Updating M (v) . Fix the other variables, and the problem to solve variable M (v) is degraded to minimize the following problem: When U (v) is fixed, Eq. (14) can be rewritten as:
J(M (v) ) =w v ||(M (v) ) 1/2 U (v) || 2 F + µ 2 ||M (v) || 2 F .(14)p i=1 dv j=1 (w v m (v) ij (u (v) ij ) 2 + µ 2 (m (v) ij ) 2 ) ⇔ p i=1 dv j=1 (m (v) ij + w v (u (v) ij ) µ ) 2 .(15)
Note that for each view, Eq. (18) is independent for different j and we can update M (v) by solving its each column m
(v) j separately as follows:
dv j=1 ||m (v) j + 1 µ k (v) j || 2 2 ,(16)
where k
(v) j is j-th column of matrix K (v) = U (v) U (v) for the v-th view.
To simplify the calculation of m (v) j , we reformulate Eq. (16) as the following Lagrangian function:
L(m (v) j , α (v) j , β (v) j ) = 1 2 ||m (v) j + 1 µ k (v) j || 2 2 − α (v) j ((m (v) j ) T 1 − 1) − (β (v) j ) T m (v) j ,(17)
where α Based on KKT condition, we can update m (v) j as follows:
m (v) j = max(α (v) j 1 − 1 µ k (v) j , 0).(18)
It is obvious that (m
(v) j ) T 1 = 1, and we can obtain α (v) j by n i=1 (α (v) j − 1 µ k (v) j ) = 1 ⇔α (v) j = 1 p + 1 pµ p i=1 k (v) j .(19)
To obtain optimal M (v) , we can update it after calculating α
(v) j .
Step 4. Updating U (v) . Fix the other variables, and the problem to solve variable U (v) is degraded to minimize the following problem: Define and Eq. (20) can be rewritten as follows:
J(U (v) ) = w v ||(M (v) ) 1/2 U (v) || 2 F + µ 2 ||Y − F (v) R (v) − U (v) + C (v) µ || 2 F .(20)H (v) = Y − F (v) R (v) + C (v) µ ,w v ||(M (v) ) 1/2 U (v) || 2 F + µ 2 ||U (v) − H (v) || 2 F ⇔ p i=1 dv j=1 (w v m (v) ij (u (v) ij ) 2 + µ 2 (m (v) ij − (h (v) ij )) 2 ) ⇔ p i=1 dv j=1 (u (v) ij − µh (v) ij µ + 2w v m (v) ij ) 2 .(21)
We can obtain the optimal solution to each element u
(v)
ij of variable U (v) by setting Eq. (21) to zero as follows:
u (v) ij = µh (v) ij µ + 2w v m (v) ij .(22)
For Step 3 and Step 4, M (v) and U (v) need to be calculated iteratively based on sub-loop in theory, which will lead to excessive computation and time consumption. To simplify computation, we only update them one time in a loop.
Step 5. Updating w v . Fix the other variables, and we can update the variable w v by
w v = ||(M (v) ) 1/2 (Y − F (v) R (v) )|| F .(23)
Step 6. Updating C (v) . Fix the other variables, the variable C (v) can be updated by
C (v) = C (v) + µ(Y − F (v) R (v) − U (v) ) µ = min(µ max , 1.1µ),(24)
where µ max is the constant [21]. The DSMC algorithm is shown in Algorithm 1.
IV. EXPERIMENTS AND ANALYSIS A. Datasets
We conduct the experiments on six real-world multi-view datasets as follows: 3 Sources 1 , ORL [22], NUS [23], 20 News Groups (20NGs) 2 , Scene [24] and BBC [25], whose important statistics summarized in Table I.
B. Compared Methods
Following [18], we compare our proposed method DSMC with the following state-of-the-art multi-view clustering methods:
(1)MMSC learns a shared graph Laplacian matrix by integrating different image features [15].
(2)RMSC learns the standard Markov chain by constructing a graph with fewer noises for each view [16].
(3)MLAN learns the local structure of multi-view data by constructing a constrained graph Laplacian [10].
(4)SwMC learns an optimal weight for each graph by constructing a Laplacian rank constrained graph [17].
(5)AWP learns optimal Procrustes for all the views adaptively [18].
Following [18], we evaluate the results by three popular metrics: Accuracy (ACC), Normalized Mutual Information (NMI) and Purity. Table II to Table IV shows the ACC, NMI and Purity values on six real-world datasets, respectively. Bold numbers denote the best result. a) Results on 3 Sources dataset:: DSMC outperforms all the other methods significantly on 3 Sources. Specifically, compared with SwMC, we improve ACC by 38.47%, NMI by 61.32%, Purity by 40.83%. Relative to the latest method AWP, DSMC improves ACC by 8.29%, NMI by 9.90%, Purity by 5.92%. One possible reason for our outstanding clustering performance is that each view of 3 Sources dataset has a large number of important features, and our methods can effectively extract these important features.
C. Experimental Results
b) Results on ORL dataset:: DSMC has better performances than the other methods on ORL. Obviously, compared with MMSC, DSMC improves ACC by 49.5%, NMI by 40.42%, Purity by 52.5%. Relative to AWP, it improves ACC by 6.00%, NMI by 2.31%, Purity by 6.75%. It is because each view of ORL has a large number of features, DSMC can distinguish the importance of these features by adaptively weighting them. In summary, MMSC and SwMC often obtain poor clustering results. It is because they only learn one weight for different views, and do not take into account the local information on different features in each view. When clustering these datasets with a large number of instances (e.g., Scene and NUS), these local information will have a great impact on our clustering results. Meanwhile, although AWP can extract local information of the data, it cannot distinguish the importance of different features. Therefore, when we cluster multi-view data with high-dimensional features (e.g., BBC), AWP cannot obtain satisfactory results. By weighting different features, our proposed DSMC can effectively integrate different features, and adaptively fuse different graphs. Finally, DSMC always achieves the best performances among the compared state-ofthe-art multi-view clustering methods.
D. Convergence Study
In Figure 1, we show the convergence curve, ACC values and NMI values w.r.t. the number of iterations 3 . The blue solid curve indicates the value of the objective function and the dashed lines represent the clustering performance (ACC and NMI) of our proposed DSMC.
Obviously, for all the datasets, DSMC converges in less than 10 steps, which proves the effectiveness and efficiency of our adaptive multiple graphs fusion. It is because after each iteration, DSMC can extract the favorable features of each view with the help of weight matrix M (v) , and integrate different features of each view by w v . When suitable weights are learned, the gradient of the objective function (Eq. (8)) is close to the ideal gradient, which accelerates the convergence of the objective function.
V. CONCLUSION
In this paper, we propose an effective dual self-weighted multi-view clustering framework, named DSMC. DSMC can remove redundant features and noises from each graph to improve clustering performance. In the framework of DSMC, we assign different weights to different features by imposing an adaptive weight factor. We design an adaptive multiple graphs fusion to fuse the features in the different views, thus integrating different graphs for clustering. We perform experiments on six real-world multi-view datasets to show the effectiveness and efficiency of our DSMC.
Data matrix X (v) ∈ R p×dv , number of clusters k.Initialize weight matrix M (v) = E, weight w v = 1/n, µ = 0.01, C (v) = 0, µ max =10 6 for each view. repeat Update Y by Eq. (11); Update R (v) by Eq. (1); Update M (v) by Eq. (19); Update U (v) by Eq. (22); Update w v by Eq. (23); Update C (v) by Eq. (24); until Eq. (8) converges. Output: M (1) , . . . , M (n) , w 1 , . . . , w n and clustering results.
is constituted by the left and right singular vectors of N T M , respectively.
TABLE I :
IStatistics of the datasets.Dataset
# of instances
# of features in each view
# of views # of clusters
3 Sources
169
3560 / 3631 / 3068
3
6
ORL
400
4096 / 3304 / 6750
3
40
NUS
2400
64 / 114 / 73 / 128 / 225 / 500
6
12
20NGs
500
2000 / 2000 / 2000
3
5
Scene
2688
512 / 432 / 256 / 48
4
8
BBC
685
4659 / 4633 / 4665 / 4684
4
5
TABLE II :
IIACCs (%) on six datasets.Method 3 Sources
ORL
NUS
20NGs Scene
BBC
MMSC
46.75
26.00 13.46
26.00
22.28
31.97
RMSC
40.16
55.17 14.40
88.69
34.57
73.25
MLAN
68.05
72.75 11.08
92.20
51.23
44.38
SwMC
39.64
70.75 15.92
32.6
25.19
24.53
AWP
69.82
69.50 30.79
95.60
67.19
69.49
DSMC
78.11
75.50 30.88
96.60
67.26
86.28
TABLE III :
IIINMIs (%) on six datasets.Method 3 Sources
ORL
NUS
20NGs Scene
BBC
MMSC
30.04
48.40
2.70
3.37
7.52
5.56
RMSC
44.55
51.59 14.26
88.64
33.92
67.41
MLAN
48.04
83.84
3.04
79.85
46.94
24.76
SwMC
11.81
83.31
6.83
16.46
15.51
4.62
AWP
63.23
86.51 18.00
86.66
54.61
59.85
DSMC
73.13
88.82 18.11
87.80
54.69
70.55
TABLE IV :
IVPurities (%) on six datasets.
Method 3 Sources
ORL
NUS
20NGs Scene
BBC
MMSC
56.21
27.75 14.29
27.00
24.93
36.50
RMSC
40.78
59.32 14.55
88.74
24.93
36.50
MLAN
71.60
77.25 11.33
92.20
53.65
47.30
SwMC
44.38
76.75 16.33
33.40
26.64
25.11
AWP
79.29
73.50 33.46
95.60
67.19
70.36
DSMC
85.21
80.25 33.54
96.00
67.26
86.28
c )
cResults on NUS dataset:: DSMC outperforms all the other methods significantly on NUS. Especially, compared with MLAN, our DSMC improves ACC by 19.8%, NMI by 15.07%, Purity by 22.21%. Relative to the latest method AWP, DSMC improves ACC by 8.29%, NMI by 9.90%, Purity by 5.92%. The main reason is that NUS has 6 views and our DSMC can effectively integrate favorable features from different graphs. d) Results on 20NGs dataset:: For 20NGs dataset, DSMC obtains better clustering results than most of the other methods. Moreover, compared with MMSC, it improves ACC by 70.6%, NMI by 84.43%, Purity by 69%. Relative to the latest method AWP, DSMC improves ACC by 8.29%, NMI by 9.90%, Purity by 5.92%. The main reason for this phenomenon is that 20NGs dataset has many unfavorable features, which plays a negative role in learning optimal graphs. But our DSMC is able to remove these unfavorable features based on the weight matrix M (v) . RMSC also achieves satisfactory clustering results. It is because each view of 20NGs has the same feature dimension (2000 features), which helps RMSC obtain the satisfactory Markov chain. e) Results on Scene dataset:: Obviously, our proposed DSMC outperforms all the other methods significantly on Scene. In addition, compared with MMSC, DSMC improves ACC by 44.98%, NMI by 47.17%, Purity by 42.33%. AWP can achieve good clustering results because each view of this dataset contains a small number of features, which makes it easy for AWP to learn optimal Procrustes. Compared with AWP, our proposed DSMC improves ACC by 7%, NMI by 8%, Purity by 7%. This is mainly because DSMC is able to weigh different features through weight matrix M (v) , thereby extracting favorable features from views with different feature dimensions. f) Results on BBC dataset:: When clustering BBC dataset, DSMC has better clustering performance than the other methods. Specifically, compared with SwMC, our proposed DSMC raises ACC by 61.75%, NMI by 65.93%, Purity by 62.17%. Relative to the latest method AWP, DSMC improves ACC by 16.79%, NMI by 10.7%, Purity by 15.92%. The main reason is that our DSMC can adaptively integrate different features multiple graphs fusion.
Fig. 1: Convergence study on six datasets.10
15
20
25
30
35
40
Interation
19
19.2
19.4
19.6
19.8
20
20.2
Obj
Fuc
Val
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Performances
Fuction Value
ACC
NMI
(a) Study on 3 Sources
5
10
15
20
25
30
Interation
25.6
25.8
26
26.2
26.4
26.6
26.8
27
27.2
Obj
Fuc
Val
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Performances
Fuction Value
NMI
ACC
(b) Study on ORL
5
10
15
20
25
30
Interation
25.5
26
26.5
27
Obj
Fuc
Val
0.1
0.15
0.2
0.25
0.3
0.35
Performances
Fuction Value
ACC
NMI
(c) Study on NUS
5
10
15
20
25
30
Interation
35
35.2
35.4
35.6
35.8
Obj
Fuc
Val
0.5
0.6
0.7
0.8
0.9
1
Performances
Fuction Value
ACC
NMI
(d) Study on 20NGs
5
10
15
20
25
30
Interation
98.8
99
99.2
99.4
99.6
99.8
100
Obj
Fuc
Val
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Performances
Fuction Value
ACC
NMI
(e) Study on Scene
5
10
15
20
25
30
Interation
48.5
49
49.5
50
Obj
Fuc
Val
0.2
0.4
0.6
0.8
1
Performances
Fuction Value
ACC
NMI
(f) Study on BBC
http://mlg.ucd.ie/datasets/3sources.html. 2 http://kdd.ics.uci.edu/databases/20newsgroups/20newsgroups.html.
"Obj Fun Val" is the abbreviation of "Objetive Function Value".
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A bayesian hierarchical model for learning natural scene categories. L Fei-Fei, P Perona, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05). IEEE2L. Fei-Fei and P. Perona, "A bayesian hierarchical model for learning natural scene categories," in 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), vol. 2. IEEE, 2005, pp. 524-531.
Practical solutions to the problem of diagonal dominance in kernel document clustering. D Greene, P Cunningham, Proceedings of the 23rd international conference on Machine learning. the 23rd international conference on Machine learningACMD. Greene and P. Cunningham, "Practical solutions to the problem of diagonal dominance in kernel document clustering," in Proceedings of the 23rd international conference on Machine learning. ACM, 2006, pp. 377-384.
|
[] |
[
"A Deep Generative Deconvolutional Image Model",
"A Deep Generative Deconvolutional Image Model"
] |
[
"Yunchen Pu \nDuke University Bell Labs Duke University Duke University Duke University\n\n",
"Xin Yuan \nDuke University Bell Labs Duke University Duke University Duke University\n\n",
"Andrew Stevens \nDuke University Bell Labs Duke University Duke University Duke University\n\n",
"Chunyuan Li \nDuke University Bell Labs Duke University Duke University Duke University\n\n",
"Lawrence Carin \nDuke University Bell Labs Duke University Duke University Duke University\n\n"
] |
[
"Duke University Bell Labs Duke University Duke University Duke University\n",
"Duke University Bell Labs Duke University Duke University Duke University\n",
"Duke University Bell Labs Duke University Duke University Duke University\n",
"Duke University Bell Labs Duke University Duke University Duke University\n",
"Duke University Bell Labs Duke University Duke University Duke University\n"
] |
[] |
A deep generative model is developed for representation and analysis of images, based on a hierarchical convolutional dictionary-learning framework. Stochastic unpooling is employed to link consecutive layers in the model, yielding top-down image generation. A Bayesian support vector machine is linked to the top-layer features, yielding max-margin discrimination. Deep deconvolutional inference is employed when testing, to infer the latent features, and the top-layer features are connected with the max-margin classifier for discrimination tasks. The model is efficiently trained using a Monte Carlo expectationmaximization (MCEM) algorithm, with implementation on graphical processor units (GPUs) for efficient large-scale learning, and fast testing. Excellent results are obtained on several benchmark datasets, including ImageNet, demonstrating that the proposed model achieves results that are highly competitive with similarly sized convolutional neural networks.
| null |
[
"https://arxiv.org/pdf/1512.07344v1.pdf"
] | 1,389,262 |
1512.07344
|
9699a0989108fafbcaf6cfa5fce4b08610493bc0
|
A Deep Generative Deconvolutional Image Model
Yunchen Pu
Duke University Bell Labs Duke University Duke University Duke University
Xin Yuan
Duke University Bell Labs Duke University Duke University Duke University
Andrew Stevens
Duke University Bell Labs Duke University Duke University Duke University
Chunyuan Li
Duke University Bell Labs Duke University Duke University Duke University
Lawrence Carin
Duke University Bell Labs Duke University Duke University Duke University
A Deep Generative Deconvolutional Image Model
A deep generative model is developed for representation and analysis of images, based on a hierarchical convolutional dictionary-learning framework. Stochastic unpooling is employed to link consecutive layers in the model, yielding top-down image generation. A Bayesian support vector machine is linked to the top-layer features, yielding max-margin discrimination. Deep deconvolutional inference is employed when testing, to infer the latent features, and the top-layer features are connected with the max-margin classifier for discrimination tasks. The model is efficiently trained using a Monte Carlo expectationmaximization (MCEM) algorithm, with implementation on graphical processor units (GPUs) for efficient large-scale learning, and fast testing. Excellent results are obtained on several benchmark datasets, including ImageNet, demonstrating that the proposed model achieves results that are highly competitive with similarly sized convolutional neural networks.
Introduction
Convolutional neural networks (CNN) (LeCun et al., 1989) are effective tools for image and video analysis (Chatfield et al., 2014;Krizhevsky et al., 2012;Mnih et al., 2013;Sermanet et al., 2013). The CNN is characterized by feedforward (bottom-up) sequential application of convolutional filterbanks, pointwise nonlinear functions (e.g., sigmoid or hyperbolic tangent), and pooling. Supervision in CNN is typically implemented via a fully-connected layer at the top of the deep architecture, usually with a softmax classifier (Ciresan et al., 2011;He et al., 2014;Jarrett et al., 2009;Krizhevsky et al., 2012).
A parallel line of research concerns dictionary learning Appearing in Proceedings of the 19 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 41. Copyright 2016 by the authors. (Mairal et al., 2008;Zhang and Li, 2010;Zhou et al., 2012) based on a set of image patches. In this setting one imposes sparsity constraints on the dictionary weights with which the data are represented. For image analysis/processing tasks, rather than using a patch-based model, there has been recent interest in deconvolutional networks (DN) (Chen et al., 2011(Chen et al., , 2013Zeiler et al., 2010). In a DN one uses dictionary learning on an entire image (as opposed to the patches of an image), and each dictionary element is convolved with a sparse set of weights that exist across the entire image. Such models are termed "deconvolutional" because, given a learned dictionary, the features at test are found through deconvolution. One may build deep deconvolutional models, which typically employ a pooling step like the CNN (Chen et al., 2011(Chen et al., , 2013. The convolutional filterbank of the CNN is replaced in the DN by a library of convolutional dictionaries.
In this paper we develop a new deep generative model for images, based on convolutional dictionary learning. At test, after the dictionary elements are learned, deconvolutional inference is employed, like in the aforementioned DN research. The proposed method is related to Chen et al. (2011Chen et al. ( , 2013, but a complete top-down generative model is developed, with stochastic unpooling connecting model layers (this is distinct from almost all other models, which employ bottom-up pooling). Chen et al. (2011Chen et al. ( , 2013 trained each layer separately, sequentially, with no final coupling of the overall model (significantly undermining classification performance). Further, in Chen et al. (2011Chen et al. ( , 2013 Bayesian posterior inference was approximated for all model parameters (e.g., via Gibbs sampling), which scales poorly. Here we employ Monte Carlo expectation maximization (MCEM) (Wei and Tanner, 1990), with a point estimate learned for the dictionary elements and the parameters of the classifier, allowing learning on large-scale data and fast testing.
Forms of stochastic pooling have been applied previously (Lee et al., 2009;Zeiler and Fergus, 2013). Lee et al. (2009) defined stochastic pooling in the context of an energybased Boltzmann machine, and Zeiler and Fergus (2013) proposed stochastic pooling as a regularization technique. Here unpooling is employed, yielding a top-down generative process.
arXiv:1512.07344v1 [stat.ML] 23 Dec 2015
To impose supervision, we employ the Bayesian support vector machine (SVM) (Polson and Scott, 2011), which has been used for supervised dictionary learning (Henao et al., 2014) (but not previously for deep learning). The proposed generative model is amenable to Bayesian analysis, and here the Bayesian SVM is learned simultaneously with the deep model. The models in Donahue et al. (2014); He et al. (2014); Zeiler and Fergus (2014) do not train the SVM jointly, as we do -instead, the SVM is trained separately using the learned CNN features (with CNN supervised learning implemented via softmax). This paper makes several contributions: (i) A new deep model is developed for images, based on convolutional dictionary learning; this model is a generative form of the earlier DN. (ii) A new stochastic unpooling method is proposed, linking consecutive layers of the deep model. (iii) An SVM is integrated with the top layer of the model, enabling max-margin supervision during training. (iv) The algorithm is implemented on a GPU, for large-scale learning and fast testing; we demonstrate state-of-the-art classification results on several benchmark datasets, and demonstrate scalability through analysis of the ImageNet dataset.
Supervised Deep Deconvolutional Model
Single layer convolutional dictionary learning
Consider N images {X (n) } N n=1 , with X (n) ∈ R Nx×Ny×Nc , where N x and N y represent the number of pixels in each spatial dimension; N c = 1 for gray-scale images and N c = 3 for RGB images. We start by relating our model to optimization-based dictionary learning and DN, the work of (Mairal et al., 2008;Zhang and Li, 2010) and (Ciresan et al., 2012;Zeiler and Fergus, 2014;Zeiler et al., 2010), respectively. The motivations for and details of our model are elucidated by making connections to this previous work. Specifically, consider the optimization problem
{D (k) ,Ŝ (n,k) } = argmin N n=1 X (n) − K k=1 D (k) * S (n,k) 2 F +λ1 K k=1 D (k) 2 F + λ2 N n=1 K k=1 S (n,k) 1(1)
where * is the 2D (spatial) convolution operator. Each D (k) ∈ R nx×ny×Nc and typically n x N x , n y N y . The spatially-dependent weights S (n,k) are of size (N x − n x + 1) × (N y − n y + 1). Each of the N c layers of D (k) are spatially convolved with S (n,k) , and after summing over the K dictionary elements, this manifests an approximation for each of the N c layers of X (n) .
The form of (1) is as in Mairal et al. (2008), with the 1 norm on S (n,k) imposing sparsity, and with the Frobenius norm on D (k) ( 2 in Mairal et al. (2008)) imposing an expected-energy constraint on each dictionary element; in Mairal et al. (2008) convolution is not used, but otherwise the model is identical, and the computational methods developed in Mairal et al. (2008) may be applied.
The form of (1) motivates choices for the priors in the proposed generative model. Specifically, consider
X (n) = K k=1 D (k) * S (n,k) + E (n) ,(2)E (n) ∼ N (0, γ −1 e I), D (k) ∼ N (0, I) (3) with S (n,k) i,j denoting element (i, j) of S (n,k) , drawn S (n,k) i,j ∼ Laplace(0, b) = 1 2b exp(−|S (n,k) i,j |/b).
We have "vectorized" the matrices E (n) and D (k) (from the standpoint of the distributions from which they are drawn), and I is an appropriately sized identity matrix. The maximum a posterior (MAP) solution to (3), with the Laplace prior imposed independently on each component of S (n,k) , corresponds to the optimization problem in (1), and the hyperparameters γ e and b play roles analogous to λ 1 and λ 2 .
The sparsity of S (n,k) manifested in (1) is a consequence of the geometry imposed by the 1 operator; the MAP solution is sparse, but, with probability one, any draw from the Laplace prior on S (n,k) is not sparse (Cevher, 2009). To impose sparsity on S (n,k) within the generative process, we consider the spike-slab (Ishwaran and Rao, 2005) prior:
S (n,k) i,j ∼ [z (n,k) i,j N (0, γ −1 s ) + (1 − z (n,k) i,j )δ 0 ], z (n,k) i,j ∼ Bern(π (n,k) ), π (n,k) ∼ Beta(a 0 , b 0 )(4)
where z (n,k) i,j ∈ {0, 1}, δ 0 is a unit point measure concentrated at zero, and (a 0 , b 0 ) are set to encourage that most π (n,k) are small (Paisley and Carin, 2009), i.e., a 0 = 1/K and b 0 = 1 − 1/K. For parameters γ e and γ s we impose the priors γ s ∼ Gamma(a s , b s ) and γ e ∼ Gamma(a e , b e ), with hyperparameters a s = b s = a e = b e = 10 −6 to impose diffuse priors (Tipping, 2001).
Generative Deep Model via Stochastic Unpooling
The model in (2) is motivated by the idea that each image X (n) may be represented in terms of convolutional dictionary elements D (k) that are shared across all N images. In the proposed deep model, we similarly are motivated by the idea that the feature maps S (n,k) may also be represented in terms of convolutions of (distinct) dictionary elements. Consider a two-layer model, with
X (n,2) = K2 k2=1 D (k2,2) * S (n,k2,2)(5)
S (n,k1,1) ∼ unpool(X (n,k1,2) ) , k 1 = 1, . . . , K 1 (6) where X (n,1) = X (n) . Dictionary elements D (k1,1) replace D (k) in (2), for representation of X (n) . The weights S (n,k1,1) are connected to X (n,2) via the stochastic operation unpool(·), detailed below. Motivated as discussed above, X (n,2) is represented in terms of convolutions with second-layer dictionary elements D (k2,2) . The forms of the priors on D (k1,1) and D (k2,2) are as above for D (k) , and the prior on E (n) is unchanged.
X (n,1) = K1 k1=1 D (k1,1) * S (n,k1,1) + E (n)(7)
The tensor X (n,2) ∈ R N (2)
x ×N (2) y ×K1 with layer/slice k 1 denoted by the matrix X (n,k1,2) ∈ R N (2)
x ×N (2) y (k 1 ∈ {1, . . . , K 1 }). Matrix X (n,k1
,2) is a pooled version of S (n,k1,1) . Specifically, S (n,k1,1) is partitioned into contiguous spatial pooling blocks, each pooling block of dimension p
(1) x ×p (1) y , with N (2) x = N x /p (1) x and N (2) y = N y /p
(1) y (assumed to be integers). Each pooling block of S (n,k1,1) is all-zeros except one non-zero element, with the non-zero element defined in X (n,k1,2) . Specifically, element (i, j) of X (n,k1,2) , denoted X (n,k1,2) i,j , is mapped to pooling block
(i, j) in S (n,k1,1) , denoted S (n,k1,1) i,j . Let z (n,k1,1) i,j ∈ {0, 1} p (1) x p (1)
y be a vector of all zeros, and a single one, and the location of the non-zero element of z (n,k1,1) i,j identifies the location of the single non-zero element of pooling block S (n,k1,1) i,j which is set as X (n,k1,2) i,j . The function unpool(·) is a stochastic operation that defines z (n,k1,1) i,j , and hence the way X (n,k1,2) is unpooled to constitute the sparse S (n,k1,2) . We impose
z (n,k1,1) i,j ∼ Mult(1, θ (n,k1,1) ) (8) θ (n,k1,1) ∼ Dir(1/(p (1) x p (1) y ))(9)
where Dir(·) denotes the symmetric Dirichlet distribution; the Dirichlet distribution has a set of parameters, and here they are all equal to the value indicated in Dir(·).
When introducing the above two-layer model, S (n,k2,2) is drawn from a spike-slab prior, as in (4). However, we may extend this to a three-layer model, with pooling blocks defined in S (n,k2,2) . A convolutional dictionary representation is similarly constituted for X (n,3) , and this is stochastically unpooled to generate S (n,k2,2) . This may continued for L layers, where the hierarchical convolutional dictionary learning learns multi-scale structure in the weights on the dictionary elements. At the top layer in the L-layer model, the weights S (n,k L ,L) are drawn from a spike-slab prior of the form in (4).
Consider an L-layer model, and assume that {{D (k l ,l) } K l k l =1 } L l=1 have been learned/specified. An image is generated by starting at the top, drawing
{S (n,k L ,L) } K L k L =1 from a spike-slab model. Then {S (n,k L−1 ,L−1) } K L−1 k L−1 =1 are constituted by convolving {S (n,k L ,L) } K L k L =1 with {D (k L ,L) } K L k L =1
, summing over the K L dictionary elements, and then performing stochastic unpooling. This process of convolution and stochastic unpooling proceeds for K L layers, ultimately yielding K1 k1=1 D (k1,1) * S (n,k1,1) at the bottom (first) layer. With the added stochastic residual E (n) , the image X (n) is specified.
We note an implementation detail that has been found useful in experiments. In (9), the unpooling was performed such that each pooling block in S (n,k l ,l) has a single non-zero element, with the non-zero element defined in X (n,k l ,l+1) . The unpooling for block (i, j) was specified by the p
(l) x p (l) y -dimensional z (n,k l ,l) i,j
vector of all-zeros and a single one. In our slightly modified implementation, we have considered a (p
(l) x p (l) y + 1)-dimensional z (n,k l ,l) i,j ,
which is again all zeros with a single one, and θ (n,k l ,l) is also p (l)
x p (l) y + 1 dimensional. If the single one in z (n,k l ,l) i,j is located among the first p x p (l) y + 1, then all elements of pooling block (i, j) are set to zero. This imposes further sparsity on the feature maps and, as demonstrated in the Supplementary Material (SM), it yields a model in which the elements of the feature map that are relatively small are encouraged to be zero. This turning off of dictionary elements with small weights is analogous to dropout (Srivastava et al., 2014), which has been used in CNN, and in our model it also has been found to yield slightly better classification performance.
Supervision via Bayesian SVMs
Assume that a label n ∈ {1, . . . , C} is associated with each of the N images, so that the training set may be denoted {(X (n) , n )} N n=1 . We wish to learn a classifier that maps the top-layer dictionary weights S (n,L) = {S (n,k l ,L) } K L k l =1 to an associated label n . The S (n,L) are "unfolded" into the vector s n . We desire the classifier mapping s n → n and our goal is to learn the dictionary and classifier jointly.
We design C one-versus-all binary SVM classifiers. For each of these classifiers, the problem may be posed as train-ing with {s n , y ( ) n } N n=1 , where s n are the top-layer dictionary weights, as discussed above, and y ( ) n ∈ {−1, 1} (a bias term is also appended to each s n , as is typical of SVMs). If n = ∈ {1, . . . , C} then y ( ) n = 1, and y ( ) n = −1 otherwise; the indicator specifies which of the C binary SVMs is under consideration. For notational simplicity, we omit the superscript ( ) for the remainder of the section, and consider the Bayesian SVM for one of the binary learning tasks, with labeled data {s n , y n } N n=1 . In practice, C such binary classifiers are learned jointly, and the value of y n ∈ {1, −1} depends on which one-versus-all classifier is being specified.
Given a feature vector s, the goal of the SVM is to find an f (s) that minimizes the objective function
γ N n=1 max(1 − y n f (s n ), 0) + R(f (s)),(10)
where max(1 − y n f (s n ), 0) is the hinge loss, R(f (s)) is a regularization term that controls the complexity of f (s), and γ is a tuning parameter controlling the trade-off between error penalization and the complexity of the classification function. The decision boundary is defined as {s : f (s) = 0} and sign(f (s)) is the decision rule, classifying s as either −1 or 1 (Vapnik, 1995). (2011) showed that for the linear classifier f (s) = β T s, minimizing (10) is equivalent to estimating the mode of the pseudo-posterior of β
Recently, Polson and Scott
p(β|S, y, γ) ∝ N n=1 L(y n |s n , β, γ)p(β|·) ,(11)
where y = [y 1 . . . y N ] T , S = [s 1 . . . s N ], L(y n |s n , β, γ) is the pseudo-likelihood function, and p(β|·) is the prior distribution for the vector of coefficients β. Choosing β to maximize the log of (11) corresponds to (10), where the prior is associated with R(f (s)). Polson and Scott (2011) showed that L(y n |s n , β, γ) admits a location-scale mixture of normals representation by introducing latent variables λ n , such that
L(y n |s n , β, γ) = e −2γ max(1−ynβ T sn,0) = ∞ 0 √ γ √ 2πλn exp − (1+λn−ynβ T sn) 2 2γ −1 λn dλ n . (12)
Note that the exponential in (12) is Gaussian wrt β. As described in Polson and Scott (2011), this encourages data augmentation for variable λ n (λ n is treated as a new random variable), which permits efficient Bayesian inference (see Polson and Scott (2011);Henao et al. (2014) for details). One of the benefits of a Bayesian formulation for SVMs is that we can flexibly specify the behavior of β while being able to adaptively regularize it by specifying a prior p(γ) as well.
We impose shrinkage (near sparsity) (Polson and Scott, 2010) on β using the Laplace distribution; letting β i denote i th element of β, we impose βi ∼ N (0, ωi), ωi ∼ Exp(κ), κ ∼ Gamma(aκ, bκ), (13) and similar to κ and λ n , a diffuse Gamma prior is imposed on γ.
For the generative process of the overall model, activation weights s n are drawn at layer L, as discussed in Sec. 2.2. These weights then go into the C-class SVM, and from that a class label is manifested. Specifically, each SVM learns a linear function of {β s} C =1 , and for a given data s, its class label is defined by (Yang et al., 2009):
n = argmax β s n .(14)
The set of vectors {β } C =1 , connecting the top-layer features s to the classifier, play a role analogous to the fullyconnected layer in the softmax-based CNN, but here we constitute supervision via the max-margin SVM. Hence, the proposed model is a generative construction for both the labels and the images.
Model Training
The previous section described a supervised deep generative model for images, based on deep convolutional dictionary learning, stochastic unpooling, and the Bayesian SVM. The conditional posterior distribution for each model parameter can be written in closed form, assuming the other model parameters are fixed (see the SM). For relatively small datasets we can therefore employ a Gibbs sampler for both training and deconvolutional inference, yielding an approximation to the posterior distribution on all parameters. Large-scale datasets prohibit the application of standard Gibbs sampling. For large data we use stochastic MCEM (Wei and Tanner, 1990) to find a maximum a posterior (MAP) estimate of the model parameters.
We consolidate the "local" model parameters (latent data-sample-specific variables) as
Φ n = {z (n,l) } L l=1 , S (n,L) , γ (n) s , E (n) , {λ ( ) n } C =1 , the "global" parameters (shared across all data) as Ψ = {D (l) } L l=1
, β , and the data as Y n = (X (n) , n ). We desire a MAP estimator
Ψ MAP = argmax Ψ n ln p(Ψ|Y n ),(15)
which can be interpreted as an EM problem: E-step: Perform an expectation with respect to the local variables, using p(Φ n |Y n , Ψ t−1 ) ∀n, where Ψ t−1 is the estimate of the global parameters from iteration (t − 1).
M-step: Maximize ln p(Ψ) + n E Φn [ln p(Y n |Φ n , Ψ)]
with respect to Ψ.
We approximate the expectation via Monte Carlo sampling, which gives
Q(Ψ|Ψt−1) = ln p(Ψ) + 1 Ns Ns s=1 n ln p(Yn|Φ s n , Ψ),(16)
where Φ s n is a sample from the full conditional posterior distribution, and N s is the number of samples; we seek to maximize Q(Ψ|Ψ t−1 ) wrt Ψ, constituting Ψ t . Recall from above that each of the conditional distributions in a Gibbs sampler of the model is analytic; this allows convenient sampling of local parameters, conditioned on specified global parameters Ψ t−1 , and therefore the aforementioned sampling is implemented efficiently (using minibatches of data, where I t ⊂ {1, . . . , N } identifies the stochastically defined subset of data in mini-batch t). An approximation to the M-step is implemented via stochastic gradient descent (SGD). The stochastic MCEM gradient at iteration t is
∇ Ψ Q = ∇ Ψ ln p(Ψ)+ 1 N s Ns s=1 n∈It ∇ Ψ ln p(Y n |Φ s n , Ψ).
(17) We solve (15) using RMSprop (Dauphin et al., 2015;Tieleman and Hinton, 2012) with the gradient approximation in (17).
In the learning phase, the MCEM method is used to learn a point estimate for the global parameters Ψ. During testing, we follow the same MCEM setup with
Φ test = {z ( * ,l) } L−1 l=1 , γ ( * ) s , E ( * ) , Ψ test = S ( * ,L)
, when given a new image X * . We find a MAP estimator: For the first five (small/modest-sized) datasets, the model is learned via Gibbs sampling. We found that it is effective to use layer-wise pretraining as employed in some deep generative models (Erhan et al., 2010;Hinton and Salakhutdinov, 2006). The pretraining is performed sequentially from the bottom layer (touching the data), to the top layer, in an unsupervised manner. Details on the layerwise pretraining are discussed in the SM. In the pretraining step, we average 500 collection samples, to obtain parameter values (e.g., dictionary elements) after first discarding 1000 burn-in samples. Following pre-training, we refine the entire model jointly using the complete set of Gibbs conditional distributions. 1000 burn-in iterations are performed followed by 500 collection draws, retaining one of every 50 iterations. During testing, the predictions are based on averaging the decision values of the collected samples.
Ψ test MAP = argmax Ψ test ln p(Ψ test |X * , D),(18)
MNIST
The MNIST data (http://yann.lecun.com/exdb/ mnist/) has 60,000 training and 10,000 testing images, each 28 × 28, for digits 0 through 9. A two-layer model is used with dictionary element size 8 × 8 and 6 × 6 at the first and second layer, respectively. The pooling size is 3 × 3 (p x = p y = 3) and the number of dictionary elements at layers 1 and 2 are K 1 = 39 and K 2 = 117, respectively. These numbers of dictionary elements are obtained by setting the initial dictionary number to a relatively large value (K 1 = 50 and K 2 = 200) in the pretraining step and discarding infrequently used elements by counting the corresponding binary indicator z -effectively inferring the number of needed dictionary elements, as in Chen et al. (2011Chen et al. ( , 2013. (Ciresan et al., 2011) and the MCDNN, which combines several deep convolutional neural net- (Ciresan et al., 2011) 0.35 MCDNN (Ciresan et al., 2012) 0.23 works (Ciresan et al., 2012). Specifically, (Ciresan et al., 2012) used a committee of 35 convolutional networks, width normalization, and elastic distortions of the data; (Ciresan et al., 2011) used elastic distortions and a single convolutional neural network to achieve the similar error as our approach.
To further examine the performance of the proposed model, we plot a selection of top-layer dictionary elements (projected through the generative process down to the data plane) learned by our supervised model, on the right of Fig. 2, and on the left we show the corresponding elements inferred by our unsupervised model. It can be seen that the elements inferred by the supervised model are clearer ("unique" to a single number), whereas the elements learned by the unsupervised model are blurry (combinations of multiple numbers). Similar results were reported in Erhan et al. (2010).
Since our model is generative, using it to generate digits after training on MNIST is straightforward, and some examples are shown in Fig. 3 (based on random draws of the top-layer weights). We also demonstrate the ability of the model to predict missing data (generative nature of the model); reconstructions are shown in Fig. 4. More results are provided in the SM.
CIFAR-10 & 100
The CIFAR-10 dataset (Krizhevsky and Hinton, 2009) is composed of 10 classes of natural 32 × 32 RGB images with 50000 images for training and 10000 images for testing. We apply the same preprocessing technique of global contrast normalization and ZCA whitening as used in the Maxout network (Goodfellow et al., 2013). A three-layer model is used with dictionary element size 5 × 5, 5 × 5, 4 × 4 at the first, second and third layer. The pooling sizes are both 2 × 2 and the numbers of dictionary elements for each layer are K 1 = 48, K 2 = 128 and K 3 = 128. If we augment the data by translation and horizontal flipping as used in other models (Goodfellow et al., 2013), we achieve 8.27% error. Our result is competitive with the state-of-art, which integrates supervision on every hidden layer (Lee et al., 2015). In constrast, we only impose supervision at the top layer. Table 2 summarizes the classification accuracy of our models and some related models.
The CIFAR-100 dataset (Krizhevsky and Hinton, 2009) is the same as CIFAR-10 in size and format, except it contains 100 classes. We use the same settings as in the CIFAR-10.
Caltech 101 & 256
To balance speed and performance, we resize the images of Caltech 101 and Caltech 256 to 128 × 128, followed by local contrast normalization (Jarrett et al., 2009). A three layer model is adopted. The dictionary element sizes are set to 7 × 7, 5 × 5 and 5 × 5, and the size of the pooling regions are 4 × 4 (layer 1 to layer 2) and 2 × 2 (layer 2 to layer 3).
The dictionary sizes for each layer are set to K 1 = 48, K 2 = 84 and K 3 = 84 for Caltech 101, and K 1 = 48, K 2 = 128 and K 3 = 128 for Caltech 256. Tables 4 and 5 summarize the classification accuracy of our model and some related models. Using only the data inside Caltech 101 and Caltech 256 (without using other datasets) for training, our results (87.82%, 66.4%) exceed the previous state-of-art results (83%, 58%) by a substantial margin (4%, 12.4%), which are the best results obtained by models without using deep convolutional models (using handcrafted features).
As a baseline, we implemented the neural network consisting of three convolutional layers and two fully-connected layers with a final softmax classifier. The architecture of three convolutional layers is the same as our model. The state-of-the-art results on these two datasets are achieved by pretraining the deep network on a large dataset, ImageNet (Donahue et al., 2014;He et al., 2014;Zeiler and Fergus, 2014). We consider similar ImageNet pretraining in Sec. 4.4. We also observe from Table 5 that when there are fewer training images, our accuracy diminishes. This verifies that the model complexity needs to be selected based on the size of the data. This is also consistent with the results reported by Zeiler and Fergus (2014), in which the classification performance is very poor without training the model on ImageNet. Fig. 5 shows selected dictionary elements learned from the unsupervised and the supervised model, to illustrate the differences. It is observed that the dictionaries without supervision tend to reconstruct the data while the dictionary elements with supervision tend to extract features that will distinguish different classes. For example, the dictionaries learned with supervision have double sides on the image edges. Our model is generative, and as an example we generate images using the dictionaries trained from the "Faces easy" category, with random top-layer dictionary weights (see Fig. 6). Similar to the MNIST example, we also show in Fig. 7 the interpolation results of face data with half the image missing. Though the background is a little noisy, each face is recovered in great detail by the third (top) layer dictionaries. More results are provided in the SM.
ImageNet 2012
We train our model on the 1000-category ImageNet 2012 dataset, which consists of 1.3M/50K/100K training/validation/test images. Our training process follows the procedure of previous work (Howard, 2013;Krizhevsky et al., 2012;Zeiler and Fergus, 2014). The smaller image dimension is scaled to 256, and a 224 × 224 crop is chosen at 1024 random locations within the image. The data are augmented by color alteration and horizontal flips (Howard, 2013;Krizhevsky et al., 2012). A five layer convolutional model is employed (L = 5); the numbers (sizes) of dictionary elements for each layer are set to 96 (5 × 5), 256 (5 × 5), 512 (3 × 3), 1024 (3 × 3) and 512(3 × 3); the pooling ratios are 4 × 4 (layer 1 to 2) and 2 × 2 (others). The number of parameters in our model is around 30 million.
We emphasize that our intention is not to directly compete with the best performance in the ImageNet challenge (Szegedy et al., 2015;Simonyan and Zisserman, 2015), which requires consideration of many additional aspects, but to provide a comparison on this dataset with a CNN with a similar network architecture (size). Table 6 summarizes our results compared with the "ZF"-net developed in Zeiler and Fergus (2014) which has a similiar architecture with ours.
The MAP estimator of our model, described in Sec. 3, achieves a top-5 error rate of 16.1% on the testing set, which is close to Zeiler and Fergus (2014). Model averaging used in Bayesian inference often improves performance, and is considered here. Specifically, after running the MCEM algorithm, we have a (point) estimate of the global parameters. Using a mini-batch of data, one can leverage our analytic Gibbs updates to sample from the posterior (starting from the MAP estimate), and therefore obtain multiple samples for the global model parameters.
We collect the approximate posterior samples every 1000 iterations, and retain 20 samples. Averaging the predictions of these 20 samples (model averaging) gives a top-5 error rate of 13.6%, which outperforms the combination of 6 "ZF"-nets. Limited additional training time (one day) is required for this model averaging. To illustrate that our model can generalize to other datasets, we follow the setup in (Donahue et al., 2014;He et al., 2014;Zeiler and Fergus, 2014), keeping five convolutional layers of our ImageNet-trained model fixed and train a new Bayesian SVM classifier on the top using the training images of Caltech 101 and Caltech 256, with each image resized to 256 × 256 (effectively, we are using ImageNet to pretrain the model, which is then refined for Caltech 101 and 256). The results are shown in Tables 4 and 5. We obtain state-of-art results (77.9%) on Caltech 256. For Caltech 101, our result (93.15%) is competitive with the stateof-the-art result (94.11%), which combines spatial pyramid matching and deep convolutional networks (He et al., 2014). These results demonstrate that we can provide comparable results to the CNN in data generalization tasks, while also scaling well. 6 More Results 6.1 Gnerated images with random weights Generated Images Figure 8: Generated images from the dictionaries trained from MNIST with random dictionary weights at the top of the two-layer model. Figure 9: Generated images from the dictionaries trained from "Faces easy" category of Caltech 256 with random dictionary weights at the top of the three-layer model. Figure 10: Generated images from the dictionaries trained from 'baseball-glove" category of Caltech 256 with random dictionary weights at the top of the three-layer model.
Conclusions
Missing data interpolation
MCEM algorithm
Algorithms 1 and 2 detail the training and testing process. The steps are explained in the next two sections.
Algorithm 1 Stochastic MCEM Algorithm
Require: Input data {X (n) , n } N n=1 . for t = 1 to ∞ do Get mini-batch (Y (n) ; n ∈ I t ) randomly. (49). end for ComputeQ(Ψ|Ψ (t) ) according to (59) for l = 1 to L do Update {δ (n,k l−1 ,l,t) } K l−1 k l−1 =1 according to (64). for k l−1 = 1 to K L−1 do for k l = 1 to K L do Update D (k l−1 ,k l ,l,t) according to (65). end for UpdateX (n,k l−1 ,l,t) := K l k l =1 D (k l−1 ,k l ,l,t) * S (n,k l ,l,t) . UpdateS (n,k l−1 ,l−1,t) = f (X (n,k l−1 ,l,t) ,Z (n,k l−1 ,l−1,t) ). end for end for for = 1 to C do Sample λ ( ) n from the distibution in (57) and compute the sample averageλ ( ,t) n . Update β ( ,t) according to (66). end for end for return A point estimator of D and β.
for s = 1 to N s do Sample {γ (n,k0) e } K0 k0=1 from the distribution in (52); sample {γ s (n,k L ) } K L k L =1 from the distribution in (51); sample {{Z (n,k l ,l) } K l k l =1 } L l=1 from the distribution in (42); sample {S (n,k L ,L) } K L k L =1 from the distribution in
Algorithm 2 Testing
Require: Input test images X ( * ) , learned dictionaires
{{D (k l ,l) } K L k l =1 } L l=1 for t = 1 to T do for s = 1 to N s do Sample {γ (n,k0) e } K0 k0=1 from the distribution in (52); sample {γ s (n,k L ) } K L k L =1 from the distribution in (51); sample {{Z (n,k l ,l) } K l k l =1 } L−1 l=1 from the distribution in (42); end for ComputeQ test (Ψ test |Ψ (t) test ) according to (73) for l = 1 to L do Update {δ ( * ,k l−1 ,l,t) } K l−1 k l−1 =1 according to (64). end for for k L = 1 to K L do
Update Z ( * ,k L ,L) according to (79). Update W ( * ,k L ,L) according to (78). end for end for Compute {S ( * ,k L ,L) } K L k L =1 and get its vector verstion s * . Predict label * =arg max β s * . return the predicted label * and the decision value β s * . 8 Gibbs Sampling
Notations
In the remainder of this discussion, we use the following definitions.
(1) The ceiling function:
ceil(x) = x is the smallest integer that is not less than x.
(2) The summation and the quadratic summation of all elements in a matrix:
if X ∈ R Nx×Ny , sum(X) = Nx i=1 Ny j=1 X ij , X 2 2 = Nx i=1 Ny j=1 X 2 ij .(19)
(3) The unpooling function:
Assume S ∈ R Nx×Ny and X ∈ R Nx/px×Ny/py . Here p x , p y ∈ N are the pooling ratio and the pooling map is
Z ∈ {0, 1} Nx×Ny . Let i ∈ {1, ..., N x /p x }, j ∈ {1, ..., N y /p y }, i ∈ {1, ..., N x }, j ∈ {1, ..., N y }, then f : R N x/px×N y/py × {0, 1} Nx×Ny → R Nx×Ny . If S = f (X, Z) S i,j = X i/px , j/py Z i,j .(20)
Thus, the unpooling process (equation (6) in the main paper) can be formed as:
S (n,k l ,l) = unpool(X (n,k l ,l+1) ) = f (X (n,k l ,l+1) , Z (n,k l ,l) ).
(4) The 2D correlation operation:
Assume B ∈ R N Bx ×N By and C ∈ R N Cx ×N Cy . If A = B C, then A ∈ R (N Bx −N Cx +1)×(N By −N Cy +1) with element (i, j) given by A i,j = N Cx p=1 N Cy q=1 B p+i−1,q+j−1 C p,q .(22)
(5) The "error term" in each layer:
δ (n,k l−1 ,l) i,j = ∂ ∂X (n,k l−1 ,l) i,j γ (n) e 2 K0 k0=1 E (n,k0) 2 2 .(23)
(6) The "generative" function:
This "generative" function measures how much the k th band of l th layer feature is "responsible" for the of input image X (n) in the current model:
g(X, n, k, l) = D (k,1) * f (X, Z (n,k,1) ) if l = 2, K l−1 m=1 g D (m,k,l−1) * f (X, Z (n,k,l−1) ), n, m, l − 1 if l > 2.(24)
It can be considered as if k th band of l th layer feature changes X (i.e. X (n,k,l) → X (n,k,l) + X), the corresponding data layer representation will change g(X, n, k, l) (i.e. X (n) → X (n) + g(X, n, k, l)). Thus, for l = 2, . . . , L, we have
X (n) = K l k=1
g(X, n, k, l) + E (n) .
Note that g() is a linear function for X, which means: g(µ 1 X 1 + µ 2 X 2 , n, k, l) = µ 1 g(X 1 , n, k, l) + µ 2 g(X 2 , n, k, l).
For convenience, we also use the following notations:
• We use Z (n,k l ,l) to represent {z (n,k l ,l) i,j
; ∀i, j}, where the vector version of the (i, j) th block of Z (n,k l ,l) is equal to z (n,k l ,l) i,j .
• 0 denotes the all 0 vector or matrix. 1 denotes the all one vector or matrix. e m denotes a "one-hot" vector with the m th element equal to 1.
Full Conditional Posterior Distribution
Assume the spatial dimension:
X (n,l) ∈ R N l x ×N l y ×K l−1 , D (k l ,l) ∈ R N l dx ×N l dy ×K l−1 , S (n,k l ,l) ∈ R N l Sx ×N l
Sy and Z (n,k l ,l) ∈ {0, 1} N l Sx ×N l Sy . For l = 0, . . . , L, we have k l = 1, . . . , K l . The (un)pooling ratio from l−th layer to (l + 1)−layer is p l x × p l y (where l = 1, . . . , L − 1). We have:
N l x = N l dx + N l Sx − 1, N l Sx = p l x × N (l+1) x ,(27)N l y = N l dy + N l Sy − 1, N l Sy = p l y × N (l+1) y .(28)
Recall that, for l = 2, . . . , L:
X (n,k l−1 ,l) = K l k l D (k l−1 ,k l ,l) * S (n,k l ,l) .(29)
Without loss of generality, we omit the superscript (n, k l−1 , l) below. Each element of X can be represent as:
X i,j = N dx p=1 N dy q=1 D p,q S (i+N dx −p,j+N dy −q) = D p,q S (i+N dx −p,j+N dy −q) + X −(p,q) i,j(30)
where X −(p,q) i,j is a term which is independent of D p,q but related by the index (i, j, p, q); so is S (i+N dx −p,j+N dy −q) . Following this, for every elements in D, we can represent X as:
X = X −(p,q) + D p,q S −(p,q)(31)
where matrices X −(p,q) and S −(p,q) are independent of D p,q but related by the index (p, q) (and the superscript (n, k l−1 , l)). Therefore:
E (n) = X (n) − K l k=1 g(X, n, k, l) (32) = X (n) − K l k=1, =k l−1 g(X, n, k, l) − g(X, n, k l−1 , l) (33) = X (n) − K l k=1, =k l−1 g(X, n, k, l) − g X −(p,q) + D p,q S −(p,q) , n, k l−1 , l (34) = X (n) − K l k=1, =k l−1
g(X, n, k, l) − g X −(p,q) , n, k l−1 , l + g S −(p,q) , n, k l−1 , l D p,q
= C p,q − D p,q F (p,q)(35)
If we add the superscripts back, we have:
E (n) = C (n,k l ,l) p,q + D (n,k l ,l) p,q F (n,k l ,l) p,q ,(37)
where matrices C (n,k l ,l) p,q and F (n,k l ,l) i,j are independent of D (n,k l ,l) p,q but related by the index (n, k l , l, p, q).
Similarly, for every elements in z, we have
E (n) = A (n,k l ,l) i,j,m + z (n,k l ,l) i,j,m B (n,k l ,l) i,j,m .(38)
1. The conditional posterior of D
(k l−1 ,k l ,l) i,j : D (k l−1 ,k l ,l) i,j |− ∼ N (µ (k l−1 ,k l ,l) i,j , σ (k l−1 ,k l ,l) i,j ),(39)
where
σ (k l−1 ,k l ,l) i,j = γ n e 2 F (n,k l ,l) i,j 2 2 + 1 −1 ,(40)µ (k l−1 ,k l ,l) i,j = σ (k l−1 ,k l ,l) i,j sum(C (n,k l ,l) i,j • F (n,k l ,l) i,j ).(41)
2. The conditional posterior of z (n,k l ,l) i,j
:
z i,j |− ∼θ 0 [z i,j = 0] + p l x p l y m=1θ m [z i,j = e m ],(42)whereθ m = θ (m) i,j η (m) i,j θ (0) i,j + pxpŷ m=1 θ (m) i,j η (m) i,j ,(43)θ 0 = θ (0) i,j θ (0) i,j + pxpŷ m=1 θ (m) i,j η (m) i,j ,(44)η (m) i,j = exp − γ e 2 A (m) i,j − B (m) i,j 2 2 − A (m) i,j 2 2 .(45)
For notational simplicity, we omit the superscript (n, k l , l). We can see that when η i,j for all m); this is mentioned in the main paper.
3. The conditional posterior of θ (n,k l ,l) θ (n,k l ,l) |− ∼ Dir(α (n,k l ,l) ),
where α (n,k l ,l) m = 1 p l x p l y + 1 + i j Z (n,k l ,l) i,j,m for m = 1, ..., p l x p l y ,
α (n,k l ,l) 0 = 1 p l x p l y + 1 + i j 1 − m Z (n,k l ,l) i,j,m .(47)
4. The conditional posterior of S
(n,k L ,L) i,j : S (n,k L ,L) i,j |− ∼ (1 − Z (n,k L ,L) i,j )δ 0 + Z (n,k L ,L) i,j N (Ξ (n,k L ,L) i,j , ∆ (n,k L ,L) i,j ),(49)
where
∆ (n,k L ,L) i,j = γ (n) e F (n,k L ,L) i,j Z (n,k L ,L) i,j 2 2 + γ λ ( ) n y ( ) n (Z (n,k L ,L) i,jβ (k L , ) i,j ) 2 + γ (n,k L ) s −1 , Ξ (n,k L ,L) i,j = ∆ (n,k L ,L) i,j Z (n,k L ,L) i,j sum(F (n,k L ,L) i,j • C (n,k L ,L) i,j ) + y ( ) nβ (k L , ) i,j (1 + λ ( ) n ) .(50)
Here we reshape the long vector β ∈ R N l sx N L sy K L ×1 into a matrixβ ∈ R N l sx ×N L sy ×K L which has the same size of S (n,L) .
The conditional posterior of γ
(n,k L ) s : γ (n,k L ) s |− ∼ Gamma a s + N L Sx × N L Sy 2 , b s + 1 2 S (n,k L ,L) 2 2 .(51)
6. The conditional posterior of γ (n) e :
γ (n) e |− ∼ Gamma a 0 + N x × N y × K 0 2 , b 0 + 1 2 K0 k0=1 E (n,k0) 2 2 .(52)
7. The conditional posterior of β :
Reshape the long vector β ∈ R N l sx N L sy K L ×1 into a matrixβ ∈ R N l sx ×N L sy ×K L which has the same size as S (n,L) . We have:β
(k L , ) i,j |− ∼ N (µ (k L , ) i,j , σ (k L , ) i,j ),(53)σ (k L , ) i,j = n γ λ ( ) n y ( ) n (S (n,k L ,L) i,j ) 2 + 1 ω (k L , ) i,j −1 ,(54)µ (k L , ) i,j = σ (n, ) i,j n y ( ) n S (n,k L ,L) i,j (1 + λ ( ) n − Γ (n,k L ,L) −(k,i,j) ) ,(55)Γ (n,k L ,L) −(k,i,j) = k k = i i =i j j =j S (n,k ,L) i ,j β (k , ) i ,j .(56)
8. The conditional posterior of λ
( ) n (λ ( ) n ) −1 ∼ IG(|1 − y n s n β ( ,t) | −1 , 1),(57)
where IG denotes the inverse Gaussian distribution.
MCEM algorithm Details
E step
Recall that we consolidate the "local" model parameters (latent data-sample-specific variables) as Φ n = {z (n,l) } L l=1 , S (n,L) , γ (n)
s , E (n) , {λ ( ) n } C =1
, the "global" parameters (shared across all data) as Ψ = {D (l) } L l=1 , β , and the data as Y n = (X (n) , n ).. At t th iteration of the MCEM algorithm, the exact Q function can be written as:
Q(Ψ|Ψ (t) ) = ln p(Ψ) + n∈It E (Φn|Ψ (t) ,Y,y) {ln p(Y n , Φ n |Ψ)} = −E (Z,γ e ,S (L) ,γ s ,λ|Y,D (t) ,β (t) ) n∈It γ e (n) 2 K0 k0=1 E (n,k0) 2 2 + C =1 (1 + λ n − y n β T s n ) 2 2λ n − 1 2 L l=1 K l−1 k l−1 =1 K l k l =1 D (k l−1 ,k l ,l) 2 2 + const,(58)
where const denotes the terms which are not a function of Ψ.
Obtaining a closed form of the exact Q function is analytically intractable. We here approximate the expectations in (58) by samples collected from the posterior distribution of the hidden variables developed in Section 8.2. The Q function in (58) can be approximated by: (1 +λ n ( ,s,t) − y n β T s n (s,t) ) 2 2λ ( ,s,t) n
Q(Ψ|Ψ (t) ) = − 1 N sNs− 1 2 L l=1 K l−1 k l−1 =1 K l k l =1 D (k l−1 ,k l ,l) 2 2 + const,(59)
whereĒ (n,k0,s,t) = X (n,k0) − K1 k1=1 D (k0,k1,1) * S (n,k1,1,s,t) ,
and for l = 2, . . . , LX (n,k l−1 ,l,s,t) = K l k l =1 D (k l−1 ,k l ,l) * S (n,k l ,l,s,t) ,
S (n,k l−1 ,l−1,s,t) = f (X (n,k l−1 ,l,s,t) ,Z (n,k l−1 ,l−1,s,t) ),
whereS (L,s,t) ,γ e (s,t) ,λ (s,t) andZ (s,t) are a sample of the corresponding variables from the full conditional posterior at the t th iteration. N s is the number of collected samples.
M step
We can maximizeQ(Ψ|Ψ (t) ) via the following updates:
1. For l = 1, . . . , L, k l−1 = 1, . . . , K L−1 and k l = 1, . . . , K L , the gradient wrt D (k l−1 ,k l ,l) is:
∂Q ∂D (k l−1 ,k l ,l,t) = n∈It δ (n,k l−1 ,l,t) S (n,k l ,l,t) + D (k l−1 ,k l ,l,t) ,
where δ (n,k0,1,t) =γ (n,k0,t) e X (n,k0) − K1 k1=1 D (k0,k1,1) * S (n,k1,1,t) , δ (n,k l−1 ,l,t) = f K l−2 k l−2 =1 (δ (n,k l−2 ,l−1,t) D (k l−2 ,k l−1 ,l−1,t) ),Z (n,k l−1 ,l−1,t) .
(64) Following this, the update rule of D based on RMSprop is: v t+1 = αv t + (1 − α)( ∂Q ∂D (k l−1 ,k l ,l,t) ) 2 , D (k l−1 ,k l ,l,t+1) = D (k l−1 ,k l ,l,t) + √ v t+1 ∂Q ∂D (k l−1 ,k l ,l,t) .
(65) 2. For = 1, . . . , C, the update rule of β is: β ( ,t+1) = (Ω ( ,t) ) −1 +s ( ,t) (Λ ( ,t) ) −1s
( ,t) −1s ( ,t) (1 + (Λ ( ,t) ) −1 ),(66)
where (Λ ( ,t) ) −1 = diag((λ ( ,t) n ) −1 ),
(Ω ( ,t) ) −1 = diag(|β ( ,t) | −1 ).
ands ( ,t) denotes a matrix with row n equal to y ns (t) n .
Testing
During testing, when given a test image X ( * ) , we treat S ( * ,L) as model parameters and use MCEM to find a MAP estimator: , γ * s , E * }. The Q function for testing can be represented as:
SQ test (Ψ test |Ψ (t) test ) = E (Φtest|Ψ (t) test , Y ( * ) , D) ln p(X ( * ) , D, Φ test , Ψ test ) .(72)
The testing also follows EM steps: E-step: In the E-step we collect the samples of γ e , γ s and {Z (l) } L−1 l=1 from conditional posterior distributions, which is similar to the training process. Q test can thus be approximated by:
Q test (Ψ test |Ψ (t) test ) = −
and for l = 2, . . . , L − 1S ( * ,k l−1 ,l−1,s,t) = f (X ( * ,k l−1 ,l,t) ,Z ( * ,k l−1 ,l−1,s,t) ),
X ( * ,k l−1 ,l,s,t) = K l k l =1 D (k l−1 ,k l ,l) * S ( * ,k l ,l,s,t) .
M-step: In the M-step, we maximizeQ test via the following updates:
1. The gradient w.r.t. W ( * ,K L ,L) is:
∂Q test ∂W ( * ,k L ,L,t) = K L k L−1 δ ( * ,k L−1 ,L,t) D (k L−1 ,k L ,L) • Z ( * ,k L ,L) +γ ( * ,k L ) s W ( * ,k L ,L,t) ,(77)
where δ ( * ,k L−1 ,L,t) is the same as (64). Therefore, the update rule of W based on RMSprop is: u t+1 = αu t + (1 − α)( ∂Qtest ∂W ( * ,k L ,L,t) ) 2 W ( * ,K L ,L,t+1) = W ( * ,K L ,L,t) + √ u t+1 ∂Qtest ∂W ( * ,k L ,L,t) 10 Bottom-Up Pretraining
Pretraining Model
The model is pretrained sequentially from the bottom layer to the top layer. We consider here pretraining the relationship between layer l and layer l + 1, and this process may be repeated up to layer L. The basic framework of this pretraining process is closely connected to top-down generative process, with a few small but important modifications. Matrix X (n,l) represents the pooled and stacked activation weights from layer l − 1, image n (K l−1 "spectral bands" in X (n,l) , due to K l−1 dictionary elements at layer l − 1). We constitute the representation X (n,l) = K l k l =1 D (k l ,l) * S (n,k l ,l) + E (n,l) ,
The features S (n,k l ,l) can be partitioned into contiguous blocks with dimension p l x × p l y . In our generative model, S (n,k l ,l) is generated from X (n,k l ,l+1) and z (n,k l ,l) , where the non-zero element within the (i, j)−th pooling block of S (n,k l ,l) is set equal to X (n,k l ,l+1) i,j
Figure 1 :
1Demonstration of the stochastic unpooling; one 2 × 2 pooling block is depicted.
location of this non-zero element identifies the location of the single non-zero element in the (i, j) pooling block, as before. However, if the non-zero element of z
Figure 2 :
2Selected layer 2 (top-layer) dictionary elements of MNIST learned by the unsupervised model (left) and the supervised model (right).
Figure 3 :
3Generated images using random dictionary weights.
Figure 4 :
4Missing data interpolation of digits. For each subfigure: (top) Original data, (middle) Observed data, (bottom) Reconstruction.
The fully-connected layers have 1024 neurons each. The results of neural network trained with dropout (Srivastava et al., 2014), after carefully parameter tuning, are also shown in Tables 4 and 5.
Figure 5 :
5Selected dictionary elements of our unsupervised model (top) and supervised model (bottom) trained from Caltech 256.
Figure 6 :
6Generated image from the dictionaries trained from the "Faces easy" category using random dictionary weigths.
Figure 7 :
7Missing data interpolation. (Top) Original data. (Middle) Observed data. (Bottom) Reconstruction.
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Figure 11 :
11Missing data interpolation of digits (left column) and Face easy (right column). For each column: (Top) Original data. (Middle) Observed data. (Bottom) Reconstruction.
j is large,θ m is large, causing the m th pixel to be activated as the unpooling location. When all of the η (m) i,j are small the model will prefer not unpooling -none of the positions m make the model fit the data (i.e., B (m) i,j is not close to A (m)
( * ,L) = argmax S ( * ,L) ln p(S ( * ,L) |X ( * ) , D). (69) Let S ( * ,k L ,L) = W ( * ,k L ,L) • Z ( * ,k L ,L) , where W ( * ,k L ,L) ∈ R N L sx ×N L sy . The marginal posterior distribution can be represented as: p(S ( * ,L) |X * , D) = p(W ( * ,L) , Z ( * ,L) |Y ( * ) ( * ) |W ( * ,L) , Z, E ( * ) , D)p(W ( * ,L) |γ ( * ) s )p(Z)p(γ ( * ) s )p(E ( * ) )dE ( * ) dγ ( * ) s , (71) where /Z (L) = {Z (l) } L−1 l=1 . Let Ψ test = {W ( * ,L) , Z ( * ,L) } and Φ test = {{Z (l) } L−1 l=1
D
(k L−1 ,k L ,L) * W ( * ,k L ,L) • Z ( * ,k L ,L) ,
The update rule Z ( * ,k L ,L) isZ ( * ,k L ,L) i,j = 1 if θ ( * ,k L ,L) η ( * ,k L ,L) i,j > 1 − θ ( * ,k L ,L) 0 otherwise(79)where η ( * ,k L ,L) i,j is the same as (45).
∼
Gamma(a e , b e ).
using MCEM (gradient wrt Ψ test ). In this form of the MCEM, all data-dependent latent variables Φ test are integrated (summed) out in the expectation, except for the toplayer feature map Ψ test , for which the gradient descent M step yields a point estimate. The top-layer features are then sent to the trained SVM to predict the label. Details for training and inference are provided in the SM.We present results for the MNIST, CIFAR-10 & 100, Caltech 101 & 256 and ImageNet 2012 datasets. The same hyperparameter settings (discussed at the end of Section 2.1) were used in all experiments; no tuning was required between datasets.4 Experimental Results
For each of these first five datasets, we show three classification results, using part of or all of our model (to illustrate the role of each component): 1) Pretraining only: this model (in an unsupervised manner) is used to extract features and the futures are sent to a separate linear SVM, yielding a 2-step procedure. 2) Unsupervised model: this model includes the deep generative developed in Sec. 2.2, but is also trained in an unsupervised manner (this is the unsupervised model after refinement). The features extracted by this model are sent to a separate linear SVM, and therefore this is also a 2-step procedure. 3) Supervised model: this is the complete refined supervised model developed in Sec. 2.2 and Sec. 2.3.ImageNet 2012 is used to assess the scalability of our model to large datasets. In this case, we learn the supervised model initialized from the priors (without layerwise pretraining). The proposed online learning method, MCEM, based on RMSProp(Dauphin et al., 2015; Tieleman and Hinton, 2012), is developed for both training and inference with mini-batch size 256 and decay rate 0.95. Our implementation of MCEM learning is based on the publicly available CUDA C++ Caffe toolbox (August 2015 branch)(Jia et al., 2014), but contains significant modifications for our model. Our model takes around one week to train on ImageNet 2012 using a nVidia GeForce GTX TITAN X GPU with 12GB memory. Testing for the validation set of ImageNet 2012 (50K images) takes less than 12 minutes. In the subsequent tables providing classification results, the best results achieved by our model are bold.
Table 1
1summarizes the classification results for MNIST.
Our 2-layer supervised model outperforms most other
modern approaches. The methods that outperforms ours
are the complicated (6-layer) ConvNet model with elas-
tic disortions
Table 1 :
1Classification Error (%) for MNIST
Method
Test error
2-layer convnet (Jarrett et al., 2009)
0.53
Our pretrained model + SVM
1.42
Our unsupervised model + SVM
0.52
Our supervised model
0.37
6-layer convnet
Table 2 :
2Classification Error (%) for CIFAR-10Method
Test error
Without Data Augmentation
Maxout (Goodfellow et al., 2013)
11.68
Network in Network (Lin et al., 2014)
10.41
Our pretrained + SVM
22.43
Our unsupervised + SVM
14.75
Our supervised model
10.39
Deeply-Supervised Nets (Lee et al., 2015)
9.69
With Data Augmentation
Maxout (Goodfellow et al., 2013)
9.38
Network in Network (Lin et al., 2014)
8.81
Our pretrained + SVM
20.62
Our unsupervised + SVM
10.22
Our supervised
8.27
Deeply-Supervised Nets (Lee et al., 2015)
7.97
Table 3
3summarizes the classification accuracy of our
model and some related models. It can be seen that our
results (34.62%) are also very close to the state-of-the-art:
(34.57%) in Lee et al. (2015).
Table 3 :
3Classification Error (%) for CIFAR-100Method
Test error
Maxout (Goodfellow et al., 2013)
38.57
Network in Network (Lin et al., 2014)
35.68
Our pretrained + SVM (2 step)
77.25
Our unsupervised + SVM (2 step)
42.26
Our supervised model
34.62
Deeply-Supervised Nets (Lee et al., 2015)
34.57
Table 4 :
4Classification Accuracy (%) for Caltech 101Training images per class
15
30
Without ImageNet Pretrain
5-layer Convnet (Zeiler and Fergus, 2014)
22.8
46.5
HBP-CFA (Chen et al., 2013)
58
65.7
R-KSVD (Li et al., 2013)
79
83
3-layer Convnet
62.3
72.4
Our pretrained + SVM (2 step)
43.24 53.57
Our unsupervised + SVM (2 step)
70.47 80.39
Our supervised model
75.37 87.82
With ImageNet Pretrain
5-layer Convnet (Zeiler and Fergus, 2014)
83.8
86.5
5-layer Convnet (Chatfield et al., 2014)
-
88.35
Our supervised model
89.1
93.15
SPP-net (He et al., 2014)
-
94.11
Table 5 :
5Classification Accuracy (%) for Caltech 256Training images per class
15
60
Without ImageNet Pretrain
5-layer Convnet (Zeiler and Fergus, 2014)
9.0
38.8
Mu-SC (Bo et al., 2013)
42.7
58
3-layer Convet
46.1
60.1
Our pretrained +SVM
13.4
38.2
Our unsupervised +SVM
40.7
60.9
Our supervised model
52.9
70.5
With ImageNet Pretrain
5-layer Convnet (Zeiler and Fergus, 2014)
65
74.2
5-layer Convnet (Chatfield et al., 2014)
-
77.61
Our supervised model
67.0
77.9
Table 6 :
6ImageNet 2012 classification error rates (%)
Method
top-1
val
top-5
val
Our supervised model
37.9
16.1
"ZF"-net (Zeiler and Fergus, 2014)
37.5
16.0
Our model averaging
35.4
13.6
6 "ZF"-net (Zeiler and Fergus, 2014)
36
14.7
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|
[] |
[
"The phenomenological cornucopia of SU(3) exotica",
"The phenomenological cornucopia of SU(3) exotica"
] |
[
"Linda M Carpenter ",
"Taylor Murphy ",
"Tim M P Tait ",
"\nDepartment of Physics\nDepartment of Physics and Astronomy\nThe Ohio State University\n191 W. Woodruff Ave43212ColumbusOHU.S.A\n",
"\nUniversity of California\n92697Irvine IrvineCAU.S.A\n"
] |
[
"Department of Physics\nDepartment of Physics and Astronomy\nThe Ohio State University\n191 W. Woodruff Ave43212ColumbusOHU.S.A",
"University of California\n92697Irvine IrvineCAU.S.A"
] |
[] |
We introduce an effort to catalog the gauge-invariant interactions of Standard Model (SM) particles and new fields in a variety of representations of the SM color gauge group SU(3) c . In this first installment, we direct this effort toward fields in the six-dimensional (sextet, 6) representation. We consider effective operators of mass dimension up to seven (comprehensively up to dimension six), featuring both scalar and fermionic color sextets. We use an iterative tensor-product method to identify the color invariants underpinning such operators, emphasizing structures that have received little attention to date. In order to demonstrate the utility of our approach, we study a simple but novel model of color-sextet fields at the Large Hadron Collider (LHC). We compute cross sections for an array of new production channels enabled by our operators, including single-sextet production and sextet production in association with photons or leptons. We also discuss dijet-resonance constraints on a sextet fermion. This example shows that there remains a wide array of fairly minimal but well motivated and unexplored models with extended strong sectors as we await the high-luminosity LHC.
|
10.1103/physrevd.105.035014
|
[
"https://export.arxiv.org/pdf/2110.11359v3.pdf"
] | 239,616,133 |
2110.11359
|
b0a309163edff985b697f8e1c12ade09c4f7e0aa
|
The phenomenological cornucopia of SU(3) exotica
Linda M Carpenter
Taylor Murphy
Tim M P Tait
Department of Physics
Department of Physics and Astronomy
The Ohio State University
191 W. Woodruff Ave43212ColumbusOHU.S.A
University of California
92697Irvine IrvineCAU.S.A
The phenomenological cornucopia of SU(3) exotica
(Dated: September 9, 2022)
We introduce an effort to catalog the gauge-invariant interactions of Standard Model (SM) particles and new fields in a variety of representations of the SM color gauge group SU(3) c . In this first installment, we direct this effort toward fields in the six-dimensional (sextet, 6) representation. We consider effective operators of mass dimension up to seven (comprehensively up to dimension six), featuring both scalar and fermionic color sextets. We use an iterative tensor-product method to identify the color invariants underpinning such operators, emphasizing structures that have received little attention to date. In order to demonstrate the utility of our approach, we study a simple but novel model of color-sextet fields at the Large Hadron Collider (LHC). We compute cross sections for an array of new production channels enabled by our operators, including single-sextet production and sextet production in association with photons or leptons. We also discuss dijet-resonance constraints on a sextet fermion. This example shows that there remains a wide array of fairly minimal but well motivated and unexplored models with extended strong sectors as we await the high-luminosity LHC.
I. INTRODUCTION
The search for physics beyond the Standard Model (bSM), spearheaded by the CERN Large Hadron Collider (LHC) [1], continues unabated, spurred on by a constellation of theoretical deficiencies and experimental anomalies that continue to bedevil the Standard Model. These issues range from the well known naturalness problems [2][3][4][5] to the collection of persistent muon [6] and flavor anomalies [7,8] and the ongoing search for particle dark matter [9,10]. While there exist a wide variety of specific, sometimes ultraviolet-complete, frameworks that seek to rectify some or all of the Standard Model's shortcomings, some -in the absence of experimental evidence for any specific bSM model -have turned to an effective field theory (EFT) approach to study new physics in a simplified and more model-independent manner.
The EFT approach, which in the simplest terms allows one to study the experimentally accessible degrees of freedom in a theory while remaining agnostic about physics at higher energy scales, has been used to great effect in many contexts. Some pertinent examples include the Standard Model Effective Field Theory (SMEFT) [11], which now includes (in at least one basis) every independent operator comprising SM fields of up to mass dimension eight [12][13][14] and has lately been used to probe experimental anomalies; and the variety of effective operators employed in supersymmetric frameworks in which supersymmetry breaking is communicated by heavy messengers from a hidden sector to the visible world [15,16].
Lying somewhere between the SMEFT and supersymmetric models on the scale of bSM theories ranked by ex- * [email protected] † [email protected] ‡ [email protected] otic particle content are simplified models in which the SM is augmented only by new matter whose only nontrivial gauge transformations are under the SM gauge group G SM = SU(3) c × SU(2) L × U(1) . A particularly simple but fruitful subset of these models feature a new color-charged SU(2) L singlet perhaps with non-vanishing weak hypercharge.
While new color-charged fields -particularly SU(3) c triplets and octets (adjoints) -occur in a panoply of bSM theories and have long been studied in those specific contexts, there has been far less attempt at a coherent and comprehensive accounting of bSM strong interactions in an EFT framework. We intend to address this gap in the literature, beginning with the present work.
We construct a catalog of effective operators governing fields in the six-dimensional (sextet) representation of SU(3) c , which remain hypothetical but can be copiously produced in proton-proton ( ) collisions and are therefore highly relevant to the ongoing LHC program. Using an iterative tensor-product method to construct new color invariants, we enumerate all possible operators of mass dimension up to six connecting sextets to Standard Model fields, and additionally identify a few potentially interesting dimension-seven operators. Several of these operators have to date received little or no attention. In order to motivate this operator catalog ex post facto, we construct two simplified models of color-sextet fields and explore the phenomenology of these particles at the LHC. These models are notably different from previous models of color sextets in that the novel particles couple not to quark pairs, like the aptly named "sextet diquarks" [17][18][19], but instead to a quark and a gluon. We find fairly light constraints on sextet fermions from a CMS search for dijet resonances [20] and propose future searches for the other interesting multijet + / or lepton signatures generated by these simple models.
This paper is organized as follows. In Section II, we show how to build a catalog of exotic-color operators starting from arXiv:2110.11359v3 [hep-ph] 7 Sep 2022 first principles, and we produce such a catalog for color-sextet fields. In Section III, we use a small subset of these operators to explore a simplified model of color-sextet scalars and fermions. We compute cross sections for a variety of production modes and discuss LHC phenomenology while surveying constraints on these novel particles. Section IV concludes. The appendix contains a thorough discussion of the representation theory of the SU(3) sextet, along with some notes about the implementation of our specific color-sextet models in public computing tools.
II. EFFECTIVE INTERACTIONS OF EXOTIC COLOR-CHARGED STATES
Our aim is to catalog the interactions of new SU(3) c -charged matter with the Standard Model. Ultimately, one could imagine considering bSM fields up to the twenty-seven-dimensional representation of SU(3) c , which is the highest representation that can be produced resonantly in a collision via fusion. In order to generate results that are novel and useful but also relatively simple, we choose to focus on particles in the six-dimensional (sextet, 6) representation of SU(3) c . The method we describe can be easily generalized to other color representations.
A. Color singlets by iteration
The goal is to find all gauge-and Lorentz-invariant operators governing exotic color-charged matter and the Standard Model. As an initial step, we need to identify all relevant gaugeinvariant contractions of SM color-charged fields with color sextets. We therefore begin by enumerating the gauge-invariant contractions of three color-charged fields that can be realized at the LHC. To do this, we recall the tensor decompositions of direct product (reducible) representations ⊗ , { , } ≤ 8, of SU(3) [21,22]:
3 ⊗ 3 =3 a ⊕ 6 s , 3 ⊗3 = 1 ⊕ 8, 6 ⊗ 3 = 8 ⊕ 10, 6 ⊗3 = 3 ⊕ 15, 6 ⊗ 6 =6 s ⊕ 15 a ⊕ 15 s , 6 ⊗6 = 1 ⊕ 8 ⊕ 27, 8 ⊗ 3 = 3 ⊕6 ⊕ 15, 8 ⊗6 = 3 ⊕6 ⊕ 15 ⊕ 24,and 8 ⊗ 8 = 1 s ⊕ 8 s ⊕ 8 a ⊕ 10 a ⊕ 10 a ⊕ 27 s ,(1)
At present, our goal is a comprehensive catalog at dimensions five and six, but we identify interesting dimension-seven operators throughout. In other words, we ignore SU(3) c invariants that can only produce dimension-eight or higher (smaller) operators.
where the subscripts s (a) indicate a symmetric (antisymmetric) contraction. There are four reducible representations of SU(3), given by the direct product of three irreducible representations (smaller than 8 and including at least one sextet) that contain a color singlet. As a shorthand, we refer to these reducible representations as invariants, and we write them as
3 ⊗ 3 ⊗6, 3 ⊗ 6 ⊗ 8, 6 ⊗ 6 ⊗ 6,and 6 ⊗6 ⊗ 8.(2)
This shorthand indicates that there exists (at least) one way to contract fields in the corresponding representations of SU(3) c that results in a color singlet. These "three-field" invariants produce a number of interesting operators by themselves, but only scratch the surface of what is possible in SU(3). In order to go deeper, we make a straightforward observation about reducible representations of SU(3) based on the simple fact that the direct product of an irreducible representation p with its conjugatep contains a gauge singlet. In particular:
Observation. If there exist invariant combinations of + 1 and + 1 fields transforming in the direct product representations r 1 ⊗ · · · ⊗ r ⊗ p and q 1 ⊗ · · · ⊗ q ⊗ p of SU(3), then there exists an invariant combination of + fields in the reducible representation r 1 ⊗· · ·⊗r ⊗q 1 ⊗· · ·⊗q .
This observation allows us to systematically identify all gaugeinvariant color structures in and beyond the Standard Model. Applying this technique to the list (1), with = = 2, yields ten independent invariants of four fields that can be used to construct effective operators of dimension seven or lower (larger) including at least one sextet and at least one SM field. We write these "four-field" invariants in order of increasing representation dimension as
3 ⊗ 3 ⊗ 6 ⊗ 6, 3 ⊗ 3 ⊗6 ⊗ 8, 3 ⊗3 ⊗3 ⊗6, 3 ⊗3 ⊗ 6 ⊗6, 3 ⊗ 6 ⊗ 6 ⊗6, 3 ⊗ 6 ⊗ 8 ⊗ 8, 3 ⊗6 ⊗6 ⊗ 8, 6 ⊗ 6 ⊗6 ⊗6, 6 ⊗ 6 ⊗ 6 ⊗ 8, and 6 ⊗6 ⊗ 8 ⊗ 8.(3)
This larger set of invariants underpins a significantly larger set of gauge-invariant operators than what is generated by the three-field invariants. The final iteration required to fit the scope of this work takes = 2, = 3, and produces the following list of ten "five-field" invariants:
3 ⊗ 3 ⊗ 3 ⊗ 3 ⊗ 6, 3 ⊗ 3 ⊗ 3 ⊗3 ⊗6, 3 ⊗ 3 ⊗ 6 ⊗6 ⊗6, 3 ⊗3 ⊗ 6 ⊗ 6 ⊗ 6, 3 ⊗3 ⊗ 6 ⊗6 ⊗ 8, 3 ⊗ 6 ⊗ 6 ⊗ 6 ⊗ 6, 3 ⊗ 6 ⊗6 ⊗6 ⊗6, 6 ⊗ 6 ⊗ 6 ⊗6 ⊗6, 6 ⊗ 6 ⊗6 ⊗6 ⊗ 8,and 6 ⊗ 6 ⊗ 6 ⊗ 8 ⊗ 8.(4)
Many of the invariants in (3) and (4) produce operators with a minimum mass dimension of seven, but several work at dimension six. At any rate, the three preceding lists provide all the group-theoretic ingredients of dimension-seven or lower operators governing color-charged fields. As we implied in the Introduction, we build operators from these invariants assuming that only the sextet is novel; i.e., we take the triplets (3) to be SM quarks and the octets (8) to be SM gluons. It should therefore be noted that specific assignments of sextet weak hypercharge may be necessary in order to preserve invariance under the full SM gauge group G SM . Most of the four-and five-field invariants enumerated above permit more than one contraction, which happens when there exists more than one intermediate representation p implying the SU(3) invariant r 1 ⊗ · · · ⊗ r ⊗q 1 ⊗ · · · ⊗q (as defined in the observation on the previous page). Distinct gauge-invariant contractions of a fixed set of fields are realized with distinct sets of generalized Clebsch-Gordan coefficients. As an example, the invariant composed of three color octets can be built in a totally symmetric or a totally antisymmetric manner, and an observable associated with these two possible vertices (summed over colors) carries a factor proportional to
= 24 vs. = 40 3 for SU(3) c .(5)
In this example, the Clebsch-Gordan coefficients correspond to the structure constants and the SU(3) totally symmetric symbol, = 2 tr {t 3 , t 3 }t 3 with t r the generators of the irreducible representation r of SU(3). Other Clebsch-Gordan coefficients have been computed in prior studies of exotic color-sextet fields; these are the coefficients linking three color triplets (L , totally antisymmetric) and two triplets to an antisextet (K , symmetric under ↔ interchange) [18]. In Tables I-III, we specify all possible contractions of color indices generating the invariants listed in (2), (3), and (4). In these tables, and throughout this document, index heights in Clebsch-Gordan coefficients (associated with irreducible representations that are not self-conjugate) are fixed in order to contract correctly with fields in the representations specified in the first column, so for instance the totally antisymmetric coefficients for the invariant 3 ⊗ 3 ⊗ 3 are written so that
L ⊃ L(6)
would be a color singlet composed of three scalars in the fundamental representation of SU(3) c . Appendix A contains a more thorough discussion of Clebsch-Gordan coefficients, including further explanation of notation, a method for computing them, and some important results used later in this work. Tables I-III display a wide variety of index contractions leading to color singlets. The number of valid contractions naturally rises with the order of the invariant, since higher-order contractions are built from smaller ones. Therefore, while most three-field invariants permit only one contraction, the five-field invariant 3 ⊗3 ⊗ 6 ⊗ 6 ⊗ 6 (for example) permits six. There are in total a few dozen contractions available for all possible sets of color-charged fields relevant to our operator catalog, even before considering sextet spin(s) and electroweak representation(s)and the other fields required to form singlets under the full SM gauge group.
B. Lorentz invariance
The construction of Lorentz-invariant operators naturally depends on the spin(s) of the exotic field(s). We consider scenarios with one or two (distinct) color-sextet fields assigned as either a complex scalar or a Dirac fermion (or one of each species). In order to collect all sextet operators, we identify all possible Lorentz structures built out of the SU(3) c invariants collected above. Since it turns out that most operators in our catalog contain Dirac fermions -either quarks, leptons, or color-sextet fermions -a natural starting place is the nonvanishing Dirac bilinears, displayed in Table IV, with their Hermitian conjugates implied.
This table establishes a shorthand for non-vanishing Dirac bilinears in addition to cataloging these objects. The top block contains bilinears reminiscent of Standard Model structures, including all contractions of some generic weak doublet L = ( + L , − L ) , and a weak singlet R . This class of bilinears is useful for invariants containing direct products of the form r ⊗ q -an irreducible representation with a (possibly distinct) conjugate irreducible representation. A familiar example is the SM operator
L SM ⊃ − (¯L R ),(7)
with , ∈ {1, 2, 3} generation (flavor) indices, the coefficients of which form the down-type Yukawa matrix . The color invariant underpinning this operator is3 ⊗ 3. We have found many higher-order color invariants that include a structure like this -either between two color-charged fields or between a color-charged field and a lepton -so these Dirac bilinears are quite useful. But we also need bilinears for direct products of the form r ⊗ q -an irreducible representation with a (possibly distinct) irreducible representation. These bilinears are displayed in the middle block of Table IV. They comprise Dirac fermions and charge-conjugated fermions:
c ≡ C¯ with C satisfying C C −1 = − .(8)
While these structures do not appear in the Standard Model, some occur in a variety of bSM theories ranging from models Table I: Color index contractions yielding the three-field color invariants needed for a catalog of dimension-six or lower operators governing color sextet interactions with SM quarks and gluons. This table establishes our notation for invariant (Clebsch-Gordan) coefficients, some of which are used for phenomenology later in this work. Tables II and III extend this catalog to four-and five-field invariants.
of color-sextet scalars [19] to supersymmetric models with Dirac gauginos [23]. The lower block of Table IV displays various objects in weak or Dirac space that can be inserted in the bilinears listed above. Omitted from this particular list is the fifth Dirac matrix 5 , which when inserted between two Dirac fields of definite chirality, as considered in this work, modifies the bilinear only by a global sign. Crucially, some of these objects are themselves Standard Model fields, including any of the SM field-strength tensors , , and the
The SU(2) L (weak-isospin) field-strength tensor must be accompanied by t 2 = /2, ∈ {1, 2, 3}, the generators of the fundamental representation 2 of SU(2). These generators must furthermore be contracted with SU(2) L doublets L . SM Higgs doublet, which in the unitary gauge takes the form
= 1 √ 2 0 + ℎ .(9)
As is well known, operators containing insertions of the Higgs doublet generate operators of lower "effective dimension" when the Higgs is replaced by its vacuum expectation value . A complete operator catalog follows from cycling through all of the bilinears and insertions that satisfy our criteria for gauge and Lorentz invariance. The desired catalog of operators governing color-sextet fields interacting with the Standard Model can be constructed from the set of Dirac bilinears and color invariants. The number of dimension-five and -six operators containing at least one sextet and at least one SM field is of O (10 2 ), and thus is quite large. Consequently, we limit the listings in Tables V-VII to Invariant Clebsch-Gordan coefficients Notes Table II: Color index contractions yielding the four-field color invariants needed for the operator catalog, ordered by the dimension of the representation whose index is summed over.
3 ⊗ 3 ⊗ 6 ⊗ 6 K S [Π 3366 ] 3 ⊗ 3 ⊗6 ⊗ 8 LJ K [t 3 ] K [t 6 ] Q V [Π 3368 ] 3 ⊗3 ⊗3 ⊗6L K [t 3 ]J [Π 3336 ] 3 ⊗3 ⊗ 6 ⊗6 [t 3 ] [t 6 ] [Π 3366 ] 3 ⊗ 6 ⊗ 6 ⊗6 J [t 6 ] Q W [Π 3666 ] 3 ⊗ 6 ⊗ 8 ⊗ 8 [t 3 ] J J [t 6 ] J { , } E G V X [Π 3688 ] 3 ⊗6 ⊗6 ⊗ 8 KJ JS [Π 3668 ] 6 ⊗ 6 ⊗6 ⊗6 [t 6 ] [t 6 ] [Π 6666 ] 6 ⊗ 6 ⊗ 6 ⊗ 8 S [t 6 ] W X [Π 6668 ] 6 ⊗6 ⊗ 8 ⊗ 8 [t 6 ] [t 8 ] F H [Π 6688 ]
Invariant
Clebsch-Gordan coefficients Notes Table IV: Fermion bilinears that can be used, possibly by themselves or paired with an appropriate second bilinear, to build all Lorentz-invariant operators of mass dimension up to seven including at least one color-sextet field. Hermitian conjugates are allowed where distinct. Stipulations exist on the use of some bilinears listed above; some comments are offered where appropriate. Extra insertions of | | 2 may be allowed in dimension-five operators where gauge invariant. The dual Higgs field˜= i 2 † could be used in place of anywhere to e.g. allow a specific sextet weak hypercharge. CP-odd bilinears replacing 1 with 5 are also allowed in principle.
3 ⊗ 3 ⊗ 3 ⊗ 3 ⊗ 6 [Π 3366 ] K [Υ 33336 ] 3 ⊗ 3 ⊗ 3 ⊗3 ⊗6 [Π 3366 ] K [Π 3368 ] [t 3 ] [Υ 33336 ] 3 ⊗ 3 ⊗ 6 ⊗6 ⊗6 [Π 3366 ]S [Π 3368 ] [t 6 ] [Υ 33666 ] 3 ⊗3 ⊗ 6 ⊗ 6 ⊗ 6 [Π 3366 ] S [Π 3668 ] J [Π 6668 ] [t 3 ] [Υ 33666 ] 3 ⊗3 ⊗ 6 ⊗6 ⊗ 8 [Π 3688 ]J [Π 6688 ] [t 3 ] [Υ 33668 ] 3 ⊗ 6 ⊗ 6 ⊗ 6 ⊗ 6 [Π 3666 ] S [Υ 36666 ] 3 ⊗ 6 ⊗6 ⊗6 ⊗6 [Π 3668 ] [t 6 ] [Π 6668 ] J [Υ 36666 ] 6 ⊗ 6 ⊗6 ⊗6 ⊗ 8 [Π 6666 ] [t 6 ] [Π 6688 ] [t 6 ] [Υ 66668 ] 6 ⊗ 6 ⊗ 6 ⊗ 8 ⊗ 8 [Π 6668 ] { , } [Υ 66688 ] 6 ⊗6 ⊗6 ⊗6 ⊗6 [Π 6666 ] S [Υ 66666 ](¯ ) (¯ ) (3 ⊗ 3), (ΨΨ) (6 ⊗ 6), (¯ℓ) (3 ⊗ 1)¯L Γ R L Ω L only if half of four-fermion operator with second R R ( ) ( ) (3 ⊗ 3), (ΨΨ) (6 ⊗ 6), ( ℓ) (3 ⊗ 1) c R Γ R Γ = non-vanishing only if ≠ c L Ω Γ L c L R needs second again Operator Notes Γ ∈ {1, } must be accompanied by or a field-strength tensor , ∈ { , t 2 , } Ω ∈ { † , i 2 }
using the shorthand Dirac bilinears introduced in Table IV, with the generic Dirac fermions , in that table replaced by quarks, leptons, and weak singlet color-sextet fermions. Here we use a similar shorthand notation in which the familiar SM quarks and leptons with quantum numbers (SU(3) c , SU(2) L , U(1) ) are denoted by
∈ L ∼ (3, 2, 1 6 ) R ∼ (3, 1, 2 3 ) R ∼ (3, 1, − 1 3 ) and ℓ ∈ L ∼ (1, 2, − 1 2 ) ℓ R ∼ (1, 1, −1),(10)
where , ∈ {1, 2, 3} are generation indices; and in which
Φ ∼ (6, 1, Φ ) and Ψ ∼ (6, 1, Ψ )(11)
respectively denote a color-sextet scalar and fermion. The gluon and weak-hypercharge field strength tensors appear explicitly as and , as does the SM Higgs doublet . Each entry in Tables V-VII, therefore, represents a set of operators given by appropriately combining all valid Lorentz structures with all available color index contractions as displayed in Tables I-III. SU(2) L invariance generally has to be ensured by judicious choices of quark/lepton bilinears, but it is straightforward to preserve U(1) by fixing the sextet hypercharge(s) after all of the other ingredients are specified. Most operators can be generalized beyond the minimal field content (to even higherdimensional operators) by insertions of Higgs or gauge boson in their Dirac bilinears. These tables represent a comprehensive catalog at dimensions five and six, and also include a variety of interesting dimension-seven operators as well.
C. Examples
It is clear upon inspection of Tables V-VII that this catalog contains a formidable variety of gauge-invariant interactions for color-sextet fields. Not only do we recoup the fairly small set of interactions that have previously been investigated between sextet scalars and quark pairs [17][18][19] -but we find many sextet interactions with quarks and a lepton, and many of these permit gauge bosons pursuant to either a color invariant or a Dirac bilinear. The operators become increasingly spectacular for the higher-order invariants: even at dimension six, for instance, there are triple-sextet interactions with quark pairs in Table VII. In order to demonstrate how it can be used to develop a concrete model, we expand two subsets of the catalog and provide the associated operators explicitly. The sections we expand correspond to the three-field invariants 3 ⊗ 3 ⊗6 and 3 ⊗ 6 ⊗ 8, which are schematically cataloged in the second and third sections of Table IV and that include at least one color-sextet field. We consider scenarios with a sextet scalar, a sextet (Dirac) fermion, and at least one of each species. Note that operators requiring a single gluon field-strength tensor must be made Lorentz invariant by judicious choice of fermion bilinear(s) or a weak-hypercharge field strength . Lists marked with † have indicated min once accompanied by minimal set of SM fields.
4 † Φ † Φ 5 † (ΨΨ) 4 (Ψℓ)Φ (Ψℓ)Φ † (lΨ)Φ † 3 ⊗ 3 ⊗6 4 ( )Φ † 6 ( )(Ψℓ) 6 ( )| | 2 Φ † (Ψ )( ℓ) (Ψ )(l ) 3 ⊗ 6 ⊗ 8 6 ( ℓ)Φ 5 ( Ψ) (l )Φ 7 ( Ψ)| | 2 6 ⊗ 6 ⊗ 6 5 † ΦΦΦ 6 (ΨΨ)(Ψℓ) 5 (Ψℓ)ΦΦ (ΨΨ)(lΨ) (lΨ)ΦΦ 6 † (ΨΨ)Φ 6 ⊗6 ⊗ 8 6 Φ † Φ 5 (ΨΨ) 6 (Ψℓ)Φ 7 (ΨΨ)| | 2 (Ψℓ)Φ † (lΨ)Φ †
These tables show all the details hidden in the schematic operator lists by fully specifying the wide range of operators with color indices, Clebsch-Gordan coefficients, couplings , and EFT cutoffs Λ {Φ,Ψ} made explicit. In addition, these tables specify the lepton numbers and weak hypercharges the sextet field must assume in order to preserve gauge invariance and the accidental lepton number conservation of the Standard Model. SU(2) L (weak) indices are suppressed throughout, and Dirac indices are contracted between objects within parentheses.
In the interest of generality, the scalar couplings Table VIII, which targets the invariant 3 ⊗ 3 ⊗6, is fairly large, even though there exists only one way to contract color indices to form this invariant. This structure minimally couples a color sextet to quark pairs , which historically motivated the term "sextet diquarks". The first and third rows of Table VIII reproduce the gauge-invariant interactions cataloged for weak-singlet color-sextet scalars in [17,19]. In addition to these known couplings, we find numerous operators with various quark chiralities and extra SM gauge and Higgs bosons. The operators become even more exotic for sextet fermions in this color structure, with leptons being necessary in every case to form gauge and Lorentz singlets.
7 ( )(Ψℓ)Φ 7 ( )Φ| | 2 Φ ( Ψ)( Ψ) ( ℓ)( Ψ)Φ 3 ⊗ 3 ⊗6 ⊗ 8 6 ( )Φ † 3 ⊗3 ⊗3 ⊗6 7 (¯ )(¯ ℓ)Φ 6 (¯ )(¯ Ψ) ( ) † ( ℓ)Φ ( ) † (¯ Ψ) 3 ⊗3 ⊗ 6 ⊗6 5 (¯ )Φ † Φ 6 (¯ )(ΨΨ) 7 * (¯ )(Ψℓ)Φ 7 (¯ )Φ † | | 2 Φ (¯Ψ)(Ψ ) (¯Ψ)( ℓ)Φ † 3 ⊗ 6 ⊗ 6 ⊗6 6 ( ℓ)|Φ| 2 Φ 6 ( Ψ)(ΨΨ) 5 ( Ψ)Φ † Φ (l )|Φ| 2 Φ (Ψ )(ΨΨ) (Ψ )ΦΦ 7 * ( Ψ)(Ψℓ)Φ † ( ℓ)(ΨΨ)Φ 3 ⊗ 6 ⊗ 8 ⊗ 8 7 ( Ψ) 3 ⊗6 ⊗6 ⊗ 8 7 ( ℓ)Φ † Φ † 6 (Ψ )Φ † (l )Φ † Φ † 6 ⊗ 6 ⊗6 ⊗6 6 † |Φ| 4 6 (Ψℓ)|Φ| 2 Φ (Ψℓ)|Φ| 2 Φ † (lΨ)|Φ| 2 Φ † 7 (ΨΨ)|Φ| 2 | | 2 6 ⊗ 6 ⊗ 6 ⊗ 8 7 ΦΦΦ 6 (ΨΨ)Φ 7 (Ψℓ)ΦΦ (lΨ)ΦΦ 6 ⊗6 ⊗ 8 ⊗ 8 6 |Φ| 2 7 (ΨΨ)3 ⊗ 3 ⊗ 3 ⊗ 3 ⊗ 6 7 ( )( )Φ 3 ⊗ 3 ⊗ 3 ⊗3 ⊗6 7 ( )(¯ )Φ † 3 ⊗ 3 ⊗ 6 ⊗6 ⊗6 6 ( )|Φ| 2 Φ † 7 * ( )(ΨΨ)Φ † (Ψ )( Ψ)Φ † 3 ⊗3 ⊗ 6 ⊗ 6 ⊗ 6 6 (¯ )ΦΦΦ 7 (¯ )(ΨΨ)Φ (¯Ψ)( Ψ)Φ 3 ⊗3 ⊗ 6 ⊗6 ⊗ 8 7 (¯ )|Φ| 2 3 ⊗ 6 ⊗ 6 ⊗ 6 ⊗ 6 7 ( ℓ)ΦΦΦΦ (l )ΦΦΦΦ 3 ⊗ 6 ⊗6 ⊗6 ⊗6 7 ( ℓ)|Φ| 2 Φ † Φ † (l )|Φ| 2 Φ † Φ † 6 ⊗ 6 ⊗6 ⊗6 ⊗ 8 7 (ΨΨ)Φ † Φ 6 ⊗ 6 ⊗ 6 ⊗ 8 ⊗ 8 7 ΦΦΦ 6 ⊗6 ⊗6 ⊗6 ⊗6 7 ΦΦ † Φ † Φ † Φ † | | 2
III. SEXTETS AT THE LHC
In the previous section we introduced a wide variety of operators governing the production and decay of exotic colorcharged states in the six-dimensional representation (6) of the Standard Model SU(3) c . In this section we exploit the operator catalog to investigate models of color-sextet fields based on a subset of these operators. In particular, we consider color-sextet fermions and scalars coupling to gluons, up-or down-type quarks, and sometimes leptons and the U(1) gauge boson(s) ; these couplings are enabled by the color invariant 3 ⊗ 6 ⊗ 8. The sextet fermion models are defined by for ∈ { , } (so for instance Ψ couples to up-type quarks). The models for sextet scalars are analogously given by
L ⊃Ψ (i / − Ψ )Ψ + 1 Λ Ψ [ J ( c R Ψ ) + H.c.] + 1 Λ 3 Ψ [ J ( c R Ψ ) + H.c.] (12) 3 ⊗ 3 ⊗6 Singlet (Lorentz + G SM ) Generic Specific Coupling Scalar Φ ( )Φ † K Φ † ( c R R ) 0 {− 2 3 , 1 3 , 4 3 } K Φ † ( c R R ) 1 Λ 2 Φ K Φ † ( c L i 2 L ) 1 3 K Φ † ( c L i 2 L ) 1 Λ 2 Φ K Φ † ( c L † L ) 1 Λ 2 Φ K Φ † ( c L t 2 L ) ( )| | 2 Φ † K Φ † ( c R R ) | | 2 {− 2 3 , 1 3 , 4 3 } Dirac Ψ ( )(Ψℓ) K ( c R R )(Ψ ℓ R ) 1 Λ 2 Ψ 1 {− 5 3 , − 2 3 , 1 3 } K ( c R R )(Ψ † L ) 1 Λ 3 Ψ K ( c R R )(Ψ † L ) K ( c L i 2 L )(Ψ ℓ R ) 1 Λ 2 Ψ 4 3 K ( c L i 2 L )(Ψ † L ) 1 Λ 3 Ψ − 2 3 K ( c L R )(Ψ ℓ R ) {− 2 3 , 1 3 } (Ψ )( ℓ) K (Ψ R )( c R ℓ R ) 1 Λ 3 Ψ {− 5 3 , − 2 3 , 1 3 } K (Ψ R )( c L ℓ R ) 1 Λ 2 Ψ {− 2 3 , 1 3 } (Ψ )(l ) K (Ψ R )(¯L i 2 L ) −1 { 1 3 , 4 3 } K (Ψ R )(l R R ) { 1 3 , 4 3 , 7 3 }L ⊃ ( Φ ) † Φ − 2 Φ Φ † Φ + 1 Λ 2 Φ [ J Φ ( c R ℓ R ) + H.c.].(13)
In both Lagrange densities (12) and (13), as elsewhere, spinor indices are contracted within parentheses. The couplings and cutoff scales are taken with light modification from Table IX. The Clebsch-Gordan coefficients J (with Hermitian conjugatesJ ) providing the gauge-invariant contraction of a color sextet with a quark and a gluon were introduced in Table I. We discuss these novel coefficients in detail and explicitly provide them in a useful basis in Appendix A. As we noted in Section II, these color sextets must have particular weak hypercharges and lepton numbers in order to preserve the symmetries of the Standard Model. We summarize the quantum numbers of the fields in (12) and (13) in Table X.
3 ⊗ 6 ⊗ 8 Singlet (Lorentz + G SM ) Generic Specific Coupling Scalar Φ ( ℓ)Φ J Φ ( c R ℓ R ) 1 Λ 2 Φ −1 { 1 3 , 4 3 } (l )Φ J Φ (¯L R ) 1 Λ 3 Φ 1 {− 5 3 , − 2 3 } Dirac Ψ ( Ψ) J ( c R Ψ ) 1 Λ Ψ 0 {− 2 3 , 1 3 } J ( c R Ψ ) 1 Λ 3 Ψ ( Ψ)| | 2 J ( c R Ψ ) | | 2G SM Scalars Φ (6, 1, 1 3 ) −1 Φ (6, 1, 4 3 ) Fermions Ψ (6, 1, − 2 3 ) 0 Ψ (6, 1, 1 3 )
We implement these simplified models in version 2.3.43 of F R [24,25], a package for M © version 12.0 [26]. Some notes on the implementation of the Clebsch-Gordan coefficients J are offered in Appendix A. We have used F R to generate a Universal Feyn-Rules Output (UFO) [27] for leading-order (LO) event generation in M G 5_ MC@NLO (MG5_ MC) version 3.2.0 [28,29]. For both cross section computations and event simulation, hard-scattering amplitudes have been convolved with the NNPDF 2.3 LO set of parton distribution functions [30]. The renormalization and factorization scales have been set to R = F = Ψ or Φ .
A. Cross sections and LHC signatures
At a high-energy hadron collider, color sextets can be produced in pairs (predominantly through their SU(3) c gauge interactions) as well as singly, often in association with SM leptons or gauge bosons. Representative diagrams for pair production are displayed in Figure 1 and proceed via gluon fusion and quark-antiquark annihilation. Contributions to ΨΨ production from the higher dimensional operators in (12) are typically negligible for reasonable choices of Λ Ψ . The cross sections of sextet fermion and scalar pair production at the LHC with √ = 13 TeV are displayed in Figure 2, with systematic errors estimated by adding the scale and PDF variations reported by MG5_ MC in quadrature. Because pair production is dominated by gauge interactions, the cross sections are not only essentially independent of and Λ Ψ , but are also nearly identical for sextets coupling to up-and down-type quarks. The spin of the sextet is important, with the scalar cross section(s) hovering a bit less than an order of magnitude below those of the fermions. All cross sections are of O (1-10) pb for masses below the TeV scale but fall quite steeply with increasing sextet mass. It is worth noting that these results are consistent with the small existing literature for sextet fermions [31,32] and scalars [18,19,33].
The novel interactions between sextet fermions, quarks, and gluons in the second line of (12) allow sextet fermions to be singly produced in quark-gluon fusion. Here the quantum numbers of the sextet are significant, and we display the cross sections for both sextet fermions and their antiparticles in Figure 3. We have chosen a simple benchmark in which all first-and second-generation quarks couple to sextets with equal strength: = = 0.05 ∀ ∈ {1, 2} and Λ Ψ = Λ Ψ = 1 TeV. These choices correspond to single-production cross sections for all sextet fermions which are comparable to those for pair production. The differences in cross section for the two fermions and their conjugates are largely the result of the significant difference between quark and antiquark parton luminosities at LHC. (One can see in (12) that the initial state forΨ production is essentially , whereas Ψ is produced by¯.) The situation would be quite different at a¯collider like the Tevatron, but here the differences amount to factors of around five. As expected, these cross sections fall more gently with increasing Ψ than that of pair production. A third interesting production mode involving a sextet fermion is enabled by the final line of (12). We find that processes with up to two photons or bosons in the final larger than that of →Ψ + . On the other hand,Ψ + is the largest of the two-boson associated production modes, followed by and . For the same reasons as for unaccompanied single fermion production, other sextet fermions produced in association with photons and bosons exhibit smaller cross sections.
(a) Ψ Ψ = Ψ Ψ + Ψ Ψ Ψ +¯Ψ Ψ + c Ψ Ψ + c c Ψ Ψ (b) Φ Φ † = Φ Φ † + Φ Φ Φ † + Φ Φ † +¯Φ Φ †
While no operator in (13) allows for the production of a lone color-sextet scalar at LHC, the production of such particles in association with SM leptons is allowed by the second line. These again are due to quark-gluon fusion and exhibit the same behavior with respect to /¯initial states. We display in Figure 5 some suggestive cross sections for sextet scalar production with an electron or its antiparticle. Here we adopt a benchmark in which the coupling matrices , are diagonal in generation space, so for instance Ψ + is produced only by¯fusion. We specifically let , = 0.1 × and again choose cutoffs of Λ Φ = Λ Φ = 1 TeV. In this scenario, these cross sections are up to a few times larger than those of sextet fermion with / production.
B. Constraining color sextets at LHC
As we have seen, color-sextet particles can be produced at the LHC singly and in pairs, sometimes in association with leptons and bosons. They will subsequently decay to a variety of two or more SM particles. Many of these decay products will hadronize in a detector, ultimately producing final states with possibly large jet multiplicities, possibly accompanied by leptons and electrically neutral bosons. This rich phenomenology makes color-sextet models ripe for exploration at the LHC, and indeed a small subset of these models, mostly corresponding to the top row of Table VIII, have received attention in the literature [18,19,34,35].
However, most of the important signatures generated by the example models (12) and (13) are fringe cases that have not been directly targeted by either experimental collaboration. This ability to evade LHC searches by producing exotic signatures is typical of our expanded color-sextet catalog; viz. Tables VIII and IX. In this discussion, we enumerate the most interesting signatures worthy of future study, which in principle arise both from leading-order color-sextet decays to SM particles [20,36] and from sextet loop contributions to meson-antimeson mixing [35].
We first examine the single conventional signal with existing constraints and map some experimental results onto our EFT parameter space. In particular, we note that the second line of (12) -which enables single sextet fermion production via quark-gluon fusion -also allows sextet fermions to decay to a quark and a gluon. The full process → Ψ →( etc.) allows us to constrain the sextet fermions Ψ as dijet resonances. Both experimental collaborations have conducted a number of dijet-resonance searches during Run 2 of the LHC. The search easiest to interpret within our model framework was conducted by the CMS collaboration using up to L = 36 fb −1 of collisions at √ = 13 TeV [20]. This analysis targets dijet resonances over a wide mass range ( ∈ [0.6, 8.0] TeV) and is specifically used to constrain (among others) a benchmark model of excited first-generation quarks decaying to a gluon and a same-flavor quark ( * → , ∈ { ,¯, ,¯}). We can use the model-independent limits at 95% confidence level (CL) [37] on the fiducial cross section ( → ) × BF( → ) × A computed by CMS to estimate constraints on our sextet fermions decaying to a gluon and a first-generation quark. Our estimates for the fermion that couples to up-type quarks, Ψ , are displayed in Figure 6. Figure 6 is in the ( Ψ , Λ Ψ ) plane. We provide no results in the region where Ψ > 2Λ Ψ , since we take Ψ = 2Λ Ψ as a threshold past which the effective field theory's applicability is dubious. Next, there are two (four) exclusion regions in the plot; solid curves denote observed limits and dashed curves indicate expected limits. We have provided two sets of results Figure 3; but red regions assume unit branching fraction to¯, while green regions take more realistic branching fractions of around 26%. In the gray region, where Ψ > 2Λ Ψ , the EFT is unlikely to be consistent.
in order to show the importance of the branching fraction of the sextet fermion to¯. The green region(s) correspond closely to the results displayed in Figures 3 and 4: all couplings and cutoffs except for Λ Ψ take the same values as in the previous plots. Recall that in these benchmarks, the couplings , take the same values of O (10 −1 ) for each generation . This choice ultimately produces BF(Ψ →¯) ≈ 0.26. We emphasize that these limits apply to the up-type sextet antifermion decay,Ψ → , since -viz. Figure 3 -the single-production process →Ψ enjoys the largest cross section at LHC. Similar reinterpretations for the other sextet fermions would yield looser constraints, all else being equal. Finally, the red region(s) use the same cross sections but take BF(Ψ →¯) = 1. These choices could be made consistent by appropriately adjusting 1 , 1 relative to the couplings to heavier up-type quarks. These results serve as a worst-case (high cross section accompanied by high branching fraction) scenario for our sextet fermions from an integrated luminosity of L ≤ 36 fb −1 . In general, we find that light sextet fermions can still be accommodated by these data if Λ Ψ is in the low multi-TeV range. The limits on this cutoff weaken to nearly 1 TeV in the "realistic" scenario plotted in green. Dijet-resonance searches are the only searches of which we are aware that currently target a signature produced by these sextet models. Even for these searches, a notable gap exists for sextets decaying to a heavy quark (especially a top quark) and a gluon (again, the CMS search [20] targets resonances decaying to first-generation quarks). A search tailored to fill this gap may be a good avenue of future study. In the interest of completeness, we note that there exists a CMS search [36], using L ≈ 37 fb −1 of Run 2 data, for pair-produced spin-3 2 color triplets * ("excited top quarks") each decaying promptly to a top quark and a gluon. This search, finding no signal of physics beyond the Standard Model, was used to exclude excited top quarks of around * = 1 TeV with fiducial cross sections ( → * ¯ * ) × BF 2 ( * → ) ≈ 100 fb. A similar lower limit would be imposed by this search on a pair-produced up-type sextet fermion Ψ given identical acceptances, but a detailed reinterpretation or a dedicated experimental analysis would be required to obtain credible constraints on spin-0 or spin-1 2 sextets producing this final state. A search for final states involving top quarks from singly produced color sextets would also be welcome.
The other important signals that can be produced by our model catalog are exotic and likely also require dedicated reinterpretations or novel search strategies. The sextet fermions, per the last line of (12), undergo a suppressed decay to a quark, a gluon, and a photon or boson. This decay minimally (in the case of single fermion production) produces an interesting dijet resonance + / signature. The sextet scalars, which we have neglected so far, generate another "dijet resonanceadjacent" signature by decaying to a quark, a gluon, and a lepton. Finally, we note that the pair production of any sextet -which occurs copiously at LHC -potentially generates an array of interesting signatures, particularly for the fermion. For instance, since one fermion could undergo the two-body decay while the other decays to three or even four SM particles, one could expect signatures comprising at least four jets and up to four electrically neutral bosons (and the decays of up to four bosons would render these signatures even more complex). Searches for any of these signals would be excellent ways to leverage the higher luminosity of the next planned run of the LHC.
On the other hand, an alternative set of constraints could potentially be imposed on some color-sextet models by searches for flavor-changing neutral currents (FCNCs), which do not exist at tree level in the Standard Model and are very tightly constrained [38]. It has been observed [35,39], for instance, that color-sextet scalars coupling to quark pairs (which, recall, is a renormalizable interaction; viz. Table VIII) can generate flavor-changing processes at tree and one-loop level. The strongest limits can be expressed in our notation as
| 11 * 22 | ≤ O (10 −6 ) Φ TeV 2(14)
from tree-level sextet scalar contributions to mixing of neutral kaons ( 0 -¯0, with down-type scalars Φ ) and -mesons ( 0 -¯0, with up-type scalars Φ ) , and | 23 * 12 | ≤ O (10 −2 ) and similar (15) from flavor-changing nonleptonic -meson decays mediated at tree level by down-type scalars [35]. Similar limits could apply to other models with color sextets. For example, the non-renormalizable sextet fermion model (12) we sought to constrain in Figure 6 can generate meson-mixing diagrams at effective one-loop order. It could therefore be instructive to estimate the size of these FCNC contributions. The superficial degree of ultraviolet (UV) divergence of these box diagrams in = 4 dimensions turns out to be
= 4 − − 2 + ∑︁ ℎ = 2,(16)
where , denote the number of fermion and boson propagators in the loop and the diagram contains vertices of type with ℎ derivatives each. But the reality is more complicated: these are really multi-loop diagrams, and our ignorance of the UV physics is reflected by a factor of a UV cutoff Λ −1 Ψ for each of the four vertices forming the boxes. We could therefore estimate the UV divergence of these loops as ∼ [loop momentum] 2 Λ −4 Ψ → −2, supposing that the EFT cutoff itself is used to regularize the superficially divergent loop integrals. This degree of divergence matches that of the sextet scalar box contributions to meson mixing discussed above [35], so it may indeed be necessary to suppress the quarksextet couplings in this model to evade FCNC constraints . On the other hand, models of sextet scalars or fermions that lack a "pure" sextet-quark-quark or sextet-quark-gluon vertex (unaccompanied by leptons or bosons) appear to evade these constraints. But it must be emphasized that this line of thinking is inconclusive without knowledge of the microscopic physics: there may exist diagrams associated with the degrees of freedom that have been integrated out that enhance or interfere with the loops we can compute within the EFT framework. We therefore leave a more detailed investigation of FCNC constraints on our effective operators to future work.
IV. CONCLUSIONS
In this work we have taken a new look at the possible interactions of beyond-the-Standard Model particles that are charged under the SM color gauge group SU(3) c . Such states can be copiously produced at the LHC, and it is important to understand the space of their possible interactions in order to understand how LHC data constrains their existence. In this work, we have explored the gauge-invariant interactions of fields in the six-dimensional (sextet) representation of SU(3) c , producing a (large) catalog of operators, many of which have received little Loop contributions yield lighter constraints of O (10 −2 ) Φ TeV −1 or larger. Such constraints would apply to all couplings , since unlike for tree-level diagrams there is no "flavor texture" that will suppress the box diagrams. or no attention in the literature to date but may produce distinct phenomenology worthy of investigation. We have focused on color sextets in this work because they transform in the lowest-dimensional representation of SU (3) c not yet observed in Nature. We have specifically focused on higher-dimensional interactions linking color-sextet fermions and scalars to a SM quark and a gluon (and possibly additional particles), computing cross sections for a variety of production modes and surveying existing LHC constraints on these particles.
Much of what we have done here is intended to set up future work. On one hand, one could undertake a much more thorough investigation of the specific color-sextet models we introduced above; such a study could compute next-to-leading order (NLO) corrections within the EFT framework or could propose an ultraviolet completion for one or more of the operators we consider . It could also be worthwhile to rigorously compute sensitivity projections for any of the non-standard signatures we described in Section III.B at the high-luminosity LHC. Such projections would be derived from a tailored selection strategy, which would be interesting to develop. It would also be natural to complete the color-sextet operator catalog at mass dimension seven, or to allow for extended sectors comprising sextets with distinct weak hypercharges or non-trivial transformations under SU(2) L .
On the other hand, as we hopefully have demonstrated with the example of color sextets, the method we have employed to build our operator catalog can be used to build a variety of phenomenological models containing higher representations of SU(3) c . Similar catalogs could be built for other representations, including the more frequently studied triplets and octets but also higher-dimensional fields; we plan to investigate several such catalogs going forward. The interactions we unearth in these catalogs could also be embedded in richer or more complete theories, for instance via the incorporation of a SM gauge singlet as a dark matter candidate.
Appendix A: An excursion in SU(3) representation theory
The concrete realization of a model incorporating any of the operators cataloged in Section II of this work requires explicit knowledge of the gauge-invariant combinations of the relevant fields. While some of our operators contain fields charged under SU(2) L , all of the exotic gauge singlets we have constructed belong to SU(3) c . Enumerating the gauge-invariant contractions of a given set of SU(3) c multiplets amounts to computing the Clebsch-Gordan coefficients connecting the irreducible (color) representations in which each multiplet transforms. While there exist some works that confront this problem in various contexts [40][41][42][43], explicit results suitable for fundamental particle physics are difficult to find. In this appendix, we construct the two minimal gauge-invariant combinations of a color sextet with two other color-charged fields. We review and extend For example, the operator on the second line of (12) could straightforwardly be generated by a loop involving SM quarks and a color-triplet scalar, à la squarks. some known basis-independent results, and we provide for the first time a new set of Clebsch-Gordan coefficients in a familiar basis well suited for integration into public computing tools.
Let a field -for definiteness, a Dirac fermion -in the sextet representation of SU(3) c be indexed by , ∈ {1, . . . , 6}. In analogy with a quark transforming in the fundamental representation of SU(3) c , a lowered index corresponds to the representation in question (6), while a raised index (e.g.¯) denotes the conjugate representation (6). Two of the product decompositions of two irreducible representations of SU (3) we studied in the body of this work are 3 ⊗ 3 = 6 ⊕3 and 3 ⊗ 8 = 3 ⊕6 ⊕ 15.
(A1)
We noted in Section II that these decompositions imply the existence of the three-field invariants 3 ⊗ 3 ⊗6 and 3 ⊗ 6 ⊗ 8.
We displayed a number of operators in Tables VIII and IX based on these invariants that couple sextets to (respectively) quark pairs and a quark and a gluon. While the first family of couplings has received some attention [18,19], the latter (to our knowledge) has not. This appendix introduces the explicit group-theoretical objects required for our novel analysis while making contact with known mathematical results.
We work in the basis where the generators of the fundamental (3) representation of SU(3) are proportional to the Gell-Mann matrices: 2t 3 = , ∈ {1, . . . , 8}. We take the generators of the adjoint (8) representation to be [t 8 ] = −i , where are the structure constants appearing in the SU(3) algebra
[t 3 , t 3 ] = i t 3 .(A2)
product of irreducible representations r 1 and r 2 are given by t r 1 ⊗r 2 = t r 1 ⊕ t r 2 ≡ t r 1 ⊗ 1 r 2 + 1 r 1 ⊗ t r 2 ,
where the Kronecker sum ⊕ is defined in terms of the Kronecker products (suggestively denoted by the same symbol ⊗ as the direct product of representations) between the generators of {r 1 , r 2 } and the identity matrices with the dimensions of {r 2 , r 1 }. In order to elucidate this point, and to make contact with more familiar notation, we rewrite the last line of (A4) as with { , } indexing the representation r , ∈ {1, 2}. The resulting generators t r 1 ⊗r 2 are again traceless and Hermitian, but now of dimension dim r 1 × dim r 2 . The operation (A4) can be iterated upon, so for instance the generators of a direct product of three representations are given by t r 1 ⊗r 2 ⊗r 3 = 3 =1 t r = t r 1 ⊕ t r 2 ⊕ t r 3 = (t r 1 ⊗r 2 ) ⊕ t r 3 .
(A6)
We provide all of this exposition because any gauge-invariant linear combination I r 1 ⊗ ··· ⊗r of fields in ≤ (not necessarily distinct) irreducible representations {r 1 , . . . , r } of SU(3) (or any semisimple Lie group), which can be written modulo index height as [44] I r 1 ⊗ ··· ⊗r = 1 ,...,
Our final remark concerns the relationship between the two sets of Clebsch-Gordan coefficients K and J . We find that the latter set can be constructed using a particular combination of the former set with other group-theoretical objects. In particular, we have that
J = −i √ 2 L [t 3 ]K andJ = i √ 2 K [t 3 ]L ,(A18)
where L are the Clebsch-Gordan coefficients governing the gauge-invariant contraction of three SU(3) triplets, which is well known to be totally antisymmetric . We mention the relations (A18) because the popular model-building and Monte Carlo simulation tools F R and M -G 5_ MC@NLO have for some time now handled colorsextet fields interacting with quark pairs by defining the Clebsch-Gordan coefficients K and L in terms of (anti-)symmetric combinations of two QCD triplets. Therefore, whereas the ability to directly handle the new coefficients J -given by (A15) -would require some significant additions to both public codes, we are able to construct our novel interactions with suitable combinations of existing semi-hard-coded objects. This strategy does not necessarily work for color-sextet interactions with higher-dimensional QCD multiplets, and may in fact be a unique exploit.
are matrices in quark (and sometimes lepton) generation space.
VIII: Gauge-invariant operators coupling color sextets to quark pairs, based on the SU(3) invariant 3 ⊗ 3 ⊗6 with Clebsch-Gordan coefficients K (viz. Appendix A). Hermitian conjugates also exist where distinct. Quark generation indices { , }, lepton generation indices { , }, and all color indices are kept explicit, while Dirac spinor indices and SU(2) L indices are suppressed. and are respectively the sextet lepton numbers and weak hypercharges in each scenario.
F 1 :F 2 :
12Representative diagrams for pair production of color-sextet (a) fermions and (b) scalars. Blobs mark vertices corresponding to an effective operator with some cutoff scale. Quarks coupling directly to sextets must have appropriate hypercharge; viz.(12) and(13). These contributions are negligible for realistic Λ Ψ , Λ Ψ . Leading order cross sections for pair production of color-sextet fermions and scalars at the LHC as a function of sextet mass.
Cross sections of color-sextet fermion single production. These are comparable to fermion pair-production cross sections for indicated couplings/cutoffs. Conjugate fermion (Ψ) cross sections dominate because quarks have greater parton luminosity than antiquarks at LHC. state may have cross sections of O (1-10) fb, which is on the margins of what is observable at the LHC. We display these cross sections for the optimal case of a sextet antifermion coupling to up-type quarks inFigure 4. These results correspond to a benchmark with (light) flavor-universal couplings ( = 0.10 ∀ ∈ {1, 2}) and a cutoff scale of Λ = 1 TeV.As expected, the cross section for →Ψ + is a few times sections ofΨ single production in association with up to two photons and/or bosons. Cross sections for Ψ and Ψ ,Ψ are smaller in analogy with single production (viz.Figure 2and discussion).
Cross sections of sextet scalar single production in association with an electron or positron. Conjugate scalar production dominates in a fashion similar to sextet fermion single production. Cross sections for associated production with , are orders of magnitude smaller if all else is equal.
Parameter space excluded for sextet antifermion coupling to up-type quarks (Ψ ) based on a CMS dijet resonance search at √ = 13 TeV. All limits are computed assuming ( →Ψ ) as displayed in
For
reference, the other non-negligible branching fractions in this benchmark are roughly as follows: BF(Ψ →¯) ≈ 0.27, BF(Ψ →¯) ≈ 0.21, BF(Ψ →¯) ≈ 0.18, BF(Ψ →¯) ≈ 0.06, and BF(Ψ → ) ≈ 0.001.
K
= [K ] † = K andJ = [J ] † .
ACKNOWLEDGMENTS T.T. is grateful for conversations with Rohini Godbole, Kirtimaan Mohan, and Daniel Whiteson. L.M.C. and T.M. are In the Gell-Mann basis, these coefficients are proportional to the Levi-Civita symbol: √ 2 L = and √ 2L = .
Table III :
IIIColor index contractions yielding the required five-field color invariants.the minimal field content at each order in the EFT expansion
for each color invariant. These "schematic" operators are built
Examples
Bilinears
Notes
Table V .
VThe resulting list of explicit operators are displayed inTables VIII and IX.Scalar sextet Φ only Dirac sextet Ψ only
≥ 1 of each
SU(3) c invariant
min
Structure
min
Structure
min Structure
6 ⊗6
Table V :
VSchematic table of three-field invariant operators, plus the unique two-field invariant 6 ⊗6, of minimum mass dimension min ≤ 7 that can be constructed using the fermion bilinears in
Table IX concerns
IXthe invariant 3 ⊗ 6 ⊗ 8, which despite being a simple three-field invariant has received scant attentionScalar sextet Φ only Dirac sextet Ψ only
≥ 1 of each
SU(3) c invariant
min
Structure
min
Structure
min
Structure
3 ⊗ 3 ⊗ 6 ⊗ 6
5
( )ΦΦ
6
( )(ΨΨ)
Table VI :
VISchematic table of four-field invariant operators. Fields are left blank if operators exist only with min = 8. Lists marked with * are not exhaustive. Lists marked with † have indicated min once accompanied by minimal set of SM fields.in the literature. This invariant couples a color sextet to a
quark and a gluon, and while no operators can be built at mass
Scalar sextet Φ only
≥ 1 of each
SU(3) c invariant
min
Structure
min
Structure
Table VII :
VIISchematic table of five-field invariant operators. Here we consider scenarios with a sextet scalar and at least one of each species, since for these invariants there are no suitable fermion-only operators. Lists marked with * are not exhaustive. Higgs bosons. Interestingly, the situation with respect to leptons is flipped relative to Table VIII, with the scalar sextet interactions requiring leptons. In short, this table depicts a fairly minimal but rich portal between the Standard Model and color-sextet scalars and fermions. We highlight some of the operators in this table in a phenomenological investigation in Section III.dimension four, it is potentially very important in LHC searches
for color sextets. Here, as for the other three-field invariant, we
find basic
couplings, as well as interactions containing extra
or
Table
Table IX :
IXGauge-invariant operators coupling color sextets to quarks and gluons, based on the SU(3) invariant 3 ⊗ 6 ⊗ 8 with
Clebsch-Gordan coefficients J
(viz. Appendix A). Conventions are similar to those of Table VIII.
Quantum numbers
Table X :
XExotic field content of color-sextet models considered in Section III. Representations in SM gauge group G SM and charges under accidental symmetries are noted.
The commutation relations (A2) are satisfied by the generators of every representation of SU(3), which are also traceless and Hermitian. A set of eight 6 × 6 matrices t 6 satisfying these criteria, which are therefore valid generators of the sextet representation in the Gell-Mann basis, is: It is useful to consider generators of reducible ("product") representations of SU(3). The objects relevant to our discussion can be constructed systematically using the generators we have provided above. In particular, the generators t r 1 ⊗r 2 of the direct The coefficients 1 ,..., appearing in the linear combinations (A7), (A8) are closely related to the Clebsch-Gordan coefficients connecting the irreducible representations. The coefficients we want simply have to be extracted and interpreted in a specific way. (A8) implies that these coefficients can be systematically computed in any basis by finding the kernel of the (potentially large) matrixThe number of independent gauge-invariant combinations of the fields is given by dim ker M r 1 ⊗···⊗r . The elements of the (dim r 1 ×· · ·×dim r ) ×1 vectors in ker M r 1 ⊗···⊗r are (up to a global factor) the desired Clebsch-Gordan coefficients. Per our notation in (A8), these are read off in an order determined by the construction of the product-representation generators. As an intuitive example, the first two non-vanishing elements (in the Gell-Mann basis) of the only vector in ker M 6⊗3⊗8 in SU(3)-which is clearly relevant to our phenomenological study of color sextets -are elements (10, 1) and (13, 1). These we label as J 1,2,2 and J 1,2,5 , elements of the first of six 3 × 8 matrices J ( ∈ {1, . . . , 6}, ∈ {1, 2, 3}, ∈ {1, . . . , 8}). This is essentially the method employed by the recently published M ©[26]package G M , which however works in what is sometimes called the Chevalley-Serre basis[45]. We have performed these calculations in the Gell-Mann basis in order to obtain basis-dependent results compatible with the literature and the popular computer tools F R and M G 5_ MC@NLO[24,25,28,29]. It should be noted that the method described above in fact yields some multiple of the Clebsch-Gordan coefficientswith 0 0 0 0 0 0 0 0 ,The first set (A14) of Clebsch-Gordan coefficients, which connect the 6 to the 3 ⊗ 3 of SU(3), are exactly what were computed in the only group-theoretical discussion similar to this appendix of which we are aware[18]. The second set, (A15), which connect the6 to the 3 ⊗ 8, have not been published before as far as we know. Both sets of coefficients are similarly normalized -
in part by the United States Department of Energy under grant DE-SC0011726. T.T. is supported in part by the United States National Science Foundation under grant no. PHY-1915005in part by the United States Department of Energy under grant DE-SC0011726. T.T. is supported in part by the United States National Science Foundation under grant no. PHY-1915005.
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|
[] |
[
"Charged Boson Stars in AdS and a Zero Temperature Phase Transition",
"Charged Boson Stars in AdS and a Zero Temperature Phase Transition"
] |
[
"Sanle Hu ",
"James T Liu ",
"Leopoldo A Pando Zayas ",
"\nDepartment of Modern Physics\nMichigan Center for Theoretical Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n",
"\nUniversity of Michigan\n48109Ann ArborMIUSA\n"
] |
[
"Department of Modern Physics\nMichigan Center for Theoretical Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"University of Michigan\n48109Ann ArborMIUSA"
] |
[] |
We numerically construct charged boson stars in asymptotically AdS spacetime. We find an intricate phase diagram dominated by solutions whose main matter contribution are alternately provided by the scalar field or by the gauge field.
| null |
[
"https://arxiv.org/pdf/1209.2378v1.pdf"
] | 118,652,361 |
1209.2378
|
0eb3573084576cc81fcbdc3719100a177ea654c6
|
Charged Boson Stars in AdS and a Zero Temperature Phase Transition
11 Sep 2012
Sanle Hu
James T Liu
Leopoldo A Pando Zayas
Department of Modern Physics
Michigan Center for Theoretical Physics
University of Science and Technology of China
230026HefeiAnhuiChina
University of Michigan
48109Ann ArborMIUSA
Charged Boson Stars in AdS and a Zero Temperature Phase Transition
11 Sep 2012(Dated: May 2, 2014)
We numerically construct charged boson stars in asymptotically AdS spacetime. We find an intricate phase diagram dominated by solutions whose main matter contribution are alternately provided by the scalar field or by the gauge field.
Introduction
The study of boson stars dates back to the late 1960's with the work of Kaup [1] (see also [2]) who, inspired by ideas of Wheeler, constructed Klein-Gordon geons. These objects and their generalizations have found numerous applications in and beyond general relativity. Some of the classical reviews in the subject include: [3][4][5][6][7].
In this letter we construct charged boson stars in asymptotically Anti-de-Sitter (AdS) spacetimes and investigate their properties. Namely, we consider particlelike solutions of a complex scalar field coupled to gravity and a Maxwell field in the presence of a negative cosmological constant. We have three main motivations to study these objects.
First, it is intrinsically interesting to understand particle-like solutions in asymptotically AdS space times to enhance and test our intuition of highly symmetric solutions of Einstein gravity with a negative cosmological constant. For example, the role of boundary conditions necessary to define dynamics in AdS is one aspect that constantly challenges our Minkowski-based intuition.
Our second motivation is also linked to understanding dynamics in AdS but it is more concretely related to gravitational collapse in AdS. It has recently been established that AdS is unstable under arbitrarily small perturbations with respect to black hole formation [8]. This result has been discussed and elaborated upon for massless fields [9][10][11][12][13]. Given previous history with critical collapse of massive [14] and Yang-Mills fields [15], it is possible that the phase diagram of critical collapse in asymptotically AdS spacetimes gets modified by the introduction of mass and other fields. In particular, the existence of boson stars might prevent a direct channel to black hole formation.
Thirdly, solutions of a scalar field in Einstein-Maxwell gravity in asymptotically AdS spacetimes have proven a fruitful ground for applications of the AdS/CFT correspondence to various situations in condensed matter physics (see, for example, [16][17][18]).
Charged boson stars
We consider a massive charged complex scalar field, φ, interacting with electromagnetism and minimally coupled to Einstein gravity with a negative cosmological constant,
S = d 4 x √ −g 1 16πG (R − Λ) − 1 4 F µν F µν −(D µ φ)(D µ φ * ) − m 2 φ * φ , (1) D µ = ∇ µ − iq A µ , F µν = ∇ µ A ν − ∇ ν A µ ,(2)
where G is Newton's constant. Since we are interested in stationary, spherically symmetric solutions, we write the metric in Schwarzschild form
ds 2 = −e 2u dt 2 + e 2v dρ 2 + ρ 2 dΩ 2 2 .(3)
Here ρ is the radial coordinate, and we take u = u(ρ) and v = v(ρ). We then take the scalar to be time-harmonic
φ = 1 √ 2 e iωt σ(ρ),(4)
where σ(ρ) is a real function of ρ. Since φ is electrically charged, it will source the electromagnetic field, and hence we turn on a scalar potential, A t = A t (ρ). At this point, a few quick comments are in order. Firstly, the Maxwell equation arising from (1) takes the standard form ∇ µ F µν = qJ ν , where
J µ = i(φ * D µ φ − φD µ φ * )(5)
is the conserved particle number current. As a result, the total charge of the boson star is given by Q = qN where N is the conserved particle number. Secondly, the time-dependence of φ may be removed by a gauge transformation of the form A t → A t − ω/q along with φ → e −iωt φ, so only the combination ω − qA t is physical. This is in contrast with the standard (ungauged) boson stars, where ω has an intrinsic meaning. It is straightforward to derive the coupled equations of motion corresponding to the above spherically symmetric ansatz. When developing the numerical solutions, we take as input the mass m and charge q of the scalar field as well as the cosmological constant Λ, and scale by Newton's constant G when appropriate in order to work with dimensionless quantities. We also introduce the gauge invariant combination
A(ρ) = (ω − qA t (ρ))/(ω − qA t (0)).(6)
To define our boundary value problem, we first demand regularity near the origin, ρ = 0. This leads to the following conditions:
u(0) = u 0 , v(0) = 0, σ(0) = σ 0 , σ ′ (0) = 0 A(0) = 1, ′ (0) = 0.(7)
At asymptotic infinity, ρ → ∞, AdS boundary conditions imply that the matter fields behave as
σ(ρ) = σ 1 ρ −∆ + σ 2 ρ ∆−3 , ∆ = 3 2 − 9 4 + (mL) 2 , A t (ρ) = a 0 + a 1 ρ −1 .(8)
Our task is to look for solutions with normalizable modes at infinity, namely σ 2 = 0. Therefore we have a boundary value problem which we solve numerically using shooting techniques. We determine σ 2 = 0 with a precision of 10 −20 . In our minimizing algorithm we shoot by changing σ 0 of the initial data for a given u 0 ; we use (mL) = 10 throughout.
Properties of the solutions
In the absence of a cosmological constant, there is a critical charge, q crit , above which the star does not exist in flat space. In the Newtonian limit the value of q crit is obtained by comparing the gravitational attraction with the electrostatic repulsion of an elementary particle interacting with the star; the result is Gm 2 = q 2 crit /(4π). In the context of general relativity this result is only slightly modified. However, for boson stars in AdS, the value of q crit above which a star does not exist is substantially larger. Moreover, the phase space of solutions is quite different.
Intuitively we can understand the increase of q crit in the presence of a negative cosmological constant as follows. Once the electrostatic force has overcome the gravitational pull, the only force standing in the way of charged particles flying away is the pull generated by the cosmological constant. Therefore for large enough charges the main mechanism supporting the star is the balance between the gauge forces and the cosmological constant.
For small charge q, the boson stars in AdS resemble those in flat space. The regular zero-node solutions have a smooth scalar profile, with maximum particle density at the core, and a gradual fall off as a function of ρ. Since the scalar field dominates the energy density, we denote these regular solutions as scalar dominated. For sufficiently large charge q, on the other hand, we find a new type of solution where the gauge field contribution dominates the total energy of the system. We naturally denote these as gauge field dominated.
An example of the regular and new solutions is given in Fig. 1. The regular scalar dominated solution is shown on the left, while the new gauge field dominated solution is on the right. There are three key properties that distinguish the new solution from the regular one: (i) The new solution has a sharper surface, as defined by the profile of the scalar field; (ii) The gauge field is concentrated near the surface of the star, instead of distributed in the interior as in the standard case; (iii) The contribution to the total mass of the star is dominated by the gauge field.
We now consider the mass of the boson star as a function of its core density σ(0) and as a function of the particle number N . Since the interplay between regular and new solutions depends on the charge q, we define two critical charges, q 1 and q 2 . For q < q 1 only the scalar dominated solution is possible. For q 1 < q < q 2 , we enter a transition region where both the scalar dominated and gauge field dominated solutions exist as distinct branches. Finally, for q > q 2 , the scalar and gauge field dominated solutions merge.
The mass as a function of σ(0) and as a function of N for an intermediate value of the charge, q 1 < q < q 2 , is shown in Fig. 2. The green curve represents the regular scalar dominated solution, while the red curve corresponds to the gauge field dominated solution. We have also shown the one-node solution as the blue curve. While this is ordinarily discarded as being an excited state, here we wish to note its proximity to the gauge field dominated solution. For each of these solutions, the right panel shows three curves of the same color. These three curves correspond to the total mass and the separate scalar and gauge field contributions to the mass. For the scalar dominated (green) curves, the scalar contribution to the mass is the larger one, while for the gauge field dominated (red) curves, the gauge field contribution to the mass is larger. Note that there is a gap on the right panel of Fig. 2. This suggests some sort of transition between scalar dominance on the left and gauge field dominance on the right. When the charge q is increased beyond the critical value q 2 we observe a very different behavior of the mass as a function of σ(0) and as a function of N , as shown in Fig. 3. There is no longer a gap in the right panel and we see clearly the change in the contribution to the total energy coming from the scalar-dominated region prevalent at small values of N and the gauge-dominated region at large values of N . The blue curve again corresponds to a one-node solution. This figure should be read as a merging of the previous one in the cases where the charge has increased, that is, increasing the charge narrows the gap between the two kind of solutions which is evident in the region of q 1 < q < q 2 .
The three green curves on the right panel of Fig. 3 are total energy, the contribution from the scalar field and the contribution from the gauge field respectively. This graph should be read as a merging of branches: now the green curve is a mixture of the normal branch for low N and gauge-dominated branch for large N ; the red curves are only in the areas of overlap in the previous graph. The most prominent feature of Fig. 3 is the crossing of the scalar and gauge contributions to the total energy.
In the left panels of Figs. 2 and 3 we have also plotted a branch corresponding to solutions with one node in the amplitude of the scalar field profile, σ(ρ); these are traditionally considered to be excited solutions. What can be seen by plotting their mass is that they play an important role in the presence of a gauge field and a negative cosmological constant. In particular, the one-node branch in Fig. 3 seems to interpolate smoothly between the scalar dominated and gauge field dominated solutions. More importantly, as can be seen from Fig. 2, the fact that the one-node and zero-node solutions come so close in terms of their energies seems to suggests that there is a mech-anism taking place at large values of the charge whereby the oscillatory (usually excited) solutions come very close to the minimal energy (ground state) solution.
FIG. 3: Mass as a function of scalar field at origin (left) and as a function of particle number (right) in the large q regime.
A Zero Temperature Phase Transition
The main property of a quantum phase transition [19] is the existence of a value of a coupling g c at which there can be a level-crossing where an excited level becomes the ground state. Usually this creates a point of nonanalyticity of the ground state energy as a function of g. Our previous graphs in Figs. 2 and 3 show this nonanalyticity and level crossing behavior. To emphasize this point, we plot the critical mass of a boson star as a function of charge in Fig. 4; note the previously defined q 1 are q 2 are the two inflexion points in this graph. Allowed values for boson star masses lie below and to the right of the curve.
In order to interpret Fig. 4, we note that, for a given scalar charge q, boson stars exist for a range values of M . What we plot is the critical mass, namely the largest mass for the scalar dominated branch, and the minimum and the maximum masses for the gauge field dominated branch. To be more graphic, the right panel in Fig. 2 contains three distinguished points: the largest mass for the green branch and the smallest and largest mass for the red branch. These are precisely the three points in the phase diagram in the region q 1 < q < q 2 . For q < q 1 there is only one type of solution and we simply plot its maximum mass. Similarly for q > q 2 , we plot only the maximum mass. In this work we are not going to be concerned with the existence of a maximum charge, q max , above which no regular solution exists. However, preliminary investigations suggest that such a point exists, and we will discuss its determination elsewhere. A very important role in phase transitions is played by the order parameter. In this case we identify it as the value of the scalar at asymptotic infinity, σ 1 . In particular, in the case of the holographic superconductors [20], σ 1 corresponds to the expectation value of an operator of mass dimension ∆. In Fig. 6 we plot σ 1 for critical mass boson stars (those following the curve of Fig. 4), and we see a sharp drop around the value of q 2 .
Conclusions
In the context of a charged scalar field minimally coupled to Einstein-Maxwell gravity with a negative cosmological constant, we have constructed explicit solutions and established a phase diagram of boson stars. The new type of solutions contain a gauge-field-dominated branch that represents a sort of "Geon" as originally envisioned by Wheeler, that is, a particle-like solution from mostly the smooth, classical fields of electromagnetism coupled to general relativity.
Let us conclude by commenting on some open questions stemming from our work. The main difference in gravitational collapse in asymptotically AdS spaces is the presence of a timelike boundary at spatial and null infinities. Under these conditions, the question of stability of our configurations is a particularly important one. In the phase space of initial conditions for gravitational collapse in AdS it is important to consider that a small perturbation does not escape to spatial infinity and actually returns to the center of AdS. We postpone such detailed study to a separate publication but will speculate on the role that boson stars might play.
Already in the context of asymptotically Minkowski gravitational collapse, boson stars [21] play a particularly important role; they are the basis for a different type of critical collapse [14]. It is natural to expect that in asymptotically AdS spaces they will figure prominently. It will be interesting to pursue the construction and investigation of properties of real boson stars, similar to those of asymptotically Minkowski spacetimes [21], to asymptotically AdS. One would expect that real boson stars, if they exist, get similarly destroyed after a certain number of bounces of the scalar field at spatial infinity. Recently, however, the conjecture that boson stars might be nonlinearly stable has been advanced in [22] based on the observation that a mass scale might prevent the resonant turbulent mechanism of [8] from being realized. Scalar boson stars have been previously considered in asymptotically AdS spacetimes by [23][24][25]. Our work has concentrated on charged objects but a systematic anal-ysis of those configurations with the corresponding interpretation and extension to the AdS/CFT applications ought to be presented. We will report on such studies, including the neutral case, in a separate publication.
By exploring the form of the boson star phase diagram, we have argued in favor of a gravity dual of a zero temperature transition for the dual field theory. It will be interesting, using the holographic dictionary, to compute explicitly various transport properties of such dual condensed matter systems.
FIG. 1 :
1The scalar field profile σ(ρ) (top) and energy contributions (bottom) for regular and new solutions. The left panel (q = 0.56) is for a standard charged boson star; the right panel (q = 0.6) is for a new solution.
FIG. 2 :
2Mass as a function of scalar field at origin (left) and as a function of particle number (right) in the intermediate q regime.
From a holographic point of view, a plot of mass versus the time component of the Maxwell field represents the FIG. 4: The phase diagram of charged boson stars in AdS.free energy as a function of chemical potential. While we use the gauge invariant defined in (6), our asymptotic boundary conditions relate this to the value of ω. We thus plot the mass versus ω inFig. 5for solutions in the intermediate and large charge regimes.
FIG. 5 :
5The Mass as a function of the frequency ω, which is a holographic proxy for chemical potential. The left panel corresponds to q1 < q < q2, while the right panel corresponds to q > q2.
FIG. 6 :
6The behavior of the order parameter σ1 as a function of the charge q.
AcknowledgmentsWe acknowledge clarifying discussions with K. Sun and
is thankful to the USTC/Michigan UROP program that made possible this collaboration. L.A.P.Z. is thankful Aspen Center for Physics for hospitality during various stages of this work. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915 (KITP), grant No. 1066293 (Aspen) and by Department of Energy under grant DE-SC0007859 to the University of Michigan. C S L Keeler, * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]. Keeler. S.L. is thankful to the USTC/Michigan UROP program that made possible this collaboration. L.A.P.Z. is thankful Aspen Center for Physics for hospitality dur- ing various stages of this work. This research was sup- ported in part by the National Science Foundation un- der Grant No. NSF PHY11-25915 (KITP), grant No. 1066293 (Aspen) and by Department of Energy under grant DE-SC0007859 to the University of Michigan. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
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F Schunck, E Mielke, arXiv:0801.0307General relativistic boson stars. 20F. Schunck and E. Mielke, General relativistic boson stars, Class.Quant.Grav. 20 (2003) R301-R356, [arXiv:0801.0307].
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On weakly turbulent instability of anti-de Sitter space. P Bizon, A Rostworowski, arXiv:1104.3702Phys.Rev.Lett. 10731102P. Bizon and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space, Phys.Rev.Lett. 107 (2011) 031102, [arXiv:1104.3702].
A Comment on AdS collapse of a scalar field in higher dimensions. J Jalmuzna, A Rostworowski, P Bizon, arXiv:1108.4539Phys.Rev. 84850213 pages, 2 figuresJ. Jalmuzna, A. Rostworowski, and P. Bizon, A Comment on AdS collapse of a scalar field in higher dimensions, Phys.Rev. D84 (2011) 085021, [arXiv:1108.4539]. 3 pages, 2 figures.
O J Dias, G T Horowitz, J E Santos, arXiv:1109.1825.*Temporaryentry*Gravitational Turbulent Instability of Anti-de Sitter Space. O. J. Dias, G. T. Horowitz, and J. E. Santos, Gravitational Turbulent Instability of Anti-de Sitter Space, arXiv:1109.1825. * Temporary entry *.
Rapid Thermalization in Field Theory from Gravitational Collapse. D Garfinkle, L A Pando Zayas, arXiv:1106.2339Phys.Rev. 8466006D. Garfinkle and L. A. Pando Zayas, Rapid Thermalization in Field Theory from Gravitational Collapse, Phys.Rev. D84 (2011) 066006, [arXiv:1106.2339].
On Field Theory Thermalization from Gravitational Collapse. D Garfinkle, L A Pando Zayas, D Reichmann, arXiv:1110.5823JHEP. 1202D. Garfinkle, L. A. Pando Zayas, and D. Reichmann, On Field Theory Thermalization from Gravitational Collapse, JHEP 1202 (2012) 119, [arXiv:1110.5823].
H Oliveira, L A Pando Zayas, C A Terrero-Escalante, arXiv:1205.3232Turbulence and Chaos in Anti-de-Sitter Gravity. H. de Oliveira, L. A. Pando Zayas, and C. A. Terrero-Escalante, Turbulence and Chaos in Anti-de-Sitter Gravity, arXiv:1205.3232.
Phases of massive scalar field collapse. P R Brady, C M Chambers, S M Goncalves, gr-qc/9709014Phys.Rev. 56P. R. Brady, C. M. Chambers, and S. M. Goncalves, Phases of massive scalar field collapse, Phys.Rev. D56 (1997) 6057-6061, [gr-qc/9709014].
Critical behavior in gravitational collapse of a Yang-Mills field. M W Choptuik, T Chmaj, P Bizon, gr-qc/9603051Phys.Rev.Lett. 77M. W. Choptuik, T. Chmaj, and P. Bizon, Critical behavior in gravitational collapse of a Yang-Mills field, Phys.Rev.Lett. 77 (1996) 424-427, [gr-qc/9603051].
Lectures on holographic methods for condensed matter physics. S A Hartnoll, arXiv:0903.3246Class.Quant.Grav. 26S. A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class.Quant.Grav. 26 (2009) 224002, [arXiv:0903.3246].
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Building a Holographic Superconductor. S A Hartnoll, C P Herzog, G T Horowitz, arXiv:0803.3295Phys. Rev. Lett. 10131601S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601, [arXiv:0803.3295].
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Boson stars with negative cosmological constant. D Astefanesei, E Radu, gr-qc/0309131Nucl.Phys. 665D. Astefanesei and E. Radu, Boson stars with negative cosmological constant, Nucl.Phys. B665 (2003) 594-622, [gr-qc/0309131].
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|
[] |
[
"Primordial Black Holes With Variable Gravity",
"Primordial Black Holes With Variable Gravity"
] |
[
"Arbab I Arbab 1e-mail:[email protected] \nDepartment of Physics\nFaculty of Science\nUniversity of Khartoum\nP.O. Box 32111115KhartoumSUDAN\n"
] |
[
"Department of Physics\nFaculty of Science\nUniversity of Khartoum\nP.O. Box 32111115KhartoumSUDAN"
] |
[] |
We have studied the evolution of primordial black holes (PBHs) in a universe with a variable gravitational constant and bulk viscosity. We have found that the strength of gravity has changed appreciably in the early universe. The gravitational constant attained its greatest value at t = 10 −23 sec after the Big Bang. PBHs formed at the GUT and electroweak epochs would have masses about 1.85 × 10 9 g and 1.8 × 10 −7 g respectively. Their temperatures when they explode are 6 × 10 8 K and 6K respectively. PBHs formed during nuclear epoch are hard to detect at the present time. The gravitational constant (G) is found to increase as G ∝ t 2 in the radiation epoch. The gamma rays bursts (GRBs) may have their origin in the evaporation of the PBHs formed during GUT time.PACS no(s): 98.80.Dr., 98.80.Hw,
| null |
[
"https://export.arxiv.org/pdf/astro-ph/9811422v4.pdf"
] | 119,539,281 |
astro-ph/9811422
|
8591acb3bcf215dd67fea4a37b9ad37030136041
|
Primordial Black Holes With Variable Gravity
14 Jun 1999
Arbab I Arbab 1e-mail:[email protected]
Department of Physics
Faculty of Science
University of Khartoum
P.O. Box 32111115KhartoumSUDAN
Primordial Black Holes With Variable Gravity
14 Jun 19991
We have studied the evolution of primordial black holes (PBHs) in a universe with a variable gravitational constant and bulk viscosity. We have found that the strength of gravity has changed appreciably in the early universe. The gravitational constant attained its greatest value at t = 10 −23 sec after the Big Bang. PBHs formed at the GUT and electroweak epochs would have masses about 1.85 × 10 9 g and 1.8 × 10 −7 g respectively. Their temperatures when they explode are 6 × 10 8 K and 6K respectively. PBHs formed during nuclear epoch are hard to detect at the present time. The gravitational constant (G) is found to increase as G ∝ t 2 in the radiation epoch. The gamma rays bursts (GRBs) may have their origin in the evaporation of the PBHs formed during GUT time.PACS no(s): 98.80.Dr., 98.80.Hw,
INTRODUCTION
As well as being formed in the course of natural stellar or galactic evolution, black holes may also have been produced primordially, i.e., at the very earliest epochs of cosmological time. The process of a primordial black hole (PBH) formation in a cosmological model with variable gravitational constant G creates an interesting problem. If one considers a Schwartzschild's black hole formed in the very early Universe at time t f when G had the value G f , which may be different from the present gravitational constant G 0 . The horizon size of a black hole is given by (1)
R f = 2G f M c 2 ∼ ct f ,
Thus as long as G is increasing the entropy of the black hole increases. However, in the case of a decreasing G one needs a further remedy to account for an increased entropy. Barrow [1] has conjectured the idea of a gravitational memory of the black hole (i.e., a black hole remembers the value of G at the time of its formation). This will have a dramatic implications for the Hawking evaporation of PBHs as the temperature and the life time of the evaporating black holes are determined by the value of G f and not by G 0 . These are given by [2] τ
∼ G 2 f M 3 and T ∼ G −1 f M −1 .(2)
Black holes which explode today are those whose Hawking's life time is equal to t 0 (the present age of the universe). Consequently one obtains
M ex ≃ 4 × 10 14 ( G 0 G f ) 2 3 g ,(3)
and their temperature when they explode is therefore given by,
T ex ≃ 24 × ( G 0 G f ) 1 3 MeV ,(4)
where the suffix "ex" stands for explode.
EVAPORATION OF PRIMORDIAL BLACK HOLES
The radiated particles from a PBH have temperature T given by [2] kT =h
c 3 8πGM ∼ 10 26 M −1 g(5)
and the entire mass is radiated away in a time given by
τ ∼ 3 × 10 −27 M 3 g sec,(6)
where M g is the mass in grams. It is clear that as M decreases T increases and the mass loss increases until finally reaches a catastrophic limit (explosion or evaporation). Note that the stellar black hole (M g > 10 23 ) is unlikely to explode in the life time of the Universe [3]. Carr [4] has investigated the PBH formation and evaporation in order to see whether the presently observed nucleon density as well as the microwave background radiation (MBR) can be explained in terms of emission of baryons, leptons, photons and so on, by a low mass black hole. PBHs formed from inhomogeneities at time t must have an initial mass (M i ) of the order of the particle horizon mass (M H ) [5]:
M i ≈ M H = c 3 G −1 t = 10 5 (t/s)M ⊙(7)
where M ⊙ is the solar mass. PBHs forming at Planck time (10 −43 sec) would have the Planck's mass (10 −5 g), whereas those formed at 10 −23 sec would have a mass 10 15 g required for PBHs which evaporate at the present epoch. The size of the PBH at any given time is limited by the size of the particle horizon. PBHs can radiate either elementary particles, e.g., quarks, gluons, which later emit particles such as baryon, meson and leptons; or composite particles directly (baryon, mesons, leptons). A PBH does not matter whether the emitted particle is a particle or an anti-particle. So the baryon number is not necessarily conserved. This possibility can be used to account for the observed baryon-to-photon ratio.
MACH'S PRINCIPLE AND THE VARIABLE GRAVITY
The inertial forces observed locally in an accelerated laboratory may be interpreted as gravitational effects having their origin in distant matter accelerated relative to the laboratory [6,7]. Einstein has tried to incorporate this principle in the formulation of his theory of general relativity (GR). Brans and Dicke have developed a theory which incorporates Mach's principle. A model incorporating the elements of Mach's principle was given by Sciama [8]. He, from dimensional argument, concluded that the gravitational constant G is related to the mass distribution in a uniform expanding Universe through GM
Rc 2 ∼ 1(8)
where R and M are the radius and the mass of the visible Universe, respectively. This relation suggests that either the ratio M to R should be fixed or the gravitational constant G observed locally should be variable and determined by the mass distribution. Only mass ratio can be compared at different points, but not masses. It should be stated that the strong equivalence principle upon which GR stands is incompatible with variable G. In 1937 Dirac postulated the existence of very large numbers and constructed a cosmological model in which G decreases with time as G ∝ t −1 in order not to change the atomic physics [12]. Unfortunately, his model could not resist the observational data.
We have recently presented a cosmological model with variable G and bulk viscosity. The gravitational constant G is found to vary with time as [9],
G ∝ t 2n−1 1−n ,(9)
where n is the viscosity 'index', 0 ≤ n ≤ 1, and G ∝ exp(Bt) , where B = const., during inflation (n = 1).
STRONG GRAVITY
Salam [10,11] has considered the gravitational interaction mediated via heavy mesons and found that the gravitational forces are very strong. He remarked that the nuclear physics should better be called strong gravity. Sivaram and Sinha [11] identified the strong f-gravity metric with Dirac's atomic metric and the large value of the coupling constant (that is G f = 10 40 G 0 ) provided the physical basis for the Large Number Hypothesis (LNH). So if one considers the earliest era when the Universe consisted of an extremely hot compact gas of hadrons, the epoch 10 −23 sec, then if G varied according to LNH right down to the present epoch, it would have a very large value G = 10 40 G 0 , at the beginning of the hadron era. This value is precisely the value G f found in considering the short range f-gravity mediated by massive 2 + − f mesons. Thus in a region of strong curvature, nuclear physics is analogous to gravity. It is evident from eq.(8) that if one considers the Planck epoch, the nuclear epoch and the present epoch we will get G N G Pl
M N M Pl = R N R Pl(10)
and
G 0 G Pl M 0 M Pl = R 0 R Pl(11)
where the G N (G Pl ), M N (M Pl ), R N (R Pl ) are the values of the gravitational constant, mass and the radius of the Universe at the nuclear (Planck) epoch, respectively. It has been shown that throughout all epochs in the early universe, the relation [7] Gm 2 =hc = const. (12) was valid, i.e., we had
G N m 2 N = G Pl m 2 Pl = G W m 2 W = G GUT m 2 GUT =hc ,(13)
This behavior of G can not be interpreted by Dirac or Brans-Dicke model [12]. Barrow and Carr [5] have considered the evolution of the PBHs in the context of scalar-tensor theories and in particular to Brans-Dicke theory. The cosmological considerations investigated by them restrict the value of ω (the coupling constant) to the value −4/3 > ω > −3/2. The case ω = −4/3 was, however, excluded. Comparing these results with our model [9], we obtain the same constraint for ω in addition to the physical significance of the case ω = −4/3. This case corresponds, in our model, to the inflationary solution. In fact, our temporal behavior of G and the scale factor R is defined for all values of n (the viscosity index). The constraint made by Barrow and Carr on ω would imply an increasing gravitational constant. From eq.(9), it can be seen that G increases for n > 1/2, decreases for n < 1/2, remains constant for n = 1/2, and increases exponentially during inflation, i.e., when n = 1 [9]. Our model predicts that in the radiation epoch, for n = 3 4 , R ∝ t, T ∝ R −1 , G ∝ t 2 and ρ ∝ t −4 . A similar variation is found by Abdel Rahman [13] in the radiation epoch. Thus for Planck, GUT, nuclear and electroweak epochs, one has
G N /G Pl = (t N /t P ) 2 = 10 40 , G N /G W = (t N /t W ) 2 = 10 8 , G N /G GUT = (t N /t GUT ) 2 = 10 32 (15) and R N /R Pl = t N /t Pl , R N /R W = t N /t W , R N /R GUT = t N /t GUT ,(16)or R N = 10 −13 cm , R W = 10 −17 cm , R GUT = 10 −29 cm , R Pl = 10 −33 cm ,(17)
where t N = 10 −23 sec, t P = 10 −43 sec, t W = 10 −27 sec and t GUT = 10 −39 sec, are the nuclear time, Planck's time, electroweak time and GUT time. Thus both Abdel Rahman's model and the present model predict the behavior of G quoted in eq. (14). The relation G ∝ t 2 is equivalent to eqs. (8) and (12). Note that in the Standard Model R ∝ t 1/2 , T ∝ R −1 , G = const. This relation can't account for the above relations. Thus it is suggestive to use this equation to predict a fundamental mass at any epoch in the early universe. If one combines this law with eqs. (8) and (12), one finds
m ∝ t −1 ,(18)
a relation that was valid in the early universe. It has been suggested by several authors that the PBH formed at the nuclear epoch (10 −23 sec) would have a mass of 10 15 g. However, if one considers the 'correct' value of G we would obtain a value of 10 −24 g. This is, in fact, equal to the mass of the proton. PBHs formed in the early universe would have temperatures and masses, when exploded, given by (eqs. (3), (4) and (14))
T N ex = 1.29 × 10 −2 K and M N ex = 8.6 × 10 −13 g ,(19)
T GUT ex = 6 × 10 8 K and M GUT ex = 1.85 × 10 9 g ,
and T W ex = 6 K and M W ex = 1.8 × 10 −7 g .
One would therefore expect to observe PBHs emitting x-rays. These PBHs were formed during the GUT phase transition. The remnant of PBHs that formed during the nuclear epoch would be difficult to detect since they have a temperature below the cosmic background radiation. It is interesting to note that PBHs forming during Planck's epoch are not affected, since G Pl = G 0 . Recently, Cline [14] has considered the PBH evaporation during the quark-gluon phase transition. He has shown that short gamma rays burst (GRB) occurs when the mass of the PBH is either 10 14 or 10 9 g. Thus eq.(20) may indicate the emission of short GRBs. Hence, the spectra of these PBHs are different from those with constant G. We would, therefore, expect to observe PBHs at a lower temperature. Thus a possible variation of G would alter the picture of the PBHs previously known.
ACKNOWLEDGMENTS
I would like to thank the Abdus Salam International Center for Theoretical Physics for hospitality and the Associate Scheme for financial support.
where M is the mass of the black hole. The horizon area (A) and the entropy (S) of the black hole are given by A ∝ M 2 G 2 and S ∝ GM 2 .
where G N , G W , G GUT refer to the strong (nuclear) gravitational, weak and GUT coupling constants respectively and m p , m Pl , m W and m GUT refer to the nucleon mass, the Planck's mass, the intermediate boson and GUT unification mass respectively. Inserting the numerical values: R 0 = 10 28 cm, R N = 10 −13 cm, R Pl = 10 −33 cm, M 0 = 10 56 g, m Pl = 10 −5 g, m W = 10 3 GeV and m GUT = 10 15 GeV: eqs.(10),(11) and (13) yield G GUT = 10 8 G 0 , G W = 10 32 G 0 , G N = 10 40 G 0 , and G Pl = G 0 .
. -J D Barrow, gr-qc /9711084-J.D.Barrow, gr-qc /9711084
. S W Hawking, Nature. 24830S.W. Hawking, Nature 248(1974) 30
. G Kang, Phy. Rev. 547483G.Kang, Phy. Rev.D54(1996) 7483
. B J Carr, Astrophy. Journal. 2011B.J.Carr, Astrophy. Journal 201(1975) 1
. -B J Carr, S W Hawking, ; J D Barrow, B J Carr, Mon.Not. Roy. Astr. Soc. 168308Nature-B.J. Carr and S.W.Hawking, Mon.Not. Roy. Astr. Soc,168 (1974) 399, J.D. Barrow and B.J.Carr, Phys. Rev.D54(1996) 3920, J.H. MacGibbon, Nature 329(1987) 308
. T Singh, L N Rai, Gen. Rel. Gravit. 15875T.Singh and L.N. Rai, Gen. Rel. Gravit.15 (1983) 875
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. V De Sabbata, C Sivaram, Astrophys. Spc. Sci. 158347V. de Sabbata and C. Sivaram, Astrophys. Spc. Sci.158 (1989) 347
. D W Sciama, Mon. Not. Roy. Astr. Soc. 11334D.W.Sciama, Mon. Not. Roy. Astr. Soc. 113 (1953) 34
. A I Arbab, Gen Rel, A.I.Arbab, Gen. Rel..
. Gravit. 2961Gravit.29 (1997) 61
. A Salam, J. Strathdee,Lett.Nouvo. Cimento. 4101A. Salam, J. Strathdee,Lett.Nouvo. Cimento. 4 (1970) 101
. -C Sivaram, K P Sinha, Phy. Lett. 60181-C. Sivaram and K.P.Sinha Phy. Lett.60B (1976) 181
. -P A M Dirac, ; C Brans, R H Dicke, Gen. Rel. Gravit. M.M. Abdel Rahman139655Phys. Rev.-P.A.M. Dirac, Nature 139 (1937) 323, C.Brans and R.H.Dicke, Phys. Rev.124 (1961) 925 13-A.-M.M. Abdel Rahman, Gen. Rel. Gravit.22 (1990) 655
. -D B Cline, Nuclear Physics. 610500-D.B. Cline, Nuclear Physics A610 (1996) 500c
|
[] |
[
"Encoding prior knowledge in the structure of the likelihood",
"Encoding prior knowledge in the structure of the likelihood"
] |
[
"Jakob Knollmüller \nMax-Planck-Institut für Astrophysik\nLudwig-Maximilians-Universität München\nKarl-Schwarzschildstr. 1, Geschwister-Scholl-Platz 185748, 80539Garching, MunichGermany, Germany\n",
"Torsten A Enßlin [email protected] \nMax-Planck-Institut für Astrophysik\nLudwig-Maximilians-Universität München\nKarl-Schwarzschildstr. 1, Geschwister-Scholl-Platz 185748, 80539Garching, MunichGermany, Germany\n"
] |
[
"Max-Planck-Institut für Astrophysik\nLudwig-Maximilians-Universität München\nKarl-Schwarzschildstr. 1, Geschwister-Scholl-Platz 185748, 80539Garching, MunichGermany, Germany",
"Max-Planck-Institut für Astrophysik\nLudwig-Maximilians-Universität München\nKarl-Schwarzschildstr. 1, Geschwister-Scholl-Platz 185748, 80539Garching, MunichGermany, Germany"
] |
[
"Journal of Machine Learning Research"
] |
The inference of deep hierarchical models is problematic due to strong dependencies between the hierarchies. We investigate a specific transformation of the model parameters based on the multivariate distributional transform. This transformation is a special form of the reparametrization trick, flattens the hierarchy and leads to a standard Gaussian prior on all resulting parameters. The transformation also transfers all the prior information into the structure of the likelihood, hereby decoupling the transformed parameters a priori from each other. A variational Gaussian approximation in this standardized space will be excellent in situations of relatively uninformative data. Additionally, the curvature of the log-posterior is well-conditioned in directions that are weakly constrained by the data, allowing for fast inference in such a scenario. In an example we perform the transformation explicitly for Gaussian process regression with a priori unknown correlation structure. Deep models are inferred rapidly in highly and slowly in poorly informed situations. The flat model show exactly the opposite performance pattern. A synthesis of both, the deep and the flat perspective, provides their combined advantages and overcomes the individual limitations, leading to a faster inference.
| null |
[
"https://arxiv.org/pdf/1812.04403v1.pdf"
] | 54,557,344 |
1812.04403
|
713826c45b789be73d9c01381760712a4b297240
|
Encoding prior knowledge in the structure of the likelihood
Jakob Knollmüller
Max-Planck-Institut für Astrophysik
Ludwig-Maximilians-Universität München
Karl-Schwarzschildstr. 1, Geschwister-Scholl-Platz 185748, 80539Garching, MunichGermany, Germany
Torsten A Enßlin [email protected]
Max-Planck-Institut für Astrophysik
Ludwig-Maximilians-Universität München
Karl-Schwarzschildstr. 1, Geschwister-Scholl-Platz 185748, 80539Garching, MunichGermany, Germany
Encoding prior knowledge in the structure of the likelihood
Journal of Machine Learning Research
00000Submitted 12/18; Published 00/00Editor:Information TheoryProbability TheoryVariational InferenceGaussian ProcessesBayesian Inference
The inference of deep hierarchical models is problematic due to strong dependencies between the hierarchies. We investigate a specific transformation of the model parameters based on the multivariate distributional transform. This transformation is a special form of the reparametrization trick, flattens the hierarchy and leads to a standard Gaussian prior on all resulting parameters. The transformation also transfers all the prior information into the structure of the likelihood, hereby decoupling the transformed parameters a priori from each other. A variational Gaussian approximation in this standardized space will be excellent in situations of relatively uninformative data. Additionally, the curvature of the log-posterior is well-conditioned in directions that are weakly constrained by the data, allowing for fast inference in such a scenario. In an example we perform the transformation explicitly for Gaussian process regression with a priori unknown correlation structure. Deep models are inferred rapidly in highly and slowly in poorly informed situations. The flat model show exactly the opposite performance pattern. A synthesis of both, the deep and the flat perspective, provides their combined advantages and overcomes the individual limitations, leading to a faster inference.
Introduction
Hierarchical Bayesian models make it possible to express the complex relations in real systems by combining a priori domain knowledge with data. The prior knowledge is updated by the observed data to obtain information about the system at hand. Such models can exhibit deep hierarchies, relating a large number of conceptually distinct parameters in nontrivial fashions. The inference of the posterior parameters can be extremely problematic, especially for large and complex models due to strong dependencies between the quantities and the resulting numerical stiffness. A way to overcome such limitations is to perform coordinate transformations of the parameters to less interdependent ones. In the context of Hamiltonian Monte Carlo (HMC) techniques it was proposed to perform a linear coordinate transformation (Kuss and Rasmussen, 2005) to decouple the parameters, which in the discussed case leads to a white, standard Gaussian prior for the new parameters. This way the numerical performance was increased. Another, more general transformation scheme was proposed by Betancourt and Girolami (2015) to flatten the deep hierarchical structure, to decouple the parameters by introducing auxiliary parameters or performing a reparametrization of existing ones. The same kind of transformation is also known as reparametrization trick (Kingma and Welling, 2013) to learn the parameters of an approximate distribution. It is used in Automatic Differentiation Variational Inference to transform the original model into a standardized space, where a variational approximation is conducted. In this paper we discuss this transformation for general hierarchical Bayesian models and numerical implications of performing variational approximations in these transformed parameters in terms of fidelity and convergence.
We derive this standardizing transformation in two steps. First we transform the original parameters of the deep hierarchy model to independent, uniformly distributed parameters, using the multivariate distributional transform (Rüschendorf, 2009). This step already removes the deep hierarchy and introduces a uniform prior. The uniform prior is problematic for many inference schemes as it limits the parameter space to the unit interval but does not provide further gradient information for the parameters. Thus, in a second step we then transform the uniform parameters to a Gaussian parameters with unit variance and vanishing mean. The overall transformation is then a non-linear, deterministic machinery which relates uniform, white Gaussian parameters to the original parameters of the deep model. The prior information of the deep model is stored in the structure of the transformation itself. The Gaussian prior allows us to make quantitative statements about the conditioning of the curvature of the log-posterior in different scenarios, which largely determines the difficulty numerical inference schemes face. The transformation leads to an optimal conditioning in parameter directions that are only poorly constrained by the data. Directions that are highly constrained by the data result in bad conditioning in this transformed space. Although the transformation discussed does not change the statistical model, inference schemes that involve approximations do depend on the choice of the coordinate system.
As an illustrating numerical example we discuss a Gaussian process regression with also unknown correlation structure. To infer the correlation structure as well statistical homogeneity and isotropy is assumed. The correlation structure can then be expressed in terms of a one-dimensional power spectrum that fully specifies it and has to be inferred together with the signal. To be specific about the statistical process generating the signal we assume the signal to be the result of a zero mean Gaussian process with a kernel as implied by the unknown power spectrum. The log-power spectrum is assumed to be generated by a Gaussian process as well, this time with a known smoothness enforcing kernel. In this scenario we can conduct the identical approximation in both coordinate systems, allowing us to investigate the effect of the transformation on the numerics. We compare the convergence behavior of this deep and highly coupled model with the transformed, flat model in situations with different amounts of data. Here we find that the deep model performs well in highly informed cases and struggles in low information scenarios. The flat model behaves the other way round. Alternating between the two perspectives in a numerical scheme overcomes their individual limitations and provides an overall increased performance.
Related works
Inverse transform sampling (Devroye, 1986) is used to generate random variables according to an arbitrary distribution from uniform samples. This is done via the inverse of the cumulative density function (CDF), or quantile function. Its multivariate generalization is the multivariate distributional transform (Rüschendorf, 2009), which encodes all the internal hierarchical dependencies of the model. We will use it to reparametrize the original model to obtain the flat formulation.
The reparametrization trick allows to infer parameters of complex approximate posterior distributions via variational inference. The problem is to take gradients with respect to the parameters of the approximation as the Kullback-Leibler divergence is only estimated statistically via samples from the approximation (Salimans et al., 2013). Reparametrizing the coordinates of the problem in a certain way allows to separate the randomness of drawing samples from a deterministic modification by the parameters. This can, for example, be achieved analogously to the inverse transform sampling by using the inverse CDF. The advantage of the reparametrization of the distribution is, that now gradients can be calculated with respect to the parameters of the distribution, which only appear in the deterministic part. It is introduced in the Auto Encoding Variational Bayes (AEVB) algorithm (Kingma and Welling, 2013) to make the parameters of the approximate distribution part of the network architecture. Samples to approximate the variational bound can then be drawn from some simple distribution, allowing the inference of all parameters.
Automatic Differentiation Variational Inference (ADVI) performs a transformation of the problem to a set of standardized, real-space parameters where then a variational approximation is conducted . We construct this transformation in terms of the multivariate distributional transform and investigate theoretical and numerical properties of performing approximations in this transformed space.
Normalizing flows are used for non-parametric density estimation (Tabak et al., 2010;Tabak and Turner, 2013). To apply those one tries to find a set of transformations of the parameters of a simple distribution, such that the transformed distribution matches the target distribution as closely as possible. Here it is important to keep track of the functional determinants introduced by the transformation, in order to keep the resulting distribution normalized. The learned transformation stores all the complexity of the approximate posterior in a deterministic way, whereas the randomness originates from a simple distribution. Similarly, the transformation captures the complex structure of the deep model and relates it back to a simple prior distribution. Instead of learning a suitable transformation, the standardizing transformation makes use of the structure of the hierarchical model. A method that makes use of both, the reparametrization trick, as well as normalizing flow is the variational inference with normalizing flows (Rezende and Mohamed, 2015).
Basics and notation
Bayesian inference
In Bayesian inference the prior knowledge P(θ), expressed as a probability density function (PDF) of some quantity θ should be updated after we obtained data d. This data is related to the quantity of interest by a likelihood P(d|θ) of observing the data, given θ. The updated knowledge is the posterior density P(θ|d) and is calculated according to Bayes theorem:
P(θ|d) = P(d|θ)P(θ) P(d)(1)
In order to do so, one has to normalize the joint density P(d, θ) = P(d|θ)P(θ) with respect to θ, which is done by division with the evidence P(d).
An alternative way how to describe probabilities is in terms of their information. Information is the negative logarithm of the distribution, H(.) = −ln P(.). Compared to the distributions, which are multiplicative, information is additive. This makes it a more convenient quantity to deal with and they will be adapted later in this work. Bayes theorem in terms of information reads:
H(θ|d) = −ln(P(θ|d)) (2) = H(d|θ) + H(θ) − H(d)(3)
The task of Bayesian inference usually comes down to dealing with the normalization term H(d). In many cases it is not accessible analytically and sampling techniques or approximate inference has to be applied that avoid the calculation of this term. Those only require all terms depending explicitly on the parameters. In order to shorten the notation we will absorb all terms of constant information, therefore parameter independent, into one single information term H 0 and introduce the symbol = that indicates equality up to parameter independent, constant terms.
H(θ|d) = H 0 + H(d|θ) + H(θ) (4) = H(d|θ) + H(θ)(5)
Variational Inference
Variational inference is a powerful tool to approximate an intractable posterior P(θ|d) distribution with simpler distribution P(θ|ϕ) parametrized by a set of parameters ϕ. This is done by minimizing the variational Kullback-Leibler divergence (KL) (Kullback and Leibler, 1951) that is defined as:
D KL P(θ|ϕ)||P(θ|d) ≡ Dθ P(θ|ϕ) ln P(θ|d) P(θ|ϕ) (6) = H(d, θ) P(θ|ϕ) − H(θ|ϕ) P(θ|ϕ)(7)
In the second line parameter independent terms are dropped, as they are irrelevant for the minimization. This includes the normalization constant of the posterior, which typically is the origin of the intractability of the posterior distribution. This term is also equivalent to the negative evidence lower bound (ELBO) (Bishop, 2006). In order to perform the optimization, we do not have to be able to compute the expectation value explicitly, it is sufficient to be capable of drawing samples from the approximate distribution (Salimans et al., 2013).
We will focus on two kinds of variational approximations, point-like, as well as Gaussian approximations. It can be argued whether fitting delta distributions P(θ|ϕ) = δ(θ − θ * ) are truly a variational method, however minimizing the variational KL and maximizing the posterior distribution results in the same procedure. A point estimate on certain parameters might be justified, especially if they are strongly constrained by the problem and they can be inferred significantly faster compared to more sophisticated approaches.
In cases the uncertainty should not be neglected, a variational Gaussian approximation is a powerful tool to infer posterior quantities, as well as their correlations and uncertainties. Gaussian distributions make it simple to sample from them, and derivatives with respect to their parameters can also be calculated easily, as the following expressions hold (Opper and Archambeau, 2009):
dD KL dθ = dH(d, θ) dθ G(θ−θ,Θ)(8)Θ −1 = d 2 H(d, θ) dθdθ † G(θ−θ,Θ)(9)
We will make use of these properties in the example.
Transforming probability densities
Probability densities are differential quantities and to turn them into probabilities they have to be equipped with the differential of their arguments, which are often not stated explicitly, and be integrated over. Keeping track of the differential is relevant for coordinate transformations, as one has to take the differential volume change into account in form of the Jacobian determinant. Assume some transformation θ = f (θ) of θ to a new set of parameters θ . The probability distribution P(θ) then transforms as follows:
θ = f (θ) (10) P(θ)dθ = df −1 (θ ) dθ P(θ )dθ (11) = P (θ )dθ .(12)
The vertical lines | . | indicate the absolute value of the determinant of the matrix expression inside. The new distribution P (θ ) is the combination of the functional determinant of the transformation and the old distribution of the transformed argument. This way the new distribution is properly normalized.
Multivariate distributional transform and standard Gaussian priors
We want to focus on the transformation that transforms a continuous distribution to a uniform distribution on the unit interval. This is achieved by the quantile function, also known as the inverse cumulative density function (CDF). From the uniformly distributed variables we can then transform to white, standard Gaussian coordinates with the CDF of this Gaussian. In the one dimensional case, the quantile transformation reads:
θ 1 = F −1 P(θ 1 ) (u 1 ) ,where(13)F P(θ 1 ) (θ 1 ) ≡ θ 1 −∞ dθ 1 P(θ 1 ) .(14)
The first equation is equivalent to inverse transform sampling (Devroye, 1986) and the second equation defines the CDF. The derivative of the CDF with respect to its argument is therefore again the original PDF.
In general the prior P(θ) with θ = (θ 1 . . . , θ n ) T can be expressed in terms of its hierarchical structure P(θ) = P(θ n |θ 1 . . . θ n−1 ) . . . P(θ 2 |θ 1 )P(θ 1 ).
From this separation we can build up iteratively the the transformation to the hierarchical parameters θ from a set of uniformly distributed parameters u.
θ 1 = F −1 P(θ 1 ) (u 1 )(16)θ 2 = F −1 P(θ 2 |θ 1 ) (u 2 )(17)
. . .
θ n = F −1 P(θn|θ 1 ,θ 2 ...,θ n−1 ) (u n )(18)
This is the multivariate distributional transform (Rüschendorf, 2009) for u i being drawn from the uniform distribution U(u i ) within the interval [0, 1]. From this parametrization we can then change to the white, standard Gaussian distribution in a second step. G(ξ, 1) with uniform, diagonal covariance and vanishing mean. In order to be able to find this transformation in practice, it is necessary that one has explicit access to the hierarchical structure of the model and that CDF's either are available or can be approximated efficiently. This limits its applicability to some extent, but even deep hierarchical models are typically constructed by combining simple distributions and transformations.
We will now perform the above stated coordinate transformation to the uniformly distributed parameters within the joint PDF P(d, θ) of the data and parameters for a given likelihood P(d|θ) and prior distribution P(θ) via P(d, θ) = P(d|θ)P(θ). P(d|θ)P(θ)dθ = P(d|θ) [P(θ 1 ) . . . P(θ n |θ 1 . . . θ n−1 )] dθ du du
(19) = P(d|θ) d dθ 1 F P(θ 1 ) (θ 1 ) . . . d dθ n F P(θn|θ 1 ...θ n−1 ) (θ n ) dθ du du (20) = P d|F −1 P(θ) (u) d dθ 1 F P(θ 1 ) F −1 P(θ 1 ) (u 1 ) . . . d dθ n F P(θn|θ 1 ...θ n−1 ) F −1 P(θn|θ 1 ...θ n−1 ) (u n ) dθ du du (21) = P d|F −1 P(θ) (u) du 1 dθ 1 . . . du n dθ n dθ du du (22) = P d|F −1 P(θ) (u) du dθ dθ du du (23) = P d|F −1 P(θ) (u) du(24)
Here, we first expanded the prior probability into a hierarchical structure and substituted to an integral over the uniform parameters. We then expressed those individual prior probabilities in terms of derivatives of the CDF. Inserting the transformations for each parameter we obtained identity operations by construction. What remained is the product of the derivatives of the new parameters with respect to the old ones that exactly canceled the Jacobian determinant of the transformation. The uniform prior is implicitly present in the last expression, as the parameters u are only defined on the unit interval, where the uniform distribution takes a value of one. For numerical purposes this coordinate space is inconvenient to perform inference due to the compact support of the uniform distribution. To avoid this, we will transform from these uniform parameters into a set of independent Gaussian parameters of unit variance and zero mean. This coordinate transformation is again done via the CDF. It takes the following form:
u =F G(ξ,1) (ξ) (25) = 1 2 + 1 2 erf ξ √ 2 ,(26)
where the error function is defined as
erf(x) = 2 √ π x 0 e −t 2 dt ,(27)
and we adopt the convention that scalar functions are applied to vectors component-wise.
Performing the transformation explicitly yields
P d|F −1 P(θ) (u) du = P d|F −1 P(θ) (u) du dξ dξ (28) = P d|F −1 P(θ) • F G(ξ,1) (ξ) d dξ F G(ξ,1) (ξ) dξ(29)= P d|F −1 P(θ) • F G(ξ,1) (ξ) G(ξ, 1)dξ .(30)
The overall standardizing transformation C P(θ) (ξ) of the deep hierarchical parameters θ to the white, standard Gaussian parameters ξ is therefore the composition, indicated by •, of the two individual transformations.
θ =F −1 P(θ) • F G(ξ,1) (ξ) (31) ≡ C P(θ) (ξ)(32)
Finally, we can rewrite the original probability in terms of the new parameters.
P(d|θ)P(θ)dθ =P d|C P(θ) (ξ) G(ξ, 1)dξ(33)
Explicit examples for a simple hierarchical model and multivariate Gaussian prior distributions are given in Appendices A and B, reproducing the reparametrization trick.
Approximations of the transformed distributions
Approximating posterior distributions allows to infer posterior quantities even for large models within reasonable computational effort. In general it matters in which coordinate system the approximation is conducted. We will discuss the impact of this transformation on two popular approximations in the standardized parameters. The first one will be the maximum posterior (MAP) estimate that is obtained by minimizing the information of the posterior with respect to the parameters. The second approach is to perform a variational approximation with a Gaussian distribution in the standardized coordinates.
Maximum Posterior
A maximum posterior approximation is cheap to compute and can provide meaningful results, if the parameters are constrained reasonably well and uncertainties are small. Conceptually, the true posterior distribution is approximated by a delta distribution at a location that has to be inferred. Minimizing the KL divergence in this case is identical to minimizing the information H(θ|d) with respect to the parameters θ. Performing the approximation in the standardized coordinate system will not necessarily maximize the posterior in the original parametrization. We can discuss the two limiting cases of uninformative likelihood and extremely constraining data.
In the case of an uninformative likelihood, maximizing the posterior will be close to maximizing the white, standard Gaussian information. The result will be a delta distribution peaked close to the the origin. This location splits the probability mass in any direction in half. Transforming this distribution back into the original coordinates this location corresponds to the median of the prior distribution rather than the (or a) mode that would be the result of MAP in the original space. Especially for heavily skewed or multi-modal prior distributions the results differ substantially.
In the limiting case of highly informative data the true posterior distribution will be narrowly peaked around the true parameter value. In this situation the essential features of the posterior can be captured by an approximation with a delta distribution, neglecting uncertainty. The true posterior then also transforms almost like a delta distribution, which will also be narrowly peaked around the transformed maximum. In the highly informed case it therefore does not matter much in which coordinates the approximation is conducted.
Any situation in between the extreme cases will exhibit characteristics of both. In general, performing a MAP approximation in the standard coordinates will push towards median prior configurations, unless the data tells otherwise. This is different to the approximation in original coordinates, which favors maximum prior configurations in the absence of additional information. Towards more conclusive data it becomes irrelevant in which coordinates the posterior is maximized.
Variational Gaussian
A natural choice to approximate the true posterior in the transformed coordinates is the Gaussian distribution. This is demonstrated for ADVI . The transformed prior distribution is just a standard Gaussian. If the data updates the prior only slightly, the true posterior will still be close to a Gaussian distribution, which is captured well by the approximation. The strength of variational inference is to take the uncertainty of the problem into account and thereby prevents over-fitting to some extent. This is especially important if the uncertainty is high, which is the case for uninformative data. This is exactly the situation where the variational Gaussian approximation in the standardized space captures the true uncertainty of the actual posterior the best.
A variational Gaussian approximation will also approximate the true posterior well if the likelihood in the transformed coordinates is close to a Gaussian distribution in the parameters, as combining a Gaussian likelihood and prior results in another Gaussian.
For well constrained parameters this approximation will behave similarly to the MAP approximation, as discussed before. If the likelihood introduces multi-modality into the posterior, the Gaussian approximation will choose one mode and approximate the true posterior locally. In these situations one might consider more flexible distributions to approximate the posterior, for example a Gaussian mixture, as demonstrated in .
Overall, the variational Gaussian approximation of the true posterior in the standardized space will be excellent if the data modifies the posterior only slightly.
Optimization and Conditioning
In order to perform the approximate inference, we do have to optimize a target functional numerically. This problem might be a non-convex optimization and we are not guaranteed to find a global optimum. Here we will discuss the convergence properties of local optimization procedures of the MAP and variational Gaussian approximation in the standardized coordinates. To perform the optimization we do only have access to local information in terms of derivatives of the loss function at the current position in parameter space. Here the negative gradient shows in the direction of the steepest descent and the curvature informs about how the terrain is changing along the different directions. This information is used in a number of Newton and quasi-Newton algorithms to minimize the target functional. The numerical difficulty is encoded in the condition number of the curvature, the ratio of the absolutes of its largest and smallest eigenvalue. The larger the spread of the eigenvalues, the harder the problem is to solve numerically. For non-convex problems the curvature might exhibit negative eigenvalues, especially far from a minimum. In this case a Newton optimization breaks down, as the step will be performed in the wrong direction. A way to overcome this is to approximate the true curvature with a quadratic approximation to the curvature with the Gauss-Newton method that is always positive definite. For our discussion we will consider only convex curvature. Otherwise the same argumentation holds for an approximate Gauss-Newton curvature.
Conditioning of general models: The general curvature of the information for a deep hierarchical model reads:
C = d 2 dθdθ † H(d|θ) + d 2 dθdθ † H(θ)(34)
The conditioning of this curvature will entirely depend on the concrete model, but a number of general properties influence the conditioning strongly. Large absolute entries in the curvature matrix will contribute to a bad conditioning, wherever they occur. The second derivative with respect to two parameters will be large if the relevant terms themselves are highly informative, as well as if the interaction of the parameters within these terms is strong. Highly informative and strongly coupled terms will therefore contribute to bad conditioning. Not only individual, large entries are problematic, but also a large number of relatively small entries associated with one individual parameter. This case is discussed in Betancourt and Girolami (2015) and illustrated there in form of a high dimensional funnel.
Preconditioning: A common technique to reduce the condition number of a linear problem is preconditioning (Shewchuk et al., 1994). The idea is to have an approximation of the original problem that has an easily accessible inverse. This approximation is pulled out of the initial matrix, taking already care of the largest and smallest eigenvalues. It remains to solve a better conditioned problem. For example, we want to have the inverse of a matrix that consists of a simple, invertible, and dominant contribution B plus a small modification A. The condition number is therefore dominated by the matrix B. A way to precondition this problem is to pull B out of it:
(A + B) −1 = B −1 (AB −1 + 1) −1(35)
Now it remains to numerically invert (AB −1 + 1) −1 instead of the initial problem. Here the smallest eigenvalue is bounded by 1 and the largest eigenvalue relates to the largest eigenvalue of AB −1 (plus one), which is smaller than the product of the largest eigenvalues of A and B −1 . As A is only a small modification, its largest eigenvalue will also be small, and therefore pulling out the dominant contribution B leads to a better conditioned problem.
Curvature of MAP and Variational Gaussian:
The curvature of the MAP approximation, which is used in the Laplace approximation to obtain an uncertainty estimate, is the second derivative of the information. It reads in the standardized coordinates:
C H = d 2 dξdξ † H(d|ξ) + 1(36)
For a variational Gaussian approximation we can pull derivatives with respect to the mean inside the expectation value and take the derivative with respect to the original parameters (Opper and Archambeau, 2009):
C D KL = d 2 dξdξ † H(d|ξ) + 1 G(ξ−m ξ ,D ξ )(37)
The curvature for the mean of the Gaussian approximation is the mean of the information curvature over the approximate distribution. This structure allows us to investigate the overall conditioning of the curvature in terms of the largest and smallest eigenvalues introduced by the likelihood. The structure of both curvatures above correspond to the one of the preconditioned problem in the previous paragraph and we can discuss them in the same way.
Conditioning: The above considerations allow us to make statements on the conditioning of the problem, as the condition number is given by
κ = λ max + 1 λ min + 1 .(38)
Here λ max and λ min are the largest and smallest eigenvalues of the likelihood information curvature. The magnitude of the eigenvalues relates to the uncertainty in the corresponding eigendirections of the posterior. This property is used for the Laplace approximation.
Large eigenvalues indicate low posterior variance, and therefore well constrained parameter directions, and vice versa. Certain directions might not be constrained by the data at all and the posterior uncertainty is the prior one, which is indicated by λ min = 0. Because of this, the smallest eigenvalue of the full curvature cannot become smaller than 1. The overall conditioning of the problem therefore mainly depends on λ max . The larger it is, the worse the overall conditioning. The inference of the model will therefore be faster for less informative data. If the smallest eigenvalue λ min is significantly above one, the data constrains all parameters well and the prior will not have much influence on the conditioning.
The bad conditioning for highly informative data can be explained by the structure of this likelihood. All parameters are directly involved into explaining the data. Within the likelihood the parameters are to some extent degenerate, as several might explain the same features. The prior breaks this degeneracy, favoring certain parameter configurations above others. If the likelihood is now extremely strong, the influence of the prior almost vanishes. The first goal of the algorithm will be to minimize the likelihood, irrespective of the prior plausibility. To restore this plausibility, the optimization has to also minimize the prior contributions. This is now only possible by following a trajectory that keeps the likelihood almost constant. This quasi-hard-constraint introduces narrow, high-dimensional valleys into the information function the optimization has to navigate through.
For well constrained parameters we might prefer a MAP approximation. As previously discussed, the resulting estimate should not depend significantly on the coordinates chosen. A way to circumvent the bad conditioning in the highly informed case might be to perform the optimization in the original parameter space with the deep hierarchy.
Numerical example
To illustrate the above considerations we present a numerical example. We will explore the inference of a linear Gaussian process s with unknown kernel operator S from incomplete data with additive, Gaussian noise n . This is an important problem, as Gaussian processes (Krige, 1951) are widely used to model continuous functions or auto-correlated quantities (Rasmussen, 2004). They are also well-suited for image reconstruction, for example in an astrophysical context (Kitaura and Enßlin, 2008;Enßlin et al., 2009;Selig et al., 2015;Junklewitz et al., 2016). The auto-correlation of the process is the result of the properties of the observed system, which might not be known a priori, so it also has to be learned from the data. Parametric (Williams and Rasmussen, 1996), as well as non-parametric (Wilson and Adams, 2013) models have been proposed to infer the correlation structure. The latter work assumes a mixture of Gaussian profiles, which, in principle, can represent any viable spectral density for a sufficiently large number of basis functions. Alternatively the spectrum itself can also be described using a Gaussian process for the logarithmic spectrum, ensuring positive definiteness (Enßlin and Frommert, 2011). This log-normal prior on the spectral density can, for example, enforce spectral smoothness . We will base our discussion on this latter description, but the results should hold for any parametrization of the correlation function.
Gaussian process with spectral smoothness
Gaussian processes are defined over continues spaces, and therefore the involved quantities will be functions and linear operators instead of vectors and matrices. We will additionally consider the presence of a general, linear response function R, which can, for example, select out individual locations where we measure the signal, resulting in our data-points. This operator is necessary to relate the continuous signal s to discrete-valued data. The data is generated according to
d = Rs + n.(39)
This results in a Gaussian likelihood with information of the form:
H(d|s) = 1 2 (d − Rs) † N −1 (d − Rs) + 1 2 ln|2πN |(40)
The information of a Gaussian process prior reads:
H(s|S) = 1 2 s † S −1 s + 1 2 ln|2πS|(41)
The kernel S(x, x ) should be homogeneous, or stationary, and it therefore only depends on the relative distance of two points S(x − x ) that allows us to express the correlation as a diagonal operator in the harmonic basis. The additional assumption of isotropy allows for one dimensional kernel functions that only depend on the relative distance S(|x − x |). The correlation structure is then a diagonal operator in the harmonic basis and fully characterized by its spectral density, according to the Wiener-Khintchin theorem (Wiener, 1949;Khintchin, 1934). The isotropy assumption implies that the spectral density only depends on the absolute values of the coordinates in the harmonic space. This can be expressed via an isotropy operator P, that distributes a one dimensional power spectrum into the diagonal of the covariance operator in the harmonic space. The relation between the correlation structure in position space S and a one dimensional power spectrum p therefore reads:
S = F † (Pp)F(42)
The hat over (Pp) indicates the transformation into a diagonal operator and the harmonic transformation is expressed in terms of F. For flat geometries this is the Fourier transformation. In order to enforce the positive definiteness of the correlation structure, p has to be strictly positive, but its values can vary strongly. In many cases one can assume a smooth power spectrum. A suitable choice of a prior to implement these characteristics is a log-normal Gaussian process prior LN (p, T ). The kernel T implements the degree of desired smoothness. As this contains the hard constraint of positivity, we will instead reparametrize the power spectrum in terms of the logarithmic power spectrum p = e τ , which transforms the hyper-prior to the Gaussian Process prior G(τ, T ). The total information is obtained by adding up all likelihood, prior, and hyper-prior terms. Disregarding all parameter independent terms, it is:
H(d, s, τ ) = 1 2 (d − Rs) † N −1 (d − Rs) + 1 2 s † F † (Pe −τ )Fs + 1 2 Tr (Pτ ) + 1 2 τ † T −1 τ(43)
We will now apply the standardizing transformation to flatten the hierarchy of the Bayesian model. Due to the assumed statistical homogeneity of the signal, we have access to the eigenbasis of the prior correlation structure. Here F is the Fourier transformation, which allows us to take the square root of the eigenvalues to standardize the s parameter as outlined in Appendix B .
s = F Pe 1 2 τ ξ(44)
Performing this substitution introduces the dependency on τ into the likelihood and removes the prior terms depending on τ . The same procedure can be applied to standardize τ as well, which is also described by a Gaussian process. In order to do so we have to express the smoothness kernel T in terms of its eigenbasis and eigenvalues. Smoothness should be a lack of curvature of τ on a logarithmic scale. We can therefore describe the inverse kernel as T −1 = 1 σ 2 ∆ † ∆. The ∆ operator implements the Laplace operator on a logarithmic coordinate system and σ the expected deviations from a smooth spectrum. The larger it is, the more curvature is accepted and vice versa. The Laplace operator is diagonal in the associated harmonic domain and the diagonal elements contained the squared harmonic coordinate l of this logarithmic space.
∆ = V † l 2 V(45)
This V operator is the harmonic transformation in the space of the one dimensional logarithmic power spectrum in logarithmic coordinates. We can now express τ in terms of the standard parameters ζ:
τ = V σ l 2 ζ(46)
With both transformation in place, the transformed information of the full problem reads:
H(d, ξ, ζ) = 1 2 d − RF Pe 1 2 V σ l 2 ζ ξ † N −1 d − RF Pe 1 2 V σ l 2 ζ ξ + 1 2 ξ † 1ξ + 1 2 ζ † 1ζ(47)
Now the entire prior knowledge, such as the concepts of homogeneity, isotropy, spectral and spatial smoothness, and positivity, are absorbed into the likelihood. This now implements the problem of the inference of a Gaussian process with unknown correlation structure in form of a characteristic and generative sequence of linear and nonlinear operations between a priori white parameters. The steps to perform the transformation were technical, but straight forward.
Inference
So far we only formulated the identical problem in two equivalent ways, a deep hierarchical model and a flat hierarchical model with a standard Gaussian white prior, but we still have to perform the inference. In this case the performed coordinate transformations were fully linear. This way we can perform the same approximation in both coordinates systems and the resulting distributions will be transformed versions of each other. We will minimize the variational KL divergence between the true posterior distribution and an approximate distribution. The approximation will be a product of a Gaussian for the signal and a pointestimate for the power spectrum. This illustrates a variational Gaussian approximation, as well as point estimates.
The approximate distribution reads:
P(s, τ |m, D, τ * ) = P(s, τ ) ≡ G(s − m, D)δ(τ − τ * )(48)
The task is to adjust the parameters m, D and τ * such that the KL divergence between this distribution and the true posterior is minimized. Up to parameter independent, constant terms the KL divergence reads:
D KL P(s, τ |d)|| P(s, τ ) = H(s, τ |d) P(s,τ ) − H(s, τ ) P(s,τ ) (49) = 1 2 (d − Rs) † N −1 (d − Rs) + 1 2 s † F † (Pe −τ * )Fs G(s−m,D) + 1 2 Tr (Pτ * ) + 1 2 τ * † T −1 τ * − Tr ln [2πeD](50)
The last term corresponds to the entropy of the Gaussian distribution and the delta distribution is already integrated out. Because the information is fully quadratic in s, we can directly solve for the posterior covariance, as well as the mean for a given τ * (Opper and Archambeau, 2009):
D = R † N −1 R + F † (Pe −τ * )F −1 (51) j = R † N −1 d (52) m = Dj .(53)
P(s|d, τ * ) = G(s − m, D) is the Wiener filter solution for a given correlation structure τ * and it minimizes the KL divergence without the need of a dedicated optimization. The minimization with respect to τ * requires the evaluation of the KL divergence and its gradient with respect to these parameters. In order to estimate the Gaussian expectation value we will draw samples from the Gaussian distribution for the current value of τ * and perform the optimization on a stochastic estimate of the KL divergence (Salimans et al., 2013). The inference procedure now iterates between estimating m and D for a given τ * and minimizing a stochastic estimate of the KL-divergence with respect to τ * for the current parameters of m and D.
Analogous terms can be calculated for the standardized coordinates and the inference procedure will be identical.
Implementation and results
Setup: In the concrete example we will consider a two dimensional Gaussian process drawn from the prior distribution. We generate data by measuring the process at random locations and we additionally add white Gaussian noise. We will present three times the identical setup, varying only the amount of data points. We choose a resolution of 128 × 128 pixel. The first scenario will consist of a measurements of all locations, in the second scenario we randomly sample 10%, or roughly 1600 locations, and in the last case we only take 0.5%, or 83 data points.
The problem is implemented in python using the NIFTy package Steininger et al., 2017). We start the optimization of both models with equivalent initial states. All hyper-parameters of the involved minimizers are the same as well.
Using the true underlying kernel, we can compute the posterior mean m wf of this simpler problem by evaluating the Wiener filter for the correct spectrum. This better informed estimator will serve as our reference point and we compare both methods in terms of the root mean squared errors (RMS), which are defined as:
= (m − m wf ) † (m − m wf )(54)
By monitoring this quantity we can discuss the convergence behavior of the different methods in the different scenarios. We do not track the KL divergence as measure for convergence, as it would require the calculation of the entropy term Tr ln [2πeD]. The posterior covariance D has 128 2 × 128 2 entries and the eigenbasis is not explicitly available. Evaluating the matrix logarithm scales with the third order in the dimension, requiring 128 6 computations in every step, making this term numerically not accessible. In order to perform the inference itself it is sufficient to have the operator implicitly available.
The overall convergence of the algorithm is determined by the convergence of the power spectrum parameters, as we immediately have the mean and variance of the Gaussian given this spectrum. In order to discuss the convergence behavior it is important to identify which parts of the power spectrum are prior dominated and which are strongly constrained by the likelihood. In order to do so we have to explore how the data contains information on the correlations of different scales. Our knowledge on the true process realization is limited in two ways, namely white Gaussian noise and incomplete coverage.
This noise affects all scales equally, and the relevant quantity to look at is the signalto-noise ratio. Scales with high power will not be affected much, and are therefore well constrained by the data. If the signal power drops significantly below the noise power, the prior information will be the dominant term.
The random sampling behaves differently. If we randomly select a small number of positions to measure, their average distance is large and we are mainly informed about the largest scales. We are informed about the small scales by having data points close to each other. The more data points we obtain, the more likely it becomes for them to lie close to another one, providing information on the small scale fluctuations.
In our case both effects will be superimposed, but the sparse sampling of data will be the main contribution. With it we will stir between prior and likelihood dominance.
The results can be seen in Fig. 1. The three columns correspond to the three different cases. The first row shows the actual data, the second row the posterior mean for the correct kernel. The third row illustrates the RMS error of the current estimate after each algorithmic update during the minimization. The last row shows the progression of the spectra during the inference.
Observations:
1. The flat hierarchy model converges faster in all cases in terms of RMS error.
2. In recovering the true spectrum, both models have complementary strengths and weaknesses. The deep hierarchy is superior in the data dominated case and the flat hierarchy in the prior dominated case.
3. Alternating both methods improves convergence with respect to each and the recovery of the true spectrum.
Convergence:
The convergence results can be seen in the third row in Fig 1. The flat hierarchy model converges faster in all three cases in terms of the RMS error. In the full data case both methods converge to the identical error, but the flat hierarchy requires one order of magnitude less iterations. In the scattered data cases, the deep model is faster at the beginning, but after a certain amount of steps, the flat model outpaces the other one significantly. In the sparse data case, the deep hierarchy seemingly stops converging after an initial drop off, whereas the flat model reaches a significantly lower error. This illustrates the advantages of the approximation in the standardized space for relatively uninformative data. In the scattered data case, the convergence behavior is similar, with slight advantage for the flat model. The problem in this case is that the large scales are well constrained by the data, whereas the small scales should still be prior dominated. Because the deep hierarchy converges fast for well constrained parameters and struggles for the less constrained parameters, the over-all convergence is bottle-necked by the most uninformative parameters. The flat model shows exactly the opposite behavior in the different regimes, and is therefore bottle-necked by the most informed parameters. These behaviors become especially apparent in the convergence of the power spectra.
Spectra: We will now discuss the evolution of the power spectra during the minimization. These are depicted in the last row of Fig. 1. Both methods start with spectra at the data full data scattered data sparse data
Wiener Filter The column corresponds to the data set with full coverage, the second one to the scattered data set, and the last one to the sparse data set. The first row shows the data in these cases, the second row depicts the Wiener filter reconstruction given the correct correlation structure. The third row illustrates the RMS error of the current reconstruction to the Wiener Filter solution for both coordinates and the last row shows the progression of the power spectra during the optimization. same horizontal line. From this location the spectra move into the direction of the true correlation structure. We can track fast movement by a low density of lines and slow movement by a high density. Convergence can be identified by small fluctuations around a common spectrum. We also observe the evolution to slow down and almost stopping at some point. This behavior appears as a color gradient in the figure, which progressively gets filled stronger. The deep hierarchy model is extremely fast in picking up scales, which are well constrained by the data. This can be seen by the density of green lines at the largest scales. In the full data case it immediately jumps close to the true spectrum. For sparser data this is a bit delayed, but still fast. This slowing down cannot only be observed between the cases, but also within the different scales in one setting. The smaller scales are less constraint by the data and for those there are a large number of intermediate lines. In the scattered data case this behavior can be seen very well. There the small scales just slowly drop down from the initial position towards the correct spectrum, bottle-necking the convergence.
In the sparse data case, this slow down behavior is even more dominant. Everything except the largest scales did not move away significantly from the initial position and the minimization is incapable to provide a result within a reasonable amount of time.
The overall behavior can be explained by the deep hierarchical structure of the model. We use the current correlation structure to estimate the mean of the process. For parameters the data is good, this current prior correlation structure does not affect its estimate strongly and if data are sparse, the posterior remains close to the prior estimate. This updated process is then used to adjust the correlation structure within the prior distribution. Therefore well constraint scales are immediately set to reasonable values, whereas the weakly constraint modes are already consistent with the current estimate of the power spectrum. The algorithm gets caught in a loop of self-fulfilling prophecies, which cripples down any progress on scales with little data. The flat hierarchy model behaves to some extent in an opposite manner. It is fast to recover scales which are weakly constraint by the data, but struggles for well informed scales. The more data it is available, the slower the spectrum approaches the true solution. In situations where the deep hierarchy has problems, this flat model recovers the spectra reasonably. Especially in the sparse data case it is capable on capturing the basic features of the true spectrum. Besides the clearly identified secondary bump, it also estimates the slope of the true spectrum correctly. Spectral features on even smaller scales could not be identified due to the lack of information. Here the correct spectrum is smoothly interpolated, as we would expect for a Gaussian process in weakly informed scenarios. The flat model is excellent in combining small amounts of data with prior knowledge. Interestingly, its incapability of recovering the correct spectrum in the high data density case does not reflect in the reduction of the RMS error, where the flat model consistently out-competes its competitor.
This behavior can be explained by a multiplicative degeneracy within the likelihood. We re-parametrized the original signal in terms of an amplitude and an excitation:
s = Aξ(55)
This allows to multiply one quantity with some factor, if the other one is divided by it accordingly. In the algorithm we first update the excitation ξ for a given amplitude. In the case of a strong likelihood, the white prior on ξ is almost irrelevant and the excitation will pick up any access power, which is not explained by the initial guess of the amplitude. In the consecutive update of the amplitude itself, this power is hidden inside the excitation and the amplitude cannot pick it up. The likelihood is immediately satisfied and the problem remains in the separation between excitation and amplitude, which is only governed by the priors. The high values in the excitation are incompatible with the prior, which wants to push them down, but it has to act against the extremely strong likelihood. This way, the prior will only slightly push down the values, which releases some power to be picked up by the amplitude. The stronger the likelihood, the slower this process will be and the push from the excitation prior becomes weaker, the progression of the power spectrum will slow down.
Alternating coordinates: Both parametrizations have severe limitations, which is best depicted in the spectra for the scattered data. On large scales, the flat model struggles with internal degeneracy, whereas for small scales the deep model is fighting with self-fulfilling prophecies. Both methods do also have their advantages. The deep model is excellent in the high density data regime, whereas the flat hierarchy allows to recover features even in low information environments.
In this case we can alternate between the coordinates without altering the approximation, due to the linearity of the transformation. In general this is not possible, but as discussed in Sec. 4.1 delta approximations for well constrained parameters can also be optimized in transformed coordinates.
Alternating between both methods might allow to overcome the limitations of either one, leading to overall faster convergence and more reasonable spectra. We will restrict our investigation to the scattered data case, where the limitations of both methods were most apparent. The results can be seen in Fig. 2. In terms of convergence, alternating the procedures allows to exceed the performance of both methods on their own. The alternating method follows the initial drop of the deep hierarchy model and for the rest it behaves as the flat model, providing the overall lowest error. This can also be seen in the progression of the spectra. The large scales behave like the deep model, rapidly jumping to the correct value, but also match the flat model behavior on the smallest scales.
Overall, the variational approximation in the standardized coordinates demonstrates its numerical superiority in the case of a weak likelihood, or more general, in directions the likelihood is relatively uninformative. The overall convergence is limited by the best informed directions, but the other directions converge fast nevertheless. The deep hierarchical parametrization shows the opposite behavior and a MAP estimate for well constrained parameters allows us to alternate between the parametrization, allowing for overall faster convergence.
Conclusion
We showed how the multivariate distributional transform can be used to transform a general Bayesian model over continuous quantities into a standardized coordinate system in which the prior becomes a standard Gaussian distribution. We discussed the behavior of popular approximations in these new coordinates. A maximum posterior approximation in this space will be oriented towards median prior configurations in the original space, instead of the maximum. For a strong likelihood, the resulting distributions will be transformed versions of each other. A variational Gaussian approximation in the standardized coordinates will be excellent in cases of a weak likelihood, as the shape of the posterior will only be slightly modified compared to the prior, and therefore a Gaussian approximation can capture the true posterior well. This is also the case if the likelihood, containing the standardizing transformation, is still close to a Gaussian.
The simple structure of the model in the standardized coordinates allowed us to investigate the numerical behavior of optimization schemes in terms of the conditioning of the curvature. Its smallest eigenvalue is larger or equal one, and the overall conditioning mainly depends on the largest eigenvalue of the likelihood. This eigenvalue is large if the data contains large amounts of information on certain parameter directions, and small otherwise. This makes the inference of the approximation parameters easier for small amounts, or uninformative data. This makes a Gaussian approximation in the standardized coordinates, as proposed for ADVI a fast and accurate inference procedure especially in cases of sparse data.
We explored numerically the inference of a Gaussian process with unknown correlation structure, which was described by a smooth log-Gaussian process, this time with known, smoothness enforcing kernel. The linearity of the standardizing transformation allowed us to perform the identical approximation in the original, as well as the standardized coordinates. The approximation was a variational Gaussian with full covariance for the Gaussian process, and a point estimate for the power spectrum. We found, as expected, that the flat hierarchical model was superior for less informed parameters, whereas the deep hierarchy outperformed for well-constrained parameters. Both scenarios occur also within the same inference problem, and the convergence speed of either approximation is bottle-necked by its sub-optimal directions. We solved this problem by alternating the optimization between the two coordinate systems, which harnessed the strength of both and lead to an overall faster convergence. We proposed that parameters well constrained by the data should be approximated by delta distributions, which allows their inference in either coordinates. This should improve the inference speed of large, hierarchical Bayesian models.
Acknowledgment
We acknowledge Philipp Arras, Philipp Frank, Maksim Greiner, Sebastian Hutschenreuter, Reimar Leike and Martin Reinecke for fruitful discussions and remarks on the manuscript.
Appendix A. A simple transformation example
We perform this transformation explicitly to a one dimensional example with a hierarchical Bayesian model with two parameters. We consider some likelihood P(d|α) that depends on a parameter α. The prior distribution on this is a Gaussian with the standard deviation σ and a known mean µ as hyper-parameters. The hyper-prior on this will be an exponential distribution depending on a known constant λ. A similar example is discussed in Betancourt and Girolami (2015).
P(α|σ) = G(α − µ, σ 2 ) α, µ ∈ R (56) P(σ) = λe −λσ σ, λ ∈ R +(57)
To calculate the transformations onto white priors we require the CDF of a white Gaussian, which is stated in Eq. 26, as well as the inverse CDF's of the prior distributions:
F −1 P(α|σ) (u) = µ + √ 2σ erf −1 (2u − 1) (58) F −1 P(σ) = − 1 λ ln(1 − u)(59)
We can then express our encoding of the prior structure in the likelihood as α = C P(α|σ) (ξ 1 ) = F −1 P(α|σ) • F G(ξ,1) (ξ 1 )
= µ + √ 2σ erf −1 2 1 2 + 1 2 erf ξ 1 √ 2 − 1 (61) = µ + σξ 1 .
This is just a simple re-scaling and shifting of the white Gaussian ξ parameter within the likelihood. This is exactly the example given for the reparametrization trick (Kingma and Welling, 2013) and the proposed transformation for hierarchical HMC (Betancourt and Girolami, 2015). Analogously we perform the transformation for the second parameter σ:
σ = C P(σ) (ξ 2 )(63)= − 1 λ ln 1 2 − 1 2 erf ξ 2 √ 2(64)
With this we can substitute σ in the likelihood to obtain full white prior distributions. We can relate back to the original parameter of the likelihood via
α = µ − 1 λ ln 1 2 − 1 2 erf ξ 2 √ 2 ξ 1(65)
and the posterior can be written as P(ξ 1 , ξ 2 |d) = P(d|ξ 1 , ξ 2 )P(ξ 1 )P(ξ 2 ) P(d)
instead of P(α, σ|d) = P(d|α)P(α|σ)P(σ) P(d) .
A graphical representation of the of the conditional dependence of the variables within the two models, the initial and the one re-parametrized to have white Gaussian random variables, is depicted in Fig. 3. With this transformation we flattened down the hierarchy of the original, deep Bayesian model. Both models contain the identical information. This is now encoded in the nonlinear structure of the parameters within the likelihood. One could continue adding higher levels of prior hierarchies to the problem, but flattening them as well is straight forward, as long as the CDF's are available. As it is pointed out in Betancourt and Girolami (2015), the parameters are now independent, conditioned on the data.
Appendix B. Transforming multivariate Gaussians
Multivariate Gaussian distributions and their generalizations to infinite dimensions, Gaussian processes (Krige, 1951;Rasmussen, 2004) are an important class of distributions, especially to express prior knowledge. They take the following form:
Here we do not have to perform the transformation explicitly, as we can find the white, standard Gaussian parametrization through a set of linear transformations. First, we have to express the correlation structure in terms of its eigenbasis.
S = F † SF(69)
Here the unitary transformation F is built from the normalized eigenvectors and the diagonal matrix S consists of the corresponding eigenvalues. The inverse in this basis can be calculated easily by inverting each individual eigenvalue and it reads
S −1 = F † S −1 F.(70)
With this we can rewrite the exponent of the Gaussian distribution as
1 2 s † S −1 S = 1 2 s † F † S −1 Fs ≡ 1 2 s † S −1 s.(71)
The eigenvalues of S encode the variance of the process along their corresponding eigendirections and it is the squared standard deviation. We can split up this diagonal covariance into two amplitude matrices by taking the matrix square root:
1 2 s † S −1 † S −1 s = 1 2 s † S −1 † 1 S −1 s(72)
Finally, we can introduce the new variable ξ = S −1 s and the exponential of this Gaussian distribution becomes
1 2 s † S −1 s = 1 2 ξ † 1ξ(73)
with the full transformation
s = F Sξ ≡ Aξ .(74)
This is therefore analogous to the one dimensional case as given in Eq. 62. We weight white, random excitations ξ in the eigenbasis of the correlation structure with the corresponding amplitudes contained within A in terms of the standard deviation and transform it back to the space we are interested in. The original correlation structure is expressed in terms of it as S = AA † .
Figure 1 :
1The setup and results in the three cases.
Figure 2 :
2The results for alternating between the two parametrizations during the inference. Left is the progression of the power spectra and right the RMS error to the Wiener filter.
Figure 3 :
3The graphical structure of the original model with a deep hierarchy and the flattened structure of the transformed model.
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|
[] |
[
"SOME GENERALIZATIONS OF SCHUR FUNCTORS",
"SOME GENERALIZATIONS OF SCHUR FUNCTORS"
] |
[
"Steven ",
"Sam And ",
"Andrew Snowden "
] |
[] |
[] |
The theory of Schur functors provides a powerful and elegant approach to the representation theory of GL n -at least to the so-called polynomial representationsespecially to questions about how the theory varies with n. We develop parallel theories that apply to other classical groups and to non-polynomial representations of GL n . These theories can also be viewed as linear analogs of the theory of FI-modules.
|
10.1090/proc/14205
|
[
"https://arxiv.org/pdf/1708.06410v1.pdf"
] | 119,713,422 |
1708.06410
|
2c7c247bfa03145dae62bb19869dabef2a66f9bc
|
SOME GENERALIZATIONS OF SCHUR FUNCTORS
21 Aug 2017
Steven
Sam And
Andrew Snowden
SOME GENERALIZATIONS OF SCHUR FUNCTORS
21 Aug 2017arXiv:1708.06410v1 [math.RT]
The theory of Schur functors provides a powerful and elegant approach to the representation theory of GL n -at least to the so-called polynomial representationsespecially to questions about how the theory varies with n. We develop parallel theories that apply to other classical groups and to non-polynomial representations of GL n . These theories can also be viewed as linear analogs of the theory of FI-modules.
Introduction
The theory of Schur functors is a major achievement of representation theory. One can use these functors to construct the irreducible representations of GL n -at least those whose highest weight is positive (the so-called "polynomial" representations). This point of view is important because it gives a clear picture of how the irreducible representation of GL n with highest weight λ varies with n, which is crucial to understand when studying problems in which n varies. Unfortunately, this theory only applies to GL n and not other classical groups, and, as mentioned, only to the polynomial representations of GL n . The purpose of this paper is to develop parallel theories that apply in other cases.
To explain our results, we focus on the orthogonal case. Let C = C O be the category whose objects are finite dimensional complex vector spaces equipped with non-degenerate symmetric bilinear forms and whose morphisms are linear isometries. An orthogonal Schur functor is a functor from C to the category of vector spaces that is algebraic, in an appropriate sense. We denote the category of such functors by A = A O . The main results of this paper elucidate the structure of this category, and its analogs in other cases.
Let S be the category of (classical) Schur functors. Given an object of S, one can evaluate on C ∞ and obtain a polynomial representation of GL ∞ . The resulting functor S → Rep pol (GL ∞ ) is an equivalence of categories. This is a useful perspective since representations of a group are more tangible than functors: it is not difficult to show that Rep pol (GL ∞ ) is semi-simple and that the simple objects are indexed by partitions; from this, one deduces the structure of S.
We take a similar approach to study A. Given an object of A, one can again evaluate on C ∞ (equipped with its standard symmetric bilinear form) and obtain a representation of O ∞ . (Technically, we cannot evaluate an object of A on C ∞ , but we can evaluate on C n and take the direct limit.) This representation is algebraic in the sense of [SS3]. We thus have a functor There are two important differences from the previous paragraph that complicate our task. First, the category Rep(O) is not semi-simple. And second, the functor T is not an equivalence. To see this, note that all maps in C are injections. Given a representation V of O n , we can therefore build a functor F : C → Vec such that F (C n ) = V and F (C k ) = 0 for k = n. This F is a non-zero object of A but T (F ) = 0.
The first issue was addressed by [SS3], which describes the algebraic representation theory of O ∞ quite clearly. (See [DPS, PSe, PSt] for similar work.) The real work in this paper goes in to addressing the second issue.
We now summarize our main results. We first show that T induces an equivalence between the Serre quotient A/A tors and Rep(O). Here A tors denotes the category of torsion objects in A; these are (essentially) those functors F for which F (C n ) = 0 for n ≫ 0, such as the functor F constructed above. We thus see that A is more or less built out of two pieces: Rep(O), which we understand from [SS3], and A tors , which is not difficult to understand directly.
We next prove the somewhat technical result that the section functor S (the right adjoint to T ) and its derived functors have amenable finiteness properties. Using this, we deduce that A is locally noetherian. This is the most important finiteness property of A.
We then go on to prove a few more results on the structure of A: we classify the indecomposable injectives, compute the Grothendieck group, define a theory of local cohomology, and introduce a Hilbert series and prove a rationality theorem for it.
All of the above work takes place in §2. In §3, we explain how the theory works in other situations (e.g., for symplectic groups). We also see in §3 that we can interpret orthogonal Schur functors as a sort of linear analog of the FI-modules of Church-Ellenberg-Farb [CEF]. In fact, many of the results in this paper are analogous to results on FI-modules obtained in [CEF] and [SS1].
Orthogonal Schur functors
2.1. First definitions and results. Let C = C O be the following category. An object is a pair (V, ω) consisting of a finite dimensional vector space V and a non-degenerate symmetric bilinear form
ω on V . A morphism (V, ω) → (V ′ , ω ′ ) is a linear map f : V → V ′ such that ω ′ (f (v 1 ), f (v 2 )) = ω(v 1 , v 2 ) for all v 1 , v 2 ∈ V
. This category appears in [SS3,(4.4.7)] as T 1 . A C-module is a functor C → Vec. We denote the category of C-modules by Mod C . For n ≥ 0, we let T n be the C-module given by T n (V ) = V ⊗n . For a partition λ, we let S λ be the C-module defined by the Schur functor S λ ; note that T n decomposes as a direct sum of S λ 's. A C-module is algebraic if it is a subquotient of a (possibly infinite) direct sum of T n 's. We denote the category of algebraic C-modules by A = A O . We call this the category of orthogonal Schur functors. Both A and Mod C are Grothendieck abelian categories.
Suppose that M is a C-module. Then M n = M(C n ) is a representation of the orthogonal group O n (C), regarded as a discrete group. (Here we let C n denote the object (C n , ω) of C, where ω is the standard form on C n for which the standard basis vectors e i are orthonormal.) The standard inclusion C n → C n+1 defines a morphism in C, and thus a linear map M n → M n+1 . This map is clearly O n (C)-equivariant. Furthermore, the image of the induced map M n → M n+m is contained in the space of H n+m,n -invariants, where H n+m,n ⊂ O n+m is the subgroup of g such that gi = i where i : C n → C n+m is the standard inclusion.
Let B be the category of sequences as above: an object of B is a pair ({M n } n≥0 , {i n } n≥0 ) where M n is a representation of O n (C) and i n : M n → M n+1 is an O n (C)-equivariant linear map such that the image of the induced map M n → M n+m is contained in the H n+m,ninvariants. Morphisms in B are the obvious things. The previous paragraph shows that M → {M(C n )} defines a functor Mod C → B.
Proposition 2.1. The functor Mod C → B is an equivalence.
Proof. Without loss of generality, we may replace C with its skeletal subcategory on the objects C n for n ≥ 0. Suppose we are given a sequence M = ({M n }, {i n }) that is an object of B. We define an associated C-module F . On objects, we put F (C n ) = M n . Given a morphism f : C n → C n+m , we can find g ∈ O n+m such that gf is the standard inclusion, and we set F (f ) = gi n+m−1 · · · i n+1 i n . Furthermore, given any other g ′ ∈ O n+m with the same property, g −1 g ′ ∈ H n+m,n , so F (f ) is independent of the particular choice of g. The construction M → F defines a functor B → Mod C that is quasi-inverse to Mod C → B.
There is a notion of finite generation for C-modules. Let M be a C-module, and suppose that we have elements {x i } i∈I where x i ∈ M(V i ). The submodule generated by the x i is the smallest C-submodule of M containing the x i . We say that M is finitely generated if it can be generated by a finite set of elements. We note that if M is finitely generated and algebraic then M(C n ) is a finite dimensional algebraic representation of O n for all n.
Let M be a C-module. An element
x ∈ M(C n ) is torsion if f * (x) = 0 for some morphism f : C n → C m in C.
The module M is torsion if all of its elements are. We let A tors be the category of torsion algebraic C-modules. We note that a finitely generated torsion object of A tors has finite length.
Algebraic representations of O
∞ . Let O ∞ (C) = n≥1 O n (C). For n ≤ ∞, let Rep(O n (C)
) be the category of all representations of the discrete group O n (C). Let T n ∞ = (C ∞ ) ⊗n , which is naturally a representation of O ∞ (C). We say that a representation of O ∞ (C) is algebraic if it appears as a subquotient of a direct sum of T n ∞ 's. We write Rep(O) for the category of algebraic representations of O ∞ , and Rep(O n ) for the category of algebraic representations of O n . The following theorem summarizes some of the main results from [SS3] (see [DPS, PSe, PSt] Let V be a representation of O ∞ (C). Let G n and H n be the subgroups of O ∞ consisting of matrices of the form
* 0 0 1 , 1 0 0 * ,
where the top left block is n×n and the bottom right block is (∞−n)×(∞−n). Put Γ n (V ) = V Hn . Since G n ∼ = O n (C) commutes with H n , we see that Γ n (V ) carries a representation of O n (C). We let Γ n be the restriction of Γ n to Rep(O). It follows from Theorem 2.2(b) and Theorem 2.3(b) below that Γ n takes values in Rep(O n ). We thus have a left-exact functor
Γ n : Rep(O) → Rep(O n ),
called the specialization functor. The following theorem summarizes some of its main properties:
Theorem 2.3. We have the following: (a) Γ n is a left-exact symmetric monoidal functor. [SS3,(4.4.4), (4.4.5)]
(b) Γ n (T r ∞ ) = (C n ) ⊗r and Γ n (S λ (C ∞ )) = S λ (C n ). (c) Γ n (L λ ∞ ) is the irreducible representation of O n with highest weight λ if n ≥ λ † 1 + λ † 2 , and 0 otherwise. (d) Suppose V ∈ Rep(O) has finite length. Then R i Γ n (V ) is finite dimensional for all i and n, and vanishes for i ≫ 0. Furthermore, for n ≫ 0 we have R i Γ n (V ) = 0 for all i > 0. (e) There is an explicit combinatorial rule to compute R i Γ n (L λ ∞ ). Proof. (a)
(b) For r = 1 of the first statement, this follows from the construction of Γ n in [SS3,(4.4.4)], the rest of the statement follows since Γ n is a symmetric monoidal functor. (c) This is a special case of (e). (d) For V simple this follows from (e). The general case follows from dévissage. (e) See [SS3,(4.4.6)].
We will also require the following property of derived specialization:
Proposition 2.4. The functor R i Γ n commutes with filtered colimits.
Proof. We first treat the case n = 0, where Γ n is simply the functor that assigns to V the invariant subspace V O∞ . We thus see that Γ 0
(V ) = Hom Rep(O) (C, V ), where C is the trivial representation, and so R i Γ 0 (V ) = Ext i Rep(O) (C, V ).
In [SS3,(4.3
.1)], we show that Rep(O) is equivalent to the category Mod 0
A of modules over the twisted commutative algebra A = Sym(Sym 2 (C ∞ )) that are supported at 0 (i.e., locally annihilated by a power of the maximal ideal A + ). Under this equivalence, C corresponds to the module C = A/A + . It thus suffices to show that Ext i Mod 0 A (C, −) commutes with filtered colimits. We claim that injective objects of Mod 0 A remain injective in Mod A . Suppose that I is a finite length injective object of Mod 0 A , and consider an injection I → M in Mod A . Every A-module is canonically graded. Let M ≥n be the submodule i≥n M i of M, and let M ≤n be the quotient M/M ≥n . The formation of M ≤n is clearly exact in M. Moreover, M ≤n is supported at 0, being annihilated by the nth power of A + . Since I has finite length, we have I ≤n = I for n ≫ 0. Thus, for appropriate n, we have an injection I = I ≤n → M ≤n in Mod 0 A . Since I is injective in Mod 0 A , this injection splits; composing with the canonical surjection M → M ≤n splits the original injection. Thus I is injective in Mod A . For the general case, simply observe that all injectives of Mod 0 A are direct sums of finite length injectives (this follows from Theorem 2.2(c), for instance), and arbitrary direct sums of injective objects of Mod A are injective, since this category is locally noetherian [NSS,Theorem 1.1].
By the previous paragraph, we have Ext i
Mod 0 A (−, −) = Ext i Mod A (−, −)
, and so it suffices to show that Ext i Mod A (C, −) commutes with filtered colimits. Let P • → C be the Koszul resolution of C as an A-module; thus P i = A ⊗ i (Sym 2 (C ∞ )). Note that each P i is finitely generated as an A-module. We thus see that Ext • Mod A (C, M) is computed by the complex Hom A (P • , M). Since each P • is finitely presented (being finitely generated and projective), formation of this complex commutes with filtered colimits. The result now follows from the fact that filtered colimits are exact.
We now treat the case of arbitrary n. Ignoring the O n -structure, the functor Γ n factors as
Rep(O) Rn / / Rep(O) Γ 0 / / Vec, where R n is the functor R n (V ) = V | Hn (note that H n ∼ = O ∞ ).
The functor R n obviously is exact and commutes with colimits. Furthermore, it takes injective objects to injective objects, as can be seen from the explicit description of injectives in Theorem 2.2(c). We thus see that R i Γ n = R i Γ 0 • R n , and so the result follows.
The relation between A and Rep(O). We define functors
T : Mod C → Rep(O ∞ (C)), S : Rep(O ∞ (C)) → Mod C by T (M) = lim − → n→∞ M(C n ), S(V ) = { Γ n (V )} n≥0 .
In the definition of S, the transition maps are the obvious inclusions Γ
n (V ) ⊂ Γ n+1 (V ). It is clear that the image of Γ n (V ) in Γ n+m (V ) is contained in the H n+m,n -invariants.
We note that T is exact, as direct limits are exact, and S is left-exact, as each Γ n is. Moreover, S commutes with arbitrary filtered colimits. We let T be the restriction of T to A and S the restriction of S to Rep(O).
Theorem 2.5. We have the following: (a) T (T n ) = T n ∞ and S(T n ∞ ) = T n , and similarly for S λ .
(b) T takes values in Rep(O) and S takes values in A.
(c) (T, S) is an adjoint pair and the unit id → ST is an isomorphism.
(d) We have (R i S)(V ) = {R i Γ n (V )} n≥0 with obvious transition maps. (e) T induces an equivalence of categories A/A tors → Rep(O). Proof. (a) The statement for T is clear. It is clear that T n (C k ) ⊂ (T n ∞ )
H k for all k, and the these inclusions are compatible with transition maps. By Theorem 2.3(b), these inclusions are equalities. Thus S(T ∞ n ) = T n . (b) Since T is exact and takes T n to T n ∞ , it takes subquotients of sums of T n 's to subquotients of sums of T n ∞ 's. Let V be an algebraic representation of Rep(O). By Theorem 2.2(b), V embeds into a direct sum of T n ∞ 's. Since S is left-exact and commutes with arbitrary direct sums, we see that V embeds into a direct sum of T n 's and is therefore algebraic.
( Conversely, suppose that g : M → S(V ) is a map of C-modules. Applying T , we obtain a map T (M) → T (S(V )). Since V embeds into a sum of T n 's, every element of V is invariant under H n for n sufficiently large, and so T (S(V )) = V . We thus have a map T (M) → V of GL ∞ representations. One easily checks that this construction is inverse to the one in the previous paragraph.
c) Let M be a C-module and V a representation of O ∞ (C). Suppose that f : T (M) → V is a GL ∞ -equivariant map. There is a canonical O n (C)-equivariant map M n = M(C n ) → T(
(
d) Let F i (V ) = {R i Γ n (V )} n≥0
. This is a well-defined C-module for the same reason S is well-defined. Both F • and R • S are cohomological δ-functors. The latter is universal, by definition. However, the former is also universal, since F i (V ) = 0 for i > 0 if V is injective. Thus the two are isomorphic.
(e) Since T is exact and kills A tors , it induces an functor T : A/A tors → Rep(O), by the universal property of Serre quotient categories. Let S : Rep(O) → A/A tors be the composition of S with the localization functor A → A/A tors . From (c), we have that ST = id. We now show that the canonical map id → ST is an isomorphism. For this, it is enough to show that for every M ∈ A the kernel and cokernel of the canonical map M → S(T (M)) is torsion. This is clear for the kernel; we now prove it for the cokernel.
Let X be the class of objects M ∈ A for which the cokernel of M → S(T (M)) is torsion. Then X contains that objects T n by (a), and therefore all direct sums of such objects since S and T commute with direct sums. It is therefore enough to show that any subquotient of an object in X again belongs to X. Thus suppose we have an exact sequence
0 → M 1 → M 2 → M 3 → 0 in A with M 2 ∈ X. Applying ST , we obtain a commutative diagram 0 / / M 1 / / f 1 M 2 / / f 2 M 3 / / f 3 0 0 / / ST (M 1 ) / / ST (M 2 ) / / ST (M 3 ) / / R 1 ST (M 1 )
Let K i = ker(f i ) and C i = coker(f i ). The snake lemma gives an exact sequence
0 → K 1 → K 2 → K 3 → C 1 → C 2 → C 3 → R 1 ST (M 1 )
We have already explained that the K i 's are torsion. Furthermore, C 2 is torsion by assumption. Thus C 1 is torsion, and so M 1 ∈ X. To show that C 3 is torsion (and thus obtain M 3 ∈ X), it suffices to show that R 1 ST (M 1 ) is torsion. We show, generally, that R 1 S(V ) is torsion for any V ∈ Rep(O). First suppose that V has finite length. Then R 1 S(V ) n = 0 for n ≫ 0 by Theorem 2.3(d), and so R 1 S(V ) is torsion. Every object of Rep(O) is the union of its finite length subobjects. Since R 1 S commutes with filtered colimits by Proposition 2.4 and any colimit of torsion modules is torsion, the result follows.
Corollary 2.6. The objects T n and S λ are injective in A.
Proof. Since T is exact, its right adjoint S carries injectives to injectives. Since T n ∞ is injective in Rep(O) (Theorem 2.2(b)) and T n = S(T n ∞ ), we see that T n is injective. Since S λ is a summand of T n , it too is injective.
Corollary 2.7. The category A has Krull-Gabriel dimension 1.
Proof. The subcategory of A consisting of objects of Krull-Gabriel dimension 0 is A tors , and A/A tors = Rep(O) has Krull-Gabriel dimension 0 by Theorem 2.2(a). Thus A has Krull-Gabriel dimension 1 by the recursive characterization of this dimension.
Remark 2.8. It follows from Theorem 2.5(e) that S induces an equivalence of categories between Rep(O) and its image in A, which consists of the saturated objects in A. This statement was also made in [SS3,§1.2.5]. While this image is an abelian category, it is not an abelian subcategory of A.
2.4. Finiteness properties of the section functor. Our goal in this section is to prove the following finiteness theorem:
Theorem 2.9. Let V ∈ Rep(O) have finite length. (a) S(V ) is a finitely generated object of A. (b) R i S(V )
is a finitely generated torsion object of A, for all i ≥ 1.
We begin with a lemma. Recall that L λ ∞ is the simple object of Rep(O) indexed by λ. We let L λ be the object of A given by S(L λ ∞ ). Lemma 2.10. Every subobject of L λ is finitely generated.
Proof. Let M = 0 be a subobject of L λ , let n be minimal so that M(C n ) = 0, and let v ∈ M(C n ) be any non-zero element. Consider a morphism f : C n → C m in C. Then f * (v) is non-zero since L λ is torsion-free, being in the image of S. Since By Lemma 2.10, the image of the map S(V ) → S(L) is finitely generated. By the inductive hypothesis, S(W ) is finitely generated. Thus S(V ) is finitely generated.
(b) This follows immediately from Theorem 2.3(d).
2.5. Noetherianity. We now prove a fundamental finiteness result about A:
Theorem 2.11. The category A is locally noetherian: any subobject of a finitely generated object is again finitely generated.
Lemma 2.12. Let N ∈ A be finitely generated and let N be a subobject such that N /N has finite length. Then N is also finitely generated.
Proof. For M ∈ A let M ≥n be the subobject of M defined by (M ≥n ) k = M k if k ≥ n and (M ≥n ) k = 0 for k < n. If M is finitely generated then so is M ≥n , for any n: indeed, the generators of M of degree > n together with a basis for the space M n generate M ≥n . Now, let n be sufficiently large so that ( N/N) k = 0 for k ≥ n. Then N ≥n = N ≥n , and is thus finitely generated. Consider the exact sequence
0 → N ≥n → N → N/N ≥n → 0.
The left term is finitely generated and the right term has finite length. We thus see that N is finitely generated.
Proof of Theorem 2.11. We first show that M = T n is noetherian. Thus let N be a subobject. Since T (N) is a subobject of the finite length object T (M), it is finite length, and so S(T (N)) is finitely generated by Theorem 2.9. The map N → S(T (N)) is injective, since N is torsion-free, and has torsion cokernel by Theorem 2.5. Since S(T (N)) is finitely generated, the cokernel has finite length. Thus N is finitely generated by the lemma, and so M is noetherian. Suppose now that M is an arbitrary finitely generated object of A. By definition, we can express M as a quotient of a subobject K of a direct sum of i∈I F i , where each F i has the form T n . For a finite subset J of I, let F J = j∈J F j and K J = K ∩ F J . Since M is finitely generated and K is the union of the K J 's, it follows that there is a J such that K J → M is surjective. Since each F j is noetherian by the previous paragraph, so is the finite sum F J . Since noetherianity descends to subquotients, we see that M is noetherian.
Remark 2.13. Patzt [Pa] has proved a similar, but more general, result in the general linear and symplectic cases; see §3. The symplectic analog likely applies in the present case with minor modifications.
2.6. Saturation and local cohomology. We now apply the theory of [SS5,§4]. Let Σ : A → A be the composition ST , the saturation functor. Let Γ : A → A be the functor that associates to M the torsion submodule M tors . This is a left-exact functor, and we refer to its derived functor RΓ as local cohomology. To apply the theory of [SS5,§4], we must verify the technical hypothesis (Inj):
Proposition 2.14. Injective objects of A tors remain injective in A. Now let I be an arbitrary injective object of A tors . Since A tors is locally noetherian, we have a decomposition I = α I α where each I α is indecomposable. Since finitely generated objects of A tors have finite length, I α is the injective envelope of some simple, and thus is of finite length itself [SS3,(2.1.5)]. Thus each I α is injective in A by the previous paragraph. Since A is locally noetherian (Theorem 2.11), any sum of injectives is injective, and so I is injective.
Proof. Let I be a finite length injective object of
Proposition 2.15. The indecomposable injectives of A are the Schur functors S λ and the injective envelopes of simple objects (which are finite length).
Proof. By [SS5,Proposition 4.5], the indecomposable injectives of A are exactly those of A tors and those of the form S(I) with I ∈ Rep(O) indecomposable injective. The indecomposable injectives of Rep(O) are the S λ (C ∞ ) (Theorem 2.2(c)) and S(S λ (C ∞ )) = S λ (Theorem 2.5(a)). Proof. The existence of the triangle is [SS5,Proposition 4.6]. Now suppose M ∈ D b fg (A). We have already shown finiteness of RΣ(M) (Theorem 2.9), while finiteness of RΓ(M) follows from this and the triangle. For the more precise statements, we argue as follows. Since T (M) is a finite length complex whose cohomology groups are finite length objects, Theorem 2.2(c,d) ensures it is quasi-isomorphic to a complex N • whose terms are sums of representations of the form S λ (C ∞ ). We thus have RΣ(M) = S(N • ). The claim now follows from the identity S(S λ (C ∞ )) = S λ . The claim about RΓ(M) is immediate.
Corollary 2.17. A finitely generated object of A has finite injective dimension.
2.7. The Grothendieck group. We now discuss the Grothendieck group K(A) of A.
Theorem 2.18. We have a canonical isomorphism
K(A) = K(Rep(O)) ⊕ n≥0 K(Rep(O n )).
Proof. From [SS5,Proposition 4.17], we obtain a canonical isomorphism K(A) = K(A/A tors )⊕ K(A tors ). We have K(A/A tors ) = K(Rep(O)) by Theorem 2.5. Every finitely generated object of A tors admits a finite filtration where the graded pieces are supported in degree. The category of objects of A supported in degree n is equivalent to Rep(O n ), and so the result follows. Remark 2.20. The Grothendieck group of SO n is a polynomial ring in ⌊n/2⌋ variables, generated by exterior powers of the standard representation. The Grothendieck group of O n is slightly more complicated, since O n is not connected.
We define the Hilbert function of M by HF M (n) = dim M(C n ). We define the Hilbert series of M by
H M (t) = n≥0 dim C M(C n )t n .
Both constructions are additive in short exact sequences, and thus factor through the Grothendieck group K(A).
Theorem 2.21. Let M be a finitely generated object in A. Then there exists a polynomial p ∈ Q[t] such that
p(n) = dim M n − i≥0 (−1) i dim R i Γ(M) n
holds for all integers n ≥ 0. In particular, HF M (n) = p(n) for n ≫ 0, and H M (t) is a rational function whose denominator is a power of 1 − t.
Proof. It suffices to verify the theorem for classes [M] that span K(A). First suppose that
M is a torsion module. Then Γ(M) = M (obvious) and R i Γ(M) = 0 for i > 0 (see [SS5,Proposition 4.2]). It follows that the identity holds with p = 0. Now suppose that M = S λ ; note that the classes [S λ ] and the torsion classes span K(A). Then Σ(M) = M (Theorem 2.5(a)) and R i Σ(M) = 0 for i > 0 since M is injective (Corollary 2.6), and so RΓ(M) = 0 by Proposition 2.16. We thus take p(n) = dim S λ (C n ). This is well-known to be a polynomial function of n.
Remark 2.22. In words, the above theorem shows that the Hilbert function coincides with a polynomial (called the Hilbert polynomial) for large values of n, and that the discrepancy between the Hilbert function and Hilbert polynomial at small values of n is calculated by local cohomology.
2.8. Pointwise algebraic C-modules. A C-module M is pointwise algebraic if M n is an algebraic representation of O n for each n. Every algebraic representation is pointwise algebraic: this follows immediately from the definitions. We now show that the converse is not true.
Let M (k) n = Sym 2k (C n ). This is an algebraic representation of O n . The obvious transition maps M n 's, taken in the category of algebraic O n representations. Explicitly, Sym 2k (C n ) decomposes as a sum of k + 1 distinct irreducible representations of O n , and the inverse limit is simply the direct sum of all of these irreducibles. One easily sees that M = (M n ) n≥0 is a C-module, and it is pointwise algebraic by definition. We now show that it is not algebraic.
Each M (k) is generated in degrees ≤ 1. Indeed, considering M n for all k, and is therefore equal to M n by the explicit description given in the previous paragraph. Since M 1 is one-dimensional, we see that M is finitely generated as a C-module. Since M n is not finite dimensional for n > 1, it therefore cannot be algebraic.
Similar examples can be constructed for the other versions of C defined in §3. In particular, this shows that the expectations outlined in [SS3,(3.4.10)] are not correct.
Remark 2.23. Let A be the tca Sym(Sym 2 (C ∞ )), so that Rep(O) = Mod 0 A (see the proof of Proposition 2.4). The section functor S can be regarded as a functor from Mod 0 A to the category of algebraic C-modules. Given an arbitrary A-module N, the quotient N ≤n (as defined in Proposition 2.4) is supported at 0, and so S(N ≤n ) is an algebraic C-module. Define S(N) to be the inverse limit of these modules, taken in the category of pointwise algebraic C-modules (this category is Grothendieck, and thus complete). This construction extends S to a continuous functor from Mod A to the category of pointwise algebraic C-modules. The module M constructed above is simply S(A/a 1 ), where a 1 is the first determinantal ideal. It would be interesting to more closely study the connection between Mod A and the category of pointwise algebraic C-modules. For instance, is the former the Serre quotient of the latter by a category of torsion modules, with S being the section functor? 3. Other cases 3.1. Symplectic groups. The theory in §2 applies almost without modification to the symplectic case. We define C Sp to be the category of finite dimensional vector spaces equipped with non-degenerate skew-symmetric forms. We let A Sp be the category of algebraic C Spmodules, using an analogous definition of algebraic. The related category of algebraic representations of the infinite symplectic group is studied in [SS3,§4]. The obvious analogs of Theorems 2.5, 2.9, 2.11, and their corollaries hold for A Sp , with exactly analogous proofs. The material in § §2.6, 2.7 carries over as well. We mention one slight difference: the description of the Grothendieck group is now
K(A Sp ) = K(Rep(Sp)) ⊕ n≥0 K(Rep(Sp 2n )).
The group K(Rep(Sp 2n )) is a polynomial ring in n variables on the classes of exterior powers of the standard representation.
The category C Sp appears in [SS3,§4.4], as T ′ 1 . It also appeared in [PSa] and [Pa] under the notation SI(C). See [Pa] for some applications. We note that [Pa, Theorem D] proves the category of pointwise algebraic C Sp -modules is locally noetherian, which encompasses our Theorem 2.11.
3.2. General linear groups. The theory in §2 also carries over to the case of algebraic representations of the general linear group, with very minor modifications. Let C GL be the following category. Objects are triples (V, W, ω) where V and W are finite dimensional and ω : V ⊗ W → C is a perfect pairing. A morphism (V, W, ω) → (V ′ , W ′ , ω ′ ) is a pair of linear maps V → V ′ and W → W ′ that respects the pairing. We let T n,m be the C GL -module taking (V, W ) to V ⊗n ⊗ W ⊗m . We say that a C GL -module is algebraic if it is a subquotient of a direct sum of T n,m 's, and let A GL be the category of algebraic representations. We note that the analog of the C O -module S λ in the present setting is the C GL -module given by
(V, W ) → S λ (V ) ⊗ S µ (W ).
The category C GL has an alternate description that is worth pointing out. Let C ′ GL be the following category. Objects are finite dimensional complex vector spaces. A morphism V → W is a pair (i, p) where i : V → W and p : W → V are linear maps such that pi = id V . The functor C ′ GL → C GL taking V to (V, V * , ω), with ω the canonical pairing, is an equivalence. The C GL -module T n,m corresponds to the C ′ GL -module given by V → V ⊗n ⊗ (V * ) ⊗m . Thus C ′ GL -modules are like Schur functors but allow one to use duals. We note that the related theory of algebraic representations of the infinite general linear group is studied in [SS3,§3].
Theorem 2.5 applies to A GL with obvious changes (e.g., replace T n with T n,m ). Theorems 2.9 and 2.11 apply without change. The material in §2.6. Theorem 2.18 applies with the obvious modifications: we obtain K(A GL ) = K(Rep(GL)) ⊕ n≥0 K(Rep(GL n )).
The Grothendieck group of Rep(GL) can naturally be identified with Λ ⊗ Λ, where Λ is the ring of symmetric functions, see [Ko,§2]. Furthermore, the Grothendieck group of GL n is isomorphic to Z[z 1 , . . . , z n−1 , z n , z −1 n ] a polynomial ring in n variables with the last variable inverted, where z i is the class of the exterior power representation i C n . Finally, the material on Hilbert functions holds in the present setting: the Hilbert function is defined to be the function n → dim M(C n , C n ).
The category C GL appears in [SS3,(3.4.6)] as T ′ 1 . The category C ′ GL appears there as well, without name. The category C ′ GL also appears in [PSa] and [Pa] as VIC (or VIC(C, C × )). We note that [Pa, Theorem C] proves that the category of pointwise algebraic C GL -modules is locally noetherian, which encompasses our Theorem 2.11.
3.3. Symmetric groups. There is also a symmetric group analog of the theory in this paper. Let C S be the category whose objects are tuples (A, m, ∆, η) where A is a finite dimensional vector space, m : A ⊗ A → A is an associative commutative multiplication on A (unit not required), ∆ : A → A⊗A is a coassociative cocommutative coalgebra structure on A with counit η such that m∆ = id and ∆m = (m ⊗ 1)(1 ⊗ ∆). The motivation for considering this category comes from considering the partition algebra (or really, a categorical form of it) as in [SS3,§6]. We show in [SS3,(6.4.7)] that C S is in fact equivalent to the category FI of finite sets and injections. Let A S be the category of C S -modules (or equivalently, the category of FI-modules). There is no algebraicity requirement now. We let T n be the free C S -module generated in degree n. This has an action of the symmetric group S n , and its λ-isotypic piece is the analog of S λ .
Let Rep(S) be the category of algebraic representations of the infinite symmetric group, as defined in [SS3,§6]. The analogs of the material in §2.2, including the theory of specialization, is developed in [SS3,§6] (which relies on some results from [SS1]). The obvious analogs of Theorems 2.5, 2.9, 2.11, and their corollaries hold for A S . The material in § §2.6, 2.7 carries over as well. We note that the result on the Grothendieck group can now be stated as K(A S ) = Λ ⊕ Λ as K(Rep(S)) and n≥0 K(Rep(S n )) are both isomorphic to Λ.
The category of FI-modules was introduced in [CEF], which proved some of the above results. It was studied in greater detail (and from a different perspective) in [SS1], where all of the above results were proved. The follow-up paper [SS5] generalizes many of these results to FI d -modules.
T
: A → Rep(O). Date: August 20, 2017. 2010 Mathematics Subject Classification. 15A69, 20G05. SS was supported by NSF grant DMS-1500069. AS was supported by NSF grants DMS-1303082 and DMS-1453893 and a Sloan Fellowship.
M), and so composing with f induces a map f n : M n → V that is O n (C)-equivariant. Since M n maps into the H n invariants of T (M), we see that f n maps into S(V ) n = V Hn . It is clear that the maps f n are compatible with transition maps, and so yield a map of C-modules M → S(V ).
L λ (C m ) is an irreducible representation of O m by Theorem 2.3(c), it follows that f * (v) generates every element of L λ (C m ) as a O m -representation, and thus every element of M(C m ) = L λ (C m ). The result follows. Proof of Theorem 2.9. (a) We proceed by induction on length. Let V = 0 be given, and consider a short exact sequence 0 → W → V → L → 0 with L simple. We obtain an exact sequence 0 → S(W ) → S(V ) → S(L).
A tors . Suppose M ⊂ N are objects of A and we have a map f : M → I. Let k be maximal so that I k is non-zero. Let M ′ ⊂ M be the subobject of M defined by M ′ n = M n for n > k and M ′ n = 0 for n ≤ k, and define N ′ ⊂ N similarly. Clearly, f (M ′ ) = 0, and so f factors through M/M ′ . The map M/M ′ → N/N ′ is injective, and both objects are torsion, so f extends to a map g : N/N ′ → I. The composition N → N/N ′ → I extends the original map f to N. This shows that I is injective in A.
Proposition 2 . 16 .
216For M ∈ D + (A) we have a canonical exact triangle RΓ(M) → M → RΣ(M) → . Furthermore, if M ∈ D b fg (A), then so are RΓ(M) and RΣ(M). More precisely, RΣ(M) is represented by a finite length complex whose terms are finite sums of Schur functors, while RΓ(M) is represented by a finite length complex whose terms are finite length injective objects.
Remark 2 . 19 .
219The isomorphism in Theorem 2.18 can be described explicitly as follows. The class of [V ] ∈ K(Rep(O)) corresponds to [RS(V )] ∈ K(A). In particular, when V = S λ (C ∞ ) we have [RS(V )] = [S λ ], and so the the classes [S λ ] span summand of K(A) corresponding to K(Rep(O)). The class of [V ] ∈ K(Rep(O n )) corresponds to the class [M] ∈ K(A), where M is the C-module with M n = V and M k = 0 for k = n.
give M (k) the structure of a C-module, and as such it is clearly algebraic. There is a natural surjection M the O n -coinvariant in Sym 2 (C n ). Let M n be the inverse limit of the M (k)
the space of degree 2k polynomials in variables x 1 , . . . , x n , the image of the transition map M representation of O n . It follows that M is generated in degrees ≤ 1; indeed, the O n -subrepresentation of M n generated by M 1 surjects on M (k)
for similar results):Theorem 2.2. The following statements hold in Rep(O):
(a) The objects T n
∞ have finite length.
(b) The objects T n
∞ are injective, and every object of Rep(O) embeds into a direct sum of
T n
∞ 's.
(c) The indecomposable injectives are exactly the Schur functors S λ (C ∞ ).
(d) Let L λ
∞ be the socle of S λ (C ∞ ). Then the L λ
∞ 's are exactly the simple objects.
(e) Every finite length object of Rep(O) has finite injective dimension.
Proof. (a) [SS3, Proposition 4.1.5]
(b) Follows from (c) since T n
∞ is a finite direct sum of Schur functors.
(c) [SS3, Proposition 4.2.9]
(d) [SS3, Proposition 4.1.4]
(e) Combine [SS3, (2.1.5)] and [SS3, Theorem 4.2.6].
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[
"THE STRUCTURES OF HIGHER RANK LATTICE ACTIONS ON DENDRITES",
"THE STRUCTURES OF HIGHER RANK LATTICE ACTIONS ON DENDRITES"
] |
[
"Enhui Shi ",
"Hui Xu "
] |
[] |
[] |
Let Γ be a higher rank lattice acting on a nondegenerate dendrite X with no infinite order points. We show that there exists a nondegenerate subdendrite Y which is Γ-invariant and satisfies the following items:(1) There is an inverse system of finite actions(2) The first point map r : X → Y is a factor map from (X, Γ) to (Y, Γ|Y ); if x ∈ X \ Y , then r(x) is an end point of Y with infinite orbit; for each y ∈ Y , r −1 (y) is contractible, that is there is a sequence g i ∈ Γ with diam(g i r −1 (y)) → 0.2010 Mathematics Subject Classification. 54H20, 37B20.
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[
"https://arxiv.org/pdf/2207.00979v1.pdf"
] | 250,265,071 |
2207.00979
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fd87ff6d7e8be6e733e92da4ffca758b10d9dec3
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THE STRUCTURES OF HIGHER RANK LATTICE ACTIONS ON DENDRITES
Jul 2022
Enhui Shi
Hui Xu
THE STRUCTURES OF HIGHER RANK LATTICE ACTIONS ON DENDRITES
3Jul 2022
Let Γ be a higher rank lattice acting on a nondegenerate dendrite X with no infinite order points. We show that there exists a nondegenerate subdendrite Y which is Γ-invariant and satisfies the following items:(1) There is an inverse system of finite actions(2) The first point map r : X → Y is a factor map from (X, Γ) to (Y, Γ|Y ); if x ∈ X \ Y , then r(x) is an end point of Y with infinite orbit; for each y ∈ Y , r −1 (y) is contractible, that is there is a sequence g i ∈ Γ with diam(g i r −1 (y)) → 0.2010 Mathematics Subject Classification. 54H20, 37B20.
INTRODUCTION
It was conjectured that every continuous action on the circle by a higher rank lattice must factor through a finite group action (the so called 1-dimensional Zimmer's rigidity conjecture). Burger-Monod [4] and Ghys [10] proved independently the existence of finite orbits for higher rank lattice actions on the circle, which translates the conjecture into the equivalent form: no higher rank lattice admits a total ordering which is invariant by left translations. The latter was answered affirmatively by Witte-Morris for arithmetic groups of higher Q-rank [21] and by Deroin-Hurtado for general higher rank lattices [5]. Recently there has been a great progress on the Zimmer program for smooth higher rank lattice actions on manifolds with dimensions ≥ 2. We do not plan to list all the related results here and just suggest the readers to consult [3,9] for the surveys.
There has been intensively studied around group actions on dendrites very recently. One motivation is that dendrites can appeare as the limit sets of some Klein groups, the structures of which are closely related to the geometric properties of 3-dimensional hyperbolic manifolds (see e.g. [2,16]). Also, the compactifications of the Cayley graphs of free groups are dendrites, which are important for understanding the algebraic properties of free groups. Group actions on the circle have been systematically investigated during the past few decades [11,17]. However, group actions on general curves lack of the same depth of understanding. Dendrites and the circle lie on two opposite ends of Peano curves in topologies. So, studying group actions on dendrites is the starting point for better understanding group actions on curves or continua with higher dimensions. Some people studied the Ghys-Margulis' alternative for dendrite homeomorphism groups ( [18,15,8]). One may consult [1,12,20] for the discussions around the structures of minimal sets for group actions on dendrites. The algebraic structures of dendrite homeomorphism groups were investigated in [7] Duchesne-Monod proved the existence of finite orbits for higher rank lattice actions on dendrites ( [8]). This motivates us to consider further the structures of higher rank lattice actions on dendrites. Though there are examples indicating that the exact analogy to the rigidity results by Witte-Morris and Deroin-Hurtado mentioned above do not hold anymore for higher rank lattice actions on dendrites, the authors showed in [19] that such actions are very restrictive. The aim of the paper is to give a more detailed description of the structures of higher rank lattice actions on dendrites. Explicitly, we obtained the following theorem. Theorem 1.1. Let Γ be a higher rank lattice acting on a nondegenerate dendrite X with no infinite order points. Then there exists a nondegenerate subdendrite Y which is Γinvariant and satisfies the following items:
(1) There is an inverse system of finite actions {(Y i , Γ) : i = 1, 2, 3, · · ·} with monotone bonding maps φ i : Y i+1 → Y i and with each Y i being a dendrite, such that (Y, Γ|Y ) is topologically conjugate to the inverse limit (lim
←− (Y i , Γ), Γ). (2) The first point map r : X → Y is a factor map from (X , Γ) to (Y, Γ|Y ); if x ∈ X \ Y , then r(x) is an end point of Y with infinite orbit; for each y ∈ Y , r −1 (y) is contractible, that is there is a sequence g i ∈ Γ with diam(g i r −1 (y)) → 0..
The proof of Theorem 1.1 relies on the non-left-orderability of higher rank lattices by Deroin-Hurtado, the elementarity of higher rank lattice actions on dendrites by Duchesne-Monod, and a connection between semilinear-left-orderability of a group and its actions on dendrites observed by the authors.
We will not introduce the related notions and properties around group actions and dendrites. One may consult [19] for the details.
GENERAL FACTS ABOUT SEMILINEAR PREORDERS
Recall that a binary relation on a set X is a preorder if it satisfies that (O1) x x for any x ∈ X ; (O2) if x y and y z then x z for any x, y, z ∈ X .
If, in addition, satisfies that (O3) if x y and y x then x = y for any x, y ∈ X , then is a partial order on X .
A preorder (resp. partial order) on X is called a total preorder (resp. total order) on X if (O4) any x, y ∈ X are comparable; that is either x y or y x.
Let G be a group with e being the unit. A preorder/ partial order/ total order on G is said to be left-invariant if (O5) for any x, y ∈ X and g ∈ G, gx gy whenever x y.
Finally, we say is a semilinear left preorder (resp. semilinear left partial order) if it satisfies (O1), (O2), (O5) (resp. (O1),(O2),(O3),(O5)) and (O6) for any x ∈ G, any two elements of {y ∈ G : y x} are comparable; (O7) for any x, y ∈ G, there is some z ∈ G with z x and z y.
The following characterization of semilinear left preorder is similar to [14, Theorem 1.1].
Proposition 2.1. Let be a semilinear left preorder on a group G. Then the positive cone P := {g ∈ G : e g} of has the following properties: To prove (P3), let g ∈ G. Take u ∈ G with u e and u g by (O7). Then g = u(u −1 g) and e u −1 g. So, G ⊂ P −1 · P. The inclusion P −1 · P ⊂ G is clear.
(P1) P is a semigroup; (P2) P ∩ P −1 is a subgroup of G; (P3) G = P −1 · P; (P4) P · P −1 ⊆ P ∪ P −1 . If is further a partial order, then (P2') P ∩ P −1 = {e}; is further a total preorder, then (P5) G = P ∪ P −1 . Conversely, a subset P of G satisfying (P1)-(P4) (resp. (P1)(P2')(P3)(P4))
To prove (P4), let x, y ∈ P. Then x −1 e and y −1 e. From (O6), either x −1 y −1 or y −1 x −1 , which implies xy −1 ∈ P ∪ P −1 . For (O6), let x, y, z ∈ G be such that x z and y z. Then x −1 z ∈ P and z −1 y ∈ P −1 . By (P4), x −1 y = x −1 z · z −1 y ∈ P ∪ P −1 . This means either x y or y x.
For (O7), let x, y ∈ G. By (P3), we have x −1 y = a · b, where a ∈ P −1 and b ∈ P.
Then a e and a x −1 y, which implies xa x and xa y.
According to Proposition 2.1, we also say that a subsemigroup P of G satisfying (P1)-(P4) is a semilinear left preorder on G.
Lemma 2.2. Let P be a semilinear left preorder on G.
(1) For any q ∈ P, we have q(P ∪ P −1 )q −1 ⊂ P ∪ P −1 .
(2) For any g ∈ G, we have {x ∈ G : x g} = gP −1 .
(3) Let w be a nontrivial word composed by some elements of P. If w is in P ∩ P −1 then every letter occurring in w is in P ∩ P −1 .
Proof. (1) For any x ∈ P, we have qxq −1 ∈ qPq −1 ⊂ P · P −1 ⊂ P ∪ P −1 by (P4). For any x ∈ P −1 , we have xq −1 ∈ P −1 . Thus qP −1 q −1 ⊂ P · P −1 ⊂ P ∪ P −1 .
(2) If x g then g −1 x ∈ P −1 and hence x = g(g −1 x) ∈ gP −1 . For any y ∈ P −1 , we have gy g. Thus (2) holds.
(3) Set H = P ∩ P −1 and w = g 1 · · · g n with g 1 , · · · , g n ∈ P. To the contrary, assume that g i / ∈ H for some i ∈ {1, · · · , n}. Then g i ≻ e and w = g 1 · · · g n g 1 · · · g n−1 · · · g 1 · · · g i ≻ g 1 · · · g i−1 e.
This contradicts that w is in H. Hence each g i is in H.
LEFT ORDERABILITY AND SEMILINEAR LEFT PREORDERS
Definition 3.1. Let be a semilinear preorder on a set X . A subset F ⊂ X is coinitial if for any x ∈ X there is some y ∈ F with y x.
For a subset S of a group G, we use sgr(S) to denote the semigroup generated by S. It was shown in [14, Theorem 2.7] that a group admitting a semilinear left partial order is left-orderable. Now we generalize it to the case of preorder. (2) if G is finitely generated and H = G, then H = G;
(3) the quotient G/ H is left-orderable. Proof. (1) From (P2), H is a subgroup of G; thus x ∈ H implies x −1 ∈ H by the definition. Let x, y ∈ H. Suppose that x ∈ q −1 p −1 1 q −1 Hq and y ∈ q −1 p −1 2 q −1 Hq for some p 1 , p 2 ∈ P.
Since p −1 1 and p −1 2 are comparable by (O6), we may assume that p −1
1 p −1 2 . Thus y ∈ q −1 p −1 2 q −1 Hq ⊆ q −1 p −1 1 q −1 Hq and hence xy ∈ q −1 p −1 1 q −1 Hq ⊂ H. So H is a group.
For each p ∈ P and g ∈ G, from Lemma 2.2 (2),
g ∩ q −1 p −1 q −1 Hq g −1 = g ∩ x∈p −1 P −1 xHx −1 g −1 = ∩ x∈p −1 P −1 gxH(gx) −1 = ∩ x∈gp −1 P −1 xHx −1 = ∩ x gp −1 xHx −1 ⊂ ∩ x y −1 xHx −1 ⊂ H,
where y −1 is some element in P −1 with y −1 gp −1 . Thus H is normal in G.
(2) Suppose that G is finitely generated and {g 1 , · · · , g n } is a finite set of generators. To the contrary, assume that G = H. Then, for each i ∈ {1, · · · , n}, there is some p i ∈ P such that g i ∈ ∩ q −1 p −1 i q −1 Hq. Take some p ∈ P with p −1 p −1 i for each i ∈ {1, · · · , n}. Then we have
{g 1 , · · · , g n } ⊂ ∩ q −1 p −1 q −1 Hq ⊂ p −1 H p.
Thus G ⊂ p −1 H p and hence G = H. So (2) holds.
(3) We may assume that H = G; otherwise, H = G and the conclusion is trivial.
Claim 1. For any finitely many x 1 , · · · , x n ∈ G \ H, there are some ε 1 , · · · , ε n ∈ {−1, 1} such that sgr(x ε 1 1 , · · · , x ε n n ) ∩ H = / 0.
We show the Claim 1 by induction on n. Given x ∈ G \ H, suppose that there is a positive integer k such that x k ∈ H. Then x k ∈ q −1 p −1 q −1 Hq, by the definition of H, for some p ∈ P with p −1 x. Let q −1 p −1 be given. Then q −1 x and hence qxq −1 ∈ P ∪ P −1 by (P4). WLOG, we may assume that qxq −1 ∈ P. Then qx k q −1 ∈ H implies that qxq −1 ∈ H, by Lemma 2.2 (3). Thus x ∈ q −1 p −1 q −1 Hq whence x ∈ H. This contradicts the assumption. So the Claim 1 holds for n = 1. Now assume that n ≥ 2 and the Claim 1 holds for any y 1 , · · · , y m ∈ G \ H with m < n. By (O7), there is some p ∈ P with p −1 x 1 , · · · , p −1 x n .
Then qx 1 , · · · , qx n ∈ P, for each q −1 p −1 . According to (P4) in Proposition 2.1, we have {qx 1 q −1 , · · · , qx n q −1 } ⊂ P ∪ P −1 . Thus there are some ε(q) = (ε 1 (q), · · · , ε n (q)) ∈ {−1, 1} n such that {qx
ε 1 (q) 1 q −1 , · · · , qx ε n (q) n q −1 } ⊂ P. For each ε = (ε 1 , · · · , ε n ) ∈ {−1, 1} n , let Q( ε) = q −1 p −1 : {qx ε 1 1 q −1 , · · · , qx ε n n q −1 } ⊂ P . Then {q −1 : q −1 p −1 } = ε∈{−1,1} n Q( ε).
Let P ++ = {g ∈ G : g ≻ e} and P −− = {g ∈ G : g ≺ e} be the strictly positive and negative cones respectively. Now we discuss into two cases. Case 1. There is an ε = (ε 1 , · · · , ε n ) ∈ {−1, 1} n and a coinitial set Q ⊂ Q(ε) such that for any q −1 ∈ Q, {qx ε 1 1 q −1 , · · · , qx ε n n q −1 } ⊂ P ++ . Noting that any word composed of qx ε 1 1 q −1 , · · · , qx ε n n q −1 will lie in P ++ for any q −1 ∈ Q and Q is a coinitial set, the semigroup sgr(x ε 1 1 , · · · , x ε n n ) has empty intersection with H. Then the Claim holds.
Case 2. For any ε = (ε 1 , · · · , ε n ) ∈ {−1, 1} n , there is some p −1 ( ε) p −1 such that for any q −1 ∈ {q −1 ∈ Q( ε) : q −1 p −1 ( ε)}, we have i ∈ {1, · · · , n} : qx ε i i q −1 ∈ H = / 0.
Thus, for each q −1 ∈ Q( ε), there is a partition {1, · · · , n} = A(q, ε) ∪ B(q, ε) such that
{qx ε i i q −1 : i ∈ A(q, ε)} ⊂ H and {qx ε i i q −1 : i ∈ B(q, ε)} ⊂ P ++ .
Since H = G by the assumption, we have P ++ = P \ H = / 0 and hence P ++ is infinite. Note that there are only finitely many partitions of {1, · · · , n}. So there is an ε = (ε 1 , · · · , ε n ) ∈ {−1, 1} n , a coinitial set Q ⊂ Q( ε) and a partition {1, · · · , n} = A ∪ B with A = / 0, B = / 0 such that
{qx ε i i q −1 : i ∈ A} ⊂ H and {qx ε i i q −1 : i ∈ B} ⊂ P ++ , for each q −1 ∈ Q.
WLOG, we may assume that A = {1, · · · , k} and B = {k + 1, · · · , n} for some k ∈ {1, · · · , n − 1}. Now by the induction hypothesis, there is some η = (η 1 , · · · , η k ) ∈ {−1, 1} k such that
sgr(x η 1 1 , · · · , x η k k ) ∩ H = / 0.
Then we conclude that (η 1 , · · · , η k , ε k+1 , · · · , ε n ) ∈ {−1, 1} n satisfies
sgr(x η 1 1 , · · · , x η k k , x ε k+1 k+1 , · · · , x ε n n ) ∩ H = / 0.
Indeed, let w be a word composed of x
η 1 1 , · · · , x η k k , x ε k+1 k+1 , · · · , x ε n n . If x ε k+1
k+1 , · · · , x ε n n do not occur in w, then the choice of η implies that w is not in H. If there are some letters of x ε k+1 k+1 , · · · , x ε n n occur in w, then for each q −1 ∈ Q, qwq −1 ∈ P ++ and hence w is not in H by the coinitiality of Q. Thus we complete the proof of the Claim 1. Now (3) is followed from some standard arguments (see [13,Lemma 2.2.3]). For convenience of the readers, we afford a detailed proof here.
By the principe of compactness, the Claim 1 implies the following directly.
Claim 2.
There is a map ε : G \ H → {−1, 1} such that for any finite g 1 , · · · , g n ⊂ G \ H, sgr g
ε(g 1 ) 1 , · · · , g ε(g n ) n ∩ H = / 0. Let P = {g ∈ G \ H : ε(g) = 1}
. Then it is easy to verify that P is a subsemigroup of G and G = P ∪ H ∪ P −1 .
Claim 3. P = H P H.
First P = e Pe ⊂ H P H. To show the converse, we conclude that for any h ∈ H and q ∈ P, hq ∈ P. Otherwise, ε(hq) = −1. Then h −1 ∈ sgr((hq) −1 , q) ∩ H, which contradicts the choice of ε. Similarly, qh ∈ P. Thus for any h 1 , h 2 ∈ H and q ∈ P, h 1 qh 2 ∈ P and hence H P H ⊂ P. Thus Claim 3 holds.
We define an order ≤ on the quotient group G/ H = {g H : g ∈ G} by
f H < g H if and only if f −1 g ∈ P.
It is well defined by Claim 3 and is a total order by the equality G = P ∪ H ∪ P −1 . It is obvious that ≤ is G-invariant, i.e. for any g, x, y ∈ G, gx H < gy H whenever x H < y H. Thus the quotient G/ H is left-orderable.
EXISTENCE OF POINTWISE FIXED ARCS
The following is the well-known Margulis' Normal Subgroup Theorem (see e.g. [22,Theorem 8.1.2]). The following is key to establish the structure theorem. Proof. Now take a finite set {g 1 , · · · , g n } of generators of Γ. For each i ∈ {1, · · · , n}, by Lemma 4.2, we can take x i ∈ X \ {z} fixed by g i . Let t be the point such that [z,t] = ∩ n i=1 [z, x i ]. Define a positive cone P of Γ by
P = g ∈ Γ : g −1 (t) ∈ [z,t] .
Then P leads to a preorder on Γ. Let H = P ∩ P −1 and H = p∈P q −1 p −1 q −1 Hq. Note that H is just the stabilizer of t in Γ.
Claim. There is some point s ∈ (z,t] fixed by Γ.
We discuss into two cases for the proof of the Claim.
Case 1. There is a sequence ( f i ) in P −1 such that f i (t) → z as i → ∞.
Then P gives rise to a semilinear left preorder on Γ: for g, h ∈ Γ,
g h if and only if [z, g(t)] ⊂ [z, h(t)].
By Proposition 3.2, Γ admits a left-orderable quotient Γ/ H. By Lemma 4.1, either H is contained in the center of G or Γ/ H is a finite group, where G is the ambient Lie group of Γ. In the former case, Γ/ H is also a higher rank lattice, which contradicts Deroin-Hurtado's theorem in [5]. In the latter case, Γ/ H is a finite group. Then the orderability implies that it is trivial and hence H = Γ. Then we have H = Γ, by Proposition 3.2 (2). Thus t is fixed by Γ and take s = t.
Case 2. There is no sequence ( f i ) in P −1 with f i (t) → z as i → ∞. Fix a canonical ordering < on [z,t] with z < t. Let s = inf g −1 ∈P −1 g(t). Then s ∈ (z,t] and we claim that s is fixed by Γ. Indeed, let g −1 ∈ P −1 be given. Suppose that s = lim n→∞ γ −1 n (t) for some sequence (γ −1 n ) in P −1 . Then g −1 (s) ≥ s with respect to the canonical ordering < on [z,t]. If g −1 (s) = s then g −1 γ −1 n (t) > γ −1 n (t), for all sufficiently large n. Then gγ −1 n (t) < γ −1 n (t) ≤ t and hence gγ −1 n ∈ P −1 . Thus s ≤ lim n→∞ gγ −1 n (t) ≤ lim n→∞ γ −1 n (t) = s.
So g(s) = s and hence g −1 (s) = s. This contradiction shows that s is fixed by each element in P. Since for each i, either g i ∈ P or g −1 i ∈ P, s is fixed by Γ. Thus the Claim holds. Now, by Denrion-Hurtado's theorem, Γ fixes the [z, s] pointwise.
PROOF OF THE MAIN THEOREM
Now we are ready to prove the main theorem.
determines a semilinear left preorder (resp. semilinear left partial order) on G. Proof. (=⇒) (P1) and (P2) are direct from the definition of semilinear left preorder.
(
P2') and (P5) are clear. (⇐=) Define x y if x −1 y ∈ P. We only check (O6) and (O7), the others are direct.
Proposition 3. 2 .
2Let G be a group admitting a semilinear left preorder and let P be its positive cone. Set H = P ∩ P −1 and H = p∈P q −1 p −1 q −1 Hq. Then
( 1 )
1H is a normal subgroup of G;
Lemma 4. 1 .
1Let G be a connected real semisimple Lie group with finite center and no compact factors, let Γ be an irreducible lattice of G. Assume that R-rank(G) ≥ 2. Then every normal subgroup of Γ either is contained in the center of G and hence is finite or has finite index in Γ.The following is the Lemma 2.1 in[19]
Lemma 4 . 2 .
42Let X a nondegenerate dendrite and h : X → X be a homeomorphism. If h fixes an end point e ∈ X , then h fixes another point o = e.
Theorem 4 . 3 .
43Let Γ be a higher rank lattice acting on a nondegenerate dendrite X . If Γ fixes some end point z of X , then there is another point s ∈ X such that Γ fixes the arc [z, s] pointwise.
]
Let G be a finitely generated group. For any n ∈ N, there are finitely many subgroups of G with indices less than n. Proof of the main theorem. Along the same lines of the proof of Theorem 1.2 in [19], Theorem 1.1 follows from Theorem 4.3 and Lemma 5.1 (We need only use Theorem 4.3 and Lemma 5.1 instead of Proposition 5.3 and Lemma 5.5 in [19] respectively).
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Partially ordered groups. A Glass, World ScientificA. Glass, Partially ordered groups. World Scientific, 1999.
Semilinearly ordered groups. Selected surveys presented at two conferences on Advances in algebra and model theory. V Kopytov, V. Kopytov, Semilinearly ordered groups. Selected surveys presented at two conferences on Advances in algebra and model theory, Pages 149-170, 1997.
On groups acting on dendrons. A Malyutin, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). 12transl. J. Math. Sci.A. Malyutin, On groups acting on dendrons. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 415(Geometriya i Topologiya. 12):62-74, 2013. transl. J. Math. Sci. (N.Y.) 212 (2016), 558-565.
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|
[] |
[
"Socially-Compatible Behavior Design of Autonomous Vehicles with Verification on Real Human Data",
"Socially-Compatible Behavior Design of Autonomous Vehicles with Verification on Real Human Data"
] |
[
"Letian Wang ",
"Liting Sun ",
"Masayoshi Tomizuka ",
"Wei Zhan "
] |
[] |
[] |
As more and more autonomous vehicles (AVs) are being deployed on public roads, designing socially compatible behaviors for them is becoming increasingly important. In order to generate safe and efficient actions, AVs need to not only predict the future behaviors of other traffic participants, but also be aware of the uncertainties associated with such behavior prediction. In this paper, we propose an uncertainaware integrated prediction and planning (UAPP) framework. It allows the AVs to infer the characteristics of other road users online and generate behaviors optimizing not only their own rewards, but also their courtesy to others, and their confidence regarding the prediction uncertainties. We first propose the definitions for courtesy and confidence. Based on that, their influences on the behaviors of AVs in interactive driving scenarios are explored. Moreover, we evaluate the proposed algorithm on naturalistic human driving data by comparing the generated behavior against ground truth. Results show that the online inference can significantly improve the human-likeness of the generated behaviors. Furthermore, we find that human drivers show great courtesy to others, even for those without right-of-way. We also find that such driving preferences vary significantly in different cultures. * The authors are equally contributed.
|
10.1109/lra.2021.3061350
|
[
"https://arxiv.org/pdf/2010.14712v8.pdf"
] | 225,094,427 |
2010.14712
|
538f97d36125b704f91894839107c8f29c129a9d
|
Socially-Compatible Behavior Design of Autonomous Vehicles with Verification on Real Human Data
Letian Wang
Liting Sun
Masayoshi Tomizuka
Wei Zhan
Socially-Compatible Behavior Design of Autonomous Vehicles with Verification on Real Human Data
As more and more autonomous vehicles (AVs) are being deployed on public roads, designing socially compatible behaviors for them is becoming increasingly important. In order to generate safe and efficient actions, AVs need to not only predict the future behaviors of other traffic participants, but also be aware of the uncertainties associated with such behavior prediction. In this paper, we propose an uncertainaware integrated prediction and planning (UAPP) framework. It allows the AVs to infer the characteristics of other road users online and generate behaviors optimizing not only their own rewards, but also their courtesy to others, and their confidence regarding the prediction uncertainties. We first propose the definitions for courtesy and confidence. Based on that, their influences on the behaviors of AVs in interactive driving scenarios are explored. Moreover, we evaluate the proposed algorithm on naturalistic human driving data by comparing the generated behavior against ground truth. Results show that the online inference can significantly improve the human-likeness of the generated behaviors. Furthermore, we find that human drivers show great courtesy to others, even for those without right-of-way. We also find that such driving preferences vary significantly in different cultures. * The authors are equally contributed.
I. INTRODUCTION
More and more autonomous vehicles (AVs) are being deployed on public roads, sharing the space with other traffic participants. In order to safely and efficiently interact with other entities (e.g, human drivers and pedestrians), AVs need to generate behaviors that are not only optimal in terms of their own interests, but also showing respect to others in a human-like way [1]. Such socially-compatible behaviors are of critical importance for the safe deployment of AVs in mixed autonomy. For instance, cautious but unnecessary stops at intersections might cause rear-end accidents [2], and aggressive ramp merging might lead to collisions.
To design socially-compatible behaviors, the first two questions to answer are 1) what social factors should be considered, and 2) how we should quantify them and integrate them into the behavior generation framework of AVs. There have been many works in the domain of human-robot interaction (HRI) trying to tackle one or both questions. In terms of methodology, we categorize them into three groups. The first group explicitly imitates some well-established social behaviors of humans via imitation learning or modelbased behavior generation [3]- [6]. One popular example of U.S.A. Corresponding email: [email protected] model-based approaches is to generate robot behaviors via the social force model [7] as in [3] [4]. Another example is the social perception behavior in [6] which imitates humans' capability of extracting useful information from others' actions. The second group, on the other hand, quantifies social factors as reward features for the robots and uses inverse reinforcement learning (IRL) to learn the appropriate weights of such features from real human data [8]- [12]. Some social factors that have been quantified include active information gathering [8], deterministic courtesy [9], and selfishness or altruism via Social Value Orientation [11]. Finally, the third group integrates social factors with deep learning to encourage socially-compliant behaviors [13]- [16]. Typical examples include Social LSTM [13], Social GAN [14], and social attention [16]. Our work belongs to the second group for its good interpretability and the ability to learn from data. In particular, we focus on how robot cars should balance their own interests and their impacts on others in the presence of inevitable behavior uncertainties. Note that behaviors of humans and robots are mutually influenced as a closed-loop system. Hence, when behavior uncertainty occurs, the closed-loop dynamics becomes more complicated. First, the distribution of such behavior uncertainty is not fixed but can be changed with different actions of the robots. Second, beyond selfinterests and others', the robots (AVs) should also take into consideration the confidence towards uncertain predictions and the corresponding responsive actions in their policies.
Hence, in this work, we propose an uncertain-aware integrated prediction and planning (UAPP) framework to generate socially-compatible behaviors for autonomous vehicles.
Our key insight is that when behavior uncertainty exists, AVs should optimize for behaviors that consider not only selfexpected rewards, but also the impact on the action distributions of others and the self-confidence of such consequences 1 . Moreover, the preferences over the three aspects should be updated online for different individuals.
Our contributions can be summarized as follows: 1) establish an uncertainty-aware integrated prediction and planning (UAPP) framework; 2) formulate the courtesy and confidence factors as rewards in the UAPP framework, and provide a systematic analysis of their influences on AVs' behaviors with varying initial relative relationships between the AVs and other cars, as illustrated in Fig. 1; 3) develop an online algorithm to estimate the reward weights of each human individual and verify it on real data; and 4) provide inspiring findings on the statistics of how humans drive, such as how often humans switch their preferences among the three policies during one interaction, and how the dominance of driving policy varies with different right-of-ways, traffic rules, and cultures.
II. PROBLEM FORMULATION
We consider a two-agent interaction system to investigate the socially-compatible behavior. The two agents include an ego car (denoted by (·) E ) and an obstacle car (denoted by (·) O ). They have different right-of-ways. Let x i and u i denote, respectively, the state and control input of the ego car (i=E) and the interacting car (i=O). x = (x T E , x T O ) T represents the states of the interaction system, satisfying an overall dynamics given by:
x t+1 = f (x t , u t E , u t O ).(1)
We assume that both vehicles are noisy optimizers, i.e., their control inputs are optimizing their cumulative reward functions R E and R O over the planning horizon. We further assume that the two cars are running a Stackelberg game where the ego car is the leader and the other car is a naive follower, as in [8] [9]. Let N be the planning horizon. Based on such assumptions and the Model Predictive Control (MPC) framework, at each time step, in the ego car's view, the other vehicle generates its optimal action sequence
u O =(u 0 O , u 1 O , ..., u N −1 O ) T via u * O = arg max u O R O (x 0 , u O , u E ),(2)
where x 0 denotes the current state, and u E is a candidate action of the ego car. Thus, considering the behavior uncertainty, the distribution of other vehicle's future behavior u O 1 Note that many other factors can influence the behavior generation of AVs. For instance, other social factors such as trustworthiness and uncertainties from other functional modules such as perception, localization, and control also impact the behaviors. A comprehensive analysis of all these factors is beyond the scope of this work. We focus on developing an algorithm to quantify some of the social factors emerging from interactions. Hence, we assume perfect conditions for other functional modules. Fig. 2: The proposed uncertainty-aware integrated prediction and planning (UAPP) framework consists of several parts: a spatial-temporal sampler to construct discrete joint behavior, a socially compatible reward, and an online algorithm for estimating reward parameter λ E .
can be expressed as a function of the ego car's actions based on the Boltzmann-rational model [17]:
u O (x 0 , u E )∼P (u O |x 0 , u E ) ∝ e βR O (x 0 ,u E ,u O ) . (3)
β controls the degree of rationality of the other vehicle. Without loss of generality, we set β = 1 and omit it for brevity.
With (3) and the assumption that the ego car is a leader in the game, the optimal behavior of the ego car u *
E = (u 0 E , u 1 E , ..., u N −1 E ) T satisfies u * E = arg max u E R E x 0 , u E , u O ∼ P (u O |x 0 , u E ) (4) = arg max u E R E x 0 , u E
To generate socially compliant behaviors, social factors such as courtesy and confidence should be quantified and formulated as additional features in reward R E . Similar to [18] [19], we propose to construct a socially compatible reward for the AVs as a linear combination of three aspects including the regular selfish reward term, the courtesy reward term, and the confidence reward term. Namely, we have
R E (x 0 , u E ; λ E )=λ Eegoism R Eegoism (x 0 , u E ) (5) + λ Ecourtesy R Ecourtesy (x 0 , u E ) + λ E conf idence R E conf idence (x 0 , u E )
where R Eegoism , R Ecourtesy , R E conf idence are, respectively, the ego car's egoism reward, courtesy reward, and confidence reward. Each reward encourages the generation of driving behaviors with different styles. Detailed definitions of them will be introduced in Section III. The reward parameter λ E =(λ Eegoism , λ Ecourtesy , λ E conf idence ) captures the tradeoff among them.
III. THE INTEGRATED PREDICTION AND PLANNING FRAMEWORK WITH SOCIALLY COMPATIBLE REWARDS
A. The Uncertain-Aware Integrated Prediction and Planning Framework
The structure of the uncertainty-aware integrated prediction and planning (UAPP) framework is shown in Fig. 2. It includes a spatial-temporal sampling module to construct the discrete joint behavior space, a socially compatible reward function as defined in (5), and finally an online algorithm for estimating reward parameters λ E . In the discrete joint behavior space, the paired future trajectories of both cars are generated via spatial-temporal sampling [20] [21]. One example of such paired trajectories is shown in Fig. 3(a).
B. Egoism Reward
Egoism reward is defined to capture an agent's expected utilities in the presence of behavior uncertainties of others. With the current state x 0 and a pair of candidate actions (u E , u O ), the utility-related reward of an agent is defined as
R(x 0 , u E , u O ) = θ T N −1 t=0 φ(x t , u t E , u t O ),(6)
where φ∈R 3 denotes the three self-interested utilities: efficiency, comfort, and safety 2 . θ∈R 3 denotes their relative weights, which can be manually tuned or learned via Inverse Reinforcement Learning [22] [18] [23]. In order to evaluate the egoism reward, i.e., the expected utilities for the ego vehicle, we have to consider all possible responsive actions u O from the other agent. Recalling (3), we have
P (u O |x 0 , u E ) = e R O (u O |x 0 ,u E ) u O ∈U O e R O (u O |x 0 ,u E ) ,(7)
2 In this paper, we designed efficiency as keeping the desired speed and staying close to the reference line. Comfort is defined as the longitudinal and lateral acceleration and jerk. Safety is quantified as the relative distance of two cars, and two cars' distance to the intersection point of reference lines.
where R O (u O |x 0 , u E ) quantifies the other car's rewards as in (6). Thus, the egoism reward of the ego car's action u E is given by:
R Eegoism (x 0 , u E )=E u O ∼P (u O |x 0 ,u E ) R E (x 0 , u E , u O ) .(8)
C. Courtesy Reward
There are several ways of modeling courtesy [11] [9] [1]. In this paper, we model courtesy as the effort of a driver to avoid interrupting other's original plan, i.e., minimizing the difference of other car's impacted behavior distribution in presence of ego car (Fig. 3(c)) and other car's original behavior distribution in absence of the ego car ( Fig. 3(d)).
Intuitively, the ego car's action could change the other car's environment and influence the other car's decision. Thus, as in Fig. 3(c), the other car's behavior distribution in the presence of the ego car is exactly given as in (7).
Similarly, we can write out the other car's behavior distribution in the absence of the ego car as follows:
P (u O |x 0 O ) = e R O (x 0 O ,u O ) u O ∈U O e R O (x 0 O ,u O ) ,(9)
where the utility function
R O (x 0 O , u O )
captures the other car's utilities (efficiency and comfort) in the absence of the ego car, as demonstrated in Fig. 3(d).
With (7) and (9), we are able to model the courtesy as the difference between the two distributions. We adopt the Kullback-Leibler divergence [24] as a distance metric. Therefore, we have the courtesy reward of ego car's action u E defined as
R Ecourtesy (x 0 , u E ) = e −D KL[ P (u O |x 0 O )||P (u O |x 0 ,u E )] (10) = e − u O ∈U O P (u O |x 0 O ) log P (u O |x 0 O ) P (u O |x 0 ,u E ) .
D. Confidence Reward
The confidence reward is defined to capture an agent's preference for certainty rather than randomness in others' behavior. Agents prefer to gain confidence by constraining other's behavior to a concentrated distribution.
Several measures can be used to quantify confidence, including maximum model [25] (maximum probability), entropy model [24] (negative entropy of the distribution), and difference model [26] (the difference between the two highest probabilities). Though the three models are consistent for two-option tasks, it is proved that the difference model can better measure confidence in multiple-option decisionmaking tasks [26]. Interactive driving with a discrete behavior space is exactly a multiple-option decision-making task. Hence, we adopt the difference model and define the confidence under ego car's action u E as
Conf (x 0 , u E ) = P (u 1 O |x 0 , u E ) − P (u 2 O |x 0 , u E ) (11) where u 1
O and u 2 O are the two responsive actions from the other agent which, respectively, takes the highest and secondhighest probabilities.
Therefore, as in Fig. 3(e), the confidence reward of the ego car's action u E can be expressed as:
R E conf idence (x 0 , u E ) = e Conf (x 0 ,u E ) .(12)
IV. ONLINE ESTIMATE ON REWARD PARAMETER
In Section III, we have designed a socially compatible reward to capture how humans drive. However, humans are diverse in terms of their preferences over the three reward terms. Hence, we should infer each individual's preference by estimating the reward parameter from historical observations.
The key idea of the online identification is based on the principle of maximum entropy, i.e., the probability of one candidate reward parameter λ is proportional to the probability of the observed trajectories under this reward parameter. Hence, if we look r steps to the past from current time step k, then the posterior probability of λ satisfies p(λ|x k−r:k ) ∝ p(ˆx k−r:k |λ) × p(λ)
where p(λ) is the prior probability of the candidate reward parameter. The probability of the observed trajectorieŝ x k−r:k under the candidate reward parameter is proportional to its normalized collected reward:
p(x k−r:k |λ) ≈ e R(û k−r:k ,λ) u∈U e R(u k−r:k ,λ)
where U is the set of all possible actions, andû is the action generating trajectory closest (determined by mean squared error) to the observed historical trajectory. Based on (13) and (14), we develop a Bayesian inference algorithm to estimate the reward parameter online. The algorithm is outlined in Algorithm 1. We first sample N reward parameters λ i in Line 2 and initialize their weights ω i (λ i ) randomly or based on the prior knowledge Θ 0 ( ω). After updating the weights in Line 5 and normalizing the weights in Line 7, Line 8 estimates the reward parameter λ k−r as the mean of the posterior distribution.
V. EXPERIMENTAL RESULTS
Two sets of experiments were conducted. First, we analyzed the influences of the three reward terms on interactive behaviors with varying relative relationships between the two vehicles. Second, we deployed the online inference
Update ω k−r i ← ω k−r−1 i × p( x k−r:k |λ i ), Eq 14 6: end for 7: Normalize ω k−r ← ω k−r / N i=1 ω k−r i 8: Computeλ k−r ← N i=1 λ i ω k−r i
algorithm on real human driving data, and compared the re-generated trajectories via the proposed UAPP framework against the ground truth. The statistical results of humans' driving behaviors in different scenarios, traffic rules, and countries were collected and discussed.
Experiment Setting: We selected three roundabout scenarios from the INTERACTION dataset [27]: USA Roundabout FT, USA Roundabout SR, and DEU Roundabout OF. Totally we chose 219 pairs of interactive trajectories (168, 25, and 26 from each scenario). Each pair contains two cars, with one car trying to merge into a roundabout while the other car driving in the roundabout. The two cars drove towards a conflict point (the intersection point on the two cars' reference paths). The experiments were conducted in Robot Operation System (ROS) on a computer with a 3.60GHz Intel Core i7 processor of 16GM RAM. To focus on the socialcompatible planning, throughout the simulation, we assume that both cars have perfect conditions on map information, perception, localization, and control.
A. Influences of Courtesy and Confidence to Behaviors
In this experiment, we simulated the interaction between two vehicles in the USA Roundabout FT scenario. The ego car was set as the car merging into the roundabout, and the other car was in the roundabout. Both cars were AVs, while the ego car was the leader and the other car was a follower. The ego car drove actively via three different policies: 1) an egoism policy that only cares about its egoism reward, 2) a courtesy policy that cares only about courtesy, i.e., its potential influence on others, and 3) a confidence policy that generates behaviors purely for higher confidence. The other car was assumed to precisely knew the ego car's actions and was simply reacting by an egoism policy as a pure follower.
1) Quantitative Results: In the USA Roundabout FT scenario, we generated interactive trajectories by setting the initial states of both cars as given by the ground truth. Two metrics are calculated: averaged relative distance (ARE) between the two vehicles during the interaction, and the averaged interaction time (AIT, the time span from start to the time instant when one car passed the conflict point). shows the rewards of three policies at the first planning step. Darker color means a higher reward. The egoistic car run at a medium speed, distance, and interaction period. The courteous car would speed up to give more room, which generated the largest relative distance and shortest interaction period. The confident car would slow down to block the other car, resulting in a smaller relative distance and a longer interaction period. As shown in Tab. I, courteous cars generated the safest and most efficient interaction. Though it appears counterintuitive, it proved that via cooperation, more efficient interactions would happen. Confidence-driven cars could result in dangerous situations and were the most time-consuming because many unnecessary acceleration-deceleration were generated to assure safety. Egoistic cars performed between confident-driven cars and courteous cars.
2) Case Analysis: With Fig. 4-6, we provide a detailed analysis of such influences in three representative situations where the two vehicles started the interaction with different initial positions: Case I -the ego car initially preceded (i.e., closer to the conflict point) in Fig. 4, Case II -the other car initially preceded (Fig. 5), and Case III -the two cars initially equally positioned (Fig. 6). We can see from (e) that in all three cases, the egoism policy preferred medium speeds, while the influence of the courtesy policy and the confidence policy varied. In Case I where the ego car preceded, the courtesy policy encouraged the ego car to speed up to leave more space for the other car, while the confidence policy preferred slower speeds to reduce the variance of the other car's possible actions. However, in Case II and III when the other car preceded or the two cars were equally positioned, the courtesy policy enabled the ego car to give way to the other car, and the confidence policy drove the ego car to chase for the other car so that it was less likely to decelerate.
B. Online Estimation of Reward Parameters
The second study we conducted was to infer human drivers' policies in the interactive human data of the USA Roundabout FT scenario. By implementing the Bayesian inference algorithm in Algorithm 1, we can online estimate the weights on the three rewards (egoism, courtesy, and confidence) for each individual and re-generate human-like behaviors via the UAPP framework. Figure 8 shows the estimate process in one example. We run the algorithm on each car separately. We can see that with more and more observations, the samples representing the reward weights converge.
Several exemplary results are shown in Fig. 7. In each subfigure, we show, respectively, the estimates of the two cars' reward parameters, their distances to the conflict point. Car 1 and car 2 denote, respectively, the drivers outside and inside the roundabout.
Egoism Policy Dominates. In Fig. 7(a), both cars took mediate speeds. So both cars are inferred as being egoistic.
Courtesy Policy Dominates. In Fig. 7(b), car 1, which entered the roundabout, almost braked thoroughly with a strong yielding intention, while car 2 took a high speed trying to finish the interaction. Both cars were interpreted as being courteous enough to not change others' original plans.
Confidence Policy Dominates. In Fig. 7(c), though running after car 2, car 1 still took a high speed, which was inferred as being confident that car 2 would constantly keep a high speed ahead. Car 2, on the other hand, was identified as running an egoistic policy for its mediate speed.
Policy Switch Exists. We also observed that agents switch policies. As in Fig. 7(d), initially, car 1 and car 2 both took mediate speeds with an egoism policy. But later car 1 speeded up, being confident that car 2 would keep a high speed to leave the roundabout. Meanwhile, car 2 decelerated, confident that car 1 would not chase it. Thus the confidence policy outperformed the egoism policy, and a policy switch happened.
C. Statistical Results of Humans' Driver behaviors
To investigate how humans' driving behaviors differ according to traffic rules and cultures, we run the inference algorithm on three scenarios sampled from three different locations in the INTERACTION dataset, i.e., the USA Roundabout FT, the USA Roundabout SR, and the DEU Roundabout OF.
1) Metrics: We calculated two metrics to draw a general understanding of human drivers' policies.
Policy switching frequency (PSF) counts how many times a human driver switches his/her policies during one interaction. Figure 9(a) shows that, both drivers inside and outside the roundabout mostly switched policies no more than 3 times. Such results can serve as prior knowledge that we should not design behavior with frequent policy switch and should not expect humans might switch policy too often during one interaction.
Dominance of policy (DOP) quantifies how long each policy is serving as a dominant policy.
2) Influence of traffic rules and right-of-ways: Traffic rules and right-of-ways for road users could significantly affect humans' driving behaviors. In the USA Roundabout FT, all the entrances set a stop sign, requiring a full stop before entering the roundabout. Hence, we can see that in Fig. 9(b), cars trying to merge-in are highly courteous (49%) and much less egoistic (27%) and confidence (23%) because they do not have the right-of-way. Meanwhile, drivers inside the roundabout were mostly driving with an egoism policy (46%) since they hold the right-of-way. However, they were also showing a great percentage of courtesy (43%) because they tended to leave the roundabout faster to let drivers outside the roundabout merge in sooner. They were less confident (9%).
On the contrary, in the USA Roundabout SR where yield signs are set at each entrance, outside cars are driving with a much lower courtesy weight (31%), as shown in Fig. 9(c). Cars inside the roundabout behave similarly in the two scenarios because the two roundabouts set the same traffic rules for them.
3) Influence of cultures: Human behavior is also significantly impacted by cultural factors [28]. To investigate such impacts, we inferred humans' driving preferences on two scenarios located in different countries. The USA Roundabout SR (in the US) and the DEU Roundabout OF (in Germany) scenarios were selected in the INTERACTION dataset. Both roundabouts have the same traffic rules (yield sign at each entrance). In Fig. 9(c)(d), we can see that when outside drivers are entering the roundabout with a yield sign, drivers in the USA are driving on the three policies almost equally(egoism 30.9%, courtesy 31.0%, confidence 38.2%), while about half of the drivers in Germany are driving with high confidence (50.6%) and much less care about courtesy (14.4%). The distributions of driving styles inside the roundabout are similar in the two countries.
These results indicate a practical application of our algorithm on the traffic rule scheduling and culture study. More applications such as responsibility assessment in traffic accidents, usage-based vehicle insurance could also be expected.
D. Human-like Behavior Generation
With the estimated reward parameter, we are also able to re-generate the interactive behaviors via the proposed integrated prediction and planning framework. To evaluate the human-likeness, we compare the difference between the re-generated trajectories and the ground-truth. To show the influence of different policies, we re-generated four sets of behaviors, i.e., 1) under egoism policy, 2) under courtesy policy, 3) under confidence policy, and 4) under the mixed policy with online estimated reward weights.
Several initialization strategies were explored, including initializing with a higher egoism, a higher courtesy, a higher confidence policy, uniform initialization, and the statistical results given by the dominance of policy in Section V-C. We calculated the mean squared error (MSE) between the ground-truth and predicted trajectories.
From Table II(a), we can see that for cars outside the roundabout where non-egoistic behaviors were frequent, all the re-generated behaviors under the mixed policy with online estimated reward weights were more human-like, achieving the smallest MSE compared to the ground truth. From Table II(b), we can tell that for cars inside the roundabout who were mostly egoistic, different initialization had a significant impact on the performance of human-likeness. Among the five initialization settings, the one given by the statistical results (DOP in Section V-C) generated the best performance.
Furthermore, we selected cases where non-egoistic policies, namely the courtesy policy, and the confidence policy, strongly dominated (weight of the policy λ policy exceeds 0.9 for at least half of the interaction time). As shown in Table. II(c)(d), for all different initializations, the behavior generation algorithm with online-estimated reward weights performed much better than typically using an egoism policy. The initialization with statistical results (DOP in Section V-C) reduced the MSE by 60% and 53% respectively in cases dominated by courtesy and confidence.
VI. DISCUSSION
Summary. In this paper, we proposed an integrated prediction and planning framework considering behavior uncertainties. We quantified two social factors, i.e., courtesy and confidence as reward features, and integrated them with the self-interest of autonomous vehicles to generate socially compatible behaviors. Via extensive simulations and systematic analysis, we found that such social factors can encourage the generation of different driving styles. We also proposed an online reward estimation algorithm to infer an individual's preference over the three social factors. Verification on real human data showed that the proposed estimation algorithm and the planning framework can not only help us gain more knowledge about humans' driving behaviors in different traffic conditions and cultures, but also can help generate more human-like behaviors.
Future Work. We will further extend the work in multiple directions, such as 1) extending the algorithm to multiagent scenarios and relaxing assumptions in game theory, 2) exploring the conditions of policy switch, and 3) comparing the algorithm with other deep learning approaches.
ACKNOWLEDGEMENT
We thank Wilko Shwarting and Qingyun Wang for insightful discussions.
Fig. 1 :
1Driving behaviors vary via social policies. (a) Egoistic drivers care about their only utilities. (b) Courteous drivers try to avoid impacting others. (c) Confident drivers prefer to generate interactive results with a highly concentrated distribution rather than a random one.
(
Fig. 3 :
3(a) By combining the two car's actions, the agents' joint behavior space is constructed. It captures both discrete decisions and continuous trajectories. (b) The egoism reward of the ego car. (c) and (d) are the probabilities over the joint action spaces with and without the ego car. (e) The socially compatible reward. As shown in (c), the probabilities of the other car's actions are significantly influenced by the ego car's action.
Figure 3 (
3b)-(d) demonstrate, respectively, the different reward maps and probabilities for different trajectory samples, andFig. 3(e) summarizes the weighted sum of all reward maps.
Fig. 4 :
4The interaction when the ego car initially preceded. (a)(b)(c) show the interaction trajectories when the ego car takes different policies and the other car runs on the egoism policy. (d) shows the two cars' distances to the conflict point. (e)
Fig. 7 :
7Exemplary results of estimating reward parameter of real driving data, where the human driver is dominated by one policy in (a)(b)(c) and the policy switch happens in (d). For each example, we show how the reward parameter evolved and two cars' distances to conflict point.
Fig. 8 :
8Visualizations of the process on estimating reward parameters from historical observations. The parameters converge with observations.
Fig. 9 :
9(a) shows the policy switch frequency(PSF) on real data in USA Roundabout FT. Most drivers switched no more than 3 times. (b)(c)(d) show the dominance of policy(DOP) of real data in three scenarios. Driving preference varies with right-of-ways, traffic rules, and cultures.
Algorithm 1 Policy Bayesian InferenceInput: N reward parameter samples λ i , corresponding
weights ω i
k−r−1 , and observationsx k−r:k
Output: Estimated reward parameterλ k−r , updated weights
for each reward parameter sample ω i
k−r ,
1: if k-r-1=0 then
2:
Initialize N reward parameter samples λ i and their
corresponding weights ω 0
i ∼ Θ 0 ( ω)
3: end if
4: for all N reward parameter samples do
5:
Fig. 5: The interaction when the other car initially preceded. Compared to an egoistic car, a courteous ego car would brake to give way to the other car. Contrarily, an overly confident car would speed up to chase the other car, which resulted in the smallest relative distance and longest interaction period.(a) Trajectory under
egoism policy
(c) Trajectory under
confidence policy
(b) Trajectory under
courtesy policy
Ego car
Other car
Time(s) Time(s) Time(s)
Distance to Exit Point(m)
Egoism Courtesy Confidence
Ego car's end speed(m/s)
(e) Reward of policies
Speed
Up
Brake
Medium
Speed
Egoism Courtesy Confidence
5.2s
5.1s
5.6s
(d) Distance to Conflict Point
Egoism Courtesy Confidence
ARE (m)
8.74
10.26
5.74
AIT (s)
4.63
4.38
5.24
TABLE I :
IThe quantitative results for the influence of different policies. Courtesy is the safest and most efficient. Confidence is the most dangerous and time-consuming. Egoism is between the other two policies.
TABLE II :
IIThe MSE between the re-generated behaviors and the ground-truth under different policies. Setting the error of egoism policy as the baseline, we show how much percent the errors of other policies increase or decrease by. Arrows indicate whether the error increases or decreases. * denotes the behaviors via the online estimated parameter with different initialization. *DOP denotes the initialization based on the dominance of policy. Compared to the traditional fixed-egoismpolicy method, our method with DOP initialization improved the performance by 19% and 3% for drivers outside and inside the roundabout, and by 60% and 53% for courteous and confident drivers.
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|
[] |
[
"NIELSEN REALIZATION FOR SPHERE TWISTS ON 3-MANIFOLDS",
"NIELSEN REALIZATION FOR SPHERE TWISTS ON 3-MANIFOLDS"
] |
[
"Chen Lei ",
"Bena And ",
"Tshishiku "
] |
[] |
[] |
For a 3-manifold M , the twist group Twist(M ) is the subgroup of the mapping class group Mod(M ) generated by twists about embedded 2-spheres. We study the Nielsen realization problem for subgroups of Twist(M ). We prove that a group G < Twist(M ) is realized by diffeomorphisms if and only if G is cyclic and M is a connected sum of lens spaces, including S 1 × S 2 .
| null |
[
"https://arxiv.org/pdf/2204.04820v1.pdf"
] | 248,085,768 |
2204.04820
|
8ec9bd1aa95f1caae72693c3c54e114176dcb70f
|
NIELSEN REALIZATION FOR SPHERE TWISTS ON 3-MANIFOLDS
Chen Lei
Bena And
Tshishiku
NIELSEN REALIZATION FOR SPHERE TWISTS ON 3-MANIFOLDS
For a 3-manifold M , the twist group Twist(M ) is the subgroup of the mapping class group Mod(M ) generated by twists about embedded 2-spheres. We study the Nielsen realization problem for subgroups of Twist(M ). We prove that a group G < Twist(M ) is realized by diffeomorphisms if and only if G is cyclic and M is a connected sum of lens spaces, including S 1 × S 2 .
Introduction
The mapping class group Mod(M ) of a smooth, closed oriented manifold M , is the group of isotopy classes of orientation-preserving diffeomorphisms of M . Denoting Diff + (M ) the group of orientation-preserving diffeomorphisms, there is a natural projection map [Ker83]. For other manifolds, only sporadic results have been obtained; see [Par21,§2] for a summary of results in dimension 3, [FL21,BK19,Lee22,Kon22] for results in dimension 4, and [FJ90,BW08,BT22] for higher dimensions.
When M is a 3-manifold, the twist group Twist(M ) < Mod(M ) is the subgroup generated sphere twists (defined in §2). McCullough [McC90] proved that Twist(M ) ∼ = (Z/2Z) d for some d ≥ 0. We address the Nielsen realization problem for subgroups of Twist(M ). This problem was studied by Zimmermann [Zim21] for M = # k (S 1 × S 2 ), but there is an error in his argument; see [Zim22]. Nevertheless, using some of the same ideas, we correct the argument, and we generalize from # k (S 1 × S 2 ) to all 3-manifolds, giving a precise condition for which subgroups can be realized or not.
To state the main result, recall that for any pair of coprime integers p, q, there is a lens space L(p, q). Every lens space is covered by S 3 with the exception of L(0, 1) ∼ = S 1 × S 2 .
Main Theorem. Fix a closed, oriented 3-manifold M , and fix a nontrivial subgroup 1 = G < Twist(M ). Then G is realizable if and only if G is cyclic and M is diffeomorphic to a connected sum of lens spaces. [Cer59], who proved that Diff + (M ) and Homeo + (M ) are homotopy equivalent. Surprisingly, the realization problem for finite groups is also the same in the topological and smooth categories in dimension 3. For dimension 4, even the connected components are different by work of Ruberman [Rub98]. The work of Baraglia-Konno [BK19] gives a mapping class that can be realized as homeomorphism but not as diffeomorphism and Konno [Kon22] generalizes the results to more manifolds.
Theorem 1.2 (Pardon, Kirby-Edwards). Let M be a closed oriented 3-manifold. A finite subgroup G < Mod(M ) is realizable by homeomorphisms if and only if it is realizable by diffeomorphisms.
In particular, this allows us to strengthen the conclusion of the Main Theorem. We emphasize that the group G in Theorem 1.2 is finite; however, we do not know an example of a 3-manifold M and an infinite group G < Mod(M ) that is realizable by homeomorphisms but not diffeomorphisms.
Proof of Theorem 1.2. Let ρ : G → Homeo + (M ) be a realization of G < Mod(M ). By Pardon [Par21], ρ can be approximated uniformly by a smooth action ρ : G → Diff + (M ). Since Homeo + (M ) is locally path-connected by Kirby-Edwards [EK71], we know that ρ (g) and ρ(g) are isotopic in Homeo + (M ) for each g ∈ G, and hence also isotopic in Diff + (M ) since Mod H (M ) = Mod(M ).
The following remarks relate the Main Theorem to some previous work. Remark 1.3 (Twist group for S 1 × S 2 ). The group Twist(S 1 × S 2 ) is isomorphic to Z/2Z. We construct a realization of this group in §5. This example seems to be overlooked in some of the literature on finite group actions on geometric 3-manifolds. It is a folklore conjecture of Thurston that any finite group action on a geometric 3-manifold is geometric (i.e. acts isometrically on some geometric structure). It is easy to see that our realization of the twist group, which also appears in work of Tollefson [Tol73], does not preserve any geometric structure on S 1 × S 2 , so it is a simple counterexample to Thurston's conjecture.
Apparently, Thurston proved some cases of his conjecture, but these results were never published. Meeks-Scott [MS86] proved Thurston's conjecture for manifolds modeled on H 2 × R, SL 2 (R), Nil, E 3 , and Sol. In [MS86,Thm. 8.4] it is asserted (incorrectly) that Thurston's conjecture also holds for 3-manifolds modeled on S 2 ×R (in particular S 2 ×S 1 ); they give an argument, but in the case when some g ∈ G has positive-dimensional fixed set (as is the case for our realization of Twist(S 1 × S 2 )), they cite a preprint of Thurston that never appeared.
Remark 1.4 (Sphere twists in dimension 4). For a 4-manifold W , for each embedded 2-sphere S ⊂ W with self-intersection S · S = −2, there is a sphere twists T S ∈ Mod(W ), which has order 2. There are several results known about realizing the subgroup generated by a sphere twist, both positive and negative; see Farb About the proof of the Main Theorem. The proof is divided into two parts: construction and obstruction (corresponding to the "if" and "only if" directions in the theorem statement). For the obstruction part of the argument, we prove the following constraint on group actions on reducible 3-manifolds.
Theorem 1.5. Let M be a closed, oriented, reducible 3-manifold. Let G < Diff + (M ) be a finite subgroup that acts trivially on π 1 (M ). Then G is cyclic. If G is nontrivial, then every prime component of M is a lens space.
Section outline. In §2, we recall results about sphere twists and the twist group. In §3 we explain results from minimal surface theory that allow us to decompose a given action into actions on irreducible 3-manifolds. In §4 and 5 we prove the "obstruction" and "construction" parts of the Main Theorem, respectively.
Acknowledgement. The authors are supported by NSF grants DMS-2203178 and DMS-2104346.
Sphere twists and the twist subgroup
In this section we recall some results we will use about sphere twists and the group they generate. In §2.1 we recall the definition of sphere twists and recall that they act trivially on π 1 (M ) and π 2 (M ). In §2.2 we give a generating set for the twist group in general, and we give a more precise generating set in the case when M is a connected sum of lens spaces. Lemma 2.1 (Action of sphere twists on homotopy groups). Let M be a closed, oriented 3-manifold with a 2-sided embedded sphere S ⊂ M . Then τ S acts trivially on π 1 (M ) and π 2 (M ).
Remark 2.2 (Action on π 1 (M ) vs. π 1 (M, * )). When we refer to the action of a diffeomorphism f ∈ Diff(M ) on π 1 (M ), we mean as an outer automorphism. Technically, this is not an action, but there is a well-defined homomorphism Diff(M ) → Out(π 1 (M )). When f has a fixed point * ∈ M , the action of f on π 1 (M, * ) refers to the induced automorphisms f * : π 1 (M, * ) → π 1 (M, * ). This distinction will be important in later sections. If f acts trivially on π 1 (M ) it can be isotoped to f ∈ Diff(M, * ) that acts on π 1 (M, * ) by conjugation (generally nontrivial).
Lemma 2.1 is well-known. That sphere twists act trivially on π 1 (M ) is implicit in [McC90]. This fact may be proved as follows. Let * ∈ M be a fixed point of the diffeomorphism T S defined in (1). After choosing a prime decomposition of M , one can show that each element of π 1 (M, * ) is represented by a loop contained entirely in the fixed set of T S . That sphere twists act trivially on π 2 (M ) follows from [McC90, Lem. 1.1]. It can also be deduced from a general result of Laudenbach that says that if f ∈ Diff(M, * ) acts trivially on π 1 (M, * ), then it also acts trivially on π 2 (M, * ); see [BBP22, Thm. 2.4].
2.2. Generating the twist group and some computations. An explicit generating set for Twist(M ) can be described as follows. As always, we assume M is orientable. Let M = # k (S 1 × S 2 )#P 1 # · · · #P be the prime decomposition of M , and construct M from S 3 by removing 2k + disjoint 3-balls and attaching k (
[−1, 1] × S 2 ) j (P j \ D 3 ),(2) (Z/2Z) k+ Twist(M ).
Interestingly, this map need not be injective. Lemma 2.3, together with the preceding generating set for Twist(M ) yields the following. For the manifold S 1 × S 2 , we call a sphere of the form * × S 2 a belt sphere (we use this terminology because this sphere can be viewed as the belt sphere of a handle attachment).
S 3 P 1 P 2 P 3 [−1, 1] × S 2 [−1, 1] × S 2 Figure 1. Construction of M = # k (S 1 × S 2 )#P 1 # · · · #P .Corollary 2.5. Let M = # k (S 1 × S 2 )#P 1 # · · · #P , where each P i is a lens space. Then Twist(M )
is generated by twists about the belt spheres of the S 1 × S 2 summands. In particular, there is a surjection (Z/2Z) k Twist(M ).
Remark 2.6. We suspect that if M is as in Corollary 2.5, then Twist(M ) is isomorphic to (Z/2Z) k , i.e. the twists about the belt spheres are linearly independent. Ultimately, we do not need this for our proof.
Decomposing finite group actions on 3-manifolds
In this section we explain some general structural results for certain finite group actions on 3-manifolds, which will allow us to decompose a G-manifold M 3 into simpler G-invariant pieces. For our application to the Main Theorem we are particularly interested in actions that are trivial on π i (M ) for i = 1, 2.
3.1. Equivariant sphere theorem. The main result of this section is Theorem 3.1. In order to state it, we introduce some notation. Let (1) There exists a G-invariant collection S of disjoint embedded spheres in M such that the components of M S are irreducible.
(2) If M = S 1 × S 2 and G acts trivially on π 1 (M ) and π 2 (M ), then G preserves every element in S.
Remark 3.2. Without loss of generality, one can assume that no S ∈ S bounds a ball in M by removing any sphere that bounds a ball from S. Similarly, if G preserves every element of S, then we can also assume that no pair S = S ∈ S bound an embedded
S 2 × [0, 1] in M . Part (1) of Theorem 3.1 is due to Meeks-Yau; see [MY80, c.f. Thm. 7
]. An alternate approach was given by Dunwoody [Dun85,Thm. 4.1]. The main tools used in these works are minimal surface theory (in the smooth and PL categories). For the proof of Theorem 3.1(2), we use the following lemmas. Proof of Lemma 3.4. Set A = S 2 × [0, 1]. Consider the arc α = * × [0, 1] and the sphere β = S 2 × 0. After orienting α and β, we view them as homology classes α ∈ H 1 (A, ∂A) and β ∈ H 2 (A), which generate these groups. Since h interchanges the components of ∂A, h(α) = −α. Since h is orientation-preserving,
α · β = h(α) · h(β) = −α · h(β). This implies h(β) = −β because the intersection pairing H 1 (A, ∂A) × H 2 (A) → Z is a perfect pairing by Poincaré-Lefschetz duality.
Proof of Theorem 3.1(2). Let S be a G-invariant collection of embedded spheres as in Theorem 3.1(1). Fix S ∈ S and g ∈ G. We want to show that g(S) = S. Suppose for a contradiction that g(S) is disjoint from S. Fix an embedding f : First assume k ≥ 3. Since we can find a G-invariant metric on M , the interiors of A and g(A) are either equal or disjoint. Since k ≥ 3, g 2 (S) = S, so g(A) = A. Consequently,
S 2 → M with f (S 2 ) = S.A ∪ g(A) is diffeomorphic to S 2 × [0, 1]. Similarly, we conclude that A ∪ g(A) ∪ · · · ∪ g k−1 (A)
is diffeomorphic to S 2 × S 1 (this is the only S 2 -bundle over S 1 with orientable total space). This contradicts the assumption that M = S 2 × S 1 . Now assume that k = 2. As in the preceding paragraph, if A and g(A) have disjoint interiors, then M = S 2 × S 1 . Therefore, g(A) = A. By Lemma 3. If S ⊂ M is separating, then M \ A is a union of two components M 1 M 2 that are interchanged by g. This contradicts the fact that G acts trivially on π 1 (M ) ∼ = π 1 (M 1 ) * π 1 (M 2 ).
3.2.
Decomposing an action along invariant spheres. Here we explain how we use Theorem 3.1 to decompose an action G M into smaller pieces. We also prove a result about the action on the fundamental group of the pieces under the assumption that G acts trivially on π 1 (M ).
Fix a finite subgroup G < Diff + (M ) and assume G acts trivially on π 1 (M ). Let S be a G-invariant collection of embedded spheres such that M S has irreducible components, as in Theorem 3.1. We will always assume that S has the additional properties discussed in Remark 3.2: no S ∈ S bounds a ball and no pair S, S ∈ S bound an embedded S 2 × [0, 1].
Observe that there is an induced action of G on M S . To construct it, recall a classical result of Brouwer, Eilenberg, and de Kerékjártó [Bro19, dK19, Eil34] that every finite subgroup of Homeo + (S 2 ) is conjugate to a finite subgroup of SO(3), hence extends from the unit sphere S 2 ⊂ R 3 to the unit ball D 3 ⊂ R 3 . In this way the action of G on M \ S∈S S extends to an action on M S , which can be made smooth as well.
Remark 3.5 (global fixed points). Since G acts trivially on S, then the center of each of the added 3-balls in M S contains a global fixed point for the G-action; we call these canonical fixed points. Each component of M S contains at least one canonical fixed point.
The following proposition will be important for our proof of the Main Theorem.
Proposition 3.6. Fix G < Diff + (M ) acting trivially on π 1 (M ) and fix a G-invariant collection S of disjoint, embedded spheres such that M S has irreducible components. Let N be a component of M S , and let p ∈ N be a canonical fixed point, as defined in Remark 3.5. Then G acts trivially on π 1 (N, p).
Proof. Fix g ∈ G. We show that the action of g on π 1 (N, p) is trivial. The statement is only interesting when N is not simply connected, so we assume this.
Let k be the number of elements of S that meet N . We separate the argument into the cases k = 1, k = 2, and k ≥ 3.
Case: k = 1. Let S ∈ S be the sphere that meets N . The sphere S is separating and gives a description of M as a connected sum M ∼ = N #N . Since g ∈ G has finite order and preserves S, there is a fixed point q ∈ S g . By van Kampen's theorem, π 1 (M, q) ∼ = π 1 (N, q) * π 1 (N , q). By Lemma 2.1, the action of g on π 1 (M, q) is by conjugation by some element π 1 (M, q). Since g preserves the decomposition M = N #N , it also preserves the factors in the splitting π 1 (N, q) * π 1 (N , q). Note that π 1 (N , q) is nontrivial, since otherwise S bounds a ball in M , contrary to our assumption. The only conjugation of A * A that preserves both A = 1 and A = 1 is the trivial conjugation, so g acts trivially on π 1 (M, q). Consequently, g acts trivially on π 1 (N, q), and also on π 1 (N, p). (In general, changing the basepoint can change the automorphism to a nontrivial conjugation, but in this does not happen here since e.g. the points p, q ∈ N are connected by an arc contained in the fixed set N g .)
Case: k = 2. Let S, S ∈ S denote the spheres the meet N . Observe that these sphere are either both separating or both nonseparating. Let N be the closed 3-manifold such that M S = N N .
If S and S are both separating, then the argument is similar to the case k = 1. We first apply the Case: k = 1 to the first sphere, and we obtain the the action is identity reference to the first point, implying the the action is by conjugation referencing other points. Then we apply the Case: k = 1 to the second sphere, we obtain that the action is identity reference to the second point. Since this does not depend on the order we take, we know that reference to both fixed points, the action is trivial.
Assume then that both S and S are nonseparating; in particular, this implies that N is connected. Then M is obtained from N N by removing balls B 1 , B 2 ⊂ N and B 1 , B 2 ⊂ N and gluing ∂B i to ∂B i . Choose a fixed points q ∈ S g and q ∈ (S ) g . There is an isomorphism π 1 (M ) ∼ = π 1 (N, q) * π 1 (N , q) * Z. (The Z factor is not important for this part of the argument, but will play a role when k ≥ 3.)
By assumption g acts on π 1 (M, q) by conjugation and preserves the free factors π 1 (N, q) and π 1 (N , q). Both of these groups is nontrivial, by our assumption that no two spheres in S are parallel. Then as before, we conclude that g acts trivially on π 1 (M, q), hence also on π 1 (N, q) and π 1 (N, p).
Case k ≥ 3. Let S 0 , . . . , S k−1 ∈ S denote the spheres that meet N . If some S i separates, then we can proceed similar to the case k = 1, so we can assume each S i is nonseparating. Then M S = N N , where N is connected, and M is obtained from N N by removing balls B 0 , . . . , B k−1 ⊂ N and B 0 , . . . , B k−1 ⊂ N and gluing B i and B i along their boundary (which is identified with S i ⊂ M ). Choose fixed points q i ∈ (S i ) g . There is an isomorphism π 1 (M, q 0 ) ∼ = π 1 (N, q 0 ) * π 1 (N , q 0 ) * F k−1 .
If π 1 (N , q 0 ) = 1, then we can argue similar to the case k = 2. Therefore, we assume that N is simply connected, which means π 1 (M, q 0 ) ∼ = π 1 (N, q 0 ) * F k−1 .
The free group F k−1 is generated by loops γ i = η i * η i , where η i is a path in N from q 0 to q i (and disjoint from the interiors of the balls B 0 , . . . , B k−1 ), and η i is a path in N from q i to q 0 .
On the one hand, 2
g(γ i ) ∼ g(η i ) * η i ∼ g(η i ) * η i * η i * η i = (g(η i ) * η i ) * γ i ,
so g acts on γ i by left multiplication by the element β i = g(η i ) * η i ∈ π 1 (N, q 0 ). On the other hand, g acts on γ i by conjugation by an element α ∈ π 1 (M, q 0 ). The only way these actions are equal is if both are trivial. If we use the word length on π 1 (M, q 0 ) given by generating sets
S = {s : s ∈ π 1 (N, q 0 ) or s ∈ F k−1 },
then the word length of αγ i α −1 is an odd number, but the word length for β i γ i is 2 unless β i = 1. This implies that β i = 1. Then we know that γ i = αγ i α −1 for every i, which implies that α = 1.
In particular, this implies that g acts trivially on π 1 (M, q 0 ) and hence also on π 1 (N, q 0 ) and π 1 (N, p).
Obstructing realizations
In this section we prove the "only if" direction of the Main Theorem. This can be deduced quickly from the following more general statements.
Theorem 4.1. Let N be a closed, oriented, irreducible 3-manifold with basepoint p ∈ N . Suppose there exists a nontrivial, finite-order element f ∈ Diff + (N, p) that acts trivially on π 1 (N, p). Then N is a lens space.
Theorem 4.2. Let M be a closed, oriented, reducible 3-manifold. Let G < Diff + (M ) be a finite subgroup that acts trivially on π 1 (M ). Then G is cyclic.
Proof of Main Theorem: obstruction. Suppose 1 = G < Twist(M ) is realizable. The fact that Twist(M ) = 1 implies that either M = S 2 × S 1 or M is reducible. In the former case, there is nothing to prove, so we assume M is reducible. This assumption together with Lemma 2.1 allow us to apply Theorem 4.2 and conclude that G is cyclic.
It remains to show M is a connected sum of lens spaces, or, equivalently, that each component of M S is a lens space. This is implied directly by Proposition 3.6 and Theorem 4.1.
Next we use Theorem 4.1 to deduce Theorem 4.2. Then we prove Theorem 4.1.
Proof of Theorem 4.2. Let S be a G-invariant collection of disjoint embedded spheres such that M S has irreducible components (Theorem 3.1). We also assume that no S ∈ S bounds a ball and that no two spheres S, S ∈ S bound an embedded S 2 × [0, 1] (Remark 3.2).
Since G acts trivially on π 1 (M ), Proposition 3.6 and Theorem 4.1 combine to show that each component of M S is a lens space.
Suppose that there exists a component N of M S that is diffeomorphic to S 3 . Let k be the number of elements of S that meet N . If k = 1, then N is obtained from M by cutting along a sphere that bounds a ball. Similarly, if k = 2, then N is obtained by cutting M along two parallel spheres. Both of these are contrary to our assumption about S. Therefore k ≥ 3, which implies that N G has at least 3 points. By the Smith conjecture [Mor84], the action of G on N ∼ = S 3 is conjugate into SO(4), and the fact that |N G | ≥ 3 implies that G is conjugate into SO(2). Therefore G is cyclic.
The remaining case is that every component of M S is a lens space different from S 3 and S 1 × S 2 . In this case Twist(M ) is the trivial group by Corollary 2.5.
We proceed to the proof of Theorem 4.1. Our argument is inspired by an argument of Borel [Bor83] that shows that a finite group G acting faithfully on a closed aspherical manifold N and π 1 (N ) has trivial center, then G also acts faithfully on π 1 (N ) (by outer automorphisms).
Before starting the proof, we recall some facts about lifting actions to universal covers. Let N be a closed manifold. Recall that N can be defined as the set of paths α : (N, * )).
Proof of Theorem 4.1. As observed above, we can lift f to a finite-order diffeomorphism F that commutes with the deck group π 1 (N, * ) and has a global fixed point.
First we show that π 1 (N ) is finite. Suppose for a contradiction that π 1 (N ) is infinite. This implies N is contractible. 3 By Smith theory [Smi34], the fixed set ( N ) F is connected, and simply connected. Since F acts smoothly, ( N ) F is a smooth 1-dimensional manifold, hence it is homeomorphic to R. Since π 1 (N ) commutes with F , it acts on ( N ) F ∼ = R, and this action is free and properly discontinuous since the action of π 1 (N, * ) on N has these properties. This implies that π 1 (N, * ) ∼ = Z, which contradicts the fact that N is a closed, aspherical 3-manifold (Z is not a 3-dimensional Poincaré duality group).
Since π 1 (N ) is finite, its universal cover is diffeomorphic to either S 3 by the Poincareconjecture. As in the preceding paragraph, consider the action of F on N ∼ = S 3 . By Smith theory and smoothness of the action, the fixed set is a smooth, connected 1-dimensional manifold with nontrivial fundamental group. Hence ( N ) F ∼ = S 1 . Since π 1 (N ) acts freely on ( N ) F this implies π 1 (N ) is cyclic, which implies that N is a lens space.
Constructing realizations
In this section we prove the "if" direction of the Main Theorem. We state this as the following theorem. Fix M as in Theorem 5.1, and write the prime decomposition
M = # k (S 1 × S 2 )#P 1 # · · · #P ,
where each P i is a lens space different from L(0, 1) ∼ = S 1 × S 2 .
To prove Theorem 5.1, given a nontrivial element g ∈ Twist(M ) we define γ ∈ Diff(M ) such that γ 2 = id and [γ] = g in Mod(M ). The basic approach is to define an order-2 diffeomorphism of (3) k (S 1 × S 2 ) P 1 · · · P in such a way that the diffeomorphisms on the components can be glued to give an order-2 diffeomorphism of M . On each component of (3) we perform one of the following diffeomorphisms.
• (constant π rotation) Define
R 0 : S 1 × S 2 → S 1 × S 2
by id × r, where r : S 2 → S 2 is any π rotation (choose one -the particular axis is not important). • (nonconstant π rotation) Let c : [0, 1] → RP 2 be a closed path that generates π 1 (RP 2 ), and let α : RP 2 → SO(3) be the map that sends ∈ RP 2 to the πrotation whose axis is . Now define
R 1 : S 1 × S 2 → S 1 × S 2 by (t, x) → (t, α(c(t))(x)).
Since α • c : [0, 1] → SO(3) defines a nontrivial element of π 1 (SO(3)), the diffeomorphism R 1 represents the generator of Twist(S 1 × S 2 ) ∼ = Z/2Z. This shows that Twist(S 1 × S 2 ) is realized. This involution appears in [Tol73,§1] in a slightly different form.
• (lens space rotation) Fix p, q relatively prime and with p ≥ 2. View L(p, q) as the quotient of S 3 ⊂ C 2 by the Z/pZ action generated by (z, w) → (e 2πi/p z, e 2πiq/p w). Define R p,q : L(p, q) → L(p, q) as the involution induced by (z, w) → (z, −w) on S 3 (which descends to L(p, q) since it commutes with the Z/pZ action). Each of the diffeomorphisms R 0 , R 1 , and R p,q has 1-dimensional fixed set. The representation in the normal direction at a fixed point is the antipodal map on R 2 (there is no other option since these diffeomorphisms are involutions). Lemma 5.2 below allows us to glue these actions along their fixed sets. Remark 5.3. The condition that the isomorphism T x M ∼ = T x M be orientation-reversing appears because the connected sum of two oriented manifolds is defined by deleting an open ball from each and identifying the boundaries of these balls by an orientation-reversing diffeomorphism. This condition is always satisfied if each tangent space contains a copy of the trivial representation (choose an appropriate reflection).
Remark 5.4 (Useful isotopies). To prove that γ ∈ Diff(M ) is in the isotopy class of g ∈ Twist(M ), the following observation will be useful. The fixed set of R p,q acting on L(p, q) contains 4 the image C of the circle {(z, 0) : |z| = 1} ⊂ S 3 . The isotopy h t (z, w) = (z, e πi(1−t) w), 0 ≤ t ≤ 1, descends to L(p, q) to give an isotopy between R p,q and the identity, and h t fixes C for each t.
Similarly, it's possible to isotope R 1 to R 1 , which is a constant π-rotation (say about the z-axis) on a neighborhood of * × S 2 , for some fixed basepoint * ∈ S 1 (observe that R 1 is still an involution). Furthermore, we can isotope R 1 to a diffeomorphism that is the identity near * × S 2 and in such a way that the isotopy at time t ∈ [0, 1] is a rotation by angle π(1 − t) (about the z-axis) on each sphere in a regular neighborhood of * × S 2 . The fixed set restricted to a neighborhood of * × S 2 remains constant during this isotopy.
Finally, we can isotope R 0 to the identity so that at time t the diffeomorphism is a constant rotation by angle π(1 − t) (about the fixed axis).
On a neighborhood of a fixed point, the local picture of the isotopies of R p,q , R 1 , and R 0 looks the same. This will allow us to perform these isotopies equivariantly on connected sums.
We proceed now to the proof of Theorem 5.1. First we warm up with the case M = # k (S 1 × S 2 ) and then we do the general case. 5.1. Realizations for connected sums of S 1 × S 2 . Fix k ≥ 1 and consider
M k := # k (S 1 × S 2 ).
Let S i be the belt sphere in the i-th connect summand, and denote the sphere twists about S i by τ i . The twists τ 1 , . . . , τ k form a basis for Twist(M k ) ∼ = (Z/2Z) k by a result of Laudenbach [Lau73]; see also [BBP22].
Fix a nonzero element g = a 1 τ 1 + · · · + a k τ k in Twist(M k ). We start by defining an involutionγ of k (S 1 × S 2 ). For ease of exposition, let W i = S 1 × S 2 denote the i-th component of k (S 1 × S 2 ). Defineγ on W i to be R 0 or R 1 , depending on whether the coefficient a i is 0 or 1, respectively. Next we glue using Lemma 5.2 to obtain an involution γ of M k = W 1 # · · · #W k ∼ = # k (S 1 × S 2 ). There are multiple ways to describe the gluing; for example, choose k − 1 distinct fixed points x 1 , . . . , x k−1 ∈ W k , and for 1 ≤ i ≤ k − 1, glue W i to W k along regular (equivariant) neighborhoods of x i and an arbitrary fixed point y i ∈ W i (the neighborhoods of x 1 , . . . , x k should be chosen to be small enough so that they are disjoint).
To see that γ ∈ Diff(M k ) is in the isotopy class g, recall the short exact sequence of Laudenbach 1 → Twist(M k ) → Mod(M k ) → Out(π 1 (M k )) → 1. It's easy to check that γ acts trivially on π 1 (M k ), so γ represents a mapping class in Twist(M k ). The particular isotopy class is determined by the action on trivializations of the tangent bundle of M k , and in this way one can check that [γ] = g in Twist(M k ). We do not spell out the details of this because we give an alternate argument in the next section in the general case.
Remark 5.5. We cannot realize a non-cyclic subgroup of Twist(M n ) using this construction because it is not possible to choose the axis for R 0 so that (1) R 0 and R 1 have a common fixed point and (2) R 0 and R 1 commute. Indeed, §4 proves no non-cyclic subgroup of Twist(M n ) is realized.
5.2.
Realizations for connected sum of lens spaces. Now we treat the general case M = # k (S 1 × S 2 )#L(p 1 , q 1 )# · · · #L(p , q ), where each L(p j , q j ) is a lens space different from L(0, 1) ∼ = S 1 × S 2 . Our approach is similar to the preceding section.
Recall from Corollary 2.5 that Twist(M ) is generated by twists τ 1 , . . . , τ k in the belt spheres of the S 1 × S 2 summands.
Fix a nonzero element g = a 1 τ 1 + · · · + a k τ k in Twist(M ). We start by defining an involutionγ of k (S 1 × S 2 ) L(p 1 , q 1 ) · · · L(p , q ). Let W i denote the i-th component diffeomorphic to S 1 × S 2 . Defineγ on L(p j , q j ) to be R p j ,q j , and on W i to be R 0 or R 1 , depending on whether the coefficient a i is 0 or 1, respectively. (Recall that R 1 is similar to R 1 , but it has a product region.)
Next we glue using Lemma 5.2 to obtain an involution γ of M . We glue by the following pattern. First we glue W 1 , . . . , W k . Choose k − 1 distinct fixed points x 1 , . . . , x k−1 ∈ W k , and for 1 ≤ i ≤ k − 1, glue W i to W k along regular (equivariant) neighborhoods of x i and an arbitrary fixed point y i ∈ W i (as was done in the preceding section). Next glue L(p j , q j ) to W k in a region whereγ acts as a product (we can choose x 1 , . . . , x k−1 and the regular neighborhoods of these points to ensure that there is room to do this). In this way we obtain an involution γ ∈ Diff(M ).
We need to check that γ is in the isotopy class of g. Using the isotopies defined in Remark 5.4, we can isotopeγ to a map that is the identity on each L(p j , q j ) component and each component W i such that a i = 0, and is a sphere twist on each component W i such that a i = 1. By construction these isotopies glue to give an isotopy of γ to a product of sphere twists representing g.
This completes the proof of Theorem 5.1.
Question 5.6. Are any two realizations of g ∈ Twist(M ) conjugate in Diff(M )?
π : Diff + (M ) → Mod(M )sending a diffeomorphism to its isotopy class. We say that a subgroup i :G → Mod(M ) is realizable if there is a homomorphism of ρ : G → Diff + (M ) such that π • ρ = i.realization problem asks which finite G < Mod(M ) are realizable. When M is a surface, every finite subgroup of Mod(M ) is realizable by work of Kerkchoff
Remark 1 . 1 (
11Smooth vs. topological Nielsen realization). The topological mapping class group Mod H (M ) is defined as the group of isotopy classes of orientation-preserving homeomorphism Homeo + (M ) of M . There is a natural projection map Homeo + (M ) → Mod H (M ), and the Nielsen realization problem can also be asked for subgroups of Mod H (M ). For 3manifolds, the smooth and topological mapping class groups coincide Mod(M ) ∼ = Mod H (M ) by Cerf
2. 1 .
1Sphere twists and their action on homotopy groups. Fix a closed oriented 3manifold M . We recall the definition of a sphere twist. Fix an embedded 2-sphere S ⊂ M with a tubular neighborhood U ∼ = S × [0, 1] ⊂ M , and fix a closed path φ : [0, 1] → SO(3) based at the identity that generates π 1 (SO(3)). Define a diffeomorphism of U by (1) T S (x, t) = φ(t)(x), t and extend by the identity to obtain a diffeomorphism T S of M . The isotopy class τ S ∈ Mod(M ) of T S is called a sphere twist. The twist subgroup of Mod(M ), denoted Twist(M ), is the subgroup generated by all sphere twists.
as pictured in Figure 1; see [McC90, §1] for a precise description. By [McC90, Lem. 1.1 and Prop. 1.2], Twist(M ) is generated by k + sphere twists: one for the sphere {0} × S 2 in each copy of [−1, 1] × S 2 and twists, and one for the boundary component of each P j \ D 3 . In particular, there is a surjection
Figure 2 .
2S be a collection of disjoint embedded spheres in a 3-manifold M . Define M S as the result of removing an open regular neighborhood of each S ∈ S and capping each boundary component with a 3-ball. The 3-manifold M S is a closed, but usually not connected. This process is pictured in Figure 2. Cutting and capping along spheres M M S . Theorem 3.1. Let M be a closed oriented 3-manifold and let G be a finite subgroup of Diff + (M ).
Lemma 3 . 3 .
33Let S 0 , S 1 ⊂ M be disjoint embedded spheres. If S 0 and S 1 are ambiently isotopic, then they bound an embedded S 2 × [0, 1] in M . Lemma 3.3 follows from [Lau73, Lem. 1.2] and the Poincaré conjecture (Laudenbach proves that homotopic spheres bound an h-cobordism, and every h-cobordism is trivial by Perelman's resolution of the Poincaré conjecture).
Lemma 3 . 4 .
34Let h be an orientation-preserving homeomorphism of S 2 × [0, 1]. If h that interchanges the two boundary components, then h acts on H 2 (S 2 × [0, 1]) ∼ = Z by −1.
Since g acts trivially on π 2 (M ), the maps f and g • f are homotopic, hence isotopic by a result of Laudenbach and the Poincaré conjecture; c.f. [Lau73, Thm. 1]. By Lemma 3.3, the spheres S and g(S) bound a submanifold A ∼ = S 2 × [0, 1]. Let k ≥ 2 be the smallest power of g so that g k (S) = S.
4, [g(S)] = −[S] in H 2 (M ). By assumption that G acts trivially on π 2 (M ), we know [g(S)] = [S]. Therefore, 2[S] = 0 in H 2 (M ). If S ⊂ M is non-separating, then there is a closed curve γ ⊂ M so that [S] · [γ] = 1; this implies that [S] has infinite order in H 2 (M ), which is a contradiction.
[0, 1] → N with α(0) = * , up to homotopy rel endpoints. Using this description, there is a left action π 1 (N, * ) × N → N given by pre-concatenation of paths [γ].[α] = [γ * α], and there is a left action Diff(N, * ) × N → N given by post-composition f.[α] = [f • α]. If f ∈ Diff(N, * ) acts trivially on π 1 (N, * ), then F ([α]) = [f • α] is a lift of f that commutes with the deck group action and fixes the homotopy class of the constant path (as well as every other homotopy class corresponding to an element of π 1
Theorem 5 . 1 .
51Let M be a connected sum of lens spaces. Then every cyclic subgroup of Twist(M ) is realizable.
Lemma 5. 2 .
2Suppose M, M are oriented manifolds, each with a smooth action of a finite group G. Assume that x ∈ M and x ∈ M are fixed points of G, and that the representations T x M and T x M are isomorphic by an orientation reversing map. Then M and M can be glued along regular neighborhoods B and B of x and x so that there is a smooth action of G on M #M that restricts to the given action on M \ B and M \ B .
-Looijenga [FL21, Cor. 1.10], Konno [Kon22, Thm. 1.1], and Lee [Lee22, Rmk. 1.7]. It would be interesting to determine precisely when a sphere twist is realizable in dimension 4.
Let M be a closed oriented manifold, and let S ⊂ M be an embedded sphere that bounds a lens space on one side. Then the sphere twist τ S is trivial in Mod(M ).The red set
represents spheres that generate Twist(M ).
Lemma 2.3. Lemma 2.3 can be proved by constructing an explicit isotopy; see [FW86, Lem. 3.5].
Remark 2.4 (Homotopy vs. isotopy of sphere twists). When S ⊂ M bounds a prism
manifold 1 on one side, then it is possible for the sphere twist T S to be homotopic but not
isotopic to the identity by Friedmann-Witt [FW86, Cor. 2.2]. Hendriks [Hen77] gives a
characterization for when a sphere twist is homotopic to the identity; see [FW86, Cor. 2.1].
These are quotients of S 3 by a subgroup Γ < SO(4) that is a cyclic extension of a dihedral group.
Here the symbol ∼ indicates homotopic loops based at q0. For the first homotopy, note that the paths g(η i ) and η i are homotopic rel endpoints because N is simply connected.
By Hurewicz, π3( N ) ∼ = H3( N ). Since π1(N ) is infinite, N is noncompact, so H3( N ) = 0. Similarly, all higher homotopy groups vanish by Hurewicz's theorem.
It's possible that the fixed set is larger (this is true for L(2, 1) ∼ = RP 3 ), but this is not important.
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Involutions on S 1 × S 2 and other 3-manifolds. J L Tollefson, Trans. Amer. Math. Soc. 183J. L. Tollefson. Involutions on S 1 × S 2 and other 3-manifolds. Trans. Amer. Math. Soc., 183:139- 152, 1973.
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4176 Campus Drive, College Park, MD 20742. Lei Chen, 151 Thayer St., Providence, RI2912Department of Mathematics, University of Maryland ; Department of Mathematics, Brown [email protected] Bena Tshishiku. bena [email protected] Chen, Department of Mathematics, University of Maryland, 4176 Campus Drive, Col- lege Park, MD 20742, [email protected] Bena Tshishiku, Department of Mathematics, Brown University, 151 Thayer St., Provi- dence, RI, 02912, bena [email protected].
|
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[
"Surface Enhanced Raman Spectroscopy of Graphene",
"Surface Enhanced Raman Spectroscopy of Graphene"
] |
[
"F Schedin \nDepartment of Physics and Astronomy\nManchester University\nManchesterUK\n",
"E Lidorikis \nDepartment of Materials Science and Engineering\nUniversity of Ioannina\nIoanninaGreece\n",
"A Lombardo \nDepartment of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUK\n",
"V G Kravets \nDepartment of Physics and Astronomy\nManchester University\nManchesterUK\n",
"A K Geim \nDepartment of Physics and Astronomy\nManchester University\nManchesterUK\n",
"A N Grigorenko \nDepartment of Physics and Astronomy\nManchester University\nManchesterUK\n",
"K S Novoselov \nDepartment of Physics and Astronomy\nManchester University\nManchesterUK\n",
"A C Ferrari \nDepartment of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUK\n"
] |
[
"Department of Physics and Astronomy\nManchester University\nManchesterUK",
"Department of Materials Science and Engineering\nUniversity of Ioannina\nIoanninaGreece",
"Department of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUK",
"Department of Physics and Astronomy\nManchester University\nManchesterUK",
"Department of Physics and Astronomy\nManchester University\nManchesterUK",
"Department of Physics and Astronomy\nManchester University\nManchesterUK",
"Department of Physics and Astronomy\nManchester University\nManchesterUK",
"Department of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUK"
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[] |
Surface enhanced Raman scattering (SERS) exploits surface plasmons induced by the incident field in metallic nanostructures to significantly increase the Raman intensity. Graphene provides the ideal prototype two dimensional (2d) test material to investigate SERS. Its Raman spectrum is well known, graphene samples are entirely reproducible, height controllable down to the atomic scale, and can be made virtually defect-free. We report SERS from graphene, by depositing arrays of Au particles of well defined dimensions on graphene/SiO2(300nm)/Si. We detect significant enhancements at 633nm. To elucidate the physics of SERS, we develop a quantitative analytical and numerical theory. The 2d nature of graphene allows for a closed-form description of the Raman enhancement. This scales with the nanoparticle cross section, the fourth power of the Mie enhancement, and is inversely proportional to the tenth power of the separation between graphene and the nanoparticle. One consequence is that metallic nanodisks are an ideal embodiment for SERS in 2d.
|
10.1016/j.molstruc.2013.03.011
|
[
"https://arxiv.org/pdf/1005.3268v1.pdf"
] | 118,516,263 |
1005.3268
|
b02374964c647edd668a19155a403a6e515330c2
|
Surface Enhanced Raman Spectroscopy of Graphene
18 May 2010
F Schedin
Department of Physics and Astronomy
Manchester University
ManchesterUK
E Lidorikis
Department of Materials Science and Engineering
University of Ioannina
IoanninaGreece
A Lombardo
Department of Engineering
University of Cambridge
CB3 0FACambridgeUK
V G Kravets
Department of Physics and Astronomy
Manchester University
ManchesterUK
A K Geim
Department of Physics and Astronomy
Manchester University
ManchesterUK
A N Grigorenko
Department of Physics and Astronomy
Manchester University
ManchesterUK
K S Novoselov
Department of Physics and Astronomy
Manchester University
ManchesterUK
A C Ferrari
Department of Engineering
University of Cambridge
CB3 0FACambridgeUK
Surface Enhanced Raman Spectroscopy of Graphene
18 May 2010arXiv:1005.3268v1 [cond-mat.mes-hall]
Surface enhanced Raman scattering (SERS) exploits surface plasmons induced by the incident field in metallic nanostructures to significantly increase the Raman intensity. Graphene provides the ideal prototype two dimensional (2d) test material to investigate SERS. Its Raman spectrum is well known, graphene samples are entirely reproducible, height controllable down to the atomic scale, and can be made virtually defect-free. We report SERS from graphene, by depositing arrays of Au particles of well defined dimensions on graphene/SiO2(300nm)/Si. We detect significant enhancements at 633nm. To elucidate the physics of SERS, we develop a quantitative analytical and numerical theory. The 2d nature of graphene allows for a closed-form description of the Raman enhancement. This scales with the nanoparticle cross section, the fourth power of the Mie enhancement, and is inversely proportional to the tenth power of the separation between graphene and the nanoparticle. One consequence is that metallic nanodisks are an ideal embodiment for SERS in 2d.
INTRODUCTION
Graphene is at the center of a significant research effort [1][2][3][4][5][6]. Near-ballistic transport at room temperature and high mobility [5][6][7][8][9][10][11] make it interesting for nanoelectronics [12][13][14][15], especially for high frequency applications [16]. Furthermore, its transparency and mechanical properties are ideal for micro and nanomechanical systems, thin-film transistors, transparent and conductive electrodes, and photonics [17][18][19][20][21][22][23][24]. Graphene is also an ideal test-bed for some long-standing problems, such as the Raman spectra of carbon materials. Here we show that this conceptually simple material (due to its low-dimensionality) can be helpful in understanding the basics of Surface Enhanced Raman Scattering (SERS).
Graphene layers can be identified by inelastic [25] and elastic light scattering [26,27]. Raman spectroscopy allows monitoring of doping, defects, strain, disorder, chemical modifications and edges [25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41]. The Raman signal can be enhanced for flakes deposited on certain substrates, such as the common Si+SiO 2 , due to interference in the SiO 2 layer, resulting into enhanced field amplitudes within graphene [42][43][44][45].
Another way to increase the Raman signal is to perform SERS experiments [46,47]. SERS is widely used [48][49][50], and enhancements as large as 14 orders of magnitude can be achieved (enough for single-molecule detection [51]). However, even though the technique is more than 30 years old [50], the exact nature of SERS is still debated [48]. Furthermore, the particular mechanism might be different, depending on whether the Raman processes involved are resonant or not. In principle, even a single metallic nanostructure, e.g., a metallic nanotip, can induce SERS at its apex, giving rise to the so-called tip-enhanced Raman scattering (TERS) [52][53][54]. The most important feature that makes TERS so attractive is its capability of optical sensing with high spatial resolution beyond the light diffraction limits [55,56].
Most SERS-active systems studied to-date are based on random nanostructures, whose properties vary from experiment to experiment making quantitative comparison between theory and experiment difficult. Graphene offers a unique model system where SERS effects could be studied in detail. Its Raman spectrum is well known, being investigated in several hundreds papers in the past 4 years. Graphene samples are very reproducible and offer an atomic-precision control on the number of layers, thus allowing a smooth transition from a purely 2d case to a 3d one. Furthermore, as both resonant and non-resonant Raman scattering can be in principle possible, such as in chemically modified graphene [33], a distinction between different enhancement mechanisms could be made. Here we focus on the resonant case, where we believe the enhancement is mostly due to near-field plasmonic effects in the vicinity of metal particles [46,52].
EXPERIMENTAL
Graphene flakes are prepared on Si+300nm SiO 2 by micromechanical cleavage[1]. Single layer graphene (SLG) is identified by a combination of optical contrast [26,27] and Raman spectroscopy [25]. Electron beam lithography in combination with thin metallic film deposition (5nm Cr+ 80nm Au) and lift-off are utilized to prepare three sets of metallic dots, as well as a set of contacts for transport measurements, Fig 1. One set is placed directly on top of graphene, one partially covers graphene and partially rests on SiO 2 , and the last completely on SiO 2 . The dots sizes and the configurations of the arrays can be seen on Fig.1. During lift-off, metallic dots are slightly shifted from their lithographically defined positions, probably by capillary forces. This indicates poor adhesion of Cr/Au dots onto graphene. Note that the dots on SiO 2 still occupy the positions defined by the lithography procedure.
Raman spectra are recorded with a Renishaw RM1000 spectrometer, equipped with a piezoelectric stage (PI) able to shift the sample at nanometer steps. Line scans are recorded across the patterned arrays, as shown in Figs 2,3, for 488, 514 and 633nm excitation.
RESULTS AND DISCUSSION
The Raman enhancement is defined as the ratio of the Raman intensity measured on the graphene covered by dots, compared to that measured outside the dots, but still on the graphene layer, Fig. 2. Fig. 3 shows representative Raman spectra measured at 633nm. A clear enhancement is seen when comparing the patterned and the unpatterned graphene.
The Raman spectrum of graphene consists of a set of distinct peaks. The G and D appear around 1580 and 1350 cm −1 , respectively. The G peak corresponds to the E 2g phonon at the Brillouin zone center (Γ point). The D peak is due to the breathing modes of six-atom rings and requires a defect for its activation [40,57,58]. It comes from TO phonons around the K point [40,58], is active by double resonance (DR) [57], and is strongly dispersive with excitation energy due to a Kohn Anomaly at K [31]. DR can also happen as intra-valley process, i. e. connecting two points belonging to the same cone around K (or K ′ ). This gives the so-called D'peak, which is at∼ 1620 cm −1 in defected graphite measured at 514nm. The 2D peak is the second order of the D peak. This is a single peak in single layer graphene (SLG), whereas it splits in four in bilayer graphene (BLG), reflecting the evolution of the band structure [25]. The 2D' peak is the second order of D'. Since 2D and 2D' originate from a process where momentum conservation is satisfied by two phonons with opposite wavevectors, no defects are required for their activation, and are thus always present. Each Raman peak is characterized by its position, width, height, and area. The frequency-integrated area under each peak represents the probability of the whole process.
It is more robust with respect to various perturbations of the phonon states than width and height [59]. Indeed, for an ideal case of dispersionless undamped phonons with frequency ω ph , the shape of the n-phonon peak is a Dirac δ distribution ∝ δ(ω − nω ph ), with zero width, in- finite height, but well-defined area. If the phonons decay (e. g, into other phonons, due to anharmonicity, or into electron-hole pairs, due to electron-phonon coupling), the δ lineshape broadens into a Lorentzian, but the area is preserved, as the total number of phonon states cannot be changed by such perturbations. If phonons have a weak dispersion, states with different momenta contribute at slightly different frequencies. This may result in an overall shift and a non-trivial peak shape, but frequency integration across the peak means counting all phonon states, as in the dispersionless case. Thus, the peak area is preserved, as long as the Raman matrix element itself is not changed significantly by the perturbation. The latter holds when the perturbation (phonon broadening or dispersion) is smaller than the typical energy scale determining the matrix element. Converting this into a time scale using the uncertainty principle we have that, if the Raman process is faster than the phonon decay, the total number of photons emitted within a given peak (i. e., integrated over frequency across the peak), is not affected by phonon decay, although their spectral distribution can be. Even if the graphene phonons giving rise to the D and D' peaks are dispersive [31], their relative change with respect to the average phonon energy is at most a few %, thus we are in the weakly dispersive case. The phonon decay in graphene is in the ps timescale, while the Raman process is faster, in the fs timescale [30,60,61]. We thus consider both the area, A(2D)/A(G), and height, I(2D)/I(G), ratios [59]. We model our experiment with the calculation box shown in Fig. 5. Starting from the bottom, this consists of a semi-infinite Si substrate, a 300nm SiO 2 and SLG of effective thickness 0.335nm. On SLG, we have a Au/Cr disk with thickness 80nm/5nm, with the diameter set to either 140 or 210nm, according to the experimental Au dot size. We time-integrate Maxwell's equations using the finite-difference time-domain method(FDTD) [62] as implemented in Refs [63,64] (see Methods). For the absorbing boundary conditions in the vertical direction we use the perfectly-matched-layer method (PML) [65], while in the lateral directions we use periodic boundary conditions simulating an infinite two-dimensional square array of Au/Cr nanodisks. We previously investigated the plasmonic resonances of similar nanoparticles (prepared in exactly the same conditions as in the present work) [66,67]. This allowed us to extract the Au and Cr optical constants as obtained in our evaporators [68]. We also recently measured the optical constants of graphene by spectroscopic ellipsometry [69]. The dispersive materials Au, Cr and graphene are described here by Drude-Lorentz models, each fitted to our experimental data. These are shown in Fig. 11 in Methods. Finally, for simplicity we use n=1.46 for SiO 2 , n=4 for Si [70].
We only consider normal incidence and emission, relevant for our Raman backscattering experiments. The incident field is a wide spectrum plane wave (i.e. a narrow Gaussian temporal profile) coming from the top. We monitor the electric fields E x (r, t) and E y (r, t) at each grid point on the graphene plane, and upon Fourier transform we get the tangential-field amplitude E (r, ω) in frequency domain. This is enhanced compared to the incident field, due to substrate interference, and the Surface Assuming the graphene absorption at a particular point to be proportional to the incident tangential field intensity at that point, the Raman emission from a particular point to be proportional to the corresponding Stokes-shifted intensity, and the emission from points underneath the nanodisks to be reabsorbed and lost, we may approximate the total Raman signal as [46]:
I SERS ∝ |E (r, ω)| 2 |E (r, ω s )| 2 dS ′ (1)
where the integration is performed over the area not directly underneath the nanodisks, and ω s = ω − δω is the Stokes shifted frequency. δω = 2πcν, with ν is the Raman shift (in cm −1 ). The outcome of Eq. 1 is normalized by the corresponding calculation for suspended graphene (where the integral is over all the area since there are no nondisks to cover the emission). We note that, given the large absorption of ∼2.3% per graphene layer [71], this approximation becomes questionable for local absorption enhancements greater than∼43. To amend this we cut above 43. Emission saturation, on the other hand, cannot be reached because the Raman efficiency, even though larger than other materials due to the process being always resonant, is in absolute terms very small [45], and would require ∼ 10 11 enhancement to exceed 100%. In order to account for the graphene layer thickness and still keep the simulation reasonably fast, we employ an anisotropic grid with 0.335nm spacing in the vertical and 2nm in the lateral dimension. By calculations on smaller cells we verified that such a grid introduces small errors, typically less than 5% and never exceeding 10%. Considering our approximations up to this point (normal incidence and the approximation in Eq. 1), we find this to be well within the overall simulation errors.
The 300nm SiO 2 /Si substrate interferometrically increases not only the visibility [26,27] but also the Raman signal of graphene [42,44,72]. Here we expect an additional enhancement due to the Au nanodisks SPR near field. In order to distinguish between the two effects (substrate interference and SPR), we separately calculate both cases for unpatterned SLG on SiO 2 /Si and SLG patterned with the Au nanodisks, still on SiO 2 /Si. We define the interference enhancement factor F as the ratio of the Raman signal from unpatterned SLG on SiO 2 /Si to that of suspended SLG, F = I unpatt /I susp , and the total enhancement factor F ′ as the ratio of the Raman signal from patterned SLG on SiO 2 /Si to that of suspended SLG: F ′ = I patt /I susp ≡ I SERS /I 0 . Fig. 6 plots such factors for the G and 2D peaks. This gives a maximum interference-related enhancement F at 550nm of∼2.5 for the G peak and ∼2 for the 2D. However, this is very modest compared to the total enhancement F ′ when the nanodisks are taken into account. The total enhancement reaches up to 50, and is maximum at different wavelengths depending on the nanodisk diameter. The shoulder at 550nm is likely related to interference, because (i) it is at the same frequency as the interference peak and (ii) it is stronger the further the plasmon peak.
There is a different enhancement for the two disk di- ameters, not only in peak value, but also in wavelength. This is a size effect: the nanodisks are comparable to the incident laser wavelength. Thus, retardation effects and higher order multipole terms become important, modifying the plasmon response as a function of disk size. Fig. 7 plots the distribution of the tangential intensity enhancement in the graphene plane at 633nm, which is approximately proportional to the absorption enhancement at that wavelength. Different patterns appear for the two sizes, demonstrating the point made above. Fig. 8 compares F ′ /F to the corresponding experimental Raman intensity ratios. Considering all approximations made, and the fact that some of the measurements where on top of distorted parts of the nanodisk arrays, the agreement is good. We find low enhancement for 488 and 514nm, and large for 633nm. Also, we reproduce the higher enhancement for the G peak in the small disks at 633nm, and for the 2D peak in the large disks. The quantitative agreement for the 2D peak is not as good, however, this is expected given that its intensity significantly depends on electron-electron interactions, which could change in presence of gold [59].
1 . 0 × 1 × 10 × 100 × |E || | 2 enhancement (a) (b)
The agreement between experiment and simulation is encouraging. It also allows to get new physical insights into the SERS process of an extended 2d system, like graphene. The basic physics and detailed theory of the electromagnetic contribution to SERS on adsorbed molecules is well known [46]: both absorption and emission are enhanced due to interaction with surface plasmons, with an expected overall dependence on the fourth power of the SPR-mediated field enhancement [46,73]. However, a detailed theory for 2d systems is lacking. Such formulation is challenging for our experiment: the Au particles do not have a regular spherical or ellipsoidal shape, they are large and so multipoles higher than dipole contribute, there is a thin Cr layer, we are on SiO 2 /Si giving additional interference and enhancement effects. We thus consider the simplified case of regular-shaped small Au nanoparticles inside a uniform medium. This will provide all the new physics for SERS in graphene, or generally any 2d system. The final connection between experiment and theory still relies on simulations, the only versatile tool that can move across different scales and experimental embodiments. The generic theoretical system under study is depicted in Fig. 9: at normal incidence, a plane wave of frequency ω excites a point dipole in the nanoparticle:
p ∝ α np (ω)(2)
where the polarizability a np is described by the Mie theory [74,75]. The poles of α np define the optimal SERS frequencies. The re-radiated near-field from this dipole scales as r −3 , and is responsible for the enhanced absorption. This will excite a Raman dipole:
p ′ ∝ α R (ω s , ω)r −3 α np (ω)(3)
where α R (ω s , ω) is the Raman polarizability and ω s the Stokes-shifted emission frequency. This dipole near-field will in turn excite a secondary dipole in the nanoparticle:
p ′′ ∝ α np (ω s )r −3 α R (ω s , ω)r −3 α np (ω)(4)
now at the emission frequency ω s . Thus, the additional surface-enhanced Raman signal is:
∆I SERS ∝ ω 4 s |p ′′ | 2 dS(5)
where the integration is over the SLG area. Assuming a square array of nanoparticles with spacing L and normalizing to the corresponding Raman signal in the absence of the nanoparticles, we get the SERS enhancement:
∆I SERS I 0 ≈ 1 9 σQ(ω) 2 Q(ω s ) 2 a h 10(6)
where h is the separation between the nanoparticle center and the SLG plane, σ = πa 2 /L 2 is the relative cross sectional area covered by the nanoparticles, and Q(ω) = |α np (ω)| /4πa 3 is the the Mie enhancement. In with Eq. 6 (lines) for G peak enhancement. The nanoparticle radius a and elevation h in Eq. 6 are adjusted to match the simulation peak enhancement. These are 2a=118nm, h=32.2nm for the 140nm disks and 2a=130nm, h=38nm for the 210nm disks. Note that the total enhancement factor F ′ is just F ′ = ∆ISERS/I0 + 1.
general, Q depends on particle size and shape [75] (for spherical particles α np (ω) is given by Eq. 8 in Methods). In the limiting case of spherical particles with a ≪ λ the size dependence in the Mie enhancement is removed and
Q(ω) ∼ = |[ǫ(ω) − 1]/[ǫ(ω) + 2]|, where ǫ(ω) is the nanoparticle's dielectric function.
This is our main theoretical result: the Raman enhancement scales with the metallic nanoparticle cross section, with the fourth power of the Mie enhancement, and inversely with the tenth power of the separation between the nanoparticle centre and graphene.
Since Eq. 6 does not take into account multiple reflections in the substrate, we compare it with FDTD simulations of Au nanodisks suspended in air, shown in Fig. 10 for the G peak (where there is a small blueshift due to the smaller effective refractive index below the nanoparticles without a substrate). In the absence of an analytical expression for the polarizability of a nanodisk, we use the polarizability of a sphere, shown in Eq. 8 in Methods. To fit the simulation (symbols in Fig. 10) we adjust the radius a (which determines the peak wavelength) and separation h (which determines the peak value) in Eq. 6. As a result, the 140nm nanodisk is best represented by a sphere of diameter 2a=118nm elevated at h=32.2nm above the SLG plane, while for the 210nm nanodisk a sphere of diameter 2a=130nm at h=38nm is needed (lines in Fig. 10). These numbers are close to the actual values for the nanodisks. The agreement is good, considering the difference between disks and spheres, confirming that our theory captures the essential physics of SERS in 2d. Equation 6 is a valuable optimization/design tool. In particular, it identifies 3 steps to further improve SERS: 1) larger nanoparticle coverage σ, 2) larger Mie enhancement Q, 3) smaller nanoparticle-graphene separation h. The first is straightforward. The second is shape related, e.g., ellipsoids have a different Q, which, for certain orientations, is stronger and red-shifted compared to a sphere [75]. This also influences the third step since, e.g., a flat oblate spheroid has a smaller distance h between its center and graphene. Thus, SERS enhancements can be much larger for ellipsoids and disks. This can already be seen in the comparisons made in Fig. 10. There, in order to match the disk simulation results, the representative spheres had to be placed at an elevation h < a (i.e. so that they "cut" into the SGL plane). If we, however, re-evaluate Eq. 6 for the same representative spheres at h = a (i.e. so that they "rest" on the SLG plane), we lose more than two orders of magnitude enhancement. This is confirmed by detailed FDTD calculations.
Finally, we note that the analytical expressions derived in Methods can be extended to flakes of increasing number of layers, by vertical integration. Note, however, that if the number of layers is large reflection becomes important (it is 1% for 7 layers but exceeds 5% above 18 layers) and has to be taken into account.
CONCLUSIONS
We studied SERS in graphene patterned with a square array of Au nanodisks on SiO 2 (300nm)/Si. Significant enhancements were measured for both G and 2D bands at 633nm. Similar results were obtained for both disk sizes. Large-scale FDTD simulations reproduce well the experiments. To elucidate the physics of SERS in 2d, we derived analytic expressions, and showed that taking into account the SPR near-fields only, a simple closed-form expression is found, where the Raman enhancement scales with the nanoparticle cross section, the fourth power of the Mie enhancement, and inversely with the tenth power of the separation between the nanoparticle centre and graphene. This points to thin nanodisks to achieve the highest SERS for 2d systems like graphene.
Let us consider a generic configuration as in Fig. 9, comprising a Au spherical nanoparticle of radius a at a distance h from SLG, normal plane wave incidence, with field amplitude E 0 , frequency ω and polarization along x. The re-radiated fields from the nanoparticle are due to an induced electric dipole at its center:
p = ǫ 0 α np (ω)E ≈ ǫ 0 α np (ω)E 0x(7)
where E is the local field at the nanoparticle, modified from the incident one due to the presence of the graphene layer. The SLG normal incidence reflectance is almost zero [71] and, due to continuity, the local field can be taken approximately the same as the incident one. This is further corroborated from calculations of the depolarization matrix of single wall nanotubes, where for polarization along the nanotube axis, no depolarization is found [52,76]. The nanoparticle polarizability a np is described for a sphere in the Mie theory [74,75]:
α np (ω) = 6πi k 3 ·ñ ψ 1 (ñka)ψ ′ 1 (ka) − ψ 1 (ka)ψ ′ 1 (ñka) nψ 1 (ñka)ξ ′ 1 (ka) − ξ 1 (ka)ψ ′ 1 (ñka)(8)
where ψ 1 and ξ 1 are the Riccati-Bessel functions, k = ω/c andñ ≡ñ(ω) is the nanoparticle complex index of refraction. In general, the refractive index of metallic nanoparticles differs from the bulk value due the reduced free electron relaxation caused by electron surface scattering [63,77]. We have relatively large particles, where this correction is negligible. Thus we use the bulk Au refractive index. If the nanoparticle is much smaller than the wavelength, Eq. 8 simplifies [75]:
α np (ω) ≈ 4πa 3 ǫ(ω) − 1 ǫ(ω) + 2(9)
with ǫ(ω) =ñ 2 (ω) the nanoparticle dielectric function. The total field at at position r on SLG is [78]:
E t (r, ω) ≈ E 0 (r, ω)x + e ikr 4πǫ 0 k 2 rr × (p ×r) + 1 r 3 − ik r 2 [3r(r · p) − p](10)
The first term in the square bracket is the radiationfield, while the second is the near-field. They scale with distance and wavelength as (λ 2 r) −1 and r −3 , (λr 2 ) −1 respectively. Assuming the nanoparticle distance from SLG to be much smaller than the wavelength, r ≪ λ, and the resonant near-field much larger than the incident field E 0 , then the near-field r −3 has the dominant contribution. Its transverse component drives the SLG enhanced absorption. Simultaneously, the Raman emitted field also interacts with the Au particle SPR, further enhancing the total Raman emission. To fully explore both processes, we evaluate two quantities: 1) total absorption enhancement; 2) total Raman enhancement.
Absorption enhancement
The additional absorption in graphene is due to the enhanced near field. Absorption is defined as current×field [78], thus approximated as:
∆A = G 0 |E nf t | 2 dS(11)
where G 0 = e 2 /4 is graphene's dynamical (optical) sheet conductance [71,[79][80][81][82]. Combining Eqs. 7,10, the near-field transverse component of the driving field is:
|E nf t (ω)| = E 0 Q(ω)a 3 r −3 |[3r(r ·x) −x] |(12)
with Q(ω) = |α np (ω)| /4πa 3 . In the above we ignore cross terms between incident and re-radiated fields. This is justified when strong near-field enhancements are expected (as it should be for SERS), but less so when the near fields are of the same order as the incident field. From Fig. 9, x = r sin θ cos φ, y = r sin θ sin φ and z = r cos θ, while r = h/ cos θ. The integration surface element is ρdρdφ = h 2 sin θ cos −3 θdφdθ, where ρ = h tan θ is r's projection on the plane. Eq. 11 then becomes: (13) with f (θ, φ) = 9 sin 4 θ cos 2 φ − 6 sin 2 θ cos 2 φ + 1. Then:
∆A = G 0 E 2 0 Q 2 a 6 h 4 π 2 0 2π 0 cos 3 θ sin θf (θ, φ)dφdθ∆A = 3πG 0 E 2 0 Q 2 (ω)a 6 8h 4(14)
For nanoparticles arranged on a square lattice with spacing L, the absorption enhancement becomes:
∆A A 0 = 3πQ 2 (ω)a 6 8h 4 L 2 = 3 8 σQ 2 (ω) a h 4(15)
where σ = πa 2 /L 2 is the nanoparticle relative cross section. For spheres directly placed on SLG, i.e. h = a, Eq. 15 simplifies to:
∆A A 0 = 3 8 σQ 2 (ω)(16)
We remind here that in the a ≪ λ limit, the Mie enhancement is Q(ω) = |[ǫ(ω) − 1]/[ǫ(ω) + 2]|.
Raman enhancement
Going back to Eq. 12, E nf t (ω) will excite a dipole field on SLG at the Raman frequency ω s :
p ′ = α R (ω s , ω)|E nf t |p ′(17)
The polarization of the Raman dipole is not necessarily the same as that of the driving field. Thus, for generality we assume this to be randomly polarized on the SLG plane. In this case it suffices to take the average of two dipoles, one polarized alongx, and another alongŷ. The one alongx will emit as the dipole term of Eq. 10. We are again interested in the dominant near-field term that decays as r −3 . This will get coupled to the nanoparticle and thus SPR enhanced. That is, it will excite a secondary dipole at the nanoparticle:
p ′′ = Q(ω s )α R (ω s , ω)Q(ω)E 0 a 6 r −6 |[3r(r·x)−x] | 2 (18)
where we again consider the projection of the dipole given the backscattering geometry considered here. The radiated flux can be taken as the additional surface-enhanced Raman signal:
∆I SERS = ck 4 s 24πǫ 0 p ′′2 dS(19)
where k s = ω s /c and we multiply by a factor 1/2 since we only consider the upper half flux. Using the angular relationships for x, y, z and dS we get:
∆I SERS = ck 4 s E 2 0 Q 2 (ω)Q 2 (ω s )a 12 24πǫ 0 h 10 |α R (ω s , ω)| 2 × π 2 0 2π 0 cos 9 θ sin θf x (θ, φ)dφdθ(20)
where f x (θ, φ) = 81 sin 8 θ cos 4 φ − 108 sin 6 θ cos 4 φ + 18 sin 4 θ cos 2 φ(1 + 2 cos 2 φ) − 12 sin 2 θ cos 2 φ + 1. The angular integration yields 33π/280. For the calculation with the Raman dipole alongŷ, the angular part of Eq. 18 becomes |[3r(r ·x) −x] ||[3r(r ·ŷ) −ŷ] | with a slightly different f y (θ, φ), but a similar angular integration value of 27π/280. The average angular contribution is 3π/28 ≈ π/9. Thus the additional Raman signal can thus be written as:
∆I SERS ≈ π 9 ck 4 s E 2 0 Q 2 (ω)Q 2 (ω s )a 12 24πǫ 0 h 10 |α R (ω s , ω)| 2 (21)
To evaluate the enhancement factor, we normalize to the expected signal I 0 in the absence of the nanoparticles. Considering a square unit cell of side equal to the nanoparticle spacing L, we get:
I 0 ≈ L 2 ck 4 s E 2 0 24πǫ 0 |α R (ω s , ω)| 2(22)
The Raman enhancement factor is ∆I SERS I 0 ≈ 1 9 σQ 2 (ω)Q 2 (ω s ) a h 10 (23)
The Finite-Difference Time-Domain Method
In the FDTD method, Maxwell's equations are timeintegrated on a computational grid:
∇ × E = −µ∂ t H (24) ∇ × H = ǫ 0 ǫ ∞ ∂ t E + ∂ t P 0 + N j=1 ∂ t P j(25)
where material polarization is taken into account through the polarizabilities P:
∂ 2 t P 0 + γ∂ t P 0 = ω 2 p ǫ 0 E(26)
∂ 2 t P j + Γ j ∂ t P j + Ω 2 j P j = ∆ǫ j Ω 2 j ǫ 0 E
This gives a Drude-Lorentz model for the dielectric function [83]:
ǫ(ω) = ǫ ∞ − ω 2 p ω 2 + iωγ + N j=1 ∆ǫ j Ω 2 j Ω 2 j − ω 2 − iωΓ j(28)
where the first term is the Drude free-electron contribution and the second contains Lorentz oscillators corresponding to interband transitions. ω p and 1/γ are the free electron plasma frequency and relaxation time, Ω j , ∆ǫ j , Γ m are transition frequency, oscillator strength and decay rate for the Lorentz terms. To accurately reproduce the experimental dielectric functions (Au and Cr from Ref. [68], SLG from Ref. [69]) we treat these as fit parameters. For Au we use N =4, and ǫ ∞ =3.454, ∆ǫ j =(0.376, 0.63, 1.208, 1.124), ω p =8.73eV, γ=0.046eV, Ω j =(2.72, 3.13, 3.88, 4.95)eV, and Γ j =(0.39, 0.655, 1.16, 1.67)eV. For Cr we use N =3, and ǫ ∞ =1, ∆ǫ j =(9.54, 15.5, 1.1), ω p =5.51eV, γ=0.731eV, Ω j =(1.43, 2.36, 3.64)eV, and Γ j =(1.19, 1.94, 1.41)eV. Finally, for SLG we use N =3, and ǫ ∞ =1.964, ∆ǫ j =(6.99, 1.69, 1.53), ω p =6.02eV, γ=4.52eV, Ω j =(3.14, 4.03, 4.59)eV, and Γ j =(7.99, 2.01, 0.88)eV. [68,69].
FIG. 1 :
1Scanning Electron Microscopy images (in false colours) of our SERS sample. False colors denote: purple -SiO2; bluish -graphene; yellow -Au electrodes and dots. a) An overall image of the sample. b,c) Golden dots on SiO2
FIG. 3 :
3Representative Raman spectra measured across a line scan moving from outside to inside the Au dot patterned area for 633 excitation. A clear enhancement of all peaks is seen
FIG. 4 :
4Ratio of height and areas of the G and 2D peaks measured on the patterned regions compared to those measured outside, as a function of the excitation energy
Fig 4 plots them for the two dot sizes and as a function of excitation energy.
FIG. 5 :
5Simulation box: 80nm/5nm Au/Cr nanodisks on SLG sitting on a SiO2/Si. The lateral periodicity is 320nm in both directions. We consider nanodisk diameters of 210 and 140nm, corresponding to the large and small dots Plasmon Resonance (SPR) near field of the nanodisks.
FIG. 6 :
6The total (patterned) enhancement factors for a) the G and b) 2D peaks. The dotted line is the corresponding interference (unpatterned) enhancement factor.
FIG. 7 :FIG. 8 :
78Tangential field intensity distribution at 633 nm for the a) 210 nm and b) 140 nm nanodisk diameters. Normalized enhancement factors F ′ /F for a) the Gband and b) the 2D-band. Squares and crosses are the experimental Raman intensities and areas.
FIG. 10: FDTD simulations of suspended nanodisks (symbols) compared with
FIG. 11 :
11Fig. 11 plots our model dielectric functions along with the experimental ones, showing an excellent agreement. * Electronic address: [email protected] † Electronic address: [email protected] [1] Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Science, 2004, 306, 666. [2] Geim, A. K.; Novoselov, K. S. Nature Mater. 2007, 6, 183. [3] Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. Rev. Mod. Phys. 2009, 81, 109. [4] Charlier, J. C.; Eklund, P.C.; Zhu, J.; Ferrari, A. C. Topics Appl. Phys. 2008, 111, 673. [5] Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature (London) 2005, 438, 197. The refractive indices used in the calculations for a) Au, b) Cr and c) SLG. Symbols are for the corresponding experimental data
ACKNOWLEDGEMENTS ACF acknowledges funding from the Royal Society, the ERC grant NANOPOTS, and EPSRC grant EP/G042357/1. The authors acknowledge computing time at the Research Center for Scientific Simulations (RCSS) at the University of Ioannina.METHODS
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[] |
[
"WELL-POSEDNESS OF TRANSONIC CHARACTERISTIC DISCONTINUITIES IN TWO-DIMENSIONAL STEADY COMPRESSIBLE EULER FLOWS",
"WELL-POSEDNESS OF TRANSONIC CHARACTERISTIC DISCONTINUITIES IN TWO-DIMENSIONAL STEADY COMPRESSIBLE EULER FLOWS"
] |
[
"Gui-Qiang Chen ",
"Vaibhav Kukreja ",
"Hairong Yuan "
] |
[] |
[] |
In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right-wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (characteristic discontinuity), the Euler equations are of elliptic-hyperbolic compositemixed type. In this paper, we further prove that such a transonic characteristic discontinuity solution is unique and L 1 -stable with respect to the small perturbation of the incoming supersonic flow in Lagrangian coordinates.
|
10.1007/s00033-013-0312-6
|
[
"https://arxiv.org/pdf/1209.3806v1.pdf"
] | 17,629,421 |
1209.3806
|
41ba5fda26936147f058759cfe99dd8a00871c65
|
WELL-POSEDNESS OF TRANSONIC CHARACTERISTIC DISCONTINUITIES IN TWO-DIMENSIONAL STEADY COMPRESSIBLE EULER FLOWS
17 Sep 2012
Gui-Qiang Chen
Vaibhav Kukreja
Hairong Yuan
WELL-POSEDNESS OF TRANSONIC CHARACTERISTIC DISCONTINUITIES IN TWO-DIMENSIONAL STEADY COMPRESSIBLE EULER FLOWS
17 Sep 2012
In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right-wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (characteristic discontinuity), the Euler equations are of elliptic-hyperbolic compositemixed type. In this paper, we further prove that such a transonic characteristic discontinuity solution is unique and L 1 -stable with respect to the small perturbation of the incoming supersonic flow in Lagrangian coordinates.
Introduction
We are concerned with the well-posedness of transonic characteristic discontinuities that separate supersonic flows from a static gas in two-dimensional steady compressible Euler flows. The governing equations are the following Euler system that consists of conservation laws of mass, momentum, and energy: where E = 1 2 (u 2 + v 2 ) + e. The unknowns ρ, p, e, and (u, v) represent the density, pressure, internal energy, and velocity of the fluid, respectively. Specifically, for a polytropic gas, the constitutive relations are
p = ρ γ exp S c ν , e = 1 γ − 1 p ρ , (1.5)
where S is the entropy, c ν is a positive constant, and γ > 1 is the adiabatic exponent. The sonic speed c is determined by
c = ∂p ∂ρ = γp ρ .
(1. 6) In our previous work [5], we have established the existence of a weak entropy solution to the following initial-boundary value problem:
(1.1) − (1.4), in x > 0, y > g(x), U = U 0 , on x = 0, y > 0, p = p, v u = g ′ (x), on x > 0, y = g(x),(1.7)
provided that the incoming supersonic flow U 0 is close in BV to a reference state U + , where U = (u, v, p, ρ), and U + = (u, 0, p, ρ + ) is a uniform supersonic flow; that is, it is a constant vector and u > c + = γp ρ + . The unknowns in problem (1.7) are the supersonic flow U = U (x, y) and the free boundary D U = {y = g(x), x ≥ 0}, which is a Lipschitz curve passing through the origin.
The free boundary is actually a transonic characteristic discontinuity (vortex sheet and/or entropy wave) separating the supersonic flow U from the unperturbed static gas U − = (0, 0, p, ρ − ) that is subsonic (cf. Figure 1). Thus, problem (1.7) is a free boundary problem (in the Euler coordinates) and, across the free boundary, the Euler equations are of elliptic-hyperbolic composite-mixed type. sheet/entropy wave is generated to separate the static gas below from the supersonic flow above.
To further study the L 1 stability, we run into a difficulty to define the L 1 -distance between two solutions, since two solutions U and V describing the supersonic flows are generically defined in different domains depending on the location of their respective free boundaries D U and D V : For some given "time" x * > 0 and at some point y * ∈ R, it may happen that, at (x * , y * ), it is a static state for one solution, but is a supersonic flow for the other. To this end, we employ the special structure of the Euler equations of the two-dimensional steady flows, and introduce the Lagrangian coordinates (ξ, η) associated to a solution, as done in the study of transonic shocks [8,11,16], to transform the free boundary to the positive ξ-axis, so that the solutions U and V are then defined in the same domain {ξ > 0, η > 0} in Lagrangian coordinates. Then the initial data functions U 0 (0, y)
and V 0 (0, y) are transformed to the corresponding initial data functions U 0 (0, η) and V 0 (0, η) in their respective Lagrange coordinates. Similar to the analysis in [5] for the construction of approximate solutions, we construct the approximate front tracking solutions U (ξ, η) and V (ξ, η) in the domain {ξ > 0, η > 0} with the initial data U 0 (0, η) and V 0 (0, η) on ξ = 0, and the boundary data p = p on η = 0 respectively, and study the L 1 -stability property of these solutions for the initial-boundary value problem (2.17) in Lagrangian coordinates, as proposed accurately below. With this merit of Lagrangian coordinates, we can show the characteristic discontinuity solutions are both unique for the given incoming supersonic flow in the Euler coordinates and L 1 -stable with respect to the small perturbations of the incoming supersonic flows in the Lagrangian coordinates.
In Section 2, we present the Lagrangian coordinates associated to a solution to problem (1.7) and formulate this problem in the Lagrangian coordinates as problem (2.17 ordinates. This is clarified in Section 6. The well-posedness results are summarized in Theorem 6.2. These results on the existence, uniqueness, and L 1 -stability in Lagrangian coordinates, together with the existence theorem established in [5], may be considered as a complete mathematical well-posedness theory of such transonic characteristic discontinuity solutions.
Formulation of the Problem in Lagrangian Coordinates
In this section we reformulate the problem in Eulerian coordinates to the problem in Lagrangian coordinates and show the existence of entropy solutions for the problem in Lagrangian coordinates.
2.1. Lagrangian coordinates. For a piecewise C 1 smooth flow, we may introduce the Lagrangian transformation as used in [9,16] to formulate problem (1.7) in Lagrangian coordinates, which enables us to straighten the streamlines and hence treat a strict hyperbolic system derived from
(1.1)-(1.4).
Let U ∈ L ∞ (R + ) and g ∈ Lip (R + ) be a piecewise C 1 weak entropy solution to problem (1.7).
Define
η = η(x, y; x 0 , y 0 ) = (x,y) (x 0 ,y 0 ) ρu(s, t) dt − ρv(s, t) ds,(2.1)
where (x 0 , y 0 ) is a fixed point on the transonic characteristic discontinuity D U , and the integration is on any smooth curve Γ connecting (x 0 , y 0 ) with (x, y) and lies in the upper side of D U . Since the domain {x > 0, y > g(x)} is simply connected, by the conservation of mass, as well as the Rankine-Hugoniot (R-H) conditions for crossing the shock-front, η is a well-defined function of (x, y) with y ≥ g(x), independent of the choice of Γ. Clearly, we have
∂η ∂x = −ρv, ∂η ∂y = ρu. (2.2)
We also note that, by the last condition in (1.7), η is independent of (x 0 , y 0 ) on D U . In fact, for (x 0 , y 0 ) and (x ′ 0 , y ′ 0 ) on the characteristic discontinuity, we have
η(x, y; x 0 , y 0 ) − η(x, y; x ′ 0 , y ′ 0 ) = (x ′ 0 ,y ′ 0 ) (x 0 ,y 0 ) ρu(s, t) dt − ρv(s, t) ds = x ′ 0 x 0 (ρu(s, g(s)) · g ′ (s) − ρv(s, g(s))) ds = 0. (2.3)
Since the characteristic discontinuity g(s) is in Lip (R + , R), it is differentiable almost everywhere, so that the integrand is discontinuous only at these points in a set of Lebesgue measure zero, which is harmless for the above calculation. Hence, in the following, we may write η = η(x, y) with η = 0 whenever y = g(x).
If u, v, and ρ belong to L ∞ , and (1.1) is satisfied in the sense of distributions, we can also find a unique Lipschitz continuous function η so that η(x 0 , y 0 ) = 0 and (2.2) holds. As a matter of fact, using standard mollification, we may approximate ρu and ρv in weak convergence of distributions by C ∞ functions (ρu) ǫ and (ρv) ǫ , for which the equality ∂ x (ρu) ǫ +∂ y (ρv) ǫ = 0 still holds. Furthermore, by the Young inequality, we have (ρu) ǫ L ∞ ≤ C ρu L ∞ and (ρv) ǫ L ∞ ≤ C ρv L ∞ . Then, as above, we can solve η ǫ ∈ C 1 so that η ǫ (x 0 , y 0 ) = 0 and hence, in any bounded domain, {η ǫ } is a family of C 1 functions that are uniformly bounded and equicontinuous. Thus, there is a subsequence {η k } that converges uniformly to some η; by taking a diagonal subsequence, we can find η that is defined in the whole domain and η k converges uniformly to η in any compact subregion, hence η k → η in D ′ . It is obvious that η(x 0 , y 0 ) = 0, and (2.2) holds by uniqueness of limits in the sense of distributions. This then ensures that η is Lipschitz continuous. The uniqueness follows from the well-known fact that a distribution with zero derivatives must be a constant. Differentiating η along a streamline (which is Lipschitz continuous) and using (2.2) yield that it is constant along the stream line. Now we introduce the following Lagrangian transformation (x, y) → (ξ, η):
(ξ, η) = (x, η(x, y)).
(2.4)
Then we have
∂(ξ, η) ∂(x, y) = 1 0 −ρv ρu , (2.5) thus ∂ x = ∂ ξ − ρv∂ η , ∂ y = ρu∂ η . (2.6)
This transform is Lipschitz continuous and one-to-one provided ρu > 0. A simple computation shows that equations (1.1)-(1.3) may be written in divergence form:
∂ ξ 1 ρu − ∂ η v u = 0, ∂ ξ (u + p ρu ) − ∂ η pv u = 0, ∂ ξ v + ∂ η p = 0, (2.7)
or, as a symmetric system for U = (u, v, p) ⊤ ,
A∂ ξ U + B∂ η U = 0, (2.8) with A = u 0 1 ρ 0 u 0 1 ρ 0 u ρ 2 c 2 , B = 0 0 −v 0 0 u −v u 0 .
(2.9)
For ρu = 0, the conservation law of energy becomes ∂ ξ ( u 2 +v 2 2 + c 2 γ−1 ) = 0, that is,
u 2 + v 2 2 + c 2 γ − 1 = b(η). (2.10)
This is the well-known Bernoulli law. As b(η) is given by the initial data, in the following we focus on system (2.7) with ρ determined by U = (u, v, p) through (2.10).
The eigenvalues λ of (2.8) (i.e. |λA − B| = 0) are
λ 1 = ρc 2 u u 2 − c 2 v u − M 2 − 1 , (2.11) λ 2 = 0, (2.12) λ 3 = ρc 2 u u 2 − c 2 v u + M 2 − 1 , (2.13) where M = √ u 2 +v 2 c
is the Mach number of the flow. Then, for u > c, system (2.8) is strictly hyperbolic. The associated right-eigenvectors are
r 1 = κ 1 ( λ 1 ρ + v, −u, −λ 1 u) ⊤ ,(2.
14)
r 2 = (u, v, 0) ⊤ , (2.15) r 3 = κ 3 ( λ 3 ρ + v, −u, −λ 3 u) ⊤ ,(2.16)
where κ j can be chosen that r j · ∇λ j ≡ 1, since the jth-characteristics fields, j = 1, 3, are genuinely nonlinear. Note that the second characteristic field is always linearly degenerate: r 2 · ∇λ 2 = 0.
2.2.
Formulation of the problem in Lagrangian coordinates. As we noted that the transonic characteristic discontinuity becomes the positive ξ-axis in Lagrangian coordinates, we formulate the problem in Eulerian coordinates into the following initial-boundary value problem for equations (2.7):
(2.7) in ξ > 0, η > 0, U (0, η) = U 0 (η) on ξ = 0, η > 0, p = p on ξ > 0, η = 0.
(2.17)
Once we solved U from this problem, we then obtain the free boundary in Eulerian coordinates
g(x) = x 0 v u (ξ, 0) dξ. (2.18)
These are the corresponding forms in Lagrangian coordinates for problem (1.7).
2.3.
Existence of entropy solutions. The entropy solutions of problem (2.17) can be defined in the standard way via integration by parts. We note that the existence of entropy solutions to problem (2.17) can be constructed easily by using the front tracking method as carried out in [5] provided that U 0 − U + BV(R + ) is sufficiently small. In particular, the following lateral Riemann problem with the boundary data p = p on the characteristic boundary {η = 0} is uniquely solvable.
Lemma 2.1. Consider the following lateral Riemann problem:
(2.7) in ξ > 0, η > 0, U = U + on ξ = 0, η > 0, p = p on ξ > 0, η = 0.
There exists ε > 0 so that, if U + lies in the ball O ε (U + ) with center U + and radius ε, then there is a unique admissible solution that contains only a 3-wave.
Proof. 1. Note that system (2.7) is strictly hyperbolic for u > c. For each point U with u > c, in its small neighborhood, we can obtain
C 2 -wave curves Φ j (α; U ), j = 1, 2, 3, so that Φ j (0; U ) = U and dΦ j dα | α=0 = r j (U ): Φ j (α; U )
is connected to U from the upper side by a simple wave of j-family with strength |α|; for α > 0 and j = 1, 3, this wave is a rarefaction wave, while, for α < 0 and j = 1, 3, this wave is a shock. For j = 2, the wave is always a characteristic discontinuity.
For our purpose, we note that there is also a C 2 -curve Ψ 3 (β; U ) which consists of those states that can be connected to U from the lower side by a 3-wave of strength β. We have Ψ 3 (0; U ) = U and dΦ j dβ | β=0 = −r 3 (U ). 2. We set U = (u, v, p) ⊤ in Lagrangian coordinates and use U 3 to represent the third argument of the vector U . Then, to solve the lateral Riemann problem, it suffices to show that there exists a unique β so that (Ψ 3 (β; U + )) 3 = p. Therefore, we consider the following function:
L(β; U + ) = Ψ 3 (β; U + )) 3 − Ψ 3 (0; U + )) 3 .
It is clear that L(0; U + ) = 0, and
∂L(0; U + ) ∂β = −r 3 (U + ) 3 = κ 3λ3 u = 0.
By the implicit function theorem, there exists ε > 0 such that, for U + ∈ O ε (U + ), there is a function β = β(U + ) so that L(β(U + ); U + ) = 0. Then, similar to Lemma 2.2 in [5], by using the Taylor expansion up to second order (recall that Ψ j is C 2 ), we obtain the following estimate:
β = K|p + − p| + O(1)|U + − U + | 2 ,(2.U m = Φ 1 (α 1 ; U l ) = Φ 3 (α 3 ; U r ).
Then
α 3 = −K 2 α 1 + M 2 |α 1 | 2 , (2.20) with the constant K 2 > 0 and the quantity M 2 bounded in O ǫ (U + ). Furthermore, for U l = (u l , v l , p l ), |K 2 | > 1, |K 2 | < 1, and |K 2 | = 1 when v l < 0, v l > 0, and v l = 0, respectively.
The proof here follows the similar argument for Lemma 2.4 in [5].
For our current problem, we have to modify slightly the Glimm functional as done in [5]. In particular, this will allow us to show later that the Lyapunov functional Φ (see Section 3) decreases when a weak wave of 1-family is reflected off the boundary. We now introduce the following version of the Glimm functional:
G(ξ) = V(ξ) + κQ(ξ),
where κ > 0 is a large constant to be chosen. The terms V and Q are explained below:
•The weighted strength term V(ξ). We define the total (weighted) strength of weak waves in
U δ (ξ, ·) as V(ξ) = α |b α |. (2.21)
Here, for a weak wave α of i α -family, we define its weighted strength as
b α = k + α if i α = 1, α if i α = 2, 3,
where k + > K 2 is a positive constant which has been fixed by considering the reflection coefficient • The interaction potential term Q(ξ). The interaction potential term we used here is a modification to the one introduced by Glimm, that is,
Q(ξ) = (α,β)∈A |b α b β | + β∈A b |b β | = Q A + Q b , (2.22)
where the set A (ξ) is of all the couples (b α , b β ) of approaching wave-fronts. The term Q b in our wave interaction potential is an additional term, in comparison with the Cauchy problem; and if a weak wave α of 1-family is approaching the boundary, we write α ∈ A b . Now, for given initial data U 0 , we may construct an approximate solution U δ by a front tracking algorithm introduced by [12], where δ is a small parameter measuring the accuracy of the solution, which controls the following four types of errors generated by the algorithm:
• Errors in the approximation of initial data;
• Errors in the speeds of shock, characteristic discontinuities (vortex sheet and entropy wave), and rarefaction fronts;
• Errors by approximating the rarefaction waves by piecewise constant rarefaction fronts;
• Errors from removing all the fronts with generation higher than N ∈ Z + (N depends on δ).
The construction of a Glimm functional as above provides the necessary uniform estimates that guarantee the existence of a subsequence of U δ which converges to a bounded entropy solution of
(2.17) in C([0, T ]; L 1 (R + )) for any T > 0.
The following sections are devoted to the L 1 -stability and uniqueness properties for problem (2.17).
A Lyapunov functional and L 1 stability
This section is devoted to establishing the L 1 -stability of two entropy solutions U and V of problem (2.17) with respective initial data U 0 and V 0 obtained by the front tracking method:
U (ξ, ·) − V (ξ, ·) L 1 (R + ) ≤ C U 0 − V 0 L 1 (R + ) , (3.1)
with a constant C depending only on the equations and the reference state U + . To this end,
following an idea in [4], we introduce a Lyapunov functional Φ(U δ 1 , V δ 2 ) by incorporating additional new waves generated from the weak wave interactions with the characteristic boundary {η = 0}, which is equivalent to the L 1 (R + )-distance:
C −1 1 U (ξ, ·) − V (ξ, ·) L 1 ≤ Φ(U, V ) ≤ C 1 U (ξ, ·) − V (ξ, ·) L 1 (3.2)
and decreases as ξ increases:
Φ(U δ 1 (ξ 2 , ·), V δ 2 (ξ 2 , ·)) − Φ(U δ 1 (ξ 1 , ·), V δ 2 (ξ 1 , ·)) ≤ C 2 max(δ 1 , δ 2 )(ξ 2 − ξ 1 ), ∀ ξ 2 > ξ 1 > 0, (3.3)
for some constants C i , i = 1, 2, where U δ 1 and V δ 2 are two approximate solutions corresponding to the initial data U 0 and V 0 obtained by the wave-front tracking, with accuracy δ 1 and δ 2 , respectively.
Estimate (3.3) has many important implications. Firstly, taking U 0 = V 0 and ξ 1 = 0 in (3.3) and using (3.2), we obtain lim δ 1 ,δ 2 →0 U δ 1 (ξ, ·) − U δ 2 (ξ, ·) L 1 (R + ) = 0. Thus, the whole sequence of approximate solutions {U δ } converges to the same limit. Secondly, taking directly δ 1 , δ 2 → 0 yields (3.1). This again implies that the entropy solution of problem (2.17) obtained by the front tracking method is unique.
In the following, we first define specifically the Lyapunov functional and then verify (3.2) and (3.3).
3.1. Definition of the Lyapunov functional and its equivalence to the L 1 -distance. Similar to [4,13,14], when ξ is fixed, for each η ≥ 0, we solve a "Riemann problem" with the below-state U (ξ, η), the upper-state V (ξ, η), by moving along the Hugoniot curves S 1 , C 2 , and S 3 of system (2.7) in the phase space. We use h i (ξ, η) to denote the strength of the i-Hugoniot wave in this "Riemann solver", which is totally determined by U (ξ, η) and V (ξ, η). As in the standard Riemann
solver, 3 i=1 |h i (ξ, η)| is equivalent to |U (ξ, η) − V (ξ, η)|,
with a constant depending only on U + . Now we define the weighted L 1 -strengths:
q i (ξ, η) = c a i h i (ξ, η),(3.4)
where the constants c a i are to be determined later, depending on the estimates on the wave interactions and reflections off the characteristic boundary {η = 0} (as done in Section 2).
Then we define the Lyapunov functional:
Φ(U (ξ, ·), V (ξ, ·)) = 3 i=1 ∞ 0 |q i (ξ, η)|W i (ξ, η) dη, (3.5)
with the weight
W i (ξ, η) = 1 + κ 1 A i (ξ, η) + κ 2 Q(U )(ξ) + Q(V )(ξ) . (3.6)
Here κ 1 and κ 2 are two constants to be specified later; Q(U )(ξ) and Q(V )(ξ) are the Glimm's total wave interaction potentials in U and V at "time" ξ, respectively; A i (ξ, η) is the total strength of waves in U and V which approach the i-wave q i (ξ, η) at "time" ξ, defined by
A i (ξ, η) = B i (ξ, η) + C i (ξ, η) (3.7)
and
B i = α∈J (U )∪J (V ) ηα<η,i<kα≤3 + α∈J (U )∪J (V ) ηα>η,1≤kα<i
|α|,
C i = ( α∈J (U ),ηα<η,kα=i + α∈J (V ),ηα>η,kα=i )|α| if q i (η) < 0, ( α∈J (V ),ηα<η,kα=i + α∈J (U ),ηα>η,kα=i )|α| if q i (η) > 0,
where J (U ) and J (V ) are the sets of fronts at "time" ξ in U and V , respectively, η α is the position of the front α, and k α is the characteristic family associated to the front α. In the following, we sometimes drop the dependence of ξ and write simply q i (η), W i (η), A i (η), etc. when no confusion arises.
Since, for any U (ξ, ·), V (ξ, ·) ∈ BV with T V U (·) + T V V (·) sufficiently small (depending only on κ 1 , κ 2 , and c a i ), we have
C −1 0 U (ξ, ·) − V (ξ, ·) L 1 (R + ) ≤ 3 i=1 ∞ 0 |q i (η)| dη ≤ C 0 U (ξ, ·) − V (ξ, ·) L 1 (R + ) , 1 ≤ W i (η) ≤ 2, i = 1, 2, 3,
for some constant C 0 depending essentially only on the system and U + . Therefore, for any ξ ≥ 0,
C −1 1 U (ξ, ·) − V (ξ, ·) L 1 (R + ) ≤ Φ(U, V ) ≤ C 1 U (ξ, ·) − V (ξ, ·) L 1 (R + ) ,(3.8)
where C 1 is independent of ξ, U , and V .
3.2.
Decreasing of the Lyapunov functional. From now on, we examine how the Lyapunov functional Φ(U δ 1 (ξ, ·), V δ 2 (ξ, ·)) evolves in the flow direction ξ > 0, where U δ 1 and V δ 2 are approximate solutions obtained by the front tracking method from the initial data U 0 and V 0 , respectively, and we also set δ = max(δ 1 , δ 2 ) to control the errors. For simplicity, in the following, we also write U δ 1 and V δ 2 as U and V , respectively.
Denote λ i (η) the speed of the i-wave q i (η) along the Hugoniot curve in the phase space (it is not the characteristic speed except for the characteristic discontinuity). At ξ > 0 which is not an
interaction "time" of the waves either in U or V , for J = J (U ) ∪ J (V ) (it is a finite set), it holds that d dξ Φ (U (ξ), V (ξ)) = α∈J 3 i=1 q i (η − α ) W i (η − α ) − q i (η + α ) W i (η + α ) η α + 3 i=1 |q i (b)| W i (b)η b = α∈J 3 i=1 q i (η − α ) W i (η − α ) η α − λ i (η − α ) − q i (η + α ) W i (η + α ) η α − λ i (η + α ) + 3 i=1 |q i (b)| W i (b) η b + λ i (b) ,
whereη α is the speed of the Hugoniot wave α ∈ J , b = 0 + stands for the points near the boundary η = 0, andη b is the slope of the boundary, which is actually zero. For the second equality, we used the fact that, when U 0 − V 0 ∈ L 1 (R + ), if α ′ is the uppermost front in J , then λ i (η + α ′ ) = 0 for i = 1, 2, 3. Also, suppose that the front β ′ ∈ J is the closest to η = 0 so that λ
i (b) = λ i (η − β ′ ). Define E α,i = q + i W + i λ + i −η α − q − i W − i λ − i −η α , (3.9) E b,i = |q i (b)| W i (b) η b + λ i (b) , (3.10) where q ± i = q i (η ± α ), W ± i = W i (η ± α ), and λ ± i = λ i (η ± α ). Then d dξ Φ (U (ξ), V (ξ)) = α∈J 3 i=1 E α,i + 3 i=1 E b,i . (3.11)
Our main goal is to establish the following bounds:
3 i=1 E b,i ≤ 0 (near the boundary), (3.12) 3 i=1 E α,i ≤ O(1)δ |α| (when α is a weak wave in J ). (3.13)
If these are established, from (3.12)-(3.13), we conclude
d dξ Φ (U (ξ), V (ξ)) ≤ O(1)δ. (3.14)
If the constant κ 2 in the Lyapunov functional is chosen large enough, by the Glimm interaction estimates, all the weight functions W i (η) decrease at each "time" where two fronts of U or two fronts of V interact. By the self-similarity of the Riemann solutions, Φ decreases at this "time".
Integrating (3.14) over the interval [0, ξ], we obtain
Φ (U (ξ), V (ξ)) ≤ Φ (U (0), V (0)) + O(1)δξ (3.15)
as desired. Actually, (3.13) has been proved in [4] or [12] via a case by case analysis.
We also need to consider what happens to the Lyapunov functional Φ if a weak wave reflected off the boundary. Even in this case, the weights W i (η) can be made to decrease across "time" ξ = τ when a 1-wave interacts with the boundary. Using the modified Glimm interaction potential Q and Lemma 2.2 above, it holds that, across "time" ξ = τ when a weak 3-wave reflected off the characteristic boundary {η = 0}, both G and Q decrease, if k + is chosen sufficiently large (see Section 2). One sees that the result holds if κ 2 ≫ κ 1 in (3.6). What remains to be proved is (3.12) near the boundary.
3.3.
Estimates near the boundary. In this section, we focus on the estimate of (3.12) near the boundary. It is different from those for the Cauchy problem. We exploit the exclusive property of Recalling thatη b = 0, we conclude
E b,2 ≡ 0. (3.16)
To estimate E b,1 + E b,3 , we consider the following two cases. characteristic Hugoniot curve to reach U 2 , and the 3-Hugoniot curve to reach V (b). Furthermore, among the first-family and third-family waves, they must be of distinct type, that is, there can be a 1-compressive shock (Lax shock) and a 3-decompressive shock (non Lax shock) or vice versa in the Riemann solution to keep the pressure unchanged in the approximate solutions U and V . We need to show that
|h 3 (b)| = O(1)|h 1 (b)|. (3.17)
If this is true, observing that λ 1 < 0 and λ 3 > 0 at U + , so that q i (b) = c a i h i (b) with the constants c a i > 0 to be chosen, we have
E b,1 + E b,3 = c a 1 |h 1 (b)|W 1 (b)λ 1 (b) + c a 3 |h 3 (b)|W 3 (b)λ 3 (b) = |h 1 (b)| − c a 1 W 1 (b)|λ 1 (b)| + c a 3 O(1)W 3 (b)|λ 3 (b)| ≤ |h 1 (b)| − c a 1 |λ 1 (b)| + 2c a 3 O(1)|λ 3 (b)| ≤ 0, by using the fact that λ 3 (b) λ 1 (b)
is bounded (depending only on U + ) and then choosing c a 1 quite large. This completes the proof of (3.12).
Then what left is to prove (3.17). We know U 1 = S 1 (h 1 )(U b ) and V b = S 3 (h 3 )(U 2 ). For U close to U + , since dS 1 (α)(U ) dα | α=0 = r 1 (U ) and especially the third argument of r 1 is κ 1 (−λ 1 u) = 0 (it is nonzero at the reference state U + ), by the inverse function theorem, we infer that |h 1 (b)| ∽ |p 1 − p|, and similarly for the third-family, |h 3 (b)| ∽ |p 2 − p|. Since p 2 = p 1 , we conclude that |h 1 (b)| ∽ |h 3 (b)|. Here, for positive quantities a and b, we use a ∽ b to mean that there is a positive constant
C 1 depending only on U + so that C −1 1 b ≤ a < C 1 b.
Existence of a Semigroup
Using the existence and uniqueness results established in the earlier sections, we can now establish the existence of the semigroup S of solutions generated by the wave-front tracking algorithm.
Proposition 4.1. Assume that TV(U (·)) is sufficiently small. Then, for δ > 0, the map (U (·), ξ) → U δ (ξ, ·) := S δ ξ (U (·)) produced by the wave-front tracking algorithm is a uniformly Lipschitz continuous semigroup such that
(i) S δ 0 U = U , S δ ξ 1 S δ ξ 2 U = S δ ξ 1 +ξ 2 U ; (ii) S δ ξ U − S δ ξ V L 1 (R + ) ≤ C U − V L 1 (R + ) + Cδξ.
Proof. Property (i) follows immediately because S δ is produced by the wave-front tracking method.
We now prove property (ii).
Let U δ and V δ be the front tracking δ-approximate solutions of (2.17) with the initial data U = U 0 (·) and V = V 0 (·), respectively. By (3.8) and (3.15), we obtain that, for any ξ ≥ 0,
U δ (ξ) − V δ (ξ) L 1 (R + ) ≤ CΦ(U δ (ξ), V δ (ξ)) ≤ CΦ(U δ (0), V δ (0)) + C 1 O(1)δξ ≤ C 1 C 2 U − V L 1 (R + ) + C 1 O(1)δξ.
This establishes the Lipschitz continuity of the δ-semigroup with respect to the initial data and time. The semigroup S generated by the wave-front tracking method is given by the subsequent theorem.
Theorem 4.1. Let TV(U (·)) be sufficiently small. Then the sequence S δ generated from the front tracking algorithm is a Cauchy sequence in the L 1 -norm, and the sequence S δ ξ (U ) converges to a unique limit S ξ (U ) as δ → 0. The map S : [0, ∞) × D → D is a uniformly continuous semigroup.
In particular, there exists a constant L such that, for all ξ 1 , ξ 2 > 0 and U , V ∈ D,
(i) Semigroup Property: S 0 U = U , S ξ 1 S ξ 2 U = S ξ 1 +ξ 2 U ; (ii) Lipschitz Continuity: S ξ U − S ξ V L 1 (R + ) ≤ L U − V L 1 (R + ) ;
(iii) Each trajectory ξ → S ξ U yields a weak solution to the initial-boundary value problem (2.17);
(iv) Consistency with Riemann Solver: For any piecewise constant initial data U ∈ D,
there exists a small θ > 0 such that, for all ξ ∈ [0, θ], the trajectory U (ξ, ·) = S ξ U agrees with the solution of (2.17) obtained by piecing together the standard entropy solutions of the Riemann problems.
Theorem 4.1 implies that a sequence of δ-approximate front tracking solutions to (2.17) converges to a unique entropy solution whose value is in D, and this unique limit is L 1 -stable. More precisely, we have the following immediate consequence from Theorem 4.1:
Z(T ) − S T Z L 1 ≤ L · Z(0) − Z L 1 + T 0 lim µ→0 + Z(ξ + µ) − S µ Z(ξ) L 1 µ dξ . (4.1)
Using the essential estimates shown in the earlier sections, Theorem 4.1 is shown along a similar line of arguments to the one followed by Bressan-Colombo in [3]. The only significant difference here between the wave-front tracking by Bressan (cf. [2]) and the version of the method (cf. Holden and
Risebro [12]) we employ is on how to control the number of fronts from growing to infinity within finite time. In [3], the accurate Riemann solver (ARS) and simplified Riemann solver (SRS) are used to construct the approximate solutions. With (ARS), waves of each family can be possibly introduced in the Riemann solution where every rarefaction wave is divided into equal parts to obtain a rarefaction fan of wave-fronts, while with (SRS), all new waves are lumped together as a single non-physical front, traveling faster than all wave speeds. Furthermore, in [3], the simplified Riemann solver, rather than the accurate Riemann solver, is employed when the interaction term is less than a cut-off function in the order of √ δ. In the front tracking method we use, the approximate Riemann solution is changed by removing weak fronts. The point of this procedure is that the fronts of high generation are quite weak. Particularly, in our arguments, when the interaction term is less than δ, this corresponds to new and removed fronts with generation higher than N (here N is a suitably large positive integer computed using the initial approximation parameter δ, see [5] and [12] for further details). We also refer to Chen-Li [6] for a related analysis with the full Euler system where (SRS) is used in place of (ARS) when the interaction term is less than δ.
Thus, using an argument similar to [3], one concludes that S δn ξ Z n is a Cauchy sequence, converging in the L 1 -sense as long as the error from removing weak fronts tends zero as the initial error parameter δ → 0 (this has been proved in [5]). Consequently, the map S : [0, ∞) × D → D defined as the limit of the approximate solutions produced by the front tracking algorithm is well-defined. Now, the proofs of statements (i) to (iv) in Theorem 4.1 are as follows. Statements (i), (ii), and (iv) are immediate since S is the limit of front tracking approximate solutions S δ . It is similar to prove (iii) as [3], but, as discussed above, the wave-front tracking method we employ here is slightly different. Lastly, we observe that the entropy solution fulfills the boundary condition of the initial-boundary value problem (2.17) due to the construction of our approximate solutions. This completes the proof of Theorem 4.1.
Uniqueness in the class of viscosity solutions
In this section, we analyze an arbitrary Lipschitz semigroup defined on the domain D of BV functions. In particular, we prove that the semigroup S is uniquely determined when the local flow is assigned in connection with the initial data, a piecewise constant function. Based on the results in Section 4, we first show that the semigroup S generated by the wave-front tracking method is the canonical trajectory of the standard Riemann semigroup (SRS). Then the uniqueness of entropy solutions is shown to extend to a broader class, namely the class of viscosity solutions as introduced by Bressan in [1]. The essential part here is to show that, in the viscosity class, the entropy solution matches the semigroup trajectory generated by the front tracking method.
(i) R 0 U = U , R ξ 1 R ξ 2 U = R ξ 1 +ξ 2 U ; (ii) R ξ U − R ξ V L 1 ≤ L U − V L 1 ;
(iii) If U ∈ D is piecewise constant, then, for all ξ ∈ [0, ε 0 ] sufficiently small, the trajectory U (ξ, ·) = S ξ U coincides with the solution of (2.17) obtained by patching together the standard entropy solutions of the Riemann problems produced by the discontinuities of U . In what follows, we discuss some necessary and sufficient conditions for a function ξ → U (ξ) ∈ D to coincide with a semigroup trajectory. Following Bressan [1], there are two types of local approximate parametrices for system (2.7).
The first approximate parametrice comes from the self-similar solution of the Riemann problem.
To that end, consider a function U : [0, ∞) × R + → R 3 and a fixed point (τ, ζ) in the domain of U .
Suppose that U (τ, ·) ∈ D. The boundedness of the total variation implies that the following limits exist:
U − = lim η→ζ − U (τ, η), U + = lim η→ζ + U (τ, η).
Let ϑ = ϑ(ξ, η) be the solution of the Riemann problem with piecewise constant data U − and U + and, for ξ > τ , define the following function:
H ♯ (U,τ,ζ) (ξ, η) = ϑ(ξ − τ, η − ζ) if |η − ζ| ≤λ(ξ − τ ), U (τ, η) if |η − ζ| >λ(ξ − τ ).
Hereλ is an upper bound for all wave speeds, sup U |λ k (U )| <λ, k = 1, 2, 3.
(5.1)
The other required parametrice is obtained from the corresponding quasilinear hyperbolic system
(2.8) A(U )∂ ξ U + B(U )∂ η U = 0
by "freezing" the coefficients of the matrices A(U ) and B(U ) in a neighbourhood of the state U (τ, ζ).
For ξ > τ , define H ♭ (U,τ,ζ) as the solution of the linear Cauchy problem with constant coefficients: With the estimates in Sections 2-3, the proof here follows along a similar line of reasoning to the one presented in [1]. The only difference is a straight boundary and physical domain restricted to the positive η-axis; nonetheless, we can still proceed with the proof provided that the convergence of the wave-front tracking method is obtained which has been outlined in Section 2 and carried out in Eulerian coordinates in [5].
A∂ ξ M +B∂ η M = 0, M (τ, η) = U (τ, η).
Remark 5.1. Note that, in simpler cases such as the potential flow, isentropic or isothermal Euler flow, as far as the L 1 -stability problem is concerned, we achieve the same results as the full Euler equations.
Uniqueness of Solutions to the Free Boundary Problem in Eulerian
Coordinates
In this section we apply the following Wagner's Theorem [15,Theorem 2] to show the uniqueness of entropy solution to the free boundary problem (1.7).
Theorem 6.1 (Wagner [15]). Consider
∂ t U + ∂ x F (U ) = 0, (6.1)
where (t, x) ∈ R 2 , U (t, x) = (u 1 , · · · , u n ) ⊤ , and F (U ) = (f 1 (U ), · · · , f n (U )) ⊤ ∈ R n is a smooth vector-function. For any bounded measurable solutions of (6.1), with u 1 (t, x) ≥ 0, let (t, y) satisfy
∂y ∂x = u 1 (t, x), ∂y ∂t = −f 1 (U (t, x))
in the sense of distributions. Then T : (t, x) → (t, y(t, x)) is a Lipschitz-continuous transformation, which induces a one-to-one correspondence between L ∞ weak solutions of (6.1) on R + ×R satisfying 0 < ε ≤ u 1 (t, x) ≤ M < ∞ for some ε and M , and L ∞ weak solutions of
∂ t 1 u 1 − ∂ y f 1 (U ) u 1 = 0,
∂ t 1 u 1 (u 2 , · · · , u n ) ⊤ + ∂ y (f 2 (U ), · · · , f n (U )) ⊤ − 1 u 1 f 1 (U )(u 2 , · · · , u n ) ⊤ = 0, (6.2)
on R + × R satisfying ε ≤ u 1 (t, x) ≤ M .
If η(U ) is any convex extension of (6.1), i.e., there is a flux q(U ) such that ∇η∇F = ∇q, so that ∂ t η + ∂ x q = 0 for classical solutions, then any solution of (6.1), satisfying ∂ t η(U ) + ∂ x q(U ) ≤ 0 (6.3)
in the sense of distributions in (t, x) corresponds to a solution of (6.2), satisfying ∂ tη (V ) + ∂ yq (V ) ≤ 0, (6.4) in the sense of distributions in (t, y), where V = (v 1 , · · · , v n ) ⊤ ,η(V ) = η(U ) u 1 , andq(V ) = q(U ) − f 1 (U )η(V ). Furthermore, η is convex if and only ifη is convex as a function of V . Thus, the Lax's entropy inequality holds for a solution of (6.1) if and only if it holds for the corresponding solution of (6.2).
As indicated by Wagner [15, p.123], this theorem may also be applied to initial-boundary value problems by slight modification. Thus, we can also use it in our case. Suppose that U 1 0 (y) and U 2 0 (y) are two initial data functions of problem (1.7) in Eulerian coordinates, and (U 1 , g 1 ) and (U 2 , g 2 ) are the corresponding weak entropy solutions obtained by the front tracking method. We have known that U 1 and U 2 must be bounded. We now show that U 1 = U 2 and g 1 = g 2 , provided U 1 0 = U 2 0 . Suppose that V 1 and V 2 are the solutions of (2.17) in Lagrangian coordinates that correspond to U 1 and U 2 , respectively. By Theorem 6.1, it suffices to show that V 1 = V 2 . Then, from (2.18), we may find g 1 = g 2 . Therefore, by the uniqueness results of problem (2.17) (Corollary 4.1), it suffices to prove the following lemma. Lemma 6.1. If U 1 0 (y) = U 2 0 (y), then V 1 (0, η) = V 2 (0, η).
Proof. For j = 1, 2, let T j : (x, y) → (ξ, η) be the Lagrangian transform (2.4) associated with the solution U j . They are Lipschitz-continuous and one-to-one. Furthermore, we know U j (x, ·) is of bounded variation and lim x→x 0 U j (x, ·) − U j (x 0 , ·) L 1 = 0. Thus, we can not only solve η from (2.2), but also those equations make sense on each line x = constant.
We first show that, if U 1 0 (y) = U 2 0 (y), then T 1 | x=0 = T 2 | x=0 . Suppose (ξ j , η j ) = T j (x, y). Recall by definition that ξ = x. Then, at x = 0, we have both ξ 1 = ξ 2 = 0. Note that η j satisfies η j (0, 0) = 0 and solves ∂η j (x, y) ∂y | x=0 = (ρu) j (0, y).
Since (ρu) 1 (0, y) = (ρu) 2 (0, y) from U 1 0 (y) = U 2 0 (y), we conclude that η 1 (0, y) = η 2 (0, y) as well. This shows that T 1 (0, y) = T 2 (0, y), which implies that (T 1 ) −1 (0, η) = (T 2 ) −1 (0, η) and lies on {x = 0}. Since V j (ξ, η) = U j ((T j ) −1 (ξ, η)), we conclude V 1 (0, η) = U 1 ((T 1 ) −1 (0, η)) = U 1 ((T 2 ) −1 (0, η)) = U 2 ((T 2 ) −1 (0, η)) = V 2 (0, η).
Finally, summarizing all the analysis above, we state the main theorem of this paper. Theorem 6.2. For the initial data U 0 close to the reference state U + in the sense that U 0 − U + BV is sufficiently small, there is one and only one weak entropy solution (U, g) to problem (1.7) constructed by the front tracking method. Furthermore, reformulating this problem in Lagrangian coordinates as problem (2.17), then the L 1 -stability holds in the sense that
V 1 (ξ) − V 2 (ξ) L 1 (R + ) ≤ C V 1 (0) − V 2 (0) L 1 (R + )
for any two solutions V 1 and V 2 of (2.17). The solution is also unique in the class of viscosity solutions in Lagrangian coordinates.
∂ x (ρu) + ∂ y (ρv) = 0, (1.1) ∂ x (ρu 2 + p) + ∂ y (ρuv) = 0, (1.2) ∂ x (ρuv) + ∂ y (ρv 2 + p) = 0, (1.3) ∂ x (ρuE + pu) + ∂ y (ρvE + pv) = 0,(1.4)
Figure 1 . 1 .
11For supersonic flow passing the corner O with a static gas U − on the right of the solid right wedge (i.e., with velocity zero), then a combined vortex
K 2
2in the strength α 3 of the reflected 3-waves in the interaction between weak 1-wave and the characteristic boundary {η = 0} as given in Lemma 2.2.
the boundary condition in (2.17): The flows U and V have the same pressure near the boundary. Then we construct a piecewise constant weak solution only along the Hugoniot curves determined by the Riemann date U (b) and V (b), which are the states of U and V near the boundary (that is, the point b is some fixed point (ξ b , η b ) near the positive ξ-axis with η b > 0). Let h i (b) be the strength of the i-th shock in the Riemann problem determined by U (b) and V (b), and λ i be the corresponding wave speed. Then, as the second-family is linearly degenerate, we infer that λ 2 ≡ 0.
Case 1 :
1h 1 (b) = 0. If h 1 (b) = 0, then there is no first-family wave in the Riemann solution to the Riemann problem (U (b), V (b)). Since the second-family waves are the characteristic discontinuities, the pressure of the middle-state U m keeps unchanged, i.e. p m = p. As the third-family is genuinely nonlinear, there must be a jump of pressure across a wave of the third-family. Thus, in this case, there must be no wave of the third-family, that is, h 3 (b) = 0. Hence, q 1 (b) = 0 = q 3 (b). We conclude in this case that E b,i = 0, i = 1, 3. Case 2: h 1 (b) = 0. For this case, as analyzed in Case 1, one concludes that h 3 (b) = 0 as well. Starting from U (b), go along the 1-Hugoniot curve to reach U 1 , then possibly along the 2-
Definition 4. 1 .
1Given δ 0 > 0, define a domain D as the closure of the set consisting of the functions U : R + → R 3 such that U belongs to L 1 (R + ; R 3 ) by modulo a constant and BV(U − U + )(R + ) ≤ ε 0 .
Corollary 4. 1 .
1Let TV(U (·)) be sufficiently small. Then the entropy solution to the initialboundary value problem (2.17) produced by the wave-front tracking algorithm is L 1 -stable and unique. The proof of Theorem 4.1 relies on the subsequent crucial estimate (also used in [3]) about the approximation of Lipschitz flows: Suppose that S : [0, ∞) × D → D is a global semigroup with a Lipschitz constant L. Let T > 0, Z ∈ D, and Z : [0, T ] → D be a continuous mapping taking values in piecewise constant functions, with jumps along finitely many polygonal lines in the (ξ, η)-plane. Then
Definition 5 . 1 .
51Problem (2.17) is said to admit a standard Riemann semigroup if, for some small ε 0 > 0, there is a continuous map R : [0, ∞) × D → D and a constant L such that for every U , V ∈ D and ξ 1 , ξ 2 ≥ 0 we have:
Theorem 5 . 1 .
51Suppose that the initial-boundary value problem (2.17) admits a standard Riemann semigroup R : [0, ∞)×D → D. Consider S the semigroup generated by the front tracking algorithm,that is, S ξ (U ) = lim δ→0 S δ ξ (U ) with U ∈ D.Then, for all ξ ≥ 0 and U ∈ D, R ξ U = S ξ U.Theorem 5.1 is proved using similar arguments as in[1] based on the fundamental estimate (4.1) and the essential feature of the local flow in the ξ-direction, that is, the wave-front tracking method and the standard Riemann semigroup both have the structure of the Riemann solutions.
|U|U
= A(U (τ, ζ)) andB := B(U (τ, ζ)) the matrices evaluated at the fixed state U (τ, ζ).Note that the functions H # and H ♭ depend on the values U (τ, ζ) and U (τ, ζ±). In the remaining, we introduce the class of viscosity solutions and discuss how these solutions share the same local characterization as H # and H ♭ .Definition 5.2. Assume that U : [0, T ] → D is a continuous map with respect to the L 1 -norm. Then the function U is called a viscosity solution of problem (2.17) if there are constants C andλ providing bound (5.1) such that, with β and δ small enough, we have, for every(τ, ζ) ∈ [0, T ) × R (τ + δ, η) − H # (U,τ,ζ) (ξ, η)| dξ ≤ CTV{U (τ ) : (ζ − β, ζ) ∪ (ζ, ζ + β)(τ + δ, η) − H ♭ (U,τ,ζ) (ξ, η)| dξ ≤ C TV{U (τ ) : (ζ − β, ζ + β)} 2 .Theorem 5.2. Suppose that problem (2.17) admits a standard Riemann semigroup R. Then a continuous mapping U : [0, T ] → D is said to be a viscosity solution of (2.17) if and only if U (ξ, ·) = R ξ U at every ξ ∈ [0, T ]. (5.3) Using Theorem 5.2, one concludes that, for problem (2.17), the entropy solution is unique in the viscosity class of solutions, which coincides with the semigroup trajectory S ξ U generated by the front tracking method. More precisely, we have Corollary 5.1. A continuous mapping U : [0, T ] → D is a viscosity solution if and only if U (ξ, ·) = S ξ U for any ξ ∈ [0, T ]. (5.4)
). Then we briefly review how the existence of a weak entropy solution is shown by using the front tracking method.weak solution of a hyperbolic system of conservation laws and a bounded measurable weak solution of the corresponding equations in Lagrangian coordinates. Thus, the uniqueness results established for problem (2.17) imply the uniqueness of transonic characteristic discontinuities in Eulerian co-Section 3 is devoted to establishing the L 1 -stability of problem (2.17). Then, in Section 4, we show
the existence of a semigroup associated with problem (2.17). Finally, in Section 5, we show the
uniqueness of solutions to problem (2.17) in the larger class of viscosity solutions. By Wagner's
Theorem [15, Theorem 2], there is a one-to-one correspondence between a bounded measurable
19 )
19with a suitable constant K depending only on U + .For the reflection of weak 1-waves off the characteristic boundary {η = 0}, we have
Lemma 2.2. Suppose that U l , U m , and U r are three states in O ǫ (U
+ ) for sufficiently small ǫ, with
The unique limit of the Glimm scheme. A Bressan, Arch. Ration. Mech. Anal. 130A. Bressan, The unique limit of the Glimm scheme, Arch. Ration. Mech. Anal. 130 (1995), 205-230.
Hyperbolic Systems of Conservations Laws: The One-Dimensional Cauchy Problem. A Bressan, Oxford Univ. PressOxfordA. Bressan, Hyperbolic Systems of Conservations Laws: The One-Dimensional Cauchy Problem, Oxford Univ. Press, Oxford, 2000.
The semigroup of 2 × 2 conservation laws. A Bressan, R M Colombo, Indiana Univ. Math. J. 44A. Bressan and R. M. Colombo, The semigroup of 2 × 2 conservation laws, Indiana Univ. Math. J. 44 (1995), 677-725.
L 1 stability estimates for n × n conservation laws. A Bressan, T.-P Liu, T Yang, Arch. Ration. Mech. Anal. 149A. Bressan, T.-P. Liu, and T. Yang, L 1 stability estimates for n × n conservation laws, Arch. Ration. Mech. Anal. 149 (1999), 1-22.
G.-Q Chen, V Kukreja, H Yuan, arXiv:1208.5183Stability of transonic characteristic discontinuities in two-dimensional steady Euler flows. PreprintG.-Q. Chen, V. Kukreja, and H. Yuan, Stability of transonic characteristic discontinuities in two-dimensional steady Euler flows, Preprint arXiv:1208.5183, 2012.
Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge. G.-Q Chen, T.-H Li, J. Differential Equations. 244G.-Q. Chen and T.-H Li, Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge, J. Differential Equations, 244 (2008), 1521-1550.
Stability of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls. G.-Q Chen, Y.-Q Zhang, D.-W Zhu, SIAM J. Math. Anal. 38G.-Q. Chen, Y.-Q. Zhang, and D.-W. Zhu, Stability of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls. SIAM J. Math. Anal. 38 (2007), 1660-1693.
Stability of a Mach configuration. S Chen, Comm. Pure. Appl. Math. 59S. Chen, Stability of a Mach configuration, Comm. Pure. Appl. Math. 59 (2006), 1-35.
Stability of reflection and refraction of shocks on interface. S Chen, B Fang, J. Differential Equations. 244S. Chen and B. Fang, Stability of reflection and refraction of shocks on interface, J. Differential Equations, 244 (2008), 1946-1984.
C M Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third Ed. Berlin HeidelbergSpringer-VerlagC. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third Ed., Springer-Verlag: Berlin Heidelberg, 2010.
Reflection and refraction of shocks on an interface with a reflected rarefaction wave. B Fang, Y.-G Wang, H Yuan, J. Math. Phys. 5214B. Fang, Y.-G. Wang, and H. Yuan, Reflection and refraction of shocks on an interface with a reflected rarefaction wave, J. Math. Phys. 52 (2011), 14 pp.
Front Tracking for Hyperbolic Conservation Laws. H Holden, N Risebro, Springer-VerlagNew YorkH. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag: New York, 2002.
On the L 1 well posedness of systems of conservation laws near solutions containing two large shocks. M Lewicka, K Trivisa, J. Differential Equations. 179M. Lewicka and K. Trivisa, On the L 1 well posedness of systems of conservation laws near solutions containing two large shocks, J. Differential Equations, 179 (2002), 133-177.
Well-posedness theory for hyperbolic conservation laws. T.-P Liu, T Yang, Comm. Pure Appl. Math. 52T.-P. Liu and T. Yang, Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (1999), 1553-1586.
Equivalence of the Euler and Lagrangian equations of gas dynamics and weak solutions. D Wagner, J. Differential equations. 68D. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics and weak solutions, J. Differential equations, 68 (1987), 118-136.
On transonic shocks in two-dimensional variable-area ducts for steady Euler system. H Yuan, SIAM J. Math. Anal. 38H. Yuan, On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J. Math. Anal. 38 (2006), 1343-1370.
. G.-Q Chen, Shanghai; Oxford, OX1 3LB, UK; Evanston, IL 60208, USA E-mail addressSchool of Mathematical Sciences, Fudan University ; China; Mathematical Institute, University of Oxford ; Department of Mathematics, Northwestern UniversityG.-Q. Chen, School of Mathematical Sciences, Fudan University, Shanghai 200433, China; Mathe- matical Institute, University of Oxford, Oxford, OX1 3LB, UK; Department of Mathematics, North- western University, Evanston, IL 60208, USA E-mail address: [email protected]
USA E-mail address: [email protected]. Vaibhav Kukreja, 60208Evanston, IL; ShanghaiInstituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil; Department of Mathematics, Northwestern University ; East China Normal [email protected] H. Yuan, Department of Mathematics. China E-mail address: [email protected]; [email protected] Kukreja, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil; De- partment of Mathematics, Northwestern University, Evanston, IL 60208, USA E-mail address: [email protected]; [email protected] H. Yuan, Department of Mathematics, East China Normal University, Shanghai 200241, China E-mail address: [email protected]; [email protected]
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"Lehn-Sorger example via Cox ring and torus action",
"Lehn-Sorger example via Cox ring and torus action"
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"Maksymilian Grab "
] |
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We present a new approach to effective construction of Cox rings of crepant resolutions of quotient singularities based on the action of an algebraic torus. In the present paper we deal with a symplectic resolution of a quotient of C 4 by a reducible symplectic representation of the binary tetrahedral group which admits an action of rank two algebraic torus (C * ) 2 . We use the Lefschetz-Riemann-Roch theorem for the equivariant Euler characteristic as well as the Kawamata-Viehweg vanishing and the multigraded Castelnuovo-Mumford regularity for a verification that the construction method established in[15], [13], [24] leads effectively to the Cox ring of a resolution of the singularity under the consideration.R(X) =D∈Cl(X)H 0 (X, D).
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"https://arxiv.org/pdf/1807.11438v1.pdf"
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1807.11438
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c766868783546a89bf440ee9bcc31e7358302a69
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Lehn-Sorger example via Cox ring and torus action
Jul 2018
Maksymilian Grab
Lehn-Sorger example via Cox ring and torus action
Jul 2018
We present a new approach to effective construction of Cox rings of crepant resolutions of quotient singularities based on the action of an algebraic torus. In the present paper we deal with a symplectic resolution of a quotient of C 4 by a reducible symplectic representation of the binary tetrahedral group which admits an action of rank two algebraic torus (C * ) 2 . We use the Lefschetz-Riemann-Roch theorem for the equivariant Euler characteristic as well as the Kawamata-Viehweg vanishing and the multigraded Castelnuovo-Mumford regularity for a verification that the construction method established in[15], [13], [24] leads effectively to the Cox ring of a resolution of the singularity under the consideration.R(X) =D∈Cl(X)H 0 (X, D).
Introduction
In this article we inspect the interaction between three themes -torus actions, crepant resolutions of quotient singularities and Cox rings -in the study of crepant resolutions of a quotient variety C 4 /G for a reducible symplectic representation G of a binary tetrahedral group generated by the matrices:
i 0 0 0 0 −i 0 0 0 0 i 0 0 0 0 −i , − 1 2 (1+i)ǫ (−1+i)ǫ 0 0 (1+i)ǫ (1−i)ǫ 0 0 0 0 (1+i)ǫ 2 (−1+i)ǫ 2 0 0 (1+i)ǫ 2 (1−i)ǫ 2 .
By the classical theorem of Hilbert and Noether, an invariant ring C[x 1 , . . . , x n ] G , where G is a finite group acting linearly on polynomial ring, is a finitely generated algebra. Quotient singularities are the corresponding singular varieties of the form C n /G = Spec C[x 1 , . . . , x n ] G . The study of their crepant resolutions generalizes the theory of minimal resolutions of du Val singularities. It shows an interplay between geometry and the theory of finite groups celebrated in the McKay correspondence [22]. The Cox ring of a normal algebraic variety X with a finitely generated class group is a Cl(X)-graded ring (see [4] for a precise construction and a detailed exposition):
If R(X) is a finitely generated C-algebra it gives a powerful tool to study the geometry of X and its small modifications. In particular one may recover X as a GIT quotient of Spec R(X) by the action of the Picard (quasi)torus T = Hom(Cl(X), C * ). This is the case in our setting, i.e. if X is a crepant resolution of a four-dimensional symplectic quotient singularity, then R(X) is a finitely generated algebra, see [3,Theorem 3.2] and [23,Theorem 1.2]. Varieties with an action of an algebraic torus (C * ) r form an important class of objects in algebraic geometry due to their relations with combinatorics and convex geometry. Well-studied examples include toric varieties [10], but not only, see also [2]. Of the most importance for us will be the recent methods presented in [8].
They enable us to describe combinatorially local data of the torus action around fixed points and to connect them with the cohomology of line bundles via the Lefschetz-Riemann-Roch theorem.
The quotient C 4 /G was an object of study in [6], where it was shown that the crepant resolution exists, and in [21], where such resolution was constructed as a sequence of two blow-ups. Here however we take a different approach, originated in [12] and [15], and developed in [13] and [24], aiming at generalizations to the study of crepant resolutions of other quotient singularities. The idea is to study the geometry of crepant resolution via the algebraic and combinatorial properties encoded by its Cox ring. In particular one would like to find the presentation of the Cox ring and construct a resolution as a GIT quotient. One of the previous attempts to give a general framework for computing Cox rings of crepant resolutions of quotient singularities was based on the embedding of the Cox ring into the ring of Laurent polynomials over the invariant ring of the commutator subgroup. Then, using the McKay correspondence one constructs a finitely generated subring R which is a candidate to be the whole Cox ring of a resolution. The nontrivial step is to check if this ring is the actual Cox ring. See [12], [15], [13], [24] for examples of this approach. In fact the (non-minimal) list of generators of the Cox ring of the crepant resolution of C 4 /G were originally found by Yamagishi by an algorithm presented in [24, §4]. In this paper we present an alternative, more geometric method to deal with this difficulty, using a torus action on the resolution. We first study the geometry of a GIT quotient Spec R//T. By a general result of Kaledin [19, § 4] a crepant resolution of a four-dimensional symplectic quotient singularity admit a stratification whose closed strata are the exceptional divisors in dimension 3 and a fibre over point 0 in dimension 2. This fibre encodes some important geometric properties of the resolution, in particular in our case it contains all the fixed points of the torus action. In section 3, assuming that the GIT quotient Spec R//T is a resolution we describe in detail its central fibre, see theorems 3.4, 3.5 and 3.7. Then, using methods of [8] we give conditionally the local description of a torus action around the fixed points, see theorem 4.5. This gives us an open cover by smooth affine varieties. We then prove unconditionally that the elements of this open cover are indeed smooth in theorem 5.1. As a consequence the GIT quotient X = Spec R//T is a crepant resolution and the results of sections 3 and 4 are unconditionally valid see 5.3. As the study of toric varieties and T -varieties of complexity one shows, a torus action on a variety may have significant consequences for the structure of its Cox ring. Inspired by the methods of [8] we use the Lefschetz-Riemann-Roch formula [5] for the equivariant Euler characteristic and the Kawamata-Viehweg vanishing to calculate the generating function for dimensions of movable linear systems on a resolution X → C 4 /G in theorem 6.3. Finally, in section 7 we show that one can use the multigraded Castelnuovo-Mumford regularity and the Kawamata-Viehweg vanishing to reduce the proof of equality R(X) = R to checking equalities between finite number of graded pieces. Having a formula like one of theorem 6.3 and the ideal of relations between the generators of the candidate ring R this can be done by a computer calculation as we do in the case of singularity C 4 /G. proposing the research topic and for many helpful discussions. He also would like to express gratitude toward Maria Donten-Bury for the help with the computation of the central fibre and the stable locus and sharing many valuable remarks on the article. The author was supported by the Polish National Science Center project of number 2015/17/N/ST1/02329.
The singularity, its resolutions and their Cox ring
In this section we collect known facts on the group G ⊂ Sp 4 (C) investigated by Bellamy and Schedler in [6] and by Lehn and Sorger in [21], the corresponding symplectic quotient singularity C 4 /G and its symplectic resolution X. In particular we introduce two Z 2 -gradings on the Cox ring of X, one induced by the Picard torus action and one induced by the action of a two-dimensional torus on X.
Let G ⊂ Sp 4 (C) be the symplectic representation of binary tetrahedral group generated by the matrices:
i 0 0 0 0 −i 0 0 0 0 i 0 0 0 0 −i , − 1 2 (1+i)ǫ (−1+i)ǫ 0 0 (1+i)ǫ (1−i)ǫ 0 0 0 0 (1+i)ǫ 2 (−1+i)ǫ 2 0 0 (1+i)ǫ 2 (1−i)ǫ 2 ,
where ǫ = e 2πi/3 is a third root of unity.
Proposition 2.1.
(1) There are 7 conjugacy classes of elements of G among which two consist of symplectic reflections. (2) The commutator subgroup [G, G] has order 8 and it is isomorphic to the quaternion group. In particular the abelianization G/[G, G] is cyclic of order 3. (3) The representation G defined above is reducible. It decomposes into two two-dimensional representations V 1 ⊕ V 2 . In particular there is a natural (C * ) 2 -action on C 4 /G induced by the multiplication by scalars on V i .
Proof. Points (1) and (2) can be quickly verified with GAP Computer Algebra System [16]. Point (3) follows directly by the definition of G.
Let Σ ⊂ C 4 /G be the singular locus of the quotient. It can be described as follows.
Proposition 2.2 ([21, § 1]). The preimage of Σ via the quotient map C 4 → C 4 /G consists of four planes, each of which maps onto Σ. Outside the image of 0 the singular locus is a transversal A 2 -singularity.
Let π : X → C 4 /G be a projective symplectic resolution which exists by [6] or by [21]. Using the symplectic McKay correspondence [20] and propositions 2.1(1) and 2.2 we obtain the following facts about the geometry of X. Proposition 2.3. There are two exceptional divisors E 1 , E 2 of X each of which is mapped onto Σ. The central fibre π −1 ([0]) consist of four surfaces. The fibre of π over any point in Σ \ [0] consists of two curves isomorphic to P 1 intersecting in one point and each of which is contained in exactly one of two exceptional divisors.
By a theorem of Kaledin [19,Thm 1.3] and by proposition 2.1(3) we have also:
Corollary 2.4.
There is a natural T := (C * ) 2 -action on X making π an equivariant map.
Remark 2.5. Beware that throughout this article we consider two different twodimensional tori -one is the Picard torus T := Hom(Cl(X), C * ) and the other one is the torus T = C * acting on C 4 = C 2 × C 2 by multiplication of scalars on each component C 2 .
Let C i be the numerical class of a complete curve which is a generic fibre of the morphism π| Ei : E i → Σ. We may describe the generators of the Picard group of X in terms of their intersections with curves C i . Proposition 2.6 (cf. [15, 2.D]). The Picard group of X is a free rank two abelian group generated by line bundles L 1 , L 2 such that the intersection matrix (L i .C j ) i,j is equal to identity matrix.
One may see the Cox ring R(X) of X as a subring of [13, § 2.1] for details). It follows that the action of the two-dimensional torus T = (C * ) 2 on X induces the action on Spec R(X). The algorithm given in [24, § 4] allows one to compute the presentation of R(X). Denote the following elements of the (Laurent) polynomial ring C[x 1 , y 1 , x 2 , y 2 ][t ±1 1 , t ±1 2 ]:
C[x 1 , x 2 , x 3 , x 4 ] [G,G] [t ±1 1 , t ±1 2 ] (seew 01 = y 1 x 2 − x 1 y 2 , w 02 = x 5 2 y 2 − x 2 y 5 2 , w 03 = x 5 1 y 1 − x 1 y 5 1 , w 04 = x 4 1 + (−4b + 2)x 2 1 y 2 1 + y 4 1 , w 05 = x 4 2 + (4b − 2)x 2 2 y 2 2 + y 4 2 , w 06 = x 1 x 3 2 + (−2b + 1)y 1 x 2 2 y 2 + (−2b + 1)x 1 x 2 y 2 2 + y 1 y 3 2 , w 07 = x 3 1 x 2 + (2b − 1)x 1 y 2 1 x 2 + (2b − 1)x 2 1 y 1 y 2 + y 3 1 y 2 , w 11 = (−3bx 2 1 x 2 2 + (−b + 2)y 2 1 x 2 2 + (−4b + 8)x 1 y 1 x 2 y 2 + (−b + 2)x 2 1 y 2 2 − 3by 2 1 y 2 2 )t 1 , w 12 = (x 4 2 + (−4b + 2)x 2 2 y 2 2 + y 4 2 )t 1 , w 13 = (x 3 1 x 2 + (−2b + 1)x 1 y 2 1 x 2 + (−2b + 1)x 2 1 y 1 y 2 + y 3 1 y 2 )t 1 , w 14 = (−5x 4 1 y 1 x 2 + y 5 1 x 2 − x 5 1 y 2 + 5x 1 y 4 1 y 2 )t 1 , w 15 = (x 1 y 1 x 4 2 + 2x 2 1 x 3 2 y 2 − 2y 2 1 x 2 y 3 2 − x 1 y 1 y 4 2 )t 1 , w 21 = ((3b − 3)x 2 1 x 2 2 + (b + 1)y 2 1 x 2 2 + (4b + 4)x 1 y 1 x 2 y 2 + (b + 1)x 2 1 y 2 2 + (3b − 3)y 2 1 y 2 2 )t 2 , w 22 = (x 4 1 + (4b − 2)x 2 1 y 2 1 + y 4 1 )t 2 , w 23 = (x 1 x 3 2 + (2b − 1)y 1 x 2 2 y 2 + (2b − 1)x 1 x 2 y 2 2 + y 1 y 3 2 )t 2 , w 24 = (y 1 x 5 2 + 5x 1 x 4 2 y 2 − 5y 1 x 2 y 4 2 − x 1 y 5 2 )t 2 , w 25 = (−2x 3 1 y 1 x 2 2 − x 4 1 x 2 y 2 + y 4 1 x 2 y 2 + 2x 1 y 3 1 y 2 2 )t 2 ,
w 3 = (9x 2 1 y 1 x 3 2 + (−2b + 1)y 3 1 x 3 2 + 9x 3 1 x 2 2 y 2 + (6b − 3)x 1 y 2 1 x 2 2 y 2 + (−6b + 3)x 2 1 y 1 x 2 y 2 2 − 9y 3 1 x 2 y 2 2 + +(2b − 1)x 3 1 y 3 2 − 9x 1 y 2 1 y 3 2 )t 1 t 2
s = t −2 1 t 2 , t = t 1 t −2 2 ,
where b is a primitive root of unity of order 6.
Theorem 2.7. The Cox ring R(X) of X is generated by 20 generators: w 01 , . . . , w 07 , w 11 , . . . , w 15 , w 21 , . . . , w 25 , w 3 , s, t.
The degree matrix of this generators with respect to the generators L 1 , L 2 of Pic(X) (first two rows) and with respect to the T -action (remaining two rows) is: We will reprove this theorem in section 7. From now on we will denote the ring generated by elements from the statement of theorem 2.7 by R.
It is a general principle that using the degrees of generators of the Cox ring of X we can describe the movable cone Mov(X) of X and find the number of resolutions and the corresponding subdivision of Mov(X) into the nef cones of resolutions of X (see [4,Prop. 3.3.2.9]). However, since we will be using this two results to prove that R is a Cox ring of X we give independent proofs.
Proposition 2.8.
(1) The cone Mov(X) of movable divisors of X is the cone generated by the line bundles L 1 and L 2 . (2) There are two symplectic resolutions of C 4 /G. The chambers in Mov(X)
corresponding to the nef cones of these resolutions are cone(L 1 , L 1 + L 2 ) and cone(L 2 , L 1 + L 2 ). The Mori cones of corresponding resolutions are cone(C 2 , C 1 − C 2 ) and cone(C 1 , C 2 − C 1 ).
Proof.
(1) This follows from [3, Theorem 3.5].
(2) We will prove the first part of the claim in section 6 as proposition 6.2. The part on Mori cones then follows by taking dual cones. if L is globally generated, then marked vertices of this polyhedron correspond to T -fixed points of X where X is the resolution on which L is relatively ample. We will see in 4.2 that indeed fixed points of this polytope correspond to points in X T and in lemma 3.8 that L is globally generated.
The structure of the central fibre
In this section we will study the structure of the central fibre π −1 ([0]) of such a resolution π : X → C 4 /G using the ideal of relation of generators of the ring R, under the assumption that X = Spec R//T. The results of this section will be useful in the next one, where we will be investigating the action of the two-dimensional torus T on X with the fixed point locus X T contained in the central fibre. The additional assumption that X = Spec R//T will be dealt with in section 5.
Lemma 3.1. We have an isomorphism Spec R T ∼ = C 4 /G. In particular the inclusion of invariants R T ⊂ R induce map p : Spec R → C 4 /G.
Proof. This follows from the fact that
R T = C[x 1 , x 2 , x 3 , x 4 ] G via the embed- ding R ⊂ C[x 1 , x 2 , x 3 , x 4 ] [G,G] [t ±1 1 , t ±1 2 ]. Let Z = p −1 ([0]
). Decomposing the ideal of relations from the presentation of Spec R one obtains the decomposition of Z into irreducible components. We consider the closed embedding Spec R ⊂ C 20 given by the generators of R from statement of theorem 2.7.
Proposition 3.2. The components of Z are the following subvarieties of C 20 :
Zu = V (w 3 , w ij | (i, j) ∈ (0, 1), . . . , (0, 7)
, (1, 1), . . . , (1, 5), (2, 1), . . . , (2, 5)),
Z 0 = V (s, t,
w 25 , w 24 , w 15 , w 14 , w 07 , w 06 , w 05 , w 04 , w 03 , w 02 , w 01 , 15 , w 14 , w 13 , w 11 , w 07 , w 06 , w 05 , w 04 , w 03 , w 02 , w 01 ,
w 12 w 22 − w 13 w 23 , w 11 w 21 − 9w 13 w 23 , w 3 21 − 27w 22 w 2 23 , w 13 w 2 21 − 3w 11 w 22 w 23 , w 12 w 2 21 − 3w 11 w 2 23 , 3w 2 13 w 21 − w 2 11 w 22 , 3w 12 w 13 w 21 − w 2 11 w 23 , w 3 11 − 27w 12 w 2 13 ) Z P = V (s, w 25 , w 24 , w 23 , w 22 , w 21 , w 15 , w 14 , w 07 , w 06 , w 05 , w 04 , w 03 , w 02 , w 01 , w 3 11 − 27w 12 w 2 13 − i √ 3w 2 3 t), Z ′ P = V (t, w 25 , w 24 , w 15 , w 14 , w 13 , w 12 , w 11 , w 07 , w 06 , w 05 , w 04 , w 03 , w 02 , w 01 , w 3 21 − 27w 22 w 2 23 + i √ 3w 2 3 s), Z 1 = V (s, w 25 , w 24 , w 23 , w 21 , w 15 , w 12 , w 07 , w 06 , w 05 , w 04 , w 03 , w 02 , w 01 ,2w 3 w 22 t + ζ 3 w 11 w 14 , 2w 2 11 w 22 + ζ 7 12 √ 3w 3 w 14 , w 3 11 − i √ 3w 2 3 t, 4w 11 w 2 22 t + ζ 5 12 √ 3w 2 14 ), Z 2 = V (t, w 25 , w 22 , w2w 3 w 12 s + ζ 6 w 21 w 24 , w 3 21 + i √ 3w 2 3 s, 2w 12 w 2 21 + ζ 11 12 √ 3w 3 w 24 , 4w 2 12 w 21 s + ζ 7 12 √ 3w 2 24 ),
where ζ 3 , ζ 6 , ζ 12 are primitive 3rd, 6th and 12th roots of unity. The component Z u is contained in the locus of unstable points with respect to any linearization of Picard torus via character from movable cone. Points in the component Z ′ P are unstable with respect to any linearization by a character (2, 1) and points in the component Z P are unstable with respect to any linearization by a character (1, 2). Lemma 3.3. The unstable locus of Spec R with respect to a linearization of the trivial line bundle by a character (2, 1) is cut out by equations:
w 12 s = w 12 w 23 = w 11 w 3 = w 12 w 3 = w 13 w 3 = w 13 w 22 = w 22 t = 0.
Moreover all the semistable points of Z are stable and have trivial isotropy groups.
Proof. This can be done by a computer calculation, using the Singular library gitcomp by Maria Donten-Bury (see www.mimuw.edu.pl/marysia/gitcomp.lib).
The following theorem gives a description of components of the central fibre. Let W be the locus of stable points of Spec R with respect to the T-action linearized by a character (2, 1) (the case (1, 2) is analogous) and consider the quotient map W → X. Denote by S 0 , S 1 , S 2 , P the images of sets of stable points of the components Z 0 , Z 1 , Z 2 , Z P . Note that these are precisely the components of the central fibre of X.
Theorem 3.4.
(a) S 0 is a non-normal toric surface whose normalization is isomorphic to the Hirzebruch surface H 6 . The action of T on the normalization of S 0 is given by characters in the columns of the matrix 1 −1 −1 −1 . (b) S 1 is a non-normal toric surface whose normalization is the toric surface of a fan spanned by rays: (0, 1), (1, 0), (1, −1), (−1, −2). The action of T on the normalization of S 1 is given by characters in the columns of the matrix 3 −1 1 −1 . (c) S 2 is a non-normal toric surface whose normalization is the toric surface of a fan spanned by rays: (0, 1), (1, −1), (−1, −2). The action of T on the normalization of S 2 is given by characters in the columns of the matrix
−1 3 −1 1 . (d) P is isomorphic to P 2 .
The action of T on P in homogeneous coordinates is given by the matrix ( 2 0 3 2 4 1 ).
Proofs of (a)-(c) follow a fairly standard procedure of division of a toric variety by a torus action. We outline an argument in the case of S 0 , and then we comment on (d).
Proof of (a).
Claim 1. By rescaling variables we may assume that Z 0 is the toric variety embedded into C 7 with coordinates w 11 , w 12 , w 13 , w 21 , w 22 , w 23 , w 3 defined by the toric ideal generated by binomials:
w 12 w 22 − w 13 w 23 , w 11 w 21 − w 13 w 23 , w 3 21 − w 22 w 2 23 , w 13 w 2 21 − w 11 w 22 w 23 , w 12 w 2 21 − w 11 w 2 23 , w 2 13 w 21 − w 2 11 w 22 , w 12 w 13 w 21 − w 2 11 w 23 , w 3 11 − w 12 w 2 13 .
Proof of the claim. This is a general argument (over an algebraically closed field) to reduce a prime binomial ideal that does not contain monomials to a toric ideal: one takes a point (a ij , a 3 ) with all a ij = 0 = a 3 in the zero set of the original ideal and then set new coordinates
w ′ ij = 1 aij w ij , w ′ 3 = 1 a3 w 3 . Claim 2. S is the affine variety of the cone σ ∨ = cone(−2e 2 + 3e 3 , e 1 , e 1 + 3e 2 − 3e 3 , e 2 , e 4 ) in space R 4 spanned by the character lattice M = Z 4 of four-dimensional torus.
Proof of the claim. By a calculation as in proof of [10, Thm 1.1.17] Z 0 is the affine toric variety defined by lattice points in M = Z 4 :
v 11 = e 1 + e 2 − e 3 , v 12 = e 1 + 3e 2 − 3e 3 , v 13 = e 1 , v 21 = e 3 , v 22 = −2e 2 + 3e 3 , v 23 = e 2 , v 08 = e 4 .
In other words there is a short exact sequence 0 → L → Z 7 ϕ → M = Z 4 → 0, where ϕ sends the canonical base e i to the vectors v ij in order written as above and L is a subgroup of Z 7 defined by the binomial ideal of Z 0 , i.e. L is generated by:
(0, 1, −1, 0, 1, −1, 0), (−1, 1, 0, 2, 0, −2, 0), (1, 0, −1, 1, 0, −1, 0), (−2, 0, 2, 1, −1, 0, 0), (0, 0, 0, 3, −1, −2, 0), (−2, 1, 1, 1, 0, −1, 0), (−1, 0, 1, 2, −1, −1, 0), (3, −1, −2, 0, 0, 0, 0).
One checks that the vectors v 11 and v 21 are not needed to generate σ ∨ . Proof of the claim. The i-th column corresponds to e i ∈ M, i = 1, 2, 3, 4 and these correspond to w 13 , w 23 , w 21 , w 3 respectively.
To obtain S = S 0 from Z 0 we have to remove unstable orbits of Z 0 and divide the remaining open subset by the action of the Picard torus. By lemma 3.3 we have:
Claim 4. The unstable locus of Z 0 is the union of the closures of two orbits:
O 1 = {w 11 = w 12 = w 13 = 0}, O 2 = {w 21 = w 22 = w 23 = w 3 = 0}.
One checks by [10, 3.2.7] that:
Claim 5. The orbits O 1 , O 2 correspond respectively to the following faces of the cone σ = cone(e 4 , e 2 + e 3 , 3e 2 + 2e 3 , e 1 , 3e 1 + e 3 ) dual to σ ∨ :
τ 1 = cone(e 1 ), τ 2 = cone(e 4 , e 2 + e 3 ).
To obtain the fan of the toric variety S = S 0 we consider the fan of Z 0 , remove cones τ 1 and τ 2 together with all the cones containing them and take the family of the images of the remaining cones via the dual of the kernel of the matrix from claim 3, i.e. by:
Q := 0 −1 1 0 −1 −1 0 1 . Now Q(e 4 ) = (0, 1), Q(e 2 + e 3 ) = (0, −1), Q(3e 2 + 2e 3 ) = (−1, −3), Q(3e 1 + e 3 ) = (1, −3) and changing coordinates in Z 2 by (1, 0) → (1, −3), (0, 1) → (0, −1)
we obtain the standard fan for Hirzebruch surface H 6 . The matrix of the T -action on S 0 is given by the product of the matrix J of embedding of T into the big torus of Z 0 with the transpose of the matrix Q. Here, inspecting the construction of the generators of R, we obtain:
J = 3 1 2 3 1 3 2 3 .
Proof of (d). Denote x := w 11 , y := w 12 , z := w 13 , w := w 3 . Rescaling coordinates we may assume that Z P is embedded in C 5 as the hypersurface x 3 − yz 2 + w 2 t = 0. Then the unstable locus is described by equations xw = yw = zw = wt = 0, see lemma 3.3. This gives an open cover of the set of semistable points, given by the union of three open sets {xw = 0} ∪ {yw = 0} ∪ {zw = 0} (note that wt = 0 together with the equation x 3 − yz 2 + w 2 t = 0 implies that one of the variables x, y, z does not vanish). Gluing the quotients of these open sets coincides with the standard construction of P 2 (with coordinates x, y, z) by gluing three affine planes. Finally, note that T acts on w 11 , w 12 , w 13 with weights (2, 2), (0, 4), (3, 1) respectively.
We can also describe the non-normal locus of components of the central fibre. Proof. We use lemma 3.6 and the description of T-stable orbits of Z i analogous to the one in the proof of theorem 3.4 for Z 0 . Altogether, T-stable orbits of Z i which consist of normal points correspond to the cones which do not contain the cones τ j and are not contained in ω k , where τ j 's are defined analogously as in the proof of 3.4(a) and ω k 's are:
• For i = 0: ω 1 = cone(e 4 , e 2 + e 3 ) and ω 2 = cone(e 1 , e 4 ).
• For i = 1, 2: ω := ω 1 = cone(e 2 + e 3 , e 4 ).
In each case one easily finds all such cones and their images via the map Q (again, notation after proof of 3.4) turn out to be the cones parametrizing orbits in the statement. We conclude since the non-normal points of S i = Z i //T are precisely the images of the non-normal points of Z i which are T-stable. Here we use the fact that all semistable points of Z i are stable and the isotropy groups of the T-action are trivial by lemma 3.3, so that the quotient Z s i → Z i //T is a torsor. to the line y = 0 on P with homogeneous coordinates x, y, z (note that this is the flex tangent of the cuspidal cubic curve S 0 ∩ P ). (e) S 2 ∩P is the point corresponding to the cone spanned by rays (0, 1), (1, −1) onthe normalization of S 2 and to the point x = z = 0 on P with homogeneous coordinates x, y, z (note that this is the cusp of the cubic curve S 0 ∩ P ).
We give an argument in the case (c) using the notation from the proof of theorem 3.4.
Proof of (c). On Z 0 the intersection Z 0 ∩ Z P is cut out by equations w 21 = w 22 = w 23 = 0. Hence it contains as a dense subset the orbit of the toric variety Z 0 which corresponds to the one-dimensional cone τ = cone(e 2 + e 3 ), since σ ∨ \ τ ⊥ ∋ v 21 , v 22 , v 23 . Then we have Q(e 2 + e 3 ) = (0, −1). On Z P the intersection Z 0 ∩Z P is cut out by the equation t = 0. By the construction of the isomorphism P ∼ = P 2 this equation yields the curve x 3 − yz 2 = 0 on P 2 .
The next lemma shows that all nef line bundles on X are globally generated, which will be important in the next sections. Proof. Since L 1 + L 2 and L 1 are invariant with respect to the T -action and the base point locus of a linear system is closed for both linear systems |L 1 + L 2 | and |L 1 | it either has to be empty or it has a nontrivial intersection with the central fibre. The assertion follows by inspecting the weights of the generators of the Cox ring with respect to the Picard torus action and the equations of components of the fibre p −1 ([0]) where p : Spec R → C 4 /G is as in proposition 3.2. It turns out that the intersections of the zero sets of elements of each of these two linear systems with p −1 ([0]) are contained in the unstable locus.
Compasses of fixed points
In this section we obtain a local description of the action of the two-dimensional torus T on a symplectic resolution X → C 4 /G at fixed points of this action, under the assumption that X = Spec R//T. In fact the conditional results of this section will serve as a guide in the section 5, where we will prove that indeed X = Spec R//T. Assume that the GIT quotient X = Spec R//T is a resolution on which the line bundle L = 2L 1 + L 2 is relatively ample (i.e. X is a GIT quotient of Spec R with respect to a character (2, 1), and we assume that it is smooth). The torus T acts on X. Let x be a fixed point of this action. Torus T acts then also on the tangent space T x X. This gives weight-space decomposition T x X = 4 i=1 V νi , where ν i are in the character lattice of T , which we will identify with Z 2 .
The following definition was first introduced in [8, § 2.3]
Definition 4.1. The weights ν 1 , . . . , ν 4 are called the compass of x in X with respect to the action of T .
We will now work to find compasses of all fixed points of the action T on X. The black dots correspond to the weights of sections of L restricted to P , blue ones to S 0 , green ones to S 1 and red ones to S 2 (note that lattice points marked by multiple colours correspond to weights occuring in restriction to more than one component).
Proof. This follows by a computer calculation (using Macaulay2 [17]) of dimensions of appropriate graded pieces of the coordinate rings of Z i and Z P from proposition 3.2. 3)]. In particular one obtains the weights of the action of T on the tangent space to S at fixed points. Here P is an exception, since the T -action on P is not faithful -P consists of the fixed points for C * -action given by the homothety which coincides with the action of C * embedded into T with the weight (1, 1). Proof of theorem 4.5. First we calculate the weights of the T -action. Taking into account the natural inclusions of the tangent spaces to the components of the central fibre into the tangent space of X most of them can be deduced from the fact that the action of T on all components except P is toric. The polytopes spanned by the points marked by one colour in the lemma 4.2 are the polytopes of those toric varieties. This gives their affine covers and the weights of the T -action on the tangent spaces to their T -fixed points. In the case of weights at (3,7), (5,5) and (6,4) which come from the action of T on P we use the explicit description of P from the proof of theorem 3.4(d). Altogether the weights calculated up to this point are the ones from the assertion except ν 1,3 , ν 1,4 , ν 3,4 , ν 7,3 , ν 7,4 . Now the calculation of all the weights for the homothety action is easy, since this action is compatible with the symplectic form in the sense of lemma 4.6 below, and we can compute at least two weights at each T -fixed point, by summing components of each known weight ν i,j . For the remaining weights of the T -action we combine claims 1 and 2 below. For example we know that ν 1,1 = (−1, 3) and ν 1,2 = (1, −5), which gives the weights of the homothety action −2 and −4 at point corresponding to (0, 16). By lemma 4.6 the remaining weights for homothety are equal to 4 and 6. By the claim 1 the remaining weights ν 1,3 is of the form (4, 0) or (0, 4) and ν 1,4 is of the form (6, 0) or (0, 6). Since ν 1,1 and ν 1,2 yield two positive weights for the C * -action considered in the claim 2 and so do (6, 0) and (4, 0) we have ν 1,3 = (0, 4) and ν 1,4 = (0, 6). The remaining weights are computed analogously. Proof of the claim. First note that every orbit of the T -action on C 4 \ 0 is either two-dimensional or has the isotropy group equal to C * × 1 or 1 × C * . Now take any x ∈ X T , any remaining weight λ of the T -action on T x X and a onedimensional eigenspace V λ with this weight which is not contained in the tangent space to the central fibre. By the Luna slice theorem such an eigenspace corresponds to the closure of an orbit O of the T -action via an equivariant localétale map U → T x X, where U is an invariant neighbourhood of x. In particular dim O = 1, and dim π(O) = 1, as O is not contained in the central fibre of the resolution π : X → C 4 /G. Therefore π(O) as well as O and V λ are stabilized by either C * × 1 or 1 × C * and the claim follows.
For the next claim, consider C * as a subtorus of T embedded with the weight (1, −1). Clearly it acts on C 4 , C 4 /G and X.
Claim 2. Let x ∈ X C * . Among the weights of the induced C * -action on T x X two weights are positive and two are negative.
Proof of the claim. Let
X + x = {x ∈ X : lim t→0 tx ∈ X x }, X − x = {x ∈ X : lim t→0 t −1 x ∈ X x },
where X x ⊂ X C * is the connected component containing x. We will use the fact that X ± x are irreducible, locally closed subsets of X (see [7, § 4] and [9, § 4.1]). If at least three weights at x were nonnegative then dim X + x ≥ 3 by the Luna slice theorem. Similarly if at least three weights at x are nonpositive then dim X − x ≥ 3. Suppose that dim X + x ≥ 3. On the other hand (C 4 ) + 0 = C 2 × 0 is two-dimensional and hence also (C 4 /G) + 0 = (C 2 × 0)/G is two-dimensional. Since dim π −1 ([0]) = 2 and the fibres of π : X \ π −1 ([0]) → C 4 /G are of dimension at most one then dim π(X + x ) ≥ dim X + x −1 = 2. As π(X + x ) ⊂ (C 4 /G) + 0 we know that dim π(X + x ) = 2 and hence X + x has to be an exceptional divisor of the resolution π : X → C 4 /G. But the image of such an exceptional divisor is contained in the singular locus of C 4 /G which consists of the image of four planes in C 4 that have one-dimensional intersection with C 2 × 0 and so, dim π(X + x ) = 1, a contradiction. The case dim X − x ≥ 3 is completely analogous.
Lemma 4.6. The symplectic form ω on X is of weight 2 with respect to the C *action induced by the homothety. In particular at each fixed point of this action we may order weights λ 1 , λ 2 , λ 3 , λ 4 of the action on T x X, so that λ 1 +λ 2 = λ 3 +λ 4 = 2.
Smoothness of the GIT quotient
Let R be a subring of the Cox ring of the crepant resolution generated by the elements from the statement of theorem 2.7. In this section we show that the GIT quotient Spec R//T with respect to the linearization of the trivial line bundle by the character 2L 1 + L 2 . In consequence we will see that Spec R//T → C 4 is a crepant resolution. This will make the results on geometry of crepant resolutions of C 4 /G in the previous sections unconditional and will help to conclude that R is the whole Cox ring in the final section 7.
We consider Spec R as a closed subvariety of C 20 via the embedding given by generators from statement of Theorem 2.7. Let T = Hom(Cl(X), C * ) be a Picard torus of X.
Theorem 5.1. The stable locus of Spec R with respect to a linearization of the trivial line bundle by a character (a, b), a > b > 0 is covered by seven T × Tinvariant open subsets U 1 , . . . , U 7 such that U i /T ∼ = C 4 . More precisely:
(1) U 1 = {w 12 s = 0} and (R w12s ) T = C w 02 , w 05 , w23 w12s , w24 w12s ,
(2) U 2 = {w 12 w 23 = 0} and (R w12w23 ) T = C w 2 12 s w23 , w21 w23 , w24 w23 , w3 w12w23 , (3) U 3 = {w 12 w 3 = 0} and (R w12w3 ) T = C w11 w12 , w13 w12 , w12w23 w3 , w12w24 w3 , (4) U 4 = {w 11 w 3 = 0} and (R w11w3 ) T = C w12 w11 , w13 w11 , w11w22 w3 , w11w23 w3 , (5) U 5 = {w 13 w 3 = 0} and (R w13w3 ) T = C w11 w13 , w12 w13 , w12w21 w3 , w12w22 w3 , (6) U 6 = {w 13 w 22 = 0} and (R w13w22 ) T = C w 2 22 t w13 , w11 w13 , w14 w13 , w3 w13w22 , (7) U 7 = {w 22 t = 0} and (R w22t ) T = C w 03 , w 04 , w13 w 2 22 t , w14
w12s .
In particular the GIT quotient Spec R//T with respect to the linearization of the trivial line bundle by a character (a, b), a > b > 0 is smooth.
Proof. Lemma 3.3 implies that {U i } i=1,...7
form an open cover of the quotient. It remains to prove equalities from points (1)- (7). Note that then in each case the four generators of the ring on the right-hand side of the equality have to be algebraically independent as the GIT quotient Spec R//T is irreducible and of dimension four. By symmetry it suffices to consider only U i for i = 1, 2, 3. In each case we calculate the invariants of the localization of the coordinate ring of the ambient C 20 , with help of 4ti2 [1] obtaining in consequence:
(1) Spec R T w12s = C w 0i , (2) Spec R T w12w23 = C w 0i , w1j w12 , w2j w23 ,
w 2 12 s w23 , w 2 23 t w12 , w3
w12w23 i=1,...,7, j=1,...,5
(3) Spec R T w3w12 = C w 0i , w1j w12 , w12w2j w3 , w 3 12 s w3 , w 2 3 t w 3 12 i=1,.
..,7, j=1,...,5
Then, using the Gröbner basis of the ideal of relations among generators of R with respect to an appropriate lexicographic order, we verify with Singular [11] in each case that each of the generators can be expressed as a polynomial of the four generators from the assertion.
Remark 5.2. To find isomorphisms U i ∼ = C 4 we used the argument from the section 4 with its a priori assumption on smoothness of the quotient Spec R//T as a heuristic. To guess the coordinates on each invariant open subset U i we picked an element f ∈ R of a degree corresponding to ith fixed point and four elements of (R f ) T of degrees equal to the predicted weights of the action on a tangent space.
In fact theorem 4.5 follows immediately by theorem 5.1 when we note that U i ∼ = C 4 is a T -invariant neighbourhood of ith T -invariant point (we order points as in rows of the table from the assertion). Nevertheless, since to guess an open cover U i we used the statement of theorem 4.5, we decided to give its conditional proof before theorem 5.1 to preserve the logical consequence of our considerations.
By the inclusion of invariants R T ⊂ R (see lemma 3.1) we have the induced projective map Spec R//T → Spec R/T ∼ = C 4 /G.
Corollary 5.3. The map π : Spec R//T → Spec R/T ∼ = C 4 /G is a crepant resolution.
Proof. Denote E 1 = {s = 0} ⊂ Spec R//T and E 2 = {t = 0} ⊂ Spec R//T. These are irreducible divisors on Spec R//T. By the construction of R the map π is an isomorphism outside E 1 ∪ E 2 . Hence π is a resolution and it has to be crepant since there are two crepant divisors over C 4 /G by symplectic McKay correspondence and they have to be present on each resolution.
Dimensions of movable linear systems
In this section we use a torus T action on X to give a formula for dimensions of these graded pieces of R(X) which correspond to the movable linear systems on some of the resolutions. Let X → C 4 /G be the resolution corresponding to the linearization of the Picard torus action by a character (2, 1). Let X ′ → C 4 /G be the resolution corresponding to the linearization (1, 2). Let T be the two-dimensional torus which acts naturally on C 4 , C 4 /G and X via homothety on each irreducible component of C 4 viewed as a representation of G. Denote by P i the fixed points of the T -action on X as in the table from theorem 4.5. Let {ν i,j } 4 j=1 denote the compass of P i in X. Let us also denote by µ i (L) the weight of the T -action on the fibre of L over P i . Note that µ i is linear i.e. µ i (A + B) = µ i (A) + µ i (B).
Remark 6.1. By lemma 3.8 we may compute the weights µ i for line bundles L 1 +L 2 and L 1 similarly as for 2L 1 + L 2 in section 4 to obtain (the last column is calculated from the first two ones by the linearity of µ i ):
i µi(L1) µi(L1 + L2) µi(L2) 1 (0, 4) (0, 12) (0, 8) 2 (0, 4) (1, 7) (1, 3) 3 (0, 4) (3, 3) (3, −1) 4 (2, 2) (3, 3)
(1, 1) 5 (3, 1) (3, 3) (0, 2) 6 (3, 1) (7, 1) (4, 0) 7 (8, 0) (12, 0) (4, 0)
We may now prove the promised observation on the subdivision of the cone of movable divisors on X, see proposition 2.8.
Proposition 6.2. There are two symplectic resolutions of C 4 /G. The chambers in Mov(X) corresponding to the nef cones of these resolutions are cone(L 1 , L 1 + L 2 ) and cone(L 2 , L 1 + L 2 ).
Proof. Consider the homomorphisms µ i : N 1 (X) → R 2 . The two walls of the chamber C containing 2L 1 + L 2 are corresponding to the contractions of X, in particular they identify some T -fixed points of X. Hence each wall has to be spanned by an element v ∈ Mov(X) satisfying µ i (v) = µ j (v) for some i = j. Now the only such elements in cone(L 1 , L 1 + L 2 ) are lying on the rays spanned by L 1 + L 2 and L 1 . Therefore C = cone(L 1 , L 1 + L 2 ). The analogous argument, using the homomorphisms µ ′ i : N 1 (X ′ ) → R 2 corresponding to the T -fixed points of X ′ , shows that the chamber containing L 1 + 2L 2 is equal to cone(L 1 + L 2 , L 2 ). Theorem 6.3. If h 0 (X, pL 1 + qL 2 ) (a,b) is the dimension of the subspace of sections H 0 (X, pL 1 + qL 2 ) on which T acts with the weight (a, b), then we have the following generating function for such dimensions for line bundles inside the movable cone:
a,b,p,q≥0 h 0 (X, pL 1 + qL 2 ) (a,b) y p 1 y q 2 t a 1 t b 2 = = 7 i=1 1 (1 − t µi(L1) y 1 )(1 − t µi(L2) y 2 ) 4 j=1 (1 − t νi,j )
.
Proof. By a corollary of Lefschetz-Riemman-Roch theorem [8, Corollary A.3] we have:
χ T (X, L) = 7 i=1 t µi(L) 4 j=1 (1 − t νi,j )
.
Using the linearity of µ i : p,q≥0 χ T (X, pL 1 + qL 2 )y p 1 y q 2 = p,q≥0
7 i=1 t pµi(L1) · t qµi(L2) 4 j=1 (1 − t νi,j ) y p 1 y q 2 = 7 i=1 1 (1 − t µi(L1) y 1 )(1 − t µi(L2) y 2 ) 4 j=1 (1 − t νi,j )
.
The assertion follows now by Kawamata-Viehweg vanishing, which implies
χ T (X, pL 1 +qL 2 ) = a,b≥0 h 0 (X, pL 1 +qL 2 ) (a,b) t a 1 t b 2 = a,b≥0 h 0 (X ′ , pL 1 +qL 2 ) (a,b) t a 1 t b 2 .
if p ≥ q ≥ 0 and likewise for q ≥ p ≥ 0 on X ′ .
Example 6.4. The dimensions of the weight spaces corresponding to the lattice points in a head of the polyhedron spanned by weights for the line bundle 2L 1 + L 2 considered in remark 2.10 and in section 4 can be depicted on the following diagram: H 0 (X, pL 1 + qL 2 ).
Proof. Every effective Weil divisor on X is of the form D + nE 1 + mE 2 where D is the strict transform of a Weil divisor on C 4 /G and m, n ≥ 0. Now note that D ∼ pL 1 + qL 2 for some p, q ≥ 0 as D.C i ≥ 0. Then the divisor D corresponds to a section of H 0 (X, pL 1 + qL 2 ) ⊂ R ≥0 and E 1 , E 2 correspond to s, t respectively.
Thus to prove that R = R(X) it suffices to show that R contains R(X) ≥0 . The following lemma reduces the problem further, to finitely many graded pieces with respect to Z 2 -grading by characters of the Picard torus of X.
Lemma 7.2. R(X) ≥0 is generated by the sections of all linear spaces H 0 (X, L) for L ∈ S ∪ S ′ where S := {O X , L 1 , L 1 + L 2 , 2L 1 , 2L 1 + L 2 , 2L 1 + 2L 2 , 3L 1 + L 2 , 3L 1 + 2L 2 , 4L 1 + 2L 2 } and S ′ := {L 2 , 2L 2 , L 1 + 2L 2 , L 1 + 3L 2 , 2L 1 + 3L 2 , 2L 1 + 4L 2 }.
Proof. We use the Kawamata-Viehweg vanishing theorem to show that if L is a line bundle and L ∈ S ∪ S ′ then L is O X -regular with respect to an appropriate finite family of globally generated line bundles on X. Then we use the surjectivity of the multiplication map of sections for Castelnuovo-Mumford regular sheaves in the sense of Hering-Schenck-Smith [18, Theorem 2.1 (2)] to show that the sections corresponding to the elements of linear systems of line bundles in S ∪ S ′ generate R(X) ≥0 .
First note that H i (X, L) = 0 for every i > 2 and every line bundle L on X since C 4 /G is affine and π : X → C 4 /G has fibres of dimension at most 2. We will use [18, Theorem 2.1(2)] several times for various families {B 1 , . . . , B l } of globally generated line bundles. Denote by X → C 4 /G the resolution on which 2L 1 + L 2 is relatively ample. Then the line bundles L 1 and L 1 +L 2 on X are globally generated by lemma 3.8. Let l = 1 and B 1 = L 1 . By the Kawamata-Viehweg vanishing theorem on X every line bundle L = mL 1 + nL 2 with m ≥ n + 2 and n = 0, 1 is O X -regular. By [18, Theorem 2.1(2)] the global sections of line bundles L 1 , 2L 1 and 3L 1 + L 2 generate all spaces H 0 (X, L) for such L. Let l = 1 and B 1 = L 1 + L 2 . By the Kawamata-Viehweg vanishing theorem on X every line bundle L = mL 1 + nL 2 with n ≥ 2 and m = n, n + 1 is O X -regular. By [18, Theorem 2.1(2)] the global sections of line bundles L 1 + L 2 , 2L 1 + 2L 2 and 3L 1 + 2L 2 generate all spaces H 0 (X, L) for such L. Let l = 2, B 1 = L 1 and B 2 = L 1 + L 2 . By the Kawamata-Viehweg vanishing theorem on X a line bundle mL 1 + nL 2 with m ≥ n + 2 and n ≥ 2 is O X -regular. By [18, Theorem 2.1(2)] the global sections of line bundles L 1 , L 1 +L 2 and 4L 1 +2L 2 generate all spaces H 0 (X, L) for such L. Changing the roles of L 1 and L 2 in the argument above (in particular considering the resolution on which L 1 + 2L 2 is ample) we see similarly that all the remaining spaces H 0 (X, L) for L ∈ S ∪ S ′ are generated by the global sections of line bundles in S ∪ S ′ .
Hence we reduced the problem to showing that R contains spaces of global sections only for these finitely many line bundles which are elements of S ∪ S ′ in lemma 7.2. This, with the help of computer algebra, can be done with the use of the previous section, namely by theorem 6.3 in which we computed the Hilbert function of R(X) ≥0 . Note that, by the symmetry of the generators of R with respect to the Picard torus action, we may restrict our attention to line bundles in S. Lemma 7.3. R contains H 0 (X, L) for each L ∈ S, where S is as in the lemma 7.2.
Proof. We calculate the Hilbert function of R in Macaulay2 [17]. Then, using Singular [11] we extract from it Hilbert functions for each of the vector spaces R L , L ∈ S, graded by characters of T . The main difficulty here is the fact that the Z 2 -grading of R corresponding to the Picard torus action is not a positive grading, i.e. it is not in N 2 . Then, we observe that for each L ∈ S this function agrees with the Hilbert function for R(X) L which we calculated in theorem 6.3. Since R ⊂ R(X) it means that R = R(X).
Remark 7.4. The above consideration about re-obtaining Cox ring suggests the following geometric approach to the verification whether the set of elements of R(X) of the form as in [14,Theorem 2.2] generates of the Cox ring of a crepant resolution X for a quotient singularity with the use of a torus H acting on the resolution.
Step 1 Calculate the fixed point locus X H and the invariants needed to compute the equivariant Euler characteristic of the resolution with Lefschetz-Riemman-Roch formula. In case when X H is discrete it suffices to know the compass at each point of X H and the weights of the H-action on fibres over fixed points of line bundles generating Pic(X).
Step 2 Use the Lefschetz-Riemman-Roch formula combined with the Kawamata-Viehweg vanishing theorem to compute the Hilbert function of non-negatively graded part of the Cox ring R(X) ≥0 . For the case when dim X H ≤ 1 one may use [8,Corollary A.3].
Step 3 Use the multigraded Castelnuovo-Mumford regularity and the Kawamata-Viehweg vanishing to obtain a finite set S of line bundles whose global sections generate the Cox ring.
Step 4 Compute the Hilbert function for R and extract from it Hilbert functions of the vector spaces R L , L ∈ S, graded by characters of H. Check whether they are equal to Hilbert functions R(X) L calculated in Step 2.
Remark 7.5. Note that if we know the Hilbert function for R(X) ≥0 calculated in step 1 then the above procedure suggests also where to look for additional generators of R if we do not have equality in step 4.
Remark 7.6. The actual implementation of the procedure proposed above is challenging, since one has to obtain enough information on the geometry of a resolutions without knowing the Cox ring. The required data are of two kinds -in step 1, one has to find data associated with the torus action. And in step 3, to successfully use regularity, one has to find sufficiently many line bundles which are globally generated on one of the resolutions.
Corollary 2. 9 .
9Taking a GIT quotient of Spec R by the Picard torus action with respect to the linearization given by character (a, b) with a > b > 0 and with b > a > 0 one obtains the two symplectic resolutions of C 4 /G.
Remark 2 . 10 .
210The weights of the T -action on global sections of the fixed line bundle L on X are lattice points in Z 2 . Taking a convex hull one obtains a lattice polyhedron in R 2 . For example fixing a line bundle L = 2L 1 + L 2 one gets a polyhedron with tail equal to the positive quadrant of R 2 and with a head spanned by the lattice points from the picture below:
Claim 3 .
3The Picard torus acting on Z 0 may be viewed as (C * ) 2 embedded into T M = (C * ) 4 by characters in the columns of the matrix:
Theorem 3. 5 .
5If ν i : S i → S i is the normalization of the component S i of the central fibre i = 0, 1, 2 and N i ⊂ S i is the locus of the non-normal points of S i , then (a) ν −1 0 (N 0 ) is the sum of the closures of the orbits corresponding to (−1, −3) and (1, −3) i.e. it is the sum of invariant fibres of H 6 . (b) ν −1 1 (N 1 ) is the sum of the closures of the orbits corresponding to (−1, −2) and (1, −1). (c) ν −1 2 (N 2 ) is the sum of the closures of the orbits corresponding to (−1, −2) and (1, −1).
Lemma 3. 6 .
6Let S be a subsemigroup of the lattice M and let U = Spec C[S] be the corresponding (not necessarily normal) affine toric variety. Let σ = cone(S). Let U = Spec C[M ∩ σ] be the toric normalization of U . Then for each subcone τ ≺ σ the orbit O(τ ) ⊂ U consists of the normal points of U if and only if we have the equality of semigroups M ∩ τ + S = M ∩ τ + M ∩ σ, where τ is the linear span of τ . Proof. The toric variety U (τ ) = Spec C[M ∩ τ + S] is the open subvariety of Spec C[S] obtained by removing these torus orbits which does not contain O(τ ) in their closure. Similarly U (τ ) = Spec C[M ∩ τ + M ∩ σ] is the open subvariety of Spec C[M ∩ σ] obtained by removing all the torus orbits whose closure does not contain the orbit corresponding to τ . Moreover U (τ ) → U (τ ) is the normalization. Hence the equality M ∩ τ + S = M ∩ τ + M ∩ σ holds if and only if U (τ ) is normal, which is precisely the case when O(τ ) consists of the normal points, since it is contained in the closure of each orbit of U (τ ) and the non-normal locus of a variety is closed. By analyzing intersections of stable loci of Z i and Z P one can systematically describe the intersections of the components of the central fibre in terms of the identifications from theorem 3.4.
a) S 0 ∩ S 1 is the curve corresponding to (−1, −3) on the normalization of S 0 and to (1, −1) on a normalization of S 1 . (b) S 0 ∩ S 2 is the curve corresponding to (1, −3) on the normalization of S 0 and to (1, −1) on a normalization of S 2 . (c) S 0 ∩P is the curve corresponding to (0, −1) on the normalization of S 0 and the cuspidal cubic curve x 3 − yz 2 = 0 on P with homogeneous coordinates x, y, z. (d) S 1 ∩ P is the curve corresponding to (1, 0) on the normalization of S 1 and
Lemma 3 . 8 .
38L 1 + L 2 and L 1 are globally generated line bundles on X.
Lemma 4 . 2 .
42The following diagram shows the weights of the action of T on the space of sections of H 0 (X, L) for L = 2L 1 + L 2 which are nonzero after the restriction to some irreducible component of the central fibre.
Remark 4 . 3 .
43Lattice points from lemma 4.2 are contained in the polyhedron from remark 2.10. Moreover their convex hull form a minimal head of this polyhedron.
Remark 4 . 4 .
44Considering the polytope which is a convex hull of weights marked by one colour in lemma 4.2 we get the polytope of the line bundle L pulled back to the corresponding component S of the central fibre viewed as a toric variety. As L is globally generated (see lemma 3.8) the vertices correspond to the fixed points of the action of T on S, cf. [8, Lemma 2.3(
Theorem 4 . 5 .
45The fixed points of the T -action correspond to the vertices of the polytopes which are convex hulls of weights marked by fixed colour in lemma 4.2. The compasses of the points corresponding to the vertices of these polytopes are as in the table below:Point Compass P1 ↔ (0, 16) ν1,1 = (1, −3), ν1,2 = (1,The following picture illustrates the weights of the T -action calculated in the theorem. It is a directed graph. The points correspond to the sections of H 0 (X, L) for L = 2L 1 + L 2 which are nonzero after the restriction to the central fibre together with vectors, as in lemma 4.2. The directed edges are the vectors from the compasses attached to the points which correspond to fixed points of T -action. In case when two vertices are connected by the two edges pointing in both ways we depict them by a single edge without any arrow.
Claim 1 .
1The remaining weights for the T -action on the central fibre are of the form (a, 0) or (0, a).
i=1,...,7, j=1,...,5
Re-obtaining the Cox ringUsing the generating function from theorem 6.3 we may prove theorem 2.7 i.e. we reconfirm that the generators calculated by algorithm in[24, § 4] (see theorem 2.7) are indeed sufficient to generate the Cox ring of X. In this section we outline the argument using the notion of multigraded Castelnuovo-Mumford regularity [18,§ 2]. The section concludes with general remarks about computing Cox rings of crepant resolutions of quotient singularities. Denote by R(X) the Cox ring of X and by R the subring of R(X) generated by the elements from the statement of theorem 2.7. Lemma 7.1. The Cox ring of X is generated by s, t and by R(X) ≥0 := p,q≥0
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|
[] |
[
"Interdependency and hierarchy of exact and approximate epidemic models on networks",
"Interdependency and hierarchy of exact and approximate epidemic models on networks"
] |
[
"Timothy J Taylor ",
"Istvan Z Kiss ",
"\nCentre for Computational Neuroscience and Robotics\nSchool of Mathematical and Physical Sciences\nDepartment of Mathematics\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n",
"\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n"
] |
[
"Centre for Computational Neuroscience and Robotics\nSchool of Mathematical and Physical Sciences\nDepartment of Mathematics\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK",
"University of Sussex\nBN1 9QHFalmer, BrightonUK"
] |
[] |
Over the years numerous models of SIS (susceptible → infected → susceptible) disease dynamics unfolding on networks have been proposed. Here, we discuss the links between many of these models and how they can be viewed as more general motif-based models. We illustrate how the different models can be derived from one another and, where this is not possible, discuss extensions to established models that enables this derivation. We also derive a general result for the exact differential equations for the expected number of an arbitrary motif directly from the Kolmogorov/master equations and conclude with a comparison of the performance of the different closed systems of equations on networks of varying structure. * Timothy Taylor is funded by a PGR studentship from the MRC, and the Departments of Informatics and Mathematics at the University of Sussex. † Corresponding Author. Electronic address: [email protected] 1 bifurcation structure of the model and gain a greater understanding of the full spectrum of behaviour. The challenge is then finding the set of equations that best approximate the solution of the Kolmogorov equations.Given that here we focus on epidemic models, usually such models are formulated in terms of the expected values of the number of infected and/or susceptible individuals or some other motif in the network such as the expected number of infected and/or susceptible individuals of different degrees (the number of connections a node has). Such models range from classic meanfield [1] to pairwise[15], heterogenous pairwise [7], effective degree[17,18]and individual-level models[22]to name a few. Whilst these models seem to use different approaches their derivation is based on the same conceptual framework, namely they begin by choosing a base-motif (e.g. a node, a link and the two nodes it connects, a node and all its links). These base-motifs are then used to formulate equations for the different possible states that they can achieve (e.g. for the expected number of motifs in different states or the probability that a specified motif in the network is in a certain state). These equations generally involve not only the base-motif itself, but larger or extended motifs of which they are usually part of. These larger motifs in turn depend on more complex motifs and a closure is needed in order to obtain a self-contained system of equations of reasonable size. Importantly the base-motif determines not only the complexity of the model (the larger the motif the greater the number of states it can be in) but also how much of the network topology can be captured. Interestingly differential equations for smaller motifs that are part of the base-motif should, in theory, be recoverable from the original differential equation. To this end the main focus of the paper is the consideration of various simple models of disease dynamics and the relations between them. We also consider which models are derivable directly (subject to a suitable closure) from the Kolmogorov/master equations and can thus be referred to as exact.We begin in section 2 with an introduction of some of the more common approaches to modelling disease dynamics on networks, considering meanfield ([1]), pairwise ([15]), heterogeneous pairwise ([7]) and the effective degree ([17]) model formulations. In section 3 we formulate an exact version of the effective degree model and then illustrate how the pairwise model can then be recovered from this new set of equations. We are, however, unable to recover the heterogenous pairwise model from the exact effective degree and this motivates, in section 4, an extension of this which incorporates further network topology into the ODEs. From this extension we then show how it is then possible to recover the heterogeneous pairwise equations. Once the links between the models have been established, in section 5 we show how the unclosed version of the models can be derived directly from the Kolmogorov equations. This is done by proving that as long as the heuristic equations for any motif are written following a certain set of rules they will always be exact. We conclude, in section 6 with a brief comparison of the models and discuss under what circumstances they perform best, in the sense of being close to simulation results.
|
10.1007/s00285-013-0699-x
|
[
"https://arxiv.org/pdf/1212.3124v2.pdf"
] | 16,175,140 |
1212.3124
|
bc9c8fd372f5e604445f0e5ff0dc54445e8fa371
|
Interdependency and hierarchy of exact and approximate epidemic models on networks
9 Apr 2013
Timothy J Taylor
Istvan Z Kiss
Centre for Computational Neuroscience and Robotics
School of Mathematical and Physical Sciences
Department of Mathematics
University of Sussex
BN1 9QHFalmer, BrightonUK
University of Sussex
BN1 9QHFalmer, BrightonUK
Interdependency and hierarchy of exact and approximate epidemic models on networks
9 Apr 2013
Over the years numerous models of SIS (susceptible → infected → susceptible) disease dynamics unfolding on networks have been proposed. Here, we discuss the links between many of these models and how they can be viewed as more general motif-based models. We illustrate how the different models can be derived from one another and, where this is not possible, discuss extensions to established models that enables this derivation. We also derive a general result for the exact differential equations for the expected number of an arbitrary motif directly from the Kolmogorov/master equations and conclude with a comparison of the performance of the different closed systems of equations on networks of varying structure. * Timothy Taylor is funded by a PGR studentship from the MRC, and the Departments of Informatics and Mathematics at the University of Sussex. † Corresponding Author. Electronic address: [email protected] 1 bifurcation structure of the model and gain a greater understanding of the full spectrum of behaviour. The challenge is then finding the set of equations that best approximate the solution of the Kolmogorov equations.Given that here we focus on epidemic models, usually such models are formulated in terms of the expected values of the number of infected and/or susceptible individuals or some other motif in the network such as the expected number of infected and/or susceptible individuals of different degrees (the number of connections a node has). Such models range from classic meanfield [1] to pairwise[15], heterogenous pairwise [7], effective degree[17,18]and individual-level models[22]to name a few. Whilst these models seem to use different approaches their derivation is based on the same conceptual framework, namely they begin by choosing a base-motif (e.g. a node, a link and the two nodes it connects, a node and all its links). These base-motifs are then used to formulate equations for the different possible states that they can achieve (e.g. for the expected number of motifs in different states or the probability that a specified motif in the network is in a certain state). These equations generally involve not only the base-motif itself, but larger or extended motifs of which they are usually part of. These larger motifs in turn depend on more complex motifs and a closure is needed in order to obtain a self-contained system of equations of reasonable size. Importantly the base-motif determines not only the complexity of the model (the larger the motif the greater the number of states it can be in) but also how much of the network topology can be captured. Interestingly differential equations for smaller motifs that are part of the base-motif should, in theory, be recoverable from the original differential equation. To this end the main focus of the paper is the consideration of various simple models of disease dynamics and the relations between them. We also consider which models are derivable directly (subject to a suitable closure) from the Kolmogorov/master equations and can thus be referred to as exact.We begin in section 2 with an introduction of some of the more common approaches to modelling disease dynamics on networks, considering meanfield ([1]), pairwise ([15]), heterogeneous pairwise ([7]) and the effective degree ([17]) model formulations. In section 3 we formulate an exact version of the effective degree model and then illustrate how the pairwise model can then be recovered from this new set of equations. We are, however, unable to recover the heterogenous pairwise model from the exact effective degree and this motivates, in section 4, an extension of this which incorporates further network topology into the ODEs. From this extension we then show how it is then possible to recover the heterogeneous pairwise equations. Once the links between the models have been established, in section 5 we show how the unclosed version of the models can be derived directly from the Kolmogorov equations. This is done by proving that as long as the heuristic equations for any motif are written following a certain set of rules they will always be exact. We conclude, in section 6 with a brief comparison of the models and discuss under what circumstances they perform best, in the sense of being close to simulation results.
Introduction
Modeling the spread of infectious diseases requires an understanding of not only disease characteristics but also an understanding of the community (be it a hospital, school, town, etc) in which it pervades. An important consideration in modelling the spread of diseases is thus the contact structure on which disease transmission happens. Whereas traditional approaches ( [2,6]) assume little or no topological structure, recent work ( [15,16,17]) has tried to incorportate the underlying linkages between entities in the population and study how these links facilitate the spread of the disease. For a continuous-time stochastic disease transmission model on an arbitrary network it is possible ( [13]), to write down the relevant Kolmogorov/master equations and thus model it as a continuous time Markov chain that fully describes the movement between all possible system states. Unfortunately the complexity of the model comes from the size of the state space and the number of equations scales exponentially as a N , where a is the number of different states a node can be in and N is the network size. One widely used resolution to this complexity is to create individual-based simulation models and investigate the system behaviour directly. Even though increasing computational power makes simulations an increasingly attractive proposition they lack analytic tractability. Whilst this is not always a hindrance, when the system displays a rich range of behaviour (e.g. oscillations, bistability) it may not be feasable to obtain a global overview of the effects of different parameter values and thus the more analytic approach is needed. For this reason, low-dimensional systems of differential equations ( [15,7,17]) are sought provided that these can approximate the exact solution. By reducing the problem to a smaller system of equations it is easier to study the
Models of disease dynamics
In this paper we focus on susceptible → infected → susceptible (SIS) disease dynamics on networks but note that all of the following models can be adapted for other disease (e.g. SIR and/or contact tracing) or non-disease (e.g. evolutionary [10]) dynamics. With this in mind we use τ as the per-link transmission rate between susceptible and infected nodes and γ as the recovery rate of an infected individual. Both infection and recovery are modelled as independent poisson processes. As a starting point we give a short summary of ODE-based models that are either exact or an approximation of the true dynamics resulting from the full system based on the Kolmogorov/master equations, where these are solvable, or based on simulation.
Pairwise and the resulting simple compartmental model
In order to focus on the underlying network of contacts, we introduce the pairwise model first ( [15,21]). The main idea of this model is to develop the hierarchical dependence of lower order moments (e.g. expected number of susceptible [S] and infected [I] nodes) on higher ones (e.g. expected number of pairs with one susceptible and one infected node, [SI]) and to derive appropriate models that correctly account for these. As already suggested, the expected number of pairs will depend on larger motifs, in this case these being the expected number of triples denoted by [ABC], where A, B,C ∈ {S, I} and B is connected to A and C. Using this notation the equations governing the evolution of the disease dynamics at the level of singles and pairs are given by d dt
[I] = −γ [I] + τ [SI] ,(1)
where the widely used transmission rate from the compartmental model, [1], is β = τn. It is also important to note that the unclosed equations above (Eqs. (1-4)) can be derived directly from the statebased Kolmogorov equations and for this reason we refer to these equations as exact. Whilst a proof for the exactness of these equations was given in [23], in section 5 we provide a more general proof that allows us to write down exact equations for, not just pairs, but any motif structure. We also note that an alternative approach was used by Sharkey in [22], to prove that the standard pairwise equations were exact for models with susceptible → infected → recovered (SIR) disease dynamics.
Heterogeneous pairwise model
Whilst the pairwise equations perform well in capturing disease dynamics on networks that are well described by their average degree, the closure assumption fails when greater heterogeneity is introduced. More precisely, whilst the pairwise equations above are exact for an arbitrary network before a closure, these do not guarantee that with the current choice of singles and pairs (i.e. [S] could be further divided to account for heterogeneity in degree) a valid closure could be found for any network. Indeed, to account for greater heterogeneity Eames et al. [7] further developed the pairwise model by taking into account not just the state of nodes and pairs but also the degrees of the nodes. By using [A n ] to represent expected number of nodes of type A with degree n and with similar notation for pairs and triples, they were able to formulate the following set of unclosed equations
d dt [S n ] = γ [I n ] − τ q [S n I q ],(6)d dt [I n ] = −γ [I n ] + τ q [S n I q ],(7)d dt [S n S m ] = −τ q ([S n S m I q ] + [I q S n S m ]) + γ ([S n I m ] + [I n S m ]) ,(8)d dt [S n I m ] = τ q ([S n S m I q ] − [I q S n I m ]) − τ [S n I m ] − γ [S n I m ] + γ [I n I m ] ,(9)d dt [I n I m ] = τ q ([I n S m I q ] + [I q S n I m ]) + τ [I n S m ] + τ [S n I m ] − 2γ [I n I m ] .(10)
Again assuming the statistical independence of pairs and absence of clustering, Eames et. al, [7], suggest the following approximations of triples
[B n C m D p ] ≈[B n C m ](m − 1) [C m D p ] m[C m ] .
The effective degree model
In [17], Lindquist et al. introduced the effective degree model for SIS (and also SIR) dynamics on a network (an equivalent model formulation was also proposed by Marceau et al. [18]). In this model they consider not only the state of a node (S or I), but also the number of the immediate neighbours in the various potential states. This is done by writing the following set of equations for all the possible star-like motifs in the network where S s,i (I s,i ) represents the expected number of susceptible (infected) nodes with s susceptible and i infected neighbours,
S s,i = − τiS s,i + γI s,i + γ[(i + 1)S s−1,i+1 − iS s,i ] + τ M k=1 j+l=k jlS j,l M k=1 j+l=k jS j,l [(s + 1)S s+1,i−1 − sS s,i ],(11)I s,i =τiS s,i − γI s,i + γ[(i + 1)I s−1,i+1 − iI s,i ] + τ M k=1 j+l=k l 2 S j,l M k=1 j+l=k jI j,l [(s + 1)I s+1,i−1 − sI s,i ],(12)with 1 ≤ s + i ≤ M,
where M is the maximum degree and the equations are suitably adjusted on the boundaries. It is important to note that this model is not exact as a closure has been already applied. Namely the infection of a node's susceptible neighbours is based on a population-level approximation. To illustrate this more precisely we borrow the notation of the pairwise model and make two observations These means that the infection pressure on the susceptible neighbours of the central node is equal to the population level average taken from all the possible star-like configurations rather then from the extended star structures that would account exactly for these infections.
Recovering the pairwise model from the effective degree
Whilst the pairwise and effective degree models seem different they are based on a similar approach. Both models work on approximating the evolution of different motifs in the network; individuals and links in the pairwise model and star-like structures in the effective degree. For both models, but more clearly for the pairwise, the models begin with a starting or base motif (e.g. nodes) for which an evolution equation is required. This will of course depend on an extended motif, typically the base motif extended by the addition of an extra node (e.g. pairs). This dependency on higher order motifs continues, for example, with pairs depending on triples, and then triples depending on quadruplets (four nodes connected by a line, i.e. A − B −C − D, or a star with a centre and three spokes, i.e. A − B − C D). Hence, the models only differ in the choice of the base motif and then potentially in the way in which the systems are closed to curtail the dependency on higher order motifs. Since, here we are mainly interested in exact models, that is before a closure is applied, we begin by conjecturing an exact version of the effective degree model and show how starting from this the exact pairwise model can be derived.
Exact effective degree
Based on the ideas presented above, we extend the star-like base motif to reveal the dependence on higher order motifs and conjecture that this unclosed version of the effective degree model is exact. We begin by introducing a variable to count the expected number of infecteds connected to a node's susceptible neighbours.
S s,i = − τiS s,i + γI s,i + γ[(i + 1)S s−1,i+1 − iS s,i ] + τ [ISS s+1,i−1 ] − τ [ISS s,i ] ,(13)I s,i =τiS s,i − γI s,i + γ[(i + 1)I s−1,i+1 − iI s,i ] + τ [ISI s+1,i−1 ] − τ [ISI s,i ] .(14)
Recovering the pairwise equations
The star-like composition of the effective degree model allows us to recover the pairwise equations via careful summations. The full derivation of the pairwise model is given in Appendix 2, whilst here we only illustrate the derivation of the individuals (trivial but given for completeness) and the [II] pairs,
d dt [S] = s,iṠ s,i = γ [I] − τ [SI] , d dt [I] = s,iİ s,i = −γ [I] + τ [SI],γ(i + 1)S s−1,i+1 −γiS si τ [ISS s+1,i−1 ] −τ [ISS s,i ]d dt [II] = s,i iİ s,i =τ i 2 S s,i − γ iI s,i + γ i(i + 1)I s−1,i+1 − γ i 2 I s,i + τ i[ISI s+1,i−1 ] − τ i[ISI s,i ] =τ i(i − 1)S s,i + τ iS s,i − γ[II] + γ[III] − γ i(i − 1)I s,i − γ iI s,i + τ (i − 1)[ISI s+1,i−1 ] + τ [ISI s+1,i−1 ] − τ i[ISI s,i ] =τ[ISI] + τ[IS] − γ[II] + γ[III] − γ[III] − γ[II] + τ[ISI] + τ[IS] =2τ ([ISI] + [IS])− 2γ[II],
where we have used that s,
i iI si = [II], s,i (i − 1)[ISI s+1,i−1 ] = i[ISI s,i ] and that [ISI s+1,i−1 ] = [ISI] + [SI]
. These all follow from the definition of the pairwise model and the definition of the new extended motifs from the exact effective degree model. We note that this result does indeed correspond to that of the given pairwise model.
Higher order models
Whilst we can recover the pairwise equations from the exact effective degree model we note that the same is not possible with the heterogeneous pairwise equations. This motivates an extension of the effective degree model where the degrees of neighbouring nodes are also taken in to account. Again we conjecture that this model can, in theory, be derived from the exact Kolmogorov equations and thus refer to it as exact.
Exact effective degree with neighbourhood composition
We extend the exact effective degree model to include the number of neighbours of the central nodes' neighbours. We begin by defining the following notation
s ′ = (s 1 , s 2 , . . . , s M ), i ′ = (i 1 , i 2 , . . . , i M ), |s ′ | = s 1 + s 2 + . . . + s M , |i ′ | = i 1 + i 2 + . . . + i M ,
where s j (i j ) represents the number of susceptible (infective) neighbours of degree j. We now define S s ′ i ′ , (I s ′ i ′ ) as the number of susceptible (infective) nodes with neighbouring nodes whose own degrees are given by the entries in s ′ and i ′ . We can now write the extended ODEs in the following forṁ
S s, ′ i ′ = − τ|i ′ |S s, ′ i ′ + γI s ′ ,i ′ + γ M k=1 (i k + 1)S s ′ k− ,i ′ k+ − γ|i ′ |S s ′ ,i ′ + τ M k=1 IS k S s ′ k+ ,i ′ k− − τ ISS s ′ ,i ′ ,(15)I s ′ ,i ′ =τ|i ′ |S s ′ ,i ′ − γI s ′ ,i ′ + γ M k=1 (i k + 1)I s ′ k− ,i ′ k+ − γ|i ′ |I s ′ ,i ′ + τ M k=1 IS k I s ′ k+ ,i ′ k− − τ ISI s ′ ,i ′ .(16)
Here s ′ k− = (s 1 , s 2 , . . . , s k − 1, . . . , s M ) and s ′ k+ = (s 1 , s 2 , . . . , s k + 1, . . . , s M ) with a similar definition for i ′ k− and i ′ k+ . With a small modification to the exact effective degree notation terms such as IS k S s ′
k+ ,i ′ k−
are taken to represent number of infectious contacts of the susceptible neighbours of degree k.
Model recovery
Here we show how, from the extended effective degree model, we can recover the heterogenous pairwise model. It is also straightforward to show, and thus omitted here, that the extended effective degree leads to the simpler exact effective degree. In turn, it also follows easily that both the exact effective degree and heterogenous pairwise models reduce to the standard pairwise model. This hierarchy of recovery is illustrated in Fig. 2.
Recovering the heterogeneous pairwise model from the extended effective degree
As earlier we make use of careful summation to recover the model. The full derivation is provided in Appendix 3 so here we just provide the derivation at the individual level and of the [I l I n ] pairs. For singles the following identities hold,
d dt [S n ] = |s ′ |+|i ′ |=nṠ s ′ ,i ′ = γ [I n ] − τ [S n I] , d dt [I n ] = |s ′ |+|i ′ |=nİ s ′ ,i ′ = −γ [I n ] + τ [S n I] ,
where most terms from the original effective degree cancel and we have used that
|s ′ |+|i ′ |=n I s ′ ,i ′ = [I n ] and |s ′ |+|i ′ |=n |i ′ |S s ′ ,i ′ = [S n I].
For the [I l I n ] pair we obtain
d dt I l I n = |s ′ |+|i ′ |=n i lİs ′ ,i ′ =τ i l |i ′ |S s ′ ,i ′ − γ i l I s ′ ,i ′ + γ i l M k=1 (i k + 1)I s ′ k− i ′ k+ − γ i l |i ′ |I s ′ ,i ′ + τ i l M k=1 IS k I s ′ k+ ,i ′ k− − τ i l ISI s ′ ,i ′ =τ i l |i ′ | − 1 S s ′ ,i ′ + τ i l S s ′ ,i ′ − γ I l I n + γ I l I n I − γ i l |i ′ | − 1 I s ′ ,i ′ − γ i ′ l I s ′ ,i ′ + τ i l k =l IS k I s ′ k+ ,i ′ k− + τ (i l − 1) IS l I s ′ l+ ,i ′ l− + τ IS l I s ′ l+ ,i ′ l− − τ i l ISI s ′ ,i ′
=τ I l S n I + τ I l S n − 2γ I l I n + γ I l I n I − γ I l I n I + τ IS l I n + τ S l I n =τ I l S n I + τ I l S n − 2γ I l I n + τ IS l I n + τ S l I n =τ q I l S n I q + I q S l I n + τ I l S n + τ S l I n − 2γ I l I n .
Again, we note that this result corresponds to previously given heterogenous pairwise model.
Exactness of the models
In the previous sections we have at times referred to a set of ODEs as being exact. This terminology implies that the ODEs can be derived directly from the Kolmogorov equations which describe the evolution of the epidemic through the full state space S (on a network of size N, S = {S, I} N ). In [23] the exactness of the pairwise equations was rigorously proven but no other motif structures were considered. In section 3.1, we conjectured that the newly defined exact effective degree model is derivable from the Kolmogorov equations. Due to the structure of the motifs used in the effective degree model a mechanistic proof (as in [23]) may be difficult and intricate to implement. Instead we will prove that a heuristic formulation of the ODEs for any motif structure is indeed exact providing they are written following rigorous bookkeeping. This derivation of the evolution equations for an arbitrary motif, directly from the Kolmogorv equations, will be based on an extension of ideas presented in [13] and [23] and using the notation defined in Tables 1 and 2.
We should note that in what follows a motif of connected nodes will only ever be counted once. In a network of size N and considering a motif, m, with k nodes this singular counting can be understood in the following way. We consider each of the N k unique sets of k nodes between 1 and N. Then for each set whose nodes are isomorphic in topological structure and status to the motif m, we simply increase the counter of such motifs by one. This formalism is unlike that used in the standard pairwise model where an SS link would contribute a value of two to the [SS] count. However, the two resultant sets of equations are equivalent in the sense that the different ways of counting can easily be recovered by using a simple mapping between the two. For this reason, whilst we prove that the following theorem is correct, it's intricacy and generality means a certain amount of care is needed when interpreting the resultant terms. Using the notation defined in Table 2 the result for a general motif is then given in the following theorem.
G = (g i j ) ∈ {0, 1} N 2 , i, j = 1,X k (t) X k (t) = X k 1 (t), X k 2 (t), . . . , X k c k (t) T .
Theorem 1
The equation for the expected number (|M|) of motifs of typem, given bẏ
|M| =τN SI in (m − ,m) + τN SI ex (m − ,m) − τ|M|N SI in (m) − τN SI ex (m) + γN I (m + ,m) − γ|M|N I (m)(17)
is derivable directly from the exact Kolmogorov equations.
Proof of Theorem 1
For a detailed description of writing the Kolmogorov equations for an arbitrary graph we refer the reader to [13]. Here we only provide a brief description making use of the notation defined in Table 1. The 2 N elements of the state space, S = {S, I} N , can be divided into N + 1 subsets where, for 0 ≤ k ≤ N, S k is the subset of all states with k infected nodes. Necessarily each subset contains c k = N k distinct configurations, i.e. S k = (S k 1 , S k 2 , . . . , S k c k ). The state of the system can only ever change in one of two ways, either via the infection of a node or via the recovery of a node. We can describe the evolution in the state space by a continuous time Markov-process. Setting X k j (t) as the probability of the system being in state S k j at time t and letting X k (t) = (X k 2 (t), X k 2 (t), . . . , X k c k (t) we can then write the Kolmogorov The set of motifs, in configuration state S k j , with the same topology asm but with 1 more infective and 1 less susceptible. Defining the i th element of M + k, j asm i+ k, j gives
M + k, j = {m 1+ k, j ,m 2+ k, j , . . . ,m |M + k, j |+ k, j }. M − k, j
The set of motifs, in configuration state S k j , with the same topology asm but with 1 less infective and 1 more susceptible. Defining the i th element of
M − k, j asm i− k, j we have M − k, j = {m 1− k, j ,m 2− k, j , . . . ,m |M − k, j |− k, j }.
Nm(S k j )
Number ofm motifs in state S k j , with k = 0, 1, . . . , N and j = 1, 2, . . . , c k . N I (ĥ, k) Number of I nodes within motifĥ, whose recovery lead to a motif of type k.
N I (ĥ, k) Expected total number of Is within motifs of typeĥ, whose recovery lead to a motif of type k. equations that capture the two possible transitions in the following matrix and vector form,
X k (t) = B 0 X 0 + C 0 X 1 if k = 0, A k X k−1 + B k X k + C k X k+1 for 1 ≤ k ≤ N − 1, A N X N−1 + B N X N if k = N
Here the matrices A k capture the transitions into S k via infection, C k capture the transitions into S k via recovery and B k captures transitions within S k . Their entries are given as follows:
• A k i, j is the rate of transition from S k−1 j to S k i , where k = 0, 1, . . . , N, i = 1, 2, . . . , c k and j = 1, 2, . . . , c k−1 . Note that none-zero entries of the matrix represent the transitions where only one individual is changing from susceptible to infected and the corresponding entrance will then equal τ multiplied by the number of infectious neighbours of the susceptible. These matrices encode the topological structure of the network.
• C k i, j is the rate of transition from S k+1 j to S k i , where k = 0, 1, . . . , N, i = 1, 2, . . . , c k and j = 1, 2, . . . , c k+1 . Note that none-zero entries of the matrix represent the transitions where only one individual is changing from infected to susceptible and the corresponding entrance will then equal γ.
• B k i, j is the rate of transition from S k j to S k i where B k i, j = 0 if i = j with k = 0, 1, . . . , N and i, j = 1, 2, . . . , c k .
Letting X(t) = (X 0 (t), X 1 (t), . . . , X N (t)) T , we then write Kolmogorov equations in the following block tridiagonal form,Ẋ = PX, where
P = B 0 C 0 0 0 0 0 A 1 B 1 C 1 0 0 0 0 A 2 B 2 C 2 0 0 0 0 A 3 B 3 C 3 0 0 0 . . . . . . . . . 0 0 0 0 0 A N B N .
From [13], we also know that the entries of the matrix B are zero except on the diagonals, where we find that
B k j j = − c k+1 i=1 A k+1 i, j − c k−1 i=1 C k−1 i, j = −τN SI (S k j ) − kγ.(18)
Where [13] focussed on individual and edge motifs here we focus on the derivation of evolution equations for the expected number of an arbitrary motif,m. We begin by writing the exact equations for an arbitrary motifm based on the transition and recovery matrices. Using the notation from Table 2 this yields,
|M| = N k=0 Nm(S k )Ẋ k =Nm(S 0 ) B 0 X 0 + C 0 X 1 + N−1 k=1 Nm(S k ) A k X k−1 + B k X k + C k X k+1 + Nm(S N ) A N X N−1 + B N X N = N k=1 Nm(S k )A k X k−1 + N k=0 Nm(S k )B k X k + N−1 k=0 Nm(S k )C k X k+1 = N−1 k=0 Nm(S k+1 )A k+1 X k + N k=0 Nm(S k )B k X k + N k=1 Nm(S k−1 )C k−1 X k = Nm(S 1 )A 1 + Nm(S 0 )B 0 X 0 + N−1 k=1 Nm(S k+1 )A k+1 + Nm(S k )B k + Nm(S k−1 )C k−1 X k + Nm(S N )B N + Nm(S N−1 )C N−1 X N .(19)
Before continuing we note the following
B N = B N 1,1 = − N i=1 C N−1 i,1 = −γN, B 0 = B 0 1,1 = − N i=1 A 1 i,1 = −τN SI (S 0 1 ) = 0.
Taking these and (18) into account and using the fact that B is only none zero on it's diagonal, we then obtain the following equation,
|M| =Nm(S 1 )A 1 X 0 + N−1 k=1 Nm(S k+1 )A k+1 − τ Nm(S k ) * N SI (S k ) − γkNm(S k ) + Nm(S k−1 )C k−1 ] X k + Nm(S N−1 )C N−1 − γNNm(S N ) X N = N−1 k=1 Nm(S k+1 )A k+1 − τ Nm(S k ) * N SI (S k ) X k − N k=1 γkNm(S k ) − Nm(S k−1 )C k−1 X k .(20)
We note that the term containing X 0 vanishes because A 1 is a column vector with all zero entries. We now consider the summations involving the A and C matrices and the state S k j . In this state there are k infected and N − k susceptible individuals. Without loss of generality the susceptible individuals are numbered 1 to N − k and the infected numbered from N − k + 1 to N. Defining r t to be the number of infective neighbours of the node numbered t we then obtain:
Nm(S k+1 )A k+1 j = c k+1 i=1 Nm(S k+1 i )A k+1 i, j =r 1 τ Nm(S k j ) + (Nm(S k+1 )A k+1 j =τN SI (S k j )Nm(S k j ) + τ |M − k, j | i=1 N SI in (m i− k, j ,m) + N SI ex (m i− k, j ,m) −τ|M k, j |N SI in (m) − τ |M k, j | i=1 N SI ex (m i k, j ) .
Similarly,
Nm(S k−1 )C k−1 j = c k−1 i=1 Nm(S k−1 i )C k−1 i, j
=γ Nm(S k j ) + (number ofm gained by node (N − k + 1) recovering ) −(number ofm lost by node (N − k + 1) recovering )] +γ Nm(S k j ) + (number ofm gained by node (N − k + 2) recovering ) −(number ofm lost by node (N − k + 2) recovering )] + . . .
Nm(S k−1 )C k−1 j =γkNm(S k j ) + γ |M + k, j | i=1 N I (m i+ k, j ,m) − γ|M k, j | N I (m) . Defining A k+1 j = τ |M − k, j | i=1 N SI in (m i− k, j ,m) + N SI ex (m i− k, j ,m) − τ|M k, j |N SI in (m) − τ |M k, j | i=1 N SI ex (m i k, j ) C k−1 j = γ |M + k, j | i=1 N I (m i+ k, j ,m) − γ|M k, j | N I (m) and setting A k+1 = [A k+1 1 , A k+1 j , . . . , A k+1 c k ] and C k−1 = [C k−1 1 ,C k−1 j , . . . ,C k−1 c k−1 ] yields, |M| = N−1 k=1 Nm(S k+1 )A k+1 − τ Nm(S K ) * N SI (S k ) X k − N k=1 γkNm(S k ) − Nm(S k−1 )C k−1 X k = N−1 k=1 τ Nm(S K ) * N SI (S k ) + A k+1 − τ Nm(S K ) * N SI (S k ) X k − N k=1 γkNm(S k ) − kNm(S k ) + C k−1 X k = N−1 k=1 A k+1 X k + N k=1 C k−1 X k = N−1 k=1 c k j=1 A k+1 j X k j + N k=1 c k j=1 C k−1 j X k j = N−1 k=1 c k j=1 τ |M − k, j | i=1 N SI in (m i− k, j ,m) + N SI ex (m i− k, j ,m) − τ|M k, j |N SI in (m) − τ |M k, j | i=1 N SI ex (m i k, j ) X k j + N k=1 c k j=1 γ |M + k, j | i=1 N I (m i+ k, j ,m) − γ|M k, j | N I (m) X k j =τN SI in (m − ,m) + τN SI ex (m − ,m) − τ|M|N SI in (m) − τN SI ex (m) + γN I (m + ,m) − γ|M|N I (m).
Which matches equation 17 from Theorem 1. It is worth noting that our result is related to the equation for the "expectation of some average quantity" given in [21]. However, whilst the result in [21] is very general here we provide a proof by construction that, for a given motif, pinpoints the events that influence these motif and their rates.
Using the Theorem to prove the conjectured exact effective degree model is derivable from the Kolmogorov equations
Lettingm be an S s,i -type motif from the effective degree model earlier and using Theorem 1, we find that the exact equations can be written as
dS s,i dt =τN SI in (m − ,m) + τN SI ex (m − ,m) − τ|M|N SI in (m) − τN
Comparison of the closed models
In comparing the models the obvious question to ask is when does one model perform better than another, i.e. which model approximates better or more accurately the simulation results or the solution of the Kolmogorov/master equations where solvable. As discussed earlier, the pairwise model is known to perform well on networks that are well characterised by the average degree (i.e. regular random and Erdős-Rényi graphs). What is less known is under what circumstances do the heterogenous pairwise and effective degree models outperform one another.
To assess the performance of the three closed models we compared individual simulations to the solutions of the ODE's on four different types of undirected network. For each of the different types of networks we used the Gillespie algorithm, [8], to run 500 simulations on networks of size N = 500 (1 simulation on 500 different randomly generated networks). The results of these simulations were then averaged to obtain an expected value to compare to the solution of the various ODE's. We began by considering regular random networks where all nodes have the same number of randomly chosen neighbours and then Erdős-Rényi random networks where the distribution of degrees converges to a Poisson distribution. Figure 3 plots simulation results against the different solutions of the ODEs for these two networks. On the regular network, whilst the two different pairwise models and the effective degree offer an improvement in performance over the standard meanfield equations, there is little to distinguish between the improved approaches. As expected, on the Erdős-Rényi random networks, the pairwise model improves on the meanfield model and, in turn, the effective degree and heterogeneous pairwise models improve even further on this. Again, however, there is little to distinguish between effective degree and the heterogeneous pairwise models.
To investigate further we ran simulations on networks exhibiting greater heterogeneity in their degree distribution. Firstly we considered networks with degrees between 1 and 25 chosen from a powerlaw degree distribution (p(x) = Ax −1.5 ) and generated by the configuration model algorithm [20]. Networks with scale-free like degree distributions may be more closely related to those of real world networks ( [4]) and may thus be of greater use in understanding the applicability of more theoretical modelling approaches. Secondly we considered graphs with the same power law degree distributions as before but this time rewired based on the "greedy" assortativity algorithm (discussed in [25]). This rewiring leads to an increase in the assortativity coefficient ( [19]) which measures the propensity of nodes of similar degrees to attach to one another. In theory, we should be able to capture this correlation with the heterogenous pairwise equations as they explicitly take the degree of connected nodes into account within the initial conditions. The results are illustrated in Figure 4.
Whilst on the powerlaw network network there is little difference between heterogeneous pairwise and effective degree when the assortativity is increased, there is a clear improvement in the performance of the heterogenous pairwise model over the effective degree. Any performance benefit must, however, be considered in terms of the model complexity given in table 3 (note that in this table M is the maximum possible degree in the network and we given the minimum number of equations needed to implement the ODEs).
meanfield 1 O(1) pairwise 3 O(1) effective degree M(M + 3) − 1 O(M 2 ) heterogeneous pairwise 2M(M + 1) − 1 O(M 2 ) Kolmogorov equations 2 N O(2 N )
A final comparison between the performance of the different closed models is to look at their rate of convergence to the solution of the Kolmogorov equations on a complete (fully connected) network. On a complete network it is possible (see [13]) to reduce the full system of 2 N equations to just N + 1 equations. This allows us to compare the true solution to the approximate solution of the meanfield, pairwise (equivalent to heterogenous pairwise on a complete graph) and effective degree models. Interestingly we find that all three exhibit O(1/N) convergence, where although both pairwise and effective degree bring an improvement on meanfield, the difference between the convergence of the two is neglible and almost indecernible (see figure 5).
Discussion
In this paper we set out to achieve a greater understanding of the relation between some of the more common approaches to modelling disease dynamics. In doing so we conjectured an exact version of the effective degree model [17] and showed how this model could be used to recover the pairwise model [15]. We then extended this model to incorporate greater network structure and illustrated how, from this extension, we could then recover the heterogeneous pairwise model [7]. We then proved that the conjectured exact effective degree model was indeed exact by proving that a heuristic derivation of an ODE model for an arbitrary motif was derivable directly from the Kolmogorov equations and noting that the exact effective degree model was just a particular case of this heuristic model. Finally we considered the performance of the different models on four different type of networks and have analysed numerically the rate of convergence to the lumped Kolmogorov equations on a complete network. These comparisons suggest a performance hierarchy of models as illustrated in Figure 6 and it is worth noting that the performance benefit of the heterogenous pairwise model on networks exhibiting susceptible → infectious → removed (SIR) disease dynamics was also touched upon in [5].
Whilst we have shown how current models can be extended in a way that can capture more network topology, these extensions have a more theoretical rather than practical motivation as their added complexity makes them not only less tractable but also more resource intensive in their solving, thus making the use of simulations more of an attractive proposition. As the links between these models are better understood, future work will likely focus on the following three areas. Firstly, a more realistic network will have a more clique-like structure. For example an individual is likely a member of a household in which he has regular contacts within and less regular contacts outside. Being able to incorporate this household structure within epidemic models is thus important in understanding the outbreak and necessary curtailment of an infectious disease (see [3,11,24]). Secondly, a network of individuals is not well represented by a static network. An individual may have regular contact with few individuals but may create or break contacts with others in ways that a static network representation cannot capture. For this reason it is important to take into consideration not only the dynamics of the disease but also the dynamics of the network and how the two impact on one another (see [9,14]). Thirdly, assuming we can write down exact differential equations we have to close them in some way. Understanding the performance of current, and also the derivation of new closures, is arguably the most important task ahead as it is the closures that limit the performance of any system of ODEs.
Fig. 1
1shows the possible transitions captured by this model.
Figure 1 :
1Illustration of the transitions into and out of the S 2,1 class. Susceptible nodes are given in blue and infective nodes in red. Transitions into and out of the class are shown in grey and green, respectively. The corresponding terms of the general equation are also given. In Appendix 1 a similar illustration is given for a configuration with a centrally infectious node.
Figure 2 :
2Illustration of the hierarchial structure of model recovery. Links that are known are given by lines and knowledge of a nodes status is given by circles. Susceptible and infective nodes are shown in blue and red respectively. The upper level (a) represents the extended effective degree ODEs. The status of the central node is known along with that of it's neighbours and also their degrees. The secondary level is given by (b), the effective degree model where there is no knowledge of neighbours' degrees and (c), the heterogenous pairwise model where the number of pairs of nodes and their relative degree is know. The final level shown, (d), is known as the standard pairwise model,[15], where the status of individual nodes and pairs is used.
2, . . . , N Adjacency matrix with g i j = 1 if nodes i and j are connected and g i j = 0 otherwise. The network is bi-directional and has no self loops such that G = G T and G ii = 0, ∀ i. τ Rate of infection per (S, I) edge. γ Rate of recovery. S = {S, I} N State space of the network, with nodes either susceptible, S, or infected, I and |S| = 2 N . S k = {S k 1 , S k 2 , . . . , S k c k } The c k = N k states with k infected individuals in all possible configurations, with k = 0, 1, . . . , N. X k j (t) Probability of being in state S k j at time t, where k = 0, 1, . . . , N and j = 1, 2, . . . , c k .
Rate of transition fromS k−1 j to S k i , where k = 0, 1, . . . , N, i = 1, 2, .. . , c k and j = 1, 2, . . . , c k−1 . Note that only one individual is changing (i.e. in this case an S node changes to an I through infection).
Rate of transition fromS k+1 j to S k i , where k = 0, 1, . . . , N, i = 1, 2, .. . , c k and j = 1, 2, . . . , c k+1 . Note that only one individual is changing (i.e. in this case an I node changes to an S through recovery).
Rate of transition fromS k j to S k i , where B k i, j = 0 if i = j with k = 0, 1, . . . ,N and i, j = 1, 2, .. . , c k .
number ofm gained by node 1 becoming infected ) −(number ofm lost by node 1 becoming infected )] +r 2 τ Nm(S k j ) + (number ofm gained by node 2 becoming infected ) −(number ofm lost by node 2 becoming infected )] + . . .+r N−k τ Nm(S k j ) + (number ofm gained by node (N − k) becoming infected ) −(number ofm lost by node (N − k) becoming infected )] =r 1 τ Nm(S k j ) + (number of elements of M − k, j where node 1 is susceptible and where node 1 ′ s infection would lead to a motif of typem) −(number of elements of M k, j where node 1 is susceptible ) +r 2 τ Nm(S k j ) + (number of elements of M − k, j where node 2 is susceptible and where node 2 ′ s infection would lead to a motif of typem) −(number of elements of M k, j where node 2 is susceptible ) + . . . +r N−k τ Nm(S k j ) + (number of elements of M − k, j where node (N − k) is susceptible and where node (N − k) ′ s infection would lead to a motif of typem) −(number of elements of M k, j where node (N − k) is susceptible ) , grouping the terms we obtain,
+γ
Nm(S k j ) + (number ofm gained by node (N) recovering ) −(number ofm lost by node (N) recovering )] =γ Nm(S k j ) + (number of elements of M + k, j where node (N − k + 1) is infective and where node (N − k + 1) ′ s recovery would lead to a motif of typem) −(number of elements of M k, j of which node (N − k + 1) belongs ) +γ Nm(S k j ) + (number of elements of M + k, j where node (N − k + 2) is infective and where node (N − k + 2) ′ s recovery lead to a motif of typem) −(number of elements of M k, j of which node (N − k + 2) belongs ) + . . . +γ Nm(S k j ) + (number of elements of M + k, j where node (N) is infective and where node N ′ s recovery would lead to a motif of typem) −(number of elements of M k, j of which node (N) belongs ) , grouping the terms we obtain
+
γN I (m + ,m) − γ|M|N I (m) =τ × (the total expected number of SI connections within S s+1,i−1 -type motifs where if infection occurs we obtain a S s,i -type motif ) + τ × (the total expected number of SI connections where S lies within S s+1,i−1 -type motifs and the I is external to the given motif and where, were an infection to occur, we obtain a S s,i -type motif ) − τS s,i × (number of SI connections within an individual S s,i -type motifs) − τ × (the total expected number of SI connections where S belongs to S s,i -type motifs and the I is external to the given motif ) + γ × (the total expected number I's within S s−1,i+1 -type and I s,i -type motifs where there recovery would give a S s,i -type motif ) − γS s,i × (number of I within an individual S s,i-type motif) =τ [ISS s+1,i−1 ] − τiS s,i − τ [ISS s,i ] + γI si + γ(i + 1)S s−1,i+1 − γiS s,iwhich is indeed the conjectured exact equation for S s,i (similar derivation holds for I s,i ). To clarify the above derivation we note that a term such as τN SI in (m − ,m) will make no contribution to the resultant equation as there are no internal SI connections within S s−1,i+1 -type motifs along which an infection would lead to an S s,i -type motif. However other terms, such as τN SI ex (m − ,m), have a direct correspondence with the resultant output (in this case the τ [ISS s+1,i−1 ] term).
Figure 3 :
3ODE performance on different networks Each network is of size N = 500 and with disease parameters given by γ = 1 and τ = 0.5. Average prevalence was calculated from individual simulations on 500 different networks. Initial conditions for the ODEs were obtained by averaging the initial conditions from each of the simulations. (a) Regular random network, each node having degree 7. (b) Erdős-Rényi random network with average degree 7.
Figure 4 :Figure 5 :
45ODE performance on different networks. Each network is of size N = 500 and with disease parameters given by γ = 1 and τ = 0.5. Average prevalence was calculated from individual simulations on 500 different networks. Initial conditions for the ODEs were obtained by averaging the initial conditions from each of the simulations. (a) Network with degrees chosen from a power law distribution (b) Networks with degrees chosen from a powerlaw distribution but rewired to have assortativity coefficient r ≈ 0.Convergence to exact solution on a complete graph. Absolute difference between the exact steady state solution of the percentage of infected individuals and those calculated from three different ODE models for 10 different network sizes and initial prevalence of 40 percent. Black triangles represent meanfield, blue circles effective degree and red squares the pairwise equations. Linear lines of best fit are also shown. This shows that the error(N) appears to be of O(1/N) as N tends to infinity.
Figure 6 :
6Model performance hierarchy. Model performance hierarchy based on our observations. Here K represents the Kolmogorov equations and HetPW the heterogeneous pairwise equations.
Figure 7 :
7Illustration of the transitions into and out of the I 2,1 class. Susceptible nodes are given in blue and infective nodes in red. Transitions into and out of the class are shown in grey and green, respectively. The corresponding terms of the general equation are also given.
Here we consider ordered pairs and triples, meaning they are counted in both directions. For pairs (with similar remarks for triples) this means [IS] = [SI] and links of type S − S and I − I have a double contribution to the [SS] and [II] counts. Importantly we note that these equations are unclosed as no equations are given for the evolution of the triples. The standard closure (in the absence of clustering) makes the assumption that the status of pairs are statistically independent of one another and then where n is the average degree of the network. When we use this closure we say we have closed "at the level of triples" . In order to derive the classic mean-field model a closure at the level of paris can be applied, namely, [SI] can be approximated asd
dt
[SS] = −2τ[ISS] + 2γ[IS],
(2)
d
dt
[SI] = τ ([ISS] − [ISI] − [IS])+ γ ([II] − [IS]),
(3)
d
dt
[II] = 2τ ([ISI] + [SI])− 2γ[II].
(4)
[ABC] ≈[AB](n − 1)
[BC]
n[B]
,
[SI] ≈ n[S]
[I]
N
and upon using Eq. (1), the classic mean-field model can be recovered
d
dt
[I] = −γ [I] + τn [S]
[I]
N
,
This is done by introducing two new terms, [ISS s ′ ,i ′ ] and [ISI s ′ ,i ′ ]. For the term [ISI s ′ ,i ′ ] (and similarly for [ISS s ′ ,i ′]) the S in the middle is actually used to represent the susceptible neighbours of the central I from the motif with composition I s ′ i ′ (i.e. the I node with neighbourhood (s ′ , i ′ ) is the centre of the star, while S is a susceptible spoke). The I (on the left-hand side), in turn, represents the infective neighbours of these susceptibles' and within this count, in the case of [ISI s ′ ,i ′ ], we also include the originating central I. The exact effective degree model can then be written aṡ
Table 1 :
1Notation for matrix representation of the Kolmogorov equations (Table from[23]).Variable
Definition
N
Number of nodes in the network
Table 2 :
2Additional notation for matrix representation of the Kolmogorov equations VariableDefinition m An arbitrary motif encompassing both topology and status of nodes (e.g. an S − I edge or a star like structure such as I 3,0 ). The arbitrary motif we are consdering which will encompass both topology and status of nodes.m +
Represents the different motifs with the same structure asm but with a susceptible
node ofm having become infected.
m −
Represents the different motifs with the same structure asm but with with an in-
fective node ofm having become susceptible.
M k, j
Set ofm motifs in configuration state S k
j . Defining the i th element of M k, j asm i
k, j
gives M k, j = {m 1
k, j ,m 2
k, j , . . . ,m
|M|
k, j }.
M +
k, j
Number of SI links within the motifĥ. Expected total number of SI links within all motifs of typeĥ, along which, were an infection to occur, would result in a motif of type k. Number of SI links where the S is contained within the motifĥ and the I is external to it. N SI ex (ĥ) Expected total number of SI links to all motifs with structureĥ, where the S is contained within the motifĥ and the I external to it. Number of SI links where the S is contained within the motifĥ and the I is external to it, along which, were an infection to occur, would result in a motif of type k. Expected total number of SI links to all motifs with structureĥ, where the S is contained within the motifĥ and the I external to it, along which, were an infection to occur, would result in a motif of type k.N SI
in (ĥ)
N SI
in (ĥ)
Expected total number of SI links within all motifs of typeĥ
N SI
in (ĥ, k) Number of SI links within the motifĥ, along which, were an infection to occur,
would result in a motif of type k.
N SI
in (ĥ, k) N SI
ex (ĥ)
N SI
ex (ĥ, k) N SI
ex (ĥ, k) N I (ĥ)
Number of I nodes within motifĥ.
Table 3 :
3Complexity of closed ODEs Model # equations complexity
Appendix 1Illustration of the exact effective degree transitions where the central node is infective.Appendix 3Derivation of the heterogeneous pairwise equations from the effective degree with neighbourhood composition model. For singles and pairs the following identities hold,=τ IS n S l − γ S l I n + γ S l I n I + γ I l I n − γ II n S l − τ IS l I n − τ S l I n =τ IS n S l − τ IS l I n − τ S l I n + γ I l I n − γ S l I n , dt I l I n = |s ′ |+|i ′ |=n=τ I l S n I + τ I l S n − 2γ I l I n + γ I l I n I − γ I l I n I + τ IS l I n + τ S l I n =τ I l S n I + τ I l S n − 2γ I l I n + τ IS l I n + τ S l I n .
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Detection of C2, CN and CH radicals in the spectrum of the transiting hot Jupiter HAT-P-1b B.E. Zhilyaev, M. V. Andreev, S. N. Pokhvala, I. A. Verlyuk M ainAstronomicalObservatory, N AS of U kraine, Zabalotnoho 27, 03680, Kyiv, U kraine [email protected] this paper we report spectroscopy of the transiting hot Jupiter HAT-P-1b. The HAT-P-1b is a giant (R = 1.2RJ), low-mean density transiting extrasolar planet in a visual binary system, composed of two sun-like stars. The host star HAT-P-1b known as ADS 16402 B is a G0V C dwarf (V = 9.87). We revealed optical emission of C2, CN and CH radicals in the spectrum of the hot Jupiter HAT-P-1b. We discovered radial pulsation of the hot Jupiter HAT-P-1b with a period of about 1900 sec.
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[
"https://arxiv.org/pdf/2202.11803v1.pdf"
] | 247,084,135 |
2202.11803
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d4baa36e37d67f93e5c3bf3b7557215f18fd94fa
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23 Feb 2022
23 Feb 2022methods: observationalstars: individual: ADS 16402 Btechniques: imaging spectroscopy
Detection of C2, CN and CH radicals in the spectrum of the transiting hot Jupiter HAT-P-1b B.E. Zhilyaev, M. V. Andreev, S. N. Pokhvala, I. A. Verlyuk M ainAstronomicalObservatory, N AS of U kraine, Zabalotnoho 27, 03680, Kyiv, U kraine [email protected] this paper we report spectroscopy of the transiting hot Jupiter HAT-P-1b. The HAT-P-1b is a giant (R = 1.2RJ), low-mean density transiting extrasolar planet in a visual binary system, composed of two sun-like stars. The host star HAT-P-1b known as ADS 16402 B is a G0V C dwarf (V = 9.87). We revealed optical emission of C2, CN and CH radicals in the spectrum of the hot Jupiter HAT-P-1b. We discovered radial pulsation of the hot Jupiter HAT-P-1b with a period of about 1900 sec.
introduction
The C 2 Swan bands were observed in the spectra of some stars [9] and comets [5]. The C 2 in the emission spectrum of a comet are excited through resonance fluorescence with the sunlight [12]. Stockhausen & Osterbrock [15] predicted from calculations with a simple molecule model that the vibrational temperature for cometary C 2 should be close to the solar color temperature and independent of the comet's heliocentric distance.
Spectroscopy of hot Jupiter HAT-P-1b during transits with the grating spectrograph STIS with R = λ/∆λ = 500 aboard the Hubble Space Telescope revealed strong optical absorbers (such as the Na I doublet at λ = 5893 [14]. The authors speculated that the best fit for the average dayside temperature of HAT-P-1b is 1500 ± 100 K. Optical emission of C 2 and other radicals were not detected.
Ground-based optical spectroscopy of hot Jupiter HAT-P-1b during transits had been observed with the Gemini Multi-Object Spectrograph (GMOS) instrument on the Gemini North telescope [16]. The authors found that the resulting transit spectrum is consistent with previous Hubble Space Telescope observations. However, the authors do not detect the Na resonance absorption line at λ = 5893 . Their 520 -950 nm observations reach a precision comparable to that of HST transit spectra in a similar wavelength range. However, GMOS transit between 320 -800 nm suffers from strong systematic effects and yields larger uncertainties.
Transmission spectroscopy has proven a powerful method to study the atmospheres of transiting exoplanets. This technique uses the differing wavelength-dependence of absorption and scattering processes in planetary atmospheres, resulting in a wavelength-dependent planetary radius [1].
Our goal is to study the emission spectrum of the planet. Emission spectra are observed in the atmospheres of comets excited by solar radiation.
An atom or molecule in open space can be detected by means of resonant absorption and reemission of electromagnetic waves, known as resonance fluorescence, which is a fundamental phenomenon of quantum optics. In three-dimensional (3D) space, experimentally achieved extinction can reach 12% of transmitted power [3], [7], [13].
In this work, optical emission of the C 2 , CN and CH radicals in the spectrum of the hot Jupiter HAT-P-1b was discovered. Section 2 describes a series of spectra obtained with a low-resolution spectrograph. Our study of hot Jupiter HAT-P-1b during transit focused on the structure of the C 2 bands. In the following sections, we describe the analysis techniques used. The observed sequence C 2 ∆ν = −1 was used in Section 3 to estimate the excitation temperature. The features of the spectra and the main results are discussed in Section 4.
OBSERVATIONS AND DATA PROCESSING
The hot-Jupiter HAT-P-1b is one of the first known exoplanet. Planetary parameters are [14]: Period P = 4.46529976 ± (55) days. A semi-major axis 0.05561 AU. Mass M P = 0.525M J. Radius RP = 1.319RJ. Density ρ ∼ 0.282 g cm −3 . Equilibrium temperature T eq 1322 ± 15 K. Incident flux F = 0.699 · 10 9 erg s −1 cm −2 . Total transit duration 0.11875 days.
HAT-P-1b was observed at the Terskol Observatory (North Caucasus, 3100 m at sea level) with the Carl Zeiss 0.6-meter telescope with an imaging slitless spectrograph [18]. The spectrograph provides a spectral resolution of R ≃ 150 for 10 m stars with a moderate signal-to-noise ratio. A series of 900 measurements were obtained with a temporal resolution of 6.6 s (Fig. 1).
The
Diagnostics of radical emissions
The spectra of the flame of carbon compounds were studied in Johnson [10]. It was concluded that Swan bands are clearly visible in the flame of burning hydrocarbons. The author gives detail of all the structures at a wavelength of the head of the λλ4382, 4737, 5165, 5635, 6191 groups. Our task is to find own variability in the spectral lines, using the difference in the noise spectrum and the spectrum of the useful signal. The solution of the problem is carried out in stages:
• Elimination of interference caused by changes in atmospheric transparency using the automatic gain control method.
• Alignment of data using the cross-correlation method to eliminate shifts caused by instrument bending and atmospheric refraction.
• Calculation of the spectra map, the matrix whose rows are wavelengths, and the columns are the power spectra of the intensity variations at each spectrum wavelength.
Detecting the emission component in the spectrum is complicated, especially if its intensity is comparable to that of noise. The task is somewhat simplified for broadband emission. The spectra have been averaged to detect very weak emission components. A special fitting procedure based on the algorithm developed by the authors was used.
To eliminate the absorption spectrum, the high pass FIR filter was used. We use a Kaiser window to design the filter with pass band frequency 0.2 px −1 , pass band ripple 0. 5 26 dB [11]. The filter transmission curves are shown in Figure 2. The spatial frequency is in pixels, the pixel is equal to 3 Angstroms in the wavelength scale.
The digital filtering method makes it possible to find a weak emission component in the spectrum. A significant amount of data (900 spectra) allowed us to detect an emission of several percent of the intensity of the continuum. Figure 3 demonstrates the operation of the algorithm on the entire array of HAT-P-1b spectra. The top figure saw the low-resolution absorption spectrum of HAT-P-1b. The Balmer hydrogen lines H α , H β , H γ are clearly seen, as well as the Na resonance absorption line at λ = 5893 . The bottom figure saw the emission in the spectrum of HAT-P-1b obtained as a result of filtration. It is seen that the emission peaks significantly exceed the measurement errors. Figure 4 demonstrates the averaged emission spectrum of HAT-P-1b with the measurement errors. Even a short examination of the emission flux distribution reveals several spectral features of HAT-P-1b (Fig. 3, 4). Three features due to C 2 are seen, corresponding to the Swan-band sequences with ∆ν = 1 (λ4425 -λ4629), C 2 λ5450 -λ5650 (∆ν = −1); C 2 λ5700 -λ5800 (∆ν = −1). Three features of CN are seen also, corresponding to λ3458, λ3510, and λ9258 (1,0). Fullerton [6] developed a method for detecting line variations, called "The Temporal Variance Spectrum" (TVS). This method compares statistically the deviations in the spectral lines with the deviations in nearby regions of the continuum. If the deviations in the spectral line are larger than the deviations in the continuum, then it can be argued that a variation is detected at a certain level of statistical significance.
The TVS technique with a signal-to-noise ratio (SNR) in spectra allows detection of variations in lines. The intensities of the pulsed flows are shown in Fig. 5. HAT-P-1b exhibit variations at the wavelengths of the λλ4382, 5165, 5635 C 2 groups, C 2 5700 -5800 .
The picture also demonstrates the activity of radicals at λλ3510 and 3800 wavelengths associated with CN (∆ν = 0) [4] and CH (as Solar CH λλ3557, 3779 ) [8]. Figure 6 shows variations in the map spectra of HAP-1b and companion star HAT-P-1a. One can see developed activity in the signal-to-noise ratio scale. Bottom picture reveals the activity in the wavelengths of the λλ4382, 5165, 5635 C 2 groups. The islands of activity (SNR > 10) with a period of about 1900 sec are probably associated with oscillation of the hot Jupiter HATP-1b. From Fig. 6 (top picture), we can conclude that the spectrum of the comparison star HAT-P-1a does not demonstrate intensity variations in the wavelengths of the C 2 groups. Low-frequency variations in the continuum is probably associated with self-variability. It can also be assumed that there is intrinsic variability in the lines of the Balmer series. Figure 8 and Figure 9 allows us to compare our data with a combination of the optical spectra of comet 109P/Swift-Tuttle obtained November 26, 1992, and the spectra of comets Shoemaker-Levy (1991a1) and Hartley-Good (1985).
One can see almost complete coincidence of the spectral features of C 2 ∆ν = −1 , wavelength range This similarity practically confirms our conclusions about the detection of radicals in the spectrum of the transiting hot Jupiter HAT-P-1b.
A well-known relationship that determines the pulsation period of stars is [17]:
P = Q(ρ ⊙ /ρ) 1 2
(1) where P denotes the pulsation period, and ρ ⊙ and ρ denote the mean solar and stellar densities. Essentially, this is free fall time. The theoretical value of Q for the standard model with a polytrope index n = 3 is 0.039 days [2]. For hot Jupiter HAT-P-1b, the average density is 0.282 g/cm −3 [14]. The pulsation period P is equal to 2122 sec and practically coincides with the observed one.
To determine the excitation temperature T ex we use the estimates of the spectrum intensity in the Swan band. We use the intensity ratio of the C 2 Swan system d 3 g ← a 3 u . Only pure vibrational transitions with ∆ν = −1 were considered. We have chosen two lines λλ5341 and 5494 as successive vibrational levels with intensity ratio 1.19 (Fig. 7, 0.64/0.55). The intensity ratio is then the population ratio for j=0 and j=1: n j=1 n j=0 = g 1 e −E1/kT g 0 e −E0/kT = 3 e −∆E/kT =
I 1 I 0(2)
where g 0 = 1, g 1 = 3 are the statistical weights, E i =1.99e-8/λ i is the energy of photon [2], k is the Boltzmann constant. This leads to the temperature
T ex = ∆E/k/ ln(3/(I 1 /I 0 ) = 814 K(3)
From the intensity ratio of the C 2 lines with j=0 and j=1 the excitation temperature of 814 K follows.
Conclusions
Spectroscopy of hot Jupiter HAT-P-1b during transit with the grating spectrograph with R = λ/∆λ ≃ 150 reveals the spectrum of non-resolved Swan bands C 2 λλ4425 -4629 (∆ν = 1); C 2 λλ5000 -5200 (∆ν = 0); C 2 λλ5450 -5650 (∆ν = −1); C 2 λλ5700 -5800 (∆ν = −1); CN λλ3448 -3509 and CH λ 3779 .
The TVS technique with a signal-to-noise ratio (SNR) in spectra allowed to detect variations in lines of radicals.
Calculation of the spectra map, that are the power spectra of the intensity variations at each spectrum wavelength allowed to find the islands of activity with a period of about 1900 sec associated, probably, with oscillation of the hot Jupiter HAT-P-1b.
For the standard model with a polytrope index n = 3 and the average density of 0.282 g cm −3 , the pulsation period of hot Jupiter HAT-P-1b P was appreciated equal to 2122 sec what practically coincides with the observed within 10%.
To determine the excitation temperature T ex we use the estimates of the spectrum intensity in the Swan band. We use the intensity ratio of the C 2 Swan system d 3 g ← a 3 u for sequence ∆ν = −1. From the intensity ratio of the C 2 lines with j=0 and j=1 a vibrational temperature of 814 K was appreciated.
We discovered radial pulsation of the hot Jupiter HAT-P-1b with a period of about 1900 sec.
references
start of exposure was on 27.11.2009, 17:22:16 UT. The observed minimum was at 17:48:40 UT. The calculated minimum from ephemeris is at JD = 2455163.24239 and corresponds to 17:49 UT on 27.11.2009. O -C = 20 sec.
Fig. 2 :
2The filter transmission curves in px −1 and px scale.
Fig. 3 :
3The absorption spectrum of HAT-P-1b (topfigure). The emission spectra of HAT-P-1b (bottomfigure).
Fig. 4 :
4The averaged emission spectrum of HAT-P-1b.
Fig. 5 :Fig. 6 :
56Variations in the emission spectrum of HAT-P-1b. The SNR map spectra of HAT-P-1a (top picture) and HAT-P-1b (bottom picture).
Fig. 7 :
7Normalized light curves of two lines λλ5341 and 5494 .
λλ5450 -5650 , (∆ν = −3), λ 6600 ; CN (2 -0), (3 -1) wavelength range λλ 7820 -8039 , CN (1 -0), (2 -1) wavelength range λλ 9109 -9223 .
Fig. 8 :
8Combination of the optical spectra of comet 109P/Swift-Tuttle and the spectra of comets Shoemaker-Levy (1991a1) andHartley-Good (1985). Arbitrary scale.
Fig. 9 :
9The emission spectrum of HAT-P-1b. Arbitrary scale.
dB, and stop band attenuation of4000
4500
5000
5500
6000
6500
0.97
0.975
0.98
0.985
0.99
0.995
1
Residual intensity
H γ
H
β
H
α
Mg I b
Na I 5890
4000
4500
5000
5500
6000
6500
0.96
0.98
1
1.02
1.04
1.06
1.08
λ Angstrom
Relative intensity
. D Alexander, J Ryan, K Joanna, arXiv:2104.10688v1[astro-ph.EPAlexander D., Ryan J.,Joanna K.,et al., 2021, arXiv:2104.10688v1 [astro-ph.EP] 21 Apr 2021
C W Allen, Astrophysical Quantities. LondonAthlone Press2d ed.Allen C. W., 1963, Astrophysical Quantities (2d ed.; London: Athlone Press).
. O Astafiev, A M Zagoskin, A A Abdumalikov, arXiv:1002.4944v1cond-mat.supr-conAstafiev O., Zagoskin A.M., Abdumalikov A.A., et al., 2010, arXiv:1002.4944v1 [cond-mat.supr-con] 26 Feb 2010
. J S A Brooke, ApJS. 21023Brooke J. S. A., 2014, ApJS, 210, 23
. K I Churyumov, L S Chubko, V O Ponomarenko, Astronomical Shool's Report. 72208Churyumov K.I., Chubko L.S., Ponomarenko V.O., 2011, Astronomical Shool's Report, 7, No 2, 208
. A Fullerton, Toronto Univ. (Ontario). Dept. of AstronomyPh.D. ThesisFullerton A., 1990, Ph.D. Thesis, Toronto Univ. (Ontario). Dept. of Astronomy
. Gerhardt I Wrigge, G Bushev, P , Phys. Rev. Lett. 9833601Gerhardt I., Wrigge G., Bushev P., et al., 2007,Phys. Rev. Lett. 98, 033601
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. P Mayer, C R Dell, ApJ. 153951Mayer P. and O'Dell C. R., 1968, ApJ., 153, 951
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. N Nikolov, D K Sing, F Pont, arXiv:1310.0083v1[astro-ph.SR]30preprintNikolov N., Sing D. K., Pont F.,et al., 2013, preprint(arXiv:1310.0083v1 [astro-ph.SR] 30 Sep 2013)
. R E Stockhausen, D E Osterbrock, ApJ. 141287Stockhausen R. E. and Osterbrock D. E., 1965, ApJ., 141, 287S
. K O Todorov, J M Desert, C M Huitson, A&A. 631169Todorov K.O., Desert J.M., Huitson C.M., et al., 2019, A&A, 631A, 169
Nonradial oscillation of srars. W Unno, J Osaki, H Ando, University of Tokio PressUnno W., Osaki J., Ando, H., et al., 1979, Nonradial oscillation of srars. University of Tokio Press.
. B E Zhilyaev, A V Sergeev, M V Andreev, KFNT. 29120Zhilyaev B.E., Sergeev A.V., Andreev M.V. et al. , 2013, KFNT, 29, 120
|
[] |
[
"Assessing generalisability of deep learning-based polyp detection and segmentation methods through a computer vision challenge",
"Assessing generalisability of deep learning-based polyp detection and segmentation methods through a computer vision challenge"
] |
[
"Sharib Ali \nInstitute of Biomedical Engineering\nDepartment of Engineering Science\nUniversity of Oxford\nOX3 7DQOxfordUnited Kingdom\n\nOxford National Institute for Health Research Biomedical Research Centre\nOX4 2PGOxfordUnited Kingdom\n\nBig Data Institute\nLi Ka Shing Centre for Health Information and Discovery\nUniversity of Oxford\nOX3 7LFOxfordUnited Kingdom\n",
"Noha Ghatwary \nComputer Engineering Department\nArab Academy for Science and Technology\n1029AlexandriaEgypt\n",
"Debesh Jha \nSimulaMet\n0167OsloNorway\n\nDepartment of Computer Science\nUiT The Arctic University of Norway\nHansine Hansens veg 189019TromsøNorway\n",
"Ece Isik-Polat \nGraduate School of Informatics\nMiddle East Technical University\n06800AnkaraTurkey\n",
"Gorkem Polat ",
"Chen Yang \nGraduate School of Informatics\nMiddle East Technical University\n06800AnkaraTurkey\n\nCity University of Hong Kong\nKowloon, Hong Kong\n",
"Wuyang Li \nCity University of Hong Kong\nKowloon, Hong Kong\n",
"Adrian Galdran \nDept. of Information and Communication Technologies\nBCN MedTech\nUniversitat Pompeu Fabra\n08018BarcelonaSpain\n",
"Miguel-Ángel González Ballester \nDept. of Information and Communication Technologies\nBCN MedTech\nUniversitat Pompeu Fabra\n08018BarcelonaSpain\n",
"Vajira Thambawita \nSimulaMet\n0167OsloNorway\n",
"Steven Hicks \nSimulaMet\n0167OsloNorway\n",
"Sahadev Poudel \nDepartment of IT Convergence Engineering\nGachon University\n13120SeongnamRepublic of Korea\n",
"Sang-Woong Lee \nDepartment of IT Convergence Engineering\nGachon University\n13120SeongnamRepublic of Korea\n",
"Ziyi Jin \nCollege of Biomedical Engineering and Instrument Science\nZhejiang University\n310027HangzhouChina\n",
"Tianyuan Gan \nCollege of Biomedical Engineering and Instrument Science\nZhejiang University\n310027HangzhouChina\n",
"Chenghui Yu \nTsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenChina\n",
"Jiangpeng Yan \nDepartment of Automation\nTsinghua University\n100084BeijingChina\n",
"Doyeob Yeo \nSmart Sensing & Diagnosis Research Division\nKorea Atomic Energy Research Institute\n34057Republic of Korea\n",
"Hyunseok Lee \nDaegu-Gyeongbuk Medical Innovation Foundation\nMedical Device Development centre\n427724Republic of Korea\n",
"Nikhil Kumar Tomar \nNepAL Applied Mathematics and Informatics Institute for Research (NAAMII)\nKathmanduNepal\n",
"Mahmood Haithmi \nComputer Science Department\nUniversity of Nottingham\nMalaysia Campus43500SemenyihMalaysia\n",
"Amr Ahmed \nComputer Science Department\nUniversity of Nottingham\nMalaysia Campus43500SemenyihMalaysia\n",
"Michael A Riegler \nSimulaMet\n0167OsloNorway\n\nDepartment of Computer Science\nUiT The Arctic University of Norway\nHansine Hansens veg 189019TromsøNorway\n",
"Christian Daul \nCRAN UMR 7039\nUniversité de Lorraine\nCNRS\nF-54500Vandoeuvre-Lès-NancyFrance\n",
"Pål Halvorsen \nSimulaMet\n0167OsloNorway\n\nOslo Metropolitan University\nPilestredet 460167OsloNorway\n",
"Jens Rittscher \nInstitute of Biomedical Engineering\nDepartment of Engineering Science\nUniversity of Oxford\nOX3 7DQOxfordUnited Kingdom\n\nBig Data Institute\nLi Ka Shing Centre for Health Information and Discovery\nUniversity of Oxford\nOX3 7LFOxfordUnited Kingdom\n",
"Osama E Salem \nFaculty of Medicine\nUniversity of Alexandria\n21131AlexandriaEgypt\n",
"Dominique Lamarque \nUniversité de Versailles St-Quentin en Yvelines\nHôpital Ambroise Paré\n\n",
"Renato Cannizzaro \nCRO Centro Riferimento Oncologico IRCCS Aviano Italy\nVia Franco Gallini, 233081AvianoPNItaly\n",
"Stefano Realdon \nVeneto Institute of Oncology IOV-IRCCS\nVia Gattamelata, 6435128PaduaItaly\n\nAv. Charles de Gaulle\n92100Boulogne-BillancourtFrance\n",
"Thomas De Lange \nMedical Department\nSahlgrenska University Hospital-Mölndal\nBlå stråket 5413 45GöteborgSweden\n\nDepartment of Molecular and Clinical Medicine\nSahlgrenska Academy\nUniversity of Gothenburg\n41345GöteborgSweden\n\nAugere Medical\nNedre Vaskegang 60186OsloNorway\n",
"James E East \nOxford National Institute for Health Research Biomedical Research Centre\nOX4 2PGOxfordUnited Kingdom\n\nDepartment of Medicine\nExperimental Medicine Div\nTranslational Gastroenterology Unit\nNuffield\n\nJohn Radcliffe Hospital\nUniversity of Oxford\nOX3 9DUOxfordUnited Kingdom\n"
] |
[
"Institute of Biomedical Engineering\nDepartment of Engineering Science\nUniversity of Oxford\nOX3 7DQOxfordUnited Kingdom",
"Oxford National Institute for Health Research Biomedical Research Centre\nOX4 2PGOxfordUnited Kingdom",
"Big Data Institute\nLi Ka Shing Centre for Health Information and Discovery\nUniversity of Oxford\nOX3 7LFOxfordUnited Kingdom",
"Computer Engineering Department\nArab Academy for Science and Technology\n1029AlexandriaEgypt",
"SimulaMet\n0167OsloNorway",
"Department of Computer Science\nUiT The Arctic University of Norway\nHansine Hansens veg 189019TromsøNorway",
"Graduate School of Informatics\nMiddle East Technical University\n06800AnkaraTurkey",
"Graduate School of Informatics\nMiddle East Technical University\n06800AnkaraTurkey",
"City University of Hong Kong\nKowloon, Hong Kong",
"City University of Hong Kong\nKowloon, Hong Kong",
"Dept. of Information and Communication Technologies\nBCN MedTech\nUniversitat Pompeu Fabra\n08018BarcelonaSpain",
"Dept. of Information and Communication Technologies\nBCN MedTech\nUniversitat Pompeu Fabra\n08018BarcelonaSpain",
"SimulaMet\n0167OsloNorway",
"SimulaMet\n0167OsloNorway",
"Department of IT Convergence Engineering\nGachon University\n13120SeongnamRepublic of Korea",
"Department of IT Convergence Engineering\nGachon University\n13120SeongnamRepublic of Korea",
"College of Biomedical Engineering and Instrument Science\nZhejiang University\n310027HangzhouChina",
"College of Biomedical Engineering and Instrument Science\nZhejiang University\n310027HangzhouChina",
"Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenChina",
"Department of Automation\nTsinghua University\n100084BeijingChina",
"Smart Sensing & Diagnosis Research Division\nKorea Atomic Energy Research Institute\n34057Republic of Korea",
"Daegu-Gyeongbuk Medical Innovation Foundation\nMedical Device Development centre\n427724Republic of Korea",
"NepAL Applied Mathematics and Informatics Institute for Research (NAAMII)\nKathmanduNepal",
"Computer Science Department\nUniversity of Nottingham\nMalaysia Campus43500SemenyihMalaysia",
"Computer Science Department\nUniversity of Nottingham\nMalaysia Campus43500SemenyihMalaysia",
"SimulaMet\n0167OsloNorway",
"Department of Computer Science\nUiT The Arctic University of Norway\nHansine Hansens veg 189019TromsøNorway",
"CRAN UMR 7039\nUniversité de Lorraine\nCNRS\nF-54500Vandoeuvre-Lès-NancyFrance",
"SimulaMet\n0167OsloNorway",
"Oslo Metropolitan University\nPilestredet 460167OsloNorway",
"Institute of Biomedical Engineering\nDepartment of Engineering Science\nUniversity of Oxford\nOX3 7DQOxfordUnited Kingdom",
"Big Data Institute\nLi Ka Shing Centre for Health Information and Discovery\nUniversity of Oxford\nOX3 7LFOxfordUnited Kingdom",
"Faculty of Medicine\nUniversity of Alexandria\n21131AlexandriaEgypt",
"Université de Versailles St-Quentin en Yvelines\nHôpital Ambroise Paré\n",
"CRO Centro Riferimento Oncologico IRCCS Aviano Italy\nVia Franco Gallini, 233081AvianoPNItaly",
"Veneto Institute of Oncology IOV-IRCCS\nVia Gattamelata, 6435128PaduaItaly",
"Av. Charles de Gaulle\n92100Boulogne-BillancourtFrance",
"Medical Department\nSahlgrenska University Hospital-Mölndal\nBlå stråket 5413 45GöteborgSweden",
"Department of Molecular and Clinical Medicine\nSahlgrenska Academy\nUniversity of Gothenburg\n41345GöteborgSweden",
"Augere Medical\nNedre Vaskegang 60186OsloNorway",
"Oxford National Institute for Health Research Biomedical Research Centre\nOX4 2PGOxfordUnited Kingdom",
"Department of Medicine\nExperimental Medicine Div\nTranslational Gastroenterology Unit\nNuffield",
"John Radcliffe Hospital\nUniversity of Oxford\nOX3 9DUOxfordUnited Kingdom"
] |
[] |
Polyps are well-known cancer precursors identified by colonoscopy. However, variability in their size, location, and surface largely affect identification, localisation, and characterisation. Moreover, colonoscopic surveillance and removal of polyps (referred to as polypectomy ) are highly operator-dependent procedures. There exist a high missed detection rate and incomplete removal of colonic polyps due to their variable nature, the difficulties to delineate the abnormality, the high recurrence rates, and the anatomical topography of the colon. There have been several developments in realising automated methods for both detection and segmentation of these polyps using machine learning. However, the major drawback in most of these methods is their ability to generalise to out-of-sample unseen datasets that come from different centres, modalities and acquisition systems. To test this hypothesis rigorously we curated a multi-centre and multi-population dataset acquired from multiple colonoscopy systems and challenged teams comprising machine learning experts to develop robust automated detection and segmentation methods as part of our crowd-sourcing Endoscopic computer vision challenge (EndoCV) 2021. In this paper, we analyse the detection results of the four top (among seven) teams and the segmentation results of the five top teams (among 16). Our analyses demonstrate that the top-ranking teams concentrated on accuracy (i.e., accuracy > 80% on overall Dice score on different validation sets) over real-time performance required for clinical applicability. We further dissect the methods and provide an experiment-based hypothesis that reveals the need for improved generalisability to tackle diversity present in multi-centre datasets.Detection and localisationWhile classification methods are frame-based classifiers for polyps[17][18][19], detection methods provide both classification and localisation of polyps 3, 4 which can direct clinicians to the site of interest, and can be additionally used for counting polyps to assess disease burden in patients. With the advancements of object detection architectures, recent methods are end-to-end networks providing better detection performance and improved speed. The state-of-the-art methods are broadly divided into two categories: multi-stage detectors and single-stage detectors. The multi-stage detector methods include Region proposals-Based Convolutional Neural Network (R-CNN) 20 , Fast R-CNN 21 , Faster R-CNN 22 , Region-based fully convolutional networks (R-FCN) 23 , Feature Pyramid Network (FPN) 24 and Cascade R-CNN 25 . On the other hand, the One-stage detectors directly provide the predicted output (bounding boxes and object classification) from input images without the region of interest (ROI) proposal stage. The One-stage detector methods include Single-Shot Multibox Detector (SSD) 26 , Yolo 27 , RetinaNet 28 and Efficientdet 29 . Different studies have been conducted in the literature that focused on polyp detection by employing both multi-stage detectors and single-stage detectors. Multi-stage Detectors: Shin et al. 30 used a transfer learning strategy based on Faster R-CNN architecture with the Inception ResNet backbone to detect polyps. Qadir et al. 4 adapted Mask R-CNN 31 to detect colorectal polyps and evaluate its performance with different CNN including ResNet50 32 , ResNet101 32 and Inception ResNetV2 33 as its feature extractor. Despite the speed limitation, multi-stage detectors are widely used in the detection task of endoscopy data challenges due to their competitive performance on evaluation metrics. Single-stage Detectors: Urban et al. 3 used YOLO to detect polyps in real-time, which also resulted in high detection performance. Lee et al. 34 employed YOLOv2 35 and validated the proposed approach on four independent dataset. They reported a real-time performance and high sensitivity and specificity on all datasets. Zhang et al.36proposed the ResYOLO network that adds residual learning modules into the YOLO architecture to train deeper networks. They reported a near-real-time performance for the ResYOLO network depending on the hardware used. Zhang et al.5proposed an enhanced SSD named SSD for Gastric Polyps (SSD-GPNet) for real-time gastric polyp detection. SSD-GPNet concatenates feature maps from lower layers and deconvolves higher layers using different pooling techniques. YOLOv3 37 with darknet53 backbone and YOLOv4 showed IOU and average precision (AP) over 0.80% and real-time FPS over 45. Moreover, there exist methods that relied on anchor-free detectors to locate the polyps where they claim to detect polyps without the definition of anchors such as CornerNet 38 and ExtremeNet 39 . Zhou et al.40proposed the CenterNet, which treats each object as a point and increases the speed significantly while ensuring the accuracy is acceptable. While Wang et al. 41 achieved state-of-the-art results on automatic polyp detection in real-time situations using anchor-free object detection methods. In addition to these works, Multi-stage, Single-stage and other types of detectors have been widely used by participants teams in different polyp detection datasets and challenges such as MICCAI'15 42 , ROBUST-MIS 43 , EAD2019 15 and EndoCV2020 9 .SegmentationSemantic segmentation is the process of grouping related pixels in an image to an object of the same category. Deep learning has been very successful in the field of the medical domain, convolutional neural networks (CNN) based techniques were suggested to generate complete and precise segmentation outputs without requiring any post-processing. In deep learning, medical segmentation methods can be categorized into four categories: Models based on fully convolutional networks, Models based on Encoder-Decoder architecture, Models based on Pyramid-based architecture and Models based on Dilated Convolution Architecture.Models based on fully convolutional networks: Brandao et al.44proposed three different FCN-based architectures for detection and segmentation of polyps from colonoscopy images. Zhang et al.45proposed multi-step practice for the polyp segmentation. The former step includes region proposal generation using FCN, and the latter step uses spatial features and a random forest classifier for the refinement process. A similar method was introduced by Akbari et al.46which uses patch selection while training FCN and Otsu thresholding to find the accurate location of polyp. Guo et al. 7 describe two methods based on FCN for Gastrointestinal ImageANALysis (GIANA) polyp segmentation sub-challenge.Models based on encoder-decoder architecture: Nguyen and Slee 6 proposed multiple deep encoder-decoder networks to capture multi-level contextual information and learn rich features during training. Zhou et al.47proposed UNet++, a deeply supervised encoder-decoder network that showed improved performance on polyp segmentation task. Similarly, PraNet 48 aggregated deep features in their parallel partial decoder to form initial guidance area maps. Mahmud et al.49integrated dilated
| null |
[
"https://arxiv.org/pdf/2202.12031v1.pdf"
] | 247,084,187 |
2202.12031
|
a23b4ae7d078e92dd5420fe7eebc720d896658a4
|
Assessing generalisability of deep learning-based polyp detection and segmentation methods through a computer vision challenge
24 Feb 2022
Sharib Ali
Institute of Biomedical Engineering
Department of Engineering Science
University of Oxford
OX3 7DQOxfordUnited Kingdom
Oxford National Institute for Health Research Biomedical Research Centre
OX4 2PGOxfordUnited Kingdom
Big Data Institute
Li Ka Shing Centre for Health Information and Discovery
University of Oxford
OX3 7LFOxfordUnited Kingdom
Noha Ghatwary
Computer Engineering Department
Arab Academy for Science and Technology
1029AlexandriaEgypt
Debesh Jha
SimulaMet
0167OsloNorway
Department of Computer Science
UiT The Arctic University of Norway
Hansine Hansens veg 189019TromsøNorway
Ece Isik-Polat
Graduate School of Informatics
Middle East Technical University
06800AnkaraTurkey
Gorkem Polat
Chen Yang
Graduate School of Informatics
Middle East Technical University
06800AnkaraTurkey
City University of Hong Kong
Kowloon, Hong Kong
Wuyang Li
City University of Hong Kong
Kowloon, Hong Kong
Adrian Galdran
Dept. of Information and Communication Technologies
BCN MedTech
Universitat Pompeu Fabra
08018BarcelonaSpain
Miguel-Ángel González Ballester
Dept. of Information and Communication Technologies
BCN MedTech
Universitat Pompeu Fabra
08018BarcelonaSpain
Vajira Thambawita
SimulaMet
0167OsloNorway
Steven Hicks
SimulaMet
0167OsloNorway
Sahadev Poudel
Department of IT Convergence Engineering
Gachon University
13120SeongnamRepublic of Korea
Sang-Woong Lee
Department of IT Convergence Engineering
Gachon University
13120SeongnamRepublic of Korea
Ziyi Jin
College of Biomedical Engineering and Instrument Science
Zhejiang University
310027HangzhouChina
Tianyuan Gan
College of Biomedical Engineering and Instrument Science
Zhejiang University
310027HangzhouChina
Chenghui Yu
Tsinghua Shenzhen International Graduate School
Tsinghua University
518055ShenzhenChina
Jiangpeng Yan
Department of Automation
Tsinghua University
100084BeijingChina
Doyeob Yeo
Smart Sensing & Diagnosis Research Division
Korea Atomic Energy Research Institute
34057Republic of Korea
Hyunseok Lee
Daegu-Gyeongbuk Medical Innovation Foundation
Medical Device Development centre
427724Republic of Korea
Nikhil Kumar Tomar
NepAL Applied Mathematics and Informatics Institute for Research (NAAMII)
KathmanduNepal
Mahmood Haithmi
Computer Science Department
University of Nottingham
Malaysia Campus43500SemenyihMalaysia
Amr Ahmed
Computer Science Department
University of Nottingham
Malaysia Campus43500SemenyihMalaysia
Michael A Riegler
SimulaMet
0167OsloNorway
Department of Computer Science
UiT The Arctic University of Norway
Hansine Hansens veg 189019TromsøNorway
Christian Daul
CRAN UMR 7039
Université de Lorraine
CNRS
F-54500Vandoeuvre-Lès-NancyFrance
Pål Halvorsen
SimulaMet
0167OsloNorway
Oslo Metropolitan University
Pilestredet 460167OsloNorway
Jens Rittscher
Institute of Biomedical Engineering
Department of Engineering Science
University of Oxford
OX3 7DQOxfordUnited Kingdom
Big Data Institute
Li Ka Shing Centre for Health Information and Discovery
University of Oxford
OX3 7LFOxfordUnited Kingdom
Osama E Salem
Faculty of Medicine
University of Alexandria
21131AlexandriaEgypt
Dominique Lamarque
Université de Versailles St-Quentin en Yvelines
Hôpital Ambroise Paré
Renato Cannizzaro
CRO Centro Riferimento Oncologico IRCCS Aviano Italy
Via Franco Gallini, 233081AvianoPNItaly
Stefano Realdon
Veneto Institute of Oncology IOV-IRCCS
Via Gattamelata, 6435128PaduaItaly
Av. Charles de Gaulle
92100Boulogne-BillancourtFrance
Thomas De Lange
Medical Department
Sahlgrenska University Hospital-Mölndal
Blå stråket 5413 45GöteborgSweden
Department of Molecular and Clinical Medicine
Sahlgrenska Academy
University of Gothenburg
41345GöteborgSweden
Augere Medical
Nedre Vaskegang 60186OsloNorway
James E East
Oxford National Institute for Health Research Biomedical Research Centre
OX4 2PGOxfordUnited Kingdom
Department of Medicine
Experimental Medicine Div
Translational Gastroenterology Unit
Nuffield
John Radcliffe Hospital
University of Oxford
OX3 9DUOxfordUnited Kingdom
Assessing generalisability of deep learning-based polyp detection and segmentation methods through a computer vision challenge
24 Feb 2022† these authors contributed equally to this work
Polyps are well-known cancer precursors identified by colonoscopy. However, variability in their size, location, and surface largely affect identification, localisation, and characterisation. Moreover, colonoscopic surveillance and removal of polyps (referred to as polypectomy ) are highly operator-dependent procedures. There exist a high missed detection rate and incomplete removal of colonic polyps due to their variable nature, the difficulties to delineate the abnormality, the high recurrence rates, and the anatomical topography of the colon. There have been several developments in realising automated methods for both detection and segmentation of these polyps using machine learning. However, the major drawback in most of these methods is their ability to generalise to out-of-sample unseen datasets that come from different centres, modalities and acquisition systems. To test this hypothesis rigorously we curated a multi-centre and multi-population dataset acquired from multiple colonoscopy systems and challenged teams comprising machine learning experts to develop robust automated detection and segmentation methods as part of our crowd-sourcing Endoscopic computer vision challenge (EndoCV) 2021. In this paper, we analyse the detection results of the four top (among seven) teams and the segmentation results of the five top teams (among 16). Our analyses demonstrate that the top-ranking teams concentrated on accuracy (i.e., accuracy > 80% on overall Dice score on different validation sets) over real-time performance required for clinical applicability. We further dissect the methods and provide an experiment-based hypothesis that reveals the need for improved generalisability to tackle diversity present in multi-centre datasets.Detection and localisationWhile classification methods are frame-based classifiers for polyps[17][18][19], detection methods provide both classification and localisation of polyps 3, 4 which can direct clinicians to the site of interest, and can be additionally used for counting polyps to assess disease burden in patients. With the advancements of object detection architectures, recent methods are end-to-end networks providing better detection performance and improved speed. The state-of-the-art methods are broadly divided into two categories: multi-stage detectors and single-stage detectors. The multi-stage detector methods include Region proposals-Based Convolutional Neural Network (R-CNN) 20 , Fast R-CNN 21 , Faster R-CNN 22 , Region-based fully convolutional networks (R-FCN) 23 , Feature Pyramid Network (FPN) 24 and Cascade R-CNN 25 . On the other hand, the One-stage detectors directly provide the predicted output (bounding boxes and object classification) from input images without the region of interest (ROI) proposal stage. The One-stage detector methods include Single-Shot Multibox Detector (SSD) 26 , Yolo 27 , RetinaNet 28 and Efficientdet 29 . Different studies have been conducted in the literature that focused on polyp detection by employing both multi-stage detectors and single-stage detectors. Multi-stage Detectors: Shin et al. 30 used a transfer learning strategy based on Faster R-CNN architecture with the Inception ResNet backbone to detect polyps. Qadir et al. 4 adapted Mask R-CNN 31 to detect colorectal polyps and evaluate its performance with different CNN including ResNet50 32 , ResNet101 32 and Inception ResNetV2 33 as its feature extractor. Despite the speed limitation, multi-stage detectors are widely used in the detection task of endoscopy data challenges due to their competitive performance on evaluation metrics. Single-stage Detectors: Urban et al. 3 used YOLO to detect polyps in real-time, which also resulted in high detection performance. Lee et al. 34 employed YOLOv2 35 and validated the proposed approach on four independent dataset. They reported a real-time performance and high sensitivity and specificity on all datasets. Zhang et al.36proposed the ResYOLO network that adds residual learning modules into the YOLO architecture to train deeper networks. They reported a near-real-time performance for the ResYOLO network depending on the hardware used. Zhang et al.5proposed an enhanced SSD named SSD for Gastric Polyps (SSD-GPNet) for real-time gastric polyp detection. SSD-GPNet concatenates feature maps from lower layers and deconvolves higher layers using different pooling techniques. YOLOv3 37 with darknet53 backbone and YOLOv4 showed IOU and average precision (AP) over 0.80% and real-time FPS over 45. Moreover, there exist methods that relied on anchor-free detectors to locate the polyps where they claim to detect polyps without the definition of anchors such as CornerNet 38 and ExtremeNet 39 . Zhou et al.40proposed the CenterNet, which treats each object as a point and increases the speed significantly while ensuring the accuracy is acceptable. While Wang et al. 41 achieved state-of-the-art results on automatic polyp detection in real-time situations using anchor-free object detection methods. In addition to these works, Multi-stage, Single-stage and other types of detectors have been widely used by participants teams in different polyp detection datasets and challenges such as MICCAI'15 42 , ROBUST-MIS 43 , EAD2019 15 and EndoCV2020 9 .SegmentationSemantic segmentation is the process of grouping related pixels in an image to an object of the same category. Deep learning has been very successful in the field of the medical domain, convolutional neural networks (CNN) based techniques were suggested to generate complete and precise segmentation outputs without requiring any post-processing. In deep learning, medical segmentation methods can be categorized into four categories: Models based on fully convolutional networks, Models based on Encoder-Decoder architecture, Models based on Pyramid-based architecture and Models based on Dilated Convolution Architecture.Models based on fully convolutional networks: Brandao et al.44proposed three different FCN-based architectures for detection and segmentation of polyps from colonoscopy images. Zhang et al.45proposed multi-step practice for the polyp segmentation. The former step includes region proposal generation using FCN, and the latter step uses spatial features and a random forest classifier for the refinement process. A similar method was introduced by Akbari et al.46which uses patch selection while training FCN and Otsu thresholding to find the accurate location of polyp. Guo et al. 7 describe two methods based on FCN for Gastrointestinal ImageANALysis (GIANA) polyp segmentation sub-challenge.Models based on encoder-decoder architecture: Nguyen and Slee 6 proposed multiple deep encoder-decoder networks to capture multi-level contextual information and learn rich features during training. Zhou et al.47proposed UNet++, a deeply supervised encoder-decoder network that showed improved performance on polyp segmentation task. Similarly, PraNet 48 aggregated deep features in their parallel partial decoder to form initial guidance area maps. Mahmud et al.49integrated dilated
Introduction
Colorectal cancer (CRC) is the third leading cause of cancer deaths, with reported mortality rate of nearly 51% 1 . CRC can be characterised by early cancer precursors such as adenomas or serrated polyps that may over time lead to cancer. While polypectomy is a standard technique to remove polyps 2 by placing a snare (thin wire loop) around the polyp and closing it to cut though the polyp tissue either with diathermy (heat to seal vessels) or without (cold snare polypectomy), identifying small or flat polyps (e.g. lesion less than 10 mm) can be extremely challenging. This is due to complex organ topology of the colon and rectum that make the navigation and treatment procedures difficult and require expert-level skills. Similarly, the removal of polyps can be very challenging due to constant organ deformations which make it sometimes impossible to keep track of the lesion boundary making the complete resection difficult and subjective to experience of endoscopists. Computer-assisted systems can help to reduce operator subjectivity and enables improved adenoma detection rates (ADR). Thus, computer-aided detection and segmentation methods can assist to localise polyps and guide surgical procedures (e.g. polypectomy) by showing the polyp locations and margins. Some of the major requirements of such system to be utilised in clinic are the real-time performance and algorithmic robustness.
Machine learning, in particular deep learning, together with tremendous improvements in hardware have enabled the possibility to design networks that can provide real-time performance despite their computational complexity. However, one major challenge in developing these methods is the lack of comprehensive public datasets that include diverse patient population, imaging modalities and scope manufacturers. Incorporating real-world challenges in the dataset can only be the way forward in building guaranteed robust systems. In the past, there has been several attempts to collect and curate gastrointestinal (GI) datasets that include other GI lesions and polyps. A summary of existing related datasets with polyps are provided in Supplementary Table 1. A major limitation of these publicly available datasets is that they consists of either single center or data cohort representing a single population. Additionally, most widely used public datasets have only single frame and single modality images. Moreover, even though conventional white light endoscopy (WLE) is used in standard colonoscopic procedures, narrow-band imaging (NBI), a type of virtual chromo-endoscopy, is widely used by experts for polyp identification and charaterisation. Most deep learning-based detection [3][4][5] and segmentation [6][7][8][9] methods are trained and tested on the same center dataset and WLE modality only. These supervised deep learning techniques has a major issue in not being able to generalise to an unseen data from a different center population 10 or even different modality from the same center 11 . Also, the type of endoscope used also adds to the compromise in robustness. Due to selective image samples provided by most of the available datasets for method development, the test dataset also comprise of similarly collected set data samples 9,[12][13][14] . Similar to the most endoscopic procedures, colonoscopy is a continuous visualisation of mucosa with a camera and a light source. During this process live videos are acquired which are often corrupted with specularity, floating objects, stool, bubbles and pixel saturation 15 . The mucosal scene dynamics such as severe deformations, view-point changes and occlusion can be some major limiting factors for algorithm performance as well. It is thus important to cross examine the generalisability of developed algorithms more comprehensively and on variable data settings including modality changes and continuous frame sequences.
With the presented Endoscopic Computer Vision (EndoCV) challenge in 2021 we collected and curated a multicenter dataset 16 that is aimed at generalisability assessment. For this we took an strategic approach of providing single modality (white light endoscopy modality, WLE) data from five hospitals (both single frame and sequence) for training and validation while the test data consisted of four different configurations -a) mixed center unseen data with WLE modality with samples from centers in training data, b) a different modality data (narrow-band imaging modality, NBI) from all centers, c) a hidden sixth center single frame data and d) a hidden sixth center continuous frame sequence data. While, unseen data with centers included in training assesses the traditional way of testing the supervised machine learning methods on held-out data, unseen modality and hidden center data testing gauge the algorithm's generalisability. Similarly, sequence test data mimics the occurrence of polyps in data as observed in routine clinical colonoscopy procedure.
inception blocks into each unit layer and aggregate the features of the different receptive fields to capture better-generalized feature representations. Huang et al. 50 proposed a low memory traffic, fast and accurate method for the polyp segmentation achieving 86 frames per second (FPS). Later, Zhang et al. 8 proposed a hybrid method combining both transformer-based network and CNN to capture global dependencies and the low-level spatial features for the segmentation task. Inspired by high-resolution network 51 , Srivastava et al. 10 proposed multi-scale residual fusion network (MSRF-Net) that allows information exchange across multiple scales and showed improved generalisability on unseen datasets. All of these encoder-decoder architectures were evaluated only on still images. Ji et al. 52 proposed a progressively normalised self-attention network (PNS-Net) for video polyp segmentation.
Models based on pyramid-based architecture: Jia et al. 53 proposed a pyramid-based model named PLPNet for automated pixel-level polyp classification in colonoscopy images. Also, Guo et al. 54 employed the Pyramid Scene Parsing Network (PSPNet) 55 with SegNet 56 and U-Net 57 as an ensemble deep learning model. The proposed model achieved a improvement upto 6.38% compared with a single basic trainer.
Models based on dilated convolution architecture: Sun et al. 58 used dilated convolution in the last block of the encoder while Safarov et al. 59 used in all encoder blocks. Though 59 used a mesh of attention blocks and residual block as a decoder, both methods tested there model on CVC-ClinicDB achieving F1-score of 96.106 and 96.043, respectively. Furthermore, nested dilation network (NDN) 60 was designed to segment lesions and tested on the GIANA2018 dataset achieving improvements on Dice upto 3% compared to other methods.
The EndoCV challenge: dataset, evaluation and submission
In this section, we present the dataset compiled for the polyp generalisation challenge, the protocol used to obtain the ground truth, evaluation metrics that were defined to assess the participants methods and a brief summary on the challenge setup and ranking procedure.
Dataset and challenge tasks Dataset:
The EndoCV2021 challenge addresses the generalisability of polyp detection and segmentation tasks in endoscopy frames. The colonoscopy video frames utilised in the challenge are collected from six different centres including two modalities (i.e. WL and NBI) with both sequence and non-sequence frames (see Figure 1 a). The challenge includes five different types of data and participants were allowed to combine accordingly for their train-validation splits: i) multi-centre video frames from five centres for training and validation, ii) polyp size-based, iii) single frame and sequence data split, iv) modality split (i.e. only for testing phase) and v) one hidden centre test (test phase only). The training dataset consisted of a total of 3242 WL frames from five centers (i.e. C1-C5) with both single and sequence frames. The test dataset, consists of: a) dataset with unseen modality, NBI (data 1), b) dataset with single frames from unknown center (data 2), c) frame sequences from the mixed centers (C1-C5, data 3), and iv) the unseen center sequence frames (C6, data 4). A total of 777 frames were used and data 3 was picked as base dataset against which generalisability of methods were assessed. Polyp size distribution (see Figure 1, left (b)) and its size in log-scale on resized images of the same resolution (540 × 720 pixels) (see Figure 1 (b), right) in both training and test sets are presented. These sizes were divided into null (for no polyp in frames), small (< 100 × 100 pixels bounding box), medium (≤ 200 × 200 pixels polyp bounding box) and large (> 200 × 200 pixels polyp bounding box). These numbers were 534, 1129, 1224 and 705, respectively, for null, small, medium and large size polyps (accounting for 3058 polyp instances) in the training set. Similarly, for the test set the numbers were 134, 144, 296 and 261, respectively, for null, small, medium and large size polyps (in total 701 polyp instances). The size distribution in both of these datasets are nearly identical (see Figure 1 (b), right) which is due to the defined range for categorically representing their occurance.
Challenge tasks: EndoCV2021 included two tasks for which the generalisability assessment was also conducted (see Figure 2): 1) detection and localisation task and 2) pixel-level segmentation task. For the detection task, participants were provided both single and sequence frames with manually annotated ground truth polyp labels and their corresponding bounding boxes. Participants were required to train their model for predicting class labels, bounding box co-ordinates and confidence scores for localisation. For the segmentation task, we provided the pixel level ground truth segmentation from experts that included the same data as provided for the detection task. Here, the participants were challenged to obtain close to ground truth segmentation binary map prediction. Both of these challenge tasks were assessed rigorously to understand the generalisability of the developed methods. In this regard, the test data consisted of four different categories, here we call it as data 1, data 2, data 3 and data 4. Data 1 consisted of unseen modality with NBI data widely used in colonoscopy, data 2 comprises of single frames of unseen center C6, data 3 consisted of mixed seen center (C1-C5) sequence data whereas data 4 included sequence data from unseen center C6. For generalisabilty, we compared the scores between data 3 (seen center data) with the other unseen data categories. All test results were evaluated on a common NVIDIA Tesla V100 GPU. Further details on metric computation is provided in Section Evaluation metrics. Trained models were then tested on both seen and unseen center datasets and on unseen data modality (widely used narrow-band imaging). Generalisability assessment is obtained by computing deviations between these unseen samples w.r.t. seen samples. Task outputs included bounding box prediction with confidence and class label for detection task and binary mask prediction for polyp segmentation.
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Ethical and privacy aspects of the data
The Data for EndoCV2021 was gathered from 6 different centers located in five different countries (i.e. UK, Italy, France, Norway and Egypt). The ethical, legal and privacy of the relevant data was handled by each responsible center. The data collected from each center included a minimum of two essential steps as described below:
• Patient consenting procedure at each institution (required).
• Review of the data collection plan by a local medical ethics committee or an institutional review board.
• Anonymization of the video or image frames (including demographic information) before sending to the organizers (required). Table 1 illustrates the ethical and legal processes fulfilled by each center along with the details of the endoscopic equipment and recorders used for the collected data.
Annotation protocol
The annotation process was conducted by a team of three experienced researchers using an online annotation tool called Labelbox 1 . Each annotation was cross-validated by the team and by the center expert for accurate boundaries segmentation. At least one senior gastroenterologist was assigned for an independent binary review process. A set of protocols for manual annotation of polyp has been designed as follows: Table 1. Data collection information for each center: Data acquisition system and patient consenting information.
• Clear raised polyps: Boundary pixels should include only protruded regions. Precaution has to be taken when delineating along the normal colon folds
• Inked polyp regions: Only part of the non-inked appearing object delineation
• Polyps with instrument parts: Annotation should not include instrument and is required to be carefully delineated and may form more than one object
• Pedunculated polyps: Annotation should include all raised regions unless appearing on the fold • Flat polyps: Zooming the regions identified with flat polyps before manual delineation. Also, consulting center expert if needed.
The annotated masks where examined by experienced gastroenterologists who gave a binary score indicating whether a current annotation can be considered clinically acceptable or not. Additionally, some of the experts provided feedback on the annotation where these images were placed into an ambiguous category for further refinement based on the experts feedback. A detailed process along with the number of annotations conducted and reviewed is outline in Supplementary Figure 1.
Evaluation metrics
Polyp detection
For the polyp detection task, we have computed standard computer vision metrics such as average precision (AP) and intersection-of-union (IoU) 61 .
• IoU: The IoU metrics measures the overlap between two bounding boxes A and B as the ratio between the target mask and predicted output.
IoU(A,B) = A ∩ B A ∪ B(1)
Here, ∩ represents intersection and ∪ represents the union.
• AP: AP is computed as the Area Under Curve (AUC) of the precision-recall curve of detection sampled at all unique recall values (r1, r2, ...) whenever the maximum precision value drops:
AP = ∑ n (r n+1 − r n ) p interp (r n+1 ) ,(2)
with p interp (r n+1 ) = max r≥r n+1 p(r). Here, p(r n ) denotes the precision value at a given recall value. This definition ensures monotonically decreasing precision. AP was computed as an average APs at 0.50 and 0.95 with the increment of 0.05. Additionally, we have calculated AP small , AP medium , AP large . More description about the detection evaluation metrics and their formulas can be found here 2 .
H d (G, E) = 1 G ∑ g∈G min e∈E d(g, e) + 1 E ∑ e∈E min g∈G d(g, e) /2.(3)
The mean H d is normalised between 0 and 1 by dividing it by the maximum value H d max for a given test set.
Polyp generalisation metrics
We define the generalisability score based on the stability of the algorithm performance on seen white light modality and center dataset (data 3) with unseen center split (data 2 and data 4) and unseen modality (data 1) in the test dataset. We conducted the generalisability assessment for both detection and segmentation approaches separately. For detection, the deviation in score between seen and unseen data types are computed over different AP categories, k ∈ {mean, small, medium, large} with tolerance, (tl = 0.1):
dev_g = 1 |k| ∑ k |AP k seen − AP k unseen |, if AP k seen − tl * AP k seen ≤ AP k unseen ≥ AP k seen + tl * AP k seen 0, otherwise.(4)
Similarly, for segmentation, the deviation in score between seen and unseen data types are computed over different segmentation metric categories, k ∈ {DSC, F2, p, r, H d } with tolerance, (tl = 0.05):
dev_g = 1 |k| ∑ k |S k seen − S k unseen |, for S k seen − tl * S k seen ≤ S k unseen ≥ S k seen + tl * S k seen 0, otherwise.(5)
Challenge setup, and ranking procedure We set-up challenge website 3 with an automated docker system for metric-based ranking procedure. Challenge participants were required to perform inference on our cloud-based system that incorporated NVIDIA Tesla V100 GPU and provided test dataset with instructions of using directly on GPU without downloading the data for round 1 and round 2. However, we added an additional round 3 where the challenge participant's trained model were used for inference by the organisers on an additional unseen sequence dataset. Thus, the challenge consisted of three rounds. All provided test frames were from unseen patient data to avoid any data leakage. Further details on data samples in each round are summarised below:
• Round 1: Test subset-I consisted of: a) 50 samples of each data 1 (unseen modality), data 2 (unseen single sample, C6) and data 3 (mixed center C1-C5 sequence data)
• Round 2: Test subset-II comprised of: a) all 88 samples of each data 1 (unseen modality), 86 samples of data 2 (unseen single sample, C6) and 124 samples of data 3 (mixed C1-C5)
• Round 3: Inference on round 3 data was performed by the organisers on the same GPU. This round comprised of: a) 135 samples of each data 1 (unseen modality), 86 samples of data 2 (unseen single sample, C6), 124 samples of data 3 (mixed center C1-C5 sequence data) and an additional set of 432 sequence samples from unseen center C6.
We conducted elimination for both round 1 and round 2 which was based on the metric scores on the leaderboard and timely submission. In round 2, we eliminated those with very high computational time for inference and metric score consistency. The chosen participants were requested for the method description paper at the EndoCV proceeding 62 to allow transparent reporting of their methods. All accepted methods were eligible for round 3 evaluation and have been reported in this paper.
Detection ranking was performed as an aggregated score between the average precision and deviation scores between data 3 (seen C1-C5) w.r.t. other unseen data in the test set. Similarly, for team ranking on segmentation, we used segmentation metrics and deviation scores for segmentation (between seen and unseen data). Please refer to Section Evaluation metrics for details. An aggregated rank was used to announce winner. In our final ranking reported in this paper, we have additionally used inference time as well.
Method summary of the participants
Below, we summarise the EndoCV2021 generalisability assessment challenge for polyp detection and segmentation methods using deep learning. Tabulated summaries are also provided highlighting the nature of the devised methods and basis of choice in-terms of speed and accuracy for detection (see Table 2) and segmentation (see Table 3). Methods are detailed in the compiled EndoCV2021 challenge proceeding 62 .
Detection Task
• AIM_CityU: 63 The team used one-stage anchor-free FCOS 67 as the baseline detection algorithm and adopted ResNeXt-101-DCN with FPN as their final feature extractor. For the model optimization, both online (random flipping and multi-scale training) and offline (random rotation, gamma contrast, and brightness transformation, etc.) data augmentation strategies are performed to improve the model generalization.
• HoLLYS ETRI: 64 The team used Mask R-CNN 31 for detection and segmentation task. All the weights were initialized with pre-trained weights. An ensemble learning method based on 5-fold cross-validation was used to improve the generalization performance. While training a single Mask R-CNN, the data acquired from all data centers were not used. Instead, only the data acquired from four centers were used for training and, the data from remained center was used for validation. Ensemble inference was performed by combining the inference results of 5 models. For the detection task, weighted box fusion technique 68 was used to combining results of detection. For segmentation task, segmentation masks from 5 models were averaged with IoU threshold of 0.6.
• JIN_ZJU: 65 The team used the YoloV5 69 as the baseline detection algorithm. To improve the generalisation ability of the standard Yolov5, different data augmentation methods were applied that included hue adjustment, saturation adjustment, value adjustment, rotating, translation, scaling, up-down flipping, left-right flipping, mosaic and mixup.
• GECE_VISION: 66 The team proposed an ensemble-based polyp detection architecture using the EfficientDet 29 as the base model family with EfficientNet as backbone network. The bootstrap aggregating (bagging) was utilized to aggregate different versions of the predictors (EfficientDet D0, D1, D2, D3) which are trained on bootstrap replicates of the training set. In order to increase the variance and improve generalization capability of the model, data augmentation have been used (i.e., scale jittering with 0.2-2.0, horizontal flipping, and rotating between 0 • -360 • ). Adam optimiser and the scheduling learning rate were used with decreasing factor of 0.2 whenever validation loss did not change in the last 10 epochs.
Segmentation Task
• aggcmab: The team 70 improved their previously developed framework cascaded double encoder-decoder convolutional neural network 76 by increasing the encoder representation capability and adapting to a multi-site sampling technique. The first encoder-decoder generates an initial attempt to segment the polyp by extracting features and downsampling spatial resolutions while increasing the number of channels by learning convolutional filters. The output from the first network acts as an input for the second encoder-decoder along with the original image. Cross-entropy loss was minimized using the stochastic gradient descent with a batch-size of 4 and a learning rate of lr = 0.01 with rate decay of 1e-8 every 25 epochs. The training images were resized to 640×512, and data augmentation (e.g. random rotations, vertical/horizontal flipping, contrast, saturation and brightness changes) was applied. Four versions were generated from the image (i.e. horizontal and vertical flipping), and the average result was calculated on the test set.
• AIM_CityU: The team 63 adopted HRNet 51 as the backbone to maintain the high-resolution representations in multiscale feature fusion mechanism. To further eliminate noisy information in segmentation predictions and enhance model generalization, the team proposed a low-rank module to distribute feature maps in the high dimensional space to a low dimensional manifold. For the model optimization, various data augmentation strategies, including random flipping, rotation, color shift (brightness, color, sharpness, and contrast) and Gaussian noise, were performed to improve the model generalization further. Cross entropy and dice loss are utilized to optimize the whole model.
• HoLLYS_ETRI: The team 64 proposed an ensemble inference model based on 5-fold cross-validation to improve the performance of polyp detection and segmentation. The Mask R-CNN was used to generate the output segmentation mask. Ensemble inference was used to generate the final segmentation mask by averaging the results from the 5 models. After averaging the masks, if the inference results were greater than the threshold (0.6) then the output mask is considered as a polyp otherwise was counted as a background. Data augmentation was performed based on the techniques provided in Detectron2. The model was trained for 50,000 steps and checkpoints were save for every 1,000 steps with learning rate lr=0.001 that changes with a warm-up scheduler • sruniga: The team 72 suggested a lightweight deep learning-based algorithm to meet the real-time clinical need. The proposed network applied the HarDNet-MSEG 50 as the backbone network as it has a low inference speed due to reduced shortcuts. Moreover, they proposed an augmentation strategy for realising improved generalizable model. The data augmentation was applied according to a certain probability. For training the model, the dataset was split into 80% training and 20% validation using adam optimizer and setting the learning rate lr of 0.00001 for all the experiments.
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Images were resized to 352×352, and data augmentation has been applied according to the proposed algorithm by the team.
• Mah_UNM: The team 74 proposed modifying the SegNet 56 by embedding Gated recurrent units (GRU) units 79 within the convolution layers to improve its performance in segmenting polyps. The hyperparameters were set as the original SegNet with learning rate lr of 0.005 and batch size of 4. The multiplicative factor of gamma of 0.8 was used for the learning rate decay with adam optimizer and weighted cross-entropy loss. The provided dataset was split into 80% training and 20% validation.
• NDS_MultiUni: The team 75 suggested building a cascaded ensemble model made of MultiResUNet 80 architectures. The input image was fed to four different MultiResUNet models in the proposed model, and each model generated an output mask. Afterwards, the four predicted outputs were averaged together to produce the final segmentation mask. Each model was trained for 100 epochs with the same setting of hyperparameters. The input images were resized to 256×256 with a batch size of 8, binary cross-entropy as loss function and using Adam optimizer. The learning rate is set to lr of 1e − 3 and using the Reduce LROnPlateau callback.
• YCH_THU: The team 73 used existing parallel reverse attention network (PraNet) 48 . They extracted multi-level features from colonoscopy images utilizing a parallel res2Net-based network. Moreover, the segmentation results are postprocessed to remove uncertain pixels and enhance the boundary. The images were resized to 512×512 and the dataset was split into 80% training and 20% validation. The model was trained for 300 epochs with batch size 18 and learning rate lr of 1e-4 which was reduced every 50 epochs.
Results
The EndoCV2021 challenge focus on detection and segmentation of polyps with different sizes from endoscopic frames. The endoscopy video frames are gathered from six worldwide centers including two different modalities (i.e. White Light and Narrow Band Imaging). The frames were annotated by clinical experts in the challenge team for the purpose of detection and localization. The training dataset consisted of total 3242 frames from five centers only with the release of binary masks for the segmentation task and bounding box coordinates for detection task. For the test dataset, frames from center six was include to provide an overall of 777 frames from the six centers with a variation between single and sequence frames. There was a variation in the polyp size in both the training and testing set as shown in Fig. 1b. Table 4 represents the average precision (AP) computed at three different IoU thresholds and AP at different scales for the participant teams on the four datasets. Moreover, results from baseline methods YOLOv4 81 , RetinaNet (ResNet50) 28 and EfficientNet-D2 are provided. Methods presented by teams HoLLYS_ETRI and JIN_ZJU outperform against the other teams in terms of AP values for the single frame datasets (i.e. both data 1 (NBI) and data 2 (WLE). The results by both teams on data 1 had an increased difference for AP mean (>20%), AP 50 (>15%) and AP 75 (>18%) when compared to the other teams. However, for data 2, team AIM_CityU produced comparable results leading them to third place with a small difference of 0.88% for AP mean score when compared to team HoLLYS_ETRI . For the seen sequence dataset (Data 3), team JIN_ZJU maintained the top performance for AP mean (i.e. higher than secondbest team AIM_CityU by 4.19%) and AP 75 (i.e. higher than second-best team HoLLYS_ETRI by 3.29%). Team HoLLYS_ETRI maintained their top performance with highest result for AP 50 with a greater difference of 2.10% when compared to AIM_CityU that comes in second place. Furthermore, the method by HoLLYS_ETRI surpassed the results of other teams and baseline methods on the unseen sequence (Data 4) where the second teams take place with a difference of (>0.037) AP mean , (>0.04) AP 50 and (>0.055) AP 75 . In general, as concluded from table 4, results by teams HoLLYS_ETRI, JIN_ZJU and AIM_CityU had the best performance while results of the baselines method did not show better performance compared to any proposed method. Table 5 shows the ranking of detection task of polyp generalisation challenge after calculating the average detection, average deviation scores and time. Team AIM_CityU ranks the first place with inference time of 0.10 second per frame and lowest deviation scores of dev_g 2−3 (0.1339), dev_g 4−3 (0.0562) and dev_g (0.0932). Followed by team HoLLYS_ETRI in second place with an increased inference time of 0.69 s per frame and a top score of 0.4914 for average detection. In third place, team JIN_ZJU takes place with 1.9 s per frame for the inference time and the second-best average detection result of 0.4783. Figure 4 (a) demonstrate the boxplots for each teams and baseline methods. It can be observed that the median values for all area-based metrics (dice, precision, recall and F2) are above 0.8 for most teams when compared on all 777 test samples. However, a greater variability can be observed for all teams and baselines that is represented by large number of outlier samples. Figure 4 (a) where dot and box plots are provided, teams MLC_SimulaMet and aggcmab obtained the best scores demonstrating least deviation and with most samples concentrated in the interquartile range (IQR). It can be observed that paired aggcmab and MLC_SimulaMet; DeepLabV3+(ResNet50) and ResNetUNet(ResNet34); and HoLLYS_ETRI and PSPNet have similar performances since their quartiles Q1, Q2, and Q3 scores are very close to each other. Although the mean DSC score of team aggcmab is slightly higher than the MLC_SimulaMet, there was no observed statistically significant difference between these two teams. However, both of these teams reported Tables 6 present the JC, DSC, F2, PPV, Recall, Accuracy and HDF acquired by top five participanting teams and baseline methods (i.e. FCN8, PSPNet, DeepLabV3+ and ResNetUNet) using data 1 to data 4 respectively. As shown in the table for data 1 (NBI still images), the method suggested by teams sruniga and AIM_CityU outperformed against the other teams an baseline methods in terms of JC (>65%), DSC (> 74%) and F2 (> 73%). The team sruniga had an outstanding performance in segmenting fewer false-positive regions achieving a PPV result of 81.52 % which is higher than other methods by atleast 5%. Nevertheless, the top recall value for team MLC_SimulaMet and HoLLYS_ETRI (> 86%) proving their ability in detecting more true positive regions. The accuracy results on this data were comparable between all teams and baseline methods ranging from 95.78% to 97.11% with the best performance by team AIM_CityU. The results on Data 2 (WLE still images) are also presented in Table 6. For this data, the methods developed by teams MLC_SimulaMet and aggcmab produced the top values for JC (>0.77), DSC (>0.82) and F2 (>0.81) with comparable results between two teams. The PPV value was maintained with the method proposed by team sruniga(i.e. as discussed for data 1 ) with value of 0.8698 ± 0.21 followed by team MLC_SimulaMet in second place with a value of 0.8635 ± 0.26. Additionally, the method by team MLC_SimulaMet surpassed the results for all evaluation measures when compared to the other teams and baseline methods on data 3 as shown in the table. Moreover, the method proposed by team aggcmab comes in second place with more the 5% reduction of results for the JC, DSC and HDF. For this dataset, the baseline method DeepLabV3+ (ResNet50) showed improved performance compared to results on previously discussed data (i.e. data 1 and data 2) where it acquires second place for the F2 and accuracy with a result of 82.66% and 95.99% respectively. Similarly to the performance of the teams on data 3, as shown in Table 6 (i.e. on Data 4 (unseen sequence)) methods by teams MLC_SimulaMet and aggcmab produce the best results for most of the evaluation measures JC (>0.68), DSC (>0.73), F2(>0.71), ACC (>0.97) and HDF (<0.34). Generally, throughout the evaluation process for all tables on the different datasets, team sruniga provided a high PPV value on data 1, data 2 and data 4. Furthermore, the baseline methods showed low performance in terms of final score compared to the methods proposed by the participants especially with data 1, data 2 and data 4.
Aggregated performance and ranking on detection task
Aggregated performance and ranking on segmentation task
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To understand the behaviour of each method for provided test data splits we plotted DSC values each separately and compared the ability of methods to generalise on these. From Figure 4 (c-d) it can be observed that difference in data setting affect almost all methods. It can be observed that there is nearly upto 20% gap in performance of the same methods when tested on WLE and NBI. Similarly, for single and sequence frame case and unseen center data. However, it could be observed that those methods that had very close values (e.g., HoLLYS_ETRI) suffered in performance compared to other methods.
To assess generalisability of each method, we also computed deviation scores for semantic segmentation referred to as dev_g (see Table 7 and Figure 4 (f)). For this assessment, team aggcmab ranked the first on both average segmentation scores R seg and deviation score R dev . Even though team sruniga was only third on R seg , they were second on R dev and ranked at the 1st position for their computation time with average inference time of only 17 ms per second. Team MLC_SimulaMet only was ranked third due to their large computational time of 120 ms per frame and larger deviations (lower generalisation ability). We provide the results of teams with performance below baseline and poor ranking compared to top five teams analysed in the paper in the Supplementary Table 2 for completeness. It is to be noted that these teams were selected in the round 3 of the challenge as well but have not been analysed in this paper due to their below baseline scores. Figure 4. Generalisation assessment on segmentation task: a) Box plots for all segmentation metrics (dice coefficient, DSC; precision, PPV; recall, Rec; F2, type-II error; and Hausdorff distance, Hd) used in the challenge for all test data samples. b) Boxplots representing descriptive statistics over all cases (median, quartiles and outliers) are combined with horizontally jittered dots representing individual data points in all test data. A red-line represent the best median line. It can be observed that teams aggcmab and MLC_SimulaMet have similar results and with Friedman-Nemenyi post-hoc p value < 0.05 denoting significant difference with the best performing baseline DeepLabv3+ method. c) White light endoscopy, WLE versus narrow-band imaging, NBI d) single versus sequence data , e) seen centers, C1-C5 versus unseen center, C6 and f) deviation scores. Table 6. Team results for the polyp segmentation methods proposed by the participating teams as well as for the baseline methods. All results are given for data 1, data 2, data 3 and data 4. Top evaluation criteria are highlighted in bold.
Data type
Teams/Method JC ↑ DSC ↑ F2 ↑ PPV ↑ Recall ↑ ACC ↑ H d ↓Data
Discussion
While polyp detection and segmentation using computer vision methods, in particular deep learning, have been widely studied in the past, rigorous assessment and benchmarking on centerwise split, modality split and sequence data have not been comprehensively studied. In our EndoCV2021 edition, we challenged participants to address generalisability issues in polyp detection and segmetation methods by providing multicenter and diverse data. For polyp detection and localisation, 3/4 teams chose feature pyramid-based networks while one team used YOLOV5 ensemble paradigm. Unlike most of these methods that require anchors to detect multiple objects of different scales, and overlap, team AIM_CityU 63 used an anchor free fully convolution one-stage object detection (FCOS) method. HoLLYS_ETRI 64 focused mostly on accuracy and used an ensemble to train five different models, i.e., one model per center, and an aggregated model output was devised for the test inference. Even though, the HoLLYS_ETRI team led the leaderboard ranking on the average detection score on almost all data type, the observed detection speed (0.69 sec.) and the deviation in generalisation score only put them on the second rank (see Table 5). On contrary, AIM_CityU team with their anchor free single stage network performed consistently well in almost all data with the fastest inference (0.1 sec.) and the least deviation score (see Figure 3 and Table 5) between teams, and hence leading the leaderboard.
Hypothesis I: It can be hypothesised that anchor free detection methods can better generalise compared to methods that require anchors in heterogeneous multi-center dataset. This is strictly true as the polyp sizes in the dataset is varied (see Figure 1 b) and also the image sizes ranged from 388 × 288 pixels to 1920 × 1080.
Since, all methods trained their algorithm on single frame images provided in this challenge, it can be observed in Table 4 that the detection scores for all methods are relatively higher for the data 2 (WLE-single) , compared to other data categories, despite they came from unseen data center 6. However, performance drop can be observed for both seen (centers, C1-C5) and unseen (center, C6) sequence data that consisted of WLE images only. In addition, change in modality has detrimental effect on the performance for all methods even on single frames (see for data 1, NBI-single, Table 4).
Hypothesis II: It can be hypothesised that methods trained on single frame produce sub-optimal and inconsistent detection in videos as image-based object detection cannot leverage the rich temporal information inherent in video data. The scenario gets worsen when applied on different center to that on which it was trained. To overcome this, Long Short-Term Memory (LSTM) based methods can be used to keep the temporal information for pruning predictions 87 .
For segmentation task, while most teams used ensemble technique targeting to win on the leaderboard (MLC_SimulaMet 71 , HoLLYS_ETRI 64 , aggcmab 70 ), there were some teams who worked towards model efficiency network (e.g., team sruniga 72 ) or modifications for faster inference and improved accuracy (e.g., team AIM_CityU 63 ). Light weight model using HarDNet68 backbone with aggregated maps across scales (team sruniga) and use of multi-scale feature fusion network (HRNet) with low-rank disentanglement by team AIM_CityU outperformed all other methods on narrow band imaging modality (data 1) including the baseline segmentation methods (see Table 6). These methods showed acceptable performance for single frames on unseen data (data 2, WLE-single) as well. However, on sequence data (both for seen sequence data 3 and unseen sequence data 4), both of these methods performed poorly compared to ensemble-based techniques (see Figure 4 (nearly 6 times higher than the fastest method).
Hypothesis III: It can be hypothesised that on single frame data multi-scale feature fusion networks perform better irrespective of their modality changes. This is without requiring to ensemble same or multiple models for inference which ideally increases both model complexity and inference time. However, on sequence data we advice to incorporate temporal information propagation in the designed networks. Furthermore, to improve model generalisation on unseen modality, domain 17/26 adaptation techniques can be applied 11 .
HoLLYS_ETRI 64 used instance segmentation approach with five separate models trained on C1 to C5 training data separately. It can be observed that this scheme provided better generalisation ability in most cases leading to the least deviation on average dice score (see Figure 4 (f)). However, reported dice metric values were lower than most methods especially ensemble techniques that are targeted towards higher accuracy but are less generalisable (in terms of consistency in test inference across multiple data categories). This is also evident in Figure 5 where proportion of samples from data 1 for top performing teams aggcmab and MLC_SimulaMet are mostly ranked on the third and fourth ranks.
Hypothesis IV: It can be hypothesised that pretext tasks can lead to improved generalisability. However, to boost model accuracy, modifications are desired that could include feature fusion blocks and other aggregation techniques.
Conclusion
We provided a comprehensive dissection of widely used deep learning baseline methods and methods devised by top participants in crowd-sourcing initiative of EndoCV2021 challenge. Through our experimental design, provided multi-center dataset and holistic comparisons, we demonstrate the need of generalisable methods to tackle real-world clinical challenges required for robust polyp detection and segmentation tasks. While most methods provided improvement over several widely used baseline methods, their design adversely impacted algorithmic robustness and/or real-time capability when provided unseen sequence data and different modality. A better trade-off in both inference time and generalisability is the key take away of this work. We provide experimental-based hypothesis to encourage future research towards innovating more applicable methods that can work effectively in multi-center data and diverse modalities that are widely used in colonoscopic procedures.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Supplementary
Figure 1 .
1Multi-center training and test samples: a. Colonoscopy video frames for which the annotation samples were reviewed and released as training (left) and test (right) is provided. Training samples included nearly proportional frames from five centers (C1-C5) while test samples consist of majority of single and sequence frames from unseen center (C6); b. Number of polyp or non-polyp samples-based on polyp sizes on resized image frames of 540 × 720 pixels (left) and their intra-size variability (right) for training (top) and testing data (bottom).
Figure 2 .
2Deviation C1-C5: Seen data centres C6: Unseen data centre Deviation: computed between test on C1-C5 with unseen modality and centre C6 C1-C5 EndoCV2021 challenge tasks: Participants performed model training on white light imaging data collected from five centers (C1-C5). The tasks included detection and segmentation.
Figure 3 .
3Generalisation assessment on detection task: (left) mean average precision (mAP) on all data versus deviation computed between seen center with unseen modality and unseen center. Least deviation with larger mAP is desired. (right) Comparison of teams and baseline methods on seen center sequence data versus unseen center sequence data, C6. Higher values along both axes is desired. significant difference with p < 0.05 when compared to the best performing baseline DeepLabV3+(ResNet50).
Figure 5 .
5(c-e)). Several network conjoint by MLC_SimulaMet and dual UNet network used by the team aggcmab have disadvantage of large inference time Algorithmic rank-based on across bootstrap samples 86 are displayed with colours according to the data categories: Histogram bars show how much proportion (in %) of each data contribute to the ranking of each team and baseline method. Here, for ranking we have only considered dice similarity coefficient values.
Author contributions S. Ali conceptualised the work, led the challenge and workshop, prepared the dataset, software and performed all analyses. S. Ali, N. Ghatwary and D. Jha contributed in data annotations. T. de Lange, J.E. East, S. Realdon, R. Cannizzaro, D. Lamarque were involved providing colonoscopy data and in the validation and quality checks of the annotations used in this challenge. Challenge participants (E. Isik-Polat, G. Polat, C. Yang, S. Poudel, S. Hicks, Z. Jin, T. Gan, C. Yu, D. Yeo, M. Haitimi) were involved in method summary and compilation of the related work. E. Isik-Polat performed the statistical tests conducted in this paper. S. Ali wrote most of the manuscript with inputs from N. Ghatwary and all co-authors. All authors participated in the revision of this manuscript, provided input and agreed for submission.
Table 2 .
2Summary of the participating teams detection task for EndoCV2021 Challenge. All test was done on NVIDIA V100 GPU provided by the organisers.FCOS: Fully Convolutional One-Stage Object Detection; FPN: Feature Pyramid Network; ATSS: Adaptive Training Sample Selection YOLO: You Only Look Once; SGD: Stochastic Gradient Descent; [d1]-[d4]: hyperlinked GitHub repos.Team Name
Algorithm
Backbone
Nature
Choice
Basis
Data
Aug.
Loss
Opt.
Code
AIM_CityU 63
FCOS
FPN, ResNeXt
-101-DCN
ATSS
Accuracy
speed
No
Generalised
Focal Loss
SGD
[d1]
HoLLYS_ETRI 64 Mask R-CNN
FPN
ResNet34
Ensemble Accuracy++ No
Smooth L1
SGD
[d2]
JIN_ZJU 65
YOLOV5
CSPdarknet
SPP
Ensemble speed++
Yes
BECLogits SGD
[d3]
GECE_VISION 66 EfficientDet 29 EfficientNet
D0-D3
Ensemble Accuracy
Yes
-
Adam [d4]
Table 3 .
3Summary of the participating teams algorithm for segmentation task EndoCV2021 Challenge. Top five teams are shown above horizontal line and worse performing team in round 3 are provided below this line.Team Name
Algorithm
Backbone
Nature
Choice
Basis
Data
Aug.
Loss
function
Optimizer Code
aggcmab 70
DPN92-FPN
DPN92-FPN Cascaded
Accuracy++ Yes
BCE
SGD
[s1]
AIM_CityU 63
HRNet + LRM HRNet
MSFF
Accuracy
speed
Yes
BCE,
DSC
SGD
[s2]
HoLLYS_ETRI 64
Mask R-CNN
ResNet50
Ensemble
Accuracy+
speed+
Yes
Smooth
L1
SGD
[s3]
MLC_SimulaMet 71 DivergentNet
TriUNet
Ensemble Accuracy++ No
BCE,
DSC
Adam
[s4]
sruniga 72
HarDNet68
HarDNet68
Multiscale
Accuracy+
speed++
No
BCE
Adam
[s5]
YCH_THU 73
PraNet
Res2Net
Reverse
attention
Accuracy
speed
No
W. IOU
W. BCE
Adam
[s6]
Mah_UNM 74
SegNet
VGG16
GRU
Accuracy
speed
No
BCE
Adam
[s7]
NDS_MultiUni 75
MultiResUnet
ResUnet
Ensemble Accuracy
No
BCE
Adam
NA
LRM: Low-rank module; MSFF: Multi-scale feature fusion; DPN: Dual path network; FPN: Feature pyramid network; BCE: Binary cross entropy
BCE: Binary cross entropy; DSC: Dice similarity coefficient; IoU: Intersection over union; W: weighted; SGD: Stochastic gradient descent
architectures in their TriUNet ensemble model. The novel TriUNet model takes a single image as input, which is passed through two separate UNet models with different randomized weights. The output of both models was then concatenated before being passed through a third UNet model to predict the final segmentation mask. The whole TriUNet network was trained as a single unit. Thus, the proposed DivergentNet included five segmentation models.• MLC_SimulaMet: The team 71 developed two ensemble models using well-known segmentation models; namely
UNet++ 47 , FPN 24 , DeepLabv3 77 , DeepLabv3+ 78 and novel TriUNet for their DivergentNet ensemble model, and three
UNet 57
Table 4 .
4Team results for the detection task with average precision AP computed at IoU thresholds 50 (AP 50 ), 75 (AP 75 ), and [0.50 : 0.05 : 0.95] mean AP (AP mean ). Size wise AP values are also presented.Average precision, AP
AP across scales
Data type Teams/Method
AP mean AP 50
AP 75
AP small AP medium AP large
Data 1
(NBI-single)
AIM_CityU 63
0.351
0.5378 0.3988 0.0802
0.3213
0.4076
GECE_VISION 66
0.3182
0.5266 0.3498 0.0506
0.1863
0.3984
HoLLYS_ETRI 64
0.4743
0.6931 0.5517 0.13
0.3961
0.5502
JIN_ZJU 65
0.4461
0.6587 0.498
0.0378
0.2963
0.5866
YOLOv4 81
0.3099
0.4472 0.3719 0.0688
0.2546
0.3711
RetinaNet (ResNet50) 28 0.3145
0.5625 0.2673 0.0475
0.223
0.3805
EfficientNet-D2 29
0.2009
0.3092 0.2279 0.0297
0.1591
0.2409
Data 2
(WLE-single)
AIM_CityU 63
0.5733
0.7847 0.6058 0.2799
0.4835
0.6595
GECE_VISION 66
0.5327
0.7859 0.5357 0.1556
0.4595
0.6235
HoLLYS_ETRI 64
0.5784
0.7908 0.6808 0.3854
0.4981
0.6552
JIN_ZJU 65
0.6049
0.8095 0.6643 0.3071
0.6148
0.721
YOLOv4 81
0.4194
0.5996 0.4636 0
0.3337
0.5237
RetinaNet 28 (ResNet50) 0.4076
0.7355 0.4607 0
0.2743
0.5246
EfficientNet-D2 29
0.4204
0.6135 0.4647 0
0.3828
0.5127
Data 3
(seen seq.)
AIM_CityU 63
0.5296
0.7804 0.5484 0.0035
0.404
0.5784
GECE_VISION 66
0.3725
0.6589 0.3844 0.005
0.0269
0.4363
HoLLYS_ETRI 64
0.5287
0.7972 0.5759 0.0174
0.0236
0.599
JIN_ZJU 65
0.5528
0.7253 0.5955 0
0.1515
0.6515
YOLOv4 81
0.2987
0.4362 0.3544 0
0
0.3283
RetinaNet (ResNet50) 28 0.3119
0.4897 0.3564 0
0.2525
0.341
EfficientNet-D2 29
0.2933
0.4035 0.3758 0
0.0757
0.3238
Data 4
(unseen seq.)
AIM_CityU 63
0.3464
0.4725 0.3767 0
0.2723
0.5225
GECE_VISION 66
0.3146
0.4997 0.3302 0
0.2787
0.4718
HoLLYS_ETRI 64
0.3843
0.5402 0.4318 0.0002
0.3096
0.5802
JIN_ZJU 65
0.3094
0.4259 0.3301 0.0001
0.3154
0.6568
YOLOv4 81
0.2363
0.3105 0.2805 0
0.2083
0.3498
RetinaNet (ResNet50) 28 0.2487
0.4491 0.2657 0.001
0.1763
0.385
EfficientNet-D2 29
0.2787
0.3818 0.3365 0
0.2647
0.4078
Table 5 .
5Ranking of detection task of polyp generalisation challenge For the mean distance-based normalised metric (1 − H d ), only marginal change can be seen for which top teams have higher values as expected. On observing closely only the dice similarity metric inTeam/Method
Avg_det ↑
Avg. deviation scores ↓
Time ↓ Rank
↓
dev_g 1−3 dev_g 2−3 dev_g 4−3 dev_g
(in s)
AIM_CityU 63
0.4501
0.0895
0.1339
0.0562
0.0932 0.10
1
GECE_VISION 66
0.3845
0.0563
0.2537
0.0699
0.1266 0.32
5
HoLLYS_ETRI 64
0.4914
0.1227
0.2116
0.0988
0.1444 0.69
2
JIN_ZJU 65
0.4783
0.0623
0.23
0.0918
0.128
1.9
3
YOLOv4 81
0.3161
0.0998
0.1781
0.0602
0.1127 0.13
6
RetinaNet (ResNet50) 28 0.3207
0.0313
0.0861
0.0405
0.0526 0.27
4
EfficientDetD2
0.2984
0.0586
0.1731
0.0782
0.1033 0.20
7
↑: best increasing ↓: best decreasing
Table 7 .
7Ranking of segmentation task of polyp generalisation challenge: Ranks are provided based on a) semantic score aggregation, R seg ; b) average deviation score, R dev ; and c) overall ranking (R all ) that takes into account R seg , R dev and time. For ties in the final ranking (R all ), segmentation score is taken into account. For time, ranks are provided into three categories: teams with < 50 ms, between 50 − 100 ms and > 100 ms. Top two values are in bold. best increasing ↓: best decreasingTeam/Method
Average Seg_score ↑
Average Dev_score ↓
Time ↓ R seg ↓ R dev ↓ R all ↓
Data 1 Data 2 Data 4 dev_g 1−3 dev_g 2−3 dev_g 4−3 (ms)
(avg.)
(avg.) (avg.)
aggcmab 70
0.7461 0.8496 0.7889 0.1199
0.0244
0.0994
107
1
1
1
AIM_CityU 63
0.7621 0.7777 0.5899 0.1289
0.1322
0.1072
80
4
3
5
HoLLYS_ETRI 64
0.7146 0.7777 0.746
0.0453
0.0864
0.0499
84
5
1
4
MLC_SimulaMet 71
0.7411 0.8586 0.7813 0.151
0.0508
0.1425
120
2
3
3
sruniga 72
0.7716 0.8303 0.6107 0.0357
0.0703
0.1411
17
3
2
2
Baselines
DeepLabV3+ (ResNet50) 82 0.6693 0.8387 0.7269 0.1847
0.0422
0.1305
19
NA
NA
NA
PSPNet 82, 83
0.5931 0.8325 0.7104 0.2356
0.0504
0.1558
45
NA
NA
NA
FCN8 84
0.6508 0.7878 0.6845 0.1376
0.0248
0.1152
27
NA
NA
NA
ResNetUNet-ResNet34 85
0.6587 0.8236 0.7293 0.1627
0.0485
0.0903
13
NA
NA
NA
↑:
Table 2 .
2Semantic segmentation results for teams ranking below 5th place on out-of-sample data 1, data 2, data 3 and data 4.Data
type
Teams/Method
JC ↑
DSC ↑
F2 ↑
PPV ↑
Recall ↑
ACC ↑
H
d ↓
Data 1
(NBI-single)
YCH_THU 73
0.2625 ±0.2917
0.3403 ±0.3314
0.3831 ±0.3563
0.5181 ±0.4149
0.3426 ±0.3473
0.8767 ±0.0971
0.5443 ±0.1831
Mah_UNM 74
03275 ±0.3121
0.4127 ±0.3488
0.4045 ±0.3588
0.4307 ±0.3870
0.6962 ±0.3496
0.9463 ±0.0626
0.4119 ±0.1501
NDS_MultiUni 75
0.1765 ±0.2569
0.2371 ±0.2944
0.2598 ±0.3145
0.3165 ±0.3855
0.5905 ±0.4158
0.9116 ±0.0740
0.4968 ±0.1758
Data 2
(WLE-single)
YCH_THU 73
0.5136 ±0.3444
0.5986 ±0.3566
0.6402 ±0.3568
0.7668 ±0.3265
0.5749 ±0.3707
0.9337 ±0.0845
0.4868 ±0.2035
Mah_UNM 74
0.4726 ±0.3166
0.5698 ±0.3403
0.5885 ±0.3467
0.6425 ±0.3575
0.6523 ±0.3424
0.9471 ±0.0770
0.4599 ±0.1805
NDS_MultiUni 75
0.3373 ±0.2784
0.4404 ±0.3134
0.4593 ±0.3200
0.5426 ±0.3643
0.5480 ±0.3764
0.9182 ±0.0954
0.5533 ±0.2026
Data 3
(seen seq.)
YCH_THU 73
0.4994 ±0.3158
0.5988 ±0.3252
0.6494 ±0.3359
0.7958 ±0.3197
0.5857 ±0.3535
0.8925 ±0.0909
0.5926 ±0.1622
Mah_UNM 74
0.4271 ±0.3491
0.5092 ±0.3712
0.5364 ±0.3978
0.5895 ±0.4320
0.7131 ±0.3317
0.8652 ±0.1329
0.5488 ±0.2409
NDS_MultiUni 75
0.5264 ±0.3234
0.6240 ±0.3150
0.6744 ±0.3011
0.7920 ±0.2973
0.6509 ±0.3300
0.9149 ±0.0612
0.6708 ±0.1963
Data 4
(unseen seq.)
YCH_THU 73
0.3285 ±0.3204
0.4096 ±0.3577
0.4443 ±0.3820
0.7475 ±0.3417
0.4143 ±0.3720
0.8802 ±0.1179
0.8802 ±0.1179
Mah_UNM 74
0.2491 ±0.3219
0.3064 ±0.3635
0.3023 ±0.3715
0.5304 ±0.4321
0.4596 ±0.4260
0.9281 ±0.0742
0.5244 ±0.1970
NDS_MultiUni 75
0.2491 ±0.3386
0.3046 ±0.3537
0.2941 ±0.3543
0.4297 ±0.4139
0.5290 ±0.4182
0.9202 ±0.0749
0.9202 ±0.0749
↑:
best increasing ↓:
best decreasing
21/26
https://github.com/sharibox/EndoCV2021-polyp_det_seg_gen/blob/main/evaluationMetrics/coco_evaluator.py7/26Polyp segmentationFor polyp segmentation task, we have used widely accepted computer vision metrics that include Sørensen-Dice Coefficient (DSC = 2·t p 2·t p+ f p+ f n ), Jaccard Coefficient (JC = t p t p+ f p+ f n ), precision (p = t p t p+ f p ), and recall (r = t p t p+ f n ), overall accuracy (Acc = t p+tn t p+tn+ f p+ f n ), and F2 (= 5p×r 4p+r ). In addition to the performance metrics, we have also computed frame per second (FPS= # f rames sec ). Here, tp, fp, tn, and fn represent true positives, false positives, true negatives, and false negatives, respectively. Another commonly used segmentation metric that is based on the distance between two point sets, here ground truth (G) and estimated or predicted (E) pixels, to estimate ranking errors is the average Hausdorff distance (H d ) and defined as:
https://endocv2021.grand-challenge.org 8/26
AcknowlgedgmentsThe research was supported by the National Institute for Health Research (NIHR) Oxford BiomedicalSupplementary Figure 1. Annotation workflow: 600 patients (N=600) data was used that consisted of both videos and frames. First 5593 relevant frames for polyp detection and segmentation were extracted. These frames comprised of both single and sequence data. For details please seeFigure 1. Review of annotations was done by at least one expert and the frames were either relabeled or immediately rejected. A second review was conducted by at least one expert. Here expert refers to a senior consultant gastroenterologist. Overall, 3762/5593 frames were annotated of which 520 frames from center 6 was directly embedded in the test set. For testing set, a similar strategy was taken for which 257 samples out-of 317 samples were accepted during the review phase.
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"Aegaeon (Saturn LIII), a G-ring object 1",
"Aegaeon (Saturn LIII), a G-ring object 1"
] |
[
"M M Hedman \nDepartment of Astronomy\nCornell University\n14853IthacaNYUSA\n",
"N J Cooper \nSchool of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK\n",
"C D Murray \nSchool of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK\n",
"K Beurle \nSchool of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK\n",
"M W Evans \nDepartment of Astronomy\nCornell University\n14853IthacaNYUSA\n\nSchool of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK\n",
". S Tiscareno \nDepartment of Astronomy\nCornell University\n14853IthacaNYUSA\n",
"J A Burns \nDepartment of Astronomy\nCornell University\n14853IthacaNYUSA\n\nDepartment of Theoretical and Applied Mechanics\nCornell University\n14853IthacaNYUSA\n"
] |
[
"Department of Astronomy\nCornell University\n14853IthacaNYUSA",
"School of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK",
"School of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK",
"School of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK",
"Department of Astronomy\nCornell University\n14853IthacaNYUSA",
"School of Mathematical Sciences\nAstronomy Unit\nQueen Mary University of London\nMile End RoadE1 4NSLondonUK",
"Department of Astronomy\nCornell University\n14853IthacaNYUSA",
"Department of Astronomy\nCornell University\n14853IthacaNYUSA",
"Department of Theoretical and Applied Mechanics\nCornell University\n14853IthacaNYUSA"
] |
[] |
Aegaeon (Saturn LIII, S/2008 S1
|
10.1016/j.icarus.2009.10.024
|
[
"https://arxiv.org/pdf/0911.0171v2.pdf"
] | 118,559,643 |
0911.0171
|
b7af732ba58ae38b3880f2b7b1cfa6da7d6b37c0
|
Aegaeon (Saturn LIII), a G-ring object 1
M M Hedman
Department of Astronomy
Cornell University
14853IthacaNYUSA
N J Cooper
School of Mathematical Sciences
Astronomy Unit
Queen Mary University of London
Mile End RoadE1 4NSLondonUK
C D Murray
School of Mathematical Sciences
Astronomy Unit
Queen Mary University of London
Mile End RoadE1 4NSLondonUK
K Beurle
School of Mathematical Sciences
Astronomy Unit
Queen Mary University of London
Mile End RoadE1 4NSLondonUK
M W Evans
Department of Astronomy
Cornell University
14853IthacaNYUSA
School of Mathematical Sciences
Astronomy Unit
Queen Mary University of London
Mile End RoadE1 4NSLondonUK
. S Tiscareno
Department of Astronomy
Cornell University
14853IthacaNYUSA
J A Burns
Department of Astronomy
Cornell University
14853IthacaNYUSA
Department of Theoretical and Applied Mechanics
Cornell University
14853IthacaNYUSA
Aegaeon (Saturn LIII), a G-ring object 1
Proposed Running Head: Aegaeon (Saturn LIII), a G-ring object 1 This paper is dedicated to the memory of Kevin Beurle.Saturn, SatellitesSaturn, ringsSatellites, dynamicsResonances, orbital
Aegaeon (Saturn LIII, S/2008 S1
Introduction
Beginning in early 2004, images from the cameras onboard the Cassini spacecraft revealed the existence of several previously unknown small Saturnian satellites: Methone, Pallene, Polydeuces, Daphnis and Anthe (Porco et al. 2005;Murray et al. 2005;Spitale et al. 2006;Cooper et al. 2008). Two of these moons -Anthe and Methone-are in meanmotion resonances with Saturn's moon Mimas. Specifically, they occupy the 10:11 and 14:15 co-rotation eccentricity resonances, respectively (Spitale et al. 2006;Cooper et al. 2008;Hedman et al. 2009). Both of these moons are also embedded in very faint, longitudinallyconfined ring arcs (Roussos et al. 2008;Hedman et al. 2009). This material probably represents debris that was knocked off the relevant moons at low velocities and thus remains trapped in the same co-rotation resonance as its source body.
Images from Cassini also demonstrated that a similar arc of material exists within Saturn's G ring, around 167,500 km from Saturn's center (Hedman et al. 2007). Images of this structure taken over the course of several years showed that it was also confined by a (7:6) corotation eccentricity resonance with Mimas. Furthermore, in-situ measurements of the plasma environment in the vicinity of the arc suggested that it contains a significant amount of mass in particles larger than the dust-sized grains that are the dominant source of scattered light observed in images (Hedman et al. 2007).
In late 2008, during Cassini's Equinox Mission (2008-2010, images of the arc taken at lower phase angles and higher resolutions than previously possible revealed a small, discrete object. Since the object was most visible at low phase angles and could be tracked over a period of roughly 600 days, it is almost certainly not a transient clump of dust but instead a tiny moonlet that represents the largest of the source bodies populating the arc. The discovery of this object was therefore announced in an IAU circular, where it was designated S2008/S1 ). More recently the International Astronomical Union has given it the name Saturn LIII/Aegaeon. As will be shown below, Aegaeon, like Anthe and Methone, occupies a corotation eccentricity resonance with Mimas, and all three of these small moonlets are associated with arcs of debris. These three objects therefore represent a distinct class of satellites and comparisons among the ring-moon systems have the potential to illuminate the connection between moons and rings.
Section 2 below describes the currently available images of Aegaeon and how they are processed to obtain estimates of the brightness and position of this object. Section 3 presents a preliminary analysis of the photometric data, which indicate that this object is approximately 500 m in diameter. Section 4 describes the orbital solutions to the astrometric data, which demonstrate that Aegaeon's orbit is indeed perturbed by the 7:6 corotation eccentricity resonance with Mimas. However,we also find that a number of other resonances, including the 7:6 Inner Lindblad Resonance, strongly influence Aegaeon's orbital motion. Finally, Section 5 compares the various resonantly-confined moon/ring-arc systems to one another in order to clarify the relationship between Aegaeon and the G ring.
Observational Data
The images discussed here were obtained with the Narrow-Angle Camera (NAC) of the Imaging Science Subsystem (ISS) onboard the Cassini spacecraft (Porco et al. 2004). All images were initially processed using the CISSCAL calibration routines (Porco et al. 2004) that remove backgrounds, flat-field the images, and convert the raw data numbers into I/F , a standardized measure of reflectance. I is the intensity of the scattered radiation while πF is the solar flux at Saturn, so I/F is a unitless quantity that equals unity for a perfect Lambert surface viewed at normal incidence.
Image Selection
The object was first noticed in two images taken on August 15 (Day-of-year 228), 2008 (see Fig. 1). These images were part of a sequence designed to image the arc in the G ring for the purposes of refining its orbit. Compared with previous imaging of the G-ring arc, the images used in this campaign were taken at lower phase angles and had better spatial resolution. This was more a result of the constraints imposed by the orbit geometry than a conscious effort to search for discrete objects in this region. When these images were taken, Cassini was in a highly inclined orbit with the ascending node near apoapse on the sunward side of the planet close to Titan's orbit. During these ring-plane crossings, the faint rings could be imaged at high signal-to-noise, and the low-phase angles were considered desirable because this geometry was comparatively rarely observed prior to this time. However, this geometry also turned out to be useful for detecting small objects in the G ring.
Two images from this sequence (Fig. 1) contained the core of the arc and also showed The arrows point to this object, which appears as a small streak within the core of the G ring due to its orbital motion through the field of view over the course of these long-exposure images. Both images are rotated so that Saturn's north pole would point towards the top of the page. a short, narrow streak in the G ring. The streaks are aligned with the local orbital motion of the arc and are clearly not aligned with the streaks associated with stars in the field of view. The lengths of the streaks are consistent with the expected movement of an object embedded in the arc over the exposure time, and the positions of the streaks in the two images are consistent with such an object's motion over the ∼30 minutes between the two images.
Since this sequence was part of a larger campaign designed to track the arc and refine its orbit, this object was quickly recovered in subsequent image sequences targeted at the arc with comparable viewing geometries, yielding 17 additional images of the object (Fig. 2). With these data, a preliminary orbit fit was used to search for earlier images of the object. However, only two images from the prime mission turned out to provide clear detections of Aegaeon (Fig. 3). This paucity of pre-discovery images is because this object is both extremely faint and embedded in the G-ring arc. While the object's faintness means that it cannot be clearly detected in images where the exposure times are too short, its proximity to the G-ring arc means that its signal cannot be isolated if the image resolution is too low or the phase angle is too high. Table 1 lists the 21 NAC images used in this analysis, which are all the images prior to February 20 (DOY 051), 2009 in which Aegaeon has been securely identified. These images cover a time interval of almost 600 days and a range of phase angles from 13 • to 43 • .
Image Data Reduction
Since Aegaeon is not resolved in any of the images listed in Table 1, the only data we can extract from each image are its position in the field of view and its total integrated brightness. However, estimating even these parameters from these images is challenging because the light from Aegaeon is smeared out into a streak and because the light from the object must be isolated from the background signal from the G ring arc. The following procedures were used to obtain the required photometric and astrometric data.
In order to isolate the moon's signal from that of the G ring, each image was first roughly navigated based on stars within the field of view. Then, the radius and longitude in the ringplane observed by each pixel was computed. Based on visual inspection of the image, a region of the image containing the arc was selected (in general these regions are 10-20 pixels across). A second region extending 10 pixels beyond this zone on either side along the ring was then used to construct a radial profile of the G ring and arc in the vicinity of the moon. A background based on this profile was then subtracted from the pixels in the selected region, which removes the signal from the G ring and arc, leaving behind only the signal from Aegaeon itself.
Two images were handled slightly differently because they were taken in a nearly-edgeon viewing geometry (N1563866776 and N1603168767). In these cases instead of computing radius and longitude for each pixel, we compute the radius and vertical height above the ringplane and remove a vertical brightness profile from the region around the object. After separating Aegaeon's signal from the G ring, the total brightness of the object in each image is estimated in terms of an effective area, which is the equivalent area of material with I/F = 1 required to account for the observed brightness:
A ef f = x y I/F (x, y) * Ω pixel * R 2 ,(1)
where x and y are the line and sample numbers of the pixels in the selected region,I/F (x, y) is the (background-subtracted) brightness of the streak in the x, y pixel, Ω pixel = (6µrad) 2 is the assumed solid angle subtended by a NAC pixel, and R is the distance between the spacecraft and the object during the observation. The assumed values for R (given in Table 1) are based on the best current orbital solution (see below).
Similarly, the object's mean position in the field of view was determined by computing the coordinates (in pixels) of the streak's center of light x c and y c :
x c = x y x * I/F (x, y) x y I/F (x, y) ,(2)
y c = x y y * I/F (x, y)
x y I/F (x, y)
.
(
For purposes of deriving the object's orbit, these estimates of Aegaeon's position within the camera's field of view are converted into estimates of its right ascension and declination on the sky as seen by Cassini. This is accomplished by comparing the center-of-light coordinates of Aegaeon to the center-of-light coordinates of various stars in the field of view. 3. Photometric analysis and the size of Aegaeon Table 1 includes 19 measurements of Aegaeon's brightness through the NAC's clear filters over a range of phase angles between 13 • and 43 • . 2 In the absence of disk-resolved images of this object, these photometric data provide the only basis for estimating its size.
For the above range of phase angles α, the effective area A ef f of a spherical object is usually well approximated by the following form:
A ef f = p ef f A phys 10 −βα/2.5 ,(4)
where A phys is the physical cross-sectional area of the object, p ef f is the effective geometric albedo (neglecting the opposition surge) and β is the linear phase coefficient (Veverka 1977).
Even if the object is not spherical, we still expect that < A ef f (α) > -the effective area at a given phase angle averaged over object orientations-will have the same basic form:
< A ef f >= p ef f < A phys > 10 −βα/2.5(5)
where < A phys > is the average physical cross-section of the object.
Fitting the photometric data over a sufficiently broad range of phase angles to Equation 5 can provide estimates of the linear phase coefficient β and the product p ef f < A phys >. However, to convert the latter into an estimate of the object's size requires additional information about p ef f , which can be obtained from comparisons with similar objects. For Aegaeon, the best points of comparison are Pallene, Methone and Anthe, three small Saturnian moons whose orbits lie between those of Mimas and Enceladus. These moons are the closest in size to Aegaeon and are in similar environments (Pallene, Methone, Anthe and Aegaeon are all embedded in faint rings or arcs of material).
To quantitatively compare the photometric characteristics of these various moons, we computed the effective areas A ef f of Pallene, Methone and Anthe from a series of images taken over a similar range of phase angles as the Aegaeon images. Tables 2, 3 and 4 list the images of Pallene, Methone, and Anthe used in this analysis. Since the goal here is to make comparisons between different moons and not to do a complete photometric analysis of these objects, the images used in the current study are only a selected subset of NAC clear-filter images that were expected to give the most reliable brightness data based on the spacecraft range and exposure duration. All these images were taken from within about 2 million kilometers of the target moon and had exposure durations that were long enough for the moon's signal to be measured accurately but short enough that there was no chance of saturation.
For each image, we computed the total integrated brightness in a 14-by-14 pixel wide zone containing the moon above the average background level in a 5-pixel wide annulus surrounding the selected region. These total brightness measurements were then converted into effective areas using the range between the spacecraft and the moon as described in Equation 1. Figure 4 shows the resulting estimates of A ef f as a function of phase angle for Pallene, Methone, Anthe and Aegaeon. The data for Pallene, Methone and Anthe all show significant scatter around the main trends. In all three cases, this scatter can be attributed to variations in the orientation of a non-uniform or non-spherical object relative to the spacecraft (As will be discussed in a future work, all three moons appear to have significant ellipticities with the long axis pointing towards Saturn). The Aegaeon data are divided into two groups in this plot based on whether the observation had a ring opening angle |B| greater or less than 1 • . Because the contrast of the moon against the background G ring is reduced at lower ring opening angles due to the increased surface brightness of the ring material, the |B| > 1 • data are considered to be more reliable measurements of A ef f .
Despite the scatter, it is clear that the data from all four objects can be fit to a mean trend of the form given in Equation 5. The lines in the plots show the resulting best-fit trend, while Table 5 gives the resulting fit parameters (note only the |B| > 1 • data are used for the Aegaeon fit). Because the scatter in the data points from each moon is not random error, but instead systematic variations associated with viewing geometry, error bars on these parameters are not reported here.
The phase coefficients of Anthe, Pallene and Methone are reasonable values for small airless objects (compare with values for asteroids in Bowell and Lumme 1979), while the coefficient for Aegaeon is somewhat on the low side, which may be because a residual unsubtracted G-ring signal adds a slightly forward-scattering component to its phase curve. Alternatively, Aegaeon may have a smoother surface than the other moons (Veverka 1971).
Of all of these moons, only Pallene has been observed with sufficient resolution to obtain a well-defined mean radius of 2.2.± 0.3 km . Given the observed value of p ef f < A phys >= 7.38 km 2 , this would imply that p ef f = 0.49 for this moon. Assuming that all four objects have roughly the same geometric albedo, we obtain estimates of the mean radii of Methone and Anthe of 1.6 km and 1.1 km, respectively. The estimated size of Methone matches the estimate derived from crudely resolved images (1.6 ± 0.6 km, Porco et al. 2007), and the radius of Anthe matches previous estimates based on its brightness relative to Pallene (Cooper et al. 2008). Applying this same albedo to Aegaeon suggests a radius of 240 m. Assuming geometric albedos between 0.1 and 1.0 gives a range of radii between 160 and 520 m, so although the size of the object is still uncertain, it is almost certainly less than 1 km across. Note that most of the scatter in the Pallene, Methone and Anthe around the trendline are due to variations in the orientation of the moon relative to the spacecraft. Two sub-sets of the Aegaeon data are highlighted. The stars are data from images with ring opening angles above 1 • , which are considered more reliable than those obtained at lower ring-opening angles (marked as plusses) where the contrast of the object against the ring is weaker. Since the scatter in the data for each moon is dominated by systematic effects, statistical error bars are not included in this plot.
Orbital Solutions
The methodology used to derive the orbital solution for Aegaeon follows the same basic procedures used by Cooper et al. (2008) with Anthe and Murray et al. (2005) with Polydeuces. As in those works, the solution is computed in a planetocentric reference frame where the x-axis corresponds to the direction of the ascending node of Saturn's equatorial plane on the equator of the International Celestial Reference Frame (ICRF); the z axis is directed along Saturn's spin axis at epoch (pointing north); and the y-axis is orthogonal to x and z and oriented as required to produce a right-handed coordinate system. The chosen epoch for the orbital solution is 2008-228T06:45:07.972 UTC (the time of the first discovery image). The assumed values for Saturn's pole position and gravitational field are given in Table 6, while Table 7 lists the SPICE kernels (Acton 1996) used in the orbit determination and numerical modeling.
As with Anthe and other small Saturnian satellites, the orbit of Aegaeon cannot be accurately fit with a simple precessing elliptical model (see below). Thus the data were fit to a numerical integration of the full equations of motion in three dimensions, solving for the initial state of Aegaeon at epoch. This model included perturbations from the Sun, Saturn, Jupiter, the eight major satellites of Saturn (Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion and Iapetus), as well as the smaller moons Prometheus, Pandora, Janus and Epimetheus. The forces from these perturbers were calculated using position vectors extracted from the JPL ephemerides listed in Table 7. These position vectors were rotated from the ICRF frame to the saturn-centric reference frame using the pole position given in Table 6, obtained by precessing the pole position of Jacobson (2004) to the chosen epoch, using rates of -0.04229 • /cy in right acension and -0.00444 • /cy in declination (Jacobson 2004). Terms up to J 6 in Saturn's gravitational field were taken into account. The adopted GM values for the satellites, etc. are given in Table 8.
Numerical integration of both the equations of motion and the variational equations was performed using the 12th-order Runge-Kutta-Nyström RKN12(10)17M algorithm of Brankin et al. (1989). For more details on the fitting procedures, see Murray et al. (2005). The final solution for the state vector at epoch in the planetocentric frame, from a fit to the full time-span of observations, is given in Table 9. All the observations listed in Table 1 were included in this fit with equal weights. Figure 5 shows the orbital coverage of the available observations, based on the numerically integrated positions. Note the data fall in two clusters, which correspond to the two ansae of the orbit when the rings are viewed at low phase angles during the observation epoch.
Fit residuals are displayed as a function of time in Fig. 6. The overall rms fit residual is 0.468 pixels for the 21 NAC images, which is equivalent to 0.578 arcsecond. This is comparable to the residuals for the NAC observations of Anthe (Cooper et al. 2008), which is remarkable given that in most of the images used here Aegaeon forms a streak several pixels long. This suggests that our methods for deriving the position of the moon are accurate, and that the systematic errors in the modeled orbit are small. The final rms uncertainty in the fitted position vector in the frame of integration is 4.3 km. Table 10 lists the planetocentric orbital elements derived by fitting a uniformly precessing ellipse to the numerically integrated orbit of Aegaeon over a one-day time-span, using a fine grid of regularly-spaced position vectors. These parameters include the semi-major axis a calc , eccentricity e, inclination i, longitude of ascending node Ω, longitude of pericenter , longitude at epoch λ o and mean motion n. Note that the calculated semi-major axis a calc was obtained by first fitting for n and then converting to semi-major axis using the standard equations involving Saturn's gravitational harmonics (Nicholson and Porco 1988). The apsidal and nodal rates were calculated using the expressions in Cooper and Murray (2004), incorporating terms up to J 6 . It should be emphasized that the fitted elements in Table 10 represent only a snapshot of the orbit at the time epoch of fit (2008-001T12:00:00 UTC, chosen to be a time when Aegaeon was near the center of its librations, see below). In reality, the orbital elements show significant periodic variations due to resonant perturbations from Mimas, so a uniformly precessing ellipse will provide a poor approximation of the orbit over a time span of more than a few days. Figure 7 shows the variations in the geometrical orbital elements over a period of 10 years. These plots were generated by integrating the initial state vector from Table 9, and state vectors were generated at 0.15-day intervals and converted into geometric orbital elements using standard methods (Borderies and Longaretti 1994;Renner 2004;Renner and Sicardy 2006). Unlike conventional osculating elements, these geometric elements are not contaminated by short-period terms caused by planetary oblateness. There are clear periodic variations in all the orbital elements. The semi-major axis varies by ± 4 km around a mean value of 167494 km. The eccentricity ranges between nearly zero and 0.00047, and the inclination ranges from 0.0001 to 0.0019 degrees. The mean values of a, e and i and the amplitude of their periodic variations are also given in Table 10. Note that when the eccentricity and inclination both periodically approach zero, the longitudes of node and pericenter change rapidly.
Since the G-ring arc appears to be confined by the 7:6 corotation eccentricity resonance with Mimas (Hedman et al. 2007), we expected that Aegaeon would also be trapped in this resonance. Figures 8a and b shows the time evolution of the resonant argument for the 7:6 corotation eccentricity resonance
ϕ CER = 7λ M imas − 6λ Aegaeon − M imas .(6)
These data indicate that the argument librates, confirming that Aegaeon indeed occupies the 7:6 corotation eccentricity resonance with Mimas. This analysis also demonstrates that the dominant libration period is 1264 ± 1 days, consistent with the estimated libration periods of particles in the G ring (Hedman et al. 2007).
The amplitude of the librations in this resonant argument is only ∼ 10 • , so one might expect that Aegaeon's longitude would only deviate by a few degrees from its expected value assuming a constant mean motion. In reality, Aegaeon's longitude can drift by tens of degrees from its expected position assuming a constant mean motion (Fig. 9). These long-period drifts have a characteristic period of 70 years, comparable to the 70.56 year libration in Mimas' longitude caused by its resonance with Tethys (Vienne and Duriez Fig. 7.-Geometric orbital elements between 2004 and 2014 derived from the numerical integration, including perturbations from the eight major satellites of Saturn plus Prometheus, Pandora, Janus and Epimetheus. Linear background trends have been subtracted from the mean longitude, pericenters and nodes prior to plotting (the rates being 445.482 • /day, 1.146 • /day and -1.098 • /day, respectively). 1995) These variations therefore likely arise because the longitude of Mimas is itself perturbed by a resonance with Tethys. Note that over the course of the Cassini Mission, the residual longitude of Aegaeon has drifted backwards at a rate of approximately 0.01 • /day. This probably explains why the best-fit mean-motion of the arc from the Cassini data was 445.475 ± 0.007 • /day instead of the expected value of 445.484 • /day (Hedman et al. 2007).
The corotation eccentricity resonance should primarily affect Aegaeon's orbital mean motion and computed semi-major axis, and have little effect on its eccentricity and inclination. However, there are clearly large fractional variations in both Aegaeon's eccentricity and inclination. Furthermore, these variations seem to be coupled, such that the eccentricity and inclination rise and fall together. This strongly suggests that additional resonances are influencing Aegaeon's orbit. In particular, the correlation between the moon's eccentricity and inclination suggests a Kozai-like mechanism may be involved. However, unlike a classical Kozai Resonance (Kozai 1962) where the correlation between the eccentricity and inclination is negative, in this case the correlation between these two parameters is positive.
To further explore these aspects of Aegaeon's orbital evolution, we looked at the time evolution of the fourteen valid resonant arguments to fourth degree in the eccentricities and the inclinations of the form ϕ = 7λ M imas − 6λ Aegaeon + ... In addition to the corotation eccentricity resonance, we found 7 other resonant arguments that exhibited interesting behavior; they are listed in Table 11. These include the resonant argument of the 7:6 Inner Lindblad Resonance ϕ ILR , two resonant arguments ϕ x and ϕ y that can be written as linear combinations of ϕ CER and ϕ ILR , and four resonant arguments involving the nodes of Mimas and Aegaeon (ϕ a − ϕ d ). Integrations of a few test cases where the initial state vector was shifted within the error bars showed the same fundamental behavior. While a detailed investigation of all of these resonant terms is beyond the scope of this paper, we will briefly discuss the behavior of a few of these resonant arguments. Figure 8c,d shows the time evolution of the resonant argument of the Inner Lindblad Resonance ϕ ILR = 7λ M imias − 6λ Aegaeon − Aegaeon . This resonant argument appears to spend most of its time librating within ±90 • of zero with a period of 824 ± 1 days, interspersed with brief episodes where the resonant argument circulates around 360 • . The dominant libration period of this argument equals the synodic period of the difference between Aegaeon's and Mimas' pericenters, which is reasonable since ϕ ILR = ϕ CER + M imas − Aegaeon . The alternations between libration and circulation imply that Aegaeon's orbit lies at the boundary of the ILR, and its free eccentricity is almost equal to the forced eccentricity from the Lindblad resonance. As noted previously, the total eccentricity periodically approaches zero and the pericenter longitude changes rapidly, as seen in Figure 7. During these episodes, ϕ ILR could either librate through zero or circulate through 180 • depending on whether the forced eccentricity (which will vary with time as Aegaeon's orbit librates around the corotation eccentricity resonance) is slightly larger or smaller than the free eccentricity.
By way of comparison, it is interesting to note that Saturn's small moon Methone also appears to occupy both a corotation eccentricity resonance and an inner Lindblad reso- nance with Mimas (Spitale et al. 2006;Hedman et al. 2009). However, the 14:15 resonances occupied by Methone are separated by less than 4 km in semi-major axis, while the 7:6 resonances affecting Aegaeon's orbit are separated by more than 18 km. The forced eccentricity from the Lindblad resonance is thus larger for Methone than for Aegaeon, which means Aegaeon needs to have a much smaller free eccentricity to be trapped in both resonances than Methone does. ϕ a − ϕ d appear to be examples of secondary resonances i.e. secular resonances existing inside their respective primary mean motion resonances. The coupling between Aegaeon's eccentricity and inclination mentioned above is also typical of this type of resonant motion (for general discussion of secondary resonant behavior, see Morbidelli, 2002). Of particular interest are the resonant arguments ϕ a = ϕ CER + M imas + Ω M imas − Aegaeon − Ω Aegaeon and ϕ b = ϕ CER + M imas − Ω M imas − Aegaeon + Ω Aegaeon . The former is equivalent to a resonant argument which was found to be librating for Anthe (φ 2 in Cooper et al. 2008). In Aegaeon, this argument typically stays within ±20 • of either 0 • or 180 • , but can abruptly switch from one state to the other during periods when the eccentricity and inclination are small (see Fig. 10a). To better understand the significance of this behavior, note that ϕ CER is already approximately constant, and that since˙ M imas −Ω M imas , M imas + Ω M imas is also a constant to good approximation. Therefore, if ϕ a remains constant, then Aegaeon − Ω Aegaeon must also be constant, which implies that˙ Aegaeon −Ω Aegaeon . This is true for the pericenter precession and nodal regression due to Saturn's oblateness, but is not obviously true for the precession and regression due to perturbations from Mimas. For such perturbations, the Lagrange equations indicate that the pericenter precession rate goes inversely with the eccentricity, while the nodal regression rate goes inversely with the inclination. Thus the only way to have˙ Aegaeon −Ω Aegaeon is for the eccentricity and inclination vary in step with one another, which is indeed the case for Aegaeon (see Fig. 7). ϕ b , like ϕ a , also exhibits periods of circulation and libration, although its libration amplitude is far greater than for ϕ a (see Fig. 10b), suggesting that Aegaeon is located further from the centre of this particular resonance. This resonant argument is of particular interest because it can be expressed as ϕ b = ϕ CER + ω M imas − ω Aegaeon . ϕ b is therefore the resonant argument of the CER plus the difference in the arguments of pericenter of Mimas and Aegaeon, which suggests that this resonance has some similarities with Kozai resonances. Although classical Kozai resonances exist only at high inclinations, Kozai-type secondary resonances can occur inside primary mean motion resonances, even in systems which have small eccentricity and inclination (Morbidelli 2002). A detailed investigation of the secondary resonances represented by ϕ a , ϕ b , etc. and their implications for the orbital properties and evolution of Aegaeon is beyond the scope of this paper, but the number of resonant arguments showing interesting behavior indicates that additional work on the detailed orbital properties of this moon should be quite rewarding..
Comparisons of moon/ring-arc systems
Aegaeon, like Anthe and Methone, is a small moon embedded in an arc of debris confined by a first-order corotation eccentricity resonance with Mimas. However Aegaeon also appears to be a special case, since it the smallest of these objects while the G-ring arc is brighter than the arcs surrounding the other moons. Thus the relationship between Aegaeon and the G-ring arc may differ from that between the other moons and their arcs. We therefore compare these systems' dynamical and optical properties.
While Aegaeon, Anthe and Methone are all trapped in corotation eccentricity resonances with Mimas (7:6, 10:11 and 14:15, respectively), their libration amplitudes within those resonances are quite different. As shown in Figure 11, the libration amplitudes of Anthe and Methone are both between 70 • and 80 • , while the libration amplitude of Aegaeon is much smaller, only around 10 • . Aegaeon is therefore more tightly trapped in its resonance than Anthe and Methone are in theirs.
These differences in the moons' libration amplitudes could explain some of the differ-ences in the gross morphology of the various arcs. These morphological differences are most visible in longitudinal brightness profiles of the arcs' radially integrated brightness, which is expressed in terms of the normal equivalent width:
W = µ (I/F )dr,(7)
where µ is the cosine of the emission angle. Note that for low optical depth rings, this quantity (with units of length) is independent of the viewing geometry and the resolution of the images.
Longitudinal brightness profiles of the Anthe and Methone arcs were computed in Hedman et al. (2009), and longitudinal profiles of the G-ring arc are derived in Hedman et al. (2007). However, the Anthe and Methone arc profiles are derived from low-phase-angle (∼ 23 • ) images, while the published G-ring arc profiles are derived from high-phase-angle (> 80 • ) images, so these published data sets are not truly comparable to each other. Fortunately, the same observations that contain Aegaeon also provide images of the arc at lower phase angles. In particular, the series of images N1597471047-N1597486437 (the sequence in which Aegaeon was first noticed) captured the entire arc at phase angles ∼ 28 • , which is comparable to the phase angles of the Anthe and Methone arc observations. A longitudinal brightness profile of the G-ring arc was derived from these images following procedures similar to those used in Hedman et al. (2007) and Hedman et al. (2009). First, the relevant imaging data were re-projected onto a grid of radii and longitudes relative to the predicted location of Aegaeon. To isolate the arc signal from the rest of the ring, a radial brightness profile of the background G ring was computed by averaging the data over longitudes between -40 • and -50 • from Aegaeon, where the arc signal was absent. After subtracting this background, the normal equivalent width at each longitude was computed by integrating the brightness over the radial range of 167,000-168,000 km. Figure 12 displays the longitudinal brightness profiles of the various arcs. Note the x-axis on these plots is the resonant argument ϕ of the appropriate corotation eccentricity resonances instead of actual longitudes, so that the curves can be compared more easily. Intriguingly, the G-ring profile has a distinct peak near ϕ = 0, while the Anthe arc is broad with a nearly constant brightness over a broad range of longitudes. In all likelihood, this difference in the morphology of the arcs is directly related to the differences in the moons' libration amplitudes described above. Aegaeon has a relatively small libration amplitude, and so never strays far from ϕ = 0, while Anthe has a large libration amplitude and thus moves through a wider range of longitudes within the pocket containing the particles. Thus we might expect that material shed from Anthe would be more evenly distributed in longitude than the material derived from Aegaeon. Furthermore, Anthe should be better able to stir and scatter debris throughout the arc as it moves back and forth through the arc.
The Methone arc presents a more complicated situation, since the amplitude of Methone's libration is comparable to Anthe's, but its arc is not as wide. This could possibly be attributed to the fact that Methone occupies both the 14:15 corotation resonance and the 14:15 Lindblad resonance, and so the dynamics of the particles in this region may be more complicated than those in the Anthe arc.
In addition to the differences in the morphology of the arcs associated with the various moons, the arcs' overall brightnesses show some interesting trends. In analogy to the normal equivalent width given above, one can define a normal equivalent area A as the total integrated brightness of a ring over radius and longitude λ:
A = r o Wdλ,(8)
where r o is the effective mean radius of the ring. Note that this quantity has units of area and provides a measure of the total surface area of material in the ring.
Integrating each arc's equivalent width over all longitudes (and interpolating the Anthe and Methone arcs over the region dominated by the signal from the moons), we obtain normal equivalent areas of the G-ring, Anthe and Methone arcs of 50, 1.0 and 0.3 km 2 , respectively. The G-ring arc's integrated brightnes is therefore about 2 orders of magnitude higher than the arcs associated with Anthe and Methone. This difference becomes more striking if we compare these numbers to the effective areas of the moons at comparable phase angles. Inserting the values in Table 5 into Equation 5, we find that the effective areas of Aegaeon, Anthe and Methone at 25 • phase are 0.07, 0.84 and 2.21 km 2 . For Anthe and Methone, the normal equivalent areas of the arcs are comparable to the effective area of the moons, which implies that the debris in these arcs have comparable surface areas as the moons. Since the particles in the arcs are likely significantly smaller than the moons, this means that the mass in the Anthe and Methone arcs are much less than the mass in the moons themselves. By contrast, the normal equivalent area of the G ring arc is between 10 3 and 10 4 times the effective area of Aegaeon.
Most of the visible material in the faint rings likely originates from clouds of debris knocked off the larger particles and moons by micrometeoroids, so one possible explanation for the distinctive characteristic of the Aegaeon/G-ring system is that Aegaeon is more efficient at generating dust than the larger moons Anthe and Methone. Smaller moons do have lower surface gravity, so a given micrometeoroid impact will yield a larger fraction of ejecta that will escape into the ring. However, smaller moons also have lower cross-sections and thus have lower impact rates, and theoretical calculations suggest that the optimal moon size for dust production is around 10 km (Burns et al. 1984(Burns et al. , 1999. Even though this optimal size depends on the assumed surface properties of the source bodies, it is larger than any of the moons considered here, so this model predicts that Aegaeon would actually be less efficient at generating dust than Anthe or Methone.
An alternative explanation arises from the realization that the normal equivalent area of the G ring and the arc are orders of magnitude higher than the physical area of Aegaeon, which is not the case for any of the other ring-moon systems. Aegaeon therefore does not dominate the cross-section of its ring to the same extent as the other moons, so it is quite likely that Aegaeon shares the arc with a number of other objects 1-100 meters across that act as additional sources of the visible G ring. Such objects would be difficult to see in the available images because they would be smeared out into streaks by the long exposure times, which makes them hard to detect against the background brightness of the G-ring arc. However, in-situ measurements provide evidence that additional source bodies do reside in the G-ring arc. Using in-situ data from the Voyager spacecraft, van Allen (1983) computed the total cross-sectional area of large (> 10 cm) particles in the G ring to be 20 km 2 . This is comparable to the normal equivalent area of the arc derived above and is much larger than the area of Aegaeon, implying that there is indeed a significant population of large objects in the vicinity of the G ring. More recently, the MIMI instrument onboard Cassini detected a ∼250-km wide electron microsignature associated specifically with the G-ring arc. The depth of this microsignature required a total mass of material equivalent to a roughly 100meter wide ice-rich moonlet, orders of magnitude greater than the mass in dust-sized grains inferred from images (Hedman et al. 2007). Furthermore, the signature is too wide to be explained by a single moon like Aegaeon, which suggests that the arc contains a substantial population of electron-absorbing source bodies. The G-ring arc therefore appears to contain debris with a broad range of sizes, perhaps the remains of a shattered moon, while the Anthe and Methone arcs are just the latest small particles knocked off of the relevant moons.
If Aegaeon does share the G-ring arc with a population of source bodies 1-100 meters across, this could influence its dynamics. As Aegaeon librates within the arc, it will collide with these smaller objects. Collisions within dense arcs of debris confined by corotation resonances are expected to increase the libration amplitudes of particles and ultimately allow them to escape the resonance because collisions dissipate energy and the stable points of corotation resonances are potential energy maxima (Porco 1991;Namouni and Porco 2002). However, this situation is slightly different, because we have a single large body moving through a sea of smaller bodies that should have no average net velocity relative to the stable point of the resonance. Hence collisions will act against any motion of Aegaeon relative to the resonance, and therefore cause Aegaeon's free inclination, free eccentricity, and libration amplitude to decay over time.
A crude estimate of the dissipation timescales due to collisions can be computed by assuming an object of radius R and mass M moves at a velocity v through a background medium consisting of a population of small particles. Say the mass density of the background medium is ρ b , then the mass encountered by the object in a time dt is ρ b πR 2 vdt. The momentum imparted to this material is Cρ b πR 2 v 2 dt, where C is a dimensionless constant of order unity. This must equal the corresponding decrease in the momentum of the object M dv, so the acceleration of the object due to collisions with the medium is given by:
dv dt = Cρ b πR 2 v M v.(9)
Assuming the object has an initial velocity v i at time t = 0, the velocity will decay with time as follows:
v(t) = v i 1 + Cπρ b R 2 v i M t −1 .(10)
Thus the characteristic timescale over which the velocity falls by a factor of 1/2 is:
t c = 1 CπR 2 v i M ρ b .(11)
-25 -Now, since the mass density of the medium (i.e., the arc) is the most uncertain variable, let us re-express that parameter in terms of the arc's mass m a and its spatial volume V a .
t c = 1 C M m a V a πR 2 v i .(12)
We may now attempt to estimate this characteristic timescale for moons like Aegaeon and Anthe. Libration amplitudes of ∼ 10 • and finite eccentricities of ∼ 10 −3 (both reasonable for moons like Aegaeon or Anthe) lead to typical velocities relative to the resonance's stable point of order 1 m/s. The G-ring arc has a longitudinal extent of ∼ 20 • or ∼ 6 * 10 4 km and a radial width of ∼250 km (Hedman et al. 2007). Assuming its vertical thickness is comparable to its radial width, the volume of the G-ring arc is of order 4 * 10 18 m 3 . Inserting these numbers into Equation 12, the critical timescale can be expressed as follows:
t c 6 * 10 5 years C M m a V a 4 * 10 18 m 3 250m R 2 1m/s v i ,(13)
where all of the terms in parentheses should be of order unity for Aegaeon.
Assuming that Aegaeon has a mass density of about 0.5 g/cm 3 , its mass would be M 3 * 10 10 kg. Based on the depth of an electron microsignature observed in the arc's vicinity, Hedman et al. (2007) estimated that the arc's total mass was between 10 8 and 10 10 kg (the width of the microsignature was more consistent with it being associated with the arc than with the moon). We can therefore estimate m a /M to be between 0.003 and 0.3, which would imply damping timescales between 10 6 and 10 8 years. By contrast, Anthe's radius is four times larger than Aegaeon's, so its mass is ∼ 64 times larger than Aegaeon's. Furthermore, assuming the total integrated brightness scales with the total mass, then the mass of the Anthe arc is at least ∼ 50 times smaller than that of the G-ring arc. The characteristic damping time for Anthe should therefore be at least ∼ 200 times longer than for Aegaeon, or 10 8 to 10 10 years.
Anthe's characteristic damping time is comparable to the age of the solar system, which implies that collisional damping has had relatively little effect on Anthe's orbit. Aegaeon's characteristic damping time is much shorter, so collisional damping may be significant for this moon. However, the above values for the damping time will only apply as long as the moon and the arc have their present masses. Since hypervelocity impacts with objects on heliocentric orbits will steadily erode or fragment small moons (cf. Colwell et al. 2000), it is likely that Aegaeon was larger in the past than it is today. Thus collisional damping can only be effective on Aegaeon if its collision damping time is less than the appropriate erosion or fragmentation time-scale.
In lieu of a detailed analysis of Aegaeon's fragmentation history, we can roughly estimate how long Aegaeon may have had its current size by computing the frequency of catastrophic impacts into the moon. The specific energy required for catastrophic fragmentation (i.e. the largest remaining fragment is less than one-half the mass of the original target) of an ice-rich object is of order 2 * 10 5 erg/g (Giblin et al. 2004, see also sources cited in Colwell et al. 2000). Assuming typical impact velocities of order 40 km/s, this means catastrophic fragmentation will occur when the ratio of the impactor's mass to the moon's mass is above about 2.5 * 10 −8 . If we again assume that Aegaeon has a mass of about 3 * 10 10 kg, then any impactors with a mass more than 1000 kg would be able catastrophically disrupt the moon. The present flux of such objects is quite uncertain, but 10 −20 /m 2 /s is consistent with previous estimates and extrapolations (Ip 1984;Colwell and Esposito 1992). This flux gives a catastrophic impact rate into a 500-m wide Aegaeon of order 1 per 10 7 years. This is comparable to the characteristic damping time derived above, so these calculations indicate that Aegaeon could have been close to its present size over a long enough period of time for collisional damping to significantly change its orbit. Clearly, more detailed analyses are needed to clarify and quantify the possible interactions between Aegaeon and the G-ring arc, but these rough calculations do suggest that collisional damping could provide a reasonable explanation for Aegaeon's distinctive dynamical properties
Conclusions
Even though the currently available data on Aegaeon are sparse, they are sufficient to demonstrate that it is a interesting object worthy of further investigation. With a photometrically estimated diameter of less than a kilometer, Aegaeon is the smallest isolated moon of Saturn yet observed, and may be comparable in size to the largest particles in Saturn's main rings, which form the so-called "giant-propellers" in the A ring ). Aegaeon occupies a corotation eccentricity resonance with Mimas, like Anthe and Methone, and all three of these moons are associated with resonantly-confined arcs of debris. However, Aegaeon also appears to be a special case in terms of its orbital properties and its relationship with its arc. Its eccentricity and inclination are both extremely low, and the large number of resonant arguments on the boundary between circulation and libration lead to some interesting dynamical behavior. At the same time, the mass in the G-ring arc is probably a significant fraction of (and may even be comparable to) Aegaeon's mass, unlike the other arcs associated with small moons, opening up the possibility that interactions between the moon and the material in the arc could be responsible for some of Aegaeon's unusual orbital characteristics. Future analysis of this system could therefore provide insights into the orbital evolution of satellites coupled to disks of debris.
Acknowledgements
We thank P.D. Nicholson and J. Veverka for useful conversations, and we also wish to thank two anonymous reviewers for their comments on earlier versions of this manuscript. We acknowledge the support of NASA, the Cassini Project and the Imaging Team. We also wish to thank the VIMS team, who designed several of the observations discussed here. N.C., C.M., K.B. and M.E. acknowledge the financial support of the UK Science and Technology Facilities Council. f Images N1603831280 and N1603831616 taken through RED and IR1 filters, respectively.
All other images taken through clear filters. Kliore et al. (1980). Zonal harmonics and GM from cpck19Sep2007.tpc Name Argument Argument in terms of ϕ CER and ϕ ILR ϕ CER 7λ M imas − 6λ Aegaeon − M imas ϕ CER ϕ ILR 7λ M imas − 6λ Aegaeon − Aegaeon ϕ CER + M imas − Aegaeon ϕ x 7λ M imas − 6λ Aegaeon + M imas − 2 Aegaeon ϕ CER + 2 M imas − 2 Aegaeon = 2ϕ ILR − ϕ CER ϕ y 7λ M imas − 6λ Aegaeon − 2 M imas + Aegaeon ϕ CER − 2 M imas + 2 Aegaeon = 2ϕ CER − ϕ ILR ϕ a 7λ M imas − 6λ Aegaeon − Aegaeon + Ω M imas − Ω Aegaeon ϕ CER + M imas + Ω M imas − Aegaeon − Ω Aegaeon =ϕ ILR + Ω M imas − Ω Aegaeon ϕ b 7λ M imas − 6λ Aegaeon − Aegaeon − Ω M imas + Ω Aegaeon ϕ CER + M imas − Ω M imas − Aegaeon + Ω Aegaeon =ϕ CER + ω M imas − ω Aegaeon =ϕ ILR − Ω M imas + Ω Aegaeon ϕ c 7λ M imas − 6λ Aegaeon − M imas + Ω M imas − Ω Aegaeon ϕ CER + Ω M imas − Ω Aegaeon ϕ d 7λ M imas − 6λ Aegaeon − M imas − Ω M imas + Ω Aegaeon ϕ CER − Ω M imas + Ω Aegaeon
Fig. 1 .
1-The pair of images taken on August 15 (DOY 228), 2008 in which Aegaeon was first noticed.
Fig. 2 .
2-Other low-phase, high-resolution images of Aegaeon obtained from late 2008 through February 20 (DOY 051), 2009. In each image the object's location is highlighted with an arrow. All images are rotated so Saturn's north pole would point upwards. Note the bright feature in the upper left corner of image N1598075119 is due to Tethys being in the camera's field of view.
Fig. 3 .
3-The only two clear images of Aegaeon obtained prior to mid-2008 found to date. In each image the location of the object is highlighted with an arrow. Both images are rotated so Saturn's north pole would point roughly up.
Fig. 4 .
4-The effective areas of Pallene, Methone, Anthe and Aegaeon as functions of phase angle.
Fig. 5 .
5-Observational coverage of Aegaeon projected onto the equatorial plane of Saturn, with superimposed circle of radius 167490 km.
Fig. 6 .
6-Numerical-integration fit residuals in pixel units : (a) line (b) sample.
Fig. 8 .
8-(a) Resonant argument of the 7:6 CER (ϕ = 7λ M imas − 6λ Aegaeon − M imas ) versus time, derived from the numerical integration, showing that it librates about 180 • . (b) Fourier spectrum of resonant argument, showing a dominant period of approximately 1260 days and amplitude 10 • (c) Resonant argument of the 7:6 ILR (ϕ = 7λ M imas − 6λ Aegaeon − Aegaeon ) versus time, showing periods of libration around 0 • interspersed with brief periods of circulation. (d) Fourier spectrum of the resonant argument, showing a libration period of approximately 820 days and amplitude of 35 • .
Fig. 9 .
9-Aegaeon's residual mean longitude (after removing a constant mean motion of 445.482 • /day) with and without Tethys included in the integration. Without Tethys, the residual longitude oscillates about zero with an amplitude of about 3 degrees, solely due to the effects of the 7:6 CER with Mimas (dashed curve). The Mimas:Tethys 4:2 resonance causes the additional large amplitude modulation of tens of degrees when Tethys is included in the model (solid curve).
Fig. 10 .
10-Resonant arguments ϕ a = 7λ M imas − 6λ Aegaeon − Aegaeon − Ω Aegaeon + Ω M imas and ϕ b = 7λ M imas − 6λ Aegaeon − Aegaeon + Ω Aegaeon − Ω M imas versus time, derived from the numerical integration. Note that both these resonant arguments seem to librate around either 0 • or 180 • .
Fig. 11 .
11-Corotation eccentricity resonant arguments versus time for Aegaeon Anthe and Methone. The specific resonant arguments are (a) Aegaeon ϕ CER = 7λ M imas − 6λ Aegaeon − M imas (b) Anthe ϕ CER = 11λ Anthe − 10λ M imas − M imas and (c) Methone. ϕ CER = 15λ M ethone − 14λ M imas − M imas In (a) coarse vertical dashed lines represent the extent of observational coverage, fine vertical line on the right corresponds to 2010-027 and that on the left to 2008-228. In (b) and (c) vertical dashed lines correspond to 2007-302.
Fig. 12 .
12-Longitudinal profiles of the arcs in the G ring (top), Anthe ring (middle) and Methone ring (bottom). Each profile shows the normal equivalent width of the arc versus the appropriate resonant argument of the appropriate corotation resonance. Note the top axis refers to the G-ring arc (which is interior to Mimas) while the bottom axis refers to both the Anthe and Methone arcs (both of which are exterior to Mimas). In all cases, the right sides of the plots lead the relevant moons.
d
The origin of the image line and sample coordinate system is at the center of the top left pixel, with line increasing downwards and sample to the right, when the image is displayed in its normal orientation. The spacecraft −X axis points in the direction of increasing line and −Z axes in the increasing sample direction. Estimated measurement uncertainties ∼ 0.5 pixel in line and sample. e RA and DEC refer to right ascension and declination in the International Celestial Reference Frame (ICRF).
Table 1
1lists all derived parameters for each of the relevant images.
Table 1 :
1Images of Aegaeon 294T04:37:25.937 1172420 14.9 +0.43 0.081 526.5 513.0 151.95312 +1.9205764 N1603172124 2008-294T04:56:04.929 1156757 14.9 +0.60 0.077 713.8 493.2 151.94754 +1.7380537 N1603831170 2008-301T19:59:56.266 1197708 30.6 +4.81 0.077 254.7 414.0 137.60681 -4.0823444 N1603831280 2008-301T20:01:46.273 1199390 30.6 +4.82 0.101 f 123.4 217.6 137.61323 -4.0912850 N1603831616 2008-301T20:07:22.279 1204524 30.6 +4.87 0.094 f 104.9 Spacecraft Event Time b Ring opening angle c Effective area of the object (see text).Image
Midtime
Range Phase
B b
A c
ef f
Line d Sample d
RA e
Dec. e
(SCET) a
(km)
(deg.) (deg.) (km 2 )
(deg)
(deg)
N1560588018 2007-166T08:05:49.180 1708692 42.9 +0.46 0.045 521.5
396.3
191.35064 +5.1764862
N1563866776 2007-204T06:51:37.403 1432779 14.5 +0.01 0.038 500.4
523.1
60.26121 +3.2032395
N1597476237 2008-228T06:45:07.972 1188766 28.2 +4.89 0.074 150.7
232.0
138.26865 -4.1721804
N1597477967 2008-228T07:13:57.959 1215278 28.2 +5.14 0.072 227.9
555.0
138.32963 -4.3106590
N1598073885 2008-235T04:46:01.799 1171032 12.9
-0.75
0.072 533.4
593.5
151.92415 +3.1237490
N1598075119 2008-235T05:06:35.775 1154016 13.0
-0.56
0.097 745.8
619.6
151.93067 +2.7766773
N1598104211 2008-235T13:11:17.572 1179499 28.4 +3.75 0.062 138.0
244.5
137.76804 -3.0640496
N1598106121 2008-235T13:43:07.559 1209028 28.4 +4.02 0.074 321.2
512.2
137.80784 -3.2486713
N1600657200 2008-265T02:20:48.735 1205305 15.2 +4.34 0.079 659.9
377.9
153.49124 -1.8808037
N1600659110 2008-265T02:52:38.706 1177863 15.5 +4.62 0.082 977.0
56.1
153.49796 -2.2741652
N1603168767 2008-294T04:00:07.953 1203664 15.0 +0.09 0.082 856.5
522.8
151.72360 +2.2458309
N1603169886 2008-294T04:18:46.945 1188111 14.9 +0.26 0.105 576.5
525.5
151.87725 +2.0895161
N1603171005 2008-241.5
137.63718 -4.1180591
N1611860574 2009-028T18:22:32.246 1180159 35.0
-1.49
0.072 510.1
500.8
133.96654 +1.8936283
N1611861868 2009-028T18:44:06.221 1199209 34.8
-1.22
0.059 273.8
403.9
134.17857 +1.6322662
N1613784711 2009-051T00:51:05.547 1186785 20.5 +13.6 0.070 291.2
516.0
158.59803 -10.684674
N1613784773 2009-051T00:52:28.255 1185601 20.6 +13.6 0.063 289.0
475.1
158.61153 -10.715706
a
Table 5 :
5Summary of photometric properties of the small moons Moon β p ef f < A phys > < r phys > a Assuming all four moons have p ef f = 0.49 (required to match mean radius of Pallene). Fit to only the |B| > 1 • data.Pallene
0.017 mag/degree
7.38 km 2
2.2 km
Methone 0.023 mag/degree
3.76 km 2
1.6 km
Anthe
0.032 mag/degree
1.76 km 2
1.1 km
Aegaeon b 0.007 mag/degree
0.084 km 2
0.24 km
a b
Table 6 :
6Saturn constants used in orbit fitting and numerical modeling Constant Value a units Pole (RA,Dec) (40.5837626692582, 83.5368877051669) deg Pole position from SPICE kernel cpck19Sep2007.tpc, precessed to the fit epoch. Reference radius fromGM
37931207.1585
km 3 s −2
Radius, R s
60330
km
J 2
0.016290543820
J 4
−0.000936700366
J 6
0.000086623065
a
Table 7 :
7SPICE kernels used in orbit fitting and numerical modeling Kernels are available by anonymous ftp from ftp://naif.jpl.nasa.gov/pub/naif/CASSINI/kernelsKernel name a
pck00007.tpc
naif0009.tls
cas00130.tsc
cpck19Sep2007.tpc
cpck rock 01Oct2007 merged.tpc
de414.bsp
jup263.bsp
sat286.bsp
080806AP SCPSE 08138 10182.bsp
081211AP SCPSE 08346 08364.bsp
090120AP SCPSE 09020 09043.bsp
090202BP SCPSE 09033 09044.bsp
090209AP SCPSE 09037 09090.bsp
090305AP SCPSE 09064 09090.bsp
081125AP RE 90165 18018.bsp
070416BP IRRE 00256 14363.bsp
070727R SCPSE 07155 07170.bsp
070822R SCPSE 07170 07191.bsp
071017R SCPSE 07191 07221.bsp
071127R SCPSE 07221 07262.bsp
080117R SCPSE 07262 07309.bsp
080123R SCPSE 07309 07329.bsp
080225R SCPSE 07329 07345.bsp
080307R SCPSE 07345 07365.bsp
080327R SCPSE 07365 08045.bsp
080428R SCPSE 08045 08067.bsp
080515R SCPSE 08067 08078.bsp
080605R SCPSE 08078 08126.bsp
080618R SCPSE 08126 08141.bsp
080819R SCPSE 08141 08206.bsp
080916R SCPSE 08206 08220.bsp
081031R SCPSE 08220 08272.bsp
081126R SCPSE 08272 08294.bsp
081217R SCPSE 08294 08319.bsp
090120R SCPSE 08319 08334.bsp
090202R SCPSE 08334 08350.bsp
090225R SCPSE 08350 09028.bsp
090423R SCPSE 09028 09075.bsp
a
Table 8 :
8GM values for other perturbing bodies used in orbit fitting and numerical modeling Values from SPICE kernels cpck19Sep2007.tpc and cpck rock 10Oct2007 merged.tpc.Body
GM a (km 3 s −2 )
Sun
132712440044.2083
Jovian system 126712764.8582231
Prometheus
0.01058
Pandora
0.00933
Janus
0.12671
Epimetheus
0.03516
Mimas
2.50400409891677
Enceladus
7.20853820010930
Tethys
41.2103149758596
Dione
73.1128918960295
Rhea
153.941891174579
Titan
8978.13867043253
Hyperion
0.370566623898283
Iapetus
120.504895547942
a
Table 9 :
9Solution for the planetocentric state vector of Aegaeon, from a fit to Cassini ISS data in the ICRF frame. Epoch is-228T06:45:07.972 UTC (2008 Aegaeon
Table 10 :
10Planetocentric Orbital Elements Parameter a 001T12:01:05.183 TDB TDB JD 2454467.00075444 TDB TDB Semi-major axis, a calc Inclination, i mean 0.0010 ± 0.0009 deg Resonant argument (CER) 7λ M imas − 6λ Aegaeon − M imas Resonant argument's libration period (CER) 1264 ± 1 days Resonant argument (ILR) 7λ M imas − 6λ Aegaeon − Aegaeon Resonant argument's libration period(ILR) 824 ± 1 days a All longitudes measured directly from ascending node of Saturn's equator on the ICRF equator. Inclination measured relative to Saturn's equatorial plane. Quoted uncertainties in the upper half of the table are the formal 1σ values from the fit. Note the values in the upper half of the table are values at a particular point in time and are provided as a guide. They are not suitable as starting conditions in integrations for the equation of motion. Mean values and their librations in the lower half of the table were obtained from a numerical integration of the period 2004-001 to 2014-001, taking into account the resonant behavior.Fitted Value
Units
Table 11 :
11Resonant Arguments of Interest
Two images (N1603831280 and N1603831616) were obtained using the RED (λ ef f = 649 nm) and IR1(λ ef f =751 nm) filters, respectively. While Aegaeon's brightness is the same at both these wavelengths at the 5% level, it is premature to make any definite conclusions about Aegaeon's color based on such limited data.
This preprint was prepared with the AAS L A T E X macros v5.2.
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|
[] |
[
"In situ resonant photoemission and X-ray absorption study of the BiFeO 3 thin film",
"In situ resonant photoemission and X-ray absorption study of the BiFeO 3 thin film"
] |
[
"Abduleziz Ablat \nSchool of physical Science and Technology\nXinjiang University\n830046UrumqiChina\n",
"Mamatrishat Mamat \nSchool of physical Science and Technology\nXinjiang University\n830046UrumqiChina\n",
"Yasin Ghupur \nSynchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina\n",
"Rong Wu \nSynchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina\n",
"Emin Muhemmed \nSynchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina\n",
"Jiaou Wang \nSynchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina\n",
"Haijie Qian \nSynchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina\n",
"Rui Wu \nSynchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina\n",
"Kurash Ibrahim \nSynchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina\n"
] |
[
"School of physical Science and Technology\nXinjiang University\n830046UrumqiChina",
"School of physical Science and Technology\nXinjiang University\n830046UrumqiChina",
"Synchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina",
"Synchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina",
"Synchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina",
"Synchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina",
"Synchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina",
"Synchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina",
"Synchrotron Radiation Facility\nInstitute of High Energy Physics\nCAS\n100049Beijing, BeijingChina"
] |
[] |
Multiferroic bismuth ferrite (BiFeO 3 ) thin films were prepared by pulsed laser deposition (PLD) technique. Electronic structures of the film have been studied by in situ photoemission spectroscopy (PES) and x-ray absorption spectroscopy (XAS). Both the Fe 2p PES and XAS spectra show that Fe ion is formally in +3 valence state. The Fe 2p and O K edge XAS spectra indicate that the oxygen octahedral crystal ligand field splits the unoccupied Fe 3d state to t 2g ↓ and e g ↓ states. Valence band Fe 2p-3d resonant photoemission results indicate that hybridization between Fe 3d and O 2p plays important role in the multiferroic BiFeO 3 thin films.
|
10.1016/j.ceramint.2016.03.157
|
[
"https://arxiv.org/pdf/1511.07747v1.pdf"
] | 118,608,812 |
1511.07747
|
5236e89daa0d0a73bc022201540b83fb1bde820c
|
In situ resonant photoemission and X-ray absorption study of the BiFeO 3 thin film
Abduleziz Ablat
School of physical Science and Technology
Xinjiang University
830046UrumqiChina
Mamatrishat Mamat
School of physical Science and Technology
Xinjiang University
830046UrumqiChina
Yasin Ghupur
Synchrotron Radiation Facility
Institute of High Energy Physics
CAS
100049Beijing, BeijingChina
Rong Wu
Synchrotron Radiation Facility
Institute of High Energy Physics
CAS
100049Beijing, BeijingChina
Emin Muhemmed
Synchrotron Radiation Facility
Institute of High Energy Physics
CAS
100049Beijing, BeijingChina
Jiaou Wang
Synchrotron Radiation Facility
Institute of High Energy Physics
CAS
100049Beijing, BeijingChina
Haijie Qian
Synchrotron Radiation Facility
Institute of High Energy Physics
CAS
100049Beijing, BeijingChina
Rui Wu
Synchrotron Radiation Facility
Institute of High Energy Physics
CAS
100049Beijing, BeijingChina
Kurash Ibrahim
Synchrotron Radiation Facility
Institute of High Energy Physics
CAS
100049Beijing, BeijingChina
In situ resonant photoemission and X-ray absorption study of the BiFeO 3 thin film
Multiferroic bismuth ferrite (BiFeO 3 ) thin films were prepared by pulsed laser deposition (PLD) technique. Electronic structures of the film have been studied by in situ photoemission spectroscopy (PES) and x-ray absorption spectroscopy (XAS). Both the Fe 2p PES and XAS spectra show that Fe ion is formally in +3 valence state. The Fe 2p and O K edge XAS spectra indicate that the oxygen octahedral crystal ligand field splits the unoccupied Fe 3d state to t 2g ↓ and e g ↓ states. Valence band Fe 2p-3d resonant photoemission results indicate that hybridization between Fe 3d and O 2p plays important role in the multiferroic BiFeO 3 thin films.
Introduction
Multiferroics are a group of materials that a same system simultaneously having different forms of ferroic properties such as ferroelectricity, ferromagnetism, and ferroelasticity [1]. In multiferroics, magnetization can be tuned by applied electric field, and electron polarization by applied magnetic field. These effects called magnetoelectric effect [2]. Few multiferroic materials, YMnO 3 , PbVO 3 , BiCrO 3 , BiMnO 3 and BiFeO 3 (BFO), exhibit as of natural occurring phases [3][4][5]. Among them, the BFO is the only material that shows magnetoelectric effects at room temperature [6]. The BFO is considered to be a promising candidate for electronic device application benefitting from its high Curie (T C ～825 °C) and Néel (T N ～370 °C) phase transition temperatures [4]. Compared to its bulk form，BFO thin film shows apparently improved ferroelectric and magnetization characters [7].
Understanding the multiferroic properties of BFO in terms of its electronic structure is important, because both the electric and magnetic properties closely relate to it. An open problem with BFO is that it loses easily the ferroic characters with high current leakage. The current leakage was attributed to small amount of Fe 2+ ion or oxygen vacancies.
To solve this problem, there has been investigation with different growth techniques and various doping methods [8][9][10]. Decrease or elimination of the Fe 2+ ion impurity is usually judged through the trends of decreased current leakage and enhanced ferroelectric hysteresis loops. Experimental investigations in terms of valence band electronic structure are rare [11,12]. On the other hand, results in the electronic band structure of BFO by theoretical calculation are controversial [13,14]. Density of state analysis [13] shows that the Bi 6p state locates above 4eV of the gap, and the origin of the ferroelectricity is ascribed as O 2p-Bi 6p dynamic hybridization. Other studies [14] show the Bi 6p state has a contribution on both valence and conduction bands through hybridization with both Fe 3d and O 2p states. In the current work we aim at the investigation of the valence band states of the BFO by means of photoemission method to uncover possible contributing routes in regard to the controversial conclusions.
We first prepared BFO thin films by PLD method for in situ PES and XAS measurement. The advantage of thin film over bulk state is at least threefold. Firstly, thin film can free from charging effect by controlling film thickness that is a main unwanted factor in the PES measurements. Secondly, thin films can be grown by thickness controlling, as well as with slightly varied stoichiometry by regulating ambient oxygen partial pressure. In the last, the in situ prepared thin films can free from surface contamination that is crucial for PES measurements, due to the measurement itself is highly surface sensitive. The results show the films prepared under higher oxygen partial pressure is near to stoichiometry, and the Fe ion is formally in valence-three state. Oxygen K edge XAS and valence band resonant photoemission spectroscopy (RPES) through Fe 2p core level excitation indicate the conduction and valence bands are mainly of Fe 3d and O 2p hybridization state.
Experiment
Ceramic BFO target of 20 mm diameter with 2 mm thickness has been prepared by sintering 1 [15].
Prior to deposition, the Pt (111)/TiO 2 /SiO 2 /Si (100) substrates were preheated at 700°C to eliminate surface contaminations. To grow near stoichiometric BFO films, the substrates were kept at 550°C and the films were grown at different oxygen partial ambient pressures. The laser fluence was 2.3J/cm 2 with pulse repetition rate 1.5Hz. A stoichiometric film was obtained at oxygen partial pressure Po 2 =5.6 Pa, consistent with reported results [16,17]. After deposition, the sample was transferred to the PES chamber under a background pressure of ~10 -8 Pa. The overall energy resolution was 0.2-0.7 eV, depending on the selected monochromatic photon energies. Photoemission spectra were calibrated to the in-substrate Pt 4f 7/2 peak at 71.2 eV. The XAS measurement carried out with total electron yield detection mode.
Results and Discussion
PES characterization of the films
Usually, PES measurement can be used to check whether a sample is stoichiometric or not and to estimate relative atomic ratios in the system. The O 1s narrow range scan spectrum (Fig.2 a) with a higher energy resolution and better statistics show asymmetric peaks near 530 eV. The spectrum can be fitted by two peaks locating at lower and higher binding energies (LBE and HBE), respectively. The LBE peak is attributed to the O 1s of perfect BFO phase, while the HBE one relates with oxygen defects in samples [18]. The peak area of the LBE is larger than that of HBE, indicating the sample was in low oxygen vacancy, the BFO is mainly with perfect stoichiometric constituent.
The Fe 2p core level photoemission spectrum (Fig.2 [19]. But recent literature report suggests that decomposing the Fe 2p into symmetric components is not feasible [20]. The way to fix the formal valence state is by satellite peak of Fe 3+ and Fe 2+ in PES, because of different d orbital electron configurations, relaxation of the Fe 2+ and Fe 3+ show satellite peaks at 6 eV and 8 eV, respectively, above their 2p 3/2 main peaks [16]. The measured spectrum in [21]. There are no shoulders appearing either at higher or lower binding energy sides as being clues for the existence of higher than Bi 3+ oxidized or lower than Bi 3+ reduced metallic bismuth states [22]. The estimated molar ratio of Bi to Fe from the peak area of the Bi 4f and Fe 2p, which are normalized to respective atomic orbital cross sections at the specific excitation energy [23], is near to 1:1 indicates that composition of the thin film under proper Po 2 pressure maintained almost the same with its target precursor.
O K edge XAS and Valance band photoemission
The XAS provides information on the excitation of a core electron into unoccupied states as function of photon energy, that is about a cross sectional variation of measured photoelectron DOS versus photon energy.
It implies unoccupied density of state and crystal field splitting effect, and most importantly provide signature on hybridization between atoms.
The left and right panels of Fig.3 show the valence band PES and O K XAS spectra of the BFO thin film. Valence band PES represents the occupied DOS distribution below Fermi level. In Fig 3(a) the VB spectrum can be divided into two main block structures labeled as α, β, γ, δ from Fermi edge to below ~9 eV and a broad peak at ~12 eV. The feature around 12 eV originates from Bi 6s states according to the DOS result by first principle calculation [24]. In the coming section, we will discuss an enhanced feature of the four α, β, γ, δ structures by resonant photoemission at Fe L edge.
The O K-edge XAS spectra in Fig 3( between the a / and a peaks in the O 1s XAS, a similar value to the crystal-field splitting 10Dq induced between the two empty states t 2g ↓ and e g ↓, is almost the same as observed in the Fe 2p XAS spectra in Fig.4.
These imply that they are of Fe 3d character and provide a solid base for the concept of dynamic charge transfer through hybridization.
A common point for the peak b is to ascribe it as of Bi 6sp character [14], that is due to interaction between the full filled O 2p and partial occupied Bi 6sp states allow to form a certain degree of covalent bonding between the O 2p and Bi 6sp. There exists also controversial points of view regarding the b's origin [27], where refers the peak b to as result of the interactions between the unoccupied O 2p and Bi 5d states.
The peak c at ~542 eV derives from hybridization of O 2p and Fe 4sp [28]. consist of four main structure labeled as α, β, γ and δ, the same as those observed in Fig.3(a).
Fe 2p XAS and Fe 2p-3d resonant photoemission
An overall fuzzy enhancement in spectral intensity of the peaks and then decreasing when the excitation energy pass through the Fe 2p 3/2 threshold region is clear in the spectra shown in Fig.5(b), that reflect the sensitiveness of the Fe 3d characteristic partial DOS to the threshold excitation. A fuzzy broad block of peaks, which simultaneously varies all together in lack of independently enhanced peak shapes below 0-14 eV
Fermi level, reaches maximum at least with a factor of ten at excitation energy 710.5 eV relative to those excited below and above at this energy.
The valence band peaks photoemission intensity increase with such fuzziness as whole, instead of observing a specific peak enhancement, reveals at least two kinds of information. The first is that the photoemission in the on-resonance region has apparently additional contribution source from that of photoemission in the off-resonance region. The second remarkable point reveals that the resonantly enhanced intensity shape of valence bands strongly relates to surviving environment of Fe 3d electrons, instead of the Fe 3d partial density of state per se.
The resonant photoemission enhancement in an order of magnitude at threshold excitation is a generic situation for most elements in condensed matter systems, and its origin can differ for different systems. One can say at least one common cause about the significant difference between the off-resonance and on-resonance is that the former mainly results in the single-photon single-channel process, and the latter through single-photon multiple-channel processes. Here the single-photon is referred to as monochromatized single energetic photons, single-channel process means the measured DOS corresponds to single-photon single-channel (or direct) photoemission process. While the single-photon multiple-channel (or indirect) processes make an additional aggregative contribution from multiple-channel de-excitations against single photon excitation, which are in addition to the DOS results in the single-photon single-channel process.
The above discussed processes are schematically shown in Fig.6.
Among them, the single-photon single-channel process refers to as the direct photoemission ①. In this process, whenever the photon energy is below or above the threshold region, the measured intensity of the valence band comes only contribution from direct photoionization of the valence state electrons, no other processes are countable for it. When the excitation photon energy match with the potential energy of certain inner shell, the resonantly excited electron from that shell leads to the processes ② or ③. These last two processes are indirect photoemissions, their contributions add up to that of the normal process ①, at the end they give the measured results where the intensity has an enormous increase. behind play the role as of pump-probe, by which one is able to inspect the electronic structural interaction of constituent elements in the valence region. As shown in Fig.5(a), the four feature through enhanced in order scale, they are with as whole but without a single peak with apparent sharpening, seemingly if they were a single body. It implies there have exist a strong hybridization between Fe 3d with its surrounding environment, such as O 2p and Bi 6p states. One can approximately assign the four feature after carefully compare the on and off resonance spectrum in Fig.5(b). According to the energy diagram of Fe 3d 5 states [31], the resonantly enhanced feature α is assigned to e g ↑(↑ denotes the majority spin) states of Fe 3d, feature β to the t 2g ↑ states with a weakly hybridized O 2p state (mostly of Fe 3d like character). The energy separation between these two states consistent with the energy separation between the unoccupied e g ↓ and t 2g ↓ states as shown in
Conclusion
In conclusion, the electronic structure of multiferroic BFO has been investigated by using the PES and XAS. The measured PES survey spectrum, Fe 2p PES and XAS spectra show that the Fe ions are formally trivalent (3d 5
Determination of ionic valence state is through indirect way by measuring energy positions of concerned elements' main peaks and related satellite peaks, by inspecting the ratios of peaks intensity of constitute atoms.
Fig. 1
1illustrates a wide range scan PES spectrum of the BFO film at 900 eV that gives overall status information about the constituent elements of the system. The dominating signals are of core-level photoemission peaks and Auger lines of Bi, Fe, and O atoms. Absence of the C 1s signal at about 285 eV binding energy indicates that the surface is free from contamination, thanks to the in situ operations. Characteristic core-level photoemission peaks of Fe 2p 3/2 711.3 eV, Fe 2p 1/2 725.4 eV, Bi 4p 3/2 680.7 eV, Bi 4d 5/2 441.4 eV, Bi 4d 3/2 465.7 eV, Bi 4f 7/2 159.3 eV, Bi 4f 5/2 164.6 eV, O 1s 530.3 eV and Pt 4f at 71.2 eV from substrate, and Auger decay channels of Fe L 3 M 23 M 23 , Fe L 3 M 23 M 45 , Fe L 3 M 45 M 45 , O KVV are labeled on the spectrum.
Fig. 2
2(b), a satellite peak 8eV above the Fe 2p 3/2 711.3 eV, confirms that the iron atom is formally in Fe 3+ state. In the limits of the PES measurement sensitivity to molar portion in system, there is no apparent evidence showing up for the existence of Fe 2+ ion in the underlying BFO thin film.
b) provides information on the hybridization of O 2p with Fe 3d states. Under ionic picture, the O 2p is considered fulfilled with six electrons. Then following the definition of XAS, we should observe any absorption effect at the O 1s region, but actually we do observe a copious structure in this region. The observed results imply a certain portion of empty O 2p orbitals with nontrivial probability due to hybridization between the Fe 3d [25], the hybridization results in dynamic charge transfer from O 2p to Fe 3d. The oxygen K edge further manifests an extended structure up to at least 17 eV above the threshold, implying an electronic structure resulted in various effects such as ligand field splitting, hybridization of the oxygen 2p and transition metal 3d states. The region near to the absorption edge, labeled as a / , a and b, the most sensitive region to those effects, is expanded and shown as inset in Fig.3. Among them the features a / and a have the key importance for understanding inter-atomic interactions. The a / at ~530.8 eV is attributed to the O 2p+Fe 3d t 2g ↓ characteristic states, the peak a to the O 2p + Fe 3d e g ↓ states, where ↓ denotes the minority spin states. Reason for how that O 2p + Fe 3d hybridization leads to the status quo a / , a structure is twofold. Firstly, the XAS is a local process in which an electron is promoted to an unoccupied electronic state, which couples to the original core level restricted by the electric dipole selection rule by parity consideration, which states that the change in the angular momentum quantum number can only take ∆L=±1 values, while the spin keeps unchanged [26]. For the excitation of an O 1s electron of l =0, the ∆L=±1 requirement means that only O 2p character l =1 can be reached, and an inter-atomic O 1s-Fe 3d transition is not allowed. In terms of ionic picture for oxygen in oxides, this prohibits, theoretically, occurrence of O 1s-O 2p absorption. These apparent idealized situations do not hold regarding the experimental reality, by showing up a wide peak envelope in 529~535 eV region with fine internal structures as of complicated interaction effects. These experimental facts imply that the rigid ionic picture is not appropriate for understanding the observed results, and the results reveal that there exist strong interactions of O ions with the central transition metal Fe ions. The interactions open up channels available for the transition between O 1s and Fe 3d-like orbitals. Such interaction (hybridization) brings about transfer of certain amount of electron density, i.e., dynamic charge transfer from O 2p to Fe 3d orbital that creates hole-state in the O 2p. The hole-state creation on the ligand side is instrumental for understanding the experimental features observed at the O K edge. Secondly, the observed ~1.4 eV distance
Fig. 4
4shows the Fe 2p XAS spectrum of BFO thin film together with those of reference oxides of formal trivalent Fe 2 O 3 (Fe 3+ ), divalent FeO (Fe 2+ )[29], and plus the atomic multiplet calculation result. The XAS data of Fe 2 O 3 and FeO were shifted by ～3 eV to align with the measured one. Calculation performed with atomic multiplet model in octahedral symmetry for different 10Dq values varying from 1.0 to 1.8eV, a multiplet splitting energy range 10Dq suited for the 3d transition metal 2p edges[30]. The Slater integrals were scaled to 80% of the associated atomic values. For simulate lifetime effects and instrument resolution, discrete multiplets were broadened with a Lorentzian of 0.3 eV and a Gaussian of 0.3 eV at both L 3 and L 2 edges. A good agreement achieved between the calculation and experiment for a value at 10Dq~1.6 eV.The measured energy separation between t 2g ↓ and e g ↓ states, both forL 3 and L 2 levels, about 1.4 eV corresponds to the crystal field splitting energy 10Dq. It leads the Fe 3d 5 electrons to high-) [31]. Resemblance of the BFO line shapes to that of Fe 2 O 3 , and a significant deviation from FeO indicate the Fe ions are formally in trivalent (Fe 3+ :3d 5 ) state, confirming the evidence drawn from the PES. Resonant photoemission (RPES) spectra shown in Fig.5(a) recorded between photon energy 702 eV and 718 eV spanning the Fe 2p 3/2 threshold region of XAS in Fig.4 to look at the valence electronic structure relation of Fe with its surrounding ligand O atoms. The most resonantly enhanced spectrum at 710.5 eV and two off-resonance spectra below and above the threshold are shown in Fig.5(b) to contrast resonant and off-resonant effects. The spectra are normalized to the storage ring current and the spectrum sweeping numbers. The valence band spectra
33]. These are the main origins for the observed intensity increase at threshold excitations in photoemission measurement. Unravel electronic structural origin of the density of state enhancement as a whole with the threshold excitation in the valence band region, where the valence band states marked as α, β, γ and δ, requires to understand the relationship between Fe n d 3 and the rest vb electrons in the region. In other words, need to have an appreciation of Fe n d 3 surviving environment. The resonant valence band photoemission measured by excitation of inner-shell electron which leaves a core-hole
Fig. 3
3(b). The feature γ to the hybridization of Fe 3d-O 2p bonding states since the intensity of this feature increased through Fe 2p-3d threshold and reach a maximum at on resonance region. The above discussed resonant photoemission investigation results show that the whole ranges of n d vb3 bands have mixed with Fe 3d, O 2p and Bi 6p states. In our previous work about BiFe 1-x Mn x O 3 , also support there have a strong hybridization between the Fe 3d states with O 2p states in pure BFO and the hybridization decreased with increasing Mn content[34]. The resonate photoemission spectra in Fig.5 strongly support our assumption that whole the marked region (α,β,γ and δ) have a Fe 3d characters through hybridization with other orbitals.
The measured valence band RPES and O K-edge XAS spectra show that the valence band and conduction band mainly consists of Fe 3d and O 2p states through hybridization, and d-d transition between the valance band and conduction band play a key role in electronic properties of BFO.
Figure
Caption
Fig. 1 .
1PES survey spectrum of BFO films onto a silicon substrate.
Fig. 2 .
2PES survey spectrum of (a)O 1s, (b)Fe 2p and (c) Bi 4f lines of the BFO films. The molar ratio of Bi and Fe was calculated by normalized peak area of Fe Fe 2p 3/2 and Bi 4f 7/2 (see text for details).
Fig. 3
3Valence band photoelectron and O 1s X-ray absorption spectrum of the BFO films. The insets at right panel shows an expended image of the XAS spectrum.
Fig. 4
4Comparison of the Fe 2p XAS of BFO films to those of FeO[29], Fe 2 O 3[29] and atomic multiplet calculation.
Fig. 5
5(a) Valence band spectrum of BFO recorded with photon energy across the Fe 2p-3d resonance. (b) On resonance (hv=710.5eV) and off resonance (hv=702.3eV and 717.5eV) PES spectra of BFO.
Fig. 6
6Schematic expression of (1) direct photoemission and (2), (3) indirect photoemission processes that make major contributions to the measured valence band DOS distribution at around Fermi level upon excitation of a core level electron in threshold region.
Fig. 1
1Fig.1
Fig. 2
2Fig.2
Fig. 3
3Fig.3
Fig. 3
3Fig.3
Fig. 4
4Fig.4
Fig. 5
5(a) and (b)
.01:1 mixtures of Bi 2 O 3 (99.99 at.%) and Fe 2 O 3 (99.99 at.%) at 820 °C, following the routine way of solid state reactions. Slight excess of Bi 2 O 3 has the role to compensate the preferential loss of bismuth during sintering. X-ray diffraction (XRD) result of the target material indicates that the sample is in single phase of BFO. The BFO films are prepared by PLD method on the Pt (111)/TiO 2 /SiO 2 /Si (100) and quartz substrates in a chamber connected to the photoemission system at 4B9B beam line of Beijing Synchrotron Radiation Facility
b) subjects to be applied to determine the valence state of Fe ions, where it splits into Fe 2p 3/2 at 711.3 eV and Fe 2p 1/2 at 725.4 eV as result of spin-orbit coupling effects. The Fe 2p 3/2 peak shape resulted in fitting is normally inspected to determine if the cation is in uni-valence or in multi-valence state, where the Fe 2p 3/2 is decomposed into symmetric superposition components of Fe 2+ and Fe 3+
refers to all the rest valence band electrons except that of Fe 3d n , the FeThe single-photon multiple-channel processes at the threshold region are
actually the summation of ① + ② + ③ +
. . . . Here in the both
processes ② and ③, the neutral initial state [
n
d
vb
p 3
2 6
], where the vb
2p state electron absorbs a photon and excites into intermediate
[
1
5
3
2
+
n
d
vb
p
] * or [
1
2
5
3
3
2
+
n
d
vb
s
p
] * excited neutral states by creating a
primary vacancy in the Fe
5
2 p core shell and promoting one electron to
1
3 +
n
d
vb
state. These states then de-excite through corresponding channels
into respective final states. The
*
1
5
]
3
2
[
+
n
d
vb
p
state de-excites into
−
− ]
3
2
[
1
6
n
d
vb
p
final state through super-Coster-Kronig channel [32], and
the
*
1
2
5
]
3
3
2
[
+
n
d
vb
s
p
state into
−
]
3
3
2
[
AcknowledgementsThis study was financially supported by National Natural Science
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|
[] |
[
"Probing Phase Fluctuations in a 2D Degenerate Bose Gas by Free Expansion",
"Probing Phase Fluctuations in a 2D Degenerate Bose Gas by Free Expansion"
] |
[
"Jae-Yoon Choi \nCenter for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea\n",
"Sang Won Seo \nCenter for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea\n",
"Jin Woo \nCenter for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea\n",
"Yong-Il Kwon \nCenter for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea\n",
"Shin \nCenter for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea\n"
] |
[
"Center for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea",
"Center for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea",
"Center for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea",
"Center for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea",
"Center for Subwavelength Optics\nDepartment of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea"
] |
[] |
We measure the power spectrum of the density distribution of a freely expanding 2D degenerate Bose gas, where irregular density modulations gradually develop due to the initial phase fluctuations in the sample. The spectrum has an oscillatory shape, where the peak positions are found to be independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the normalized spectral peak strength and observed to decrease at lower temperatures, confirming the thermal nature of the phase fluctuations. We investigate the relaxation dynamics of nonequilibrium states using the power spectrum. Free vortices are observed with ring-shaped density ripples in a perturbed sample after a long relaxation time. 03.75.Hh, 03.75.Lm Phase coherence is one of the main characteristics of superfluidity and critically affected by the dimensions of the system. In low dimensional systems, large thermal and quantum phase fluctuations prohibit the establishment of long-range phase coherence which is a typical order parameter for a three-dimensional superfluid[1,2]. Nevertheless, a two-dimensional (2D) interacting system can undergo a superfluid phase transition at a finite critical temperature with an algebraically decaying coherence. This transition is successfully described in the Berezinskii-Kosterlitz-Thouless (BKT) theory [3, 4] as a topological phase transition, where the critical point is associated with spontaneous pairing of free vortices with opposite circulations.Recent experiments with 2D atomic Bose gases have demonstrated that this BKT physics can be studied in a finite-size trapped sample[5][6][7][8][9][10][11]. Phase coherence and thermodynamic properties have been investigated using matter-wave interference and by detailed analysis of the in situ density and momentum distributions of trapped samples, observing the algebraic decay of coherence [5], a presuperfluid regime[7][8][9][10], and the scale invariance of the equation of state[9,11]. It is now highly desirable to have quantitative probes directly sensitive to phase fluctuations in order to study the topological nature of the phase transition. In particular, nonequilibrium phase dynamics near the critical point would provide valuable insights on the BKT transition. Relaxation dynamics in the 2D XY model have been under intense theoretical investigation[12,13]and recently an experimental scheme to study a dynamic BKT transition in 2D Bose gases was proposed[14].In this paper, we demonstrate a new quantitative probe for phase fluctuations in a 2D degenerate Bose gas using the density correlations in a freely expanding sample. Irregular density modulations gradually develop in the coherent part of the sample during expansion. We observe that the power spectrum of the density distribution has an oscillatory shape and find that the peak positions are independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the spectral peak strength normalized with the central density of the coherent part and we show that it decreases at lower temperature in thermal equilibrium. This confirms the thermal nature of phase fluctuations in the 2D system. In addition, we investigate the relaxation dynamics of nonequilibrium states by measuring the time evolution of the relative intensity of phase fluctuations. Interestingly, ring-shaped density ripples stochastically appear in a perturbed sample after a long relaxation time, which we identify with free vortices having a long lifetime in a 2D sample.We prepare a 2D degenerate Bose gas of 23 Na atoms in a single pancake-shape optical dipole trap[15,16]. Thermal atoms in the |F = 1, m F = −1 state are loaded from a plugged magnetic trap [17] into the optical trap and evaporative cooling is applied by reducing the trap depth. The sample temperature is controlled by the final trap depth in the evaporation, resulting in 0.6 to 1.3×10 6 atoms in a sample. Finally, the optical trap depth ramps up and the trapping frequencies (ω x , ω y , ω z ) = 2π×(3.0, 3.9, 370) Hz. The cooling procedure is intentionally set to be slow over 15 s, ensuring thermal equilibrium. The lifetime in the optical trap is over 50 s. In the Thomas-Fermi approximation, the chemical potential is about h × 260 Hz less than the confining energy ω z , so we expect 2D physics in the phase coherence of the sample at low temperature. The dimensionless interaction strength g = a 8πmω z / 0.013, where a is the 3D scattering length and m is the atomic mass. The in-plane density distribution n(x, y) is measured by taking an absorption image after an expansion time t e which is initiated by suddenly turning off the trapping potential.Expansion has been a conventional and powerful method in quantum gas experiments to study coherence properties of a sample[7,10]. In our experiment, we are interested in the short expansion regime where t e 1/ω x,y and the phase coherence information would arXiv:1206.2232v2 [cond-mat.quant-gas]
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10.1103/physrevlett.109.125301
|
[
"https://arxiv.org/pdf/1206.2232v2.pdf"
] | 17,381,789 |
1206.2232
|
07e55ee1a2046d7fd008fe1e9953f48a3d538e10
|
Probing Phase Fluctuations in a 2D Degenerate Bose Gas by Free Expansion
(Dated: May 1, 2014) 9 Aug 2012
Jae-Yoon Choi
Center for Subwavelength Optics
Department of Physics and Astronomy
Seoul National University
151-747SeoulKorea
Sang Won Seo
Center for Subwavelength Optics
Department of Physics and Astronomy
Seoul National University
151-747SeoulKorea
Jin Woo
Center for Subwavelength Optics
Department of Physics and Astronomy
Seoul National University
151-747SeoulKorea
Yong-Il Kwon
Center for Subwavelength Optics
Department of Physics and Astronomy
Seoul National University
151-747SeoulKorea
Shin
Center for Subwavelength Optics
Department of Physics and Astronomy
Seoul National University
151-747SeoulKorea
Probing Phase Fluctuations in a 2D Degenerate Bose Gas by Free Expansion
(Dated: May 1, 2014) 9 Aug 2012
We measure the power spectrum of the density distribution of a freely expanding 2D degenerate Bose gas, where irregular density modulations gradually develop due to the initial phase fluctuations in the sample. The spectrum has an oscillatory shape, where the peak positions are found to be independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the normalized spectral peak strength and observed to decrease at lower temperatures, confirming the thermal nature of the phase fluctuations. We investigate the relaxation dynamics of nonequilibrium states using the power spectrum. Free vortices are observed with ring-shaped density ripples in a perturbed sample after a long relaxation time. 03.75.Hh, 03.75.Lm Phase coherence is one of the main characteristics of superfluidity and critically affected by the dimensions of the system. In low dimensional systems, large thermal and quantum phase fluctuations prohibit the establishment of long-range phase coherence which is a typical order parameter for a three-dimensional superfluid[1,2]. Nevertheless, a two-dimensional (2D) interacting system can undergo a superfluid phase transition at a finite critical temperature with an algebraically decaying coherence. This transition is successfully described in the Berezinskii-Kosterlitz-Thouless (BKT) theory [3, 4] as a topological phase transition, where the critical point is associated with spontaneous pairing of free vortices with opposite circulations.Recent experiments with 2D atomic Bose gases have demonstrated that this BKT physics can be studied in a finite-size trapped sample[5][6][7][8][9][10][11]. Phase coherence and thermodynamic properties have been investigated using matter-wave interference and by detailed analysis of the in situ density and momentum distributions of trapped samples, observing the algebraic decay of coherence [5], a presuperfluid regime[7][8][9][10], and the scale invariance of the equation of state[9,11]. It is now highly desirable to have quantitative probes directly sensitive to phase fluctuations in order to study the topological nature of the phase transition. In particular, nonequilibrium phase dynamics near the critical point would provide valuable insights on the BKT transition. Relaxation dynamics in the 2D XY model have been under intense theoretical investigation[12,13]and recently an experimental scheme to study a dynamic BKT transition in 2D Bose gases was proposed[14].In this paper, we demonstrate a new quantitative probe for phase fluctuations in a 2D degenerate Bose gas using the density correlations in a freely expanding sample. Irregular density modulations gradually develop in the coherent part of the sample during expansion. We observe that the power spectrum of the density distribution has an oscillatory shape and find that the peak positions are independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the spectral peak strength normalized with the central density of the coherent part and we show that it decreases at lower temperature in thermal equilibrium. This confirms the thermal nature of phase fluctuations in the 2D system. In addition, we investigate the relaxation dynamics of nonequilibrium states by measuring the time evolution of the relative intensity of phase fluctuations. Interestingly, ring-shaped density ripples stochastically appear in a perturbed sample after a long relaxation time, which we identify with free vortices having a long lifetime in a 2D sample.We prepare a 2D degenerate Bose gas of 23 Na atoms in a single pancake-shape optical dipole trap[15,16]. Thermal atoms in the |F = 1, m F = −1 state are loaded from a plugged magnetic trap [17] into the optical trap and evaporative cooling is applied by reducing the trap depth. The sample temperature is controlled by the final trap depth in the evaporation, resulting in 0.6 to 1.3×10 6 atoms in a sample. Finally, the optical trap depth ramps up and the trapping frequencies (ω x , ω y , ω z ) = 2π×(3.0, 3.9, 370) Hz. The cooling procedure is intentionally set to be slow over 15 s, ensuring thermal equilibrium. The lifetime in the optical trap is over 50 s. In the Thomas-Fermi approximation, the chemical potential is about h × 260 Hz less than the confining energy ω z , so we expect 2D physics in the phase coherence of the sample at low temperature. The dimensionless interaction strength g = a 8πmω z / 0.013, where a is the 3D scattering length and m is the atomic mass. The in-plane density distribution n(x, y) is measured by taking an absorption image after an expansion time t e which is initiated by suddenly turning off the trapping potential.Expansion has been a conventional and powerful method in quantum gas experiments to study coherence properties of a sample[7,10]. In our experiment, we are interested in the short expansion regime where t e 1/ω x,y and the phase coherence information would arXiv:1206.2232v2 [cond-mat.quant-gas]
We measure the power spectrum of the density distribution of a freely expanding 2D degenerate Bose gas, where irregular density modulations gradually develop due to the initial phase fluctuations in the sample. The spectrum has an oscillatory shape, where the peak positions are found to be independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the normalized spectral peak strength and observed to decrease at lower temperatures, confirming the thermal nature of the phase fluctuations. We investigate the relaxation dynamics of nonequilibrium states using the power spectrum. Free vortices are observed with ring-shaped density ripples in a perturbed sample after a long relaxation time. Phase coherence is one of the main characteristics of superfluidity and critically affected by the dimensions of the system. In low dimensional systems, large thermal and quantum phase fluctuations prohibit the establishment of long-range phase coherence which is a typical order parameter for a three-dimensional superfluid [1,2]. Nevertheless, a two-dimensional (2D) interacting system can undergo a superfluid phase transition at a finite critical temperature with an algebraically decaying coherence. This transition is successfully described in the Berezinskii-Kosterlitz-Thouless (BKT) theory [3,4] as a topological phase transition, where the critical point is associated with spontaneous pairing of free vortices with opposite circulations.
Recent experiments with 2D atomic Bose gases have demonstrated that this BKT physics can be studied in a finite-size trapped sample [5][6][7][8][9][10][11]. Phase coherence and thermodynamic properties have been investigated using matter-wave interference and by detailed analysis of the in situ density and momentum distributions of trapped samples, observing the algebraic decay of coherence [5], a presuperfluid regime [7][8][9][10], and the scale invariance of the equation of state [9,11]. It is now highly desirable to have quantitative probes directly sensitive to phase fluctuations in order to study the topological nature of the phase transition. In particular, nonequilibrium phase dynamics near the critical point would provide valuable insights on the BKT transition. Relaxation dynamics in the 2D XY model have been under intense theoretical investigation [12,13] and recently an experimental scheme to study a dynamic BKT transition in 2D Bose gases was proposed [14].
In this paper, we demonstrate a new quantitative probe for phase fluctuations in a 2D degenerate Bose gas using the density correlations in a freely expanding sample. Irregular density modulations gradually develop in the coherent part of the sample during expansion. We observe that the power spectrum of the density distribution has an oscillatory shape and find that the peak positions are independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the spectral peak strength normalized with the central density of the coherent part and we show that it decreases at lower temperature in thermal equilibrium. This confirms the thermal nature of phase fluctuations in the 2D system. In addition, we investigate the relaxation dynamics of nonequilibrium states by measuring the time evolution of the relative intensity of phase fluctuations. Interestingly, ring-shaped density ripples stochastically appear in a perturbed sample after a long relaxation time, which we identify with free vortices having a long lifetime in a 2D sample.
We prepare a 2D degenerate Bose gas of 23 Na atoms in a single pancake-shape optical dipole trap [15,16]. Thermal atoms in the |F = 1, m F = −1 state are loaded from a plugged magnetic trap [17] into the optical trap and evaporative cooling is applied by reducing the trap depth. The sample temperature is controlled by the final trap depth in the evaporation, resulting in 0.6 to 1.3×10 6 atoms in a sample. Finally, the optical trap depth ramps up and the trapping frequencies (ω x , ω y , ω z ) = 2π×(3.0, 3.9, 370) Hz. The cooling procedure is intentionally set to be slow over 15 s, ensuring thermal equilibrium. The lifetime in the optical trap is over 50 s. In the Thomas-Fermi approximation, the chemical potential is about h × 260 Hz less than the confining energy ω z , so we expect 2D physics in the phase coherence of the sample at low temperature. The dimensionless interaction strength g = a 8πmω z / 0.013, where a is the 3D scattering length and m is the atomic mass. The in-plane density distribution n(x, y) is measured by taking an absorption image after an expansion time t e which is initiated by suddenly turning off the trapping potential.
Expansion has been a conventional and powerful method in quantum gas experiments to study coherence properties of a sample [7,10]. In our experiment, we are interested in the short expansion regime where t e 1/ω x,y and the phase coherence information would be revealed as density correlations. Since the sample initially expands fast along the tight direction, the atom interaction effects are rapidly reduced and the subsequent evolution in other directions can be described as free expansion. When the sample contains phase fluctuations, self interference would result in density modulations in the expanding sample. This method has been exploited in previous studies of phase fluctuations in elongated Bose-Einstein condensates [18] and 1D Bose gases [19].
We observe that density fluctuations develop in an expanding 2D Bose gas (Fig. 1), where the characteristic size and the visibility of the density lumps increases with the expansion time. Density fluctuations appear discernible only when the sample shows a bimodal density distribution so we refer the center part as the coherent part of the sample in the following [6,7]. In order to obtain the density correlation information, we measure the power spectrum of the density distribution as the square of the magnitude of its Fourier transform, P ( q) = | dxdye i q· r n( r)| 2 . Although the spatial pattern of the density modulations appears random in each realization, the power spectrum clearly reveals a multiple ring structure that scales down with the expansion time [ Fig. 1(c) and (f)]. For quantitative analysis, we obtain an 1D spectrum P (q) by azimuthally averaging the 2D spectrum ( Fig. 2) [20]. The strong signal around q = 0 corresponds to the finite size of the coherent part.
The scaling behavior of the spectrum can be qualitatively understood in terms of the Talbot effect [21,22]. It is well known in near-field diffraction that when a grating is illuminated by monochromatic waves, the identical selfimage of the grating is formed at a distance L T = 2d 2 /λ away from the grating, where d is the grating period and λ is the wavelength of the incident radiation. The same effect occurs with a phase grating [23]. In matter wave optics, λ = h/mυ, where υ is the incident speed of atoms [24], so the propagation time for self-imaging is defined as t T = L T /υ = 2md 2 /h independent of υ. If we consider a 2D Bose gas as a macroscopic matter wave containing phase fluctuations at all length scales, it is expected that the component of wavenumber q satisfying the Talbot condition q 2 = 4πm/ t e will emerge predominantly in the density distribution at a given expansion time t e . The multiple peaks in P (q) can be accounted for by the fractional Talbot effect where self-images with smaller periods d/n (n > 0 is an integer) are produced at L T /2n [22].
Recently, theoretical calculations on the spectrum of density modulations have been performed for a homogeneous 2D Bose gas at low temperatures [25], showing that the nth peak position q n closely satisfies q 2 n t e /2πm ≈ (color online) Temperature dependence of phase fluctuations at thermal equilibrium. The relative intensity of phase fluctuations is estimated from the normalized strength of the first spectral peakP (q1) and the relative temperature is parameterized with the coherent fraction. The upper insets show the density profiles for different temperatures. The red dashed lines are gaussian fits to the thermal wings, from which temperatures are estimated (lower inset). Each data point consists of ten independent measurements and error bars indicate standard deviation.
(n − 1/2). In particular, they predict that for sufficiently long expansion times the spectrum remains self-similar during expansion and its shape is determined only by the exponent of the power-law decay of the first order coherence function.
In our experiment, we observe that the spectrum preserves its oscillatory shape during expansion and that the peak positions q n 's are independent of temperature ( Fig. 2), which are in qualitative agreement with the theoretical prediction. However, we find different scaling behavior of q n 's in the measured spectra, which is well described as q 2 n t e /2πm = αn γ with 0.2 < α < 0.45 and 0.7 < γ < 1 for t e = 10 ∼ 25 ms [ Fig. 2(d)]. Furthermore, the phase of the spectral oscillation is opposite to the theoretical prediction, suggesting that an additional peak is present at q 0 = 0, which is also hinted by the shoulder-like hump in q < q 1 / √ 2. We rule out the finite size effects by seeing no dependence of q n 's on sample size as well as temperature, implying that the observed scaling behavior might be intrinsic to the expansion dynamics. We note that the long expansion time condition t e /m ξ 2D for the validity of the theoretical prediction is marginally fulfilled in our experiment ( t e /m/ξ 2D ∼), where ξ 2D = / √ mµ is the 2D healing length, µ being the chemical potential [26]. The universality of the spectral peak positions suggests that the spectral peak strength can be used as a measure of the magnitude of phase fluctuations in a sample. In order to quantify the relative intensity of phase fluctua-tions, we normalize the strength of the first spectral peak with the square of the central density n c of the coherent part in the sample,P (q 1 ) = P (q 1 )/n 2 c , where n c is determined from a fit of two gaussian curves to the density distribution [6,7,[27][28][29].
UsingP (q 1 ), we first investigate the temperature dependence of phase fluctuations in thermal equilibrium. To estimate the relative temperature to the critical point in a model-independent way [8,11], we use the coherent fraction η that is defined as the ratio of the atom number of the coherent part to the total atom number. The value of η was constant within 5% for our expansion times. Fig. 3 shows thatP (q 1 ) is suppressed at lower temperature (higher η), confirming the thermal nature of phase fluctuations.
For a weakly interacting 2D Bose gas, especially for our smallg = 0.013, the BKT critical temperature T c is close to the BEC critical temperature for a trapped ideal Bose gas T c,BEC = 0.94 (ω x ω y ω z N ) 1/3 /k B [7,27,28]. We estimate T c ≈ 80 nK for N ≈ 1.3×10 6 . Since k B T c ≈ 4 ω z , thermal populations in the tight direction is not negligible, accounting for the gaussian-like profile of the saturated thermal cloud [27,28]. The rapid increase ofP (q 1 ) at η < 0.2 might indicate the behavior in the proximity of the critical point. In Ref. [7], the critical point was identified at η ≈ 0.1 with the abrupt change in the width of the coherent part. Since the spatial extent of the coherent part becomes small, the signal-to-noise ratio is poor when η ≤ 0.05 so we cannot study the presuperfluid regime where the decay of the coherence function changes from algebraic to exponential, which might be reflected in the spectral shape.
The power spectrum can be used to study nonequilibrium dynamics in a 2D Bose gas. For this study, we prepare a 2D sample in a nonequilibrium state by transferring a condensate instead of thermal atoms from the plugged magnetic trap into the optical trap. The induced perturbations are small enough that the density profile in the optical trap is quite close to that at equilibrium. In Fig. 4(a), we plot the time evolutions of samples in various initial conditions in the plane ofP (q 1 ) and η (Fig. 4f), clearly showing that the nonequilibrium states decay to equilibrium. The decay time of the excess phase fluctuations with respect to the equilibrium value is measured to ∼ 4 s, corresponding to ∼ 10 collision times in our typical condition. Note that the hottest sample first decays and then moves along the equilibrium line with increasing η because of the evaporation cooling due to the finite trap depth. This verifies that our previous measurements are indeed for phase fluctuations in thermal equilibrium.
Remarkably, ring-shaped density ripples are observed in the perturbed samples after long relaxation times (Fig. 4). We believe that this corresponds to vortex excitations generated in the sample transferring procedure. Since a vortex in 2D can decay only via pairing with another vortex with opposite circulation or drifting out of the finite sample, we may expect a metastable state with vortices having a long lifetime. Recently, it has been reported that vortex excitations survive longer in an oblate condensate because Kelvin mode excitations on a vortex line are suppressed [30,31]. The ring pattern appears more often at lower temperature with stronger perturbations. In the two lowest temperature cases (circle and triangle in Fig. 4), the appearance probability is about 60% at t h = 18 s of hold time, where the spectral strength of the samples with vortices is about 10% higher than without them. In Fig. 4(b), the decay rates of the two coldest samples become slightly slower after t h > 5 s, which might be attributed to the long lifetime of vortices. Fig. 4(d)-(f) shows the expansion dynamics of the ringshape density ripples. In the thermal equilibrium case, the ring pattern was never seen for η > 0.2.
In conclusion, we have demonstrated the power spectrum of the density fluctuations in a freely expanding 2D Bose gas as a new quantitative probe for phase fluctuations. Together with more controlled perturbations [32,33], we expect this method to be extended for studying nonequilibrium phenomena in BKT physics such as critical exponents [12,13] and dynamic transitions [14,34].
This work was supported by the NRF grants funded by the Korea government (MEST) (Nos. 2010-0010172, 2011-0017527, 2008-0062257, and WCU-R32-10045). JC, SSW, and YS acknowledge support from the Global PhD Fellowship, the Kwanjeong Scholarship, and the TJ Park Science Fellowship, respectively.
PACS numbers: 67.85.-d, 03.75.Hh, 03.75.Lm
online) Emergence of density fluctuations in a freely expanding 2D Bose gas. Density distributions after (a) te = 14 ms and (d) 23 ms of time-of-flight. Density fluctuations gradually develop during expansion, increasing their length scale and visibility. (b,e) The horizontal density profiles in the center of the samples. The red dashed lines indicate the averaged profiles over 10 individual realizations of the same experiment. The coherent fraction η ≈ 30% (see text for details). The power spectrum of the density distribution is measured with the magnitude square of its Fourier transform. (c,f) The averaged power spectra corresponding to (a,d).
FIG. 2 :
2(color online) The power spectrum of density fluctuations. (a) 1D power spectra P (q) are obtained by azimuthally averaging the averaged 2D spectra for various expansion times te. Each spectrum is displayed with an offset for clarity. The dashed lines are the corresponding spectra for the averaged density distributions over 10 individual realizations. (b) P (q) at te = 17 ms for various temperatures. The spectral peak positions qn's are determined from a fit of multiple gaussian curves to each spectrum (n is the peak order number). The vertical dashed lines indicate the average peak positions. (c) The empirical function q 2 n te/2πm = αn γ are fit to the peak positions in (a) for each te. (d) The expansion time dependence of the two free parameters α and γ.
FIG. 3: (color online) Temperature dependence of phase fluctuations at thermal equilibrium. The relative intensity of phase fluctuations is estimated from the normalized strength of the first spectral peakP (q1) and the relative temperature is parameterized with the coherent fraction. The upper insets show the density profiles for different temperatures. The red dashed lines are gaussian fits to the thermal wings, from which temperatures are estimated (lower inset). Each data point consists of ten independent measurements and error bars indicate standard deviation.
FIG. 4 :
4(color online) Relaxation of a nonequilibrium 2D Bose gas. (a) Time evolutions of various nonequilibrium states in the plane ofP (q1) and η (te = 17 ms). The dashed line is the interpolation for the thermal equilibrium data with te = 17 ms in Fig. 3. (b) The decay curves of the excess phase fluctuationsP (q1) −Peq(q1), wherePeq(q1) is the equilibrium value corresponding to η at t h = 18 s of hold time. Density distributions of the lowest temperature sample at (c) t h = 0 s and (d) 18 s with te = 17 ms, and at t h = 18 s with (e) te = 8 ms and (f) 23 ms. The samples in (e) and (f) were strongly perturbed to see more ring-shaped density ripples.
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|
[] |
[
"Generative Adversarial Networks for text using word2vec intermediaries",
"Generative Adversarial Networks for text using word2vec intermediaries"
] |
[
"Akshay Budhkar [email protected] \nDepartment of Computer Science\nUniversity of Toronto\n\n\nVector Institute\n\n",
"Krishnapriya Vishnubhotla \nDepartment of Computer Science\nUniversity of Toronto\n\n",
"Safwan Hossain [email protected] \nDepartment of Computer Science\nUniversity of Toronto\n\n\nVector Institute\n\n",
"Frank Rudzicz \nDepartment of Computer Science\nUniversity of Toronto\n\n\nVector Institute\n\n\nSt Michael's Hospital\n4 Georgian Partners\n\nSurgical Safety Technologies Inc\n\n"
] |
[
"Department of Computer Science\nUniversity of Toronto\n",
"Vector Institute\n",
"Department of Computer Science\nUniversity of Toronto\n",
"Department of Computer Science\nUniversity of Toronto\n",
"Vector Institute\n",
"Department of Computer Science\nUniversity of Toronto\n",
"Vector Institute\n",
"St Michael's Hospital\n4 Georgian Partners",
"Surgical Safety Technologies Inc\n"
] |
[
"Proceedings of the 4th Workshop on Representation Learning for NLP (RepL4NLP-2019)"
] |
Generative adversarial networks (GANs) have shown considerable success, especially in the realistic generation of images. In this work, we apply similar techniques for the generation of text. We propose a novel approach to handle the discrete nature of text, during training, using word embeddings. Our method is agnostic to vocabulary size and achieves competitive results relative to methods with various discrete gradient estimators.
|
10.18653/v1/w19-4303
|
[
"https://www.aclweb.org/anthology/W19-4303.pdf"
] | 102,353,164 |
1904.02293
|
2f32f6776a3030af78e0eb9186bfea6626e69a6d
|
Generative Adversarial Networks for text using word2vec intermediaries
August 2
Akshay Budhkar [email protected]
Department of Computer Science
University of Toronto
Vector Institute
Krishnapriya Vishnubhotla
Department of Computer Science
University of Toronto
Safwan Hossain [email protected]
Department of Computer Science
University of Toronto
Vector Institute
Frank Rudzicz
Department of Computer Science
University of Toronto
Vector Institute
St Michael's Hospital
4 Georgian Partners
Surgical Safety Technologies Inc
Generative Adversarial Networks for text using word2vec intermediaries
Proceedings of the 4th Workshop on Representation Learning for NLP (RepL4NLP-2019)
the 4th Workshop on Representation Learning for NLP (RepL4NLP-2019)Florence, ItalyAugust 215
Generative adversarial networks (GANs) have shown considerable success, especially in the realistic generation of images. In this work, we apply similar techniques for the generation of text. We propose a novel approach to handle the discrete nature of text, during training, using word embeddings. Our method is agnostic to vocabulary size and achieves competitive results relative to methods with various discrete gradient estimators.
Introduction
Natural Language Generation (NLG) is often regarded as one of the most challenging tasks in computation (Murty and Kabadi, 1987). It involves training a model to do language generation for a series of abstract concepts, represented either in some logical form or as a knowledge base. Goodfellow introduced generative adversarial networks (GANs) (Goodfellow et al., 2014) as a method of generating synthetic, continuous data with realistic attributes. The model includes a discriminator network (D), responsible for distinguishing between the real and the generated samples, and a generator network (G), responsible for generating realistic samples with the goal of fooling the D. This setup leads to a minimax game where we maximize the value function with respect to D, and minimize it with respect to G. The ideal optimal solution is the complete replication of the real distributions of data by the generated distribution.
GANs, in this original setup, often suffer from the problem of mode collapse -where the G manages to find a few modes of data that resemble real data, using them consistently to fool the D. Workarounds for this include updating the loss function to incorporate an element of multidiversity. An optimal D would provide G with the information to improve, however, if at the current stage of training it is not doing that yet, the gradient of G vanishes. Additionally, with this loss function, there is no correlation between the metric and the generation quality, and the most common workaround is to generate targets across epochs and then measure the generation quality, which can be an expensive process.
W-GAN rectifies these issues with its updated loss. Wasserstein distance is the minimum cost of transporting mass in converting data from distribution P r to P g . This loss forces the GAN to perform in a min-max, rather than a max-min, a desirable behavior as stated in (Goodfellow, 2016), potentially mitigating modecollapse problems. The loss function is given by:
L critic = min G max D∈D (E x∼pr(x) [D(x)] −Ex ∼pg(x) [D(x)])(1)
where D is the set of 1-Lipschitz functions and P g is the model distribution implicitly defined bỹ x = G(z), z ∼ p(z). A differentiable function is 1-Lipschtiz iff it has gradients with norm at most 1 everywhere. Under an optimal D minimizing the value function with respect to the generator parameters minimizes the W(p r , p g ), where W is the Wasserstein distance, as discussed in (Vallender, 1974). To enforce the Lipschitz constraint, the authors propose clipping the weights of the gradient within a compact space [−c, c]. (Gulrajani et al., 2017) show that even though this setup leads to more stable training compared to the original GAN loss function, the architecture suffers from exploding and vanishing gradient problems. They introduce the concept of gradient penalty as an alternative way to enforce the Lipschitz constraint, by penalizing the gradient norm directly in the loss. The loss function is given by:
L = L critic + λEx ∼p(x) [(||∇xD(x)|| 2 − 1) 2 ] (2)
wherex are random samples drawn from P x , and L critic is the loss defined in Equation 1.
Empirical results of GANs over the past year or so have been impressive. GANs have gotten stateof-the-art image-generation results on datasets like ImageNet (Brock et al., 2018) and LSUN (Radford et al., 2015). Such GANs are fully differentiable and allow for back-propagation of gradients from D through the samples generated by G. However, if the data is discrete, as, in the case of text, the gradient cannot be propagated back from D to G, without some approximation. Workarounds to this problem include techniques from reinforcement learning (RL), such as policy gradients to choose a discrete entity and reparameterization to represent the discrete quantity in terms of an approximated continuous function (Williams, 1992;Jang et al., 2016).
Techniques for GANs for text
SeqGAN (Yu et al., 2017) uses policy gradient techniques from RL to approximate gradient from discrete G outputs, and applied MC rollouts during training to obtain a loss signal for each word in the corpus. MaliGAN (Che et al., 2017) rescales the reward to control for the vanishing gradient problem faced by SeqGAN. RankGAN (Lin et al., 2017) replaces D with an adversarial ranker and minimizes pair-wise ranking loss to get better convergence, however, is more expensive than other methods due to the extra sampling from the original data. (Kusner and Hernández-Lobato, 2016) used the Gumbel-softmax approximation of the discrete one-hot encoded output of the G, and showed that the model learns rules of a contextfree grammar from training samples. (Rajeswar et al., 2017), the state of the art in 2017, forced the GAN to operate on continuous quantities by approximating the one-hot output tokens with a softmax distribution layer at the end of the G network.
MaskGAN (Fedus et al., 2018) uses policy gradient with REINFORCE estimator (Williams, 1992) to train the model to predict a word based on its context, and show that for the specific blank-filling task, their model outperforms maximum likelihood model using the perplexity metric. LeakGAN allows for long sentence generation by leaking high-level information from D to G, and generates a latent representation from the features of the already generated words, to aid in the next word generation. TextGAN adds an element of diversity to the original GAN loss by employing the Maximum Mean Discrepancy objective to alleviate mode collapse.
In the latter half of 2018, introduced Texygen, a benchmarking platform for natural language generation, while introducing standard metrics apt for this task. surveys all these new methods along with other baselines, and documents model performance on standard corpus like EMNLP2017 WMT News 1 and Image COCO 2 .
Motivation
Problems with the Softmax Function
The final layer of nearly all existing language generation models is the softmax function. It is usually the slowest to compute, leaves a large memory footprint and can lead to significant speedups if replaced by approximate continuous outputs (Kumar and Tsvetkov, 2018). Given this bottleneck, models usually limit the vocabulary size to a few thousand and use an unknown token (unk) for the rare words. Any change in the allowed vocabulary size also means that the researcher needs to modify the existing model architecture.
Our work breaks this bottleneck by having our G produce a sequence (or stack) of continuous distributed word vectors, with n dimensions, where n << V and V is the vocabulary size. The expectation is that the model will output words in a semantic space, that is produced words would either be correct or close synonyms (Mikolov et al., 2013;Kumar and Tsvetkov, 2018), while having a smaller memory footprint and faster training and inference procedures.
GAN2vec
In this work, we propose GAN2vec -GANs that generate real-valued word2vec-like vectors (as opposed to discrete one-hot encoded outputs). While this work mainly focuses specifically on word2vec-based representation, it can be easily extended to other embedding techniques like GloVe and fastText.
Expecting a neural network to generate text is, intuitively, expecting it to learn all the nuances of natural language, including the rules of grammar, context, coherent sentences, and so on. Word2vec has shown to capture parts of these subtleties by capturing the inherent semantic meaning of the words, and this is shown by the empirical results in the original paper (Mikolov et al., 2013) and with theoretical justifications by (Ethayarajh et al., 2018). GAN2vec breaks the problem of generation down into two steps, the first is the word2vec mapping, with the following network expected to address the other aspects of sentence generation. It also allows the model designers to swap out word2vec for a different type of word representation that is best suited for the specific language task at hand.
As a manifestation of the similar-context words getting grouped in word embedding space -we expect GAN2vec to have synonymic variety in the generation of sentences. Generating real-valued word vectors also allows the G architecture to be vocabulary-agnostic, as modifying the training data would involve just re-training the word embedding with more data. While this would involve re-training the weights of the GAN network, the initial architectural choices could remain consistent through this process. Finally, as discussed in Section 2.1, we expect a speed-up and smaller memory footprint by adapting this approach.
All the significant advances in the adaptation of GANs since its introduction in 2016, has been focused in the field of images. We have got to the point, where sometimes GAN architectures have managed to generate images even better than real images, as in the case of BigGAN (Brock et al., 2018). While there have been breakthroughs in working with text too, the rate of improvement is no-where close to the success we have had with images. GAN2vec attempts to bridge this gap by providing a framework to swap out image representations with word2vec representations.
The Architecture
Random normal noise is used as an input to the G which generates a sequence of word2vec vectors.
We train the word2vec model on a real text corpus and generate a stack word vector sequences from the model. The generated and the real samples are then sent to D, to identify as real or synthetic. The generated word vectors are converted to text at regular intervals during training and during inference for human interpretation. A nearest-neighbor approach based on cosine similarity is used to find the closest word to the generated embedding in the vector space.
The Algorithm
The complete GAN2vec flow is presented in Algorithm 1. Send minibatch of G generated data, G(z), to D 8:
Update D using gradient descent 9:
Update G using gradient ascent 10: end for 11: G(z) = Sample random normal z and feed to G 12: w generated = argmin w {d(ê, e(w))}, for everŷ e in G(z) and every w in the corpus
Conditional GAN2vec
We modify GAN2vec to measure the adaptability of GAN2vec to conditions provided a priori, as seen in (Mirza and Osindero, 2014). This change can include many kinds of conditions like positive/negative, question/statement or dementia/controls, allowing for the ability to analyze examples from various classes on the fly during inference. Both the G (and D) architectures get passed the condition at hand as an input, and the goal of G now is to generate a realistic sample given the specific condition.
Environmental Setup
All the experiments are run using Pytorch (Paszke et al., 2017). Word2vec training is done using the gensim library (Řehůřek and Sojka, 2010). Unless specified otherwise, we use the default parameters for all the components of these libraries, and all Figure 1: Structure of the GAN2vec model. Random normal noise is given as input to the generator network G. The discriminator network D is responsible for determining whether a sample originated from G or from the training set. At inference time, we use a nearest-neighbor approach to convert the output from G into human-readable text. our models are trained for 100 epochs. The word embedding dimensions are set to 64. The learning rate for the ADAM optimizers for D and G are set to 0.0001, with the exponential decay rates for the first and second moments set to 0.5 and 0.999 respectively.
All our Ds take the word2vec-transformed vectors as an input and apply two 2-D convolutions, followed by a fully connected layer to return a single value. The dimensions of the second 2-D convolution are the only things varied to address the different input dimensions. Similarly, our Gs take a random normal noise of size 100 and transform it to the desired output by passing it through a fully-connected layer, and two 2-D fractionallystrided convolution layers. Again, the dimensions of the second fractionally-strided convolution are the only variables to obtain different output dimensions.
Normalizing word vectors after training them has no significant effect on the performance of GAN2vec, and all the results that we present do not carry out this step. Keeping in punctuation helped improve performance, as expected, and none of the experiments filter them out.
To facilitate stable GAN training, we make the following modifications, covered by (Chintala et al., 2016), by running a few preliminary tests on a smaller sample of our dataset:
• Use LeakyRELU instead of RELU
• Send generated and real mini-batches to D in separate batches
• Use label smoothing by setting the target labels to 0.9 and 0.1 instead of 1 and 0 for real and fake samples respectively (for most of our experiments).
6 Metrics 6.1 BLEU BLEU (Papineni et al., 2002) originated as a way to measure the quality of machine translation given certain ground truth. Many text generation papers use this as a metric to compare the quality of the generated samples to the target corpus. A higher n-gram coverage will yield a higher BLEU score, with the score reaching a 100% if all the generated n-grams are present in the corpus. The two potential flaws with this metric are: 1) It does not take into account the diversity of the text generation, this leads to a situation where a mode-collapsing G that produces the same one sentence from the corpus gets a score of 100%. 2) It penalizes the generation of grammatically coherent sentences with novel n-grams, just because they are absent from the original corpus. Despite these problems, we use BLEU to be consistent with other GANs for text papers. We also present generated samples for the sake of qualitative evaluation by the reader.
Self-BLEU
Self-BLEU is introduced as a metric to measure the diversity of the generated sentences. It does a corpus-level BLEU on a set of generated sentences, and reports the average BLEU as a metric for a given model. A lower self-BLEU implies a higher diversity in the generated sentences, and accordingly a lower chance that the model has mode collapsed. It is not clear from 's work on how many sentences Texygen generates to calculate Self-BLEU. For purposes of GAN2vec's results, we produce 1000 sentences, and for every sentence do a corpus-level BLEU on remaining 999 sentences. Our results report the average BLEU across all the outputs.
Chinese Poetry Dataset
The Chinese Poetry dataset, introduced by (Zhang and Lapata, 2014) presents simple 4-line poems in Chinese with a length of 5 or 7 tokens (henceforth referred to Poem 5 and Poem 7 respectively). Following previous work by (Rajeswar et al., 2017) and (Yu et al., 2017), we treat every line as a separate data point. We modify the Poem 5 dataset to add start and end of tokens, to ensure the model captures (at least) that pattern through the corpus (given our lack of Chinese knowledge). This setup allows us to use identical architectures for both the Poem 5 and Poem 7 datasets. We also modify the GAN2vec loss function with the objective in Eq. 2, and report the results below.
Rajeswar et al.
GAN2vec
GAN2vec ( The better performance of the GAN2vec model with the wGAN objective is in-line with the im-age results in Gulrajani et al. (2017)'s work. We were not able to replicate (Rajeswar et al., 2017)'s model on the Chinese Poetry dataset to get the reported results on the test set. This conclusion is in-line with our expectation of lower performance on the test set, given the small overlap in the bigram coverage between the provided train and test sets. also point out that this work is unreliable, and that their replicated model suffered from severe mode-collapse. On 1000 generated sentences of the Poem-5 dataset, our model has a self BLEU-2 of 66.08% and self BLEU-3 of 35.29%, thereby showing that our model does not mode collapse.
CMU-SE Dataset
CMU-SE 3 is a pre-processed collections of simple English sentences, consisting of 44,016 sentences and a vocabulary of 3,122-word types. For purposes of our experiments here, we limit the number of sentences to 7, chosen empirically to capture a significant share of the examples. For the sake of simplicity in these experiments, for the real corpus, sentences with fewer than seven words are ignored, and those with more than seven words are cut-off at the seventh word. Table 1 presents sentences generated by the original GAN2vec model. Appendix A.2 includes additional examples. While this is a small subset of randomly sampled examples, on a relatively simple dataset, the text quality appears competitive to the work of (Rajeswar et al., 2017) on this corpus. Rajeswar et al. (2017) <s> will you have two moment ? </s> <s> how is the another headache ? </s> <s> what s in the friday food ? ? </s> <s> i d like to fax a newspaper . </s> GAN2vec <s> i dropped my camera . </s> <s> i 'd like to transfer it <s> i 'll take that car , <s> prepare whisky and coffee , please Table 2: Example sentences generated by the original GAN2vec. We report example sentences from Rajeswar et al. (2017) and from our GAN2vec model on CMU-SE.
Conditional GAN2vec
We split the CMU-SE dataset into questions and sentences, checking for the presence of a question mark. We modify the original GAN2vec, as seen in Section A.1, to now include these labels. Our conditional GANs learn to generate mainly coherent sentences on the CMU-SE dataset, as seen in Table 3. Figure 2 shows the loss graphs for our GAN2vec and conditional GAN2vec trained for ∼300 epochs. As seen above, the conditional GAN2vec model generates relatively atypical sentences. This is supported by the second loss curve in Figure 2. The G loss follows a progression similar to the normal GAN2vec case, but the loss is about 16% more through the 100 epochs.
Hyperparameter Variation Study
We study the effects of different initial hyperparameters for GAN2vec by reporting the results in Table 4. All the experiments were run ten times, and we report the best scores for every configuration. It must be noted that for conditional GAN2vec training for this experiment, we randomly sample points from the CMU-SE corpus to enforce a 50-50 split across the two labels (question and sentence).
The overall performance of most of the models is respectable, with all models generating gram-matically coherent sentences. GAN2vec with wGAN objective outperforms original GAN2vec, and is inline with the results of (Gulrajani et al., 2017) and our results in Section 7. Sense2vec does not have a significant improvement over the original word2vec representations. In agreement with (Goodfellow, 2016), providing labels in the conditional variant leads to better performance. During training, we map our generated word2vec vectors to the closest words in the embedding space and measure the point-wise cosine similarity of the generated vector and the closest neighbour's vector. Figure 3 shows these scores for the first, third, fourth and seventh word of the 7-word generated sentences on the CMU-SE dataset for about 300 epochs. The model immediately learns that it needs to start a sentence with <s> and gets a cosine similarity of around 1. For the other words in that sentence, the model tends to get better at generating word vectors that are close to their real-valued counterparts of the nearest neighbours. It seems as if the words close to the start of the sentence follow this trend more strongly (as seen with words 1 and 3) and it is relatively weaker for the last word of the sentence.
Word2vec cosine similarity
Coco Image Captions Dataset
The Coco Dataset is used to train and generate synthetic data as a common dataset for all the best-performing models over the last two years. In Texygen, the authors set the sentence length to 20. They train an oracle that generates 20,000 sentences, with one half used as the training set and the rest as the test set. All the models in this benchmark are trained for 180 epochs.
Questions
Sentences <s> can i get you want him <s> i bring your sweet inexpensive beer <s> where 's the hotel ? <s> they will stop your ship at <s> what is the fare ? </s> <s> i had a pocket . </s> <s> could you buy the timetable ? <s> it 's ten at detroit western Figure 4 shows the distribution of the sentence lengths in this corpus. For purposes of studying the effects of longer training sentences on GAN2vec, we set the sentence lengths to 7, 10 and 20 (with the respective models labeled as GAN2vec-7, GAN2vec-10, GAN2vec-20 going forward). Any sentence longer than the predefined sentence length is cut off to include only the initial words. Sentences shorter than this length are padded with an end of sentence character to fill up the remaining words (we use a comma (,) for purposes of our experiments as all the sentences in the corpus end with either a full stop or a word). We tokenize the sentences us-ing NLTK's word tokenizer 4 which uses regular expressions to tokenize text as in the Penn Treebank corpus 5 . We also report the results of a naive split at space approach for the GAN2vec-20 architecture (GAN2vec-20-a), to compare different ways of tokenizing the corpus. We only use the objective from Equation 2, given its superior performance to original GAN2vec, as seen in the previous sections.
The results are summarized in the tables below: On the train set (Table 5), GAN2vec models have BLEU-2 scores comparable to its SOTA counterparts, with the GAN2vec-20 model having better bigram coverage that TextGAN. The BLEU-3 scores, even though commendable, do not match up as well, possibly signaling that our models cannot keep coherence through longer sentences. The increase in the cut-off sentence length, surprisingly, does not degrade performance. As expected, a trained word tokenizer outperforms its space-split counterpart. The performance of the GAN2vec models on the test set (Table 6) Table 7 reports the self-BLEU scores, and all the GAN2vec models significantly outperform the SOTA models, including MLE. This implies that GAN2vec leads to more diverse sentence generations and is less susceptible to mode collapse.
Discussions
Overall, GAN2vec can generate grammatically coherent sentences, with a good bi-gram and trigram coverage from the chosen corpus. BLEU does not reward the generation of semantically and syntactically correct sentences if the associated n-grams are not present in the corpus, and coming up with a new standard evaluation metric is part of on-going work. GAN2vec seems to have comparable, if not better, performance compared to Rajeswar et al. (2017)'s work on two distinct datasets. It depicts the ability to capture the critical nuances when trained on a conditional corpus. While GAN2vec performs slightly worse than most of the SOTA models using the Texygen benchmark, it can generate a wide variety of sentences, possibly given the inherent nature of word vectors, and is less susceptible to mode collapse compared to each of the models. GAN2vec provides a simple framework, with almost no overhead, to transfer state of the art GAN research in computer vision to natural language generation.
We observe that the performance of GAN2vec gets better with an increase in the cut-off length of the sentences. This improvement could be because of extra training points for the model. The drop from BLEU-2 to BLEU-3 scores is more extreme than the other SOTA models, indicating that GAN2vec may lack the ability to generate long coherent sentences. This behavior could be a manifestation of the chosen D and G architectures, specifically the filter dimensions of the convolution neural networks. Exploration of other structures, including RNN-based models with their ability to remember long term dependencies, might be good alternatives to these initial architecture choices. Throughout all the models in the Texygen benchmark, there seems to be a mild negative correlation between diversity and performance. GAN2vec in its original setup leans more towards the generation of new and diverse sentences, and modification of its loss function could allow for tilting the model more towards accurate NLG.
Conclusion
While various research has extended GANs to operate on discrete data, most approaches have approximated the gradient in order to keep the model end-to-end differentiable. We instead explore a different approach, and work in the continuous domain using word embedding representations. The performance of our model is encouraging in terms of BLEU scores, and the outputs suggest that it is successfully utilizing the semantic information encoded in the word vectors to produce new, coherent and diverse sentences.
A CMU-SE
A.1 Conditional Architecture
While designing GAN2vec to support conditional labels, as presented in Mirza and Osindero (2014), we used the architecture in Figure 5 for our G. The label is sent as an input to both the fully connected and the de-convolution neural layers. The same change is followed while updating D to support document labels. A.2 Examples of Generated Sentences
Figure 2 :
2The minimax loss of D and G, with increasing iterations for the GAN2vec model (top) and the conditional GAN2vec (bottom).
Figure 3 :
3Cosine similarities of the first, third, fourth, and seventh words to the closest words from sentences generated by GAN2vec trained on the CMU-SE dataset.
Figure 4 :
4Distribution of sentence lengths in the Coco dataset. Most of the captions have less than 20 words, the cut-off set by Texygen.
Figure 5 :
5Generator Architecture for Conditional GAN2vec.
Table 1 :
1Chinese Poetry BLEU-2 scores.
Table 3 :
3Examples of sentences generated by the conditional GAN. We report examples of sentences with our model conditioned on sentence type, i.e., question or sentence.Architecture Conditional Vector Type Loss function BLEU-2 BLEU-3
R.1
No
Sense2vec
Original
0.743
0.41
R.1
No
Sense2vec
wgan
0.7933
0.4728
R.1
No
Word2Vec
wgan
0.74
0.43
C.1
Yes
word2vec
Original (Real) 0.717
0.412
C.1
Yes
word2vec
Original
0.743
0.4927
C.1
Yes
word2vec
wgan
0.7995
0.5168
C.1
Yes
word2vec
wgan
0.821
0.51
C.2
Yes
word2vec
wgan
0.8053
0.499
Table 4 :
4Performance of different models on the CMU-SE train dataset. R.1 is the original GAN2vec, C.1 is R.1
modified with addition of labels, C.2 adds batch normalization on the CNN layer of G. Original (Real) sets the
real label to 0.9, the rest use 1.
Table 5 :
5Model BLEU scores on Train Set of the Coco Dataset (higher is better).
follows the same trends as that on the train set.Model
BLEU-2 BLEU-3
LeakGAN
0.746
0.816
SeqGAN
0.745
0.53
MLE
0.731
0.497
TextGAN
0.593
0.645
GAN2vec-7
0.429
0.196
GAN2vec-10
0.527
0.245
GAN2vec-20-a 0.484
0.206
GAN2vec-20
0.551
0.232
Table 6 :
6Model BLEU scores on Test Set of the Coco Dataset (higher is better).Model
BLEU-2 BLEU-3
LeakGAN
0.966
0.913
SeqGAN
0.95
0.84
MLE
0.916
0.769
TextGAN
0.942
0.931
GAN2vec-7
0.537
0.254
GAN2vec-10
0.657
0.394
GAN2vec-20-a 0.709
0.394
GAN2vec-20
0.762
0.518
Table 7 :
7Self BLEU scores of the models trained on the Coco dataset (lower is better).
http://www.statmt.org/wmt17/ 2 http://cocodataset.org/
https://github.com/clab/sp2016. 11-731/tree/master/hw4/data
https://www.nltk.org/api/nltk. tokenize.html#nltk.tokenize.word_ tokenize 5 https://catalog.ldc.upenn.edu/docs/ LDC95T7/cl93.html
AcknowledgementsThis study was partially funded by the Vector Institute for Artificial Intelligence. Rudzicz is an Inaugural CIFAR Chair in artificial intelligence.<s> what 's the problem ? </s> <s> i have a hangover dark . <s> i come on the monday . <s> i appreciate having a sushi . <s> the bus is busy , please <s> i dropped my camera . </s> <s> i want to wash something . <s> it 's too great for <unk> <s> i have a driver in the <s> it is delicious and <unk><unk> <s> please leave your luggage . . <s> i had alcohol wow , </s> <s> it is very true . </s> <s> where 's the hotel ? </s> <s> will you see the cat ? </s> <s> where is this bus ? </s> <s> how was the spirits airline warranty <s> i would n't sunburn you <s> prepare whisky and coffee , please
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|
[
"https://github.com/clab/sp2016."
] |
[
"Giant magnetoresistance in the junction of two ferromagnets on the surface of diffusive topological insulators",
"Giant magnetoresistance in the junction of two ferromagnets on the surface of diffusive topological insulators"
] |
[
"Katsuhisa Taguchi \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n",
"Takehito Yokoyama \nDepartment of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan\n",
"Yukio Tanaka \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n"
] |
[
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Department of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan"
] |
[] |
We reveal the giant magnetoresistance induced by the spin-polarized current in the ferromagnet (F1)/topological insulator (TI)/ferromagnet (F2) junction, where two ferromagnets are deposited on the diffusive surface of the TI. We can increase and reduce the value of the giant magnetoresistance by tuning the spin-polarized current, which is controlled by the magnetization configurations. The property is intuitively understood by the non-equilibrium spin-polarized current, which plays the role of an effective electrochemical potential on the surface of the TI.PACS numbers: 73.43.Qt,
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10.1103/physrevb.89.085407
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[
"https://arxiv.org/pdf/1309.4195v1.pdf"
] | 118,428,200 |
1309.4195
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c19d56f5432faf0db349e726e9cd287cb14ed138
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Giant magnetoresistance in the junction of two ferromagnets on the surface of diffusive topological insulators
17 Sep 2013 (Dated: May 11, 2014)
Katsuhisa Taguchi
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Takehito Yokoyama
Department of Physics
Tokyo Institute of Technology
152-8551TokyoJapan
Yukio Tanaka
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Giant magnetoresistance in the junction of two ferromagnets on the surface of diffusive topological insulators
17 Sep 2013 (Dated: May 11, 2014)
We reveal the giant magnetoresistance induced by the spin-polarized current in the ferromagnet (F1)/topological insulator (TI)/ferromagnet (F2) junction, where two ferromagnets are deposited on the diffusive surface of the TI. We can increase and reduce the value of the giant magnetoresistance by tuning the spin-polarized current, which is controlled by the magnetization configurations. The property is intuitively understood by the non-equilibrium spin-polarized current, which plays the role of an effective electrochemical potential on the surface of the TI.PACS numbers: 73.43.Qt,
Spintronics aims to control the charge transport by using spin degrees of freedom [1][2][3][4][5], and also to manipulate the spin-polarization by using the charge current [6][7][8][9][10][11]. The former is applicable for the giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) in the junction of a metal or an insulator sandwiched by two ferromagnets. The resistance depends on the direction of the magnetizations. The latter, the control of the spin-polarization, has been discussed in metals, semiconductors, in the presence of the spin-orbit interaction, which can easily induce spin polarization by the electric field. Spin-orbit interaction, which plays a dominant role in the manipulation of the spin polarization, can also play an important role in the control of the charge transport.
Topological insulators (TIs) are candidate materials where the spin-orbit interaction enhances the magnitude of both charge and spin currents, where we can also expect anomalous electromagnetic phenomena stemmed from topological features of bulk electronic properties [12][13][14]. Nowadays, to clarify the electromagnetic phenomena specific to TIs is believed to serve as a guide to fabricate future magneto-electronic devices. TIs are gapped in bulk and have gapless surface state, which is dubbed as the helical surface state, in which the spin configuration and momentum are locked by spin-orbit interaction [12][13][14].
In the helical surface state, persistent pure spinpolarized current without dissipation is generated. This spin-polarized current could be useful for GMR and TMR, which lead to novel magnetic devices. Preexisting works [15][16][17][18] have predicted anomalous charge transport features by the spin-polarized current in the junction of ferromagnets on the surface of TIs in the ballistic transport regime. However, the actual charge transport on the surface of TIs may be in the diffusive regime with impurity scattering. Thus, a study of charge transport due to the helical surfaces state is necessary in a realistic situation.
In this letter, we study charge transport between two-dimensional ferromagnet (F 1 )/topological insulator (TI)/ferromagnet (F 2 ) junction, where two ferromagnets are deposited on the surface of TI. We find anomalous properties of GMR, which have never appeared in conventional metallic ferromagnet junctions. The GMR includes not only the Ohmic resistance term but also anomalous one (defined in Eq.(16)) depending on two magnetizations. The later term is proportional to the spin polarization, which stems from the injected and extracted spin-polarized current. By tuning the configuration of magnetizations in the F 1 and F 2 , these two terms cancel each other and gigantic reduction of magnetoresistance becomes possible. The present feature may serve as a guide to fabricate future magneto-electronic devices based on TIs.
Hereafter, we will consider a ferromagnet F 1 /TI/ferromagnet F 2 junction on the surface of TI in the presence of the applied charge current ( Fig.1 (a)). We assume that the directions of magnetizations of F 1 and F 2 are along the y-axis and the direction of the charge current is along the x-axis. Then, the charge current induces the spin polarization s y by the spin-orbit interaction on the surface of TI, and s y obeys the spin-diffusion equation as [19,20]
s y = 3D 2 ∇ 2 x s y − s y τ + v F 2e ∇ x N e ,(1)
where N e , D ≡ v F ℓ/2, v F , ℓ, and τ are charge density, diffusion constant, Fermi velocity, mean free path, and transport relaxation time, respectively. The charge current drives the spatial gradient of the charge density ∇ x N e as
j = −D∇N e + ev F (ẑ × s).(2)
In the following discussion, we simply assume that the applied current does not have spatial dependence and is independent of x (j = Ix). The charge current, in addition, generates the spin polarization on the surface of TI by the spin-polarized current. The spin polarization is determined by the boundary conditions at the interface of the left-side (x = −L/2) between F 1 and TI and the right side (x = L/2) between TI and F 2 , where L is the distance between the two electrodes. We use the boundary conditions at both interfaces as follows: The spin-polarized current η1I e flows from F 1 into TI at the left interface x = −L/2, where η 1 is the the degree of spin polarization of the magnetization of F 1 [19]. η 1 = 1(−1) is defined such that the direction of magnetization is positive (negative) along the y-axis and the current is fully spin-polarized. At the right interface, the spinpolarized current η2I e is extracted from TI into F 2 . η 2 represents the direction of the magnetization F 2 and η 2 is defined in the same way. On the other hand, the spin current j y s,x on the surface of TI is given by
j y s,x = − 3 2 D∇ x s y − 1 2e v F N e .(3)
The first and the second terms represent the nonequilibrium and equilibrium component of spin-polarized currents, respectively. The equilibrium component does not appear explicitly in the boundary condition [21,22]. Thus, the boundary conditions are given by [19] −
3 2 D∇ x s y x=− L 2 = η 1 I e ,(4)− 3 2 D∇ x s y x= L 2 = η 2 I e .(5)
The spin polarization is obtained from Eqs. (1)-(5) as
s y = s y 1 + s y 2 ,(6)
with
s y 1 = I 2ev F κℓ η 1 cosh [κ( L 2 − x)] − η 2 cosh [κ( L 2 + x)] sinh(κL) ,(7)s y 2 = − I 2ev F ,(8)
and κ ≡ 2 ℓ 2 3 . s y 1 depends on the position x (−L/2 ≤ x ≤ L/2), η 1 and η 2 , and it decays exponentially with the diffusion length scale of κ −1 from the boundaries. s y 2 is only proportional to I independent of x. When η 2 = 0, the spin polarization is consistent with that in Ref. 19[23]. The physical meaning of s y 1 is the spin polarization due to the spin-polarized current, which is given by − 3 2 D∇ x s y [Eqs. (4)- (5)]. The expression of s y 2 means that the spin polarization generated by the spin-orbit interaction on the surface of TI[24], which is independent of the magnetizations.
The spin polarization s y is plotted in Fig. 1 (b)-(c). First, we discuss the case where magnetizations of F 1 and F 2 are antiparallel to each other with (η 1 , η 2 ) = (−1, 1). In the antiparallel magnetization configuration, the magnitudes of s y decay from the interface at x = ±L/2, and s y has a nonzero value for all x. s y has a strong spatial dependence for −L/2 < x < L/2 with L/ℓ ∼ 1 and L/ℓ ∼ 2, while it becomes almost constant except for the vicinity of x = ±L/2 with L/ℓ ∼ 10. s y is an even function of x for all L in the antiparallel magnetization configuration (see Fig. 1 (b)), while, in the parallel case [(η 1 , η 2 ) = (−1, −1)], s y is not an even function of x (see Figs. 1 (c)), because s y 1 and s y 2 (=const) are an odd and even function of x, respectively. s y contributes to the voltage drop V between F 1 and F 2 on the surface of TI due to the spin-orbit interaction [19].
V is defined by [25] V = − 1 e 2 ν L/2 −L/2 dN e dx dx,(9)
where ν = µ 2πv 2 F is the density of states on the surface state of TI at chemical potential µ = v F k F . From Eq.
(2), the voltage drop can be expressed by
V = V O + V s ,(10)
with
V O = 2 e 2 ℓνv F L/2 −L/2 dx(I + ev F s y 2 ),(11)V s = 2 eℓν L/2 −L/2 dxs y 1 .(12)
In the above, the voltage drop V O , which is given by the charge current and constant spin polarization due to I, obeys the Ohm's law. The spin polarization s y 2 in the second term of V O is negative in the presence of the charge current flowing towards positive x-axis, and hence s y 2 has an opposite contribution to the first term of V O in Eq. (11). On the other hand, the voltage drop V s in Eq. (12) is given by the spin polarization s y 1 , which depends on the magnetization configuration [see Eq. (7)]. Since V s is determined by the magnetization configurations, the resulting V s only depends on the boundary condition of spin-polarized current and no more obeys the Ohm's law. V s can be regarded as a boundary term and is obtained as V s = η1−η2 e 2 νvF I. Finally, the total voltage drop V is given by
V = 1 e 2 νv F L ℓ + η 1 − η 2 I.(13)
The first term in Eq. (13) obeys the Ohm's law and is proportional to the length L of TI. The second (third) term is independent of L, and is proportional to η 1 (η 2 ). The coefficient 1 e 2 νvF is proportional to the Sharvin resistance in the two-dimension [26].
The magnetoresistance V /I = 1 e 2 νvF ( L ℓ + η 1 − η 2 ) depends on the configurations of the magnetizations in F 1 and F 2 :
R σσ ′ = R O + R s,σσ ′ ,(14)R O = L e 2 νv F ℓ ,(15)R s,σσ ′ = 1 e 2 νv F (η 1 − η 2 )(16)
where σ and σ ′ denote spin indices with σ =↑ (↓) and σ ′ =↑ (↓). η 1 = 1(−1) and η 2 = 1(−1) correspond to σ =↑ (↓) and σ ′ =↑ (↓), respectively. The magnetoresistance has two terms: Ohmic resistance term (R O ) and the boundary term (R s,σσ ′ ). R O does not depend on the magnetization configurations. R s,σσ ′ depends on the magnetization configurations. For the parallel configuration, the boundary term is absent. On the other hand, for the antiparallel configurations, the boundary term exists. Moreover, in the antiparallel configuration, the boundary term has a negative (positive) sign when the direction of the magnetization F 1 and F 2 are negative (positive) and positive (negative) along the y-axis, respectively. Whether R s,σσ ′ is zero or nonzero can be seen from the difference of the spatial dependences of s y 1 for L/ℓ = 2 in Fig. 2 (a)-(d). s y 1 is even (odd) function of x in antiparallel (parallel) configuration (see inset in Fig. 2 (a)-(d)). Since R s,σσ ′ is obtained by V s /I ∝ L/2 −L/2 dxs y 1 , it is nonzero (zero) in the antiparallel (parallel) configurations of the magnetizations. The sign of the R s,σσ ′ term can be seen from the sign of s y 1 in Fig. 2 (c) and (d), and is positive (negative) in R ↑↓ (R ↓↑ ).
These properties of the magnetoresistance are quite different from those of the conventional magnetoresistance [1][2][3][4][5]:
The magnetoresistance is only proportional to the relative angle between the directions of the two magnetizations (i.e., R ↑↑ = R ↓↓ and R ↑↓ = R ↓↑ ). The magnitude of the magnetoresistance is usually given by R ↑↑ < R ↑↓ . On the other hand, the magnetoresistance in Eqs. (14)-(16) behaves in a drastically different way. There are two conditions to determine the magnitudes of magnetoresistance R σσ ′ : Whether the magnetization configuration is parallel or antiparallel, and whether the direction of the magnetizations F 1 and F 2 are positive and negative or negative and positive along the y-axis in the antiparallel configuration, respectively. R σσ ′ satisfies R ↑↑ = R ↓↓ and R ↓↑ < R ↑↓ , and we can obtain R ↓↑ < R ↑↑ < R ↑↓ from Eqs. (14)- (16). Thus, the relation R ↓↑ < R ↑↑ is different from that of the conventional case, although R ↑↑ = R ↓↓ and R ↑↑ < R ↑↓ are the same properties as that of the conventional magnetoresistance. Moreover, R ↓↑ is lower than the Ohmic resistance, and becomes zero for L/ℓ = 2 [19]. The anomalous property of the magnetoresistance, R ↓↑ , could be applicable for an charge transport with low Joule heating.
We will consider the magnetoresistance ratio, MR ↓ (MR ↑ ), when the direction of the magnetization F 2 is free and F 1 is fixed along the negative (positive) y-axis. These magnetoresistance ratios are given by
MR ↓ ≡ R ↓↓ − R ↓↑ R ↓↑ = 2 L/ℓ − 2 ,(17)MR ↑ ≡ R ↑↑ − R ↑↓ R ↑↓ = −2 L/ℓ + 2 .(18)
MR ↓ and MR ↑ depend on L/ℓ. The value of MR ↓ is negative for L < 2ℓ and is positive for L > 2ℓ, and diverges at L/ℓ = 2 (see Fig. 2(b)) since R ↓↑ is zero. The value of MR ↑ is negative for all L/ℓ (see the inset of Fig. 2(b)). The magnitude of MR ↑ and MR ↓ are 49% and 1000% in L/ℓ = 2.1, respectively. The gigantic magnetoresistance ratio MR ↓ stems from the boundary term R s,σσ ′ in the surface state of the TI. Finally, we show that the value of V s can be intuitively understood from the boundary conditions of the spinpolarized current. s y 1 of V s can be rewritten by Eqs. (1) and (2)
s y 1 = 3 4 τ D∇ 2 x s y 1 .(19)
By using the above equation, V s in Eq. (12) is given by the integration of the spatial gradient of the nonequilibrium spin-polarized current − 3 2 D∇ x s y , and the value of V s is proportional to the difference of the nonequilibrium spin-polarized current at the interfaces:
V s = −1 eνv F L/2 −L/2 ∇ x − 3 2 D∇ x s y 1 dx(20)= −1 eνv F − 3 2 D∇ x s y 1 x→ L 2 x→− L 2 = η 1 − η 2 e 2 νv F I.(21)
In the electromagnetism, the voltage drop is generated by electrochemical potential φ as dx∇ x φ and the electric field is given by −∇ x φ. Therefore, we can regard the nonequilibrium spin-polarized current as an electrochemical potential and its gradient as an effective electric field, which drives the anomalous voltage drop in the diffusive surface of TIs.
In conclusion, we studied the anomalous magnetoresistance in the F 1 /TI/F 2 junction by using the spindiffusion equation on the surface of TIs. It is found that the magnetoresistance includes the boundary term, which is absent in the usual ferromagnetic junction. The boundary resistance is triggered by the spin polarization of the current, which is induced by both the spinpolarized current injected from F 1 into the TI and extracted from the TI into F 2 . The boundary resistance plays the role of the reduction (enhancement) to the Ohmic term when the directions of the two magnetizations are negative (positive) and positive (negative) along the y-axis. Recently, a fine quality film of the TI comes to be able to fabricate [28,29]. Also, it has been demonstrated that the ferromagnetism at ambient temperature can be induced in the Mn-doped Bi 2 Te 3 by the magnetic proximity effect through deposited Fe overlayer [30]. By using the film of the TI, the boundary resistance can be experimentally demonstrated and would be applicable for a charge transport with low energy consumption in future. (2) as 3D 2 ∇ 2 x s y − 2 τ s y − I eℓ = 0. To substitute s y = s ′y − Iτ 2eℓ into the above equation, we obtain 3D 2 ∇ 2 x s ′y = 2 τ s ′y . Now s ′y is equal to s y 1 , because of − Iτ 2eℓ = s y 2 and s ′y = s y +
FIG. 1 :
1(Color online) (a) A schematic illustration of the F1/TI/F2 junction, where the arrow from F1 to F2 shows the applied charge current and the blue arrows represent the direction of the magnetization in F1 and F2. (b)-(c) The spin polarization s y for various magnetization configurations with L/ℓ = 1 (solid line), 2 (dashed line), and 10 (dotted line),where L is the length between two ferromagnets and ℓ is the mean free path on the surface of TI. The direction of the magnetization F2 is antiparallel (b) or parallel (c) to that of F1 (inset (b) or (c)).
FIG
. 2: (Color online) The spin polarization, s y 1 , for antiparallel [(a) and (b)] and parallel [(c) and (d)] magnetization configurations with L/ℓ = 2. The schematic magnetization configurations in F1/TI/F2 junctions are shown in the insets. (e) Magnetoresistance (MR) ratio as a function L. The blue solid (red dashed) line represents MR ↓ = R ↓↓ −R ↓↑ R ↓↑ (MR ↑ = R ↑↑ −R ↑↓ R ↑↓ ). The enlarged plot of MR ↑ is shown by the dashed line in the inset.
This work was supported by a Grant-in-Aid for Young Scientists (B) (No. 22740222, No. 23740236)) and by a Grant-in-Aid for Scientific Research on Innovative Areas "Topological Quantum Phenomena" (No. 22103005, No. 25103709) from the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT). K.T. acknowledges the support by the JSPS.
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|
[] |
[
"Exploiting Chain Rule and Bayes' Theorem to Compare Probability Distributions",
"Exploiting Chain Rule and Bayes' Theorem to Compare Probability Distributions"
] |
[
"Huangjie Zheng [email protected] \nDepartment of Statistics & Data Science\nMcCombs School of Business The University of Texas at Austin Austin\nThe University of Texas at Austin Austin\n78712, 78712TX, TX\n",
"Mingyuan Zhou [email protected] \nDepartment of Statistics & Data Science\nMcCombs School of Business The University of Texas at Austin Austin\nThe University of Texas at Austin Austin\n78712, 78712TX, TX\n"
] |
[
"Department of Statistics & Data Science\nMcCombs School of Business The University of Texas at Austin Austin\nThe University of Texas at Austin Austin\n78712, 78712TX, TX",
"Department of Statistics & Data Science\nMcCombs School of Business The University of Texas at Austin Austin\nThe University of Texas at Austin Austin\n78712, 78712TX, TX"
] |
[] |
To measure the difference between two probability distributions, referred to as the source and target, respectively, we exploit both the chain rule and Bayes' theorem to construct conditional transport (CT), which is constituted by both a forward component and a backward one. The forward CT is the expected cost of moving a source data point to a target one, with their joint distribution defined by the product of the source probability density function (PDF) and a sourcedependent conditional distribution, which is related to the target PDF via Bayes' theorem. The backward CT is defined by reversing the direction. The CT cost can be approximated by replacing the source and target PDFs with their discrete empirical distributions supported on mini-batches, making it amenable to implicit distributions and stochastic gradient descent-based optimization. When applied to train a generative model, CT is shown to strike a good balance between modecovering and mode-seeking behaviors and strongly resist mode collapse. On a wide variety of benchmark datasets for generative modeling, substituting the default statistical distance of an existing generative adversarial network with CT is shown to consistently improve the performance. PyTorch code is provided.IntroductionMeasuring the difference between two probability distributions is a fundamental problem in statistics and machine learning[1][2][3]. A variety of statistical distances, such as the Kullback-Leibler (KL) divergence [4], Jensen-Shannon (JS) divergence [5], maximum mean discrepancy (MMD) [6], and Wasserstein distance [7], have been proposed to quantify the difference. They have been widely used for generative modeling with different mode covering/seeking behaviors[8][9][10][11][12][13]. The KL divergence, directly related to both maximum likelihood estimation and variational inference[14][15][16], requires the two probability distributions to share the same support and is often inapplicable if either is an implicit distribution whose probability density function (PDF) is unknown[17][18][19][20]. Variational auto-encoders (VAEs) [8], the KL divergence based deep generative models, are stable to train, but often exhibit mode-covering behaviors in its generated data, producing blurred images. The JS divergence is directly related to the min-max loss of a generative adversarial net (GAN) when the discriminator is optimal [9], while the Wasserstein-1 distance is directly related to the min-max loss of a Wasserstein GAN [11], whose critic is optimized under the 1-Lipschitz constraint. However, it is difficult to maintain a good balance between the updates of the generator and discriminator/critic, making (Wasserstein) GANs notoriously brittle to train. MMD [6] is an RKHS-based statistical distance behind21,22], which have also shown promising results in generative modeling when trained with a min-max loss. Different from VAEs, these GAN-based models often exhibit mode dropping and face the danger of mode collapse if not well tuned during the training.
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"https://arxiv.org/pdf/2012.14100v5.pdf"
] | 235,670,276 |
2012.14100
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4232080024e257445c654f741aaeaa656e7e7cb2
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Exploiting Chain Rule and Bayes' Theorem to Compare Probability Distributions
Huangjie Zheng [email protected]
Department of Statistics & Data Science
McCombs School of Business The University of Texas at Austin Austin
The University of Texas at Austin Austin
78712, 78712TX, TX
Mingyuan Zhou [email protected]
Department of Statistics & Data Science
McCombs School of Business The University of Texas at Austin Austin
The University of Texas at Austin Austin
78712, 78712TX, TX
Exploiting Chain Rule and Bayes' Theorem to Compare Probability Distributions
To measure the difference between two probability distributions, referred to as the source and target, respectively, we exploit both the chain rule and Bayes' theorem to construct conditional transport (CT), which is constituted by both a forward component and a backward one. The forward CT is the expected cost of moving a source data point to a target one, with their joint distribution defined by the product of the source probability density function (PDF) and a sourcedependent conditional distribution, which is related to the target PDF via Bayes' theorem. The backward CT is defined by reversing the direction. The CT cost can be approximated by replacing the source and target PDFs with their discrete empirical distributions supported on mini-batches, making it amenable to implicit distributions and stochastic gradient descent-based optimization. When applied to train a generative model, CT is shown to strike a good balance between modecovering and mode-seeking behaviors and strongly resist mode collapse. On a wide variety of benchmark datasets for generative modeling, substituting the default statistical distance of an existing generative adversarial network with CT is shown to consistently improve the performance. PyTorch code is provided.IntroductionMeasuring the difference between two probability distributions is a fundamental problem in statistics and machine learning[1][2][3]. A variety of statistical distances, such as the Kullback-Leibler (KL) divergence [4], Jensen-Shannon (JS) divergence [5], maximum mean discrepancy (MMD) [6], and Wasserstein distance [7], have been proposed to quantify the difference. They have been widely used for generative modeling with different mode covering/seeking behaviors[8][9][10][11][12][13]. The KL divergence, directly related to both maximum likelihood estimation and variational inference[14][15][16], requires the two probability distributions to share the same support and is often inapplicable if either is an implicit distribution whose probability density function (PDF) is unknown[17][18][19][20]. Variational auto-encoders (VAEs) [8], the KL divergence based deep generative models, are stable to train, but often exhibit mode-covering behaviors in its generated data, producing blurred images. The JS divergence is directly related to the min-max loss of a generative adversarial net (GAN) when the discriminator is optimal [9], while the Wasserstein-1 distance is directly related to the min-max loss of a Wasserstein GAN [11], whose critic is optimized under the 1-Lipschitz constraint. However, it is difficult to maintain a good balance between the updates of the generator and discriminator/critic, making (Wasserstein) GANs notoriously brittle to train. MMD [6] is an RKHS-based statistical distance behind21,22], which have also shown promising results in generative modeling when trained with a min-max loss. Different from VAEs, these GAN-based models often exhibit mode dropping and face the danger of mode collapse if not well tuned during the training.
Chain rule and Bayes' theorem based conditional transport
Exploiting the chain rule and Bayes' theorem, we can constrain π(x, y) with both p X (x) and p Y (y) in two different ways, leading to the forward CT and backward CT, respectively. To define the forward CT, we use the chain rule to factorize the joint distribution as
π(x, y) = p X (x)π Y (y | x),
where π Y (y | x) is a conditional distribution of y given x. This construction ensures π(x, y)dy = p X (x) but not π(x, y)dx = p Y (y). Denote d φ (h 1 , h 2 ) ∈ R as a function parameterized by φ, which measures the difference between two vectors h 1 , h 2 ∈ R H of dimension H. While allowing π(x, y)dx = p Y (y), to appropriately constraint π(x, y) by p Y (y), we treat p Y (y) as the prior distribution, view e −d φ (x,y) as an unnormalized likelihood term, and follow Bayes' theorem to define
π Y (y | x) = e −d φ (x,y) p Y (y)/Q(x), Q(x) := e −d φ (x,y) p Y (y)dy,(1)
where Q(x) is a normalization term that ensures π Y (y | x)dy = 1. We refer to π Y (y | x) as the forward "navigator," which specifies how likely a given x will be mapped to a target point y ∼ p Y (y). We now define the cost of the forward CT as
C(X → Y ) = E x∼p X (x) E y∼π Y (· | x) [c(x, y)].(2)
In the forward CT, we expect large c(x, y) to typically co-occur with small π Y (y | x) as long as p Y (y) provides a good coverage of the density of x. Thus minimizing the forward CT cost is expected to encourage p Y (y) to exhibit a mode-covering behavior w.r.t. p X (x). Such kind of behavior is also expected when minimizing the forward KL divergence as KL(p X ||p Y ) = E x∼p X ln p X (x) p Y (x) , which calls for p Y (x) > 0 whenever p X (x) > 0.
Reversing the direction, we construct the backward CT, where the joint is factorized as π(x, y) = p Y (y)π X (x | y) and the backward navigator is defined as π X (x | y) = e −d φ (x,y) p X (x)/Q(y), Q(y) := e −d φ (x,y) p X (x)dx.
(
This ensures π(x, y)dx = p Y (y); while allowing π(x, y)dy = p X (x), it constrains π(x, y) by treating p X (x) as the prior to construct π X (x | y). The backward CT cost is now defined as
C(X ← Y ) = E y∼p Y (y) E x∼π X (· | y) [c(x, y)].(4)
In the backward CT, we expect large c(x, y) to typically co-occur with small π X (x | y) as long as p X (x) has good coverage of the density of y. Thus minimizing the backward CT cost is expected to encourage p Y (y) to exhibit a mode-seeking behavior w.r.t. p X (x). Such kind of behavior is also expected when minimizing the reverse KL divergence as KL(p Y ||p X ) = E x∼p Y ln p Y (x) p X (x) , which allows p Y (x) = 0 when p X (x) > 0 and it is fine for p Y to just fit some portion of p X .
In comparison to the forward and revers KLs, the proposed forward and backward CT are more broadly applicable as they don't require p X and p Y to share the same distribution support and have analytic PDFs. For the cases where the KLs can be evaluated, we introduce
D(X, Y ) = KL(p X ||p Y ) − KL(p Y ||p X )
as a formal way to quantify the mode-seeking and mode-covering behavior of p Y w.r.t. p X , with D(X, Y ) > 0 implying mode seeking and with D(X, Y ) < 0 implying mode covering.
Combining both the forward and backward CTs, we now define the CT cost as
C ρ (X, Y ) := ρC(X → Y ) + (1 − ρ)C(X ← Y ),(5)
where ρ ∈ [0, 1] is a parameter that can be adjusted to encourage p Y (y) to exhibit w.r.t. p X (x) mode-seeking (ρ = 0), mode-covering (ρ = 1), or a balance of two distinct behaviors (ρ ∈ (0, 1)). By definition we have C ρ (X, Y ) ≥ 0, where the equality can be achieved when p X = p Y and the navigator parameter φ is optimized such that e −d φ (x,y) is equal to one if and only if x = y and zero otherwise. We also have C ρ=0.5 (X, Y ) = C ρ=0.5 (Y, X). We fix ρ = 0.5 unless specified otherwise.
Conjugacy based analytic conditional distributions
Estimating the forward and backward CTs involves π Y (y | x) and π X (x | y), respectively. Both conditional distributions, however, are generally intractable to evaluate and sample from, unless p X (x) and p Y (y) are conjugate priors for likelihoods proportional to e −d(x,y) , i.e., π X (x | y) and π Y (y | x) are in the same probability distribution family as p X (x) and p Y (y), respectively. For example, if d(x, y) = x − y 2 2 and both p X (x) and p Y (y) are multivariate normal distributions, then both π X (x | y) and π Y (y | x) will follow multivariate normal distributions.
To be more specific, we provide a univariate normal based example, with x, y, φ, θ ∈ R and
p X (x) = N (0, 1), p Y (y) = N (0, e θ ), d φ (x, y) = (x − y) 2 /(2e φ ), c(x, y) = (x − y) 2 . (6) Here we have D(X, Y ) = KL[N (0, 1)||N (0, e θ )] − KL[N (0, e θ )||N (0, 1)] = θ − sinh(θ)
, which is positive when θ < 0, implying mode-seeking, and negative when θ > 0, implying mode-covering. As shown in Appendix C, we have analytic forms of the forward and backward navigators as
π Y (y | x) = N (σ(θ − φ)x, σ(θ − φ)e φ ), π X (x | y) = N (σ(−φ)y, σ(φ)),
where σ(a) = 1/(1 + e −a ) denotes the sigmoid function, and forward and backward CT costs as
C(X → Y ) = σ(φ − θ)(e θ + σ(φ − θ)), C(X ← Y ) = σ(φ)(1 + σ(φ)e θ ).
As a proof of concept, we illustrate the optimization under CT using the above example, for which θ = 0 is the optimal solution that makes p X = p Y . Thus when applying gradient descent to minimize the CT cost C ρ=0.5 (X, Y ), we expect the generator parameter θ → 0 with proper learning dynamic, as long as the learning of the navigator parameter φ is appropriately controlled. This is confirmed by C(X, Y) (e = 1) Figure 1: Illustration of minimizing the CT cost C φ,θ (X, Y ) between N (0, 1) and N (0, e θ ). Left: Evolution of CT cost, its parameters, and forward and backward costs; Right: 4 CT cost curves against θ as e φ is being optimized to a small value to jointly show the optimized φ provides better learning dynamic for the learning of θ. Fig. 1, which shows that as the navigator φ gets optimized by minimizing CT cost, it is more obvious that θ will minimize the CT cost at zero. This suggests that the navigator parameter φ mainly plays the role in assisting the learning of θ. The right four subplots describe the log-scale curves of forward cost, backward cost and bi-directional CT costs w.r.t. θ as φ gets optimized to four different values. It is worth noting that the forward cost is minimized at e θ > 1, which implies a mode-covering behavior, and the backward cost is minimized at e θ → 0, which implies a mode-seeking behavior, while the bi-directional cost is minimized at around the optimal solution e θ = 1; the forward CT cost exhibits a flattened curve on the right hand side of its minimum, adding to which the backward CT cost not only moves that minimum left, making it closer to θ = 0, but also raises the whole curve on the right hand side, making the optimum of θ become easier to reach via gradient descent.
C(X, Y) C(X Y) C(X Y) 4 2 0 2 4 2 1 0 1 2 C(X, Y) (e = 2) C(X, Y) C(X Y) C(X Y) 4 2 0 2 4 2 1 0 1 2 C(X, Y) (e = 5) C(X, Y) C(X Y) C(X Y) 4 2 0 2 4 2 1 0 1 2 C(X, Y) (e = 20) C(X, Y) C(X Y) C(X Y)
To apply CT in a general setting where the analytical forms of the distributions are unknown, there is no conjugacy, or we only have access to random samples from the distributions, below we show we can approximate the CT cost by replacing both p X (x) and p Y (y) with their corresponding discrete empirical distributions supported on mini-batches. Minimizing this approximate CT cost, amenable to mini-batch SGD based optimization, is found to be effective in driving the target (model) distribution p Y towards the source (data) distribution p X , with the ability to control the mode-seeking and mode-covering behaviors of p Y w.r.t. p X .
Approximate CT given empirical samples
Below we use generative modeling as an example to show how to apply the CT cost in a general setting that only requires access to random samples of both x and y. Denote x as a data taking its value in R V . In practice, we observe a finite set
X = {x i } |X | i=1
, consisting of |X | data samples assumed to be iid drawn from p X (x). Given X , the usual task is to learn a distribution to approximate p X (x), for which we consider a deep generative model (DGM) defined as y = G θ ( ), ∼ p( ), where G θ is a generator that transforms noise ∼ p( ) via a deep neural network parameterized by θ to generate random sample y ∈ R V . While the PDF of the generator, denoted as p Y (y; θ), is often intractable to evaluate, it is straightforward to draw y ∼ p Y (y; θ) with G θ .
While knowing neither p X (x) nor p Y (y; θ), we can obtain discrete empirical distributions pX N and pŶ M supported on mini-batches x 1:N and y 1:M , as defined below, to guide the optimization of G θ in an iterative manner. With N random observations sampled without replacement from X , we define
pX N (x) = 1 N N i=1 δ(x − x i ), {x 1 , . . . , x N } ⊆ X(7)
as an empirical distribution for x. Similarly, with M random samples of the generator, we define
pŶ M (y) = 1 M M j=1 δ(y − y j ), y j = G θ ( j ), j iid ∼ p( ) .(8)
Substituting p Y (y; θ) in (2) with pŶ M (y), the continuous forward navigator becomes a discrete one asπ
Y (y | x) = M j=1π M (y j | x, φ)δ y j ,π M (y j | x, φ) := e −d φ (x,y j ) M j =1 e −d φ (x,y j ) .(9)
Thus given pŶ M , the cost of a forward CT can be approximated as
C φ,θ (X →Ŷ M ) = E y 1:M iid ∼ p Y (y;θ) E x∼p X (x) M j=1 c(x, y j )π M (y j | x, φ) ,(10)
which can be interpreted as the expected cost of following the forward navigator to stochastically transport a random source point x to one of the M randomly instantiated "anchors" of the target distribution. Similar to previous analysis, we expect this approximate forward CT to stay small as long as p Y (y; θ) exhibits a mode covering behavior w.r.t. p X (x).
Similarly, we can approximate the backward navigator and CT cost aŝ
π X (x | y) = N i=1π N (x i | y, φ)δ xi ,π N (x i | y, φ) := e −d φ (x i ,y) N i =1 e −d φ (x i ,y) , C φ,θ (X N ← Y ) = E x 1:M iid ∼ p X (x) E y∼p Y (y;θ) N i=1 c(x i , y)π N (x i | y, φ) .(11)
Similar to previous analysis, we expect this approximate backward CT to stay small as long as p Y (y; θ) exhibits a mode-seeking behavior w.r.t. p X (x).
Combining (10) and (11), we define the approximate CT cost as
C φ,θ,ρ (X N ,Ŷ M ) = ρC φ,θ (X →Ŷ M ) + (1 − ρ)C φ,θ (X N ← Y ),(12)
an unbiased sample estimate of which, given mini-batches x 1:N and y 1:M , can be expressed as
L φ,θ,ρ (x 1:N , y 1:M ) = N i=1 M j=1 c(x i , y j ) ρ Nπ M (y j | x i , φ) + 1−ρ Mπ N (x i | y j , φ) = N i=1 M j=1 c(x i , y j ) ρ N e −d φ (x i ,y j ) M j =1 e −d φ (x i ,y j ) + 1−ρ M e −d φ (x i ,y j ) N i =1 e −d φ (x i ,y j ) . (13) Lemma 1. Approximate CT in (12) is asymptotic as lim N,M →∞ C φ,θ,ρ (X N ,Ŷ M ) = C φ,θ,ρ (X, Y ).
Cooperatively-trained or adversarially-trained feature encoder
To apply CT for generative modeling of high-dimensional data, such as natural images, we need to define an appropriate cost function c(x, y) to measure the difference between two random points. A naive choice is some distance between their raw feature vectors, such as c(x, y) = x − y 2 2 , which, however, is known to often poorly reflect the difference between high-dimensional data residing on low-dimensional manifolds. For this reason, with cosine similarity [27] as cos(h 1 ,
h 2 ) := h T 1 h2 √ h T 1 h1 √ h T 2 h2
, we further introduce a feature encoder T η (·), parameterized by η, to help redefine the point-to-point cost and both navigators as
c η (x, y) = 1 − cos(T η (x), T η (y)), d φ Tη(x)
Tη(x) , Tη(y) Tη(y)
.
To apply the CT cost to train a DGM, we find that the feature encoder T η (·) can be learned in two different ways: 1) Cooperatively-trained: Training them cooperatively by alternating between two different losses: training the generator under a fixed T η (·) with the CT loss, and training T η (·) under a fixed generator with a different loss, such as the GAN discriminator loss, WGAN critic loss, and MMD-GAN [10] critic loss. 2) Adversarially-trained: Viewing the feature encoder as a critic and training it to maximize the CT cost, by not only inflating the point-to-point cost, but also distorting the feature space used to construct the forward and backward navigators' conditional distributions.
To be more specific, below we present the details for the adversarial way to train T η . Given training data X , to train the generator G θ , forward navigator π φ (y | x), backward navigator π φ (x | y), and encoder T η , we view the encoder as a critic and propose to solve a min-max problem as
min φ,θ max η E x 1:N ⊆X , 1:M iid ∼ p( ) [L φ,θ,ρ,η (x 1:N , {G θ ( j )} M j=1 )],(15)
where L φ,θ,ρ,η is defined the same as in (13), except that we replace c(x i , y j ) and d φ (·, ·) with their corresponding ones shown in (14) and use reparameterization in (8) to draw y 1:
M := {G θ ( j )} M j=1 . With SGD, we update φ and θ using ∇ φ,θ L φ,θ,ρ,η (x 1:N , {G θ ( j )} M j=1 )
) and, if the feature encoder is adversarially-trained, update η using −∇ η L φ,θ,ρ,η (x 1:N , {G θ ( j )} M j=1 )). We find by experiments that both ways to learn the encoder work well, with the adversarial one generally providing better performance. It is worth noting that in (Wasserstein) GANs, while the adversarially-trained discriminator/critic plays a similar role as a feature encoder, the learning dynamics between the discriminator/critic and generator need to be carefully tuned to maintain training stability and prevent trivial solutions (e.g., mode collapse). By contrast, the feature encoder of the CT cost based DGM can be stably trained in two different ways. Its update does not need to be well synchronized with the generator and can be stopped at any time of the training.
Related work
In practice, variational auto-encoders [8], the KL divergence based deep generative models, are stable to train, but often exhibit mode-covering behaviors and generate blurred images [28][29][30][31][32]. By contrast, both GANs and Wasserstein GANs can generate photo-realistic images, but they often suffer from stability and mode collapse issues, requiring the update of the discriminator/critic to be well synchronized with that of the generator. This paper introduces conditional transport (CT) as a new method to quantify the difference between two probability distributions. Deep generative models trained under CT not only allow the balance between mode-covering and mode-seeking behaviors to be adjusted, but also allow the encoder to be pretrained or frozen at any time during cooperative/adversarial training.
As the JS divergence requires the two distributions to have the same support, the Wasserstein distance is often considered as more appealing for generative modeling as it allows the two distributions to have non-overlapping support [24][25][26]. However, while GANs and Wasserstein GANs in theory are connected to the JS divergence and Wasserstein distance, respectively, several recent works show that they should not be naively understood as the minimizers of their corresponding statistical distances, and the role played by their min-max training dynamics should not be overlooked [33][34][35]. In particular, Fedus et al. [34] show that even when the gradient of the JS divergence does not exist and hence GANs are predicted to fail from the perspective of divergence minimization, the discriminator is able to provide useful learning signal. Stanczuk et al. [35] show that the dual form based Wasserstein GAN loss does not provide a meaningful approximation of the Wasserstein distance; while primal form based methods could better approximate the true Wasserstein distance, they in general clearly underperform Wasserstein GANs in terms of the generation quality for high-dimensional data, such as natural images, and require an inner loop to compute the transport plan for each mini-batch, leading to high computational cost [12,[35][36][37][38]. See previous works for discussions on the approximation error and gradient bias when estimating the Wasserstein distance with mini-batches [10,23,39,40].
MMD-GAN [10,21,22] that calculates the MMD statistics in the latent space of a feature encoder is the most similar to the CT cost in terms of the actual loss function used for optimization. In particular, both the MMD-GAN loss and CT loss, given mini-batches x 1:N and y 1:M , involve computing the differences of all N M pairs (x i , y j ). Different from MMD-GAN, there is no need in CT to choose a kernel and tune its parameters. We provide below an ablation study to evaluate both 1) MMD generator + CT encoder and 2) MMD encoder + CT generator, which shows 1) performs on par with MMD, while 2) performs clearly better than MMD and on par with CT.
Experimental results
Forward and backward analysis: To empirically verify our previous analysis of the mode covering (seeking) behavior of the forward (backward) CT, we train a DGM with (12) and show the corresponding interpolation weight from the forward CT cost to the backward one, which means CT ρ reduces from forward CT (ρ = 1), to the CT in (12) (ρ ∈ (0, 1)), and to backward CT (ρ = 0). We consider the squared Euclidean (i.e.
L 2 2 ) distance to define both cost c(x, y) = x − y 2 2 and d φ (x, y) = T φ (x) − T φ (y) 2 2 ,
where T φ denotes a neural network parameterized by φ. We consider a 1D example of a bimodal Gaussian mixture p X (x) = 1 4 N (x; −5, 1) + 3 4 N (x; 2, 1) and a 2D example of 8-modal Gaussian mixture with equal component weight as in Gulrajani et al. [41]. We use an empirical sample set X , consisting of |X | = 5, 000 samples from both 1D and 2D cases, and illustrate in Fig. 2 the KDE of 5000 generated samples y j = G θ ( j ) after 5000 training epochs. For the 1D case, we take 200 grids in [−10, 10] to approximate the empirical distribution ofp X and p Y , and report the corresponding forward KL
(KL[p X ||p Y ]), reverse KL (KL[p Y ||p X ]), and their difference D(X, Y ) = KL[p X ||p Y ] − KL[p X ||p Y ] below each corresponding sub-figure in Fig. 2.
Comparing the results of different ρ in Fig. 2, it suggests that minimizing the forward CT cost only encourages the generator to exhibit mode-covering behaviors, while minimizing the backward CT cost only encourages mode-seeking behaviors. Combining both costs provides a user-controllable balance between mode covering and seeking, leading to satisfactory fitting performance, as shown in Columns 2-4. Note that for a fair comparison, we stop the fitting at the same iteration; in practice, we find if training with more iterations, both ρ = 0.75 and ρ = 0.25 can achieve comparable results as ρ = 0.5 in this example. Allowing the mode covering and seeking behaviors to be controlled by adjusting ρ is an attractive property of CT ρ . Resistance to mode collapse: We continue to use a 8-Gaussian mixture to empirically evaluate how well a DGM resists mode collapse. Unlike the data in Fig. 2, where 8 modes are equally weighted, here the mode at the left lower corner is set to have weight γ, while the other modes are set to have the same weight of 1−γ 7 . We set X with 5000 samples and the mini-batch size as N = 100. When γ is lowered to 0.05, its corresponding mode is shown to be missed by GAN, WGAN, and SWD-based DGM, while well kept by the CT-based DGM. As an explanation, GANs are known to be susceptible to mode collapse; WGAN and SWD-based DGMs are sensitive to the mini-batch size, as when γ equals to a small value, the samples from this mode will appear in the mini-batches less frequently than those from any other mode, amplifying their missing mode problem. Similarly, when γ is increased to 0.5, the other modes are likely to be missed by the baseline DGMs, while the CT-based DGM does not miss any modes. The resistance of CT to mode dropping can be attributed to its forward component's mode-covering property. The backward's mode-seeking property further helps distinguish the density of each mode component to avoid making components of equal weight.
CT for 2D toy data and robustness in adversarial feature extraction: To test CT with more general cases, we further conduct experiments on 4 representative 2D datasets for generative modeling evaluation [41]: 8-Gaussian mixture, Swiss Roll, Half Moons, and 25-Gaussian mixture. We apply the vanilla GAN [9] and Wasserstein GAN with gradient penalty (WGAN-GP) [41] as two representatives of min-max DGMs that require solving a min-max loss. We then apply the generators trained under the sliced Wasserstein distance (SWD) [42] and CT cost as two representatives of min-max-free DGMs. Moreover, we include CT with an adversarial feature encoder trained with (14) to test the robustness of adversary and compare with the baselines in solving the min-max loss.
On each 2D data, we train these DGMs as one would normally do during the first 5k epochs. We then only train the generator and freeze all the other learnable model parameters, which means we freeze the discriminator in GAN, critic in WGAN, the navigator parameter φ of the CT cost, and both (φ, η) of CT with an adversarial feature encoder, for another 5k epochs. Figs. 7-10 in Appendix E.1 illustrate this training process on each dataset, where for both min-max baseline DGMs, the models collapse after the first 5k epochs, while the training for SWD remains stable and that for CT continues to improve. Compared to SWD, our method covers all data density modes and moves the generator much closer to the true data density. Notably, for CT with an adversarially trained feature encoder, although it has switched from solving a min-max loss to freezing the feature encoder after 5k epochs, the frozen feature encoder continues to guide the DGM to finish the training in the last 5k epochs, which shows the robustness of the CT cost.
Ablation of cooperatively-trained and adversarially-trained CT: As previous experiments show the adversarially-trained feature encoder could provide a valid feature space for CT cost, we further study the performance of the encoders cooperatively trained with other losses. Here we leverage, as two alternatives, the space of an encoder trained with the discriminator loss in GANs and the empirical Wasserstein distance in sliced 1D spaces [43]. We test these settings on both 8-Gaussian, as shown Fig. 4, and CIFAR-10 data, as shown in Table 1. It is confirmed these encoders are able to cooperatively work with CT, in general producing less appealing results with those trained by maximizing CT. From this view, although CT is able to provide guidance for the generators in the feature space learned with various options, maximizing CT is still preferred to ensure the efficiency. Moreover, as observed in Figs. 4b-4e, CT clearly improves the fitting with sliced Wasserstein distance.
To explain why CT helps improve in the sliced space, we further provide a toy example in 1D to study the properties of CT and empirical Wasserstein distance in Appendix E.3.
Ablation of MMD and CT: As MMD also compares the pair-wise sample relations in a mini-batch, we study if MMD and CT can benefit each other. The feature space of MMD-GAN can be considered as T η •k, where k is the rational quadratic or distance kernel in Bińkowski et al. [10]. Here we evaluate the combinations of MMD/CT as the generator/encoder criterion to train DGMs. On CIFAR-10, shown in Table 2, combining MMD and CT generally has improvement over MMD alone in FID.
It is interesting to notice that for MMD-GAN, learning its generator with the CT cost shows more obvious improvement than learning its feature encoder with the CT cost. We speculate the estimation of MMD relies on a supremum of its witness function, which needs to be maximized w.r.t T η • k and cannot be guaranteed by maximizing CT w.r.t T η . In the case of MMD-dist, using CT for witness function updates shows a more clear improvement, probably because CT has a similar form as MMD when using the distance kernel. From this view, CT and MMD are naturally able to be combined to compare the distributional difference with pair-wise sample relations. Different from MMD, CT does not involve the choice of kernel and its navigators assist to improve the comparison efficiency. Below we show on more image datasets, CT is compatible with many existing models, and achieve good results to show improvements on a variety of data with different scale.
Adversarially-trained CT for natural images: We conduct a variety of experiments on natural images to evaluate the performance and reveal the properties of DGMs optimized under the CT cost. We consider three widely-used image datasets, including CIFAR-10 [44], CelebA [45], and LSUN-bedroom [46] for general evaluation, as well as CelebA-HQ [47], FFHQ [48] for evaluation in high-resolution. We compare the results of DGMs optimized with the CT cost against DGMs trained with their original criterion including DCGAN [49], Sliced Wasserstein Generative model (SWG) [42], MMD-GAN [10], SNGAN [50], and StyleGAN2 [51]. For fair comparison, we leverage the best configurations reported in their corresponding paper or Github page. The detailed setups can be found in Appendix D. For evaluation metric, we consider the commonly used Fréchet inception distance (FID, lower is preferred) [52] on all datasets and Inception Score (IS, higher is preferred) [53] on CIFAR-10. Both FID and IS are calculated using a pre-trained inception model [54].
The summary of FID and IS on previously mentioned model is reported in Table 3. We observe that trained with CT cost, all the models have improvements with different margin in most cases, suggesting that CT is compatible with standard GANs, SWG, MMD-GANs, WGANs and generally helps improve generation quality, especially for data with richer modalities like CIFAR-10. CT is also compatible with advanced model architecture like StyleGAN2, confirming that a better feature space could make CT more efficient to guide the generator and produce better results.
The qualitative results shown in Fig. 5 are consistent with quantitative results in Table 3. To additionally show how CT works for more complex generation tasks, we show in Fig. 6 example higher-resolution images generated by CT-SNGAN on LSUN bedroom (128x128) and CelebA-HQ (256x256), as well as images generated by CT-StyleGAN2 on LSUN bedroom (256x256), FFHQ (256x256), and FFHQ (1024x1024). On the choice of ρ for natural images: In previous experiments, we fix ρ = 0.5 by default when we prefer neither mode-covering nor mode-seeking. We further tune ρ as an additional ablation study on CIFAR-10 dataset with both the CT + DCGAN backbone and CT + SNGAN backbone to see its affects in terms of certain metrics, such as the FID score. The results shown in Table 4 suggest that CT is not sensitive to the choice of ρ as long as 0 < ρ < 1, and the FID score could be further improved if we choose a smaller ρ to bias towards mode-seeking.
Conclusion
We propose conditional transport (CT) as a new criterion to quantify the difference between two probability distributions, via the use of both forward and backward conditional distributions. The forward and backward expected cost are respectively with respect to a source-dependent and targetdependent conditional distribution defined via Bayes' theorem. The CT cost can be approximated with discrete samples and optimized with existing stochastic gradient descent-based methods. Moreover, the forward and backward CT possess mode-covering and mode-seeking properties, respectively. By combining them, CT nicely incorporates and balances these two properties, showing robustness in resisting mode collapse. On complex and high-dimensional data, CT is able to be calculated and stably guide the generative models in a valid feature space, which can be learned by adversarially maximizing CT or cooperatively deploying existing methods. On various benchmark datasets for deep generative modeling, we successfully train advanced models with CT. Our results consistently show improvement over the original ones, justifying the effectiveness of the proposed CT loss.
Discussion: Note CT brings consistent improvement to these DGMs by neither improving their network architectures nor gradient regularization. Thus it has great potential to work in conjunction with other state-of-the-art architectures or methods, such as BigGAN [55], self-attention GANs [56], partition-guided GANs [57], multimodal-DGMs [58], BigBiGAN [59], self-supervised learning [60], and data augmentation [61][62][63], which we leave for future study. As the paper is primarily focused on constructing and validating a new approach to quantify the difference between two probability distributions, we have focused on demonstrating the efficacy and interesting properties of the proposed CT on toy data and benchmark image data. We have focused on these previously mentioned models as the representatives in GAN, MMD-GAN, WGAN under CT, and we leave to future work using the CT to optimize more choices of DGMs, such as VAE-based models [8] and neural-SDE [64].
Exploiting Chain Rule and Bayes' Theorem to Compare Probability Distributions: Appendix A Broader impact
This paper proposes to quantify the difference between two probability distributions with conditional transport, a bidirectional cost that we exploit to balance the mode seeking and covering behaviors of a generative model. The generative models trained with the proposed CT and datasets used in the experiments are classic in the area. Thus the capacities of these models are similar to existing ones, where we can see both positive and negative perspectives, depending on how the models are used. For example, good generative models can generate images for datasets that are expensive to collect, and be used to denoise and recover images. Meanwhile, they can also be misused to generate fake images for malicious purposes.
B Proof of Lemma 1
Proof. According to the strong law of large numbers, when M → ∞, 1 (10) converges almost surely to (11) converges almost surely to the backward CT C φ,θ (X ← Y ) defined in (4) when N → ∞.
M M j=1 e −d φ (x,y j ) , where y j iid ∼ p Y (y), converges almost surely to e −d φ (x,y) p Y (y)dy and 1 M M j=1 c(x, y j )e −d φ (x,y j ) converges almost surely to c(x, y)e −d φ (x,y) p Y (y)dy. Thus when M → ∞, the term M j=1 c(x, y j )π M (y j | x, φ) inc(x,y)e −d φ (x,y) p Y (y)dy e −d φ (x,y) p Y (y)dy = c(x, y)π Y (y | x)dy. Therefore, C φ,θ (X →Ŷ M ) defined in (10) converges almost surely to the forward CT cost C φ,θ (X → Y ) defined in (2) when M → ∞. Similarly, we can show that C φ,θ (X N ← Y ) defined in
C Additional details for the univariate normal toy example shown in (6) For the toy example specified in (6), exploiting the normal-normal conjugacy, we have an analytical conditional distribution for the forward navigator as
π φ (y | x) ∝ e − (x−y) 2 2e φ N (y; 0, e θ )
∝ N (x; y, e φ )N (y; 0, e θ ) = N e θ e θ + e φ x, e φ e θ e θ + e φ , and an analytical conditional distribution for the backward navigator as
π φ (x | y) ∝ e − (x−y) 2 2e φ N (x; 0, 1) ∝ N (y; x, e φ )N (x; 0, 1) = N y 1 + e φ , e φ 1 + e φ .
Plugging them into (2) and (4), respectively, and solving the expectations, we have 1) e φ e θ + e φ e θ + e φ e θ + e φ x 2 = e φ e θ + e φ e θ + e φ e θ + e φ ,
C φ,θ (µ → ν) = E x∼N (0,C φ,θ (µ ← ν) = E y∼N (0,e θ ) e φ 1 + e φ 1 + e φ 1 + e φ y 2 = e φ 1 + e φ 1 + e φ 1 + e φ e θ .
D Experiment details
Preparation of datasets We apply the commonly used training set of MNIST (50K gray-scale images, 28 × 28 pixels) [65], Stacked-MNIST (50K images, 28 × 28 with 3 channels pixels) [66], CIFAR-10 (50K color images, 32 × 32 pixels) [44], CelebA (about 203K color images, resized to 64 × 64 pixels) [45], and LSUN bedrooms (around 3 million color images, resized to 64 × 64 pixels) [46]. For MNIST, when calculate the inception score, we repeat the channel to convert each gray-scale image into a RGB format. For high-resolution generation, we use CelebA-HQ (30K images, resized to 256 × 256 pixels) [67] and FFHQ (70K images, with both original size 1024 × 1024 and resized size 256 × 256) [48]. All image pixels are normalized to range [−1, 1].
Experiment setups
To avoid a large increase in model complexity, the navigator is parameterized as d φ (x, y) := d φ ((x − y) • (x − y)), where • denotes the Hadamard product, i.e., the element-wise product. To be clear, we provide a Pytorch-like pseudo-code in Algorithm 1. For the toy datasets, we apply the network architectures presented in Table 5, where we set H = 100 for generator, navigator and feature encoder. For navigator, we set input dimension V = 2 and output dimension d = 1. If apply a feature encoder, we have V = 2, d = 10 for feature encoder and V = 10, d = 1 for navigator. The input dimension of generator is set as 50. The slopes of all leaky ReLU functions in the networks are set to 0.1 by default. We use the the Adam optimizer [68] with learning rate α = 2 × 10 −4 and β 1 = 0.5, β 2 = 0.99 for the parameters of the generator, and discriminator/critic. The learning rate of navigator is divided by 5. Typically, 5, 000 training epochs are sufficient. However, our experiments show that the DGM optimized with the CT cost can be stably trained at least over 10, 000 epochs (or possibly even more if allowed to running non-stop) regardless of whether the navigators are frozen or not after a certain number of iterations, where the GAN's discriminator usually diverges long before reaching that many training epochs even if we do not freeze it after a certain number of iterations.
For image experiments, to make the comparison fair, we strictly adopt the architecture of DCGAN Tables 7-12. To adapt the navigator, we apply the backbone of the discriminator in these GAN models as feature encoder and suppose the output dimension as m. The navigator is an MLP with architecture shown in Table 5 by setting V = m, H = 512, and d = 1. All models are able to be trained on a single GPU, Nvidia GTX 1080-TI/Nvidia RTX 3090 in our CIFAR-10, CelebA, LSUN-bedroom experiments. For high-resolution experiments, all experiments are done on 4 Tesla-V100-16G GPUs. We visualize the results on the 8-Gaussian mixture toy dataset and other three commonly-used 2D toy datasets: Swiss-Roll, Half-Moon and 25-Gaussian mixture. As shown in Figs. 7-10, in the first 5k epochs, all DGMs are normally trained and the generative distributions are getting close to the true data distribution, while on 8-Gaussian and 25-Gaussian data, Vanilla GANs show mode missing behaviors. After 5k epochs, as the discriminator/navigator/feature encoder components in all DGMs are fixed, we can observe GAN and WGAN that solve min-max loss appear to collapse. This mode collapse issue of both GAN and WGAN-GP becomes more severe on the Swiss-Roll, Half-Moon, and 25-Gaussian datasets, since they rely on an optimized discriminator/critic to guide the generator. SWG relies on the slicing projection and is not affected, while its generated samples only cover the modes and ignore the correct density, indicating the effectiveness of slicing methods rely on the slicing [69]. The proposed CT cost show consistent good performance on the fitting of all these toy datasets, even after the navigator and the feature encoder are fixed after 5k epochs. This justifies our analysis about the robustness of CT cost.
E.2 Additional results of cooperative vs. adversarial encoder training
Here we provide additional results to the cooperative experiments, where we minimize CT in the feature encoder spaces trained by: 1) maximizing discriminator loss in GANs, 2) using random slicing projections, 3) maximizing MMD and 4) maximizing CT cost. Fig. 11 shows the results analogous to Fig. 4 on other three synthetic datasets: Swiss-Roll, Half-Moon and 25-Gaussian mixture. Fig. 12 provide qualitative results of Table 1 and Table 2. Table 1, Fig. 4, and Fig. 11 we notice the proposed CT can improve the fitting with SWG [70] in the sliced 1D space. Considering SWG applies random slicing projections to project high-dimensional data to several 1D spaces, since the empirical Wasserstein distance has a close form in 1D case and can be calculated with ordered statistics, here we compare the empirical Wasserstein loss and empirical CT cost with a 1D toy experiments.
Let's consider the same 1D Gaussian mixture data used in Fig. 2, where the bimodal Gaussian mixture has a density form p X (x) = 1 4 N (x; −5, 1) + 3 4 N (x; 2, 1). We use an empirical sample set X , consisting of |X | = 5, 000 samples, and train a generative model with the Wasserstein loss and CT cost estimated with these empirical data and generated samples. We vary the training mini-batch size from small to large. Fig. 13 shows the training curve w.r.t. each training epoch and the fitting results with mini-batch size 20, 200 and 5000. We can observe when the mini-batch size N is as large as 5000, both Wasserstein and CT lead to a well-trained generator. However, as shown in the left and middle columns, when N is getting much smaller, the generator trained with Wasserstein under-performs that trained with ACT, especially when the mini-batch size becomes as small as N = 20. While the Wasserstein distance W(X, Y ) in theory can well guide the training of a generative model, the sample Wasserstein distance W(X N ,Ŷ N ), whose optimal transport plan is locally re-computed for each mini-batch, could be sensitive to the mini-batch size N , which also explains why in practice the SWG are difficult to fit desired distribution. By contrast, CT shows better robustness across mini-batches, leading to a well-trained generator whose performance has low sensitivity to the mini-batch size.
E.4 Additional results on mode-covering/mode-seeking study
The mode covering and mode seeking behaviors discussed in Figs. 2 also exist in the real image case. For illustration, we use the Stacked-MNIST dataset [66] and fit CT in three configurations: normal, forward only, and backward only. DCGAN [49], VEEGAN [66], PacGAN [71], and PresGAN [72] are applied here as the baseline models to evaluate the mode-capturing capability. We calculate the captured mode number of each model, as well as the Kullback-Leibler (KL) divergence of the predicted label distributions between the generated samples and true data samples. For Stacked-MNIST data, there are 1000 modes in total. The results in Table 6 justify CT using only forward or using both forward and backward can almost capture all the modes, thus we do not suffer from the mode collapse problem. Using backward only can only encourages the mode seeking/dropping behavior. Fig. 14 provides the visual justification of this experiment, where the observations is consistent with those on toy datasets: if we only apply forward CT, the generator is encouraged to cover all the modes; if we only apply the backward CT for optimization, we can observe the mode seeking behavior of the generator.
F Architecture summary
Figure 2 :
2Forward and backward analysis: (top) Fitting 1D bi-modal Gaussian. Quantitative results of estimated forward KL (KL[pX ||pY ]), reverse KL (KL[pY ||pX ]), and the difference between the forward and reverse KL (D=KL[pX ||pY ]-KL[pY ||pX ]) are shown below each sub-figure. (bottom) 2D 8-Gaussian mixture by interpolating between the forward CT (ρ = 1) and backward CT (ρ = 0).
Figure 3 :
3Experiments on the resistance to model collapse: Comparison of the generation quality on 8-Gaussian mixture data: one of the 8 modes has weight γ and the rest modes have equal weight as 1−γ 7 .
Figure 4 :
4Ablation of fitting results by minimizing CT in different spaces: (a) CT calculated with adversarially trained encoder. (b-c) GAN vs. CT with feature space cooperatively trained with discriminator loss. (d-f) Sliced Wasserstein distance and CT in the sliced space.
Figure 5 :
5Generated samples of the deep generative model that adopts the backbone of SNGAN but is optimized with the CT cost on CIFAR-10, CelebA, and LSUN-Bedroom. See Appendix E for more results.
Figure 6 :
6Generation results in higher-resolution cases, with SNGAN and StyleGAN2 architecture. Top:LSUN-Bedroom (128x128) and CelebA-HQ (256x256), done with CT-SNGAN. Bottom: LSUN-Bedroom (256x256) and FFHQ (256x256/1024x1024), done with CT-StyleGAN2.
[ 49 ] 1 ,
491Sliced Wasserstein Generative model (SWG) [42] 2 , MMD-GAN [10] 3 , SNGAN [50] 4 , and StyleGAN2 [51] 5 , and follow their experiment setting: DCGAN and SWG apply CNN architecture on all datasets; MMD-GAN applies CNN on CIFAR-10 and ResNet architecture on other datasets; SN-GAN and StyleGAN2 apply their modified ResNet architecture. A summary of CNN and ResNet architecture is presented from
Figure 7 :
7On a 8-Gaussian mixture data, comparison of generation quality and training stability between two min-max deep generative models (DGMs), including vallina GAN and Wasserstein GAN with gradient penalty (WGAN-GP), and two min-max-free DGMs, whose generators are trained under the sliced Wasserstein distance (SWD) and the proposed CT cost, respectively. The critics of GAN, WGAN-GP, the navigators of CT and the adversarially trained feature encoders of AdvCT are fixed after 5k training epochs. The last column shows the true data density.
Figure 8 :
8Analogous plot toFig. 7for the Swiss-Roll dataset.
Figure 9 :
9Analogous plot toFig. 7for the Half-Moon dataset.
Figure 10 :
10Analogous plot to Fig. 7 for the 25-Gaussian mixture dataset.
Figure 11 :
11Analogous plot to Fig. 4 on Swiss roll, half-moon and 25 Gaussians datasets. Ablation of fitting results by minimizing CT in different spaces E.3 Empirical Wasserstein loss vs empirical CTFrom
Figure 12 :
12Analogous plot to Fig. 4 and Fig. 11 on image datasets. Ablation of fitting results by minimizing CT in different spaces.
Figure 13 :
13Top: Plot of the sample Wasserstein distance W2(X5000,Ŷ5000) 2 against the number of training epochs, where the generator is trained with either W2(XN ,ŶN ) 2 or the CT cost betweenXN andŶN , with the mini-batch size set as N = 20 (left), N = 200 (middle), or N = 5000 (right); one epoch consists of 5000/N SGD iterations. Bottom: The fitting results of different configurations, where the KDE curves of the data distribution and the generative one are marked in red and blue, respectively.
Figure 14 :
14Visual results of the generated samples produced by DCGAN, VEEGAN, PacGAN, PresGAN, and ACT-DCGAN on the Stacked-MNIST dataset.
Figure 15 :
15Analogous plot to Fig. 5, with additional generated samples. Top: samples generated with CNN backbone; Bottom: samples generated with ResNet backbone. (a) CIFAR-10. (b) CelebA. (c) LSUN-Bedroom.
Figure 16 :
16Analogous plot toFig. 15.
Table 1 :
1FID comparison with differ-ent cooperative training on CIFAR-10
(lower FID is preferred).
Critic space FID ↓
Discriminator 29.7
Slicing
32.4
Adversarial CT 22.1
Table 2 :
2FID Comparison with using MMD (Rational quadratic kernel/distance kernel) and CT loss in training critic/generator on CIFAR-10 (lower FID is preferred).MMD-rq
Generator loss MMD-dist
Generator loss
MMD CT
MMD CT
Critic MMD 39.9 24.1 Critic MMD 40.3 28.8
loss
CT
41.4 23.9 loss
CT
30.9 29.4
Table 3 :
3Results of CT with different deep generative models on CIFAR-10, CelebA and LSUN. Base model results are quoted from corresponding paper or github page.Method
Fréchet Inception Distance (FID ↓)
Inception Score (↑)
CIFAR-10
CelebA
LSUN-bedroom
CIFAR-10
DCGAN [49]
30.2±0.9
52.5±2.2
61.7±2.9
6.2±0.1
CT-DCGAN
22.1±1.1
29.4±2.0
32.6±2.5
7.5±0.1
SWG [42]
33.7±1.5
21.9±2.0
67.9±2.7
-
CT-SWG
25.9± 0.9 18.8 ± 1.2
39.0 ± 2.1
6.9 ± 0.1
MMD-GAN [10] 39.9±0.3
20.6±0.3
32.0±0.3
6.5±0.1
CT-MMD-GAN 23.9 ± 0.4 13.8 ± 0.4
38.3 ± 0.3
7.4 ± 0.1
SNGAN [50]
21.5±1.3
21.7±1.5
31.1±2.1
8.2±0.1
CT-SNGAN
17.2±1.0
9.2±1.0
16.8±2.1
8.8±0.1
StyleGAN2 [51]
5.8
5.2
2.9
10.0
CT-StyleGAN2
2.9 ± 0.5
4.0 ± 0.7
6.3 ± 0.2
10.1 ± 0.1
Table 4 :
4FID of generation results on CIFAR-
10, trained with different ρ.
ρ
1 0.75 0.5 0.25 0
CT-DCGAN 25.1 22.1 22.1 21.4 72.1
CT-SNGAN 23.2 17.5 17.2 17.2 33.2
Table 5 :
5Network architecture for toy datasets (V , H and d indicate the dimensionality).(a) Generator G θ
∈ R 50 ∼ N (0, 1)
50 → H, dense, BN, lReLU
H → H
2 , dense, BN, lReLU
H
2 → V , dense, linear
(b) Navigator d φ / Feature encoder Tη
x ∈ R V
V → H, dense, BN, lReLU
H → H
2 , dense, BN, lReLU
H
2 → d, dense, linear
1 DCGAN architecture follows: https://github.com/pytorch/examples/tree/master/dcgan
2 SWG architecture follows: https://github.com/ishansd/swg
3 MMD-GAN architecture follows: https://github.com/mbinkowski/MMD-GAN
4 SN-GAN architecture follows: https://github.com/pfnet-research/sngan_projection
5 StyleGAN2 architecture follows: https://github.com/NVlabs/stylegan2. We use their config-f.
Table 6 :
6Assessing mode collapse on Stacked-MNIST. The true total number of modes is 1,000. DCGAN, VEEGAN, and CT (Backward only) all suffer from collapse. The other models capture nearly all the modes of the data distribution. Furthermore, the distribution of the labels predicted from the images produced by these models is closer to the data distribution, which shows lower KL scores.Method
Mode Captured KL
DCGAN [49]
392.0 ± 7.376
8.012 ± 0.056
VEEGAN [66]
761.8 ± 5.741
2.173 ± 0.045
PacGAN [71]
992.0 ± 1.673
0.277 ± 0.005
PresGAN [72]
999.4 ± 0.80
0.102 ± 0.003
CT
999.07 ± 0.162 0.181 ± 0.003
CT (Foward only)
999.18 ± 0.9
0.124 ± 0.003
CT (Backward only) 192 ± 1.912
9.166 ± 0.06
Table 7 :
7DCGAN architecture for the CIFAR-10 dataset.(a) Generator G θ ∈ R 128 ∼ N (0, 1) 128 → 4 × 4 × 512,dense, linear 4 × 4, stride=2 deconv. BN 256 ReLU 4 × 4, stride=2 deconv. BN 128 ReLU h × w × 512 → m, dense, linear4 × 4, stride=2 deconv. BN 64 ReLU
3 × 3, stride=1 conv. 3 Tanh
(b) Feature encoder Tη
x ∈ [−1, 1]
32×32×3
3 × 3, stride=1 conv 64 lReLU
4 × 4, stride=2 conv 64 lReLU
3 × 3, stride=1 conv 128 lReLU
4 × 4, stride=2 conv 128 lReLU
3 × 3, stride=1 conv 256 lReLU
4 × 4, stride=2 conv 256 lReLU
3 × 3, stride=1 conv. 512 lReLU
Table 8 :
8DCGAN architecture for the CelebA and LSUN datasets. (a) Generator G θ ∈ R 128 ∼ N (0, 1) 128 → 4 × 4 × 1024, dense, linear 4 × 4, stride=2 deconv. BN 512 ReLU 4 × 4, stride=2 deconv. BN 256 ReLU 4 × 4, stride=2 deconv. BN 128 ReLU 4 × 4, stride=2 deconv. BN 64 ReLU 3 × 3, stride=1 conv. 3 Tanh (b) Feature encoder Tη x ∈ [−1, 1] 64×64×3 4 × 4, stride=2 conv 64 lReLU 4 × 4, stride=2 conv BN 128 lReLU 4 × 4, stride=2 conv BN 256 lReLU 3 × 3, stride=1 conv BN 512 lReLU h × w × 512 → m, dense, linear, Normalize
Table 9 :
9ResNet architecture for the CIFAR-10 dataset.(a) Generator G θ ∈ R 128 ∼ N (0, 1) 128 → 4 × 4 × 256, dense, linear BN, ReLU, 3 × 3 conv, 3 Tanh (b) Feature encoder Tη x ∈ [−1, 1] ReLU Global sum pooling h = 128 → m, dense, linear, NormalizeResBlock up 256
ResBlock up 256
ResBlock up 256
32×32×3
ResBlock down 128
ResBlock down 128
ResBlock 128
ResBlock 128
Table 10 :
10ResNet architecture for the CelebA and LSUN datasets. (a) Generator G θ ∈ R 128 ∼ N (0, 1) 128 → 4 × 4 × 1024, dense, linear BN, ReLU, 3 × 3 conv, 3 Tanh (b) Feature encoder Tη x ∈ [−1, 1] ReLU Global sum pooling h = 1024 → m, dense, linear, NormalizeResBlock up 512
ResBlock up 256
ResBlock up 128
ResBlock up 64
64×64×3
ResBlock down 128
ResBlock down 256
ResBlock down 512
ResBlock down 1024
Table 11 :
11ResNet architecture for the LSUN-128 dataset. (a) Generator G θ ∈ R 128 ∼ N (0, 1) 128 → 4 × 4 × 1024, dense, linear ResBlock up 1024 BN, ReLU, 3 × 3 conv, 3 Tanh (b) Feature encoder Tη x ∈ [−1, 1] ReLU Global sum pooling h = 1024 → m, dense, linear, NormalizeResBlock up 512
ResBlock up 256
ResBlock up 128
ResBlock up 64
128×128×3
ResBlock down 128
ResBlock down 256
ResBlock down 512
ResBlock down 1024
ResBlock 1024
Table 12 :
12ResNet architecture for the CelebA-HQ dataset. (a) Generator G θ ∈ R 128 ∼ N (0, 1) 128 → 4 × 4 × 1024, dense, linear ResBlock up 1024 BN, ReLU, 3 × 3 conv, 3 Tanh (b) Feature encoder Tη x ∈ [−1, 1] ReLU Global sum pooling h = 1024 → m, dense, linear, NormalizeResBlock up 512
ResBlock up 512
ResBlock up 256
ResBlock up 128
ResBlock up 64
256×256×3
ResBlock down 128
ResBlock down 256
ResBlock down 512
ResBlock down 512
ResBlock down 1024
ResBlock 1024
AcknowledgmentsThe authors acknowledge the support of NSF IIS-1812699, the APX 2019 project sponsored by the Office of the Vice President for Research at The University of Texas at Austin, the support of a gift fund from ByteDance Inc., and the Texas Advanced Computing Center (TACC) for providing HPC resources that have contributed to the research results reported within this paper.
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FFHQ (256x256). FFHQ (256x256).
. Ffhq, FFHQ (1024x1024).
Analogous plot to Fig. 6: additional high-resolution samples. Figure. 17Figure 17: Analogous plot to Fig. 6: additional high-resolution samples.
|
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"https://github.com/pytorch/examples/tree/master/dcgan",
"https://github.com/ishansd/swg",
"https://github.com/mbinkowski/MMD-GAN",
"https://github.com/pfnet-research/sngan_projection",
"https://github.com/NVlabs/stylegan2."
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[
"Isospin Effect on the Process of Multifragmentation and Dissipation at Intermediate Energy Heavy Ion Collisions",
"Isospin Effect on the Process of Multifragmentation and Dissipation at Intermediate Energy Heavy Ion Collisions"
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"Jian-Ye Liu \nCCAST (World Lab.)\nP.O.Box 8730100080Beijing\n\nCenter of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator Lanzhou 730000\nP. R. China\n\nInstitute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China\n",
"Yan-Fang Yang \nInstitute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China\n",
"Wei Zuo \nCCAST (World Lab.)\nP.O.Box 8730100080Beijing\n\nCenter of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator Lanzhou 730000\nP. R. China\n\nInstitute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China\n",
"Shun-Jin Wang \nCCAST (World Lab.)\nP.O.Box 8730100080Beijing\n\nCenter of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator Lanzhou 730000\nP. R. China\n\nInstitute of Modern Physics\nsouthwest Jiaotong University\n610031ChenduP. R. China\n",
"Qiang Zhao \nInstitute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China\n",
"Wen-Jun Guo \nInstitute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China\n",
"Bo Chen \nInstitute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China\n"
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[
"CCAST (World Lab.)\nP.O.Box 8730100080Beijing",
"Center of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator Lanzhou 730000\nP. R. China",
"Institute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China",
"Institute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China",
"CCAST (World Lab.)\nP.O.Box 8730100080Beijing",
"Center of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator Lanzhou 730000\nP. R. China",
"Institute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China",
"CCAST (World Lab.)\nP.O.Box 8730100080Beijing",
"Center of Theoretical Nuclear Physics\nNational Laboratory of Heavy Ion Accelerator Lanzhou 730000\nP. R. China",
"Institute of Modern Physics\nsouthwest Jiaotong University\n610031ChenduP. R. China",
"Institute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China",
"Institute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China",
"Institute of Modern Physics\nChinese Academy of Sciences\nP. O. Box 31 Lanzhou 730000P. R. China"
] |
[] |
In the simulation of intermediate energy heavy ion collisions by using the isospin dependent quantum molecular dynamics, the isospin effect on the process of multifragmentation and dissipation has been studied. It is found that the multiplicity of intermediate mass fragments N imf for the neutron-poor colliding system is always larger than that for the neutron-rich system, while the quadrupole of single particle momentum distribution Q zz for the neutron-poor colliding system is smaller than that of the neutron-rich system for all projectile-target combinations studied at the beam energies from about 50MeV/nucleon to 150MeV/nucleon. Since Q zz depends strongly on isospin dependence of in-medium nucleon-nucleon cross section and weakly on symmetry potential at the above beam energies, it may serve as a good probe to extract the information on the in-medium nucleon-nucleon cross section. The correlation between the multiplicity N imf of intermediate mass fragments and the total numer of charged particles N c has the behavior similar to Q zz , which can be used as a complementary probe to the in-medium nucleon-nucleon cross section.PACs Number(s): 25·70·pg, 02·70· Ns, 24·10·Lx
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10.1103/physrevc.63.054612
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"https://arxiv.org/pdf/nucl-th/0102052v1.pdf"
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nucl-th/0102052
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a6acf192d4e686253f6d505d2d6ed89b6ec8af30
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Isospin Effect on the Process of Multifragmentation and Dissipation at Intermediate Energy Heavy Ion Collisions
arXiv:nucl-th/0102052v1 22 Feb 2001
Jian-Ye Liu
CCAST (World Lab.)
P.O.Box 8730100080Beijing
Center of Theoretical Nuclear Physics
National Laboratory of Heavy Ion Accelerator Lanzhou 730000
P. R. China
Institute of Modern Physics
Chinese Academy of Sciences
P. O. Box 31 Lanzhou 730000P. R. China
Yan-Fang Yang
Institute of Modern Physics
Chinese Academy of Sciences
P. O. Box 31 Lanzhou 730000P. R. China
Wei Zuo
CCAST (World Lab.)
P.O.Box 8730100080Beijing
Center of Theoretical Nuclear Physics
National Laboratory of Heavy Ion Accelerator Lanzhou 730000
P. R. China
Institute of Modern Physics
Chinese Academy of Sciences
P. O. Box 31 Lanzhou 730000P. R. China
Shun-Jin Wang
CCAST (World Lab.)
P.O.Box 8730100080Beijing
Center of Theoretical Nuclear Physics
National Laboratory of Heavy Ion Accelerator Lanzhou 730000
P. R. China
Institute of Modern Physics
southwest Jiaotong University
610031ChenduP. R. China
Qiang Zhao
Institute of Modern Physics
Chinese Academy of Sciences
P. O. Box 31 Lanzhou 730000P. R. China
Wen-Jun Guo
Institute of Modern Physics
Chinese Academy of Sciences
P. O. Box 31 Lanzhou 730000P. R. China
Bo Chen
Institute of Modern Physics
Chinese Academy of Sciences
P. O. Box 31 Lanzhou 730000P. R. China
Isospin Effect on the Process of Multifragmentation and Dissipation at Intermediate Energy Heavy Ion Collisions
arXiv:nucl-th/0102052v1 22 Feb 2001isospin effectmultifragmentationdissipationnucleon-nucleon cross sec- tion 1
In the simulation of intermediate energy heavy ion collisions by using the isospin dependent quantum molecular dynamics, the isospin effect on the process of multifragmentation and dissipation has been studied. It is found that the multiplicity of intermediate mass fragments N imf for the neutron-poor colliding system is always larger than that for the neutron-rich system, while the quadrupole of single particle momentum distribution Q zz for the neutron-poor colliding system is smaller than that of the neutron-rich system for all projectile-target combinations studied at the beam energies from about 50MeV/nucleon to 150MeV/nucleon. Since Q zz depends strongly on isospin dependence of in-medium nucleon-nucleon cross section and weakly on symmetry potential at the above beam energies, it may serve as a good probe to extract the information on the in-medium nucleon-nucleon cross section. The correlation between the multiplicity N imf of intermediate mass fragments and the total numer of charged particles N c has the behavior similar to Q zz , which can be used as a complementary probe to the in-medium nucleon-nucleon cross section.PACs Number(s): 25·70·pg, 02·70· Ns, 24·10·Lx
Introduction
In recent years, the establishment of radioactive beam facilities at many laboratories over the world and the use of radioactive beams with large neutron or proton excess have offered an excellent opportunity to investigate the isospin-dependence of heavy ion collision (HIC) dynamics [1−4] . This kind of study has made it possible to obtain the information about the equation of state (EOS) of asymmetric nuclear matter ranging from symmetric nuclear matter to pure neutron matter and the information on isospin-dependence of in-medium nucleon-nucleon (N-N) cross section, which are significantly important not only in understanding nuclear properties but also in exploring the explosion mechanism of supernova and the cooling rate of neutron stars.
However two essential ingredients in HIC dynamics, the symmetry potential of the mean field and the isospin-dependent in-medium N-N cross section, have not been well determined so far. Recently, Li et al. [1,5,6] made use of the isospin dependent transport theory to investigate nuclear symmetry energy and showed that the rate of pre-equilibrium neutron-proton emitted in intermediate energy HIC is sensitive to the density dependence of nuclear symmetry potential, but insensitive to the incompressibility of symmetric nuclear matter and the in-medium N-N cross section. Also, R.Pak and Bao-An Li et al. have found that the isospin dependence of collective flow and balance energy are mainly originated from the isospin-dependent in-medium N-N cross section [5], [7−11] . Recently we have found that nuclear stopping can be used as a good probe for exploring the in-medium N-N cross section in HIC in the beam energy ranging from above Fermi energy to 150MeV/nucleon. However, it is still not clear how the nuclear stopping depends on the neutron-proton ratio of the colliding system [12] .
In viewing that little information is known about the in-medium N-N cross section and its isospin dependence up to now, it is thus very desirable to find an efficient way to gain such kind of knowledge.
It is known that many intermediate mass fragments (IMF) are emitted in the process of intermediate energy HIC and that the element distribution and fragment multiplicity indicate a strong correlation between the multiplicity of intermediate mass fragments N imf and the total number of charged particles N c [7] . The question is whether the multifragmentation and dissipation, especially the N imf and Q zz are sensitive to the neutron-proton ratio of the colliding system. To answer this question, we have investigated the isospin effect on the process of multifragmentation and dissipation in HIC for the colliding systems with different ratios of neutron to proton by using isospin dependent quantum molecular dynamics (IQMD). To increase the efficiency of detectors and the statistics of N imf and Q zz , the reversible kinetic reactions with heavy projectiles on light targets are suggested to have more intermediate mass fragments emitted towards forward angles.
The calculated results show prominent isospin effect for the multifragmentation N imf and the quadrupole of single particle momentum distribution Q zz for the colliding systems with different neutron or proton excesses. The multiplicity of intermediate mass fragments N imf of neutron-poor systems is always larger than that of neutronrich systems, while the quadrupole of single particle momentum distribution Q zz for neutron-poor systems is smaller than that of neutron-rich systems at the beam energies from 50MeV/nucleon to 150MeV/nucleon for all the reversible colliding systems studied here. The mechanism responsible for the above isospin effect can be found from the fact that the mean N-N cross section for a neutron-poor system is larger than that of the corresponding neutron-rich system with the same masses of projectile and target. The calculated results also show that the correlation between the multiplicity of intermediate mass fragments N imf and the total number of charged particles N c , depends strongly on the isospin dependence of in-medium N-N cross section and weakly on the symmetry potential in the chosen beam energy region. In Ref. [12], it is found that Q zz is very sensitive to the isospin dependence of in-medium N-N cross section and insensitive to the symmetry potential. In the present paper, the calculations show that the conclusion drawn in Ref. [12] about Q zz remains true for the reversible kinetic reactions with heavy projectiles on light targets, but it is found that Q zz increases slightly as increasing the neutron-proton ratio of the colliding system. Therefore, Q zz can be a good probe for extracting information on in-medium N-N cross section in HIC in the beam energies ranging from above Fermi energy to 150MeV/nucleon [12] , while the correlation between N imf and N c may serve as a complementary one.
Theoretical Model
The dynamics of intermediate energy HIC described by QMD [13−14] contains three ingredients: density dependent mean field, in-medium N-N cross section and Pauli blocking. To describe isospin effect appropriately, QMD should be modified properly: the density dependent mean field should contain correct isospin terms including symmetry energy and Coulomb potential, the in-medium N-N cross section should be different for neutron-neutron ( proton-proton ) and neutron-proton collisions, and the Pauli blocking should be counted by distinguishing neutrons and protons. In addition, the initial condition of the ground state of two colliding nuclei should also contain isospin information. In the present calculation, the ground state of each colliding nucleus is prepared by using the initial code of IQMD, according to its density distribution obtained from the Skyrme-Hatree-Fock calculation with the parameter set of SKM * [15] .
The interaction potential is as follows,
U(ρ) = U Sky + V c (1 − τ z ) + U sym + V yuk + U M DI + U P auli(1)
U Sky is the density-dependent Skyrme potential,
U Sky = α( ρ ρ 0 ) + β( ρ ρ 0 ) γ(2)
V c is Coulomb potential. U Y uk is the Yukawa potential [13] ,
U Y uk j = t 3 i =j e L/m 2 r ij /2m {e −r ij /m [1 − Φ( √ L/m − r ij /2 √ L)]− e r ij /m [1 − Φ( √ L/m + r ij /2 √ L)]}(3)
where Φ is the error function. U M DI is the momentum dependent interaction ( MDI ) [16] ,
U M DI = t 4 ln 2 [t 5 ( − → p 1 − − → p 2 ) 2 + 1] ρ ρ 0(4)
U P auli is the Pauli potential [17−18] ,
U P auli = V p {h p 0 q 0 ) 3 exp(− ( − → r i − − → r j ) 2 2q 2 0 − ( − → p i − − → p j ) 2 2p 2 0 }δ p i p j(5)
δ p i p j = 1 for neutron-neutron or proton-proton 0 for neutron-proton U sym is the symmetry potential. In the present paper, three different kinds of U sym have been used [1] ,
U sym 1 = cF 1 (u)δτ z (6) U sym 2 = cF 2 (u)δτ z + 1 2 cF 2 (u)δ 2 (7) U sym 3 = cF 3 (u)δτ z − 1 4 cF 3 (u)δ 2 (8) with τ z = 1 for neutron −1 for proton
Here c is the strength of the symmetry potential, taking the value of 32MeV. F 1 (u) = u,
F 2 (u) = u 2 and F 3 (u) = u 1/2 , u ≡ ρ ρ 0 , δ is the relative neutron excess δ = ρn−ρp ρn+ρp = ρn−ρp ρ .
ρ and ρ 0 , ρ n and ρ p are the nuclear density and its normal value, neutron density and and proton density, respectively. The parameters of the interaction potentials are given in table 1. Table 1. The parameters of the interaction potentials It is worth mentioning that recent studies of collective flow in HIC at intermediate energies have indicated a reduction of the in-medium N-N cross sections [19−21] . An empirical expression of the in-medium N-N cross section [21] is used:
α β γ t 3 m t 4 t 5 V p p 0 q 0 (MeV) (MeV) (MeV) (fm) (MeV) (MeV −2 ) (MeV) (MeV/c) (fm) −σ med N N = (1 + α ρ ρ 0 )σ f ree N N (9)
with the parameter α ≈ −0.2 which has been found to better reproduce the flow data [19−20] . Here σ f ree N N is the experimental N-N cross section [22] . The neutron-proton cross section is about 3 times larger than the proton-proton or neutron-neutron cross section below 300 MeV. Q zz will be used to describe the nuclear stopping:
Q ZZ = A i (2P z (i) 2 − P x (i) 2 − P y (i) 2 ).(10)
In order to make the isospin effect on the multifragmentation process in HIC more prominent, comparable study is carried out for two pairs of reversible reaction systems.
For each pair of comparable reaction systems the same mass of heavy projectiles and light targets, the same incident energy and the same impact parameter have been used Ca with the neutron-proton ratios 1.42 and 1.07. Hence the differences of N imf and Q zz for each pair of the comparable colliding systems are mainly due to the isospin effect on the process of multifragmentation and dissipation.
By means of the modified isospin-dependent coalescence model [23] , we construct clusters within which the particle relative momentum is smaller than p 0 = 300MeV/c and the relative distance is smaller than R 0 = 3.5fm. To avoid the nonphysical clusters, the restructured aggregation model [24] is used until no nonphysical cluster is produced.
Results and Discussions
N imf and N c are calculated event by event with the charge number of IMF from Z=3 to 13 for the heavy colliding systems, and Z=3 to 8 for the medium mass colliding systems. As is well known that nuclear reaction mechanism and reaction yields are sensitive to both impact parameter and incident energy. Denoted by the same lines as in Fig.1 Fig.1, at small impact parameters, the lines labeled 2 for the neutron-poor colliding systems are always above the corresponding lines labeled 1 for the neutron-rich systems.
The isospin effect on the process of multifragmentation and dissipation in HIC
The difference between them disappears gradually as increasing impact parameter. MeV/nucleon at b=0 fm for the above two pairs of colliding systems with the line labels as in Fig.1. The relative locations between the two lines in each panel of the figure are also similar to those in Fig.1, namely, N imf for the two neutron-poor colliding systems 120 Xe + 40 Ca and 76 Kr + 40 Ca are larger than those for the corresponding neutronrich systems 120 Cd + 40 Ar and 76 Zn + 40 Ar for the beam energy above 50MeV/nucleon. However, as the beam energy decreases to below 50MeV/nucleon the collision dynamics is governed by both the mean field and the nucleon-nucleon collisions. In this case, the difference between the two lines of the colliding systems in each pair vanishes gradually.
From Figs.1, 2, and 3 we can see that the intermediate mass fragment multiplicity N imf for the neutron-poor colliding system are larger than that of the corresponding neutron-rich system with the same mass projectile on the same mass target and with the same entrance channel conditions except that the ratios of neutron to proton of the colliding systems are different.
The correlation between N imf and N c for the central collision of the above two pairs of colliding systems with the same beam energies as above is shown in Fig.4. It is seen from the figure that the correlation between N imf and N c for the two pairs of colliding systems displays a clear isospin effect, i.e., the N imf -N c correlation for the neutron-poor systems is different from that of the neutron-rich systems. Considering the total yield of N imf (namely the integral area of each curve in the figure), one can reach the same conclusion as from Fig.1, Fig.2, and Fig.3. This means that the N imf for neutron-poor systems (solid line), on average, is larger than that for the corresponding neutron-rich systems (dash line). The difference comes mainly from the isospin effect on multifragmentation in intermediate energy HIC.
The mechanism of the above fragmentation process can be understood as follows.
Experimentally the free neutron-proton cross section is about three times larger than the free neutron-neutron or proton-proton cross section below 300 MeV. The mean N-N cross section is defined as: (11) where N np , N nn and N pp are the collision numbers for neutron-proton, neutron-neutron, and proton-proton, respectively, and N = N np + N nn + N pp . σ np , σ nn , and σ pp are the free N-N cross sections for neutron-proton, neutron-neutron, and proton-proton, respectively. In general, σ nn = σ pp = σ. Because the total collision numbers for two colliding systems with the same mass projectile and the same mass target are the same, and the number of neutron-proton collisions for neutron-poor colliding system is larger than that of the neutron-rich colliding system, the mean total N-N cross section < σ > of neutron-poor system is thus larger than that of neutron-rich system. Due to the effect of Pauli blocking, the effective collision numbers become smaller. But after considering the Pauli blocking, the above conclusion remains unchanged. Therefore, the neutron-poor system will have more effective N-N collisions and lead to stronger compression-expansion, resulting in a large number of multifragmentation N imf for the neutron-poor system in comparison to the neutron-rich system in the above 50MeV region where the N-N collisions are dominant.
< σ >= N np σ np + (N nn + N pp )σ pp N np + N pp + N nn = (3N np + N nn + N pp ) N np + N nn + N pp σ pp = (1 + 2N np N )σ
In ) at E=80, 100, 150MeV/nucleon and b = 0.0fm. In the figure, Q zz for the neutron-poor system is always smaller than that of the corresponding neutron-rich system. Smaller Q zz indicates larger dissipation of the initial longitudinal collective motion into the internal chaotic motion and the subsequent thermalization of the system.
A good probe and a complementary probe to in-medium N-N cross section
Since the reaction dynamics of HIC is mainly governed by both nuclear EOS and in-medium N-N cross section, to understand the collision dynamics in details, both ingredients should be studied carefully. As is well known that the effects of both ingredients are usually mixed in the dynamics and the main uncertainty of the information about the nuclear EOS extracted from HIC is due to our poor knowledge of the N-N cross section in medium. If one can find an experimental probe which can distinguish the contribution of EOS from that of in-medium N-N cross section, that will be very desirable. In this paper we have found that Q zz may serve as a good probe to in-medium N-N cross section.
In Fig.6 is given Q zz as a function of neutron-proton ratio for seven colliding sys- It is clear to see that Q zz depends strongly on the isospin dependence of in-medium N-N cross section and weakly on the symmetry potential ( namely, line 1 is located near lines 3 and 4, but far away from line 2 in Fig.6), though Q zz increases slightly with increasing neutron-proton ratio of asymmetry colliding system. In this case, Q zz is a good probe for extracting information on the isospin dependence of in-medium N-N cross section. We have discussed this in more details in Ref. [12], but it is not clear in Ref. [12] how Q zz depends on the neutron-proton ratio of colliding system. In addition, we shall report a complementary probe for extracting the information on in-medium N-N cross section: the correlation between the multiplicity of intermediate mass fragments N imf and the total number of charged particles N c , based on the fact that it is also sensitive to the in-medium N-N cross section and insensitive to the symmetry potential in the chosen energy region. To study the contributions to N imf from different ingredients separately, we consider four cases as the same as in Fig.6.
In Fig.7 time. This implies that the isospin effect of in-medium N-N cross section on N imf is more important than that of the symmetry potential in the energy region studied here.
As is well known that nuclear reaction products and reaction mechanism sensitively depend on impact parameter. Fig.8 shows the multiplicity of intermediate mass fragments N imf as a function of impact parameter for the four cases as illustrated in Fig.6 for the reactions 76 Zn + 40 Ar at E = 80MeV/nucleon ( left panel ) and 120 Xe + 40 Ca at E = 100MeV/nucleon ( right panel ). As in Fig.6, lines 1 in the figures locate near lines 3 and 4, but far away from lines 2 at small impact parameters. This indicates again that the isospin effect of in-medium N-N cross section on N imf is larger in comparison with that of symmetry potential. With increasing impact parmeter this isospin effect disappears gradually.
The correlations between N imf and N c for the four cases are plotted in Fig.9 for the reactions 76 Zn + 40 Ar at E = 80MeV/nucleon ( left panel ) and 120 Xe + 40 Ca at E = 100MeV/nucleon ( right panel ). The behaviors of the four kinds of lines in each panel are similar to those in Fig.6, namely, the integral area of the curve 1 is always close to those of the curves 3 and 4, but larger than that of the curve 2. From Fig.7, Fig.8, and Fig.9 one can see that the isospin effect of in-medium N-N cross section on the correlation between N imf and N c is more important than that of symmetry potential. Here it should be stressed that we have to choose the incident energy for certain asymmetric colliding system carefully to get the above feature, namely, in this energy region the N-N collisions should be dominant. In this case we may conclude that the correlation between the multiplicity of intermediate mass fragments N imf and the total number of charged particles N c can be used as a complementary probe to the isospin dependence of the in-medium N-N cross section in HIC.
Summary and conclusions
Starting from the simulation of the intermediate energy HIC by using IQMD, the calculated results have shown prominent isospin effects on the process of multifragmentation and dissipation, i.e., the intermediate mass fragment multiplicity N imf for a neutron-poor colliding system is always larger than that of the corresponding neutronrich system, while the quadrupole of single particle momentum distribution Q zz for a neutron-poor system is smaller than that of the neutron-rich system with the same masses of projectile and target, and the same entrance channel conditions for small impact parameters. We also can see that Q zz increases slight with increasing ratio of neutron to proton in the colliding system.
Fig. 1
1shows the time evolution of N imf for the central reactions 76 Zn + 40 Ar (line 1) and 76 Kr + 40 Ca (line 2) at E=80 MeV/nucleon (left panel), and 120 Cd + 40 Ar (line 1) and 120 Xe + 40 Ca (line 2) at E=100 MeV/nucleon (right panel). From Fig.1 one can see that N imf for the two neutron-poor colliding systems 120 Xe+ 40 Ca and 76 Kr + 40 Ca are larger than those for the neutron-rich systems 120 Cd + 40 Ar and 76 Zn + 40 Ar. This difference are mainly due to the isospin effect on the multifragmentation because other reaction conditions, except the ratios of neutron to proton, are the same.
Fig. 3 shows
3N imf as a function of beam energy from 15 MeV/nucleon to 200
Fig. 5
5is plotted the time evolution of the quadrupole of single particle momentum distribution Q zz for the reactions 76 Zn + 40 Ar ( dash line ) and 76 Kr + 40 Ca ( solid line ) ( bottom panel), 120 Cd+ 40 Ar ( dash line ) and 120 Xe+ 40 Ca ( solid line)(top panel
E=100MeV/nucleon and b=0.0 fm for four cases: (1) the symmetry potential U sym1 being employed and in-medium N-N cross section σ med N N being isospin-dependent, namely, U sym 1 +σ iso ; (2) U sym 1 and N-N cross section σ med N N being isospin-independent, denoted by U sym 1 + σ noiso ; (3) U sym 2 and σ med N N being isospin-dependent, denoted by U sym 2 + σ iso ; (4) U sym 3 and σ med N N being isospin-dependent, denoted by U sym 3 + σ iso . In fig.6, lines labeled 1, 2, 3 and 4 correspond to the above four cases.
is plotted the time evolution of N imf for two different asymmetric colliding systems, 76 Zn + 40 Ar at E = 80MeV/nucleon and b = 0.0fm (left panel), 120 Xe+ 40 Ca at E = 100MeV/nucleon and b = 0.0fm (right panel). It is noted that lines 1 are always located near lines 3 and 4, but far away from lines 2 with increasing colliding
FromFigure 1 :
1the theoretical simulation, it is clear to see that Q zz depends strongly on the isospin dependence of in-medium N-N cross section and weakly on the symmetry potential in the beam energies ranging from about 50 MeV/nucleon to 150MeV/nucleon. And the correlation between N imf and N c has the same properties as Q zz in the chosen energy region. So we would suggest that Q zz may serve as a good probe and N imf a complementary probe for extracting the information on the isospin dependent NThe time evolution of N imf for the central collisions 120 Cd + 40 Ar (line 1) and 120 Xe + 40 Ca (line 2) at E = 100MeV/nucleon (right panel), 76 Zn + 40 Ar (line 1) and 76 Kr + 40 Ca (line 2) at E = 80MeV/nucleon (left panel).
Figure 2 :Figure 3 :
23The multiplicity of intermediate mass fragments N imf as a function of impact parameter for the reactions 120 Cd + 40 Ar ( line 1 ) and 120 Xe + 40 Ca ( line 2 ) at E = 100MeV/nucleon ( left panel ), and the reactions 76 Zn + 40 Ar ( line 1 ) and 76 Kr + 40 Ca ( line 2 ) at E = 80MeV/nucleon ( right panel). The N imf as a function of incident energies from 15MeV/nucleon to 200MeV/nucleon at b=0 fm for the reactions 120 Cd + 40 Ar ( line 1 ) and 120 Xe + 40 Ca ( line 2 ) at E = 100MeV/nucleon ( left panel ), and the reactions 76 Zn + 40 Ar ( line 1 ) and 76 Kr + 40 Ca ( line 2 ) at E = 80MeV/nucleon ( right panel ).
Figure 4 :
4The correlations between N imf and N c for the reactions 120 Cd + 40 Ar ( dash line ) and 120 Xe + 40 Ca ( solid line ) at E = 100MeV/nucleon ( left panel ), and the reactions 76 Zn+ 40 Ar ( dash line ) and 76 Kr+ 40 Ca ( solid line ) at E = 80MeV/nucleon ( right panel ).
Figure 5 :
5The time evolution of the quadrupole of single particle momentum distribution Q zz for the reactions 76 Zn + 40 Ar ( dash line ) and 76 Kr + 40 Ca ( solid line ) ( bottom panel ), and the reactions 120 Cd + 40 Ar ( dash line ) and 120 Xe + 40 Ca ( solid line )( top panel ) at E=80,100 and 150MeV/nucleon and b = 0.0fm.
Figure 6 :
6The quadrupole of single particle momentum distribution Q zz as a function of the neutron-proton ratio for seven colliding systems 76 Kr + 40 Ca, 120 Xe + 40 Ca, 64 Ni + 40 Ar, 86 Kr + 40 Ar, 76 Zn + 40 Ar, 85 Ge + 40 Ar, and 74 Ni + 47 Ar at E = 100MeV/nucleon and b = 0.0fm for the four cases ( see text ).
Figure 7 :
7The time evolution of N imf for the reactions 76 Zr + 40 Ar at E = 80MeV/nucleon and b = 0.0fm ( left panel ), 120 Xe + 40 Ca at E = 100MeV/nucleon and b = 0.0fm ( right panel ). Lines 1, 2, 3, and 4 correspond to the four cases as inFig.6.
Figure 8 :Figure 9 :
89The multiplicity of intermediate mass fragments N imf as a function of impact parameter for the reactions 76 Zn + 40 Ar at E = 80MeV/nucleon ( left panel ) and 120 Xe + 40 Ca at E = 100MeV/nucleon ( right panel ) in the four cases as inFig.6. The correlations between N imf and N c for the reactions 76 Zn + 40 Ar at E = 80MeV/nucleon ( left panel ) and 120 Xe + 40 Ca at E = 100MeV/nucleon (right panel) for the four cases as inFig.6.
,
Fig.2 depicts the multiplicity of intermediate mass fragments N imf as a function of impact parameter for the two pairs of colliding systems, 120 Ca. The incident energy E is 100 MeV/nucleon for the heavy systems (left panel), and 80 Mev/nucleon for the medium systems (right panel). As in48 Cd+ 40
18 Ar and 120
54 Xe+ 40
20 Ca,
76
30 Zn+ 40
18 Ar and 76
36 Kr + 40
20
tems 76 Kr + 40 Ca, 120 Xe + 40 Ca, 64 Ni + 40 Ar, 86 Kr + 40 Ar, 76 Zn + 40 Ar, 85 Ge + 40 Ar,74 Ni + 47 Ar with the neutron-proton ratios 1.07, 1.16, 1.26, 1.33, 1.42, 1.5 and 1.56 at
AcknowledgmentThis work was supported in part by 973 Project Grant No.G2000077400, 100 person Project of the Chinese Academy of Sciences, the National Natural Foundation of China under Grants No. 19775057 and No. 19775020, No. 19847002, No. 19775052 and KJ951-A1-410, and by the Foundation of the Chinese Academy of Sciences
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|
[] |
[
"PHOTON-PHOTON TOTAL INELASTIC CROSS-SECTION",
"PHOTON-PHOTON TOTAL INELASTIC CROSS-SECTION"
] |
[
"R M Godbole ",
"G Pancheri ",
"\nA. CORSETTI Physics Department\nCenter for Theoretical Studies\nNortheastern University\nBostonUSA\n",
"\nINFN -Laboratori Nazionali di Frascati\nIndian Institute of Science\nBangalore, FrascatiIndia and, Italy\n"
] |
[
"A. CORSETTI Physics Department\nCenter for Theoretical Studies\nNortheastern University\nBostonUSA",
"INFN -Laboratori Nazionali di Frascati\nIndian Institute of Science\nBangalore, FrascatiIndia and, Italy"
] |
[] |
We discuss predictions for the total inelastic γγ cross-section and their model dependence on the input parameters. We compare results from a simple extension of the Regge Pomeron exchange model as well as predictions from the eikonalized mini-jet model with recent LEP data.
| null |
[
"https://arxiv.org/pdf/hep-ph/9707360v1.pdf"
] | 15,246,894 |
hep-ph/9707360
|
b6f236b73ea66766e537416ef8915208f6d7bf70
|
PHOTON-PHOTON TOTAL INELASTIC CROSS-SECTION
Jul 1997 April 7, 2018 May 1997
R M Godbole
G Pancheri
A. CORSETTI Physics Department
Center for Theoretical Studies
Northeastern University
BostonUSA
INFN -Laboratori Nazionali di Frascati
Indian Institute of Science
Bangalore, FrascatiIndia and, Italy
PHOTON-PHOTON TOTAL INELASTIC CROSS-SECTION
Jul 1997 April 7, 2018 May 1997arXiv:hep-ph/9707360v1 16 * Talk presented at Photon'97, Egmond aan Zee, 1
We discuss predictions for the total inelastic γγ cross-section and their model dependence on the input parameters. We compare results from a simple extension of the Regge Pomeron exchange model as well as predictions from the eikonalized mini-jet model with recent LEP data.
It is by now established that all total cross-sections, including photoproduction, rise as the c.m. energy of the colliding particles increases. So far a successful description of total cross-sections is obtained in the Regge/Pomeron exchange model [1], in which a Regge pole and a Pomeron are exchanged and total cross-sections are seen to first decrease and subsequently rise according to the expression σ tot ab = Y ab s −η + X ab s ǫ where ǫ and η are related to the intercept at zero of the leading Regge trajectory and of the Pomeron, respectively η ≈ 0.5 and ǫ ≈ 0.08. This parametrization applies successfully [1] to photoproduction, as shown in Fig. 1, and to the lower energy data on γγ [2]. Assuming the hypothesis of factorization at the poles, one can make a prediction for γγ total inelastic crosssection, using Y 2 ab = Y aa Y bb X 2 ab = X aa X bb and extracting the coefficients X and Y from those for the fit to photoproduction and hadron-hadron data. In particular, using for η and ǫ the average values from the Particle Data Group compilation [3] and averaging among the pp andpp coefficients, one can have a first check of the factorization hypothesis. Noticing that the coefficient Y from photoproduction data has a large error and that prediction from the Regge/Pomeron exchange model refer to total cross-sections rather than the inelastic ones, these predictions can be enlarged into a band as shown in Fig.2.
An alternative model for the rise of all total cross-sections, relies on hard parton-parton scattering. It was suggested [4] that hard collisions between elementary constituents of the colliding hadrons, the partons, could be responsible for this rise which starts around √ s ≥ 10÷20 GeV . This suggestion has subsequently evolved into mini-jet models [5], whose eikonal formulation satisfies unitarity while embodying the concepts of rising total cross-sections with rising jet cross-sections. For processes involving photons, the model has to incorporate [6] the hadronisation probability P had γ for the photon to fluctuate itself into a hadronic state. The eikonalised mini-jet cross-section is then
σ inel ab = P had ab d 2 b[1 − e n(b,s) ](1)
with the average number of collisions at a given impact parameter b given by
n(b, s) = A ab (b)(σ sof t ab + 1 P had ab σ jet ab )(2)
In eqs. (1,2), P had ab is the probability that the colliding particles a, b are both in a hadronic state, A ab (b) describes the transverse overlap of the partons in the two projectiles normalised to 1, σ sof t ab is the non-perturbative part of the cross-section from which the factor of P had ab has already been factored out and σ jet ab is the hard part of the cross-section. The basic statement of the mini-jet model for total cross-sections is that the rise in σ jet ab drives the rise of σ inel ab with energy. Letting
P had γp = P had γ and P had γγ ≈ (P had γ ) 2(3)
one can extrapolate the model from photoproduction to photon-photon collisions. The issue of total γγ cross-sections assumes an additional significance in view of the large potential backgrounds that Beamstrahlung photons could cause at future Linear Colliders [7]. Because the hadronic structure of the photon involves both a perturbative and nonperturbative part, it has been proposed [2,8] to use a sum of eikonalized functions instead of eq.(1) in processes involving photons. The predictions of the eikonalised mini-jet model for photon induced processes thus depend on 1) the assumption of one or more eikonals 2) the hard jet cross-section σ jet = p tmin d 2σ dp 2 t dp 2 t which in turn depends on the minimum p t above which one can expect perturbative QCD to hold viz. p tmin and the parton densities in the colliding particles a and b, 3) the soft cross-section σ sof t ab 4) the overlap function A ab (b), defined as
A(b) = 1 (2π) 2 d 2 qF 1 (q)F 2 (q)e i q· b(4)
where F is the Fourier transform of the b-distribution of partons in the colliding particles and 5) last, but not the least, P had ab . In this note we shall restrict ourselves to a single eikonal. The hard jet cross-sections are calculated in LO perturbative QCD and use photonic parton densities GRV [9] calculated to the leading order. We determine σ sof t γγ from σ sof t γp which in turn is determined by a fit to the photoproduction data. From inspection of the photoproduction data, one can assume that σ sof t should contain both a constant and an energy decreasing term. Following the suggestion [8]
σ sof t γp = σ 0 + A √ s + B s(5)
we then calculate values for σ 0 , A and B from a best fit [10] to the low energy photoproduction data, starting with the Quark Parton Model ansatz σ 0 γp ≈ 2 3 σ 0 pp . For γγ collisions, we repeat the QPM suggestion and propose σ sof t γγ = 2 3 σ sof t γp , i.e. σ 0 γγ = 20.8mb, A γγ = 6.7 mb GeV 3/2 , B γγ = 25.3 mb GeV (6) Whereas the effect of the uncertainties in the above three quantities on the predictions of the inelastic photoproduction and γγ cross-sections has been studied in literature to a fair extent [2,8,11] the effect of the other two has not been much discussed. In the original use of the eikonal model, the overlap function A ab (b) of eq.(4) is obtained using for F the electromagnetic form factors. For protons this is given by the dipole expression
F prot (q) = [ ν 2 q 2 + ν 2 ] 2(7)
with ν 2 = 0.71 GeV 2 . For photons a number of authors [8,12], on the basis of Vector Meson Dominance, have assumed the same functional form as for pion, i.e. the pole expression
F pion (q) = k 2 0 q 2 + k 2 0 with k 0 = 0.735 GeV.(8)
There also exists another possibility, i.e. that the b-space distribution of partons is the Fourier transform of their intrinsic transverse momentum distributions [13]. While for the proton this would correspond to use a Gaussian distribution instead of the dipole expression, eq. (7), for the photon one can argue that the intrinsic transverse momentum ansatz [14] would imply the use of a different value of the parameter k o [15] in the pole expression for the form factor. By varying k o one can then explore both the intrinsic transverse distribution case and the form factor cum VMD hypothesis. Notice that the region most important to this calculation is for large values of the parameter b, where the overlap function changes trend, and is larger for smaller k o values. Let us now look at P had γ . This is clearly expected to be O(α em ). Based on Vector Meson Dominance one expects, Although in principle, P had γ is not a constant, for simplicity, we adopt here a fixed value [12] of 1/204, which includes a non-VMD contribution of ≈ 20%. Notice that a fixed value of P had can be absorbed into a redefinition of the parameter k o through a simple change of variables [16].
P had γ = P V M D = V =ρ,ω,φ
Having thus established the range of variability of the quantities involved in the calculation of total inelastic photonic cross sections, we can proceed to compare the predictions of the eikonalized minijet model with data. We use GRV (LO) densities and show the mini-jet result in Fig.1, using the form factor model for A(b), i.e. eq.(4) with k o = 0.735 GeV . In the figures, we have not added the direct contribution, which will slightly increase the cross-section in the 10 GeV region. We observe that it is possible to include the high energy points using GRV densities and p tmin = 2 GeV , but the low energy region would be better described by a smaller p tmin . This is the region where the rise, according to some authors, notably within the framework of the Dual Parton Model, is attributed to the so-called soft Pomeron.
We now apply the same criteria and parameter set used in γp collisions to the case of photon-photon collisions, i.e. P h/γ = 1/204, p tmin = 2 GeV and A(b) from eq.(4). A comparison with γγ data shows that although the value k o = 0.735, corresponding to the pion-factor, is compatible with the low energy data up to 10 GeV [17] within the limits established by the large errors involved, at higher energies [18] the best fit is obtained using a slightly larger value, i.e. k 0 = 1 GeV , and this is the one used in Fig.2. For comparison, we have also added mini-jet model predictions with SAS1 photon densities [19]and predictions (Pomeron/SaS) based on a Pomeron/Regge type parametrization [2].
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|
[] |
[
"Symmetry Protected Topological States of Interacting Fermions and Bosons",
"Symmetry Protected Topological States of Interacting Fermions and Bosons"
] |
[
"Yi-Zhuang You \nDepartment of physics\nUniversity of California\n93106Santa BarbaraCAUSA\n",
"Cenke Xu \nDepartment of physics\nUniversity of California\n93106Santa BarbaraCAUSA\n"
] |
[
"Department of physics\nUniversity of California\n93106Santa BarbaraCAUSA",
"Department of physics\nUniversity of California\n93106Santa BarbaraCAUSA"
] |
[] |
We study the classification of interacting fermionic and bosonic symmetry protected topological (SPT) states. We define a SPT state as whether or not it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short range interactions can reduce the classification of free fermion SPT states, and we demonstrate these results by making connection between fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for all the known SPT states, with or without interaction, then we will generalize our method to SPT states that involve the spatial inversion symmetry. arXiv:1409.0168v1 [cond-mat.str-el]
|
10.1103/physrevb.90.245120
|
[
"https://arxiv.org/pdf/1409.0168v2.pdf"
] | 119,197,627 |
1409.0168
|
01afb928e3249ff5f5f79dd5ae0e47c2f0dea93d
|
Symmetry Protected Topological States of Interacting Fermions and Bosons
Yi-Zhuang You
Department of physics
University of California
93106Santa BarbaraCAUSA
Cenke Xu
Department of physics
University of California
93106Santa BarbaraCAUSA
Symmetry Protected Topological States of Interacting Fermions and Bosons
(Dated: September 2, 2014)
We study the classification of interacting fermionic and bosonic symmetry protected topological (SPT) states. We define a SPT state as whether or not it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short range interactions can reduce the classification of free fermion SPT states, and we demonstrate these results by making connection between fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for all the known SPT states, with or without interaction, then we will generalize our method to SPT states that involve the spatial inversion symmetry. arXiv:1409.0168v1 [cond-mat.str-el]
I. INTRODUCTION
A symmetry protected topological (SPT) state 48 is usually defined as a state with completely trivial bulk spectrum, but nontrivial (e.g. gapless or degenerate) boundary spectrum when and only when the system including the boundary preserves certain symmetry 1,2 . The most well-known SPT states include the Haldane phase of spin-1 chain 3,4 , quantum spin Hall insulator 5,6 , topological insulator [7][8][9] , and topological superconductor such as Helium-3 B-phase. So far all the free fermion SPT states have been well understood and classified in Ref. [10][11][12], and recent studies suggest that interaction may not lead to new SPT states, but it can reduce the classification of fermionic SPT states [13][14][15][16][17][18][19][20] . Unlike fermionic systems, bosonic SPT states do need strong interaction to overcome its tendency to form a Bose-Einstein condensate. Most bosonic SPT states can be classified by symmetry group cohomology 1,2 , Chern-Simons theory 21 and semiclassical non-linear σ-model 22 .
The definition for SPT states we gave above is based on the most obvious phenomenology of the SPT states, and it gives SPT states a convenient experimental signature, which is their boundary state. Indeed, the quantum spin Hall insulator and 3d topological insulator were verified experimentally by directly probing their boundary properties. [23][24][25][26] However, if a SPT state needs certain spatial symmetry, 27 its boundary may be trivial because this spatial symmetry can be explicitly broken by its boundary. In this work we will study SPT states both with and without spatial symmetries, thus in our current work, a SPT state is simply defined as a gapped and nondegenerate state that must be separated from the trivial direct product state defined on the same Hilbert space through one or more bulk phase transitions, as long as the Hamiltonian always preserves certain symmetry.
In this work we study both strongly interacting fermionic and bosonic SPT states. We will deduce the classification of interacting fermionic SPT (iFSPT) states by making connection to bosonic SPT (BSPT) states with the same symmetry, and we will argue that the classification of BSPT states implies the classification of their fermionic counterparts. More specifically, since BSPT states always need strong interaction, their classification tells us how interaction affects the classification of FSPT states. When describing BSPT states, we will adopt the formalism developed in Ref. 22, namely we describe a d-dimensional BSPT state using an O(d + 2) nonlinear sigma model (NLSM) field theory with a topological Θterm, and we only focus on the stable "fixed point" states with Θ = 2πk. Depending on integer k, these fixed points can correspond to either trivial or BSPT state. This formalism fits well with our definition of SPT states: it very naturally tells us whether two "fixed point" states can be connected with or without a phase transition. This is an advantage that we will fully exploit in our work.
We will first demonstrate our method in section II with well-known examples such as 1d Kitaev's Majorana chain with Z T 2 symmetry 13,14 , 2d p ± ip topological superconductor (TSC) with Z 2 symmetry [15][16][17][18] , and 3d topological superconductor 3 He-B with Z T 2 symmetry 19,20 . Previous studies show that although all these states have Z classification without interaction, their classifications will reduce to Z 8 , Z 8 , Z 16 under interaction. We will demonstrate that these interaction-reduced classifications naturally come from the Z 2 classification of 1d Haldane spin chain, 2d Levin-Gu paramagnet, and 3d bosonic SPT state with Z T 2 symmetry. More precisely, the Z 2 classification of the BSPT states give us a necessary condition for interaction reduced classification of their fermionic counterparts. Moreover, the analysis of BSPT states can generate the specific four-fermion interaction that likely gaps out the bulk critical point between free fermion SPT (fFSPT) state and the trivial state in the noninteracting limit. In section III-V we will generalize our method to SPT states that involve the spatial inversion symmetry. By making connection to BSPT states, we will show that interaction reduced classification occurs very generally for inversion SPT states too.
II. SPT STATES WITHOUT SPATIAL SYMMETRY
A. From Kitaev Chain to Haldane Chain
Lattice Model and Bulk Theory
The Kitaev's Majorana chain is a 1d free fermion SPT (fFSPT) state protected by the time-reversal symmetry Z T 2 with T 2 = +1 (symmetry class BDI) 28 . For generality, we consider ν copies of the Majorana chain. The model is defined on a 1d lattice with ν flavors of Majorana fermions χ iα (α = 1, · · · , ν) on each site i,
H ×ν = ν α=1 ij iu ij χ iα χ jα ,(1)
with the bond strength u ij = u(1 + δ(−) i ) alternating along the chain. Each unit cell contains two sites, labeled by A and B, as shown in Fig. 1. The Hamiltonian is invariant under the time-reversal symmetry Z T 2 , T : χ Aα → χ Aα , χ Bα → −χ Bα , i → −i, which flips the sign of the Majorana fermions on the B sublattice followed by the complexed conjugation. Fourier transform to the momentum space and introduce the basis χ kα = (χ kAα , χ kBα ) , the Hamiltonian Eq. (1) becomes
H ×ν = 1 2 ν α=1 k χ −kα 0 −iu * k iu k 0 χ kα ,(2)
with iu k = iu(1 + δ) − iu(1 − δ)e −ik , and the time-reversal symmetry acts as T : χ kα → σ 3 χ −kα . In this paper, we use σ 1 , σ 2 , σ 3 to denote the Pauli matrices. When δ 1, in the long-wave-length limit (k → 0), iu k → u(−k+2iδ), so the low-energy effective Hamiltonian reads
H ×ν = 1 2 ν α=1 dx χ α (i∂ x σ 1 + mσ 2 )χ α .(3)
Here we have set u = 1 and introduced m = 2δ as the topological mass. The time-reversal symmetry operator may be written as T = Kσ 3 , where K −1 iK = −i implements the complex conjugation by flipping the imaginary unit.
On the free fermion level, the 1d BDI class FSPT states are Z classified, 10,11 as indexed by a bulk topological integer
N = dk 4πi Tr σ 3 G(k)∂ k G(k) −1 ,(4)
where G(k) = − χ k χ −k is the fermion Green's function at zero frequency. Given the model in Eq. (2), the topological number N = ν(1 − sgn δ)/2 is identical to the fermion flavor number ν when δ < 0. Every topological number N ∈ Z indexes a distinct fFSPT phase, which, in respect of the symmetry, can not be connected to each other without going through a bulk transition.
Corresponding BSPT state
However with interaction, the classification can be reduced from Z to Z 8 , meaning that eight copies of the Majorana chain (N = 8) can be smoothly connect to the trivial state (N = 0) without closing the bulk gap in the presence of interaction. This interaction reduced classification was discovered in Ref. 13,14. But we will show it again by making connection to the boson SPT (BSPT) states, as our approach can be generalized to higher spacial dimensions.
Let us start by showing that four copies of the Majorana chain (ν = 4) can be connected to the Haldane spin chain, 29 a Z T 2 BSPT state in 1d. In this case, we have four flavors of Majorana fermion per site, denoted χ iα (α = 1, · · · , 4), from which we can define a spin-1/2 object on each site, as 49 One can couple the staggered component of the spin to an O(3) order parameter n i on each site, as H BF = − i (−) i n i · S i . Then the low-energy effective Hamiltonian for four copies of the Majorana chain coupled to the n field reads (which we call fermionic σ-model, or FSM)
S i = (S i1 , S i2 , S i3 ) = χ i (σ 12 , σ 20 , σ 32 )χ i in the basis χ i = (χ i1 , χ i2 , χ i3 , χ i4 ) .H ×4 = 1 2 dx χ h ×4 χ,
h ×4 = i∂ 1 σ 100 + mσ 200 + n 1 σ 312 + n 2 σ 320 + n 3 σ 332 ,
where χ = (χ A , χ B ) . The time-reversal symmetry operator T = Kσ 300 necessarily requires to flip the order parameters n → −n under the time-reversal. Following the calculation in Ref. 30, after integrating out the fermion field χ, we arrive at the effective theory for the boson field n, which is a non-linear σ-model (NLSM) with a topological Θ term at Θ = 2π, given by the following action
S[n] = dτ dx 1 g (∂ µ n) 2 + iΘ 4π abc n a ∂ τ n b ∂ x n c .(6)
g −1 (∂ ν n) 2 describes the remaining dynamics in the bosonic sector. Presumably we work in the large g → ∞ limit, such that the n field is deep in its disordered phase. With Θ = 0, the Hamiltonian of Eq. (6) reads H = dx gL 2 + 1 g (∇ x n) 2 (where L(x) is the canonical conjugate variable of n(x) at each spatial position x), and since g flows to +∞ under coarse-graining, in the longwave-length limit the ground state wave function of this theory is a trivial direct product state |Ω = x |l = 0 31 with a fully gapped and nondegenerate spectrum in the bulk and at the boundary (on each coarse grained spatial point, L 2 = l(l + 1)). However, with Θ = 2π, Eq. (6) describes a non-trivial BSPT state for the n field, and is equivalent 3,4,32 to the Haldane phase of spin-1 chain protected by the spin-flipping time-reversal symmetry T : n → −n. In fact, the spatial boundary of Eq. (6) with Θ = 2π is a (0 + 1)d O(3) NLSM with a Wess-Zumino-Witten (WZW) term at level k = 1, and by solving this theory exactly, we can demonstrate explicitly that the ground state of the spatial boundary of Eq. (6) with Θ = 2π is doubly degenerate 22,32 , which is equivalent to the boundary ground state of four copies of Kitaev's chain under interaction. In the low-energy limit, the boundary of four copies of the FSPT states is faithfully captured by the bosonic field n. Thus we have established a connection between four copies of Majorana chain and a single copy of Haldane chain, bridging the FSPT and BSPT states in 1d.
Using the knowledge of the better understood BSPT states, we can gain insight of the interacting fermion SPT (iFSPT) states. If eight copies of the iFSPT states is a trivial phase, then necessarily the bosonic theory of eight copies of the FSPT states derived using the same method must also be a trivial state. Indeed, because the Haldane phase has a well-known Z 2 classification 2 , it is expected that two copies of the Haldane chain can be smoothly connected to the trivial state without breaking the symmetry. This can be shown by coupling two layers of the Haldane chain with a large inter-layer anti-ferromagnetic interaction (which preserves the Z T 2 symmetry), as described by the action S = S[n (1) ] + S[n (2) ] + S cp ,
S cp = dτ dx An (1) · n (2) ,(7)
when A → +∞, n (1) and n (2) are locked into opposite directions, i.e. n (1) = −n (2) = n. Then the effective NLSM for n has Θ = 0 due to the cancellation of the Θ angles between the two layers. So two copies of the Haldane chain can be trivialized by the A coupling. Also, when the two Haldane phases in Eq. (7) are decoupled from each other (A = 0), both Haldane phases in Eq. (7) are separated from the trivial phase (Θ = 0) with a critical point at Θ = π. However, with A = 0, this critical point is also gapped out by the A coupling, 50 thus with A > 0, the entire phase diagram of Eq. (7) has only one trivial phase. This observation already suggests that 8 copies of Kitaev's Majorana chain is trivial under interaction.
Bulk transition
Now let us carefully investigate the interactions in the fermion model. Two copies of the Haldane chain would correspond to eight copies of the Majorana chain. Recall the relation n i ∼ (−) i S i on the mean-field level, the inter-layer coupling A can be immediately ported to the fermion model as an on-site interaction among eight flavors of Majorana fermions
H int = J 4 i S (1) i · S (2) i = J i (−χ i1 χ i2 χ i5 χ i6 + χ i1 χ i2 χ i7 χ i8 − χ i1 χ i3 χ i5 χ i7 − χ i1 χ i3 χ i6 χ i8 − χ i1 χ i4 χ i5 χ i8 + χ i1 χ i4 χ i6 χ i7 + χ i2 χ i3 χ i5 χ i8 − χ i2 χ i3 χ i6 χ i7 − χ i2 χ i4 χ i5 χ i7 − χ i2 χ i4 χ i6 χ i8 + χ i3 χ i4 χ i5 χ i6 − χ i3 χ i4 χ i7 χ i8 ),(8)
with J > 0. We should expect that eight copies of the Majorana chain can be connected to the trivial state under this interaction, as the same interaction can trivialize the BSPT in the NLSM. Such an expectation is obvious in the spin sector. In the free fermion limit, depending on the sign of δ, the ν = 8 fFSPT states may correspond to two different spinsinglet dimerization patterns: the intra-unit-cell dimerization (δ > 0 trivial state) or the inter-unit-cell dimerization (δ < 0 SPT state), as shown in Fig. 2. While the strong on-site interaction in Eq. (8) will lead to a third pattern, i.e. the on-site (inter-layer) dimerization, see Fig. 2. The three patterns are connected by the ring exchange of the dimmers. However, it is known that the ring exchange is a smooth deformation and will not close the spin gap, so at least in the spin sector, the δ < 0 and the δ > 0 SPT states can be smoothly connected.
A B A B A B 0 ∆ J FIG
To show that the charge gap also remains open, we can perform an explicit calculation based on the lattice model Eq. (1) in the strong dimerization limit δ = ±1, such that the 1d chain is decoupled into independent twosite segments. In each segment, the interacting fermion system can be exact diagonalized. Then it can be shown that the charge gap indeed persists as u is tuned to zero in the present of the interaction J, as shown in Fig. 3. So one can smoothly connect the N = 8 fFSPT state to the N = 0 fFSPT state in three steps: (i) turn on J and turn off u, (ii) change the sign of δ, (iii) turn on u and turn off J. The bulk gap will never close during this process. Thus the whole phase diagram in Fig. 2 is actually one phase. to the strong interaction (right) limit, the many-body gap never closes.
In conclusion, we have demonstrated that the classification of the 1d FSPT states with the (BDI class) timereversal symmetry is reduced from Z to Z 8 under interation. We obtain the iFSPT classification by making connection to the BSPT classification. This approach can be readily generalized to higher spacial dimensions in the following. Moreover, the way that the BSPT state can be trivialized in the NLSM naturally provide us the correct fermion interaction that is needed to trivialize the FSPT states, which can be much more general than the currently known Fidkowski-Kitaev type of interaction. And this interaction can gap out the critical point m = 0 in Eq. (3), which is 8 copies of nonchiral 1d Majorana fermions. This bulk analysis is particularly suitable to study the crystalline SPT states, which may not have symmetry protected physical boundary modes.
B. From 2d TSC to Levin-Gu Paramagnet
Lattice Model and Bulk Theory
Now we turn to the 2d example of the p ± ip topological superconductor (TSC) protected by a Z 2 symmetry (symmetric class D) 28 . The p x ± ip y TSC can be viewed as both layers of the p x + ip y and the p x − ip y TSC's [33][34][35] stacked together with the Z 2 symmetry acts only in the p x − ip y layer by flipping the sign of the fermion operator (i.e. the fermion parity transform). On a 2d lattice, the model Hamiltonian can be written as
H = k,l ξ k c † kl c kl + 1 2 (∆ k c −kl c kl + h.c.), ξ k = −2t(cos k 1 + cos k 2 ) − µ, ∆ kl = −∆(sin k 1 + (−) l i sin k 2 ),(9)
where l = 0, 1 labels the two opposite layers of the chiral TSC's. The Z 2 symmetry acts as c kl → (−) l c kl which prevents the mixing of fermions from different layers, such that the fermion parity is conserved in each layer independently. Depending on the chemical potential µ, the model has two phases: the |µ| > 4t strong pairing trivial superconductor phase and the |µ| < 4t weak pairing topological superconductor phase. Switching to the Majorana basis
χ k = (c k0 + c † −k0 , c k1 + c † −k1 , ic k0 − ic † −k0 , ic k1 − ic † −k1 ) / √
2 and in the long-wave-length limit, the effective Hamiltonian reads
H ×1 = 1 2 k χ −k (−k 1 σ 30 − k 2 σ 13 + mσ 20 )χ k ,(10)
where we have set ∆ = 1 as our energy unit, and defined the topological mass m = −4t − µ (assuming µ < 0). The Z 2 symmetry acting on the Majorana fermions as χ k → σ 03 χ k . The trivial (m > 0) and the topological (m < 0) phases are separated by the phase transition at m = 0 where the bulk gap closes. This bulk criticality is protected by the Z 2 symmetry. The above p ± ip TSC is an example of the 2d D class fFSPT states, which are known to be Z classified, 10,11 and are indexed by the topological number
N = d 3 k 8π 2 Tr σ 03 G∂ iω G −1 G∂ k1 G −1 G∂ k2 G −1 ,(11)
where G(k) = − χ k χ −k with k = (iω, k) is the fermion Green's function in the frequency-momentum space. The p ± ip TSC in Eq. (9) corresponds to N = 1. While the other topological states in this Z classification may be realized by considering multiple copies of such p ± ip TSC's, which can be described by the following effective field theory Hamiltonian
H ×ν = 1 2 ν α=1 d 2 x χ α (i∂ 1 σ 30 +i∂ 2 σ 13 +mσ 20 )χ α ,(12)
with the Z 2 : χ α → σ 03 χ α symmetry protection. ν-copy p ± ip TSC would correspond to the topological number N = ν.
Corresponding BSPT state
However with interaction, the classification of the Z 2 p ± ip TSC is reduced from Z to Z 8 15-18 , meaning that eight copies of the p ± ip TSC (N = 8) can be smoothly connected to the trivial state (N = 0) in the presence of interaction. This interaction reduced classification was discussed in Ref. [15][16][17][18], but here we will provide another argument for it by making connection to 2d BSPT states.
Let us start by showing that four copies of the p ± ip TSC (ν = 4) can be connected to the Levin-Gu topological paramagnet, 36 a Z 2 BSPT state in 2d. We first introduce a set of inter-layer s-wave pairing terms (with l ≡ 1 − l and α, α = 1, 2, 3, 4 labeling the 4 copies)
∆ 1 = l,α,α cl α iσ 12 αα c lα + h.c. = χ σ 1112 χ, ∆ 2 = l,α,α
cl α iσ 20 αα c lα + h.c. = χ σ 1120 χ,
∆ 3 = l,α,α cl α iσ 32 αα c lα + h.c. = χ σ 1132 χ, ∆ 4 = l,α cl α (−) l c lα + h.c. = χ σ 1200 χ,(13)
where χ = (χ 1 , χ 2 , χ 3 , χ 4 ) ; and couple them to an O(4) order parameter field n = (n 1 , n 2 , n 3 , n 4 ). The lowenergy effective Hamiltonian of this FSM reads
H ×4 = 1 2 d 2 x χ h ×4 χ,
h ×4 = i∂ 1 σ 3000 + i∂ 2 σ 1300 + mσ 2000 + n 1 σ 1112 + n 2 σ 1120 + n 3 σ 1132 + n 4 σ 1200 .
Because the inter-layer pairing mixes the fermions between the p x + ip y and the p x − ip y TSC's, they will gain a minus sign under the Z 2 symmetry transform. To preserve the Z 2 symmetry, we must require the order parameters to change sign as well, i.e. n → −n, under the Z 2 symmetry action. After integrating out the fermion field χ, we arrive at the effective theory for the boson field n, which is a NLSM with a topological Θ term at Θ = 2π, given by the following action (d 3 x = dτ d 2 x)
S[n] = d 3 x 1 g (∂ µ n) 2 + iΘ 2π 2 abcd n a ∂ 0 n b ∂ 1 n c ∂ 2 n d ,(15)
which describes a non-trivial BSPT state for the n field 22,37 , and is equivalent to the Levin-Gu state 22 protected by the Z 2 symmetry n → −n. This can be understood from the wave function perspective. We first reparameterize n = (m cos α, φ sin α) where m = (m 1 , m 2 , m 3 ) is an O(3) unit vector and φ = ±1. Suppose the system energetically favors m (i.e. α = 0), then the wave function for the m field in its paramagnetic phase (g → ∞) can be derived from the action Eq. (15) as 37
|Ψ ∼ D[m]e d 2 x iπ 4π abc ma∂1m b ∂2mc |[m] = D[m](−) Ns[m] |[m] ∼ D[m 3 ](−) N d [m3] |[m 3 ] ,(16)
which is a superposition of all m configurations with a sign factor (−) Ns[m] counting the parity of the Skyrmion number N s of the m field. In the Ising limit where m 3 is energetically favored, the Skyrmion number N s becomes the domain-wall number N d of the Ising spin m 3 , so the wave function becomes the superposition of Ising configurations with the domain-wall sign 37 , which is exactly the wave function of the Levin-Gu state 36 . Thus we have established a connection from four copies of the p ± ip TSC to a single copy of the Levin-Gu paramagnet, bridging the FSPT and BSPT states in 2d. Now we can discuss the iFSPT states using the knowledge about the BSPT states: if eight copies of the TSC is trivial, then the bosonic theory derived using the same method above must necessarily be trivial. Indeed, on the BSPT side, we know that two copies of the Levin-Gu paramagnets can be smoothly connected to the trivial state without breaking the symmetry, which can be realized by coupling two layers of the Levin-Gu paramagnet with a large inter-layer anti-ferromagnetic interaction, such that the domain-wall configuration in both layers will become identical, and the domain-wall sign from both layers will cancel out, so that the resulting wave function is just a trivial Ising paramagnetic state. At the field theory level, it can be described by the following action with inter-layer coupling
S =S[n (1) ] + S[n (2) ] + S cp , S cp = d 3 x A(n (1) 1 n (2) 1 + n (1) 2 n (2) 2 + n (1) 3 n (2) 3 ) − Bn (1) 4 n (2) 4 .(17)
It is easy to check that the coupling S cp respects the Z 2 symmetry. When A, B → +∞, n (1) and n (2) are locked anti-ferromagnetically for their first three components and ferromagnetically for their last components, i.e. n Then the effective NLSM for the combined field n has Θ = 0 due to the cancellation of the Θ angles between the two layers. So two copies of the Levin-Gu paramagnet can be trivialized by the A, B → +∞ coupling 51 . This suggests that eight copies of the original p ± ip TSC is trivial.
Boundary modes and Bulk transition
Recall the relation n a ∼ ∆ a (a = 1, 2, 3, 4) on the mean-field level, the inter-layer coupling S cp can be immediately ported to the fermion model as the following four-fermion interaction (with A, B > 0)
H int = d 2 x A a=1,2,3 ∆ (1) a ∆ (2) a − B∆ (1) 4 ∆ (2) 4 ,(18)
where ∆ a is defined in Eq. (13). Without any interaction, eight copies of p ± ip TSC with the Z 2 symmetry is separated from the trivial state through a critical point that has 16 copies of 2d massless Majorana fermions in the bulk (m = 0 in Eq. (12)). We should expect that the bulk criticality can be gapped out by the interaction Eq. (18), and eight copies of the p ± ip TSC can be smoothly connected to the trivial state, as the same interaction can trivialize the BSPT in the NLSM.
Admittedly, in 2d (and higher dimensions), it is hard to explicitly demonstrate how the interaction gaps out the gapless bulk fermion at the critical point. Nevertheless we can show that, on an open manifold, the interaction Eq. (18) can gap out the 1d boundary states of eight copies of the p ± ip TSC (N = 8) without breaking the symmetry, and hence there should be no obstacle to tune the bulk system smoothly from the N = 8 state to the N = 0 state under interaction. The "transition" between N = 8 and N = 0 states can be viewed as growing N = 0 domains inside the N = 8 state, which is equivalent to sweeping the interface between the two states through the entire bulk (this is essentially the picture of Chalker-Coddington model 38 for the quantum Hall plateau transition), then as long as the interface is gapped out by interaction, the bulk gap never has to close during this "transition", namely the bulk phase transition can be gapped out by the interaction. Thus all we need to show here is that the interaction Eq. (18) induces an effective interaction at the 1d boundary, which will gap out the boundary states.
Let us consider a boundary of the 2d system along the x 2 axis, i.e. the topological mass m ∼ x 1 changes sign across x 1 = 0. For four copies of the p ± ip TSC as described in Eq. (14), the boundary states are given by the projection operator P = (1 − σ 3000 σ 2000 )/2, such that the effective FSM Hamiltonian along the boundary is given by
H ×4 = 1 2 dx 2 η h ×4 η, h ×4 = i∂ 2 σ 300 + n 1 σ 112 + n 2 σ 120 + n 3 σ 132 + n 4 σ 200 ,(19)
where η denotes the Majorana edge modes, and the Z 2 symmetry acts as η → σ 300 η. Under a basis transformation η → exp(− iπ 4 σ 200 )η, the boundary FSM Hamiltonian can be reformulated as h ×4 = −i∂ 2 σ 100 + n 1 σ 312 + n 2 σ 320 + n 3 σ 332 + n 4 σ 200 , (20) which, at the field theory level, is equivalent to four copies of 1d (critical) Majorana chain described by Eq. (5), with the transformed Z 2 symmetry η → −σ 100 η. (n 1 , n 2 , n 3 ) is the analogue of the O(3) order parameter of the Majorana chain introduced in the previous section. All these order parameters are forbidden to condense by the Z 2 symmetry, i.e. n = 0, so that the edge is gapless at the free fermion level. Now we consider the boundary of eight copies of the p ± ip TSC, which is simply a doubling of Eq. (20). The field theory of this 1d boundary is equivalent to eight copies of the critical Kitaev's Majorana chain. The bulk interaction Eq. (18) will induce the interaction between Majorana surface modes, which corresponds to the coupling of n (1) and n (2) at the boundary:
S cp = dτ dx 2 A a=1,2,3 n (1) a n (2) a − B n (1) 4 n(2)
4 . (21) The A term corresponds to exactly the same fermion interaction that trivialized eight copies of Majorana chain in the previous section, and this coupling can gap out the critical point in the previous 1d case. This means that the A term can also gap out the boundary of the 8 copies of 2d p ± ip TSC without degeneracy. Once the boundary is gapped and nondegenerate, a weak B term in Eq. (21) will not close the gap of the boundary. Since the boundary coupling Eq. (21) is induced by the bulk interaction Eq. (18), this implies that the interaction in Eq. (18) (with strong enough strength) can gap out the bulk criticality (with 16 copies of 2d massless Majorana fermions) in 2d.
C. From 3 He-B to 3d Bosonic SPT
Lattice Model and Bulk Theory
Let us go one dimension higher, and consider the 3 He superfluid B phase 39-42 (will be denoted as 3 He-B) which is a 3d TSC protected by the Z T 2 symmetry with T 2 = −1 (symmetry class DIII) 28 . The 3 He-B TSC is described by the following Hamiltonian
H = k k 2 2m He −µ c † k c k − ∆ 2 (c −k iσ 2 k·σc k +h.c.),(22)
where c k = (c k↑ , c k↓ ) is the fermion operator for the 3 He atom, and ∆ ∈ R is the p-wave pairing strength.
The Hamiltonian is invariant under the time-reversal Z T 2 symmetry, which acts as T : c k → iσ 2 c −k followed by the complex conjugation. 3 He-B TSC corresponds to the µ > 0 topological phase of the model, while for µ < 0 the model describes a trivial superconductor. Switching to the Majorana basis
χ k = (c k↑ + c † −k↑ , −c k↓ − c † −k↓ , ic k↑ − ic † −k↑ , −ic k↓ + ic † −k↓ ) / √
2 and in the long-wave-length limit (to the first order in k), the effective Hamiltonian reads
H ×1 = 1 2 k χ k (−k 1 σ 33 − k 2 σ 10 − k 3 σ 31 + mσ 20 )χ k ,(23)
where we have set ∆ = 1 as our energy unit, and defined the topological mass m = −µ (which should not be confused with the mass of the 3 He atom m He ). The timereversal operator acting on the Majorana basis is given by T = Kiσ 32 . The trivial (m > 0) and the topological (m < 0) phases are separated by the phase transition at m = 0 where the bulk gap closes. This bulk criticality is protected by the Z T 2 symmetry. The 3 He-B TSC belongs to the 3d DIII class fFSPT states, which is known to be Z classified, 10,11 and are indexed by the topological number
N = d 3 k 8π 2 Tr σ 32 G∂ k1 G −1 G∂ k2 G −1 G∂ k3 G −1 ,(24)
where G(k) = − χ k χ −k is the fermion Green's function at zero frequency iω = 0. The 3 He-B TSC in Eq. (22) corresponds to N = 1. While the other topological states in this Z classification may be realized by considering multiple copies of the 3 He-B TSC's, which can be described by the following effective field theory Hamiltonian
H ×ν = 1 2 ν α=1 d 3 xχ α (i∂ 1 σ 33 + i∂ 2 σ 10 + i∂ 3 σ 31 + mσ 20 )χ α ,(25)
with the Z T 2 symmetry protection (T = Kiσ 32 ). ν-copy 3 He-B TSC would correspond to the topological number N = ν.
Corresponding BSPT state
However with interaction, the classification of the 3d DIII class FSPT states is reduced from Z to Z 16 , meaning that sixteen copies of the 3 He-B TSC (N = 16) can be smoothly connected to the trivial state (N = 0) in the presence of interaction. This interaction reduced classification was discussed in Ref. 19,20, but here we will provide another argument for it by making connection to the 3d BSPT states.
Let us start by showing that eight copies of the 3 He TSC (ν = 8) can be connected to the 3d BSPT state with Z T 2 symmetry. Similar to our previous approach in 1d and 2d, here we should introduce five fermion pairing terms and couple them to an O(5) order parameter field n = (n 1 , n 2 , n 3 , n 4 , n 5 ), the low-energy effective FSM Hamiltonian reads
H ×8 = 1 2 d 3 x χ h ×8 χ,
h ×8 = i∂ 1 σ 33000 + i∂ 2 σ 10000 + i∂ 3 σ 31000 + mσ 20000 + n 1 σ 32212 + n 2 σ 32220 + n 3 σ 32232 + n 4 σ 32300 + n 5 σ 32100 ,
where χ = (χ 1 , χ 2 , · · · , χ 8 ) . It turns out that these order parameters are spin-singlet s-wave (time-reversal broken) imaginary pairing among the eight copies of fermions. The particular form of the pairing terms given here is not a unique choice. We only require that the pairing terms anti-commute with each other, and also anti-commute with the momentum and the topological mass terms. However any other set of such pairing terms are related to the above choice by basis transformation among the eight copies of fermions, so we may stick to our current choice without losing any generality. On this ν = 8 Majorana basis, the time-reversal operator is extended to T = Kiσ 32000 , from which, it is easy to see that all five s-wave pairing terms change sign under T . To preserve the Z T 2 symmetry, we must require the order parameters to change sign as well, i.e. n → −n, under the Z T 2 transform. After integrating out the fermion field χ, we arrive at the effective theory for the boson field n, which is a NLSM with a topological Θ term at Θ = 2π, given by the following action (d 4 x = dτ d 3 x)
S[n] = d 4 x 1 g (∂ µ n) 2 + iΘ Ω 4 abcde n a ∂ 0 n b ∂ 1 n c ∂ 2 n d ∂ 3 n e ,(27)
where Ω 4 = 8π 2 /3 is the volume of S 4 . Eq. (27) describes a non-trivial 3d BSPT state 22,29,31 protected by the Z T 2 symmetry n → −n. Thus we have established a connection from eight copies of the 3 He-B TSC to a single copy of the 3d Z T 2 BSPT state, bridging the FSPT and BSPT states in 3d. Now we can discuss the iFSPT states using the knowledge about the BSPT states. On the BSPT side, based on the well-known Z 2 classification of this state 2,43 , it is expected that two copies of the Z T 2 BSPT state can be smoothly connected to the trivial state without breaking the symmetry. In our NLSM formalism, this conclusion can be drawn by the following inter-layer coupling
S =S[n (1) ] + S[n (2) ] + S cp , S cp = d 4 x A a=1,2,3 n (1) a n (2) a − Bn (1) 4 n (2) 4 − Cn (1) 5 n (2) 5 .(28)
It is easy to check that the coupling S cp respects the Z T 2 symmetry. When A, B, C → +∞, n (1) and n (2) are locked anti-ferromagnetically for their first three components and ferromagnetically for their last two components, i.e. n (1) a = −n (2) a = n a (a = 1, 2, 3) and n
(1) b = n (2) b = n b (b = 4, 5)
. Then the effective NLSM for the combined field n has Θ = 0 due to the cancelation of the Θ angles between the two layers. So two copies of the Z T 2 BSPT state can be trivialized by the A, B, C → +∞ coupling. Again, this is the necessary condition for interaction to reduce the classification for 3 He-B phase to Z 16 .
Bulk Phase Transition under Interaction
Now we would like to argue that the quantum critical point in the noninteracting limit can be gapped out by interaction for 16 copies of 3 He-B states. We start from the critical point m = 0 in the FSM Eq. (26), where the bulk gap is closed on the free fermion level. The field theory Eq. (26) at m = 0 has an extra inversion symmetry P = −Iiσ 32000 (where the space inversion operator I sends x → −x), besides the original time-reversal symmetry T = Kiσ 32000 . Fermion interactions will be generated after integrating out dynamical field n. We will argue that in this particular field theory Eq. (26), interaction can gap out the critical point, without driving the system into either m < 0 or m > 0 state.
We can first gap out the fermions in the bulk by setting up a fixed configuration of the order parameter field n at the cost of breaking the time-reversal symmetry. Then we restore the symmetry by proliferating the topological defects of the n field, which is an approach adopted by Ref. 20,44. Here we consider the point defect, namely the monopole configuration of n, which is described by n a ∼ x a (for a = 1, 2, 3) and n 4 = n 5 = 0 near the monopole core. This monopole breaks both T and P, but it preserves the combined symmetry T = PT . After proliferating this monopole, all the symmetries will be restored.
However the potential obstacle is that the monopole may trap Majorana zero modes and is therefore degenerated. Proliferating such defect will not result in a gapped and non-degenerated ground state, and hence fails to gap out the bulk criticality. So we must analyze the fermion modes at the monopole core carefully. By solving the BdG equation for a single copy of the FSM Eq. (26), it can be shown that the monopole will trap four Majorana zero modes, which transforms under T as T : γ a → γ a , with a = 1, · · · 4, followed by complex conjugation and space inversion. Thus for two copies of FSM Eq. (26), the monopole will trap eight Majorana zero modes, and the T symmetry will guarantee the spectrum of the monopole is degenerate at the noninteracting level. Nevertheless the degeneracy can be completely lifted by interaction 13,14 without breaking T . So after the monopole proliferation, all the symmetries of Eq. (26) are restored, and the system will enter a fully gapped state which still resides on the line m = 0. Therefore with two copies of the FSM, the iFSPT state can be smoothly connected to the trivial state via strong interaction, resulting in the Z 16 classification, which is consistent with the NLSM analysis.
Later we will show that this analysis of bulk phase transition using topological defects can be naturally generalized to all higher dimensions.
Boundary Modes and Bulk transition
Similar to what has been discussed in the 1d and 2d cases, the inter-layer coupling S cp in Eq. (28) can be immediately ported to the fermion model as a four-fermion local interaction (with A, B, C > 0)
H int = d 3 x A a=1,2,3 ∆ (1) a ∆ (2) a −B∆ (1) 4 ∆ (2) 4 −C∆ (1) 5 ∆ (2) 5 ,(29)
where ∆ = χ (σ 32212 , σ 32220 , σ 32232 , σ 32300 , σ 32100 )χ are defined for both layers of the ν = 8 fermions. We should expect that sixteen copies of the 3 He-B TSC can be connected to the trivial state under this interaction, as the same interaction can trivialize the BSPT in the NLSM.
Following the same idea of the 2d case, we argue that the interaction can remove the 3d bulk criticality by showing that its 2d boundary states can be symmetrically gapped out under interaction.
Let us consider a boundary of the 3d system along the x 1 -x 3 plane, i.e. the topological mass m ∼ x 2 changes sign across x 2 = 0. For eight copies of the 3 He-B TSC as described in Eq. (26), the boundary states are given by the projection operator P = (1 − σ 10000 σ 20000 )/2, such that the effective FSM Hamiltonian along the boundary is given by
H ×8 = 1 2 dx 1 dx 3 η h ×8 η,
h ×8 = i∂ 1 σ 3000 + i∂ 3 σ 1000 + n 1 σ 2212 + n 2 σ 2220 + n 3 σ 2232 + n 4 σ 2300 + n 5 σ 2100 ,
the boundary Hamiltonian can be reformulated as h ×8 = i∂ 1 σ 3000 + i∂ 3 σ 1300 + n 1 σ 1112 + n 2 σ 1120 + n 3 σ 1132 + n 4 σ 1200 + n 5 σ 2000 ,
which, at the field theory level, is equivalent to four copies of 2d (critical) p ± ip TSC described by Eq. (14), with the transformed Z T 2 symmetry T = Kiσ 2300 . (n 1 , n 2 , n 3 , n 4 ) is the analogue of the O(4) order parameter of the p ± ip TSC. So once again, the problem is reduced by one dimension, to the critical 2d iFSPT states.
If we consider the boundary of sixteen copies of the 3 He-B TSC, it will simply be a doubling of Eq. (32), which is analogous to eight copies of (critical) p ± ip TSC at the field theory level. The bulk interaction Eq. (29) will induce the interaction between Majorana surface modes, which corresponds to the coupling of n (1) and n (2) at the boundary:
S cp = dτ dx 1 dx 3 A a=1,2,3 n (1) a n (2) a − B n (1) 4 n(2)4 − C n (1) 5 n(2)
5 .
As we already argued, the A and B term together can gap out the quantum critical point for 8 copies of 2d p±ip TSC, this means that the same interaction in the field theory can also gap out the 2d boundary of 16 copies of 3d 3 He-B phase. And after the boundary is gapped out, adding a C term will not close the gap at the boundary. So there should be no obstacle to smoothly connect sixteen copies of 3 He-B TSC to the trivial state, under the interaction that is ported from inter-layer coupling for the corresponding 3d Z T 2 BSPT states, i.e. the 3d bulk interaction in Eq. (29) can gap out the SPT to trivial state quantum critical point of 16 copies of 3 He-B.
As one can see clearly now, the same pattern of logic will appear again and again in every spatial dimension. Using the dimension reduction argument, the boundary of d-dimensional iFSPT state can be viewed, at the field theory level, as the (d − 1)-dimensional (critical) iFSPT. If the (d−1)-dimensional criticality can be gapped out by a (d − 1)-dimensional interaction, then the same kind of interaction will be able to trivialize the boundary of the d−dimensional iFSPT state, and the d−dimensional bulk interaction that induces this (d − 1)-dimensional boundary interaction likely gaps out the d−dimensional bulk criticality. Of course, one should be reminded that we are not saying that the d-dimensional iFSPT boundary has a (d − 1)-dimensional lattice realization, our induction is only based on the effective field theory description of the long-wave-length physics. Following this induction approach, a class of the iFSPT states and their interaction reduced classification can be studied systematically in all dimensions.
S[n] = d d+1 x 1 g (∂ µ n) 2 + iΘ Ω d+1 a1a2a3···a d+2 n a1 ∂ 0 n a2 ∂ 1 n a3 · · · ∂ d n a d+2 ,(34)
where x 0 ≡ τ is the time coordinate and the rest of x i 's (i = 1, · · · , d) are space coordinates, and Ω d+1 = 2π d+2 2 /Γ( d+2 2 ) is the volume of a (d+1)-hypersphere with unit radius. The action of the inversion symmetry Z P 2 inverts the space and flips all components of n, P :
x i → −x i for i = 1, · · · , d n a → −n a for a = 1, · · · , d + 2 .
It is straight forward to check that the action Eq. (34) is invariant under this inversion. In the g → ∞ regime, the model has a unique gapped disordered ground state, which is a non-trivial SPT state when Θ = 2π. 22,31,45 This BSPT state is Z 2 classified, meaning that two copies of such state can be smoothly connected to the trivial state without breaking the symmetry. To show this, we first make two copies of the model in Eq. (34), with n-vectors denoted by n (1) and n (2) in each copy respectively, such that the total action reads S = S[n (1) ] + S[n (2) ]. Then we are allowed to turn on the following inversion symmetric coupling between the two copies,
S cp = d d+1 x An (1) 1 n (2) 1 − B d+2 a=2 n (1) a n (2) a ,(36)
In the limit of A, B → +∞, n (1) and n (2) are locked together, and the final theory has effectively Θ = 0, and it is a trivial state. Thus the BSPT with inversion symmetry is classified by Z 2 within the framework of NLSM. However in two dimensional space (perhaps in some higher dimensions as well), there are additional Z classified BSPT states beyond NLSM, such as the E 8 state in 2d. So the classifications in d = 2 mod 4 dimensions should be extended to Z 2 × Z.
Free Fermion SPT with Z P 2
Free fermion SPT (fFSPT) state with inversion symmetry also exists in all dimensions, described by the quadratic Majorana Hamiltonian
H = 1 2 d d x χ h ×1 χ, h ×1 = d i=1 i∂ i α i + mβ 0 ,(37)
where χ denotes the Majorana fermion operator. α i are symmetric matrices while β 0 is anti-symmetric, and they all anti-commute with each other. The action of the inversion symmetry is given by the operator P = Iiβ 0 , where I is the space inversion operator such that I −1 x i I = −x i for all i = 1, · · · , d, and it is followed by an orthogonal transform iβ 0 in the Majorana basis, where β 0 is just the mass matrix. Note that this inversion symmetry acts as P 2 = −1 on the Majorana fermions. The Z P 2 fFSPT state belongs to the symmetry class D, and is Z classified in general (see Table II in Ref. 46). Because P = Iiβ 0 rules out all the other additional mass terms that anti-commute with the topological mass mβ 0 , so one has to go through a bulk phase transition (by closing the single-particle gap) to drive the SPT state trivial (i.e. to change the sign of m). The exception rests in d = 2 mod 4 dimensions, where the classification is extended to Z × Z, which was pointed out in Ref. 46, and will be discussed in more details later.
Although the field theory Hamiltonian in Eq. (37) only describes the low-energy physics, it can be immediately cast into lattice models by the substitution i∂ i → sin k i and m → d i=1 cos k i − d + m, with k i being the quasimomentum of the fermion on the lattice. Some lattice models has been explicitly constructed in Ref. 46.
Interacting Fermion SPT with Z P
2
The interacting fermion SPT (iFSPT) states can be obtained by introducing inversion symmetric interaction terms to the free fermion Hamiltonian in Eq. (37). As we discussed previously, interaction can reduce the classification of FSPT states, the same phenomenon is expected here. To study the interaction reduced classification, we still make use of the BSPT states discussed in the last section, and connect the iFSPT to BSPT by introducing bosonic n degrees of freedoms:
S = d d+1 x 1 2 χ (i∂ 0 + h ×ν )χ + 1 g (∂ µ n) 2 + · · · , h ×ν = d i=1 i∂ i α i + mβ 0 + d+2 a=1 n a β a ,(38)
where h ×ν describes ν copies of the fFSPT in Eq. (37) coupling to the bosonic fields n a . Here β a are anticommuting anti-symmetric matrices, and they also anticommute with the all matrices α i and β 0 (which has been enlarged from those in Eq. (37) by tensor product with ν × ν identity matrix). Of course, we need enough flavors of fermions (by making enough ν copies of the fFSPT states) in order to support the d + 2 additional β a matrices. Integrating out the boson field n, Eq. (38) gives a pure fermionic model with interaction. While integrating out the fermion field χ, Eq. (38) becomes the NLSM as in Eq. (34). So Eq. (38) establishes a connection between the iFSPT and the BSPT phases 29 . The inversion symmetry act as P :
x → −x χ → iβ 0 χ n → −n .(39)
It is straight forward to verify that the action in Eq. (38) respects this inversion symmetry. The inversion symmetry satisfies P 2 = +1 on the bosonic n vector, but acts projectively as P 2 = −1 on the Majorana fermion. With this set up, we can study the classification of iFSPT by resorting to the classification of BSPT states, which are much better understood.
B. Examples in Each Dimension
1. d = 1
The Z P 2 fFSPT phase in d = 1 is classified by Z, and its root state (Kitaev's Majorana chain) is described by the lattice model Eq. (1) (at ν = 1). The Hamiltonian is invariant under a bond centered inversion symmetry Z P 2 (see Fig. 1), which acts on the Majorana fermions as P : χ A → χ B , χ B → −χ A . Because the bond is directed due to the imaginary hopping iu ij , the inversion not only takes the fermion from A sites to B sites and vice versa, but must also be followed by a gauge transformation, and hence we can have P 2 = −1 here. The inversion operator can be also written as P = Iiσ 2 in the basis χ = (χ A , χ B ) with I −1 xI = −x implements the inversion of the spacial coordinate.
The low-energy effective Majorana Hamiltonian for the root state is given by Eq. (3) (at ν = 1), and we repeat here
h ×1 = i∂ 1 σ 1 + mσ 2 ,(40)
with P = Iiσ 2 . As long as this inversion symmetry is preserved, without interaction, the two sides of the phase diagram m > 0 and m < 0 are always separated by a gapless critical point at m = 0, no matter how many copies of the system we make.
To incorporate an O(3) order parameter, the model must be copied four times h ×4 = i∂ 1 σ 100 + mσ 200 + n 1 σ 312 + n 2 σ 320 + n 3 σ 332 , (41) with P = Iiσ 200 acting on the Majorana basis. Under P, we must also require n → −n. Then if we integrate out the fermions, the effective theory becomes the O(3) NLSM at Θ = 2π (Haldane spin chain). If we double the model h ×4 again to h ×8 , eight copies of this FSPT root state can be trivialized by interaction (see section II for detailed analysis), as the corresponding BSPT state is trivial due to its Z 2 classification. Thus the Z 2 classification of the BSPT suggests that the Z P 2 iFSPT in d = 1 is Z 8 classified.
d = 2
The Z P 2 fFSPT phase in d = 2 is classified by Z × Z, which has two root states. They are p + ip and p − ip topological superconductors (TSC) respectively. The free fermion p + ip TSC in 2d is already Z classified without any symmetry protection. However with the inversion symmetry, the p + ip and p − ip TSC's are not allowed to trivialize with each other, thus we will have two independent Z topological indices ν 1 and ν 2 labelling the copies of the p + ip and p − ip TSC's respectively.
To study the interaction reduced classification of this FSPT, we start from the special case when ν 1 = ν 2 (i.e. the non-chiral p ± ip TSC). The low-energy effective Majorana Hamiltonian for the non-chiral root state is given by Eq. (12) (at ν = 1), and we repeat here
h ×1 = i∂ 1 σ 30 + i∂ 2 σ 13 + mσ 20 ,(42)
with P = Iiσ 20 . In this case, the iFSPT can be studied by making connection to the BSPT within the scope of FSM. To incorporate an O(4) order parameter, the model must be copied four times h ×4 =i∂ 1 σ 3000 + i∂ 2 σ 1300 + mσ 2000 + n 1 σ 1112 + n 2 σ 1120 + n 3 σ 1132 + n 4 σ 1200 ,
with P = Iiσ 2000 acting on the Majorana basis. Under P, we must also require n → −n. Then if we integrate out the fermions, the effective theory becomes the O(4) NLSM at Θ = 2π. If we double the model h ×4 again to h ×8 , eight copies of this FSPT root state can be trivialized by interaction (as was discussed in section II), as the corresponding BSPT state is trivial due to its Z 2 classification. When ν 1 = ν 2 , the iFSPT state cannot be connected to a BSPT as discussed above, as no order parameter can be embedded no matter how many copies of the fFSPT state we make. However we can consider such FSPT state as attaching additional layers of chiral p + ip TSC (or p−ip TSC) to the non-chiral p±ip TSC's. It is known that interaction can not reduce the classification of the p + ip TSC, and the only possible effect of the interaction is to drive 16 copies of the p + ip TSC to a BSPT state known as the E 8 state 29 . So we can simply extend the Z 8 classification of non-chiral iFSPT by attaching the Z classified chiral iFSTP, and as a result, the Z P 2 iFSPT in d = 2 is Z 8 × Z classified. Correspondingly the BSPT in d = 2 is Z 2 × Z, in which the Z index labels the number of E 8 states.
d = 3
The Z P 2 fFSPT phase in d = 3 is classified by Z. The low-energy effective Majorana Hamiltonian for the root state is given by Eq. (25) (at ν = 1), and we repeat here
h ×1 = i∂ 1 σ 33 + i∂ 2 σ 10 + i∂ 3 σ 31 + mσ 20 ,(44)
with P = Iiσ 20 . To incorporate an O(5) order parameter, the model must be copied eight times h ×8 = i∂ 1 σ 33000 + i∂ 2 σ 10000 + i∂ 3 σ 31000 + mσ 20000 + n 1 σ 32212 + n 2 σ 32220 + n 3 σ 32232 + n 4 σ 32300 + n 5 σ 32100 ,
with P = Iiσ 20000 acting on the Majorana basis. Under P, we must also require n → −n. Then if we integrate out the fermions, the effective theory becomes the O(5) NLSM at Θ = 2π, which is equivalent to the BSPT with Z P 2 symmetry. If we double the model h ×8 again to h ×16 , sixteen copies of this FSPT root state can be trivialized by interaction, as the corresponding BSPT state is trivial due to its Z 2 classification. Therefore the Z P 2 iFSPT in d = 3 is Z 16 classified.
Higher Dimensions
The above examples can be systematically generalized to higher dimensions using the representation of Clifford algebras. In d-dimensional space, the non-chiral root state of the Z P 2 fFSPT phase is described by the following Majorana Hamiltonian at low-energy
h ×1 = d i=1 i∂ i α i + mβ 0 ,(46)
with d symmetric matrices α i and one anti-symmetric matrix β 0 , which are taken from the generators of the real Clifford algebra C d,1 , i.e. {α 1 , · · · , α d ; iβ 0 } (see Appendix A for definitions). The inversion symmetry acts as P = Iiβ 0 . To construct the FSM in d-dimension, we must make enough copies of the fFSPT root state to incorporate the O(d + 2) order parameter. Suppose ν is the minimal copies that should be made, we can write down the following Majorana Hamiltonian
h ×ν = d i=1 i∂ i α i + mβ 0 + d+2 a=1 n a β a ,(47)
in which the symmetric matrices α i (i = 1, · · · , d) and the anti-symmetric matrices β a (a = 1, · · · , d + 2) can be taken from the generators of the real Clifford algebra C d,d+2 , i.e. {α 1 , · · · , α d ; iβ 1 , · · · , iβ d+2 } (see Appendix A for definitions), and the mass matrix β 0 is chosen to be the pseudo scalar of
C d,d+2 , i.e. iβ 0 = d i=1 α i d+2 a=1 (iβ a )
. It is straight forward to verify that the matrix β 0 is anti-symmetric by definition of C d,d+2 , and hence qualified as a mass term. The inversion symmetry still acts as P = Iiβ 0 , and under P, we require n → −n as well, such that h ×ν is inversion symmetric.
If we integrate out the fermions in the FSM H = d d xχ h ×ν χ, the effective theory becomes a NLSM of n at Θ = 2π. If we double h ×ν again to h ×2ν , 2ν copies of this d-dimensional FSPT root state can be trivialized by interaction, as the corresponding d-dimensional BSPT is also trivial due to its Z 2 classification. So the non-chiral Z P 2 iFSPT is classified by Z 2ν . The minimal copy number ν will be determined in the following. However, we recall that for d = 2 mod 4, we also have the chiral FSPT states, which fall outside the FSM-based classification. So the Z P 2 iFSPT states in d = 2 mod 4 dimension will be Z 2ν × Z classified, where Z index labels the chiral FSPT states.
C. The Classification Table
Counting Minimal Copy Number
The minimal copies ν that one should make to go from the iFSPT root state to the FSM model can be simply determined from the Majorana fermion flavor numbers of both models. Given that h ×1 and h ×ν are constructed using the irreducible representations of C d,1 and C d,d+2 respectively, the minimal copy number ν follows from
ν = dim C d,d+2 dim C d,1 ,(48)
where dim C p,q denotes the dimension of the irreducible real representation of the real Clifford algebra C p,q . As concluded in Appendix A, C d,d+2 ∼ = H(2 d ), so dim C d,d+2 = 2 d+2 . Therefore we have ν = 2 d+2 / dim C d,1 , from which we can conclude the Z 2ν classification of Z P 2 iFSPT states as in Tab. I. The classification of iFSPT also shows the 8-fold Bott periodicity. iFSPT states in each dimension d. The data of C d,1 (also see Appendix A) and the minimal copy number ν are also listed.
d mod 8 dim C d,1 ν classification 0 dim C(2 d 2 ) = 2 d+2 2 2 d+2 2 Z 2 d+4 2 1 dim R(2 d+1 2 ) = 2 d+1 2 2 d+3 2 Z 2 d+5 2 2 dim 2R(2 d 2 ) = 2 d+2 2 2 d+2 2 Z 2 d+4 2 × Z 3 dim R(2 d+1 2 ) = 2 d+1 2 2 d+3 2 Z 2 d+5 2 4 dim C(2 d 2 ) = 2 d+2 2 2 d+2 2 Z 2 d+4 2 5 dim H(2 d−1 2 ) = 2 d+3 2 2 d+1 2 Z 2 d+3 2 6 dim 2H(2 d−2 2 ) = 2 d+4 2 2 d 2 Z 2 d+2 2 × Z 7 dim H(2 d−1 2 ) = 2 d+3 2 2 d+1 2 Z 2 d+3 2
In d mod 4 = 2 dimensions, the chiral FSPT states are not included in this classifying scheme [10][11][12] . As the chiral FSPT states can not be trivialize by the fermion interaction, therefore they provide an additional Z classification. So we conclude that the Z P 2 iFSPT states in d mod 4 = 2 dimensions are Z 2ν × Z classified.
Bulk Phase Transition under Interaction
Finally we would like to mention that the same classification for the non-chiral states can be obtained by the same argument as in section II. C. 3. The idea is that if 2ν copies of the FSPT root state can be trivialized by fermion interaction, then there must be a way to gap out its bulk phase transition with the trivial FSPT state without breaking the symmetry.
Again we start from the critical point, which corresponds to m = 0 in the FSM Eq. (47). We first gap out the fermions in the bulk by setting up a fixed configuration of the order parameter field n at the cost of breaking the inversion symmetry. Then we restore the symmetry by proliferating the inversion-symmetric topological defects of the n field. Here we choose to focus on the point defect, namely the monopole configuration of n, because such defect generally exists in all dimensions. The monopole configuration is described by n a ∼ x a (for a = 1, · · · , d) and n d+1 = n d+2 = 0 near the monopole core. Under inversion, both n and x changes sign, so the above monopole configuration is indeed inversionsymmetric. Thus if we can proliferate such monopoles, the inversion symmetry will be restored.
Again by solving the BdG equation for a single copy of the FSM, it can be shown that the monopole will always trap four Majorana zero modes no matter in which dimension d. This general property can be simply verified by counting the fermion flavors. The d-dimensional FSM has dim C d,d+2 = 2 d+2 flavors of Majorana fermions. Confining them to the core of a d-dimensional monopole will reduce the fermion flavor number by 2 d , so the remaining flavor number is 2 d+2 /2 d = 4. Thus for two copies of FSM, the monopole will trap eight Majorana zero modes, whose degeneracy is protected on the free fermion level by the inversion symmetry left in the monopole core, together with the assumption of m = 0 at the critical point. Nevertheless the degeneracy can be completely lifted by interaction, 13,14 such that the monopole can be trivialized. So after the monopole proliferation, the inversion symmetry is restored, and we are left with a gapped symmetric state at m = 0. Therefore with two copies of the FSM, the iFSPT state can be smoothly connected to the trivial state via strong interaction, resulting in the Z 2ν classification, which is consistent with the NLSM analysis.
IV. SPT STATES WITH Z P 2 COMBINED WITH OTHER SYMMETRIES A. U(1) × Z P 2 SPT States 1. BSPT with U(1) × Z P 2
The U(1)×Z P 2 BSPT states can be studied similarly as the Z P 2 BSPT states under the framework of the O(d + 2) NLSM. The inversion symmetry still flips all components of n as in Eq. (35). One remains to specify the U(1) symmetry action as well. Based on our experiences from lower dimensional cases 22,43 , the different ways of imposing the U(1) symmetry in the NLSM correspond to different BSPT root states.
In odd dimension d, there are (d + 3)/2 ways to impose the U(1) symmetry transformation, labeled by k = 0, · · · , (d + 1)/2 as
U k :(n 2a−1 + in 2a ) → e iθ (n 2a−1 + in 2a )
for a = 1, 2, · · · , k,
whereas k = 0 labels the case that the U(1) symmetry has no action on n. Each assignment U k of the symmetry action leads to a Z 2 classification of the BSPT states, as two layers of the BSPT root states can be trivialized via the inter-layer coupling Eq. (36) as argued previously. So the U(1) × Z P 2 BSPT states in odd dimension is Z (d+3)/2 2 classified.
In even dimension d, there are (d + 4)/2 ways to impose the U(1) symmetry transformation, labeled by k = 0, · · · , (d + 2)/2 following the same definition in Eq. (49). For the first (d + 2)/2 implementations U k (k = 1, · · · , d/2), each leads to a Z 2 classification respectively. However the last implementation U (d+2)/2 leads to a Z classification, as the coupling term Eq. (36) would necessary breaks the U (1) symmetry, and is therefore forbidden, so that there is no way to reduce the Z classifica-tion. Thus the U(1)×Z P 2 BSPT states in even dimension is Z (d+2)/2 2 × Z classified. We therefore conclude the classification of U(1) × Z P 2 BSPT states in Tab. II. However, this classification is not complete. The chiral states, such as E 8 states in 2d, are not covered by the NLSM classification.
× Z 1 Z (d+3)/2 2 2. Free Fermion SPT with U(1) × Z P 2
With the U(1) symmetry, the Majorana fermions χ can be paired up to Dirac fermions ψ = χ + iχ , such that ψ → e iθ ψ under the action of U(1). The non-chiral U(1) × Z P 2 fFSPT root state can be described by the following Dirac Hamiltonian at low-energy
H = d d x ψ †h ×1 ψ, h ×1 = d i=1 i∂ i γ i + mγ d+1 ,(50)
in which γ i (i = 1, · · · , d + 1) are Hermitian complex matrices which anti-commute with each other. They can be taken from the generators of the complex Clifford algebra C d+1 (in the complex representation). The action of inversion symmetry is given by the operator P = Iiγ d+1 (if P 2 = −1) or by P = Iγ d+1 (if P 2 = +1). In the presence of the U(1) symmetry, there is no essential difference between P 2 = −1 and P 2 = +1 (as they only differed by a U(1) rotation which is part of the symmetry), so we will focus on the former case. The U(1) × Z P 2 fFSPT state belongs to the symmetry class A, and is Z classified in odd dimensions and Z × Z classified in even dimensions (see Table I, II in Ref. 46). Because the U(1) symmetry rules out any fermion pairing terms, and within the fermion hopping terms, the inversion symmetry P forbids all the other possible mass terms that anti-commute with the topological mass mγ d+1 , so one has to go through a bulk transition (by closing the single-particle gap) to drive the SPT state trivial (i.e. to change the sign of m). This explains the Z classification in odd dimensions, and one of the Z classification in even dimensions. While the other Z classification in even dimensions comes from the chiral state, whose root state is described by the Dirac Hamiltoniañ
h ×1 = d i=1 i∂ i γ i + m γ ch ,(51)
where γ ch ≡ d i=1 γ i is the chiral matrix (which exists only for even d). The chiral mass m γ ch also preserves the U(1) × Z P 2 symmetry. It is impossible to find any additional U(1) preserving mass term that anti-commute with the chiral mass, so the chiral states lead to the other Z classification in even dimensions.
Interacting Fermion
SPT with U(1) × Z P 2
The non-chiral U(1) × Z P 2 iFSPT states can be also studied by extending the fFSPT model to the FSM. In each dimension, the FSM is still given by Eq. (38) in the Majorana fermion basis, with the matrices α i (i = 1, · · · , d) and β a (a = 1, · · · , d + 2) taken form the generators of the real Clifford algebra C d,d+2 , and the mass matrix β 0 chosen to be the pseudo scalar of C d,d+2 . We must make enough copies of the U(1) × Z P 2 fFSPT root states described in Eq. (50) to obtain the FSM. To count the number ν of copies correctly, we first rewrite the Hamiltonian in Eq. (50) in the Majorana basis, which takes the form of Eq. (37), with the matrices α i (i = 1, · · · , d) and β 0 still taken from the generators of the complex Clifford algebra C d+1 but using its real representation. Then the minimal copy number ν is given by
ν = dim C d,d+2 dim C d+1 ,(52)
where dim C n denotes the dimension of the irreducible real representation of the complex Clifford algebra C n .
Given that dim C d,d+2 = 2 d+2 (see Appendix A), ν in each dimension d can be calculated in Tab. III. We conclude that the non-chiral U(1)×Z P 2 fFSPT root state can be made into a FSM incorporating the O(d+2) order parameters by copying ν times, such that their corresponding iFSPT states can be classified by making connection to the BSPT classifications. However, the chiral fFSPT root state (which appears in even dimensions) can not be connected to the FSM without breaking the U(1) symmetry, and should be classified separately. iFSPT states in each dimension d. The data of C d+1 (also see Appendix A) and the minimal copy number ν are also listed.
d mod 2 dim C d+1 ν classification 0 dim 2C(2 d 2 ) = 2 d+4 2 2 d 2 Z 2 d+2 2 × Z 1 dim C(2 d+1 2 ) = 2 d+3 2 2 d+1 2 Z 2 d+3 2
Integrating out the fermions in the FSM, we arrive at the O(d + 2) NLSM in Eq. (34) at Θ = 2π, with the symmetry action inherited from the FSM, such that the inversion symmetry flips all components of n, while the U(1) symmetry rotates two and only two components of n, say (n 1 + in 2 ) → e iθ (n 1 + in 2 ) (see Appendix B for examples). The action of U(1) here corresponds to the U 1 implementation as defined in Eq. (49), which gives a Z 2 classification of the BSPT states, meaning that two copies of the FSM can be trivialized by the interaction which also trivialize the corresponding BSPT states. As have been counted in Eq. (52), each copy of the FSM corresponds to ν copies of the fFSPT root states, so the non-chiral U(1) × Z P 2 iFSPT states are Z 2ν classified (see Tab. III). For the chiral fFSPT states, it is not possible to extend them to the FSM without breaking the U(1) symmetry, thus their classification can not be reduced by the interaction which is still Z. In conclusion, the U(1) × Z P 2 iFSPT states are Z × Z classified in even d dimensions.
B. Z T 2 × Z P 2 SPT States 1. Boson SPT with Z T 2 × Z P 2
We study the Z T 2 × Z P 2 BSPT states under the framework of the O(d + 2) NLSM. The inversion symmetry Z P 2 always flips all components of n as in Eq. (35), while the time-reversal symmetry Z T 2 must flips odd number of n components to keep the Θ-term invariant. Based on our experiences gained from lower dimentional cases 22 , the different ways of imposing the Z T 2 symmetry in the NLSM correspond to different of BSPT root states.
In odd dimension d, there are (d + 3)/2 ways to impose the Z T 2 symmetry transformation, labeled by k = 0, · · · , (d + 1)/2 as T k : n i → −n i for i = 1, · · · , 2k + 1,
which flips the first (2k + 1)-components of n, leaving the rest of the components unchanged. Each assignment T k of the symmetry action leads to a Z 2 classification of the BSPT states, as two layers of the BSPT root states can be trivialized via the inter-layer coupling Eq. (36) as argued previously. So the Z T 2 × Z P 2 BSPT states in odd dimension is Z (d+3)/2 2 classified. In even dimension d, there are (d+2)/2 ways to impose the U(1) symmetry, labeled by k = 0, · · · , d/2 following the same definition in Eq. (53). Each assignment T k still leads to a Z 2 classification. Thus the Z T 2 × Z P 2 BSPT states in even dimension is Z (d+2)/2 2 classified. We therefore conclude the classification of Z inversion P commute with each other. For our purpose to study the interaction reduced classification of FSPT states, we wish to start with Z classified fFSPT states. Therefore we consider a peculiar setting where T and P do not commute, and the symmetry group (acting on the fermions) is defined by
T 2 = P 2 = −1, T PT P = −1.(54)
This is a projective representation of the Z T 2 × Z P 2 symmetry, which can be realized as a projective symmetry group 47 if the fermions are coupled to a Z 2 gauge field (such as spinons in the Z 2 spin-liquid).
In the presence of the time-reversal symmetry, the chiral SPT states are ruled out. The Z T 2 × Z P 2 fFSPT root state is non-chiral, and can be described by the following Majorana Hamiltonian at low-energy
H = 1 2 d d x χ h ×1 χ, h ×1 = d i=1 i∂ i α i + mβ 1 ,(55)
with the inversion symmetry P = Iiβ 1 and time-reversal symmetry T = Kiβ 2 . Here the symmetric matrices α i (i = 1, · · · , d) and the anti-symmetric matrices β 1 , β 2 are taken from the generators of the real Clifford algebra C d,2 , i.e. {α 1 , · · · , α d ; iβ 1 , iβ 2 } (see appendix A for definitions). It worth mention that at dimensions d = 3, 7 (mod 8), the representations C 3,2 ∼ = R(4) ⊕ R(4) and C 7,2 ∼ = H(8) ⊕ H(8) can be split into two sub-algebras. Each sub-algebra is sufficient to faithfully represent the anti-commutation relations among the generators. So the minimal fermion flavor is only half of dim C d,2 when d = 3, 7 (mod 8). For later convenient, we define the reduced dimension rdim as the dimension of the minimal representation of the anti-commuting generators (but not the whole algebra), which follows
rdim C p,q ≡ 1 2 dim C p,q p − q = 1, 5(mod 8), dim C p,q otherwise.(56)
Thus in terms of the reduced dimension, the Majorana fermion flavor number of the Z T 2 × Z P 2 root state in Eq. (55) is simply given by rdim C d,2 in dimension d.
Because the inversion symmetry P has ruled out all the other possible mass terms that anti-commute with the topological mass mβ 1 , which already leads to the Z classification, and the additional time-reversal symmetry will not change the classification. So the Z T 2 × Z P 2 symmetry defined in Eq. (54) fFSPT states are Z classified.
Interacting Fermion
SPT with Z T 2 × Z P 2
The (projective) Z T 2 × Z P 2 iFSPT states can be also studied by extending the fFSPT model to the FSM. In each dimension, the FSM is still given by Eq. (38) in the Majorana fermion basis, with the matrices α i (i = 1, · · · , d) and β a (a = 1, · · · , d + 2) taken form the generators of the real Clifford algebra C d,d+2 , and the mass matrix β 0 chosen to be the pseudo scalar of C d,d+2 . We must make enough copies of the Z T 2 × Z P 2 fFSPT root states described in Eq. (55) to obtain the FSM. Then the minimal copy number ν is given by
ν = dim C d,d+2 rdim C d,2 ,(57)
where rdim C d,2 is the reduced dimension defined in Eq. (56). Given that dim C d,d+2 = 2 d+2 (see Appendix A), ν in each dimension d can be calculated in Tab. V. We conclude that the Z T 2 × Z P 2 fFSPT root state can be made into a FSM incorporating the O(d+2) order parameters by copying ν times, such that their corresponding iFSPT states can be classified by making connection to the BSPT classifications.
Z 2 d+2 2 1 rdim C(2 d+1 2 ) = 2 d+3 2 2 d+1 2 Z 2 d+3 2 2 rdim R(2 d+2 2 ) = 2 d+2 2 2 d+2 2 Z 2 d+4 2 3 rdim 2R(2 d+1 2 ) = 2 d+1 2 2 d+3 2 Z 2 d+5 2 4 rdim R(2 d+2 2 ) = 2 d+2 2 2 d+2 2 Z 2 d+4 2 5 rdim C(2 d+1 2 ) = 2 d+3 2 2 d+1 2 Z 2 d+3 2 6 rdim H(2 d 2 ) = 2 d+4 2 2 d 2 Z 2 d+2 2 7 rdim 2H(2 d−1 2 ) = 2 d+3 2 2 d+1 2 Z 2 d+3 2
Integrating out the fermions in the FSM, we arrive at the O(d + 2) NLSM in Eq. (34) at Θ = 2π, with the symmetry action inherited from the FSM, such that the inversion symmetry flips all components of n, while the Z T 2 symmetry always flips odd number of n components (see Appendix B for examples). Such an implementation of the symmetry action always results in the Z 2 classified the BSPT states, meaning that two copies of the FSM can be trivialized by the interaction which also trivialize the corresponding BSPT states. As have been counted in Eq. (57), each copy of the FSM corresponds to ν copies of the fFSPT root states, so the Z T 2 × Z P 2 iFSPT states are Z 2ν classified (see Tab. V).
V. SUMMARY
In this paper we systematically studied the classification of strongly interacting fermionic and bosonic SPT states in all dimensions. And for all the examples we considered in this paper, we argue that the classification of BSPT states implies that short range interactions can reduce the classification of FSPT states with the same symmetry. Further more, using different methods, we argue that certain interaction can gap out the critical point between the FSPT state and trivial state in the noninteracting limit, which implies that under interaction some FSPT states are driven trivial, and it can be connected to the trivial state without bulk phase transition.
where α i (i = 1, · · · , d) are transpose-symmetric Hermitian matrices and β a (a = 0, · · · , d + 2) are transposeantisymmetric Hermitian matrices. We consider that the inversion symmetry always act as P = Iiβ 0 . We will provide the explicit examples of these matrices in the fermion Hamiltonian h (d) . In general, the Majorana fermion χ is of 2 d+2 flavors, meaning that the dimensions of the matrices α i and β a are 2 d+2 . We will use the notation σ ijk··· = σ i ⊗ σ j ⊗ σ k ⊗ · · · to denote the direct product of Pauli matrices σ 1 , σ 2 , σ 3 and as well as the 2 × 2 identity matrix σ 0 . Each Pauli matrix σ i acts on a 2-dimensional single-particle Hilbert space, which is also the size of a qubit. We may thus count the dimension of the single-particle Hilbert space (which is also the Majorana fermion flavor number) by qubits. A Hamiltonian made of matrices like σ ijk··· with n indices acts in the Hilbert space of n qubit which is of the dimension 2 n . In the following, we will use examples to demonstrate both the FSM and the FSPT root state model with various symmetries.
In d = 1 spacial dimension, the Hamiltonian h (1) is defined on a 3-qubit single-particle Hilbert space, h (1) = i∂ 1 σ 100 + mσ 200 + n 1 σ 312 + n 2 σ 332 + n 3 σ 320 .
With the Z P 2 : χ → iσ 200 χ, n → −n symmetry only, the root state Hamiltonian takes the first 1-qubit subspace, and must be 4-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 020 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 2-qubit subspace, and must be doubled to form the FSM. With the additional Z T 2 : χ → iσ 320 χ, n 3 → −n 3 symmetry, the root state Hamiltonian takes the first 2-qubit subspace, and must be doubled to form the FSM.
In d = 2 spacial dimension, the Hamiltonian h (2) is defined on a 4-qubit single-particle Hilbert space, h (2) = i∂ 1 σ 1000 + i∂ 2 σ 3100 + mσ 2000 + n 1 σ 3312 + n 2 σ 3332 + n 3 σ 3320 + n 4 σ 3200 .
With the Z P 2 : χ → iσ 2000 χ, n → −n symmetry only, the root state Hamiltonian takes the first 2-qubit subspace, and must be 4-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 0020 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 3-qubit subspace, and must be doubled to form the FSM. With the additional Z T 2 : χ → iσ 3200 χ, n 4 → −n 4 symmetry, the root state Hamiltonian takes the first 2-qubit subspace, and must be 4-multiplied to form the FSM.
In d = 3 spacial dimension, the Hamiltonian h (3) is defined on a 5-qubit single-particle Hilbert space, h (3) = i∂ 1 σ 10000 + i∂ 2 σ 31000 + i∂ 3 σ 33000 + mσ 20000 + n 1 σ 32100 + n 2 σ 32300 + n 3 σ 32212 + n 4 σ 32232 + n 5 σ 32220 . (B4)
With the Z P 2 : χ → iσ 20000 χ, n → −n symmetry only, the root state Hamiltonian takes the first 2-qubit subspace, and must be 8-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 00200 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 3-qubit subspace, and must be 4-multiplied to form the FSM. With the additional Z T 2 : χ → iσ 32000 χ, n → −n symmetry, the root state Hamiltonian takes the first 2-qubit subspace, and must be 8-multiplied to form the FSM.
In d = 4 spacial dimension, the Hamiltonian h (4) is defined on a 6-qubit single-particle Hilbert space, h (4) = i∂ 1 σ 100000 + i∂ 2 σ 310000 + i∂ 3 σ 330000 + i∂ 4 σ 322000 + mσ 200000 + n 1 σ 321100 + n 2 σ 321300 + n 3 σ 321212 + n 4 σ 321232 + n 5 σ 321220 + n 6 σ 323000 .
With the Z P 2 : χ → iσ 200000 χ, n → −n symmetry only, the root state Hamiltonian takes the first 3-qubit subspace, and must be 8-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 000200 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 4-qubit subspace, and must be 4-multiplied to form the FSM. With the additional Z T 2 : χ → iσ 323000 χ, n 6 → −n 6 symmetry, the root state Hamiltonian takes the first 3-qubit subspace, and must be 8-multiplied to form the FSM.
In d = 5 spacial dimension, the Hamiltonian h (5) is defined on a 7-qubit single-particle Hilbert space, h (5) = i∂ 1 σ 1000000 + i∂ 2 σ 3100000 + i∂ 3 σ 3300000 + i∂ 4 σ 3212000 + i∂ 5 σ 3232000 + mσ 2000000 + n 1 σ 3201000 + n 2 σ 3203000 + n 3 σ 3222100 + n 4 σ 322300 + n 5 σ 3222212 + n 6 σ 3222232 + n 7 σ 3222220 .
With the Z P 2 : χ → iσ 2000000 χ, n → −n symmetry only, the root state Hamiltonian takes the first 4-qubit subspace, and must be 8-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 0002000 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 4-qubit subspace, and must be 8-multiplied to form the FSM. With the additional Z T 2 : χ → iσ 3201000 χ, n 1 → −n 1 symmetry, the root state Hamiltonian takes the first 4-qubit subspace, and must be 8-multiplied to form the FSM.
In d = 6 spacial dimension, the Hamiltonian h (6) is defined on a 8-qubit single-particle Hilbert space, h (6) = i∂ 1 σ 10000000 + i∂ 2 σ 31000000 + i∂ 3 σ 33000000 + i∂ 4 σ 32120000 + i∂ 5 σ 32320000 + i∂ 6 σ 32222000 + mσ 20000000 + n 1 σ 32221000 + n 2 σ 32223000 + n 3 σ 32010100 + n 4 σ 32010300 + n 5 σ 32010212 + n 6 σ 32010232 + n 7 σ 32010220 + n 8 σ 32030000 .
(B7)
With the Z P 2 : χ → iσ 20000000 χ, n → −n symmetry only, the root state Hamiltonian takes the first 5-qubit subspace, and must be 8-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 00002000 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 5-qubit subspace, and must be 8-multiplied to form the FSM. With the additional Z T 2 : χ → iσ 32030000 χ, n 6 → −n 6 symmetry, the root state Hamiltonian takes the first 5-qubit subspace, and must be 8-multiplied to form the FSM.
In d = 7 spacial dimension, the Hamiltonian h (7) is defined on a 9-qubit single-particle Hilbert space, h (7) = i∂ 1 σ 100000000 + i∂ 2 σ 310000000 + i∂ 3 σ 330000000 + i∂ 4 σ 321200000 + i∂ 5 σ 323200000 + i∂ 6 σ 320120000 + i∂ 7 σ 320320000 + mσ 200000000 + n 1 σ 322010002 + n 2 σ 322030002 + n 3 σ 322221200 + n 4 σ 322223200 + n 5 σ 322220120 + n 6 σ 322220320 + n 7 σ 322222010 + n 8 σ 322222030 + n 9 σ 322222220 .
(B8)
With the Z P 2 : χ → iσ 200000000 χ, n → −n symmetry only, the root state Hamiltonian takes the first 5-qubit subspace, and must be 16-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 000020000 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 5-qubit subspace, and must be 16-multiplied to form the FSM. With the additional Z T 2 : χ → iσ 322200000 χ, n → −n symmetry, the root state Hamiltonian takes the first 5-qubit subspace, and must be 16-multiplied to form the FSM.
In d = 8 spacial dimension, the Hamiltonian h (8) is defined on a 10-qubit single-particle Hilbert space, h (8) = i∂ 1 σ 1000000000 + i∂ 2 σ 3100000000 + i∂ 3 σ 3300000000 + i∂ 4 σ 3212000000 + i∂ 5 σ 3232000000 + i∂ 6 σ 3201200000 + i∂ 7 σ 3203200000 + i∂ 8 σ 3222200000 + mσ 2000000000 + n 1 σ 3220310002 + n 2 σ 3220330002 + n 3 σ 3220101200 + n 4 σ 3220103200 + n 5 σ 3220100120 + n 6 σ 3220100320 + n 7 σ 3220102010 + n 8 σ 3220102030 + n 9 σ 3220102220 + n 10 σ 3220320000 .
(B9)
With the Z P 2 : χ → iσ 2000000000 χ, n → −n symmetry only, the root state Hamiltonian takes the first 5-qubit subspace, and must be 32-multiplied to form the FSM. With the additional U(1) : χ → exp(iϕσ 0000020000 )χ, (n 1 + in 2 ) → e 2iϕ (n 1 + in 2 ) symmetry, the root state Hamiltonian takes the first 6-qubit subspace, and must be 16-multiplied to form the FSM. With the additional Z T 2 : χ → iσ 3220320000 χ, n 10 → −n 10 symmetry, the root state Hamiltonian takes the first 6-qubit subspace, and must be 16-multiplied to form the FSM. a=1 n aβ a , the extension is given by α i =α i ⊗ σ 00000000 , (i = 1, · · · , d − 9) β a =β a ⊗ σ 00000000 , (a = 0, · · · , d − 7) α d−8 =α d−8 ⊗ σ 22220000 , β d−6 =β d−6 ⊗ σ 00002222 , α d−7 =α d−8 ⊗ σ 10000000 , β d−5 =β d−6 ⊗ σ 00001000 , α d−6 =α d−8 ⊗ σ 30000000 , β d−4 =β d−6 ⊗ σ 00003000 , α d−5 =α d−8 ⊗ σ 21200000 , β d−3 =β d−6 ⊗ σ 00002120 , α d−4 =α d−8 ⊗ σ 23200000 , β d−2 =β d−6 ⊗ σ 00002320 , α d−3 =α d−8 ⊗ σ 20120000 , β d−1 =β d−6 ⊗ σ 00002012 , α d−2 =α d−8 ⊗ σ 20320000 , β d =β d−6 ⊗ σ 00002032 , α d−1 =α d−8 ⊗ σ 22010000 , β d+1 =β d−6 ⊗ σ 00002201 , α d =α d−8 ⊗ σ 22030000 , β d+2 =β d−6 ⊗ σ 00002203 .
(B10)
Every symmetry transform matrix O is extended to O⊗σ 00000000 . The structure of the Hamiltonian and the symmetry action remains the same under the extension. The single-particle Hilbert space of every root state is enlarged by 4 qubits, while that of the FSM is enlarged by 8 qubits, so the minimal copy number to obtain the FSM from the root state is always 16 times multiplied in every 8 dimensions higher.
FIG. 1 :
1Lattice model of Kitaev's Majorana chain.
FIG. 3 :
3Many-body energy levels of the two-site segment by exact diagonalization. From the weak interaction limit (left)
where η denotes the Majorana surface modes, and the Z T 2 symmetry act as T = Kiσ 2000 on η.
SPT (BSPT) state with inversion symmetry exists in all dimensions. The construction is based on the O(d + 2) non-linear σ model (NLSM) in (d + 1)dimensional space-time with a topological Θ-term at Θ = 2π
For higher spacial dimensions (d > 8), due to the Bott periodicity of the Clifford algebra, the Hamiltonian h (d) can be extended systematically from the Hamiltonian h (d−8) with 8 dimensions lower. Suppose h (d) = d i=1 i∂ i α i + mβ 0 + d+2 a=1 n a β a and h (d
. 2: Phase diagram of the interacting Majorana chain at ν = 8. The bulk criticality at the origin can be avoided if interaction is allowed. Each horizontal chain is four copies of the Majorana chain, equivalent to a Haldane chain. The vertical bound is the on-site spin-spin interaction. A and B labels the sites in a unit cell. Gray ovals mark out the spinsinglet dimers.
TABLE I :
IThe classification of Z P 2
TABLE II :
IIThe classification of U(1) × Z P
2 BSPT States
d mod 2 classification
0
Z
(d+2)/2
2
TABLE III :
IIIThe classification of U(1) × Z P 2
TABLE IV :
IVThe classification of Z T
2 × Z P
2 BSPT States
d mod 2 classification
0
Z
(d+2)/2
2
1
Z
(d+3)/2
2
TABLE V :
VThe classification of Z T 2 ×Z P 2 iFSPT states in each dimension d. The data of C d,2 (also see Appendix A) and the minimal copy number ν are also listed.d mod 8 rdim C d,2
ν
classification
0
rdim H(2
d
2 ) = 2
d+4
2
2
d
2
X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 87, 155114 (2013).
AcknowledgmentsWe would like to acknowledge the helpful discussions with Yuan-Ming Lu, Alexei Y. Kitaev and Xiao-Gang Wen. The authors are supported by the the David and Lucile Packard Foundation and NSF Grant No. DMR-1151208.Appendix A: Irreducible Representation of Clifford AlgebraThe generators of the real Clifford algebra C p,q can be represented by a set of real matrices {α 1 , · · · , α p ; iβ 1 , · · · , iβ q } anti-commuting with each otheramong which the α i matrices square to 1 (as α i α i = 1), and the iβ i matrices squares to −1 (as iβ i iβ i = −1). We adopt this notation such that both α i and β j matrices are Hermitian, and can be expressed as the direction product of Pauli matrices. α i 's are real and transpose symmetric, and there are p of them in the generators of C p,q ; while β i 's are imaginary and transpose anti-symmetric, and there are q of them in the generators of C p,q . The real Clifford algebras are isomorphic to the matrix algebras of real numbers R, complex numbers C or quaternions H. The first several examples include: C 0,0 ∼ = R whose irreducible representation is one-dimensional (just a real number), C 0,1 ∼ = C generated by {iσ 2 } giving a two-dimensional irreducible representation (2 × 2 real matrix), and C 0,2 ∼ = H generated by {iσ 12 , iσ 32 } giving a four-dimensional irreducible representation (4 × 4 real matrix). Here The complex Clifford algebra C n is much simpler, whose generators can be represented by a set of complex matrices {γ 1 , · · · , γ n }, satisfyingThe complex Clifford algebras are isomorphic to the matrix algebras of complex numbers C. For even n, C 2m ∼ = C(2 m ); and for odd n, C 2m+1 ∼ = C(2 m ) ⊕ C(2 m ) (or shorthanded as 2C(2 m )). Their real irreducible representations are of the dimensions: dim C(2 m ) = 2 m+1 and dim 2C(2 m ) = 2 m+2 .Appendix B: Fermion σ-Model in Each DimensionHere we enumerate the examples of fermion σ-model (FSM) in each dimension with explicit matrix representation, and show how the various symmetry action is embedded in the FSM. The action of the FSM in d-dimension takes the following general form S = d d+1 x 1 2 χ (i∂ 0 + h (d) )χ + 1 g (∂ µ n) 2 + · · · ,
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Sometimes this kind of states are also called "symmetry protected trivial" states in literature, depending on the taste and level of terminological rigor of authors. Sometimes this kind of states are also called "symmetry protected trivial" states in literature, depending on the taste and level of terminological rigor of authors.
use the notation σ ijk··· ≡ σ i ⊗ σ j ⊗ σ k ⊗ · · · for the Kronecker product (direct product) of the Pauli matrices. Through out this paper, we. where σ 1 , σ 2 , σ 3 stands for the three Pauli matrices respectively while σ 0 denotes the 2×2 identity matrixThrough out this paper, we use the notation σ ijk··· ≡ σ i ⊗ σ j ⊗ σ k ⊗ · · · for the Kronecker product (direct prod- uct) of the Pauli matrices, where σ 1 , σ 2 , σ 3 stands for the three Pauli matrices respectively while σ 0 denotes the 2×2 identity matrix.
can be viewed as the low energy field theory of spin-1/2 chains. Then the antiferromagnetic inter-chain coupling A will drive the system into a fully gapped state which is a direct product of inter-chain spin singlet on every site. When Θ = Π, n (1) ] and S[n (2)When Θ = π, both S[n (1) ] and S[n (2) ] can be viewed as the low energy field theory of spin-1/2 chains. Then the antiferromagnetic inter-chain coupling A will drive the system into a fully gapped state which is a direct product of inter-chain spin singlet on every site.
As one can see, the design of the coupling is not unique, any inter-layer coupling that locks odd number of n components anti-ferromagnetically will do the job to trivialize the BSPT state (for example A, B → −∞ is also a choice). but here let us stick to our current design and focus on the A, B → +∞ couplingAs one can see, the design of the coupling is not unique, any inter-layer coupling that locks odd number of n com- ponents anti-ferromagnetically will do the job to trivialize the BSPT state (for example A, B → −∞ is also a choice), but here let us stick to our current design and focus on the A, B → +∞ coupling.
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[
"On non-separated zero sequences of solutions of a linear differential equation",
"On non-separated zero sequences of solutions of a linear differential equation"
] |
[
"Igor Chyzhykov ",
"Jianren Long "
] |
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[] |
Let (z k ) be a sequence of distinct points in the unit disc D without limit points there. We are looking for a function a(z) analytic in D and such that f ′′ + a(z)f = 0 possesses a solution having zeros precisely at the points z k , and the resulting function a(z) has 'minimal' growth. We focus on the case of non-separated sequences (z k ) in terms of the pseudohyperbolic distance when the coefficient a(z) is of zero order, but sup z∈D (1 − |z|) p |a(z)| = +∞ for any p > 0. We established a new estimate for the maximum modulus of a(z) in terms of the functions n z (t) = |z k −z|≤t 1 and N z (r) = r 0 (nz(t)−1) + t dt. The estimate is sharp in some sense. The main result relies on a new interpolation theorem.
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10.1017/s0013091521000122
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[
"https://arxiv.org/pdf/2001.06378v1.pdf"
] | 210,713,843 |
2001.06378
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b0616364dd97f8fa6d60f6952e78ca8ff22aa0d5
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On non-separated zero sequences of solutions of a linear differential equation
15 Jan 2020 January 20, 2020
Igor Chyzhykov
Jianren Long
On non-separated zero sequences of solutions of a linear differential equation
15 Jan 2020 January 20, 2020interpolationunit discanalytic functionoscillation of solutiondifferential equationprescribed zeros MathSubjClass: 34C1030C1530H9930J99
Let (z k ) be a sequence of distinct points in the unit disc D without limit points there. We are looking for a function a(z) analytic in D and such that f ′′ + a(z)f = 0 possesses a solution having zeros precisely at the points z k , and the resulting function a(z) has 'minimal' growth. We focus on the case of non-separated sequences (z k ) in terms of the pseudohyperbolic distance when the coefficient a(z) is of zero order, but sup z∈D (1 − |z|) p |a(z)| = +∞ for any p > 0. We established a new estimate for the maximum modulus of a(z) in terms of the functions n z (t) = |z k −z|≤t 1 and N z (r) = r 0 (nz(t)−1) + t dt. The estimate is sharp in some sense. The main result relies on a new interpolation theorem.
Introduction
Let (z n ) be a sequence of different complex numbers in the unit disc D = {z : |z| < 1}, and let U(z, t) = {ζ ∈ C : |ζ − z| < t}. In the sequel, the symbol C stands for positive constants which depend on the parameters indicated, not necessarily the same at each occurrence.
The aim of the paper is twofold. On one hand, we are interested in zeros of solutions of f ′′ + a(z)f = 0, (1) where a(z) is an analytic function in D. On the other hand it leads us to some interpolation problems for corresponding classes of analytic functions in D.
Oscillation of solution
It was proved byŠeda [15] that given a sequence of distinct complex numbers (z n ) with no finite limit points there exists an entire function a(z) such that the equation (1) has an entire solution f with the zero sequence (z n ). This result was recently generalized for an arbitrary domain G ⊂ C when the condition f (z n ) = 0 is replaced with
f (z k ) = b k ,(2)
(b n ) being an arbitrary sequence [17]. We are interested in description of zero sequences (z n ) of solutions of (1) where a(z) belongs to some growth class. Let A −p be the Banach space of analytic functions in D with the norm a A −p := sup |z|<1 (1 − |z| 2 ) p |a(z)|.
In [10] the authors described behavior of the coefficient when all zero-free solutions belong to some space. We restrict ourselves to the case when f has infinitely many zeros is in the focus of the paper.
Let ϕ(z, w) = z−w 1−zw . Let σ(z, w) = |ϕ z (w)| denote the pseudohyperbolic distance in D. A sequence (z n ) in the unit disc is called uniformly separated if inf j n =j σ(z n , z j ) > 0.
A positive Borel measure µ is a Carleson measure if and only if there exists a constant K such that µ(Q δ ) ≤ Kδ for any Carleson box Q δ = {ζ ∈ D : |ζ| ≥ 1 − δ, | arg ζ − ϕ| ≤ πδ}.
The following result describes coefficients a(z) such that zero sequence of a solution (1) is uniformly separated.
Theorem A. A sequence Z in the unit disc is the zero-sequence of a nontrivial solution of (1) such that |a(z)| 2 (1 − |z| 2 ) 3 dm(z) is a Carleson measure if and only if Z is uniformly separated.
The necessity is proved by Gröhn, Nicolau, Rättyä in [10]. The sufficiency is established recently by Gröhn in [8].
The following problem was formulated in [12]. Problem. Let (z k ) be a sequence of distinct points in D without limit points there. Find a function a(z), analytic in D such that (1) possesses a solution having zeros precisely at the points z k . Estimate the growth of the resulting function a(z).
The next result is closely connected to the problem. In order to formulate it we need more notation. A sequence Z = (z n ) in the unit disc is called separated if inf n =j σ(z n , z j ) > 0. The uniform density of a sequence (z n ) ( [16]) is defined by
D + (Z) = lim sup r→1− sup z∈D 1 2 <σ(z,z j )<r log 1 σ(z,z j ) log 1 1−r . Theorem B ([8, Theorem 1]). If Z = (z k )
is a separated sequence in the unit disc with D + (Z) < 1 then there exists a ∈ A −2 such that (1) admits a nontrivial solution that vanishes on Z.
Conversely, if a ∈ A −2 and f is a nontrivial solution of (1) whose zerosequence is Z, then Z is separated and contains at most one point if
a A −2 ≤ 1, while D + (Z) ≤ (2π + 1) C (1−C) 2 , where C = 1 − 2 √ a A −2 a A −2 +1 .
Remark 1. The proof of Theorem B uses essentially the interpolation result by K. Seip. In the first part of Theorem B, Z is a subset of the zero set of f .
For an analytic function f in D we denote M(r, f ) = max{|f (z)| : |z| = r}, r ∈ (0, 1). Let n ζ (t) = |z k −ζ|≤t 1 be the number of the members of the sequence (z k ) satisfying |z k − ζ| ≤ t. We write
N ζ (r) = r 0 (n ζ (t) − 1) + t dt.
Let ψ : [1, +∞) → R + be a nondecreasing function. We definẽ
ψ(x) = x 1 ψ(t) t dt.
Remark 2. In the case ψ(x) = log p x, p ≥ 0 we getψ(x) = 1 p+1 log p+1 x, and in the case
ψ(x) = x ρ , ρ > 0 we haveψ(x) = 1 ρ (x ρ − 1).
Let, in addition, ψ have finite order in the sense of Pólya ( [7]), i.e.
ψ(2x) = O(ψ(x)), x → +∞.(3)
Theorem C ( [3]). Let (z n ) be a sequence of distinct complex numbers in D.
Assume that for some nondecreasing unbounded function ψ :
[1, +∞) → R + satisfying (3) we have ∃C > 0 : ∀n ∈ N N zn 1 − |z n | 2 ≤ Cψ 1 1 − |z n | .(4)
Then there exists an analytic function a in D satisfying
∃C > 0 : log M(r, a) ≤ Cψ 1 1 − r , r ∈ (0, 1)
such that (1) possesses a solution f having zeros precisely at the points z k , k ∈ N.
Corollary D. If for some ρ > 0 a sequence (z k ) satisfies the condition
∃C > 0 : N z k 1 − |z k | 2 ≤ C 1 1 − |z k | ρ ,
then there exists a function a analytic in D satisfying log M(r, a) = O((1 − r) −ρ ), r ∈ (0, 1) such that possesses a solution f having zeros precisely at the points z k , k ∈ N.
The following theorem is based on an example due to J. Gröhn and J. Heittokangas [9]. It shows that the statement of the corollary is sharp.
Theorem E. For arbitrary ρ > 0 there exists a sequence of distinct numbers {z n } in D with the following properties:
i) N z k 1−|z k | 2 ≤ C 1 1−|z k | ρ , k ∈ N;
ii) (z k ) cannot be the zero sequence of a solution of (1), where log M(r, a) = O((1 − r) −ρ+ε 0 ) for any ε 0 > 0.
In [9] the case when the coefficient a ∈ A −p , p > 2, is considered. Some other problems on zeros of solutions are considered in a survey [13].
The aforementioned results give a complete solution to the Problem in the cases when a ∈ A −2 and the order a is finite and positive. In the intermediate cases, when a is of zero order, but outside of A −2 , zero sets of solutions of (1) is not described completely. The aim of the paper is to fill this gap. In particular, we improve Theorem C and obtain sharp, in some sense, estimates of a(z) in terms of the zero distribution of a solution of (1). Our proof relies on a new interpolation result.
Interpolation in the unit disc
For the space A −n , n > 0, an interpolation set is defined by the condition that for every sequence (2). These sets were described by K. Seip in [16]. Namely, necessary and sufficient that (z k ) be an interpolation set for A −n is that (z n ) be separated, i.e. inf j =k σ(z k , z j ) > 0, and D + (Z) < n.
(b k ) with (b k (1 − |z k |) n ) ∈ l ∞ there is a function f ∈ A −n satisfying
In 2002 A. Hartmann and X. Massaneda [11] proved that condition
∃δ ∈ (0, 1) ∃C > 0 ∀n ∈ N : N zn (δ(1 − |z n |)) ≤ η C 1 − |z n | .
is necessary and sufficient for Z to be an interpolation set for a class of growth functions η containing all power functions. They also describe interpolation sequences in the unit ball in C n in the similar situation. Note that the proof of sufficiency in [11] is based on L 2 -estimate for the solution to a∂-equation and is non-constructive.
The following theorem was established in [3]. It gives sufficient conditions for interpolation sequences in classes of analytic functions of moderate growth in the unit disc and complements Hartmann and Massaneda's result when ψ(t) grows slowly than any power function.
Theorem F. Let (z n ) be a sequence of distinct complex numbers in D. Assume that for some nondecreasing unbounded function ψ : [1, +∞) → R + satisfying (3) the condition (4) is valid. Then for any sequence (b n ) satisfying
∃C > 0 : log |b n | ≤ Cψ 1 1 − |z n | , n ∈ N,(5)
there exists an analytic function f in D with the property (2) and
∃C > 0 : log M(r, f ) ≤ Cψ 1 1 − r .(6)
The class R consists of functions ψ : [1, +∞) → R + which are nondecreasing, and such thatψ(r) = O(ψ(r)) as r → +∞. We note that the power function x ρ , ρ > 0, belongs to R. The previous theorem becomes a criterion if ψ ∈ R ([3, Theorem 5]).
In 2007 A. Borichev, R. Dhuez and K. Kellay [2] solved an interpolation problem in classes of functions of arbitrary growth in both the complex plane and the unit disc. Following [2]
let h : [0, 1) → [0, +∞) such that h(0) = 0, h(r) ↑ ∞ (r → 1−).
Denote by A h and A p h , p > 0 the Banach spaces of analytic functions on D with the norms
f h = sup z∈D |f (z)|e −h(z) < +∞, f h,p = D |f (z)| p e −ph(|z|) dm 2 (z) 1 p ,
respectively. We then suppose that h ∈ C 3 ([0, 1)), ρ(r) := (∆h(r)) − 1 2 ց 0, and ρ ′ (r) → 0 as r → 1−, for all K > 0:
ρ(r + x) ∼ ρ(r) for |x| ≤ Kρ(r), r → 1−(7)
provided that Kρ(r) < 1 − r, and either ρ(r)(1 − r) −c increases for some finite c or ρ ′ (r) log ρ(r) → 0 as r ↑ 1. Note that these assumptions imply h(r)/ log 1 1−r → +∞ (r → 1−). Given such an h and a sequence Z = (z k ) in D denote by
D + ρ (Z) = lim sup R→∞ lim sup |z|→1− card(Z ∩ U(z, Rρ(z))) R 2 . Theorem G (Theorem 2.3 [2]). A sequence Z is an interpolation set for A h (D) if and only if D + ρ (Z) < 1 2 and inf k =n |z k − z n | min{ρ(|z k |), ρ(|z n |)} > 0.(8)
The similar description holds for interpolation sets for the classes A p h (D), p > 0 ( [2]).
Main results
In this paper we are mostly interested in the case where ψ(r) is a slowly growing function unbounded with respect to log 1 1−r as r → 1−, in particular, ψ ∈ R. Theorem F seems no longer to be sharp for such functions ψ.
For s = [ρ] + 1, where ρ = ρ * [ψ], we consider a canonical product of the form
P (z) = P (z, Z, s) = ∞ n=1 E 1 − |z n | 2 1 −z n z , s ,(9)
where
E(w, 0) = 1 − w, E(w, s) = (1 − w) exp{w + w 2 /2 + · · · + w s /s}, s ∈ N.
This product is an analytic function in D with the zero sequence Z = (z n ) provided
zn∈Z (1 − |z n |) s+1 < ∞. Remark 3. The Pólya order ρ * [ψ] ([7]
) characterized by the condition that for any ρ > ρ * [ψ], we have
ψ(Cx) ≤ C ρ ψ(x), x, C → ∞.(10)
The following result allows to relax the assumption on N zn (t) in comparison with Theorem F. Theorem 1. Let (z n ) be a sequence of distinct complex numbers in D. Assume that for some nondecreasing unbounded function ψ : [1, +∞) → R + satisfying (3) we have that (14) and either
∃C > 0 : ∀n ∈ N N zn 1 − |z n | 2 ≤ Cψ 1 1 − |z n | ,(11)
or
∀n ∈ N : − log (1 − |z n |)|P ′ (z n )| ≤ Cψ 1 1 − |z n | ,(12)
or
∀n ∈ N : − log |B n (z n )| ≤ Cψ 1 1 − |z n | ,(13)
holds, where B n (z) = P (z)/E( 1−|zn| 2 1−znz , s), P is the canonical product defined by (9), s ≥ [ρ] + 1, where ρ is Polýa order of ψ. Then for any sequence (b n ) satisfying (5) there exists an analytic function f in D with the properties (2) and (6).
Hypotheses similar to (12) are frequently used in interpolation problems (e.g. [1]) The next theorem addresses the problem formulated in the introduction. such that (1) possesses a solution f having zeros precisely at the points z k , k ∈ N.
Corollary 3. If for some ρ > 0 and β > 0 a sequence (z k ) satisfies the conditions This corollary is sharp in the following sense Theorem 4. For arbitrary η 1 , η 2 > 0 there exists a sequence of distinct numbers (z n ) in D with the following properties:
∃C > 0 : n z k 1 − |z k | 2 ≤ C log β 1 1 − |z k | , ∃C > 0 : N z k 1 − |z k | 2 ≤ C log β+1 1 1 − |z k | ,i) ∃C > 0 : n z k 1−|z k | 2 ≤ C log η 1 1 1−|z k | , k ∈ N; ii) ∃C > 0 : N z k 1−|z k | 2 ≤ C log 1+η 1 +η 2 1 1−|z k | , k ∈ N;
iii) (z k ) cannot be the zero sequence of a solution of (1) where log M(r, a) = O(log 1+η 1 1−r ), η < η 2 . Since Theorem B effectively uses the notion of the uniform density, one may ask whether it is possible to use D + ρ density to solve the Problem. The next theorem gives an estimate of the growth of a(z) under an assumption in terms of the density introduced by Borichev, Dhuez, and Kellay [2]. provides that h(r)(1 − r) q is bounded for some finite q > 0.
Proofs of the results
Proof of Theorem 1. We follow the scheme of the proof of Theorem F from [3]. It follows from the estimate (14) and [3,Lemma 9
] that ∞ n=1 1 − |z n | 2 1 −z n z s+1 ≤ C(s)ψ 1 1 − |z| , z ∈ D.
The following two lemmas are important in our arguments.
Lemma H ([3]).
For an arbitrary δ ∈ (0, 1), any sequence Z in D satisfying
z k ∈Z (1 − |z k |) s+1 < ∞, s ∈ Z + there exists a positive constant C(δ, s) | log |B k (z k )| + N z k (δ(1 − |z k |))| ≤ C(δ, s) ∞ n=1 1 − |z n | 2 1 −z n z s+1 , k → +∞.
The next proposition compares some conditions frequently used in interpolation problems (cf. [1]).
Lemma I ([3, Proposition 11]). Given a function ψ ∈ R for
∃C > 0 ∀z ∈ D : N z 1 − |z| 2 ≤ Cψ 1 1 − |z|
it is necessary and sufficient that
∃C > 0 ∀z ∈ D : n z 1 − |z| 2 ≤ Cψ 1 1 − |z| ,(14)
and
∀n ∈ N : | log (1 − |z n |)|P ′ (z n )| | ≤ Cψ 1 1 − |z n | ,(15)
where P is the canonical product defined by (9), s = [ρ * ] + 1, where ρ * is Polya's order of ψ.
Lemma H directly implies that under the assumption (14) the conditions (11) and (13) are equivalent.
Moreover, one has
P ′ (z n )(1 − |z n | 2 ) z n = −B n (z n ) exp 1 + 1 2 + · · · + 1 s ,(16)
which yields equivalence of (15) and (13). Therefore, under assumption (14) any of the hypotheses (11)-(13) imply the other two. Hence, we may assume that the conditions (11)-(13) hold. Taking into account [3, Lemma 9], which gives an upper estimate of a canonical product, and Lemma H we deduce
| log |B k (z k )|| ≤ C(δ, s)ψ 1 1 − |z k | , k → +∞.(17)
Consider the interpolating function
f (z) = ∞ n=1 b n z − z n P (z) P ′ (z n ) 1 − |z n | 2 1 −z n z sn−1 ,
where is an appropriate increasing sequence of natural numbers (s n ) (see [3]
1 −z n z z n P (z) z − z n ≤ exp Cψ 1 1 − |z| .
Therefore, using our assumption on (b n ) and the above estimates we deduce
|f (z)| = ∞ n=1 b n P (z)(1 −z n z) z n (z − z n )z n (1 − |z n | 2 )P ′ (z n ) 1 − |z n | 2 1 −z n z sn ≤ ≤ ∞ n=1 exp Cψ 1 1 − |z n | exp Cψ 1 1 − |z| 1 |B n (z n )| 1 − |z n | 2 |1 −z n z| sn ≤ ≤ exp Cψ 1 1 − |z| ∞ n=1 exp Cψ 1 1 − |z n | 1 − |z n | 2 |1 −z n z| sn .(18)
The last estimate coincides with inequality (24) from [3], and the end of the proof are in lines of that from [3, pp. 328-329].
Proof of Theorem 4. Let η 1 , η 2 > 0 be given. Let ε n = e −(n log 3) 1+η 2 , m n = [(n log 3) η 1 ], n ∈ N. Let (z n,k ) be the sequence defined by z n,k = 1 − 3 −n + kε n /m n , 0 ≤ k ≤ m n − 1.
Then n z n,k (t) = m n ≍ log η 1
1 1 − |z n,k | , ε n ≤ t ≤ 1 − |z n,k | 2 , 0 ≤ k ≤ m n −1, n ∈ N,
and the assertion i) holds. Further,
1−|z n,k | 2 εn (n z n,k (t) − 1) + t dt ≍ m n log 1 − |z n,k | 2ε n ≍ (n log 3) η 1 +1+η 2 ,(19)
for 0 ≤ k ≤ m n − 1, n ∈ N. On the other hand, for the same range of n and k we have
εn 0 (n z n,k (t) − 1) + t dt ≤ εn 0 2tm n ε n t dt = 2m n ≍ (n log 3) η 1 .(20)
Combining (19) and (20) we deduce
N z n,k 1 − |z n,k | 2 = εn 0 + 1−|z n,k | 2 εn (n z n,k (t) − 1) + t dt ≍ (n log 3) η 1 +1+η 2 ∼ ∼ log η 1 +1+η 2 1 1 − |z n,k | , 0 ≤ k ≤ m n − 1, n → ∞.
Thus, assertion ii) is proved.
To prove assertion iii) we assume on the contrary that there exists a solution f = Be g of (1) having the zero sequence (z n ), where B is the Blaschke product, and such that
log M(r, a) ≤ C 0 log 1+η 1 1 − r , r ∈ [0, 1),(21)
where η 2 > η > 0 and C 0 is a positive constant. The function ϕ(r) = exp log 1+η 1 1−r has infinite logarithmic order, that is
lim sup r→1− log ϕ(r) log log 1 1−r = ∞.
Using the notation from [4] σ ϕ (M 1/2 (r, a) 1/2 (1 − r)) := lim sup Then by [4,
Theorem 1] lim sup r→1− log T (r,f ) log ϕ(r) ≤ C 0 2 . Hence log log M(r, f ) ≤ log T 1 + r 2 , f 2 1 − r ≤ ≤ C 0 2 + o(1) log ϕ(r) ∼ C 0 2 log 1+η 1 1 − r , r → 1 − .(22)
We write R n = 1 − 2 · 3 −n−1 and δ = = 0, and applying Lemma H we deduce that
ℜg(R n e iθ ) ≤ log M(R n , f ) + | log |B(R n e iθ )|| ≤ ≤ log M(R n , f ) + C 1 4 , 1 j,k 1 − |z j,k | 2 |1 − z j,k R n e iθ | ≤ ≤ e C 0 2 +o(1)) log 1+η 1 1−Rn + C 1 4 , 1 |z j,k |≤Rn 1 + 1 1 − R n |z j,k |>Rn (1 − |z j,k |) ≤ ≤ exp C 0 2 + o(1) log η 1 1 − R n + O log 1+η 1 1 1 − R n ∼ ∼ exp C 0 2 + o(1) log η 1 1 − R n , n → +∞.(23)
Since ℜg is harmonic, B(r, ℜg) = max{ℜg(re iθ ) : θ ∈ [0, 2π]} is an increasing function. It follows from (23) and the relation 1
− R n ≍ 1 − R n+1 that B(r, ℜg) ≤ exp C 0 2 + o(1) log 1+η 1 1 − r , r → 1 − .M(r, g) ≤ exp C 0 2 + o(1) log 1+η 1 1 − r , r → 1 − .
Then, applying Cauchy's integral formula, we obtain
M(r, g ′ ) ≤ C 1 − r exp C 0 2 + o(1) log 1+η 1 1 − r , r ↑ 1, which yields log M(r, g ′ ) ≤ C 0 2 + o(1) log 1+η 1 1 − r , r → 1 − .(24)
We write (cf. [12])
−g ′ (z n,0 ) = B ′′ (z n,0 ) 2B ′ (z n,0 ) = mn−1 k=1 1 z n,0 − z n,k 1 − |z n,k | 2 1 −z n,k z n,0 + + j =n m j −1 k=0 1 z j,k − z n,0 1 − |z j,k | 2 1 −z j,k z n,0 =: I 1 + I 2 .(25)
It is easy to see that
|I 1 | ≥ 1 2 m n ε n mn−1 k=1 1 k ≥ ≥ C exp log η 2 +1 1 1 − |z n,0 | log η 1 1 1 − |z n,0 | log log 1 1 − |z n,0 | , n → +∞.(26)
Then
|I 2 | ≤ j =n m j −1 k=0 1 |z j,k − z n,0 | 1 − |z j,k | 2 |1 −z j,k z n,0 | ≤ ≤ n−1 j=1 m j −1 k=0 4 1 − |z j,0 | + ∞ j=n+1 m j −1 k=0 4(1 − |z j,k |) (1 − |z n,0 |) 2 ≤ ≤ 4 n−1 j=1 3 j+1 (j log 3) η 1 + 4 (1 − |z n,0 |) 2 ∞ j=n+1 3 −j (j log 3) η 1 ≤ ≤ C3 n (n log 3) η 1 + C (1 − |z n,0 |) 2 (n log 3) η 1 3 n ≤ C log η 1 1 1−|z n,0 | 1 − |z n,0 | . Therefore |g ′ (z n,0 )| = B ′′ (z 2n ) 2B ′ (z 2n ) ≥ ≥ C exp log η 2 +1 1 1 − |z n,0 | log η 1 1 1 − |z n,0 | log log 1 1 − |z n,0 | , n → +∞.
Hence,
log M(|z n,0 |, g ′ ) ≥ (1 + o(1)) log η 2 +1 1 1 − |z n,0 | , n → +∞.(27)
This contradicts to (24). The theorem is proved.
Proof of Theorem 5. Let D + ρ (Z) < D < ∞. It follows from the definition of D + ρ (Z) that under the assumption of Theorem G that there exist R 0 > 0 and r 0 ∈ (0, 1) such that n z (Rρ(z)) < DR 2 , R > R 0 and |z| ∈ (r 0 , 1). Separation condition (8) and the property (7) imply that each disk U(z, ρ(z)/3) contains at most one point z k , and there exists a constant D 1 ≥ D such that n z (Rρ(z)) ≤ D 1 R 2 for 0 < R ≤ R 0 and |z| ∈ (r 1 , 1) for some r 1 ∈ (r 0 , 1). Therefore n z Rρ(z) < D 1 R 2 , R > 0, |z| ∈ (r 1 , 1).
The last estimate directly implies
n z 1 − |z| 2 ≤ D 1 4 (1 − |z|) 2 ρ 2 (z) = D 1 4 (1 − |z|) 2 ∆h(r), |z| = r ∈ (r 1 , 1).
We write ψ(t) = t −2 ∆h(r)
r=1−t −1
. It follows from the definition of σ(r) that ψ is a nondecreasing function on [1, ∞) and ψ(2t) = O(ψ(t)), t ≥ 2. Let us estimateψ(T ) = T 1 ψ(t) t dt. We havẽ
ψ(T ) = T 1 ∆h(r) r=1−t −1 t 3 dt = 1−T −1 0 (1 − r)∆h(r) dr = = 1−T −1 0 (1 − r) (rh ′ (r)) ′ r dr = = (1 − r)h ′ (r) 1−T −1 0 + 1−T −1 0 h ′ (r) r dr ≤ ≤ h ′ (1 − T −1 ) T + 2h(1 − T −1 ), T ≥ 2. Since h ′ (r) h(r) = O((1 − r) −1 ), r → 1−, we deduce that ψ 1 1 − r = O h(r) , r → 1 − .
We then estimate N z k (t) for t ≤ (1 − |z k |)/2. Using the separation condition (8) and estimate (28) we deduce
N z k ((1 − |z k |)/2) = (1−|z k |)/2 0 (n ζ (t) − 1) + t dt ≤ (1−|z k |)/2 ρ(z k )/2 n ζ (t) t dt = = (1−|z k |)/(2ρ(z k )) 1/2 n ζ (ρ(z k )τ ) τ dτ ≤ (1−|z k |)/(2ρ(z k )) 1/2 D 1 τ dτ ≤ ≤ D 1 8 (1 − |z k |) 2 ρ 2 (z k ) = 1 2 ψ 1 1 − r .
Since n z ( 1−|z| 2 ) = O(ψ( 1 1−|z| )), |z| ∈ (r 1 , 1), applying Lemma 9 from [3] we get the following estimate of the canonical product log |P (z)| ≤ Cψ
1 1 − |z| ≤ Ch(|z|).(29)
Any analytic function f in D with the zero sequence Z = (z k ) can be written in the form f (z) = P (z)e g(z) , where g is analytic in D. If f is a solution of (1), then
P ′′ + 2P ′ g ′ + (g ′2 + g ′′ + a)P = 0,(30)
and, consequently
g ′ (z k ) = − P ′′ (z k ) 2P ′ (z k ) =: b k , k ∈ N.(31)
Therefore, in order to find a solution of (1) with the zero sequence Z we have to find an analytic function h = g ′ solving the interpolation problem h(z k ) = b k , k ∈ N. Using Cauchy's integral theorem and (29) we deduce
|P ′′ (z k )| ≤ 8 (1 − |z k |) 2 max |z|= 1+|z k | 2 |P (z)| ≤ 8 (1 − |z k |) 2 e Cψ( 2 1−|z k | ) .
On the other hand, (16) and (17) becauseψ(t)/ log t → +∞ (t → +∞). Since the assumptions of Theorem F are satisfied, there exists a function h analytic in D such that h(z k ) = b k and log M(r, h) ≤ Cψ( 1 1−r ), r → 1−, i.e. log M(r, g ′ ) ≤ Cψ( 1 1−r ), r → 1−. Then, applying Cauchy's integral theorem once more, we get that
M(r, g ′′ ) ≤ 2 1 − r M 1 + r 2 , g ′ ≤ e Cψ( 1 1−r ) , r → 1 − .
From (30) we obtain |a(z)| ≤ P ′′ (z) P (z) + 2|g ′ (z)| P ′ (z) P (z) + |g ′ (z)| 2 + |g ′′ (z)|.
It follows from results of [5] (or [6]) that for any δ > 0 there exists a set E δ ⊂ [0, 1) such that
max |P ′′ (z)| |P (z)| , |P ′ (z)| |P (z)| ≤ 1 (1 − |z|) q , |z| ∈ [0, 1) \ E δ ,
where q ∈ (0, +∞), and m 1 (E δ ∩ [r, 1)) ≤ δ(1 − r) as r ↑ 1. Thus,
|a(z)| ≤ eCψ ( 1 1−|z| ) , |z| ∈ [0, 1) \ E.(32)
Since M(r, a) increases, condition (3) and Lemma 4.1 from [6] imply that inequality (32) holds for all z ∈ D for an appropriate choice ofC.
Remark 5. Though the hypotheses of Theorem 5 are similar to those of Theorem G we do not use the interpolating function constructed in [2]. We need an interpolating function f to have the property f (z) ≍ dist(z, Z f )p(z), where Z f is the zero set of F , p(z) some nonvanishing continuous function in D. But it seems that it is not the case.
Theorem 2 .
2Let conditions of Theorem 1 be satisfied. Then there exists an analytic function a in D satisfying ∃C > 0 : log M(r, a) ≤ Cψ 1 1 − r , r ∈ (0, 1)
then there exists a function a analytic in D and satisfying log M(r, a) = O log β+1 1 1 − r , r ∈ (0, 1)such that possesses a solution f having zeros precisely at the points z k , k ∈ N.
Theorem 5 .
5Let h ∈ C 2 [0, 1) be an increasing function with h(0) = 0 and such that for ρ(r) = (∆h(r)) − 1 2 (7) holds and(1−r)h ′ (r) h(r) is bounded. Let the function σ(r) = (1 − r) 2 /ρ 2 (r) ր ∞ as r ր 1−, and satisfy σ((1 + r)/2) = O(σ(r)), r ∈ [1/2, 1). Suppose that D + ρ (Z) < ∞ and(8)holds. Then there exists an analytic function a in D satisfying ∃C > 0 : log M(r, a) ≤ Ch(r), r ∈ (0, 1) such that (1) possesses a solution f having zeros precisely at the points z k , k ∈ N.
Remark 4 .
4Note that the assumption (1 − r)/ρ(r) → ∞ as r → 1− implies that h(r)/ log 1 1−r → ∞ as r → 1−. On the other hand, (1−r)h ′ (r) h(r)
|z k | ) .Hence|b k | = P ′′ (z k ) 2P ′ (z k ) |z k | ) , k ∈ N,
The last estimate and Caratheodory's inequality ([14, Chap.1, §6]) imply
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On the finiteness of ϕ-order of solutions of linear differential equations in the unit disc. I Chyzhykov, J Heittokangas, J Rättyä, J. d'Analyse Math. 1091Chyzhykov I., Heittokangas J., Rättyä J., On the finiteness of ϕ-order of solutions of linear differential equations in the unit disc, J. d'Analyse Math. 109 (1), 163-196 (2010)
Linear differential equations and logarithmic derivative estimates. I Chyzhykov, G Gundersen, J Heittokangas, Proc. London Math. Soc. 863Chyzhykov I., Gundersen G. , Heittokangas J. , Linear differential equa- tions and logarithmic derivative estimates, Proc. London Math. Soc. 86 (3), 735-754 (2003)
Sharp logarithmic derivative estimates with applications to ODE's in the unit disc. I Chyzhykov, J Heittokangas, J Rättyä, J. Australian Math. Soc. 88Chyzhykov I., J. Heittokangas, J. Rättyä, Sharp logarithmic derivative estimates with applications to ODE's in the unit disc, J. Australian Math. Soc. 88, 145-167 (2010)
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|
[] |
[
"Logarithmic corrections to 2 scaling in lattice QCD with Wilson and Ginsparg-Wilson quarks Logarithmic corrections to 2 scaling in lattice QCD with Wilson and GW quarks",
"Logarithmic corrections to 2 scaling in lattice QCD with Wilson and Ginsparg-Wilson quarks Logarithmic corrections to 2 scaling in lattice QCD with Wilson and GW quarks"
] |
[
"Nikolai Husung [email protected] ",
"Peter Marquard [email protected] ",
"Rainer Sommer [email protected] ",
"Nikolai Husung ",
"\nDeutsches Elektronen-Synchrotron DESY\nPhysics and Astronomy\nInstitut für Physik\nUniversity of Southampton\nPlatanenallee 615738, SO17 1BJZeuthen, SouthamptonGermany, UK\n",
"\nHumboldt-Universität zu Berlin\nNewtonstr. 1512489BerlinGermany\n"
] |
[
"Deutsches Elektronen-Synchrotron DESY\nPhysics and Astronomy\nInstitut für Physik\nUniversity of Southampton\nPlatanenallee 615738, SO17 1BJZeuthen, SouthamptonGermany, UK",
"Humboldt-Universität zu Berlin\nNewtonstr. 1512489BerlinGermany"
] |
[
"The 38th International Symposium on Lattice Field Theory"
] |
We analyse the leading logarithmic corrections to the 2 scaling of lattice artefacts in QCD, following the seminal work of Balog, Niedermayer and Weisz in the O(n) non-linear sigma model. Limiting the discussion to contributions from the action, the leading logarithmic corrections can be determined by the anomalous dimensions of mass-dimension 6 operators. These operators form a minimal on-shell basis of the Symanzik Effective Theory. We present results for non-perturbatively O( ) improved Wilson and Ginsparg-Wilson quarks.
| null |
[
"https://arxiv.org/pdf/2111.04679v2.pdf"
] | 243,847,641 |
2111.04679
|
67995f38bc63e1700312a7636c22f1e5ea7c23a8
|
Logarithmic corrections to 2 scaling in lattice QCD with Wilson and Ginsparg-Wilson quarks Logarithmic corrections to 2 scaling in lattice QCD with Wilson and GW quarks
LATTICE2021 26th-30th July, 2021
Nikolai Husung [email protected]
Peter Marquard [email protected]
Rainer Sommer [email protected]
Nikolai Husung
Deutsches Elektronen-Synchrotron DESY
Physics and Astronomy
Institut für Physik
University of Southampton
Platanenallee 615738, SO17 1BJZeuthen, SouthamptonGermany, UK
Humboldt-Universität zu Berlin
Newtonstr. 1512489BerlinGermany
Logarithmic corrections to 2 scaling in lattice QCD with Wilson and Ginsparg-Wilson quarks Logarithmic corrections to 2 scaling in lattice QCD with Wilson and GW quarks
The 38th International Symposium on Lattice Field Theory
LATTICE2021 26th-30th July, 2021Zoom/Gather@Massachusetts Institute of Technology * Speaker Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ arXiv:2111.04679v2 [hep-lat]
We analyse the leading logarithmic corrections to the 2 scaling of lattice artefacts in QCD, following the seminal work of Balog, Niedermayer and Weisz in the O(n) non-linear sigma model. Limiting the discussion to contributions from the action, the leading logarithmic corrections can be determined by the anomalous dimensions of mass-dimension 6 operators. These operators form a minimal on-shell basis of the Symanzik Effective Theory. We present results for non-perturbatively O( ) improved Wilson and Ginsparg-Wilson quarks.
Introduction
Symanzik Effective Field Theory (SymEFT) [2][3][4][5] can be used to describe the lattice artifacts of lattice QCD for asymptotically small lattice spacings . In contrast to the (classical) 2 ansatz commonly used in continuum extrapolations 0, the leading asymptotic lattice spacing dependence is actually of the form 2 [¯2(1/ )]Γ due to quantum corrections, whereΓ is a real constant and¯is the running coupling. We assume here the use of fully O( ) improved lattice actions throughout. KnowingΓ is required to put continuum extrapolations on more solid grounds and to rule out any trouble arising from distinctly negative values forΓ as there is no theoretical lower bound on this value. A particularly problematic example was found for the O (3) non-linear sigma model, where such an analysis [6,7] was performed for the first time yieldingΓ = −3. To highlight the impact a non-zeroΓ can have, we added the oversimplified sketch in fig. 1 for the case of threeflavour QCD.
Symanzik Effective Theory
For a more complete picture of SymEFT see [8] as we give here only a short summary of the main concepts. To describe the lattice artifacts we start from the effective Lagrangian
ℒ Sym = − 1 2 2 0 tr( ) +Ψ + Ψ + 2 ∑︁ O + O( 3 ),(1)
which is just the (Euclidean) continuum QCD Lagrangian for f quark flavours Ψ with additional O( 2 ) corrections. The matching coefficients depend on the choice for the lattice discretisation.
(Only) For tree-level matching it suffices to naively expand the lattice action in the lattice spacing. The basis of operators O must be chosen such that it parametrises all lattice artifacts originating from the lattice action up to higher order corrections in the lattice spacing. Being interested in either Ginsparg-Wilson (GW) or Wilson [9,10] quarks for the lattice discretisation yields the following symmetry constraints on our minimal operator basis • SU( ) gauge symmetry,
• invariance under Euclidean reflections,
• invariance under charge conjugation,
• H(4) lattice symmetry, i.e. continuum O(4) symmetry is broken due to reduced rotational symmetry,
• flavour symmetries, SU( f ) L × SU( f ) R × U(1) for massless GW quarks and U( f ) V for massless Wilson quarks.
Notice that SU( f ) L × SU( f ) R × U(1) ⊂ U( f ) V such that the minimal basis of GW quarks is a subset of the full minimal basis needed for Wilson quarks. Due to being only interested in on-shell physics we can make use of the continuum equations of motion to reduce the operator basis further [11]. The minimal on-shell operator basis for the massless case (or sufficiently small quark masses) then is the following [12][13][14]
O 1 = 1 2 0 tr( ), O 2 = 1 2 0 ∑︁ tr( ), O 3 = ∑︁Ψ 3 Ψ, O ≥4 = 2 0 (ΨΓ Ψ) 2 ,(2)
where
Γ 4−7 ∈ { , 5 } ⊗ {1, } and Γ 8−13 ∈ {1, 5 , } ⊗ {1, } with = 2 [ , ]
. The operators O 2 and O 3 both break O(4) symmetry. For massless GW quarks we only need O ≤7 , while massless Wilson quarks require the entire set of operators listed here. For the general massive case we get additional massive operators, that are listed and discussed in [15,16].
Leading powers in the coupling
For an arbitrary Renormalisation Group invariant (RGI) spectral quantity P we may use the operator basis to write the leading lattice artifacts as
P ( ) = P (0) − 2 ∑︁¯O P O (1/ ) × 1 + O(¯2(1/ )) + O( 3 ),(3)
where¯O is the tree-level matching coefficient and P O contains the matrix elements of interest with an additional insertion of ∫ d 4 O ( ). The remaining scale dependence of P O (1/ ), where 1/ is the relevant renormalisation scale for lattice artifacts, is governed by the renormalisation group equation
P O ( ) d = − O 0¯2 ( ) + O(¯4) P O ( ),(4)
where O 0 is the 1-loop coefficient of the anomalous dimension matrix. In general O 0 is not diagonal, but in our case we can make a change of basis O → B such that B 0 = diag ( 0 ) 1 , . . . , ( 0 ) becomes diagonal. In turn this allows to introduce the RGI, where all scale dependence is absorbed into some perturbatively known prefactor
P B (1/ ) = [2 0¯2 (1/ )]ˆP B ;RGI × 1 + O(¯2(1/ )) ,ˆ= ( B 0 ) 2 0 ,(5)
where 0 is the 1-loop coefficient of the -function and the factor 2 0 in front of¯2(1/ ) is the common choice for the normalisation. Taking the leading order matching B (¯2) = For a non-spectral quantity also corrections from the local fields involved must be taken into account, which cancel out for spectral quantities.
1PI p 1 p 2 q = 0 (a) 1PI p 1 p 2 q = 0 (b) 1PI p 1 p 2 p 3 q = 0 (c) 1PI p 1 p 3 p 2 q = 0 (d)
which has precisely the form we mentioned in the beginning. Of course there are now multipleˆ. Those must be computed to give a lower bound on these powers and to sort out, which one gives the leading contribution, if anyˆis actually dominant.
Renormalisation strategy
Our strategy to compute the 1-loop anomalous dimensions is based on the background field gauge [17][18][19][20] in which we compute the one-particle-irreducible (1PI) graphs as depicted in fig. 2. This particular choice allows us to easily perform the renormalisation of the inserted operator at zero momentum, which then allows us to ignore any mixing from total divergence operators. Since we perform our operator renormalisation off-shell we have to take EOM vanishing operators E into account, i.e. the desired mixing matrix O can be extracted from
O E MS = O O E 0 E O E ,(7)
where the subscript MS indicates that we are using the MS renormalisation scheme working in = 4 − 2 dimensions. The 1-loop coefficient of the anomalous dimension matrix can then be easily obtained from the mixing matrix
O = 1 + O 0¯2 + O(¯4).(8)
Leading powersF
ollowing the strategy described before, we are left with a range of valuesΓ and the (unknown) constants =ˆB P B
;RGI . If a matching coefficient B vanishes at tree-level, we assume the 1-loop order to be the first non-vanishing contribution -of course those contributions could still be further suppressed. For an in-depth discussion ofˆB for commonly used lattice discretisations, see [16]. We will rather focus here on the spectrumΓ and try to make statements about the leading lattice artifacts ignoring potential hierarchies between differentˆB. The plots in figure 3 show all powerŝ Γ for O( ) improved Wilson and GW quarks respectively up to N 3 LO contributions. This is done to indicate the large spread ofΓ at leading order, while anything beyondΓ ≤ 1 + minΓ will be hard to distinguish from e.g. the NLO contributions of the truly leading powers. Also the very dense spectrum at subleading orders becomes more apparent this way.
Γ = 0 LO NLO N 2 LO N 3 LO O(a) improved
Conclusion
We find a very dense spectrumΓ for both Wilson and GW quarks due to the presence of four fermion operators at mass-dimension 6. This will make it hard to decide, which contributions actually dominate the O( 2 ) lattice artifacts due to potentially complicated cancellations and pileups of the various contributions. Nonetheless, ignoring any hierarchy between the matching coefficients, we find e.g. for f = 3 the leading asymptotic dependence for spectral quantities (orderingΓ ≤Γ +1 )
P ( ) P (0) = 1 − 2 [2 0¯2 (1/ )]Γ min 1 + 2 [2 0¯2 (1/ )] ΔΓ + . . . , massless massivê Γ min 0.25 −0.11 ΔΓ 0.42 0.36 ,(9)
which is universal for O( ) improved Wilson and Ginsparg-Wilson quarks. The asymptotic form for the massless case should also be a good approximation for f = 2 and may still work at f = 2 + 1 at physical quark masses. Once the physical charm quark is added the contributions from massive operators will certainly not be small any longer and may actually be the dominant contributions.
For the massless case and f = 2, 3, 4 the convergence towards the continuum limit should be slightly improved due toΓ > 0, while both f = 8 and the massive case have slightly negativê Γ −0.2, such that the convergence might be worse. In contrast to the O(3) non-linear sigma model [6,7] all leading powers are very close to the classical zero and not distinctly negative, i.e.Γ −3, which is good news. When the different constants have a similar magnitude, the leading power in the coupling dominates the 2 effects. However, as analysed in some detail in [16], common lattice actions can haveˆB which differ very much. For example for an O( ) improved fermion action and an improved gauge action, a single term dominates and it does not have the leading power. Such information should be incorporated when continuum extrapolations are performed and checks on contaminations of O( 3 ) or O( 4 ) contributions are advisable as well. Necessary extensions to this work are amongst others the inclusion of contributions from local fields to go beyond spectral quantities and staggered quarks, which require an enlarged operator basis due to flavour changing interactions.
Figure 1 :
1Sketch of the asymptotic lattice spacing dependence 2 [¯2 (1/ )]Γ for various values ofΓ compared to the plain 2 . We used here the 5-loop perturbative running of the QCD coupling in MS[1].
Figure 2 :
21PI graphs considered to perform the 1-loop renormalisation of the minimal operator basis at zero momentum. The double line indicates the operator insertion at zero momentum. Graph (e) is only needed to renormalise the 4-fermion opreators, while the graphs (a) and (c) would suffice for the case of pure gauge theory.B [2 0¯2 (1/ )] I × 1 + O(¯2(1/ )) into account, we eventually arrive at the central formula for the leading asymptotic lattice spacing dependence P ( ) = P (0) − 2 ∑︁ [2 0¯2 (1/ )]ΓˆB P B ;RGI × 1 + O(¯2(1/ )) + O( 3 ),Γ =ˆ+ I ,
Figure 3 :
3Spectra ofΓ forWilson (left) and Ginsparg-Wilson quarks (right). All powers up to N 3 LO have been plotted to highlight the spread of the leading powers and the density at subleading powers. While the solid lines correspond to the contributions from the massless operator basis in eq. (2), the dash-dotted lines correspond to contributions from massive operators. The number of flavours is chosen as f = 2, 3, 4 for the conventional lattice simulations and as f = 8 to highlight the approach to the conformal window. Notice that due to the dense spectrum some contributions are hard to distinguish.
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[] |
[
"Disease spreading with social distancing: a prevention strategy in disordered multiplex networks",
"Disease spreading with social distancing: a prevention strategy in disordered multiplex networks"
] |
[
"Ignacio A Perez ",
"Matías A Di Muro ",
"Cristian E La Rocca ",
"Lidia A Braunstein ",
"\nInstituto de Investigaciones Físicas de Mar del Plata (IFIMAR)\nDepartamento de Física\nFCEyN\nInstituto de Investigaciones Físicas de Mar del Plata (IFIMAR\nDepartamento de Física\nFCEyN, Universidad Nacional de Mar del Plata-CONICET\nInstituto de Investigaciones Físicas de Mar del Plata (IFIMAR\nDepartamento de Física\nFCEyN, Universidad Nacional de Mar del Plata-CONICET\nArgentina and Physics Department\nUniversidad Nacional de Mar del Plata-CONICET\nDéan Funes 3350, 7600) Mar del Plata, Déan Funes 3350, (7600) Mar del Plata, Déan Funes 3350, (7600) Mar del PlataArgentina., Argentina\n",
"\nBoston University\n590 Commonwealth Av. (02215) BostonMAUSA\n"
] |
[
"Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)\nDepartamento de Física\nFCEyN\nInstituto de Investigaciones Físicas de Mar del Plata (IFIMAR\nDepartamento de Física\nFCEyN, Universidad Nacional de Mar del Plata-CONICET\nInstituto de Investigaciones Físicas de Mar del Plata (IFIMAR\nDepartamento de Física\nFCEyN, Universidad Nacional de Mar del Plata-CONICET\nArgentina and Physics Department\nUniversidad Nacional de Mar del Plata-CONICET\nDéan Funes 3350, 7600) Mar del Plata, Déan Funes 3350, (7600) Mar del Plata, Déan Funes 3350, (7600) Mar del PlataArgentina., Argentina",
"Boston University\n590 Commonwealth Av. (02215) BostonMAUSA"
] |
[] |
The continue emerging of diseases that have the potential to become threats at local and global scales, such as influenza A(H1N1), SARS, MERS, and COVID-19, makes it relevant to keep designing models of disease propagation and strategies to prevent or mitigate their effects in populations. Since isolated systems are very rare to find in any context, specially in human contact networks, here we examine the susceptible-infected-recovered model of disease spreading in a multiplex network formed by two distinct networks or layers that are interconnected through a fraction q of shared individuals. We model the interactions between individuals in each layer through a weighted network, because person-to-person interactions are diverse (or disordered ); weights represent the contact times of these interactions and we use a distribution of contact times to assign an individual disorder to each layer. Using branching theory supported by simulations, we study a social distancing strategy where we reduce the average contact time in one layer (or in both of them, if necessary). We find a set of disorder parameters -associated with average contact times -that prevents a disease from becoming an epidemic. When the disease is very likely to spread, the system is always in an epidemic phase, regardless of the disorder parameters and the fraction of shared nodes. However we find that it is still possible to protect a giant component of susceptible individuals, which is crucial to keep the functionality of the system composed by the two interconnected layers. * [email protected] 2
|
10.1103/physreve.102.022310
|
[
"https://arxiv.org/pdf/2004.10593v1.pdf"
] | 216,056,482 |
2004.10593
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adee9d1e685be17b6c0dbe1573d859b7605e51a2
|
Disease spreading with social distancing: a prevention strategy in disordered multiplex networks
22 Apr 2020
Ignacio A Perez
Matías A Di Muro
Cristian E La Rocca
Lidia A Braunstein
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)
Departamento de Física
FCEyN
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR
Departamento de Física
FCEyN, Universidad Nacional de Mar del Plata-CONICET
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR
Departamento de Física
FCEyN, Universidad Nacional de Mar del Plata-CONICET
Argentina and Physics Department
Universidad Nacional de Mar del Plata-CONICET
Déan Funes 3350, 7600) Mar del Plata, Déan Funes 3350, (7600) Mar del Plata, Déan Funes 3350, (7600) Mar del PlataArgentina., Argentina
Boston University
590 Commonwealth Av. (02215) BostonMAUSA
Disease spreading with social distancing: a prevention strategy in disordered multiplex networks
22 Apr 20201
The continue emerging of diseases that have the potential to become threats at local and global scales, such as influenza A(H1N1), SARS, MERS, and COVID-19, makes it relevant to keep designing models of disease propagation and strategies to prevent or mitigate their effects in populations. Since isolated systems are very rare to find in any context, specially in human contact networks, here we examine the susceptible-infected-recovered model of disease spreading in a multiplex network formed by two distinct networks or layers that are interconnected through a fraction q of shared individuals. We model the interactions between individuals in each layer through a weighted network, because person-to-person interactions are diverse (or disordered ); weights represent the contact times of these interactions and we use a distribution of contact times to assign an individual disorder to each layer. Using branching theory supported by simulations, we study a social distancing strategy where we reduce the average contact time in one layer (or in both of them, if necessary). We find a set of disorder parameters -associated with average contact times -that prevents a disease from becoming an epidemic. When the disease is very likely to spread, the system is always in an epidemic phase, regardless of the disorder parameters and the fraction of shared nodes. However we find that it is still possible to protect a giant component of susceptible individuals, which is crucial to keep the functionality of the system composed by the two interconnected layers. * [email protected] 2
Abstract
The continue emerging of diseases that have the potential to become threats at local and global scales, such as influenza A(H1N1), SARS, MERS, and COVID-19, makes it relevant to keep designing models of disease propagation and strategies to prevent or mitigate their effects in populations. Since isolated systems are very rare to find in any context, specially in human contact networks, here we examine the susceptible-infected-recovered model of disease spreading in a multiplex network formed by two distinct networks or layers that are interconnected through a fraction q of shared individuals. We model the interactions between individuals in each layer through a weighted network, because person-to-person interactions are diverse (or disordered ); weights represent the contact times of these interactions and we use a distribution of contact times to assign an individual disorder to each layer. Using branching theory supported by simulations, we study a social distancing strategy where we reduce the average contact time in one layer (or in both of them, if necessary). We find a set of disorder parameters -associated with average contact times -that prevents a disease from becoming an epidemic. When the disease is very likely to spread, the system is always in an epidemic phase, regardless of the disorder parameters and the fraction of shared nodes. However we find that it is still possible to protect a giant component of susceptible individuals, which is crucial to keep the functionality of the system composed by the two interconnected layers.
I. INTRODUCTION
Due to the increasing levels of population migration, travel, and relocation from rural to urban regions, infectious diseases that were once under control, such as Ebola [1] and measles [2], have begun to resurface [3,4]. An outbreak of such a disease may become an epidemic if no preventive measures are undertaken, nor mitigation strategies are implemented. Worse still, the high interconnection degree between cities or countries extremely favors the spreading of a disease throughout the entire world, which may become a pandemic in a matter of months, weeks, or even days. This is what recently happened with the COVID-19 disease, declared as a pandemic on March 11, 2020, by the World Health Organization (WHO) [5,6]. Thus, nowadays researchers across multiple disciplines model disease propagation to develop strategies that could prevent or at least curtail epidemics (and pandemics, in the worst cases). Infectious diseases usually propagate through physical contacts among individuals [3,7], and researchers have found that modeling these contact patterns [8,9] is best achieved using complex networks [10][11][12][13], in which individuals and their interactions are represented by nodes and links, respectively. Numerous disease propagation models have made use of complex networks, including the susceptible-infected-susceptible (SIS) [7,10,13] and susceptible-infected-recovered (SIR) [3,10,[13][14][15][16] models. In these epidemic models, individuals can be in different states. For example, infected (I) individuals carry the disease and can transmit it to susceptible (S) neighbors that are not immune to the disease, while recovered (R) individuals do not participate in the propagation process because they have either recovered from a previous infection or because they have died.
The SIR model, in which individuals acquire permanent immunity after recovering from an illness, is the simplest and most used to study non-recurrent diseases. In the discrete-time version of this model [7], at each time step I individuals spread the disease to their S neighbors with the same probability β ∈ [0, 1], and switch to the R state t r time steps after being infected, where t r is the recovery time of the disease. The propagation reaches the final stage when the number of I individuals goes to zero. At this stage, the fraction R of recovered individuals indicates the extent of the infection, since all recovered individuals were once infected. In this model, the spreading is controlled by the effective probability of infection T = 1 − (1 − β) tr , the transmissibility, with T ∈ [0, 1]. When T is below a critical value T c , also called the epidemic threshold, the fraction R of recovered individuals, which is the order parameter of a continuous phase transition, is negligible compared to the system size N , and the system is in a non-epidemic phase. On the other hand, above T c the fraction R is comparable to N and thus it is said that the disease becomes an epidemic [16][17][18][19].
Different strategies have been proposed for preventing or mitigating the impact of diseases in healthy populations, the most common being vaccination [13,[20][21][22][23][24][25][26][27] and partial or complete lockdown. Vaccination is a pharmacological strategy that provides immunity against a disease to individuals, who are then unable to get infected or infect neighbors.
However in emergency situations such as the COVID-19 pandemic, a new disease rapidly spreads throughout the population and, since to this date a vaccine is yet to be developed, it is necessary to take other types of measures. In contrast, lockdown is a non-pharmacological strategy which consists in isolating individuals to prevent disease propagation. Nevertheless, a major drawback of this strategy is that it may cause a significant disruption in the economy of a region, because it implies a massive cut of social interactions. In addition, large-scale implementation of this mitigation strategy is challenging. On the other hand, there are other less extreme measures than lockdown with the same spirit. "Social distancing" strategies [28][29][30][31][32][33][34] are a set of measures for reducing contact between individuals (keep a minimum physical distance, work from home when possible, avoid crowded places, etc.), thus decreasing the probability of disease transmission. This approach assumes that an infection is more likely to be transmitted between people if they spend more time together, and thus the goal is to shorten the duration of interaction times. Social distancing strategies, along with partial or complete lockdown, have been undertaken in many countries to face the COVID-19 pandemic due to the absence of a vaccine and the fast propagation of the disease. These kind of measures respond to the direct transmission mechanism of the virus: it is expelled in the form of droplets from an infected individual through mouth and nose (when talking, exhaling, or coughing), and can get into a healthy individual (located within a 1 meter range, approximately, and facing the infected individual) through the mucous (eyes, nose, and mouth) [35]; as individuals spend more time toghether the probability of infection increases.
Contact times in real-world networks usually span a broad distribution [9,36,37]. These kind of systems are known as disordered networks, which are characterized by the existing diversity in the strength or intensity of the interactions among the different parts of the system, and have been receiving much attention recently [11,31,33,34]. In Ref. [34] disorder is implemented using a weighted complex network [11], in which the weights associated with links represent the normalized contact times ω of the interaction between two individuals, and they follow a power law distribution, W (ω) = 1/(aω) that resembles experimental results [9,36,37]. The parameter a is the disorder intensity that controls the range of allowable contact times in the distribution and their average value. Also, the interactions between individuals are categorized as either close (larger average contact time) or distant (shorter average contact time), each representing complementary fractions (f 1 and 1 − f 1 , respectively) of the total number of interactions. This is carried out by controlling the corresponding disorder intensity of the contact times distributions. Researchers have found that for a system in an epidemic phase (R > 0), when the fraction f 1 of close contacts is sufficiently small, increasing the disorder intensity of the distant interactions to decrease their average contact time may switch the system to a non-epidemic phase (R = 0).
A very relevant magnitude to also consider is the size of the giant cluster of susceptible individuals (GCS), or biggest connected cluster, at the final stage. This cluster is formed by all the remaining susceptible (healthy) individuals that are connected with each other and it is the network that sustains the functionality of a society, e.g., the economy of a region. Using a generating function formalism, Newman [38] showed that in the SIR model there exists a second threshold T * above which the giant cluster of susceptible individuals vanishes at the final stage. On the other hand, Valdez et al. [32] showed that T * is an important parameter to determine the efficiency of a mitigation or control strategy, because any strategy that manages to decrease the transmissibility below T * can protect a large and connected cluster of susceptible individuals, even when the system is in an epidemic state.
The previously mentioned studies were carried out using isolated networks, i.e., networks that do not interact with other different networks. Researchers have noted that isolated network models ignore the "external" connections that real-world systems use to communicate with their environment, which usually affect the behavior of the dynamical processes that take place on complex systems [39][40][41][42]. Thus, the modeling of interconnected networks, i.e., networks of networks (NoN) or multilayer networks, has become utterly relevant as it allows a more accurate representation of real systems. The ubiquity of the NoN has encouraged researchers to use them in the study of several topics such as cascading failure [43][44][45], social dynamic [46,47], and disease propagation [48][49][50]. Particularly, the SIR model was simulated and solved theoretically in an overlapped multiplex network [51] system consisting of two individual networks or layers, in which a fraction q of shared nodes (q overlapping) is present in both layers. These shared individuals connect the different layers, and their presence makes diseases more likely to spread as q increases [51].
In this paper we study a disease spreading process using the SIR model in an overlapped two-layer multiplex network. The layers have particular disorder intensities, which represent the average contact time between individuals in each layer, and are connected through a fraction q of shared nodes. We use the branching theory, supported by extensive simulations, to study a social distancing strategy in which we increase the disorder intensity in one of the layers (or in both of them) to reduce the average contact time between individuals, with the purpose of preventing the onset of an epidemic. In addition, the ultimately goal is to protect the total giant component of susceptible individuals, formed by the healthy individuals which are in contact through both layers, that will keep the economy of a region running.
II. MODEL AND RESULTS OF THE STOCHASTIC SIMULATION
We use an overlapped multiplex network formed by two layers, A and B, in which a fraction q of nodes are present in both layers, called shared nodes. For the construction of the layers, both of which are of size N , we use the Molloy-Reed algorithm [52]. Each layer has its own uncorrelated degree distribution P i (k), which gives the probability that a node has k neighbors in layer i = A, B. Here we use a Poisson distribution P (k) = e − k k k /k! which is homogeneous, being the average connectivity k the most probable value, and allows to easily obtain some analytical results, and a power law distribution with exponential cutoff k c , P (k) ∝ k −λ e −k/kc , which is heterogeneous and is more representative of real-word networks, where some individuals may have a high number of contacts while the majority has only a few. We consider the exponential cutoff k c in the power law distribution as real-world networks are finite and the maximum number of connections of a node is limited by the size of the system [16]. Additionally, each layer is a weighted network in which links have associated weights ω, where ω is a normalized contact time that defines the intensity of the interaction between two individuals. The ω values are taken from the theoretical distribution W i (ω) = 1/(a i ω) [31,34], ω [e −a i , 1], where a i is the disorder intensity of layer i = A, B.
Therefore, each link of the network is assigned a weight ω = e −a i r [53], where r is a random number uniformly distributed within the interval [0, 1].
To simulate the disease spread, we use the SIR model taking into account that the probability of infection depends not only on the type of disease, but also on the disorder intensity. All individuals are initially susceptible, except for one that randomly becomes infected, called the patient zero. At each time step, infected individuals spread the disease to susceptible neighbors with a probability βω, where β is the intrinsic infectivity of the disease, and infected individuals recover after t r time steps. The propagation stops when the number of infected individuals is zero in both layers. By using the model described above, we can write the transmissibility of the disease in layer [54]. Note that T a i is a decreasing function of the disorder intensity a i because, since ω [e −a i , 1], for higher a i values shorter contact times become more probable, and hence the disease is less likely to propagate. This allows us to implement a social distancing strategy by increasing the disorder intensity a i in layer i. On the other hand, in the limit a i → ∞ we have that T a i → 0, which is a complete lockdown scenario where each individual is isolated from the rest. On the other hand, in the limit a i → 0 there is no disorder in layer i, then the infection probabilities throughout layer i are all simply equal to β, recovering the original SIR model in which the transmissibility is T a i → T = 1 − (1 − β) tr .
i = A, B as T a i = tr t=1 [(1 − βe −a i ) t −(1−β) t ]/(a i t)
Next, we examine the disease parameters β and t r so that the system enters an epidemic phase (i.e., R > 0, where R is the fraction of recovered individuals in the multiplex network) without disorder, i.e., a A = a B = 0 (thus the interactions in both layers have the largest temporal duration). We want to analyze whether or not we can find a pair of disorder intensity values (a A , a B ) (or a set of pairs of values) that prevent the disease from becoming an epidemic, and examine how they depend on the fraction q of shared nodes. In addition, we look for disorder intensity values that can prevent the total giant component of susceptible individuals GCS (composed of S nodes connected through both layers) to fall apart, which would certainly cause a significant disruption in the economy of a region or country. We remark that the social distancing strategy is applied before the disease starts spreading and during its evolution. This could represent an optimistic scenario, where the authorities of a particular region (e.g., city or country) are well informed about the existence of an infectious disease, and they immediately undertake strict measures to get the most out of them. In order to implement social distancing strategies, the structure of a population may have to be taken into consideration. For instance, the population could be divided into two layers: one layer representing essential workers (food supply and distribution systems, sanitary system, public transport, etc.), where social distancing is certainly difficult to apply (lower disorder intensity), while the other layer could represent people who stay at their homes and only go out to make necessary purchases, such as food and cleaning supplies, for medical care or to do paperwork, where it is usually easier for the individuals to keep a safe distance from the others (higher disorder intensity). In this structured system, it is reasonable that a fraction q of individuals may belong to both groups of people.
In Fig. 1 we show the simulation results for the total fraction R of recovered individuals and the size GCS of the giant component of susceptible individuals that span the entire multiplex network as functions of the disorder intensity a B in layer B. Note that in Fig. 1 (a) the system can be either in an epidemic phase, where R > 0, or in a non-epidemic phase, where only small outbreaks take place and hence it is considered that R = 0. When a B is below a critical value a Bc , so that T B > T Bc , the average contact time in layer B is longer and hence the interactions between individuals in that layer. Therefore, the disease is more likely to propagate and switch the system to an epidemic phase. Conversely, when a B ≥ a Bc (T ≤ T Bc ), shortening the average contact time limits the spread of the disease down to a negligible fraction of individuals, which avoids an epidemic (non-epidemic phase). Also, we observe that GCS increases with a B , as the giant cluster of susceptible individuals is less affected due to the curtail of the spread of the disease. On the other hand, in Fig. 1 (b) the distancing between individuals is not enough to prevent an epidemic, but it can preserve the GCS if a B is above a critical value a B * .
In this section we see that it is definitely possible to switch the system from an epidemic phase to a non-epidemic phase by increasing the social distancing between individuals in one layer, i.e., by controlling the temporal duration of their interactions. Besides, if there is an epidemic in the system, social distancing may prevent the GCS to fall apart, thus keeping the system functional. In addition to the computational simulations, in the next section we introduce a theoretical approach that facilitates the analysis of the phase space for R and GCS, at the final stage of the process. It has been demonstrated that in isolated networks the final stage of the SIR model [3,10,[13][14][15][16] maps exactly into link percolation [16,53,55] in which links between nodes are occupied with probability p. Thus, the relevant magnitudes of this model can be obtained theoretically. The mapping holds in the thermodynamic limit, where N → ∞, and considering that the number of recovered individuals is zero unless they are above a threshold s c [19], which distinguishes between an epidemic and a small outbreak. In isolated complex networks, the critical transmissibility for which the system switches from an epidemic to a non-epidemic phase is T c = 1/(κ − 1), where κ = k 2 / k is the branching factor of the network, and k and k 2 are the first and second moments of the degree distribution P (k), respectively [15,16].
Next, we proceed to map the final stage of our social distancing model into link percolation using the branching theory and the generating functions framework [16,53,[55][56][57]. Given a two-layer multiplex network with overlapping 0 < q ≤ 1, we can write a system of trascendental coupled equations for f i (T A , T B ) ≡ f i , which is the probability that a branch of infection (formed by recovered individuals) that originates from a random link in layer i = A, B expands infinitely through any of the layers,
f A = (1 − q)(1 − G A 1 (1 − T A f A )) + q(1 − G A 1 (1 − T A f A )G B 0 (1 − T B f B )),(1)f B = (1 − q)(1 − G B 1 (1 − T B f B )) + q(1 − G B 1 (1 − T B f B )G A 0 (1 − T A f A )),(2)
where G i 0 (x) = k P i (k)x k and G i 1 (x) = k (kP i (k)/ k i )x k−1 = G i 0 (x)/G i 0 (1) (with G ≡ dG/dx and G 0 (1) = k ) are the generating functions of the degree and the excess degree distributions, respectively [16,53,[55][56][57]. Note that the factor G i 1 (x), with x = 1 − T i f i , is the probability that in layer i a branch of infection reaches a node with connectivity k, so that it cannot keep extending through its k − 1 remaining connections. In a similar way, G i 0 (x) is the probability that a randomly chosen node is not reached by a branch of infection through its k connections in layer i. Thus, f A is the sum of two main terms. First, the probability of reaching an individual only present in layer A (with probability 1 − q) so that the branch of infection expands through any of the k − 1 remaining connections of the individual in that layer, and second, the probability of reaching one of the shared nodes (with probability q) so that the branch of infection expands through any of its k − 1 contacts in layer A or through any of its k connections in layer B (see Fig. 2). An analogous interpretation holds for f B . Once we calculate the non-trivial roots of Eqs. (1) and (2), then the fractions R A , R B and R of recovered individuals (i.e., those reached by the branches of
infection) can be obtained from
R A = (1 − q)(1 − G A 0 (1 − T A f A )) + ξ R ,(3)R B = (1 − q)(1 − G B 0 (1 − T B f B )) + ξ R ,(4)R = (R A + R B − ξ R )/(2 − q),(5)
where
ξ R = q(1−G A 0 (1−T A f A )G B 0 (1−T B f B ))
is the fraction of shared recovered nodes at the final stage. The factor 2−q in Eq. (5) is present because the total number of individuals in the system is (2 − q)N . Fig. 1 shows the results for the total fraction R of recovered individuals obtained from Eq. (5) (full lines), which agree with the simulation results (symbols). We observe that the system undergoes a phase transition between epidemic (R > 0) and non-epidemic (R = 0) phases when we vary the disorder intensity a B . Generally, if there is a critical disorder intensity a Bc for a given a A value, it can be computed by solving the equation det(J f − I) = 0 evaluated at f A = f B = 0 (since at the critical point none of the branches of infection expands infinitely). Here I is the identity matrix and J f is the Jacobian matrix of the system of Eqs. (1) and (2), J f i,k = ∂f i /∂f j ,
J f | f A =f B = 0 = T A (κ A − 1) qT B k B qT A k A T B (κ B − 1) ,
where κ i and k i are the branching factor and the average connectivity in layer i = A, B.
Then, the critical transmissibility T Bc is given by
T Bc = T A (κ A − 1) − 1 [T A (κ A − 1) − 1](κ B − 1) − q 2 T A k A k B .(6)
This result differs from the one obtained in Ref. [51], where both layers of the multiplex network have the same transmissibility T , which yields a quadratic equation for T c with only one stable solution. Inverting T Bc (and T A ) from Eq. (6), we can calculate the critical disorder intensity a Bc for a given a A value.
In Fig. 3 we show the phase diagram for R on the plane (a A , a B ), for two layers with power law degree distribution and exponential cutoff, and a disease with β = 0.1 and t r = 5. Each curve shows the critical value a Bc as a function of a A for a given fraction q of shared nodes, separating the epidemic phase (R > 0 below the curve) from the epidemic-free phase (R = 0 above the curve). As a A increases a Bc decreases, indicating that for shorter contacts in layer A, the critical average duration of interactions in layer B becomes larger. The effect of the shared nodes is reflected on the increase of the a Bc values for fixed a A as q increases, because these nodes facilitate the propagation between layers, which in addition widens the epidemic phase. Note that in the limit a A → ∞, a Bc approximates to the critical disorder intensity in an isolated network, a B∞ , as T Bc → 1/(κ B − 1)-see Eq. (6)-for all q values (in Ref. [34], this result corresponds to a single layer with f 1 = 0). This is because as a A → ∞ there is complete isolation between the individuals in layer A, and thus the spread of the disease in only possible in layer B. An analogous analysis holds for a A∞ , which is the critical disorder intensity in layer A when it is isolated. Note also that for a A < a A∞ there is no critical value a Bc (gray-colored region in Fig. 3), and therefore an epidemic cannot be avoided even if we completely isolate the individuals in layer B. This analysis shows that the negative effect of the interconnection between layers means that, to prevent an epidemic in the system, the disorder intensities in both layers must be above the critical values corresponding to the isolated-layer case. while it is in a non-epidemic phase above (R = 0). Note that the epidemic phase widens as q increases, since the shared nodes ease the spreading of the disease. The results correspond to two layers with a power law degree distribution with λ A = λ B = 2.35, exponential cutoff k c = 20, k min = 2, and k max = 250, so that the single layer critical disorder intensity is a A∞ = a B∞ ≈ 2.3.
B. Phase space for GCS
In what follows, we present a set of equations that allow us to calculate the size GCS of the giant component of susceptible individuals of the multiplex network at the final stage of the process. It is straightforward to write the fraction of nodes of layers A and B that belong to the GCS as
GCS A = (1 − q)(G A 0 (1 − T A f A ) − G A 0 (ν A )) + ξ S ,(7)GCS B = (1 − q)(G B 0 (1 − T B f B ) − G B 0 (ν B )) + ξ S .(8)
The first terms takes into account the probability that a node that is part of only one of the layers (with probability 1 − q) belongs to the GCS. This can be written as the probability that a node of layer i is susceptible (G i 0 (1 − T i f i )) minus the probabilty of the node being susceptible but not belonging to the GCS (G i 0 (ν i )). On the other hand,
ξ S = q(G A 0 (1 − T A f A )G B 0 (1 − T B f B ) − G A 0 (ν A )G B 0 (ν B )
) is the fraction of shared nodes that belong to the GCS. A node is susceptible and does not belong to the GCS if none of its links lead to susceptible nodes that do belong to the GCS. But, if one of these links connects to a R node, in order to be susceptible, the node cannot have been infected by this R node.
Thus, we define ν i as the probability that a random link from layer i leads to a susceptible node that does not belong to the GCS, or to a R node. However, note that in the last case the link must be unoccupied, with probability 1 − T i . Note that similar to Eqs. (1) and (2), we can write a set of coupled trascendental equations for the probabilities
ν A (T A , T B ) ≡ ν A and ν B (T A , T B ) ≡ ν B , ν A = (1 − T A )f A + (1 − q)G A 1 (ν A ) + qG A 1 (ν A )G B 0 (ν B ),(9)ν B = (1 − T B )f B + (1 − q)G B 1 (ν B ) + qG B 1 (ν B )G A 0 (ν A ).(10)
From left to right, the first term is the probability that a random link in layer i leads to a recovered node, which is f i but considering that the link is unoccupied, with probability 1 − T i . The second is the probability that the random link connects to a node that only belongs to layer i (with probability 1−q), so that none of its outgoing links lead to susceptible nodes belonging to the GCS, nor any of its outgoing links leads to a recovered node and is ocuppied. Finally, the last term is similar to the second, but the random link in layer i leads to a shared node (with probability q) so that besides its outgoings links in layer i, none of its links in layer j connect to susceptible nodes that are part of the GCS, nor any of its links connects to a recovered node and is ocuppied. Once the values of ν i and consequently GCS i for i = A, B are obtained from Eqs. (7)(8)(9)(10), the size of the GCS can be computed
GCS = (GCS A + GCS B − ξ S )/(2 − q),(11)
where the factor 2−q accounts for the total number of nodes in the system, which is (2−q)N . To study the phase space for the GCS, we define µ i as the probability that a random link in layer i = A, B connects to a node belonging to the GCS (similar to what was done in Sec. III A with f i for the recovered individuals). Recall that ν i is the probability that a random link connects to an S node which does not belong to the GCS or that it connects to an R node but considering that the link is unoccupied with probability 1 − T i . Then we have that µ i = 1 − (ν i + T i f i ) and we obtain a system of equations for µ A and µ B
µ A = 1 − f A − (1 − q)G A 1 (u A − µ A ) − qG A 1 (u A − µ A )G B 0 (u B − µ B ),(12)µ B = 1 − f B − (1 − q)G B 1 (u B − µ B ) − qG B 1 (u B − µ B )G A 0 (u A − µ A ),(13)
where u i ≡ 1 − T i f i . Given a disorder intensity value a A , the condition for the existence of a critical value a * B for the GCS is that det(J µ − I) = 0 evaluated at µ A = µ B = 0, where J µ is the Jacobian matrix of the system of Eqs. (12) and (13), J µ i,k = ∂µ i /∂µ j ,
J µ | µ A =µ B = 0 = (1 − q)G A 1 (u A ) + qG A 1 (u A )G B 0 (u B ) qG A 1 (u A )G B 1 (u B ) k B qG B 1 (u B )G A 1 (u A ) k A (1 − q)G B 1 (u B ) + qG B 1 (u B )G A 0 (u A ) .
In this case we obtain an implicit equation for f A , f B and a * B , with a A fixed. Then we proceed to solve it together with Eqs. (1) and (2). Fig. 4 shows the phase diagram for GCS on the plane (a A , a B ), for a disease with β = 0.1 and t r = 5. Each curve shows the critical value a * B as a function of a A for a given fraction q of shared nodes, separating the non-functional phase (GCS = 0 below the curve) from the functional phase (GCS > 0 above the curve).
We observe that as a A increases a Bc decreases, indicating that for shorter contacts times in layer A, the critical average duration of interactions in layer B grows. The effect of the shared nodes is reflected on the increase of a * B values for fixed a A as q increases, because these nodes facilitate the propagation between layers, which makes the GCS to decrease.
Thus, the non-functional phase widens for larger q values. In this case, regardless of the fraction q of shared nodes present in the multiplex network, a B decreases until it becomes zero. From there on, for any a A value the system is in a functional phase independently of a B . This means that if the average contact time is short enough in one layer, then the GCS will not vanish even if the duration of interactions in the other layer is extremely high.
In Fig. 5 we show the critical curve for the GCS together with the critical curve for R.
We observe that the phase space projected in the (a A , a B ) plane is now divided into three regions, which is the most general scenario. Region I corresponds to a non-epidemic and functional phase (NE-F), the best posible scenario, in which the disorder intensities in both layers are high enough, i.e., social distancing measures are undertaken intensively. In region II the distancing is moderate and the epidemic can not be avoided, but there is still a GCS, which means that the system remains functional (E-F). Finally, in region III the disease is certainly likely to spread because the disorder intensities are quite low, which not only fails to prevent an epidemic but also makes the GCS to fall apart, causing the system to collapse (E-NF). Considering this, even though the social distancing efforts may not prevent that a particular disease extends through a significant portion of the population, they could serve to the purpose of keeping the integrity of the system. It is important to note that, if a GCS remains functional after the end of an epidemic, it is highly recommended not to relax the set of measures undertaken to prevent the transmission of the disease, since there is always the possibility of a second outbreak (originating, for instance, from an imported or undetected case).
We want to point out that our overlapped multiplex network model is a simplification of what usually happens in real social networks. For instance, it is known that nowadays passenger traffic is one of the main causes of the disemination of a disease across different regions (cities, states, countries). A way to tackle this issue is to isolate individuals once they arrive to its destination, which is currently being implemented during the COVID-19
pandemic. This way, it is expected that the passengers cannot spread the disease within the region. To take into account this possibility, instead of using an overlapped multiplex network, we could devise a model in which shared nodes do not belong to both layers, but rather they connect to nodes from other layers according to an inter-layer degree distribution.
These nodes would represent individuals that travel to other regions or countries, which are isolated with a certain probability. Such a model may provide more realistic and widely applicable results, and may encourage researchers to devise more suitable and efficient strategies for preventing/mitigating the spread of a disease. In addition, it would be relevant to study the temporal evolution of a disease in this scenario, and how it would respond to mitigation measures that are undertaken with some delay.
IV. CONCLUSIONS
We apply the SIR model to study the spread of a disease in a partially overlapped multiplex network composed of two layers, in which a fraction q of individuals belong to both layers and also each layer has its own distribution of contact times. We propose a social distancing strategy that reduces the average contact time of the interactions between individuals within each layer, by increasing its respective disorder intensity. When the disorder intensity in a layer is below its critical value for q = 0 (the isolated-layer case), an epidemic cannot be avoided, even when the individuals in the other layer are completely isolated. However, we find that it is still possible to protect the functional network of susceptible individuals through social distancing policies, i.e., by increasing the disorder intensity. This is fundamental to keep running the economy of a society. In the best case scenario, when the disorder intensity in a layer is above its critical value for q = 0, we find that there is a critical disorder intensity in the other layer that can reduce the epidemic size to zero. All the critical values increase with the overlapping q because the individuals that are shared by the layers ease the spread of disease, so that more social distancing measures must be undertaken to prevent an epidemic. All in all, the control of contact times between individuals can serve as a prevention strategy that overcomes the overlapping effect in multiplex networks, preventing not only an epidemic, but also the economic collapse of a region or country, which might be equally bad.
ACKNOWLEDGMENTS
We acknowledge UNMdP and CONICET (PIP 00443/2014) for financial support. CEL, MAD and IAP acknowledge CONICET for financial support. Work at Boston University is supported by NSF Grant PHY-1505000 and by DTRA Grant HDTRA1-14-1-0017.
FIG. 1 :
1Total fraction R of recovered individuals (red circles) and the size GCS of the giant component of susceptible individuals (green squares) at the final stage, as functions of the disorder intensity a B . (a) We observe that the increase of a B , which decreases the transmissibility T B by reducing the average contact time in layer B, can bring the system from an epidemic phase (R > 0) to a non-epidemic phase (R = 0), while keeping the integrity of the largest connected cluster of susceptible individuals -β = 0.25, t r = 1, a A = 1 and q = 0.3-. (b) The system always seems to be in an epidemic phase (R > 0) and despite starting off with a null GCS, the increase of the disorder intensity can make the system functional (GCS > 0) -β = 0.1, t r = 5, a A = 0 and q = 0.4-. The results correspond to two layers with Poisson degree distributions, with k A = k B = 4, k min = 0 and k max = 40, where k min and k max are the minimum and maximum connectivity of a node, respectively. Simulations, shown in symbols, were averaged over 10 4 realizations and using layers of size N = 10 5 . We only consider realizations where the number of recovered individuals in each layer is above a threshold s c = 200[19]. The full black lines correspond to theoretical results.III. THEORY AND RESULTS AT THE FINAL STAGEA. Phase space for R
FIG. 2 :
2Scheme of a disordered multiplex network formed by two partially overlapped layers, A and B. The size of the layers is N A = N B = 15, and the fraction of nodes present in both layers is q = 3/15 = 0.2 (vertical lines are used as a guide to show the shared nodes, which are represented by boxes). The thickness of the segments represents the diversity of the normalized contact times ω between individuals. (a) Initially, all the individuals are in the susceptible (S) state, except for an infected (I) node, which kickstarts the propagation of the disease. (b) At the final stage, the recovered (R) individuals are connected by the branches of infection, which originate from the link denoted by a red arrow. One of the branches, denoted by dotted lines, corresponds to the spread of the disease only through layer A, and is represented by the first term in Eq. (1).The other branch, denoted by dash-dotted lines, is a branch of infection that spreads through both layers and is represented by the second term in Eq. (1).
FIG. 3 :
3Phase space that shows the extent of an epidemic on the (a A , a B ) plane for a disease with β = 0.1 and t r = 5. The black full curves represent the critical intensity a Bc as a function of a A , for different fractions of shared nodes q (q goes from 1 to 0.1 at intervals of 0.1, from top to bottom). Below each curve the system is in an epidemic phase (R > 0)
Fig. 1
1shows the results of the size GCS of the total giant component of susceptible individuals obtained from Eq. (11) (full lines), which agree with the simulation results (symbols). We observe that the system undergoes a phase transition between functional (GCS > 0) and non-functional (GCS = 0) phases when we vary the disorder intensity a B .
nf u n c t i o n a l P h a s e FIG. 4: Phase space that shows the integrity of the giant component of susceptible individuals (GCS) on the (a A , a B ) plane for a disease with β = 0.1 and t r = 5. The black full curves represents the critical intensity a * B as a function of a A , for different fractions of shared nodes q (q goes from 1 to 0.4 at intervals of 0.1, from top to bottom). The system is in a non-functional phase (GCS = 0) below each curve, while is in a functional phase above it (GCS > 0). Note that the non-functional phase widens as q increases, because the shared nodes facilitate the propagation of the disease. The results correspond to two layers with a power law degree distribution, with λ A = λ B = 2.35, exponential cutoff k c = 20, k min = 2, and k max = 250.
FIG. 5 :
5Division of the phase space on the (a A , a B ) plane, according to the outcomes of R and GCS in the final stage. The full and dashed lines correspond to the critical curves for R and GCS, respectively. A strict social distancing policy (i.e., high disorder intensities in both layers) ensures the system to lie on a non-epidemic and functional phase (region I, NE-F phase). As the distancing policies are relaxed, the emergence of an epidemic becomes more likely (region II, E-F phase), and if the policies are rather week even the GCS falls apart, disrupting the functionality of the system (region III, E-NF phase). The results correspond to two layers with a power law degree distribution, with λ A = λ B = 2.35, exponential cutoff k c = 20, k min = 2, k max = 250, and q = 0.3. Also we consider β = 0.1 and t r = 7.
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[
"MODULI INTERPRETATION OF EISENSTEIN SERIES",
"MODULI INTERPRETATION OF EISENSTEIN SERIES"
] |
[
"Kamal Khuri-Makdisi "
] |
[] |
[] |
Let ℓ ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(ℓ), which make sense over any field k in which 6ℓ = 0 and that contains the ℓth roots of unity. Over the complex numbers, these forms include all holomorphic Eisenstein series on Γ(ℓ) in all weights, in a natural way. The graded ring R ℓ that is generated by our special modular forms turns out to be generated by certain forms in weight 1 that, over C, correspond to the Eisenstein series on Γ(ℓ). By a combination of algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions, we show that when k = C, the ring R ℓ , which is generated as a ring by the Eisenstein series of weight 1, contains all modular forms on Γ(ℓ) in weights ≥ 2. Our results give a straightforward method to produce models for the modular curve X(ℓ) defined over the ℓth cyclotomic field, using only exact arithmetic in the ℓ-torsion field of a single Q-rational elliptic curve E 0 .
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10.1142/s1793042112500418
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[
"https://arxiv.org/pdf/0903.1439v6.pdf"
] | 15,127,723 |
0903.1439
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329d31b8600c4f755faa5822ed5b7df9f25566f3
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MODULI INTERPRETATION OF EISENSTEIN SERIES
3 Jul 2009
Kamal Khuri-Makdisi
MODULI INTERPRETATION OF EISENSTEIN SERIES
3 Jul 2009
Let ℓ ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(ℓ), which make sense over any field k in which 6ℓ = 0 and that contains the ℓth roots of unity. Over the complex numbers, these forms include all holomorphic Eisenstein series on Γ(ℓ) in all weights, in a natural way. The graded ring R ℓ that is generated by our special modular forms turns out to be generated by certain forms in weight 1 that, over C, correspond to the Eisenstein series on Γ(ℓ). By a combination of algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions, we show that when k = C, the ring R ℓ , which is generated as a ring by the Eisenstein series of weight 1, contains all modular forms on Γ(ℓ) in weights ≥ 2. Our results give a straightforward method to produce models for the modular curve X(ℓ) defined over the ℓth cyclotomic field, using only exact arithmetic in the ℓ-torsion field of a single Q-rational elliptic curve E 0 .
Introduction
Let L be a lattice in C, and consider elliptic functions with respect to L. A standard formula (see, e.g., equation IV.3.6 of [Cha85]), which we reprove in Corollary 3.6 of this article, states that if α, β, γ ∈ C − L and α + β + γ = 0, then
(1.1) −1 2 · ℘ ′ (α) − ℘ ′ (β) ℘(α) − ℘(β) = ζ(α) + ζ(β) + ζ(γ).
Here ℘ and ζ are the Weierstrass ℘ and zeta functions with respect to L; our notation for elliptic functions follows [Cha85], unless otherwise specified. Let us temporarily call the above expression λ = λ α,β,γ,L . In terms of the projective embedding of the elliptic curve E = C/L as a plane cubic using ℘ and ℘ ′ , this essentially says that λ is the slope of the line (in the affine part of the plane) joining the images of α and β. After a short calculation, we obtain that λ can be written as the absolutely convergent series
(1.2) λ = ζ(α) + ζ(β) + ζ(γ) = ω∈L ′ 1 ω + α + 1 ω + β + 1 ω + γ − 3 ω ,
where the notation ′ means that one omits the term 3/ω from the summand when ω = 0. Note that the individual sums such as ω 1/(ω + α) do not converge; however, if α, β, γ ∈ 1 ℓ L for some integer ℓ, then the sums can be regularized by Hecke's method to obtain Eisenstein series of weight 1 on the congruence subgroup Γ(ℓ). After overcoming some analytic hurdles, we indeed show in Section 2 below that λ is the value of a suitable Eisenstein series of weight 1. As for Eisenstein series in weights 2 and 3, these can be related to values of the ℘ and ℘ ′ functions, in other words to the affine coordinates of the torsion points of E corresponding to α, β, and γ. This means that the values of Eisenstein series of weights up to 3 can be computed from the Weierstrass model of the varying elliptic curve E and its ℓ-torsion, in other words from the moduli problem that is parametrized by the modular curve X(ℓ). This is the "moduli interpretation" referred to in our title.
More generally, we can search for other "moduli-friendly" modular forms on Γ(ℓ) that have an agreeable modular interpretation; this allows us to define such modular forms over more general base fields than C. In truth, the first paragraph above reverses the order in which this author came across the family of moduli-friendly forms treated in this article. The realization that λ was a modular form came first, from different considerations. Indeed, the expression of λ as a slope quotient shows that it is the quotient of an Eisenstein series of weight 3 by one of weight 2; thus viewing λ as a function of the varying L (as well as α, β, γ), we obtain that λ is a meromorphic modular form of weight 1. Holomorphy of λ on the upper half plane H and at the cusps then follows from the addition formula on the elliptic curve, namely, from the formula λ 2 = ℘(α) + ℘(β) + ℘(γ), which expresses λ 2 as a holomorphic modular form. It was in this way that we first collected a family of moduli-friendly forms (Definition 3.1, equation (3.23) and Theorem 3.9 below), defined more generally than in (1.1) by the coefficients in the Laurent expansions of certain elliptic functions, or rather the algebraic Laurent expansions of certain elements of the function field of E. It was only later in our investigations that we made the connection from these forms to the Weierstrass ζ function and Eisenstein series (Theorem 2.8, which now comes earlier in our treatment).
In particular, this article provides a moduli-friendly, algebraic treatment of all holomorphic Eisenstein series of arbitrary weight j on Γ(ℓ), and gives a natural way to express these Eisenstein series as polynomials in the Eisenstein series of weight 1. This occupies Sections 2 and 3 of this article. In fact, all the modular forms that we obtain belong to a ring R ℓ , which turns out to be generated by the algebraic version of the Eisenstein series in weight 1 for ℓ ≥ 3 (Theorem 3.13). This result is similar to the results proved in [BG01a], where Borisov and Gunnells define and study "toric modular forms" on Γ 1 (ℓ), and prove that the ring of toric modular forms is generated by certain Eisenstein series in weight 1, and that it is stable under the Hecke operators T n for Γ 1 (ℓ); their proofs rely on q-expansions of modular forms. Thus the results in this article include a generalization to Γ(ℓ) of the ring of toric modular forms introduced in [BG01a]. (See also [Cor97] for a study of the ring generated by weight 1 Eisenstein series in the Drinfeld modular case.) The above article [BG01a], as well as the subsequent articles [BG01b,BG03,BGP01], were a definite inspiration for several of the results in this article, even though our proofs tend to proceed along different lines (most notably, without any q-expansions).
Sections 4 and 5 contain the main technical results of this article. Continuing the analogy with [BG01a], we also prove invariance of our ring R ℓ under the Hecke algebra. We first use an algebraic method to prove this in weights 2 and 3 (Propositions 4.6, 4.8, and 4.11, with whose proofs we are rather pleased). We then combine the Hecke invariance in these low weights with analytic techniques (Rankin-Selberg and nonvanishing of L-functions), along with some algebraic geometry of sufficiently positive line bundles on curves, to prove that over C, the ring R ℓ contains all modular forms of weights j ≥ 2 (Theorem 5.1). In weight 1, our ring contains precisely the Eisenstein series. Thus our ring is of necessity stable under Hecke operators, this time for the full (noncommutative) Hecke algebra of all double cosets for Γ(ℓ).
Our result that R ℓ contains all modular forms in higher weights is analogous to the results in [BG01b,BG03] for toric modular forms on Γ 1 (ℓ). Borisov and Gunnells prove there that the cuspidal part of the toric modular forms in weight 2 consists of all cusp forms with nonvanishing central L-value, while in weight j ≥ 3, the cuspidal part is all of S j (Γ 1 (ℓ)). Their approach also uses nonvanishing of L-functions, but is otherwise somewhat different; see the introduction to Section 5 below. In that section, we use our results so far to study models of the modular curve X(ℓ). We use our moduli-friendly interpretation of the elements of R ℓ to show the final result of this article (Theorem 5.5), which can be stated in the following striking manner: for ℓ ≥ 3, the slopes of lines joining the ℓ-torsion points of any one elliptic curve over Q with j = 0, 1728 (for example, E 0 : y 2 = x 3 + 3141x + 5926) contain enough information to deduce equations for X(ℓ), which parametrizes the ℓ-torsion of all elliptic curves. Moreover, the computations involved to find the equations for X(ℓ) are all exact computations in the number field Q(E 0 [ℓ]), and yield a model for X(ℓ) over the cyclotomic field Q(µ ℓ ). In particular, no infinite series or other approximations in C are necessary.
Since our results are moduli-friendly and largely algebraic as opposed to analytic (except for Theorem 5.1), the approach in this article has the advantage that large parts of the theory work over more general fields k than C, provided that 6ℓ is invertible in k, and that the ℓth roots of unity are contained in k. Our approach also has the benefit of yielding a more direct connection to Eisenstein series and to moduli of elliptic curves without using q-expansions at any stage. We thus hope that the techniques we have developed can be of use in the study of modular forms over indefinite quaternion algebras and of Shimura curves.
To summarize, here are the main results in this article:
• A purely algebraic way to evaluate any Eisenstein series at a noncuspidal point p ∈ X(ℓ), in terms of a Weierstrass equation for the elliptic curve E p corresponding to p in the moduli interpretation, along with the coordinates of the ℓ-torsion E p [ℓ] (this is in Sections 2 and 3, which also include effectively computable expressions for Eisenstein series of any weight as polynomials in Eisenstein series of weight 1) • An expression for Eisenstein series of weights 1 and 2 as absolutely convergent sums, without the need for Hecke's method of analytically continuing c,d (cτ + d) −j |cτ + d| −2s in the parameter s ∈ C (Section 2) • Several relations between the moduli-friendly forms, proved algebraically by a consideration of the moduli of elliptic curves (simpler relations in Section 3, and deeper relations in Section 4, which include the action of Hecke operators in weights 2 and 3) • A proof that R ℓ contains all modular forms of weights ≥ 2; thus the only modular forms that are missed by R ℓ are the cusp forms of weight 1 (Theorem 5.1), which is in some sense not surprising, since these correspond to Galois representations of Artin type, and are the most intractable from an arithmetic viewpoint • A systematic method to produce models for the curve X(ℓ) (Theorem 5.5; this model of a curve was called "Representation B" in [KM07], and allows for efficient computation in the Jacobian of X(ℓ)). The idea is to use one fixed elliptic curve E 0 to produce sufficiently many points on a projective model for X(ℓ), so that only one curve X(ℓ) can reasonably interpolate through all these points. This involves lengthy but purely algebraic computations in the field Q(E 0 [ℓ]), and generalizes directly to all modular curves.
Acknowledgements. This research was partially supported by the University Research Board at the American University of Beirut, and the Lebanese National Council for Scientific Research, through the grants "Equations for modular and Shimura curves". The author is grateful to L. Merel for helpful discussions about the Hecke action, and to R. Ramakrishna for useful comments on the manuscript.
Eisenstein series and Laurent expansions of elliptic functions
Our first goal in this section is to describe a rearrangement of the sum in Eisenstein series that converges absolutely for all weights j, not just for j ≥ 3. Let τ ∈ H, where H is the complex upper half plane, and consider the lattice L = L τ = Z+Zτ . Recall the definition of Eisenstein series on the principal congruence subgroup Γ(ℓ), with ℓ ≥ 1.
Definition 2.1. For a 1 , a 2 ∈ Z, let α = α τ = (a 1 τ + a 2 )/ℓ ∈ 1 ℓ L τ . For an integer j ≥ 1 and for s ∈ C, we define, following [Hec27], the Eisenstein series of weight j on Γ(ℓ):
G j (τ, α; s) = ω∈Lτ ′ 1 (α + ω) j |α + ω| 2s = (m,n)∈Z 2 ′ (m + a 1 /ℓ)τ + n + a 2 /ℓ −j (m + a 1 /ℓ)τ + n + a 2 /ℓ −2s , (2.1) (2.2)
G j (τ, α) = G j (τ, α; 0), by analytic continuation.
Here the notation ′ ω omits ω = −α in case we have α ∈ L τ ; similarly for ′ (m,n) . The sum for G j (τ, α; s) converges for Re s > 1 − j/2, and hence when j ≥ 3 we have the absolutely convergent series G j (τ, α) = ′ ω (α + ω) −j . For j ≥ 1, Hecke showed that G j (τ, α; s) can be analytically continued to all s ∈ C, and that G 1 (τ, α) is a holomorphic function of τ , while G 2 (τ, α) is the sum of −2πi/(τ − τ ) and a holomorphic function of τ . Since G j (τ, α; s) depends only on the class of α modulo L τ , we can view α as an ℓ-torsion point on the elliptic curve E = E τ = C/L τ . We shall nonetheless take care to distinguish between α ∈ C and its image P α ∈ E.
We reformulate our Eisenstein series in terms of divisors on C and on E. We establish the following notation to distinguish the notation for the formal sums of points appearing in divisors from sums in C and from the group operation on E:
• A divisor on C will be writtenD = α m α (α), and its image in E is D = α m α (P α ). Note that the α need not be distinct modulo L, so some cancellation can occur in the formal sum for D. We callD a lift of D.
• The group operations of addition, inversion, and multiplication by an integer n ∈ Z on points P, Q ∈ E are given by
(2.3) P, Q → P ⊕Q, P → ⊖P = [−1]P, P → [n]P = P ⊕· · ·⊕P, if n ≥ 1.
We denote by P 0 ∈ E the additive identity in that group.
Definition 2.2. Let D be a divisor on E that is supported on the ℓ-torsion points E[ℓ], and choose any liftD = α m α (α) of D to C. We then define the following Eisenstein series on Γ(ℓ):
(2.4) G j (τ, D; s) = α m α G j (τ, α; s), G j (τ, D) = G j (τ, D; 0).
It is immediate that the definition does not depend on the choice of liftD. We remind the reader that the values α ∈ 1 ℓ L τ (and corresponding points P α ∈ E[ℓ]) vary with τ , as in Definition 2.1.
Our observation is that suitable choices of the liftD lead to series for G j (τ, D; s) with good convergence for all j ≥ 1. We motivate our discussion with the classical fact that a divisor D = α m α (P α ) on E is principal if and only if
(2.5) deg D := α m α = 0, D := α [m α ]P α = P 0 .
The latter sum above is evaluated in E.
Definition 2.3. Let D be a principal divisor on E. A principal lift of D is a divisor D = α m α (α) on C satisfying (2.6) α m α = 0, α m α α = 0 (both sums evaluated in C).
An arbitrary liftD would a priori merely satisfy α m α α ∈ L.
It is easy to see that principal lifts always exist. For example, if α = (a 1 τ +a 2 )/ℓ, then the divisor D = ℓ(P α )−ℓ(P 0 ) is principal, and all of the following are principal lifts of D:D
1 = ℓ(α) − (ℓ − 1)(0) − (a 1 τ + a 2 ), D 2 = (ℓ − 1)(α) + (α − a 1 τ − a 2 ) − ℓ(0), D 3 = (ℓ + 1)(α) − (α + a 1 τ + a 2 ) − ℓ(0).
(2.7)
Proposition 2.4. Given a principal divisor D supported on E[ℓ], choose a principal liftD satisfying (2.6). Then
(2.8) α m α (α + ω) j |α + ω| 2s = O 1 |ω| 2s+j+2
, for large |ω|.
We hence obtain for all j ≥ 1 the following convergent double series (where the notation ′ α means that we omit α = −ω if it appears in the inner sum):
(2.9) G j (τ, D) = ω∈L α ′ m α (α + ω) j |α + ω| 2s s=0 = ω∈L α ′ m α (α + ω) j .
Note that the outer sum over ω is absolutely convergent for Re s > −j/2, even though the double sum converges only conditionally.
Proof. Define the C ∞ function F (u) = 1 (u+ω) j |u+ω| 2s = (u + ω) −j−s (u + ω) −s , upon taking suitable branches for the powers. By Taylor's formula with respect to u and u, we have
F (α) = F (0) + ∂F ∂u u=0 α + ∂F ∂u u=0 α + α u=0 (α − u) ∂ 2 F ∂u 2 du + (α − u) ∂ 2 F ∂u∂u du + (α − u) ∂ 2 F ∂u∂u du + (α − u) ∂ 2 F ∂u 2 du .
(2.10)
The integral is along any suitable path in the complex plane from 0 to α, say for example a line segment. We hence obtain an expansion valid for |ω| > 2|α|:
(2.11) 1
(α + ω) j |α + ω| 2s = 1 ω j |ω| 2s − (s + j)α ω j+1 |ω| 2s − s α ω j−1 |ω| 2s+2 +O 1 |ω| 2s+j+2 .
Here the implied constant depends on α, j, and s, and is uniform in τ when τ is restricted to a compact subset of H. Now multiply (2.11) by m α , and sum over α to obtain (2.8) and (2.9). (Note that the sum ′ α does not omit any α once |ω| is sufficiently large).
Remark 2.5. Note that we always obtain holomorphic functions of τ above. In the setting of weight j = 2, this arises because we have always taken deg D = 0, so the nonholomorphic terms cancel.
Proposition 2.4 allows us to rederive Hecke's second definition of weight 1 Eisenstein series as "division values" of the Weierstrass ζ function in Section 6 of [Hec26], as well as Corollary 3.4.24 of [Kat76]; we reprove those results in (2.14) below. Recall the absolutely and uniformly convergent series for ζ(z) for z in a compact subset of C − L τ :
(2.12) ζ(z) = 1 z + 0 =ω∈L 1 z − ω + 1 ω + z ω 2 = 1 z + 0 =ω∈L z 2 (z − ω)ω 2 .
It is a standard fact that ζ(z + mτ + n) = ζ(z) + 2mη 2 + 2nη 1 for m, n ∈ Z (with "constants" η i = η i (L) satisfying 2η 1 τ − 2η 2 = 2πi). Here we follow the notation of Chapter IV of [Cha85]; note that Hecke and other authors use η i for what we have called 2η i . Moreover, ζ is an odd function of z, and in fact its Laurent expansion near 0 is
ζ(z) = z −1 + O(z 3 ).
Corollary 2.6. Let D be a principal divisor supported on E[ℓ], and take a principal liftD = α m α (α) for which every instance of P 0 in D is lifted to α = 0. Then
(2.13) G 1 (τ, D) = α =0 m α ζ(α).
Moreover, let P α ∈ E[ℓ] − {P 0 }, with any choice of lift α = (a 1 τ + a 2 )/ℓ with a 1 , a 2 ∈ Z. Then
G 1 (τ, P α ) = ζ(α) + 1 ℓ [ζ(α) − ζ(α + a 1 τ + a 2 )] = ζ a 1 τ + a 2 ℓ − a 1 ℓ · 2η 2 − a 2 ℓ · 2η 1 .
(2.14)
Proof. WriteD = m 0 (0)+ α =0 m α (α), with α = 0 =⇒ α / ∈ L by our assumption onD. Changing the sign of ω in (2.12), we obtain
(2.15) α =0 m α ζ(α) = α =0 m α α + ω =0 α =0 m α α + ω − m α ω + m α α ω 2 .
The change of order of summation is justified by the good convergence of the series for ζ and because the sum over α is finite. SinceD satisfies (2.6), we have α =0 m α = −m 0 and α =0 m α α = 0, which allows us to rewrite the above sum in the form of (2.9) (at the cost of replacing absolute convergence with conditional convergence), and hence to obtain (2.13). Now apply this result in the case D = ℓ(α) − ℓ(0), using the principal liftD 3 from (2.7). This yields (2.14), because G 1 (τ, ℓ(α) − ℓ(0)) = ℓG 1 (τ, α) − ℓG 1 (τ, 0) and G 1 (τ, 0) = 0 (more generally, G j (τ, −β; s) = (−1) j G j (τ, β; s)).
We now turn to the second goal of this section, which is to relate Eisenstein series on Γ(ℓ) to Laurent expansions of elliptic functions.
Definition 2.7. Let D be a principal divisor on E, and let m 0 be the multiplicity of P 0 in D. We define an element f D of the function field of E, which we also view as an elliptic function on C with respect to L, by the requirements
(2.16) div(f D ) = D, f D = z m0 (1 + O(z)), near z = 0.
Here the first requirement determines f D up to a nonzero constant factor, and the second requirement normalizes the constant so as to fix our choice of f D . Our normalization ensures that for principal divisors D and E,
(2.17) f D+E = f D · f E .
We remark that the precise normalization of the constant factor in f D will be needed in later sections of this article; it is not significant in this section, since we will mainly consider the logarithmic differential df D /f D .
Theorem 2.8. Let D be a principal divisor, and take a principal liftD = α m α (α). Make the same assumption onD as in Corollary 2.6. Then
(2.18) df D f D = α m α ζ(z − α) dz = ω∈L α m α z − α − ω dz,
where the last series has similar convergence properties to the series of (2.9). Furthermore, if D is supported on E[ℓ], then the Laurent series expansion of df D /f D near z = 0 is
(2.19) df D f D = m 0 z − j≥1 G j (τ, D)z j−1 dz.
Proof. It is classical (see, for example, Section IV.3 of [Cha85]) that we can express f D up to a nonzero constant C = C τ in terms of the Weierstrass σ function, provided that we have taken a principal liftD:
(2.20) f D (z) = C α σ(z − α) mα .
Taking logarithmic differentials yields the first equality in (2.18), since σ ′ /σ = ζ. The second equality now follows from substituting the series for ζ and using the fact that α [m α /ω + m α (z − α)/ω 2 ] = 0. We can now prove (2.19). The first term in the Laurent expansion is easy, and the other terms are equivalent to showing that Res z=0 z −j dfD fD = −G j (τ, D) for j ≥ 1. This residue can be computed by a contour integral on a small circle enclosing z = 0. Since the sum over ω in (2.18) converges well, we are justified in computing the residue term-by-term, using the expansion 1 z−β = − 1 β − z β 2 − z 2 β 3 −· · · for β = 0 to compute residues for each inner sum over α that occurs as a term in the sum over ω. Comparing with (2.9) yields the desired result.
Remark 2.9. Note that (2.18) nicely confirms the fact (a simple consequence of (2.16)) that the differential form df D /f D is periodic with respect to L, has only simple poles, and has residue m α at all points α + ω. We would have liked to use this fact to give a different proof of (2.19), by taking the contour integral of z −j dfD fD around a large circle with center at 0 and radius R (or perhaps using a large parallelogram). At least for j ≥ 2, this approach works, since the contour integral tends to zero as R → ∞. This explains the minus sign in our results, as well as the summation ′ for Eisenstein series, which gives special treatment to the pole at z = 0 due to the presence of z −j . However, we were not able to push through this argument for the important case j = 1. This is because an argument based only on the locations and residues of the poles of the differential form df D /f D cannot distinguish it from any other differential form df D /f D + C dz where C is a constant.
The above theorem appears to relate Laurent expansions of elliptic functions only to those Eisenstein series G j (τ, D) where D is principal. On the other hand, G j (τ, D) depends Z-linearly on D (in fact, so does df D /f D , by (2.17)), so we are led to consider linear combinations of Eisenstein series. Proof. For all P ∈ E[ℓ], the divisor ℓ(P ) − ℓ(P 0 ) is principal, so the C-span of our Eisenstein series contains all Eisenstein series of the form G j (τ, P )− G j (τ, P 0 ). (If j is odd, then G j (τ, P 0 ) = 0 as we have already noted in the proof of Corollary 2.6, so we are done. But we will not use this fact). We conclude that our C-span contains all combinations P ∈E[ℓ] c P G j (τ, P ) for which P c P = 0. If j = 2, then this is the space of all holomorphic Eisenstein series, since we want the nonholomorphic terms 2πi/(τ − τ ) in G 2 to cancel. If j = 2, then it suffices to show that we can obtain G j (τ, P 0 ) (which is of course an Eisenstein series on Γ(1)). To this end, consider the principal divisor D = P ∈E[ℓ] (P ) − ℓ 2 (P 0 ). We obtain G j (τ, D) = (ℓ j − ℓ 2 )G j (τ, P 0 ), so we are done, since ℓ j − ℓ 2 = 0 by our assumptions on ℓ and j.
Remark 2.11. The insistence on restricting to the case D principal is in fact a red herring, for deeper reasons than the above proposition. Take a more general D, which we assume for convenience is supported on E[ℓ] (although we can often manage with the weaker assumption that D ∈ E[ℓ], in the notation of (2.5)).
We can canonically replace D with D − (deg D)(P 0 ), which does not change D but now gives us a divisor of degree zero.
We thus assume in this discussion that deg D = 0, but that D ∈ E[ℓ] need not be trivial. Now the divisor ℓD is principal, and we can formally define f D = (f ℓD ) 1/ℓ for compatibility with (2.17). Note that if D = P 0 , then f D cannot be an elliptic function with respect to L; its formal logarithmic derivative is nonetheless always periodic with respect to L, and we can simply take df D /f D = (1/ℓ)df ℓD /f ℓD as a definition. With this convention, (2.19) continues to hold, and we can obtain an analog of (2.18) as a series with good convergence properties, similarly to our derivation of (2.14).
We can however be more ambitious. Since f ℓD has zeros and poles with multiplicity everywhere divisible by ℓ, we see that f D makes sense as a meromorphic function on C. We use this to normalize the choice of ℓth root f D so that its Laurent series begins with z m0 , as in (2.16). With our above conventions (especially in light of (2.17)), the f D that we consider are products of (positive and negative) powers of the f P = (f ℓ(P )−ℓ(P0) ) 1/ℓ , for P ∈ E[ℓ] − {P 0 }. For such a "basic" f P , Theorem 2.8 then states that
df P f P = z −1 −1 − j≥1 G j (τ, P ) − G j (τ, P 0 ) z j dz = z −1 −1 − G 1 (τ, P )z + −G 2 (τ, P ) + G 2 (τ, P 0 ) z 2 + · · · dz.
(2.21)
Note that if P = P 0 , then f P0 = 1, so (2.21) no longer holds (indeed, df P0 /f P0 = 0, so the first coefficient in the Laurent expansion is now 0 instead of −1). Even when D is nonprincipal as above, one can show that f D is still an elliptic function, however with respect to the sublattice ℓL of L. When D = P , the behavior of f P under translations by L is described by a Weil pairing; see Definition 4.1 in Section 4 below, where we work instead with the function g P (z) = f P (ℓz), which is elliptic with respect to the original lattice L. One can similarly analyze the behavior of an arbitrary f D under translations by L in terms of suitable Weil pairings. The approach of working with f P that are periodic with respect to ℓL is used in the work of Borisov and Gunnells on toric modular forms [BG01a]. They use the function ϑ = ϑ 11 to write down what amounts to the same function as f P when P = a/ℓ + L is in the subgroup of E[ℓ] generated by P 1/ℓ . They then use the expansion of df P /f P at z = 0 to define their toric modular forms s (k) a/ℓ (see Section 4.4 of [BG01a]). Thus their s (k) a/ℓ are a special case of our G j (τ, D), where the divisor D is of the form [a]P 1/ℓ − P 0 . This means that the s (k) a/ℓ are Eisenstein series with respect to Γ 1 (ℓ) instead of Γ(ℓ); Borisov and Gunnells recognize this from the q-expansions, while our approach is more direct. Another advantage of our generalization to Γ(ℓ) is that for ℓ ≥ 2, we obtain the full space of holomorphic Eisenstein series of level Γ(ℓ), in all weights, by Proposition 2.10; see also Theorems 3.9 and 3.13 below. In contrast, the ring of toric modular forms on Γ 1 (ℓ) does not always contain all Eisenstein series on that group: see Remark 4.13 of [BG01b].
Remark 2.12. One can find the Laurent expansion of f D by formally exponentiating the integral of df D /f D . Keeping track of the algebra, one obtains that f D has an expansion of the following form near z = 0:
(2.22) f D = z m0 (1 + F 1 (τ )z + F 2 (τ )z 2 + · · · ),
where F j is a modular form on Γ(ℓ) of weight j, expressible as a polynomial in the G j (τ, D). This approach is used extensively in [BG01a]. In the next section, we study the Laurent series of f D directly in a purely algebraic setting over a more general field k, and reformulate and extend the results of this section algebraically. For now, we simply note the result for f P , obtained from (2.21):
(2.23)
f P = z −1 1 − G 1 z + (G 2 1 −G 2 ) 2 z 2 − G 3 3 − G 1G2 2 + G 3 1 6 z 3 + · · · where we wrote G 1 = G 1 (τ, P ),G 2 = G 2 (τ, P ) − G 2 (τ, P 0 ), and G 3 = G 3 (τ, P ) to save space. For the "genuine" elliptic function f ℓ(P )−ℓ(P0) = f ℓ P , we have the expansion (2.24) f ℓ(P )−ℓ(P0) = z −ℓ 1 − ℓG 1 (τ, P )z + · · · .
Analogous results to (2.23) and (2.24) hold for arbitrary D.
Algebraic reformulation and the ring R ℓ of modular forms
Our first step in "algebrizing" the results of the previous section is to normalize the equation of our elliptic curve E. We embed E into the projective plane P 2 as follows (note the factor 1/2):
(3.1) z → P z = [℘(z; L) : (1/2)℘ ′ (z; L) : 1] = [x(z) : y(z) : 1].
As usual, P 0 = [0 : 1 : 0] is the identity element. The affine algebraic equation of E and the invariant differential ω on E are
(3.2) E : y 2 = x 3 + ax + b, ω = dx/(2y) = dz.
Here a = a(τ ) and b = b(τ ) are, up to constant factors, the Eisenstein series of level 1 and weights 4 and 6, respectively:
(3.3) a(τ ) = −15G 4 (τ, 0) = −15 0 =ω∈Lτ ω −4 , b(τ ) = −35G 6 (τ, 0).
The symbol ω in (3.3) denotes an element of L, but for the rest of this article it will refer almost exclusively to the invariant differential, as in (3.2). We now regard the family {E τ | τ ∈ H} as a single elliptic curve E over the rational function field C(a, b) in two independent transcendental variables. We can work with more general fields k instead of C; in that case, E is a curve over the field K = k(a, b). It is convenient to define the following graded rings, where a and b have weights 4 and 6, respectively:
(3.4) R 1 = k[a, b], R 1,Z = the image of Z[a, b] inside R 1 .
Here R 1 is of course an algebraic analog of the graded ring C[a(τ ), b(τ )] of modular forms on the full modular group Γ(1). Since we wish to use Weierstrass normal form for E, and also need to consider the ℓ-torsion throughout, we require 6ℓ to be invertible in k, and for k to contain the group µ ℓ of ℓth roots of unity (so as to accommodate the Weil pairing later). We extend scalars so that E is now defined over the ℓ-torsion extension field K ℓ , a subfield of the algebraic closure K of K:
(3.5)
K ℓ = K(E[ℓ]) = K x P , y P | P = (x P , y P ) ∈ E[ℓ](K) − {P 0 } .
Over C, it is classical (Section 2 of [Hec27], especially equations (12-14)) that x P and y P are Eisenstein series of weights 2 and 3, respectively, when viewed as functions of τ . Specifically, let P = P α for α = α τ ∈ 1 ℓ L τ − L τ . Then the usual series for ℘ and ℘ ′ , along with (2.9), immediately give us
(3.6) x P = ℘(α; L τ ) = G 2 (τ, α) − G 2 (τ, 0), y P = (1/2)℘ ′ (α; L τ ) = −G 3 (τ, α).
We now turn to the algebraic Laurent expansions of meromorphic functions on E (i.e., of elements of the function field K ℓ (E), but we also view these as elliptic functions with respect to L τ when k = C). We fix an algebraic uniformizer t at P 0 :
(3.7) t = −x/y (= z − 2az 5 /5 + O(z 7 ) when k = C).
We also writeÔ for the completion of the local ring of E at P 0 ; henceÔ is canonically isomorphic to the power series ring K ℓ [[t]] -we occasionally tacitly extend scalars to work in K[[t]] -and we can view
R 1,Z [[t]
] as a subring ofÔ. (When k has characteristic zero, we can still make sense of the analytic uniformizer z as an element ofÔ, since the relation ω = dz means that z = ω = t + 2at 5 /5 + · · · , from (3.8) below.) The meromorphic functions x, y ∈ K ℓ (E[ℓ]) then have the following algebraic Laurent expansions:
x = t −2 − at 2 + · · · = t −2 1 − at 4 + · · · ∈ t −2 R 1,Z [[t]], −tx = y = −t −3 + at + · · · = t −3 −1 + at 4 + · · · , ∈ t −3 R 1,Z [[t]], ω = (1 + 2at 4 + · · · )dt ∈ R 1,Z [[t]]dt.
(3.8)
Moreover, the coefficient of t j in the power series inside each pair of parentheses above is always a weight j homogeneous element of the graded ring R 1,Z . For all this, see for example Section IV.1 in [Sil86], as well as Lemma 3.8 below; alternatively, one can proceed starting from the usual analytic expansion of ℘ in case k = C to obtain expansions of x, y, and t in terms of z, and then revert the series t(z) to obtain series for z, x, and y in terms of t.
Our goal is now to study the algebraic Laurent expansions of the meromorphic functions f D ∈ K(E) of Definition 2.7. The second requirement in (2.16), normalizing the constant factor in f D , now becomes f D = t m0 (1 + O(t)) ∈ t m0 (1 + tÔ). This is compatible with our previous normalization when k = C, since t = z + O(z 2 ) by (3.7).
Definition 3.1. Let D be a principal divisor supported on E[ℓ], with m 0 the multiplicity of P 0 in D as before. For j ≥ 1, we define λ (j) D to be the following coefficient in the Laurent expansion of f D at P 0 :
(3.9) f D = t m0 (1 + λ (1) D t + λ (2) D t 2 + · · · ) = t m0 (1 + λ D t + µ D t 2 + ν D t 3 + · · · ).
In the above equation, we have also introduced the useful abbreviations
(3.10) λ D = λ (1) D , µ D = λ (2) D , ν D = λ (3)
D . We extend the above definitions to arbitrary D supported on E[ℓ] by the method of Remark 2.11: we form the degree zero divisor D − (deg D)(P 0 ), and multiply it by ℓ to obtain a principal divisor
D ′ = ℓ · [D − (deg D)(P 0 )]. We then define (3.11) f D = (f D ′ ) 1/ℓ = t m0−deg D (1 + λ D t + · · · ) ∈ t m0−deg DÔ ,
using the formal ℓth root of the power series, and use this expansion to define the λ
D }, most notably (3.12) λ D+E = λ D + λ E .
By the discussion in Remark 2.12, each λ (j) D is a modular form of weight j on Γ(ℓ) when k = C; the fact that the expansions in (2.22)-(2.24) are with respect to z instead of t does not affect this statement. We nonetheless prefer to give an independent self-contained algebraic formulation and proof of this result. It is sufficient for this article to work with the following naive algebraic definition of modular forms; in contrast to the standard definition in, e.g., Section 2 of [Kat76], we evaluate our modular forms only on the pair (E, ω) with ℓ-torsion over the base field K ℓ .
Definition 3.2. An algebraic modular form of level Γ(ℓ) and weight j ≥ 0 is an element f ∈ K ℓ satisfying the two properties:
(1) We can write f = g({x P , y P })/h({x P , y P }) as a quotient of isobaric polynomials (with coefficients in the graded ring R 1 ) in the variables {x P , y P | P ∈ E[ℓ] − {P 0 }}, where x P has weight 2 and y P has weight 3, so that the resulting weight of f is j;
(2) f satisfies an equation of graded integral dependence over the graded ring R 1 . (Over C, this requirement would ensure that we only select weight j elements of K ℓ that are holomorphic at all τ ∈ H and at all the cusps of the modular curve X(ℓ).)
Now in light of (3.6), we expect that the {x P } and {y P } will turn out to be modular forms of weights 2 and 3 by the above definition. We see that this is indeed the case for the {x P }, since the equation of graded integral dependence that they satisfy is the square 1 of the ℓ-division polynomial ψ 2 [Sil86]). Similarly, the {y P } are integrally dependent over R 1 by transitivity, using y 2 P = x 3 P + ax P + b. We note for later use a consequence of the above discussion. Since the division polynomial ψ 2 ℓ is isobaric, the coefficient of x ℓ 2 −2 is a weight 2 element of R 1 , and must therefore vanish. This implies that (3.13)
ℓ (x) = ℓ 2 P ∈E[ℓ]−{P0} (x − x P ) = ℓ 2 x ℓ 2 −1 + · · · ∈ R 1,Z [x] (see, for example, Exercise III.3.7 of
Remark 3.4. The weight of a homogeneous element of K ℓ can be defined intrinsically by considering, for each u ∈ k × , the automorphism of K ℓ and corresponding isomorphism of elliptic curves given by:
a → u 4 a, b → u 6 b, ω → u −1 ω, t → u −1 t, (x, y) ∈ E : y 2 = x 3 + ax + b → (u 2 x, u 3 y) ∈ E ′ : y 2 = x 3 + u 4 ax + u 6 b. (3.14)
This automorphism naturally sends x P → u 2 x P and y P → u 3 y P , and is compatible with the grading on R 1 ; hence a modular form f of weight j is sent by this automorphism to u j f . Incidentally, u = −1 corresponds to inversion on E, since ⊖P = (x P , −y P ). This easily distinguishes modular forms of odd and even weight. In particular, we have
(3.15) λ ⊖P = −λ P , µ ⊖P = µ P , x ⊖P = x P , ν ⊖P = −ν P , y ⊖P = −y P .
The above equations are for P ∈ E[ℓ] − {P 0 }. When P = P 0 , we of course have f P0 = 1, and so λ P0 = µ P0 = ν P0 = 0. For convenience, we shall also define x P0 = y P0 = 0 in this case, even though the point P 0 , being at infinity, does not have affine coordinates. With this convention, we have the further identities
(3.16) P ∈E[ℓ] λ P = P ∈E[ℓ] µ P = P ∈E[ℓ] x P = P ∈E[ℓ] ν P = P ∈E[ℓ]
y P = 0.
The above equations are obvious for the odd weights (λ P , ν P , y P ), while x P = 0 is (3.13). We can however give a uniform proof of all of these results, including the fact that µ P = 0. The motivation for the uniform proof is that each sum over all P ∈ E[ℓ] in (3.16) gives a modular form on Γ(1) of weight 1, 2, or 3, which can only be zero. To see this algebraically, note that such a sum is invariant under the Galois group of the extension K ℓ /K; this group acts on the points of E[ℓ] in a way that preserves the Weil pairing (since µ ℓ ⊂ k), and is easily seen to be isomorphic to SL(2, Z/ℓZ). Thus each such sum is a weight j element of K = k(a, b) for some j ∈ {1, 2, 3}. Now by Theorem 3.9 below (the reader can check that no circular reasoning is involved), the above sums are all modular forms, and hence are integral over the subring R 1 = k[a, b]. But R 1 is integrally closed, and so the above sums actually belong to R 1 , which means that they must vanish due to their weight.
We remark incidentally that an alternative proof of µ P = 0 is contained in the proof of Proposition 4.3.
The following proposition is an easy consequence of the expansions in (3.8) and standard facts on elliptic curves:
Proposition 3.5. (1) Let P = (x P , y P ) ∈ E[ℓ]− {P 0 }. Then the divisor (P )+ (⊖P ) − 2(P 0 ) is principal, and we have f (P )+(⊖P ) = f (P )+(⊖P )−2(P0) = x − x P = t −2 (1 − x P t 2 − at 4 + · · · ) ∈ t −2 R 1 [x P ][[t]].f D = f D−3(P0) = −y + λ D x + ν D = t −3 (1 + λ D t + ν D t 3 − at 4 + · · · ) ∈ t −3 R 1 [λ D , ν D ][[t]]. (3.18) In particular, µ D = 0, and we have λ D = (y P − y Q )/(x P − x Q ) if P = Q (whence x P = x Q , since we cannot have P = ⊖Q due to R = P 0 ). On the other hand, if P = Q, then λ D = (3x 2 P + a)/2y P ; here again, y P = 0 since we again cannot have P = Q ∈ E[2]
. In both cases, the following standard identity (whose first equality follows from (3.12)) shows that λ D satisfies the integrality condition in part (2) of Definition 3.2, and is therefore a modular form of weight 1:
(3.19) (λ P + λ Q + λ R ) 2 = λ 2 D = x P + x Q + x R . Finally, we also record the trivial identity (3.20) ν D = y P − λ D x P = y Q − λ D x Q = y R − λ D x R .
As mentioned before in Remark 3.3, we also include a direct proof in case k = C that the form λ D in part (2) of the above proposition is a modular form:
Corollary 3.6. If k = C, then in part (2) of the above proposition take a principal
liftD = (α) + (β) + (γ) − 3(0) of D − 3(P 0 ). We then obtain (3.21) λ D (τ ) = −G 1 (τ, α) − G 1 (τ, β) − G 1 (τ, γ) = −ζ(α) − ζ(β) − ζ(γ).
More generally, we have λ D+E = λ D + λ E from (2.17), and so for all D supported on E[ℓ], we conclude that
(3.22) λ D (τ ) = −G 1 (τ, D).
Proof. From (3.18), we have that df D /f D = t −1 (−3+λ D t+· · · ). By (3.7), we know that t and z agree up to O(z 4 ), so we obtain the desired result from (2.19) (recall that G 1 (τ, 0) = 0) and (2.13). The more general result now follows from (3.12).
We shall now define R ℓ for ℓ ≥ 2, generalizing our previous definition R 1 = k[a, b]. Namely, we let R ℓ be the graded k-subalgebra of the ring of all modular forms on Γ(ℓ) that is generated by:
• The forms a and b, in weights 4 and 6, • All coordinates x P , y P , in weights 2 and 3, • All slopes λ D for D = (P ) + (Q) + (R) as in Proposition 3.5, in weight 1.
(We do not need to include the ν D in weight 3, since they already belong to R ℓ by (3.20).) In other words,
(3.23) R ℓ = k a, b, {x P , y P | P ∈ E[ℓ]−{P 0 }}, {λ D | D = (P )+(Q)+(R) ∼ 3P 0 } . We easily have R ℓ ′ ⊂ R ℓ for ℓ ′ a divisor of ℓ (including ℓ ′ = 1).
We observe that the coefficients in the formal Laurent expansions (3.17) and (3.18) belong to R ℓ for all ℓ ≥ 2; moreover, the expansions in (3.17) and (3.18) respect the weights of the modular forms, in the sense that each series has the form t m (1 + c 1 t + c 2 t 2 + · · · ) where c j is a modular form of weight j. This observation motivates the following definition:
Definition 3.7. Let R be any graded subalgebra of the ring of modular forms (say on Γ(ℓ)). An R-balanced Laurent series in t is a series of the form
(3.24) t m 1 + ∞ j=1 c j t j , c j ∈ R of weight j.
An analogous definition holds for series expressed in terms of the analytic uniformizer z (when k has characteristic zero); as the following elementary lemma observes, the condition of being R ℓ -balanced does not depend on whether one expands with respect to t or z.
Lemma 3.8. Let R be a graded algebra as above. Then
(1) If f (t) and g(t) are R-balanced Laurent series, then so are f (t)g(t) and
f (t)/g(t). (2) If f (t) = t m (1 + c 1 t + · · · ) is R-balanced, with ℓ|m, then the "principal branch" of the ℓth root f (t) 1/ℓ = t m/ℓ (1 + c 1 t/ℓ + · · · ) is again R-balanced. (3) Assume that k has characteristic 0. Then z = z(t) = t + 2at 5 /5 + · · · and t = t(z) = z − 2az 5 /5 + · · · are both R 1 -balanced series. It follows that whenever R 1 ⊂ R, then a series f (t) is R-balanced if and only if f (t(z)) is. (4) If f (t) = t m (1 + c 1 t + · · · ) is R-balanced, then the logarithmic differential df /f has the expansion df /f = t −1 (m + j≥1 d j t j )dt with d j a weight j element of R.
Proof. The first two assertions are elementary; for the second, recall that ℓ is invertible in k by assumption. The third follows because the invariant differential ω = dx/(2y) = dz has, by the first assertion, an R 1 -balanced expansion ω = (1 + 2at 4 + · · · )dt; now integrate to obtain that z = z(t) is balanced. The rest is immediate.
We can now state the first main result of this section.
Theorem 3.9.
(1) Let D be a divisor supported on E[ℓ], as in Definition 3.1. Then f D (t) is an R ℓ -balanced Laurent series, and hence for all j ≥ 1, λ (j) D is a modular form of weight j on Γ(ℓ); furthermore, λ
f D (t) is R ℓ -balanced.
The result for f D (z) is also immediate from the above lemma.
Remark 3.10. Statement (2) above can also be proved as in Sections 10.2-10.5 of [Shi07], by expressing the higher derivatives of ℘ in terms of ℘, ℘ ′ , and a(τ ); this relates Eisenstein series of weights 4 and above to the forms x P , y P , and a.
For the cases D = (P ) + (⊖P ) and D = (P ) + (Q) + (R) as in Proposition 3.5, we note that the corresponding f D are polynomials in x and y: namely,
f D = x − x P and f D = −y + λ D x + ν D . In this case, the value of f D at a point T ∈ E[ℓ] − {P 0 } is either x T − x P or −y T + λ D x T + ν D ,
which is a weight 2 or 3 element of R ℓ . More generally, we have the following result:
Corollary 3.11. Assume in the setting of Theorem 3.9 that D is an effective divisor supported on E[ℓ] − {P 0 }, and assume that ⊕D = P 0 , so that D − (deg D)(P 0 ) is principal. Then f D is a polynomial in x and y, whose coefficients all belong to R ℓ . We can in fact expand
f D = x n − H (1) D x n−2 y + H (2) D x n−1 − H (3) D x n−3 y + · · · , if deg D = 2n ≥ 2; f D = −x n y + H (1) D x n+1 − H (2) D x n−1 y + H (3) D x n + · · · , if deg D = 2n + 3 ≥ 3.
(3.25) (The choice of signs above ensures that the monomials x N = t −2N + · · · and −x N y = t −2N −3 + · · · for varying N are normalized R 1 -balanced Laurent series in t.) Moreover, H Remark 3.12. We do not use the results of the above corollary in this article, but we anticipate that they will be useful in other places. For example, the translation τ * T f D of a function f D by an element T ∈ E[ℓ] has as divisor the translation D ′ = τ ⊖T (D) of D by ⊖T ; here τ * T f D will not be normalized, but if T does not belong to the support of D we can still write τ * T f D = f D (T )f D ′ and deduce useful formulas. Another interesting example is the case when we take a principal divisor D = (P 1 ) + (P 2 ) + (P 3 ) + (P 4 ) − 4(P 0 ), such that D = ⊖D. Then f D = x 2 − λ D y + µ D x + H (4) D , and λ D cannot equal zero because D is not an "even" divisor. This means that, over C, λ D is then a weight 1 modular form that cannot vanish at any point of H, but that can only vanish at the cusps; hence it is a kind of generalized modular unit constructed from weight 1 Eisenstein series. A simple example of this is the case P 0 = P 1 = P , P 2 = ⊖P ⊖Q, and P 3 = ⊖P ⊕Q, for P ∈ E[2]∪{Q, ⊖Q}. In this case we obtain λ D = λ (P )+(Q)+(⊖P ⊖Q) + λ (P )+(⊖Q)+(⊖P ⊕Q) = 2y P /(x P − x Q ); this expression appears below in the proof of Theorem 3.13. The numerator and denominator in this expression are modular forms that are well known to vanish only at the cusps, as seen in [KL81]. Our methods have just shown that the ratio of these two forms is also a modular form (i.e., it does not have any poles, even at the cusps), and that this ratio is in fact λ D , an Eisenstein series of weight 1.
Our second main result in this section is the fact that R ℓ is generated by its elements {λ P } of weight 1, in other words (over C) by Eisenstein series of weight 1. This result holds only for ℓ ≥ 3. Indeed, if ℓ = 2, then write as usual E[2] = {P 0 , P 1 , P 2 , P 3 } with P i = (e i , 0) for 1 ≤ i ≤ 3. Hence x Pi = e i and y Pi = 0 for 1 ≤ i ≤ 3, and all the λ P are zero in this case; moreover, (x − e 1 )(x − e 2 )(x − e 3 ) = x 3 + ax + b, as usual. We easily obtain that R 2 is the full ring of modular forms on Γ(2), namely R 2 = k[e 1 , e 2 , e 3 | e 1 + e 2 + e 3 = 0], which is generated by Eisenstein series of weight 2 when k = C.
Theorem 3.13. Assume that ℓ ≥ 3. Define the subring R ′ of R ℓ to be the subring generated by all λ D , where D = (P )+(Q)+(R) is a divisor supported on E[ℓ]−{P 0 } with ⊕D = P 0 as in part (2) of Proposition 3.5. Then the forms a, b, {x P }, {y P }, for P ∈ E[ℓ]−{P 0 }, all belong to R ′ . In particular, R ℓ = R ′ and is hence generated by the λ D of the above form.
Proof. We begin by showing that all the {x P } belong to R ′ . This boils down to a judicious use of (3.19), and involves three cases, depending on ℓ:
(1) If ℓ ≥ 5, let P be a point of exact order ℓ, and consider the following four elements of R ′ (recall also that x ⊖P = x P ):
(λ (P )+(P )+([−2]P ) ) 2 = x P + x P + x [−2]P = 2x P + x [2]P (λ (P )+([2]P )+([−3]P ) ) 2 = x P + x [2]P + x [−3]P = x P + x [2]P + x [3]P (λ (P )+([3]P )+([−4]P ) ) 2 = x P + x [3]P + x [4]P (λ ([2]P )+([2]P )+([−4]P ) ) 2 = 2x [2]P + x [4]P .
(3.26)
Here the determinant det In that case, the points P ′ = (⊖P )⊕R = [−d]Q⊕R and P ′′ = ⊖R both have exact order ℓ, so x P ′ and x P ′′ both belong to R ′ . The points P, P ′ , P ′′ are collinear, and so (λ (P )+(P ′ )+(P ′′ ) ) 2 = x P + x P ′ + x P ′′ belongs to R ′ , whence x P ∈ R ′ . (Alternatively, we can deal with a point P = [d]Q of order less than ℓ by using identities analogous to (3.26) to see that x Q +x [n]Q +x [n+1]Q ∈ R ′ , and to deduce inductively that the x-coordinates of all multiples [n]Q belong to R ′ whenever Q has exact order ℓ.) (2) If ℓ = 3, we simply note that (λ 3(P ) ) 2 = 3x P for all P ∈ E[3] − {P 0 }.
(3) If ℓ = 4, let {Q, R} be a basis for E[4] ∼ = (Z/4Z) 2 . By the same technique as in the first case above, we see that the following sums belong to R ′ , being squares of suitable λ's:
(3.27) 2x Q +x [2]Q , 2x R +x [2]R , x Q +x R +x Q⊕R , x Q +x R +x Q⊖R , x [2]Q +x Q⊕R +x Q⊖R , x [2]R +x Q⊕R +x Q⊖R .
(For example, the fourth sum above is (λ (Q)+(⊖R)+(R⊖Q) ) 2 .) The corresponding determinant is −12, again invertible, so we deduce in particular that x Q , x [2]Q ∈ R ′ . Now any P ∈ E[4] − {P 0 } has exact order either 4 or 2. So we can choose our basis {Q, R} so as to have P = Q in the former case, and P = [2]Q in the latter case, thereby concluding that x P ∈ R ′ . Now that we have shown that all the x P belong to R ′ , let us show that all the y P also belong to R ′ . Fix P ∈ E[ℓ] − {P 0 }, and take any Q ∈ E[ℓ] − {P 0 , P, ⊖P }. Then (y P − y Q )/(x P − x Q ) and (y P + y Q )/(x P − x Q ) are among our λ's (the latter being the slope of the line through P and ⊖Q), and so their sum 2y P /(x P − x Q ) belongs to R ′ . Multiplying by x P − x Q ∈ R ′ shows that y P ∈ R ′ . Observe that at this point we know by (3.20) that the forms {ν D } for D = (P ) + (Q) + (R) also belong to R ′ . Finally, take any P ∈ E[ℓ] − E[2]. Then a = 2y P λ (P )+(P )+([−2]P ) − 3x 2 P also belongs to R ′ , as does b = y 2 P −x 3 P −ax P . Alternatively, we can deduce that a, b ∈ R ′ from the polynomial identity (
x − x P )(x − x Q )(x − x R ) = x 3 − (λ D x + ν D ) 2 + ax + b
whenever D = (P ) + (Q) + (R) with P, Q, R collinear as usual.
Remark 3.14. We can also define a subring R ′ A of R ′ , corresponding to a subgroup A ⊂ E[ℓ]: let R ′ A be generated by the forms λ (P )+(Q)+(R) , for P, Q, R ∈ A − {P 0 } with P ⊕Q⊕R = P 0 . Assume that A ∼ = Z/mZ⊕Z/ℓZ with m|ℓ and ℓ ≥ 5 (possibly m = 1). Then our methods of proof show that a, b, {x P , y P | P ∈ A − {P 0 }} all belong to R ′ A , as do the appropriate ν's coming from points in A. Compare this to Proposition 4.9 in [BG01a].
Remark 3.15. The above two theorems show that when ℓ ≥ 3, all the modular forms that we have constructed through Laurent expansions can be expressed as polynomials in the λ D , which are special Eisenstein series of weight 1 when k = C. It is equally useful to consider a different set of generators of R ℓ , namely the {λ P | P ∈ E[ℓ] − {P 0 }}. We have the relation λ (P )+(Q)+(R) = λ P + λ Q + λ R , which shows that the {λ P } for single points generate the {λ D } as above. Our proof above gives a rather indirect proof of the converse statement, that the {λ D } generate the {λ P }. One can also see this converse directly by observing that ℓ is invertible in k and that ℓλ P = ℓ−2 n=1 λ (P )+([n]P )+([−n−1]P ) . Alternatively, one can express λ P as a linear combination of O(log ℓ) different λ D s using values of n starting from 1 and increasing by a "double-and-add" approach until we reach n = ℓ − 1. This is left to the reader.
We conclude this section by noting a couple of useful algebraic relations between the modular forms in R ℓ . We note that (3.30) below has already appeared for Γ 1 (ℓ) in [BG01b,BGP01]. The approach of obtaining relations by taking a sum of residues over all points of E is taken from [BG01a].
Lemma 3.16.
(1) Let P ∈ E[ℓ] − {P 0 }. Then the Laurent expansion of the logarithmic differential df P /f P begins with
(3.28) df P /f P = t −1 [−1 + λ P t − x P t 2 + y P t 3 + · · · ]dt.
(This is the algebraic analog of (2.21), taking into account (3.6), (3.7), and (3.22).) We deduce the following equations, which over C can also be seen from (2.23):
(3.29) x P = λ 2 P − 2µ P , y P = 3ν P − 3µ P λ P + λ 3 P .
(2) Let D = (P ) + (Q) + (R) be as usual a divisor supported on E[ℓ] − {P 0 } with ⊕D = P 0 . Then
(3.30) λ P λ Q + λ Q λ R + λ P λ R + µ P + µ Q + µ R = 0.
Proof. For (3.28) and (3.29), consider the meromorphic differential form df P /f P on E. Recall that f P = (f ℓ(P )−ℓ(P0) ) 1/ℓ exists inÔ but is not a meromorphic function on E; however, its logarithmic differential is globally defined since f ℓ(P )−ℓ(P0) is a global meromorphic function on E, and df P /f P has simple poles at each of P 0 and P , with residues −1 and 1, respectively. Now use the fact that the sum of the residues of the global meromorphic differential x df P /f P (respectively, y df P /f P ) at all points of E(K) is zero. Taking into account the fact that x = t −2 (1 + O(t 4 )) and y = −t −3 (1 + O(t 4 )), this yields the coefficients x P and y P in (3.28). On the other hand, we can directly compute the logarithmic differential of f P = t −1 (1 + λ P t + µ P t 2 + ν P t 3 + · · · ), and this yields the coefficient λ P in (3.28), as well as (3.29). Finally, to see (3.30), combine the equations x P = λ 2 P − 2µ P for P , Q, and R with (3.19).
Relations involving the Weil pairing and Hecke operators
In this section, we prove deeper algebraic relations between the modular forms λ (j) D than those in Lemma 3.16. The first few relations owe their existence to the Weil pairing on the ℓ-torsion group E[ℓ] of our elliptic curve. Others are related to the action of the full Hecke algebra of Γ(ℓ) on modular forms in R ℓ . We eventually obtain enough relations to be able to show in essence that, over C, the weight 2 and 3 parts of R ℓ are stable under the action of the Hecke algebra. (Actually, in the case of weight 3 we obtain only a partial result at this stage of the proof.) We use this in Section 5 to conclude over C that the ring R ℓ contains all modular forms of weights 2 and above. This of course implies Hecke stability in all weights, and supersedes the previous result. Thus the only modular forms that do not appear in R ℓ are the cusp forms of weight 1; all other modular forms of all weights are expressible as polynomials in the λ P , or equivalently as polynomials in the λ (P )+(Q)+(R) which are slopes of lines through torsion points of the Weierstrass model of E.
The overall shape of the formulas giving the action of the Hecke operators is similar to the results in the articles of Borisov and Gunnells [BG01a,BG01b,BG03]. The treatment in those articles concerns only the group Γ 1 (ℓ), and proceeds via qexpansions and periods of modular forms (the reader is referred also to [Paş06]). Our formulation in terms of Γ(ℓ) involves neither of the above techniques, but focuses instead on the modular parametrization given by the modular curve. We hope to treat some of the connections between this article and those previous articles in later work; it would also be desirable to understand the Hecke action better by directly relating our relations coming from Laurent expansions of elliptic functions to the geometry of toric varieties used in [BG01a].
In order to introduce the Weil pairing on E[ℓ], we also need to discuss pullbacks (i.e., composition) of elementsÔ by the multiplication map [n] : E → E; our main concern is to define the element f Q • [n] ∈Ô, in the sense of controlling its algebraic Laurent expansion in terms of t. This can be done entirely inside the formal group, since we have an expansion of the form t•[n] = nt+2at 5 (n−n 5 )/5+O(t 7 ) ∈ R 1,Z [[t]], so we can formally obtain f Q • [n] = n −1 t −1 (1 + λ Q nt + · · · ). At the same time, we can identify f Q • [n] by its formal zeros and poles as below, and normalize it by a constant factor so that its expansion begins with n −1 t −1 . We thus obtain the first part of the following definition. Here f Q • [n] is usually not an element of the function field of E, but we have the Laurent expansion
(4.1) f Q • [n] = n −1 t −1 (1 + λ Q nt + µ Q n 2 t 2 + ν Q n 3 t 3 + O(t 4 )),
where the remaining terms after t 3 do not follow the simple initial pattern.
(In case Q = P 0 , we have f P0 = f P0 • [n] = 1.) (2) In the special case n = ℓ, we introduce the notation g Q = f Q • [ℓ]. In this setting, g Q is a genuine element of K ℓ (E), since the divisor D of part (1) is now principal. Remark 4.2. If k = C, consider the case when Q = P 1/ℓ and R = P τ /ℓ . One can then show that our normalization gives e ℓ (P 1/ℓ , P τ /ℓ ) = e 2πi/ℓ . (The easiest way to do this calculation is to avoid the Weierstrass σ-function; instead, begin by showing that g P 1/ℓ (z) = C · ϑ(ℓz − 1/ℓ)/ϑ(ℓz) for some nonzero constant C, where ϑ = ϑ 11 .)
We are now ready for the relations arising from the Weil pairing. In weight 1, they imply a subtle symmetry between the {λ P }, essentially a duality under the Fourier transform on E[ℓ] induced by the pairing e ℓ . When k = C, this subtle symmetry motivates Hecke's result that the dimension of the space of Eisenstein series of weight 1 on Γ(ℓ) is half the number of cusps of X(ℓ) (see the end of Section 2 of [Hec27]). This symmetry is usually expressed in terms of q-expansions of weight 1 Eisenstein series; see the second identity at the beginning of Section 7 of [Hec26], which is also derived in Sections 3.4 and 3.5 of [Kat76].
λ R = −1 ℓ Q∈E[ℓ]
λ Q e ℓ (Q, R),
x R = − Q∈E[ℓ]
µ Q e ℓ (Q, R),
y R = −ℓ Q∈E[ℓ]
ν Q e ℓ (Q, R).
(4.3)
By Fourier inversion on the finite group E[ℓ]
, we obtain from (4.3) the identities
(4.4) µ R = −1 ℓ 2 Q∈E[ℓ]
x Q e ℓ (Q, R), ν R = −1 ℓ 3
Q∈E[ℓ]
y Q e ℓ (Q, R).
Proof. Let Q ∈ E[ℓ] − {P 0 }, and consider the meromorphic function g Q = f Q • [ℓ] ∈ K ℓ (E), whose Laurent expansion is of the form (4.1). Define the global meromorphic differential form η Q = g Q ω on E, where ω = (1 + O(t 4 ))dt is the invariant differential; the only singularities of η Q are simple poles at the points of
E[ℓ]
. Now the residue of η Q at P 0 is ℓ −1 , and (4.2) says that τ * R η Q = e ℓ (Q, R)η Q , where τ R : E → E is translation by R. Thus the residue of η Q at any R ∈ E[ℓ] is ℓ −1 e ℓ (Q, R). Now define the differential form η = −ℓ Q∈E[ℓ]−{P0} η Q . We see that η has simple poles at all the points of E[ℓ], and that the residue of η at P 0 is −ℓ 2 + 1, while the residue at R ∈ E[ℓ] − {P 0 } is 1, by the nondegeneracy of the Weil pairing. More precisely, the series expansions of η and τ * R η for R = P 0 have the following form (the sums below are over Q ∈ E[ℓ] − {P 0 }):
η = t −1 (−ℓ 2 + 1) + Q λ Q ℓt + Q µ Q ℓ 2 t 2 + Q ν Q ℓ 3 t 3 + · · · dt = t −1 (−ℓ 2 + 1) + O(t 4 ) dt, τ * R η = t −1 1 − Q λ Q e ℓ (Q, R)ℓt − Q µ Q e ℓ (Q, R)ℓ 2 t 2 − Q ν Q e ℓ (Q, R)ℓ 3 t 3 + · · · dt.
(4.5)
The second equality above in the expansion of η follows from (3.16) (which incidentally yields (4.3) in the special case R = P 0 ; see however the upcoming footnote in this proof). The expansion of τ * R η holds because τ * R η = −ℓ Q e ℓ (Q, R)η Q . We now relate the differential form η to the function f D , corresponding to the
divisor D = Q∈E[ℓ] (Q). The divisor of f D is (f D ) = Q∈E[ℓ] (Q) − ℓ 2 (P 0 ) = Q∈E[ℓ]−{P0} (Q) + (−ℓ 2 + 1)(P 0 )
. Thus η and df D /f D have poles at the same locations, with the same residues. We claim that in fact η = df D /f D , since the difference is not only everywhere holomorphic, but also vanishes at P 0 , by looking beyond the first term in the Laurent expansions at P 0 . Indeed, f D = ±ℓ −1 ψ ℓ (x, y) where ψ ℓ is the ℓth division polynomial, and hence f D has an R 1 -balanced Laurent expansion of the form f D = t −ℓ 2 +1 (1 + O(t 4 )), which implies that df D /f D = t −1 [(−ℓ 2 + 1) + O(t 4 )]; on the other hand, η has a similar expansion by (4.5), and our claim follows. 2 We now consider the translation of the identity η = df D /f D by the point R, when R = P 0 . This gives us τ *
R η = τ * R df D /f D = d(τ * R f D )/τ * R f D .
We shall compare the expansion of τ * R η from (4.5) to the expansion of the logarithmic differential of τ * R f D . Comparing the locations of zeros and poles, we see that τ * R f D = C · f D · (f (⊖R) ) −ℓ 2 for some nonzero constant C. (Here f (⊖R) is not a genuine meromorphic function on E, but its ℓth power is, so (f ⊖R ) −ℓ 2 is also a genuine meromorphic function.)
We obtain that d(τ * R f D )/τ * R f D = df D /f D − ℓ 2 df ⊖R /f ⊖R .
However, from (3.28) and (3.15), we have
(4.6) df ⊖R /f ⊖R = t −1 [−1 − λ R t − x R t 2 − y R t 3 + · · · ]dt.
Combining all this and comparing the Laurent expansions, we obtain (4.3) as desired. Equation (4.4) then follows immediately.
2 The alert reader will note that we needed only the simple result P Q λ Q = 0 of (3.16) to deduce that η = df D /f D . The form of the expansion of f D then allows us to conclude the identity P Q µ Q = 0 -as well as the simple identity P Q ν Q = 0 -thereby giving a second way to complete the proof of (3.16), and hence of (4.3) when R = P 0 .
The relations (4.4), when combined with (3.6), imply that the forms {µ P , ν P } are Eisenstein series of weights 2 and 3, when k = C. It will be useful for us to formalize this algebraically, while also taking into account (3.22).
Definition 4.4. For j ∈ {1, 2, 3}, we define the algebraic space E j of Eisenstein series of weight j by (4.7)
E 1 = span{λ P | P ∈ E[ℓ]}, E 2 = span{x P }, E 3 = span{y P }.
(If we wish to draw attention to the level ℓ, we will write E ℓ j .) We deduce from (4.4) and (3.29) that for all P ∈ E[ℓ],
(4.8) µ P , λ 2 P ∈ E 2 , ν P ∈ E 3 .
From (3.30), we also obtain that for P, Q, R ∈ E[ℓ] with P ⊕ Q ⊕ R = P 0 ,
(4.9) λ P λ Q + λ Q λ R + λ P λ R ∈ E 2 .
Note that in the above equation, the points P, Q, R are allowed to take the value P 0 ; for example, if Q = P 0 , then λ R = −λ P , in which case (4.9) becomes the statement −λ 2 P ∈ E 2 that we know from (4.8). (The result that µ P and λ 2 P are Eisenstein series, as well as the result (4.9), were already observed for Γ 1 (ℓ) in [BG01b]).
In our treatment of Hecke operators, we shall need the following identities, which are related to the fact that the trace from Γ(nℓ) to Γ(ℓ) of an Eisenstein series on Γ(nℓ) is again an Eisenstein series. Proof. Over C, equation (4.10) is immediate from the definition of G j in (2.1) and (2.2) -in verifying this, the reader should bear in mind that x P is a difference between two G 2 s. Let us however give a proof in our algebraic setting. Now (4.10) is trivial for P = P 0 . If P = P 0 , we begin by noting the following identity, which is obtained by comparing zeros and poles, as well as the leading coefficient of the Laurent expansion:
(4.12) f [n]P • [n] = n −1 T ∈E[n]
f P ⊕T /f D .
Here D = T ∈E[n] (T ) is the divisor supported on the n-torsion points. Hence the principal divisor of f D is (f D ) = D − n 2 (P 0 ), similarly to the proof of Proposition 4.3; this also implies the expansion f D = t −n 2 +1 (1 + O(t 4 )). Now taking the logarithmic differential of both sides of (4.12) and comparing the first few coefficients yields (4.10), as desired.
As for (4.11), we prove it using the Fourier duality of Proposition 4.3. (This approach also yields a different proof of (4.10).) For instance, use (4.4) to express each µ in the first sum in (4.11) in terms of an x. This yields (4.13)
T ∈E[n] µ P ⊕T = T ∈E[n] −1 n 2 ℓ 2 A∈E[nℓ]
x A e nℓ (A, P ⊕ T ).
Rearrange the sum as A T , and use the property of the Weil pairing (4.14)
A
µ P ⊕T = −n 2 n 2 ℓ 2 A∈E[ℓ]
x A e nℓ (A, P ) = −1 ℓ 2
A∈E[ℓ]
x A e ℓ (A, [n]P ),
where the last equality is analogous to (4.14). This implies the first part of (4.11). The second part, involving ν, is proved similarly.
We are now ready for the main ingredient in the proof that the degree 2 part of R ℓ is stable under the Hecke algebra. This proof involves an interesting induction on the level. One starts with forms on Γ(nℓ), "raises the level" to rewrite them in terms of forms of higher level Γ(snℓ) with s < n, then "lowers the level" back to level Γ(sℓ). Repeating this process reduces the value of s, and one eventually reaches s = 0, which can be dealt with using Lemma 4.5. where the linear combination above is over finitely many (a, b, c
, d) ∈ Z 4 satisfying (4.17) det a b c d = ±n, a − sb ≡ c − sd ≡ 0 (mod n).
Proof. The proof is by induction on n, the case n = 1 (so T = P 0 ) being trivial. Note that the value of s only matters modulo n, so we henceforth assume that 0 ≤ s < n. If s = 0, then the sum over T is nλ [n]A λ B by (4.10), so we are done. In general, we shall invoke an inductive step analogous to the Euclidean algorithm, reducing (4.16) for the pair (n, s) to the analogous statement for (s, n), which amounts to the same as (s, n mod s).
λ A⊕T λ B ′ ⊖T ⊕U ≡ T,U λ A⊕T λ A⊕B ′ ⊕U − T,U λ ⊖B ′ ⊕T ⊖U λ A⊕B ′ ⊕U (mod E snℓ 2 ), (4.19)
where the congruence is obtained from (4.9) with P = A ⊕ T , Q = B ′ ⊖ T ⊕ U , and R = ⊖A⊖B ′ ⊖U ; we have also used (3.15). Now the first sum on the right hand side of equation (
λ [−n]B ′ ⊖[n]U λ A⊕B ′ ⊕U .
By the inductive hypothesis, the above sum is congruent modulo E s!nℓ 2 to a linear combination of terms of the form
(4.21) λ [a ′ ](A⊕B ′ )⊕[−nb ′ ]B ′ λ [c ′ ](A⊕B ′ )⊕[−nd ′ ]B ′ = λ [a ′ ]A⊕[ a ′ −nb ′ s ]B λ [c ′ ]A⊕[ c ′ −nd ′ s ]B
where (a ′ , b ′ , c ′ , d ′ ) satisfy (4.17) with the roles of s and n interchanged; in particular, a ′ −nb ′ s , c ′ −nd ′ s ∈ Z, and we get that each term is of the form λ Remark 4.7. The element of E n!ℓ 2 above actually belongs to E nℓ 2 , but we shall not prove this in our algebraic context; it is obvious over C, since it is an Eisenstein series with level n!ℓ that happens to transform under Γ(nℓ). (Similarly, if A, B ∈ E[ℓ], then the element of E 2 above actually belongs to E ℓ 2 .) It is possible to specify this element more precisely by applying (3.30) instead of (4.9) in the above proof. Typically, this yields an element of E 2 that is a linear combination of terms µ [a]A+[b]B where a − sb ≡ 0 (mod n), after one also invokes (4.11). However, one must be careful not to apply (3.30) when one of the torsion points P, Q, R is P 0 .
On another topic, we observe that the linear combination in (4.16) is Z-linear, and the coefficients are all divisible by n. This we leave to the reader.
We need a few more standard observations before we prove the Hecke stability of the weight 2 part of R ℓ . Starting from this point, we shall for convenience work exclusively over C; also, since R 1 and R 2 are the full rings of modular forms on Γ(1) and Γ(2), we can restrict to ℓ ≥ 3. We use the standard notation for the spaces of cusp forms and modular forms of weight j on a congruence subgroup Γ: We also make use of the usual group action f → f | j γ of γ ∈ Γ(1) on the space M j (Γ(ℓ)). This action of course preserves the spaces of Eisenstein series and cusp forms in all weights. We interchangeably view γ as an element of Γ(1) or of Γ(1)/Γ(ℓ) ∼ = SL(2, Z/ℓZ). This group also acts on the torsion group E[ℓ] while preserving the Weil pairing, and on our ring R ℓ via ring isomorphisms. Indeed, for γ ∈ SL(2, Z/ℓZ), we have P → P · γ, where
z P = a 1 τ + a 2 ℓ =⇒ z P ·γ = a 1 ′ τ + a 2 ′ ℓ with (a 1 ′ a 2 ′ ) = (a 1 a 2 )γ, P ∈ E[ℓ] =⇒ λ P | 1 γ = λ P ·γ .
(4.23)
We briefly review the well-known interpretation of Hecke operators in terms of a trace between congruence subgroups. Given a Hecke operator described as a double coset Γ(ℓ)αΓ(ℓ) with α ∈ GL + (2, Q), we can harmlessly multiply α by a scalar to obtain a primitive integral matrix; then composing this double coset on the left and right by the action of elements γ 1 , γ 2 ∈ Γ(1) allows us to assume without loss of generality that α = ( n 1 ) for some n ≥ 1. We then have, for f (τ ) ∈ M j (Γ(ℓ)):
(4.24) f | j Γ(ℓ) n 1 Γ(ℓ) = C γ∈Γ(nℓ)\Γ(ℓ) f (nτ ) | j γ,
where C = C n,ℓ,j is a suitable normalizing constant. Note that if f (τ ) ∈ R ℓ , then f (nτ ) ∈ R nℓ ; indeed, the map f → f (nτ ) respects multiplication of forms, so it is enough to check the above statement for the weight 1 Eisenstein series λ P = −G 1 (τ, P ) that generate R ℓ . This is just the identity (4.25)
G 1 (nτ, a 1 τ + a 2 ℓ ) = n −1 k mod n G 1 (τ, a 1 nτ + a 2 + kℓ nℓ ).
The sum over representatives γ ∈ Γ(nℓ)\Γ(ℓ) in (4.24) is a trace from M j (Γ(nℓ)) to M j (Γ(ℓ)), and we shall henceforth work with it instead of with double cosets. With these preliminaries out of the way, we can state and prove our result for weight 2, which will be superseded later when we show that R ℓ contains all of M 2 (Γ(ℓ)).
Proposition 4.8. Let k = C. Then the trace of a weight 2 element of R nℓ from M 2 (Γ(nℓ)) to M 2 (Γ(ℓ)) actually belongs to R ℓ . ( A priori, this trace merely belongs to R nℓ ∩ M 2 (Γ(ℓ)).) Corollary 4.9. Over C, the weight 2 part of R ℓ is stable under the action of the Hecke algebra for Γ(ℓ).
Proof of Proposition 4.8. By the observation immediately following Remark 4.7, we can assume that ℓ ≥ 3; it is enough to show in that case that the trace of any product λ P λ Q = G 1 (τ, P )G 1 (τ, Q) with P, Q ∈ E[nℓ] − {P 0 } belongs to R ℓ . Now R ℓ already contains all the Eisenstein series on Γ(ℓ) in weight 2 (indeed, in all weights j, by Theorem 3.9), so we can work modulo Eisenstein series in our proof. As we mentioned in Remark 4.7, this can be done even if we encounter Eisenstein series of higher level in some intermediate steps. Furthermore, the trace down from level nℓ to level ℓ can be done one prime factor at a time, so we may harmlessly assume that n is a prime number. There are two cases to consider: (i) n is prime and n | ℓ, and (ii) n is prime and n|ℓ.
In case (i), we take a direct sum decomposition E[nℓ] = E[ℓ] E[n], and note that Γ(nℓ)\Γ(ℓ) is isomorphic to SL(2, Z/nZ) and that it affects only the E[n] part.
Decompose P = A ⊕ T 0 and Q = B ⊕ U 0 , with A, B ∈ E[ℓ] and T 0 , U 0 ∈ E[n].
We may suppose that one of {T 0 , U 0 } -let us say, T 0 -is not equal to P 0 , since otherwise λ P λ Q ∈ R ℓ already. Then there are two subcases: (i.a) there exists s ∈ Z/nZ (s = 0 is allowed) such that U 0 = [s]T 0 , and (i.b) {T 0 , U 0 } are a basis for E[n]. In subcase (i.a), the trace of λ A⊕T0 λ B⊕[s]T0 is equal to a multiple of T ∈E[n]−{P0} λ A⊕T λ B⊕[s]T . By Proposition 4.6, this is congruent modulo E 2 to an element of R ℓ (the "missing term" in the sum, corresponding to T = P 0 , is λ A λ B , which already belongs to R ℓ ). Hence the trace itself belongs to R ℓ , as we have observed before.
In subcase (i.b), let ζ = e n (T 0 , U 0 ), which is a primitive nth root of unity. Then the trace that we wish to compute is For fixed T and V , we must hence study the sum over those U for which e n (T, U ) = ζ. Such a U exists if and only if T = P 0 (here we use the facts that n is prime and ζ = 1), in which case U ranges over the set of torsion points {U T ⊕ [t]T | t ∈ Z/nZ} for some particular choice of U T (depending on T ) with e n (T, U T ) = ζ. The sum over U thus contains a factor t∈Z/nZ e n ([ℓ]V, U T ⊕ [t]T ), which vanishes unless V belongs to the cyclic subgroup generated by T (recall that ℓ is relatively prime to n). We obtain that (4.27) is equal to We now turn to case (ii). We write ℓ = Ln k with n | L and k ≥ 1, and decompose P = A ⊕ T 0 and Q = B ⊕ U 0 with A, B ∈ E[L] and T 0 , U 0 ∈ E[n k ]. We wish to compute a trace using representatives for Γ(Ln k+1 )\Γ(Ln k ). Such representatives again do not affect A or B, and their action on T 0 and U 0 can be described by matrices in SL(2, Z/n k+1 Z) that are congruent to the identity modulo n k ; thus such matrices have the form (4.29) 1 + n k α n k β n k γ 1 − n k α = I + n k M,
M = α β γ −α ∈ M trace 0 2 (Z/nZ).
The reader should note that we view the entries of M as being in Z/nZ, but that multiplying them by n k yields elements of n k Z/n k+1 Z, not zero. We also point out that we shall feel free to use other bases for E[n k+1 ] ∼ = (Z/n k+1 Z) 2 than the standard basis {P τ /n k+1 , P 1/n k+1 }; even if the change of basis does not have determinant 1 (and hence changes the Weil pairing), our description of M in (4.29) remains valid. Let us writeT 0 = [n k ]T 0 andÛ 0 = [n k ]U 0 . We haveT 0 ,Û 0 ∈ E[n], and the trace that we wish to compute is then (4.30)
M∈M trace 0 2 (Z/nZ) λ A⊕T0⊕(T0·M) λ B⊕U0⊕(Û0·M) ,
where the action of M is analogous to that in (4.23). Once again we may suppose thatT 0 andÛ 0 are not both P 0 (otherwise, T 0 , U 0 ∈ E[n k ] and we are already in R n k L = R ℓ ), and that without loss of generalityT 0 = P 0 . We face analogous subcases: Modulo Eisenstein series, this last expression is a linear combination of terms of the form
(4.32) λ [a](A⊕T0)⊕[b](B⊕U0) λ [c](A⊕T0)⊕[d](B⊕U0) , for a + sb ≡ c + sd ≡ 0 (mod n). We observe that [n k ]([a]T 0 ⊕ [b]U 0 ) = [a]T 0 ⊕ [b]Û 0 = [a + sb]T 0 = P 0 , whereas [a]A ⊕ [b]B ∈ E[L]
, so the first factor in (4.32) involves torsion points in E[n k L] = E[ℓ]; an analogous statement holds for the second factor, and we obtain an element of R ℓ , as desired.
In subcase (ii.b), we write M in terms of the basis {T 0 ,Û 0 }, and obtain that we wish to study (4.33)
α,β,γ∈Z/nZ λ A⊕T0⊕[α]T0⊕[β]Û0 λ B⊕U0⊕[γ]T0⊕[−α]Û0 .
Similarly to subcase (i.b), we rewrite the factor λ B⊕U0⊕[γ]T0⊕[−α]Û0 in terms of the Weil pairing and λ C⊕V , for C ∈ E[L] and V ∈ E[n k+1 ]. We obtain a linear combination of terms of the following form (here the triples (α, β, γ) ∈ (Z/nZ) 3 are analogous to the pairs {(T, U ) ∈ E[n] × E[n] | e n (T, U ) = ζ} of (4.27)):
(4.34) Writing the sum in the order X T , we see that the inner sum overT is now exactly analogous to (4.31). In our setting, C and [s]T 0 ⊕ X play the roles of B and U 0 from (4.31) 3 , and we obtain by an identical argument to the subcase (ii.a) that our final expression is congruent modulo Eisenstein series to an element of R ℓ . This completes the proof.
α,β,γ∈Z/nZ V ∈E[n k+1 ] λ A⊕T0⊕[α]T0⊕[β]Û0 λ C⊕V e n k+1 ([L]V, U 0 ⊕ [γ]T 0 ⊕ [−α]Û 0 ).
Having disposed of weight 2, we now turn our attention to weight 3. We shall prove weight 3 analogs of Propositions 4.6 and 4.8, but only for modular forms of the form x P λ Q , i.e., for products of an Eisenstein series of weight 2 with an Eisenstein series of weight 1. We shall continue to work modulo Eisenstein series, i.e., modulo the space E 3 . In this context, the analog of (4.9) is the following statement, which holds whenever P ⊕ Q ⊕ R = P 0 :
(4.38) (x P − x R )(λ P + λ Q + λ R ) ∈ E 3 .
To see this, first observe by Proposition 3.5 that if none of P , Q, and R is equal to P 0 , then the above expression is equal to y P − y R , which is in E 3 . On the other hand, if one of the points is P 0 , then λ P + λ Q + λ R = 0 by our conventions, and the above expression is equal to 0. The next lemma is the weight 3 analog of the key computational step that we did in (4.19). We note incidentally that we could have applied the techniques of this lemma to the weight 2 identity (λ P + λ Q + λ R ) 2 = x P + x Q + x R . This would have yielded a slightly weaker result than Proposition 4.6 (analogous to the proof below in weight 3) that would also have been sufficient for our purposes.
x A⊕T λ B⊖T ≡ −nx [n]A λ [n]A + n 2 x [n]A λ A⊕B + nx A⊕B λ [n]A + nx A⊕B λ [n]B − n 2 x A⊕B λ A⊕B .
(4.40)
Proof. To show (4.39), we take P = A⊕T , Q = ⊖A⊕U , and R = ⊖T ⊖U in (4.38), and we sum the result over all T, U ∈ E[n], knowing that the final result will be ≡ 0 modulo E 3 . We now observe that
T,U∈E[n] x A⊕T λ A⊕T = n 2 T x A⊕T λ A⊕T , T,U x A⊕T λ ⊖A⊕U = n 2 x [n]A · nλ [−n]A = −n 3 x [n]A λ [n]A , T,U x A⊕T λ ⊖T ⊖U = T,V ∈E[n] x A⊕T λ V = 0, T,U x ⊖T ⊖U λ A⊕T = T,V x V λ A⊕T = 0; similarly, T,U x ⊖T ⊖U λ ⊖A⊖U = 0, T,U x ⊖T ⊖U λ ⊖T ⊖U = n 2 V x V λ V = n 2 V x ⊖V λ ⊖V = −(itself) = 0, (4.41)
where we have used (4.10) and (3.16) as needed.
For the proof of (4.40), we take the sum over all T in E[n] of (4.38) with P = A ⊕ T , Q = B ⊖ T , and R = ⊖A ⊖ B (so λ R = −λ A⊕B and x R = x A⊕B ). We then proceed as in the proof of (4.39), while using (4.39) at one point, to obtain the desired result.
At this point, the generalization of Propositions 4.6 and 4.8 to weight 3 is straightforward.
Proposition 4.11. Make the same hypotheses as Lemma 4.10, and let s ∈ Z. Then we have the following congruence modulo E 3 :
T ∈E[n]
x A⊕T λ B⊖[s]T ≡ a linear combination of terms of the form
x [a]A⊕[b]B λ [c]A⊕[d]B ,
with a − sb ≡ c − sd ≡ 0 (mod n).
(4.42) (An analogous statement holds for sums T x A⊖[s]T λ B⊕T , in which case the congruence condition modulo n becomes −sa + b ≡ −sc + d ≡ 0.) We remark incidentally that the determinant of a b c d above need not be equal to n. Furthermore, if P, Q ∈ E[nℓ], then the trace of the weight 3 element x P λ Q ∈ R nℓ down to level Γ(ℓ) is congruent modulo E 3 to a linear combination of terms
x R λ S ∈ R ℓ , with R, S ∈ E[ℓ].
Proof. The proof of (4.42) follows the same lines as the proof of Proposition 4.6, with the same type of induction on s. For s = 0, it follows as usual from (4.10), and we have already proved the case s = 1 in (4.40). The key step in the induction (analogous to (4.19)) amounts to applying (4.40) to the T -part of the sum T,U x A⊕T λ B ′ ⊖T ⊕U . The ideas are essentially the same as before, with the use of (4.39) thrown in for good measure. (It is worth pointing out that while carrying out the same proof in the case of T As for the proof of the statement about the trace of x P λ Q , it follows the argument of Proposition 4.8 with only trivial changes. The only point worth mentioning is that the roles of T 0 and U 0 are no longer symmetric, so we cannot simply assume that T 0 in case (i) (respectively,T 0 in case (ii)) is not equal to P 0 . However, if T 0 (respectively,T 0 ) is equal to P 0 , then P ∈ E[ℓ] already, and the trace is then equal to x P tr(λ Q ), which is easy to analyze using (4.10), or for that matter by noting that the trace of the Eisenstein series λ Q is again an Eisenstein series.
5. Generating all modular forms in weights ≥ 2, and a model for X(ℓ)
We are now ready to use Propositions 4.8 and 4.11 to show that the ring of modular forms over C generated by the Eisenstein series of weight 1 contains all modular forms in weights 2 and above. This is Theorem 5.1 below. We then apply the result to obtain a convenient method to find explicit models for the modular curve X(ℓ), in Theorem 5.5 below.
We prove Theorem 5.1 via relating the result to the nonvanishing of a special value of an L-function, which is also the strategy of [BG01b,BG03]. Our proof brings in the L-function via a Rankin-Selberg integral, in contrast to the calculations in the articles by Borisov and Gunnells, which use q-expansions whose coefficients are modular symbols. It is worth noting that one can give a much simpler proof of the (rather weaker) fact that R ℓ contains all modular forms in sufficiently high weights. To see this, note that the ring of all modular forms is the graded integral closure of R ℓ in its own field of fractions, by Definition 3.2 and Remark 3.3. Hence X(ℓ) = Proj R ℓ ; since X(ℓ) is nonsingular, it is then a standard fact that the graded components of the two rings (R ℓ and the ring of modular forms) agree in sufficiently high weights -see for example [Har77], Section II.5.19 and Exercises II.5.9, II.5.14. Precise but large bounds for the meaning of "sufficiently high" for arbitrary curves are given in [GLP83], but they of course grow with the genus of the curve, which for X(ℓ) is O(ℓ 3 ). The interest of our results, as well as those of Borisov-Gunnells, is that they give a fixed value for "sufficiently high": 2 in our result for Γ(ℓ), and 3 for their result for Γ 1 (ℓ) (where they obtain all cusp forms modulo Eisenstein series, but potentially miss some Eisenstein series).
Theorem 5.1. Let k = C. Then R ℓ contains all modular forms on Γ(ℓ) of weight 2 and above. In other words, R ℓ "misses" precisely the cusp forms in weight 1.
Proof. Since R ℓ contains all modular forms for ℓ ≤ 2, we as usual restrict to the case ℓ ≥ 3. Our first claim is that it is enough to show that R ℓ contains all of M 2 (Γ(ℓ)) and M 3 (Γ(ℓ)). To see this claim, observe that Γ(ℓ) has no elliptic elements or irregular cusps; hence there exists a line bundle L on X(ℓ) such that for all weights j, we have M j (Γ(ℓ)) = H 0 (X(ℓ), L ⊗j ). Moreover, elements of M 2 can be viewed as 1-forms on X(ℓ) with at worst a simple pole at each cusp. Hence the degree of L ⊗2 is equal to 2g − 2 + κ, where g is the genus of X(ℓ), and κ is the number of cusps. Since κ ≥ 4 for ℓ ≥ 3, by standard formulas for modular curves (e.g., Section 1.6 of [Shi71]), we obtain that 2 deg L ≥ 2g + 2. This is enough to show that the multiplication map M j (Γ(ℓ)) ⊗ M j ′ (Γ(ℓ)) → M j+j ′ (Γ(ℓ)) is surjective for j, j ′ ≥ 2, since the degrees of L ⊗j and L ⊗j ′ are both ≥ 2g + 1 (for a sketch of this standard result, see Lemma 2.2 of [KM04]; the survey in Section 1 of [Laz89] is also a particularly useful reference). This implies that any ring of modular forms that contains M 2 (Γ(ℓ)) and M 3 (Γ(ℓ)) must contain all forms in higher weights, thereby establishing our claim.
We therefore turn to the situation in weights 2 and 3, which we study using a result of Shimura [Shi76]. This result states that a suitable Rankin-Selberg convolution of a newform F with an Eisenstein series gives a product of two special values of Hecke L-functions associated to F and to Dirichlet characters ξ, ψ. More precisely, Theorem 2 (with r = 0) of [Shi76], and equation (4.3) of the same article (with k = j ≥ 2, l = 1, and m = j − 1) imply the following statement, which holds for any j ≥ 2: let F ∈ S j be a newform with character χ, and let ξ, ψ be Dirichlet characters with (ξψ)(−1) = −1; then there exists a product GG ′ of two Eisenstein series, with G ∈ E 1 and G ′ ∈ E j−1 , such that the Petersson inner product of F with GG ′ gives
(5.1) F, GG ′ = C · L(j − 1, F, ξ)L(j − 1, F, ψ)
with an explicit nonzero constant C. (Here, if j = 3, we must have χξψ = 1 in order for G ′ ∈ E 2 to be holomorphic.) Note that we have normalized the Petersson inner product so that it is insensitive to the choice of common congruence subgroup Γ with respect to which F , G, and G ′ are all invariant. We deduce from (5.1) that for a given F , we can choose ξ and ψ (and, with them, G and G ′ ) so as to make the above inner product nonzero. Indeed, when j ≥ 3, then, regardless of ξ and ψ, the L-functions on the right side are nonzero, since they are evaluated outside the critical strip if j ≥ 4, and at the edge of the critical strip if j = 3 (see, e.g., Proposition 2 of [Shi76], or [JS77] for a more general result). Thus we can also ensure that χξψ = 1 as needed in the special case j = 3. On the other hand, if j = 2, then, by Theorem 2 of [Shi77], there exist ξ and ψ for which the right side of (5.1) is nonzero.
We can now show that R ℓ contains all of M 2 (Γ(ℓ)) and M 3 (Γ(ℓ)). Since R ℓ contains all Eisenstein series on Γ(ℓ), we are reduced to checking whether R ℓ contains all of S j (Γ(ℓ)) for j ∈ {2, 3}, or alternatively to checking that the orthogonal complement [R ℓ ∩ S j (Γ(ℓ))] ⊥ in S j (Γ(ℓ)) is zero. Let 0 = f ∈ S j (Γ(ℓ)) be any nonzero cuspform in this orthogonal complement. Then there exist constants c 1 , . . . , c N ∈ C and matrices α 1 , . . . , α N ∈ GL + (2, Q) such that the linear combination F = i c i f |α i is actually a newform (for instance, use an element of the Hecke algebra to project to a single automorphic representation, and then move around within it to reach the newform). We can find G, G ′ as above such that F, GG ′ = 0. But this means that
(5.2) 0 = i c i f |α i , GG ′ = i c i f, (GG ′ )|α −1 i .
In the above expression, each form (GG ′ )|α −1 i = (G|α −1 i )(G ′ |α −1 i ) is still the product of an Eisenstein series of weight 1 with an Eisenstein series of weight j − 1 ∈ {1, 2}; hence it can be written as a linear combination of modular forms of the form λ P λ Q or λ P x Q , with P, Q ∈ E[nℓ] for some (possibly rather large) n. We obtain a linear combination of inner products of the form f, λ P λ Q or f, λ P x Q , which can in turn be reexpressed (up to a constant factor) as an inner product of the form f, tr Γ(nℓ) Γ(ℓ) λ P λ Q or f, tr Γ(nℓ) Γ(ℓ) λ P x Q , and the traces belong to R ℓ by Propositions 4.8 and 4.11. Furthermore, the Eisenstein part of each such trace, and therefore also the cuspidal part, must then belong to R ℓ . Thus the inner products must all be zero if f belongs to the orthogonal complement [R ℓ ∩ S j (Γ(ℓ))] ⊥ in question. This contradicts the fact that F, GG ′ = 0, and we deduce that the orthogonal complement is zero after all. This concludes our proof.
Theorem 5.1 makes it possible to compute nice models for the modular curve X(ℓ). These models are defined over Q(µ ℓ ) (we suspect that a more careful investigation would yield models over Q), and are in the form called "Representation B" in [KM07]. The basic idea is to work implicitly with the projective embedding of X given by a line bundleL, with degL ≥ 2g + 2, for which it is a standard fact that the ideal of equations defining the image of X is generated by quadrics (see, for example, Section 1 of [Laz89]). (In our setting, we will have X = X(ℓ) andL = L ⊗2 , in the notation of the proof of Theorem 5.1.) We then define V = H 0 (X,L) and V ′ = H 0 (X,L ⊗2 ); in our setting, this means that V = M 2 (Γ(ℓ)) and V ′ = M 4 (Γ(ℓ)). Let µ be the multiplication map µ : V ⊗ V → V ′ , and note that µ factors through a map µ : Sym 2 V → V ′ . Then the kernel of µ describes exactly the quadric equations that define X in its projective embedding, and hence X can be recovered from a knowledge of the spaces V and V ′ and of the multiplication map µ. Now it is possible to represent µ by a multiplication table in terms of bases for V and V ′ (this was called "Representation A" in [KM07]), but a superior method is to take a collection of points p 1 , . . . , p N of points on X, with N > 2 degL, and to represent elements of V and V ′ by their "values" at the points p i ; this presupposes some fixed choice of trivialization ofL in a neighborhood of each p i . It turns out that the points p i need not be distinct, provided we replace the value of an element s ∈ V (or V ′ ) by its nth order Taylor expansion at a point that appears with multiplicity n. A better point of view is to replace the points p i by the effective divisor D = (p 1 ) + · · · + (p N ) on X, and reformulating the value of s ∈ V = H 0 (X,L) at the points of D in terms of the image of s in H 0 (X,L)/H 0 (X,L(−D)) ∼ = H 0 (X,L/L(−D)). The local trivialization ofL then amounts to fixing once and for all an isomorphism between the sheaves of O X -modulesL/L(−D) and O X /O X (−D) = O D , which are supported on the possibly nonreduced zero-dimensional subscheme D of X. Thus our "values" in H 0 (X,L/L(−D)) are interpreted as elements of the finite-dimensional algebra A = H 0 (X, O D ). A similar assertion works for the values of an element in V ′ (using the induced isomorphism betweenL ⊗2 /L ⊗2 (−D) and O D ), and in this case the multiplication map µ amounts to the multiplication in A. All our ingredients are now in place to give the definition of Representation B.
Definition 5.2. Let X be a smooth projective curve over a base field F , and choose a line bundleL and an effective divisor D on X that are both F -rational. Assume moreover that degL ≥ 2g +2 and that deg D > 2 degL, as discussed above. Then Representation B of the curve X is given by the finite-dimensional F -algebra A = H 0 (X, O D ), along with F -subspaces V, V ′ ⊂ A. Here we have replaced V by its image under the F -linear map H 0 (X,L) → H 0 (X,L/L(−D)) ∼ = A, and similarly for V ′ with respect toL ⊗2 . (The condition on deg D ensures that these two Flinear maps are injections.) The multiplication map µ is simply the restriction to V of the multiplication in A, and this yields sufficient information to deduce the set of quadric equations that define the image of X in projective space defined by the embedding associated toL.
We point out that the precise definition of Representation B in [KM07] also specifies that A is represented as a product of rings that are explicitly given in the form F [x]/(f (x)); this shall not concern us here.
Remark 5.3. In our setting, where V and V ′ are spaces of modular forms, we can take D to be a multiple of the cusp at infinity; then the values at D are q-expansions of modular forms up to O(q N ), and these q-expansions give rise to equations for modular curves via the approach sketched above. This approach has already appeared in the literature; see [Gal96] and Section 2 of [BGJGP05], whereL is replaced by the canonical bundle, and the projective embedding is then replaced with the canonical embedding (if the curve is not hyperelliptic), with some modifications since the ideal of a canonical curve occasionally requires generators going up to degree 4.
One novel aspect of our approach is that we evaluate the modular forms at noncuspidal points; we hope that this approach, suitably developed, can eventually also yield equations of Shimura curves.
Remark 5.4. Here is a more concrete way to describe what is going on in Representation B. Let {s 0 , . . . , s L } be a basis for V ; then each vector of values p ′ i = [s 0 (p i ) : · · · : s L (p i )] ∈ P L gives the image of the point p i ∈ X under the projective embedding. (For convenience, suppose in this remark that the points p i are all distinct, and that the field F is perfect; it is easy to modify the argument for the general case). We obtain sufficiently many points to be able to identify X as the unique projective curve that interpolates the {p ′ i }, in the sense that X is defined by all the quadric equations vanishing at the {p ′ i }. The quadrics that generate the ideal of X are of the form j,k c jk X j X k , and can be found by solving for the c ij in the linear system { j,k c jk s j (p i )s k (p i ) = 0 | 1 ≤ i ≤ N }. Now the individual p i need not be defined over F , even though the divisor D is F -rational; still, the set of points {p ′ i } is stable under Gal(F /F ), and so the linear system of equations for the c jk is unaffected by the Galois group. This implies that X can be defined by quadrics with F -rational coefficients; for example, take an echelon basis for the solution space of the linear system.
We are now ready for the last result of this article.
Theorem 5.5. Let ℓ ≥ 3. Fix a number field F ⊂ C and an elliptic curve E 0 over F given by a Weierstrass equation y 2 = x 3 + a 0 x + b 0 , with a 0 , b 0 ∈ F − {0}. Then consider all torsion points {(x 0,P , y 0,P ) | P ∈ E 0 [ℓ](F ) − {P 0 }}, and the slopes λ 0,(P )+(Q)+(⊖P ⊖Q) = (y 0,P − y 0,Q )/(x 0,P − x 0,Q ) ∈ F (E 0 [ℓ]) of lines through pairs of torsion points (with the appropriate modification when P = Q). These slopes for the one elliptic curve E 0 contain enough information to reconstruct the projective embedding of X(ℓ) coming from the linear system M 2 (Γ(ℓ)). This embedding is defined over F (µ ℓ ).
Proof. We first observe that the condition a 0 , b 0 = 0 implies that E 0 does not have nontrivial automorphisms, and hence does not correspond to an elliptic point for Γ(1) in the upper half plane H. Thus the projection map π : X(ℓ) → X(1) is unramified over the point q 0 ∈ X(1)(F ) corresponding to E 0 , and hence the preimages {p 1 , . . . , p N } = π −1 ({q 0 }) are distinct points of X(ℓ), which are rational over F (E 0 [ℓ]). We claim that N (which is incidentally |P SL(2, Z/ℓZ)|) is large enough that we can identify modular forms of weight < 12 via their "values" at the {p i }. To see this claim, either use standard formulas for the degree of the line bundle L ⊗j , whose global sections are M j (Γ(ℓ)), or note that one section of the line bundle L ⊗12 is the Γ(1)-invariant modular form b 2 0 a(τ ) 3 − a 3 0 b(τ ) 2 , which vanishes precisely to order 1 at each point p i ; this last statement holds because modular forms in M 12 (Γ(1)) have precisely one zero (counted appropriately) in the fundamental domain for the Γ(1)-action on H. Thus N = 12 deg L, and our claim is proved.
Hence, as mentioned in our discussion preceding the theorem, we can represent X(ℓ) in Representation B using the line bundleL = L ⊗2 and the divisor D = i (p i ); this amounts to representing the spaces V and V ′ by the "values" of modular forms of weights 2 and 4 at the points {p i }. Concretely, such a point p i corresponds to a choice of symplectic basis {T 0 , U 0 } for the ℓ-torsion E 0 [ℓ], with e ℓ (T 0 , U 0 ) = e 2πi/ℓ ∈ F (E 0 [ℓ]). We know how to "evaluate" an Eisenstein series of weight 1 at p i ; this amounts to computing slopes between the torsion points to get the λ 0 s appearing in the statement of the theorem. Here, the local trivialization of each line bundle L ⊗j near p i corresponds to the particular choice of Weierstrass model of E 0 and of its global differential ω 0 . To define this trivialization more precisely, let τ 1 ∈ H be such that the elliptic curve E 1 = E τ1 = C/L τ1 and its symplectic ℓ-torsion basis {P 1/ℓ , P τ1/ℓ } are isomorphic to our given triple (E 0 , T 0 , U 0 ). Then there exists a unique u ∈ C × such that a 0 = u 4 a(τ 1 ) and b 0 = u 6 b(τ 1 ), and which is also compatible with the level structures. Hence each λ 0 is equal to uλ 1 (τ 1 ) for a corresponding classical modular form λ 1 (τ ) ∈ E ℓ 1 , and similarly for other weights j. It follows that our trivialization of L ⊗j near p i is u j times the trivialization induced by evaluating modular forms in a neighborhood of τ 1 .
At this point, we see that if we work over the field F ℓ = F (E 0 [ℓ]), which contains the values of all the λ 0 s, then our algebra A is isomorphic to the direct product F N ℓ , and our space V (respectively, V ′ ) can be obtained as the span of all products of the values of two (respectively, four) of the λ 0 s at each p i . This follows from Theorems 5.1 and 3.13. We thus obtain equations for X(ℓ) from the kernel of µ : Sym 2 V → V ′ . These equations are actually defined over the smaller cyclotomic extension F (µ ℓ ), because our whole setup is invariant under the full SL(2, Z/ℓZ)-action on modular forms, which permutes the possible symplectic bases for E 0 [ℓ]. Now the action of Gal(F ℓ /F (µ ℓ )) arises from a subgroup H of SL(2, Z/ℓZ) (in fact, when E 0 does not have complex multiplication, then for almost all ℓ, H = SL(2, Z/ℓZ)), so the equations that we obtain can be set up over the smaller field F (µ ℓ ), as mentioned in Remark 5.4.
We note in closing that an analog of Theorem 5.5 holds for the projective embedding of X(ℓ) coming from the (usually incomplete) linear system E ℓ 1 ⊂ M 1 (Γ(ℓ)). By Theorem 5.1 and a computation of Castelnuovo-Mumford regularity, that projective model is defined by equations in degrees 2 and 3.
2000
Mathematics Subject Classification. 11F11, 14H52, 14K10, 11F67, 11F25, 11G18. July 3, 2009.
Proposition 2 . 10 .
210Let ℓ ≥ 2. Then for all j ≥ 1, the C-span of the Eisenstein series {G j (τ, D) | D principal, supported on E[ℓ]} consists of all holomorphic Eisenstein series of weight j on Γ(ℓ).
In particular, λ (P )+(⊖P ) = ν (P )+(⊖P ) = 0 and µ (P )+(⊖P ) = −x P .(2) Let P, Q, R ∈ E[ℓ] − {P 0 } satisfy P ⊕ Q ⊕ R = P 0 ; in other words, they are collinear in the affine Weierstrass model of E. Then λ (P )+(Q)+(R) is the slope of the line joining the three points. Specifically, the equation of the line is y = λ D x + ν D , and we have the following for D = (P ) + (Q) + (R):
D
∈ R ℓ . (2) The same result holds if we expand f D with respect to the analytic uniformizer z in characteristic zero, as well as if we expand the logarithmic derivative df D /f D . Thus if k = C, this theorem combined with Theorem 2.8 and Proposition 2.10 imply that all Eisenstein series on Γ(ℓ) belong to R ℓ . Proof. Proposition 3.5 already shows that f D (t) is R ℓ -balanced in the two cases (i) D = (P ) + (⊖P ) and (ii) D = (P ) + (Q) + (R) with ⊕D = P 0 . A more general D that is supported on E[ℓ] − {P 0 }, but that still satisfies ⊕D = P 0 , can be written as a Z-linear combination of divisors D of types (i) and (ii). We can thus use the multiplicativity of the f D from (2.17) and the first part of Lemma 3.8 to conclude that f D (t) is R ℓ -balanced in this case. For general D supported on E[ℓ], use if needed the second part of Lemma 3.8 to also conclude that
D
is a weight j element of R, and we have H but not for j ≥ 4. Finally, for everyT ∈ E[ℓ] − {P 0 }, we have f D (T ) is an element of R ℓ of weight deg D.Proof. The coefficients H (j) D above can be computed from the Laurent expansion of f D by successively subtracting multiples of x N and −x N y for N going from n down to 0.
invertible in k, and so each of x P , x [2]P , x [3]P , x [4]P can be expressed in terms of λ's, and so belongs to R ′ . Now if P ∈ E[ℓ] is a point of order less than ℓ, we can find a basis {Q, R} for E[ℓ] ∼ = (Z/ℓZ) 2 , such that P = [d]Q for some d > 1.
Q ∈ E[ℓ] − {P 0 } and let 1 ≤ n ∈ Z, with n invertible in k. Choose a point Q ′ ∈ E[nℓ] such that [n]Q ′ = Q. Then define the element f Q •[n] := n −1 f D ∈Ô, where D = T ∈E[n] (Q ′ ⊕T )− T ∈E[n] (T ).
( 3 )
3The Weil pairing e ℓ : E[ℓ] × E[ℓ] → µ ℓ is given (as usual) by the behavior of the functions g Q under translation by elements of E[ℓ]: namely,(4.2) g Q (P ⊕ R) = e ℓ (Q, R)g Q (P ), where Q, R ∈ E[ℓ]and P ∈ E(K).
Proposition 4 . 3 .
43The following identities hold for all R ∈ E[ℓ], where we use the conventions of Remark 3.4 (thus the sums over Q below are unchanged if we sum instead over Q ∈ E[ℓ] − {P 0 }):
Lemma 4 . 5 .
45Let n ≥ 1 be invertible in k. Let P ∈ E[nℓ] (typically, P ∈ E[ℓ]), and let T ∈ E[n]. Consider the modular forms λ P ⊕T , x P ⊕T , and y P ⊕T on Γ(nℓ). We then have (4.10) T ∈E[n] λ P ⊕T = nλ [n]P , T ∈E[n] x P ⊕T = n 2 x [n]P , T ∈E[n]y P ⊕T = n 3 y [n]P .
Proposition 4 . 6 .
46Let n ≥ 1 and assume that n! is invertible in k. Let A, B ∈ E[nℓ] (as before, typically A, B ∈ E[ℓ]), and let s ∈ Z. Then T ∈E[n] λ A⊕T λ B⊖[s]T = a linear combination of terms of the form λ [a]A⊕[b]B λ [c]A⊕[d]B + an element of E n
[a]A⊕[b]B λ [c]A⊕[d]B , satisfying the original requirements of (4.17). Finally, we remark that E snℓ 2 and E s!nℓ 2 are both subspaces of E n!ℓ 2 .
( 4 .
422) S j (Γ) = {cusp forms} ⊂ M j (Γ) = {holomorphic modular forms over C}.
λ
A⊕T λ C⊕V e nℓ (C ⊕ V, B ⊕ U ). Here we have invoked (4.3), where we let C ⊕ V range over the elements of E[nℓ]. Now e nℓ (C ⊕ V, B ⊕ U ) = e ℓ ([n]C, B)e n ([ℓ]V, U ), so the quantity in (4.26) is a linear combination of terms (indexed by C) of the form (4.27) T,U,V ∈E[n] en(T,U)=ζ λ A⊕T λ C⊕V e n ([ℓ]V, U ).
Tλ
∈E[n]−{P0} V of the form V =[s]T λ A⊕T λ C⊕[s]T · ne n ([ℓ][s]T, A⊕T λ C⊕[s]T , (4.28) which brings us back to subcase (i.a).
(ii.a) there exists s ∈ Z/nZ such thatÛ 0 = [s]T 0 and (ii.b) {T 0 ,Û 0 } are a basis for E[n]. In subcase (ii.a), the pointsT 0 · M cover all of E[n] (including P 0 ), each point T ∈ E[n] occurring n times. (The easiest way to see this is to write M with respect to a basis for E[n] that includesT 0 .) Hence we obtain that (4.30) is a multiple of (4.31) T ∈E[n] λ A⊕T0⊕T λ B⊕U0⊕[s]T .
W
Now perform the sum over γ first: the inner factor γ e n k+1 ([L]V, [γ]T 0 ) can be rewritten as γ e n [L]([n k ]V ), [γ]T 0 with [n k ]V ∈ E[n]. Thus, as in case (i.b), the only terms that survive are those where [n k ]V = [s]T 0 = [sn k ]T 0 for some s ∈ Z/nZ. Equivalently, we can write V = [s]T 0 ⊕ W for some s and for some W ∈ E[n k ]. In such a situation, we have e n k+1 ([L]V, [−α]Û 0 ) = e n ([n k L]V, [−α]Û 0 ) = e n ([Ls]T 0 , [−α]Û 0 ) = e n [−Ls]([α]T 0 ⊕ [β]Û 0 ),Û 0 . At this point, we note that as α and β vary, the pointT := [α]T 0 ⊕ [β]Û 0 runs over all points of E[n]. Putting all this together, we obtain that our expression is a linear combination of terms (indexed by C and s) ∈E[n k ] λ A⊕T0⊕T λ C⊕[s]T0⊕W e n k+1 ([Ls]T 0 ⊕ [L]W, U 0 )e n ([−Ls]T ,Û 0 ). The above expression contains a common factor e n k+1 ([Ls]T 0 , U 0 ). We can also rewrite (4.36) e n k+1 ([L]W, U 0 )e n ([−Ls]T ,Û 0 ) = e n k+1 [L](W ⊖ [s]T ), U 0 . We define X := W ⊖ [s]T ∈ E[n k ]; this yields a bijection from the set E[n] × E[n k ] to itself, sending the pair (T , W ) to the pair (T , X). Our expression (4.35) then becomes (4.37) e n k+1 ([Ls]T 0 , U 0 ) T ∈E[n] X∈E[n k ] λ A⊕T0⊕T λ C⊕[s]T0⊕X⊕[s]T e n k+1 ([L]X, U 0 ).
Lemma 4 . 10 .x
410Let k = C, let n ≥ 1, and let A, B ∈ E[nℓ] (typically with A, B ∈ E[ℓ]). Then we have the following congruences modulo E 3 : A⊕T λ A⊕T ≡ nx [n]A λ [n]A , 3 Recall that U 0 in (4.31) had the property that [n k ]U 0 =Û 0 = [s]T 0 . The analogous observation in our setting is that [n k ]([s]T 0 ⊕ X) = [s]T 0 . T ∈E[n]
x A⊖[s]T λ B⊕T , we encounter at one stage the sum U∈E[s] x [n]A ′ ⊕[n]U λ [n]A ′ ⊕[n]U , where [s]A ′ = A. Write d = gcd(n, s) and s = s/d; then our sum becomes d 2 · Û ∈E[ŝ] x [n]A ′ ⊕Û λ [n]A ′ ⊕Û , which we simplify using (4.39).)
P ∈E[ℓ]−{P0}x P = 0.Remark 3.3. Definition 3.2 implies that the graded ring of modular forms is the (graded) integral closure of R 1 in K ℓ . It is a pleasant exercise to verify that, over C, this produces the usual graded ring of modular forms. (Part of the proof involves observing that K ℓ contains a, b, and all the x P s and y P s, which, by Proposition 6.1 of[Shi71], suffice to generate the function field of X(ℓ) via weight 0 meromorphic ratios of elements of R ℓ .) We reassure the reader who wishes to avoid this verification that in any case we only use a certain subring R ℓ of the ring of modular forms, given in Definition 3.7, that is generated by Eisenstein series of weights ≤ 6 in case k = C, and which turns out to be generated by the Eisenstein series of weight 1 (Theorems 3.9 and 3.13).1 We have used the square ψ 2 ℓ here so as to avoid encountering a factor of y when ℓ is even; if we did not take the square, we would need to write ψ ℓ (x, y) to allow for the presence of y.
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[] |
[
"HAT-P-4b: A METAL-RICH LOW-DENSITY TRANSITING HOT JUPITER †",
"HAT-P-4b: A METAL-RICH LOW-DENSITY TRANSITING HOT JUPITER †"
] |
[
"G Kovács ",
"G Á Bakos ",
"G Torres ",
"A Sozzetti ",
"D W Latham ",
"R W Noyes ",
"R P Butler ",
"G W Marcy ",
"D A Fischer ",
"J M Fernández ",
"G Esquerdo ",
"D D Sasselov ",
"R P Stefanik ",
"A Pál ",
"J Lázár ",
"I Papp ",
"P Sári "
] |
[] |
[] |
We describe the discovery of HAT-P-4b, a low-density extrasolar planet transiting BD+36 2593, a V = 11.2 mag slightly evolved metal-rich late F star. The planet's orbital period is 3.056536 ± 0.000057 d with a mid-transit epoch of 2, 454, 245.8154 ± 0.0003 (HJD). Based on high-precision photometric and spectroscopic data, and by using transit light curve modeling, spectrum analysis and evolutionary models, we derive the following planet parameters: M p = 0.68 ± 0.04 M J , R p = 1.27 ± 0.05 R J , ρ p = 0.41±0.06 g cm −3 and a = 0.0446±0.0012 AU. Because of its relatively large radius, together with its assumed high metallicity of that of its parent star, this planet adds to the theoretical challenges to explain inflated extrasolar planets.
|
10.1086/524058
|
[
"https://arxiv.org/pdf/0710.0602v1.pdf"
] | 14,966,730 |
0710.0602
|
d8a897edf876ff1c78820e33a90e351dddf6e780
|
HAT-P-4b: A METAL-RICH LOW-DENSITY TRANSITING HOT JUPITER †
2 Oct 2007 Draft version February 2, 2008 Draft version February 2, 2008
G Kovács
G Á Bakos
G Torres
A Sozzetti
D W Latham
R W Noyes
R P Butler
G W Marcy
D A Fischer
J M Fernández
G Esquerdo
D D Sasselov
R P Stefanik
A Pál
J Lázár
I Papp
P Sári
HAT-P-4b: A METAL-RICH LOW-DENSITY TRANSITING HOT JUPITER †
2 Oct 2007 Draft version February 2, 2008 Draft version February 2, 2008Preprint typeset using L A T E X style emulateapj v. 10/09/06Subject headings: planetary systems: individual: HAT-P-4b -stars: individual: BD+36 2593 - techniques: photometric -techniques: spectroscopic
We describe the discovery of HAT-P-4b, a low-density extrasolar planet transiting BD+36 2593, a V = 11.2 mag slightly evolved metal-rich late F star. The planet's orbital period is 3.056536 ± 0.000057 d with a mid-transit epoch of 2, 454, 245.8154 ± 0.0003 (HJD). Based on high-precision photometric and spectroscopic data, and by using transit light curve modeling, spectrum analysis and evolutionary models, we derive the following planet parameters: M p = 0.68 ± 0.04 M J , R p = 1.27 ± 0.05 R J , ρ p = 0.41±0.06 g cm −3 and a = 0.0446±0.0012 AU. Because of its relatively large radius, together with its assumed high metallicity of that of its parent star, this planet adds to the theoretical challenges to explain inflated extrasolar planets.
INTRODUCTION
In the course of our ongoing wide field planetary transit search program HATNet (Bakos et al. 2004), we have discovered a large radius and low density planet orbiting an 11th magnitude star BD+36 2593. This planet is the fifth member of a group of low-density transiting exoplanets. The combination of its low mass and the relatively high metallicity and age of the parent star makes theoretical interpretation of its large radius difficult. In this Letter we describe the observational properties of the system and derive the physical parameters both for the host star and for the planet. We also briefly comment on the theoretical status of inflated extrasolar planets.
THE PHOTOMETRIC DISCOVERY AND FOLLOW-UP OBSERVATIONS
The star BD+36 2593 (also GSC 02569-01599) at α = 15 h 19 m 57. s 92, δ = +36 • 13 ′ 46. ′′ 7, is contained in field G191 of HATNet, centered at α = 15 h 28 m , δ = +37 • 30 ′ . As we show in the remainder of the paper, the star is orbited by a planetary companion, and so we label the host star as HAT-P-4 and the planet as HAT-P-4b. Field Table 3 for the resulting transit parameters). The out-of-transit level of the HATNet data corresponds to I = 10.537 mag, phase is zero at the transit center.
June by the HAT-7 telescope at the Fred Lawrence Whipple Observatory (FLWO) of the Smithsonian Astrophysical Observatory (SAO). Nearly 90% of the data points were gathered after February 2005. The field is relatively sparse, with ∼ 14000 objects brighter than I ≈ 14 mag. Most of the light curves have some 5300 data points. The sampling cadence is 5.5 min, slightly longer than the 5 min integration time.
A first look at the compact subset collected in the 98 d time span between March and June, 2005, revealed the presence of a transit signal with a nearly integer 3 d period. The detection was made possible by the application of the Trend Filtering Algorithm (TFA; Kovács, Bakos, & Noyes 2005). However, assuming the primary is a main-sequence F or G type star, the relative length of the transit is some 20 − 30% greater than expected for a Jupiter-size companion with the above orbital period. Although both the lack of detection in the raw data and the length of the transit made us suspicious about the viability of this candidate, we left some room for the possibility that the primary was slightly evolved (an assumption that proved to be true in the subsequent investigations -see § 4). Therefore, we proceeded with rejection-mode spectroscopy, which showed no sign of radial velocity (RV) variation at the km s −1 level.
An improved reduction of all available frames of the field was completed by February 2007. In addition to using a new data pipeline that employs refined astrometry (Pál & Bakos 2006) and aperture photometry, we detrended the light curves before signal search with the aid of an External Parameter Decorrelation technique (EPD, see also Bakos et al. 2007). This technique utilizes the fact that various "external parameters" that are specific to the star, such as sub-pixel position on the frame, point spread function properties (e.g., width and elongation), or specific to the frame, such as telescope position, are correlated with the deviations of the star's brightness from the median. The technique is optimal for stars that are otherwise non-variable most of the time (e.g. transit candidates). EPD derives the correlation between brightness deviations and the underlying external parameters, and subsequently corrects them for each individual star. This is different from what is done during the application of TFA, where we consider the full time history of the light variation and use the light curves to "cure themselves" by recognizing the hidden systematics in each other. The two methods are complementary and we use both (EPD followed by TFA). All these led to a powerful confirmation of the earlier detection.
The folded light curve constructed from the HAT-Net data is shown in the upper panel of Fig. 1. The best period was obtained from the BLS analysis (Kovács, Zucker, & Mazeh 2002) after applying TFA with some 700 template stars. 11 As seen, the transit is clearly visible in the light curve even though no binning was applied. There is no sign of periodic out-of-transit variation, indicating that this is a near "textbook" transit signal.
To update the photometric ephemeris and obtain a precise light curve for model fitting, we observed our target with KeplerCam on the 1.2 m telescope at FLWO. We observed two full transits on May 24 and 27, 2007. The folded light curve in the Sloan z band for all 985 data points is shown in the lower panel of Fig. 1. The followup observations verify the discovery light curve and are accurate enough to make transit modeling possible.
THE SPECTROSCOPIC VERIFICATION AND
EXCLUSION OF BLEND SCENARIOS
As mentioned in § 2, basic rejection mode spectroscopy to exclude stellar binary and certain blend types was already initiated after the first detection in the fall of 2005. By June 2006 we had collected 9 spectra with the CfA Digital Speedometer (DS; Latham 1992) on the FLWO 1.5 m telescope, covering a 45Å window centered at the Mg b triplet at 5187Å. We derived spectroscopic parameters by comparing the observed spectra against synthetic spectra, as described in detail by Torres et al. (2002). The initial stellar parameters from this analysis were: T eff = 5500 K, log g= 3.5 [cgs] and v sin i= 5.9 km s −1 , indicating an evolved G-type pri- mary 12 . The DS spectra were also used to derive radial velocities by cross-correlating the spectra with synthetic spectra based on Kurucz model atmospheres (e.g. Latham 2002). The rms scatter of the RV data was 0.71 km s −1 around the mean velocity of −1.58 km s −1 . The velocities did not show any correlation with the orbital phase and we found no sign of a second stellar component in the spectra. These spectroscopic results, together with the secure detection in the HATNet data, made a viable case for obtaining high-precision RV measurements to seek evidence for orbital motion and to obtain refined stellar parameters. Accordingly, highresolution spectroscopy was conducted with the HIRES instrument (Vogt et al. 1994) on the Keck I telescope between 2007 March 27 and May 29. See, e.g., Torres et al. (2007) for the details of the observational procedure. The nine resulting RV measurements are listed in Table 1. They are relative velocities in the Solar System barycentric frame of reference (see Butler et al. 1996).
The data were fitted by a circular 13 orbit with period and mid transit epoch constrained by the photometric data (see Table 3). The result of this two-parameter (systematic offset and semi-amplitude) fit is displayed in Fig. 2. The unbiased estimate of the standard deviation of the residuals is 4.1 m s −1 , larger than the internal errors listed in Table 1. A part of this scatter may come from the last data point that was taken under unfavorable weather conditions (leaving out this point we get 2.7 m s −1 for the standard deviation of the residuals). Nevertheless, it is also possible that there is a "velocity jitter" due to stellar activity. The size of this jitter is estimated to be about 3 m s −1 , falling in the expected range for an inactive late F-type star (Wright 2005). The low activity level is supported by the absence of emission features in the Ca II H and K lines.
A crucial part of the verification of the sub-stellar nature of the companion is testing various blend scenarios. There are at least two basic ways of such testing: (i) light curve modeling, combined with the information available from the spectra (e.g., maximum flux contribution by the blended binary); (ii) checking spectral line asymmetry by searching for bisector span variations (e.g., Santos et al. 2002;Torres et al. 2005). Our experience shows that while (i) requires several pieces of information (good quality full light curve, running many binary models and performing a fairly deep spectrum analysis), (ii) is more sensitive to hidden components and simpler to perform. Therefore, we settled on (ii) and searched for a variation in the differences measured between the velocities at the top and the bottom of the correlation profiles of the individual spectra. The result is shown in the lower panel of Fig. 2. For a hidden stellar binary blended by our target we would expect a variation in the bisector span in phase with the radial velocity, and with an amplitude comparable to that of the measured RV variation itself (Torres et al. 2005). However, the standard deviation of the bisector variation is 6.5 m s −1 , which is only 8% of the RV amplitude (the same numbers are 2.9 m s −1 and 4%, respectively, if we leave out the last data point). Furthermore, no correlation with the RV variation is seen. Therefore, we are confident that the source of the RV variation is stellar wobble due to the gravitational pull of a sub-stellar companion.
STELLAR AND PLANETARY PARAMETERS
The cornerstone of the derivation of the absolute parameters of a planet discovered by radial velocity and transit observations is the accurate estimation of the stellar mass and radius.
The procedure often involves determination of T eff and [Fe/H], as well as log g, from high-resolution spectroscopic analysis. If an accurate distance is available, for example from a Hipparcos parallax, then the absolute magnitude can be used to improve the value for the stellar radius. In general, none of the methods yield better than ∼ 10% accuracy in the derived stellar parameters (implying corresponding limits to the accuracy of the absolute planet parameters). An improvement can be achieved by using the value of a/R ⋆ derived from the transit light curve to measure directly the mean stellar 310 ± 30 from M V c a SME = "Spectroscopy Made Easy" package to generate synthetic spectra and to fit the observed ones (Valenti & Piskunov 1996). b Y 2 +LC+SME = Yonsei-Yale isochrones (Demarque et al. 2004), transit light curve modeling and SME analysis. c Using V = 11.22 ± 0.12 of Droege (2006) and assuming zero reddening.
density (Sozzetti et al. 2007). Since this latter parameter is directly linked to log g, entering in the spectroscopic analysis, we followed an iterative determination of the stellar and planetary parameters. First, for the determination of [Fe/H] and T eff , we used the iodine-free template spectrum from Keck. The modeling was performed using the SME software (Valenti & Piskunov 1996) incorporating the same method and atomic data as given in Valenti & Fischer (2005). We obtained T eff = 6032 ± 80 K, [Fe/H]= +0.32 ± 0.08 [dex] and log g= 4.36 ± 0.11. Next, for the computation a/R ⋆ , we fitted the high precision KeplerCam light curve (see Fig. 1) by using the formulae of Mandel & Agol (2002) with quadratic limb darkening coefficients from Claret (2004). We set e = 0.0 as a result of our test on the RV data with non-zero eccentricity. Fitted parameters were the center of transit T c , the radius ratio R p /R ⋆ , the normalized relative semimajor axis a/R ⋆ , and the impact parameter b.
Next, to compute the stellar mass and radius we relied on current stellar evolution models. As in our earlier papers, we compared the observational properties of the host star with a finely interpolated grid of model isochrones from Demarque et al. (2004). The inferred stellar properties are based on the best match to the measured values of T eff , [Fe/H], and a/R ⋆ within their observational errors, in a χ 2 sense. This procedure led to a better approximation of the stellar gravitational acceleration with log g= 4.12 ± 0.04.
In the second loop of iteration we used the newly determined value of log g in the SME analysis and redid the above sequence of computation. This led to a slightly modified set of stellar and planetary parameters with somewhat smaller errors than in the first loop. Since the changes were, in general, fairly small (e.g., log g has changed to 4.14 ± 0.03), we decided to stop the iteration after this second loop. The final stellar and planetary parameters are shown in Table 2 and Table 3, respectively.
Concerning the derived parameters and their errors, we note the following. The dependence of the result on the evolutionary models was tested by using the isochrones given for solar-scaled Z = 0.03 models with core overshooting by Pietrinferni et al. (2006). As noted by the authors, their models are hotter by some 200 K The γ velocity is −1.58 ± 0.24 km s −1 , from the DS data. c Adopted than those of Demarque et al. (2004). Therefore, we used an effective temperature of 6060 K. With these input parameters we got nearly the same stellar and planet parameters as from the Yonsei-Yale isochrones (i.e., ρ p = 0.41 g cm −3 , age= 4.5 Gyr). By using models without overshooting or of lower effective temperature, we got larger ages and also slightly larger densities, up to ρ p = 0.43 g cm −3 . The stability of the planet density is mainly related to the correlated change of stellar mass and radius when models or input parameters are changed. These age ranges and the derived metallicity fit reasonably to the relation recently given by Reid et al. (2007). We also note that the derived radius is 34% larger than the one corresponding to an unevolved mainsequence star. This explains the longer than expected transit duration by which we were puzzled at the early phase of the discovery.
DISCUSSION AND CONCLUSIONS
We presented the discovery data and derived the physical parameters of HAT-P-4b, an inflated planet orbiting BD+36 2593. Among the 20 transiting planets discovered so far, there are five with ρ p 0.4 g cm −3 . All others have at least 50% higher densities. For ease of comparison, Table 4 lists the relevant properties of the five inflated planets. It is remarkable how similar these planets are (except for TrES-4 that has distinctively low density). With its Safronov number of 0.036, HAT-P-4b belongs to the Class II planets according to the recent classification of Hansen & Barman (2007) and (together with TrES-4) further strengthens the mysterious dichotomy of the known transiting planets in this parameter. The parent star of HAT-P-4b is among the largest radii, largest mass, lowest gravity and highest metallicity transiting planet host stars. Current models of irradiated giant planets are able to match the observed radii of most of the planets with- Mandushev et al. (2007) and Winn et al. (2007). From top to bottom, metallicities for the parent stars are: 0.23 (Stempels et al. 2007), 0.24, 0.02 0.0 (adopted) and 0.13. out invoking any additional heating mechanism. Higher metallicity cases, such as the present one, however, may pose problems (assuming that the planet and star have similar metallicities). More metals imply two opposite effects on the radius: (i) inflating it due to higher opacities in the envelope; (ii) shrinking it due to the higher molecular weight of the interior and the possible development of a large high density core. These effects have been discussed recently by Burrows et al. (2007). Since WASP-1b is similar in several aspects (i.e., irradiance, metallicity) to HAT-P-4b, we consider the coreless models of WASP-1b as shown in Fig. 7 of Burrows et al. (2007). It seems that HAT-P-4b can be fitted by near solar metallicity coreless models, assuming that its age is not too much greater that 4 Gyr. We also refer to the layered convective mechanism of Chabrier & Baraffe (2007) that gives an alternative explanation for planets with inflated radii. We conclude that more definite statements on the relation of the observations and planet structure theories can be made only by reaching higher accuracy in the observed star/planet parameters. Nevertheless, HAT-P-4b (together with WASP-1b) does not seem to support the existence of a simple relation between host star metallicity and planet's core mass (see Guillot et al. 2006;Burrows et al. 2007).
Operation of the HATNet project is funded in part by NASA grant NNG04GN74G. Work by G.Á. B. was supported by NASA through Hubble Fellowship Grant HST-HF-01170.01-A. G. K. wishes to thank support from Hungarian Scientific Research Foundation (OTKA) grant K-60750. We acknowledge partial support from the Kepler Mission under NASA Cooperative Agreement NCC2-1390 (D. W. L., PI). G. T. acknowledges partial support from NASA Origins grant NNG04LG89G. The Keck Observatory was made possible by the generous financial support of the W. M. Keck Foundation. D. A. F is a Cottrell Science Scholar of the Research Corporation. We acknowledge support from NASA grant NNG05G164G to DAF.
Fig. 1 .
1-Folded and unbinned light curves of HAT-P-4. The HATNet and FLWO 1.2 m data are plotted in the upper and lower panels, respectively. The transit model fit to the FLWO data is shown by continuous line (see
Fig. 2 .
2-Relative radial velocity and bisector span variations from the Keck observations of HAT-P-4. Panel (a) shows the nine Keck RV measurements with a zero-eccentricity orbital fit. In panel (b) we show the bisector spans (from nine iodine exposures and one template spectrum) displaying a much smaller scatter. Error bars include the 3 m s −1 estimated velocity jitter.
d) a . . . . . . . . . . . . . . 3.056536 ± 0.000057 Tc (HJD) a . . . . . . . . . . . . . . 2,454,245.8154 ± 0.0003 2,454,248.8716 ± 0.0006 K (m s −1 ) . . . . . . . . . . . . . . . 81.1 ± 1.9 Offset velocity (m s −1 ) b . 12.1 ± 0.9 e c . . . . . . . . . . . . . . . . . . . . . . . 0.0 Light curve parameters: Transit duration (day) . . . 0.1760 ± 0.0003 a/R⋆ . . . . . . . . . . . . . . . . . . . . 6.04 [−0.18, 0.03] Rp/R⋆ . . . . . . . . . . . . . . . . . . 0.08200 ± 0.00044 b ≡ a cos i/R⋆ . . . . . . . . . . . . 0.01 [−0.01, 0.23] Planet parameters: Mp(M J ) . . . . . . . . . . . . . . . . . 0.68 ± 0.04 Rp(R J ) . . . . . . . . . . . . . . . . . . 1.27 ± 0.05 ρp(g cm −3 ) . . . . . . . . . . . . . . 0.41 ± 0.06 log gp (cm s −2 ) . . . . . . . . . . 3.02 ± 0.02 a (AU) . . . . . . . . . . . . . . . . . . 0.0446 ± 0.0012 ip(deg) . . . . . . . . . . . . . . . . . . 89.9 • [−2.2, 0.1] a Taken from the photometry (HATNet and 2 nights of FLWO 1.2 m ). b
TABLE 1 HIRES
1Relative Radial Velocities for HAT-P-4.BJD
RV
σ RV
O-C
Phase a
(2,400,000+) (m s −1 ) (m s −1 ) (m s −1 )
54186.98522
101.0
2.1
2.7
0.753
54187.11241
92.6
2.1
−2.6
0.794
54188.01160 −25.4
2.0
0.3
0.088
54188.07150 −35.8
2.0
−2.0
0.108
54189.00174 −28.1
2.2
−2.8
0.412
54189.08262 −13.9
2.0
−0.7
0.439
54189.13222
−3.1
3.3
2.3
0.455
54249.93769 −52.5
2.8
−3.6
0.349
54279.86357 −36.8
2.9
8.4
0.139
a Relative to the center of transit.
TABLE 2
2Stellar parameters for HAT-P-4.Parameter
Value
Source
T eff (K)
5860 ± 80
SME a
[Fe/H] (dex)
+0.24 ± 0.08
SME
v sin i (km s −1 ) 5.5 ± 0.5
SME
Mass (M ⊙ )
1.26 [−0.14, 0.06] Y 2 +LC+SME b
Radius (R ⊙ )
1.59 [−0.07, 0.07] Y 2 +LC+SME
log g (cgs)
4.14 [−0.04, 0.01] Y 2 +LC+SME
L⋆ (L ⊙ )
2.68 [−0.34, 0.39] Y 2 +LC+SME
M V (mag)
3.74 [−0.16, 0.16] Y 2 +LC+SME
Age (Gyr)
4.2 [−0.6, 2.6]
Y 2 +LC+SME
Distance (pc)
TABLE
TABLE 4
4Comparison of the properties of inflated planets. aName
P
a
M
R
ρ
log g
(d)
(AU) (M J ) (R J ) (cgs) (cgs)
WASP-1b
2.52 0.038
0.87
1.40
0.39
3.04
HAT-P-4b
3.06 0.045
0.68
1.27
0.41
3.02
HD 209458b 3.53 0.045
0.64
1.32
0.35
2.96
TrES-4
3.55 0.049
0.84
1.67
0.22
2.87
HAT-P-1b
4.47 0.055
0.53
1.20
0.38
2.96
a Data from Shporer et al. (2007), this paper, Burrows et al.
(2007),
The EPD technique played the main role in the detection; TFA subsequently led to an increase of 20% in the signal-to-noise ratio.
These parameters were derived with [Fe/H]= 0.0. By using [Fe/H]= 0.24 from the Keck spectra we get a slightly better match with the synthetic spectra that yields T eff = 6000 K and log g= 4.0, in better agreement with the Keck results described below.
A fit with free-floating eccentricity yields e = 0.055 ± 0.035, supporting the assumption of a circular orbit.
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|
[] |
[
"Griffiths Singularities in the Disordered Phase of a Quantum Ising Spin Glass",
"Griffiths Singularities in the Disordered Phase of a Quantum Ising Spin Glass"
] |
[
"H Rieger ",
"A P Young ",
"\nInstitut für Theoretische Physik\nHLRZ\nForschungszentrum Jülich\n52425JülichGermany\n",
"\nDepartment of Physics\nUniversität zu Köln\n50937KölnGermany\n",
"\nUniversity of California\n95064Santa CruzCA\n"
] |
[
"Institut für Theoretische Physik\nHLRZ\nForschungszentrum Jülich\n52425JülichGermany",
"Department of Physics\nUniversität zu Köln\n50937KölnGermany",
"University of California\n95064Santa CruzCA"
] |
[] |
We study a model for a quantum Ising spin glass in two space dimensions by Monte Carlo simulations. In the disordered phase at T = 0, we find power law distributions of the local susceptibility and local non-linear susceptibility, which are characterized by a smoothly varying dynamical exponent z. Over a range of the disordered phase near the quantum transition, the local non-linear susceptibility diverges. The local susceptibility does not diverge in the disordered phase but does diverge at the critical point. Approaching the critical point from the disordered phase, the limiting value of z seems to equal its value precisely at criticality, even though the physics of these two cases seems rather different
|
10.1103/physrevb.54.3328
|
[
"https://arxiv.org/pdf/cond-mat/9512162v1.pdf"
] | 17,782,147 |
cond-mat/9512162
|
ddcab71d7605872223eff2a75b4e48704f59382f
|
Griffiths Singularities in the Disordered Phase of a Quantum Ising Spin Glass
arXiv:cond-mat/9512162v1 22 Dec 1995 (March 19, 2018)
H Rieger
A P Young
Institut für Theoretische Physik
HLRZ
Forschungszentrum Jülich
52425JülichGermany
Department of Physics
Universität zu Köln
50937KölnGermany
University of California
95064Santa CruzCA
Griffiths Singularities in the Disordered Phase of a Quantum Ising Spin Glass
arXiv:cond-mat/9512162v1 22 Dec 1995 (March 19, 2018)numbers: 7110Nr7510Jm7540Mg
We study a model for a quantum Ising spin glass in two space dimensions by Monte Carlo simulations. In the disordered phase at T = 0, we find power law distributions of the local susceptibility and local non-linear susceptibility, which are characterized by a smoothly varying dynamical exponent z. Over a range of the disordered phase near the quantum transition, the local non-linear susceptibility diverges. The local susceptibility does not diverge in the disordered phase but does diverge at the critical point. Approaching the critical point from the disordered phase, the limiting value of z seems to equal its value precisely at criticality, even though the physics of these two cases seems rather different
I. INTRODUCTION
A feature of disordered systems which has no counterpart in pure systems, is that rare regions, which are more strongly correlated than the average, can play a significant role. For classical magnetic systems, Griffiths 1 showed that such regions lead to a free energy which is a non-analytic function of the magnetic field at temperatures below the transition temperature of the pure system. However, for the static properties of a classical system, the Griffiths singularities are very weak; just essential singularities 2 . By contrast, Griffiths singularities are much more spectacular for quantum phase transitions at T = 0, especially for systems with a broken discrete symmetry. One model where these effects can be worked out in great detail, and where Griffiths singularities dominate not only the disordered phase but also the critical region, is the one-dimensional random transverse-field Ising model [3][4][5][6] . In that model, one finds very broad distributions of various quantities including the local susceptibility and the energy gap, and a dynamical exponent, z, which is infinite at the critical point. In the disordered phase, the distribution of local susceptibilities is found to be a power law, in which the power can be related 6 to a continuously varying dynamical exponent 5 , which diverges at criticality. The average susceptibility diverges when z > 1, i.e. over a finite region of the disordered phase in the vicinity of the critical point, a result first found many years ago by McCoy and Wu 3 .
Many of these surprising results, such as the power law distribution of susceptibilities in the disordered phase, are expected to hold more generally 7 . However, it is not clear whether the average susceptibility will diverge in the disordered phase for dimension, d, greater than 1, or whether this is a special feature of d = 1. Here we inves-tigate Griffiths singularities for a two-dimensional quantum Ising spin glass system by Monte Carlo simulations. Additional motivation for our study comes from experimental work 8 on a quantum spin glass system, which did not find the expected strong divergence in the nonlinear susceptibility at the quantum phase transition. By contrast, subsequent numerical simulations 9,10 did find a rather strong divergence, comparable with that at the classical spin glass transition. Hence it seems worth investigating whether this discrepancy might be due to Griffiths singularities causing the non-linear susceptibility to diverge even in the disordered phase, thus making the location of the transition difficult in the experiments. Somewhat less detailed results on the two-dimensional spin glass have also been reported in parallel work by Guo et al. 11 , who, additionally, performed calculations in three dimensions.
II. THE MODEL
The two-dimensional quantum Ising spin glass in a transverse field 12 is defined via the following quantum mechanical Hamiltoniañ
H QM = − i,j J ij σ z i σ z j − Γ i σ x i ,(1)
where the {σ α i } are Pauli spin matrices, theJ ij are quenched random interaction strengths and Γ is an external transverse field. A system described by this Hamiltonian undergoes a quantum phase transition at zero temperature, T = 0, from a paramagnetic (or spin liquid) phase to a spin glass phase for some critical field strength Γ c 9 . As is described elsewhere 9,10 this model can be mapped onto an effective classical Hamiltonian in two space plus one imaginary time dimensions, with disorder that is perfectly correlated along the imaginary time axis. This classical Hamiltonian is
H = − τ i,j J ij S i (τ )S j (τ ) − τ,i S i (τ )S i (τ + 1), (2)
where the Ising spins, S i (τ ), take values ±1, i and j refer to the sites on an L × L spatial lattice, while τ denotes a time slice, τ = 1, 2, . . . , L τ . The number of time slices, L τ , is proportional to the inverse of the temperature, T , of the original quantum system in Eq. (1). Periodic boundary conditions are applied in all directions. The model is simulated at an effective classical "temperature" T cl , which controls the amount of order in the spins. Of course, T cl , is not the real temperature T , which is zero at the transition, but is rather a measure of the strength of the quantum fluctuations. Increasing T cl therefore corresponds to increasing the transverse field Γ, in the quantum Hamiltonian, Eq. (1). The nearest neighbor interactions J ij are independent of τ , because they are quenched random variables, and are chosen independently from a Gaussian distribution with zero mean and standard deviation unity, i.e.
[J ij ] av = 0,
[J 2 ij ] av = 1 ,(3)
where [· · ·] av denotes an average over the disorder. Statistical mechanics averages for a given sample will be denoted by angular brackets, i.e. · · · . The interactions between time slices are ferromagnetic and taken to be non-random with strength unity. The phase diagram of the model is sketched in Fig. 1. Because we are dealing with a two-dimensional lattice, there is no finite-temperature spin glass transition 13 , and the region with spin glass order therefore lies entirely along the T = 0 axis in the region 0 ≤ T cl < T cl c , where T cl c denotes the critical point. In earlier work 9 we found that
T cl c = 3.275 ± 0.025 .(4)
III. THEORY
Griffiths singularities arise from localized regions which are more correlated than the average. They do not have large spatial correlations, but give rise to singularities because of large correlations in imaginary time. To focus on these time correlations, it is simplest to study quantities that are completely local in space, i.e. are just on a single site. We shall be particularly interested in the local susceptibility, there is no spin glass order but there are Griffiths singularities. In the region T cl c < T cl < T cl χ nl the Griffiths singularities are sufficiently strong that the average non-linear susceptibility diverges as the (real) temperature tends to zero. For T cl > T cl χ nl the average non-linear susceptibility stays finite in the zero temperature limit.
χ (loc) = ∂ ∂h i σ z i ,(5)
where h i is a local field on site i. In the imaginary time formalism this can be evaluated from
χ (loc) = Lτ τ =1 S i (0)S i (τ ) .(6)
Since the divergent response function at a conventional spin glass transition is the non-linear susceptibility 13 , it is also interesting to study the local non-linear susceptibility, given by
χ (loc) nl = ∂ 3 ∂h 3 i σ z i ,(7)
which can be determined in the simulations from
χ (loc) nl = − 1 6L τ m 4 i − 3 m 2 i 2 ,(8)
where
m i = Lτ τ =1 S i (τ ) .(9)
We consider distributions of χ (loc) and χ (loc) nl obtained both by measuring at different sites in a given sample, and by taking many samples with different realizations of the disorder.
A. Disordered Phase
In the disordered phase the distributions of χ (loc) and χ (loc) nl will be very broad with a power law variation at large values. Physically this comes from regions which are locally ordered. The probability of having such a region is exponentially small in its volume, V , but, when it occurs, there is an exponentially large relaxation time 7 , because, to invert the spins in this region at some imaginary time one has to insert a domain wall of size V , for which the Boltzmann factor is exponentially small in V , as is sketched in Fig. 2. The combination of an exponentially large result happening with exponentially small probability gives a broad distribution of results in complete analogy to the effect of the Griffiths phase on the dynamics in classical random magnets 14,15 . In the latter case the volume to surface ratio determines the resulting probability distribution for the logarithm of relaxation times. In contrast to this the extra dimension present in the quantum problem gives rise to a volume to volume ratio instead (cf. Fig. 2), which leads to a power law distribution of correlation lengths in the imaginary time direction. The power depends on the microscopic details and so is expected to vary smoothly throughout the Griffiths phase.
E~ V σ + d V~L τ ξ τ E/T e FIG. 2.
The strongly coupled space region (cluster) of volume V ∼ L d tends to order (locally) the spins along the imaginary time (τ ) direction, indicated by the plus and minus sign meaning a spin orientation parallel (plus) or anti-parallel (minus) with respect to the ground state configuration of the isolated cluster. The insertion of a domain wall costs an energy E ∼ σV , where σ is a surface tension (note that the couplings in the τ -direction are all ferromagnetic). This event occurs with a probability exp(−E/T cl ), resulting in the exponentially large (imaginary) correlation time ξτ ∼ exp(σL d /T cl ).
We can relate the power in the distribution to a dynamical exponent, defined in the disordered phase, as follows 6 . The excitations which give rise to a large χ (loc) at T = 0 are well localized and so we assume that their probability is proportional to the spatial volume, L d . These excitations have a very small energy gap,
∆E = E 1 − E 0 ,
where E 0 is the ground state energy of the quantum system and E 1 is the first excited state. This small gap is responsible for the large susceptibility because the latter is essentially proportional to the inverse of the gap because the matrix elements which enter χ (loc) do not have very large variations. Since there is no characteristic energy gap it is most sensible to use logarithmic variables. Hence, if the power in the distribution of ln ∆E is λ, say, then we have
P (ln ∆E) ≡ ∆EP (∆E) ∼ L d ∆E λ = L∆E 1/z d ,(10)
where the last line defines the dynamical exponent, z, in the conventional way as the power relating a time scale to a length scale. Comparing the last two expressions we see that
λ = d z .(11)
Hence the tail of the distribution of ∆E has the form
ln [P (ln ∆E)] = d z ln ∆E + const.(12)
Since the local susceptibility is proportional to the inverse of the gap, the power law tail of its distribution should be given by
ln P (ln χ (loc) ) = − d z ln χ (loc) + const.(13)
The non-linear susceptibility involves three integrals over time, whereas the linear susceptibility only involves one. Hence we assume that the distribution of χ (loc) nl is similar to that of (χ (loc) ) 3 , which leads to the following power law tail in the distribution:
ln P (ln χ (loc) nl ) = − d 3z ln χ (loc) nl + const.(14)
Hence there should be a factor of 3 between the powers in the distributions of ln χ (loc) and ln χ (loc) nl . We shall see that this prediction is confirmed by the numerics. As a result, one can characterize all the Griffiths singularities in the disordered phase by a single exponent, z.
The average uniform susceptibility will diverge, in the disordered phase, at the same point as the average local susceptibility because spatial correlations are short range and so cannot contribute to a divergence. From Eq. (13) we see that this happens when
z > d .(15)
Similarly, according to Eq. (14), the average non-linear susceptibility will diverge when
z > d 3 .(16)
One can also infer the nature of the divergence of χ (loc) nl and χ (loc) as the (real) temperature T tends to zero, see Fig. 1. For χ (loc) we expect that the distribution in Eq. (13) will be cut off at
χ (loc) ∼ T −1 which gives [χ (loc) ] av ∼ T d/z−1 .(17)
For χ (loc) nl we expect that the cutoff will be at of order T −3 , which, together with Eq. (14) gives
[χ (loc) nl ] av ∼ T d/z−3 .(18)
The global non-linear susceptibility will diverge in the same way, possibly with logarithmic corrections, as occurs in the one-dimensional random ferromagnet 5 .
The dynamical exponent will tend to some limit as the critical point is approached. It is interesting to ask whether this limit will be the same as the value of z precisely at criticality. On the face of it, there does not seem any reason why they should be equal, since z in the disordered phase is determined by rare compact clusters, whereas z at criticality is determined by fluctuations on large length scales of order of the (divergent) correlation length. Nonetheless, for the 1-d random ferromagnet, they are both equal (to infinity). For the 2-d spin glass we shall also find that these two values are numerically close, and may well be equal (though finite).
B. The Critical Point
We expect that the distribution of χ (loc) will also have a power law at the critical point just as it does in the disordered phase. To deduce the exponent, note that the average time dependent correlation function at criticality is given by scaling as 9,10
[ S i (0)S i (τ ) ] av ∼ 1 τ β/(νz) .(19)
where
β ν = d + z − 2 + η 2(20)
is the order parameter exponent and ν is the correlation length exponent.
Since the average local susceptibility is just the integral of this over τ , it follows that the distribution of ln χ (loc) must have the same power, i.e. ln P (ln χ (loc) ) = − β νz ln χ (loc) + const.
at criticality. In earlier work 9 we found z = 1.5, η = 0.5, so numerically β/(νz) is about 2/3. Integrating Eq. (19) over τ from 0 to T −1 , one sees that the average susceptibility (which is the same as the average local susceptibility for a model with a symmetric distribution of interactions, such as that used here) diverges as T → 0 like
[χ] av ∼ T β/zν−1 ,(22)
at criticality. Similarly the average local non-linear susceptibility will diverge like 9,10
[χ
(loc) nl ] av ∼ T 2β/zν−3 ,(23)
at criticality. The global non-linear susceptibility will have a stronger divergence at criticality 9 :
[χ nl ] av ∼ T (2β−dν)/zν−3 .(24)
Note that Eqs. (22) to (24) refer to the situation in which T cl is set to the critical value T cl c , and the real temperature tends to zero, see Fig. 1. There is no significant dependence on L and the data is also independent of Lτ at small χ (loc) . Increasing Lτ seems to simply extend the range over which the data lies on a straight line. The solid line is a fit to the straight line region of the data and has slope −3.92 which gives z = 0.51 from Eq. (13).
IV. RESULTS IN THE DISORDERED PHASE
We use standard Monte Carlo methods to simulate the model in Eq. (2). Except where noted, 2560 realizations of the disorder were averaged over. The simulations were done on parallel computers: a Parsytec GCel1024 with 1024 nodes (T805 transputers) and a Paragon XP/S10 14), in quite good agreement with the fit to the data in Fig. 3. Since the slope is more negative than −1, or equivalently z < 2/3, the average non-linear susceptibility does not diverge at this point.
with 140 nodes (i860XP microprocessors). Massively parallel machines with many medium-sized nodes (in terms of memory) are ideal for the problem considered here: as long as one physical system fits into the RAM of one processor one only has to set up a farm topology to distribute the initial seed for the random number generators and to collect the results for the different realizations at the end of the simulation. Apart from that, no communication between processors is needed, so the parallelization is perfectly efficient; the gain in speed is directly proportional to the number of nodes.
The distributions of ln χ (loc) and ln χ (loc) nl at T cl = 3.7 and 3.5 (both in the disordered phase) are shown in Figs. 3-6. There is a straight line region for large values as expected, which is independent of L, and the only dependence on L τ is that the tail extends further for larger L τ . This is not surprising since there is a cutoff due to the finite number of time slices at χ (loc) = L τ and χ (loc) nl = L 3 τ . At both temperatures one sees that the values of z obtained from χ (loc) and χ (loc) nl are in reasonably good agreement with each other. At T cl = 3.7, we find that z ≃ 0.51, from the data for χ (loc) and z ≃ 0.54 from data for χ does not diverge at T cl = 3.7 because z < 2/3. At T cl = 3.5, we find z ≃ 0.71 from the data for χ (loc) and z ≃ 0.76 from data for χ Fig. 5. Since the slope is greater than −1, or equivalently z > 2/3, the average non-linear susceptibility does diverge at this point. Fig. 7 shows the values of z at various points in the disordered phase. In all cases there is good agreement between the estimates from the data for χ (loc) and χ (loc) nl . From this data we find that the average non-linear sus- to Eqs. (13) and (14), is plotted for different values of T cl . The estimates obtained from data for χ (loc) are shown by the triangles and the estimates from the data for χ ceptibility diverges, in the disordered phase, for
T cl c ≤ T cl ≤ T cl χ nl ,(25)
where T cl χ nl ≃ 3.56 .
and T cl c ≃ 3.275 from earlier work 9 . Note that, according to Fig. 7 and Eq. (15), the average linear susceptibility does not diverge anywhere in the disordered phase. It appears that the value of z precisely at criticality may equal the value as the critical point is approached from the disordered phase, even though it is not clear that they have to be equal.
V. RESULTS AT THE CRITICAL POINT
From finite size scaling, the average uniform susceptibility (which is the same as the average local susceptibility for a model with a symmetric distribution of interactions, such as that used here) varies with L and L τ at the critical point according to 9,10
[χ] av = L xχ L τ L z ,(27)
where
x = z − β ν = z − d + 2 − η 2 .(28)
In earlier work 9 we found z ≃ 1.5, η ≃ 0.5, ν ≃ 1.0 and β ≃ 1.0 so (z − d + 2 − η)/2 ≃ 1/2. The earlier work concentrated on an fixed value of L τ /L z , which we call the "aspect ratio". Here we investigate whether Eq. (27) is satisfied for a range of aspect ratios. The data, shown in Fig. 8, does indeed collapse well with the expected values of the exponents. For L τ /L z ≪ 1, which corresponds to a large system at finite temperature, the dependence on L should drop out, and so, from L τ ∝ T −1 , one recovers Eq. (22). Using the numerical values of the exponents gives
[χ] av ∼ T −1/3 .(29)
Thus the average susceptibility diverges at the critical point, though we have seen above that it does not diverge in the disordered phase.
FIG. 8. A scaling plot of the data for the average susceptibility at T cl = 3.30, which is the critical point, to within our errors, see Eq. (4). The plot assumes the form in Eq. (27) with the exponent values deduced in earlier work 9 , i.e. z = 1.5, β/ν = 1. It is seen that the plot works well. The solid line, which fits the data for Lτ /L z < 0.8 has slope 0.5/1.5 = 1/3 as expected, since the average susceptibility should be independent of L in this limit. The power 1/3 gives the divergence as T → 0 at criticality, see Eq. (29),
The average local non-linear susceptibility is expected to vary at the critical point as A finite size scaling plot of the average local non-linear susceptibility, according to Eq. (30), at T cl = 3.30, very close to the critical point. The solid line, which fits the data for small Lτ /L z has slope 2.7/1.5 = 1.8 as expected, since the average local non-linear susceptibility should be independent of L in this limit. The power 1.8 gives the divergence as T → 0, and this value agrees well with estimates from the earlier estimates of exponent 9 , see Eq. (32),
[χ (loc) nl ] av = L yχ (loc) nl L τ L z ,(30)
where 16
y = 3z − 2β ν .(31)
With the numerical values of the exponents found earlier 9 , one has y ≃ 2.5. For L τ /L z ≪ 1, which corresponds to a large system at finite temperature, the dependence on L should drop out, and so one recovers Eq. (23). Using the numerical values of the exponents gives
[χ (loc) nl ] av ∼ T −5/3 .(32)
The data, shown in Fig 9, collapses well for L τ /L z not too large provided y/z ≃ 1.8, which gives a T −1.8 divergence, close to the prediction in Eq. (32). However, the data collapse is not as good for larger values of L τ with z = 1.5. Data in this region is difficult to equilibrate, which may be the cause of the discrepancy. It should be noted, though, that a better data collapse is obtained for larger values of z. However, the data for [χ] av does not scale well with a significantly larger value of z. We show results for the distribution of χ (loc) at the critical point, T cl = 3.3, in Fig. 10. Unlike the data in the disordered phase, shown in Figs. 3 and 5, there is here a significant size dependence, with the slope of the tail becoming less negative with increasing L. Asymptotically, the slope should be given by Eq. (21), which has the value −2/3 using the exponent values obtained earlier 9 . The L = 6 data has a slope of −1.7 and the L = 12 a slope of −0.95 so it is possible that the slope would tend to −2/3 for L → ∞. However, it is also possible that the slope might be less negative then this, which would imply a value of z larger than 3/2.
FIG. 10.
High precision data with large Lτ for the distribution of the local susceptibility at the critical point, T cl = 3.30. The number of samples was 25600 for L = 6 and 10240 for L = 8 and 12. The dashed line, which is a fit to the L = 6 data, has a slope of −1.7, and the solid line, which is a fit to the L = 12 data, has a slope of −0.95. Using the values of exponents found earlier 9 the slope, given by Eq. (21), is expected to be about −2/3 in the thermodynamic limit, and it is certainly plausible that data for larger sizes would extrapolate to this value.
VI. GLOBAL NON-LINEAR SUSCEPTIBILITY
Experimentally 8 one measures the global non-linear susceptibility, for which the local magnetization σ z i in (7) and the local external field h i have to be replaced by the mean global magnetization, L −d i σ z i and a uniform field, H, respectively. For the effective classical model this means that one has to consider
χ nl = − 1 6L τ L d M 4 − 3 M 2 2(33)
with M = i,τ S i (τ ). In earlier work 9 we found that this quantity diverges at criticality (for fixed aspect ratio) like χ nl ∼ L y+d , with y given by (31), so for arbitrary aspect ratio, finite size scaling gives
[χ nl ] av = L y+dχ nl L τ L z ,(34)
at criticality. Here, we have looked at scaling of various moments of the global non-linear susceptibility at the critical point, for a range of aspect ratios. For example the average is shown in Fig. 11. Since there are two exponents which can be adjusted, y and z, the data is unable to determine them both with precision. However, in the limit L τ /L z ≪ 1, where the dependence on L drops out and
[χ nl ] av ∼ L (d+y)/z τ ∼ T −(d+y)/z ,(35)
the data constrains (d + y)/z to be about 3, in agreement with the T −3 divergence at criticality found in earlier work 9 . Fig. 11 assumes the previously determined value of z, i.e. z = 1.5.
FIG. 11. A plot of the average global non-linear susceptibility at criticality, for a range of sizes and aspect ratios. The exponents used in this fit are z = 1/5 and y + d = 4.7. In earlier work 9 , we found z ≃ 1.5, y + d ≃ 4.5 so the present results are consistent with these estimates. In the limit Lτ ≪ L z , the average global non-linear susceptibility varies as L (d+y)/z τ ∼ T −(d+y)/z giving a strong divergence of roughly T −3.1 . This behavior is shown by the solid line, which is a fit to the data for small Lτ /L z and has slope 3.1.
In addition we have evaluated the typical global nonlinear susceptibility defined by
χ typ nl = exp[log χ nl ] av ,(36)
at the critical point and show the data in Fig. 12. As with the data for the average in Fig 11, the ratio (d + y)/z is more tightly constrained than either z or d+y separately. A good fit is obtained with (d+y)/z ≃ 2.4, which leads to a divergence of roughly T −2.4 , not quite so strong as from the average non-linear susceptibility. The difference may well reflect corrections to scaling for the range of sizes that we were able to study. Note that even the typical value has a strong divergence with T at criticality, in contrast to the experiments 8 .
We have also studied the probability distribution P (χ nl ) in the disordered phase. This shows a slightly more complicated behavior than the probability distribution of local quantities presented above. We find that the power describing the tail in the same distribution is the same as that of the local non-linear susceptibility. This is reasonable since these unusually large values come from correlations which are very long ranged in time, whereas the spatial correlation length is small and so does not give a significant extra effect. These spatial correlations do, however, cause the peak in the distribution to shift to larger values with increasing system size, though presumably the peak position would eventually settle down to a constant for sizes greater than the correlation length.
VII. CONCLUSIONS
One can characterize Griffiths singularities in the disordered phase of a quantum system undergoing a T = 0 transition with discrete broken symmetry, by a single, continuously varying, dynamical exponent, z. Average response functions may or may not diverge in part of the disordered phase near the critical point, depending on the value of z, see Eqs. (15) and (16). The numerical results, summarized in Fig. 7 and Eqs. (25) and (26), indicate that the average linear susceptibility does not diverge in the disordered phase of the 2-d quantum Ising spin glass, though it diverges at the critical point. The average nonlinear susceptibility does, however, diverge in part of the disordered phase.
Numerically, as one approaches the critical point from the disordered phase, the value of z is close to the value obtained precisely at the critical point, see Fig. 7. Since the same result is known to hold exactly in 1-d, where they are both equal to infinity 5 , one might speculate that they are equal in general, though we are not aware of any proof of this. Presumably the detailed dependence of z with T cl in the disordered phase, shown in Fig. 7, is nonuniversal. However, it is interesting to ask whether the answer to the question "Does the non-linear susceptibility diverge in the disordered phase" is universal or not. From Eq. (16) this depends on whether z > d/3 as the critical point is approached. If this limit for z is precisely the same as the value of z at criticality, then the answer to the question is universal. However, as we just mentioned, we are not aware of any argument which shows that these two values of z should be equal in general.
A related study has also been carried out recently by Guo et al. 11,17 Their results for the 2-d spin glass are consistent with ours, and they also performed some simulations for the 3-d spin glass. In three dimensions the classical model has a finite temperature transition 13 , so the spin glass phase would exist for a finite range of T , which shrinks to zero as T cl → T cl c − , see Fig. 1. The results of Guo et al. 11 indicate that, in three dimensions, the range of the Griffiths phase over which the non-linear susceptibility diverges, i.e. the region between T cl c and T cl χnl in Fig. 1, is very small but apparently non-zero. It is interesting to speculate on whether the possible divergence of the non-linear susceptibility in part of the disordered phase might be related to the difference between the experiments 8 which apparently do not find a strong divergence of χ nl at the quantum critical point, and the simulations 9,10 which do.
The data presented here is consistent with our earlier results 9 in finding a dynamic exponent at the critical point of about 1.5. This is rather different from the situation in one-dimension 4,5 where z = ∞, and one might ask whether the true dynamical exponent might not be larger than 1.5 in two-dimensions, and possibly infinite. While the data for the modest range of sizes that can be studied by Monte Carlo simulations is consistent with a small value of z, we cannot completely rule out the possibility that this estimate would increase if one could study larger sizes. Unfortunately, it does not seem feasible to study very much larger sizes with current computer power, un-less a more sophisticated algorithm can be found than the single spin-flip approach used here. Assuming that z is indeed finite, the critical scaling in the two-dimensional quantum Ising spin glass is of a fairly conventional, but anisotropic, type, with z playing the role of an anisotropy exponent. The difference from a classical magnet with anisotropic scaling is that Griffiths singularities give additional singularities in various scaling functions in the limit L τ ≫ L z , or equivalently T L z ≪ 1.
ACKNOWLEDGMENTS
The work of APY has been supported by the National Science Foundation under grant No. DMR-9411964. The work of HR was supported by the Deutsche Forschungsgemeinschaft (DFG) and he thanks the Physics Department of UCSC for the kind hospitality. We should like to thank R. N. Bhatt, M. Guo, D. A. Huse and D. S. Fisher for helpful discussions.
FIG. 1 .
1The phase diagram of the two-dimensional quantum Ising spin glass. The horizontal axis can be thought of as T cl if one is using the effective classical Hamiltonian in Eq.(2) or Γ if one is using the original quantum Hamiltonian in Eq. (1). There is a critical point at T cl = T cl c . For T cl < T cl c there is a spin glass ordered phase. For T cl > T cl c
FIG. 3 .
3The log of the distribution of the log of the local susceptibility for T cl = 3.7 for different values of L and Lτ .
FIG. 4 .
4Similar toFig. 3but for the local non-linear susceptibility. The straight line has slope −1.23 which gives z = 0.54 from Eq. (
. Hence, the average χ (loc) nl does diverge at T cl = 3.5.
FIG. 5 .
5Similar to Fig. 3 but for T cl = 3.5. The straight line has slope −2.78 which gives z = 0.71 from Eq. (13). FIG. 6. Similar to Fig. 5 but for the local non-linear susceptibility. The straight line has slope −0.87 which gives z = 0.76 from Eq. (14), in fair agreement with the fit to the data in
FIG. 7 .
7The dynamical exponent z, obtained by fitting the distributions of χ (loc) and χ (loc) nl
by the hexagons. The two are in good agreement. The dotted vertical line indicates the critical point, obtained in Ref.9 and the solid square indicates the estimate of z at the critical point. The dashed line is z = 2/3; the average non-linear susceptibility diverges for z larger than this, i.e. T cl > T cl χ nl ≃ 3.56. The solid curve is just a guide to the eye.
FIG. 9 .
9FIG. 9.
FIG. 12 .
12A plot of the typical global non-linear susceptibility, defined in Eq. (36), at criticality, for a range of sizes and aspect ratios. In the limit Lτ ≪ L z , the typical global non-linear susceptibility varies as L (d+y)/z τ ∼ T −(d+y)/z giving a quite strong divergence of roughly T −2.4 . This behavior is shown by the solid line, which is a fit to the data for small Lτ /L z and has slope 2.4.
. R B Griffiths, Phys. Rev. Lett. 2317R. B. Griffiths, Phys. Rev. Lett. 23. 17 (1969).
. A B Harris, Phys. Rev. B. 12203A. B. Harris, Phys. Rev. B 12, 203 (1975).
. B M Mccoy, T T Wu, Phys. Rev. B. 176982B. M. McCoy and T. T. Wu, Phys. Rev. B 176, 631 (1968); 188, 982 (1969).
. R Shankar, G Murphy, Phys. Rev. B. 36536R. Shankar and G. Murphy, Phys. Rev. B 36, 536 (1987).
. D S Fisher, Phys. Rev. Lett. 69534D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992);
. Phys. Rev. B. 516411Phys. Rev. B 51, 6411 (1995).
. A P Young, H Rieger, cond-mat/9510027A. P. Young and H. Rieger, cond-mat/9510027.
. M J Thill, D A Huse, Physica A. 15321M. J. Thill and D. A. Huse, Physica A, 15, 321 (1995).
. W Wu, B Ellmann, T F Rosenbaum, G Aeppli, D H Reich, Phys. Rev. Lett. 672076W. Wu, B. Ellmann, T. F. Rosenbaum, G. Aeppli and D. H. Reich, Phys. Rev. Lett. 67, 2076 (1991);
. W Wu, D Bitko, T F Rosenbaum, G Aeppli, Phys. Rev. Lett. 711919W. Wu, D. Bitko, T. F. Rosenbaum and G. Aeppli; Phys. Rev. Lett. 71, 1919 (1993).
. H Rieger, A P Young, Phys. Rev. Lett. 724141H. Rieger and A. P. Young, Phys. Rev. Lett. 72, 4141 (1994).
. M Guo, R N Bhatt, D A Huse, Phys. Rev. Lett. 724137M. Guo, R. N. Bhatt and D. A. Huse, Phys. Rev. Lett. 72, 4137 (1994).
. M Guo, R N Bhatt, D A Huse, unpublishedM. Guo, R. N. Bhatt and D. A. Huse, (unpublished).
B K Chakrabarti, A Dutta, P Sen, Quantum Ising Phases and Transitions in Transverse Ising Models. and references thereinFor a review on transverse field Ising models see B. K. Chakrabarti, A. Dutta and P. Sen: Quantum Ising Phases and Transitions in Transverse Ising Models, to be pub- lished in Lecture Notes in Physics (1995), and references therein.
. K Binder, A P Young, Rev. Mod. Phys. 58801K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).
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See reference 18 in Ref. 9See reference 18 in Ref. 9 .
. M Guo, Princeton University thesisM. Guo, Princeton University thesis, (1995).
|
[] |
[
"Insights on unconventional superconductivity in HfV 2 Ga 4 and ScV 2 Ga 4 from first principles electronic structure calculations",
"Insights on unconventional superconductivity in HfV 2 Ga 4 and ScV 2 Ga 4 from first principles electronic structure calculations"
] |
[
"P P Ferreira \nMaterials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil\n",
"F B Santos \nMaterials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil\n",
"A J S Machado \nMaterials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil\n",
"H M Petrilli \nInstituto de Física\nUniversidade de São Paulo\n66318, 05315-970São Paulo -SPCPBrazil\n",
"L T F Eleno \nMaterials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil\n"
] |
[
"Materials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil",
"Materials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil",
"Materials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil",
"Instituto de Física\nUniversidade de São Paulo\n66318, 05315-970São Paulo -SPCPBrazil",
"Materials Engineering Department\nEscola de Engenharia de Lorena da Universidade de São Paulo\nLorena -SP\nBrazil"
] |
[] |
The HfV 2 Ga 4 compound was recently reported to exhibit unusual bulk superconducting properties, with the possibility of multiband behavior. To gain insight into its properties, we performed ab-initio electronic structure calculations based on the Density Functional Theory (DFT). Our results show that the density of states at the Fermi energy is mainly composed by V-d states. TheMcMillan formula predicts a superconducting critical temperature (T c ) of approximately 3.9 K, in excellent agreement with the experimental value at 4.1 K, indicating that superconductivity in this new compound may be explained by the electron-phonon mechanism. Calculated valence charge density maps clearly show directional bonding between Hf and V atoms with 1D highly populated V-chains, and some ionic character between Hf-Ga and V-Ga bonds. Finally, we have shown that there are electrons occupying two distinct bands at the Fermi level, with different characters, which supports experimental indications of possible multiband superconductivity. Based on the results, we propose the study of a related compound, ScV 2 Ga 4 , showing that it has similar electronic properties, but probably with a higher T c than HfV 2 Ga
|
10.1103/physrevb.98.045126
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[
"https://arxiv.org/pdf/1805.08285v1.pdf"
] | 118,959,430 |
1805.08285
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49bcde61d08866c6ebbda8efb9b13503be1f4887
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Insights on unconventional superconductivity in HfV 2 Ga 4 and ScV 2 Ga 4 from first principles electronic structure calculations
May 2018
P P Ferreira
Materials Engineering Department
Escola de Engenharia de Lorena da Universidade de São Paulo
Lorena -SP
Brazil
F B Santos
Materials Engineering Department
Escola de Engenharia de Lorena da Universidade de São Paulo
Lorena -SP
Brazil
A J S Machado
Materials Engineering Department
Escola de Engenharia de Lorena da Universidade de São Paulo
Lorena -SP
Brazil
H M Petrilli
Instituto de Física
Universidade de São Paulo
66318, 05315-970São Paulo -SPCPBrazil
L T F Eleno
Materials Engineering Department
Escola de Engenharia de Lorena da Universidade de São Paulo
Lorena -SP
Brazil
Insights on unconventional superconductivity in HfV 2 Ga 4 and ScV 2 Ga 4 from first principles electronic structure calculations
May 2018
The HfV 2 Ga 4 compound was recently reported to exhibit unusual bulk superconducting properties, with the possibility of multiband behavior. To gain insight into its properties, we performed ab-initio electronic structure calculations based on the Density Functional Theory (DFT). Our results show that the density of states at the Fermi energy is mainly composed by V-d states. TheMcMillan formula predicts a superconducting critical temperature (T c ) of approximately 3.9 K, in excellent agreement with the experimental value at 4.1 K, indicating that superconductivity in this new compound may be explained by the electron-phonon mechanism. Calculated valence charge density maps clearly show directional bonding between Hf and V atoms with 1D highly populated V-chains, and some ionic character between Hf-Ga and V-Ga bonds. Finally, we have shown that there are electrons occupying two distinct bands at the Fermi level, with different characters, which supports experimental indications of possible multiband superconductivity. Based on the results, we propose the study of a related compound, ScV 2 Ga 4 , showing that it has similar electronic properties, but probably with a higher T c than HfV 2 Ga
I. INTRODUCTION
Although superconductivity has attracted the attention of the scientific community for a long time, the understanding of the phenomenon, which started with the model proposed by Bardeen, Cooper and Schrieffer (BCS) [1], is still very challenging. The BCS theory, although useful for a large class of superconducting materials, fails to correctly explain other experimentally observed superconducting elements or compounds [2] and a plethora of different behaviors demands new approaches.
First-principles electronic structure calculations, within the framework of the Density Functional Theory (DFT), has proven to be an important tool to study superconducting materials. Although strongly correlated systems are beyond the scope of the Kohn-Sham scheme of the DFT, many successful attempts have been made to deal with the description of superconducting materials. In particular, some specific properties of the normal state, e.g. electronic band dispersions and electronic density of states, are very useful to elucidate aspects of the superconducting mechanism and to predict relevant parameters, such as the critical temperature T c and the electron-phonon coupling constant λ. In the last few years, an increasing number of studies appeared using this methodology, either as support for experimental discoveries [3][4][5] or fully theoretical investigations [6][7][8][9].
Superconductivity was recently experimentally reported, by some of the present authors, for the HfV 2 Ga 4 compound, with a critical temperature (T c ) of 4.1 K [10]. The investigators observed some deviations from the more conventional BCS theory signatures, such as an unusual inflection near T c in lower and upper critical field as a function of reduced temperature, and a second jump in the specific heat vs. temperature curve. The authors speculated that the experimental results could be either due to sample inhomogeneity or to the presence of more than one superconducting gap at the Fermi surface, resulting in a two-band superconductivity [11].
These recent experimental results for the bulk HfV 2 Ga 4 point to a new promising class of materials to study unconventional superconducting behavior. Motivated by these results, here we perform ab-initio electronic structure calculations for HfV 2 Ga 4 . We focus our attention on the analysis of the possible mechanisms behind the superconducting properties. The theoretical study was extended to a new (possibly) bulk superconducting compound with the same prototype structure, ScV 2 Ga 4 , as a way to manipulate the electronic structure aiming at enhancing the superconducting transition temperature.
II. COMPUTATIONAL METHODS
The ab-initio electronic structure calculations were performed in the framework of the Kohn-Sham scheme [12] within Density Functional Theory (DFT) using the Full Potential -Linearized Augmented Plane Wave plus local orbitals (FP-LAPW+lo) method [13], as implemented in the WIEN2k computational code [14]. The Exchange and Correlation (XC) functional was described by the Generalized Gradient Approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) formulation [15], taking relativistic corrections and spin-orbit coupling (SOC) effects into account. We used muffin-tin spheres with radius R MT = 2.0 a 0 (Bohr's radius) for all atoms, with R MT K max = 9.0, where K max is related to the basis set size [14]. Self-consistent-field (SCF) calculations were carried out with a 32 × 32 × 32
Monkhorst-Pack [16] shifted k-point mesh discretization in the first Brillouin zone. All lattice parameters and internal degrees of freedom were relaxed in order to guarantee a ground state convergence to about 10 −5 Ry in the total energy, 10 −4 e for electron density and 0.5 mRy/a 0 for forces acting on the nuclei. The Birch-Murnaghan equation of state [17] was used to fit the total energy as a function of the unit cell volume (keeping c/a constant) at several c values in order to obtain the ground state lattice constants and bulk modulus.
Finally, six different lattice distortions, with 15 intensities for each one (a total of 90 different structures), were used to provide data for the determination of the elastic properties with the ElaStic code [18], using Quantum Espresso [19] for DFT calculations of deformed structures. The Quantum Espresso calculations were performed using PBE SG15 Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials [20], with a cutoff energy of 220 Ry and 1728 k-points in the first Brillouin zone. Anderson's simplified method [21] was then employed for the calculation of the Debye temperature. sites, at 2a (0, 0, 0) Wyckoff positions, are surrounded by V and Ga sites at 4d (0, 1/2, 1/4) and 8h (0.303, 0.303, 0), respectively [22], as schematically shown in Figure 1.
The calculated optimized lattice parameters are in excellent agreement with the experimental data reported in the literature [23], as seen in Table I. There is a slight difference of at most 1% with respect to the experimental values, which is commonly related to the inherent imprecision of the approximations required by the computational method [24][25][26]. The calculated total DOS at the Fermi level is N (E F ) = 2.29 states/eV. This quantity is related to the linear coefficient of the electronic specific heat γ, known as Sommerfeld coefficient, given by
γ = π 2 3 k 2 B N (E F ) ,(1)
where k B is the Botzmann constant.The calculated N (E F ) leads to a value of 5.41 mJ mol −1 K −2 for the theoretical γ calc . From the value of the Sommerfeld coefficient γ calc resulting from the ab-initio calculations and the experimentally measured value
(γ exp = 8.263 mJ mol −1 K −2 ) [10]
, we can estimate reasonably well the electron-phonon coupling constant λ using the well-known approximation [27,28]
λ = γ exp γ calc − 1 ,(2)
which stems from the fact that the calculations give static (0 K) results. Following Eq. (2), we arrive at λ = 0.53. This value can be used to calculate the superconducting transition temperature T c using the empirical McMillan formula [29],
T c = Θ D 1.45 exp − 1.04(1 + λ) λ − µ * (1 + 0.62λ) ,(3)
where µ * is the Coulomb pseudopotential, which measures the strength of the electronelectron Coulomb repulsion [30]. A typical value of µ * is 0.12, as used in many previous works [27,28,31]. For the HfV 2 Ga 4 compound, using the calculated Θ D and the above values for λ and µ * , we arrive at an estimated critical temperature T c = 3.9 K, in excellent agreement with the experimental (4.1 K) value. This indicates that the electron-phonon interaction may be the mechanism behind superconductivity in HfV 2 Ga 4 . The V states (100) and (001) planes. This reveal that Hf atoms, which are "locked" in the center of a cage-like structure, are not simply passive electron donors: they stabilize the charge transfer to the V atoms (as also observed in Figure 2) that, in turn, commands the electronic properties.
Furthermore, it is interesting to note that the charge density gives rise to a kind of electron sharing channel in the lattice, composed by directional, strongly-bonded, highly populated V chains in the (100) and (010) crystallographic planes. The Hf nuclei are weakly bonded with the two V atoms within adjacent unit cells in these 1D chains. As a consequence of these V chains that concentrate most of the electronic states that will give rise to Cooper pairs, this electronic configuration may lead to a high anisotropy that could be identified via transport measurements. Finally, despite the small difference in electronegativity between the atomic species, Hf-Ga and V-Ga bonds exhibit some ionic character. In Figure 3 There are two bands crossing the Fermi energy, with very different Hf and V characters.
The fact that there are electrons occupying two distinct bands in disconnected sheets of the Fermi surface (corresponding to the two bands crossing the Fermi level) supports the experimental evidence of a possible two-gap superconductivity [10]. These results open a promising scenario for a possible multiband behaviour, so a superconducting gap calculation [32,33] would be a very interesting test for that hypothesis.
In the band character plots for HfV 2 Ga 4 shown in Figure 4 Several compounds that crystallize in the same body-centered tetragonal prototype YbMo 2 Al 4 , such as RTi 2 Ga 4 (R = Ho, Er, Dy) and RV 2 Ga 4 (R = Sc, Zr, Hf), have been reported in the literature. These compounds are poorly investigated, most efforts having been focused exclusively on magnetic properties in rare-earth compounds [36,37].
The results reported above for HfV 2 Ga 4 led us to consider an effective way to manipulate the electronic structure of such compounds, aiming at enhancing superconducting properties.
In Figure 2 we can observe that the Fermi level is situated down a deep valley in the total DOS. As a consequence, the density of states at E F is extremely sensitive. So, considering The nature of atomic bonding is the same showed for the HfV 2 Ga 4 in Figure 3, with high populated 1D covalent V-chains and Sc atoms acting to stabilize the transfer of charge to the V atoms. Therefore, the DOS overall appearance for HfV 2 Ga 4 is qualitatively identical to ScV 2 Ga 4 , as can be verified in Figure 5(a). Hence, the contribution of each orbital in the density of states for ScV 2 Ga 4 is also very similar to HfV 2 Ga 4 , with a higher contribution due to Sc-d states in the unoccupied bands. The calculated value of N (E F ) is 3.62 states/eV, that leads to γ calc = 8.53 mJ mol −1 K −2 using Eq. (1). Confirming our hypothesis, the presence of Sc atoms instead of Hf in the 2a sites causes the Fermi level to shift to a higher DOS value, an increase of about 60%, escaping from the bottom of the well.
In Figures 5(b) and 5(c) we show the calculated band structure without and with SOC effects, respectively. It may be seen that the band structure is related to that presented for HfV 2 Ga 4 (Fig. 4), with similar features in the vicinity of the Fermi energy. Fermi bands in level, attached to the fact that there are electrons originated from two distinct bands in the Fermi surface, strongly suggest that ScV 2 Ga 4 could be a new example of two-band electronphonon superconducting material with a considerable higher critical temperature than the one reported for the HfV 2 Ga 4 compound.
IV. CONCLUSIONS
In this work we presented ab-initio calculations for the bulk superconductor HfV 2 Ga 4 .
The McMillan formula predicts a T c of 3.9 K, in excellent agreement with experimental reported values (4.1 K), indicating that superconductivity can be readily explained in an electron-phonon framework. From the signature of the DOS in the vicinity of the Fermi energy, we have proposed to improve the superconducting critical temperature by investigating the ScV 2 Ga 4 compound. Theoretically, we have shown that the presence of Sc instead of Hf in the crystal structure causes the Fermi level to shift to a higher DOS value.
The band structure around the Fermi level, which comes mainly from V-d states, and the DOS overall appearance, are qualitatively very similar for both compounds. Valence electron density plots unveil Hf(Sc)-V shared bonding and 1D highly populated V-chains, while
Hf(Sc)-Ga and V-Ga bonds have a partially ionic character. It was found that there are electrons derived from two distinct bands in disconnected sheets of the Fermi surface for both compounds, in agreement with the experimental evidence [10] of a possible two-gap superconductivity for HfV 2 Ga 4 . Finally, we argue that ScV 2 Ga 4 is presumably a new candidate for two-band electron-phonon superconductivity with a higher T c than HfV 2 Ga 4 , a result that should be confirmed experimentally.
III. RESULTS AND DISCUSSIONA. HfV 2 Ga 4 electronic structure calculations HfV 2 Ga 4 crystallize in the YbMo 2 Al 4 prototype (space group I4/mmm, Pearson symbol tI14), a body-centered tetragonal structure composed by a cage-like structure, where Hf
Figure 1 :
1HfV 2 Ga 4 body-centered tetragonal unit cell (conventional setting). Hf (gray), V (blue) and Ga (red) sites are at the 2a (0, 0, 0), 4d (0, 1/2, 1/4) and 8h (0.303, 0.303, 0) Wyckoff positions, respectively.
The calculated bulk modulus is 134.75 GPa, with a Poisson ratio of 0.24, resulting in 416.3 K for the Debye temperature (Θ D ). Our ab initio calculations for Θ D reproduce with great accuracy the 418.97 K value obtained through experimental measurements[10].The total density of states (DOS), as well as the site and orbital projected density of states (PDOS), are shown in Figures 2a-d. Both occupied and unoccupied states involve considerable hybridization, as seen in Figure 2a. In the lowest energy region Ga orbitals are dominant, with some contribution from V; in the region around the Fermi level (from −2.5 eV to 3 eV), V states are prevailing, mainly due to V-d character (notice the different PDOS scales onFigures 2b-d), with some Hf-d and Ga-p contributions; in the higher (above 3 eV), unoccupied energy region, Hf and V states contribute equally. Almost half of the total DOS at the Fermi level is due to V, although these states are extended along the whole studied energy region.
Figure 2 :Figure 3 :
23(color online) (a) Total and site projected density of states for HfV 2 Ga 4 . The orbital-projected (s, p and d ) contributions at each site are also shown: (b) Hf, (c) V, and (d) Ga . The Fermi level is set at 0 eV in all figures. dominate the N (E F ) and therefore have the major contribution for the pairing. The nature of the atomic bonding can be further elucidated with the help of valence electron density plots such as those shown in Figure 3, in which the electron density is plotted, with an appropriate logarithmic scale in a (001) plane, passing through the center of Hf and Ga nuclei within a unit cell (Figure 3a), and a (100) plane, passing through the center of Hf and V nuclei (Figure 3b). It should be noted that, in Figure 3a, the non-labelled Valence electron density plot in (001) plane (a) and (100) plane (b) for HfV 2 Ga 4 . Along a contour the electron density is constant. high-density regions are V nuclei not centered on the (100) plane. The plots clearly show a directional shared bond between Hf and V atoms, evidenced by the density contours in the
observe isolated clusters containing four Ga atoms within a unit cell, forming weak bonds with adjacent Hf atoms.
Figure 4
4shows the resulting band character plots along high symmetry points in the first Brillouin zone, not including (Fig. 4a) and including (Fig. 4b) spin-orbit coupling (SOC) effects in the calculations. In the band character plots, stronger colors mean stronger character due to the respective orbital projection. Indeed, the cage-like symmetry of the lattice gives rise to complex dispersive metallic bands in the vicinity of the Fermi level.
Figure 4 :
4, we can see that a hole pocket develops in the M point, with a maximum at ≈ 0.4 eV, originated mainly from the Hf-d states containing some mixing with the V-d states. In particular, the electron band crossing the P point just below E F is made up mostly by V-d states. Notice that, near the Fermi level, the band plot unveils dispersive cones with zero gap at M and also along the M-Γ direction, as well as one such feature at P. However, when SOC effects are considered, these features are gapped. Indeed, SOC leads to a visible lifting of some band degeneracies, mainly at M and M-Γ, and less-pronounced at P (just a few meV). Moreover, although these compounds are metallic, SOC broken degeneracy creates a continuous pseudo-gap around the Fermi energy, (color online) Band character plots along high symmetry points in the first Brillouin zone of HfV 2 Ga 4 , without (a) and with (b) SOC effects. Colors give a picture about the band character, with color intensity indicating qualitatively the strengh of the contribuition of a given state. Only the Hf-d (red) and V-d (blue) states are represented (the Fermi level is set at 0 eV). although the gap almost closes at P (not visible in the scale of Figure 4b). This kind of signature also occurs in a few nontrivial topological materials like Bi 14 Rh 3 In 9 , PbTaSe 2 and Cu x ZrTe 2−y [5, 34, 35]. Therefore, more detailed experimental and theoretical studies about the possibility of nontrivial topological effects in HfV 2 Ga 4 could be an interesting subject for futures investigations. B. Theoretical predictions for ScV 2 Ga 4
ScV 2 Figure 5 :
25Ga 4 are well described as coming from hybridization between mainly V-d and some Sc-d states. However, SOC in ScV 2 Ga 4 plays only a marginal role, making nontrivial topological effects unlikely. Nevertheless, the important point here resides on the fact that, similar to HfV 2 Ga 4 , there are two bands crossing the Fermi level, opening again the possibility for a multiband scenario.The large contribution of V-d state electrons and the higher DOS value at the Fermi Total and projected density of states for ScV 2 Ga 4 (a) and band character plots along high symmetry points in first Brillouin zone without (b) and with (c) SOC effects.
Table I :
ICalculated lattice parameters and optimized 8h (Ga) atomic position for the
HfV 2 Ga 4 tetragonal compound, compared to experimental values [23].
calc.
exp.
a, b (Å) 6.459
6.45
c (Å) 5.197
5.20
8h (Ga) (0.303, 0.303, 0) (0.303, 0.303, 0)
Table II :
IIa rigid band model, it is a reasonable to assume that an element with a different valence configuration in the 2a site of HfV 2 Ga 4 could shift the Fermi level to higher states. Based on what has been presented, we also have carried out first principles electronic structure calculations for ScV 2 Ga 4 , to test this hypothesis.Calculated lattice parameters and optimized 8h (Ga) atomic position for the
ScV 2 Ga 4 , compared to experimental values [23].
calc.
exp.
a, b (Å) 6.497
6.432
c (Å) 5.200
5.216
8h (Ga) (0.3004, 0.3004, 0) (0.303, 0.303, 0)
Table II
IIshows the relaxed calculated lattice parameters, together with experimental reported values for ScV 2 Ga 4 . Following the same methodology applied in the previous section, we reached Θ D = 447.8 K for ScV 2 Ga 4 . Unfortunately, in this case, there are no experimental data for comparison.
AcknowledgmentsWe gratefully acknowledge the financial support of the Conselho Nacional de Desen-
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|
[] |
[
"Correcting Presbyopia with Autofocusing Liquid-Lens Eyeglasses",
"Correcting Presbyopia with Autofocusing Liquid-Lens Eyeglasses"
] |
[
"Mohit U Karkhanis \nDepartment of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA\n",
"Chayanjit Ghosh \nDepartment of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA\n",
"Aishwaryadev Banerjee \nDepartment of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA\n",
"Nazmul Hasan \nIntel Corporation\nPortlandORUSA\n",
"Rugved Likhite \nIntel Corporation\nPortlandORUSA\n",
"Tridib Ghosh \nNewEyes Inc\nSalt Lake CityUTUSA\n",
"Hanseup Kim \nDepartment of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA\n",
"Carlos H Mastrangelo \nDepartment of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA\n"
] |
[
"Department of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA",
"Department of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA",
"Department of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA",
"Intel Corporation\nPortlandORUSA",
"Intel Corporation\nPortlandORUSA",
"NewEyes Inc\nSalt Lake CityUTUSA",
"Department of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA",
"Department of Electrical and Computer Engineering\nUniversity of Utah\nSalt Lake CityUTUSA"
] |
[] |
Presbyopia, an age-related ocular disorder, is characterized by the loss in the accommodative abilities of the human ocular system and afflicts more than 1.8 billion people world-wide. Conventional methods of correcting presbyopia fragment the field of vision, inherently resulting in significant vision impairment. We demonstrate the development, assembly and evaluation of autofocusing eyeglasses for restoration of accommodation without vision field loss. The adaptive optics eyeglasses consist of two variable-focus piezoelectric liquid lenses, a time-of-flight range sensor and low-power, dual microprocessor control electronics housed within an ergonomic frame. Patientspecific accommodation deficiency models were utilized to demonstrate a high-fidelity accommodative correction. Each accommodation correction calculation was performed in ~67 ms requiring 4.86 mJ of energy. The optical resolution of the system was 10.5 cycles/degree, featuring a restorative accommodative range of 4.3 D. This system can run for up to 19 hours between charge cycles and weighs ~132 g, allowing comfortable restoration of accommodative function.
|
10.1109/tbme.2021.3094964
|
[
"https://arxiv.org/pdf/2101.08782v1.pdf"
] | 231,693,267 |
2101.08782
|
a3ab4f86162a896a6d9ecf8f21d7605dd03b591f
|
Correcting Presbyopia with Autofocusing Liquid-Lens Eyeglasses
Mohit U Karkhanis
Department of Electrical and Computer Engineering
University of Utah
Salt Lake CityUTUSA
Chayanjit Ghosh
Department of Electrical and Computer Engineering
University of Utah
Salt Lake CityUTUSA
Aishwaryadev Banerjee
Department of Electrical and Computer Engineering
University of Utah
Salt Lake CityUTUSA
Nazmul Hasan
Intel Corporation
PortlandORUSA
Rugved Likhite
Intel Corporation
PortlandORUSA
Tridib Ghosh
NewEyes Inc
Salt Lake CityUTUSA
Hanseup Kim
Department of Electrical and Computer Engineering
University of Utah
Salt Lake CityUTUSA
Carlos H Mastrangelo
Department of Electrical and Computer Engineering
University of Utah
Salt Lake CityUTUSA
Correcting Presbyopia with Autofocusing Liquid-Lens Eyeglasses
Index Terms-Autofocusing eyeglassespresbyopiaaccommodationadaptive opticssmart eyeglassesvariable focus lens
Presbyopia, an age-related ocular disorder, is characterized by the loss in the accommodative abilities of the human ocular system and afflicts more than 1.8 billion people world-wide. Conventional methods of correcting presbyopia fragment the field of vision, inherently resulting in significant vision impairment. We demonstrate the development, assembly and evaluation of autofocusing eyeglasses for restoration of accommodation without vision field loss. The adaptive optics eyeglasses consist of two variable-focus piezoelectric liquid lenses, a time-of-flight range sensor and low-power, dual microprocessor control electronics housed within an ergonomic frame. Patientspecific accommodation deficiency models were utilized to demonstrate a high-fidelity accommodative correction. Each accommodation correction calculation was performed in ~67 ms requiring 4.86 mJ of energy. The optical resolution of the system was 10.5 cycles/degree, featuring a restorative accommodative range of 4.3 D. This system can run for up to 19 hours between charge cycles and weighs ~132 g, allowing comfortable restoration of accommodative function.
I. INTRODUCTION
RESBYOPIA is an age-related condition, which stems from the gradual loss in the eye's ability to change its optical power [1]. It manifests as an inability to focus on objects, often leading to symptoms like visual discomfort, eye strain and headache, after around 45 years of age [2]. The amount of accommodation loss as a function of age is characterized by Duane's curve [3], shown in Fig. 1. This curve shows the accommodative range (or accommodative amplitude AA) of the eye crystalline lens versus age. A young person's crystalline lens has about 12 D of accommodative range, but that range is reduced to about 1 D by age 45 (physically, this translates into being able to focus objects located between 0 m and 1/AA m). A young individual can thus see objects clearly between infinity up to 8 cm in front of them. An older, 60-year-old presbyope can only see objects between infinity to 1 m or worse as loss of accommodation often is accompanied by a fixed power offset. Manuscript [3]. The amplitude of accommodation progressively reduces to <1.0 D as a person enters their fifth decade. Note that the curve flattens out but does not become zero indicating that some degree of remnant accommodation is present.
The precise mechanism underpinning the development and progression of presbyopia is still debated, but the principal reasons behind this condition are the loss of power in the ciliary muscles and the morphological changes in the crystalline lens of the eye [3]- [22]. In 2018, globally, more than 1.8 billion people were afflicted with this condition [23]. With increasing life expectancies, the number of people affected by this debilitating condition is expected to keep on rising [24]. Currently presbyopia can be partially corrected using several optical devices and methods listed below, none of which can restore normal vision.
Multifocal Corrective Eyeglasses: Traditional methods used to correct vision loss due to presbyopia include the use bifocal, trifocal, multifocal and progressive eyeglasses [25]. These devices feature lenses which divide the visual field into zones of different optical powers. The lens zoning enables a user to focus on near or far objects by adjusting their viewing angle. However, these mature technologies have serious drawbacks. Bifocal and trifocal eyeglasses provide a very fragmented field of view per zone which causes "image jumps" when the user adjusts their viewing angle [26]. Design of highly customized progressive lenses is now possible using free-form surfacing and sophisticated computer-aided modelling [27]- [29]. However, such lenses are expensive, have a steep learning curve and come with a penalty of unwanted astigmatism which distorts the user's peripheral visual field [30]- [32]. Presbyopes who use multifocal eyeglasses are also more likely to experience accidents and mobility issues due to dioptric blurs in their visual field [33]- [39].
Monovision and Multifocal Contact Lenses: The idea of correcting presbyopia using contact lenses has been around since the last five decades [40] but it has enjoyed limited success [26], [41], [42]. Presbyopic contact lens technologies have been extensively reviewed in literature [43]- [48] and can be divided into two main categories-monovision and multifocal contact lens correction. In monovision correction, the dominant eye is usually corrected for distance vision, while the other is corrected for near vision [49]- [54]. Many clinical trials have shown that after an adaptation period to monovision correction, presbyopes experienced an improved range of clear focus, at the expense of reduced contrast sensitivity and stereopsis [55]- [64]. There are also hard limitations on the optimum optical power of the near addition lens in monovision correction, as higher optical powers negatively impact stereopsis [49]- [51], [65]- [67]. Studies report that presbyopes successfully adapt to multifocal contact lenses despite the presence of optical aberrations [68]- [74]. Despite such successes, contact lens correction of presbyopia remains unpopular due to discomfort and inconvenience, poor visual experience, and cost [75], [76].
Intra-Ocular Lens Replacement: Many surgical approaches have been suggested for the correction of presbyopia, which involve replacing the natural crystalline lens of the eye with an artificial intra-ocular lens (IOL). These IOLs generally follow the same principles of refraction as the monovision and multifocal contact lenses mentioned above [47], [77]. However, this correction strategy requires detailed knowledge of the patient's ocular system, involves very precise lens calibrations before surgery, and a very delicate surgical procedure to implant the lens in the eye. Successes of this correction strategy are debatable as patients have reported haloes and glare in their vision and low contrast sensitivity [78].
Accommodating Lenses: A major problem with all the above methods is that they only provide a partial and unnatural tiled or warped correction of vision. To restore normal vision, the accommodation must be corrected actively, which means it must be dependent on the observed object distance, without any losses in the visual field or viewing angle adjustments. Restoring active accommodation in such patients, requires a variable optical power change of at least 3.0 D [26]. The earliest practical attempts at developing variable-focus eyeglasses were spearheaded by B. F. Edwards with his "polyfocal spectacles" in the 1950s [79]. These eyeglasses featured variable-focus, fluid-filled lenses [80] made from deformable membranes, a reservoir to store the optical fluid and a mechanism to "actuate" or control the amount of fluid in the variable-focus lens, thereby changing its optical power. Improvements in the actuation mechanism of such variable-focus lenses have been suggested by multiple inventors and researchers [81]- [84]. J. D. Silver's pioneering work to provide low-cost, variable-focus eyeglasses [85] was taken further by Adlens Ltd. with their adaptive eyewear products [86]. In addition to liquid-filled lenses, such variable-focus lenses can be implemented utilizing other technologies such as the sliding Alvarez lenses [87], electro-wetting lenses [88] and liquid crystal (LC) lenses [89]- [92], each holding its own set of advantages and pitfalls [93], [94]. The fabrication of slim, lightweight variable-focus lenses with 30-50 mm apertures necessary for eyeglasses, however, still remains a challenge.
Smart Autofocusing Wearables: Recent advances in fluidfilled lenses [95]- [98] have enabled successful integration of large aperture variable-focus lenses into eyeglasses [98]- [101] integrated with object range sensors which measure the distance between the eyeglasses and the object a patient is trying to focus on. The range measured is utilized to automatically configure the power of the variable-focus lenses that corrects the accommodative deficiency [98], [99], as shown in the schematic of Fig. 2. Previously described unique designs [100], [101] feature binocular eye tracking systems to analyze the gaze of the wearer and vary the optical power of the variable-focus lens. Such systems require investments in expensive computational devices and smartphones. However, authors in these articles show through clinical trials, that variable-focus eyeglasses outperform the traditional methods of correcting presbyopia.
A successful implementation of variable-focus eyeglasses requires the autofocusing system to reliably tune correct optical powers with minimal delay, while at the same time, crucial factors such as the system's electrical power consumption, affordability and aesthetics cannot be ignored. This paper attempts to provide a solution to the drawbacks of the existing methods of alleviating presbyopia. We utilize the adaptive optics technology developed by Hasan et.al. [95] to implement autofocusing eyeglasses. We demonstrate the performance of this system in terms of its ability to provide automatic, adequate optical correction to a presbyopic eye at 5 different distances ranging from 1 m to 30 cm, as presbyopes usually find it difficult to focus on objects placed within 1m. We discuss the electrical performance of this system and its ability to operate without an external power source. We also demonstrate the optical performance of this system using a 1951 USAF optical resolution test chart and we compare the modulation transfer function (MTF) of our variable-focus lens to that of an average human eye.
II. MATERIALS AND METHODS
A. System Assembly
The housing for the eyeglasses was designed in Blender 2.79b, an open source, 3D creation platform. This design was 3D printed at a 3D printing service provider (Xometry) using a nylon-based polymer and selective laser sintering (SLS) process. Electronic PCBs were designed in the electronic design automation (EDA) software, Eagle 9.6.2. These designs were manufactured on low profile, two-layered PCBs. Surface mount technology (SMT) components were used while designing the PCBs to keep their height as small as possible. Flat pack connectors (FPC) and cables were used to facilitate communications between different PCBs. Software necessary for the proper functioning of the eyeglasses, including patientspecific algorithms were programmed into the microcontrollers with Arduino integrated development environment (IDE). Finally, the lenses and the PCBs were fixed in their respective housing with the means of micro screws.
B. Optical Measurements
The performance of the optical components and subsystems of the set were measured using the following methods.
Accommodation vs Object Distance: The performance of the variable-focus lenses, being the most critical subsystem within the autofocusing eyeglasses, were evaluated for their optical response with respect to different actuator voltages. The optical power of each lens was measured using a Shack-Hartman wavefront sensor (Thorlabs WFS150-7AR) while varying the bimorph actuator voltage from -200V to +200V. This process was repeated 5 times.
To evaluate the system response (i.e., lens optical response) in terms of the accommodation stimulus (reciprocal of the object distance), an object was placed at 5 different distances (1 m, 70 cm, 50 cm, 40 cm, 30 cm) in front of the autofocusing system, as shown in Fig. 3. The ToF distance sensor and control electronics were attached to the same lens-holder which held the variable-focus lens. This allowed the distance sensor to gather real time distance data of an object placed directly in its line of sight and communicate this distance to the control electronics which set the corrective optical powers on the variable-focus lens. The relationship between the object distance and the lens corrective power was determined using patient-specific empirical accommodation deficiency models [102] programmed into the control electronics. Patient data necessary to test these eyeglasses were obtained from a recent, unpublished clinical trial study undertaken by authors of this article [102], [103] We used a 633 nm light source and the Shack-Hartman wavefront sensor to measure the various optical powers generated by this system at various object distances, as shown in the schematic of Fig. 3. The distance sensor and the control electronics were attached to the variable-focus lens holder. The object was moved towards or away from the distance sensor while the autofocusing system automatically changed the optical power of the lens in response to the object distance. The minimum and maximum object distances were 30 cm and 1 m, respectively.
Optical Resolution: To evaluate the approximate optical resolution of the variable-focus lens undergoing automatic focus correction, we repeated the experiment described above while utilizing a 1951 USAF optical resolution test chart. This test chart was placed at the 5 different distances from the eyeglasses, mentioned above, and a DSLR camera (Canon EOS 1200D) was used to obtain raw images of the resolution test chart through the tuned variable-focus lenses. Test chart illumination was kept constant using a diffused, LED studio lighting system. The correlated color temperature of the lighting system was kept constant at 5000 K. The DSLR camera's ISO, aperture and shutter speed were fixed at 1600, f/22 and 0.5 s, respectively. Fig. 4 shows a schematic of the setup used for this experiment. The distance sensor and control electronics were attached to the variable-focus lens holder. The board was moved towards and away from the distance sensor, with minimum and maximum board-distance sensor distances being 30 cm and 1 m, respectively. The autofocusing system automatically tuned the variable-focus lens in response to the board distance.
Modulation Transfer Function (MTF): The MTF of an optical system is its intensity response as a function of spatial frequency [104] and a measure of the system optical quality. We estimated the 1-D MTF along the X-axis and Y-axis of the variable-focus lens used in the autofocusing eyeglasses directly from the aberration Zernike polynomial coefficients up to vertical quadrafoil (OSA/ANSI index Z14) [105], [106], which were measured with the help of the Shack-Hartman sensor. These estimated MTFs were then compared to those of an average human eye with different pupil diameters.
δ = 0 δ = x δ = -x
III. RESULTS AND DISCUSSION
A. System Design
It is important to note that as a person ages, their accommodative ability is reduced, but not completely lost [107], [108]. Hence, remnant accommodation plays a crucial role when a presbyope is trying to achieve clear focus.
In order to restore normal vision, the variable-focus lens must reproduce the accommodation deficiency (AD) curve [102] as shown in Fig. 5, which is the difference between the expected ideal response of the accommodative system (clear focus) and the degraded accommodation response (AR) of the presbyopic eye.
The AD must be provided by the accommodative optics in order to restore normal vision. Therefore, in practical terms, the patient's accommodation deficiency data should be programmed into an individual's autofocusing eyeglasses. Such data are a function of the accommodation stimulus (AS), which is the reciprocal of the distance between the eyes and the object plane where clear focus is expected.
In order to compensate for the AD, the autofocusing eyeglasses thus require a distance or a depth sensor that measures the range between the observer and the object being imaged. The autofocusing eyeglasses discussed in this paper thus consists of four key parts: (1) variable-focus lenses, (2) distance/ depth sensor, (3) control electronics, and (4) patientspecific AD models. Each of these subsystems is discussed below.
1) Patient-Specific AD Models
The AD curves vary quite a bit between presbyopes and, while fundamentally one can perform a full measurement of the AD and program it into the autofocusing eyeglasses, the cost of such a test can be prohibitive and a barrier to the use of the autofocusing system. Therefore, we chose to use model functions for the AD that require just a few parameters (and measurements), to describe the AD more economically. Multiple accommodation response models, which express the accommodation response of an ocular system in terms of the object distance, have been suggested so far [102], [109].
We use a newly developed model described in [102], which relates the accommodation response to the object distance using a sigmoid function:
, , , 1(1)
where AR is the accommodation response, AS is the accommodation stimulus (reciprocal of the object distance), a determines the accommodative amplitude of the individual i.e., the useful linear range of the experimental accommodation response curve, k determines range of stimuli for which the AR curve exhibits a linear response and parameter b corresponds to the midpoint of the non-flat accommodation stimulus range. It should be noted that the parameters a, b and k are patientspecific and vary from eye to eye. This model allows us to predict how a patient's accommodation system behaves as an object is moved within their visual field. The solid curve of Fig. 5 shows a fit of the model to an experimental AD data. Accommodation can then be restored in patients by simply adding the deficiency in their accommodation system's behavior with the help of the distance data from the distance sensor and properly tuning lenses thereafter. Fig. 5. Example presbyope accommodation deficiency versus accommodative stimulus curve. Accommodation stimulus is the reciprocal of the object distance from the eyes, while the accommodation deficiency is the defocus error present in the human ocular system. The solid line represents a typical AD model, developed in a recent study [102].
2) Variable-focus lenses
We utilize two squeezable-type liquid-filled lenses which do not require external fluid reservoirs. This significantly reduces the weight and profile of the variable-focus lens subsystem. Each lens consists of a cylindrical liquid-filled cavity, bound by two flexible polydimethylsiloxane (PDMS) membranes. Additional details of this lens, including its structure and electro-optical performance, have been described in [95]. 6 (a) and (b) show the working principle and a simplified cross-section of this variable-focus lens without the actuators, respectively. Three curved piezoelectric bimorphs, placed along the periphery of the lens, form the actuators. One end of these bimorphs is fixed to the lens rim, while the other is connected to a hollow piston with three extended arms. The piston is attached to one of the deformable membranes. When voltage is applied to the actuators, they move the piston into or out of the lens, depending on the applied voltage amplitude and polarity. This makes a concave or a convex profile on the other membrane, changing the optical power of the lens. Fig. 7 (a) shows a photograph of a 30 mm aperture diameter tunable liquid lens. The lens weighs less than 15 g, making them suitable for eyeglasses applications, much lighter than other piezoelectric liquid lenses (eg. Optotune EL-16-40-TC [110]). Fig. 7 (b) shows the optical response of the lens as a function of the applied voltage across the bimorph actuator. This lens features a repetitive linear optical response to the applied voltage, which greatly simplifies the electronics system required to control the lens. The response time of this lens is ~40 ms with a variable-focus range of ~4.3 D.
3) Object Distance Sensor and Range Cutoff
The role of a distance sensor in this system is crucial and requires precise measurement of the distance between the eyeglasses and the object to be focused. Distance sensors are characterized by their resolution, refresh rates (time taken per measurement) and the maximum range of reliable operation. IR (infrared) time-of-flight (ToF) distance sensors (eg. RF Digital Simblee RFD77402) provide very fine resolution measurements, with rapid refresh rates and a reasonable working range of up to 2 m, corresponding to an accommodation stimulus of 0.5 D. The autofocusing system regards objects located at distances > 2 m as being reasonably well focused at the 0.5 D cut off. This is a reasonable first order approximation as the average RMS aberration of the human eye [111]- [115] translates to an equivalent defocus error of 0.05 D-0.49 D; thus lens adjustments smaller than about 0.5 D are marginally noticed. Fig. 8 shows a block diagram of the feedback control electronics. As evident from Fig. 7 (b), the variable-focus lenses used within this system require a variable voltage between +200 V and -200 V. Efficient high-voltage DC-DC converters are necessary to convert the on-board battery voltage of 3.3 V to +200 V. We use a high voltage, ultra-miniature DC-DC piezo transformer for this operation. The bimorph actuators of each lens are connected to the high voltage converter in a H-bridge configuration with the help of four high-voltage optoelectronic switches (Ixys LAA100P). These optoelectronic switches help in setting the proper polarity of the voltage across the lenses. The high-voltage converter is operated within a feedback loop, such that it does not need to be switched on all the time, which increases the electrical power efficiency of this system.
4) High-Voltage Circuits and Feedback Control Electronics
We use two microcontrollers in a master-slave configuration to control the autofocusing eyeglasses. The master microcontroller continuously fetches distance data from the ToF distance sensor, through an I 2 C bus and calculates the optical powers for the two lenses, using the distance data and the patient-specific correction algorithm. These two optical powers are then mapped to the corresponding voltage values, using the optical power-actuation voltage relation of the variable-focus lens, shown in Fig. 7 (b). The master microcontroller sends these two voltage values to the slave microcontroller through a standard UART bus, operating at 19,200 baud. The slave microcontroller is responsible for continuously converting the voltage data to pulse-widthmodulated (PWM) control signals, which digitally control the analog output of the ultra-miniature DC-DC piezo transformer. A resistor-divider feedback network provides real-time measurements of the lens voltage, allowing our system to be precisely and power-efficiently controlled.
Prescriptions of patients are bound to change as time and presbyopia progress within their ocular systems. It has also been shown that visual acuity and accommodation response are dependent on illumination [102], [116]- [118]. We incorporated a Bluetooth Low Energy (BLE) module within our system, which works in conjunction with a smartphone application. This allows the patients to easily enter their updated prescriptions and user settings into the eyeglasses' control electronics. This BLE subsystem is connected to the master microcontroller through its principal hardware SPI bus, enabling over-the-air (OTA) upgrades for the entire systemcontrolling software, when required.
B. Power Consumption
In practice, bulky power packs or batteries to reliably power sophisticated presbyopia correction systems for long hours, are unfeasible, inconvenient, and unacceptable to users. Presbyopes require continuous accommodation correction and such technologies must stay operational, preferably without charging, to allow the users unimpeded functionality. We specifically utilize low power microcontrollers (Microchip ATmega32U4) and BLE SoC (Nordic nRF51822) in our design to facilitate better power efficiency. Key systems in the master microcontroller which are not used, like the analog-digital converter (ADC) and a few timers have been disabled to conserve power. The microcontrollers run with a clock frequency of 8 MHz to save power, without affecting the computation speed. The entire system is operated with a 3.7 V, 400 mAh, rechargeable lithium-ion polymer battery. We have also incorporated a battery charger (Microchip MCP73831) and a fuel gauge (Texas Instruments bq27441) which measures the remaining battery capacity and communicates battery status with the master microcontroller through a high speed I 2 C bus. Power is routed to the entire electronics system through a 3.3 V, low-noise, low dropout voltage regulator (MaxLinear SPX3819). A low power on/off switch (Maxim Integrated MAX16054) in conjunction with a push button constitute the power switching subsystem. We measured the current consumption of our system at 3.3 V with a 6 ½ digit multimeter (Agilent 34401A), to be ~21 mA. This value increased to 22 mA, during BLE communications with a smartphone application. However, software and prescription upgrades, which require BLE functionality, are rare and hence, the average steady-state current consumption of 21 mA can be safely approximated for this system. With the 400 mAh capacity lithium-ion polymer battery we use in our design, this ideally yields an average operational time of ~19 hours between every charge cycle, when the system is configured to perform 1 correction per second. However, battery age and frequency of use will determine the practical limits of this operational time.
C. Accommodation Correction and Image Quality
The autofocusing system's ability to provide adequate optical correction in presbyopes was evaluated by comparing their AD before and after optical correction with our system. Fig. 9 (a) shows the deficiency models (solid lines) and deficiency data (markers) for 5 different eyes of uncorrected presbyopes, presented in [102]. The increase in their AD corresponding to increasing stimulus (i.e., decreasing object distance) can be clearly seen from the deficiency curves in Fig. 9 (a). The AD increases from ~0 D to >3.0 D as the stimulus is progressively increased from 1.0 D (object distance = 1.0 m) to 3.3 D (object distance = 30 cm). Ideally, optical corrections using autofocusing eyeglasses should reduce this stimulus-dependent deficiency in presbyopes to 0 D. Fig. 9 (b) shows the deficiency in the same 5 presbyopia patients after correction using our autofocusing system. The curves represent difference in the optical correction determined by the patient-specific AD models and our system's response measured at 5 object distances (1 m, 70 cm, 50 cm, 40 cm and 30 cm). Our results indicate that the average corrected AD for the 5 patients ranged from -0.021 D to 0.016 D over the 5 object distances, using our autofocusing system. The maximum corrected deficiency in presbyopes did not exceed 0.4 D. The system was able to repeatedly generate the optical powers as determined by the presbyopia correction algorithms for different patients.
Optical resolution plays an important part in determining the effectiveness of autofocusing systems to be used for presbyopia correction. To that extent, we also measured the approximate, subjective optical resolution of this system with the help of a 1951 USAF optical resolution test chart. Fig. 10 (a)-(c) show the images of the resolution test chart placed at different distances away from variable-focus lens. The autofocusing system was running a patient-specific algorithm, developed in [102], and automatically set the appropriate optical powers (necessary to correct that patient's AD) on the variable-focus lens. We observed the smallest, clearly visible elements on the test chart. Our observations show that the resolution of this system was between 0.75 -1.00 lp/mm.
An autofocusing system intended to provide optical correction to a presbyope should also have optical quality comparable to that of an average human eye. We measured the 1-D vertical and horizontal MTFs for the variable-focus lens and compared it to that of a best-corrected human eye. An analytical formula constructed by Watson [119] was used to model the pupil-diameter dependent MTFs for a human eye. Fig. 11 compares the 1-D MTFs, measured for our lens to those modelled for a best-corrected human eye. We report an average MTF-50 (i.e., spatial frequency at 50% MTF) value of 10.5 cycles/degree for our variable-focus lens.
The green shaded area represents the range of the MTFs for a best corrected human eye.
D. Response Time
The electronics control system and the lenses introduce inherent time delays in the autofocusing system. As reported before, the lenses have a response time of ~40 ms, i.e., it takes cm. The accommodation deficiency model for patient no. 3 was used in conjunction with the autofocusing system for these images [102]. The photos highlight the smallest, clearly visible element on the optical resolution test chart. ~40 ms from the moment a particular voltage is applied to the actuators of the lenses till the point where the lenses produce the expected optical power.
Our measurements show that the average time taken by the electronics subsystem to produce the required voltages for the lenses is ~67 ms. This response time takes into account the time required by the distance sensor to complete its ranging operation, the time required by the master microcontroller to extrapolate the correct optical powers and their related voltages, and the time required by the slave microcontroller to set these voltages across the lens actuators. Hence, the total time taken by this system from the moment distance ranging begins, till the point where the lenses produce the appropriate optical power, is 107 ms. This enables us to have a maximum refresh rate of ~10 Hz, i.e., the system can change the optical powers of the lenses up to 10 times a second.
E. Ergonomics, Aesthetics and Portability
Autofocusing systems are meant to be worn all the time, throughout the daily routine of a presbyope, in order to provide seamless accommodation. In practical terms, this requires the system to be lightweight, comfortable and fashionable enough to be worn during daily engagements. Engineering limitations impose strict restrictions on the sizes of various components within our system, for example, the lenses cannot be thinned down any further, without sacrificing their electro optical performance. Similarly, the housing for electronics cannot be any thinner without sacrificing battery capacity. Despite such limitations, our system attempts to provide a frame design which is much sleeker compared to recent similar technologies [98], [100], [101]. Fig 12 (a)-(d) show the mechanical dimensions of the autofocusing eyeglasses. Fig. 13 (a) shows the arrangement of the internal components in these eyeglasses. The maximum thickness of the frame is 14.46 mm, while the maximum height of the electronics housing is 28.00 mm. The diameter of the lens housing is 51 mm to house lenses with diameter 50 mm. The 0.5 mm annular gap between the lens and its housing helps in snugly fitting FPC communications cables and the HV wires for the lenses. The total weight of these autofocusing eyeglasses is 132.06 g. Our system has been designed to automatically focus on the object which is directly in the line-of-sight of the ToF distance sensor. Some limitations exist for this autofocusing system. The eyeglasses do not change their optical power if the user changes their gaze direction, without moving their head. This can often lead to "jumps" in focus when the object distance changes drastically, in a short amount of time. However, such issues can be addressed with a low-profile, low-power digital oculometer designed for this system [120]. It should be noted that such upgrades will consume more power, leading to a faster battery discharge.
IV. CONCLUSION
We demonstrated the design, assembly, and performance of an autofocusing eyeglasses system which can potentially restore accommodation in presbyopes. These autofocusing eyeglasses consist of two, variable-focus, liquid-filled lenses, a ToF distance sensor and a rechargeable battery-powered electronics control system, which utilizes patient-specific accommodation deficiency models to restore pre-presbyopic levels of accommodation in them. This system was evaluated on its ability to perform high fidelity and accurate optical corrections based on presbyopic patient data from a clinical trial. The optical resolution (MTF-50) of our autofocusing eyeglasses is 10.5 cycles/degree. Our system can provide up to 4.3 D in accommodation correction.
Fig. 2 .
2Schematic of a smart eyeglasses system. The adaptive lenses continuously adjust the lens power to bring the object in focus for the observer along the line of sight. The lenses must provide the accommodation deficiency, a function of the object distance.
Fig. 3 .
3Schematic of the experimental setup used to measure the power -distance relation of the variable-focus lens.
Fig. 4 .
4Schematic of the experimental setup to measure the optical resolution of the variable-focus lens. The 1951 USAF resolution test chart was mounted on a board.
Fig. 6 .
6(a) Working principle of liquid-filled variable-focus lens. (b) Simplified schematic of the variable-focus lens without the piezoelectric bimorph actuators[95].
Fig.
Fig. 6 (a) and (b) show the working principle and a simplified cross-section of this variable-focus lens without the actuators, respectively. Three curved piezoelectric bimorphs, placed along the periphery of the lens, form the actuators. One end of these bimorphs is fixed to the lens rim, while the other is connected to a hollow piston with three extended arms. The piston is attached to one of the deformable membranes. When voltage is applied to the actuators, they move the piston into or out of the lens, depending on the applied voltage amplitude and
Fig. 7 .
7(a) Photograph of the variable-focus lens[95]. (b) Optical response of the variable-focus lens as a function of bimorph actuator voltage. Bars represent the standard deviation. The optical power of the lens is a linear function of the applied voltage.
Fig. 8 .
8Block diagram of the entire autofocusing system.
Fig. 9 .
9(a) Accommodation deficiencies of 5 patients before correction, plotted using models developed in[102]. (b) Average corrected accommodation deficiencies of the same 5 patients after our autofocusing system correction. Bars represent the standard deviation. Corrected AD does not exceed 0.2 D.
Fig. 10 .
10Photos of the USAF optical resolution test chart through the variable-focus lens, placed at (a) 30 cm, (b) 40 cm and (c) 50
Fig. 11 .
11Comparison of the 1-D MTF of variable-focus lens used in the autofocusing system and the radial MTF of the human eye. The solid red and blue lines represent the 1-D MTFs of the variable focus lens along the X (horizontal) and Y (vertical) axes, respectively.
Fig. 13 .
13(a) Autofocusing eyeglasses from the inside. This photo shows the various subsystems and their placements within the eyeglasses. (b) Ergonomic fit of autofocused eyeglasses resembles oversized sunglasses. The reflective covers hide the variable-focus lenses.
Fig. 12 .
12Dimensions of the lens and electronics housing in the autofocusing eyeglasses system. (a) and (b) show the top and front views of the lens and ToF distance sensor housing, respectively, while (c) and (d) show the side and front views of the electronics housing, respectively. Dimensions are in mm.
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|
[] |
[
"BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation",
"BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation",
"BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation",
"BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation"
] |
[
"Mingguo He [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n",
"Zhewei Wei [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n",
"Zengfeng Huang [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n",
"Hongteng Xu [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n",
"Mingguo He [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n",
"Zhewei Wei [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n",
"Zengfeng Huang [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n",
"Hongteng Xu [email protected] \nRenmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n\n"
] |
[
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n",
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n",
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n",
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n",
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n",
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n",
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n",
"Renmin University of China\nRenmin University of China\nFudan University\nRenmin University of China\n"
] |
[] |
Many representative graph neural networks, e.g., GPR-GNN and ChebNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose BernNet, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-K Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks. Code is available at https://github.com/ivam-he/BernNet.
| null |
[
"https://arxiv.org/pdf/2106.10994v3.pdf"
] | 235,490,093 |
2106.10994
|
bbed89eee0a43baf17dd5eab8354deecadda8acf
|
BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation
Mingguo He [email protected]
Renmin University of China
Renmin University of China
Fudan University
Renmin University of China
Zhewei Wei [email protected]
Renmin University of China
Renmin University of China
Fudan University
Renmin University of China
Zengfeng Huang [email protected]
Renmin University of China
Renmin University of China
Fudan University
Renmin University of China
Hongteng Xu [email protected]
Renmin University of China
Renmin University of China
Fudan University
Renmin University of China
BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation
Many representative graph neural networks, e.g., GPR-GNN and ChebNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose BernNet, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-K Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks. Code is available at https://github.com/ivam-he/BernNet.
Introduction
Graph neural networks (GNNs) have received extensive attention from researchers due to their excellent performance on various graph learning tasks such as social analysis [24,17,29], drug discovery [12,25], traffic forecasting [18,3,6], recommendation system [38,32] and computer vision [39,4]. Recent studies suggest that many popular GNNs operate as polynomial graph spectral filters [7,13,5,16,2,35]. Specifically, we denote an undirected graph with node set V and edge set E as G = (V, E), whose adjacency matrix is A. Given a signal x = [x] ∈ R n on the graph, where n = |V | is the number of nodes, we can formulate its graph spectral filtering operation as K k=0 w k L k x, w k 's are the filter weights, L = I − D −1/2 AD −1/2 is the symmetric normalized Laplacian matrix of G, and D is the diagonal degree matrix of A. Another equivalent polynomial filtering operation is K k=0 c k P k x, where P = D −1/2 AD −1/2 is the normalized adjacency matrix and c k 's are the filter weights.
BernNet < l a t e x i t s h a 1 _ b a s e 6 4 = " a r O / D a 0 q o g P 8 u Z a y 7 t v m / c / t s w A = " > A A A C b n i c d V H L S g M x F M 2 M 7 / q q C i 4 s Y r A I F a Q k Y 5 + 7 o h u X C r Y V 2 q F k 0 r Q N z T x I M k I Z Z u k P u v M b 3 P g J p i / Q o h c C h 3 P P P f c R L x J c a Y Q + L H t t f W N z a 3 s n s 7 u 3 f 3 C Y P T p u q T C W l D V p K E L 5 4 h H F B A 9 Y U 3 M t 2 E s k G f E 9 w d r e + H 6 a b 7 8 y q X g Y P O t J x F y f D A M + 4 J R o Q / W y b 0 l 3 Z t K R Q 8 9 N U B E 5 t X L J u U F F p 4 z q u G 5 A G e F 6 p Z S O C u m K t F I r V 3 H N K N A s D M A Y 3 e J S 2 h W m f 5 + s 6 v + 1 v k 5 7 2 f z S D y 7 9 4 N I P 4 g W T B 4 t 4 7 G X f u / 2 Q x j 4 L N B V E q Q 5 G k X Y T I j W n g q W Z b q x Y R O i Y D F n H w I D 4 T L n J b J w U X h m m D w e h N C / Q c M b + r E i I r 9 T E 9 4 z S J 3 q k V n N T 8 q 9 c J 9 a D m p v w I I o 1 C + i 8 0 S A W U I d w e n v Y 5 k S E W m C g d Z k G H s P g p + p + 0 K m X L L l d u q q X 6 Z R Z H H g 7 g E I 7 B g n O o w z U 0 o A k E 7 u E R n u H F e D C e j F f j b W 7 N G V n P P v y A 8 f E F l U K c 7 Q = = < / l a t e x i t > Eigenvalue Bernstein Basis Filter Graph Spectral Filtering < l a t e x i t s h a 1 _ b a s e 6 4 = " F Y k 0 B u + J T Y 7 D q L q D G J 0 l 5 E P 9 K 5 Y = " > A A A C K H i c b V D L S g M x F M 3 4 r P V V d e l m s A g u p M w U U T d i 0 Y 3 g p o J 9 w M w 4 Z N K 0 D Z N 5 k N w R S p j P c e O v u B F R p F u / x P Q h a O u B w O G c c 5 O b E 6 S c S b C s o b G w u L S 8 s l p Y K 6 5 v b G 5 t l 3 Z 2 m z L J B K E N k v B E t A M s K W c x b Q A D T t u p o D g K O G 0 F 4 f X I b z 1 S I V k S 3 8 M g p V 6 E e z H r M o J B S 3 7 p U r n j S x z R C z x l V a w x j u d I 7 i o X + h S w H 7 q 5 r 8 I L O 3 9 Q t 3 n u l 8 o / E X O e 2 F N S R l P U / d K b 2 0 l I F t E Y C M d S O r a V g q e w A E Y 4 z Y t u J m m K S Y h 7 1 N E 0 x h G V n h r v m J u H W u m Y 3 U T o E 4 M 5 V n 9 P K B x J O Y g C n Y w w 9 O W s N x L / 8 5 w M u u e e Y n G a A Y 3 J 5 K F u x k 1 I z F F r Z o c J S o A P N M F E M L 2 r S f p Y Y A K 6 2 6 I u w Z 7 9 8 j x p V i v 2 a a V 6 d 1 K u X U 3 r K K B 9 d I C O k I 3 O U A 3 d o D p q I I K e 0 A t 6 R x / G s / F q f B r D S X T B m M 7 s o T 8 w v r 4 B M p q j F Q = = < / l a t e x i t >
J R L S Y G E C q m R X S E Z G E a v N D G X O E a b w f 9 B y i r h S d J K + c b d 4 h z b I A c u Q Q F g U A U N 8 A A e Q R N Q 8 G k d W W d W z v q y T + 1 z + 2 I u t a 1 F z Q n 4 F X b h G 6 H 1 s N s = < / l a t e x i t
{✓ k } K k=0 Model Parameter < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 V V F g Q a N 0 x M K I j X p n b u f 7 M V d E q M = " > A A A C I n i c b V B J S 8 N A G J 3 U r d Y t 6 t F L s A g V p C R F X G 5 F L 4 K X C n a B J I b J Z N o O n U z C z E Q o I b / F i 3 / F i w d F P Q n + G K d p B G 1 9 M P B 4 7 9 v m + T E l Q p r m p 1 Z a W F x a X i m v V t b W N z a 3 9 O 2 d j o g S j n A b R T T i P R 8 K T A n D b U k k x b 2
Y Y x j 6 F H f 9 0 e X E 7 9 5 j L k j E b u U 4 x m 4 I B 4 z 0 C Y J S S Z 5 + n j r 5 E J s P f D c 1 6 2 a O o z m S + Z 5 5 l 1 5 n N Y e q 4 Q E 8 z D y 9 + m M a 8 8 Q q S B U U a H n 6 u x N E K A k x k 4 h C I W z L j K W b Q i 4 J o j i r O I n A M U Q j O M C 2 o g y G W L h p f l 1 m H C g l M P o R V 4 9 J I 1 d / d 6 Q w F G I c + q o y h H I o Z r 2 J + J 9 n J 7 J / 5 q a E x Y n E D E 0 X 9 R N q y M i Y 5 G U E h G M k 6 V g R i D h R t x p o C D l E U q V a U S F Y s 1 + e J 5 1 G 3 T q p N 2 6 O q 8 2 L I o 4 y 2 A P 7 o A Y s c A q a 4 A q 0 Q B s g 8 A C e w A t 4 1 R 6 1 Z + 1 N + 5 i W l r S i Z x f 8 g f b 1 D S P U n 7 w = < / l a t e x i t > b K 0 ( ) < l a t e x i t s h a 1 _ b a s e 6 4 = " P b a a 2 z e P L A Z c f m b s 8 j A O C 9 l D F 7 Q = "
> A A A C I n i c b V B J S 8 N A G J 3 U r d Y t 6 t F L s A g V p C R F X G 5 F L 4 K X C n a B J I b J Z N o O n U z C z E Q o I b / F i 3 / F i w d F P Q n + G K d p B G 1 9 M P B 4 7 9 v m + T E l Q p r m p 1 Z a W F x a X i m v V t b W N z a 3 9 O 2 d j o g S j n A b R T T i P R 8 K T A n D b U k k x b 2 Y Y x j 6 F/ 5 q a E x Y n E D E 0 X 9 R N q y M i Y 5 G U E h G M k 6 V g R i D h R t x p o C D l E U q V a U S F Y s 1 + e J 5 1 G 3 T q p N 2 6 O q 8 2 L I o 4 y 2 A P 7 o A Y s c A q a 4 A q 0 Q B s g 8 A C e w A t 4 1 R 6 1 Z + 1 N + 5 i W l r S i Z x f 8 g f b 1 D S V m n 7 0 = < / l a t e x i t > b K 1 ( ) < l a t e x i t s h a 1 _ b a s e 6 4 = " x T / z d L k p Q 2 g X Z H u V + O 2 h a i v w L z 4 = " > A A A C J n i c b V D L S s N A F J 3 U V 6 2 v q E s 3 w S J U 0 J I U U T d C 0 Y 3 Q T Q X 7 g C S G y W T S D p 0 8 m J k I J e R r 3 P g r b l x U R N z 5 K U 7 T C N p 6 Y O B w 7 r m P O W 5 M C R e 6 / q m U l p Z X V t f K 6 5 W N z a 3 t H X V 3 r 8 u j h C H c Q R G N W N + F H F M S 4 o 4 g g u J + z D A M X I p 7 7 u h m W u 8 9 Y s Z J F N 6 L c Y z t A A 5 C 4 h M E h Z Q c 9 S q 1 8 i E m G 7 h 2 q t f 1 H C c L J H O d t H V q Z A 9 p K 6 t Z V C 7 w 4 H H m q N U f g 7 Z I j I J U Q Y G 2 o 0 4 s L 0 J J g E O B K O T c N P R Y 2 C l k g i C K s 4 q V c B x D N I I D b E o a w g B z O 8 0 v z L Q j q X i a H z H 5 Q q H l 6 u + O F A a c j w N X O g M o h n y + N h X / q 5 m J 8 C / t l I R x I n C I Z o v 8 h G o i 0 q a Z a R 5 h G A k 6 l g Q i R u S t G h p C B p G Q y V Z k C M b 8 l x d J t 1 E 3 z u u N u 7 N q 8 7 q I o w w O w C G o A Q N c g C a 4 B W 3 Q A Q g 8 g R c w A W / K s / K q v C s f M 2 t J K X r 2 w R 8 o X 9 8 u c 6 F V < / l a t e x i t > b K K 1 ( ) < l a t e x i t s h a 1 _ b a s e 6 4 = " k t Q b q a o o 3 5 C i 0 T 5 c J 9 i I T Q t W R o 4 = " > A A A C I n i c b V B J S 8 N A G J 2 4 1 r p F P X o Z L E I F K U k R l 1 v R i 9 B L B b t A E s N k M m 2 H T h Z m J k I J + S 1 e / C t e P C j q S f D H O E 0 j a O u D g c d 7 3 z b P i x k V 0 j A + t Y X F p e W V 1
d J a e X 1 j c 2 t b 3 9 n t i C j h m L R x x C L e 8 5 A g j I a k L a l k p B d z g g K P k a 4 3 u p r Figure 1: An illustration of the proposed BernNet.
4 3 X v C B Y 3 C W z m O i R O g Q U j 7 F C O p J F e / S O 1 8 i M U H n p M a N S P H 8 R z J P L d 5 l z a z q s 3 U c B 8 d Z a 5 e + T H h P D E L U g E F W q 7 + b v s R T g I S S s y Q E J Z p x N J J E Z c U M 5 K V 7 U S Q G O E R G h B L 0 R A F R D h p f l 0 G D 5 X i w 3 7 E 1 Q s l z N X f H S k K h B g H n q o M k B y K W W 8 i / u d Z i e y f O y k N 4 0 S S E E 8 X 9 R M G Z Q Q n e U G f c o I l G y u C M K f q V o i H i C M s VV L + U y T g x K N v 8 o S A Q x E c n O J w O u k B k x s Y Q y x W 1 W w k Z U U W Z s S U V b g r d 4 8 j J p V i v e R a V 6 d 1 6 u X e d 1 F O A Y T u A M P L i E G t x C H R r A Q M I z v M K b o 5 0 X 5 9 3 5 m I + u O P n O E f y B 8 / k D / 4 O R I w = = < / l a t e x i t > x Laplacian Matrix Signal
We can broadly categorize the GNNs applying the above filtering operation into two classes, depending on whether they design the filter weights or learn them based on observed graphs. Some representative models in these two classes are shown below.
• The GNNs driven by designing filters: GCN [13] uses a simplified first-order Chebyshev polynomial, which is proven to be a low-pass filter [1,31,34,41]. APPNP [14] utilizes Personalized PageRank (PPR) to set the filter weights and achieves a low-pass filter as well [15,41]. GNN-LF/HF [41] designs filter weights from the perspective of graph optimization functions, which can simulate high-and low-pass filters.
• The GNNs driven by learning filters: ChebNet [7] approximates the filtering operation with Chebyshev polynomials, and learns a filter via trainable weights of the Chebyshev basis. GPR-GNN [5] learns a polynomial filter by directly performing gradient descent on the filter weights, which can derive high-or low-pass filters. ARMA [2] learns a rational filter via the family of Auto-Regressive Moving Average filters [21].
Although the above GNNs achieve some encouraging results in various graph modeling tasks, they still suffer from two major drawbacks. Firstly, most existing methods focus on designing or learning simple filters (e.g., low-and/or high-pass filters), while real-world applications often require much more complex filters such as band-rejection and comb filters. To the best of our knowledge, none of the existing work supports designing arbitrary interpretable spectral filters. The GNNs driven by learning filters can learn arbitrary filters in theory, but they cannot intuitively show what filters they have learned. In other words, their interpretability is poor. For example, GPR-GNN [5] learns the filter weights w k 's but only proves a small subset of the learnt weight sequences corresponds to low-or high-pass filters. Secondly, the GNNs often design their filters empirically or learn the filter weights without any necessary constraints. As a result, their filter weights often have poor controllability. For example, GNN-LF/HF [41] designs its filters with a complex and non-intuitive polynomial with difficult-to-tune hyperparameters. The multi-layer GCN/SGC [13,31] leads to "ill-posed" filters (i.e., those deriving negative spectral responses). Additionally, the filters learned by GPR-GNN [5] or ChebNet [7] have a chance to be ill-posed as well.
To overcome the above issues, we propose a novel graph neural network called BernNet, which provides an effective algorithmic framework for designing and learning arbitrary graph spectral filters. As illustrated in Figure 1, for an arbitrary spectral filter h : [0, 2] → [0, 1] over the spectrum of the symmetric normalized Laplacian L, our BernNet approximates h by a K-order Bernstein polynomial approximation, i.e., h(λ)
= K k=0 θ k b K k (λ). The non-negative coefficients {θ k } K k=0
of the Bernstein basis {b K k (λ)} K k=0 work as the model parameter, which can be interpreted as h(2k/K), k = 0, . . . , K (i.e., the filter values uniformly sampled from [0, 2]). By designing or learning the θ k 's, we can obtain various spectral filters, whose filtering operation can be formulated as
K k=0 θ k 1 2 K K k (2I − L) K−k L k x, where
x is the graph signal. We further demonstrate the rationality of our BernNet from the perspective of graph optimization -any valid polynomial filers, i.e., those polynomial functions mapping [0, 2] to [0, 1], can always be expressed by our BernNet, and accordingly, the filters learned by our BernNet are always valid. Finally, we conduct experiments to demonstrate that 1) BernNet can learn arbitrary graph spectral filters (e.g., band-rejection, comb, low-band-pass, etc.), and 2) BernNet achieves superior performance on real-world datasets.
BernNet
Bernstein approximation of spectral filters
Given an arbitrary filter function h : [0, 2] → [0, 1], let L = UΛU T denote the eigendecomposition of the symmetric normalized Laplacian matrix L, where U is the matrix of eigenvectors and Λ = diag[λ 1 , ..., λ n ] is the diagonal matrix of eigenvalues. We use
h(L)x = Uh(Λ)U T x = Udiag[h(λ 1 ), ..., h(λ n )]U T x(1)
to denote a spectral filter on graph signal x. The key of our work is approximate h(L) (or, equivalently, h(λ)). For this purpose, we leverage the Bernstein basis and Bernstein polynomial approximation defined below.
p K (t) := K k=0 θ k · b K k (t) = K k=0 f k K · K k (1 − t) K−k t k .(2)
Here, for k = 0, ...,
K, b K k (t) = K k (1 − t) K−k t k is the k-th Bernstein base, and θ k = f ( k K ) is the function value at k/K, which works as the coefficient of b K k (t). Lemma 2.1 ( [10]). Given an arbitrary continuous function f (t) on t ∈ [0, 1], let p K (t) denote the Bernstein approximation of f (t) as defined in Equation (2). We have p K (t) → f (t) as K → ∞. For the filter function h : [0, 2] → [0, 1], we let t = λ 2 and f (t) = h(2t), so that the Bernstein polynomial approximation becomes applicable, where θ k = f (k/K) = h(2k/K) and b K k (t) = b K k ( λ 2 ) = K k (1 − λ 2 ) K−k ( λ 2 ) k for k = 1, ..., K. Consequently, we can approximate h(λ) by p K (λ/2) = K k=0 θ k K k (1 − λ 2 ) K−k λ 2 k = K k=0 θ k 1 2 K K k (2 − λ) K−k λ k , and Lemma 2.1 ensures that p K (λ/2) → h(λ) as K → ∞. Replacing {h(λ i )} n i=1 with {p K (λ i /2)} n i=1
, we approximate the spectral filter h(L) in Equation (1) as Udiag[p K (λ 1 /2), ..., p K (λ n /2)]U T and derive the proposed BernNet. In particular, given a graph signal x, the convolutional operator of our BernNet is defined as follows:
z = Udiag[p K (λ 1 /2), ..., p K (λ n /2)]U T BernNet x = K k=0 θ k 1 2 K K k (2I − L) K−k L k x(3)
where each coefficient θ k can be either set to h(2k/K) to approximate a predetermined filter h, or learnt from the graph structure and signal in an end-to-end fashion. As a natural extension of Lemma 2.1, our BernNet owns the following proposition.
= h(2k/K), k = 0, . . . , K, the z in Equation (3) satisfies z → h(L)x as K → ∞.
Proof. According to the above derivation, we have
p K (λ/2) = K k=0 θ k K k (1 − λ 2 ) K−k λ 2 k = K k=0 θ k 1 2 K K k (2 − λ) K−k λ k , and Lemma 2.1 ensures that p K (λ/2) → h(λ) as θ k = h(2k/K) and K → ∞. Consequently, we have z = Udiag[p K (λ 1 /2), ..., p K (λ n /2)]U T x → Udiag[h(λ 1 ), ..., h(λ n )]U T x = h(L)
as θ k = h(2k/K) and K → ∞.
Realizing existing filters with BernNet.
As shown in Proposition 2.1, our BernNet can approximate arbitrary continuous spectral filters with sufficient precision. Below we give some representative examples of how our BernNet exactly realizes existing filters that are commonly used in GNNs.
Filter types
Filter h(λ) θ k for k = 0, . . . , K Bernstein approximation p K ( λ 2 ) BernNet All-pass 1
θ k = 1 1 I Linear low-pass 1 − λ/2 θ k = 1 − k/K 1 − λ/2 I − 1 2 L Linear high-pass λ/2 θ k = k/K λ/2 1 2 L Impulse low-pass δ 0 (λ) θ 0 = 1 and other θ k = 0 (1 − λ/2) K 1 2 K (2I − L) K Impulse high-pass δ 2 (λ) θ K = 1 and other θ k = 0 (λ/2) K 1 2 K L K Impulse band-pass δ 1 (λ) θ K/2 = 1 and other θ k = 0 K K/2 (1 − λ/2) K/2 (λ/2) K/2 1 2 K K K/2 (2I − L) K/2 L K/2
• All-pass filter h(λ) = 1. We set θ k = 1 for k = 0, . . . , K, and the approximation p K ( λ 2 ) = 1 is exactly the same with h(λ). Accordingly, our BernNet becomes an identity matrix, which realizes the all-pass filter perfectly.
• Linear low-pass filter h(λ) = 1−λ/2. We set θ k = 1−k/K for k = 0, . . . , K and obtain
p K ( λ 2 ) = 1 − λ/2. The BernNet becomes K k=0 (K−k) K 1 2 K K k (2I − L) K−k L k = I − 1 2 L,
which achieves the linear low-pass filter exactly. Note that I − 1 2 L = 1 2 (I + P) is also the same as the graph convolutional network (GCN) before renormalization [13].
• Linear high-pass filter h(λ) = λ/2. Similarly, we can set θ k = k/K for k = 0, . . . , K to get a perfect approximation p K ( λ 2 ) = λ 2 , and the BernNet becomes 1 2 L.
Note that even for those non-continuous spectral filters, e.g., the impulse low/high/band-pass filters, our BernNet can also provide good approximations (with sufficient large K).
• Impulse low-pass filter h(λ) = δ 0 (λ). † We set θ 0 = 1 and θ k = 0 for k = 0, and p K ( λ 2 ) = (1 − λ 2 ) K . Accordingly, the BernNet becomes 1 2 K (2I − L) K , deriving an K-layer linear low-pass filter.
• Impulse high-pass filter h(λ) = δ 2 (λ). We set θ K = 1 and θ k = 0 for k = K, and p K ( λ 2 ) = ( λ 2 ) K . The BernNet becomes 1 2 K L K , i.e., an K-layer linear high-pass filter. • Impulse band-pass filter h(λ) = δ 1 (λ). Similarly, we set θ K/2 = 1 and θ k = 0 for k = K/2, and p K ( λ 2 ) = K K/2 (1 − λ/2) K/2 (λ/2) K/2 . The BernNet becomes 1 2 K K K/2 (2I − L) K/2 L K/2 , which can be explained as stacking a K/2-layer linear low-pass filter and a K/2-layer linear high-pass filter. Obviously, K should be an even number in this case. Table 1 summarizes the design of the BernNet for the filters above. We can find that an appealing advantage of our BernNet is that its coefficients are highly correlated with the spectral property of the target filter. In particular, we can determine to pass or reject the spectral signal with λ ≈ 2k K by using a large or small θ k because each Bernstein base b K k (λ) corresponds to a "bump" located at 2k K . This property provides useful guidance when designing filters, which enhances the interpretability of our BernNet.
Learning complex filters with BernNet
Besides designing the above typical filters, our BernNet can express more complex filters, such as band-pass, band-rejection, comb, low-band-pass filters, etc. Moreover, given the graph signals before and after applying such filters (i.e., the x's and the corresponding z's), our BernNet can learn their approximations in an end-to-end manner. Specifically, given the pairs {x, z}, we learn the coefficients {θ k } K k=0 of the BernNet by gradient descent. More implementation details can be found at the experimental section below. Figure 2 illustrates the four complex filters and the approximations we learned (The low-band pass filter is h(λ) = I [0,0.5] (λ) + exp (−100(λ − 0.5) 2 )I (0.5,1) (λ) + exp (−50(λ − 1.5) 2 )I [1,2] (λ), where I Ω (λ) = 1 when λ ∈ Ω, otherwise I Ω (λ) = 0). In general, our BernNet can learn a smoothed approximation of these complex filters, and the approximation precision improves with the increase of the order K. Note that although the BernNet cannot pinpoint the exact peaks of the comb filter or drop to 0 for the valleys of comb or low-band-pass filters due to the limitation of K, it still significantly outperforms other GNNs for learning such complex filters. † The impulse function δx(λ) = 1 if λ = x, otherwise δx(λ) = 0
BernNet in the Lens of Graph Optimization
In this section, we motivate BernNet from the perspective of graph optimization. In particular, we show that any polynomial filter that attempts to approximate a valid filter has to take the form of BernNet.
A generalized graph optimization problem
Given a n-dimensional graph signal x, we consider a generalized graph optimization problem
min z f (z) = (1 − α)z T γ(L)z + α z − x 2 2(4)
where α ∈ [0, 1) is a trade-off parameter, z ∈ R n denotes the propagated representation of the input graph signal x, and γ(L) denotes an energy function of L, determining the rate of propagation [28]. Generally, γ(·) operates on the spectral of L, and we have γ(L) = Udiag[γ(λ 1 ), ..., γ(λ n )]U T .
We can model the polynomial filtering operation of existing GNNs with the optimal solution of Equation (4). For example, if we set γ(L) = L, then the optimization function (4) becomes f (z) = (1 − α)z T Lz + α z − x 2 2 , a well-known convex graph optimization function proposed by Zhou et al. [40]. f (z) takes the minimum when the derivative ∂f (z) ∂z = 2(1 − α)Lz + 2α (z − x) = 0, which solves to
z * = α (I − (1 − α)(I − L)) −1 x = ∞ k=0 α(1 − α) k (I − L) k x = ∞ k=0 α(1 − α) k P k x.
By taking a suffix sum K k=0 α(1 − α) k P k x, we obtain the polynomial filtering operation for APPNP [14]. Zhu et al. [41] further show that GCN [13], DAGNN [19], and JKNet [36] can be interpreted by the optimization function (4) with γ(L) = L.
The generalized form of Equation (4) allows us to simulate more complex polynomial filtering operation. For example, let α = 0.5 and γ(L) = e tL − I, a heat kernel with t as the temperature parameter. Then f (z) takes the minimum when the derivative ∂f (z) ∂z = e tL − I z + z − x = 0, which solves to
z * = e −tL x = e −t(I−P) x = ∞ k=0 e −t t k k! P k x.
By taking a suffix sum K k=0 e −t t k k! P k x, we obtain the polynomial filtering operation for the heat kernal based GNN such as GDC [15] and GraphHeat [34].
Non-negative constraint on polynomial filters
A natural question is that, does an arbitrary energy function γ(L) correspond to a valid or ill-posed spectral filter? Conversely, does any polynomial filtering operation K k=0 w k L k x correspond to the optimal solution of the optimization function (4) for some energy function γ(L)?
As it turns out, there is a "minimum requirement" for the energy function γ(L); γ(L) has to be positive semidefinite. In particular, if γ(L) is not positive semidefinite, then the optimization function f (z) is not convex, and the solution to ∂f (z) ∂z = 0 may corresponds to a saddle point. Furthermore, without the positive semidefinite constraint on γ(L), f (z) may goes to −∞ as we set z to be a multiple of the eigenvector corresponding to the negative eigenvalue.
Non-negative polynomial filters. Given a positive semidefinite energy function γ(L), we now consider how the corresponding polynomial filtering operation K k=0 w k L k x should look like. Recall that we assume γ(L) = Udiag[γ(λ 1 ), ..., γ(λ n )]U T . By the positive semidefinite constraint, we have γ(λ) ≥ 0 for λ ∈ [0, 2]. Since the objective function f (z) is convex, it takes the minimum when ∂f (z) ∂z = 2(1 − α)γ(L)z + 2α (z − x) = 0. Accordingly, the optimum z * can be derived as
α (αI + (1 − α)γ(L)) −1 x = Udiag α α + (1 − α)γ(λ 1 ) , ..., α α + (1 − α)γ(λ n ) U T x. (5)
Let h(λ) = α α+(1−α)γ(λ) denote the exact spectral filter, and g(λ) = K k=0 w k λ k denote a polynomial approximation of h(λ) (e.g. the suffix sum of h(λ)'s taylor expansion). Since γ(λ) ≥ 0 when λ ∈ [0, 2], we have 0 ≤ h(λ) ≤ α α+(1−α)·0 = 1 for λ ∈ [0, 2]. Consequently, it is natural to assume the polynomial filter g(λ) = K k=0 w k λ k also satisfies 0 ≤ g(λ) ≤ 1. Constraint 3.1. Assuming the energy function γ(L) is positive semidefinite, a polynomial filter g(λ) = K k=0 w k λ k approximating the optimal solution to Equation (4) has to satisfy
0 ≤ g(λ) = K k=0 w k λ k ≤ 1, ∀ λ ∈ [0, 2].(6)
While Constraint 3.1 seems to be simple and intuitive, some of the existing GNN may not satisfies this constraint. For example, GCN uses z = Px = (I − L) x, which corresponds to a polynomial filter g(λ) = 1 − λ that takes negative value when λ > 1, violating Constraint 3.1. As shown in [31], the renormalization trickP = (I + D) −1/2 (I + A) (I + D) −1/2 shrinks the spectral and thus reliefs the problem. However, g(λ) may still take negative value as the maximum eigenvalue ofL = I −P is still larger than 1.
Non-negative polynomials and Bernstein basis
Constraint 3.1 motivates us to design polynomial filters g(λ) = K k=0 w k λ k such that 0 ≤ g(λ) ≤ 1 when λ ∈ [0, 2]. The g(λ) ≤ 1 part is trivial, as we can always rescale each w k by a factor of K k=0 |w k |2 k . The g(λ) ≥ 0 part, however, requires more elaboration. Note that we can not simply set w k ≥ 0 for each k = 0 . . . , K, since it is shown in [5] that such polynomials only correspond to low-pass filters.
As it turns out, the Bernstein basis has the following nice property: a polynomial that is non-negative on a certain interval can always be expressed as a non-negative linear combination of Bernstein basis. Specifically, we have the following lemma.
p(x) = K k=0 θ k b K k (x) = K k=0 θ k K k (1 − x) K−k x k
Lemma 3.1 suggests that to approximate a valid filter, the polynomial filter g(λ) has to be a nonnegative linear combination of Bernstein basis. Specifically, by setting x = λ/2, the filter g(λ) that satisfies g(λ) ≥ 0 for λ ∈ [0, 2] can be expressed as
g(λ) := p λ 2 = K k=0 θ k 1 2 K K k (2 − λ) K−k λ k .
Consequently, any valid polynomial filter that approximate the optimal solution of (4) with positive semidefinite energy function γ(L) has to take the following form: z = K k=0 θ k
1 2 K K k (2I − L) K−k L k x.exp(−10λ 2 ) 1 − exp(−10λ 2 ) exp(−10(λ − 1) 2 ) 1 − exp(−10(λ − 1) 2 ) | sin(πλ)|GCN
Related Work
Graph neural networks (GNNs) can be broadly divided into spectral-based GNNs and spatial-based GNNs [33].
Spectral-based GNNs design spectral graph filters in the spectral domain. ChebNet [7] uses Chebyshev polynomial to approximate a filter. GCN [13] simplifies the Chebyshev filter with the first-order approximation. GraphHeat [34] uses heat kernel to design a graph filter. APPNP [14] utilizes Personalized PageRank (PPR) to set the filter weights. GPR-GNN [5] learns the polynomial filters via gradient descent on the polynomial coefficients. ARMA [2] learns a rational filter via the family of Auto-Regressive Moving Average filters [21]. AdaGNN [9] learns simple filters across multiple layers with a single parameter for each feature channel at each layer. As aforementioned, these methods mainly focus on designing low-or high-pass filters or learning filters without any constraints, which may lead to misspecified even ill-posed filters.
On the other hand, spatial-based GNNs directly propagate and aggregate graph information in the spatial domain. From this perspective, GCN [13] can be explained as the aggregation of the one-hop neighbor information on the graph. GAT [30] uses the attention mechanism to learn aggregation weights. Recently, Balcilar et al. [1] bridge the gap between spectral-based and spatial-based GNNs and unify GNNs in the same framework. Their work shows that the GNNs can be interpreted as sophisticated data-driven filters. This motivates the design of the proposed BernNet, which can learn arbitrary non-negative spectral filters from real-world graph signals.
Experiments
In this section, we conduct experiments to evaluate BernNet's capability to learn arbitrary filters as well as the performance of BernNet on real datasets. All the experiments are conducted on a machine with an NVIDIA TITAN V GPU (12GB memory), Intel Xeon CPU (2.20 GHz), and 512GB of RAM.
Learning filters from the signal
We conduct an empirical analysis on 50 real images with the resolution of 100×100 from the Image Processing Toolbox in Matlab. We conduct independent experiments on these 50 images and report the average of the evaluation index. Following the experimental setting in [1], we regard each image as a 2D regular 4-neighborhood grid graph. The graph structure translates to an 10, 000 × 10, 000 adjacency matrix while the pixel intensity translates to a 10, 000-dimensional signal vector. Nodes 2708 3327 19717 13752 7650 2277 5201 7600 183 183 Edges 5278 4552 44324 245861 119081 31371 198353 26659 279 277 Features 1433 3703 500 767 745 2325 2089 932 1703 1703 Classes 7 6 5 10 8 5 5 5 5 5 For each of the 50 images, we apply 5 different filters (low-pass, high-pass, band-pass, band-rejection and comb) to the spectral domain of its signal. The formula of each filter is shown in Table 2. Recall that applying a low-pass filter exp(−10λ 2 ) to the spectral domain L = Udiag [λ 1 , . . . , λ n ] U means applying Udiag exp(−10λ 2 1 ), . . . , exp(−10λ 2 n ) U to the graph signal. Figure 3 shows the one of the input image and the corresponding filtering results.
In this task, we use the original graph signal as the input and the filtering signal to supervise the training process. The goal is to minimize the square error between output and the filtering signal by learning the correct filter. We evaluate BernNet against five popular GNN models: GCN [13], GAT [30], GPR-GNN [5], ARMA [2] and ChebNet [7]. To ensure fairness, we use two convolutional units and a linear output layer for all models. We train all models with approximately 2k trainable parameters and tune the hidden units to ensure they have similar parameters. Following [1], we discard any regularization or dropout and simply force the GNN to learn the input-output relation. For all models, we set the maximum number of epochs to 2000 and stop the training if the loss does not drop for 100 consecutive times and use Adam optimization with a 0.01 learning rate without decay. Models do not use the position information of the picture pixels. We use a mask to cover the edge nodes of the picture, so the problem can be regarded as a simple regression problem. For BernNet, we use a two-layer model, with each layer sharing the same set of θ k for k = 0, . . . , K and set K = 10. For GPR-GNN, we use the officially released code (see the supplementary materials for URL and commit numbers) and set the order of polynomial filter K = 10. Other baseline models are based on Pytorch Geometric implementation [11]. The more detailed experiments setting can be found in the Appendix. Table 2 shows the average of the sum of squared error (lower the better) and the R 2 scores (higher the better). We first observe that GCN and GAT can only handle low-pass filters, which concurs with the theoretical analysis in [1]. GPR-GNN, ARMA and ChebNet can learn different filters from the signals. However, BernNet consistently outperformed these models by a large margin on all tasks in terms of both metrics. We attribute this quality to BernNet's ability to tune the coefficients θ k 's, which directly correspond to the uniformly sampled filter values.
Node classification on real-world datasets
We now evaluate the performance of BernNet against the competitors on real-world datasets. Following [5], we include three citation graph Cora, CiteSeer and PubMed [27,37], and the Amazon co-purchase graph Computers and Photo [20]. As shown in [5] these 5 datasets are homophilic graphs on which the connected nodes tend to share the same label. We also include the Wikipedia graph Chameleon and Squirrel [26], the Actor co-occurrence graph, and webpage graphs Texas and Cornell from WebKB ‡ [22]. These 5 datasets are heterophilic datasets on which connected nodes tend to have different labels. We summarize the statistics of these datasets in Table 3.
Following [5], we perform full-supervised node classification task with each model, where we randomly split the node set into train/validation/test set with ratio 60%/20%/20%. For fairness, we generate 10 random splits by random seeds and evaluate all models on the same splits, and report the average metric for each model.
We compare BernNet with 6 baseline models: MLP, GCN [13], GAT [30], APPNP [14], ChebNet [7], and GPR-GNN [5]. For GPR-GNN, we use the officially released code (see the supplementary materials for URL and commit numbers) and set the order of polynomial filter K = 10. For other models, we use the corresponding Pytorch Geometric library implementations [11]. For BernNet, we ‡ http://www.cs.cmu.edu/afs/cs.cmu.edu/project/theo-11/www/wwkb/ use the following propagation process:
Z = K k=0 θ k 1 2 K K k (2I − L) K−k L k f (X) ,(7)
where f (X) is a 2-layer MLP with 64 hidden units on the feature matrix X. Note that this propagation process is almost identical to that of APPNP or GPR-GNN. The only difference is that we substitute the Generalized PageRank polynomial with Bernstein polynomial. We set the K = 10 and use different learning rate and dropout for the linear layer and the propagation layer. For all models, we optimal leaning rate over {0.001, 0.002, 0.01, 0.05} and weight decay {0.0, 0.0005}. More detailed experimental settings are discussed in Appendix.
We use the micro-F1 score with a 95% confidence interval as the evaluation metric. The relevant results are summarized in Table 4. Boldface letters indicate the best result for the given confidence interval. We observe that BernNet provides the best results on seven out of the ten datasets. On the other three datasets, BernNet also achieves competitive results against SOTA methods.
More interestingly, this experiment also shows BernNet can learn complex filters from real-world datasets with only the supervision of node labels. Figure 4 plots some of the filters BernNet learnt in the training process. On Actor, BernNet learns an all-pass-alike filter, which concurs with the fact that MLP outperforms all other baselines on this dataset. On Chameleon and Squirrel, BernNet learns two comb-alike filters. Given that BernNet outperforms all competitors by at least 1% on these two datasets, it may suggest that comb-alike filters are necessary for Chameleon and Squirrel. Figure 5 shows the Coefficients θ k learnt from real-world datasets by BernNet. When comparing Figures 4 and 5, we observe that the curves of filters and curves of coefficients are almost the same. This is because BernNet's coefficients are highly correlated with the spectral property of the target filter, which indicates BernNet Bernnet has strong interpretability.
Finally, we present the train time for each method in Table 5. BernNet is slower than other methods due to its quadratic dependence on the degree K. However, compared to the SOTA method GPR-GNN, the margin is generally less than 2, which is often acceptable in practice. In theory, both ChebNet [7] and GPR-GNN [5] are linear time complexity related to propagation step K, but BernNet is quadratic time complexity related to K. Delgado et al. [8] show that Bernstein approximation can be evaluated in linear time related to K using the corner cutting algorithm. However, BernNet can not use this algorithm directly, because we need to multiply signal x. How to convert BernNet to linear complexity will be a problem worth studying in the future.
Conclusion
This paper proposes BernNet, a graph neural network that provides a simple and intuitive mechanism for designing and learning an arbitrary spectral filter via Bernstein polynomial approximation. Compared to previous methods, BernNet can approximate complex filters such as band-rejection and comb filters, and can provide better interpretability. Furthermore, the polynomial filters designed and learned by BernNet are always valid. Experiments show that BernNet outperforms SOTA methods in terms of effectiveness on both synthetic and real-world datasets. For future work, an interesting direction is to improve the efficiency of BernNet.
Broader Impact
The proposed BernNet algorithm addresses the challenge of designing and learning arbitrary spectral filters on graphs. We consider this algorithm a general technical and theoretical contribution, without any foreseeable specific impacts. For applications in bioinformatics, computer vision, and natural language processing, applying the BernNet algorithm may improve the performance of existing GNN models. We leave the exploration of other potential impacts to future work.
A Additional experimental details
A.1 Learning filters from the signal (Section 5.1)
For all models, we use two convolutional layers and a linear output layer that projects the final node representation onto the single output for each node. We train all models with approximately 2k trainable parameters and tune the hidden units to ensure they have similar parameters. We discard any regularization or dropout and simply force the GNN to learn the input-output relation. We stop the training if the loss does not drop for 100 consecutive epochs with a maximum limit of 2000 epochs and use Adam optimization with a 0.01 learning rate without decay. For GPR-GNN, we use the officially released code (URL and commit number in Table 6), and other baseline models are based on Pytorch Geometric implementation [11].
For GCN, we set the hidden units to 32 Regarding the analysis experiment in section 2.3, BenNet's settings are the same as above and the image used is Figure 3.
A.2 Node classification on real-world datasets (Section 5.2)
In this experiment, we also use the officially released code (URL and commit number in Table 6) for GPR-GNN and Pytorch Geometric implementation [11] for other models. For all models, we use two convolutional layers and use early stopping 200 with a maximum of 1000 epochs for all datasets. We use the Adam optimizer to train the models and optimal leaning rate over {0.001, 0.002, 0.01, 0.05} and weight decay {0.0, 0.0005}.
For GCN, we use 64 hidden units for each GCN convolutional layer. For MLP, we use the 2-layer full connected network with 64 hidden units. For GAT, we make the first layer have 8 attention heads and each heads have 8 hidden units, the second layer have 1 attention head and 64 hidden units. For APPNP, we use 2-layer MLP with 64 hidden units and set the propagation steps K to be 10. We search the optimal α within {0.1, 0.2, 0.5, 0.9}. For ChebNet, we set the propagation steps K to be 2 and use 32 hidden units for each layer, which the number of equivalent hidden units is 64 for this case. For GPR-GNN, we use 2-layer MLP with 64 hidden units and set the propagation steps K to be 10, and use PPR initialization with α ∈ {0.1, 0.2, 0.5, 0.9} for the GPR weights. For BernNet, we use 2-layer MLP with 64 hidden units and set the order K = 10, and we optimize the learning rate for the linear layer and the propagation layer. We fix the dropout rate for the convolutional layer or the linear layer to be 0.5 for all models, and optimize the dropout rate for the propagation layer for GPR-GNN and BernNet. The Table 7 shows the hyperparameters of BernNet on real-world datasets. Figure 6 plots the filters learnt from real-world datasets by BernNet. Besides some special filters discussed in section 5.2, we find that BernNet learns a low-pass filter on Cora and CiteSeer which concurs the analysis in [1]. Figure 7 shows the Coefficients θ k learnt from real-world datasets by BernNet. We observe that the curves of filters and curves of coefficients are almost the same, this is because BernNet's coefficients are highly correlated with the spectral property of the target filter which indicates BernNet Bernnet has strong interpretability.
Definition 2.1 ( [10]). (Bernstein polynomial approximation) Given an arbitrary continuous function f (t) on t ∈ [0, 1], the Bernstein polynomial approximation (of order K) for f is defined as
Figure 2 :
2Illustrations of four complex filters and their approximations learnt by BernNet.
Lemma 3.1 ([23]). Assume a polynomial p(x) = K k=0 θ k x k satisfies p(x) ≥ 0 for x ∈ [0, 1].Then there exists a sequence of non-negative coefficients θ k , k = 0, . . . , K, such that
This observation motivates our BernNet from the perspective of graph optimizationany valid polynomial filers, i.e., the g : [0, 2] → [0, 1], can always be expressed by BernNet, and accordingly, the filters learned by our BernNet are always valid.
Figure 3 :
3A input image and the filtering results.
Figure 4 :
4Filters learnt from real-world datasets by BernNet.
Figure 5 :
5Coefficients θ k learnt from real-world datasets by BernNet.
Figure 6 :
6Filters learnt from real-world datasets by BernNet.
Figure 7 :
7Coefficients θ k learnt from real-world datasets by BernNet.
Proposition 2.1. For an arbitrary continuous filter function h : [0, 2] → [0, 1], by setting θ k
Table 1 :
1Realizing commonly used filters with BernNet.
Table 2 :
2Average sum of squared error and R 2 score in parentheses.Low-pass
High-pass
Band-pass
Band-rejection
Comb
Table 3 :
3Dataset statistics.Cora CiteSeer PubMed Computers
Photo
Chameleon Squirrel Actor Texas Cornell
Table 4 :
4Results on real world benchmark datasets: Mean accuracy (%) ± 95% confidence interval.
GCN
GAT
APPNP
MLP
ChebNet
GPR-GNN
BernNet
Cora
87.14±1.01 88.03±0.79 88.14±0.73 76.96±0.95 86.67±0.82
88.57 ±0.69
88.52±0.95
CiteSeer
79.86±0.67 80.52 ±0.71 80.47±0.74 76.58±0.88 79.11±0.75
80.12±0.83
80.09±0.79
PubMed
86.74±0.27 87.04±0.24 88.12±0.31 85.94±0.22 87.95±0.28
88.46±0.33
88.48 ±0.41
Computers 83.32±0.33 83.32±0.39 85.32±0.37 82.85±0.38 87.54±0.43
86.85±0.25
87.64 ±0.44
Photo
88.26±0.73 90.94±0.68 88.51±0.31 84.72±0.34 93.77±0.32
93.85 ±0.28
93.63±0.35
Chameleon 59.61±2.21 63.13±1.93 51.84±1.82 46.85±1.51 59.28±1.25
67.28±1.09
68.29 ±1.58
Actor
33.23±1.16 33.93±2.47 39.66±0.55 40.19±0.56 37.61±0.89
39.92±0.67
41.79 ±1.01
Squirrel
46.78±0.87 44.49±0.88 34.71±0.57 31.03±1.18 40.55±0.42
50.15±1.92
51.35 ±0.73
Texas
77.38±3.28 80.82±2.13 90.98±1.64 91.45±1.14 86.22±2.45
92.95±1.31
93.12 ±0.65
Cornell
65.90±4.43 78.21±2.95 91.81±1.96 90.82±1.63 83.93±2.13
91.37±1.81
92.13 ±1.64
Table 5 :
5Average running time per epoch (ms)/average total running time (s). Chameleon 4.93/0.99 13.11/2.66 7.93/1.62 3.14/0.63 10.92/2.25 10.93/2.41 22.54/4.GCN
GAT
APPNP
MLP
ChebNet
GPR-GNN
BernNet
Cora
4.59/1.62
9.56/2.03
7.16/2.32 3.06/0.93
6.25/1.76
9.94/2.21
19.71/5.47
CiteSeer
4.63/1.95
9.93/2.21
7.79/2.77 2.95/1.09
8.28/2.56
11.16/2.37 22.36/6.32
PubMed
5.12/1.87 16.16/3.41 8.21/2.63 2.91/1.61 18.04/3.03 10.45/2.81 22.02/8.19
Computers 5.72/2.52 30.91/7.85 9.19/3.48 3.47/1.31 20.64/9.64 16.05/4.38 28.83/8.69
Photo
5.08/2.63 19.97/5.41 8.69/4.18 3.67/1.66 13.25/7.02 13.96/3.94 24.69/7.37
75
Actor
5.43/1.09 11.94/2.45 8.46/1.71 3.82/0.77
7.99/1.62
11.57/2.35 23.34/5.81
Squirrel
5.61/1.13 22.76/4.91 8.01/1.61 3.41/0.69 38.12/7.78
9.87/5.56
25.58/9.23
Texas
4.58/0.92
9.65/1.96
7.83/1.63 3.19/0.65
6.51/1.34
10.45/2.16 23.35/4.81
Cornell
4.83/0.97
9.79/1.99
8.23/1.68 3.25/0.66
5.85/1.22
9.86/2.05
22.23/5.26
and set the linear units to 64. For GAT, the first layer has 4 attention heads and each head has 8 hidden; the second layer has 8 attention heads and each head has 8 hidden. And we set the linear units to 64. For ARMA, we set the ARMA layer and ARMA stacks to 1, and set the hidden units to 32 and set the linear units to 64. For ChebNet, we use 3 steps propagation for each layer with 32 hidden units and set the linear units to 64. For GPR-GNN, we set the hidden units to 32 with 10 steps propagation and set the linear units to 64, and use the random initialization for the GPR part. For BernNet, we use same set of θ k for k = 0, . . . , K in each layer and set K = 10.
Table 6 :
6URL and commit number of GPR-GNN codes URL Commit GRP-GNN https://github.com/jianhao2016/GPRGNN 2507f10
Table 7 :
7Hyperparameters for BernNet on real-world datasets.Datasets
Linear layer
learning rate
Propagation layer
learning rate
Hidden
dimension
Propagation layer
dropout
Linear layer
dropout
K
Weight
decay
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|
[
"https://github.com/ivam-he/BernNet.",
"https://github.com/jianhao2016/GPRGNN"
] |
[
"A REMARK ON GLOBAL W 1,p BOUNDS FOR HARMONIC FUNCTIONS WITH LIPSCHITZ BOUNDARY VALUES",
"A REMARK ON GLOBAL W 1,p BOUNDS FOR HARMONIC FUNCTIONS WITH LIPSCHITZ BOUNDARY VALUES"
] |
[
"Nikos Katzourakis "
] |
[] |
[] |
In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in L p for all finite p ≥ 1.
|
10.3336/gm.52.1.08
|
[
"https://arxiv.org/pdf/1601.00190v2.pdf"
] | 11,832,470 |
1601.00190
|
3339b33807f73d0fc82836197b741dc515512cd2
|
A REMARK ON GLOBAL W 1,p BOUNDS FOR HARMONIC FUNCTIONS WITH LIPSCHITZ BOUNDARY VALUES
Nikos Katzourakis
A REMARK ON GLOBAL W 1,p BOUNDS FOR HARMONIC FUNCTIONS WITH LIPSCHITZ BOUNDARY VALUES
In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in L p for all finite p ≥ 1.
Introduction
Kellogg in [K] pioneered the study of the boundary behaviour of the gradient of Harmonic functions on a bounded domain. Roughly speaking, he established that in a domain of R 3 near a boundary region which can represented as the graph of a planar function, the gradient of any Harmonic function is continuous up to the boundary provided that the gradient of the boundary function and of the Harmonic function are Dini continuous themselves on the boundary. The celebrated theory of Schauder estimates [GT] establishes strong relevant results for general uniformly elliptic PDEs, providing interior and global Hölder bounds for solutions and their derivatives in terms of the Hölder norms of the boundary values of the solution and the right hand side of the PDE. The Schauder theory has been improved and extended by many authors, but typically for second order elliptic PDEs with boundary values of the solutions and right hand sides of the PDEs in the Hölder spaces C 2,α or C 1,α , in order to obtain uniform estimates for the solutions in the respective Hölder spaces.
In [GH] Gilbarg-Hörmander have extended Schauder theory to include hypotheses of lower regularity of the boundary values of the solution, of the boundary of the domain and of the coefficients of the equations. Troianiello [T] relaxed further some conditions of Gilbarg-Hörmander [GH]. In the paper [HS] Hile-Stanoyevitch, extending an older result of Hardy-Littlewood [HL], proved that the gradient of a Harmonic function with Lipschitz continuous boundary values is pointwise bounded up to a constant by the logarithm of a multiple of the inverse of the distance to the boundary.
However, it appears that in none of these results, even for the special case of the Laplacian, there is an explicit global bound in L p for the gradient of Harmonic functions which have just Lipschitz boundary values and not C 1,α . In this note establish the following consequence of the result of Hile-Stanoyevitch:
Theorem 1. Let n ≥ 2, Ω ⊆ R n a bounded open set with C 2 boundary. Let also g : ∂Ω → R with g ∈ Lip(∂Ω), that is g ∈ C 0 (∂Ω) and
Lip(g, ∂Ω) := sup
x,y∈∂Ω, x =y |g(x) − g(y)| |x − y| < ∞.
(1) There exists a positive function f Ω,n : Ω → (0, ∞) depending on Ω, n such that
(1.1) f Ω,n ∈ p∈[1,∞) L p Ω) ∩ C 0 (Ω)
and if u ∈ C 2 (Ω) ∩ C 0 (Ω) is the Harmonic function solving
(1.2) ∆u = 0, in Ω, u = g, on ∂Ω, then we have the estimate
(1.3) Du(x) ≤ Lip(g, ∂Ω) f Ω,n (x), x ∈ Ω. (2) Let (g m ) ∞ 1 ⊆ Lip(∂Ω) satisfy for some C > 0 (1.4) Lip(g m , ∂Ω) + max ∂Ω |g m | ≤ C, m ∈ N.
Let also (u m ) ∞ 1 ∈ C 2 (Ω) ∩ C 0 (Ω) be the Harmonic functions solving
(1.5) ∆u m = 0, in Ω, u m = g m , on ∂Ω. Then, (u m ) ∞ 1 is strongly precompact in ∞ p=1 W 1,p (Ω) and if (1.6) g m k −→ g in C 0 (Ω), as k → ∞, then there is a unique limit point u ∈ C 2 (Ω) ∩ C 0 (Ω) of the subsequence (u m k ) ∞ 1 such that along perhaps a further subsequence (1.7) u m k −→ u in W 1,p (Ω) ∀ p ≥ 1, as k → ∞,
and the limit function u solves
∆u = 0, in Ω, u = g, on ∂Ω.
The motivation to derive the above integrability result and its consequences comes from certain recent advances in generalised solutions of nonlinear PDE and vectorial Calculus of Variations in the space L ∞ ( [Ka4] and [Ka2,Ka3]). The vectorial counterparts of Harmonic functions provide useful energy comparison maps since they are "stable" in L p for all 1 < p < ∞.
Proofs
Our notation is either self-explanatory or otherwise standard as e.g. in [E], [Ka]. The starting point of our proof is the following estimate of Hile-Stanoyevitch: under the hypotheses of Theorem 1, the gradient Du of a Harmonic function u ∈ C 2 (Ω) ∩ C 0 (Ω) which solves (1.2) with g ∈ Lip(∂Ω) satisfies the logarithmic estimate
(2.1) Du(x) ≤ C(Ω, n) Lip(g, ∂Ω) ln diam(Ω) dist(x, ∂Ω) , x ∈ Ω.
for some C depending just on Ω (and the dimension). In (2.1), diam(Ω) is the diameter of the domain and dist(x, ∂Ω) the distance of x from the boundary:
diam(Ω) := sup |x − y| : x, y ∈ Ω , dist(x, ∂Ω) := inf |x − z| : z ∈ ∂Ω .
Proof of (1) of Theorem 1. Fix ε > 0 smaller than the diameter of Ω and consider the inner open ε neighbourhood of Ω:
Ω ε := x ∈ Ω : dist(x, ∂Ω) > ε .
It is well known that (see e.g. [GT]) dist(·, ∂Ω) ∈ W 1,∞ loc (R n ) and
(2.2) D dist(·, ∂Ω) = 1, a.e. on Ω.
Let p ∈ [1, ∞). By the Co-Area formula (see e.g. [ [EG], Proposition 3, p. 118]) applied to the function
R n x −→ χ Ω ε (x) ln diam(Ω) dist(x, ∂Ω) p ∈ R (where χ Ω ε is the characteristic function of Ω ε ), we have Ω ε ln diam(Ω) dist(x, ∂Ω) p dx = = diam(Ω) ε {dist(·,∂Ω)=t} ln diam(Ω) dist(z, ∂Ω) p D dist(z, ∂Ω) dH n−1 (z) dt (2.3)
where H n−1 is the (n − 1)-dimensional Hausdorff measure. By using (2.2), (2.3) simplifies to
Ω ε ln diam(Ω) dist(x, ∂Ω) p dx = = diam(Ω) ε {dist(·,∂Ω)=t} ln diam(Ω) dist(z, ∂Ω) p dH n−1 (z) dt
Further, since dist(z, ∂Ω) = t, for all z ∈ {dist(·, ∂Ω) = t}, by setting and hence the inequality (2.5) gives
(2.4) I ε,p := Ω ε ln diam(Ω) dist(x, ∂Ω) p dx we obtain I ε,p = diam(Ω) ε {dist(·,∂Ω)=t} ln diam(Ω) t p dH n−1 (z) dt = diam(Ω) ε ln diam(Ω) t p H n−1 {dist(·, ∂Ω) = t} dt.(2.6) I ε,p ≤ C(Ω) diam(Ω) ε ln diam(Ω) t p dt.
By the change of variables ω := diam(Ω) t we can rewrite the estimate (2.6) as
I ε,p ≤ C(Ω) diam(Ω) diam(Ω)/ε 1 (ln ω) p ω 2 dω
and by enlarging perhaps the constant C(Ω), we rewrite this as
(2.7) I ε,p ≤ C(Ω) diam(Ω)/ε 1 ln ω ω 2/p p dω.
Claim 2. We have that lim ε→0 I ε,p ≤ C(Ω, n, p) < ∞.
First proof of Claim 2 (proposed by one of the referees): By using the following known property of the Gamma function ∞ 1 ln p x x 2 dx = Γ(1 + p)
we readily conclude.
Second proof of Claim 2: We now give a direct argument without quoting special functions. Consider now the function g(ω) := ln ω ω 2/p , g : (1, ∞) → (0, ∞). Since g (ω) = 1 − (2/p) ln ω ω (2/p)+1 , we have that g is strictly increasing on (1, e 2/p ) and strictly decreasing on (e 2/p , ∞). Further, note that t → g p (t) also enjoys the exact same monotonicity properties since s → s p is strictly increasing. Moreover, since e 2/p ≤ 10 for all p ∈ [1, ∞) and by using that ε → I ε,p is decreasing (in view of (2.4)), we have lim ε→0 I ε,p ≤ C(Ω)
I ε,p ≤ C(Ω) 10 (2/p) p e 4/p + ∞ k=10 k+1 k (ln ω) p ω 2 dt ≤ C(Ω) 10 (2/p) p e 4/p + ∞ k=10 sup k<ω<k+1 (ln ω) p ω 2 which gives (2.8) lim ε→0 I ε,p ≤ C(Ω) 10 (2/p) p e 4/p + ∞ k=10 (ln k) p k 2 .
Now we show that the series
S := ∞ k=10 (ln k) p k 2
converges.
Method 1: Since the sequence (ln k) p k 2 , k = 10, 11, 12, ...
is decreasing, by the Cauchy condensation test the series S converges if and only if
∞ m=10 2 m A 2 m < ∞, A k := (ln k) p k 2 . Since 2 m+1 A 2 m+1 2 m A 2 m = (m + 1) p (ln 2) p 2 −m−1 m p (ln 2) p 2 −m = 1 2 1 + 1 m p −→ 1 2 ,
as m → ∞, by the Ratio test we have that
∞ k=10 (ln k) p k 2 = C(p) < ∞ since ∞ m=10 2 m A 2 m converges.
Method 2 (proposed by one of the referees): By repeated applications of the del Hospital rule (p ∈ N), we have
lim k→∞ (ln k) p k 2 1 k 3/2 = lim k→∞ p! k = 0
and hence (ln k) p k 2 ≤ 1 k 3/2 for k ∈ N large enough. Since the series ∞ k=10 1 k 3/2 converges and hence so does S by the comparison test.
In either case, by (2.4) and (2.8) we have that there is a constant C(Ω, n, p) depending only on Ω, n, p such that
Ω ln diam(Ω) dist(x, ∂Ω) p dx = lim ε→0 I ε,p ≤ C(Ω, n, p).
(2.9) By combining (2.9) with (2.1), we see that by setting
f Ω,n (x) := C(Ω, n) ln diam(Ω) dist(x, ∂Ω) , x ∈ Ω
(1) of Theorem 1 is established.
Proof of (2) of Theorem 1. Let u m solve (1.5). By standard interior bounds on the derivatives of Harmonic functions in terms of their boundary values (see e.g. [GT]) and (1.4), we have that the Hessians (D 2 u m ) ∞ 1 are bounded in C 0 (Ω, R n×n ), that is uniformly over the compact subsets of Ω. The same is true for the 3rd order derivatives as well; thus, for any Ω Ω, there is C(Ω ) such that
3 k=1 D k u m C 0 (Ω ) ≤ C(Ω ) u m C 0 (Ω)
and by the Maximum Principle we have
u m C 0 (Ω) ≤ max ∂Ω |g m | ≤ C.
As a consequence,
D k u m (x) − D k u m (y) ≤ C(Ω )|x − y|, D k u m (x) ≤ C(Ω ),
x, y ∈ Ω , k = 0, 1, 2, 3, m ∈ N and by the Ascoli-Arzela theorem, the sequence
u m , Du m , D 2 u m ∞ m=1
is precompact uniformly over the compact subsets of Ω. Again by (1.4), we have
g m (x) − g m (y) ≤ C|x − y|, g m (x) ≤ C,
x, y ∈ ∂Ω, m ∈ N which gives that (g m ) ∞ 1 is bounded and equicontinuous on ∂Ω. Thus, by the Ascoli-Arzela theorem and by the lower semicontinuity of the Lipschitz seminorm with respect to uniform convergence, there is a subsequence (g m k ) ∞ 1 and g ∈ Lip(∂Ω) such that g m k −→ g, as k → ∞ in C 0 (∂Ω).
Along perhaps a further subsequence, by the above bounds on (u m ) ∞ 1 ⊆ C 2 (Ω) ∩ C 0 (Ω), there is u ∈ C 2 (Ω) such that (2.10)
u m k −→ u, in C 0 (Ω),
Du m k −→ Du, in C 0 (Ω, R n ),
D 2 u m k −→ D 2 u, in C 0 (Ω, R n×n ),
as k → ∞. By passing to the limit in the equation ∆u m = 0 we get that ∆u = 0. Since the measure of Ω is finite, for any p ∈ [1, ∞) by Hölder inequality we have that u m L p (Ω) ≤ |Ω| 1/p u m C 0 (Ω) ≤ C(Ω, p). By item (1) of the theorem and by (1.4), we have that Du m L p (Ω) ≤ C(Ω, n, p). Hence, we have the bound u m W 1,p (Ω) ≤ C(Ω, n, p), p ≥ 1. By the Morrey embedding theorem, by choosing p > n we have that (u m k ) ∞ 1 is precompact in C 0 (Ω) and hence by (2.10) we have that (2.11) u m k −→ u, in C 0 (Ω) as k → ∞.
Hence, u = g on ∂Ω and as a consequence u solves the limit Dirichlet problem. Finally, if E ⊆ Ω is a measurable subset, by the Hölder inequality we have that
E Du m (x) p dx ≤ |E| 1− p p+1 E Du m (x) p+1 dx p p+1 = |E| 1− p p+1 Du m L p+1 (Ω) p ≤ |E| 1− p p+1 C(Ω, n, p).
Hence, the sequence of gradients (Du m k ) ∞ 1 is p-equi-integrable on Ω. By (2.10), we have Du m k −→ Du in measure on Ω, as k → ∞. Since Ω has finite measure, the Vitali Convergence theorem (e.g. [FL]) implies that Du m k −→ Du in L p (Ω), as k → ∞.
Item (2) of Theorem 1 has been established.
consequence of the regularity of the boundary, standard results regarding the equivalence between the Hausdorff measure and the Minkowski content for rectifiable sets (see e.g. [[AFP], Section 2.13, Theorem 2.106]) imply that there is a C = C(Ω) such that ess sup 0<t<diam(Ω) H n−1 {dist(·, ∂Ω) = t} ≤ C(Ω)
=
C(Ω) 10 (ln ω) p ω 2 ω=e 2/p + ∞ 10 (ln ω) p ω 2 dt and hence lim ε→0
Acknowledgement. The author would like to thank the referees of this paper most warmly for the careful reading of the manuscript and for providing thoughtthrough alternative simpler proofs of certain of the original arguments.
Functions of Bounded Variation and Free Discontinuity Problems. L Ambrosio, N Fusco, D Pallara, Oxford Mathematical Monographs. 1st EditionL. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, 1st Edition, 2000.
L C Evans, Partial Differential Equations. AMS19L.C. Evans, Partial Differential Equations, AMS, Graduate Studies in Mathematics Vol. 19, 1998.
Measure theory and fine properties of functions. L C Evans, R Gariepy, Studies in advanced mathematics. CRC pressL.C. Evans, R. Gariepy, Measure theory and fine properties of functions, Studies in advanced mathematics, CRC press, 1992.
Modern methods in the Calculus of Variations: L p spaces. I Fonseca, G Leoni, Springer Monographs in MathematicsI. Fonseca, G. Leoni, Modern methods in the Calculus of Variations: L p spaces, Springer Monographs in Mathematics, 2007.
Intermediate Schauder estimates. D Gilbarg, L Hörmander, Arch. Rational Mech. Anal. 74D. Gilbarg, L. Hörmander, Intermediate Schauder estimates, Arch. Rational Mech. Anal., Vol. 74 (1980), 297-318.
Elliptic Partial Differential Equations of Second Order. D Gilbarg, N S Trudinger, Springer-VerlagBerlin-HeidelbergD. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg, 1983.
Theorems concerning mean values of analytic or harmonic functions. G H Hardy, J E Littlewood, Quart. J. of Math. (Oxford). 3G. H. Hardy and J. E. Littlewood, Theorems concerning mean values of analytic or harmonic functions, Quart. J. of Math. (Oxford) 3, 221-256 (1932).
Gradient bounds for harmonic functions Lipschitz on the boundary. G Hile, A Stanoyevitch, Applicable Analysis. 73G. Hile, A. Stanoyevitch, Gradient bounds for harmonic functions Lipschitz on the boundary, Applicable Analysis 73, Issue 1-2 (1999).
An Introduction to viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞. N Katzourakis, DOI10.1007/978-3-319-12829-0Springer Briefs in Mathematics. N. Katzourakis, An Introduction to viscosity Solutions for Fully Nonlinear PDE with Ap- plications to Calculus of Variations in L ∞ , Springer Briefs in Mathematics, 2015, DOI 10.1007/978-3-319-12829-0.
Generalised solutions for fully nonlinear PDE systems and existenceuniqueness theorems. N Katzourakis, ArXiv preprint. N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence- uniqueness theorems, ArXiv preprint, http://arxiv.org/pdf/1501.06164.pdf.
N Katzourakis, A New Characterisation of ∞-Harmonic and p-Harmonic Mappings via Affine Variations in L ∞ , ArXiv preprint. N. Katzourakis, A New Characterisation of ∞-Harmonic and p-Harmonic Mappings via Affine Variations in L ∞ , ArXiv preprint, http://arxiv.org/pdf/1509.01811.pdf.
Existence of Vectorial Absolute Minimisers in Calculus of Variations in L ∞. N Katzourakis, manuscript in preparationN. Katzourakis, Existence of Vectorial Absolute Minimisers in Calculus of Variations in L ∞ , manuscript in preparation.
On derivatives of harmonic functions at the boundary. O D Kellogg, Trans. Amer. Math. Soc. 33O. D. Kellogg, On derivatives of harmonic functions at the boundary, Trans. Amer. Math. Soc., Vol. 33 (1931), 486-510.
Estimates of the Caccioppoli-Schauder type in weighted function spaces. G M Troianiello, Trans. Amer. Math. Soc. 334G. M. Troianiello, Estimates of the Caccioppoli-Schauder type in weighted function spaces, Trans. Amer. Math. Soc. 334 (1992), 551-573.
|
[] |
[
"Multivariate analysis of short time series in terms of ensembles of correlation matrices OPEN",
"Multivariate analysis of short time series in terms of ensembles of correlation matrices OPEN"
] |
[
"Manan Vyas \nInstituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\n62210CuernavacaMexico\n",
"T Guhr \nFakultät für Physik\nUniversität Duisburg-Essen\nLotharstraβe 1D-47048DuisburgGermany\n",
"T H Seligman \nInstituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\n62210CuernavacaMexico\n"
] |
[
"Instituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\n62210CuernavacaMexico",
"Fakultät für Physik\nUniversität Duisburg-Essen\nLotharstraβe 1D-47048DuisburgGermany",
"Instituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\n62210CuernavacaMexico"
] |
[
"SCIEntIFIC REPORTS |"
] |
When dealing with non-stationary systems, for which many time series are available, it is common to divide time in epochs, i.e. smaller time intervals and deal with short time series in the hope to have some form of approximate stationarity on that time scale. We can then study time evolution by looking at properties as a function of the epochs. This leads to singular correlation matrices and thus poor statistics. In the present paper, we propose an ensemble technique to deal with a large set of short time series without any consideration of non-stationarity. Given a singular data matrix, we randomly select subsets of time series and thus create an ensemble of non-singular correlation matrices. As the selection possibilities are binomially large, we will obtain good statistics for eigenvalues of correlation matrices, which are typically not independent. Once we defined the ensemble, we analyze its behavior for constant and block-diagonal correlations and compare numerics with analytic results for the corresponding correlated Wishart ensembles. We discuss differences resulting from spurious correlations due to repetitive use of time-series. The usefulness of this technique should extend beyond the stationary case if, on the time scale of the epochs, we have quasi-stationarity at least for most epochs.
|
10.1038/s41598-018-32891-4
| null | 52,908,709 |
1801.07790
|
6d91e0d7c65abaa731a19bc08ea52118b49d801e
|
Multivariate analysis of short time series in terms of ensembles of correlation matrices OPEN
2018
Manan Vyas
Instituto de Ciencias Físicas
Universidad Nacional Autónoma de México
62210CuernavacaMexico
T Guhr
Fakultät für Physik
Universität Duisburg-Essen
Lotharstraβe 1D-47048DuisburgGermany
T H Seligman
Instituto de Ciencias Físicas
Universidad Nacional Autónoma de México
62210CuernavacaMexico
Multivariate analysis of short time series in terms of ensembles of correlation matrices OPEN
SCIEntIFIC REPORTS |
814620201810.1038/s41598-018-32891-4Received: 24 January 2018 Accepted: 17 September 20181 3 Centro Internacional de Ciencias, 62210, Cuernavaca, Mexico. Correspondence and requests for materials should be addressed to M.V.
When dealing with non-stationary systems, for which many time series are available, it is common to divide time in epochs, i.e. smaller time intervals and deal with short time series in the hope to have some form of approximate stationarity on that time scale. We can then study time evolution by looking at properties as a function of the epochs. This leads to singular correlation matrices and thus poor statistics. In the present paper, we propose an ensemble technique to deal with a large set of short time series without any consideration of non-stationarity. Given a singular data matrix, we randomly select subsets of time series and thus create an ensemble of non-singular correlation matrices. As the selection possibilities are binomially large, we will obtain good statistics for eigenvalues of correlation matrices, which are typically not independent. Once we defined the ensemble, we analyze its behavior for constant and block-diagonal correlations and compare numerics with analytic results for the corresponding correlated Wishart ensembles. We discuss differences resulting from spurious correlations due to repetitive use of time-series. The usefulness of this technique should extend beyond the stationary case if, on the time scale of the epochs, we have quasi-stationarity at least for most epochs.
Non-equilibrium stationary states (NESS) have attracted large amounts of attention in recent years 1-6 but more recently increasing attention is given to non-stationary situations, as they actually cover a wide range of observational as well as of experimental data. Such data cover diverse fields including astronomy, financial markets and meteorology or chemical engineering, fractures and colloids, as well as numerical results for models of such systems and dynamical systems and many others. Among such systems the ones that have several near stationary states with more or less abrupt transitions are of particular interest. Such systems are wide spread and of relevance. They include bi-stable, and multi-stable systems with smooth transitions as well as systems that might run into catastrophic instability. We can think of both types occurring as first order phase transitions under temperature change depending on conditions. Beyond that, we may hope that non-stationary systems may be quasi-stationary over sufficiently short time periods. Yet abrupt non-stationarities may occur and we may hope to obtain either warnings or at least post-event learning from a correlation analysis of known facts over a short time period before the abrupt events.
For the sake of illustration, let us think of a chemical reactor that should produce certain end products in a stationary fashion, but in fact the state is only quasi-stationary. This reactor may have other states that produce less of the desired and more undesirable products and a transition might prove costly. Yet this might get much worse if breaking stationarity may lead to explosions with release of toxic substances, that may in addition cause great cost of lives and health, such as in Bhopal 1984 7 . To use a Wishart model as a model for non-stationarity was first put forward in 8 and used for credit risk analysis in 9,10 .
Our interest was triggered by studies of financial markets, where the very attempt to define states of quasi-stationary evolution is relatively new 11 . In this paper, the correlation matrix of short time series was detected to be a good basis to specify the states and clustering techniques were used to identify these states. An attempt to detect conditions under which change may occur was not made and may also be futile in this context, as the clustering technique by definition assigned each correlation matrix to a state and thus borders become unclear. One could use different clustering algorithms to detect larger differences in clusterings, but this might depend very much on the definition of distance we use 12 .
Time series analysis is rather straight forward for stationary systems, even if these are out of equilibrium (NESS). It can also be extended to analyze non-stationary states using standard detrending techniques to eliminate log time trends and periodic oscillations. These tasks are more challenging if we have many time series. Nonetheless, it becomes much more difficult to identify state changes 11 or have some early signals of catastrophic events 13 . The most usual way out in complicated non-stationary situations is to assume stationarity over short time intervals. Early attempts in this direction analyze the entire correlation matrix. This matrix is not invariant and thus the ordering, i.e. the labeling of the time series, becomes very relevant. While for financial markets there is some ordering that has long standing merits in other cases (e.g. for two dimensional array of detectors) this is no longer the case. If we detect different properties, say pressure and temperature, it is entirely unclear if we should give preference to the spatial distribution or to the different types of measurements to assigned indices in the matrix. The use of sophisticated data handling tools will not remove the arbitrariness of basis dependence. For these reasons (and probably because of heritage from the physics background of the authors), it seems reasonable to choose invariant quantities (not only under permutations but also under orthogonal or unitary transformations) that correspond to linear changes in the measuring devices. The logical choices are eigenvalues and eigenvectors. Obviously, the eigenvectors are basis dependent but they may provide relevant information if the preferred basis is reasonable.
We believe that in some sense eigenvalues do indicate very relevant aspects of dynamics and recently it was shown, that this is also true for the correlation matrices. Using Metropolis dynamics the larger eigenvalues of the correlation matrix of a 2-D Ising model at critical temperature, display a power law, that can be directly derived from the power law of space correlations in this system 14 ; it was further shown that such a power law will survive if a sufficiently large random subset of time series is used. Yet long time series are essential to see this effect because the number of large eigenvalues rapidly becomes too small as the correlation matrix becomes more and more singular with shorter time series.
We define a short time correlation matrix as the one for which the time horizon T (length of the time series) is much smaller than the number N of time series. This obeys to the idea that we can have information on a complex dynamical system if we measure more properties. By definition, the correlation matrix will have only T-1 (except for T = 1) non-zero eigenvalues. Thus, increasing N but not T will not affect the number of non-zero eigenvalues. It is important to mention that, given a N × T data matrix for N time series of length T with N ≫ T, one can always diagonalize the T × T dimensional correlation matrix of the position series A t A/N rather than the N × N correlation matrix of the time series AA t /T as they have same non-zero eigenvalues. However, as we obtain only T non-zero eigenvalues, we do not obtain good statistics and the analysis is inconclusive -whether the spectral features are noise based or have correlations. Importantly, the correlation matrix of the position series A t A/N is capable of detecting lead-lag and other non-Markovian effects [15][16][17][18] . One can also increase both N and T keeping N/T fixed. While this limit is nice for theoretical purposes, financial markets, chemical reactions, neuron processes etc. have time scales which determine T. Thus, we have to either return to the idea of treating the entire correlation matrix or to use some technique to obtain a large enough set of eigenvalues to get good statistics. One option is to use the power map technique. The use of the power map originally introduced for noise reduction 19,20 was suggested in 21 as an appropriate tool to detect correlations if powers very near to identity are used, indeed in 14 this can be explicitly seen. Yet while the power map does detect correlations efficiently, a detailed understanding of the correlations via the additional emerging spectrum is not yet available. Importantly, information as to the nature of the correlation at this point has not been given and the power map is not transparent due to its nonlinearity.
The method proposed in the present paper is a more natural one. We shall replace the large but singular (N ≫ T) correlation matrix by an ensemble of smaller non-singular correlation matrices. This is achieved by first defining an ensemble of data matrices, i.e. sets of m(≪N) time series, chosen from a large number of random selections among the given N time series (corresponding to the large but singular correlation matrix). While making selections, we ensure that no time series is repeated in a given data matrix and no two data matrices are the same in the ensemble. There are non-trivial dependencies in the resulting matrices due to random choices. Using this ensemble of the data matrices, we construct the corresponding ensemble of non-singular correlation matrices. In principle, the number of members in the ensemble is given by ( ) N m and thus, the number of non-zero eigenvalues we have can now be increased dramatically. Thus, the method provides higher statistical significance by retrieving information from the given singular data matrix. In this way, we explore the entire position space and thus, obtain the distribution for each of the eigenvalues rather than a single number. While probably we don't retrieve the full information of eigenvectors we thus reach simultaneously two goals -we obtain smooth distributions and obtain the information about the distribution of each of the eigenvalues. This is particularly relevant for outliers. It implicitly gives information about the outliers among the time series, as these will only appear in some of the subsets. We shall focus on the largest eigenvalues in our examples. Keeping in mind that finally there is no more information available than there is in the original data matrix, we arbitrarily choose the size of the ensemble in order to produce good statistics i.e. we smooth the curves. One can also construct an ensemble of completely independent data matrices in the very large N limit to obtain the ensemble of non-singular correlation matrices and we mention this limit on occasions. It avoids some deviations due to the dependencies but reduces the smoothing.
In the next section, we describe in detail the construction of the ensemble. In the following section, we will present basic results obtained from supersymmetric calculations to derive the formula for correlated Wishart ensembles with arbitrary correlations. We treat in some detail the case of constant correlation Wishart matrices including the zero correlation case that provides the unbiased a priori hypothesis to which experimental data can be compared to obtain clarification of the data. In situations where average correlations are important, one can also use correlated matrices as a priori hypothesis. We compare numerics with the result obtained from supersymmetric calculations. Here, we will also discuss the differences between the proposed random choice selections resulting in dependencies and completely independent choice for which plenty of analytical results are known. As block structures are important, at least in econophysics, we analyze the special case of block-wise correlated subsets of time series. We compare the supersymmetric results with numerical calculations where we restrict the block situation to two blocks of different size and block-wise constant correlations. We see that the bulk of the spectra is well-described by the analytics while the outliers will only be approximated as far as their average position is concerned. The shape is different as the analytic result we present is for independent time series with large N, T and a fixed ratio κ = N/T. Finally, we give conclusions and an outlook.
Construction of the Ensemble
For short time series, the correlation matrices will be strongly singular i.e. the number of non-zero eigenvalues will be greatly reduced and the eigenfunctions corresponding to zero eigen sub-space are arbitrary. We will now introduce a method to overcome these shortcomings.
The building blocks for the correlation(covariance) matrices are rectangular N × T data matrices A = [Aij], with = … i N 1, 2, , and = … j T 1, 2, , . Each row in the data matrix A is a time series of length T, measured at usually equidistant times. It can be obtained from observations or experimental measurements of observables like stock prices, temperature, intensity, astronomical observations and so on. The matrix C = AA t /T, with A t denoting the transpose of matrix A, is the N × N covariance matrix. Wishart matrices are random matrix models used to describe universal features of covariance matrices 22 . We consider the case for real entries
∈ A ij
, known in the literature as Wishart orthogonal ensemble (WOE). For WOE, the matrix elements of A are real independent Gaussian variables with fixed mean μ and variance σ 2 i.e.
μ σ ∈ A N ( , ) ij 2 .
In order to arrive at correlation matrices, one needs to normalize μ = 0 and σ 2 = 1. In the context of time series, C may be interpreted as the correlation matrix, calculated over stochastic time series of time horizon T for N statistically independent variables. By construction, C is a real symmetric positive semidefinite matrix. For T < N, C is singular and has exactly (N − T − 1) zero eigenvalues. Note that, stationarity improves when short time series are used. In real applications, one needs to understand the role of correlations and thus, correlated WOE (CWOE) models provide the null hypothesis. CWOE is defined by real-symmetric
matrices C AA = T / t , with χ = A 1/2 .
Here, χ is a real symmetric positive definite non-random N × N matrix that accounts for the correlations in time series (rows) of data matrix and
∈ A N (0,1) ij . On ensemble average, χ =
. We analyze highly singular correlation matrices (N ≫ T) by constructing ensembles of correlation matrices from a given correlation matrix by randomly selecting short observational time series. By randomly choosing m rows out of N given rows of A() such that m = aT with a being a real number close but smaller than unity, we construct an ensemble of m × T dimensional matrices. While making selections, we ensure that no two rows are same in a given matrix and no two matrices are same in the ensemble. Using these, we obtain an ensemble of m × m non-singular correlation matrices and analyze eigenvalue distribution.
If the number N of time series available is large compared to the number of entries T in each time series, the discussion of eigenvalues becomes statistically unsatisfactory. A typical example would be financial time series of increments of 40 consecutive closing prices for a selection of N = 400 shares from some index (say a selection from Standard and Poors 500). In this case we would obtain but 39 non-zero eigenvalues (40 for covariance matrices) from the 400 × 400 correlation matrix, which might be all over the place. We propose to select m time series (experimental, observational or computational) with m < T at random. If we allow all different choices, we would end up with a very large ensemble of correlation matrices (in our example, we might choose m = 36 leading to ≈ . × ( ) 2 5 10 400 36 51 choices, which is an unpractically large number). So we choose a random subset of a few thousand and get excellent statistics for eigenvalues. Having more members in the ensemble would increase the amount of spurious information, which enters unavoidably if we allow repeated time series in different members of the ensemble. If on the other hand we do not allow repetitions the results would depend very much on the selection we make and statistics would be less adequate. An alternative may be to make an ensemble of ensembles with different but totally independent choices, and calculate averages and variances of specific statistical quantities obtained for the lower level ensemble. We choose not to go this more complicated route. The question arises, how stable and informative the corresponding results are. The purpose of the present paper is to take this simple idea and compare it to cases where analytic results can be derived from well-known results 23,24 . We start by analyzing white noise time series and the resulting correlation matrices known as the Wishart ensemble 22,25 as well as for correlated Wishart ensembles with constant correlations 26 . Here, the level densities are known analytically and the n-point correlation function converges to the universal result 25 . Because the case of constant correlations will mimic real situations only very roughly, we shall study in more detail the situations where subsets of time series are more correlated among each other than with the time series of other subsets. This will be the typical case of market sectors of stock exchanges. To emphasize the characteristics of such a block structure, we shall restrict ourselves in graphical displays to two blocks in this paper.
We shall see that clear signatures of the correlations (or lack thereof) can be obtained with very good statistics. This distinguishes the present linear method both from the clustering techniques 11 and the power-map technique 21 , which are inherently non-linear. The first is a transparent standard technique but requires considerable previous insight into the problem on hand, while the second turns out to be quite stable but interpretation is an open problem.
Supersymmetry Approach
Time series analysis is an imperative tool to study dynamics of variety of complex systems. Wishart correlation matrices are standard models employed for statistical analysis of ensembles of time series. We provide here a brief sketch of the derivation using standard supersymmetric steps; for further details refer to 23 In multivariate analysis, it is desirable to derive a "null hypothesis" from a statistical ensemble to understand the measured eigenvalue density of the given correlation matrix. The random matrix ensemble we consider is CWOE with arbitrary correlations that gives the 'empirical' (population) correlation matrix C 0 upon averaging over the probability density function P(A|C 0 ) (normalized to unity),
π | = − . − − P A C C A C A ( ) [2 det( )] exp 1 2 tr( ) (1) T t 0 0 /2 0 1 By construction, ∫ | = d A P A C AA T C [ ] ( ) / t 0 0 , with measure = ∏ ∏ = = d A d A [ ] i N j T ij 1 1
being product of differentials of all independent elements in A. It is important to mention that, in the supersymmetric approach, one assumes T ≥ N to ensure invertibility of C 0 . In order to be able to derive the ensemble averaged eigenvalue density (one-point function), we may replace
C 0 by diagonal matrix Λ of its eigenvalues Λ Λ … Λ { , , , } N 1 2 since the domain × N T of A is orthogonally invariant.
In terms of resolvent, the ensemble averaged eigenvalue density for correlation matrix AA t is defined by
∫ π ε = |Λ − − . ε→ + I S x N d A P A tr x i AA ( ) 1 lim [ ] ( ) ( ) (2) N N t 0
In case of CWOE (also WOE), the eigenvalue density for the correlation matrices is derived using the supersymmetry technique 23,24 . In this approach, the eigenvalue density is written as the derivative of the generating function. The generating function in turn is mapped onto a suitable superspace which leads to drastic reduction in degrees of freedom. Then, the eigenvalue density is derived by introducing eigenvalue coordinates for the supermatrix and integrating over the anti-commuting Grassmann variables.
The generating function Z as a function of source variable J is the starting point of this approach, Note that x + = x + iε and Z(0) = 1. The one-point function is then computed by the derivative,
∫ = | Λ + − − ∈ . + + + Z J d A P A x J AA det x AA x ( ) [ ] ( ) det( ) ( ) ;(3)π = − ∂ ∂ . = S x N Z J J ( ) 1 ( ) (4) J 0
The generalized Hubbard-Stratanovich transformation 29,30 and superbosonization formula 31 have been used to express the generating function as an integral over a suitable superspace. In fact, these are equivalent 32 . The determinant in the denominator of Equation (3) can be expressed as a Gaussian integral over a vector in ordinary commuting variables. Similarly, the determinant in the numerator can be expressed as a Gaussian integral over a vector in anti-communting variables. Combining these expressions, we obtain a Gaussian integral over a rectangular supermatrix which is n × (2|2) dimensional, (3) and performing the Gaussian integral over A, we apply the duality relation between ordinary spaces and
ζ ζ ζ ζ = = − . * * ≤ ≤ u v u v [ , , , ],(5)π ζ ζ = ∑ ∂ ∂ − = ⁎ d du dv [ ] (2 ) N i N i i i i 1 , ∈ u v , i i in Equationsuperspaces + Λ = + Λ i i det(1 ) sdet(1 ) N t t 4
, one can then rewrite the determinant as a superdeterminant. Importantly, the supermatrix Λ t is 4 × 4 dimensional and the original matrix t is N × N dimensional. This dimensional reduction is the advantage of the supersymmetry technique. The left upper block (boson-boson block) of supermatrix Λ t is a Hermitian matrix. We now use the generalized Hubbard-Stratonovich transformation to replace the supermatrix Λ t by a supermatrix σ with independent matrix elements. For the required power of superdeterminant in the expression for the generating function, we write a super-Fourier representation
∫ ρ ρ + Λ = − Λ . ρ − i d I i sdet ( ) [ ] ( ) exp 2 str ( ) (6) T t t /2 4
The Fourier transform gives a supersymmetric Ingham-Siegel distribution,
∫ ρ σ σ σ ρ = + . − I d i i ( ) [ ]sdet ( )exp 2 str ( )(7)
T/ 2 4 Here, , where σ 0aa , σ 0bb are diagonal and σ 0ab is the off-diagonal elements of σ 0 . The measure d[ρ] is defined in a similar fashion. Using these and integrating over the supermatrix , the generating function is a supermatrix integral,
∫ ∏ ρ ρ γ ρ = − − Λ . = − + Z J d I x J ( ) [ ] ( ) sdet 1 2 (8) i N i 1 1/2 4
Here, the matrix γ = − diag(0 , ) 2 2 is diagonal. For arbitrary small J, using Equations (7) and (8), we have This is the main analytic result of the paper which we test with numerics for different WOE models in the following section. The one-point function is then given in terms of the complex solution, say ρ x ( ) 0 , of this saddle point equation,
∫ ∫ σ ρ = Z J d d e ( ) [ ] [ ] with a Lagrangian given by ∑ ρ ρ ρ = − − Λ + − = + x T 1 2 str ln 1 2 2 str ln str ,(9)ρ π = − I S N x x ( ) 2 (x)/ ,(11)
Numerical Results
For the random selections, we have two choices: (a) 'Non-Singular Random Selection Ensemble' (NSRSE) in which a given time series can appear many times but at most once in the construction of any correlation matrix to avoid singularities. As mentioned above, we will have binomially many choices but the members of the ensemble are not entirely independent. We will usually not have N and T very large, but even so we will find that the behavior of the bulk is not significantly affected although the outliers are. Alternatively, for sufficiently large N and T, we could use a random matrix model 'Exclusive Random Selection Ensemble' (ERSE) that constructs an ensemble that excludes any repetition of time series in its construction. We can use this ensemble to calculate the expectation values of the quantities we are interested in and average those over all or a subset of possible selections. In this case, we expect to a large extent coincidence with correlated Wishart ensembles but the procedure is rather complicated and we will thus focus on the first choice namely NSRSE.
We now proceed to analyze two special cases. First, we consider the case of constant correlations where we, in addition to the spectrum bulk, have an outlier that should be described. Here, we also consider the case of zero correlations i.e. uncorrelated time series, where we reproduce the Marčenko-Pastur distribution 33,37 . Then, we proceed to the block structure which we illustrate by using two blocks of time series which have constant internal correlation and relatively small correlation between the two blocks. Note that our results for NSRSE need not agree with theory for Wishart matrices because starting with a single representative of this ensemble in the large space, we select the smaller matrices from that space and repetitions of the selection will turn out to be important. We compare the distribution of outliers for NSRSE with ERSE in terms of the first four moments.
Correlated Non-Singular Random Selection Ensemble With Constant Linear Correlations.
We
consider correlated NSRSE χ = A 1/2 with constant linear correlations defined by χ δ υ δ = + − (1 ) j k j k j k , ,
, ; υ being the correlation coefficient. NSRSE of correlation matrices will then be obtained from the data set of correlated white noise time series by selecting m time series in L samples from the ( ) N m possible selections. The corresponding eigenvalues will be obtained numerically below and compared to the solution of the polynomial equation in Equation (10). The parameters used in the calculations are L = 5000, N = 1000, κ = 10 and a = 0.9. We choose constant linear correlations defined by υ = 0, 0.1, 0.5 and 0.9. For Monte-Carlo simulations, we start with a singular data matrix of dimension 1000 × 100 (κ = 10). One can normalize these 1000 time series in two ways: (1) by rescaling each time series by its respective mean and standard deviation (micro-canonical normalization) and (2) by rescaling all the time series by their average mean and average standard deviation (canonical normalization). Then, by randomly selecting the rows of this data matrix as explained above, we construct a 5000 member ensemble of 90 × 100 (a = 0.9) data matrices (κ = 0.9). Using these, we construct the L = 5000 members of NSRSE and diagonalize these to obtain the eigenvalues.
The choice c = 0 corresponds to uncorrelated NSRSE and average eigenvalues Λ = 1 i for = … i N 1, , . Using these in Equation (10) results in a quadratic equation which can be solved analytically to obtain Here, κ = N T / and κ = ± ± y T(1 )/4 define the spectral support of the eigenvalue density. This describes the distribution of non-zero eigenvalues for WOE in the limit → ∞ N T , with fixed κ. Hence, in order to be able to compare with the Marčenko-Pastur distribution and numerics, one needs to re-scale the variables as → ′ + x x T/4 and ρ ρ → ′ + x x T ( ) 4 ( )/ in Equation (10). We compare numerical NSRSE eigenvalue densities with the analytical result given by Equation (12) in Fig. 1. In Fig. 1(a), we show the numerical histogram for the 1000 eigenvalues of the correlation matrix corresponding to the initial 1000 × 100 data matrix obtained using microcanonical normalization and similarly for canonical normalization in Fig. 1(b). The spectral bounds are in agreement with the . Similarly, the lower panels [(e-h)] gives the distribution of second largest eigenvalue. In the upper panel, the largest eigenvalues are normalized with respect to their centroids (μ) and widths (σ) i.e.
ρ κ π = − − . + − y y y y y y ( ) 2 ( ) ( )(12)μ σ = − E E ( ) / max . Similarly, μ σ = − − E E (( 1) )/ max
with the corresponding μ and σ in the lower panel. The solid histograms correspond to NSRSE and the empty histograms correspond to ERSE. Corresponding first four moments are given in Table 1.
SCIEntIFIC REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4 solid curve obtained using Equation (12). However, as we have a single copy of correlation matrix, there are a lot of fluctuations in numerics. We do not find any significant differences between microcanonical and canonical normalizations for NSRSE. Then, we apply ensemble technique and eigenvalue histograms for microcanonical and canonical normalizations respectively are shown in Fig. 1(c,d). The agreement with the solid curves obtained using Equation (12) is excellent. Again, we do not observe any significant differences in the microcanonical and canonical normalizations for NSRSE using the ensemble technique.
The numerical histograms obtained for correlated NSRSE with constant linear correlations defined by υ = . 0 1, 0.5 and 0.9 are shown respectively in Fig. 2(a-c). The solid histograms correspond to microcanonical normalization and empty histograms correspond to canonical normalization. The solid curves are obtained by numerically solving Equation (10)
with υ Λ = − 1 i for = … − i N 1, , 1 and υ υ Λ = + − N 1 N
(a third order polynomial equation). Insets in each of these pictures show the distribution of the outlier Λ N . The agreement of the polynomial equation solution in the bulk of the spectrum is excellent except for small deviations in the tails with increasing correlation coefficient υ. Notice the increasing difference between the bulk and the outlier along with shrinking of spectral bounds for the bulk distribution with increasing υ. The histograms for microcanonical and canonical normalizations are similar for the bulk distribution while there are differences in outliers noticeable with increasing υ.
The shape of the farthest peak (outlier) is Gaussian for the numerical histograms whereas it resembles a semicircle for the respective solutions from the polynomial equation. The saddle point approximation must be good where many peaks overlap. It must be worse for individual peaks (outliers). But as seen from Fig. 2, the saddle point approximation reproduces the position of the outliers not too far from reality. However, it cannot reproduce the shape of the peaks. In the saddle point approximation, the bulk of the spectrum is order N correction and if the outlier is far away from the bulk, it is only order 1 correction term. The exact problem is highly complex and one cannot expect to get all the features by a simple polynomial equation.
It is now well established that the distribution of the largest eigenvalue separated from the bulk for a correlated covariance matrix converges to a Gaussian distribution 38 . As ERSE should produce results close to Wishart ensembles, we compare the largest eigenvalue distributions for NSRSE and ERSE in Fig. 3. The corresponding moments are given in Table 1. As can be seen from these results, the largest eigenvalue distributions for NSRSE are also Gaussian, however the moments are different. Thus, the repetition of time series in the construction of NSRSE strongly affects the outliers.
Block Non-Singular Random Selection Ensemble. As is usual in financial market analysis, one deals with approximate block matrices where each block represents a sector. For instance, energy, utility and technology are a few sectors in stocks. Inspired by this, we consider a simple 2 × 2 model for a two sector NSRSE model. The corresponding data matrix has the structure = . These are the number of time series randomly chosen from each sector.
One can also make the random permutations without any weights. This does not affect the structure of the correlation matrices in NSRSE. Figure 4 shows the structure of ensemble averaged correlation matrices constructed using canonical normalization with (a) υ υ = = . . Here random permutations were carried out without any weights. Thus, the block structure remains intact even without the weighted random permutations. This is obvious as the χ matrix is invariant under permutations.
In Fig. 5, we compare the eigenvalue histograms (solid ones corresponding to microcanonical normalization and empty ones corresponding to canonical normalization) of block NSRSE for (a) υ υ = = . with the solid curve obtained using Equation (10)
with υ Λ = − 1 i 1 f o r = … − i N 1, , 1 1 , υ υ Λ = + − N 1 N 1 1 1 1 , υ Λ = − 1 i 2 f o r = + … − i N N 1, ,1 1 2, υ υ Λ = + − N 1 N 2 2 2 2
(fifth order polynomial equation). We find good agreement in the bulk distributions with deviations in the tails for larger correlation coefficients υ 1 and υ 2 . Insets show the distributions of the two outliers (Λ N 1 and Λ N 2 ). It can be single peaked, overlapping peaks or double peaked as the positions depend on correlation coefficients υ 1 and υ 2 . The choice of normalization generates differences in the distribution of outliers. The saddle point approximation gives the approximate positions of the peaks but not the shape.
We compare the distributions of the outliers (largest and second largest eigenvalues) for NSRSE with those corresponding to ERSE in Fig. 6. The corresponding moments are given in Table 1. The distributions of outliers separated from the bulk are well approximated by Gaussians for both NSRSE and ERSE while the moments are different. The convergence to Gaussian distribution also depends on the separation of the outliers from the bulk distribution. Thus, the repetition of time series in the construction of block NSRSE strongly affects the outliers.
Conclusions and Outlook
We have presented an entirely new way to treat large numbers of short time series pertaining to the same system and therefore likely to display some correlation. Basically the proposition consists in dividing the entire set of time series in different ways, thus obtaining the Non-Singular Random Selection Ensemble from the data. This allows to obtain a spectral distribution for an ensemble of correlation or covariance matrices and also to get distributions of particular eigenvalues, importantly the largest or the smallest one. Using our technique, we obtain a large set of eigenvalues for a singular data matrix and thus, get information about the bulk eigenvalue distribution along with the outliers, which is otherwise not possible. It also allows analysis of two and three point functions which is impossible for a small set (total number T with T ≪ N) of non-zero eigenvalues for a single data matrix. The SCIEntIFIC REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4 same will hold for eigenfunctions. If the matrix elements are not Gaussian distributed, the shapes and moments of distributions of eigenvalues will change with the strength of the correlation coefficient and there will be effects in the tails of the distributions. We expect that our technique will be sensitive to tails of distributions of eigenvalues as the probability of randomly selecting the outliers increases with our technique. Finally we may note that selection of such subsets gives valuable insight into the consequences of having incomplete sets of time series. This is illustrated in Ising model calculations 14 where random selections of subsets of very long time series were used to test the effect of having incomplete measurements, which showed that a power law remaines visible using only 10% of the time series associated with the lattice. This opens a new alley for investigations of systems that are not stationary on longer time scales but quasi-stationary on a short time scale as defined by the length of the epochs we choose. We can then study the temporal evolution of such an ensemble. This in turn may give hints to instabilities emerging in the system which might be sufficiently strong to be used to give an early warning.
The next step will be to show how such an ensemble behaves, when at or near a critical transition. At this point we are studying this in financial markets and in two dynamical systems, namely the TASEP 6 and the 2-D Ising model near criticality 21 . The range of potential applications is very wide and in the present paper we have performed the first tests using correlated random matrices as a model where analytic results are available. The case we generically discuss is a set of time series, which are strongly correlated within each of two subsets leading to a block structure in the correlation matrix. This is a toy model for financial markets with its traditional division into market sectors. Preliminary results on financial markets can be viewed in a master thesis 39 and further work in this direction is in progress.
Data Availability Statement
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
SCIEntIFIC
REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4
Figure 1 .
1Density of non-zero eigenvalues of a singular correlation matrix C obtained from a data matrix A of dimension 1000 × 100; κ = 10 with (a) micro-canonical and (c) canonical normalization. Ensemble averaged eigenvalue density for a 5000 member NSRSE of correlation matrices constructed using 0.97T × T (κ = 0.9) dimensional A matrices with (b) micro-canonical and (d) canonical normalization. Numerical results are histograms and solid curves are obtained from Equation (12). SCIEntIFIC REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4
Grassmann variables. Using this and
Figure 2 .
2Ensemble averaged eigenvalue density for a 5000 member ensemble of 90 × 90 dimensional correlated NSRSE matrices with constant linear correlations defined by (a) υ = 0.1, (b) υ = 0.5 and (c) υ = 0.9. Here, κ = 10. Numerical results are histograms and solid curves are obtained from Equation (10). The solid histograms correspond to microcanonical normalization and empty histograms correspond to canonical normalization. Insets give the distribution of the outlier. SCIEntIFIC REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4
Figure 3 .
3Probability distribution of the outliers for NSRSE and ERSE. The largest eigenvalues are normalized with respect to their centroids (μ) and widths (σ) i.e.
Figure 4 .SCIEntIFIC
4end up with a scalar polynomial equation resulting from the saddle point equation that can be solved numerically, Ensemble averaged non-singular block correlation matrices constructed using 90 × 100 dimensional data matrices A with constant block correlations: (a) REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4
ratio κ = N/T. Note that the eigenvalue density is normalized to unity. Equation (10) is valid for CWOE with arbitrary correlations and the structure of the correlation matrix enters via its eigenvalues Λ i (1 ≤ i ≤ N). Equation(10)is another version of a classical result[33][34][35][36] .
Figure 5 ..
5Ensemble averaged eigenvalue density for a 5000 member block NSRSE of correlation matrices constructed using 0.9T × T (κ = 0.9) dimensional A matrices with constant block correlations defined by (a) Numerical results are histograms and solid curves are obtained from Equation (10). The solid histograms correspond to microcanonical normalization and empty histograms correspond to canonical normalization. Insets give the distribution of the outliers. SCIEntIFIC REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4
Figure 6 .
6Probability distribution of the outliers for NSRSE and ERSE. Upper panels [(a-d)] gives the distribution of largest eigenvalue for block correlation matrices constructed using 90 × 100 dimensional data matrices A with constant block correlations: (a)
.
A 1 and A 2 representing data matrix in each sector with respective dimensions N 1 × T and N 2 × T; In each sector, we consider constant linear correlations with correlation coefficients υ 1 and υ 2 .For numerics, we construct a L = 5000 member block NSRSE with N = 1000, κ = 10, = . To generate the ensemble, for each member, random selections of time series from the given A matrix are done depending on the weights (say, these are p 1 and p 2 ):
,24,27,28 . SCIEntIFIC REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4
The solid histograms correspond to NSRSE and the empty histograms correspond to ERSE. Corresponding first four moments are given inTable 1.Table 1. Moments for outliers with constant correlations. All values are listed as (NSRSE/ERSE).Case
Mean
Width
Skewness
Kurtosis
Largest eigenvalue,
correlated NSRSE/
ERSE
υ = 0.1
(10.69/11.94)
(1.23/0.61)
(0.21/0.05)
(−0.12/0.05)
υ = 0.5
(45.53/48.90)
(3.18/0.93)
(−0.10/0.01)
(−0.12/0.12)
υ = 0.9
(80.97/82.19)
(1.17/0.53)
(−0.44/−0.01)
(0.32/0.13)
Largest eigenvalue,
block NSRSE/ERSE
υ 1 = 0.1,
υ 2 = 0.1
(8.96/10.27)
(1.03/0.53)
(0.25/0.08)
(0.03/−0.03)
υ 1 = 0.1,
υ 2 = 0.5
(36.66/41.07)
(3.09/0.79)
(−0.04/0.03)
(−0.07/−0.04)
υ 1 = 0.5,
υ 2 = 0.1
(10.83/10.85)
(1.05/0.45)
(0.31/0.16)
(0.13/−0.03)
υ 1 = 0.5,
υ 2 = 0.5
(36.71/40.92)
(3.12/0.79)
(−0.04/0.03)
(−0.04/−0.03)
Second largest
eigenvalue, block
NSRSE/ERSE
υ 1 = 0.1,
υ 2 = 0.1
(3.87/3.87)
(0.32/0.28)
(0.47/0.30)
(0.16/−0.02)
υ 1 = 0.1,
υ 2 = 0.1
(3.41/3.21)
(0.41/0.28)
(0.36/0.22)
(0.27/0.01)
υ 2 = 0.5,
υ 2 = 0.1
(8.3/9.71)
(0.85/0.44)
(0.06/−0.04)
(0.04/−0.06)
υ 1 = 0.5,
υ 2 = 0.5
(9.91/9.48)
(1.31/0.42)
(0.29/0.08)
(0.15/0.003)
SCIEntIFIC REPORTS | (2018) 8:14620 | DOI:10.1038/s41598-018-32891-4
are supermatrices of dimension 4 × 4 with real-symmetric 2 × 2 diagonal blocks. The off-diagonal blocks are
Grassmann variables with the structure
τ
η η
ξ ξ
τ
η
ξ
η
ξ
=
=
− −
*
*
*
*
*
and
(similarly, for ω). The super-integration measure is σ
π
σ σ σ σ η η ξ ξ
=
∂ ∂ ∂ ∂
−
⁎
⁎
d
d d d d
[ ] (2 )
aa
ab
bb
2
0
0
0
1
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, 62210, Cuernavaca, Mexico. 2 Fakultät für Physik, Universität Duisburg-Essen, Lotharstraβe 1, D-47048, Duisburg, Germany. 3 Centro Internacional de Ciencias, 62210, Cuernavaca, Mexico. Correspondence and requests for materials should be addressed to M.V. (email: [email protected]) or T.G. (email: [email protected]) or T.H.S. (email: [email protected])
© The Author(s) 2018
AcknowledgementsWe thank F. Leyvraz and M. Kieburg for useful discussions. Authors acknowledge financial support from UNAM/ DGAPA/PAPIIT research grant IA104617 and CONACyT FRONTERAS 201.Author ContributionsManan Vyas, Thomas Guhr and Thomas H. Seligman contributed to the entire development and writing of the paper in all aspects, except for numerics carried out exclusively by Manan Vyas.Additional InformationCompeting Interests: The authors declare no competing interests.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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CLT for linear spectral statistics of large-dimensional sample covariance matrices. Z D Bai, J W Silverstein, Ann. Prob. 32Bai, Z. D. & Silverstein, J. W. CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Prob. 32, 553-605 (2004).
J Morales, Master thesis, UNAM) Técnicas nuevas en el análisis de mercados de valores. Morales, J. (Master thesis, UNAM) Técnicas nuevas en el análisis de mercados de valores (2016).
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"Claude Roux [email protected] \nXerox Research Centre Europe 6\nchemin de Maupertuis\n38 240MeylanFrance\n"
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External linguistic resources have been used for a very long time in information extraction. These methods enrich a document with data that are semantically equivalent, in order to improve recall. For instance, some of these methods use synonym dictionaries. These dictionaries enrich a sentence with words that have a similar meaning. However, these methods present some serious drawbacks, since words are usually synonyms only in restricted contexts. The method we propose here consists of using word sense disambiguation rules (WSD) to restrict the selection of synonyms to only these that match a specific syntactico-semantic context. We show how WSD rules are built and how information extraction techniques can benefit from the application of these rules.
| null |
[
"https://arxiv.org/pdf/cs/0506048v1.pdf"
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cs/0506048
|
662b14c53ecce3326e1795873637eff50966992e
|
Enriching a Text by Semantic Disambiguation for Information Extraction
12 Jun 2005
Bernard Jacquemin [email protected]
Xerox Research Centre Europe 6
chemin de Maupertuis
38 240MeylanFrance
Caroline Brun [email protected]
Xerox Research Centre Europe 6
chemin de Maupertuis
38 240MeylanFrance
Claude Roux [email protected]
Xerox Research Centre Europe 6
chemin de Maupertuis
38 240MeylanFrance
Enriching a Text by Semantic Disambiguation for Information Extraction
12 Jun 2005
External linguistic resources have been used for a very long time in information extraction. These methods enrich a document with data that are semantically equivalent, in order to improve recall. For instance, some of these methods use synonym dictionaries. These dictionaries enrich a sentence with words that have a similar meaning. However, these methods present some serious drawbacks, since words are usually synonyms only in restricted contexts. The method we propose here consists of using word sense disambiguation rules (WSD) to restrict the selection of synonyms to only these that match a specific syntactico-semantic context. We show how WSD rules are built and how information extraction techniques can benefit from the application of these rules.
Introduction
In today's world, the society of communications is gaining in importance every day. The amount of electronic documents -mainly by Internet, but not only -grows more and more. With this increase, no one is able to read, classify and structure those documents so that the requested information can be reached when it is needed. Therefore we need tools that reach a shallow understanding of the content of these texts to help us to select the requested data.
The process of understanding a document consists in identifying the concepts of the document that correspond to requested information. This operation can be performed with linguistic methods that permit the extraction of various components related to the data that are requested.
Since the beginning of the '90s, several research projects in information extraction from electronic text have been using linguistic tools and resources to identify relevant elements for a request. The first ones, based on domainspecific extraction patterns, use hand-crafted pattern dictionaries (CIRCUS (Lehnert, 1990)). But systems were quickly designed to build extraction pattern dictionaries automatically. Among these systems, AutoSlog (Riloff, 1993;Riloff and Lorenzen, 1999) builds extraction pattern dictionaries for CIRCUS. CRYSTAL (Soderland et al., 1995) creates extraction patterns lists for BADGER, the successor of CIRCUS. These learners use hand-tagged specific corpora to identify structures containing the relevant information. The syntactic structure used by CRYSTAL is more subtle than the one used by AutoSlog. CRYSTAL is able to make the most of semantic classes. WHISK (Soderland, 1999) is one of the most recent information extraction system. WHISK has been designed to learn which data to extract from structured, semi-structured and free text 1 . A parser and a semantic tagger have been implemented for free text. This system is the only one to process all of these three categories of text.
These methodologies need domain-specific pattern dic- 1 We use the term "structured text" to refer to what the database community calls semi-structured text; "semi-structured text" is ungrammatical and often telegraphic text that does not follow any rigid format; "free text" is simply grammatical text (Soderland, 1999). tionaries that must be built for each different kind of information. However, none of these methods can be directly applied to generic information. Thus we decide to bypass these two obstacles: our approach is based on the utilization of an existing electronic dictionary, in order to expand the data in a document to equivalent forms extracted from that dictionary.
Our method deals with the identification of semantic contents in documents through a lexical, syntactic and semantic analysis. It then becomes possible to enrich words and multi-word expressions in a document with synonyms, synonymous expressions, semantic information etc. extracted from the dictionary.
Problems and Prospects
As for a lot of methodologies developed for natural language processing, the results of a method of information extraction are evaluated by two measures: precision and recall. Precision is the ratio of correctly extracted items to the number of items both correctly and erroneously extracted from the text; noise is the ratio of the faulty extracted items to all the achieved extractions. Recall is the ratio of correctly extracted items to the number of items actually present in the text. The problem consists in improving both precision and recall.
Recall improvement
A usual technique to improve the recall consists of enriching a text with a list of synonyms or near-synonyms for each word of that text. For example, all the synonyms of "climb" would be added to the document, even though some of those meanings have a remote semantic connection to the text. By this kind of enrichment, all the ways to express the same token (but not the same meaning) are taken into account.
This type of enrichment can be extended to synonymous expressions with a robust parser: syntactic dependencies and their arguments (the tokens belonging to the selected expression) are enlarged to dependencies that are generated out of the corresponding synonymous expressions.
The recall is usually optimised to the detriment of the precision with those techniques, since most words within a set of synonyms are themselves polysemous and are seldom equivalent for each of their meanings. Thus, a simply adding of all those polysemous synonyms in a document introduces meaning inconsistencies. Noise may stem from these inconsistencies.
Reduction of noise -Precision improvement
We notice that improving the recall using synonyms may often increase the noise. Although identified in the domain of IE, this problem is not yet solved and it has a negative influence on the system effectiveness. Our purpose is to use the linguistic context of the polysemous tokens to identify their meanings and select contextual synonyms or synonymous expressions. This approach should improve the precision in comparison with adding all the synonyms.
Sentences in the text: For example, the dictionary 2 entry for the word grimper contains a set of 5 synonyms. If we use these synonyms to enrich the original text, we obtain five variations of the original sentence. Only the second and the fourth of the enriching variations are accurate in this context. The meteorological context associated with the word température in the dictionary should correctly discriminate the synonyms in this context: in the dictionary, each synonym of a lemma is associated with a meaning of this lemma and with the typical linguistic context of the lemma in this sense.
Consequently, we decided to use the linguistic context of the words that can be enriched to discriminate which synonyms should be used and which should not. The synonyms are stored in the dictionary according to the sense of 2 The dictionary we use is a French electronic one (Dubois and Dubois-Charlier, 1997). We will give a more detailed information about it later. each lemma. So, the task amounts to performing a lexical semantic disambiguation of the text and using synonymous expressions in the selected meanings to enrich the document.
Enrichment method by WSD
Our experience in WSD
We previously have developed a range of tools and techniques to perform Word Sense Disambiguation (WSD), for French and English. The basic idea is to use a dictionary as a tagged corpus in order to extract semantic disambiguation rules, Brun, 2000;Dini et al., 1998;Dini et al., 2000). Since electronic dictionaries exist for many languages and they encode fine-grained reliable sense distinctions, be they monolingual or bilingual, we decided to take advantage of this detailed information in order to extract a semantic disambiguation rule database 3 . The disambiguation rules associate each word with a sense number taking the context into account. For bilingual dictionaries the sense number is associated with a translation, for monolingual dictionaries with a definition. WSD is therefore performed according to sense distinctions of a given dictionary. The linguistic rules have been created using functional dependencies provided by an incremental shallow parser (IFSP, (Aït-Mokhtar and Chanod, 1997)), semantic tags from an ontology (45 classes from WordNet (Fellbaum, 1998) for English) as well as information encoded in SGML tags of dictionaries. This method comprises two stages, rule extraction and rule application.
• Rule extraction process: for each entry of the dictionary, and then for each sense of the entry, examples are parsed with the IFSP shallow parser. The shallow parsing task includes tokenization, morphological analysis, tagging, chunking, extraction of functional dependencies, such as subject and object (SUBJ(X, Y), DOBJ (X, Y)), etc. For instance, parsing the dictionary example attached to one particular sense S i of drift :
1)The country is drifting towards recession.
Gives as output the following chunks and dependencies :
[ Using both the output of the shallow parser and the sense numbering from the dictionary we extract the following semantic disambiguation rule: When the ambiguous word "drift" has country as subject and/or toward recession as modifier, it can be disambiguated with its sense S i . We repeat this process as all dictionary example phrases in order to extract the word level rules, so called because they match the lexical context.
Finally, for each rule already built, we use semantic classes from an ontology in order to generalize the scope of the rules. In the above example the subject "country" is replaced in the semantic disambiguation rule by its ambiguity class. We call ambiguity class of a word, the set of WordNet tags associated with it. Each word level rule generates an associated class level rule, so called because it matches the semantic context: when the ambiguous word "drift" has a word belonging to the WordNet ambiguity class noun.location and noun.group as subject and/or a word belonging to the WordNet ambiguity class noun.shape, noun.act, and noun.state as modifier, it disambiguates with its sense S i . Once all entries are processed, we can use the disambiguation rule database to disambiguates new unseen texts. For French, semantic classes (69 distinctive characteristics) provided by the AlethDic dictionary (Gsi, 1993) have been used with the same methodology.
• Rule application process: The rule applier matches rules of the semantic database against new unseen input text using a preference strategy in order to disambiguate words on the fly. Suppose we want to disambiguate the word drift, in the sentence:
2) In November 1938, after Kristallnacht, the world drifted towards military conflict.
The dependencies extracted by the shallow parser, which might lead to a disambiguation, i.e., which involve drift, are:
SUBJ(world, drift) VMODOBJ(drift, towards, conflict)
The next step tries to match these dependencies with one or more rules in the semantic disambiguation database. First, the system tries to match lexical rules, which are more precise. If there is no match, then the system tries the semantic rules, using a distance calculus between rules and semantic context of the word in the text 4 . In this particular case, the two rules previously extracted match the semantic context of drift, because world and country shares semantic classes according to WordNet, as well as conflict and recession.
The methodology attempts to avoid the data acquisition bottleneck observed in WSD techniques. Thanks to this methodology, we built all-words (within the limits of the used dictionary) unsupervised Word Sense Disambiguator for French (precision: 65%, recall: 35%) and English (precision: 79%, recall: 34%).
Xerox Incremental Parser (XIP)
IFSP, which was used in the first experiments on semantic disambiguation at Xerox, has been implemented with transducers. Transducers proved to be an interesting formalism to implement quickly an efficient dependency parser, as long as syntactic rules would only be based on POS. The difficulty of using more refined information, such as syntactic features, drove us to implement a specific platform that would keep the same strategies of parsing as in IFSP, but would no longer rely on transducers.
This new platform (Aït-Mokhtar et al., 2001;Roux, 1999) comprises different sorts of rules that chunk and extract dependencies from a sequence of linguistics tokens, which is usually but not necessarily a sentence. The grammar of French that has been developed computes a large number of dependencies such as Subject, Object, Oblique, NN etc. These dependencies are used in specific rules, the disambiguation rules, to detect the syntactic and semantic information surrounding a given word in order to yield a list of words that are synonyms according to that context. Thus, a disambiguation rule manipulates together a list of semantic features originating from dictionaries, and a list of dependencies that have been computed so far. The result is a list of contextual synonyms.
If (Dependency 0 (t, t 0 ) & . . . & Dependency n (t,t k ) & . . . attribute p (t j )=v u )
synonym(t) = s 0 ,. . . ,s n . where t 0 ,. . . ,t n is a list of token s 0 ,. . . ,s n a list of synonyms.
Example:
• La température grimpe.
(the temperature is climbing)
• La température augmente.
(the temperature is rising)
• L'alpiniste grimpe le mont Ventoux. The contextual synonymy between grimper and augmenter can be defined with the following rule. The feature MTO is one of the semantic features that are associated with the entries of the Dubois dictionary. This feature is associated with each word that is connected to meteorology, such as chaleur, froid, température (heat, cold, température).
if (Subject(grimper, X) AND feature(X, domain)=MTO) synonym(grimper) = augmenter.
This rule applies on the above first example, La température grimpe, but fails to apply on the third sentence, L'alpiniste grimpe le mont Ventoux, since the subject does not bear the MTO feature.
Which WSD for which enrichment? 3.3.1. A very rich dictionary information
The new robust parser offers a flexible formalism and the possibility to handle semantic or other features. In addition to this parser, the semantic disambiguation now uses a monolingual French dictionary (Dubois and Dubois-Charlier, 1997). This dictionary contains many kind of information in the lexical field as well as in the syntactic or the semantic one. From the 115 229 entries of this dictionary, we can only use the 38 965 ones that are covered by the morphological analyser. These entries represent 68 588 senses, ie a polysemy of 1.76.
We build lexico-syntactic WSD rules using the methodology presented above (cf. section 3.1.): examples of the dictionary are parsed; extracted syntactic relations and their arguments are used to create the rules. We also make the most of the domain indication (171 different domains) to generalize the example rules (see later for details) -as previously done using WordNet for the English WSD and by AlethDic for the French one .
We use the specificity of the dictionary to improve the disambiguation task as far as possible in order to maximize the enrichment of the documents. The information of this dictionary is divided into several fields: domain, example, morphological variations, derived or root words, synonyms, POS, meaning, estimate of use frequency in the common language; in the verbal part of the dictionary only, syntactico-semantic class and subcategorization patterns of the arguments of the verb. Resulting WSD rules are spread over three levels reflecting the abstraction register of the dictionary fields.
Disambiguation rules at various levels
We build a disambiguation rule database at three levels: rules at word level (23 986), rules at domain level (22 790) and rules at syntactico-semantic level (40 736).
Word level rules use lexical information from the examples. They correspond to the basic rules in the previous system, which use constraints on words and syntactic relations. These dependencies are extracted from the illustrative examples from the dictionary. L'avion de la société décrit un large cercle avant de (. . . ) (The company's plane describes a wide circle before (. . . )) SUBJECT(décrire,avion) OBJECT(décrire,cercle) Example in the dictionary for the entry "décrire": L'avion décrit un cercle. Rules at domain level are generalized from word level rules: instead of using the words of the examples as ar-guments of the syntactic relations in the rules, we replace them by the domains they belong to. These rules correspond to the class level rules in the previous system, but an improvement in comparison with them is that in some cases, we can discriminate the right domain if the argument is polysemous. This is mainly due to the internal consistency of the dictionary that enables the correspondences of domain across different arguments of a dependency. The consistency should help to reduce the noise. We don't rule out the possibility of using other lexicosemantic resources to generalize or expand this kind of rules, as we did previously using French EuroWordNet or AlethDic. These lexicons present the advantage of a hierarchical structure that doesn't exist for the domain field in the Dubois dictionary. Nevertheless, we will encounter the problem of the mapping of the various resources used by the system to avoid inconsistencies between them, as shown in (Ide and Véronis, 1990;Lux et al., 1999;.
The third level of the rules currently in use in the semantic disambiguator is the syntactico-semantic one. The abstraction level of these rules is even higher than in the domain level. They are built from a syntactic pattern of subcategorization that indicates the typical syntactic construction of the current entry in its current meaning. Although the distinction between the arguments is very general -they are differentiated from human, animal and inanimate -our examination of the verbal dictionary indicates that, for 30% of the polysemous entries, this kind of rules is sufficient to choose the appropriate meaning.
Enrichment at various levels
WSD is not an end in itself. In our system, it is a means to select appropriate information in the dictionary to enrich a document. The quality and the variety of this enrichment vary according to the quality and the richness of the information in the dictionary. The variety of information allows several kind of enrichment.
For the specific task of information extraction, an index of the documents whose information is likely to be extracted is built. It allows the classification of all the linguistic realities extracted from text analysis. These realities are L'escadrille décrit son approche vers l'aéroport where (. . . ) (The squadron describes its approach to the airport where(. . listed according to the XIP-formalism: syntactic relations, arguments, and features attached to the arguments. The enrichment is done inside the index because dependencies can be added without affecting the original document.
Lexical level
Replacing a word by its contextual synonyms is the easiest way to perform enrichment. This method of recall improvement is very common in IE, but in our system, the enrichment is targeted according to the context thanks to the semantic disambiguation. This process often reduces the noise. The enrichment is achieved by copying the dependencies containing the disambiguated word and by replacing this word by one of its synonyms.
La température grimpe. (The temperature is climbing.)
Original index: SUBJECT(grimper,température) Set of targeted synonyms: monter, augmenter.
Enriched index: SUBJECT(grimper,température) SUBJECT(monter,température) SUBJECT(augmenter,température) Figure 6: Enrichment at lexical level.
Lexico-syntactic level
The lexico-syntactic level of enrichment is more complex to achieve. The task consists in replacing a word by a multi-word expression (more than 14 000 synonyms are multi-word expressions in our dictionary) or in replacing a multi-word expression by a word, taking into account the words (lexical) and the dependencies between them (syntactic): -Parse the multi-word expression to obtain dependencies;
-Match the corresponding dependencies in the text;
-Instantiate the missing arguments with the text arguments.
• Replacing a multi-word expression by a word:
-Identify the POS of the word;
-Select dependencies implying one and only one word of the multi-word expression;
-Eliminate dependency where this word has a different POS;
-Replace this word with its synonym in the remaining dependencies.
Le spécialiste aédité un manuscrit très abîmé. Since our work is based on the Dubois dictionarywhose entries are single words -most of the enrichment is one-to-one word. When a multi-word expression appears in the synonyms list, a single word has to be replaced by a multi-word expression, and the inverse process can be achieved if necessary. The complex case of replacing a multi-word expression by another multi-word expression could arise, but we never encounter this situation. The replacement of a multi-word expression by another is not yet implemented because of the complexity of the process. Nevertheless, the system relies on relations and arguments that are easy to handle, very simple and modular. These characteristics should allow us to bypass the inherent complexity of these structures.
A semantic level example
Syntactico-semantic fields in the dictionary allow a third enrichment level. The syntactico-semantic class structure contains very useful information that makes it possible to link verbs that are semantically related but lexically and syntactically very different. It might be interesting to semantically link vendre ("to sell", class D2a) and acheter ("to buy", class D2c) even though their respective actors are inverted. For example, le marchand vend un produit au client (the trader sells a product to the customer) bears the same meaning as le client achète un produit au marchand (the customer buys a product from the trader). The semantic class gives a general meaning of the verb(D2, meaning donner, obtenir, to give, to obtain), while the syntactic pattern (a for vendre: fournir qc qn, to supply so with sth, transitive with a oblique compliment, c for acheter: prendre qc qn, to take sth to so, transitive with a oblique compliment) yields the semantic realization. In a same perspective, a syntactico-semantic class constitutes another synonym set. Since this set is too general and too imprecise, it cannot be used to enrich a document. Still, it can be used as a last resort to enrich the query side when other methods have failed. We will not use this set as enrichment, but only to match a query by the class if the enrichment fails.
Evaluation
Though the method presented in this article is based on previous works, the use of other tools and lexical resource may have extended the potential of WSD rules. In particular, it is possible that the number of domains increase preci-sion, and the use of subcategorization patterns may ensure more general rules to increase recall.
The partial evaluation we performed concerns 604 disambiguations in a corpus of 82 sentences from the French newspaper Le Monde. Precision in WSD is ratio of correct disambiguations to all disambiguations performed; recall is ratio of correct disambiguations to all possible disambiguations in the corpus. We distinguish the mistakes due to the method and the ones linked to our analysis tools in order to identify what we have to improve in order to increase the performance. These results are promising since both precision and recall are better than in the previous system. We note some remarks about this evaluation:
1. The lexicon used to perform tokenization has been modified in order to include additional information from the dictionary. We noticed during this evaluation some problems of coverage;
2. For this first prototype, we do not yet establish a strategy for cases in which multiple rules match. If more than one rule can be applied to the context, the sense is randomly chosen among the ones suggested by the matching rules 5 ;
3. Conversely, we do not yet try a strategy using the domain of disambiguated words as a general context to choose the corresponding meaning of a word to disambigate.
During the evaluation, we also notice that when a result was correct, the suggested synonymous expressions were always correct for the disambiguated word in this context. Our method for an optimized enrichment is validated.
Conclusion
In this paper, we present an original method for processing documents, preparing the text for information extraction. The goal of this processing is to expand each concept by the largest list of contextualy synonymous expressions in order to match a request corresponding to this concept. Therefore, we implement an enrichment methodology applied to words and multi-word expressions. In order to perform the enrichment task, we have decided to use WSD to contextually identify the appropriate meaning of the expressions to expand. Inconsistent enrichment by synonyms is currently known as a major cause of noise in Information Extraction systems. Our strategy lets the system target the enriching synonymous expressions according to the semantic context. Moreover, this enrichment is achieved not only with single synonymous words, but also with multiword expressions that might be more complex than simple synonyms.
The WSD task and the resulting enrichment stage are achieved using syntactic dependencies extracted by a robust parser: the WSD is performed using lexico-semantic rules that indicate the preferred meaning according to the context. The linguistic information extracted from the analysis of the documents is indexed for the IE task. This index also stores additional new dependencies stemming from the enrichment process.
The utilization of a unique, all-purpose dictionary to achieve WSD and enrichment ensures the consistency of the methodology. Nevertheless, the information quality and richness of the dictionary might determine the system effectiveness.
The evaluation validates the quality of our method, which allows a great deal of lexical enrichment with less noise than is introduced by other enrichment methods. We have also indicated some ways our method could be expanded and our analysis tools could be improved. Our next step will be to test the effect of the enrichment in an IE task.
The method is designed to achieve a generic IE task, and the tools and resources are developed to process text data at a lexical level as well as at a syntactic or semantic level.
Figure 1 :
1Enrichment by a list of synonyms.
Figure 2 :
2(the alpinist climbs the mount Ventoux)• ???L'alpiniste augmente le mont Ventoux.(???the alpinist raises the mount Ventoux) Application of a disambiguation rule for enrichment.
Figure 3 :
3(The plane describes a circle.) SUBJECT(décrire,avion) OBJECT(décrire,cercle) WSD at word level.
LFigure 4 :
4'escadrille décrit son approche vers l'aéroport où (. . . ) (The squadron describes its approach to the airport where (. . . )) SUBJECT(décrire,escadrille[dom:AER]) OBJECT(décrire,approche[dom:LOC]) Example in the dictionary for the entry "décrire": L'avion décrit un cercle. (The plane describes a circle.) SUBJECT(décrire,avion[dom:AER]) OBJECT(décrire,cercle[dom:LOC]) WSD at domain level.
Figure 5 :
5WSD at lexico-semantic level.
•
Replacing a word by a multi-word expression (see figure 7):
(Figure 7 :
7The specialist published a very damaged manuscript.) Original index: SUBJECT(éditer,spécialiste) OBJECT(éditer,manuscrit) Targeted synonymous expression: etablir l'édition critique de Extracted dependencies from the expression: SUBJECT(établir,?) OBJECT(établir,édition) EPITHET(édition,critique) PP(édition,de,?) Enriched index: SUBJECT(éditer,spécialiste) OBJECT(éditer,manuscrit) SUBJECT(établir,spécialiste) OBJECT(établir,édition) EPITHET(édition,critique) PP(édition,de,manuscrit) Enrichment at lexico-syntactic level.
Figure 8 :
8Enrichment at semantic level.
OBJECT(offrir,cadeau) OBLIQUE(offrir,fille) offrir 01: D2a (to give sth to sb) D2a corresponds to D2e (receive, obtain sth from sb). recevoir 01: D2e Enriched index: SUBJECT(offrir,papa) OBJECT(offrir,cadeau) OBLIQUE(offrir,fille) SUBJECT(recevoir,fille) OBJECT(recevoir,cadeau) ????(recevoir,de,papa)Le papa offre un cadeauà sa fille.
(The father is giving a present to his
daughter.)
Original index:
SUBJECT(offrir,papa)
Table 1: WSD method evaluation.Tokenization mistakes
44
7.28%
Tagging mistakes
19
3.15%
Parsing mistakes
9
1.49%
WSD mistakes
84 13.91%
Precision
448 74.17%
Recall
43.61%
The English dictionary contained 39 755 entries and 74 858 senses, ie a polysemy of 1.88; the French dictionary contained 38 944 entries and 69 432 senses, ie a polysemy of 1.78
The first parameter of this metric is the intersection of the rule classes and the context classes; the second one is the union of the rule classes and the context classes. Distance equals the ratio of intersection to union.
This random choice is only performed for this evaluation and not in a IE perspective, since noise is better than silence in this field.
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|
[] |
[
"Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics",
"Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics"
] |
[
"Hui Sun \nFlorida State University\n\n",
"Nicholas J Moore \nUnited States Naval Academy\n\n"
] |
[
"Florida State University\n",
"United States Naval Academy\n"
] |
[] |
The truncated Korteweg-De Vries (TKdV) system -a shallow-water wave model with Hamiltonian structure that exhibits weakly turbulent dynamics -has been found to accurately predict the anomalous wave statistics observed in recent laboratory experiments. Majda et al. (2019) developed a TKdV statistical mechanics framework based on a mixed Gibbs measure that is supported on a surface of fixed energy (microcanonical) and takes the usual macroconical form in the Hamiltonian. This paper reports two rigorous results regarding the surface-displacement distributions implied by this ensemble, both in the limit of the cutoff wavenumber Λ growing large. First, we prove that if the inverse temperature vanishes, microstate statistics converge to Gaussian as Λ → ∞. Second, we prove that if nonlinearity is absent and the inverse-temperature satisfies a certain physically-motivated scaling law, then microstate statistics converge to Gaussian as Λ → ∞. When the scaling law is not satisfied, simple numerical examples demonstrate symmetric, yet highly non-Gaussian, displacement statistics to emerge in the linear system, illustrating that nonlinearity is not a strict requirement for non-normality in the fixed-energy ensemble. The new results, taken together, imply necessary conditions for the anomalous wave statistics observed in previous numerical studies. In particular, non-vanishing inverse temperature and either the presence of nonlinearity or the violation of the scaling law are required for displacement statistics to deviate from Gaussian. The proof of this second theorem involves the construction of an approximating measure, which we find also elucidates the peculiar spectral decay observed in numerical studies and may open the door for improved sampling algorithms. *
| null |
[
"https://arxiv.org/pdf/2010.02970v3.pdf"
] | 222,177,990 |
2010.02970
|
a088d437d6f49a3195dd5e347332390b00ab728b
|
Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics
May 10, 2022
Hui Sun
Florida State University
Nicholas J Moore
United States Naval Academy
Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics
May 10, 2022
The truncated Korteweg-De Vries (TKdV) system -a shallow-water wave model with Hamiltonian structure that exhibits weakly turbulent dynamics -has been found to accurately predict the anomalous wave statistics observed in recent laboratory experiments. Majda et al. (2019) developed a TKdV statistical mechanics framework based on a mixed Gibbs measure that is supported on a surface of fixed energy (microcanonical) and takes the usual macroconical form in the Hamiltonian. This paper reports two rigorous results regarding the surface-displacement distributions implied by this ensemble, both in the limit of the cutoff wavenumber Λ growing large. First, we prove that if the inverse temperature vanishes, microstate statistics converge to Gaussian as Λ → ∞. Second, we prove that if nonlinearity is absent and the inverse-temperature satisfies a certain physically-motivated scaling law, then microstate statistics converge to Gaussian as Λ → ∞. When the scaling law is not satisfied, simple numerical examples demonstrate symmetric, yet highly non-Gaussian, displacement statistics to emerge in the linear system, illustrating that nonlinearity is not a strict requirement for non-normality in the fixed-energy ensemble. The new results, taken together, imply necessary conditions for the anomalous wave statistics observed in previous numerical studies. In particular, non-vanishing inverse temperature and either the presence of nonlinearity or the violation of the scaling law are required for displacement statistics to deviate from Gaussian. The proof of this second theorem involves the construction of an approximating measure, which we find also elucidates the peculiar spectral decay observed in numerical studies and may open the door for improved sampling algorithms. *
Introduction
Abnormally large water waves, known variously as rogue, freak, or anomalous waves, have been the subject of intense scientific scrutiny over recent decades [19,15,17,53,13,12,23,14]. A recent line of experimental, numerical, and theoretical work demonstrates that anomalous waves can be triggered by abrupt changes in water depth [56,5,36,37,45,20,54,41,62,46,31,29,30,25,40,2]. In particular, a step in bottom topography is sufficient to generate highly non-normal surface-displacement statistics in a field of randomized, unidirectional water waves [56,5,36,41]. This arrangement offers a paradigm system: one in which the conditions are sufficient to generate anomalous waves, yet simple enough to offer the hope for rigorous analysis.
In particular, [36] created a statistical mechanics framework to help explain the experimental observations made by Bolles et al. (2019) [5]. The framework is based on statistical and dynamical analysis of the truncated Korteweg-de Vries (TKdV) equations, with exploitation of the Hamiltonian structure to construct a particular invariant measure that is consistent with physical constraints. The invariant measure corresponds to a mixed Gibbs ensemble that is macrocanonical in Hamiltonian and microcanonical in energy [1,36,41] (see Eq. (13) of the current paper). The latter condition mitigates the far-field, sign-indefinite divergence of the Hamiltonian, as recognized by Abromov et al. (2003) [1] for the truncated Burgers-Hopf system. As reported in [36,37,41], the TKdV model accurately predicts the anomalous behavior observed in experiments, including the heightened skewness of the outgoing wave-field, elevated levels of extreme events and intermittency, the amplification of high frequencies in the spectrum, and even the detailed shape of the outgoing surface-displacement distribution. Other studies have demonstrated the ability to predict these, and related, extreme events through machine learning or stochastic parameterization [45,16,8].
The above studies provide ample numerical evidence for the creation of anomalous wave statistics within the TKdV framework [36,37,41] and offer a relatively transparent statistical description in terms of the system's Hamiltonian, which is a macrostate quantity. The simplicity of this statistical description offers several advantages, for example: (1) it illustrates how the depth change enhances nonlinearity and thus promotes positive skewness in the outgoing wave field [36,41]; (2) it leads to an explicit formula linking the outgoing surface-displacement skewness to the change in slope variance. This formula, in fact, has been verified by experimental measurements [41]. However, the detailed distributions of surface displacements, or the microstates, implied by the theory have remained more elusive. So far, these microstate distributions have required numerical computation from either dynamical simulations or numerical sampling of the Gibbs ensemble. It is precisely these surface-displacement distributions that are of direct interest to practitioners who seek to quantify and predict anomalous waves in the field. This paper takes a major step towards providing the precise, mathematical link between the Gibbsbased macrostate description and the microstate statistics implied by it. Our main focus is to determine which conditions lead to Gaussian statistics of the surface displacement and which may result in anomalous statistics. We first prove that the zero inverse-temperature case, β = 0, of the Majda's mixed Gibbs ensemble implies Gaussian surface-displacement statistics in the limit of the cutoff wavenumber, Λ, growing large. The proof is elementary and it provides the foundation for subsequent analysis. Second, we examine the special case of positive inverse temperature, β > 0, with vanishing nonlinearity; i.e. the linear TKdV system, also known as the (truncated) Airy equation. From the form of the Hamiltonian, it is easy to see that nonlinearity promotes skewness in surface-displacement statistics. Furthermore, with nonlinearity absent, one can show that the surface-displacement distributions must be symmetric. However, for certain values of Λ and β, simple numerical sampling experiments demonstrate that the linear TKdV system can produce displacement statistics that, while they remain symmetric, are strongly non-Gaussian and exhibit significant kurtosis. Laboratory experiments often observe near Gaussian statistics when nonlinearity is sufficiently small, thus raising the question of what conditions are needed to restore normality. We resolve this question by identifying a scaling relationship between the inverse temperature, the cutoff wavenumber, and the dispersive coefficient that guarantees convergence to Gaussian statistics in the limit Λ → ∞. This proof is more involved than the β = 0 case and requires the construction of a particular approximating measure. In addition to its value in the proof, we find this approximating measure lends additional physical insight into the TKdV system, as it elucidates a peculiar spectral decay that has been observed in previous numerical studies [36,37,41]. Taken together, our rigorous results outline sufficient conditions for Gaussian statistics or, equivalently, necessary conditions for anomalous statistics. In particular, for surface-displacement statistics to deviate from Gaussian, the inverse temperature must be non-zero and either nonlinearity must be present or the scaling-law must be violated.
Related studies have performed statistical analysis on the nonlinear Schrödinger equation [27,28,6,7], a partial differential equation (PDE) with Hamiltonian structure similar to KdV. In particular, both PDEs exhibit the same far-field, sign-indefinite divergence of the simple canonical measure. To avoid this divergence, many studies impose an upper bound on energy, E < E 0 , which offers a considerable simplification in that one component of the invariant measure reduces to Weiner measure [6,7]. By contrast, the mixed microcanonical-macrocanonical ensemble of [36] -whose validity has been borne out by careful comparison to laboratory experiments -fixes the value of energy, E = E 0 . Geometrically, dynamics are confined to the surface of a hypersphere in spectral space, rather than its interior. Other studies have explicitly recognized this fixed-energy ensemble as the 'physically preferred choice' [27,28], but choose the bounded-energy formalism for technical convenience. In the case of fixedenergy, we will show that the analogous component of the mixed Gibbs measure does not reduce to Weiner measure and exhibits a markedly different spectral structure. We therefore cannot rely on the simple 1/k spectral decay of Weiner measure [6,7], nor do we artificially impose an empirical spectrum with randomly sampled multipliers [13,12]. Instead, we construct a non-trivial measure that converges to the Gibbs measure of the linear system as Λ → ∞, and the corresponding spectrum is allowed to emerge naturally. Thus, rather than being rooted in an empirical JONSWAP spectrum, our study firmly links the observable statistics to the fundamental physical principles of an entropy-maximizing Gibbs ensemble.
Interestingly, we find that this approximating measure, initially constructed as a theoretical tool, also appears to capture the spectral decay of the nonlinear system fairly well. This observation may prove valuable for future importance-sampling algorithms, as discussed further in Section 3.3.
There exist wide range of complex geophysical problems for which the extraction of statistical features is a driving interest. Examples abound from climate science [50,24,49], atmospheric science [42,58,9,52], morphology formation [33,21,10,47,59,39], and thermal convection [18,38,34,61,32,60]. The statistical analysis presented in this paper may ultimately prove valuable for such applications.
The outline of the paper is follows. In Section 2, we provide the relevant physical and mathematical background, including the Hamiltonian structure of TKdV and the fixed-energy, mixed Gibbs ensemble. In Section 3, we report the main results of the paper, including the numerical experiments that demonstrate nonlinearity is not a strict requirement for non-Gaussian statistics, and Theorems 3.1 and 3.2, each of which outline conditions that guarantee convergence to Gaussian statistics. Central to the proof of Theorem 3.2 is the construction of the approximating measure mentioned above. We discuss some salient properties of this measure in Section 3.3 and provide concluding remarks in Section 4. [5] provide the physical motivation for our study.
In these experiments, a randomized field of unidirectional water waves propagates through a region of constant depth, encounters an abrupt depth change (ADC) in the bottom topography, and then continues into a region of shallower depth. Bolles et al. (2019) [5] discovered that an initial Gaussian distribution of surface waves can become highly skewed upon encountering the ADC, with an elevated level of extreme events and enhanced intermittency a short distance downstream. Inspired by these observations, [36] developed the statistical mechanical TKdV framework discussed above and demonstrated its ability to recover a wide range of the experimental measurements [36,41].
The theory exploits the Hamiltonian structure of the TKdV system to construct invariant Gibbs measures that describe wave statistics upstream and downstream of the ADC. Importantly, the upstream and downstream measures are furnished with distinct inverse temperatures. This theory views the upstream state as an incoming wave-field with inverse temperature that is set externally, for example by the experimental apparatus or, in the ocean, by the strength and character of the wind or tidal forcing. The downstream state, however, is slave to the upstream one as determined by a statistical matching condition enforced at the ADC [36,41]. Ultimately, this matching condition yields the outgoing inverse temperature as a function of the specified incoming inverse temperature. By modifying the coefficients that enter the Hamiltonian and thus the Gibbs measure, the depth change can dramatically alter the character of the randomized waves and produce anomalous statistics.
The present study limits attention to the outgoing dynamics, supposing that its inverse temperature has already been set by enforcing the statistical matching condition. Our starting point is therefore the constant-depth KdV equation [26,22,57] for surface displacement u depending on horizontal location ξ and time t:
u t + C 3 uu ξ + C 2 u ξξξ = 0 for ξ ∈ [−π, π](1)
This equation holds in a moving reference frame that travels with the leading-order wave speed. Boundary conditions are periodic on the normalized domain ξ ∈ [−π, π]. All variables u, ξ, t are assumed to be dimensionless already, and the dimensionless coefficients C 3 and C 2 characterize the strength of nonlinearity and dispersion respectively (see Moore et al. 2020 [41] for how these dimensionless parameters relate to physical ones). The effect of the depth change is to increase the value of C 3 and decrease that of C 2 , thereby simultaneously promoting nonlinearity and suppressing dispersion. Once again, the present study limits attention to the downstream state, so that C 3 and C 2 remain constant. The KdV equation (1) enjoys a Hamiltonian structure, most easily defined by introducing the components
H 3 = 1 6 π −π u 3 dξ , H 2 = 1 2 π −π ∂u ∂ξ 2 dξ .(2)
Then the Hamiltonian can be expressed as
H = C 2 H 2 − C 3 H 3(3)
This expression makes clear the notational choices for the coefficients C 2 and C 3 above. The KdV equation (1) can then be written as
∂u ∂t = ∂ ∂ξ δH δu (4)
where ∂ ξ is a symplectic operator. Hence KdV (1) is a Hamiltonian system and, consequently, the Hamiltonian (3) is conserved during evolution. In addition, momentum and energy are conserved under KdV dynamics
M[u] ≡ π −π u dξ = 0, E[u] ≡ 1 π π −π u 2 dξ = 1(5)
The momentum vanishes, as indicated above, because u is measured as displacement from equilibrium. Meanwhile, the energy has been normalized to unity due to the choice of characteristic wave amplitude. We remark that, compared to Moore et al. (2020) [41], we have rescaled E by a factor of 2/π to simplify the forthcoming analysis. [36], we perform a finite Galerkin truncation of (1). To this end, consider a spatial Fourier representation of the state variable
The truncated KdV system and Hamiltonian structure
u(ξ, t) = ∞ k=−∞û k (t) e ikξ = ∞ k=1 a k (t) cos(kξ) + b k (t) sin(kξ) ,(6)u k = 1 2 (a k − ib k ) = 1 2π π −π u(ξ, t) e −ikξ dξ .(7)
For convenience, we have recorded both the real and the complex Fourier representations,û k ∈ C and a k , b k ∈ R. Hereafter, we will usually suppress the time dependence of these coefficients. Note that û −k =û * k since u(ξ, t) is real valued andû 0 = 0 due to momentum vanishing. Next, consider the Galerkin truncation at wavenumber Λ
u Λ (ξ, t) = P Λ u = |k|≤Λû k e ikξ = Λ k=1 a k cos(kξ) + b k sin(kξ) ,(8)
where P Λ is a projection operator and (7) still holds. Inserting the projected variable, u Λ , into the KdV equation and applying the projection operator, P Λ , again where necessary produces the truncated KdV equation (TKdV) [4,36,41] ∂u Λ ∂t
+ 1 2 C 3 ∂ ∂ξ P Λ (u Λ ) 2 + C 2 ∂ 3 u Λ ∂ξ 3 = 0 for ξ ∈ [−π, π](9)
Equation (9) represents a finite-dimensional dynamical system. The quadratic nonlinearity, ∂ ξ P Λ (u Λ ) 2 , mixes the modes during evolution, and the additional projection operator in this term removes the aliased modes of wavenumber larger than Λ. Typical values of the cutoff wavenumber used in the previous studies are Λ = 8-32 [36,41], whereas Majda & Qi (2019) found exact solutions for more severe truncations, as low as Λ = 2 [37]. The rigorous results obtained in the current study, however, hold in the limit of large cutoff-wavenumber, Λ → ∞. We remark that this analysis is not necessarily the same as direct analysis of the continuous KdV system (1). The TKdV equation (9) enjoys nearly the same Hamiltonian structure as KdV, with the only modification being the inclusion of the projection operator,
H Λ = C 2 H 2 [u Λ ] − C 3 H 3 [u Λ ] ,(10)∂ ∂t u Λ = ∂ ∂ξ P Λ δH Λ δu Λ(11)
where now ∂ ξ P Λ is the symplectic operator of interest. The system's microstate can either by described in physical space u Λ (ξ, t), in complex spectral space, (û 1 ,û 2 , · · · ,û Λ ) ∈ C Λ , or in real spectral space (a 1 , a 2 , · · · , a Λ , b 1 , b 2 , · · · , b Λ ) ∈ R 2Λ . All are equivalent through (7)- (8), and this paper primarily uses the real spectral representation. The momentum and energy defined in (5) are also conserved in the truncated system and have the same normalized values
M[u Λ ] = 0 and E[u Λ ] = 1. Parseval's identity implies E[u Λ ] = 4 Λ k=1 |û k | 2 = Λ k=1 a 2 k + b 2 k = 1(12)
Thus, in real spectral space, the TKdV dynamics of interest
x(t) := (a 1 (t), · · · , a Λ (t), b 1 (t), · · · , b Λ (t)) are confined to the unit hypersphere, S 2Λ−1 = {x ∈ R 2Λ : |x| = 1}
, which is a compact set. This geometric interpretation is of central importance to the present study. , we employ a mixed canonical-microcanonical Gibbs measure G to examine the statistical mechanics of (9). Similar to that used in the truncated Burgers-Hopf system [?, ?, 1], this ensemble is microcanonical in energy and canonical in the Hamiltonian. The basic idea can be gleaned through the abstract representation [4] dG ∝ exp(−βH)δ(E − 1)
The mixed canonical-microcanonical Gibbs ensemble
where β is the system's inverse temperature. The exponential dependence with respect to the Hamiltonian is the well-known canonical distribution, which, under suitable conditions, maximizes entropy [35]. The role of the Dirac-delta term δ(E − 1) is to confine the distribution to the compact set E = 1, thus avoiding the far-field, sign-indefinite divergence of the cubic component H 3 and thereby creating a normalizable distribution [1,4,36,41]. Previous studies have found that a positive inverse temperature, β > 0, yields a physically realistic decaying spectrum that agrees with laboratory experiments [36,41]. We remark that, when equipped with measure (13), the TKdV dynamical system is not strictly ergodic, but numerical evidence suggests that weak thermostatting of just the largest mode is sufficient to produce ergodic properties [4].
To gain a practical understanding of the mixed Gibbs measure (13), first consider the special case of zero inverse temperature, β = 0, for which (13) reduces to the uniform measure on S 2Λ−1 . The uniform measure can be identified with integration over R 2Λ by sampling from any rotationally invariant distribution and then normalizing to the unit hypersphere S 2Λ−1 [1]. More specifically, consider a random vector X = (X 1 , X 2 , · · · , X 2Λ ) ∈ R 2Λ drawn from any rotationally invariant distribution on R 2Λ . Identify this vector with a specific microstate by defining the real coefficients,
a k = X k / |X| , b k = X Λ+k / |X| , for k = 1, 2, · · · Λ (14) where |X| = 2Λ k=1 X 2 k 1/2
is the standard 2-norm. This normalization guarantees that (12) is satisifed, and the microstates corresponding to (14) are uniformly distributed on S 2Λ−1 [1]. For simplicity, we use the standard normal distribution on R 2Λ as the rotationally invariant measure:
dγ(x) := 2Λ k=1 1 √ 2π e −x 2 k /2 dx k(15)
Then, for any measurable function on the unit hypersphere φ : S 2Λ−1 → R, integration with respect to the uniform measure dG 0 on S 2Λ−1 can be written as
S 2Λ−1 φ dG 0 = R 2Λ φ(x) dγ(x)(16)
wherex = x/ |x| is a unit vector.
We can now precisely define the Gibbs measure (13) for arbitrary β. For a fixed location ξ and time t, and for any measurable function φ : (14) and (8). Meanwhile, Z β is the partition function, i.e. a normalization constant that depends on β and Λ. The analysis in this paper relies only on the mixed Gibbs ensemble (17), and not on any dynamics. Hence, in the remainder of the paper, we fix an arbitrary location in space ξ and time t, and analyze the corresponding microstates u Λ (ξ, t) sampled from (17). We first establish a simple Lemma that will be helpful in the analysis: Proof. The temporal independence is trivial since the Hamiltonian (10) is independent of time. Spatial independence follows from the fact that translation of the periodic function u Λ (ξ, t) in ξ preserves both components H 2 and H 3 , given in Eq. (2), and therefore preserves the Hamiltionian that is defined in Eq. (10) and used in the Gibbs measure (17).
S 2Λ−1 → R, let S 2Λ−1 φ dG = Z −1 β R 2Λ φ(x) exp(−β H[U Λ (x)]) dγ(x)(17)Above, U Λ (x) := u Λ (ξ, t) identifies a unit vectorx ∈ S 2Λ−1 with a particular microstate u Λ through
Results
In this section, we report the two main theorems of the paper. Both concern the statistical distributions of surface displacement, u Λ , implied by the mixed Gibbs ensemble (17) with inverse temperature β. Once again, we have fixed an arbitrary position in space ξ and time t, with the aid of Lemma 2.1, and so we have suppressed the dependence of u Λ on these variables. The first theorem treats the case of β = 0 with arbitrary C 3 and C 2 , while the second treats the case of β > 0 with nonlinearity absent, C 3 = 0.
TKdV with zero inverse temperature
We now introduce the first theorem:
Theorem 3.1 (Zero Inverse Temperature).
Consider the TKdV mixed-Gibbs ensemble (17) with zero inverse temperature, β = 0, and arbitrary choices of C 2 and C 3 . Under these conditions, the surface displacement, u Λ , convergences in probability to a normal random variable with mean zero and variance 1/2.
Proof. Consider the vector of the real Fourier basis elements evaluated at location ξ B := (cos ξ, cos 2ξ, · · · , cos Λξ, sin ξ, sin 2ξ, · · · , sin Λξ) ∈ R 2Λ
As suggested by (15), let X = (X 1 , X 2 , · · · , X 2Λ ) ∈ R 2Λ be a standard normal random vector X ∼ N (0, I 2Λ ) (i.e. the components X k are i.i.d. with X k ∼ N (0, 1)), and letX = X/ |X| be the corresponding unit vector. Then the corresponding microstate u Λ (ξ, t) = U Λ (X) defined by (14) can be written as
U Λ (X) = B ·X(19)
where · is the standard dot product on R 2Λ . This is nothing more than a vector representation of the real Fourier series (8), with standard normal i.i.d. coefficients, a k and b k , that have been normalized to satisfy the unit-energy constraint (12). LetB = B/ |B| and note that |B| = √ Λ. Then the above can be rewritten as
U Λ = |B| |X|B · X = √ Λ |X|B · X(20)
Consider the scalar random variable
|X| 2 2Λ = 1 2Λ 2Λ k=1 X 2 k(21)
where the components X k are i.i.d. with X k ∼ N (0, 1). By the law of large numbers (LLN),
|X| 2 2Λ p → 1 as Λ → ∞(22)
where p → indicates convergence in probability [55]. Now, returning to (20), the scalar productB · X is a finite sum of independent normal random variables, hence is a normal random variable. Furthermore, sinceB is a unit vector, the variance is unity, B · X ∼ N (0, 1). Substituting the limiting value (22) into (20) implies that, in the limit Λ → ∞, u Λ converges to a normal random variable with variance 1/2:
u Λ p → N (0, 1/2) as Λ → ∞(23)
3.2 Linear TKdV with positive inverse temperature Theorem 3.1 indicates that zero inverse temperature, β = 0, leads to Gaussian surface-displacement statistics, consistent with previous numerical results [36,41]. Those studies found that positive inverse temperatures, β > 0, can produce non-Gaussian displacement statistics, often with a high degree of skewness and sometimes even a bimodal structure [37]. Through the Hamiltonian, it is easy to see that nonlinearity, represented by the component H 3 in (2), promotes skewness in Gibbs ensemble (13), and is therefore at least partially responsible for the non-Gaussian statistics. This observation raises the question of whether nonlinearity is strictly required for non-Gaussian statistics. In short, we find the answer to be no, with some simple numerical experiments demonstrating strongly non-Gaussian features to arise in the linear TKdV Gibbs ensemble provided that the inverse temperature is sufficiently large. In particular, Fig. 1 shows three histograms of the surface displacement, u Λ , numerically sampled from (17) with C 3 = 0. Each of these histograms exhibits a significant departure from Gaussian -either enhanced flatness near the middle as in Fig. 1(b) or bimodality as in Fig. 1(a) and (c) -despite nonlinearity being completely absent. Figure 1(a) demonstrates non-Gaussian features under the simplest possible circumstances of a single complex mode Λ = 1. For this case, a simple thought experiment is enough to realize that ensemble statistics cannot be Gaussian. Indeed, since the energy is fixed, E = 1, in Gibbs ensemble (13), all sampled microstates are unimodal sine-waves of the same amplitude (though different phases). Therefore, the ensemble statistics follow the so-called arcsine distribution, characterized by sharp peaks at the extrema as evident in Fig. 1(a). This situation contrasts with the simpler, bounded-energy construct E < E 0 , for which the random amplitudes of sampled microstates permit near-Gaussian ensemble statistics even for Λ = 1. Furthermore, this example illustrates the subtlety of the fixed-energy ensemble -a construct that other authors have explicitly recognized as the 'physically preferred choice', but instead chose the bounded energy ensemble for technical convenience [27,28]. Here, we aim to confront some of the technical challenges presented by the the fixed-energy ensemble, rather than avoid them.
In this first example with Λ = 1, the Gibbs ensemble (17) is independent of the inverse temperature β, since each microstate consists of a single mode whose magnitude becomes normalized to satisfy E = 1. For Λ > 1, however, the inverse temperature plays a strong role. We introduce a normalized inverse temperature,
β = πβC 2 Λ 2 /2 ,(24)
which will prove convenient in the theoretical analysis. Figure 1(b)-(c) shows microstate histograms for the case Λ = 4 and two choices of β . These figures show non-Gaussian features to persist even when the number of modes is increased. As seen in Fig. 1(b), β = 10 produces a visibly flat distribution as compared to a Gaussian of the same variance (green dotted curve). The excess kurtosis is -0.7, indicating a significant departure from Gaussian statistics. Meanwhile, Fig. 1(c) shows that increasing β to 50 causes the bimodal structure seen previously to reemerge. This observation sheds light on previous numerical simulations, which showed skewed, bimodal distributions to arise dynamically under conditions of strong nonlinearity and large inverse temperature [37]. Such distributions were associated with phase-change behavior, but little explanation was given for why they emerge, other than observing their formation in direct numerical simulations. Our simple, sampling experiments suggest such states result from the combination of two separate effects: bimodality results from a large ratio of inverse temperature to cutoff wavenumber (i.e. large β ), while skewness results from strong nonlinearity as evident from the Hamiltonian structure.
Experimental studies, meanwhile, have typically observed near Gaussian statistics to arise under conditions of sufficiently weak nonlinearity, raising the question of how to reconcile the above numerical examples. The resolution is that all of these examples feature a moderate or small number of modes and a relatively large inverse temperature, giving β 1. Below, we will prove that if the scaling law is satisfied, then ensemble statistics of the linear TKdV system converge to Gaussian in the limit of large Λ. In terms of primitive quantities, this scaling law is βC 2 = O(Λ −2 ), which has two possible physical interpretations. First, one can consider fixed β and C 2 ∼ Λ −2 , which is a scaling that was justified on physical grounds by Moore et al. (2020) [41] and was found to produce consistency with laboratory measurements. In essence, this scaling requires the dispersive coefficient to decay with the cutoff wavenumber so that the peak spectral frequency is resolved in the TKdV model and, furthermore, lies near the logarithmic center of the resolved modes. Alternatively, one could consider C 2 fixed and β ∼ Λ −2 , which would be the interpretation most appropriate for relating to the continuous KdV system (1) with a non-vanishing dispersive coefficient C 2 . It is interesting, and perhaps meaningful, that our results suggest a particular scaling of the inverse temperature in order to make that connection. Interpretation aside, we henceforth assume that the scaling relation (25) holds to prove converge to Gaussian statistics in the linear system. For linear TKdV (C 3 = 0) the Hamiltonian reduces to
β = O(1) for Λ 1(25)H = 1 2 C 2 π −π ∂u ∂ξ 2 dξ = πC 2 2 |x| 2 Λ k=1 k 2 x 2 k + x 2 Λ+k(26)
where we have used Parseval's identify. Below, we will prove the surface displacement, u Λ (ξ, t) = U Λ (x), converges in distribution to a normal random variable. That is, for any bounded, continuous function
f ∈ C b (R → R), we will show Ef (u Λ ) → Ef (Z) as Λ → ∞, where Z is a normal random variable. Computing the expected value of such a function f ∈ C b (R → R) gives Ef (u Λ ) = S 2Λ−1 f (U Λ (x)) dG = Z −1 β R 2Λ f (U Λ (x)) exp − β Λ 2 |x| 2 Λ k=1 k 2 x 2 k + x 2 Λ+k dγ(x) (27)
We remark that, with the above formula, it is easy to see that the linear system always produces a symmetric surface-displacement distribution, regardless of the value of β . Indeed, the one-to-one mapping x → −x preserves the Hamiltonian (26) of linear TKdV, making any displacement value u Λ and its negative −u Λ equally likely. This observation is confirmed by the numerical examples from Fig. 1, which though strongly non-Gaussian, are clearly symmetric distributions. We next prove that if scaling law (25) is satisfied, not only is the surface-displacement distribution symmetric, but it also converges to a normal distribution.
Theorem 3.2 (Linear TKdV). Consider the TKdV mixed-Gibbs ensemble (17) with nonlinearity absent, C 3 = 0, with positive inverse temperature β > 0, and with scaling law (25) satisifed. Under these conditions, the surface displacement, u Λ , converges in distribution to a normal random variable with mean zero and variance 1/2.
Intuitive argument (non-rigorous). The essence of the proof is fairly intuitive, and so we provide this main part of the argument before going back to fill in some technical details. As seen in the proof of Theorem 3.1, for a standard normal vector X ∼ N (0, I 2Λ ), the LLN gives |X| 2 /(2Λ) p → 1 as Λ → ∞. This relationship motivates the naive substitution |x| 2 ≈ 2Λ in (27), or more generally |x| 2 ≈ 2Λ/α, where α > 0 is simply an extra degree of freedom. After making this substitution, some straightforward calculation produces
Ef (u Λ ) ≈ R 2Λ f (U Λ (ŷ)) dγ Σ (y)(28)
where γ Σ is a modified Gaussian measure with covariance matrix
Σ = diag σ 2 1 , σ 2 2 , · · · , σ 2 Λ , σ 2 1 , σ 2 2 , · · · , σ 2 Λ (29) σ 2 k = 1 1 + αβ k 2 /Λ 3 for k = 1, 2, · · · , Λ(30)
More explicitly
dγ Σ (y) = Λ k=1 1 2πσ 2 k exp − y 2 k + y 2 Λ+k 2σ 2 k dy k dy Λ+k(31)
The above is a change of measure from the standard, isotropic Gaussian dγ with covariance I 2Λ , to an anisotropic Gaussian with covariance Σ. Hence, we can follow the same reasoning as in the proof of Theorem 3.1, except with a random vector Y ∈ R 2Λ drawn from the altered normal distribution Y ∼ N (0, Σ). Notice that in (28) the normalization constant Z β has dropped out, which can be seen by simply taking φ ≡ 1 as the test function. Also, note that σ k is bounded above and below
1 1 + αβ /Λ < σ 2 k < 1 for k = 1, 2, · · · , Λ(32)
Mirroring the proof of Theorem 3.1, we fix an arbitrary location ξ and consider the vector of basis functions B ∈ R 2Λ from (18), which has corresponding microstate
u Λ = U Λ (Ŷ ) = |B| |Y |B · Y = √ Λ |Y |B · Y(33)
where Y ∼ N (0, Σ). The scalar productB · Y is a finite sum of independent normal random variables (though not identically distributed) and hence is a normal variable [55]. Considering the scalar random variable
|Y | 2 2Λ = 1 2Λ 2Λ k=1 Y 2 k .(34)
The LLN gives
|Y | 2 2Λ p → 1 2Λ Λ k=1 2σ 2 k as Λ → ∞(35)where 1 1 + αβ /Λ < 1 2Λ Λ k=1 2σ 2 k < 1(36)
as a result of (32). The convergence in probability in (35) is guaranteed because Var |Y | 2 /2Λ is finite as Λ → ∞ (as will be verified in the next section) [55]. Hence, the pre-factor √ Λ/ |Y | in (33) converges to a finite limit, implying that u Λ behaves asymptotically like a normal random variable.
What is the variance? The variance of u Λ can be computed by brute force (i.e. summing the variance σ 2 k of each random variable Y k ) or by simply recalling the normalization (12) and invoking Lemma 2.1 (spatial independence) to get Var (u Λ ) = 1/2 for any Λ. Therefore,
u Λ ∼ N (0, 1/2) approximately as Λ → ∞(37)
where the sense of convergence will be made more precise in the next section.
Rigorous proof. The preceding argument gives the main intuition underlying Theorem 3.2, but it is not rigorous since the naive substitution |x| 2 ≈ 2Λ/α that produced (28) was not justified. We will now determine the specific value of α for which this substitution can be justified, and we will bound the error that is incurred. The key is to perform the change of measure to dγ Σ while controlling the integrand carefully. We break the proof into a few parts, the first of which is a domain truncation that will eventually enable the change of measure.
Domain truncation: Consider a random vector X ∼ N (0, I 2Λ ), and the associated random variable
X 2 2Λ = 1 2Λ 2Λ k=1 X 2 k = |X| 2 2Λ(38)
The random variable X 2 2Λ has an expected value E X 2 2Λ = 1 and a variance of
Var X 2 2Λ = 1 4Λ 2 2Λ k=1 Var X 2 k = 1 Λ (39)
since Var X 2 k = 2 for each k. Consider the set
A = X 2 2Λ − 1 < Λ −p(40)
where p > 0 will be chosen later. Applying Chebyshev's inequality [55] with = Λ −p gives
P[A c ] = P X 2 2Λ − 1 ≥ Λ −p ≤ Λ 2p Var X 2 2Λ = Λ −1+2p(41)
which is equivalent to the measure bound γ(A c ) ≤ Λ −1+2p . To make the measure of A c asymptotically small, it suffices to chose any p in the range 0 < p < 1/2. We will now bound the integrand in (27) in order to show convergence as Λ → ∞. In particular, the normalization constant Z β must be included in these bounds. First, the bound
1 |x| 2 Λ k=1 k 2 Λ 2 x 2 k + x 2 Λ+k < 1 (42) immediately gives exp − β Λ 2 |x| 2 Λ k=1 k 2 x 2 k + x 2 Λ+k ≥ exp −β(43)
Therefore, the normalization constant Z β is bounded below by
Z β := R 2Λ exp − β Λ 2 |x| 2 Λ k=1 k 2 x 2 k + x 2 Λ+k dγ ≥ exp −β(44)
since γ is a probability measure. Now, the argument of the exponential in (27) is non-positive, and f ∈ C b (R → R) is bounded |f | < M . Therefore, (44) gives an overall bound on the integrand in (27),
Z −1 β f (U Λ (x)) exp − β Λ 2 |x| 2 Λ k=1 k 2 x 2 k + x 2 Λ+k ≤ M exp β(45)
Importantly, this bound is independent of Λ, since by assumption β = O(1). The bound (41), shows that γ(A c ) → 0 as Λ → ∞, and therefore
A c Z −1 β f (U Λ (x)) exp − β Λ 2 |x| 2 Λ k=1 k 2 x 2 k + x 2 Λ+k dγ → 0 as Λ → ∞(46)
Of course, the same holds for any subset of A c . In particular, taking a radius of R 2 = 2Λ(1 + Λ −p ) gives the containment |x| 2 ≥ R 2 ⊂ A c , and thus
Ef (u Λ ) − |x|<R Z −1 β f (U Λ (x)) exp − β Λ 2 |x| 2 Λ k=1 k 2 x 2 k + x 2 Λ+k dγ(x) → 0 as Λ → ∞ .(47)
Change of measure: So far, we have successfully truncated the integration (27) to a finite domain and shown that the expected value Ef is unaffected asymptotically as Λ → ∞. The next step is to perform a change of measure to the anistropic dγ Σ . First note that
1 |x| 2 = α 2Λ + 1 2Λ 2Λ |x| 2 − α(48)
where α > 0. Therefore, (47) can be rewritten as
Ef (u Λ ) − |y|<RZ β −1 f (U Λ (ŷ)) exp − β 2Λ 3 2Λ |y| 2 − α Λ k=1 k 2 y 2 k + y 2 Λ+k dγ Σ (y) → 0 as Λ → ∞ .
(49) Above, dγ Σ is the anisotropic Gaussian measure given by (31), andZ β is a modified partition function that can be calculated explicitly in terms of Z β if desired. The variables x and y are simply dummy integration variables, and we choose to write (49) in terms of y to distinguish the notation for the random variables that will be introduced soon. Now, the argument of the exponential in (49) is no longer non-positive over R 2Λ . In particular, it can grow arbitrarily large in the far-field, |y| 1, which is problematic for bounding the integrand. However, on the truncated domain, |y| < R, the argument can be bounded as follows
− β 2Λ 3 2Λ |y| 2 − α Λ k=1 k 2 y 2 k + y 2 Λ+k ≤ αβ 2Λ Λ k=1 k 2 Λ 2 y 2 k + y 2 Λ+k ≤ αβ 2Λ |y| 2 ≤ αβ 1 + Λ −p(50)
Additionally, the normalization constantZ β −1 ≤ exp(β ) can be bounded by the exact same reasoning as before, owing to the fact that dγ Σ is a probability measure. These results give the overall bound on the integrand in (49)
Z β −1 f (U Λ (ŷ)) exp − β 2Λ 3 2Λ |y| 2 − 1 Λ k=1 k 2 y 2 k + y 2 Λ+k ≤ M exp β + 2αβ(51)
which holds within the truncated domain |y| < R. Importantly, this bound is independent of Λ by the scaling assumption (25). We now apply the Chebyshev-inequality argument used before, except this time with the random variable Y ∼ N (0, Σ) corresponding to the measure dγ Σ . Consider the associated random variable
Y 2 2Λ = 1 2Λ 2Λ k=1 Y 2 k = |Y | 2 2Λ(52)
This variable has expected value
µ := E Y 2 2Λ = 1 2Λ Λ k=1 2σ 2 k < 1(53)
The inequality above relies on the fact that σ 2 k < 1 for all k as expressed in (32). Meanwhile, the variance is given by
Var Y 2 2Λ = 1 4Λ 2 2Λ k=1 Var Y 2 k = 1 4Λ 2 Λ k=1 4σ 4 k < 1 Λ(54)
which relies on the fact that Var[X 2 ] = 2σ 4 for a normal random variable with variance σ 2 . Now consider the set
B = Y 2 2Λ − µ < Λ −q(55)
for some q > 0 to be chosen later. Chebyshev's inequality gives
P [B c ] ≤ Λ 2q Var Y 2 2Λ < Λ −1+2q(56)
where the bound (54) has been applied. As before, this bound in probability is equivalently to the bound in measure γ Σ (B c ) < Λ −1+2q , suggesting that q should be chosen in the range 0 < q < 1/2. Furthermore, if q ≥ p, then µ < 1 implies the containment B ⊂ {|y| < R}. In that case, (51) provides a bound for the integrand in (49), and γ Σ (B c ) → 0 as Λ → ∞, which implies
Ef (u Λ ) − BZ β −1 f (U Λ (ŷ)) exp − β 2Λ 3 2Λ |y| 2 − α Λ k=1 k 2 y 2 k + y 2 Λ+k dγ Σ (y) → 0 as Λ → ∞ .(57)
Next, we would like to choose the degree of freedom α to make the argument of the exponential in (57) small on the set B. Recall that, on the set B,
|y| 2 2Λ − µ < Λ −q(58)
Comparing (57) and (58) makes it clear that α = 1/µ should be chosen, which implies, through (53) and (30), that α is a root of the nonlinear function
F (α) := 1 − α Λ Λ k=1 1 1 + αβ k 2 /Λ 3(59)
We note that F (0) = 1 and lim α→∞ F (α) = 1 − Λ 3 (Λ + 1)(2Λ + 1)/(6β ). The latter formula implies lim α→∞ F (α) < 0 for sufficiently large Λ as long as the scaling law β = O(Λ 0 ) holds, in which case the intermediate value theorem guarantees a root to exist on the interval 0 < α < ∞.
Setting α equal to the root of (59), implies that 2Λ
|y| 2 − α = O(Λ −q )(60)
holds on the set B. Furthermore,
1 2Λ 3 Λ k=1 k 2 y 2 k + y 2 Λ+k ≤ 1 2Λ Λ k=1 k 2 Λ 2 y 2 k + y 2 Λ+k ≤ |y| 2 2Λ(61)
and the right-hand side is bounded on the set B. Therefore, on the set B, the argument of the exponential in (57) is O(Λ −q ), which simplifies (57) to
Ef (u Λ ) − BZ β −1 f (U Λ (ŷ)) 1 + O(Λ −q ) dγ Σ (y) → 0 as Λ → ∞ .(62)
Finally, since γ Σ (B c ) → 0 as Λ → ∞ and the integrand in the above is bounded, the domain of integration can once again be expanded,
Ef (u Λ ) − R 2Λ f (U Λ (ŷ)) dγ Σ (y) → 0 as Λ → ∞ .(63)
where the vanishing O(Λ −q ) contribution has been removed. In (63),Z β has been replaced by its limiting valueZ β → 1 as Λ → ∞, which can be seen by simply taking φ ≡ 1 as the test function. The only requirements on p and q are that 0 < p ≤ q < 1/2, so it suffices to choose p = q = 1/3 for example. Importantly, the measure γ Σ in (63) is Gaussian. Therefore, the steps between (33)- (35) in the nonrigorous proof apply, with (54) providing the justification for (35). If Z = U Λ (Ŷ ), where Y ∼ N (0, Σ), then Z converges in probability to a normal random variable by the LLN. Furthermore, (63) shows that Ef (u Λ ) − Ef (Z) → 0 as Λ → ∞, hence u Λ converges in distribution to a normal random variable. The variance can be obtained by recalling the normalization (12) and invoking spatial independence, Lemma 2.1. We have therefore established the desired convergence in distribution:
u Λ d → N (0, 1/2) as Λ → ∞(64)
An improved importance distribution
Central to the proof of Theorem 3.2 is the construction of the approximate measure γ Σ from (31), with standard deviations σ 2 k given by (30) and α a root of (59). Not only is this measure useful as a theoretical tool, but it may pave the way for new numerical algorithms designed to efficiently sample from the Gibbs ensemble (17).
In particular, preliminary numerical tests indicate that the projection of γ Σ onto the hypersphere, E = 1, accurately captures the spectral decay of Gibbs measure dG from (17). Figure 2(a) illustrates this idea in the case of linear TKdV (C 3 = 0), with Λ = 16 and β = 10. In this figure, the true spectrum of dG, shown by the dots, is obtained numerically by the sampling-importance resampling (SIR) algorithm [51], where the instrumental distribution is taken to be the uniform measure dG 0 from (16). That is, microstates {û k , k = 1, 2, · · · , Λ} are sampled from dG 0 , then resampled with probability determined by their Hamiltonian. Roughly 10 7 microstate samples are used. This is essentially a bruteforce computation requiring significant resources in CPU time and memory. Meanwhile, Fig. 2(a) shows that the simple formula (30) for σ 2 k (solid curve) accurately predicts the decay structure, with the only numerical requirement being the computation of α as a root of (59). This close agreement suggests that γ Σ may serve as an improved instrumental distribution that samples the important regions of dG The numerically computed spectral density, E |û k | 2 (dots), is well predicted by the variances σ 2 k (solid) from Eq. (30). The dotted curve shows the naive Weiner-measure decay rate, 1/k 2 , which is not a good approximation to the true spectrum. (b) The surface-displacement histogram shows nearly Gaussian statistics for the linear system. (c) For nonlinear TKdV with C 3 /C 2 = 600 the spectrum is still well predicted by Eq. (30). (d) The histogram, however, shows a sensitive response to nonlinearity and exhibits a strong positive skewness. more faithfully than the uniform measure. For comparison, we also show the naive prediction 1/k 2 associated with Weiner measure (dotted curve), which clearly disagrees with the true spectrum. This simple decay structure, and the accompanied convenience of working with Weiner measure, applies only to the bounded-energy formalism, E < E 0 , used in other works [27,28,6,7]. In the figure, all spectra have been normalized to have unit mean so as to facilitate comparison.
For this case of linear TKdV, Theorem 3.2 suggests that the corresponding microstate statistics should be nearly Gaussian, provided that the scaling law on β is reasonably satisfied. To offer numerical confirmation of this important fact, Fig. 2(b) shows a histogram of surface displacements, u Λ , obtained by inverse discrete Fourier transform of each sample {û k , k = 1, 2, · · · , Λ}. The histogram, shown on a log-scale, is seen to agree closely with a normal distribution of the same variance (green dotted curve).
How does the picture change when nonlinearity is reintroduced? Figure 2(c)-(d) shows the corresponding spectra and histogram for the same set of parameters (Λ = 16, β = 10), but with C 3 /C 2 = 600 so as to introduce nonlinearity into the system (due to the normalization of the inverse temperature, only the ratio C 3 /C 2 needs to be specified). Interestingly, the spectral decay seen in Fig. 2(c) is nearly unchanged and remains well-predicted by formula (30). Evidently, the spectrum is controlled primarily by the contribution H 2 associated with the linear term, and is relatively insensitive to the contribution H 3 from the nonlinear term. Moreover, the decay rate of the spectrum is rather gradual, much more gradual than the 1/k 2 Weiner-measure decay for instance. This observation sheds light on previous dynamical simulations of KdV, which, when run until equilibrium, produced a broad spectrum with a gradual decay rate (up until the sudden drop-off near the largest resolved wave-numbers due to numerical truncation) [44,36,37,41]. A similar gradual decay of the spectrum has been observed in statistical and dynamical studies of the truncated Burgers-Hopf system [1,4].
While the spectrum is relatively insensitive to nonlinearity, the histogram shown in Fig. 2 a visible response. What was previously a symmetric histogram has become conspicuously skewed in the direction of positive u Λ . The slower decay of the tail as u Λ → +∞ corresponds to events of large surface displacement, e.g. rogue waves, occurring on a more frequent basis, as has been borne out numerically by other studies [36,37,41]. The fact that the spectrum remains well approximated by the variances σ 2 k calculated from formulas (30) and (59), even with the reintroduction of nonlinearity, suggests that the measure γ Σ from (31) could serve as the foundation for a highly efficient importance sampling algorithm [43]. In particular, this measure generates microstates with the proper spectral decay that could then be input into an acceptancerejection step to generate independent samples from the Gibbs ensemble. Roughly speaking, since the spectral decay has already been accounted for, there would only remain one criterion for acceptance, namely sufficient skewness. This would suggest a relatively high acceptance rate. Unlike the commonly used Markov-Chain Monte Carlo (MCMC), which produces correlated samples and does not have good parallelization properties, the proposed algorithm would generate independent samples and would be easy to parallelize. Though beyond the scope of the current paper, this is an exciting avenue for future work.
The parameters in the above numerical example, Λ = 16 and β = 10, were chosen to showcase the accuracy of our theory when the scaling law (25) is reasonably satisfied. A natural question is whether these results extend to realistic experimental parameters. To address this question, we now test the theory with the experimental parameters used by Moore et al. [41]. In particular, direct measurements of the wave amplitude, along with input parameters such as the water depth and peak forcing frequency, yield the ratio C 3 /C 2 = 140. Meanwhile, by matching the surface-displacement skewness observed downstream, Moore et al. [41] was able estimate the system's inverse temperature. With Λ = 16 modes, this estimate produces, in the notation of the current paper, the value β = 40. Interestingly, this value does not obviously satisfy the scaling condition β = O(Λ 0 ), thus providing a challenging test for our theory. Figure 3(a)-(b) shows the spectrum and surface-displacement histogram for the linear system (C 3 = 0) with Λ = 16 and β = 40. Somewhat surprisingly, the formula (30) still captures the spectral decay accurately, even with this relatively large value of β . Figure 3(b) shows that the surface-displacement statistics are nearly Gaussian in this linear case. Meanwhile, Figure 3(c)-(d) shows the spectrum and histogram for the nonlinear system, with the true experimental parameters of C 3 /C 2 = 140 and β = 40. Even with the reintroduction of nonlinearity and the large value of β , the formula (30) still approximates the numerically-sampled spectrum well. Thus, formula (30) seems to provide a good approximation even outside the strict asymptotic regime in which it was derived. Figure 3(d) shows that the strength of nonlinearity is sufficient to severely skew the surface-displacement statistics, as is consistent with the experimental measurements of Moore et al. [41] .
Conclusions
This paper reports two fundamental results, Theorems 3.1 and 3.2, on the surface-displacement distributions implied by a canonical-microcanonical Gibbs ensemble of the truncated KdV system. Theorem 3.1 establishes that vanishing inverse temperature, β = 0, implies Gaussian displacement statistics in the limit of large cutoff-wavenumber, Λ → ∞. For β > 0, our numerical sampling experiments demonstrate that the linear TKdV system can produce symmetric, yet highly non-Gaussian, statistics, contrasting with the common intuition that linearity leads to a Gaussian state. To offer resolution, we identify a precise scaling law, βC 2 = O(Λ −2 ), that guarantees convergence to Gaussian statistics in the linear system, as established by Theorem 3.2. In particular, the fixed-energy formalism, which has been recognized by others as the most physically relevant [27,28], renders the convergence to Gaussian non-trivial. The scaling law clearly delineates two parameter regimes: that in which the intuitive association of linearity with Gaussian statistics holds, and that in which it does not. Taken together, the new rigorous results imply necessary conditions for anomalous wave behavior, namely non-zero inverse temperature and either nonlinearity or the violation of scaling law (25).
In the regime of stronger nonlinearity, the form of H 3 in (2) shows that nonlinearity promotes skewed microstates in the Gibbs ensemble (17), in agreement with both laboratory and numerical experiments. However, determining the precise skewed distribution to which surface-displacement statistics converge under those conditions remains an open question. One candidate suggested by empirical evidence is a mean-zero gamma distribution [5,36]. In future work, we hope to extend the foundation established here to examine such possibilities.
In addition to the end results, the proofs of Theorems 3.1 and 3.2 provide important insights into TKdV statistical mechanics that will likely prove valuable in future studies. First, it is a remarkable coincidence that the same scaling relationship, βC 2 = O(Λ −2 ), obtained from physical reasoning in earlier work [41] serves a central role in the proof of Theorem 3.2. Without this relationship, the argument of the exponential in (27) cannot be controlled, thus making it difficult to bound the error incurred by the change of measure. Given this coincidence, it is possible that the scaling law (25) plays a more fundamental role in the statistical mechanics of TKdV and KdV than has been previously recognized. In particular, if one were to extend similar analysis to the continuous KdV system with non-vanishing dispersion, the scaling law (25) suggests taking β → 0 at a particular rate. There exist complementary results from statisticalmechanics analysis of continuous PDE systems [48,11,3] versus discrete counterparts [1,35,37], but it is not always clear how to reconcile the two. For the case of KdV, perhaps scaling law (25) can bridge these different perspectives.
Lastly, our proof of Theorem 3.2 relies on the construction of an approximating measure γ Σ , given in (31). For linear TKdV, we proved that the projection of this measure onto the hypersphere approximates the Gibbs measure dG as Λ → ∞. Moreover, numerical experiments suggest this measure to accurately capture the spectral decay rate of dG, even upon the reintroduction of nonlinearity. This observation suggests that γ Σ may serve as the foundation for an efficient and easily parallelized importance sampling algorithm to generate independent samples of the Gibbs ensemble. This possibility will be explored in future work.
KdV system and Hamiltonian structure The laboratory experiments of Bolles et al. (2019)
Following
Following
[36]
Lemma 2 . 1 (
21Spatiotemporal Independence). Microstate statistics of u Λ (ξ, t) corresponding to measure(17) are independent of location ξ and time t.
Figure 1 :
1Non-Gaussian statistics in linear TKdV: Numerical experiments demonstrate strongly non-Gaussian features in microstates sampled from Gibbs measure (17), despite nonlinearity being completely absent. (a) For a single mode, Λ = 1, statistics follow the arcsine distribution. (b) For Λ = 4 and β = 10, the distribution is visible flat compared to a Gaussian of the same variance (green dotted curve). (c) For higher β , the bimodality reemerges.
Figure 2 :
2Spectra and histograms for linear TKdV (top) and nonlinear TKdV (bottom) with Λ = 16 and β = 10. (a)
Figure 3 :
3Application of the theory to the experimental parameters used by Moore et al.[41]. Even with a value of β = 40 that lies outside the region of asymptotic validity, the formula (30) still captures the proper spectral decay for both linear (a) and nonlinear (c) TKdV. (b) The linear system exhibits nearly Gaussian microstate statistics. (d) The nonlinear system, with value C 3 /C 2 = 140 taken from laboratory experiments, shows a strong positive skewness as is consistent with the experimental measurements. In all cases the number of modes is Λ = 16.
(d) displaystrue spectrum
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AcknowledgementsN.J. Moore would like to acknowledge support from the National Science Foundation, DMS-2012560, and from the Simons Foundation, award 524259.
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[
"JET DIFFERENTIALS ON TOROIDAL COMPACTIFICATIONS OF BALL QUOTIENTS",
"JET DIFFERENTIALS ON TOROIDAL COMPACTIFICATIONS OF BALL QUOTIENTS"
] |
[
"Benoît Cadorel \nUniversité Grenoble Alpes\n\n"
] |
[
"Université Grenoble Alpes\n"
] |
[
"Ann. Inst. Fourier, Grenoble"
] |
We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.Résumé. -On donne des estimées explicites pour le volume des différentielles de jets de Green-Griffiths à tout ordre sur une compactification toroïdale d'un quotient de la boule. Pour ce faire, on détermine tout d'abord l'ordre de croissance des différentielles de jets de Green-Griffiths logarithmiques sur ces objets, en utilisant une déformation naturelle de l'espace des jets logarithmiques d'un ordre fixé, vers un fibré projectivisé à poids adéquat. Ensuite, on estime la croissance du nombre de conditions d'annulation au bord qu'une différentielle de jets logarithmique doit satisfaire pour être une différentielle de jets standard.(1) on it, such that for any m, E GG k,m Ω X = (π k ) * O k (m) is a vector bundle whose
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10.5802/aif.3356
|
[
"https://aif.centre-mersenne.org/article/AIF_2020__70_6_2331_0.pdf"
] | 119,599,467 |
1707.07875
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23b5d6671bfc136fb771c89ea2d25548dab01483
|
JET DIFFERENTIALS ON TOROIDAL COMPACTIFICATIONS OF BALL QUOTIENTS
2020
Benoît Cadorel
Université Grenoble Alpes
JET DIFFERENTIALS ON TOROIDAL COMPACTIFICATIONS OF BALL QUOTIENTS
Ann. Inst. Fourier, Grenoble
702020ANNALES DE L'INSTITUT FOURIER Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverteJet differentials, Ball quotients
We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.Résumé. -On donne des estimées explicites pour le volume des différentielles de jets de Green-Griffiths à tout ordre sur une compactification toroïdale d'un quotient de la boule. Pour ce faire, on détermine tout d'abord l'ordre de croissance des différentielles de jets de Green-Griffiths logarithmiques sur ces objets, en utilisant une déformation naturelle de l'espace des jets logarithmiques d'un ordre fixé, vers un fibré projectivisé à poids adéquat. Ensuite, on estime la croissance du nombre de conditions d'annulation au bord qu'une différentielle de jets logarithmique doit satisfaire pour être une différentielle de jets standard.(1) on it, such that for any m, E GG k,m Ω X = (π k ) * O k (m) is a vector bundle whose
Benoît C
Jet di erentials on toroidal compacti cations of ball quotients Tome , n o ( ), p.
-.
<http://aif.centre-mersenne.org/item/AIF_2020__70_6_2331_0> © Association des Annales de l'institut Fourier, , Certains droits réservés.
Cet article est mis à disposition selon les termes de la licence C C -. F . http://creativecommons.org/licenses/by-nd/3.0/fr/
Introduction
When dealing with complex hyperbolicity problems, finding global jet differentials on manifolds is an important question, since they permit to give restrictions on the geometry of the entire curves. Let us recall a few basic facts concerning Green-Griffiths jet differentials, which can be found in [9]. Let X be a complex projective manifold. Then, for any k, there exists a (singular) variety X GG k π k −→ X, and an orbifold line bundle O X GG k sections are holomorphic differential equations of order k, and degree m, for some suitable notion of weighted degree.
In [8], Demailly proves that if V is a complex projective manifold of general type, then for any k large enough, the Green-Griffiths jet differentials of order k will have maximal growth, or equivalently, O X GG k (1) is big. Finding an effective k for which this property holds is an interesting question, whose answer depends on the context: in [8], Demailly uses his metric techniques to give an effective lower bound on k in the case of hypersurfaces of P n .
We propose here a method to obtain a similar effective result in the case of toroidal compactifications of ball quotients (see [14] for the main properties of these manifolds). Specifically, we will find a combinatorial lower bound on the volume of E GG k,• Ω X , valid for any k :
Theorem 1.1. -Let X be a toroidal compactification of a ball quotient by a lattice with only unipotent parabolic isometries. Then, for any k ∈ N, we have the following lower bound on the volume of the k-th order Green-Griffiths jet differentials:
(1.1) vol(E GG k,• Ω X ) 1 (k!) n (K X + D) n (n + 1) n {u1 ··· un}⊂S k,n 1 u 1 . . . u n + (−D) n 1 i1 ··· in k 1 i 1 . . . i n ,
where S k,n is the ordered set S k,n = {1 1 < · · · < 1 n+1 < 2 1 < · · · < 2 n+1 < · · · < k 1 < · · · < k n+1 } .
Here, the fractions 1 u1...un are to be computed by forgetting the indexes on the integers in the set S k,n . In [5], the particular case where k = 1 (i.e. the case of symmetric differentials) was already proved, under the additional assumption that Ω X is nef. Our formula removes this hypothesis, and extends the result to any order k.
Using the results of [3], it is not hard to derive explicit orders k for E GG k,
• Ω X to have maximal growth:
Corollary 1.2.
-Let X = B n Γ be a toroidal compactification of a ball quotient. Let k ∈ N. Then, under any of the following hypotheses:
(1) n ∈ [|4, 5|] and k > e −γ e −(−D) n ((n−2)n!+1) ;
(2) n 6 and k > e −γ e π 2 6 (n−2)n!+1 The starting point to prove Theorem 1.1 consists in giving a more algebraic interpretation of the central metric construction of [8]. Let us give the main ideas about this construction. For any complex manifold X, the Green-Griffiths jet differential spaces X GG k can be deformed into a weighted projective bundle, using the standard construction of the Rees algebra. More specifically, there exists a family X GG k −→ X × C such that for any λ ∈ C * , the specialization (X GG k ) λ −→ X × {λ} is isomorphic to X GG k , and the specialization (X GG k ) 0 −→ X×{0} is isomorphic to the weighted projective bundle P(T (1)
X ⊕· · ·⊕T (k) X ). This last bundle is defined to be the quotient of T X ⊕ · · · ⊕ T X −→ X by the C * -action λ · (v 1 , . . . , v k ) = (λv 1 , . . . , λ k v k ). Moreover, there is a natural orbifold line bundle O X GG k
(1) on the family X GG k whose restriction to the fibers X GG k λ gives the tautological bundles of X GG k and P(T
(1) X ⊕ · · · ⊕ T (k) X )
. The metric used in [8] can actually be seen as a singular metric on O X GG k (1); it is constructed in such a way that its specialization to the zero fiber P(T (1)
X ⊕ · · · ⊕ T (k) X ) is induced by some metric on T X .
One convenient feature about this family X GG k is the fact that it permits to interpret the intersection products on the jet spaces X GG k in terms of the intersection theory on P(T (1) X ⊕ · · · ⊕ T (k) X ). When dealing with Chow groups computations, these last spaces share many properties with the usual weightless projective spaces. In the first part of our work, we will recall some results about the intersection theory with rational coefficients for a weighted projective spaces P(E
(a1) 1 ⊕ · · · ⊕ E (ap) p )
, which were proved first by Al-Amrani [1]. Since we work with rational coefficients instead of integer ones, the study is somewhat simplified; for the reader's convenience, we will explain how we could prove these results by following [11] in a standard way.
The other reason why studying the family X GG k is interesting is the fact that the main positivity properties (e.g. nefness, ampleness) of the tautological line bundle O(1) −→ P(T (1) X ⊕ · · · ⊕ T (k) X ) can be extended TOME 70 (2020), FASCICULE 6 from the fiber over 0 to other fibers over λ ∈ C * , i.e. to the line bundle O X GG k (1). Moreover, the positivity properties of O(1) on P(T (1)
X ⊕· · ·⊕T (k) X )
are directly related to the ones of the vector bundle T X . More generally, we will show in Section 3 that if E * 1 , . . . , E * p are ample (resp. nef), then the
orbifold line bundle O(1) −→ P(E (a1) 1 ⊕ · · · ⊕ E (ap) p
) is ample (resp. nef) in the orbifold sense for any choice of weights a 1 , . . . , a p . This will imply in particular the following result:
Proposition 1.3.
-Let X be a complex projective manifold. Assume that Ω X is ample (resp. nef). Then for any k ∈ N * , O X GG k (1) is ample (resp. nef) in the orbifold sense.
Getting back to the case of a ball quotient, we will use the logarithmic version of the previous discussion, and Riemann-Roch theorem in the orbifold case (see [17]) to obtain an estimate for the volume of the Green-Griffiths logarithmic jets differentials E GG k,• Ω X (log D) in terms of the Segre class of the weighted direct sum T X (− log D) (1)
⊕ · · · ⊕ T X (− log D) (k) .
This last Segre class can be in turn expressed in terms of the standard Segre class s • (T X (− log D)). An application of Hirzebruch proportionality principle in the non-compact case (see [15]) will give our final estimate on vol(E GG k,• Ω X (log D)), which will be the first member of the estimate (1.1). Finally, it will remain to relate the growth of the logarithmic jet differentials to the growth of the standard ones. To do this, we will simply bound from above the sections of the coherent sheaves Q k,m , defined for any k and m by the exact sequence
0 −→ E GG k,m Ω X −→ E GG k,m Ω X (log D) −→ Q k,m −→ 0.
We will find a suitable filtration on the sheaves Q k,m , in such a way that the graded terms are locally free above the boundary D, and can be expressed in terms of the vector bundles Ω D and N D/X . Then, using Riemann-Roch computations and the fact that D is a disjoint union of abelian varieties, we will be able to bound h 0 (Q k,m ) from above, for a fixed k, and m going to +∞. This will give the second term in the estimate (1.1).
Acknowledgments
I would like to thank my advisor Erwan Rousseau for his guidance and his support, and Julien Grivaux for many helpful and enlightening discussions. I thank also the anonymous referee for their suggestions which I hope have permitted to improve the quality and clarity of this article.
Segre classes of weighted projective bundles
We will now recall some results, first proved by Al-Amrani [1], permitting to construct Chern classes of weighted projective bundles. We will state the results in the simpler setting of Chow rings with rational coefficients.
E i , a i ) is the projectivized scheme of the graded O X - algebra Sym(E (a1) 1 ⊕ · · · ⊕ E (ap) p ) * , defined as Sym(E (a1) 1 ⊕ · · · ⊕ E (ap) p ) * = Sym E * 1 (a1) ⊗ O X · · · ⊗ O X Sym E * p (ap) ,
where, for any i, Sym E * i (ai) is the graded O X -algebra generated by sections of E * i (ai) in degree a i . We will denote this scheme by P(E (a1) 1 ⊕ · · · ⊕ E (ap) p ); remark that we use here the geometric convention for projectivized bundles.
We will say, by abuse of language, that E
(a1) 1 ⊕ · · · ⊕ E (ap) p
is a weighted direct sum, or even a weighted vector bundle.
Proposition 2.2. -The variety P(E (a1) 1 ⊕ · · · ⊕ E (ap) p
) has a natural orbifold structure (or a structure of Deligne-Mumford stack), for which the tautological line bundle O(1) is naturally defined as an orbifold line bundle. Moreover, this orbifold line bundle is locally ample, in the sense that the local isotropy groups of the orbifold structure act transitively on the fibres of O(1) (see for example [16]). Besides, if lcm(a 1 , . . . , a p )|m, the bundle O(m) can be identified to a standard line bundle on P(E
(a1) 1 ⊕ · · · ⊕ E (ap) r ).
Proof. -We can naturally endow P(E (a1) 1
⊕· · ·⊕E
(ap) r ) with a structure of Artin stack P, since it can be considered as a quotient stack
E 1 ⊕ · · · ⊕ E p C * , where C * acts by λ · (v 1 , . . . , v p ) = (λ a1 v 1 , . . . , λ ap v p ).
Locally on X, the weighted projectivized bundle can be trivialized as a product of the base with a weighted projectivized space P(a 1 , . . . , a p ), where each a i appears rk E i times. Consequently, the Artin stack P has locally an orbifold structure, which makes it an orbifold stack. The claims on O(1) are local, and they can be proved directly using [10] and [16].
Let us start our review of the properties of the Chow groups with rational coefficients of the weighted projectivized bundles.
Proposition 2.3. -Let E (a1) 1 ⊕ · · · ⊕ E (ap) p
be a weighted direct sum. Let us denote the natural projection by p :
P X (E (a1) 1 ⊕ · · · ⊕ E (ap) p ) −→ X.
For any k, there is an isomorphism
(2.1) A k P X E (a1) 1 ⊕ · · · ⊕ E (ap) p Q ∼ = r j=0 (A k−r+j X) Q , where r = p j=1 rk E j − 1.
To prove this result, we can start by checking it in the case where X is an affine scheme. In that case, the weighted projective bundle is a quotient of a standard (trivial) projective bundle by a finite group, and it suffices to use the fact that such a quotient induces an isomorphism on the Chow rings with rational coefficients. We can then use the localization exact sequence to prove the general result.
Using the isomorphism (2.1), we can now define the Segre classes associated with a weighted direct sum E
(a1) 1 ⊕ · · · ⊕ E (ap) p .
Definition 2.4. -Let X be a projective algebraic variety of dimension n, and let E
(a1) 1 ⊕ · · · ⊕ E (ap) p be a weighted direct sum on X. Let q : P(E (a1) 1 ⊕ · · · ⊕ E (ap) p ) −→ X be the natural projection. If k ∈ [|0, n|], the k-th Segre class of E (a1) 1 ⊕ · · · ⊕ E (ap) p is defined as an endomorphism of (A * X) Q . If α ∈ (A l X) Q , let s k E (a1) 1 ⊕ · · · ⊕ E (ap) p ∩ α = 1 m k+r q * c 1 O(m) r+k ∩ q * α .
where r = i rk E i − 1, and m = lcm(a 1 , . . . , a p ).
Remark 2.5. -In Definition 2.4, we could have replaced m by any integer divisible by lcm(a 1 , . . . , a p ). The important fact used here is that O(m) is a standard line bundle, which allows us to define its first Chern class in the usual way.
There is a Whitney formula for the weighted projective bundles, which permits to express the Segre classes s j (E
(a1) 1 ⊕ · · · ⊕ E (ap) p
) in terms of the s • (E j ) and of the weights (a j ):
Proposition 2.6. -Let E (a1) 1 ⊕ · · · ⊕ E (ap) p be a weighted projective sum. We have (2.2) s • E (a1) 1 ⊕ · · · ⊕ E (ap) p = gcd(a 1 , . . . , a p ) a 1 . . . a p 1 j p s • E (aj ) j ,
where, for any vector bundle E and any weight a ∈ N, we have
s • E (a) = 1 a rk E−1 j 0 sj (E) a j .
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To prove this result, we can use the "splitting principle" to get back to the case where the E i are all line bundles L i . Now, denote P = P(L (a0) 0 ⊕ · · · ⊕ L (ar) r ), and p : P −→ X the canonical projection. Then, for some m ∈ N, there exists a section of (p * L 0 ) ⊗l0 ⊗ O P (m) cutting out the subvariety P(L (a1) 1 ⊕ · · · ⊕ L (ar) r ) with some computable multiplicity. As in [11], we can use this fact to relate the Segre class s • (L
(a1) 1 ⊕ · · · ⊕ L (ar) r ) with the classes s • (L (a0) 0 ⊕ · · · ⊕ L (ar) r
) and c • (L 0 ). The formula then follows by induction.
Positivity of weighted vector bundles
We now study the extension of the usual positivity properties of vector bundles to the case of weighted vector bundles.
Definition 3.1. -Let E = E (a1) 1 ⊕ · · · ⊕ E (ap) p be a weighted direct sum. We say that E * = E * 1 (a1) ⊕ · · · ⊕ E * p (ap) is ample (resp. nef) if for any m ∈ N divisible enough, the (standard) line bundle O(m) is ample (resp. nef) on P(E). Remark 3.2.
-With the terminology of [16], saying that E * is ample amounts to saying that O(1) is orbi-ample on P X (E), the tautological orbifold line bundle being locally ample by [10].
We will see that the positivity properties of weighted vector bundles are exactly similar to the ones of the usual vector bundles, and can be proved in the same manner, following [13].
Proposition 3.3. -Assume that E * 1 ,. . . , E * p are ample on X.
Then, (1) For any coherent sheaf F on X, there exists m 1 ∈ N such that, for any m m 1 , the sheaf
F ⊗ a1l1+···+aplp=m S l1 E * 1 ⊗ . . . S lp E * p is globally generated. (2) For any ample divisor H on X, there exists m 2 ∈ N such that for any m m 2 , the sheaf a1l1+···+aplp=m S l1 E * 1 ⊗ · · · ⊗ S lp E * p is a quotient of a direct sum of copies of O X (H). (3) If lcm(a 1 , . . . , a p )|m, then O(m) is ample on P(E (a1) 1 ⊕ · · · ⊕ E (ap) p ). In particular, O(1) is orbi-ample.
Proof. -(1) This is easy to prove by induction on p using the similar characterization of ample vector bundles.
(2) It suffices to apply the point 1. to the sheaf F = O(−H).
(3) Because of (2), there exist m, N ∈ N and a surjective morphism
O X (H) ⊕N −→ a1l1+...aplp=m S l1 E * 1 ⊗ · · · ⊗ S lp E * p .
Besides, because of Lemma 3.4, increasing m if necessary, we can suppose that for any q ∈ N, the following natural morphism of vector bundles on X is surjective:
S q a1l1+...aplp=m S l1 E * 1 ⊗ · · · ⊗ S lp E * r −→ a1l1+···+aplp=mq S l1 E * 1 ⊕ · · · ⊕ S lp E * p .
We obtain a surjective morphism of graded O X -algebras
q 0 S q O X (H) ⊕N −→ q 0 a1l1+···+aplp=mq S l1 E * 1 ⊗ · · · ⊗ S lp E * p .
which determines an embedding
P E (a1) 1 ⊕ · · · ⊕ E (ap) p → P O X (−H) ⊕N , with O P(O X (−H) ⊕N ) (1)| P(E (a 1 ) 1 ⊕···⊕E (ap) p ) ∼ = O(qm).
Since the tautological line bundle on P(O X (−H) ⊕N ) is ample (cf. [13]), this ends the proof.
Lemma 3.4. -Let E 1 , .
. . , E p be C-vector spaces, and let a 1 , . . . , a p ∈ N * . Then, for any m ∈ N divisible enough, the natural linear maps
S q a1l1+···+aplp=m S a1 E 1 ⊗ · · · ⊗ S ap E p −→ a1l1+···+aplp=mq S a1 E 1 ⊗ · · · ⊗ S ap E p are onto for all q 1.
Proof. -Because of [10], if m is sufficiently large and divisible by all a 1 , . . . , a r , the (standard) line bundle O(m) on the weighted projective space P pt (E * 1 (a1) ⊕ · · · ⊕ E * p (ap) ) is very ample. Consequently, there exists
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is onto for all p 1, which gives the result.
We will now study the case of nef line bundles. We will prove the following result.
Proposition 3.5. -Let E (a1) 1 ⊕ · · · ⊕ E (ap) p be a weighted direct sum. Assume that E * 1 , . . . , E * p are nef. Then, if m is sufficiently divisible, the line bundle O(m) is nef on P(E (a1) 1 ⊕ · · · ⊕ E (ap) p ).
To this aim, we will use the formalism of vector bundles twisted by rational classes (see [13] for the definition and the positivity properties of these objects). As in the weightless case, we naturally define the notion of ampleness for a weighted sum of twisted vector bundles: Definition 3.6. -We say that a weighted direct sum of twisted vector bundles of the form
E 1 a 1 δ (a1) ⊕ · · · ⊕ E p a p δ (ap) is ample, if for any m, divisible by lcm(a 1 , . . . , a p ) the Q-line bundle O(m)⊗ π * O X (mδ) is ample on P(E * 1 (a1) ⊕ . . . E * p (ap) ).
Remark 3.7. -In Definition 3.6, we consider twists of the form a 1 δ, . . . , a r δ with δ ∈ N 1 (X) Q . This is related to the fact that if E 1 , . . . , E r are vector bundles, and if L is a line bundle, we have, for any m,
a1l1+···+arlr=m S l1 (E * 1 ⊗ L ⊗a1 ) ⊗ · · · ⊗ (E * r ⊗ L ⊗ar ) = L ⊗m ⊗ a1l1+···+arlr=m S l1 E * 1 ⊗ · · · ⊗ E * r ,
which implies in particular that the weighted projective bundle P = P (
E 1 ⊗ L * ⊗a1 ) (a1) ⊗ · · · ⊗ (E p ⊗ L * ⊗ap ) (ap) is identified to the variety P = P(E (a1) 1 ⊕ · · · ⊕ E (ar) r ), with O P (m) ∼ = O P (m) ⊗ p * L ⊗m . Lemma 3.8. -Let E 1 a 1 δ , . . . , E r a r δ be twisted vector bundles on X. Assume that each E * i −a i δ is ample. Then the weighted direct sum E * 1 −a 1 δ (a1) ⊕ · · · ⊕ E * r −a r δ (ar)
is ample.
Proof. -We follow directly the proof presented in [13]. Because of Bloch-Gieseker theorem about ramified covers (see [13,Theorem 4.1.10]), TOME 70 (2020), FASCICULE 6 there exist a finite, surjective, flat morphism f : Y −→ X, where Y is a variety, and a divisor A such that f * δ ≡ lin A. We have a fibered diagram
P = P Y (f * E * 1 (a1) ⊕ ··· ⊕ f * E * r (ar) ) g / / P X (E * 1 (a1) ⊕ ··· ⊕ E * r (ar) ) = P Y f / / X. Let Q = P Y (f * E * 1 ⊗ O(a 1 A)) (a1) ⊕ · · · ⊕ (f * E * r ⊗ O(a r A)) (ar) . Then, we have a canonical identification Q ∼ = P , which leads to identifying the line bundle O Q (m) with O P (m) ⊗ π * Y O Y (mA), as mentioned in Re- mark 3.7.
Besides, the Q-line bundle
(3.1) g * (O P (m) ⊗ π * O X (mδ)) is canonically identified to O P (m)⊗π * Y O Y (mA), thus to O Q (m). However, since each E * i −a i δ is ample, and since f is finite, each vector bundle f * E * i ⊗ O(−a i A) is ample. Because of Proposition 3.3, the line bundle O Q (m) is ample, so the line bundle (3.1) is ample. But g is finite and surjective, so O P (m)⊗π * O X (mδ) is ample on P ,
An example of combinatorial application
We present a simple example of application of the previous discussion, which will turn out to be useful in Section 5, where we deal with jet bundles on a toroidal compactification of a quotient of the ball. Proposition 3.9. -Let k, n ∈ N. We have the following asymptotic upper bound, as r −→ +∞ :
j1+2j2+···+kj k =r (j 1 + · · · + j k ) n n! 1 k! 1 i1 ··· in k 1 i 1 . . . i n r n+k−1 (n + k − 1)! + O(r n+k−2 ).
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Proof. -Let X be an abelian variety of dimension n, endowed with an ample line bundle L. Because of Proposition 3.3, the weighted direct sum L (1) ⊕ · · · ⊕ L (k) is ample on X. This means that the orbifold line bundle O(1) is orbi-ample on P = P X L * (1) ⊕ · · · ⊕ L * (k) . By orbifold asymptotic Riemann-Roch theorem ( [17], see also [16]), we have then, for any m ∈ N,
h 0 orb (P, O(m)) P c 1 O(1) n+k−1 m n+k−1 (n + k − 1)! + O(m n+k−1 ).
However, because of Definition 2.4 and Proposition 2.6, X c 1 O(1) n+k−1 can be computed as
P c 1 O(1) n+k−1 = X s n L * (1) ⊕ · · · ⊕ L * (k) = 1 k! X i H i . . . i H i l i . . . i H i k i n ,
where H = c 1 (L). Expending the computation yields
P c 1 O(1) n+k−1 = (L n ) k! l1+···+l k =n 1 1 l1 . . . k l k = (L n ) k! 1 i1 ··· in k 1 i 1 . . . i k .
To obtain the result, it suffices to remark that we can identify the vector space H 0 (X, j1+2j2+···+kj k =m L ⊗(j1+···+j k ) ) to a subspace of the orbifold global sections of O(m). Thus :
h 0 X, j1+2j2+···+kj k =m L ⊗(j1+···+j k ) h 0 orb (P, O(m)).
Besides, a direct application of Riemann-Roch-Hirzebruch theorem and Kodaira vanishing theorem on X gives
h 0 (X, L ⊗(j1+···+j k ) ) = (j 1 + · · · + j k ) n n! (L n ) if j 1 + . . . j k = 0.
Combining all these equations, we get the inequality.
We can also get back the following classical result. TOME 70 (2020), FASCICULE 6 Proposition 3.10. -Let a 0 , . . . , a n ∈ N * . Let X = P(a 0 , . . . , a n ) be the associated weighted projective space, endowed with its tautological orbifold line bundle O X (1). We then have the asymptotic estimate
h 0 orb (X, O X (m)) = gcd(a 0 , . . . , a n ) j a j m n n! + O(m n−1 ).
Proof. -It is clear that the weighted direct sum C (a0) ⊕ · · · ⊕ C (an) −→ Spec C is ample, which means that O X (1) is orbi-ample. Then, using [16] and Definition 2.4,
h 0 orb (X, O X (m)) = X c 1 O(1) n · m n n! + O(m n−1 ) = s 0 (C (a0) ⊕ · · · ⊕ C (an) ) m n n! + O(m n−1 ).
Besides, because of Proposition 2.6, we have s 0 (C (a0) ⊕ · · · ⊕ C (an) ) = gcd(a 0 , . . . , a n )
j a j ,
which gives the result.
Green-Griffiths jet bundles
Deformation of the jet spaces
We first remark that for any projective complex manifold, there is a natural deformation of its Green-Griffiths jets spaces to a weighted projectivized bundles, which will permit us to apply the previous discussion to the study of jet differentials.
Let X be a projective complex manifold. For k ∈ N, we consider the Green-Griffiths jet differentials algebra E GG k,• Ω X . Recall (cf. [9]) that E GG k,• Ω X is endowed with a natural filtration, which can be described as follows.
For each (n 1 , . . . , n k ) ∈ N k , and any coordinate chart U ⊂ X, we define the (n 1 , . . . , n k )-graded term as the following space of local jet differentials:
F (l1,...,l k ) E GG k,m Ω X (U ) = I=(I1,...,I k ) a I (f ) I1 . . . (f (k) ) I k (|I 1 |, . . . , |I k |) (l 1 , . . . , l k ) .
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where for each I l = (p 1 , . . . , p n ), we write (f (l) ) I l = (f
(l) 1 ) r1 . . . (f (l)
n ) rn . In the above formula, the lexicographic order on N k is defined so that (p 1 , . . . , p k ) < (n 1 , . . . , n k ) means that either p k < n k , or p k = n k and (p 1 , . . . , p k−1 ) < (n 1 , . . . , n k−1 ) in the lexigraphic order for N k−1 .
The formula of derivatives of composed maps implies that these local definitions glue together to give a well defined N k -filtration F • E GG k,m Ω X , compatible with the O X -algebra structure on E GG k,• Ω X , and which is increasing with respect to the lexicographic order. Moreover, the graded terms occur only for (l 1 , . . . , l k ) such that l 1 + 2l 2 + · · · + kl k = m, and we have, in this case:
Gr (l1,...,l k ) F E GG
k,m = Sym l1 Ω X ⊗ · · · ⊗ Sym l k Ω X . By Definition 2.1, this means exactly that, as an O X -algebra,
(4.1) Gr F E GG k,• ∼ = Sym Ω (1) X ⊗ · · · ⊗ Sym Ω (k) X .
We can use the Rees deformation construction (see, e.g., [4]
), to construct a O X×C -algebra E GG k,• on X × C, such that for any λ ∈ C * , E GG k,• | X×{λ} is identified to E GG k,• Ω X , and E GG k,• | X×{0} is identified to Gr F (E GG k,• )
. Let us give a few details about the construction. We first define a graded O X×C k -algebra E GG k,• , as follows:
E GG k,m (U ×C k ) = l1,...,l k u l1,...,l k t l1 1 . . . t l k k u l1,...,l k ∈ F (l1,...,l k ) E GG k,m Ω X (U )
where (t 1 , . . . , t k ) are coordinates on C k . It is easy to check that for any (λ 1 , . . . , λ k ) ∈ C k such that each λ j = 0, we have E GG k,m | X×{(λ1,...,λ k )} ∼ = E GG k,m Ω X , and that E GG k,m | X×{(0,...,0)} ∼ = Gr F (E GG k,m Ω X ). Now, define E GG k,• to be the pullback of E GG k,• by the embedding (x, t) ∈ X × C → (x, t, . . . , t) ∈ X × C k .
Remark 4.1. -While E GG k,• seems to be the natural object arising in the construction above, it is more tractable to work over X × C with the sheaf E GG k,• . To define the latter, we could have used any embedding t ∈ C −→ (tα 1 , . . . , tα k ) ∈ C k , with α i = 0, so our choice (α 1 , . . . , α k ) = (1, . . . , 1) is rather arbitrary. The same phenomenon occurs in [8], where the metric on O X GG k (m) constructed by Demailly depends on some auxiliary parameters 1 , . . . , k ∈ R * + .
Applying the Proj functor, we obtain the following result. (1) on X GG k , such that :
(1) for any λ ∈ C * , X GG k λ is identified to X GG k , and O X GG k (1) λ iden- tified to O X GG k (1) ; (2) the fibre X GG k 0 is identified to the variety Proj X Gr F E GG k,• ∼ = P X T (1) X ⊕ · · · ⊕ T (k) X , and O X GG k (1) 0
is identified to the tautological line bundle of this weighted projective bundle. Proof. -Let us prove the second point, the first one being similar. By construction, we have a natural identification between
E GG k,• | X×{0} = m 0 E GG k,m | X×{0} and Gr F (E GG k,• Ω X ) = m 0 Gr F (E GG k,m Ω X )
Moreover, this identification is compatible with the grading in m. Besides, by (4.1), the latter sheaf is identified, as a sheaf of graded algebras, with
Sym Ω
(1)
X ⊗ · · · ⊗ Sym Ω (k) X = m 0 l1+2l2+···+kl k =m Sym l1 Ω X ⊗ · · · ⊗ Sym l k Ω X .
Now, by Definition 2.1, the projectivized bundle associated to the latter sheaf of graded algebras, with respect to the grading in m, is P(T (1)
X ⊕ · · · ⊕ T (k) X )
. This implies immediately the identifications of varieties and orbifold line bundles mentioned in the second point.
We can now show that some usual positivity properties of the cotangent bundle can be transmitted to the higher order jet differentials.
Proposition 4.4. -
If Ω X is ample (resp. nef), then for any k ∈ N * , E GG k,• Ω X is ample (resp. nef), meaning that O X GG k (1) is ample (resp. nef) as an orbifold line bundle.
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Proof. -Let X GG k −→ X × C be the variety given by Proposition 4.2, endowed with its orbifold line bundle O X GG k (1). Assume first that Ω X is ample. Then, because of Proposition 3.3, a suitable power of the tautological line bundle O(m) is ample on P(T (1)
X ⊕ · · · ⊕ T (k) X ) if m is sufficiently divisible. Because of Proposition 4.2, it means that O X GG k
(1)| 0 is ample. By semi-continuity of the ampleness property, for any λ ∈ C * in a Zariski neighborhood of 0, the orbifold line bundle O X GG k (1)| λ is ample. Again because of Proposition 4.2, this means exactly that E GG k,• Ω X is ample.
The case where Ω X is nef is dealt with in the same manner, using Proposition 3.5, and the fact that if O X GG k (1)| 0 is nef, then O X GG k (1)| λ is nef for any very general λ ∈ C (see [12]).
The previous discussion extends naturally to the case of logarithmic jet differentials. We then have the following proposition.
(1) for any λ ∈ C * , X GG,log k | λ is identified to X GG,log k , and O X GG,log k (1)| λ identified to O X GG k (1) ;
(2) the fibre X GG,log k | 0 is identified to (1) ⊕ · · · ⊕ T X (− log D) (k) , and O X GG,log k (1)| 0 is identified to the tautological orbifold line bundle of this weighted projectivized bundle.
Proj X Gr F (E GG k,• ) ∼ = P X T X (− log D)
Proof. -As before, it suffices to use the fact that E GG k,m Ω X (log D) admits a filtration whose graded algebra is
Sym Ω X (log D) (1) ⊗ · · · ⊗ Sym Ω X (log D) (k) .
In this setting, Proposition 4.4 extends naturally:
Proposition 4.6. -Let (X, D) be a smooth log-pair. If Ω X (log D) is nef (resp. ample), then for any k ∈ N * , E GG k,• Ω X (log D) is nef (resp. ample), meaning that O X GG,log k (1) is nef (resp. ample) as orbifold line bundle. and the restriction of this map to D shows that Ω X (log D)| D has a trivial quotient. This implies that Ω X (log D) is not ample.
The following result, combining two theorems of Campana and Păun [7], and Demailly [8], shows that the bigness of the canonical orbifold line bundle O(1) on P(T (1) X is of general type;
(2) for large k, E GG k,• X is big, meaning that the usual line bundle
O(m) −→ X GG k is big for m sufficiently divisible ; (3) for large k, the orbifold line bundle O(1) −→ P X (T (1) X ⊗ · · · ⊗ T (k) X ) is big, i.e. the line bundle O(m) is big for m sufficiently divisible.
Proof.
(1) ⇒ (2). -This is the main result of [8].
(2) ⇒ (3). -Let k ∈ N * large enough, and consider a sufficiently divisible m ∈ N * . Let X GG k −→ X × C be the variety given by Proposition 4.2, endowed with its tautological orbifold line bundle
O X GG k (1). For any λ ∈ C * , O X GG k (m)| λ is identified to O X GG k (m)
, which is big. Consequently, there exists a constant C > 0, such that for any λ ∈ C * ,
h 0 (X GG k | λ , O X GG k (m)| λ ) Cm n+nk−1 .
Since O X GG k (m) is flat on the base C, we deduce by semi-continuity that
h 0 (X GG k | 0 , O X GG k (m)| 0 ) Cm n+nk−1 .
Besides, X GG k | 0 and O X GG k (1)| 0 are identified with P X (T
X ⊕ · · · ⊕ T (k) X ) and to its tautological line bundle, so the previous inequality means exactly that O(1) is big on P(T (1)
X ⊕ · · · ⊕ T (k) X ). (3) ⇒ (1). -This result is proved in [7].
Application to the toroidal compactifications of ball quotients
Let Γ ∈ Aut(B n ) be a lattice with only unipotent parabolic isometries. Then, by [2] and [14], we can compactify the quotient X = B n Γ into a toroidal compactification X = X D, where D is a disjoint union of abelian
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varieties. From now, on, X will always denote such a toroidal compactification of a ball quotient.
Let k ∈ N * , and let X GG,log k −→ X × C be the family given by Proposition 4.5. Denote by P GG,log k ⊂ X GG,log k the fibre above 0 ⊂ C, which is isomorphic to P X (T X (− log D) (1) ⊕· · ·⊕T X (− log D) (k) ). Let us also denote by O P (1) the orbifold tautological line bundle on this weighted projective bundle.
Proposition 5.1. -The orbifold line bundle O X GG,log k (1) −→ X GG,log k is nef.
Proof. -The vector bundle Ω X (log D) is nef because of [5]. Thus, the result comes from Proposition 4.4.
If m 0 = lcm(1, . . . , k), the standard line bundle O X GG,log k (m 0 ) is nef. This gives the following asymptotic expansion:
(5.1) h 0 (X, E GG k,lm0 Ω X (log D)) = h 0 (X GG k , O X GG,log k (lm 0 )) = χ(X GG k , O X GG,log k (lm 0 )) + O(l n+nk−2 ) = X GG,log k c 1 O(m 0 ) m+nk−1 l n+nk−1 + O(l n+nk−2 ).
By Proposition 4.5, X GG,log k and P GG,log k are members of the same flat family X GG,log k −→ C. Thus, since the first Chern class is a topological invariant, we can compute the leading coefficient of this last expansion, as follows:
X GG,log k c 1 O(m 0 ) m+nk−1 = P GG,log k c 1 O P (m 0 ) m+nk−1 .
Then, using Definition 2.4, we find
X GG,log k c 1 O(m 0 ) m+nk−1 = m n+nk−1 0 X s n (T X (− log D) (1) ⊕ · · · ⊕ T X (− log D) (k) ).
TOME 70 (2020), FASCICULE 6 If we insert this equation in (5.1), we see that if m ∈ N is divisible by m 0 , we have
h 0 (X, E GG k,m Ω X (log D)) = X s n (T X (− log D) (1) ⊕ · · · ⊕ T X (− log D) (k) ) m n+nk−1 (n + nk − 1)! + O(m n+nk−2 ).
This gives the following value for the volume of E GG k,• Ω X (log D):
(5.2) vol E GG k,• Ω X (log D) = X s n (T X (− log D) (1) ⊕ · · · ⊕ T X (− log D) (k) ).
Combinatorial expression of the volume. Uniform lower bound in k
The volume (5.2) can be expressed as a certain universal polynomial with rational coefficients in the Chern classes of T X (− log D). The same polynomial, applied to the Chern classes of T P n over P n , permits to compute P n s n (T (1) P n ⊕ · · · ⊕ T (k) P n ), and Hirzebruch proportionality principle in the non-compact case (see [15]) implies X s n (T X (− log D) (1) ⊕ · · · ⊕ T X (− log D) (k) ) = (−1) n (K X + D) n (n + 1) n P n s n (T (1)
P n ⊕ · · · ⊕ T (k) P n )
Using Proposition 2.6, we can give an explicit combinatorial expression of this last quantity. Indeed, if we let we see that choosing exponents (l i,j ) 1 i n+1,1 j k such that i,j l i,j = n amounts to choosing a non-decreasing sequence u 1 · · · u n of elements of the ordered set
H = c 1 O P n (1), since s • (T P n ) = n i=1 (−1) i H i n+1 , we find (−1) n P n s n (T (1) P n ⊕ · · · ⊕ T (k) P n ) = (−1) n (k!) n n i=1 (−1) i H i n+1 n i=1 (−1) i H i 2 i n+1 · · · · · n i=1 (−1) i H i k i n+1 · [P n ] 0 = 1 (k!) n l1,1+l1,2+···+ln+1,k=n 1 1 l1,1+l2,1+···+ln+1,1 · . . . · k l1,k+···+ln+1,k (H n · [P n ]).S k,n = {1 1 < · · · < 1 n+1 < 2 1 < · · · < 2 n+1 < · · · < k 1 < · · · < k n+1 } ,
where each integer between 1 and k is repeated n + 1 times. The bijection between the set of choices of (l i,j ) and the set of sequences u 1 · · · u n can easily be made explicit : to (l i,j ), we associate the sequence (u i ), where the element j m is repeated l m,j times. Thus, we find
(−1) n P n s n T (1) X ⊕ · · · ⊕ T (k) X = 1 (k!) n {u1 ··· un}⊂S k,n 1 u 1 . . . u n ,
where, in the quotient appearing on the right hand side, we compute the product by treating the elements of S as ordinary integers (we forget their indexes). We then find an explicit combinatorial formula for the volume of logarithmic jet differentials of order k :
(5.3) vol E GG k,• Ω X (log D) = K X + D n (n + 1) n (k!) n {u1 ··· un}⊂S k,n 1 u 1 . . . u n .
It is not hard to use this formula to obtain a more tractable lower bound on the volume. Indeed, we have: where, in the first inequality, we use the fact that for any ordered set {u 1 · · · u n }, the number of distinct n-uples (v 1 , . . . , v n ) having the same elements is at least n!. The letter γ represents the Euler-Mascheroni constant.
We obtain the following lower bound, valid for any k 1:
vol E GG k,• Ω X (log D) K X + D n (log k + γ) n (k!) n n! .
This formula can be seen as an effective version of the asymptotic estimates of [8], in the case of logarithmic jet differentials on a toroidal compactification of a ball quotient.
Upper bound on the vanishing conditions on the boundary
We will now study the number of vanishing conditions on the boundary that a logarithmic jet differential must satisfy to be a standard one.
For any k ∈ N * , and any m ∈ N, we define a sheaf Q k,m , supported on D, in the following manner:
(6.1) 0 −→ E GG k,m Ω X −→ E GG k,m Ω X (log D) −→ Q k,m −→ 0
. Then, we have:
(6.2) h 0 (X, E GG k,m Ω X ) h 0 (X, E GG k,m Ω X (log D)) − h 0 (X, Q k,m ).
Filtration on the quotient Q k,m
Our goal is to obtain an upper bound on h 0 (Q k,m ), as m −→ +∞, with fixed k ∈ N. To do this, we will produce a sufficiently sharp filtration on the sheaf Q k,m , so that the graded terms are locally free O D -modules. We will then the bound from above the number of global sections of these graded terms. Consequently, Q k,m admits a induced filtration F 1 , whose graded terms can be written as a quotient of the corresponding graded terms in E GG k,m Ω X and E GG k,m Ω X (log D) :
(6.3) Gr F1 • (Q k,m ) = l1+2l2+···+kl k =m S l1 Ω X (log D) ⊗ · · · ⊗ S l k Ω X (log D) Im S l1 Ω X ⊗ · · · ⊗ S l k Ω X .
We will now produce successive refinements of the filtration F 1 , until we obtain a filtration whose graded terms are all locally free O D -modules. We
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can already simplify the quotient appearing in (6.3), using the following elementary result. Lemma 6.2. -Let E 1 , . . . E l be O X -modules. For any i, we consider a sub-module E i → E i . Then the quotient E 1 ⊗ · · · ⊗ E l Im (E 1 ⊗ · · · ⊗ E l ) admits a filtration whose i-th graded term can be identified with
E 1 ⊗ · · · ⊗ E i−1 ⊗ E i E i ⊗ E i+1 ⊗ · · · ⊗ E l .
Proof. -It suffices to consider the filtration induced on the quotient sheaf E 1 ⊗ . . . E l Im (E 1 ⊗ · · · ⊗ E l ) by the images of any of the sheaves appearing in the sequence of morphisms
E 1 ⊗· · ·⊗E l −→ . . . −→ E 1 ⊗· · ·⊗E i−1 ⊗E i ⊗· · ·⊗E l −→ . . . −→ E 1 ⊗· · ·⊗E l .
We deduce from this proposition and the previous one the existence of a filtration F 2 on Q k,m , whose graded module can be written
Gr F2 • (Q k,m ) = l1+2l2+···+kl k =m k i=1 S l1 Ω X ⊗ · · · ⊗ S li ⊗ · · · ⊗ S l k Ω X (log D), where S l = S l Ω X (log D) S l Ω X .
The O X -modules S l can be in turn filtered in O D -modules, using a filtration that was first introduced in [6]. For completeness, we will describe this filtration in our special case. Proposition 6.3. -For any l ∈ N, S l is endowed with a filtration, whose graded terms are O D -modules, written
Gr • (S l ) = l j=0 j s=0 N * D/X ⊗s ⊗ S l−j Ω D .
Proof. -According to [14], each boundary component T b admits a tubular neighborhood U , quotient of its universal cover U ⊂ C n−1 × C by a lattice Λ ⊂ C n−1 . The component T b can be identified to the quotient of
C n−1 × {0} by Λ. Let D • = C n−1 × {0} ⊂ U .
The elements a ∈ Λ act on Ω U (log D 0 ) in the following way:
a · d zn zn = d zn zn + n−1 i=1 γ i (a) d z i ; a · d z i = d z i if 1 i n − 1,
where the γ i : C n−1 −→ C are R-linear maps. The natural filtration by the degree of d zn zn in S l Ω U (log D • ) is consequently preserved by Λ, and induces a filtration G l on S l Ω U (log D)) whose graded terms are globally trivial and can be written
Gr G l j (S l Ω U (log D)) = d z n z n j · S l−j Ω D .
This expression in local coordinates shows that the induced filtration by G l on S l Ω U admits as general graded term
Gr G l ∩S l Ω U j (S l Ω U ) = I jD ⊗ O U d z n z n j · S l−j Ω D ,
where I jD is the sheaf of ideals of the divisor jD. Consequently, G l induces a new filtration on the quotient S l Ω U (log D) S l Ω U , whose graded terms are
Gr • (S l ) = O jD ⊗ O U S l−j Ω D .
To obtain the proposition, it suffices to refine this last filtration, remarking that O jD = O X I jD is itself filtered by 0 ⊂ I (j−1)D I jD ⊂ · · · ⊂ I lD I jD ⊂ · · · ⊂ O jD , whose successive quotients can be identified to
I lD I (l+1)D N * D/X ⊗l .
We can consequently refine the filtration F 2 , to obtain a new one F 3 , whose graded module is
Gr F3 • (Q k,m ) = k i=1 l1+2l2+···+kl k =m li ji=0 ji si=0 S l1 Ω X ⊗ · · · ⊗ S li−1 Ω X ⊗ N * D/X ⊗si ⊗ S li−ji Ω D ⊗ S li+1 Ω X (log D) ⊗ . . . ⊗ S l k Ω X (log D) .
Each one of the terms of this direct sum can be seen as an O D -module. Besides, we have seen in the proof of Proposition 6.3 that S l Ω X (log D)| D admits a natural filtration whose graded terms are trivial:
Gr • (S l Ω X (log D)) = l j=0 S j Ω D .
On the other hand, since each boundary component admit a tubular neighborhood, we have
Ω X | D = N * D/X ⊕ Ω D , ANNALES DE L'INSTITUT FOURIER so S l Ω X | D l j=0 N * D/X j ⊗ S l−j Ω D .
We can consequently refine a last time the filtration on Q k,m , to obtain the following proposition. Proposition 6.4. -For any k, m ∈ N * , there exists a filtration F • Q k,m , whose graded module is an O D -module written
Gr F • (Q k,m ) (6.4) = k i=1 l1+2l2+···+kl k =m li j1=0 . . . l k j k =0 ji si=0 N * D/X ⊗(j1+···+ji−1+si) ⊗ S l1−j1 Ω D ⊗ · · · ⊗ S l k −j k Ω D .
where all tensor products are taken over O D .
Upper bound on the graded terms of the filtration
We want to obtain an asymptotic upper bound on h 0 D, Gr F • (Q k,m ) when m −→ 0. We start by changing the indexing of the direct sums, so that we sum over r = j 1 + 2j 2 + · · · + kj k . If we proceed to the substitution l i ← l i − j i , we find:
Gr F • (Q k,m ) = m r=0 j1+2j2+···+kj k =r k i=1 ji si=0 N * D/X ⊗(j1+···+ji−1+si) ⊗ l1+2l2+···+kl k =m−r S l1 Ω D ⊗ · · · ⊗ S l k Ω D
The term on the right is a trivial vector bundle, because D is made of disjoint abelian varieties. Consequently, we have . TOME 70 (2020), FASCICULE 6 Recall that the line bundle N * D/X is ample (cf. [14]). Consequently, since the boundary is made of abelian varieties, Kodaira vanishing theorem yields χ(D, (N * D/X ) ⊗(j1+j2+···+si) ) = h 0 (D, (N * D/X ) ⊗(j1+j2+···+si) ), as soon as j 1 + j 2 + · · · + s i = 0. Besides, still because the boundary is a union of abelian varieties, Hirzebruch-Riemann-Roch theorem gives [(j 1 + · · · + j i ) n − (j 1 + · · · + j i−1 ) n ] + O((j 1 + · · · + j i ) n−1 ),
where we use the multinomial formula at the second and fourth lines. If we sum over i, and using the fact that for a fixed i, there is only one term for which j 1 + · · · + j i−1 + s i = 0, we finally find Applying Proposition 3.9, we can sum (6.6) over the j 1 , . . . , j k such that j 1 + 2j 2 + · · · + kj k = r, to find If we put these two asymptotic expressions in (6.5), we obtain the following final estimate on h 0 (Q k,m ), when m −→ +∞ :
(6.7) h 0 (Q k,m ) h 0 (Gr F • (Q k,m ))
Uniform lower bound in k on vol(E GG k,m Ω X )
Combining (6.2) with (5.3) and (6.7), we finally obtain the lower bound (1.1) on vol(E GG k,• Ω X ), which proves Theorem 1.1. The expression (1.1) being valid for any k, we can use the results of [3] to determine an order k after which the algebra E GG k,
• Ω X has maximal growth. For example, it is not hard to obtain an asymptotic expansion of (1.1), with leading coefficient 1 n!(k!) n (log k) n (K X + D) n + (−D) n = 1 n!(k!) n (log k) k (K X ) n . TOME 70 (2020), FASCICULE 6 Thus,
Definition 2.1. -Let X be a complex algebraic projective variety. Consider a family (E i , a i ) 1 i p , where the E i are vector bundles on X, and the a i are positive integers. The weighted projectivized bundle associated with the datum (
which gives the result. Proof of Proposition 3.5. -It suffices to show that for any ample class h ∈ N 1 (X) Q , the class O(m) ⊗ π * O(mh) is ample. Let h be such a class. Then since each E * i is nef, the twisted vector bundles E * i a i h are ample for any i. Consequently, by Lemma 3.8 and Definition 3.6, if lcm(a 1 , . . . , a r )|m, the line bundle O(m) ⊗ π * O X (mh) is ample on P(E * 1 (a1) ⊕ · · · ⊕ E * r (ar) ). This gives the result.
For any complex projective manifold X, and for any k ∈ N * , there exists a morphism of varieties X GG k −→ X × C, and an orbifold line bundle O X GG k
By construction, the identifications mentioned above already occur at the level of sheaves of algebras. To obtain the identifications when taking Proj functors, we just need to check that the gradings on these sheaves of algebras are compatible under these identifications.
Proposition 4.5. -Let (X, D) be a smooth log-pair. For any k ∈ N * , there exists a morphism X GG,log k −→ X × C and an orbifold line bundle O X GG,log k (1) on X GG,log k such that
Remark 4.7. -Proposition 4.6 is actually only relevant for the nef property. Indeed, except when X is a curve, the bundle Ω X (log D) cannot be ample in general: if D is smooth and dim X 2, we have the residue map Ω X (log D) −→ O D −→ 0, TOME 70 (2020), FASCICULE 6
X
) for k large enough suffices to characterize the manifolds of general type.
Proposition 4.8.-Let X be a projective smooth manifold. The following assertions are equivalent.
where each index l i,j (i ∈ [|1, n + 1|], j ∈ [|1, k|])represents a possible choice of power for H in the i-th factor of the product n l=1 (−1) l H l j l n+1 . Thus, ANNALES DE L'INSTITUT FOURIER
n (log k + γ) n
.
Proposition 6.1. -The inclusion of (6.1) preserves the natural filtrations on E GG k,m Ω X and E GG k,m Ω X (log D). Proof. -We only need to check this locally: this inclusion sends an jet differential equation of the form i,l (f The exponents of the different f (l) i are then preserved by the inclusion, so the natural filtrations are also preserved.
S
l1 Ω D ⊗ · · · ⊗ S l k Ω D .For a fixed (j 1 , . . . , j k ), we now compute
nnn
χ(D, (N * D/X ) ⊗(j1+j2+...ji−1+si) 1 + · · · + j i−1 + s i ) n−1 [−(−D) n ].We can sum this last term on s i , to findji si=0 (j 1 + · · · + j i−1 + s i ) − 1 l 1 , . . . , l i j l1 1 . . . j − 1 l 1 , . . . , l i j l1 1 . . . j − 1 l 1 , . . . , l i j l1 1 . . .
(j 1 + · · · + j n ) n [−(−D) n ] n! + O i (j 1 + · · · + j i ) n−1 .ANNALES DE L'INSTITUT FOURIER6.3. Final asymptotic estimate overh 0 (Q k,m ) as m −→ +∞
··· in k 1 i 1 . . . i n [−(−D) n ] + O(r n+k−2 ).Moreover, according to Proposition 3.10, we haverk l1+2l2+···+kl k =r S l1 Ω D ⊗ · · · ⊗ S l k Ω D (m nk−2 ).
··· in k 1 i 1 . . . i k m n+nk−1 (n + nk − 1)! + O(m n+nk−2 ).
is big. For the first values of n 4, this yields the lower bounds for log k displayed in Table 1.1. Table 1.1. Effective lower bounds on log k to have O X GGn+1
2π
−1
,
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the line bundle O X GG
k
(1) k
(1) big
TOME 70 (2020), FASCICULE 6
When K X is nef and big, we get back the asymptotic lower bound of[8].vol(E GG k,m Ω X ) (log k) n n!(k!) n vol(K X ) + O((log k) −1 ) .Explicit orders k to have a big E GG k,• Ω XIn this section, we prove Corollary 1.2. We will use (1.1) to determine an effective k after which E GG k,• Ω X is big. Let us begin by determining anthe datum of n integers 1 i 1 · · · i n k in non-decreasing order being equivalent to the one of an integer p giving the number of distinct i j , of p integers 1 j 1 < · · · < j p k, and of positive exponents l 1 , . . . , l p such that k l k = n. Now, for any p 1, we have:Let p n − 1, and choose l 1 , . . . , l p such that l 1 + · · · + l p = n and l i = 0 for any i. Necessarily, at least one l i is larger than 2, soIt is easy to see that l 1 +···+lp =n ∀i, li> 0 1 = n−1 p−1 (choosing the integers l i amounts to choosing p − 1 cuts in the set [|1, n|], i.e. among n − 1 possible cuts). Consequently, we findWe can use the following upper bound:where we used the mean value inequality in the last line. Thus,. TOME 70 (2020),FASCICULE 6Inserting (5.4) and (6.8), in (1.1), we find a lower bound of the formfor a certain C k ∈ R * , andLet us first deal with the case where n 6. According to[3], we have (K X + D) n + α(−D) n > 0 for all α ∈ ]0, ( n+1 2π ) n [. The only thing left now is to determine an integer k such that A(k, n) < n+1 2π n .Let j = log k + γ. We haveWe see that if j > π 2 6 (n−2)n!+1 Besides, if n ∈ [|4, 5|], then (K X ) n = K X + D n + (−D) n > 0. Consequently, since (K X ) n is an integer, (K X + D) n + (−D) n 1, and (K X + D) n + λ(−D) n > 0 for any λ ∈ ]0, 1 +
. E Gg K,• Ω X, Thus. > 0 as soon as A(k, n) <−(−D) n [. Thus, vol(E GG k,• Ω X ) > 0 as soon as A(k, n) <
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. Lorraine Benoît Cadorel Institut Élie Cartan De, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex (France) [email protected]é de Lorraine, Site de NancyBenoît CADOREL Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine, Site de Nancy, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex (France) [email protected]
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[
"Fundamental solutions for the super Laplace and Dirac operators and all their natural powers",
"Fundamental solutions for the super Laplace and Dirac operators and all their natural powers"
] |
[
"H De Bie ",
"F Sommen ",
"\nDepartment of Mathematical Analysis Faculty of Engineering\nClifford Research Group\nGhent University Galglaan\n\n",
"\n9000GentBelgium\n"
] |
[
"Department of Mathematical Analysis Faculty of Engineering\nClifford Research Group\nGhent University Galglaan\n",
"9000GentBelgium"
] |
[] |
The fundamental solutions of the super Dirac and Laplace operators and their natural powers are determined within the framework of Clifford analysis.
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10.1016/j.jmaa.2007.06.015
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[
"https://arxiv.org/pdf/0707.2862v1.pdf"
] | 16,567,332 |
0707.2862
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ba14cd09ab9d0ae6652d7824c84b5d6113fadf8f
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Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
19 Jul 2007 MSC 2000 : 30G35
H De Bie
F Sommen
Department of Mathematical Analysis Faculty of Engineering
Clifford Research Group
Ghent University Galglaan
9000GentBelgium
Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
350819 Jul 2007 MSC 2000 : 30G35Clifford analysisDirac operatorSuperspaceFundamental solution
The fundamental solutions of the super Dirac and Laplace operators and their natural powers are determined within the framework of Clifford analysis.
Introduction
In a previous set of papers (see [1,2,3,4]) we have developed the basic framework for Clifford analysis in superspace. Clifford analysis in standard Euclidean space is a function theory of the so-called Dirac operator and as such a generalization of the theory of holomorphic functions of one complex variable. Basic references are [5,6,7]. A superspace on the other hand is a generalization of the classical concept of space, where not only commuting variables are considered, but also a set of anti-commuting ones. These superspaces find their use in theoretical physics. We refer the reader to e.g. [8,9,10,11,12].
We have first established the basic algebraic framework necessary for developing a theory of Clifford analysis in superspace (see [1] and [2]). We have constructed the fundamental differential operators such as the Dirac and Laplace operators, the Euler and Gamma operators, etc. Next we have constructed a theory of spherical monogenics in superspace (see [3]), which was then used to introduce an integration over the supersphere and over superspace (see [4]), inspired by similarities with the classical theory of harmonic analysis. Moreover it turned out that this integration is equivalent with the one introduced in the work of Berezin (see [8,9]), although our approach offers better insight. Indeed, the definition given by Berezin is not motivated by any connection with classical types of integration, whereas our approach connects the integral over superspace with the well-known theory of integration in Euclidean space (see the discussion in [4]).
All previous work is situated on the level of polynomial functions. A next step is to consider larger algebras and to start the study of the function-theoretical properties of our differential operators. Thus is the aim of the present paper. More precisely we will determine the fundamental solutions for all natural powers of the super Dirac operator.
The paper is organized as follows. We start with a short introduction to Clifford analysis on superspace. Then we recapitulate the results on polyharmonic functions needed later on. In the next section we derive the fundamental solutions for the super Laplace and Dirac operators and compare them with an ad hoc approach, inspired by the theory of radial algebra. In the following section we extend this technique to construct fundamental solutions for all natural powers. Finally we discuss how the present technique is valid for a still larger class of differential operators.
Clifford analysis in superspace
We first consider the real algebra P = Alg(x i , e i ; x`j, e`j), i = 1, . . . , m, j = 1, . . . , 2n generated by
x i x j = x j x i x`ix`j = −x`jx`i x i x`j = x`jx i and
e j e k + e k e j = −2δ jk e`2 j e`2 k − e`2 k e`2 j = 0 e`2 j−1 e`2 k−1 − e`2 k−1 e`2 j−1 = 0 e`2 j−1 e`2 k − e`2 k e`2 j−1 = δ jk e j e`k + e`ke j = 0 and where moreover all elements e i , e`j commute with all elements x i , x`j. If we denote by Λ 2n the Grassmann algebra generated by the anti-commuting variables x`j and by C the algebra generated by all the Clifford numbers e i , e`j, then we clearly have that
P = R[x 1 , . . . , x m ] ⊗ Λ 2n ⊗ C.
The most important element of the algebra P is the vector variable
x = x + x`with x = m i=1 x i e i x`= 2n j=1 x`je`j.
One calculates that
x 2 = x`2 + x 2 = n j=1 x`2 j−1 x`2 j − m j=1 x 2 j .
The super Dirac operator is defined as
∂ x = ∂ x`− ∂ x = 2 n j=1 e`2 j ∂ x`2 j−1 − e`2 j−1 ∂ x`2 j − m j=1 e j ∂ x j .
Its square is the super Laplace operator:
∆ = ∂ 2 x = 4 n j=1 ∂ x`2 j−1 ∂ x`2 j − m j=1 ∂ 2 x j .
The bosonic part of this operator is
∆ b = − m j=1 ∂ 2 x j , which is the classical Laplace operator. The fermionic part is ∆ f = 4 n j=1 ∂ x`2 j−1 ∂ x`2 j .
For the other important operators in super Clifford analysis and their commutation relations we refer the reader to [1]. If we let ∂ x act on x we find that
∂ x x = m − 2n = M
where M is the so-called super-dimension. This numerical parameter gives a global characterization of our superspace and will be used in remark 1. We will also need the following basic formulae (see [1])
∂ x (x 2s ) = 2sx 2s−1 (1) ∂ x (x 2s+1 ) = (M + 2s)x 2s .(2)
In the case where m = 1, n = 0 this reduces to the familiar formula d dx x k = kx k−1 . Now we can consider several generalizations of the algebra P. This leads to the introduction of the function-spaces:
F(Ω) m|2n = F(Ω) ⊗ Λ 2n ⊗ C where F(Ω) stands for D(Ω), C k (Ω), L p (Ω), L loc 1 (Ω), . . .
with Ω an open domain in R m . Finally the space of harmonic functions, i.e. null-solutions of the super Laplace operator, will be denoted by H(Ω) m|2n ⊆ C 2 (Ω) ⊗ Λ 2n . Similarly we denote by M(Ω) m|2n ⊆ C 2 (Ω) m|2n the space of monogenic functions, i.e. null-solutions of the super Dirac operator. We have that M(Ω) m|2n ⊆ H(Ω) m|2n ⊗ C. Now we have the following theorem, which generalizes a classical result in harmonic analysis.
H(Ω) m|2n ⊗ C ⊆ C ∞ (Ω) m|2n M(Ω) m|2n ⊆ C ∞ (Ω) m|2n .
Proof. It suffices to give the proof for harmonic functions, as monogenic functions are also harmonic. So we consider a function f ∈ H(Ω) m|2n . Such a function can be written as
f = (α) f (α) x`α 1 1 . . . x`α 2n
2n with (α) = (α 1 , . . . , α 2n ), α i ∈ {0, 1} and f (α) ∈ C 2 (Ω). Expressing that ∆f = ∆ b f + ∆ f f = 0 leads to a set of equations of the following type
∆ b f (α) = (β) c (β) f (β) , |(β)| = |(α)| + 2 with c (β) ∈ R. We conclude that for every (α) there exists a k ∈ N (k ≤ n + 1) such that ∆ k b f (α) = 0. Hence f (α)
is polyharmonic and thus an element of C ∞ (Ω). We end this section with a few words on integration in superspace. The proper integral to consider is the so-called Berezin integral B (see [8,9] and [4]) which has the following formal definition
B = R m dV (x) ∂ x`2 n . . . ∂ x`1 , with dV (x) the Lebesgue measure in R m .
One can also define a super Dirac distribution as
δ(x) = δ(x)x`1 . . . x`2 n = δ(x) x`2 n n! with δ(x) the classical Dirac distribution in R m . We clearly have that < δ(x − y), f (x) > = B δ(x − y)(x`1 − y`1) . . . (x`2 n − y`2 n )f (x) = f (y) with f ∈ D(Ω) m|2n .
Fundamental solutions in R m
The fundamental solutions for the natural powers of the classical Laplace operator ∆ b are very well known, see e.g. [13]. We denote by ν m|0 2l , l = 1, 2, . . . a sequence of such fundamental solutions, satisfying
∆ j b ν m|0 2l = ν m|0 2l−2j , j < l ∆ l b ν m|0 2l = δ(x).
Their explicit form depends both on the dimension m and on l. More specifically, in the case where m is odd we have that
ν m|0 2l = r 2l−m γ l−1 , γ l = (−1) l+1 (2 − m)4 l l! Γ(l + 2 − m/2) Γ(2 − m/2) 2π m/2 Γ(m/2)(3)
with r = −x 2 . The formulae for m even are more complicated and can be found in [13].
Concerning the refinement to Clifford analysis, we clearly have that ν
m|0 2l+1 = ∂ x ν m|0 2l+2 is a fundamental solution of ∆ l b ∂ x .
Fundamental solution of ∆ and ∂ x
From now on we restrict ourselves to the case where m = 0. The purely fermionic case will be discussed briefly in section 7.
Our aim is to construct a function ρ such that in distributional sense ∆ρ = δ(x).
We propose the following form for the fundamental solution:
ρ = n k=0 a k (∆ n−k b φ)x`2 n−2k ,
with φ and a k ∈ R to be determined. Now let us calculate ∆ρ
∆ρ = (∆ b + ∆ f )ρ = n k=0 a k (∆ n−k+1 b φ)x`2 n−2k + n k=1 a k (2n − 2k)(2n − 2k − 2 − 2n)(∆ n−k b φ)x`2 n−2k−2 = a 0 (∆ n+1 b φ)x`2 n + n k=1 [a k − 2k(2n − 2k + 2)a k−1 ] (∆ n−k+1 b φ)x`2 n−2k .
So ρ is a fundamental solution if and only if a 0 (∆ n+1 b φ) = δ(x) 1 n! and a k satisfies the recurrence relation a k = 4k(n − k + 1)a k−1 .
The first equation leads to
φ = ν m|0 2n+2 , a 0 = 1 n! .
We then immediately find the following expression for the a k a k = 4 k k! (n − k)! , k = 0, . . . , n.
Summarizing we obtain the following theorem
= Γ(m/2) 2(2 − m)π m/2 n k=0 (−1) k+1 (n − k)! Γ(2 − m/2) Γ(k + 2 − m/2) r 2k+2−m x`2 n−2k ,(4)
where we have used formula (3).
As we have that ∆ν
= n−1 k=0 2 4 k k! (n − k − 1)! ν m|0 2k+2 x`2 n−2k−1 − n k=0 4 k k! (n − k)! ν m|0 2k+1 x`2 n−2k
is a fundamental solution for the operator ∂ x .
Remark 1. We could propose the following form
g = 1 (x 2 ) M −2 2
for the fundamental solution of ∆, where we have replaced m by the super-dimension M in the classical expression. This technique is inspired by radial algebra (see [14]), which gives a very general framework for constructing theories of Clifford analysis, based on the introduction of an abstract dimension parameter (in this case the superdimension). This leads partially to the correct result (see also [15]). Indeed, formally we can expand this as
g = 1 (x 2 ) M −2 2 = 1 (x 2 + x`2) M −2 2 = 1 (x 2 ) M −2 2 1 + x`2 x 2 1− M 2 = n k=0 1 − M 2 k x`2 k (x 2 ) M 2 −1+k .
The coefficients in this development are proportional to the ones obtained in theorem 2 (see also formula (4)), so this yields the correct result. This expansion is however only valid if m is odd.
The fundamental solution can of course be used to determine solutions of the inhomogeneous equation ∆f = ρ. We have for example the following Proposition 1. Let ρ ∈ D(Ω) m|2n , then a solution of ∆f = ρ is given by
f (x) = ν m|2n 2 * ρ = B ν m|2n 2 (x − y)ρ(y).
5 Fundamental solution of ∆ k and ∆ k ∂ x A similar technique as in section 4 can be used for the polyharmonic case. First we expand ∆ k as
∆ k = k j=0 k j ∆ k−j b ∆ j f .
This expansion is valid as ∆ b commutes with ∆ f . Now we propose the following form for its fundamental solution:
ρ = n l=0 a l ∆ n−l b φ x`2 n−2l
with φ and a l ∈ R still to be determined. We calculate that
∆ k ρ = n l=0 a l k j=0 k j ∆ n−l+k−j b φ ∆ j f x`2 n−2l .
As we have that, using formulae (1) and (2) in the case where m = 0, M = −2n,
∆ j f x`2 n−2l = 4 j (−1) j (n − l)! (n − l − j)! (l + j)! l! x`2 n−2l−2j , j ≤ n − l this yields ∆ k ρ = n l=0 a l k j=0 k j 4 j (−1) j (n − l)! (n − l − j)! (l + j)! l! ∆ n−l+k−j b φ x`2 n−2l−2j .
Putting ∆ k ρ = δ(x) leads to the following set of equations
a 0 ∆ n+k b φ = δ(x) 1 n! (5) k j=0 a l−j k j 4 j (−1) j (n − l + j)! (n − l)! l! (l − j)! = 0 (6) a −1 = a −2 = a −3 = . . . = 0.(7)
We immediately have that a 0 = 1/n! and that φ = ν m|0 2n+2k . Equation (6) can be simplified by the substitution
a l = 4 l l! (n − l)! b l to k j=0 b l−j k j (−1) j = 0, b 0 = 1
which has the solution (see the subsequent lemma 1)
b l = l + k − 1 l .
We conclude that
a l = 4 l (l + k − 1)! (n − l)!(k − 1)! , l = 0, . . . , n.
We can summarize the previous results in the following theorem.
4 l (l + k − 1)! (n − l)!(k − 1)! ν m|0 2l+2k x`2 n−2l ,
is a fundamental solution for the operator ∆ k .
In a similar vein we obtain the fundamental solution ν
l (l + k)! (n − l − 1)!k! ν m|0 2l+2k+2 x`2 n−2l−1 − n l=0 4 l (l + k)! (n − l)!k! ν m|0 2l+2k+1 x`2 n−2l ,
is a fundamental solution for the the operator ∆ k ∂ x .
We still have to prove the technical lemma we used in the derivation of theorem 4. Proof. Define the polynomial R l (x) by
R l (x) = min(k,l) j=0 (−1) j k j (x + k − j − 1)! (x − j)! = min(k,l) j=0 (−1) j k j (x + k − 1 − j) . . . (x + 1 − j).
We then have to prove that R l (l) = 0. We distinguish between three cases.
1) l ≤ k − 2 We claim that for all t ≤ k − 2 R t (x) = (−1) t k − 1 t (x + k − t − 1) . . . (x + 1)(x − 1) . . . (x − t).
This can be proven using induction. The case where t = 1 is easily checked. So we suppose the formula holds for t − 1, then we calculate
R t−1 (x) + (−1) t k t (x + k − 1 − t) . . . (x + 1 − t) = (−1) t (x + k − t − 1) . . . (x + 1)(x − 1) . . . (x − t + 1) × k t x − k − 1 t − 1 (x + k − t) = (−1) t k − 1 t (x + k − t − 1) . . . (x + 1)(x − 1) . . . (x − t) = R t (x)
which proves the hypothesis. Now clearly R l (l) = 0.
2) l = k − 1 Using the previous results, it is shown that in this case R k−1 (x) equals
R k−1 (x) = −(−1) k (x − 1) . . . (x + 1 − k) so R k−1 (k − 1) = 0. 3) l ≥ k Now we have that R l (x) = R k (x) = 0
so this case is also proven.
A larger class of differential operators
The technique used above can be extended to a larger class of differential operators. Suppose we consider an operator of the following form
P = L(x, ∂ x ) + ∆ f
with L(x, ∂ x ) an elliptic operator in R m and ∆ f the fermionic Laplace operator. Note that L(x, ∂ x ) and ∆ f clearly commute. An interesting operator in this class is the super Helmholtz operator
∆ − λ 2 , λ ∈ R with L(x, ∂ x ) = ∆ b − λ 2 .
Denoting by µ m|0 2k (k = 1, 2, . . .) a set of fundamental solutions for the operators L(x, ∂ x ) k such that
L(x, ∂ x ) j µ m|0 2k = µ m|0 2k−2j , j < k L(x, ∂ x ) k µ m|0 2k = δ(x)
we can now use the same technique as in section 5 to obtain a fundamental solution µ
Conclusions
In this paper we have developed a technique to construct fundamental solutions for certain differential operators in superspace. In particular we have constructed the fundamental solutions of the natural powers of the super Dirac operator ∂ k x . We envisage to use these fundamental solutions in a further development of the function theory of Clifford analysis in superspace. There is no doubt that they will play an important role in the generalization of e.g. the Cauchy and Hilbert transform to superspace.
• m commuting variables x i and m orthogonal Clifford generators e i • 2n anti-commuting variables x`i and 2n symplectic Clifford generators e`i subject to the multiplication relations
Theorem 1 .
1Null-solutions of the super Laplace and the super Dirac operator are C ∞ -functions, i.e.
Theorem 2 .
2The as in section 3, is a fundamental solution for the operator ∆.
Theorem 4 .
4The
Lemma 1 .
1The sequence (b l ), l = 0, 1, . . ., recursively defined by explicitly byb l = l + k − 1 l .
purely fermionic case In this case there is no fundamental solution. Indeed, determining the fundamental solution of ∆ f requires solving the algebraic equation ∆ f ν 0|2n 2 = x`1 . . . x`2 n which clearly has no solution, since there are no polynomials of degree higher than 2n.
Proof. It is clear that νm|2n
2
∈ L loc
1 (R m ) m|2n . Moreover, we have that ν
m|2n
2
∈
H(R m − {0}) m|2n and that ∆ν
m|2n
2
= δ(x) in distributional sense.
Now suppose that m is odd, then the previous formula simplifies to
ν
m|2n
2
AcknowledgementThe first author is a research assistant supported by the Fund for Scientific Research Flanders (F.W.O.-Vlaanderen). He would like to thank Liesbet Van de Voorde for a discussion concerning section 5.
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[] |
[
"The equilibrium landscape of the Heisenberg spin chain",
"The equilibrium landscape of the Heisenberg spin chain"
] |
[
"Enej Ilievski [email protected] \nInstitute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands\n",
"Eoin Quinn †[email protected] \nInstitute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands\n\nLPTMS\nCNRS\nUniv. Paris-Sud\nUniversité Paris-Saclay\n91405Orsay cedexFrance\n"
] |
[
"Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands",
"Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands",
"LPTMS\nCNRS\nUniv. Paris-Sud\nUniversité Paris-Saclay\n91405Orsay cedexFrance"
] |
[
"SciPost Phys"
] |
We characterise the equilibrium landscape, the entire manifold of local equilibrium states, of an interacting integrable quantum model. Focusing on the isotropic Heisenberg spin chain, we describe in full generality two complementary frameworks for addressing equilibrium ensembles: the functional integral Thermodynamic Bethe Ansatz approach, and the lattice regularisation transfer matrix approach. We demonstrate the equivalence between the two, and in doing so clarify several subtle features of generic equilibrium states. In particular we explain the breakdown of the canonical Y-system, which reflects a hidden structure in the parametrisation of equilibrium ensembles.
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10.21468/scipostphys.7.3.033
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[
"https://scipost.org/SciPostPhys.7.3.033/pdf"
] | 135,465,348 |
1904.11975
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eba5f134fac40724a9f242db8bb560ecf6ae70df
|
The equilibrium landscape of the Heisenberg spin chain
2019
Enej Ilievski [email protected]
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
University of Amsterdam
Science Park 9041098 XHAmsterdamThe Netherlands
Eoin Quinn †[email protected]
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
University of Amsterdam
Science Park 9041098 XHAmsterdamThe Netherlands
LPTMS
CNRS
Univ. Paris-Sud
Université Paris-Saclay
91405Orsay cedexFrance
The equilibrium landscape of the Heisenberg spin chain
SciPost Phys
733201910.21468/SciPostPhys.7.3.033Received 21-05-2019 Accepted 25-07-2019Check for updates
We characterise the equilibrium landscape, the entire manifold of local equilibrium states, of an interacting integrable quantum model. Focusing on the isotropic Heisenberg spin chain, we describe in full generality two complementary frameworks for addressing equilibrium ensembles: the functional integral Thermodynamic Bethe Ansatz approach, and the lattice regularisation transfer matrix approach. We demonstrate the equivalence between the two, and in doing so clarify several subtle features of generic equilibrium states. In particular we explain the breakdown of the canonical Y-system, which reflects a hidden structure in the parametrisation of equilibrium ensembles.
Introduction
The equilibration phenomena of quantum many-body systems have become a vigorous research topic for both theoretical and experimental studies of condensed matter systems in recent years. For generic interacting systems a central role has been played by the Eigenstate Thermalisation Hypothesis [1][2][3], which offers a unifying framework for characterising ergodic behaviour. The unconventional equilibration exhibited by (nearly) integrable systems has also drawn substantial interest, leading to the notion of the Generalized Gibbs Ensemble (GGE) [4][5][6]. The anomalous behaviour of integrable systems is due to the existence of infinitely many local conservation laws, whose essence is to protect quasi-particle excitations against decay [7]. The majority of the literature on equilibration in integrable systems has focused on non-interacting models, where the concept of a generalised Gibbs ensemble is synonymous to prescribing the occupations of single-particle modes [8,9]. Interacting integrable systems on the other hand exhibit rich spectra of stable excitations which undergo non-trivial completely factorizable scattering [10]. This places interacting integrable systems in a distinguished position, and raises the question whether interactions induce physically discernible features among equilibrium states. The objective of this paper is to establish a framework to address this.
We focus our study on the isotropic Heisenberg spin-1/2 chain, a paradigmatic model of exactly solvable quantum many-body dynamics, due to both its simplicity and physical relevance. Our main result is an explicit construction of the entire manifold of equilibrium states, which helps expose a rich structure intrinsically linked to inter-particle interactions. We term this the 'equilibrium landscape'.
Studies of the thermodynamic properties of exactly solvable models have been traditionally focused on canonical Gibbs equilibrium. Only in recent years has interest shifted towards the Generalized Gibbs ensembles, predominantly discussed in the context of quantum quench dynamics in several physically relevant models such as the anisotropic Heisenberg model and Lieb-Liniger Bose gas [4,[11][12][13]. Here prominence was given to simple initial states of potential experimental relevance, while more recent works have considered more generally a special class of 'integrable' product states [14][15][16][17]. In the present work we pursue a general and systematic characterisation of the entire space of equilibrium ensembles without appealing to any initial state specific considerations.
Throughout the work we shall employ a range of techniques from the integrability toolbox, combining the algebraic, thermodynamic, and functional Bethe ansatz approaches. We begin by formulating an explicit algebraic construction of the GGE, and then proceed to analyse two complementary routes for evaluating equilibrium partition sums. On the one hand, the celebrated Thermodynamic Bethe Ansatz (TBA) approach [18][19][20] casts the partition function as a functional integral, invoking a spectral resolution through coupled interacting quasi-particle modes. A saddle-point of the functional integral yields an infinite set of coupled integral equations encoding the equilibrium state. On the other hand, we achieve a regularisation of a general partition function as a two-dimensional classical vertex model where, similarly as in the Gibbs canonical ensemble [21][22][23][24][25][26][27], the main subject of study is the dominant eigenvalue of a column transfer matrix. To our knowledge, no previous work on the statistical mechanics of exactly solvable models, including a large body of work on solvable classical vertex models, has achieved a similarly comprehensive description of the entire equilibrium manifold of a model.
For the case of canonical Gibbs equilibrium, compatibility of the two approaches to thermodynamics has been already demonstrated previously in [26]. By means of an integrable Trotterisation of the density operator, usually referred to as the Quantum Transfer Matrix, the Gibbs free energy can be expressed as a solution to the non-linear integral equation [24,25,27,28].
Section 3 equilibrium ensembles
Section 4 Thermodynamic Bethe Ansatz
Section 5 lattice regularisation Section 6 Figure 1: Outline of the paper. In Section 2 we introduce the Heisenberg spin-1/2 chain and define the main objects of the integrability framework. In Section 3 we discuss generic equilibrium ensembles and specify the general density matrix. In Section 4 we cover the TBA approach and systematically discuss analytic properties of generic macrostates. In Section 5 we regularise the general density matrix and recast it as a two-dimensional classical vertex model. In Section 6 we define the mirror system and employ functional Bethe ansatz to demonstrate equivalence with TBA.
In this regard, a special and seemingly non-generic analytic structure of the transfer matrix spectrum turns out to be crucial. In this work we demonstrate that typical macrostates from the equilibrium landscape have a much richer structure which necessitate going beyond the conventional Trotterisation techniques. Through achieving this we establish compatibility with the TBA formalism on the general grounds, and find that this yields a clear and instructive picture of the emergence of the equilibrium landscape.
The Heisenberg spin chain
In this work we outline our construction for perhaps the most widely studied interacting integrable quantum system, the one-dimensional isotropic spin-1/2 Heisenberg model [19,20,29,30]
H = J L i=1 1 4 − S i · S i+1 ,(2.1)
with exchange coupling J (here J > 0 corresponds to a ferromagnetic ground state) and periodic boundary conditions S L+1 = S 1 . The S ≡ (S x , S y , S z ) are the local generators of the su(2) algebra, [S α , S β ] = i ε αβγ S γ . The model possesses a manifest global su(2) symmetry,
H, S α tot = 0, (2.2) with S α tot = L i=1 S α i .
In this section we summarise the integrable structure of the model, introducing the concepts and notations which form the foundation of the work.
Spectrum. The degrees of freedom of the spin chain are magnon excitations, corresponding to spin waves with respect to a reference fully polarised ferromagnetic vacuum. A key feature of integrability is that the magnons undergo non-trivial yet non-diffractive scattering, implying that any interaction process can be reduced to a sequence of two-particle scatterings. With respect to the vacuum state, the quantisation conditions for the magnons are known as the Bethe equations [29]
e ik(u i )L M j=1, j =i S(u i − u j ) = 1. (2.3)
Here the magnons are conveniently parametrised by a rapidity variable u, through which their momentum is 4) and the two-magnon scattering amplitude is given by
k(u) = 1 i log u + i 2 u − i 2 ,(2.S(u) = u − i u + i ,(2.E = J M i=1 1 − cos k(u i ) . (2.6)
The descendant states in a multiplet are obtained by adding zero momentum magnons, i.e. rapidities with u = ∞, for which the scattering amplitude trivialises.
Bound states. The scattering between magnons induces bound state formation. These correspond to the collections of complex magnon rapidities aligning into 'string' patterns in the complex rapidity plane. In the large-L limit the Bethe roots are classified according to the 'string hypothesis',
M i=1 {u i } −→ ∞ j=1 M j i=1 j a=1 u j,i + ( j + 1 − 2a) i 2 ,(2.7)
with all u j,i ∈ . Bound states of j magnons are accordingly called j-strings, and the set of j-strings provide the thermodynamic particle content of the model, i.e. M = ∞ j=1 M j . The string rapidities are subject to the 'string Bethe equations',
e ik j (u j,i )L ∞ j =1 M j i =1 S j, j (u j,i − u j ,i ) = −1,(2.8)
valid up to corrections which are suppressed in system size L. Here the bare momentum of a j-string is 9) and the scattering amplitudes between a j-string and a -string are
k j (u) = 1 i log u + j i 2 u − j i 2 ,(2.S j, (u) = j a=1 b=1 S u + ( j − − 2a + 2b) i 2 .
(2.10)
The factor −1 on the right-hand side of Eq. (2.8) compensates the self-scattering factor S j, j (0) = −1 from the left-hand side.
Macrostates. In the thermodynamic limit, defined as L, M → ∞ with ratio M /L kept fixed, the string rapidities distribute densely along the real line. Macrostates are characterised by the complete set of the string rapidity densities {ρ j (u)}, with Lρ j (u)du being the number of occupied j-string modes in an infinitesimal rapidity interval du around u. These obey the log-differential form of Eq. (2.8), the Bethe-Yang equations,
ρ j +ρ j = 1 2π dk j du − K j, ρ ,(2.11)
whereρ j (u) denotes the corresponding density of holes, i.e. unoccupied modes, and the scattering kernels
K j, (u) = 1 2πi ∂ u log S j, (u),(2.12)
are the differential scattering phases. Here we use the following short-hand notation for matrix convolutions
F j, f ≡ ∞ =1 ∞ −∞ dwF j, (u − w) f (w),(2.13)
and adopt the summation convention for the repeated indices. In addition, we also use a short-hand notation for scalar integrations as follows
f g ≡ ∞ −∞ dw f (u − w)g(w), f • g ≡ ∞ −∞ dw f (w)g(w).
(2.14)
For each macrostate there is an associated entropy, which is the logarithm of the number corresponding microstates. This is expressed through the entropy density functional [18,20] s[ρ j ,ρ j ] = (ρ j +ρ j ) log(ρ j +ρ j ) − ρ j log ρ j −ρ j logρ j , (2.15) where exp Ls ρ j (u),ρ j (u) du counts the number of ways of distributing Lρ j (u) du particles between the L ρ j (u) +ρ j (u) du many j-string mode numbers on an infinitesimal rapidity interval du centred at u.
Kernel identities. The scattering kernels K j, are differential scattering phase shifts which encode interactions at the level of macrostates. They exhibit a rich structure which we will exploit throughout this work. Firstly, the Fredholm operator (1 + K) admits a pseudo-inverse
(1 − R) through (1 − R) j, (1 + K) ,k = 1, (2.16) with 1 ≡ δ j,k δ(u), where the Fredholm resolvent, R j, (u) = I j, s(u),(2.17)
is defined through the s-kernel
s(u) = 1 2 cosh (πu)
, (2.18) and the nearest-neighbour incidence matrix I,
I j, = δ j−1, + δ j+1, .(2.19)
Here (1 − R) admits a non-trivial nullspace and is thus not the true inverse of (1 + K). In particular, the relation (1 − R) j, n = 0, with boundary condition n 0 = 0, has a one-parameter solution n j = h j. Similarly the s-kernel admits a pseudo-inverse s −1 , which is a left-inverse under convolution, i.e. s −1 s f = f , and which also possesses a non-trivial nullspace. Explicitly it is, (2.20) where ε ≡ 0 + is an essential positive infinitesimal which prescribes the avoidance of the poles of s(u) at u = ± i 2 . The nullspace of s −1 , generated by functions ζ obeying s −1 ζ = 0, is a linear span of basis functions log τ(u; w) of the form
(s −1 f )(u) = f (u + i 2 − iε) + f (u − i 2 + iε),log τ(u; w) ≡ log tanh π 2 (u − w) , w ∈ P, (2.21)
where P is a strip in the complex plane defined as
P = u ∈ : |Im(u)| ≤ 1 2 − ε ,(2.22)
commonly referred to as the 'physical strip'. We highlight the explicit dependence on the infinitesimal regulator ε here, which ensures that the boundaries Im(u) = 1 2 are excluded from the strip, as it will prove useful in later sections. The functions log τ(u; w) are related back to the s-kernel through the identity
s(u) = ∓ 1 2πi ∂ u log τ(u; ± i 2 ). (2.23)
A related object is the discrete d'Alembertian operator 24) or explicitly,
= s −1 (1 − R) = s −1 − I,(2.( f ) j (u) = f j (u + i 2 − iε) + f j (u − i 2 + iε) − f j−1 (u) − f j+1 (u), (2.25)
for a set of functions f j (u). The associated Green's function, obeying G = 1, is given by
G j,k = (1 + K) j,k s. (2.26)
The d'Alembertian inherits a non-trivial nullspace from both (1 − R) and s −1 . From (1 − R) the functions n j = h j obey n = 0, while given a set of functions ζ j in the nullspace of s −1 , there exist related functions ν j = (1 + K) j, ζ which also satisfy ν = 0. It is further useful to define kernels K j and their associated amplitudes S j as follows
K j (u) = 1 2πi ∂ u log S j (u) = 1 2πi 1 u − j i 2 − 1 u + j i 2 , S j (u) = u − j i 2 u + j i 2 , (2.27)
with the kernels obeying the identities 1 2π
dk j du = K j (u), (1 − R) j, K = δ j,1 s. (2.28)
These provide convenient explicit expressions for the matrix elements of the Green's function G j,k and its associated amplitude Ψ j,k as follows (2.29) and in turn for the scattering kernels and their amplitudes through
G j,k (u) = 1 2πi ∂ u log Ψ j,k (u) = min( j,k) a=1 K j+k+1−2a (u), Ψ j,k (u) = min( j,k) a=1 S j+k+1−2a (u),K j,k (u) = I j, G ,k (u), log S j,k (u) = I j, log Ψ ,k (u). (2.30)
Finally, we emphasise that the ε regulator is tied to the pseudo-inverse s −1 , and in particular does not appear in the string compounds in Eq. (2.7), nor in the general definitions of kernels and amplitudes, e.g. Eqs. (2.10), (2.27). In the following we employ a compact notation for half-unit imaginary shifts not involving a regulator
f ± (v) = f (v ± i 2 ). (2.31)
Transfer matrices. The algebraic formulation of integrability is founded upon the Lax representation [31][32][33], see also [34][35][36][37] and references therein. The Lax matrices are a family of
operators L k,1 (v, u) : V k ⊗ V 1 → V k ⊗ V 1 ,
where V k denotes the (k + 1)-dimensional irreducible unitary representation of su (2), of the form
L k,1 (v, u) = (v − u)1 ⊗ 1 + 2i α=x,y,z S α ⊗ S α . (2.32)
These provide local building blocks for the transfer matrices, a commuting family of operators acting on the Hilbert space H ∼ = V ⊗L 1 , which are given as traces over path-ordered products of operators L k,1 ,
T k (v) = Tr V k L k,1 (v, 0) ⊗ L k,1 (v, 0) ⊗ · · · ⊗ L k,1 (v, 0), (2.33)
where the trace is taken over the common auxiliary space
V k , with k ∈ . Trivially, T 0 (v) = v L .
Integrability ensures that the transfer matrices T k (v) mutually commute,
T k (v), T k (v ) = 0, (2.34)
for all values of k, k ∈ and v, v ∈ .
Fusion hierarchy. The eigenvalues T k (v) of the transfer matrices T k (v) are called T -functions.
They are polynomial and satisfy the Hirota equation 1
T + k (v)T − k (v) = φ k (v)φ k (v) + T k−1 (v)T k+1 (v), k ≥ 0, (2.35) with initial conditions T −1 ≡ 0, T 0 (v) = v L , and boundary 'scalar potentials', φ k (v) = v + (k + 1) i 2 L ,φ k (v) = v − (k + 1) i 2 L . (2.36)
The Hirota equation exhibits a gauge freedom corresponding to the overall normalisation of the Lax matrix. There however exist Y -functions,
Y k (v) = T k−1 (v)T k+1 (v) φ k (v)φ k (v) = T + k (v)T − k (v) φ k (v)φ k (v) − 1, (2.37)
which are gauge-invariant quantities satisfying the canonical Y -system hierarchy [41,42]
Y + k (v)Y − k (v) = 1 + Y k−1 (v) 1 + Y k+1 (v) , k ≥ 1, (2.38) with initial condition Y 0 (v) = 0.
Thermodynamic inversion identity. In the large-L limit, the entire family of transfer matrices T k (v) satisfy the useful identity [43] lim
L→∞ T + k (v)T − k (v) φ k (v)φ k (v) = 1, (2.39)
which allows for their inversion. This property can equivalently be expressed as the large-L decay of the physical Y -functions
lim L→∞ Y k (v) = 0, v ∈ P. (2.40)
Local charges. The transfer matrices serve as generating operators for the local charges through their logarithmic derivatives [4,43],
X k (v) = 1 2πi ∂ v log T + k (v) φ k (v) . (2.41)
These charges are well-defined only on the physical strip, v ∈ P, specified above in Eq. (2.22).
When v approaches the boundary of the strip, the charges acquire a divergent localisation length [43,44] and thus become singular at the boundaries ∂ P = Im(v) = ± 1 2 . The Hamiltonian Eq. (2.1) is given by H = πJ X 1 (0).
String charge-duality. In the thermodynamic limit, the eigenvalues of the charges X k (v) are expressible as a linear functional of the rapidity densities [7]
X k = G k, j ρ j , (2.42)
where G, given in Eq. (2.26), is the Green's function of the d'Alembertian . The inverted relation, promoted to the level of operators,
ρ = X, (2.43)
bears the name 'string-charge duality'. We highlight that this defines the mode operators ρ j in a manner which is independent of the of the orientation of the reference ferromagnetic vacuum.
Even though the positive infinitesimal ε does not appear in the definition of the charges X k (v), the mode operators ρ j inherit the ε-prescription through the left-inverse s −1 which enters in the d'Alembertian , Eq. (2.25). The important consequence of the regulator ε is that the boundaries ∂ P at Im(u) = 1 2 are avoided, ensuring a finite localisation length of the ρ j , i.e. as ε is strictly positive the localisation lengths are strictly finite, cf. [43,44]. Thus ε admits a physical interpretation as a regulator which governs the notion of locality in the large-L limit.
Equilibrium ensembles
The purpose of this article is to characterise the equilibrium landscape of an interacting integrable model, specifically the Heisenberg spin chain. Equilibrium ensembles emerge dynamically in the long-time limit of unitary evolution from generic initial states in thermodynamically large systems. These can be viewed equivalently in either the canonical sense as density matrices governing the expectation values of all local observables, or in the microcanonical sense as unbiased collections of eigenstates sharing the same values of all local charge densities. There are two key mechanisms underlying local equilibration: (i) decoherence, causing the dynamical phases between individual eigenstates to average out during the relaxation process, and (ii) the eigenstate thermalisation hypothesis which supposes the local equivalence of distinct eigenstates from the same microcanonical shell [1][2][3]45]. Local correlation function are thus expressible as functionals of the quasi-particle densities characterising a macrostate, see e.g. [46][47][48].
For the Heisenberg spin chain, a microcanonical shell corresponds to the set of microstates associated with a given macrostate, parametrised by the full set of occupied mode distributions ρ j (u). In this context, the statement of the eigenstate thermalisation hypothesis is substantiated by the Yang-Yang form of the entropy density, Eq. (2.15), see e.g. [6,49,50]. The corresponding (unnormalised) density matrix is [51]
= exp − µ j • ρ j + h · S tot ,(3.1)
where µ j (u) provide a general set of chemical potentials for the mode operators ρ j . The term h · S tot incorporates a general Cartan charge of the global su(2) symmetry, which serves to specify the polarisation direction h = (h x , h y , h z ) with respect to which the Bethe magnons are defined, i.e. with respect to a ferromagnetic vacuum oriented in the h direction. For consistency, the chemical potentials must not diverge with j, i.e.
lim j→∞ µ j (u) j = 0. (3.2)
The above functional parametrisation of the density matrix differs from the more commonly used definition involving a formal infinite sum over a discrete basis of local conservation laws (see e.g. [4][5][6]52]) or a 'truncated' GGE [53][54][55]. We argue however that gauge-invariant formulation (3.1) not only clearly conveys the physical picture underlying the GGE concept, it moreover provides a natural and convenient starting point for analysis of the ensemble as developed in the following sections.
Thermodynamic Bethe Ansatz
In this section we revisit the formalism of Thermodynamic Bethe Ansatz, a functional integral formulation of thermodynamics [18][19][20]. Partition sums are cast in the basis of Bethe eigenstates, which in the thermodynamic limit translates to functional integration in the space of macrostates. In particular, the (generalized) free energy density,
f = − lim L→∞ 1 L log Tr H , (4.1)
is reformulated as a functional integral over the string rapidity distributions,
f = D[{ρ j }] µ j + h j • ρ j − j s[ρ j ,ρ j ] ,(4.2)
with h = | h|, and the entropy functional s is specified by Eq. (2.15). Identifying the saddle point of the equilibrium partition sum through δ f /δρ j (u) = 0, subject to the constraints (2.11), yields the celebrated TBA equations,
log Y j = µ j + h j + K j, log 1 + 1/Y , (4.3) expressed in terms of the 'thermodynamic Y -functions', Y j (u) =ρ j (u) ρ j (u) . (4.4)
The equilibrium free energy density then becomes
f = − 1 2π dk j du • log(1 + 1/Y j ),(4.5)
which can be equivalently expressed in the form The TBA equations provide the link between the chemical potentials µ j and the thermodynamic functions of general equilibrium states. To further elucidate the underlying structure, we next bring the TBA equations to a local form by convolving with the left-inverse (1 − R) of the Fredholm operator (1 + K), resulting in
f = s • µ 1 − s • log(1 + Y 1 ),(4.log Y j = d j + I j, s log(1 + Y ), (4.7)
with Y 0 ≡ 0, and source terms
d j = (1 − R) j, µ . (4.8)
Here information about h, which has dropped from the source term as it belongs to the nullspace of (1 − R), instead appears implicitly through the large-j asymptotics
lim j→∞ log Y j (u) j = h, (4.9)
which must be supplemented to Eqs. (4.7) in order to unambiguously fix a solution. The information stored in µ j is preserved by (1 − R), as condition Eq. (3.2) forbids a nullspace contribution, and so Eq. (4.8) can be readily inverted
µ j = (1 + K) j, d . (4.10)
The TBA equations are integral equations defined on the real rapidity axis. We now analytically continue them to the complex rapidity plane, obtaining an equivalent set of functional relations called the 'thermodynamic Y -system' [51]. This is achieved by convolving both sides of Eqs. (4.7) with the pseudo-inverse s −1 and subsequently exponentiating, resulting in
Y + j (u − iε)Y − j (u + iε) = e λ j (u) 1 + Y j−1 (u) 1 + Y j+1 (u) , (4.11) with λ j = s −1 d j . (4.12)
This provides a natural decomposition of the source terms,
d j = s λ j + ζ j ,(4.13)
where ζ j satisfy s −1 ζ j = 0. (4.14)
Owing to Eq. (2.21), we adopt the following generic parametrisation
ζ j (u) = i α j,i log[τ(u; w j,i )τ(u;w j,i )],(4.15)
in terms of parameters α j,i ∈ , and complex-conjugate w j,i andw j,i located in P. Although the sum involves a finite number of terms, infinite convergent sequences of simple poles and zeros (α j,i = ±1) which condense along certain contours are also permissible and can be understood as limits of Padé approximants for complex functions with branch points. We adopt the convention that all branch cuts in P extend vertically away from the real axis. The decomposition Eq. (4.13) can also be expressed at the level of the chemical potentials
µ j = G j, λ + ν j ,(4.16)
where here λ = µ, and ν j encodes the nullspace of µ j inherited from s −1 through
ν j = (1 + K) j, ζ . (4.17)
Indeed the thermodynamic Y -system can be obtained directly from the TBA equations Eq. (4.3) by applying the d'Alembertian and exponentiating.
To this point the analysis appears completely formal. It however reveals a structure in the space of equilibrium states. The information from the nullspace of s −1 , Eq. (2.21), which appears to have dropped out from Eq. (4.11), enters instead implicitly via analytic properties of functions Y j . Specifically, w j,i andw j,i are nothing but branch points of Y j of degree α j,i . To establish this, we now re-obtain Eq. (4.7) from Eq. (4.11). In order to undo the complex shifts and transform Eq. (4.11) to the real axis, we introduce adapted Y -functions,
Y adp j (u) = Y j (u) i τ(u; w j,i )τ(u;w j,i ) −α j,i , (4.18) which obey Y adp j (u + i 2 − iε)Y adp j (u − i 2 + iε) = Y j (u + i 2 − iε)Y j (u − i 2 + iε) due to τ + τ − = 1
, and are (by construction) analytic on P with constant large-u asymptotics. Substituting these into Eq. (4.11), taking the logarithm and convolving with s we readily recover Eq. (4.7). The analytic properties of the thermodynamic Y -functions in P are therefore completely determined by the data w j,i andw j,i and α j,i . If all α j,i ∈ then the Y -functions are meromorphic on P, while more generally they possess non-integer branch points.
Integrable lattice regularisation
In this section we derive an explicit lattice regularisation of a generic equilibrium ensemble. This is achieved this by re-expressing the density matrix as a product of transfer matrices, thereby casting it as a two-dimensional classical vertex model on a cylinder.
The mode operators ρ j are related to the transfer matrices via string-charge duality, Eq. (2.43). The density matrix Eq. (3.1) can thus be presented in the form
= exp − µ j • ( X) j + h · S tot ,(5.1)
where the charges X j are the logarithmic derivatives of the transfer matrices, Eq. (2.41). To proceed it is necessary to be careful about the potential nullspace of µ j under action of the d'Alembertian . As described in Section 2, this nullspace is spanned by functions n j in the nullspace of (1 − R), along with functions ζ j in the nullspace of s −1 . The condition Eq. (3.2) forbids a contribution n j , and so the chemical potential decomposes as
µ j = G j, λ + ν j ,(5.2)
where λ = µ, and ν j = (1 + K) j, ζ encode the nullspace inherited from s −1 , ν = 0. This essentially repeats the derivation of Eq. (4.16) in Section 4. In the following we will adopt the same generic parametrisation of ζ j as in Eq. (4.15). The decomposition of µ j induces a factorisation of the density matrix Eq. (3.1) into three commuting factors
= λ · ν · h ,(5.3)
given respectively by
λ = exp − λ k • X k , ν = exp − ν j • ρ j , h = exp h · S tot ,(5.4)
where the first factor is obtained through (G λ)• X = G λ •X = λ•X. For convenience we will refer to λ as encoding the 'node data', ν as encoding the 'analytic data', and h as encoding the Cartan charge. We now proceed to analyse each factor in turn.
The node data. The first factor of the ensemble encodes the node data λ k ,
λ = exp − k λ k • X k . (5.5)
To begin we regularise the λ k , by invoking a large-rapidity cut-off Λ ∞ , and casting λ k (u) as a discrete sum on the remaining interval. First it is useful to introduce, for each k separately,
two disjoint domains {−Λ ∞ , Λ ∞ } = I (+) k ∪ I (−) k , such that λ k (v) ≥ 0 on I (+) k and λ k (v) < 0 on I (−)
k . That is, we decompose the λ k as
λ k (v) = λ (+) k (v) + λ (−) k (v),(5.6)
where λ
(±) k (v) = 0 on I (∓)
k . Each part can then be regularised via a discrete sum of Dirac δ-distributions,
λ (±) k (v) = lim n k →∞ Λ (±) k n (±) k n k i=1 δ v − x (±) k,i ,(5.7)
where n (±) k ∈ control the resolution of the discretisation, the parameters x (±)
k,i ∈ are chosen uniformly from the distributions λ (±) k (v), and the overall normalisation is fixed by
Λ (±) k = I (±) k dv λ (±) k (v). (5.8)
This readily translates to a further factorisation λ =
(+) λ · (−) λ , with (±) λ = k n (±) k i=1 exp − Λ (±) k n (±) k X k (x (±) k,i ) . (5.9)
Referring back to the definition of the charges Eq. (2.41), each exponential factor is written as
X k v = lim ∆v→0 1 ∆v 1 2πi log T + k v + ∆v 2 φ k (v + ∆v 2 ) − log T + k (v − ∆v 2 ) φ k (v − ∆v 2 )
. (5.10)
Using the inversion identity Eq. (2.39), which is exact in the large-L limit, this becomes
X k v = lim ∆v→0 1 ∆v 1 2πi log T + k v + ∆v 2 φ k (v + ∆v 2 ) T − k (v − ∆v 2 ) φ k (v − ∆v 2 ) . (5.11)
Now, for each factor in Eq. (5.9) we couple the finite difference to the corresponding discretisation parameters n
(±) k through the identifications v → x (±) k,i and ∆v → i ξ (±) k , with parameters ξ (±) k = Λ (±) k 2π n (±) k ,(5.12)
resulting in
(±) λ = k n (±) k i=1 T + k x (±) k,i + ξ (±) k i 2 φ k x (±) k,i + ξ (±) k i 2 T − k x (±) k,i − ξ (±) k i 2 φ k x (±) k,i − ξ (±) k i 2 . (5.13)
Let us highlight that while Eq. (5.13) may in a sense be viewed as an 'integrable Trotterisation', the Suzuki-Trotter formula [56,57] has not been invoked. Instead, we coupled the resolution of the discretisation of λ k (u) in Eq. (5.7) with the finite-difference approximation ∆v of the derivative in the definition of the charges.
The analytic data. We next analyse the factor (5.14) where ν = (1 + K) ζ, with ζ j given generically by Eq. (4.15). Here we substitute ρ = X, bringing it to the form
ν = exp − j ν j • ρ j ,ν = j,k exp dz ν j (z)I j,k X k (z) − dz ν j (z)δ jk X k (z + i 2 − iε) + X k (z − i 2 + iε) .
(5.15) Then exploiting the null property ν = 0, the exponent becomes recast as a sum of two contour integrals
ν = j,k exp C + dz ν − j (z + iε)δ j,k X k (z) − C − dz ν + j (z − iε)δ j,k X k (z) ,(5.16)
where contours C + and C − encircle the upper and lower half of the physics strip P in the counter-clockwise direction, respectively. Here we have shifted the integration contours using and integrating by parts, we obtain
dz f (z)X ± k (z ∓ iε) = dz f ∓ (z ± iε)X k (z) ∓ C ± dz f ∓ (z ± iε)X k (z),(5.ν = j,k exp C + dz γ − j (z + iε)δ jk log T + k (z) φ k (z) − C − dz γ + j (z − iε)δ jk log T + k (z) φ k (z) . (5.19)
Specifically, the functions γ j are given through
γ ± j (u) = ±α j,i k i=1 G j,k (u − w k,i ) + G j,k (u −w k,i ) ,(5.20)
using identities (2.23) and (2.26). Now we evaluate the above contour integrals.
Recalling the explicit expression for G from Eq. (2.29), and noting that the poles of kernels K j (u) are of the form, 2πi Res u=0 K j (u ± i 2 k) = ±δ jk , we observe that the poles of γ ± j (u) with residues ±α j,i are located at the null components shifted by integer multiples of i 2 . Given that all the null components lie within the physical strip P, the only contribution which does not shift the poles outside the contours then comes from the term K 1 (u) which appears only in G j, j (u). In effect, the analytic data gets shifted by ± i 2 , giving a pole in C − (C + ) for each null component in C + (C − ). Thus Eq. (5.19) simplifies to
ν = k i exp α k,i log T + k (w k,i + iε − i 2 ) φ k (w k,i + iε − i 2 ) T − k (w k,i − iε + i 2 ) φ k (w k,i − iε + i 2 ) ,(5.21)
upon using the inversion identity Eq. (2.39).
Finally, Eq. (5.21) must be further regularised in order to convert to a product of transfer matrices. To achieve this we note the following property: any term α log[τ(u; w)τ(u;w)] with 0 < α < 1 and w,w ∈ P can be systematically approximated by a sum
a log τ(u; w z a ) τ(u;w z a ) − b log τ(u; w p b ) τ(u;w p b ) ,(5.22)
where the sets of zeros {ω z a ,ω z a } and poles {ω p b ,ω p b } are obtained as the zeros and poles lying within the physical strip P of a Padé approximation of (u−w) α (u−w) α about w 0 = Re(w) = Re(w). Applying this to each contribution of ζ j individually we obtain a regularised form
ζ reg j (u) = n z j a=1 log τ(u; ω z j,a ) τ(u;ω z j,a ) − n p j b=1 log τ(u; ω p j,b ) τ(u;ω p j,b ) ,(5.23)
where both the distinction between distinct zero modes and dependence on the degree of the Padé approximation are left implicit. In this way we obtain the factor ν of the ensemble as a product of transfer matrices
ν = k n z k a=1 T + k (ω z k,a + iε − i 2 ) φ k (ω z k,a + iε − i 2 ) T − k (ω z k,a − iε + i 2 ) φ k (ω z k,a − iε + i 2 ) × k n z p b=1 T + k (ω p k,b − iε + i 2 ) φ k (ω p k,b − iε + i 2 ) T − k (ω p k,b + iε − i 2 ) φ k (ω p k,b + iε − i 2 )
.
(5.24)
The Cartan charge. The final factor of the ensemble is simply given by
h = exp h · S tot ≡ L j=1 D( h), D( h) = exp h · S . (5.25)
There is no trace over an auxiliary space associated with this factor.
The two-dimensional vertex model
Now recombining the three factors the ensemble takes the compact form
= h k N k i=1 T + k (θ k,i ) φ k (θ k,i ) T − k (θ k,i ) φ k (θ k,i ) ,(5.26)
where we have collected the impurity parameters as follows the row transfer matrices are equivalent. Let us emphasise that the resulting vertex model is not the common six-vertex model, but can be regarded instead as a fused variant thereof. Indeed, in general there is no upper bound on the 'number of vertices', as a generic equilibrium state requires arbitrary large spin representation labels. The contribution h appears as an additional factor. It plays the role of a boundary twist when the cylinder is wrapped to a torus (i.e. when the ensemble is traced over).
{θ k,i } ≡ x (+) k,i + ξ (+) k i 2 ∪ x (−) k,i + ξ (−) k i 2 ∪ ω z k,i + iε − i 2 ∪ ω p k,i − iε + i 2 ,(5.
The mirror system
The vertex-model regularisation of a generic equilibrium ensemble achieved in the previous section allows for a description of equilibrium states in manner which is complementary to the TBA analysis of Section 4. Here the free energy density is
f = − lim L→∞ lim N →∞ 1 L log Tr H h k N k i=1 T + k (θ k,i ) φ k (θ k,i ) T − k (θ k,i ) φ k (θ k,i ) , (6.1)
which can be viewed as an inhomogeneous vertical iteration of row transfer matrices T k . Alternatively this can be re-expressed as a horizontal iteration of an inhomogeneous column transfer matrix, as illustrated in Figure 3. As the two-dimensional vertex model is homogeneous in the horizontal (i.e physical) direction, the iteration of the column transfer matrix is homogeneous, with the consequence that its dominant eigenvalue yields the free energy density. In this section we analyse this process in detail. We then pay particular attention to the re-emergence of the thermodynamic Y -system, Eq. (4.11), in the large-N limit. Given a two-dimensional vertex model, we introduce the corresponding 'mirror system' as
k, θ k,i k,θ k,i j, 0 T + k (θ k,i )/φ k (θ k,i ) T − k (θ k,i )/φ k (θ k,i )
T j (0) Figure 3: The equilibrium partition function is computed by wrapping the cylinder of Figure 2 to a torus, i.e. tracing over the physical sites. This can be viewed as an inhomogeneous iteration of the homogeneous row transfer matrices T ± k (red) of the physical system as in Eq. (6.1). Alternatively it can be expressed as a homogeneous iteration of the inhomogeneous column transfer matrices T j (blue) of the mirror system. The latter admits a dominant eigenvalue as given in Eq. (6.3).
the combination of the auxiliary spaces:
H M ∼ = k V ⊗2N k k
. The fundamental column transfer matrix 2 acts on the mirror system as follows
T 1 (u) = Tr V 1 D( h) k N k i=1 L + k,1 (θ k,i , u) θ k,i − u + (k + 1) i 2 L − k,1 (θ k,i , u) θ k,i − u − (k + 1) i 2 , (6.2)
with spectral parameter u, and the trace taken over the common fundamental space V 1 of all the L k,1 inherited from a lattice site of the original spin chain. That is, in switching to the column transfer matrix the role of physical and auxiliary spaces is interchanged. We emphasise that unlike the row (physical) transfer matrices defined in Eq. (2.33), the column transfer matrix is inhomogeneous in terms of both impurities and representation labels, and has a boundary twist encoded by the factor D( h) which also acts on V 1 . For notational convenience we proceed by suppressing explicit dependence of T 1 (u) on impurities θ k,i , twist h and dimension N . The fundamental column transfer matrix T 1 (u) allows for computation of the free energy density through its dominant (i.e. largest) eigenvalue according to the prescription
f = − lim N →∞ lim L→∞ 1 L log Tr H M T 1 (0) L = − lim N →∞ log T 1 (0), (6.3)
where T 1 (u) denotes the dominant eigenvalue. Care must be taken when interchanging the thermodynamic L → ∞ and the scaling N → ∞ limits (cf. [21,22]), and in our analysis we justify this step by establishing full consistency, as summarised in Figure 1.
One way to proceed to determine T 1 (0) is to obtain the Bethe equations which diagonalise T 1 (u) as a route towards obtaining their spectrum T 1 (u), and thereby its dominant eigenvalue. Here we instead take a different route and employ the framework of functional Bethe ansatz [24,35,39,60], which conveniently allows for the mirror-system Bethe equations to be completely bypassed.
With aid of the fusion rule for the Lax matrices, we proceed by embedding T 1 (u) into the infinite fusion hierarchy of commuting transfer matrices T j (u). To this end, we extend the family of Lax matrices L k, 1
(v, u) to L k, j (v, u) : V k ⊗ V j → V k ⊗ V j .
These are readily obtained through fusion [34,38,61,62]
L k, j (v, u) = N −1 k, j (v, u) P j j a=1 L (a) k,1 v, u + ( j + 1 − 2a) i 2 P j ,(6.4)
where each of j copies L (a)
k,1 acts on V k ⊗ V (a) 1 , P j : j a=1 V (a)
1 → V j is the totally-symmetric projection operator,
P j = j a=1 S · S − a(a − 1) j( j + 1) − a(a − 1) , (6.5) with S = j a=1 S (a) , and N k, j (v, u) = j−k a=1 (v − u + (k − j − 1 + 2a) i 2 )
is the common polynomial produced by fusion. Correspondingly, the higher-spin column transfer matrices are given explicitly as
T j (u) = Tr V j D( h) k N k i=1 L + k, j (θ k,i , u)L − k, j (θ k,i , u) min( j,k) a=1 θ k,i − u + (k − j + 2a) i 2 θ k,i − u − (k − j + 2a) i 2 , (6.6) where here D( h) acts on V j .
In particular, analogously to the physical T -functions Eq. (2.35), the hierarchy of column T -functions T j (u) satisfy the Hirota equation
T + j T − j = ϕ jφ j + T j−1 T j+1 , j ≥ 0,(6.7)
with initial conditions T −1 ≡ 0, T 0 = 1, and the scalar potentials
ϕ j = k N k i=1 Ψ −1 j,k (u − θ k,i ),φ j = k N k i=1 Ψ j,k (u −θ k,i ),(6.8)
with Ψ j,k given by Eq. (2.29). By construction, T j (u) are meromorphic functions of rapidity variable u. The boundary twist manifests itself as non-trivial large-u asymptotics
lim |u|→∞ T j (u) = sinh (( j + 1)h/2) sinh (h/2) ≡ χ j (h),(6.9)
parametrised by h = | h| through the su(2) characters χ j (h) obeying the 'classical limit' of the Hirota equation
χ 2 j (h) = 1 + χ j−1 (h)χ j+1 (h). (6.10)
The associated 'gauge'-invariant Y-functions
Y j = T j−1 T j+1 ϕ jφ j = T + j T − j ϕ jφ j − 1,(6.Y + j Y − j = 1 + Y j−1 1 + Y j+1 ,(6.12)
with large-u asymptotics
Y ∞ j ≡ lim |u|→∞ Y j (u) = χ 2 j (h) − 1. (6.13)
The above functional relations, Eqs. (6.7), (6.11) and (6.12), are valid for the full spectrum of T j (u) at finite N . To specify a particular eigenvalue, it is necessary to identify their analytic data in the physical strip P. This is made transparent by transforming the functional relations to integral equations on the real rapidity axis. To perform this step, it is useful to introduce adapted T -functions,
T adp j (u) = T j (u) i τ(u; t z j,i )
, (6.14) obtained by factoring out their (possible) zeros t z j,i in P, so that log T adp j (u) are analytic on P with constant large-u asymptotics. The absence of poles of T j (u) in P can be seen from the denominator in Eq. (6.6). Now manipulating Eq. (6.11), by taking the logarithm and convolving with s, we obtain
log T j = i log τ u; t z j,i + s log ϕ jφ j + s log 1 + Y j . (6.15)
We similarly transform the Y-system relations, Eq. (6.12), to the real axis. The procedure is analogous to that described in Section 4, specialised to meromorphic data only. Introducing adapted Y-functions The spectrum of the T -functions is thus given by Eq. (6.15), where the Y-functions obey Eq. (6.17), and eigenvalues are specified by the locations of the analytic data t z j,a , y z j,a , y p j,a . To determine this state-dependent data, one has in general to resort to the Bethe equations. The Y-functions however inherit certain zeros and poles directly from the impurity data through the scalar potentials in Eq. (6.11). To identify these we need the following properties which can verified from the definition of Ψ j,k in Eq. (2.29): if w belongs to the lower-half of the strip P then Ψ j,k (u − w) has a zero at u = w + i 2 if and only if j = k, and has no poles in P, while if w belongs to the upper-half of the strip P then Ψ j,k (u − w) has a pole at u = w − i 2 if and only if j = k, and has no zeros in P. Then from Eq. (6.8) one deduces that zeros of the Y-functions are located at x (−) (6.19) along with their complex conjugates, and we remind that ξ (−) k < 0. Correspondingly, the poles of the Y-functions appear at (6.20) along with their complex conjugates, with ξ
Y adp j (u) = Y j (u) b τ(u; y p j,b ) a τ(u; y z j,a ) ,(6.k,i + 1 + ξ (−) k i 2 ∪ ω z k,i + iε ,x (+) k,i + 1 − ξ (+) k i 2 ∪ ω p k,i + iε ,(+) k > 0.
The eigenstate for which these are the only zeros and poles of the Y-functions inside P is a distinguished state of the mirror system. In particular, the corresponding T -functions T j (u) are analytic and free of zeros in the strip P for this state, a further consequence of Eq. (6.11). As any zero of a T -function in P yields a negative definite contribution to Eq. (6.15), i.e. log τ(u; w) < 0, u ∈ , w ∈ P, (6.21) it is natural to anticipate that this state gives the dominant eigenvalue determining the free energy density through Eq. (6.3). In the following we establish this assertion by taking the large-N limit, and demonstrating that it reproduces precisely the TBA description of Section 4.
The scaling limit
In the preceding analysis we considered the mirror system at finite N . At this level, there is no strict distinction between the impurities encoding the node and analytic data from Eq. (5.27). Now we take the large-N scaling limit in which this distinction becomes manifest. We focus on the distinguished state identified above for which all the T -functions are analytic and free of zeros in the physical strip P.
The task is to inspect the large-N scaling limit for both Eq. (6.15) and Eq. (6.17). For Eq. (6.15) the non-trivial N -dependent term is log ϕ jφ j . For clarity, we consider below the contributions coming from the node and analytic data separately, that is we split log ϕ jφ j = log ϕ jφ j λ + log ϕ jφ j ν . (6.22) The node data λ j involves a sum over pairs of impurities
x (±) k,i + ξ (±) k i 2 , x (±) k,i − ξ (±) k i 2 , (6.23) yielding [log ϕ jφ j ] λ = σ=+,− k n (σ) k i=1 log Ψ j,k u − x (σ) k,i + ξ (σ) k i 2 Ψ j,k u − x (σ) k,i − ξ (σ) k i 2 . (6.24)
Individual contributions, which can be deduced with aid of Eq. (2.29), are of the form
− Λ (±) k n (±) k G j,k u − x (±) k,i . (6.25)
In the large-N limit, the net contribution of Eq. (6.24) yields
lim N →∞ [log ϕ jφ j ] λ = −G j,k λ k ,(6.26)
upon removing the large-rapidity cut-off Λ ∞ and converting sums to convolution-type integrals. In the process, the impurities re-condense in accordance with Eq. (5.7). The contribution from the analytic data stemming from zeros
{ω z k,i + iε − i 2 ,ω z k,i − iε + i 2 } and poles {ω p k,i + iε + i 2 ,ω p k,i − iε − i 2 } inside P, reads [log ϕ jφ j ] ν = − k n z k a=1 log Ψ + j,k u − ω z k,a − iε Ψ − j,k u −ω z k,a + iε) − k n p k b=1 log Ψ − j,k u − ω p k,b − iε Ψ + j,k u −ω p k,b + iε . (6.27)
Employing the identity log Ψ ± j,k = ±(1+K) j,k log τ which follows from Eqs. (2.23) and (2.26), this becomes [log ϕ jφ j ] ν = −(1 + K) j, ζ reg , (6.28) with ζ reg j given by Eq. (5.23). Here the infinitesimal regulator ε can be safely dropped. As the degrees of the Padé approximations tend to infinity with N , we then obtain
lim N →∞ [log ϕ jφ j ] ν = −(1 + K) j, ζ . (6.29)
Thus combining the contributions from the node and analytic data we recover the TBA chemical potentials (4.16), lim
N →∞ log ϕ jφ j = −µ j . (6.30)
Next we take the large-N scaling limit of Eq. (6.17). Here we must examine the source terms d j given by Eq. (6.18). The node data consists of pairs of zeros and poles, cf. Eq. (6.19),
x (±) k,i + 1 ∓ ξ (±) k i 2 , x (±) k,i − 1 ∓ ξ (±) k i 2 , (6.31) each contributing to d j a term log τ u; x (±) k,i + 1 ∓ ξ (±) k i 2 τ u; x (±) k,i − 1 ∓ ξ (±) k i 2 . (6.32)
Since there are n k terms, in the n k → ∞ limit we are left with
Λ (±) k n (±) k s u − x (±) k,i ,(6.33)
employing Eq. (2.23). The full contribution to d j in the large-N scaling limit then retrieves the node data, that is s λ j . The contribution from the regularised analytic data, which is for finite N straightforwardly given by ζ reg j , and in the large-N limit converges to ζ j . In conclusion, Figure 4: Schematic depiction of the analytic data for a typical meromorphic Yfunction associated with the dominant eigenvalue of the fundamental column transfer matrix T 1 (u) at finite N . The analytic data comprises zeros (red circles) and poles (blue crosses) in the complex strip P. The integration contours C + and C − , encircling the upper and lower halves of the physical strip P, are separated from the lines u = ± i 2 by the regulator ε. The contributions from the regularised node data λ j appear separated by distance ξ
lim N →∞ d j = d j = s λ j + ζ j ,(6.i/2 −i/2 C + C − Y j ( y z ) = 0 Y j ( y p ) = ∞(±) j = Λ (±) j /2πn (±) j
from the lines u = ± i 2 . The analytic data lie within the contours C ± , with clusters of zeros and poles appearing as Padé approximants of non-integer branch points.
Discussion
In the preceding sections we considered generic equilibrium states and developed the dual approaches of TBA and lattice regularisation, as summarised in Figure 1. The central role in establishing their equivalence is played by the mirror system, which provides a compact regularisation of a given equilibrium state. The mirror system exhibits a 'universal' canonical structure, seen here in Eqs. (6.7)-(6.13), which is deep-rooted in the fusion properties of the underlying quantum symmetry algebra. One of the key findings of this work is however that in thermodynamic/large-N limit the canonical structure is superseded by an emergent equilibrium landscape, encoded in non-trivial node terms λ j and generically non-meromorphic thermodynamic Y -functions involving branch points inside P. There are several discernible aspects which deserve a brief discussion.
Breakdown of the canonical Y-system. The thermodynamic Y -system generically contains state-dependent node terms λ j , see Eq. (4.11). This non-canonical form, which is seemingly conflict with the expected universality of the canonical Y-system [42], has been previously observed the context of quantum quenches where several explicit examples have been identified [7,16,63]. Here we clarify this aspect.
The Y-functions at finite N obey the canonical Y-system, Eq. (6.12), with n (±) j zeros and poles stemming from the regularised λ j residing are finite distance ξ
(±) j = |Λ (±) j |/2πn
(±) j from the boundary of P. A subtlety however arises in the thermodynamic scaling limit N → ∞, where ξ ± j invariably become smaller than the positive infinitesimal regulator ε which controls the width of the physical strip P as in Eq. (2.22). 3 In this event, a subset of the analytic data leaves the nullspace of s −1 by escaping from the integration contours C ± and finally collapsing onto the boundary of P at Im(u) = 1 2 . This piece of information emerges through the recondensed λ j as in Eq. (6.34), rendering the thermodynamic Y -system of the non-canonical form. Indeed, a simple example of a non-canonical Y -system is provided by the canonical Gibbs equilibrium state, specified by λ j (u) = πJ β δ j,1 δ(u) and ζ j (u) = 0. The corresponding TBA source term thus reads d j = s λ j = πJ β δ j,1 s(u), in agreement with [26].
Non-meromorphic Y -functions. As observed in the TBA analysis, in general the thermodynamic Y -functions are non-meromorphic complex functions. This is to be contrasted with the canonical Y-functions of the mirror system describing regularised macrostates which are manifestly meromorphic on P. The non-meromorphic structure appears in the large-N scaling limit, when the analytic data from the interior of P form macroscopic condensates. A degree-N Padé approximation of a generic non-meromorphic Y -function in terms of a canonical (i.e. meromoprhic) Y-function provides a particular algorithm of information compression.
Non-canonical asymptotics. A third subtlety concerns the large-u asymptotics of the thermodynamic Y -functions. At finite N the canonical Y-functions possess a one-parameter family of large-u asymptotics parametrised by h, cf. Eq. (6.13), expressed through the solution to the classical limit of the Hirota equation Eq. (6.10). In the large-N limit however, upon removing the regulator Λ ∞ → ∞, these asymptotics may decouple, with the Y -functions acquiring non-canonical large-u asymptotics provided the large-u asymptotics of the chemical potentials µ j (u) are non-trivial. A particular instance of this is an infinite-parametric family of 'dispersionless' states, that is the class of states characterised by constant µ j .
Conclusion
In this work we have undertaken a comprehensive study of the equilibrium landscape of the isotropic Heisenberg spin-1/2 chain. We have developed a robust and unified framework which encompasses both the Thermodynamic Bethe Ansatz and the two-dimensional vertexmodel regularisation approaches to thermodynamics. In particular we have explained the emergence of a splitting of the chemical potentials µ j into two contributions: the node data λ j which determine the thermodynamic Y -system, and the analytic data ζ j which encode the branch points of the Y -functions in the physical strip P. These characterise equilibrium states in distinct ways, endowing the equilibrium landscape with a structure that is deserving of further exploration.
There are several novel features of the work worth highlighting. Firstly, we express the generic density matrix Eq. (3.1) in a form which reflects the underlying su(2) symmetry of the model, which clarifies how the polarisation of the mode operators ρ j is set. In our TBA analysis, we demonstrate on general grounds that the equilibrium Y -functions are generically non-meromorphic in the physical strip P. Our lattice regularisation of a generic ensemble treats the node and analytic data separately. For the node data we invoke a discretisation of the λ j (u), and achieve a variant of 'Trotterisation' without actually appealing to the Suzuki-Trotter formula. For the analytic data we develop a contour integration procedure, and intro-duce Padé approximants to regularise generic branch points of Y -functions. In Section 6 we reconnect the TBA and vertex model approaches by identifying a distinguished eigenvalue of the mirror system, and use it to demonstrate complete equivalence of the descriptions. The interconnected nature of our analysis is summarised in Figure 1.
A technical point emerging from our study is an explanation of the breakdown of the canonical Y-system, i.e. the emergence of the thermodynamic Y -system. Put simply, this is due to competition between the infinitesimal regulator ε ≡ 0 + which is tied to locality of the ρ j , and the regularisation parameter N which is finite for any vertex model approximation of the ensemble and diverges in the scaling limit. At finite N all the poles and zeros of the Y j lie on the physical strip P of Eq. (2.22) and the corresponding Y-system is canonical, Eq. (6.12). The poles and zeros resulting from the node data λ j however approach the boundary of the strip as 1/N , so that in the large-N limit they escape P, resulting in the appearance of node terms in the thermodynamic Y -system, Eq. (4.11).
Throughout the work we have adopted a universal language, which should facilitate extensions to other models solvable through the Bethe ansatz framework. The simplest generalisation of the model considered here is its uniaxial anisotropic deformation [19]. The extension to the 'hyperbolic' (easy-axis) regime, with quantum deformation parameter q ∈ , is straightforward and readily follows from an analytic continuation of the local degrees of freedom (i.e. the scattering matrix) of the isotropic model considered here. In distinction, in the 'trigonometric' (easy-plane) regime with the deformation parameter q = e iγ at the rootsof-unity γ = m/ ∈ , the total number of local degrees of freedom depends quite intricately on the values of co-prime integers m and , see [20,40,64]. To implement our approach, the compete set of unitary local charges is required, already identified previously in [7]. The closure of the functional hierarchy and the string-charge duality requires here an extra family of non-unitary charges [44,65,66].
Going forward, an important open question is whether there exist physically discernible features associated with the structures identified here. Addressing this will require analysing the correlation functions of local observables [47,59,[67][68][69]. We anticipate that it will be interesting to examine the splitting between node and analytic data in this context, and hope that the framework we provide offers a solid foundation upon which to proceed.
6) obtained by inserting Eq. (2.16) in Eq. (4.5), and making use of the identities Eq. (2.28), the TBA equations Eq. (4.3), and that h j belongs to the nullspace of (1 − R).
17) along with the asymptotic behaviour lim |u|→∞ ν j (u) = 0. Introducing the functions γ j (u)
kFigure 2 :
2their complex conjugates {θ k,i }. The regularised equilibrium ensemble is naturally cast as a two-dimensional classical vertex model wrapped around a cylinder of circumference L and height 2N + n z k + n p k , (5.28) as illustrated in Figure 2. The 'Boltzmann weights' are formally identified with the matrix elements of Lax matrices, through the Lax representation of transfer matrices provided in Eq. (2.33). By virtue of the involution property of Eq. (2.34), all stacking configurations of k, θ An illustration of the integrable two-dimensional vertex model corresponding to a regularised equilibrium macrostate, defined on a cylinder of circumference L and height 2N . The 'Boltzmann weights' are given by the matrix elements of the Lax matrices attached at each vertex. The square vertices attached at the vertical boundary incorporate the h-dependent twist, and the dashed gray lines denote traces over the auxiliary spaces.
16) by again multiplying out all (possible) zeros y z j,a and poles y p j,b on the physical the strip P, so that log Y adp j (u) are analytic on P with constant large-u asymptotics. Substituting this into Eq. (6.12), taking the logarithm and convolving with s, we end up with the coupled integral equations log Y j = d j + I j, s log(1 + Y ),
34 )
34are precisely the source terms (4.15) of the local form of the TBA equations Eq. (4.7). energy. Having taken the continuum scaling limit, we can now determine the eigenvalues of T -functions T j (u) encoding the distinguished state. In particular, employing Eq. (6.30) in Eq. (6.15) we obtainlog T 1 (0) = −s • µ 1 + s • log 1 + Y 1 . (6.35)Finally, substituting this into Eq. (6.3) as the dominant eigenvalue we re-obtain precisely the expression for the free energy given by Eq. (4.6). As the large-N scaling limit of the canonical Y-functions Y j satisfy the local TBA equations Eq. (4.7) due to Eq. (6.34), we have fully recovered the TBA description of Section 4.In conclusion, for a general equilibrium state of the Heisenberg spin-1/2 chain, the thermodynamic Y -functions coincide with the large-N scaling limit of the distinguished canonical Y-functions associated with the dominant eigenvalue of the mirror system.Free u
Y j (u)
The Hirota equation(2.35) can be understood as the 'quantum' counterpart of the fusion identities for characters of the 'classical' Lie algebra su(2), i.e. fusion identities amongst unitary irreducible representations of su(2). Additional details on fusion identities can be found in e.g.[24,34,35,[38][39][40].
The canonical version of the column transfer matrix appears under several different names in the literature, including the 'virtual-space'[21], 'crossing'[23], 'column-to-column'[58], 'quantum'[27] and 'Matsubara'[59] transfer matrix.
We remind that in distinction to N , ε is not an adjustable parameter of the regularisation scheme.
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[
"Hermenegildo Borges De Oliveira [email protected]. \nFCT -Universidade do Algarve\n\n\nCMAF -Universidade de Lisboa\nPortugal\n"
] |
[
"FCT -Universidade do Algarve\n",
"CMAF -Universidade de Lisboa\nPortugal"
] |
[
"MSC 2010: 76D03, 76D05, 35J60, 35Q30"
] |
In this work the existence of weak solutions for a class of non-Newtonian viscous fluid problems is analyzed. The problem is modeled by the steady case of the generalized Navier-Stokes equations, where the exponent q that characterizes the flow depends on the space variable: q = q(x). For the associated boundary-value problem we show that, in some situations, the log-Hölder continuity condition on q can be dropped and the result of the existence of weak solutions still remain valid for any variable exponent q ≥ α > 2N N+2 , where α = ess inf q.
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"https://arxiv.org/pdf/1203.6799v1.pdf"
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1203.6799
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b13f09f8870bec835d46364ed2227ff3399f12db
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30 Mar 2012 Date: March 30, 2012.
Hermenegildo Borges De Oliveira [email protected].
FCT -Universidade do Algarve
CMAF -Universidade de Lisboa
Portugal
MSC 2010: 76D03, 76D05, 35J60, 35Q30
30 Mar 2012 Date: March 30, 2012.A NOTE ABOUT EXISTENCE FOR A CLASS OF VISCOUS FLUID PROBLEMSand phrases: steady flowsnon-Newtonianvariable exponentexistencelocal decomposition of the pressureLipschitz truncation
In this work the existence of weak solutions for a class of non-Newtonian viscous fluid problems is analyzed. The problem is modeled by the steady case of the generalized Navier-Stokes equations, where the exponent q that characterizes the flow depends on the space variable: q = q(x). For the associated boundary-value problem we show that, in some situations, the log-Hölder continuity condition on q can be dropped and the result of the existence of weak solutions still remain valid for any variable exponent q ≥ α > 2N N+2 , where α = ess inf q.
Introduction
In this article we study the steady motion of an incompressible and homogeneous viscous fluid in a bounded domain Ω ⊂ R N , N ≥ 2, with the boundary denoted by ∂Ω. We assume the motion is described by the following boundary-value problem for the generalized Navier-Stokes equations:
(1.1) div u = 0 in Ω;
(1.2) div(u ⊗ u) = f − ∇p + div S in Ω;
(1.3) u = 0 on ∂Ω.
Here, u is the velocity field, p stands for the pressure divided by the constant density and f is the external forces field. We assume the extra stress tensor S has a variable q-structure in the following sense: Here, M N sym is the vector space of all symmetric N × N matrices, which is equipped with the scalar product A : B and norm |A| = √ A : A. The existence of weak solutions to the problem (1.1)-(1.3) with a constant qstructure was established by [12] and [14] for q ≥ 3N N +2 , by [9] and [15] for q > 2N N +1
and, finally and again, by [10] for q > 2N N +2 . These results were obtained in the class (1.4) V q := closure of V in W 1,q (Ω), where V := {v ∈ C ∞ 0 (Ω) : div v = 0} . The proofs in [12,14] use the theory of monotone operators together with compactness arguments, whereas in [9,15] and [10] are used, in addition, the L ∞ and the Lipschitz-truncation methods, respectively. Each one of these results improves the previous one in the sense that the convective term u ⊗ u : D(ϕ) is in L 1 (Ω) for an increasingly smaller lower limit of q.
The mathematical analysis of the problem (1.1)-(1.3), with the deviatoric stress tensor satisfying to (A)-(D) with a variable q-structure, must be done in the context of Orlicz spaces. These spaces resemble many of the aspects of classical Lebesgue and Sobolev spaces, but there are some important differences which must be pointed out (see Section 2). Existence results for the problem (1.1)-(1.3), with the deviatoric stress tensor satisfying to (A)-(D) with a variable q-structure, are due to [16], [11] and [8] and were obtained in the class
(1.5) W q (Ω) := closure of V in the D(v) L q(·) (Ω) -norm ,
where q ∈ P(Ω), the set of all measurable functions q : Ω → [1, ∞], satisfies to
(1.6) 1 < α := ess inf x∈Ω q(x) ≤ q(x) ≤ ess sup x∈Ω q(x) := β < ∞.
The proofs in [16] and in [11] are valid for α > 3N N +2 and α > 2N N +1 , respectively. Moreover they follow the same approach of [12,14] and [9,15], respectively, and use the fact that W q (Ω) is continuously imbedded into V α . The proof of [8] is valid for α > 2N N +2 , provided the variable exponent q is globally log-Hölder continuous in the sense of (2.3) below. The proof here follows the same approach of the result for constant q in [8] and uses results on Lipschitz truncations of functions in Orlicz-Sobolev spaces performed still in [8]. See also [3,4,15] for concrete fluid models with a variable q-structure.
Our goal in this work is to show that the log-Hölder continuity condition (2.3) is not necessary to show the existence of weak solutions to the problem (1.1)-(1.3) with the deviatoric stress tensor satisfying to (A)-(D) with a variable q-structure. As one can sees in the proof of [8,Theorem 5.1], assumption (2.3) is fundamental to achieve the existence result by the method proposed there. Firstly, we shall seek for a different condition that assures the existence of weak solutions for this problem in the case of α > 2N N +2 . At the end, we shall give an example to which neither this new condition nor (2.3) are needed.
Weak Formulation
The notation used in this work is largely standard in Mathematical Fluid Mechanics (see e.g. [14]). In this article, the notations Ω or ω stand always for a domain, i.e., a connected open subset of R N , N ≥ 1. Given k ∈ N, we denote by C k (Ω) the space of all k-differentiable functions in Ω. By C ∞ 0 (Ω) we denote the space of all infinity-differentiable functions with compact support in Ω. In the context of distributions, the space C ∞ 0 (Ω) is denoted by D(Ω) instead. The space of distributions over D(Ω) is denoted by D ′ (Ω). If X is a generic Banach space, its dual space is denoted by X ′ . Let 1 ≤ q ≤ ∞ and Ω ⊂ R N , with N ≥ 1, be a domain. We use the classical Lebesgue spaces L q (Ω), whose norm is denoted by · L q (Ω) . For any nonnegative k, W k,q (Ω) denotes the Sobolev space of all functions u ∈ L q (Ω) such that the weak derivatives D α u exist, in the generalized sense, and are in L q (Ω) for any multi-index α such that 0 ≤ |α| ≤ k. In particular, W 1,∞ (Ω) stands for the space of Lipschitz functions. The norm in W k,q (Ω) is denoted by · W k,q (Ω) . We define W k,q 0 (Ω) as the closure of C ∞ 0 (Ω) in W k,q (Ω). For the dual space of W k,q 0 (Ω), we use the identity (W k,q 0 (Ω)) ′ = W −k,q ′ (Ω), up to an isometric isomorphism. Vectors and vector spaces will be denoted by boldface letters.
We denote by P ( Given q ∈ P(Ω), we denote by L q(·) (Ω) the space of all measurable functions f in Ω such that its semimodular is finite:
(2.1) A q(·) (f ) :=ˆΩ |f (x)| q(x) d x < ∞.
The space L q(·) (Ω) is called Orlicz space and is also known by Lebesgue space with variable exponent. Equipped with the norm
(2.2) f L q(·) (Ω) := inf λ > 0 : A q(·) f λ ≤ 1 ,
L q(·) (Ω) becomes a Banach space. If q + < ∞, L q(·) (Ω) is separable and the space C ∞ 0 (Ω) is dense in L q(·) (Ω). Moreover, if 1 < q − ≤ q + < ∞, L q(·) (Ω) is reflexive. One problem in Orlicz spaces is the relation between the semimodular (2.1) and the norm (2.2). If (1.6) is satisfied, one can shows that
f q − L q(·) (Ω) − 1 ≤ A q(·) (f ) ≤ f q + L q(·)
(Ω) + 1 . In Orlicz spaces, there holds a version of Hölder's inequality, called generalized Hölder's inequality. Given q ∈ P(Ω), the Orlicz-Sobolev space W 1,q(·) (Ω) is defined as the set of all functions f ∈ L q(·) (Ω) such that D α f ∈ L q(·) (Ω) for any multiindex α such that 0 ≤ |α| ≤ 1. In W 1,q(·) (Ω) is defined a semimodular and the correspondent induced norm analogously as in (2.1)-(2.2). For this norm, W 1,q(·) (Ω) is a Banach space, which becomes separable and reflexive in the same conditions as L q(·) (Ω). The Orlicz-Sobolev space with zero boundary values is defined by:
W 1,q(·) 0 (Ω) := f ∈ W 1,q(·) (Ω) : supp f ⊂⊂ Ω · W 1,q(·) (Ω) .
In contrast to the case of classical Sobolev spaces, the set C ∞ 0 (Ω) is not necessarily dense in W 1,q(·) 0
(Ω) -the closure of C ∞ 0 (Ω) in W 1,q(·) (Ω) is strictly contained in W 1,q(·) 0
(Ω). The equality holds only if q is globally log-Hölder continuous, i.e., if exist positive constants C 1 , C 2 and q ∞ such that
(2.3) |q(x) − q(y)| ≤ C 1 ln(e + 1/|x − y|) , |q(x) − q ∞ | ≤ C 2 ln(e + |x|) ∀ x, y ∈ Ω.
See the monograph [7] for a thorough analysis on Orlicz and Orlicz-Sobolev spaces.
In order to introduce the notion of weak solutions we shall consider in this work, let us recall the well-known function spaces of Mathematical Fluid Mechanics defined at (1.4). Due to the presence of the variable exponent q(·) in the structure of the tensor S, we need to consider the weak solutions to the problem (1.
1)-(1.3) in some Orlicz-Sobolev space. Since the set C ∞ 0 (Ω) is not necessarily dense in W 1,q(·) 0
(Ω), we define the analogue of V q by (1.5). It is a easy task to verify the space W q (Ω) satisfies to the following continuous imbeddings:
V β ֒→ W q (Ω) ֒→ V α .
Moreover, W q (Ω) is a closed subspace of V α and therefore it is a reflexive and separable Banach space for the norm
v Wq(Ω) := D(v) L q(·) (Ω) . Definition 2.1.
Let Ω be a bounded domain of R N , with N ≥ 2, and let q ∈ P(Ω) be a variable exponent satisfying to (1.6). Let also f ∈ L 1 (Ω) and assume that conditions (A)-(D) are fulfilled with a variable exponent q. A vector field u is a weak solution to the problem (1.
1)-(1.3), if: (1) u ∈ W q (Ω); (2) For every ϕ ∈ C ∞ 0 (Ω), with div ϕ = 0, Ω (S(D(u)) − u ⊗ u) : D(ϕ) dx =ˆΩ f · ϕ d x.
The main goal of this work is to seek for the condition(s) we have to impose in the problem (1.
(2.4) f = −div F, F ∈ M N sym , F ∈ L q ′ (·) (Ω), (2.5) 2N N + 2 < α ≤ β < ∞ .
Is it possible to find a distinct condition from the log-Hölder continuity property (2.3) that assures the existence of a weak solution to the problem (1.1)-(1.3) in the sense of Definition 2.1?
The answer to Question 2.1 will be the aim of next sections. For that, we shall prove an existence result for the problem (1.1)-(1.3) under the conditions stated in Question 2.1. We will see that the validity of such an existence result will demand a new and different condition.
The regularized problem
Let Φ ∈ C ∞ ([0, ∞)) be a non-increasing function such that 0 ≤ Φ ≤ 1 in [0, ∞), Φ ≡ 1 in [0, 1], Φ ≡ 0 in [2, ∞) and 0 ≤ −Φ ′ ≤ 2. For ǫ > 0, we set (3.1) Φ ǫ (s) := Φ(ǫs), s ∈ [0, ∞).
We consider the following regularized problem:
(3.2) div u ǫ = 0 in Ω, (3.3) div(u ǫ ⊗ u ǫ Φ ǫ (|u ǫ |)) = f − ∇p ǫ + div S(D(u ǫ )) + ǫ|D(u ǫ )| β−2 D(u ǫ )
in Ω,
(3.4) u ǫ = 0 on ∂Ω. A vector function u ǫ ∈ V β is a weak solution to the problem (3.2)-(3.4), if (3.5) Ω S(D(u ǫ )) + ǫ|D(u ǫ )| β−2 D(u ǫ ) − u ǫ ⊗ u ǫ Φ ǫ (|u ǫ |) : D(ϕ) dx =ˆΩ F : D(ϕ) d x for all ϕ ∈ V.
Under the assumptions stated in Question 2.1, it can be proved that, for each ǫ > 0, there exists a weak solution u ǫ ∈ V β to the problem (3.2)-(3.4).
The proof of this result is based on the Schauder fixed point theorem. The map construction is done by putting the convective term on the right hand side and by solving a nonlinear equation via the monotone operator theory. Moreover, it can be proved that every weak solution satisfies to the following energy equality:
(3.6)ˆΩ S(D(u ǫ )) : D(u ǫ )dx + ǫˆΩ |D(u ǫ )| β dx =ˆΩ F : D(u ǫ )dx. Now, let u ǫ ∈ V β be a weak solution to the problem (3.2)-(3.4). From (3.6) we can prove that (3.7)ˆΩ |D(u ǫ )| q(x) dx + ǫˆΩ |D(u ǫ )| β dx ≤ C,
where, by the assumption (2.4), C is a positive constant and, very important, does not depend on ǫ. Then we can prove from (3.7) that
(3.8) D(u ǫ ) L q(·) (Ω) ≤ C, (3.9) u ǫ Vα ≤ C, (3.10) S(D(u ǫ )) L q ′ (·) (Ω) ≤ C, (3.11) S(D(u ǫ )) L β ′ (Ω) ≤ C.
On the other hand, by using (3.9) and Sobolev's inequality, we have
(3.12) u ǫ L α * (Ω) ≤ C,
where α * denotes the Sobolev conjugate of α. As a consequence of (3.12) and (3.1),
(3.13) u ǫ ⊗ u ǫ Φ ǫ (|u ǫ |) L α * 2 (Ω) ≤ C.
From (3.9), (3.11) and (3.13), there exists a sequence of positive numbers ǫ m such that ǫ m → 0, as m → ∞, and
(3.14) u ǫm → u weakly in V α , as m → ∞, (3.15) S(D(u ǫm )) → S weakly in L β ′ (Ω), as m → ∞, (3.16) u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) → G weakly in L α * 2 (Ω), as m → ∞, (3.17) ǫ m |D(u ǫm )| q−2 D(u ǫm ) → 0 weakly in L β ′ (Ω), as m → ∞.
Now we observe that, due to (3.14), the application of Sobolev's compact imbedding theorem implies (3.18) u ǫm → u strongly in L γ (Ω), as m → ∞, for any γ : 1 ≤ γ < α * .
Since 2 < α * , it follows from (3.18) that (3.19) u ǫm → u strongly in L 2 (Ω), as m → ∞.
Using (3.1) and (3.19), we can prove that
(3.20) u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) → u ⊗ u strongly in L 1 (Ω), as m → ∞.
Then gathering the information of (3.16) and (3.20), we see that G = u ⊗ u. Finally, using the convergence results (3.14)-(3.17) and observing (3.20), we can pass to the limit m → ∞ in the following integral identity, which results from (3.5), (3.21)
Ω S(D(u ǫm )) + ǫ m |D(u ǫm )| β−2 D(u ǫm ) − u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − F : D(ϕ) dx = 0,
valid for all ϕ ∈ V, to obtain
(3.22)ˆΩ(S − u ⊗ u − F) : D(ϕ) dx = 0 ∀ ϕ ∈ V.
4. Decomposition of the pressure.
Since we shall use test functions which are not divergence free, we first have to determine the approximative pressure from the weak formulation (3.21). First, let ω ′ be a fixed but arbitrary open bounded subset of Ω such that
(4.1) ω ′ ⊂⊂ Ω and ∂ω ′ is Lipschitz,
where ω ′ ⊂⊂ Ω means that ω ′ is compactly contained in Ω, and let us set
(4.2) Q ǫm := S(D(u ǫm )) + ǫ m |D(u ǫm )| β−2 D(u ǫm ) − u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − F.
Using assumption (2.4) and the results (3.11), (3.13) and (3.17), we can prove that
(4.3) Q ǫm ∈ L r (Ω), where 1 < r ≤ r 0 := min β ′ , α * 2 .
Note that r 0 = min{β ′ , N α 2(N −α) } if α < N and r 0 = β ′ if N ≥ α. Then we define a linear functional
(4.4) Π ǫm : W 1,r ′ 0 (ω ′ ) → W −1,r (ω ′ ) , (4.5) Π ǫm , ϕ W −1,r (ω ′ )×W 1,r ′ 0 (ω ′ ) :=ˆω ′ Q ǫm : D(ϕ) dx.
Using (4.4)-(4.5), we can prove, owing to (4.3), that
(4.6) Π ǫm (V r ′ ) ′ ≤ C ,
where C is a positive constant independent of m. Note that here V r ′ is taken over ω ′ . Moreover, since V is dense in V r ′ , we can see that (3.21), (4.2) and (4.5) imply
(4.7) Π ǫm , ϕ (V r ′ ) ′ ×V r ′ = 0 ∀ ϕ ∈ V r ′ .
By virtue of (4.4)-(4.7) and due to assumption (4.1), we can apply a version of de Rham's Theorem to prove the existence of a unique function
(4.8) p ǫm ∈ L r ′ (ω ′ ), withˆω ′ p ǫm dx = 0, such that (4.9) Π ǫm , ϕ W −1,r (ω ′ )×W 1,r ′ 0 (ω ′ ) =ˆω ′ p ǫm divϕ dx ∀ ϕ ∈ W 1,r ′ 0 (ω ′ ) , (4.10) p ǫm L r ′ (ω ′ ) ≤ Π ǫm (V r ′ ) ′ .
Then, gathering the information of (3.21), (4.2), (4.5) and (4.9), we obtain
ω ′ S(D(u ǫm )) : D(ϕ) dx + ǫ mˆω ′ |D(u ǫm )| β−2 D(u ǫm ) : D(ϕ) dx = ω ′ F : D(ϕ) dx +ˆω ′ u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) : D(ϕ) dx +ˆω ′ p ǫm divϕ dx (4.11)
for all ϕ ∈ W 1,r ′ 0 (ω ′ ). On the other hand, due to (4.6) and (4.10) and by means of reflexivity, we get, passing to a subsequence, that (4.12) p ǫm → p 0 weakly in L r ′ (ω ′ ), as m → ∞.
Next, passing to the limit m → ∞ in the integral identity (4.11) by using the convergence results (3.15), (3.16) together with (3.20), using also (3.17) and (4.12), we obtainˆω
′ (S − u ⊗ u − F) : D(ϕ) dx =ˆω ′ p 0 divϕ dx (4.13)
for all ϕ ∈ W 1,r ′ 0 (ω ′ ). Next, we shall decompose the pressure found in the first part of this section. With this in mind, let ω be a fixed but arbitrary domain such that (4.14)
ω ⊂⊂ ω ′ ⊂⊂ Ω and ∂ω is C 2 .
To simplify the notation in the sequel, let us set A s (ω) := {a ∈ L s (ω) : a = △u, u ∈ W 2,s 0 (ω)}, 1 < s < ∞.
Here we shall use some results due to [17] that allow us to locally decompose the pressure. Applying [17, Lemma 2.4], with s = β ′ first and then with s = α * 2 , and using (3.15) and (3.17) by one hand and (3.16) and (3.20) on the other, we can infer that exist unique functions
(4.15) p 1 ǫm ∈ A β ′ (ω), (4.16) p 2 ǫm ∈ A α * 2 (ω) such that (4.17)ˆω p 1 ǫm △φ dx =ˆω S(D(u ǫm )) + ǫ m |D(u ǫm )| β−2 D(u ǫm ) − S : ∇ 2 φ dx, (4.18)ˆω p 2 ǫm △φ dx = −ˆω(u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − u ⊗ u) : ∇ 2 φ dx for all φ ∈ C ∞ 0 (ωp 1 ǫm L β ′ (ω) ≤ C 1 S(D(u ǫm )) − S + ǫ m |D(u ǫm )| β−2 D(u ǫm ) L β ′ (ω) , (4.20) p 2 ǫm L α * 2 (ω) ≤ C 2 u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − u ⊗ u L α * 2 (ω)
.
where C 1 and C 2 are positive constants depending on β ′ , α * and on ω.
Now, combining (4.11) and (4.13), and using the definition of the distributive derivative, we obtain
div S(D(u ǫm )) − S + ǫ m |D(u ǫm )| β−2 D(u ǫm ) − div (u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − u ⊗ u) = ∇(p ǫm − p 0 ) in D ′ (ω). (4.21)
Then, testing (4.21) by ∇φ, with φ ∈ C ∞ 0 (ω), integrating over ω and comparing the resulting equation with the one resulting from adding (4.17) and (4.18), we obtain
p ǫm − p 0 = p 1 ǫm + p 2 ǫm . Inserting this into (4.21), it follows that div S(D(u ǫm )) − S + ǫ m |D(u ǫm )| β−2 D(u ǫm ) − div (u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − u ⊗ u) = ∇ p 1 ǫm + p 2 ǫm in D ′ (ω).
The Lipschitz truncation
To start this section, let us set
(5.1) w ǫm := (u ǫm − u)χ ω ,
where χ ω denotes the characteristic function of the set ω introduced in (4.14).
Having in mind the extension of (4.22) to R N , here we shall consider that
(5.2) Υ ǫm := Υ 1 ǫm + Υ 2 ǫm is extended from ω to R N by zero, where (5.3) Υ 1 ǫm := − S(D(u ǫm )) − S + ǫ m |D(u ǫm )| β−2 D(u ǫm ) + p 1 ǫm I, (5.4) Υ 2 ǫm := u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − u ⊗ u + p 2
ǫm I, and I denotes the identity tensor. Now, due to the definition (5.1) and by virtue of (3.14) and (3.18), we have (5.5) w ǫm → 0 weakly in W 1,α (R N ), as m → ∞, (5.6) w ǫm → 0 strongly in L γ (R N ), as m → ∞, for any γ : 1 ≤ γ < α * .
Moreover, due to (3.15), (3.17) and (4.19) by one hand, and due to (3.16), (3.20) and (4.20) on the other, we have
(5.7) Υ 1 ǫm L β ′ (R N ) ≤ C, (5.8) Υ 2 ǫm L α * 2 (R N ) ≤ C.
In addition to (5.8), we see that, due to (3.18) and (4.20),
(5.9) Υ 2 ǫm → 0 strongly in L γ 2 (R N )
, as m → ∞, for any γ : 1 ≤ γ < α * . Next, let us consider the Hardy-Littlewood maximal functions of |w ǫm | and |∇w ǫm | defined by
M(|w ǫm |)(x) := sup 0<R<∞ 1 L N (B R (x))ˆB R (x)
|w ǫm (y)| dy,
M(|∇w ǫm |)(x) := sup 0<R<∞ 1 L N (B R (x))ˆB R (x)
|∇w ǫm (y)| dy;
where B R (x) denotes the ball of R N centered at x and with radius R > 0, and L N (ω) is the N -dimensional Lebesgue measure of ω. Arguing as in [8, p. 218] and using the boundedness of the Hardy-Littlewood maximal operator M, we can prove that for all m ∈ N and all j ∈ N 0 there exists
(5.10) λ m,j ∈ 2 2 j , 2 2 j+1
such that
(5.11) L N (F m,j ) ≤ 2 −j λ −γ m,j w ǫm L γ (R N )
, for any γ : 1 ≤ γ < α * ,
(5.12) L N (G m,j ) ≤ 2 −j λ −α m,j ∇w ǫm L α (R N ) , where F m,j := x ∈ R N : M(|w ǫm |)(x) > 2λ m,j , G m,j := x ∈ R N : M(|∇w ǫm |)(x) > 2λ m,j . Setting (5.13) R m,j := F m,j ∪ G m,j ∪ x ∈ R N : x is not a Lebesgue point of |w ǫm | ,
we can see that, by virtue of (5.11)-(5.13),
(5.14) L N (R m,j ) ≤ 2 −j λ −α m,j w ǫm W 1,α (R N )
. In addition, due to (5.5)-(5.6) and (5.10),
(5.15) lim sup m→∞ L N (R m,j ) ≤ C2 −j λ −α m,j .
Then, by [1] together with (5.1),
(5.16) ∃ z m,j ∈ W 1,∞ (R N ), z m,j = w ǫm in ω \ A m,j 0 R N \ ω , where (5.17) A m,j := {x ∈ ω : z m,j (x) = w ǫm (x)}, such that (5.18) z m,j L ∞ (ω) ≤ 2λ m,j , (5.19) ∇z m,j L ∞ (ω) ≤ Cλ m,j , C = C(N, ω).
Moreover, by [13, Proposition 2.2] and using (5.11)-(5.13) and (5.17),
(5.20) A m,j ⊂ ω ∩ R m,j .
As a consequence of (5.20) together with (5.14) and (5.15),
(5.21) L N (A m,j ) ≤ 2 −j λ −α m,j w ǫm W 1,α (R N ) , (5.22) lim sup m→∞ L N (A m,j ) ≤ C2 −j λ −α m,j .
On the other hand, due to (5.18)-(5.19) and (5.5) together with (5.21)-(5.22), we can prove that for any j ∈ N 0 (5.23) z m,j → 0 weakly in W 1,α 0 (ω), as m → ∞. Then by Sobolev's compact imbedding theorem, we get for any j ∈ N 0 z m,j → 0 strongly in L γ (ω), as m → ∞, for any γ : 1 ≤ γ < α * .
Using this information, (5.18) and interpolation, we prove that for any j ∈ N 0 (5.24) z m,j → 0 strongly in L s (ω), as m → ∞, for any s : 1 ≤ s < ∞.
Finally, as a consequence of (5.23) and (5.24), we obtain for any j ∈ N 0 (5.25) z m,j → 0 weakly in W 1,s 0 (ω), as m → ∞, for any s : 1 ≤ s < ∞.
Convergence of the approximated extra stress tensor
Let us first observe that, using the notations (5.2)-(5.4), we can write (4.22) as
(6.1) divΥ ǫm = 0 in D ′ (ω).
On the other hand, due to (5.7)-(5.8), Υ ǫm ∈ L r (R N ) for r satisfying to (4.3). Then, using this information and (5.25), we infer, from (6.1), that for any j ∈ N 0 (6.2)ˆω Υ ǫm : ∇z m,j dx = 0.
Expanding Υ ǫm in (6.2) through the notations (5.2)-(5.4) and subtracting and adding the integral of S(D(u)) : D(z m,j ) to the left hand side of the resulting equation, we obtain for any j ∈ N 0
ω (S(D(u ǫm )) − S(D(u))) : D(z m,j ) dx =ˆω (S − S(D(u))) : D(z m,j ) dx −ˆω ǫ m |D(u ǫm )| β−2 D(u ǫm ) : D(z m,j ) dx +ˆω p 1 ǫm div z m,j dx+ ω u ǫm ⊗ u ǫm Φ ǫm (|u ǫm |) − u ⊗ u + p 2 ǫm I : D(z m,j ) dx := J 1 m,j + J 2 m,j + J 3 m,j + J 4 m,j . (6.3)
We claim that, for a fixed j, To prove this, we will carry out the passage to the limit m → ∞ in all absolute values |J i m,j |, i = 1, . . . , 4. • lim sup m→∞ |J 1 m,j | = 0. By (5.25), with s = β, this is true once we can justify that S − S(D(u)) is uniformly bounded in L β ′ (ω). But this is a consequence of (3.15), the continuous imbedding L q ′ (·) (ω) ֒→ L β ′ (ω) and (3.10).
• lim sup m→∞ |J 2 m,j | = 0. Indeed, by Hölder's inequality, (5.19), (5.10) and (3.7), we have successively
|J 2 m,j | ≤ ǫ m |D(u ǫm )| β−2 D(u ǫm ) L 1 (ω) ∇z m,j L ∞ (ω) ≤ C 1 λ m,j ǫ 1 β m ˆω ǫ m |D(u ǫm )| β dx β−1 β ≤ C 2 ǫ 1 β m → 0, as m → ∞.
• lim sup m→∞ |J 3 m,j | ≤ C2 − j β . In fact, by Hölder's inequality and (4.19) together with (3.15) and (3.17), and using (5.16)) together with (5.1),
|J 3 m,j | ≤ C 1 div z m,j L β (ω) ≤ C 1 ∇z m,j L ∞ (ω) L N (A m,j ) 1 β .
The result follows by the application of (5.19), (5.10) and (5.22), provided that
(6.5) λ 1− α β m,j
is uniformly bounded in m.
• lim sup m→∞ |J 4 m,j | = 0. Using Hölder's inequality and (5.4), we have
|J 4 m,j | ≤ Υ 2 ǫm L 1 (ω) ∇z m,j L ∞ (ω) ≤ C 1 Υ 2 ǫm L 1 (ω) → 0, as m → ∞.
The last inequality and the conclusion follow, respectively, from (5.19) and (5.10), and (5.9) with γ = 2, observing here that assumption (2.5) implies 2 < α * .
Gathering the estimates above we just have proven (6.4).
We proceed with the proof by using an argument due to [6,Theorem 5]. Firstly, observing the definition of z m,j (cf. (5.16)), we have Then (6.4) and (6.6) imply that
(6.7) lim sup m→∞ I m,j ≤ lim sup m→∞ |II m,j | + C2 − j β .
For the term II m,j , we have by applying successively Hölder's inequality, (3.15), the continuous imbedding L q ′ (·) (ω) ֒→ L β ′ (ω) and (3.10) altogether with (5.19),
|II m,j | ≤ C 1 S(D(u ǫm )) − S(D(u)) L β ′ (Am,j) ∇z m,j L β (Am,j ) ≤ C 2 λ m,j L N (A m,j ) 1 β .
Then, (5.10) and (5.22) yield that for any j ∈ N 0
(6.8) lim sup m→∞ |II m,j | ≤ C2 − j β λ 1− α β m,j .
As a consequence of (6.7) and (6.8), we obtain for any j ∈ N 0
(6.9) lim sup m→∞ |I m,j | ≤ C2 − j β 1 + λ 1− α β m,j .
Arguing as we did to prove (6.8)-(6.9) and using (5.16) and (5.22), we have for any θ ∈ (0, 1) (6.10)
lim sup m→∞ˆω g θ ǫm dx ≤ C 1 2 −θ j β 1 + λ 1− α β m,j θ + C 2 2 −θ j β −(1−θ)j λ (1− α β )θ−(1−θ)α m,j ,
where g ǫm := (S(D(u ǫm )) − S(D(u))) : (D(u ǫm ) − D(u)).
Since β > 1, θ ∈ (0, 1) and j ∈ N 0 is arbitrary, 2 −θ j β → 0 and 2 −θ j β −(1−θ)j → 0, as j → ∞. This and (6.10) imply that for any θ ∈ (0, 1) lim sup m→∞ˆω g θ ǫm dx = 0 provided that (6.5) holds. Then, passing to a subsequence, (6.11) g ǫm → 0 a.e. in ω, as m → ∞. Finally, (3.11) and (6.12) allow us to use Vitali's theorem together with (3.15) to conclude that S = S(D(u)).
Answer to Question 2.1
From Section 3 until Section 6 we have proven the existence of, at least, a weak solution to the problem (1.1)-(1.3) in the sense of Definition 2.1 and satisfying to the conditions stated in Question 2.1, provided condition (6.5) is fulfilled. A simple analysis shows us that condition (6.5) is equivalent to assume that α ≥ β. But this cannot happen unless α = β. In this situation, we would fall in the case of a constant exponent q studied in [10]. If we go further behind, we see that condition (6.5) came as a result of (5.22) and this in turn had its origin in (5.5)-(5.6). Therefore the best way to assure that (6.5) is fulfilled is to assume that (5.5)-(5.6) are satisfied with α replaced by β, i.e.
(7.1) w ǫm → 0 weakly in W 1,β (R N ), as m → ∞, (7.2) w ǫm → 0 strongly in L γ (R N ), as m → ∞, for any γ : 1 ≤ γ < β * .
We observe that (5.5)-(5.6) came as a result of (3.9). In consequence, (7.1)-(7.2) hold if we had
(7.3) u ǫ V β ≤ C
instead of (3.9). Finally, condition (7.3) is satisfied if
(7.4)ˆΩ |D(u ǫ )| β dx ≤ C .
In consequence, if we go back to Sections 5 and 6 and replace all the exponents α by β, than we have λ 1− α β m,j = 1 in condition (6.5), and the existence result follows. Condition (7.4) can be seen as a consequence of the following higher integrability condition: assume that exists δ > 0 such that (7.5)ˆω ′ |D(u)| q(x)(1+δ) dx < ∞ for any subdomain ω ′ ⊂⊂ Ω. Then, under all the assumptions of Question 2.1, we can prove the existence of weak solutions for the problem (1.1)-(1.3) with q = q(x), which a priori satisfy condition (7.5). This property is crucial to control the gradients of velocity in the space L β (Ω) and with the technique we used we can control them only in the spaces L α (Ω) and L q(·) (Ω). Despite assumption (7.5) is so strong that weakens very much such an existence result, we observe that this property is satisfied by the weak solutions to some fluid problems. In fact, fluids with viscosity dependence described using non-standard growth conditions have been treated, in the stationary case, in various settings (see [2,5] and the references therein). We think that the approach followed in [2,5] can be potentially useful to extend theses results to our problem. Therefore we are let to believe the higher integrability property expressed by assumption (7.5) is satisfied by every weak solution to the problem (1.1)-(1.3) with q = q(x). In this case, to prove an existence result for our problem we do not need the log-Hölder continuity condition (2.3) on q. On the other hand, we can realize that for models of generalized fluid flows in which the stress tensor brings itself this higher regularity, the existence result follows without assuming (2.3) and (7.5). An example of this situation is the problem with the stress tensor defined by
S = µ + τ |D| q(x)−2 D ,
where µ and τ are positive constants related with the viscosity. In this case, an existence result can be proved for 2N N +2 < α < β ≤ 2 proceeding as in the above sections.
A similar analysis can be done for the parabolic version of the problem (1.1)-(1.3). In this case all the reasoning is identical and we just have to use the parabolic versions of the results considered from Section 3 to Section 6. The impact of our work in the transient problem is in fact more important, because the parabolic extension of the work [8] is, to the best of our knowledge, still not proved. A thorough analysis of these problems is being written and it will be published elsewhere.
(
A) S : Ω × M N sym → M N sym is a Charathéodory function; (B) |S(x, A)| ≤ C|A| q(x)−1 for all A in M N sym and a.a. x in Ω; (C) S(x, A) : A ≥ C|A| q(x) for all A in M N sym and a.a. x in Ω; (D) (S(x, A) − S(x, B)) : (A − B) > 0 for all A = B in M N sym and a.a. x in Ω.
Ω) the set of all measurable functions q : Ω → [1, ∞] and define q − := ess inf x∈Ω q(x), q + := ess sup x∈Ω q(x).
1)-(1.3) that assure(s) the existence of weak solutions to this problem in the sense of Definition 2.1 and without any further restriction on q besides (1.6) above and (2.4)-(2.5) below. Question 2.1. Let Ω be a bounded domain in R N , N ≥ 2. Assume that conditions (A)-(D) are fulfilled with a variable exponent q ∈ P(Ω) satisfying to (1.6), and
(
S(D(u ǫm )) − S(D(u))) : D(z m,j ) dx ≤ C2 − j β
ˆω (S(D(u ǫm )) − S(D(u))) : D(z m,j ) dx = I m,j + II m,j , where I m,j :=ˆω \Am,j (S(D(u ǫm )) − S(D(u))) : (D(u ǫm ) − D(u)) dx, II m,j :=ˆA m,j (S(D(u ǫm )) − S(D(u))) : D(z m,j ) dx.
ǫm ) → D(u) a.e. in ω, as m → ∞.
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On stationary thermo-rheological viscous flows. S N Antontsev, J F Rodrigues, Ann. Univ. Ferrara Sez. VII Sci. Mat. 521S.N. Antontsev and J.F. Rodrigues. On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), no. 1, 19-36.
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Lebesgue and Sobolev spaces with variable exponents. L Diening, P Harjulehto, P Hästo, M Ružička, SpringerHeidelbergL. Diening, P. Harjulehto, P. Hästo and M. Ružička. Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg, 2011.
On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. L Diening, J Málek, M Steinhauer, ESAIM Control Optim. Calc. Var. 142L. Diening, J. Málek and M. Steinhauer. On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optim. Calc. Var. 14 (2008), no. 2, 211-232.
An existence result for fluids with shear dependent viscosity-steady flows. J Frehse, J Málek, M Steinhauer, Nonlinear Anal. 305J. Frehse, J. Málek and M. Steinhauer. An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear Anal. 30 (1997), no. 5, 3041-3049.
On analysis of steady flows of fluids with shear dependent viscosity based on the Lipschitz truncation method. J Frehse, J Málek, M Steinhauer, SIAM J. Math. Anal. 345J. Frehse, J. Málek and M. Steinhauer. On analysis of steady flows of fluids with shear dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34 (2003), no.5, 1064-1083.
The divergence equation in weighted-and L p(·) -spaces. A Huber, Math. Z. 267A. Huber The divergence equation in weighted-and L p(·) -spaces. Math. Z. 267 (2011), 341- 366.
New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problem for them. O A Ladyzhenskaya, Proc. Steklov Inst. Math. 102O.A. Ladyzhenskaya. New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problem for them. Proc. Steklov Inst. Math. 102 (1967), 95-118.
Quasimonotone versus pseudomonotone. R Landes, Proc. Roy. Soc. Edinburgh Sect. A. 126R. Landes. Quasimonotone versus pseudomonotone. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 705-707.
Quelques mèthodes de résolution des problèmes aux limites non liniaires. J.-L Lions, DunodParisJ.-L. Lions. Quelques mèthodes de résolution des problèmes aux limites non liniaires. Dunod, Paris, 1969.
A note on steady flow of fluids with shear dependent viscosity. M Ružička, Nonlinear Anal. 305M. Ružička. A note on steady flow of fluids with shear dependent viscosity. Nonlinear Anal. 30 (1997), no. 5, 3029-3039.
Electrorheological fluids: modeling and mathematical theory. M Ružička, Lecture Notes in Mathematics. 1748Springer-VerlagM. Ružička. Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin, 2000.
Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J Wolf, J. Math. Fluid Mech. 91J. Wolf. Existence of weak solutions to the equations of non-stationary motion of non- Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9 (2007), no. 1, 104-138.
|
[] |
[
"Generalizable Physics-constrained Modeling using Learning and Inference assisted by Feature-space Engineering",
"Generalizable Physics-constrained Modeling using Learning and Inference assisted by Feature-space Engineering"
] |
[
"Vishal Srivastava ",
"Karthik Duraisamy ",
"\nPh.D. Candidate\nDepartment of Aerospace Engineering\nDepartment of Aerospace Engineering\nUniversity of Michigan\n48105Ann Arbor\n",
"\nUniversity of Michigan\n48105Ann Arbor\n"
] |
[
"Ph.D. Candidate\nDepartment of Aerospace Engineering\nDepartment of Aerospace Engineering\nUniversity of Michigan\n48105Ann Arbor",
"University of Michigan\n48105Ann Arbor"
] |
[] |
This work presents a formalism to improve the predictive accuracy of physical models by learning generalizable augmentations from sparse data. Building on recent advances in data-driven turbulence modeling, the present approach, referred to as Learning and Inference assisted by Featurespace Engineering (LIFE), is based on the hypothesis that robustness and generalizability demand a meticulously-designed feature space that is informed by the underlying physics, and a carefully constructed features-to-augmentation map. The critical components of this approach are:(1)Maintaining consistency across the learning and prediction environments to make the augmentation case-agnostic; (2) Tightly-coupled inference and learning by constraining the augmentation to be learnable throughout the inference process to avoid loss of inferred information (and hence accuracy); (3) Identification of relevant physics-informed features in appropriate functional forms to enable significant overlap in feature space for a wide variety of cases to promote generalizability;(4)Localized learning, i.e. maintaining explicit control over feature space to change the augmentation function behavior only in the vicinity of available datapoints. To demonstrate the viability of this approach, it is used in the modeling of bypass transition. The augmentation is developed on skin friction data from two flat plate cases from the ERCOFTAC dataset. Piecewise linear interpolation on a structured grid in feature-space is used as a sample functional form for the augmentation to demonstrate the capability of localized learning. The impact of using a different function class (neural network) is also assessed. The augmented model is then applied to a variety of flat plate cases which are characterized by different freestream turbulence intensities, pressure gradients, and Reynolds numbers. The predictive capability of the augmented model is also tested on single-stage high-pressure-turbine cascade cases, and the model performance is analyzed from the perspective of information contained in the feature space. The results show consistent improvements across these cases, as long as the physical phenomena in question are well-represented in the training.
|
10.1103/physrevfluids.6.124602
|
[
"https://arxiv.org/pdf/2103.16042v2.pdf"
] | 232,417,757 |
2103.16042
|
d19cef2c16fd9cdfe5f0a61db89ee8ea8db619b4
|
Generalizable Physics-constrained Modeling using Learning and Inference assisted by Feature-space Engineering
26 Jul 2021
Vishal Srivastava
Karthik Duraisamy
Ph.D. Candidate
Department of Aerospace Engineering
Department of Aerospace Engineering
University of Michigan
48105Ann Arbor
University of Michigan
48105Ann Arbor
Generalizable Physics-constrained Modeling using Learning and Inference assisted by Feature-space Engineering
26 Jul 2021(Dated: July 28, 2021)* Electronic address: [email protected] † Electronic address: [email protected]
This work presents a formalism to improve the predictive accuracy of physical models by learning generalizable augmentations from sparse data. Building on recent advances in data-driven turbulence modeling, the present approach, referred to as Learning and Inference assisted by Featurespace Engineering (LIFE), is based on the hypothesis that robustness and generalizability demand a meticulously-designed feature space that is informed by the underlying physics, and a carefully constructed features-to-augmentation map. The critical components of this approach are:(1)Maintaining consistency across the learning and prediction environments to make the augmentation case-agnostic; (2) Tightly-coupled inference and learning by constraining the augmentation to be learnable throughout the inference process to avoid loss of inferred information (and hence accuracy); (3) Identification of relevant physics-informed features in appropriate functional forms to enable significant overlap in feature space for a wide variety of cases to promote generalizability;(4)Localized learning, i.e. maintaining explicit control over feature space to change the augmentation function behavior only in the vicinity of available datapoints. To demonstrate the viability of this approach, it is used in the modeling of bypass transition. The augmentation is developed on skin friction data from two flat plate cases from the ERCOFTAC dataset. Piecewise linear interpolation on a structured grid in feature-space is used as a sample functional form for the augmentation to demonstrate the capability of localized learning. The impact of using a different function class (neural network) is also assessed. The augmented model is then applied to a variety of flat plate cases which are characterized by different freestream turbulence intensities, pressure gradients, and Reynolds numbers. The predictive capability of the augmented model is also tested on single-stage high-pressure-turbine cascade cases, and the model performance is analyzed from the perspective of information contained in the feature space. The results show consistent improvements across these cases, as long as the physical phenomena in question are well-represented in the training.
I. INTRODUCTION
Traditional modeling of transition and turbulence has generally relied on theory and expert knowledge to determine the model-form, and on data for the calibration of resulting parameters. Recent advances [1-3] have explored the possibility of using data to introduce complex functional forms within existing models to improve their accuracy and applicability. In the context of turbulence modeling, Oliver and Moser [4] were among the first to quantify model-form uncertainties by introducing discrepancies in the Reynolds stresses as spatially-dependent Gaussian random fields. Dow and Wang [5] applied a similar approach to introduce discrepancies in the eddy viscosity field, and assimilated data from multiple flow fields. These precursors to the current machine learning approaches in RANS modeling are relevant as they treat model inadequacies as fields instead of parameters.
Tracey et al., [6] in 2013, proposed the idea to transform such inadequacies from a field variable (a function of space) to a function of features (local functions of states and modeled quantities). This was applied to learn the discrepancies in the eigenvalues of the Reynolds anisotropy tensor between RANS and DNS results. A kernel regression ML model used the following features -eigenvalues of the anisotropy tensor, ratio of production-to-dissiptaion rate of turbulent kinetic energy and a marker function to mask the free shear layer regions in the flow. Xiao and co-workers [7][8][9][10] modeled the discrepancy in not just the eigenvalues of the anisotropy tensor but also its eigenvectors and turbulent kinetic energy while using a broader feature set. Since the values of features and the inadequacy term used to train the ML models are extracted from the DNS data, this approach is inherently inconsistent and unsuitable for predictive use via the baseline RANS models. In addition to this, the requirement of full field DNS data to create such ML models takes away from the robustness and versatility of this method.
Ling and Templeton [11] introduced Tensor Basis Neural Networks (TBNNs) to learn the coefficients of a Tensor Basis expansion (w.r.t. mean strain rate and vorticity tensors) intended to approximate the Reynolds stress term. The features used for this work included five invariants based on the mean strain rate and vorticity. TBNNs have, since, been used to model turbulent heat and scalar fluxes [12,13].
Another class of approaches, which attempts to optimize the functional form of the featureto-augmentation map, is symbolic regression. Weatheritt and Sandberg [14] employed ge-netic programming to construct models for Reynolds stress anisotropy in terms of invariants of the velocity gradient tensor. Schmelzer et al. [15] used sparse regression over a library of candidate functions to obtain an algebraic Reynolds stress model. An advantage of such approaches is that the resulting model form is simple, and presents the added benefit of interpretability.
Techniques to extract model inadequacy mentioned above do not explicitly enforce model consistency, i.e. the inadequacy information embedded in the modified model is extracted from higher-fidelity data directly and can hence be inconsistent with the lower-fidelity model. The importance of model consistency is discussed in a recent review [3]. To address model consistency, Duraisamy and co-workers [16][17][18][19] introduced the field inversion and machine learning framework which aims to learn the inadequacy in the same environment as used for predictive purposes. To achieve this, one needs to solve an additional inverse problem prior to the machine learning step to obtain a model-consistent inadequacy field which is optimal in the sense that the corresponding outputs from the model match the target high-fidelity data as accurately as possible. While this framework mitigates inconsistency, it also facilitates the use of sparse high-fidelity data from experiments in contrast to full fields of direct numerical simulation data. To solve this so-called field inversion problem, an adjoint-driven gradient-based optimization technique is used. The following subsection focuses on the developments and applications that this framework has witnessed since its inception. Other techniques and frameworks introduced later (e.g. [20][21][22]) also suggest the use of model-consistent training, and present opportunities of bypassing the requirement for adjoint-driven gradient computations by pursuing weakly-coupled inference.
The original formulation of FIML, hereafter referred to as the classic FIML, has been used by several research groups, with applications including, but not limited to, predictive modeling of adverse pressure gradients flows [23,24], separated flows [19,[25][26][27], bypass transition modeling [28], natural transition modeling [29], hypersonic aerothermal prediction for aerothermoelastic analysis [30], turbomachinery flows [31], shock-turbulent boundary layer interactions [32], etc. Matai et al. [24] proposed a zonal version of FIML, where the augmentation field obtained from field inversion was quantized into a set number of clusters, following which a decision tree based architecture was used to classify corresponding features into appropriate clusters. The quantization reduced the number of classes that needed to be accounted for by the learning architecture and hence improved the training performance across a wide range of flow problems. A significant evolution in the framework was proposed by Holland et al. [26,27] in the context of RANS, and Sirignano et al. [33] in the context of LES. These techniques integrate the learning step in the inference process. This is an improvement over classic FIML for two reasons -(1) the augmentation fields obtained using classic FIML may be non-unique which means that two near-optimal augmentation fields might have completely different functional representations in the feature space which might result in convergence problems when learning augmentations from several field inversion problems simultaneously; and (2) significant information loss might occur when learning an augmentation field (relatively high-dimensional vector) to obtain ML-model parameters (relatively low-dimensional vector) which might result in a considerable loss in predictive accuracy. This tight coupling between the two inverse problems, however, requires additional infrastructure for gradient evaluation, and presents convergence challenges due to the additional non-linearity of the learning algorithm. This framework is, hereafter, referred to as integrated inference and learning.
In this work, we introduce another step in this evolution, viz. Learning and Inference assisted by Feature-space Engineering (LIFE). The approach is developed on the hypothesis that robustness and generalizability demand a low-dimensional feature space that is informed by the underlying physics, and that the features-to-augmentation map has to be carefully constructed. LIFE uses tightly coupled inference and learning to use data in order to locally update regions of feature space and can potentially make the augmented model more generalizable, robust and reliable by inherently suppressing spurious behavior and offering the modeler more control over the feature space. Note that, spurious behaviour, in the context of this work, refers to the augmented model predicting non-baseline behaviour by virtue of extrapolation to physical conditions which have not been encountered during training. In addition, this work also introduces new constructs (guiding principles and techniques) across different stages in the inference process that contribute towards boosting the efficacy and generalizability of the resulting augmentation while underlining the importance of human intuition and expert knowledge in the process.
A. RANS models for bypass transition
To demonstrate the capability of LIFE, we choose the problem of developing a data-driven bypass transition model. Transition modeling has been studied over the past two decades, and several models are in existence. In this work, however, the aim is not to create a universally accurate transition model. Rather, the goal is to develop and demonstrate LIFE and examine the possibility of generalization in the low data limit using bypass transition as an example.
The phenomenon of laminar-to-turbulent transition is of vital importance in the design of aerospace vehicles and energy systems. Transition can occur either by the amplification and non-linear interactions of 2-D instability waves (also known as Tollmien-Schlichting waves) or via the perturbation of an otherwise laminar boundary layer by freestream turbulence, surface roughness, impinging wakes, etc. The latter is referred to as the bypass mode of transition, and is typically the preferred route for transition when freestream turbulence intensities are more than 1% [34]. While Reynolds-Averaged Navier-Stokes (RANS) simulations have remained the workhorse in the industry for design and optimization and a subject of active research for decades, bypass transition models for RANS turbulence closures are relatively in their infancy.
Two major categories of approaches exist in the literature to model bypass transition induced by freestream turbulence, viz. (1) Data correlation based models, where the model predicts the transition onset location based on some criteria (usually a correlation between momentum thickness Reynolds number Re θt and freestream turbulence intensity T u ∞ ) and switches from laminar to turbulent computation [34][35][36][37]; and (2) transport equation based models, where the turbulence model is modified by introducing a new transport scalar, which is solved for using an additional equation. Further, there are two major approaches within transport equation-based models -laminar fluctuation models and intermittency function models: a) Laminar fluctuation models are based on the modeling of the so-called laminar kinetic energy, k L , i.e. the energy of fluctuations in laminar boundary layers corresponding to Klebanoff modes; and the transfer of energy between the laminar disturbances and turbulent eddies by introducing a transfer term in the transport equations of k L and k (modeled turbulent kinetic energy) [38][39][40]; b) Intermittency function models, on the other hand, model the intermittency γ (which is a statistical quantity referring to the fraction of the time the flow is turbulent and varies from 0 in laminar flow to 1 in fully turbulent flow) which is usually multiplied to the production term in the transport equation for k or directly to the eddy viscosity, thus, attenuating the production of turbulence in the regions supposed to be laminar or transitional [41][42][43][44][45][46][47][48].
In this work, we utilize the idea of LIFE to create a data-driven intermittency-based bypass transition model using data from only two flat plate cases, and predict the bypass transition triggered by freestream turbulence across different flow conditions and geometries.
The outline of this paper is as follows: The formal problem description of model augmentation, and a background of the Field Inversion and Machine Learning Framework is laid out in section II. In section III, the methodology of LIFE, which includes its philosophy, the required infrastructure and related algorithms is presented. Section IV begins with relevant background of the bypass transition modeling problem necessary for, and followed by, the description of baseline model-form used in this work, how the augmentation is introduced and what features are used to express the functional form of the augmentation. In section V, the training and testing results of the transition model augmentation is shown and analyzed.
Section VI summarizes the work. The appendices details the impact of modeling choices on the results of the modeling augmentation.
II. PROBLEM STATEMENT & BACKGROUND
A. Premise 1. The closure problem Consider a physical system described on a spatial domain Ω, the governing equations for which can be represented with appropriate boundary conditions as follows
∂q ∂t + R(q) = 0 ∀ x ∈ Ω (1)
where q(x, t) represents the highly resolved spatio-temporal field of state variables. For practical problems, like flow over an aircraft wing or inside a gas turbine combustor, direct simulations of the underlying physics with the required spatio-temporal resolution are presently infeasible. In turbulence modeling, the state variables are decomposed into coarse-grained and unresolved parts, represented by q and q, respectively. Performing the coarse-graining operation on the equations representing the high-fidelity system results in an unknown closure term N (q), arising from any non-linearities in the operator R and the unaccounted contribution from q. Note that the following representation is not an approximation.
∂ q ∂t + R(q) = 0 ⇒ ∂ q ∂t + R( q) + N (q) = 0 (2)
In a generic sense, a closure model can be written as N m ( q m , s m ), where q m refers to the state variables for the reduced-fidelity model which will be different from the coarse-grained variables q because of a modeled closure term. s m represents the vector of any secondary variables (e.g. turbulent kinetic energy and other scale providing variables) which might need to be solved for by using an additional system of equations as shown below.
∂ s m ∂t + G m ( s m , q m ) = 0 ∀ x ∈ Ω (3)
where Ω represents the appropriately discretized version of the original domain Ω. In this work, the focus is restricted to steady-state reduced-fidelity models, which, using the terminology described above, can be represented as
R( q m ) + N m ( q m , s m ) = 0 ; G m ( s m , q m ) = 0 (4)
For ease of notation, we shall refer to this system of equations in a compact manner , and the inputs to the model (discretized domain, boundary conditions, etc.) embedded into the notation via ξ.
A surrogate measure of model inadequacy
Reduced-fidelity models of turbulence and transition can be insufficiently accurate for use in predictive simulations that aid design and optimization owing to model-form inadequacies in the representation of the operators N m and/or G m . To address these inadequacies, one can "augment" the model which involves appropriately introducing an inadequacy field, β(x) in the model-form of R m , giving rise to the following augmented model.
R m ( u m ; β(x), ξ) = 0(5)
Examples of how such inadequacy terms have been introduced into turbulence models in the available literature include multiplication with the production term of the Spalart-Allmaras model by Singh et al. in [19], addition to the eigenvalues of the Reynolds anisotropy tensor by Xiao et al. [6,7]. Indeed, β can represent the vector of coefficients in the tensor basis expansion for the Reynolds anisotropy tensor as proposed by Ling et al. [11].
Model consistency and data
The task of quantifying the model inadequacy now reduces to inferring some ideal β(x)
field, that when used in this augmented model, minimizes a chosen measure of discrepancy (e.g. L2 norm) between the data and corresponding predictions for quantities of interest.
Note that, even if the full high-fidelity field of state variables is available, data alone should not be used to obtain β values by simply plugging values into the reduced-fidelity model at different locations, as this could result in a field which is inconsistent with the model and in a general case may lead to prediction inaccuracies [3]. In other words, the inferred β(x) field is only a surrogate measure of the inadequacy, and that the true model inadequacy might be slightly different than this field, as q and particularly s may be different from q m and s m , respectively.
For a dataset (a dataset in the present context refers to a unique combination of domain geometry and flow conditions) referred to by the index i, a discrepancy between the vector of high-fidelity data for these observables, y i , and the corresponding vector of predictions using the augmented model, y i m ( u i m ), can be used to formulate the cost function for this case as C i (y i , y i m ( u i m )). Since the field u i m necessarily satisfies R m ( u i m ; β i , ξ i ) = 0, it implicitly depends on the augmentation field, β i (x), and so, the cost function is ultimately a function of β i (x). Thus, a gradient-based optimization can, potentially, be used to infer β i (x). A rudimentary candidate for the cost function can be the square of L 2 -norm of the difference between the two vectors. This process of inference will be explained in greater detail in the section II B.
Representing inadequacies via augmentation functions
While the augmentation field, β(x), provides information about the local inadequacy in the model, it cannot be used for predictive improvements on its own. However, if a consistent functional relationship could be extracted between β(x) and some carefully-chosen features, η( u m , ζ), this function could replace the β(x) field in the reduced-fidelity model equations for predictive use. Here ζ denotes local quantities independent of the state or secondary variables which are used to design features. Note that, for the augmentation to be usable in a predictive setting with a wide range of applicability, it should ideally have no dependence on the domain geometries or boundary conditions used for inference. The augmentation is assumed to be dependent only on local quantities in the present work as including non-local information in the features requires a more sophisticated, and potentially less general treatment. The features must be invariant [3, 10] to appropriate transformations (e.g. rotation, as they must not depend on the orientation of the coordinate axes.) Given the features-to-augmentation map, the model can be written as
R m ( u m , β(η( u m , ζ); w); ξ) = 0.(6)
It should be mentioned here that, once the model class of the augmentation (e.g. a neural network) is chosen, the goal is to infer the parameters w. The section II B focuses on variants of the FIML framework to obtain the parameters w from sparse data.
B. Field Inversion and Machine Learning (FIML)
The field inversion and machine learning framework, originally proposed by Duraisamy and co-workers, [16,17], was formulated to reduce inadequacies in a given reduced-fidelity model by inferring optimal augmentations such that the predictive accuracy of the model is improved. Point-wise and integrable data of variable sparsity across multiple cases can be used from experiments and higher-fidelity simulations. The two main variants of this framework are been discussed in the following sections.
Classic FIML
FIML seeks the optimal augmentation into two steps: Firstly, a "field inversion" problem is solved separately over each dataset to infer optimal augmentation fields, β(x), in the respective discretized domains. Then, the step of machine learning uses these inferred augmentation values and respective features for all spatial locations across all datasets and infers the optimal augmentation function parameters w. These two steps are described below.
a. Field Inversion
β * i (x) = arg min β i (x) C i (y i , y i m ( u i m )) + λ i β T i β (β i (x)) s.t. R m ( u i m ; β i (x), ξ i ) = 0 ∀ i = 1, 2, . . . , N(7)
Equation 7 represents the field inversion problem which is solved individually over each of the N training cases (cases for which available high-fidelity data is to be used to infer the augmentation function parameters) to obtain optimal augmentation fields β * i (x) in the respective domains, where i is the case index. The augmentation field is optimal in the sense that it minimizes an objective function which consists of a cost function, C i , and a regularization term, T i β , with λ i β as the regularization constant.
C i = y i − y i m ( u i m ) 2 2 ; T i = β i (x) − β 0 2 2(8)
Note here that an additive inadequacy term which is non-dimensionalized with the same turbulent length and time scales as the source term, can also be viewed as a multiplicative inadequacy term.
Given the very high-dimensional nature of the augmentation fields that need to be inferred, the field inversion problem is usually solved using a gradient-based optimization technique.
b. Supervised Learning
Once the field inversion problem is solved for all available datasets, the supervised learning problem can be solved using the augmentation fields and correspondingly calculated feature fields from all datasets to optimize for the function parameters w as w = arg min w L(β * , β(η * ; w )) where, β * refers to the stacked vector containing optimal augmentation fields across all datasets, and η * contains the respectively stacked feature values.
Note that, while the inferred augmentation from the field inversion step is fully consistent with the underlying model, the augmentation field provided by the field inversion process is not necessarily learnable as a function of the chosen features [27]. This can lead to a loss of information extracted in the field inversion step. A simple measure of learnability can be defined as follows.
Learnability = 1.0 − 1 n N i=0 |β(η i ; w * ) − β * i | |β(η i ; w * )| + |β * i |
where w * is the globally optimal set of parameters that minimizes the discrepancy between the augmentation values obtained using field inversion and corresponding machine learning predictions, and n is the number of datapoints. This measure can assume values between 0 and 1 such that higher value refers to higher learnability. As discussed previously, the augmentation field obtained from field inversion is not unique, it is possible that the field inversion results obtained from two different datasets correspond to different features-toaugmentation mappings. This inconsistency can also degrade the learnability of the model.
The more recent variant of FIML, referred to as integrated inference and learning addresses these concerns.
Integrated Inference and Learning
The integrated inference and learning approach, proposed by Holland et al. [26,27] and Sirignano et al. [33] replaces the two-step inference problem introduced in classic FIML by just one inverse problem, mentioned as follows
w = arg min w N i=1 C i (y i , y i m ( u i m )) + λ i β T i β (β(η i ( u i m , ζ); w )) s.t. R m ( u i m , β(η( u i m , ζ i ); w ); ξ i ) = 0 ∀ i = 1, 2, . . . , N,(9)
where is a generic assembly operator to build a composite objective function using the individual objective functions calculated for each dataset. It can be as simple as a sum, if all datasets are equally important for the inference, or it could be a weighted sum, if some datasets are to be assigned more importance than others, or it could be something even more complex as designed/needed by the user. Notice here that using several cases at once adds an implicit regularization to the problem. Similar to the field inversion problem, the required sensitivities can be obtained using discrete adjoints as described in Appendix A.
Benefits over classic FIML
• Classic FIML can suffer from a loss of information during the machine learning step owing to the inability of the chosen functional form to accurately approximate the augmentation inferred during the field inversion step as a function of the specified features. Integrated inference and training bypasses this problem as the weights are directly updated and consequently, the obtained augmentation field remains consistent with the functional form of the augmentation. Thinking in terms of the augmentation field, this constrains the optimization to minimize the objective function to find an augmentation field which is realizable w.r.t. its functional form.
• Since integrated inference and learning can infer the function parameters w while simultaneously assimilating data from multiple datasets, the augmentation fields for all datasets is constrained to be realizable w.r.t. the functional form of the augmentation as explained in the previous point. This has an added advantage that this procedure, unlike classic FIML, is by design prevented from learning augmentation fields from different datasets that behave differently in the feature space. In other words, consistency in the features-to-augmentation mapping across datasets is automatically enforced when using integrated inference and learning.
• Building on the previous point, this means that if we have even a handful of DNS(Direct Numerical Simulation) fields from which true augmentation values can be extracted (or are readily available), they can be used to enforce a near-physical relationship between the features and augmentation by simultaneously using a plethora of other sparse field data from experiments or higher fidelity simulations. This is critical from the viewpoint of generalizable and physical augmentations.
Limitations
• Unlike the classic FIML approach, the task of designing an appropriate feature space must precede the inference from data. This, at times, could be difficult for the modeler and demonstrates the preliminary need of an independent field inversion step which can lend crucial information about the quantities correlated to the augmentation field that may then be used to formulate features.
• In its original form, integrated inference and learning does not consider the significance of localized learning. For practical problems, the data being used might not populate the entire feature space, which means that if the optimization problem is not constrained to change the augmentation only in the vicinity of the feature space locations for which data is available, it might lead to spurious predictions in other regions, which might not only result in worse accuracy compared to the baseline model but also severely affect the stability of the numerical solver.
• When using complex learning algorithms such as neural networks or decision trees, the augmented model inherits non-linearities from the baseline model and the learning algorithm. This, combined with the previous point, can lead to a disorderly optimization behavior when solving the inference problem, in addition to the aforementioned potential deterioration in accuracy and/or numerical stability with every successive optimization iteration.
III. MODEL-CONSISTENT LEARNING AND INFERENCE ASSISTED BY
FEATURE-SPACE ENGINEERING (LIFE)
To alleviate the limitations in integrated inference and learning, and to make the augmen- This section is structured as follows. Section III A and III B propose guiding principles to take into consideration when designing the augmentation and features. Section III C deals with the notion of localized learning and provides three different ideas on how to make it work. Finally, III D deals with the algorithm and practical concerns of implementing the LIFE framework.
A. How should a model augmentation be designed?
The following aspects could be considered to decide how to introduce an augmentation in the baseline model:
1. Efficacy: The augmentation should be capable of effectively addressing the inadequacy in consideration. For instance, augmenting the coefficient of an isotropic diffusion term in a model cannot correct for an inadequacy caused as a consequence of the anisotropically diffusive behaviour in the corresponding true system.
Spurious behavior:
The augmentation should be introduced such that it does not corrupt model behavior in regions where the inadequacy under consideration is not a concern.
For instance, a RANS model augmented to predict transition should not change the model behavior outside the boundary layer or in regions inside the boundary layer in which the flow is fully turbulent.
Sensitivity:
In practice, the predicted augmentation values might have small errors which can have significant adverse effects on the stability and accuracy of the model if it is highly sensitive to the augmentation. Hence, the augmentation should be introduced in a manner that mitigates this as much as possible.
B. What features should the augmentation depend on?
To decide the features that the augmentation function would depend on, one should use the following guiding principles.
Choice of features using expert knowledge
The choice of features has a major role to play in how effective the augmentation function can be at addressing the model inadequacy in consideration. While automated feature selection techniques exist in literature, they heavily (if not prohibitively) depend on data to find the best features from amongst hundreds/thousands of possible candidate functions of local quantities for a complex problem like transition or turbulence. The authors, hence, believe that human intuition and expert knowledge can lead the way in feature selection as it has proven effective for traditional modeling. This is one of the major reasons why the LIFE framework is aimed towards use by expert modelers. While choosing features, it is desired that there exists a causal relationship between the chosen features and the inadequacy targeted by the augmentation function. However, for steady-state models, the augmentation in turn influences the features by virtue of feedback, and hence quantities which do not share a causal relationship with the inadequacy and, rather are only correlated with it, can also serve as features.
Physics-based non-dimensionalization
Once the quantities to be used as features are chosen, they must be non-dimensionalized in order to be generalizably used for prediction. This is because similar physical phenomena can occur for significantly different magnitudes of dimensional model quantities and, since the LIFE framework aims at using as little data as possible to discover generalizable augmentation functions, properly non-dimensionalization of the features becomes imperative.
While statistics from the training datasets can be used to non-dimensionalize features, the available data might not be sufficiently representative of the complete range of values that could be encountered during prediction on an unseen case. Thus, it is better to make use of model quantities to non-dimensionalize features.
A simple example can be used to illustrate this point. Let us consider that the eddy viscosity ν t is intended for use as a feature. Technically speaking, 0 ≤ ν t < ∞. With limited data, it seems hopeless that the value of ν t can be used as a feature in a generic predictive setting.
However, simply using the turbulent Reynolds number ν t /ν can immediately improve the quality of results. This is because we take some burden of discerning physical conditions for a given combination of features off of the learning algorithm and make the integrated inference and learning problem better-posed. Note that one can also non-dimensionalize ν t as ν t |Ω|/|U | 2 , but the physical conditions represented here will be different than the ones represented by ν t /ν. Which non-dimensionalization is chosen depends on its relevance w.r.t.
the inadequacy to be alleviated. While the example shown here seems trivial, physicsbased non-dimensionalization can prove to be a challenge in some scenarios which might require domain-specific theoretical knowledge and/or experimentally obtained correlations to resolve.
Effectively bounded feature-space
While non-dimensionalization is critical, it should be remembered that a nondimensionalized feature can still be unbounded (e.g. ν t /ν). Since the data used to learn the augmentation is limited, this can lead to extrapolated predictions by the augmentation function for features outside the range of values available from the training data. To circumvent this, the functional form of the non-dimensionalized feature must be chosen such that either the feature is mathematically bounded or it takes non-baseline values (baseline value is 0 for an additive augmentation and 1 for a multiplicative augmentation) only in a bounded part of the feature space. We shall call such a feature as "effectively" bounded, hereafter.
Effective boundedness minimizes extrapolation errors and makes the augmentation more robust and generalizable.
Appropriate functional forms for features
Several functional forms of a non-dimensionalized feature can offer effective boundedness.
But, only a few of these forms might address the inadequacy in a major part of the feature space. For instance, the part of the feature space covered by the feature ν/(ν t + ν) between the values of 0 and 0.3 reduces to being covered between the values 0 and 0.000729 for a different form of the same feature, viz. (ν/(ν t + ν) 6 ). Note that, both features have the same mathematical range between 0 and 1. However, by the virtue of different functional forms, the regions denoting the same physical conditions span very differently in the feature space.
For an augmentation which is predominantly affected within the said range, the feature ν/(ν t + ν) offers a better-conditioned learning problem. Hence, the choice of which among different effectively bounded non-dimensional functional forms of a feature to use can play a significant role in setting up a well-conditioned learning problem. Choosing how fine/coarse the discretization is (in case of method 1), or the strategy used to sample points from existing augmentation field (in case of method 2), or the support width (in case of method 3) in order to control the influence of datapoints in the feature space requires the user to make a trade-off. If, for a given datapoint, the region of influence is very small, the augmentation would behave as an over-fitted function which, while performing accurately on the training cases, will not be able to generalize well. On the other hand, if the region of influence is too large, the augmentation will be over-regularized and the predictive accuracy will suffer, again resulting in a loss of generalizability. Clearly, there exists ample scope to explore different localized learning techniques (along with strategies to optimally choose regions of influence). However, our main motive in this work w.r.t. localized learning is to introduce it in the LIFE framework and emphasize on its importance in creating robust and generalizable augmentations, and so we choose to deal with these explorations in future works.
Algorithm 1 Simultaneous Integrated Inference and Learning
Inputs: w 0 , y 1 , . . . , y N , n, α Outputs: w n
Solve for u i m s.t. R m ( u i m ; β(η( u i m , ζ i ); w 0 ), ξ i ) = 0 [Forward problem]
3: for i = 1 : N do 8:
J i := C i (y i , y i m ( u i m )) + λ i β T i β (β(η( u i m , ζ i ); w 0 )) [ψ T := dC i d u i m dR m d u m ξ i −1 where d d u i m = ∂ ∂ u i m β + ∂dJ i dw k−1 = dJ i dw k−1 + λ i β ∂T i β ∂β ∂β ∂w k−1 i j − ψ T ∂R m ∂β i ∂β ∂w kw k = w k−1 − α dJ dw k−1 dJ dw k−1 −1 ∞ [SteepestSolve for u i m s.t. R m ( u i m , β(η( u i m , ζ i ; w k )); ξ i ) = 0 [Forward problem]
18: Praisner and Clark [37], in 2004, presented a correlation that can approximate the transition momentum thickness Reynolds number over a wide range of pressure gradients, freestream turbulence intensities and Mach numbers. This correlation is given as follows.
J i = C i (y i , y i m ( u i m )) + λ i β T i β (β(η i ( u i m , ζ); w k )) [Objective evaluation]Re θ,t = A T u ∞ θ t λ ∞ B(10)
where A = 8.52 and B = −0.956. Approximating B to be -1, recasting this correlation and assuming C µ ω = u /λ for the Wilcox k − ω model one can write this correlation as
θ 2 t = A 1 νλ ∞ u ∞ = A 1 C µ ν ω ∞(11)
where A 1 = 0.07 ± 0.011. The universality of this estimate across such a varied set of flow conditions, even though the dataset comes predominantly from turbomachinery setups, is adopted in the model presented in this work.
A brief review of recent intermittency-based models
Langtry and Menter [46] proposed one of the most widely used intermittency-based transition models with transport equations for intermittency (γ) and an estimate of transition momentum thickness Reynolds number ( Re θ,t ) to supplement the SST turbulence model equations. The additional transport equation for Re θ,t is used to diffuse the values calculated using algebraic correlations from outside the boundary layer to regions inside the boundary layer. This quantity is then used to evaluate the critical momentum thickness
Reynolds number, Re θc , which when compared with the vorticity Reynolds number, d 2 Ω/ν, can be used to predict the transition onset and hence trigger intermittency production. The intermittency takes effect in the solver by scaling the production term of the TKE transport equation.
In 2012, Durbin [47] proposed a simple transition model with a single production term which is directly proportional to the local vorticity, Ω and (γ max − γ) √ γ, where γ max = 1.1, to supplement the Wilcox's 1988 k − ω turbulence model. The (γ max − γ) term would give a trivial solution of γ = γ max in the entire field if the model is used as it is. For this reason, the author proposed to set the intermittency to zero for all mesh locations where the eddy viscosity was very small compared to the local molecular viscosity and the vorticity Reynolds number is less than a pre-defined threshold, after every solver iteration. After every solver iteration, the values of γ is set as min(γ, 1.0) as γ max > 1 helps the intermittency to rapidly increase, but at the same time, results in a non-physical growth in turbulent production. As in the Langtry-Menter model, the intermittency is multiplied to the production term in the transport equation for k.
Ge and Durbin [48] proposed another transition model based on the structure of the previous model in 2014 which attempts to predict bypass transition due to freestream turbulence and flow separation. Separation effects were accounted for by multiplying the production term in the transport equation for k with a separation modification. It is noteworthy, here, that the presence of a dissipation term in this equation removes the requirement of setting the intermittency to zero, externally.
In the models mentioned above, one can see a common theme. All of these models attenuate the production of turbulent kinetic energy using the intermittency function rather than attenuating the eddy viscosity directly. This is justified as turbulent kinetic energy should be very low in the laminar regions of the flow, something which is hard to implement by just altering the eddy viscosity term. Secondly, one of the most important terms in all these models is the vorticity Reynolds number, the maximum value of which approximates the momentum thickness Reynolds number fairly well for a zero pressure gradient boundary layer, as shown by van Driest and Blumer [49]. Even in the presence of favorable/adverse pressure gradients, the approximation still provides an excellent tool for modeling the transition onset as a function of a local quantity, as opposed to the momentum thickness Reynolds number which is an integral quantity. These ideas were instrumental in designing the transition model presented in this work.
B. Chosen baseline model and proposed augmentation
The transport equations that constitute the chosen baseline model have been shown below. Wilcox's 1988 turbulence model containing a vorticity-based production term is used for this work, in contrast to the strain-rate based production term.
Dρk Dt = ∇ · µ + µ T σ k ∇k + µ t Ω 2 − 2 3 ρk∇ · u − C µ kω (12) Dρω Dt = ∇ · µ + µ T σ ω ∇ω + C ω1 ω k µ t Ω 2 − 2 3 ρk∇ · u − C ω2 ω 2(13)Dργ Dt = ∇ · µ σ l + µ T σ γ ∇γ + ρ(1.0 − γ) √ γ|Ω|,(14)
with the eddy viscosity given by µ t = ρk/ω. Clearly, this model will give a fully turbulent
Dργ Dt = ∇ · µ σ l + µ T σ γ ∇γ + ρ(β − γ) √ γ|Ω|.(15)
This particular form of baseline model and augmentation was chosen for the following reasons:
• For most cases, the range for this augmentation is bounded between 0 and 1.
• Any discontinuities in the predicted augmentation field will be smoothed over by the transport equation and will not require special attention. Indeed, this is the reason why the intermittency transport equation was retained in contrast to augmenting the production term in the k-equation directly.
• The knowledge of correlations between the intermittency field and other flow quantities can be readily used to formulate this augmentation, as the converged intermittency field would closely resemble a smoothed augmentation field.
C. Choice of features and limiters for the augmentation
A clear understanding of the effect of intermittency term γ on the modeled turbulent kinetic energy k played a vital role in the choice of features. A brief explanation of this effect is given as follows.
From equation 12, it can be observed that the production of k is directly proportional to eddy viscosity (ν t ) and vorticity (Ω), and inversely proportional to the modeled specific dissipation (ω). Ω, for external flows, is usually the highest at the wall and decreases away from the wall. Also, the freestream eddy viscosity (which is usually the initial condition for ν t in RANS solvers) is very small (around an order of magnitude higher than molecular viscosity). Hence, the production of k starts in a region very close to the wall at a distance where ω has sufficiently decreased and Ω remains high enough to enable a net production of k (for a fully turbulent boundary layer, this region lies within the inner layer but outside the viscous sublayer). By the virtue of diffusion, k starts increasing slightly farther from the wall relative to where it is being produced, thus increasing local ν t and triggering a net production of k. This process keeps spreading away from the wall until Ω drops so low that a net production is no longer possible. Also, the convective term in equation 12 ensures that a downstream production of k has minimal effect on upstream locations. Using all this information, it can be concluded that in order to laminarize the upstream of the transition location, the production of k needs to be attenuated by sufficiently reducing γ only in a region close to the wall. Doing this requires -(1) feature(s) that can indicate the transition onset (the location up to which the augmentation should be effective); (2) feature(s) to uniquely identify the regions close to the wall where the augmentation should be active.
The following sections describe the rationale behind the choice of three such features.
Feature 1: Ratio of Vorticity Reynolds Number and an estimate of Re θ,t
Traditional models have used experimentally obtained correlations for transition momentum thickness Reynolds number (Re θ,t ) to predict transition by comparing it with the local momentum thickness Reynolds number Re θ . As mentioned in section IV A 2, the vorticity Reynolds number, Re Ω , provides a good surrogate to use instead of Re θ , which is an integral quantity. The vorticity Reynolds number, as defined in [47], is given as follows.
Re Ω = d 2 Ω 2.188ν (16) The transition momentum thickness Reynolds number, Re θ,t , is given as follows using the Praisner-Clark estimate [37] of transition momentum thickness described in section IV A 1.
Re θ,t = U ∞ θ t ν ≈ U ∞ ν 7ν 9ω ∞ = Re θ,t(17)
Note that this correlation does not depend on k ∞ which is physically inconsistent and might render it ineffective in certain scenarios. However, in their study, Praisner and Clark found this correlation seems to work quite well on a range of turbomachinery configurations across different Reynolds numbers, Mach numbers, pressure gradients and freestream turbulence intensities. Hence, this correlation is currently being used as a physics-based non-dimensionalization for the first feature which is formulated as the ratio Re Ω /Re θ,t . A conservative upper-bound for this ratio is assumed to be 3. Hence, the feature can be reformulated as follows to strictly bound the feature space as follows.
η 1 = min d 2 Ω 2.188νRe θ,t , 3.0(18)
Note here, that the augmentation resulting from this feature would be considerably sensitive to the far-field boundary condition of ω, and hence the boundary conditions for ω must be set carefully to simulate the decay of the freestream turbulence intensity as accurately as possible.
The remaining issue is to extract Re θ,t from the freestream. In the current work, this is done as follows. A preset distance interval [r − δr, r + δr] is chosen for every case. For every discrete element on a wall boundary, ∂ Ω w k , the volume element inside the domain located at the minimum normal distance within the preset distance interval, is chosen to be the corresponding freestream cell, Ω ∞ k . If no such element is present, δr needs to be increased. Now for every volume element, Ω j inside the domain which is at a minimum distance from the wall boundary element ∂ Ω w k among all such wall boundary elements, any freestream quantity is set as the corresponding quantity in Ω ∞ k . While this can be inaccurate at large distances from the wall, it works fairly well inside the boundary layer with appropriate choice of the distance interval. Features 2 and 3 distinguish between the volume elements within the boundary layer where the augmentation needs to be significantly lower than the baseline value and volume elements elsewhere. Hence, the freestream estimate of quantities being poor far from walls is of no concern. The distance interval is usually chosen to be as close to the wall and as small as possible, such that the maximum boundary layer thickness should remain within the lower bound of this interval. It was also noted that the augmentation is not very sensitive to small changes in the choice of this interval as demonstrated in appendix C.
Note that although ω ∞ and U ∞ seem to be non-local quantities at a first glance, but since the augmentation mainly affects the flow only within the boundary layer, the freestream quantities virtually remain constant across LIFE iterations. Thus, ω ∞ and U ∞ can be thought of as frozen quantities local to each volume element in the discretized domain.
Finally, while U ∞ can be used in moderate pressure gradient problems, the quantity could be ambiguous for very high pressure gradients. Also, note that the maximum value of Re Ω in the wall-normal direction approximates Re θ for zero pressure gradients and hence, max dw Re Ω would also poorly estimate Re θ for very high pressure gradients.
Feature 2: Ratio of wall distance and turbulent length scale
To identify laminar regions within the boundary layer, one can compare the local turbulent length scale √ k/ω with the wall distance d. A predominantly laminar region is characterized by an overwhelmingly larger wall distance compared to the turbulent length scale. A modified functional form for the ratio of these two quantities which is mathematically bounded between the numerical values of 0 and 1 can be written as,
η 2 = d d + √ k/ω(19)
The laminar regions in the flow are denoted by values of η 1 that are significantly close to 1.
Note here that, the value of this feature will be close to 1 for regions outside the boundary layer as well.
Feature 3: Ratio of laminar viscosity and eddy viscosity
Since the magnitude of ν t in the freestream is usually greater than ν, the following modified functional form for the ratio of these two quantities would be less than 0.5 in the freestream while being close to 1 in the laminar boundary layer and the viscous sublayer in the turbulent boundary layer.
η 3 = ν ν t + ν(20)
Together with feature 2, this feature can distinguish regions within the boundary layer for locations upstream of the transition location. Note here, that the feature design in this case is inspired from the augmentation being correlated to (and not caused by) the features.
Limiter for flows without separation
As we are not considering flows with separation, where the intermittency might be allowed to attain higher values in order to match the physical behavior, the resulting intermittency is simply limited to values between the theoretical bounds of 0 and 1.
D. Augmentation function as Piecewise Linear Interpolation on Uniformly Structured Grid in Feature-space
One of the simplest class of functions that can perform localized learning while approximating sensitivities w.r.t. features is linear interpolation on a uniform grid. While this is not the most sophisticated class of functions to approximate an augmentation, it certainly demonstrates the capability of localized learning. Note that the discretization need not be uniform, or structured for that matter, but these assumptions have been made to simplify calculations, implementation and embedding of the augmentation function within the numerical solver.
In this method, the effectively bounded feature space is discretized into a uniformly spaced structured grid. An augmentation value can be attributed to the center of each grid cell.
These values can then be linearly interpolated using second order finite difference approximations inside every cell, to obtain the augmentation for any feature space location. Hence, these cell-centered values serve as the augmentation function parameters, w.
Given a vector of feature values, η, and a value of the augmentation function at the center of each grid cell β c , a linearized approximation of the augmentation can be obtained as described below. Note that the vector of values β c , viz. β c , here represents the function parameter vector w mentioned in the previous sections.
1. Find the grid cell which η belongs to. Let the features at the center of this grid cell be η c .
2. Based on the augmentation value at the centers of the neighboring grid cells, calculate a gradient approximation ∇ F D η β for the th grid cell using a second order accurate central finite difference scheme along each feature.
Return the following linearized approximation of the augmentation function
β (η) = β c + (η − η c ) · ∇ F D η β(21)
The sensitivity approximation of the augmentation function in a locally linear fashion is sufficient for use in method of adjoints which is used to evaluate the sensitivities , which are used in the algorithm given for integrated inference and learning in Algorithm 1 (also refer Appendix A for the adjoint method), are given as follows:
∂β ∂η = ∇ F D η β ∂β ∂w m = ∂β ∂β cm = 1 if = m η j − η c ,j 2∆η j if η cm,j = η c ,j + ∆η j η c ,j − η j 2∆η j if η cm,j = η c ,j − ∆η j 0 otherwise(22)
For instance, given a feature pair (η 1 , η 2 ) for the above schematic, such that the feature pair lies in the cell with center (1.5∆η 1 , 1.5∆η 2 ), the augmentation value can be calculated as Training data: The augmented model presented below is obtained by inferring the augmentation from the T3A and T3C1 cases only. Hence, the following results should be observed in the context of generality of the augmentation obtained in the low data limit. To examine sensitivity of the results to the training data, a model trained on just T3A is also shown in appendix B. A. Training the model on T3A and T3C1 simultaneously
β(η 1 , η 2 ) = β 11 + β 12 − β 10 2∆η 1 (η 1 − 1.5∆η 1 ) + β 21 − β 01 2∆η 2 (η 2 − 1.5∆η 2 )(23)
Mesh for T3A and T3B
The length of the flat plate is 1.5 m with the inlet being 0.04 m upstream of the leading edge of the flat plate. The mesh resolution next to the wall is on the order of y + ≈ 1.
U ∞ (x) = 2 (p in + (1/2)ρU 2 in − p w (x)) /ρ
The mesh resolution next to the wall is on the order of y + ≈ 1.
Inflow conditions
The information needed to characterize the inflow for both the training cases has been presented in table I. The turbulence intensity decay plots shown in figure 5 verify the ω boundary conditions.
LIFE on T3A and T3C1
To obtain the augmentation, LIFE was used to infer the functional parameters (see IV D)
in the feature space using the T3A and T3C1 cases simultaneously. The cost function for this inference is defined as follows. where n wall is the number of faces corresponding to the flat plate in the mesh and C f refers to the local skin friction coefficient. The two different problems and the discretization of the feature space acts as an implicit regularizer. Figures 6 and 7 show the optimization and residual histories. Figure 8 shows the C f of the baseline (β = 1 throughout the domain) and optimal skin friction coefficient distributions compared to data. The initial residual convergence for the direct solver is much better compared to the subsequent optimization iterates. This is due to the change in the feature-to-augmentation relationship, which affects the convergence. However, the magnitude of the residual suggests sufficient convergence.
C = n wall i wall =0 C f,i wall − C f,data,i wall 2 2(24)
Also, one does not see a significant drop in the residuals because each optimization iterate is restarted from the converged state of the previous iterate. The effectively bounded feature space was uniformly discretized into a cartesian grid with 30, 10 and 10 cells along the features η 1 , η 2 and η 3 , respectively. It should be noted that the discretization of the feature space presented here is not necessarily the most efficient one. The objective rather is to demonstrate the capability of localized learning. Further, the discretization need not be in the form of a uniform grid as shown here and could be made adaptive. Finally, as mentioned in section IV D, the discretization needs to be just enough to characterize the augmentation satisfactorily. We found that the discretization presented here predicts the transition location well. As can be seen in the plots above, the transition locations are inferred with a reasonable accuracy for both the cases. Notice, that in the T3C1 case, the skin friction coefficient deviates from the data in the vicinity of transition. The same happens for T3A, though the effects there are much subtler. Also, it can be noticed that the laminar part of the flow does not show fully laminar friction coefficients as can be seen when compared to data. This can The contours of augmentation w.r.t. the feature space (slices of η 1 ) are shown in Figure 10.
As can be seen in the feature space contours, the augmentation varies w.r.t. η 1 until only For the purpose of comparing different learning techniques, three different set of results obtained -(1) using only the T3A case for training, (2) without using localized learning (i.e.
using Neural Networks), and (3) using a finer discretization have been shown in appendices B, D and E, respectively. Looking at these results, we can make the following observations:
1. As seen from the augmentation obtained by only using data from the T3A dataset for training in appendix B which performs worse compared to the augmentation obtained using both the cases T3A and T3C1, adding more datasets which exhibit different physical phenomena compared to the already existing training datasets consistently improves predictive accuracy.
2. Reducing spurious behavior using localized learning can not only make the augmentation more robust but is at times essential to infer a usable augmentation from available data. This is clearly evident when comparing the results presented in this section with the training results presented in appendix D (which do not use localized learning and use neural networks as the functional form for the augmentation and fail miserably by predicting partially/fully laminar flow at all locations along the flat plate).
3. During localized learning, correctly setting the range of influence that the datapoints have in the feature space is crucial in order to obtain a generalizable augmentation as mentioned in section III C. Also, as mentioned in section III C, this work does not deal with a method to optimally set the range of influence and instead focuses on why localized learning is necessary to improve generalizability and reduce spurious behavior arising from extrapolation within the bounded feature space. Nevertheless, the comparison between the test results for augmentations using different grid resolutions in the feature space (see appendix E where a finer grid leads to the augmentation being learnt in a smaller region of the feature space which results in limited generalizability) demonstrate the importance of setting an optimal range of influence.
B. Testing the model
Mesh for VKI turbine cascade cases
The mesh used for the turbine cascade cases can be seen in Figure 11. Notice here, that MUR116 is a case with high pressure differential and low T u ∞ , MUR129
is a case with low pressure differential and low T u ∞ , MUR241 is a case with high pressure differential and high T u ∞ , and finally MUR224 is a case with low pressure differential and high T u ∞ . This distinction will be important later.
The comparison of velocity profiles at three different locations on the turbine blade along the wall normal direction is shown in figure 13 for verification. To diagnose the behavior in the MUR116 case, Figure 16 shows contours of two features and the intermittency near the transition location. The intermittency is seen to grow rapidly in the region 0.1 ≤ η 1 ≤ 0.2 and η 2 ≤ 0.8. This corresponds to a region in the feature space in which the augmentation does not learn significantly from the available data, leading to the augmentation reverting to its baseline behavior of β = 1. It should be recognized however, that there is a strong feedback loop between the features, augmentation and transport processes . This means that all these quantities can influence each other in the model irrespective of the physical cause-effect relationships. This behavior can be improved by bringing in more data to populate a larger part of the feature space. For comparison purposes, η 1 , η 2 and γ contours for MUR241 are shown in figure 17 which exhibit a physically intuitive growth of the intermittency in a region of high η 1 and η 2 . In addition, it can also be observed that -while the coefficient values are close to the data in laminar regions -there are large discrepancies between the two in the fully turbulent regions.
This is because the turbulence model is imperfect These discrepancies are not related to the transition phenomena and hence, are beyond the scope of the current augmentation.
(a)η 1 = min Additional measures, such as preventing a region in the feature space from representing different regions in the physical space where the augmentation must assume different values (i.e. ensuring a one-to-one features-to-augmentation mapping) by selecting an appropriate combination of features, also helps improve robustness. The LIFE framework offers a framework along with guiding principles and techniques that are intended for use by modelers to develop data-driven augmentations for low-fidelity models using high-fidelity data. Predictions on T3 cases: It can be seen in Figure 21 that the prediction is completely wrong for T3C1 and T3B, while it appears reasonable for the other cases. Another important observation is that the transition locations for T3C1, T3C2 and T3C3 are over-predicted.
Re Ω Re θ,t , 3 (b)η 2 = ν ν + ν t
This is due to the fact that the augmentation remains at a lower value for a longer distance spuriously as the augmentation has no way of differentiating between how η 1 changes for different pressure gradients as the training is performed only for a zero pressure gradient case.
Prediction on VKI cases As can be seen in Figure 22, training a model only on the T3A data results in significantly inaccurate transition location predictions on at least one side of the blade except MUR224 when compared to the results presented in the main text where both T3A and T3C1 cases were used for training. This is in accordance with the explanation provided in the last section. Since the transition model has little information on how the features behave in the presence of non-zero pressure gradients, additional data which can highlight such behavior (in the main text as the T3C1 case) is required to extract information about the behavior of the augmentation in feature space.
Appendix C: Results with varying preset distance intervals for Re θ,t extraction As shown in Figure 23, we found that varying the preset distance usually has a small effect on the predictions for the turbine blades. A minor discrepancy is observed in MUR116 and 60 nodes respectively were tried out but yielded similarly poor results as shown in figure 24. The number of parameters (weights and biases) for the latter two architectures was kept around 3000 to match the number of parameters used in the main text during localized learning via feature-space discretization. The results demonstrate that localized learning is essential to both, better condition the optimization problem and also to aid the optimization trajectory in the correct direction in the augmentation space. The second point is especially important, as in several applications (including transition prediction), changing the augmentation behavior in regions where no data is available can in fact lead to spurious model behavior as these parts of the feature space can be accessed before the solver converges. This is further worsened by any feedback present between features and augmentation. It has to be mentioned that using the same software framework, neural networks have been used in the context of integrated inference and learning more successfully in Refs. [26,27] on a less challenging problem. The authors
R
m ( u m ; ξ) = 0, with the state variables and secondary variables combined into a single vector of model variables u m = q
tation robust and generalizable we present a set of guiding principles to choose features along with the notion of localized learning in the feature space which requires controlling how learning takes place in different parts of the feature space. Since a significant effort is put into how features are chosen and how the learning takes place in different parts of the feature space, we call this version of integrated inference and learning as "Learning and Inference assisted by Feature-space Engineering (LIFE)".
Finally
, the augmentation should be a function of as small a number of features as possible to maintain simplicity, as a simpler augmentation could mean better generalizability when training on limited data. On the other hand, the features should be chosen such that, there exists a one-to-one feature-to-augmentation mapping. While a perfect one-to-one mapping might be exceedingly difficult to achieve in the entire feature space, the property should be virtually preserved (i.e. if more than one, all possible augmentation values should be close to each other) in as large a region of the effectively bounded feature space as possible.More features imply a better chance of ensuring such a mapping. Note that "mapping" here refers to the true relation between features and corresponding optimal augmentation values for all locations in the feature space. It does not refer to the augmentation function (which is one-to-one by definition and tries to approximate this mapping). If the mapping is not one-to-one, then during the inference and learning process, sensitivities from two different datapoints might try to pull the augmentation value at some location in the feature space in opposite directions and the so-obtained optimal augmentation function would predict a compromise between two significantly different values which are optimal w.r.t. each of the two datapoints. Physically, this translates to the feature space being inadequate to uniquely represent distinct physical phenomena corresponding to significantly different augmentation values. Thus, a balance has to be attained by choosing just enough features to ensure a virtually one-to-one mapping in most of the effectively bounded feature space.C. Which class of functions should be used to model this augmentation? Depending on the complexity of the behavior of the augmentation in feature space, there exist different alternatives to choose the functional form of the augmentation prior to the inference and learning procedure. If the behavior is very simple and the feature space is very low-dimensional, one can hand-fit a function on the inferred set of augmentation values. If the behavior is not intractably complex, a user-defined functional form can be used. Lastly, if the behavior is highly non-linear, a class of functions available in the machine learning literature (decision trees, neural networks, etc.) could be used. This choice becomes even more important when the available data does not span/represent all parts of the effectively bounded feature space. In such a scenario the augmentation function might extrapolate to predict values for feature space regions corresponding to which no data was available. This could lead to spurious model behavior and could make the prediction capabilities of the augmented model worse than its baseline counterpart. To preempt this complication the idea of localized learning can be used. Localized Learning, here, refers to the class of learning techniques that modifies the augmentation function behavior only in the vicinity of available data without perturbing augmentation values in regions far from available datapoints. Thus, the augmentation value at any point in the feature space could be either "influenced" by one or more datapoints, or remain "uninfluenced". Three different ways to perform localized learning are mentioned as follows: 1. Discretizing the feature space into subdomains and representing the augmentation function as a piecewise combination of local augmentation functions corresponding to each of these subdomains. Here, a datapoint influences only those points in the feature space that lie in its own subdomain. 2. Introducing artificial datapoints in the learning process by sampling from the existing augmentation field in regions where no datapoints are available. Here, the sampling strategy determine the influenced regions in the feature space.3. Superposing finite support functions (e.g., truncated Gaussians) centered at available datapoints can be added on top of the existing augmentation to update it. Here the support width of these individual functions characterize the influence of the datapoints in the feature space.
Localized learning forces the augmentation function to retain the baseline behavior for feature space regions which remain uninfluenced throughout the LIFE process. The certainty that the augmented model reverts back to its baseline behavior when faced with feature combinations (physical conditions) not encountered during the LIFE process means that the performance of the augmented model would always be better than or equal to the baseline version.Also note that, for steady-state models like RANS, the feature values obtained in one solver iteration influence the feature values obtained in the next iteration. The inference and learning process, however, works with the feature values obtained at solver convergence only. Hence, uninfluenced feature space regions (according to the converged solutions) could be accessed by the solver when the residuals are not converged. So, if the augmentation function is modified in these regions, it could consequently affect the converged result, which is undesirable. Thus, it is even more important that in uninfluenced feature space regions, the augmentation function does not learn from data, and rather predict from the unaltered existing augmentation behavior.
To be able to evaluate different functional forms for the augmentation, the solver has to be made agnostic of how the augmentation is implemented. To do so, the solver must interact with only a linearized version of the augmentation function, β . In other words, given feature values η 0 , an implementation of the augmentation function should internally compute the numerical values β 0 = β(η 0 ; w) and g β = ∂β ∂η η 0 . These numerical values should then be passed to the solver which should construct the linearized augmentation using AD variables η and non-AD type numerical values for η 0 as β (η) = β 0 + g β (η − η 0 ). Notice here, that the value of β at η = η 0 (which is the value the direct solver uses in the model) remains β 0 and the second term involving g β is present only to record ∂β ∂η η 0 in the AD tape. Notice that this also presents the advantage of making the AD tape shorter.The detailed algorithm for the LIFE formulation is presented in Algorithm 1. The jacobians presented in the algorithm can be calculated using hard-coded analytic derivatives or using an automatic differentiation (AD) package. In the transition modeling demonstration presented in this work (section IV), the in-house solver makes use of the open-source automatic differentiation package ADOL-C to calculate these jacobians.IV. AUGMENTING A BYPASS TRANSITION MODELA. Relevant ideas from bypass transition modeling literature 1. The Praisner-Clark estimate of transition momentum-thickness Reynolds number
This model is a variation of the model proposed by Durbin in 2012 [47] which used Wilcox's 1988 k-ω model as the underlying turbulence model. The source term for the intermittency transport equation in Durbin's model is expressed as F γ |Ω|(γ max − γ) √ γ, where |Ω| is the local vorticity magnitude, γ max = 1.1 and F γ is a piecewise linear function that modulates the source term. Instead, we consider the baseline source term as ρ(1.0 − γ) √ γ|Ω| (the additional ρ is attributed to the compressible formulation of the model). The version of
solution, as the intermittency equation has a trivial solution with the value 1.0 throughout the domain. Multiplying the value 1.0 in the intermittency transport equation with the augmentation β, we have an augmented intermittency transport equation as follows
dJ dw to perform integrated inference and learning as sensitivities by definition only require the linearized functional form. Note here, that the chosen structure of the feature space, i.e. the uniform discretization characterized by ∆η 1 and ∆η 2 , and the functions parameters w = {β 00 , β 01 , . . . , β 21 , β 22 } are sufficient to determine the linearized approximation of the augmentation as shown above, at any point in the feature space. The sensitivities dβ dη and dβ dw
FIG. 1 :
1Schematic of feature space discretizationFor this specific function class, a trade-off is required in deciding the feature space grid resolution while learning the augmentation. If the resolution is too high, the augmentation might not be learned for a majority of grid cells as they might not be used during the optimization process, which can lead to a loss of generality of the augmentation. If the resolution is too low, the augmentation would not have enough resolution in the feature space and might lead to sub-optimal learning of the augmentation. This is further complicated by the curse of dimensionality, which restricts the resolution per feature with increasing number of features. Thus, this particular technique of approximating local augmentations using linear interpolation cannot be practically used for a "large" number of features.V. RESULTSWe use the datasets for the T3 series of experiments conducted by ERCOFTAC[50] to study bypass transition over flat plates across a range of inflow freestream turbulence intensities and pressure gradients. The T3A and T3B cases from this dataset have zero pressure gradient along the length of the flat plate. The freestream turbulence intensity at the inflow for T3B is significantly higher when compared to T3A. Other cases in the dataset include T3C1, T3C2, T3C3, T3C4 and T3C5. Out of these, T3C4 exhibits flow separation and since we are not considering separation-induced transition in this work, this case has been ignored. All of the T3C cases have a favorable pressure gradient near the leading edge which gradually decreases in magnitude along the length of the plate and turns into an increasingly adverse pressure gradient. T3C1 and T3C5 exhibit bypass transition in the favorable pressure gradient region of the flow, while T3C2 and T3C3 exhibit transition in regions with mild and strong adverse pressure gradients respectively.
FIG. 2 :
2Mesh used for RANS simulation of T3A and T3B cases 2. Mesh for T3C cases The length of the flat plate is 1.7 m with the inlet being 0.15 m upstream of the leading edge of the flat plate. The contouring of the upper wall follows the correlations given in the work by Suluksna et al [52], which results in the trends of U ∞ /U in along the length of the plate as shown in figure 4. Here U ∞ is calculated using the surface pressure, p w , at the flat plate wall assuming incompressible flow as
FIG. 3 :
3Mesh used for RANS simulation of T3C cases FIG. 4: U ∞ /U in vs x for T3C1
FIG. 5 :
5Decay of freestream turbulence intensity
FIG. 7 :
7ρE Residual histories for all optimization iterates
FIG. 8 :
8Skin friction coefficients along the length of the flat plate be attributed to the predicted intermittency having low values in a very narrow region in the flow field. Better results may be obtained by using good limiters on features and/or the predicted augmentation values. For instance, the augmentation value can be set to zero ifit is below a certain threshold (e.g., 0.3). Given that such a threshold would have to be extracted using training cases and might not work for certain test cases, these limiters might be ad hoc.
FIG. 9 :
9Intermittency contours (White line marks U = 0.95U ∞ ), 40x scaling in the y-direction
FIG. 10 :
10Augmentation contours on feature space slices after training on T3A and T3C1 datasets around η 1 ≈ 0.75 above which it has a virtually constant value of 1.0 (Baseline). This happens because the augmentation is trained using cases involving transition occurring in regions of zero/favorable pressure gradients. Training the cases for the adverse pressure gradient regions would potentially populate some region with higher values of η 1 . The augmentation assumes small values only in regions of high η 3 because low intermittency correlates to laminar regions in the flow where η 3 will be high. However, this is also the case in the viscous sub-layer and some parts of the buffer layer in the fully turbulent parts of the flow. In such a scenario, η 2 distinguishes between the laminar and turbulent parts of the flow. This is a clear example of how adding an extra feature can help in differentiating physical regions where the augmentation must take different values.
Re
The blade chord is 0.067 m in length and makes a 55 • angle with the streamwise direction. The inlet is situated 0.055 m upstream of the leading edge, while the outlet is located 0.242 m downstream of the leading edge. The mesh resolution next to the wall is on the order of y + ≈ 1.FIG. 11: Mesh used for RANS simulation of VKI turbine cascade2. Inflow conditions for T3 test casesThe information needed to characterize the inflow conditions for the T3 test cases has been mentioned in table II. The turbulence intensity decay plots shown infigure 12verify the ω L,in 940000 550000 418000 946000 ω in (in s −1 ) 7.943 11.4083 10.98218.70738
FIG. 13 :
13Verification of velocity profiles for MUR224 using LES data[53] (C ax refers to chord length in the x-direction) 4. Predictions using the augmented model on the same geometriesFigure 14shows results obtained by applying the model trained on T3A and T3C1 to the T3B, T3C2, T3C3, and T3C5 cases. The transition locations for T3B, T3C2 and T3C5 cases are predicted reasonably well in the results shown below. This is because the transition occurs in all of these cases in either zero, favorable or very mild adverse pressure gradient regions. The model fails to predict the transition location correctly for T3C3 for this reason and instead assumes the baseline value for β = 1 in the adverse pressure gradient regions, thus predicting premature transition. Once again, it can be noticed that the laminar part of the flow does not show fully laminar friction coefficients as can be seen when compared to data. This can be attributed, as explained before, to the predicted intermittency having low values in a very narrow region of the flow field.
FIG. 14 :
14Predicted Skin friction coefficients for flat plate cases using the model trained on T3A and T3C1.5. Predictions using the augmented model on a different geometryFigure 18shows the heat transfer coefficients predicted by the model trained on T3A and T3C1 on the VKI high pressure turbine cases, which are not only geometrically different from the training set, but also involve combinations of pressure gradients and freestream turbulence intensities not seen in the training data. Predictions with two different preset distance intervals to extract Re θ,t information from the freestream are shown in Appendix C to assess the sensitivity of the solution to such modeling choices. For MUR224 and MUR129, the transition location is predicted fairly accurately and consistently with both the preset distance intervals. For MUR241, the transition location on the suction side is predicted very well. On the pressure side, however, the data shows a gradual transition to turbulence, while the model predicts a sharp transition at different locations within this gradual transition range when using different preset distance intervals. For MUR116, transition is premature for on either sides of the blade.
FIG. 15 :
15Zoomed out views to indicate predicted transition locations on upper surface
FIG. 17 :FIG. 18 :
1718Contours near the transition location on suction surface for MUR241VI. SUMMARYThis work presents a methodology that can be used to build data-driven augmentations to physics-based models. Building on recent work in data-driven turbulence modeling, this approach develops generalizable features-to-augmentation mapping across multiple datasets in a model-consistent manner. The proposed framework, referred to as "Learning and Inference assisted by Feature-space Engineering (LIFE)" places particular emphasis on the design of the feature space and control of the augmentation behavior in the feature space. Choosing features (along with appropriate physics-based non-dimensionalization) to create an "effectively" bounded feature space is indispensable as this boundedness ensures the tractability of the learning problem. In an ideal scenario, if an augmentation is learnt in this entire bounded region, an extrapolation in geometry/boundary conditions would translate to an Predicted heat transfer coefficients for VKI turbine cases using model trained on T3A and T3C1.interpolation in feature space. Given the availability of a limited number of datasets, human intuition and expert knowledge is essential, in the authors' opinion to formulate such features. Although automated feature selection techniques do exist, they are heavily (and often prohibitively) expensive to find a parsimonious combination of features that can address the augmentation, especially for complex physical problems like transition and turbulence. To address the issue of the possible lack of data in certain significantly sized regions of the feature space, the notion of localized learning becomes critical to avoid spurious learning outputs. Localized learning refers to the modification of the augmentation function behavior at every learning step only in the vicinity of available datapoints in the feature space.
FIG. 20 :
20Feature maps (x-axis: η 2 , y-axis: η 3 , uniform color-bar range [0,1] across all plots)
FIG. 21 :
21Skin friction coefficients for T3C cases (characterized by a small bump in the heat transfer coefficient), whereas a major discrepancy, resulting in considerable different transition behavior, is seen for MUR241.Appendix D: Results without localized learning (Neural Networks) Three different fully connected neural network architectures were utilized as shown in table IV. Owing to the simple nature of augmentation behavior in feature space, a simple neural network with two hidden layers containing seven nodes each was tested. The results obtained during training, as shown in figure 24, are clearly poor as they partially laminarize the entire flow rather than laminarizing the flow only before the transition location. Further, two other architectures with 1 hidden layer containing 600 nodes, and 2 hidden layers containing 45
FIG. 22: Heat transfer coefficients for VKI cases
FIG. 23 :
23Comparison between different preset distance intervals acknowledge that more sophisticated training methods may yield better results.Appendix E: Results with a finer discretization in the feature space For comparison purposes, a finer feature-space discretization was also used to obtain the augmentation function, the training results and augmentation contours on feature-space slices for which have been shown in figures 25 and 26. The feature space was divided into subdomains of size 1/30 along all three feature space directions (90, 30, and 30 cells along the first, second, and third features respectively). As can be noticed infigure 26, the influence of the changes made be the data have been restricted to smaller regions owing to the smaller cell sizes.Figures 27 and 28show the results from testing the augmentation on the T3B, T3C2, T3C3, T3C5, MUR116, MUR129, MUR224, and MUR241 cases. As can be seen from the results, while the augmentation learned on the finer grid seems to predict the transition locations for T3 cases with nearly similar accuracy (with little laminarization in the T3B case and slightly premature transition in T3C5) as its counterpart trained on the coarser grid, almost all results from the VKI cases exhibit premature transition. This
Objective reduction (T3C1) (d)Skin friction profile (T3C1) FIG. 24: Training results with Different Neural Network Architectures happens because the limited region of influence that the available data has in the feature space allow some feature space locations to exhibit baseline behavior (which did not happen when using the augmentation learned on a coarser grid as the region of influence covered a larger part of the feature space).
FIG. 28 :
28)η 1 = 0.45 (f)η 1 = 0.55 (g)η 1 = 0.65 (h)η 1 = 0.75 FIG. 26: Augmentation contours on feature-space slices Heat transfer coefficient profiles for the VKI test cases
for all j s.t. x j ∈ Ω i do∂β
∂β
∂η
∂η
∂ u i
m
[Adjoints]
9:
dJ i
dw k−1 := 0
[Set sensitivities to zero]
10:
11:
gradient descent update]16:
for i = 1 : N do
17:
TABLE II :
IIInflow conditions for the T3 test cases(a)T3B
(b)T3C2
(c)T3C3
(d)T3C5
FIG. 12: Decay of freestream turbulence intensity
3. Inflow, wall and outflow conditions for VKI test cases
The information needed to characterize the inflow conditions for the VKI turbine cascade
test cases along with pressure at outflow boundary and temperature at the isothermal wall
is presented in table III. The boundary condition for ω in was set to ensure that the viscosity
ratio ν t in /ν in remains between the range from 1.5 to 50.
Cases
MUR116 MUR129 MUR224 MUR241
T u in
0.008
0.008
0.06
0.06
ν t in /ν in
3
1.556
43.537
15.465
p 0,in (in bar)
3.269
1.849
0.909
3.257
T 0,in (in K)
418.9
409.2
402.6
416.4
p out (in bar)
1.550
1.165
0.522
1.547
T wall (in K)
300.0
300.0
300.0
300.0
ω in (in s −1 ) 1.5 × 10 4 1.5 × 10 4 1.5 × 10 4 1.5 × 10 5
TABLE III :
IIIInflow, wall and outflow conditions for VKI test cases
To demonstrate LIFE in practice, a simple intermittency-based bypass transition model-form based on Wilcox's k-ω turbulence model was augmented appropriately with a careful choice of features. This augmentation function is inferred from two benchmark cases of flat plate transition from the T3 series of experiments. These two cases are characterized by transition under zero pressure gradient and favorable pressure gradient regions respectively, with an inflow freestream turbulence intensity significantly greater than 1%. Linear interpolation on uniformly structured feature-space discretization was used as the functional form for the augmentation in order to implement a very crude variant of localized learning. Though more sophisticated ways to perform localized learning are possible, this implementation is showcased to demonstrate how effective and capable the notion of localized learning can be. This augmented model is then used to predict transition on four other flat plate cases and four single-stage high-pressure-turbine cascade cases. The augmented model is shown to be able to predict the transition locations for problems in which similar physics was encountered in the dataset. It is noted that the focus of this work is not to develop the ultimate bypass transition model, rather to present a formalism that can be used to develop physics-constrained, data-enabled models. More comprehensive training datasets, especially involving higher pressure gradients and turbulence intensities can help in improving the accuracy and applicability of the model.Currently, there are three major challenges in the LIFE framework -(1) Feature design for a given application; (2) Deciding the range of the vicinity in which a datapoint updates the augmentation; and (3) Formal uncertainty quantification. Designing features that can capture the behavior of an augmentation function in the simplest possible manner is a challenge that modelers have been dealing with for decades now. While there exist numerical diagnostics and techniques to assist and/or automate feature selection and engineering, the authors feel that domain expertise and intuition is the best way forward for problems as complex as turbulence and transition, at least given the prevailing constraints on data and computational power which render automatic feature selection infeasible. As far as the range of influence that a datapoint has in the feature space is concerned, a trade-off needs to be made between over-fitting and predictive accuracy. If the influence is very limited, the augmented model will perform very well on the training cases, but can revert to baseline behavior for test cases which lie outside the influence of these datapoints. On the other hand, if the influence is too wide, the augmentation behavior might not be sufficiently resolved and hence could adversely affect predictive accuracy. Finally, formal uncertainty quantificationtechniques need to be developed for use in design of experiments to extend the capability of the LIFE framework. Initial funding for this work came from the Office of Naval Research (ONR) under the project Advancing Predictive Strategies for Wall-Bounded Turbulence by Fundamental Studies and Data-driven Modeling (Tech. Monitor: Dr. Thomas Fu). Currently, it is being funded by Advanced Research Projects Agency-Energy (ARPA-E) DIFFERENTIATE program under the project Multi-source Learning-accelerated Design of High-efficiency Multi-stage Compressor (MULTI-LEADER) (Award no. DE-AR0001201) led by Raytheon Technologies Research Center (RTRC).[1] Karthik Duraisamy, Gianluca Iaccarino, and Heng Xiao. Turbulence modeling in the age of data. Annual Review of Fluid Mechanics, 51:357-377, 2019. [2] Heng Xiao and Paola Cinnella. Quantification of model uncertainty in rans simulations: A review. Progress in Aerospace Sciences, 108:1-31, 2019.[3] Karthik Duraisamy. Perspectives on machine learning-augmented reynolds-averaged and large eddy simulation models of turbulence. Physical Review Fluids, 2021.[4] Todd A Oliver and Robert D Moser. Bayesian uncertainty quantification applied to RANS turbulence models. Journal of Physics: Conference Series, 318(4):042032, dec 2011.[5] Eric Dow and Qiqi Wang. Quantification of structural uncertainties in the k -w turbulence model. In 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference.[6] Brendan Tracey, Karthik Duraisamy, and Juan Alonso. Application of supervised learning and using AD for this linearized function is a better and efficient alternative to evaluate this Jacobian. Any other jacobians w.r.t. the augmentation can be converted to the jacobians w.r.t. function parameters w using ∂β ∂w , which the learning framework being used must be able to calculate. The sensitivity calculation for an integrated inference and learning approach can, thus, be calculated similar to what has been shown above for field inversionVII. ACKNOWLEDGEMENTS
dJ
dw
=
∂J
∂β um
−
∂J
∂ u m w
∂R
∂ u m
−1
w,ξ
∂R
∂β um,ξ
∂β
∂w
.
TABLE IV :
IVDescription of neural network architectures
Appendix A: Sensitivity evaluation using discrete adjoints In a field inversion problem, the objective function J = C( u m ) + λ β T β (β) is a function of both u m and β, and since u m is dependent on β via R( u m ; β, ξ) = 0, we can write theNow, using the above representations one can write, variables. An algorithmic differentiation (AD) package is used to evaluate the different jacobians involved in the above discrete adjoint approach. The in-house solver[54]used in this work makes use of the ADOL-C package for AD.The integrated inference and learning framework, however requires additional infrastructure for this to work. Since, there is an additional dependence of β(η( u m , ζ); w) on u m to account for, this pathway needs to be included in the calculation of any jacobians w.r.t. states The main issue with training only with T3A is that the behavior in the augmentation is learnt only on the basis of data from a zero pressure gradient case. The more cases are added to the mix, the stronger is the consistency of an optimal augmentation for different problems.Optimization convergence and feature space contours: Although the convergence (Figure 19(a)) appears similar to that observed when both T3A and T3C1 were simultaneously used to learn the augmentation, the skin friction predictions(Figure 19(b)) andaugmentation contours in the feature space (figure 20) have significant differences between the two cases. The effect of this difference between the two augmentations can be seen in the following sections. It is however, noteworthy how a single additional case with unseen physical behavior can make a considerable difference in the predictions. This suggests the ability of the framework to effectively extract information about the features-to-augmentation functional relationship.
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[] |
[
"Quantum Information Processing A Poisson Model for Entanglement Optimization in the Quantum Internet",
"Quantum Information Processing A Poisson Model for Entanglement Optimization in the Quantum Internet"
] |
[
"Laszlo Gyongyosi ",
"Sandor Imre "
] |
[] |
[] |
We define a nature-inspired model for entanglement optimization in the quantum Internet. The optimization model aims to maximize the entanglement fidelity and relative entropy of entanglement for the entangled connections of the entangled network structure of the quantum Internet. The cost functions are subject of a minimization defined to cover and integrate the physical attributes of entanglement transmission, purification, and storage of entanglement in quantum memories. The method can be implemented with low complexity that allows a straightforward application in the quantum Internet and quantum networking scenarios.IntroductionQuantum entanglement and the entangled network structure serve as fundamental concepts of the quantum Internet [1-5], long-distance quantum networks and future quantum communications[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Since the no-cloning theorem makes it impossible to use the "copy-and-resend" mechanisms of traditional repeaters[16,23], in a quantum Internet scenario the quantum repeaters have to transmit correlations in a different way L. Gyongyosi: Parts of this work were presented in conference proceedings[8].
|
10.1007/s11128-019-2335-1
|
[
"https://eprints.soton.ac.uk/431945/1/Gyongyosi_Imre2019_Article_APoissonModelForEntanglementOp.pdf"
] | 3,736,531 |
1803.02469
|
77a4b98d262bf1427ee2b8481d4aef9749ea0b6e
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Quantum Information Processing A Poisson Model for Entanglement Optimization in the Quantum Internet
(2019) 18:233
Laszlo Gyongyosi
Sandor Imre
Quantum Information Processing A Poisson Model for Entanglement Optimization in the Quantum Internet
(2019) 18:23310.1007/s11128-019-2335-1Received: 3 April 2018 / Accepted: 21 May 2019Quantum Internet · Quantum repeaters · Quantum entanglement · Quantum communication · Quantum Shannon theory
We define a nature-inspired model for entanglement optimization in the quantum Internet. The optimization model aims to maximize the entanglement fidelity and relative entropy of entanglement for the entangled connections of the entangled network structure of the quantum Internet. The cost functions are subject of a minimization defined to cover and integrate the physical attributes of entanglement transmission, purification, and storage of entanglement in quantum memories. The method can be implemented with low complexity that allows a straightforward application in the quantum Internet and quantum networking scenarios.IntroductionQuantum entanglement and the entangled network structure serve as fundamental concepts of the quantum Internet [1-5], long-distance quantum networks and future quantum communications[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Since the no-cloning theorem makes it impossible to use the "copy-and-resend" mechanisms of traditional repeaters[16,23], in a quantum Internet scenario the quantum repeaters have to transmit correlations in a different way L. Gyongyosi: Parts of this work were presented in conference proceedings[8].
[1][2][3][4]24]
. In the entangled network structure of the quantum Internet, the main task of quantum repeaters is to distribute quantum entanglement between distant points that will then serve as a fundamental base resource for quantum teleportation and other quantum protocols [1]. Since in an experimental scenario [25][26][27][28][29][30][31] the quantum links between nodes are noisy and entanglement fidelity decreases as hop distance increases, entanglement purification is applied to improve the entanglement fidelity between nodes [1][2][3][4][5]. Quantum nodes also perform internal quantum error correction that is a requirement for reliability and storage in quantum memories [1,4,5,29,[32][33][34][35][36][37][38][39][40][41][42]. Both entanglement purification and quantum error correction steps in local nodes are high-cost tasks that require significant minimization [1][2][3][4][5][6][7][8][9]25,26,[29][30][31].
The shared entangled states between nodes form entangled connections. Significant attributes of these entangled connections are entanglement fidelity [1,4,5] and correlation in terms of relative entropy entanglement [68,69]. Entanglement fidelity is a crucial parameter. It serves as the primary objective function in our model, which is a subject of maximization. Maximizing the relative entropy of entanglement is the secondary objective function. Minimizing the cost of classical communications, which is required by the entanglement optimization method as an auxiliary objective function, is also considered.
Besides these attributes, the entangled connections are characterized by the entanglement throughput that identifies the number of transmittable entangled systems per sec at a particular fidelity. In our model, the nodes are associated with an incoming entanglement throughput [1], that serves as a resource for the nodes to maximize the entanglement fidelity and the relative entropy of entanglement. The nodes receive and process the incoming entangled states. Each node performs purification and internal quantum error correction, and it stores the entangled systems in local quantum memories. The amount of input entangled systems in a node is therefore connected to the achievable maximal entanglement fidelity and correlation in the entangled states associated with that node. The objective of the proposed model is to reveal this connection and to define a framework for entanglement optimization in the quantum nodes of an arbitrary quantum network. The required input information for the optimization without loss of generality is the number of nodes, the number of fidelity types of the received entangled states, and the node characteristics. In a realistic setting, these cover the incoming entanglement throughput in a node and the costs of internal entanglement purification steps, internal quantum error corrections, and quantum memory usage.
In this work, an optimization framework for quantum networks is defined. The method aims to maximize the achievable entanglement fidelity and correlation of entangled systems, in parallel with the minimization of the cost of entanglement purification and quantum error correction steps in the quantum nodes of the network. The problem model is therefore defined as a multiobjective optimization. This paper aims to provide a model that utilizes the realistic parameters of the internal mechanisms of the nodes and the physical attributes of entanglement transmission. The proposed framework integrates the results of quantum Shannon theory, the theory of evolutionary multiobjective optimization algorithms [77,78], and the mathematical modeling of seismic wave propagation [77][78][79][80][81][82].
Inspired by the statistical distribution of seismic events and the modeling of wave propagations in nature, the model utilizes a Poisson distribution framework to find optimal solutions in the objective space. In the theory of earthquake analysis and spatial connection theory [77][78][79][80][81][82], Poisson distributions are crucial in finding new epicenters. Motivated by these findings, a Poisson model is proposed to find new solutions in the objective space that is defined by the multiobjective optimization problem. The solutions in the objective space are represented by epicenters with several locations around them that also represent solutions in the feasible space [77,78]. The epicenters have a magnitude and seismic power operators that determine the distributions of the locations and fitness [77,78] of locations around the epicenters. Epicenters with low magnitude generate high seismic power in the locations, whereas epicenters with high magnitude generate low seismic power in the locations. Epicenters are generated randomly in the feasible space, and each epicenter is weighted from which the magnitude and power are derived. By a general assumption, epicenters with lower magnitude produce more locations because the locations are closer to the epicenter. The locations are placed within a certain magnitude around the epicenters in the feasible space. The optimization framework involves a set of solutions to the Pareto optimal front [77,78] by combining the concept of Pareto dominance and seismic wave propagations. The new epicenters are determined by a Poisson distribution in analogue to prediction theory in earthquake models. The mathematical model of epicenters allows us to find new solutions iteratively and to find a global optimum. The framework has low complexity that allows an efficient practical implementation to solve the defined multiobjective optimization problem.
The multiobjective optimization problem model considers the fidelity and correlation of entanglement of entangled states available in the quantum nodes. The resources for the nodes are the incoming entangled states from the quantum links and the already stored entangled quantum systems in the local quantum memories. In the optimization procedure, both memory consumptions and environmental effects, such as entanglement purification and quantum error correction steps, are considered to develop the cost functions. In particular, the amount of resource, in terms of number of available entangled systems, is a coefficient that can be improved by increasing the incoming number of entangled systems, such as the incoming entanglement throughput in a node. In the proposed model, the incoming entanglement fidelity is further divided into some classes, which allows us to differentiate the resources in the nodes with respect to their fidelity types. Therefore, the fidelity type serves as a quality index for the optimization procedure. The optimization aims to find the optimal incoming entanglement throughput for all nodes that leads to a maximization of entanglement fidelity and correlation of entangled states with respect to the relative entropy of entanglement, for all entangled connections in the quantum network.
The novel contributions of our manuscript are as follows: This paper is organized as follows. Section 2 presents the problem statement. Section 3 details the optimization method. Section 4 provides the problem resolution. Section 5 proposes numerical evidence. Finally, Sect. 6 concludes the paper. Supplemental material is included in the Appendix.
Problem statement
The problem to be solved is summarized in Problem 1.
Problem 1
For a given quantum network with N nodes, for all nodes x i , i = 1, . . . , N , the entanglement fidelity and relative entropy of entanglement for all entangled connections are maximized, and the cost of optimal purification and quantum error correction and the cost of memory usage for all nodes are minimized.
The network model is as follows. Let B F (x) be the incoming number of received entangled states (incoming entanglement throughput) in a given quantum node x, measured in the number of d-dimensional entangled states per sec at a particular entanglement fidelity F [1][2][3].
Let N be the number of nodes in the network, and let T be the number of fidelity types F j , j = 1, . . . , T of the entangled states in the quantum network.
Let B j F (x i ) be the number of incoming entangled states in an ith node x i , i = 1, . . . , N , from fidelity type j. In our model, B j F (x i ) represents the utilizable resources in a particular node x i . Thus, the task is to determine this value for all nodes in the quantum network to maximize the fidelity and relative entropy of shared entanglement for all entangled connections. Let X be an N × T matrix
X = B j F (x i ) N ×T .(1)
The matrix describes the number of entangled states of each fidelity type for all nodes in the network, B j F (x i ) ≥ 0 for all i and j.
Objective functions
For a given node x i , let F (x i ) be the primary objective function that identifies the cumulative entanglement fidelity (a sum of entanglement fidelities in x i ) after an entanglement purification P (x i ) and an optimal quantum error correction C (x i ) in x i . In our framework, F i (X) for a node x i is defined as
F i (X) = T j=1 T k=1 A i jkB j F (x i )B k F (x i ) + T j=1 R i jB j F (x i ) + c i ,(2)
where A i jk is the quadratic regression coefficient, R i j is the simple regression coefficient, c i is a constant, andB
j F (x i ) is defined as B j F (x i ) = B j F (x i ) + B j F (x i ) ,(3)
where
B j F (x i ) is an initialization value for B j F (x i ) in a particular node x i . Then let E (D i (X)
) be the secondary objective function that refers to the expected amount of cumulative relative entropy of entanglement (a sum of relative entropy of entanglement) in node x i , defined as
E (D i (X)) = T j=1 T k=1 A * i jkB j F (x i )B k F (x i ) + T j=1 R * i jB j F (x i ) + c * i ,(4)
where A * i jk , R * i j , and c * i are some regression coefficients, by definition. Therefore, the aim is to find the values of B j F (x i ) for all i and j in (1), such that F i (X) and E (D i (X)) are maximized for all i.
Assuming that the fidelity of entanglement is dynamically changing and evolves over time, the w j (x i ) quantum memory coefficient is introduced for the storage of entangled states from the jth fidelity type in a node x i as follows:
w j (x i ) = η j B j F (x i ) + κ j B j F (x i ) ,(5)
where η j and κ j are coefficients that describe the storage characteristic of entangled states with the jth fidelity type.
Cost functions
The cumulative entanglement fidelity (2) and cumulative relative entropy of entanglement (4) in a particular node x i are associated with a f C (P (x i )) cost entanglement purification P (x i ) and a f C (C (x i )) cost of optimal quantum error correction C (x i ) in x i , where f C (·) is the cost function.
Then let C (X) be the total cost function for all of the T fidelity types and for all of the N nodes, as follows:
C (X) = N i=1 f C (P (x i )) + f C (C (x i )) = N i=1 T i=1 f j B j F (x i ) ,(6)
where f j is a total cost of purification and error correction associated with the jth fidelity type of entangled states.
Let F * be a critical fidelity on the received quantum states. The entangled states are then decomposable into two sets S low and S high with fidelity bounds S low (F) and S high (F) as
S low (F) : max ∀i F i < F * ,(7)
and
S high (F) : min ∀i F i ≥ F * .(8)
For the quantum systems of S low , the highest fidelity is below the critical amount F * , and for set S high , the lowest fidelity is at least F * . Then let X S low and X S high identify the set of nodes for which condition (7) or (8) holds, respectively. Let S i (X) be the cost of quantum memory usage in node x i , defined as
S i (X) = λ T j=1 α i 1 ϒ i B j F (x i ) ,(9)
where λ is a constant, α i is a quality coefficient that identifies set (7) or (8) for a given node x i , and ϒ i is the capacity coefficient of the quantum memory. The main components of the network model are depicted in Fig. 1.
Multiobjective optimization
The optimization problem is as follows. The entanglement fidelity and the relative entropy of entanglement for all types of fidelity of stored entanglement for all nodes are maximized, while the cost of entanglement purification and quantum error correction is minimal, and the memory usage cost (required storage time) is also minimal. These requirements define a multiobjective optimization problem [77,78]. Utilizing functions (2) and (4), the function subject of a maximization to yield maximal entanglement fidelity and maximal relative entropy of entanglement in all nodes of the network is defined via main objective function G (X):
F i (X), E (D i (X)), and F j (X), E D j (X) .
The maximum of the received entanglement fidelity in the nodes allows the classification of the nodes to sets X S low and X S high : node x i belongs to set X S low , whereas node x j belongs to set X S high (depicted by dashed frames)
G (X) = max N i=1 F i (X) E (D i (X)) .(10)
Function G (X) should be maximized while cost functions (6) and (9) are minimized via functions F 1 (N ) and F 2 (N ):
F 1 (N ) = min C (X) = N i=1 T i=1 f j B j F (x i ) ,(11)
and
F 2 (N ) = min S (X) = N i=1 S i (X),(12)
with the problem constraints [77,78] C 1 , C 2 , and C 3 for all i and j. Constraint C 1 is defined as
C 1 : ζ (X) ≥ γ,(13)
where γ is a cumulative lower bound on the required entanglement fidelity for all nodes, while ζ (X) is
ζ (X) = N i=1 F i (X) ,(14)
and constraint C 2 is
C 2 : X ≤ ,(15)
where is an upper bound on the total cost function C (X), while X is
X = N i=1 T i=1 f j B j F (x i ) .(16)
For constraint C 3 , let τ j (X) be a differentiation of storage characteristic of entangled states from the jth fidelity type:
τ j (X) = N i=1 w j (x i ) − 2 ,(17)where = N i=1 w j (x i ) N .(18)
Then, C 3 is defined as
C 3 : ν (X) ≤ ,(19)
where is an upper bound on the storage characteristic of entangled states from the jth fidelity type, while ν is evaluated via (17) as
ν = N j=1 τ j (X) .(20)
System model
This section defines the Poisson entanglement optimization method, and it is applied to the solution of the multiobjective optimization problem of Sect. 2.
Motivation and utility of the mathematical model in the quantum internet
The quantum Internet is defined as a complex network model with quantum and classical layers that involve several optimization criteria and objectives. An optimization problem model of the quantum Internet therefore induces a multiobjective optimization problem model that considers the special requirements of the environment of the quantum Internet. These requirements cover the entanglement transmission procedure, processing of quantum entanglement in the quantum nodes, and auxiliary communication through the classical links that support the entangled network structure. The quantum transmission procedure models the generation of the entangled quantum network with quantitative and qualitative measures. In this manner, a quantitative measure is the relative entropy of entanglement between the quantum nodes, while the entanglement fidelity is a qualitative measure. Classical communication could also cause an overhead in the entanglement distribution mechanism of the quantum Internet. Thus, a multiobjective optimization framework should consider the attributes of both quantum and classical layers.
To address the multiple criteria and several objectives of the quantum Internet, a multiobjective optimization framework is defined. The multiple criteria of the quantum Internet are defined as diverse objective functions that should be satisfied in parallel. The problem is therefore analogous to finding solutions in an objective space such that the objective space is defined and spanned by the input problems induced by the environment of the quantum Internet. The multiobjective optimization framework should evolve a set of solutions to the Pareto optimal front. In our model, these solutions are evolved via the mathematical model of epicenters that provide a naturally inspired answer to the multiobjective problem defined via the environment of the quantum Internet. The mathematical model of epicenters utilizes the theory of Pareto dominance in the problem resolution such that the selection and evaluation processes in the objective space that are required to identify a global optimal solution are controlled via our nature-inspired model. The proposed Poisson model ensures a robust randomization and efficient convergence in the objective space such that the solutions determined by utilizing the epicenters in the objective space will converge to a global optimal solution. The global optimal solution in the objective space represents the parallel satisfaction of the multiple criteria and objective functions defined by the quantum Internet. The randomness injected by the Poisson distribution not just avoids early convergence to a local optimal solution but also induces a fast convergence for the global optimal solution in the objective space.
Since the multiple objectives and optimization criteria of the mathematical framework are motivated by practical assumptions and considerations of the quantum Internet, the proposed mathematical model of epicenters is strongly connected with a quantum Internet scenario. As follows, the utility of the proposed multiobjective optimization framework represents an effective solution for the practical optimization problems induced by the quantum Internet.
Poisson operators
The attributes of the Poisson operator are as follows.
Dispersion
The D (E) dispersion coefficient of an epicenter E (solution in the feasible space S F ) determines the number of affected L j , j = 1, . . . , D (E), locations around an epicenter E. The random locations around an epicenter also represent solutions in S F that help in increasing the diversity of population P (a set of possible solutions) to find a global optimum. The diversity increment is therefore a tool to avoid an early convergence to a local optimum [77,78].
The dispersion D (E i ) operator for an ith epicenter E i is defined as D (E i ) = m f ( E ) −f (E i ) + ϑ |P| i=1 f ( E ) −f (E i ) + ϑ ,(21)
where m is a control parameter, E i is an ith individual (epicenter) from the |P| individuals (epicenters) in population P, |P| is the size of population P, functionf (·) is the fitness value (see Sect. A.2.1),f ( E ) is a maximum objective value among the |P| individuals, and ϑ is a residual quantity. Without loss of generality, assuming |P| epicenters, the q total number of locations is as
q = |P| i=1 D (E i ) .(22)
Seismic power and magnitude
Assume that L j is a random location around E i . For L j , the Euclidean distance d E i , l j between an ith epicenter E i and the projection point l j of a jth location point L j , j = 1, . . . , D (E) on the ellipsoid around E i is as follows:
d E i , l j = dim 1 l j 2 + dim 2 l j 2 = 1 + tg 2 α E i l j a −2 + tg 2 α E i l j ,(23)
where dim i (·) is the ith dimension of l j , and
dim 1 l j 2 a 2 + dim 2 l j 2 b 2 = 1,(24)
where coefficients a and b define the shape of the ellipse around epicenter E i (see Fig. 2), while α E i l j is an angle:
tgα E i l j = dim 2 l j dim 1 l j .(25)
The seismic power P E i , L j operator for an ith epicenter E i in a jth location point
L j , j = 1, . . . , D (E i ) is defined as P E i , L j = 1 d E i , l j M E i , L j b 1 b 0 e σ ln P ( E i ,L j ) ,(26)
where b 0 and b 1 are regression coefficients, σ ln P(E j ) is the standard deviation [82], M E i , L j is the seismic magnitude in a location L j , and l j is the projection of L j onto the ellipsoid around E i [82]. Thus, at a given L j with d E i , l j ((23)), from P E i , L j (see (26)), the magnitude M E i , L j between epicenter E i and location L j is evaluated as
M E i , L j = P E i , L j 1 b 0 e σ ln P ( E i ,L j ) 1 b 1 d E i , l j .(27)
Cumulative magnitude
Let L E i j be the location point where the seismic power P E i , L E i j is maximal for a given epicenter E i . Let P * (E i ) be the maximal seismic power,
P * (E i ) = max ∀ j P E i , L E i j .(28)
Assuming that |P| epicenters, E 1,...,|P| exist in the system, let identify by P max E the epicenter E with a maximal seismic power among as
P max E = max ∀i P * (E i ) ,(29)with magnitude M E , L E j , where L E j is the location point where the seismic power P max E is maximal yielded for E .
Then the C (E i ) cumulative magnitude for an epicenter E i is defined as
C (E i ) = M f (E i ) −f E + ϑ |P| i=1 f (E i ) −f (E ) + ϑ ,(30)
where E is the highest seismic power epicenter with magnitude M E , L E j ,f E is the minimum objective value among the |P| epicenters, and M is a control parameter defined as
M = |P| i=1 M E i , L E i j ,(31)
where L E i j provides the maximal seismic power for an ith epicenter E i , functionsf (E i ) andf E are the fitness values (see Sect. A.2.1) for the current epicenter E i and for the highest seismic power epicenter E , and ϑ is a residual quantity.
Distribution of epicenters
Assume that E i is a current epicenter (solution) and R k and R l are two random reference points around E i . Using the C (E i ) cumulative seismic magnitude (see (30)) of an epicenter E i , the generation of a new epicenter E p is as follows:
Let (E i , R k , R l ) be a Poisson range identifier function [80,81] for E i using R k and R l as random reference points:
(E i , R k , R l ) = d (E i , R k ) c w (R k , R l ) cos θ E i ,R k , R k ,R l · d (R k , R l ) c w (E i , R k ) ,(32)
where E i is a current epicenter, R k and R l are random reference points, d (·) is the Euclidean distance function, c w (E i , R k ) and c w (R k , R l ) are weighting coefficients between epicenters E i and R k and between R k and R l , and
θ E i ,R k , R k ,R l is the angle between lines E i ,R k and R k ,R l : θ E i ,R k , R k ,R l = cos −1 d (E i , R k ) 2 + d (E k , R l ) 2 − d (E i , R l ) 2 2d (E i , R k ) d (R k , R l ) .(33)
Without loss of generality, using (32), a Poissonian distance function D E p for the finding of new epicenter E p is defined via a P Poisson distribution [80,81] as follows:
D E p = P (k, λ) ,(34)
where
k = (E i , R k , R l ) ,(35)
with mean
λ = E [ (E i , R k , R l )] .(36)
Therefore, the resulting new epicenter E p is a Poisson random epicenter E p with a Poisson range identifier D E p . For a large set of reference points, only those reference points that are within the r (E i ) radius around the current solution E i are selected for the determination of the new solution E p . This radius is defined as
r (E i ) = χ 10 Q 1 2M −Q 2 ,(37)
whereM is the average magnitude,
M = 1 |P| M = 1 |P| |P| i=1 M E i , L E i j ,(38)
Q 1 and Q 2 are constants, and χ is a normalization term. Motivated by the corresponding seismologic relations of the Dobrovolsky-Megathrust radius formula [81], the constants in (37) are selected as Q 1 = 0.414 and Q 2 = 1.696.
In the relevance range r (E i ) of (37), the weights of reference points are determined by the seismic power function (26).
Population diversity
Hypocentral
The hypocentral of an epicenter is aimed to increase the diversity of population by a randomization.
Let dim k (E i ) be an actual randomly selected kth dimension and k = 1, (30)):
. . . , dim (E i ) be a current epicenter E i , i = 1, . . . , |P|. The H (dim k (E i )) hypocentral provides a random displacement [80,81] of dim k (E i ) using C (E i ) (seeH (dim k (E i )) = dim k (E i ) = 1 M dim k (E i ),L dim k (E i ) j dim k (E i ) 2 + (U (−C (E i ) , C (E i ))) 2 (39) where U (−C (E i ) , C (E i )) is a uniform random number from the range of [−C (E i ) , C (E i )] to yield the displacement dim k (E i ), M dim k (E i ) , L dim k (E i ) j is the magni- tude, and L dim k (E i ) j is a location point where P dim k (E i ) , L dim k (E i ) j is maximal for dim k (E i ).
The D (E i ) locations around the cumulative magnitude C (E i ) of E i are generated by (39) through all the randomly selected Y dimensions, where Y is as follows [77,78]:
Y = U (1, dim (E i )) .(40)
The process is repeated for all E i .
Poisson randomization
To generate random locations around dim k (E i ), a Poisson distribution is also used to increase the diversity of the population. A random location in the kth dimension L dim k (E i ) r around dim k (E i ) is generated as follows:
L dim k (E i ) r = dim k (E i ) w,(41)
where w ∈ P (X = k, λ)
is a Poisson random number with distribution coefficients k and λ. Given that it is possible that using (41) some randomly generated locations will be out of the feasible space S F , a normalization operator N (·) of L dim k (E i ) r is defined to keep the new locations around dim k (E i ) in S F , as follows [77,78]:
L dim k (E i ) r = L dim k (E i ) r mod B k up − B k low + B k low ,(43)
where B k low and B k up are lower and upper bounds on the boundaries of locations in a kth dimension, and mod(·) is to a modular arithmetic function. The procedure is repeated for the randomly selected t = U (1, dim (E i )) dimensions of E i , for ∀i.
Iterative convergence
The method of convergence of solutions in the Poisson optimization is summarized in Method 1.
Method 1 Convergence of Solutions
Step 1. Generate |P| epicenters, E 1 , . . . , E |P| , with D (E i ) random locations around a given i th epicenter E i .
Step 2. Select an epicenter E i , and determine the seismic operators D (
E i ), P E i , L j ,M E i , L j .
Step 3. Determine the D E p Poisson distance function using references R k and R l to yield a new solution E p .
Step 4. Repeat steps 1-3, until the closest epicenter to the E optimal epicenter is not found or other stopping criteria are not met.
An epicenter E i and the generation of a new solution E p in the objective space S O are depicted in Fig. 2. The ellipsoid around E i and the projection point l k of the reference location R k are serving the determination of power function P (E i , R k ) in the reference location R k .
A new epicenter E p is determined via the Poisson function D E p . Locations with low power function (26) values have high magnitudes (27) from the epicenter, whereas locations with high power function values have low magnitudes from the epicenter.
Framework
The algorithmical framework that utilizes the Poisson entanglement optimization method for the problem statement presented in Sect. 2 is defined in Algorithm 1. (27)). Notation dim i (·) refers to the ith dimension of l k , and coefficients a and b define the shape of the ellipse (yellow) around epicenter E i . The H (dim k (E i )) hypocentral of E i is determined via the range of the C (E i ) cumulative magnitude (depicted by the green circle). The new epicenter E p (depicted by the green dot) is determined by the D E p Poisson distance function using R k and R l , with angle θ E i ,R k , R k ,R l between lines E i ,R k and R k ,R l (Color figure online)
R k is P (E i , R k ) (see (26)), while the magnitude is M (E i , R k ) (see
Algorithm 1 Poisson Entanglement Optimization for Quantum Networks
Step 0. In an initial phase, a random population P of |P| feasible solutions is generated [77,78] Let G be an upper bound on the number of generations, n G .
Step 1. For each epicenter
x i = E i in P, define D (E i ) random locations around E i . For a diversity increment, determine the H (dim k (E i )) hypocentral displacement function (39) for dim k (E i ), for k = 1, . . . , dim (E i ).
Step 2. Determine the seismic power P E i , L j operator via (26) for an i th epicenter E i in a j th location point L j , j = 1, . . . , D (E i ). Determine the L E i j , the location point where the seismic power P E i , L E i j is maximal for a given epicenter E i , via (28).
Step 3. Determine epicenter E with a maximal seismic power P max E via (29). Compute seismic (27), and determine the sum of all N seismic magnitudes M via (31).
magnitude M E , L E j via
Step 4. Compute the D (E i ) dispersion via (21) and the C (E i ) cumulative seismic magnitude via (30). Select non-dominated solutions from the P population set to the set N P of non-dominated
solutions. Identify ϕ k as ϕ k = L E i k , where L E k
k is a k th location around E i . Update N P with the non-dominated solutions.
Step 5. Create set P of epicenters by selecting p feasible solutions from P using the Pr (ϕ i ) selection probability as Pr (ϕ i ) =f (ϕ i ) r ∈Pf (ϕ r ). Apply Sub-procedure 1.
Step 6. If n G ≥ G, then stop the iteration; otherwise, repeat steps 1-4.
Sub-procedure 1 of step 5 is discussed in the Appendix.
Optimization of classical communications
To achieve the minimization of classical communications required by the entanglement optimization, the S-metric (or hypervolume indicator) is integrated, which is a quality measure for the solutions or a contribution of a single solution in a solution set [77,78]. By definition, this metric identifies the size of dominated space (size of space covered).
By theory, the S (R) S-metric for a solution set R = {r 1 , . . . , r n } is as follows:
S (R) = L r ∈R x re f ∠x∠ x| r ,(44)
where L is a Lebesgue measure, notation b∠a means a dominates b (or b is dominated by a), and x re f is a reference point dominated by all valid solutions in the solution set [77,78]. For a given solution r i , the S-metric identifies the size of space dominated by r i but not dominated by any other solution, without loss of generality as:
S (r i ) = S (R, r i ) = S (R) − S (R\{r i }) .(45)
In the optimization of classical communications, the existence of two objective functions is assumed. The first objective function, f 1 , is associated with the minimization of the cost of the first type of classical communications related to the reception and storage of entangled systems in the quantum nodes. (It covers the classical communications related to the required entanglement throughput by the nodes, fidelity of received entanglement, number of stored entangled states, and fidelity parameters.) Thus,
f 1 : min ∀i C 1 (x i ) ,(46)
where C 1 (x i ) is the cost associated with the first type of classical communications related to a x i . The second objective function, f 2 , is associated with the cost of the second type of classical communications that is related to entanglement purification:
f 2 : min ∀i C 2 (x i ) ,(47)
where C 2 (x i ) is the cost associated with the second type of classical communications with respect to x i .
Assuming objective functions f 1 and f 2 , the S (r i ) of a particular solution r i is as follows:
S (r i ) = ( f 1 (r i ) − f 1 (r i−1 )) ( f 2 (r i ) − f 2 (r i+1 )) .(48)
Given that the S-metric is calculated for the solutions, a set of nearest neighbors that restrict the space can be determined. Since the volume of this space can be quantified by the hypervolume, the solutions that satisfy objectives f 1 and f 2 can be found by utilizing (48).
Computational complexity
The computational complexity of the Poissonian optimization method is derived as follows. Given that |P| epicenters are generated in the search space and that the number of locations for an ith epicenter E i is determined by the dispersion operator D (E i ), the resulting computational complexity at a total number of locations q =
|P| i=1 D (E i ) (see (22)) is O (|P| + q) d/2 log (|P| + q) ,(49)
since after a sorting process the locations for a given epicenter E i can be calculated with complexity O (D (E i )), where d is the number of objectives.
Considering that in our setting d = 2, the total complexity is
O ((|P| + q) log (|P| + q)) .(50)
Problem resolution
The resolution of the problem shown in Sect. 2 using the Poissonian entanglement optimization framework of Sect. 3 is as follows. Let X S low be a set of nodes for which condition (7) holds for the fidelity of the received entangled states in the nodes, and let X S high be a set of nodes for which condition (8) holds for the received fidelity entanglement.
Then let X S low and X S high be the cardinality of X S low and X S high , respectively. Specifically, function (10) for the X S low -type nodes is rewritten as
G X S low (X) = max X S low i=1 F X S low i (X) E D X S low i (X) ,(51)
where F X S low i (X ) is the entanglement fidelity function for an ith X S low -type node x i ,
x i ∈ X S low , and E D X S low i (X) is the expected relative entropy of entanglement in an ith X S low -type x i .
Similarly, for the X S low -type nodes, function (10) is as follows:
G X S high (X) = max X S high i=1 F X S high i (X) E D X S high i (X) .(52)
From (51) and (52), a cumulative G X S high ⊗X S high (X) is defined as
G X S low ⊗X S high (X) = X S high i=1 A i F X S high i (X) E D X S high i (X) + X S low + X S high i= X S high +1 A i F X S low i (X) E D X S low i (X) F 1 (X) ,(53)
where A i refers to the number of received entangled systems in an ith node, while
F 1 (X) = min C (X) = N i=1 T i=1 f j B j F (x i ).(54)
The fidelity types of the available resource states in the nodes should be further divided into T classes. The final function is then evaluated as
G X S low ⊗X S high (X) = F 1 (X) F 2 (X) = X S low + X S high i=1 T j=1 f j B j F (x i )F 2 (X) ,(55)
where
F 2 (X) = min S (X) = X S low + X S high i=1 S i (X).(56)
Thus,
G X S low ⊗X S high (X) = X S low + X S high i=1 S i (X),(57)
such that [77,78]
X S low + X S high i=1 F i (X) ≥ γ F 1 (N ) = γ X S low + X S high i=1 T j=1 f j B j F (x i ) ≤ ν X (ϕ i ) ≤ B j F (x i ) ,(58)
where ν X (ϕ i ) = Z j=1 τ j (ϕ i ), γ is given by the constraint of (13), while is given by the constraint of (19).
Convergence of solutions
Let F i (X) ∈ [0, 1] be the objective function that refers to the resulting entanglement fidelity in a particular node x i , after purification and quantum error correction with per-node cost functions F i 1 (X), and F i 2 (X), respectively. Precisely, a current ith epicenter E i identifies a solution in the objective space S O ,
S O : F i 1 (N ) , F i 2 (N ) , F i (X) .(59)
The random locations around E i also represent possible solutions. Let E * be an optimal solution in the S O subject space, which maximizes F i (X) and minimizes F i 1 (X) and F i 2 (X). From E i , the algorithm determines a new solution (epicenter) E p via the D E p Poisson distance function, using the connection model between the locations around E i . To improve the diversity, locations around E p are generated. The new epicenter E p converges to an optimal solution E * . The iterations are repeated until E * is not found or until a stopping criterion is met.
The iteration from a current solution E i to a new solution E p toward a global optimal E * in S O is illustrated in Fig. 3.
Numerical evidence
In this section, a numerical evidence is proposed to demonstrate the Poisson entanglement optimization method.
Decision making
To demonstrate the results of Sect. 4, let F i (X) be the object function subject to maximize. The problem is to determine a matrix X that maximizes F i (X), and also (N ). Thus, for each node N , the optimal number of received and stored entangled systems should be determined, with high and low fidelity classes.
Particularly, finding an optimal solution E * in S O with the assumptions given in Sect. 4 therefore means the selection of the optimal objective function (e.g., maximizing the entanglement fidelity F i (X) or maximizing the relative entropy of entanglement E D N S low i (X) ), in particular node types X S low and X S high , while all cost functions are minimized in the quantum network.
A solution set in S O is depicted in Fig. 4. An optimal solution E * in S O therefore yields the maximization of entanglement fidelity F N (X ) if a particular node N belongs to the class N S high , whereas it maximizes the relative entropy of entanglement E D N S low i (X) if N belongs to the class N S low .
Increasing B j F (x i ) for a N S high -class node and then performing an optimal purification and quantum error correction could significantly improve the fidelity of entanglement. On the other hand, for a N S low -class node, the fidelity improvement at an optimal purification and quantum error correction is insignificant. Thus, incrementing B j F (x i ) does not lead to a significant improvement in the fidelity. The optimal solution for these nodes is to focus on improving the relative entropy of entanglement, which requires lower cost function values.
This decision strategy provides a global optimal with respect to all quantum nodes of the quantum network.
The decision making is illustrated in Fig. 5. In Fig. 5a, the F entanglement fidelity is depicted in function of F i 1 (N ) for N S low and N S high nodes. In Fig. 5b, the D relative entropy of entanglement is depicted in function of F i 1 (N ) for N S low and N S high nodes. The initial values of F and D are assumed to be equal for a given class, while the value of F i 2 (N ) is set to constant for illustration purposes. For an N S high node, the increment of F i 1 (N ) leads to significant improvement in F, while the increment in D is moderate. For an N S low node, the increment of F i 1 (N ) leads to moderate improvement in F, while the improvement in D is significant. As a corollary, the increment of the entanglement throughput is a useful approach to increase the entanglement fidelity for the X S high set, and to boost the relative entropy of entanglement in the X S low set.
Distribution of solutions
First, we analyze the distribution of solutions in the feasible space S F focusing on the magnitudes associated to the locations around epicenters. Let us assume that the total number of q locations (see (22)) can be divided into m magnitude ranges [79], such that
m i=1 n i = q = |P| i=1 D (E i ) ,(60)
where n i is the number of locations belonging to an ith magnitude range, |P| is the population size. Then let M i be the magnitude associated to the ith magnitude range. Then añ i approximation of n i is evaluated as
n i = f (M i ) ,(61)
where f (·) is a fitting function. To give an estimate on n i at a particular magnitude M i , we utilize a power law distribution [79] function B (n i ) for a log-scaled n i , as
B (n i ) : log 10 (n i ) = a − bM i ,(62)
whereM i is a log-scaled M i , while a and b are constants [79]. Then, theñ i Poisson estimate is yielded as
n i = σ 2 i = λ i ,(63)
where σ 2 i is the observational variance, while λ i is the mean of a Poisson distribution. Since the sum of independent Poisson variables is also a Poisson variable with mean equals to the sum of the components means, where λ (q) is the mean total number, while λ i is an ith component mean. Using the Poisson property σ 2 = λ, the σ 2 q estimated uncertainty is yielded as [79]
λ (q) = m i=1 λ i ≈ q,(64)σ 2 q = λ (q) = m i=1 f M i .
Thus using a corresponding fitting function f (·), the mean and the variance of the total number of events are equal to the sum of the fitted values.
In our model the distribution of the log-scaledñ i = λ i values in function of M i is well approachable by the power law distribution B (λ i ) : log 10 (λ i ) = a − bM i , while the distribution of the λ (q) total number (64) of locations is approachable by a N λ (q) , σ 2 N , Gaussian distribution with variance σ 2 N = λ (q) as λ (q) → ∞, by theory.
The distributions of B (λ i ) in function of the magnitude M i and coefficient b are illustrated in Fig. 6.
The distributions of λ (q) (see (64)) for k it iterations are depicted in Fig. 7. In Fig. 7a, λ (q) = 10 2 , while Fig. 7b illustrated the distribution at λ (q) = 10 6 . As
Conclusions
We defined an optimization framework for the transmission and processing of quantum entanglement in the entangled network structure of the quantum Internet. The proposed Poissonian entanglement optimization framework fuses the fundamental concepts of quantum Shannon theory with the theory of evolutionary algorithms and seismic wave propagations. Two objective functions are defined, with primary focus on the entanglement fidelity and secondary focus on the relative entropy of entanglement. As an additional objective function, the minimization of classical communications required by the entanglement optimization procedure is considered. The cost functions are defined to cover the physical attributes of entanglement transmission, purification, and storage in quantum memories. This method can be implemented with low complexity that allows a straightforward application in future quantum Internet and quantum networking scenarios.
Compliance with ethical standards
Ethics statement This work did not involve any active collection of human data.
Data accessibility statement This work does not have any experimental data.
Conflict of interest We have no competing interests.
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Appendix
Definitions
Entanglement Fidelity
Let
|β 00 = 1 √ 2 (|00 + |11 ) (A.1)
be the target Bell state subject to be created at the end of the entanglement distribution procedure. The entanglement fidelity F at an actually created noisy quantum system σ is F (σ ) = β 00 |σ |β 00 ,
(A.2)
where F is a value between 0 and 1, F = 1 for a perfect Bell state and F < 1 for an imperfect state. The fidelity for two pure quantum states is defined as
F(|ϕ , |ψ ) = | ϕ|ψ | 2 . (A.3)
The fidelity of quantum states can describe the relation of a pure channel input state |ψ and the received mixed quantum system σ = n−1 i=0 p i ρ i = n−1 i=0 p i |ψ i ψ i | at the channel output as
F(|ψ , σ ) = ψ|σ |ψ = n−1 i=0 p i | ψ|ψ i | 2 . (A.4)
Fidelity can also be defined for mixed states σ and ρ
F(ρ, σ ) = Tr √ σ ρ √ σ 2 = i p i Tr √ σ i ρ i √ σ i 2 .
(A.5)
Relative Entropy of Entanglement
By definition, the E(ρ) relative entropy of entanglement function of a joint state ρ of subsystems A and B is defined by the D(· ·) quantum relative entropy function, without loss of generality as
E(ρ) = min ρ AB D(ρ ρ AB ) = min ρ AB Tr(ρ log ρ) − Tr(ρ log(ρ AB )), (A.6)
where ρ AB is the set of separable states
ρ AB = n i=1 p i ρ A,i ⊗ ρ B,i .
Evaluation of Solutions
Fitness Function
To evaluate the performance of the epicenters we utilize a mathematical apparatus based on the Pareto strength and fitness assignment [77,78]. Let Pr (E i ) be the probability of selection of an epicenter E i , defined as
Pr
(E i ) = κ (E i ) l∈K κ (E l ) , (A.7)
where κ (E i ) is the sum of d (·) Euclidean distances between E i and the other epicenters, as
κ (E i ) = K l=1 d (E i , E l ) = K l=1 E i − E l , (A.8)
where K is a set with cardinality
|K | = |P| i=1 D (E i ) + N i=1 dim(E i ) k=1 R (i, k) , (A.9)
where D (E i ) is given in (21), and l ∈ K refers to that the position of E j belongs to set K , and |P| is the population size. Let N P refer to the non-dominated solution archive, and let
ϕ i = E i (A.10)
refer to the selected epicenter, i.e, to an individual solution in P or in N P. Let (ϕ i ) be a strength coefficient for solution ϕ i , defined as
(ϕ i ) = ϕ k ∈ P N P ϕ k ∠ϕ i | , (A.11)
where ∠ refers to the Pareto dominance relation between ϕ i and ϕ k = E k . As follows, (A.11) depends on the number of individuals it dominates, by theory [77,78]. By definition, a decision vector A dominates a vector B, i.e., B∠A, if
f i (A) ≤ f i (B) (A.12)
for ∀i, i = 1, . . . , m and for at least one j with i, j = 1, . . . , n,
f j (A) ≤ f j (B) , (A.13)
where f : R m → R n . The set of non-dominated decision vectors in R n is called a Pareto optimal set, while the image under f in the solution space is called the Pareto front [77,78]. In a multiobjective optimization the aim is to achieve the best Pareto front, by theory. Using (A.11), let α (ϕ i ) be the raw fitness value of ϕ i evaluated by the (·) strength function (see (A.11)) of its dominators as
α (ϕ i ) = (ϕk∈P N P)∧ (ϕ i ∠ϕ k )| (ϕ k ) , (A.14)
with an inverse distance function (referred to as the density value of ϕ i ), ρ (ϕ i ) as
ρ (ϕ i ) = 1 d g (ϕ i ) , (A.15)
where d g (ϕ i ) is the distance from solution ϕ i to its g th nearest individual, where g is initialized as the square root of the sample size P N P , by theory [77,78]. Using (A.15), a for a random solution r (ϕ i ) thef (·) fitness function of ϕ i is as
f (ϕ i ) = α (ϕ i ) + ρ (ϕ i ) . (A.16)
Then let p refer to the number of selected ϕ i solutions in P. Using (A.16), the selection probability of each solution is yielded as
Pr (ϕ i ) =f (ϕ i ) r ∈Pf (ϕ r )
.
(A.17)
Constraints
As a solution ϕ i does not satisfy the problem constraints C 1 , C 2 , C 3 , a H C z (ϕ i ), z = 1, 2, 3 degrees of violation are defined for the constraints. For constraint C 1 (see (13)), the H C 1 (ϕ i ) violation function [77,78] is as
H C 1 (ϕ i ) = γ − ζ (ϕ i ) , if ζ (ϕ i ) ≤ γ 0, otherwise, (A.18) where ζ (ϕ i ) = N i=1 F i (ϕ i ) . (A.19)
For constraint C 2 (see (15)), the H C 2 (ϕ i ) violation function is as follows
H C 2 (ϕ i ) = F 1 (ϕ i ) − , if F 1 (ϕ i ) ≥ 0, otherwise, (A.20) where F 1 (ϕ i ) = N i=1 T i=1 f j B j F (ϕ i ) . (A.21)
For constraint C 3 (see (19)), the H C 3 (ϕ i ) violation function is as
H C 3 (ϕ i ) = ν (ϕ i ) − , if ν (ϕ i ) ≥ 0, otherwise, (A.22) where ν (ϕ i ) = N j=1 τ j (ϕ i ) .(ϕ i ) is defined as ∂ (ϕ i ) = w 1 H C 1 (ϕ i ) + w 2 H C 2 (ϕ i ) + w 3 H C 3 (ϕ i ) , (A.24)
where w i -s are weighting coefficients [77,78].
Selection Condition
Assuming that there are χ number of selected random solutions such that the selection probabilities are proportional to their fitness values. The selection of a solution ϕ i is as follows. First from the selected random solutions a mutant solution i is generated as
i = ϕ r a + ϑ ϕ r b − ϕ r c , (A.25)
where r i ∈ {a, . . . p} are the random indexes, while ϑ > 0 is a coefficient. From the components of i a trial solution T i is defined with a j th component T
( j) i as T ( j) i = ( j) i if r (0, 1) < P cr oss , or j = r (i) ϕ ( j) i , otherwise, (A.26)
where r (0, 1) is a random number from the range [0, 1], r (i) is a random integer within (0, X ] for each i, while P cr oss is the crossover probability ranged in (0, 1).
Then the selection of the solution ϕ i using the trial solution T i is as
ϕ i = T i , iff (T i ) ≤f (ϕ i ) ϕ i , otherwise, (A.27)
where functionf (·) is given in (A.16).
Sub-Procedure 1
The Sub-procedure 1 of Algorithm 1 is as follows [77,78].
Sub-procedure 1 Convergence of Solutions
Apply feasible space exploration (41) through the dimensions L dim k (E i ) r around dim k (E i ) of the epicenters. For i = 1, . . . , p obtain a T i trial solution (A.26) for ϕ i . Determine the best solution between ϕ i and T i via (A.27). Iff (T i ) ≤f (ϕ i ) and T i is a non-dominated solution, then update N P with T i . Then, update P with the best solution, and with other p − 1 randomly selected solutions, ϕ q , q = 1, . . . , p − 1, using the selection probability function (A.7) as
Pr ϕ q =f ϕ q p−1 i=1f (ϕ i ).
Notations
The notations of the manuscript are summarized in Table 1.
E (D i (X))
A secondary objective function. It refers to the expected amount of cumulative relative entropy of entanglement (a sum of relative entropy of entanglement) in node
x i , w j (x i )
Quantum memory coefficient for the storage of entangled states from the jth fidelity type in a node x i , evaluated as:
w j (x i ) = η j B j F (x i ) + κ j B j F (x i ),
where η j and κ j are coefficients to describe the storage characteristic of entangled states with the jth fidelity type τ j (X)
Differentiation of storage characteristic of entangled states from the jth fidelity type, defined as
τ j (X) = N i=1 w j (x i ) − 2 , where = N i=1 w j (x i ) N f C (P (x i )) Cost of entanglement purification P (x i ) in x i f C (C (x i )) Cost of optimal quantum error correction C (x i ) in x i C (X)
Total cost function, defined as
C (X) = N i=1 f C (P (x i )) + f C (C (x i )) = N i=1 T i=1 f j B j F (x i ) ,
where T is the number of fidelity types, N is the number of nodes, f j is a total cost of purification and error correction associated to the jth fidelity type of entangled states
(X) = λ T j=1 α i 1 ϒ i B j F (x i ),
where λ is a constant, α i is a quality coefficient, while ϒ i is a capacity coefficient of the quantum memory G (X)
Main objective function,
G (X) = max N i=1 F i (X) E (D i (X))P E i , L j = 1 d E i ,l j M E i , L j b 1 b 0 e σ ln P E i ,L j ,
where b 0 and b 1 are regression coefficients, σ ln P E j is the standard deviation,
while M E i , L j is the seismic magnitude in a location L j , while l j is the projection of L j onto the ellipsoid around E i M E i , L j Magnitude between epicenter E i and location L j is evaluated as
M E i , L j = ⎛ ⎝ P E i , L j 1 b 0 e σ ln P E i ,L j ⎞ ⎠ 1 b 1 d E i ,(E i , R k , R l )
Poisson range identifier function of E i , where R k and R l are random reference points
c w (E i , R k ), c w (R k , R l )
Weighting coefficients between epicenters E i and R k , and between R k and R l D E p Poissonian distance function D E p , where E p is a new solution Hypocentral, provides a random displacement of dim k (E i ) using C (E i )
L dim k (E i ) r A random location in the kth dimension L dim k (E i ) r around dim k (E i ) N (·) Normalization operator N (·) of L dim k (E i )
Fig. 1
1Illustration of the network model components. The quantum nodes x i and x j are associated with current input values B j F (x i ) and B l F x j (blue and green arrows), where j and l identify the fidelity types of received entangled states. The nodes have several entangled connections (depicted by gray lines) in the network. The nodes are associated with subject functions
Fig. 2
2Iteration step of the Poisson optimization model in the objective space S O . An ith epicenter, E i (depicted by the red dot), with a projected point l k of random reference location R k . Reference locations R k and R l (blue dots) identify locations L k and L l , respectively. The power in
Fig. 3
3Distribution of solutions for entanglement fidelity maximization in the objective space S O :F i 1 (N ) , F i 2 (N ) , F i (X) ofthe entanglement optimization problem (cost functions F 1 (N ) and F 2 (N ) and the objective function F i (X) are normalized onto the range of [0, 1]). a A random epicenter E i refers to a current solution (depicted by the red dot) with the Poisson distributed reference locations (the reference points are not real solutions). The random reference locations are clustered into two classes: (1) reference locations within radius r (E i ) around E i and (2) reference locations outside the radius (depicted by the gray dots). Reference locations outside the range are neglected in the iteration. b A new epicenter E p (depicted by the green dot) is determined via the connection model of relevant reference points (e.g., lie inside the range of r (E i )), which yields the D E p Poisson distance function. The new solution, E p , converges toward an optimal solution E * (depicted by the purple dot). The reference locations inside the relevance region are weighted by the seismic power function (Color figure online) E D N S low i (X) , and minimizes the cost functions F i 1 (N ) and F i 1
Fig. 4
4Solution set in S O , with an optimal epicenter E * ,
Fig. 5
5Illustration of the decision making. a The F entanglement fidelity values in function of F i 1 (N ) for N S low (red line) and N S high (blue line) nodes. The value of F i 2 (N ) is set to constant. b The D relative entropy of entanglement values in function of F i 1 (N ) for N S low (red line) and N S high (blue line) nodes. The value of F i 2 (N ) is set to constant (Color figure online)
Fig. 6 Fig. 7
67Distribution of B (λ i ) for different magnitudes M i , M i = 1, .. . , 10 and coefficient b, for a a = 10, and b a Distribution of λ (q) for k it iterations, k it = 0, . . . , 1000, for a λ (q) = 10 2 , and b λ (q) = 10 6
Fig. 8
8The distributions of B (λ i ) for k it iterations, k it = 0, . . . , 1000, a λ (q) = 10 2 , and b λ (q) = 10 6λ (q) → ∞, the distributions of λ (q) can be approximated by a N λ (q) , σ 2 N , σ 2 N = λ (q) Gaussian distribution.Theassociated distributions of B (λ i ) for the values of λ (q) are depicted in Fig. 8. The maximum value of B (λ i ) is selected to B (λ i ) ≈ 10 in each cases which values are picked up at λ (q), where λ (q) = 10 2 in Fig. 8a, and λ (q) = 10 6 in Fig. 8b. The B (λ i ) values approximates to a Gaussian distribution. The statistical distribution of B (λ i ) is therefore constitutes a similar pattern for arbitrary λ (q).
A.18), (A.20) and (A.22) a penalty coefficient ∂
FFF
Fidelity of entanglement N Number of nodes in the network T Number of fidelity types F j , j = 1, . . . , T of the entangled states S O Objective space S F Feasible space L l An l-level entangled connection. For an L l link, the hopdistance is 2 l−1 d (x, y) L l Hop-distance of an l-level entangled connection between nodes x and y E L l (x, y) entangled connection E L l (x, y) between nodes x and y B F E L l (x, y) Entanglement throughput of an L l -level entangled connection E L l (x, y) between nodes (x, (x i ) Number of incoming entangled states in an ith node x i , with fidelity-type j, i = 1, . . . , N X An N × T matrix, X = B j F (x i ) N ×T , it describes the number of resource entangled states injected into the nodes from each fidelity-type in the network, B j (x i ) ≥ 0 for all i and j F (x i ) A primary objective function. It identifies the cumulative entanglement fidelity (a sum of entanglement fidelities in x i ) after an entanglement purification P (x i ) and an optimal quantum error correction C (x i ) in x i P (x i ) Entanglement purification in x i C (x i ) Optimal quantum error correction in x i B j F (x i ) An initialization value for B j F (x i ) in a particular node x i
F 1 (
1N )Minimization function for cost C (X)F 2 (N )Minimization function for cost S (X)C 1 , C 2 , C 3 Problem constraints EEpicenter, represents a solution in the feasible spaceL j A random location around epicenter E Dispersion coefficient of an epicenter E (solution in the feasible space). It determines the number of affected L j , j = 1, . . . , D (E), locations (also represent solutions in the feasible space) around an epicenter E P Population P (a set of possible solutions) m Control parameter E i An ith individual (epicenter) from the |P| individuals (epicenters) in the population P f (·) Fitness functioñ f ( E ) A maximum objective value among the |P| individuals ϑA residual quantityf R (·) Rounding function q Total number of locations, q = |P| i=1 D (E i ) D (E i ) Upper bound on D (E i ) for a given epicenter E i d E i , l j Euclidean distance d E i ,l j between an ith epicenter E i and the projection point l j of a jth location point L j , j = 1, . . . , D (E) on the ellipsoid around E i dim i (·) An ith dimension of l j P E i , L j Seismic power P E i , L j operator for an ith epicenter E i . Measures the power in a jth location point L j , j = 1, . . . , D (E i ), as
.
It keeps the new locations around dim k (E i ) in S F , where B k low and B k up are lower and upper bounds on the boundaries of locations in a kth dimension S-metric Hypervolume indicator. A quality measure for the solutions or a contribution of a single solution in a solution set S (R) S-metric for a solution set R = {r 1 , . . . , r n } is asS (R) = L r ∈R x re f ∠x∠ x| r ,where L is a Lebesgue measure, notation b∠a refers to that a dominates b (or b is dominated by a), while x re f is a reference point dominated by all valid solutions in the solution setf 1 , f 2 Objective functions C 1 (x i )Cost results from the first-type classical communications related to a x iC 2 (x i )Cost results from the second-type classical communications with respect to x i E * Global optimam Number of magnitude ranges n i Number of locations belonging to an ith magnitude range B (n i ) Power law distribution function for a log-scaled n i , B (n i ) : log 10 (n i ) = ab tildeM i , whereM i is a log scaled M i , while a and b are constants n i Poisson estimate of n i , asn i = σ 2 i = λ i , where σ 2i is the observational variance, while λ i is the mean of a Poisson distributionσ 2 q Estimated uncertainty, σ 2 q = λ (q) = m i=1 f M i , where f (·) is a fitting function λ (q) Mean total number, λ (q) = m i=1 λ i ≈ q, where λ i is an ith component mean B (λ i )Power law distribution function for λ i =ñ i k it Number of iterations
The method fuses the results of quantum Shannon theory and theory of evolutionary multiobjective optimization algorithms. 4. The model maximizes the entanglement fidelity and relative entropy of entanglement for all entangled connections of the network. It minimizes the cost functions to reduce the costs of entanglement purification, error correction, and quantum memory usage. 5. The optimization framework allows a low-complexity implementation.1. A nature-inspired, multiobjective optimization framework is conceived for the
quantum Internet.
2. The model considers the physical attributes of entanglement transmission and
quantum memories to provide a realistic setting (realistic objective functions and
cost functions).
3.
Table 1
1Summary of notationsNotation
Description
l
Level of entanglement
Table 1 continued
1NotationDescription
f j
fTotal cost of purification and error correction associated to the jth fidelity type of entanglement fidelity F * Critical fidelity coefficient S low , S high Sets with fidelity bounds S low (F) and S high (F) as S low (F) : max X S low Set of nodes for which condition S low (F) : max ∀i F i < F * holds X S high Set of nodes for which condition S high (F) : min ∀i F i ≥ F * holdsS i (X)Cost of quantum memory usage in node x i , defined as S i∀i
F i < F * ,
and
S high (F) : min
∀i
F i ≥ F *
l j P * (E i ) Maximal seismic power for a given epicenter E i C (E i ) Cumulative magnitude for an epicenter E i E Highest seismic power epicenter with magnitude M E , L E E i j provides the maximal seismic power for an ith epicenter E ij
f E
Minimum objective values among the |P| epicenters
M
Control parameter,
M =
|P|
i=1 M E i , L
E i
j ,
where L
Table 1 continued
1NotationDescriptionr (E i ) Radius around a current solution E i , defined as r (E i ) = χ 10 Q 1 2M −Q 2 ,whereM is the average magnitudẽ M = 1 |P| M = 1 , while Q 1 and Q 2 are constants, while χ is a normalization termdim k (E i ) Randomly selected kth dimension, k = 1, . . . , dim (E i ) of a current epicenter E i , i = 1, . . . , |P| H (dim k (E i ))|P|
|P|
i=1 M E i , L
E i
j
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Acknowledgements Open access funding provided by Budapest University of Technology and Economics (BME). This work was partially supported by the European Research Council through the Advanced FellowAuthor contributionsLGY designed the protocol and wrote the manuscript. LGY and SI analyzed the results. All authors reviewed the manuscript.
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[] |
[
"Probability-driven scoring functions in combining linear classifiers",
"Probability-driven scoring functions in combining linear classifiers"
] |
[
"Pawel Trajdos \nDepartment of Systems and Computer Networks\nWroclaw University of Science and Technology\nWybrzeze Wyspianskiego 2750-370WroclawPoland\n",
"Robert Burduk \nDepartment of Systems and Computer Networks\nWroclaw University of Science and Technology\nWybrzeze Wyspianskiego 2750-370WroclawPoland\n"
] |
[
"Department of Systems and Computer Networks\nWroclaw University of Science and Technology\nWybrzeze Wyspianskiego 2750-370WroclawPoland",
"Department of Systems and Computer Networks\nWroclaw University of Science and Technology\nWybrzeze Wyspianskiego 2750-370WroclawPoland"
] |
[] |
Although linear classifiers are one of the oldest methods in machine learning, they are still very popular in the machine learning community. This is due to their low computational complexity and robustness to overfitting. Consequently, linear classifiers are often used as base classifiers of multiple ensemble classification systems. This research is aimed at building a new fusion method dedicated to the ensemble of linear classifiers. The fusion scheme uses both measurement space and geometrical space. Namely, we proposed a probability-driven scoring function which shape depends on the orientation of the decision hyperplanes generated by the base classifiers. The proposed fusion method is compared with the reference method using multiple benchmark datasets taken from the KEEL repository. The comparison is done using multiple quality criteria. The statistical analysis of the obtained results is also performed. The experimental study shows that, under certain conditions, some improvement may be obtained.
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10.3897/jucs.80747
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[
"https://arxiv.org/pdf/2109.07815v1.pdf"
] | 237,532,200 |
2109.07815
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aa1fbf6bbbd56fff1ad0fff1f2deb27f778b4367
|
Probability-driven scoring functions in combining linear classifiers
Pawel Trajdos
Department of Systems and Computer Networks
Wroclaw University of Science and Technology
Wybrzeze Wyspianskiego 2750-370WroclawPoland
Robert Burduk
Department of Systems and Computer Networks
Wroclaw University of Science and Technology
Wybrzeze Wyspianskiego 2750-370WroclawPoland
Probability-driven scoring functions in combining linear classifiers
Linear ClassifierPotential FunctionEnsemble of ClassifiersScore Function
Although linear classifiers are one of the oldest methods in machine learning, they are still very popular in the machine learning community. This is due to their low computational complexity and robustness to overfitting. Consequently, linear classifiers are often used as base classifiers of multiple ensemble classification systems. This research is aimed at building a new fusion method dedicated to the ensemble of linear classifiers. The fusion scheme uses both measurement space and geometrical space. Namely, we proposed a probability-driven scoring function which shape depends on the orientation of the decision hyperplanes generated by the base classifiers. The proposed fusion method is compared with the reference method using multiple benchmark datasets taken from the KEEL repository. The comparison is done using multiple quality criteria. The statistical analysis of the obtained results is also performed. The experimental study shows that, under certain conditions, some improvement may be obtained.
Introduction
The concept of a linear classifier is one of the oldest machine learning methods. They are dated back to the 1950s [1]. For decades many algorithms for building linear classifiers have been developed [2,3,4,5,6,7]. Today they are still used by the machine learning society [8]. This is due to the relatively low computational complexity of training and predicting phases [9]. What is more, after a suitable data preprocessing, linear classifiers may offer classification quality comparable to nonlinear ones [10]. Additionally, for some linear classifiers, it is possible to obtain nonlinear decision boundaries using the kernel trick [7]. Nonlinear decision boundaries may also be obtained using ensembles of classifiers [8,11].
is, the greater is belief that the sample belongs to the given class. The class support values are usually normalized within the [0; 1] interval and sum up to one [34].
• Geometric space. The geometric combiners use properties of the geometrical space in which the decision boundary is placed [33]. For example, the method proposed in [33] uses characteristic boundary points and weight estimation to provide a piecewise linear classifier. On the other hand, the methods proposed in [36,37] constructs a multiconlitron that separates the classes.
In this paper, we extended the probability-based scoring function proposed in [38]. The scoring function combines measurement-level fusion with geometric space fusion. That is, it uses geometric properties of the space combined with the probabilistic framework. We have proposed three extended scoring functions. The scoring functions harness additional probabilistic information about the data distribution in the input (geometric) space. Namely, they utilize information about point distribution along the decision plane. The function proposed in [38] uses only information about the point spread along the normal vector of the decision plane.
The main objectives of this work can be summarized as follows:
• A proposal of new scoring functions that better utilize the probabilistic information available.
• Harnessing the proposed scoring functions in the task of building a homogeneous ensemble of classifiers.
• An experimental setup to compare the proposed scoring functions with the reference methods. The comparison is done in terms of the selected quality criteria. During the experiments, different base classifiers are used.
The outline of the paper is as follows: In the next section (Section 2), related works are outlined. The proposed methods are presented in Section 3. In Section 4, the research questions are formulated, and the experimental setup is described. The experimental results are presented and discussed in Section 5. Finally, the paper is concluded in section 6.
Related Work
This section describes the previous work related to the problem of building ensembles of linear classifiers. We begin with the definition of a linear classifier and then switch to the topic of building ensembles of such classifiers.
Linear Classifier
Let us begin with the definition of a linear classifier. The linear classifier assigns points taken from the feature space X, which in this work is assumed to be a d − dimensional Euclidean space X = R d , to two possible classes M = {−1; 1} [39]. To separate the two classes of feature space points, the classifier utilizes a hyperplane π defined by the following equation:
π : n; x + b = 0,(1)
where n is a unit normal vector of the decision hyperplane, b is the distance from the hyperplane to the origin and ·; · is a dot product defined as follows [40]:
a; b = d i=1 a i b i ∀a, b ∈ X.(2)
The norm of the vector x is defined using the dot product:
x = x; x .(3)
For each instance x, the linear classifier ψ produces the discriminant function [34]:
ω(x) = n; x + b,(4)
which absolute value equals the perpendicular distance from the decision hyperplane π to the point x. The sign of the value returned by the discriminant function depends on the site of the plane where the instance x lies. The decision of the linear classifier is thus determined by checking the sign of the discriminant function:
ψ(x) = sign (ω(x)) .
During the training phase of the linear classifier, the proper decision plane is found using the training set T . which consists of |T | (where |·| is the cardinality of a set) pairs of feature space vectors x and their corresponding class labels m:
T = (x (1) , m (1) ), (x (2) , m (2) ), . . . , (x (|T |) , m (|T |) ) ,(6)
where x (k) ∈ X and m (k) ∈ M. The literature contains various procedures for obtaining the decision plane [10]. Among the others, we may mention such algorithms as: FLDA [2], Logistic Regression [3], Perceptron [4], Nearest Centroid Classifier [5], and SVM [6]. The procedures used in the experiments are listed in Section 4.1.
Despite their simplicity, linear classifiers are often used to solve practical classification tasks [10]. First of all, they are useful due to their low computational complexity. Additionally, due to their simplicity, they are also less overfitting prone [9]. What is more, they can obtain a classification quality comparable to nonlinear classifiers when the dimensionality of the input space is high [10]. However, for some classification problems with nonlinear classification boundaries they are insufficient [7]. One solution may be to tailor linear classifiers to find non-linear decision boundary. This may be done by applying the kernel trick [10,7]. The other way is to build a structure that consists of multiple linear classifiers that are trained together. An example of such a technique is to build a multilayer neural network [11], a deep neural network in particular [41]. Another way is to use the multi-classifier approach and build an ensemble of linear classifiers [8].
An Ensemble of Linear Classifiers
Generally speaking, an ensemble of classifiers (a multiclassifier) is a set of classifiers that work together to deliver more robust results [34]. Throughout this paper, the ensemble of classifiers is denoted by:
Ψ = ψ (1) , ψ (2) , · · · , ψ (N )(7)
In this paper, we are focused on the ensemble combination methods dedicated to linear classifiers. The proposed weighting methods are trainable and dynamic ones that combine the base classifiers in the geometric space. That is, the weights depend on the orientation of the decision plane.
In the literature, we may find multiple methods of combining linear classifiers. Now, we list these methods starting from the simplest one. The most straightforward way to combine the results of several classifiers is to use model averaging [42]. The model averaging approach is to simply calculate the mean value of the classifier-specific discriminant functions:
ω(x) = 1 N N i=1 ω (i) (x),(8)
where ω (i) (x) is the value of the discriminant function provided by the classifier ψ (i) for the point x. As we said before, the value taken by the discriminant function of the linear classifier is proportional to the distance from the given point x to the decision plane. In general, this distance is unbounded, which poses a major disadvantage of the model averaging approach. That is, when one of the base classifiers has produced a misplaced decision boundary, the high value of the discriminant function coming from this boundary may significantly change the response of the entire ensemble. This issue may be easily addressed by ignoring the exact value of the discriminant function and taking only the sign of the value. This approach is called majority voting, and the response of the ensemble is given by following formula [43]:
ω(x) = N i=1 sign ω (i) (x) ,(9)
Although this approach is robust to misplaced decision boundaries, it loses the information related to the exact value of the discriminant function.
The aforementioned disadvantage can be partly eliminated by the application of a type of a sigmoid transformation [34]. The sigmoid function, also called S-shaped function, is an increasing one that has finite upper and lower bounds. An example of such a function is the softmax function:
ω (i) (x) = 1 + exp −ω (i) (x) −1 .(10)
Applying this kind of transformation ensures that distance-specific information is not lost, and it also reduces the impact of misplaced hyperplanes. Employing a simple sigmoid function is a simplified version of the probability calibration task.
In this task, we want to provide an estimation of the class posterior probability distribution [44]. This distribution may be obtained using various techniques such as Plat scaling (logistic calibration) [45,46], sigmoid fitting [47], or beta scaling [44] To produce the final outcome of the ensemble, the transformed values are averaged:
ω(x) = 1 N N i=1 ω (i) (x).(11)
As it was said before, one of the problems with combining the linear classifiers is that the discriminative function grows monotonically with the distance to the decision plane. This poses no problem when a single classifier is queried. However, it may cause a situation where the absolute value of the discriminant function is high, but the training set of a classifier contains no instances so far from the decision boundary. In other words, the classifier returns a high value of the discriminant function outside its region of competence, which may distort the final prediction of the ensemble. The application of a sigmoid function or more generally a sort of posterior probability scaling mitigates the problem, but does not resolve it. The reason is that at a great distance from the decision boundary, the calibrated discriminant function approaches its upper (lower) limit. The values close to the limits still express relatively high class-specific support outside the competence region of the base classifier. Our previous research has shown that reducing the value of the discriminant function outside the competence region of the classifier may significantly improve the classification quality achieved by the ensemble [48,38]. Our first attempt was to provide a simple non-monotonic parametric function [48]:
g ω(x) = ω(x) exp −ζ ω(x) 2 + 0.5 2ζ,(12)
where ζ is a coefficient that controls the position and steepness of peaks. Unfortunately, we have not proposed a closed-form formula for finding the good value of this coefficient. Consequently, the proper value of this coefficient must be found using cross-validation. The translation constant 0.5 and the scaling factor √ 2ζ assure that the maximum and positive and negative peaks of the discriminant functions are 1 and −1 respectively. The final value of the discriminant function of the ensemble is calculated by averaging the transformed values given by the base classifiers:
ω(x) = 1 N N i=1 g ω (i) (x) .(13)
The conducted experimental evaluation showed that applying this kind of nonmonotonic transformation causes a gain in the classification quality obtained by heterogeneous ensembles. Unfortunately, the practical applications of this method are limited since it is very sensitive to imbalanced class distribution. What is more the ζ coefficient has to be tuned for each dataset separately.
To eliminate these drawbacks, we proposed an approach that models the data spread along the plane vector using kernel probability estimators [38]. The conducted experimental evaluation showed that the previously proposed method offers some improvement over the formerly proposed and reference methods. The discriminant function created by a linear classifier uses only the information about the distance between an object and the decision hyperplane. However, the information about the data distribution along the basis of the decision hyperplane may also be useful when determining the competence region of the base classifier. The basis of the decision hyperplane is a set of linearly independent vectors that span the plane [40]:
B = {b 1 , b 2 , · · · , b d−1 } .(14)
The plane base for the two-dimensional classification problem is shown in Figure 1.
With that in mind, we proposed a method that incorporates this information into the ensemble classifier.
Proposed Improvements of the Probability-based Potential function
Let us begin with a more detailed description of the method proposed in [49]. This description is needed since the methods proposed in this paper are extensions of the above-mentioned methods.
Potential Functions
The potential function defined in [49] is defined using a probabilistic framework. It means that x and m are realizations of random variables X and M, respectively. The joint distribution P (X, M) is also known. Then, the value of the discriminant function ω(x) is also a realisation of a random variable defined as follows:
W = n; X + b.(15)
This one-dimensional distribution describes the data spread along the line defined by the normal vector n of the decision plane. This random variable is also jointly distributed with M: P (W, M). We denote its probability density function of this random variable by w(ω).
Under these assumptions, we may define the conditional probability of class m = 1 given ω(x):
P M = 1|ω(x) = w ω(x)|M = 1 P (M = 1) m∈M w ω(x)|M = m P (M = m) .(16)
The potential function is then defined to be proportional to the probability:
β ω(x) = exp w ω(x)|M = 1 P (M = 1) m∈M exp w ω(x)|M = m P (M = m) − 0.5.(17)
Using the softmax transformation allows avoiding numerical problems in areas with low point density. Subtracting 0.5 from the expression puts the result into
[−0.5; 0.5] interval.
In this work, we employed probability density estimations that describe the point distribution along the vectors of the plane basis B. To do so, we define a new multidimensional random variable which elements are defined as the projection coefficients of the random variable X onto the base vectors of the decision plane:
Y i = X; b i b i .(18)
Consequently, the random variable describes the distribution of points projected onto the basis of the decision plane. The probability density function of this random variable is denoted by y(x). Our first approach is to define the modified potential function using y(x) solely. This potential function uses the information about the point distribution regardless the class assigned to each of the points. We assumed that Y is normally distributed with the expected value µ and the covariance matrix Σ: Y ∼ N (µ, Σ). This assumption is made because the normal distribution is a unimodal one. Consequently, the max value of the probability density function can be easily determined. The potential function is then defined as:
(x) = 1 z(x) P M = 1|ω(x) exp (y(x)) exp (y(x))+exp (y(µ)) − 0.5,(19)
where z(x) is a normalization factor that guarantees these potentials comming from P M = 1|ω(x) and P M = −1|ω(x) sum up to zero. Exponent in the equation (19) is a softmax between the highest pdf value of the distribution y(µ) and y(x). For low values of y(x) the value of the potential function tends to 0. Consequently, in the areas where the concentration of samples is low, the discriminant function of the base classifier is close to zero. In other words, in those areas, the classifier cannot definitely say which class to choose. An example plot of the potential function is shown in Figure 2 .
Another strategy is to use conditional probabilities y(x|M = m) and use them to calculate P M = 1|ω(x) . To employ these probabilities, we made a naive assumption that y(x|M = m) and w ω(x)|M = m are conditionally independent given M = m. The same assumption is done in the Naive Bayes classifier [50]. Taking this into account, the conditional class probability is calculated using the following formula:
P M = 1|ω(x) = w ω(x)|M = 1 y x|M = 1 P (M = 1) m∈M w ω(x)|M = m y x|M = m P (M = m)
. (20) Consequently, the potential is calculated as follows:
2 (x) = exp w ω(x)|M = 1 y x|M = 1 P (M = 1) m∈M exp w ω(x)|M = m y x|M = m P (M = m) − 0.5. (21)
Probability Estimation
In the previous section, a set of potential functions has been presented. For readability purposes, all necessary probability distributions were assumed to be known. Unfortunately, in real-world classification problems, these distributions remain unknown, and they have to be estimated using the training data. This section describes the techniques used to estimate the probabilities needed by the potential function.
Prior class probabilities P (M = m) are estimated using the following formula:P
(M = m) = |T (m) | |T | ,(22)
where T (m) is a subset of the training set containing objects which belong to class m:
T (m) = (x (k) , m (k) )|m (k) = m .(23)
To calculate the potential function (x), we need to estimate the probability distribution y(x). Random variable Y has also been assumed to follow the multivariate normal distribution. Consequently, we use the maximum likelihood estimator to find the parameters of the distribution [51].
For the conditional distribution y(x|M = m), we considered two estimation procedures to be compared during the experimental study:
• We assumed that the underlying random variable follows the multivariate Gaussian distribution. To estimate the conditional probability density function, we used the maximum likelihood estimator [51].
• To make no assumptions about the shape of the distribution, we employed a nonparametric kernel estimator. To avoid using multidimensional kernels, we used the Naive Bayes assumption about the variables [52,53]. In our work, we also decided to select the bandwidth using Silverman's rule of thumb [54]
Toy Examples
In this section, the process of potential function building is visualized using a simple two-dimensional data set shown in Figure 3. The decision boundary, shown in Figure 3, is generated using the Nearest Centroid classifier [5]. After obtaining the decision boundary, the conditional probability density functions w ω(x)|M = m are estimated. The result is shown in Figure 4. Now, the process of calculating the potential function (x) is shown. First, the probability density function y(x) is estimated. The result is shown in Figure 5. Then the potential function (x) is calculated. It is visualised Figure 6. Now, the process of calculating the potential function 2 (x) is shown. First the conditional probability density functions f (X|M = m) are estimated. The reault is shown in Figure 7. The potential function 2 (x) is then calculated according to (12). It is visualised in Figure 8.
Experimental Evaluation
The main goal of the experimental evaluation is to answer the following research questions: RQ1: Does the utilization of the information about the point distribution along the basis of the decision plane significantly impact the classification quality achieved by the ensemble?
RQ2: Do the new formulated potential functions allow to improve the classification quality achieved by the ensemble?
RQ3: How is the ensemble utilizing the newly proposed potential functions doing compared to the ensemble created using Naive Bayes classifier which uses the class conditional probability calculated in a similar way. Table 1 displays the collection of the 70 benchmark sets that were used during the experimental evaluation of the proposed methods. The table is divided into two sections. Each section is organized as follows. The first column contains the names of the datasets. The remaining ones contain the set-specific characteristics of the benchmark sets: the number of instances in the dataset |S|; dimensionality of the input space d; the number of classes C and the average imbalance ratio IR, respectively.
Setup
The datasets were taken from the Keel 1 repository. The datasets are also available in our repository 2 During the dataset preprocessing stage, a few transformations on the datasets were applied. The PCA method [55] was applied and the percentage of covered variance was set to 0.95. The attributes were also normalized to have zero mean and unit variance.
In the experimental study we conducted, the proposed potential functions were used to combine the predictions produced by a homogeneous ensemble of classifiers. The homogeneous ensembles were created using a bagging approach [42]. The generated ensembles consist of 11 classifiers learned by using the bagging method. Each bagging sample contains 80% of the number of instances from the original dataset.
For each of the kernel estimators used, the kernel bandwidth was selected using the Silverman rule [54]. The Gaussian kernel is used.
During the experiment, the following ensembles were considered:
• ψ NB -The ensemble created using Naive Bayes classifier [50].
• ψ KE -The ensemble in which base classifiers are combined according to approach proposed in [38]. See also equation (17). • ψ KA -The ensemble in which base classifiers are combined using the potential function defined in (19).
• ψ KB -The ensemble in which base classifiers are combined using the potential function defined in (21). The parametric gaussian estimator is used.
• ψ KC -The ensemble in which base classifiers are combined using the potential function defined in (21). The kernel estimator is used.
The following base classifiers were used to build the above-mentioned ensembles (Except for Naive Bayes ensemble):
• ψ FLDA -Fisher LDA [2],
• ψ LR -Logistic regression classifier [3],
• ψ MLP -single layer MLP classifier [4],
• ψ NC -nearest centroid (Nearest Prototype) [5] with the class-specific Euclidean distance,
• ψ SVM -SVM classifier with linear kernel (no kernel) [6].
The classifiers used were implemented in the WEKA framework [56]. If not stated otherwise, the classifier parameters were set to their defaults. The multiclass problems were dealt with using One-vs-One decomposition [57]. The source code of the proposed algorithms is available online 3 To evaluate the proposed methods, six classification quality criteria are used:
• Macro-averaged: Macro and micro-averaged measures were used to assess the performance for the majority and minority classes. This is because the macro-averaged measures are more sensitive to the performance for minority classes [58]. The criteria are bounded in the interval [0, 1], where zero denotes the best classification quality.
To maintain consistency, the results obtained using the MCC criterion are also transformed to fit the above-mentioned properties. The experimental procedure was conducted using the ten-fold cross-validation procedure. The data folds were generated using methods implemented in WEKA software. The random seed used to generate them is zero.
Following the recommendation of [59] the statistical significance of the obtained results was assessed using the two-step procedure. The first step was to perform the Iman-Davenport test [59] for each quality criterion separately. Since multiple criteria were employed, the family-wise errors (FWER) should be controlled [60]. To do so, the Bergmann-Hommel [60] procedure of controlling FWER of the conducted Iman-Davenport tests was employed. When the Iman-Davenport test shows that there is a significant difference within the group of classifiers, the Bergmann-Hommel post hoc test is applied [59,60]. For all tests, the significance level was set to α = 0.05.
Results and Discussion
To compare multiple algorithms on multiple benchmark sets, the average rank approach is used. In this approach, the winning algorithm achieves a rank equal to '1', the second achieves a rank equal to '2', and so on. In the case of ties, the ranks of algorithms that achieve the same results are averaged. To provide a visualization of the average ranks, radar plots are employed. In the radar plot, each of the radially arranged axes represents one quality criterion. In the plots, the data is visualized in such a way that the lowest ranks are closer to the centre of the graph. Consequently, higher ranks are placed near the outer ring of the graph. Graphs are also scaled so that the inner ring represents the lowest rank recorded for the analyzed set of classifiers, and the outer ring is equal to the highest recorded rank. The radar plots are presented in Figures 9 -13.
The numerical results are given in Table 2 to 6. Each table is structured as follows. The first row contains the names of the investigated algorithms. Then, the table is divided into six sections -one section is related to a single evaluation criterion. The first row of each section is the name of the quality criterion investigated in the section. The second row shows the p-value of the Iman-Davenport test. The third one shows the average ranks achieved by algorithms. The following rows show p-values resulting from the post hoc test. The p-value equal to .000 informs that the p-values are lower than 10 −3 and p-value equal to 1.00 informs that the value is higher than 0.999. P-values lower (or equal) than α are bolded. Consequently, the bolded results show that there is a significant difference between classifiers.
Let us begin with the analysis of differences between ψ KE and its modifications that allow us to incorporate the information about points spread along the decision plane basis (ψ KA , ψ KB , and ψ KC respectively). The conducted statistical analysis reports only a few significant differences between these methods.
Most of the differences are observed for the macro-averaged FNR criterion. In terms of this criterion ψ KB and ψ KC tend to be better than ψ KE . For this criterion, the average ranks achieved by the proposed methods tend to be lower than the ranks achieved by ψ KE . For the macro-averaged FDR measure, almost no significant differences have been reported. For this criterion, only one significant difference is reported for ψ NC base classifier. What is more, the order (according to the average ranks) of classifiers depends on the base classifier used. It means that for the minority classes, the modified methods (ψ KB and ψ KC ) improve the recall without harming the precision. Unfortunately, the overall classification quality expressed in terms of the macro-averaged MCC criterion has not been significantly improved. However, we may observe that the averaged ranks for the proposed methods tend to be lower for this criterion. This is especially for ψ KB and ψ KC . This trend is observed for all base classifiers. It means that including the information about the point spread along the hyperplane basis allows obtaining some improvement. Utilizing the class-specific densities y(x|M = m) causes higher differences in average ranks. It means that using class-specific densities gives better results than using a global density P (X = x).
For the micro-averaged criteria, significant differences are observed only for a sole base classifier. The difference is observed for the ψ MLP base classifier and all micro-averaged criteria. However, in this case, the result is not so strong because the p-value resulted from the Iman Davenport test is above the significance level α. Apart from that, the results of the post-hoc test show that the ψ KE classifier is significantly better than ψ KB classifier. What is more, the ψ KB classifier tends to achieve higher ranks than ψ KE for all micro averaged criteria and base classifiers. It means that ψ KB classifier may be weaker when classifying the examples from the majority classes. For ψ KA and ψ KC , on the other hand, the average ranks for micro-averaged criteria are lower than the ranks calculated for ψ KE . This result shows that for the proposed method, the choice of the probability estimation method is fairly important. The nonparametric estimator seems to be a better choice than a parametric one related to the arbitrary chosen distribution (The Gaussian one in this study). This is likely due to the ability of the kernel estimator to provide a better estimation of the multimodal probability density.
Finally, let us compare ψ KA , ψ KB and ψ KC classifies and the ensemble built using the Naive Bayes algorithm (ψ NB ). This comparison needs to be made since the algorithms use a similar approach to estimating the multidimensional probability distribution as the Naive Bayes algorithm does.
First of all, for the macro-averaged FNR and FDR measures, the ψ NB classifier significantly outperforms the remaining classifiers for three out of five base classifiers. What is more, the ψ NB is also better in terms of the macro-averaged MCC classifier for the nearest centroid base classifier. This is also true for the ψ KC classifier that uses almost the same procedure for estimation probability. The probable reasons for these differences are twofold. The first reason is that the ψ KC classifier uses one-vs-one decomposition to deal with multiclass problems whereas ψ NB can handle multiclass problems directly. The second reason is that ψ NB estimates the probabilities using original attributes which are, due to applied PCA transformation, uncorrelated. ψ KC on the other hand uses an input space spanned by the normal vector and decision hyperplane basis. The literature shows that applying ψ NB on uncorrelated attributes gives better results [61].
For the micro-averaged measures, on the other hand, no significant difference is observed. It means that, when dealing with the majority classes, the investigated classifiers offer comparable classification quality.
Conclusions
In this paper, we proposed a few modifications of the algorithm proposed in [38]. The modifications allow the aforementioned algorithm to utilize the information about the point spread along the decision plane basis. This information should allow the created ensemble to better view the competence regions of the employed base classifiers. Identifying competence regions should allow the ensemble to achieve better classification quality than the ensemble that does not use this information.
We conducted a set of experiments using different base classifiers and a set of different quality measures to answer the formulated research questions. The experiments were conducted using 70 publicly available benchmark sets.
The experimental study carried out allowed us to provide the following answers to the research questions raised. RQ1: The utilization of the information about the point distribution along the basis of the decision plane has some impact on the classification quality obtained by the ensemble.
RQ2: The utilization of the new formulated potential functions improves the ensemble's classification quality only for the macro-averaged FNR criterion. Figure 13: The radar plot for the ensembles based on ψ SVM RQ3: The ensemble using the proposed new potential functions is comparable to the ensemble constructed using the Naive Bayes classifier in terms of four out of six criteria. For the remaining criteria, they are worse.
The proposed modifications do not significantly outperform the initial approach. Consequently, our future research should be aimed at other techniques of improving the ensembles of linear classifiers. For example, a different ensemble building technique may be proposed.
Figure 1 :
1The linear decision boundary for binary, two-dimensional data. Objects belonging to the first class have been marked using red circles, whereas points belonging to the other class have been marked using green triangles. The plot also shows the normal vector of the decision plane n and a base vector of the plane b 1 .
Figure 2 :
2The plot of potential functions for different values of P M = 1|ω(x) . The y(µ) value is set to 1.0.
Figure 3 :
3The linear decision boundary created by the Nearest Centroid classifier for binary, two-dimensional, banana-shaped data. Objects belonging to the first class have been marked using red circles, whereas points belonging to the other class have been marked using green triangles. The plot also shows the decision plane generated by the Nearest Centroid classifier.
Figure 4 :Figure 5 :
45Class conditioned probability density functions estimated using the kernel estimator The probability density function f (X) estimated using the maximum likelihood estimator. The histogram of values is also shown.
Figure 6 :
6The visualisation of the potential function (X).
Figure 7 :
7Probability density functions f (X|M = m) estimated using the kernel estimator.
Figure 8 :
8The visualisation of the potential function 2 (X).
-
false discovery rate (1 − precision, FDR); -false negative rate (1 − recall, FNR); -Matthews correlation coefficient(MCC) • Micro-averaged: -false discovery rate (1 − precision, FDR); -false negative rate (1 − recall, FNR); -Matthews correlation coefficient(MCC)
Figure 9 :
9The radar plot for the ensembles based on ψ FLDA
Figure 10 :Figure 11 :
1011The radar plot for the ensembles based on ψ LR The radar plot for the ensembles based on ψ MLP
Figure 12 :
12The radar plot for the ensembles based on ψ NC
Table 1 :
1The characteristics of the benchmark setsName
|S|
d C
IR
Name
|S|
d C
IR
abalone
4174 10 28 162.59
mammographic
830
5 2
1.03
adult
45222 103 2 2.02
marketing
6876 13 9
1.80
appendicitis
106
7 2 2.52
monk-2
432
6 2
1.06
australian
690
18 2 1.12
movement_libras 360 90 15 1.00
automobile
159
61 6 4.30
mushroom
5644 92 2
1.31
balance
625
4 3 2.63
newthyroid
215
5 3
3.43
banana
5300
2 2 1.12
nursery
12960 26 5 435.25
bands
365
19 2 1.35
optdigits
5620 64 10 1.02
breast
277
38 2 1.71
page-blocks
5472 10 5 58.12
bupa
345
6 2 1.19
penbased
10992 16 10 1.04
car
1728 21 4 10.08
phoneme
5404 5 2
1.70
chess
3196 38 2 1.05
pima
768
8 2
1.43
cleveland
297
13 5 5.08
post-operative
87
21 3 21.86
coil2000
9822 85 2 8.38
ring
7400 20 2
1.01
connect-4
67557 126 3 3.52
saheart
462
9 2
1.44
contraceptive
1473
9 3 1.37
satimage
6435 36 6
1.66
crx
653
42 2 1.10
segment
2310 19 7
1.00
dermatology
358
34 6 2.43
shuttle
57999 9 7 1326.03
ecoli
336
7 8 23.56
sonar
208 60 2
1.07
fars
100968 362 8 610.12
spambase
4597 57 2
1.27
flare
1066 37 6 2.90
spectfheart
267 44 2
2.43
german
1000 59 2 1.67
splice
3190 287 3
1.77
glass
214
9 6 3.91
tae
151
5 3
1.03
haberman
306
3 2 1.89
texture
5500 40 11 1.00
hayes-roth
160
4 3 1.37
thyroid
7200 21 3 19.76
heart
270
13 2 1.13
tic-tac-toe
958 27 2
1.44
hepatitis
80
19 2 3.08
titanic
2201 3 2
1.55
housevotes
232
16 2 1.07
twonorm
7400 20 2
1.00
ionosphere
351
33 2 1.39
vehicle
846 18 4
1.03
iris
150
4 3 1.00
vowel
990 13 11 1.00
kr-vs-k
28056 40 18 20.96
wdbc
569 30 2
1.34
led7digit
500
7 10 1.16
wine
178 13 3
1.23
letter
20000 16 26 1.06
wisconsin
683
9 2
1.43
lymphography 148
38 4 15.77
yeast
1484 8 10 17.08
magic
19020 10 2 1.42
zoo
101 21 7
4.84
Table 2 :
2Statistical evaluation: the post-hoc test for the ensembles based on the FLDA classifier.ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KCNam.
MaFDR
MaFNR
MaMCC
ImD.
1.000e+00
1.253e-03
4.155e-01
Rank 2.929 3.236 3.029 3.007 2.800 2.957 3.636 3.200 2.593 2.614 2.971 3.336 3.157 2.921 2.614
ψ NB
1.00 1.00 1.00 1.00
.044 .727 .519 .519
.727 1.00 1.00 .727
ψ KE
1.00 1.00 1.00
.412 .001 .001
1.00 .727 .069
ψ KA
1.00 1.00
.139 .139
1.00 .253
ψ KB
1.00
.936
.727
Nam.
MiFDR
MiFNR
MiMCC
ImD.
1.000e+00
1.000e+00
1.000e+00
Rank 3.136 2.986 2.836 3.207 2.836 3.136 2.986 2.836 3.207 2.836 3.136 2.986 2.836 3.207 2.836
ψ NB
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
ψ KE
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
ψ KA
1.00 1.00
1.00 1.00
1.00 1.00
ψ KB
1.00
1.00
1.00
Table 3 :
3Statistical evaluation: the post-hoc test for the ensembles based on the LR classifier.ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KCNam.
MaFDR
MaFNR
MaMCC
ImD.
1.000e+00
4.966e-03
1.000e+00
Rank 2.864 3.193 3.236 2.900 2.807 2.950 3.557 3.236 2.543 2.714 2.993 3.279 3.179 2.800 2.750
ψ NB
1.00 1.00 1.00 1.00
.092 .570 .511 .570
1.00 1.00 1.00 1.00
ψ KE
1.00 1.00 1.00
.511 .001 .010
1.00 .480 .480
ψ KA
1.00 1.00
.057 .153
.653 .653
ψ KB
1.00
.570
1.00
Nam.
MiFDR
MiFNR
MiMCC
ImD.
1.000e+00
1.000e+00
1.000e+00
Rank 3.164 2.936 2.879 3.129 2.893 3.164 2.936 2.879 3.129 2.893 3.164 2.936 2.879 3.129 2.893
ψ NB
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
ψ KE
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
ψ KA
1.00 1.00
1.00 1.00
1.00 1.00
ψ KB
1.00
1.00
1.00
Table 4 :
4Statistical evaluation: the post-hoc test for the ensembles based on the MLP classifier.ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KCNam.
MaFDR
MaFNR
MaMCC
ImD.
1.000e+00
4.497e-01
1.000e+00
Rank 2.993 3.093 2.921 3.057 2.936 3.086 3.279 3.150 2.764 2.721 3.150 3.007 2.921 3.100 2.821
ψ NB
1.00 1.00 1.00 1.00
1.00 1.00 .691 .691
1.00 1.00 1.00 1.00
ψ KE
1.00 1.00 1.00
1.00 .371 .371
1.00 1.00 1.00
ψ KA
1.00 1.00
.653 .653
1.00 1.00
ψ KB
1.00
1.00
1.00
Nam.
MiFDR
MiFNR
MiMCC
ImD.
8.367e-02
8.367e-02
8.367e-02
Rank 3.279 2.657 2.743 3.414 2.907 3.279 2.657 2.743 3.414 2.907 3.279 2.664 2.736 3.414 2.907
ψ NB
.120 .135 1.00 .329
.120 .135 1.00 .329
.129 .129 1.00 .329
ψ KE
1.00 .046 1.00
1.00 .046 1.00
1.00 .050 1.00
ψ KA
.072 1.00
.072 1.00
.067 1.00
ψ KB
.231
.231
.231
Table 5 :
5Statistical evaluation: the post-hoc test for the ensembles based on the NC classifier.NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KCRank 2.214 3.643 3.379 2.879 2.886 2.236 3.829 3.493 2.607 2.836 2.214 3.514 3.407 2.979 2.886 Rank 2.379 3.157 3.150 3.350 2.964 2.379 3.157 3.150 3.350 2.964 2.379 3.157 3.150 3.350 2.964ψ Nam.
MaFDR
MaFNR
MaMCC
ImD.
1.866e-06
1.084e-09
1.184e-05
ψ NB
.000 .000 .048 .048
.000 .000 .329 .099
.000 .000 .017 .048
ψ KE
.645 .025 .025
.418 .000 .001
1.00 .135 .112
ψ KA
.184 .184
.003 .028
.153 .153
ψ KB
.979
.418
1.00
Nam.
MiFDR
MiFNR
MiMCC
ImD.
9.557e-03
9.557e-03
9.557e-03
ψ NB
.021 .021 .003 .114
.021 .021 .003 .114
.021 .021 .003 .114
ψ KE
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
ψ KA
1.00 1.00
1.00 1.00
1.00 1.00
ψ KB
.894
.894
.894
Table 6 :
6Statistical evaluation: the post-hoc test for the ensembles based on the SVM classifier.ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KC ψ NB ψ KE ψ KA ψ KB ψ KCRank 2.414 3.314 3.193 3.321 2.757 2.529 3.536 3.171 2.864 2.900 2.557 3.329 3.150 3.064 2.900Nam.
MaFDR
MaFNR
MaMCC
ImD.
6.298e-03
1.400e-02
1.908e-01
ψ NB
.007 .014 .007 .798
.002 .097 .658 .658
.039 .159 .231 .798
ψ KE
1.00 1.00 .208
.658 .072 .097
1.00 .968 .653
ψ KA
1.00 .208
.751 .751
1.00 1.00
ψ KB
.208
.894
1.00
Nam.
MiFDR
MiFNR
MiMCC
ImD.
5.739e-01
5.739e-01
5.739e-01
Rank 3.014 2.764 2.914 3.386 2.921 3.014 2.764 2.914 3.386 2.921 3.014 2.764 2.914 3.386 2.921
ψ NB
1.00 1.00 .658 1.00
1.00 1.00 .658 1.00
1.00 1.00 .658 1.00
ψ KE
1.00 .201 1.00
1.00 .201 1.00
1.00 .201 1.00
ψ KA
.466 1.00
.466 1.00
.466 1.00
ψ KB
.466
.466
.466
best
worst
MaFDR
MaFNR
MaMCC
MiFDR
MiFNR
MiMCC
Algorithms:
ψNB
ψKE
ψKA
ψKB
ψKC
https://sci2s.ugr.es/keel/category.php?cat=clas 2 Removed due to the double-blind review.
Removed due to the double-blind review.
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"New results on Mesonic Weak Decay of p-shell Λ-Hypernuclei The FINUDA Collaboration",
"New results on Mesonic Weak Decay of p-shell Λ-Hypernuclei The FINUDA Collaboration"
] |
[
"M Agnello \nDipartimento di Fisica\nPolitecnico di Torino\nCorso Duca degli Abruzzi 24TorinoItaly\n\nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n",
"A Andronenkov \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n",
"G Beer \nUniversity of Victoria\nFinnerty RdVictoriaCanada\n",
"L Benussi \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"M Bertani \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"H C Bhang \nDepartment of Physics\nSeoul National University\n151-742SeoulSouth Korea\n",
"G Bonomi \nDipartimento di Meccanica\nUniversita' di Brescia\nvia Valotti 9BresciaItaly\n\nINFN Sezione di Pavia\nvia Bassi 6PaviaItaly\n",
"E Botta \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n\nDipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly\n",
"M Bregant \nDipartimento di Fisica\nUniversita' di Trieste\nvia Valerio 2TriesteItaly\n\nINFN Sezione di Trieste\nvia Valerio 2TriesteItaly\n",
"T Bressani \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n\nDipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly\n",
"S Bufalino \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n\nDipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly\n",
"L Busso \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n\nDipartimento di Fisica Generale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly\n",
"D Calvo \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n",
"P Camerini \nDipartimento di Fisica\nUniversita' di Trieste\nvia Valerio 2TriesteItaly\n\nINFN Sezione di Trieste\nvia Valerio 2TriesteItaly\n",
"B Dalena \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n\nDipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly\n",
"F De Mori \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n\nDipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly\n",
"G D'erasmo \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n\nDipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly\n",
"F L Fabbri \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"A Feliciello \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n",
"A Filippi \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n",
"E M Fiore \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n\nDipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly\n",
"A Fontana \nINFN Sezione di Pavia\nvia Bassi 6PaviaItaly\n",
"H Fujioka \nDepartment of Physics\nKyoto University\nSakyo-kuKyotoJapan\n",
"P Genova \nINFN Sezione di Pavia\nvia Bassi 6PaviaItaly\n",
"P Gianotti \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"N Grion \nINFN Sezione di Trieste\nvia Valerio 2TriesteItaly\n",
"O Hartmann \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"B Kang \nDepartment of Physics\nSeoul National University\n151-742SeoulSouth Korea\n",
"V Lenti \nDipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly\n",
"V Lucherini \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"S Marcello \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n\nDipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly\n",
"T Maruta \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n",
"N Mirfakhrai \nDepartment of Physics\nShahid Behesty University\n19834TeheranIran\n",
"P Montagna \nINFN Sezione di Pavia\nvia Bassi 6PaviaItaly\n\nDipartimento di Fisica Teorica e Nucleare\nUniversita' di Pavia\nvia Bassi 6PaviaItaly\n",
"O Morra \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n\nINAF-IFSI\nSezione di Torino\nCorso Fiume 4TorinoItaly\n",
"T Nagae \nDepartment of Physics\nKyoto University\nSakyo-kuKyotoJapan\n",
"D Nakajima \nDepartment of Physics\nUniversity of Tokyo\n113-0033BunkyoTokyoJapan\n",
"H Outa \nRIKEN\n351-0198WakoSaitamaJapan\n",
"E Pace \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"M Palomba \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n",
"A Pantaleo \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n",
"A Panzarasa \nINFN Sezione di Pavia\nvia Bassi 6PaviaItaly\n",
"V Paticchio \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n",
"S Piano \nINFN Sezione di Trieste\nvia Valerio 2TriesteItaly\n",
"F Pompili \nLaboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly\n",
"R Rui \nDipartimento di Fisica\nUniversita' di Trieste\nvia Valerio 2TriesteItaly\n\nINFN Sezione di Trieste\nvia Valerio 2TriesteItaly\n",
"A Sanchez Lorente \nInstitüt fur Kernphysik\nJohannes Gutenberg-Universität Mainz\nGermany\n",
"M Sekimoto \nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan\n",
"G Simonetti \nINFN Sezione di Bari\nvia Amendola 173BariItaly\n\nDipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly\n",
"A Toyoda \nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan\n",
"R Wheadon \nINFN Sezione di Torino\nvia P. Giuria 1TorinoItaly\n",
"A Zenoni \nDipartimento di Meccanica\nUniversita' di Brescia\nvia Valotti 9BresciaItaly\n\nINFN Sezione di Pavia\nvia Bassi 6PaviaItaly\n",
"A Gal \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n"
] |
[
"Dipartimento di Fisica\nPolitecnico di Torino\nCorso Duca degli Abruzzi 24TorinoItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"University of Victoria\nFinnerty RdVictoriaCanada",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"Department of Physics\nSeoul National University\n151-742SeoulSouth Korea",
"Dipartimento di Meccanica\nUniversita' di Brescia\nvia Valotti 9BresciaItaly",
"INFN Sezione di Pavia\nvia Bassi 6PaviaItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica\nUniversita' di Trieste\nvia Valerio 2TriesteItaly",
"INFN Sezione di Trieste\nvia Valerio 2TriesteItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica Generale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica\nUniversita' di Trieste\nvia Valerio 2TriesteItaly",
"INFN Sezione di Trieste\nvia Valerio 2TriesteItaly",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"Dipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"Dipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"Dipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly",
"INFN Sezione di Pavia\nvia Bassi 6PaviaItaly",
"Department of Physics\nKyoto University\nSakyo-kuKyotoJapan",
"INFN Sezione di Pavia\nvia Bassi 6PaviaItaly",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"INFN Sezione di Trieste\nvia Valerio 2TriesteItaly",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"Department of Physics\nSeoul National University\n151-742SeoulSouth Korea",
"Dipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Fisica Sperimentale\nUniversita' di Torino\nVia P. Giuria 1TorinoItaly",
"Department of Physics\nTohoku University\n980-8578SendaiJapan",
"Department of Physics\nShahid Behesty University\n19834TeheranIran",
"INFN Sezione di Pavia\nvia Bassi 6PaviaItaly",
"Dipartimento di Fisica Teorica e Nucleare\nUniversita' di Pavia\nvia Bassi 6PaviaItaly",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"INAF-IFSI\nSezione di Torino\nCorso Fiume 4TorinoItaly",
"Department of Physics\nKyoto University\nSakyo-kuKyotoJapan",
"Department of Physics\nUniversity of Tokyo\n113-0033BunkyoTokyoJapan",
"RIKEN\n351-0198WakoSaitamaJapan",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"INFN Sezione di Pavia\nvia Bassi 6PaviaItaly",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"INFN Sezione di Trieste\nvia Valerio 2TriesteItaly",
"Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi\n40FrascatiItaly",
"Dipartimento di Fisica\nUniversita' di Trieste\nvia Valerio 2TriesteItaly",
"INFN Sezione di Trieste\nvia Valerio 2TriesteItaly",
"Institüt fur Kernphysik\nJohannes Gutenberg-Universität Mainz\nGermany",
"High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan",
"INFN Sezione di Bari\nvia Amendola 173BariItaly",
"Dipartimento di Fisica Universita' di Bari\nvia Amendola 173BariItaly",
"High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan",
"INFN Sezione di Torino\nvia P. Giuria 1TorinoItaly",
"Dipartimento di Meccanica\nUniversita' di Brescia\nvia Valotti 9BresciaItaly",
"INFN Sezione di Pavia\nvia Bassi 6PaviaItaly",
"Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael"
] |
[] |
The FINUDA experiment performed a systematic study of the charged mesonic weak decay channel of p-shell Λ-hypernuclei. Negatively charged pion spectra from mesonic decay were measured with magnetic analysis for the first time for 7 Λ Li, 9 Λ Be, 11 Λ B and 15 Λ N. The shape of the π − spectra was interpreted through a comparison with pion distorted wave calculations that take into account the structure of both hypernucleus and daughter nucleus. Branching ratios Γ π − /Γtot were derived from the measured spectra and converted to π − decay rates Γ π − by means of known or extrapolated total decay widths Γtot of p-shell Λ-hypernuclei. Based on these measurements, the spin-parity assignment 1/2 + for 7 Λ Li and 5/2 + for 11 Λ B ground-state are confirmed and a spin-parity 3/2 + for 15 Λ N ground-state is assigned for the first time.
|
10.1016/j.physletb.2009.09.061
|
[
"https://arxiv.org/pdf/0905.0623v2.pdf"
] | 55,573,063 |
0905.0623
|
5a7b9044d101cce33976de211a2416acc705bac3
|
New results on Mesonic Weak Decay of p-shell Λ-Hypernuclei The FINUDA Collaboration
25 Sep 2009 25 September 2009
M Agnello
Dipartimento di Fisica
Politecnico di Torino
Corso Duca degli Abruzzi 24TorinoItaly
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
A Andronenkov
INFN Sezione di Bari
via Amendola 173BariItaly
G Beer
University of Victoria
Finnerty RdVictoriaCanada
L Benussi
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
M Bertani
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
H C Bhang
Department of Physics
Seoul National University
151-742SeoulSouth Korea
G Bonomi
Dipartimento di Meccanica
Universita' di Brescia
via Valotti 9BresciaItaly
INFN Sezione di Pavia
via Bassi 6PaviaItaly
E Botta
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
Dipartimento di Fisica Sperimentale
Universita' di Torino
Via P. Giuria 1TorinoItaly
M Bregant
Dipartimento di Fisica
Universita' di Trieste
via Valerio 2TriesteItaly
INFN Sezione di Trieste
via Valerio 2TriesteItaly
T Bressani
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
Dipartimento di Fisica Sperimentale
Universita' di Torino
Via P. Giuria 1TorinoItaly
S Bufalino
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
Dipartimento di Fisica Sperimentale
Universita' di Torino
Via P. Giuria 1TorinoItaly
L Busso
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
Dipartimento di Fisica Generale
Universita' di Torino
Via P. Giuria 1TorinoItaly
D Calvo
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
P Camerini
Dipartimento di Fisica
Universita' di Trieste
via Valerio 2TriesteItaly
INFN Sezione di Trieste
via Valerio 2TriesteItaly
B Dalena
INFN Sezione di Bari
via Amendola 173BariItaly
Dipartimento di Fisica Universita' di Bari
via Amendola 173BariItaly
F De Mori
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
Dipartimento di Fisica Sperimentale
Universita' di Torino
Via P. Giuria 1TorinoItaly
G D'erasmo
INFN Sezione di Bari
via Amendola 173BariItaly
Dipartimento di Fisica Universita' di Bari
via Amendola 173BariItaly
F L Fabbri
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
A Feliciello
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
A Filippi
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
E M Fiore
INFN Sezione di Bari
via Amendola 173BariItaly
Dipartimento di Fisica Universita' di Bari
via Amendola 173BariItaly
A Fontana
INFN Sezione di Pavia
via Bassi 6PaviaItaly
H Fujioka
Department of Physics
Kyoto University
Sakyo-kuKyotoJapan
P Genova
INFN Sezione di Pavia
via Bassi 6PaviaItaly
P Gianotti
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
N Grion
INFN Sezione di Trieste
via Valerio 2TriesteItaly
O Hartmann
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
B Kang
Department of Physics
Seoul National University
151-742SeoulSouth Korea
V Lenti
Dipartimento di Fisica Universita' di Bari
via Amendola 173BariItaly
V Lucherini
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
S Marcello
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
Dipartimento di Fisica Sperimentale
Universita' di Torino
Via P. Giuria 1TorinoItaly
T Maruta
Department of Physics
Tohoku University
980-8578SendaiJapan
N Mirfakhrai
Department of Physics
Shahid Behesty University
19834TeheranIran
P Montagna
INFN Sezione di Pavia
via Bassi 6PaviaItaly
Dipartimento di Fisica Teorica e Nucleare
Universita' di Pavia
via Bassi 6PaviaItaly
O Morra
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
INAF-IFSI
Sezione di Torino
Corso Fiume 4TorinoItaly
T Nagae
Department of Physics
Kyoto University
Sakyo-kuKyotoJapan
D Nakajima
Department of Physics
University of Tokyo
113-0033BunkyoTokyoJapan
H Outa
RIKEN
351-0198WakoSaitamaJapan
E Pace
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
M Palomba
INFN Sezione di Bari
via Amendola 173BariItaly
A Pantaleo
INFN Sezione di Bari
via Amendola 173BariItaly
A Panzarasa
INFN Sezione di Pavia
via Bassi 6PaviaItaly
V Paticchio
INFN Sezione di Bari
via Amendola 173BariItaly
S Piano
INFN Sezione di Trieste
via Valerio 2TriesteItaly
F Pompili
Laboratori Nazionali di Frascati dell'INFN, via. E. Fermi
40FrascatiItaly
R Rui
Dipartimento di Fisica
Universita' di Trieste
via Valerio 2TriesteItaly
INFN Sezione di Trieste
via Valerio 2TriesteItaly
A Sanchez Lorente
Institüt fur Kernphysik
Johannes Gutenberg-Universität Mainz
Germany
M Sekimoto
High Energy Accelerator Research Organization (KEK)
305-0801TsukubaIbarakiJapan
G Simonetti
INFN Sezione di Bari
via Amendola 173BariItaly
Dipartimento di Fisica Universita' di Bari
via Amendola 173BariItaly
A Toyoda
High Energy Accelerator Research Organization (KEK)
305-0801TsukubaIbarakiJapan
R Wheadon
INFN Sezione di Torino
via P. Giuria 1TorinoItaly
A Zenoni
Dipartimento di Meccanica
Universita' di Brescia
via Valotti 9BresciaItaly
INFN Sezione di Pavia
via Bassi 6PaviaItaly
A Gal
Racah Institute of Physics
The Hebrew University
91904JerusalemIsrael
New results on Mesonic Weak Decay of p-shell Λ-Hypernuclei The FINUDA Collaboration
25 Sep 2009 25 September 2009Preprint submitted to Physics Letters Bp-shell Λ-hypernucleimesonic decayground-state spin assignment PACS: 2180+a1375Ev
The FINUDA experiment performed a systematic study of the charged mesonic weak decay channel of p-shell Λ-hypernuclei. Negatively charged pion spectra from mesonic decay were measured with magnetic analysis for the first time for 7 Λ Li, 9 Λ Be, 11 Λ B and 15 Λ N. The shape of the π − spectra was interpreted through a comparison with pion distorted wave calculations that take into account the structure of both hypernucleus and daughter nucleus. Branching ratios Γ π − /Γtot were derived from the measured spectra and converted to π − decay rates Γ π − by means of known or extrapolated total decay widths Γtot of p-shell Λ-hypernuclei. Based on these measurements, the spin-parity assignment 1/2 + for 7 Λ Li and 5/2 + for 11 Λ B ground-state are confirmed and a spin-parity 3/2 + for 15 Λ N ground-state is assigned for the first time.
Introduction
A Λ-hypernucleus in its ground-state decays to non-strange nuclear systems through the mesonic (MWD) and non-mesonic (NMWD) weak decay mechanisms. In MWD the Λ hyperon decays to a nucleon and a pion in the nuclear medium, similarly to the weak decay mode in free space: Λ f ree → p + π − + 37.8 MeV (64.2%) (1) n + π 0 + 41.1 MeV (35.8%) (2) in which the emitted nucleon (pion) carries a momentum q ≈ 100 MeV/c. For a Λhypernucleus, the total decay width (or equivalently the decay rate) Γ tot ( A Λ Z) is given by the sum of the mesonic decay width (Γ m ) and the non-mesonic decay width (Γ nm ), where the first term can be further expressed as the sum of the decay widths for the emission of negative (Γ π − ) and neutral (Γ π 0 ) pions:
Γ tot ( A Λ Z) = Γ π − + Γ π 0 + Γ nm ,(3)
with Γ tot ( A Λ Z) expressed in terms of the hypernuclear lifetime as:
Γ tot ( A Λ Z) = /τ ( A Λ Z).(4)
MWD is suppressed in hypernuclei with respect to the free-space decay due to the Pauli principle, since the momentum of the emitted nucleon is by far smaller than the nuclear Fermi momentum (k F ≃ 270 MeV/c) in all nuclei except for the lightest, s-shell ones.
The theory of hypernuclear MWD was initiated by Dalitz [1,2], based on a phenomenological Lagrangian describing the elementary decay processes (1) and (2), and motivated by the observation of MWD reactions in the pioneering hypernuclear physics experiments with photographic emulsions that provided means of extracting hypernuclear groundstate spins and parities; see Ref. [3] for a recent summary. Following the development of counter techniques for use in (K − , π − ) and (π + , K + ) reactions in the 1970s and 1980s, a considerable body of experimental data on Γ π − and/or Γ π 0 is now available on light Λ-hypernuclei up to 12 Λ C: 4 Λ H [4], 4 Λ He [5], 5 Λ He [6], 11 Λ B and 12 Λ C [6,7,8,9]. Comprehensive calculations of the main physical entities of MWD were performed during the 1980s and 1990s for very light s-shell [10,11], p-shell [11,12,13] and sd-shell hypernuclei [11,13]. The basic ingredients of the calculations are the Pauli suppression effect, the enhancement of MWD owing to the pion-nuclear polarization effect in the nuclear medium as predicted for MWD in Refs. [14,15], the sensitive final-state shellstructure dependence, and the resulting charge dependence of the decay rates.
An important ingredient of MWD calculations is the choice of pion-nucleus potential which generates pion-nuclear distorted waves that strongly affect the magnitude of the pionic decay rates. Indeed, for low-energy pions, the pion-nucleus potential has been studied so far through π-nucleus scattering experiments [16] and measurements of X-rays from pionic atoms [17]; the study of MWD in which a pion is created by the decay of a Λ hyperon deep inside the nucleus offers important opportunities to investigate in-medium pions and to discriminate between different off-shell extrapolations inherent in potential models. For this reason MWD continues to be an interesting item of hypernuclear physics, and precise and systematic determinations of Γ π − and Γ π 0 are very welcome.
In the present work we report on new measurements by the FINUDA experiment of MWD of hypernuclei in the p-shell, comparing the measured π − spectra and decay rates with the calculations by one of the authors [18] that update the calculations by Motoba et al. [11,12,13]. These two sets of spectroscopic calculations agree reasonably well with each other for all hypernuclei considered in the present report, except for 15 Λ N which is discussed in detail below. The measured spectra are consistent with the observation, made in these shell-model calculations, that the partial decay contributions from the high-lying continuum of the daughter nuclear system outside the 0 ω p-shell configuration are unimportant in this mass range. The level of agreement between the reported measurements and the calculations allows us to confirm the previous spin-parity assignments made for 7 Λ Li and 11 Λ B, and to assign J π = 3/2 + to 15 Λ N ground-state.
Experimental and analysis techniques
FINUDA is a hypernuclear physics experiment, with cylindrical symmetry, installed at one of the two interaction regions of the DAΦN E e + e − collider, the INFN-LNF Φ-factory. A description of the experimental apparatus can be found in [19,20]. Here we briefly sketch its main components, moving outwards from the beam axis: the interaction/target region, composed by a barrel of 12 thin scintillator slabs (TOFINO), surrounded by an octagonal array of Si microstrips (ISIM) facing eight target tiles; the external tracking device, consisting of four layers of position sensitive detectors (a decagonal array of Si microstrips (OSIM), two octagonal layers of low mass drift chambers (LMDC) and a stereo system of straw tubes (ST)) arranged in coaxial geometry; the external time of flight detector (TOFONE), a barrel of 72 scintillator slabs. The whole apparatus is placed inside a uniform 1.0 T solenoidal magnetic field; the tracking volume is immersed in He atmosphere to minimize the multiple scattering effect.
The scientific program of the experiment is focussed on the study of spectroscopy and decay of Λ-hypernuclei produced by means of the (K − , π − ) reaction with K − 's at rest:
K − stop + A Z → π − + A Λ Z(5)
by stopping in very thin targets the low energy (∼ 16 MeV) K − 's coming from the Φ → K − K + decay channel. In (5) A Z indicates the target nucleus and A Λ Z the produced Λ-hypernucleus. Λ-hypernuclei decay through both the mesonic weak decay processes:
A Λ Z → A (Z + 1) + π − (6) A Λ Z → A Z + π 0(7)
and the non-mesonic weak decay processes:
A Λ Z → A−2 (Z − 1) + p + n (8) A Λ Z → A−2 Z + n + n(9)
where the final nuclear states in (6-9) are not necessarily particle stable. In contrast to the mesonic decays, the non-mesonic decays are not Pauli blocked, producing highmomentum nucleons (≤ 600 MeV/c). The thinness of the target materials needed to stop the K − 's, the high transparency of the FINUDA tracker and the very large solid angle (∼ 2π sr) covered by the detector ensemble make the FINUDA apparatus suitable to study the formation and the decay of Λ-hypernuclei by means of high resolution magnetic spectroscopy of the charged particles emitted in the processes (5) [19], (6) and (8) [21]; the features of the apparatus give also the possibility to investigate many other final states produced in the interaction of stopped kaons with nuclei [22].
In this paper results are presented obtained by analyzing the data collected by the FINUDA experiment from 2003 to 2007 with a total integrated luminosity of 1156 pb −1 . Only targets leading to the formation of the p-shell hypernuclei 7 Λ Li, 9 Λ Be, 11 Λ B and 15 Λ N are here considered, namely 7 Li (2×, 4 mm thick, natural isotopic composition), 9 Be (2×, 2 mm thick, natural isotopic composition), 12 C (3×, 1.7 mm thick, natural isotopic composition, mean density 2.265 g cm −3 ) and D 2 O (mylar walled, 1×, 3 mm thick), together with 6 Li targets (2×, 4 mm thick, 90% enriched) leading to the production of 5 Λ He; 5 Λ He is reported for the sake of completeness. To investigate the MWD process (6) events where analyzed in which two π − 's were detected in coincidence. One π − , with a momentum as high as 260 − 290 MeV/c, gives the signature of the formation of the ground-state of the hypernuclear system or of a low lying excited state decaying to it by electromagnetic emission. The second π − , with a momentum lower than 115 MeV/c, gives the signature of the decay. By requiring this coincidence, negative pions are the only negative particles originating from the K − 's stopping point in the targets that enter the tracking volume of the apparatus. Nevertheless, to get a cleaner data sample, only tracks identified as π − 's by the FINUDA detectors were considered. In particular, the information of the specific energy loss in both OSIM and the LMDC's and the mass identification from the time of flight system (TOFINO-TOFONE), if present, were used to obtain a multiple identification selection.
In the present analysis we required good quality tracks to determine the momentum of the formation π − . These tracks must originate in a properly defined fiducial volume around the primary K − vertex and are identified by four hits, one in each of the FINUDA tracking detectors (long tracks), and are selected with a quite strict requirement on the χ 2 from the track fitting procedure (corresponding to a 90% confidence level). They have a resolution ∆p/p ∼ 1% FWHM in the region 260−280 MeV/c; this resolution is about twice worse than the best value obtained with top quality tracks [19] for spectroscopy studies. The worsening was due to the more relaxed quality criteria applied to increase the statistics of the sample available for the coincidence measurement. In particular, no cut has been made to select the direction of the outgoing tracks. Table 1 Summary of the momentum and binding energy (B.E.) intervals selected to identify the formation of various hypernuclear systems. First column: target nucleus A Z; second column: weakly decaying final hypernucleus; third column: production-pion momentum interval; fourth column: A Λ Z binding energy interval; fifth column: references to previous missing mass spectroscopy experiments. Table 1 reports the binding energy intervals selected to identify the formation of the different hypernuclei. The intervals have been determined by comparing our experimental inclusive formation spectra with the known values of binding energies for ground-states and low lying excited states, as deduced from the references indicated in the last column. The interval width takes into account our experimental resolution, σ p ∼ 1 MeV/c and σ B.E. ∼ 1 MeV for a typical pion momentum of about 270 MeV/c. For 7 Λ Li a sharp cut was set at an excitation energy of 3.94 MeV, corresponding to the threshold for the 7 Λ Li → 5 Λ He + d fragmentation. As it is well known, for 11 Λ B and 15 Λ N, produced on 12 C and 16 O (D 2 O) targets respectively, the production momentum region partially overlaps the higher part of the momentum spectrum of π − 's emitted in the Λ quasi-free (Λ qf ) production. This holds particularly for 15 Λ N which is expected to be dominantly formed by proton emission from the two peaks of 16 Λ O observed at B.E.≃ 2 MeV and B.E.≃ −4 MeV [30]. However, in order to minimize the contamination in the 15 Λ N decay spectrum by decays of other hypernuclear species, which may be formed in the opening of higher energy emission channels, events were selected corresponding only to the positive B.E. 16 Λ O peak, as indicated in Table 1. A likely source of contamination is provided by the production of 12 Λ C and its subsequent mesonic decay:
target final p π − B.E. ( A Λ Z) references A Z hypernucleus (MeV/c) (MeV) 6K − + 16 O → π − + α + 12 Λ C 12 Λ C → 12 N + π − (p π − max ≃ 91 MeV/c).(10)
For 11 Λ B, on the other hand, it was enough to focus on the sizable excitation peak of 12 Λ C at B.E.≃ 0 MeV which is known to lead, upon proton emission, to several excited states of 11 Λ B [31]. Moreover, it should be noted that the contribution to the inclusive spectra due to the reaction chain:
K − + (np) → Σ − + p Σ − → n + π −(11)
constitutes the only physical background below the hypernuclear formation peaks. It was evaluated by simulating a sample of background events and applying to the simulated data the same selection criteria as for the real ones. The background spectra were then normalized to the experimental ones above the kinematical limits for the hypernuclear ground-state formation and subtracted. A detailed description of such a procedure is available in Ref. [21]. Figure 1 shows the inclusive binding energy spectra for formation π − from 12 C and 16 O targets, after subtraction of the K − (np) background. The continuous line is the best fit to the spectra, while the dashed curves represent the contributions from the known hypernuclear states and the dot-dashed curve represents the polynomial background, due to Λ qf production in the negative B.E. region. In the positive B.E. region a background contribution from K − 's decay in flight, shown separately in the figure for 16 O, is also considered. This background affects differently the targets placed in different positions with respect to the e + e − interaction region due to the fact that the (e + , e − ) crossing beams collide with an angle of 12.5 mrad in order to increase the luminosity; the effect has been very well studied in [20]. On the other hand, this background affects only the inclusive B.E. spectra and does not give any contribution to the low energy π − spectra from MWD, for which a two π − coincidence is required. Table 2 reports the binding energy values of the hypernuclear states obtained from the global fitting procedure in the B.E. regions indicated in Table 1, as mean values of the corresponding gaussians; the χ 2 /ndf values are also indicated. It must be noted that these binding energy values can be different from the ones obtained in analyses dedicated to spectroscopy studies, due to the relaxation of the quality cuts applied to the tracks: indeed, the requirement of the coincidence with a low momentum π − acts as a filter that allows to untighten the quality selections on the long tracks, as will be shown in the next section. The precise choice of peaks does not change the MWD spectra at any qualitative level, except for reducing the sample statistics. For the decay π − momentum measurement only tracks not reaching the ST system (short tracks) have been used. The lower threshold for the detection momentum of these π − 's is ∼ 80 MeV/c. These tracks correspond mainly to particles backward emitted from the targets, crossing the whole interaction/target region before entering the tracker; their momentum resolution is ∆p/p ∼ 6% FWHM at 110 MeV/c. The acceptance for low energy π − 's, ǫ, was evaluated for each target, taking into account the geometrical layout, the efficiency of the FINUDA pattern recognition algorithm, the trigger and the efficiency of the quality cuts applied in the analysis procedure. The acceptance function, R = 1/ǫ, for the momentum features a negative quadratic exponential behaviour in the 80 − 160 MeV/c range and flattens above 90 MeV/c; for the kinetic energy the behaviour is similar in the 20 − 70 MeV range, as shown in Fig. 2 for 7 Λ Li with a dot-dashed line. The error on the acceptance function is always < 5%.
peak 7 Λ Li 9 Λ Be 12 Λ C 16 Λ O (MeV) (MeV) (MeV)(MeV)
MWD π − 's spectra
It is worth recalling that the information available up to now on the charged MWD of light hypernuclei consists almost entirely of Γ π − /Γ Λ values obtained by means of counting measurements in coincidence with the hypernuclear formation π − detection, with no magnetic analysis of the decay meson; π − kinetic energy spectra have been obtained for 12 Λ C MWD only [9]. The π − spectra presented here allow to have a more careful confirmation of the elementary mechanism that is supposed to underlie the decay process, as well as to have information on the spin-parity of the initial hypernuclear ground state. In this respect the study of pion spectra from MWD can be regarded as an indirect spectroscopic investigation tool.
Due to the π − momentum detection threshold of the apparatus (∼ 80 MeV/c), only MWD spectra of 7 Λ Li, 9 Λ Be, 11 Λ B ( 12 C targets) and 15 Λ N ( 16 O target) were investigated. Spectra from 12 Λ C and 16 Λ O could not be observed. Background coming from Λ qf decay was simulated taking into account the Fermi momentum of the neutron in the target: it was found that the spectrum of the decay π − momentum extends up to ∼ 160 MeV/c, well above the stopping point of the hypernucleus MWD contribution at ∼ 110 MeV/c. This background was then subtracted from the 11 Λ B spectrum by normalizing the area of the simulated spectra, after reconstruction, to the experimental ones in the 110 − 160 MeV/c decay-pion region, populated only by Λ qf decays. Each spectrum was corrected by means of the acceptance function, R, described in the previous section. The decay π − momentum spectra show interesting structures whose meaning can be better understood by considering the corresponding kinetic energy spectra that are directly related to the excitation function of the daugther nucleus. Kinetic energy spectra, background subtracted and acceptance corrected, were evaluated for MWD of 7 Λ Li, 9 Λ Be, 11 Λ B and 15 Λ N. Kinetic energy spectrum of MWD π − for 7 Λ Li before acceptance correction. The dot-dashed curve is the acceptance function, R, to be applied to the data.
In Fig. 2, the kinetic energy spectrum for 7 Λ Li, before applying the acceptance correction, is reported. The acceptance correction is represented by the dot-dashed curve. The errors are statistical only. It is evident that the low energy π − spectrum is practically background free and that the 50 − 70 MeV contribution is negligible and compatible with zero. This demonstrates the effectiveness of the coincidence requirement. The small residual background, which is similar also for the other targets, has been evaluated after acceptance correction and subtracted to calculate the decay ratio (see next section). In the following only the 16 − 60 MeV region of the MWD spectra will be shown. Upper part: kinetic energy spectrum of MWD π − for 7 Λ Li after acceptance correction. The solid curve is a gaussian fit to the peak in the spectrum, to compare with theoretical predictions in the lower part. Lower part: calculated major decay rates to final 7 Be states from [18], in red bars for 7 Λ Li ground-state spin-parity 1/2 + , and in blue bars for 7 Λ Li ground-state spin-parity 3/2 + (see text)
. In Fig. 3 the acceptance corrected spectrum for 7 Λ Li is shown in the upper part and compared with calculated decay ratios (Γ π − /Γ Λ ) to final 7 Be states [18] shown in the lower part. These calculated rates are close to those calculated by Motoba et al. [13]. The errors in the spectrum of the upper part are inclusive of both the statistical and the acceptance contributions. Only major contributions are shown in the lower part with a common, arbitrary 1 MeV width, although each of the two bars in Fig. 3 between 25 and 27 MeV stands for 7 Be states spread over roughly 2 − 3 MeV. The correspondence of the structures observed in the experimental spectra with the rates of decay to different excited states of the daugther nucleus, assuming initial spin-parity 1/2 + , is clear. The peak structure corresponds to the production of 7 Be in its 3/2 − ground-state and in its only bound 1/2 − excited state, at 429 keV. Due to the FINUDA experimental resolution these close levels are not resolved and the gaussian fit superimposed on the data points yields a FWHM ∼ 4.5 MeV, compatible with the intrinsic resolution of the apparatus: ∆T /T ∼ 11% FWHM at 38 MeV. The part of the spectrum at lower energies is due to three body decays.
The portion of the spectrum that cannot be measured due to our experimental detection threshold (∼ 22 MeV) is negligible if compared with the errors affecting the counts in the spectrum, according to the calculated excitation spectra shown in [13]. The same holds also for the other hypernuclei studied in this paper. It is then reasonable to compare the total area of the spectrum with the decay rates summed over the whole excitation energy interval, as done in the next section.
The shape of the spectrum confirms the spin assigned to the hypernuclear ground state of 7 Λ Li [32]. Indeed, only a spin-parity 1/2 + for 7 Λ Li ground state, shown by red bars, reproduces the fitted peak at ∼ 36 MeV due to the 7 Be ground state and excited state at 429 keV. A spin-parity 3/2 + for 7 Λ Li ground state would imply a radically different spectrum shape [13,18], as indicated in Fig. 3 by the blue bars, short of any peak about 7 Be ground state and its 429 keV excited state. Upper part: kinetic energy spectrum of MWD π − for 9 Λ Be after acceptance correction. The solid curve is a gaussian fit to the peak in the spectrum, to compare with theoretical predictions in the lower part. Lower part: calculated major decay rates to final 9 B states from [18], in red bars for 9 Λ Be ground-state spin-parity 1/2 + .
In Fig. 4 the acceptance corrected spectrum for 9 Λ Be is shown in the upper part and compared with calculated decay ratios (Γ π − /Γ Λ ) to final 9 B states [18] shown in the lower part. The errors in the spectrum of the upper part are again inclusive of both the statistical and the acceptance contributions. Only major contributions are shown in the lower part with a common, arbitrary 1 MeV width. In this case too, as for 7 Λ Li above, the calculated rates are very close to those calculated by Motoba et al. [13].
In the 9 Λ Be spectrum our energy resolution does not allow a separation between the two components predicted to dominate the spectrum [13,18], the 9 B ground-state 3/2 − and the excited state 1/2 − at 2.75 MeV. As a consequence, the gaussian fit superimposed on the data points yields a FWHM ∼ 7.5 MeV. The correspondence between the experimental spectrum and the calculated rates of decay to different excited states of the daugther nucleus is clear. Our spectrum is consistent with the interpretation from (π + , K + ) reactions [25] according to which the 9 Λ Be ground state is dominantly a 1s-Λ coupled to 8 Be(0 + ) ground state. Upper part: kinetic energy spectrum of MWD π − for 11 Λ B after acceptance correction. The solid curve is a two-gaussian fit to the peaks in the spectrum, to compare with theoretical predictions in the lower part; dashed curves are the single components. Lower part: calculated major decay rates to final 11 C states from [18], in red bars for 11 Λ B ground-state spin-parity 5/2 + , and in blue bars for 11 Λ B ground-state spin-parity 7/2 + .
In Fig. 5 the spectrum for 11 Λ B is shown and compared with calculated decay rates to final 11 C states [18]. The errors in the spectra are again inclusive of both the statistical and the acceptance contributions. Major and secondary contributions are shown in the lower part with a common, arbitrary 1 MeV width. Assuming ground-state spin-parity 5/2 + , it is possible to identify two major contributions in the 11 Λ B spectrum due to 11 C ground-state 3/2 − and its 7/2 − excited state at 6.478 MeV, both shown by red bars.
A third contribution due to the 3/2 − excited state at 8.10 MeV is considerably weaker than the former ones, its main effect being to introduce some asymmetry to the overall spectrum towards lower kinetic energies. Additional contribution of a similar strength [11] in this energy range arises from transitions to several sd states within 7 − 10 MeV excitation energy in 11 C (not shown in Fig. 5). It is clear from the figure that the shape of the spectrum is well reproduced by assigning spin-parity 5/2 + to 11 Λ B g.s. . Assuming ground-state spin-parity 7/2 + , the 11 C ground-state peak is missing and the dominant decay is to the 5/2 − excited state at 8.420 MeV shown by a blue bar. A secondary contribution due to the 7/2 − excited state at 6.478 MeV is also considerably weaker than that arising under the assumption of spin-parity 5/2 + . We note that the major contributions to the 11 Λ B spectrum discussed above for both 11 Λ B g.s. possible spin-parity values are also borne out by the calculation of Ref. [11].
A 5/2 + assignment for 11 Λ B ground-state, first made by Zieminska studying emulsion spectra [33], was experimentally confirmed by the KEK measurement [9] comparing the derived value of the total π − decay rate with the total π − decay rate calculated in [11]. The present measurement of the decay spectrum shape provides a confirmation of J π ( 11 Λ B g.s. ) = 5/2 + by a different observable.
15 Λ N
In Fig. 6 the spectrum for 15 Λ N is shown and compared with calculated decay rates to final 15 O states [18]. The errors in the spectra are again inclusive of both the statistical and the acceptance contributions. The contribution from Λ qf was evaluated to be less than 5% and then neglected, taking into account the overwhelming importance of the statistical errors. In the experimental spectrum, the 15 O ground-state 1/2 − contribution stands out clearly, along with a hint for a secondary structure separated by about 6 MeV. The gaussian fit of the ground state component yields a FWHM of ∼ 6 MeV, larger than our standard value ∼ 4.5 MeV due to the already mentioned malfunctioning of the apparatus (see caption of Table 2) and to the limited statistics. The fit to the lower-energy secondary structure is strongly influenced by the substantial error affecting the lowest energy point. According to Refs. [12,18], this secondary structure derives most of its strength from sd states scattered around 6 MeV excitation while the contribution of the 15 O p −1 3/2 p 1/2 excited state at 6.176 MeV is negligible. We note that, in the upper part of Fig. 6, the channel at about 37 MeV kinetic energy might get contribution from the reaction chain K − + 16 O → π 0 + α + 12 Λ B, 12 Λ B → π − + 12 C g.s. , with a non negligible pion charge exchange in the D 2 O target.
The ground-state spin has not been determined experimentally. The most recent theoretical study of Λ-hypernuclear spin dependence [30] predicts J π ( 15 Λ N g.s. ) = 3/2 + , setting the 1/2 + excited ground-state doublet level about 90 keV above the 3/2 + ground-state. The spin ordering, however, cannot be determined from the γ-ray de-excitation spectra measured recently on a 16 O target at Brookhaven [34]. As for MWD, the prominence of 15 O g.s. in the spectrum of Fig. 6 supports this J π ( 15 Λ N g.s. ) = 3/2 + theoretical assignment since the decay 15 Λ N(1/2 + ) → π − + 15 O g.s. is suppressed according to the following simple argument. Recent shell-model calculations suggest that the nuclear-core 14 N g.s. wavefunction is very close to a 3 D 1 wavefunction [30], which for J π ( 15 Λ N) = 1/2 + leads in the weak coupling limit to a single LS hypernuclear component Upper part: kinetic energy spectrum of MWD π − for 15 Λ N after acceptance correction. The solid curve is a two-gaussian fit to the peaks in the spectrum, to compare with theoretical predictions in the lower part; dashed curves are the single components. Lower part: calculated major decay rates to final 15 O states from [18], in red bars for 15 Λ N ground-state spin-parity 3/2 + , and in blue bars for 15 Λ N ground-state spin-parity 1/2 + . Note that in this case the arbitrary bar width was reduced to 0.5 MeV to avoid overlap and to facilitate the comparison between the two spin hypotheses: indeed, the energies of the produced final states are practically the same. transition 15 Λ N( 4 D 1/2 ) → 15 O( 2 P 1/2 ) requires spin-flip, it is forbidden for the dominant π − -decay s-wave amplitude. In a more realistic calculation, for spin-parity 3/2 + , the 15 O ground-state main peak is expected to dominate over the secondary peak at about 6 MeV by a ratio close to 3:1, as shown in the lower part of Fig. 6 by the red bars [18]. This is in rough agreement with the fitted gaussians shown by dashed lines in the upper part of Fig. 6, where the relative contribution of the 15 O g.s. gaussian amounts to (67 ± 18)%. In contrast, for spin-parity 1/2 + the calculation in Ref. [18] produces a ratio close to 1:1 with respect to the ∼ 6 MeV excitation, as shown by the blue bars in the lower part. Thus, the shape of the measured spectrum slightly favors J π ( 15 Λ N g.s. ) = 3/2 + . 2
MWD decay ratios and total decay rates
In general, due to the quite large errors affecting the spectra and to the lack of energy resolution at low values, the assignment of distinct MWD transitions to the daughter 2 We note that the calculations in Refs. [12,13] suggest that neither the shape of the decay spectrum nor the total π − decay rate of 15 Λ N are sensitive to the assumed value of the ground-state spin. However, it has been shown recently [18] that these older results for 15 Λ N violate a model-independent sum rule and, therefore, are not used here further to suggest interpretations of the present experimental results. nucleus in our experimental spectra is somewhat tentative, except for the two-body component of 7 Λ Li. However, by considering complementarily the total area of each spectrum it is possible to infer decay rates with a reasonable statistical significance.
The branching ratios of the MWD reaction, b π − = Γ π − /Γ tot , were evaluated for each hypernucleus as:
b π − = N π − decay N hyp(12)
where N π − decay is the number of the π − MWD reactions and N hyp is the number of the produced hypernuclei. The number of MWD reactions was obtained from the counts in the momentum and the kinetic energy spectra after subtracting the residual background discussed in the previous section. In both cases, counts were considered up to the kinematical limit for a pure two-body decay, folded by our experimental resolution. 1.37 ± 0.16 [37] (0.116) [11] 0.23 ± 0.06 ± 0.03 [8] 0.196 [18] 0.212 ± 0.036 ± 0.045 [9] (0.101) [18] linear fit (0.074) [38] 0.080 [18] (0.040) [18] Table 3 Branching ratios b π − , total hypernuclear weak decay rates Γtot/Γ Λ mostly from a linear fit in A, and total decay rates Γ π − /Γ Λ evaluated for charged MWD. Total decay rates are given in units of Γ Λ . In the second and fourth columns the first quoted error is statistical, the second one is systematic. Comparison with previous measurements and theoretical predictions is reported. The calculated total rates are for ground state spin-parity 1/2 + for 7 Λ Li, 5/2 + for 11 Λ B and 3/2 + for 15 Λ N (in brackets 3/2 + for 7 Λ Li, 7/2 + for 11 Λ B and 1/2 + for 15 Λ N).
b π − = Γ π − /Γtot Γtot/Γ Λ Γ π − /Γ Λ previous
To evaluate the number of formed hypernuclei the area of the inclusive binding energy spectra was evaluated within the intervals reported in Table 1, after subtraction of the K − np background and of the K − in flight decay contribution, as described above.
The obtained values of the branching ratios, b π − , are reported in Table 3 with statistical and systematic errors. The latter ones are due to the different techniques used to evaluate the areas and the background in the inclusive spectra, while the systematic error due to the detection threshold of the apparatus has been estimated to be less than 2% and has been neglected. A preliminary account of the results presented here can be found in [39]. Total decay rates were calculated, using known Γ tot /Γ Λ values or relying on a linear fit to the known values of all measured Λ-hypernuclei in the mass range A = 4 − 12 [32] as shown in Table 3:
Γ tot /Γ Λ (A) = (0.990 ± 0.094) + (0.018 ± 0.010) · A, χ 2 /ndf = 5.317/6.
A good agreement holds among the present results and previous measurements, when existing, and among the present results and theoretical calculations assuming ground state spins as listed in the caption. The calculated total decay rates for the other choice of ground state spins (in brackets) are substantially lower and disagree with the experimentally derived values. In particular, the total π − decay rate of 15 Λ N and its decay spectrum shape, as evaluated here, agree with calculations by Motoba et al. [13] and by Gal [18] assuming a ground-state spin-parity assignment 3/2 + . These two calculations disagree for a 1/2 + spin-parity assignment, but as argued in Section 3 (footnote) the new calculation [18] for 15 Λ N corrects the older calculations [12,13]. The calculation by Gal [18] finds a significantly smaller total decay rate for 1/2 + spin-parity than for a 3/2 + spin-parity assignment, by ∼ 2σ lower than the rate evaluated by us as listed in the table. Based on this argument a spin-parity 1/2 + is excluded and a spin-parity J π ( 15 Λ N g.s. ) = 3/2 + assignment is made.
Conclusions
We have reported a systematic study of MWD of p-shell Λ-hypernuclei by the FI-NUDA experiment, performing for the first time a magnetic analysis of spectra of π − 's from MWD of 7 Λ Li, 9 Λ Be, 11 Λ B and 15 Λ N. MWD decay rates Γ π − /Γ Λ have been evaluated and compared with previous measurements and theoretical calculations. The spin-parity assignments J π ( 7 Λ Li g.s. ) = 1/2 + and J π ( 11 Λ B g.s. ) = 5/2 + were confirmed and a new assignment, J π ( 15 Λ N g.s. ) = 3/2 + , was made based on the shape of the MWD spectra and the evaluated decay rates.
Aknowledgements
Fig. 1 .
1Inclusive binding energy spectra for good quality π − tracks coming from 12 C (left) and 16 O (right) targets. The continuous line is the best fit curve to the spectra; the dashed curves represent the contributions from the known hypernuclear states and the dot-dashed curve the Λ qf background. For 16 O the dotted curve parametrizes the K − decay in flight.
Fig. 2. Kinetic energy spectrum of MWD π − for 7 Λ Li before acceptance correction. The dot-dashed curve is the acceptance function, R, to be applied to the data.
Fig. 3 .
3Fig. 3. Upper part: kinetic energy spectrum of MWD π − for 7 Λ Li after acceptance correction. The solid curve is a gaussian fit to the peak in the spectrum, to compare with theoretical predictions in the lower part. Lower part: calculated major decay rates to final 7 Be states from [18], in red bars for 7 Λ Li ground-state spin-parity 1/2 + , and in blue bars for 7 Λ Li ground-state spin-parity 3/2 + (see text)
Fig. 4 .
4Fig. 4. Upper part: kinetic energy spectrum of MWD π − for 9 Λ Be after acceptance correction. The solid curve is a gaussian fit to the peak in the spectrum, to compare with theoretical predictions in the lower part. Lower part: calculated major decay rates to final 9 B states from [18], in red bars for 9 Λ Be ground-state spin-parity 1/2 + .
Fig. 5 .
5Fig. 5. Upper part: kinetic energy spectrum of MWD π − for 11 Λ B after acceptance correction. The solid curve is a two-gaussian fit to the peaks in the spectrum, to compare with theoretical predictions in the lower part; dashed curves are the single components. Lower part: calculated major decay rates to final 11 C states from [18], in red bars for 11 Λ B ground-state spin-parity 5/2 + , and in blue bars for 11 Λ B ground-state spin-parity 7/2 + .
Fig. 6 .
6Fig. 6. Upper part: kinetic energy spectrum of MWD π − for 15 Λ N after acceptance correction. The solid curve is a two-gaussian fit to the peaks in the spectrum, to compare with theoretical predictions in the lower part; dashed curves are the single components. Lower part: calculated major decay rates to final 15 O states from [18], in red bars for 15 Λ N ground-state spin-parity 3/2 + , and in blue bars for 15 Λ N ground-state spin-parity 1/2 + . Note that in this case the arbitrary bar width was reduced to 0.5 MeV to avoid overlap and to facilitate the comparison between the two spin hypotheses: indeed, the energies of the produced final states are practically the same.
Mean values of the gaussians representing hypernuclear states in global best fits to binding energy inclusive spectra for 7 Λ Li, 9 Λ Be, 12 Λ C and 16 Λ O: only the peaks contributing to the B.E. selections of Table 1 are considered. The FWHM of the peaks for 7 MeV, due to the malfunctioning of the outer drift chamber directly facing the target. Values of χ 2 /ndf for global fits (hypernuclear states and polynomial background) are also reported. See the references in the fifth column of Table 1 for comparison with previous measurements.1 g.s. B.E.
5.85 ± 0.13 6.30 ± 0.10 11.57 ± 0.04 12.42 fixed
2
B.E
3.84 ± 0.15 3.45 ± 0.10 8.4 fixed
6.800 ± 0.017
3
B.E.
1.9 ± 0.3
0.25 ± 0.22 5.9 fixed
1.85 fixed
4
B.E
0.39 ± 0.20
3.9 fixed −4.100 ± 0.004
5
B.E −2.000 ± 0.047
1.6 fixed
6
B.E
0.27 fixed
χ 2 /ndf
1.10
1.00
1.72
1.78
Table 2
Λ Li, 9
Λ Be and 12
Λ C is 2.31 MeV, while for 16
Λ O is 4.48
Corresponding author. Fax: +39 011 6707324; e-mail address: [email protected]
The authors are grateful to Prof. Toshio Motoba for providing details on the calculations reported in Refs.[11,12,13]and for the interest demonstrated in the FINUDA results.
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[] |
[
"Curve graphs for Artin-Tits groups of type B, A and C are hyperbolic",
"Curve graphs for Artin-Tits groups of type B, A and C are hyperbolic"
] |
[
"Matthieu Calvez ",
"Bruno A Cisneros De La Cruz "
] |
[] |
[] |
For n 3, we prove that the graphs of irreducible parabolic subgroups of the Artin-Tits groups of type An and Bn are both isomorphic to the curve graph of an (n + 1)-times punctured disk. For n 2, we show that the graphs of irreducible parabolic subgroups of the Artin-Tits groups of type An and Cn are isomorphic to some subgraphs of the curve graph of the (n + 2)-times punctured disk which are not quasi-isometrically embedded. We prove nonetheless that all these graphs are hyperbolic. the group of isotopy classes of orientation-preserving automorphisms of D n+1 which induce the identity on the boundary of D n+1 . The curve graph CG(D n+1 ) of the (n + 1)-times punctured disk D n+1 is the graph whose vertices are the isotopy classes of essential simple closed curves in D n+1 (closed curves without autointersection and enclosing at least 2 and at most n punctures) and where two vertices are joined by an edge if the corresponding isotopy classes of curves are distinct and admit disjoint representatives. The graph CG(D n+1 ) is equipped with the combinatorial metric d CG(Dn+1) defined by declaring each edge to have length one. There is a natural action of the Artin-Tits group A An on the set of isotopy classes of essential simple closed curves in D n+1 ; this action preserves disjointness, hence A An acts by isometries on the curve graph of D n+1 . A celebrated theorem of Masur and Minsky states that the curve graph of a surface is hyperbolic [15, Theorem 1.1]. The mapping class group of a surface Σ, together with the projections to the curve graphs of the subsurfaces of Σ constitute the typical example of what is now called a hierarchically hyperbolic space [20, 2, 3]. A natural and challenging question is whether any irreducible Artin-Tits group A Γ (not necessarily of spherical type) admits such a hierarchical structure. A first step forward is to define a hyperbolic space on which A Γ acts in the same way as the braid group acts on the curve graph of the punctured disk. For spherical type A Γ , the graph of irreducible parabolic subgroups C parab (Γ) was recently introduced by Cumplido, Gebhardt, González-Meneses and Wiest [10, Definition 2.3] as a candidate for this purpose. The graph of irreducible parabolic subgroups, however, can be defined for arbitrary Artin-Tits groups:Definition 1.1. [17, Definition 4.1] Let A Γ be an Artin-Tits group. Two distinct proper irreducible parabolic subgroups P and Q are called adjacent if one of the following conditions holds.• P ⊂ Q or Q ⊂ P ,• P ∩ Q = {1} and pq = qp for all p ∈ P and q ∈ Q.The graph of irreducible parabolic subgroups of A Γ is the graph C parab (Γ) whose vertices are the proper irreducible parabolic subgroups of A Γ and where two vertices are connected by an edge if and only if they correspond to adjacent parabolic subgroups.Note that C parab (Γ) is empty if Γ consists of a single vertex (A Γ is cyclic). If A Γ is dihedral, then by [17, Lemma 5.2, Theorem 5.3], C parab (Γ) has infinite diameter and is not connected, unless Γ consists of two vertices with no edge, in which case C parab (Γ) consists of two vertices and a single edge between them. Also, if Γ is not connected, C parab (Γ) is easily shown to have diameter 2. In this paper, we will always assume that Γ is connected and has at least 3 vertices. The graph C parab (Γ) is equipped with a metric, declaring each edge to have length one. We denote by d Γ the distance on C parab (Γ). There is a natural simplicial action of A Γ on C parab (Γ), by conjugation on parabolic subgroups. This action will be denoted on the right: given a proper irreducible parabolic subgroup P of A Γ and g ∈ A Γ , the parabolic subgroup g −1 P g will be denoted P g . Accordingly, we will always use the exponent notation for conjugacy in a group G: given g, h ∈ G, h g = g −1 hg. Most of the known properties of the graph C parab (Γ) are gathered in[7]and[17]. For example, if A Γ is irreducible and of spherical type (Γ with at least 3 vertices), then C parab (A Γ ) is connected and has infinite diameter ([17, Lemma 5.2] and [7, Corollary 4.13]). As noted in[17], hyperbolicity of C parab (Γ) is currently "the major challenge for research moving forward in the area". So far, the only known affirmative result concerns Artin's braid groups (type A n ). This can be synthetized as follows:Theorem 1.2.[10] Let n 3. The graph of irreducible parabolic subgroups C parab (A n ) is isomorphic to the curve graph of the (n + 1)-times punctured disk D n+1 . Therefore C parab (A n ) is hyperbolic.In this paper, we study the graphs of irreducible parabolic subgroups of three infinite families
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10.1112/tlm3.12029
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"https://arxiv.org/pdf/2003.04796v1.pdf"
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2003.04796
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ca2bfd13e2c0677fc7ed2f71ed96b6f22e2a63eb
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Curve graphs for Artin-Tits groups of type B, A and C are hyperbolic
March 11, 2020
Matthieu Calvez
Bruno A Cisneros De La Cruz
Curve graphs for Artin-Tits groups of type B, A and C are hyperbolic
March 11, 2020
For n 3, we prove that the graphs of irreducible parabolic subgroups of the Artin-Tits groups of type An and Bn are both isomorphic to the curve graph of an (n + 1)-times punctured disk. For n 2, we show that the graphs of irreducible parabolic subgroups of the Artin-Tits groups of type An and Cn are isomorphic to some subgraphs of the curve graph of the (n + 2)-times punctured disk which are not quasi-isometrically embedded. We prove nonetheless that all these graphs are hyperbolic. the group of isotopy classes of orientation-preserving automorphisms of D n+1 which induce the identity on the boundary of D n+1 . The curve graph CG(D n+1 ) of the (n + 1)-times punctured disk D n+1 is the graph whose vertices are the isotopy classes of essential simple closed curves in D n+1 (closed curves without autointersection and enclosing at least 2 and at most n punctures) and where two vertices are joined by an edge if the corresponding isotopy classes of curves are distinct and admit disjoint representatives. The graph CG(D n+1 ) is equipped with the combinatorial metric d CG(Dn+1) defined by declaring each edge to have length one. There is a natural action of the Artin-Tits group A An on the set of isotopy classes of essential simple closed curves in D n+1 ; this action preserves disjointness, hence A An acts by isometries on the curve graph of D n+1 . A celebrated theorem of Masur and Minsky states that the curve graph of a surface is hyperbolic [15, Theorem 1.1]. The mapping class group of a surface Σ, together with the projections to the curve graphs of the subsurfaces of Σ constitute the typical example of what is now called a hierarchically hyperbolic space [20, 2, 3]. A natural and challenging question is whether any irreducible Artin-Tits group A Γ (not necessarily of spherical type) admits such a hierarchical structure. A first step forward is to define a hyperbolic space on which A Γ acts in the same way as the braid group acts on the curve graph of the punctured disk. For spherical type A Γ , the graph of irreducible parabolic subgroups C parab (Γ) was recently introduced by Cumplido, Gebhardt, González-Meneses and Wiest [10, Definition 2.3] as a candidate for this purpose. The graph of irreducible parabolic subgroups, however, can be defined for arbitrary Artin-Tits groups:Definition 1.1. [17, Definition 4.1] Let A Γ be an Artin-Tits group. Two distinct proper irreducible parabolic subgroups P and Q are called adjacent if one of the following conditions holds.• P ⊂ Q or Q ⊂ P ,• P ∩ Q = {1} and pq = qp for all p ∈ P and q ∈ Q.The graph of irreducible parabolic subgroups of A Γ is the graph C parab (Γ) whose vertices are the proper irreducible parabolic subgroups of A Γ and where two vertices are connected by an edge if and only if they correspond to adjacent parabolic subgroups.Note that C parab (Γ) is empty if Γ consists of a single vertex (A Γ is cyclic). If A Γ is dihedral, then by [17, Lemma 5.2, Theorem 5.3], C parab (Γ) has infinite diameter and is not connected, unless Γ consists of two vertices with no edge, in which case C parab (Γ) consists of two vertices and a single edge between them. Also, if Γ is not connected, C parab (Γ) is easily shown to have diameter 2. In this paper, we will always assume that Γ is connected and has at least 3 vertices. The graph C parab (Γ) is equipped with a metric, declaring each edge to have length one. We denote by d Γ the distance on C parab (Γ). There is a natural simplicial action of A Γ on C parab (Γ), by conjugation on parabolic subgroups. This action will be denoted on the right: given a proper irreducible parabolic subgroup P of A Γ and g ∈ A Γ , the parabolic subgroup g −1 P g will be denoted P g . Accordingly, we will always use the exponent notation for conjugacy in a group G: given g, h ∈ G, h g = g −1 hg. Most of the known properties of the graph C parab (Γ) are gathered in[7]and[17]. For example, if A Γ is irreducible and of spherical type (Γ with at least 3 vertices), then C parab (A Γ ) is connected and has infinite diameter ([17, Lemma 5.2] and [7, Corollary 4.13]). As noted in[17], hyperbolicity of C parab (Γ) is currently "the major challenge for research moving forward in the area". So far, the only known affirmative result concerns Artin's braid groups (type A n ). This can be synthetized as follows:Theorem 1.2.[10] Let n 3. The graph of irreducible parabolic subgroups C parab (A n ) is isomorphic to the curve graph of the (n + 1)-times punctured disk D n+1 . Therefore C parab (A n ) is hyperbolic.In this paper, we study the graphs of irreducible parabolic subgroups of three infinite families
Introduction and background
This presentation can be encoded by a Coxeter graph Γ: the vertices are in bijection with the set S and two distinct vertices a, b are connected by an edge labeled m ab if m ab > 2 and labeled by ∞ if a, b satisfy no relation. The Artin-Tits group defined by the Coxeter graph Γ will be denoted A Γ . Because all the relations in the given presentation of A Γ are balanced, there is a homomorphism Γ : A Γ −→ Z assigning to each element g ∈ A Γ the exponent sum of any word on S representing it. The group A Γ is said to be irreducible if Γ is connected and dihedral if Γ has two vertices. The quotient by the normal subgroup generated by the squares of the elements in S is a Coxeter group denoted by W Γ . The Artin-Tits group A Γ is said to be of spherical type if W Γ is finite. A proper subset ∅ = X S generates a proper standard parabolic subgroup of A Γ which is naturally isomorphic to the Artin-Tits group A Ξ defined by the subgraph Ξ of Γ induced by the vertices in X [21]. A subgroup P of A Γ is parabolic if it is conjugate to a standard parabolic subgroup. The flagship example of an Artin-Tits group (of spherical type) is the braid group on (n + 1) strands -i.e. the Artin-Tits group defined by the graph A n shown in Figure 1(a). This group is isomorphic to the Mapping Class Group of an (n + 1)-times punctured closed disk D n+1 , that is Figure 1: Coxeter graphs: (a) of type A n ; (b) of type B n ; (c) of type A n ; (d) of type C n . As usual, we omit the label m ab whenever m ab = 3. The given labeling of the standard generators will be used throughout the paper.
of Artin-Tits groups closely related to Artin's braid groups, whose defining Coxeter graphs are depicted in Figure 1 (b)-(c)-(d). Here is a brief summary of the results in the paper. Firstly, we recall that the Artin-Tits group of (spherical) type B n can be realized as a finite index subgroup of Artin's braid group on (n + 1) strands A An [13]. We shall prove that this inclusion induces a graph isomorphism between the respective graphs of irreducible parabolic subgroups: Theorem 1.3. For n 3, the graphs C parab (B n ) and C parab (A n ) are isomorphic; hence both are hyperbolic.
Secondly, we focus on the Artin-Tits group of type A n . This group can be embedded in the Artin-Tits group of type B n+1 and A Bn+1 can be decomposed as a semi-direct product A An Z [13]. From this we obtain the following (ii) The graph C parab ( A n ) is isomorphic to a subgraph of C parab (B n+1 ).
(iii) The graph C parab ( A n ) has infinite diameter.
It turns out that the inclusion of C parab ( A n ) in C parab (B n+1 ) is not a quasi-isometric embedding -see Proposition 4.8; however using Theorems 1.3 and 1.2, we are able to see C parab ( A n ) as a subgraph of the curve graph of D n+2 . This allows us to prove, using results from [22]: Theorem 1.5. Let n 2. The graph C parab ( A n ) is hyperbolic.
Finally, we turn attention to the Artin-Tits group of type C n . We recall that it embeds as a finite index subgroup of Artin's braid group on (n + 2) strands A An+1 in a very similar way as A Bn embeds in A An [1]. We prove the following Theorem 1.6. Let n 2.
(i) The graph C parab ( C n ) is connected.
(ii) The graph C parab ( C n ) is isomorphic to a subgraph of C parab (A n+1 ).
(iii) The graph C parab ( C n ) has infinite diameter.
By contrast with the relation between C parab (B n ) and C parab (A n ), it turns out (Remark 5.10) that the inclusion of C parab ( C n ) in C parab (A n+1 ) is not a quasi-isometric embedding. We nevertheless will show, again seeing C parab ( C n ) as a subgraph of the curve graph of D n+2 and using results from [22]: Theorem 1.7. Let n 2. The graph C parab ( C n ) is hyperbolic.
It is important to note that our proofs of the hyperbolicity of C parab (B n ), C parab ( A n ) and C parab ( C n ) strongly depend on the hyperbolicity of curve graphs; it would be highly desirable to obtain independent algebraic proofs. The paper is arranged as follows. In Section 2 we review some results on parabolic subgroups of Artin-Tits groups, we explain the isomorphism given by Theorem 1.2 and we introduce a result from [22] which allows to establish the hyperbolicity of some subgraphs of the curve graph. Theorem 1.3 is proved in Section 3. Section 4 is devoted to the proofs of Theorems 1.4 and 1.5 while Theorems 1.6 and 1.7 are established in Section 5. Finally, Section 6 contains some closing remarks and open questions; in particular it is shown in Corollary 6.9 that the union of the normalizers of standard parabolic subgroups (the union of standard parabolic subgroups and the center, respectively) are hyperbolic structures on A Bn and that these structures are not equivalent -see [7].
Prerequisites
Artin-Tits groups and Coxeter groups
Let A Γ be any Artin-Tits group with standard generators S. Let W Γ = A Γ / s 2 | s ∈ S be the associated Coxeter group. The canonical projection π : A Γ W Γ admits a set section τ defined as follows -see for instance [4, Theorem 3.3.1(ii)]. For s ∈ S, denote bys its image in W Γ and S = {s | s ∈ S}. Let w ∈ W Γ and lets 1 . . .s r be a reduced expression for w, meaning a shortest word representative for w onS; then τ (w) = s 1 . . . s r . The kernel of the projection π is called the pure Artin-Tits group (or coloured Artin-Tits group) and is denoted P A Γ . Given a subset X of S, the standard parabolic subgroup of A Γ generated by X is denoted by A X . In the rest of this section, we assume that A Γ is of spherical type. In this case, W Γ contains a unique longest element w 0 (see for instance [11,Lemma 4.6.1]). Denote its lift τ (w 0 ) in A Γ by ∆ Γ . Whenever Γ is connected, it is known that A Γ has cyclic center generated by ∆ Γ or ∆ 2 Γ [5, Theorem 7.2]. Any proper irreducible parabolic subgroup P of A Γ is itself an irreducible Artin-Tits group of spherical type. The center of P is a cyclic group generated by an element z P (actually we have the generators z P and z −1 P and we choose z P so that its exponent sum Γ (z P ) is positive). We will always refer to this particular element as the central element of P . The following two results will be used throughout. The first one says in particular (with g = 1) that the element z P determines completely the subgroup P (and conversely). Proposition 2.3. [19, Theorem 5.2] Let A Γ be an Artin-Tits group of spherical type. Let P, Q be two irreducible parabolic subgroups of A Γ and let g ∈ A Γ . Then Q = P g if and only if z Q = z g P .
The second result reduces the definition of adjacency in the graph of irreducible parabolic subgroups to a very simple commutation condition between the respective central elements.
Correspondence between curves and parabolic subgroups
Recall that the braid group on (n + 1) strands -or Artin-Tits group A An -can be identified with the Mapping Class Group of a closed disk with (n + 1) punctures D n+1 . Assume that D n+1 is the closed disk in the complex plane of radius n+2 2 centered at n+2 2 and the punctures are at the integer numbers 1 i n + 1. For 1 i n, the standard generator σ i of A An corresponds to a clockwise half-Dehn twist along the horizontal segment [i, i + 1]. The group A An naturally acts -on the right-on the set of isotopy classes of essential simple closed curves in D n+1 . In the sequel we will simply write "essential curve" or even "curve" instead of "isotopy class of essential simple closed curve"; accordingly we will say that two curves are disjoint if the corresponding isotopy classes admit disjoint representatives. The result of the action of a braid y on a curve C will be denoted by C y . Finally, note that a curve C in D n+1 divides the disk in two connected components naturally seen as the interior and the exterior of C. Let I be a proper subinterval of [n] = {1, . . . , n}, that is
∅ = I [n], [(i < j < k) ∧ (i, k ∈ I)] =⇒ j ∈ I.
This defines a proper irreducible standard parabolic subgroup of A An , generated by {σ i | i ∈ I}; denote this subgroup by A I . As explained in [10,Section 2], there is a one-to-one correspondence
{curves in D n+1 } f − −−−− → {proper irreducible parabolic subgroups of A An }
which induces the graph isomorphism of Theorem 1.2. To a curve C in D n+1 , we associate the subgroup f(C) of A An consisting of all isotopy classes of automorphisms of D n+1 whose support is enclosed by C; this is a proper irreducible parabolic subgroup. In particular, given a proper subinterval I of [n], with m = min(I) and k = #I, the standard parabolic subgroup A I is the image f(C I ) of the circle C I surrounding the k + 1 punctures m, . . . , m + k -such a curve is called standard or round. The inverse correspondence is given by the -well-defined-formula A y I → C y I , for any proper subinterval I of [n] and any y ∈ A An . Let us see that the adjacency condition given in Proposition 2.4 turns f into a graph isomorphism. Let C be a curve in D n+1 , let P = f(C) and let z P be the central element of P . If C surrounds at least three punctures, then z P is the Dehn twist around the curve C. Otherwise, z P is the half-Dehn twist along an arc connecting the two punctures in the interior of C and which does not intersect C. Now, given two parabolic subgroups P 1 = f(C 1 ) and P 2 = f(C 2 ), z P1 and z P2 commute if and only if C 1 and C 2 are disjoint.
Hyperbolicity for some graphs of curves
In this section, we present a specialization of a theorem of Kate Vokes, which we will use as a criterion for proving the hyperbolicity of some subgraphs of the curve graph of the punctured disk. Consider the n-times punctured disk D n . A subsurface of D n is (the isotopy class of) a connected subsurface X of D n so that every boundary component of X is either ∂D n or an essential curve in D n . A simple closed curve in X is essential (in X) if it cannot be isotoped in X to a point, a puncture or a boundary component of X. By an annulus in D n we mean the subsurface consisting of a tubular neighborhood of some essential curve in D n . Given a curve C in D n and a subsurface X of D n we say that C and X are disjoint if they admit disjoint representatives; X is said to be a witness for C if C and X are not disjoint. In particular, X is not a witness for any of its boundary components. Two subsurfaces are disjoint if they admit disjoint representatives.
Theorem 2.5. [22, Corollary 1.5] LetĈ be a family of curves in D n ; let K be the subgraph of CG(D n ) induced byĈ, equipped with the combinatorial metric d K (each edge has length one). Let X be the set of witnesses for K, that is the set of all subsurfaces of D n which are a witness for every element ofĈ. Suppose:
(i) K is connected,
(ii) The action of P A An−1 on D n induces an isometric action of P A An−1 on K.
(iii) X contains no annulus.
(iv) No two elements of X are disjoint.
Then K is hyperbolic.
Remark 2.6. Let us check that the hypothesis of Theorem 2.5 match the hypothesis of [22,Corollary 1.5], namely that K is a twist-free multicurve graph having no pair of disjoint witnesses. The second half is exactly our clause (iv). The definition of a twist-free multicurve graph ([22, Definition 2.1]) consists of clauses (1)- (5). To see clauses (2) and (4), observe that each vertex of K being a curve in D n is in particular a multicurve and each pair of adjacent vertices in K are disjoint curves. Clauses (1) and (5) correspond to (i) and (iii) of Theorem 2.5, respectively. Clause (3) is adapted into clause (ii) of Theorem 2.5. The results in [22] work for compact surfaces (possibly with boundary). However, according to [16,Section 2.3], punctures can be treated as boundary components, so the results of [22] apply to punctured surfaces as well. In the case of the punctured disc, we have to replace the whole braid group by the pure braid group since mapping classes in [22] are required to fix the boundary pointwise. In the sequel we will find it more convenient to maintain the difference between punctures of the disk and "real" boundaries, thinking of punctures as "distinguished boundary components".
3 The graph C parab (B n ) A proper irreducible standard parabolic subgroup of A Bn is determined by a proper subinterval of [n]: for any proper subinterval I of [n], we denote by B I the proper irreducible standard parabolic subgroup of A Bn generated by {τ i | i ∈ I}. Notice that in A Bn , by Lemma 2.2, the standard generators fall into two conjugacy classes, namely each τ i , i 2 is conjugate to each other and not conjugate to τ 1 . In view of Lemma 2.1, each proper irreducible parabolic subgroup P of A Bn is exactly one of the following types:
• Type A if P is conjugate to B I for I ⊂ {2, . . . , n},
• Type B if P is conjugate to B I with 1 ∈ I.
There is a monomorphism
η : A Bn −→ A An τ i −→ σ 2 1 if i = 1, σ i if 2 i n.
The image of η is the subgroup P 1 of (n + 1)-strands 1-pure braids, that is the subgroup of all (n + 1)-strands braids in which the first strand ends in the first position. In other words, a braid y on (n + 1) strands belongs to P 1 if and only if π y (1) = 1, where π y = π(y) is the permutation in S n+1 = W An associated to y. A presentation for P 1 was given by Wei-Liang Chow [8] in 1948; for a proof that η defines an isomorphism between A Bn and P 1 , the reader may consult [13].
z A I = σ m I ((σ m I . . . σ m I +k I −1 ) . . . (σ m I σ m I +1 )σ m I ) 2 if k I = 1, if k I 2, z B I = τ m I ((τ m I . . . τ m I +k I −1 ) . . . (τ m I τ m I +1 )τ m I ) 2 (τ k I . . . τ 2 τ 1 τ 2 . . . τ k I ) . . . (τ 2 τ 1 τ 2 )τ 1 if k I = 1, if k I 2 and 1 / ∈ I, if k I 2 and 1 ∈ I.
The proof of the next lemma follows from an easy computation left to the reader.
Lemma 3.1. Let I be a proper subinterval of [n]. We have (i) η(B I ) = A I ∩ P 1 . (ii) η(z B I ) = z A I , except if I = {1}, in which case η(z B I ) = z 2 A I . Proposition 3.2.
Let I, J be proper subintervals of [n] and let g ∈ A Bn . The following are equivalent:
(i) B g I = B J , (ii) A η(g) I = A J .
Proof. Assume (i). According to Proposition 2.3, we have z g
B I = z B J . Assume first that I = {1} (hence J = {1}, by Lemmas 2.1 and 2.2) so that τ g 1 = τ 1 , which yields (σ 2 1 ) η(g) = σ 2 1 after applying the monomorphism η. Then [12, Theorem 2.2] ensures that also σ η(g) 1 = σ 1 whence A η(g) I = A J . If on the contrary I, J = {1}, Lemma 3.1(ii) yields that z η(g) A I = z A J from which (ii) follows using Proposition 2.3.
Conversely, assume (ii). By Proposition 2.3, we get z η(g)
A I = z A J . By Lemma 2.1, again I = {1}
if and only if J = {1}, as η(g) is 1-pure. In this case it follows that (σ 2 1 ) η(g) = σ 2 1 ; as η is injective, we get τ g 1 = τ 1 , which is to say B g I = B J . If on the contrary I = {1}, we can immediately pull back to A Bn the relation z η(g) A I = z A J using Lemma 3.1(ii) and we get z g B I = z B J which, by Proposition 2.3, implies (i). Now, observe that P 1 has index (n + 1) in A An . For 1 i n, define a i = σ i . . . σ 1 and a 0 is the trivial braid. Notice that π ai (i + 1) = 1, for all 0 i n. Given y ∈ A An , there is a unique i ∈ {0, . . . , n} so that ya i ∈ P 1 . In Figure 2(i)-(iii) are depicted a 2 , a 8 , a 5 ∈ A A9 , respectively.
(i) If i 0 + 1 < m I , i.e. if the puncture i 0 + 1 is to the left of C I , then C ai 0 I = C I . (ii) If i 0 + 1 > m I + k I , i.e. if the puncture i 0 + 1 is to the right of C I , then C ai 0 I = C I , where I = {i + 1 | i ∈ I}. (iii) If i 0 +1 ∈ [m I , m I +k I ], i.e. if the puncture i 0 +1 is in the interior of C I , then C ai 0 I = C am I −1 I .
Proof. The contents of Lemma 3.3 are depicted in Figure 2(i)-(iii). Only the third case might need a short proof: it suffices to observe that the crossings σ i0 , . . . , σ m I fix the curve C I as they are inner to it, so it only remains the action of σ m I −1 · · · σ 1 = a m I −1 .
For each I, we define C I = C am I −1 I
. Observe that C I is not a standard curve whenever m I > 1 (see the bottom part of Figure 2(iii)); however it can be transformed into a standard curve by the action of a 1-pure braid. To be precise, the action of the 1-pure braid transforms the curve C I into the round curve C [1,k I ] which surrounds the k I + 1 first punctures -see an example in Figure 2(iv).
ξ I = (σ m I . . . σ m I +k I −1 ) . . . (σ 2 . . . σ k I +1 ) if m I > 1, Id if m I = 1;
Using the correspondence f between curves and proper irreducible parabolic subgroups (Section 2.2), we have shown that
A ai 0 I = A I if i 0 + 1 < m I , A {i+1 | i∈I} if i 0 + 1 > m I + k I , A am I −1 I if i 0 + 1 ∈ [m I , m I + k I ]
is a standard parabolic subgroup in the first two cases and that in the third case, there is ξ I ∈ P 1 such that (A ai 0 Proof. Let ζ be any braid such that P ζ is a standard parabolic subgroup, say A J , and write m = min(J) and k = #J. [1,k] is, and we can take I = [1, k], α = ζa i0 ξ J , which is 1-pure as ζa i0 and ξ J are 1-pure.
I ) ξ I = A [1,k I ] is standard.Let i 0 = π ζ (1) − 1, in such a way that ζa i0 is 1-pure. If i 0 + 1 < m, then P ζai 0 = A ai 0 J = A J (Lemma 3.3 (i)) and we can take I = J, α = ζa i0 . If i 0 + 1 > m + k, then P ζai 0 = A ai 0 J = A J , where J = {j + 1 | j ∈ J} (Lemma 3.3 (ii)) and we can take I = J , α = ζa i0 . Finally, if i 0 + 1 ∈ [m, m + k], then P ζai 0 = A am−1 J (Lemma 3.3(iii)) is not necessarily standard but P ζai 0 ξ J = (A ai 0 J ) ξ J = AProposition 3.5. The formula B g I → A η(g) I
, where I is a proper subinterval of [n] and g ∈ A Bn , defines a bijective map H between the respective sets of vertices of C parab (A n ) and C parab (B n ).
Proof. By Proposition 3.2, the map H is well-defined and injective; by Proposition 3.4, it is surjective. According to [13], there is a monomorphism
θ : A An −→ A Bn+1 σ i −→ τ i+1 if i 1, τ −1 n+1 . . . τ −1 3 τ 1 τ 2 τ −1 1 τ 3 . . . τ n+1 if i = 0. Proposition 4.1. [13] Let ρ = (τ 1 τ 2 . . . τ n+1 ) −1 ∈ A Bn+1 . (i) ρ −(n+1) is the central element of A Bn+1 . (ii) θ( σ i ) ρ = θ( σ i+1 ) for 0 i n modulo n + 1. (iii) The group A Bn+1 can be decomposed as the semi-direct product A Bn+1 = θ( A n ) ρ ,
where the action of ρ is given by conjugation, as in (ii). k + 1 < l n, then (θ( A I )) ρ n−l+2 = τ 2 , . . . , τ k+n−l+3 , whence θ( A I ) is again conjugate to a proper irreducible standard parabolic subgroup. Finally, if P is not standard, there is some g ∈ A An such that P g = P 0 is standard; by the above discussion, θ(P 0 ) is a proper irreducible parabolic subgroup of A Bn+1 and from θ(P ) θ(g) = θ(P 0 ), we deduce that θ(P ) is itself a proper irreducible parabolic subgroup of A Bn+1 . Proposition 4.2 allows us to define a map Θ from the set of vertices of C parab ( A n ) to the set of vertices of C parab (B n+1 ), which sends a proper irreducible parabolic subgroup of A An to its image under the monomorphism θ. We first describe the image of Θ. Recall from Section 3 that each proper irreducible parabolic subgroup P of A Bn+1 is either of type A (P is conjugate to B I with 1 / ∈ I) or of type B (P is conjugate to B I with 1 ∈ I). If P is not cyclic, P is of type A if and only if P is an Artin-Tits group of type A k , k 2 and P is of type B if and only if P is an Artin-Tits group of type B k , k 2. (i) There exists a proper irreducible parabolic subgroup P of A An such that Q = θ(P ),
(ii) Q is of type A.
Proof. Suppose that Q = θ(P ), for some proper irreducible parabolic subgroup P of A An . If P is cyclic, P is conjugate in A An to σ i for some 0 i n; by definition of θ, we then see that Q is conjugate to τ 2 (recall that in A Bn+1 all τ i , i 2, are conjugate) so Q is of type A. If P is not cyclic, P is isomorphic to a braid group A A k for 2 k n. Note that Q = θ(P ) and P are isomorphic; therefore Q must be of type A. Conversely, suppose that Q is of type A. Assume first that Q is standard; that is Q = B I , for
I ⊂ {2, . . . , n + 1}. We see that Q = θ( A I ), where I = {i − 1 | i ∈ I}. If Q is not standard, let ζ ∈ A Bn+1 be such that Q ζ = B I is standard, for I ⊂ {2, . . . , n} and let I = {i − 1 | i ∈ I} so that Q ζ = B I = θ( A I ). If ζ = θ(x) for some x ∈ A An we are done as Q = θ( A I ) θ(x) −1 = θ( A x −1 I )
is the image of a proper irreducible parabolic subgroup of A An . If on the contrary ζ is not in the image of θ, there is some r ∈ Z and x ∈ A An such that ζρ r = θ(x) -see Proposition 4.1(iii). Using
Proposition 4.1(ii), we have Q ζρ r = (Q ζ ) ρ r = (θ( A I )) ρ r = θ( A {i+r, | i∈I } ),
where in the last term all indices are taken modulo n + 1. But this is equivalent to saying that Q = θ( A x −1 {i+r | i∈I } ), which achieves the proof.
Let K be the subgraph of C parab (B n+1 ) induced by the vertices of the form θ(P ), where P is a proper irreducible parabolic subgroup of A An ; equivalently, according to Lemma 4.3, K is the subgraph of C parab (B n+1 ) induced by those proper irreducible parabolic subgroups of type A.
Proposition 4.4. The map Θ defines an isomorphism between C parab ( A n ) and K.
Proof. The injectivity of Θ at the level of vertices follows from the injectivity of θ; by Lemma 4.3, Θ is surjective onto the vertices of K. Applying the monomorphism θ, it is easily seen that whenever P and Q are adjacent proper irreducible parabolic subgroups of A An (Definition 1.1), then θ(P ) and θ(Q) are adjacent as well. It follows that the map Θ induces a graph homomorphism. It remains to show that the inverse map is also a graph homomorphism, that is for every proper irreducible parabolic subgroups P, Q of A An , the condition that θ(P ) and θ(Q) are adjacent in C parab (B n+1 ) implies that P and Q are adjacent in C parab ( A n ). But this again follows from the injectivity of the homomorphism θ.
The last ingredient for Theorem 1.4 is to prove that the subgraph K is 1-dense in C parab (B n+1 ), namely:
Lemma 4.5. For every proper irreducible parabolic subgroup Q of A Bn+1 , there exists a proper irreducible parabolic subgroup P of A An so that d Bn+1 (Q, θ(P )) 1.
Proof. By Lemma 4.3, we may assume that Q is of type B, otherwise Q is already the image of a proper irreducible parabolic subgroup of A An . Suppose first that Q is standard, that is Q = τ 1 , . . . , τ k , for 1 k n − 1. We shall see that d Bn+1 (Q, θ( σ e )) = 1, for e = 1 or e = 2. Indeed, we have d Bn+1 ( τ 1 , θ( σ 2 ) = 1 and for k 2, d Bn+1 ( τ 1 , . . . , τ k , θ( σ 1 )) = 1. Suppose now that Q is not standard; let ζ ∈ A Bn+1 such that Q ζ is standard. As we have just seen, by taking e = 1 or 2, we have d Bn+1 (Q ζ , θ( σ e )) = 1. Let r ∈ Z and x ∈ A An be such that ζρ r = θ(x) (Proposition 4.1(iii)). Taking P = σ e+r x −1 , we obtain (using Proposition 4.1(ii) for the second equality),
d Bn+1 (Q, θ(P )) = d Bn+1 (Q, θ( σ e+r ) ρ −r ζ −1 ) = d Bn+1 (Q, θ( σ e ) ζ −1 ) = d Bn+1 (Q ζ , θ( σ e )) = 1.
We are now in position to prove Theorem 1. It would be conceivable that the subgraph K is quasi-isometric to C parab (B n+1 ), which would imply at once, in view of Theorem 1.3, the hyperbolicity of C parab ( A n ). However, this is not the case, as we will now see. In the sequel, we shall identify C parab (B n+1 ) with CG(D n+2 ), the curve graph of an (n + 2)-times punctured disk, thanks to Theorems 1.3 and 1.2:
CG(D n+2 ) f − −−−− → C parab (A n+1 ) H −1 −−−−−−−→ C parab (B n+1 ).
The following describes the curves which correspond to vertices of K under this identification. (ii) C does not surround the first puncture.
Proof. Suppose that C = f −1 H(P ), where P is a proper irreducible parabolic subgroup of type A of A Bn+1 . There is g ∈ A Bn+1 such that P = B g I , for I ⊂ {2, . . . , n + 1}. We have
C = f −1 H(P ) = f −1 (A η(g) I ) = C η(g) I .
Notice that C I does not surround the first puncture and that η(g) is 1-pure. It follows that C does not surround the first puncture either. Conversely, suppose that C does not surround the first puncture. Write Q = f(C). By Proposition 3.4, there is a 1-pure braid α and a proper subinterval I of [n + 1] such that Q α = A I is standard. We then also have C α = C I . As α is 1-pure, C I does not surround the first puncture, so that I does not contain 1. Let x ∈ A Bn+1 be such that η( Let K be the subgraph of CG(D n+2 ) induced by the curves which do not surround the first puncture; by Proposition 4.4 and Lemma 4.6, we have graph isomorphisms
C parab ( A n ) Θ − −−−− → K f −1 H − −−−−−− → K .
Our next task is to show that K matches hypothesis (i)-(iii) of Theorem 2.5. (iii) Given an essential curve C in D n+2 , we will see that there always exists some curve c in K which is disjoint from the annulus determined by C. Assume first that C does not surround the first puncture, so that C is a curve in K ; then C itself can be isotoped so that it does not intersect the annulus it determines. Suppose on the contrary that C surrounds the first puncture; if the exterior of C contains at least 2 punctures, we can take c to be any curve in the exterior of C. Otherwise the interior of C contains n + 1 3 punctures and we can choose c to be any curve surrounded by C and not enclosing the first puncture.
Denote by d CG the distance in the curve graph of D n+2 , by d K the distance in the graph K and by d K the distance in the graph K . As the next proposition is not needed in the sequel, its proof is only sketched and we refer the reader to [22] for a precise statement of the results used throughout.
Proposition 4.8. The subgraph K is not quasi-isometrically embedded in C parab (B n+1 ). More precisely, given any M > 0, there exists a pair of parabolic subgroups P, Q of type A of A Bn+1 with the following properties:
• P, Q are adjacent to B [1] so that d Bn+1 (P, Q) = 2; We are now ready for the proof of Theorem 1.5. As C parab ( A n ) is isomorphic to K it suffices to prove that K is hyperbolic. We need to check the remaining hypothesis of Theorem 2.5. The next lemma describes all possible witnesses for K ; its proof will achieve the demonstration of Theorem 1.5. Throughout, p 1 denotes the first puncture of D n+2 . Lemma 4.9. Let X be a subsurface of D n+2 . Then X is a witness for K if and only if one of the following holds.
(i) X = D n+2 or X = D n+2 \ D,
where D is the interior of an essential curve surrounding p 1 and exactly one other puncture.
(ii) X is the interior of an essential curve surrounding p 1 and n other punctures.
(iii) X = X \ D, where X is the interior of an essential curve surrounding p 1 and n other punctures and D is the interior of an essential curve surrounding p 1 and exactly one other puncture.
We will say that X is a witness of type (i), (ii) or (iii). Two witnesses for K are never disjoint.
Proof. The three types of subsurfaces in Lemma 4.9 are depicted in Figure 3. First, we check that all subsurfaces (i)-(iii) are witnesses for K : we see that the only curves which can fail to be witnessed by X must surround the first puncture. Conversely, let X be a witness for K . We shall distinguish two cases. First case. Suppose that ∂D n+2 is a boundary component of X. Assume that X = D n+2 . Therefore, there is at least some essential curve C of D n+2 which is a boundary component of X.
Assume that C is another essential curve of D n+2 which is a boundary component of X. Notice that C and C cannot be nested as X has to be connected. Then at least one of C or C does not surround p 1 and this provides a particular curve of K for which X is not a witness, a contradiction. Therefore X has exactly one essential curve C of D n+2 as a boundary component and C must surround p 1 . Moreover, C must surround exactly 2 punctures, otherwise there would exist a curve c in the interior of C, not surrounding p 1 , and X would fail to be a witness for this curve c. Letting D be the interior of C, we have shown that whenever X has ∂D n+2 as a boundary component, X = D n+2 \ D has to be of type (i).
Second case. Suppose that the boundary ∂D n+2 is not a boundary component of X. As X is connected, X has exactly one outermost boundary component which is an essential curve C of D n+2 . Again, C must surround p 1 , otherwise C is a curve from K which is disjoint from X and X fails to be a witness for K . Moreover, C must surround n + 1 punctures, otherwise the outer component of D n+2 \ C would contain at least two punctures and there would exist a curve c from K entirely contained in D n+2 \ X, contradicting that X is a witness for K . If X has no other boundary component, we have shown that X is of type (ii). Finally, suppose that X has another boundary component. This must be an essential curve C of D n+2 which is nested in C. Let C be another putative boundary component of X nested in C.
Then C and C cannot be nested, as X is connected. Therefore only one of C , C can surround p 1 : one of C , C is a curve from K for which X is not a witness, contradiction. Therefore there is exactly one boundary component C of X nested in C and C must surround p 1 . Moreover, C must surround exactly 2 punctures, for the same reasons as in the first case. Taking X to be the interior of C and D to be the interior of C , we have shown that X = X \ D is of type (iii). Finally, the last claim follows by a direct case-by-case inspection.
5 The graph C parab ( C n )
We start with a description of the proper irreducible standard parabolic subgroups of A Cn . Such a subgroup is determined by a proper subinterval of [n + 1]: if I is a proper subinterval of [n + 1], we denote by C I the proper irreducible standard parabolic subgroup of A Cn generated by
{ τ i | i ∈ I}.
Notice that, by Lemma 2.2, the standard generators of A Cn fall in three conjugacy classes, namely τ 1 ( τ n+1 , respectively) is the unique standard generator in its conjugacy class while each τ i , 2 i n, is conjugate to each other.
The following facts can be found in [1,Section 4]. There is a monomorphism
λ : A Cn −→ A An+1 τ i −→ σ 2 i if i = 1 or i = n + 1, σ i if 2 i n,
The image of λ is the subgroup P of (n + 2)-strands braids in which the first strand ends in the first position and the (n + 2)nd strand ends in the (n + 2)nd position. In other words, a braid y on (n + 2) strands belongs to P if and only if π y (1) = 1 and π y (n + 2) = n + 2, where π y = π(y) is the permutation in S n+2 = W An+1 associated to y.
Although A Cn is not of spherical type, we observe that each proper irreducible parabolic subgroup of A Cn is an irreducible Artin-Tits group of spherical type. This allows to associate to each proper irreducible parabolic subgroup P of A Cn its central element z P , as in Section 2.1. Given I a proper subinterval of [n + 1], the formula for z C I is very similar to the formulae given in Section 3 and we do not write it explicitly. The following is the analogue of Lemma 3.1:
Lemma 5.1. Let I be a proper subinterval of [n + 1]. Then (i) λ( C I ) = A I ∩ P, (ii) λ(z C I ) = z A I if I = {1}, {n + 1} z 2 A I if I = {1} or I = {n + 1}.
As A Cn is not of spherical type, we do not know a priori the analogues of Propositions 2.3 and 2.4. However, these analogues hold, as it will follow from the next Propositions 5.2 and 5.3.
Proposition 5.2. Let I, J be proper subintervals of [n + 1] and let g ∈ A Cn . The following are equivalent.
(i) C g I = C J , (ii) z g C I = z C J , (iii) A λ(g) I = A J .
Proof. Suppose (i). Then Z( C I ) g = Z( C J ); the exponent sum being invariant under conjugation, we get z g
C I = z C J , which is (ii).
Let us show that (ii) implies (iii). Suppose first that I = {1}. Then z g C I = τ g 1 . Recall that = Cn is the exponent sum of words on { τ 1 , . . . , τ n+1 }. Note that (z g C I ) = 1. Now, observe that for any subinterval K of [n + 1], (z C K ) = 1 if and only if #K = 1. From z g C I = z C J we deduce (z C J ) = 1 and #J = 1. Therefore z C J = τ i for some i ∈ [n + 1]. We then have τ g 1 = τ i . By Lemma 2.2, i = 1 and we deduce τ g 1 = τ 1 . It follows that (σ 2 1 ) λ(g) = σ 2 1 ; from [12, Theorem 2.2] we deduce that σ Finally, assume (iii) and let us show (i). We have, as λ(g) ∈ P and using Lemma 5.1(i),
λ( C g I ) = (λ( C I )) λ(g) = (A I ∩ P) λ(g) = A λ(g) I ∩ P λ(g) = A J ∩ P = λ( C J )
and the injectivity of λ ensures that C g I = C J , as required. Proposition 5.3. Let I, J be proper subintervals of [n + 1] and let g ∈ A Cn . The following are equivalent.
(i) C g I and C J are adjacent in C parab ( C n ). Proof. Suppose (i). If C g I ⊂ C J , then z g C I must commute with z C J , which is central in C J ; similarly if C J ⊂ C g I . Otherwise, C g I ∩ C J = {1} and uv = vu for each u ∈ C g I and v ∈ C J ; in particular z g C I and z C J commute. This is (ii). Suppose (ii). By Lemma 5.1(ii) (and [12, Theorem 2.2] for the case when I or J = {1} or {n + 1}), we obtain that z λ(g) A I and z A J commute, which is to say, according to Proposition 2.4, that A λ(g) I and A J are adjacent in C parab (A n+1 ), whence (iii) Suppose (iii) and let us show (i). Suppose first that A λ(g) I ⊂ A J . Then we have, as λ(g) ∈ P and using Lemma 5.1(i),
λ( C g I ) = λ( C I ) λ(g) = (A I ∩ P) λ(g) = A λ(g) I ∩ P ⊂ A J ∩ P = λ( C J ).
It follows that λ( C g I ) ⊂ λ( C J ) and injectivity of λ shows that C g I ⊂ C J . The proof when A J ⊂ A λ(g) I is similar. Assume finally that A λ(g) I ∩ A J = {1} and that both subgroups commute. Then using Lemma 5.1(i) and the fact that λ(g) ∈ P,
λ( C g I ∩ C J ) = λ( C g I ) ∩ λ( C J ) = (A λ(g) I ∩ P) ∩ (A J ∩ P) = A λ(g) I ∩ A J ∩ P = {1}.
By injectivity of λ, it follows that C g
I ∩ C J = {1}. As λ( C g I ) is a subgroup of A λ(g) I
, λ( C J ) is a subgroup of A J and A λ(g) I and A J commute mutually, we obtain that λ( C g I ) and λ( C J ) commute mutually and again by injectivity of λ, C g I and C J commute. Therefore we have shown that C g I and C J are adjacent.
Proposition 5.4. The formula Λ( C g I ) = A λ(g) I
, where I is a proper subinterval of [n + 1] and g ∈ A Cn , defines an injective map Λ from the set of proper irreducible parabolic subgroups of A Cn to the set of proper irreducible parabolic subgroups of A An+1 . Moreover, the map Λ induces a graph isomorphism from C parab ( C n ) onto its image.
Proof. By Proposition 5.2, the given formula provides a well-defined map Λ which is injective. By Proposition 5.3, Λ is a graph isomorphism onto its image.
Contrary to what happened with the embedding of C parab (B n ) in C parab (A n ), the map Λ is not surjective, as shows the following example. . In other words, we would have C λ(g) −1 = C I . As C surrounds both the first and the last punctures and λ(g) ∈ P, we deduce that the essential standard curve C I must surround both the first and the last punctures; this is a contradiction.
This example suggests a description of the image of Λ : C parab ( C n ) −→ C parab (A n+1 ). This is stated in the next proposition. Proof. Suppose first that P is in the image of Λ; that is P = A λ(g) I for some proper subinterval I of [n+1] and some g ∈ A Cn . We have f −1 (P ) = f −1 (A I ) λ(g) = C λ(g) I
. The curve C I cannot surround both the first and the last punctures as it is standard; assume for instance that it does not surround the first puncture (the other case is similar). Then as λ(g) ∈ P, we see that C λ(g) I = f −1 (P ) does not surround the first puncture either. Conversely, suppose that C = f −1 (P ) does not surround both the first and the last punctures. Suppose for instance that C does not surround the first puncture.We must show that P can be transformed into a standard parabolic subgroup (equivalently, that C can be transformed into a standard curve) by the action of some braid β in P. By Proposition 3.4, we know that there exists a 1-pure braid α and a proper subinterval I of [n + 1] such that P α = A I ; in other words C α = C I is a standard curve surrounding punctures m, . . . , m + k, for some m 2, k 1. Let j 0 = π α (n + 2) ∈ [2, . . . , n + 2], in such a way that α(σ j0 . . . σ n+1 ) ∈ P. The following analysis is analogue to Lemma 3.3. Suppose first that j 0 < m, that is the puncture numbered j 0 is to the left of C I . Then C α(σj 0 ...σn+1) = C
ξ I = (σ m+k−1 . . . σ m ) . . . (σ n . . . σ n−k+1 ) if m + k < n + 2, Id otherwise,
we obtain that ξ I ∈ P and that C α(σj 0 ...σn+1)ξ I = C [n−k+2,n+1] is standard and we can choose β = α(σ j0 . . . σ n+1 )ξ I ∈ P. The proof when C does not surround the last puncture is similar.
Lemma 5.7. The image of Λ is 1-dense in C parab (A n+1 ); that is, for every proper irreducible parabolic subgroup Q of A An+1 , there exists a proper irreducible parabolic subgroup P of A Cn such that d An+1 (Q, λ(P )) 1.
Proof. Using the identification from Section 2.2, we can show the lemma in the curve graph of D n+2 . By Proposition 5.6, it suffices to show that given a curve C surrounding both the first and the last punctures, it is possible to find another curve c disjoint from C and such that c does not surround both the first and the last punctures. If the exterior of C contains at least 2 punctures, we take any curve c in the exterior of C. Otherwise, the interior of C contains n + 1 3 punctures and we can choose in the interior of C any curve which does not surround both the first and the last punctures.
From now on, we denote by G the subgraph of CG(D n+2 ) induced by those vertices which are curves not enclosing both the first and the last punctures. We are now able to prove Theorem 1.6.
Proof of Theorem 1.6. (i) The same argument as given in [17,Lemma 5.2] shows the connectivity of C parab ( C n ). (ii) This follows from Proposition 5.4. More specifically, by Proposition 5.6, the graph C parab ( C n ) is isomorphic to the subgraph f(G) of C parab (B n+1 ). (iii) This follows from Lemma 5.7 in the same lines as Theorem 1.4(iii) follows from Lemma 4.5.
To show Theorem 1.7 it will suffice to show that G is hyperbolic. In the next two lemmas, we show that G satisfies the hypothesis of Theorem 2.5, from which we conclude that G is hyperbolic. This achieves the proof of Theorem 1.7.
Lemma 5.8. The graph G is connected. The natural action of the pure braid group P A An+1 on D n+2 induces an action by isometries on G. No annulus in D n+2 is a witness for G.
Proof. In view of Theorem 1.6(i), the proof is identical to the proof of Lemma 4.7.
For the following lemma, we denote by p 1 the first puncture and by p n+2 the last puncture.
Lemma 5.9. Let X be a subsurface of D n+2 . Then X is a witness for G if and only if one of the following holds.
(i) X = D n+2 or X = D n+2 \ D,
where D is the interior of an essential curve surrounding p 1 and p n+2 and no other puncture.
(ii) X is the interior of an essential curve surrounding p 1 , p n+2 and exactly n−1 other punctures.
(iii) X = X \ D, where X is the interior of an essential curve surrounding p 1 , p n+2 and exactly n − 1 other punctures and D is the interior of an essential curve surrounding p 1 and p n+2 and no other puncture.
We will say that X is a witness of type (i), (ii) or (iii). Two witnesses for G are never disjoint.
Proof. Mutatis mutandis, the proof is the same as the proof of Lemma 4.9.
Remark 5.10. The same argument as in the proof of Proposition 4.8 shows that the embedding of C parab ( C n ) in C parab (A n+1 ) is not quasi-isometric.
Miscellaneous
Additional results on parabolic subgroups of A An and A Cn
The work presented so far allows us to generalize to the Artin-Tits groups of type A n and C n some results which, to the best of our knowledge, were only known in the framework of Artin-Tits groups of spherical type. If P is a proper irreducible parabolic subgroup of A An or A Cn , P is an irreducible Artin-Tits group of spherical type so that we can define as in Section 2.1 the central element z P of P . The following two results are the analogues for A An of Propositions 2.3 and 2.4.
Proposition 6.1. Let P, Q be two proper irreducible parabolic subgroups of A An ; let g ∈ A An . Then P g = Q if and only if z g P = z Q .
Proof. The direct implication is obvious, considering the centers. Let us prove the converse. Given any proper irreducible parabolic subgroup P of A An , Proposition 4.2 says that θ(P ) is a proper irreducible parabolic of A Bn+1 . Note that θ induces an isomorphism between P and θ(P ); moreover, in the formulae defining θ we see that Bn+1 (θ(x)) = An (x) for every x ∈ A An . Therefore θ(z P ) = z θ(P ) , for every proper irreducible parabolic subgroup P of A An . Now assume z g P = z Q . We have θ(z g P ) = θ(z P ) θ(g) = z θ(g) θ(P ) = z θ(P ) θ(g) = z θ(P g ) , while θ(z Q ) = z θ(Q) . By Proposition 2.3, it follows that θ(P g ) = θ(Q). Finally, as θ is injective, P g = Q as desired.
Proposition 6.2. Let P, Q be two distinct proper irreducible parabolic subgroups of A An . Then P and Q are adjacent if and only if z P and z Q commute.
Proof. The argument given in the (i)⇒ (ii) part of the proof of Proposition 5.3 works for the direct implication. Conversely, suppose that z P and z Q commute. Then θ(z P ) = z θ(P ) and θ(z Q ) = z θ(Q) commute, whence by Proposition 2.4, θ(P ) and θ(Q) are adjacent in C parab (B n+1 ). By Proposition 4.4, this implies that P and Q are adjacent in A An .
We close this section with two statements about simultaneous standardization. This generalizes [14,Proposition 4.4] and [10, Section 11]. Proposition 6.3. Let P, Q be adjacent proper irreducible parabolic subgroups of A An . Then there exists ς ∈ A An so that P ς and Q ς are standard. and J of {τ 2 , . . . , τ n+1 } so that θ(P ) ζ = B I and θ(Q) ζ = B J . Let I = {i − 1 | i ∈ I} and J = {j − 1 | j ∈ J} so that θ(P ) ζ = θ( A I ) and θ(Q) ζ = θ( A J ).
Suppose first that ζ = θ(ς) for some ς ∈ A An . Then we have θ(P ς ) = θ(P ) ζ = θ( A I ) and
θ(Q ς ) = θ(Q) ζ = θ( A J ). We deduce from the injectivity of θ, P ς = A I and Q ς = A J , showing our claim. If on the contrary ζ / ∈ Imθ, in view of Proposition 4.1(iii), we can write ζ = ζ 0 ρ r for some r ∈ Z and ζ 0 = θ(ς) for some ς ∈ A An . In view of Proposition 4.1(ii), we then have
θ(P ς ) = θ(P ) ζ0 = θ( A I ) ρ −r = θ( A I ) and θ(Q ς ) = θ(Q) ζ0 = θ( A J ) ρ −r = θ( A J ),
where I = {i − r | i ∈ I }, J = {j − r | j ∈ J } and the indices are taken modulo n + 1. We obtain that P ς = A I and Q ς = A J are standard, as needed.
Proposition 6.4. Let P, Q be adjacent proper irreducible parabolic subgroups of A Cn . Then there exists ς ∈ A Cn so that P ς and Q ς are standard.
Proof. By Proposition 5.3, Λ(P ) and Λ(Q) are adjacent in A An+1 . By Proposition 5.6, C 1 = f −1 (Λ(P )) and C 2 = f −1 (Λ(Q)) are disjoint essential curves in D n+2 which do not surround both the first and the last punctures. We shall prove that there exists ζ 0 in P = λ(A Cn ) and I, J
proper subintervals of [n + 1] such that C ζ0 1 = C I and C 2 ζ0 = C J are standard curves; in other words Λ(P ) ζ0 = A I and Λ(Q) ζ0 = A J are standard parabolic subgroups of A An+1 . In this way, we will obtain ζ 0 = λ(ς), for ς ∈ A Cn , Λ(P ς ) = Λ(P ) ζ0 = Λ( C I ) and Λ(Q ς ) = Λ(Q) ζ0 = Λ( C J ), whence by injectivity of Λ (Proposition 5.4), P ς = C I and Q ς = C J are both standard as wanted. By [14,Proposition 4.4], there exist ζ ∈ A An+1 and I, J proper subintervals of [n + 1] such that C ζ 1 = C I and C 2 ζ = C J are standard curves. Let i 0 = π ζ (1) and j 0 = π ζ (n + 2). Notice that none of the curves C I , C J can surround the two punctures numbered i 0 and j 0 . If i 0 = 1 and j 0 = n + 2, we have nothing to do and we can take ζ 0 = ζ. If i 0 = 1 and j 0 < n + 2 (or j 0 = n + 2 and i 0 > 1, respectively) then define ζ = ζ(σ j0 . . . σ n+1 ) (or ζ = ζ(σ i0−1 . . . σ 1 ), respectively). In any case, ζ ∈ P. We follow the proof of Proposition 5.6 (Proposition 3.4, respectively). If C ζ e is not standard (e = 1, 2), which can happen only if the first puncture (the last puncture, respectively) is not enclosed by C ζ e , we can find a braid ξ ∈ P, with the first strand (the last strand, respectively) straight, so that C ζ ξ e is standard and we can choose ζ 0 = ζ ξ. We can therefore assume that i 0 > 1 and j 0 < n + 2. We then define
ζ = ζ(σ i0−1 . . . σ 1 )(σ j0 . . . σ n+1 ) if i 0 < j 0 , ζ(σ i0−1 . . . σ 1 )(σ j0+1 . . . σ n+1 ) if j 0 < i 0 .
Note that ζ ∈ P. The reader may check that, as none of C I and C J surrounds both punctures i 0 and j 0 , even if C ζ 1 or C ζ 2 is not standard we can always find ξ ∈ P so that C ζ ξ e is standard, for e = 1, 2, and ζ 0 = ζ ξ does the job as required.
Other graphs
In this section we show somme connections between the previous reIn [7] were described three different (infinite) generating sets of an Artin-Tits group A (of spherical type) and the corresponding Cayley graph of A was conjectured to be hyperbolic. A generating set X of a group G such that the associated Cayley graph Cay(G, X) is hyperbolic is called an hyperbolic structure for G. Suppose that A is of spherical type; these three sets are the following:
• X N P (A) is the union of the normalizers of the proper irreducible standard parabolic subgroups of A.
• X P (A) is the union of the union of the proper irreducible standard parabolic subgroup of A and the cyclic subgroup generated by the square of the element ∆ (recall that ∆ is the lift of the longest element of the corresponding Coxeter group).
• X abs (A) is the set of absorbable elements as described in [6].
The definition of X abs (A) rests on the Garside structure of A and can be defined only for A of spherical type; this is the only one of the three sets which is known to be a hyperbolic structure for all A of spherical type [6,Theorem 1]. The definition of X P (A) and X N P (A) can be easily extended to any Artin-Tits group (simply dropping the powers of ∆ in the definition of X P ). When A is of type A n , both Cayley graphs Cay(A An , X N P (A An )) and Cay(A An , X P (A An )) are hyperbolic. Indeed, Cay(A An , X N P (A An )) is quasi-isometric to the curve graph of the (n + 1)times punctured disk D n+1 [7, Proposition 3.2] while Cay(A An , X P (A An )) is quasi-isometric to A ∂ (D n+1 ), the graph of arcs in D n+1 both of whose endpoints lie in the boundary ∂D n+1 [7, Proposition 3.4]. More generally, for each A of spherical type (except dihedral), Cay(A, X N P (A)) is quasi-isometric to the graph of irreducible parabolic subgroups of A [7,Proposition 4.4]. From the work done so far, we can deduce that this still holds true for A of type A n or C n : Proposition 6.5. Assume that Γ is either A n or C n (n 2). Then C parab (Γ) is quasi-isometric to Cay(A Γ , X N P (A Γ )).
Proof. This will be a consequence of [7, Lemma 2.5]. By Theorems 1.4(i) and 1.6(i), C parab (Γ) is connected. The finite set of standard parabolic subgroups of A Γ is a set of representatives of the orbits of vertices under the action of A Γ . By Propositions 6.3 and 6.4, the finite set of edges bounded by two standard parabolic subgroups is a set of representatives of the orbits of edges under the action of A Γ . We conclude applying [7, Lemma 2.5].
Our results thus give a partial answer to [7, Conjecture 4.7]: Proposition 6.6. Suppose that Γ is either B n , n 3, A n or C n , n 2. Then X N P (A Γ ) is a hyperbolic structure for A Γ .
We now focus on the graph Cay(A Bn , X P (A Bn )). Recall that the monomorphism η : A Bn −→ A An expresses A Bn as the subgroup P 1 of 1-pure braids on (n+1) strands and that P 1 has index (n+1) in A An . For 1 i n, recall that a i = σ i . . . σ 1 and a 0 = Id; for each y ∈ A An , there is a unique i such that ya i ∈ P 1 . We first prove a lemma (see Lemma 3.3): Lemma 6.7. Let I be a proper subinterval of [n]; let m = min(I) and k = #I. Let 0 i 0 , j 0 n. Let I = {i + 1 | i ∈ I} (whenever max(I) < n). Let g ∈ A I . Suppose that z = a −1 i0 ga j0 is 1-pure.
1) If i 0 + 1 < m, then z = g ∈ A I .
2) If i 0 + 1 > m + k, then z = sh(g) ∈ A I 3) If i 0 + 1 ∈ [m, m + k], then z ξ I ∈ A {1,...,k} .
Here, ξ I ∈ P 1 is the braid defined in Section 3 which satisfies A am−1ξ I I = A [1,k] and sh denotes the shift homomorphism σ i → σ i+1 .
Proof. First observe that as z is 1-pure, we must have π g (i 0 + 1) = j 0 + 1. Moreover, as g ∈ A I , π g (i) = i for all i ∈ [1, m − 1] ∪ [m + k + 1, n + 1]. In particular, if i 0 + 1 < m or i 0 + 1 > m + k, we must have i 0 + 1 = j 0 + 1, whence i 0 = j 0 . 1) Suppose that i 0 + 1 < m; as we have just seen, z = a −1 i0 ga i0 . But a i0 commutes with all letters σ i , i ∈ I whence z = g.
2) Suppose that i 0 + 1 > m + k; again z = a −1 i0 ga i0 . We have, for all i ∈ I, where g = (σ −1 m . . . σ −1 i0 )g(σ j0 . . . σ m ) ∈ A I (the first and third factor may be trivial if i 0 + 1 = m or j 0 + 1 = m) so that z ∈ A
a −1 i0 σ i a i0 = (σ −1 1 . . . σ −1 i0 )σ i (σ i0 . . . σ 1 ) = σ −1 1 . . . σ −1 i (σ −1 i+1 σ i σ i+1 )σ i . . . σ 1 = σ −1 1 . . . σ −1 i (σ i σ i+1 σ −1 i )σ i . . . σ 1 = σ i+1 ,
An
Artin-Tits group is a group defined by a presentation involving a finite set of generators S (the standard generators) and where all the relations are as follows: every pair of generators satisfies at most one balanced relation of the form Π(a, b; m ab ) = Π(b, a; m ab ), with m a,b 2 and where for k 2if k is odd.
Theorem 1 . 4 .
14Let n 2.
(i) The graph C parab ( A n ) is connected.
For the rest of this section, given a proper subinterval I of [n], we shall denote m I = min(I) and k I = #I. The central elements z A I of A I and z B I of B I are given by the following formulae (see [18, Lemmas 3.1 and 4.1]):
Lemma 3. 3 .
3Let I be a proper subinterval of [n], m I = min(I) and k I = #I, so that the circle C I in D n+1 surrounds the punctures m I to m I + k I . Let 0 i 0 n.
Figure 2 :
2(i)-(iii) shows the action of a i on a standard curve C I for distinct values of i, illustrating the cases (i)-(iii) of Lemma 3.3. (iv) depicts the action of the 1-pure braid ξ I on C am I −1 I .
Proposition 3 . 4 .
34Let P be a proper irreducible parabolic subgroup of A An . There exists α ∈ P 1 and a proper subinterval I of [n] such that P α = A I .
Proof of Theorem 1.3. It suffices to observe that the maps H and H −1 from Proposition 3.5 preserve adjacencies. Let B g I and B J be two adjacent parabolic subgroups in C parab (B n ). Then by Proposition 2.4, z g B I and z B J commute. By Lemma 3.1(ii), z η(g) A I and z A J commute (notice that σ 2 1 and σ 1 have the same centralizer in A An by [12, Theorem 2.2]). Using Proposition 2.4 again, we see that H(B g I ) = A η(g) Iand H(B J ) = A J are adjacent in C parab (A n ). The proof that H −1 preserves edges is similar. Hyperbolicity of C parab (B n ) follows immediately from Theorem 1.2 and the hyperbolicity of the curve graph.
4
The graph C parab ( A n ) Let us start with a description of the proper irreducible standard parabolic subgroups of A An . We say that a proper subset I of {0, . . . , n} is a proper cyclic subinterval if I is either a proper subinterval of {0, . . . , n} or the union of two proper subintervals of {0, . . . , n} of the form [l, n] and [0, k], for some k, l with 1 k + 1 < l n. Every proper irreducible standard parabolic subgroup of A An is of the form A I = σ i | i ∈ I , for some proper cyclic subinterval I of {0, . . . , n}.
Proposition 4 . 2 .
42If P is a proper irreducible parabolic subgroup of A An , then θ(P ) is a proper irreducible parabolic subgroup of A Bn+1 . Proof. Suppose first that P is standard: P = A I for some proper cyclic subinterval of {0, . . . , n}. If I is a subinterval of {0, . . . , n} which does not contain 0, then θ(P ) is the subgroup of A Bn+1 generated by {τ i+1 | i ∈ I}, which is a proper irreducible standard parabolic subgroup. If I is a subinterval of {0, . . . , n} which contains 0, then in view of Proposition 4.1(ii), θ( A I ) ρ = τ i+2 , i ∈ I , whence ρ conjugates θ( A I ) to a proper irreducible standard parabolic subgroup. Similarly, if I is of the form [l, n] ∪ [0, k], with 1
Lemma 4 . 3 .
43Let Q be a proper irreducible parabolic subgroup of A Bn+1 . The following are equivalent.
4 .
4Proof of Theorem 1.4. Part (i), the connectivity of C parab ( A n ), can be proved with literally the same arguments as[17, Lemma 5.2]. Part (ii) is exactly Proposition 4.4. Let us show (iii). Let P be any vertex of C parab ( A n ) and let M > 0. We shall find a vertex P of C parab ( A n ) so that d An (P, P ) > M . We know that C parab (B n+1 ) has infinite diameter [7, Corollary 4.13, Proposition 4.4]; in particular there exists a vertex Q of C parab (B n+1 ) so that d Bn+1 (θ(P ), Q) > M + 1. By Lemma 4.5, there is a proper irreducible parabolic subgroup P of A An such that d Bn+1 (Q, θ(P )) 1. It follows that d Bn+1 (θ(P ), θ(P )) > M and by Proposition 4.4, we deduce that d An (P, P ) > M .
Lemma 4. 6 .
6Let C be a curve in D n+2 . The following are equivalent.(i) There exists a proper irreducible parabolic subgroup P of type A of A Bn+1 such that C = f −1 H(P ),
x) = α; then P = H −1 (Q) = H −1 (A α −1 I ) = B x −1 I is a proper irreducible parabolic subgroup of type A of A Bn+1 and we have C = f −1 H(P ).
Lemma 4. 7 .
7(i) K is connected. (ii) The action of the pure braid group P A An+1 on D n+2 induces an isometric action on K . (iii) No annulus in D n+2 can be a witness for all vertices of K . Proof. (i) Immediate in view of Theorem 1.4(i). (ii) The restriction to the pure braids of the natural action of A An+1 on D n+2 provides a simplicial action of P A An+1 on K .
•
d K (P, Q) > M .Proof. Using the isomorphism f −1 H, we restate our goal in terms of curves in D n+2 . Given any M > 0, we look for two curves a, b in D n+2 not surrounding the first puncture, disjoint from the circle C {1} and such that d K (a, b) > M . In view of Lemma 4.7 (see also Remark 2.6) and according to[22, Corollary 1.2], we have a distance formula in K from which we can deduce the claim. Fix M > 0. Fix any curve a not surrounding the first puncture and disjoint from the circle C {1} . Let D be the subdisk in D n+2 enclosed by C {1} and let X = D n+2 \D. Note that the subsurface X is homeomorphic to a disk with n+1 punctures and is a witness for every vertex of K . Note also that a is a curve in X. Let C 0 be the constant associated to K by [22, Corollary 1.2], let C > C 0 and let K 1 = K 1 (C), K 2 = K 2 (C) as given by[22, Corollary 1.2]. Define an element f of A An+1 by choosing a pseudo-Anosov mapping class of X which fixes each puncture of X (a pseudo-Anosov pure braid on (n + 1) strands) and doubling its first strand. Then f acts loxodromically on the curve graph of X[15, Proposition 4.6] and, choosing b as the image of a under a sufficiently high power of f , we can arrange that the distance d X (a, b) between a and b in the curve graph of X is bigger than max{C, M K 1 + K 2 }. Notice that b is disjoint from C {1} and that b does not surround the first puncture in D n+2 . The distance formula [22, Corollary 1.2] then says in particular that K 1 .d K (a, b) + K 2 is bounded from below by a sum of positive terms to which d X (a, b) contributes. It follows in particular that d K (a, b) d X (a,b)−K2K1> M , as desired. This finishes the proof of Proposition 4.8.
Figure 3 :
3Shaded, the different types of witnesses for K , the big dot represents p 1 .
=
= A {1} as desired. The proof is similar if we assume I = {n + 1} or J ∈ {{1}, {n + 1}}. Suppose then that I, J = {1}, {n + 1}. By Lemma 5.1(ii) we have z A J as desired.
(ii) z g C I and z C J commute. (iii) A λ(g) I and A J are adjacent in C parab (A An+1 ).
Example 5 . 5 .
55Consider a curve C in D n+2 which surrounds both the first and the last punctures. Let P = f(C) be the proper irreducible parabolic subgroup of A An+1 corresponding to C under the isomorphism of Section 2.2. Then P = Λ(Q) for every proper irreducible parabolic subgroup Q of A Cn . Otherwise, there would exist a proper subinterval I of [n + 1] and an element g ∈ A Cn such that P = Λ( C g I ) = A λ(g) I
Proposition 5 . 6 .
56Let P be a proper irreducible parabolic subgroup of A An+1 . Then P is in the image of Λ if and only if the curve f −1 (P ) in D n+2 does not surround both the first and the last punctures.
=
C I , where I = {i − 1 | i ∈ I} and we can choose β = α(σ j0 . . . σ n+1 ). Suppose that j 0 > m + k, i.e. the puncture numbered j 0 is to the right of C I . Then C α(σj 0 ...σn+1) = C σj 0 ...σn+1 I = C I and we can choose again β = α(σ j0 . . . σ n+1 ). Finally, suppose that j 0 ∈ [m, m + k], that is the puncture numbered j 0 is enclosed by C I . ThenC α(σj 0 ...σn+1) = C σj 0 ...σn+1 I = C σ m+k ...σn+1 Ineed not be standard. However, setting
and the claim follows.3) Suppose that i 0 + 1 ∈ [m, m + k]; then also j 0 + 1 ∈ [m, m + k]. We have z = a −1 i0 ga j0 = (σ −1 1 . . . σ −1 m−1 )(σ −1 m . . . σ −1 i0 gσ j0 . . . σ m )(σ m−1 . . . σ 1 ) = (σ −1 1 . . . σ −1 m−1 )g (σ m−1 . . . σ 1 ),
Lemma 2.1. [19, Theorem 4.1] Let X, Y ⊂ S. The following are equivalent. (i) The subgroups A X and A Y are conjugate in A Γ , (ii) the sets X, Y are conjugate in A Γ . Lemma 2.2. [19, Corollary 4.2] Let s, t ∈ S; then s and t are conjugate in A Γ if and only if there is a path in Γ which connects s and t and follows only edges with odd labels.
Proposition 2.4. [10, Theorem 2.2] Let A Γ be an Artin-Tits group of spherical type. Let P, Q be two distinct irreducible parabolic subgroups of A Γ . Then P, Q are adjacent (Definition 1.1) if and only if z P and z Q commute.
Proof. By Proposition 4.4, θ(P ) and θ(Q) are adjacent in C parab (B n+1 ). As A Bn+1 is of spherical type, by[10, Section 11], there is ζ ∈ A Bn+1 so that θ(P ) ζ and θ(Q) ζ are standard parabolic subgroups of A Bn+1 . By Lemma 4.3, θ(P ), θ(Q) and their respective standard conjugates θ(P ) ζ and θ(Q) ζ are parabolic subgroups of type A of A Bn+1 . Therefore there exist proper subintervals I
Acknowledgements This work builds on an early draft by the first author; he thanks Juan González-Meneses for pointing a mistake in that preliminary version. Most of the work was done during a visit of the second author at Universidad de Valparaíso supported by FONDECYT Regular 1180335. The first author is supported by FONDECYT Regular 1180335, MTM2016-76453-C2-1-P and FEDER. Heartfelt thanks to Bert Wiest for bringing the reference[22]to our attention.
then by definition of ξ I , z ξ I ∈ A [1,k] as claimed. then by definition of ξ I , z ξ I ∈ A [1,k] as claimed.
The monomorphism η : A Bn −→ A An induces a quasi-isometry between the respective Cayley graphs Cay(A Bn , X P (A Bn )) and Cay(A An. Proposition 6.8. Let n 3Proposition 6.8. Let n 3. The monomorphism η : A Bn −→ A An induces a quasi-isometry between the respective Cayley graphs Cay(A Bn , X P (A Bn )) and Cay(A An , X P (A An )).
Throughout the proof, the notation ||x|| X P (Bn) means the word length of x ∈ A Bn with respect to the generating set X P. B nProof. Throughout the proof, the notation ||x|| X P (Bn) means the word length of x ∈ A Bn with respect to the generating set X P (B n ).
1(i)) that given a proper subinterval I of [n], η(B I ) = A I ∩ P 1 ; therefore, whenever x, y ∈ A Bn differ by an element of B I , η(x) and η(y) differ by an element of A I . Similarly, if x, y ∈ A Bn differ by ∆ 2k Bn for some k ∈ Z. We know (Lemma. 3then η(x) and η(y) differ by η(∆ 2kWe know (Lemma 3.1(i)) that given a proper subinterval I of [n], η(B I ) = A I ∩ P 1 ; therefore, whenever x, y ∈ A Bn differ by an element of B I , η(x) and η(y) differ by an element of A I . Similarly, if x, y ∈ A Bn differ by ∆ 2k Bn for some k ∈ Z, then η(x) and η(y) differ by η(∆ 2k
. An Bn ) = ∆ 4k, Bn ) = ∆ 4k An .
Therefore, η induces a 1-Lipschitz map from Cay(A Bn , X P (A Bn )) to Cay(A An , X P (A An )). Therefore, η induces a 1-Lipschitz map from Cay(A Bn , X P (A Bn )) to Cay(A An , X P (A An )).
Let us see that ψ is a quasi-inverse for η. Indeed, we have by construction, for x ∈ A Bn , ψ • η(x) = x. Conversely, for y ∈ A An , y and η • ψ(y) differ by a i , for some 0 i n; however a i can be written as a product of at most 2 elements in X P (A An ), whence the distance between y and η • ψ(y) in Cay. . ∈ {0, A An −→ A Bn in the following way. Given y ∈ A An. , n} (it is unique!) be such that ya i ∈ P 1 = Im(η), and define ψ(y) = η −1 (ya i ). A An , X P (A An )) is at most 2We define a map ψ : A An −→ A Bn in the following way. Given y ∈ A An , let i ∈ {0, ..., n} (it is unique!) be such that ya i ∈ P 1 = Im(η), and define ψ(y) = η −1 (ya i ). Let us see that ψ is a quasi-inverse for η. Indeed, we have by construction, for x ∈ A Bn , ψ • η(x) = x. Conversely, for y ∈ A An , y and η • ψ(y) differ by a i , for some 0 i n; however a i can be written as a product of at most 2 elements in X P (A An ), whence the distance between y and η • ψ(y) in Cay(A An , X P (A An )) is at most 2.
We show finally that the map ψ is Lipschitz. Let y, y ∈ A An be adjacent in Cay(A An. P (A An ). We show finally that the map ψ is Lipschitz. Let y, y ∈ A An be adjacent in Cay(A An , X P (A An ));
We have unique i 0 , j 0 so that ya i0 ∈ P 1 and y a j0 ∈ P 1 . Then also a −1 i0 ga j0 is 1-pure. Let x = η −1 (ya i0 ) and x = η −1 (y a j0 ); we must estimate the distance in Cay(A Bn , X P (A Bn )) between x and x . If g = ∆ 2k An , g is pure. As a −1 i0 ga j0 is 1-pure, we must have i 0 + 1 = π g (i 0 + 1) = j 0 + 1, whence , the distance between x and x is at most ||∆ Bn || X P (A Bn ) + 1 (and this bound is 1 if k is even). standard parabolic subgroup of A An . Then by Lemma 3.1(i), x −1 x either belongs to a standard parabolic subgroup of A Bn or to the conjugate by η −1 (ξ −1 I ) of a standard parabolic subgroup of A Bn. that is g = ∆ 2k An for k ∈ Z or g ∈ A I for some proper subinterval I of. It follows that the distance between x and x in Cay(A Bn , X P (A Bn )) is at most 1 + 2||η −1 (ξ I )|| X P (A Bnwrite g = y −1 y . This means that g ∈ X P (A An ), that is g = ∆ 2k An for k ∈ Z or g ∈ A I for some proper subinterval I of [n]. We have unique i 0 , j 0 so that ya i0 ∈ P 1 and y a j0 ∈ P 1 . Then also a −1 i0 ga j0 is 1-pure. Let x = η −1 (ya i0 ) and x = η −1 (y a j0 ); we must estimate the distance in Cay(A Bn , X P (A Bn )) between x and x . If g = ∆ 2k An , g is pure. As a −1 i0 ga j0 is 1-pure, we must have i 0 + 1 = π g (i 0 + 1) = j 0 + 1, whence , the distance between x and x is at most ||∆ Bn || X P (A Bn ) + 1 (and this bound is 1 if k is even). standard parabolic subgroup of A An . Then by Lemma 3.1(i), x −1 x either belongs to a standard parabolic subgroup of A Bn or to the conjugate by η −1 (ξ −1 I ) of a standard parabolic subgroup of A Bn . It follows that the distance between x and x in Cay(A Bn , X P (A Bn )) is at most 1 + 2||η −1 (ξ I )|| X P (A Bn ) .
A Bn ) is a hyperbolic structure for A Bn : Cay(A Bn , X P (A Bn )) is hyperbolic. (ii) The identity map from Cay(A Bn , X P (A Bn )) to Cay(A Bn. X , Bn )) is Lipschitz but not a quasi-isometry. X P (A Bn ) is a hyperbolic structure for A Bn : Cay(A Bn , X P (A Bn )) is hyperbolic. (ii) The identity map from Cay(A Bn , X P (A Bn )) to Cay(A Bn , X N P (A Bn )) is Lipschitz but not a quasi-isometry.
(ii) follows immediately from Proposition 6.8, Theorem 1.3 and the corresponding fact for A An. i) is a direct consequence of Proposition 6.8 and [7, Proposition 4.1(ii)].Proof. (i) is a direct consequence of Proposition 6.8 and [7, Proposition 4.1(ii)]. (ii) follows im- mediately from Proposition 6.8, Theorem 1.3 and the corresponding fact for A An [7, Proposi-
A Γ )) for Γ = A n or C n . Are these graphs hyperbolic?. A Γ , Describe the graphs Cay. Does [7, Conjecture 4.11(i)] still hold for these groupsDescribe the graphs Cay(A Γ , X P (A Γ )) for Γ = A n or C n . Are these graphs hyperbolic? Does [7, Conjecture 4.11(i)] still hold for these groups?
Is there a quasiisometry between C AL (A An ) and C AL (A Bn )? A negative answer would yield a negative answer to [7, Question 4.14] which asked whether C parab (A) and C AL (A) are quasi-isometric. A) is a graph quasiisometric to the Cayley graph of A built on the generating set X abs (A). This would provide a strong evidence for disproving the conjecture that the additional length graph of the braid group is quasi-isometric to the curve graph of the punctured disk [6, Conjecture 1Recall that for A of spherical type, the additional length graph C AL (A) is a graph quasi- isometric to the Cayley graph of A built on the generating set X abs (A). Is there a quasi- isometry between C AL (A An ) and C AL (A Bn )? A negative answer would yield a negative answer to [7, Question 4.14] which asked whether C parab (A) and C AL (A) are quasi-isometric. This would provide a strong evidence for disproving the conjecture that the additional length graph of the braid group is quasi-isometric to the curve graph of the punctured disk [6, Conjecture 1].
Is it possible to show that C parab (D n ) is hyperbolic, using the same kind of techniques? This would likely yield also the hyperbolicity of C parab ( B n ) and C parab. D nIs it possible to show that C parab (D n ) is hyperbolic, using the same kind of techniques? This would likely yield also the hyperbolicity of C parab ( B n ) and C parab ( D n ).
Braid pictures for Artin groups. D Allcock, Trans. Amer. Math. Soc. 3549D. Allcock, Braid pictures for Artin groups, Trans. Amer. Math. Soc. 354 (9), 2002, 3455- 3474.
Hierarchically hyperbolic spaces I: curve complexes for cubical groups. J Behrstock, M Hagen, A Sisto, Geom. Top. 21J. Behrstock, M. Hagen, A. Sisto, Hierarchically hyperbolic spaces I: curve complexes for cubical groups, Geom. Top. 21, 2017, 1731-1804.
J Behrstock, M Hagen, A Sisto, arXiv:1509.00632Hierarchically hyperbolic spaces II: Combination theorems and the distance formula. J. Behrstock, M. Hagen, A. Sisto, Hierarchically hyperbolic spaces II: Combination theorems and the distance formula, arXiv:1509.00632
A Bjorner, F Brenti, Combinatorics of Coxeter groups. Springer231A. Bjorner, F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Vol. 231, Springer, 2005.
. E Brieskorn, K Saito, Artin-Gruppen Und Coxeter-Gruppen, Invent. Math. 17E. Brieskorn, K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17, 1972, 245- 271.
Curve graphs and Garside groups. M Calvez, B Wiest, Geom. Ded. 1881M. Calvez, B. Wiest, Curve graphs and Garside groups, Geom. Ded. 188 (1), 2017, 195-213.
M Calvez, B Wiest, arXiv:1904.02234Hyperbolic structures on Artin-Tits groups of spherical type. M. Calvez, B. Wiest, Hyperbolic structures on Artin-Tits groups of spherical type, arXiv:1904.02234
On the algebraical braid group. W.-L Chow, Ann. of Math. 2W.-L. Chow, On the algebraical braid group, Ann. of Math. (2) 49, 1948, 654-658.
On the minimal positive standardizer of a parabolic subgroup of an Artin-Tits group. M Cumplido, J. Alg. Comb. 493M. Cumplido, On the minimal positive standardizer of a parabolic subgroup of an Artin-Tits group, J. Alg. Comb. 49 (3), 2019, 337-359.
On parabolic subgroups of Artin-Tits groups of spherical type. M Cumplido, V Gebhardt, J González-Meneses, B Wiest, Adv. Math. 352M. Cumplido, V. Gebhardt, J. González-Meneses, B. Wiest, On parabolic subgroups of Artin- Tits groups of spherical type, Adv. Math. 352, 2019, 572-610.
The geometry and topology of Coxeter groups. M Davis, London Math. Soc. Monographs. 32Princeton University PressM. Davis, The geometry and topology of Coxeter groups, London Math. Soc. Monographs 32, Princeton University Press, 2007.
Centralisers in the braid group and singular braid monoid, L'enseignement mathématique 42. R Fenn, D Rolfsen, J Zhu, R. Fenn, D. Rolfsen, J. Zhu, Centralisers in the braid group and singular braid monoid, L'enseignement mathématique 42, 1996, 75-96.
A geometric and algebraic description of annular braid groups. A Kent, D Peifer, Int. J. Alg. Comp. 12A. Kent, D. Peifer, A geometric and algebraic description of annular braid groups, Int. J. Alg. Comp. 12, 2002, 85-97.
A Garside theoretic approach to the reducibility problem in braid groups. E.-K Lee, S.-J Lee, J. Algebra. 3202E.-K. Lee, S.-J. Lee, A Garside theoretic approach to the reducibility problem in braid groups, J. Algebra 320 (2), 2008, 783-820.
H Masur, Y Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138. H. Masur, Y. Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138, 1999, 103-149.
Geometry of the complex of curves II: Hierarchical structure. H Masur, Y Minsky, Geom. Funct. Anal. 104H. Masur, Y. Minsky, Geometry of the complex of curves II: Hierarchical structure, Geom. Funct. Anal. 10 (4), 2000, 902-974.
R Morris-Wright, arXiv:1906.07058Parabolic Subgroups of FC-Type Artin groups. R. Morris-Wright, Parabolic Subgroups of FC-Type Artin groups, arXiv:1906.07058
Centralizers of parabolic subgroups of Artin groups of type A, B and D. L Paris, J. Algebra. 1962L. Paris, Centralizers of parabolic subgroups of Artin groups of type A, B and D, J. Algebra 196 (2), 1997, 400-435.
Parabolic subgroups of Artin groups. L Paris, J. Algebra. 196L. Paris, Parabolic subgroups of Artin groups, J. Algebra 196, 1997, 369-399.
What is a hierarchically hyperbolic space?. A Sisto, arXiv:1707.00053A. Sisto, What is a hierarchically hyperbolic space?, arXiv:1707.00053
The Homotopy Type of Complex Hyperplane Complements. H Van Der Lek, NijmegenPh.D. thesisH. Van der Lek, The Homotopy Type of Complex Hyperplane Complements, Ph.D. thesis (1983), Nijmegen
K Vokes, arXiv:1711.03080Hierarchical hyperbolicity of graphs of multicurves. K. Vokes, Hierarchical hyperbolicity of graphs of multicurves, arXiv:1711.03080.
|
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[
"Three Random Tangents to a Circle",
"Three Random Tangents to a Circle"
] |
[
"Steven R Finch "
] |
[] |
[] |
Among several things, we find the side density for random triangles circumscribing the unit circle and calculate that its median is 5.5482.... An analogous exact computation for perimeter density remains open.
| null |
[
"https://arxiv.org/pdf/1101.3931v1.pdf"
] | 118,147,022 |
1101.3931
|
074cb0842a34b1074e5bcd9cd76bc3baaa090947
|
Three Random Tangents to a Circle
20 Jan 2011 January 20, 2011
Steven R Finch
Three Random Tangents to a Circle
20 Jan 2011 January 20, 2011
Among several things, we find the side density for random triangles circumscribing the unit circle and calculate that its median is 5.5482.... An analogous exact computation for perimeter density remains open.
Let us initially discuss two random tangents to the unit circle. Without loss of generality, let one of the tangents be the vertical line passing through the point (−1, 0). Let the other tangent pass through the point (cos(θ), sin(θ)), where θ is uniformly distributed on the interval [0, π]. Hence it has slope − cot(θ) and is the unique such line touching the upper semicircle. The two lines cross at (x, y), where x = −1, y − sin(θ) = − cot(θ) (x − cos(θ)) thus h = cot(θ/2) is the (positive) height of the intersection point. We wish to determine the probability density of h. Since
d dθ cot θ 2 = − 1 2 csc θ 2 2 the density is [1] 1 π 1 2 csc θ 2 2 θ=2 arccot(h) = 2 π cot θ 2 2 + 1 θ=2 arccot(h) = 2 π 1 h 2 + 1
for h > 0, that is, the one-sided Cauchy distribution. The mean of h is infinite, as is well-known; its median is 1.
Let us now discuss three random tangents to the unit circle, incorrectly modeled. Take first and second lines exactly as in the preceding, and define similarly a third line to touch the lower semicircle, independent of the second. We study h + k, where h is as before and k is the (positive) depth of the intersection between first and third lines. Let ℓ = h + k. The density of ℓ is the convolution [
1] 4 π 2 ℓ 0 1 (ℓ − k) 2 + 1 1 k 2 + 1 dk = 8 π 2
ℓ arctan (ℓ) + ln (ℓ 2 + 1) (ℓ 2 + 4) ℓ for ℓ > 0. Why is this model incorrect? Clearly E (θ) = π/2, not π/3, hence the three contact points are not equidistant (on average). Consider also the triangle T determined by the three lines: ℓ is the vertical side of T , but cannot be regarded as an "arbitrary" side. The assumption that θ ∼ Uniform[0, π] requires change.
Another change (less a requirement than a preference) involves the relationship between T and the unit circle C. Clearly C is an incircle of T if and only if there is no semicircle containing all three contact points. Otherwise C is an excircle of T . We wish to refine our model (which presently incorporates both incircles and excircles) so that the density of ℓ is based on incircles alone. Naturally ℓ > 2; the infimum 2 occurs in the limit as second and third lines both become horizontal. The density for this refined model is given in the next section; a related optimization problem appears at the end. As far as we know, these results have not appeared in the random triangle literature before [2].
Unit Inradius
Without loss of generality, let the first tangent be the vertical line passing through the point (−1, 0). Let the second tangent pass through the point (cos(α), sin(α)); let the third tangent pass through the point (cos(β), − sin(β)). It is assumed that the bivariate density for angles α, β is 2/π 2 if 0 < α < π, 0 < β < π and α + β < π, 0 otherwise.
It is best to think of α being measured in a counterclockwise direction (as is customary) and β being measured in a clockwise direction. The condition α+β < π prevents contact points from all crowding onto any semicircle (think of what happens when α + β = π). Dependency between α and β makes our analysis more complicated than earlier.
As a check, the univariate density for α is
2(π − α)/π 2 if 0 < α < π, 0
otherwise.
Thus points on C far away from (−1, 0) are favored (that is, small angles α are weighted more heavily than large α) and E (α) = π/3. We wish to determine the bivariate density of h = cot(α/2), k = cot(β/2). Via a Jacobian determinant argument, the density is [1] Let ℓ = h + k. The density of ℓ is the convolution
8 π 2 b(ℓ) a(ℓ) 1 (ℓ − k) 2 + 1 1 k 2 + 1 dk
where limits of integration a(ℓ), b(ℓ) are found from (ℓ−k)k > 1, hence k 2 −ℓ k+1 < 0; the zeroes of the quadratic are
a(ℓ) = ℓ − √ ℓ 2 − 4 2 > 0, b(ℓ) = ℓ + √ ℓ 2 − 4 2 < ℓ
and these are real because ℓ > 2. Integrating, we obtain the density of ℓ to be 16
π 2 f (ℓ) + g(ℓ) (ℓ 2 + 4) ℓ for ℓ > 2, where f (ℓ) = ℓ arctan ℓ + √ ℓ 2 − 4 2 − ℓ arctan ℓ − √ ℓ 2 − 4 2 , g(ℓ) = ln ℓ + √ ℓ 2 − 4 − ln ℓ − √ ℓ 2 − 4 .
The mean of ℓ is infinite; its median 5.5482039188784452776442997... can be computed to high numerical precision as a consequence of our exact density formula. See [3] for experimental confirmation of our work. Having derived the density of an arbitrary side, let us briefly mention other properties. The triangle T , under the condition that it circumscribes the circle C, is acute with probability 1/4 [4,5]. Since the inradius of T is 1, the area of T is one-half the perimeter of T [6,7] Unfortunately the perimeter density of T (as well as a trivariate density for sides) remains analytically intractible.
Optimization Problem
Of all triangles circumscribing the unit circle, an equilateral triangle minimizes the perimeter p. The minimum value for p is 6 √ 3 = 10.39230.... Let s denote a side of a triangle of unit inradius. If the other two sides are nearly parallel and infinite, then s approaches 2 from above. The infimum for s is 2.
Let u, v denote two sides of a triangle of unit inradius. What is the minimum value for u + v? It is surprising that this question is not better known, especially since the answers for one side (s) and for the sum of three sides (p) are clear.
By symmetry, the minimizing triangle is isosceles and u = v. Let w denote the remaining side of the triangle. By Heron's formula,
u + v + w = 1 2 (u + v + w)(−u + v + w)(u − v + w)(u + v − w) hence 4(u + v + w) = (−u + v + w)(u − v + w)(u + v − w) hence 4(2v + w) = w 2 (2v − w) hence v = (w 2 + 4) w 2(w 2 − 4) .
Differentiating with respect to w, we find that
w = 8 + 4 √ 5
is a zero of the derivative. Substituting into the expression for v, we deduce that the minimum value for u + v is 2v = 22 + 10 √ 5 = 6.66038....
The angle θ at the apex of the minimizing triangle is also interesting. By the Law of Cosines, u 2 + v 2 − 2u v cos(θ) = w 2 hence 2v 2 − w 2 = 2v 2 cos(θ) hence cos(θ) = 1 − 1 2 w 2 v 2 = −2 + √ 5 = 1 ϕ 3 where ϕ is the Golden mean [8]. Finally, θ = 1.33247... ≈ 76.34 • . This material constitutes a (very small) first step toward characterizing the density for the sum of two arbitrary sides of T .
Copyright c 2011 by Steven R. Finch. All rights reserved.
A Papoulis, Probability, Random Variables, and Stochastic Processes. McGraw-HillA. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw- Hill, 1965, pp. 125-137, 187-191, 202-205;
MR0176501 (31 #773). MR0176501 (31 #773).
Random triangles. I-VI. S R Finch, unpublished essaysS. R. Finch, Random triangles. I-VI, unpublished essays (2010), http://algo.inria.fr/bsolve/.
S R Finch, Simulations in R involving triangles and tetrahedra. S. R. Finch, Simulations in R involving triangles and tetrahedra, http://algo.inria.fr/csolve/rsimul.html.
Mathematical Questions and Solutions from the "Educational Times. T C Simmons, S Aiyar, v. 45, ed. W. J. Miller, Francis Hodgson825548T. C. Simmons and S. Aiyar, Problem 8255, Mathematical Questions and Solu- tions from the "Educational Times", v. 45, ed. W. J. Miller, Francis Hodgson, 1886, p. 48; available online at http://books.google.com/.
Problems 8055 & 8101, Mathematical Questions and Solutions from the "Educational Times. T C Simmons, A Gordon, G Heppel, D Biddle, v. 44, ed. W. J. Miller, Francis HodgsonT. C. Simmons, A. Gordon, G. Heppel and D. Biddle, Problems 8055 & 8101, Mathematical Questions and Solutions from the "Educational Times", v. 44, ed. W. J. Miller, Francis Hodgson, 1886, p. 56-57; available online at http://books.google.com/.
The remarkable incircle of a triangle. I Fine, T J Osler, Mathematics and Computer Education. 35I. Fine and T. J. Osler, The remarkable incircle of a triangle, Math- ematics and Computer Education 35 (2001) 44-50; available online at http://www.rowan.edu/open/depts/math/osler/.
Some unusual expressions for the inradius of a triangle. T J Osler, T R Chandrupatla, AMATYC Review, v. 26T. J. Osler and T. R. Chandrupatla, Some unusual expressions for the inradius of a triangle, AMATYC Review, v. 26 (2005) n. 2, 12-17; available online at http://www.rowan.edu/open/depts/math/osler/.
The Golden mean, Mathematical Constants. S R Finch, Cambridge Univ. PressS. R. Finch, The Golden mean, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 5-11;
. R Finch Steven, ; Dept, Ma, Steven, [email protected]. of Statistics Harvard University CambridgeSteven R. Finch Dept. of Statistics Harvard University Cambridge, MA, USA [email protected]
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[] |
[
"Operations on fuzzy ideals of Γ−semirings",
"Operations on fuzzy ideals of Γ−semirings"
] |
[
"T K Dutta ",
"Sujit Kumar Sardar ",
"Sarbani Goswami sarbani7−[email protected] ",
"\nDepartment of Pure Mathematics\nDepartment of Mathematics\nCalcutta University\nKolkataIndia\n",
"\nLady Brabourne College\nJadavpur University\nKolkata, KolkataIndia, India\n"
] |
[
"Department of Pure Mathematics\nDepartment of Mathematics\nCalcutta University\nKolkataIndia",
"Lady Brabourne College\nJadavpur University\nKolkata, KolkataIndia, India"
] |
[] |
The purpose of this paper is to introduce different types of operations on fuzzy ideals of Γ−semirings and to prove subsequently that these oprations give rise to different structures such as complete lattice, modular lattice on some restricted class of fuzzy ideals of Γ−semirings. A characterization of a regular Γ−semiring has also been obtained in terms of fuzzy subsets.
| null |
[
"https://arxiv.org/pdf/1101.4791v1.pdf"
] | 118,859,071 |
1101.4791
|
2880e7f8d4db37f906cf9e754474963ad680b0de
|
Operations on fuzzy ideals of Γ−semirings
25 Jan 2011
T K Dutta
Sujit Kumar Sardar
Sarbani Goswami sarbani7−[email protected]
Department of Pure Mathematics
Department of Mathematics
Calcutta University
KolkataIndia
Lady Brabourne College
Jadavpur University
Kolkata, KolkataIndia, India
Operations on fuzzy ideals of Γ−semirings
25 Jan 2011Mathematics Subject Classification[2000]:16Y6016Y9903E72 Key Words and Phrases: Γ-semiringregular Γ-semiringhemiringsemiringfuzzy ideal
The purpose of this paper is to introduce different types of operations on fuzzy ideals of Γ−semirings and to prove subsequently that these oprations give rise to different structures such as complete lattice, modular lattice on some restricted class of fuzzy ideals of Γ−semirings. A characterization of a regular Γ−semiring has also been obtained in terms of fuzzy subsets.
Introduction
If we remove the restriction of having additive inverse of each element in a ring then a new algebraic structure is obtained what we call a semiring. Semiring has found many applications in various fields. In this regard we may refer to Golan's [5] and Weinert's [6] monographs. Semiring arises very naturally as the nonnegative cone of a totally ordered ring. But the nonpositive cone of a totally ordered ring fails to be a semiring because the multiplication is no longer defined. One can provide an algebraic home, called Γ−semiring, to the nonpositive cone of a totally ordered ring. The notion of Γ−semiring was introduced by M.M.K.Rao [9] in 1995 as a generalization of semiring as well as of Γ−ring. Subsequently by introducing the notion of operator semirings of a Γ−semiring Dutta and Sardar enriched the theory of Γ−semirings. In this connection we may refer to [3]. The motivation for this paper is the fact that Γ−semiring is a generalization of semiring as well as of Γ−ring and fuzzy concepts of Zadeh [10] has been successfully applied to Γ−rings and semirings by Jun et al [7] and Dutta et al [2], [1]. We define here some compositions of fuzzy ideals in a Γ−semiring and study the structures of the set of fuzzy ideals of a Γ−semiring. Among other results we have deduced that sets of fuzzy left ideals and fuzzy right ideals form a zero-sum free semiring with infinite element. We have also deduced that fuzzy ideals of a Γ−semiring is a complete lattice which is modular if every fuzzy ideal is a fuzzy k-ideal.
Preliminaries
Definition 2.1 [9]Let S and Γ be two additive commutative semigroups. Then S is called a Γ−semiring if there exists a mapping S × Γ × S → S (images to be denoted by aαb for a, b ∈ S and α ∈ Γ) satisfying the following conditions:
(i) (a + b)αc = aαc + bαc, (ii) aα(b + c) = aαb + aαc, (iii) a(α + β)b = aαb + aβb,(
iv) aα(bβc) = (aαb)βc for all a, b, c ∈ S and for all α, β ∈ Γ.
Further, if in a Γ−semiring, (S, +) and (Γ, +) are both monoids and (i) 0 S αx = 0 S = xα0 S (ii) x0 Γ y = 0 S = y0 Γ x for all x, y ∈ S and for all α ∈ Γ then we say that S is a Γ−semiring with zero.
Throughout this paper we consider Γ−semiring with zero. For simplification we write 0 instead of 0 S and 0 Γ which will be clear from the context.
Definition 2.2 [10]
Let S be a non empty set. A mapping µ : S → [0, 1] is called a fuzzy subset of S.
Definition 2.3 [4]
Let µ be a non empty fuzzy subset of a Γ−semiring S (i.e. µ(x) = 0 for some x ∈ S). Then µ is called a fuzzy left ideal [ fuzzy right ideal] of S if (i) µ(x + y) ≥ min[µ(x), µ(y)] and (ii) µ(xγy) ≥ µ(y) [resp. µ(xγy) ≥ µ(x)] for all x, y ∈ S, γ ∈ Γ.
A fuzzy ideal of a Γ−semiring S is a non empty fuzzy subset of S which is a fuzzy left ideal as well as a fuzzy right ideal of S. Definition 2.4 [5]Let S be a non empty set and '+' and '.' be two binary operations on S, called addition and multiplication respectively. Then (S, +, .) is called a hemiring (resp. semiring) if (i) (S, +) is a commutative monoid with identity element 0; (ii) (S, .) is a semigroup (resp. monoid with identity element 1);
(iii) a.(b + c) = a.b + a.c and (b + c).a = b.a + c.a for all a, b, c ∈ S. (iv) a.0 = 0.a = 0 for all a ∈ S; (v) 1 = 0. A hemiring S is said to be zero-sum free if a + b = 0 implies that a = b = 0 for all a, b ∈ S.
An element a of a hemiring S is infinite iff a + s = a for all s ∈ S.
For more on preliminaries we may refer to the references and their references.
Operations on fuzzy ideals
Throughout this paper unless otherwise mentioned S denotes a Γ-semiring with unities [3] and F LI(S), F RI(S) and F I(S) denote respectively the set of all fuzzy left ideals, the set of all fuzzy right ideals and the set of all fuzzy ideals of the Γ-semiring S. Also in this section we assume that µ(0) = 1 for a fuzzy left ideal (fuzzy right ideal, fuzzy ideal) µ of a Γ−semiring (Γ−hemiring) S. . Then the sum µ 1 ⊕ µ 2 , product µ 1 Γµ 2 and composition µ 1 • µ 2 of µ 1 and µ 2 are defined as follows:
(µ 1 ⊕ µ 2 )(x) = sup x=u+v [min[µ 1 (u), µ 2 (v)] : u, v ∈ S] = 0 if for any u, v ∈ S, u + v = x. (µ 1 Γµ 2 )(x) = sup x=uγv [min[µ 1 (u), µ 2 (v)] : u, v ∈ S; γ ∈ Γ]
= 0 if for any u, v ∈ S and for any γ ∈ Γ, uγv = x.
(µ 1 • µ 2 )(x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ] = 0 otherwise.
Note. Since S contains 0, in the above definition the case x = u + v for any u, v ∈ S does not arise. Similarly since S contains left and right unity, the case x = i u i γ i v i for any u i , v i ∈ S, γ i ∈ Γ does not arise. In case of product of µ 1 and µ 2 if S has strong left or right unity [i.e., there exists e ∈ S, δ ∈ Γ such that eδa = a for all a ∈ S] then the case x = uγv for any u, v ∈ S and for any γ ∈ Γ does not arise. i.e., in otherwords there are u, v ∈ S and γ ∈ Γ such that x = uγv. Proof
. (µ 1 ⊕ µ 2 )(0) = sup 0=u+v [min[µ 1 (u), µ 2 (v)] : u, v ∈ S] ≥ min[µ 1 (0), µ 2 (0)] = 1 = 0. Thus µ 1 ⊕ µ 2 is non empty and (µ 1 ⊕ µ 2 )(0) = 1. Let x, y ∈ S and γ ∈ Γ. Then (µ 1 ⊕ µ 2 )(x + y) = sup x+y=p+q [min[µ 1 (p), µ 2 (q)] : p, q ∈ S] ≥ sup x = u + v y = s + t [min[µ 1 (u + s), µ 2 (v + t)]: u, v, s, t ∈ S] ≥ sup x = u + v y = s + t [min[min[µ 1 (u), µ 1 (s)], min[µ 2 (v), µ 2 (t)]]: u, v, s, t ∈ S] = sup x = u + v y = s + t [min[min[µ 1 (u), µ 2 (v)], min[µ 1 (s), µ 2 (t)]]: u, v, s, t ∈ S] = min[ sup x=u+v [min[µ 1 (u), µ 2 (v)]], sup y=s+t [min[µ 1 (s), µ 2 (t)]]] = min[(µ 1 ⊕ µ 2 )(x), (µ 1 ⊕ µ 2 )(y)]. Again (µ 1 ⊕ µ 2 )(xγy) = sup xγy=p+q [min[µ 1 (p), µ 2 (q)]] ≥ sup y=u+v [min[µ 1 (xγu), µ 2 (xγv)]] [ Since xγy = xγ(u + v) = xγu + xγv] ≥ sup y=u+v [min[µ 1 (u), µ 2 (v)]] = (µ 1 ⊕ µ 2 )(y). Hence µ 1 ⊕ µ 2 ∈ F LI(S). Proposition 3.3 Let µ 1 , µ 2 , µ 3 ∈ F LI(S)[F RI(S), F I(S)]. Then (i) µ 1 ⊕ µ 2 = µ 2 ⊕ µ 1 . (ii) (µ 1 ⊕ µ 2 ) ⊕ µ 3 = µ 1 ⊕ (µ 2 ⊕ µ 3 ). (iii) θ ⊕ µ 1 = µ 1 = µ 1 ⊕ θ where θ is a fuzzy ideal of S, defined by, θ(x) = 1 if x = 0 0 if x = 0 (iv) µ 1 ⊕ µ 1 = µ 1 . (v) µ 1 ⊆ µ 1 ⊕ µ 2 and (vi) µ 1 ⊆ µ 2 implies that µ 1 ⊕ µ 3 ⊆ µ 2 ⊕ µ 3 .
Proof. (i) We leave it as it follows easily.
(ii) Let x ∈ S. ((µ 1 ⊕ µ 2 ) ⊕ µ 3 )(x) = sup x=u+v [min[(µ 1 ⊕ µ 2 )(u), µ 3 (v)] : u, v ∈ S] = sup x=u+v [min[ sup u=p+q [min[µ 1 (p), µ 2 (q)] : p, q ∈ S]], µ 3 (v)] = sup x=u+v sup u=p+q [min[min[µ 1 (p), µ 2 (q)], µ 3 (v)]] = sup x=p+q+v [min[µ 1 (p), µ 2 (q), µ 3 (v)]].
Similarly we can deduce that (
µ 1 ⊕(µ 2 ⊕µ 3 ))(x) = sup x=p+q+v [min[µ 1 (p), µ 2 (q), µ 3 (v)]]. Therefore (µ 1 ⊕ µ 2 ) ⊕ µ 3 = µ 1 ⊕ (µ 2 ⊕ µ 3 ). (iii) For any x ∈ S, (θ ⊕ µ 1 )(x) = sup x=u+v [min[θ(u), µ 1 (v)], for u, v ∈ S] = min[θ(0), µ 1 (x)] = µ 1 (x). Thus θ ⊕ µ 1 = µ 1 . From (i) µ 1 ⊕ θ = θ ⊕ µ 1 = µ 1 . (iv) Let x ∈ S. Then (µ 1 ⊕ µ 1 )(x) = sup x=u+v [min[µ 1 (u), µ 1 (v)], for u, v ∈ S] ≤ sup x=u+v µ 1 (u + v) = µ 1 (x) So µ 1 ⊕ µ 1 ⊆ µ 1 Again µ 1 (x) = min[µ 1 (0), µ 1 (x)] ≤ sup x=u+v [min[µ 1 (u), µ 1 (v)], for u, v ∈ S] = (µ 1 ⊕ µ 1 )(x).
Therefore
µ 1 ⊆ µ 1 ⊕ µ 1 . Consequently, µ 1 = µ 1 ⊕ µ 1 . (v) Let x ∈ S. Then (µ 1 ⊕ µ 2 )(x) = sup x=u+v [min[µ 1 (u), µ 2 (v)], for u, v ∈ S] ≥ min[µ 1 (x), µ 2 (0)] = µ 1 (x). Thus µ 1 ⊆ µ 1 ⊕ µ 2 . (vi) Let µ 1 ⊆ µ 2 . and x ∈ S. Then (µ 1 ⊕ µ 3 )(x) = sup x=u+v [min[µ 1 (u), µ 3 (v)], for u, v ∈ S] ≤ sup x=u+v [min[µ 2 (u), µ 3 (v)], for u, v ∈ S] = (µ 2 ⊕ µ 3 )(x). Hence µ 1 ⊕ µ 3 ⊆ µ 2 ⊕ µ 3 . Proposition 3.4 Let µ 1 , µ 2 ∈ F LI(S)[F RI(S), F I(S)]. Then µ 1 • µ 2 ∈ F LI(S) [resp. FRI(S), FI(S)].
Proof.
Since (µ 1 • µ 2 )(0) = sup 0= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] ≥ min[µ 1 (0), µ 2 (0)] = 1 = 0 [Since µ 1 (0) = µ 2 (0) = 1], it follows that µ 1 • µ 2 is nonempty and (µ 1 • µ 2 )(0) = 1. Now, for any x, y ∈ S, (µ 1 • µ 2 )(x + y) = sup x+y= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] ≥ sup[ min 1 ≤ i ≤ m 1 ≤ k ≤ l [min[min[µ 1 (u i ), µ 2 (v i )], min[µ 1 (p k ), µ 2 (q k )]]] : x = m i=1 u i γ i v i , y = l k=1 p k γ k q k , u i , v i , p k , q k ∈ S; γ i ∈ Γ; m, l ∈ Z + ] = min[ sup x= m i=1 u i γ i v i [ min 1≤i≤m [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, m ∈ Z + ], sup y= l k=1 p k γ k v k [ min 1≤k≤l [min[µ 1 (p k ), µ 2 (q k )]] : p k , q k ∈ S, γ k ∈ Γ, l ∈ Z + ]] = min[(µ 1 • µ 2 )(x), (µ 1 • µ 2 )(y)]. Now (µ 1 • µ 2 )(xγy) = sup xγy= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] ≥ sup y= m j=1 s j δ j t j [ min 1≤j≤m [min[µ 1 (xγs j ), µ 2 (t j )]]] ≥ sup y= m j=1 s j δ j t j [ min 1≤j≤m [min[µ 1 (s j ), µ 2 (t j )]]] = (µ 1 • µ 2 )(y) Hence µ 1 • µ 2 ∈ F LI(S) Proposition 3.5 Let µ 1 , µ 2 ∈ F LI(S)[F RI(S), F I(S)]. Then µ 1 Γµ 2 ⊆ µ 1 • µ 2 .
Proof. If for any u, v ∈ S and for any γ ∈ Γ, uγv = x then
µ 1 Γµ 2 ⊆ µ 1 • µ 2 . Now for any x ∈ S, (µ 1 • µ 2 )(x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] ≥ sup x=uγv [min[µ 1 (u), µ 2 (v)]] = (µ 1 Γµ 2 )(x).
Thus
µ 1 Γµ 2 ⊆ µ 1 • µ 2 .
Proposition 3.6 Let µ 1 be a fuzzy right ideal and µ 2 be a fuzzy left ideal of S. Then µ 1 Γµ 2 ⊆ µ 1 ∩ µ 2 .
Proof. Let µ 1 be a fuzzy right ideal and µ 2 be a fuzzy left ideal of S. For
x ∈ S, (µ 1 Γµ 2 )(x) = sup x=uγv [min[µ 1 (u), µ 2 (v)] : u, v ∈ S] ≤ sup x=uγv [min[µ 1 (uγv), µ 2 (uγv)]] ≤ sup x=uγv (µ 1 ∩ µ 2 )(uγv) = (µ 1 ∩ µ 2 )(x). Thus µ 1 Γµ 2 ⊆ µ 1 ∩ µ 2 .
The following is a characterization of a regular Γ−semiring in terms of fuzzy subsets.
Theorem 3.7 A Γ−semiring S is multiplicatively regular [9] if and only if µ 1 Γµ 2 = µ 1 ∩ µ 2 for every fuzzy right ideal µ 1 and every fuzzy left ideal µ 2 of S.
Proof. Let S be a multiplicatively regular Γ−semiring and µ 1 be a fuzzy right ideal and µ 2 be a fuzzy left ideal of S. Then by Proposition 3.6, µ 1 Γµ 2 ⊆ µ 1 ∩ µ 2 . Let c ∈ S. Since S is multiplicatively regular, there exists an element x in S and γ 1 , γ 2 ∈ Γ such that c = cγ 1 xγ 2 c.
Now (µ 1 Γµ 2 )(c) = sup c=aγb [min[µ 1 (a), µ 2 (b)] : a, b ∈ S; γ ∈ Γ] ≥ min[µ 1 (cγ 1 x), µ 2 (c)]
[Since c = (cγ 1 x)γ 2 c] ≥ min[µ 1 (c), µ 2 (c)] = (µ 1 ∩ µ 2 )(c). Therefore (µ 1 ∩ µ 2 ) ⊆ µ 1 Γµ 2 and hence µ 1 Γµ 2 = µ 1 ∩ µ 2 . Conversely, let S is a Γ−semiring and for every fuzzy right ideal µ 1 and every fuzzy left ideal µ 2 of S, µ 1 Γµ 2 = µ 1 ∩ µ 2 . Let L and R be a left ideal and a right ideal of S respectively and let x ∈ L ∩ R.
So λ L (x) = 1 = λ R (x). Thus (λ L ∩ λ R )(x) = 1. Now since λ R Γλ L = λ R ∩ λ L , so (λ R Γλ L )(x) = 1. Therefore sup x=yγz [min[λ R (y), λ L (z)] : y, z ∈ S; γ ∈ Γ] = 1.
Thus there exists some r, s ∈ S and γ 1 ∈ Γ such that λ L (s) = 1 = λ R (r) for x = rγ 1 s. Then r ∈ R and s ∈ L and so x = rγ 1 s ∈ RΓL. Therefore L ∩ R ⊆ RΓL. Also L ∩ R ⊇ RΓL. Thus RΓL = R ∩ L. Consequently, S is multiplicatively regular.
Proposition 3.8 Let µ 1 , µ 2 ∈ F I(S). Then
µ 1 Γµ 2 ⊆ µ 1 • µ 2 ⊆ µ 1 ∩ µ 2 ⊆ µ 1 , µ 2 . Proof. By Proposition 3.5, µ 1 Γµ 2 ⊆ µ 1 • µ 2 . For any x ∈ S, if (µ 1 • µ 2 )(x) = 0 then obviously µ 1 • µ 2 ⊆ µ 1 ∩ µ 2 . Now for any x ∈ S, (µ 1 • µ 2 )(x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] ≤ sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i γ i v i ), µ 2 (u i γ i v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] ≤ min[µ 1 (x), µ 2 (x)] = (µ 1 ∩ µ 2 )(x). Therefore µ 1 • µ 2 ⊆ µ 1 ∩ µ 2 . Again (µ 1 ∩ µ 2 )(x) = min[µ 1 (x), µ 2 (x)] ≤ µ 1 (x). Thus µ 1 ∩ µ 2 ⊆ µ 1 .
Similarly it can be shown that µ 1 ∩ µ 2 ⊆ µ 2 . Hence the proposition.
Proposition 3.9 Let µ 1 , µ 2 , µ 3 ∈ F LI(S)[F RI(S), F I(S)]. Then µ 1 Γµ 2 ⊆ µ 3 if and only if µ 1 • µ 2 ⊆ µ 3 . Proof. Since µ 1 Γµ 2 ⊆ µ 1 • µ 2 it follows that µ 1 • µ 2 ⊆ µ 3 implies that µ 1 Γµ 2 ⊆ µ 3 . Assume that µ 1 Γµ 2 ⊆ µ 3 . Let x ∈ S and x = n i=1 u i γ i v i , u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + . Then µ 3 (x) = µ 3 ( n i=1 u i γ i v i ) ≥ min[µ 3 (u 1 γ 1 v 1 ), µ 3 (u 2 γ 2 v 2 ), ........., µ 3 (u n γ n v n )] ≥ min[(µ 1 Γµ 2 )(u 1 γ 1 v 1 ), (µ 1 Γµ 2 )(u 2 γ 2 v 2 ), ........., (µ 1 Γµ 2 )(u n γ n v n )] ≥ min[min[µ 1 (u 1 ), µ 2 (v 1 )], ......., min[µ 1 (u n ), µ 2 (v n )]. µ 3 (x) ≥ sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]]] = (µ 1 • µ 2 )(x).
Thus
µ 1 • µ 2 ⊆ µ 3 .i) (µ 1 • µ 2 ) • µ 3 = µ 1 • (µ 2 • µ 3 ). (ii) µ 1 ⊆ µ 2 implies that µ 1 • µ 3 ⊆ µ 2 • µ 3 . (iii) µ 1 • µ 2 = µ 2 • µ 1 , if S is commutative Γ−semiring. (iv) 1 • µ 1 = µ 1 where 1 ∈ F LI(S) is defined by 1(x) = 1 for all x ∈ S [resp. µ 1 • 1 = µ 1 , 1 • µ 1 = µ 1 • 1 = µ 1 ].
Proof. Proof of (i) follows from the definition.
(ii) Let µ 1 ⊆ µ 2 . Now (µ 1 • µ 3 )(x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 3 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] ≤ sup[ min 1≤i≤n [min[µ 2 (u i ), µ 3 (v i )]]] = (µ 2 • µ 3 )(x).
Thus
µ 1 • µ 3 ⊆ µ 2 • µ 3 . (iii) (µ 1 • µ 2 )(x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] = sup x= n i=1 v i γ i u i [ min 1≤i≤n [min[µ 2 (v i ), µ 1 (u i )]]] if S is commutative Γ−semiring = (µ 2 • µ 1 )(x). Hence µ 1 • µ 2 = µ 2 • µ 1 . (iv) As S is with left unity i [e i , δ i ] ∈ L which is defined by i e i δ i x = x(cf. Definition 5.1[3])
for every x ∈ S we have,
(1 • µ 1 )(x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[1(u i ), µ 1 (v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] = sup[ min 1≤i≤n [min[1, µ 1 (v i )]]] = sup[ min 1≤i≤n µ 1 [(v i )]] ≤ sup[ min 1≤i≤n [µ 1 (u i γ i v i )]] ≤ µ 1 ( n i=1 u i γ i v i ) = µ 1 (x). Therefore (1 • µ 1 ) ⊆ µ 1 . Again (1 • µ 1 )(x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[1(u i ), µ 1 (v i )]] : u i , v i ∈ S; γ i ∈ Γ, n ∈ Z + ] ≥ min 1≤i≤n [min[1(e i ), µ 1 (x)]] [Since j e i δ i x = x] = µ 1 (x) So µ 1 ⊆ 1 • µ 1 and hence 1 • µ 1 = µ 1 .
The following result shows that '.' distributive over '⊕' from both sides.
(i) µ 1 • (µ 2 ⊕ µ 3 ) = µ 1 • µ 2 ⊕ µ 1 • µ 3 , and (ii) (µ 2 ⊕ µ 3 ) • µ 1 = µ 2 • µ 1 ⊕ µ 3 • µ 1 . Proof. Since µ 2 ⊆ µ 2 ⊕ µ 3 therefore µ 1 • µ 2 ⊆ µ 1 • (µ 2 ⊕ µ 3 ). Similarly µ 1 • µ 3 ⊆ µ 1 • (µ 2 ⊕ µ 3 ). Thus (µ 1 • µ 2 ) ⊕ (µ 1 • µ 3 ) ⊆ (µ 1 • (µ 2 ⊕ µ 3 )) ⊕ (µ 1 • (µ 2 ⊕ µ 3 )) = (µ 1 • (µ 2 ⊕ µ 3 )). Now let x ∈ S be arbitrary.Then [µ 1 • (µ 2 ⊕ µ 3 )](x) = sup x= n i=1 u i γ i v i [ min 1≤i≤n [min[µ 1 (u i ), (µ 2 ⊕ µ 3 )(v i )]] : u i , v i ∈ S, γ i ∈ Γ, n ∈ Z + ] = sup[ min 1≤i≤n [min[µ 1 (u i ), sup v i =r i +s i [min[µ 2 (r i ), µ 3 (s i )]]]]] = sup x= n i=1 (u i γ i r i + u i γ i s i ) [ min 1≤i≤n [min[µ 1 (u i ), µ 2 (r i ), µ 3 (s i )]] ≤ sup x= n j=1 p j δ j q j + m k=1 p ′ K δ ′ k q ′ k [min[min[min 1≤j≤ [µ 1 (p j ), µ 2 (q j )]], min[ min 1≤k≤m [µ 1 (p ′ k ), µ 3 (q ′ k )]]]] = sup[min[(µ 1 • µ 2 )(u), (µ 1 • µ 3 )(v)] : u = n j=1 p j δ j q j and v = m k=1 pk ′ δ ′ k q ′ k ] = ((µ 1 • µ 2 ) ⊕ (µ 1 • µ 3 ))(x). Thus µ 1 • (µ 2 ⊕ µ 3 ) ⊆ (µ 1 • µ 2 ) ⊕ (µ 1 • µ 3 ). Hence we conclude that µ 1 • (µ 2 ⊕ µ 3 ) = (µ 1 • µ 2 ) ⊕ (µ 1 • µ 3 ).
Proof of (ii) follows similarly. Proof. It is easy to see that θ ∈ F LI(S). Now by using Propositions 3.2, 3.3, 3.4, 3.10, 3.11 for any µ 1 , µ 2 , µ 3 ∈ F LI(S), we easily obtain
(i) µ 1 ⊕ µ 2 ∈ F LI(S), (ii)µ 1 • µ 2 ∈ F LI(S), (iii) µ 1 ⊕ µ 2 = µ 2 ⊕ µ 1 , (iv) θ ⊕ µ 1 = µ 1 , (v) µ 1 ⊕ (µ 2 ⊕ µ 3 ) = (µ 1 ⊕ µ 2 ) ⊕ µ 3 , (vi) µ 1 • (µ 2 • µ 3 ) = (µ 1 • µ 2 ) • µ 3 , (vii) µ 1 • (µ 2 ⊕ µ 3 ) = (µ 1 • µ 2 ) ⊕ (µ 1 • µ 3 ), (viii) (µ 2 ⊕ µ 3 ) • µ 1 = (µ 2 • µ 1 ) ⊕ (µ 3 • µ 1 ).
Cosequently, FLI(S) is a hemiring under the operations of sum and composition of fuzzy ideals of S. Now by Proposition 3.3(v), 1 ⊆ 1 ⊕ µ for µ ∈ F LI(S). [1(y), µ(z)] : y, z ∈ S] ≤ 1 = 1(x) for all x ∈ S. Therefore 1 ⊕ µ ⊆ 1 and hence 1 ⊕ µ = 1 for all µ ∈ F LI(S). Thus 1 is an infinite element of FLI(S). Now let µ 1 ⊕ µ 2 = θ for µ 1 , µ 2 ∈ F LI(S). Then
Also (1 ⊕ µ)(x) = sup x=y+z [minµ 1 ⊆ µ 1 ⊕ µ 2 = θ ⊆ µ 1 . Consequently, µ 1 = θ.
Similarly it can be shown that µ 2 = θ. Hence the hemiring FLI(S) is zero-sum free.
In analogous manner we can proof the result for FRI(S).
Remark. If S is a commutative Γ−semiring then FLI(S) and FRI(S) are semirings.
Corollary 3.13 FI(S) is a zero-sum free simple semiring under the operations of sum and composition of fuzzy ideals.
Proof. By Proposition 3.10(iv) we have 1 • µ = µ • 1 = µ for all µ ∈ F I(S). Hence the result follows from the above theorem.
Lemma 3.14 Intersection of a nonempty collection of fuzzy left ideals (resp. fuzzy right ideals, fuzzy ideals ) is a fuzzy left ideal ( resp. fuzzy right ideal, fuzzy ideal) of S.
Proof. Let {µ i : i ∈ I} be a nonempty family of fuzzy ideals of S. Let x, y ∈ S. Then (
i∈I µ i )(x + y) = inf i∈I [µ i (x + y)]≥ inf i∈I [min[µ i (x), µ i (y)]] = min[inf i∈I [µ i (x)], inf i∈I [µ i (y)]] = min[( i∈I µ i )(x), ( i∈I µ i )(y)]. Again ( i∈I µ i )(xγy) = inf i∈I [µ i (xγy)]≥ inf i∈I [µ i (y)] = ( i∈I µ i )(y).
Thus i∈I µ i is a fuzzy left ideal of S.
Similarly we can prove the other statements.
Theorem 3.15 Let µ 1 and µ 2 be two fuzzy left ideals (fuzzy right ideals, fuzzy ideals) of a Γ−semiring S. Then µ 1 ⊕ µ 2 is the unique minimal element of the family of all fuzzy left ideals (resp. fuzzy right ideals, fuzzy ideals) of S containing µ 1 and µ 2 and µ 1 ∩ µ 2 is the unique maximal element of the family of all fuzzy left ideals (resp. fuzzy right ideals, fuzzy ideals) of S contained in µ 1 and µ 2 .
Proof. Letµ 1 , µ 2 ∈ F LI(S). Then µ 1 , µ 2 ⊆ µ 1 ⊕ µ 2 [cf. ≤ sup ψ(y + z) = ψ(x) Thus µ 1 ⊕ µ 2 ⊆ ψ. Again µ 1 ∩ µ 2 ⊆ µ 1 , µ 2 . Let us suppose that φ ∈ F LI(S) be such that φ ⊆ µ 1 and φ ⊆ µ 2 . Then for any x ∈ S,
(µ 1 ∩ µ 2 )(x) = min[µ 1 (x), µ 2 (x)] ≥ min[φ(x), φ(x)] = φ(x).
Thus φ ⊆ µ 1 ∩ µ 2 . Uniqueness of µ 1 ⊕ µ 2 and µ 1 ∩ µ 2 with the stated properties are obvious. Proofs of other cases follow similarly. Proof. We define a relation '≤' on FLI(S) as follows: µ 1 ≤ µ 2 if and only if µ 1 (x) ≤ µ 2 (x) for all x ∈ S. Then FLI(S) is a poset with respect to '≤'.By Theorem 3.15, every pair of elements of FLI(S) has lub and glb in FLI(S). Thus FLI(S) is a lattice. Now 1 ∈ F LI(S) and µ ≤ 1 for all µ ∈ F LI(S). So 1 is the greatest element of FLI(S). Let {µ i : i ∈ I} be a non empty family of fuzzy left ideals of S. Then by Lemma 3.14, it follows that i∈I µ i ∈ F LI(S).
Also it is the glb of {µ i : i ∈ I}. Hence FLI(S) is a complete lattice. Proofs of other cases follow similarly. Proof. Let us assume that every member of FLI(S) is a fuzzy left k-ideal and µ 1 , µ 2 , µ 3 ∈ F LI(S) such that µ 2 ∩ µ 1 = µ 2 ∩ µ 3 , µ 2 ⊕ µ 1 = µ 2 ⊕ µ 3 and µ 1 ⊆ µ 3 . Then for any x ∈ S, µ 1 (x) = (µ 1 ⊕ µ 1 )(x) = sup
Definition 3. 1
1Let S be a Γ-semiring and µ 1 , µ 2 ∈ F LI(S) [F RI(S), F I(S)]
Proposition 3. 2
2Let µ 1 , µ 2 ∈ F LI(S)[F RI(S), F I(S)]. Then µ 1 ⊕ µ 2 ∈ F LI(S)[ resp. F RI(S), F I(S)].
Proposition 3. 10
10Let µ 1 , µ 2 , µ 3 ∈ F LI(S)[F RI(S), F I(S)]. Then (
Proposition 3.11 Let µ 1 , µ 2 , µ 3 ∈ F LI(S)[F RI(S), F I(S)]. Then
Theorem 3 .
312 Let S be a Γ−semiring. Then FLI(S) and FRI(S)both are zero-sum free hemiring having infinite element 1 under the operations of sum and composition of fuzzy left ideals and fuzzy right ideals respectively.
Proposition 3.3(v)]. Suppose µ 1 ⊆ ψ and µ 2 ⊆ ψ where ψ ∈ F LI(S). Now for any x ∈ S, (µ 1 ⊕ µ 2 )(x) = sup x=y+z [min[µ 1 (y), µ 2 (z)] : y, z ∈ S] ≤ sup[min[ψ(y), ψ(z)]]
Theorem 3.16 FLI(S) [resp. FRI(S), FI(S)] is a complete lattice.
Proposition 3.17 If S is a Γ−semiring then the lattice (F LI(S), ⊕, ∩) [(F RI(S), ⊕, ∩), (F I(S), ⊕, ∩)] is modular if each of its member is a fuzzy left k-ideal [resp. fuzzy right k-ideal, fuzzy k-ideal].
x=u+v [min[µ 1 (u), µ 1 (v)] : u, v ∈ S] ≥ sup[min[µ 1 (u), (µ 2 ∩ µ 1 )(v)]] = sup[min[µ 1 (u), (µ 2 ∩ µ 3 )(v)]] = sup[min[µ 1 (u), min[µ 2 (v), µ 3 (v)]]] = sup[min[min[µ 1 (u), µ 2 (v)], µ 3 (v)]] ≥ sup[min[min[µ 1 (u), µ 2 (v)], min[µ 3 (u+v), µ 3 (u)]]] [Since µ 3 is a left k-ideal]. ≥ sup[min[min[µ 1 (u), µ 2 (v)], min[µ 3 (u + v), µ 1 (u)]]] = sup[min[min[µ 1 (u), µ 2 (v)], µ 3 (u + v)]] = min[sup[min[µ 1 (u), µ 2 (v)]], sup[µ 3 (u + v)]] = min[(µ 1 ⊕ µ 2 )(x), µ 3 (x)] = min[(µ 1 ⊕ µ 3 )(x), µ 3 (x)] = µ 3 (x) [Since µ 3 ⊆ µ 1 ⊕ µ 3 ].Thus µ 3 ⊆ µ 1 and hence µ 1 = µ 3 . Hence (F LI(S), ⊕, ∩) is modular.
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On the Operator Semirings of a Γ−semiring. T K Dutta, S K Sardar, Southeast Asian Bull. of Math. 26Dutta, T.K. and Sardar, S.K.: On the Operator Semirings of a Γ−semiring; Southeast Asian Bull. of Math. 26(2002), 203-213.
An introduction to fuzzy ideals of Γ−semirings. T K Dutta, S K Sardar, S Goswami, Proceedings of National Seminar on Algebra. Analysis and Discrete MathematicsDutta, T.K., Sardar, S.K. and Goswami, S.: An introduction to fuzzy ideals of Γ−semirings; (To appear) Proceedings of National Seminar on Algebra, Analysis and Discrete Mathematics.
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Y B Jun, C Y Lee, Fuzzy Γ−rings. Jun, Y.B. and Lee, C.Y. : Fuzzy Γ−rings;
. Pusom Kyongnam Math. J. 82Pusom Kyongnam Math. J. 8(2) (1992), 63-170.
Fuzzy Algebra; University of Delhi Publication Division. R Kumar, Kumar, R.: Fuzzy Algebra; University of Delhi Publication Division, 1993.
Γ−semiring-1. M M K Rao, Rao, M.M.K.: Γ−semiring-1 ;
. Southeast Asian Bull. of Math. 19Southeast Asian Bull. of Math. 19 (1995), 49-54.
Fuzzy sets. L A Zadeh, Information and Control. 8Zadeh, L.A.: Fuzzy sets; Information and Control 8( 1965 ), 338-353.
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[
"GLOBAL WELL-POSEDNESS OF SHOCK FRONT SOLUTIONS TO ONE-DIMENSIONAL PISTON PROBLEM FOR COMBUSTION EULER FLOWS",
"GLOBAL WELL-POSEDNESS OF SHOCK FRONT SOLUTIONS TO ONE-DIMENSIONAL PISTON PROBLEM FOR COMBUSTION EULER FLOWS"
] |
[
"Kai Hu ",
"Jie Kuang "
] |
[] |
[] |
This paper is devoted to the well-posedness theory of piston problem for compressible combustion Euler flows with physical ignition condition. A significant combustion phenomena called detonation will occur provided the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of BV ∩ L 1 space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure, and from the characteristic-boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary and exothermic reaction, then obtain the L 1 −stability of combustion wave solutions. Our results reveal that one dimensional ZND detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in BV sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.
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"https://arxiv.org/pdf/2202.12554v2.pdf"
] | 247,154,868 |
2202.12554
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002f56993efd16da459d4db807dd1448a9ad535b
|
GLOBAL WELL-POSEDNESS OF SHOCK FRONT SOLUTIONS TO ONE-DIMENSIONAL PISTON PROBLEM FOR COMBUSTION EULER FLOWS
3 Mar 2022
Kai Hu
Jie Kuang
GLOBAL WELL-POSEDNESS OF SHOCK FRONT SOLUTIONS TO ONE-DIMENSIONAL PISTON PROBLEM FOR COMBUSTION EULER FLOWS
3 Mar 2022
This paper is devoted to the well-posedness theory of piston problem for compressible combustion Euler flows with physical ignition condition. A significant combustion phenomena called detonation will occur provided the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of BV ∩ L 1 space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure, and from the characteristic-boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary and exothermic reaction, then obtain the L 1 −stability of combustion wave solutions. Our results reveal that one dimensional ZND detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in BV sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.
In Eulerian coordinates, the one-dimensional compressible Euler equations for combustible fluids can be written in the following form
∂ t ρ + ∂ x (ρu) = 0,
∂ t (ρu) + ∂ x (ρu 2 + p) = 0, ∂ t ρ( 1 2 u 2 + e) + ∂ x ρu( 1 2 u 2 + e) + pu = q 0 ρY φ(T ), ∂ t (ρY ) + ∂ x (ρuY ) = −ρY φ(T ), (1.1) where ρ, u, p, T and e are respectively the density, velocity, pressure, temperature and specific internal energy of the fluid, Y denotes the mass fraction of the reactant in mixed gas, and function φ(T ) stands for the combustion reaction rate. Positive constant q 0 is the specific binding energy of reactant.
If we choose the density ρ and entropy S as two independent thermodynamical variables, then p, e and T can be seen as the functions of (ρ, S), i.e., p = p(ρ, S), e = e(ρ, S), T = T (ρ, S) through thermodynamical relation:
T dS = de − p ρ 2 dρ. (1.2)
For ideal polytropic gas, the constitutive relations are represented by
p = κρ γ e S cv , e = κ γ − 1 ρ γ−1 e S cv , T = κ (γ − 1)c v ρ γ−1 e S cv ,(1.3)
where γ > 1 is the adiabatic exponent, c v > 0 is the specific heat at constant volume, and κ > 0 is any constant under scaling. Throughout this paper, we suppose that reaction rate φ(T ) for combustion flows satisfies the following Arrhenius Law with physical ignition condition, namely
φ(T ) = T ϑ e − R T if T > T i , 0 if T ≤ T i ,(1.4)
where T i > 0 stands for the ignition temperature of reactant, and constants ϑ ≥ 0, R > 0. In terms of (1.4), the combustion process is switched on-off by reactant temperature. System (1.1) with the reaction rate (1.4) is also called Zeldovich-von Neumann-Döring model (i.e. ZND ) which is widely applied in combustion theory and related numerical simulation (see [30] for more details). In this paper, we are concerned with the combustion process caused by piston motion in a tube, and aim to investigate the well-posedness of combustion waves for the ZND model. Specifically, a fuel-filled long tube is closed by a piston at one end, and open at the other end. If we push the piston with a speed u p (t) > 0 towards the combustible gas at rest, then a shock front appears and departs from the piston (see Fig.1). Since the temperature of the flow increases across the shock front, chemical reaction will be initiated behind the leading shock. Consequently, a kind of fierce combustion phenomena, called detonation, may occur in the tube. Its dynamics research has a significant application of detonation engine for hypersonic aircrafts. Mathematically, for given piston velocity u p (t), the path of moving piston can be determined by χ p (t) = t 0 u p (τ )dτ . Thus, we can define the domain and its boundary respectively by
Ω p . = {(t, x) ∈ R 2 : t > 0, x > χ p (t)}, Γ p . = {(t, x) ∈ R 2 : t > 0, x = χ p (t)}.
See Fig.1. Set U = (ρ, u, p, Y ) ⊤ . Then we prescribe the initial state of the flow in the tube as (1.5) and along the path of piston, the flow satisfies u(t, x) = u p (t) on Γ p . (1.6) With this set-up, the piston problem described above can be formulated as the initialboundary value problem to equations (1.1) with shock front as a free boundary. In this paper, we study the entropy solutions to problem (1.1) and (1.5)-(1.6) defined as follows. (ii) U (t, x) satisfies the entropy inequality
U (0, x) = U 0 (x) = (ρ 0 , u 0 , p 0 , Y 0 ) ⊤ (x) for x > 0,∂ t (ρS) + ∂ x (ρuS) ≥ q 0 ρY φ(T ) T in the distribution sense in Ω p .
When the piston moves with a constant speed, that is u p (t) =ū p > 0, into the static gas with constant stateŪ b,r = (ρ r , 0,p r , 0) ⊤ in the tube (see Fig.2), then we follow the arguments in [13] and know that the initial-boundary value problem (1.1) and (1.5)-(1.6) admits a unique piecewise constant solutionŪ b = (ρ b ,ū b ,p b ,Ȳ b ) ⊤ which contains a strong 4-shock front x =χ s (t) with constant speedχ s (t) =s b for t ≥ 0. Precisely,
U b (t, x) = Ū b,l .
= (ρ l ,ū l ,p l , 0) ⊤ for x ∈ (ū p t,s b t), U b,r . = (ρ r , 0,p r , 0) ⊤ for x ∈ (s b t, +∞), (1.7) where (ρ l ,ū l ,p l , 0) and (ρ r , 0,p r , 0) are both constant states satisfyinḡ ρ l >ρ r > 0,ū l =ū p ,s b >ū l > 0.
Thus, the corresponding temperaturesT l andT r (determined by the relations in (1.3)) respectively forŪ l andŪ r satisfyT l >T r . In this case, we call the solutionŪ b with shock front x =s b t as the background solution to problem (1.1) and (1.5)-(1.6). Then, the main purpose in this paper is to investigate the well-posedness of entropy solutions with more general data (U 0 , u p ) near the background state (Ū b,r ,ū p ) in the BV ∩ L 1 framework. We begin with some basic assumptions on the initial data U 0 (x) and velocity u p (t) of moving piston. Assume that (A1) the initial data U 0 defined in (1.5) satisfies U 0 −Ū b,r ∈ (BV ∩ L 1 )(R + ; R 4 ), inf x∈R + ρ 0 (x) > 0 and Y 0 ∈ [0, 1]; (A2) the velocity u p of the moving piston defined on Γ p in (1.6) satisfies u p −ū p ∈ (BV ∩ L 1 )(R + ; R). For notational convenience, denote the initial-boundary value problem (1.1) and (1.5)-(1.6) by (IBVP ). Our main result in this paper is stated as follows. The solution U (t, x) is the unique limit of approximations obtained by fractional-step wave front tracking algorithm, which contains a strong shock front x = χ s (t) ∈ Lip(R + ) such that χ s (0) = 0, χ p (t) < χ s (t) andχ s (t) ∈ BV (R + ). Furthermore, there holds where the constant C 0 > 0 depends solely onŪ b . Assume V (t, x) = (ρ,ũ,p,Ỹ ) ⊤ (t, x) is another global entropy solution to (IBV P ) with the initial-boundary data (V 0 (x), v p (t)) satisfying the similar condition to (U 0 (x), u p (t)) from (C1)-(C3) for V 0 (x) = (ρ 0 ,ũ 0 ,p 0 ,Ỹ 0 ) ⊤ (x). Then for any t >t ≥ 0, there holds the L 1 -stability estimate U (t, ·) − V (t, ·) L 1 (R) ≤ L U (t, ·) − V (t, ·) L 1 (R) + u p (·) − v p (·) L 1 ([t,t]) , (1.12) where L > 0 is a Lipschitz constant dependent ofŪ b and Y 0 L 1 (R + ) , Ỹ 0 L 1 (R + ) .
Remark 1.1. Throughout this paper, U (t, ·) L 1 (R) is defined by
U (t, ·) L 1 (R) . = R |U ex (t, x)|dx,
where the extension U ex (t, x) is given by
U ex (t, x) = U (t, x) if x ≥ χ p (t), 0 if x < χ p (t).
Remark 1.2. The solutions under conditions (C1)-(C3) correspond to entirely unburnt flow, partially ignited flow and completely ignited flow, respectively. For non-reacting fluid, the variation of Y 0 can be arbitrarily large, because the reactant is just transported along particle path, but never amplifies the oscillation of fluid. For reacting fluid, we require only smallness condition on T.V.{Y 0 ; R + }, but allow the total reactant Y 0 L 1 (R + ) and ρ 0 Y 0 L 1 (R + ) suitably large. Thus Theorem 1.1 shows that one-dimensional ZND detonation waves are nonlinearly stable provided the leading shock is strong enough.
Remark 1.3.
Comparing with the results in [17,21] that require the adiabatic constant γ satisfies 1 < γ ≤ 3, we only need γ > 1 for Theorem 1.1 in this paper. Besides, we also consider more realistic reaction rate φ(T ) which has a discontinuity at the ignition temperature T i , and establish the global well-posedness of the entropy solution that contains a strong combustion wave front. Therefore, the results in [17,21] can be seen as special cases for Theorem 1.1 from this point of view.
There are many literatures on the mathematical theories of piston problem for compressible inviscid flow, which involve the existence and structural stability of shock waves or rarefaction waves in the framework of BV or L ∞ space. When the piston moves to static gas in a tube, the global existence of shock front solutions to one-dimensional piston problem for Euler equations was established in BV space by Wang [29] and Ding [17]. For multidimensional case, the authors in [9,10] considered the axially symmetric Euler flow, and established the global stability of multidimensional shock front solutions for both weak and strong shocks induced by axially symmetric piston motion. On the other hand, when the piston is withdrawn in the tube, the global stability of rarefaction wave solutions to one-dimensional piston problem was completed in [18] through a modified wave-front tracking scheme. In the L ∞ framework, the global entropy solution for spherically symmetric piston problem was constructed in [8] by means of compensated compactness and shock capturing.
For exothermically reacting Euler flow, by developing a fractional-step Glimm scheme, Chen and Wagner [7] firstly established the global existence of entropy solutions for onedimensional Cauchy problem without smallness assumption on initial data in BV norm. We refer to [25,28] for more details about the BV solutions in gas dynamics with large data. After that, Ding [16] considered the piston problem of reacting fluid and obtained the global existence and asymptotic behavior of the entropy solutions that contain a strong rarefaction wave. Recently, the authors in [21] also studied shock front solution to one-dimensional piston problem with reaction, and established the structural stability of strong shock wave as well as its asymptotic behavior by fractional-step wave front tracking scheme. Particularly, all the previous literatures on combustion equations require that the reaction rate φ(T ) is continuous and positive everywhere in order to derive the exponential decay of exothermic source.
Finally, we remark that there are a lot of important results on well-posedness theory for general hyperbolic conservation or balance laws in one dimension situation. For more details, we can refer to [23,24] for Cauchy problem with initial data containing large profile, [1,15] for initial-boundary value problem of conservation laws, [2,3,11,12,14,26] for Cauchy problem or initial-boundary value problem with inhomogeneous term, and the references therein.
In this paper, we establish the well-posedness theory of one-dimensional piston problem for compressible and combustible Euler flows within the BV ∩ L 1 framework. To the best of our knowledge, it is the first work on BV ∩ L 1 theory for inviscid combustible fluids with physical ignition condition (1.4). Mathematically, this problem can be formulated as an initial-boundary value problem for one-dimensional hyperbolic balance laws with a discontinuous source. There are two main technical difficulties arising from ignition temperature and characteristic boundary. For the presence of ignition temperature T i in (1.4), the partial reaction phenomena will occur such that the decrease rate of exothermic source varies greatly within reaction zone. It leads to the loss of uniform dissipation structures (for instance exponential decay), so that the arguments in [7,16,19,20,21] are invalid here. Besides, nonuniform heat release definitely amplifies the oscillation of combustion waves, and probably induces the instability of free boundary of flame. The characteristic boundary of multiple degenerate fields is another difficulty of proving well-posedness. Concerning the time derivative of stability functional near characteristic boundary, the linear terms of distance index q i of two solutions (see Definition 2.1) cannot be completely canceled by means of direct weight manipulation as in [4,5,23,12]. This is the major reason why standard Lyapunov functional does not work for the characteristic boundary value problems of many hyperbolic systems.
In dealing with the obstacles stated above, we employ a fractional-step wave front tracking scheme to construct approximate solutions, and then devise a modified Glimmtype functional G(t) for seeking their uniform BV bound. Observe that the product of reflection coefficients on the strong shock and on the piston boundary is less than one (see Propositions 2.2-2.3 below). Thus we can choose appropriate weights in G(t) to prove its monotonicity in non-reaction process, which indicates the global existence of the entropy solutions to (IBVP ) without combustion. Based on this, we further consider the reacting flows dominated by ignition temperature. The crux of this matter is how to calculate the growth of oscillation within reaction zone if the source fails to uniformly decay. To achieve this, we utilize nonuniform spatial estimates in present paper instead of the temporal decay of dissipation in earlier framework. Insert a special consumption term L by (t) into functional G(t), which accounts for the consumption of reactant along the strong shock front and piston boundary. This crucial term finally offsets the growth of combustion waves, and enables us to establish the existence of combustion solutions.
We further apply the technique of Lyapunov functional and quasi-characteristics analysis to L 1 -stability argument. Precisely, take two approximate solutions U ε and V ε with data (U ε 0 , u ε p ) and (V ε 0 , v ε p ) respectively, and consider the maximal curve x = χ m p (t) of distinct piston paths for U ε and V ε . We design a weighted Lyapunov functional L (U ε , V ε ), which is a equivalent metric to ||U ε − V ε || L 1 ([χ m p ,+∞)) . For the purpose of stability estimates, a concise weights distribution is proposed to balance the nonlinear effects arising from large shock, characteristic boundary and exothermic reaction. Then, we will show that L (U ε , V ε ) is strictly decreasing in reaction process. To cope with the difficulty caused by characteristic boundary, we first establish a new accurate estimate on different distance indices of genuinely nonlinear fields (see (5.19) in section 5.2) by employing the degeneracy feature of contact discontinuity. As a result, all of the linear terms of index q i can be successfully canceled near the piston boundary. It implies that the boundary parts of d dt L (U ε , V ε ) are dominated by the velocities difference of U ε and V ε at χ m p . Subsequently, by introducing a tool of quasi-characteristic curve, we establish the velocities comparison estimate along the characteristic boundary and maximal curve. Based on linear terms cancellation and velocities comparison, we obtain the stability estimate on piston boundaries, then eventually establish the L 1 −stability result for both non-reacting and reacting flows.
The remainder of this paper will be arranged as follows. In section 2, we consider the elementary wave curves for homogeneous system, and then study the Riemann problems including perturbations of strong shock front and piston boundary. Moreover, some local estimates of interaction between weak wave and strong shock, as well as the weak wave reflection on piston boundary are also presented. We then construct the approximate solutions to (IBVP ) in section 3 by proposing a fractional-step wave front tracking algorithm. Section 4 is devoted to the global existence of entropy solutions for both non-reacting and reacting flows by means of modified Glimm-type functional. In addition, the notion of quasi-characteristic curve is introduced, and the velocities comparison in L 1 norm on distinct curves is shown. In section 5, we further develop the weighted Lyapunov functional and derive L 1 -stability estimates of entropy solutions. As a byproduct, we can show the uniqueness of entropy solution for non-reacting flow. In the end, section 6 exhibits the uniqueness of limit solution for reacting flow obtained by fractional-step wave front tracking scheme.
Homogenous system
In this section, as a preliminary, we first present some elementary wave curves and their properties for the homogeneous system reduced from (1.1), and then study the Riemann problems which involve the piston boundary as well as the strong shock front. Based on these facts, we can further show some local estimates on interaction between weak waves, weak wave and strong shock, as well as the weak wave reflection on the boundary.
2.1.
Elementary wave curves for homogeneous system. Let
E(U ) = ρ, ρu, ρ 1 2 u 2 + e , ρY ⊤ , F (U ) = ρu, ρu 2 + p, ρu 1 2 u 2 + e + p ρ , ρuY ⊤ , G(U ) = 0, 0, q 0 ρY φ(T ), −ρY φ(T ) ⊤ .
Then equations (1.1) can be rewritten as the following hyperbolic balance laws:
∂ t E(U ) + ∂ x F (U ) = G(U ). (2.1)
When G(U ) ≡ 0, equations (2.1) become one-dimensional hyperbolic system of conservation laws, i.e.,
∂ t E(U ) + ∂ x F (U ) = 0. (2.
2) The system (2.2) admits four eigenvalues:
λ 1 (U ) = u − c, λ 2 (U ) = λ 3 (U ) = u, λ 4 (U ) = u + c,
and the corresponding linearly independent eigenvectors are
r 1 (U ) = 2 (γ + 1)c − ρ, c, −γp, 0 ⊤ , r 2 (U ) = (1, 0, 0, 0) ⊤ ,r 3 (U ) = (0, 0, 0, 1) ⊤ , r 4 (U ) = 2 (γ + 1)c ρ, c, γp, 0) ⊤ ,
where c is the sound speed defined by c = γp ρ . Moreover, a direct computation shows that
∇ U λ i (U ) · r i (U ) ≡ 1 for i = 1, 4, ∇ U λ j (U ) · r j (U ) ≡ 0 for j = 2, 3.
This means that the ith characteristic fields are genuinely nonlinear for i = 1, 4, while the jth characteristic fields are linearly degenerate for j = 2, 3. Given constant state U * = (ρ * , u * , p * , Y * ) ⊤ away from vaccum, the Rankine-Hugoniot condition with wave speed s, i.e.,
F (U ) − F (U * ) = s E(U ) − E(U * ) ,(2.3)
for system (2.2) determines the Hugoniot loci through U * in state space as follows:
S 1 (U * ) : p p * = ηρ − ρ * ηρ * − ρ , u − u * c * = − 2 γ − 1 ρ − ρ * (ηρ * − ρ)ρ , Y = Y * , ρ = ρ * ; S 2 (U * ) : ρ = ρ * , u = u * , p = p * , Y = Y * ; S 3 (U * ) : ρ = ρ * , u = u * , p = p * , Y = Y * ; S 4 (U * ) : p p * = ηρ − ρ * ηρ * − ρ , u − u * c * = 2 γ − 1 ρ − ρ * (ηρ * − ρ)ρ , Y = Y * , ρ = ρ * ,(2.4)
where constant η = γ+1 γ−1 . Let the Hugoniot locus S i (U * ) be parameterized by mapping
α i → S i (α i )(U * ) with α i = λ i (S i (α i )(U * )) − λ i (U * ) if i = 1, 4, ρ − ρ * if i = 2, Y − Y * if i = 3.
(2.5)
The isentropes through U * for system (2.2) read
R 1 (U * ) : u + 2c γ − 1 = u * + 2c * γ − 1 , p ρ γ = p * ρ γ * , Y = Y * ; R 4 (U * ) : u − 2c γ − 1 = u * − 2c * γ − 1 , p ρ γ = p * ρ γ * , Y = Y * ,(2.6)
which are parameterized by mapping
α i → R i (α i )(U * ) with α i = λ i (R i (α i )(U * )) − λ i (U * ), i = 1, 4.
Notice that the last equation for Y in (2.2) can be decoupled from the first three equations (i.e., non-isentropic Euler equations). Henceforth, we can rewrite any state
U = (U e , Y ) ⊤ with U e . = (ρ, u, p) ⊤ .
Then the states along parameterized curves S i (U * ) and R i (U * ) in (2.4)(2.6) are decomposed into Eulerian part and mass fraction part as follows,
S i (α i )(U * ) = S e i (α i )(U e * ), Y * ⊤ (i = 1, 2, 4), R i (α i )(U * ) = R e i (α i )(U e * ), Y * ⊤ (i = 1, 4), S 3 (α 3 )(U * ) = (U e * , S y 3 (α 3 )(Y * )) ⊤ .
Treat U l = (U e l , Y l ) ⊤ and U r = (U e r , Y r ) ⊤ as the left and right states of Riemann problem (U l , U r ). Thus, according to Lax's entropy condition, the elementary wave curves projected into U e −subspace are defined by
Φ e i (α i )(U e l ) = S e i (α i )(U e l ) for α i < 0, R e i (α i )(U e l ) for α i ≥ 0, (i = 1, 4) Φ e 2 (α 2 )(U e l ) = S e 2 (α 2 )(U e l ). It is clear that Φ e i (α i )(U e l ) α i =0 = U e l , dΦ e i (α i )(U e l ) dα i α i =0 = r e i (U e l ) (i = 1, 2, 4).
Here r e i (U e ) consists of the first three components of eigenvector r i (U ). Accordingly, denote the projected elementary curves through U e r by
Ψ e i (α i )(U e r ) = S e i (α i )(U e r ) for α i > 0, R e i (α i )(U e r ) for α i ≤ 0, (i = 1, 4) Ψ e 2 (α 2 )(U e r ) = S e 2 (α 2 )(U e r ).
With regard to background shock in (1.7), we claim that
U e b,r = Φ e 4 (ᾱ s 4 )(Ū e b,l ),Ū e b,l = Ψ e 4 (−ᾱ s 4 )(Ū e b,r ) (2.7)
for a negativeᾱ s 4 = λ 4 (Ū e b,r ) − λ 4 (Ū e b,l ). Let r e i (U e * , α i ) be the tangent vector for curve S e i (α i )(U e * ), while r e i (R e i (α i )(U e * )) be the tangent vector for curve R e i (α i )(U e * ). Taking U e * = U e r when i = 1, 4, we define the tangent field along curve Ψ e i (α i )(U e r ) bỹ
r e i (U e r , α i ) . = r e i (U e r , α i ) for α i > 0, r e i (R e i (α i )(U e r )) for α i ≤ 0. (2.8)
In particular, one has r e i (U e r , 0) = r e i (U e r ). Henceforth, we write the derivatives˙. = d dt and ′ . = d dα 4 for brevity.
The previous parametrization of elementary curves will provide us some technical advantages to analyze the solvability, wave-interaction and stability, etc. For instance, the following lemma exhibits the monotonicity of fluid state along the shock curve.
Lemma 2.1. Assume state U e r = (ρ r , u r , p r ) ⊤ is away from vacuum. Along the shock curve corresponding to the compressive branch of Ψ e 4 (α 4 )(U e r ), the density, velocity, pressure and shock speed are all increasing with respect to parameter α 4 ; i.e.,
ρ ′ (α 4 ) > 0, u ′ (α 4 ) > 0, p ′ (α 4 ) > 0, s ′ (α 4 ) > 0, where (ρ, u, p) = U e = S e 4 (α 4 )(U e r ) with α 4 > 0.
Proof. Suppose state U = (ρ, u, p, Y ) ⊤ = S 4 (α 4 )(U r ). One has α 4 = λ 4 (U ) − λ 4 (U r ) > 0 iff ρ ∈ (ρ r , ηρ r ), where constant η = (γ + 1)/(γ − 1). Clearly, Hugoniot locus S 4 (U r ) in (2.4) can be treated as a curve of single parameter ρ. Since U ∈ S 4 (U r ), we differentiate u and p with respect to ρ and obtain dp dρ
= p r ρ r · η 2 − 1 (ηρ r − ρ) 2 > 0, du dρ = c r ρ r · √ η − 1 2 · (η − 2)ρ + ηρ r (ηρ r ρ − ρ 2 ) 3/2 > 0 (2.9)
for any ρ ∈ (ρ r , ηρ r ); furthermore,
dc dρ = c r √ ρ r · (ρ − ρ r ) 2 + (η − 1)(ρ 2 + ρ 2 r ) 2ρ 3/2 (ηρ r − ρ) 3/2 (ηρ − ρ r ) 1/2 > 0, d 2 dρ 2 (ρu) = c r ρ 2 r · √ η − 1 4 · ηρ (3η − 4)ρ + ηρ r (ηρ r ρ − ρ 2 ) 5/2 > 0. (2.10) They immediately give dα 4 dρ = du dρ + dc dρ > 0. (2.11)
From (2.9) and (2.11), we see that
ρ ′ > 0, p ′ = dp dρ · dρ dα 4 > 0, u ′ = du dρ · dρ dα 4 > 0. The Hugoniot-Rankine condition s E(U ) − E(U r ) = F (U ) − F (U r ) determines the 4−shock speed by s = [ρu]/[ρ], where [ρ] = ρ − ρ r > 0.
Note that inequality (2.10) implies the fact momentum ρu is convex in ρ. Hence we have
s ′ = ds dρ · dρ dα 4 = ρ ′ [ρ] · d(ρu) dρ − [ρu] [ρ] > 0.
2.2. Riemann problem and local interaction estimates for homogeneous system. First, we consider the Riemann problem of (2.2) with the data
U | t=t = U l = (U e l , Y l ) ⊤ , x <x, U r = (U e r , Y r ) ⊤ , x >x,(2.12)
where U e l = (ρ l , u l , p l ) ⊤ and U e r = (ρ r , u r , p r ) ⊤ . Then, its Riemann solver involving only weak waves is given by the following lemma.
Lemma 2.2. There exists δ > 0 sufficiently small such that for U l , U r ∈ N δ (Ū b,l ) (or N δ (Ū b,r )), the Riemann problem (2.2)(2.12) admits a unique admissible solution that consists of at most four constant states separated by four elementary waves. Moreover, U l and U r satisfy
U e r = Φ e 4 (α 4 ) • Φ e 2 (α 2 ) • Φ e 1 (α 1 )(U e l ), Y r = Y l + α 3 , and 4 j=1 |α j | ≤ O(1) |U e r − U e l | + |Y r − Y l | .
Throughout this paper, symbol O(1) denotes the bounded quantity that depends solely onŪ b .
By Lemma 2.2 and the arguments in [4], we have the following the local estimates on interaction between weak waves. Proposition 2.1. Assume that the solution U (t, x) to system (2.2) contains three adjacent
constant states U l = (U e l , Y l ) ⊤ , U m = (U e m , Y m ) ⊤ , U r = (U e r , Y r ) ⊤ ∈ N δ (Ū b,r ) (or N δ (Ū b,l )
) with δ > 0 sufficiently small. They are separated by two incoming waves with strengths α * i , α * j respectively.
(i) If 1 ≤ j ≤ i ≤ 4 and i, j = 3 such that U e m = Φ e i (α * i )(U e l ), U e r = Φ e j (α * j )(U e m ), Y l = Y m = Y r , then for the outgoing waves determined by equality U e r = Φ e 4 (α 4 ) • Φ e 2 (α 2 ) • Φ e 1 (α 1 )(U e l ), there hold |α i − α * i | + |α j − α * j | + k =i,j,3 |α k | = O(1)|α * i α * j | if i > j, |α i − α * i − α * j | + k =i,3 |α k | = O(1)|α * i α * j | if i = j. (2.13) (ii) If i = 1 and j = 3 such that U e m = U e l , Y m = Y l + α * 3 and U e r = Φ e 1 (α * 1 )(U e m ), Y r = Y m , then for α k (1 ≤ k ≤ 4) determined by U e r = Φ e 4 (α 4 ) • Φ e 2 (α 2 ) • Φ e 1 (α 1 )(U e l ), Y r = Y l + α 3 ,(2.
14)
there hold Fig. 4. Weak wave reflection on the piston Next we turn to study the Riemann problem for the system (2.2) near the piston with the following initial-boundary data:
α 1 = α * 1 , α 3 = α * 3 , α 2 = α 4 = 0. Similarly, if i = 4 and j = 3 such that U e m = Φ e 4 (α * 4 )(U e l ), Y m = Y l and U e r = U e m , Y r = Y m + α * 3 , then for α k (1 ≤ k ≤ 4) determined by (2.14), there hold α 1 = α 2 = 0, α 3 = α * 3 , α 4 = α * 4 . α 4 U l U r (t, χ ε p (t))U + b U r U − b (t, χ ε p (t)) α 4 α * 1u = u p (t + 0) on (t, x) : x = χ ε p (t), t >t , U = U r on (t, x) : x > χ ε p (t), t =t ,(2.15)
where U r = (U e r , Y r ) ⊤ for U e r = (ρ r , u r , p r ) ⊤ , and χ ε p (t) represents the polygonal approximate boundary that will be defined in section 3. Then we have the following solvability of Riemann problem (2.2) and (2.15). Lemma 2.3. There exists a sufficiently small parameter δ > 0 such that if u r = u p (t − 0) and u l = u p (t + 0) with u p (t ± 0) ∈ N δ (ū p ) and U r ∈ N δ (Ū b,l ), then the Riemann problem (2.2) and (2.15) admits a unique admissible solution which includes a 4-wave with strength α 4 (see Fig.3). Moreover, there holds the estimate
α 4 = O(1)|u p (t + 0) − u p (t − 0)|. (2.16)
The result below shows us an estimate for weak wave reflection on the boundary.
Proposition 2.2. Suppose that constant states U r = (U e r , Y r ) ⊤ , U − b = (U e,− b , Y − b ) ⊤ ∈ N δ (Ū b,l ) and velocity u p ∈ N δ (ū p ) satisfy U e,− b = Ψ e 1 (−α * 1 )(U e r ), Y − b = Y r , u − b = u p ,(2.
17)
and that constant state Fig.4). Then, for δ > 0 sufficiently small, it holds the estimate
U + b = (U e,+ b , Y + b ) ⊤ ∈ N δ (Ū b,l ) satisfies U e,+ b = Ψ e 4 (−α 4 )(U e r ), Y + b = Y r , u + b = u p , (2.18) where U e,± b = (ρ ± b , u ± b , p ± b ) ⊤ , α * 1 = λ(U r ) − λ(U − b ) and α 4 = λ(U r ) − λ(U + b ) (seeα 4 = α * 1 + O(1)|α * 1 | 2 . (2.19) Proof. By (2.17) and (2.18), we have the relation Ψ e 4 (−α 4 )(U e r ) · n = Ψ e 1 (−α * 1 )(U e r ) · n, where vector n = (0, 1, 0) ⊤ . To evaluate α 4 , let's consider the function f b (α 4 , α * 1 ) . = Ψ e 4 (−α 4 )(U e r ) · n − Ψ e 1 (−α * 1 )(U e r ) · n. Notice that f b (0, 0) = 0 and ∂f b ∂α 4 α 4 =α * 1 =0 = −r e 4 (U e r ) · n = 0.
Then, by implicit function theorem, we know that α 4 can be solved as C 2 -function of α * 1 from the equation f b (α 4 , α * 1 ) = 0 provided δ > 0 sufficiently small. Moreover, a direct computation shows that
∂α 4 ∂α * 1 α * 1 =0 = r e 1 (U e r ) · n r e 4 (U e r ) · n = 1.
By Taylor formula, we finally derive the estimate (2.19).
We further consider the Riemann problem for system (2.2) with the data (2.12) in different regions, and obtain the following solvability of this problem involving strong shock.
Lemma 2.4. Given two constant sates U l , U r in (2.12) satisfying U l ∈ N δ (Ū b,l ) and U r ∈ N δ (Ū b,r ),
there exists a small constant δ > 0 such that the Riemann problem (2.2)(2.12) admits a unique admissible solution which consists of at most four constant states separated by three weak waves α k (k = 1, 2, 3) and a strong shock α s 4 . Moreover, U l and U r satisfy
U e r = S e 4 (α s 4 ) • Φ e 2 (α 2 ) • Φ e 1 (α 1 )(U e l ), Y r = Y l + α 3 . (2.20) Proof. It is obvious that α 3 = Y r − Y l by definition (2.5).
To prove (2.20) for U e , we need to investigate the equivalent equation
U e l = Ψ e 1 (−α 1 ) • Ψ e 2 (−α 2 ) • S e 4 (−α s 4 )(U e r ). Set the intermediate state U e m . = Ψ e 2 (−α 2 ) • S e 4 (−α s 4 )(U e r ).
Then define a function
f s (α 1 , α 2 , α s 4 ; U e l , U e r ) = U e r + −α s 4 0 r e 4 (U e r , σ)dσ − α 2 r e 2 + −α 1 0r e 1 (U e m , σ)dσ − U e l ,
where vectorr e 1 is given by (2.8). The fact (2.7) and Lemma 2.1 directly yield that
f s (0, 0,ᾱ s 4 ;Ū e b,l ,Ū e b,r ) ≡ 0 and det ∇ (α 1 ,α 2 ,α s 4 ) f s (0, 0,ᾱ s 4 ;Ū e b,l ,Ū e b,r ) = − det r e 1 (Ū e b,l ), r e 2 (Ū e b,l ), r e 4 (Ū e b,r , −ᾱ s 4 ) = 2 c l p ′ (−ᾱ s 4 ) + γp l u ′ (−ᾱ s 4 ) (γ + 1)c l > 0.
If choosing δ > 0 small enough, then by implicit function theorem, f s = 0 has a unique solution ( Fig.5) with δ > 0 sufficiently small, and they satisfy
α 1 , α 2 , α s 4 ) near (0, 0,ᾱ s 4 ). Finally, relation (2.20) is a direct conclusion of f s = 0. α s 4 α 2(3) α 1 α * 4 α s, * 4 U m U m1 U r U l Fig. 5. Weak-strong waves interaction U r U m U m1 U l α s 4 α 2(3) α 1 α s, * 4 α * k Fig. 6. Strong-weak waves interaction Proposition 2.3. Suppose that constant states U l , U m ∈ N δ (Ū b,l ) and U r ∈ N δ (Ū b,r ) (seeU e m = Φ e 4 (α * 4 )(U e l ), U e r = S e 4 (α s, * 4 )(U e m ), Y r = Y m = Y l . (2.21) If U e r = S e 4 (α s 4 ) • Φ e 2 (α 2 ) • Φ e 1 (α 1 )(U e l ), Y r = Y l + α 3 , (2.22)
then there hold that
α 1 = κ s α * 4 + O(1)|α * 4 | 2 , α 2 = O(1)|α * 4 |, α 3 = 0,(2.
23)
and
α s 4 = α s, * 4 + O(1)|α * 4 |, (2.24)
where the reflection coefficient κ s satisfies |κ s | < 1. To estimate α 1 , α 2 and α s 4 , we rewrite the equation (2.25) as
U e r + −α s 4 0 r e 4 (U e r , σ)dσ − α 2 r e 2 + −α 1 0r e 1 (U e m1 , σ)dσ = U e r + −α s, * 4 0 r e 4 (U e r , σ)dσ + −α * 4 0r e 4 (U e m , σ)dσ, where U e m1 = Ψ e 2 (−α 2 ) • S e 4 (−α s 4 )(U e r )
. Differentiating this with respect to α * 4 , and letting α *
4 = 0, α s, * 4 =ᾱ s 4 and (U e m , U e r ) = (Ū e b,l ,Ū e b,r ), one derives that r e 1 (Ū e b,l ) ∂α 1 ∂α * 4 α * 4 =0,α s, * 4 =ᾱ s 4 +r e 2 (Ū e b,l ) ∂α 2 ∂α * 4 α * 4 =0,α s, * 4 =ᾱ s 4 +r e 4 (Ū e b,r , −ᾱ s 4 ) ∂α s 4 ∂α * 4 α * 4 =0,α s, * 4 =ᾱ s 4 = r e 4 (Ū e b,l ),
which gives
∂α 1 ∂α * 4 α * 4 =0,α s, * 4 =ᾱ s 4 = det r e 4 (Ū e b,l ), r e 2 (Ū e b,l ), r e 4 (Ū e b,r , −ᾱ s 4 ) det r e 1 (Ū e b,l ), r e 2 (Ū e b,l ), r e 4 (Ū e b,r , −ᾱ s 4 ) = γp l u ′ (−ᾱ s 4 ) −c l p ′ (−ᾱ s 4 ) γp l u ′ (−ᾱ s 4 ) +c l p ′ (−ᾱ s 4 )
.
With the help of Lemma 2.1, we can further deduce that
∂α 1 ∂α * 4 α * 4 =0,α s, * 4 =ᾱ s 4 = γp l u ′ (−ᾱ s 4 ) −c l p ′ (−ᾱ s 4 ) γp l u ′ (−ᾱ s 4 ) +c l p ′ (−ᾱ s 4 ) < 1, and ∂α s 4 ∂α * 4 α * 4 =0,α s, * 4 =ᾱ s 4 = det r e 1 (Ū e b,l ), r e 2 (Ū e b,l ), r e 4 (Ū e b,l ) det r e 1 (Ū e b,l ), r e 2 (Ū e b,l ), r e 4 (Ū e b,r , −ᾱ s 4 ) = 4γc lpl (γ + 1)c l γp l u ′ (−ᾱ s 4 ) +c l p ′ (−ᾱ s 4 ) > 0.
By Taylor formula and the regularity of α 1 , we finially obtain that for any U l , U m ∈ N δ (Ū b,l ) and U r ∈ N δ (Ū b,r ) with δ sufficiently small, there always holds Similarly, if a weak wave interacts with the strong 4-shock from right, we also have
α 1 (α * 4 , α s, * 4 ) = α 1 (0, α s 4 ) + ∂α 1 ∂α * 4 α * 4 =0 · α * 4 + O(1)|α * 4 | 2 = κ s α * 4 + O(1)|α * 4 | 2 with coefficient |κ s | < 1.Proposition 2.4. Suppose constant states U l ∈ N δ (Ū b,l ) and U m , U r ∈ N δ (Ū b,r ) (see Fig.6) with δ > 0 sufficiently small. (i) If these states satisfy U e m = S e 4 (α s, * 4 )(U e l ), U e r = Φ e k (α * k )(U e m ), Y r = Y m = Y l , k = 3, then, for α i (1 ≤ i ≤ 3) and α s 4 determined by U e r = S e 4 (α s 4 ) • Φ e 2 (α 2 ) • Φ e 1 (α 1 )(U e l ), Y r = Y l + α 3 ,(2.
26)
there holds
α 1 = O(1)|α * k |, α 2 = O(1)|α * k |, α 3 = 0, α s 4 = α s, * 4 + O(1)|α * k |;
(ii) If these states satisfy
U e r = U e m = S e 4 (α s, * 4 )(U e l ), Y m = Y l , Y r = Y m + α * 3 , then, for α i (1 ≤ i ≤ 3) and α s 4 determined by (2.26), there holds α 1 = α 2 = 0, α 3 = α * 3 , α s 4 = α s, * 4 .
For the purpose of L 1 -stability analysis in section 5, we state the following result on solvability of Riemann problems in the sense of states connection by Hugoniot loci.
Lemma 2.5. Assume constant states U l , U r ∈ N δ (Ū b,l ) ∪ N δ (Ū b,r ) with δ > 0 sufficiently small. Then there exists a unique vector q = (q 1 , q 2 , q 3 , q 4 ) such that U r = S 4 (q 4 ) • S 3 (q 3 ) • S(q 2 ) • S 1 (q 1 )(U l ). When U l ∈ N δ (Ū b,l ) and U r ∈ N δ (Ū b,r ), the component q 4 is very close toᾱ s 4 .
The proof of above lemma is completely based on the technique of tangent field analysis and implicit function theorem, as done in Lemma 2.4. So we skip the details here.
Definition 2.1. For given states U l and U r , if there is a unique vector q = (q 1 , q 2 , q 3 , q 4 ) such that U r = H(q)(U l ) . = S 4 (q 4 ) • S 3 (q 3 ) • S(q 2 ) • S 1 (q 1 )(U l ),
we say that q i (1 ≤ i ≤ 4) are the distance indices of two states U l and U r .
Construction of the approximate solutions to (IBVP )
In this section, we will construct the approximate solutions to (IBVP ) via a modified fractional-step wave front tracking scheme. The detailed algorithm is stated as follows. Under assumptions (A1)(A2), we can construct the approximations for given initialboundary data (U 0 , u p ). More precisely, choose a small parameter ε > 0, and then take two piecewise constant functions U ε 0 (x) and u ε p (t) such that: • both U ε 0 (x) and u ε p (t) have finitely many discontinuities;
x t O Γ ε p Γ ε s Ω ε p• U ε 0 (·) − U 0 (·) L 1 (R + ) < ε, T.V.{U ε 0 (·); R + } ≤ T.V.{U 0 (·); R + }, u ε p (·) − u p (·) L 1 (R + ) < ε, T.V.{u ε p (·); R + } ≤ T.V.{u p (·); R + }.
Accordingly, the approximate piston boundary for Γ p and the approximate domain for Ω p are defined by
Γ ε p = {(t, x) ∈ R 2 : t > 0, x = χ ε p (t)}, Ω ε p = {(t, x) ∈ R 2 : t > 0, x > χ ε p (t)}, where x = χ ε p (t) . = t 0 u ε p (τ )dτ . We divide time interval [0, ε −1 ] into N subintervals, such that division points t 0 = 0, t 1 = ε, t 2 = 2ε, · · · t N = N ε,
and additionally set t N +1 = +∞. Then the whole domain Ω ε p is divided into some strips
Ω ε k = {(t, x) ∈ R 2 : t k−1 ≤ t < t k , x > χ ε p (t)} for k = 1, · · · , N.
We start to construct the ε-approximate solution U ε in Ω ε p in the following way. At time t 0 , the Riemann problem for homogenous system (2.2) at each discontinuous point of U ε 0 is solved as stated in section 2, such that the solution consists of weak or strong shocks, rarefaction waves and contact discontinuities. As done in [4], rarefaction waves need to be partitioned into several small central rarefaction fans with strength less thanδ, which is called the accurate Riemann solver denoted by (ARS) for simplicity. Another is the simplified Riemann solver denoted by (SRS), in which all the new waves are lumped into a single non-physical wave traveling with a fixed speedλ larger than all the characteristic speeds (see [4, pp.129-pp.132] for more details). At initial time t 0 , we always use the (ARS).
The piecewise constant approximate solution U ε can be prolonged until waves interactions occur in Ω ε 1 . This refers to (i) interactions between weak waves; (ii) interactions between weak waves and strong shock; (iii) wave reflections at non-corner points on Γ ε p (see Fig.4); (iv) weak wave appearance from the boundary corner (see Fig.3). Then we can solve the Riemann problem again as stated in section 2. For cases (iii)(iv) on the boundary, we always use the (ARS) as in [1,12]. To decide which Riemann solver is used for cases (i)(ii), we introduce a threshold parameter ̺ > 0. If the strengths α, β of two weak waves satisfy |αβ| > ̺, the (ARS) is used; otherwise, we use the (SRS). If a weak physical wave front with strength |α| > ̺ hits the strong shock, then we use the (ARS); otherwise, we use the (SRS).
At time t 1 , we treat the combustion process by linear approximation, and hence prescribe the fluid state U ε (t 1 , x) by the following equations
E(U ε (t 1 , x)) = E(U ε (t 1 − 0, x)) + εG(U ε (t 1 − 0, x)), where U ε (t 1 − 0, x) = lim t→t − 1 U ε (t, x).
After that, solve the homogeneous system (2.2) with initial data U ε (t 1 , x). In this process, if a physical weak wave or strong shock wave hits the line t = t 1 , we use the (ARS); if a non-physical wave hits the line t = t 1 , we use the (SRS).
Inductively, assume that the approximate solution U ε has been constructed for 0 < t ≤ t k−1 (k > 1), and contains the jumps of weak shock fronts, strong shock front, contact discontinuities, rarefaction fronts and non-physical fronts, which are denoted by
J (U ε ) . = S ∪ S b ∪ C ∪ R ∪ N P.
Then, in strip Ω ε k , we construct the approximate solution U ε for homogeneous system (2.2) with initial data U ε (t k−1 , x) by repeating the above processes. Finally, considering the combustion process at time t k , we approximately solve the nonhomogeneous system (2.1) with initial data U ε (t k − 0, x) by the algebraic euqations
E(U ε (t k , x)) = E(U ε (t k − 0, x)) + εG(U ε (t k − 0, x)).
(3.1)
Remark 3.1. In each step of construction, we can slightly perturb the speed of a single wave front with its strength error less than ε, so that every interaction exactly involves two waves, and that every reflection on boundary Γ ε p includes only one incident wave. In addition to the four characteristic fields of system (2.2), we treat N P waves as the fifth family. Then assign their speed λ 5 .
=λ > λ i for 1 ≤ i ≤ 4.
Remark 3.2. The parametersδ and ̺ are functions of ε, i.e.,δ =δ(ε), ̺ = ̺(ε), and satisfy thatδ(ε) → 0, ̺(ε) → 0 as ε → 0.
Based on the above algorithm, we have the following properties for ε-approximate solution U ε , whose proofs are similar to those for Lemmas 7.1-7.2 in [4].
Proposition 3.1. Suppose α is a wave front of ε-approximation U ε to the system (2.2), which is located at (t, x α (t)). Then the following local error estimates are satisfied:
x α (t)[E(U ε )] − [F (U ε )] = O(1)ε|α|, α ∈ J (U ε ) \ N P, x α (t)[E(U ε )] − [F (U ε )] = O(1)|α|, α ∈ N P, where U ε,± = U ε (t, x α (t) ± 0) and [E(U ε )] = E(U ε,− ) − E(U ε,+ ), [F (U ε )] = F (U ε,− ) − F (U ε,+ ). Proposition 3.2. Suppose that U ε contains three adjacent constant states U l = (U e l , Y l ) ⊤ , U m = (U e m , Y m ) ⊤ , U r = (U e r , Y r ) ⊤ ∈ N δ (Ū b,r ) (or N δ (Ū b,l )
) with δ > 0 sufficiently small, and they are separated by two incoming waves α * i , α * j ∈ J (U ε ). Then the (SRS) generated from Riemann problem (U l , U r ) satisfies the following properties.
(i) If α * i , α * j / ∈ N P for i ≥ j but i, j = 3, such that U e m = Φ e i (α * i )(U e l ), U e r = Φ e j (α * j )(U e m ), Y l = Y m = Y r , then the (SRS) includes an auxiliary stateÛ r = (Û e r , Y r ) ⊤ witĥ U e r . = Φ e i (α * i ) • Φ e j (α * j )(U e l ) for i > j, Φ e i (α * i + α * j )(U e l ) for i = j,
and non-physical
α 5 . = |Û e r − U e r |. Moreover, there holds α 5 = O(1)|α * i α * j |. (ii) If α * i / ∈ N P and α * j ∈ N P for i = 3, j = 5, such that U e r = Φ e i (α * i )(U e m ), Y l = Y m = Y r and α * 5 . = |U e m − U e l |, then the (SRS) includes an auxiliary stateÛ r . = (Φ e i (α * i )(U e l ), Y r ) ⊤ and non-physical α 5 . = |Φ e i (α * i )(U e l ) − U e r |. It satisfies that α 5 = α * 5 + O(1)|α * i α * 5 |. (iii) If α * i / ∈ N P and α * j ∈ N P for i = 3, j = 5, such that U e r = U e m , Y r = Y m + α * 3 and α * 5 . = |U e l − U e m |, Y m = Y l , then the (SRS) includes an auxiliary stateÛ r . = (U e l , Y l + α * 3 ) ⊤ and non-physical α 5 . = |U e l − U e r |.
It is obvious that
α 5 = α * 5 . Proposition 3.3. Suppose that U ε contains three adjacent constant states U l = (U e l , Y l ) ⊤ and U m = (U e m , Y m ) ⊤ ∈ N δ (Ū b,l ), while U r = (U e r , Y r ) ⊤ ∈ N δ (Ū b,r ) with δ > 0
sufficiently small, and they are separated by two incoming waves α * 5 ∈ N P, α s,
* 4 ∈ S b . If U e r = S e 4 (α s, * 4 )(U e m ), Y l = Y m = Y r and α * 5 . = |U e l − U e m |, then the (SRS) of Riemann problem (U l , U r ) includes an auxiliary stateÛ r . = (S e 4 (α s, * 4 )(U e l ), Y r ) ⊤ and non-physical α 5 . = |S e 4 (α s, * 4 )(U e l ) − U e r |. It satisfies that α 5 = O(1)|α * 5 |.
Global existence of entropy solutions to (IBVP )
On account of ignition condition (1.4), the global solutions to combustion equations (2.1) may evolve in different trajectories in state space. We make a heuristic analysis on the background solutionŪ b . If temperature behind the leading shock is below ignition one (i.e.T l < T i ), then reaction rate φ(T ) = 0 everywhere; the entirely unburnt fluid is dominated by homogeneous Euler equations and reactant transport equation. IfT r > T i , then φ(T ) > 0 everywhere; and combustion reaction may take place in the whole tube. We claim that the chemical dissipative term G(U ) decays exponentially as t → +∞. It implies the global well-posedness of combustion solutions. However, ifT l > T i >T r , then only the fuel behind the large shock is ignited; an exothermic reaction occurs within the region enclosed by piston and shock front. Thus any uniform decay rate of reaction source fails to hold. The global well-posedness of combustion solutions becomes very delicate. In this section, we intend to establish the global existence of BV solutions for non-reacting and reacting flows respectively.
Since the perturbation of initial-boundary data is sufficiently small, there exist positive d and d such that
d < |λ i (U ) − λ i+1 (V )| < d (i = 1, 3), |λ j (U ) − λ j (V )| < d, d < |λ j (U ) − s| < d (1 ≤ j ≤ 4), (4.1) for states U, V in the identical domain N δ (Ū b,l ) or N δ (Ū b,r ).
Recall that s is the speed of a large 4-shock wave connecting states from
N δ (Ū b,l ) to N δ (Ū b,r ). Suppose α i (i = 1, 4) is an i-wave which connects states U + , U − ∈ N δ (Ū b,l ). Then there is a positive a such that 1 a |α i | ≤ [u] + [p] ≤ a|α i |,
where [u] and [p] represent the jumps of velocity and pressure across the wave α i , respectively.
4.1.
Interaction potential and modified Glimm-type functional.
Definition 4.1 (Approaching waves). We say that k α -wave α and k β -wave β in the approximate solution U ε are approaching at time t, if one of the following conditions holds.
• k α > k β and x α < x β .
• k α = k β and the k α -family field is genuinely nonlinear . At least one of α and β is shock wave.
Introduce the sets A = (α, β) neither α nor β is the large 4-shock, and they are approaching , A(α) = β β is not the large 4-shock, and it is approaching given wave α .
Obviously A includes all couples of approaching small waves at time t. In particular, let symbol α s 4 denote the large 4-shock wave in solution U ε (t, x), and curve x = χ ε s (t) be the path of this large shock.
Then we define the modified Glimm-type functional for approximate solution U ε to (IBVP ) by
G(t) . = L(t) + KQ(t) + KL b (t) (4.2) where functionals L(t) = L e (t) + L Y (t) . = |α s 4 −ᾱ s 4 | + i =3 |α i | + θ 3 |α 3 |, Q(t) = Q w (t) + Q p (t) + Q s (t) . = A |α i α j | + θ 1 |α 1 | + A(α s 4 ) i<3 |α i | + θ 4 |α 4 | + θ 5 |α 5 | , L b (t) = L bp (t) + L by (t) . = T.V.{u ε p (τ ); (t, +∞)} + kε≥t Υ ε,p k + kε≥t Υ ε,s k .
We remark that functional Q(t) is a variant of quadratic term in classic Glimm-type functional, which includes the sum Q p (t) of 1-waves approaching the piston boundary Γ ε p , and the sum Q s (t) of weak waves approaching the large 4-shock. Functional L b (t) manifests the boundary effect caused by piston motion L bp (t) and by reactant consumption L by (t) at piston and large leading shock. We require that constant K in (4.2) is suitably large according to the standard BV theory in [4,27]. Coefficient θ 1 is suitably small, and θ 4 , θ 5 satisfy 0 < θ 1 |κ s | < θ 4 < θ 1 , 0 < θ 5 < θ 1 , (4.3) where κ s given by Proposition 2.3 is the coefficient of weak waves reflection on strong 4-shock front. We place θ 1 and θ 4 in Q(t) for the purpose of controlling the reflection effect arising from weak-strong waves interaction. The significant coefficient θ 3 is specified according to different conditions in Theorem 1.1. Under condition (C1) for entirely unburnt fluid, choose θ 3 so small that for k ≥ 1.
By the scheme in section 3 and definition of G(t), we observe the fact that for given small δ > 0, there exists a small constantδ > 0 such that if G(t) <δ at some t > 0, then the solution U ε (t, x) contains a unique large shock α s 4 . Since |α s 4 −ᾱ s 4 | <δ and
T.V.{U ε (t, ·); (χ ε p (t), χ ε s (t))} + T.V.{U ε (t, ·); (χ ε s (t), +∞)} ≤ O(1)G(t), there hold U ε (t, x) ∈ N δ (Ū b,l ), ∀ x ∈ (χ ε p (t), χ ε s (t)); U ε (t, x) ∈ N δ (Ū b,r ), ∀ x ∈ (χ ε s (t), +∞). (4.4)
Our aim is to show that functional G(t) is decreasing with respect to t. If this monotonicity holds, we naturally suppose G(0) is a priori upper bound of G(t). The proof involves two parts respectively for non-reacting and reacting fluids.
4.2.
Existence of the entropy solutions for non-reaction process. We first investigate the non-reacting flow for equations (1.1) under condition (C1), which is the basis of further discussion on combustion phenomena in section 4.3.
SinceT r <T l < T i under condition (C1), combustion does not occur at all. Thus
∆L Y . = L Y (t + 0) − L Y (t − 0) = 0, L by (t) ≡ 0, for all t > 0.
We will prove the monotonicity of G(t) in different cases of local wave interactions. Henceforth, employ the notation α * i + α * j → α i + α j + α k to denote the process that collision of waves α * i and α * j produces a set of outgoing waves α i , α j , α k . Similarly, process α * 1 → α 4 stands for the wave reflection on piston boundary. We have the following lemma on monotonicity. Proof. The argument consists of several cases as follows. Case 4.1.1. Reflection α * 1 → α 4 on boundary. Suppose that a incident weak α * 1 hits the piston boundary, and produces a reflection weak wave α 4 . Then the estimate (2.19) in Proposition 2.2 gives
α 4 = α * 1 + O(1)|α * 1 | 2 .
Based on this, we obtain
∆L(t) = |α 4 | − |α * 1 | = O(1)|α * 1 | 2 ≤ O(1)G(t − 0)|α * 1 |,∆Q(t) = ∆Q w (t) + ∆Q p (t) + ∆Q s (t) = |α 4 | A(α 4 ) |β| − |α * 1 | A(α * 1 ) |β| − θ 1 |α * 1 | + θ 4 |α 4 | ≤ θ 4 − θ 1 + G(t − 0) + O(1)G(t − 0) G(t − 0) + θ 4 |α * 1 | ≤ − 1 2 (θ 1 − θ 4 )|α * 1 |, which leads to ∆G(t) = ∆L(t) + K∆Q(t) ≤ O(1)G(t − 0) − 1 2 K(θ 1 − θ 4 ) |α * 1 | < − 1 4 K(θ 1 − θ 4 )|α * 1 | < 0,(4.6)
providedδ is sufficiently small. Case 4.1.2. Appearance of single α 4 from boundary. If a weak wave α 4 issues from a corner point (t, χ ε p (t)) with the left and right velocities satisfying u l = u ε p (t + 0) and u r = u ε p (t − 0), then by estimate (2.16) in Lemma 2.3, we know that the strength |α 4 | is equivalent to |u ε
p (t + 0) − u ε p (t − 0)|, i.e., 1 a |α 4 | ≤ |u ε p (t + 0) − u ε p (t − 0)| ≤ a|α 4 |.
Thus it implies that
∆L(t) = |α 4 |, ∆L b (t) = −|u ε p (t + 0) − u ε p (t − 0)| ≤ − 1 a |α 4 |, ∆Q(t) = ∆Q w (t) + ∆Q s (t) = |α 4 | A(α 4 ) |β| + θ 4 |α 4 | ≤ θ 1 + G(t − 0) |α 4 |,
and
∆G(t) = ∆L(t) + K∆Q(t) + K∆L b (t) ≤ |α 4 | + K θ 1 + G(t − 0) − 1 a |α 4 | ≤ |α 4 | − 1 2a K|α 4 | ≤ − 1 4a K|α 4 | < 0,(4.7)
provided both θ 1 andδ are small, while constant K suitably large. Case 4.1.3. Interaction α * 4 + α s, * 4 → α 1 + α 2 + α s 4 . If a weak wave α * 4 interacts with strong shock α s, * 4 from left, then Proposition 2.3 directly gives the estimates
α 1 = κ s α * 4 + O(1)|α * 4 | 2 , α 2 = O(1)|α * 4 |, α s 4 = α s, * 4 + O(1)|α * 4 |.
From these and coefficient condition (4.3), we deduce that
∆L(t) = |α 1 | + |α 2 | + |α s 4 −ᾱ s 4 | − |α * 4 | − |α s, * 4 −ᾱ s 4 | = O(1)|α * 4 |, ∆Q(t) = ∆Q w (t) + ∆Q p (t) + ∆Q s (t) = |α 1 | A(α 1 ) |β| + |α 2 | A(α 2 ) |β| − |α * 4 | A(α * 4 ) |β| + θ 1 |α 1 | − θ 4 |α * 4 | ≤ |κ s α * 4 | + O(1)|α * 4 | 2 G(t − 0) + θ 1 + O(1)G(t − 0)|α * 4 | − θ 4 |α * 4 | = θ 1 |κ s | − θ 4 + O(1)G(t − 0) |α * 4 | ≤ − 1 2 (θ 4 − θ 1 |κ s |)|α * 4 |,
which gives
∆G(t) = ∆L(t) + K∆Q(t) ≤ − 1 4 K(θ 4 − θ 1 |κ s |)|α * 4 | < 0,(4.|α 1 | + |α 2 | = O(1)|α * i |, α s 4 = α s, * 4 + O(1)|α * i |. Then it follows that ∆L(t) = |α 1 | + |α 2 | + |α s 4 −ᾱ s 4 | − |α * i | − |α s, * 4 −ᾱ s 4 | = O(1)|α * i |, ∆Q(t) = ∆Q w (t) + ∆Q p (t) + ∆Q s (t) = |α 1 | A(α 1 ) |β| + |α 2 | A(α 2 ) |β| − |α * i | A(α * i ) |β| + θ 1 |α 1 | − (2 − i)|α * i | − |α * i | ≤ −|α * i | + O(1)(G(t − 0) + θ 1 )|α * i | ≤ − 1 2 |α * i |.
Thus we can get
∆G(t) = ∆L(t) + K∆Q(t) ≤ O(1)|α * i | − 1 2 K|α * i | < 0,(4.∆Q(t) = ∆Q w (t) + ∆Q p (t) + ∆Q s (t) = |α 5 | A(α 5 ) |β| − |α * 5 | A(α * 5 ) |β| − θ 5 |α * 5 | ≤ O(1)G(t − 0) − θ 5 |α * 5 | ≤ − 1 2 θ 5 |α * 5 |.
Thus we find that
∆G(t) = ∆L(t) + K∆Q(t) ≤ O(1)|α * 5 | − 1 2 Kθ 5 |α * i | < 0,(4.10)
if K is sufficiently large.
Case 4.1.6. Interactions between weak waves α * i and α * j (1 ≤ i, j ≤ 5). For instance, consider the interaction α * i + α * j → α 1 + α 2 + α 4 for i, j ∈ {1, 2, 4}. Estimates in (2.13) directly give
|α i − α * i | + |α j − α * j | + |α k | = O(1)|α * i α * j | for i = j, |α i − α * i − α * j | + k =i |α k | = O(1)|α * i α * j | for i = j.
From these estimates, we deduce that
∆L(t) = O(1)|α * i α * j |, ∆Q(t) ≤ − 1 + O(1) θ 1 + θ 4 + G(t − 0) |α * i α * j | ≤ − 1 2 |α * i α * j |,
provided θ 1 , θ 4 andδ are taken suitably small. It further implies that
∆G(t) ≤ O(1)|α * i α * j | − 1 2 K|α * i α * j | < 0,(4.11)
by choosing K > 0 sufficiently large. We turn to interaction α * i + α * j → + α 5 for i, j = 3. Here symbol stands for a pair of outgoing waves of family i or j. Recall Proposition 3.2 on interactions involving N P waves. If i = j < 5, then is actually a coalesced wave α i such that α i = α * i + α * j . If i = j and i, j < 5, then contains two waves α i and α j such that α i = α * i , α j = α * j . If i = 5 and j < 5, then preserves a single α j after interaction, which satisfies α j = α * j . In any case, we always have
∆L(t) = O(1)|α * i α * j |, ∆Q(t) ≤ −|α * i α * j | + O(1)(G(t − 0) + θ 1 )|α * i α * j | ≤ − 1 2 |α * i α * j |.
It follows that
∆G(t) ≤ O(1)|α * i α * j | − 1 2 K|α * i α * j | < 0. (4.12)
Consider α * 3 + α * j → α 3 + α j for j = 2, 3. In this case, transport of reactant does not influence the motion of non-reacting fluid. So α 3 = α * 3 and α j = α * j , which leads to
∆L(t) = 0, ∆Q(t) = −|α * 3 α * j |.
It gives that
∆G(t) ≤ −K|α * 3 α * j | < 0. (4.13)
Finally, combining the estimates (4.6)-(4.13) altogether, we derive the conclusion (4.5), and complete the proof.
sup x∈(χ ε p (t),χ ε s (t)) |U ε (t, ·) −Ū b,l | + sup x∈(χ ε s (t),+∞) |U ε (t, ·) −Ū b,r | + sup t>0 |χ ε s (t) −s b | + T.V.{U ε (t, ·); (χ ε p (t), χ ε s (t))} + T.V.{U ε (t, ·); (χ ε s (t), +∞)} < C 1 ǫ,(4.14)
and
U ε (t, ·) − U ε (t, ·) L 1 (R) + |χ ε s (t) − χ ε s (t)| ≤ C 1 (t −t) for t >t > 0. (4.15)
Moreover, there holds that
Y ε (t, ·) L 1 (R) ≤ sup ρ ε inf ρ ε Y ε (t, ·) L 1 (R) + C 2 (t −t)ε for t >t > 0. (4.16)
Proof. Estimate (4.14) is a direct result based on assumption (A1), Lemma 2.1 and the fact
|χ ε s (t) −s b | + T.V.{U ε (t, ·); (χ ε p (t), χ ε s (t))} + T.V.{U ε (t, ·); (χ ε s (t), +∞)} ≤ O(1)G(t)
. See also [21] for detailed argument. One can readily verify (4.15) according to finite propagation speed for homogenous system (2.2) under condition (C1). Now we continue to consider the estimate (4.16). To this end, let's define the amount of reactant by
Z ε (t) = +∞ χ ε p (t) ρ ε (t, x)Y ε (t, x)dx.
It is clear that Z ε (t) is globally Lipshcitz continuous, and differentiable everywhere except finitely many times related to wave-interaction and angular points on piston boundary. Let wave front α ∈ J (U ε ) be located at x α (t) in (χ ε p (t), +∞). Then, by Rankine-Hugoniot conditions and Proposition 3.1, we geṫ
Z ε (t) = α∈J (U ε ) [ρ ε Y ε ]ẋ α (t) − ρ ε (t, χ ε p (t))Y ε (t, χ ε p (t))u ε p (t) = α∈J (U ε )\N P [ρ ε u ε Y ε ] + O(1)ε|α| + α∈N P [ρ ε u ε Y ε ] + O(1)|α| − ρ ε (t, χ ε p (t))Y ε (t, χ ε p (t))u ε p (t) = O(1)ε α∈J (U ε )\N P |α| + O(1) α∈N P |α| = O(1)ε,(4.17)
where the notation
[f ] = f (t, x α (t) − 0) − f (t, x α (t) + 0) for f = ρ ε Y ε or f = ρ ε u ε Y ε . Integrate (4.17) fromt to t , then derive that +∞ χ ε p (t) (ρ ε Y ε )(t, x)dx − +∞ χ ε p (t) (ρ ε Y ε )(t, x)dx = O(1)ε(t −t), for t >t > 0.
Sinceρ l >ρ r > 0, it follows from (4.14) and the smallness of ǫ that
Y ε (t, ·) L 1 (R) ≤ sup ρ ε inf ρ ε Y ε (t, ·)|| L 1 (R) + O(1)ε(t −t),
which leads to the estimate (4.16).
Proof of Theorem 1.1 under condition (C1). By Helly's compactness theorem and Proposition 4.1, there is a sequence {ε k } such that as ε k → 0 + , the approximation U ε k converges almost everywhere in R + × R to some function U satisfying (1.11). Moreover, following the standard methods as in [4], we can show that U is an entropy solution to (IBVP ), and complete the proof on existence and structural stability under condition (C1).
4.3.
Existence of the entropy solutions for reaction process. In this subsection, we are concerned with the existence of large amplitude combustion solutions to piston problems under condition (C2) or (C3). Assume that U ε is an ε-approximate solution constructed by fractional-step front tracking scheme as presented in section 3. We aim to establish the uniform bound on the total variation of U ε for reacting flow. Suppose G(t) <δ for every t ∈ (0, t k ). Hence (4.4) holds. The piecewise continuity of φ(T ) implies that there exist two positive constants φ andφ such that = (χ ε p (t), χ ε s (t)) under condition (C3). It suffices to prove G(t k − 0) < G(t k + 0). Then the decrease of G(t) ensures that the properties (4.4)(4.18) further hold for t ∈ (0, t k+1 ).
φ ≤ φ(T ε (t, x)) ≤φ for t ∈ (0, t k ), x ∈ I.
For brevity, we always use superscript * in this subsection to mark the states before reaction. According to the scheme (3.1), the states transition in combustion process is given by
U = U * + εG 0 (U * ) at time t k ,(4.19)
where U . = U ε (t k , x), U * . = U ε (t k − 0, x) and
G 0 (U * ) = 0, 0, (γ − 1)q 0 ρ * φ(T * )Y * , −φ(T * )Y * ⊤ .
When a wave α * i hits the grid line t = t k , there are diverse cases in which wave α * i splits into several new fronts after combustion (see Fig 8). Here again we use the notation α * i → α i + α j + α k + α m at x α to represent the process of wave splitting caused by combustion reaction at position x α . Lemma 4.2. Suppose thatδ and ε are small enough, and that G(t) <δ for t ∈ (0, t k ). Then, in reaction process at t = t k , we have the estimates on wave strengths as follows.
(
i) If α * 3 → α 1 + α 2 + α 3 + α 4 at x α , then there hold |α 3 | ≤ |α * 3 | − |α * 3 |φε, j =3 |α j | = O(1)|α * 3 |ε. (ii) If α * i → α 1 + α 2 + α 3 + α 4 at x α for i = 3, then there holds |α i − α * i | + j =i |α j | = O(1)ε|α * i |Y * α , where Y * α = Y ε (t k − 0, x α ).
Proof. In order to distinguish the states on distinct sides of x α , we set
Y * α± . = Y ε (t k − 0, x α ± 0), Y α± . = Y ε (t k , x α ± 0), T * α± . = T ε (t k − 0, x α ± 0), T α± . = T ε (t k , x α ± 0), Υ α± . = |Y α± − Y * α± |. (i) Since T * α+ = T * α−
in this case, the definitions of α 3 and Υ α± directly yield that
|α 3 | = |(Y * α+ − Y * α− )(1 − φ(T * α− )ε)| ≤ |α * 3 |(1 − φε), |Υ α+ − Υ α− | = |Y * α+ − Y * α− |φ(T * α− )ε ≤ |α * 3 |φε.
(4.20)
Notice that the outgoing waves α 1 , · · · , α 4 are uniquely determined by α * 3 , Υ α− and Υ α+ . Thus set function α j = f j (α * 3 , Υ α− , Υ α+ ). It follows from (4.20) that
α j = f j (α * 3 , Υ α− , Υ α− ) + O(1)|Υ α− − Υ α+ | = f j (α * 3 , 0, 0) + O(1)(|α * 3 Υ α− | + |Υ α− − Υ α+ |) = δ 3j · α * 3 + O(1)|α * 3 |ε,(4.21)
where δ ij is the Kronecker delta. The above estimate in fact gives the conclusion in (i).
(ii) In this case, we have Y
* α+ = Y * α− = Y * α . Hence |α 3 | = |Υ α+ − Υ α− | = Y * α |φ(T * α+ ) − φ(T * α− )|ε = O(1)|α * i |Y * α ε.
We employ the similar argument in (4.21) to derive that
α j = δ ij · α * i + |α * i Υ α− | + |Υ α− − Υ α+ | = δ ij · α * i + O(1)|α * i |Y * α ε.
Next, according to the fractional-step scheme, we drop the exothermic reaction terms if t ∈ (t k−1 , t k ). Thus,
∆L Y (t) = L Y (t + 0) − L Y (t − 0) = 0, ∆L by (t) = L by (t + 0) − L by (t − 0) = 0.
which implies that when t ∈ (t k−1 , t k ), functional G(t) is decreasing as proved in section 4.2. It suffices to check its monotonicity in the combustion process at time t k with k ≥ 1.
Lemma 4.3. Suppose thatδ and ε are small enough, and that G(t) <δ for t ∈ (0, t k ). It holds that
G(t k + 0) < G(t k − 0). (4.22)
Proof. We mainly investigate the partial reaction phenomenon under condition (C3), since the global existence of completely ignited flow (i.e. T (t, x) > T i for every x ≥ χ p (t)) under condition (C2) can be seen as a byproduct which will be discussed at the end of this proof. In fact the condition (C3) and property (4.4) imply that
T ε (t, x) > T i if χ ε p (t) ≤ x < χ ε s (t), T ε (t, x) < T i if x > χ ε s (t).
Then, under the circumstance of partial reaction, we can show that the total consumption satisfies
∞ k=1 Υ ε,p k ≤ ∞ k=1 Y 0 ∞ e −φkε ·φε ≤ 2φ||Y 0 || ∞ φ on χ ε p , and ∞ k=1 Υ ε,s k ≤ 2φ Y 0 L 1 (R) minχ ε s (t) < +∞ on χ ε s .
Based on these, we now focus on the pointwise changes of G(t k ) within the reaction zone [χ ε p (t k ), χ ε s (t k )]. Case 4.2.1. Combustion process at χ ε p (t k ). Concerning the combustion process near the piston boundary, we have
∆G(t k ) x=χ ε p (t k ) = K∆L b (t k ) x=χ ε p (t k ) = −KΥ ε,p k < 0. (4.23) t = t k α 4 α 2(3) α 1 α * i Fig. 8. Combustion pro- cess in (χ ε p (t), χ ε s (t)) t = t kα * i → α 1 + α 2 + α 3 + α 4 at x α .
Similarly, β * j → β 1 + β 2 + β 3 + β 4 represents waves splitting arising from the reaction at other position x β . First of all, consider the situation i = 3. It follows from Lemma 4.2 that
∆L(t k ) x=xα = (∆L e (t k ) + ∆L Y (t k )) x=xα = i =3 |α i | + θ 3 (|α 3 | − |α * 3 |) ≤ − 1 2 θ 3 φε|α * 3 |, (4.24) provided θ 3 > 0 is sufficiently large. Notice that i =3 |α i | A(α i ) |β| ≤ O(1)G(t k − 0)|α * 3 |ε,
and
|α 3 | − |α * 3 | A(α 3 ) |β| + |α * 3 | A(α 3 ) |β| − A(α * 3 ) |β * | ≤ O(1)|α * 3 | x β <χ ε s (t k ) x β =xα |β * 3 | + |β * j |Y * β ε + Υ ε,s k .
Subsequently, we have that
∆Q w (t k ) x=xα = i =3 |α i | A(α i ) |β| + |α 3 | A(α 3 ) |β| − |α * 3 | A(α * 3 ) |β * | ≤ O(1)G(t k − 0)|α * 3 |ε + O(1)|α * 3 | x β <χ ε s (t k ) x β =xα |β * 3 | + |β * j |Y * β ε + Υ ε,s k .
On the other hand, by Lemma 4.2 (i) and (4.3), we also get
∆Q p + ∆Q s (t k ) x=xα = θ 1 |α 1 | + θ 4 |α 4 | ≤ O(1)θ 1 |α * 3 |ε. So it follows that ∆Q(t k ) x=xα = (∆Q w + ∆Q p + ∆Q s )(t k ) x=xα ≤ O(1) G(t k − 0) + θ 1 |α * 3 |ε + O(1)|α * 3 | x β <χ ε s (t k ) x β =xα |β * 3 | + |β * j |Y * β ε + Υ ε,s k .
(4.25)
By (4.24) and (4.25), we thus obtain the local estimate
∆G(t k ) x=xα = (∆L + K∆Q)(t k ) x=xα ≤ − 1 2 θ 3 φ + O(1)(G(t k − 0) + θ 1 )K |α * 3 |ε + O(1)K|α * 3 | x β <χ ε s (t k ) x β =xα |β * 3 |ε + |β * j |Y * β ε + O(1)K|α * 3 |Υ ε,s k .
(4.26)
Next, we investigate the situation i = 3, and take i = 1 for instance. It is deduced from Lemma 4.2 (ii) that
∆L(t k ) x=xα = i =3 |α i | + θ 3 |α 3 | − |α * 1 | ≤ O(1) 1 + θ 3 |α * 1 |Y * α ε, and ∆Q w (t k ) x=xα = |α 1 | A(α 1 ) |β| − |α * 1 | A(α * 1 ) |β * | + i =1 |α i | A(α i ) |β| ≤ −|α * 1 |φε x β <xα |β * 3 | + O(1)G(t k − 0)|α * 1 |Y * α ε + O(1)|α * 1 | x β <χ ε s (t k ) x β =xα |β * 3 |ε + |β * j |Y * β ε + Υ ε,s k , ∆Q p + ∆Q s (t k ) x=xα = θ 1 |α 1 | − |α * 1 | + θ 4 |α 4 | ≤ O(1)θ 1 |α * 1 |Y * α ε.
We take advantage of a significant spatial estimate
x β <xα |β * 3 | ≥ Y * α − Y (t k − 0, χ ε p (t k ) + 0).
Substituting this into the estimate ∆Q w (t k ) x=xα , we derive that
∆Q(t k ) x=xα = ∆Q w + ∆Q p + ∆Q s (t k ) x=xα ≤ −|α * 1 |φε Y * α − Y (t k − 0, χ ε p (t k ) + 0) + O(1)G(t k − 0)|α * 1 |Y * α ε + O(1)|α * 1 | x β <χ ε s (t k ) x β =xα |β * 3 | + |β * j |Y * β ε + Υ ε,s k + O(1)θ 1 |α * 1 |Y * α ε ≤ O(1) G(t k − 0) + θ 1 − φ |α * 1 |Y * α ε + O(1)|α * 1 | x β <χ ε s (t k ) x β =xα |β * 3 | + |β * j |Y * β ε + |α * 1 |Υ ε,p k + O(1)|α * 1 |Υ ε,s k .
Combining the estimates on ∆L(t k ) x=xα and ∆Q(t k ) x=xα , we thus obtain
∆G(t k ) x=xα ≤ O(1) 1 + θ 3 + O(1) G(t k − 0) + θ 1 K − Kφ |α * 1 |Y * α ε + O(1)K|α * 1 | x β <χ ε s (t k ) x β =xα |β * 3 | + |β * j |Y * β ε + K|α * 1 |Υ ε,p k + O(1)K|α * 1 |Υ ε,s k . (4.27)
Likewise, when wave α * i (1 < i = 3) hits the grid line t = t k , we observe that
x β <xα |β * 3 | ≥ Y * α − Y (t k − 0, χ ε p (t k ) + 0) if i = 2, x β >xα |β * 3 | ≥ Y * α − Y (t k − 0, χ ε s (t k ) − 0) if i > 3,
and then figure out
∆G(t k ) x=xα ≤ O(1)(1 + θ 3 ) + O(1)K G(t k − 0) + θ 1 − Kφ |α * i |Y * α ε + O(1)K|α * i | x β <χ ε s (t k ) x β =xα |β * 3 |ε + |β * j |Y * β ε + K|α * i |Υ ε,p k + O(1)K|α * i |Υ ε,s k .
(4.28)
Case 4.2.3. Combustion process at χ ε s (t k ). Suppose the strong shock α s, * 4 hits the grid line t = t k . Then the local reaction gives rise to α s, * 4 → α 1 + α 2 + α 3 + α s 4 at χ ε s (t k ). See Fig.9. According to the relations 29) and
|α 1 | + |α 2 | + |α s 4 − α s, * 4 | = O(1)Υ ε,s k , |α 3 | = Υ ε,s k . we deduce that ∆L(t k ) x=χ ε s (t k ) = |α 1 | + |α 2 | + θ 3 |α 3 | + |α s 4 −ᾱ s 4 | − |α s, * 4 −ᾱ s 4 | = O(1) + θ 3 Υ ε,s k ,(4.∆Q w (t k ) x=χ ε s (t k ) = 3 i=1 |α i | A(α i ) |β| ≤ O(1)G(t k − 0)Υ ε,s k , ∆Q p (t k ) x=χ ε s (t k ) = θ 1 |α 1 | = O(1)θ 1 Υ ε,s k , ∆Q s (t k ) x=χ ε s (t k ) = θ 4 A(α s 4 ) |β 4 | + θ 5 x β <χ ε s (t k ) |β 5 | − θ 4 A(α s, * 4 ) |β * 4 | − θ 5 x β <χ ε s (t k ) |β * 5 | ≤ O(1)θ 1 x β <χ ε s (t k ) |β * 3 | + |β * j |Y * β ε.
Thus the previous estimates yield
∆Q(t k ) x=χ ε s (t k ) = ∆Q w + ∆Q p + ∆Q s (t k ) x=χ ε s (t k ) ≤ O(1) G(t k − 0) + θ 1 Υ ε,s k + O(1)θ 1 x β <χ ε s (t k ) |β * 3 | + |β * j |Y * β ε. (4.30)
On the other hand, we obviously have
∆L b (t k ) x=χ ε s (t k ) = ∆L bp (t k ) x=χ ε s (t k ) = −Υ ε,s k . (4.31)
Therefore, combining the estimates (4.29)-(4.31) altogether, we find that
∆G(t k ) x=χ ε s (t k ) ≤ O(1) + θ 3 + O(1)K G(t k − 0) + θ 1 − K Υ ε,s k + O(1)Kθ 1 x β <χ ε s (t k ) |β * 3 | + |β * j |Y * β ε.
(4.32)
Finally, taking summation of the pointwise estimates (4.23), (4.26)-(4.28) and (4.32) on ∆G(t k ) over region [χ ε p (t k ), χ ε s (t k )], and using the symmetry of ∆G(t k ) with respect to x α and x β , we conclude that
∆G(t k ) = ∆G(t k ) x≤χ ε s (t k ) ≤ K 1 2 G(t k − 0) − 1 Υ ε,p k + O(1) + θ 3 + O(1)K G(t k − 0) + θ 1 − K Υ ε,s k + − 1 2 θ 3 φ + O(1)K G(t k − 0) + θ 1 xα<χ ε s (t k ) |α * 3 |ε + O(1) 1 + θ 3 + O(1)K G(t k − 0) + θ 1 − 1 2 Kφ i =3 xα<χ ε s (t k ) |α * i |Y * α ε ≤ − 1 2 KΥ ε,p k − 1 2 KΥ ε,s k − 1 4 θ 3 φ xα<χ ε s (t k ) |α * 3 |ε − 1 4 Kφ i =3 xα<χ ε s (t k ) |α * i |Y * α ε < 0, (4.33)
provided thatδ and θ 1 are sufficiently small, while K is suitably large. This gives the proof of (4.22) under condition (C3).
In the light of previous argument on partial reaction, one can easily verify the situation of completely ignited flow under condition (C2), i.e., T ε (t,
x) > T i for x ≥ χ ε p (t). Since φ(T ε ) > φ > 0 everywhere, the argument for χ ε p (t k ) < x α < χ ε s (t k ) in Case 4.2.2 remains valid for x α > χ ε s (t k )
. Under condition (C2), we have the estimates
3 i=1 |α i | = O(1)|α s, * 4 |Y * ε = O(1)Υ ε,s k at x = χ ε s (t k ).
Then, similar to the proof of (4.33), we see that
∆G(t k ) = ∆G(t k ) x≥χ ε p (t k ) ≤ − 1 2 θ 3 φ + O(1)K G(t k − 0) + θ 1 |α * 3 |ε + O(1) 1 + θ 3 + O(1)K G(t k − 0) + θ 1 − 1 2 Kφ i =3 xα =χ ε s (t k ) |α * i |Y * α ε + O(1) 1 + θ 3 + O(1)K G(t k − 0) + θ 1 − K Υ ε,s k + 1 2 G(t k − 0) − 1 KΥ ε,p k ≤ − 1 2 KΥ ε,p k − 1 2 KΥ ε,s k − 1 4 θ 3 φ |α * 3 |ε − 1 4 Kφ i =3 xα =χ ε s (t k ) |α * i |Y * α ε < 0, (4.34)
This completes the proof of (4.22) under condition (C2).
By Lemma 4.3, we can also obtain the following estimates on the approximate solution U ε for reacting flow. for t > 0, and
sup x∈(χ ε p (t),χ ε s (t)) |U ε (t, ·) −Ū b,l | + sup x∈(χ ε s (t),+∞) |U ε (t, ·) −Ū b,r | + sup t>0 |χ ε s (t) −s b | + T.V.{U ε (t, ·); (χ ε p (t), χ ε s (t))} + T.V.{U ε (t, ·); (χ ε s (t), +∞)} < C 3 ǫ,U ε (t, ·) − U ε (t, ·) L 1 (R) + |χ ε s (t) − χ ε s (t)| ≤ C 4 t −t + ε , (4.36)
for any t >t > 0.
Proof. The argument of (4.35) is completely analogous to that of (4.14) under condition (C1). It remains to verify the estimate (4.36) for U ε . According to (4.15) in Proposition 4.1, we have
U ε (t, ·) − U ε (t, ·) L 1 (R) ≤ O(1)|t −t|, for anyt, t ∈ [t k−1 , t k ).
On the other hand, by (4.19)(4.16), we obtain that
U ε (t k + 0, ·) − U ε (t k − 0, ·) L 1 (R) ≤ O(1)ε +∞ χ ε p (t k ) Y ε (t k − 0, x)φ(T ε (t k − 0, x))dx ≤ O(1)ε Y ε (t k − 0, ·) L 1 (R) ≤ O(1) Y 0 L 1 (R) + 1 ε.
Then, combining the above two estimates altogether, we can derive the approximate continuity of U ε (t, ·) L 1 (R) with respect to t, i.e.,
U ε (t) − U ε (t) L 1 (R) ≤ O(1) t −t + ⌈(t −t)/ε⌉ · ε ≤ O(1) t −t + ε ,
for any t >t > 0.
Remark 4.1. Review the existence argument by means of Glimm-type functional in this section. Basically, it is allowed that the initial positions of piston and large shock are separated, namely χ ε p (0) < χ ε s (0). Therefore, the initial restriction on T.V.{U 0 (·) −Ū b,r ; R + } in Theorem 1.1 can be relaxed to a smallness condition of Likewise, one can obtain the information of the flow on any time-like curve. For this purpose we devise a new geometric object called quasi-characteristic, which is a generalization of characteristics in the classic theory of hyperbolic partial differential equations. Furthermore, it becomes an essential tool to establish L 1 stability of weak solutions in sections 5.
T.V.{U 0 (·) −Ū b,l ; (χ ε p (0), χ ε s (0))} + T.V.{U 0 (·) −Ū b,r ; (χ ε s (0), +∞)}.
Definition 4.2. Assume that a generic hyperbolic system
∂ t E(U ) + ∂ x F (U ) = G(U ) (4.37)
has n eigenvalues such that
λ 1 (U ) ≤ · · · ≤ λ i (U ) ≤ · · · ≤ λ n (U ).
Let U (t, x) be a solution to system (4.37), which takes values in domain D * . We say a Lipschitz continuous curve x = χ(t) is an i-quasi-characteristic associated with U (t, x), if there exists a constant d > 0 such that
sup U ∈D * λ j (U ) + d <χ(t) < inf U ∈D * λ k (U ) − d, ∀ t > 0,
for any indices j, k satisfying j < i < k and λ j (U ) < λ i (U ) < λ k (U ).
As pointed out previously, the class of quasi-characteristics includes not only classic characteristics, but their small perturbations in Lipschitz sense. Now we suppose χ(t) Fig. 10. Quasi-characteristic curve is a 2-quasi-characteristic associated with (exact or approximate) solution U to (IBVP ). Moreover, it is required that χ(t) satisfies
α 1 x = χ(t) α 4 α 5 α * 4 α * 1 (u * , p * ) (u, p)χ p (t) ≤ χ(t) < χ s (t)
for any t > 0, (4.38) where χ p (t) is the piston path and χ s (t) is the large shock front path. Set χ(I) = {(t, χ(t)) : x = χ(t), t ∈ I}. Let notation T.V.{f ; χ(I)} stand for the variation of f along the Lipschitz continuous curve x = χ(t) over interval I. Then we introduce the following functionals which are used to investigate the variation of (u, p) on quasi-characteristic x = χ(t). Set
L χ w (t) = i =2,3 xα<χ(t) l i |α i | + i =2,3 xα>χ(t) r i |α i | + θ 3 |α 3 |, and L χ b (t) = T.V.{(u, p); χ(I t )} + a t k ≥t Υ χ+ k + a t k ≥t Υ χ− k , I t = interval (0, t),
where the transient consumption of reactant on χ(t) are
Υ χ+ k = Y (t k − 0, χ(t k ) + 0)φ(T ((t k − 0, χ(t k ) + 0)))ε, Υ χ− k = Y (t k − 0, χ(t k ) − 0)φ(T ((t k − 0, χ(t k ) − 0)))ε.
Here the weights satisfy
r 1 > l 1 > l 4 > r 4 > 0, l 5 > r 5 > 0. (4.39)
Define the redistributed Glimm-type functional for U by
G χ (t) = L χ w (t) + KQ(t) + KL b (t) +ǫL χ b (t) with 0 <ǫ ≪ 1.
The assumption (4.39) on coefficients readily gives the fact that functional G χ (t) is decreasing when any single α i (i = 2, 3) goes across the curve x = χ(t). Next, it suffices to prove the monotonicity of G χ (t) at time t of waves interaction and of reaction step. Hence we show the following lemma that covers the situations of non-reacting and reacting flows. Proof. Since G χ (t) and G(t) have the same term Q(t), the argument here is highly analogous to that in sections 4.2-4.3. So we only need to show some key estimates to prove the lemma. (i) Non-reacting flow under condition (C1). Case 4.3.1. Reflection α * 1 → α 4 at χ p (t). Recall that α 4 = α * 1 + O(1)|α * 1 | 2 , then by the coefficients restriction (4.39), we have
∆L χ w (t) = l 4 |α 4 | − l 1 |α * 1 | = l 4 |α 4 | − |α * 1 | + (l 4 − l 1 )|α * 1 | ≤ O(1)|α * 1 | 2 ≤ O(1)G(t − 0)|α * 1 |. Therefore, ∆G χ (t) ≤ O(1)G(t − 0)|α * 1 | − 1 2 K θ 1 − θ 4 |α * 1 | ≤ − 1 4 K(θ 1 − θ 4 )|α * 1 | < 0.∆G χ (t) ≤ l 4 |α 4 | − 1 2a K|α 4 | ≤ − 1 4a K|α 4 | < 0. Case 4.3.3. Interaction α * i + α * j → α 1 + α 2 + α 4 (i, j = 3) at χ(t)
. It suffices to verify the typical case i = 4, j = 1 at x = χ(t). The other cases can be treated likewise. Since
∆L χ w (t) = l 1 |α 1 | + r 4 |α 4 | − r 1 |α * 1 | − l 4 |α * 4 | = (l 1 − r 1 )|α * 1 | + (r 4 − l 4 )|α * 4 | + O(1)|α * 1 α * 4 |, ∆L χ b (t) = |u − u * | + |p − p * | ≤ a(|α 1 | + |α * 4 |) = a(|α * 1 | + |α * 4 | + O(1)|α * 1 α * 4 |), it follows from (4.39) that ∆G χ (t) ≤ l 1 − r 1 +ǫa |α * 1 | + r 4 − l 4 +ǫa |α * 4 | + O(1)|α * 1 α * 4 | − 1 2 K|α * 1 α * 4 | ≤ − 1 4 K|α * 1 α * 4 | < 0. Case 4.3.4. Interaction α * i + α * j → + α 5 (i, j = 3) at χ(t)
. When a non-physical wave α 5 appears at x = χ(t), we also figure out the estimates for redistributed functionals similar to that in Case 4.1.6. For instance, we consider the typical situation i = 4, j = 1. See Fig.10. Notice that
α 1 = α * 1 , α 4 = α * 4 , α 5 = O(1)|α * 1 α * 4 |. Hence a direct computation shows that ∆L χ w (t) = l 1 − r 1 |α * 1 | + r 4 − l 4 |α * 4 | + O(1)|α * 1 α * 4 |, ∆L χ b (t) ≤ a |α 1 | + |α * 4 | , which yield ∆G χ (t) ≤ l 1 − r 1 +ǫa |α * 1 | + r 4 − l 4 +ǫa |α * 4 | + O(1)|α * i α * j | − 1 2 K|α * 1 α * 4 | < − 1 4 K|α * 1 α * 4 | < 0.
Analogously, for every i, j < 5, there holds
∆G χ (t) < − 1 4 K|α * i α * j | < 0.
If i = 5 and j = 1, then the estimates α 1 = α * 1 and
α 5 = α * 5 + O(1)|α * 1 α * 5 | imply ∆G χ (t) ≤ l 1 − r 1 +ǫa |α * 1 | + r 5 − l 5 +ǫa |α * 5 | + O(1)|α * 1 α * 5 | − 1 2 K|α * 1 α * 5 | < − 1 4 K|α * 1 α * 5 | < 0.
Similarly, if i = 5 and j > 1, there holds
G χ (t) < − 1 4 K|α * i α * j | < 0.
The rest of cases for waves collision at x = χ(t) are so trivial and easy to verify. This completes the proof for non-reacting flow.
(ii) Reacting flow under condition (C2) or (C3). Consider the ε-approximation solution U ε for partially ignited or completely ignited flow. Based on the previous argument, functional G χ (t) is decreasing for t ∈ (t k−1 , t k ). Our purpose now is verifying the monotonicity for reaction process. In fact, it suffices to establish the local estimate on ∆G χ (t k ) x=χ(t k ) . We first consider that wave α * 1 hits the grid line t = t k and then splits into new waves by reaction, i.e., α * 1 → α 1 + α 2 + α 3 + α 4 at x = χ(t k ). A straightforward calculation gives that
∆L χ w (t k ) x=χ(t k ) = l 1 |α 1 | − |α * 1 | + l 1 − r 1 |α * 1 | + r 4 |α 4 | + θ 3 |α 3 | ≤ O(1)|α * 1 |Y * α ε + (l 1 − r 1 )|α * 1 |, ∆L χ b (t k ) x=χ(t k ) ≤ a |α 1 | + Υ χ− k − aΥ χ+ k − aΥ χ− k ≤ a|α * 1 | + O(1)|α * 1 |Y * α ε. Hence, we have ∆L χ w +ǫ∆L χ b (t k ) x=χ(t k ) ≤ O(1)|α * 1 |Y * α ε + l 1 − r 1 +ǫa |α * 1 | ≤ O(1)|α * 1 |Y * α ε, ifǫ is small enough. Generally, for every i = 3, reaction α * i → α 1 + α 2 + α 3 + α 4 at x = χ(t k ) always implies ∆L χ w +ǫ∆L χ b (t k ) x=χ(t k ) ≤ O(1)|α * i |Y * α ε.
When reaction α * 3 → α 1 + α 2 + α 3 + α 4 takes place at x = χ(t k ), we readily derive the estimate
∆L χ w +ǫ∆L χ b (t k ) x=χ(t k ) ≤ − 1 2 θ 3 |α * 3 |φε + O(1)ǫ|α * 3 |ε ≤ − 1 4 θ 3 |α * 3 |φε.
Note that L χ w (t) merely extracts part of waves from L(t), particularly preserves all the 3-waves with respect to mass fraction. Replace ∆L(t) in sections 4.2-4.3 with estimates on ∆L χ w (t) +ǫ∆L χ b (t). By the argument analogous to (4.33)-(4.34), we can also show that ∆G χ (t k ) < 0 regardless of whetherT r > T i . This concludes the proof.
The decrease of G χ (t) implieŝ ǫT.V.{(u, p); χ(I ∞ )} ≤ G χ (+∞) ≤ G χ (0) ≤ O(1)G(0),(4.40)
where I ∞ = (0, +∞). Hence we conclude that the total variation of components (u, p) in U on any quasi-characteristic x = χ(t) satisfying (4.38) is bounded provided G(0) is small enough. We next compare the state distribution on distinct quasi-characteristics. To this end, suppose that U = (ρ, u, p, Y ) ⊤ is a solution to system (2.1), and x =χ(t) is its 2-quasicharacteristic curve satisfying (4.38) instead of χ(t). Then, we have the following property on the difference of fluid velocities.
u(τ, χ p (τ )) − u(τ,χ(τ )) L 1 ([t,t]) ≤ O(1) T.V.{U 0 (·); R + } + T.V.{u p (·); R + } χ p (τ ) −χ(τ ) C([t,t]) , (4.41) where x = χ p (t) is the path of piston.
Proof. It suffices to technically treat U as the ε-approximate solution which includes finitely many constant states. Suppose x = χ(t) is some 2-quasi-characteristic associated with U and satisfies (4.38). Let h(t) be a non-negative function which is a small perturbation of χ(t) with h C 0,1 ≪ 1 so that • there exists a positive constant d such that
sup N δ (Ū b,l ) λ 1 (U ) + d <χ(t) < inf N δ (Ū b,l ) λ 4 (U ) − d, sup N δ (Ū b,l ) λ 1 (U ) + d <χ(t) +ḣ(t) < inf N δ (Ū b,l ) λ 4 (U ) − d;
• there is not any wave-colliding point (t, x) in the region χ(t) < x < χ(t) + h(t) for every t > 0;
• there are finitely many wave fronts {x α } in U which divide the region between χ(t) and χ(t) + h(t) into many subregions with constant states.
Next we only focus on the fronts of family i with i = 1, 4, 5. Assume x = x α (t) is an i-wave front that moves from point (τ 0 , x 0 ) ∈ χ (resp. χ + h) to point (τ 1 , x 1 ) ∈ χ + h (resp. χ). Then it satisfies
|χ(t) −ẋ α (t)| ≥ d, |χ(t) +ḣ(t) −ẋ α (t)| ≥ d,
for a.e. t ∈ (τ 0 , τ 1 ). This implies that
d∆t α ≤ τ 1 τ 0 |χ(τ ) +ḣ(τ ) −ẋ α (τ )|dt ≤ h C([τ 0 ,τ 1 ]) ,
where ∆t α . = τ 1 − τ 0 . Let u α+ , u α− stand for the velocities at two banks of discontinuity x α . Then from above inequality we can deduce for every t >t ≥ 0 that
t t |u(τ, χ(τ )) − u(τ, χ(τ ) + h(τ ))|dτ = α |u α+ − u α− |∆t α ≤ 1 d h C([t,t]) α |u α+ − u α− | = O(1) h C([t,t]) T.V.{(u, p); χ((t, t))}.
(4.42)
Define a homotopic mapping
χθ(t) =θχ p (t) + (1 −θ)χ(t) forθ ∈ [0, 1].
Decompose [0, 1] into finitely many small intervals [θ i ,θ i+1 ] with 0 =θ 0 <θ 1 < · · · < θ n−1 <θ n = 1, so that the wave interactions do not occur in each region between the curves χθ i and χθ i+1 . Then, by (4.40) and (4.42), we obtain
t t |u(τ, χ p (τ )) − u(τ,χ(τ )|dτ ≤ i t t |u(τ, χθ i (τ )) − u(τ, χθ i+1 (τ ))|dτ ≤ O(1)T.V.{(u, p); χθ i ((t, t))} i χθ i (τ ) − χθ i+1 (τ ) C([t,t]) ≤ O(1) T.V.{U 0 (·); R + } + T.V.{u p (·); R + } χ p (·) −χ(·) C([t,t]) ,
which gives the estimate (4.41).
L 1 -stability for the entropy solutions to (IBVP )
In this section, we will further study the L 1 -stability of entropy solutions U to (IBVP ). Suppose that U ε = (U e,ε , Y ε 1 ) and V ε = (V e,ε , Y ε 2 ) with U e,ε = (ρ ε 1 , u ε 1 , p ε 1 ) ⊤ and V e,ε = (ρ ε 2 , u ε 2 , p ε 2 ) ⊤ are two ε-approximate solutions corresponding to the initial-boundary data (U ε 0 , u ε p ) and (V ε 0 , v ε p ), respectively. In the following discussion, we will drop the superscript ε in U ε , V ε as well as the corresponding data for simplicity.
For given boundary data u p and v p , we define the corresponding boundary curves by
χ p,1 (t) . = t 0 u p (τ )dτ, χ p,2 (t) . = t 0 v p (τ )dτ
for t ≥ 0. Then denote their minimal and maximal curves respectively by
χ m p (t) . = min χ p,1 (t), χ p,2 (t) , χ m p (t) . = max χ p,1 (t), χ p,2 (t) .
Indeed, χ m p (t) is a 2-quasi-characteristic associated with the solutions U and V . Besides, let's denote the paths of large 4-shocks in U and V by χ s,1 (t), χ s,2 (t), respectively. Then define another two curves χ m s (t)
.
= min χ s,1 (t), χ s,2 (t) , χ m s (t)
.
= max χ s,1 (t), χ s,2 (t) .
5.1.
Construction of the weighted Lyapunov functional. For any (t, x) ∈ {(t, x) ∈ R 2 : x ≥ χ m p (t), t > 0}, we make an orientation rule by
(Ũ l ,Ũ r ) = (U, V ) if both U e and V e ∈ N δ (Ū e b,l ) or N δ (Ū e b,r ), (U, V ) if U e ∈ N δ (Ū e b,l ) and V e ∈ N δ (Ū e b,r ), (V, U ) if V e ∈ N δ (Ū e b
,l ) and U e ∈ N δ (Ū e b,r ).
(5.1)
Then, according to Lemma 2.5, statesŨ l andŨ r can be connected by the Hugoniot curves of system (2.2). That is, there exists a group of distance indices q = (q 1 , q 2 , q 3 , q 4 ) such thatŨ
r = H(q)(Ũ l ) = S 4 (q 4 ) • S 3 (q 3 ) • S 2 (q 2 ) • S 1 (q 1 )(Ũ l )
. Set ω 0 =Ũ l and ω 4 =Ũ r . Define the intermediate states
ω i . = S i (q i )(ω i−1 ) (i = 1, 2, 3).
Obviously, the total strength 4 i=1 |q i | is equivalent to |U −V |. Henceforth, we will employ notation λ i .
= λ i (ω i−1 , ω i ) (1 ≤ i ≤ 4)
for brevity. Now we define Lyapunov functional for (IBVP ) by ) is a function to balance the distinct speeds of large shocks. Assume further ℓ i is piecewise constant in x and assigned values as in Tables 1 and 2. Table 2. Distribution of ℓ i (t, x) under condition (C2) or (C3)
L (U (t), V (t)) = 4 i=1 +∞ χ M p (t) ℓ i W i |q i |dx, where ℓ i = ℓ i (t, xx (χ m p , χ m s ) (χ m s , χ m s ) (χ m s , +∞) ℓ 1 ℓ ℓ 2 ℓ 3 ℓ 2 1 ℓ 2 ℓ 3 ℓ 3 1/K ℓ/K ℓ 2 /K ℓ 4 κ 0 ℓ 1 ℓ 3 Table 1. Distribution of ℓ i (t, x) under condition (C1) x (χ m p , χ m s ) (χ m s , χ m s ) (χ m s , +∞) ℓ 1 ℓ ℓ 2 ℓ 3 ℓ 2 1 ℓ 2 ℓ 3 ℓ 3 ℓ 4 ℓ 5 ℓ 6 ℓ 4 κ 0 ℓ 1 ℓ 3
We require that the constants K and ℓ are chosen so large that
K ≫ ℓ 6 ≫ 1, ℓ > 1 θ 3 . (5.2)
The different strategies of ℓ 3 distribution in two tables are caused by lack of smallness restriction on T.V.{Y 0 ; R + } in condition (C1). So we require ℓ 3 is small enough for nonreacting flow, while ℓ 3 is much larger than other ℓ i for reacting flow. The positive coefficient κ 0 will be specified later. W i is the key weight to retrieve the monotonicity of L (U (t), V (t)). It is originally introduced by Bressan-Liu-Yang [5] for general small BV data and by [23] for large BV data. Specifically,
W i (x, t) . = 1 + K A i (x) + K Q(U (t)) + Q(V (t)) + KLb(t) ,(5.3)
where K is the large constant from Glimm functional G(t), and the linear terms A i only includes part of waves selected from U and V . Precisely, let α be a wave of the k α th family, with location x α and strength |α|. Given time t, introduce notations |α|.
J e . = J e (U ) ∪ J e (V ), J y 3 . = J y 3 (U ) ∪ J y 3 (V ), J e l . = J e l (U ) ∪ J e l (V ), J e r . = J e r (U ) ∪ J e r (V ), where J e (U ) = {α | α is a small wave of U, k α = 3}, J e l (U ) = {α | α ∈ J (U ) and component U e (t, x α ± 0) ∈ N δ (Ū e b,l )}, J e r (U ) = {α | α ∈ J (U ) and component U e (t, x α ± 0) ∈ N δ (Ū e b,r )}, J y 3 (U ) = {α | α is a 3-wave of U }, and similarly define J e (V ), J e l (V ), J e r (V ) and J y 3 (V ). For i = 1, 4, we define A i (t, x) = i<kα≤4, xα<x, α∈J e + 1≤kα<i, xα>x, α∈J e |α| + θ 3 i=1, xα<x, α∈J y 3 + i=4, xα>x, α∈J y 3 + α∈J y 3 |α| + kα=i,
In addition, the term Q in (5.3) remains the amount of potential interactions in G(t); and the term Lb(t) is given by
Lb(t) . = L b (U (t)) + L b (V (t)) = T.V.{u p ; (t, +∞)} + kε≥t Υ p k + kε≥t Υ s k U + T.V.{v p ; (t, +∞)} + kε≥t Υ p k + kε≥t Υ s k V .
Finally, we claim that if the initial-boundary data (U 0 , u p ) and (V 0 , v p ) are sufficiently small in BV sense, then there holds
1 ≤ W i ≤ W (1 ≤ i ≤ 4) (5.4)
for some constant W very close to 1.
5.2. L 1 -stability and uniqueness for non-reacting flow. By means of Lyapunov functional L , we now proceed to show the stability of global solutions for non-reacting flow. Owing to condition (C1), the functional L (U (t), V (t)) is Lipschitz continuous in t and differentiable at any time except a finite subset of (0, +∞). We will consider how L (U (t), V (t)) evolves at the time t without waves interaction away from χ m p (t) and without waves emergence on χ m p (t). To this end, let's suppose that J denotes the set of all the waves in U and V at time t. And assume that α ∈ J which belongs to k α -familiy. Differentiating L with respect to t gives thaṫ
L (U (t), V (t)) = α∈J 4 i=1 |q α− i |ℓ α− i W α− i − |q α+ i |ℓ α+ i W α+ i ẋ α − 4 i=1 |q p+ i |ℓ p+ i W p+ iχ m p = α∈J 4 i=1 |q α+ i |ℓ α+ i W α+ i (λ α+ i −ẋ α ) − |q α− i |ℓ α− i W α− i (λ α− i −ẋ α ) + 4 i=1 |q p+ i |ℓ p+ i W p+ i λ p+ i −χ m p . = α∈J 4 i=1 E i,α + 4 i=1 E i,p ,
where the symbols q α− i , ℓ α− i , W α− i , λ α− i (resp. q α+ i , ℓ α+ i , W α+ i , λ α+ i ) stand for the quantities on the left-hand side (resp. right-hand side) of a front x α , while the terms q p+ i , ℓ p+ i , W p+ i , λ p+ i on curve x = χ m p (t) can be interpreted in the similar way. In the sequel, our main purpose is to verify that
4 i=1 E i,α < O(1)ε|α|, α ∈ J \ N P, 4 i=1 E i,α ≤ O(1)|α|, α ∈ N P, 4 i=1 E i,p ≤ O(1)|∆u m p |,(5.5)
where ∆u m p = u 1 (t, χ m p (t) + 0) − u 2 (t, χ m p (t) + 0). Given wave α ∈ J , we rewrite E i,α in the concise form
E i = |q + i |ℓ + i W + i (λ + i −ẋ α ) − |q − i |ℓ − i W − i (λ − i −ẋ α ), 1 ≤ i ≤ 4,(5.2 (γ + 1)c − γpu ′ − cp ′ .
Then, from Lemma 2.1, it follows that
|κ s | = γpu ′ − cp ′ γpu ′ + cp ′ < 1.
By (5.7)(5.8) and Taylor formula, we obtain the estimate
q − 1 = q + 1 + κ s q − 4 + O(1) |q + 1 | 2 + |q + 2 | 2 + |q − 4 | 2 (5.9)
with |κ s | < 1.
|κ s | · |λ 1 (V − ) −ẋ α | |λ 4 (V − ) −ẋ α | < 1 (5.10)
for the large 4-shock α s 4 joining two states V − ∈ N δ (Ū b,l ) and V + ∈ N δ (Ū b,r ). Proof. Let s =ẋ α be the speed of large 4-shock α s 4 . A straightforward computation gives
|κ s | · |λ 1 (V − ) −ẋ α | |λ 4 (V − ) −ẋ α | = γpu ′ − cp ′ γpu ′ + cp ′ · c + (s − u) c − (s − u) = c γpu ′ − p ′ (s − u) − c 2 p ′ − γpu ′ (s − u) c γpu ′ − p ′ (s − u) + c 2 p ′ − γpu ′ (s − u) .
Therefore, it suffices to prove
γpu ′ − (s − u)p ′ > 0, c 2 p ′ − γ(s − u)pu ′ > 0. (5.11) By the Rankine-Hugoniot condition F (V − ) − F (V + ) = s E(V − ) − E(V + ) , we have [u] 2 = [ρ][p] ρρ + , p ′ − ρu ′ (s − u) = s ′ ([ρu] − u[ρ]), γpu ′ − p ′ (s − u) = s ′ [p] − γ − 1 2 · ρ + [u] 2 .
(5.12)
The equality (5.12) 1 together with 1 < ρ/ρ + < γ+1 γ−1 gives
[p] − γ − 1 2 · ρ + [u] 2 = 1 2 [p] 3 − γ + (γ − 1) · ρ + ρ > 0.
By (5.12) 2 -(5.12) 3 and s ′ > 0, we see that the first inequality in (5.11) holds. Equality (5.12) 1 also yields the second inequality in (5.11), i.e., Assume that α = α s 4 is the large 4-shock of U or V (see . In this case, we have the estimates
c 2 p ′ − γ(s − u)pu ′ = c 2 p ′ − (s − u)ρu ′ = c 2 s ′ ([ρu] − u[ρ]) = c 2 s ′ ρ + [u] > 0. < 0, provided that ℓ ≫ 1 ≫ ǫ and κ 0 > κ 1 W.|q − 1 | + |q − 2 | = O(1) i =3 |q + i |, |q + 3 | = |q − 3 |, |λ − 4 −ẋ α | = O(1) i =3 |q + i |, which yield that 4 i=1 E i ≤ −|q + 1 |ℓ + 1 W + 1 |λ + 1 −ẋ α | + O(1) i =3 |q + i |ℓ − 1 W − 1 |λ − 1 −ẋ α | − |q + 2 |ℓ + 2 W + 2 |λ + 2 −ẋ α | + O(1) i =3 |q + i |ℓ − 2 W − 2 |λ − 2 −ẋ α | − |q + 3 |ℓ + 3 W + 3 |λ + 3 −ẋ α | + |q − 3 |ℓ − 3 W − 3 |λ − 3 −ẋ α | − |q + 4 |ℓ + 4 W + 4 |λ + 4 −ẋ α | + O(1)|q − 4 |ℓ − 4 W − 4 i =3 |q + i | ≤ 4 i=1 |q + i |ℓ + i |λ + i −ẋ α | − 1 + O(1) ℓ −1 + ℓ −3 < 0. χ m s x α χ m s x α U V Fig. 15. Flow compari- son for Case 5.1.3 • • • U e,+ U e,− q + q − α V e N δ (Ū e b,l ) N δ (Ū e b,|q + i − q − i | = O(1)|α|, |λ + i − λ − i | = O(1)|α|, and |W + i − W − i | = K|α| if i = k α , K|α| if i = k α = 1 and q + i q − i > 0, 0 if i = k α = 2. If q + i q − i > 0, then E i = |q + i |ℓ i W + i − W − i λ + i −ẋ α + ℓ i W − i |q + i | − |q − i | λ + i −ẋ α + |q − i | λ + i − λ − i ≤ |q + i |ℓ i K|α|d + ℓ i W |q + i − q − i |d + |q − i ||λ + i − λ − i | ≤ O(1)ℓ 2 |α|. If q + i q − i ≤ 0, then the relation |q + i | + |q − i | = |q + i − q − i | = O(1)|α| implies E i = |q + i |ℓ i W + i (λ + i −ẋ α ) − |q − i |ℓ i W − i (λ + i −ẋ α ) ≤ ℓ 2 W |q + i ||λ + i −ẋ α | + |q − i ||λ − i −ẋ α | ≤ O(1)ℓ 2 |α|. For i = 3, we have W + 3 − W − 3 = K|α|sgn(k α − 2) and |q + 3 | = |q − 3 |. Then E 3 = ℓ 3 |q + 3 | W + 3 − W − 3 λ + 3 −ẋ α + W − 3 λ + 3 − λ − 3 ≤ ℓ 3 |q + 3 | Kd|α| + O(1)W|α| = O(1)ℓ|α|,(5.4 i=1 E i ≤ −Kd|α||q + 4 | + O(1)ℓ 2 |α| < 0.
When k α = 3, it is clear that
q + i = q − i , λ + i = λ − i , W + i − W − i = Kθ 3 |α|sgn(2 − i), for every i = 3. Note that |q + 3 − q − 3 | = |α| and λ + 3 = λ − 3 . Then we have E 1 ≤ ℓ 1 Kθ 3 |α||q + 1 |d = O(1)ℓ 1 |α|, E 2 = 0, E 3 = ℓ 3 W + 3 (|q + 3 | − |q − 3 |)(λ + 3 −ẋ α ) = O(1)ℓ 3 |α|,
E 4 ≤ −ℓ 4 Kθ 3 |α||q + 4 |d. From the above estimates, it follows that
4 i=1 E i ≤ O(1)ℓ 2 |α| − K ℓ d|α||q + 4 | < 0,
due to the coefficients condition (5.2). When k α = 5, that is α ∈ N P, then
W + i = W − i , q + i − q − i = O(1)|α|, λ + i − λ − i = O(1)|α|, and E i = ℓ i W i (|q + i | − |q − i |)(λ + i −ẋ α ) + |q − i |(λ + i − λ − i ) = O(1)|α| for 1 ≤ i ≤ 4. Therefore 4 i=1 E i ≤ O(1)|α|.
Case 5.1.4. Front x α ∈ (χ m p (t), χ m s (t)) or (χ m s (t), +∞). This case is similar to the small initial data problems in [5], since projected states U e , V e are located in the identical domain N δ (Ū e b,l ) or N δ (Ū e b,r ) (see . If k α = 1 or k α = 4, we deduce that
E 3 = ℓ 3 |q + 3 | W + 3 − W − 3 λ + 3 −ẋ α + W − 3 λ + 3 − λ − 3 = ℓ 3 |q + 3 | − K|α|d + O(1)|α| ≤ − 1 2 Kdℓ 3 |α||q + 3 | < 0.
By the same argument in [4,5], it is trivial that i =3 E i < O(1)ε|α| which leads to estimate (5.5) 1 .
If
k α = 2, then λ + 3 − λ − 3 = O(1)|α| |q + 1 | + |q + 4 | which gives E 3 = ℓ 3 |q + 3 |W + 3 λ + 3 − λ − 3 ≤ O(1)ℓ 3 |α||q + 3 | |q + 1 | + |q + 4 | .
On the other hand, based on the estimates (8.59) and (8.60) in [4], we see that
i =3 E i ≤ − 1 2 Kd|α| |q + 1 | + |q + 4 | + O(1) i =3 ℓ i |α| |q + 1 | + |q + 4 | < 0.
Thus, combining the above two estimates altogether and taking K ≫ 4 i=1 ℓ i , we obtain
4 i=1 E i < 0.
If k α = 3, then there hold
q + i = q − i , λ + i = λ − i , W + i − W − i = Kθ 3 |α|sgn(2 − i) for i = 3, and |q + 3 − q − 3 | = |α|, λ + 3 = λ − 3 , λ + 3 −ẋ α = O(1) |q + 1 | + |q + 4 | .
Owing to condition (5.2) and above estimates, we have
E 1 ≤ −ℓ 1 Kθ 3 |α||q + 1 |d ≤ −K|α||q + 1 |d, E 2 = 0, E 3 ≤ O(1)ℓ 3 W|α| |q + 1 | + |q + 4 | , E 4 ≤ −ℓ 4 Kdθ 3 |α||q + 4 | ≤ −Kκ 0 d|α||q + 4 |. Therefore, by choosing K ≫ ℓ 3 , it follows that 4 i=1 E i ≤ O(1)ℓ 3 |α| |q + 1 | + |q + 4 | − Kκ 0 d|α| |q + 1 | + |q + 4 | < 0.
In addition, it is obvious that the distance of two pistons satisfies
χ p,1 (·) − χ p,2 (·) C([t,t]) ≤ |χ p,1 (t) − χ p,2 (t)| + t t |u p (τ ) − v p (τ )|dτ.(5.26)
and lett = 0 in (5.27). Then, it yields that
U µ (t, ·) − U ν (t, ·) L 1 (R) ≤ O(1) U µ 0 − U ν 0 L 1 (R + ) + u µ p (τ ) − u ν p (τ ) L 1 ([0,t]) + O(1) max{µ, ν}t ≤ O(1)(µ + ν) + O(1) max{µ, ν}t.
Passing to the limits as µ, ν → 0, we see that
U µ (t, ·) − U ν (t, ·) L 1 (R) → 0.
This means the approximations {U ε } constructed by fractional-step front tracking algorithm is indeed a Cauchy sequence which has a unique limit U . In terms of the strong convergence, we suppose that
U ε → U, V ε → V, u ε p → u p , v ε p → v p as ε → 0.
Then estimate (5.27) obviously implies that
U (t, ·) − V (t, ·) L 1 (R) ≤ O(1) U (t, ·) − V (t, ·) L 1 (R) + u p (·) − v p (·) L 1 ([t,t])
for any t >t ≥ 0. This eventually established the L 1 -stability of non-reacting flow under condition (C1).
5.3. L 1 -stability for reacting flow. Consider the ε-approximate solutions U and V to the (IBVP ) under condition (C2) or (C3) constructed by fractional-step wave front tracking scheme as stated in section 3. Then, for any t ∈ (t k−1 , t k ) with k ≥ 1, the analysis ofL (U (t), V (t)) in section 5.2 remains valid by only slight modifications on ℓ 3 distribution in Table 2. More precisely, since T.V.{Y 1 (0, ·); (0, +∞)}+T.V.{Y 2 (0, ·); (0, +∞)} < 2ǫ ≪ 1, we can use K|q 3 | < 1 instead of Kℓ 3 = ℓ in (5.16) and deduce again thaṫ
L (U (t), V (t)) ≤ O(1)(ε + |∆u m p |),
for a.e. t ∈ (t k−1 , t k ). (5.28) So our purpose now is to show the changes of L when crossing the time t = t k . Denote the states before reaction by
U * = (ρ * 1 , u * 1 , p * 1 , Y * 1 ) ⊤ , V * = (ρ * 2 , u * 2 , p * 2 , Y *
2 ) ⊤ , and states after reaction by
U = (ρ 1 , u 1 , p 1 , Y 1 ) ⊤ , V = (ρ 2 , u 2 , p 2 , Y 2 ) ⊤ .
They satisfy the relations (4.19). Besides, we always use the notations
q i = q i (t k , x), W i = W i (t k , x), q * i = q i (t k − 0, x), W * i = W i (t k − 0,
x) for brevity. Above all, we consider ∆L (U (t k ), V (t k )) for partially ignited flow in three cases as below.
Case 5.3.1. Difference ∆L at x ∈ (χ m p (t k ), χ m s (t k )). We begin with the estimates on distance indices in combustion process.
Lemma 5.2. For sufficiently small ε, there hold
|q 3 | − |q * 3 | ≤ −|Y * 1 − Y * 2 |φε + O(1) j =3 |q * j |(Y * 1 + Y * 2 )ε,(5.29)
and
|q i | − |q * i | = O(1) |Y * 1 − Y * 2 | + j =3 |q * j |(Y * 1 + Y * 2 ) ε for i = 3,(5.30)
− Let χ p,1 (τ ) and χ s,1 (τ ) be respectively the positions of the piston and large shock corresponding to U τ , while χ p,2 (τ ) and χ s,2 (τ ) corresponding to V τ . Then define the initial intervals at time τ by I j (τ ) = (χ p,j (τ ), χ s,j (τ )), J j (τ ) = (χ s,j (τ ), +∞), for j = 1, 2.
It is required that I 1 (τ ) ∩ I 2 (τ ) = ∅. We also demand that the new initial-boundary data satisfy
U e τ −Ū e b L 1 (R) + V e τ −Ū e b L 1 (R) + u p −ū p L 1 (R + ) + v p −ū p L 1 (R + ) < +∞,
T.V.{U e τ −Ū e b,l ; I 1 (τ )} + T.V.{U e τ −Ū e b,r ; J 1 (τ )} Thus we can choose a positivet suitably small, so that T.V. S ε t (U 0 , u p ) + εG 0 S ε t (U 0 , u p ) ; (χ p,1 , +∞) < 2C 5 ǫ, ∀ ε ∈ (0,t).
The smallness of variation ensures the global stability after perturbation. So we can utilize all the results proved in section 5.3 to establish the following lemma on operators commutation.
Lemma 6.1 (Commutation estimate). Assume (U τ , u p ) is the initial-boundary data under condition (C1) or (C2). Then the operators S ε t and T ε satisfy S ε t • T ε (U τ , u p ) − T ε • S ε t (U τ , u p ) L 1 (R) ≤ O(1)εt, for any t > 0 and 0 < ε ≪t.
Proof. SetÛ (t) = S ε t • T ε (U τ , u p ),V (t) = S ε t (U τ , u p ) andŵ(t) = εG 0 (S ε t (U τ , u p )). Obviously,Û ,V andV +ŵ have the same boundary curve x = χ p,1 (t) and boundary velocity u p . ConnectÛ withV +ŵ by Hugoniot curves according to the orientation rule in (5.1). Then there is a unique vector q = (q 1 , q 2 , q 3 , q 4 ) such thatV +ŵ = H(q)(Û ), or converselŷ U = H(q)(V +ŵ). Here the mapping H is given by Definition 2.1. We further define a perturbed Lyapunov functional
+ 4 i=1 |q p+ i |ℓ p+ i W p+ i (λ p+ i − u p ) . = α∈J 4 i=1 E i,α + 4 i=1 E i,p .
By identical argument in Case 5.1.5, it is readily derived that
4 i=1 E i,p < 0.
We only need to evaluate E i,α . For every wave α ofÛ , sum 4 i=1 E i,α satisfies the estimates (5.5) 1 -(5.5) 2 by the standard argument in section 5.2. Now it remains to consider the wave α ofV , which is certainly associated with the front ofV +ŵ. If 1 ≤ k α ≤ 4, one hasV + = S kα (α)(V − ) and the front speedẋ
α = λ kα (V − ) if α > 0, λ kα (V − ,V + ) if α < 0. = 4 i=1 ℓ i W + i (|q + i | − |q − i |)(λ − i −ẋ α ) + |q − i |(λ + i − λ − i ) = O(1)|α|.
Combining the above four cases altogether, we eventually obtaiṅ Therefore L p (Û (t), (V +ŵ)(t)) − L p (Û (t), (V +ŵ)(t)) ≤ O(1)ε(t −t), ∀ t >t ≥ 0.
L p (Û ,V +ŵ) = α 4 i=1 E i,α + 4 i=1 E i,
Takingt = 0 and using the equivalence between L p and L 1 metric, we conclude that
S ε t • T ε (U τ , u p ) − T ε • S ε t (U τ , u p ) L 1 (R) = Û −V −ŵ L 1 (R) ≤ O(1)εt.
Next we proceed to establish the tangent estimate which exhibits that P ε t and S ε t • T t have an identical tangent operator at (U τ , u p ). Lemma 6.2 (Tangent estimate). Given initial-boundary data (U τ , u p ) under condition (C1) or (C2), there holds P ε t (U τ , u p ) − S ε t • T t (U τ , u p ) L 1 (R) = O(1) 1 + Y 1,τ L 1 (R) t 2 for every t <t and ε ≪ min{t, t 2 }.
Proof. Since operator T t does not affect the velocity of piston, we claim that P ε t (U τ , u p ), S ε t • T t (U τ , u p ) have the same boundary χ p,1 . Set n = ⌈t/ε⌉. We first consider the partially ignited flow under condition (C2). Recall that χ s,1 is the flame front in U τ . Then the fluid temperature dramatically exceeds its ignition temperature behind the front, i.e., T > T i for x ∈ (χ p,1 , χ s,1 ), T < T i for x ∈ (χ s,1 , +∞).
This implies
G 0 (T ε ) n (U τ ) = 0, G 0 (T εn (U τ )) = 0 for x ∈ (χ p,1 , χ s,1 ), and G 0 (T ε ) n (U τ ) = G 0 (T εn (U τ )) = 0 for x ∈ (χ s,1 , +∞),
provided nε is small, where (T ε ) n stands for composition of n operators T ε . Furthermore, G 0 (T ε ) n (U τ ) − G 0 (T nε (U τ )) L 1 (R) ≤ L G 0 χ s,1
χ p,1 |(T ε ) n (U τ ) − T nε (U τ )|dx ≤ L G 0 (T ε ) n (U τ ) − T nε (U τ ) L 1 (R)
where L G 0 denotes the local Lipschitz constant of function G 0 . By induction, we derive that (T ε ) n (U τ ) − T nε (U τ ) L 1 (R) = O(1)(n − 1) 2 ε 2 Y 1,τ L 1 (R) . (6.7)
The above equality also holds under condition (C1). The flow estimates (6.3)(6.4)(6.7) and Lemma 6.1 yield that P ε nε (U τ , u p ) − S ε nε • T nε (U τ , u p ) L 1 (R)
≤ P ε nε (U τ , u p ) − S ε nε • (T ε ) n (U τ , u p ) L 1 (R) + S ε nε • (T ε ) n (U τ , u p ) − S ε nε • T nε (U τ , u p ) L 1 (R) ≤ n k=1 P ε (n−k)ε • T ε • S ε kε • (T ε ) k−1 (U τ , u p ) − P ε (n−k)ε • S ε kε • (T ε ) k (U τ , u p ) L 1 (R) + S ε nε • (T ε ) n (U τ , u p ) − S ε nε • T nε (U τ , u p ) L 1 (R) ≤O(1) n k=1 T ε • S ε kε • (T ε ) k−1 (U τ , u p ) − S ε kε • (T ε ) k (U τ , u p ) L 1 (R) + (n − k)ε 2 + O(1) (T ε ) n (U τ ) − T nε (U τ ) L 1 (R) + O(1)nε 2 ≤O(1) n k=1 kε 2 + (n − k)ε 2 + O(1)(n − 1) 2 ε 2 Y 1,τ L 1 (R) + O(1)nε 2
=O(1) 1 + Y 1,τ L 1 (R) n 2 ε 2 .
(6.8)
We consequently deduce that P ε t (U τ , u p ) − S ε t • T t (U τ , u p ) L 1 (R) ≤ P ε t (U τ , u p ) − P ε nε (U τ , u p ) L 1 (R) + P ε nε (U τ , u p ) − S ε nε • T nε (U τ , u p ) L 1 (R) + S ε nε • T nε (U τ , u p ) − S ε t • T t (U τ , u p ) L 1 (R) ≤O(1)(t − nε) + O(1) 1 + Y 1,τ L 1 (R) n 2 ε 2
+ O(1) T nε (U τ ) − T t (U τ ) L 1 (R) + O(1)nε 2 ≤O(1) 1 + Y 1,τ L 1 (R) t 2
by estimates (6.8) and the Lipschitz continuity of S ε t in (6.4)(6.5).
Finally, we establish the time-additivity of reacting flow P ε t (U τ , u p ). Lemma 6.3 (Additivity estimate). Given initial-boundary data (U τ , u p ) under condition (C1) or (C2), there holds P ε t • P ε t (U τ , u p ) − P ε t+t (U τ , u p ) L 1 (R) = O(1) 1 + ε + t +t + Y 1,τ L 1 (R) ε, for every t,t > 0 and ε ≪t.
Proof. Fix ε ≪t. There exist two integers k and K such that (k − 1)ε < t ≤ kε, Kε ≤ t +t < (K + 1)ε, Now we proceed to discuss the uniqueness of limit solution. Assume that (U µ 0 , u µ p ) and (U ν 0 , u ν p ) are two couples of piecewise constant functions, which satisfy that T.V.{(U ε 0 , u ε p ); R + } ≤ T.V.{(U 0 , u p ); R + } for ε = µ, ν, and U ε 0 − U 0 L 1 (R + ) < ε, u ε p − u p L 1 (R + ) < ε, for ε = µ, ν.
We claim that for every t ≥ 0, there holds P µ t (U µ 0 , u µ p ) − P ν t (U ν 0 , u ν p ) L 1 (R) → 0 as µ, ν → 0. (6.10)
To prove the claim, let's divide the interval [0, t] equally into m parts such that ∆t = t/m is small. Let the points of such division satisfy 0 =t 0 <t 1 < · · · <t m = t. If µ, ν < min{t, (∆t) 2 }, it is deduced from Lemma 6.3 that
P µ t (U µ 0 , u µ p ) − P ν t (U µ 0 , u µ p ) L 1 (R) ≤ m i=1 P µ t−t i • P ν t i (U µ 0 , u µ p ) − P µ t−t i−1 • P ν t i−1 (U µ 0 , u µ p ) L 1 (R) ≤ m i=1 P µ t−t i • P ν t i (U µ 0 , u µ p ) − P µ t−t i • P μ t i −t i−1 • P ν t i−1 (U µ 0 , u µ p ) L 1 (R) + O(1) m i=1 1 + µ + t −t i−1 + Y µ 0 L 1 (R) µ . = Σ 1 + Σ 2 .
(6.11)
Using flow estimates (6.3)(6.9), Proposition 4.1 and Lemma 6.3, we derive
Σ 1 = m i=1 P µ t−t i • P ν t i (U µ 0 , u µ p ) − P µ t−t i • P μ t i −t i−1 • P ν t i−1 (U µ 0 , u µ p ) L 1 (R) ≤ O(1) m i=1 P ν t i (U µ 0 , u µ p ) − P μ t i −t i−1 • P ν t i−1 (U µ 0 , u µ p ) L 1 (R) + O(1) m i=1 (t −t i )µ ≤ O(1) m i=1 ||P ν t i −t i−1 • P ν t i−1 (U µ 0 , u µ p ) − P μ t i −t i−1 • P ν t i−1 (U µ 0 , u µ p ) L 1 (R) + O(1) m i=1 (1 + ν +t i + Y µ 0 L 1 (R) )ν + O(1) m i=1 (t −t i )µ ≤ O(1) m i=1 1 + Y µ 0 L 1 (R) (∆t) 2 + O(1) 1 + ν + t + Y µ 0 L 1 (R) mν + O(1)tmµ ≤ O(1) 1 + µ + Y 0 L 1 (R) t∆t + O(1) 1 + µ + ν + t + Y 0 L 1 (R) mν + O(1)tmµ,(6.12)
and
Σ 2 = O(1) m i=1 1 + µ + t −t i−1 + Y µ 0 L 1 (R) µ ≤ O(1) 1 + µ + t + Y 0 L 1 (R) mµ.
(6.13)
The above estimates (6.11)-(6.13) yield P µ t (U µ 0 , u µ p ) − P ν t (U µ 0 , u µ p ) L 1 (R) ≤ O(1) 1 + µ + ν + t + Y 0 L 1 (R) t∆t + m(µ + ν) . As a result, it follows that
P µ t (U µ 0 , u µ p ) − P ν t (U ν 0 , u ν p ) L 1 (R) ≤ P µ t (U µ 0 , u µ p ) − P ν t (U µ 0 , u µ p ) L 1 (R) + P ν t (U µ 0 , u µ p ) − P ν t (U ν 0 , u ν p ) L 1 (R) ≤ O(1) 1 + µ + ν + t + Y 0 L 1 (R) t∆t + m(µ + ν) + O(1) U µ 0 − U ν 0 L 1 (R) + O(1)tν ≤ O(1) 1 + µ + ν + t + Y 0 L 1 (R) t∆t + m(µ + ν) .
Passing to limits as µ, ν → 0, one has lim µ,ν→0
P µ t (U µ 0 , u µ p ) − P ν t (U ν 0 , u ν p )|| L 1 (R) ≤ O(1) 1 + t + Y 0 L 1 (R) t∆t,
which implies (6.10) owing to the arbitrariness of ∆t. Subsequently, applying this strong convergence to estimate (5.39), we establish (1.12), namely the Lipschitz continuity of combustion solution U (t, x) . = P t (U 0 , u p ) with respect to initial-boundary data.
problem and local interaction estimates for homogeneous system 10 3. Construction of the approximate solutions to (IBVP ) 15 4. Global existence of entropy solutions to (IBVP ) 18 4.1. Interaction potential and modified Glimm-type functional 18 4.2. Existence of the entropy solutions for non-reaction process 20 4.3. Existence of the entropy solutions for reaction process 25 4.4. Some further properties of the entropy solution 32 5. L 1 -stability for the entropy solutions to (IBVP ) 37 5.1. Construction of the weighted Lyapunov functional 38 5.2. L 1 -stability and uniqueness for non-reacting flow 40 5.3. L 1 -stability for reacting flow 51 6. Uniqueness of limit solution to (IBVP ) for combustion flow 56 1. Introduction and main result
Fig. 2 .
2Shock front for piston problem with constant speed
Definition 1. 1 (
1Entropy solution). A vector-valued function U = (ρ, u, p, Y ) ⊤ ∈ BV loc (Ω p ) is called an entropy solution to the initial-boundary value problem (1.1) and (1.5)-(1.6) provided that (i) U (t, x) is a weak solution to equations (1.1) in the distribution sense in Ω p , and satisfies (1.5)-(1.6) in the trace sense;
Theorem 1 . 1 .
11Under the assumptions (A1)-(A2), there exists a constant ǫ > 0 depending solely onŪ b such that the (IBV P ) has a global entropy solution U (t, x) if the initial-boundary data (U 0 (x), u p (t)) satisfy one of the following conditions:(C1)T l < T i andT.V.{U e 0 (·) −Ū e b,r ; R + } + T.V.{u p (·); R + } < ǫ, (1.8) where U e 0 = (ρ 0 , u 0 , p 0 ) ⊤ andŪ e b,r = (ρ r , 0,p r ) ⊤ ;(C2)T r > T i and T.V.{U 0 (·) −Ū b,r ; R + } + T.V.{u p (·); R + } < ǫ; (1.9) (C3)T l > T i >T r and T.V.{U 0 (·) −Ū b,r ; R + } + T.V.{u p (·); R + } +ū −1 p Y 0 L 1 (R + ) < ǫ. (1.10)
|U
(t, ·) −Ū b,l | + supx>χs(t) |U (t, ·) −Ū b,r | + sup t>0 |χ s (t) −s b | + T.V.{U (t, ·); (χ p (t), χ s (t))} + T.V.{U (t, ·); (χ s (t), +∞)} ≤ C 0 ǫ,(1.11)
Fig. 3 .
3Riemann problem near the corner point of the piston
Remark 2. 1 .
1The fact |κ s | < 1 in (2.23) is a key point in proving the global existence of entropy solution to (IBVP).
Proof.
First, by relations (2.21) and (2.22), we have α 3 = 0 and Ψ e 1 (−α 1 ) • Ψ e 2 (−α 2 ) • S e 4 (−α s 4 )(U e r ) = Ψ e 4 (−α * 4 ) • S e 4 (−α s, * 4 )(U e r ). (2.25) Then, from Lemma 2.4, we know that α 1 , α 2 and α s 4 can be solved from equation (2.25) as C 2 -functions of variables α * 4 and α s, * 4 in the vicinity of point (0,ᾱ s 4 ).
Fig. 7 .
7Approximate solutions to the (IBVP )
Lemma 4. 1 .
1Assume that positive t is a time when interaction occurs in domain Ω ε p or on boundary Γ ε p , and that constantδ is sufficiently small. If G(t − 0) <δ, then there holds G(t + 0) < G(t − 0).(4.5)
Lemma 4.1 implies the fact that G(t) ≤ G(0) ≤ O(1)ǫ for every t > 0, provided ǫ is small enough. Consequently, we obtain the following properties on the approximate solutions.
Proposition 4 . 1 .
41Under the assumptions (A1)-(A2), there exist constants C 1 > 0, C 2 > 0 depending solely onŪ b such that if condition (C1) holds for ǫ > 0 sufficiently small, then the approximate solution U ε to (IBVP) satisfies the estimates
p (t), +∞) under condition (C2), while I .
Fig. 9 .
9Combustion process near strong shock
Combustion process at x α ∈ (χ ε p (t k ), χ ε s (t k )). Suppose a front of wave α * i hits the grid line t = t k .See Fig 8.Then combustion process gives rise to
Proposition 4 . 2 .
42Under the assumptions (A1)-(A2), there exist constants C 3 , C 4 > 0 which depend only onŪ b , such that if the condition (C2) or (C3) holds for sufficiently small ǫ, then the fractional-step wave front tracking scheme yields a global in time approximate solution U ε to the (IBVP) satisfying the estimates
The global existence conclusion in Theorem 1.1 remains true. This observation will be used to analyze the trajectories of reacting flow in section 6.Proof of Theorem 1.1 under condition (C2) or (C3). Similar to the proof under condition (C1), we follow Proposition 4.2 and apply Helly's compactness theorem to get the global existence of combustion solution U to (IBVP ) under condition (C2) or (C3). 4.4. Some further properties of the entropy solution. Given initial-boundary data, one can derive the variation of the flow on any space-like curve by Glimm functional G(t).
Lemma 4 . 4 .
44Redistributed Glimm-type functional G χ (t) is decreasing in t.
Case 4 .3. 2 .
42Appearance of single α 4 from boundary. It is clear that
Proposition 4 . 3 (
43Velocity comparison). For any t >t ≥ 0, there holds
Fig. 11 .Fig. 12 .. 1 .
111216) and prove the estimates(5.5) in different cases. Flow Projected Front x α = χ m s (t). Suppose that α = α s 4 is the large 4-shock of V which satisfies V e,− = (ρ, u, p) ⊤ = S e where det r e 4 (V e,− ), r e 2 , r e 4 (V e,+ , ′ − cp ′ , det r e 1 (V e,− ), r e 2 , r e 4 (V e,+ , −α s 4 ) =
Lemma 5. 1 (
1Stability condition). Assume parameter ǫ in the conditions (C1)-(C3) is small enough. Then there holds
Fig. 13 .Fig. 14 .
1314Flow Projected phase space for Case 5.1.2 Case 5.1.2. Front x α = χ m s (t).
Fig. 18 .
18Projected phase space for Case 5.1.4
W i |q i |dx, where W i = 1 + K A i (Û ,V ) + K Q(Û ) + Q(V ) + KLb(Û ,V )follows the definition in (5.3). Compute the time derivative of L p to geṫ L p (Û ,V +ŵ)
ε + ŵ L ∞ + T.V.{ŵ; (χ p (τ ), +∞)} = O(1) ε + (γ − 1)εT.V.{Y φ(T ); (χ p (τ ), +∞)} = O(1)ε.
This gives the estimate for α 1 in (2.23). The other estimates in (2.23)(2.24) can be obtained by analogous argument.
r )
rFig. 16. Projected phase space for Case 5.1.3Case 5.1.3. Front x α ∈ (χ m s (t), χ m s (t)). We start with the case k α ∈ {1, 2, 4} (see. For i < 3, there hold
16 )
16≤ −K|α||q + 4 |d + O(1)|α|.In summary, we conclude that if k α ∈ {1, 2, 4}, there holds
< θ 3 T.V.{Y 0 (·); (0, +∞)} ≪ 1.It ensures that G(0) is small enough. In contrast, for reacting fluid under condition (C2) or (C3), we demand θ 3 is suitably large such that the decrease of 3-waves may offset the growth of nonlinear waves caused by exothermic reaction. Hence we introduce the transit consumption of reactant at χ ε p and χ ε s , respectively asΥ ε,p k = Y ε,p k · φ(T ε,p k )ε, Υ ε,s k = Y ε,s k · φ(T ε,s k )ε, where Y ε,p k = Y ε (t k − 0, χ ε p (t k ) + 0), T ε,p k = T ε (t k − 0, χ ε p (t k ) + 0), Y ε,s k = Y ε (t k − 0, χ ε s (t k ) − 0), T ε,s k = T ε (t k − 0, χ ε s (t k ) − 0),
(−α s 4 )(V e,+ ) for V e,+ = (ρ + , u + , p + ) ⊤ (see. For given strength |α s
U (t, ·) − V (t, ·) L 1 (R) ≤ O(1) U 0 (·) − V 0 (·) L 1 (R) + u p (·) − v p (·) L 1 ([0,t]) ,
t (U τ , u p ) − S ν t (U τ , u p ) L 1 (R) ≤ O(1) max{µ, ν}t, (6.6)for any t >t ≥ 0 and µ, ν ≤ ε.According to the equivalence of total variation and Glimm-type functional, there exists a constant C 5 > 0 dependent ofŪ b such that condition T.V.{U 0 (·); R + }+T.V.{u p (·); R + } < ǫ implies T.V.{S ε t (U 0 , u p ); (χ p,1 , +∞)} < C 5 ǫ ∀ t > 0.
AcknowledgementsThe research of Kai Hu4 |, the variables q − 1 , q − 2 , q + 4 are uniquely determined by q + 1 , q + 2 , q − 4 , but independent of q − 3 , q + 3 . In particular q − 3 = q Assume q − 1 = f 1 (q + 1 , q + 2 , q − 4 ). By direct computation, we immediately find that ∂ ∂q + 1 f 1 (q + 1 , 0, 0) = 1, ,(5.8)Now we proceed to prove the estimate (5.5) for x α = χ m s (t). Observe thatThis together with (5.10) implies thatfor some positive constant κ 1 < 1. On the other hand, we obviously haveFrom (5.6)(5.9)(5.13)(5.14), the assumption (4.1) on distinct characteristics fields, weight condition (5.4) and the ℓ i -distribution inTable 1, we deduceprovided that ℓ sufficiently large, the variation of initial-boundary data sufficiently small, and κ 0 > κ 1 W.(5.15)Suppose α = α s 4 is the large 4-shock of U . Similar to estimates (5.9)(5.13), we havebecause ℓ 3 K = ℓ according toTable 1.For i = 4, the weightK|α| if k α = 4 and α ∈ J e l , − K|α| if k α = 4 and α ∈ J e r , impliesIf k α = 5, the argument for α ∈ N P can be repeated as in Case 5.1.3. Hence we can also derive the estimate (5.5) 2 .= U e (t, χ m p (t) + 0) and V e p .= V e (t, χ M p (t) + 0). They satisfy = S e 2 (q p+ 2 ) • S e 1 (q p+ 1 )(U e p ). For fixed state U e p ∈ N δ (Ū e b,l ), we find that q p+ 4 can be uniquely determined by q p+ 1 , q p+ 2 and ∆u m p .= u 1 (t, χ m p (t)+ 0) − u 2 (t, χ m p (t) + 0). Next, we begin to evaluate q p+ 4 . To this end, we can rewrite (5.17) asMultiply it by n = (0, 1, 0) to getHence there holdThen it follows from (5.19) thatWe further find the relationRecall that the constant W is sufficiently close to 1. Now from (5.15)(5.23), we can choose the appropriate κ 0 such thatFinally, by the estimates (5.22)(5.24), we arrive atWe eventually verified all the conclusions in (5.5), and then derive the derivative esti-mateLIntegrate it fromt to t to getWe thus deduce thatwhere we have the estimateProof of Theorem 1.1 for the L 1 -stability under condition (C1). Suppose U µ , U ν are two approximate solutions to (IBVP ) under condition (C1). Particularly, we takeProof. First, by(4.19), one figures out thatThus above two estimates give(5.29).Consider the function q i = f i (q * , Υ 1 , Υ 2 ) where q * = (q * 1 , q * 2 , q * 4 ). It follows from (5.31) thatwhich leads to the estimate (5.30).Now we continue to study ∆L (U (t k ), V (t k )). Recall the estimate (4.33) on ∆G(t k ). We obtain(5.32) By (5.29), we deduce that(5.33) Therefore, it follows from the estimates (5.32)(5.33) and the coefficients condition (5.2) that). Without loss of generality, we assume that T 1 > T i > T 2 for this case. HenceThese imply thatThus we conclude thatbecause |q * 4 | is approximate to strength of large 4-shock and K is large enough. Case 5.3.3. Difference ∆L at x ∈ (χ m s (t k ), +∞). In this case, we know that both T 1 and T 2 are less than T i . This leads to Υ 1 = Υ 2 = 0, i.e., the chemical reaction is absent in the region (χ m s (t k ), +∞). By construction, no wave interaction takes place at t = t k . Hence the Lipschitz continuity yieldsCombining (5.36) with (5.34)(5.35), we conclude that at t = t k , there holdsFinally, we are concerned with the completely ignited flow. It follows from (4.34) thatThen, according to Lemma 5.2, there holdsTherefore at t = t k , we still haveAs a consequence, estimates (5.28)(5.37)(5.38) giveProof of Theorem 1.1 for the L 1 -stability under condition (C2) or (C3). By Proposition 4.3 for the velocities comparison, and apply the similar argument for non-reacting flow in section 5.2 to reacting flow, we deduce that(5.39) Taket = 0 in (5.39). Extracting a subsequence {ε k } if necessary, then as ε k → 0, we see that which establishes the L 1 -stability for combustion flow.Uniqueness of limit solution to (IBVP ) for combustion flowIn this section, we will consider the uniqueness of the entropy solutions obtained by fractional-step wave front tracking scheme for the combustion reacting flow under condition (C2) or (C3). To complete it, we need to establish the error estimates on distinct trajectories. For this purpose, we first rewrite the approximate solutions to the (IBVP ) with the initial-boundary data (U ε 0 , u ε p ) constructed in section 3 in the operator form:, where integer k = min{⌈t/ε⌉, N }. Here, S ε denotes the approximate solution operator to the (IBVP ) for non-reaction flowMoreover, the solution to (6.1) is defined by∈ Ω ε k for k ≥ 1. Let T be the approximate solution operator to the following Cauchy problem of ordinary differential equations:, whose approximate solution can be given by T t(U (0, x)) . = U (0, x) + tG 0 (U (0, x)).In the sequel, we always use the notations U 0 , V 0 , U τ , V τ (or u p , v p ) etc. to represent the piecewise constant approximations to the initial (or boundary) values. Then, by (5.39), we have Proposition 6.1. Given two ε-approximate initial-boundary data (U 0 , u p ) and (V 0 , v p ) satisfying condition (C2), there holdsfor any t >t ≥ 0. Moreover, the above flow estimate also holds under condition (C3).Next, we turn to discuss the evolution of reacting flows after shifting the initial time to τ > 0. Thus the initial position of large leading shock must be separated from that of piston. But we need not be concerned with what the backward trajectory is and whether the large shock front coincides with the piston in the past. Suppose that U τ = (U e τ , Y 1,τ ) and V τ = (V e τ , Y 2,τ ) are new initial data for reacting flow, which are the small perturbation of the background statesŪ b (τ, x). Moreover, the boundary data u p and v p are constructed by small perturbations of reference velocityū p on the boundary.We make the following assumptions: (C1) conditions (6.2) 1 -(6.2) 3 hold ifT r > T i ; (C2) conditions (6.2) 1 -(6.2) 4 hold ifT l > T i >T r . Upon Remark 4.1 and previous stability argument in sections 5.2-5.3, we get the following conclusions on operators P ε t and S ε t . Proposition 6.2. Suppose initial-boundary data (U τ , u p ) and (V τ , v p ) satisfy the assumption (C1) or (C2). Then the reacting flows constructed by fractional-step front tracking method satisfywhile the non-reacting flows satisfyandS µ which lead to ⌈t/ε⌉ = K − k or K − k + 1.It suffices to discuss the case ⌈t/ε⌉ = K − k. SetŨ . = P ε t (U τ , u p ). Then the Lipschitz continuity (6.3)-(6.5) and Proposition 4.1 give thatProof of Theorem 1.1 for the uniqueness under condition (C2) or (C3). Remember that (U ε 0 , u ε p ) is an approximate initial-boundary data under condition (C2) or (C3). Taking ε = µ, ν respectively for P ε t (U ε 0 , u ε p ), we have P µ t (U µ 0 , u µ p ) − P ν t (U ν 0 , u ν p ) L 1 (R) ≤ P µ t (U µ 0 , u µ p ) − P ν t (U µ 0 , u µ p ) L 1 (R) + P ν t (U µ 0 , u µ p ) − P ν t (U ν 0 , u ν p ) L 1 (R) . To prove convergence, we need a local error estimate of distinct trajectories P µ t (U µ 0 , u µ p ) and P ν t (U µ 0 , u µ p ) in advance. As shifting the initial time to τ > 0, we use the notations U µ = U µ (τ, x) and u µ p = u µ p (t + τ ) in this paragraph for convenience. Then it is deduced from Lemma 6.2 and (6.6) that≤O(1) 1 + Y µ (τ, ·) L 1 (R) t 2 + O(1) max{µ, ν}t ≤O(1) 1 + Y µ (τ, ·) L 1 (R) t 2 , (6.9)for any positive µ, ν ≪ min{t, t 2 } and t <t.
. = S Kα, α)(V − +ŵ −= S kα (α)(V − +ŵ − ).
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[] |
[
"Parameterized Complexity of Elimination Distance to First-Order Logic Properties *",
"Parameterized Complexity of Elimination Distance to First-Order Logic Properties *"
] |
[
"Fedor V Fomin ",
"Petr A Golovach ",
"Dimitrios M Thilikos "
] |
[] |
[] |
The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: For every graph property P expressible by a first order-logic formula ϕ ∈ Σ 3 , that is, of the form ϕ = ∃x 1 ∃x 2 · · · ∃x r ∀y 1 ∀y 2 · · · ∀y s ∃z 1 ∃z 2 · · · ∃z t ψ,where ψ is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Σ 3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: There are formulas ϕ ∈ Π 3 , for which computing elimination distance is W[2]-hard.
|
10.1109/lics52264.2021.9470540
|
[
"https://arxiv.org/pdf/2104.02998v1.pdf"
] | 233,169,117 |
2104.02998
|
024d2fd43047aaaa8eb7ec6b3097c238e444d50b
|
Parameterized Complexity of Elimination Distance to First-Order Logic Properties *
Fedor V Fomin
Petr A Golovach
Dimitrios M Thilikos
Parameterized Complexity of Elimination Distance to First-Order Logic Properties *
First-order logicelimination distanceparameterized complexitydescriptive com- plexity
The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: For every graph property P expressible by a first order-logic formula ϕ ∈ Σ 3 , that is, of the form ϕ = ∃x 1 ∃x 2 · · · ∃x r ∀y 1 ∀y 2 · · · ∀y s ∃z 1 ∃z 2 · · · ∃z t ψ,where ψ is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Σ 3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: There are formulas ϕ ∈ Π 3 , for which computing elimination distance is W[2]-hard.
Introduction
One of the successful concepts in parameterized complexity is the "distance from triviality" [20]. Roughly speaking, a parameter can measure the "distance" of the given instance from an instance that is solvable efficiently and then exploit such a distance algorithmically. In graph problems, a standard measure of distance from triviality is the vertex deletion distance to some specific graph property P. That is, the minimum number of vertices whose deletion results in a graph in P. An interesting alternative to vertex deletion distance, called elimination distance was introduced by Bulian and Dawar [5] in their study of the parameterized complexity of the graph isomorphism problem. The elimination distance of a graph G to graph property P is
ed P (G) = 0, if G ∈ P, 1 + min v∈V (G) ed P (G − v),
if G / ∈ P and G is connected, max{ed P (C) | C is a component of G}, otherwise.
Arguably, elimination distance can be seen as a non-deterministic version of vertex deletion distance, where the source of non-determinism is connectivity: each vertex removal creates connected components and the elimination should be applied to each one of them as an independent vertex deletion scenario. In the most simple case where P is the property of being edgeless, vertex deletion distance to P generates vertex cover, while the elimination distance to P generates tree-depth [26].
In their follow-up work, Bulian and Dawar [6] proved that deciding whether a given nvertex graph has elimination distance at most k to any minor-closed property of graphs can be done by an algorithm running in time f (k) · n O(1) (that is an FPT-algorithm), and thus is fixed-parameter tractable parameterized by k. In the same paper, Bulian and Dawar [6] asked whether computing the elimination distance to graphs of bounded degree is fixed-parameter tractable.
The problem. The question of Bulian and Dawar is the departure point of our study. Every graph property characterized by a finite set of forbidden induced subgraphs (and thus the bounded degree property as well) is first-order logic definable (in short, FOL-definable), i.e., there is a FOL formula ϕ where P = {G | G |= ϕ}. It is well-known that Model Checking for a FOL formula ϕ, that is deciding whether G |= ϕ, can be done in time n O(|ϕ|) . It is also easy to design an algorithm that, in time n O(k) · n O(|ϕ|) , decides whether the elimination distance to a property expressible by a FOL formula ϕ is at most k. Thus, for every FOL formula ϕ, the problem asking, given as input a graph G and a non-negative integer k, whether the elimination distance from G to P ϕ := {G | G |= ϕ} is k is in the parameterized complexity class XP (when parameterized by k). This brings us to the following question.
What is the parameterized complexity of computing the elimination distance to FOLdefinable properties?
Notice that the above general question could also be made for higher order logic-definable properties. In this direction, one may observe that there are formulas ϕ in existential secondorder logic (ESOL) for which Model Checking for ϕ is already intractable: such ESOLdefinable problems are Hamiltonian Cycle or 3-Coloring that are NP-complete. This means that for the corresponding ESOL formulas ϕ the problem of checking whether ed Pϕ (G) ≤ k, parameterized by k, is para-NP-complete. Motivated by this, we delimit our study to the framework of first-order logic where our parameterized problem is in XP for every FOL-formula. This permits us to set up the problem that we consider in this paper, that is to completely determine the prefix classes of FOL that demark the parametric-tractability borders of elimination distance to FOL-definable properties (that is FPT versus W-hardness).
The above question has been inspired by the study of Gottlob, Kolaitis, Schwentick in [18] who provided an analogous dichotomy result (P versus NP-complete) for ESOL-formulas, in several graph-theoretic contexts. They identified the set F of prefix classes of ESOL such that, if ϕ ∈ F, then Model Checking for ϕ is polynomially solvable, while every prefix class not in F contains some ESOL formula ϕ where Model Checking for ϕ is NP-complete.
Our results. We identify sufficient and necessary conditions on the structure of prefixes of firstorder logic formulas demarcating tractability borders for computing the elimination distance.
Our main algorithmic contribution is the proof that computing the elimination distance to any graph property defined by a formula from Σ 3 is fixed-parameter tractable. We formally define prefix classes Π i and Σ i in the next section. For the purpose of this introduction, it is sufficient to know that every formula in ϕ ∈ Σ 3 can be written in the form ϕ = ∃x 1 ∃x 2 · · · ∃x r ∀y 1 ∀y 2 · · · ∀y s ∃z 1 ∃z 2 · · · ∃z t ψ,
where ψ is a quantifier-free FOL-formula and r, s, t are non-negative integers.
Every graph property characterized by a finite set of forbidden subgraphs can be expressed by ϕ ∈ Σ 3 . Actually, for this partiuclar purpose, we can consider even more restricted formulas ϕ ∈ Π 1 ⊂ Σ 3 with only ∀ quantifications over variables. The property that the diameter of a graph is at most two cannot be expressed by using forbidden subgraphs but can easily be written as an FOL formula from Π 2 ⊂ Σ 3 :
∀u∀v∃w[(u = v) ∨ (u ∼ v) ∨ ((u ∼ w) ∧ (v ∼ w))].
Another interesting example of a property expressible in Σ 3 is the property of containing a universal vertex, and, more generally, having an r-dominating set of size at most d for constants r and d. Having a twin-pair, that is a pair of vertices with equal neighborhoods, is also the property expressible in Σ 3 .
Theorem 1 (Informal). For every ϕ ∈ Σ 3 , n-vertex graph G, and k ≥ 0, deciding whether the elimination distance from G to property P ϕ , is at most k, can be done in time f (k) · n O(|ϕ|) for some function f of k only.
Our second theorem shows that the assumptions on the prefix of the formula are necessary. Let Π 3 be the class of first-order logic formulas of the form ϕ = ∀x 1 ∀x 2 · · · ∀x r ∃y 1 ∃y 2 · · · ∃y s ∀z 1 ∀z 2 · · · ∀z t ψ,
where ψ is a FOL-formula without quantifiers and s, t, q are non-negative integers. We show that Theorem 2 (Informal). There is ϕ ∈ Π 3 such that deciding whether the elimination distance to P ϕ is at most k, is W[2]-hard parameterized by k.
Variants of elimination distance. The main reason why we give informal statements of our theorems in the introduction is due to the following issue. The definition of elimination distance is tailored to the graph properties P with the condition that G ∈ P if and only if C ∈ P for every component C of G. Graph properties defined by FOL do not necessarily satisfy such a condition. This leads to ambiguities. As an example, consider graph property P = {G | G |= ∀x∀y x = y}. Thus G ∈ P if and only if G is a single-vertex graph. Let G be an edgeless graph with n ≥ 2 vertices. Since G / ∈ P it would be a natural assumption that the elimination distance from G to P is positive. However, it is not: every connected component of G is in P and, therefore, ed P (G) = max{ed P (C) | C is a component of G} = 0.
To avoid such ambiguities, we refine the definition of the elimination distance.
Since we consider graph properties P ϕ = {G | G |= ϕ} for formulas ϕ, we define the distances with respect to formulas. Notice that the notion of elimination distance combines "connectivity" and "inclusion" in a graph class. Depending on which of these two properties we want to prioritize, we give different definitions. Let ϕ be a FOL formula.
Definition 1 (Elimination distances ed conn ϕ and ed prop ϕ ). The first definition prioritize on connectivity. For a graph G, we set if G is connected. We set ed conn ϕ (G) = max{ed conn ϕ (C) | C is a component of G} when G is not connected. The second definition prioritize on the graph property
ed prop ϕ (G) = 0, if G |= ϕ, 1 + min v∈V (G) ed prop ϕ (G − v),
if G |= ϕ and G is connected, max{1, max{ed prop ϕ (C) | C is a component of G}}, otherwise.
We assume that ed conn ϕ (G) = ed prop ϕ (G) = 0 if G = (∅, ∅) for any formula ϕ.
If ϕ is such that G ∈ P ϕ if and only if C ∈ P ϕ for every component C of G, then ed Pϕ (G) = ed conn ϕ (G) = ed prop ϕ (G). However, in general ed conn ϕ (G) and ed prop ϕ (G) may differ significantly. Consider ϕ = ∃u∃v ¬(u = v) ∧ ¬(u ∼ v) that defines the property that a graph has two nonadjacent vertices. Let G be the disjoint union of the complete n-vertex graph K n and an isolated vertex. Then G |= ϕ and, therefore, ed prop ϕ (G) = 0. On the other hand, G is not connected and it is easy to see that ed conn ϕ (K n ) = n. Given the above, a more precise statement of Theorems 1 and 2 is that they hold for both distances ed conn ϕ and ed prop ϕ . We also define another interesting type of elimination distance ed depth ϕ that focuses on the depth of the elimination set. We prove Theorems 1 and 2 hold for that distance as well. Precise statements of both theorems and their proofs are given in Sections 4 and 5.
Related work. Results of this work fit into two popular trends in logic and parameterized complexity. A significant amount of research in descriptive complexity is devoted to the study of prefix classes of certain logics. We refer to the book of Börger, Grädel, and Gurevich [4], as well as the aforementioned work of Gottlob, Kolaitis and Schwentick [19] for further references. The study of graph modification problems is a well-established trend in parameterized complexity. The books [10,13,14,27] provide a comprehensive overview of the area. In particular, Fomin, Golovach, and Thilikos [15] studied parameterized complexity of computing vertex deletion distance and edge editing to graph properties defined by first-order logic formulas; [15, Theorem 1] establishes fixed-parameter tractability for vertex removal to a graph property P ϕ for ϕ ∈ Σ 3 and shows that the problem is W[2]-hard for some ϕ ∈ Π 3 . While our Theorem 1 reaches the same tractability border for the elimination distance, the proof is significantly more complicated.
The general question on the parameterized complexity of elimination distance to graph properties was settled by Bulian and Dawar [5,6]. Properties that have been considered so far are minor-free graph classes [6], cluster graphs [1, 2], bounded degree graphs [5,23], and H-free graphs [1]. Moreover, Hols et al. [21] studied the existence of polynomial kernels for the Vertex Cover problem parameterized by the size of a deletion set to graphs of bounded elimination distance to certain graph classes. Lindermayr, Siebertz, and Vigny [23] proved that computing the elimination distance to graphs of bounded degree is fixed-parameter tractable when the input does not contain K 5 as a minor. While preparing our paper, we have learned about the very recent work of Agrawal et al.
[1]. Agrawal et al. established fixed-parameter tractability of computing an elimination distance to any graph property characterized by a finite set of graphs as forbidden induced subgraphs. Since graphs of bounded vertex degree can be characterized by a finite set of forbidden induced subgraphs, the work of Agrawal et al. answers the question of Bulian and Dawar [6] about the elimination distance to graphs of bounded degree.
Comparing with the result of Agrawal et al.
[1], our Theorem 1 is more general. First, it provides the tractability of the elimination ordering to a strictly larger family of graph properties. Every graph property described by a finite set of forbidden induced subgraphs is also definable by a formula from Σ 3 . However, properties like having a universal vertex or bounded diameter, which are expressible in Σ 3 , cannot be described by forbidden subgraphs. Second, Theorem 1 holds for three variants of the elimination distance: ed conn ϕ , ed prop ϕ , and ed depth ϕ . With this terminology, the result of Agrawal et al. is only about computing ed conn ϕ . When it comes to the proof techniques, both Theorem 1 and the result of Agrawal et al. use recursive understanding, which seems to be a very natural technique for approaching problems about elimination distances. However, the details are quite different. To deal with various types of elimination distances and FOL formulas in a uniform way, we need different combinatorial characterizations of the distances via sets of bounded elimination depths. Furthermore, while solving our problems on unbreakable graphs is done by recursive branching algorithms, similarly to Agrawal et al., we do it in different way that exploits the random separation technique to deal with our more general FOL framework. Moreover, the analysis of components of the graph obtained by the deletion of an elimination set for computing ed depth ϕ (G), and especially, ed prop ϕ (G), is great deal more challenging. In particular, this is the reason why we apply the random separation technique contrary to the more straightforward tools used by Agrawal et al.
Overview of the approach. The firsts two variants of the elimination distance that we examine are defined recursively using the containment in the graph class P ϕ as the base case. We start by providing equivalent formulations that are more suitable from the algorithmic perspective. For this, we introduce the notion of elimination set of depth at most d that is a set X ⊆ V (G) that can be bijectively mapped to a rooted tree T of depth d expressing selection of elimination vertices in recursive calls. We next prove that ed conn ϕ (G) ≤ k if and only if G has an elimination set of X depth at most k − 1 such that C |= ϕ for every component C of G − X. Similar, however more technical, equivalent formulations is given for ed prop ϕ . All alternative definitions and the proofs of their equivalences to the recursive ones are gathered in Section 3.
The new definitions allows to certify a solution by a set X of bounded elimination depth. However, the size of X could be unbounded. Moreover, there could be many connected components of G − X and the sizes of these components could be immense. We use the recursive understanding technique, introduced by Chitnis et al. in [8] to reduce the solution of the initial problem to a much more structured problem. In the reduced problem, we can safely assume that each yes-instance is certified by an elimination set X whose size, as well as the size of the union of all but one of the components of G − X, is also bounded by a function of k.
More precisely, by making use of recursive understanding, we can consider only instances that are (p(k), k)-unbreakable for some suitably chosen function p. Roughly speaking, a graph is (p(k), k)-unbreakable when it has no separator of size at most k that partitions the graph in two parts of size at least p(k) + 1 each. The application of recursive understanding uses the meta-algorithmic result of Lokshtanov et al. [24] and the fact that all variants of the elimination distance to P ϕ are expressible in monadic-second order logic (MSOL) when φ is a formula in FOL (Lemma 5).
The (p(k), k)-unbreakability permits us to assume that |X| ≤ p(k) + k. Moreover, exactly one connected component C X of G − X, is big, that is of size at least p(k) + 1, and the size of G − V (C X ) is bounded by some function of k (see Lemma 6). Given that C X is the big component corresponding to a solution X, we also consider the set S X of the neighbours of the vertices of C X in G and we set U X = V (G) \ (V (C X ) ∪ S X ). We show that |S X | ≤ k and |U X | ≤ p(k).
Our next step is to use the random separation technique, introduced by Cai, Chan, and Chan in [8]. We construct in FPT-time a family F of at most f (k) · log n partitions (R, B) of V (G) to "red" and "blue" vertices such that for every elimination set X corresponding to a potential solution, F contains some (R, B) where U X ⊆ R and S X ⊆ B. In our algorithm, we go over all these blue-red partitions and, for each one of them, we check whether there exists an elimination set X (called colorful elimination set) where all vertices in S X are blue and all vertices in U X are red.
The correct "guess" of the above red-blue partition permits us to design a recursive procedure that solves the latter problem, i.e., finds a colourful elimination set X. This procedure is different for each of the three versions of the problem and its variants are presented in Subsection 4.2.
The key task here is to identify the big component C X . It runs in FPT-time and its correctness is based on the prefix structure of the formula φ.
Organization of the paper. In Section 2 we provide the basic definitions of the concepts that we use in this paper: complexity classes graphs, and formulas. In Section 3 we prove some properties and relations between the elimination ordering variants that we consider. We also provide alternative definitions and we prove their equivalencies with the original ones. The main algorithmic result is in Section 4 where we explain how we apply the recursive understanding technique, the randomized separtion technique, and we present the branching procedure for the "colourful version" of each variant. Section 5 gives the lower bound of the paper. This uses parameterized reduction from the Set Cover problem. Finally, in Section 6, we provide some discussion on the kernelization complexity of our problems as well as some directions on further research on elimination distance problems.
Preliminaries
Sets. We use N to denote the set of all non-negative numbers. We denote by a = a 1 , . . . , a r a sequence of elements of a set A and call a an r-tuple of simply a tuple. Note that the elements of a are not necessarily distinct. We denote by ab = a 1 , . . . , a r , b 1 , . . . , r s the concatenation of tuples a = a 1 , . . . , a r and b = b 1 . . . , b s . Parameterized Complexity. We refer to the recent books of Cygan et al. [10] and Downey and Fellows [13] for the detailed introduction to the field. Formally, a parmeterized problem is a language L ⊆ Σ * × N, where Σ * is a set of strings over a finite alphabet Σ. This means that an input of a parameterized problem is a pair (x, k), where x is a string over Σ and k ∈ N is a parameter. A parameterized problem is fixed-parameter tractable (or FPT) if it can be solved in time f (k) · |x| O(1) for some computable function f . Also, we say that a parameterized problem belongs in the class XP if it can be solved in time |x| f (k) for some computable function f . The complexity class FPT contains all fixed-parameter tractable problems. Parameterized complexity theory also provides tools to disprove the existence of an FPT-algorithm for a problem under plausible complexity-theoretic assumptions. The standard way is to show that the problem is W[1] or W[2]-hard using a parameterized reduction from a known W[1] or W[2]-hard problem; we refer to [10,13] for the formal definitions of the classes W[1] and W [2] and parameterized reductions.
Graphs. We consider only undirected simple graphs and use the standard graph theoretic terminology (see, e.g., [12]). Throughout the paper we use n to denote |V (G)| if it does not create confusion.
[v] = {v} ∪ N G (v) is the closed neighborhood. For S ⊆ V (G), N G (S) = v∈S N G (v) \ S and N G [S] = v∈S N G [v]. For a vertex v, d G (v) = |N G (v)| denotes the degree of v.
A graph G is connected if for every two vertices u and v, G contains a path whose end-vertices are u and v. For a positive integer k, G is k-connected if |V (G)| ≥ k and G−S is connected for every S ⊆ V (G) of size at most k−1. A connected component (or simply a component) is an inclusion maximal induced connected subgraph of G. For two distinct vertices u and v of a graph G, a set S ⊆ V (G) is a (u, v)-separator if G − S has no (u, v)-path.
A rooted tree is a tree T with a selected node (we use the term "node" instead of "vertex" for such a tree) r called a root. The selection of r defines the standard parent-child relation on V (T ). Nodes without children are called leaves and we use L(T ) to denote the set of leaves of T . The depth depth T (v) of a node v is the distance between r and v, and the depth (or height) depth(T ) of T is the maximum depth of a node. The nodes of the (r, v)-path are called ancestors of v. We use A T (v) to denote the set of ancestors of v in T . Note that v is its own ancestor; we say that an ancestor is proper if it is distinct from v. Two nodes u and v of T are comparable if either v is an ancestor of u or u is an ancestor of v. Otherwise, u and v are incomparable. A node w of T is the lowest common ancestor of nodes u and v if w is the ancestor of maximum depth of both u and v. Note that the lowest common ancestor is unique and if u and v are incomparable then the lowest common ancestor is distinct from u and v. A node v is a descendant of u if u if u is an ancestor of v. By D T (u) we denote the set of descendants of u in T . As with ancestors, a node is its own descendant and we say that a descendant v of u is proper if u = v. For a node v, the subtree induced by the descendants of v is the subtree rooted in v.
Formulas. In this paper we deal with first-order and monadic second-order logic formulas on graphs.
The syntax of the first-order logic (FOL) formulas on graphs includes the logical connectives ∨, ∧, ¬, variables for vertices, the quantifiers ∀, ∃ that are applied to these variables, the predicate u ∼ v, where u and v are vertex variables and the interpretation is that u and v are adjacent, and the equality of variables representing vertices. It also convenient to assume that we have the logical connectives → and ↔. An FOL formula ϕ is in prenex normal form if it is written as ϕ = Q 1 x 1 Q 2 x 2 · · · Q t x t χ where each Q i ∈ {∀, ∃} is a quantifier, x i is a variable, and χ is a quantifier-free part that depends on the variables x 1 , . . . , x t . Then Q 1 x 1 Q 2 x 2 · · · Q t x t is referred as the prefix of ϕ. From now on, when we write "FOL formula", we mean an FOL formula on graphs that is in prenex normal form. Also we assume that a formula has no free, that is, non-quantified variables unless we explicitly say that free variables are permitted. For an FOL formula ϕ and a graph G, we write G |= ϕ to denote that ϕ evaluates to true on G.
We use the arithmetic hierarchy (also known as Kleene-Mostowski hierarchy) for the classification of formulas in the first-order arithmetic language (see, e.g., [28]). For this, we define prefix classes according to alternations of quantifiers, that is, switchings from ∀ to ∃ or vice versa in the prefix string of the formula. Here we allow a formula to have free variables. Let Σ 0 = Π 0 be the classes of FOL-formulas without quantifiers. For a positive integer , the class Σ contains formulas that may be written in the form
ϕ = ∃x 1 ∃x 2 · · · ∃x s ψ,
where ψ is a Π −1 -formula, s is some integer, and x 1 , . . . , x s are free variables of ψ. Respectively, Π consists of formulas ϕ = ∀x 1 ∀x 2 · · · ∀x s ψ,
where ψ is a Σ −1 -formula and x 1 , . . . , x s are free variables of ψ. Note that for > , Σ ∪ Π ⊆ Σ ∩ Π , that is, every Σ or Π formula is both a Σ and Σ -formula. For technical reasons, we extend FOL formulas on graphs to structures of a special type. We say that a pair (G, v), where v = v 1 , . . . , v r is an r-tuple of vertices of G, is an r-structure. Let ϕ be an FOL formula without free variables and let x = x 1 , . . . , x r be an r-tuple of distinct variables of ϕ. We denote by ϕ[x] the formula obtained from ϕ by the deletion of the quantification over x 1 , . . . , x r , that is, these variables become the free variables of ϕ[x]. For an r-structure (G, v) with v = v 1 , . . . , v r and ϕ[x], we write (G, v) |= ϕ[x] to denote that ϕ [x] evaluates to true on G if x i is assigned v i for i ∈ {1, . . . , r}. If r = 0, that is, v and x are empty,
then (G, v) |= ϕ[x] is equivalent to G |= ϕ.
As a subroutine in our algorithms, we have to evaluate FOL formulas on graph, that is, solve the Model Checking problem. Let ϕ be a FOL formula. The task of Model Checking is, given a graph G, decide whether G |= ϕ. It was shown by Vardi [29] that Model Checking is PSPACE-complete. The problem is also hard from the parameterized complexity viewpoint when parameterized by the size of the formula. It was proved by Frick and Grohe in [17] that the problem is AW[ * ]-complete for this parametrization (see, e.g., the book [14] for the definition of the class). Moreover, it can be noted that the problem is already W[1]-hard for formulas having only existential quantifiers, that is, for ϕ ∈ Σ 1 , by observing that the existence of an independent set of size k can be easily expressed by such a formula and Independent Set is well-known to be W[1]-complete [13]. This implies that we cannot expect an FPT algorithm for the problem. However, it is easy to see that Model Checking is in XP when parameterized by the number of variables, because the problem for a formula with s variables can be solved in O(n s ) time by the exhaustive search (the currently best algorithm is given by Williams in [30]). This explains the exponential dependence of the polynomials in running times in our algorithm on the number of variables. For referencing, we state the following observation. In monadic second-oder logic (MSOL), we additionally can quantify over sets of vertices and edges. Formally, we can use variables for sets of vertices and edges and have the predicate x ∈ X, where x is a vertex (an edge, respectively) variable and X a vertex set (an edge set, respectively) variable, denoting that x is an element of X. As with FOL formulas, we write G |= ϕ to denote that an MSOL formula ϕ evaluates true on G. We refer to the book of Courcelle and Engelfriet [9] for the details of MSOL on graphs.
Properties of elimination distance
In this section we derive the properties of the elimination distances , ed conn ϕ and ed prop ϕ that will be used in the proof of the main theorem. We also define ed depth ϕ . Observation 2. For every FOL formula ϕ and every graph G, ed prop ϕ (G) ≤ ed conn ϕ (G) + 1.
Proof. The proof is by induction on the value of ed conn ϕ (G). Suppose that ed conn ϕ (G) = 0 for a graph G. If G is connected, then G |= ϕ and ed prop ϕ (G) = 0. Hence, the inequality holds. If G is disconnected, then C |= ϕ for every component C of
G. If G |= ϕ, then ed prop ϕ (G) = 0. If G |= ϕ, then ed prop ϕ (G) = max{1, max{ed prop ϕ (C) | C is a component of G}} = 1. In both cases, ed prop ϕ (G) ≤ ed conn ϕ (G) + 1. Assume that ed conn ϕ (G) > 0 and ed prop ϕ (G ) ≤ ed conn ϕ (G ) + 1 for all G with ed conn ϕ (G ) < ed conn ϕ (G). The claim is trivial if ed prop ϕ (G) = 0. Let ed prop ϕ (G) > 0.
We have two cases.
Case 1. G is connected. By definition, there is u ∈ V (G) such that ed conn ϕ (G) = 1 + ed conn ϕ (G − u). Because ed prop ϕ (G) > 0, ed prop ϕ (G) = 1 + min v∈V (G) ed prop ϕ (G − v) ≤ 1 + ed prop ϕ (G − u).
Then by induction,
ed prop ϕ (G) ≤ 1 + ed prop ϕ (G − u) ≤ 2 + ed conn ϕ (G − u) = 1 + ed conn ϕ (G). Case 2. G is disconnected. Let C 1 , .
. . , C s be the components of G. By definition, ed conn ϕ (G) = max 1≤i≤s ed conn ϕ (C i ). In particular, we have that ed conn ϕ (C i ) ≤ ed conn ϕ (G) for every i ∈ {1, . . . , s}. Notice that by the already proved claim for connected graphs in Case 1,
ed prop ϕ (C i ) ≤ ed conn ϕ (C i ) + 1 for every i ∈ {1, . . . , s}. Because ed conn ϕ (G) > 0, G |= ϕ. Then ed prop ϕ (G) = max{1, max 1≤i≤s ed prop ϕ (C i )} ≤ max{1, max 1≤i≤s (ed conn ϕ (C i ) + 1)} = max 1≤i≤s ed conn ϕ (C i ) + 1 = ed conn ϕ (G) + 1
as required. This completes the proof.
The example of ϕ = ∀x∀y x = y and an edgeless graph G with at least two vertices shows that the inequality in Observation 2 is tight. However, ed conn ϕ (G) and ed prop ϕ (G) can be far apart. Consider ϕ = ∃u∃v ¬(u = v) ∧ ¬(u ∼ v) that defines the property that a graph has two nonadjacent vertices. Let G be the disjoint union of the complete n-vertex graph K n and an isolated vertex. Then G |= ϕ and, therefore, ed prop ϕ (G) = 0. From the other side, G is disconnected and it is easy to see that ed conn ϕ (K n ) = n. This means that ed conn ϕ (G) = n and ed conn ϕ (G) − ed prop ϕ (G) = n, that is, the difference can be arbitrary large. For algorithmic purposes, it is convenient for us to define ed conn ϕ (G) and ed prop ϕ (G) via deletions of sets of vertices of G with a special structure. Similar approach was recently exploited by Agrawal et al.
[1] but we do it in a different way, because we consider two variants of eliminations distances.
Let G be a graph and let d ≥ 0 be an integer. We say that a set of vertices X ⊆ V (G) is an elimination set of depth at most d if there is a rooted tree T of depth at most d and a bijective mapping α : V (T ) → X such that for every two distinct incomparable nodes x and y
of T , α(A T (v)) is an (α(x), α(y))-separator in G,
where v is the lowest common ancestor of x and y (recall that A T (v) denotes the set of ancestors of v). We also say that the pair (T, α) is a representation of X (or represents X). The depth of X ⊆ V (G), denoted depth(X), is the minimum d such that X is an elimination set of depth at most d. We assume that the empty set is an elimination set of depth −1.
We call a representation (T, α) of an elimination set X ⊆ V (G) nice if for every nonleaf node v ∈ V (T ) and its child x, the vertices of α(D T (x)) are in the same component of G − A T (v). The following property is useful for us.
Lemma 1. Let G be a connected graph and let d ≥ 0 be an integer. Then a nonempty X ⊆ V (G) is an elimination set of depth at most d if and only if X has a nice representation (T, α) with
depth(T ) ≤ d. Moreover, if (T, α) is a representation of X, then there is a nice representation (T , α) of X with V (T ) = V (T ) such that (i) α(L(T )) ⊆ α(L(T )) and (ii) for each v ∈ X, depth T (α −1 (v)) ≥ depth T (α −1 (v)).
Proof. Clearly, if X ⊆ V (G) has a nice representation (T, α) with depth(T ) ≤ d, then X is an elimination set of depth at most d. For the opposite direction, it is sufficient to show the second claim. Let (T, α) is a representation of X and depth(T ) ≤ d. We show the existence of (T , α) satisfying (i) and (ii) by induction on d.
The claim is trivial if |X| = 1 as T = T in this case. Assume that |X| ≥ 2 and d ≥ 1. Denote by r the root of T and let u = α(r). Consider the components C 1 , . . . , C s of G − u containing at least one vertex of X. For every i ∈ {1, . . . , s}, let
X i = V (C i ) ∩ X and U i = α −1 (X i ).
For every i ∈ {1, . . . , s}, we construct the tree T i with the set of vertices U i ∪ {r} as follows.
For every x ∈ U i such that x = r, we find a proper ancestor y ∈ U i with respect to T of maximum depth and make y the parent of x, and if x has no ancestors in U i , we make r the parent of x. Because the choice of the parent is unique, T i has no cycles, and because we assign the parent to every node distinct from r, we conclude that T i is a tree. Denote byT the union of T 1 , . . . , T s and set r be its root. Because every node ofT distinct from r got a parent from the set of its proper ancestors in T , (i) α(L(T )) ⊆ α(L(T )) and (ii) for each v ∈ X,
depth T (α −1 (v)) ≥ depthT (α −1 (v)).
We prove that (T , α) represents X. Consider incomparable nodes x and y ofT and denote by v their lowest common ancestor. We have to show that α(AT (v)) is an (α(x), α(y))-separator in G. This is trivial if α(x) and α(y) are in distinct components of G − u. Assume that α(x) and α(y) are in the same component C i for some i ∈ {1, . . . , s}, that is, x, y ∈ U i . Note that by the construction ofT , x and y are incomparable in T . Let v be their lowest common ancestor in T . Clearly, v is an ancestor of v in T . By the construction ofT ,
AT (v) ∩ V (C i ) = A T (v ) ∩ V (C i ). Because A T (v) separates α(x) and α(y), we have that A T (v)∩V (C i ) is an (α(x), α(y))-separator. Therefore, α(AT (v))
is also an (α(x), α(y))-separator. This proves that (T , α) represents X.
Consider i ∈ {1, . . . , s}. Observe that r has a unique child in U i inT . Otherwise, if x and y are distinct children of r, we have that x and y have no ancestors in U i with respect to T . Let v be the lowest common ancestor of x and y in T . Note that v = x, y and α(A T (v)) does not separate α(x) and α(y) contradicting that (T, α) represents X. Hence, r has the unique child
r i in U i . LetT i be the subtree ofT rooted in r i . We set α i (x) = α(x) for x ∈ U i . Because (T , α) represents X, it is straightforward to verify that (T i , α i ) represents X j in the graph C i . Because depth(T i ) ≤ d − 1,
we can apply the inductive assumption. We obtain that there is a nice
representation (T i , α i ) of X i in C i with V (T i ) = V (T i ) such that (i) α(L(T i )) ⊆ α(L(T i )) and (ii) for each v ∈ X i , depthT i (α −1 i (v)) ≥ depth T i (α −1 i (v)). Notice that by the second condition, T i is rooted in r i .
We construct the trees T i for all i ∈ {1, . . . , s} and then construct T from their union by making r 1 , . . . , r s the children of r. Clearly, depth(T ) ≤ d and (T , α) is a nice representation of X satisfying conditions (i) and (ii) of the lemma.
It is also useful to characterize the depths of an elimination set in a disconnected graph.
Lemma 2. Let G be a graph with components C 1 , . . . , C s and let X ⊆ V (G) such that X = ∅.
Then depth(X) = min 1≤i≤s max{depth(X i ), max{depth(X j ) | 1 ≤ j ≤ s, j = i} + 1},(1)where X i = X ∩ V (C i ) for i ∈ {1, . . . , s}.
Proof. Recall that depth(∅) = −1 by definition. This allows us to assume without loss of generality that X i = ∅ for all i ∈ {1, . . . , s}. Otherwise, we can delete each component C i such that X i = ∅ without violating the value of depth(X) and the right part of (1). To show that
depth(X) ≤ min 1≤i≤s max{depth(X i ), max{depth(X j ) | 1 ≤ j ≤ s, j = i} + 1},
assume that the minimum value of the right part of (1) is achieved for i ∈ {1, . . . , s}. For every j ∈ {1, . . . , s}, let (T j , α j ) be a representation of X j in C j , where T j is rooted in r j and depth(T j ) = depth(X j ). We construct the tree T with the root r = r i from T 1 , . . . , T s by making each r j for j ∈ {1, . . . , s} \ {i} a child of r. Clearly,
depth(T ) = max{depth(T i ), max{depth(T j ) | 1 ≤ j ≤ s, j = i} + 1} = max{depth(X i ), max{depth(X j ) | 1 ≤ j ≤ s, j = i} + 1}. We define α : V (T ) → X by setting α(x) = α i (x) whenever x ∈ X i for some i ∈ {1, . . . , s}.
It is straightforward to verify that (T, α) represents X.
To show the opposite inequality
depth(X) ≥ min 1≤i≤s max{depth(X i ), max{depth(X j ) | 1 ≤ j ≤ s, j = i} + 1},
let (T, α) be a representation of X, where T is rooted in r and depth(T ) = depth(X). By symmetry, we assume without loss of generality that r ∈ V (C 1 ). For every j ∈ {1, . . . , s}, let
U i = α −1 (X j ).
The rest of the proof is done similarly to the proof of Lemma 1. For every j ∈ {1, . . . , s}, we construct the tree T j with the set of vertices U j ∪ {r} as follows. For every x ∈ U j such that x = r, we find a proper ancestor y ∈ U j of x with respect to T of maximum depth and make y the parent of x, and if x has no ancestors in U j , we make r the parent of x. Because the choice of the parent is unique, T j has no cycles, and because we assign the parent to every node distinct from r, we conclude that T j is a tree. Denote by T the union of T 1 , . . . , T s and set r be the root. Because every node of T distinct from r got a parent from the set of its proper
ancestors in T , depth(T ) ≤ depth(T ) = depth(X).
We claim that (T , α) represents X. To show this, let x and y be incomparable nodes of T and let v be their lowest common ancestor. We show that α(A T (v)) is an (α(x), α(y))-separator in G. This is trivial if α(x) and α(y) are in distinct components of G. Assume that α(x) and α(y) are in the same component C j for some j ∈ {1, . . . , s}, that is, x, y ∈ U j . Notice that by the construction of T ,
A T (v) ∩ C j = A T (v ) ∩ C j , where v is the lowest common ancestor of x and y in T . Because A T (v ) separates α(x) and α(y), we have that A T (v ) ∩ C j is an (α(x), α(y))-separator. Therefore, α(A T (v)) is an (α(x)
, α(y))-separator as well, as required.
Let j ∈ {2, . . . , s}. Observe that r has a unique child in U j in T . Otherwise, if x and y are distinct children of r, we have that x and y have no ancestors in U j . Let v be the lowest common ancestor of x and y in T . Note that v = x, y and α(A T (v)) does not separate α(x) and α(y) contradicting that (T, α) represents X. Hence, r has the unique child
r j in U i . Let T j be the subtree of T rooted in r j . Define α j (x) = α(x) for x ∈ U j . Since (T , α) represents X, we obtain that (T j , α j ) represents X j . Then depth(X j ) ≤ depth(T j ) ≤ depth(T ) − 1 = depth(X) − 1.
It is straightforward to verify that (
T 1 , α 1 ) represents X 1 , where α 1 (x) = α(x) for x ∈ U 1 . This means that depth(X 1 ) ≤ depth(T 1 ) ≤ depth(X). Because depth(X j ) + 1 ≤ depth(X) for j ∈ {2, . . . , s}, depth(X) ≥ max{depth(X 1 ), depth(X 2 ) + 1, . . . , depth(X s ) + 1} ≥ min 1≤i≤s max{depth(X i ), max{depth(X j ) | 1 ≤ j ≤ s, j = i} + 1}.
This completes the proof.
It is sufficient for our purposes to characterize ed conn ϕ (G) for connected graphs and we do it in the following lemma.
Lemma 3. Let ϕ be an FOL formula and let G be a connected graph. Let also d ≥ 0 be an integer. Then ed conn
ϕ (G) ≤ d if and only if G contains an elimination set X of depth at most d − 1 such that C |= ϕ for every component C of G − X. Proof. First, we show that if ed conn ϕ (G) ≤ d, then G has an elimination set X of depth at most d − 1 such that C |= ϕ for every component C of G − X. The proof is by induction on d. The claim is trivial if ed conn ϕ (G) = 0 as depth(∅) = −1 by the definition. Let d ≥ ed conn ϕ (G) ≥ 1. Because G is connected and ed conn ϕ (G) > 0, there is v ∈ V (G) such that ed conn ϕ (G) = 1 + ed conn ϕ (G − v).
We construct a node r of T and set it be the root. If C |= ϕ for every component C of G − v, then the construction of X and T is completed and we define α(r) = v.
Otherwise, let C 1 , . . . , C s be the components of G − v such that C i |= ϕ for i ∈ {1, . . . , s}. Clearly, ed conn ϕ (C i ) ≤ d − 1 for i ∈ {1, . . . , s}. Let i ∈ {1, . . . , s}. By induction, there is an elimination set X i ⊆ V (C i ) of depth at most d − 2 such that H |= ϕ for every component H of C i − X i . Then there is a corresponding representation (T i , α i ) of X i in C i . Let r i be the root of T i . We define X = {v} ∪ s i=1 X i , and construct T from T 1 , .
. . , T s by making r 1 , . . . , r s the children of r. Finally, we define
α(x) = v, if x = r, α i (x), if x ∈ V (C i ) for some i ∈ {1, . . . , s}.
It is straightforward to verify that X is an elimination set of depth at most d − 1 with respect to (T, α).
For the opposite direction, we assume that X is an elimination set of minimum depth such that C |= ϕ for every component C of G − X. We assume that the depth of X is d − 1 and prove that ed conn ϕ G ≤ d. The proof is by induction on d. The claim is trivial if d = 0, that is, if X = ∅. Suppose that d = 1, that is, the depth of an elimination set X is zero and, therefore, X = {u} for some u ∈ V (G). We have that C |= ϕ for every component C of G − u. This means that ed conn ϕ (C) = 0 for every component C and, therefore, ed conn
ϕ (G−u) = 0. Then because ed conn ϕ (G) > 0, ed conn ϕ (G) = 1+min v∈V (G) ed conn ϕ (G− v) = 1 + ed conn ϕ (G − u) = 1 ≤ d. Suppose that d ≥ 2 and the claim holds for the lesser values of d. Because G is connected, by Lemma 1, there is a nice represenation (T, α) of X with depth(T ) = d − 1. Let r be the root of T and u = α(r). Because ed conn ϕ (G) > 0, ed conn ϕ (G) = 1+min v∈V (G) ed conn ϕ (G−v) ≤ 1+ed conn ϕ (G−u) and it is sufficient to show that ed conn ϕ (G − u) ≤ d − 1. For this, we have to prove that ed conn ϕ (C) ≤ d − 2 for every component C of G − u. If V (C) ∩ X = ∅ for a component C, then C is a component of G − X and we have that C |= ϕ. Then ed conn ϕ (C) = 0 ≤ d − 2. Consider the components C 1 , . . . , C s of G − u such that V (C i ) ∩ X = ∅. Because (T, α) is nice, r has s children x 1 , . . . , x s such that for every i ∈ {1, . . . , s}, α(V (T i )) ⊆ V (C i ), where T i is the subtree of T rooted in x i . Let α i : V (T i ) → V (C i ) be the restriction of α on V (T i ) for i ∈ {1, . . . , s}. Consider i ∈ {1, . . . , s}. We have that (T i , α i ) is a representation of X i = X ∩ V (C i ). Notice that for each component C of C i − X i , C |= ϕ. Clearly, depth(T i ) < depth(T )
. This implies that we can use the inductive assumption and conclude that ed conn ϕ (C i ) ≤ d − 1. Therefore, ed conn ϕ (G − u) ≤ d − 1 and this concludes the proof.
To characterize ed prop ϕ (G), we need additional definitions. Let G be a connected graph and let X ⊆ V (G) be an elimination set represented by (T, α). We say that a node x ∈ V (T ) is an anchor of a component C of G − X if x is the node of maximum depth in T such that α(x) ∈ N G (V (C)). We also say that C is anchored in x. Notice that the definition of an elimination set immediately implies the following property.
Observation 3. Let G be a connected graph and let X ⊆ V (G) be an elimination set represented by (T, α). Then for every component
C of G − X, N G (V (C)) ⊆ α(A T (x)), where x is an anchor of C.
In particular, Observation 3 implies that an anchor of each component of G − X is unique. For a node x ∈ V (T ), we denote by P x the set of components of G − X anchored in x, and G x denotes the subgraph of G induced by the vertices of the graphs of P x , that is, G x is the union of the components of G − X anchored in x. Clearly, P x and G x may be empty. Note that the anchors of the components of G − X depend on the choice of a representation. Therefore, we use the above notation only when (T, α) is fixed and clear from the context.
(i) for every nonleaf node x ∈ V (T ), C |= ϕ for every C ∈ P x , (ii) for every leaf x of T with depth T (x) ≤ d − 2, either G x |= ϕ or C |= ϕ for every C ∈ P x , (iii) for every leaf x of T with depth T (x) = d − 1, G x |= ϕ.
Proof. The lemma is proved similarly Lemma 3. We begin with showing that if ed prop ϕ (G) ≤ d, then G has an elimination set X of depth at most d − 1 with a representation (T, α) such that conditions (i)-(iii) are fulfilled. For this, we inductively construct X and (T,
α) with depth(T ) ≤ d − 1 using the definition of ed prop ϕ (G). Since G is connected and ed prop ϕ (G) > 0, there is v ∈ V (G) such that ed prop ϕ (G) = 1 + ed conn ϕ (G − v).
We construct a node r of T and set it be the root. If either G − v |= ϕ or C |= ϕ for every component C of G − v, then the construction of X and T is completed and we define α(r) = v. Note that ed prop ϕ (G) = 1 in the first case and ed prop ϕ (G) = 2 in the second. This implies that (i)-(iii) are fulfilled. Assume from now that this is not the case.
Denote by C 1 , . . . , C s the components of G − v such that C i |= ϕ for i ∈ {1, . . . , s}. By definition, ed prop ϕ (C i ) ≤ d − 1 for i ∈ {1, . . . , s}. Notice that for each component C of G − v distinct from C 1 , . . . , C s , C |= ϕ. Then we can assume inductively that for every i ∈ {1, . . . , s}, there is an elimination set X i ⊆ V (C i ) of depth at most d − 2 with respect to C i with a representation (T i , α i ) such that conditions (i)-(iii) are fulfilled. Let r i be the root of T i for i ∈ {1, . . . , s}. We define X = {v} ∪ s i=1 X i , and construct T from T 1 , .
. . , T s by making r 1 , . . . , r s the children of r. Then we set
α(x) = v, if x = r, α i (x), if x ∈ V (C i ) for some i ∈ {1, . . . , s}.
Using the inductive assumptions that (i)-(iii) are fulfilled for X i with (T i , α i ) for every i ∈ {1, . . . , s} and the observation that P r consists of the components of G − v distinct from C 1 , . . . , C s , we obtain that (i)-(iii) are fulfilled for X and the representation (T, α).
To show the implication in the opposite direction, assume that X is an elimination set of depth at most d − 1 with a representation (T, α) satisfying (i)-(iii). By the second claim of Lemma 1, we can assume that T is nice. We show that ed prop ϕ (G) ≤ d by the induction on depth(T ).
Suppose that depth(T ) = 0, that is, the depth of an elimination set X is zero and, therefore,
X = {u} for some u ∈ V (G). If d = 1, then G u |= ϕ and ed prop ϕ (G) = 1. If d ≥ 2, then either G u |= ϕ or C |= ϕ for every component C of G − u. In both cases, ed prop ϕ (G) ≤ 2 by the definition of ed prop ϕ (G). Assume that depth(T ) ≥ 1. In particular, d ≥ 2. Since G is connected and ed prop ϕ (G) > 0, ed prop ϕ (G) = 1 + min v∈V (G) ed prop ϕ (G − v) ≤ 1 + ed prop ϕ (G − u) and it is sufficient to show that ed prop ϕ (G − u) ≤ d − 1. If G − u |= ϕ, then ed prop ϕ (G) = 1 ≤ d. Assume from now that G − u |= ϕ. Then, by the definition of ed conn ϕ (G), it is sufficient to show that ed prop ϕ (C) ≤ d − 2 for every component C of G − u. If V (C) ∩ X = ∅ for a component C of G − u, then C ∈ P r and C |= ϕ. Then ed prop ϕ (C) = 0 ≤ d − 2. Consider the components C 1 , . . . , C s of G − u such that V (C i ) ∩ X = ∅. Because (T, α) is nice, r has s children x 1 , . . . , x s such that for every i ∈ {1, . . . , s}, α(V (T i )) ⊆ V (C i ), where T i is the subtree of T rooted in x i . Let α i : V (T i ) → V (C i ) be the restriction of α on V (T i ) for i ∈ {1, . . . , s}. Consider i ∈ {1, . . . , s}. We have that (T i , α i ) is a representation of X i = X ∩ V (C i ) satisfying (i)-(iii)
. Notice that depth(T i ) < depth(T ). Then by the inductive assumption ed prop ϕ (C i ) ≤ d−1. Therefore, ed prop ϕ (G−u) ≤ d−1 and this concludes the proof.
Lemmas 3 and 4 demonstrate that ed conn ϕ (G) and ed prop ϕ (G), respectively, can be defined via the deletion of an elimination set. We also use these results in order to define a third variant of the elimination distance.
Definition 2 (Elimination distance ed depth ϕ ). Let ϕ be a FOL formula. For a graph G, ed depth ϕ (G) is the minimum d such that G has an elimination set X ⊆ V (G) of depth d − 1 such that G − X |= ϕ.
Notice that if the trees in the considered representations of elimination sets are constrained to be paths, then we obtain the classical deletion distance, that is, the minimum size of a set
X ⊆ V (G) such that G − X |= ϕ.
Given a FOL formula ϕ, we define the following three variants of the Elimination Distance problems for ∈ {conn, prop, depth}:
Input:
A graph G and a nonnegative integer k. Task:
Decide whether ed ϕ (G) ≤ k.
Elimination Distance-( ) to ϕ parameterized by k
These problems may be seen as generalizations of Deletion to ϕ problem for a formula ϕ that asks, given a graph G and a nonnegative integer k, whether there is a set S of size at most k such that G − S |= ϕ. In particular, Observation 3 implies the following.
An FPT algorithm for Σ 3 -formulas
In this section, we show the main algorithmic result, Theorem 1, that Elimination Distance-( ) to ϕ is FPT for formulas from Σ 3 . Now we state this theorem formally.
Theorem 1. For every FOL formula ϕ ∈ Σ 3 , Elimination Distance-( ) to ϕ can be solved in f (k) · n O(|ϕ|) time for each ∈ {conn, prop, depth}.
We prove the theorem using the recursive understanding technique introduced by Chitnis et al. [8]. It was recently demonstrated by Agrawal et al.
[1] that this approach is useful for elimination problems. As we are interested in the quality result, we apply the meta theorem of Lokshtanov et al. [24] (see the arxiv version [25] for more details). This simplifies the arguments, but makes the proof nonconstructive. Moreover, we only show the existence of nonuniform FPT algorithms. However, it is possible to show the theorem in constructive way by giving uniform algorithm by either using the original approach of Chitnis et al. [8] or the dynamic programming scheme proposed by Cygan et al. [11].
The remaining part of the section contains the proof of Theorem 1. In Subsection 4.1, we introduce the notation and provide auxiliary results needed to apply the recursive understanding technique, and in Subsection 4.2, we prove that the Elimination Distance-( ) to ϕ is FPT for the key case when the input graphs cannot be partitioned in big parts by separators of bounded size.
Recursive understanding
Let G be a graph. A pair (A, B), where A, B ⊆ V (G) and A∪B = V (G), is called a separtion of G if there is no edge uv with u ∈ A \ B and v ∈ B \ A. In other words, A ∩ B is a (u, v)-separator for every u ∈ A \ B and v ∈ B \ A. The order of (A, B) is |A ∩ B|.
Let p, q be positive integers. A graph G is said to be (p, q)-unbreakable if for every separation (A, B) of G of order at most q, either |A \ B| ≤ p or |B \ A| ≤ p, that is, G has no separator of size at most q that partitions the graph into two parts of size at least p + 1 each. It is crucial that the considered problems may be expressed in MSOL.
Lemma 5. For every FOL formula ϕ, every ∈ {conn, prop, depth}, and every integer k ≥ 0, there is a MSOL formula ψ k such that for each graph G, G |= ψ k if and only if ed ϕ (G) ≤ k.
Proof. We use capital letter to write vertex set variables and small letters are used for vertex variables. To simplify notation, we introduce some auxiliary formulas. Notice that we can express that Z = X ∩ Y in MSOL and we write X ∩ Y for such an expression. Similarly, we write X − Y to express that Z = X \ Y , and we write X − y for X \ {y}. Also X is used for the complement of X. It is well-known that the connectivity property can be expressed in MSOL, because of the following observation: a set X ⊆ V (G) induces a connected subgraph of G if and only if for every partition (U, W ) of X, there is an edge uw ∈ E(G) such that u ∈ U and w ∈ W . Then we can observe that for every X ⊆ V (G), G[X] is a component of G if and only if X induces a connected subgraph but for every v ∈ V (G) \ X, G[X ∪ {v}] is not a connected graph. This allows us to use the MSOL formula comp(X) with a free variable X expressing the property that X induces a component. Clearly, every FOL formula is a MSOL formula. In particular, this means that we can construct the MSOL formula ϕ(X) for a free variable X expressing the property that the subgraph induced by X models ϕ.
First, we show the lemma for ∈ {conn, prop} using the definitions. For this, we inductively construct ψ conn k and ψ prop k .
It is easy to see that for k = 0, ψ conn 0 = ∀X comp(X) → ϕ(X). Now let k ≥ 1 and assume that ψ conn k−1 is constructed. Then we can define the MSOL formula ψ conn k−1 (X) for a free variable X expressing the property that the subgraph induced by X models ψ conn k−1 . Then it is straightforward to verify that
ψ conn k = ψ conn k−1 ∨ (∀X comp(X) → (∃x(x ∈ X) ∧ ψ conn k−1 (X − x))).
Next, we construct ψ prop k for k ≥ 0. It is straightforward to see that ψ prop 0 = ϕ and
ψ prop 1 = ψ prop 0 ∨ ∀X comp(X) → (ψ prop 0 (X) ∨ (∃x (x ∈ X) ∧ ψ prop 0 (X − x))) .
Then for k ≥ 2,
ψ prop k = ψ prop k−1 ∨ ∀X comp(X) → (∃x (x ∈ X) ∧ ψ prop k−1 (X − x)) ,
where ψ prop k−1 (X) for a free variable X expresses the property that the subgraph induced by X models ψ prop k−1 . Finally, we prove the claim for ψ depth k . Here, the proof is more complicated and uses Lemmas 1 and 2. We express the property that X is an elimination set of set at most d.
By Lemma 1, if G is a connected graph and d ≥ 0, then depth(X) ≤ d if and only if X has a nice representation of depth at most d. For a free variable X and an integer d ≥ −1, we define the formula ξ d (X) expressing that X has a nice representation (T, α) of depth at most d. For d = −1, ξ d (X) = (X = ∅), and for d = 0, ξ d (X) = (|X| = 1) by the definition (clearly, the property |X| = 1 can be expressed in MSOL). Assume that d ≥ 1 and ξ d−1 (X) is already constructed. Additionally, we assume that we are given the formula ξ d−1 (X, Y ) which expresses the property that X has a nice representation of depth at most d − 1 in the subgraph induced by Y . For this, we observe that ξ d−1 (X, Y ) can be constructed from ξ d−1 (X) in a straightforward way. Also we use comp(Y, x) to denote the formula expressing that Y induces a component of the subgraph obtained by the deletion of x. Then
ξ d (X) = ξ d−1 (X) ∨ ∃x (x ∈ X) ∧ (∀Y (comp(Y, x) ∧ (X ∩ Y = ∅)) → ξ d−1 (X ∩ Y, Y )) .
To see this, it is sufficient to observe that ∃x ( , Y )) expresses that G has a vertex x ∈ X such that for the root r of T , x = α(r), and in each component C of G − x containing some vertices of X, there is a subtree of T of depth at most d − 1 that can be used to represent V (C) ∩ X in C.
x ∈ X) ∧ (∀Y (comp(Y, x) ∧ (X ∩ Y = ∅)) → ξ d−1 (X ∩ Y
Now we construct the formulaξ d which expresses that X is an elimination set of depth at most d using Lemma 2. It is easy to see thatξ −1 (X) = (X = ∅) andξ d (X) = (|X| = 1). Assume that d ≥ 1,ξ d−1 (X) is already constructed, and we have a formulaξ d−1 (X, Y ) expressing that the depth of X is at most d − 1 in the subgraph induced by Y . Then by Lemma 2,
ξ d (X) =ξ d−1 (X) ∨(∃Y comp(Y ) ∧ξ d (X ∩ Y, Y ) ∧ (∀Z (comp(Z) ∧ (Z = Y )) →ξ d−1 (X ∩ Z, Z))). Usingξ d for d ≥ −1, we can write ψ depth k for k ≥ 0 as follows ψ depth k = ∃Xξ k−1 ∧ ϕ(X).
This completes the proof. Theorem 3 and Lemma 5 allows us to reduce the proof of Theorem 1 to solving Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} on unbreakable graphs. For this, we show that any elimination set in an unbreakable graph has bounded size. Lemma 6. Let G be a (p, q)-unbreakable graph for positive integers p and q with |V (G)| > (3p + 2q)(p + 1). Let also X ⊆ V (G) be an elimination set of depth at most d ≤ q − 1. Then |X| ≤ p + q. Furthermore, there is a unique component C of G − X with at least p + 1 vertices and |V (G) \ N G [V (C)]| ≤ p.
Proof. Let (T, α) be a representation of X with depth(T ) ≤ d. Denote by r the root of T .
First, we show the weaker bound |X| ≤ 3p + 2q. For the sake of contradiction, assume that |X| ≥ 3p + 2q + 1. Because T is a tree, it has a node x such that every component of T − x has at most 1 2 |V (T | nodes. Let S = A T (x) and S = α(A T (x)). Since depth(T ) ≤ d ≤ q−1, |S| = |S | ≤ q. By the definition of a representation, for every two distinct components C and C of T − S, and every x ∈ α(V (C)) and y ∈ α(V (C )), S is an (x, y)-separator in G.
We now claim that every component C of T −S has at most p nodes. Suppose to the contrary that there is a component C of T − S with at least p + 1 nodes. Consider the components we have that (A, B) is a separation of G with S = A ∩ B. In particular, (A, B) is a separation of order at most q. However, |A \ B| ≥ p + 1 and |B \ A|. This contradicts the unbreakability condition and the claim follows.
C 1 , . . . , C s of G − S such that V (C i ) ∩ α(V (C)) = ∅. Define A = S ∪ s i=1 V (C i ). Note that |A \ S| ≥ |V (C)| ≥ p + 1. Let Y = V (T ) \ (S ∪ V (C)). By the choice of x, |V (C)| ≤ 1 2 |V (T )|. Then |Y | ≥ 1 2 |V (T )| − q ≥ ( 3 2 p + q + 1 2 ) − q = 3 2 p + 1 2 ≥ p + 1. Observe that for every node y ∈ Y , α(y) / ∈ V (C i ) for i ∈ {1, . . . , s}. Then α(Y ) ⊆ V (G) \ A and |V (G) \ A| ≥ p + 1. For B = (V (G) \ A) ∪ S ,
Denote by C 1 , . . . , C s the components of T − S. Consider a set of indices I ⊆ {1, . . . , s} such that | i∈I V (C i )| ≥ p + 1 and for every proper I ⊂ I, | i∈I V (C i )| ≤ p. Such a set I exists,
because | s i=1 V (C i )| ≥ |V (T )| − q ≥ 3p + q + 1.
Since each component has at most p nodes, we have that | i∈I V (C i )| ≤ 2p. Then, because |V (T ) \ S| ≥ 3p + 1, | i∈{1,...,s}\I V (C i )| ≥ p + 1.
Consider the components C 1 , . . . , C t of G−S that contain at least one vertex of α(V (C i )) for
some i ∈ I. Define A = S ∪ t i=1 V (C i ). Note that |A\S | ≥ p+1, because | i∈I V (C i )| ≥ p+1. Let B = (V (G) \ A) ∪ S .
Since i∈{1,...,s}\I α(V (C i )) ⊆ B \ S, |B \ S| ≥ p + 1. Then we obtain that (A, B) is a separation of G of order at most q with |A \ B| ≥ p + 1 and |B \ A| ≥ p + 1; a contradiction. This concludes the proof of our claim that |X| ≤ 3p + 2q. Now we improve the obtained upper bound. Because |X| ≤ 3p + 2q and |V (G)| > (3p + 2q)(p+1), |V (G)\X| > (3p+2q)p. Observe that for the set of leaves L(T ), we have that |L(T )| ≤ 3p + 2q. By Observation 3, it holds that for every component C of G − X, N G (V (C)) ⊆ A T (x) for some x ∈ L(T ). By the pigeon hole principle, we conclude that there is x ∈ L(T ) such that for the components C 1 , . . . ,
C s of G − X with N G (V (C i )) ⊆ A T (x) for i ∈ {1, . . . , s}, it holds that | s i=1 V (C i )| ≥ p + 1. Let S = α(A T (x)). Note that |S| ≤ d + 1 ≤ q. Consider A = S ∪ s i=1 V (C i ) and B = (V (G) \ A) ∪ S.
We obtain that (A, B) is a separation of G of order at most q and |A \ B| ≥ p + 1. Since G is (p, q)-unbreakable, we have that |B| ≤ p + q. Notice that X ⊆ B. Thus, |X| ≤ p + q.
To show the second claim, note that |L(T )| ≥ p + q. In the same way as above, there is x ∈ L(T ) such that for the components C 1 , . . . ,
C s of G − X with N G (V (C i )) ⊆ A T (x) for i ∈ {1, . . . , s}, it holds that | s i=1 V (C i )| ≥ p + 1. We show that there is i ∈ {1, .
. . , s} such that |V (C i )| ≥ p + 1. For the sake of contradiction, assume that |V (C i )| ≤ p + 1 for all i ∈ {1, . . . , s}. Then there is a set of indices I ⊆ {1, . . . , s} such that | i∈I V (C i )| ≥ p + 1 and for every proper Then (A, B) is a separation of G of order at most q with |A\B| ≥ p+1 and |B\A| ≥ p+1; a contradiction with the condition that G is (p, q)-unbreakable. This implies that there is a component C of G − X with |V (C)| ≥ p + 1.
I ⊂ I, | i∈I V (C i )| ≤ p. Because each component has at most p vertices, | i∈I V (C i )| ≤ 2p. Consider A = S ∪ s i=1 V (C i ), where S = α(A T (x)) and B = (V (G) \ A) ∪ S. Note that |B \ S| ≥ |V (G)| − 2p − q ≥ p + 1.
Because G is a (p, q)-unbreakable graph and |N G (V (C))| ≤ q, we have that |V Using the notation in Lemma 6, we say that a component C of G − X with at least p + 1 vertices is big and the other components are small.
We can use backtracking to verify, given a X, whether depth(X) ≤ d. For this we combine Lemmas 1 and 2 with backtracking and obtain the following straightforward lemma. We also need the following technical lemma that will allow us to consider inclusion minimal elimination sets.
Lemma 9. Let G be a connected graph and let X be a nonempty elimination set with a nice representation (T, α). Let also C be a component of G − X anchored in x * ∈ V (T ). Suppose that S ⊆ N G (V (C)) and C is a component of G − S with V (C) ⊆ V (C ). Then there is an elimination set X ⊆ X with a nice representation (T , α ) such that V (T ) ⊆ V (T ) and the following is fulfilled:
(a) S ⊆ X and (N G (V (C)) \ S) ∩ X = ∅, (b) for every component H of G − X , either V (H) ⊆ V (C ) or H is a component of G − X, (c) for every node y ∈ V (T ), depth T (y) ≤ depth T (y), (d) if a component H of G − X distinct from C is anchored in a leaf z of T , then H in anchored in z in T and z is a leaf of T , (e) if x * is a leaf of T and x * ∈ S, then C is anchored in x * in T .
Proof. Let R = N G (V (C)). The claim is trivial if S = ∅, because C = G and we can take X = ∅. We assume that this is not the case. The proof is by induction on |R \ S|. The claim is straightforward if R = S as we can take X = X and consider the same representation (T, α). The crucial case is the case |R \ S| = 1. Let u be the unique vertex of R \ S. We consider two possibilities for u. Let v = α(r), where r is the root of T .
Case 1. u = v. Let W = V (C ). Notice that a vertex w ∈ V (G) is in W if and only if either w ∈ V (C) or w / ∈ S ∪ V (C) and G − S has a (u, w)-path. Define X = X \ W . Clearly, (a) holds for this X . Observe that for a component H of G − X, we have that V (H) ⊆ W if N G (V (H))
contains a vertex of W and V (H) ∩ W = ∅ otherwise. In particular, this implies (b).
Next, we construct T and α . We set V (T ) = α −1 (X ) and define α (x) = α(x) for every x ∈ V (T ). Because S = ∅, there is a descendant r of r of minimum depth such that α(r ) ∈ S. For every w ∈ X distinct from α(r ), we consider x = α −1 (w) and find a proper ancestor y of x in T of maximum depth such that α(y) ∈ X . Then we define y be the parent of x.
We argue that T is a tree rooted in r . We have to show that for every w ∈ X distinct from α(r ), we have an ancestor y of x = α −1 (w) in T such that α(y) ∈ X . For the sake of contradiction, assume that there is w ∈ X such that for every proper ancestors y of x = α −1 (w) in T , α(y) / ∈ X . Clearly, x is not a descendant of r in T . In particular, r and x are incomparable. Let z be the lowest proper ancestor of r and y in T . We have that α(A T (z)) is an (α(r ), w)-separator of G and, moreover, S has no vertices in the component G − α(A T (z)) containing α(x). Since (T, α) is nice, this component has an (α(z), w)-path. Because α(z) / ∈ S, G − S has a (u, α(z))-path. We conclude that G − S has a (u, w)-path and w / ∈ X ; a contradiction. This proves that T is a tree rooted in r . We prove that (T , α ) represents X . Towards a contradiction, assume that this is not the case, that is, there are distinct x, y ∈ V (T ) whose lowest common descendant z = x, y and α (x) and α (y) are in the same component of G−A T (z). By the definition of T , z has a descendant z such that z = x, y is the lowest common ancestor of x and y in T . Clearly, either x / ∈ N G (V (C)) or y / ∈ N G (V (C)). By symmetry, assume that y / ∈ N G (V (C)). Because (T, α) is a representation of X, α(A T (z )) is an (α(x), α(y))-separator. This means that every (α (x), α (y))-path in G contains a vertex of (α(x), α(y)). In particular, this implies that there is a vertex z ∈ A T (z ) such that G has an (α(z ), α (y))-path P in G such that the internal vertices of the path are in the component G − α(A T (z )) containing α (y). Because y / ∈ N G (V (C)), we have that P avoids the vertices of S. Since z / ∈ X , G − S has a (u, α(z ))-path P . Concatenating P and P , we obtain that G − S has a (u, α(y ))-path. However this contradicts that α (y) ∈ X . This proves that (T , α ) represents X .
By the construction of T , it is easy to see that T is nice, because T is nice. Also the construction of T immediately implies (c)-(e). This concludes the analysis of the first case.
Case 2. u = v. We show the claim by induction on d = depth(T ). Notice that depth(T ) ≥ 1, because X \ {v} = ∅. Let C 1 , . . . , C s be the components of G − v such that X i = X ∩ V (C i ) = ∅ for i ∈ {1, . . . , s}. Because T is nice, T has children r 1 , . . . , r s such that for every i ∈ {1, . . . , s}, the subtree T i of T rooted in r i together with α i (x) = α(x) for x ∈ V (T i ) represent X i in C i . By Observation 3, we can assume without loss of generality that V (C) ⊆ V (C 1 ) and N G (V (C)) ⊆ V (C 1 ) ∪ {v}. Because depth(T 1 ) < depth(T ), we can apply the inductive assumption and construct an elimination set X 1 ⊂ X 1 with a nice representation (T 1 , α 1 ) satisfying (a)-(e). Then we construct X = X 1 ∪ s i=1 X i . Then we construct T from T 1 and T 2 , . . . , T s by making r 1 and r 2 , . . . , r s the children of r, where r 1 is the root of T 1 . We set
α (x) = α i (x), if x ∈ X i for some i ∈ {2, . . . , s}, α 1 (x), if x ∈ X 1 .
It is straightforward to verify that X and (T , α ) satisfy (a)-(e).
This concludes the poof for the base case |R \ S| = 1. To show the claim for |R \ S| > 1, consider a vertex w ∈ R\S and apply the claim for S = S ∪{w} using the inductive assumption. We have that there is an elimination set X ⊆ X with a nice representation (T , α ) such that V (T ) ⊆ V (T ) and (a)-(e) are fulfilled with respect to S . Then we apply the claim for X and (T , α ) with respect to the component C and S. Clearly, we obtain an elimination set X ⊆ X ⊆ X with a nice representation (T , α ) such that V (T ) ⊆ V (T ) ⊆ V (T ) satisfying (a)-(e). This completes the proof.
In our algorithms, we use the random separation technique introduces by Cai, Chan and Chan in [7]. To avoid dealing with randomized algorithms, we use the following lemma stated by Chitnis et al. in [8].
Lemma 10 ([8])
. Given a set U of size n and integers 0 ≤ a, b ≤ n, one can construct in time 2 O(min{a,b} log(a+b)) · n log n a family F of at most 2 O(min{a,b} log(a+b)) · log n subsets of U such that the following holds: for any sets A, B ⊆ U , A ∩ B = ∅, |A| ≤ a, |B| ≤ b, there exists a set R ∈ F with A ⊆ R and B ∩ R = ∅.
Algorithm for unbreakable graphs
In this subsection, we give FPT-algorithms for Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} for FOL formulas ϕ ∈ Σ 3 on unbreakable graphs. Throughout the subsection, we assume without loss of generality that ϕ = ∃x 1 · · · ∃x r ∀y 1 · · · ∀y s ∃z 1 · · · ∃z t χ,
where χ is quantifier-free and r, s, t are positive integers, because we always can write a FOL formula from Σ 3 in this form by adding dummy variables if necessary. We also write x = x 1 , . . . , x r , y = y 1 , . . . , y s , and z = z 1 , . . . , z t .
Notice that ed conn ϕ (G) = 0 if and only if for every component C of G, C |= ϕ. Also ed prop ϕ (G) = 0 (ed depth ϕ (G) = 0, respectively) if and only if G |= ϕ. This implies that Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} can be solved in time n O(|ϕ|) if k = 0 by Observation 1, that is, Theorem 1 trivially holds for k = 0. Hence, throughout this subsection we assume that the parameter k in the considered instances is positive.
By Theorem 3 and Lemma 5, to prove Theorem 1, it is sufficient to demonstrate FPT algorithm for the considered problems on (p(k), k)-unbreakable graphs for a computable function p : N → N. Slightly abusing notation, we write p instead of p(k).
The algorithms for Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} are similar. However, there are differences that make it inconvenient to describe them together. Hence, we first give the details of the algorithm for Elimination Distance-(conn) to ϕ and then more briefly explain our algorithms for Elimination Distance-(prop) to ϕ and Elimination Distance-(depth) to ϕ. Then we derive Theorem 1 from Lemmas 12,15, and 17 in which we summarize the properties of the algorithms for the considered problems.
Algorithm for Elimination Distance-(conn) to ϕ. Let (G, k) be an instance of Elimination Distance-(conn) to ϕ, where G is a (p, k)-unbreakable graph. We assume without loss of generality that G is connected. Otherwise, because ed conn ϕ (G) = max{ed conn ϕ (C) | C a component of G}, we can solve the problem for each component separately. If |V (G)| ≤ (3p + 2k)(p + 1), we solve the problem in (p + k) O(k+|ϕ|) time by Lemma 8. From now we assume that |V (G)| > (3p + 2k)(p + 1).
By Lemma 3, (G, k) is a yes-instance of Elimination Distance-(conn) to ϕ if and only if G contains an elimination set X of depth at most k − 1 such that C |= ϕ for every component C of G − X. Our algorithm finds such a set X, called a solution, if it exists. We verify in n O(|ϕ|) time whether X = ∅ has the required property and return yes if this holds. Assume that this is not the case, that is, we have to find a nonempty solution.
Suppose that (G, k) is a yes-instance and let X be a solution with a representation (T, α). By Lemma 6, |X| ≤ p + k and there is a unique big component C of G − X with at least p + 1 vertices, the other components are small, and
|V (G) \ N G [V (C)]| ≤ p. By Observation 3, N G (V (C)) ⊆ α(A T (x))
, where x is an anchor of C. In particular, this means that |N G (V (C))| ≤ k. We use these properties to identify C. This is done by combining the random separation technique [7] with a recursive branching algorithm.
We use random separation to highlight the hypothetical sets S = N G (V (C)) and U = V (G)\ N G [V (C)] (if they exist). To avoid randomized algorithms, we directly use the derandomization tool from Lemma 10. By this lemma, we can construct in 2 O(min{p,k} log(p+k)) · n log n time a family F of at most 2 O(min{p,k} log(p+k)) · log n subsets of V (G) such that there is R ∈ F such that U ⊆ R and S ∩ R = ∅. In our algorithm, we go over all sets R ∈ F and for each set R, we check whether there is a solution X such that U ⊆ R and S ∩ R = ∅ for the sets S and U corresponding to X (recall that S = N G (V (C)) and U = V (G) \ N G [V (C)], where C is the unique big component of G − X with at least p + 1 vertices). Clearly, (G, k) is a yes-instance of Elimination Distance-(conn) to ϕ if and only if there is R ⊆ F and a solution X with the required property. From now on we assume that R ∈ F is given. We set B = V (G) \ R. We say that the vertices of R are red and the vertices of B are blue. We also call the components of G[R] red components of G and we use the same convention for induced subgraphs of G. A solution X is colorful if the vertices of U are red and the vertices of S are blue (see Figure 1). The crucial property of colorful solutions is that if a red vertex v is in U , then the set of vertices of the red component H containing v is a subset of U .
If G − X has a big component C and C |= ϕ, then there is an r-tuple v = v 1 , . . . , v r of vertices of C such that (C, v) |= ϕ[x] (recall that ϕ[x] is the formula with the free variables x 1 , . . . , x r obtained from ϕ by the removal of the quantification over x 1 , . . . , x r , and (C, v) |= ϕ[x] means that ϕ[x] evaluates true on G when x i is assigned v i for all i ∈ {1, . . . , r}). Using brute force, we consider all r-tuples v = v 1 , . . . , v r of vertices of G, and for each v, we explain how to check whether there is a colorful solution X with the big component C such that v i ∈ V (C) for all i ∈ {1, . . . , r}. Note that at most n r r-tuples v can be listed in n O(|ϕ|) time. The algorithm returns yes if we find a colorful solution for some choice of v, and it concludes that there is no colorful solution for the considered selection of R otherwise.
From now we assume that v = v 1 , . . . , v r is fixed. Because these vertices should be in C, we temporarily (i.e., only for the current choice of v) recolor them red to simplify further notation. We apply a recursive branching algorithm to find C and S.
By definition, we have that (C, v) |= ϕ[x] if and only if for every s-tuple u = u 1 , . . . , u s of vertices of C, (C, vu) |= ϕ[xy]. Suppose that (C, v) |= ϕ[x]. Then there is an s-tuple u = u 1 , . . . , u s of vertices such that (C, vu) |= ϕ [xy]. Notice now that, because ϕ ∈ Σ 3 , we have that for any induced subgraph C of C such that v i ∈ V (C ) for every i ∈ {1, . . . , r},
if (C , v) |= ϕ[x]
, then there is j ∈ {1, . . . , s} such that u j / ∈ V (C ). This implies that if (C, vu) |= ϕ[xy], then there is j ∈ {1, . . . , s} such that either u j ∈ S and should be deleted or u j is in a component of G − S distinct from C and this component should be deleted together with its neighborhood. Note that u j is blue in the first case, and u j is red in the second. Moreover, in the second case, we should delete the red component containing u j together with its blue neighborhood. We branch on all possible deletions of v i 's, using the following subroutine FindC(C, S, h), where we initially set C := G, S := ∅, and h := k.
Subroutine FindC(C, S, h).
• If (C, v) |= ϕ[x] and h ≥ 0, then return C, S, and stop.
• If (C, v) |= ϕ[x] and h ≤ 0, then stop.
• If h ≥ 1 and there is an s-tuple u = u 1 , . . . , u s of vertices of C such that (C, vu) |= ϕ[xy], then do the following for every j ∈ {1, . . . , s}.
- We show the following lemma. . Because the subroutine is called only for connected induced subgraphs of G, we have thatS ⊂ S and, therefore,h > 0. This implies that the subroutine does not stop in the second step. Then it proceeds to the third step and finds an s-tuple u = u 1 , . . . , u s of vertices ofC such that (C, vu) |= ϕ [xy]. Because (C, v) |= ϕ[x], there is a j ∈ {1, . . . , s} such that u j / ∈ V (C). We consider the following two cases.
If u j ∈ B and there is a component C of C − u j such that v i ∈ V (C ) for all i ∈ {1,
Case 1. u j ∈ S. Notice that because X is a colorful solution, u j is blue in this case. Observe also thatC − u j has a componentC such that V (C) ⊆ V (C ). Then the subroutine calls FindC(C ,S ,h ), whereS =S ∪ {u j } andh =h − 1. It is easy to see that (a )-(d ) are fulfilled forC ,S , andh . Case 2. u j ∈ U . As X is colorful, u j is red in this case. Let H be the red component of C containing u j and let W = V (H). Because X is a colorful solution, we have that W ⊆ U and NC(W ) ⊆ S. Then G − N G [W ] has a componentC such that V (C) ⊆ V (C ). Then the subroutine calls FindC(C ,S ,h ), whereS =S ∪ NC(W ) andh =h − |NC(W )|. We obtain that (a )-(d ) are fulfilled forC ,S , andh . This concludes the case analysis and the proof of the claim.
Observe that conditions (a)-(d) of the claim are fulfilled if C = G, S = ∅, and h = k. Then the inductive application of the claim proves that there is a leaf of the search tree for which it outputsC andS such that (a) V (C) ⊆ V (C), (b)S = N G (V (C)), and (c)S ⊆ S. Recall that X is an inclusion minimal colorful solution. Then Lemma 9 immediately implies that C =C and S =S and this concludes the proof.
Note that the number of branches of every node of the search tree produced by FindC(G, ∅, k) is at most s and the depth of the search tree is at most k. This implies that the search tree has at most s k leaves. By Lemma 11, if (G, k) has an inclusion minimal colorful solution X, then the subroutine outputs the corresponding big component C containing v 1 , . . . , v r and S. We consider all pairs (C, S) produced by FindC(G, ∅, k) and for each of these pairs, we verify whether there is a colorful solution corresponding to it. If we find such a solution we return yes (or return the solution), and we return no if we fail to find a clorful solution for each C and S. In the last case we conclude that we have no colorful solution and discard the current choice of R ∈ F.
Assume that C and S are given. Recall that v i ∈ V (C) for i ∈ {1, . . . , r}, S = N G (V (C)), and (G, v) |= ϕ[x]. First, we check whether C is a big components of G − S by verifying whether |V (C)| ≥ p + 1. Clearly, if |V (C)| ≤ p, C cannot be a big component of G − X for a solution X and we discard the considered choice of C and S. Assume that this is not the case, that is, |V (C)| ≥ p + 1. Then because G is a (p, k)-unbreakable graph, we have that |V (G) \ N G [V (C)]| ≤ p. We use brute force and consider every subset Y ⊆ V (G) \ N G [V (C)] and then verify whether (i) X = S ∪ Y is an elimination set of depth at most k − 1 and (ii) for every component C = C of G − X, C |= ϕ. Note that checking (i) can be done by Lemma 7 in (k + p) O(k) · n O(1) time and (ii) can be verified in n O(|ϕ|) time by Observation 1. If we find X = S ∪ Y satisfying (i) and (ii), then we conclude that X is a solution and return yes. Otherwise, if we fail to find such a set, we return no.
This concludes the description of the algorithm for Elimination Distance-(conn) to ϕ and its correctness proof. We summarize in the following lemma.
Lemma 12. Elimination Distance-(conn) to ϕ on (p, k)-unbreakable graphs for ϕ ∈ Σ 3 can be solved in 2 O((p+k) log(p+k)) · n O(|ϕ|) time.
Proof. Since the correctness of the algorithm was already established, it remains to evaluate the total running time. Recall that if |V (G)| ≤ (3p + 2k)(p + 1), then the problem is solved in (p + k) O(k+|ϕ|) time. Otherwise, we construct F of size at most 2 O(min{p,k} log(p+k)) · log n in 2 O(min{p,k} log(p+k)) · n log n time. Then for every R ∈ F, we try to find a colorful solution. For this, we first guess v. Clearly, we have n O(|ϕ|) possibilities for the choice of v. Then we run the subroutine FindC(C, S, h). Note that the search tree produced by the subroutine has at most |ϕ| k leaves and each call (without recursive calls) requires n O(|ϕ|) time. Then the running time of the subroutine is |ϕ| k · n O(|ϕ|) . We consider the pairs (C, S) produced by the subroutine, and for each C and S, we verify whether we have a corresponding colorful solution X. The brute force selection of X can be done in 2 O(p) time. Then checking whether X is a solution requires (k + p) O(k) · n O(1) . Then we conclude that the total running time is 2 O((p+k) log(p+k)) · n O(|ϕ|) .
Algorithm for Elimination Distance-(prop) to ϕ. Let (G, k) be an instance of Elimination Distance-(prop) to ϕ, where G is a (p, k)-unbreakable graph. We check whether G |= ϕ and immediately return yes if this is fulfilled. Assume that this is not the case and that ed prop ϕ (G) ≥ 1. Then we can assume without loss of generality that G is connected. Otherwise, because ed prop ϕ (G) = max{1, max{ed prop ϕ (C) | C a component of G}}, we can solve the problem for each component separately. In the same way as with Elimination Distance-(conn) to ϕ, we solve the problem in (p + k) O(k+|ϕ|) time by Lemma 8 if |V (G)| ≤ (3p + 2k)(p + 1). Therefore, from now on, we may assume that |V (G)| > (3p + 2k)(p + 1).
Let (T, α) be a representation of an elimination set X. Recall that P x denotes the set of components of G − X anchored in x, where x is a node of T . Also G x denotes the subgraph of G induced by the vertices of the graphs of P x , that is, G x is the union of the components of G − X anchored in x. By Lemma 4, (G, k) is a yes-instance of Elimination Distance-(prop) to ϕ if and only if G contains an elimination set X of depth at most k − 1 with a representation (T, α) such that (i) for every nonleaf node x ∈ V (T ), C |= ϕ for every C ∈ P x , (ii) for every leaf x of T with depth T (x) ≤ k − 2, either G x |= ϕ or C |= ϕ for every C ∈ P x , and
(iii) for every leaf x of T with depth T (x) = k − 1, G x |= ϕ.
We call such a set X a solution. We observe that, given a set X, we can decide whether X is a solution.
Lemma 13. Let X ⊆ V (G) be nonempty. It can be decided in |X| k · n O(|ϕ|) time whether X has a representation (T, α) satisfying (i)-(iii).
Proof. Because G is connected and k ≥ 1, it is sufficient to verify the existence of a nice representation. We do it by a recursive algorithm that for a given x ∈ X finds a nice representation (T, α) such that α(r) = x, where r is the root of T . More precisely, given a graph G, a nonempty X ⊆ V (H), a vertex x ∈ X, and a positive integer k, the algorithm find a nice representation (T, α) of X satisfying (i)-(iii) such that α(r) = x if such a representation exists. Suppose that |X| = 1, that is, X = {x}. If G − x |= ϕ, the algorithm returns a single-vertex tree rooted in r with α(r) = x. If G − x |= ϕ and k ≥ 2, we check whether C |= ϕ for every component C − x. If this holds, then again, the algorithm returns a single-vertex tree rooted in r with α(r) = x. In all other cases, the algorithm returns no.
Suppose from now that |X| ≥ 2. If k = 1, then we immediately return no and stop. Also if there is a component C of G − x such that V (C) ∩ X = ∅ and C |= ϕ, the the algorithm returns no and stops. Assume that these are not cases. Let C 1 , . . . , C s be the components of
G − x such that X i = X ∩ V (C i ) = ∅.
For every i ∈ {1, . . . , s}, we call the algorithm recursively for C i , X i , every y ∈ X i , and k − 1. If there is i ∈ {1, . . . , s} such that the algorithm failed to produce a representation for every choice of y ∈ X i , the algorithm returns no and stops. Otherwise, the algorithm finds for every i ∈ {1, . . . , s} a vertex x i ∈ X i and a nice representation (T i , α i ) of X i in C i satisfying (i)-(iii) (with respect to the new parameters) such that the root r i is mapped to x i by α i . We construct T from T 1 , . . . , T s by creating a root r and making it the parent of r 1 , . . . , r s . Then
α(z) = x, if z = r, α i (z), if z ∈ V (T i ) for some i ∈ {1, . . . , s}.
This completes the description of the algorithm. It is straightforward to verify its correctness using the definition of a nice representation of an elimination set. To decide whether X has a representation (T, α) satisfying (i)-(iii), we run the algorithm for all x ∈ X. Clearly, a representation exists if and only if the algorithm produces a representation for some choice of x. Since in each call of the algorithm, we make at most |X| recursive calls and the depth of the recursion is at most k, the total running time is |X| k · n O(|ϕ|) .
Suppose that (G, k) is a yes-instance and let X be a solution with a nice representation (T, α). By Lemma 6, |X| ≤ p + k and there is a unique big component C of G − X with at least p + 1 vertices, the other components are small, and |V (G) \ N G [V (C)]| ≤ p. By Observation 3, N G (V (C)) ⊆ α(A T (x)), where x is an anchor of C. In particular, this means that |N G (V (C))| ≤ k. As with the algorithm for Elimination Distance-(conn) to ϕ, our aim is to identify C. We consider two possibilities for C.
First, we try to find C assuming that one of the following holds: either (a) the anchor of C is not a leaf of T or (b) the anchor x is leaf but depth T (x) < k −1 and C |= ϕ for every C ∈ P x , or (c) G x = C. In this case, the algorithm is essentially identical to the algorithm for Elimination Distance-(conn) to ϕ. We use Lemma 10 to highlight S and U = V (G) \ N G [V (C)]. Then we guess v in C and call the subroutine FindC(G, ∅, k) to enumerate all candidate big components C and S = N G (V (S)). The difference occurs only in the last step of the algorithm, where we find a solution X. We use brute force and consider every subset Y ⊆ V (G) \ N G [V (C)] and then verify whether X = S ∪ Y is an elimination set of depth at most k − 1 satisfying (i)-(iii) using Lemma 13. If we find a required X, then we conclude that X is a solution and return yes. Otherwise, if we fail to find such a set for every candidate C, we return no for the considered set R and discard it. The correctness is proved and the running time is analysed in exactly the same way as for Elimination Distance-(conn) to ϕ.
Next, if we failed to find a solution so far, we consider the remaining possibility that the anchor x of C is a leaf of T and G x |= ϕ, where G x is a disconnected graph. Our algorithm for this case uses the same approach as the algorithm for Elimination Distance-(conn) to ϕ but the arguments are more involved, as we aim to identify C together with the other components of G x . In other words, we find G x .
Let S = N G (V (G x )). Note that S ⊆ α(A T (x)) and, therefore, |S| ≤ k. Observe that N G (V (C)) ⊆ S. Let also U = V (G) \ (V (C) ∪ S). Because C is a big component and G is (p, k)-unbreakable, |U | ≤ p.
Similarly to the algorithm for Elimination Distance-(conn) to ϕ, we use Lemma 10 to highlight hypothetical S and U . By this lemma, we can construct in 2 O(min{p,k} log(p+k)) · n log n time a family F of at most 2 O(min{p,k} log(p+k)) · log n subsets of V (G) such that there is R ∈ F such that U ⊆ R and S ∩ R = ∅. In our algorithm, we go over all sets R ⊆ F and for each set R, we check whether there is a solution X such that U ⊆ R and S ∩ R = ∅ for the sets S and U corresponding to X. Clearly, (G, k) is a yes-instance of Elimination Distance-(prop) to ϕ if and only if there is R ⊆ F and a solution X with the required property.
From now on we assume that R ⊆ F is given. We set B = V (G) \ R. In the same way as before, we say that the vertices of R are red and the vertices of B are blue. The components of G[R] are called red components of G and the same convention is used for induced subgraphs of G. A solution X is called colorful if the vertivces of U are red and the vertices of S are blue. We aim to find a colorful solution.
Assume that a colorful solution X exists. Suppose that w = α(x) for the leaf x of T that is the anchor of G x . Notice that w ∈ B. Then for every component C of G x distinct from C, we have that C is a red component and z ∈ N G (V (C )). We also observe that by the assumption for R, if C is a red component of G such that w ∈ N G (V (H)), then either V (C ) ⊆ V (C) or C is a component of G x distinct from C. Using these observations, we consider all possible choices of w in B, and decide whether there is a colorful solution X such that for the required G x , the leaf x of T is mapped to w. We say that X is a colorful solution attached to w.
From now we assume that w is given. Let W = V (H), where the union is taken over all red components H of G such that w ∈ N G (V (H)). Notice that if there is a colorful solution X attached to w for the considered choice of w, then W ⊆ V (G x ) for the corresponding graph G x .
Since we require that G x |= ϕ, then there is an r-tuple v = v 1 , . . . , v r of vertices of G x such that (G x , v) |= ϕ[x]. Using brute force, we consider all r-tuples v = v 1 , . . . , v r of vertices of G distinct from w, and for each v, we check whether there is a colorful solution X with G x such that v i ∈ V (G x ) for all i ∈ {1, . . . , r}. Note that at most n r r-tuples v can be listed in n O(|ϕ|) time. The algorithm returns yes if we find a colorful solution attached to w for some choice of v, and it concludes that there is no colorful solution for the considered selection of R otherwise.
From this point we assume that v = v 1 , . . . , v r is fixed. Because these vertices should be in G x , we temporarily (i.e., only for the current choice of v) recolor them red to simplify further notation and recompute W if necessary. We apply a recursive branching algorithm to find In the same way as with Lemma 11, we show the following. Lemma 14. If X is an inclusion minimal colorful solution attached to w for (G, k) with such that (a) X has a representation (T, α) with α(
x) = w, (b) W ⊆ V (G x ), (c) v i ∈ V (G x ) for all i ∈ {1, . . . , s} and (G x , v) |= ϕ[x]
, then there is a leaf of the search tree produced by FindC(G, {w}, k − 1) for which the subroutine outputs F and S = N G (V (F )).
Since the number of branches of every node of the search tree produced by FindF(G, {w}, k− 1) is at most s and the depth of the search tree is at most k, the search tree has at most r k leaves. By Lemma 14, if (G, k) has an inclusion minimal colorful solution X attached to w with respect to some representation (T, α) of X, then the subroutine outputs the corresponding graph F = G x containing v 1 , . . . , v r and S. We consider all pairs (F, S) produced by FindF(G, {w}, k − 1) and for each of these pairs, we verify whether there is a colorful solution corresponding to it. If we find such a solution we return yes (or return the solution), and we return no if we fail to find a colorful solution for each F and S. In the last case we conclude that we have no colorful solution and discard the current choice of R ∈ F.
Assume that F and S are given. Recall that v i ∈ V (C) for i ∈ {1, . . . , r}, S = N G (V (C)), and (G, v) |= ϕ[x]. First, we check whether F has a big components of G − S by verifying whether F has a component with at least p + 1 vertices. If we have no such a component, we discard the considered choice of F and S. Assume that this is not the case. Then because G is a (p, k)-unbreakable graph, we have that |V (F ) \ N G [V (F )]| ≤ p. We use brute force and consider every subset Y ⊆ V (G) \ N G [V (F )] and then verify whether X = S ∪ Y is a solution using Lemma 13. If we find a solution, we return yes. Otherwise, if we fail to find Y with the required properties, we return no.
This concludes the description of the algorithm for Elimination Distance-(conn) to ϕ and its correctness proof. We summarize in the following lemma that is proved in the same way as Lemma 12.
Lemma 15. Elimination Distance-(prop) to ϕ on (p, k)-unbreakable graphs for ϕ ∈ Σ 3 can be solved in 2 O((p+k) log(p+k)) · n O(|ϕ|) time.
Algorithm for Elimination Distance-(depth) to ϕ. Our final task is to construct an algorithm for Elimination Distance-(depth) to ϕ. Let (G, k) be an instance of Elimination Distance-(depth) to ϕ, where G is a (p, k)-unbreakable graph. If G |= ϕ, then we return yes. Assume that this is not the case and ed depth ϕ (G) ≥ 1. Suppose that G is disconnected. Denote by C 1 , . . . , C s the components of G. Because G is a (p, k)-unbreakable graph, at most one component can have more p vertices. Then we can assume that |V (C i )| ≤ p for every i ∈ {2, . . . , s}. For each i ∈ {2, . . . , s}, we solve Elimination Distance-(depth) to ϕ for (C i , k − 1) and (C i , k) in 2 p · p O(k+|ϕ|) time using brute force. Let i ∈ {2, . . . , s}. For each set X ⊆ V (C i ), we check whether depth(X) ≤ k − 2 (depth(X) ≤ k − 1, respectively) applying Lemma 7, and if this holds we verify whether G−X |= ϕ. This can be done in 2 p ·p O(k) ·p O(|ϕ|) time. If we find that either there is i ∈ {2, . . . , s} such that ed depth ) is a yes-instance if and only if (C 1 , k − 1) is a yes-instance by Lemma 2. If ed depth ϕ (C i ) ≤ k − 1 for every i ∈ {2, . . . , s}, then by the same lemma, (G, k) is a yes-instance if and only if (C 1 , k) is a yes-instance. Thus, we are able to reduce solving the problem on G to solving it on C 1 . This implies that we can assume without loss of generality that G is connected.
ϕ (C i ) ≥ k +1 or there are two distinct i, j ∈ {2, . . . , s} such that ed depth ϕ (C i ) = ed depth ϕ (C j ) = k, we return no by Lemma 2. If there is a unique i ∈ {2, . . . , s} with ed depth ϕ (C i ) = k and ed depth ϕ (C j ) ≤ k − 1 for j ∈ {2, . . . , s} distinct from i, (G, k
If |V (G)| ≤ (3p+2k)(p+1), we again can solve the problem using brute force in 2 (3p+2k)(p+1) · ((3p + 2k)(p + 1)) O(k+|ϕ|) time in the same way as above. Then we assume that |V (G)| > (3p + 2k)(p + 1).
Given a subset X ⊆ V (G), we can verify in |X| O(k) · n O(1) whether depth(X) ≤ k − 1 by Lemma 7 and then can check whether G − X |= ϕ using Observation 1. Based on this, we aim to find X that we call a solution in the same way as for the previously considered problems.
For Elimination Distance-(conn) to ϕ, we used the random separation technique to highlight a big component of G−X (or rather its complement), and for Elimination Distance-(prop) to ϕ, besides a big component we had to highlight some specific small components composing G x together with the big component. Now we are highlighting the small components, X and the neighborhood N G (V (C)) ⊆ X of the big component.
Suppose that (G, k) is a yes-instance with a solution X. By Lemma 6, |X| ≤ p + k and |V (G) \ N G [V (C)]| ≤ p, where C is a big component of G − X. By Lemma 10, we can construct the family F of subsets of V (G) of size at most 2 O((p+k) log(p+k)) ·log n in 2 O((p+k) log(p+k)) ·n log n time such that if (G, k) has a solution X, then F has a set R such that V (H) ⊆ R for every small component and R ∩ X = ∅. Then for every R ∈ F, we aim to find a solution X such that the vertices of the small components of G − X are in R and X ∩ R = ∅.
From this point we assume that R is given. Consider U = V (G) \ R. If C is a big component of a (hypothetical) solution X satisfying the above conditions, then N G (V (C)) ⊆ U and |N G (V (C))| ≤ k. Recall that |X \ N G (V (C))| ≤ |V (G) \ N G [V (C)]| ≤ p. Since |U | ≤ n, applying Lemma 10 for U , we construct the family F of subsets of U of size at most 2 O(min{p,k} log(p+k)) · log n in 2 O(min{p,k} log(p+k)) · n log n time such that F has a set Y with the property that X \ N G (V (C)) ⊆ Y and N G (V (C)) ∩ Y = ∅. We consider all Y ∈ F and try find a solution X such that (i) the vertices of the small components of G − X are in R,
(ii) for the big component C of G − X, X \ N G (V (C)) ⊆ Y , (iii) for the big component C of G − X, N G (V (C)) ⊆ B, where B = V (G) \ (R ∪ Y ).
If a solution X satisfies (i)-(iii), then we say that X is colorful.
We say that the vertices of R are red, the vertices of Y are yellow, and the vertices of B are blue. The components of G[R] are called red and the components of G[R ∪ Y ] are called red-yellow components of G, and we use the same term for the induced subgraphs of G.
Assume that X is a colorful solution. Notice that if H is a red component of G, then either H is a small component of . There are at most n r r-tuples v can be listed n O(|ϕ|) time. Our algorithm returns yes if we find a colorful solution for some choice of v, and it concludes that there is no colorful solution for the considered selection of R otherwise.
G − X with N G (V (H)) ⊆ X or V (H) ⊆ V (C),
From now we assume that v = v 1 , . . . , v r is fixed. Again we observe that these vertices should not belong to X and we recolor them red for the considered choice of R. We use a recursive branching algorithm to find X. The algorithm exploits the subroutine FindX(Z, h), where initially Z = ∅ and h = p + k.
Subroutine FindX(Z, h). • If u j ∈ R for every j ∈ {1, . . . , s}, then stop executing the subroutine.
Set
F := G − Z. 2. If (F, v) |= ϕ[x], depth(Z) ≤ k − 1,
• Otherwise, for every j ∈ {1, . . . , s} such that u j ∈ Y ∪ B, call FindX(Z ∪ {u j }, h − 1).
5. If depth(Z) ≥ k, then for every x ∈ Z such that there is a red-yellow component H of Notice that if the subroutine outputs Z, then we stop the algorithm and report that we found a solution. If we stop in other steps, then we only stop the execution of the subroutine for the current call. The crucial property of the subroutine are proved the following lemma. Since FindX(Z, h) substantially differs from FindC(C, S, h) and FindF(F, S, h), we provide the proof.
Lower bound for Π 3 -formulas
In this section, we complement Theorem 1 by proving that there are formulas in Π 3 for which Elimination Distance-( ) to ϕ is W[2]-hard. We state now Theorem 2 formally.
Theorem 2. For every ∈ {conn, prop, depth}, there is ϕ ∈ Π 3 such that Elimination Distance-( ) to ϕ is W[2]-hard.
Proof. We show that the problems are W[2]-hard for the formula ϕ expressing the property that for every vertex u of a graph, there is a vertex v at distance at most two from u of degree at most one. Notice that the property that a vertex v of a given graph G has degree at most one can be written as follows: for every z 1 , z 2 ∈ V (G), if v is adjacent to z 1 and z 2 , then z 1 = z 2 . Thus, we define the formula
ψ(v, z 1 , z 2 ) = [((v ∼ z 1 ) ∧ (v ∼ z 2 )) → (z 1 = z 2 )]
with three free variables and set
ϕ = ∀x∃y 1 ∃y 2 ∀z 1 ∀z 2 [ψ(x, z 1 , z 2 ) ∨ ((x ∼ y 1 ) ∧ ψ(y 1 , z 1 , z 2 )) ∨ ((x ∼ y 1 ) ∧ (y 1 ∼ y 2 ) ∧ ψ(y 2 , z 1 , z 2 ))]. Clearly, ϕ ∈ Π 3 .
To show W[2]-hardness, we reduce from the Set Cover problem. The problem asks, given a universe U , a family S of subsets of S, and a positive integer k, whether there is S ⊂ S of size at most k that covers U , that is, for every u ∈ U , there is S ∈ S such that u ∈ S. It is well-known that Set Cover is W[2]-complete when parameterized by k [13]. Let (U, S, k) be an instance of Set Cover with U = {u 1 , . . . , u n }, S = {S 1 , . . . , S m }. We also assume that n ≥ 2 and k ≤ m. We construct the following graph G (see Figure 2).
• For every i ∈ {1, . . . , n}, construct k + 2 vertices u (1) i , . . . , u (k+2) i , and then for every i, j ∈ {1, . . . , n} and all p, q ∈ {1, . . . , k + 2} such that (i, p) = (j, q), make u • For every j ∈ {1, . . . , m}, construct three vertices s j , v j , w j and edges s j v j and v j w j .
• For every i ∈ {1, . . . , n} and every j ∈ {1, . . . , m}, make s j adjacent to u We claim that G has a set cover of size at most k if and only if ed ϕ (G) ≤ k for ∈ {conn, prop, depth}. Notice that by the definition of ϕ, H |= ϕ if and only if C |= ϕ for every component C of H. Therefore, ed conn ϕ (G) = ed prop ϕ (G) = ed depth ϕ (G), and it is sufficient to prove that G has a set cover of size at most k if and only if there is an elimination set X of G with depth(G) ≤ k − 1 such that C |= ϕ for every component C of G − X.
Suppose that sets S j 1 , . . . , S j k ∈ S form a set cover. We define X = {w j 1 , . . . , w j k }. Since |X| = k, depth(X) ≤ k − 1. Notice that H = G − X is connected. We claim that H |= ϕ.
Recall that H |= ϕ if and only if for every vertex x ∈ V (H) there is a vertex y ∈ V (H) at distance at most two such that d H (y) ≤ 1. This property trivially holds if x ∈ {s j , v j , w j } \ X for j ∈ {1, . . . , m}. Consider a vertex u (p) i for some i ∈ {1, . . . , n} and p ∈ {1, . . . , k + 2}. We have that there is h ∈ {1, . . . , k} such that u i ∈ S j h . Then u i s j h ∈ E(H). Because w j h ∈ X, we obtain that d H (v j h ) = 1. Since s j h v j h ∈ E(H), v j h is at distance two from u (p) i as required. We conclude that H |= ϕ.
For the opposite direction, assume that there is an elimination set X of G with depth(X) ≤ k−1 such that C |= ϕ for every component C of G−X. Consider Z = {u (p) i | 1 ≤ i ≤ n, 1 ≤ p ≤ k + 2}. Because Z is a clique, we have that |Z ∩ X| ≤ k. To see this, consider a representation (T, α) of X with depth(T ) ≤ k − 1. Then there is a leaf x of T such that α −1 (X ∩ Z) ⊆ A T (x). Since depth(T ) ≤ k − 1, we conclude that |Z ∩ X| ≤ k. Note that the vertices of Z \ X are in the same component H of G − X. Let W = {w 1 , . . . , w m }. By Observation 3, |N G (V (H)) ∩ X| ≤ k. Hence, |N G (V (H)) ∩ (X ∩ W )| ≤ k as well. Let {w j 1 , . . . , w j } = N G (V (H)) ∩ (X ∩ W ). We claim that the sets S j 1 , . . . , S j k cover U .
Consider an arbitrary i ∈ {1, . . . , n}. Because |Z ∩ W | ≤ k, there are two distinct p, q ∈ {1, . . . , k + 2} such that u i . Hence, d H (s h ) ≥ 2. We obtain that v h ∈ V (H) and d H (v h ) ≤ 1 by the construction of G. This means that w h / ∈ H, that is, w h ∈ N G (V (H)) ∩ (X ∩ W ). We conclude that there is t ∈ {1, . . . , } such that j t = h. Finally, because u (p) i is adjacent to s jt , u i ∈ S jt and this concludes the proof.
Discussion
We established a parameterized complexity dichotomy for the elimination problems whose aim is to satisfy a FOL formula ϕ with respect to the quantification structure of the prefix. For this, we considered three variants of the elimination distance to the class of graphs modelling ϕ and defined the Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} corresponding to the considered type of distance. In Theorem 1, we proved that for every FOL formula ϕ ∈ Σ 3 , Elimination Distance-( ) to ϕ is FPT for ∈ {conn, prop, depth}. In Theorem 2, we showed that this result is tight in the sense that there are FOL formulas ϕ ∈ Π 3 such that these problems are W[2]-hard.
Notice that the above dichotomy is the same for all the considered variants of the elimination problems. Moreover, it coincides with the structural dichotomy obtained by for Deletion to ϕ by Fomin, Golovach, and Thilikos in [15]. This leads to the following natural question: is there a FOL formula ϕ such that the parameterized complexity of Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} and Deletion to ϕ differs? In particular, is there a formula ϕ such that Deletion to ϕ is FPT but one of the problem Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} turns to be, say W[1] or W[2]-hard? Note that Lemma 5 holds for every FOL formula ϕ. Thus, solving Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} can be reduced to solving these problems on unbreakable graphs by Theorem 3. Since Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} is somehow similar to Deletion to ϕ on unbreakable graphs, it may happen that Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} are FPT whenever Deletion to ϕ is FPT. However proving this would demand applying different algorithmic tools as our techniques are tailored for ϕ ∈ Σ 3 . Also it would be interesting to know whether there are FOL formulas such that Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} differ from the parameterized complexity viewpoint.
In contrast with the same behaviour of the elimination and deletion problems with respect to the inclusion in FPT, we would like to point that they behave differently with respect to kernelization (we refer to the books [10,16] for the definition of the notion). It was shown in [15] that Deletion to ϕ admits a polynomial kernel for ϕ ∈ Σ 1 ∪ Π 1 (in fact, Deletion to ϕ is polynomial for ϕ ∈ Σ 1 ) and there are formulas ϕ ∈ Π 2 and Σ 2 such that Deletion to ϕ has no polynomial kernel unless NP ⊆ coNP /poly. For the elimination problems, we can show the following lower bound. be a graph and let X ⊆ V (G). We define the torso of X as the graph H obtained from G[X] by making every two vertices u, v ∈ X adjacent if three is a component C of G − X such that u, v ∈ N G (V (C)). Then the following property can be shown by the definition of tree-depth.
Observation 5. For a set X ⊆ V (G) and an integer k, depth(X) ≤ k − 1 if and only if the tree-depth of the torso of X is at most k.
Thus, Elimination Distance-(depth) to ϕ can be stated as follows: given a graph G and a nonnegative integer k, is there X ⊆ V (G) whose torso has the tree-depth at most k such that G − X |= ϕ? In other words, we ask whether there is a set of vertices whose torso has bounded tree-depth such that the graph obtained by the deletion of this set models our formula. Then we can consider the variants of Elimination Distance-(depth) to ϕ for other "width-measures".
For example, what can be said about parameterized complexity of the variant of Elimination Distance-(depth) to ϕ, where the tree-width (see, e.g, [10] for the defintion) of the torso of X should be at most k − 1?
Finally, we believe that it could be interesting to consider yet another variant of the elimination distance. Recall that in the definitions of ed ϕ for ∈ {conn, prop, depth}, we considered properties of the components. In particular, Then we can define the respective Elimination Distance-(part) to ϕ and investigate its parameterized complexity depending of ϕ. Note that our approach for solving the elimination problems fails in this case. In particular, we cannot express the problem using MSOL.
ed conn ϕ (G) = 0, if G |= ϕ, 1 + min v∈V (G) ed conn ϕ (G − v), otherwise, 3
For a set of vertices S ⊆ V (G), we denote by G[S] the subgraph of G induced by the vertices from S. We also define G − S = G[V (G) \ S]; we write G − v instead of G − {v} for a single vertex set. For a vertex v, N G (v) denotes the open neighborhood of v, that is, the set of vertices adjacent to v, and N G
Observation 1 .
1Model Checking for an FOL formula ϕ can be solved in n O(|ϕ|) time.
Lemma 4 .
4Let ϕ be an FOL formula and let G be a connected graph with ed prop ϕ (G) > 0. Let also d be a positive integer. Then ed prop ϕ (G) ≤ d if and only if G contains an elimination set X of depth at most d − 1 with a representation (T, α) such that the following is fulfilled:
Observation 4 .
4Deletion to ϕ and Elimination Distance-( ) to ϕ for ∈ {conn, prop, depth} are equivalent on instances (G, k), where G is a (k + 1)-connected graph.
(G)\N G [V (C)]| ≤ p. To see it, it is sufficient to consider the separation (A, B) of G with A = N G [V (C)] and B = V (G) \ V (C). Clearly, |B \ A| ≤ p and, therefore, |V (G) \ N G [V (C)]| ≤ p. This also implies the uniqueness of a component of G − S with at least p + 1, because for every other component C , we have that V (C ) ⊆ B \ A. This concludes the proof.
Lemma 7 .
7Given a graph G, a set of vertices X ⊆ V (G), and an integer d ≥ −1, it can be decided in|X| O(d) · n O(1) time whether depth(X) ≤ d.We have to solve Elimination Distance-( ) to ϕ for ∈ {conn, prop} on instances of bounded size. It is straightforward to see that this also can be done by backtracking following the definitions of ed conn ϕ and ed prop ϕ .Lemma 8. Let ϕ be a FOL formula. Then Elimination Distance-( ) to ϕ can be solved in n O(k+|ϕ|) time for ∈ {conn, prop}.
Figure 1 :
1A visualization of the set X, the component C, the sets S, and U , and and the way the red and blue colors are distributed among them.
-
. . . , r}, then call FindC(C , S ∪ {u j }, h − 1). If u j ∈ R and there is a red component H of C with the set of vertices W and S = N C (W ) such that (a) u j ∈ W , (b) |S | ≤ h, and (c) there is a component C of C −N C [W ] with v i ∈ V (C ) for all i ∈ {1, . . . , r}, then call FindC(C , S ∪S , h−|S |).
Lemma 11 .
11If X is an inclusion minimal colorful solution to (G, k) with the big component C such that v i ∈ V (C) for all i ∈ {1, . . . , s} and (C, v) |= ϕ[x], then there is a leaf of the search tree produced by FindC(G, ∅, k) for which the subroutine outputs C and S = N G (V (C)). Proof. To prove the lemma, we show the following claim. If the subroutine FindC is called for (C,S,h) such that (a) V (C) ⊆ V (C), (b)S = N G (V (C)), (c)S ⊆ S, and (d)h = k − |S|, then either the subroutine outputsC andS or it recursively calls FindC(C ,S ,h ), where (a ) V (C) ⊆ V (C ), (b )S = N G (V (C )), (c )S ⊆ S, and (d )h = k − |S |. Notice thath = k − |S| ≥ 0, because |S| ≤ k. Hence, if (C, v) |= ϕ[x], then FindC(C,S,h) outputsC andS in the first step, and the claim holds. Assume that (C, v) |= ϕ[x]
F--
= G x and S. Similarly to the subroutine FindC, we construct the subroutine FindF(F, S, h),where initially F = G − w, S = {w}, and h = k − 1. Subroutine FindF(F, S, h). • If (F, v) |= ϕ[x]and h ≥ 0, then return F , S, and stop.• If (F, v) |= ϕ[x] and h ≤ 0, then stop. • If h ≥ 1 and there is an s-tuple u = u 1 , . . . , u s of vertices of F such that (F, vu) |= ϕ[xy], then do the following for every j ∈ {1, . . . , s}. If u j ∈ B and there is an induced subgraph F of F that is the disjoint union of the components of F − u j containing vertices of W and vertices v i for i ∈ {1, . . . , r}, then call FindF(F , S ∪ {u j }, h − 1). If u j ∈ R and there is a red component H of C with the set of vertices Z and S = N F [Z] such that (a) u j ∈ Z, (b) Z ∩ W = ∅ and v i / ∈ Z for all i ∈ {1, . . . , r}, (c) |S | ≤ h, and (d) there is an induced subgraph F of F that is a disjoint union of the components of F − N F [W ] containing vertices of W and vertices v i for some i ∈ {1, . . . , r}, then call FindF(F , S ∪ S , h − |S |).
where C is the big component. Also we have that if H is a red-yellow component of G, then either V (H) ⊆ V (C) or every red component of V (H) is a small component of G−X. These are the crucial properties of colorful solutions exploited by our algorithm. Because G − X |= ϕ for a solution X, it should exist an r-tuple v = v 1 , . . . , v r of vertices of G − X such that (G − X, v) |= ϕ[x]. In the same way as for the previous problem, we use brute force to list all r-tuples v = v 1 , . . . , v r of vertices of G. Then for each v, we check whether there is a colorful solution X such that v i / ∈ X for all i ∈ {1, . . . , r} and (G − X, v) |= ϕ[x]
F
with the properties (i) |N F (V (H))| ≤ k, (ii) |V (H)| ≤ p, (iii) x ∈ V (H), and (iv) N F [V (H)] ∩ (B ∪ Y ) = ∅, set S := N F [V (H)] ∩ (B ∪ Y ) and call FindX(Z ∪ S, h − |S|).
Figure 2 :
2Construction of G for n = 2 and m = 2 with S 1 = {u 1 , u 2 , u 3 } and S 2 = {u 2 , u 3 , u 4 }; for simplicity, just one copy of each u (p) i for p ∈ {0, . . . , k} is shown.
i ∈ S j .
V (H). Since H |= ϕ, there is a vertex z ∈ V (H) at distance at most two from u (p) i such that d H (z) ≤ 1. Since n ≥ 2, we have that |Z \X| ≥ 3 and, therefore,d H (u (p) i ) ≥ 2. Moreover, for every h ∈ {1, . . . , n} and r ∈ {1, . . . , k + 2}, if u (r) h ∈ V (H), then d H (u (r) h ) ≥ 2.Then the construction of G implies that there is h ∈ {1, . . . , m} such that s h ∈ V (H) and u (p) i s h ∈ E(G) with the property that either d H (s j ) ≤ 1 or s j has a neighboor in H of degree at most one. As s h is adjacent to u (p) i , this vertex is adjacent to u (q)
G |= ϕ, 1 + min v∈V (G) ed prop ϕ (G − v), if G |= ϕ and G is connected, max{1, max{ed prop ϕ (C) | C is a component of G}}, otherwise.However, we can consider unions of components instead. We say that graphsG 1 , . . . , G s form a component-partition of G if every component of G is a component of G i for some i ∈ {1, .. . , s} and G is the disjoint union of G 1 , . . . , G s . Then, we can defineed part ϕ (G |= ϕ, 1 + min v∈V (G) ed part ϕ (G − v),if G |= ϕ and G is connected, min{max{1, ed part ϕ (G 1 ), . . . , ed part ϕ (G s )} | G 1 , . . . , G s is a component partition of G} otherwise.
We state the restricted variant of the meta theorem of Lokshtanov et al.[24]. Lokshtanov et al. proved the theorem for structures and counting monadic second-order logic. For us, it is sufficient to state the theorem for graphs and MSOL.Theorem 3 ([24, Theorem 1] ). Let ψ be a MSOL formula. For all q ∈ N, there exists p ∈ N such that if there exists an algorithm that solves Model Checking for ψ on (p, q)-unbreakable graphs in O(n d ) time for some d ≥ 4, then Model Checking for ψ can be solved on general graphs in O(n d ) time.
and h ≥ 0, then return Z, and stop executing the algorithm. 3. If (F, v) |= ϕ[x] and h ≤ 0, then stop executing the subroutine. 4. If h ≥ 1 and there is an s-tuple u = u 1 , . . . , u s of vertices of F such that (F, vu) |= ϕ[xy], then do the following:
If (G, k) has a colorful inclusion minimal solution X with v i ∈ V (G) \ X for all i ∈ {1. Lemma. 16s} such that (G − X, v) |= ϕ[x], then FindX(∅, p + k) returns XLemma 16. If (G, k) has a colorful inclusion minimal solution X with v i ∈ V (G) \ X for all i ∈ {1, . . . , s} such that (G − X, v) |= ϕ[x], then FindX(∅, p + k) returns X.
Let t = p + k. We show that the algorithm maintains the following property: if the subroutine FindX is called for (Z, h) such that (a). The lemma is proved similarly to Lemma 11Proof. The lemma is proved similarly to Lemma 11. Let t = p + k. We show that the algorithm maintains the following property: if the subroutine FindX is called for (Z, h) such that (a)
s} such that u j ∈ X \ X. Because X is a colorful solution, u j ∈ B . Therefore, the subroutine calls FindX(Z , h ) for Z = Y \ {u j } and h = h − 1. It is easy to see that (a ) and (b ) are fulfilled. Assume that (F, v) |= ϕ[x]. Then depth(Z) ≥ k. Because depth(X) ≤ k − 1, X has a representation (T, α) with depth(T ) ≤ k − 1. Because depth(T ) ≤ k − 1 and depth(Z) ≥ k, there are vertices x, y ∈ Z such that the nodes x = α −1 (x) and y = α −1 (y) have the lowest common ancestor z in T such that z = x, y and it holds that α(A T (s)) \ Z = ∅ and α(A T (z)) is an (x, y)-separator in G. In particular, x and y cannot be. Z ⊆ X Z ⊆ X And (b) H = T − |z| ; G − X, V) |= Φ[x], . {1, ; Z = Z ∪ S And H = H − |s|, then either the subroutine outputs Z = X or it recursively calls FindX(Z , h ), where (a ) Z ⊆ X and (b ) h = t − |Z | In the first step, the algorithms sets F = G − Z. If (F, v) |= ϕ[x], depth(Z) ≤ k − 1, h ≥ 0, and (F, v) |= ϕ[x], then Z is a colorful solution and the algorithm returns return Z. Since X is inclusion minimal, we have that X = Z. Thus, the claim holds. As the subroutine calls FindX(Z , h ), we conclude that the claim if fulfilled. Recall that we call FindX(∅, p + k) and note that conditions (a) and (b) are trivially fulfilled for Z = ∅ and h = t. Observe also that in each recursive call of the subroutine the parameter h strictly decreases. Thus, we conclude that we output X is some recursive call of FindX(Z, hZ ⊆ X and (b) h = t − |Z|, then either the subroutine outputs Z = X or it recursively calls FindX(Z , h ), where (a ) Z ⊆ X and (b ) h = t − |Z | In the first step, the algorithms sets F = G − Z. If (F, v) |= ϕ[x], depth(Z) ≤ k − 1, h ≥ 0, and (F, v) |= ϕ[x], then Z is a colorful solution and the algorithm returns return Z. Since X is inclusion minimal, we have that X = Z. Thus, the claim holds. Assume that this is not the case. Since Z ⊆ X, we have that h ≥ 1, that is, the subroutine does not stop in step 3. Clearly, we have that (F, v) |= ϕ[x] and/or depth(Z) ≥ k. Suppose that (F, v) |= ϕ[x]. Then there is an s-tuple u = u 1 , . . . , u s of vertices of F such that (F, vu) |= ϕ[xy]. This means that the subroutine executes step 4. As (G − X, v) |= ϕ[x] and Z ⊆ X, there is j ∈ {1, . . . , s} such that u j ∈ X \ X. Because X is a colorful solution, u j ∈ B . Therefore, the subroutine calls FindX(Z , h ) for Z = Y \ {u j } and h = h − 1. It is easy to see that (a ) and (b ) are fulfilled. Assume that (F, v) |= ϕ[x]. Then depth(Z) ≥ k. Because depth(X) ≤ k − 1, X has a representation (T, α) with depth(T ) ≤ k − 1. Because depth(T ) ≤ k − 1 and depth(Z) ≥ k, there are vertices x, y ∈ Z such that the nodes x = α −1 (x) and y = α −1 (y) have the lowest common ancestor z in T such that z = x, y and it holds that α(A T (s)) \ Z = ∅ and α(A T (z)) is an (x, y)-separator in G. In particular, x and y cannot be both in N G [V (C)]. By symmetry, we can assume that x / ∈ N G [V (C)]. This means that there is a red-yellow component H of F such that properties (i)-(iv) of step 5 are fulfilled. Because X is a colorful solution, we have that S = N F [V (H)] ∩ (B ∪ Y ) ⊆ X. Thus, (a ) and (b ) are fulfilled for Z = Z ∪ S and h = h − |S|. As the subroutine calls FindX(Z , h ), we conclude that the claim if fulfilled. Recall that we call FindX(∅, p + k) and note that conditions (a) and (b) are trivially fulfilled for Z = ∅ and h = t. Observe also that in each recursive call of the subroutine the parameter h strictly decreases. Thus, we conclude that we output X is some recursive call of FindX(Z, h).
Lemma 16 concludes the description of the algorithm and its correctness proof. We summarize and evaluate the running time in the following lemma. Lemma 16 concludes the description of the algorithm and its correctness proof. We summa- rize and evaluate the running time in the following lemma.
Elimination Distance-(depth) to ϕ on (p, k)-unbreakable graphs for ϕ ∈ Σ 3 can be solved in 2 O((p+k)(log(p+k)+p)) · n O(|ϕ|) time. Lemma 17. Lemma 17. Elimination Distance-(depth) to ϕ on (p, k)-unbreakable graphs for ϕ ∈ Σ 3 can be solved in 2 O((p+k)(log(p+k)+p)) · n O(|ϕ|) time.
the problem is solved by brute force in 2 (3p+2k)(p+1) · ((3p + 2k)(p + 1)) O(k+|ϕ|) time. Assume that |V (G)| > (3p + 2k)(p + 1). Then we construct F in 2 O((p+k) log(p+k)) · n log n time. The size of F is at most 2 O((p+k) log(p+k)) · log n, and for every R ∈ F, we construct F in 2 O(min{p,k} log(p+k)) · n log n time. Proof. If |V (G)| ≤ (3p + 2kRecall that the size of F is at mostProof. If |V (G)| ≤ (3p + 2k)(p + 1), the problem is solved by brute force in 2 (3p+2k)(p+1) · ((3p + 2k)(p + 1)) O(k+|ϕ|) time. Assume that |V (G)| > (3p + 2k)(p + 1). Then we construct F in 2 O((p+k) log(p+k)) · n log n time. The size of F is at most 2 O((p+k) log(p+k)) · log n, and for every R ∈ F, we construct F in 2 O(min{p,k} log(p+k)) · n log n time. Recall that the size of F is at most
· log n. Then we consider at most n r r-tuples of vertices v that can be listed in n O(|ϕ|) time. Finally, for every. Y ∈ F O(min{p ; R ∈ F, we call FindX(∅, p + kO(min{p,k} log(p+k)) · log n. Then we consider at most n r r-tuples of vertices v that can be listed in n O(|ϕ|) time. Finally, for every R ∈ F, Y ∈ F , and every v, we call FindX(∅, p + k).
Then we can verify in (p + k) O(k) · n O(1) time whether depth(Z) ≤ k − 1 using Lemma 7. Also we can check whether (F, v) |= ϕ[x] in n O(|ϕ|) by Observation 1. Simultaneously, we find an s-tuple u of vertices of F such that (F, vu) |= ϕ[xy] if this is not the case. In step 4, we perform at most s recursive calls. In step 5, finding H can be done in polynomial time. Notice that we have at most |Z| ≤ p + k recursive calls in this step. The depth if the recursion is upper bounded by k + p. This implies that the running time of FindX(∅, p + k) is (p + k) O(p+k) · n |ϕ|. ; Thus, |z| ≤ P + K, Notice that in each call. Summarizing, we obtain that the total running time is 2 O((p+k)(log(p+k)+p)) · n O(|ϕ|Thus, it remains to evaluate the running time of FindX(∅, p + k). Notice that in each call, |Z| ≤ p + k. Then we can verify in (p + k) O(k) · n O(1) time whether depth(Z) ≤ k − 1 using Lemma 7. Also we can check whether (F, v) |= ϕ[x] in n O(|ϕ|) by Observation 1. Simultaneously, we find an s-tuple u of vertices of F such that (F, vu) |= ϕ[xy] if this is not the case. In step 4, we perform at most s recursive calls. In step 5, finding H can be done in polynomial time. Notice that we have at most |Z| ≤ p + k recursive calls in this step. The depth if the recursion is upper bounded by k + p. This implies that the running time of FindX(∅, p + k) is (p + k) O(p+k) · n |ϕ| . Summarizing, we obtain that the total running time is 2 O((p+k)(log(p+k)+p)) · n O(|ϕ|) .
Proposition 1. There are formulas ϕ ∈ Π 1 such that Elimination Distance-( ) to ϕ do not admit polynomial kernels unless NP ⊆ coNP /poly for ∈ {conn. prop, depth}Proposition 1. There are formulas ϕ ∈ Π 1 such that Elimination Distance-( ) to ϕ do not admit polynomial kernels unless NP ⊆ coNP /poly for ∈ {conn, prop, depth}.
We show the claim for the formula ϕ expressing the property that a graph has no triangles, that is, cycles of length three: ϕ = ∀x∀y∀z [(x = y) ∨ (y = z) ∨ (x = z) ∨ ¬(x ∼ y) ∨ ¬. y ∼ z) ∨ ¬(x ∼ z)Proof. We show the claim for the formula ϕ expressing the property that a graph has no triangles, that is, cycles of length three: ϕ = ∀x∀y∀z [(x = y) ∨ (y = z) ∨ (x = z) ∨ ¬(x ∼ y) ∨ ¬(y ∼ z) ∨ ¬(x ∼ z)].
Then for ∈ {conn, prop}, we have that (G, k) is a yes-instance of Elimination Distance-( ) to ϕ if and only if (G j , k) is a yes-instance of Elimination Distance-( ) to ϕ for every j ∈ {1, . . . , t}. Then by the result of Bodlaender, Jansen, and Kratsch [3] (see also [16, Part III] for the introduction to the technique), Elimination Distance-( ) to ϕ does not admit a polynomial kernel unless NP ⊆ coNP /poly. For Elimination Distance-(depth) to ϕ, consider G that is the disjoint union of G and K k+3 . Clearly, ed depth ϕ (K k+2 ) = k + 1. Then by Lemma 2, (G , k + 1) is a yes-instance of Elimination Distance-(depth) to ϕ if and only if (G j , k) is a yes-instance of Elimination Distance-(depth) to ϕ for every j ∈ {1, . . . , t}. This implies that Elimination Distance-(depth) to ϕ has no polynomial kernel unless NP ⊆ coNP /poly. Notice that Proposition 1 does no exclude existence of Turing kernels. ; . A Agrawal, L Kanesh, F Panolan, M S Ramanujan, S Saurabh, 38th International Symposium on Theoretical Aspects of Computer Science, (STACS). 18714This makes it natural to ask whether Elimination Distance-( ) to ϕ admit polynomial Turing kernels for ϕ ∈ Σ 3 for ∈ {conn, prop, depth}. We defined the depth of a set X ⊆ V (G) using a representation. However, there is an equivalent definition that uses the notion of tree-depth (see, e.g., [5] for the definition. An FPT algorithm for elimination distance to bounded degree graphs. Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2021, pp. 5:1-5:11. 4, 9It is straightforward to see that G |= ϕ if and only if G has no triangles. By the classical results of Lewis and Yannakakis [22], Deletion to ϕ is NP-complete. Then it is easy to observe that the problem remains NP on instances (G, k), where G is a (k + 1)- connected graph. For example, we can reduce from Deletion to ϕ on general graphs. Let G be an n-vertex graph. We assume that k < n − 1 as otherwise the problem is trivial. We construct the graph G from G by adding k + 1 copies of the complete bipartite graph K n,n and making each vertex of one part of the vertex partition to a unique vertex of G. Clearly, G is (k +1)-connected and it is easy to see that G−X is triangle-free if and only if G has no triangles for every X ⊆ V (G ). This proves the NP-hardness for Deletion to ϕ on (k + 1)-connected graphs. Then Observation 4 implies that Elimination Distance-( ) to ϕ is NP-complete for every ∈ {conn, prop, depth}. Let (G 1 , k), . . . (G t , k) be instances of Elimination Distance-( ) to ϕ for some ∈ {conn, prop, depth}. Let G be the disjoint union of G 1 , . . . , G t . Then for ∈ {conn, prop}, we have that (G, k) is a yes-instance of Elimination Distance-( ) to ϕ if and only if (G j , k) is a yes-instance of Elimination Distance-( ) to ϕ for every j ∈ {1, . . . , t}. Then by the result of Bodlaender, Jansen, and Kratsch [3] (see also [16, Part III] for the introduction to the technique), Elimination Distance-( ) to ϕ does not admit a polynomial kernel unless NP ⊆ coNP /poly. For Elimination Distance-(depth) to ϕ, consider G that is the disjoint union of G and K k+3 . Clearly, ed depth ϕ (K k+2 ) = k + 1. Then by Lemma 2, (G , k + 1) is a yes-instance of Elimination Distance-(depth) to ϕ if and only if (G j , k) is a yes-instance of Elimination Distance-(depth) to ϕ for every j ∈ {1, . . . , t}. This implies that Elimination Distance-(depth) to ϕ has no polynomial kernel unless NP ⊆ coNP /poly. Notice that Proposition 1 does no exclude existence of Turing kernels (we again refer to [10, 16] for the definition of the notion). This makes it natural to ask whether Elimination Distance-( ) to ϕ admit polynomial Turing kernels for ϕ ∈ Σ 3 for ∈ {conn, prop, depth}. We defined the depth of a set X ⊆ V (G) using a representation. However, there is an equivalent definition that uses the notion of tree-depth (see, e.g., [5] for the definition). Let G References [1] A. Agrawal, L. Kanesh, F. Panolan, M. S. Ramanujan, and S. Saurabh, An FPT algorithm for elimination distance to bounded degree graphs, in 38th International Symposium on Theoretical Aspects of Computer Science, (STACS), vol. 187 of LIPIcs, Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2021, pp. 5:1-5:11. 4, 9, 14
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"Adapting the Predator-Prey Game Theoretic Environment to Army Tactical Edge Scenarios with Computational Multiagent Systems"
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"Derrik E Asher \nComputational & Information Sciences Directorate\nArmy Research Laboratory\nUS\n",
"Erin Zaroukian \nComputational & Information Sciences Directorate\nArmy Research Laboratory\nUS\n",
"Sean L Barton \nComputational & Information Sciences Directorate\nArmy Research Laboratory\nUS\n"
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"Computational & Information Sciences Directorate\nArmy Research Laboratory\nUS",
"Computational & Information Sciences Directorate\nArmy Research Laboratory\nUS",
"Computational & Information Sciences Directorate\nArmy Research Laboratory\nUS"
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The historical origins of the game theoretic predator-prey pursuit problem can be traced back to Benda, et al., 1985 [1]. Their work adapted the predator-prey ecology problem into a pursuit environment which focused on the dynamics of cooperative behavior between predator agents. Modifications to the predator-prey ecology problem[2]have been implemented to understand how variations to predator [3] and prey[3][4][5]attributes, including communication [6], can modify dynamic interactions between entities that emerge within that environment[7][8][9]. Furthermore, the predator-prey pursuit environment has become a testbed for simulation experiments with computational multiagent systems[10][11][12]. This article extends the theoretical contributions of previous work by providing 1) additional variations to predator and prey attributes for simulated multiagent systems in the pursuit problem, and 2) military-relevant predator-prey environments simulating highly dynamic, tactical edge scenarios that Soldiers might encounter on future battlefields. Through this exploration of simulated tactical edge scenarios with computational multiagent systems, Soldiers will have a greater chance to achieve overmatch on the battlefields of tomorrow.
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10.29007/dlq7
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"https://arxiv.org/pdf/1807.05806v1.pdf"
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1807.05806
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6d52f009d250388bd78a00a0ab6d3fe63871508a
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Adapting the Predator-Prey Game Theoretic Environment to Army Tactical Edge Scenarios with Computational Multiagent Systems
Derrik E Asher
Computational & Information Sciences Directorate
Army Research Laboratory
US
Erin Zaroukian
Computational & Information Sciences Directorate
Army Research Laboratory
US
Sean L Barton
Computational & Information Sciences Directorate
Army Research Laboratory
US
Adapting the Predator-Prey Game Theoretic Environment to Army Tactical Edge Scenarios with Computational Multiagent Systems
The historical origins of the game theoretic predator-prey pursuit problem can be traced back to Benda, et al., 1985 [1]. Their work adapted the predator-prey ecology problem into a pursuit environment which focused on the dynamics of cooperative behavior between predator agents. Modifications to the predator-prey ecology problem[2]have been implemented to understand how variations to predator [3] and prey[3][4][5]attributes, including communication [6], can modify dynamic interactions between entities that emerge within that environment[7][8][9]. Furthermore, the predator-prey pursuit environment has become a testbed for simulation experiments with computational multiagent systems[10][11][12]. This article extends the theoretical contributions of previous work by providing 1) additional variations to predator and prey attributes for simulated multiagent systems in the pursuit problem, and 2) military-relevant predator-prey environments simulating highly dynamic, tactical edge scenarios that Soldiers might encounter on future battlefields. Through this exploration of simulated tactical edge scenarios with computational multiagent systems, Soldiers will have a greater chance to achieve overmatch on the battlefields of tomorrow.
Background
The predator-prey paradigm originally emerged from the field of ecology and was analyzed through a series of differential equations describing population dynamics among at least two species (often predator and prey) [13]. Over time, this evolved into a framework for investigating environmental and spatial contributions towards behavioral dynamics. From 1925 to 1966 modifications to the predator-prey model resulted in the emergence of functional predator behavior dependent on prey death rate and prey population density [14][15][16]. These modifications introduced the first inferred spatial component to the predator-prey paradigm. In 1985, the predator-prey model was extended to the problem of individual pursuit (i.e., focusing on individual interactions rather than population dynamics). This shift was aimed at exploring cooperative behavioral dynamics in multi-agent systems [1]. Since then, the predator-prey pursuit problem has begun to see use as a benchmark for testing multiagent algorithms, due to the inherent competitive and cooperative elements intrinsic to its design [17].
The typical predator-prey pursuit environment consists of multiple predators and a single prey moving around in a 2-D confined arena either discretely (one space at a time in either up, down, left, or right directions) or continuously (smooth continuous movement in any direction) for a fixed duration [7,11,12,[17][18][19][20][21][22]. The goals of the predators are in direct competition with those of the prey. The predators' shared goal is to come in contact with (continuous movement) or settle adjacent to (discrete movement) the prey, while the prey's goal is to avoid contact or adjacency with all predators (provided the prey moves). This creates an interesting dichotomy of competition between species (predator and prey), while promoting cooperation, coordination, and collaboration within the predator species. Although competition weighed against cooperation within the predator group has been explored [20], the majority of studies utilize the predator-prey pursuit environment to investigate team dynamics with respect to a shared goal.
The original predator-prey pursuit environment ( Figure 1) required four predator agents to surround the single prey agent from four directions in a discretized grid world [1,9,21,23]. The predator agents were guided by an algorithm and their movements were limited to one grid square per time step to an adjacent available square (not occupied by another agent or a boundary) in only the vertical or horizontal directions (no diagonal movements). The prey agent was restricted to the same criteria and guided by random movement. The goal of their simulation experiments was to show the impact that varying degrees of agent cooperation and control had on the efficiency of prey capture. This first rendition of the predator-prey pursuit problem introduced an environment to test the effectiveness of an algorithm to cooperate in a well-constrained domain. Cooperative algorithms inherently enable predators to collaborate [12,[24][25][26][27][28]. However, under certain conditions, predators using a greedy strategy may have greater success [29], though cooperative algorithms often win out when a sophisticated prey is faced [23], and in some environments prey benefit most from a mix of greedy and altruistic strategies [20]. These algorithmic explorations of multi-agent cooperation show how the predator-prey pursuit environment provides an ideal testbed for understanding collaborative agent behavior.
Since the origination of the predator-prey pursuit environment, many studies have leveraged manipulations to the pursuit environment to investigate human and artificial intelligence behavior in multiagent systems. The next section explores many of these predator-prey environmental manipulations along with the experimenters' goals to provide a foundation for simulating tactical edge scenarios.
Modifications to the Predator-Prey Pursuit Environment
Many multiagent research efforts utilize the original discretized predator-prey pursuit environment shown in Figure 1, implementing a toroidal grid world and requiring the single prey to be blocked on all sides by four predators [7,21,22]. These studies tend to focus less on the structure of the environment and more on how specific sets of predator strategies impact cooperation and teamwork in homogenous and heterogeneous groups of predators. Whereas this foundational work is important, the remainder of this section will discuss modifications to the predator-prey pursuit environment and their implications.
Changes in Task Constraints
One of the simplest modifications of the original predator-prey pursuit task is to change the constraints of the task in order to address specific questions or increase task realism. The most straight forward example of this is pursuit environments that are made more complex through the use of continuous spaces. While in grid worlds the prey are typically considered caught when surrounded on four sides by predators (Figure 1), continuous environments require a predator to come within some small distance of [30], or to "tag" (touch or overlap in center of mass) [17,31] they prey. While discrete environments are typically easier to analyze and reason over, it is worth recognizing that most real-world tasks take place in a continuous domain. Thus, the application of theoretical and empirical revelations that emerge from continuous (or other more realistic) predator-prey pursuit environments will aid our understanding of real life pursuit problems.
As an intermediate step, some authors have extended (while maintaining) the discrete environment with diagonal movements [32], or more complex cell-shapes (such as hexagons [18,32], or irregular convex cells [33]). The timing of predator and prey movement has also been manipulated, with the standard parametrization allowing agents to move synchronously per time step [32], alternating agents' movements through a sequence of time steps (e.g., turn-taking as described in [34]), or allowing more realistic unrestricted, asynchronous movement [23] such that predators and prey can react to each other at flexible time intervals.
Manipulations to environment bounding have illuminated a dependence between structure and predator pursuit strategies. Unbounded non-toroidal environments have been used, including unbounded planes where pursuers move along an unbounded curvature [35], as well as bounded environments, e.g., [36] where traffic in a police chase is restricted to a closed grid of streets or [24] where encountering the edge of the environment results in death. These manipulations have clear implications for pursuit and evasion strategies, since a non-toroidal unbounded environment will allow fast enough prey to continue indefinitely in a given direction and bounded environments contains corners in which prey can be trapped.
Changes in the Number of Agents
The most frequent predator-prey pursuit environment modifications have been to vary the number of agents. In one study, the number of pursuers (predators) was varied between one and two in a discrete environment with block obstacles to investigate how different agent learning parameters (Q-learning: learning rate, discount factor, and decay rate), implemented into both the predator and prey (in one condition), alter evader (prey) capture time [19]. In other work, the number of competitive (egoistic) and collaborative (altruistic) predator agents was varied along with the total number of predators (up to 20) to understand how different sizes of homo-and heterogeneous egoistic and altruistic groups of predators catch a single prey [20]. Together, this work demonstrates how simply changing the number of predators and including various types of obstacles can illuminate aspects of collaborative behavior while simultaneously testing the effectiveness of different algorithmic approaches.
Modifications of the External Environment
Obstacles in the pursuit environment can take on various attributes that force agents to adapt and develop more sophisticated behaviors to achieve the task goal. Typically, these obstacles are static and must be circumnavigated as was described previously [17,28,30,37], but they may take on attributes that disrupt or eliminate an agent (predator or prey) [24]. In one research effort, three pursuers (predators) needed to collaborate in a 2-D environment with complex maze-like obstacles (branching obstacles) to capture an intruder (prey) before it escaped [37]. The goal of this particular study was to identify an optimal strategy that emphasized group reward over individual reward, essentially discovering optimal collaboration within their environmental. Dynamic obstacles have also been utilized to investigate a predator-prey like task where a police chase was modulated with varying degrees of street traffic [36]. In general, these studies found that successful agent behavior was dependent on the attributes of the obstacles.
Changes in Agent Capabilities
Similar to the inclusion of obstacles in the pursuit environment, restricting how far agents (predators and prey) can see necessitates changes to agent behavior to achieve task success. Some research allows the agents to be omniscient, where all entities' positions are known at every time point [17,35], whereas other studies permit agents to see in a straight line until an occlusion is encountered [33,[38][39][40]. Some allow predators and prey to see only within a given range around themselves [21,30,41] or apply random limitations to predator vision [42]. While some research explores predators with limited sensing abilities, these studies allow information to be shared among predators [21,36], either through direct communication (e.g., police radio, [36]) or indirectly (e.g., by leaving cues such as pheromone trails in the environment) [26]. Other work has explored an imbalance between sensing abilities of the predators and prey. For example, a predator may see a prey from a greater distance than the prey can detect, reflecting a more realistic scenario where the predator is on the hunt for a non-vigilant prey [21]. Other work has further modified agent vision or sensing abilities by testing something akin to sound, where a predator agent can hide around corners [43]. Together, these studies show the importance of agents' sensing capabilities and how manipulations illuminate the dependence between pursuit strategies and these capabilities.
Limitations to agents' sensing capabilities in simulation experiments naturally extends the pursuit problem into the physical domain with robots. A common physical limitation of a robotic agent is the field of view [44]. The field of view is dependent on two main factors, 1) the distance between the visual system and the object, and 2) the degrees of visual angle the system can see (the average human can see approximately 210 degrees of visual angle). Given that robotic systems need to have some visual representation of the environment around them for obstacle avoidance and navigation, certain limitations to field of view can cause a catastrophic failure (possibly disabling or destroying the robot). Therefore, it is of critical importance to understand how manipulations to field of view effect robotic systems' pursuit behavior.
Additional Dimensions
In general, 3-D environments provide an opportunity for more complex behavioral strategies, such as those required in aviation, aquatics, and for traversing uneven surfaces [31,39]. Through investigation of these various 3-D environments, a natural emergence of behavioral strategies that depend on the terrain can be discovered. A physical example of this phenomenon is found in the attack and evasion strategies for spiders and crickets as forest leaf litter change geometries between winter and summer [45].
Adaptation of the Predator-Prey Pursuit Environment to Tactical Edge Scenarios
The predator-prey pursuit environment has been used by researchers since 1985 [1] to investigate a host of topics including various aspects of group dynamics [25,46], pursuit strategies [3,47], escape/evasion strategies [27,41,48], and multi-agent systems [18][19][20]49] to name a few. The last portion of this article discusses a set of proposed manipulations to the predator-prey pursuit environment to investigate potential simulated tactical scenarios that easily map to a physical domain.
Impact of Spatial Constraints on Strategies and Behavior
The predator-prey environment size, or the space available for agents (predators and prey) to move around in, has been manipulated to understand how environmental geometries [18,32] and sizes [50] influence agent behavior in discrete spaces. Building upon this body of work and extending into the continuous domain [17], it would be of interest to start with a trivial minimized space, such that the predators always catch the prey within a short period of time, and incrementally increase the size of the environment to gain a quantitative understanding of how environment size impacts prey catch times as a function of increasing environment size. We would expect that prey catch time would increase with a power function of the dimensions of the environment, with a certain size resulting in a near-zero probability of catch. The dimensional expansion can provide a fundamental understanding of task difficulty with respect to environment size. In a simulated tactical scenario, such an understanding would allow us to better assess the probability of acquiring a moving target given the estimated escape space available. This could also lead towards valuable insight into the type of agent capabilities needed to estimate a high probability of mission success (e.g., small and fast reconnaissance agents).
Modifying Predator Capabilities
While holding all other possible manipulations constant, modifications to agent velocity or rate of predator agent movement in a continuous space introduces many additional degrees of freedom and provides an opportunity to simulate tactical teams with homo-and heterogeneous capabilities. A systematic sweep through a range of velocities for a set of predator agents can provide a valuable mapping between agent velocity and mission success (i.e., prey catch time). However, it is important to note that changes to predator velocity are relative to prey velocity and should likely be thought of as a ratio. With that stated, a simultaneous change to all predator agents' velocities (increase or decrease) relative to the prey agent will result in an understanding of the relationship between homogeneous alterations to a team's capabilities and mission success. Similarly, heterogeneous manipulations to the predator agents' velocities relative to the prey agent in a continuous predator-prey pursuit environment can provide an estimate of mission success for the various capabilities (velocity sweep across all predators) in this simulated target acquisition tactical scenario.
On the other side of the proverbial coin, holding predator attributes constant and applying manipulations to the prey's velocity introduces additional dimensionality to the task domain and may result in the need for more complex collaborative behavior to achieve predator agent mission success. As was suggested for the predator agents, a velocity sweep for the prey would dictate task difficulty (easy for low velocities and hard for high velocities) and should result in a spectrum of competitive to collaborative predator agent behaviors for low to high velocities respectively. It is important to note that modifications to the prey's velocity would need to be relative to the predator agents' velocities. We would expect that manipulations from low to high predator to prey velocity ratio would result in the shifts from easy to hard for task difficulty and competitive to collaborative predator agent team behavior. These simulation experiments might represent the differences between having a homo-or heterogeneous teams of slow, heavy, powerful assets (e.g., tanks), camouflaged insurgent ground assets, or drones in reconnaissance or target acquisition scenarios.
Modifying Prey Capabilities
In the prey manipulation domain, introducing multiple prey in various forms could change the task goals entirely. The inclusion of a second prey expands the dimensionality of the task domain to include homo-and heterogeneous adversarial dynamics (two prey agents forming an adversarial team), a potential decoy (catching the decoy prey agent does not complete the mission), and variable temporal mission objective windows (the predator agents must coordinate to catch both prey agents within a preselected duration). The inclusion of additional prey agents (number of prey agents > 2) can further increase the complexity of the task domain, possibly to the point of which the probability of simulated tactical mission success goes to zero. Mission failure is important to explore, especially in simulated environments, in order to maximize the probability of mission success in the multi-domain battlespace.
Modifications to coordinate locations in the simulated environment, while holding all agent attributes (both predator and prey) constant, permits an investigation of degraded agent capabilities. Coordinate manipulations can take the form of static impassible barriers that represent buildings/walls/obstacles, patches that induce injury by temporarily (short duration) or permanently (remaining duration) reducing/minimizing/stopping agents' (predator and/or prey) movements or small environmental regions with simulated hidden explosives that completely remove an agent from further participation in the mission. Other coordinate manipulations are possible (e.g., teleport agents randomly around the environment), but they might not have an easily identifiable correspondence to tactical scenarios. Therefore, environmental manipulations that easily map to tactical scenarios can provide an estimate of the relationship between agent capability degradation and mission success.
Conclusion
Complexification of the predator-prey pursuit environment to include modifications that easily map to simulated tactical scenarios allows for the adaptation of computational agents to these domains. This line of predator-prey pursuit research can be extended to a physical environment that accommodates the testing of robotic platforms working with Soldiers in target acquisition training drills, for the eventual implementation on the multi-domain battlefield.
Figure 1 .
1Typical predator-prey pursuit environment utilized a discretized grid world bounded on all sides. The goal was to have four predator agents (green blocks) surround the prey agent (red block) on four sides (top, bottom, left, and right). Agents could only move one square at a time and neither could move diagonally nor catch the prey from an adjacent diagonal location.
AcknowledgementsResearch was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-17-2-0003. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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A heuristic approach for solving decentralized-pomdp: Assessment on the pursuit problem. I Chades, B Scherrer, F Charpillet, I. Chades, B. Scherrer, and F. Charpillet, "A heuristic approach for solving decentralized-pomdp: Assessment on the pursuit problem." pp. 57-62.
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[] |
[
"Swarmrobot.org -Open-hardware Microrobotic Project for Large-scale Artificial Swarms",
"Swarmrobot.org -Open-hardware Microrobotic Project for Large-scale Artificial Swarms"
] |
[
"Serge Kernbach [email protected] \nInstitute of Parallel and Distributed High-Performance Systems\nUniversity of Stuttgart\nUniversitaetsstr. 3870569StuttgartGermany\n"
] |
[
"Institute of Parallel and Distributed High-Performance Systems\nUniversity of Stuttgart\nUniversitaetsstr. 3870569StuttgartGermany"
] |
[] |
The purpose of this paper is to give an overview of the openhardware microrobotic project swarmrobot.org and the platform Jasmine for building large-scale artificial swarms. The project targets an open development of cost-effective hardware and software for a quick implementation of swarm behavior with real robots. Detailed instructions for making the robot, open-source simulator, software libraries and multiple publications about performed experiments are ready for download and intend to facilitate exploration of collective and emergent phenomena, guided self-organization and swarm robotics in experimental way.
| null |
[
"https://arxiv.org/pdf/1110.5762v1.pdf"
] | 17,915,732 |
1110.5762
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29ad8a185dda6130354b9320f7fffbaf956e8dc4
|
Swarmrobot.org -Open-hardware Microrobotic Project for Large-scale Artificial Swarms
Serge Kernbach [email protected]
Institute of Parallel and Distributed High-Performance Systems
University of Stuttgart
Universitaetsstr. 3870569StuttgartGermany
Swarmrobot.org -Open-hardware Microrobotic Project for Large-scale Artificial Swarms
The purpose of this paper is to give an overview of the openhardware microrobotic project swarmrobot.org and the platform Jasmine for building large-scale artificial swarms. The project targets an open development of cost-effective hardware and software for a quick implementation of swarm behavior with real robots. Detailed instructions for making the robot, open-source simulator, software libraries and multiple publications about performed experiments are ready for download and intend to facilitate exploration of collective and emergent phenomena, guided self-organization and swarm robotics in experimental way.
Introduction
The high miniaturization degree of the robotic systems with increasing efficiency of the hardware and software represents an important trend in the area of collective systems [1]. The subarea -swarm robotics -is an emerged research field focused on designing collective intelligent systems comprised by a large number of robots. Its fascination and theoretical foundations originate from studying and understanding a group behavior of animals and insects societies [2]. It is expected that in a similar way a group of relatively simple and cheap robots can solve complex tasks that are beyond the capabilities of a single robot [3].
Natural physical and biological systems, being a metaphor for the phenomenon of self-organization, can organize themselves to emerge complex macroscopic behavior or spatiotemporal ordered structures and formations [4]. There is no central element coordinating the system, a group behavior (macroscopic level) emerges from individual set of local rules (microscopic level) via interactions and cooperation. However, natural systems have multiple capabilities of interactions, perception and communication, which are significantly more complex than those in technical systems. Microrobots, due to a small size, are very restricted in locomotion, sensing and communication. Therefore a swarm-like behavior, expected from microrobotic systems, can "approach" a complexity of natural phenomena only when the robots are specifically developed, equipped and highly optimized for the desired collective activities [5].
Being motivated by the challenge of creating artificial self-organizing and emergent phenomena, we started open-hardware microrobotic research project arXiv:1110.5762v1 [cs.RO] 26 Oct 2011 for developing a specialized robot platform for large-scale swarms. It is based on the concept of swarm embodiment and is completely open at www.swarmrobot.org, see Fig. 1, for contributions from the domains of electrical and mechanical engineering, computer science and biology. The project consists of several parts: a development of cheap and reliable hardware platform, an integrated SDK, open-source simulator and a description of the preformed experiments. All of them should allow a quick implementation of swarm behavior with real robots, exploration of collective phenomena emerged by artificial self-organization and may be of interest for broad areas of research and education. Since the project is maintained by universities, we can neither manufacture the robots for other participants nor perform any commercial activities with the platform -the project swarmrobot.org targets primarily scientific goals.
This paper is structured in the following way: Section 2 briefly describes the concept of swarm embodiment, which underlies the development. Sections 3, 4 and 5 survey hardware, software and simulator for the robot platform. Finally, in Section 6 we overview performed experiments with Jasmine robots.
Concept of swarm embodiment
Swarm embodiment represents one of the main motivations for starting an openhardware microrobotic development. The origin of this concept lies in the phenomenon of artificial self-organization (SO). Artificial SO differs from natural one in a couple of essential points, where a purposeful character of SO (so-called guided self-organization) represents the most important aspect. It means that a developer creates artificial SO to achieve some desired emergent effects, such as a specific group behavior of robots or desired molecular self-assembled structures.
Artificial SO is created by local rules. Each participant, driven by these local rules, demonstrates an individual behavior. All individual behaviors are macroscopically observable in the form of a coherent collective behavior. From the viewpoint of an observer, the collective behavior is emerged from interactions among swarm agents, which are guided by local rules. Derivation of local rules represent a serious problem, which is analytically non-solvable as proved by Poincare for the physical case of N-bodies interaction. There are three main strategies to derive (to approximate) such rules in non-analytical way. At the bottom-up strategy, the local rules are first programmed into each agent [6]. Performing many simulations and gradually changing the local rules, a desired collective behavior can be derived. Using the top-down strategy, the derivation of local rules starts from a definition of the macroscopic pattern and the corresponding constraints. Then, using "distributing" transformation or evolutionary strategies, this macroscopic pattern can be transformed into a set of local rules, that in turn generate the desired pattern [7]. The last way consists in observing the behavior of already existed artificial and natural swarm-like systems and in trying to reproduce this behavior [2]. All these ways lead finally to the local rules that control the behavior of each swarm agent.
The researcher, once derived these local rules and intended to implement them in a real robotic or in a simulation system, can face the questions "which general degree of collective intelligence is feasible in the destined system? " and "how to implement it with the obtained local rules? " To illustrate this point, we collect in Table 1 several swarm activities that robots can collectively perform. We can roughly say, that these collective activities represent some building blocks for purposeful design of the swarm intelligence. Let us take the most simple example of spatial orientation and assume a robot has found a "food source" being relevant for the whole swarm. This robot, guided by local rules, sends the following message: "I, robot X, found Y, come to me". Other robots, receiving this message can propagate it further through a swarm, so that finally each robot knows "there is a resource Y at the robot X". However, none of robots can find it because they do not know a coordinate of this "food source". The robot X cannot provide these coordinates because it does not know its own position. In this way, even possessing corresponding local rules and capabilities to communicate, robot cannot execute the desired collective activity "find food source" without some additional efforts intended to localization. Considering the localization problem, we face in turn the next generation of questions "how to implement it? " In this way, the original problem of local rules, generating a desired self-organization, changes into the problem of their implementation in real systems.
More generally, during a derivation of local rules R k we assume some basic functionality F b , like message transmission, localization or environmental sensing. Often we do not take into account real restrictions underlying this basic functionality or they are generally unknown at this step. However, implementing later these R k , we obtain the swarm behavior, that differs from our expectations (even with correctly derived R k ), because a real functionality F b can essentially differ from the expected one.
To get round this problem, we involved the embodiment concept (see Fig. 2). This says the same functionality can be implemented in many different ways: "intelligent behavior" can even be implemented when using only some properties of materials [8]. Embodied functionally is also often implemented in some "unusual" way. For example a robot can estimate a distance to neighbors by sending an IR-impulse and measuring a reflected light. However, distances can also be obtaining during communication by measuring a signal intensity. This simple mechanism saves time and energy: such an unusual functionality is a typical sight of embodiment.
The embodiment in our context means the system possesses the desired functionality F b , but this functionality is in a latent form, "it is not appeared". This offers a way of how to get basic functionality for the local rules R k : the local rules have to influence the hardware development of a robot. The swarm embodiment takes then the following form: definition of the macroscopic pattern Ω and the corresponding microscopic/macroscopic constraints; derivation of the local rules R k ; trade off between required functionality and adjustment of hardware; change of the hardware. The local rules have always been considered as a pure software components, however now they are a combination between software and hardware. We can say that in this way the local rules for the whole swarm behavior are embodied into each individual robot. The embodiment in sense "hardware → rules" has been demonstrated e.g. in works [7] [9], where we analyzed a dependence between agent's movement and sensor data for the derived S k . Optimizing local rules to specific motion system, the "top-down"-derived rules can be 5-20% more efficient than corresponding "bottom-up" rules. The embodiment in sense "rules → hardware" has been demonstrated e.g. in works [10], [11]. In these works we considered context awareness related collective capabilities of interacting robotic group and incorporate several local rules into specific sensor system of real microrobots. The achieved results improve collective robotic behavior and reduce required communication and computational efforts.
After those experiments we came to conclusion that swarm robot differs from a general-purpose robot, even when this robot is small enough. The swarmcapable robot should not only incorporate in its own construction some specific details, such as capabilities of swarm communication or cooperative actuation, but be open for embodiment of different local rules. It should have some openhardware structure. This was the main motivation to derive new microrobotic platform for large-scale swarm experiment and make it in open-hardware manner.
Open-hardware development
The design of the robot, assembling instructions, electrical schemes and PCB can be found at www.swarmrobot.org. Here we briefly sketch several main points:
1. One of the most important requirements is that the microrobot platform for large-scale swarm experiments should be cheap. We require that the swarm robot should cost at most 100 euro. Obviously, this requirement imposes several compromises on the capabilities of the platform.
2. The robot should be small, no larger than a cube with the edge of 30 mm. This size hardly limits hardware, so that we come closely to real "limited swarm agent". 3. It should be easy in assembling so that each research institute or even private persons can produce enough robots for their own swarms. This means primarily avoiding SMD soldering and a high-precision micro-mechanical assembling. 4. The robot should have enough hardware resources (ROM, RAM, MPU frequency) to maintain at least a simple operational system with users-defined software. 5. Rich communication and perception capabilities oriented for large-scale swarms. This means: omni-directional communication system; communication radius, which is large enough for a reasonable information transfer in a swarm; support for context-based features; perception system, which allow recognition of main object features such as geometry or colors. 6. The running time without a new recharging should be at least 1-2 hours for the hardest conditions (full speed motion, continuous communication and perception); 7. The robot has to be extendable (modular) for other sensors/actuators/ communication boards with different embodiment rules. This means: standardized inter-board logical, mechanical and electrical interfaces; standardized robots size and structural construction.
General design of the microrobot is limited by the size and geometrical configurations of used components. Since we would like to have a robot that is small as possible, but still cheap and easy in assembling, the most efficient structure in this case is ether an integration of all components on one PCB (flexible PCB) or a "sandwich design" (see Fig. 3). Analyzing sensors for sensing and com- munication we come at a conclusion that SMD sensors are not really suitable. Thus, an integration of all components into one PCB is not possible and so only "sandwich design" can be used. Order of "sandwich layers" is defined by the size of accumulator: it can be placed only between the motor PCB and sensors board. All layers of this "sandwich" can be covered by a thin plastic or, more sophisticated, by metal chassis. This design is simple for assembling, cheap and protect electronics and mechanics from possible damages. The robot uses two GM15(RM-N1) motors with gear motor-wheel coupling, one Li-Po accumulator cell C=250mA/h with 3.7V, electronics of the robot works with 3V. Robot consumes about 200-220 mA in a full motion and communication modus, that is less that 1C and it has have enough capacity for running time of 1-2 hours. The sensing and communication capabilities of the Jasmine platform are already described (see e.g. for communication [10] and for perception [12]). Briefly, the robot can recognize the surface geometry (concave, convex, round, flat), color of surfaces (at least three main colors), can sense obstacles in 360 degree around itself, in a distance over 150mm, internal energy level and so on. Communication is omni-directional, half-duplex, max. communication radius is over 150mm (300mm in 15 degree opening angle). In Fig. 4 we show different developed versions of the platform.
The microrobotic platform is modular, it means the robot can be configured as main board plus different extensions boards (see Fig. 5). Currently there are a few extension systems, listed in the Table 2 that are in different state of development/testing. Table 2. Potential and already existed extension boards for Jasmine platform.
Extension
Description Ego-position system Allows determining the robot position/rotation in global coordinate system with accuracy of a few mm. It requires a beamer on the top of robot-arena, software for beamer control and specific two-sensor system on the top of the robot.
Light sensors system
Allows following the light source. It requires two light sensor on the top of the robot. Wireless communication Uses Atmel's RF chip on the top extension.
It requires a relatively large antenna.
Odometrical system
Allows determining the local robot displacement and rotation in with accuracy of a few mm. It required specific sensor system on the motor board. Auto Recharge System Allows recharging of robot' accu during a motion.
The robot can decide when its own accu needs to be recharged and does it autonomously without human participation. Electro-magnetic gripper Allows gripping small metallic objects (in development).
Finally, we summarize the main capabilities of the platform:
-low-price solution, as components less than 100 euro per robot; -size 26 x 26 x 26 mm; -RAM 2kB, Flash ROM 24kB,nonvolatile EEPROM 1kB; -2 DC motors (reversible), max. velocity over 500 mm/sec.; -autonomous work 1-2 hours; -omni-directional 6 channel robot-robot communication, half-duplex; -physically adjustable communication radius: 0-max, max. communication over 300 mm, capacity of channel over 1000 bits/sec.; -remote control and host-robot communication; -proximity and distance measurement over 300 mm; -perception of objects geometries, light, tough, internal energy sensor and so on; -extendable periphery (e.g. camera) with serial interface, external actuators; -programmable via COM-Port from PC, many open-source development tools.
(a) (b) (c) (d) (e) (f)
Since this is the open-hardware project, the hardware details can be downloaded from the site, otherwise in the community there are a few companies that can provide a complete solution with a hardware support.
Control concepts and supported software
The software is structured hierarchically, where the high-level modules are based on the low-level modules, as shown in Fig. 6. These high-level modules can be exchanged without changing basic capability of the robot. Since the high-level modules describe the behavioral scenarios, we can implement many different scenarios without overstepping hardware capabilities (flash memory) of the microrobot. The modular construction allows also a "distributed" software development by different persons, easy improvement and extension. Moreover, modules that are not required in a particular scenario, can be removed from a configuration without destroying the software architecture.
There are five main levels of the software architecture:
I/O
BIOS Module:
This module provides the main interface to hardware resources. It supports remote control, motion control, proximity sensing, TWI interface, software interruption system, collision avoiding and so represents the basic capabilities of the microrobot.
2. I/O Services: This module provide the interface to PC (serial interface) for reading/writting and writing into EEPROM.
3. Comm Module: Communication module provides a low-level half-duplex local data exchange between the robots. It supports two main protocol: with confirmation and without confirmation. The module is relatively large and can be removed from configuration (in scenarios where communication is not required). This module includes also a support for active communication (requires userdefined support).
4. Perception Module: Perception module is destined for scanning and recognition of surfaces geometry. This routine includes also basic functions for collective perception (requires user-defined modules).
5. User-defined module: This module includes primarily two functions Deci-sionProcess() and ExecutePlan() and determines the high-level behavioral patterns. Each user (working on separate scenario) creates its own user-defined module or uses them from share-library. The "operating system" of the robot executes user-defined code and provides hardware resources, timing and other resources to a user-defined code. There are two operating systems that use either the autonomy cycle or software interruptions. Independently of the used operating system, the BIOS, perception and communication modules remain the same. These components build an integrated on-board SDK, which allows a quick and efficient implementation of swarm algorithms. The C-source and libraries for AVR compiler (open-source) can be downloaded from the site.
Simulator
The simulator is used for preliminary tests and debugging the software before programming real robots. The program used to create the simulation system is called 'Breve'. It is an open-source, multiplatform software package, which makes an fast development of 3D simulations for decentralized systems and artificial life. Users define the behaviours of agents in 3D world and observe how they interact. 'Breve' includes an OpenGL engine so it is possible to visualize simulated worlds. Breve simulations are written in an object-oriented language called Steve. More information about Breve can be found in [13]. Breve executes the code written in each simulation agent in a parallel way every iteration, so all agents are independent.
Especial attention was paid to create equal hardware abstraction layers in Jasmine robots and in the simulator so that the same code can be run in both systems without essential modifications. A large effort was invested into making reality gap as minimal as possible: for instance, we compared behavior between simulated and real setups, see Fig. 7(a) and 7(b), and tuned the simulator to obtain almost identical behaviors, see more in [14]. Since all sensors are fully implemented, the simulator was employed even for preliminary tests with collective perception based on surface colors and surface geometries, see Fig. 7(c). Forasmuch as the reality gap between Jasmine robots and simulator became finally relatively low, it was also used for performing preparatory experiments with large-scale swarms, see Fig. 7(d), where the effort of performing real experiments with many attempts was especially high.
Experiments with microrobots Jasmine
Currently there are a few swarms with Jasmine robots at different institutions. In our lab, we focused primarily on exploring various ways of deriving local rules, creating artificial self-organization and different kind of emergent phenomena. There are four main approaches in developing such local rules: top-down, bottomup, evolutionary and bio-inspired, see e.g. [15], [16], [5], [17], [18], [19] and others. The same problem can be considered in light of these four approaches: in several experiments we explored and compared their output and performance (e.g. bio-inspired [20] and tech-inspired [21] approaches). Especial attention has been paid to basic problems such as collective energy management, swarm communication, collective and perception and collective awareness. Several images of the performed experiments are shown in Fig. 8. Many our projects, such as [22], [23], [24], [25], [26], [27], [28] address different aspects of the performed research.
Bottom-up approach. The local rules are first programmed into each agent and then cyclically evaluated and redesigned, see rule-based programming [6], refining sequential program into concurrent ones [29], formal definition of coop-eration and coordination [30]. There are several variations of this technique: the application of optimization [31] or probabilistic [32] approaches for finding optimal rules, geometrical [33] and functional [34] considerations, the exploration of different aspects of embodiment: properties of communication [11], [35], [36], power management [37], [38] and sensing [39].
Top-down approach. Macroscopic behavior is formally described, see for example grammatical and semantical structures [40]. By using a formal transformation, this high-level description can be converted to low-level programs in each collective agent. The top-down approach works well in different fields of nonlinear dynamics [41], and in the application of analytical approaches for controlling collective behavior [42], [43], collective decision-making [44] and similar problems. Several optimization approaches can be used to perform top-down derivation of local rules for industrial environments [44], [45], [46], [47], [48]. In robotic systems, top-down approaches have been applied to cooperative actuation [49], [50], creation of desired behavioral patterns [14], and self-assembly processes [51], [52].
Evolutionary approach. The search space of a collective system contains a desired solution, that is, local rules, which are able to create a desired behavior [53]. This desired solution is described by the fitness function [54]. Applying the principles of computational evolution, the developer can find the required rules. Such an approach has been applied to foraging problems, for example [55], simple behavioral primitives [56], genetic frameworks [57], and for evolving morphology, controllers, behavior, or strategies.
Bio-inspired approach. Observations from social insects, animals, microorganisms, or even humans are transferred to technical systems [58]. The number of bio-inspired works in the domain of collective systems, especially swarms, is very large. To give some examples, there are several attempts towards hormonebased regulation [59], artificial sexual reproduction [60], aggregation strategies inspired by bees [61], bio-inspired decision-making [62] and foraging [63], [20].
Collective perception. This problem was addressed in several works based on recognition of colors [39] and geometries [32], [64], [12]. They are related to recognition of such objects that are larger than the robot itself. For that, all robots surround an object and scan the corresponding object's surfaces (see Fig. 8(a)). The scan data provide information about surface's geometry and allow classifying the type of surface (flat, concave, convex; size of surface and so on). The classification data are exchanged between robots and matched with the distributed model of the object. However for a particular robot is important not only to recognize an object, but also to know its own position in relation to this object. This positional context cannot be obtained from the sensor data of individual robot. The idea here is that during local communication, all robots know their neighbors, and this "embodied" information can be used for estimating a position. When sequences of connected particular observations are matched with models (e.g. the model A-B-C-A-C-D and the connected particular observation A-B), these connected observations can be located in the model. In this way the robots can collectively estimate their own spatial context (see more in [32]). As [66]; (c) Experiments with tech-inspired aggregation [67]; (d) Experiments with bio-inspired aggregation [61]; (e) Experiments with docking and recharging [21]; (f ) Experiments with bio-inspired collective energy management [20].
demonstrated by experiments, even for uncomplete observations, this approach allows deriving the positions. Spatial information processing. These experiments are focused on spatial information processing, such as distance/area measurement over a swarm, collective calculation of the swarm's center of gravity [33], spatial decision making [62] and others. These works are related in many aspects to cooperative actuation [49], embodiment issues [11], [50], cooperation and functional selforganization [34]. To perform these operations, the robots have to create the dynamic communication network so that a part of them is continuously contained within the communication radius of each other. In this way they build a collective information system, which we call a "communication street".
Swarm communication. Communication approaches are integrated into other algorithms and into the software library of collective behavior. Several of these works are published in relation to IR-based [65], ZigBee [68], [57] approaches, to using context-based communication [10] and to high-level communication procedures [66], [67], [21]. Several aspects of these works are related to maintain the dynamical communication networks, optimization of energy and using the context provided by the amplitude and direction of IR signals. In this way they not only support an optimal distance for global information propagation, but also can collectively perform different spatial computations (distances, areas, centers of gravity and others).
Feedback-based aggregation. Aggregation was considered as a typical problem (task) for swarms and was solved via bio-inspired [61], [69], [62], [63], bottom-up [50], [39] [14], top-down [18], [9] and evolutionary [70], [56] methods. The example of aggregation was also used to compare scalability, performance and efficiency of different approaches.
Energy-related aspects. Energy aspects are investigated from different points of view: docking and low-level power management [37], [38], bio-inspired [20] and tech-inspired [21] behavioral strategies and energy homeostasis. We performed also several long-term experiments to explore self-sustainability of robot swarm and autonomous energy management.
Self-assembling and multi-cellularity. These issues are primarily related to the projects [25], [27], where we used another hardware platform [71], [72], [73], [74] or the in edited book [75]. However, initial experiments as well as swarm-related topics in these projets are still explored with Jasmine robots [76].
Collective awareness. These works are relatively new and related primarily to the project [28]. Here we explore mechanisms, which create awareness about a global state of the swarm on a local level without using any centralized mechanisms.
Conclusion
In this paper we presented an open-hardware project that targets a development of a cheap and reliable microrobot as well as SDK for large-scale swarms.
We demonstrated the concept of a swarm embodiment, which underlies the development of robot platform Jasmine. Its capabilities and features are briefly described and illustrated by examples. Software part and simulator are briefly described. Generally, we have shown that despite the limited hardware capabilities of microrobots, the specific construction of hardware and software parts make feasible many advanced collective properties.
The performed experiments between 2005 and 2010 are briefly listed in categories related to research topics explored in the group. However, a systematic consideration of all performed experiments and their evaluation in terms of embodiment, performance, scalability and other features still remain open and represents future works.
Acknowledgment
The swarmrobot.org project was started in a close collaboration with Marc Szymanski who worked at those time at the University of Karlsruhe. His enthusiasm and energy essentially contributed the success of this project.
Author and the team are supported by the followings grants: EU-IST FET project "I-SWARM", grant agreement no. 507006; EU-NMP project "Golem", grant agreement no. 033211; EU-IST FET project "SYMBRION", grant agreement no. 216342; EU-ICT project "REPLICATOR", grant agreement no. 216240; EU-ICT project "EvoBody", grant agreement no. 258334; EU-ICT FET project "Angels", grant agreement no. 231845; EU-ICT project "CoCoRo", grant agreement no. 270382.
Fig. 1 .
1Swarmrobot.org, which is devoted to the platform Jasmine and large-scale swarm experiments.
Fig. 2 .
2Concept of embodiment of local rules.
Fig. 3 .
3"Sandwich design" of the Jasmine platform.
Fig. 4 .Fig. 5 .
45Different versions of the microrobot Jasmine. (a) Chassis with the megabitty board; (b) The first test development for testing of sensors; (c) The first version; (d) The second version; (e) The third version; (f ) The third plus version; Modularity of the Jasmine platform: the main board (left) and the motors board (right).
Fig. 6 .
6Software structure of Jasmine platform.
Fig. 7 .
7(a) Setup in simulation; (b) The same setup in real experiments; (c) Example of a setup for exploring collective perception; (d) Example of a setup for exploring large-scale swarm.
Fig. 8 .
8(a) Experiments with collective perception[64]; (b) Experiments swarm communications[65],
Table 1 .
1Several types of collective activities performed by a swarm.Context N Swarm Capability
1 Spatial orientation
Spatial 2 Building spatial structures
3 Collective movement
4 Building informational structures
5 Collective decision making
Information 6 Collective information processing
7 Collective perception/recognition
8 Building functional structures
9 Collective task decomposition
Functional 10 Collective planning
11 Group-based specialization
S Kernbach, Handbook of Collective Robotics: Fundamentals and Challenges. SingaporePan Stanford PublishingKernbach, S., ed.: Handbook of Collective Robotics: Fundamentals and Challenges. Pan Stanford Publishing, Singapore (2011)
Swarm intelligence: from natural to artificial systems. E Bonabeau, M Dorigo, G Theraulaz, Oxford University PressNew YorkBonabeau, E., Dorigo, M., Theraulaz, G.: Swarm intelligence: from natural to artificial systems. Oxford University Press, New York (1999)
Swarm Intelligence in autonomous collective robotics: from Tools to the analysis and synthesis of distributed control strategies PhD Thesis. A Martinoli, Lausanne, EPFLMartinoli, A.: Swarm Intelligence in autonomous collective robotics: from Tools to the analysis and synthesis of distributed control strategies PhD Thesis. Lausanne, EPFL (1999)
H Haken, Advanced synergetics. BerlinSpringer-VerlagHaken, H.: Advanced synergetics. Springer-Verlag, Berlin (1983)
Generation of desired emergent behavior in swarm of micro-robots. S Kornienko, O Kornienko, P Levi, Proc. of the 16th European conference on artificial intelligence. Lopez de Mantaras, R., Saitta, L.of the 16th European conference on artificial intelligenceValencia, Spain, AmsterdamIOS PressKornienko, S., Kornienko, O., Levi, P.: Generation of desired emergent behavior in swarm of micro-robots. In Lopez de Mantaras, R., Saitta, L., eds.: Proc. of the 16th European conference on artificial intelligence (ECAI 2004), Valencia, Spain, Amsterdam: IOS Press (2004) 239-243
Formal derivation of rule-based programs. G C Roma, R F Gamble, W E Ball, IEEE Trans. Softw. Eng. 193Roma, G.C., Gamble, R.F., Ball, W.E.: Formal derivation of rule-based programs. IEEE Trans. Softw. Eng. 19(3) (1993) 277-296
About nature of emergent behavior in micro-systems. S Kornienko, O Kornienko, P Levi, Proc. of the Int. Conf. on Informatics in Control, Automation and Robotics. of the Int. Conf. on Informatics in Control, Automation and RoboticsSetubal, PortugalKornienko, S., Kornienko, O., Levi, P.: About nature of emergent behavior in micro-systems. In: Proc. of the Int. Conf. on Informatics in Control, Automation and Robotics (ICINCO 2004), Setubal, Portugal. (2004) 33-40
Embodied artificial intelligence: Trends and challenges. R Pfeifer, F Iida, Embodied artificial intelligence. et al. (Eds), I.SpringerPfeifer, R., Iida, F.: Embodied artificial intelligence: Trends and challenges. In et al. (Eds), I., ed.: Embodied artificial intelligence. Springer (2004) 1-26
Swarm embodiment -a new way for deriving emergent behaviour in artificial swarms. S Kornienko, O Kornienko, P Levi, Autonome Mobile Systeme (AMS'05. et al., P.,Kornienko, S., Kornienko, O., Levi, P.: Swarm embodiment -a new way for deriv- ing emergent behaviour in artificial swarms. In et al., P., ed.: Autonome Mobile Systeme (AMS'05). (2005) 25-32
Collective AI: context awareness via communication. S Kornienko, O Kornienko, P Levi, Proc. of the IJCAI 2005. of the IJCAI 2005Edinburgh, UK.Kornienko, S., Kornienko, O., Levi, P.: Collective AI: context awareness via com- munication. In: Proc. of the IJCAI 2005, Edinburgh, UK. (2005) 1464-1470
Exploration of embodiment in real microrobotic swarm. J Caselles, GermanyUniversity of StuttgartMaster ThesisCaselles, J.: Exploration of embodiment in real microrobotic swarm. Master Thesis, University of Stuttgart, Germany (2005)
Minimalistic approach towards communication and perception in microrobotic swarms. S Kornienko, O Kornienko, P Levi, Proc. of the International Conference on Intelligent Robots and Systems (IROS-2005). of the International Conference on Intelligent Robots and Systems (IROS-2005)Edmonton, CanadaKornienko, S., Kornienko, O., Levi, P.: Minimalistic approach towards commu- nication and perception in microrobotic swarms. In: Proc. of the International Conference on Intelligent Robots and Systems (IROS-2005), Edmonton, Canada (2005) 2228-2234
Breve Project. J Klein, Klein, J.: Breve Project. http://www.spiderland.org/breve/ (2000)
Development of cooperative behavioural patterns for swarm robotic scenarios. V Prieto, Stuttgart, GermanyUniversity ofMaster ThesisPrieto, V.: Development of cooperative behavioural patterns for swarm robotic scenarios. Master Thesis, University of Stuttgart, Germany (2006)
Top-down vs bottom-up methodologies in multi-agent system design. V Crespi, A Galstyan, K Lerman, Auton. Robots. 243Crespi, V., Galstyan, A., Lerman, K.: Top-down vs bottom-up methodologies in multi-agent system design. Auton. Robots 24(3) (2008) 303-313
Reconciling top-down and bottom-up design approaches in rmm. T Isakowitz, A Kamis, M Koufaris, SIGMIS Database. 294Isakowitz, T., Kamis, A., Koufaris, M.: Reconciling top-down and bottom-up design approaches in rmm. SIGMIS Database 29(4) (1998) 58-67
The benefits of bottom-up design. G Mcfarland, SIGSOFT Softw. Eng. Notes. 115McFarland, G.: The benefits of bottom-up design. SIGSOFT Softw. Eng. Notes 11(5) (1986) 43-51
Structural Self-organization in Multi-Agents and Multi-Robotic Systems. S Kernbach, Logos Verlag. Kernbach, S.: Structural Self-organization in Multi-Agents and Multi-Robotic Sys- tems. Logos Verlag, Berlin (2008)
A brief top-down and bottom-up philosophy on software evolution. M Pizka, A Bauer, IWPSE '04: Proceedings of the Principles of Software Evolution, 7th International Workshop. Washington, DC, USAIEEE Computer SocietyPizka, M., Bauer, A.: A brief top-down and bottom-up philosophy on software evolution. In: IWPSE '04: Proceedings of the Principles of Software Evolution, 7th International Workshop, Washington, DC, USA, IEEE Computer Society (2004) 131-136
Specialization and generalization of robotic behavior in swarm energy foraging. Mathematical and Computer Modelling of Dynamical Systems. S Kernbach, V Nepomnyashchikh, T Kancheva, O Kernbach, DOI10.1080/13873954.2011.601421Kernbach, S., Nepomnyashchikh, V., Kancheva, T., Kernbach, O.: Specialization and generalization of robotic behavior in swarm energy foraging. Mathematical and Computer Modelling of Dynamical Systems, DOI 10.1080/13873954.2011.601421 (2011)
Collective energy homeostasis in a largescale micro-robotic swarm. S Kernbach, O Kernbach, DOI10.1016/j.robot.2011.08.001Robotics and Autonomous Systems. 59Kernbach, S., Kernbach, O.: Collective energy homeostasis in a large- scale micro-robotic swarm. Robotics and Autonomous Systems, DOI 10.1016/j.robot.2011.08.001 59 (2011) 1090-1101
Intelligent Small World Autonomous Robots for Micromanipulation. European Union 6th Framework Programme Project No. I-Swarm ; I-Swarm , FP6-2002- IST-1GOLEM: Bio-inspired Assembly Process for Mesoscale Products and Systems. EU Project. 23NMP-2004-3.4.1.2-1I-Swarm: I-Swarm: Intelligent Small World Autonomous Robots for Micro- manipulation. European Union 6th Framework Programme Project No FP6-2002- IST-1 (2003-2007) 23. : GOLEM: Bio-inspired Assembly Process for Mesoscale Products and Systems. EU Project, NMP-2004-3.4.1.2-1 (2006-2009)
EU-project 231845ANGELS: ANGuilliform robot with ELectric Sense. European CommunitiesANGELS: ANGuilliform robot with ELectric Sense, EU-project 231845, 2009-2011. European Communities (2011)
SYMBRION: Symbiotic Evolutionary Robot Organisms, 7th Framework Programme Project No FP7-ICT-2007.8.2. European Communities. EVOBODY: EVOBODY: New Principles of Unbound Embodied Evolution. Project reference: 258334. European CommunitiesSYMBRION: Symbiotic Evolutionary Robot Organisms, 7th Framework Pro- gramme Project No FP7-ICT-2007.8.2. European Communities (2008-2012) 26. EVOBODY: EVOBODY: New Principles of Unbound Embodied Evolution, Project reference: 258334. European Communities (2010-2011)
REPLICATOR: Robotic Evolutionary Self-Programming and Self-Assembling Organisms. 7th Framework Programme Project No FP7-ICT-2007.2.1. European Communities. REPLICATOR: Robotic Evolutionary Self-Programming and Self-Assembling Or- ganisms, 7th Framework Programme Project No FP7-ICT-2007.2.1. European Communities (2008-2012)
CoCoRo: Collective Cognitive Robots, FP7. European Communities. CoCoRo: Collective Cognitive Robots, FP7. European Communities (2011-2013)
Stepwise refinement of action systems. R J R Back, K Sere, Structured Programming. 12Back, R.J.R., Sere, K.: Stepwise refinement of action systems. Structured Pro- gramming 12 (1991) 17-30
Distributed cooperation with action systems. R J R Back, F Kurki-Suonio, ACM Trans. Program. Lang. Syst. 104Back, R.J.R., Kurki-Suonio, F.: Distributed cooperation with action systems. ACM Trans. Program. Lang. Syst. 10(4) (1988) 513-554
Optimization of communication in a swarm of micro-robots. X Chen, GermanyUniversity of StuttgartMaster ThesisChen, X.: Optimization of communication in a swarm of micro-robots. Master Thesis, University of Stuttgart, Germany (2003)
Collective Classification in a Swarm of Microrobots. M Pradier, GermanyUniversity of StuttgartMaster ThesisPradier, M.: Collective Classification in a Swarm of Microrobots. Master Thesis, University of Stuttgart, Germany (2005)
Swarm-based computation and spatial decision making. Z Fu, GermanyUniversity of StuttgartMaster ThesisFu, Z.: Swarm-based computation and spatial decision making. Master Thesis, University of Stuttgart, Germany (2005)
Mechanism of cooperation and functional self-organization in a swarm of micro-robots. O A Warraich, GermanyUniversity of StuttgartMaster ThesisWarraich, O.A.: Mechanism of cooperation and functional self-organization in a swarm of micro-robots. Master Thesis, University of Stuttgart, Germany (2005)
Testing and Re-Implementation of Communication Protocols for the Microrobot Jasmine. F Mletzko, Stuttgart, GermanyStudienarbeit, University ofMletzko, F.: Testing and Re-Implementation of Communication Protocols for the Microrobot Jasmine. Studienarbeit, University of Stuttgart, Germany (2006)
Development of context-based communication protocols for the microrobot 'Jasmine'. Studienarbeit. R Geider, University of Stuttgart, GermanyGeider, R.: Development of context-based communication protocols for the micro- robot 'Jasmine'. Studienarbeit, University of Stuttgart, Germany (2006)
Development of a docking approach for autonomous recharging system for micro-robot "Jasmine. K Jebens, University of Stuttgart, GermanyStudient ThesisJebens, K.: Development of a docking approach for autonomous recharging system for micro-robot "Jasmine". Studient Thesis, University of Stuttgart, Germany (2006)
Development of advanced power management for autonomous micro-robots. A Attarzadeh, Stuttgart, GermanyUniversity ofMaster ThesisAttarzadeh, A.: Development of advanced power management for autonomous micro-robots. Master Thesis, University of Stuttgart, Germany (2006)
Collaborative actuation in microrobotic swarm based on collective decision making and surfacecolor identification. G Zetterström, University of Stuttgart, GermanyMaster ThesisZetterström, G.: Collaborative actuation in microrobotic swarm based on collec- tive decision making and surfacecolor identification. Master Thesis, University of Stuttgart, Germany (2006)
Interaction und Kooperation in Multiagentsystemen. M Muscholl, StuttgartUniversity of StuttgartPhD thesisMuscholl, M.: Interaction und Kooperation in Multiagentsystemen. PhD thesis, University of Stuttgart, Stuttgart (2001)
H Haken, Laser theory. BerlinSpringer-VerlagHaken, H.: Laser theory. Springer-Verlag, Berlin (1984)
Application of order parameter equation for the analysis and the control of nonlinear time discrete dynamical systems. P Levi, M Schanz, S Kornienko, O Kornienko, Int. J. Bifurcation and Chaos. 98Levi, P., Schanz, M., Kornienko, S., Kornienko, O.: Application of order parameter equation for the analysis and the control of nonlinear time discrete dynamical systems. Int. J. Bifurcation and Chaos 9(8) (1999) 1619-1634
Control of periodical motion using the synergetic concept. S Kornienko, O Kornienko, not published yetKornienko, S., Kornienko, O.: Control of periodical motion using the synergetic concept. not published yet (1999)
Collective decision making using natural self-organization in distributed systems. O Kornienko, S Kornienko, P Levi, Proc. of Int. Conf. on Computational Intelligence for Modelling, Control and Automation (CIMCA'2001). of Int. Conf. on Computational Intelligence for Modelling, Control and Automation (CIMCA'2001)Las Vegas, USAKornienko, O., Kornienko, S., Levi, P.: Collective decision making using natural self-organization in distributed systems. In: Proc. of Int. Conf. on Computational Intelligence for Modelling, Control and Automation (CIMCA'2001), Las Vegas, USA. (2001) 460-471
Application of multi-agent planning to the assignment problem. S Kornienko, O Kornienko, J Priese, Computers in Industry. 543Kornienko, S., Kornienko, O., Priese, J.: Application of multi-agent planning to the assignment problem. Computers in Industry 54(3) (2004) 273-290
Flexible manufacturing process planning based on the multi-agent technology. S Kornienko, O Kornienko, P Levi, Proc. of the 21st IASTED Int. Conf. on AI and Applications (AIA '2003). of the 21st IASTED Int. Conf. on AI and Applications (AIA '2003)Innsbruck, AustriaKornienko, S., Kornienko, O., Levi, P.: Flexible manufacturing process planning based on the multi-agent technology. In: Proc. of the 21st IASTED Int. Conf. on AI and Applications (AIA '2003), Innsbruck, Austria. (2003) 156-161
Application of distributed constraint satisfaction problem to the agent-based planning in manufacturing systems. S Kornienko, O Kornienko, P Levi, Proc. of IEEE Int. Conf. on AI Systems (AIS'03). of IEEE Int. Conf. on AI Systems (AIS'03)Divnomorsk, RussiaKornienko, S., Kornienko, O., Levi, P.: Application of distributed constraint satis- faction problem to the agent-based planning in manufacturing systems. In: Proc. of IEEE Int. Conf. on AI Systems (AIS'03), Divnomorsk, Russia. (2003) 124-140
Multi-agent repairer of damaged process plans in manufacturing environment. S Kornienko, O Kornienko, P Levi, Proc. of the 8th Conf. on Intelligent Autonomous Systems (IAS-8). of the 8th Conf. on Intelligent Autonomous Systems (IAS-8)Amsterdam, NLKornienko, S., Kornienko, O., Levi, P.: Multi-agent repairer of damaged process plans in manufacturing environment. In: Proc. of the 8th Conf. on Intelligent Autonomous Systems (IAS-8), Amsterdam, NL. (2004) 485-494
Cooperative actuation in a large robotic swarm. M Jiménez, Stuttgart, GermanyUniversity ofMaster ThesisJiménez, M.: Cooperative actuation in a large robotic swarm. Master Thesis, University of Stuttgart, Germany (2006)
Creating emergent behavior in a group of micro-robots. F Mletzko, Stuttgart, GermanyUniversity ofMaster ThesisMletzko, F.: Creating emergent behavior in a group of micro-robots. Master Thesis, University of Stuttgart, Germany (2006)
Simulation and visualization of fluid behavior in bio-inspired selfassembly processes. J Kabir, GermanyUniversity of StuttgartMaster ThesisKabir, J.: Simulation and visualization of fluid behavior in bio-inspired self- assembly processes. Master Thesis, University of Stuttgart, Germany (2008)
Application and improvement of a simulation of bio-inspired selfassembly process. R S Urien, GermanyUniversity of StuttgartMaster ThesisUrien, R.S.: Application and improvement of a simulation of bio-inspired self- assembly process. Master Thesis, University of Stuttgart, Germany (2009)
Evolutionary Robotics: The Biology, Intelligence, and Technology of Self-Organizing Machines. S Nolfi, D Floreano, The MIT PressCambridge, MA. / LondonNolfi, S., Floreano, D.: Evolutionary Robotics: The Biology, Intelligence, and Tech- nology of Self-Organizing Machines. The MIT Press, Cambridge, MA. / London (2000)
Evolutionary design of emergent behavior. J Branke, H Schmeck, Organic Computing. Würtz, R.SpringerBranke, J., Schmeck, H.: Evolutionary design of emergent behavior. In Würtz, R., ed.: Organic Computing. Springer (2008) 123-140
Genetic programming: on the programming of computers by means of natural selection. J Koza, MIT PressCambridge, Massacgusetts, London, EnglandKoza, J.: Genetic programming: on the programming of computers by means of natural selection. MIT Press, Cambridge, Massacgusetts, London, England (1992)
A model for developing behavioral patterns on multi-robot organisms using concepts of natural evolution. L Koenig, GermanyUniversity of StuttgartMaster ThesisKoenig, L.: A model for developing behavioral patterns on multi-robot organ- isms using concepts of natural evolution. Master Thesis, University of Stuttgart, Germany (2007)
Development of a Software Framework for Jasmine Robots using ZigBee Communication Protocol. A Nagarathinam, Stuttgart, GermanyUniversity ofMaster ThesisNagarathinam, A.: Development of a Software Framework for Jasmine Robots using ZigBee Communication Protocol. Master Thesis, University of Stuttgart, Germany (2007)
Evolutionary Robotics. D Floreano, P Husbands, S Nolfi, Handbook of Robotics. BerlinSpringer VerlagFloreano, D., Husbands, P., Nolfi, S.: Evolutionary Robotics. In: Handbook of Robotics. Springer Verlag, Berlin (2008)
Artificial cell differentiation in a multi-robot organisms through gene regulation. G Speidel, GermanyUniversity of StuttgartMaster ThesisSpeidel, G.: Artificial cell differentiation in a multi-robot organisms through gene regulation. Master Thesis, University of Stuttgart, Germany (2008)
Investigation of Evolutionary Reproduction in a Robot Swarm. C Schwarzer, GermanyUniversity of StuttgartMaster ThesisSchwarzer, C.: Investigation of Evolutionary Reproduction in a Robot Swarm. Master Thesis, University of Stuttgart, Germany (2008)
Re-embodiment of honeybee aggregation behavior in artificial micro-robotic system. S Kernbach, R Thenius, O Kernbach, T Schmickl, Adaptive Behavior. 173Kernbach, S., Thenius, R., Kernbach, O., Schmickl, T.: Re-embodiment of hon- eybee aggregation behavior in artificial micro-robotic system. Adaptive Behavior 17(3) (2009) 237-259
Bio-inspired approach towards collective decision making in robotic swarms. D Häbe, Stuttgart, GermanyUniversity ofMaster ThesisHäbe, D.: Bio-inspired approach towards collective decision making in robotic swarms. Master Thesis, University of Stuttgart, Germany (2007)
Adaptive role dynamics in energy foraging behavior of a real microrobotic swarm. T Kancheva, GermanyUniversity of StuttgartMaster ThesisKancheva, T.: Adaptive role dynamics in energy foraging behavior of a real micro- robotic swarm. Master Thesis, University of Stuttgart, Germany (2007)
Cognitive micro-agents: individual and collective perception in microrobotic swarm. S Kornienko, O Kornienko, C Constantinescu, M Pradier, P Levi, Proc. of the IJCAI-05 Workshop on Agents in real-time and dynamic environments. of the IJCAI-05 Workshop on Agents in real-time and dynamic environmentsEdinburgh, UK.Kornienko, S., Kornienko, O., Constantinescu, C., Pradier, M., Levi, P.: Cognitive micro-agents: individual and collective perception in microrobotic swarm. In: Proc. of the IJCAI-05 Workshop on Agents in real-time and dynamic environments, Edinburgh, UK. (2005) 33-42
IR-based communication and perception in microrobotic swarms. S Kornienko, O Kornienko, IROS 2005. Edmonton, CanadaKornienko, S., Kornienko, O.: IR-based communication and perception in micro- robotic swarms. In: IROS 2005, Edmonton, Canada. (2005)
Three Cases of Connectivity and Global Information Transfer in Robot Swarms. S Kernbach, ArXiv e-prints (2011arXiv1109.4221KKernbach, S.: Three Cases of Connectivity and Global Information Transfer in Robot Swarms. ArXiv e-prints (2011arXiv1109.4221K) (2011)
Diffusion of information in robot swarms. S Kernbach, arXiv:1110.5183v1ArXiv e-printsKernbach, S.: Diffusion of information in robot swarms. ArXiv e-prints (arXiv:1110.5183v1) (2011)
Design and tests of ZigBee communication system for robot organisms. D Beurer, GermanyUniversity of StuttgartMaster ThesisBeurer, D.: Design and tests of ZigBee communication system for robot organisms. Master Thesis, University of Stuttgart, Germany (2008)
Get in touch: cooperative decision making based on robot-torobot collisions. T Schmickl, R Thenius, C Moeslinger, G Radspieler, S Kernbach, M Szymanski, K Crailsheim, 10.1007/s10458-008-9058-5Autonomous Agents and Multi-Agent Systems. 18Schmickl, T., Thenius, R., Moeslinger, C., Radspieler, G., Kernbach, S., Szymanski, M., Crailsheim, K.: Get in touch: cooperative decision making based on robot-to- robot collisions. Autonomous Agents and Multi-Agent Systems 18 (2009) 133-155 10.1007/s10458-008-9058-5.
Stability of on-line and on-board evolving of adaptive collective behavior. L Koenig, K Jebens, S Kernbach, P Levi, Proc. of the EUROS. of the EUROSPrague, Czech RepublicKoenig, L., Jebens, K., Kernbach, S., Levi, P.: Stability of on-line and on-board evolving of adaptive collective behavior. In: Proc. of the EUROS 2008, Prague, Czech Republic. (2007)
Evolutionary robotics: The next-generation-platform for on-line and on-board artificial evolution. S Kernbach, E Meister, O Scholz, R Humza, J Liedke, L Ricotti, J Jemai, J Havlik, W Liu, IEEE Congress on Evolutionary Computation, CEC '09. Kernbach, S., Meister, E., Scholz, O., Humza, R., Liedke, J., Ricotti, L., Jemai, J., Havlik, J., Liu, W.: Evolutionary robotics: The next-generation-platform for on-line and on-board artificial evolution. In: IEEE Congress on Evolutionary Com- putation, CEC '09. (2009) 1079 -1086
Study of macroscopic morphological features of symbiotic robotic organisms. S Kernbach, L Ricotti, J Liedke, P Corradi, M Rothermel, Proceedings of the workshop on self-reconfigurable robots. the workshop on self-reconfigurable robots08Kernbach, S., Ricotti, L., Liedke, J., Corradi, P., Rothermel, M.: Study of macro- scopic morphological features of symbiotic robotic organisms. In: Proceedings of the workshop on self-reconfigurable robots, IROS08, Nice. (2008) 18-25
S Kernbach, E Meister, F Schlachter, K Jebens, M Szymanski, J Liedke, D Laneri, L Winkler, T Schmickl, R Thenius, P Corradi, L Ricotti, Proceedings of the 8th Workshop on Performance Metrics for Intelligent Systems. PerMIS '08. the 8th Workshop on Performance Metrics for Intelligent Systems. PerMIS '08New York, NY, USAACMSymbiotic robot organisms: REPLICATOR and SYMBRION projectsKernbach, S., Meister, E., Schlachter, F., Jebens, K., Szymanski, M., Liedke, J., Laneri, D., Winkler, L., Schmickl, T., Thenius, R., Corradi, P., Ricotti, L.: Symbi- otic robot organisms: REPLICATOR and SYMBRION projects. In: Proceedings of the 8th Workshop on Performance Metrics for Intelligent Systems. PerMIS '08, New York, NY, USA, ACM (2008) 62-69
On adaptive self-organization in artificial robot organisms. S Kernbach, H Hamann, J Stradner, R Thenius, T Schmickl, K Crailsheim, A C V Rossum, M Sebag, N Bredeche, Y Yao, G Baele, Y V D Peer, J Timmis, M Mohktar, A Tyrrell, A E Eiben, S P Mckibbin, W Liu, A F T Winfield, Patterns. COMPUTATIONWORLD '09Proceedings of the 2009 Computation World: Future Computing, Service Computation, Cognitive. the 2009 Computation World: Future Computing, Service Computation, CognitiveAdaptive, Content; Washington, DC, USAIEEE Computer SocietyKernbach, S., Hamann, H., Stradner, J., Thenius, R., Schmickl, T., Crailsheim, K., Rossum, A.C.v., Sebag, M., Bredeche, N., Yao, Y., Baele, G., Peer, Y.V.d., Tim- mis, J., Mohktar, M., Tyrrell, A., Eiben, A.E., McKibbin, S.P., Liu, W., Winfield, A.F.T.: On adaptive self-organization in artificial robot organisms. In: Proceedings of the 2009 Computation World: Future Computing, Service Computation, Cog- nitive, Adaptive, Content, Patterns. COMPUTATIONWORLD '09, Washington, DC, USA, IEEE Computer Society (2009) 33-43
Symbiotic Multi-Robot Organisms: Reliability, Adaptability, Evolution. P Levi, S Kernbach, Springer VerlagLevi, P., Kernbach, S., eds.: Symbiotic Multi-Robot Organisms: Reliability, Adapt- ability, Evolution. Springer Verlag (2010)
From real robot swarm to evolutionary multi-robot organism. S Kornienko, O Kornienko, A Nagarathinam, P Levi, Evolutionary Computation. Kornienko, S., Kornienko, O., Nagarathinam, A., Levi, P.: From real robot swarm to evolutionary multi-robot organism. In: Evolutionary Computation, 2007. CEC 2007. IEEE Congress on. (2007) 1483-1490
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[] |
[
"Wafer-Scale Lateral Self-assembly of Mosaic Ti3C2Tx (MXene) Monolayer Films",
"Wafer-Scale Lateral Self-assembly of Mosaic Ti3C2Tx (MXene) Monolayer Films"
] |
[
"Mehrnaz Mojtabavi \nDepartment of Bioengineering\nNortheastern University\n02115BostonMAUSA\n",
"Armin Vahidmohammadi \nInnovation Partnership Building\nUConn TechPark\nUniversity of Connecticut\n06269StorrsCTUSA\n",
"Davoud Hejazi \nDepartment of Physics\nNortheastern University\n02115BostonMAUSA\n",
"Swastik Kar \nDepartment of Physics\nNortheastern University\n02115BostonMAUSA\n",
"Sina Shahbazmohamadi \nInnovation Partnership Building\nUConn TechPark\nUniversity of Connecticut\n06269StorrsCTUSA\n",
"Meni Wanunu [email protected] \nDepartment of Physics\nNortheastern University\n02115BostonMAUSA\n\nDepartment of Chemistry and Chemical Biology\nNortheastern University\n02115BostonMAUSA\n"
] |
[
"Department of Bioengineering\nNortheastern University\n02115BostonMAUSA",
"Innovation Partnership Building\nUConn TechPark\nUniversity of Connecticut\n06269StorrsCTUSA",
"Department of Physics\nNortheastern University\n02115BostonMAUSA",
"Department of Physics\nNortheastern University\n02115BostonMAUSA",
"Innovation Partnership Building\nUConn TechPark\nUniversity of Connecticut\n06269StorrsCTUSA",
"Department of Physics\nNortheastern University\n02115BostonMAUSA",
"Department of Chemistry and Chemical Biology\nNortheastern University\n02115BostonMAUSA"
] |
[] |
Bottom-up assembly of two-dimensional (2D) materials into macroscale morphologies with emergent properties requires control of the material surroundings, so that energetically favorable conditions direct the assembly process. MXenes, a class of recently developed 2D materials, have found new applications in areas such as electrochemical energy storage, nanoscale electronics, sensors, and biosensors. In this report, we present a lateral self-assembly method for wafer-scale deposition of a mosaic-type 2D MXene flake monolayers that spontaneously order at the interface between two immiscible solvents. Facile transfer of this monolayer onto a flat substrate (Si, glass) results in high-coverage (>90%) monolayer films with uniform thickness, homogeneous optical properties, and good electrical conductivity. Multiscale characterization of the resulting films reveals the mosaic structure and sheds light on the electronic properties of the films, which exhibit good conductivity over cm-scale areas.
|
10.1021/acsnano.0c06393
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[
"https://arxiv.org/pdf/2006.12740v1.pdf"
] | 219,980,599 |
2006.12740
|
716446861b3d4ab9eef52390d1f9e1f1a578a010
|
Wafer-Scale Lateral Self-assembly of Mosaic Ti3C2Tx (MXene) Monolayer Films
Mehrnaz Mojtabavi
Department of Bioengineering
Northeastern University
02115BostonMAUSA
Armin Vahidmohammadi
Innovation Partnership Building
UConn TechPark
University of Connecticut
06269StorrsCTUSA
Davoud Hejazi
Department of Physics
Northeastern University
02115BostonMAUSA
Swastik Kar
Department of Physics
Northeastern University
02115BostonMAUSA
Sina Shahbazmohamadi
Innovation Partnership Building
UConn TechPark
University of Connecticut
06269StorrsCTUSA
Meni Wanunu [email protected]
Department of Physics
Northeastern University
02115BostonMAUSA
Department of Chemistry and Chemical Biology
Northeastern University
02115BostonMAUSA
Wafer-Scale Lateral Self-assembly of Mosaic Ti3C2Tx (MXene) Monolayer Films
Self-assemblyMXene2D materials2D titanium carbidemonolayerconductive
Bottom-up assembly of two-dimensional (2D) materials into macroscale morphologies with emergent properties requires control of the material surroundings, so that energetically favorable conditions direct the assembly process. MXenes, a class of recently developed 2D materials, have found new applications in areas such as electrochemical energy storage, nanoscale electronics, sensors, and biosensors. In this report, we present a lateral self-assembly method for wafer-scale deposition of a mosaic-type 2D MXene flake monolayers that spontaneously order at the interface between two immiscible solvents. Facile transfer of this monolayer onto a flat substrate (Si, glass) results in high-coverage (>90%) monolayer films with uniform thickness, homogeneous optical properties, and good electrical conductivity. Multiscale characterization of the resulting films reveals the mosaic structure and sheds light on the electronic properties of the films, which exhibit good conductivity over cm-scale areas.
Introduction
In recent years, various types of two-dimensional (2D) materials have gained much attention due to their unique properties, and have played significant roles in the development of new types of electrical devices 1-3 , chemical/biological sensors [4][5][6][7] , water purification membranes [8][9] , energy conversion/storage devices 1,[10][11] , and other applications. Despite widespread theoretical and experimental research into the physics and applications of 2D materials, a practical challenge common to all of these materials are related to their handling, where their extremely thin nature and ultrahigh aspect ratios (1,000-10,000) demand methods for manipulation and assembly of these materials during device manufacturing. For example, in applications where a controlled 2D material film is required for harvesting the material's electrical and optical properties over macroscale areas, techniques for large-scale assembly of 2D flakes into uniform mono-and multi-layer films are in need. Efforts to address this ongoing challenge has resulted in the development of several large-scale thin-membrane assembly methods. These methods can be categorized into two groups: 1. assembly of ultrathin films from a 2D material flake dispersion, either directly on a substrate or at an interface (liquid/liquid or liquid/air) 12-13 , 2. direct large-area Chemical Vapor Deposition (CVD) of 2D materials on a substrate, followed by use or transfer to another substrate [14][15] .
Among 2D materials, graphene oxide (GO) and its reduced counterpart (rGO) have been most widely studied for large-scale monolayer film assembly from their dispersions. Precise methods for casting uniform layers of GO/rGO on substrates include the Langmuir method [16][17][18][19][20][21] , spincoating 19,[22][23] , and layer-by-layer assembly [24][25] . The Langmuir and spin-coating methods result in well-packed films with organized structure and controlled thickness 23 while other methods result in a looser, more disordered structure 12 . For other 2D materials such as graphene, MoS2, and WS2, poor stability in water and other common solvents limits the quality of their assembly into a film from a dispersion phase [26][27][28] . Their wafer-scale CVD synthesis, however, has been shown to result in more uniform and reproducible films [29][30][31][32][33][34][35] .
2D transition metal carbides and nitrides (MXenes) as an emerging class of electrically conductive, hydrophilic, and optically active 2D materials have been subjects of numerous studies in recent years 2,36 . Their synthesis from MAX phases, in which the MXene layers are bound together by group 13 elements such as Al, Si, etc. (A layer in MAX phase structure), is well established and involves selective etching of the A elements and MXene exfoliation in aqueous fluorine-containing acidic solutions 37 . The resulting MXenes have a general formula of Mn+1XnTx where M is a transition metal, X is carbon/nitrogen, n can be 1, 2, 3, or 4, and Tx represents oxygen, hydroxyl, and fluorine groups present on the MXene basal planes after their exfoliation. Assembly and fabrication of ultrathin (i.e. monolayer) MXene films, however, has so far been challenging and limited the implementation of these materials into devices that require nanometer-thick membranes. Layer-by-layer assembly 38 has shown potential for controlled assembly of MXene monolayer and multilayer films, and other methods have been employed for MXene assembly into thin films, including spin coating 39 , spray coating [40][41] , and dip coating 42 .
Recently, an interfacial assembly method was used for assembling Ti3C2Tx MXene layers at an air/water interface, assisted by ethyl acetate 43 . However, in this work the number of layers varied significantly during deposition, possibly due to a strong convection induced by rapid ethyl acetate evaporation. Table S1 highlights the advantages and disadvantages of large-scale ultrathin film fabrication methods with single-layer precision.
We recently developed a method for lateral self-assembly of MXenes on a liquid-liquid interface to fabricate monolayer to multilayer Ti3C2Tx and Ti2CTx MXene films on small scale (with tens of microns lateral size) and used them as nanometer-thin freestanding membranes in solid-state nanopore sensing applications 44 . Herein, we have further refined our rapid interfacial selfassembly technique to enable large-scale fabrication of monolayer to few-layer Ti3C2Tx films with control over the nominal film thickness, flake density, and lateral size of the self-assembled films. We find that careful control over the MXene suspension concentration affords large-area films with monolayer coverage values that exceed 90%, as characterized by transmission electron microscopy (TEM), scanning electron microscopy (SEM), and atomic force microscopy (AFM) techniques as well as Raman mapping and ellipsometric mapping. Further, successive depositions of monolayer films result in bilayers and trilayers as indicated by optical transmission spectra and electrical sheet resistance measurements. Our lateral self-assembly method results in a mosaic-type high-coverage monolayer MXene films achieved rapidly (~30 min) and without the need for any specialized instrumentation, paving the way for applications that require largearea uniform MXene films.
Results and Discussion
We synthesized Ti3C2Tx MXene from its MAX phase (Ti3AlC2) according to the MILD synthesis method 45 (see Materials and Methods section for details). Previously, we have shown using Ti3C2Tx and Ti2CTx MXenes that liquid−liquid interfacial self-assembly can be used to generate large-area suspended flake assemblies, which facilitates transfer onto substrates without the need for alignment of flake and substrate feature (e.g., nanoaperture) 44 . In this study, we refined and scaled-up our approach to the wafer scale, as schematically illustrated in Figure 1a. After chemical exfoliation of Ti3C2Tx and delamination of the monolayer flake dispersion in water, we diluted the dispersion to an optimal concentration controlled by UV-vis absorbance spectrometry using an Eppendorf BioSpectrometer (New York, USA) for monolayer assembly with ( Figure S1). Then, we added methanol to the MXene dispersion in a 1:8 methanol:water volume ratio (step i). Next, we submerged the substrate of interest in a chloroform bath in a PTFE dish, and drop cast a few droplets (ca. 500 µl) of the prepared mixed dispersion on the chloroform to initiate the self-assembly process (step ii). Immediately following this, flakes begin to coalesce at the chloroform-water interface, forming a large-area two-dimensional film within 5-10 minutes depending on the final film dimensions. Once assembled, as visually seen by no increase in the overall size of the assembly, the film is transferred to the substrate by slowly emerging it from the chloroform bath, followed by drying on a hot plate at 100°C for 30 minutes to complete the transfer process (step iii). Depending on the desired application or characterization method, different substrate types and sizes could be used.
To investigate the properties of the fabricated films and their potential applications, we studied structure, uniformity, fractional area of coverage, optical and electrical properties of the fabricated films. Figure 1b shows an AFM image of the transferred monolayer Ti3C2Tx film on a SiNx substrate (see also Figure S2 (a-i)). These images confirm the mosaic structure and effective monolayer nature of the films with well-packed and edge-to-edge arrangements and partial overlaps at the edges. The corresponding line height profiles of monolayer flakes are shown in Figure 1b and a monolayer Ti3C2Tx thickness is measured around 1.8 nm (see also Figure S2g). Theoretically, the thickness of a Ti3C2Tx flake is 0.98 nm; however, height artifacts in AFM are common for hydrophilic materials due to presence of water beneath/above the flake, presence of functional groups on the MXene basal plane, and interactions of the bound surface groups of Ti3C2Tx flakes with the substrate, all resulting in a typical AFM-based thickness of 1.5 -2 nm for Ti3C2Tx monolayer flakes [46][47] , which our measurements agree with. To further corroborate the film thickness, we performed ellipsometric measurements. In Figure 1c, d, we present an optical image and an ellipsometric thickness map of a 3" Si wafer onto which we deposited a monolayer of Ti3C2Tx. The optical image shows a partially covered wafer before the complete evaporation of water. Around the edges of the wafer there is no coverage, which provides an optical contrast as indicated by arrows in the inset image (blue = MXene monolayer, red = bare Si). To assess the thickness variation across the wafer, we mapped the ellipsometric quantities Ψ and Δ as a function of wavelength at 81 points along the wafer and produced a color map using Voronoi interpolation that represents the MXene layer thickness. The dashed yellow circle shows the area (4 cm in diameter) within which ellipsometric mapping was carried out, and the yellow oval shape on the image roughly represents the size/shape of the ellipsometric incident light beam (~1.2 mm beam diameter). Using these data and published optical data for n, k for a Ti3C2Tx film for various wavelengths (see Figure S2) 39 , we constructed a 3-layer model of our sample (Si, SiO2, and MXene layer), and fit our mapping data to obtain a thickness map (see Materials and Methods section for more details). The areas covered with film show thickness values in the range of 0.9 -1.5 nm, which we speculate corresponds to monolayer Ti3C2Tx film along with some areas containing partial bilayer due to laterally overlapping flakes in the assembly.
For a deeper insight into the quality of the ellipsometric data, we show in Figure 2a violin plots of Ψ and Δ values measured at different points on the mapping experiment in Figure 1d and compare these with values for a bare Si substrate. The difference between Ψ and Δ values for these two substrates confirms the presence of the film over the entire area of the wafer. For each wavelength measured, we find a normally-distributed variation in Ψ and Δ values for the MXene monolayer film. An analogous map of the measured SiO2 thickness is shown in Figure S5. Moreover, the outlier points of the violin plots fit the data for bare Si substrate, which identifies a few uncovered regions on the wafer (Figure 1c, red arrow, and Figure 1d, red region).
Another method to assess the coverage and uniformity of Ti3C2Tx films on a substrate is Raman spectroscopy. Sarycheva et al 48 recently showed that the Raman spectrum of a single Ti3C2Tx flake has two characteristic peaks, and further, observed an increase in peak intensities upon multilayer stacking of these flakes. Unlike transition metal-dichalcogenides (TMDs), multilayer Ti3C2Tx flakes didn't show any peak shifts, which relates to the substantial interlayer gap that exists between MXene sheets (~5-6Å) 2 . Figure 2b shows a Raman spectrum of monolayer Ti3C2Tx film on a Si/SiO2 substrate, which shows two characteristic Ti3C2Tx peaks centered at ~300 cm -1 and ~975 cm -1 . The strong peak at 521 cm -1 corresponds to the Si substrate. Therefore, to provide some assessment of coverage we obtained Raman spectra from a 68.5 µm x 44.5 µm area and mapped the peak height intensity of MXene peak at ~300 cm -1 to the Si peak. While we cannot derive thickness values based on these spectra, we find that MXene peaks are everywhere in the region, fluctuating in relative intensity by <25%. Three spectra (a, b, and c) from three different areas (green, yellow, and orange) on the map in Figure 2b show the mean and extremities of our measurements.
Next, to gain a deeper insight into the packing density and fractional area coverage of the films we used AFM, SEM, and TEM characterization. To allow us to inspect the resulting MXene films using electron microscopy, we lifted MXene monolayers onto 5x5 mm 2 Si chips hosting freestanding 50-nm-thick silicon nitride (SiNx) membranes at their centers. Figure S6 show SEM images of monolayer Ti3C2Tx film transferred on the Si chips. The SiNx membranes appear as dark areas beneath the Ti3C2Tx film because electrons are only weakly scattered from the ultrathin membrane. We used ImageJ software 49 to calculate the fractional area covered with monolayer Ti3C2Tx films based on finding the optimum threshold intensity that differentiates the substrate from the monolayer and quantifying the substrate area. In Figure 3a-c, we show AFM, SEM, and TEM images of MXene monolayer films, respectively, along with corresponding postanalysis images below after thresholding, which were used in our coverage analysis. For the AFM measurements we used a SiNx substrate, whereas for SEM and TEM measurements we used 50-nm-thick freestanding SiNx membranes supported by a Si chip. For these images we find MXene coverage values of 93%, 91%, and 84.1%, respectively, with similar coverage values obtained in other areas we inspected (see Figures S7-S8 for other images). We conclude that our lateral assembly method produces a mosaic structure of closely-packed flakes that cover >90% of the substrate following transfer.
So far, we have shown that the lateral self-assembly method is capable of producing ultrathin, uniform monolayer and packed Ti3C2Tx films. To investigate sequential assembly of bilayers and trilayers, we repeated our transfer method by lifting a monolayer film onto a substrate, drying the substrate, re-immersing the substrate in chloroform, repeating lateral self-assembly of MXene flakes at the interface, and emerging the substrate from the chloroform phase. We examined the resulting multilayer films using optical spectroscopy and electrical conductivity measurements. Figure 4a shows the UV-vis absorbance spectra of monolayer, bilayer, and trilayer Ti3C2Tx films on a glass substrate. As seen in the spectra, absorbance increases with increasing number of layers. Inset shows the absorbance for monolayer, bilayer, and trilayer films at 550 nm, showing that although the increase is not linear, it is in qualitative agreement with previous studies 43 .
Due to the inherent in-plane conductivity of Ti3C2Tx flakes, we examined how the number of MXene layers affects the electrical properties of the films. We used the van der Pauw method (vdP) to measure the sheet resistance of the films 50 using a home-made apparatus and a Keithley 2401 source meter (Tektronix, Inc., USA). Figure 4b shows the schematic of the apparatus, which consists of a square grid of four gold electrodes separated by 10 mm. Four wires connect to each electrode at points 1, 2, 3, and 4 and attach to the Keithley source meter (see Figure S9 for an image of the experimental set-up). Figure 4b also shows a schematic topview image of the electrodes and depicts the four edges of the film, two horizontal (H1 and H2), and two vertical (V1 and V2), where conductivity measurements are carried out. First, current is sourced along each edge (horizontal and vertical) and voltage is measured along the opposite edge. This is repeated for all combinations, which results in four independent measurements. Current vs. voltage (IV) curves of these measurements for a monolayer Ti3C2Tx film transferred onto a glass substrate are shown in Figure 4b. Due to the uniformity of the film within this 10 mm x 10 mm area, the four IV curves are identical. Figure 4c shows one representative IV curve for each monolayer, bilayer, and trilayer Ti3C2Tx films measured after the sequential transfer of each layer. From the resistance values obtained by IV measurements, sheet resistance values can be calculated (see Materials and Methods section for details). Figure 4d shows the sheet resistance values for two samples of Ti3C2Tx films on a glass substrate with a thickness of 1, 2, and 3 layers. As expected, as the thickness of the film increases, the sheet resistance decreases. In general, our films show higher sheet resistance (9-11 kΩ/sq) compared to the recent study by Yun et al 43 , which showed resistance of 1.5 kΩ/sq for a monolayer film. We attribute this discrepancy to the mosaic morphology of our films, in which flake-to-flake contact is edge-to-edge with minimal overlapping. Despite the higher resistance we find that our films exhibit ohmic response and that resistance drops with increasing number of layers.
Finally, we have compared the advantages and disadvantages of the various 2D material coating methods in Table S1. Compared to other methods, one advantage of our wafer-scale lateral self-assembly is the resulting uniform film structure across the entire film area, which is unlike the spin-coating method that results in a compact monolayer film only at the center of the substrate. Moreover, compared to the Langmuir transfer method, our assembly method is much faster (under 30 minutes for film formation on a 3" wafer scale), and further, does not require any special apparatus apart from a PTFE or glass dish. It is worthwhile to mention that, as discussed in our previous study 44 , the driving force for film formation at the liquid-liquid interface is a combination of repulsion force, capillary force, and surface tension, which results in a delicate organization of the resulting mosaic film structure. Our lateral assembly method thus results in smoother and more organized films than those produced by spontaneous absorption of flakes from solution, for example.
Conclusion
In summary, we have shown here that lateral self-assembly of MXene films on a liquid-liquid interface produces uniform mosaic-type monolayers that span over large areas (we have coated up to 3" substrates). We have studied the film morphology of our coated substrates at various scales using a combination of tools that include AFM, SEM, and TEM imaging, as well as ellipsometric and Raman mapping. Moreover, the sequential transfer of single-layer films on a substrate enables the fabrication of multilayer films without peeling off the preceding layers. This was confirmed by the change in sheet resistance and optical properties of the films upon sequential deposition of multilayer films. While we have focused here on large-scale selfassembly of Ti3C2Tx flakes, our method should be compatible with a variety of other members of the MXene family. Considering the 30+ different MXene compositions experimentally synthesized so far, our method offers an attractive approach for producing large-area uniform films that may find use in applications that require hydrophilic and electrically conductive films such as transparent electrodes for bioelectronics, illuminated displays, and other devices.
Materials and Methods:
MXenes synthesis. Ti3C2Tx MXene flakes with large lateral sizes were synthesized according to previous reports in the literature 46 . Initially, for every 1 g of the MAX phase, the mixed saltacid etching solution was prepared by adding 1 g of LiF powder (98.5%, Alfa Aesar) to 20 mL of 6 M HCl solution (ACS grade, BDH) followed by stirring for 15 min to completely dissolve the LiF powder in the acid solution. The etching process was started by slowly adding 1 g of MAX phase powder (Ti3AlC2 synthesized according to previous work 51 ) to the etching solution. The etching container was placed into an ice bath during the addition of MAX phase to avoid excessive heat generation due to the exothermic nature of the reaction. The etching solution was stirred at 550 rpm (by using a PTFE-coated magnetic bar) continued for 24 h at 35 °C. After 24 h, the solution containing etched MXene multilayers was divided into four different centrifuge vials, and 45 mL DI water was added to them to start the washing process. The solutions were then shaken by hand for 1 min and centrifuged at 3,500 rpm (Eppendorf 5810R) for 3 min, after which the supernatant was poured out. The washing process was continued for several times, each time adding 45 mL DI water followed by manual shaking of the solutions for 2 min, and then centrifuging them at 3,500 rpm for 3 min until a dark green supernatant was observed (pH > 4.5).The supernatant after this stage is called delaminated Ti3C2Tx dispersion. The initial delaminated solution (the first supernatant after washing step was complete) was poured out and DI water was added to the sediments. The solutions were shaken for another 2 min and this time centrifuged at 3500 rpm for 1 h to collect the large flake size MXene solutions (pH ∼5).
Wafer-scale Lateral Self-assembly of MXenes and Transfer on Substrate. Monolayer MXene film formation was carried out by optimizing our previously reported lateral interfacial self-assembly method 44 . MXene dispersion was first diluted to the desired concentration and then mixed with methanol to a 1:8 volume ratio. The substrate of desire was submerged in a chloroform bath and dispersion was added to the chloroform until the droplet size enlarged to the size of the substrate. After flakes were self-assembled on the droplet/chloroform interface, excess chloroform was removed from the bath and the film level lowered down to reach the substrate. Afterward, the substrate was lifted and baked at 100°C for ~30 min to dry. The substrates were kept in a vacuum desiccator until use.
Imaging Techniques. For SEM and TEM imaging, monolayer Ti3C2Tx films were transferred onto Si chips with fabricated 50 mm x 50 mm 50-nm-thick SiNx membranes at their center. For AFM imaging, a film was transferred onto SiNx substrates. High-resolution TEM imaging was done using a JEOL 2010FEG operating in bright-field mode at 200 kV. Scanning electron microscopy (SEM) was done using JSM-IT200 at 3.0 kV accelerating voltage. AFM measurements were carried out by the FastScan AFM instrument (Bruker Instruments, Billerica, MA) using ScanAsyst cantilevers (Bruker Instruments) performing at the FastScan device's ScanAsyst Mode.
Ellipsometry Spectroscopy and Mapping: Ellipsometry measurements were carried out using HORIBA UVISEL 2 Ellipsometer (HORIBA Scientific, USA) on monolayer Ti3C2Tx films deposited on Si substrate. Real (n) and imaginary (k) refractive indices previously developed by Dillon et al 39 were used to make a reference file for Ti3C2Tx using DeltaPsi2 software. Using reference files for modeling, the optical constants are fixed for fitting, and only the thickness changes. We produced the model using DeltaPsi2 software based on the layered structure, consisting of a Si layer, a SiO2 layer (due to the presence of native oxides on Si surface), and a Ti3C2Tx layer ( Figure S4 inset). Figure S4 shows how well our model fits the measured Ψ and Δ values obtained from a monolayer Ti3C2Tx film. Based on these parameters, the thickness of the film is calculated to be 0.95 Å which matches the theoretical thickness of a single Ti3C2Tx layer. For ellipsometry mapping, the DeltaPsi2 software mapping recipe feature was used to design automated measurements at a spectrum range of 300 to 800 nm at the 70-degree incidence angle on a pre-designed circular grid with a 40 mm diameter and 81 points. The measurement results (Ψ and Δ values) at each point were fitted to the layered model consisting of Si/SiO2/Ti3C2Tx to calculate the thickness. Using X, Y coordinates of grid points, and thickness values, the Voronoi interpolation method was adopted, using Igor Pro software, to map the measurement results to the whole wafer area.
Raman Spectroscopy and Mapping: The Raman measurements were carried out using a Thermo Scientific DXR2xi Raman Imaging Microscope on monolayer Ti3C2Tx films deposited on 300-nm-thick SiO2 substrate. The excitation source was a 455 nm laser with 0.5 µm spot size. The laser power was kept at 2 mW with a 1 sec exposure time for 10 scans per position. Mapping was done on a 68.5 µm x 44.5 µm area using OminiCX Software.
UV-vis Absorbance Measurement: UV-vis absorbance measurements were carried out using a PerkinElmer Lambda 35 UV− vis−NIR spectrophotometer on monolayer, bilayer, and trilayer Ti3C2Tx films deposited on glass substrates in the range of 330 to 1100 nm. The effect of the glass substrates was removed by placing an identical clean glass slide in the reference beam position of the spectrophotometer, and hence, the absorbance spectra (shown in Figure 4a) are representative for the Ti3C2Tx films without any contribution from the substrate.
Conductivity Measurement: The conductivity measurements were carried out using Keithley 2401 source meter and a home-made apparatus on monolayer, bilayer, and trilayer Ti3C2Tx films deposited on glass substrates. The apparatus consists of four gold electrodes located at four corners of a square with a 10 mm length (shown in Figure 3b). To calculate sheet resistance, two sets of measurements were carried out. The first set is to measure the voltage drop across each horizontal line of the square (H1 or H2) by the sourcing current across the opposite edge, followed by calculating the resistance from the slope of the I-V curves (RH1, RH2). The second set is to measure the voltage drop across each vertical line of the square (V1or V2) by sourcing current across the opposite edge (RV1, RV2). After the four measurement sweeps were taken, the sheet resistance was calculated as follows:
RHorizontal = RVertical= + , Rs= sheet resistance
Notes
The authors declare no competing financial interest. References. Page 12. Figure S1. UV-vis absorbance of diluted Ti3C2Tx dispersion optimized for assembly of monolayer flakes at the liquid-liquid interface.
Figure 1 .
1Lateral self-assembly of monolayer MXenes on a liquid-liquid interface and transfer on a substrate. (a) Schematic illustration of self-assembly and transfer steps of monolayer Ti3C2Tx film. The atomic structure of Ti3C2Tx MXene is shown with O -, OH -, and Fas functional groups on the surface. (b) AFM image of the transferred Ti3C2Tx monolayer film on a SiNx substrate. Corresponding line profiles at the bottom of the image show thickness of ~ 1.8 nm for a monolayer film. (c) Image of the Si wafer with transferred monolayer Ti3C2Tx film on top used for ellipsometric measurement. The yellow dashed circle shows the area (with 4 cm diameter) within which ellipsometric mapping was carried out. The yellow oval shape shows the approximate shape and size of the incident beam. The magnified image of the black dashed square shows the edges of the Ti3C2Tx film on the Si substrate. The blue arrow shows the film and the red arrow shows the bare substrate. (d) Ellipsometric mapping of Ti3C2Tx film thickness on a circular area with a diameter of 4 cm.
Figure 2 .
2Characterization of monolayer Ti3C2Tx film uniformity. (a) Violin plots of obtained Ψ and Δ values from 81 points in a circular area of a transferred Ti3C2Tx film (shown with blue arrows). The red arrows show the Ψ and Δ values obtained from the bare Si wafer. (b) Raman spectrum obtained from a monolayer Ti3C2Tx film on Si/SiO2 substrate. The strong peak of Si at 521 cm −1 originates from the substrate. Inset shows the Raman mapping of peak height ratio of MXene: Si for a 68.5 µm x 44.5 µm are. The corresponding Raman spectra of points a, b, and c on the map are shown at the bottom of the image.
Figure 3 .
3Characterization of fractional coverage of monolayer Ti3C2Tx films. (a) Top: AFM image of the transferred monolayer Ti3C2Tx films on a SiNx substrate: Bottom: Analyzed image using ImageJ software showing 92.8% coverage of substrate by the film. (b) Top: SEM image of a transferred monolayer Ti3C2Tx film on 50-nmthick freestanding SiNx membrane: Bottom: Analyzed image using ImageJ software showing 91.8% coverage of substrate by the film. (c) Top: Bright-field TEM image of the transferred monolayer Ti3C2Tx films on a ~50 nm thick SiNx membrane: Bottom: Analyzed image using ImageJ software showing 84.1% coverage of substrate by the film. All scale bars = 10 µm, and black areas represent uncovered areas in all bottom images.
Figure 4 .
4Optical and electrical characterization of monolayer and multilayer Ti3C2Tx films. (a) UV-vis absorbance spectra of monolayer, bilayer, and trilayer Ti3C2Tx films. Inset shows the absorbance of the films at 550 nm. (b) Schematic of vdP apparatus used for sheet resistance measurement of the films. Bottom: Top view of the apparatus showing the location of corresponding electrodes from 1-4 and edges of the square area where electrical measurements were carried out (H1, H2, V1, and V2). Corresponding IV curves of current sweep across each edge is shown on the right. (c) Representative IV curves of monolayer, bilayer, and trilayer Ti3C2Tx films. (d) Sheet resistance of Ti3C2Tx films versus the number of Ti3C2Tx layers. Markers show the average of each sheet resistance values and the lines show the range of the values for six measurements on the sample.
Figure S1 .Figure S2 .
S1S2UV-vis absorbance spectrum of the diluted Ti3C2Tx dispersion used for monolayer film assembly (Page 3) AFM images of transferred monolayer Ti3C2Tx film on SiNx substrates (Page 4).
Figure S3 .
S3Real (n) and imaginary (k) refractive indices of Ti3C2Tx film (Page 5).
Figure S4 .
S4Ellipsometric Ψ and Δ values obtained from Ti3C2Tx film and the model produced from the layered structure (Page 6).
Figure S5 .
S5Ellipsometric mapping SiO2 layer thickness (Page 7).
Figure S6 .
S6SEM images of transferred monolayer Ti3C2Tx film on 50-nm-thick SiNx membranes (Page 8).
Figure S7 .
S7AFM images of the monolayer Ti3C2Tx films on SiNx substrates with the corresponding analyzed image (Page 9).
Figure S8 .
S8SEM images of the monolayer Ti3C2Tx films on 50-nm-thick SiNx membranes with the corresponding analyzed image (Page 10).
Figure S9 .
S9Image vdP experimental set-up for measurement of TI3C2Tx sheet resistance (Page 11).
Figure S2 .Figure S3 .
S2S3(a-i) AFM images of the transferred monolayer Ti3C2Tx film on a SiNx substrate with the corresponding Z scales. Corresponding line profiles of (g) shows a thickness of ~ 1.8 nm for a monolayer film and 1.6 nm at the overlap. (All scale bar = 4 µm) Real (n) and imaginary (k) refractive indices of a Ti3C2Tx film.
Figure S4 .
S4Ψ and Δ values obtained from a monolayer Ti3C2Tx film on a silicon substrate and Ψ and Δ values obtained from the model based on the layered structure (Si/SiO2/Ti3C2Tx) shown in the inset.
Figure S5 .
S5Ellipsometric mapping of the SiO2 layer thickness, beneath the MXene layer, over a circular area of a Si wafer with a diameter of 4 cm.
Figure S6 .Figure S7 .
S6S7(a-e) SEM images of the transferred Ti3C2Tx monolayer films on Si chips. The black area shows the free-standing 50-nm-thick SiNx membranes. (All scale bars = 20 µm) (a-c) AFM images of the transferred monolayer Ti3C2Tx films on SiNx substrates with their corresponding analyzed images (on the right) using ImageJ software. The analyzed images show substrate coverage of (a) 93.8% (b) 93.9% and (c)93.4 % by the film, respectively.
Figure S8 .
S8(a-c) SEM images of the transferred monolayer Ti3C2Tx films on 50-nm-thick SiNx membranes with their corresponding analyzed images (on the right) using ImageJ software. The analyzed images show substrate coverage of (a) 94.5% (b) 89.8% and (c) 90.1% by the film, respectively.
Figure S9 .
S9VdP experimental set-up for measurement of Ti3C2Tx sheet resistance.
Table of Contents
ofTable S1. Comparison between methods of large-scale film fabrication of 2D materials with single-layer precision (Page 2).
Table S1 .
S1Comparison between methods of large-scale film fabrication of 2D materials with single-layer precision No need for specialized apparatus Freedom in lateral size variation Requires further transfer onto a substrate low dispersibility of graphene, MoS2, and WS2 in water and common solventsInterfacial Assembly (liquidliquid and liquid-air)Method of film fabrication
2D Materials used
Advantages
Disadvantages
Spin Coating
Graphene oxide 1-3
Short fabrication time
Directly on a substrate, no
further transfer is needed
Non-uniform monolayer
coverage vs. radial distance from
center of substrate.
Langmuir-Blodgett (LB)
Graphene oxide 2, 4, 5 , 6-8
Graphene 9
Ti2CTx 10
Well-packed and
structurally organized film
Freedom in lateral size
variation
Need for specialized apparatus
and skill
Long fabrication time
Requires further transfer onto a
substrate
Layer-by-layer Assembly
Reduced graphene oxide 11-
12
Ti3C2Tx 13
No need for specialized
apparatus
Directly on a substrate, no
further transfer is needed
Spontaneous absorption (no
lateral force)
Relatively open, loose and
disordered film
Interfacial Assembly
(liquid-liquid and liquid-air)
Graphene 14
MoS2 and WS2 15
Ti3C2Tx 16
Ti3C2Tx (this work)
No need for specialized
apparatus
Freedom in lateral size
variation
Well-packed and
structurally organized film
Dispersibility of Ti3C2Tx
in water and common
solvents
Requires further transfer onto a
substrate
AcknowledgmentWe acknowledge the National Science Foundation (DMR 1710211) for funding this work. The authors thank Prof. Yuri Gogotsi and Dr. Aaron Fafarman for providing us the values for modeling ellipsometric measurement. The authors acknowledge Dr. Tzahi Cohen-Karni and Raghav Garg (Carnegie-Mellon University) for loaning us the vdP apparatus. The authors thank Dr. Joshua Gallaway and Matthew Kim for use of their Raman spectroscopy and mapping tool. We thank Dr. David Hoogerheide for providing us with the 3" Si wafer for ellipsometric measurements.
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"Thin Position through the lens of trisections of 4-manifolds",
"Thin Position through the lens of trisections of 4-manifolds"
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"Román Aranda "
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Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to {1, . . . , 1}, and give a relation between the width of M and its double M ∪ id ∂ M . In particular, we describe how to obtain genus 2g + 2 and g + 2 trisection diagrams for sphere bundles over orientable and non-orientable surfaces of genus g, respectively. By last, we study the problem of describing relative handlebodies as cyclic covers of 4-space branched along knotted surfaces from the width perspective.
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10.2140/pjm.2021.313.293
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"https://arxiv.org/pdf/1805.08857v3.pdf"
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1805.08857
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9c4f5712770f64378fe8e7197bd267539ea89eab
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Thin Position through the lens of trisections of 4-manifolds
Román Aranda
Thin Position through the lens of trisections of 4-manifolds
Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to {1, . . . , 1}, and give a relation between the width of M and its double M ∪ id ∂ M . In particular, we describe how to obtain genus 2g + 2 and g + 2 trisection diagrams for sphere bundles over orientable and non-orientable surfaces of genus g, respectively. By last, we study the problem of describing relative handlebodies as cyclic covers of 4-space branched along knotted surfaces from the width perspective.
Key words and phrases. Heegaard Splittings, Kirby diagrams, Thin position, Trisections of four manifolds.
Introduction
In 1994, M. Scharlemann and A. Thompson introduced the notion of thin position of 3-manifolds. In their work [16], they described thin position as follows:
"Any closed orientable 3-manifold M can be constructed as follows: begin with some 0-handles, add some 1-handles, then some 2-handles, then some more 1-handles, etc... and conclude by adding some 3-handles. Of course M can be built less elaborately: in the previous description, all the 1-handles can be added at once, followed by all the 2-handles. This corresponds to a Heegaard splitting of the manifold; the 0-and 1-handles comprise one handlebody of the Heegaard splitting, the 2-and 3-handles to the other. The idea of thin position is to build the manifold as first described, with a succession of 1-handles and 2-handles chosen to keep the boundaries of the intermediate steps as simple as possible."
The complexity that the position of Scharlemann and Thompson seeks to minimize is the width of a handle decomposition of a 3-manifold. It is in terms of the genera of the surfaces S between the 1-and 2-handles.
In dimension four, we can apply a similar reasoning to talk about thin position of a closed 4-manifold M if we add the necessary 4-handles at the end of the process. In this context, the action of alternating between 1-, 2and 3-handles becomes a suitable decomposition of a handle decomposition M and the question now is what do we want "as simple as possible" to mean.
In 2013, D. Gay and R. Kirby [7] showed that every closed smooth 4manifold admits a trisection. A trisection of a closed 4-manifold M is a decomposition of M into three 4-dimensional 1-handlebodies with pairwise intersection being a connected 3-dimensional handlebody and triple intersection a connected closed surface. In [7] and [9], a correspondance between trisections and certain handle decompositions was described. In such decompositions, all the 1-handles are added at once, followed by all the 2-handles and all the 3-handles. The 0-and 1-handles comprise the first 1-handlebody of the trisection, the 3-and 4-handles form the second 1-handlebody, and the third 1-handlebody is given by a suitable neighborhood of the 2-handles. With this in mind, one can think of trisections as the 4-dimensional analogue of Heegaard splittings of M with the "trisection surface" being the triple intersection of the 4-dimensional 1-handlebodies.
In this paper, we use ideas of trisections of 4-manifolds from [9] to define the width of a handle decomposition of a connected 4-manifold. This allows us to define the width of a 4-manifold M by looking at the minimum width among all possible handle decompositions of M . Similar to [16], the width of M is a multiset {c i } where c i is a function of some "trisection-like surfaces" for the attaching link of the i th 2-handles of M (see Section 3.1 for the detailed definition).
In Section 3, we describe situations when the width is not minimal. The interested reader should compare these situations with the rules to decrease the width introduced in [16]. We also determine all 4-manifolds satisfying width(M ) = {1, . . . , 1}. We show Theorem 3.11. Let M be a connected 4-manifold satisfying width(M ) = {1, . . . , 1}. Then M is diffeomorphic to a (boundary) connected sum of copies of S 1 × S 3 , S 1 × B 3 , and linear plumbings of disk bundles over the sphere.
In Section 4, we define the notion of the nerve of a 4-manifold X obtained by adding 2-handles to Y 3 × [0, 1] along Y × {1}, basically the same as the nerve of a trisection for a closed 4-manifold. In Subsection 4.2, we show how the nerves of the suitable relative handlebodies around the 2-handles carry the width information of the handle decomposition.We use this to show that the operation of turning a handle decomposition up-side-down leaves the width invariant. We also prove In Example 4.11, we give an upper bound for the width of sphere bundles over closed surfaces. In particular, we are able to obtain low genus trisection diagrams for such manifolds. Denote by X g,n , Y g,n the disk bundles of euler number n over an orientable and non-orientable surface of genus g, respectively. We describe how to obtain trisection diagrams of genus 2g + 2 and g + 2 for D(X g,n ) and D(Y g,n ), respectively (see Example 4.11). It is important to mention that trisection diagrams for such manifolds have been described before in [7] and [5]. But the existance of lower genus diagrams was proven in [2] with no explicit drawings. In Figure 4, we draw explicit lower genus diagrams for D(X 1,0 ) and D(Y 1,1 ).
In Section 5, we study the problem of describing relative handlebodies as cyclic covers of 4-space branched along knotted surfaces from the width perspective. We use ideas from [10], [11] and [12] to prove Theorem 5.10. Let M be a closed 4-manifold. Then M is the p-fold cyclic cover of S 4 branched along a knotted surface K 2 ⊂ S 4 if and only if M admits a p-symmetric trisection diagram.
In [16], M. Scharlemann and A. Thompson proved that 3-manifolds of width < {5} are 2-fold branched covers of connected sums of S 1 × S 2 . Subsection 5.2 is a digression on an attempt of lifting this result to 4-manifolds with connected boundary.
Acknowledgements. The author would like to thank the Topology group of the University of Iowa for listening and commenting on previous versions of this work. Special thanks to Maggy Tomova, Charles Frohman and Mitchell Messmore.
Preliminaries
Along this work, all manifolds will be compact, smooth, oriented and connected unless the opposite is stated. For A ⊂ B an embedded submanifold of any dimension, η(A) will denote the closed tubular neigborhood of A in B.
For 0 ≤ k ≤ n, an n-dimensional k-handle is a copy of D k × D n−k , attached to the boundary of an n-manifold X along (∂D k ) × D n−k by an embedding ϕ : (∂D k ) × D n−k → ∂X. By definition, 0-handles are attached along the empty set. In dimension 3, 1-handles are attached along pairs of disjoint disks, 2-handles along annuli, and 3-handles along 2-spheres. In dimension 4, 1-handles are attached along pairs of disjoint 3-balls, 2-handles along solid tori (hence framed knots), 3-handles along thickened 2-spheres, and 4-handles along copies of S 3 in ∂X.
A handle decomposition of X (relative to ∂ − X) is an identification of X with a manifold obtained from ∂ − X × [0, 1] by attaching handles along ∂ − X × {1}. We will usually start building X with ∂ − X = ∅ thus start by adding some 0-handles. The action on ∂X n of a k-handle addition h k = D k × D n−k is by surgery on X. In other words, the new boundary is given by
∂X − ϕ ((∂D k ) × D n−k ) ∪ ϕ D k × ∂(D n−k )
. For a more detailed review of the calculus on handlebody diagrams, Chapter 1 of [1] or Chapters 4 and 5 of [8] are good references.
The 4-manifold obtained by attaching a handle h to the 4-manifold X will be denoted by
X[h]. If Y = ∂X, we will write Y [h] to denote the surgered 3- manifold ∂(X[h]). If L is a framed link in Y 3 , Y [L]
will denote the surgered 3−manifold. For simplicity, a 1-handlebody will mean a 4-ball with some 4-dimensional 1-handles attached and the three dimensional analugue will be called just handlebody.
A Heegaard splitting of a closed 3-manifold Y is a decomposition of Y in two handlebodies with common part a connected surface of genus g. The smallest such g is the Heegaard genus of Y , HG(Y ).
A trisection of a closed 4-manifold M is a decomposition of M in three 1-handlebodies M = X 1 ∪ X 2 ∪ X 3 so that the double intersections are connected 3-dimensional handlebodies, X i ∩ X j = H i,j ; and the triple intersection is a closed connected surface of genus g, Σ. It follows from the definitions that ∂X i = H i,k ∪H i,j is a Heegaard splitting for the boundary of X i , which is homeomorphic to the connected sum of some copies of S 1 × S 2 . The triplet (Σ; H 1,2 , H 2,3 , H 3,1 ) is called the nerve of the trisection.
Let L ⊂ Y be a link in a connected closed 3-manifold. Denote by t Y (L) the smallest number of embedded arcs t ⊂ Y with t ∩ L = ∂t such that Y − η(L ∪ t) is a handlebody. t Y (L) is called the tunnel number of L in Y and t is a system of tunnels for L. Note that the surface Σ = ∂η(L ∪ t) induces a Heegaard splitting of Y = H ∪ Σ H with L ⊂ core(H). Define g(Y, L) to be the smallest genus of a Heegaard surface for Y with L being a subset of the core of one of the handlebodies. We have
t Y (L) = g(Y, L) + 1
The equation above will serve to us as an equivalent definition of tunnel number. If L = ∅, define t Y (∅) = HG(Y ) + 1. We will need the following computation of the tunnel number of split links.
Lemma 2.1. Let K 1 , K 2 be links inside closed three-manifolds X 1 , X 2 , respectively. Let X = X 1 #X 2 and K = K 1 ∪ K 2 ⊂ X. Then t X ( K) = t X 1 (K 1 ) + t X 2 (K 2 ) + 1 Furthermore, if K 1 = ∅ and K 2 = ∅, then t X ( K) = t X 1 (K 1 ) + HG(X 2 )
Proof. We will prove the equation for K 1 , K 2 = ∅, the other case is similar. One can see that the LHS is smaller by constructing a system of tunnels of cardinality t X 1 (K 1 ) + t X 2 (K 2 ) + 1.
We now prove LHS ≥ RHS. Let t ⊂ X be a system of tunnels for K in X with |t| = t X ( K), and let H = η( K ∪ t), H = X − int(H), Σ = H∩H . By construction, Σ is a Heegaard splitting of genus t X ( K)+1 for X −int(η( K)) with H a handlebody and H −int(η(K)) a compression body with inner boundary the collection of tori given by ∂η( K). An application of Haken's Lemma gives us the existance of a sphere S intersecting Σ in one simple closed loop, separating X in punc(X 1 ), K 1 and punc(X 2 ), K 2 , here punc(A) denotes A minus an open 3-ball. In other words, (Σ; H, H ) is the connected sum of Heegaard splittings for X 1 and X 2 , say (Σ i ; H i , H i ) for i = 1, 2, satisfying that H i − int(η(K i )) is a compression body with inner boundary the tori ∂η(K i ). But recall that a compression body deformation retracts to the wedge of its inner boundary with a finite collection of arcs in the interior of the compression body with endpoints on the inner boundary. Thus H i is the tubular neighborhood of the union of K i with a collection of t i arcs with endpoints on K i . This shows that t i ≥ t X i (K i ). By last, notice that t i + 1 = g(Σ i ) and, since Σ = Σ 1 #Σ 2 , we get t X ( K) + 1 = g(Σ) = g(Σ 1 ) + g(Σ 2 ) = t 1 + t 2 + 2 ≥ t X 1 (K 1 ) + t X 2 (K 2 ) + 2 Hence, LHS ≥ RHS.
The width of a Kirby diagram
In the following section, we define the notion of the width of a handle decomposition of a 4-manifold and the width of a 4-manifold. Following [16] closely, we describe specific cases when the width of a decomposition is not minimal and classify 4-manifolds with low width.
H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ C 2 ∪ D 2 ∪ E 2 ∪ . . . C N ∪ D N ∪ E N ∪ b 4 where b 0 , ∪C i , ∪D i , ∪E i , b 4
are collections of 0-handles, 1-handles, 2handles, 3-handles and 4-handles, respectively. We are thinking of building M in steps; starting with b 0 , then adding C 1 , then D 1 , then E 1 , etc.
For
each 1 ≤ i ≤ N , denote by Y i = ∂ (b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ C i ) and let L i ⊂ D i ∩ Y iY i . For a connected component Y of Y i , set L = L i ∩ Y . If L is not empty, define c(L , Y ) = 2t(L , Y ) + 1; if L = ∅ define c(L , Y ) = max{2HG(Y ) − 1, 0} where HG(Y ) is the Heegaard genus of Y . Define c i = c(L i , Y i ) := Y ⊂Y i c(L , Y ).index i ≥ 1 such that a j = b j for j ≤ i − 1 and a i < b i . We say that M is in thin position if there is M = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ . . . E N ∪b 4 with minimal width.
It is important to mention that the infimum is always achieved; so for smooth compact 4−manifolds the width always exists. For completeness, we include a proof of this fact in Lemma 3.3.
Remark 3.2. The decomposition b 0 ∪b 4 has empty width so width(S 4 ) = ∅. For non-empty diagrams, the width will be a finite multiset of odd integers c i ≥ 1, ∀i. Lemma 3.3. Let X be the set of all sequences of non-negative integers with finitely many non-zero elements endowed with the order described above. Then any non-empty set has a minimal element.
Proof. Let A ⊂ X be non-empty. For an element P ∈ X, we define P (k) ∈ N ∪ {0} to be the k−th largest element in P. Take B (0) = A and define the set A (1) = {P (1) |P ∈ B (0) }. Since A (1) ⊂ N ∪ {0} we can consider α 1 = min(A (1) ) and B (1) = {P ∈ B (0) |P (1) = α 1 }. We inductively define the sets B (n) = {P ∈ B (n−1) |P (n) = α n }, A (n) = {P (n) |P ∈ B (n−1) } and α n = min(A (n) ). By definition, the sequence (α n ) n is decreasing in N ∪ {0}, so it is stationary; say α n = α ≥ 0 for all n ≥ N . Let P ∈ B (N ) . By construction P (j) = α j for all j ≤ N . For l ≥ 0 we have the following inequalities,
α = P (n) ≥ P (N +l) ≥ α N +l = α
Thus P (m) = α for all m ≥ N . But P ∈ A, so all but finitely many components are non-zero. Hence α = 0 and P = min(A).
3.2. Ways to decrease the width.
Proposition 3.4. Let H = b 0 ∪C 1 ∪D 1 ∪E 1 ∪· · ·∪E N ∪b 4 be a thin position of M . For every i, define Z i = ∂b 0 [C 1 ∪ D 1 ∪ D 1 ∪ · · · ∪ D i ].
(1) Suppose that for some 1 ≤ i < N , one of the level 3-manifolds Z i ,
Z i [E i ], or Z i [E i ∪ C i+1
] is diffeomorphic to S 3 , say X. Then M − X has no components diffeomorphic to B 4 unless one of them is equal
to b 0 or b 4 . (2) If D i = E i = ∅ for some 1 ≤ i ≤ N , then i = N . (3) If C i = D i = ∅ for some 1 ≤ i ≤ N , then i = 1.
Proof.
(1) We will prove the case X = Z[E i ] = S 3 , the other cases are similar.
Within this case, we have two options. If M i ≈ B 4 , we can remove the handles {C l ∪D l ∪E l |l ≤ i} to reduce the width. If M − M i ≈ B 4 , we can remove the handles {C l ∪ D l ∪ E l |l > i} to reduce the width. (2) Suppose D i = E i = ∅ for some i < N . We will show that D i+1 and C i+1 are both non-empty. Suppose first D i+1 = ∅. In particular, c i+1 = c i + 2|C i+1 |. If we consider the new decomposition of M by merging (taking the union of) C i and C i+1 we will get the multiset {c j |j = i}, which has lower width, a contradiction. In a similar fashion, suppose that C i+1 is empty: By defining D i to be D i+1 , we obtain a new decomposition of M with width being the multiset {c l : l = i}, which is smaller, contradicting the minimality of the width.
We have shown that if
D i = E i = ∅, then D i+1 and C i+1 are both non-empty. Notice that Y i+1 = Y i [C i+1
]. Thus if we take the new decomposition given by merging C i and C i+1 , then the complexity c i+1 will not change. The width of this new decomposition will be the original width without the element c i ≥ 1, which is strictly lower.
Therefore, D i ∪ E i has to be non-empty if i < N . We say that a link L ⊂ Y in a 3-manifold splits if there exists a separating sphere S ⊂ Y disjoint from L. In such case, we can decompose Y = A# S B,
L = L A ∪ L B with L A ⊂ A and L B ⊂ B links in the corresponding pieces, L A , L B = ∅.
The following proposition studies how the width could change if one of the attaching links of the 2-handles splits in Y i .
Proposition 3.5. Let H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ E N ∪ b 4 be a handle decomposition for M . Suppose that there is a component Y * i ⊂ Y i so that the link L * i = Y * i ∩ D i splits in Y * i . Write Y * i = A# S B, L * i = L A i ∪ L B i and let D B i ⊂ D i be the 2-handles with attaching region L B i . Consider the new handle decomposition H of M obtained from H by separating the 2-handles of D i , adding first D i − D B i and then D B i . (1) If t B (L B i ) > HG(B) or t A (L A i ) > HG(A[L A i ]), then width(H ) < width(H). (2) If t B (L B i ) = HG(B) and t A (L A i ) = HG(A[L A i ]), then width(H ) > width(H).
Proof. In the new handle decomposition H , the i th level Y i got replaced by two levels:
Y A = Z i−1 [E i−1 ∪ C i ] and Y B = Y A [D i − D B i ]. Thus width(H ) = width(H) ∪ {c A , c B } − {c i } where c A = c(Y A , D i − D B i ) and c B = c(Y B , D B i ). We will show that c A = c i + 2 HG(B) − t B (L B i ) and c B = c i + 2 HG(A[L A i ]) − t A (L A i ) . The result follows since t B (L B i ) ≥ HG(B) and t A (L A i ) ≥ HG(A[L A i ]). By Part 1 of Lemma 2.1, c(Y * i , L * i ) = 2t A (L A i ) + 2t B (L B i ) + 1 Using the Part 2 of Lemma 2.1 and the equality Y A = Y i we get, c A = Y ⊂Y i −Y * i c(Y , L ) + c(Y * i , L A i ) = Y =Y * i c(Y , L ) + 2t Y * i (L A i ) + 1 = Y =Y * i c(Y , L ) + 2t A#B (L A i ) + 1 = Y =Y * i c(Y , L ) + 2t A (L A i ) + 2HG(B) + 1 = Y =Y * i c(Y , L ) + c(Y * i , L * i ) + 2HG(B) − 2t B (L B i ) + 1 =c i + 2 HG(B) − t B (L B i ) Similarly, since Y B = Y i − Y * i and Y * i [L A i ] ⊂ Y B , Part 2 of Lemma 2.1 gives us c A = Y ⊂Y i −Y * i [L A i ] c(Y , L ) + c(Y * i [L A i ], L B i ) = Y =Y * i [L A i ] c(Y , L ) + 2t Y * i [L A i ] (L B i ) + 1 = Y =Y * i [L A i ] c(Y , L ) + 2t A[L A i ]#B (L B i ) + 1 = Y =Y * i [L A i ] c(Y , L ) + 2t B (L B i ) + 2HG(A[L A i ]) + 1 =c(Y * i , L * i ) + 2HG(A[L A i ]) − 2t A (L A i ) + 1
The following lemma states that the width of a decomposition is invariant under certain types of sliding. More precisely, if we slide handles on latter levels along previous handles, the width does not change. It follows that if a level C i ∪ D i is a 1/2−pair of cancelling handles, we can remove such handles from the decomposition, decreasing the width. Proof. In terms of the attaching regions of the handles, sliding A along B corresponds to an isotopy of the attaching circles of A in Y j , which does not affect the values of c l . Hence the width does not change.
Lemma 3.6. Let H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ E N ∪ b 4 be a handle decomposition of M . For fixed 1 ≤ i < j ≤ N ,
Lemma 3.7. If H is a thin position of M , then C i ∪ D i does not contain a pair of cancelling 1-handle and 2-handle say, α ∪ β where α is the attaching region of a handle of C 1 and β ⊂ L i , so that α is unlinked with L i − {β} and β goes through α geometrically once. Similarly, D i ∪E i does not contain a pair of cancelling 2-handle and 3-handle
β 2 ∪ γ 3 where the attaching region of γ 3 is disjoint from L i − {β}.
Proof. We prove the contrapositive. Suppose that there is a 1/2-cancelling pair α ∪ β, β ∈ L i such that α is disjoint from L i − {β}. In order to erase the pair (α, β) from the diagram, we need to slide along β the other 2−handles of H intersecting α. By assumption such 2−handles can only be part of L k (k > i), so we can slide them along β without changing width(H) and then erase the pair (α, β), reducing the width. The case for 2/3-cancelling pairs is similar. It also follows from the fact that width(H op ) = width(H) (see Lemma 4.7), where H op is the up-side-down handle decomposition of H.
Manifolds with small width.
By taking the simplest handle decompositions of S 1 × S 3 and ±CP (2), we see that these manifolds have width equal to {1}. We will now show that no other 4−manifold has such width. We also study the equation
width(M ) = {1, 1, . . . , 1}. For a handle decomposition H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ E N ∪ b 4 , denote by Y i = ∂ (b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ . . . C i ) Z i = Y i [D i ] = ∂ (b 0 ∪ C 1 ∪ D 1 ∪ D 1 ∪ · · · ∪ D i ) Proposition 3.8. Let M be a closed 4-manifold satisfying width(M ) = {1}. Then M is diffeomorphic to either S 1 × S 3 or ±CP (2). Proof. Let M = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ b 4 be a decomposition for M of width equal to {1}, we can assume |b 0 | = 1. Suppose first D 1 = ∅, then C 1 is not empty and the Heegaard genus of Y 1 = # |C 1 | S 1 × S 2 is 1 since c 1 = 1. Thus |C 1 | = 1 and M admits a handle decomposition with only one 1−handle, i.e. M ≈ S 1 × S 3 . Suppose now that D 1 = ∅. We have that 1 = c 1 = 2t 1 + 1 = 2g 1 − 1, so t 1 = 0 and HG(Y 1 ) ≤ g 1 = 1. It follows that |C 1 | ≤ 1, giving us two options. If |C 1 | = 0, then Y 1 = S 3 and L 1 is an unknot with framing say n ∈ Z. Recall that M is closed so Y 1 [L 1 ] must be a connected sum of S 1 ×S 2 .
This forces n to be 0 or ±1, which implies that M is either S 4 or ±CP (2).
Assume the second case: D 1 = ∅ and |C 1 | = 1. Since t 1 = 0, a neighborhood of L 1 induces a genus one Heegaard splitting of S 1 × S 2 . Uniqueness of such splittings [4] implies that L 1 intersects C 1 geometrically once and that C 1 ∪ D 1 is a 1/2−cancelling pair for M . In particular we conclude M ≈ S 4 .
Proof. By definition of c(L i , Y i ), the condition 1 = c i implies HG(Y i ) ≤ 1 and HG(Y i [D i ]) ≤ 1 for all i. Since Y i [D i ] is a lens space and Z i [E i ] = Y i [D i ∪ E i ] is connected, we get |E i | ≤ 1. Note that |E i | = 1 if and only if Z i = S 1 × S 2 and E i ∩ Z i = {pt} × S 2 . Thus Z i [E i ] = SE i = ∅ ∀1 ≤ i < N .
In a similar vein,
HG(Y i ) ≤ 1 implies |C i | ≤ 1, and |C i | = 1 if and only if Z i [E i ] = Z i = S 3 ,
giving us a decomposition M = R#S as above. Thus we may assume that C i = ∅ for all 1 < i ≤ N . Moreover, if both C 1 and D 1 are non-empty, then 1 =
c 1 = 2t Y 1 (L 1 ) + 1 which forces t Y 1 (L 1 ) = 0. So L 1 is isotopic to S 1 × {pt} ⊂ S 1 × S 2 = Y 1 . Then C 1 ∪ D 1 is a cancelling pair,
contradicting the minimality of the width by Lemma 3.7. Thus by avoiding the case M = S 1 × S 3 , we may assume that
C i = ∅ ∀1 ≤ i ≤ N .
As before note that t Y i (L i ) = 0; in particular, L i is an knot for all i. Since 1) is a solid torus so that the push-off of L 1 with framing n 1 bounds a meridian disk in W 1 . Since c 2 = 1, we get t Y 2 (L 2 ) = 0 and, by uniqueness of genus one Heegaard splittings for lens spaces [4], we conclude that L 2 is isotopic (in Y 2 ) to the core of V 1 or W 1 .
L 1 ⊂ Y 1 = S 3 , L 1 is a unknot with framing n 1 ∈ Z. Y 2 = Y 1 [L 1 ] = S 3 [L 1 ] is a lens space L(n 1 , 1), with n 1 = ±1. Decompose Y 2 as V 1 ∪ W 1 where V 1 = S 3 − η(L 1 ) and W 1 ⊂ L(n 1 ,
Suppose L 2 is isotopic to the core of W 1 . Let f : ∂W 1 → ∂V 1 be an orientation reversing homeomorphism mapping the meridian of ∂W 1 , say m, to the curve with framing n 1 on L 1 in ∂V 1 . Let µ, λ ⊂ ∂V 1 be the meridian and preferred longitude of ∂V 1 induced by L 1 , λ is the boundary of the meridian of η(L 1 ). By construction, f (m) = µ + n 1 λ. We can take f and l ⊂ ∂W 1 a longitude of ∂W 1 so that f * : H 1 (∂W 1 ) → H 1 (∂V 1 ) in the ordered basis {λ, µ}, {m, l} is given by the matrix n 1 1 1 0 . We make such choice since the map f is determined up to isotopy by m → µ + n 1 λ. In particular, L 2 ⊂ W 1 is isotopic in W 1 to λ and so it can be pushed into V 1 , becoming parallel to µ. Hence we can assume that L 2 is isotopic to the core of V 1 . Then L 1 ∪ L 2 is a Hopf link with framings say (n 1 , n 2 ).
For i = 3, ∂η(L 3 ) ⊂ Y 3 = S 3 [L 1 ∪ L 2 ]
is a genus one Heegaard surface for Y 3 . We can decompose the lens space Y 3 as the union
Y 3 = W 1 ∪ S 3 − η(L 1 ∪ L 2 ) ∪ W 2 ,
where W i is the solid torus such that the push-off of L i with framing n i bounds a meridian disk in W i and F i = ∂η(L i ). L 3 ⊂ Y 3 has tunnel number one so uniqueness of genus one Heegaard splittings for lens spaces [4] forces L 3 to be isotopic in Y 3 to the core of either W 1 or W 2 . Without loss of generality, L 3 ⊂ W 2 . Using an argument analogous to that given above, we can pick an adequate attaching map f : ∂W 2 → F 2 = ∂ηL 2 so that we can isotope L 3 out of W 2 so that L 3 is parallel to a meridian of η(L 2 ) in S 3 − η(L 1 ∪ L 2 ). Let W 3 ⊂ Y 4 be the solid torus with meridian disk a curve in ∂η(L 3 ) and framing n 3 . If we sit L 3 in the core of W 2 , it is clear that W 1 and W 3 induce a Heegaard splitting for Y 4 (note that
Y 3 − (W 1 ∪ η(L 3 )) is a product region T 2 × I).
We have shown that we can perturb H (without changing the width) so that L 1 ∪ L 2 ∪ L 3 are isotopic to the link in Figure 1. Also, notice that if W i (i = 1, 2, 3) is the solid torus apearing in Y 4 after surgering S 3 along
L 1 ∪ L 2 ∪ L 3 , then Y 4 − (W 1 ∪ W 3 )
is a product region. We can proceed inductively, repeating the argument above to conclude that H is the Kirby diagram of a "linear" plumbing of disk bundles over spheres.
Proof. Let H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ E N ∪ b 4(Y [D i ]) ≤ 1 for all Y ⊂ Y i . Let Y * i be the unique component of Y i such that c(L * i , Y * i ) = 1. Suppose that Y * i = # j X j # S 1 × S 2 , where X j ⊂ Z i−1 [E i−1 ]
are some connected components. Note that X j = S 3 for all j. Suppose that
D i = ∅, then 0 = t Y * i (L * i ) = t S 1 ×S 2 (L * i )
. So there is a 1-handle C ⊂ C i so that C ∪ D i is a 1/2-cancelling pair. Lemma 3.7 contradicts the minimality of the width. We have shown that whenever D i = ∅, the 1-handles of C i only act as connected sums of distinct components of Z i−1 [E i−1 ]. In other words, L i does not intersect the attaching regions of the 1-handles of C i . Moreover, since Y * i = S 3 and t Y * i (L i ) = 0 whenever D i = ∅, it must be the case that L i is an unknot in S 3 which we can isotope in Y i (and so slide D i over previous handles) to be away from the attaching regions of
j<i (C j ∪ D j ∪ E j ) ∪ C i in ∂b 0 .
By turning H up-side-down, the same argument shows that if D i = ∅, then the attaching sphere of each 3-
handle E ⊂ E i is separating in Z i = Y i [D i ]. In particular, since HG(Y [D i ]) ≤ 1 ∀Y ⊂ Y i ,
the attaching sphere of E must bound a 3-ball in Z i . We can then isotope these attaching spheres in Z i (thus slide E i over previous handles) so that E i ∩ Y i are disjoint from ∪ j≤i L j in ∂b 0 .
We will prove the theorem by induction on the number of 2-handles of H. If H has no 2-handles the result is clear. Suppose then it has at least one 2-handle. Pick 1 ≤ i 0 ≤ N to be the smallest index with D i 0 = ∅, then Y * i 0 = S 3 . L * i 0 is an unknot in S 3 so it can be isotoped to lie inside a 3-ball G bounded by a 2-sphere F ⊂ Y * i 0 disjoint from the attaching region of the previous handles in ∂b 0 . Suppose
D i 0 +1 = ∅, then Y * i 0 +1 = # j X j,i 0 +1 for some connected components X j,i 0 +1 ⊂ Z i 0 [E i 0 ]
. By the previous paragraph, F and G can be picked to avoid the attaching spheres of E i 0 . We can then think of F and G as embedded in
Z i 0 [E i 0 ] and so in Z i 0 [E i 0 ∪ C i 0 +1 ]
; one can do this by flowing F and G using the morse map induced by H. If
L i 0 +1 ⊂ Y * i 0 +1 lies in the same component as F ∪ G ⊂ Y i 0 = Z i 0 [E i 0 ∪ C i 0 +1 ], then we can isotope in Z i 0 [E i 0 ] to lie inside G. To see this, recall that Y * i 0 +1 = # j X j,i 0 +1
where all but (at most) one X j,i 0 +1 are equal to S 3 , and the non-S 3 component must contain F ∪ G. Such isotopies correspond to handle slides of D i 0 +1 over previous handles which keep the width unchanged. As before, F and G can be picked to be disjoint from the attaching regions of
E i 0 +1 .
We can continue this process of isotopying L i 0 +l to be inside G until either D i 0 +k = ∅ for some k ≥ 1 or L i 0 +k lies in a different component of Y i 0 +k than the one containing F ∪ G. In any case, this implies that the component of Y i 0 +k containing F ∪ G must be S 3 . Focusing our attention on the attaching region of the handles in ∂b 0 , the sphere F ⊂ ∂b 0 will separate the attaching links
Relative handlebodies and generalized trisections
In dimension 3, thin position arguments reduce the problem to the study properties of compression bodies. In dimension 4, we will break our thinned 4-manifold in pieces that only contain the information of the 2-handles: the relative handlebodies X(Y, L). In Subsection 4.1, we will see two ways of representing these relative handlebodies. Similar versions of the content of this subsection can be found in [9] and [7]. We include the details here due to the slight change of setting. We will use the above in Subsection 4.2 to show that the width does not change when turning up-side-down a handle decomposition. In Subsection 4.3 we will relate the width with trisections of closed 4-manifolds. As an application, in Subsection 4.4 we give an upper bound for the width of the union of two 4-manifolds, which we use to compute an upper bound for the width of sphere bundles over connected surfaces.
4.1. Nerves of relative handlebodies. We will describe a way to decompose 4-manifolds obtained by 2-handle attachements on collars of closed 3-manifolds. The ideas in this subsection are motivated by the notion of a Heegaard-Kirby diagrams introduced in [9]. It would be interesting to see what one can say about thinned 4−manifolds using results from dimension three and lemmas 4.1 and 4.3. The interested reader can look at the proofs of Proposition 3.5 and Theorem 1.2 of [9] to see examples of this technique.
For a closed surface F and a collection of pairwise disjoint simple closed curves ε ⊂ F , denote by F |ε the closed surface resulting from compressing F along ε. In other words, F |ε is obtained by capping-off with 2-disks the boundary components of F − ε.
Let Σ be a closed surface of genus g ≥ 1 and let α, β, γ ⊂ Σ be three collections of g pairwise disjoint non-separating simple closed curves determining three handlebodies H, H and H , respectively. Suppose that H ∪ Σ H is a Heegaard splitting for # k S 1 ×S 2 , k ≥ 0. We build a 4-manifold Z(Σ; α, β, γ) as follows: Attach 2-handles to Σ × D 2 along α × {e 4πi/3 }, β × {1} and γ × {e 2πi/3 } with framings induced by Σ. The resulting 4-manifold, denoted by W 1 (Σ; α, β, γ), has three special 2-spheres on its connected boundary: Σ|α, Σ|β and Σ|γ. Attach one 3-handle along each sphere to obtain a 4manifold, denoted by W 2 (Σ; α, β, γ), with three boundary components diffeomorphic to: H ∪ Σ H , H ∪ Σ H and H ∪ Σ H . Let W (Σ; α, β, γ) be the result of capping-off with 3-and 4-handles the boundary component of
H ∪ Σ H ≈ # k S 1 × S 2 .
A schematic picture of W can be found on the right side of Figure 2. Proof. The procedure of capping-off the boundary component of W 2 given by H ∪ Σ H with 3-and 4-handles is unique by [6], so it is enough to show that W 2 (Σ; α, β, γ) is determined by the associated handlebodies H, H and H . Recall that two collections of g pairwise disjoint non-separating simple closed curves in Σ determine the same handlebody if and only if they differ by disk slides on the surface. Thus by the symmetry of the construction of W 2 , it suffices to check W 2 (Σ; α, β, γ) = W 2 (Σ; α , β, γ) when α and α differ by one disk slide. In this setup, disk slides correspond to 4-dimensional 2-handle slides so W 1 (Σ; α, β, γ) = W 1 (Σ; α , β, γ).
Label the components of α and α so that α 1 = α 1 and α i = α i for all i > 1. It follows that α 1 and α 1 are disjoint, forcing Σ − (α ∪ α 1 ) to be disconnected. Let S 1 and S 2 be its two components and label them so that S 2 is a thrice punctured sphere with ∂S 2 = α 1 ∪ α 1 ∪ α j 0 for some j 0 > 1. Write the corresponding components of Σ|(α ∪ α 1 ) as S 1 ∪ S 2 . Let W be the 4-manifold resulting from W 1 (Σ; α, β, γ) by attaching a 2-handle along α 1 × {e 4πi/3 } and 3-handles along S 1 , S 2 , Σ|β and Σ|γ.
Let b ≈ D 2 × D 2 be the 2-handle attached along α 1 and c i the 3-handle attached along S i , for i = 1, 2. Since the framing of b is given by Σ, one can check that the intersection Σ ∩ b = S 1 × [−1, 1] when written in the coordinates of the handle. Also, the belt sphere of b, which is given by {0} × S 1 , is isotopic in ∂b to a loop of the form {t 0 } × S 1 for some t 0 ∈ S 1 .
To see that, recall that the belt sphere is a component of the Hopf link {0} × S 1 ∪ S 1 × {0} in ∂b ≈ S 3 . It follows that the belt sphere of b intersects Σ in two points, one per side of α 1 in Σ. Thus the belt sphere intersects S 2 in one point. On the other hand, S 2 is the union of S 2 with the cores of the 2-handles given by α 1 , α 1 and α j 0 , which are disjoint from the belt sphere of b. Hence the belt sphere of b will intersect S 2 geometrically once. Thus the 3-handle c 2 cancels with b. Similarly, S 1 intersects the belt of b once. In order to eliminate b and c 2 , it is necessary to slide c 1 along c 2 ; this will change the attaching sphere of c 1 from S 1 to Σ|α. Therefore, W ≈ W 2 (Σ; α, β, γ). Analogously, one can show that W ≈ W 2 (Σ; α , β, γ), and so W only depends on the handlebodies H ∪ H ∪ H . There is a correspondance between 4-manifolds of the form W (H, H , H ) and X(Y, L). For the closed case, W is a trisection diagram (see Subsection 4.3), and the correspondance has been proven in [7]. The proofs can be extended to our context.
The following lemma is essentially Lemma 4.1 of [11] or Lemma 14 of [7]. For completeness and due to the slight change of setting (closed case vs relative case), we include a proof. We can use the latter region to add a copy of Σ × I to H and assume the three handlebodies to intersect simultaneously at the surface Σ (see Figure 2). By construction, H ∪ Σ H is a Heegaard splitting for a connected sum of g − |L| copies of S 1 × S 2 .
Let
∪ Σ H = # k S 1 × S 2 for some 0 ≤ k ≤ g. Then for Y := H ∪ Σ H , there is a link L ⊂ Y satisfying X(Y, L) ≈ W (H, H , H ).
Proof. Since H ∪ Σ H = # k S 1 × S 2 , there are collections of g curves α, γ ⊂ Σ determining H and H , respectively, such that α l = γ l , for all 1 ≤ l ≤ k and |α i ∩ γ j | = δ i,j , for all k < i, j ≤ g. The desired link is given by L = {γ i : k < i ≤ g}. Notice that the β curves, which induce H , are useful to determine the embedding of Σ into H ∪ H and so the embedding of L.
Relative handlebodies and width. Let
H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ E N ∪ b 4 be a handle decomposition of M . Recall that Y i = ∂ (b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ . . . C i ) and Z i = Y i [D i ]. For each 1 ≤ i ≤ N , denote by W i = X(Y i , L i ), note that W i may be disconnected. Let (Σ i ; H i , H i , H i ) be nerves for W i with smallest genus 1 satisfying W i = X(Y i , L i ) = W (Σ i ; H i , H i , H i ). M can be decomposed as [b 0 ∪ C 1 ] Y 1 W 1 Z 1 [η(Z 1 ) ∪ E 1 ∪ C 2 ] Y 2 · · · Y N W N Z N [η(Z N ) ∪ E N ∪ b 4 ] .
Note that the proofs of Lemmas 4.3 and 4.4 show that
width(H) = Σ ⊂Σ i max{2g(Σ ) − 1, 0} | i = 1, . . . , N . Let H op = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ E N ∪ b 4 be the up-side-down handle decomposition of M , where b 0 = b 4 , C i = E N −i , D i = D N −i , E i = C N −i and b 4 = b 0 .
The relative handlebodies W i satisfy (see Remark 4.6),
W i = (W N −i ) op = W (Σ i ; H i , H i , H i ) op = W (Σ i ; H i , H i , H i ).
We have proven the following
Relative handlebodies and trisections of 4-manifolds. An interesting case of the discussion in Subsection 4.2 is when M is closed and the handle decomposition is self-indexed
; i.e., H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ b 4 .
Here, the decomposition in relative handlebodies becomes One would expect closed 4-manifolds with width(M ) being a singleton to be "small". For example, Proposition 3.8 implies that the only 4-manifolds of trisection genus one are S 1 × S 3 and ±CP 2.
M = [b 0 ∪ C 1 ] Y 1 W 1 Z 1 [η(Z 1 ) ∪ E 1 ∪ b 4 ] = [b 0 ∪ C 1 ] Y 1 W (Σ; H, H , H ) Z 1 [η(Z 1 ) ∪ E 1 ∪ b 4 ] = [b 0 ∪ C 1 ] X(H, L) [E 1 ∪ b 4 ] , where Y 1 = H ∪ Σ H , Z 1 = H ∪ Σ H , L ⊂ H is
For a generic handle decomposition of M , we can think of a generalized trisection to be the collection of nerves for the relative handlebodies {X(Y i , L i )} i , together with data describing the 1-handles and 3-handles pasting them along some boundary components. With this philosophy in mind, in some part of the rest of this work we will derive results about trisections of 4-manifolds as a particular case of constructions on thin position (see Example 4.11 and Section 5.1.2). We can refine the above argument as follows. It is important to mention that if one decides to add the 2-handles L i after L i (or at the same time), the tunnel number of L i+1 in Y i+1 will be equal to the tunnel number of L i+1 in Z i [C i+1 ]; allowing the complexity c i+1 to change with no control. Hence, the upper bound on Corollary 4.9 is expected to be sharp only for special cases.
Example 4.11 (Sphere bundles over surfaces). Let g, n ∈ Z, g > 0 and denote by S g , N g the orientable and non-orientable surface of genus g, respectively. Let X g,n , Y g,n be the disk bundles over S g and N g with Euler number n, respectively. Kirby diagrams for X and Y with only one 2handle are known (Fig. 3), so stimates for the width of such 4-manifolds can be found. More explicitly, for g > 0, width(X g,n ) ≤ {4g + 1} and width(Y g,n ) ≤ {2g + 1}. Using Proposition 4.9, we obtain estimates for the width of the corresponding doubles; i.e. sphere bundles over surfaces.
width(D(X g,n )) ≤ {4g+1, 4g+1} and width(D(Y g,n )) ≤ {2g+1, 2g+1}
For this particular examples, one can add all 2-handles of D(X g,n ) (resp. D(Y g,n )) at the same time and get tunnel numbers 2g + 1 (resp. g + 1). Lemma 4.3 together with the discussion on Subsection 4.3 imply that, to draw a genus m + 1 trisection surface for M , it is enough to find a system of m tunnels for the attaching region of the 2-handles 2 . Thus we can draw diagrams for D(X g,n ) (resp. D(Y g,n )) of genus 2g + 2 (resp. g + 2). For completeness, we draw the diagrams for genus 1 case in Figure 4. . Trisection diagrams D(X 1,0 ) ≈ T 2 ×S 2 (left) and D(Y 1,1 ) ≈ RP 2 ×S 2 (right). To change the euler number it is enough to add twisting to the longer γ−curve. α (red), β (blue) and γ (green) curves correspond to curves that bound disks in H, H and H , respectivelly.
Symmetries of relative handlebodies
In [16], M. Scharlemann and A. Thompson proved that 3-manifolds of width < {5} are 2-fold branched covers of connected sums of copies of S 1 × S 2 . In this section, we relate ideas of trisections of 4-manifolds from [9], [10] and [12] to discuss an attempt of lifting this result to 4-manifolds with connected boundary. In Subsection 5.1 we study symmetries on the relative handlebodies X(Y, L). We use this in Subsection 5.1.2 to talk about symmetric trisection diagrams and to prove a "trisected version" of a Theorem of J. Birman and H. Hilden. In Subsection 5.2 we study the extension problem: how to paste symmetric pieces of M .
Symmetric nerves.
In this subsection X will denote a 4-manifold of the form X = X(Y, L) for some link L ⊂ Y 3 .
Definition 5.1. We say that a nerve T = (Σ; H, H , H ) for X is psymmetric if there is a piecewise-linear homeomorphism τ : Σ → Σ of finite period p, extending to the interior of each handlebody, satisfying (1) For each handlebody, the orbit space by the action of τ is a 3-ball.
(2) F ix(τ ) = F ix(τ k ) for all 1 ≤ k ≤ p.
(3) The image of the fix set of τ on each handlebody is an unknotted set of arcs in the quotient. Using the ideas of [10], one can show that if T is a p-symmetric nerve of X, then the finite order map τ : Σ → Σ extends to X. By definition the quotient map on each handlebody q τ : H → B 3 is a p-fold cyclic branched cover of B 3 along a collection of b boundary parallel arcs in B 3 , say θ α , θ β and θ γ .
Since, L is a subset of the core of H, each component of L is dual to a meridian disk of H, and the 3-manifold H ∪ Σ H is a connected sum of k := g − |L| copies of S 1 × S 2 . A corollary in Section 2 of [15] states that such 3-manifolds arise as p−fold cyclic branched cover of S 3 only when the branched set is an unlink. Thus θ γ ∪ θ α is an unlink in B ∪ B ≈ S 3 , which bounds a unique collection D of trivial disks in B 4 by Lemma 2.3 of [11]. By Remark 4.5, the tuple (Σ/τ ; B, B , B ) is a nerve for S 3 × I, and one can complete θ α ∪ θ β ∪ θ γ ⊂ B ∪ B ∪ B to a properly embedded surface K 2 ⊂ S 3 × I by attaching the collection D along θ γ ∪ θ α . By construction,
K ∩ (S 3 × {0}), S 3 × {0} = θ α ∪ θ β , B ∪ Σ/τ B K ∩ (S 3 × {1}), S 3 × {1} = θ γ ∪ θ β , B ∪ Σ/τ B
We can now take the p-fold cyclic covering of S 3 × I branched along K 2 and lift the nerve of S 3 ×I to a nerve for the resulting 4-manifold. Recall that Remark 5.5. Although Section 5 was written mainly to discuss the symmetries of 4-manifolds of width less than {5} (see Section 5.2.1), one can talk about constructions of more general 4-manifolds than those that appear in Definition 4.2. Take a closed orientable surface Σ; pick finitely many points
{t i } N i=1 ⊂ ∂D 2 and meridian systems {α i } N i=1 for handlebodies {H i } N i=1
. Let X be the 4-manifold obtained by attaching 2-handles to Σ×D 2 along α i × {t i } for i = 1, . . . , N and capping off with 3-and 4-handles (if desired) some boundary components corresponding to Heegaard splittings of connected sums of copies of S 2 × S 1 . The tuple (Σ; {H i } i ) will be the nerve of X and Proposition 5.3 will immediately extend to this context using the same proof. One can naively ask if every 4-manifold with disconnected boundary admits such decomposition.
Proposition 5.6. Let Σ and let T be as in Remark 5.5. If T is p-symmetric, then X is the p-fold cover of S 4 with |∂X| 4-balls removed branched along a properly embedded surface. 5.1.1. Drawing the branched set. We will briefly describe how to obtain band diagrams for the branched set of a given p-symmetric nerve. Recall the notation of Proposition 5.3 (see Fig. 5 for simplicity). In Lemma 3.3 of [10], J. Meier and A. Zupan described an algorithm to obtain a banded link diagram (also called movie presentation) for K 2 ⊂ S 3 × I from the tuple (S 2 ; θ α , θ β , θ γ ), which the authors called a bridge trisection diagram for K 2 ⊂ S 3 × I. Using our current notation, the algorithm goes as follows:
Lemma 5.7 (Lemma 3.3 of [10] rephrased). Let (S 2 ; B 3 , B 3 , B 3 ) be a nerve for S 3 × I and let (θ α , θ β , θ γ ) be three collections of b trivial arcs on B 3 with the same set of end points in S 2 such that θ γ ∪ θ α is an unlink in
S 3 = B 3 α ∪ B 3 γ .
Let θ * α ⊂ S 2 be a collection of shadows of θ α , and excise one arc of θ * α corresponding to each component of θ α ∪ θ γ . Denote by v ⊂ θ * α the remaining shadows and let C = θ β ∪ θ γ ⊂ S 3 . Then (C, v) is a banded diagram for K 2 in S 3 × I.
The bands of v correspond to saddles of K 2 ocurring inside the collection of disks D that θ α ∪ θ γ bounds in the 4-ball. One can change, if desired, the roles of γ and α in the statement of Lemma 5.7 to get the "reversed" banded diagram for K 2 .
Example 5.8. Figure 7 exemplifies the procedure of Proposition 5.3 to obtain a movie presentation for the branching surface K 2 in the case that X admits a symmetric nerve. This breaks in four steps as follows:
(1) Find a system of tunnels t for L in Y so that the corresponding nerve is p-symmetric. the tuple (θ α , θ β , θ γ ). (4) Apply the algorithm described in Lemma 5.7 to draw the banded diagram for K 2 .
An application to trisections of 4-manifolds.
We finish this subsection with an application of Proposition 5.3 to the theory of trisections of closed 4-manifolds. We will show that 4-manifolds with symmetric trisections are 2-fold covers of S 4 branched along knotted surfaces. It is important to mention that subsections 5.2 and 5.2.1 are independent of the following.
In [3], J. Birman and H. Hilden studied the relations between "p-symmetric" Heegaard splittings and branched covering representations of closed 3-manifolds. For example, they showed 3 that every genus g ≥ 3 Heegaard splitting of a closed 3-manifold may be represented as a (4g−4)-sheeted branched covering of S 3 , with branching set a 1-manifold of at most 4g − 4 components. Question 5.9. Is it possible to extend the results of J. Birman and H. Hilden in [3] to the context of p-symmetric nerves of relative handlebodies? If so, can we do the converse of such results following M. Mulazanni's ideas in [13]?
We can partially answer Question 5.9 in Theorem 5.10. The interested reader can compare this theorem in the case of p = 2 with Corollary 11.3 of [1] or with Theorem 3 of [14]. One can think of this result as a trisection analogue of Theorems 2-5 in [3]. Proof. The forward direction was discussed in Section 2.6 of [10]. For the backwards direction, recall that in a trisection the triplet (Σ; H, H , H ) has the property that the 3-manifolds H ∪ Σ H , H ∪ Σ H and H ∪ Σ H are homeomorphic to connected sums of copies of S 1 × S 2 . Thus for each pair, we can use the argument in Proposition 5.3 to extend the involution to all M .
Extending involutions. Let
H = b 0 ∪ C 1 ∪ D 1 ∪ E 1 ∪ · · · ∪ E N ∪ b 4 be a handle decomposition decomposition of a smooth 4−manifold M . Denote by N i = b 0 [C 1 ∪ D 1 ∪ · · · ∪ C i ], M i = N i [D i ], Y i = ∂N i , Z i = ∂M i and X i = M i − N i . X i is obtained by attaching 2-handles to Y i × I along L i × {1}, so let T i = (Σ i ; H i , H i , H i )
be a nerve for X i with g(Σ i ) = g i . By construction, ∂X i is divided in two parts: Y i and Z i . Suppose each T i is 2-symmetric. By Proposition 5.3, there are involutions τ i : X i → X i with fixed set a properly embedded surface K i ⊂ X i . By construction, τ i | Y i and τ i | Z i are induced by involutions on the handlebodies H i , H i , H i .
In M , Z i−1 and Y i cobound a submanifold given by adding 3-handles E i−1 and 1-handles C i to Z i−1 × I along Z i−1 × {1}. We are interested in knowing under what conditions the involution on the pair (Z i−1 , Y i ) can be extended to the interior of such cobordism, obtaining involutions on bigger pieces of M . With the setting just described, we can extend the involution when adding 1-handles. Proof. Write B 4 ≈ [−1, 1] × B 3 where B 3 ⊂ C × R has coordinates (w, t). Consider g : B 4 → [−1, 1] given by g(x, w, t) = −x 2 + |w| 2 + t 2 . The map g models a 4-dimensional 1-handle attachement. Define σ : B 4 → B 4 by σ(x, w, t) = (x, −w, t). σ is an involution of B 4 satisfying σ(g −1 (δ)) = g −1 (δ) for all δ. Notice that B 4 /σ ≈ [−1, 1] × B 3 is again a 4-ball, and the quotient map B 4 → B 4 /σ is a 2-fold cover of B 4 branched along the 2-disk
F = F ix(σ) = {(x, (0, 0), t) : |x| ≤ 1, |t| ≤ 1} . g| F is given by (x, (0, 0), t) → −x 2 + t 2 which models a 2-dimensional 1- handle attachement. Take V − , V + ⊂ Y 3 to be two disjoint closed 3-balls so that V ± ∩ F ix(τ ) is one arc. Pick coordinates for V ± so that V ± = {(v, t) ∈ C × R : |v| ≤ 1, |t| ≤ 1} and τ | V ± : (v, t) → (−v, t). By hypothesis, W ≈ [0, 1] × Y 3 ∪ f B 4 where f : {±1} × B 3 →V + ∪ V − ⊂ {1} × Y 3 (+1, w, t) →(w, t) ∈ V + (−1, w, t) →(w, t) ∈ V − .
By construction, σ and id [0,1] × τ agree on a neigborhood of V + ∪ V − ⊂ {1} × Y . We then obtain the desired involution τ = σ ∪ f τ in W .
Remark 5.12. Let C ⊂ ∂W − {0} × Y be the new boundary component of W ; we have C ≈ Y #S 1 × S 2 . Furthermore, one can check that F ix( τ | C ) = F ix(τ | Y 3 ) − int(V + ∪ V − ) ∪ {(x, (0, 0), ±1) : |x| ≤ 1}
Thus the action of τ before and after the 1-handle attachment in Lemma 5.11 is described by Figure 8. In particular, if V + and V − are neigborhoods of points of the intersection F ix(τ ) Σ where Σ ⊂ Y is a Heegaard surface fixed setwise by τ , then the involution τ | C will delete the corresponding intersection points between τ C and the new Heegaard surface. Figure 8. How the extension of the involution in Y 3 looks once we decided the attaching region of the 1-handle. The Heegaard surface (plane in the picture) can "survive" the 1-handle attachement.
5.2.1. When the width is less than {5}. In [16], M. Scharlemann and A. Thompson showed that if the width of a 3-manifold is less than {5}, then it is a 2-fold branched cover of a connected sum of copies of S 1 × S 2 . In the following subsection, we will apply the previous discussion to describe an attempt of proving the analogue of this result in dimension four, say: if M is a 4-manifold with width(M ) < {5}, then M is the 2-fold cover of connected sum of copies of S 1 × S 3 with |∂M | 4-balls removed, branched along a properly embedded surface.
The outline of the solution is analogous to the original one in [16]: we first show that the X i blocks are branched covers of copies of S 3 × I and then we study how to paste them together preserving the branched covering map. In dimension three this pasting problem resulted being a known mapping class group problem. From the 3-manifold theory perspective, it is interesting how this attempt reduces a 4-dimensional problem to the study of non-uniqueness of genus two Heegaard splittings of closed 3-manifolds.
Proposition 5.13. If width(D) < {5}, each X i admits a smooth involution τ i : X i → X i such that the projection map p i : X i → X i /τ i ≈ S 3 × I, is a 2-fold cover of S 3 × I branched along a properly embedded surface K i . Furthermore, for i = 1, n, we can take the involution to be defined on M 1 and M − N n , respectively.
Proof.
Recall that X i is obtained by attaching 2-handles to Y i × I along L i × {1}. Since {c i = 2g i − 1} = width(D) < {5}, g i ≤ 2 which gives us the existance of a Heegaard splitting H i ∪ Σ i H i of Y i of genus at most 2 such that L i ⊂ core(H i ). Then, T i = (Σ i ; H i , H i , H i = H i [L i ]) is a nerve for X i by Lemma 4.3. The hyperelliptic involution on Σ fixes the isotopy class of unoriented curves in Σ, so it extends to the interior of H i , H i and H i ; thus T i is 2-symmetric for every i. The result follows from Proposition 5.3.
When i = 1, H 1 ∪ Σ 1 H 1 is a Heegaard splitting for # |C 1 | S 1 × S 2 . By uniqueness of involutions on such manifolds, we will obtain that θ α ∪ θ β is also an unlink in # |C 1 | S 1 × S 2 , thus we can cap X 1 off with 3-and 4-handles on that side and extend the surface (with its involution) with the unique boundary parallel disks; obtaining a branched cover M i → B 4 . We proceed analugously for i = n.
We now proceed to describe how to paste the X i blocks preserving their involutions. Since width(H) < {5}, we have that HG(Y ) ≤ 2 and HG(Z ) ≤ 2 for every connected component Y ⊂ Y i and Z ⊂ Z i . Let V i = Z i [E i ]. For simplicity of the argument, assume Z i and Y i+1 are connected.
Suppose first that there is a 3-handle of E i with attaching region a nonseparating sphere in Z i . Then Z i = S 1 ×S 2 #L(p, q), with L(p, q) a (possibly trivial) lens space. It follows that Z i has a unique heegaard spliting and so a unique involution. V i is a disjoint union of copies of S 3 and one L(p, q). Take the standard involutions on then and apply the extension in Lemma 5.11 with the 1-handles E op i to obtain an involution on the cobordism between Z i and V i . Then apply Lemma 5.11 again to the 1-handles of C i+1 to obtain an involution connecting Z i and Y i+1 . In this case, Y i+1 is of the form S 1 × S 2 #L(p, q) which has a unique involution. Hence, we can paste X i and X i+1 .
Suppose now that Z i does not contain a S 1 × S 2 summand. Then the attaching regions of the 3-handles of E i induce a connected sum decomposition of Z i = # j X i,j (with possibly trivial pieces). Proposition 2 of [15] states that if Z i is the 2-fold cover of S 3 branched along J i , then J i = # j J i,j splits as a connected sum where X i,j is the 2-fold cover of S 3 along J i,j . Taking then the involutions on each X i,j induced by the cover gives us an involution on V i = Z i [E i ] = j X i,i . By Lemma 5.11 to the 1-handles E op i we get an involution on the cobordism between Z i and V i . Applying Lemma 5.11 again with the 1-handles of C i+1 gives us a involution connecting Z i with Y i+1 . The issue here is that the resulting involution in Y i+1 may not be isotopic to the involution on Y i+1 induced by the 2-symmety on X i+1 . This is due to the non-uniqueness of genus 2 Heegaard splittings for certain irreducible 3-manifolds. This motivates more the study of the problem of finding a suitable set of Montesinos moves for 2-fold branched coverings of S 3 . Question 5.14. Is the space of involutions for a irreducible 3-manifold Y of Heegaard genus 2 connected under a suitable topology? Given τ 0 , τ 1 two involutions of Y , is there a smooth one parameter family of maps {τ s : s ∈ [0, 1]} connecting τ 0 with τ 1 so that for all but finitely many values τ s is an smooth involution?
Proposition 4 . 8 .
48Let M and N be connected 4-manifolds with non-empty boundary. Suppose f : ∂M → ∂N is a diffeomorphism between their boundaries. Then width(M ∪ f N ) ≤ width(M ) ∪ width(N ) In Subsection 4.3, we explain how handle decompositions of closed 4manifolds of width equal to a single element correspond to trisections of 4-manifolds. This allows us to specialize some of our results to the theory of trisections of 4-manifolds (see Example 4.11 and Section 5.1.2).
3. 1 .
1Definition of width. Let M be a connected, compact, smooth 4manifold. Consider a handle decomposition of M
be the cores of the attaching regions of the 2-handles D i . Note that Y i might be disconnected and that L i is a link (possibly empty) in each component of
Definition 3 . 1 .
31The width of the handle decomposition H is the multiset {c i }. The width of M is the infimum of all widths among all possible handle decompositions of M . The infimum is taken with respect to the following order: Let A = {a 1 ≥ a 2 ≥ . . . } and B = {b 1 ≥ b 2 ≥ . . . } be two bounded multisets of N ∪ {0} ordered in a decreasing way. Then A < B if and only if there is an
( 3 )
3The result follows by turning H up-side-down and applying Part 2 of this proposition (seeLemma 4.7).
take connected components A ⊂ L j and B ⊂ J i ∪ L i and let H be a decomposition obtained by sliding A along B. Then D and H have the same width.
Proposition 3 . 9 .
39Let M be a connected 4-manifold and let H be a thin position of M satisfying width(H) = {1, . . . , 1}. Suppose that the level 3manifolds Y i [D i ∪ E i ] are connected for all i. Then M is diffeomorphic to a (boundary) connected sum of copies of S 1 ×S 3 , S 1 ×B 3 , and linear plumbings of disk bundles over the sphere.
3 and either i = N or M = R#S where width(R) and width(S) are collections of at most N − 1 ones. Thus we can assume that
Corollary 3 . 10 .
310For closed prime 4-manifolds, width(M ) = {1, 1} if and only if M ≈ S 2 × S 2 .
Figure 1 .
1Kirby diagram of a linear plumbing of disk bundles over a sphere.
Theorem 3 . 11 .
311Let M be a connected 4-manifold satisfying width(M ) = {1, . . . , 1}. Then M is diffeomorphic to a (boundary) connected sum of copies of S 1 × S 3 , S 1 × B 3 , and linear plumbings of disk bundles over the sphere.
be a thin position of M of width {1, . . . , 1}. The level 3-manifolds Y i , Z i and Z i [E i ] may not be connected. By definition, 1 = c i = Y ⊂Y i c(L , Y ) where the sum runs through all connected components of Y i and c(L , Y ) ≥ 0. In particular, HG(Y ) ≤ 1 and HG
0 +l from the rest of the Kirby diagram for M . This shows that M splits as a connected sum M = R#S where R has a handle decomposition given by b 0 ∪ 0≤l≤k−1 D i 0 +l . Notice here that width(R) ≤ {1, . . . , 1}, width(S) ≤ {1, . . . , 1}, R satisfies the conditions of Proposition 3.9, and S has strictly less 2-handles than H. This concludes the inductive step.
Lemma 4.1. W (Σ; α, β, γ) is determined by the embedding of H ∪ H ∪ H .
Definition 4 . 2 .
42Let H, H , H be three connected handlebodies with common boundary a surface Σ. We say that the tuple T = (Σ; H, H , H ) is a nerve of W if W ≈ W (Σ; α, β, γ) for some collections of curves α, β, γ ⊂ Σ determining H, H , H , respectively. Let L ⊂ Y be a framed link inside a closed 3-manifold Y , and let X(Y, L) be the smooth 4-manifold built from Y × [0, 1] by attaching 2-handles along L × {1}.
Lemma 4 . 3 .
43Let L ⊂ Y be a framed link inside a closed 3-manifold. Then there exist handllebodies H, H , H so that X(Y, L) ≈ W (Σ; H, H , H ). Proof. Take (Σ; H, H ) a Heegaard splitting of genus g ≥ 1 of Y so that L ⊂ H is a subset of its core; i.e. (Σ, L) is an admissible pair. Then H after surgery along L is still a 3-dimensional handlebody denoted by H . Let h : X → [0, 2] be a Morse function arising from the construction of X with h −1 (0) = Y × {0}, h −1 (2) ≈ Y [L], and all critical points of index 2 at t = 1. The flow of h restricted to H × {0} induces an injective isotopy in X between H × {0} and the handlebody Y [L] − H , thus a product region.
X 2
2be the 4-manifold given by flowing H × {0} with h. X 2 is obtained by adding 2-handles to H × [0, 1] along L × {1}. Note that H × [0, 1] is a 4-dimensional 1-handlebody of genus g with one 0-handle and g 1-handles, where the 1−handles are in correspondance with any set of g meridians determining H. Since L is a subset of the core of H, 2-handles along L cancel |L| 1-handles of H × [0, 1]. Hence X 2 is a 1-handlebody of genus g − |L|. It follows that X(Y, L) ≈ W (H, H , H ).
Figure 2 .
2Schematic diagram of the Morse function of X(Y, L) (left) and W (Σ; H, H , H ) (right).
Lemma 4 . 4 .
44Let Σ be a surface and H, H , H be handlebodies with boundary Σ satisfying H
Remark 4 . 5 .
45Notice that if L = ∅, then X ≈ Y ×I, and (Σ; H, H , H = H) is a nerve for X for any H ∪ Σ H Heegaard splitting of Y . In particular, (S 2 ; B 3 , B 3 , B 3 ) is a nerve for S 3 × I.
Remark 4 . 6 .
46We can build an up-side-down version of X(Y, L) from Y [L] by attaching 2-handles to Y [L] × [0, 1] along L × {1}, where L ⊂ Y [L] are co-cores of the 2-handles along L. In terms of nerves, one can see that if X(Y, L) = W (Σ; H, H , H ), then this up-side-down presentation for X(Y, L) can be described by W (Σ; H , H , H).
Lemma 4 . 7 .
47Let H be a handle decomposition of M . Then width(H) = width(H op ).
the attaching region of the 2-handles D 1 and X(H, L) is the cobordism between H and H[L] (see Figure 2). Note that the last expression is a trisection of M . Hence, width(H) = {max(2g(Σ) − 1, 0)}, where Σ is the smallest genus trisection surface for M inducing the handle decomposition H.
4. 4 .
4Width under specific operations. Let M be a 4-manifold with non-empty boundary and denote by D(M ) the double of M ; i.e. D(M ) ≈ M ∪ id ∂ M . If ∂M is connected, a Kirby diagram for the double of M can be built by adding an unknotted 2-handle with framming zero around each 2-handle of a Kirby diagram for M . Let H = b 0 ∪ C 1 ∪ D 1 be a self-indexed handle decomposition in thin position for M with no 3-handles. Denote by L the attaching links of the new 2-handles for D(M ). An application of the slam-dunk move [8], shows that the components of the link L are isotopic in S 3 [C 1 ∪ D 1 ] to the cores of the solid tori after surgery along L. Thus the tunnel number of L in S 3 [C 1 ∪ D 1 ] is the same as the tunnel number of L in S 3 [C 1 ]. In particular, width(D(M )) ≤ width(M ) ∪ {2t S 3 [C 1 ] (L) + 1}.
Proposition 4. 8 .
8Let M and N be connected 4-manifolds with non-empty boundary. Suppose f : ∂M → ∂N is a diffeomorphism between their boundaries. Then width(M ∪ f N ) ≤ width(M ) ∪ width(N ) Proof. Let H M and H N be a thin position for M and N , respectively, and consider H M ∪ H op N a decomposition of M ∪ f N given by doing H M first, followed by H N in the opposite order. By construction, width(M ∪ f N ) ≤ width(H M ) ∪ width(H op N ). The result follows from Lemma 4.7.
Corollary 4 . 9 .
49Let M be a 4-manifold with non-empty boundary. Then width(D(M )) ≤ width(M ) ∪ width(M ) Remark 4.10.
Figure 3 .
3Kirby diagrams for disk bundles over closed surfaces with specific systems of tunnels.
Figure 4
4Figure 4. Trisection diagrams D(X 1,0 ) ≈ T 2 ×S 2 (left) and D(Y 1,1 ) ≈ RP 2 ×S 2 (right). To change the euler number it is enough to add twisting to the longer γ−curve. α (red), β (blue) and γ (green) curves correspond to curves that bound disks in H, H and H , respectivelly.
Remark 5. 2 .
2The 2-symmetric condition of a nerve T = (Σ; H, H , H ) is equivalent to the existance of an involution τ of Σ extending to the interior of each handlebody such that τ is conjugate to the hyperelliptic involution.
Proposition 5 . 3 .
53Let L ⊂ Y be a framed link inside a closed 3-manifold and let T = (Σ; H, H , H ) be a nerve for X = X(Y, L). If T is p-symmetric, then X is a p-fold cyclic covering of S 3 × I branched along a properly embedded knotted surface with boundary in both S 3 × {0, 1}.Proof. By fixing a model handlebody of genus g, we can assume H, H , H are standard and take maps f αβ , f βγ , f γα between the corresponding boundaries codifying the pairwise intersections. Let τ be the order p homeomorphism of Σ extending to the three handlebodies. By assumption, τ commutes with the f -maps and hence the maps descend to the quotients B := H/τ , B := H /τ , B := H /τ (Figure 5).
Figure 5 .
5The nerve of T descends to a bridge trisection diagram on S 3 × I.the p-fold cyclic cover of a 4-ball branched along a collection of trivial disks is also 4-dimensional 1-handlebody. By construction, the new tuple is indeed equal to the original nerve for X (Σ; H, H , H ). Lemma 4.1 concludes that X is the p-fold cyclic branched covering of S 3 × I along K and that the map τ extends to all X.Remark 5.4. Suppose that a p-symmetric nerve T = (Σ; H, H , H ) comes from a Kirby diagram H = b 0 ∪ C 1 ∪ E 1 with 1-handles E 1 and 2-handles D 1 , that is, Y = # |C 1 | S 1 × S 2 and Σ = ∂η(L ∪ t) where t is a system of tunnels for the attaching link of the 2-handles L in Y . Since H ∪ Σ H = Y = # |C 1 | S 1 × S 2 , we can use the argument in Proposition 5.3 to obtain a Z p action on M := b 0 [C 1 ∪ D 1 ] which provides a description of M as a p-fold cyclic cover of B 4 branched along a properly embedded knotted surface.
( 2 )
2Draw three collections of g(Σ) + 1 non-separating curves in Σ = ∂η(L ∪ t) determining the handlebodies H = η(L ∪ t), H = Y − H, H = H[L]. (3) Project the curves with the finite order map τ : Σ → Σ to obtain
Figure 6 .
6The Poincare 4-manifold is a 2-fold cover of B 4 branched along a properly embedded annulus with boundary a hopf link.
Figure 7 .
7Example of how to obtain the branched surface.
Theorem 5 . 10 .
510Let M be a closed 4-manifold. Then M is the p-fold cyclic cover of S 4 branched along a knotted surface K 2 ⊂ S 4 if and only if M admits a p-symmetric trisection diagram.
Lemma 5 . 11 .
511Let τ be an involution of a closed 3-manifold Y (possibly disconnected) with 1-dimensional fixed set and let W be the 4-manifolds obtained from [0, 1]×Y by adding a 1-handle along {1}×I. Then there is an involution τ of W so that τ | {0}×Y = τ . Furthermore, W/ τ is diffeomorphic to [0, 1] × (Y /τ ) with a 1-handle attached.
For closed disconnected surfaces, the genus is the sum of the genera of the connected components.
This is Lemma 2.3 of[12].
This is Theorem 1 of[3].
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Heegaard splittings of branched coverings of S 3. Joan S Birman, Hugh M Hilden, Transactions of the American Mathematical Society. 213Birman, Joan S., and Hugh M. Hilden. Heegaard splittings of branched coverings of S 3 . Transactions of the American Mathematical Society 213 (1975): 315-352.
Scindements de Heegaard des espaces lenticulaires. Francis Bonahon, Jean-Pierre Otal, Annales scientifiques de l'cole Normale Suprieure. Elsevier16Bonahon, Francis, and Jean-Pierre Otal. Scindements de Heegaard des espaces lentic- ulaires. Annales scientifiques de l'cole Normale Suprieure. Vol. 16. No. 3. Elsevier, 1983.
Trisections of 4-manifolds via Lefschetz fibrations. Nickolas A Castro, Burak Ozbagci, arXiv:1705.09854arXiv preprintCastro, Nickolas A., and Burak Ozbagci. "Trisections of 4-manifolds via Lefschetz fibrations." arXiv preprint arXiv:1705.09854 (2017).
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[
"First-principles study of the effect of Fe impurities in MgO at geophysically relevant pressures",
"First-principles study of the effect of Fe impurities in MgO at geophysically relevant pressures"
] |
[
"Donat J Adams ",
"W M Temmerman ",
"Z Szotek ",
"\nDep. Materials Sciences\nLab. Crystallography\nETH Zürich\nSwitzerland\n",
"\nDaresbury Laboratory, Daresbury\nWA4 4ADWarringtonUnited Kingdom\n"
] |
[
"Dep. Materials Sciences\nLab. Crystallography\nETH Zürich\nSwitzerland",
"Daresbury Laboratory, Daresbury\nWA4 4ADWarringtonUnited Kingdom"
] |
[] |
The self-interaction corrected local spin density (SIC-LSD) formalism and the standard GGA treatment of the exchange-correlation energy have been applied to study the collapse of the magnetic moment of Fe impurities in MgO. The system Mg 1−x Fe x O is believed to be the second most abundant mineral in the Earth's lower mantle. We confirm the experimentally found increase of the critical pressure upon iron concentration. Our calculations using standard GGA for a fixed Fe concentration show that different arrangements of Fe atoms can remarkably shift the transition pressure of the high spin (HS) to low spin (LS) transition. This could explain the experimentally found broad transition regions. Our results indicate that the HS-LS transition in Mg 1−x Fe x O is first order. We find that SIC-LSD fails to predict the divalent Fe configuration as the lowest energy configuration and discuss possible reasons for it.
| null |
[
"https://arxiv.org/pdf/0904.2901v1.pdf"
] | 117,408,032 |
0904.2901
|
b338a3cfae10692758c647083649502b815c7b78
|
First-principles study of the effect of Fe impurities in MgO at geophysically relevant pressures
19 Apr 2009
Donat J Adams
W M Temmerman
Z Szotek
Dep. Materials Sciences
Lab. Crystallography
ETH Zürich
Switzerland
Daresbury Laboratory, Daresbury
WA4 4ADWarringtonUnited Kingdom
First-principles study of the effect of Fe impurities in MgO at geophysically relevant pressures
19 Apr 2009(Dated: April 19, 2009)arXiv:0904.2901v1 [cond-mat.str-el]PACS numbers: 7127+a, 7115Mb
The self-interaction corrected local spin density (SIC-LSD) formalism and the standard GGA treatment of the exchange-correlation energy have been applied to study the collapse of the magnetic moment of Fe impurities in MgO. The system Mg 1−x Fe x O is believed to be the second most abundant mineral in the Earth's lower mantle. We confirm the experimentally found increase of the critical pressure upon iron concentration. Our calculations using standard GGA for a fixed Fe concentration show that different arrangements of Fe atoms can remarkably shift the transition pressure of the high spin (HS) to low spin (LS) transition. This could explain the experimentally found broad transition regions. Our results indicate that the HS-LS transition in Mg 1−x Fe x O is first order. We find that SIC-LSD fails to predict the divalent Fe configuration as the lowest energy configuration and discuss possible reasons for it.
I. INTRODUCTION
The high spin (HS) to low spin (LS) transitions in magnesiowüstite, Mg 1−x Fe x O, and wüstite, Fe 1−x O, are of great geophysical importance. Magnesiowüstite is believed to be the second most abundant mineral in the Earth's mantle. 1,2 While MgO was thoroughly studied both in theory and experiment 3,4,5,6,7,8,9,10,11,12,13,14,15,16 at high pressures, few theoretical studies exist for Mg 1−x Fe x O mainly because of the difficulty to treat the effects of strong correlation for the Fe-3d orbitals. 17,18,19 The magnetic phase transition could strongly influence the partition coefficient of Fe between MgO and MgSiO 3 perovskite and postperovskite, 20 dramatically change radiative conductivities, 21 and lead to a hardening of the materials. 22 However, magnesiowüstite Mg 1−x Fe x O and wüstite Fe 1−x O have been studied in numerous experimental papers. 20,23,24,25,26,27,28 The main problem has been, that wüstite exists only as a non-stoichiometric compound. 19,29,30,31,32 There is a general consensus that its rocksalt-type structure contains a fully occupied O 2− sublattice, while Fe exists predominantly in the form of Fe 2+ , with some Fe 3+ , as well as vacancies and interstitials, exhibiting a short range order. The resulting clusters arrange with a long range order, which -depending on the density of interstitials -can be commensurate or incommensurate. 25,26,27,28 For Fe 1−x O Jacobsen et al. 33 reported a transition from a cubic to a rhombohedral phase at a pressure between 22.8 GPa and 27.7 GPa at room temperature. At low temperatures, on the other hand, Struzhkin et al. 34 found a phase transition from a cubic to a rhombohedral structure with a Néel-temperature of T N = 198 K at ambient pressure. At ambient temperature this transition is shifted to a pressure of 15 GPa.
At pressures of 85 to 143 GPa a magnetic high-spin to lowspin transition was observed in several experiments. 19,32,35 Some studies find a broad transition region, 35 while in others 32 the transition is first order. Differential stress might have smeared out the phase transition over a broad region in some studies or small compositional differences within the sample could have changed the transition behavior.
Recent developments in high pressure physics allowed to investigate the LS-HS transition in magnesiowüstite. Badro et al. 20 and Lin et al. 24 could prove what was suggested a long time before: 2 the complete electronic rearrangement of Fe in Mg 1−x Fe x O under pressure, while the NaCl-type structure remains stable across the phase transition. The transition pressures are in reasonable agreement (49 to 75 GPa for x = 0.15, 20 50 to 60 GPa for x = 0.25 24 ). The transition is linked to a hardening of the materials and a decrease of the molar volume (across the phase transition: V 0LS /V 0HS = 0.904, bulk modulus: K HS 0T = 160.7 GPa, K LS 0T = 250 GPa 24 ). In another study the HS-LS transition in Mg 1−x Fe x O was studied for different iron concentrations (x = 0.2, 0.5 and 0.8). 36 The transition pressure turns out to depend linearly on the Fe concentration and is 40, 60 and 80 GPa, respectively. For Fe 0.97 O the transition pressure is shifted to 90 GPa. At high Fe concentration (x = 17%) Badro et al. 37 in an X-ray emission study find that the HS state is stable up to 143 GPa, whereas Pasternak et al. 35 conjecture from Mössbauer spectroscopy a strong temperature dependence of the transition pressure. They found the transition to be completed only at 120 GPa (Fe 0.94 0) at 450 K.
In the present paper we study the effect of Fe impurities in MgO at geophysically relevant pressures, using selfinteraction corrected local spin density (SIC-LSD) method which allows to treat the localized d electrons of Fe on equal footing with the other itinerant electrons. In addition, we use GGA approach and the VASP code 38 to study the influence of possible Fe impurity clustering in the MgO supercells on the relevant transition pressures. The aim is to realize different Fe concentrations in MgO and investigate their effect on the experimentally observed properties.
The outline of the paper is as follows. In Section II, we briefly describe the SIC-LSD methodology. Section III concentrates on the discussion of the present SIC-LSD and GGA results, while Section IV concludes the paper.
II. METHODOLOGY
A. The self-interaction corrected LSD
The standard LSD approximation for the exchangecorrelation energy introduces an unphysical interaction of an electron with itself, the so-called self-interaction (SI). It is the aim of the self-interaction corrected local spin density approximation to construct a self-interaction free energy functional
E SIC = E LSD − α δ SIC α ,(1)
where α numbers the occupied orbitals and the self-interaction correction (SIC), δ SIC α , of the orbital α is
δ SIC α = U[n α ] + E LSD xc [n α ] .(2)
It is known that the exact exchange-correlation energy E xc [n α ], depending on the spin densityn α , for the case of a single electron orbital with electron density n α , cancels the Hartree energy U[n α ] identically:
U[n α ] + E xc [n α ] = 0 .(3)
In the case of approximate exchange-correlation energy functionals this cancelation can be guaranteed by Eq. 1.
Varying the above SIC-LSD energy functional with respect to the orbital spin densities, with the constraint that the φ α 's form a set of orthonormal functions, one gets the SIC-LSD generalized eigenvalue equations
H α | φ α > = H 0σ + V S IC α (r) | φ α >= α ′ λ αα ′ | φ α ′ >,(4)
with H 0σ being the orbital independent LSD Hamiltonian. The Lagrangian multipliers λ αα ′ are used to secure the fulfilment of the orthonormality constraint. Due to the orbital dependent SIC potential, V S IC α , the SIC energy functional is not stationary with respect to infinitesimal unitary transformations among the orbitals. The so-called localization criterion
< φ β | V S IC α − V S IC β | φ α >= 0 ∀(α, β)(5)
has to be fulfilled to ensure that the solutions of the SIC-LSD equations (4) are most optimally localized to reach the absolute minimum of the SIC-LSD functional (1). The SIC-LSD approach is fully ab initio and introduces no adjustable parameters, either for the delocalized (band-like) or localized electrons. For extended states the SIC vanishes. The SIC-LSD formalism has the advantage that it allows to compare different valence states and spin configurations of the same atom. The nominal valence, N val , in the SIC-LSD approach is defined as For realizing LS state, SIC is applied to the three majority and three minority t 2g electron states. The HS 3 stands for the trivalent Fe configuration, where all the majority d electrons are treated as localized, by applying SIC. The calculations with applying no SI-corrections have been labeled lsd. All possible electronic configurations with five or six SI-corrected orbitals namely trivalent and divalent ions have been considered in the present study. However, the ones listed below correspond to the lowest enthalpies. Here 1 means that a given state has been SI-corrected, and 0 otherwise.
N val = Z − N core − N S IC ,(6)
Majority channel
Minority channel t 2g t 2g e g t 2g e g t 2g t 2g e g t 2g e g HS 1 1 1 1
1 1 0 0 0 0 LS 1 1 0 1 0 1 1 0 1 0 HS 3 1 1 1 1 1 0 0 0 0 0
where Z is the atomic number, N core is the number of core (and semicore) states and N S IC is the number of self-interaction corrected states. The ground state valence is the one defined by the ground state energy. In our application of SIC-LSD to (Mg,Fe)O all possible electron configurations of the Fe atom with five and six SI-corrected d orbitals have been considered. 65 The most energetically relevant ones are given in Table I. The energy functional without any SIC states is simply equivalent to the standard LSD approximation and labelled lsd in the following. Thus LSD is a local minimum of the SIC-LSD functional, corresponding to the case when all the electrons are treated as itinerant and described by the Bloch wave functions. The choice of the SIC orbital configuration, i. e. the number and symmetry of localized vs. delocalized states, can significantly influence the corresponding total energy. The electronic configuration giving rise to the most negative total energy, with the most optimally localized orbitals, defines the absolute energy minimum of the SIC-LSD energy functional as well as the corresponding valency. The SIC-LSD approach used in this work has been implemented in the linear muffin-tin orbitals (LMTO) band structure method with the atomic sphere approximation (ASA). 39,40,41 The slightly overlapping ASA spheres approximate the polyhedral Wigner-Seitz cell and the sum of their volumes equals to the volume of the actual unit cell. The ASA does not allow for atomic relaxations, as different ionic arrangments will lead to different sphere overlaps. In the LMTO-ASA approach empty spheres (E) can be introduced in order to increase the space filling and to minimize the overlap.
In all the calculations performed with the LMTO-ASA method we have treated the valence states of the Mg-3s, Mg-3p, O-2s, O-2p, Fe-3d, Fe-4s, Fe-4p, E-1s as the so-called low waves, while the Mg-3d, O-3d and E-2p states have been represented as intermediate waves. 42 All the electrons in lower shells have been put in the core but allowed to relax. The separation of valence electrons into the low and intermediate waves has been done for the purpose of reducing the size of the eigenvalue problem. The secular equation that is fully di-agonalized is constituted by the low waves and provides the eigenvalues and eigenvectors. The intermediate waves enter the Hamiltonian through their tails, partly retaining their true characteristics.
For both magnesiowüstite and wüstite, we have assumed the ideal rock salt structure. The ASA radii of MgO were adjusted in order to reproduce the experimental band gaps of 7.83 eV. 43 Best agreement was found with radii of 1.34646 Å, 1.05313 Å, 0.751847 Å for Mg, O and the interstitial spheres E, with the resulting LDA band gap of 4.7 eV. The total volume of the spheres was equal to the volume of the cell for all the SIC calculations while the ratios of the ASA radii were kept constant. The ASA radius of Fe was equal to the one of Mg.
To realize different concentrations of Fe impurities in the systems studied, three supercells have been used, namely consisting of four, eight and 32 formula units of MgO. By substituting up to four Mg atoms by Fe atoms one could realize Fe concentrations (x = n Fe n cations ) of 3.125%, 12.5%, 25%, 50% and 100%. For Fe concentrations of 3.125%, 12.5%, 25%, we have chosen the smallest possible supercell of the ideal FCC/NaCl, namely replacing one Mg atom by one Fe atom respectively in the supercells consisting of 32, eight and four formula units. The concentrations of 50% and 100% have been realized using a supercell of eight atoms in total and replacing respectively two and four Mg atoms by Fe atoms. Consequently, both ferromagnetic (FM) and antiferromagnetic (AFM) orders as well as charge disproportionation could be studied. This corresponds to an Fe arrangement with maximal distances between the Fe atoms and would provide the lowest transition pressures. Regarding the AFM structure, the chosen supercells allowed to realize only the so-called AF1 order where the magnetic spins are arranged in parallel in the (001) planes, but are anti-aligned between the planes. The number of k-points has been chosen inversely proportional to the size of the supercell (eight, 16, 64 atoms) and was 16 × 16 × 16, 8 × 8 × 8 and 4 × 4 × 4, respectively. This gives energy differences precise to within 1.7×10 −6 eV/atom (1.25×10 −7 Ryd/atom).
B. Pressure determination
The pressures and enthalpies, of relevance to the present study, can be determined from the LMTO-ASA total energy, E, vs. volume, V, calculations through invoking the equation of state (EOS) and fitting to a number of calculated data points. 66 Specifically, the third-order Birch Murnaghan EOS has been used, 44,45 namely
E(x) = E 0 + 9B 0 V 0 16 B ′ (x 2/3 − 1) 3 +(x 2/3 − 1) 2 × (6 − 4x 2/3 ) ,(7)P(x) = 3B 0 2 x 7/3 − x 5/3 × 1 + 3 4 (B ′ − 4) × (x 2/3 − 1) ,(8)x = V 0 V ,(9)
allowing to compute the enthalpy H at any volume as
H(V) = E(V) − P(V) V,
with P denoting the pressure. In the above, V 0 stands for the equilibrium volume, B 0 for the bulk modulus and B ′ for its first pressure derivative. The calculation of enthalpy differences, ∆H(P), can be accomplished through the numerical inversion of the pressure in Eq. 8. This allows to compare enthalpies of two spin configurations labelled s and s ′ at a given pressure:
∆H (s,s ′ ) = H s (V s ) − H s ′ (V s ′ ) .(10)
To determine the theoretical equilibrium volumes and total energies for all the studied scenarios, the lattice parameters have been sampled in the range from -10% up to + 16% (relative to the zero-pressure value) in steps of 2%. The resulting volume increase or decrease of -33.1% to 40.7% typically corresponds to a pressure range of -20 GPa to 285 GPa. 67
III. RESULTS AND DISCUSSION
A. (Mg,Fe)O in the SIC-LSD framework
In the application of SIC-LSD to Fe-doped MgO, we have studied 29 different localization-delocalization scenarios for three different supercells, various Fe concentrations, and symmetries of localized Fe d states. Among all the cases we have identified four energetically relevant configurations, namely the three SIC configurations listed in Table I as well as the LSD solution. This finding has been independent of the actual Fe concentration. The HS configuration is the high-spin divalent scenario, where all majority Fe d states and one minority Fe t 2g state are treated as localized through invoking SIC, giving rise to an insulating state. The HS 3 , trivalent, configuration corresponds to the scenario where only the majority Fe d states are corrected for the self-interaction, and thus localized, constituting a half-metallic state. The LS state corresponds to the case where both majority and minority Fe t 2g states are localized by SIC. This naturally leads to a non-magnetic and insulating configuration.
From the enthalpy calculations we find, in disagreement with experimental evidence, that the HS 3 and LSD solutions are energetically most favorable, respectively at the low and high pressure regions. This can be seen in Fig. 1(a), where the enthalpy differences, with respect to the HS 3 configuration, are plotted as a function of pressure for all the relevant scenarios and Fe concentration of 3.125%. The trivalent HS 3 state constitutes the ground state solution for all the pressures up to about 140 GPa. However, at very low pressures the divalent HS state lies very close in energy to the HS 3 state and the enthalpy difference, at 0 GPa and 3.125% Fe concentration, is H HS − H HS 3 =0.05 eV. Note that at this Fe concentration the total energies of the HS and HS 3 configurations, evaluated at their respective theoretical volumes, are fully degenerate.
The nominal valence of the HS 3 state, being 3+, is at odds with the experimental findings, where Fe appears clearly as Fe 2+ in (Mg,Fe)O (see e.g. Ref. 20) or with a small ratio of additional Fe 3+ in Fe x O (see e.g. Ref. 32). Note, however, that the nominal valence defined in SIC-LSD not always corresponds to the chemical valence. Although the nominal valences of the trivalent and divalent states differ by one (namely 3+ vs. 2+), in terms of a simple charge counting or charge disproportionation, the two ions, corresponding respectively to the HS 3 and HS configurations, differ only by up to 0.1 electron. Similar observation was made by Szotek et al. 46 for magnetite. Despite the latter, it is still surprising to find the trivalent, instead of the divalent, Fe-state as the lowest energy configuration, in particular for Fe x O (with x=1). This is in variance to the SIC-LSD results obtained for all the other transition metal monoxides, namely MnO, CoO and NiO. 47,48 It is possible that the failure of SIC-LSD to find the divalent ground state in FeO has its origin in the fact that LSD substantially overestimates the exchange splitting for the systems in question. Consequently, the energy gained on localizing an additional electron to create a divalent Fe ion is not sufficient to overcome that exchange splitting. However, since FeO does not occur in a stoichiometric form, but as Fe 1−x O, and experiments indicate existence of both divalent and trivalent ions, one would need to perform more realistic calculations to establish the importance of the off-stoichiometry and valence fluctuations for obtaining the divalent ground state in FeO.
At high pressures, it is the experimentally indicated 36 LS configuration that is relevant, with its nominal valence of 2+ in agreement with the experimentally found Fe 2+ in wüstite and magnesiowüstite. The LS configuration competes with the LSD description of the Fe-3d states, as seen through the corresponding enthalpy difference which at 0 GPa and 3.125% Fe concentration is H lsd − H LS =0.7 eV (see Fig. 1), in favour of the LS state, while the LSD becomes energetically more favorable for pressures in the excess of 140 GPa. At such high pressures the gain in band formation energy on delocalization of all the Fe d electrons clearly wins with the localization energy of those electrons. Since the magnitude of SIC is strongly dependent on the orbitals, one needs to be guided by energetics in defining the ground state energy and configuration.
Introducing several Fe impurities into a supercell (Mg 2 Fe 2 O 4 and Fe 4 O 4 ) opens a possibility of realizing both parallel (labelled [f], ferromagnetic) and antiparallel (labelled [a], antiferromagnetic≡AF1) arrangements of the spin magnetic moments on those Fe atoms. In Fig. 1(b), the enthalpy differences are plotted for the above mentioned configurations, and both parallel and antiparallel arrangements of spins on the Fe atoms in the relevant supercell with 50% Finally, we consider the dependence of the HS-LS transition pressure on the concentration of the Fe impurities in a supercell. As can be seen in Fig. 2, our study predicts an increase of this transition pressure with rising Fe concentration. However, the absolute transition pressures for isostructural phase transitions, as calculated in this SIC-LSD study, are higher than the experimental values. 49 [50]; triangles ( ) -Mössbauer spectroscopy, [36]; arrow ( ) -Mössbauer spectroscopy [35]; circle ( ) -X-ray diffraction [32]. tion from HS 3 to lsd or from HS to LS. The second one occurs at considerably lower pressures than the first one, and is the relevant one for comparison with experiments, although leading to a phase diagram in rather poor agreement with experiment. The latter implies the existence of Fe 2+ , and a low pressure structure with one minority Fe t 2g state and five majority d states. 36 One might envisage that the HS state could possibly become favourable if in the SIC-LSD calculations the ionic relaxations had been applied. It is obvious that ASA error can substantially affect total energies, although one would like to hope that these errors would be less severe when considering only the energy differences between different configurations. Also, across the phase transition Fe is known to reduce its ionic radius and therefore the ionic relaxations could significantly reduce the enthalpies of the LS structures which in turn could further decrease the transition pressure. 69
B. Density of states from SIC-LSD
In Fig. 3, we present the densities of states (DOS) calculated for all the energetically relevant scenarios and the Fe impurity concentration of 12.5%. As can be seen in the figure, at this intermediate concentration the localized Fe-3d states form narrow bands. At higher concentrations of Fe impurities these bands extend over a considerable energy range, e.g. 3-5 eV at 25% of Fe. For the HS and HS 3 configurations, the occupied Fe 3d band states lie below the predominantly O 2p valence band. As mentioned earlier, the SI corrections for the HS and HS 3 configurations differ by one localized electron more in the former, which is reflected in the presented DOS (see Figs. 3(a) and 3(b)). The difference is that the first isolated, minority Fe t 2g band, lying just below the valence band, is moved above the valence band, coinciding with the Fermi level (see Fig. 3(b)). While for the HS configuration SIC-LSD delivers, in agreement with with experiment and other theoretical considerations, 51,52,53 an insulator of charge transfer character, the HS 3 configuration gives rise to a half-metal, with a large band gap in the majority band, and a metallic behavior in the spin-down channel, with a partially occupied minority Fe d band at the Fermi level. At high pressures the LS and LSD configurations are of relevance. The DOS of the LS is shown in Fig. 3(c). It is characterized by hybridized Fe-3d electrons with the predominantly O 2p valence band, and a large band gap to the Fe d conduction band. The lsd DOS shows a substantially reduced band gap, which is between the occupied and unoccupied Fe impurity d bands. So, while the SIC-LSD LS state is a charge transfer insulator, the LSD gives rise to a Mott-Hubbard insulator (see Fig. 3(d)).
C. GGA-PAW calculations, Fe clustering
So far we have concentrated mostly on the correlated nature of Fe d electrons in describing electronic properties of doped MgO within SIC-LSD approach. Here we consider an importance of a possible Fe impurity clustering. To study this, we have performed a number of independent calculations for low Fe concentrations in Mg 1−x Fe x O, using the standard GGA approximation 54 and the Vienna Ab-Initio Simulation Package (VASP) 38 . The latter provides the standard projector augmented wave potentials for Mg, Fe and O with core configurations 2p 6 3s 2 (Mg), 3p 6 3d 7 4s 1 (Fe) and 2s 2 2p 4 (O). 55 In the actual calculations the electronic optimization has been run with a self-consistency threshold of 10 −9 eV, ionic optimization has been done with a threshold of 10 −6 eV. The 4 × 4 × 4-Monkhorst-Pack scheme has been applied for the k-point sampling, producing four irreducible k-points. 56 The energy cut-off of the plane wave basis set has been equal to 700 eV. In order to speed up the evaluation of the non-local part of the potentials real space projections have been used. A second order Methfessel-Paxton smearing has been used (σ=0.04 eV), introducing an error in electronic entropy, T S el , of at most 0.111 meV/atom. 57 The accuracy of the energy differences between the HS and the LS state has been converged to 1.3 × 10 −6 eV/atom. Note, however, that within the GGA implementation all the electrons are treated on equal footing as delocalized, with the HS state being described as a ferromagnetic state, whilst the LS state as a non-magnetic state. Thus the two are not directly comparable to the HS and LS states calculated within SIC-LSD, as the concept of divalent and trivalent ions cannot be clearly defined within GGA.
To investigate the effects of Fe clusters, supercells containing 64 atoms have been set up, including one and four Fe impurities (x Fe = 3.125 %, x Fe =12.5 %). In the latter case, 30 distinct arrangements of Fe atoms exist. 70 We have used the most extreme cases, where the Fe atoms form a tetragonal cluster with an Fe-Fe distance of ≃ 2.69 Å and where they avoid each other at a maximal distance of ≃ 5.37 Å (at 100 GPa). The resulting three clusters (one Fe atom, four Fe atoms at minimum distance, four Fe atoms at maximum distance) have been explored performing spin-polarized and nonspin-polarized GGA calculations (in the former case, a starting magnetic moment of 4 µ B has been assigned to each Fe atom). In the spin-polarized case the Vosko-Wilk-Nusair interpolation for the correlation part of the exchange-correlation functional has been used, 58 . The lowest energy structure has been found by comparing the enthalpies of all the different structures.
The GGA-PAW calculations in the 64 atomic supercell find HS-LS transition pressures of 26 GPa (x = 3.125 %), 30.3 GPa (x = 12.5 %, diluted configuration of Fe), and 63.4 GPa (x = 12.5 %, clustered configuration of Fe), as calculated from the intersection of the enthalpy curves of the ferromagnetic and non-magnetic results. The lowest enthalpy structures always give a total magnetization of either µ = 0.0µ B (non-magnetic) or µ = 4.0µ B (ferromagnetic). Arrangements of Fe atoms with an intermediate spin magnetic moment have been found in some cases (e.g. µ = 3µ B in the case of Fe clusters between 90 GPa and 185 GPa). However they always turn out to be unstable because energetically lower lying cluster configurations exist.
The transition pressure turns out to increase with the Fe concentration. The two studied concentrations (x = 3.125% and x = 12.5%) allow to determine the linear dependence of the transition pressure as
P cr = 24.6 + 0.45 x .(11)
The relatively small difference to the experimental value of P cr = 28+0.63 x can be explained, if it is assumed that the statistical distribution of the Fe atoms in the experimental sample will not be perfectly diluted, but in some cases result in having Fe atoms at smaller distances (which dramatically raises the critical pressure). However, this can also explain the broadness of the experimentally found transition pressure: depending on the local Fe configuration the spin magnetic moments collapse at different pressures. Figure 4 indicates that at low pressures (where only the HS arrangments are of interest) the Fe atoms prefer to form clusters, while at high pressures (where only the LS arrangments are of interest) the Fe atoms prefer a diluted configuration. Upon pressure release at the pressure of about 60 GPa the LS diluted arrangement and the HS clustered arrangement coincide. The different Fe arrangments in the high pressure and low pressure regions might hamper this phase transition. Experimental HS-LS transition pressures are predicted to depend on the pressure the sample has been equilibrated at (the equilibration could be enhanced through heat, which accelerates ionic diffusion). If the equilibration is done at high pressure, upon pressure release the LS configuration would be stable up to relatively low pressures (≃30 GPa). A sample equilibrated at low pressure on the other hand would preserve its magnetization up to relatively high pressures (≃60 GPa).
IV. CONCLUSIONS
Our GGA results indicate a first order transition for all the compositions studied, both for Mg 1−x Fe x O and Fe 1−x O, in good agreement with the notion in Ref. 32 for Fe 1−x O. The electronic rearrangement of the Fe atoms leads to a discontinuity of the first derivative of the enthalpy, i.e. the volume (see Fig. 4). This affects various physical properties of the HS and LS phases, such as the zero pressure bulk modulus and the ground state volume.
In experiment the HS-LS transition appears to be smeared out over a large pressure range, 20,24,36 which in the case of Fe 1−x O has led to the conjecture that the transition might be second order. 35 We can not support this unless this is a temperature induced smearing which our T = 0 calculations do not consider. The results for various Fe arrangements, keeping the Fe concentration fixed, indicate that the local arrangement of Fe strongly influences the spin transition pressure of the Fe atoms. The statistical distribution of the Fe atoms and the short range defect order can therefore smear out the transition pressure over a large pressure range. As the HS-LS transition is isosymmetric, according to Landau theory it can only be first-order or (above the critical temperature) fully continuous. 59,60 We can confirm the experimentally found trend that the HS-LS transition pressure increases with increasing Fe concentration. This might be the result of Fe atoms which -if situated closely enough -simultaneously optimize their electronic configurations. These magnetic Fe clusters can not easily be demagnetized by pressure. The increase of the Fe concentration leads to a statistical increase of short Fe-Fe distances, which through the described mechanism could in turn lead to an increase of the critical pressure.
It has been proposed that the HS-LS transition in magnesiowüstite would lead to a discontinuity in the Earth's lower mantle. 20 Although the transition appears to be first order a discontinuity seems to be improbable: first the sensitivity of the transition pressure to the (local) Fe concentration could smear out the transition over a wide region in depth. Second, temperature could further increase this effect and shift the transition to even higher pressures, barring LS Fe 2+ from the Earth's interior. 18,61 The transition pressures predicted by SIC-LSD appear to be higher which might be a result of an "overcorrection" of the energy of the localized states. It has been argued in Ref. 62 that a weighted self-interaction correction would be desirable. The high transition pressures calculated in this work might also be due to the fact that the ASA does not allow relaxations of the atomic spheres, which might be significant in the LS phase, where the ionic radius of Fe is small. At high pressures the LSD treatment can lead to low energy structures. This could justify the treatment of Mg 1−x Fe x O at those pressures using standard approximations for the exchange-correlation energy such as the GGA. Apart from this it could also indicate a second phase transition which involves a full delocalization of the Fe-3d states. Furthermore all the transitions considered in this paper involve five or six localized electrons in the high volume phase and no localized electrons in the low volume phase. Consideration of valence fluctuations or 'intermediate' localization involving four, three, two and one localized states could plausibly lead to a reduction of the transition pressure. The SIC-LSD formalism, implemented in the multiple scattering theory, allows to consider static valence and spin fluctuations at finite temperatures, by invoking CPA (coherent potential approximation) and DLM (disordered local moments) approaches, which was successfully applied to studying phase diagrams. 63,64 This would also be useful for the further theoretical investigation of the present geophysically relevant systems. For the nearest future, it may be instructive to first apply SIC-LSD in a full potential version in order to allow for ionic relaxations to study the influence of distortions on the ground state properties of these systems.
Fe concentration. For the HS configuration the [f] and [a]
FIG. 1 :
1Enthalpy differences at different Fe concentration x. (a) x = 3.125%. The HS 3 configuration appears to be energetically most favorable up to a pressure of 145 GPa, where a phase transition is predicted to the state where all the Fe-3d delocalize (lsd). The commonly assumed HS configuration is less energetically favourable, and at the pressure of 90 GPa the electrons completely rearrange to the non-polarized configuration LS, although the localization of the electrons is not lost. (b) x = 50%. Two Fe atoms were introduced into a cell containing 8 atoms in total (two Mg, two Fe, four O). Parallel and antiparallel alignments of the spin moments is denoted in brackets [f] and [a], respectively. More than 10 different arrangements with asymmetric charge configurations on the Fe atoms were also tried out, but resulted in high enthalpies. They are omitted in the plot for more clear readability.arrangments return the same energies while for HS 3 the [f] arrangement appears always lower in energy (∆E =0.5 eV for Mg 0.5 Fe 0.5 O and ∆E =1.5 eV FeO 68 ) than the antiparallel alignment.
FIG. 2 :
2Two possible scenarios can operate at any Fe concentration: a possible transi-Mg 1−x Fe x O: Experimental and theoretical transition pressures. Squares ( ) -SIC-LSD calculations, scenario HS 3 -lsd (electronic configuration HS 3 at low pressures, no SI-corrections at high pressure); bullets ( ) -SIC-LSD calculations scenario HS-LS (electronic configuration HS at low pressures, configuration LS at high pressure); lozenges ( ) -GGA-PAW calculations (low concentrations); stars (⋆) -LDA+U calculations
FIG. 3 :
3(a) Spin-resolved density of states (DOS) for HS state as calculated from SIC-LSD at a pressure of 2 GPa and Fe concentration of 12.5%. (b) Spin-resolved density of states for the HS 3 state as calculated from SIC-LSD at a pressure of 2 GPa and Fe concentration of 12.5%. (c) Spin-resolved density of states for the LS state calculated at a pressure of 124 GPa and Fe concentration of 12.5%. (d) Spin-resolved density of states calculated with the standard LSD approximation at 102 GPa and Fe concentration of 12.5%. In all the cases the Fe DOS states is in red, while the total DOS is in black and the O 2p DOS in green.
FIG. 4 :
4GGA-PAW calculations, 64 atomic supercell including four Fe atoms: Comparison of the enthalpies of the different Fe and spin arrangements to the diluted LS setting. In the clustered configuration Fe atoms are at the largest distance possible.
TABLE I :
IThe most relevant spin configurations of Fe, labeled HS and LS according to Ref. 36. Here HS state corresponds to the case where SIC is applied to all the majority and one minority electrons.
AcknowledgementsThe authors are indebted to A.R. Oganov who contributed considerably to the concept of this work. DJA thanks A.R. Oganov and the ETH Zurich Research fund for their support of this work (Grant No. TH-27033). Supercomputers were provided by the Swiss National Supercomputing Centre (CSCS) and ETH Zürich. WMT and ZS would like to acknowledge useful discussions with Drs. Axel Svane and Leon Petit.
* Electronic address: [email protected] † CEA, DAM, DIF, F 91297. Arpajon, France* Electronic address: [email protected] † CEA, DAM, DIF, F 91297 Arpajon, France
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. = Z − N Core, − N Sic, not always correspond to the chemical valence. For example, in the case of the conventional LSD scheme the valence of Fe would be N val = Z − N core − N SIC = 26 − 18 − 0 = 8The nominal valence N val = Z − N core − N SIC does not always correspond to the chemical valence. For example, in the case of the conventional LSD scheme the valence of Fe would be N val = Z − N core − N SIC = 26 − 18 − 0 = 8.
We have used a standard procedure such as provided with the EX-CITING. We have used a standard procedure such as provided with the EX- CITING code at http://exciting.sourceforge.net.
The enthalpy differences have been fitted to the second order polynomials. However, the error resulting from these fits and the numerical inversion of Eq. 8 have given rise to total errors of the order 10 −8 -10 −12 eV/atomThe enthalpy differences have been fitted to the second order poly- nomials. However, the error resulting from these fits and the nu- merical inversion of Eq. 8 have given rise to total errors of the order 10 −8 -10 −12 eV/atom.
∆E is given for one supercell containing 2 Mg, 2 Fe and 4 O in the first case and 4 Fe and 4 O in the second. ∆E is given for one supercell containing 2 Mg, 2 Fe and 4 O in the first case and 4 Fe and 4 O in the second.
our calculations the Fe-ASA radii were fixed to those of Mg at ambient conditions. According to Anderson 2 they are comparable: r Mg = 0.72 Å, r Fe,HS = 0.77 Å. However. in the LS phase the radius of Fe dramatically reduces to 0.61 Å. Due to the LMTO-ASA approach, the reduction of the ionic radii could not be taken into accountIn our calculations the Fe-ASA radii were fixed to those of Mg at ambient conditions. According to Anderson 2 they are compara- ble: r Mg = 0.72 Å, r Fe,HS = 0.77 Å. However, in the LS phase the radius of Fe dramatically reduces to 0.61 Å. Due to the LMTO- ASA approach, the reduction of the ionic radii could not be taken into account.
The clusters can be characterized by the distance between the first Fe impurity to the following ones. Five different distances can be found in the 64 atomic cell. They can host 12, three, 12, three and one Fe atom respectively. This gives one, five, 14, 30, 53 distinct scenarios of one. Note that this is a combinatorial problem. two, three, four and five impurities, respectivelyNote that this is a combinatorial problem. The clusters can be characterized by the distance between the first Fe impurity to the following ones. Five different distances can be found in the 64 atomic cell. They can host 12, three, 12, three and one Fe atom respectively. This gives one, five, 14, 30, 53 distinct scenarios of one, two, three, four and five impurities, respectively.
|
[] |
[
"A Flexible Modeling Approach for Robust Multi-Lane Road Estimation",
"A Flexible Modeling Approach for Robust Multi-Lane Road Estimation"
] |
[
"Alexey Abramov ",
"Christopher Bayer ",
"Claudio Heller ",
"Claudia Loy "
] |
[] |
[] |
A robust estimation of road course and traffic lanes is an essential part of environment perception for next generations of Advanced Driver Assistance Systems and development of self-driving vehicles. In this paper, a flexible method for modeling multiple lanes in a vehicle in real time is presented. Information about traffic lanes, derived by cameras and other environmental sensors, that is represented as features, serves as input for an iterative expectation-maximization method to estimate a lane model. The generic and modular concept of the approach allows to freely choose the mathematical functions for the geometrical description of lanes. In addition to the current measurement data, the previously estimated result as well as additional constraints to reflect parallelism and continuity of traffic lanes, are considered in the optimization process. As evaluation of the lane estimation method, its performance is showcased using cubic splines for the geometric representation of lanes in simulated scenarios and measurements recorded using a development vehicle. In a comparison to ground truth data, robustness and precision of the lanes estimated up to a distance of 120 m are demonstrated. As a part of the environmental modeling, the presented method can be utilized for longitudinal and lateral control of autonomous vehicles.
|
10.1109/ivs.2017.7995904
|
[
"https://arxiv.org/pdf/1706.01631v1.pdf"
] | 24,917,157 |
1706.01631
|
5b0815538a9f7f6aeb1fbb7eae3eddfd083d19cd
|
A Flexible Modeling Approach for Robust Multi-Lane Road Estimation
Alexey Abramov
Christopher Bayer
Claudio Heller
Claudia Loy
A Flexible Modeling Approach for Robust Multi-Lane Road Estimation
A robust estimation of road course and traffic lanes is an essential part of environment perception for next generations of Advanced Driver Assistance Systems and development of self-driving vehicles. In this paper, a flexible method for modeling multiple lanes in a vehicle in real time is presented. Information about traffic lanes, derived by cameras and other environmental sensors, that is represented as features, serves as input for an iterative expectation-maximization method to estimate a lane model. The generic and modular concept of the approach allows to freely choose the mathematical functions for the geometrical description of lanes. In addition to the current measurement data, the previously estimated result as well as additional constraints to reflect parallelism and continuity of traffic lanes, are considered in the optimization process. As evaluation of the lane estimation method, its performance is showcased using cubic splines for the geometric representation of lanes in simulated scenarios and measurements recorded using a development vehicle. In a comparison to ground truth data, robustness and precision of the lanes estimated up to a distance of 120 m are demonstrated. As a part of the environmental modeling, the presented method can be utilized for longitudinal and lateral control of autonomous vehicles.
I. INTRODUCTION AND RELATED WORK
Advanced Driver Assistance Systems in modern vehicles are aimed to help drivers and increase comfort and safety for them and their passengers. Systems like lane departure warning, lane keeping assist, adaptive cruise control, emergency brake assist and blind spot monitoring are designed to support drivers in keeping their car within the lane and avoiding collisions with other traffic participants and static objects [1]. The next generation of driver assistance technologies and autonomous vehicles require comprehensive and extensive knowledge about the environment. One of its fundamental parts is a continuous and robust perception of the road and its lanes. This includes the reliable representation of multiple lanes on various types of roads with high detection range and availability.
Lane estimation approaches consist of several processing parts: detection of lane markings based on sensors, possible fusion and accumulation of lane information, followed by modeling of lanes. The latter presumes building a mathemat- past: straight lines, parabolic curves, clothoids and splines. The simple straight [2] and parabolic [3] lane models are computationally efficient and very robust against measurement noise, but they only can model certain road shapes and do not have sufficient flexibility for representing a wider range of lane geometries. Clothoids are traditionally used for highway scenarios, since they are utilized in the design and construction of this type of roads [4]- [6]. Despite a high precision on ordinary highway sections, a single clothoid model cannot represent highways in full detail. For instance, it is often not flexible enough to model construction zones, junctions, entrance and exit ramps. Furthermore, on rural roads the clothoid model sometimes fails or cannot be set up with desired distance and accuracy.
A common way to model curves is using piecewise defined functions such as splines, which are composed of connected polynomials. They have proven to have the capacity to approximate complex road shapes with high curvatures, e.g. double bends or sharp turns. Splines are more flexible but also more sensitive to uncertainties in measurement data. Some state-of-the-art approaches model only the ego lane [7], [8]. Others can handle roads with multiple lanes [9], [10], but do not consider semantic information, such as lane marking type, and geometric correlation between lanes are not considered.
In the focus of this paper is a generic approach for robust and continuous modeling of multiple traffic lanes on highways and rural roads. The method builds on features describing lane markings which can be provided by common lane detection techniques. Centerpiece is an iterative lane model estimation method which allows for a free choice of mathematical model. Beyond position and orientation of the measured features, the modeling procedure incorporates supplementary information, such as parallelism, tracking and continuity of lanes. By means of the presented approach, lane estimation can be performed online in a vehicle without requiring prior knowledge about course of the road or number of existing traffic lanes. The outcome of the system can be used for localization, trajectory planning and control of self-driving vehicles. Evaluation of the modeling method is performed in simulated scenarios and on real sensor data, where the modeling result is compared to ground truth.
A description of techniques for estimating the lane model can be found in section II, whereas the evaluation methods and results are presented in section III, followed by the conclusion in section IV. arXiv:1706.01631v1 [cs.RO] 6 Jun 2017 Fig. 1: Architectural overview of the iterative lane modeling method. Either the previous model L t−1 is predicted to the current time step or an initial model L t0 is derived from the current input data F t . After that the expectation maximization algorithm performs the Association and Fitting iteratively until the final solutionL t is estimated.
L t−1 PREDICTION L t L t0 F t EXPECTATION MAXIMIZATION ASSOCIATION FITTINGL t
II. LANE MODEL ESTIMATION
The goal of the method presented in this study is to obtain a robust representation of traffic lanes on a road that can be used for trajectory planning, longitudinal and lateral control of a vehicle and localization within a map. The resulting lane model (see section II-B) is composed of several mathematical functions each depicting a lane marking which separates two lanes or delimits a lane. Thus, the lane model contains information about number, position and geometry of detected lanes with respect to the vehicle as well as attributes of lane markings, such as type (dashed, solid, block) and color. To find the optimal model, its parameters are estimated in an iterative expectation-maximization (EM) process [11] that alternates between associating the input data (expectation) to the current model and fitting a new model to the associated data (maximization).
An architectural overview of the lane modeling method is shown in fig. 1. At the beginning of each processing loop, a lane model is needed for association of the input data F t . If a model L t−1 was derived at the previous time step, it serves as initial model after predicting it to the current time step as described in section II-D. If no previous model is available, an initial model L t0 is estimated based on the present input data (see section II-C). The input data
A. Input Data
The input data for the lane model estimation provides information about lane markings on the road that the vehicle is currently driving on. This information can, for instance, be obtained by a camera system and image processing methods. In the scope of this work, lane information is derived by fusion of data from various input sensors as described in [6]. Another possibility is the utilization of information provided by maps as (additional) input for the modeling. To be independent from the source and type of the lane information, the interface of the input data is defined in a general way. It is specified as a feature vector F t , which contains lane features
f i = [x i , y i , θ i , Σ i , a i ] ∈ F t ,(1)
where x i and y i constitute the position of a feature f i and θ i constitutes its heading in the vehicle coordinate system 1 .
The measurement uncertainty of each feature is given by a covariance matrix Σ i ∈ R 3×3 with respect to x, y and θ. Additional information such as color of a feature or the corresponding lane marking type is described by the attributes a i . Note that this two-dimensional feature representation of lanes is free of geometric model assumptions.
Additionally to the lane features, the odometry of the vehicle is utilized in the process of lane estimation.
B. Lane Model Description
f (x, l m n ) = ax 3 + bx 2 + cx + d, x ∈ [s m−1 n , s m n [ , where l m n = [a, b, c, d]
T are the parameters of the function. The functions of two subsequent segments are continuous at the M + 1 control points s m n with respect to position, heading and curvature. Additionally, each line has attributes for type (solid, dashed and block 2 ) and color to describe the corresponding lane marking properties.
C. Initial Lane Model
An essential part of the lane estimation method, is the association between the present lane feature input and the current best estimate of the lane model (see section II-E). As no previous estimate is available in the first algorithm loop, an initial model needs to be derived. Therefore, the input 2 In some countries entry and exit lanes are separated from usual driving lanes by wide dashed markings called block markings features in close proximity to the vehicle are projected onto the lateral axis of the vehicle coordinate system to obtain several separated distributions. If enough features contribute to such a distribution, mean lateral position and orientation of the features are computed. Assuming that roads are rather straight within a short distance, each line in the lane model is initialized as a straight line at the computed lateral offset with the corresponding heading. Additionally, this method is applied at every time step to all features that could not be associated to the previously estimated lane model for the case that new lane markings accrue.
D. Prediction
Due to the movement of the vehicle between two subsequent time steps, it is necessary to transform the lane model, obtained in the previous time step, to the current vehicle coordinate system. The prediction of L t−1 is conducted based on the change in odometry and yields a lane model L t that can be used for association of the input data F t . In order to keep the mathematical form of the lane representation when rotating to another coordinate system, an approximate prediction is performed. For a lane model composed of cubic splines, each spline segment is predicted by transforming its limiting control points to the current vehicle coordinate system and performing a fit to position and heading of the transformed points. Note that a cubic spline transformed using this approximation is not continuous in curvature anymore, but it is still sufficiently precise for association of the input data.
E. Association and Model Assumptions
In the expectation part of the EM method, the association is performed by determining the correspondences, which maximize the likelihood given the current lane model estimate L t and the current lane features F t :
c t = arg max ct p(c t |F t , L t ).
The correspondence vector c t consists of tuples f i , l m n of a lane feature and the associated line segment and can be estimated by minimizing the distance between those. Therefore, the Mahalanobis distance [12] is evaluated between a feature and a line with respect to position and orientation. Furthermore, to reject outliers, an upper limit in the Euclidean distance of 2 m is used as additional criteria 3 . If no association can be found for enough lane features, an attempt to set up a new line of the lane model using the method described in section II-C is made and, if successful, the association procedure is repeated. To prevent limiting the association to the range of the previous lane model, each line is extrapolated in longitudinal direction to also associate lane features that lie beyond that range. This means that the scope of the lane model can be increased with every EM iteration.
After association, the current range of each lane model line is determined and its attributes are derived based on the associated lane features. The range of a line is given by the associated features with the shortest and largest longitudinal distance. Lane marking type and color of the lines are determined fusing the corresponding information of the associated features according to Dempster-Shafer theory [13].
Based on the derived lane marking type, assumptions about parallelism between adjacent lines in the lane model are made. From a dashed narrow line one can conclude that it is parallel to its left and right adjacent lines. A continuous line on the other hand indicates lanes that might separate into different directions and therefore no parallelism of lines beyond the continuous one is assumed. In the case of block markings, also no parallelism is assumed beyond the corresponding line. fig. 3).
F. Lane Model Fit Using Constrained Gauss Newton Method
The goal of the maximization in the EM algorithm is to find an optimal model to the input data. In general this task can be formulated as maximizing the probability distribution
L t = arg max Lt p(L t |F t , c t ),(2)
which yields the optimized lane modelL t , given the input features F t and the association c t between input features and lines of the current model. In the following, a description for incorporating different information and constraints into the optimization problem is given.
1) Measurement optimization:
To find the lane model which represents the current lane features in the best way, eq. 2 is formulated as a quadratic minimization problem
L t = arg min Lt fi,l m n ∈ct e(f i , l m n ) T Ω i e(f i , l m n ),(3)
where c t is the correspondence vector determined in the expectation step (see section II-E). The error function
e(f i , l m n ) = f (x i , l m n ) − y i f (x i , l m n )
− θ i is defined as the distance of position and heading between a lane feature (x i , y i , θ i ∈ f i ) and the associated line segment. It is multiplied from both sides to the information matrix Ω i = diag([σ 2 y , σ 2 θ ]) −1 which corresponds to the variances of feature f i .
The sum over the non-linear quadratic equations in eq. 3 can be solved by the Gauss-Newton algorithm as shown for example for pose graph optimization [14]. Given an initial guessL t , the solution to the minimization problem can be found iteratively by solving
H∆L * t = −b (4) with H = fi,l m n ∈ct J(f i ,l m n ) T Ω i J(f i ,l m n ) (5) and b = fi,l m n ∈ct e(f i ,l m n ) T Ω i J(f i ,l m n ).(6)
Here J(f i ,l m n ) is the Jacobian of the error function e(f i ,l m n ) evaluated at the current estimateL t . After solving eq. 4 for ∆L * t , the current estimate is updated L * t =L t + ∆L * t and used in the next iteration as initial guess. After convergence (no change in the parameter update ∆L * t ), the optimized solutionL t = L * t is found. is added to the optimization (eq. 3), whereš m n corresponds to position and orientation of the predicted control point. The information matrix Ω m n regarding the lateral displacement and orientation of the control point is calculated by the inverse of the previous system matrix H t−1 (eq. 5), the Jacobian of the line function and the Jacobian of the coordinate transformation. Incorporation of these error terms into eq. 4 prevents large jumps in position and heading of the lane model between two subsequent time steps.
3) Continuity and Parallelism as equality constraints: In addition to previously formulated optimization problem, equality constraints can be added to limit the state space of the lane model. On the one hand the function defining the geometry of a line needs to be continuous in position, heading and curvature at the control points. On the other hand lanes on highways are often parallel and therefore this parallelism needs to be taken into account while solving the optimization problem.
As
where s m+1 n is the longitudinal position of the control point between the two segments.
For arbitrary line functions the equality constraints can be incorporated by extending the system of eq. 4 using the method of Lagrange multipliers [15]. In general, the quadratic function from eq. 3 subject to the constraints g(L t ) = 0 can be iteratively minimized by solving the linear equation system
H −K T −K 0 ∆L * t λ = −b g(L t ) ,(8)
for the state space update ∆L * t and λ as the Lagrange multiplier. K is the Jacobian of g(L t ) evaluated at the current state estimateL t . H and b are the matrices defined for the measurement optimization in eq. 5 and eq. 6. Using this method to integrate the equality constraints of eq. 7 into the optimization problem, g(x) is a vector of 3N (M − 1) equations.
For certain functions, the linear equation system resulting from eq. 7 can be solved explicitly. To also incorporate parallelism constraints between neighboring lines, the vector of equality equations g(L t ) can be extended. For cubic splines the degree of freedom of the model can be considered to find the number of necessary constraints. Two cubic spline lines have 8M individual parameters, respectively 2(M +3) parameters after including continuity constraints with the substitutions. If these lines should be parallel, the degree of freedom needs to be reduced to two lateral offsets for left and right line, heading, curvature and curvature derivate for the first segment and one curvature change per following segment: d 1 , a 1 , a 2 , ..., a m ] T = (M + 4). Therefore, 2(M + 3) − (M + 4) = M + 2 constraints need to be added to the system of eq. 8, three for the first segment and one per additional segment, where a constraint is defined as the equality of the orientation:
[d 1 n , d 1 n+1 , c 1 ,g(l i , l j ) = f (x, l i ) − f (x, l j ) ! = 0.
The evaluation point x corresponds either to the control points or in the first segment also to the middle of the two limiting control points. Note that the parallelism criteria is approximated by demanding equality of orientation.
III. EXPERIMENTAL EVALUATION
Several tests have been performed to evaluate the presented lane modeling method. In the first section, the modeling of a simulated double bend is analyzed comparing two different modeling functions. In the second section, the performance of the lane detection system is evaluated using sensor data collected on German highways 4 .
A. Lane Modeling on Simulated Data
To demonstrate the functioning and the flexibility of the presented method, a spline and a clothoid 5 Fig. 5 shows the measured RMSE for the resulting lanes when using clothoids and cubic splines in the modeling method. As expected, there is no difference between the two models for the straight road section. However, one can clearly see the benefit of the spline model once the curve comes into range. The maximum error for the clothoid model is approximately four times as high as the one for the spline model. Note that the cubic spline model is not able to perfectly describe the simulated scenario, as its control points are not positioned at the connection points of the simulated clothoids. Nevertheless, using the spline model the lanes are modeled at any point of the simulated scenario with an RMSE below 0.1 m.
B. Lane Estimation Evaluation on Sensor Data
In the analysis described in the following, the performance of lane feature fusion [6] and subsequent lane modeling using the proposed modeling method with cubic splines is evaluated on real data measurements. The modeled lanes are compared to a ground truth map that contains global positions of lane markings as a point vector. The analyzed route is a highway in Germany with three lanes and left and right curves 7 . The map is generated using a high precision GPS system, which is also used for localization in the map. The input data has been collected during several drives on the highway with a development vehicle corresponding to a total driving distance of 24 km. The development vehicle is equipped with camera and radar systems to detect lane markings and other traffic participants. The sensor information is fused in a GraphSLAM based process that yields the lane features which serve as input for the modeling. In the analysis, the lateral deviation of the modeled lanes to the relevant map points is computed at each time step. The result is accumulated in dependence of the longitudinal distance for all of the recorded data. As a measure of performance the RMSE is determined within distance intervals of 10 m. Fig. 6 shows the result for ego and adjacent lanes up to a longitudinal distance of 120 m. Due to higher precision 7 The analysis presented in [6] is based on the same data and ground truth map. Compared to the result presented in [6], the application of the proposed modeling approach using cubic splines provides a similar performance but offers a higher degree of flexibility as shown in section III-A and in the following.
In fig. 7
IV. CONCLUSION
In this work, a flexible real-time modeling method for robust estimation of ego and adjacent lanes is presented. An iterative expectation-maximization method is applied, which alternately associates the input data to the current model and estimates a new model by solving the corresponding constrained optimization problem. The underlying lane model is defined in a generic way as a composition of arbitrary mathematical functions. In the scope of this study, cubic splines are utilized in the approach to model traffic lanes. Evaluation of the method is shown in simulated scenarios as well as real data measurements that were collected with a development vehicle. In the latter, performance of the modeling method is analyzed by comparison of the result to ground truth data.
The results show that the method is capable of modeling multiple lanes on highways that include entry and exit lanes, transitions between roads and lanes with double bends, like construction sites. The precision and robustness achieved in modeling lanes up to a range of 120 m on highways suffice to be used in the development and testing of selfdriving vehicles. In the recent months, the presented method in combination with a lane feature fusion algorithm has been applied to drive several thousand kilometers autonomously on highways.
Enhancement of the presented modeling method could be achieved by incorporating additional information to the optimization, such as constraining the lanes within detected road boundaries. To further improve robustness of the modeled lanes, boundary conditions, such as limiting curvature of the modeled lanes could be considered. Using the road curvature from a map would be a possibility to include prior knowledge for improvement of the result. In addition, the quality of the derived lane model depends on the input data and would therefore draw benefit from improved lane detection methods.
ical model which describes course of the road and relevant traffic lanes. What kind of lane geometries can be represented accurately, depends on the choice of the lane model. A variety of models have been utilized for lane modeling in the § These authors contributed equally to this work. Alexey Abramov, Christopher Bayer, Claudio Heller and Claudia Loy are with Continental Teves AG, Chassis & Safety Division, Advanced Engineering, Guerickestrasse 7, DE-60488, Frankfurt am Main, Germany. {alexey.abramov, christopher.bayer, claudio.heller, claudia.loy} @continental-corporation.com
F t is comprised of a vector of lane features that describe the lane markings and which are defined in section II-A. During the expectation step the lane features are associated to the lane model. As described in detail in section II-F, in the maximization part a new lane model is estimated taking into account the associated input data and the result from the previous time step. This process is repeated iteratively until
modeling result (black) based on the extracted features (red) in vehicle coordinates.
Fig. 2 :
2Examplary snapshot of feature extraction and optimized lane model. there is no change in the association of the input data. The optimized resultL t for the current time step corresponds to the last estimated lane model. Fig. 2 shows an example of features extracted in a camera image (fig. 2a) and the resulting optimized lane model (fig. 2b).
Fig. 3 :
3Example of a lane model with two driving lanes and an exit ramp. This model consists of four lines (N = 4) with two segments (M = 2) and three control points s m n each. Note that s 3 3 equals s 2 3 due to the length of that line. The marking type of the lines are either solid (n = 1, n = 4), dashed (n = 2) or block (n = 3). The vertical red double lines (e.g. from s 1 1 to s 1 2 ) indicate parallelism between the connected segments. The vehicle coordinate system is denoted in gray.
For the modeling of lanes a generic representation is used and an example is shown in fig. 3. The lane model consists of n = 1...N lines each composed of m = 1...M segments. Each segment is described by a mathematical function f (x, l m n ), which depends on the parameters l m n and is computed in x. Therefore, the lane model L t is defined by the stacked parameter vector [l 1 1 , l 2 1 , ..., l M 1 , ..., l 1 N , l 2 N , ..., l M N ] T . Within the scope of this work each line is represented as a cubic spline. Thus, third order polynomials are used as functions to describe the segments
An example of grouping the lines of a lane model according to parallelism based on the lane marking type is shown in fig. 3, where the upper three lines are parallel. Inferred information about parallel lines enters the lane model fit as equality constraint (see section II-F). After this, the positions of control points connecting segments of a line are determined. In the case of parallel lines, a control point is set at the end of each line. In fig. 3 the longitudinal position of s 2 1 , s 2 2 and s 2 3 are equal and correspond to the end of the block marking. To provide flexibility to the lane model, the length of line segments is restricted by usage of additional control points where necessary (e.g. s 2 4 in
2 )
2Time filter optimization: In addition, the previous state L t−1 is considered in the optimization to ensure continuity of the result over time. As shown in section II-D the control points of the previously derived lane model can be predicted to the current time step. Therefore, an additional sum of error terms
described in section II-B, the lane model is composed of N lines with M segments each. For each pair of successive line segments m, m + 1 the continuity with respect to position, orientation and curvature is established by the equality constraints f (s m+1 n , l m n ) − f (s
This method has two advantages. First, the state space of L t in eq. 8 is reduced from 4M N parameters to 4M N −3(M − 1)N = (M + 3)N . Additionally in this case the 3(M − 1)N equations for g(L t ) are not needed, which makes the solving of the linear system computationally faster.
Fig. 4 :
4Double bend of the simulated scenario. The double bend consists of four connected 50 m long clothoid segments. Along each segment the radius of curvature changes from 1000 m to 100 m or vice versa.
model are compared using a simulated scenario. Due to the generic description of the lane model, clothoids can be modeled with the proposed method by simply using a third order polynomial as line function with one segment. The CarMaker simulation software 6 is used to generate odometry and lane features of the simulated scenario. It consists of a 200 m long straight road section followed by a double bend, which is composed of four connected clothoid segments. The sshape curve of the simulated scenario is shown in fig. 4. At each time step, lane features up to a longitudinal distance of 100 m in front of the vehicle serve, together with the current odometry, as input for the lane modeling. As a measure of performance, the Root Mean Square Error (RMSE) of the lateral distance between the estimated lane model and simulated lane features is computed.
Fig. 5 :
5RMSE of the difference between modeled and simulated lanes. The result is shown for clothoids (red) and cubic splines (blue) in a simulated scenario containing a double bend.
Fig. 6 :
6RMSE of the difference between estimated lanes and ground truth for ego (red) and neighbor lane (blue).of the input data, the result for the ego lane is better in comparison to the one for adjacent lanes. At a distance of 120 m lanes are modeled with an RMSE of less than 0.75 m.
an example of feature extraction and lane model estimation result inside a construction zone is shown. In this setting the lane markings inside the construction zone are yellow and have a double bend shape. Features are extracted along the lane markings in the camera image (fig. 7a). After accumulation and fusion of the features, the lanes are modeled using the presented method with cubic splines and three segments (fig. 7b). As one can see, the resulting lane model is properly describing the shape of the lane within the construction zone. Despite outliers in the input data, a robust estimation of the lane is obtained.
of camera features accumulated over time (red).
Fig. 7 :
7Example of lane modeling based on cubic splines inside a construction zone.
The vehicle coordinate system (x, y, z) is a right-handed coordinate system, its origin lies in the middle of the front axle (height of the road), x is identical to the driving direction, y points to the left and z points upwards.
Traffic lanes usually have a width between 3.5 m to 4 m.
A video sequence illustrating the results: https://drive.google. com/open?id=0B8eccpkD0x9nSGJEZExhTmZhUm8.5 Here, clothoids are approximated by a third order polynomial[4].6 http://ipg.de/de/simulation-software/carmaker/
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|
[] |
[
"Conformal Covariance Subalgebras",
"Conformal Covariance Subalgebras"
] |
[
"Søren Köster \nInst. f. Theor. Physik\nUniversität Göttingen Tammannstr. 1\n37077GöttingenGermany\n"
] |
[
"Inst. f. Theor. Physik\nUniversität Göttingen Tammannstr. 1\n37077GöttingenGermany"
] |
[] |
We give a direct Lie algebraic characterisation of conformal inclusions of chiral current algebras associated with compact, reductive Lie algebras. We use quantum field theoretic arguments and prove a longstanding conjecture of Schellekens and Warner on grounds of unitarity and positivity of energy. We explore the structures found to characterise conformal covariance subalgebras and coset current algebras.
|
10.1023/b:math.0000027688.52662.92
|
[
"https://arxiv.org/pdf/hep-th/0303201v2.pdf"
] | 16,578,093 |
hep-th/0303201
|
37559c4c9ac9aa569d9f05df1431f4b2c3b8f0c7
|
Conformal Covariance Subalgebras
arXiv:hep-th/0303201v2 17 Jul 2003
Søren Köster
Inst. f. Theor. Physik
Universität Göttingen Tammannstr. 1
37077GöttingenGermany
Conformal Covariance Subalgebras
arXiv:hep-th/0303201v2 17 Jul 2003AMS Subject classification (2000): 81T4081R1017B6781T05
We give a direct Lie algebraic characterisation of conformal inclusions of chiral current algebras associated with compact, reductive Lie algebras. We use quantum field theoretic arguments and prove a longstanding conjecture of Schellekens and Warner on grounds of unitarity and positivity of energy. We explore the structures found to characterise conformal covariance subalgebras and coset current algebras.
Introduction
Conformal inclusions of chiral current algebras are of interest for a large variety of reasons. Their classification was undertaken some time ago, because they are particularly relevant to string theory for their making string compactification possible without altering conformal covariance. Using general arguments this task was transferred to checking maximal inclusions of reductive Lie algebras in simple Lie algebras, for which a classification was available already. The classification of conformal inclusions was thus achieved, looking at the central charge of the respective stress-energy tensors, by several authors [AGO87,BB87,SW86].
Many of the conformal inclusions were found to correspond to symmetric spaces (cf. [GNO85,Dab96] in particular), and isotropy irreducibility of the coset space proved a useful yet neither necessary nor sufficient criterion for an inclusion being conformal. We undertake a complete characterisation of conformal inclusions by means of straightforward arguments familiar in (axiomatic) quantum field theory. On the course we prove a longstanding 1 conjecture of Schellekens and Warner [SW86].
We use properties of any Wightman quantum field theory 2 : positivity of energy, separating property of the vacuum for local quantum fields, and unitarity. Our analysis clarifies the situation in natural group theoretical terms and in direct correspondence to quantum field theoretical notions. Moreover, there is no need to specialise in maximal subalgebras and our approach is rather direct in that respect.
The point of view taken in this work arose from a more general question: how does the inner-implementing representation U A , uniquely associated with every covariant subtheory A of a chiral conformal theory B by means of the Borchers-Sugawara construction [Kös02], act on the observables of the larger theory B? An answer to the more general question is given in an independent work [Kös03].
We proceed as follows: In the next section we introduce notations and conventions, prove the conjecture of Schellekens and Warner and provide a direct argument for conformal inclusions being necessarily restricted to level 1. The third section is about studying conformal covariance subalgebras associated to Lie algebra inclusions, these being intermediate to the original inclusion, if not trivial. The section will be closed by a simple characterisation of coset currents, i.e. current subalgebras commuting with the given current subalgebra.
Characterisation of conformal inclusions
We study a current algebra associated with a simple 3 , compact Lie algebra g consisting of symmetric operator valued distributions, the currents. We will treat these as fields on the chiral light ray. Basis elements of g will be denoted by T a ; they give the colour of the corresponding current j a . The current algebra is given by the following commutation relations:
j a (x), j b (y) = if ab c j c (x)δ(x − y) + kg ab g i 2π δ ′ (x − y) .
g g denotes the Killing metric of g, f ab c its structure constants and k the current algebra's level; k is a positive integer.
By embedding a reductive Lie subalgebra h into g via an injective homomorphism ι : h ֒→ g we have an associated current subalgebra. h consists of several simple ideals, denoted (for the time being) by h α , and an abelian ideal of dimension n ≥ 0. The inclusions ι(h α ) ⊂ g are partly characterised by their 2 Wightman's axioms are manifestly fulfilled for all current algebras which are available as quark models [BH71] (corresponding to abelian Lie algebras R n and to the classical Lie algebras of type A n , B n , C n , D n and to the exceptional Lie algebra G 2 ). For the remaining four cases (corresponding to the exceptional Lie algebras of type E 6 , E 7 , E 8 , F 4 ) Wightman's axioms appear implicitly in the literature [Kac90], [GW84], [TL97].
3 General reasoning leads to an extension of the following discussion to inclusions of reductive subalgebras in reductive Lie algebras, cf. eg. [AGO87].
Dynkin index I α . Denoting the Killing metric of h α by g α we have the following commutation relations for currents associated with colours in ι(h α ):
j ι(a) (x), j ι(b) (y) = if ι(a)ι(b) ι(c) j ι(c) (x)δ(x − y) + I α kg ab α i 2π δ ′ (x − y) .
The infinitesimal conformal transformations are implemented by the adjoint action of the Sugawara stress-energy tensor Θ g . It is given by:
Θ g (x) = π k + v g g g ab : j a j b : (x) .
v g is the dual Coxeter number of g. The commutation relation of Θ g with a current reads as:
[Θ g (x), j c (y)] = ij c (x)δ ′ (x − y) .
Restricting to colours in ι(h α ) the same construction yields a stress-energy tensor Θ α having the same commutation relation with currents associated with colours in ι(h α ):
Θ α (x) = π I α k + v α g α ab : j ι(a) j ι(b) : (x) .
For the abelian ideal we adopt the following conventions:
I R n := 1, v R n := 0, g ij R n := g ι(i)ι(j) g
. Using these as input all the formulas above apply to currents associated with colours in ι(R n ). We drop the distinction between simple and abelian ideals of h and use the symbol h α for any simple or the abelian ideal from now on.
With this general notation the action of a stress-energy tensor Θ α on an arbitrary current j c reads as:
[Θ α (x), j c (y)] = π I α k + v α g α ab if ι(b)c d : j ι(a) j d + j d j ι(a) : (x) δ(x − y) +i k I α k + v α j ι(a) (x) g α ab g ι(b)c g δ ′ (x − y) +i 1 2(I α k + v α ) j d (x)(C α 2 ) d c δ ′ (x − y) .(1)
This equation is obtained by applying the current algebra and the normal ordering prescription for currents [FST89]. The matrix C α 2 stands for the second Casimir element of h α in the representation Ad g • ι| hα , if h α is a simple ideal. In any case we have:
(C α 2 ) d c = g α ab if ι(b)e d if ι(a)c e = g α ab (Ad T ι(b) Ad T ι(a) ) d c .
Taking the trace of this matrix one may readily see that it does not vanish for the abelian ideal. Now we are prepared to state and prove our main result. Schellekens and Warner conjectured it in their discussion closing [SW86].
Theorem 1 The following holds true for the weighted Casimir element C ι(h) 2 of ι(h) (P α stands for the projection onto ι(h α )):
C ι(h) 2 := α 2I α kP α + C α 2 2(I α k + v α )
1l .
(
2)
This inequality is saturated if and only if ι(h) ⊂ g yields a conformal inclusion, i.e. α Θ α =: Θ h = Θ g .
Proof: By invariance of g g the orthocomplementation g = ι(h)+ι(h) ⊥ provides a reduction of the representation
Ad g • ι. We have C α 2 | ι(h) = 2v α P α , i.e. C ι(h) 2 | ι(h) = 1l
, and the inequality only remains to be proven for colours orthogonal to ι(h), where P α | ι(h) ⊥ = 0. Because all Casimir elements commute and all are positive operators, we assume as well that T c is a common eigenvector for all linear mappings C α 2 . We prove the inequality by looking at specific expectation values of the coset
Hamiltonian L g 0 − L h 0 .
This is a positive operator, which is given by the coset stress-energy tensor Θ g − Θ h smeared with the test function ξ L 0 (x) = 1 2 (x 2 + 1). The infinitesimal action of a conformal Hamilton operator on the test function of a smeared field covariant with respect to it shall be abbreviated by l 0 , i.e. we have
[L g 0 , j c (g)] = i dxg ′ (x)ξ L 0 (x)j c (x) ≡ i dx l 0 g(x) j c (x) = i j c (l 0 g) .
Using the general commutation relation (1), calculating two and three point functions of currents (cf. [FST89]), observing that some group-theoretical tensors involved are null for reasons of permutation symmetry/ antisymmetry and carefully taking into account the normal ordering of currents [FST89] one arrives at the following formula:
0 Ω, j c (g) † (L g 0 − L h 0 )j c (g)Ω = i 1 − α C α 2 [T c ] 2(I α k + v α ) Ω, j c (g) † j c (l 0 g)Ω .(3)
The desired inequality may be established through division by i Ω, j c (g) † j c (l 0 g)Ω , which does not vanish for generic g and is positive as an expectation value of L g 0 − L h 0 ≥ 0. If we have Θ g = Θ h , (2) is saturated on ι(h) trivially and because of (3) on ι(h) ⊥ as well, hence on all of g. The conclusion in the opposite direction is, actually, a consequence of equation (4) in proposition 3: This leads to trivial commutation relations for Θ g − Θ h , especially to c g = c h , which yields, by a variant of the Reeh-Schlieder theorem 4 , Θ g − Θ h = 0.
Corollary 2 An embedding ι(h) ⊂ g can give rise to a conformal inclusion of the associated current algebras only, if the current algebra associated with g has level k = 1.
Proof: Highest-weight representations of current algebras may be characterised uniquely by a vector of lowest energy which is a highest-weight vector with respect to the horizontal subalgebra of currents j a ([1]) smeared with the constant testfunction [1](x) = 1. We look at the representation defined by the highest weight ψ g of the adjoint representation of g. Since ψ g has, by the usual convention, length 2, this representation is in accordance with the Weyl-alcove condition [FST89, (4.51)] for unitary representations of current algebras for k ≥ 2. The following argument applies, therefore, to all but level 1.
Actually, we may restrict attention to the action of L g 0 − L h 0 on gψ g , the highest-weight module of g generated from the vector with lowest energy and highest weight ψ g . Here we have:
0 L g 0 − L h 0 gψg = v g k + v g 1l − α C α 2 2(I α k + v α ) .
This implies a strictly sharper bound than inequality (2) and by theorem 1 immediately yields the desired result.
Covariant and invariant colours
After we have given a characterisation of conformal inclusions ι(h) ⊂ g, we now pursue further the structures in colour space which are associated with the action of Θ h on currents with colours in g. We find that covariant and invariant colours form reductive Lie algebras, the first being intermediate to the original embedding ι(h) ⊂ g, the second being orthogonal to and commuting with it. All these results are in terms of the weighted Casimir element C ι(h) 2 of the Lie algebra ι(h), which already appeared in the previous section.
The following is the main ingredient of the results in this section:
Proposition 3 For an arbitrary colour T c ∈ g we have:
Θ g − Θ h (f ), j c (g) Ω 2 = 8kπ 2 (1l − C h 2 )T c , C h 2 T c g ∆ 4 (f · g, f · g) +k (1l − C h 2 )T c , (1l − C h 2 )T c g ∆ 2 (f · g ′ , f · g ′ ) .
(4)
Here ., . g stands for the scalar product on g induced by the Killing form. We define
Φ c (x) := α 1 2(I α k + v α ) g α ab f ι(b)c d : j ι(a) j d + j d j ι(a) : (x) .
The two-point function of Φ c is given by:
Ω, Φ c (x)Φ c (y)Ω = 2k ∆ 4 (x − y) (1l − C h 2 )T c , C h 2 T c g . (5)
The numerical distributions in these formulae are given by:
∆ 4 (f · g, f · g) = (2π) −4 dx dy (i[(x − y) − iε]) −4 f · g(x)f · g(y)
,
∆ 2 (f · g ′ , f · g ′ ) = (2π) −2 dx dy (i[(x − y) − iε]) −2 f · g ′ (x)f · g ′ (y) .
Proof: We will not give the derivation of these formulae in detail. We rather indicate their verification. First, one may restrict attention to colours T c ∈ ι(h) ⊥ since the weighted Casimir respects the orthogonal decomposition g = ι(h) + ι(h) ⊥ with respect to Ad • ι and Θ g − Θ h commutes with all currents whose colours are in ι(h). "All" that one has to do is to apply the general commutation relation (1) restricted to colours from ι(h) ⊥ , follow carefully the normal ordering of currents, observe symmetries of group theoretical coefficients, keep in mind T c ∈ ι(h) ⊥ , calculate some n-point functions of currents following the scheme in [FST89], use Jacobi's identity a few times and recognise the second Casimir element in the adjoint representation, which amounts to twice the dual Coxeter number. With all that, it is a straightforward algebraic exercise.
Remark: Taking g as the test function of constant value 1, equation (4) implies C h 2 (1l − C h 2 ) ≥ 0, from which we immediately get inequality (2), and the second statement in theorem 1 follows from (4), too.
Definition 4 A current j c is said to transform covariantly with respect to Θ h , if and only if [Θ g (f ), j c (g)] = Θ h (f ), j c (g) ∀f, g.
Corollary 5 A current j c transforms covariantly with respect to Θ h , if and only if its colour fulfills: C ι(h) 2 T c = T c . These covariant colours form a reductive Lie algebra, k, containing ι(h) as a subalgebra. If k = ι(h), then the level of the current algebra associated with g has to be k = 1.
Proof: If we have C ι(h)
2 T c = T c , we know from the variant of the Reeh-Schlieder theorem (see footnote 4) and proposition 3 above, that j c and the coset stress-energy tensor commute. This is another way of saying: j c transforms covariantly with respect to Θ h .
Conversely: If j c is covariant with respect to Θ h , the group theoretical scalar products in equation (4) have to be zero, since the numerical distributions involved are linearly independent. The second one of these is the norm of (1l − The reductivity of k is not difficult to prove, either. k is a subspace of g, endowed with an invariant scalar product, which is given by the restriction of the Killing form on g. By invariance of this scalar product on k with respect to Ad k (this being a mere restriction of invariance under Ad g ) any invariant subspace of k has an invariant orthogonal complement. Now this is complete reducibility of Ad k and thus k is reductive [Cor89] (25.3.a).
Since one can reduce the problem of understanding all conformal inclusions to the studies of reductive inclusions in simple Lie algebras (cf. eg. [AGO87]) the last part follows immediately from corollary 2, as ι(h) ⊂ k is, by construction of k, a conformal inclusion and Dynkin indices of the simple ideals in k are greater than or equal to 1.
Corollary 6 A current j c , whose colour T c lies in ι(h) ⊥ and fulfills C ι(h) 2 T c = 0, commutes with the entire current algebra associated with ι(h). These colours form a reductive Lie algebra, the algebra of invariant colours; we call their current algebra coset current algebra.
Proof: We set V 0 := ker( C ι(h) 2 ) ∩ ι(h) ⊥ . V 0 is an invariant subspace with respect
to the action of h on g via Ad g • ι. In fact, it is the representation space for the trivial subrepresentation on ι(h) ⊥ : For any simple ideal h α we have by complete reducibility C α 2 | V 0 = Λ Λ + 2ρ, Λ hα = 0. Since both the Weyl vector ρ and the contributing highest-weight vectors Λ are dominant, we have Λ = 0. For the abelian ideal the irreducible subrepresentations on V 0 are given by common eigenvectors, such that C R n 2 v = g g ι(i)ι(j) λ i λ j v = 0. This gives the same result. This means, that all of V 0 commutes with ι(h), i.e. V 0 ⊂ ι(h) ′ . We gain directly:
V 0 = ι(h) ′ ∩ ι(h) ⊥ .
By Jacobi's identity and invariance of the Killing metric, ι(h) ′ ∩ι(h) ⊥ forms a Lie subalgebra of g. This is reductive by the same argument as in the proof of corollary 5.
Two concluding remarks: Generically, the coset theory is not generated by coset currents. Obviously ι(h) ⊕ (ι(h) ′ ∩ ι(h) ⊥ ) ⊂ g has to be a conformal inclusion for that to be the case, since the coset stress-energy tensor has to be the Sugawara stress-energy tensor of the current algebra associated with ι(h) ′ ∩ ι(h) ⊥ . Casimir elements of ι(h) ′ ∩ ι(h) ⊥ give, when transferred to the corresponding horizontal subalgebra, charge operators of the coset theory. These will, in general, fail to separate the representations of the coset theory. The same goes for the Cartan subalgebra of ι(h) ′ ∩ ι(h) ⊥ , whose spectrum defines characters of the representations of the coset theory. The coset current algebra is trivial for all inclusions with minimal coset theory: Here the coset theory is generated by the coset stress-energy tensor and this theory contains nothing but this field [Car98]. Triviality of coset current algebra ought to be regarded as the generic situation.
Currents j c with vanishing covariance field Φ c are linear combinations of covariant and coset currents. This is obvious, since a decomposition of T c into eigenvectors of C h 2 with distinct eigenvalues λ yields (cf. equation 5):
(1l − C h 2 )T c , C h 2 T c g = λ λ(1 − λ) T c λ , T c λ g .
As 0 ≤ λ ≤ 1 this scalar product vanishes, if and only if just 0 and 1 contribute. This means, that there are no currents with a "simple" intermediate transformation behaviour with respect to the action of Θ h . Typically, a current j c has Φ c = 0, i.e. a "complicated" transformation behaviour. By the analysis in [Kös03] this behaviour is known to be physically satisfactory, still.
2
)T c , which makes the equation C ι(h) 2 T c = T c valid. Now, if T a and T b are covariant colours, then so is −i[T a , T b ]. This is clear, if one observes f ab c j c (g) = −i[j a ([1]), j b (g)], where [1] is a constant test function: [1](x) = 1.
See for example[Jos65] (lemma 2, section V.3.B); for an argument directly referring to the Virasoro algebra see[GW85],[Gom86].
Acknowledgements I thank K.-H.Rehren (Göttingen) for many helpful discussions and a critical reading of the manuscript. Financial support from the Ev. Studienwerk Villigst is gratefully acknowledged.
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Absence of subsystems for the Haag-Kastler net generated by the energy-momentum tensor in two-dimensional conformal field theory. S Carpi, Lett. Math. Phys. 45S. Carpi. Absence of subsystems for the Haag-Kastler net generated by the energy-momentum tensor in two-dimensional conformal field the- ory. Lett. Math. Phys. 45 (1998) 259-267.
Supersymmetries and Infinite-Dimensional Algebras. J F Cornwell, Group Theory in Physics. IIIAcademic PressTechniques in PhysicsJ.F. Cornwell. Group Theory in Physics, Volume III: Supersymmetries and Infinite-Dimensional Algebras, volume 10 of Techniques in Physics. Academic Press, London, San Diego, 1989.
Algebraic proof of the symmetric space theorem. C , J. Math. Phys. 37C. Daboul. Algebraic proof of the symmetric space theorem. J. Math. Phys. 37 (1996) 3576-3586.
Two-dimensional conformal quantum field theory. P Furlan, G M Sotkov, I T Todorov, Riv. Nuovo Cim. 12P. Furlan, G.M. Sotkov, and I.T. Todorov. Two-dimensional conformal quantum field theory. Riv. Nuovo Cim. 12 (1989) 1-203.
Symmetric spaces, Sugawara's energy momentum tensor in two dimensions and free fermions. P Goddard, W Nahm, D Olive, Phys. Lett. 160P. Goddard, W. Nahm, and D. Olive. Symmetric spaces, Sugawara's energy momentum tensor in two dimensions and free fermions. Phys. Lett. B160 (1985) 111-116.
The triviality of representations of the Virasoro algebra with vanishing central element and L 0 positive. F Gomes, Phys. Lett. 17175F. Gomes. The triviality of representations of the Virasoro algebra with vanishing central element and L 0 positive. Phys. Lett. 171B (1986) 75.
Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. R Goodman, N R Wallach, J. Reine Angew. Math. 347R. Goodman and N.R. Wallach. Structure and unitary cocycle repre- sentations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math. 347 (1984) 69-222.
Projective unitary positive-energy representations of Diff(S 1 ). R Goodman, N R Wallach, J. Funct. Anal. 63R. Goodman and N.R. Wallach. Projective unitary positive-energy rep- resentations of Diff(S 1 ). J. Funct. Anal. 63 (1985) 299-321.
The General Theory of Quantized Fields. R Jost, Proceedings of the Summer Seminar. the Summer SeminarProvidence, RI; Boulder, ColoradoMarc KacR. Jost. The General Theory of Quantized Fields, volume IV of Lectures in Applied Mathematics. American Mathematical Society, Providence, RI, 1965. Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Marc Kac (ed.).
Infinite dimensional Lie algebras. V Kac, Cambridge University Press3rd editionV. Kac. Infinite dimensional Lie algebras. Cambridge University Press, 3rd edition, 1990.
Conformal transformations as observables. S Köster, Lett. Math. Phys. 61S. Köster. Conformal transformations as observables. Lett. Math. Phys. 61 (2002) 187-198.
Local nature of coset models. S Köster, math-ph/0303054S. Köster. Local nature of coset models, 2003. math-ph/0303054.
Conformal subalgebras of Kac-Moody algebras. A N Schellekens, N P Warner, Phys. Rev. D. 34A.N. Schellekens and N.P. Warner. Conformal subalgebras of Kac- Moody algebras. Phys. Rev. D 34 (1986) 3092-3096.
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|
[] |
[
"Learning of Generalized Low-Rank Models: A Greedy Approach",
"Learning of Generalized Low-Rank Models: A Greedy Approach"
] |
[
"Quanming Yao \nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology Hong Kong\n\n",
"James T Kwok [email protected] \nDepartment of Computer Science and Engineering\nHong Kong University of Science and Technology Hong Kong\n\n"
] |
[
"Department of Computer Science and Engineering\nHong Kong University of Science and Technology Hong Kong\n",
"Department of Computer Science and Engineering\nHong Kong University of Science and Technology Hong Kong\n"
] |
[] |
Learning of low-rank matrices is fundamental to many machine learning applications.A state-of-the-art algorithm is the rank-one matrix pursuit (R1MP). However, it can only be used in matrix completion problems with the square loss. In this paper, we develop a more flexible greedy algorithm for generalized low-rank models whose optimization objective can be smooth or nonsmooth, general convex or strongly convex. The proposed algorithm has low per-iteration time complexity and fast convergence rate. Experimental results show that it is much faster than the state-of-the-art, with comparable or even better prediction performance.
| null |
[
"https://arxiv.org/pdf/1607.08012v1.pdf"
] | 16,595,645 |
1607.08012
|
c10c1fb850a1f29e8c09a47468523be0f9bcb7a1
|
Learning of Generalized Low-Rank Models: A Greedy Approach
Quanming Yao
Department of Computer Science and Engineering
Hong Kong University of Science and Technology Hong Kong
James T Kwok [email protected]
Department of Computer Science and Engineering
Hong Kong University of Science and Technology Hong Kong
Learning of Generalized Low-Rank Models: A Greedy Approach
Learning of low-rank matrices is fundamental to many machine learning applications.A state-of-the-art algorithm is the rank-one matrix pursuit (R1MP). However, it can only be used in matrix completion problems with the square loss. In this paper, we develop a more flexible greedy algorithm for generalized low-rank models whose optimization objective can be smooth or nonsmooth, general convex or strongly convex. The proposed algorithm has low per-iteration time complexity and fast convergence rate. Experimental results show that it is much faster than the state-of-the-art, with comparable or even better prediction performance.
Introduction
In many machine learning problems, the data samples can be naturally represented as low-rank matrices. For example, in recommender systems, the ratings matrix is low-rank as users (and items) tend to form groups. The prediction of unknown ratings is then a low-rank matrix completion problem [Candès and Recht, 2009]. In social network analysis, the network can be represented by a matrix with entries representing similarities between node pairs. Unknown links are treated as missing values and predicted as in matrix completion [Chiang et al., 2014]. Low-rank matrix learning also have applications in image and video processing [Candès et al., 2011], multitask learning [Argyriou et al., 2006], multilabel learning [Tai and Lin, 2012], and robust matrix factorization [Eriksson and van den Hengel, 2012].
The low-rank matrix optimization problem is NP-hard [Recht et al., 2010], and direct minimization is difficult. To alleviate this problem, one common approach is to factorize the target m × n matrix X as a product of two low-rank matrices U and V , where U ∈ R m×r and V ∈ R n×r with r min{m, n}. Gradient descent and alternating minimization are often used for optimization [Srebro et al., 2004;Eriksson and van den Hengel, 2012;Wen et al., 2012]. However, as the objective is not jointly convex in U and V , this approach can suffer from slow convergence [Hsieh and Olsen, 2014].
Another approach is to replace the matrix rank by the nuclear norm (i.e., sum of its singular values). It is known that the nuclear norm is the tightest convex envelope of the matrix rank [Candès and Recht, 2009]. The resulting optimization problem is convex, and popular convex optimization solvers such as the proximal gradient algorithm [Beck and Teboulle, 2009] and Frank-Wolfe algorithm [Jaggi, 2013] can be used. However, though convergence properties can be guaranteed, singular value decomposition (SVD) is required in each iteration to generate the next iterate. This can be prohibitively expensive when the target matrix is large. Moreover, nuclear norm regularization often leads to biased estimation. Compared to factorization approaches, the obtained rank can be much higher and the prediction performance is inferior [Mazumder et al., 2010].
Recently, greedy algorithms have been explored for lowrank optimization [Shalev-shwartz et al., 2011a;Wang et al., 2014]. The idea is similar to orthogonal matching pursuit (OMP) [Pati et al., 1993] in sparse coding. For example, the state-of-the-art in matrix completion is the rank-one matrix pursuit (R1MP) algorithm [Wang et al., 2014]. In each iteration, it performs an efficient rank-one SVD on a sparse matrix, and then greedily adds a rank-one matrix to the matrix estimate. Unlike other algorithms which typically require a lot of iterations, it only takes r iterations to obtain a rankr solution. Its prediction performance is also comparable or even better than others.
However, R1MP is only designed for matrix completion with the square loss. As recently discussed in [Udell et al., 2015], different loss functions may be required in different learning scenarios. For example, in link prediction, the presence or absence of a link is naturally represented by a binary variable, and the logistic loss is thus more appropriate. In robust matrix learning applications, the 1 loss or Huber loss can be used to reduce sensitivity to outliers [Candès et al., 2011]. While computationally R1MP can be used with these loss functions, its convergence analysis is tightly based on OMP (and thus the square loss), and cannot be easily extended.
This motivates us to develop more general greedy algorithms that can be used in a wider range of low-rank matrix learning scenarios. In particular, we consider low-rank matrix optimization problems of the form
min X f (X) : rank(X) ≤ r,(1)
where r is the target rank, and the objective f can be smooth or nonsmooth, (general) convex or strongly convex. The proposed algorithm is an extension of R1MP, and can be reduced to R1MP when f is the square loss. In general, when f is convex and Lipschitz-smooth, convergence is guaranteed with a rate of O(1/T ). When f is strongly convex, this is improved to a linear rate. When f is nonsmooth, we obtain a O(1/ √ T ) rate for (general) convex objectives and O(log(T )/T ) for strongly convex objectives. Experiments on large-scale data sets demonstrate that the proposed algorithms are much faster than the state-of-theart, while achieving comparable or even better prediction performance. Notation: The transpose of vector / matrix is denoted by the superscript . For matrix A = [A ij ] ∈ R m×n (without loss of generality, we assume that m ≤ n), its Frobenius norm
is A F = i,j A 2 ij , 1 norm is ||X|| 1 = i,j |X ij |, nuclear norm is A * = i σ i (A), where σ i (A)'
s are the singular values, and σ max (A) is its largest singular value. For two vectors x, y, the inner product x, y = i x i y i ; whereas for two matrices A and B, A,
B = i,j A ij B ij . For a smooth function f , ∇f denotes its gradient. When f is convex but nonsmooth, g ∈ {u | f (y) ≥ f (x) + u, x − y } is its subgradient at x. Moreover, given Ω ∈ {0, 1} m×n , [P Ω (A)] ij = A ij if Ω ij = 1, and 0 otherwise.
Review: Rank-One Matrix Pursuit
The rank-one matrix pursuit (R1MP) algorithm [Wang et al., 2014] is designed for matrix completion [Candès and Recht, 2009]. Given a partially observed m × n matrix O = [O ij ], indices of the observed entries are contained in the matrix Ω ∈ {0, 1} m×n , where Ω ij = 1 if O ij is observed and 0 otherwise. The goal is to find a low-rank matrix that is most similar to O at the observed entries. Mathematically, this is formulated as the following optimization problem:
min X (i,j)∈Ω (X ij − O ij ) 2 : rank(X) ≤ r,(2)
where r is the target rank. Note that the square loss has to be used in R1MP.
The key observation is that if X has rank r, it can be written as the sum of r rank-one matrices, i.e., X =
r i=1 θ i u i v i , where θ i ∈ R and u i 2 = v i 2 = 1.
To solve (2), R1MP starts with an empty estimate. At the tth iteration, the (u t , v t ) pair that is most correlated with the current residual R t = P Ω (O − X t−1 ) is greedily added. It can be easily shown that this (u t , v t ) pair are the leading left and right singular vectors of R t , and can be efficiently obtained from the rankone SVD of R t . After adding this new u t v t basis matrix, all coefficients of the current basis can be updated as (θ 1 , . . . , θ t ) ← arg min θ1,...,θt
P Ω t i=1 θ i u i v i − O 2 F .(3)
Because of the use of the square loss, this is a simple leastsquares regression problem with closed-form solution.
To save computation, R1MP also has an economic variant. This only updates the combination coefficients of the current estimate and the rank-one update matrix as:
(µ, ρ) ← arg min µ,ρ P Ω µ t−1 i=1 θ i u i v i +ρu t v t −O 2 F . (4)
The whole procedure is shown in Algorithm 1.
R t = P Ω (O − X t−1 ); 4: [u t , s t , v t ] = rank1SVD(R t ); 5:
update coefficients using (3) (standard version), or (4) (economic version); 6:
X t = t i=1 θ i u i v i ; 7: end for 8: return X T .
Note that each R1MP iteration is computationally inexpensive. Moreover, as the matrix's rank is increased by one in each iteration, only r iterations are needed in order to obtain a rank-r solution. It can also be shown that the residual's norm decreases at a linear rate, i.e.,
R t 2 F ≤ γ t−1 P Ω (O) 2
F for some γ ∈ (0, 1).
Low-Rank Matrix Learning with Smooth Objectives
Though R1MP is scalable, it can only be used for matrix completion with the square loss. In this Section, we extend R1MP to problems with more general, smooth objectives. Specifically, we only assume that the objective f is convex and L-Lipschitz smooth. This will be further extended to nonsmooth objectives in Section 4.
Definition 1. f is L-Lipschitz smooth if f (X) ≤ f (Y ) + X − Y, ∇f (Y ) + L 2 X − Y 2 F for any X, Y .
Proposed Algorithm
Let the matrix iterate at the tth iteration be X t−1 . We follow the gradient direction ∇f (X t−1 ) of the objective f , and find the rank-one matrix M that is most correlated with ∇f (X t−1 ):
max M M, ∇f (X t−1 ) : rank(M ) = 1, M F = 1. (5)
Similar to [Wang et al., 2014], its optimal solution is given by u t v t , where (u t , v t ) are the leading left and right singular vectors of ∇f (X t−1 ). We then set the coefficient θ t for this new rank-one update matrix to −s t /L, where s t is the singular value corresponding to (u t , v t ). Optionally, all the coefficients θ 1 , . . . , θ t can be refined as
(θ 1 , . . . , θ t ) ← arg min θ1,...,θt f t i=1 θ i u i v i .(6)
As in R1MP, an economic variant is to update the coefficients as [µθ 1 , . . . , µθ t−1 , ρ], where µ and ρ are obtained as
(µ, ρ) ← arg min µ,ρ f µ t−1 i=1 θ i u i v i + ρ u t v t .(7)
The whole procedure, which will be called "greedy low-rank learning" (GLRL), is shown in Algorithm 2. Its economic variant will be called EGLRL. Obviously, on matrix completion problems with the square loss, Algorithm 2 reduces to R1MP.
Algorithm 2 GLRL for low-rank matrix learning with smooth convex objective f . 1: Initialize: X 0 = 0; 2: for t = 1, . . . , T do 3:
[u t , s t , v t ] = rank1SVD(∇f (X t−1 )); 4: X t = X t−1 − st L u t v t ; 5:
(optional:) refine coefficients using (6) (standard version) or (7) (economic version); 6: end for 7: return X T .
Note that (6), (7) are smooth minimization problems (with ≤ r and 2 variables, respectively). As the target matrix is low-rank, r should be small and thus (6), (7) can be solved inexpensively. In the experiments, we use the popular limited-memory BGFS (L-BFGS) solver [Nocedal and Wright, 2006]. Empirically, fewer than five L-BFGS iterations are needed. Preliminary experiments show that using more iterations does not improve performance.
Unlike R1MP, note that the coefficient refinement at step 5 is optional. Convergence results in Theorems 2 and 3 below still hold even when this step is not performed. However, as will be illustrated in Section 5.1, coefficient refinement is always beneficial in practice. It results in a larger reduction of the objective in each iteration, and thus a better rank-r model after running for r iterations.
Convergence
The analysis of R1MP is based on orthogonal matching pursuit [Pati et al., 1993]. This requires the condition
1 2 ∇f (X) 2 F = f (X),
which only holds when f is the square loss. In contrast, our analysis for Algorithm 2 here is novel and can be used for any Lipschitz-smooth f .
The following Proposition shows that the objective is decreasing in each iteration. Because of the lack of space, all the proofs will be omitted.
Proposition 1. If f is L-Lipschitz smooth, f (X t ) ≤ f (X t−1 ) − γ 2 t−1 2L ∇f (X t−1 ) 2 F , where γ t−1 = σ max (∇f (X t−1 )) ∇f (X t−1 ) F ∈ 1 √ m , 1 .(8)
If f is strongly convex, a linear convergence rate can be obtained.
Definition 2. f is µ-strongly convex if f (X) ≥ f (Y ) + X − Y, ∇f (Y ) + µ 2 X − Y 2 F for any X, Y . Theorem 2.
Let X * be the optimal solution of (1). If f is µ-strongly convex and L-Lipschitz smooth, then
f (X T ) − f (X * ) ≤ 1 − d 2 1 µ L T [f (X 0 ) − f (X * )] ,
where d 1 = min T t=1 γ t . If f is only (general) convex, the following shows that Algorithm 2 converges at a slower O(1/T ) rate. Theorem 3. If f is (general) convex and L-Lipschitz smooth, then (2) is only general convex and 1-Lipschitz smooth. From Theorem 3, one would expect GLRL to only have a sublinear convergence rate of O(1/T ) on matrix completion problems. However, our analysis can be refined in this special case. The following Theorem shows that a linear rate can indeed be obtained, which also agrees with Theorem 3.1 of [Wang et al., 2014].
f (X T ) − f (X * ) ≤ 2d 2 2 L [f (X 0 ) − f (X * )] d 2 1 T [f (X 0 ) − f (X * )] + 2d 2 2 L , where d 2 = max T t=1 X t − X * F . The square loss inTheorem 4. When f is the square loss, f (X T ) − f (X * ) ≤ (1 − d 2 1 ) T [f (X 0 ) − f (X * )].
Per-Iteration Time Complexity
The per-iteration time complexity of Algorithm 2 is low.
Here, we take the link prediction problem in Section 5.1 as an example. With f defined only on the observed entries of the link matrix, ∇f is sparse, and computation of ∇f (X t−1 ) in step 3 takes O(||Ω|| 1 ) time. The rank-one SVD on ∇f (X t−1 ) can be obtained by the power method [Halko et al., 2011] in O(||Ω|| 1 ) time. Refining coefficients using (6) takes O(t||Ω|| 1 ) time for the tth iteration. Thus, the total per-iteration time complexity of GLRL is O(t||Ω|| 1 ). Similarly, the per-iteration time complexity of EGLRL is O(||Ω|| 1 ). In comparison, the state-of-the-art AIS-Impute algorithm [Yao and Kwok, 2015] (with a convergence rate of O(1/T 2 )) takes O(r 2 ||Ω|| 1 ) time in each iteration, whereas the alternating minimization approach in [Chiang et al., 2014] (whose convergence rate is unknown) takes O(r||Ω|| 1 ) time per iteration.
Discussion
To learn the generalized low-rank model, Udell et al. Similar to R1MP, the greedy efficient component optimization (GECO) [Shalev-shwartz et al., 2011a] is also based on greedy approximation but can be used with any smooth objective. However, GECO is even slower than R1MP [Wang et al., 2014]. Moreover, it does not have convergence guarantee.
Low-Rank Matrix Learning with Nonsmooth Objectives
Depending on the application, different (convex) nonsmooth loss functions may be used in generalized low-rank matrix models [Udell et al., 2015]. For example, the 1 loss is useful in robust matrix factorization [Candès et al., 2011], the scalene loss in quantile regression [Koenker, 2005], and the hinge loss in multilabel learning [Yu et al., 2014]. In this Section, we extend the GLRL algorithm, with simple modifications, to nonsmooth objectives.
Proposed Algorithm
As the objective is nonsmooth, one has to use the subgradient g t of f (X t−1 ) at the tth iteration instead of the gradient in Section 3. Moreover, refining the coefficients as in (6) or (7) will now involve nonsmooth optimization, which is much harder. Hence, we do not optimize the coefficients. To ensure convergence, a sufficient reduction in the objective in each iteration is still required. To achieve this, instead of just adding a rank-one matrix, we add a rank-k matrix (where k may be greater than 1). This matrix should be most similar to g t , which can be easily obtained as:
M * ≡ arg min M :rank(M )=k M − g t 2 F = k i=1 s i u i v i ,(9)
where {(u 1 , v 1 ), . . . , (u k , v k )} are the k leading left and right singular vectors of g t , and {s 1 , . . . , s k } are corresponding singular values. The proposed procedure is shown in Algorithm 3. The stepsize in step 3 is given by
η t = c 1 /t if f is µ-strongly convex c 2 / √ t if f is (general) convex ,(10)
where c 1 ≥ 1/µ and c 2 > 0.
Convergence
The following Theorem shows that when f is nonsmooth and strongly convex, Algorithm 3 has a convergence rate of O(log T /T ).
Theorem 5. Assume that f is µ-strongly convex, and g t F ≤ b 1 for some b 1 (t = 1, . . . , T ), then
min t=0,...,T f (X t ) − f (X * ) ≤ 1 T (1 + log T ) c 1 b 2 1 2 + b 2 ,
where c 1 is as defined in (10), and b 2 is a constant (depending on X 0 , µ, ν and c 1 ).
When f is only (general) convex, the following Theorem shows that the rate is reduced to O(1/ √ T ).
Algorithm 3 GLRL for low-rank matrix learning with nonsmooth objective f . 1: Initialize: X 0 = 0 and choose ν ∈ (0, 1); 2: for t = 1, . . . , T do 3:
set η t as in (10); 4:
compute subgradient g t of f (X t−1 ), h t = 0;
5:
for i = 1, 2, . . . do 6:
[u i , s i , v i ] = rank1SVD(g t − h t ); 7: h t = h t + s i u i v i ; 8: if g t − h t 2 F ≤ ν g t−1 − h t−1 2 FthenX t = X t−1 − η t h t ; 13: end for 14: return X T . Theorem 6. Assume that f is (general) convex, g t F ≤ b 1 for some b 1 and X t − X * F ≤ b 3 for some b 3 (t = 1, . . . , T ), then min t=0,...,T f (X t ) − f (X * ) ≤ c 2 (b 2 1 + b 2 3 ) 2 √ T − 2b 1 b 3 (1 − √ ν)T ,
where c 2 is as defined in (10).
For other convex nonsmooth optimization problems, the same O(log(T )/T ) rate for strongly convex objectives and and O(1/ √ T ) rate for general convex objectives have also been observed [Shalev-Shwartz et al., 2011b]. However, their analysis is for different problems, and cannot be readily applied to our low-rank matrix learning problem here.
Per-Iteration Time Complexity
To study the per-iteration time complexity, we take the robust matrix factorization problem in Section 5.2 as an example. The main computations are on steps 4 and 6. In step 4, since the subgradient g t is sparse (nonzero only at the observed entries), computing g t takes O(||Ω|| 1 ) time. At outer iteration t and inner iteration i, g t in step 6 is sparse and h t has low rank (equal to i − 1). Thus, g t − h t admits the so-called "sparse plus low-rank" structure [Mazumder et al., 2010;Yao et al., 2015]. This allows matrix-vector multiplications and subsequently rank-one SVD to be performed much more efficiently. Specifically, for any v ∈ R n , the multiplication (g t − h t )v takes only O(||Ω|| 1 + (i − 1)n) time (and similarly for the multiplication u (g t − h t ) with any u ∈ R m ), and rank-one SVD using the power method takes O(||Ω|| 1 + (i − 1)n) time. Assuming that i t inner iterations are run at (outer) iteration t, it takes a total of O(i t ||Ω|| 1 + (i t − 1) 2 n) time. Typically, i t is small (empirically, usually 1 or 2).
In comparison, though the ADMM algorithm in [Lin et al., 2010] has a faster O(1/T ) convergence rate, it needs SVD and takes O(m 2 n) time in each iteration. As for the Wiberg algorithm [Eriksson and van den Hengel, 2012], its convergence rate is unknown and a linear program with mr + ||Ω|| 1 variables needs to be solved in each iteration. As will be seen in Section 5.2, this is much slower than GLRL.
Experiments
In this section, we compare the proposed algorithms with the state-of-the-art on link prediction and robust matrix factorization. Experiments are performed on a PC with Intel i7 CPU and 32GB RAM. All the codes are in Matlab.
Social Network Analysis
Given a graph with m nodes and an incomplete adjacency matrix O ∈ {±1} m×m , link prediction aims to recover a low-rank matrix X ∈ R m×m such that the signs of X ij 's and O ij 's agree on most of the observed entries. This can be formulated as the following optimization problem [Chiang et al., 2014]:
min X (i,j)∈Ω log(1 + exp(−X ij O ij )) : rank(X) ≤ r,(11)
where Ω contains indices of the observed entries. Note that (11) uses the logistic loss, which is more appropriate as O ij 's are binary. The objective in (11) is convex and smooth. Hence, we compare the proposed GLRL (Algorithm 2 with coefficient update step (6)) and its economic variant EGLRL (using coefficient update step (7)) with the following:
1. AIS-Impute [Yao and Kwok, 2015]: This is an accelerated proximal gradient algorithm with further speedup based on approximate SVD and the special "sparse plus low-rank" matrix structure in matrix completion; 2. Alternating minimization ("AltMin") [Chiang et al., 2014]: This factorizes X as a product of two low-rank matrices, and then uses alternating gradient descent for optimization. As a further baseline, we also compare with the GLRL variant that does not perform coefficient update. We do not compare with greedy efficient component optimization , 2014], we use 10-fold cross-validation and fix the rank r to 40. Note that AIS-Impute uses the nuclear norm regularizer and does not explicitly constrain the rank. We select its regularization parameter so that its output rank is 40. To obtain a rankr solution, GLRL is simply run for r iterations. For AIS-Impute and AltMin, they are stopped when the relative change in the objective is smaller than 10 −4 . The output predictions are binarized by thresholding at zero. As in [Chiang et al., 2014], the sign prediction accuracy is used as performance measure. Table 2 shows the sign prediction accuracy on the test set. All methods, except the GLRL variant that does not perform coefficient update, have comparable prediction performance. However, as shown in Figure 1, AltMin and AIS-Impute are much slower (as discussed in Section 3.3). EGLRL has the lowest per-iteration cost, and is also faster than GLRL.
Robust Matrix Factorization
Instead of using the square loss, robust matrix factorization uses the 1 loss to reduce sensitivities to outliers [Candès et al., 2011]. This can be formulated as the optimization problem [Lin et al., 2010]:
min X (i,j)∈Ω |X ij − O ij | s.t. rank(X) ≤ r.
Note that the objective is only general convex, and its subgradient is bounded (≤ Ω F ). Since there is no smooth component in the objective, AIS-Impute and AltMin cannot be used. Instead, we compare GLRL in Algorithm 3 (with ν = 0.99 and c 2 = 0.05) with the following: 1. Alternating direction method of multipliers (ADMM) 2 [Lin et al., 2010]: The rank constraint is replaced by the nuclear norm regularizer, and ADMM [Boyd et al., 2011] is then used to solve the equivalent problem: , 2012]: It factorizes X into U V and optimizes them by linear programming. Here, we use the linear programming solver in Matlab. Experiments are performed on the MovieLens data sets 3 (Table 3), which have been commonly used for evaluating recommender systems [Wang et al., 2014]. They contain ratings {1, 2, . . . , 5} assigned by various users on movies. The setup is the same as in [Wang et al., 2014]. 50% of the ratings are randomly sampled for training while the rest for testing. The ranks used for the 100K, 1M, 10M data sets are 10, 10, and 20, respectively. For performance evaluation, we use the mean absolute error on the unobserved entries Ω ⊥ : Results are shown in Table 4. As can be seen, ADMM performs slightly better on the 100K data set, and GLRL is more accurate than Wiberg.
min X,Y (i,j)∈Ω |X ij − O ij | + λ Y * s.t. Y = X.
The Wiberg algorithm [Eriksson and van den Hengel
MABS = 1 ||Ω ⊥ || 1 (i,j)∈Ω ⊥ |X ij − O ij |
However, ADMM is computationally expensive as SVD is required in each iteration. Thus, it cannot be run on the larger 1M and 10M data sets. Figure 2 shows the convergence of MABS with CPU time. As can be seen, GLRL is the fastest, which is then followed by Wiberg, and ADMM is the slowest.
Conclusion
In this paper, we propose an efficient greedy algorithm for the learning of generalized low-rank models. Our algorithm is based on the state-of-art R1MP algorithm, but allows the optimization objective to be smooth or nonsmooth, general convex or strongly convex. Convergence analysis shows that the proposed algorithm has fast convergence rates, and is compatible with those obtained on other (convex) smooth/nonsmooth optimization problems. Specifically, on smooth problems, it converges with a rate of O(1/T ) on general convex problems and a linear rate on strongly convex problems. On nonsmooth problems, it converges with a rate of O(1/ √ T ) on general convex problems and O(log(T )/T ) rate on strongly convex problems. Experimental results on link prediction and robust matrix factorization show that the proposed algorithm achieves comparable or better prediction performance as the state-of-the-art, but is much faster.
Proposition 1
Proof. At the tth iteration, we have
X t−1 = t−1 i=1 θ i u i v i . Constructθ = θ i ∈ R t as θ i = θ i i < t −s t /L i = t ,and letX t = t i=1θ i u i v i . Thus,X t − X t−1 = −s t u t v t /L. As f is L-Lipschitz smooth, f (X t ) ≤ f (X t−1 ) + X t − X t−1 , ∇f (X t−1 ) + L 2 X t − X t−1 2 F .(12)
Note that u t ∇f (X t−1 )v t = s t . Together with (12), we have
f (X t ) ≤ f (X t−1 ) − 1 2L ∇f (X t−1 ), u t v t 2 .(13)
For the second term on the RHS of (13),
∇f (X t−1 ), u t v t = γ t−1 ∇f (X t−1 ) F u t v t F = γ t−1 ∇f (X t−1 ) F , where γ t−1 = ∇f (Xt−1),utv t ∇f (Xt−1) F .
Note that s t is the largest singular value of ∇f (X t−1 ).
Thus, s t ≥ ∇f (X t−1 ) F / √ m, and
γ t−1 = s t ∇f (X t−1 ) F ∈ [1/ √ m, 1] Then (13) becomes f (X t ) ≤ f (X t−1 ) − γ 2 t−1 2L ∇f (X t−1 ) 2 F .
In Algorithm 2, X t is obtained by minimizing θ t over (6) or (7). Once we warm start θ usingθ, we can ensure that f (X t ) ≤ f (X t ), and thus the Proposition holds.
Theorem 2
Lemma 7. If f is µ-strongly convex, f (Y ) ≥ f (X) − 1 2µ ∇f (X) 2 F for any X, Y .
Proof. Since f is µ-strongly convex,
f (Y ) ≥ f (X) + ∇f (X), Y − X + µ 2 Y − X 2 F ≥ min Y f (X) + ∇f (X),Ȳ − X + µ 2 Ȳ − X 2 F .
The minimum is achieved atȲ = X − 1 µ ∇f (X), and
f (Ȳ ) = f (X) − 1 2µ ∇f (X) 2 F .
Proof. On optimal ∇f (X * ) = 0, using Lemma 7:
∇f (X t−1 ) − ∇f (X * ) 2 F ≥ 2µ [f (X t−1 ) − f (X * )] . (14) Combine it with Proposition 1, then f (X t ) − f (X * ) ≤ [f (X t−1 ) − f (X * )] − (γ t−1 ) 2 2L ∇f (X t−1 ) 2 F , ≤ (1 − µ(γ t−1 ) 2 L ) [f (X t−1 ) − f (X * )] , ≤ (1 − µd 2 1 L ) [f (X t−1 ) − f (X * )]
Induct from t = 1 to t = T , we then have the Theorem.
Theorem 3
First, we show that
{ X t − X * F } is upper-bounded. Proposition 8. For {X t } generated by Algorithm 2, max T t=1 X t − X * F ≤ d 2 for some d 2 . Proof. Let A t = f (X t ) − f (X * ), from Proposition 1 A t−1 − A t ≥ d 2 1 2L ∇f (X t−1 ) 2 F .(15)
Summing (15) from t = 1 to T , then
T t=1 d 2 1 2L ∇f (X t−1 ) 2 F ≤ T t=1 (A t−1 − A t ) = A 0 − A T ≤ A 0
Since f (X) is lower bounded, thus on T = +∞, we must have ∇f (X t−1 ) = 0, i.e. X t is a convergent sequence to X * and will not diverse. As a result, there must exist a constant d 2 such that X t − X * F ≤ d 2 . Now, we start to prove Theorem 3.
Proof. From convexity, and since f (X * ) is the minimum, therefore, we have
X t−1 − X * , ∇f (X t−1 ) ≥ f (X t−1 ) − f (X * ) ≥ 0.
Next, since ∇f (X * ) = 0 and use Cauchy inequality
X, Y ≤ X F Y F , then f (X t−1 ) − f (X * ) ≤ X t−1 − X * , ∇f (X t−1 ) − ∇f (X * ) ≤ X t−1 − X * F ∇f (X t−1 ) − ∇f (X * ) F(16)
Then, from Proposition 8, there exist a d 2 such that X t−1 − X * F ≤ d 2 , combining it with (16), we have:
∇f (X t−1 ) − ∇f (X * ) F ≥ 1 d 2 [f (X t−1 ) − f (X * )]
Combine above inequality with Proposition 1,
A t ≤ A t−1 − d 2 1 2L ∇f (X t−1 ) − ∇f (X * ) 2 F ≤ A t−1 − d 2 1 2Ld 2 2 A 2 t−1 .
Therefore, by Lemma B.2 of [Shalev-Shwartz et al., 2010], the above sequence converges to 0 of rate
f (X T ) − f (X * ) ≤ 2Ld 2 2 A 0 d 2 1 T A 0 + 2Ld 2 2 .
which proves the Theorem.
Theorem 4
Proof. When restrict to Ω in (2):
(X) ≤ (Y ) + P Ω (X − Y ), ∇ (Y ) + 1 2 P Ω (X − Y ) 2 F .
On optimal ∇ (X * ) = 0, using above inequality, then
(X t−1 ) ≤ (X * ) + 1 2 ∇ (X t−1 ) − ∇ (X * ) 2 F
As a result, we get
∇ (X t−1 ) − ∇ (X * ) 2 F ≥ 2 [ (X t−1 ) − (X * )](17)
Since is 1-Lipschitz smooth and ∇ (X * ) = 0, together with Proposition 1 and (17), then
(X t ) − (X * ) ≤ [ (X t−1 ) − (X * )] − d 2 1 2 ∇ (X t−1 ) 2 F , ≤ 1 − d 2 1 [ (X t−1 ) − (X * )] .
Induct from t = 1 to t = T , we then have
(X T ) − (X * ) ≤ 1 − d 2 1 T [ (X 0 ) − (X * )] .
Thus, we get the Theorem, and linear rate exists.
Theorem 5
Proof follows Theorem 5 at [Grubb and Bagnell, 2011].
Proof.
X t − X * 2 F = X t−1 − X * − η t h t 2 F = X t−1 − X * 2 F + η 2 t h t 2 F − 2η t h t , X t−1 − X * = X t−1 − X * 2 F − 2η t g t , X t−1 − X * −2η t h t − g t , X t−1 − X * + η 2 t h t 2 F . Rearranging items, we have: g t , X t−1 − X * = 1 2η t X t−1 − X * 2 F − 1 2η t X t − X * 2 F + η t 2 h t 2 F − h t − g t , X t−1 − X * .(18)
As f is µ-strongly convex, f (X * ) ≥ f (X t−1 ) + g t , X * − X t−1
+ µ 2 X t−1 − X * 2 F .(19)
Sum (19) from t = 1 to T , and using (18) T t=1 f (X * )
≥ T t=1 f (X t−1 ) + g t , X * − X t−1 + µ 2 X t−1 − X * 2 F ≥ T t=1 f (X t−1 ) − η t 2 h t 2 F − X * − X t−1 , h t − g t + 1 2 T −1 t=1 ( 1 η t − 1 η t+1 + µ) X t − X * 2 F + 1 2 (µ − 1 η 1 ) X 0 − X * 2 F + 1 2η T X T − X * F ≥ T t=1 f (X t−1 ) − η t 2 h t 2 F − X * − X t−1 , h t − g t + 1 2 T −1 t=0 ( 1 η t − 1 η t+1 + µ) X t − X * 2 F − 1 2η 0 X 0 − X * 2 F(20)
Recall, the step size is η t = c 1 /t, then (20) becomes
T t=1 f (X * ) ≥ T t=1 f (X t−1 ) − c 1 2 T t=1 1 t h t 2 F + 1 2 µ − 1 c 1 T t=1 X t−1 − X * 2 F − 1 2c 1 X 0 − X * 2 F − T t=1 X * − X t−1 , h t − g t(21)
where η 0 is can be picked up as c 1 . For the second term in (21), it is simply bounded as
c 1 2 T t=1 h t 2 F t ≤ c 1 2 T t=1 b 2 1 t ≤ c 1 b 2 1 2
(1 + log(T )) .
Letĉ = 1 2 µ − 1 c1 , since c 1 ≥ 1 µ , thusĉ ≥ 0. For last term in (21), we use A 2 F + 2 A, B ≥ − B 2 F , let A = X t − X * and B = 1 2ĉ (h t − g t ), then
c T t=1 X t − X * 2 F + 1 c X t − X * , h t − g t ≥ −ĉ T t=1 1 2ĉ (h t − g t ) 2 F = − 1 4ĉ T t=1 h t − g t 2 F(23)
By definition of h t and assumption max T t=1 g t F ≤ b 1 , for the first iteration (t = 1) h 1 − g 1 2 F = s 1 (u 1 v 1 ) + g 1 2 F = (s 1 ) 2 + 2s 1 u 1 v 1 , g 1 + g 1 2 F ≤ 4 g 1 2 F = 4b 2 1 . In Algorithm 3, we ensure
h t − g t 2 F ≤ ν h t−1 − g t−1 2 F thus (23) becomes: T t=1 h t − g t 2 F ≤ 4 T t=1 ν t−1 b 2 1 ≤ 4 +∞ t=1 ν t−1 b 2 1 = 4b 2 1 1 − ν .(24)
Now, using (22) and (24), we can bound (21) as (1 + log(T ))
− 1 2c 1 X 0 − X * 2 F − b 2 1 (1 − ν)ĉ .
Thus, we can get convergence rate as
min t=0,··· ,T −1 [f (X t ) − f (X * )] T ≤ T t=1 [f (X t−1 ) − f (X * )] ≤ (1 + log(T )) c 1 b 2 1 2 + 1 2c 1 X 0 − X * 2 F + b 2 1
(1 − ν)ĉ .
Finally, note that (1−ν)ĉ , we get the theorem.
Theorem 6
Proof follows Theorem 6 at [Grubb and Bagnell, 2011]. First, we introduce below Proposition
Algorithm 1
1R1MP [Wang et al., 2014]. 1: Initialize: X 0 = 0; 2: for t = 1, . . . , T do 3:
(2015) followed the common approach of factorizing the target matrix as a product of two low-rank matrices and then performing alternating minimization[Srebro et al., 2004; Eriksson and van den Hengel, 2012; Wen et al., 2012;Chiang et al., 2014; Yu et al., 2014]. However, this may not be very efficient, and is much slower than R1MP on matrix completion problems[Wang et al., 2014]. More empirical comparisons will be demonstrated in Section 5.1.
Figure 1 :
1Training and testing accuracies (%) vs CPU time (seconds) on the Epinions (left) and Slashdot (right) data sets.
(GECO) [Shalev-shwartz et al., 2011a], matrix norm boosting [Zhang et al., 2012] and active subspace selection [Hsieh and Olsen, 2014], as they have been shown to be slower than AIS-Impute and AltMin [Yao and Kwok, 2015; Wang et al., 2014]. Experiments are performed on the Epinions and Slashdot data sets 1 [Chiang et al., 2014] (Table 1). Each row/column 1 https://snap.stanford.edu/data/ of the matrix O corresponds to a user (users with fewer than two observations are removed). For Epinions, O ij = 1 if user i trusts user j, and −1 otherwise. Similarly for Slashdot, O ij = 1 if user i tags user j as friend, and −1 otherwise.
Figure 2 :
2Training (top) and testing (bottom) MABS vs CPU time (seconds) on the MovieLens data sets.
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min t=0,··· ,T f (X t ) − f (X * ) ≤ min t=0,··· ,T −1 f (X t ) − f (X * ) (25) Let b 2 = 1 2c1 X 0 − X
Table 1 :
1Signed network data sets used. As in [Wang et al., 2014], we fix the number of power method iterations to 30. Following [Chiang et al.#rows #columns #observations
Epinions 42,470
40,700
7.5 × 10 5
Slashdot 30,670
39,196
5.0 × 10 5
Table 2 :
2Testing sign prediction accuracy (%) on link prediction. The best and comparable results (according to the pairwise t-test with 95% confidence) are highlighted.Epinions Slashdot
AIS-Impute
93.3±0.1 84.2±0.1
AltMin
93.5±0.1 84.9±0.1
GLRL w/o coef upd 92.4±0.1 82.6±0.3
GLRL
93.6±0.1 84.1±0.4
EGLRL
93.6±0.1 84.4±0.3
,2 http://perception.csl.illinois.edu/
matrix-rank/Files/inexact_alm_rpca.zip
3 http://grouplens.org/datasets/movielens/
whereX is the predicted matrix [Eriksson and van den
Hengel, 2012]. Experiments are repeated five times with
random training/testing splits.
Table 3 :
3MovieLens data sets used in the experiments.#users #movies #ratings
100K
943
1,682
10 5
1M
6,040
3,449
10 6
10M 69,878
10,677
10 7
Table 4 :
4Testing MABS on the MovieLens data sets. The best results (according to the pairwise t-test with 95% confidence) are highlighted. ADMM cannot converge in 1000 seconds on the 1M and 10M data sets, and thus is not shown.100K
1M
10M
ADMM 0.717±0.004
-
-
Wiberg 0.726±0.001 0.728±0.006 0.715±0.005
GLRL
0.724±0.004 0.694±0.001 0.683±0.001
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boosting approach. In Advances in Neural Information
Processing Systems, pages 2906-2914, 2012.
AcknowledgmentsThanks for helpful discussion from Lu Hou. This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region (Grant 614513).Proposition 9 ([Beardon, 1996]). Given k > 0, the approximation of sum over natural numbers is:Now, we start to prove Theorem 6.Proof. For weak convex convexity (obtained from(20)withwhere the second inequality comes from the fact 1For the last term in(27), since X * − X t−1 F ≤ b 3 and h 1 − g 1 F ≤ 2b 1 , it can be bounded asThen, note that h t F ≤ g t F ≤ b 1 , use(27)and(28)Rearrange items in above inequality, from Proposition 9:Finally, using (25) we get the Theorem.
Beardon. Sums of powers of integers. Argyriou, Advances in Neural Information Processing Systems. 103Multi-task feature learningReferences [Argyriou et al., 2006] A. Argyriou, T. Evgeniou, and M. Pontil. Multi-task feature learning. In Advances in Neural Information Processing Systems, pages 41-48, 2006. [Beardon, 1996] A.F. Beardon. Sums of powers of integers. The American Mathematical Monthly, 103(3):201-213, 1996.
A fast iterative shrinkage-thresholding algorithm for linear inverse problems. A Beck, M Teboulle, SIAM Journal on Imaging Sciences. 21and Teboulle, 2009] A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1):183-202, 2009.
Eriksson and van den Hengel, 2012] A. Eriksson and A. van den Hengel. Efficient computation of robust weighted low-rank matrix approximations using the 1 norm. 11:1-11:37Proceedings of the 28th International Conference on Machine Learning (ICML-11). the 28th International Conference on Machine Learning (ICML-11)Grubb and Bagnell3Machine Learninget al., 2011] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3(1):1-122, 2011. [Candès and Recht, 2009] E.J. Candès and B. Recht. Exact matrix completion via convex optimization. Foundations of Computational Mathematics, 9(6):717-772, 2009. [Candès et al., 2011] E.J. Candès, X. Li, Y. Ma, and J. Wright. Robust principal component analysis? Journal of the ACM, 58(3):11:1-11:37, 2011. [Chiang et al., 2014] K. Chiang, C. Hsieh, N. Natarajan, I.S. Dhillon, and A. Tewari. Prediction and clustering in signed networks: A local to global perspective. Journal of Machine Learning Research, 15(1):1177-1213, 2014. [Eriksson and van den Hengel, 2012] A. Eriksson and A. van den Hengel. Efficient computation of robust weighted low-rank matrix approximations using the 1 norm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(9):1681-1690, 2012. [Grubb and Bagnell, 2011] A. Grubb and D. Bagnell. Generalized boosting algorithms for convex optimization. In Proceedings of the 28th International Conference on Machine Learning (ICML-11), pages 1209-1216, 2011.
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"Josef Dick ",
"Friedrich Pillichshammer "
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We analyze the weighted star discrepancy of so-called p-sets which go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have largely been ignored since a number of other constructions have been discovered which achieve a better convergence rate. However, it has recently been discovered that the p-sets perform well in terms of the dependence on the dimension.We prove bounds on the weighted star discrepancy of the p-sets which hold for any choice of weights. For product weights we give conditions under which the discrepancy bounds are independent of the dimension s. This implies strong polynomial tractability for the weighted star discrepancy. We also show that a very weak condition on the product weights suffices to achieve polynomial tractability.
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10.1090/proc/12636
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"https://arxiv.org/pdf/1404.0114v1.pdf"
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1404.0114
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e9f196b5f19ed0318e27f4bc68f3fcb2960962ea
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1 Apr 2014
Josef Dick
Friedrich Pillichshammer 1 Apr 2014arXiv:1404.0114v1 [math.NT] The weighted star discrepancy of Korobov's p-setsWeighted star discrepancyp-sets(strong) tractabilityquasi-Monte
We analyze the weighted star discrepancy of so-called p-sets which go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have largely been ignored since a number of other constructions have been discovered which achieve a better convergence rate. However, it has recently been discovered that the p-sets perform well in terms of the dependence on the dimension.We prove bounds on the weighted star discrepancy of the p-sets which hold for any choice of weights. For product weights we give conditions under which the discrepancy bounds are independent of the dimension s. This implies strong polynomial tractability for the weighted star discrepancy. We also show that a very weak condition on the product weights suffices to achieve polynomial tractability.
Introduction
1]
s . There one requires point sets P which are very well distributed. In many cases the quality of the distribution of a point set is measured by the star discrepancy D * N (P), which is intimately linked to the worst-case error of quasi-Monte Carlo integration via the well known Koksma-Hlawka inequality
[0,1] s f (x) dx − 1 N N −1 n=0 f (x n ) ≤ D * N (P)V (f ),
where V (f ) is the variation of f in the sense of Hardy and Krause. See, for example, [5,7,10,11,15] for more information.
At the end of the 1990s Sloan and Woźniakowski [21] (see also [1,18]) introduced the notion of weighted discrepancy and proved a "weighted" Koksma-Hlawka inequality. The idea is that in many applications some projections are more important than others and that this should also be reflected in the quality measure of the point set.
We start with some notation which goes back to the paper [21]: let [s] = {1, 2, . . . , s} denote the set of coordinate indices. For u ⊆ [s], u = ∅, let γ u be a nonnegative real number (the weight corresponding to the group of variables given by u), |u| the cardinality of u, and for a vector z ∈ [0, 1] s let z u denote the vector from [0, 1] |u| containing the components of z whose indices are in u. By (z u , 1) we mean the vector z from [0, 1] s with all components whose indices are not in u replaced by 1.
Definition 1
For an N-element point set P in [0, 1) s and given weights γ = {γ u : ∅ = u ⊆ [s]}, the weighted star discrepancy D * N,γ is given by
D * N,γ (P) = sup z∈[0,1] s max ∅ =u⊆[s] γ u |∆(z u , 1)|.
If γ u = 1 for all u ⊆ [s] (or γ [s] = 1 and γ u = 0 for u [s]), then the weighted star discrepancy coincides with the classical star discrepancy.
The most popular and studied weights in literature are so-called product weights which are weights of the form γ u = j∈u γ j , for ∅ = u ⊆ [s], where the γ j 's are positive reals, the weights associated with the jth component. See, for example, [1,21]. We assume throughout the paper that the weights γ j are non-increasing, i.e., γ 1 ≥ γ 2 ≥ γ 3 ≥ . . ..
Tractability
The dependence on the dimension is the subject of tractability studies [17,18,19]. We introduce the necessary background in the following.
For s, N ∈ N the Nth minimal weighted star discrepancy is
disc γ (N, s) = inf P⊆[0,1) s #P=N D * N,γ (P).
We would like to have a point set in the s-dimensional unit cube with weighted star discrepancy of at most ε ∈ (0, 1) and we are looking for the smallest cardinality N of a point set such that this can be achieved. For ε ∈ (0, 1) and dimension s ∈ N we define the information complexity N min (ε, s) := min{N ∈ N : disc γ (N, s) ≤ ε}, which is sometimes also called the inverse of the weighted star discrepancy.
Definition 2
1. We say that the weighted star discrepancy is polynomially tractable, if there exist nonnegative real numbers C, α and β such that
N min (ε, s) ≤ Cs α ε −β(1)
holds for all dimensions s ∈ N and for all ε ∈ (0, 1). The infima over all α, β > 0 such that (1) holds are called the s-exponent and the ε-exponent, respectively, of polynomial tractability.
2. We say that the weighted star discrepancy is strongly polynomially tractable, if there exist nonnegative real numbers C and α such that
N min (ε, s) ≤ Cε −α(2)
holds for all dimensions s ∈ N and for all ε ∈ (0, 1). The infimum over all α > 0 such that (2) holds is called the ε-exponent of strong polynomial tractability.
Polynomial tractability means that there exists a point set whose cardinality is polynomial in s and ε −1 such that the weighted star discrepancy of this point set is bounded by ε. Polynomial tractability and strong polynomial tractability for the classical star discrepancy are defined in the same manner as in the weighted case.
An excellent survey on tractability of different notions of discrepancy can be found in the paper [16] or in the books [17,18,19] which perfectly summarize the current state of the art in tractability theory.
Results on the weighted star-discrepancy
We provide an informal description of our results. The details are given in Section 3. We study three kinds of point sets in the unit cube which go back to Korobov, and Hua and Wang, and which are sometimes called the "p-sets". The advantage of these point sets is that their constructions are very easy (see Section 2 for details). As an example, one of these p-sets is given by the points ({n/p}, {n 2 /p}, . . . , {n s /p}) for n = 0, 1, . . . , p − 1, where p is a prime number, and where {x} = x − ⌊x⌋ denotes the fractional part of x for positive real numbers x.
For simplicity we restrict ourselves to product weights. Results for general weights are given in Section 3. In this paper we show that if the product weights
γ = { j∈u γ j } satisfy ∞ j=1 γ j < ∞,
then for any 0 < δ < 1/2 there exists a constant c (1) γ,δ > 0 which depends only on γ and δ but not on the number of points N and the dimension s, such that Korobov's p-sets P satisfy
D * N,γ (P) ≤ c (1) γ,δ 1 N 1/2−δ .
This implies strong polynomial tractability. If there exists a real number t > 0 such that the product weights
γ = { j∈u γ j } satisfy ∞ j=1 γ t j < ∞,
then for any 0 < δ < 1/2 there exists a constant c (2) γ,δ,t > 0 which depends only on γ, δ and t, but not on the number of points N and the dimension s, such that Korobov's p-sets P satisfy
D * N,γ (P) ≤ c (2) γ,δ,t s N 1/2−δ .
In this case we have polynomial tractability.
Literature review
To put our results into context, we provide a review of known results. From Heinrich, Novak, Wasilkowski, and Woźniakowski [12] it is known that for any number of points N and dimension s there exists a point set P N,s ⊆ [0, 1) s , such that
D * N (P N,s ) ≤ C s N (3)
for some constant C > 0. Hence the classical star discrepancy is tractable with s-exponent at most one and ε-exponent at most two. It was further shown in [12] that the inverse of the classical star discrepancy is at least cs log ε −1 with an absolute constant c > 0 for all ε ∈ (0, ε 0 ] and s ∈ N. This lower bound was improved by Hinrichs [13] to csε −1 with an absolute constant c > 0 for all ε ∈ (0, ε 0 ] and s ∈ N. From these results it follows, that the classical star discrepancy cannot be strongly polynomially tractable. We stress that all mentioned results are non-constructive. A first constructive approach is given in [6]. However here for given s and ε the authors can only ensure a running time for the construction algorithm of order C s s s (log s) s ε −2(s+2) which is too expensive for practical applications. The bound (3) can be interpreted as having product weights (γ j ) for which γ j = 1 for all j. In comparison, to achieve polynomial tractability in our result, we require that for product weights we have ∞ j=1 γ t j < ∞ for some arbitrarily large real number t > 0. Note that [12] only show an existence result, whereas our result is completely constructive.
We recall some known results for the weighted star discrepancy. The first result was shown in [14] (see also [5,18] for a summary).
Theorem 1 (Hinrichs, Pillichshammer, Schmid) There exists a constant C > 0 with the following property: for given number of points N and dimension s there exists
an N-element point set P in [0, 1) s such that D * N,γ (P) ≤ C 1 + √ log s √ N max ∅ =u⊆[s] γ u |u|.(4)
Note that the point set P from Theorem 1 is independent of the choice of weights. The result is a pure existence result. Under very mild conditions on the weights Theorem 1 implies polynomial tractability with s-exponent zero. See [14] for details. For product weights we have the following result which is taken from [3] (see also [
(γ j ) j≥1 with j γ j < ∞ one can construct (component- by-component) a p m -element point set P in [0, 1) s such that for every δ > 0 there exists a quantity C γ,δ > 0 with the property D * p m ,γ (P) ≤ C γ,δ p m(1−δ) .
Note that the point set P from Theorem 2 depends on the choice of weights. The result implies that the weighted star discrepancy is strongly polynomially tractable with ε-exponent equal to one, as long as the weights γ j are summable. See [3,4,5,14] for more details.
The next result is about Niederreiter sequences in prime-power base q. For the definition of Niederreiter sequences we refer to [5,15]. The following result is [23, Lemma 1]:
Lemma 1 (Wang) For N ∈ N let P be the first N-elements of a Niederreiter sequence in prime-power base q. For u ⊆ [s] we denote by P u the |u|-dimensional point set consisting of the projections of the elements of P onto the coordinates which belong to u. Then we have
D * N (P u ) ≤ 1 N j∈u (Cj log(j + q) log(qN)),
where C > 0 is an absolute constant which is independent of u and s.
Similar results can be shown for Sobol' sequences and for the Halton sequence (see [22,23]). From this result one obtains:
Theorem 3 For the weighted star discrepancy of the first N elements P of an s-dimensional Niederreiter sequence in prime-power base q we have
D * N,γ (P) ≤ 1 N max ∅ =u⊆[s]
γ u j∈u (Cj log(j + q) log(qN)).
In the case of product weights one can easily deduce from Theorem 3 that the weighted star discrepancy of the Niederreiter sequence can be bounded independently of the dimension whenever the weights satisfy j γ j j log j < ∞. This implies strong polynomial tractability with ε-exponent equal to one. (The same result can be shown for Sobol' sequences and for the Halton sequence).
A comparison of the results presented in this section with the new results will be given at the end of Section 3.
Korobov's p-sets
Let p be a prime number. We consider the following point sets in [0, 1) s :
• Let P p,s = {x 0 , . . . , x p−1 } with x n = n p , n 2 p , . . . , n s p for n = 0, 1, . . . , p − 1.
The point set P p,s was introduced by Korobov [9] (see also [20, Section 4.3]).
• Let Q p 2 ,s = {x 0 , . . . , x p 2 −1 } with x n = n p 2 , n 2 p 2 , . . . , n s p 2 for n = 0, 1, . . . , p 2 − 1.
The point set Q p,s was introduced by Korobov [8] (see also [20,Section 4.3]).
• Let R p 2 ,s = {x a,k : a, k ∈ {0, . . . , p − 1}} with
x a,k = k p , ak p , . . . , a s−1 k p for a, k = 0, 1, . . . , p − 1.
Note that R p 2 ,s is the multi-set union of all Korobov lattice point sets with modulus p. The point set R p 2 ,s was introduced by Hua and Wang (see [20,Section 4.3]).
Hua and Wang [20] called the point sets P p,s , Q p 2 ,s and R p 2 ,s the p-sets.
The weighted star discrepancy of the p-sets
The classical (i.e. unweighted) star discrepancy of the p-sets is studied in [20,]. Here we consider the weighted star discrepancy.
Theorem 4 Let p be a prime number. For arbitrary weights
γ = {γ u : ∅ = u ⊆ [s]} we have: D * p,γ (P p,s ) ≤ 2 √ p max ∅ =u⊆[s] γ u (max u) (4 log p) |u| , D * p 2 ,γ (Q p 2 ,s ) ≤ 3 p max ∅ =u⊆[s]
γ u (max u) (6 log p) |u| , and
D * p 2 ,γ (R p 2 ,s ) ≤ 2 p max ∅ =u⊆[s] γ u (max u) (4 log p) |u| .
The proof of Theorem 4 will be given in Section 4.2. Note that the point sets P p,s , Q p 2 ,s and R p 2 ,s from Theorem 4 are independent of the choice of weights. Now we consider product weights and study tractability properties. Let γ u = j∈u γ j where γ j > 0 for j ∈ N and γ 1 ≥ γ 2 ≥ γ 3 . . ..
Theorem 5
Assume that the weights γ j are non-increasing.
1. If ∞ j=1 γ j < ∞,
then for all δ > 0 there exist quantities c ′ γ,δ , c ′′ γ,δ , c ′′′ γ,δ > 0, which are independent of p and s, such that
D * p,γ (P p,s ) ≤ c ′ γ,δ p 1/2−δ , D * p 2 ,γ (Q p 2 ,s ) ≤ c ′′ γ,δ p 1−δ , and D * p 2 ,γ (R p 2 ,s ) ≤ c ′′′ γ,δ p 1−δ .
If there exists a real number t > 0 such that
∞ j=1 γ t j < ∞,
then for all δ > 0 there exist quantities c ′ γ,δ,t , c ′′ γ,δ,t , c ′′′ γ,δ,t > 0, which are independent of p and s, such that
D * p,γ (P p,s ) ≤ c ′ γ,δ,t s p 1/2−δ , D * p 2 ,γ (Q p 2 ,s ) ≤ c ′′ γ,δ,t s p 1−δ , and D * p 2 ,γ (R p 2 ,s ) ≤ c ′′′ γ,δ,t s p 1−δ .
The proof of Theorem 5 will be given in Section 4.3.
Note that the results in Point 1. of Theorem 5 imply strong polynomial tractability for the weighted star discrepancy. We show this for the p-set P p,s . Assume that j γ j < ∞. Fix δ > 0. For ε > 0, let p be the smallest prime number that is larger or equal to ⌈(c γ,δ ε −1 ) 2 1−2δ ⌉ =: M. Then we have D * p,γ (P p,γ ) ≤ ε and hence
N min (ε, s) ≤ p < 2M = 2⌈(c γ,δ ε −1 ) 2 1−2δ ⌉,
where we used Bertrand's postulate which tells us that M ≤ p < 2M. Hence the weighted star discrepancy is strongly polynomially tractable with ε-exponent at most 2.
In the same way, the results in Point 2. of Theorem 5 imply polynomial tractability for the weighted star discrepancy. We show this for P p,s . Assume that j γ t j < ∞ for some t > 0. Fix δ > 0. For ε > 0, let p be the smallest prime number that is larger or equal to ⌈(sc γ,δ,t ε −1 ) 2 1−2δ ⌉ =: M. Then we have D * p,γ (P p,s ) ≤ ε and hence N min (ε, s) ≤ p < 2M = 2⌈(sc γ,δ,t ε −1 )
2 1−2δ ⌉,
where we used again Bertrand's postulate which tells us that M ≤ p < 2M. Hence the weighted star discrepancy is polynomially tractable with s-exponent and ε-exponent at most 2.
With the following table we put the result from Theorem 5 into the context of the known results from Section 1. The second column "point set P" gives information about the point set, the column "P = P(γ)" indicates whether the point set depends on γ or not, the column "SPT" displays the conditions on product weights under which strong polynomial tractability is achieved and the last column "ε-exponent" displays the respective ε-exponents of strong polynomial tractability. point set P P = P(γ) SPT ε-exponent Theorem 1 existence NO not possible - Then we have
Theorem 2 CBC YES j γ j < ∞ 1 Theorem 3 explicit NO j γ j j log j < ∞ 1 Theorem 5 explicit NO j γ j < ∞ ≤ 2D * N,γ (P) ≤ max ∅ =u⊆[s] γ u |u| M + max ∅ =u⊆[s] γ u h∈C * |u| (M ) 1 r(h) 1 N N −1 n=0 exp(2πih · y n,u /M) ,
where y n,u ∈ [0, 1) |u| is the projection of y n to the coordinates given by u.
Proof. We have
D * N,γ (P) = sup z∈(0,1] s max ∅ =u⊆[s] γ u |∆ P ((z u , 1))| ≤ max ∅ =u⊆[s] γ u D * N (P u ),
where P u = {x 0,u , . . . , x N −1,u } in [0, 1) |u| consists of the points of P projected to the components whose indices are in u. For any ∅ = u ⊆ [s] we have from Lemma 2 that
D * N (P u ) ≤ |u| M + h∈C * |u| (M ) 1 r(h) 1 N N −1
The proof of Theorem 4
For the proof of Theorem 4 we use results which were already stated in the book of Hua and Wang [20]. exp(2πi(h 1 n + h 2 n 2 + · · · + h s n s )/p) ≤ (s − 1) √ p.
Proof. The result follows from a bound from A. Weil [24] on exponential sums which is widely known as Weil bound. For details we refer to [2]. ✷ Lemma 5 Let p be a prime number and let s ∈ N. Then for all h 1 , . . . , h s ∈ Z such that p ∤ h j for at least one j ∈ [s] we have
p 2 −1 n=0 exp(2πi(h 1 n + h 2 n 2 + · · · + h s n s )/p 2 ) ≤ (s − 1)p.
Proof. See [20,Lemma 4.6]. Proof.
• We consider P p,s : Here x n is of the form x n = {y n /M}, where y n = (n, n 2 , . . . , n s ) ∈ Z s and M = p.
For ∅ = u ⊆ [1 r(h) (max u − 1) √ p ≤ max u √ p (1 + S p ) |u| ,
where S p = h∈C * 1 (p) |h| −1 . A straight-forward estimate gives
S p ≤ 2 ⌊p/2⌋ h=1 1 h ≤ 2 1 + p/2 1 dt t = 2(1 + log(p/2)).(5)
Hence we obtain
h∈C * |u| (p) 1 r(h) 1 p p−1 n=0 exp(2πih · y n,u /p) ≤ max u √ p (2 + 2 log(p/2)) |u| ,
Inserting this into Lemma 3 gives
D * p,γ (P p,s ) ≤ max ∅ =u⊆[s] γ u |u| p + max ∅ =u⊆[s] γ u max u √ p (2 + 2 log(p/2)) |u| ≤ 2 √ p max ∅ =u⊆[s]
γ u (max u) (4 log p) |u| .
• We consider Q p 2 ,s : Here x n is of the form x n = {y n /M}, where y n = (n, n 2 , . . . , n s ) ∈ Z s . Let M = p 2 .
For h = (h j ) j∈u ∈ Z |u| we write p|h if p|h j for all j ∈ u and p ∤ h if this is not the case.
For ∅ = u ⊆ [s] we obtain from Lemma 5
h∈C * |u| (p 2 ) 1 r(h) 1 p 2 p 2 −1 n=0 exp(2πih · y n,u /p 2 ) ≤ h∈C * |u| (p 2 ) p|h 1 r(h) 1 p 2 p 2 −1 n=0 exp(2πih · y n,u /p 2 ) + 1 p 2 h∈C * |u| (p 2 ) p∤h 1 r(h) (max u)p.
We consider now the first sum where p | h. Let k = h/p ∈ Z u . Since y n,u = (n j ) j∈u we have from Lemma 4 that 1 p 2
p 2 −1 n=0 exp(2πih · y n,u /p 2 ) = 1 p 2 p−1 ℓ=0 ℓp+p−1 n=ℓp exp(2πik · y n,u /p) = 1 p p−1 n=0 exp(2πik · y n,u /p) ≤ max u √ p .
Further we have
h∈C * |u| (p 2 ) p|h 1 r(h) ≤ 1 p h∈C * |u| (p 2 ) 1 r(h) ≤ 1 p (1 + S p 2 ) |u| ,
where S p 2 = h∈C * 1 (p 2 ) |h| −1 . As before we find that S p 2 ≤ 2(1 + log(p 2 /2)).
which implies that for any k ∈ N we have
γ k ≤ Γ 0 k < ∞.
Thus for any finite set u ⊆ N we have
γ max u max u ≤ Γ 0 < ∞. Therefore D * p,γ (P p,s ) ≤ 2 √ p max ∅ =u⊆[s] (max u) j∈u (4γ j log p) ≤ 8Γ 0 log p √ p max ∅ =u⊆[s−1] j∈u (4γ j log p) ≤ 8Γ 0 log p √ p ℓ j=1 (4γ j log p),
where ℓ ∈ N 0 is such that γ ℓ ≥ 1 4 log p > γ ℓ+1 . If 1 4 log p > γ 1 then we set ℓ = 0 and set the empty product 0 j=1 (4γ j log p) to 1. For ℓ ≤ k 0 we have D * p,γ (P p,s ) ≤ 2(4Γ 0 log p) k 0 +1 √ p .
Now consider the case when ℓ > k 0 . For k > k 0 we have (k − k 0 )γ k ≤ k j=k 0 +1 γ j ≤ Γ k 0 and therefore γ k ≤ Γ k 0 k−k 0 . Thus we have
D * p,γ (P p,s ) ≤ 2(4Γ 0 log p) k 0 +1 √ p ℓ j=k 0 +1 (4γ j log p) ≤ 2(4Γ 0 log p) k 0 +1 √ p (4Γ k 0 log p) ℓ−k 0 (ℓ − k 0 )! .
Using Stirling's formula we obtain
(4Γ k 0 log p) ℓ−k 0 (ℓ − k 0 )! ≤ 1 2π(ℓ − k 0 ) 4Γ k 0 e log p ℓ − k 0 ℓ−k 0 ≤ 1 + 4Γ k 0 e log p ℓ − k 0 ℓ−k 0 ≤e 4Γ k 0 e log p =p 4Γ k 0 e ≤ p δ/2 .
There exists a constant c γ,δ > 0 depending on γ and δ but not on s and p, such that 2(4Γ 0 log p) k 0 +1 ≤ c γ,δ p δ/2 for all p ∈ N.
Thus we obtain D * p,γ (P p,s ) ≤ c γ,δ p 1/2−δ .
2. We use the notation from above. Assume now that for some t > 0 we have ∞ j=1 γ t j < ∞. Let h 0 be the smallest integer such that
Γ h 0 ,t ≤ δ 8e t t .
We estimate max u by s to obtain where ℓ is defined as above and where we set ℓ j=h 0 +1 (4γ j log p) = 1 if ℓ ≤ h 0 . Now we have
(h − h 0 )γ t h ≤ h j=h 0 +1 γ t j ≤ Γ t h 0 ,t and therefore γ h ≤ Γ h 0 ,t (h − h 0 ) 1/t .
For
an N-element point set P = {x 0 , . . . , x N −1 } in the s-dimensional unit cube [0, 1) s the discrepancy function ∆ is defined by∆(α 1 , . . . , α s ) := A N ( s i=1 [0, α i )) N − α 1 · · · α s for 0 < α 1 , . . . , α s ≤ 1. Here A N (E)denotes the number of indices n ∈ {0, 1, . . . , N − 1}, such that x n belongs to the set E. By taking the sup norm of this function, we obtain the star discrepancy D * N (P) = sup z∈[0,1] s |∆(z)| of the point set P. The motivation for the definition of the star discrepancy comes from quasif (x n ) ≈ [0,1] s f (x) dx of functions over the s-dimensional unit cube [0,
For
M ∈ N, M ≥ 2, put C(M) = (−M/2, M/2] ∩ Z and C s (M) = C(M) s the s-fold Cartesian product of C(M). Further we write C * s (M) = C s (M) \ {0}. For h ∈ C(M) put r(h) = max(1, |h|) and for h = (h 1 , . . . , h s ) ∈ C s (M) put r(h) = s j=1 r(h j ). The following result is from Niederreiter [15, Theorem 3.10] (in a slightly simplified form). Lemma 2 (Niederreiter) For integers M, N ≥ 2 and y 0 , . . . , y N −1 ∈ Z s , let P = {x 0 , . . . , x N −1 } be the N-element point set consisting of the fractional parts x n = {y n /M} for n = 0, . . . , N − 1. Then we have 2πih · y n /M) , where "·" denotes the usual inner-product in R s and where i = √ −1. Now we extend this result to the weighted star discrepancy: Lemma 3 For integers M, N ≥ 2 and y 0 , . . . , y N −1 ∈ Z s , let P = {x 0 , . . . , x N −1 } be the N-element point set consisting of the fractional parts x n = {y n /M} for n = 0, . . . , N − 1.
n=0 exp(2πih · y n,u /M) , and the result follows.
✷ Lemma 6 1
61Let p be a prime number and let s ∈ N. Then for allh 1 , . . . , h s ∈ Z such that p ∤ h j for at least one j ∈ [s] we have 2πik(h 1 + h 2 a + · · · + h s a s−1 )/p) ≤ (s − 1)p.Proof. Under the assumption that p ∤ gcd(h 1 , . . . , h s ), the number of solutions of the congruence h 1 + h 2 x + · · · + h s x s−1 ≡ 0 (mod p) in {0, 1, . . . , p − 1} is at most s − 2πik(h 1 + h 2 a + · · · + h s a s−1 )can give the proof of Theorem 4:
set Γ h = Γ h,1 .
4, Corollary 8] and [5, Corollary 10.30]): Theorem 2 (Dick, Leobacher, Pillichshammer) For every prime number p, every m ∈ N and for given product weights
Lemma 4 Let p be a prime number and let s ∈ N. Then for all h 1 , . . . , h s ∈ Z such that p ∤ h j for at least one j ∈ [s] we havep−1
n=0
s] we obtain from Lemma 4h∈C *
|u| (p)
1
r(h)
1
p
p−1
n=0
exp(2πih · y n,u /p) ≤
1
p
h∈C *
|u| (p)
Thus we haveexp(2πih · y n,u /p 2 ) ≤ 2 max u p (2 + 2 log(p 2 /2)) |u|Inserting this into Lemma 3 givesγ u (max u) (6 log p) |u| .• We consider R p 2 ,s : Here x a,k is of the form x a,k = {y a,k /M}, where y a,k = (ak, a 2 k, . . . , a s−1 k) ∈ Z s and M = p.where we used(5). Inserting this into Lemma 3 givesγ u (max u) (4 log p) |u| .✷The proof of Theorem 5Proof. We give the proof only for P p,s . The proofs for Q p 2 ,s and R p 2 ,s follow by the same arguments.1. Let δ > 0. Assume that γ 1 ≥ γ 2 ≥ . . . and j γ j < ∞. For k ∈ N 0 let Γ k := ∞ j=k+1 γ j < ∞ and note that lim k Γ k = 0. Let k 0 ∈ N be the smallest integer such thatwhere e = exp(1). Note that k 0 depends on γ and δ, but not on s and p.We haveAssume that ℓ > h 0 . Then, using Stirling's formula again, we obtainThe result now follows by the same arguments as in the previous case. ✷
Liberating the weights. J Dick, I H Sloan, X Wang, H Woźniakowski, J. Complexity. 20Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Liberating the weights. J. Com- plexity 20: 593-623, 2004.
Numerical integration of Hölder continuous absolutely convergent Fourier-, Fourier cosine-, and Walsh series. J Dick, Submitted for publicationDick, J.: Numerical integration of Hölder continuous absolutely convergent Fourier-, Fourier cosine-, and Walsh series. Submitted for publication, 2014.
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M Drmota, R F Tichy, Sequences, Discrepancies and Applications. BerlinSpringer-Verlag1651Drmota, M. and Tichy, R.F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997.
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Number-theoretic methods in approximate analysis. (Russian) Gosudarstv. N M Korobov, Izdat. Fiz.-Mat. Lit. Korobov, N.M.: Number-theoretic methods in approximate analysis. (Russian) Gosu- darstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963.
Uniform Distribution of Sequences. L Kuipers, H Niederreiter, WileyKuipers, L. and Niederreiter, H.: Uniform Distribution of Sequences. Wiley, 1974.
Introduction to Quasi-Monte Carlo Integration and Applications. G Leobacher, F Pillichshammer, Compact Textbooks in Mathematics. to appearLeobacher, G. and Pillichshammer, F.: Introduction to Quasi-Monte Carlo Integra- tion and Applications. Compact Textbooks in Mathematics, Birkhäuser, Basel, 2014 (to appear).
The inverse of the star discrepancy depends linearly on the dimension. S Heinrich, E Novak, G W Wasilkowski, H Woźniakowski, Acta Arith. 96Heinrich, S., Novak, E., Wasilkowski, G.W., and Woźniakowski, H.: The inverse of the star discrepancy depends linearly on the dimension. Acta Arith. 96: 279-302, 2001.
Covering numbers, Vapnik-Červonenkis classes and bounds on the stardiscrepancy. A Hinrichs, J. Complexity. 20Hinrichs, A.: Covering numbers, Vapnik-Červonenkis classes and bounds on the star- discrepancy. J. Complexity 20: 477-483, 2004.
Tractability properties of the weighted star discrepancy. A Hinrichs, F Pillichshammer, W Schmid, J. Complexity. 24Hinrichs, A., Pillichshammer, F., and Schmid, W.: Tractability properties of the weighted star discrepancy. J. Complexity 24: 134-143, 2008.
Random Number Generation and Quasi-Monte Carlo Methods. H Niederreiter, CBMS-NSF Series in Applied Mathematics. SIAM. Philadelphia63Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. No. 63 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia, 1992.
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Author's Addresses. Author's Addresses:
Josef Dick, School of Mathematics and Statistics. Sydney, NSW 2052, AustraliaThe University of New South WalesEmail: [email protected] Dick, School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia. Email: [email protected]
. Friedrich Pillichshammer, Altenbergerstraße. 69Institut für Analysis, Universität LinzEmail: [email protected] Pillichshammer, Institut für Analysis, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria. Email: [email protected]
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[] |
[
"Straining graphene by chemical vapor deposition growth on copper",
"Straining graphene by chemical vapor deposition growth on copper"
] |
[
"V Yu \nDepartment of Physics\nMcGill University\nH3A 2T8MontréalCanada\n",
"E Whiteway \nDepartment of Physics\nMcGill University\nH3A 2T8MontréalCanada\n",
"J Maassen \nDepartment of Physics\nMcGill University\nH3A 2T8MontréalCanada\n",
"M Hilke \nDepartment of Physics\nMcGill University\nH3A 2T8MontréalCanada\n"
] |
[
"Department of Physics\nMcGill University\nH3A 2T8MontréalCanada",
"Department of Physics\nMcGill University\nH3A 2T8MontréalCanada",
"Department of Physics\nMcGill University\nH3A 2T8MontréalCanada",
"Department of Physics\nMcGill University\nH3A 2T8MontréalCanada"
] |
[] |
Strain can be used as an alternate way to tune the electronic properties of graphene. Here we demonstrate that it is possible to tune the uniform strain of graphene simply by changing the chemical vapor deposition growth temperature of graphene on copper. Due to the cooling of the graphene on copper system, we can induce a uniform compressive strain on graphene. The strain is analyzed by Raman spectroscopy, where a shift in the 2D peak is observed and compared to our ab initio calculations of the graphene on copper system as a function of strain.
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10.1103/physrevb.84.205407
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[
"https://arxiv.org/pdf/1101.1884v1.pdf"
] | 119,178,090 |
1101.1884
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a491d0b37f6b1060b26f022f31b26de62a97ece7
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Straining graphene by chemical vapor deposition growth on copper
7 Jan 2011
V Yu
Department of Physics
McGill University
H3A 2T8MontréalCanada
E Whiteway
Department of Physics
McGill University
H3A 2T8MontréalCanada
J Maassen
Department of Physics
McGill University
H3A 2T8MontréalCanada
M Hilke
Department of Physics
McGill University
H3A 2T8MontréalCanada
Straining graphene by chemical vapor deposition growth on copper
7 Jan 2011(Dated: January 11, 2011)
Strain can be used as an alternate way to tune the electronic properties of graphene. Here we demonstrate that it is possible to tune the uniform strain of graphene simply by changing the chemical vapor deposition growth temperature of graphene on copper. Due to the cooling of the graphene on copper system, we can induce a uniform compressive strain on graphene. The strain is analyzed by Raman spectroscopy, where a shift in the 2D peak is observed and compared to our ab initio calculations of the graphene on copper system as a function of strain.
Strain can be used as an alternate way to tune the electronic properties of graphene. Here we demonstrate that it is possible to tune the uniform strain of graphene simply by changing the chemical vapor deposition growth temperature of graphene on copper. Due to the cooling of the graphene on copper system, we can induce a uniform compressive strain on graphene. The strain is analyzed by Raman spectroscopy, where a shift in the 2D peak is observed and compared to our ab initio calculations of the graphene on copper system as a function of strain. The discovery of the electric field effect in a few layers of graphene [1,2] led to a genuine explosion of work on graphene with the subsequent observation of the anomalous quantum Hall effect in graphene monolayers [3,4] following its theoretical prediction [5]. The most important property for electronic device applications is the possibility to tune the resistance and electronic density of the graphene sheet. This is achieved by using the substrate directly as a backgate [1,3,4] or by processing a top gate [2].
More recently, it was suggested theoretically that it is possible to tune the electronic density of graphene and create confinement [6], as well as to observe a zero-field quantum Hall effect by using strain engineering [7]. In general, strain as a way to enhance electronic and optical properties in semiconductors has been very successful [8]. Mechanically, graphene turns out to be of very high strength [9] and is stretchable up to 30% in some cases [10].
While exfoliated graphene bears very high mobilities when suspended [11], the size is limited to a few microns. Large scale epitaxial graphene on SiC is an interesting avenue but suffers from high substrate costs and difficulty to transfer [12]. Alternatively, large scale graphene can be synthesized by chemical vapor deposition (CVD) using a polycrystalline metal foil as a catalyst [10,13].
In this letter, we demonstrate that it is possible to strain graphene intrinsically and uniformly during the graphene growth process. This leads to interesting new properties, such as a change in Fermi energy and Fermi velocity in graphene. This opens new doors to the possibility of engineering devices, where the electronic properties are tuned by strain. We consider the situation depicted in figure 1, where a graphene monolayer is sitting on a Cu surface. For our ab initio calculations we consider a Cu(111) surface, where the graphene and the Cu share the same in-plane lattice constant [14]. We assume that the system has no structural defects, meaning that the graphene and the Cu match perfectly at the interface. In this case, the most stable configuration of graphene on Cu(111) corresponds to one carbon sitting at the top-site (A-site) and the other carbon located at the hollow-site (C-site) [15,16]. The same configuration also applies to graphene on other metallic surfaces [17]. To induce strain, we uniformly expand or compress the in-plane lattice constant of the whole graphene/Cu system.
Zero strain corresponds to a C-C bond length of a eq =1.415Å, which was found to minimize the total energy of an isolated graphene sheet using density functional theory. For every a, we (i) perform a structural relaxation of the system to ensure that the forces acting on each atom is less than 0.01 eV/Å, (ii) obtain the selfconsistent electronic density and Hamiltonian and (iii) calculate the band structure of the hybrid graphene/Cu system along the high-symmetry points Γ, M and K.
The most striking features induced by the strain are the change in Fermi velocity close to the Dirac point and more notably a shift in the Fermi energy with respect to the Dirac point as a function of strain as shown in figure 2 b). This is very interesting, since this allows for the tuning of the electronic density in graphene by strain.
Experimentally, we expect to be able to strain graphene on Cu simply by heating the combined system, since the thermal expansion of Cu is different from that of graphene. While graphene has a small negative thermal expansion coefficient α [18], for Cu we have α ≃ 25.8 × 10 −6 / • C [19] at 1000 • C. This would lead to a relative graphene compression of 0.5% from 1100 to 900 • C.
We synthesized graphene monolayers by CVD of hydrocarbons on 25 µm-thick commercial polycrystalline copper foils. The Cu foil is first acid-treated for 10 mins using acetic acid and then washed thoroughly with deionized water. Graphene growth is realized in similar conditions to the recently reported ones [20,21], but using a vertical quartz tube. Graphene is grown with temperatures ranging from 900 • C to 1100 • C in 0.5 Torr, with a 4 sccm H 2 flow and a 40 sccm CH 4 flow for 30 minutes. During the cooling process the methane flow is stopped while the hydrogen flow is kept on. Figure 3 illustrates the temperature effect on the synthesis of graphene. These scanning electron microscope (SEM) images show that the density and size of growth domains increase with temperature. At the lowest growth temperatures we did not observe full coverage, even for longer growth times.
In order to characterize the graphene layer further, we used Raman spectroscopy. Raman spectra are measured with a 100x objective at 514nm, having a 1800 grooves/mm grating and spectral resolution of about 2 cm −1 . The results are shown in figure 4. There are two dominant peaks, labeled G and 2D. The G peak around 1576 cm −1 arises from a first order G band phonon process. The energy corresponds to the phonon energy at the Γ point, which is degenerate. Several authors have shown that the degeneracy is lifted by the application of uniaxial strain [22], which leads to a splitting of the G peak. For uniform biaxial strain, there are no experiments on compressive strain in graphene, but we expect the strain to simply renormalize the phonon energy (ω G ) following ω G = ω G eq (1 − 2ǫγ) [23,24]. γ ≃ 1.24 is the Grüneisen parameter in carbon nanotubes [25], ǫ = (a − a eq )/a eq is the strain and the factor 2 stems from the biaxial strain in both directions. The other dominant peak is labeled 2D (also called G ′ ) and stems from a two phonon process. The two phonons of wavenumber δk and -δk scatter the electron with wavenumber δq into the other valley (K to K ′ ) and back as illustrated in figure 2 c). The phonon energy depends on the dispersion close to the Dirac points (K or K ′ ) along the M direction. Hence δk will depend on the electronic band structure, which is shown in figure 2 a) for two different values of strain. Using the calculated electronic band structure we can extract δq in the M direction for our Raman laser energy E laser = 2.412 eV (514 nm). From a geometrical consideration, we have δk = (KK ′ ) 2 + 2 · KK ′ · δq + 4 · δq 2 (see figure 2 c)), where KK ′ =K−K ′ is the distance in reciprocal space between the two valleys K and K ′ . KK ′ will simply be renormalized by a uniform strain leading to KK ′ = (KK ′ ) eq /(1 + ǫ), which is shown in figure 2 b). For small values of ǫ, the renormalization constant is approximately linear in strain. δk, on the other hand, is non-linear as it depends on δq.
In this region, the phonon band velocity can be parameterized by v iT O = 5.47 × 10 −3 v F [23], where v F ≃ 10 6 m/s is the Fermi velocity. In order to evaluate the change in phonon energy due to uniform strain for the 2D peak we simply combine the change in wavenumber δk from the linear dispersion with the renormalized total phonon energy due to the Grüneisen parameter as 2ǫγ). The factor 2 in front of the band velocity term stems from the phonon pair involved in the process. For the equilibrium (unstrained) values of ω (2D,G) eq , we use those obtained for exfoliated graphene at the same wavelength (514nm) [26], i.e., ω 2D eq = 2678cm −1 and ω G eq = 1576cm −1 . We can now evaluate the expected frequencies for the 2D and G peaks as a function of a uniform strain, which is shown in figure 5. The corresponding growth temperature dependence of the 2D peak data is then fitted to the 2D peak position obtained from the strain calculation, assuming a linear dependence. We also show the result (in dotted lines) for the 2D peak computed without the band structure effect, which is then simply a linear function of strain. By including the band structure effect, the 2D peak dependence becomes non-linear with strain. This is in contrast to the G peak, which does not depend on the electronic band structure.
ω 2D ≃ [ω 2D eq + 2v iT O (δk − δk eq )](1 −
The data fits nicely with the expected Raman shift when the difference in strain between growth temperatures of 900 and 1100 • C is -0.5%, which is the expected value from the thermal expansion of copper for this temperature range. This clearly demonstrates that it is possible to strain graphene uniformly, simply by changing the CVD growth temperature, which is one of the important results of this letter. This also shows that when graphene is grown at high temperatures it is under small compressive strain at room temperature (the Raman spectra are taken at room temperature).
The peak labeled D in figure 4 is located at about half the frequency of the 2D peak, since it involves only a one phonon process. This process, however, is only activated in the presence of disorder [23,27]. Raman spectroscopy can be employed to determine the number of graphene layers [28], which confirms that we have mainly single layered graphene as expected in CVD on Cu [20]. It is instructive to compare our results to those obtained for exfoliated graphene, where several groups have been able to detect uniaxial stain in exfoliated graphene flakes using Raman spectroscopy [29]. The reported shifts in the 2D peak vary from -7.8 cm −1 /1% to -66 cm −1 /1%. It was recently suggested that the large range of values could be due to the dependence on the polarization of the laser beam and orientation of the graphene lattice [30]. Our results, between 0 and -1% strain, give values of -66 cm −1 /1% (without band structure effect) and -79 cm −1 /1% (with band structure effect), which is the higher bound of the reported values for uniaxial strain. This is consistent with expectations, since for biaxial strain we expect approximately a doubling of the strain induced Raman shift. In a very different geometry, where stretched biaxial strain has been measured by depositing a graphene flake on a shallow depression, a very high 2D peak shift of about 200 cm −1 /1% was reported [31]. In epitaxial graphene, shifts of the principal Raman peaks were observed to be dependent on the growth time for a fixed annealing temperature [32]. For CVD on nickel, no systematic shift with growth temperature was observed [33].
The formed graphene/Cu heterostructure constitutes a very interesting system by itself. Not only can the amount of relative strain be tuned as shown above, leading to changes in the Fermi energy and Fermi velocity, but this system also opens the door to numerous applications. For instance, the added graphene layer can enhance electronic properties in Cu interconnects, for example, or greatly enhance thermal conductivities of thin Cu films. The tunability of the strain of the graphene layer, greatly increases this potential. The graphene/Cu heterostructure can also be selectively etched for partial or full transfer onto other substrates, which can lead to interesting transport properties such as a very sharp weak localization peak [34].
We would like to acknowledge help from S. Elouatik for the Raman characterization and the use of MIAM microfab facilities and GCM at Polytechnique for processing and characterization in addition to financial support from RQMP, NSERC and FQRNT.
PACS numbers: 78.30.-j, 63.22.-m, 62.20.D-, 81.05.ue, 79.60.Jv, 68.65.-k, 81.15.Gh
FIG. 1 :
1View of graphene on a Cu(111) surface. The grey and dark orange spheres represent the C and Cu atoms, respectively. The blue parallelogram delimits the area of the supercell used for the first principles calculations. The letters A, B and C depict the different Cu(111) sites. The minimal energy configuration of graphene on copper corresponds to one sub-lattice carbon sitting at the top-site (A-site) and the other sub-lattice carbon residing at the hollow-site (C-site).
structure of graphene on Cu close to the K point for two different values of strain, i.e., 0% (solid line) and -6.7% (dashed line). b) depicts the change of EF (relative to the Dirac point), δk, δq and KK ′ as a function of strain. c) Partial view of the Brillouin zone. The arrow labeled δkeq indicates the phonon process at the origin of the unstrained 2D Raman peak. The arrow labeled δqeq corresponds to the wavenumber of the Raman excited electron of the unstrained lattice and δq for the strained (compressed) lattice. The circles show the constant energy surface of the Raman excited electron in the unstrained (solid line) and strained (dashed lined) lattice.
FIG. 3 :
3SEM images of graphene on Cu grown for 30 minutes with different growth temperatures: (a) 900 • C, (b) 950 • C (c) 975 • C, and (d) 1050 • C, respectively. Each image is approximately 30µm×30µm.
FIG. 4 :
4Raman spectra for each growth temperature normalized to the 2D peak. Inset: Close-up of the 2D peaks showing the shifts of the peaks. The 975 • C spectrum shows a region of possible bilayer.
FIG. 5 :
5The Raman shift of the 2D and G peaks as a function of strain obtained from the ab initio results. The solid line and the dotted line for the 2D peak are obtained with and without considering the effect of the electronic band structure. The square symbols are the experimental data points obtained for different growth temperatures. The inset shows a zoom-in of the 2D peak data points and the line represents the best linear fit.
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[] |
[
"Spinor Bose-Einstein Condensates with Many Vortices",
"Spinor Bose-Einstein Condensates with Many Vortices"
] |
[
"T Kita \nDivision of Physics\nHokkaido University\n060-0810SapporoJapan\n",
"T Mizushima \nDepartment of Physics\nOkayama University\n700-8530OkayamaJapan\n",
"K Machida \nDepartment of Physics\nOkayama University\n700-8530OkayamaJapan\n"
] |
[
"Division of Physics\nHokkaido University\n060-0810SapporoJapan",
"Department of Physics\nOkayama University\n700-8530OkayamaJapan",
"Department of Physics\nOkayama University\n700-8530OkayamaJapan"
] |
[] |
Vortex-lattice structures of antiferromagnetic spinor Bose-Einstein condensates with hyperfine spin F = 1 are investigated theoretically based on the Ginzburg-Pitaevskii equations near Tc. The Abrikosov lattice with clear core regions are found never stable at any rotation drive Ω. Instead, each component Ψi (i = 0, ±1) prefers to shift the core locations from the others to realize almost uniform order-parameter amplitude with complicated magnetic-moment configurations. This system is characterized by many competing metastable structures so that quite a variety of vortices may be realized with a small change in external parameters.
|
10.1103/physreva.66.061601
|
[
"https://arxiv.org/pdf/cond-mat/0204025v1.pdf"
] | 54,010,302 |
cond-mat/0204025
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0de81b5d8a410223ee24d413783791342e71a747
|
Spinor Bose-Einstein Condensates with Many Vortices
1 Apr 2002 (Dated: October 29, 2018)
T Kita
Division of Physics
Hokkaido University
060-0810SapporoJapan
T Mizushima
Department of Physics
Okayama University
700-8530OkayamaJapan
K Machida
Department of Physics
Okayama University
700-8530OkayamaJapan
Spinor Bose-Einstein Condensates with Many Vortices
1 Apr 2002 (Dated: October 29, 2018)
Vortex-lattice structures of antiferromagnetic spinor Bose-Einstein condensates with hyperfine spin F = 1 are investigated theoretically based on the Ginzburg-Pitaevskii equations near Tc. The Abrikosov lattice with clear core regions are found never stable at any rotation drive Ω. Instead, each component Ψi (i = 0, ±1) prefers to shift the core locations from the others to realize almost uniform order-parameter amplitude with complicated magnetic-moment configurations. This system is characterized by many competing metastable structures so that quite a variety of vortices may be realized with a small change in external parameters.
Realizations of the Bose-Einstein condensation (BEC) in atomic gases have opened up a novel research field in quantized vortices as created recently with various techniques [1,2,3,4]. Especially interesting in these systems are vortices of spinor BEC's in optically trapped 23 Na [5] and 87 Rb [6], where new physics absent in superconductors [7], 4 He [8], and 3 He [9,10,11], may be found.
Theoretical investigations on spinor BEC's were started by Ohmi and Machida [12] and Ho [13], followed by detailed studies on vortices with a single circulation quantum [14,15,16,17,18,19,20,21,22,23]. However, no calculations have been performed yet on structures in rapid rotation where the trap potential will play a less important role. Indeed, the clear hexagonal-lattice image of magnetically trapped 23 Na [3] suggests that predictions on infinite systems are more appropriate for BEC's with many vortices. Such calculations have been carried out by Ho for the single-component BEC [24] and by Mueller and Ho for a two-component BEC [25] near the upper critical angular velocity Ω c2 at T = 0.
The purpose of the present paper is to perform detailed calculations on vortices of F = 1 spinor BEC's in rapid rotation to clarify their essential features. To this end, we focus on the mean-field high-density phase rather than the low-density correlated liquid phase [26], and use the phenomenological Ginzburg-Pitaevskii (or Ginzburg-Landau) equations near T c [27,28] instead of the Gross-Pitaevskii equations at T = 0. Since fluctuations are small in the present system, this approach will yield quantitatively correct results near T c . It should be noted that the corresponding free-energy is formally equivalent to that derived with Ho's "mean-field quantum Hall regime" near Ω c2 at T = 0 [24,25], so that the results obtained here are also applicable to that region.
Model.-The free-energy density of an F = 1 spinor BEC near T c may be expanded with respect to the order parameters Ψ i (i = 0, ±1) as
f = −αΨ * i Ψ i + Ψ * i (−i ∇ − M Ω×r) 2 2M Ψ i + β n 2 Ψ * i Ψ * j Ψ i Ψ j + β s 2 Ψ * i Ψ * j (F µ ) ik (F µ ) jl Ψ k Ψ l .(1)
Here α, β n , and β s are expansion parameters, F µ (µ = x, y, z) denotes the spin operator, M is the particle mass, and summations over repeated indices are implied. The rotation axis Ω is taken along z. The quantities β n and β s are assumed to be constant near T c with β n > 0, whereas α changes its sign at T c with α > 0 for T < T c . To simplify Eq. (1), we measure the length, the energy density, the angular velocity, and the order parameter in units of / √ 2M α, α 2 /β n , α/ , and α/β n , respectively. The corresponding free-energy density is obtained from Eq.
(1) by α → 1, β n → 1, 2 /2M → 1, M Ω → 1
2 Ω, and β s → g s ≡ β s /β n . It thus takes a simple form with only two parameters (g s , Ω). We then introduce a couple of operators by a ≡ ℓ(
∂ x +i∂ y )/ √ 2 and a † ≡ ℓ(−∂ x +i∂ y )/ √ 2 with ∂ ≡ ∇ − i 2 Ω×r and ℓ ≡ 1/ √ Ω which satisfy aa † −a † a = 1. Equation (1) now reads f = Ψ * i [(2a † a+1)Ω − 1]Ψ i + 1 2 Ψ * i Ψ * j Ψ i Ψ j + g s 2 Ψ * i Ψ * j (F µ ) ik (F µ ) jl Ψ k Ψ l ,(2)
with Ω c2 = 1, and the free-energy is given by
F ≡ f (r) dr .(3)
We can find the stable structure for each (g s , Ω) by minimizing F . We have performed extensive calculations over the whole antiferromagnetic region g s ≥ 0, where (Ψ 1 , Ψ 0 , Ψ −1 ) = e iθ U(0, 1, 0) and f = 1 2 at Ω = 0 with θ an arbitrary phase and U the spin-space rotation [13].
A major difference from superfluid 3 He [11] lies in the fact that terms such as Ψ * i aaΨ j (i = j) are absent, i.e., there are no gradient couplings between different components. Hence Ω c2 is the same for all components, whereas in 3 He only a single component becomes finite at Ω c2 to realize the polar state [11]. This degenerate feature is what characterizes the present system to bring many competing metastable structures, as seen below.
Method.-We minimize Eq. (3) with the Landau-levelexpansion method (LLX) [11,29] by expanding the order parameters as
Ψ i (r) = √ V ∞ N =0 q c (i) N q ψ N q (r) ,(4)
with N the Landau-level index, q the magnetic Bloch vector, and V the system volume. The basis functions {ψ N q } are eigenstates of the magnetic translation oper-
ator T R ≡ exp[−R · (∇ + i 2 Ω × r)]
, which can describe periodic vortex structures of all kinds [29]. Its explicit expression is given by
ψ N q (r) = N f /2 n=−N f /2+1 e i[qy(y+ℓ 2 qx/2)+na1x(y+ℓ 2 qx−na1y/2)/ℓ 2 ] ×e −ixy/2ℓ 2 −(x−ℓ 2 qy−na1x) 2 /2ℓ 2 × 2πℓ/a 2 2 N N ! √ π V H N x−ℓ 2 q y −na 1x ℓ ,(5)
with N 2 f the number of the circulation quantum κ ≡ h/M in the system, H N the Hermite polynomial, and a j the basic vectors in the xy plane with a 2 ŷ and a 1x a 2 = 2πℓ 2 . Substituting Eq. (4) into Eq. (3) and carrying out the integration in terms of (s, t) ≡ (x/a 1x , y/a 2 − xa 1y /a 1x a 2 ), the free energy is transformed into a functional of the expansion coefficients {c (i) N q }, the apex angle β ≡ cos −1 (a 1y /a 1 ), and the ratio of the two basic vectors
ρ ≡ a 2 /a 1 as F = F [{c (i) N q }, β, ρ]
. For a given Ω, we directly minimize F /V with respect to these quantities.
Search for stable structures.-We here sketch our strategy to find stable structures. To this end, we first summarize the basic features of the conventional Abrikosov lattice within the framework of LLX [29]: (i) any single q = q 1 suffices, due to the broken translational symmetry of the vortex lattice; (ii) the triangular (square) lattice is made up of N = 6n (4n) Landau levels (n = 0, 1, 2 · · · ); (iii) more general structures can be described by N = 2n levels, odd N 's never mixing up since those bases have finite amplitudes at the cores. This Abrikosov lattice has a single circulation quantum per unit cell.
With multi-component order-parameters, there can be a wide variety of vortices, which may be divided into two categories. We call the first category as "shift-core" states, where core locations are different among the three components with an enlarged unit cell. General structures with n κ circulation quanta per unit cell can be described by using n κ different q's, where the unit cell becomes n κ times as large as that of the Abrikosov lattice. For example, structures with two quanta per unit cell are given by choosing (q 1 , q 2 ) = (0, b1+b2 2 ), where b 1 and b 2 are reciprocal lattice vectors. This possibility was already considered by Mueller and Ho [25] for a two-component system and shown to yield stable structures. It also describes the mixed-twist lattice to be found in 3 He [11].
The second category may be called "fill-core" states with a single circulation quantum per unit cell (i.e., a single q is relevant). Here the cores of the conventional Abrikosov lattice are filled in by some superfluid components using odd-N wavefunctions of Eq. (5). This entry of odd-N Landau levels occurs as a second-order transition below some critical angular velocity smaller than Ω c2 /3. It has been shown that the A-phase-core vortex in the B-phase of superfluid 3 He belong to this category [11].
We have carried out an extensive search for stable structures with up to n κ = 9 circulation quanta per unit cell, including fill-core states. Since we are near T c where normal particles are also present, we have performed the minimization without specifying the value of the magnetic moment M for the superfluid components. However, all the stable states found below have M = 0. Each of the three components were expanded as Eq. (4) using n κ different q's, and the free-energy (3) is minimized with respect to c (i) N q , β, and ρ by using Powell's method [30]. To pick out stable structures correctly, we calculated Eq. (3) many times starting from different initial values for c (i) 0q , β, and ρ given randomly within 0.1 ≤ Rec (i) 0q , Imc (i) 0q ≤ 0.9, 0.1π ≤ β ≤ 0.5π, and 0.8 ≤ ρ ≤ 3, respectively. The state with the lowest energy was thereby identified as the stable structure. The spin quantization axis and an overall phase were fixed conveniently to perform efficient calculations. Thus, any structures obtained from the solutions below with a spinspace rotation and a gauge transformation are also stable.
Instability of Abrikosov lattice.-The present calculations have revealed that the Abrikosov lattice with clear core regions is never stable at any rotation drive Ω over the entire antiferromagnetic region g s ≥ 0. Thus, any optical experiments to detect vortices by amplitude reductions are not suitable for the spinor vortices.
The most stable Abrikosov lattice is given by
Ψ 0 (r) = √ V N c (0) N ψ N q1 (r) ,(6)
with c (0) N q1 real, N = 6n, β = π 3 , and ρ = 1. Here the antiferromagnetic component Ψ 0 forms a hexagonal lattice with a single circulation quantum per unit cell. Below some critical velocity Ω f smaller than Ω c2 /3, the core regions start to be filled in by
Ψ ±1 (r) = i √ V N ′ c (1) N ′ ψ N ′ q1 (r) ,(7)
with N ′ odd. The second transition for this odd-Landaulevel entry occurs at Ω f = 0.1497Ω c2 and 0.0938Ω c2 for g s = 0.1 and 0.3, respectively. However, calculations down to 0.0001Ω c2 of using 1800 Landau levels for g s ≥ 0 have clarified that the above fillcore state carries higher free energy than the following shift-core state with two circulation quanta per unit cell:
Ψ 1 (r) = √ V N c N ψ N q1 (r) ,(8)Ψ −1 (r) = √ V N c N ψ N q2 (r) ,(9)
with c N real and common to both, N = 4n, β = π 2 , ρ = 1, and (q 1 , q 2 ) = (0, b1+b2 2 ). The cores of Ψ ±1 (r) are shifted from each other by 1 2 (a 1 + a 2 ). Figure 1 displays basic features of this shift-core state. The magnetic moment is ordered antiferromagnetically along z axis, and the amplitude is almost uniform taking its maximum at each site where the moment vanishes to form the antiferromagnetic state realized in the uniform state.
Stable structures near Ω c2 .-Having seen that the conventional Abrikosov lattice is never favored in the whole antiferromagnetic domain g s ≥ 0, we now enumerate all the stable structures found near Ω c2 to clarify their essential features. This rapidly rotating domain is especially interesting, because the same free energy also becomes relevant near Ω c2 at T = 0, as shown by Ho using a "meanfield quantum Hall regime" [24]. Thus, the conclusions obtained here are also applicable to the region at T = 0. Figure 2 displays the lowest free energy per unit volume as a function of g s for Ω = 0.95Ω c2 . The value of each n κ denotes the number of circulation quanta per unit cell. A special feature to be noted is that these various structures are energetically quite close to each other; for example, the n κ = 8 state at g s = 0.02 is favored over the n κ = 3 state by a relative free-energy difference of order 10 −6 . This fact suggests that we may realize quite a variety of metastable structures by a small change of the boundary conditions, the rotation process, etc.
Details of these structures are summarized as follows. The n κ = 2 state of Eqs. (8) and (9) is stable for g s ≥ 0.0671. We have already seen its basic features above. The n κ = 3, 5, 7 states can be expressed compactly as
Ψ 0 = N nκ−1 2 ν=1 c (0) N qν ψ N qν − e −2i ν nκ π ψ N qn κ−ν ,(10)Ψ ±1 = N nκ−1 2 ν=1 c (±1) N qν ψ N qν + e −2i ν nκ π ψ N qn κ−ν +c (±1) N qn κ ψ N qn κ ,(11)
where N 's are even and q ν = ν nκ b 1 . These n κ = 3, 5, 7 states are stable for 0 ≤ g s ≤ 0.0196, 0.0313 ≤ g s ≤ 0.0613, and 0.0613 ≤ g s ≤ 0.0671, respectively. Unlike the two component system considered by Mueller and Ho [25] where each component is specified by a single-q basis function, Ψ m here is made up of multiple basis functions ψ N qν whose cores are shifted from each other by (ν/n κ )a 2 . Figure 3 displays the basic features of the n κ = 3 state. The lattice is hexagonal at g s = 0, but deforms rapidly as g s increases. The order-parameter amplitude is again almost constant, and the magnetic moment M has a complicated structure. These features are common to all the n κ ≥ 3 states discussed here, although no details are presented for the other states. The lattice parameters (β, ρ) for n κ = 5, 7 states are (0.1311π, 2.056) and (0.1830π, 2.609) for g s = 0.05 and 0.066, respectively, which change little in each relevant range of stability.
The remaining n κ = 8 state, stable over 0.0196 ≤ g s ≤ 0.0313, is given by . The parameters (β, ρ) at g s = 0.025 are (0.3317π, 1.049), and changes only slightly in the above range of g s .
Ψ 0 = N c (0) N q1 ψ N q1 + ψ N q2 − ψ N q3 + iψ N q4 ,(12)Ψ ±1 = N 8 ν=5 c (±1) N qν ψ N qν ,(13)
Concluding remarks.-We have performed extensive calculations on antiferromagnetic F = 1 spinor vortices in rapid rotation. The conventional Abrikosov lattice is shown never favored. Each stable structure has almost constant order-parameter amplitude and a complicated magnetic-moment configuration, as shown in Figs. 1 and 3, for example. This means that any optical experiments to detect vortices by amplitude reduction will not be suitable for the spinor vortices. Instead, techniques to directly capture spatial magnetic-moment configurations will be required. The system has many metastable structures which are different in the number of circulation quanta per unit cell n κ , but are quite close to each other energetically. Thus, domains to separate different structures may be produced easily. This degenerate feature is also present within the (β, ρ) space of a fixed n κ , where β is the vortex-lattice apex angle and ρ is the length ratio of the two basic vectors. Put it differently, we can deform a stable lattice structure with a tiny cost of energy. These facts indicate that the spinor BEC's can be a rich source of novel vortices realized with a small change in external parameters.
FIG. 1 :
1Spatial variations of (a) the order-parameter amplitude and (b) the magnetic moment along z, for the shift-core state (8)-(9) over |x|, |y| ≤ a1 at gs = 0.08 and Ω = 0.95Ωc2. The moment is directed along z.
FIG. 2 :
2Calculated free energy per unit volume as a function of gs at Ω = 0.95Ωc2. Five different structures have been found for gs ≥ 0, and the value of each nκ denotes the number of circulation quanta per unit cell for each stable structure.
FIG. 3 :
3(a) Variations of β/π and ρ as a function of gs for the nκ = 3 state at Ω = 0.95Ωc2. This lattice for gs = 0 is hexagonal with β = π/3 and ρ = 3.Figures 2(b)-(d) display, for gs = 0.01, spatial variations of (b) the order-parameter amplitude, (c) amplitude of the magnetic moment, and (d) the magnetic moment, over −3a1x/2 < x < 3a1x/2 and −a2/2 < y < a2/2.
This research is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.
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[] |
[
"Estimating the Direction and Radius of Pipe from GPR Image by Ellipse Inversion Model",
"Estimating the Direction and Radius of Pipe from GPR Image by Ellipse Inversion Model"
] |
[
"Xiren Zhou ",
"Qiuju Chen ",
"Shengfei Lyu ",
"Senior Member, IEEEHuanhuan Chen "
] |
[] |
[] |
Ground Penetrating Radar (GPR) is widely used as a non-destructive approach to estimate buried utilities. When the GPR's detecting direction is perpendicular to a pipeline, a hyperbolic characteristic would be formed on the GPR Bscan image. However, in real-world applications, the direction of pipelines on the existing pipeline map could be inaccurate, and it is hard to ensure the moving direction of GPR to be actually perpendicular to underground pipelines. In this paper, a novel model is proposed to estimate the direction and radius of pipeline and revise the existing pipeline map from GPR B-scan images. The model consists of two parts: GPR B-scan image processing and Ellipse Iterative Inversion Algorithm (EIIA). Firstly, the GPR B-scan image is processed with downward-opening point set extracted. The obtained point set is then iteratively inverted to the elliptical cross section of the buried pipeline, which is caused by the angle between the GPR's detecting direction and the pipeline's direction. By minimizing the sum of the algebraic distances from the extracted point set to the inverted ellipse, the most likely pipeline's direction and radius are determined. Experiments on real-world datasets are conducted, and the results demonstrate the effectiveness of the method.
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"https://arxiv.org/pdf/2201.10184v1.pdf"
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2201.10184
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8099b4d02d065c73bd7da8bf403d72ef2dd21efe
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Estimating the Direction and Radius of Pipe from GPR Image by Ellipse Inversion Model
Xiren Zhou
Qiuju Chen
Shengfei Lyu
Senior Member, IEEEHuanhuan Chen
Estimating the Direction and Radius of Pipe from GPR Image by Ellipse Inversion Model
1Index Terms-Buried pipeline detectionGround-Penetrating Radar (GPR)Data processing
Ground Penetrating Radar (GPR) is widely used as a non-destructive approach to estimate buried utilities. When the GPR's detecting direction is perpendicular to a pipeline, a hyperbolic characteristic would be formed on the GPR Bscan image. However, in real-world applications, the direction of pipelines on the existing pipeline map could be inaccurate, and it is hard to ensure the moving direction of GPR to be actually perpendicular to underground pipelines. In this paper, a novel model is proposed to estimate the direction and radius of pipeline and revise the existing pipeline map from GPR B-scan images. The model consists of two parts: GPR B-scan image processing and Ellipse Iterative Inversion Algorithm (EIIA). Firstly, the GPR B-scan image is processed with downward-opening point set extracted. The obtained point set is then iteratively inverted to the elliptical cross section of the buried pipeline, which is caused by the angle between the GPR's detecting direction and the pipeline's direction. By minimizing the sum of the algebraic distances from the extracted point set to the inverted ellipse, the most likely pipeline's direction and radius are determined. Experiments on real-world datasets are conducted, and the results demonstrate the effectiveness of the method.
I. INTRODUCTION
Underground pipeline are indispensable for the normal operation of urban cities, and part of pipelines are nearing their practical life and need to be replaced or repaired [1]. To locate the buried utilities and revise the existing pipeline map, Ground Penetrating Radar (GPR) has been widely used due to its fast speed and minimal ground intrusion [2].
When detecting underground pipelines in an area, assuming that the underground medium is uniform (or little change), the permittivity of the underground medium could be estimated by fitting a hyperbolic feature generated by the pipeline on the GPR B-scan image [3], provided that the direction of the underground pipeline is perpendicular to the detecting direction, and the depth of the underground pipeline is within the effective detecting depth of the utilized GPR [2]. Considerable efforts have been devoted to extract and fit hyperbolas on Bscan images [4]- [9], In [10], a probabilistic hyperbola mixture model is proposed to estimate buried pipelines from GPR B-scan images, where the Expectation-Maximization (EM) algorithm is upgraded to extract multiple hyperbolas, and a designed fitting algorithm is then adopted to fit these hyperbolas. In [9], Dou et al. proposed the Column-Connection Clustering (C3) algorithm. The algorithm could extract point clusters from B-scan image, which are then identified by a neural-network-based method to locate hyperbolic ones. In our previous work [11], a GPR B-scan image interpreting model has been proposed to extract and fit hyperbolic signatures on B-scan images, estimate the permittivity of the detected area, and obtain the information of buried pipelines. Experiments in different media has validated the effectiveness of the model. By applying these methods, the permittivity of the detected area could be roughly obtained.
After obtaining the permittivity of the detected area through above methods, the ordinate of the GPR B-scan image can be converted from time to depth [3]. When detecting other pipelines in this area, their depth could be inferred directly from the GPR B-scan image [11]. To revise the existing pipeline map, the follow-up work is to further estimate the direction and the radius of each pipeline from GPR data. When the pipeline's direction is not perpendicular to the GPR's detecting direction, the cross section of the pipeline in the GPR's detecting direction is elliptical, and the generated feature is not hyperbolic [2]. In this case, fitting the generated feature by hyperbolic equation would cause errors. In [12], the direction of the underground pipeline is roughly estimated by the statutory records of buried ultities, and detections at different directions are then conducted to derive the specific direction of each pipeline. In [14], a Marching-Cross-Sections (MCS) algorithm is proposed to merge the individual hypothesized pipeline segments. In this algorithm, parallel scan-lines are established, and the Kalman Filter (KF) [15] is extended to connect hypothesized points on the pipe, and infer the direction and position of each pipeline. The abovementioned methods determine the direction of each pipeline by multiple detections in different directions, or connecting two or more detected positions of a pipeline. In addition, when the pipeline's direction is not perpendicular to the GPR's detecting direction, it could be inaccurate to measure the radius of the pipeline by fitting hyperbolic signatures on the B-scan image.
In this paper, the Ellipse Inversion Model is proposed to estimate the direction and radius of pipeline from GPR Bscan image and revise the existing pipeline map. The model consists of two parts: GPR B-scan image processing and Ellipse Iterative Inversion Algorithm (EIIA). The GPR B-scan image is firstly processed with downward-opening point set extracted by extending part of our previous work [11]. As the angle between the GPR's detecting directions and the pipe direction might not be perpendicular, the cross section arXiv:2201.10184v1 [cs.CV] 25 Jan 2022 of the pipeline could be elliptical. In this case, the EIIA iteratively inverts the extracted downward-opening point set to the elliptical cross section of the pipe. By minimizing the sum of the algebraic distances from these points to the inverted ellipse, the most likely pipe direction and radius are determined.
This rest of this paper is organized as follows. The GPR B-scan image processing is discussed in Section II. Section III provides the Ellipse Iterative Inversion Algorithm. Experiments are conducted and analyzed in Section IV. Finally, conclusions are drawn in Section V.
II. GPR B-SCAN IMAGE PROCESSING
In this section, the downward-opening signature on GPR B-scan images generated from the elliptical cross section is analyzed. Then the method to extract point set with downwardopening signatures is introduced.
A. The downward-opening signatures generated by the elliptical cross section line (blue dotted line) of LQ, which indicates the distance d between the GPR and the ellipse, does not pass through the center 1 of the ellipse, as the blue and the green dashed lines do not coincide in Fig. 1. Therefore, the feature produced by the elliptical cross section on the GPR B-Scan image could not be described by the hyperbolic equation [3]. As GPR moves from L to L 0 , the distance between it and the pipeline gradually decreases, and the trajectory when the GPR moves gradually away from L 0 is symmetrical, thus the feature produced by the elliptical cross section on the GPR B-Scan image could be represented by the red downward-opening curve in Fig. 1.
B. Extract point set with downward-opening signatures
In our previous work [11], the GPR B-scan image preprocessing method and Open-Scan Clustering Algorithm (OSCA) have been proposed. In this paper, the preprocessing method firstly eliminates the discrete noises and transforms the original image into binary image. The obtained binary image is then scanned by OSCA from top to bottom with downward-opening point clusters identified and extracted.
Since the top of the pipe is closest to the surface and the generated feature on the image is also the most obvious, the column of the identified point cluster that is closest to the ground are selected with several columns on the left and right sides. The distance between each two adjacent selected columns is set to be 2cm in this paper, and the midpoints of these columns are extracted to form a point set P as
P = {P i (x i , y i ) | 0 ≤ i ≤ n}(1)
which is further inverted to the elliptical cross section of the buried pipe in the next section. Fig. 2 illustrates the process of extracting a downward-opening point set from an original GPR B-scan image.
III. ELLIPSE ITERATIVE INVERSION ALGORITHM
The EIIA aims to revert the downward-opening point set P to the elliptical cross section of the pipe, which consists of two parts: ellipse fitting algorithm, and updating P by converting the coordinates of each point. The procedure of EIIA is presented at the end of this section.
A. Ellipse fitting algorithm
The fitting of a general conic can be approached by minimizing the sum of squared algebraic distances D (P) of the curve [16] to the point set
P = {(x i , y i ) |1 ≤ i ≤ n} as Minimize D (P) = n i=1 F (A, x i ) 2 ,(2)
F (A, x) = A · x = Ax 2 + Bxy + Cy 2 + Dx + Ey + F. (3) where A = [A, B, C, D, E, F ] T , x = x 2 , xy, y 2 , x, y, 1 , and F (A, x) is the "algebraic distance" of a point (x, y) to the conic F (A, x) = 0. In our model, an ellipse could be presented as
(x − x 0 ) 2 a 2 + (y − y 0 ) 2 b 2 = 1.(4)
By transforming Equation (4) into the form of Equation (3), the following equation is obtained
b 2 x 2 + a 2 y 2 − 2b 2 x 0 x − 2a 2 y 0 y + a 2 y 2 0 + b 2 x 2 0 − a 2 b 2 = 0.(5)
Comparing Equations (5) and (3), it could be seen that B = 0. To ensure the fitted conic to be elliptical, 4AC = 1 is utilized as the constraint to limit the fitted curve to be an ellipse. Therefore, fitting the point set P to an ellipse could be formulated as
Minimize D (P) = n i=1 F (A, x i ) 2 s.t. B = 0, 4AC = −1,(6)
which is a convex optimization (CVX) problem and could be solved by the method proposed in [17].
B. Updating the coordinates of each point in P
Given the point set P = {P i (x i , y i ) | 0 ≤ i ≤ n} as Equation (1), the projection of each point P i = (x i , y i ) on the X axis is P i,X , which is the position of the GPR that collects the signal at P i . As Fig. 3(a) shows, P is firstly fitted into Fig. 3. (a) and (b) show the first and the (k+1)th coordinate conversion of P i . The gray ellipse represents the elliptical pipe section fitted before coordinate conversion. an ellipse. Then P i rotates around the point P i,X to P i,1 with the angle of θ 1 , where |P i,X P i | = |P i,X P i,1 | and the distance between P i,X and the fitted ellipse is the shortest, that is, the straight line determined by P i,X and P i,1 is perpendicular to the tangent line at the closest intersection of the line and the fitted ellipse. The coordinates of P i,1 could be obtained by
,1 i P 1 O X Y O X Y 1 k , 1 i k P , i k P , ( ,0) i X i P x i P , ( ,0) i X i P x (a) ,1 i P 1 O X Y O X Y 1 k , 1 i k P , i k P , ( ,0) i X i P x i P , ( ,0) i X i P x (b)x i,1 = x i − y i sin θ 1 , y i,1 = y i cos θ 1 .(7)
where θ 1 could be obtained by calculating the shortest distance from a P i,X outside the ellipse to the ellipse [18]. The rotation is applied on every point in P, and the coordinate of these points are updated, by which P is updated to
P 1 = {P i,1 (x i,1 , y i,1 ) | 0 ≤ i ≤ n}.
The ellipse fitting and rotation are alternately and iteratively performed on P. As Fig. 3(b) shows, after kth iteration, the coordinates of P i,k+1 at the k + 1 iteration are updated as
x i,k+1 = x i − y i sin( k+1 j=1 θ j ), y i,k+1 = y i cos( k+1 j=1 θ j ).(8)
where θ j is the jth rotating angle. The condition for stopping the above iteration of fitting and rotation is that the iterative number reaches the set threshold K, or the sum of algebraic distance D (P) from each point in P to the fitted ellipse and tends to be stable.
C. The Procedure of Ellipse Iterative Inversion Algorithm
Based on the above ellipse fitting algorithm and coordinate updating method, the procedure of the Ellipse Iterative Inversion Algorithm is presented in the following: (1), which are shown in Fig. 4(a). As Fig. 4(b) shows, for an α, there are two possible pipeline's directions as the red and blue line. Therefore, when choosing the detecting direction of GPR, we would choose a direction that is not perpendicular to the pipeline according to the existing pipeline map, such as 4 9 π or 3 8 π. The obtained pipeline's direction that has a smaller angle with the direction on the existing pipeline map is adopted as the modified pipeline's direction, as the blue line in Fig. 4(b).
1) (Input) Point set P = {P i (x i , y i ) | 0 ≤ i ≤ n}
IV. EXPERIMENTAL STUDY
To evaluate the effectiveness of the proposed model, realworld experiments are conducted in this section. After that, the analysis of the experimental results are presented.
A. Experiments on real-world datasets
Two experimental areas are identified, where pipeline excavation and repair work have been carried out within two months, but the latest pipeline maps are revised one year ago. GSSI's SIR-30 GPR with 200MHz antenna is utilized to collect GPR B-scan images. The utilized GPR is presented in Fig. 5. The two selected areas with existing pipeline maps are shown in Fig. 6 and the selected detecting positions and directions are also presented. Due to the limitation of the paper's length, the obtained GPR B-scan images could not be fully demonstrated here, and two images that illustrate the process of the proposed model are shown in Fig. 7 and Fig. 8. When extracting point sets from the obtained GPR B-scan images, the horizontal interval between each two points is 2cm, and for each pipe, 30 points are extracted. The maximum number of iterations K is set to 10, and the threshold of the sum of algebraic distance D t is set to 90cm (the average fitting error of each extracted point is less than 3cm). Specific analysis of the results are presented in the next subsection.
B. Analysis of the experimental results
By applying the proposed model, the direction of buried pipes are obtained, by which the existing pipeline maps are modified as Fig .9. To validate the results, more detections are conducted as Fig. 9, and evacuations are conducted to determine the actual direction and radius of each pipe. The average errors of Ellipse Inversion Model are presented in Table I. As the result shows, the error of direction obtained by the proposed model could be controlled at about 5% in the experimental environments of this paper. This is of great practical value in real-world applications, since only one detection is needed to confirm the general direction of the pipeline, which provides a basis for detection at the next position. In the experimental settings, the maximum number of iterations is set to 10. In the conducted detections, all extracted point sets converge and launch iterations within 10 times. Moreover, when the permittivity of the soil is known, the radius of the pipeline could be obtained through a detection that is not strictly required to be perpendicular to the pipeline. The Restricted Algebraic-Distance-based Fitting algorithm (RADF) [11] and Orthogonal-Distance-based Fitting with Constraints (ODFC) method [9] are also applied to fit the extracted point sets to estimate the radius of the pipe, and the results are shown in Table II. It could be seen that when the GPR's detecting direction is not perpendicular to the pipeline, fitting the generated features by hyperbolic equations would leads to larger errors than the proposed model. V. CONCLUSION
In this paper, a novel method to estimate the direction and radius of the buried pipeline from GPR B-scan image is proposed. The model consists of two parts: GPR B-scan image processing and Ellipse iterative inversion algorithm. The GPR B-scan image is firstly processed with downward-opening point set extracted. Then the obtained point set is iteratively inverted to the cross section of the buried pipe, that is, the elliptical cross section caused by the angle between the GPR detection directions and the pipe direction. By minimizing the sum of the algebraic distances from these points to the inverted ellipse, the most likely pipe direction and radius are determined. Experiments on real-world datasets are conducted, and the existing pipeline map is modified, which validated the effectiveness of the proposed model. In future work, we will study how to map the pipelines of an area where there is no existing pipeline map.
Fig. 1 Fig. 1 .
11illustrates the schematic diagram when the GPR's detecting direction is not perpendicular to the pipeline's direction, where the cross section of the pipeline is elliptical. It could be seen that when the GPR is at L, The case where the cross section of the pipeline in the detecting direction of the GPR is elliptical. The gray ellipse indicates the cross section of the pipe. L is the location of the GPR, and L 0 means the location where the GPR is directly above the pipeline. Q is the closest point on the ellipse to the GPR position L, and the distance from L to the ellipse is |LQ| = d. The red line indicates the downward-opening signature generated by the pipeline on the GPR B-scan image. Q is the point on the downward-opening signature directly below L generated by Q, and |LQ| = |LQ | = d.
Fig. 2 .
2The processing flow of extracting a downward-opening point set from a GPR B-scan image. (a) is the original image. (b) is the preprocessed binary image. (c) is the obtained result after OSCA. (d) shows the extracted downward-opening point set.
extracted from the GPR B-scan image; the maximum number of iterations K; the threshold of the sum of algebraic distance D t . 2) (Ellipse Fitting) Fit P into the ellipse by the proposed ellipse fitting algorithm, along which the sum of algebraic distance D (P) from each point in P to the fitted ellipse is obtained 3) (Coordinate Updating) Updating the coordinates of each point in P by Equation (8). 4) (Check for convergence) If the iterative number reaches K, or D (P) ≤ D t , output the ellipse equation with the smallest D (P). Otherwise, return to Step 2 (Ellipse Fitting). The angle α between the pipeline and the GPR's detecting direction could be calculated as arcsin b a , and b indicates the radius of the pipe. a and b are the parameters of obtained ellipse equation as Equation
Fig. 4 .
4(a) The angle α between the pipe and the GPR's detecting direction could be calculated as arcsin b a , where a and b are the parameters of obtained ellipse equation as Equation (1). (b) The black line with an arrow indicates the direction of GPR, the blue and red lines represent two possible pipeline's directions, both of which have an angle of α with the GPR's detecting direction. The dashed line indicates the direction of the pipe on the pipeline map. The blue line is adopted as the pipeline's direction, since it has smaller angle with the black dashed line compared with the red line.
Fig. 5 .Fig. 6 .
56(a) and (b) are the host and 200MHz antenna of the utilized GSSI SIR-30 GPR. The two selected areas are shown as (a) and (b). The existing pipeline maps are shown in (c) and (d), where the black lines indicate the pipeline. The blue lines with arrows indicate the GPR's detecting direction and position.
Fig. 7 .Fig. 8 .
78The processing flow of the proposed model. (a) is the B-scan image. (b) is the obtained result after preprocessing and OSCA. (c) is the extracted point set and the fitted ellipse at the beginning of the iteration. (d) is the result of the proposed model, where the extracted points are inverted to the elliptical cross section of the pipe. These four pictures indicate another example of the proposed model, and the meaning of each picture is the same as the above one.
Fig. 9 .
9More detections are conducted as the red lines with arrows in (a) and (b). The modified pipeline maps are shown as the red dotted lines.
The authors are with USTC-Birmingham Joint Research Institute in Intelligent Computation and Its Applications, School of Computer Science and Technology, University of Science and Technology of China, Hefei 230027, China (e-mail: [email protected], [email protected], [email protected], [email protected]. Corresponding author: Huanhuan Chen).
TABLE I THE
IERRORS OF THE PROPOSED MODELArea
The average error
The max error
Directions(α) Radius(b) Directions(α) Radius(b)
1
4.03%
5.23%
7.14%
7.22%
2
5.10%
5.90%
8.40%
7.41%
TABLE II THE
IIAVERAGE ERROR OF RADIUS BY RADF, ODF AND EIIA Area The average error of radiusODF
RADF
EIIA
1
15.20% 13.25% 5.23%
2
17.31% 15.59% 5.90%
The center of the ellipse is the midpoint between the two focal points of the ellipse.
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Least-squares fitting of circles and ellipses. W Gander, G Golub, R Strebel, BIT Numerical Mathematics. 344W. Gander, G. Golub, and R. Strebel, "Least-squares fitting of circles and ellipses," BIT Numerical Mathematics, vol. 34, no. 4, pp. 558-578, 1994.
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[] |
[
"Assessing thermal spike model of swift heavy ion-matter interaction via Pd 1−x Ni x /Si interface mixing",
"Assessing thermal spike model of swift heavy ion-matter interaction via Pd 1−x Ni x /Si interface mixing"
] |
[
"Paramita Patra \nDepartment of Physics\nIndian Institute of Technology Kharagpur\n721302KharagpurINDIA\n",
"S A Khan \nInter-University Accelerator Centre\nAruna Asaf Ali MargNew Delhi -110067INDIA\n",
"M Bala \nDepartment of Physics and Astrophysics\nUniversity of Delhi\nNew Delhi -110007INDIA\n",
"D K Avasthi \nAmity Institute of Nanotechnology\nAmity University\nSector 125Noida -201313INDIA\n",
"S K Srivastava \nDepartment of Physics\nIndian Institute of Technology Kharagpur\nKharagpur-721302INDIA\n"
] |
[
"Department of Physics\nIndian Institute of Technology Kharagpur\n721302KharagpurINDIA",
"Inter-University Accelerator Centre\nAruna Asaf Ali MargNew Delhi -110067INDIA",
"Department of Physics and Astrophysics\nUniversity of Delhi\nNew Delhi -110007INDIA",
"Amity Institute of Nanotechnology\nAmity University\nSector 125Noida -201313INDIA",
"Department of Physics\nIndian Institute of Technology Kharagpur\nKharagpur-721302INDIA"
] |
[] |
Thermal spike model (TSM) is presently a widely accepted mechanism of swift heavy ion (SHI) -matter interaction. It provides explanation to various SHI induced effects including mixing across interfaces. The model involves electron-phonon (e-p) coupling to predict the evolution of lattice temperature with time. SHI mixing is considered to be a result of diffusion in transient molten state thus achieved. In this work, we assess this conception primarily via tuning the e-p coupling strength by taking a series Pd1−xNix of a completely solid soluble binary, and then observing 100 MeV Au ion induced mixing across Pd1−xNix/Si interfaces. The extent of mixing has been parametrised by the irradiation induced change ∆σ 2 in variances of Pd and Ni depth profiles derived from X-ray photoelectron spectroscopy. The x-dependence of ∆σ 2 follows a curve that is concave upward with a prominent minimum. Theoretically, e-p coupling strength determined using density functional theory has been used to solve the equations appropriate to TSM, and then an equivalent quantity L 2 proportional to ∆σ 2 has been calculated. L 2 , however, increases monotonically with x without any minimum, bringing out a convincing disparity between experiment and theory. Perhaps some mechanisms more than the TSM plus the transient molten state diffusion are operative, which can not be foreseen at this point of time.
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10.1039/c9cp02052g
|
[
"https://arxiv.org/pdf/1810.04478v2.pdf"
] | 53,125,595 |
1810.04478
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704114d66ee202971fabbaae94c963bf8e616330
|
Assessing thermal spike model of swift heavy ion-matter interaction via Pd 1−x Ni x /Si interface mixing
11 Oct 2018
Paramita Patra
Department of Physics
Indian Institute of Technology Kharagpur
721302KharagpurINDIA
S A Khan
Inter-University Accelerator Centre
Aruna Asaf Ali MargNew Delhi -110067INDIA
M Bala
Department of Physics and Astrophysics
University of Delhi
New Delhi -110007INDIA
D K Avasthi
Amity Institute of Nanotechnology
Amity University
Sector 125Noida -201313INDIA
S K Srivastava
Department of Physics
Indian Institute of Technology Kharagpur
Kharagpur-721302INDIA
Assessing thermal spike model of swift heavy ion-matter interaction via Pd 1−x Ni x /Si interface mixing
11 Oct 2018(Dated: October 16, 2018)numbers: 6180Jh6182Bg6835Fx8280Pv
Thermal spike model (TSM) is presently a widely accepted mechanism of swift heavy ion (SHI) -matter interaction. It provides explanation to various SHI induced effects including mixing across interfaces. The model involves electron-phonon (e-p) coupling to predict the evolution of lattice temperature with time. SHI mixing is considered to be a result of diffusion in transient molten state thus achieved. In this work, we assess this conception primarily via tuning the e-p coupling strength by taking a series Pd1−xNix of a completely solid soluble binary, and then observing 100 MeV Au ion induced mixing across Pd1−xNix/Si interfaces. The extent of mixing has been parametrised by the irradiation induced change ∆σ 2 in variances of Pd and Ni depth profiles derived from X-ray photoelectron spectroscopy. The x-dependence of ∆σ 2 follows a curve that is concave upward with a prominent minimum. Theoretically, e-p coupling strength determined using density functional theory has been used to solve the equations appropriate to TSM, and then an equivalent quantity L 2 proportional to ∆σ 2 has been calculated. L 2 , however, increases monotonically with x without any minimum, bringing out a convincing disparity between experiment and theory. Perhaps some mechanisms more than the TSM plus the transient molten state diffusion are operative, which can not be foreseen at this point of time.
I. INTRODUCTION
While interacting with a solid, swift heavy ions (SHI's) are known to transfer a large amount of energy predominantly in the electronic subsystem of the solid. 1 At first hand, therefore, SHI's are not anticipated to cause any atomic displacements in the solid they interact with. However, irradiation of solids with SHI's has often been found to result in atomic displacements, which may cause, among various effects, mixing across interfaces in layered materials. [2][3][4] There certainly has to be a mechanism by which the electronic energy, deposited by the SHI in the solid, gets transferred to the lattice atoms to cause such atomic displacements. At present, there are essentially two established models, viz., the Coulomb spike model (CSM) and the thermal spike model (TSM), to explain such atomic displacements. [5][6][7][8] According to the CSM, a SHI, while passing through a solid material, ionizes the material in a cylindrical region around its path. The consequent strong collective electrostatic repulsion amongst the positive ions in the ionized zone leads to violent atomic displacements, resulting ultimately into a modified material in a cylindrical so-called ion track. This model, however, lacks applicability in metals, where the high mobility of conduction electrons leads to neutralization of charges much before the Coulomb explosion could occur. The TSM, on the other hand, assumes that the energy deposited initially in the electronic subsystem in a time scale of 10 −15 − 10 −14 s gets subsequently transferred to the lattice subsystem via electron-phonon (e-p) coupling in ∼ 10 −13 − 10 −12 s. This results in a rapid rise in the lattice temperature (up to ∼ 10 4 K) in a cylindrical zone of typically a few nm radius. In certain conditions a molten state is created along the ion track for a ∼ 10 −12 − 10 −11 s duration and is quenched rapidly (at a rate of ∼ 10 14 K/s), freezing the molten modified state of the cylindrical zone. The modified frozen cylindrical zone thus formed is conventionally known as a latent tack. If such melting happens across an interface between two materials, the atoms on the two sides interdiffuse while in the molten state, giving rise to mixing across the interface. 8 The mechanism suggests that the TSM must be applicable in metals, semiconductors and insulators alike. The model has acquired a wide acceptance in course of time.
The TSM is mathematically described by the following two coupled partial differential equations, which are basically the constituents of the so-called two temperature model (TTM) and govern the diffusion of the energy brought in by the ion into the electronic and lattice subsystems: 5,9 C e (T e ) ∂Te ∂t = ∇.(K e (T e )∇T e )−(T e −T l )G(T e )+A e (r, t) and
C l (T l ) ∂T l ∂t = ∇.(K l (T l )∇T l ) + (T e − T l )G(T e ).(1)
Here, C e , C l and K e , K l stand for specific heats and thermal conductivities of the electronic and lattice subsystems, and T e and T l are the electronic and lattice temperatures, respectively. G(T e ) is the electronic temperature dependent e-p coupling strength, and A e (r, t) is the energy density per unit time supplied by the incident ions to the electronic system at time t and at radius r from the ion path in such a way that the integral 2πrA e (r, t) dr dt is equal to the electronic energy loss S e , defined as the energy deposited by the ions in the electronic subsystem per unit length travelled in the solid.
A direct experimental proof of the validity of the TSM has hitherto not been possible because of the extremely short time scales involved. A number of ion fluence dependent SHI induced effects observed experimentally have been used to coarsely derive the latent track radii, which have been found comparable with those calculated roughly from the mathematical equations pertaining to the TSM. 8 These provide a highly indirect and very crude indication of the occurrence of SHI induced processes as hypothesized in the TSM. In all these reports, the free electron theory of metals, which predicts a parabolic density of electron states (eDOS), has been used to determine the electronic part of the thermophysical parameters, viz. C e , K e and G(T e ), 5,10 to be used in the TTM equations. Accordingly, C e is given by 5
C e (T e ) = π 2 g(ǫ F )k 2 B 2ǫ F T e ,(2)
K e is related with C e via electronic thermal diffusivity D e (T e ) by a relation
K e (T e ) = C e (T e )D e (T e ),(3)
and G(T e ) is determined using
G e (T e ) = π 4 [g(ǫ F )k B v s ] 2 18K e (T e ) .(4)
Here, g(ǫ F ) is the eDOS at the Fermi energy ǫ F , and k B is the Boltzmann constant. The phonon contribution to the e-p coupling strength appears in the form of the speed v s of sound in the solid. However, a couple of reports on 120 MeV Au induced mixing in Si/M/Si (M = V, Fe, Co, Mn, Nb) layered structures 11,12 suggested that the relatively more localized d-electrons, which bring in features to the eDOS over the parabolic background, also have influence on the efficiency of SHI mixing. This necessitates the consideration of exact electron density of states g(ǫ) as a function of energy ǫ, computable using f irst− principles density functional theory (DFT), to derive the thermophysical quantities required for the TTM. The following forms of C e and G(T e ), as reported by Lin et al., 13 would be more appropriate in this scenario:
C e (T e ) = ∞ −∞ g(ǫ) ∂f (ǫ, T e ) ∂T e ǫ dǫ(5)
and
G(T e ) = hk B λ ω 2 2g(ǫ F ) ∞ −∞ g 2 (ǫ) − ∂f (ǫ, T e ) ∂ǫ dǫ,(6)
where h is the Planck's constant and f (ǫ, T e ) is the Fermi-Dirac distribution function given by f (ǫ,
T e ) = 1/[1 + exp{(ǫ − ǫ F )/k B T e }].
The electron band mass enhancement factor λ 14,15 and the second moment ω 2 of the phonon spectrum 16 can be obtained from ab initio phonon bandstructure calculations. Lin et al., 13 this way, have calculated electron temperature dependent electronic specific heats and electron phonon coupling strengths for a number of noble and transition metals, and have reported a substantial difference between the free-electron and full eDOS values. In a recent work, 17 we have shown how slight variations of g(ǫ) for different orientations of a thin Bi 2 Te 3 slab result into different C e (T e ) and G(T e ) curves. If the TSM is valid, the use of equations (6) and (7), instead of (3) and (5), in the TTM equations ought to improve the predicting ability of the model for getting an outcome of a SHI-matter interaction experiment. To enact this, conducting a series of SHI-matter interaction experiments, e.g. SHI driven interface mixing across a number of thin film/substrate interfaces, with one kind of substrate and different kinds of thin films of differing C e and G values, would be helpful. As far as the rest of TTM parameters, viz. K e , C l and K l , are concerned, they could, to an appreciable extent, be predictable or obtainable from literature for each film. One possibility could be taking a series of M/Si interfaces with different metals M so that the eDOS and the resultant C e and G values could be calculated for each M. However, the K e , C l and K l values might be arbitrarily different for different M, a case which is obviously undesirable. An appropriate choice for M would be to take thin films A 1−x B x of a complete solid-soluble binary metal system with 0 ≤ x ≤ 1. For such a series, g(ǫ), C e and G would be easily computable for each x. Furthermore, the K e , C l and K l values will have a smooth (to the first approximation linear) variation with x. 18 One such system is Pd 1−x Ni x binary alloy system, which forms a complete solid solution throughout the whole composition range without any change of the crystal structure, as depicted by their equilibrium phase diagram. 19 The present work aims at convincingly assessing the thermal spike model by (i) experimental determination of the x-variation of efficiency of SHI driven mixing of Pd and Ni in Si via 100 MeV Au irradiation of Pd 1−x Ni x /Si system, (ii) computation of x-variation of C e and G using DFT, and then use of the TTM equations to qualitatively estimate the expected x-variation of extent of mixing, and (iii) a comparison between the experimental and computational results. Any slight variation in the computationally predicted efficiency of mixing should be observable also in the experimental results if the TSM is indeed the mechanism of SHI matter interaction. However, neither Pd nor Ni is known to be mixed with Si by SHI's; Pd/Si or Ni/Si mixing has only been reported to be induced by low energy ions, 20,21 where elastic collisions are responsible for the process. In the present work also, Pd/Si or Ni/Si mixing in the Pd 1−x Ni x /Si system has not been observed as an immediate effect of the irradiation; it is rather the Ar + ion sputtering process in the subsequent X-ray photoelectron spectroscopy (XPS) depth profiling that augments the effects of SHI's and leads to observable mixing. Since the sputtering conditions are the same for all the samples, any x-variation of extent of mixing should have indirectly been driven by the SHI irradiation. Such a combination of SHI irradiation and XPS depth profiling to enable one to observe SHI effects in the form of interface mixing has hitherto not been reported.
II. EXPERIMENTAL AND COMPUTATIONAL DETAILS
Prior to depositing the Pd 1−x Ni x (x = 0, 1) alloy thin films on Si substrates, the alloys were first prepared by Ar arc melting. Palladium wire of 99.9% purity and nickel foil of 99.994% purity were melted together to prepare the alloys. The alloyed ingots were flipped and remelted to improve the homogeneity. Subsequently, Pd and Ni metals, and two compositions x = 0.40 and 0.78 of Pd 1−x Ni x alloys were deposited onto pre-cleaned Si substrates by electron beam evaporation. The pressure during deposition was ∼ 1.7 × 10 −7 torr, and the deposition rate ranged from 0.1 to 0.3Å/s. The thicknesses of the four Pd 1−x Ni x (x = 0, 0.40, 0.78 and 1) films were in 25 -40 nm range, as determined from Rutherford backscattering spectra (RBS) and XPS depth profiles to be discussed in the following sections. For all the samples, 1 cm × 1 cm pieces were taken out for irradiation by 100 MeV Au ions each at 1× 10 14 ions/cm 2 fluence using the 15 UD Pelletron accelerator at Inter University Accelerator Centre (IUAC), New Delhi. The electronic (S e ) and nuclear (S n ) energy losses of the ions in Pd are 34.10 keV/nm and 0.09 keV/nm, respectively, as calculated from the SRIM software. 22 The two values for Ni are 32.40 keV/nm and 0.08 keV/nm, respectively. Thus, the electronic energy losses are dominant and hence the condition is relevant to the present study. The pristine and irradiated samples were characterized by X-ray diffraction (XRD), RBS, and XPS depth profiling. XRD patterns for all the samples were recorded using Cu K α radiation from a Philips X'Pert MRD X-ray diffractometer in the 2θ range of 20 • -80 • . The RBS measurements were performed using 2 MeV He + ions from the PARAS facility of IUAC, and were undertaken to determine the thickness and composition of the samples. The XPS depth profiles were carried out using a PHI 5000 Versaprobe II machine under DST-FIST scheme.
The DFT computations were performed using the code Wien2K, 23 which is based on a full-potential linearized augmented plane wave (FLAPW) method. Pd, Ni and all the solid solutions crystallize in an fcc lattice with space group Fm3m. 19 x = 0, 0.25, 0.5, 0.75 and 1 were taken for the computations. Pd and Ni (x = 0, 1) structures were first constructed by taking literature values of the respective lattice constants, and then by optimizing the volume, so that the equilibrium lattice constants correspond to the respective minimum energy configurations. Birch-Murnaghan equation of state 24,25 was used to fit the energy versus volume curves for the optimizations. The crystal structure for x = 0.25 (0.75) was generated by constructing a 1 × 1 × 2 supercell of Pd (Ni) and then replacing one of the four site-split Pd (Ni) atoms with Ni (Pd). For the case of x = 0.5, 2 of the four site-split Pd atoms in a 1 × 1 × 1 supercell were replaced with Ni. All these structures were separately volume-optimized by the same procedure as stated above. For the x = 0 structures, the atomic coordinates were further relaxed to limit the atomic forces to less than 1 mRy/au. The exchange-correlation functionals adopted for the calculations were taken according to the generalized gradient approximation (GGA) as introduced by Perdew, Burke and Ernzerhof. 26 The energy of separation between core and valence states was taken as -0.6 Ry. In the FLAPW method, the potential in a Muffin-tin radius R MT around each atom is taken as atomic-like, and atomic spherical wavefunctions are used as basis functions for the basis set, 27 while in the interstitials, the potential is smooth and plane waves constitute the basis functions. The R MT value for both Pd and Ni was taken as 2.5 a.u. In all the calculations, the wavevectors for plane waves were kept limited to a maximum value k max such that k max R MT = 7.0. The maximum multipolarity l max for the spherical wavefunctions was set at 10. Further, the Fourier expansion of the charge density was limited to G max = 12. The Brillouin zone was sampled using a k-mesh with 72 irreducible k-points.
From the eDOS's as computed from the DFT method described above, C e (T e ) and G(T e ) were calculated using equations (6) and (7) for each composition. For the latter, in addition, λ ω 2 values were taken from literature for pure Pd and Ni, which were then linearly interpolated to get the values for intermediate compositions. For the TTM calculations in the next step, C e (T e ) and G(T e ) values for temperature ranges taken usually in a TSM code 28 were sampled out from the derived C e (T e ) and G(T e ) curves, and used as inputs to the code. The rest of the parameters, viz. K e , C l and K l were interpolated between Pd and Ni values in the same manner as λ ω 2 values were derived for calculating G(T e ).
III. RESULTS AND DISCUSSION
A. XRD X-ray diffraction patterns of all the samples before and after irradiation are shown in Figs. 1 (a) -(d). Peaks corresponding to Pd, Ni and Si elemental solids have been identified from the corresponding Joint Committee on Powder Diffraction Standards (JCPDS) data. The peaks of the compounds, on the other hand, have been identified by a comparison with the powder diffraction pat-terns simulated using PowderCell 29 and by taking the crystal parameters for NiSi, Ni 3 Si 2 and Ni 2 Si from Ref. [30] and for PdSi 31 and Pd 2 Si 32 from SpringerMaterials online database. A look at the XRD patterns of the pristine samples indicates the presence of a combination of elemental and compound phases mentioned above. The elemental peaks are as expected, while the compounds (silicides) must have been formed during the film deposition. Silicide formation while depositing thin metal films on Si is not uncommon. 33 The common effect of irradiation of all the samples has been either a complete removal or a significant suppression or broadening of almost all but the substrate peaks. This must be due to SHI irradiation induced defect creation and amorphization of the films. Such irradiation induced effects are also not uncommon. 34 The takeaway message from the XRD patterns is that (i) the samples contain thin films of a combination of elemental and silicide phases, and (ii) SHI irradiation does produce structural modifications in the samples. Any further interpretation of the XRD patterns would perhaps become an over-interpretation.
B. RBS
RBS spectra of all the pristine and irradiated samples were recorded to determine the compositions and thicknesses of the thin films, and also to examine whether there is any interface mixing occurring as a consequence of the irradiation alone. The spectra and their fits using the code SIMNRA 35 The spectra were fitted using a resolution in the range of 20 -24 keV, which is equivalent to ∼ 12 nm. An insignificant but noticeable decrease in Pd and Ni peak intensities after irradiation for all the samples indicates that Pd/Si and Ni/Si interface mixing might have taken place as a consequence of irradiation. However, the mixing thicknesses must be too small compared to the resolution (∼ 12 nm) to show up in the spectra to any significant extent. In order to investigate whether there is indeed a SHI induced mixing, XPS depth profiles, which could provide a much better spatial resolution, have been performed on all the pristine and irradiated samples. The results are discussed in the following section.
C. XPS depth profile
All the samples have been analyzed using depth profiling XPS paired with 1 keV Ar + ion sputtering to collect Pd3d, Ni2p and Si2p high-resolution spectra. A large number of sputtering cycles were used for the study so that the film/substrate interface is reached in about 35 -40 cycles. A comparison of these many number of sput- ter cycles (to reach the interface) with the depth of the interface (i.e., the film thickness) indicates that the XPS spectra have been recorded at the interval of 0.5 -0.6 nm. This is likely to provide a (minimum) depth resolution of 0.5 -0.6 nm for elemental depth profiling, a spatial depth resolution at least an order of magnitude better than the RBS depth resolution mentioned above, and hence to enable us to observe any small amount of SHI induced interdiffusion (or intermixing) of Pd or Ni into Si and vice-versa. It should be noted here, though, that the Ar + ion sputtering process itself can induce an additional interface mixing. 36 However, since its effects are similar for all the samples, any x-variation of observed mixing can be taken essentially as the SHI irradiation effect, which is augmented further by the sputtering induced mixing equally. We will be comparing the experimentally observed x-variation of mixing with the x-variation of an equivalent quantity estimated using DFT and TTM computations. Figure 3 shows the high-resolution XPS spectra in Pd 3d 5/2 , Ni 2p 3/2 and Si 2p 3/2 regions of the pristine and irradiated Pd 0.60 Ni 0.40 /Si samples for different sputter cycles. This composition has been taken as a representative for all the samples. For both the pristine and irradiated samples, both the Pd and Ni XPS peaks (i) diminish gradually and (ii) shift to higher binding energies (BE's), while approaching the interface. The Si peak also has a gradual rise from the interface, with a less pronounced shift. The gradual changes in the peak heights are indicative of interdiffusion or intermixing of Pd and Ni in Si even in the pristine sample. The intermixed interface seems to broaden on irradiation, as can be seen from the apparently deeper interpenetrations of the peaks in the interfacial region. These observations can be made more quantitative by plotting the XPS peak position at maximum intensity (PPMI) versus sputter cycle (SC). Figure 4 displays the PPMI versus SC plots in Pd 3d 5/2 , Ni 2p 3/2 and Si 2p 3/2 regions of the pristine and irradiated Pd 0.60 Ni 0.40 /Si samples. The symbol sizes are proportional to the corresponding normalized intensities. It is to be noted that the PPMI's as plotted in the figure are not true single peak positions, as each of the Pd 3d 5/2 , Ni 2p 3/2 and Si 2p 3/2 peaks may consist of more than one sub-peaks signifying different elemental or compound phases. A deconvolution of these peaks, which will be shown in the following, would identify the phases present. Coming back to the PPMI versus SC plots, the PPMI for pristine Ni shifts continuously from 852.25 eV at SC 32 to 853.75 at SC 47 with concomitantly diminishing intensity. The trend continues beyond the 47 th cycle with the PPMI saturating at 854 eV. The concomitant PPMI shift and intensity reduction is indicative of an increase in the number of Si atoms surrounding a Ni atom, and hence suggests the occurrence of diffusion of Ni into Si substrate. 37,38 The PPMI profile with SC, thus, can be considered to represent the reverse Ni concentration profile with depth (depth profile). The middle (around SC 40) of the interfacial interdiffused region (SC 32 to SC 47) can be considered as the true film substrate interface. Equating SC 40 to 28.8 nm as obtained from the RBS results, the interdiffsuion region extends from about 23.0 nm to about 33.8 nm, i.e. in a span of about 10.8 nm. This can be considered as the standard deviation σ of the interfacial position, or in other words the interface width, for the present depth profile. This Ni-Si interdiffusion has taken place during the deposition itself, as has also been argued in the XRD subsection above. The effect of 100 MeV Au irradiation, augmented by the Ar + ion sputtering during the depth profiling, has been to broaden the PPMI profile such that the PPMI shifts from 852.25 eV at SC 30 to 853.75 at SC around 55, once again with continuously diminishing intensity. The standard deviation after irradiation, thus, becomes 25 sputter cycles, which is equivalent to 18.0 nm. As the irradiation itself has been shown to cause modifications in the sample as revealed from the XRD patterns, the irradiation can be considered as the primary source of the relative interface broadening or the enhanced interdiffusion. The enhanced interdiffusion, in turn, can be considered to represent the 100 MeV Au (i. e., SHI) in- normalized with respect to the maximum intensity which is maximum of all for each region and sample condition (pristine or irradiated). Tentative elements or compound phases corresponding to different peak positions are also shown for a guidance. The apparent film substrate interface is marked with a dashed line. duced mixing. The PPMI profile of Pd also follows the same pattern, the only difference being that the σ value changes from 11 cycles (9.8 nm) for the pristine sample to 26 cycles (19.4 nm) after irradiation. In the case of Si, the intensity starts decreasing from about the same depth till which Pd and Ni interdiffuse for both pristine and irradiated samples separately. Further, the presence of Si extends, with diminishing intensity, into the film till the depth from where Pd and Ni had started depleting. Although the Si peak shifts are small, and hence likening it to a concentration versus depth profile would not be very convincing, its simultaneous presence with Pd and Ni corroborates the conjecture of interdiffusion of Pd and Ni in Si, which increases as a result of irradiation. The interfacial broadening can be better quantified by determining the atomic fractions of the elements using the areas under the peaks and the elemental sensitivity factors, and then plotting these against the depth, scaled appropriately from the sputter cycle. These depth profiles for all the pristine and irradiated samples will be shown and discussed later. Moving on to the depth profiles, these are shown in Figs. 6 (a) -(d) as variations of atomic fractions (concentrations, m) of Pd, Ni and Si as a function of depth z for the four studied samples. Here, the SC has been converted to depth with the help of RBS analyses as discussed above. The depth profiles have been obtained using the quantification scheme provided in the MultiPak Data Reduction Software available with the PHI 5000 Versaprobe II XPS instrument used for measuring XPS spectra. Absence of sharp interfaces even in the pristine samples, as can be seen from figures, suggests that there is already an interdiffusion in the pristine samples, in line with the earlier arguments. This interdiffusion might have taken place during the film deposition, and may have further been augmented by Ar + ion sputtering during the depth profiling.
The Pd and Ni depth profiles have been roughly fitted with error function, and the fits are shown overlapping with the data. The squared interface width, or variance, σ 2 has then been calculated using 45
σ 2 = ∞ 0 z 2 m ′ (z)dz ∞ 0 m ′ (z)dz ,(7)
where m ′ (z) is the gradient of the fitted interface profile. The change ∆σ 2 in variance on irradiation for a particular depth profile is then given by the difference in the variances after and before irradiation. The value 265±30 nm of ∆σ 2 for Ni in Si for the case of x = 0.40 can be compared with (18 nm) 2 -(10.8 nm) 2 ∼ 208 nm 2 , as obtained using the PPMI versus SC plots discussed above. These values are fairly close to each other. Figure 7 shows the variation of ∆σ 2 , for Pd and Ni interdiffusion in Si, with Ni concentration x. It is noteworthy that ∆σ 2 (x) is neither flat nor monotonic; it rather has a pronounced minimum. The crudest approximation to this variation would be a curve which is concave upward and has a rather deep minimum. In the following sections, we would attempt to examine whether such a variation can be simulated computationally based on the concepts of TSM and DFT.
The strategy for doing the calculations is as follows: (i) Assume the validity of TSM. (ii) Use the TTM equations (1) and (2) to see the evolution of Pd 1−x Ni x lattice temperature, at various radial distances from the ion track, with time. At this stage, use DFT to compute C e and G appropriate to be taken as inputs to the TTM equations (6) and (7). For this, construct Pd 1−x Ni x crystal structures of various compositions close to the experimental ones and calculate electron densities of states g(ǫ). Limit the TSM calculations to bulk Pd 1−x Ni x , as Si is common substrate to all the samples and hence its influence is likely to have no x dependence. (iii) See if the Pd 1−x Ni x lattice melts. Otherwise, record the x variation of an average lattice temperature T av achievable in the lattice at a fixed r, and then derive a quantity which could be correlated with ∆σ 2 . (iv) See whether the variations of this quantity with x follows any concave upward trend. The first step for the calculations is to compute g(ǫ) using DFT. Figure 8 displays the g(ǫ) curves for Pd 1−x Ni x (x = 0, 0.25, 0.5, 0.75, 1) alloys. Without interpreting the densities of states, which themselves are rich of information like g(ǫ F ), the underlying orbital hybridizations, etc. and their variation with x, which in turn may shed light on the electronic and magnetic properties of the Pd 1−x Ni x alloys, 46 we move over to directly using them as an input to the TTM equations. We restrict the refinements of the parameters used in the TTM equations only to the electronic part, which is accessible through DFT. For the rest of the parameters, like even the phonon part λ ω 2 in Eq. (7), we take, as the first approximation, the terminal (x = 0, 1) values from literature, and linearly interpolate these values for the intermediate compositions. The λ ω 2 values taken for x = 0 and 1 are 41 meV 2 and 63 meV 2 , respectively 5,14 . Figure 9 (a) displays the variations of C e and G, respectively, with T e for the computed Ni compositions. Instead of investigating how these parameters vary with T e , finding out how they vary with x for representative electronic temperatures would be more relevant. C e (x) at 5000 K and G(x) at 300 K, the two temperatures being representative ones to be used as inputs in the TSM calculations, are shown in Fig. 9 (b). Thus comes the first milestone of comparison with the experiment: qualitatively, the two parameters increase monotonically with x almost together, do have a shallow concave upward nature, but do not possess a minimum. We now move forward to performing the TSM calculations. Table I enlists the additional input parameters that have been used in the TTM equations (1) and (2) in order to proceed with the TSM calculations. Figures 10 (a) and (b) show the evolution of lattice temperature with time at different radial distances from ion path for the terminal (x = 0 and 1, respectively) systems, taken as representative ones. As can be observed, the lattice does not reach its melting temperature 19 in either case. This is true for all the compositions, not all shown here though. Thus, occurrence of molten state diffusion, the requirement for SHI induced interface mixing according to the TSM, is out of question in the present case. If the TSM is valid, any possible interdiffusion must have taken place in the solid state itself while the lattice was hot, and then would have got enhanced by the sputtering. To investigate this aspect, we analyse the TSM results in the following way.
Since we are interested in finding out only the x variations of the derived quantities, we can look at the evolution of lattice temperature with time, at a common radial distance, say 1 nm, for all x. This is plotted in Fig. 10 (c). As far as diffusion (of Pd and Ni in Si separately) is concerned, we need to derive an appropriate average temperature T av at which the diffusion can be considered to have taken place, and the average duration ∆t for the diffusion. A suitable choice for selecting the two quantities for a particular x would be to take FWHM of the corresponding T l versus t curve at 1 nm radial distance. This sets T av at the temperature where the half maximum of the curve occurs, while ∆t is just the FWHM. Variations of T av and ∆t with x, as derived from Fig. 10 (c), are shown in Fig. 11 (a). Then the Arrhenius equation D = D 0 exp(−E a /k B T av ) for diffusion coefficient D can be used to calculate the square L 2 of the diffusion length as L 2 ∝ D∆t, which itself should be proportional to σ 2 measured experimentally, if the TSM is valid. Here, D 0 is a pre-exponential factor, E a is the activation energy for diffusion, and k B is the Boltzmann's constant. According to the literature, 47 D 0 and E a for Pd diffusion in Si are 3.13×10 −4 cm 2 /s and 1.1 eV, respectively. For Ni diffusion in Si, these values are 6×10 −4 cm 2 /s and 0.67 eV, respectively.
Plots of the diffusion coefficients thus calculated are shown in Fig. 11 (b). It can readily be inferred from the figure that although the two diffusion coefficients follow the same x variation, they are three orders of magnitude apart. Figure 7, however, suggest that if the experimentally observed change in interfacial width on irradiation is due to interdiffusion, both the diffusion coefficients, or at least the effective diffusion coefficients, must be almost equal. So, the experimental observations can not be a result of simple solid state diffusion. Liquid (molten) state diffusion has already been ruled out. In this scenario, only one proposition seems to work: SHI irradiation must have produced an enormous amount of defects, thereby reducing the activation energy for diffusion to such an extent that the effective diffusion coefficients are liquidlike. The liquid state diffusivities are of the same order of magnitude (10 −5 cm 2 /s). 8,48,49 Once the activation energy is reduced, even the Ar + ion sputtering must be able to produce the observed diffusion. So, we ignore the magnitudes of the diffusion coefficients and focus only on their variation with x. We can see, from Fig. 11 (b), that this variation is monotonic, and certainly does not follow a concave upward trend. This can be considered as the second milestone for comparison, where experimental results do not seem to be reproduced by theory. Next, since ∆t changes only by a factor of two from Pd to Ni, L 2 must also follow the same trend. The variation of L 2 with x determined this way has been plotted in Fig. 11 (c) for interdiffusion of both Pd and Ni in Si. As can be seen, this no way resembles a concave upward curve with a minimum. The last thing to check is the influence of xdependence of crystallite size, based on the reports, 50,51 on nanometric size effects on irradiation induced changes. The crystallite sizes of all the samples as determined using the Debye-Scherrer equation 52 applied to the XRD patterns ( Fig. 1) range from 20 nm to 60 nm. This size variation hardly changes the influence of SHI's, as can be seen in the references 50 -51. So, at last, it seems that the TSM, which has so far been successful in explaining the experimental outcomes of SHI -matter interaction, is not able to explain the experimental results of SHI mixing presented in this work. Perhaps some other considerations, apart from the TSM suggested molten state diffusion for SHI mixing, need to be explored.
IV. CONCLUSION
The thermal spike model of SHI -matter interaction has been assessed via its applicability in SHI mixing by comparing the results of (i) 100 MeV Au ion irradiation of Pd 1−x Ni x thin films deposited on Si, and (ii) calculations based on density functional theory and the model. The key concept behind the calculations is the well accepted notion that SHI mixing is a result of diffusion in transient molten state created by the SHI. Although no mixing has been detected in RBS spectra, a considerable amount is observable on Ar + ion sputtering based depth profiling associated with XPS characterizations. This is proposed to be possible due to both the high spatial resolution obtainable from XPS and the sputtering assisted mixing. Since the latter has to be uniform in all the samples, its role can be ignored in the study of x dependence of mixing, which has been the prime focus of the work. The irradiation induced change in the variance of the depth profile determined from the XPS spectra follows an x variation which is concave upward with a pronounced minimum. DFT has been used to derive the x dependences of electronic specific heat and electronphonon coupling strength. These data have been used as an input to the TTM equations appropriate to the thermal spike model to calculate the evolution of lattice temperature with time at various radial distances from the ion path. Finally, a quantity L 2 proportional to the variance is calculated, and its variation with x is derived. If the TSM, along with the conception that mixing is a result of molten state diffusion is valid, L 2 must also vary with x along a curve which is concave upward with minimum. However, L 2 increases monotonically with x without any minimum, and thus is not in accordance with the experimental observations. Even the crystallite size variation, as determined from the XRD patterns, can not account for the observed x-variation of σ 2 . This leads to the conjecture that the combination of the TSM and the molten state diffusion is probably an insufficient description of SHI mixing, and hence the underlying mechanism requires further considerations.
are shown in Figs. 2(a) -(d). According to the fits, the samples have configurations Pd (19.2 nm)/Si, Pd 0.60 Ni 0.40 (28.8 nm)/Si, Pd 0.22 Ni 0.78 (18.8 nm)/Si and Ni (23.0 nm)/Si.
FIG. 1 :
1XRD spectra of (a) Pd/Si (b) Pd 0.22 Ni 0.78 /Si (c) Pd 0.60 Ni 0.40 /Si and (d) Ni/Si. The steps at ∼ 67 • in (b) and (d) are experimental artefacts, and do not affect the interpretations made.
FIG. 2 :
2RBS spectra of (a) Pd/Si (b) Pd 0.22 Ni 0.78 /Si (c) Pd 0.60 Ni 0.40 /Si and (d) Ni/Si
FIG. 3 :
3High-resolution XPS spectra in Pd 3d 5/2 , Ni 2p 3/2 and Si 2p 3/2 regions of Pd 0.60 Ni 0.40 /Si pristine (a) and irradiated (b) samples for different sputter cycles. The curves corresponding to the surface and interface are indicated. For clarity, the spectra of alternate cycles only have been plotted.
FIG. 4 :
4Binding energy positions of maximum intensity versus sputter cycle in Pd 3d 5/2 , Ni 2p 3/2 and Si 2p 3/2 regions of Pd 0.60 Ni 0.40 /Si pristine and irradiated samples. The symbol sizes and colours signify the intensity of the maximum for a particular sputter cycle
FIG. 5 :
5The background corrected high-resolution XPS spectra in Si 2p 3/2 , Pd 3d 5/2 , O 1s and Ni 2p 3/2 regions (shown in top panel) and their fits for the Pd 0.60 Ni 0.40 /Si pristine and irradiated samples. O1s and Pd 3p 3/2 spectra overlap 44 . Tentative assignments of different peak positions (bottom panel) are also shown.It would be worthwhile in the meantime to examine the XPS spectra of these samples at the interface (SC 40) to see the interfacial phases present before and after irradiation. The background corrected high-resolution XPS spectra in Si 2p 3/2 , Pd 3d 5/2 , O 1s and Ni 2p 3/2 regions and their fits for the Pd 0.60 Ni 0.40 /Si pristine and irradiated samples are shown inFig. 5. Tentative assignments of different peak positions to pure 39 Pd and Ni and their silicides with random compositions (e.g., Pd x Si 1−x 40-42 ) or in compound form, like Ni 2 Si 42,43 , and to adsorbed oxygen, 44 are also shown. In brief, both the pristine and irradiated Pd 0.60 Ni 0.40 /Si samples contain elemental and silicide phases of Pd and Ni at the interface with slightly different amounts before and after irradiation. The irradiated sample has an additional Pd x Si 1−x peak. These conjectures are in agreement with the XRD observations, as discussed earlier.
FIG. 6 :FIG. 7 :
67XPS depth profile of (a) Pd/Si (b) Pd 0.22 Ni 0.78 /Si (c) Pd 0.60 Ni 0.40 /Si and (d) Ni/Si. Overlapped with Pd and Ni profiles are their error function fits. Variation of ∆σ 2 with x for Pd and Ni interdiffusion in Si. The dotted line is a guide to the eyes. The concave upward curve shown as continuous line is the crudest approximation for this variation.D. g(ǫ), Ce and G
FIG. 8 :
8The electronic densities of states of the Pd 1−x Ni x alloy system.E. TSM calculations
FIG. 9 :
9Variations of (a) specific heat (dotted curves) and e-p coupling strength (continuous lines) with electron temperature, and (b) of both with Ni composition. The lines in (b) are a guide to the eyes.
Kabiraj and S. R. Abhilash of the Target Lab of IUAC, New Delhi are acknowledged for their help in sample preparation. The help provided by S. Ojha and G. R. Umapathy of PARAS facility of IUAC, New Delhi in RBS measurements is also acknowledged.
FIG. 11 :
11(a) Variations of average lattice temperature T av at 1 nm radial distance from ion path, and of the duration ∆t (inset) of this temperature with x. (b) Pd and Ni diffusion coefficients at these temperatures as a function of x. (c) L 2 versus x curves for Pd and Ni. Lines are a guide to the eyes.
TABLE I :
ILattice specific heat (C l ), electronic thermal conductivity (K e ), lattice thermal conductivity (K) and electronic energy loss (S e ), used as inputs to the TSM calculations.Parameter
unit
x = 0
x = 1
Ref.
(T-range)
C l
J/g-K 0.12 -0.30 0.27 -0.60
[5]
(90 -10000 K)
Ke
W/cm-K
0.05
0.05
[5]
(all T)
K l
W/cm-K 2.33 -0.78 11.33 -0.50 [5]
(50 -10000 K)
Se
keV/Å
3.41
3.24
[28]
(T-independent)
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|
[] |
[
"Non-zero θ 13 and CP violation in a model with A 4 flavor symmetry",
"Non-zero θ 13 and CP violation in a model with A 4 flavor symmetry"
] |
[
"Y H Ahn [email protected]†email:[email protected] \nSchool of Physics\nInstitute of Convergence Fundamental Studies & School of Liberal Arts\nKIAS\n130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea\n",
"Kyu Sin \nSchool of Physics\nInstitute of Convergence Fundamental Studies & School of Liberal Arts\nKIAS\n130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea\n",
"Kang \nSchool of Physics\nInstitute of Convergence Fundamental Studies & School of Liberal Arts\nKIAS\n130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea\n"
] |
[
"School of Physics\nInstitute of Convergence Fundamental Studies & School of Liberal Arts\nKIAS\n130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea",
"School of Physics\nInstitute of Convergence Fundamental Studies & School of Liberal Arts\nKIAS\n130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea",
"School of Physics\nInstitute of Convergence Fundamental Studies & School of Liberal Arts\nKIAS\n130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea"
] |
[] |
Motivated by recent observations of non-zero θ 13 from the Daya Bay and RENO experiments, we propose a renormalizable neutrino model with A 4 discrete symmetry accounting for deviations from the tri-bimaximal mixing pattern of neutrino mixing matrix indicated by neutrino oscillation data.In the model, the light neutrino masses can be generated by radiative corrections, and we show how the light neutrino mass matrix can be diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix whose entries are determined by the current neutrino data including the Daya Bay result. We show that the origin of the deviations from the TBM mixing is non-degeneracy of the neutrino Yukawa coupling constants, and unremovable CP phases in the neutrino Yukawa matrix give rise to both low energy CP violation measurable from neutrino oscillation and high energy CP violation.
|
10.1103/physrevd.86.093003
|
[
"https://arxiv.org/pdf/1203.4185v3.pdf"
] | 119,219,305 |
1203.4185
|
f5477c721b9bda0b7f04a4b950053b3ed5a45d2e
|
Non-zero θ 13 and CP violation in a model with A 4 flavor symmetry
23 Oct 2012
Y H Ahn [email protected]†email:[email protected]
School of Physics
Institute of Convergence Fundamental Studies & School of Liberal Arts
KIAS
130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea
Kyu Sin
School of Physics
Institute of Convergence Fundamental Studies & School of Liberal Arts
KIAS
130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea
Kang
School of Physics
Institute of Convergence Fundamental Studies & School of Liberal Arts
KIAS
130-722, 139-743Seoul, Seoul-Tech, SeoulKorea, Korea
Non-zero θ 13 and CP violation in a model with A 4 flavor symmetry
23 Oct 2012(Dated: May 5, 2014)PACS numbers: *
Motivated by recent observations of non-zero θ 13 from the Daya Bay and RENO experiments, we propose a renormalizable neutrino model with A 4 discrete symmetry accounting for deviations from the tri-bimaximal mixing pattern of neutrino mixing matrix indicated by neutrino oscillation data.In the model, the light neutrino masses can be generated by radiative corrections, and we show how the light neutrino mass matrix can be diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix whose entries are determined by the current neutrino data including the Daya Bay result. We show that the origin of the deviations from the TBM mixing is non-degeneracy of the neutrino Yukawa coupling constants, and unremovable CP phases in the neutrino Yukawa matrix give rise to both low energy CP violation measurable from neutrino oscillation and high energy CP violation.
I. INTRODUCTION
Very recently, the Daya Bay Collaborations [1] announced 5.2σ observation of the nonzero mixing angle θ 13 with the result given by sin 2 2θ 13 = 0.092 ± 0.016(stat) ± 0.005(syst) 1 . This result is in good agreement with the previous data from the T2K, MINOS and Double Chooz Collaborations [3], and the Daya Bay and RENO progresses have led us to accomplish the measurements of three mixing angles, θ 12 , θ 23 and θ 13 from three kinds of neutrino oscillation experiments. A combined analysis of the data coming from T2K, MINOS, Double Chooz and Daya Bay experiments shows [4] that sin 2 2θ 13 = 0.089 ± 0.016(0.047) ,
at 1σ (3σ) levels and that the hypothesis θ 13 = 0 is now rejected at a significance level higher than 6σ. In addition to the measurement of the mixing angle θ 13 , the global fit of the neutrino mixing angles and mass-squared differences at 1σ (3σ) levels are given by [5] θ 12 = 34.0
in which NH and IH stand for normal hierarchical neutrino spectrum and inverted one, respectively. The data in Eqs. (2,3) strongly support that the tri-bimaximal (TBM) mixing pattern of the lepton mixing matrix [6] should be modified. There have been theoretical attempts to explain what cause the three mixing angles to be deviated from their TBM values [7].
Motivated by the measurements of θ 13 from the Daya Bay and RENO experiments, we propose in this paper a renormalizable model with A 4 discrete symmetry which gives rise to deviations from the TBM mixing indicated by the current neutrino data. In addition to the leptons and the Higgs scalar of the standard model (SM), the model we porpose contains three right handed heavy Majorana neutrinos and several scalar fields which are electroweak singlets required to construct desirable forms of the letponic mass matrices.
Although we introduce electroweak singlet heavy Majorana neutrinos, the usual seesaw mechanism does not operate because the scalar field involved in neutrino Yukawa terms can not get vacuum expectation value (VEV). However, as will be shown later, the light neutrino masses can be generated through loop corrections which is a kind of the so-called radiative seesaw mechanism [8]. In the paper, we will show how the light neutrino mass matrix generated through loop corrections can be diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix whose entries are determined by the current neutrino data.
The origin of the deviations from TBM mixing in our model is non-degeneracy of the neutrino Yukawa coupling constants among three generations, which is different from other attempts to explain the deviations from the TBM mixing [7].
Since non-trivial Dirac CP phase can exist only when the mixing angle θ 13 has non-zero value in the standard parametrization of the leptonic mixing matrix, the observations of non-zero θ 13 from the Daya Bay and RENO experiments shed light on the search for CP violation in the leptonic sector. We will show that unremovable CP phases in the neutrino Yukawa matrix are the origin of the low energy CP violation measurable from neutrino oscillation as well as high energy CP violation. Therefore, we can anticipate that there may exist some correlation between low energy CP violation and high energy CP violation.
II. A MODEL WITH A 4 SYMMETRY
The model we consider is the standard model (SM), extended to contain three righthanded SU(2) L -singlet Majorana neutrinos, N R . In addition to the usual SM Higgs doublet Φ, we newly introduce two scalar fields, χ and η, that are singlet and doublet under SU(2) L , respectively:
Φ = ϕ + , ϕ 0 T , χ , η = η + , η 0 T .(4)
In order to account for the present neutrino oscillation data, we impose A 4 flavor symmetry for leptons and scalars. In addition to A 4 symmetry, we introduce extra auxiliary Z 2 symmetry so that a radiative seesaw at around TeV scale should operate. Here we recall that A 4 is the symmetry group of the tetrahedron and the finite groups of the even permutation of four objects [9]. The group A 4 has two generators S and T , satisfying the relation
S 2 = T 3 = (ST ) 3 = 1.
In the three-dimensional unitary representation, S and T are given
by S = 1 0 0 0 −1 0 0 0 −1 , T = 0 1 0 0 0 1 1 0 0 .(5)
The group A 4 has four irreducible representations, one triplet 3 and three singlets 1,
1 ′ , 1 ′′ with the multiplication rules 3 ⊗ 3 = 3 s ⊕ 3 a ⊕ 1 ⊕ 1 ′ ⊕ 1 ′′ , 1 ′ ⊗ 1 ′′ = 1, 1 ′ ⊗ 1 ′ = 1 ′′ and 1 ′′ ⊗ 1 ′′ = 1 ′ . Let's denote two A 4 triplets as (a 1 , a 2 , a 3 ) and (b 1 , b 2 , b 3 ), then we have (a ⊗ b) 3s = (a 2 b 3 + a 3 b 2 , a 3 b 1 + a 1 b 3 , a 1 b 2 + a 2 b 1 ) , (a ⊗ b) 3a = (a 2 b 3 − a 3 b 2 , a 3 b 1 − a 1 b 3 , a 1 b 2 − a 2 b 1 ) , (a ⊗ b) 1 = a 1 b 1 + a 2 b 2 + a 3 b 3 , (a ⊗ b) 1 ′ = a 1 b 1 + ωa 2 b 2 + ω 2 a 3 b 3 , (a ⊗ b) 1 ′′ = a 1 b 1 + ω 2 a 2 b 2 + ωa 3 b 3 ,(6)
where ω = e i2π/3 is a complex cubic-root of unity. The representations of the field content of the model under SU(2) × U(1) × A 4 × Z 2 are summarized in Table-I : TABLE I: Representations of the fields
under A 4 × Z 2 and SU (2) L × U (1) Y . Field L e , L µ , L τ l R , l ′ R , l ′′ R N R χ Φ η A 4 1, 1 ′ , 1 ′′ 1, 1 ′ , 1 ′′ 3 3 1 3 Z 2 + + − + + − SU (2) L × U (1) Y (2, −1) (1, −2) (1, 0) (1, 0) (2, 1) (2, 1)
With the field content and the symmetries specified in Table I, the relevant renormalizable Lagrangian for the neutrino and charged lepton sectors invariant under SU(2)×U(1)×A 4 ×Z 2 is given by
− L Yuk = y ν 1L e (ηN R ) 1 + y ν 2L µ (ηN R ) 1 ′ + y ν 3L τ (ηN R ) 1 ′′ + M 2 (N c R N R ) 1 + λ χ 2 (N c R N R ) 3s χ + y eLe Φ l R + y µLµ Φ l ′ R + y τLτ Φ l ′′ R + h.c ,(7)
whereη ≡ iτ 2 η * with the Pauli matrix τ 2 . Here, L e,ν,τ and l While the standard Higgs scalar Φ 0 gets a VEV v = (2 √ 2G F ) −1/2 = 174 GeV, the neutral component of scalar doublet η would not acquire a nontrivial VEV because η has odd parity of Z 2 as assigned in Table I and the auxiliary Z 2 symmetry is exactly conserved even after electroweak symmetry breaking ;
η 0 i = 0 , (i = 1, 2, 3) , Φ 0 = υ = 0 .(8)
Therefore, the neutral component of scalar doublet η can be a good dark matter candidate, and the usual seesaw mechanism does not operate because the neutrino Yukawa interactions can not generate masses. However, the light Majorana neutrino mass matrix can be generated radiatively through one-loop with the help of the Yukawa interactionL L N Rη in the Lagrangian, which will be discussed more in detail in Sec.III. In our model, the A 4 flavor symmetry is spontaneously broken by A 4 triplet scalars χ. From the condition of the global minima of the scalar potential, we can obtain a vacuum alignment of the fields χ relevant to achieve our goal.
The most general renormalizable scalar potential of Φ, η and χ invariant under SU(2) L ×
U(1) Y × A 4 × Z 2 is given as V = V (η) + V (Φ) + V (χ) + V (ηΦ) + V (ηχ) + V (Φχ)(9)
where
V (η) = µ 2 η (η † η) 1 + λ η 1 (η † η) 1 (η † η) 1 + λ η 2 (η † η) 1 ′ (η † η) 1 ′′ + λ η 3 (η † η) 3s (η † η) 3s + λ η 4 (η † η) 3a (η † η) 3a + λ η 5 (η † η) 3s (η † η) 3a + h.c. , V (Φ) = µ 2 Φ (Φ † Φ) + λ Φ (Φ † Φ) 2 , V (χ) = µ 2 χ (χχ) 1 + λ χ 1 (χχ) 1 (χχ) 1 + λ χ 2 (χχ) 1 ′ (χχ) 1 ′′ + λ χ 3 (χχ) 3s (χχ) 3s + λ χ 4 (χχ) 3a (χχ) 3a + λ χ 5 (χχ) 3s (χχ) 3a + ξ χ 1 χ(χχ) 3s + ξ χ 2 χ(χχ) 3a , V (ηΦ) = λ ηΦ 1 (η † η) 1 (Φ † Φ) + λ ηΦ 2 (η † Φ)(Φ † η) + λ ηΦ 3 (η † Φ)(η † Φ) + h.c V (ηχ) = λ ηχ 1 (η † η) 1 (χχ) 1 + λ ηχ 2 (η † η) 1 ′ (χχ) 1 ′′ + λ ηχ * 2 (η † η) 1 ′′ (χχ) 1 ′ + λ ηχ 3 (η † η) 3s (χχ) 3s (χχ) 3s + λ ηχ 4 (η † η) 3s (χχ) 3a + λ ηχ 5 (η † η) 3a (χχ) 3a + ξ ηχ 1 (η † η) 3s χ V (Φχ) = λ Φχ (Φ † Φ)(χχ) 1 .(10)
Here, µ η , µ Φ , µ χ , ξ χ 1 , ξ χ 2 , ξ ηχ 1 and ξ ηχ 2 have a mass dimension, whereas λ η 1,...,5 , λ Φ , λ χ 1,...,5 , λ ηΦ 1,...,3 , λ ηχ 1,...,6 and λ Φχ are all dimensionless. In V (ηΦ), the usual mixing term Φ † η and Φ † ηχ are forbidden by the A 4 × Z 2 symmetry. The vacuum configuration is obtained by vanishing of the derivative of V with respect to each component of the scalar fields Φ and χ i but with η i = 0 (i = 1, 2, 3) as follows;
∂V ∂χ 1 <χ i >=vχ i = 2v χ 1 v 2 Φ λ Φχ + µ 2 χ + (2λ χ 1 − λ χ 2 + 4λ χ 3 )(v 2 χ 2 + v 2 χ 3 ) + 2(λ χ 1 + λ χ 2 )v 2 χ 1 + 6ξ χ 1 v χ 2 v χ 3 = 0 , ∂V ∂χ 2 <χ i >=vχ i = 2v χ 2 v 2 Φ λ Φχ + µ 2 χ + (2λ χ 1 − λ χ 2 + 4λ χ 3 )(v 2 χ 1 + v 2 χ 3 ) + 2(λ χ 1 + λ χ 2 )v 2 χ 2 + 6ξ χ 1 v χ 1 v χ 3 = 0 , ∂V ∂χ 3 <χ i >=vχ i = 2v χ 3 v 2 Φ λ Φχ + µ 2 χ + (2λ χ 1 − λ χ 2 + 4λ χ 3 )(v 2 χ 1 + v 2 χ 2 ) + 2(λ χ 1 + λ χ 2 )v 2 χ 3 + 6ξ χ 1 v χ 1 v χ 2 = 0 .(11)
From those equations, we can get 2
χ 1 ≡ υ χ = −µ 2 χ − v 2 Φ λ Φχ 2(λ χ 1 + λ χ 2 ) = 0 , χ 2 = χ 3 = 0 ,(12)
2 There exists another nontrivial solution
χ = v χ (1, 1, 1) with v χ = −3ξ χ 1 ± 9ξ χ2 1 −8(µ 2 χ +v 2 Φ λ Φχ )(3λ χ 1 +4λ χ 3 ) 4(3λ χ 1 +4λ χ 3 )
. But, it is not desirable for our purpose.
where υ χ is real. Requiring vanishing of the derivative of V with respect to Φ,
∂V ∂ϕ 0 <ϕ 0 >=v Φ = 2v Φ 2v 2 Φ λ Φ + µ 2 Φ + λ Φχ (v 2 χ1 + v 2 χ2 + v 2 χ3 ) = 0 ,(13)
and inserting the results given by Eq. (12), we obtain electroweak VEV,
v ≡ v Φ = −µ 2 Φ − v 2 χ λ Φχ 2λ Φ .(14)
In our scenario, we assume that v χ is larger than v Φ .
After the breaking of the flavor and electroweak symmetries, the vacuum alignment in Eq. (12) leads to the right-handed Majorana neutrino mass matrix expressed as
M R = M 1 0 0 0 1 κe iξ 0 κe iξ 1 ,(15)
where κ = |λ s χ υ χ /M|. In addition, the charged lepton sector has a diagonal mass matrix m ℓ = v Diag.(y e , y µ , y τ ). We note that the vacuum alignment in Eq. (12) implies that the A 4 symmetry is spontaneously broken to its residual symmetry Z 2 in the heavy neutrino sector since (1, 0, 0) is invariant under the generator S in Eq. (5).
After the scalar fields get VEVs, the Yukawa interactions in Eq. (7) and the charged gauge interactions in a weak eigenstate basis can be written as
− L = 1 2 N c R M R N R + ℓ L m ℓ ℓ R + ν L Y νη N R + g √ 2 W − µ ℓ L γ µ ν L + h.c ,(16)
whereη = Diag.(η 1 ,η 2 ,η 3 ). One can easily see that the neutrino Yukawa matrix is given as follows;
Y ν = √ 3 y ν 1 0 0 0 y ν 2 0 0 0 y ν 3 U † ω , with U ω = 1 √ 3 1 1 1 1 ω 2 ω 1 ω ω 2 .(17)
For our convenience, let us take the basis where heavy Majorana neutrino and charged lepton mass matrices are diagonal. Rotating the basis
N R → U † R N R ,(18)
the right-handed Majorana mass matrix M R becomes real and diagonal by a unitary matrix
U R ,M R = U T R M R U R = MDiag.(a, 1, b) ,(19)
where a = 1 + κ 2 + 2κ cos ξ and b = 1 + κ 2 − 2κ cos ξ with real and positive mass
eigenvalues, M 1 = Ma, M 2 = M and M 3 = Mb. The unitary matrix U R diagonalizing M R
given in Eq. (15) is
U R = 1 √ 2 0 √ 2 0 1 0 −1 1 0 1 e i ψ 1 2 0 0 0 1 0 0 0 e i ψ 2 2 ,(20)
with the phases
ψ 1 = tan −1 −κ sin ξ 1 + κ cos ξ and ψ 2 = tan −1 κ sin ξ 1 − κ cos ξ .(21)
The phases ψ 1,2 go to 0 or π as the magnitude of κ defined in Eq. (15) decreases. Due to the rotation (18), the neutrino Yukawa matrix Y ν gets modified tõ
Y ν = Y ν U R , = P † ν Diag.(|y ν 1 |, |y ν 2 |, |y ν 3 |)U † ω U R .(22)
Absorbing P ν into the neutrino field ν L and then transforming ℓ L → P * ν ℓ L , ℓ R → P * ν ℓ R , we can make P ν disappeared inỸ ν as well as the Lagrangian Eq. (7). Then, the neutrino fields ν L in the weak basis are simply transformed into the mass basis by the lepton mixing matrix, U PMNS , so-called PMNS mixing matrix.
The lepton mixing matrix U PMNS can be written in terms of three mixing angles and three CP -odd phases (one for the Dirac neutrino and two for the Majorana neutrino) as follows [10]
U PMNS = c 13 c 12 c 13 s 12 s 13 e −iδ CP Q ν ,(23)
where s ij ≡ sin θ ij and c ij ≡ cos θ ij , and Q ν = Diag.(e −iϕ 1 /2 , e −iϕ 2 /2 , 1). Here, we notice that the origin of the CP phases in U PMNS is the CP phases ψ 1 , ψ 2 (or ξ) originally coming from M R as can be seen by comparing Eqs. (15)(16)(17)(18)(19)(20)(21)(22). Thus, we expect that there can be some correlation between low energy CP violation measurable from neutrino oscillations and high energy CP violation responsible for leptogenesis in the neutrino sector.
III. NEUTRINO MASSES AND MIXING ANGLES
We now proceed to investigate the low energy neutrino observables. Due to the auxiliary Z 2 symmetry, the usual seesaw mechanism does not operate any more, and thus light neutrino masses can not be generated at tree level. However, similar to the scenario presented in [8], the light neutrino mass matrix can be generated through one loop diagram drawn in
(m ν ) αβ = i ∆m 2 η i 16π 2 (Ỹ ν ) αi (Ỹ ν ) βi M i f M 2 ī m 2 η i ,(24)
where
f (z i ) = z i 1 − z i 1 + z i ln z i 1 − z i , ∆m 2 η i ≡ |m 2 R i − m 2 I i | = 4v 2 λ Φη 3 ,(25)with z i = M 2 i /m 2 η i . The explicit expressions form 2 η i are presented in the Appendix. Here, m R i (m I i ) is the mass of the field component η 0 R i (η 0 I i ) and m 2 R i (I i ) =m 2 η i ± ∆m 2 η i /2
where m i (i = 1, 2, 3) are the light neutrino mass eigenvalues, y 1(2) = y ν 1(2) /y ν 3 , and
A = f (z 2 ) + 2e iψ 1 f (z 1 ) a , B = f (z 2 ) − e iψ 1 f (z 1 ) a , D = f (z 2 ) + e iψ 1 f (z 1 ) 2a − 3e iψ 2 f (z 3 ) 2b , m 0 = v 2 |y ν 3 | 2 λ Φη 3 4π 2 M , G = f (z 2 ) + e iψ 1 f (z 1 ) 2a + 3e iψ 2 f (z 3 ) 2b .(27)
It is worthwhile to notice that in the limit of y 2 → 1 the above mass matrix in Eq. (26) goes to µ − τ symmetry leading to θ 13 = 0 and θ 23 = −π/4. Moreover, in the limit of y 1 , y 2 → 1 the above mass matrix gives TBM angles and mass eigenvalues, respectively,
θ 13 = 0, θ 23 = − π 4 , θ 12 = sin −1 1 √ 3 , m 1 = 3m 0 f (z 1 ) a e iψ 1 , m 2 = 3m 0 f (z 2 ) , m 3 = 3m 0 f (z 3 ) b e i(ψ 2 +π) ,(28)
indicating that mass eigenvalues are divorced from mixing angles. However, recent neutrino data including the observations of non-zero θ 13 requires deviations of y 1,2 from unit. Now, let us show how deviations of y 1,2 from unit are responsible for non-vanishing θ 13 , and they are related with neutrino mass eigenvalues. To separately obtain real values for the neutrino mixing angles and masses, we diagonalize the hermitian matrix m ν m † ν with m ν given by Eq. (26),
m ν m † ν = m 2 0 Ã y 2 1 y 1 y 2 P −Q 2 − i 3(R+S) 2 y 1 P +Q 2 − i 3(R−S) 2 y 1 y 2 P −Q 2 + i 3(R+S) 2 y 2 2F +G−K 4 y 2 F −G 4 − i 3D 2 y 1 P +Q 2 + i 3(R−S) 2 y 2 F −G 4 + i 3D 2 F +G−K 4 = U PMNS Diag.(m 2 1 , m 2 2 , m 2 3 ) U † PMNS ,(29)
whereÃ,D,F ,G,K, P, Q, R and S are real :
A = (1 + 4y 2 1 + y 2 2 ) f 2 (z 1 ) a 2 + (1 + y 2 1 + y 2 2 )f 2 (z 2 ) − 2(1 − 2y 2 1 + y 2 2 ) f (z 1 )f (z 2 ) a cos ψ 1 , F = (1 + 4y 2 1 + y 2 2 ) f 2 (z 1 ) a 2 + 4(1 + y 2 1 + y 2 2 )f 2 (z 2 ) + 4(1 − 2y 2 1 + y 2 2 ) f (z 1 )f (z 2 ) a cos ψ 1 , K = 6(1 − y 2 2 ) f (z 3 ) b f (z 1 ) a cos ψ 12 + 2f (z 2 ) cos ψ 2 , G = 9(1 + y 2 2 ) f 2 (z 3 ) b 2 , D = (1 − y 2 2 ) f (z 3 ) b f (z 1 ) a sin ψ 12 − 2f (z 2 ) sin ψ 2 , P = −(1 + 4y 2 1 + y 2 2 ) f 2 (z 1 ) a 2 + 2(1 + y 2 1 + y 2 2 )f 2 (z 2 ) − (1 − 2y 2 1 + y 2 2 ) f (z 1 )f (z 2 ) a cos ψ 1 , Q = 3(1 − y 2 2 ) f (z 3 ) b f (z 1 ) a cos ψ 12 − f (z 2 ) cos ψ 2 , R = (1 − 2y 2 1 + y 2 2 ) f (z 1 )f (z 2 ) a sin ψ 1 , S = (1 − y 2 2 ) f (z 3 ) b f (z 1 ) a sin ψ 12 + f (z 2 ) sin ψ 2 ,(30)
with ψ ij ≡ ψ i − ψ j . To see how neutrino mass matrix given by Eq. (26)
θ 23 = − π 4 + ǫ 1 , θ 13 = ǫ 2 , θ 12 = sin −1 1 √ 3 + ǫ 3 .(31)
Then, the PMNS mixing matrix keeping unitarity up to order of ǫ i can be written as
U PMNS = √ 2−ǫ 3 √ 3 1+ǫ 3 √ 2 √ 3 ǫ 2 e −iδ CP − 1+ǫ 1 +ǫ 3 √ 2 √ 6 + ǫ 2 e iδ CP √ 3 √ 2+ǫ 1 √ 2−ǫ 3 √ 6 + ǫ 2 e iδ CP √ 6 −1+ǫ 1 √ 2 −1+ǫ 1 +ǫ 3 √ 2 √ 6 − ǫ 2 √ 3 e iδ CP √ 2−ǫ 3 − √ 2ǫ 1 √ 6 − ǫ 2 √ 6 e iδ CP 1+ǫ 1 √ 2 Q ν + O(ǫ 2 i ) .(32)
The small deviation ǫ 1 from maximality of atmospheric mixing angle is expressed in terms of the parameters in Eq. (30) as
tan ǫ 1 = R(1 + y 2 ) − S(1 − y 2 ) R(1 − y 2 ) − S(1 + y 2 ) .(33)
The reactor angle θ 13 and Dirac-CP phase δ CP are expressed as tan 2θ 13 ≃ y 1 |Ω| √ 2(Θ −Ã) ,
tan δ CP = 3 (R − S) 2 + y 2 2 (R + S) 2 (P + Q)(R − S) − y 2 2 (P − Q)(R + S) ,(34)
where Ω = (1 − y 2 )P + (1 + y 2 )Q + ǫ
1 {(1 + y 2 )P + (1 − y 2 )Q − 3i R(1 − y 2 ) − S(1 + y 2 ) + ǫ 1 (R(1 + y 2 ) − S(1 − y 2 )) , Θ = 1 4 (F +G −K) 1 + y 2 2 2 + ǫ 1 (1 − y 2 2 ) − y 2 (F −G) .(35)
In the limit of y 1 , y 2 → 1, the parameters Q, R, S, ǫ 1 go to zero, which in turn leads to θ 13 → 0 and δ CP → 0 as expected. Finally, the solar mixing angle is given as
tan 2θ 12 ≃ y 1 Z √ 2(Ψ 2 − Ψ 1 ) ,(36)
where the parameters Ψ 1 , Ψ 2 and Z with |ǫ i | ≪ 1 are given as
Ψ 1 ≃Ã − ǫ 2 |Ω| √ 2 , Z ≃ P (1 + y 2 ) + Q(1 − y 2 ) − ǫ 1 {P (1 − y 2 ) + Q(1 + y 2 )} , Ψ 2 ≃F +G −K 8 (1 + y 2 2 ) +F −G 4 y 2 − ǫ 1F +G −K 4 (1 − y 2 2 ) .(37)
Note that in Eq. (36) the condition P (1 + y
2 ) + Q(1 − y 2 ) ≫ |ǫ 1 {P (1 − y 2 ) + Q(1 + y 2 )} |
should be satisfied, in order for the solar mixing angle θ 12 to be lie in the allowed region from the experimental data given in Eq. (3). The squared-mass eigenvalues of three light neutrinos are given by
m 2 1 ≃ m 2 0 c 2 12 Ψ 1 + s 2 12 Ψ 2 − y 1 Z 2 √ 2 sin 2θ 12 , m 2 2 ≃ m 2 0 s 2 12 Ψ 1 + c 2 12 Ψ 2 + y 1 Z 2 √ 2 sin 2θ 12 , m 2 3 ≃ m 2 0 Θ + ǫ 2 |Ω| √ 2 .(38)
We see from Eq. (37) that the deviation ǫ 3 from tri-maximality of solar mixing angle is roughly expressed as
sin ǫ 3 ≃ y 1 3 √ 2Zm 2 0 2∆m 2 21 − 2 √ 2 .(39)
In the limit of |ǫ i | ≪ 1, the solar and atmospheric mass-squared differences are roughly given in a good approximation by Here we note that the parameter M ri in Eq. (26) can be simplified in the following limiting cases as
∆m 2 Sol ≡ m 2 2 − m 2 1 ≃ m 2 0 24 (F +G −K)(1 + y 2 2 ) + 2y 2 (F −G) − 8Ã + 16y 1 P (1 + y 2 ) + Q(1 − y 2 ) , ∆m 2 Atm ≡ m 2 3 − m 2 1 ≃ m 2 0 3 F +G −K 4 (1 + y 2 2 ) − y 2 (F −G) − 2Ã − y 1 P (1 + y 2 ) + Q(1 − y 2 ) .(40)M ri ≃ M i [ln z i − 1] −1 , for z i ≫ 1 2M i , for z i → 1 m 2 η M −1 i , for z i ≪ 1 .
(41)
IV. NUMERICAL RESULTS
As is well known, the observed hierarchy |∆m 2 Atm | ≫ ∆m 2 Sol > 0 leads to two possible neutrino mass spectrum: (i) m 1 < m 2 < m 3 (normal mass spectrum), and (ii) m 3 < m 1 < m 2 (inverted mass spectrum). Since there are many unknown parameters such as masses of heavy Majorana neutrinos and scalar fields η R , η I , we consider a particular parameter set for those parameters and show how the measured values of the mixing angle θ 13 can be accommodated in our model while keeping the other neutrino parameters such as solar and atmospheric mixing angles and mass-squared differences are satisfied with the current data.
The mass matrix in Eq. (26) contains 10 free parameters : λ Φη 3 , M, y ν 3 , z 1 , z 2 , z 3 and y 1 , y 2 , ξ, κ. The combination of the first three of them, {λ Φη 3 , M, y ν 3 }, leads to the overall neutrino scale parameter m 0 . As shown above, the elements of the mass matrix in Eq. (26) are expressed in terms of measurable neutrino parameters, θ 12 , θ 13 , θ 23 , m 1,2,3 , δ CP , ϕ 1,2 . Among them, three mixing angles and two mass squared differences are measured. For numerical analysis [11], we need to fix some parameters by hand since there are too many model parameters to be predicted. As an example, we take a case M 2 1 =m 2 η 1 , M 2 2 = 1.3m 2 η 2 M 2 3 = 1.5m 2 η 3 , and fix the overall seesaw scale M to be 1 TeV. Then, the parameters m 0 , y 1 , y 2 , κ, ξ can be determined from the experimental results of three mixing angles and two mass squared differences. In addition, the CP phases δ CP , ϕ 1,2 can be predicted after determining the model parameters. Depending on the values of the model parameters, there exist two possibilities for the light neutrino spectrum, one is normal mass hierarchy and the other is inverted hierarchy. In the following, we discuss the two cases separately.
(i) normal hierarchy of light neutrino
Based on the formulae for the neutrino mixing angles and masses, we numerically scan the parameters m 0 , y 1 , y 2 , κ, ξ and then pick up the values of those five parameters which are consistent with the experimental data given at 3σ in Eq. (3). For the mixing angle θ 13 , we a bit widely allow its value from 5 • to 15 • instead of its experimental values at 3σ 3 . In such a way, we can obtain the allowed regions of the parameters given by 1.40 < κ < 2.38 , 0.44 < y 1 < 0.89 , 0.60 < y 2 < 0.84 and 1.1 < y 2 < 1.89 ,
190 • ≤ ξ < 211 • , 0.23 ≤ y ν 3 λ Φη 3 10 −9 < 0.46 .(42)
We found that normal mass ordering of light neutrino can be achieved when M 1 M 2 < M 3 or M 2 M 1 < M 3 are satisfied for the parameter spaces given above. In the left upper panel of Fig. 2, the data points indicate how the mixing angle θ 13 is determined in terms of the ratio y 1 /y 2 . The result shows that the upper limit of y 1 /y 2 is 0.86, and the measured value of θ 13 from the Daya Bay and RENO can be achieved for two regions, 0.40 < y 1 /y 2 < 0.57 and 0.67 < y 1 /y 2 < 0.82. To see how the parameters are correlated with low energy CP violation measurable through neutrino oscillations, we consider the leptonic CP violation parameter defined by the Jarlskog invariant J CP ≡ Im[U e1 U µ2 U * e2 U * µ1 ] = 1 8 sin 2θ 12 sin 2θ 23 sin 2θ 13 cos θ 13 sin δ CP [12] which can be described in terms of the elements h = m ν m † ν [13]: .
J CP = −Im{h
The behavior of J CP is plotted in the right upper panel of Fig. 2 as a function of θ 13 . We see that the value of |J CP | lies between 0 and 0.034 for the measured value of θ 13 . In our model, since Im{h 12 h 23 h 31 } is proportional to 1 − y 2 2 , the leptonic CP violation J CP goes to zero in the limit of y 2 → 1. However, y 2 = 1 is not allowed in our analysis, and thus J CP = 0 indicates that there exists some cancelation among the terms composed of sin ψ 12 , sin(ψ 1 + ψ 2 ), sin(2ψ 1 − ψ 2 ) and sin ψ 2 multiplies by y 1,2 , f (z 1 )/a, f (z 2 ), and f (z 3 )/b even if CP phases ψ 1,2 are non zero. In the lower panel of Fig. 2, the data points indicate how the values of θ 13 depend on θ 12 and θ 23 in the allowed regions given by Eq. (3). We see that the measured values of θ 13 can be achieved for two separate regions of θ 23 : 38.6 • θ 23 43 • and 47 • θ 23 53.1 • , which indicates that the parameter set strongly prefers deviations from maximal mixing for the atmospheric neutrino oscillation. From the right lower panel of Fig. 2, we see that predictions of θ 13 does not strongly depend on θ 12 for the allowed region.
We see from the figures that θ 13 for the normal hierarchy prefers rather large values more than 5 degrees.
We also see from Fig However, deviation of y 2 from one can be associated with deviation from maximality of atmospheric mixing angle by the following relation,
tan ǫ 1 = 1 + y 2 1 − y 2 (1 − 2y 2 1 + y 2 2 ) sin ψ 1 f (z 1 )f (z 2 ) a − (1 − y 2 ) 2 f (z 3 ) b f (z 1 ) a sin ψ 12 + f (z 2 ) sin ψ 2 (1 − 2y 2 1 + y 2 2 ) sin ψ 1 f (z 1 )f (z 2 ) a − (1 + y 2 ) 2 f (z 3 ) b f (z 1 ) a sin ψ 12 + f (z 2 ) sin ψ 2 .
This formular for the parameter ǫ 1 is relevant only when y 2 = 1. In the case of y 2 → 1 while y 1 = 1 and sin ψ 1 = 0, we see from the above equation that the value of θ 23 (or ǫ 1 )
can be large but restricted by experimental data. Then, due to Eq . (26) and Ω in Eq. (35), the value of θ 13 gets smaller as y 2 → 1. On the other hand, when y 2 is much deviated from 1, two cases for θ 23 (or ǫ 1 ) are possible. One is that rather smaller values of θ 23 (or ǫ 1 ) are preferred as the value of κ (or sin ψ 1 → 0 and sin ψ 2 → 0) decreases, and the other is that the combination of two parts in numerator of the above equation can lead to wide ranges of θ 23 (or ǫ 1 ). However, when y 2 ≈ 1 which makes the above equation irrelevant, the value of θ 13 goes to 0 • (numerically 1 • ), and the value of θ 23 can approach 45 • (or ǫ 1 → 0) for y 1 → 1 or sin ψ 1 → 0 converge more faster than y 2 → 1. We have neglected this case in our paper.
By using the conventional parametrization of the PMNS matrix [10] and Eq. (32), one can deduce a expression for Dirac CP phase δ CP given by
δ CP = − arg U * e1 U e3 U τ 1 U * τ 3c
Moreover, we can straightforwardly obtain the effective neutrino mass |m ee | which is associated with the amplitude for neutrinoless double beta decay :
|m ee | ≡ | i (U PMNS ) 2 ei m i | ,(45)
where U PMNS is given in Eq. (32). The left panel of Fig. 3 shows that δ CP is predicted to be 0 • δ CP 60 • , 120 • δ CP 240 • and 300 • δ CP 360 • for the measured values of θ 13 at 3σ. In the right panel of Fig. 3, we plot the prediction of the effective neutrino mass |m ee | as a function of θ 13 , which lies between 0.014 and 0.021 in the region of the measured values of θ 13 at 3σ.
(ii) inverted hierarchy of light neutrino Now let us turn to the inverted hierarchical case. Similar to case (i), scanning the parameters m 0 , y 1 , y 2 , κ, ξ based on the formulae for the neutrino mixing angles and masses and taking the experimental data given at 3σ in Eq. (3)
We found that this case is achieved when M 1 < M 2 < M 3 is satisfied. For those parameter regions, we in turn investigate how the mixing angle θ 13 depends on other parameters and whether CP violation is realized. In the left upper panel of Fig. 4, the data points indicate how the mixing angle θ 13 is determined in terms of the ratio y 1 /y 2 . We see that the measured value of θ 13 in 3σ including the Daya Bay experiment in Eq. (2), can be achieved for two separate regions, 0.82 < y 1 /y 2 < 0.88 and 1.12 y 1 /y 2 1.3. We plot J CP vs. θ 13 in the right upper panel of Fig. 4. For 5.9 • θ 13 9.5 • , |J CP | ≃ 0.018 ∼ 0.036 and −0.02 ∼ −0.034, which indicates CP violation in the leptonic sector. In the lower panel of Fig. 4, the data points show how θ 13 is determined in the allowed regions of θ 12 and θ 23 given by Eq. (3). We see that the narrowed regions of the atmospheric mixing angle θ 23 , 38.6 • θ 23 40.5 • and 49.5 • θ 23 53.1 • are preferred, which indicates that the parameter set disfavors maximal mixing for the atmospheric mixing angles. From the lower right panel of Fig. 4, we see that determination of θ 13 does not strongly depend on θ 12 for the allowed region. We see from the figures that contrary to the case (i), θ 13 for the inverted hierarchy prefers rather lower values less than 9.5 degrees. The left panel of Fig. 5 shows that δ CP is predicted to be around 70 • , 100 • , 160 • , 250 • and 290 • . In the right panel of Fig. 5, the value of |m ee | is predicted as a function of θ 13 and we see that |m ee |[eV] lies between 0.038 and 0.049 in the allowed region of θ 13 . To have a good dark matter candidate, we imposed auxiliary Z 2 symmetry, and thus light neutrino masses at tree level are absent in our model. However, the light neutrino masses can be generated through loop diagram, and we have shown how the light neutrino mass matrix can be diagonalized by the PMNS mixing matrix whose entries are determined by the current neutrino data including the Daya Bay result. In our model, the origin of the deviations from TBM mixing is non-degenerate neutrino Yukawa coupling constants among three generations. Also, unremovable CP phases in the neutrino Yukawa matrix are the origin of the low energy CP violation measurable from neutrino oscillation as well as high energy CP violation. We have discussed some implication on leptonic CP violation. 2 , A ′ 3 ) is block diagonalized due to Z 2 symmetry and CP conservation, which is given by
M 2 neutral = m 2 h ′ m 2 h ′ χ ′ 1 0 0 0 0 0 0 0 0 m 2 h ′ χ ′ 1 m 2 χ ′ 1 0 0 0 0 0 0 0 0 0 0 m 2 χ ′ 2 m 2 χ ′ 2 χ ′ 3 0 0 0 0 0 0 0 0 m 2 χ ′ 2 χ ′ 3 m 2 χ ′ 3 0 0 0 0 0 0 0 0 0 0 m 2 h ′ 1 0 0 0 0 0 0 0 0 0 0 m 2 h ′ 2 m 2 h ′ 2 h ′ 3 0 0 0 0 0 0 0 0 m 2 h ′ 3 h ′ 2 m 2 h ′ 3 0 0 0 0 0 0 0 0 0 0 m 2 A ′ 1 0 0 0 0 0 0 0 0 0 0 m 2 A ′ 2 m 2 A ′ 2 A ′ 3 0 0 0 0 0 0 0 0 m 2 A ′ 3 A ′ 2 m 2 A ′ 3 ,(A1)
where the primed particles are not mass eigenstates, and mass parameters are given as
m 2 h ′ = 4λ Φ v 2 Φ , m 2 h ′ χ ′ 1 = 2v Φ v χ λ Φχ , m 2 χ ′ 1 = 4v 2 χ (λ χ 1 + λ χ 2 ) , m 2 χ ′ 2(3) = v 2 χ (3λ χ 2 + 4λ χ 3 ) , m 2 χ ′ 2 χ ′ 3 = 3v χ ξ χ 1 m 2 h ′ 1 = v 2 Φ (λ ηΦ 1 + λ ηΦ 2 + 2λ ηΦ 3 ) + µ 2 η + v 2 χ (λ ηχ 1 + 2Re[λ ηχ 2 ]) , m 2 A ′ 1 = v 2 Φ (λ ηΦ 1 + λ ηΦ 2 − 2λ ηΦ 3 ) + µ 2 η + v 2 χ (λ ηχ 1 + 2Re[λ ηχ 2 ]) , m 2 h ′ 2 = v 2 Φ (λ ηΦ 1 + λ ηΦ 2 + 2λ ηΦ 3 ) + µ 2 η + v 2 χ (λ ηχ 1 − Re[λ ηχ 2 ] − √ 3Im[λ ηχ 2 ]) , m 2 h ′ 3 = v 2 Φ (λ ηΦ 1 + λ ηΦ 2 + 2λ ηΦ 3 ) + µ 2 η + v 2 χ (λ ηχ 1 − Re[λ ηχ 2 ] + √ 3Im[λ ηχ 2 ]) , m 2 A ′ 2 = v 2 Φ (λ ηΦ 1 + λ ηΦ 2 − 2λ ηΦ 3 ) + µ 2 η + v 2 χ (λ ηχ 1 − Re[λ ηχ 2 ] − √ 3Im[λ ηχ 2 ]) , m 2 A ′ 3 = v 2 Φ (λ ηΦ 1 + λ ηΦ 2 − 2λ ηΦ 3 ) + µ 2 η + v 2 χ (λ ηχ 1 − Re[λ ηχ 2 ] + √ 3Im[λ ηχ 2 ]) , m 2 h ′ 2 h ′ 3 = m 2 A ′ 2 A ′
•
+1.0 • (+2.9 • ) −0.9 • (−2.7 • ) , θ 23 = 46.1 •+3.5 • (+7.0 • ) −4.0 • (−7.5 • )
SU( 2 )
2L doublets and right handed lepton SU(2) L singlets, respectively. The higher dimensional operators (d ≥ 5) driven by χ and η fields are suppressed by a cutoff scale Λ which is a very high energy scale. Thus, their contributions are expected to be very small and we do not include them in this work. In the above Lagrangian, mass terms of the charged leptons are given by the diagonal form because the Higgs scalar Φ and the charged lepton fields are assigned to be A 4 singlet. The heavy neutrinos N Ri acquire a bare mass M as well as a mass induced by a vacuum of electroweak singlet scalar χ assigned to be A 4 triplet.
−c 23 s
2312 − s 23 c 12 s 13 e iδ CP c 23 c 12 − s 23 s 12 s 13 e iδ CP s 23 c 13 s 23 s 12 − c 23 c 12 s 13 e iδ CP −s 23 c 12 − c 23 s 12 s 13 e iδ CP c 23 c 13
FIG. 1 :
1One-loop generation of light neutrino masses.
Fig. 1
1thanks to the quartic scalar interactions. After electroweak symmetry breaking, the light neutrino masses in the flavor basis where the charged lepton mass matrix is real and diagonal are written as
By 1 y 2 By 1 By 1 y 2
12where the subscripts R and I indicate real and imaginary component, respectively. WithM R = Diag(M r1 , M r2 , M r3 ) and M ri ≡ M i f −1 (z i ), the above formula Eq. (24) can be expressed as Dy
can lead to the deviations of neutrino mixing angles from their TBM values, we first introduce three small quantities ǫ i , (i = 1 − 3) which are responsible for the deviations of the θ jk from their TBM values ;
FIG. 2 :
2Plots for Case (i) displaying the reactor mixing angle θ 13 versus the ratio y 1 /y 2 (upper left panel), and the Jarlskog invariant J CP versus the reactor angle θ 13 (upper right panel). Allowed values for the atmospheric mixing angle θ 23 (lower left panel) and the solar mixing angle θ 12 (lower right panel) versus the mixing angle θ 13 , respectively. The thick line corresponds to θ 13 = 8.68 • which is the best-fit value of Eq. (2) including the Daya Bay result. And the horizontal and vertical dotted lines in both plots indicate the upper and lower bounds on θ 13 given in Eq. (2) at 3σ
FIG. 3 :
3Predictions for the Dirac CP phase δ CP versus θ 13 (left panel) and the effective mass of neutrinoless double beta decay |m ee | versus the mixing angle θ 13 (right panel) for Case (i) . The thick and dotted lines correspond to θ 13 = 8.68 • which is the best-fit value and the 3σ bounds given in Eq.(2)including the Daya Bay result, respectively.
. 2 that small deviations for θ 23 prefer to large value of θ 13 in normal hierarchical case. This can be understood by considering two relations given in Eq.(21) andEq. (26). The phases ψ 1,2 go to 0 or π as the magnitude of κ defined in Eq.(15) decreases, and in the case of y 2 = 1 the neutrino mass matrix indicates directly θ 13 = 0 and θ 23 = −π/4.
FIG. 4 :
4Same as Fig. 2 for Case (ii).
FIG. 5 :
5Same as Fig. 3 for Case (ii) V. CONCLUSION Motivated by recent observations of non-zero θ 13 from the Daya Bay and RENO experiments, in this paper, we have proposed a neutrino model with A 4 symmetry and shown how deviations from the TBM mixing indicated by the current neutrino data including the Daya Bay result can be accounted for. In addition to the leptons and the Higgs scalar of the SM, our model contains three right handed heavy Majorana neutrinos and several scalar fields which are electroweak singlets and demanded to construct desirable forms of the letponic mass matrices.
as constraints, we can obtain the allowed regions of model parameters given by1.30 < κ < 1.56 , 209 • ≤ ξ < 222 • ,0.27 ≤
y ν
3 λ Φη
3
10 −9 < 0.45 ,
0.79 < y 1 < 0.88 ,
0.60 < y 2 < 0.79 ,
and
1.12 < y 1 < 1.24 ,
1.28 < y 2 < 1.5 .
The RENO Collaboration also announced observation of the non-zero mixing angle θ 13[2] in consistent with the result from the Daya Bay Collaboration.
Note that very small mixing angle θ 13 less then 1 • can be achieved in the case that y 1 → 1 or sin ψ 1 → 0 converges more faster than y 2 → 1.
= v χ ξ ηχ 1 .(A2)
AcknowledgmentsThe work of S.K. Kang was supported in part by the National Research Foundation ofAppendix A: The Higgs massOur model contains four Higgs doublets and three Higgs singlets. Here, we present the masses of physical scalar bosons, where the standard Higgs h ′ is mixed with χ ′ 0i , not with h ′ i , A ′ i . For simplicity, we assume that CP is conserved in the scalar potential, and then the coupling λ ηΦ 3 is real and the term ξ ηχ 2 (η † η) 3a χ is neglected in the Higgs potential given in Eq.(10). The neutral Higgs boson mass matrix in the basis ofSince the matrix in Eq. (A1) is block diagonalized, it is easy to obtain the mass spectrum given as follows;where λ ηΦ 12 ≡ λ ηΦ 1 + λ ηΦ 2 . Note here that the unprimed particles denote mass eigenstates. And the charged Higgs boson mass matrix in the basis of (η ± 1 , η ± 2 , η ± 3 ) is given aswhereUsing m 2 h i , m 2 A i in Eq. (A2) and Eq. (A4), the expressions form 2 η i appeared in Eq. (24) arem
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. G C Branco, R Gonzalez Felipe, F R Joaquim, I Masina, M N Rebelo, C A Savoy, arXiv:hep-ph/0211001Phys. Rev. D. 6773025G. C. Branco, R. Gonzalez Felipe, F. R. Joaquim, I. Masina, M. N. Rebelo and C. A. Savoy, Phys. Rev. D 67, 073025 (2003) [arXiv:hep-ph/0211001].
|
[] |
[
"Conventional Superconductivity in Fe-Based Pnictides: the Relevance of Intra-Band Electron-Boson Scattering",
"Conventional Superconductivity in Fe-Based Pnictides: the Relevance of Intra-Band Electron-Boson Scattering"
] |
[
"M L Kulić \nGoethe-Universität Frankfurt\nTheor. PhysikD-60054Frankfurt/MainGermany\n",
"S.-L Drechsler \nLeibniz-Inst. f. Festkörper-u. Werkstoffforschung Dresden (IFW Dresden\nGermany\n",
"O V Dolgov \nMax-Planck-Institut für Festkörperphysik\nD-70569StuttgartGermany\n",
"\nEurophysics Letters PREPRINT\n\n"
] |
[
"Goethe-Universität Frankfurt\nTheor. PhysikD-60054Frankfurt/MainGermany",
"Leibniz-Inst. f. Festkörper-u. Werkstoffforschung Dresden (IFW Dresden\nGermany",
"Max-Planck-Institut für Festkörperphysik\nD-70569StuttgartGermany",
"Europhysics Letters PREPRINT\n"
] |
[] |
Various recent experimental data and especially the large Fe-isotope effect point against unconventional pairings, since the large intra-band impurity scattering is strongly pair-breaking for them. The strength of the inter-band impurity scattering in some single crystals may be strong and probably beyond the Born scattering limit. In that case the proposed s ± pairing (hole(h)-and electron(el-gap are of opposite signs) is suppressed but possibly not completely destroyed. The data imply that the intra-band pairing in the h-and in the el-band, which are inevitably due to some nonmagnetic el-boson interaction (EBI), must be taken into account. EBI is either due to phonons (EPI) or possibly due to excitons (EEI), or both are simultaneously operative. We discuss their interplay briefly. The large Fe-isotope effect favors the EPI and the s + pairing (the h-and el-gaps are in-phase).c EDP Sciences
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10.1209/0295-5075/85/47008
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0811.3119
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818734f41c004c835c0f223b1d19b0ad15808120
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Conventional Superconductivity in Fe-Based Pnictides: the Relevance of Intra-Band Electron-Boson Scattering
1 Dec 2008
M L Kulić
Goethe-Universität Frankfurt
Theor. PhysikD-60054Frankfurt/MainGermany
S.-L Drechsler
Leibniz-Inst. f. Festkörper-u. Werkstoffforschung Dresden (IFW Dresden
Germany
O V Dolgov
Max-Planck-Institut für Festkörperphysik
D-70569StuttgartGermany
Europhysics Letters PREPRINT
Conventional Superconductivity in Fe-Based Pnictides: the Relevance of Intra-Band Electron-Boson Scattering
1 Dec 2008
Various recent experimental data and especially the large Fe-isotope effect point against unconventional pairings, since the large intra-band impurity scattering is strongly pair-breaking for them. The strength of the inter-band impurity scattering in some single crystals may be strong and probably beyond the Born scattering limit. In that case the proposed s ± pairing (hole(h)-and electron(el-gap are of opposite signs) is suppressed but possibly not completely destroyed. The data imply that the intra-band pairing in the h-and in the el-band, which are inevitably due to some nonmagnetic el-boson interaction (EBI), must be taken into account. EBI is either due to phonons (EPI) or possibly due to excitons (EEI), or both are simultaneously operative. We discuss their interplay briefly. The large Fe-isotope effect favors the EPI and the s + pairing (the h-and el-gaps are in-phase).c EDP Sciences
Introduction. -Recently, superconductivity (SC) with a relatively high critical temperature T c was discovered in several families of Fe-pnictide materials. In the electron (el) doped (1111) system LaFeAsO 1−x F x one has T c ≈ 26 K (and 43 K at high pressure) [1]. By replacing La by rare earths T c is higher with the present record of T c ≈ 55 K in SmFeAsO 1−x F x [2] and T c ≈ 56 K in Sr 1−x Sm x FeAsF [3], ect. In the hole (h) doped (122) system Ba 0.6 K 0.4 Fe 2 As 2 one has T c = 38 K [4]. In most of the parent compounds the SDW-type magnetic ordering occurs at T SDW ∼ 150 K, which is suppressed either by el-or h-doping at high pressure. The vicinity of these systems to an antiferromagnetic (AFM) phase inspired pairing models based on spin fluctuations which are repulsive in the s-wave channel. This situation resembles the one in high-temperature superconductors (HTSC), where due to the vicinity of an AFM phase pairing scenarios based on the Hubbard and t-J models have been proposed. However, it turns out that by interpreting experimental data of HTSC in the framework of the Eliashberg-theory, phonons might be important for the pairing mechanism of cuprates, giving rise to a large el-phonon coupling constant 1 ≤ λ ep 3.5, while the coupling due to the spin fluctuations is weak: λ sf < 0.3 [5].
The density functional band structure methods (DFT-LDA) applied to the Fe-based superconductors predict at least four bands at the Fermi energy E F -two h-bands around the Γ-point and two el-bands around the M -point [6], which have been already observed in ARPES [7]. The incipient magnetism in the parent compounds of the Fe-pnictides provoked proposals for unconventional SC either gapless or nodeless singlet or triplet pairing [9], [10]. A possible excitonic-like mediated SC has been mentioned in [11], while an el-exciton mediated pairing (EEI) has been proposed in [12]. Rather large electronic polarizability α e = (9 − 12)Å 3 of the As ions is estimated in [12], which screens the Coulomb repulsion between the charge carriers. Some NMR measurements on the Knight shift and the T −1 1 relaxation rate were interpreted in terms of an unconventional (d-wave like) gapless pairing [13]. Based on an LDA analysis of the dynamical spin susceptibility, in [14] the so called s ± pairing in the two-band model has been proposed. In fact, the s ± pairing scenario, where the pairing is due to a repulsive inter-band interaction either by the direct Coulomb one or spin fluctuations and the superconducting gaps on the h-and e-Fermi surfaces exhibit opposite signs, sign(∆ h ) = −sign(∆ e ), is not quite new [15]. The proposed nonphononic pairing finds support in the DFT-LDA calculations which give small EPI coupling constant λ ep for LaO 1−x F x FeAs [16], i.e. λ ep ≈ 0.2 and T c ∼ 1 K. The total coupling constant λ tot = λ ep + λ ee ... as derived from the zero temperature penetration depth λ L (0) (adopted from µSR (muon spin rotation) data [17]) is rather small λ tot ≈ 0.5 − 1. The plasma frequency analysis (from reflectivity data) was performed within the framework of a clean limit effective single-band approximation adopting that at T = 0 all charge carriers are in the superfluid condensate. [18].
Usually, the Fe-based superconductors are studied within the minimal BCS-like two-band model which is proposed much earlier [19], [20]. In the following, this model is used in conjunction with experimental results to argue that in order to explain pairing in the Fepnictides it is necessary to take into account the electron-boson interaction (EBI), either EPI or EEI, or both, which point to the presence of a conventional intra-band pairing.
Conventional EBI pairing. -At present, there is no evidence, neither experimental nor theoretical, for a magnetic pairing mechanism. If the latter is operative in the intraband pairing it would give rise for an unconventional order parameter. Contrary, there is numerous experimental evidence against an unconventional pairing in Fe-based pnictides: (i) the presence of a large amount of impurities and defects in single crystalls and polycrystalline samples [21], (ii) rather isotropic bands and superconducting gaps in the ARPES spectra [7], (iii) the large Fe-isotope effect with α F e ≈ 0.4 for the substitution 56 F e → 54 F e [25], (iv) the s-wave like T -dependence of the penetration depth, (v) the STM conductness with a nodeless gap [22], (vi) some NMR T −1 1 spectra [31], etc. The DFT-LDA calculations show appreciable change of the electronic density of states due to the Fe-As based phonons [24], which points to the importance of EPI, while the large As polarizability may favor excitonic pairing (EEI) [12]. These two EBI pairing mechanisms favor conventional superconductivity in Fe-pnictides and will be discussed briefly below.
Effect of impurities on T c . -The large residual resistivity observed in some single crystals of Ba 1−x K x F e 2 As 2 [21] excludes gapless unconventional superconductivity. Namely, the residual conductivity σ imp = ρ −1 imp in the minimal two-band model is given by [26]
ρ −1 imp = 1 4π ( ω 2 pl,h Γ tr h + ω 2 pl,e Γ tr e ),(1)
where the electron and hole transport scattering rates are Γ tr h = Γ tr hh +Γ tr he and Γ tr e = Γ tr ee +Γ tr eh . Here, the intra-band scattering is labelled by hh and ee, while he and eh label the inter-band scattering. Since the density of states of hole-and electron-bands are similar N e (0) ≈ 1.4N h (0) it is reasonable to assume that Γ tr h ∼ Γ tr e (≡ Γ tr ) holds, where Γ tr is related to the in-plane
residual resistivity ρ imp Γ tr (K) ≈ π 2ω 2 pl (eV )ρ imp (µΩcm).(2)
The effective plasma frequencyω 2 pl ≈ ω 2 pl,h +ω 2 pl,e measures the total number of the conduction carriers. The LDA calculations give an averaged plasma frequency (Ω pl ) in the Fe conduction plane 2 eV < Ω xx pl ≤ 3 eV . The estimated ρ imp in single crystals of
Ba 1−x K x Fe 2 As 2 (T c ≈ 38 K) is in the range ρ imp = (30 − 50) µΩcm [21]. Sinceω 2 pl ≈ (Ω xx pl ) 2 > 4(eV ) 2 then Eq.(2) gives Γ tr ≥ ∼ (200 − 300) K.
There is no apparent reason that the intra-band scattering is very different from the inter-band one (contrary to MgB 2 ), then one obtains
Γ hh ∼ Γ ee ∼ Γ he ∼ Γ eh > ∼ Γ tr /2 ∼ (100 − 150)K >> T c . Since Γ hh ∼ Γ ee
T c then the intra-band impurity scattering is strongly pair-breaking [5], which means that any unconventional intra-band pairing is immediately destroyed. This is also supported by the recent ARPES measurements on single crystals of NdFeAsO 0.9 F 0.1 (with T c = 53 K) where a slightly anisotropic gap around the Γ point with ∆ ≈ 15 meV was observed. A similar conclusion holds also for the case Γ tr h = Γ tr e since for ω 2 pl,h ∼ ω 2 pl,e the resistivity is dominated by the smaller value of Γ tr h , Γ tr e , i.e. by min(Γ tr h , Γ tr e ). The s ± pairing is unaffected by the intra-band impurity (u ii ) scattering in the Born limit (N i u ij ≪ 1) [27], [28], [29] but it is sensitive to the inter-band one Γ he , Γ eh , which are pair-breaking. The pairing interaction in the weak coupling BCS limit is given by H =
H 0 + H BCS + H imp H BCS = − ij V ij dxψ † i↑ ψ † i↓ ψ j↓ ψ j↑ , H imp = ijσ=↑,↓ u ij dxψ † iσ ψ jσ + c.c.,(3)
where the pairing interaction V ij (i, j = 1, 2, i.e. h, e) can be positive for attraction, negative for repulsion. The self-consistent equations for the order parameters ∆ i (i = 1, 2, i.e. h, e) are
∆ i = πT −ωcj <ωn<ωcj j,n λ ij∆ jn ω 2 jn +∆ 2 jn (4)
where ω cj is the energy cutoff in the j-th band and the intra-and inter-band coupling constants are λ ij = N j (0)V ij . To illustrate the effect of the multiple impurity scattering beyond the Born limit we consider the case of an arbitrary strength of the inter-band potential u 12 = u 21 = u, while u ii = 0 [28]. The renormalized Matsubara frequencyω jn and the gap∆ jn are giveñ
ω 1n = ω n + Γ 1 σ (σ − 1)δ 2 1nω 2n − σω 1nδ1nδ2n D 1n(5)∆ 1n = ∆ 1 + Γ 1 σ (σ − 1)δ 2 1n∆2n − σ∆ 1nδ1nδ2n D 1n ,(6)
where
D 1n = 2(σ − 1)σδ 1n ∆ 1n∆2n +ω 1nω2n − [2(σ − 1)σ + 1]δ 2 1nδ2n andδ 2 jn =ω 2 jn +∆ 2 jn .
The equations forω 2n ,∆ 2n are obtained by replacing 1 ←→ 2. The unitary amplitude is
Γ i = c imp /πN i (0) and σ = π 2 N 1 (0)N 2 (0)u 2 /(1 + π 2 N 1 (0)N 2 (0)u 2 ).
In the unitary limit one has N i (0)u ≫ 1 and σ → 1. To illustrate the pair-breaking effect of the inter-band scattering let us consider the Born limit of Eqs.(4-6) , i.e. N i u << 1, σ ≪ 1. For small c imp one has Γ he (≡ Γ 1 ),Γ eh (≡ Γ 2 ) << T c0 , where T c0 stands for the clean system. The suppression of T c is given by (
δT c /T c0 ) ≈ (πΓ he (∆ h − ∆ e ) (N e ∆ h − N h ∆ e ) /8T c0 N e ∆ 2 h + ∆ 2 e )
, where (Γ he /Γ eh ) = N e /N h . In the weak coupling limit one has |∆ h /∆ e | ≈ N e /N h , which gives |∆ h /∆ e | ≈ 1 and δT c ≈ −Γ he . We see that s ± pairing is affected by the inter-band impurity scattering only and the pair-breaking of the inter-band scattering is maximal for ∆ h ≈ −∆ e . This result holds also in the presence of the intra-band scattering u ii = 0 in the Born limit, since u ii drops out from equations -the Anderson theorem. For Γ he ∼ T c0 then T c is drastically reduced (T c << T c0 ) in the Born limit σ ≪ 1. The above estimate Γ he ≈ 100 − 150 K means that Γ he > T c0 for the reported single crystals with T c > 30 K. At first glance a large Γ he (> T c0 ) destroys the s ± pairing. However, the theory, contained in Eqs. (4)(5)(6), predicts that in the unitary limit πN e u he ≫ 1 the effect of impurities on s ± pairing disappears, i.e. the Anderson theorem is restored [28]. This also holds in the presence of the intra-band scattering u ii = αu he with α = 1 [28]. In the unitary limit with α = 1 the inter-band pair-breaking effect is strongly amplified by the intra-band one and T c is strongly suppressed [28]. If one assumes the Born limit one has Γ he ∼ πc imp N e u 2 he ∼ (100 − 150)K, and for the DFT-LDA value N e ∼ 0.6 states/eV·spin [14] the Born limit would be realized in the reported single crystals of Ba 1−x K x Fe 2 As 2 [21] for c imp > 10% if N e u he < 0.3. However, for a realistic value c imp 1% one has πN e u he 1, σ < 0.5 and the impurity scattering in the single crystals of Ba 1−x K x Fe 2 As 2 [21] is in the intermediate regime. In this case the inter-band impurity scattering suppresses s ± pairing moderately. However, due to many-body effect the unscreened plasma-frequencies are renormalized, i.eω pl < Ω pl . Optical data analyzed in the single-band model yield 1.5 eV ≤ω pl < 2 eV [18], thus reducing all Γ ij by a factor of about 1.7, i.e. the renormalized value Γ r ij ≈ (60 − 90) K. Strong coupling effects further reduce Γ r ij (e.g., in the single-band case by the factor 1 + λ). Nevertheless, the intra-band scattering Γ r ii is sufficiently strong to destroy the intra-band unconventional pairing. At the same time the inter-band impurity scattering Γ r ij (> T c0 ) is pushed toward the Born limit, i.e. σ < 0.5 thus being more detrimental for s ± pairing than in the unitary limit [27], [28].
The ARPES study of the over-doped Ba 1−x K x Fe 2 As 2 , with x = 1 and small T c = 3 K, shows the absence of the hole band, thus destroying the inter-(in the hole band) and intraband pairings [30]. Is the inter-band interaction λ he (and λ eh ) repulsive or attractive in the Fe-pnictides? This cannot be extracted from resistivity measurements. However, the large Fe-isotope effect on the polycrystalline samples of SmFeAsO 1−x F x and Ba 1−x K x Fe 2 As 2 with α F e ≈ 0.4 [25], if confirmed, implies that the inter-band pairing is dominated by the attractive EPI. In that case, the s-wave (s + ) pairing is realized in the hole-and the electron-band with sign(∆ h /∆ e ) = +. The magnetic interaction in this scenario is pair-weakening, i.e. it is detrimental. If α F e is going to be small, then the EEI electron-boson pairing might be operative. That the repulsive part of the inter-band interaction due to spin fluctuations might be small comes out from the recent NMR studies of the 75 As relaxation rate (T 1 T ) −1 on LaFeAsO 1−x F x (0.04 ≤ x ≤ 0.14) as a function of x [31]. While(T 1 T ) −1 , which is dominated by the antiferromagnetic fluctuations, decreases by increasing doping by almost two orders of magnitude, T c is practically unchanged. The latter suggests weakness of the inter-band interaction due to spin fluctuations. It is worth to mention that the residual in-plane resistivity ρ ab imp of (Ba,K)Fe 2 As 2 in two different single crystals was different, 70 and 100 µΩcm respectively, giving Γ tr ≈ (300 − 600) K, but T c was unchanged, i.e. T c ≈ 28 K [32]. This result is an additional evidence for the robustness against impurities and the conventional intra-band SC in these materials. The conventional pairing scenario is also supported by the data for H c2 (T ) near T c as a function of enhanced disorder provided e.g. by As vacancies [33], where as a result of disorder the slope of H c2 (T ) increases, while T c is even increased in some samples or remains nearly unchanged. Similar effects have been observed also in Ref. [34]. To the best of our knowledge, only within conventional s-wave intra-band pairing the upper critical field H c2 (T ) near T c can be readily strongly enhanced due to the reduction of the well-known spin independent orbital pair-breaking effect caused by the Lorentz force on Cooper pairs. Within a dominant s ± inter-band scenario there is no room for such reduction effects. The most what can be "explained" by unconventional scenarios is the robustness of T c in adopting simply an ineffective impurity scattering mechanism (ascribed to the different orbitals involved in each of the two bands) [29] ; but by no means the reported improved H c2 and T c .
Nonmagnetic (s + ) or magnetic (s ± ) pairing. -The STM conductance [22] gives evidence for a single gap, while in ARPES spectra three gaps were observed -two hole gaps ∆ h,1 ≈ 12 meV and ∆ h,2 ≈ 6 meV and one electron gap ∆ e ≈ 12 meV [7]. Neglecting ∆ h,2 a two-band model (with ∆ h,1 ≡ ∆ 1 and ∆ e ≡ ∆ 2 ) is applicable. Let us analyze the Fe-pnictides in the weak-coupling limit and assume that the couplings λ 11 ≈ λ 22 ∼ λ 12 ∼ λ 21 . In that case one has for 0 < T ≤ T c
λ max − λ 22 λ 21 ≈ ∆ 1 ∆ 2 ≈ λ 12 λ max − λ 11 ≈ N 2 (0) N 1 (0) sign(λ 12 ),(7)
where λ max = (λ 11 + λ 22 + (λ 11 − λ 22 ) 2 + 4λ 12 λ 21 )/2 is the maximal eigen-value of the matrix λ ij which determines T c = 1.13ω c exp{−1/λ max }. It gives (∆ h /∆ e ) ≈ sign(λ he ). Note, the couplings λ ′ s are effective ones due to the presence of attractive and repulsive interactions. So, for λ he > 0 the s + pairing is realized, while for λ he < 0 the s ± pairing is more stable. The first case is dominated by the conventional EBI pairing, while the latter one by a repulsive interaction. The obtained ratio |∆ h /∆ e | is similar to the experimental value |∆ h /∆ e | ≈ 1 [7]. This result suggests that the intra-and inter-band couplings are of the same order. In this case the specific heat jump at T c is given by (∆C S /C N ) ≈ 1.43[(N h + N e δ 2 ) 2 /(N h + N e )(N h + N e δ 4 )] < ∼ 1.43, with δ = ∆ e /∆ h . The above analysis is based on the weak coupling limit and it serves for obtaining a qualitative picture only, while a full quantitative analysis should be based on the Eliashberg (strong coupling) theory which is important even in the weak coupling limit.
Discussion. -The available experimental data do not allow us to extract the intraand inter-band couplings and their signs. But some qualitative estimates of λ 11 , λ 22 , λ 12 and λ 21 (note 1 ≡ h, 2 ≡ e) are possible in the weak coupling limit. In the over-doped Ba 1−x K x F e 2 As 2 , with x = 1 and T over c = 3 K [30], the hole band is absent and λ 22 = λ 12 = λ 21 = 0, while in the optimally doped system T opt c ≈ 40 K one expects λ 11 ≈ λ 22 ∼ λ 12 ≈ λ 21 . From these data one obtains ω c ≈ 260 K, λ 11 ≈ λ 22 ≈ 0.22 and λ 12 ≈ λ 21 ≈ 0.28. The Debye temperature in the Fe-pnictides is ω D ≈ 280 K(≈ ω c ) and the magnitudes of the λ ′ s are in the range of the LDA value for the EPI coupling constant. By approaching the overdoped region one expects that λ 12 and λ 22 decreases and vanishes, as it is observed in ARPES spectra of Ba 1−x K x Fe 2 As 2 . This scenario could be tested in clean systems by measuring the collective Leggett mode which is specific for clean two(or multi)-band) systems with an internal interband Josephson effect. In this case the contribution to the total energy, for small λ 12 , λ 21 , is E inter = −E 0 cos(θ 1 − θ 2 ). For s + pairing with λ 12 (λ 21 ) > 0 one has E 0 > 0 and the ground state is realized for θ 1 − θ 2 = 0 -the 0-contact, while for s ± pairing E 0 < 0 and θ 1 − θ 2 = π -the π-contact. In both cases the relative phase (θ 1 − θ 2 ) of the two order parameters (∆ 1,2 = |∆ 1,2 | exp{−iθ 1,2 }) oscillates with the Leggett-frequency
ω 2 L = 4 |λ 12 + λ 21 | |∆ 1 ∆ 2 | λ 11 λ 22 − λ 12 λ 21 .(8)
The condition for the existence of the Leggett mode is D ≡ λ 11 λ 22 − λ 12 λ 21 > 0 and it is undamped for ω L < 2 |∆ 1 |. This can be realized in low doped F e-pnictides where it is expected that |λ 12 | ≪ λ 11 and D > 0, while by doping toward the optimal doping it may disappear or become overdamped. This proposal can be tested by studying the corresponding tunnelling and electronic Raman spectra -like in MgB 2 [35]. What is the possible origin of the EBI pairing in Fe-pnictides? The isotope effect, if confirmed, points to the importance of EPI. However, the effective Coulomb interaction µ * (see below) suppresses EPI and in this case an excitonic pairing (EEI) might be helpful to compensate the Coulomb interaction. In Fe-pnictides the As ions posses large polarizability α e = (9−12)Å 3 [12] which can partly compensate or overcompensate the Coulomb interaction, thus changing the electronic dielectric function ε el (q, ω = 0) = ε c (q, 0) + ε ex (q, 0), where ε c is the dielectric function of conduction carriers. Since in a multi-band model this mechanism can be obscured by numerous parameters we elucidate it in the single-band model by assuming that the effective pairing interaction V pair (ξ, ξ ′ ) contains three dominant parts, two attractive -EPI with the potential V epi and EEI with V eei , and one repulsive with V c
V pair (ξ, ξ ′ ) = V epi (ξ, ξ ′ ) + V eei (ξ, ξ ′ ) − V c (ξ, ξ ′ ),(9)
where
V epi (ξ, ξ ′ ) = V epi > 0 for |ξ| , ξ ′ ω ph ; V eei (ξ, ξ ′ ) = V eei > 0 for |ξ| , ξ ′ ω ex and V c (ξ, ξ ′ ) = V c > 0 |ξ| , ξ ′ ω c .
The characteristic phonon, exciton and Coulomb energies are related by ω ph ≪ ω ex , ω c . In the limit when the effective exciton and Coulomb interactions are small, i.e. when λ * ex µ * ≪ λ * ex , µ * , the critical temperature is given by
T c ≈ 1.14ω ph exp{−1/(λ ep − µ * + λ * ex },(10)
where µ * = µ/(1 + µ ln(ω c /ω ph )), and λ * ex = λ ex /(1 − λ ex ln(ω ex /ω ph )). The corresponding coupling constants are λ ep = N (0)V epi , λ ex = N (0)V eei and µ = N (0)V c . For λ ex < µ the exciton pairing may compensate the Coulomb repulsion giving λ * ex > µ * . In the single band case it is difficult to realize an excitonic mechanism of pairing since the electronic (not the total!) dielectric function ε el (q, ω = 0) must be positive (ε el (q, 0) > 0) in order to keep the lattice stability [5]. In a multi-band case the physics can be different and the restrictions for an excitonic mechanism can be relaxed.
Conclusions. -The recent experimental results in single crystals of the Fe-pnictides give evidence for: (i) the multi-band character of pairing; (ii) the presence of a large amount of nonmagnetic impurities (and other defects) which are strong pair-breakers for unconventional pairing. The result (ii) strongly favors conventional nodeless intra-band pairings with λ hh ,λ ee > 0, which are due to a non-magnetic electron-boson interaction (EBI), either to phonons or eventually to excitons, or both are operative. If the large Fe-isotope effect α F e ≈ 0.4 is confirmed, then EPI is favored. The EBI inter-band coupling with λ he (λ eh ) > 0 favors s + pairing with sign(∆ h /∆ e ) = +. The s ± pairing with sign(∆ h /∆ e ) = − is favored by repulsive inter-band interactions with λ he < 0. This pairing is less favorable due to the detrimental effect of disorder present in single crystals and polycrystals and due to EBI scattering. Even, if excitonic effects would not give solely a high T c they may nevertheless strengthen a pairing due to EPI. *** We thank Zlatko Tešanović for useful discussions.
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At variance to LaFeAsO0.9F0.1 considered there, a similar analysis for the HTSC reveals strong coupling: λtot ∼ 2-3, in accord with Ref. S-L Drechsler, ibid. 101in pressDrechsler S-L, et al., ibid. 101 (2008) in press . At variance to LaFeAsO0.9F0.1 considered there, a similar analysis for the HTSC reveals strong coupling: λtot ∼ 2-3, in accord with Ref. [5].
. H Suhl, ibid. 3552Suhl H et al., ibid. 3 (1959) 552;
. Moskalenko V, Fiz. Met. Metallov. 8503Moskalenko V, Fiz. Met. Metallov. 8 (1959) 503
Moskalenko V Electromagnetic and kinetic properties of superconducting alloys with overlapping bands. KishinevMoskalenko V Electromagnetic and kinetic properties of superconducting alloys with overlap- ping bands, Kishinev (1976)
. H Luo, arXiv:0807.0759v3Luo H et al., arXiv:0807.0759v3 (2008)
. T Y Chen, Z Tešsanović, R H Liu, X H Chen, C L Chien, Nature. 4531224Chen TY, Tešsanović Z, Liu RH, Chen XH and Chien CL, Nature 453 (2008) 1224
. Y Nakai, It, arXiv:0810.3569v1Nakai Y it et al., arXiv:0810.3569v1 (2008)
. M Zbiri, arXiv:0807.4429v1Zbiri M et al., arXiv:0807.4429v1 (2008)
The observation of an isotope effect also for TSDW of the parent compound reported there remains a puzzle especially in view of the expected small amplitude of zero-point motion related to the relatively heavy Fe-nucleii. However, an anomalous upturn of the mean-square relative displacement of the Fe-As. R H Liu, arXiv:0810.2694v1bond has been detected at T <70Liu RH et al., arXiv:0810.2694v1 (2008) . The observation of an isotope effect also for TSDW of the parent compound reported there remains a puzzle especially in view of the expected small amplitude of zero-point motion related to the relatively heavy Fe-nucleii. However, an anomalous upturn of the mean-square relative displacement of the Fe-As bond has been detected at T <70
using x-ray absorption spectroscopy and considered as evidence for a strong el-lattice interaction. C J Zhang, arXiv:0811.3268LaFeAsO0.03F0.07 by. K (but suppressed below Tc) in LaFeAsO0.03F0.07 by C.J. Zhang et al. arXiv:0811.3268 using x-ray absorption spectroscopy and considered as evidence for a strong el-lattice interaction.
. P B Allen, Phys. Rev. B. 3305Allen PB, Phys. Rev. B 3 (1971) 305
. A A Golubov, I I Mazin, Phys. Rev. B. 55305Golubov AA and Mazin II, Phys. Rev. B 55 (1997) 305
) 13062; these results are confirmed in Ohashi Y. M L Kulić, Ov ; 41 Dolgov, Y Senga, H Kontani, arXiv:0809.0374v1J. Phys. Soc. Jpn. 71Physica C412-414Kulić ML and Dolgov OV, ibid. 60 (2008) 13062; these results are confirmed in Ohashi Y, J. Phys. Soc. Jpn. 71 (2002) 1978; Physica C412-414 (2004) 41 and Senga Y and Kontani H, arXiv:0809.0374v1 (2008)
For the vicinity of the very special 'unitary limit' point in the scattering parameter space the authors [28] find a minor suppresion of the s± pairing by the interband impurity scattering. Note that even in this special case with preserved s± symmetry of the order parameter in the presence of dominant repulsive interband interaction, our main statement of s-wave intraband scattering being relevant for the enhanced slope of Hc2(T ) and the resistivity in the normal state does hold. In this case Tc might be considerably enhanced by the repulsive interband scatteringFor the vicinity of the very special 'unitary limit' point in the scattering parameter space the authors [28] find a minor suppresion of the s± pairing by the interband impurity scattering. Note that even in this special case with preserved s± symmetry of the order parameter in the presence of dominant repulsive interband interaction, our main statement of s-wave intraband scattering being relevant for the enhanced slope of Hc2(T ) and the resistivity in the normal state does hold. In this case Tc might be considerably enhanced by the repulsive interband scattering.
. T Sato, arXiv:0810.3047v1Sato T et al., arXiv:0810.3047v1 (2008)
. Y Nakai, arXiv:0810.3569v1Nakai Y et al., arXiv:0810.3569v1 (2008)
. H Q Yuan, arXiv:0807.3137Yuan HQ et al., arXiv:0807.3137 (2008)
. G Fuchs, S-L Drechsler, N Kozlova, Phys. Rev. Lett. 101in pressFuchs G, Drechsler S-L, Kozlova N et al., Phys. Rev. Lett. 101 (2008) in press
. S J Singh, arXiv:0810.2362Singh SJ et al. arXiv:0810.2362
. Ponomarev Yag, Sol. St. Comm. 12985Ponomarev YaG et al. Sol. St. Comm. 129 (2004) 85;
. G Blumberg, Phys. Rev. Lett. 99227002Blumberg G et al., Phys. Rev. Lett. 99 (2007) 227002
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In this article, we define and study a geometry on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter N . Then we emulate the theory of random matrices in a combinatorial framework: for any parameter N , we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory.A renormalization of the partition algebras allows us to consider the weak convergence as a simple convergence in a fixed space. This leads us to the definition of a deformed partition algebra for any integer parameter N and to the definition of two transforms: the cumulants transform and the exclusive moments transform. Using an improved triangular inequality for the distance defined on partitions, we prove that the deformed partition algebras, endowed with a deformation of the linear forms converge as N go to infinity. This result allows us to prove combinatorial properties about geodesics and a convergence theorem for semi-groups of functions on partitions.At the end we study a sub-algebra of functions on infinite partitions with finite support : a new addition operation and a notion of R-transform are defined. We introduce the set of multiplicative functions which becomes a Lie group for the new addition and multiplication operations. For each of them, the Lie algebra is studied.The appropriate tools are developed in order to understand the algebraic fluctuations of the moments and cumulants for converging sequences. This allows us to extend all the results we got for the zero order of fluctuations to any order.
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1503.02792
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10 Mar 2015
10 Mar 2015arXiv:1503.02792v1 [math.CO] COMBINATORIAL THEORY OF PERMUTATION-INVARIANT RANDOM MATRICES I: PARTITIONS, GEOMETRY AND RENORMALIZATION. by Franck Gabriel
In this article, we define and study a geometry on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter N . Then we emulate the theory of random matrices in a combinatorial framework: for any parameter N , we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory.A renormalization of the partition algebras allows us to consider the weak convergence as a simple convergence in a fixed space. This leads us to the definition of a deformed partition algebra for any integer parameter N and to the definition of two transforms: the cumulants transform and the exclusive moments transform. Using an improved triangular inequality for the distance defined on partitions, we prove that the deformed partition algebras, endowed with a deformation of the linear forms converge as N go to infinity. This result allows us to prove combinatorial properties about geodesics and a convergence theorem for semi-groups of functions on partitions.At the end we study a sub-algebra of functions on infinite partitions with finite support : a new addition operation and a notion of R-transform are defined. We introduce the set of multiplicative functions which becomes a Lie group for the new addition and multiplication operations. For each of them, the Lie algebra is studied.The appropriate tools are developed in order to understand the algebraic fluctuations of the moments and cumulants for converging sequences. This allows us to extend all the results we got for the zero order of fluctuations to any order.
9
. Algebraic fluctuations. . . . 45 10. An introduction to the general R-transform. .
Introduction
This article is the first of a self-contained set of three articles [8], [9] and [10] on a combinatorial method in random matrices theory based on a geometry on partitions and a new point of view on usual/free cumulants based on dualities between groups and subalgebras of partitions. This general method allows to work with random matrices which are invariant by conjugation by the symmetric group instead of the unitary or orthogonal group, besides, no more assumption about the factorization of moments is needed. The first article is about the combinatorial framework based on the partition algebra. In the second article we will apply this framework to random matrices, and the third one will put the emphasis on the random walks on the symmetric group and the link with the S ∞ -Yang-Mills measure.
This set of articles has to be considered as the continuation of what could be called the Gauge Theory School in random matrices. The article of F. Xu [18] is one of the pioneer work about this point of view on random matrices. Later, this point of view was developed by A. Sengupta [17], then highly improved by T. Lévy [12], [13], then it was used by two students of T. Lévy: A. Dahlqvist in [7] and G. Cébron [5], [4].
We wrote these articles as a lesson for graduate students with the intention that no special requirement is needed to understand them. The reader will find a new presentation and introduction to the random matrices theory. To achieve this, we only used the Gauge Theory School's papers, the seminal article for partition algebras [11], and the book [16] which, in some sense, we tried partially to generalize. Another point of view on random matrices which are invariant by conjugation by the symmetric group was given first by C. Male in his paper on traffics [14]. Yet, the goal was to develop the ideas of the Gauge Theory School and thus we did not use this article. In the forthcoming article [6], the author and his coauthor build connections between the notions developed here and the notions developped in [14]. In some sense, these articles can be seen also a bridge to go from the book [16] to the traffic interpretation of [14], traffics which have shown their importance in the study of random graphs [15]. At the moment the author was finishing these articles, he was informed of M. Capitaine and M. Casalis's work, [3], on their Schur-Weyl's interpretation of non-commutative free cumulants for unitary and orthogonal invariant random matrices.
The point of view developed in the three articles [8], [9] and [10] allows us to recover in a very simple way some famous theorems. The reader will also find in these articles a simple tool box in order to prove new convergences of random matrices (for example random walks on the symmetric group). He will also find the tools in order to understand the algebraic fluctuations of moments of random matrices. Besides, this point of view allows to define a general notion of freeness for matrices which are invariant by conjugation by the symmetric group and we construct the first non-commutative multiplicative Lévy processes for this notion of freeness. We will formulate two equivalent definitions of this freeness: one based on cumulants, and the other on moments. This freeness notion is linked with a new Rtransform which generalizes the old known R-transform. A Kreweras complement is defined for partitions: this generalizes the notion already set for permutations. Amongst others, we will state a matricial Wick's theorem, which allows to recover the Wick law for Gaussian Hermitian or symmetric matrices. We will also recover theorems about convergence of Hermitian Lévy processes proved in [2], [1] and unitary Lévy processes proved in [4]: we extend them to the symmetric and the orthogonal case. A new central limit theorem will be stated, which generalizes the non-commutative and the commutative central limit theorem. In the article [9], convergences of random walks on the symmetric group will be proved, and will be used in order to show that the Wilson loops of the S N -Yang-Mills measure converge in probability when N goes to infinity. This will imply the first result known about convergence of ramified coverings on the disk. We will also see how to inject the usual theory of probabilities in this framework. This last assertion shows that one could, in this framework, study the probabilistic fluctuations. 1. 1. Renormalization and a physical point of view. -In this article, we emulate the theory of random matrices in a combinatorial framework. Given a partition p of a number of points, and an integer N , we consider (p, N ) as a physical system involving N particles. When the number of points is even, by polarizing the points in two sets, we can consider (p, N ) as a discrete time transformation operation. A partition p can be seen as an elementary evolution of a system of size N : we can define the composition of two partitions. Later in the paper, we consider these discrete-time transformations also as the Hamiltonian of continuous time transformations.
An evolution of a system of size N is a linear combination of elementary evolutions of size N . Thus, every transformation is uniquely characterized by a size N and by a finite number of coefficients which, as we will see in the article, are bare quantities. Two questions arise: how to describe a system of infinite size and how to renormalize the bare quantities. As one does for perturbative renormalization, the important idea is to consider observables: we define some observables, one for each partition. In Theorem 4.1, we show how the bare coefficient must be renormalized in order to have finite observables at the limit N = ∞.
Then, we show that, by using the same renormalization, the composition of two evolutions converge also: this is proved in Theorems 6.1 and 7. 1. In Theorem 7.2, we consider continuous-time evolution transformations: we show that if the Hamiltonian is renormalized as we did for discrete time transformations, then the evolution converges. In Theorem 10.2, we characterize the Hamiltonian so that the factorization property of large N holds.
We study also the development in power of 1/N of systems of size N which converge to a continuous system.
The main novelty is to show that, even if one knows how to renormalize the bare constants, it does not seem interesting to define a vector space of infinite systems since all systems considered are defined in the same vector space whose basis is the set of partitions of 2k elements. In order to have an interesting space of infinite systems, one has to consider a renormalization of the algebras in which are defined the N -dimensional systems: the limit defines a non-trivial algebra in which one can study continuous evolutions of continuous systems.
Let us remark that a consequence of our results is that, in our toy-model, given a continuous system, one has canonically a sequence of approximations by systems involving N particles. 1.2. Layout of the article. -Using the set P k of partitions of 2k elements as basis, one can define an algebra known as the partition algebra which definition depends on a parameter N ∈ N: the partition algebra C[P k (N )]. For a comprehensive study of this algebra, we recommend the article [11]. The main definitions are set in Section 2.
In Section 3, we define a geometry on the set of partitions of 2k elements which generalizes a well-known geometry on the symmetric group S k . Using this new geometry, in Section 4 we define two notions of convergence of sequences which are shown to be equivalent. We define the notion of coordinate numbers, normalized moments, exclusive coordinate numbers and exclusive normalized moments. One of the results that we prove is that exclusive coordinate numbers and exclusive normalized moments are equal. In Section 5, a new deformed partition algebra is defined: C[P k (N, N )]. These algebras are shown to converge to a new algebra: this is obtained by an improvement of the triangle inequality proved in Section 6 for the distance defined on the set of partitions of 2k elements. Let us remark that we define in the same section a Kreweras complement for partitions which generalizes the notion for permutations. We use these results in Section 7 in order to show that the multiplication is continuous for the notion of convergence of elements of N ∈N C[P k (N )]. We also study the convergence of semi-groups in N ∈N C[P k (N )]. In Section 8, using the convergence of sequences defined in Section 4, we show how one can prove combinatorial results, for exemple, a new proof of the improved triangle inequality is given.
In Section 9, we develop the notion of algebraic fluctuations, and extend the results already proved for the zero order of fluctuations to any order.
In Section 10, we construct an algebra E[P] which elements are functions on ∪ k∈N P k . This algebra can be endowed with two special laws: ⊞ and ⊠. We study two subgroups of E[P] associated with the operations ⊞ and ⊠, the group of multiplicative invertible elements. These groups are Lie groups, the Lie algebras of these groups are studied. We also define the R A -transform, which generalizes the usual R-transform and we define two others transformations linked with the notion of exclusive moments. To finish the article, we extend these definitions to the setting of higher order fluctuations. 1. 3. Acknowledgements. -This work has been realized during my PhD at the LPMA which offered me the necessary liberty to complete this article. Many thanks to the researchers and administrative staff of the LPMA. I am grateful to my PhD advisor Thierry Lévy for supporting this research, for his helpful comments and corrections which led to improvements in this manuscript and for the useful discussions about mathematics and other subjects. I would like to express my special gratitude to Terence Tao, particularly for his blog which, during a period of doubts, made me enjoy maths again. This project began when I wanted to understand the link between the work of T. Lévy and the formulation given at the Pims summer school by David Brydges of Wick's theorem, I am really thankful to him for this. Also many thanks are due to Antoine Dahlqvist who explained me the duality between permutations and partitions. I would like to thank Tom Halverson and Arun Ram for answering my questions about partitions. I am also grateful to Guillaume Cébron since I learned a lot thanks to his papers. Thanks to Patrick Gabriel for his interest in my work and the discussions related to this work that we had together, Marie-Françoise Gabriel and Catherine Lam for trying to correct the English in this manuscript. At last, a thought to all the people which are supporting researchers and whose names never appear in the acknowledgements.
Partition algebra
2.1. First definitions. -Let k and N be two integers. We will consider three differ-
ent algebras C [S k ] , C [B k (N )] , C [P k (N )]
: respectively the symmetric algebra, the Brauer algebra, and the partition algebra. These algebras satisfy the inclusions:
C [S k ] ⊂ C [B k (N )] ⊂ C [P k (N )] .
Thus, we will first construct C [P k (N )] and we will see the two others algebras as subalgebras of C [P k (N )]. The reference article for the partition algebra is the article [11] of T.Halverson and A. Ram. Let us consider 2k elements which we denote by: 1, . . . , k and 1 ′ , . . . , k ′ . We define P k as the set of set partitions of {1, . . . , k} ∪ {1 ′ , . . . , k ′ }. If k = 0, we consider P k = {∅}. Let p be an element of P k . We will denote by p 1 , . . . , p r the blocks in p. The number of connected components nc(p), the propagating number pn(p) and the length l(p) of p are defined respectively by: Any partition p ∈ P k can be represented by a graph. For this we consider two rows: k vertices are in the top row, labeled by 1 to k from left to right and k vertices are in the bottom row, labeled from 1 ′ to k ′ from left to right. Any edge between two vertices means that the labels of the two vertices are in the same block of the partition p. Examples are given in Figure 1 and 2.
nc(p) = r,
The notion of tensor product of partitions will be also very useful.
Definition 2.1.
-Let k and l be two integers. Let p be an element of P k and let p ′ be an element of P l . Let us consider two diagrams: one associated with p, another with p ′ . Let
1 = {1 ′ , 1}{2 ′ }{2, 3 ′ , 5 ′ }{3, 4, 4 ′ }{5} . Figure 2. Partition p 2 = {1 ′ , 2 ′ }{1, 2, 3 ′ , 5}{3}{4 ′ , 4}{5 ′ } .
p ⊗ p ′ be the partition in P k+l associated with the diagram where one has put the diagram associated with p on the left of the diagram associated with p ′ . Let k be an integer. Let p 1 and p 2 be two elements of P k . We say that p 1 is coarser than p 2 if any two elements which are in the same block of p 2 are also in the same block of p 1 . This order is directed: for any partitions p 1 and p 2 in P k there exists a third partition p 3 which is coarser than p 1 and p 2 : for example, one can consider the partition p 1 ∨ p 2 defined as follows.
Definition 2.2. -Let k be an integer. Let p 1 and p 2 be two elements of P k . We define p 1 ∨ p 2 as the partition in P k such that for any i, j ∈ {1, . . . , k } ∪ {1 ′ , . . . , k ′ }, i and j are in the same block of p 1 ∨ p 2 if and only if there exists i = x 0 , x 1 , . . . , x l = j with x n ∈ {1, . . . , k } ∪ {1 ′ , . . . , k ′ } such that x n and x n+1 are in the same block of p 1 or p 2 for any n ∈ {0, . . . , l − 1}.
It is always interesting to have a graphical representation for the operations defined on partitions. One can recover a diagram representing p 1 ∨ p 2 by putting a diagram representing p 2 over one representing p 1 .
Let us play a little with the graphical representation of p 1 and p 2 in order to define other natural operations on the set of partitions.
We will use later the transposition of a partition: it is the partition obtained by permuting the role of {1, . . . , k} and {1 ′ , . .
. , k ′ }. For example if k = 3, let p = {1, 1 ′ , 3 ′ }, {2, 3}, {2 ′ } , then t p = {1 ′ , 1, 3}, {2 ′ , 3 ′ }, {2} . For every diagram∨ p 2 = {{1, 1 ′ , 2, 2 ′ , 3 ′ , 5, 5 ′ }, {3, 4, 4 ′ }}.
associated with p, the diagram obtained by flipping it according to a horizontal axis is a diagram associated with t p. One can find an example in Figure 5 An other thing we can do is to put one diagram representing p 2 above one diagram representing p 1 . Let us identify the lower vertices of p 2 with the upper vertices of p 1 . We obtain a graph with vertices on three levels, then erase the vertices in the middle row, keeping the edges obtained by concatenation of edges passing through the deleted vertices. Any connected component entirely included in the middle row is then removed. Let us denote by κ(p 1 , p 2 ) the number of such connected components. We obtain an other diagram associated with a partition denoted by p 1 • p 2 . For any elements p 1 and p 2 of P k , the partition p 1 • p 2 does not depend on the choice of diagrams representing the partitions p 1 and p 2 . The set of Brauer elements and the set of partitions will be stable by this operation of concatenation.
Definition 2.3. -
The set of Brauer elements B k is the set of pair partitions in P k . The set of permutation S k is the set of pair partitions in P k whose propagating number is equal to k.
For any p 1 and p 2 in B k (resp. S k ), p 1 • p 2 ∈ B k (resp. S k ). Let us define the three algebras
C [S k ] , C [B k (N )] and C [P k (N )].
Definition 2. 4. -Let k and N be two integers. The partition algebra C [P k (N )] is the associative algebra over C with basis P k endowed with the multiplication defined by:
∀p 1 , p 2 ∈ P k , p 1 p 2 = N κ(p 1 ,p 2 ) (p 1 • p 2 ).
The Brauer algebra C [B k (N )] (resp. symmetric algebra C [S k ]) is the sub-algebra of C [P k (N )] generated by the elements of B k (resp. the elements of S k ). Notation 2. 1. -In all the paper, A k will stand either for P k or B k or S k . Thus for any
= N
N ∈ N, C[A k (N )] will stand for C[P k (N )], C[B k (N )] or C[S k (N )].
Let us remark that actually, the algebra C[S k (N )] does not depend on N . We can see any permutation σ ∈ S k as a bijection from {1, . . . , k} to itself: for any i ∈ {1, . . . , k} there exists a unique j ∈ {1 ′ , . . . , k ′ } such that {i, j ′ } ∈ σ, we set σ(i) = j. For any permutations σ 1 and σ 2 , the bijection associated with σ 1 σ 2 is the composition of the two bijections associated with σ 1 and σ 2 .
We can extend the operations of transposition, tensor product and multiplication on the partition algebra, by linearity or bi-linearity.
The sub-algebra C[S k ] is not only stable for the • operation. It also satisfies the following property which can be proved by looking at the propagating number.
Lemma 2.1. -Let p, p ′ ∈ P k , if p • p ′ ∈ S k then p and p ′ are in S k .
Besides, for any partition σ ∈ S k and any p ∈ A k , κ(σ, p) = κ(p, σ) = 0. Let us remark that, for any integer N , the algebras C[A k (N )] have the same neutral element, denoted by id k or id, for the product operation: . . , k} , whose diagram for k = 5 is drawn in figure 8. A consequence of Lemma 2.1 is that, as id k ∈ S k , the only invertible elements of A k (N ), for the multiplication operation, are the permutations. The inverse of a permutation σ is σ −1 = t σ.
id k = {i, i ′ }, i ∈ {1, .
We will later need some special permutations. . . , i l } with i 1 < · · · < i l . We define σ I the permutation which sends i j on j for any j ∈ {1, . . . , l} and i / ∈ I on l + i − #{n, i n < i}. This is the partition:
σ I = {i j , j ′ }, j ∈ {1, . . . , l} ∪ {i, (l + i − #{n, i n < i}) ′ }, i / ∈ I . Definition 2.6. -The transposition (1, 2) is the partition σ {2} in P 2 defined by: (1, 2) = {1, 2 ′ }, {2, 1 ′ } .
The Weyl contraction is the Brauer element in P 2 defined by:
[1, 2] = {1, 2}{1 ′ , 2 ′ } .
These partitions are drawn in Figure 9.
(i, j) = σ −1 {i,j} (1, 2) ⊗ Id k−2 σ {i,j} = {{i ′ , j}, {i, j ′ }} ∪ {{l, l ′ }, l / ∈ {i, j}}.
The set of transpositions on k elements is:
T k = (i, j), i, j ∈ {1, . . . , k}, i = j .
The Weyl contraction [i, j] in B k is:
[i, j] = σ −1 {i,j} [1, 2] ⊗ Id k−2 σ {i,j} = {{i, j}, {i ′ , j ′ }} ∪ {{l, l ′ }, l / ∈ {i, j}}.
Due to the remark we made after Lemma 2.1, the product does not depend on which C[B k (N )] one considers to define the product. We denote by W k the set of Weyl contractions in B k :
W k = [i, j], i, j ∈ {1, . . . , k}, i = j .
A notion linked with the tensor operation, which will be central in the asymptotic freeness results in the article [9], is the notion of irreducibility of partitions. Let k be an integer. Let p ∈ P k . Definition 2. 8. -A cycle of p is a block of p ∨ id. The set of cycles of p is denoted by C(p). The number of cycles of p is denoted by c(p). The partition p is composed if c(p) > 1. The partition p is irreducible if it is not composed. By convention, the empty partition is irreducible.
Let us consider the set of irreducible partitions. Definition 2. 9. -For any integer k, we will denote by A
(i) k the set of irreducible parti- tions of A k .
It has to be noted that for any integer k:
S (i) k = {σ(1, . . . , k)σ −1 , σ ∈ S k }.
The partition p ∈ P k is composed if and only if there exist p 1 and p 2 two partitions non equal to the empty partition, and I a subset of {1, . . . , k} such that #I = l(p 1 ), l(p 2 ) = k − #I and:
σ −1 I (p 1 ⊗ p 2 )σ I = p.
Let us define the decomposition of p into two partitions.
Definition 2. 10. -The set of decompositions of p into two partitions is:
F 2 (p) = (p 1 , p 2 , I) , σ −1 I (p 1 ⊗ p 2 )σ I = p .
Let us remark that for any partition, even the irreducible partitions, F 2 (p) = ∅. For example, if p is irreducible:
F 2 (p) = {(p, ∅, {1, . . . , k}), (∅, p, ∅)}.
Let also remark that F 2 (∅) = { (∅, ∅, ∅)}.
We will need a notion of weak irreducibility later: this is based on the notion of extraction and restriction. For any partition p we have a lot of choice in order to represent p as a graph: the complete graph which represents p is the graph such that i and j, two elements of {1, ..., k} ∪ {1 ′ , ..., k ′ } are linked if and only if i and j are in the same block of p.
Definition 2.11. -Let k be an integer, let p be in P k . Let J be a subset of {1, . . . , k} ∪ {1 ′ , . . . , k ′ }. Let us denote by J s the symmetrization of J:
J s = J ∪ {j ∈ {1 ′ , . . . , k ′ }, ∃i ∈ J ∩ {1, . . . , k}, j = i ′ } ∪ {i ∈ {1, . . . , k}, i ′ ∈ J}.
We define:
-The extraction of p to J, denoted p J . Let us take the complete graph which represents p, let us erase all the vertices which are not in J s and all the edges which are not from two vertices in J s and at last let us label, from left to right the vertices. This is the graph of p J . -The restriction of p to J, denoted p |J . Let us take the complete graph which represents p, let us erase all the edges which are not from two vertices in J and let us connect each i / ∈ J with i ′ . This is the graph of p |J .
By convention, if J s = {1, . . . , k} ∪ {1 ′ , . . . , k ′ }, then p J = ∅ and p |J = id.
Definition 2. 12.
-The support of p is:
S(p) = {1, . . . , k} \ {i ∈ {1, . . . , k}, {i, i ′ } ∈ p}.
The partition p is weakly irreducible if p S(p) is irreducible. In particular Id k is weakly irreducible.
2.2.
Partitions and representation. -In this section, we define a natural action of the partition algebra (and by restriction of the Brauer and of the symmetric algebra) on C N ⊗k . This action will be useful in order to translate combinatorial properties into linear algebraic properties.
Let N and k be two integers. Let k and N be any integers. We can now define the action of the partition algebra C[P k (N )] on C N ⊗k . Let (e 1 , . . . , e N ) be the canonical basis of C N .
Definition 2.14. -The action of the partition algebra C[P k (N )] on C N ⊗k is defined by the fact that for any p ∈ P k , for any (i 1 , . . . , i k ) ∈ {1, . . . , N } k :
p.(e i 1 ⊗ · · · ⊗ e i k ) = (i 1 ′ ,...,i k ′ )∈{1,...,N } k p i 1 ,...,i k i 1 ′ ,...,i k ′ e i 1 ′ ⊗ · · · ⊗ e i k ′ .
This action defines a representation of the partition algebra C[P k (N )] on C N ⊗k which we denote by ρ P k N :
ρ P k N : C[P k (N )] → End C N ⊗k .
Let us define E j i be the matrix which sends e j on e i . Let p be a partition in P k . We can write the matrix of ρ P k N (p) in the basis (e i 1 ⊗ · · · ⊗ e i k ) (i l ) k l=1 ∈{1,...,N } k :
ρ P k N (p) = (i 1 ,...,i k ,i 1 ′ ,...,i k ′ )/p i 1 ,...,i k i 1 ′ ,...,i k ′ =1 E i 1 ′ i 1 ⊗ . . . ⊗ E i k ′ i k .(1)
For example, if p is the transposition (1, 2), then:
ρ P 2 N ((1, 2)) = N a,b=1 E b a ⊗ E a b . Figure 10. i1,i2,i3,i4,i5 E i1 i1 ⊗ E i3 i2 ⊗ E i4 i3 ⊗ E i4 i4 ⊗ E i5 i3 = ρ P5 N (p 1 ).
Let us suppose that N ≥ 2k. Using the Theorem 3.6 in [11], the application ρ P k N is injective. Actually, if one considers only its restriction to the symmetric algebra or the Brauer algebra, it is enough to ask for N ≥ k. For N = k − 1 this result does not hold, this is a consequence of the Mandelstam's identity which asserts that:
σ∈S k (−1) ǫ(σ) ρ P k k−1 (σ) = 0, where ǫ(σ) is the signature of σ.
Let us remark that the natural action of C[P k (N )] on C N ⊗k behaves well under the operation of product tensor.
Lemma 2.2. -Let k and k ′ be two integers. Let p ∈ C[P k ] and p ′ ∈ C[P k ′ ].
We have for any integer N :
ρ P k+k ′ N (p ⊗ p ′ ) = ρ P k N (p) ⊗ ρ P k N (p ′ ).
The exclusive basis of
C[P k ]. -
The basis used to define the partition algebra is quite natural, yet, it is not always very easy to work with. Indeed, if we look at the representation ρ P k N of a partition, we see that the condition we used to define the delta function is not exclusive. It means that we did not use the following exclusive delta function: By changing in Definition 2.14 the delta function defined in Definition 2.13 by this new exclusive delta function, we define a new function:
(p i 1 ,...,i k i 1 ′ ,...,i k ′ ) ex = 1,ρ k N : C[P k (N )] → End C N ⊗k .
Does it exist, for any partition p ∈ P k an element p c ∈ C[P k ] such that for any integer N , ρ P k N (p c ) =ρ P k N (p) ? The answer is given by the following definition, as explained by Equation (2.3) of [11].
Definition 2. 15. -For any k ∈ N. We define the family (p c ) p∈P k as the only family of elements in C[P k ] defined by the relation:
p = p ′ coarser than p p ′c .
The notion of being coarser defines a partial order on P k : the relation can be inverted. The family (p c ) p∈P k is well defined and it is a basis of the partition algebra C[P k ]. We will call (p c ) p∈P k the exclusive partition basis, it satisfies the following proposition.
ρ P k N (p c ) =ρ P k N (p).
Geometry on the set of partitions
Let k be an integer. In this section, we define a new geometry on the set of partitions P k which generalizes some well-known geometry on the symmetric group. We will see three ways to construct a distance on P k : one will allow us to work with linear algebra, another to compute the distance in a combinatorial way, and the last one will use a graph which we will consider as the generalized Cayley graph of P k .
Depending on the context, we will consider a partition either as an element of P k or as an element of End C N ⊗k via the action defined in Definition 2.14. We remind the reader that (e 1 , . . . , e N ) is the canonical base of C N . The family {e i 1 ⊗ · · · ⊗ e i k , (i 1 , . . . , i k ) ∈ {1, . . . , N } k } is a basis of C N ⊗k : let T r k be the trace with respect to this canonical basis. We do not renormalize it, thus T r k Id (C N ) ⊗k = N k . We can define the trace of a partition.
Definition 3.1. -Let k and N be two integers, let p be a partition in P k . We define:
T r N (p) = T r k ρ P k N (p) . For any integer N , we extend T r N by linearity to C[P k (N )].
Let us remark that, if one does not want to use the representation ρ P k N , one could have also define the trace by defining for any partition p ∈ P k ,
T r N (p) = N nc(p∨id) .(2)
We can now define a distance on P k . Proposition 3.1. -Let k and N be two integers, let p and p ′ be two elements of P k . The number:
d(p, p ′ ) = −log N T r N ( t pp ′ ) T r N ( t pp)T r N ( t p ′ p ′ )
does not depend on N : it is called the distance between p and p ′ .
The fact that d(p, p ′ ) does not depend on N is a consequence of Lemma 3. 1. Actually we have not prove yet that it is a distance, even if it is fairly easy to see that it satisfies the strict positivity property: it is a consequence of the Cauchy-Schwarz's inequality.
The easiest way to prove that d(p, p ′ ) does not depend on N is to show that it is a combinatorial object.
Lemma 3.1. -Let k and N be two integers, for any p and p ′ in P k :
d(p, p ′ ) = 1 2 nc(p) + nc(p ′ ) − nc(p ∨ p ′ ).
Proof. -This is a consequence of the following equality which holds for any p and p ′ in P k and any positive integer N :
T r N ( t pp ′ ) = N nc( t p•p ′ ∨id)+κ( t p,p ′ ) = N nc(p∨p ′ ) ,(3)
which is a consequence of Equality 2 and the combinatorial equality:
nc( t p • p ′ ∨ id) + κ( t p, p ′ ) = nc(p ∨ p ′ ).
This latter equality can be understood by flipping the diagram of t p over the one of p ′ : the flip transposes t p thus we get the two diagrams of p and p ′ one over the other. By definition, the diagram constructed by putting a diagram representing p ′ over one representing p is associated with p ∨ p ′ .
It remains to show that d satisfies the triangular inequality on the set of partitions P k . For that we will show that it is a geodesic distance on a graph. -Let k be an integer. We define the weighted graph G k = (V k , E k , w k ) such that:
the set of vertices V k is P k , -there exists an edge e in E k between p and p ′ two elements of P k if and only if:
• one can go from one to the other by gluing two blocks. Let us suppose that we can go from p to p ′ . If p is the partition {p 1 , . . We gave this definition so that the reader can understand easily why this graph is a generalization of the usual Cayley graph. Yet, there is an other graph which will be used in Proposition 3. 2
. Let us define
G ′ k = (V ′ k , E ′ k , w ′ k )
such that: -the set of vertices V ′ k is P k , -there exists an edge in E ′ k between p and p ′ two elements of P k if and only if one can go from one to the other by gluing two blocks, -the weight function w ′ k is constant equal to 1/2. The graphs G k and G ′ k are interesting as they allow us to better understand the distance d.
Proposition 3.2. -Let k be an integer. Let p and p ′ be two elements of P k . Let us define C G k (p, p ′ ) (resp. C G ′ k (p, p ′ )) the set of paths π in G k (resp. G ′ k ) which begin in p and finish in p ′ . Let us define the geodesic distance on G k and on G ′ k between p and p ′ by:
d G k (p, p ′ ) = min π∈C G k (p,p ′ ),π=e 1 ...e l
w(e 1 ) + · · · + w(e l ),
d G ′ k (p, p ′ ) = min π∈C G ′ k (p,p ′ ),π=e 1 ...e l
w(e 1 ) + · · · + w(e l ).
We have the equalities:
d(p, p ′ ) = d G k (p, p ′ ) = d G ′ k (p, p ′ ). Proof. -Let p and p ′ be two elements of P k . It is enough to prove that d G k (p, p ′ ) = d G ′ k (p, p ′ ) and d(p, p ′ ) = d G ′ k (p, p ′ ). First, let us show that d G k (p, p ′ ) = d G ′ k (p, p ′distance d G k (p, p ′ ), it is enough to look only at paths in G ′ k : d G k (p, p ′ ) = d G ′ k (p, p ′ ). Then, let us show that d(p, p ′ ) = d G ′ k (p, p ′ ).
For this, let us see what happens to the distance d(p, p ′ ) between p and p ′ when one moves from p ′ to one neighborhood of p ′ in G ′ k . Suppose first that we glue two blocks of p ′ , then nc(p) is constant, nc(p ′ ) decreases by 1 and nc(p ∨ p ′ ) stays constant or decreases by 1. In this case d(p, p ′ ) will increase or decrease by 0. 5. If we cut one block of p ′ , then nc(p) is constant, nc(p ′ ) increases by 1 and nc(p ∨ p ′ ) stays constant or increases by 1. In this case d(p, p ′ ) will also increase or decrease by 0. 5.
Thus a gluing/cutting can at most increase the value of d(p, p ′ ) by 0. 5
. It implies that d(p, p ′ ) ≤ d G ′ k (p, p ′ ). We have to show now that d G ′ k (p, p ′ ) ≤ d(p, p ′ ).
Let us remark that p ∨ p ′ is coarser than p: we can go from p to p ∨ p ′ by doing nc(p) − nc(p ∨ p ′ ) gluing of blocks of p. The same holds for p ′ : we can go from p ′ to p ∨ p ′ by doing nc(p ′ ) − nc(p ∨ p ′ ) gluing of blocks of p ′ . Thus one can go from p to p ∨ p ′ and then from p ∨ p ′ to p in
nc(p) + nc(p ′ ) − 2nc(p ∨ p ′ ) steps in G ′ k . Thus d G ′ k (p, p ′ ) ≤ 1 2 [nc(p ′ ) + nc(p ′ ) − 2nc(p ∨ p ′ )] = d(p, p ′ ).
The function d G ′ k is a geodesic distance on a graph: it is thus a distance. As we have just shown that d = d G ′ k , the next corollary is immediately proved.
Corollary 3.1. -The function d : P k × P k → R + is a distance.d(σ, σ ′ ) = k − nc(σ −1 σ ′ ),
for any σ, σ ′ ∈ S k . This distance is in fact the geodesic distance on the Cayley graph S k of S k .
By Lemma 6.26 of [12], the restriction of the distance d to B k is also the geodesic distance on the Cayley graph B k of B k .
Using this distance, we can define a notion of set-geodesic for any of the three sets of partitions we are interested in. We remind the reader that the notation A k was settled in Notation 2.1.
Definition 3.3. -Let p ∈ A k , the set-geodesic [id, p] A k is defined by: [id, p] A k = p ′ ∈ A k , d(id, p) = d(id, p ′ ) + d(p ′ , p) .
A geodesic in a graph between two vertices p and p ′ is a path in this graph which length is equal to the geodesic distance. Using Proposition 3.2 and Lemma 3.2, one shows that for any p ∈ A k , the set-geodesic [id, p] A k is the union of the geodesics between id and p in the Cayley graph of A k .
The distance on A k allows to define a new partial order on A k .
Definition 3.4. -Let p and p ′ be elements of A k , we write that p ≤ p ′ if and only if d(id, p) = d(id, p ′ ) + d(p, p ′ ).
This is a partial order as the restriction of d to A k × A k is a distance. In the following lemma, we show that the geodesic in the Cayley graph of P k between two permutations either stay in the set of permutations or intersect P k \ B k . Using the fact that [id, p] A k is the union of the geodesics between id and p in the Cayley graph of A k , we get an equality between [id, σ] B k and [id, σ] S k .
Lemma 3.3. -Let k be an integer. Let σ ∈ S k , then: [id, σ] B k = [id, σ] S k .
Proof. -Let k be an integer. We will do a proof by contradiction. Let S ⊂ S k be the set of permutations such that:
[id, σ] B k = [id, σ] S k . Let σ ∈ S be a permutation such that d(id, σ) = min σ ′ ∈S d(id, σ ′ ). Let b be an element of B k \S k such that b ∈ [id, σ] B k .
There exists a geodesic in B k which goes through b and goes from id to σ.
Thus, there exists b ′ ∈ B k such that d(id, b ′ ) = 1 and b ′ ∈ [id, σ] B k . The element b ′ can not be a permutation. Indeed, if b ′ was a permutation, then [b ′ , σ] B k = [b ′ , σ] S k and thus, [id, b ′−1 σ] B k = [id, b ′−1 σ] S k . Yet d(id, b ′−1 σ) = d(b ′ , σ) = d(id, σ) − 1. This would contradict the fact that d(id, σ) = min σ ′ ∈S d(id, σ ′ ). Thus b ′ must be an element of B k \ S k . As d(id, b ′ ) = 1, there exist i and j in {0, . . . , k} such that: b ′ = [i, j], where [i, j] is the Weyl contraction in B k . Thus there exist i and j in {0, . . . , k} such that [i, j] ∈ [id, σ] B k .
Recall that c(σ) is the number of cycles of σ. We have:
d(id, [i, j]) + d([i, j], σ) − d(id, σ) = nc([i, j]) + nc(id ∨ σ) − nc([i, j] ∨ id) − nc([i, j] ∨ σ) = k + c(σ) − (k − 1) − nc([i, j] ∨ σ) = 1 + c(σ) − c(σ) + δ i
and j not in the same cycle of σ = 1 + δ i and j not in the same cycle of σ > 0.
Thus [i, j] / ∈ [id, σ] B k : this yields the contradiction.
This lemma is the key point which will allow us to explain in the second article [9] why processes on U (N ) and O(N ) have the same limit when one only considers usual moments.
The last property, known in the S k and B k case, is still true for P k : a geodesic between id and p 1 ⊗ p 2 must be the tensor product of the geodesic between id and p 1 and the geodesic between id and p 2 .
Lemma 3.4. -Let p ∈ A k , we have: [id, p] A k ≃ C∈C(p) id #C 2 , p C A #C 2 .
In particular if p 1 and p 2 are two partitions,
p 1 ∈ A k 1 and p 2 ∈ A k 2 , then p ′ ∈ [id, p 1 ⊗ p 2 ] A k 1 +k 2 if and only if there exist p ′ 1 ∈ A k 1 and p ′ 2 ∈ A k 2 such that p ′ = p ′ 1 ⊗ p ′ 2 .
Let us finish this section with two propositions on geodesics. Let us define a notion of default in order to simplify the proofs.
Definition 3.5. -Let p and p ′ be two elements of A k . We define the default of p ′ not being on the geodesic [id, p] A k by: df(p ′ , p) = d(id, p ′ ) + d(p ′ , p) − d(id, p).
A simple but useful lemma is the following.
Lemma 3.5. -Let k be an integer, let p ∈ P k and p ′ ∈ P k such that p is coarser than p ′ . Then:
df(p ′ , p) = nc(p ′ ) − nc(p ′ ∨ id) − nc(p) + nc(p ∨ id).
Proof. -This is a simple calculation, where one has to use the fact that nc(p ∨ p ′ ) = nc(p) since p is coarser than p ′ .
Proposition 3.3. -Let p and p ′ be two partitions in P k . Then p ′ ∈ [id, p] A k if and only if nc(p ∨ p ′ ∨ id) = nc(p ∨ id) and p ′ ∈ [id, p ∨ p ′ ] A k .
One can see this proposition as a direct consequence of the forthcoming Theorem 10.4, by considering the element of E[A] which is equal to p, with p ∈ P k . Yet, we give a direct proof: the proof is simple yet, without Theorem 10.4, it would have been trickier to guest the following proposition.
Proof of Proposition 3. 3. -Let p and p ′ be two partitions in P k . Using the Lemma 3.4, we see that
p ′ ∈ [id, p] P k if and only if nc(p ∨ p ′ ∨ id) = nc(p ∨ id) and p ′ ∈ [id, p] P k , thus, if and only if nc(p ∨ p ′ ∨ id) = nc(p ∨ id) and: df(p ′ , p) = nc(p ′ ) − nc(p ′ ∨ id) − nc(p ∨ p ′ ) + nc(p ∨ id) = 0, which is equivalent to nc(p ∨ p ′ ∨ id) = nc(p ∨ id) and: df(p ′ , p ∨ p ′ ) = nc(p ′ ) − nc(p ′ ∨ id) − nc(p ∨ p ′ ) + nc(p ∨ p ′ ∨ id) = 0, which is again equivalent to nc(p ∨ p ′ ∨ id) = nc(p ∨ id) and p ′ ∈ [id, p ∨ p ′ ] P k .
For the last geometric proposition, we need to define the left and right parts of a partition p.
Definition 3. 6. -Let k and l be two integers. Let p ∈ A k+l , we denote by p g k the extraction of p to {1, ..., k} and p d k the extraction of p to {k + 1, ..., k + l}. The partition p g k is in P k and p d k is in P l .
Convergence of elements of
N ∈N C [P k (N )]
Coordinate numbers and moments. -
4.1.1. Definitions. -Let k be an integer, recall the notation A k defined in Notation 2.1. For each integer N , we have defined an algebra C[A k (N )]. Let (E N ) N ∈N be a sequence such that for any integer N , E N ∈ C[A k (N )]. For each integer N , the algebra C[A k (N )]
, seen as a vector space has the same basis A k . Thus, we could study the convergence of (E N ) N ∈N only from the vector space point of view by saying that the sequence (E N ) N ∈N converges if and only if the coordinates of E N in the basis A k converge. Actually, this convergence forgets the fact that C[A k (N )] is an algebra which depends on an integer N . In order to define a better definition of convergence, we have to define the coordinate numbers of
E in C[A k (N )]. Definition 4.1. -Let N be an integer. Let E be an element of C[A k (N )].
We define the numbers κ p (E) p∈A k as the only numbers such that:
E = p∈A k κ p (E) N −k+nc(p) 2 +d(id,p) p.
The family (κ p (E)) p∈A k (N ) is called the coordinate numbers of E.
After Definition 4.4, we will explain how we get this definition, and why this definition is in fact the most natural thing one can do. We will need to use the following equality: for any integer k, for any p ∈ A k ,
−k + nc(p) 2 + d(id, p) = nc(p) − nc(p ∨ id).(4)
This implies the following remark.
E = p∈A k κ p (E) N nc(p)−nc(p∨id) p.
We will consider the coordinate numbers as linear applications from C[A k (N )] to R:
κ p : C[A k (N )] → R E → κ p (E).
The notion of coordinate numbers allows us to define a strong convergence for any
sequence (E N ) N ∈N ∈ N ∈N C[A k (N )]. Definition 4.2. -Let (E N ) N ∈N be an element of N ∈N C[A k (N )].
The sequence (E N ) N ∈N converges strongly if the coordinate numbers of E N converges when N goes to infinity: for any p ∈ A k , κ p (E N ) converges when N goes to infinity.
The goal now is to give a dual definition of convergence. We have seen in Definition 2.14 that any element of C[A k (N )] can be seen as an element of End C N ⊗k and we defined in Definition 3.1 the trace of any element C[A k (N )]. Using this trace and the structure of algebra of C[A k (N )], we define, for any element of C[A k (N )] and any element p ∈ A k , the p-normalized moment of E.
Definition 4.3. -Let N ∈ N, let p ∈ A k and E ∈ C[A k (N )]. The p-normalized moment of E is: m p (E) = 1 T r N (p) T r N (E t p).
Using these normalized moments, we can define a weak notion of convergence for any
sequence (E N ) N ∈N ∈ N ∈N C[A k (N )].
Coordinate numbers-moments transformation.
-We can now explain how we ended up with Definition 4.1 and we had the idea to define the distance on the set of partitions. The idea behind these definitions is that we want to know, given a sequence of E N ∈ C[A k (N )], how the usual coordinates of E N in the basis A k must scale so that for any p ∈ A k , m p (E N ) converges when N goes to infinity. Let N be an integer, we have
E N = p∈A k a p N p. Thus m p 0 (E N ) = p∈A k T r N (p t p 0 ) T r N (p 0 ) a p N .
Thus the vector m N = (m p 0 (E N )) p 0 and a N = (a p N ) p are linked by the relation
m N = M N a N where M N = T r N (p t p 0 ) T r N (p 0 ) p 0 ,p .
There are then two possible possibilities: to invert M N for N big enough. This is the usual way, which leads to the Weingarten function. Or, one can make the following Ansatz: if we write the system, we see that for any p, (a N ) p is going to be multiplied by (M N ) p 0 ,p for any p 0 ∈ A k . Thus we make the assumption that (m N ) p must grow as the inverse of the maximum of (M N ) p 0 ,p over p 0 . That is a p N ∼ a p N −η(p) , where η p is given by:
η p = sup p 0 lim N →∞ log N T r N (p t p 0 ) T r N (p 0 ) .
The goal now is to know in which p 0 the supremum is obtained. It is more than tempting, seeing the scalar product T r N (p 0 t p) to write what is inside the log N as:
T r N (p t p 0 ) T r N (p 0 ) = T r N (p t p 0 ) T r N (p t p)T r N (p 0 t p 0 ) T r N (p 0 t p 0 )T r N (p t p) T r N (p 0 ) = T r N (p t p 0 ) T r N (p t p)T r N (p 0 t p 0 ) T r N (p 0 t p 0 )T r N (id k t id k ) T r N (p 0 t id k ) T r N (p t p) T r N (id k t id k )
.
We recognize thus the distance that we defined. In fact the intuition that is should be a distance comes from the fact that one can write:
η p = sup p 0 [−d(p, p 0 ) + d(p 0 , id k )] + 1 2 (−k + nc(p)).
If d was a distance, then by the triangle inequality, for any p 0 ,
d(p 0 , id k ) − d(p 0 , p) ≤ d(p, id k ).
This shows that the supremum is obtained at p 0 = p, and thus the Ansatz tells us that:
a p N ∼ a p N −[ 1 2 (−k+nc(p))+d(id,p)]
, to be compared with the Definition 4. 1.
The first main result is given by Theorem 4.1 which shows the equivalence between strong and weak convergence.
Theorem 4.1. -Let (E N ) N ∈N be a sequence such that for any N ∈ N, E N ∈ C[A k (N )].
It converges strongly if and only if it converges in moments. Let us suppose that (E N ) N ∈N converges in moments or strongly, for any p ∈ A k :
lim N →∞ m p (E N ) = p ′ ∈[id,p] A k lim N →∞ κ p ′ (E N ). (5) Proof. -Let (E N ) N ∈N be an element of N ∈N C[A k (N )]
, let p ∈ A k and let N be an integer.
Using the coordinate numbers of E N , we can calculate the p-normalized moments of E:
m p (E N ) = 1 T r N (p) T r N (E N t p) = 1 T r N (p) T r N p ′ ∈A k (N ) κ p ′ (E N ) N −k+nc(p ′ ) 2 +d(id,p ′ ) p ′ t p = p ′ ∈A k κ p ′ (E N ) T r N (p ′ t p) T r N (p)N −k+nc(p ′ ) 2 +d(id,p ′ )
.
Using the definition of the distance, in Proposition 3.1, one has:
T r N (p ′ t p) = N −d(p,p ′ )+ nc(p)+nc(p ′ ) 2 , T r N (p) = N −d(id,p)+ nc(p)+k 2 .
Thus:
m p (E N ) = p ′ ∈A k κ p ′ (E N )N −d(p,p ′ )+ nc(p)+nc(p ′ ) 2 +d(id,p)− nc(p)+k 2 + k−nc(p ′ ) 2 −d(id,p ′ ) .
Hence:
m p (E N ) = p ′ ∈A k κ p ′ (E N )N d(id,p)−d(id,p ′ )−d(p,p ′ ) .(6)
Let us suppose that (E N ) N ∈N converges strongly. The triangular inequality for d shows that for any p ∈ A k (N ) converges when N goes to infinity. Besides, it allows us to write that for any p ∈ A k (N ):
lim N →∞ m p (E N ) = p ′ ∈[id,p] A k lim N →∞ κ p ′ (E N ).
Now, let us suppose that it converges in moments. We can write (6) as:
m N = G N κ N ,
where:
m N = (m p (E N )) p∈A k (N ) , κ N = (κ p (E N )) p∈A k (N ) , G N = N d(id,p)−d(id,p ′ )−d(p,p ′ ) p,p ′ ∈A k (N )
.
Thus the sequence (G N ) N ∈N converges to the matrix of the partial order ≤ defined in Definition 3.4: lim
N →∞ G N = G, where G p,p ′ = δ p≤p ′ . This last matrix is invertible, thus κ N = G −1 N m N converges to G −1 m where m = (lim N →∞ m p (E N )) p∈A k .
Let us take some notations in order to simplify our up-coming discussions.
Notation 4.1. -Let (E N ) N ∈N be an element of N ∈N C[A k (N )]
. From now on, we will say that (E N ) N ∈N converges if and only if it converges either strongly or in moments. Besides, let suppose that (E N ) N ∈N converges, then we will set, for any partition p ∈ A k and any P ⊂ A k :
m p (E) = lim N →∞ m p (E N ), κ p (E) = lim N →∞ κ p (E N ) κ P (E) = p∈P κ p (E).
Consequences of
S or B. Let (E N ) N ∈N be an element of N ∈N C[A k (N )]
which converges in moments, then for any p ∈ P k , the limit of m p (E N ) exists. Besides, for any p ∈ P k , the following equality holds:
m p (E) = p ′ ∈A k ,p ′ ∈[id,p] P k κ p ′ (E).
In the case where A = B, one can also prove that, under some hypothesis, the convergence of the S-moments is equivalent to the convergence of the S-coordinate numbers.
Theorem 4.3. -Let (E N ) N ∈N be an element of N ∈N C[B k (N )]
and let us suppose that
for any p ∈ B k , (m p (E N )) N ∈N is bounded.
The following assertions are equivalent: -for any σ ∈ S k , κ σ (E N ) converges when N goes to infinity, -for any σ ∈ S k , m σ (E N ) converges when N goes to infinity. and if one of the condition is satisfied, then for any σ ∈ S N ,
m σ (E) = σ ′ ∈[id,σ] S k κ σ ′ (E). Proof. -Let (E N ) N ∈N be an element of N ∈N C[B k (N )]
which satisfies the hypothesis of the theorem. First of all, using the same notations of the proof of Theorem 4.1, we know that, for N big enough κ N = G −1 N m N . As the sequence (m N ) N ∈N is bounded and as G −1 N converges to G −1 when N goes to infinity, we deduce that (κ N ) N ∈N is also bounded. Let σ ∈ S k . Using the Equation (6), for any integer N ,
m σ (E N ) = p ′ ∈B k κ p ′ (E N )N d(id,σ)−d(id,p ′ )−d(σ,p ′ ) . Yet, if p ′ ∈ B k \ S k , using Lemma 3.3, d(id, σ) − d(id, p ′ ) − d(σ, p ′ ) < 0.
Let us suppose that for any σ ′ ∈ S k , κ σ ′ (E N ) converges, then m σ (E N ) converges as N goes to infinity, and:
lim N →∞ m σ (E N ) = p ′ ∈[id,σ] S k κ p ′ (E N ).
Let us suppose now that for any σ ∈ S k , m σ (E N ) converges when N goes to infinity, then for any increasing sequence (i N ) N ∈N of integers such that for any
σ ′ ∈ S k , κ σ ′ (E i N ) converges, we have: lim N →∞ m σ (E N ) = p ′ ∈[id,σ] S k lim N →∞ κ p ′ (E i N ).
Hence, for any p ′ ∈ S k , lim N →∞ κ p ′ (E i N ) does not depend on the sequence (i N ) N ∈N : this shows that for any σ ′ ∈ S k , κ p ′ (E N ) converges when N goes to infinity. Again we get also:
lim N →∞ m σ (E N ) = p ′ ∈[id,σ] S k κ p ′ (E N ).
This finishes the proof. 4. 3. Exclusive coordinate numbers and exclusive moments.
E = p∈P k κ p c (E) N d(id,p)+ −k+nc(p) 2 p c = p∈P k κ p c (E) N nc(p)−nc(p∨id) p c .
The family (κ p c (E)) p∈P k is called the exclusive coordinate numbers of E.
The next proposition shows that one can choose to work either with the exclusive basis or with the usual basis of C[P k ] in order to study the convergence of ( Besides, if (E N ) N ∈N converges then for any p ∈ P k , κ p c (E N ) converges as N goes to infinity, and for any p ∈ P k :
E N ) N ∈N ∈ N ∈N C[A k (N )].lim N →∞ κ p c (E N ) = p ′ ∈A k ,p ′ finer than p,p ′ ∈[id,p] P k lim N →∞ κ p ′ (E N ) Proof. -Let k be an integer, let (E N ) N ∈N be an element of N ∈N C[A k (N )]
. Then for any integer N :
E N = p∈A k κ p (E N ) N nc(p)−nc(p∨id) p = p∈A k κ p (E N ) N nc(p)−nc(p∨id) p ′ coarser than p,p ′ ∈P k p ′c = p∈A k ,p ′ coarser than p,p ′ ∈P k κ p (E N )N −nc(p)+nc(p∨id)+nc(p ′ )−nc(p ′ ∨id) p ′c N nc(p ′ )−nc(p ′ ∨id) ,
and using Lemma 3.5:
E N = p ′ ∈P k p∈A k ,p finer than p ′ κ p (E N )N −df(p,p ′ ) p ′c N nc(p ′ )−nc(p ′ ∨id) .
Thus, for any integer N , for any p ′ ∈ P k
κ p ′ c (E N ) = p∈A k ,p finer than p ′ κ p (E N )N −df(p,p ′ ) .(7)
The result follows from this equality, and the usual arguments already explained in Theorem 4.1.
Let us remark that, using the Equality (7), one has the following proposition.
Proposition 4.1. -Let A be either S or B. Let N be an integer, let E ∈ C[A k (N )], for any p ∈ A k : κ p c (E) = κ p (E).
Proof. -This is a consequence of Equality (7) and the fact that p ′ in P k is finer than
p ′ ∈ A k implies that p ′ / ∈ A k .
Exclusive moments.
-As we did for the coordinate numbers, one can define exclusive normalized moments.
Definition 4.6. -Let N ∈ N, let p ∈ P k and E ∈ C[A k (N )]. The p-exclusive normalized moment of E is: m p c (E) = 1 T r N (p) T r N (E t (p c )).
One can also give a combinatorial definition of the p-exclusive normalized moment.
Lemma 4.1.
-Let p and p ′ be in P k , then:
T r N (p t (p ′c )) = δ p ′ coarser than p N ! (N − nc(p ′ ))! .
The easiest way to prove this lemma is to do it graphicaly: we see that p ′ must be coarser than p, if not the trace is equal to zero, and if p ′ is coarser than p, it is equal to
N ! (N −nc(p ′ ))! .
Definition 4.7. -Let p and p ′ be in P k . We say that p ′ is coarser-compatible than p if and only if p ′ is coarser than p and nc(p ∨ id) = nc(p ′ ∨ id) and p ′ is coarser than p.
The condition p ′ coarser compatible with p just means that one can glue only blocks of p which are in the same cycle in order to get p ′ . Similarly to what we proved for coordinate numbers, we prove the following proposition. Let us consider (
E N ) N ∈N ∈ N ∈N C[A k (N )].lim N →∞ m p (E N ) = p ′ ∈P k ,p ′ coarser-compatible than p lim N →∞ m p ′c (E N ). Proof. -It is enough to consider (E N ) N ∈N an element of N ∈N C[P k (N )]. By computa- tion: m p (E N ) = p ′ coarser than p N nc(p ′ ∨id)−nc(p∨id) m p ′c (E N ).
We are in the same setting as for the proof of Theorem 4.1: we can write this equality as:
m N = G N m c,N , where (m N ) p = m p (E N ), (m c,N ) p = m p c (E N )
and G N converges to the matrix of the partial order of being coarser-compatible. With the same arguments than in the proof of Theorem 4.1, we get that m N converges to infinity if and only if m c,N converges to infinity: the sequence (E N ) N ∈N converges in P k −exclusive normalized moments if and only if it converges in normalized moments and in that case:
lim N →∞ m p (E N ) = p ′ coarser-compatible than p lim N →∞ m p ′c (E N ).
This finishes the proof. Proof. -We will prove that for any integer N , any p ∈ A k , seen as an element of C[A k (N )], for any p ′ ∈ P k ,
κ p ′ c (p) = nc(p ′ )−1 i=0 N N − k m p ′c (p).
Indeed by the Equality 7, we get that for any p ′ ∈ P k :
κ p ′ c (p) = δ p ′ coarser than p N nc(p ′ )−nc(p ′ ∨id) .(8)
Let p ′ ∈ P k , by Lemma 4.1:
m p ′c (p) = 1 N nc(p ′ ∨id) T r N (p t (p ′c )) = δ p ′ coarser than p N ! (N − nc(p ′ ))! N −nc(p ′ ∨id) .
The theorem is now a simple consequence of a linearity argument and taking N going to infinity.
Let us remark that one can prove Theorem 4.5 also by a purely combinatorial argument using Proposition 3.3, we give the proof below.
Combinatorial proof of 4.5. -It is enough to show that lim N →∞ κ p (E N ) satisfies the Equality in Proposition 4.2: for any p ∈ P k , lim N →∞
m p (E N ) = p ′ ∈P k ,p ′ coarser-compatible than p lim N →∞ κ p ′ c (E N ).
Using the fact that for any p ∈ P k :
lim N →∞ κ p c (E N ) = p ′ ∈A k ,p ′ finer than p,p ′ ∈[id,p] P k lim N →∞ κ p ′ (E N ),
we only have to prove that for any p ∈ P k :
lim N →∞ m p (E N ) = p ′ ∈P k ,p ′′ ∈A k ,p ′ coarser-compatible than p,p ′′ finer than p ′ ,p ′′ ∈[id,p ′ ] P k lim N →∞ κ p ′′ (E N ).
Using a slight modification of Proposition 3.3, there exists p ′ ∈ P k coarser than p ∨ p ′′ , coarser compatible than p and such that p ′′ ∈ [id, p ′ ] P k if and only if p ′′ ∈ [id, p] P k . Thus we only have to prove that for any p ∈ P k :
lim N →∞ m p (E N ) = p ′′ ∈A k ,p ′′ ∈[id,p] P k lim N →∞ κ p ′′ (E N ).
Using the Theorem 4.2, we can conclude.
Using
m p c (E N ) = p ′ ∈A k ,p ′ finer than p,p ′ ∈[id,p] P k lim N →∞ κ p ′ (E N ).
Besides, let us suppose until the end of the theorem that A is equal either to S or B, then for any N ∈ N and any p ∈ A k :
κ p (E N ) = nc(p ′ )−1 i=0 N N − k m p c (E N ).
In particular, for any p ∈ A k :
lim N →∞ κ p (E N ) = lim N →∞ m p c (E N ).
At the beginning of this section, we have argued that the simplest notion of convergence of elements of N ∈N C[A k (N )] was not interesting as it did not take into account the fact that C[A k (N )] is an algebra which depends on the parameter N . In the following section, we will slightly modify the product defined on C[A k (N )] in order to define a new algebra C[A k (N, N )]. In this new algebra the strong convergence will be the usual notion of convergence in vector spaces.
The deformed partition algebra
Let us define a deformation of the partition algebra by modifying the multiplication which was set in Definition 2.4.
Definition 5.1. -Let k and N be two integers. We define the application:
M N k : A k → A k p → 1 N d(id,p)+ −k+nc(p) 2 p.
This application can be extended as an isomorphism of vector spaces from C[A k ] to itself.
Let k and N be two integers. Seen as a vector space, the algebra
C[A k (N )] is isomorphic to C[A k ]. Thus, we can see M N k as an isomorphism of vector spaces from C[A k ] to C[A k (N )].
Let us endow C[A k ] with a structure of associative algebra by taking the pullback of the structure of algebra of C[A k (N )] by M N k : for any p 1 , p 2 in A k the new product of p 1 with p 2 is given by:
p 1 . N p 2 = M N k −1 M N k (p 1 )M N k (p 2 ) .
This is the deformed algebra C[A k (N, N )]. Using the definition of M N k , one gets the following proposition. (N, N )] is the associative algebra over C with basis P k , endowed with the multiplication defined by the fact that for any p 1 , p 2 ∈ A k :
p 1 . N p 2 = N κ(p 1 ,p 2 ) N d(id,p 1 •p 2 )−d(id,p 1 )−d(id,p 2 )+ k+nc(p 1 •p 2 )−nc(p 1 )−nc(p 2 ) 2 (p 1 • p 2 ).
One can write the exponent in an other form so that it looks like a triangle inequality.
Lemma 5.1. -Let p and p ′ in A k , let N be an integer. We have the equality:
d(id, p • p ′ ) − d(id, p) − d(id, p ′ ) + k + nc(p • p ′ ) − nc(p) − nc(p ′ ) 2 + κ(p, p ′ ) = d( t p ′ , p) − d(id, p) − d(id, p ′ ) + k + nc(p • p ′ ) − nc(p) − nc(p ′ ) + 2κ(p, p ′ ).
Proof. -For this, we consider N to the power to the r.h.s and the l.h.s. and we use the following equality:
N −d( t p ′ ,p) = T r(pp ′ ) N nc(p)+np(p ′ ) 2 = N κ(p,p ′ ) T r(p • p ′ ) N nc(p)+np(p ′ ) 2 = N κ(p,p ′ ) N −d(id,p•p ′ )+ k+nc(p•p ′ ) 2 N nc(p)+np(p ′ ) 2 .
This allows to prove Lemma 5.1.
Using the definition of the deformed algebra C[A k (N, N )], we have the straightforward proposition. Actually, the application M N k is not only compatible with the multiplication, but also with the ⊗ operation defined in Definition 2.1.
Lemma 5.2. -Let k, k ′ and N be any integers. Let p ∈ A k and p ′ ∈ A k ′ . The following equality holds:
M N k+k ′ (p ⊗ p ′ ) = M N k (p) ⊗ M N k ′ (p ′ ).(9)
The definition of the morphism M N k was not chosen randomly: it was set so that the following lemma holds.
Lemma 5.3. -Let E ∈ C[A k (N )], we have: (M N k ) −1 (E) = p∈A k κ p (E)p.
Thus, one can see that we will be able to formulate the strong convergence in
(M N k ) −1 (E N )
converges when N goes to infinity in C[A k ] for the usual convergence in finite dimensional vector spaces.
Refined geometry of the partition algebra
In the last section, we defined the deformed algebra C[A k (N, N )] and we explained that the strong convergence can be seen as the natural notion of convergence in finite dimensional vector space as soon as one works in the deformed algebra. In this section, we will study the convergence of the algebras C[A k (N, N )].
The core of Section 3 was to prove the triangular inequality for the function d defined on A k in Definition 3. 1. The study of the convergence of the algebras C[A k (N, N )] will use intensively the following improved triangular inequality for A k .
Proposition 6.1. -Let k be an integer. Let p and p ′ be two elements of P k . We have the following improved triangular inequality:
d(p ′ , p) ≤ d(p ′ , id) + d(p, id) − k − nc(p • t p ′ ) + nc(p) + nc(p ′ ) − 2κ(p, t p ′ ).
The restriction of the improved triangle inequality to the permutations is obvious as it is a consequence of the usual triangle inequality. Indeed, for any permutations σ and σ ′ , nc(σ) = 0 and κ(p, p ′ ) = 0. Yet, this is indeed an improved triangular inequality as soon as one considers elements on B k : let us suppose that p and p ′ are equal to the Weyl contraction [1,2]. The triangular inequality asserts that 0 ≤ 2, since d(id, [1, 2]) = 1. Yet, in this case:
d(p ′ , id) + d(p, id) − k − nc(p • t p ′ ) + nc(p) + nc(p ′ ) − 2κ(p, t p ′ ) = 0.
The improved triangular inequality asserts thus the stronger fact that 0 ≤ 0.
In fact, we can see this improved triangular inequality as a consequence of the usual triangular inequality and an inequality between d(p, p • p ′ ) and d(id, p ′ ). If we consider p and p ′ in the symmetric group, then we know that d(p, p • p ′ ) = d(p, pp ′ ) = d(id, p ′ ). Yet, this equality does not hold any more in the general case, we only get the following inequality.
Proposition 6.2. -Let k be an integer. Let p and p ′ in P k . We have the following inequality:
d(p, p • p ′ ) ≤ d(id, p ′ ) − k + nc(p • p ′ ) − nc(p) − nc(p ′ ) 2 − κ(p, p ′ ).
Proof. -Let k be an integer. Let p and p ′ in P k . Let us define τ ∈ S k :
τ k = (1, k + 1)(2, k + 2) . . . (k, 2k).
Let us apply the triangular inequality:
d p ⊗ id k , (p • p ′ ) ⊗ id k τ ≤ d(p ⊗ id k , p ⊗ p ′ ) + d p ⊗ p ′ , (p • p ′ ) ⊗ id k τ .(10)
The goal is to understand each of these three terms. The term d(p ⊗ id k , p ⊗ p ′ ) is simple:
d(p ⊗ id k , p ⊗ p ′ ) = d(id, p ′ ).
Let us study d (p ⊗ id k , ((p • p ′ ) ⊗ id k ) τ ). Using the definition of the distance in Proposition 3.1, and the Equality 3:
N −d(p⊗id k ,((p•p ′ )⊗id k )τ ) = T r N (p ⊗ id k ) t (((p • p ′ ) ⊗ id k ) τ ) N nc(p)+k 2 N nc(p•p ′ )+k 2 , since nc(p ⊗ id k ) = nc(p) + k and nc(((p • p ′ ) ⊗ id k ) τ ) = nc((p • p ′ ) ⊗ id k ) = nc((p • p ′ ) + k. Yet: T r N (p ⊗ id k ) t (p • p ′ ) ⊗ id k τ = T r N p t (p • p ′ ) .
Thus, using again Proposition 3.1:
d p ⊗ id k , (p • p ′ ) ⊗ id k τ = d(p, p • p ′ ) + k. Let us consider d (p ⊗ p ′ , ((p • p ′ ) ⊗ id k ) τ ).
Using the same arguments:
N −d(p⊗p ′ ,((p•p ′ )⊗id k )τ ) = T r N pp ′ t (p • p ′ ) N nc(p)+nc(p ′ ) 2 N nc(p•p ′ )+k 2 .
Using the definition of κ(p, p ′ ) and the Equality 3:
N −d(p⊗p ′ ,((p•p ′ )⊗id k )τ ) = N κ(p,p ′ ) T r N p • p ′ t (p • p ′ ) N nc(p)+nc(p ′ ) 2 N nc(p•p ′ )+k 2 = N κ(p,p ′ )+ 1 2 [nc(p•p ′ )−nc(p)−nc(p ′ )−k] . Thus: d p ⊗ p ′ , (p • p ′ ) ⊗ id k τ = −κ(p, p ′ ) − 1 2 [nc(p • p ′ ) − nc(p) − nc(p ′ ) − k].
Let us come back to the triangular inequality 10. This shows that:
d(p, p • p ′ ) + k ≤ d(id, p ′ ) − κ(p, p ′ ) − 1 2 [nc(p • p ′ ) − nc(p) − nc(p ′ ) − k],
and thus:
d(p, p • p ′ ) ≤ d(id, p ′ ) − nc(p • p ′ ) + k − nc(p) − nc(p ′ ) 2 − κ(p, p ′ ).
This is the inequality we wanted to prove.
Proof of Proposition 6.1. -Let k be an integer. Let p and p ′ be two elements of A k . Using the triangular inequality:
d(id, p • p ′ ) ≤ d(id, p) + d(p, p • p ′ ).
And an application of Proposition 6.2 implies that:
d(id, p • p ′ ) ≤ d(id, p) + d(id, p ′ ) − k + nc(p • p ′ ) − nc(p) − nc(p ′ ) 2 − κ(p, p ′ ).(11)
And using Lemma 5.1:
d( t p ′ , p) ≤ d(id, p) − d(id, p ′ ) + k + nc(p • p ′ ) − nc(p) − nc(p ′ ) + 2κ(p, p ′ ).
The result follows then from the fact that nc( t p ′ ) = nc(p ′ ).
We can generalize the inequality (11) to a n-uple of elements of A k . Lemma 6.1. -Let k be an integer. For any integer n, for any (p i ) n i=1 ∈ A n k :
d(id, • n i=1 p i ) ≤ n i=1 d(id, p i )− 1 2 (n − 1)k + nc(• n i=1 p i ) − n i=1 nc(p i ) − n−1 i=1 κ(p i , p i+1 ),
where we have used the notation • n i=1 p i = p 1 • . . . • p n . In fact, the best way to understand the improved triangular inequality is to work with the equivalent inequality (11). This formulation of the improved triangular inequality leads us to the next notion.
Definition 6.1. -Let p and p ′ be two elements of A k . We will say that p ≺ p • p ′ if and only if:
d(id, p • p ′ ) − d(id, p) − d(id, p ′ ) + k + nc(p • p ′ ) − nc(p) − nc(p ′ ) 2 + κ(p, p ′ ) = 0.
Let p 0 ∈ A k . We will write that p ≺ p 0 if there exists p ′ ∈ A k such that p 0 = p • p ′ and p ≺ p • p ′ .
Definition 6.2. -Let us suppose that p ≺ p 0 . We define for any p ≺ p 0 :
K p 0 (p) = {p ′ ∈ A k , p • p ′ = p 0 }.
Let us suppose that p ≺ p • p ′ . We recall that:
d(id, p • p ′ ) ≤ d(id, p) + d(p, p • p ′ ) ≤ d(id, p) + d(id, p ′ ) − k + nc(p • p ′ ) − nc(p) − nc(p ′ ) 2 − κ(p, p ′ ).
Thus, if the first term and the third one are equal, then p ∈ [id, p • p ′ ] A k . We have shown the following lemma. Let us remark that {σ ′ ∈ S k , σ ′ ≺ σ} = [id, σ] S k . This is due to the fact that κ(σ, σ ′ ) = 0 for any couple of permutations, the fact that nc is constant on the set of permutations and the fact that any permutation is invertible. Using Lemma 2.1 one can have the better result. Lemma 6.3. -Let k be an integer. Let σ ∈ S k , then:
{p ∈ P k , p ≺ σ} = [id, σ] S k .
Let us state a consequence of Lemma 6.2: the factorization property for ≺. Lemma 6.4. -Let k and l be two integers. Let a ∈ P k and b ∈ P l . For any p ∈ P k+l such that p ≺ a ⊗ b, there exist p 1 ≺ a and p 2 ≺ b such that p = p 1 ⊗ p 2 .
This lemma is a consequence of Lemma 6.2 and the factorization property for the geodesics stated in Lemma 3. 4.
Let p and p 0 in A k such that p ≺ p 0 . Let us have a little discussion on K p 0 (p): by definition this is not empty but it is not reduced to a unique partition. For example, one can show that if p = {{1, 2, 1 ′ , 2 ′ }} and p 0 = {{1 ′ , 2 ′ }, {1}, {2}} then:
K p 0 (p) = {1}, {2}, {1 ′ }, {2 ′ } , {1}, {2}, {1 ′ , 2 ′ } .
Let k be an integer. Let (1, . . . , k) be the k-cycle in S k . It is well-known that the set of non-crossing partition over {1, . . . , k} is isomorphic to [id, (1, . . . , k)] S k . From now on, we will consider any non-crossing partition over {1, . . . , k} as an element of [id, (1, .., k)] S k . The following lemma is straightforward. We are going now to see one of the main results of the paper, namely the fact that the improved triangle inequality implies the convergence of the deformed algebras C[A k (N, N )]) N ∈N stated in the forthcoming Theorem 6.1. Before doing so, we need to define the notion of convergence of algebras. We say that L N converges to the algebra L ∞ when N goes to infinity if for any x and y in C,
x. N y −→ N →∞ x. ∞ y in C[C],
for the usual notion of convergence in finite dimensional linear spaces.
Let us state the convergence of the deformed algebras C[A k (N, N )]) N ∈N . Theorem 6.1. -As N goes to infinity, the deformed algebra C[A k (N, N )] converges to the deformed algebra C[A k (∞, ∞)] which is the associative algebra over C with basis A k endowed with the multiplication defined by:
∀p, p ′ ∈ P k , pp ′ = δ p≺p•p ′ p • p ′ . Proof. -Let k be an integer. For any N ∈ N ∪ {∞}, A k is a linear basis of C[A k (N, N )].
By bi-linearity of the product, it is enough to prove that for any p and p ′ in A k , p. N p ′ converges to δ p≺p•p ′ p • p ′ .
Let p and p ′ be two elements of P. We have:
p. N p ′ = N d(id,p•p ′ )−d(id,p)−d(id,p ′ )+ k+nc(p•p ′ )−nc(p)−nc(p ′ ) 2 +κ(p,p ′ ) (p • p ′ ).
By the version of the improved triangle inequality stated in Proposition 6.1 or in the inequality (11), we have:
d(id, p • p ′ ) − d(id, p) − d(id, p ′ ) + k + nc(p • p ′ ) − nc(p) − nc(p ′ ) 2 + κ(p, p ′ ) ≤ 0.
According to Definition 6.1, we have p. N
p ′ −→ N →∞ δ p≺p•p ′ p • p ′ .
To conclude this section, let us remark that for any integer k, we have the inclusion of algebras:
C[S k (∞, ∞)] ⊂ C[B k (∞, ∞)] ⊂ C[P k (∞, ∞)].κ p 0 (EF ) = p∈A k ,p≺p 0 κ p (E)κ Kp 0 (p) (F ),(12)m p 0 (EF ) = p∈A k ,p∈[id,p 0 ] A k κ p (E)mt p•p 0 (F ).(13)(M N k ) −1 (E N F N ) = (M N k ) −1 (E N ). N (M N k ) −1 (F N ).
We know, by Lemma 5.4 Besides, using Lemma 5.3, we have:
(M N k ) −1 (E N F N ) = p∈A k κ p 0 (E N F N )p 0 , (M N k ) −1 (E N ). N (M N k ) −1 (F N ) = p∈A k ,p ′ ∈A k κ p (E N )κ p ′ (F N )p. N p ′ .
Using the formula for the limit of . N shown in Theorem 6.1, for any p 0 ∈ P k :
κ p 0 (EF ) = p∈A k ,p≺p 0 κ p (E)κ Kp 0 (p) (F ).
For the second equality, one could use the link, between A k -moments and coordinate numbers when N → ∞ given by Equality (5). Yet, this happens to be more difficult than a direct proof. Indeed, by bi-linearity, we have only to show that the equality (13) holds when, for any integer N :
E N = 1 N −k+nc(p) 2 +d(id,p) p.
Let N be an integer, let us suppose that E N is of this form. Let p 0 ∈ A k , we have:
m p 0 (E N F N ) = 1 N −k+nc(p) 2 +d(id,p) T r(F N t p 0 p) T r(p 0 ) = N κ( t p 0 ,p) N −k+nc(p) 2 +d(id,p) T r(F N t p 0 • p) T r(p 0 ) = N κ( t p 0 ,p) N −k+nc(p) 2 +d(id,p) T r( t p • p 0 )m t p•p 0 (F N ) T r(p 0 ) = 1 N −k+nc(p) 2 +d(id,p) T r( t pp 0 ) T r(p 0 ) m t p•p 0 (F N ).
We remind the reader that, for any p and p 0 in A k :
1 N −k+nc(p) 2 +d(id,p) T r( t pp 0 ) T r(p 0 ) = N d(id,p 0 )−d(id,p)−d(p,p 0 ) .
This equality allows to finish the proof, as:
m p 0 (E N F N ) = N d(id,p 0 )−d(id,p)−d(p,p 0 ) m t p•p 0 (F N ).
Using the triangular inequality, one gets finally that m p 0 (E N F N ) converges when N goes to infinity to δ p∈[id,p 0 ] A k m t p•p 0 (F N ).
m p 0 (EF ) = p∈A k ,p∈[id,p 0 ] A k m p 0 • t p (E)κ p (F ).(14)
7.2. Semi-groups. -Let k be an integer. In this subsection, we will study convergence of sequences of semi-groups in C[A k (N )]. Semi-groups in different algebras will appear in the paper: for this paper, a family (a t ) t≥0 is a semi-group if there exists h, called the generator, such that for any t 0 ≥ 0: d dt |t=t 0 a t = ha t 0 .
If we consider the algebra N ∈N C[A k (N )], we are led to the next definition.
Definition 7.1. -The family (E N t ) N t≥0 is a semi-group in N ∈N C[A k (N )] if there exists (H N ) N ∈N ∈ N ∈N C[A k (N )]
, called the generator, such that for any t ≥ 0, for any integer N :
d dt |t=t 0 E N t = H N E N t 0 .
From now on, let us suppose that (E N t ) N t≥0 is a semi-group in N ∈N C[A k (N )] whose generator is (H N ) N ∈N . Let us define the convergence for semi-groups in N ∈N C[A k (N )]. Besides, we have the two differential systems of equations:
∀p ∈ A k , ∀t 0 ≥ 0, d dt |t=t 0 κ p (E t ) = p 1 ∈A k ,p 1 ≺p κ p 1 (H)κ Kp(p 1 ) (E t 0 ). (15) ∀p ∈ A k , ∀t 0 ≥ 0, d dt |t=t 0 m p (E t ) = p 1 ∈[id,p] A k κ p 1 (H)mt p 1 •p (E t 0 ).(16)d dt |t=t 0 p 0 ∈A k κ p 0 (E N t )p 0 = p∈A k κ p (H N )p . N p ′ ∈A k κ p ′ (E N t 0 )p ′ .
Then the following equality must hold for any integer N , any t 0 ≥ 0 and any p 0 ∈ A k :
d dt |t=t 0 κ p 0 (E N t ) = p,p ′ ∈A k ,p•p ′ =p 0 κ p (H N )κ p ′ (E N t )N d(id,p•p ′ )−d(id,p)−d(id,p ′ )+ k+nc(p•p ′ )−nc(p)−nc(p ′ ) 2 +κ(p,p ′ ) .
Let us take N going to infinity. Because of the hypothesis and because of the improved triangular inequality, this differential system converges: κ p (E N t ) must converge for any p ∈ A k and any real t ≥ 0. Besides, we get for any t 0 ≥ 0:
∀p ∈ A k , d dt |t=t 0 κ p (E t ) = p 1 ∈A k ,p 1 ≺p κ p 1 (H)κ Kp(p 1 ) (E t 0 ).
Since the semi-group converges, using the usual notations, we can write that for any p ∈ A k and any t 0 ≥ 0:
d dt |t=t 0 m p (E t ) = m p (HE t 0 ) ,
and using equality (13), one has:
lim N →∞ m p (H N E N t 0 ) = p 1 ∈[id,p] A k κ p 1 (H)mt p 1 •p (E t 0 ).
Hence we recover the second system of differential equations.
Of course one also has, by using equality (14) instead of (13), that for any p ∈ P k and any t 0 ≥ 0:
d dt |t=t 0 m p 0 (E t ) = p∈[id,p 0 ] A k m p 0 • t p (H)κ p (E t 0 ).
Moreover, Theorem 7.2 can be very easily generalized for any semi-group with time dependent generator. In order to finish the section, let us prove a consequence of Lemma 3.3.
Theorem 7.3. -Let (E N t ) N t≥0 be a semi-group in N ∈N C[B k (N )].
Let us suppose that the sequence (E N 0 ) N ∈N converges as N goes to infinity. Let us suppose that for any σ ∈ S k , κ σ (H N ) converges when N goes to infinity. Then for any σ ∈ S k , for any positive real t, κ σ (E N t ) converges as N goes to infinity. Besides for any σ ∈ S k and any t 0 ≥ 0:
d dt |t=t 0 κ σ (E t ) = σ∈S k ,σ 1 ≺σ κ σ 1 (H)κ Kσ(σ 1 ) (E t 0 ). (17) Proof. -Let (E N t ) N t≥0 be a semi-group in N ∈N C[B k (N )]
which satisfies the hypothesis of the theorem. Let σ ∈ S k and let N be an integer. We have seen in the last proof that for any t 0 ≥ 0:
d dt |t=t 0 κ σ (E N t ) = p,p ′ ∈B k ,p•p ′ =σ κ p (H N )κ p ′ (E N t 0 )N d(id,p•p ′ )−d(id,p)−d(id,p ′ )+ k+nc(p•p ′ )−nc(p)−nc(p ′ )
Yet, by Lemma 2.1, if p • p ′ = σ, then p and p ′ are in S k . Thus,
d dt |t=t 0 κ σ (E N t ) = p,p ′ ∈S k ,p•p ′ =σ κ p (H N )κ p ′ (E N t 0 )N d(id,p•p ′ )−d(id,p)−d(id,p ′ ) .
Thus, we see that (κ σ (E N t )) σ∈S k t≥0 satisfies a linear differential system whose coefficients converge by hypothesis. Thus, for any σ ∈ S k , for any positive real t, κ σ (E N t ) converges as N goes to infinity. The Equation 17 is obtained by taking N going to infinity in the last equation.
A new way to get combinatorial properties
In Section 6, we showed new inequalities on the set of partitions P k . The proofs were quite combinatorial, and used only the notion of distance. In this section, we want to show that one can prove new inequalities or equalities, by using Theorem 4.1 as a black box.
p N = M N k (p), p ′ N = M N k (p ′ ).
Using Lemma 5.4, (p N ) N ∈N and (p ′ N ) N ∈N converge strongly. Let N be an integer. Applying the equality (9), we have:
p N ⊗ p ′ N = M N 2k (p ⊗ p ′ ).
Thus, using Lemma 5.4, p N ⊗ p ′ N converges strongly when N goes to infinity. An application of Theorem 4.1 shows that it converges in moments: for anyp ∈ A 2k ,
mp(p N ⊗ p ′ N ) converges when N → ∞.
For any partitionp ∈ A k , we define P (p) be the partition in A 2k :
P (p) = (p ⊗ id k )(1, k + 1)(2, k + 2) . . . (k, 2k).
Then for any E ∈ C[A k (N )] and F ∈ C[A k (N )], and any p 0 ∈ A k , we have:
m P (p 0 ) (E ⊗ F ) = m p 0 (EF ).
Thus for any p 0 ∈ A k , m p 0 (p N p ′ N ) which is equal to m P (p 0 ) (p N ⊗ p ′ N ) converges as N goes to infinity. Using again the Theorem 4.1, we have that p N p ′ N converges strongly as N goes to infinity. It implies, because of Lemma 5.4 that (M N k ) −1 (p N p ′ N ) converges in C[A k ] when N goes of infinity. We can calculate this last expression:
(M N k ) −1 (p N p ′ N ) = (M N k ) −1 (M N k (p)M N k (p ′ )) = p. N p ′ = N d( t p ′ ,p)−d(id,p)−d(id,p ′ )+k+nc(p•p ′ )−nc(p)−nc(p ′ )+2κ(p,p ′ ) (p • p ′ ),
where we used Lemma 5. 1. Thus we must have that for any p and p ′ in A k :
d( t p ′ , p) ≤ d(id, p) + d(id, p ′ ) − k − nc(p • p ′ ) + nc(p) + nc(p ′ ) − 2κ(p, p ′ ).
The improved inequality is just a consequence of the last inequality as soon as we see that for any p ∈ A k , nc( t p) = nc(p), and d(id, p) = d(id, t p).
Again, using the same ideas, one can show the following interesting property. We have: Let us calculate, using two ways, the limit of m (p 0 ⊗id k )τ p 1 N ⊗ p 2 N , where τ = (1, k + 1)(2, k + 2) . . . (k, 2k).
δ p 1 ⊗p 2 ∈[id,(p 0 ⊗id k )τ ] A 2k = δ p 1 •p 2 ∈[id,p 0 ] A k δ p 1 ≺p 1 •p 2 .
First, using Theorem 4.1 and the Equation (5), we get that:
lim N →∞ m (p 0 ⊗id k )τ p 1 N ⊗ p 2 N = p∈[(p 0 ⊗id k )τ ] A 2k lim N →∞ κ p p 1 N ⊗ p 2 N .
Yet, for any p ∈ A 2k , κ p p 1 N ⊗ p 2 N = δ p=p 1 ⊗p 2 , thus:
lim N →∞ m (p 0 ⊗id k )τ p 1 N ⊗ p 2 N = δ p 1 ⊗p 2 ∈[id,(p 0 ⊗id k )τ ] A 2k .
Then, using the fact that m (p 0 ⊗id k )τ p 1 N ⊗ p 2 N = m p 0 p 1 N p 2 N , and using again Theorem 4.1 and the Equation (5):
lim N →∞ m (p 0 ⊗id k )τ p 1 N ⊗ p 2 N = p∈[id,p 0 ] A k lim N →∞ κ p p 1 N p 2 N . Let p ∈ A k , κ p p 1 N p 2 N is the coefficient of p in the expression M N k −1 (p 1 N p 2 N ). Let us remark that M N k −1 (p 1 N p 2 N ) = M N k −1 (M N k (p 1 )M N k (p 2 )) = p 1 . N p 2 which converges in C[A k ] to δ p 1 ≺p 1 •p 2 p 1 • p 2 . Thus, lim N →∞ κ p p 1 N p 2 N = δ p 1 ≺p 1 •p 2 δ p=p 1 •p 2 .
This implies that:
lim N →∞ m (p 0 ⊗id k )τ p 1 N ⊗ p 2 N = δ p 1 ≺p 1 •p 2 δ p 1 •p 2 ∈[id,p 0 ] A k .
Using the two ways to compute lim N →∞ m (p 0 ⊗id k )τ p 1 N ⊗ p 2 N , we get:
δ p 1 ⊗p 2 ∈[id,(p 0 ⊗id k )τ ] A 2k = δ p 1 •p 2 ∈[id,p 0 ] A k δ p 1 ≺p 1 •p 2 .
which was the desired equality.
In fact, one can always prove the results by a combinatorial argument: the ideas we present are more an automatic way to get combinatorial results that one can prove after by combinatorial means. For exemple, let us consider Definition 9.1. Using Proposition 8.1, one can now expect that df(p 1 ⊗ p 2 , (p ⊗ id k )τ ) = df(p 1 • p 2 , p) + η(p 1 , p 2 ). Indeed, we have the following proposition.
τ = (1, k + 1)(2, k + 2) . . . (k, 2k).
We have:
df(p 1 ⊗ p 2 , (p 0 ⊗ id k )τ ) = df(p 1 • p 2 , p 0 ) + η(p 1 , p 2 ).
Proof. -The proof is only based on calculations. Let p and p ′ be two partitions in A k , then:
df(p ′ , p) = nc(p ′ ) − nc(p ′ ∨ id) − nc(p ′ ∨ p) + nc(p ∨ id),
and η(p, p ′ ) is equal to:
nc(p) + nc(p ′ ) − nc(p • p ′ ) − nc(p ∨ id) − nc(p ′ ∨ id) + nc(p • p ′ ∨ id) − κ(p, p ′ ).
Thus:
df(p 1 • p 2 , p 0 ) + η(p 1 , p 2 ) − df(p 1 ⊗ p 2 , (p 0 ⊗ id k )τ ) = nc(p 1 • p 2 )−nc((p 1 • p 2 ) ∨ id)−nc((p 1 • p 2 ) ∨ p 0 )+ nc(p 0 ∨ id)+ nc(p 1 )+ nc(p 2 ) − nc(p 1 • p 2 ) − nc(p 1 ∨ id) − nc(p 2 ∨ id) + nc((p 1 • p 2 ) ∨ id) − κ(p 1 , p 2 ) − nc(p 1 ⊗ p 2 ) + nc((p 1 ⊗ p 2 ) ∨ id) + nc([(p 0 ⊗ id k )τ ] ∨ [p 1 ⊗ p 2 ]) − nc([(p 0 ⊗ id k )τ ] ∨ id 2k ).
Using the following equalities:
nc(p 1 ⊗ p 2 ) = nc(p 1 ) + nc(p 2 ), nc([p 1 ⊗ p 2 ] ∨ id) = nc(p 1 ∨ id) + nc(p 2 ∨ id),
we get:
df(p 1 • p 2 , p 0 ) + η(p 1 , p 2 ) − df(p 1 ⊗ p 2 , (p 0 ⊗ id k )τ ) = −nc((p 1 • p 2 ) ∨ p 0 ) + nc(p 0 ∨ id) − κ(p 1 , p 2 ) + nc([(p 0 ⊗ id k )τ ] ∨ [p 1 ⊗ p 2 ]) − nc([(p 0 ⊗ id k )τ ] ∨ id 2k ).
The equalities:
T r N (p 0 ) = T r N ((p 0 ⊗ id k )τ ), N κ(p 1 ,p 2 ) T r N ((p 1 • p 2 ) t p 0 ) = T r N ((p 1 p 2 ) t p 0 ) = T r N ((p 1 ⊗ p 2 ) t [(p 0 ⊗ id k )τ ]),
allow us to prove, as an application of Equations (2) and (3), that: Proof. -Let us consider (m p ) p∈A k a family of complex numbers. Let us consider (κ p ) p∈A k , the unique family of real such that for any p ∈ A k :
nc(p 0 ∨ id) = nc([(p 0 ⊗ id k )τ ] ∨ id 2k ), nc((p 1 • p 2 ) ∨ p 0 ) + κ(p 1 , p 2 ) = nc([(p 0 ⊗ id k )τ ] ∨ [p 1 ⊗ p 2 ]). Thus df(p 1 • p 2 , p 0 ) + η(p 1 , p 2 ) − df(p 1 ⊗ p 2 , (p 0 ⊗ id k )τ ) = 0.m p = p ′ ∈[id,p] A k κ p ′ .
Let us consider then:
E N = M N k p∈A k κ p p .
According to Lemma 5.4, (E N ) N ∈N ∈ N ∈N C[A k (N )] converges strongly. Thus, by Theorem 4.1 it converges in moments and for any p ∈ A k :
lim N →∞ m p (E N ) = p ′ ∈[id,p] A k lim N →∞ κ p ′ (E N ).
Yet, using Lemma 5.3, κ p ′ (E N ) is equal to κ p . Thus:
lim N →∞ m p (E N ) = p ′ ∈[id,p] A k κ p ′ = m p .
This concludes the proof.
This theorem is very important, as actually, it shows that, in order to understand the transformation between moments and coordinate numbers, we have an approximation setting in which one can work with: the space of convergent sequences in N ∈N C[A k (N )]. Let us show some exemples of propositions that one can get using this point of view. For this, we need the notion of cumulants and exclusive moments. Let us consider (m p ) p∈A k a family of complex numbers.
Definition 8. 1. -The cumulants of (m p ) p∈A k is the unique family of complex numbers (κ p ) p∈A k such that for any p ∈ A k :
m p = p ′ ∈[id,p] A k κ p ′ .
The exclusive moments of (m p ) p∈A k is the only family (m p c ) p∈P k of complex numbers such that:
m p = p ′ ∈P k ,p ′ coarser-compatible than p m p ′c .
Let us consider the cumulants (κ p ) p∈A k and the exclusive moments (m p c ) p∈P k of (m p ) p∈A k .
δ p∈[id,p 0 ] A k mt p•p 0 = p ′ ∈[id,p 0 ] A k δ p≺p ′ κ K p ′ (p) .
where for any P ⊂ A k , κ P = p∈P κ p .
By specifying p = id in Proposition 8.4, we get back the Equation (5). Besides, one can get a similar formula for m p 0 • t p (E) by using the Equation (14). Using Theorem 8.1, the last proposition is a consequence of Proposition 8. 4.
δ p∈[id,p 0 ] A k mt p•p 0 (E) = p ′ ∈[id,p 0 ] A k δ p≺p ′ κ K p ′ (p) (E).
Proof. -Let p and p 0 be two elements of A k . Let us consider for any N ,
M N k (p) ∈ C[A k (N )]. The sequence M N k (p) N ∈N ∈ N ∈N C[A k (N )]κ p ′ (M N k (p)E N ) = δ p≺p ′ κ K p ′ (p) (E).
Let us use the Equation (5):
lim N →∞ m p 0 (M N k (p)E N ) = p ′ ∈[id,p 0 ] A k δ p≺p ′ κ K p ′ (p) (E).
Yet, according to Equation (13),
lim N →∞ m p 0 (M N k (p)E N ) = δ p∈[id,p 0 ] A k mt p•p 0 (E),
hence the equality stated in Proposition 8. 4.
Let us show one more exemple. Using Theorem 8.1, one can translate Theorem 4. 6.
Theorem 8.2. -For any p ∈ A k : m p c = p ′ finer than p,p ′ ∈[id,p] A k κ p ′ .
8. 3. Convergence of the modified observables. -In Section 5, we have defined a deformed partition algebra, by deforming the multiplication. Yet, we have not defined any deformed linear form m p on the algebra C[A k (N, N )]. In fact, on C[A k (N, N )], for any p ∈ P k we define:
m N p : C[A k (N, N )] → C E → m p M N k (E) .
A consequence of Theorem 4.1 is that for any E ∈ A k , for any p ∈ P k , m N p (E) converges as N goes to infinity: let us denote the limit by m ∞ p (E). We already know that the algebra C[A k (N, N )] converges to C[A k (∞, ∞)] when N goes to infinity. Thus, we have that:
Theorem 8.3. -For any integer k, C[A k (N, N )], (m N p ) p∈P k converges to C[A k (∞, ∞)]
, (m ∞ p ) p∈P k as N goes to infinity. This means that:
1. the algebra C[A k (N, N )] converges to C[A k (∞, ∞)] as N goes to infinity, 2. for any E ∈ C[A k (N, N )], for any p ∈ P k , m N p (E) converges to m ∞ p (E) as N goes to infinity, where m ∞ p (E) is defined below. Besides, let E = p∈A k E p p and F = p∈A k F p p in C[A k (∞, ∞)], then EF = p 1 ,p 2 ∈A k E p 1 F p 2 δ p 1 ≺p 1 •p 2 p 1 • p 2 . And if p 0 ∈ A k : m ∞ p 0 (E) = p∈A k δ p∈[id,p 0 ] A k E p .
In fact, this theorem has to be read in the other way: given the algebra with linear forms (C[A k (∞, ∞)], (m ∞ p ) p∈P k ), one can find an approximation given by C [A k (N, N )], (m N p ) p∈P k .
Algebraic fluctuations
In this section, we generalize Sections 4, 5 and 7 in order to study the asymptotic developments of the coordinate numbers and normalized moments. The proofs will be either omitted or simplified as they will use the same arguments as we have seen in Sections 4, 5 and 7.
In order to study the asymptotic developments, we need to introduce two notions of default. One already seen is linked with the triangular inequality and the other to the improved triangular inequality. Let k be an integer.
Definition 9.1. -Let p and p ′ be two elements of A k . We define the default of p ′ not being on the geodesic [id, p] A k by:
df(p ′ , p) = d(id, p ′ ) + d(p ′ , p) − d(id, p).
We define also the default η(p, p ′ ) that p ≺ p • p ′ is not satisfied by:
d(id, p) + d(id, p ′ ) − d(id, p • p ′ ) − k + nc(p • p ′ ) − nc(p) − nc(p ′ ) 2 − κ(p, p ′ ).
We warn the reader that, in general:
df(p, p • p ′ ) = η(p, p ′ ),
even if this equality is true when one considers p, p ′ ∈ S k . Let us remark that if p and p 0 are elements of A k , p ≺ p 0 holds if and only if there exists p ′ such that p 0 = p • p ′ and η(p, p ′ ) = 0. Let us define the N -development algebra of order m of A k . This algebra is the good setting in order to study fluctuations of the coordinate numbers and moments. (N )], is the associative algebra generated by the elements of the form: p X i , where p ∈ A k and i ∈ {0, . . . , m}. The product is defined such that, for any p and p ′ in A k , and any i and j in {0, . . . , m}:
p X i . p ′ X j = 1 N max(i+j+η(p,p ′ )−m,0) p • p ′ X min(i+j+η(p,p ′ ),m) .
This product is well defined: indeed the improved triangle inequality, Proposition 6.1 or Lemma 6.1, assert that for any p, p ′ ∈ A k , η(p, p ′ ) ≥ 0, thus, for any i, j ∈ {0, . . . , m}, any p, p ′ ∈ A k , we have min(i + j + η(p, p ′ ), m) ≥ 0. This implies that:
p • p ′ X min(i+j+η(p,p ′ ),m)
is an element of the canonical basis of the N -development algebra of order m of A k .
There is an important remark to be done: once one has defined the N -development algebra of order m of A k , we will not have any energy to speend in oder to get interesting results.
Let us also remark that for any integers k, N and m in N,
C (m) [S k (N )] ⊂ C (m) [B k (N )] ⊂ C (m) [P k (N )]
, where these inclusions are inclusions of algebras.
L N k : A k → C (0) [A k (N )] p → p X 0 can be extended as an isomorphism of algebra between C[A k (N, N )] and C (0) [A k (N )].
Proof. -Let us show that for any p,
p ′ in A k , L N k (p. N p ′ ) = L N k (p)L N k (p ′ ). As for any p, p ′ ∈ A k , η(p, p ′ ) ≥ 0, L N k (p)L N k (p ′ ) is equal to: p X 0 p ′ X 0 = 1 N η(p,p ′ ) p • p ′ X 0 = 1 N η(p,p ′ ) L N k (p • p ′ ).
Yet, looking at the definition of η(p, p ′ ) given in Definition 9.1, for any integer N the following equation holds in C[A k (N, N )]:
p. N p ′ = 1 N η(p,p ′ ) p • p ′ .
This allows to conclude.
Using this remark, we define, for any i ≤ m, the coordinate numbers of order i of any element of C (m) [A k (N )] as following.
E = p∈A k m i=0 κ p i (E) p X i .
Let p ∈ A k and i ≤ m. The number κ p i (E) is the coordinate number of E on p of order i.
We define also a notion of convergence for (E N ) N ∈N ∈ N ∈N C (m) [A k (N )]. In order to do so, we must not forget that, when m = 0, C (m) [A k (N )] is isomorphic to the deformed algebra C [A k (N, N )
p ∈ A k , κ p i (E N )
is independent of N , and for any p ∈ A k , κ p m (E N ) converges when N goes to infinity.
Notation 9.1. -Let (E N ) N ∈N ∈ N ∈N C (m) [A k (N )].
Let us suppose that (E N ) N ∈N converges as N goes to infinity. We denote, for any i ∈ {0, . . . , m}, and any p ∈ A k :
κ p i (E) = lim N →∞ κ p i (E N ).p X i p X j = δ i+j+η(p,p ′ )≤m p • p ′ X i+j+η(p,p ′ ) .
Let us recall Definition 6.3, where we defined the convergence of algebras. We then have the following proposition. . Since for any p, p ′ ∈ A k , any i, j ∈ N:
p X i p ′ X i ′ = 1 N max(i+j+η(p,p ′ )−m,0) p • p ′ X min(i+j+η(p,p ′ ),m) −→ N →∞ δ i+j+η(p,p ′ )≤m p • p ′ X i+j+η(p,p ′ ) ,
where the first product is seen in C (m) [A k (N )], the algebra C (m) [A k (N )] converges to C (m) [A k (∞)] as N goes to infinity.
Let us write the first easiest consequence of the Proposition 9.1, which can be proved by using a bi-linearity argument, Proposition 9.1 and Definition 9.4.
κ p 0 i 0 (EF ) = p,p ′ ∈A k ,η(p,p ′ )≤i 0 ,p•p ′ =p 0 i∈{0,...,i 0 −η(p,p ′ )} κ p i (E)κ p ′ i 0 −η(p,p ′ )−i (F ).
As for Section 7.2, the good behavior of the product, given by Proposition 9. We have the following proposition, whose proof relies on the ideas behind the proof of Proposition 9.2.
d dt |t=t 0 κ p 0 i 0 (E t ) = p,p ′ ∈A k ,η(p,p ′ )≤i 0 ,p•p ′ =p 0 i∈{0,...,i 0 −η(p,p ′ )} κ p i (H)κ p ′ i 0 −η(p,p ′ )−i (E t 0 ).
In order to finish this section, let us introduce the evaluation morphism: it is a morphism which allows to inject an element from C (m) [A k (N )] in C[A k (N )]. Let N and m be two integers. The function eval N is defined by:
eval N (m) : C (m) [A k (N )] → C[A k (N )] p∈A k m i=0 κ p i (E) p X i → p∈A k m i=0 κ p i (E) 1 N i p N − k 2 + nc(p) 2 +d(id,p)
. Lemma 9.2. -For any integers N and m, eval N (m) is a morphism of algebra.
Proof. -Let N and m be two integers, let i, j ∈ {0, . . . , m} and p, p ′ ∈ A k . Then:
eval N (m) p X i p ′ X j = eval N (m) 1 N max(i+j+η(p,p ′ )−m,0) p • p ′ X min(i+j+η(p,p ′ ),m) = 1 N i+j+η(p,p ′ )− k 2 + nc(p•p ′ ) 2 +d(id,p•p ′ ) p • p ′ = 1 N i p N − k 2 + nc(p) 2 +d(id,p) 1 N j p ′ N − k 2 + nc(p ′ ) 2 +d(id,p ′ ) = eval N (m) p X i eval N (m) p X j .κ p i (E) = lim N →∞ κ p i (E N ).
When one works in N ∈N C[A k (N )], one has to be aware that the coordinate numbers of higher order of fluctuations are only defined for a sequence (E N ) N ∈N which converges strongly. Thus, one must not forgot that the notation κ p i (E N ) means that we are looking at the coordinate numbers of E N seen as an element of the sequence (E N ) N ≥0 .
The Definition 9.7 might seem strange as it only uses once the notion of convergence. Yet, it is easy to see that an equivalent definition is the following one.
Let m be an integer, let (E N ) N ∈N ∈ N ∈N C[A k (N )]. It converges strongly up to the m th order of fluctuations if and only if there exists a family (κ p i ) i∈{0,...,m},p∈A k of real numbers such that for any i ∈ {0, . . . , m},
N i κ p (E N ) − i−1 j=0 κ p j N j −→ N →∞ κ p i ,
with the convention
E N = p∈A k m−1 i=0 κ p i p X i + κ p m,N p X m .
The following lemma is then straightforward. We are going to define a weak notion of convergence up to the m th order of fluctuations and we will show that this notion is equivalent to the strong convergence notion we defined in Definition 9.7.
-∀p ∈ A k , m p (E N ) = m−1 i=0 m i p N i + m m p N m , -∀p ∈ A k , m mm i p (E) = lim N →∞ m i p (E N ).
We can state a remark for the fluctuations of the p-normalized moments of E N similar to the one explained just after Notation 9.2 about the coordinate numbers of E N on p of order i.
The next theorem shows that the strong convergence up to the m th order of fluctuations is equivalent to the convergence in moments up to the m th order of fluctuations.
m i 0 p (E) = p ′ ∈A k ,df(p ′ ,p)≤i 0 κ p ′ i 0 −df(p ′ ,p) (E).(18)E N = p∈A k m i=0 κ p i (E N ) N i p N − k 2 + nc(p) 2 +d(id,p)
.
Besides, for any p ∈ A k and any i ≤ m − 1, κ p i (E N ) does not depend on N and κ p m (E N ) converges when N goes to infinity.
We can compute the p-normalized moments of E N , using the same arguments as for the proof of Theorem 4.1. For any N ∈ N and any p ∈ A k :
m p (E N ) = 1 T r N (p) T r N (E N ) = p ′ ∈A k m i=0 κ p ′ i (E N ) 1 N i+df(p ′ ,p) = m−1 j=0 (p ′ ,i)∈A k ×{0,...,m−1},i+df(p,p ′ )=j κ p ′ i (E N ) 1 N j + (p ′ ,i)∈A k ×{0,...,m},i+df(p,p ′ )≥m κ p ′ i (E N ) N i+df(p,p ′ )−m 1 N m .
Let us define for any N ∈ N, any j ∈ {0, . . . , m − 1} and any p ∈ A k :
m j p (E N ) = (p ′ ,i)∈A k ×{0,...,m−1},i+df(p,p ′ )=j κ p ′ i (E N ) and m m p (E N ) = (p ′ ,i)∈A k ×{0,...,m},i+df(p,p ′ )≥m κ p ′ i (E N ) N i+df(p,p ′ )−m ,
so that, for any p ∈ A k and any N ∈ N:
m p (E N ) = m−1 j=0 m i p (E N ) N j + m m p (E N ) N m .
For any p ∈ A k and any i ≤ m − 1, m i p (E N ) does not depend on N and for any p ∈ A k , κ p m (E N ) converges when N goes to infinity. Thus m m p (E N ) converges when N goes to infinity to
p ′ ∈A k ,df(p ′ ,p)≤m κ p ′ m−df(p ′ ,p) (E).
By Definition 9.9, this shows that (E N ) N ∈N converges in moments up to the m th order of fluctuations and the Equation (18)
E N = p∈P k l−1 j=0 κ p j (E) N j + κ p l (E N ) N l p,
where, for any p ∈ P k , κ p l (E N ) is converging when N goes to infinity to a number κ p l (E). We can use the computation, that we already did, of the normalized moments of E N . For any partition p ∈ A k :
m p (E N ) = l−1 j=0 (p ′ ,i)∈A k ×{0,...,l−1},i+df(p,p ′ )=j κ p ′ i (E) 1 N j + (p ′ ,i)∈A k ×{0,...,l},i+df(p,p ′ )≥l κ p ′ i (E N ) N i+df(p,p ′ )−l 1 N l .
Thus, using the same notations than those used in the first part of the proof, we get:
m p (E N ) = l j=0 m j p (E) N j + p ′ ∈A k ,df(p,p ′ )=0 κ p ′ l (E N ) − κ p ′ l (E) N l + (p ′ ,i)∈A k ×{0,...,l},i+df(p,p ′ )−l=1 κ p ′ i (E N ) N l+1 + o 1 N l+1 .
Let us use the fact that (E N ) N ∈N converges in moments up to the order l+1 of fluctuations:
for any p ∈ A k , N l+1 m p (E N ) − l j=0 m j p (E) N j
converges as N goes to infinity. This implies that for any p ∈ A k ,
p ′ ∈[id,p] A k N (κ p ′ l (E N ) − κ p ′ l (E))
converges as N goes to infinity. We are thus in the same setting as for the order 0 of fluctuations: for any p ∈ A k ,
N (κ p ′ l (E N ) − κ p ′ l (E)
) converges as N goes to infinity: this is equivalent to say that (E N ) N ∈N converges strongly up to order l + 1 of fluctuations. This implies by recurrence that (E N ) N ∈N converges strongly up to order m of fluctuations.
Multiplication and convergence of fluctuations in
N ∈N C[A k (N )]. -
The results in Section 9.3 were only algebraic: we will now give the similar results for elements in N ∈N C[A k (N )]. The main ingredients used in order to do so are Lemma 9.2, Lemma 9.3 and Lemma 9.4 which respectively assert that eval N (m) is a morphism of algebra, compatible with the strong convergence notion and, in some sense, can be inverted. Besides, using the Notations 9.2 and 9.3, for any i 0 ∈ {0, . . . , m} and for any p 0 ∈ A k : In order to prove the equality (20), the best way is to come back to the definitions, and do a proof similar to the one for (13) in Theorem 7.1.
κ p 0 i 0 (EF ) = p,p ′ ∈A k ,η(p,p ′ )≤i 0 ,p•p ′ =p 0 i 0 −η(p,p ′ ) i=0 κ p i (E)κ p ′ i 0 −η(p,p ′ )−i (F ). (19) m i 0 p 0 (EF ) = p 1 ∈A k i+j+df(p 1 ,p 0 )=i 0 κ p 1 i (E)m j t p 1 •p 0 (F ). (20)κ p 0 i 0 (ẼF ) = p,p ′ ∈A k ,η(p,p ′ )≤i 0 ,p•p ′ =p 0 i∈{0,...,i 0 −η(p,p ′ )} κ p i (Ẽ)κ p ′ i 0 −η(p,p ′ )−i (F ).
Let us consider the implication of Proposition 9.3 for the semi-groups in N ∈N C[A k (N )]. From now on, let us suppose that E N t N t≥0 is a semi-group in N ∈N C[A k (N )] whose generator is (H N ) N ∈N . We would like to state a theorem for the fluctuations of E N t N t≥0 similar to Theorem 7.2. For this, we need the following definition. We can now state the theorem about the convergence to the m th order of fluctuations of a semi-group in N ∈N C[A k (N )]. The proof will not be given, as it is a direct consequence of Proposition 9.3 with a lift-argument as for the last proof. Besides, we have the two differential systems of equations:
∀p 0 ∈ A k , ∀t 0 ≥ 0, ∀i 0 ∈ {0, . . . , m}, d dt |t=t 0 κ p 0 i 0 (E t ) = p,p ′ ∈A k ,η(p,p ′ )≤i 0 ,p•p ′ =p 0 i∈{0,...,i 0 −η(p,p ′ )} κ p i (H)κ p ′ i 0 −η(p,p ′ )−i (E t 0 ). ∀p 0 ∈ A k , ∀t 0 ≥ 0, ∀i ∈ {0, . . . , m}, d dt |t=t 0 m i 0 p 0 (E t ) = p 1 ∈A k i+j+df(p 1 ,p 0 )=i 0 κ p 1 i (H t )m j t p 1 •p (E t 0 ).
10. An introduction to the general R-transform 10. 1. The zero order. -Up to now, we only worked with partitions which have a fixed length: we worked in A k for a fixed integer k. Yet, we could have worked with A ∞ = ∪ k∈N A k endowed with the product: pp ′ = δ l(p)=l(p ′ ) pp ′ where we recall that l(p) is the length of p. With this definition, we see that all the results hold when one changes k by k = ∞. For example C[A ∞ (N, N )] converges when N goes to infinity to an algebra C[A ∞ (∞, ∞)]. We could have studied this algebra, yet, in the theory of random matrices, we will see that the first elements E ∈ C[A ∞ (∞, ∞)] we naturally obtain are the elements which, seen as elements of C[A ∞ ], are invariant and such that E 0 = 1. The invariance of E means that for any integer k and any σ ∈ S k :
σE k σ −k = E k ,
where E k is the restriction of E on A k . This definition allows to make the link with the usual theory of R-transform. In fact, we will not use the invariance by S in most of what we will do. Yet, we prefered to write it like this since it is the setting in which one works when one consider the limit of one sequence of random matrices (A N ) N ∈N . Yet, we will need similar results in [9] for the non-conjugation invariant case and we will use them by refering to the theorem proved in the conjugation invariant case. 10
E g [A] = ∞ k=0 C S [A k (∞, ∞)] ,
where, for any integer k, C S [A k (∞, ∞)] is the algebra of elements of C [A k (∞, ∞)] which, seen as elements of C[A k ] are invariant by conjugation by any element of S k . Actually, two subspaces of E g [A] will be interesting for us:
E[A] = {E ∈ E g [A], E ∅ = 1}, e[A] = {E ∈ E g [A], E ∅ = 0}. Any element E ∈ E g [A]
is of the form:
p∈A k (E k ) p p k∈N .
In order to simplify the notations, we will use the following convention: for any integer k, for any p ∈ A k ,
E p = E(p) = (E k ) p ,
and for any positive integer k:
E k = p∈A k E p p.
The algebra E g [A] is naturally endowed with a natural addition and multiplication given, for any E, F ∈ E g [A] and any k ∈ N * by:
(E + F ) k = E k + F k (E ⊠ F ) k = E k F k . By convention (E ⊠ F ) ∅ = E ∅ F ∅ .(E ⊞ F ) p = (p 1 ,p 2 ,I)∈F 2 (p) E(p 1 )F (p 2 ),
where F 2 (p) was defined in Definition 2. 10.
In fact, the two operations ⊠ and ⊞ are convolution operations.
p ∈ A k , (0 E ) p = 0.
The operation ⊠ is not commutative and the set of invertible elements in E[A] is the set of elements E such that E id k = 0 for any k ≥ 1, we denote it by GE[A]. We denote by 1 E the neutral element for ⊠ which is the only element such that for any k ≥ 1, (1 E ) k = id k .
Let us consider an interesting sub-vector space of E g [A]: the sub-vector space of irreducible partitions.
E (i) g [A] = ∞ k=0 C S A (i) k .
Actually, two subspaces of E g [A] will be interesting for us:
E (i) [A] = E (i) g [A] ∩ E[A], e (i) [A] = E (i) g [A] ∩ e[A]
. When A = S, we have already seen after Definition 2.9 that:
A (i) k = {σ(1, . . . , k)σ −1 , σ ∈ S k }. Proposition 10.1. -The affine space E (i) [S]
can ben identified, by the following isomorphism, with the affine space C 1 [[z]] of formal power series which constant term is equal to 1:
E (i) [S] → C 1 [[z]] E → k∈N E (1,...,k) z k .
Any element E in E g [A] can be restricted in order to obtain an element of E such that for any integer k, any p ∈ A k ,
(M(E)) p = C∈C(p) E p C .
Any element of the image of the application:
M : E (i) [A] → E[A] E → M(E)
is called multiplicative and we denote ME
[A] = M [E[A]] .
Let us remark that 0 E and 1 E are multiplicative elements. This is not the only property satisfied by ME[A].
(E ⊞ F ) p 1 ⊗p 2 = (E ⊞ F ) p 1 (E ⊞ F ) p 2 .(22)
Yet, by definition:
(E ⊞ F ) p 1 ⊗p 2 = (a 1 ,a 2 ,I)∈F 2 (p 1 ⊗p 2 ) E a 1 F a 2 ,
and:
(E ⊞ F ) p 1 (E ⊞ F ) p 2 = (a 1 1 ,a 1 2 ,I 1 )∈F 2 (p 1 ),(a 2 1 ,a 2 2 ,I 2 )∈F 2 (p 2 ) E a 1 1 E a 2 1 F a 1 2 F a 2 2 .
Using the fact that E and F are multiplicative, that E ∅ = 1 = F ∅ and using the fact that for any (a 1 , a 2 , I) ∈ F 2 (p 1 ⊗ p 2 ), a 1 and a 2 can be decomposed into two parts in order to get two 3-tuples (a 1 1 , a 1 2 , I 1 ) ∈ F 2 (p 1 ) and (a 2 1 , a 2 2 , I 2 ) ∈ F 2 (p 2 ), one gets the equality (22). Let us show that E ⊠ F is multiplicative. Let p 1 and p 2 be two partitions, we have to show that:
(E ⊠ F ) p 1 ⊗p 2 = (E ⊠ F ) p 1 (E ⊠ F ) p 2 .
By definition:
(E ⊠ F ) p 1 ⊗p 2 = a,b/a•b=p 1 ⊗p 2 ,a≺p 1 ⊗p 2 E a F b .
Yet, using Lemma 6.4, any partition a such that a ≺ p 1 ⊗ p 2 can be decomposed as a 1 ⊗ a 2 such that a 1 ≺ p 1 and a 2 ≺ p 2 . Then if b is a partition such that a 1 ⊗ a 2 • b = p 1 ⊗ p 2 , b can be also decomposed as b = b 1 ⊗ b 2 with a 1 ⊗ b 1 = p 1 and a 2 ⊗ b 2 = p 2 . Using the multiplicative property of E and F , one gets:
(E ⊠ F ) p 1 ⊗p 2 = a 1 ,a 2 ,b 1 ,b 2 /a 1 •b 1 =p 1 ,a 2 •b 2 =p 2 ,a 1 ≺p 1 ,a 2 ≺p 2 E a 1 E a 2 F b 1 F b 2 = a 1 ,b 1 /a 1 •b 1 =p 1 ,a 1 ≺p 1 E a 1 F b 1 a 2 ,b 2 /a 2 •b 2 =p 2 ,a 2 ≺p 2 E a 2 F b 2 = (E ⊠ F ) p 1 (E ⊠ F ) p 2 .
This ends the proof. Let us justify our notation ⊞. If we consider the pull-back of the ⊞ operation from ME[A] to E (i) [A] and if one consider only the coefficients for the non-empty partitions, one simply obtains the usual additive law on E (i) [A]. We will also see in the article [9] that ⊞ is the natural operation which appears when one is working with sum of free elements.
We believe that the inverse of a multiplicative element for the ⊞ and ⊠ is still multiplicative, but we have not yet written the proof. It is natural to wonder, as we have two semi-groups (ME[A], ⊞) and (ME[A] ∩ GE A , ⊠) on which one can define differentiable one-parameter semi-groups, what are the "Lie algebras" of these two semi-groups. Let us remark that ME[A] ∩ GE A is only the set of elements E of ME[A] such that E id 1 = 0. We need to define two ways to inject e (i) [A] in e[A], the first of which is the natural injection. such that, for any integer k, any weakly irreducible p = id k in A k :
(J(E)) p = E(p S(p) ),
and (J(E)) id k = k (J(E)) id 1 and for any other p ∈ A k , (J(E)) p = 0. We define me
⊠ [A] = J(E (i) [A]).
This might look strange that we change the definition for (J(E)) id k . It is easier to understand it by using an other equivalent definition of the weakly irreducible notion. The partition p is weakly irreducible if there exist p 0 irreducible and I ⊂ {1, . . . , k} such that p = σ −1 I (p 0 ⊗ Id k−l(p 0 ) )σ I . The partition p 0 is unique if and only if p = id k . If p 0 is unique then (J(E)) p = E(p 0 ). If p = id k , then id k = σ −1 {l} (id 1 ⊗ id k−1 )σ {l} for any integer l ∈ {1, . . . , k}. We do not choose and we prefer to sum all the values: (J(E)) id k = kE id 1
d dt |t=t 0 e tE ⊠ = E ⊠ e t 0 E ⊠ , e 0E ⊠ = 1 E .
We defined e tE ⊞ and e tE ⊠ as a one-parameter semi-group for two reasons: it will appear later in this formulation, and it allows to have a Lie group/Lie algebra formalism. An equivalent definition is given by the next proposition.
E(p ′ 1 )E t 0 (p ′ 2 ).
Yet, we must not forget that E is in me ⊞ [A]: for any integer k, any p ∈ P k , if p is not irreducible or if p = ∅, then E(p) = 0. Thus the sum we are considering can be taken over the (p ′ 1 , p ′ 2 , I) ∈ F 2 (p 1 ⊗ . . . ⊗ p n ) such that p ′ 1 is irreducible and not equal to ∅: this means in particular that p ′ 1S(p ′ 1 ) is one of the (p i ) n i=1 . Thus:
d dt |t=t 0 E t p 1 ⊗...⊗pn = n i=1 E(p i )E t 0 p 1 ⊗···⊗p i−1 ⊗p i+1 ⊗···⊗pn .
On the other hand,
d dt |t=t 0 E t p 1 . . . E t pn = n i=1 d dt |t=t 0 E t p i j =i E t 0 p j = n i=1 E(p i ) j =i E t 0 p j .
This allows to conclude that E t ∈ ME[A] for any real t ≥ 0. Let (E t ) t≥0 be a differentiable one-parameter semi-group for the ⊞ operation which is in ME[A] and such that E 0 = 0 E . Then using the same calculation that we did, for any integer n and any irreducible partitions p 1 , . . . , p n in ∪ k∈N A (i) k , for any real t 0 ≥ 0, we have:
d dt |t=t 0 (E t p 1 ⊗...⊗pn ) = d dt t=t 0 (E t p 1 ...E t pn ) = n i=1 d dt |t=t 0 E t p i j =i E t 0 p j .
Yet p i is irreducible, thus d dt |t=t 0 E t p i = d dt |t=0 E t ⊞ E t 0 p i = d dt |t=0 E t p i , and thus:
d dt |t=t 0 (E t p 1 ⊗...⊗pn ) = n i=1 d dt |t=0 E t p i j =i E t 0 p j = I d dt |t=0 E t |E (i) [A]
⊞ E t 0 p 1 ⊗. ..⊗pn and thus there exists E ∈ me ⊞ [A] such that for any t ≥ 0, e tE ⊞ = E t . Now, let E ∈ me ⊠ [A]. For any t ≥ 0 we consider E t = e tE ⊠ . Let n be an integer and let us consider n irreducible partitions p 1 , . . . , p n in ∪ k∈N A Yet, we must not forget that E is in me ⊠ [A]: for any integer k, any p ∈ P k , if p is not weakly irreducible then E(p) = 0. Thus the sum we are considering can be taken over the (a, b) ∈ F 2 (p 1 ⊗ · · · ⊗ p n ) such that a is weakly irreducible. Besides, E id l = lE id 1 for any integer l. Thus:
E id n i=1 l(p i ) E t 0 p 1 ⊗···⊗pn = n i=1 E id l(p i ) E t 0 p 1 ⊗···⊗pn .
Thus, we get: On the other hand,
d dt |t=t 0 E t p 1 . . . E t pn = n i=1 d dt |t=t 0 E t p i j =i E t 0 p j = n i=1 a,b/a•b=p i ,a≺p i E a E t 0 b j =i E t 0 p j .
This allows to conclude that E t ∈ ME[A] for any real t ≥ 0. Let (E t ) t≥0 be a differentiable one-parameter semi-group for the ⊠ operation which is in ME[A] and such that E 0 = 1 E . Then using the same calculation that we did, for any integer n and any irreducible partitions p 1 , . . . , p n in ∪ k∈N A (i) k , for any real t 0 ≥ 0, we have:
d dt |t=t 0 (E t p 1 ⊗...⊗pn ) = d dt t=t 0 (E t p 1 ...E t pn ) = n i=1 d dt |t=t 0 E t p i j =i E t 0 p j .
Yet p i is irreducible, thus d dt |t=t 0 E t p i = a,b/a•b=p i ,a≺p i d dt |t=0 E t p i E t 0 b , and thus:
d dt |t=t 0 (E t p 1 ⊗...⊗pn ) = n i=1 a,b/a•b=p i ,a≺p i d dt |t=0 E t p i E t 0 b j =i E t 0 p j = J d dt |t=0 E t |E (i) [A]
⊠ E t 0 p 1 ⊗. ..⊗pn and thus there exists E ∈ me ⊠ [A] such that for any t ≥ 0, e tE ⊠ = E t . Remark 10.2. -In fact, e[A] is endowed with two structures of Lie algebras. Indeed, it is a vector space for the addition and multiplication by a scalar, and we can define two Lie brackets on it, one named [., .] ⊞ which comes from the ⊞ operation and the other named [., .] ⊠ which comes from the ⊠ operation. In order to know which bracket is considered on e[A], we will denote it either by e ⊞ [A] or by e ⊠ [A].
Since the operation ⊞ is commutative, the bracket [., .] ⊞ is trivial. Thus me ⊞ is a sub-Lie algebra of e ⊞ .
Since the operation ⊠ is not commutative, the bracket [., .] ⊠ is not trivial and for any E and F in e ⊠ [A],
[E, F ] ⊠ = E ⊠ F − F ⊠ E.
Then, it is not difficult to see directly that me ⊠ [A] is a sub-Lie algebra of e ⊠ [A]. 10. 1.2. The R A -transform. -We will define the notion of R A -transform. This application will be defined as the inverse of the M A -transform whose definition lies on the Equation (5). such that for any E ∈ E[A], for any integer k, any p ∈ A k :
(M A (E)) p = p ′ ∈[id,p] A k E p ′ .
This application is a bijection. Thus we can consider its inverse.
Definition 10. 10. -The R A -transform is the inverse of the M A -transform:
R A = M −1 A .
We will often forget about the indices A when we will work with the R-transforms. One can show that the R A -transform is a bijection from ME[A] to itself. This concludes the proof.
It is well-known in the literature that there exists a transformation on C 1 [[z]] which we will call the R u -transform. In order to finish this section, we make the link between our R A -transform and the R u -transform. Let C(z) be the formal power serie C(z) = 1 + ∞ n=1 k n z n such that C[zM (z)] = M (z). The R u -transform of M is C.
The R A -transform is a generalization of the usual R u -transform. Indeed, we have the following theorem.
E[P]
M →c / / M P " " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ E[P] M c→ E[P]
Proof. -This is a straithforward application of Proposition 8.2.
10.2.
Higher order. -In Definition 9.5, we defined the ∞-development algebra of order m of A k . Thus, one can also define a higher order R-transform: we will only give definitions in this section. Let m ∈ N be the higher order of fluctuations which we are working with. Again, in order to simplify the notations, we will use the following convention: for any p ∈ ∪ ∞ k=0 A k and any i ∈ {0, . . . , m}: E p,i = E i (p) = (E l(p) ) p,i , and for any integer k:
E k = p∈A k ,i∈{0,...,m} E p,i p X i .
As for E g [A], the algebra E g,(m) [A] is naturally endowed with a natural addition and multiplication given, for any E, F ∈ E g,(m) [A], by:
(E + F ) k = E k + F k , (E ⊠ F ) k = E k F k .
Besides, we can also construct an other law on E g,(m) [A].
Definition 10. 15.
-Let E and F be two elements of E g,(m) [A]. We denote by E ⊞ F the element of E g,(m) [A] such that for any positive integer k, any p ∈ A k and any i ∈ {0, . . . , m}:
(E ⊞ F ) i (p) = (p 1 ,p 2 ,I)∈F 2 (p) i i 1 =1 E i 1 (p 1 )F i−i 1 (p 2 ),
where F 2 (p) was defined in Definition 2. 10.
Again, the subset E (m) [A] is stable by the ⊠ and ⊞ operations. Besides, E (m) [A] is an affine space.
The operation ⊞ is commutative, it defines a structure of group on E (m) [A]. The neutral element is the element 0 E (m) ∈ E (m) [A] such that, for any positive integer k, any p ∈ A k and any i ∈ {0, . . . , m}, (E) p,i = 0.
The operation ⊠ is not commutative and the set of invertible elements in E (m) [A] is the set of elements E such that E id k ,0 = 0 for any k ≥ 1. We denote by 1 E (m) the neutral element for ⊠ which is the only element in E (m) [A] such that for any k ≥ 1, (E k ) k = id k X 0 . We can also define a R This application is a bijection: we can consider its inverse.
Conclusion
We have defined a geometry on partitions, and new notions of convergence for elements of N ∈N C[A k (N )]. Using Schur-Weyl's duality and similar results, we will link the study of random matrices with the study of elements in N ∈N C[A k (N )] and in E[A]. In the article [9], we apply the results proved in this article to the theory of random matrices invariant in law by conjugation by the symmetric group. We also study additive and multiplicative unitary or orthogonal invariant Lévy processes. In the article [10], we apply the results of the first two articles to the study of random walks on the symmetric group and the study of the S ∞ -Yang-Mills theory.
pn(p) = # i, p i contains both an element of {1, . . . , k} and one of {1 ′ , . . . , k ′ } , l(p) = k.
Figure 1 .
1Partition p
Figure 3 .
3Partition p 1 ⊗ p 2 .
Figure 4 .
4Two diagrams which represent p 1
Figure 5 .
5Partition t p 2
Figure 6 .
6Partition p 1 • p 2 .
Figure 7 .
7Example of a product which involves the counting of loops.
Figure 8 .
8The neutral element id 5 . Definition 2.5. -Let k be an integer. Let I ⊂ {1, . . . , k}: I = {i 1 , .
Figure 9 .
9The transposition (1, 2) and the Weyl contraction [1, 2]. Definition 2.7. -Let k be an integer. Let i, j be two distinct integers in {1, . . . , k}. The transposition (i, j) in S k is:
Definition 2 .
213. -For any p ∈ P k and any k-uples (i 1 , . . . , i k ) and (i 1 ′ , . . . , i k ′ ) of elements of {1, . . . , N }, we set:p i 1 ,...,i k i 1 ′ ,...,i k ′ = iffor any two elements r and s ∈ {1, . . . , k} ∪ {1 ′ , . . . , k ′ } which are in the same block of p, one has i r = i s , 0, otherwise.
if for any two elements r and s ∈ {1, . . . , k} ∪ {1 ′ , . . . , k ′ }, i r = i s if and only if r and s are in the same block of p, 0, otherwise.
Proposition 2.1. -For any integers k and N , for any partition p ∈ P k ,
Remark 3. 1 .
1-The graph G k plays the role of the Cayley graph of P k . Actually, if one considers the subgraph S k obtained by restraining it to the vertices which are permutations, one really obtains the Cayley graph of the symmetric group S k . The Cayley graph B k of B k is defined as the restriction of G k to the vertices which are in B k .
Figure 11 .
11The graph G ′ 2 .
Lemma 3.2.-The restriction of d to the permutation group is quite usual:
Remark 4.1. -For any integer k, any integer N , for any E ∈ C[P k (N )]:
Definition 4.4. -The sequence (E N ) N ∈N converges in moments if the normalized moments of E N converges when N goes to infinity: for any p ∈ A k , m p (E N ) converges when N goes to infinity.
Exclusive coordinate numbers. -In Section 2.3, we defined an other basis of C[P k ], namely the exclusive basis. In the case we are working with an element E ∈ C[A k (N )] we can also define the exclusive coordinate numbers.
Definition 4.5. -Let k and N be two integers. Let E be an element of C [A k (N )].We define the numbers κ p c (E) p∈P k as the only numbers such that:
Theorem 4.4. -Let k be an integer. Let (E N ) N ∈N be an element of N ∈N C[A k (N )]. The exclusive coordinate numbers (κ p c (E N )) p∈A k converge as N goes to infinity if and only if (E N ) N ∈N converges.
The sequence (E N ) N ∈N converges in exclusive normalized moments if and only if for any p ∈ P k , (m p c (E N )) N ∈N converges. Besides, if (E N ) N ∈N converges in normalized moments then for any p ∈ P k :
the exclusive world, coordinate numbers and moments are equal. -We will prove that the limit of exclusive normalized moments are in fact equal to the limit of the exclusive coordinate numbers. Let (E N ) N ∈N ∈ N ∈N C[A k (N )].
Theorem 4.5. -Let us suppose that (E N ) N ∈N converges in normalized moments. Then, for any p ∈ P k , lim N →∞ m p c (E N ) = lim N →∞ κ p c (E N ).
Theorem 4. 5 ,
5Theorem 4.4 and Proposition 4.1, one can give an expression of the exclusive moments which involves the coordinate numbers, and one can link the exclusive normalized moments with the coordinate numbers.
Theorem 4.6. -Let (E N ) N ∈N ∈ N ∈N C[A k (N )].Let us suppose that (E N ) N ∈N converges in normalized moments. Then, for any p ∈ P k , lim N →∞
Proposition 5.1. -Let N be an integer. The deformed algebra C[A k
Proposition 5.2. -Let k and N be two integers. The application M N k can be extended as an isomorphism of algebra from C[A k (N, N )] to C[A k (N )]. Its extension will be also denoted by M N k . For any integer N , the deformed algebra C[A k (N, N )] is isomorphic to C[A k (N )].
N
∈N C[A k (N )] by using the morphisms (M N k ) N ∈N and the usual notion of convergence in vector spaces. Indeed, for any integers N and k, any element in C[A k (N, N )] can be considered as an element of C[A k ]. This allows to state the following lemma.
Lemma 5. 4 .
4-Let (E N ) N ∈N be an element of N ∈N C[A k (N )]. The sequence (E N ) N ∈Nconverges strongly if and only if:
Lemma 6. 2 .
2-Let k be an integer. Let p and p 0 in A k . If p ≺ p 0 then there exists p ′ ∈ A k such that p 0 = p • p ′ and p ∈ [id, p 0 ] A k .
Lemma 6.5. -The notion of K p 0 (p) generalizes the notion of Kreweras complement for non-crossing partitions over {1, . . . , k} and p 0 = (1, . . . , k).
Definition 6.3. -Let C be a finite set of elements. For any N ∈ N ∪ {∞}, let L N be an algebra such that C is a linear basis of L N . For any elements x and y of C, for each N ∈ N ∪ {∞}, we denote the product of x with y in L N by x. N y.
7 .
7Consequences of the convergence of the deformed algebras. 7.1. Convergence of a product. -Let k be an integer. As usual, let A k be S k , B k or P k . Let us give the first consequence of Theorem 6.1 for the product of two elements of N ∈N C[A k (N )]. Recall the Notation 4.1. Theorem 7.1. -Let (E N ) N ∈N , (F N ) N ∈N be elements of N ∈N C[A k (N )]. Let us suppose that (E N ) N ∈N and (F N ) N ∈N converge, then the sequence E N F N N ∈N converges. Besides, for any p 0 ∈ A k :
Proof. -Let (E N ) N ∈N , (F N ) N ∈N elements of N ∈N C[A k (N )]. Let us suppose that (E N ) N ∈N and (F N ) N ∈N converge. We have by definition:
, that (M N k ) −1 (E N ) and (M N k ) −1 (F N ), seen as elements of C[A k ], converge when N → ∞. Besides, the algebra C[A k (N, N )] converges to C[A k (∞, ∞)], as it was proved in Theorem 6.1. Thus (M N k ) −1 (E N F N ) converges when N goes to infinity. Again, by Lemma 5.4 and Theorem 4.1, this shows that (E N F N ) N ∈N converges.
Definition 7.2. -The semi-group (E N t ) N t≥0 converges if and only if for any t ≥ 0, E N t converges as N goes to infinity. The next theorem, one of the main theorems of the paper, shows that a semi-group in N ∈N C[A k (N )] converges if the initial condition and the generator converge. Recall the Notation 4.1.
Theorem 7.2. -The semi-group (E N t ) Nt≥0 converges if the sequences (E N 0 ) N ∈N and (H N ) N ∈N converge as N goes to infinity.
Proof. -Let us suppose that (H N ) N ∈N converges. For any integer N and any t ≥ 0, we define:Ẽ N t = (M N k ) −1 (E N t ), H N = (M N k ) −1 (H N ). As for any integer N , M N k is a morphism of algebra, the family (Ẽ N t ) N ∈N t≥0 is a semi-group in N ∈N C[A k (N, N )] and its generator is H N N ∈N . An application of Lemma 5.3 allows us to write the condition of semi-group in the basis A k ; for any t 0 ≥ 0:
8. 1 .
1Geometric consequences of Theorem 4.1. -First, let us give a new proof of the improved triangular inequality. Proof of Proposition 6.1. -Let k be an integer. Let p and p ′ be two elements of A k . Let us consider (p N ) N ∈N and (p ′ N ) N ∈N such that for any integer N :
Proposition 8.1. -Let p 0 , p 1 and p 2 be three partitions in A k . Let τ be the partition in A 2k defined by: τ = (1, k + 1)(2, k + 2) . . . (k, 2k).
Proof. -Let p 0 , p 1 and p 2 be three partitions in A k . Let us consider (p 1 N ) N ∈N and (p 2 N ) N ∈N such that for any integer N , p 1 N = M N k (p 1 ) and p 2 N = M N k (p 2 ). Using Lemma 5.4, (p 1 N ) N ∈N and (p 2 N ) N ∈N converge strongly. Thus, (p 1 N ⊗ p 2 N ) N ∈N converges strongly, and by Theorem 4.1 it converges in moments.
Proposition 8.2.-Let p 0 , p 1 and p 2 be three partitions in A k . Let τ be the partition in A 2k defined by:
8. 2 .
2Combinatorial consequences of Theorem 4.1. -Let us remark the following important, yet simple theorem.
Theorem 8.1. -Let (m p ) p∈A k be a family of complex numbers. There exists a sequence (E N ) N ∈N ∈ N ∈N C[A k (N )] which converges and such that: lim N →∞ m p (E N ) = m p .
Proposition 8.3. -Let p and p 0 be two elements of A k . Then:
Proposition 8.4. -For any integer N , let us consider E N an element of C[A k (N )]. Let us suppose that (E N ) N ∈N converges. Let p and p 0 be two elements of A k . Then:
Definition 9. 2 .
2-Let N, k and m be integers, let X be a formal variable. The Ndevelopment algebra of order m of A k , C (m) [A k
9. 1 .
1Coordinate numbers. -Let us remark that for any integer N , the Ndevelopment algebra of order 0 of A k is canonically isomorphic to C[A k (N, N )]. Lemma 9.1. -Let N be an integer, the application:
Definition 9. 3 .
3-Let N and m be two integers.Let E ∈ C (m) [A k (N )].The coordinate numbers of E up to the order m are the elements (κ p i (E)) i∈{0,...,m},p∈P k such that:
] and not the algebra C[A k (N )]. Definition 9.4. -Let m ∈ N. The sequence (E N ) N ∈N converges if and only if for any i ∈ {0, . . . , m − 1}, and any
9. 2 .
2Convergences: C (m) [A k (N )] and multiplication. -Using Lemma 9.1, Theorem 6.1, as the algebra C[A k (N, N )] is isomorphic to C (0) [A k (N )] by an isomorphism which sends the canonical base of the first algebra on the canonical base of the second algebra, we know that the algebra C (0) [A k (N )] converges as N tends to infinity. In fact, the result holds for any m ∈ N. Definition 9.5. -Let N, k and m be three integers. Let X be a formal variable. The ∞-development algebra of order m of A k , denoted by C (m) [A k (∞)] is the associative algebra generated by the elements of the form: p X i , where p ∈ A k and i ∈ {0, . . . , m}. The product is defined such that, for any p and p ′ in A k , and any i and j in {0, . . . , m},
Proposition 9.1. -Let k and m be two integers. When N goes to infinity, the Ndevelopment algebra of order m of A k , C (m) [A k (N )] converges to the ∞-development algebra of order m of A k , namely C (m) [A k (∞)]. Proof. -Let k be an integer. The algebras C (m) [A k (N )] have, for any integer N , the same linear basis p X i i∈{0,...,m},p∈A k
-
Let m be an integer, let (E N ) N ∈N and (F N ) N ∈N be elements of N ∈N C (m) [A k (N )]. Let us suppose that the two sequences (E N ) N ∈N and (F N ) N ∈N converge. The sequence (E N F N ) N ∈N converges and, using Notations 9.1, for any i 0 ∈ {0, . . . , m} and for any p 0 ∈ A k :
2, implies a criteria for the convergence of semi-groups in N ∈N C (m) [A k (N )]. Definition 9.6. -Let m and k be two integers. Let (E N t ) N t≥0 be a semi-group in N ∈N C (m) [A k (N )]. The semi-group (E N t ) N t≥0 converges if and only if for any t ≥ 0, E N t N ∈N converges.
Proposition 9.3. -Let m ∈ N. Let us consider E N t N t≥0 a semi-group in N ∈N C (m) [A k (N )]which generator is denoted by (H N ) N ∈N . It converges if the sequences (E N 0 ) N ∈N and (H N ) N ∈N converge. Besides, using Notation 9.1, for any p ∈ A k , for any t 0 ≥ 0 and any i ∈ {0, . . . , m},
The other properties are easily verified. The function eval N (m) has an inverse if and only if m = 0. This will motivate us in order to define a notion of convergence up to order m of fluctuations for sequences in N ∈N C[A k (N )]. Then, given a linear or multiplicative problem in C[A k (N )], one can try to find a similar problem in C (m) [A k (N )], solve this last problem, and push by eval N (m) the solution on a solution of the first problem.
9. 3 .
3Convergence at any order of fluctuations in N ∈N C[A k (N )]. -We are interested in elements in C[A k (N )] and we want to define a notion of strong convergence up to the m th order of fluctuations.Definition 9.7. -Let m be an integer, let (E N ) N ∈N ∈ N ∈N C[A k (N )]. The sequence (E N ) N ∈N converges strongly up to the m th order of fluctuations if and only if there exist two families of real (κ p i ) i∈{0,...,m−1},p∈A k and (κ p m,N ) p∈A k ,N ∈N such that: -∀p ∈ A k , κ p (E N ) = , -∀p ∈ A k , κ p m,N converges as N goes to infinity. The families (κ p i ) i∈{0,...,m−1},p∈A k and (κ p m,N ) p∈A k are uniquely defined. For any p ∈ A k , any integer N and any i ∈ {0, . . . , m − 1}, κ p i is the coordinate number of E N on p of order i, and κ p m,N is the coordinate number of E N on p of order m. Notation 9.2. -Let m be an integer. Let (E N ) N ∈N in N ∈N C[A k (N )] such that (E N ) N ∈N converges strongly up to the m th order of fluctuations. From now on, the coordinate numbers of E N on p of order i will be denoted by κ p i (E N ). For any p ∈ A k and any i ∈ {0, . . . , m}, we will define:
= 0 .
0This definition explains why the families (κ p i ) i∈{0,...,m−1},p∈A k and (κ p m,N ) N ∈N,p∈A k defined in Definition 9.7 are uniquely defined. The next lemma makes a link between the convergence of elements of N ∈N C (m) [A k (N )] and the convergence up to the m th order of fluctuations of elements ofN ∈N C[A k (N )]. Lemma 9.3. -Let m ∈ N. Let (E N ) N ∈N ∈ N ∈N C (m) [A k (N )]. Let us suppose that (E N ) N ∈N converges. Then eval N (m) (E N ) N∈N converges strongly up to the m th order of fluctuations. The notion of strong convergence to the m th order of fluctuations allows to inject canonically an element of N ∈N C[A k (N )] which converges strongly up to the m th order of fluctuations into N ∈N C (m) [A k (N )]. Definition 9.8. -Let m be an integer, let (E N ) N ∈N ∈ N ∈N C[A k (N )]. Let us suppose that (E N ) N ∈N converges strongly up to the m th order of fluctuations. For any p ∈ A k , any integer N , let (κ p i ) i∈{0,...,m−1} and κ p m,N be the coordinate numbers of E N on p. We define the lift of the sequence (E N ) N ∈N as (Ẽ N ) N ∈N ∈ N ∈N C (m) [A k (N )] defined by:
Lemma 9.4. -Let m ∈ N, let (E N ) N ∈N ∈ N ∈N C[A k (N )]and let us suppose that (E N ) N ∈N converges strongly up to the m th order of fluctuations. Let (Ẽ N ) N ∈N be its canonical lift in N ∈N C (m) [A k (N )]. Then (Ẽ N ) N ∈N converges as N goes to infinity and for any N ∈ N, one has eval N (m) (Ẽ N ) = E N .
-
Let m be an integer, let (E N ) N ∈N ∈ N ∈N C[A k (N )]. The sequence (E N ) N ∈N converges in moments up to the m th order of fluctuations if and only if there exist two families (m i p ) i∈{0,...,m−1},p∈A k and (m m p,N ) N ∈N,p∈A k such that:
N goes to infinity. The families (m i p ) i∈{0,...,m−1},p∈A k and (m m p,N ) N ∈N,p∈A k are uniquely defined. For any p ∈ A k , any integer N , and any i ∈ {0, . . . , m − 1}, m i p is the i th -order fluctuations of the p-normalized moment of E N , and m m p,N is the m th -order fluctuations of the p-normalized moment of E N . Notation 9.3. -Let m be an integer. Let (E N ) N ∈N ∈ N ∈N C[A k (N )] such that (E N ) N ∈N converges in moments up to the m th order of fluctuations. From now on, the i th -order fluctuations of the p-normalized moment of E N will be denoted by m i p (E N ). For any p ∈ A k and any i ∈ {0, . . . , m}, we define:
Theorem 9.1. -Let m ∈ N, let (E N ) N ∈N ∈ N ∈N C[A k (N )]. The sequence (E N ) N ∈N converges strongly up to the m th order of fluctuations if and only if it converges in moments up to the m th order of fluctuations. We will say that (E N ) N ∈N converges up to the m th order of fluctuations. Let us suppose that (E N ) N ∈N ∈ N ∈N C[P k (N )] converges up to the m th order of fluctuations. Using Notations 9.2 and 9.3 and the Definition 9.1, we have that, for any i 0 ∈ {0, . . . , m} and any p ∈ A k :
Proof. -Let m be an integer and let (EN ) N ∈N ∈ N ∈N C[A k (N )]. Let us consider p in A k .Let us suppose that (E N ) N ∈N converges strongly up to the m th order of fluctuations. The coordinate numbers of E N are defined up to order m of fluctuations and:
holds. Let us suppose now that (E N ) N ∈N converges in moments up to the m th order of fluctuations. Then, by Theorem 4.1, it converges strongly up to order 0 of fluctuation. Let us suppose that (E N ) N ∈N converges strongly up to order l of fluctuations with l < m. Thus, the coordinate numbers of E N up to order l of fluctuations are well defined and we can write:
-
Let m ∈ N. Let (E N ) N ∈N and (F N ) N ∈N be elements of N ∈N C[A k (N )]. Let us suppose that the sequences (E N ) N ∈N and (F N ) N ∈N converge up to the m th order of fluctuations. Then, the sequence (E N F N ) N ∈N converges up to the m th order of fluctuations.
Proof. -Let (E N ) N ∈N and (F N ) N ∈N be elements of N ∈N C[A k (N )]. Let us suppose that the sequences (E N ) N ∈N and (F N ) N ∈N converge strongly or in moments up to the m th order of fluctuations. By Lemma 9.4, let us consider the canonical lifts of (E N ) N ∈N (resp. (F N ) N ∈N ) in N ∈N C (m) [A k (N )]: (Ẽ N ) N ∈N (resp. (F N ) N ∈N ). The two sequences (Ẽ N ) N ∈N and (F N ) N ∈N converge. According to Proposition 9.2, the sequence (Ẽ NFN ) N ∈N converges. For any i 0 ∈ {0, . . . , m} and for any p 0 ∈ A k :
of Lemma 9.3 shows that the sequence eval N (m) (Ẽ NFN ) N ∈N converges up to the m th order of fluctuations. As eval N (m) is a morphism of algebra, Lemma 9.2, for any N ∈ N, eval N (m) (Ẽ NFN ) = eval N (m) (Ẽ N )eval N (m) (F N ) = E N F N . We deduce that (E N F N ) N ∈N converges strongly up to the m th order of fluctuations. The equality (19) is deduced from (21).
Definition 9.10. -Let m ∈ N. The semi-group E N t N t≥0 converges to the m th order of fluctuations if and only if for any t ≥ 0, E N t N ∈N converges up to the m th order of fluctuations.
-
Let m ∈ N. The semi-group E N t N t≥0 converges to the m th order of fluctuations if the sequences (E N 0 ) N ∈N and (H N ) N ∈N converge up to the m th order of fluctuations.
by conjugation by any element of S k . We define:
denote by E |E i [A] . Conversely, given an element of E (i) g [A], one can inject it non-trivially in E g [A] in a natural way. Recall the definition of the extraction of p in Definition 2.11, and the definition of cycles given in Definition 2.8. We only consider the injection of an element of E (i) [A] in E[A]. Definition 10.4. -For any E ∈ E (i) [A], we denote by M(E) the unique element of E[A]
Definition10.5. -For any E ∈ e (i) [A], we denote by I(E) the unique element of e[A] such that, for any positive integer k, any irreducible p ∈ A k ,(I(E)) p = E p , and for any non-irreducible p ∈ A k , (I(E)) p = 0. We define me ⊞ [A] = I(E (i) [A]).The second injection uses the notion of support of a partition and the notion of weakly irreducible partitions defined in Definition 2.12. Recall also the notion of extraction defined in Definition 2.11.Definition 10.6. -For any E ∈ e (i) [A], we denote by J(E) the unique element of e[A]
For any real t 0 ≥ 0, we have:d dt |t=t 0 E t p 1 ⊗...⊗pn = E ⊠ E t 0 p 1 ⊗...⊗pn = a,b/a•b=p 1 ⊗···⊗pn,a≺p 1 ⊗···⊗pn E a E t 0 b .
,b/a•b=p i ,a≺p i E a E t 0 p 1 ⊗···⊗p i−1 ⊗b⊗p i+1 ⊗···⊗pn .
Definition10.9. -The M A -transform is the application:M A : E[A] → E[A] E → M A (E)
Proposition 10.3. -The R A -transform is a bijection from ME[A] to itself. Proof. -It is only a consequence of the fact that: commutative. Indeed, using Lemma 3.3, if E ∈ E[B], and if σ ∈ S k :r [M B (E)] (σ) = (M B (E)) (σ) = p∈[id,σ] B k E p = p∈[id,σ] S k E p = [M S • r(E)] (σ).
Definition 10.11. -Let M (z) be a formal power serie in C 1 [[z]], that is a formal power serie of the form:
Theorem 10.3. -Using the identification E (i) [S] ≃ C 1 [[z]] explained in Proposition 10.1, the following diagram is commutative:Proof. -Let E be an element of E (i) [S] ≃ C 1 [[z]].Using Theorem 2.7 of[16], and using the bijection between non-crossing partitions of k elements and the set [id, (1, . . . , k)] S k , we know that R u (E) is characterized by the fact that for any integer k > 0:E (1,...,k) =p∈[id,(1,...,k)] S k c cycle of p Theorem 10.4. -The following diagram is commutative.
) [A k (∞)]where, for any integer k,C S (m) [A k (∞)] is the algebra of elements of C (m) [A k (∞)]which, seen as elements of C[A k ], are invariant by conjugation by any element of S k . We also consider the subspace of E g,(m) [A] defined by:E (m) [A] = {E ∈ E g,(m) [A], E ∅,0 = 1, E ∅,i = 0, ∀i ≥ 1}. Let us remark that E (0) [A] = E[A].Any element E ∈ E[A] is of the form:
A
: E (m) [A] → E (m) [A] E → M (m) A (E) such that for any E ∈ E (m) [A], for any positive integer k, any p ∈ A k and any i ∈ {0, . . . , m}: ∈A k ,df(p ′ ,p)≤i E p ′ ,i−df(p ′ ,p) .
A
-transform is the inverse of the M A -transform:
. , p r } then there exist i and j, distinct, such that p ′ = {p s , s ∈ {1, . . . , r} \ {i, j}} ∪ { p i ∪ p j }. The weight of the edge e is set to 0.5: w k (e) = 0.5. • one can go from one to the other by permuting two elements of {1, . . . , k} ∪ {1 ′ , . . . , k ′ } which are in distinct blocks. Let us suppose that we can go from p to p ′ by permuting two elements. In this case, if p is the partition {p 1 , . .. , p r },
there exist s, t ∈ {1, . . . , k, 1 ′ , . . . , k ′ } distinct and i, j ∈ {1, . . . , r} distinct, such
that s ∈ p i , t ∈ p j and p ′ = {p s , s ∈ {1, . . . , r} \ {i, j}} ∪ {(p i \ {s}) ∪ {t}, (p i \
{t}) ∪ {s}}. The weight of the edge e is set to 1: w k (e) = 1.
). This assertion comes from the fact that one can permute two elements of {1, . . . , k} ∪ {1 ′ , . . . k ′ } in the partition p by gluing two blocks of p and then splitting one block of the resulting partition. Indeed, let us suppose that p = {p 1 , . . . , p r }. Let s, t ∈ {1, . . . , k, 1 ′ , . . . , k ′ }, distinct, and let i, j ∈ {1, . . . , r}, distinct, such that s ∈ p i and t ∈ p j . Then p ′ = {p s , s ∈ {1, . . . , r} \ {i, j}} ∪ {(p i \ {s}) ∪ {t}, (p i \ {t}) ∪ {s}}can be obtained by:
1. gluing p i and p j ,
2. splitting p i ∪ p j in two: (p i \ {s}) ∪ {t} and (p i \ {t}) ∪ {s}.
The weight of this path is equal to 0.5 + 0.5 = 1. Thus, to compute the
Theorem 4.1.-We have already an interesting corollary of Theorem 4.1. Theorem 4.2. -For this theorem, let us suppose that A is equal either to
Remark 7.1. -We can show the similar result that, under the same assumptions:
.1.1. Order zero: general definitions and Lie algebras. -Recall that A is either S, B or P. Definition 10.1. -Let us define the algebra:
Besides, one can construct an other law on E g [A]. Definition 10.2. -Let E and F be two elements of E g [A]. We denote by E ⊞ F the element of E g [A] such that for any p ∈ A l(p) :
Remark 10.1. -The sets E[A] and e[A] are stable by the ⊞ and ⊠ operations. Besides, E[A] is an affine space whose underlying vector space is e[A]. The operation ⊞ on E[A] is commutative, it defines a structure of group on E[A]. The neutral element 0 E is the only element in E[A] such that for any positive integer k, any
Theorem 10.1. -The set ME[A] is stable by the operations ⊞ and ⊠. Proof. -Let E and F be two elements of M(E). Let us show that E ⊞ F is multiplicative. Let p 1 and p 2 be two partitions, we have to show that:
. Due to the definitions, it is obvious that the sets me ⊠ [A] and me ⊞ [A] are vector spaces. Let us define the exponentiation of any element of e[A] associated with the operation ⊞. Definition 10.7. -Let E ∈ e[A]. The ⊞-semi group associated with E is the family (e tE ⊞ ) t≥0 of elements of E[A] such that for any t 0 ≥ 0: d dt |t=t 0 e tE ⊞ = E ⊞ e t 0 E ⊞ , e 0E ⊞ = 0 E . Due to the commutativity of ⊞, one has that for any E, F ∈ e[A], e E ⊞ ⊞ e F ⊞ = e E⊞F ⊞ . Let us define the exponentiation associated with the operation ⊠. Definition 10.8. -Let E ∈ e[A]. The ⊠-semi group associated with E is the family (e tE ⊠ ) t≥0 of elements of E[A] such that for any t 0 ≥ 0:
Proposition 10.2. -Let E ∈ e[A]. For any t ∈ R + ,e tE ⊞ = prove that there exists E ∈ me ⊞ [A] (resp. E ∈ me ⊠ [A]) such that for any t ≥ 0, e tE ⊞ = E t (resp. e tE ⊠ = E t ), it is enough to show that:(E t )t≥0 and satisfy the same differential linear equations. Let E ∈ me ⊞ [A]. For any t ≥ 0 we consider E t = e tE ⊞ . Let n be an integer and let us consider n irreducible partitions p 1 , . . . , p n in ∪ k∈N A (i) k . For any real t 0 ≥ 0, we have: d dt |t=t 0 E t p 1 ⊗···⊗pn = E ⊞ E t 0 p 1 ⊗...⊗pn ,I)∈F 2 (p 1 ⊗...⊗pn)
e
tI
d
dt |t=0 Et
|E (i) [A]
⊞
t≥0
resp. (E t ) t≥0 and
e
tJ
d
dt |t=0 Et
|E (i) [A]
⊠
t≥0
=
(p ′
1 ,p ′
2
Proposition 3.4. -Let k 1 and k 2 be two positive integers and let k = k 1 + k 2 . Let p be an element of P k . Let p 1 and p 2 be respectively in P k 1 and P k 2 . We have equivalence between:1. p is coarser than p 1 ⊗ p 2 and p 1 ⊗ p 2 ∈ [id, p] P k , 2. p g k 1 is coarser than p 1 , p 1 is in [id, p g k 1 ] P k 1 , p d k 1 is coarser than p 2 , p 2 is in [id, p d k 2 ] P k 2 and p g k 1 ⊗ p d k 1 ∈ [id, p] P k .Proof. -Let k 1 and k 2 be two positive integers and let k = k 1 + k 2 . Let p be an element of P k . Let p 1 and p 2 be respectively in P k 1 and P k 2 .First of all, it is easy to see that p is coarser than p 1 ⊗p 2 if and only if p g k 1 is coarser than p 1 and p d k 1 is coarser than p 2 . Let us suppose that p is coarser than p 1 ⊗ p 2 , let us show thatp 1 ⊗ p 2 ∈ [id, p] A k if and only if p 1 ∈ [id, p g k 1 ] P k 1 , p 2 ∈ [id, p d k 1 ] P k 2 and p g k ⊗ p d k ∈ [id, p] A k .Since for any partitions the default between two partitions is always positive, this is equivalent to show that:df(p 1 ⊗ p 2 , p) = df(p 1 , p g k ) + df(p 2 , p d k ) + df(p g k ⊗ p d k , p).Yet, using Lemme 3.5:df(p 1 ⊗ p 2 , p) − df(p 1 , p g k ) − df(p 2 , p d k ) − df(p g k ⊗ p d k , p) = nc(p 1 ⊗ p 2 ) − nc((p 1 ⊗ p 2 ) ∨ id) − nc(p) + nc(p ∨ id) − nc(p 1 ) + nc(p 1 ∨ id) + nc(p l k ) − nc(p g k ∨ id) − nc(p 2 ) + nc(p 2 ∨ id) + nc(p r k ) − nc(p d k ∨ id) − nc(p g k ⊗ p d k ) + nc((p g k ⊗ p d k ) ∨ id) + nc(p) − nc(p ∨ id) = 0, since: nc(p 1 ⊗ p 2 ) = nc(p 1 ) + nc(p 2 ), nc((p 1 ⊗ p 2 ) ∨ id) = nc(p 1 ∨ id) + nc(p 2 ∨ id),nc(p g k ⊗ p d k ) = nc(p g k ) + nc(p d k ), nc((p g k ⊗ p d k ) ∨ id) = nc(p g k ∨ id) + nc(p d k ∨ id) This ends the proof.
+κ(p,p ′ ) .
R u (E)(1, . . . , #c).
Proof.-Using the multiplicativity of E, and Lemma 3.4, we see that E being in ME[A], the familysatisfies in fact that for any integer k, any p ∈ A k :We can also translate the Lemma 3.3 in terms of R-transform. For this, we need the restriction function defined for any integer k by:
5 3. Geometry on the set of partitions. 13 4. Convergence of elements of N ∈N C [P k (N )Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Partition algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Geometry on the set of partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4. Convergence of elements of N ∈N C [P k (N )] . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Refined geometry of the partition algebra. The deformed partition algebraThe deformed partition algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6. Refined geometry of the partition algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Consequences of the convergence of the deformed algebras. . . . . . . . . . , 35Consequences of the convergence of the deformed algebras.. . . . . . . . . . . . 35
A new way to get combinatorial properties. A new way to get combinatorial properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
. Or. with our notations: E (1,...,k) = p∈[id,(1,...,k)] S k M [R u (E)Or, with our notations: E (1,...,k) = p∈[id,(1,...,k)] S k M [R u (E)] .
By the factorization property of the geodesics, Lemma 3.4, for any σ ∈ S k. M (E)] (σ) =By the factorization property of the geodesics, Lemma 3.4, for any σ ∈ S k : [M (E)] (σ) =
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"\nLIANE GABORA University of British Columbia\n\n"
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"LIANE GABORA University of British Columbia\n"
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The speed and transformative power of human cultural evolution is evident from the change it has wrought on our planet. This chapter proposes a human computation program aimed at (1) distinguishing algorithmic from non-algorithmic components of cultural evolution, (2) computationally modeling the algorithmic components, and amassing human solutions to the non-algorithmic (generally, creative) components, and (3) combining them to develop human-machine hybrids with previously unforeseen computational power that can be used to solve real problems. Drawing on recent insights into the origins of evolutionary processes from biology and complexity theory, human minds are modeled as self-organizing, interacting, autopoietic networks that evolve through a Lamarckian (non-Darwinian) process of communal exchange. Existing computational models as well as directions for future research are discussed.
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"https://arxiv.org/pdf/1310.6342v1.pdf"
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1310.6342
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00e056d4c683cc08213d77fd766bee9aa7a735e0
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LIANE GABORA University of British Columbia
Reference Gabora, L. (2013). Cultural evolution as distributed human computation. In P. Michelucci (Ed.) Handbook of Human Computation. Berlin: Springer. Cultural Evolution as Distributed Computation
The speed and transformative power of human cultural evolution is evident from the change it has wrought on our planet. This chapter proposes a human computation program aimed at (1) distinguishing algorithmic from non-algorithmic components of cultural evolution, (2) computationally modeling the algorithmic components, and amassing human solutions to the non-algorithmic (generally, creative) components, and (3) combining them to develop human-machine hybrids with previously unforeseen computational power that can be used to solve real problems. Drawing on recent insights into the origins of evolutionary processes from biology and complexity theory, human minds are modeled as self-organizing, interacting, autopoietic networks that evolve through a Lamarckian (non-Darwinian) process of communal exchange. Existing computational models as well as directions for future research are discussed.
Introduction
The origin of life brought about unprecedented change to our planet; new forms emerged creating niches that paved the way for more complex forms, completely transforming the lands, skies, and oceans. But if biological evolution is effective at bringing about adaptive change, human cultural evolution is arguably even more effective, and faster. Cultural change doesn't take generations; it works at the speed of thought, capitalizing on the strategic, intuitive creative abilities of the human mind.
This chapter outlines current and potential future steps toward the development of a human computation program inspired by the speed and effectiveness of how culture evolves. The overarching goal of the kind of research program outlined in this chapter is to develop a scientific framework for cultural evolution by abstracting its algorithmic structure, use this algorithmic structure to develop human-machine hybrid structures with previously unforeseen computational power, and to apply it to solving real problems. The proposed approach can be thought of as a "repeatable method" or "design pattern" for fostering cultural emergence, defined by specific computational methods for modeling interactions at the conceptual level, the individual level, and the social level, and their application to the accumulation of adaptive, open-ended cultural novelty.
Two Approaches to a Scientific Framework for Culture
Cultural evolution entails the generation and transmission of novel behavior and artifacts within a social group, both vertically from one generation to another, and horizontally amongst members of a generation. Like biological evolution, it relies on mechanisms for both introducing variation 2008a,b). EVOC (for EVOlution of Culture) consists of neural network based agents that invent new actions and imitate actions performed by neighbors. The assemblage of ideas changes over time not because some replicate at the expense of others, as in natural selection, but because they transform through inventive and social processes. Agents can make generalizations concerning what kinds of actions are fittest, and use this acquired knowledge to modify ideas for actions between transmission events. EVOC exhibits typical evolutionary patterns, e.g., cumulative increase in fitness and complexity of cultural elements over time, and an increase in diversity as the space of possibilities is explored, followed by a decrease as agents find and converge on the fittest possibilities. EVOC has been used to model how the mean fitness and diversity of cultural elements is affected by factors such as leadership, population size and density, borders that affect transmission between populations, and the proportion and distribution of creators (who acquire new ideas primarily by inventing them) versus imitators (who acquire new ideas primarily by copying their neighbors) (Gabora, 1995(Gabora, , 2008aGabora, & Firouzi, 2012;Gabora & Leijnen, 2009;Leijnen & Gabora, 2010).
A communal exchange inspired method for organizing artifacts into historical lineages has also been developed. Worldview Evolution, or WE for short, uses both superficial (e.g., 'beveled edge') and abstract (e.g., 'object is thrown') attributes, as well as analogical transfer (e.g., of 'handle' from knife to cup) and complementarity (e.g., bow and arrow) (Gabora, Leijnen, Veloz, & Lipo, 2011). It represents objects not in terms of a convenient list of discrete measurable attributes, but in terms of how they are actually conceptualized, as a network of interrelated properties, using a perspective parameter that can be weighted differently according to their relative importance. Preliminary analyses show that the conceptual network approach can recover previously unacknowledged patterns of historical relationship that are more congruent with geographical distribution and temporal data than is obtained with an alternative cladistic approach that is based on the assumption that cultural evolution, like biological evolution, is Darwinian.
These two computational models, EVOC and WE, show that a communal exchange approach to cultural evolution is computationally tractable. However such models will not begin to approach the open-ended ingenuity and complexity of human cultural evolution until they incorporate certain features of the cognitive process by which cultural novelty is generated.
The Generation of Cultural Novelty
We said that cultural evolution is a depth-first evolution strategy. A depth-first evolution strategy entails processes that adaptively bias the generation of novelty. A number of key, interrelated processes have been identified that, in addition to learning, accomplish this in cultural evolution. We now look briefly at some of these processes, as well as efforts to model them.
Recursive Recall and Restorative Restructuring
Recursive recall (RR) is the capacity for one thought to trigger another, enabling progressive modification of an idea. Donald's (1991) hypothesis that cultural evolution was made possible by onset of the capacity for RR has been tested using EVOC (Gabora & Saberi, 2011;Gabora & DiPaola, 2012). A comparison was made of runs in which agents were limited to single-step actions to runs in which they could recursively operate on ideas, and chain them together, resulting in more complex actions. While RR and no-RR runs both converged on optimal actions, without RR this set was static, but with RR it was in constant flux as ever-fitter actions were found. In RR runs there was no ceiling on mean fitness of actions, and RR enhanced the benefits of learning.
Although these findings support Donald's hypothesis, the novel actions generated with RR were predictable. They did not open up new cultural niches in the sense that, for example, the invention of cars created niches for the invention of things like seatbelts and stoplights. EVOC in its current form could not solve insight problems, which require restructuring the solution space (Boden, 1990;Kaplan & Simon, 1990, Ohlsson, 1992. Restructuring can be viewed as a form of RR that entails looking at the problem from a new context or perspective, and that this is driven by the mind's self-organizing, restorative capacity.
Contextual Focus (CF) and Divergent versus Associative Thought
It has been proposed that restorative restructuring is aided by contextual focus (CF): the capacity to spontaneously and temporarily shift to a more divergent mode of thought (Gabora, 2003). Divergent thought entails an increase in activation of the associates of a given item (Runco, 2010). Thus for example, given the item TABLE, in a convergent mode of thought you might call to mind accessible associates such as CHAIR, but in a divergent mode of thought you might also call to mind more unusual associates such as PICNIC or MULTIPLICATION TABLE. CF has been implemented in EVOC (the computational model of cultural evolution). Low fitness of ideas induces a temporary shift to a more divergent processing mode by increasing the 'reactivity', α, which determines the degree to which a newly invented idea can differ from the idea on which it was based.
Current research on the architecture of memory suggests that creative thought is actually not divergent but associative, as illustrated in Figure 1 (Gabora, 2010;Gabora & Ranjan, 2013). While divergent thought refers to an increase in activation of all associates, associative thought increases only activation of those relevant to the context. Because memory is distributed and content-addressable, associations are forged by way of shared structure. In associative thought, items come together that, though perhaps seemingly different, share properties or relations, and are thus more likely than chance to be relevant to one another, perhaps in a previously unnoticed but useful way. Divergent thought (centre) activates not just key properties but also peripheral (less salient) properties, represented by both grey dots and black triangles. The grey dots represent peripheral properties that are relevant to the current context (goal or situation); the black triangles represent peripheral properties that are irrelevant to the current context. Associative thought (right) activates key properties and context-relevant peripheral properties.
A processing mode that is not just divergent but associative could be simulated in a model such as EVOC capitalizing on the ability to learn generalizations (e.g., symmetrical movements tend to be fit) to constrain changes in α. It would also be interesting to investigate the topological and dynamical properties of fitness landscapes for which divergent versus associative forms of CF is effective. CF is expected to be most beneficial for fitness landscapes that are rugged and subject to infrequent, abrupt change, with associative CF outperforming divergent CF.
Concept Interaction
Since creative processes such as restructuring involve putting concepts together in new contexts, a model of cultural evolution should be built upon a solid theory of concepts and how they interact. However, people use conjunctions and disjunctions of concepts in ways that violate the rules of classical logic; i.e., concepts interact in ways that are non-compositional (Osherson & Smith, 1981;Hampton, 1987;Aerts, 2009;Aerts, Aerts, & Gabora, 2009;Aerts, Broekaert & Gabora, 2010;Kitto, Ram, Sitbon, & Bruza, 2011). This is true both with respect to properties (e.g., although people do not rate 'talks' as a characteristic property of PET or BIRD, they rate it as characteristic of PET BIRD), and exemplar typicalities (e.g., although people do not rate 'guppy' as a typical PET, nor a typical FISH, they rate it as a highly typical PET FISH). Because of this, concepts have been resistant to mathematical description.
This non-compositionality can be modeled using a generalization of the formalisms of quantum mechanics (QM) Gabora & Aerts, 2002a,b;Kitto, Ramm, Sitbon, & Bruza, 2011). The reason for using the quantum formalism is that it allows us to describe the chameleon-like way in which concepts interact, spontaneously shifting their meanings depending on what other concepts are nearby or activated. The following formal exposition, though not essential for grasping the underlying concepts, is provided for the mathematically inclined reader. In QM, the state÷yñ of an entity is written as a linear superposition of a set of basis states {÷f i ñ} of a complex Hilbert space H. Hence ÷yñ = S i c i ÷f i ñ where each complex number coefficient c i of the linear superposition represents the contribution of each component state ÷f i ñ to the state ÷yñ. The square of the absolute value of each coefficient equals the weight of its component basis state with respect to the global state. The choice of basis states is determined by the observable to be measured. The basis states corresponding to this observable are called eigenstates. Upon measurement, the state of the entity collapses to one of the eigenstates. In the quantum inspired State COntext Property (SCOP) theory of concepts, the basis states represent states (instances or exemplars) of a concept, and the measurement is the context that causes a particular state to be evoked. SCOP is consistent with experimental concept data on concept combination (Aerts, 2009;Aerts, Aerts, & Gabora, 2009;Aerts, Broekaert, Gabora, & Veloz, 2012;Aerts, Gabora, & Sozzo, submitted;Hampton, 1987), and with findings that a compound's constituents are not just conjointly activated but bound together in a contextspecific manner that takes relational structure into account (Gagné & Spalding, 2009). The model is being expanding to incorporate larger conceptual structures (Gabora & Aerts, 2009), and different modes of thought (Veloz, Gabora, Eyjolfson, & Aerts, 2011). This theoretical work is complemented by empirical studies aimed at establishing that (i) some concept combinations involve interference and entangled states, and (ii) creative products are external evidence of an internal self-organization process aimed at resolving dissonance and restoring equilibrium through the recursive actualization of potentiality (Gabora, 2011;Gabora, O'Connor, & Ranjan, 2012;Gabora & Saab, 2011).
Harnessing the Computational Power of Cultural Evolution
We have looked at some of the key milestones that have been crossed in the development of a scientific framework for how culture evolves. These milestones include a crude but functional computational model of cultural evolution, research into the cognitive mechanisms underlying the generation of cultural novelty, and preliminary efforts to computationally model these mechanisms. The rest of this chapter presents new, untested, yet-to-be-implemented ideas for how to go about harnessing the speed and power of cultural evolution in the development of a human computation research program.
Computational Model of Restorative Restructuring
A first step is to develop a model of problem restructuring using a "reaction network" inspired model that has as its basic unit, not catalytic molecules, but interacting concepts. There are various methods for going about this, for example using Concat, or Holographic Reduced Representations to computationally model the convolution or 'twisting together' of mental representation (Aerts, Czachor, & De Moor, 2009;Eliasmith & Thagard, 2001;Thagard & Stewart, 2011). Another promising route is to use a quantum-inspired theory of concepts such as SCOP that incorporates the notion of context-driven actualization of potential (Aerts & Gabora, 2005a,b;Gabora & Aerts, 2002a,b). A concept is defined in terms of (1) its set of states or exemplars S, each of which consists of a set L of relevant properties, (2) set M of contexts in which it may be relevant, (3) a function n that describes the applicability or weight of a certain property for a specific state and context, and (4) a function µ that describes the transition probability from one state to another under the influence of a particular context.
The procedure is best explained using an example, such as the idea of using a tire to make a swing, i.e., the invention of a tire swing (from Gabora, Scott, & Kauffman, in press). The concept TIRE consists of the set S of states of TIRE, and in the context 'winter', TIRE might collapse to SNOW TIRE. Suppose that the network's initial conception of TIRE, represented by vector |pñ of length equal to 1, is a superposition of only two possibilities (Fig. 2). The possibility that the tire has sufficient tread to be useful is denote by unit vector |uñ. The possibility that it should be discarded as waste is denoted by unit vector, |wñ. Their relationship is given by the equation |pñ = a 0 |uñ + a 1 |wñ, where a 0 and a 1 are the amplitudes of |uñ and |wñ respectively. If a tire us useful only for transportation, denoted |tñ then, |uñ = |tñ. States are represented by unit vectors and all vectors of a decomposition such as |uñ and |wñ have unit length, are mutually orthogonal and generate the whole vector space, thus |a 0 | 2 + |a 1 | 2 = 1. Figure 2. Graphical depiction of a vector |pñ representing particular state of TIRE, specifically, a state in which the tread is worn away. In the default context, the state of tire is more likely to collapse to the projection vector |wñ which represents wasteful than to its orthogonal projection vector |uñ which represents useful. This can be seen by the fact that subspace a0 is smaller than subspace a1. Under the influence of the context playground equipment, the opposite is the case, as shown by the fact that b0 is larger than b1. Also shown is the projection vector after renormalization.
The conception of TIRE changes when activation of the set L of properties of TIRE, e.g. Gabora -Cultural Evolution as Distributed Computation 7 'weather resistant', spreads to other concepts in the network for which these properties are relevant. Contexts such as playground equipment that share properties with TIRE become candidate members of the set M of relevant contexts for TIRE. The context playground equipment, denoted e, consists of the concepts SWING, denoted |s e ñ, and SLIDE, denoted |l e ñ. The restructured conception of TIRE in the context of playground equipment, denoted |p e ñ, is given by b 0 |u e ñ + b 1 |w e ñ, where u e ñ = b 2 |t e ñ + b 3 |t e s e ñ + b 4 |t e l e ñ, and where |t e s e ñ stands for the possibility that a tire functions as a swing, and |t e l e ñ stands for the possibility that a tire functions as a slide. The amplitude of |w e ñ, |b 1 |, is less than |a 1 |, the amplitude of |wñ. This is because |b 0 | > |a 0 |, since |b 0 | consists of the possibility of a tire being used not just as a tire, but as a swing or slide. Because certain strongly weighted properties of SLIDE, such as 'long' and 'flat', are not properties of TIRE, |b 4 | is small. That is not the case for SWING, so |b 3 | is large. Therefore, in the context playground equipment, the concept TIRE has a high probability of collapsing to TIRE SWING, an entangled state of the concepts TIRE and SWING. Entanglement introduces interference of a quantum nature, and hence the amplitudes are complex numbers (Aerts, 2009). If this collapse takes place, TIRE SWING is thereafter a new state of both concepts TIRE and SWING.
This example shows that a formal approach to concept interactions that is consistent with human data (Aerts, 2009;Aerts, Aerts, & Gabora, 2009;Aerts, Broekaert, Gabora, & Veloz, 2012;Aerts, Gabora, & Sozzo, submitted;Hampton, 1987) can model the restructuring of information (e.g., TIRE) under a new context (e.g., playground equipment). Note how in the quantum representation, probability is treated as arising not from a lack of information per se, but from the limitations of any particular context (even a 'default' context).
The limitations of this approach are as interesting as its strengths. It is not possible to list, or even develop an algorithm that will list, all possible uses or contexts for any item such as a tire or screwdriver (Longo, Montevil, & Kaufman, 2012). This is what has been referred to as the frame problem. As a consequence, human input is particularly welcome at this juncture to define the relevant contexts, e.g., the possible uses of a tire. Studies would be run using data collected from real humans to determine the extent to which the model matches typicality ratings and generation frequencies of exemplars of concepts in particular contexts by human participants, as per (Veloz, Gabora, Eyjolfson, & Aerts, 2011). SCOP models of individual concepts can be embedded into an associative "reaction network". Concept interactions are then modeled as reactions that generate products. Chemical Organization Theory (Dittrich & Speroni di Fenizio, 2008;Dittrich & Winter, 2007;Dittrich, Ziegler, & Banzhaf, 2001), which provides an algebraic means of solving nonlinear, coupled differential equations in reaction networks, or some other such theory, can be used to model the associative structure of interrelated sets of concepts a whole, and study the conditions under which it restores equilibrium in response to the introduction of new states of concepts that results from placing them in new contexts.
Using this SCOP-based cognitive "reaction network" it would be possible to test the hypothesis that contextual focus (the ability to shift between different modes of thought depending on the context− increases cognitive efficiency. If the amplitude associated with |wñ for any concept becomes high-such as for TIRE if the weight of the property 'tread' is low-this signals that the potentiality to re-conceptualize the concept is high. This causes a shift to a more associative mode by increasing α, causing activation of other concepts that share properties with this concept, as described previously.
Enhanced Computational Model of Cultural Evolution
Gabora -Cultural Evolution as Distributed Computation 8
Let us now examine how a model of restorative restructuring such as the SCOP-based one we just looked at could be used to develop a cognitively sophisticated computational model of cultural evolution. We will refer to this 'new and improved' model as EVOC2.
So that the EVOC2 agents have something to make artifacts from, their world would contain resource bases from which objects are extracted and wastes are generated. Extracted objects can be joined (lego-style) to construct other objects. Agents have mental representations of resources and objects made from resources. Objects derived from the same resource are modeled in their conceptual networks as states of a concept. Newly extracted or constructed objects have a fitness that defines how useful or wasteful they are with respect to the other objects an agent has encountered. Thus existing objects provide contexts that affect the utility of new objects, and an agent's knowledge of existing objects defines its perspective.
The artificial culture can now evolve as follows: Invent. Agents invent as in EVOC, except that they invent not actions but objects, using resources in adjacent cells. Extracting an object from a resource creates waste objects.
Detect and Actualize Potential for Adaptive Change. If a waste object p is accumulating adjacent to A1, A1 recursively modifies p by considering it from A1's perspective. This continues until p is in a new less wasteful state p A1* which is an eigenstate with respect to A1's perspective. This process may modify not just p, but A1's perspective. Perspectives change in response to the ideas and objects an agent interacts with; thus a perspective can encompass more than one context.
Contextual focus. The previous step may involve temporarily assuming a more associative processing mode in response to the magnitude of potential for adaptive change.
Transmission. Modified object, p A1* , becomes input to the associative networks of adjacent agents.
Context-dependent Restructuring. If p A1* is wasteful (has potential to change) with respect to the perspective of another agent, A2, then A2 recursively modifies p A1* until it is an eigenstate with respect to A2's perspective, at which point it is referred to as p A1*A2* . Since A1's perspective is reflected in p A1* , assimilation of p A1* modifies A2's perspective in a way that reflects exposure to (though not necessarily incorporation of or agreement with) A1's perspective. This continues until p settles on stable or cyclic attractor, or we terminate after a set number of iterations (since a chaotic attractor or limit cycle may be hard to distinguish from a non-stable transient).
Evaluate. The user assesses the usefulness of the culturally evolved objects for the agents, as well as object diversity, and wastefulness.
EVOC2 will be deemed a success if it not only evolves cultural novelty that is cumulative, adaptive, and open-ended (as in EVOC with RR), but also (a) restructures conceptions of objects by viewing them from different perspectives (new contexts), (b) generates inventions that open up niches for other inventions, and (c) exhibits contextual focus, i.e., shifts to an associative mode to restructure and shifts back to fine-tune. It is hypothesized that these features will increase the complexity of economic webs of objects and recycled wastes.
Elucidating the Algorithmic Structure of Biological versus Cultural Evolution
The design features that made EVOC2 specific to the problem of waste recycling can eventually be replaced by general-purpose counterparts, resulting in a cultural algorithm (CAL 1 ). It will be interesting to compare the performance of a CAL with a GA on standard problems (e.g., the Rosenbrock function) as well as on insight tasks such as real-world waste recycling webs that require restructuring. Waste recycling is a particularly appropriate application because it explicitly requires considering how the same item offers a different set of constraints and affordances when considered with respect to a different goal, a different demographic, or a different aesthetic sensibility (one person's trash is another person's treasure). In general the CAL is expected to outperform the GA on problems that involve not just multiple constraints but multiple perspectives, e.g., economic and environmental.
A long-term objective is to develop an integrated framework for evolutionary processes that encompasses natural selection, cultural evolution, and communal exchange theories of early life. Finally, it can advance knowledge of how systems evolve. Early efforts toward a general cross-disciplinary framework for evolution Processes were modeled as context-dependent actualization of potential: an entity has potential to change various ways, and how it does change depends on the contexts it interacts with . These efforts focused on distinguishing processes according to the degree of non-determinism they entail, and the extent to which they are sensitive to, internalize, and depend upon a particular context. With the sorts of tools outlined here, it will be possible to compare the effectiveness of communal exchange, Darwinian, and mixed strategies in different environments (simple versus complex, static versus fluctuating, and so forth. This will result in a more precise understanding of the similarities and differences between biological and cultural evolution, and help us recognize other evolutionary processes that we may discover as science penetrates ever deeper into the mysteries of our universe.
Summary and Conclusions
Culture evolves with breathtaking speed and efficiency. We are crossing the threshold to an exciting frontier: a scientific understanding of the process by which cultural change occurs, as well as the means to capitalize on this understanding. The cultural evolution inspired human computation program of research described in this chapter is ambitious and interdisciplinary, but it builds solidly on previous accomplishments.
We examined evidence that culture evolves through a non-Darwinian communal exchange process, and discussed a plan for modeling the autopoietic structures that evolve through biological and cultural processes-i.e., metabolic reaction networks and associative networks. This will make it possible to undertake a comparative investigation of the dynamics of communally exchanging groups of these two kinds of networks. This research is necessary to achieve a unification of the social and behavioral sciences comparable to Darwin's unification of the life sciences.
Efforts are underway toward the development of a computational model of cultural evolution that incorporates the kind of sophisticated cognitive machinery by which cultural novelty evolves. These include the combining of concepts to give rise to new concepts sometimes with emergent properties, and the capacity to shift between different modes of thought depending on the situation. An important step is to embed formal models of concepts in a modified "reaction network" architecture, in order to computationally model how clusters of interrelated concepts modify one another to achieve a more stable lower energy state, through a process we referred to as context-driven restorative restructuring. Efforts are also underway toward the development of a computer program for identifying patterns of historical relationship amongst sets of artifacts. Human input is used to define contexts-perspectives or situations that define which features or attributes are potentially relevant. One long-term objective of this kind of research program is to develop a cultural algorithm: an optimization and problem-solving tool inspired by cultural evolution. This will allow us to investigate how strategies for recursively re-processing and restructuring information, or shifting between different processing modes, affect the capacity to evolve cumulative, adaptive, open-ended novelty.
The ideas presented in this chapter are speculative, ambitious, and innovative both conceptually and methodologically, but they have far-reaching implications and potentially diverse applications. The human computation program proposed here could promote a scientific understanding of the current accelerated pace of cultural change and its transformative effects on humans and our planet. It may foster cultural developments that are healthy and productive in the long term as well as the short term, and help us find solutions to complex crises we now face.
Figure 1 .
1Convergent thought (left) activates key properties only, represented by black dots.
Cultural algorithm is abbreviated CAL because CA customarily refers to cellular automaton.
AcknowledgementsThis research was conducted with the assistance of grants from the National Science and Engineering Research Council of Canada, and the Fund for Scientific Research of Flanders, Belgium.
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] |
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We model a sheared disordered solid using the theory of Shear Transformation Zones (STZs). In this mean-field continuum model the density of zones is governed by an effective temperature that approaches a steady state value as energy is dissipated. We compare the STZ model to simulations by Shi, et al. [Y. Shi et al. PRL 98, 185505 (2007)], finding that the model generates solutions that fit the data, exhibit strain localization, and capture important features of the localization process. We show that perturbations to the effective temperature grow due to an instability in the transient dynamics, but unstable systems do not always develop shear bands. Nonlinear energy dissipation processes interact with perturbation growth to determine whether a material exhibits strain localization. By estimating the effects of these interactions, we derive a criterion that determines which materials exhibit shear bands based on the initial conditions alone. We also show that the shear band width is not set by an inherent diffusion length scale but instead by a dynamical scale that depends on the imposed strain rate.
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10.1103/physreve.76.056106
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"https://arxiv.org/pdf/0706.1078v2.pdf"
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0706.1078
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eeb2130f750249706c580b9ef44b43d9775a08b1
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Strain localization in a shear transformation zone model for amorphous solids
14 Jun 2007 (Dated: February 1, 2008)
M L Manning
Department of Physics
University of California
93106Santa Barbara
J S Langer
Department of Physics
University of California
93106Santa Barbara
J M Carlson
Department of Physics
University of California
93106Santa Barbara
Strain localization in a shear transformation zone model for amorphous solids
14 Jun 2007 (Dated: February 1, 2008)arXiv:0706.1078v2 [cond-mat.soft]
We model a sheared disordered solid using the theory of Shear Transformation Zones (STZs). In this mean-field continuum model the density of zones is governed by an effective temperature that approaches a steady state value as energy is dissipated. We compare the STZ model to simulations by Shi, et al. [Y. Shi et al. PRL 98, 185505 (2007)], finding that the model generates solutions that fit the data, exhibit strain localization, and capture important features of the localization process. We show that perturbations to the effective temperature grow due to an instability in the transient dynamics, but unstable systems do not always develop shear bands. Nonlinear energy dissipation processes interact with perturbation growth to determine whether a material exhibits strain localization. By estimating the effects of these interactions, we derive a criterion that determines which materials exhibit shear bands based on the initial conditions alone. We also show that the shear band width is not set by an inherent diffusion length scale but instead by a dynamical scale that depends on the imposed strain rate.
I. INTRODUCTION
Amorphous materials often comprise or lubricate sheared material interfaces and require more complicated constitutive equations than simple fluids or crystalline solids. They flow like a fluid under large stresses, creep or remain stationary under smaller stresses, and have complex, history-dependent behavior. Bulk metallic glasses, granular materials, and bubble rafts are just some of the disordered materials that exhibit these features [2,3,4].
In this paper, we focus on strain localization, the spontaneous development of coexisting flowing and stationary regions in a sheared material. Strain localization has been identified and studied experimentally in granular materials [5,6], bubble rafts [3,7], complex fluids [8,9], and bulk metallic glasses [4,10]. Shear banding may play an important role in the failure modes of structural materials and earthquake faults. Localization is a precursor to fracture in bulk metallic glasses [4] and has been cited as a mechanism for material weakening in granular fault gouge on faults [11].
We model sheared material interfaces using the theory of Shear Transformation Zones (STZs). Spaepen first postulated that plastic flow could be modeled by tracking localized zones or regions of disorder [12]. Argon and Bulatov developed mean-field equations for these localized regions [13,14]. Building on this work, Falk and Langer [15] introduced a mean-field theory which postulates that the zones, which correspond to particle configurations that are more susceptible to plastic rearrangements than the surrounding particles, could reversibly switch between two orientational states in response to stress. Additionally, these zones are created and annihilated as energy is dissipated in the system. This theory captures many of the features seen in experiments, such as work-hardening and yield stress [16,17,18]. * Electronic address: [email protected] There are several complementary approaches to understanding disordered solids. On the smallest scales, molecular dynamics and quasi-static simulations generate a wealth of information about particle interactions and emergent macroscopic behavior -including shear banding [1,19,20] -but they are limited to smaller numbers of particles and time scales. On the largest scales, phenomenological models such as viscoelasticity and the Dieterich-Ruina friction law [21,22], which has been studied extensively in the context of rock mechanics, describe stress-strain step and frequency responses, but to date these laws have not been derived from microscopic dynamics.
A third approach focuses on the dynamics of collections of particles and their configurations. Examples of this approach include the Spot Model for granular materials [23], Soft Glassy Rheology (SGR) [24], and the theory of Shear Transformation Zones. The Spot Model generalizes dislocation dynamics in crystalline solids by postulating that a spot, a region of extra free volume shared among a collection of particles, executes a random walk through the material. Both SGR and STZ theories postulate that there are meso-scale configurational soft spots which are more susceptible than the surrounding material to yield under shear stress.
Recently, Langer postulated that an effective temperature should govern the density of shear transformation zones [25]. Mehta and Edwards [26] were perhaps the first to point out that although thermal temperature does not determine statistical distributions for macroscopic particles such as powders, the statistical properties of these systems could still be characterized by a small number of macroscopic state variables, such as free volume. Cugliandolo, Kurchan, and Peliti [27] show that heat flow in these slowly stirred systems is determined by an effective temperature. Several studies have confirmed that effective temperature can be used to characterize slowly sheared granular packings and bubble rafts [28,29].
For many systems thermal temperature is not sufficient to cause configurational rearrangements, but slow shearing causes the particles to ergodically explore configuration space. Although we make no attempt to calculate the entropy from first principles, we postulate that because the material is being sheared slowly, configurational (as opposed to kinetic) degrees of freedom are the dominant contributions to the entropy. The statistical distribution of steady state configurations should maximize this entropy and be described by an effective temperature.
The effective temperature is an important macroscopic state variable in this STZ description. It governs the properties of statistical distributions for configurational degrees of freedom in the material, such as density fluctuations. Langer hypothesized that shear transformation zones are unlikely, high-energy density fluctuations, and therefore the number of such zones in a system should be proportional to a Boltzmann factor which here is a function of the effective temperature instead of the usual thermal temperature [25]. In this manner, the density of shear transformation zones is related to the effective temperature and therefore the plastic flow is coupled to the local disorder. This is a mechanism for strain localization because a region with higher disorder is more susceptible to flow under stress and flowing regions become more disordered.
The effective temperature mechanism for strain localization is different from other proposed mechanisms governed by thermal temperature or free volume. Griggs and Baker [30] proposed that the strong temperature dependence of viscosity near the glass transition leads to thermal softening and strain localization. In that formulation the heat generated by plastic work raises the thermal/kinetic temperature, instead of the configurational/effective temperature. Lewandowski and Greer [31] have shown that in a bulk metallic glass the thermal temperature diffuses too quickly to control shear localization in an adiabatic description, though Braeck and Podladchikov [32] have shown that a non-adiabatic thermal theory can explain the data.
Alternatively, several authors have suggested that the free volume governs strain localization [10,33]. In these descriptions, plastic deformation in the solid is accompanied by a larger free volume, which in turn softens the material. However, the exact relationship between the free volume and plastic deformation is difficult to determine. In this paper we suggest an alternative description: the heat generated by plastic deformation is dissipated in the configurational degrees of freedom, according to the first law of thermodynamics. The local free volume is related to the effective temperature (for example, in a Lennard-Jones glass the particles can become more disordered by moving away from their equilibrium separations), but the effective temperature is the relevant macroscopic state variable governing particle dynamics.
The STZ formulation in this paper is athermal -the thermal temperature T does not cause any configurational rearrangements. This is an approximation which is clearly appropriate for bubble rafts and granular mate-rials and may be relevant for metallic glasses well below the glass transition temperature. The athermal approximation significantly simplifies the STZ equations and clarifies the nature of the instability that leads to localization, but the resulting description does not capture features such as thermally activated creep or relaxation. As we discuss in Section IV D, some behaviors of real materials and simulations are likely due to thermal activation or relaxation and will not be accurately represented in this athermal STZ formulation.
In this paper we show how athermal STZ theory with effective temperature generates solutions with strain localization. This paper is divided into sections as follows: Section II develops the constitutive equations for athermal STZ theory with effective temperature in a simple shear geometry. Section III compares numerical results from the continuum STZ theory to simulations of Lennard-Jones glasses by Shi, et al. We compare macroscopic stress-strain curves for different initial conditions and compare the strain rate and potential energy as a function of position and time inside the sheared material. Section IV investigates the stability properties of the STZ equations, and presents a description of how shear bands form and remain intact for long times. We show that an instability in the transient dynamics amplifies perturbations to the effective temperature, and that this process interacts with energy dissipation processes to determine the long-time behavior of the strain rate field. Section V concludes the paper with a discussion of our results and future directions.
II. REVIEW OF STZ EQUATIONS
A mean field theory for shear transformation zones, first derived by Argon [13], has been developed in a series of papers [15,16,17,25]. For completeness, this section reviews the athermal STZ equations and emphasizes their similarity to other models for elastic and plastic flow. Section II A introduces and motivates the STZ equations. Section II B tailors the equations to describe a particular geometry -an infinite strip of material driven from the boundaries at a velocity V 0 .
A. General equations
When describing systems that undergo plastic and elastic deformation, it is convenient to break up the total stress tensor into hydrostatic and deviatoric components, σ ij = −pδ ij + µs ij , because hydrostatic stress generally does not cause yield. To simplify notation, the deviatoric stress has been nondimensionalized by an effective shear modulus µ that characterizes the stiffness of the STZs. In STZ theory, the stress scale µ is important because it characterizes the stress at which the material begins to yield. In the slowly sheared materials we are modeling, the speed of sound in the material is very fast compared to the rate of plastic deformation. In this case the stress gradients equilibrate very quickly, and we take the zero density limit of the momentum conservation equations. This results in static elastic equations for the stress:
∂σ ij ∂x j = 0.(1)
We further assume that the rate of deformation tensor is the sum of elastic and plastic parts:
D total ij = 1 2 ∂v i ∂x j + ∂v j ∂x i = D Dt − p 2K δ ij + µ 2µ s ij + D plast ij ,(2)
where D/Dt is the material or corotational derivative defined as:
DA ij Dt = ∂A ij ∂t + v k ∂A ij ∂x k + A ik ω kj − ω ik A kj ,(3)
and ω ij = 1/2(∂v i /∂x j − ∂v j /∂x i ). We also use the no-tationȦ ij = ∂A ij /∂t. The plastic rate of deformation tensor can be written in terms of dynamical variables from STZ theory. We postulate that under shear stress, each STZ deforms to accommodate a certain amount of shear strain, and cannot deform further in the same direction. This is modeled by requiring that each STZ be in one of two states: oriented along the principle stress axis in the direction of applied shear, which we will denote "+", or in the perpendicular direction, "−".
Under applied strain, the STZ will flip in the direction of strain, from "−" to "+". Under shear stress in the opposite direction, the STZs can revert to their original configurations, which corresponds to a flip from "+" to "−". These rearrangements or flips occur at a rate which depends on the stress R(s) and a characteristic attempt frequency, τ 0 . Because each STZ can flip at most once in the direction of applied strain, STZs must be created and annihilated to sustain plastic flow. Based on these considerations, the number density of STZs in each direction, n ± obeys the following differential equation
τ 0ṅ± = R(±s)n ∓ −R(∓s)n ± +Γ n ∞ 2 e −1/χ − n ± ,(4)
where R(±s)/τ 0 is the rate of switching per STZ as a function of stress, Γ is the rate at which energy is dissipated per STZ, and n ∞ e −1/χ is the steady state density of STZs in equilibrium. The first two terms in Eq. (4) correspond to STZs switching from "+" to "−" states and vice-versa, while the last term shows that the STZs are created at a rate proportional to n ∞ e −1/χ and annihilated at a rate proportional to their density. The plastic rate of deformation tensor is given by the rate at which STZs flip:
D pl = λ τ 0 (R(s)n − − R(−s)n + ) .(5)
Pechenik [18] generalized Eqs. (4) and (5) to the case where the principal axes of the STZ orientation tensor n ij are not aligned with principal axes of the stress tensor s ij . These generalized equations can be written in terms of two new variables. The first is Λ ≡ n tot /n ∞ , where n tot is the tensorial generalization of (n + +n − ), and corresponds to the total density of zones in a sample. The second is m ij ≡ n ij /n ∞ , where n ij is the tensorial generalization of (n + − n − ) and corresponds to the STZ orientational bias.
The rate at which STZs are created/destroyed, denoted by Γ, is an important, positive quantity. Falk and Langer [15] first proposed that Γ should be proportional to the rate of plastic work done on the system. Pechenik [18] refined this idea, noting that the rate of plastic work could be negative since energy could be stored in the plastic degrees of freedom. He proposed an alternative definition for Γ: the rate at which energy is dissipated per STZ, which is always positive. We adopt this important assumption in our STZ model.
In previous papers, [18,25], the functional form of Γ was necessarily complicated because thermally activated switching can release energy stored in the plastic degrees of freedom. As noted by Bouchbinder et. al [16], the functional form of Γ is considerably simplified in an athermal description because an STZ can never flip in a direction opposite to the direction of applied stress. Therefore no energy can be stored in the plastic degrees of freedom. The rate at which energy is dissipated per STZ is the rate at which plastic work is done on the system, Q, divided by the volume (area in 2D) density of STZs, ǫ 0 Λ, where ǫ 0 = λn ∞ . In terms of these new variables, the rate of deformation tensor can be written as:
D pl ij = ǫ 0 τ 0 C(s)( s ij s − m ij )Λ,(6)
and the energy dissipated per STZ is
Γ = Q ǫ 0 Λ = s ij D pl ij ǫ 0 Λ ,(7)
where s 2 = 1/2 s ij s ij , and C(s) = 1/2 (R(s) + R(−s)). Based on physical considerations, C(s) is symmetric function of the stress that approaches zero as the stress approaches zero and approaches a line with a slope of unity as the stress becomes large. Bouchbinder et al. [16] describe a one-parameter family of functions with the correct properties, and we will use the simplest of these: C(s) = −2 + |s| + exp(−|s|)(2 + |s|). Together, Eqs. (6), (7), and the tensorial generalization of Eq. (4) describe the evolution of the density of STZs and the orientational bias of STZs. The density Λ is driven towards a Boltzmann distribution as energy is dissipated in the system:
Λ = Γ e −1/χ − Λ .(8)
The orientational bias of the STZs m ij is incremented to accommodate plastic strain, and reset as energy is dissipated in the system:
Dm ij Dt = 2 τ 0 C(s)( s ij s − m ij ) − Γm ij e −1/χ Λ .(9)
To close the system of equations we require an equation of motion for the effective temperature, χ. As mentioned in the introduction, we postulate that the effective temperature describes the energy in the disordered configurational degrees of freedom. Ono et al. [28] show that at low strain rates the system is driven towards a specific steady state of disorder that corresponds to a specific value for the steady state effective temperature, χ ∞ . It is assumed that the effective temperature is proportional to the potential energy per particle, with a constant specific heat c 0 . This implies that the steady state effective temperature is also proportional to the energy dissipated by the configurational degrees of freedom in steady state. Energy balance requires that the effective temperature change at a rate proportional to the rate at which energy is dissipated in the system. However, this change in configurational energy must be limited as the effective temperature approaches its steady state. As χ approaches χ ∞ , the heat generated by plastic work is dissipated by other mechanisms, which we do not track. Additionally, we assume that the configurational energy can be transferred between neighboring particles, so that there is a flux in the effective temperature which is proportional to its gradient. The resulting equation for effective temperature is:
c 0χ = Q (χ ∞ − χ) + D χ ∂ 2 χ ∂y 2 ,(10)
where D χ is a diffusion constant with units of area per unit time.
In the athermal STZ formulation there is no mechanism for relaxation of the effective temperature to the thermal temperature T , so that the effective temperature everywhere tends towards χ ∞ . This is in contrast to a thermal formulation that permits the effective temperature to relax in regions undergoing minimal plastic deformation. In Lennard-Jones and bulk metallic glasses well below the thermal glass temperature we expect any relaxation to be very slow, so that the athermal description is a good approximation. However, even very slow relaxation processes may have a significant impact on localization and we re-evaluate the athermal assumption in Section V.
B. STZ equations in an infinite strip geometry
We will now derive the STZ equations of motion in the geometry of a two-dimensional infinite strip driven by the boundaries in simple shear. Let x be the direction of the velocity at the boundaries, and y be normal to these boundaries. The speed at the boundaries is V 0 and the width is L. Therefore the average strain rate inside the material isγ ≡ V 0 /L, while the local strain rate is denotedγ ≡ ∂v x /∂y ≡ 2D total xy . Assuming that there are no pressure gradients, Eq. (2) requires that the pressure field remain constant in time.
The stress s and STZ bias m are traceless tensors. Let the off-diagonal elements of the tensors be denoted by s = s xy = s yx and m = m xy = m yx and the diagonal elements by s 0 = s xx = −s yy and m 0 = m xx = −m yy . By symmetry, all fields are constant in x, and Eq. (1) shows the stress tensor is constant in the y direction as well. Substituting this into Eq. (6) we find that the plastic rate of deformation tensor then has two independent components:
D pl = ǫ 0 τ 0 C(s)( s s − m)Λ;(11)D pl 0 = ǫ 0 τ 0 C(s)( s 0 s − m 0 )Λ,(12)
and the energy dissipation per STZ, Γ, is given by:
Γ = 2 D pl 0 s 0 + D pl s ǫ 0 Λ = 2 τ 0 C(s)(s − s 0 m 0 − sm).(13)
Because s ij is independent of position, the tensor equation of motion for the stress (Eq. (2)) can be integrated in y over the interval [y = −L, y = +L]. This results in simplified equations for s and s 0 :
s = µ µ V 0 L − 2ǫ 0 τ 0 C(s)( s s − m)Λ − s 0 V 0 L ,(14)s 0 = −2 µ µ 2ǫ 0 τ 0 C(s)( s 0 s − m 0 )Λ + s V 0 L ,(15)
where (·) denotes an average over a field in the ydirection. For example, Λ is:
Λ = 1 2L L −L Λ(y)dy.(16)
and D total xy = V 0 /(2L). Eq. (9) can be simplified to equations for m and m 0 as follows:
m + m 0γ = −Γm e −1/χ Λ + 2 τ 0 C(s)( s s − m);(17)m 0 − mγ = −Γm 0 e −1/χ Λ + 2 τ 0 C(s)( s 0 s − m 0 ). (18)
In slowly sheared experiments and simulations, the strain rateγ is always much smaller than the inherent attempt frequency 1/τ 0 . Therefore the complicated rotation terms, m 0γ and mγ, are very small and can be neglected. We rewrite the equations of motion so that times are in units of the inverse average strain rate, 1/γ = L/V 0 , and lengths are in units of L. Note that stresses are already in units of the yield stress µ.
The resulting system of equations is given by:
s = µ * 1 − 2ǫ 0 q 0 C(s) s s − m Λ − s 0 ;(19)s 0 = −µ * 2ǫ 0 q 0 C(s) s 0 s − m 0 Λ + s;(20)m = −Γm e −1/χ Λ + 2 q 0 C(s)( s s − m) ;(21)m 0 = −Γm 0 e −1/χ Λ + 2 q 0 C(s)( s 0 s − m 0 ) ; (22) Λ = Γ e −1/χ − Λ ;(23)χ = 1 c 0 Γǫ 0 Λ (χ ∞ − χ) + D χ ∂ 2 χ ∂y 2 ,(24)
where
Γ = 2 q 0 C(s)(s − s 0 m 0 − sm).(25)
As noted in [16] the density of STZs, ǫ 0 Λ, is necessarily small. Eqs. (19), (20) and (24) each contain this factor in their numerators and they equilibrate very slowly compared to m, m 0 and Λ, which are governed by Eqs. (21), (22) and (23), respectively. Therefore we replace Λ, m and m 0 by their steady state values:
Λ = Λ ss (χ) = e −1/χ ; (26) m = m(s, s 0 ) = s/s, s ≤ 1, s/s 2 , s > 1,(27)m 0 = m 0 (s, s 0 ) = s 0 /s, s ≤ 1, s 0 /s 2 , s > 1.(28)
Below the yield stress the solid deforms only elastically because all the existing STZs are already flipped in the direction of stress. Three simple partial differential equations remain:
s = µ * 1 − 2ǫ 0 q 0 C(s) s s − m Λ − s 0 ,(29)s 0 = −µ * 2ǫ 0 q 0 C(s) s 0 s − m 0 Λ + s,(30)χ = 2ǫ 0 C(s) c 0 q 0 (s − s 0 m 0 − sm) e −1/χ (χ ∞ − χ) + D * χ ∂ 2 χ ∂y 2 .(31)
This set of equations has several dimensionless parameters. The volume (area in 2D) fraction of the strip covered by STZs or the probability of finding an STZ at a given point in the strip is ǫ 0 Λ. Λ is the probability that a particle participates in an STZ. Assuming that STZs have a characteristic size of a few particle diameters, ǫ 0 = λn ∞ is the characteristic number of particles in an STZ. The parameter µ * = µ/µ is the ratio of average material stiffness to STZ stiffness, and corresponds to the slope of the linear elastic portion of a stress-strain curve. The specific heat in units of the STZ formation energy, c 0 , determines how fast the effective temperature changes compared to plastic rearrangements. The parameter q 0 = τ 0 V 0 /L is the ratio of the two physical timescales in the system, the inverse STZ attempt frequency and the average inverse strain rate. Finally, χ ∞ is the steady state effective temperature in units of the STZ formation energy divided by the Boltzmann constant.
The effective temperature diffusivity is the remaining parameter and requires some discussion. The dimensionless parameter D * χ in Eq. (10) can be written as follows:
D * χ = (D χ /c 0 )(L/V 0 )(1/L 2 ),(32)
where D χ = a 2 /τ D is a diffusivity with units of length scale squared per unit time. In STZ theory there are two candidates for the attempt frequency 1/τ D : the local strain rateγ loc and the inherent material attempt frequency 1/τ 0 . The first possibility implies that the local potential energy diffuses at a rate proportional to the rate of local particle rearrangements, while the second implies that the energy flux depends on an inherent vibrational/frictional timescale. Naively, one might think that the inherent frequency 1/τ 0 is too high and results in a diffusion constant much too large to be physically reasonable. However, if one assumes that configurational energy diffusion is an activated process with large energy barriers (τ D = τ 0 exp(∆E/kT )), then the resulting diffusion time τ D could be large and yet remain independent of strain rate.
We have examined solutions to the STZ equations with an effective temperature diffusivity D * χ (a) proportional to the local strain rate and (b) independent of strain rate. In case (a) multiple shear bands are more likely, the effective temperature takes much longer to reach its peak value, and the dynamics are more complicated. In case (b) a single shear band is more likely and the effective temperature quickly reaches its plateau. While further study is required to determine which diffusion frequency best describes the data, an effective temperature diffusivity that is independent of strain rate results in simpler dynamic equations, and we will use this assumption for the remainder of this paper.
III. NUMERICAL INTEGRATION OF THE STZ EQUATIONS
In this section we show that numerical integration of the simple, macroscopic STZ equations (Eqs. (29-31)) qualitatively and quantitatively match molecular dynamics simulations of a Lennard-Jones glass by Shi, Katz, Li and Falk [1]. The simulations were performed in a simple shear geometry with periodic Lees-Edwards boundary conditions. The parameter range for the simulations was chosen to ensure that the glass exhibits strain localization, and stress-strain curves were computed for a variety of quenches and strain rates.
The investigators calculated the potential energy per atom and strain rate as a function of position for several different strains. They used the data to investigate the hypothesis (found in the STZ model and other models) that the local plastic strain rate is related to an effective temperature or free volume. In [1], they show that a value for the steady state effective temperature,χ ∞ , can be rigorously extracted from a thorough analysis of the simulation data. In this paper we numerically integrate the STZ equations in the same infinite strip geometry, and compare the results to simulation data. Building on the work of Shi, et al., we choose parameters for the STZ theory to match the conditions found in the simulations.
In experiments and simulations, boundary conditions on the particles at the top and bottom of the strip impact the dynamics and may influence strain localization. Shi, et al. chose periodic boundary conditions for their simulations, while other investigators [19] have chosen rigid rough walls. While both sets of simulations exhibit strain localization, features of the localization process are different in each case and it is unclear whether the differences are due to the dissimilar boundary conditions or other features of the simulations.
Similarly, the STZ equations require boundary conditions for the effective temperature field at the top and bottom of the strip. We have studied the STZ equations with both periodic and no conduction (∂χ/∂y = 0) boundary conditions and found qualitatively and quantitatively similar results. For the remainder of this paper we will describe solutions to the STZ equations with periodic boundary conditions for comparison to the results of Shi, et al.
A. Matching parameter values to simulations
The first task is to choose values for the parameters in STZ theory consistent with those in the simulations by Shi, et al. The simulation data for the stress-strain curves at a particular strain rate can be fit by a range of values for χ ∞ , q 0 , and the specific heat c 0 . However, a much narrower range of parameters fit both the stressstrain data and the data for the strain-rate as a function of position.
We estimate an order of magnitude for each parameter motivated by physical considerations. Table I lists the specific parameter values used in the numerical integration. The steady state effective temperature χ ∞ determines the steady state density of STZs:
Λ ∞ = exp[−1/χ ∞ ].
To ensure that STZs are rare, χ ∞ should be around 0.1. Shi, et al., extracted χ ∞ = 0.15 from the data using the hypothesis that the strain rate is proportinal to e −1/χ , and we will use this value here. The parameter ǫ 0 is the characteristic number of particles in a shear transformation zone. Studies of non-affine particle
Parameter Value χ∞ 0.15 ǫ0 10 c0 1 D * χ 0.01 µ * 70 q0 1 × 10 −6 χ0
0.68, 0.69, 0.74 motion in a 2D Lennard-Jones glass [15] and a 2D system of hard spheres [34] suggest that an STZ has a radius of a few particle diameters. This implies that the number of particles in an STZ is of order 10 for 2D simulations.
Because c 0 is a dimensionless constant and does not depend on other scales in the problem, c 0 should be of order unity. The parameter µ * is the slope of the linear elastic part of the stress-strain curve plotted in units of the yield stress. It corresponds to the ratio of the elastic material stiffness to the STZ stiffness, and should be much greater than one. In the MD simulations, µ * is about 70.
Eqs. (29)(30)(31) can be further simplified because µ * is large. For s ≤ 1, the rate of plastic deformation is zero and and Eq. (29) indicates that the stress increases proportionally to µ * . During this same time, Eq. (30) requires that s 0 increases proportionally to s, which is less than unity. During the linear elastic response, the offdiagonal component of the stress is order µ * larger than the on-diagonal stress, which matches our physical intuition for simple shear. Numerical integration of Eqs. (29)(30)(31) confirms that s 0 remains about two orders of magnitude smaller than s. We therefore use the approximation s = s and s 0 = 0. This results in two simple equations for the STZ dynamics:
s = µ * 1 − 2ǫ 0 q 0 C(s) (1 − m(s)) Λ ,(33)χ = 2ǫ 0 C(s) s c 0 q 0 (1 − m(s)) e −1/χ (χ ∞ − χ) + D * χ ∂ 2 χ ∂y 2 .(34)
The remaining parameters are D * χ and q 0 . As mentioned in the previous section, we postulate that the effective temperature diffusion D * χ is a constant independent of strain rate which we determine from the long-time diffusion of the shear band in the simulations. We find that D * χ ≃ 0.01 matches the simulation data. The Lennard-Jones glass has a natural time scale t 0 = σ SL M/ǫ SL , where σ SL is the equilibrium distance between small and large particles, ǫ SL is the depth of the potential energy well between the two species and M is the mass. The average strain rate in the simulations varies from 2 × 10 −5 t −1 0 to 5 × 10 −4 t −1 0 .
If we assume that the STZ attempt frequency τ 0 is approximately t 0 , then q 0 ≃ 2 × 10 −5 and the resulting numerical solutions to the STZ equations never exhibit localization. This is inconsistent with simulation data. In order to match the localization seen in the simulations, we are required to choose a value for τ 0 that is an order of magnitude smaller than t 0 . We find that q 0 ≃ 1 × 10 −6 results in STZ solutions where strain rate matches results from the MD simulations. We will discuss the implications of this in Section IV D.
Initial conditions for s and χ must also be determined. Because the stress is constant as a function of position and the simulations begin in an unstressed state, s(t = 0) = 0. The initial effective temperature χ(y, t = 0) is a function of position with many more degrees of freedom. In the MD simulations, the initial potential energy is nearly constant as a function of position with fluctuations about the mean. Therefore, the initial effective temperature is also nearly constant with small amplitude perturbations about its mean. We expect fluctuations in the effective temperature to occur on the scale of a few particle radii in dimensionless units. The simulation box of Shi, et al. has a width of about 300 particle radii, so perturbations which span five particle radii have a nondimensionalized wavelength w = 1/60.
The amplitude of the initial perturbations, δχ 0 , can be approximated empirically from the initial potential energy per atom as a function of position, shown in Figure 2(b). The potential energy at 0% strain exhibits small perturbations with standard deviation ∼ 0.02 ǫ SL about a mean value of −2.51 ǫ SL , while the system reaches a maximum of −2.42 ǫ SL in the shear band at larger strains. Assuming the maximum potential energy in the band corresponds to χ ∞ and the initial mean potential energy per atom corresponds to χ 0 , the amplitude of initial perturbations is about δχ = 0.02. This is consistent with a thermodynamic calculation [35] for the magnitude of fluctuations about the effective temperature,
< (δχ) 2 >= χ 2 /(c 0 ) ≃ 0.01.
We use two different methods for generating an initial effective temperature distribution for the STZ equations. The first is to use a deterministic function with a single peak that serves as a nucleation point for shear bands: χ(y, t = 0) = χ 0 + δχ 0 sech(y/w). The second method generates a random number from a uniform distribution, and smooths those values using a simple moving average of width w. The resulting function is normalized so that its standard deviation from the mean value is δχ 0 , and the mean is set at χ 0 .
The first type of initial condition is less physically realistic but more tractable because it generates at most a single shear band. The second type of initial condition can generate solutions with varying numbers of shear bands, depending on the system parameters. We find that for the parameter range which best fits the simulation data, the STZ equations with random initial conditions generate a solution with a single shear band, which agrees with observations from the simulations and further validates our choice of STZ parameters.
B. Comparison of macroscopic stress-strain behavior
We numerically integrate Eqs. (33,34) with parameters values discussed above, for many different values of χ 0 . Figure 1(a) is a plot of stress vs. strain for three different initial preparations of a material starting from rest and driven in simple shear. Dashed lines correspond to MD simulation data from samples permitted to relax for three different amounts of time before being sheared. Solid lines are numerical solutions to the simplified STZ model (Eqs. (33) and (34)) for three different average initial effective temperatures.
All of the samples (for both the MD simulations and the STZ solutions) exhibit strain localization, which influences the stress-strain curves shown in Figure 1. We will discuss how localization affects the macroscopic stress-strain curves later in this section, and analyze the localization process in Section IV.
The STZ solutions shown differ only in the initial mean value for the effective temperature, χ 0 , and are the best least-squares fit as a function of χ 0 for each quench. Note that general features of the stress-strain curves match: there is a linear elastic segment, followed by a decrease in the stress slope as the material begins to deform plastically, a peak at about 2 % strain, and stress relaxation as the material softens and stored elastic energy is released.
The glass in the MD simulations behaves differently depending on how long the system was quenched before shearing, and the STZ model captures this behavior. A longer quench results in a more ordered solid and corresponds to a lower initial effective temperature. Numerical integration of the STZ equations indicate that lower initial effective temperatures generically result in higher peak stresses. This matches both the MD simulations and physical intuition: a more ordered solid takes more time and stress to plastically deform because more STZs must be created to permit the deformation.
In Figure 1(b) the slope (µ * ) of the linear elastic segment of the most disordered glass (Quench III) is smaller than in the samples that were quenched for longer times before shearing. A linear fit to these data shows that µ * ≃ 70 for Quenches I and II, while µ * ≃ 60 for Quench III.
We model the material surrounding the STZs as an elastic medium, and the parameter µ * is the ratio of the elastic material stiffness to the STZ stiffness. We assume that the STZ stiffness is constant for different sample preparations. Therefore a variation in µ * between samples indicates that the surrounding elastic medium is less stiff in the more disordered material. This is consistent with the work of Maloney and Lemaître, who have shown that the elastic shear modulus is smaller in more disordered materials due to non-affine particle motion [36].
Although we could better fit the MD simulation data by allowing µ * to vary between samples, in order to limit the adjustable parameters in the theory we fix µ * = 70 for all samples. As expected, Figure 1(a) shows that the numerical STZ results closely match the simulation data for Quenches I and II, while the best STZ fit systematically deviates from the simulation data for Quench III.
C. Comparison of strain localization inside the material
One feature of the infinite strip geometry is that it permits the system to achieve very large strains without fracturing, so that both theory and simulation can track system evolution over very large strains. The STZ solutions not only match the short-time macroscopic stressstrain behavior, but also match the long-time dynamics of strain localization within the strip. Figure 2 shows the strain rate as a function of position for various values of the strain (corresponding to different times) for (a) the MD simulations and (b) STZ theory. This figure also shows (c) the simulation potential energy per atom and (d) theoretical effective temperature as functions of position. The simulation data is averaged over increments of 100 % strain, while the STZ model is evaluated for specific values of the strain corresponding to the midpoint of each binning range.
Localization is evident in both the simulation data and numerical STZ solutions. The strain rate as a function of position inside the material exhibits very slow relaxation over 800 % strain. This is in contrast to the much faster stress dynamics that attain steady state in less than 10% strain, as seen in Fig. 1. In the numerical STZ solutions, the effective temperature attains a maximum in the same physical location within the strip as the strain rate. This is remarkably similar to the dynamics of the potential energy per atom in the simulations, and completely consistent with the assumption that χ is proportional to the potential energy. Shi, et al. were the first to systematically check this hypothesis and they used it to extract various STZ parameters, such as χ ∞ [1].
2 Strain Rate (L/V 0 ) (a) 0 -1 +1
D. Implications for constitutive laws
Strain localization has a large effect on the short-time macroscopic stress-strain behavior of the system. For ex-ample, Fig. 3 (a) shows two possible initial conditions for the initial effective temperature field. The average initial effective temperature is the same in both cases. One initial condition for the effective temperature varies with position while the other is constant as a function of position. Figure 3 (b) shows the resulting stress-strain curves for each case. When χ 0 varies as a function of position, the system localizes and the resulting stress-strain curve does not reach as high a peak value, because the material releases elastic energy more quickly. The steady state stress is the same in each case. This is just one example of the general sensitivity of macroscopic state variables to microscopic details. The initial effective temperature distribution is either homogeneous (blue) or slightly perturbed (red), though the average value χ0 is the same in both cases. (b) The stress vs. strain predictions are significantly different for these two different initial conditions. The perturbed system localizes, as shown in Figure 2(d), and this changes the resulting macroscopic behavior.
The fact that localization influences the macroscopic stress-strain curves has important implications for constitutive laws. Constitutive laws provide a relationship between the strain rate, stress, and a set of state variables that characterize the internal structure of a material interface. They describe how features of microscopic particle dynamics determine macroscopic frictional properties, and are used extensively in models of earthquakes [21,22], granular flows [37], and machine control [38]. Several investigators have adapted STZ theory to describe the macroscopic behavior of lubricated surfaces [39], dense granular flows [34], and earthquake faults [40], but all of these formulations assume that the STZ variables do not vary as a function of position across the sheared interface. Deriving new constitutive laws based on localized dynamics is beyond the scope of this paper. However, our analysis shows that localization is possible in STZ models and that the corresponding constitutive behavior is altered. Therefore it is important for localization to be included in modeling efforts, and this will be a direction of future research.
IV. STABILITY ANALYSIS
Continuum models for amorphous solids create a framework for understanding the instability that leads to localization in amorphous materials. In the previous section we showed that STZ theory with effective temperature exhibits strain localization. In the following subsections we investigate the stability properties of the dynamical system given by Eqs. (33) and (34).
A. Steady state linear stability
The first step in understanding how the simplified STZ model given by Eqs. (33) and (34) responds to perturbations is to perform a linear stability analysis around the steady state solution. Settingṡ = 0 in Eq. (33), results in the following equation for the steady state flow stress s f :
1 = 2ǫ 0 q 0 C(s f )(1 − 1/s f ) exp[−1/χ ∞ ].(35)
For the values of q 0 and χ ∞ in Table I, s f = 1.005the yield stress is nearly equal to the steady state stress. The steady state solution for χ is spatially homogeneous:
χ(y) = χ ∞ .
We now show that the perturbed system, s = s f + s, χ = χ ∞ + δχ(y), is linearly stable. To simplify notation, we define the functions f and g:
f (s, χ) ≡ ∂s ∂t = µ * 1 − 2 ǫ 0 C(s) q 0 (1 − m) L −L e −1/χ 2L dy ;(36)
g(s, χ) = ∂χ ∂t
= 2sǫ 0 C(s) q 0 c 0 (1 − m)e −1/χ (χ ∞ − χ) + D χ ∂ 2 χ ∂y 2 .(37)
The perturbation to χ can be written as a sum of normal modes that satisfy the boundary conditions:
δχ(y) = ∞ k=−∞ δχ k e iky , k = nπ L .(38)
The operator ∂ 2 /∂y 2 is diagonalized in the basis of normal modes and therefore the dynamics of each normal mode are decoupled. In the limit of infinitesimal perturbations, terms of order δχ 2 and s 2 can be neglected. This results in the following linear equations for each mode:
s = ∂f ∂s (s f , χ ∞ ) s + ∂f ∂s (s f , χ ∞ )δχ k ,(39)δχ k = ∂g ∂s (s f , χ ∞ ) s + ∂g ∂χ (s f , χ ∞ )δχ k .(40)
The second term in Eq. (39) requires some additional explanation, because the operator ∂f /∂s includes an integral. The action of this operator on a normal mode δχ k e iky is given by:
∂f ∂χ δχ k e iky = F (s f , χ ∞ ) 1 2L +L −L δχ k e inπy/L dy(41) = 0, k = 0, F (s f , χ ∞ ), k = 0,(42)
where
F (s f , χ ∞ ) = − 2e −1/χ∞ ǫ 0 µ * 1 − 1 s f C(s f ) q 0 χ 2 ∞ δχ k . (43)
This analysis reveals a particularly interesting and important feature of the STZ model dynamics which we will return to in the next section. There is a fundamental difference between the dynamics of spatially homogeneous perturbations to χ and perturbations with zero mean. Because the stress is always spatially homogeneous, to linear order the stress dynamics depend only on the average value of χ and are completely unaffected by zero mean perturbations to χ.
In order to determine the linear stability of each mode, we calculate the two eigenvalues (as a function of k) for the steady state linear operator A ss (k), which is the twoby-two matrix given by:
˙ ṡ δχ k ≡ A ss (k) s δχ k (44) ≡ A 11 (k) A 12 (k) A 21 (k) A 22 (k) s δχ k .(45)
The two diagonal terms can be written:
A 11 = 2e −1/χ∞ ǫ 0 µ * q 0 1 s f − 1 C ′ (s f ) − C(s f ) s 2 × Θ(s f − 1),(46)A 22 = 2ǫ 0 s f e −1/χ∞ c 0 q 0 × −C(s f )(1 − 1 s f ) Θ(s f − 1) − D * χ k 2 ,(47)
where Θ is the unit step function. If either of the two eigenvalues of A ss has a positive real part for a particular value of k, then that mode grows exponentially and the system is unstable with respect to perturbations with that wavenumber k.
For k = 0, A 12 (k) is zero and the matrix is lower triangular, so the eigenvalues are simply A 11 and A 22 . Because C(s) is monotonically increasing and positive for s > 0, both eigenvalues are negative for all values of k = 0. The eigenvalues calculated for k = 0 are negative also. Therefore, the STZ model is stable with respect to perturbations in steady state. An analysis by Foglia of the operator A 22 for an STZ model that included thermal effects resulted in the same conclusion [41].
B. Time-varying stability analysis
In simulations and numerical integration of the STZ equations, localization of strain first occurs before the stress reaches a steady state. Transient localization is also seen in numerical simulations of Spaepen's free volume model by Huang et al. [42], and in the Johnson-Segalman model for complex fluids [43]. This motivates us to study the stability of transient STZ dynamics.
The field χ in the STZ model given by Eqs. (33,34) can be rewritten as the sum of normal modes. As discussed in Section IV A, for small perturbations with wavenumber k, the governing equations for the k = 0 mode are fundamentally different from all other modes. This motivates us to write the field χ at each point in time as the sum of a spatially homogeneous field χ(t) and zero-mean perturbations:
χ(y, t) = χ(t) + k =0 δχ k (t)e iky , k = nπ L .(48)
To analyze the transient stability of this system, we permit the k = 0 mode, χ(t), to be arbitrarily large but constrain the zero mean perturbations, δχ k , to be small. This is slightly different from the normal mode analysis in Section IV A where both k = 0 and k = 0 modes were small. Substituting Eq. (48) into Eq. (36) and neglecting the second order terms in δχ k results in the following equation:
s = f (s, χ + k =0 δχ k (t)e iky ) = µ * 1 − 2 ǫ 0 q 0 C(s)(1 − m 0 )e −1/χ × 1 + 1 χ 2 k =0 1 2L L −L δχ k (t)e iky dy = µ * 1 − 2 ǫ 0 q 0 C(s)(1 − m 0 )e −1/χ = f (s, χ).(49)
This indicates that to linear order, zero-mean perturbations to χ do not affect the stress. Substituting Eq. (48) into Eq. (37) results in the following linearized equation for each perturbation mode:
δχ k (t) = ∂g ∂χ (s(t), χ(t), k) δχ k .(50)
This is a linearized equation for the dynamics of small perturbations about a spatially homogeneous, time varying solution. It is valid whenever the perturbations are small, even if the magnitude of χ(t) is large. Physically, this corresponds to the following experiment: the system is started from a spatially homogeneous initial condition and the system remains spatially homogeneous until a small, zero-mean perturbation in introduced at time τ . The time-varying linear operator ∂g ∂χ (s(τ ), χ(τ )) describes the growth or decay of these small perturbations. The functions s(t), χ(t) are the solutions to the STZ equations with spatially homogeneous initial conditions s(t = 0) = s 0 , χ(y, t = 0) = χ 0 . If the initial conditions are homogeneous, the STZ equations are simply ordinary differential equations, which are much easier to solve than the inhomogeneous equations.
This description for the linearized start-up dynamics is relevant to many experiments and simulations. For example, the simulations by Shi, et al. begin from a state where the initial potential energy, which corresponds to the initial effective temperature, is roughly constant as a function of position. This corresponds to a spatially homogeneous initial condition for the effective temperature, χ(t = 0, y) ≡ χ 0 = c. Additionally, the MD samples are started from an unstressed state, which corresponds to s 0 = 0. The potential energy does vary slightly as a function of position, which corresponds to small, zeromean perturbations to χ introduced at time t = 0.
We define the stability exponent ω c (k, t) in the following manner. Let s c (t), χ c (t) be the unique spatially homogeneous solution to the STZ equations starting from the initial condition s 0 = 0, χ 0 = c. Then the exponent at time τ is defined as:
ω c (k, τ ) = ∂g ∂χ (s c (τ ), χ c (τ ), k) = 2e −1/χ c ǫ 0 s c c 0 q 0 × χ ∞ − χ c χ 2 c − 1 C(s c )(1 − 1 s c ) × Θ(s c − 1) − D * χ k 2 ,(51)
where the functions s c (t) and χ c (t) are understood to be evaluated at t = τ and Θ(s) is the unit step function. This exponent describes the rate of growth or decay of a small perturbation with wavenumber k introduced at time τ to the solution s c (t), χ c (t). If the real part of ω c (k, t) is greater than zero then the perturbations grow exponentially and the system is unstable, and otherwise the system is stable with respect to perturbations.
Note that ω c (k, t) contains a single term that depends on the wavenumber k -this term is proportional to −k 2 and corresponds to the diffusion of effective temperature. Because diffusion can only act to stabilize perturbations to χ, the most dangerous mode corresponds to k = ±π/L.
A plot of ω c (k, t) for c = χ 0 = 0.6 and k = π/L is shown in Fig. 4. The system is marginally stable with respect to perturbations during the linear elastic response of the system, highly unstable at intermediate times, and becomes stable with respect to perturbations as χ(t) approaches χ ∞ . A lower bound for stable χ is determined from Eq. (51) as follows. Because the diffusion term is always negative, an upper bound on the stability exponent for all values of k is given by:
ω c (k, t) < H(s)e −1/χ c (t) χ ∞ − χ c (t) χ 2 c (t) − 1(52)
where H(s) is zero for s ≤ 1 and positive for s > 1. Therefore the stability exponent is less than zero when χ(t) satisfies the following inequality:
0 > χ ∞ − χ(t) χ(t) 2 − 1,(53)
which defines (54)
χ(t) > 1 2 −1 + 1 + 4χ ∞ ≡ χ crit .(55)
This analysis indicates if the initial effective tempera- The stability exponent ωc(k, t) with c = 0.06 and k = π/L as a function of total strain for spatial perturbations to the effective temperature in the limit k → 0. The system is linearly unstable with respect to spatial perturbations to χ when ωc(k, t) is greater than zero, and stable otherwise. The figure contains two scales for the average strain in order to simultaneously illustrate both the short-and long-time behavior.
ture is above a critical value, χ 0 > χ crit , then small amplitude perturbations are stable and the system will not localize. However, if the initial effective temperature is less than this value (plus a diffusive term that depends on the wavenumber), there exists a period of time where small perturbations to the effective temperature will grow. A time-varying stability analysis of thermal STZ equations by Foglia shows that the effective temperature is transiently unstable to perturbations in that context as well [41]. This description of perturbations to a spatially homogeneous, time-varying, "start-up" trajectory is remarkably similar that given by Fielding and Olmsted [43] in a study of shear banding in the Johnson-Segalman model for complex fluids. The stability of the solution to perturbations is simpler to analyze in the STZ formulation than in the complex fluid model because the linearized evolution of perturbations to χ are decoupled from the stress, as shown in Eq. (49). This suggests that the mechanism for shear band nucleation in "start-up" flows may be similar in a variety of materials.
C. Finite amplitude effects and a criterion for localization
While linear stability analysis determines when perturbations grow, by itself it does not provide enough information to determine whether or not the integrated STZ equations exhibit strain localization. Many of the numerical STZ solutions for an unstable initial effective temperature do not localize if the initial perturbations are "too small". For example, numerical integration of the STZ equations with an unstable initial value for the effective temperature χ 0 = 0.09 < χ crit ≃ 0.13 and a small initial perturbation δχ 0 = 0.001, result in a solution that is virtually indistinguishable from a homogeneous system. The purpose of this section is to derive a criterion based only on initial conditions (the mean value of the initial effective temperature χ 0 and the amplitude of initial perturbations δχ 0 ) that determines which materials will exhibit long-time strain localization.
At first glance, the fact that a finite perturbation is required to generate localized solutions seems to contradict our assertion that the system is linearly unstable. However, linear stability equations are only valid for short times when perturbations are small -they give no information about the long-time behavior of the unstable perturbed/inhomogeneous states. Nonlinear system dynamics can enhance or inhibit localization in the inhomogeneous states. Upon further examination, we recognize that the "finite-amplitude effect" is due to nonlinear dynamics involving the interaction of two dynamical processes: perturbation growth and energy dissipation. Based on this understanding we derive a criterion for which initial conditions result in shear banding.
We first show that rate of energy dissipation as a function of position determines whether perturbed states remain inhomogeneous or become uniform. Our guiding intuition is that perturbations that grow quickly permit most of the energy to be dissipated in the incipient shear band, which enhances localization. Otherwise the energy is dissipated throughout the material, which inhibits localization.
Eq. (34) can be rewritten in terms of the rate of plastic work dissipated by the system, Q:
s = 1 − Q s ,(56)
where Q is spatial average of the rate of plastic work Q(y). Q can be resolved into a product of two terms -the rate of plastic work per STZ, Γ(s), that depends only on the stress and is spatially invariant, and the density of STZs, Λ(χ), which depends only on the effective temperature and is a function of position:
Q = 2ǫ 0 q 0 sC(s)(1 − m)e −1/χ ≡ Γ(s)Λ(χ).(57)
In steady state the average Q = Γ(s)Λ is constrained to be a constant, but the value Q(y) = Γ(s) exp [−1/χ(y)] is not similarly constrained. This admits the possibility that the plastic work dissipated is very small at some positions and large at others, and this must occur for the system to remain localized.
To see this, first note that if the effective temperature is below χ ∞ , Eq. (10) requires that the effective temperature must always increase proportional to Q(χ ∞ − χ) or diffuse -there is no relaxation towards thermal temperature in this approximation. Therefore, a spatially heterogeneous effective temperature distribution will be nearly stationary only if Q is very close to zero whereever χ = χ ∞ . In other words, strain localization will only persist if the rate of plastic work dissipated outside the shear band is very small.
The rate at which plastic work is dissipated is proportional to exp [−1/χ(y)]. A rough assumption about the nonlinear, inhomogeneous effective temperature dynamics is that the largest initial perturbation of amplitude δχ 0 continues to grow at the rate we derived using linear stability analysis: ω(t)δχ(t), while the effective temperature in regions outside the incipient shear band grows at the rate of a homogeneous system,χ = f (χ(t), s(t)). This assumption is consistent with behavior seen in numerical solutions to the STZ equations. Figure 5 shows the behavior of the effective temperature as a function of time for (a) a system that localizes and (b) a system that does not localize, illustrating the two different initial stages. In both plots the upper solid line is the maximum value of χ(y) inside the shear band, the lower solid line is the minimum value of χ(y) outside the shear band, and the dashed line is a homogeneous solution with the same mean value, χ. In plot (a), the perturbation grows quickly compared to the homogeneous solution for χ, while in (b) the perturbation grows slowly.
We are interested in whether the system dynamics focus energy dissipation inside the shear band or spread energy dissipation evenly throughout the material. Ideally, we would like to compare energy dissipation inside the band (Q in ) to energy dissipation outside of the band (Q out ) at various times t. However, we only have information about the initial conditions, χ 0 and δχ 0 . Therefore we compare the time derivatives of Q inside and outside the band:
∂Q in ∂t = Γ(s) e −1/χin (χ in ) 2 ∂χ in ∂t ≃ Γ(s) e −1/χ0 (χ 0 ) 2 ω χ0 (t) δχ(t);(58)∂Q out ∂t = Γ(s) e −1/χout (χ out ) 2 ∂χ out ∂t ≃ Γ(s) e −1/χ0 (χ 0 ) 2 f (χ(t), s(t)).(59)
We approximate χ in and χ out as the initial condition, χ 0 . This approximation is valid only for short times when χ has not changed appreciably from χ 0 inside or outside the band. The ratio of the derivatives is given by:
Q iṅ Q out (δχ, χ 0 , t) = ω χ0 (k, t)δχ(t) f (χ(t), s(t)) .(60)
Equation (60) indicates that for short times when χ has not changed appreciably, the energy dissipation is determined by comparing the growth rate of perturbations to the growth rate of the mean effective temperature. Equation (60) must be evaluated at a particular time and wavenumber. We evaluate ω χ0 (k, t) for the most unstable mode, which corresponds to k = π/L. For notational simplicity this value of k is assumed in the following derivation. In writing Eqs. (58) and (59), we required that the time t is small enough that χ remains close to χ 0 . Numerical solutions confirm that χ changes very little as s increases from zero to its maximum shear stress. This stress, s m , can be approximated as the solution to Eq. (37) withṡ = 0 and χ = χ 0 :
0 = 1 − 2 ǫ 0 q 0 C(s m )(1 − 1 s m )e −1/χ0 (61)
We must also approximate the numerator of Eq. 60 at the time when the perturbations are growing most rapidly. Naively, one might expect this to be the maximum of ω χ0 (t) times the amplitude of the initial perturbation, δχ 0 . This underestimates the numerator because the amplitude of the perturbation increases with time.
A better approximation can be found by noting that in numerical solutions, ω χ0 (t) rises almost linearly from zero at the time the material reaches the yield stress to a maximum at nearly the same time as the material attains its maximum stress, s m . This behavior is seen in Fig. 4. Equation (51) shows that ω χ0 (t) depends on t only through s(t) and χ(t), so we evaluate the stability exponent at (s m , χ 0 ). Therefore we can approximate the linearized equation for the perturbations, Eq. (50), as:
δχ(t) = ω χ0 (s m , χ 0 ) t m − t y (t− t y )δχ(t), t m > t > t y ,(62)
where t y is the time when the material first reaches the yield stress and t m is the time at the maximum stress. The solution to this differential equation, evaluated at s m , is
δχ(t m ) = δχ 0 exp[ ω(s m , χ 0 ) (t m − t y ) 2 ].(63)
Therefore a lower bound on the numerator is given by:
max t [ω χ0 (t)δχ(t)] > ω(s m , χ 0 )δχ 0 exp[ ω(s m ) ∆t 2 ],(64)
where ∆t = t m −t y is nearly constant in all the numerical STZ solutions and is about 0.03 in units of strain. While better than the first guess, this estimate is likely to be low because ω χ0 (t) is generally not sharply peaked about t m , so the maximum growth rate occurs at later times than t m . To account for this and to match the numerical solutions, we choose ∆t = 0.10.
Substituting these approximations into Eq. 60, we define localization ratio R as follows:
R(δχ, χ 0 ) = ω(s m , χ 0 ) δχ 0 e ω(sm,χ0)0.05 µ * 1 − 2 ǫ0C(sm) q0 (1 − 1 sm )e −1/χ0 ,(65)
where ω(s m , χ 0 ) is defined by Eq. (51). R is large if the system tends to focus energy dissipation inside the band and small otherwise. We now systematically compare the value of the localization ratio to the degree of localization in a numerical STZ solution, and find that the ratio is an excellent criterion for localization.
For each set of initial conditions, (χ 0 , δχ 0 ), we compute a numerical solution to the STZ equations and then calculate the Gini coefficient for the strain rate as a function of position for each point in time. The Gini coefficient, φ(t), is a measure of localization and is defined as follows [44,45]:
φ(t) = 1 2n 2 D pl i j |D pl (y i , t) − D pl (y j , t)| = 1 2n 2 Λ i j |e −1/χ(yi,t) − e −1/χ(yj ,t) |,(66)
where the function χ is evaluated at n points in space, {y n }. A discrete delta function has a Gini coefficient equal to unity, and a constant function has a Gini coefficient equal to zero. We define the Numerical localization number Φ as the maximum value of φ(t) over t. If Φ is close to unity the strain rate is highly localized (as in Figure 2(b)) and when Φ is close to zero the strain rate remains homogeneous. Figure 6 shows that the numerical localization number Φ (rectangles) and the theoretical ratio R (contour plot) exhibit the same behavior as a function of χ 0 and δχ 0 . This indicates that the theoretical R is a good criterion for the numerically computed strain localization Φ.
The shading of the rectangles corresponds to the value of Φ for a given set of initial conditions: red corresponds to highly localized strain rate distributions (Φ > 0.8), yellow to partially localized solutions (0.8 ≥ Φ ≥ 0.3), and dark blue to homogeneous solutions (Φ < 0.3). The contour plot corresponds to values of R: light pink for R > 0.6, darker shades for R ≤ 0.6. The data in figure 6 show that to a good approximation, R > 0.6 corresponds to solutions with strain localization, and R ≤ 0.6 corresponds to homogeneous solutions.
The localization ratio R is a function only of the STZ parameters and the initial conditions for the function χ(y). This is potentially a very useful tool for incorporating effects of localization into constitutive laws. It determines an important material property without requiring computation of the full inhomogeneous system dynamics.
D. Long-time behavior
The localization ratio compares the energy dissipated inside the shear band to the energy dissipated outside the shear band, and determines whether the nonlinear dynamics enhance or inhibit shear band formation. An interesting and more difficult question is what determines whether the system dynamics result in a single shear band or multiple shear bands. The MD simulations generally have a single shear band except at the highest strain rates (see Figure 2 in [1]). In the parameter range used to match the MD simulations, the STZ solutions also exhibit a single shear band. However, outside this Figure 6 corresponds to a solution to the STZ equations for a particular choice of (χ0, δχ0). The shading of the rectangle corresponds to the value of Φ for a numerical solution with the indicated initial conditions: red corresponds to highly localized strain rate distributions, 1 > Φ > 0.8, yellow corresponds to partially localized solutions 0.8 > Φ > 0.3 and dark blue corresponds to homogeneous solutions 0.3 > Φ > 0. The background shading corresponds to a contour plot for R. Light pink corresponds to values of R > 0.6, dark purple corresponds to values of R < 0.1, and contours are at 0.1 intervals in between. The yellow rectangles consistently overlay the yellow contour, showing that the theory and numerics agree on the localization transition.
parameter range it is possible for the STZ equations to generate solutions with multiple shear bands.
It appears that the STZ equations generate a single shear band when the system is in a parameter range where perturbations to χ are highly unstable. In this case the largest amplitude perturbation grows rapidly, resulting in a large rate of energy dissipation at that position. Consequently the energy dissipation rates outside the shear band are smaller, which inhibits the growth of smaller amplitude perturbations. A full analysis of these dynamics is beyond the scope of this paper.
However, we use the fact that there is a single shear band to determine the width of that band. As discussed above, most of the strain must be accommodated in the shear band if the strain is to remain localized. χ ∞ sets the steady state of disorder in the system, and we assume that χ attains this value in the shear band. We can estimate the width of the band as a function of time, w E (t), by postulating that all of the shear band strain is accommodated plastically in the shear band:
0 =γ − 1 2L L −L D pl (y)dy; (67) 0 = 1 − w E 2L 2 q 0 C(s(t)) 1 − 1 s(t) e −1/χ∞ ;(68) w E (t) 2L = q 0 exp[1/χ ∞ ] 2C(s(t))(1 − 1 s(t) ) ,(69)
where s(t) is the numerical solution to the STZ equation for the stress as a function of time.
We also evaluate the width of the shear band in the numerical STZ solution for the strain rate. The numerical shear band width (w N (t)) is computed as the width of the strain rate vs. position curve at a value corresponding to the imposed strain rate, 1 = V 0 /L. Figure 7 compares the numerical shear band width to the estimated width, as a function of time. The two are in agreement. The numerically computed width is systematically larger than the estimated width because we chose a relatively low cutoff, 1 = V 0 /L, for the strain rate in the band.
The estimated width is not a prediction for the width of the shear band because it depends on the numerical solution for the stress as a function of time. However, it does suggest that because the STZ equations contain no natural length scale for the width of a shear band, the system dynamically sets the width based on the imposed strain rate.
The shear band has a well-defined effective temperature (χ ∞ ) and therefore accommodates a fixed rate of plastic strain per unit width at a given stress. Together, the stress and the imposed strain set the shear band width: the shear band must grow to the width required (at a fixed stress) to accommodate all of the imposed strain. While we cannot compute the shear band width analytically, we have recast the problem in terms of a potentially simpler one to solve: determining the stress relaxation as a function of time.
The value of the steady state stress has a large effect on strain localization. While the spatial dependence of Q(y) is through χ(y), the magnitude of Q depends on the value of the stress through Γ(s). When the system is in the flowing regime, the nondimensionalized stress s is always greater than one. The energy dissipation per STZ, Γ(s) ∝ sC(s)(1 − 1/s), approaches zero as the stress approaches one from above. Therefore, if the steady state stress is very close to unity, the effective temperature dynamics become very slow -they are "frozen in" by the stress dynamics. In this parameter range the rate of plastic work can become very small at some positions as the system approaches steady state and this is exactly what permits a localized strain state to be long-lived. In contrast, when the steady state stress is too large the magnitude of Q is also large (regardless of χ(y)) and the material will not remain localized.
Numerical integration of the STZ equations confirms that localization occurs only when the steady state stress is close to unity. STZ solutions exhibit localization when q 0 is O(10 −6 ) and χ ∞ is O(1/10), which corresponds to a steady state stress that is very close to the yield stress (s f = 1.005). However, they do not exhibit long-lived shear bands if the system is driven more quickly (q 0 ∼ O(10 −5 )), which corresponds to a steady state stress of s f ≃ 1.1. This helps us interpret MD simulation data at higher strain rates. In addition to the data shown in Figure 1, Shi, et al. have computed stress-strain curves for two higher strain strain rates (five and 25 times greater, respectively, than the strain rate used to generate Fig. 1) [46].
The lowest strain rate is denoted V1 and the higher strain rates are denoted V2 and V3. In this paper we defined the timescale ratio q 0 using V1. Therefore V2 corresponds to 5q 0 and V3 corresponds to 25q 0 .
We numerically integrate the STZ equations at 5q 0 and 25q 0 and find that the resulting stress-strain curves do not match the simulation data. The theoretical STZ steady state flow stress is the same for all three strain rates, while in the MD simulations the flow stress for a system driven at V3 is significantly (∼ 15%) higher.
One way to eliminate this discrepancy is to assume that we have misidentified the original timescale ratio q 0 . For a given value of q 0 (and χ ∞ ), we can calculate the flow stress s f using Eq. (35). To match the simulation data, the flow stress that corresponds to 25q 0 should be 15% higher than the flow stress that corresponds to q 0 . In Table II we have calculated the flow stress ratios for several values of q 0 . This table shows that q 0 ≃ 1 × 10 −5 better fits the MD simulation data for different strain rates than q 0 ≃ 1 × 10 −6 . The higher value for q 0 also makes sense physically because it corresponds to an STZ timescale τ 0 which is simply the natural timescale for the Lennard-Jones glass, t 0 .
Our athermal model requires a small value for q 0 to q0 s f (5q0)/s f (q0) s f (25q0)/s f (q0) generate solutions with long-lived localized states, but a larger value of q 0 to be consistent with MD simulations data for different strain rates. This disparity is likely an indication that the athermal approximation does not adequately capture long-time behavior of Lennard-Jones glasses.
In an athermal description, the effective temperature always tends towards χ ∞ because there is no thermal relaxation. Alternatively, if the effective temperature can relax towards a thermal bath temperature in regions where there is little plastic deformation, localized states can persist even when s f is large. It seems likely that a thermal description would generate localized solutions with higher values for q 0 . Although the athermal STZ formulation reproduces many aspects of the MD simulations, a model with thermal relaxation may be required to capture more details of the MD simulations. This will be a direction of future research.
V. CONCLUDING REMARKS
We have investigated the athermal STZ model with effective temperature for amorphous materials in a simple shear geometry. In contrast to other studies, the effective temperature field varies as a function of position and by numerically integrating the STZ equations we have shown that the resulting solutions can exhibit strain localization. The numerical STZ solutions match the stress-strain curves for MD simulations of Lennard-Jones glasses by Shi, et al. and exhibit strain rate and effective temperature fields that are consistent with those seen in simulations. The continuum STZ model provides a framework for understanding how shear bands nucleate and persist for long times. We have shown that the model is unstable with respect to small perturbations of the effective temperature during the transient dynamics, though it is stable to perturbations in steady state. Interestingly, shear localization does not always occur when the system is transiently unstable because shear bands can only form if the perturbations grow quickly compared to the homogeneous solution. This is a nonlinear effect which is driven by uneven energy dissipation in the system. Using rough approximations to these nonlinear dynamics we were able to derive a localization criterion that depends only on the initial conditions for the effective temperature.
In the parameter range studied, STZ theory predicts that the sheared material attains a state with a single shear band. The effective temperature field reaches its steady state value χ ∞ in the shear band and remains close to its initial value outside the band. To a good approximation, all of the strain imposed at the boundaries is accommodated in the band, and we determine the width of the shear band using the STZ expression for the plastic strain rate when χ = χ ∞ . This analysis implies the shear band width is not determined by an inherent length scale in the material, but instead by a dynamical scale set by the imposed strain rate.
Interesting questions remain concerning the interaction between shear bands and material boundaries. We evaluate the STZ equations with simple, periodic boundary conditions on the effective temperature field. In our numerical solutions, the location of the shear band depends on the details of the initial fluctuations in χ and therefore seems arbitrary on macroscopic scales. This is similar to what is seen in Shi, et al. for simulations with periodic boundary conditions. However, in other MD simulations and in experiments on bulk metallic glasses the boundary conditions are non-trivial and most likely play an important role in nucleating shear bands. For example, in simulations by Varnik et al. [19] the system is driven at the boundary by a rough, rigid layer and the material consistently develops shear bands at the boundary. Additional simulations by Falk and Shi [47] show that shear bands are more likely to occur in the presence of surface defects.
Boundary conditions may also influence the long-time behavior of shear bands. The long-lived, localized solutions to the STZ equations are not truly stationary states -at long times the shear band will diffuse to cover the entire strip and the system will become homogeneous. Figure 8 is a plot of the steady state stress vs. the natural log of the imposed strain-rate for STZ solutions. This function is not multivalued, indicating that the only steady state solution to the STZ equations with periodic boundary conditions is the spatially homogeneous solution χ(y) = χ ∞ , s = s f . Even excluding diffusion, the STZ equations will evolve to a homogeneous solution because the plastic dissipation rate is positive everywhere, which means the effective temperature is everywhere driven towards χ ∞ , albeit at different rates. However, if the effective temperature field is specified at the boundaries (i.e, the boundary causes ordering and fixes the disorder temperature there) then it is possible for solutions to have inhomogeneous steady states. This is similar to results for simulations with rough, rigid boundaries where the material develops stationary localized states [19]. More work is needed to understand the relationship between shear band development and material boundaries, and STZ theory should provide an excellent framework for these investigations.
The athermal STZ description provides a simple starting point for understanding the dynamics of shear band formation. As discussed in the previous section, a thermal description is likely a better model for the MD simulations at different strain rates. Adding thermal relax- ation to the STZ equations generates a picture of localization similar to those found in complex fluids, where the stress vs. strain rate curve is multi-valued [48]. As discussed by Langer [25] and Foglia [41], relaxation terms in the equation for the effective temperature generate solutions where the steady state stress is a multi-valued function of the strain-rate. In steady state, two regions with different strain rates can coexist: one region which is nearly jammed except for thermal rearrangements and a second region where STZs are constantly created and flipped, resulting in plastic flow. This is different from the athermal theory where all of the material deforms due to shear-induced transitions, though the transitions occur at different rates. While athermal analysis provides a simple framework for understanding mechanisms leading to shear band nucleation, an STZ formulation with thermal relaxation may further enhance our understanding of steady-state dynamics and connect mechanisms for localization in amorphous solids to those in complex fluids.
VI. ACKNOWLEDGMENTS
FIG. 1 :
1(color online) (a)Simulation data and STZ theory for stress vs. strain from Ref.[1]. Dashed lines are simulation data for three different initial preparations. The red curve corresponds to a sample quenched for the longest time (100,000 t0)before the sample was sheared and has an initial average potential energy per atom of -2.507. The green and purple curves were quenched for 50,000 t0 and 10,000 t0 respectively, and have initial PE/atom of -2.497 and -2.477. The solid lines correspond to STZ solutions with values for the initial effective temperature (χ0) that best fit the simulation data: 0.062, 0.063 and 0.067, respectively. (b)Linear elastic regime of the simulation data for the three different quenches.
FIG
. 2: (color online) (a) Simulation data for strain rate vs. position at various strains from Ref.[1]. Strain rates are averaged over bins of 100 % strain. The dashed line corresponds to the imposed average strain rate. (b) Theoretical STZ solutions for strain rate vs. position at various strains. Strain rate is evaluated for specific values of the strain corresponding to the midpoint of each binning range for simulation data. The dashed line is the average strain rate. (c) Simulation data for potential energy as a function of position[1] and (d) Theoretical solution for the effective temperature as a function of position. The dashed lines correspond to the initial values of the potential energy per atom and effective temperature.
FIG
. 3: (color online) Macroscopic stress-strain curves are sensitive to localization. (a)
FIG. 4: The stability exponent ωc(k, t) with c = 0.06 and k = π/L as a function of total strain for spatial perturbations to the effective temperature in the limit k → 0. The system is linearly unstable with respect to spatial perturbations to χ when ωc(k, t) is greater than zero, and stable otherwise. The figure contains two scales for the average strain in order to simultaneously illustrate both the short-and long-time behavior.
online) Plot of effective temperature χ as a function of time for (a) an STZ solution for δχ0 = 0.01 that localizes and (b) an STZ solution for δχ0 = 0.001 that does not localize. The dashed blue line indicates the homogeneous solution for χ as a function of time. The red(orange) solid lines corresponds to the function χ(y) evaluated at a point located within (outside of) the shear band. The shaded region indicates the range of values spanned by χ(y) at every time, so that a large shaded region for long times indicates a long-lived localized state. Dashed vertical lines correspond to calculated values for the time the material first reaches the yield stress, ty, and the time the material reaches its maximum stress tmax.
online) Comparison of numerical localization Φ to theoretical R as a function of χ0 and log 10 (δχ0). Each rectangle in
online) Data for shear band width as a function of strain. The red dashed line is the numerical width wN calculated from the STZ solution for the strain rate. The solid green line is the estimated shear band width wE calculated from the STZ solution for the stress only.
FIG. 8 :
8STZ solution for the steady state stress vs. Log10 of the imposed total strain rate. The curve is single-valued throughout, indicating that the system does not posses steady state coexisting regions with two different phases unless imposed by the boundary conditions.
TABLE I :
IList of parameter values used in the numerical integration of the STZ model, Eqs.(33,34).
TABLE II :
IIComparison of the STZ steady-state flow stress for different values of q0.
The authors whould like to thank Yunfeng Shi and Michael Falk for numerous useful discussions and also for generously sharing their data from reference[1]and additional simulations[46].
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[
"A general existence result for stationary solutions to the Keller-Segel system",
"A general existence result for stationary solutions to the Keller-Segel system"
] |
[
"Luca Battaglia "
] |
[] |
[] |
We consider the following Liouville-type PDE, which is related to stationary solutions of the Keller-Segel's model for chemotaxis:where Ω ⊂ R 2 is a smooth bounded domain and β, ρ are real parameters. We prove existence of solutions under some algebraic conditions involving β, ρ. In particular, if Ω is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.(1.1)Since both (P β,ρ ) and (1.1) are invariant under addition of constants, it will not be restrictive to look for solutions to (P β,ρ ) satisfyingˆΩ u = 0; equivalently, we will consider J β,ρ not on its naturalIn particular, we will study the topology and the homology of very low energy sub-levelsthen will deduce existence of solutions using Morse theory. This argument has been introduced in [17] for a fourth-order problem in Riemannian geometry and it has become a rather classical tool in the study of Liouville-type equation (see for instance the surveys [26, 25]). With respect to most previous results, the main new difficulties are given by the Neumann boundary conditions, rather than Dirichlet, and the presence of the extra linear term βu.Neumann conditions may cause concentration on ∂Ω, which was excluded in the case of Dirichlet conditions, whereas concentration on the interior is similar in the two cases. The main difference when concentration occurs at the boundary is due to the fact that, roughly speaking, a shrinking ball centered at a point p ∈ ∂Ω is asymptotically half of a shrinking ball contained in Ω. The argument to fix such an issue is inspired by[28], which deals with a similar problem on higherdimensional manifolds with boundary.Another issue may be given by linear part −∆ + β not being positive definite, if β < 0, which is a new feature in second order Liouville equations. This naturally leads to consider the projection of u into positive and negative sub-spaces of the operator −∆ + β. Precisely, we take an orthonormal frame {ϕ i } i∈N of eigenfunctions of −∆ with associated positive non-decreasing eigenvalues {λ i } i∈N (counted with multiplicity), so thatNow, if −β is not an eigenvalue of −∆, then −λ I+1 < β < −λ I for some I ∈ N, therefore we can define the projection Π I on a finite dimensional subspace, on which orthogonal −∆ + β is positive definite:Arguing as in[17], we can show that, if J β,ρ (u) ≪ 0, then either e ú Ω e u is concentrated around a finite number of points or Π I u is large (or both occur). To express this alternative we will use the join, which has been used in the variational study of Liouville system of two equations, where one has an alternative between the concentration of each component (see[6,4,20,7,5]). Given two topological spaces X and Y , its join X × Y is defined as the product between the two spaces and the unit interval, with identifications at each endpoint. We set
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10.3934/dcds.2019038
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[
"https://arxiv.org/pdf/1802.02551v2.pdf"
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1802.02551
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d39c28e109e460b453361e08faa95cb2463e9abe
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A general existence result for stationary solutions to the Keller-Segel system
9 Dec 2018
Luca Battaglia
A general existence result for stationary solutions to the Keller-Segel system
9 Dec 2018
We consider the following Liouville-type PDE, which is related to stationary solutions of the Keller-Segel's model for chemotaxis:where Ω ⊂ R 2 is a smooth bounded domain and β, ρ are real parameters. We prove existence of solutions under some algebraic conditions involving β, ρ. In particular, if Ω is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.(1.1)Since both (P β,ρ ) and (1.1) are invariant under addition of constants, it will not be restrictive to look for solutions to (P β,ρ ) satisfyingˆΩ u = 0; equivalently, we will consider J β,ρ not on its naturalIn particular, we will study the topology and the homology of very low energy sub-levelsthen will deduce existence of solutions using Morse theory. This argument has been introduced in [17] for a fourth-order problem in Riemannian geometry and it has become a rather classical tool in the study of Liouville-type equation (see for instance the surveys [26, 25]). With respect to most previous results, the main new difficulties are given by the Neumann boundary conditions, rather than Dirichlet, and the presence of the extra linear term βu.Neumann conditions may cause concentration on ∂Ω, which was excluded in the case of Dirichlet conditions, whereas concentration on the interior is similar in the two cases. The main difference when concentration occurs at the boundary is due to the fact that, roughly speaking, a shrinking ball centered at a point p ∈ ∂Ω is asymptotically half of a shrinking ball contained in Ω. The argument to fix such an issue is inspired by[28], which deals with a similar problem on higherdimensional manifolds with boundary.Another issue may be given by linear part −∆ + β not being positive definite, if β < 0, which is a new feature in second order Liouville equations. This naturally leads to consider the projection of u into positive and negative sub-spaces of the operator −∆ + β. Precisely, we take an orthonormal frame {ϕ i } i∈N of eigenfunctions of −∆ with associated positive non-decreasing eigenvalues {λ i } i∈N (counted with multiplicity), so thatNow, if −β is not an eigenvalue of −∆, then −λ I+1 < β < −λ I for some I ∈ N, therefore we can define the projection Π I on a finite dimensional subspace, on which orthogonal −∆ + β is positive definite:Arguing as in[17], we can show that, if J β,ρ (u) ≪ 0, then either e ú Ω e u is concentrated around a finite number of points or Π I u is large (or both occur). To express this alternative we will use the join, which has been used in the variational study of Liouville system of two equations, where one has an alternative between the concentration of each component (see[6,4,20,7,5]). Given two topological spaces X and Y , its join X × Y is defined as the product between the two spaces and the unit interval, with identifications at each endpoint. We set
Introduction
We are interesting in the study of the following partial differential equation:
−∆u + βu = ρ e ú Ω e u − 1 |Ω| in Ω ∂ ν u = 0 on ∂Ω .
(P β,ρ )
Here, Ω ⊂ R 2 is a smooth bounded open domain in the plane, β and ρ are real parameters and |Ω| is the Lebesgue measure of Ω.
Problem (P β,ρ ) is related to a model introduced by Keller and Segel in [22] to study chemotaxis in biology, namely the movement of organisms according to the the presence of chemicals in the environment.
In particular, (P β,ρ ) models stationary solutions in Keller and Segel's model.
In the case β > 0, solutions to (P β,ρ ) have been found via a mountain-pass argument in [30], whereas families of blowing-up solutions have been constructed in [29,16,1,9,10].
Here we allow the parameter β to have any sign. We will tackle problem (P β,ρ ) variationally; in fact, solutions are all and only the critical points of the energy functional domain H 1 (Ω), but rather on its subspace
X ⋆ Y := X × Y × [0, 1] ∼ ,(1.3)
with ∼ being defined by (x, y, 0) ∼ (x, y ′ , 0) ∀ x ∈ X, ∀ y, y ′ ∈ Y, (x, y, 1) ∼ (x ′ , y, 1) ∀ x, x ′ ∈ X, ∀ y ∈ Y.
We will suitably choose X and Y as objects to model each alternative. When t = 0, the whole Y is collapsed, which means only the first alternative occurs, similarly at t = 1 only Y is left hence we have the second alternative; if 0 < t < 1 we see both spaces because both alternatives occur.
To be in position to apply such methods, we need some compactness assumptions on the energy functional J β,ρ .
Unfortunately, Palais-Smale condition is not known to hold for problem (P β,ρ ), nor for similar Liouville-type PDEs. Anyway, such problem can be by-passed thanks to a a deformation lemma by [23] and some compactness of solutions to (P β,ρ ) holding locally uniformly in ρ.
As in most known results, such a result holds true provided ρ is not an integer multiple of 4π. This time we also need −β not to be an eigenvalue of −∆, in order for the linear operator to be non-degenerate (see Section 2).
Finally, we need to verify that the solution found using these tools is not the trivial one u ≡ 0. This is equivalent to evaluate the Morse index of the trivial solution.
We will get non-trivial solutions under some algebraic condition involving the parameters β, ρ and the eigenvalues λ i . In the case when Ω is multiply connected, we are allowed to take more cases, since the topology of low sublevels J −L β,ρ is more involved. Precisely, the main result of this paper is the following:
Theorem 1.1. Assume β = −λ i = β − ρ |Ω|
for any i ∈ N and ρ ∈ 4πN and let I ≤ J, K be non-negative integers such that
4Kπ < ρ < 4(K + 1)π − λ I+1 < β < −λ I − λ J+1 < β − ρ |Ω| < −λ J .
If Ω is simply connected and 2K + I = J, then the problem (P β,ρ ) has non-trivial solutions.
If Ω is not simply connected and (K, I) = (0, J), then the problem (P β,ρ ) has non-trivial solutions.
Remark 1.2.
Since Theorem 1.1 is proved via Morse theory, then the same arguments would also give multiplicity of solutions, provided the energy functional J β,ρ is a Morse function, as was done with similar problems in [14,3,4,19,5,15]. However, it is not clear under which conditions on Ω this occurs, although we suspect it is a somehow generic conditions. One easily see that, assuming u ≡ 0 to be the only solution to (P β,ρ ), J β,ρ is Morse if and only if β and ρ satisfy an algebraic relation (see Proposition 4.1); this fact implies that the solution found in Theorem 1.1 is not trivial.
The same arguments as Theorem 1.1 are also useful, with minor modifications, to find a solution to
−∆u + βu = ρ he ú Σ he u − 1 |Σ| in Σ ∂ ν u = 0 on ∂Σ ,
with Σ being a compact surface with boundary and h ∈ C ∞ (Σ) being strictly positive.
Here, if h is not constant, we can also cover the case in which Theorem 1.1 gave a trivial solution. Moreover, by arguing as in the previous references, one can show that J β,ρ satisfies the Morse property under a generic assumption on h and/or on the metric g on Σ.
The plan of this paper is the following. Section 2 is devoted to the study of compactness properties of the equation (P β,ρ ); Section 3, which is divided in three sub-sections, deals with the analysis of energy sublevels of J β,ρ , and finally in Section 4 Theorem (1.1) is proved.
We will denote as d(x, y) the distance between two points x, y ∈ Ω and, similarly, for Ω ′ , Ω ′′ ⊂ Ω,
d(x, Ω ′ ) := inf{d(x, y) : y ∈ Ω ′ } d(Ω ′ , Ω ′′ ) := inf{d(x, y) : x ∈ Ω ′ , y ∈ Ω ′′ }.
We will denote as B r (x) the open ball of radius r centered at p. The symbol
Ω ′ f := 1 |Ω ′ |ˆΩ′ f
will stand for the average of f ∈ L 1 (Ω ′ ) on some Ω ′ ⊂ Ω. The letter C will denote large constant which may vary among different formulas and lines.
Compactness issues
This section is devoted to the proof of the following concentration-compactness result with quantization of blow-up limits:
Proposition 2.1.
Let (u n ) n∈N be a sequence of solutions to (P β,ρ ) with ρ n → n→+∞ ρ and −β = λ i for all i's.
Then, up to sub-sequences, one of the following alternatives occur:
(Compactness) (u n ) n∈N is compact in H 1 (Ω);
(Concentration) The blow-up set S, defined by
S := x ∈ Ω : ∃ x n → n→+∞ x such that u n (x n ) → n→+∞ +∞ ,
is non-empty and finite.
Moreover, ρ n e uń Ω e un ⇀ n→+∞ x∈S σ(x)δ x as measures, with σ(x) := lim r→0 lim n→+∞ ρ n´B r (x) e uń Ω e un = 4π if x ∈ S ∩ ∂Ω 8π if x ∈ S ∩ Ω . (2.1)
In particular, if ρ ∈ 4πN then (Compactness) occurs.
When Compactness occurs, a standard consequence is the following: since the set of solutions is compact, then its energy is uniformly bounded from above, hence the whole space H 1 (Ω) can be retracted on a suitable sublevel containing all solutions.
Corollary 2.2.
Assume ρ ∈ 4πN and −β = λ i for all i's. Then, there exists L > 0 such that J L β,ρ is a deformation retract of H 1 (Ω). In particular, it is contractible.
Proof.
Since we can write the energy functional as J β,ρ (u) =
u 2 H 1 (Ω) 2 − K 1 (u) − λK 2 (u)
, we are in position to apply Proposition 1.1 in [23]. From Proposition 2.1, there are no solutions u n to (P β,ρ ) with L ≤ J β,ρn ≤ L + 1, if L is large enough; therefore, arguing as in [23], J L β,ρ is a deformation retract of J L+1 β,ρ . Being L arbitrary, we find that J L β,ρ is a retract of the whole space H 1 (Ω).
Proposition 2.1 is rather classical for Liouville-type equations like (P β,ρ ). It was first given by [11] and, in the case of Neumann conditions, in [30]. With respect to the latter reference, the presence of the extra term βu in the linear part, which may cause the maximum principle to fail, can be dealt by just moving it to the right-hand side.
A key tool is a "minimal mass" lemma. The proof in [30] also works in the case f n ≡ 0. with (f n ) n∈N bounded in L q (Ω) for some q > 1 withˆΩ f n = 0 ∀n ∈ N.
Then, there exists σ 0 = σ 0 (Ω) > 0 such that if lim sup r→0 ρ n´B r (x) e vń Ω e vn ≤ σ 0 for all x ∈ Ω, then (u n ) n∈N is compact in H 1 (Ω).
Roughly speaking, the idea to prove Proposition 2.1 will be the following. If Concentration occurs, then we have blow-up at a finite number of points, thanks to Lemma 2.3. Then, the local mass at each blow-up point x ∈ S is found via a Pohožaev identity based on the asymptotic behavior of solution, and this fact excludes the presence of a residual mass.
Proof of Proposition 2.1.
Let (u n ) n∈N be a sequence of solutions with sup Ω u n ≤ C. Then, Jensen's inequality givesˆΩ e un ≥ |Ω|e ffl Ω un = |Ω|, therefore
| − ∆u n + βu n | ≤ ρ n e uń Ω e un + 1 |Ω| ≤ (ρ + 1) e C |Ω| + 1 |Ω|
is uniformly bounded, hence by standard regularity u n is bounded in W 2,2 (Ω) and compact in H 1 (Ω). Suppose now sup Ω u n → n→+∞ +∞, namely S = ∅. This time, we just have − ∆u n + βu n L 1 (Ω) ≤ 2(ρ + 1); since −β is not an eigenvalue of −∆, this gives ∇u n L q (Ω) + u n L q (Ω) ≤ C for any q < 2. Therefore, u n will solve (2.
2) with f n = −βu n ∈ L q (Ω), hence we can apply Lemma 2.3 to get
|S|σ 0 ≤ x∈Si σ(x) ≤ ρ.
This means that S is finite and we easily get ρ n e uń Ω e un ⇀ n→+∞ x∈S
σ(x)δ x + f for some f ∈ L 1 (Ω) ∩ L ∞ loc (Ω \ S), while u n → n→+∞ x∈S σ(x)G x + w in W 1,q (Ω) ∩ C 1,α loc (Ω \ S) for q < 2, α < 1, with G x and w solving, respectively, −∆G x + βG x = δ x − 1 |Ω| in Ω ∂ ν G x = 0 on ∂Ω −∆w + βw = f − Ω f in Ω ∂ ν w = 0 on ∂Ω .
We need to show that f ≡ 0, which will be done arguing as in [8] (Lemmas 2.2 and 2.3). If f ≡ 0, then one easily sees that
f = V e w , with V := ρ lim n→+∞ˆΩ e un =+∞ e x∈S σ(x)Gx ∈ L ∞ loc (Ω \ S) whereas, due to the behavior of G x , V ∼ | · −x| − σ(x) 2π if x ∈ S ∩ Ω and V ∼ | · −x| − σ(x) π if x ∈ S ∩ ∂Ω. Now, since −∆w ≥ −βw − Ω f ∈ L q (Ω), then w is bounded from below, namely V ≤ Cf ∈ L 1 (Ω),
which in particular means σ(x) < 4π for any x ∈ S. This contradicts (2.1), which can be deduced arguing as in [30] (Lemma 3.4), hence it must be f ≡ 0. Finally, if Concentration occurs then
ρ = lim n→+∞ˆΩ ρ n e uń Ω e un = lim n→+∞ˆΩ x∈S σ(x)δ x = x∈S σ(x) = x∈S∩∂Ω 4π + x∈S∩Ω 8π ∈ 4πN.
Analysis of sublevels
In this Section, which is the largest of the paper, we will study topologically the energy sublevels of J β,ρ .
In the first sub-section, we will introduce a topological space which will be later "compared" to energy sublevels, and we will compute some of its homology groups. Then, we will construct maps from this topological space to low sublevels and vice-versa and we will deduce that J −L β,ρ has nontrivial homology.
The space (Ω ∂ ) K,I and its homology
Let us introduce a set of barycenters on Ω, namely a set of finitely-supported probability measures on Ω. With respect to most previous works, we will not give a constraint on the cardinality of the support, basically because points in Ω and ∂Ω have to be treated differently, for reasons which will be discussed in the forthcoming sub-sections. Roughly speaking, points in the interior will count twice as much as points in the boundary.
(
Ω ∂ ) K := ⌊ K 2 ⌋ l=0 Ω l,K−2l Ω l,m := l k=1 t k δ x k + m k ′ =1 t ′ k ′ δ x ′ k ′ ; l k=1 t k + m k ′ =1 t ′ k ′ = 1, x k ∈ Ω, x ′ k ′ ∈ ∂Ω
On such spaces, we will consider the distance induced by the Lip ′ norm, that is the norm on the space of signed measures induced by duality with Lipschitz functions:
µ Lip ′ (Ω) := sup h∈Lip(Ω), h Lip(Ω) ≤1 ˆΩ hdµ .
As a first result, we see that such barycenters spaces are Euclidean deformation retracts. The proof has been given in [28] in the case when Ω is replaced by a 4-dimensional compact manifold with boundary, but the same proof holds in any dimension and in particular for planar domains. There exist ǫ 0 > 0 and a continuous retraction
Ψ : µ ∈ M Ω : d Lip ′ (Ω) (µ, (Ω ∂ ) K ) < ǫ 0 → (Ω ∂ ) K .
Among all the "layers" which compose (Ω ∂ ) K , a special role will be played by the first one, consisting of measures supported on ∂Ω. In [28] it was shown that it is a deformation retract within (Ω ∂ ) K ; again, their proof is also valid in our case.
Lemma 3.2. ([28], Proposition 4.5) For any K ≥ 1 the set Ω 0,K = (∂Ω) K ⊂ (Ω ∂ ) K is a deformation retract of some its open neighbor- hood U in (Ω ∂ ) K .
Since ∂Ω is homotopically equivalent to a disjoint union of g circles, we can use a result from [15] to compute the homology of Ω 0,K . The homology groups of Ω 0,K = (∂Ω) K are given by
H q ((∂Ω) m ) = Z ( g+q−K+1 g )( g 2K−q−1 ) max{K − 1, 2K − g − 1} ≤ q ≤ 2K − 1 0 q < max{K − 1, 2K − g − 1}, q > 2K − 1
The space (Ω ∂ ) K will be used in the analysis of sublevels to express the fact that, if J β,ρ (u) ≪ 0, then u may concentrates at a finite number of points. Anyway, it J β,ρ is very low, it may also happen that the projection Π I on the space of negative eigenvalues for −∆ + β, defined by (1.2), is very large in norm. Since Π I u ∈ R I , this naturally leads to consider, after a normalization, the sphere S I−1 to deal with phenomena. As anticipated in the introduction, the alternative between concentration and large Π I will be modeled by the join (1.3), therefore we will be interested in the following space:
(
Ω ∂ ) K,I = (Ω ∂ ) K ⋆ S I−1 . (3.1) Remark 3.4.
In [17], the authors used a space of the kind
X × B I−1 ∼ , with B I−1 indicating the unit ball in R I−1
and ∼ defined by (x, y) ∼ (x ′ , y) for any x, x ′ ∈ X and y ∈ S I−1 . Actually, this space is homeomorphic to the join X ⋆ S I−1 , with one possible homeomorphism given by the map
X ⋆ S I−1 ∋ (x, y, t) ←→ (x, ty) ∈ X × B I−1 ∼ .
Since we are interested in the homology of (Ω ∂ ) K,I , we will use a well-known result concerning the homology of a join. Then, its homology group are
H q (X ⋆ Y ) = q q ′ =0 H q ′ (X) ⊗ H q−q ′ −1 (Y ),
where H q denotes the reduced homology groups:
H q (X) = H q (X) ⊕ Z if q = 0 H q (X) if q ≥ 1 .
We are now able to get some information on the homology on (Ω ∂ ) K,I . In particular, we will compute its maximal dimensional homology group, which is not trivial.
Proposition 3.6. Let g be the genus of Ω.
The homology groups of (Ω ∂ ) K,I satisfy
H 2K+I−1 ((Ω ∂ ) K,I ) = Z ( K+g g ) .
Proof.
We start with the case I = 0. If K = 1, then (Ω ∂ ) 1 = ∂Ω so its homology is computed immediately; otherwise, we write (
Ω ∂ ) K = U ∪ V , with U as in Lemma 3.2 and V := (Ω ∂ ) K \ (∂Ω) K . The space V = ⌊ K 2 ⌋
l=0 Ω l,K−2l is a stratified set whose maximal dimension equals 2K − 3, and the same holds
true for U ∩ V , hence H q (U ∩ V ) = H(U ∩ V ) = 0 for any q ≥ 2K − 2.
Therefore, the Mayer-Vietoris exact sequence gives:
0 = H 2K−1 (U ∩ V ) → H 2K−1 (U ) ⊕ H 2K−1 (V ) → H 2K−1 ((Ω ∂ ) K ) → H 2K−2 (U ∩ V ) = 0, that is H 2K−1 ((Ω ∂ ) K ) = H 2K−1 (U ) ⊕ H 2K−1 (V ) = H 2K−1 ((∂Ω) K ) = Z ( K+g g ) .
Finally, if I ≥ 1, then Lemma 3.5 gives
H 2K+I−1 ((Ω ∂ ) K,I ) = H 2K+I−1 (Ω ∂ ) K ⋆ S I−1 = 2K+I−1 q ′ =0 H q ′ ((Ω ∂ ) K ) ⊗ H 2K+I−q ′ −2 S I−1 = H 2K−1 ((Ω ∂ ) K ) = Z ( K+g g ) .
Remark 3.7.
As pointed out by the referee, one can compute the Euler characteristic of (Ω ∂ ) K using the results in [21,2]:
χ((Ω ∂ ) K ) = χ Ω ⌊ K 2 ⌋,0 = 1 − 1 K 2 ! ⌊ K 2 ⌋ k=1 (k − χ(Ω)) = 1 − K 2 + g − 1 g − 1 . 3.2 The map Φ Λ : (Ω ∂ ) K,I → J −L β,ρ
In this Subsection we will build a map from the space (Ω ∂ ) K,I , whose properties have just been discussed, into a suitably low energy sublevel J −L β,ρ . Precisely, we will construct a family Φ Λ of maps with J β,ρ Φ Λ → Λ→+∞ −∞ uniformly, so that for Λ large enough the image of J β,ρ is contained in J −L β,ρ . The choice of L, hence of Λ, will be made in the following subsection. Consistently with the previous discussion, the family of test functions defined by Φ Λ will have the following property: as Λ goes to 0, either it concentrates at a finite number of points, according to the definition of (Ω ∂ ) K (if t = 1), or its projection Π I will be large (if t = 0). Proposition 3.8. Let (Ω ∂ ) K,I be defined by (3.1) and let Φ Λ : (Ω ∂ ) K,I → H 1 (Ω) be defined, for L ≫ 0, in the following way:
ζ = (µ, ς, t) = k t k δ x k , (ς 1 , . . . , ς I ), t −→ Φ Λ (ζ) := φ Λ(1−t) − Ω φ Λ(1−t) + ψ Λt φ Λ(1−t) = φ Λ(1−t) (µ) := log k t k (1 + (Λ(1 − t)) 2 | · −x k | 2 ) 2 ψ Λt = ψ Λt (ς) := log + (Λt) I i=1 ς i ϕ i If ρ > 4Kπ and β < −λ I , then J β,ρ Φ Λ (ζ) → Λ→+∞ −∞ independently on ζ.
To prove this result, we will estimate separately the three parts defining J β,ρ : the Dirichlet integral, the L 2 norm and the nonlinear term. Each estimate is contained in a separate lemma.
Lemma 3.9.
Let Φ Λ be as in Proposition 3.8. Then,ˆΩ
∇Φ Λ (ζ) 2 ≤ 16Kπ log + (Λ(1 − t)) + λ I log + (Λt) + C log + (Λt).
Proof.
Since, by definition, we have
∇Φ Λ (ζ) 2 = ∇φ Λ(1−t) 2 + 2∇φ Λ(1−t) · ∇ψ Λt + ∇ψ Λt 2 ,
we will suffice to show the following estimates:
Ω ∇φ Λ(1−t) 2 ≤ 16Kπ log + (Λ(1 − t)) + C; (3.2) Ω ∇φ Λ(1−t) · ∇ψ Λt ≤ C log + (Λt); (3.3) Ω ∇ψ Λt 2 ≤ λ I log + (Λt). (3.4)
The first estimate can be obtained similarly as [24] (Proposition 4.2), the main difference being that we have to take care of the points x k lying on ∂Ω. The estimate is trivial if Λ(1 − t) is bounded from above, otherwise we get:
∇φ Λ(1−t) = k −4t k (Λ(1−t)) 2 (·−x k ) (1+(Λ(1−t)) 2 |·−x k | 2 ) 3 k t k (1+(Λ(1−t)) 2 |·−x k | 2 ) 2 ≤ k 4t k (Λ(1−t)) 2 |·−x k | 2 (1+(Λ(1−t)) 2 |·−x k | 2 ) 3 k t k (1+(Λ(1−t)) 2 |·−x k | 2 ) ≤ max k 4(Λ(1 − t)) 2 | · −x k | 1 + (Λ(1 − t)) 2 | · −x k | 2 ≤ min 4Λ(1 − t), 4 min k | · −x k | .
Now, we divide Ω in some regions Ω k depending on which point x k is the closest:
Ω k := x ∈ Ω : |x − x k | = min k ′ |x − x k ′ | ; therefore, we get Ω ∇φ Λ(1−t) 2 ≤ kˆΩ k ∇φ Λ(1−t) 2 ≤ kˆΩ k \B 1 Λ(1−t) 16 | · −x k | 2 + kˆB 1 Λ(1−t) 16(Λ(1 − t)) 2 ≤ 16 kˆΩ k \B 1 Λ(1−t) 1 | · −x k | 2 + C.
To evaluate the last integral we distinguish the cases x k ∈ Ω and x k ∈ ∂Ω. In the former, we are basically integrating the function 1 | · | 2 on an annulus whose internal radius
1 Λ(1 − t)
is shrinking, plus negligible terms, hence its asymptotical value will be 2π log(Λ (1 − t)). On the other hand, if x k ∈ ∂Ω, then we are actually integrating on a domain asymptotically resembling a half-annulus with its internal radius shrinking, therefore we only get half of before, namely π log(Λ(1 − t)). Following these considerations, we get (3.2):
∇φ Λ(1−t) ≤ 16 x k ∈Ω (2π log(Λ(1 − t)) + C) + x k ∈∂Ω π(log(Λ(1 − t)) + C) + C ≤ 16K log(Λ(1 − t)) + C.
Concerning (3.3), by the construction of ψ Λ we get
Ω ∇φ Λ(1−t) · ∇ψ Λt = log + (Λt) I i=1 s iˆΩ ∇φ Λ(1−t) · ∇ϕ i = log + (Λt) I i=1 s i λ iˆΩ φ Λ(1−t) − Ω φ Λ(1−t) ϕ i ≤ C log + (Λt) I i=1 s i λ i ˆΩ φ Λ(1−t) − Ω φ Λ(1−t) 2 ˆΩ ϕ 2 i ≤ C log + (Λt) ˆΩ φ Λ(1−t) − Ω φ Λ(1−t) 2 ,
therefore we suffice to show that the last integral is uniformly bounded.
To this purpose, we first estimate the average of φ Λ(1−t) :
Ω φ Λ(1−t) = Ω log 1 (1 + (Λ(1 − t)) 2 min k | · −x k | 2 ) 2 + O(1) = Ω\ k B 1 Λ(1−t) log 1 (Λ(1 − t) min k | · −x k |) 4 + O(1) = −4 log + (Λ(1 − t)) + O(1).
(3.5) Now, (3.3) will follow by estimating φ Λ(1−t) + 4 log + (Λ(1 − t)) in L 2 (Ω), which can be done similarly as before:ˆΩ
φ Λ(1−t) + 4 log + (Λ(1 − t)) 2 =ˆΩ log k t k max{1, Λ(1 − t)} (1 + (Λ(1 − t)) 2 | · −x k | 2 ) 2 2 Ω log k t k | · −x k | 4 2 ≤ˆΩ log 1 min k | · −x k | 8 ≤ C. (3.6)
Finally, (3.4) follows easily by the properties of the φ i 's:
Ω ∇ψ Λt 2 = log + (Λt)ˆΩ I i=1 ς i ∇ϕ i 2 = log + (Λt) I i=1 ς 2 iˆΩ |∇ϕ i | 2 = log + (Λt) I i=1 ς 2 i λ i ≤ log + (Λt)λ I . Lemma 3.10.
Let Φ Λ be as in Proposition 3.8. Then,ˆΩ Φ Λ (ζ) 2 ≤ log + (Λt) + C log + (Λt).
Proof.
By expanding the square of the sum and using (3.5), (3.6), we havê
Ω Φ Λ (ζ) 2 =ˆΩ φ Λ(1−t) − Ω φ Λ(1−t) 2 + 2ˆΩ φ Λ(1−t) − Ω φ Λ(1−t) ψ Λt +ˆΩ ψ Λt 2 ≤ˆΩ φ Λ(1−t) − Ω φ Λ(1−t) 2 + 2 ˆΩ φ Λ(1−t) − Ω φ Λ(1−t) 2 ˆΩ (ψ Λt ) 2 +ˆΩ ψ Λt 2 ≤ C 1 + ˆΩ (ψ Λt ) 2 +ˆΩ ψ Λt 2 .
Therefore, we only need a suitable estimate for ψ Λt , which in turn follows from the very definition of the ϕ i 's:
Ω ψ Λt 2 = log + (Λt)ˆΩ I i=1 ς i ϕ i 2 = log + (Λt) I i=1 ς 2 iˆΩ ϕ 2 i = log + (Λt) I i=1 ς 2 i ≤ log + (Λt).
By combining the two estimates the proof is complete.
Lemma 3.11.
Let Φ Λ be as in Proposition 3.8. Then, logˆΩ e Φ Λ (ζ) ≥ 2 log + (Λ(1 − t)) − C log + (Λt).
Proof.
We first notice that, since ψ Λt belongs to a finite-dimensional space, all of its norms are equivalent, and in particular L 2 and L ∞ , therefore by Lemma (3.10),
ψ Λt ≤ ψ Λt L ∞ (Ω) ≤ C ψ Λt L 2 (Ω) ≤ C log + (Λt).
Moreover, in view of the asymptotical behavior (3.5) of the average of φ λ(1−t) , we are reduce to show that logˆΩ e φ Λ(1−t) ≥ −2 log + (Λ(1 − t)) − C; this follows from the simple calculations:
Ω e φ Λ(1−t) = k t kˆΩ 1 (1 + (Λ(1 − t)) 2 | · −x k | 2 ) 2 ≥ k t kˆB 1 Λ(1−t) (x k ) 1 (1 + (Λ(1 − t)) 2 | · −x k | 2 ) 2 ≥ k t kˆB 1 Λ(1−t) (x k ) 1 2 ≥ C max{1, Λ(1 − t)} 2 .
By putting together these three lemmas, Proposition (3.8) may be proved easily.
Proof of Proposition 3.8. By Lemmas 3.9, 3.10, 3.11, we get
J β,ρ Φ Λ (ζ) = 1 2ˆΩ ∇Φ Λ (ζ) 2 + β 2ˆΩ Φ Λ (ζ) 2 − ρ logˆΩ e Φ Λ (ζ) ≤ (8Kπ − 2ρ) log + (Λ(1 − t)) + λ I + β 2 log + (Λt) + C log + (Λt) ≤ − min 2ρ − 8Kπ, − λ I + β 2 max log + (Λ(1 − t)), log + (Λt) + C log + (Λt) ≤ − min 2ρ − 8Kπ, − λ I + β 2 log Λ 2 + C log Λ → Λ→+∞ −∞,
uniformly on ζ ∈ (Ω ∂ ) K,I .
The map
Ψ : J −L β,ρ → (Ω ∂ ) K,I
We will now show the existence of "counterpart" to the map Φ defined in the previous subsection. Precisely, we will build a map Ψ from a low sub-level J −L β,ρ to (Ω ∂ ) K,I which is somehow compatible with Φ, in the sense that their composition is homotopically equivalent to the identity on (Ω ∂ ) K,I . The existence of such maps Φ and Ψ easily gives, via the functorial properties of homology, the following information on the homology groups of energy sublevels.
Corollary 3.13.
Under the assumptions of Propositions 3.8 and 3.12, the homology groups of the sublevel J −L β,ρ satisfy
Z ( K+g g ) ֒→ H 2K+I−1 J −L β,ρ .
The main tool to prove Proposition 3.12 is a so-called improved Moser-Trudinger inequality. Roughly speaking, such inequalities state that, under some spreading conditions on u, the best constant in the classical Moser-Trudinger inequality can be improved. We recall here the well-known Moser-Trudinger inequalities, in two forms depending whether we consider only compactly supported function or also function which may touch the boundary. We stress that, in the two cases, the constant multiplying the Dirichlet integral is different. If instead u ∈ H 1 0 (Ω), then logˆΩ e u ≤ 1 16πˆΩ |∇u| 2 + C.
(3.8)
We will now prove the improved Moser-Trudinger inequality, a classical result in variational Liouvilletype problems (see [17,24,12,6,4,5]). Basically, if u is somehow spread in some regions, then the constant 8π in (3.7) can almost be multiplied by an integer number. This time, the integer will depend not only on the number of regions but also on how many of them touch the boundary; moreover, we need to take account of the negative projection Π I . To prove such a result, we will take cutoff functions on the regions where u is spread and apply to each cutoff either (3.7) or (3.8). We will also use some splitting in Fourier modes, which we need to deal with Π I .
Lemma 3.15. Let δ > 0, {Ω 1k } l k=1 , {Ω 2k ′ } m k ′ =1 and u ∈ H 1 (Ω) satisfying d(Ω ik , Ω i ′ k ′ ) ≥ 2δ ∀ (i, k) = (i ′ , k ′ ); d(Ω 1k , ∂Ω) ≥ δ ∀ k = 1, . . . , l; Ω ik e ú Ω e u ≥ δ ∀ i, k; Π I u ≤ 1.
Then, for any ε > 0 there exists C = C(ε, δ, β, I, l, m) > 0 such that
logˆΩ e u ≤ 1 + ε 8π(2l + m)ˆΩ |∇u| 2 + βu 2 + C.
Proof.
First of all, for any i, k we take cutoff functions η ik ∈ Lip Ω satisfying
0 ≤ η ik ≤ 1 in Ω ik η ik | Ω ik ≡ 1 spt(η ik ) ⊂ B δ (Ω ik ) |∇η ik | ≤ 1 δ in B δ (Ω ik )\Ω ik .
Then, we split u = u 1 + u 2 + u 3 via truncation in Fourier modes:
u 1 = I i=1 ˆΩ uϕ i ϕ i u 2 = Nε−1 i=I+1 ˆΩ uϕ i ϕ i u 3 = +∞ i=Nε ˆΩ uϕ i ϕ i ,
with N ε so large that
1 λ Nε 1 + 1 ε 1 δ 2 ≤ ε λ Nε ≤ (1 + ε)(λ Nε + β). (3.9)
By applying (3.7) to each η 1k u − η 1k u we get:
logˆΩ e u ≤ logˆΩ 1k e u + log 1 δ ≤ logˆΩ e η 1k u + log 1 δ ≤ η 1k u 1 L ∞ (Ω) + η 1k u 2 L ∞ (Ω) + logˆΩ e η 1k u3 + log 1 δ ≤ u 1 L ∞ (Ω) + u 2 L ∞ (Ω) + η 1k u 3 + 1 8πˆΩ |∇(η 2k u 3 )| 2 + C.
Similarly, since η 2k u ∈ H 1 0 (Ω), we can apply (3.8):
logˆΩ e u ≤ logˆΩ 2k e u + log 1 δ ≤ logˆΩ e η 2k u + log 1 δ ≤ η 2k u 1 L ∞ (Ω) + η 2k u 2 L ∞ (Ω) + logˆΩ e η 2k u3 + log 1 δ ≤ u 1 L ∞ (Ω) + u 2 L ∞ (Ω) + 1 16πˆΩ |∇(η 1k u 3 )| 2 + C.
The terms involving u 1 and u 2 can be estimated because, since each belongs to a finite-dimensional space, all of its norms are equivalent on the respective space. We can use the L 2 norm for u 1 , which is uniformly bounded by hypotheses:
u 1 L ∞ (Ω) ≤ C u 1 L 2 (Ω) = C Π I u ≤ C.
As for u 2 , since we got rid of low Fourier coefficients, we can choose as a norm ˆΩ (|∇u 2 | 2 + βu 2 2 ); since we took an orthogonal decomposition, we get
u 2 L ∞ (Ω) ≤ C ˆΩ (|∇u 2 | 2 + βu 2 2 ) ≤ εˆΩ |∇u 2 | 2 + βu 2 2 + C ≤ εˆΩ |∇u 2 | 2 + βu 2 2 + εˆΩ |∇u 3 | 2 + βu 2 3 + εˆΩ |∇u 1 | 2 + C = εˆΩ |∇u| 2 + βu 2 − εβ Π I u 2 + C ≤ εˆΩ |∇u| 2 + βu 2 + C.
We then estimate the average of η 1k u 3 via Poincaré-Wirtinger inequality:
|η 1k u 3 | ≤ u 3 L 1 (Ω) ≤ ∇u 3 L 2 (Ω) ≤ εˆΩ |∇u 3 | 2 + C.
Concerning the last term, we expand the square and use the properties of the η ik 's:
Ω |∇(η ik u 3 )| 2 =ˆΩ |η ik ∇u 3 + u 3 ∇η ik | 2 ≤ (1 + ε)ˆΩ η 2 ik |∇u 3 | 2 + 1 + 1 ε ˆΩ u 2 3 |∇η ik | 2 ≤ (1 + ε)ˆB δ (Ω ik ) |∇u 3 | 2 + 1 + 1 ε 1 δ 2ˆB δ (Ω ik ) u 2 3 .
Since by hypothesis
B δ (x ik ) ∩ B δ (x i ′ k ′ ) = ∅ for (i, k) = (i ′ , k ′ )
, then putting together all these estimates and summing on i = 1, 2 and all k's we get (2l+m)ˆΩ e u ≤ (2l+m)εˆΩ |∇u| 2 + βu 2 +mεˆΩ |∇u 3 | 2 + 1 + ε 8πˆΩ
|∇u 3 | 2 + 1 8π 1 + 1 ε 1 δ 2ˆΩ u 2 3 +C.
At this point, we need the conditions (3.9) defining u 3 : the former gives
1 + 1 ε 1 δ 2ˆΩ u 2 3 ≤ 1 λ Nε 1 + 1 ε 1 δ 2ˆΩ |∇u 3 | 2 ≤ εˆΩ |∇u 3 | 2 .
on the other hand, the latter implieŝ
Ω |∇u 3 | 2 = +∞ j=Nε λ i ˆΩ uϕ i 2 ≤ (1 + ε) +∞ j=Nε (λ i + β) ˆΩ uϕ i 2 = (1 + ε)ˆΩ |∇u 3 | 2 + βu 2 3 ≤ (1 + ε)ˆΩ |∇u 3 | 2 + βu 2 3 + (1 + ε)ˆΩ |∇u 1 | 2 + (1 + ε)ˆΩ |∇u 2 | 2 + βu 2 2 ≤ (1 + ε)ˆΩ |∇u| 2 + βu 2 + C.
Therefore, we get
(2l + m)ˆΩ e u ≤ (2l + m)εˆΩ |∇u| 2 + βu 2 + 1 + ε 8π + ε 8π + mε ˆΩ |∇u 3 | 2 + C ≤ (2l + m)ε + (1 + ε) 1 + ε 8π + ε 8π + mε ˆΩ |∇u| 2 + βu 2 + C,
which, up to re-labeling ε, concludes the proof.
We need a few more technical steps to prove Proposition 3.12.
First of all, we need a covering lemma basically saying that, if concentration does not occur, then one has spreading in the sense of Lemma 3.15.
Lemma 3.16.
Let f ∈ L 1 (Ω) be non-negative a.e., satisfyingˆΩ f = 1 and such that, for any ε > 0 and
x 11, , . . . , x 1l ∈ Ω, x 21 , . . . , x 2m ∈ ∂Ω with 2l + m ≤ K for some K ∈ N, i,k Bε(x ik ) f < 1 − ε.
Then, there exist ε = ε(ε, Ω), r = r(ε, Ω) > 0 and x 11 , . . . , x 1 l , x 21 , . . . , x 2 m satisfying
2 l + m ≥ K + 1; d( x 1k , ∂Ω) ≥ r ∀ k = 1, . . . , l | x ik − x i ′ k ′ | ≥ 4 r ∀ (i, k) = (i ′ , k ′ )ˆB r ( x ik ) f ≥ ε ∀ i, k
Proof.
We will mostly argue as in [17] (Lemma 2.3) and [24] (Lemma 3.3), with minor modifications. Fix r := ε 6 and take the finite cover of Ω given by {B ε (y n )} N n=1 for some L = L r,Ω , then set ε := ε L .
One easily sees that there exists some n such thatˆB
r (yn)
f ≥ ε; up to re-labeling, we can assume that this hold true if and only if n ≤ N ′ , for some N ′ ≤ N . Now, we choose recursively the points { y j } ⊂ {y n } N ′ n=1 : we set y 1 := y 1 and
Ω 1 := N ′ n=1 B r (y n ) : |y n − y 1 | < 4 r ⊂ B 5 r ( y 1 ).
If there is some l 0 such that |y n0 − y 1 | ≥ 4 r, then we set y 2 = y n0 and
Ω 2 := N ′ n=2 B r (y n ) : |y n − y 2 | < 4 r ⊂ B 5 r ( y 2 ).
Inductively, we find a finite number of points y j and closed set Ω j . Among the y j 's, some of them will be at a distance less than δ from ∂Ω; we denote the number of such points as m and the number of the other y j 's as l, then we denote the former points as x 1k and the latter as x 2k ′ , so that { y j } j = x 11 , . . . , x 1 l , x 21 , . . . , x 2 m and we call Ω ik the set Ω j corresponding to the point x ik . To complete the proof, we only need to show that 2 l + m > K, since we already verified that the other required properties are satisfied. Assume by contradiction that 2 l+ m ≤ K, set x 1k := x 1k for k = 1, . . . , l and take, for k = 1, . . . , m, some x 2k ′ ∈ ∂Ω such that d (x 2k ′ , x 2k ) ≤ r. Then, by hypothesis,
Ω\ i,k Bε(x ik ) f ≥ ε.
However, due to our construction,
N ′ n=1 B r (y n ) ⊂ i,k Ω ik ⊂ i,k B 5 r ( x ik ) ⊂ i,k B ε (x ik ),
which leads to a contradiction:
Ω\ i,k Bε(x ik ) f ≤ˆΩ \ N ′ n=1 B r (yn) f ≤ˆ N ′ n=1 B r (yn) f ≤ (N − N ′ ) ε < ε
Now we see that either concentration at a finite number of points or large Π I must occur in very low sublevels. In fact, if it does not, then by the previous lemma one has spreading and small Π I , hence by the improved Moser-Trudinger inequality the energy J β,ρ cannot be too low. This is explained in details by the following lemma.
Lemma 3.17.
For any ε > 0 there exists L = L(ε) > 0 such that, if J β,ρ (u) ≤ −L and Π I u > 1, then there exists x 11 , . . . , x 1l ∈ Ω, x 21 , . . . , x 2m ∈ ∂Ω with 2l + m ≤ K such that
i,k Bε(x ik ) e ú Ω e u ≥ 1 − ε.
Proof.
Assume that the statement is not true. Then, there exists ε > 0 and (u n ) n∈N such that Π I u n ≤ 1,
J β,ρ (u n ) → n→+∞ −∞ and´ i,k Bε(x ik ) e uń Ω e un < 1 − ε.
for any x 11 , . . . , x 1l ∈ Ω, x 21 , . . . , x 2m ∈ ∂Ω satisfying 2l + m ≤ K.
We apply lemma 3.16 to f = e uń Ω e un and we find ε, r, not depending on n, and x 11 , . . . , x 1 l , x 21 , . . . , x 2 m as in the lemma. One can easily see that Lemma 3.15 can be applied to Ω ik = B r ( x ik ) with δ = min{ ε, r} and ε ′ = 4π ρ 2 l + m − 1. This leads to the following contradiction:
−∞ ← n→+∞ J β,ρ (u n ) = 4π 2 l + m 1 + ε ′ 1 + ε ′ 8π 2 l + m ˆΩ |∇u n | 2 + βu 2 n − ρ logˆΩ e un ≥ −C
Since e ú Ω e u tends to concentrates in very low sublevels, provided Π I is not too large, then it will be very close to an element of (Ω ∂ ) K . This will be essential to later use the retraction Ψ defined in Lemma 3.1.
Proof of Proposition 3.12.
Take ε 0 as in Lemma 3.1 and L = L(ε 0 ) as in Lemma 3.17. We define the map Ψ : J −L β,ρ → (Ω ∂ ) K,I as Ψ(u) = (µ(u), ς(u), t(u)) := Ψ e ú Ω e u ,
Π I u Π I u , min{1, Π I u } .
We need to verify that it is well-posed, namely that µ(u) is well-defined if t = 1 and ς(u) is welldefined if t = 0.
Assume t = 1: this means Π I u < 1 so, since J β,ρ (u) ≤ −L, Lemma 3.18 will give d Lip ′ (Ω) e ú Ω e u , (Ω ∂ ) K ≤ ε 0 ; hence, Lemma 3.1 ensures that Ψ is well-defined, hence µ(u) is.
On the other hand, if t = 0, then Π I u = 0, hence one can define Π I u Π I u .
As for second part of the lemma, consider the map Φ := Φ Λ0 as defined in Proposition 3.8, with
Λ 0 ≫ 1 so large that Φ Λ0 ((Ω ∂ ) K,I ) ⊂ J −L β,ρ .
To get a homotopical equivalence, we let Λ go to +∞. One immediately sees that
e φ Λ(1−t) (µ) Ω e φ Λ(1−t) (µ) ⇀ Λ→+∞
µ for any µ ∈ (Ω ∂ ) K and t = 1, hence being e ψ Λt negligible with respect toˆΩ e φ Λ(1−t) (see proof of Lemma 3.11) one also has e Φ Λ (ζ)
Ω e Φ Λ (ζ) ⇀ Λ→+∞ µ. Similarly, since φ Λ(1−t) − Ω φ Λ(1−t) is bounded
in L 2 (Ω), its projection will be negligible with respect to ψ Λt , therefore Π I Φ Λ → Λ→+∞ ς as long as t = 0. The scalar parameter t in the join will be more delicate to handle, because by the proof of Lemma 3.10 one gets t Φ Λ (ζ) ∼ min 1, log + (Λt) ; moreover, it is forced to be either 0 or 1 if either element in the join is not defined. Therefore, before letting Λ go to +∞, we need to properly rescale such a parameter, taking into account when it is allowed to be different from 0 and/or 1. To this purpose, we notice that, since e φ Λ(1−t) (µ) Ω e φ Λ(1−t) (µ) gets closer to (Ω ∂ ) K,I as Λ(1 − t) is larger, we can assume that µ Φ Λ (ζ) is welldefined for Λ(1 − t) ≥ Λ 0 3 and similarly that t Φ Λ (ζ) is well-defined for Λt ≥ Λ 0 3 . Therefore, we will construct an intermediate parameter t ′ , which we set to be either 0 or 1 if t is outside the previous range, which fills the whole (0, 1) as Λ goes to +∞: as a first step, we fix Λ 0 and interpolate linearly between t Φ Λ (ζ) and t ′ , and everything is well defined because Λ = Λ 0 is fixed; then, we pass to the limit as Λ → +∞ and, by the previous considerations, it is still well-posed and in the limit we recover the identity map. Precisely, a continuous homotopical equivalence between Ψ • Φ Λ0 and Id (Ω ∂ )K,I is given by
F (ζ, s) = µ Φ Λ0 (ζ) , ς Φ Λ0 (ζ) , (1 − 2s)t Φ Λ0 (ζ) + 2st ′ (t, 1) if 0 ≤ s < 1 2 µ Φ Λ 0 2−2s (ζ) , ς Φ Λ 0 2−2s (ζ) , t ′ (t, 2 − 2s) if 1 2 ≤ s < 1 ζ if s = 1 with t ′ (t, r) := 0 if t < r 3 3t − r 3 − 2r if r 3 ≤ t ≤ 1 − r 3 1 if t > 1 − r 3
Remark 3.19.
All the result shown in this section hold true, also when K and/or I equals zero. In each case, the space (Ω ∂ ) K,I is replaced by
X = (Ω ∂ ) K if I = K = 0 S I−1 if K = I = 0 ∅ if I = K = 0 .
One can easily see that all the proofs are still valid in all these cases. When I = 0, in Proposition 3.8 we just consider Φ Λ (µ, −, 0) and when K = 0 we take Φ λ (−, ς, 1); in Proposition 3.12 we just set F (ζ, s) = Φ • Φ Λ 1−s . If I = K = 0, then Propositions 3.8 and 3.12 make no sense, but Lemma 3.15 applied with l = 0, m = 1, Ω 21 = Ω implies that J β,ρ is coercive, namely J −L β,ρ = ∅ for L large. For this reason, the proof of Theorem 1.1 can be adapted also to this case.
Proof of the main result
We need one last lemma concerning the Morse property of the functional J β,ρ . Then J β,ρ is a Morse functional and the Morse index J of the trivial solution u ≡ 0 is such that
λ J+1 < β − ρ |Ω| < λ J .
which contradicts Proposition 3.6. In fact, if 2K +I = J, Corollary 3.13 gives a non-trivial homology group for q = 2K + I − 1 = J − 1; moreover, if Ω is not simply connected and K > 0, then even when 2K + I = J we get a bigger homology group: Z Z ( K+g g ) ֒→ H J−1 J −L β,ρ
Lemma 2. 3 .
3([30], Lemma 3.2) Let (v n ) n∈N be a sequence of solutions to −∆u n = ρ n e uń Ω e un − 1 |Ω| + f n in Ω ∂ ν u n = 0 on ∂Ω (2.2)
Lemma 3. 1 .
1([28], Lemma 4.10)
Lemma 3.3. ([15], Proposition 5.1)
Lemma 3. 5 .
5([18], Theorem 3.21) Let X and Y be two CW-complexes and X ⋆ Y their join as defined by (1.3).
Let (Ω ∂ ) K,I be defined by (3.1). If ρ < 4(K + 1)π and β > −λ I+1 , then there exists L ≫ 0 and a map Ψ : J −L β,ρ → (Ω ∂ ) K,I . Moreover, if ρ > 4Kπ and β < −λ I , then there exists a map Φ : (Ω ∂ ) K,I → J −L β,ρ such that the composition Ψ • Φ is homotopically equivalent to the identity on (Ω ∂ ) K,I .
Proposition 3 .
314. ([27], Theorem 2, [13], Corollary 2.5) There exists C > 0 such that for any u ∈ H 1 (Ω) logˆΩ e u ≤ 1 8πˆΩ |∇u| 2 + C. (3.7)
Assume (P β,ρ ) has no non-trivial solutions and β − ρ |Ω| = −λ j for any j ∈ N.
AcknowledgmentsThe author wishes to thank Professor Angela Pistoia for the discussions concerning the topics of the paper.Lemma 3.18.For any ε > 0 there exists L = L(ε) > 0 such that any u ∈ J −L β,ρ satisfies either of the following condition:Proof.Fix ε > 0, apply Lemma 3.17 with ε 3 and take L = L ε 3 as in the lemma. For any u ∈ J −L β,ρ satisfying Π I u ≤ 1, take µ(u) = ik t ik δ x ik with x ik as in the lemma and t ik defined byTo conclude the proof, we suffice to show thatWe split the integral between the union of the balls of radius ε 3 and its complement: on the latter,On the union of balls, we have:The proof is now complete.We are now in condition to prove the main result of this subsection. We will construct Ψ : J −L β,ρ → (Ω ∂ ) K,I in the following way. The element in (Ω ∂ ) K will be given by the retraction Ψ, while the element in S I−1 is just the normalization of Π I u ∈ R I . The choice of the third parameter in the join will be more delicate, especially in the homotopy, because we need to be sure that everything is well-defined outside the endpoints of the interval.Proof.One immediately sees that the second derivative of J β,ρ is given byhence in u ≡ 0 its quadratic form isAssume the only solution to (P β,ρ ) is the trivial one. Then, the J β,ρ is a Morse functional if and only if the previous quadratic form is nondegenerate. One immediately sees that this depends on the relative position of ρ |Ω| − β and the λ j (Ω)'s as we required.Now we are finally in position to prove the main result of the paper.Proof of Theorem 1.1. Assume, by contradiction, that u ≡ 0 is the only solution to (P β,ρ ). By Lemma 4.1, J β,ρ is a Morse functional and the Morse index of the solution is J, therefore by Morse theory the relative homology of sublevels satisfiesany L > 0. Moreover, by Corollary 2.2, J L β,ρ is contractible if L is large enough; therefore, the exactness of the sequence (see[18], Theorem 2.13 and Proposition 2.
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[
"Factorization Machines Leveraging Lightweight Linked Open Data-enabled Features for Top-N Recommendations",
"Factorization Machines Leveraging Lightweight Linked Open Data-enabled Features for Top-N Recommendations"
] |
[
"Guangyuan Piao [email protected] \nInsight Centre for Data Analytics\nNational University of Ireland Galway IDA Business Park\nLower DanganGalwayIreland\n",
"John G Breslin [email protected] \nInsight Centre for Data Analytics\nNational University of Ireland Galway IDA Business Park\nLower DanganGalwayIreland\n",
"Guangyuan Piao \nIntroduction\n\n",
"John G Breslin \nIntroduction\n\n"
] |
[
"Insight Centre for Data Analytics\nNational University of Ireland Galway IDA Business Park\nLower DanganGalwayIreland",
"Insight Centre for Data Analytics\nNational University of Ireland Galway IDA Business Park\nLower DanganGalwayIreland",
"Introduction\n",
"Introduction\n"
] |
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With the popularity of Linked Open Data (LOD) and the associated rise in freely accessible knowledge that can be accessed via LOD, exploiting LOD for recommender systems has been widely studied based on various approaches such as graph-based or using different machine learning models with LOD-enabled features. Many of the previous approaches require construction of an additional graph to run graphbased algorithms or to extract path-based features by combining useritem interactions (e.g., likes, dislikes) and background knowledge from LOD. In this paper, we investigate Factorization Machines (FMs) based on particularly lightweight LOD-enabled features which can be directly obtained via a public SPARQL Endpoint without any additional effort to construct a graph. Firstly, we aim to study whether using FM with these lightweight LOD-enabled features can provide competitive performance compared to a learning-to-rank approach leveraging LOD as well as other well-established approaches such as kNN-item and BPRMF. Secondly, we are interested in finding out to what extent each set of LOD-enabled features contributes to the recommendation performance. Experimental evaluation on a standard dataset shows that our proposed approach using FM with lightweight LOD-enabled features provides the best performance compared to other approaches in terms of five evaluation metrics. In addition, the study of the recommendation performance based on different sets of LOD-enabled features indicate that propertyobject lists and PageRank scores of items are useful for improving the performance, and can provide the best performance through using them together for FM. We observe that subject-property lists of items does not contribute to the recommendation performance but rather decreases the performance.
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10.1007/978-3-319-68786-5_33
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"https://arxiv.org/pdf/1707.05651v2.pdf"
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1707.05651
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e35ce9b2c8ce6b4a33f92720b3c0a4c3ef911a5c
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Factorization Machines Leveraging Lightweight Linked Open Data-enabled Features for Top-N Recommendations
28 Jul 2017
Guangyuan Piao [email protected]
Insight Centre for Data Analytics
National University of Ireland Galway IDA Business Park
Lower DanganGalwayIreland
John G Breslin [email protected]
Insight Centre for Data Analytics
National University of Ireland Galway IDA Business Park
Lower DanganGalwayIreland
Guangyuan Piao
Introduction
John G Breslin
Introduction
Factorization Machines Leveraging Lightweight Linked Open Data-enabled Features for Top-N Recommendations
28 Jul 2017Linked DataRecommender SystemDBpediaFactoriza- tion Machines
With the popularity of Linked Open Data (LOD) and the associated rise in freely accessible knowledge that can be accessed via LOD, exploiting LOD for recommender systems has been widely studied based on various approaches such as graph-based or using different machine learning models with LOD-enabled features. Many of the previous approaches require construction of an additional graph to run graphbased algorithms or to extract path-based features by combining useritem interactions (e.g., likes, dislikes) and background knowledge from LOD. In this paper, we investigate Factorization Machines (FMs) based on particularly lightweight LOD-enabled features which can be directly obtained via a public SPARQL Endpoint without any additional effort to construct a graph. Firstly, we aim to study whether using FM with these lightweight LOD-enabled features can provide competitive performance compared to a learning-to-rank approach leveraging LOD as well as other well-established approaches such as kNN-item and BPRMF. Secondly, we are interested in finding out to what extent each set of LOD-enabled features contributes to the recommendation performance. Experimental evaluation on a standard dataset shows that our proposed approach using FM with lightweight LOD-enabled features provides the best performance compared to other approaches in terms of five evaluation metrics. In addition, the study of the recommendation performance based on different sets of LOD-enabled features indicate that propertyobject lists and PageRank scores of items are useful for improving the performance, and can provide the best performance through using them together for FM. We observe that subject-property lists of items does not contribute to the recommendation performance but rather decreases the performance.
Introduction
The term Linked Data, indicates a new generation of technologies responsible for the evolution of the current Web from a Web of interlinked documents to a Web of interlinked data [8]. Thanks to the Semantic Web's growth and the more recent Linked Open Data (LOD) initiative [1], a large amount of RDF 1 data has been published in freely accessible datasets. These datasets are connected with each other to form the so-called Linked Open Data cloud 2 . DBpedia [16] which is a 1st-class citizen in this cloud, has become one of the most important and interlinked datasets on the LOD cloud. DBpedia provides cross-domain background knowledge about entities which can be accessible via its SPARQL Endpoint 3 . For example, Figure 1 shows pieces of background knowledge about the movie dbr 4 :The Godfather in RDF triples, which can be obtained from DBpedia. A RDF triple consists of a subject, a property and an object. As we can see from the figure, there can be incoming knowledge, e.g., dbr:Carlo Savina→dbo 5 :knownFor→dbr:The Godfather where dbr:The Godfather is used as an object, as well as outgoing knowledge such as dbr:The Godfather→dbo:director→dbr:Francis Ford Coppola where dbr:The Godfather is a subject. In the context of the great amount of freely accessible information, many researches have been conducted in order to consume the knowledge provided by LOD for adaptive systems such as recommender systems [2,6]. There have been many approaches for LOD-enabled recommender systems (LODRecSys) such as semantic similarity/distance measures, graph-based approaches, and learning-to-rank approaches by consuming LOD-enabled features.
Some previous studies compared their LODRecSys approaches against wellestablished collaborative filtering approaches such as kNN and matrix factorization models such as BPRMF [30], and have shown the benefits of consuming background knowledge powered by LOD. On the other hand, matrix factorization models such as BPRMF, which do not exploit LOD-enabled features, have shown competitive performance even compared to some LODRecSys approaches [17,20]. This has in turn motivated us to investigate factorization models consuming LOD-enabled features.
In this paper, we investigate the use of Factorization Machines (FMs), which can mimic other well-known factorization models such as matrix factorization, by leveraging LOD-enabled features. Previous works require increased effort to maintain an additional graph based on user-item interactions and background knowledge about items from LOD in their approaches (We will discuss this in detail in Section 2). In this work, we especially focus on lightweight LODenabled features for FM. We define lightweight LOD features as features that can be directly obtained via a public SPARQL Endpoint.
The contributions of this work are summarized as follows.
-We investigate lightweight LOD-enabled features, which can be directly obtained via the public DBpedia Endpoint, for FM to provide the top-N recommendations. Therefore, there is no need to construct a graph which combines user-item interactions (e.g., likes, dislikes) and background knowledge about items. In addition, we investigate to what extent different sets of these features contribute to FM in terms of recommendation performance.
-We comprehensively evaluate our approach by comparing it to other approaches such as PopRank, kNN, BPRMF, and a state-of-the-art LODRec-Sys approach SPRank [20] in terms of five different evaluation metrics.
The organization of the rest of the paper is as follows. Section 2 gives some related work, and Section 3 describes our proposed approach using FM with lightweight LOD-enabled features. In Section 4, we describe our experimental setup including the dataset and evaluation metrics. Experimental results are presented in Section 5. Finally, Section 6 concludes the paper with some brief ideas for future work.
Related Work
The first attempts to leverage LOD for recommender systems were by [10,25]. Heitmann et al. [10] proposed a framework using LOD for open collaborative recommender systems. The Linked Data Semantic Distance (LDSD) measure [25] was one of the first works to use LOD for recommender systems in the music domain [24]. This distance measure considers direct links between two entities/nodes. In addition, it also considers that the same incoming and outgoing nodes via the same properties of two nodes in a graph such as DBpedia. Piao et al. [26,27] extended LDSD by investigating different normalization strategies for the paths between two entities. These measures have been designed to work directly on LOD without considering the collaborative view of users. Based on the nature of the graph structure of DBpedia, graph-based approaches have been proposed [17,19]. For instance, Musto et al. [17] presented apersonalized PageRank algorithm [7] using LOD-enabled features for the top-N recommendations. Nguyen et al. [19] investigated SimRank [12] and PageRank, and their performance for computing similarity between entities in RDF graphs and investigated their usage to feed a content-based recommender system. Di Noia et al. [3] adapted the Vector Space Model (VSM) to a LOD-based setting, and represented the whole RDF graph as a matrix. On top of the VSM representation, they used the Support Vector Machine (SVM) as a classifier to predict if a user would like an item or not. Using the same representation, they also proposed to assign a weight to each property that represents its worth with respect to the user profile [4]. In this regard, they used a Genetic Algorithm (GA) to learn the weights of properties that minimize the misclassification errors. More recently, Di Noia et al. [20,22] proposed SPRank, which is a semantic path-based approach using learning-to-rank algorithms. This approach first constructed a graph based on user-item interactions and thebackground knowledge of items from LOD. Afterwards, features, called semantic paths, were extracted based on the number of paths between a user and an item with min-max normalization. The extracted features were then fed into existing learning-to-rank algorithms such as LMART [33] provided by RankLib 6 . The common requirement for graphbased approaches as well as SPRank is that a graph has to be built based on user-item interactions and background knowledge from LOD. Our approach is different as we only consider lightweight LOD-enabled features which can be directly obtained through a public SPARQL Endpoint, and without any additional effort to build a graph. This also makes our model consume updated background knowledge of DBpedia easier when compared to other approaches such as graphbased ones which require downloading a DBpedia dump and building a graph by adding user-item interactions.
There have also been some other interesting directions related to LODenabled recommender systems such as the practical LODRecSys [21], explaining using LOD [18], rating predictions based on matrix factorization with semantic categories [31], and cross-domain recommendations [9,11]. For example, Oliveira et al. [21] presented a recommender system in the movie domain that consumes LOD (not restricted to DBpedia), which was evaluated with comparison to seevl (ISWC challenge winner at 2011). Different types of evaluation metrics have been used such as accuracy, novelty etc. The authors from [18] presented ExpLOD -a framework which can generate explanations in natural language based on LOD cloud. Musto et al. [17] investigated various feature (property) selection strategies and their influences on recommendation performance in terms of accuracy and diversity in movie and book domains. Lalithsena et al. [14] proposed a novel approach using type-and path-based methods to extract a subgraph for domain specific recommendation systems. They presented that their approach can decrease 80% of the graph size without losing accuracy in the context of recommendation systems in movie and book domains. These, although interesting, are however beyond the scope of this paper and we aim to explore them in future work.
Proposed Method
In this section, we first briefly introduce FMs and the optimization criteria we used in this study (Section 3.1). Next, we will describe our features from useritem interactions as well as background knowledge from DBpedia (Section 3.2).
Factorization Machines
Factorization Machines (FMs) [28], which can mimic other well known factorization models such as matrix factorization, SVD++ [13], have been widely used for collaborative filtering tasks [29]. FMs are able to incorporate the high-prediction accuracy of factorization models and flexible feature engineering. An important advantage of FMs is the model equation
y F M (x) = w 0 + p i=1 w i x i + p i=1 p j>i < v i , v j > x i x j(1)
where w 0 ∈ R, x and w ∈ R p , v i ∈ R m . The first part of the FM model captures the interactions of each input variable x i , while the second part of it models all pairwise interactions of input variables x i x j . Each variable x i has a latent factor v i , which is a m-dimensional vector allows FMs work well even in highly sparse data.
Optimization. In this work, we use a pairwise optimization approach -Bayesian Personalized Ranking (BPR). The loss function was proposed by Rendle et al. [30].
l(x 1 , x 2 ) = x1∈C + u x2∈C − u (− log[δ(ŷ F M (x 1 ) −ŷ F M (x 2 ))])(2)
where δ is a sigmoid function: δ(x) = 1 1+e −x , and C + u and C − u denote the set of positive and negative feedbacks respectively. L2-regularization is used for the loss function.
Learning. We use the well-known stochastic gradient descent algorithm to learn the parameters in our model. To avoid overfitting on the training dataset, we adopt an early stopping strategy as follows. User and item index. The first two sets of features indicate the indexes of the user and item in a training example. A feature value equals 1 for the corresponding user/item index, e.g., val(U i ) = 1 and val(I j ) = 1 denote an example about the i -th user and j -th item.
1 0 … 1 0 … 0.2 0.2 … 0.1 0 … 0.1 0 1 … 0 1 … 0.3 0.5 … 0 0.3 … 0.2 … … … … … … … … … … … …
Features
Property-Object list (PO). This set of features denotes all property-objects of an item i when i is a subject in RDF triples. This set of features can be obtained easily by using a SPARQL query as shown below via the DBpedia SPARQL Endpoint. An intuitive way of giving feature values for PO might be to assign 1 for all property-objects of an item i (P O i ). However, it can be biased as some entities in DBpedia have a great number of property-objects while others do not. Therefore, we normalize the feature values of P O i based on the size of P O i so that all the feature values of P O i sum up to 1. Formally, the feature value of j -th property-object for an item i is measured as val(P O i (j)) = 1 |P Oi| . Take the graph in Figure 1 as an example, as we have two property-objects for the movie dbr:The Godfather, each property-object of the movie will have a feature value of 0.5, respectively (see Figure 3). PageRank score (PR). PageRank [23] is a popular algorithm with the purpose of measuring the relative importance of a node in a graph. In order to capture the importance of an entity in Wikipedia/DBpedia, Thalhammer et al. [32] proposed providing PageRank scores of all DBpedia entities, which are based on links using dbo:wikiPageWikiLink among entities. A PageRank score of an item (entity) might be a good indicator of the importance of an entity for recommendations in our case. The PageRank score of a DBpedia entity can be obtained by using the SPARQL as shown below. The scale of PageRank scores is different from other feature values, which can delay the convergence of learning parameters for our model. In this regard, we normalize the PageRank scores by their maximum value.
val(P R i ) = P ageRank i max(P ageRank j , j ∈ I)
where P ageRank i denotes the original PageRank score of i which is obtained from the SPARQL Endpoint, and max(P ageRank j , j ∈ I) denotes the maximum PageRank score of all items.
Experimental Setup
In this section, we introduce the dataset for our experiment (Section 4.1) and five evaluation metrics for evaluating the performance of the recommendations (Section 4.2). Afterwards, we describe four methods that have been used for comparison with our approach for evaluation (Section 4.3).
Dataset
We used the Movielens dataset from [20]. The dataset was originally from the Movielens dataset 7 , which consists of users and their ratings about movie items. To facilitate LODRecSys, each of the items in this dataset has been mapped into DBpedia entities if there is a mapping available 8 . In the same way as [20], we consider ratings higher than 3 as positive feedback and others as negative one. Table 1 shows details about the dataset. The dataset consists of 3,997 users and 3,082 items with 695,842 ratings where 56% of them are positive ratings. We split the dataset into training (80%) and test (20%) sets for our experiment.
Evaluation Metrics
We use five different evaluation metrics to measure the quality of recommendations provided by different approaches.
-P@N: The Precision at rank N represents the mean probability that retrieved items within the top-N recommendations are relevant to the user.
P @N = |{relevant items}| ∩ |{retrieved items@n}| |{retrieved items}| (4)
-R@N: The Recall at rank N represents the mean probability that relevant items are successfully retrieved within the top-N recommendations.
R@N = |{relevant items}| ∩ |{retrieved items@n}| |{relevant items}| (5)
-nDCG@N: Precision and recall consider the relevance of items only. On the other hand, nDCG takes into account the relevance of items as well as their rank positions.
nDCG@N = 1 IDCG@N N k=1 2r uk − 1 log 2 (1 + k)(6)
Here,r uk denotes the rating given by a user u to the item in position k in the top-N recommendations, and IDCG@N denotes the score obtained by an ideal or perfect top-N ranking and acts as a normalization factor.
AP = N n=1 P @n × like(n) |I| (8)
where n is the number of items, |I| is the number of liked items of the user, and like(n) is a binary function to indicate whether the user prefers the n-th item or not.
The bootstrapped paired t-test, which is an alternative to the paired t-test when the assumption of normality of the method is in doubt, is used for testing the significance where the significance level was set to 0.01 unless otherwise noted.
Compared Methods
We use four approaches including a baseline PopRank and other methods which have been frequently used in the literature [17,20] to evaluate our proposed method.
-PopRank: This is a non-personalized baseline approach which recommends items based on the popularity of each item.
-kNN-item: This is a collaborative filtering approach based on the k most similar items. We use a MyMedialiite [5] implementation for this baseline where k = 80.
-BPRMF [30]: This is a matrix factorization approach for learning latent factors for users and items. We use a MyMedialiite [5] implementation for this baseline where the dimensionality of the factorization m = 200.
-SPRank [20]: This is a learning-to-rank approach for LODRecSys based on semantic paths extracted from a graph including user-item interactions (e.g., likes, dislikes, etc.) as well as the background knowledge obtained from DBpedia. In detail, semantic paths are sequences of properties including likes and dislikes based on user-item interactions. For example, given the graph information user1→likes→item1→p1→item2, a semantic path (likes, p1) can be extracted from user1 to item2. Another difference between SPRank [20] and our approach in terms of features is that the authors considered property-objects for each item including the property dbo:wikiPageWikiLink which cannot be queried via the DBpedia Endpoint but requires settings up a local endpoint using a DBpedia dump. On the other hand, we only considers sets of LOD-enabled features which can be obtained from a public DBpedia Endpoint. We use LMART [33] as the learning algorithm for SPRank as this approach overall provides the best performance compared to other learning-to-rank algorithms in [20]. We used the author's implementation 9 which has been optimized for nDCG@10.
Results
In this section, we first compare our approach to the aforementioned methods in terms of five evaluation metrics (Section 5.1). We denote our approach as LODFM, and the results of LODFM are based on best tuned parameters, i.e., m = 200 using PO and PR as LOD-enabled features. We discuss self comparison by using different sets of features, as well as different dimensionality m for factorization, in detail in Section 5.2.
Comparison with Baselines
The results of comparing our proposed approach with the baselines are presented in Table 2 in terms of MRR, MAP, nDCG@N, P@N and R@N.
Overall, LODFM provides the best performance in terms of all evaluation metrics. In line with the results from [20], SPRank does not perform as well on the Movielens dataset compared to other collaborative filtering approaches such as kNN and BPRMF. On the other hand, we observe that LODFM significantly outperforms SPRank as well as other baseline methods. Among baselines, kNNitem is the best performing method in terms of P@5 and P@10 while BPRMF is the best performing baseline in terms of other evaluation metrics. A significant improvement of LODFM over BPRMF in MRR (+5.3%), MAP (+14.9%), nDCG@10 (+4.6%), P@10 (+12.9%) and R@10 (+8%) can be noticed. The results indicate that LOD-enabled features are able to improve the recommendation performance for factorization models. Compared to kNN-item, LODFM improves the performance by 8.2% and 2.4% in terms of P@5 and P@10, respectively. It is also interesting to observe that factorization models such as BPRMF and LODFM have much better performance especially in terms of recall compared to kNN-item. For example, LODFM improves the performance by 103%, 90% and 76.9% in terms of recall when N = 1,5 and 10, respectively.
Model Analysis
Analysis of features. To better understand the contributions of each feature set for recommendations, we discuss the recommendation performance with different sets of features for FM in this section. Table 3 shows the recommendation Overall, using a property-object list (PO) and the PageRank score (PR) of items provides the best performance compared to other strategies. As we can see from Table 3, PO+PR improves the recommendation performance compared to PO in terms of most of the evaluation metrics. Similar results can be observed by comparing PO+SP+PR against PO+SP, which shows the importance of PageRank scores of items. On the other hand, the performance is decreased by including SP, e.g., PO+SP vs. PO and PO+SP+PR vs. PO+PR. This shows that incoming knowledge about movie items is not helpful in improving recommendation performance.
Analysis of dimensionality m for factorization. The dimensionality of factorization plays an important role in capturing pairwise interactions of input variables when m is chosen large enough [29]. Figure 4 illustrates the recommendation performance using different values for the dimensionality of factorization (The results of P@1 are equal to nDCG@1 and therefore omitted from Figure 4(b)) using PO and PR as LOD-enabled features. As we can see from the figure, the performance consistently increases with higher values of m until m = 200 in terms of five evaluation metrics. For example, the performance is improved by 7.5% and 11.4% in terms of MRR and MAP with m = 200 compared to m = 10. There is no significant improvement with values higher than 200 for m.
Conclusions
In this paper, we investigated using FM with lightweight LOD-enabled features, such as property-object lists, subject-property lists, and PageRank scores of items which can be directly obtained from the DBpedia SPARQL Endpoint, for top-N recommendations. The results show that our proposed approach significantly outperforms compared approaches such as SPRank, BPRMF. In addition, we analyzed the recommendation performance based on different combinations of features. The results indicate that using the property-object list and the PageRank scores of items can provide the best performance. On the other hand, including the subject-property list of items is not helpful in improving the quality of recommendations but rather decreases the performance. In the future, we plan to evaluate our approach using other datasets in different domains. Furthermore, we aim to investigate other lightweight LOD-enabled features which might be useful to improve the recommendation performance.
Fig. 1 :
1An example of background knowledge about the movie dbr:The Godfather from DBpedia.
Figure 2
2presents the overview of features for our FM. The details of each set of features are described below.
Fig. 3 :
3An example for PO values for the movie dbr:The Godfather in Figure 1. Subject-Property list (SP). Similar to the PO, we can obtain incoming background knowledge about an item i where i is an object in RDF triples. This set of features can be obtained by using a SPARQL query as shown below. PREFIX dbo:<http://dbpedia.org/ontology/> SELECT DISTINCT ?s ?p WHERE { ?s ?p <itemURI> . FILTER REGEX(STR(?p), ''^http://dbpedia.org/ontology'') . FILTER (STR(?p) NOT IN (dbo:wikiPageRedirects, dbo:wikiPageExternalLink, dbo:wikiPageDisambiguates) } In the same way as we normalized feature values of P O i for an item i, we normalize the feature values of SP i based on the size of SP i so that all the feature values of SP i sum up to 1. The feature value of the j -th SP for an item i is measured as val(SP i (j)) = 1 |SPi| .
-
-MRR: The Mean Reciprocal Rank (MRR) indicates at which rank the first link relevant to the user occurs (denoted by rank k ) on average. MAP: The Mean Average Precision (MAP) measures the average of the average precision (AP) of all liked items for all users. For each user, the average precision of the user is defined as:
Fig. 4 :
4Recommendation performance based on different values for the dimensionality m of FM using PO+PR in terms of different evaluation metrics.
Fig. 2: Overview of features for Factorization Machine. PO denotes all propertyobjects, and SP denotes all subject-property for items in the dataset. PR denotes the PageRank scores of items. 1. Split the dataset into training and validation sets. 2. Measure the current loss on the validation set at the end of each epoch. 3. Stop and remember the epoch if the loss has increased. 4. Re-train the model using the whole dataset.…
user
item
PO
SP
PR
1
0
…
x1
Feature vector x
Target y
x2
Table 1 :
1Movielens dataset statistics# of users
3,997
# of items
3,082
# of ratings
695,842
avg. # of ratings
174
sparsity
94.35%
% of positive ratings
56%
Table 2 :
2Recommendation performance compared to baselines in terms of five different evaluation metrics. The best performing strategy is in bold.PopRank kNN-item BPRMF
SPRank
LODFM
MRR
0.4080
0.5756
0.5906
0.3013
0.6218
MAP
0.1115
0.2037
0.2018
0.0612
0.2318
nDCG@1
0.2459
0.4086
0.4269
0.1758
0.4685
P@1
0.2459
0.4086
0.4269
0.1758
0.4685
R@1
0.0064
0.0132
0.0258
0.0082
0.0268
nDCG@5
0.2809
0.4049
0.4176
0.2195
0.4537
P@5
0.2240
0.3538
0.3393
0.1287
0.3829
R@5
0.0305
0.0553
0.0977
0.0291
0.1052
nDCG@10
0.3664
0.4753
0.5000
0.2845
0.5231
P@10
0.2104
0.3179
0.2883
0.1068
0.3256
R@10
0.0580
0.0978
0.1602
0.0488
0.1730
Table 3 :
3Recommendation performance of LODFM using different sets of features such as property-object list (PO), subject-property list (SP) and PageRank scores (PR). The best performing strategy is in bold.performance of LODFM using different features with m = 10. The two fundamental features -user and item indexes are included by default and omitted from the table for clarity.PO
PO+SP
PO+PR
PO+SP+PR
MRR
0.5769
0.5403
0.5783
0.5561
MAP
0.2096
0.1957
0.2080
0.2008
nDCG@1
0.4224
0.3788
0.4236
0.3971
P@1
0.4224
0.3788
0.4236
0.3971
R@1
0.0237
0.0210
0.0241
0.0223
nDCG@5
0.4152
0.3861
0.4214
0.3963
P@5
0.3459
0.3222
0.3479
0.3280
R@5
0.0931
0.0841
0.0934
0.0866
nDCG@10
0.4904
0.4627
0.4945
0.4743
P@10
0.2973
0.2805
0.2975
0.2860
R@10
0.1558
0.1436
0.1541
0.1476
https://sourceforge.net/p/lemur/wiki/RankLib/
https://grouplens.org/datasets/movielens/1m/ 8 http://sisinflab.poliba.it/semanticweb/lod/recsys/datasets/
https://github.com/sisinflab/lodreclib
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|
[
"https://github.com/sisinflab/lodreclib"
] |
[
"Data-driven model order reduction of linear switched systems",
"Data-driven model order reduction of linear switched systems"
] |
[
"I V Gosea ",
"M Petreczky ",
"A C Antoulas ",
"† "
] |
[] |
[] |
The Loewner framework for model reduction is extended to the class of linear switched systems. One advantage of this framework is that it introduces a trade-off between accuracy and complexity. Moreover, through this procedure, one can derive state-space models directly from data which is related to the input-output behavior of the original system. Hence, another advantage of the framework is that it does not require the initial system matrices. More exactly, the data used in this framework consists in frequency domain samples of inputoutput mappings of the original system. The definition of generalized transfer functions for linear switched systems resembles the one for bilinear systems. A key role is played by the coupling matrices, which ensure the transition from one active mode to another. † Data-Driven System Reduction and Identification Group,
| null |
[
"https://arxiv.org/pdf/1712.05740v1.pdf"
] | 119,323,915 |
1712.05740
|
3d51e5c170142434f96ad0d9626a60be292d1096
|
Data-driven model order reduction of linear switched systems
I V Gosea
M Petreczky
A C Antoulas
†
Data-driven model order reduction of linear switched systems
The Loewner framework for model reduction is extended to the class of linear switched systems. One advantage of this framework is that it introduces a trade-off between accuracy and complexity. Moreover, through this procedure, one can derive state-space models directly from data which is related to the input-output behavior of the original system. Hence, another advantage of the framework is that it does not require the initial system matrices. More exactly, the data used in this framework consists in frequency domain samples of inputoutput mappings of the original system. The definition of generalized transfer functions for linear switched systems resembles the one for bilinear systems. A key role is played by the coupling matrices, which ensure the transition from one active mode to another. † Data-Driven System Reduction and Identification Group,
Introduction
Model order reduction (MOR) seeks to transform large, complicated models of time dependent processes into smaller, simpler models that are nonetheless capable of representing accurately the behavior of the original process under a variety of operating conditions. The goal is an efficient, methodical strategy that yields a dynamical system evolving in a substantially lower dimension space (hence requiring far less computational resources for realization), yet retaining response characteristics close to the original system. Such reduced order models could be used as efficient surrogates for the original model, replacing it as a component in larger simulations.
Hybrid systems are a class of nonlinear systems which result from the interaction of continuous time dynamical sub-systems with discrete events. More precisely, a hybrid system is a collection of continuous time dynamical systems. The internal variable of each dynamical system is governed by a set of differential equations. Each of the separate continuous time systems are labeled as a discrete mode. The transitions between the discrete states may result in a jump in the continuous internal variable. Linear switched systems (in short LSS) constitute a subclass of hybrid systems; the main property is that these systems switch among a finite number of linear subsystems. Also, the discrete events interacting with the sub-systems are governed by a piecewise continuous function called the switching signal.
Hybrid and switched systems are powerful models for distributed embedded systems design where discrete controls are routinely applied to continuous processes. However, the complexity of verifying and assessing general properties of these systems is very high so that the use of these models is limited in applications where the size of the state space is large. To cope with complexity, abstraction and reduction are useful techniques. In this paper we analyze only the reduction part.
In the past years, hybrid and switched systems have received increasing attention in the scientific community. For a detailed characterization of this relatively new class of dynamical systems, we refer the readers to the books [24], [37], [38] and [16]. Such systems are used in modeling, analysis and design of supervisory control systems, mechanical systems with impact, circuits with relays or ideal diodes.
The study of the properties of hybrid systems in general and switched systems in particular is still the subject of intense research, including the problems of stability (see [13] and [37]), realization including observability/controllability (see [30] and [31]), analysis of switched DAE's (see [26] and [39]) and numerical solutions (see [17]). Recently, considerable research has been dedicated to the problem of MOR for linear switched systems. The most prolific method that has been applied is balanced truncation (or some sort of gramian based derivation of it). Techniques that are based on balancing have been considered in the following: [15], [11], [8], [36], [27], [33] and [29]. Also, another class of methods involve matching of generalized Markov parameters (known also as time domain Krylov methods) such as the ones in [7] and [6]; H ∞ type of reduction methods were developed in [41], [9] and [42]. Finally, we mention some publications that are focused on the reduction of discrete LSS, such as [40] and [10]. A linear switched system involves switching between a number of linear systems (the modes of the LSS). Hence, to apply balanced truncation techniques to a switched linear system, one may seek for a basis of the state space such that the corresponding modes are all in balanced form. It may happen that some state components of the LSS are difficult to reach and observe in some of the modes yet easy to reach and observe in others. In that case, deciding how to truncate the state variables and obtain a reduced order model is not trivial. A solution to this problem is proposed in [27] where it turns out that the average gramian can be used to obtain a reduced order model. This method will be used as a comparison tool for our new MOR method.
In the sequel we exclusively consider interpolatory MOR methods and in particular the Loewner framework applied to LSS. Roughly speaking, in the linear case, interpolatory methods seek reduced models whose transfer function matches that of the original system at selected frequencies.
For the nonlinear case, these methods require appropriate definitions of transfer functions.
In this paper, we focus on generalizing the Loewner Framework for reducing linear switched systems. The presentation is tailored to emphasize the main procedure for a simplified case of LSS (i.e only two modes and LTI's in SISO format activating in both modes). The paper is organized as follows. In the next section, we review the formal definition of continuous-time linear switched systems. Furthermore, we introduce the generalized transfer functions for LSS as input-output mappings in frequency domain. Section 3 includes a brief introduction of the Loewner framework for linear systems. In Section 4, we introduce the Loewner framework for LSS with two modes. In section 5, we generalize most of the results in the previous section for the case of LSS with D 2 modes. Finally, in Section 6, we discuss the applicability of the new introduced method for reducing LSS. In this sense, by means of three numerical examples (one of which large scale), we compare the results obtained by applying the Loewner method against the method in [27]. In Section 7, we present a summary of the findings and the conclusions.
2 Linear switched systems Definition 2.1 A continuous time linear switched system (LSS) is a control system of the form:
Σ : E σ(t)ẋ (t) = A σ(t) x(t) + B σ(t) u(t), x(t) = x 0 , y(t) = C σ(t) x(t),(1)
where Q = {1, 2, . . . , D}, D > 1, is a set of discrete modes, σ(t) is the switching signal, u is the input, x is the state, and y is the output.
The system matrices E q , A q ∈ R nq×nq , B q ∈ R nq×m , C q ∈ R p×nq , where q ∈ Q, correspond to the linear system active in mode q ∈ Q, and x 0 is the initial state. We consider the E q matrices to be invertible. Furthermore, the transition from one mode to another is made via the so called switching or coupling matrices K q 1 ,q 2 ∈ R nq 2 ×nq 1 where q 1 , q 2 ∈ Q.
Remark 2. 1 The case for which the coupling is made between identical modes is excluded, Hence, when q 1 = q 2 = q, consider that the coupling matrices are identity matrices, i.e. K q,q = I nq .
The notation Σ = (n 1 , n 2 , . . . , n D , {(E q , A q , B q , C q )|q ∈ Q}, {K q i ,q i+1 |q i , q i+1 ∈ Q}, x 0 ) is used as a short-hand representation for LSS's described by the equations in (1). The vector n = n 1 n 2 · · · n D is the dimension (order) of Σ. The linear system which is active in the q th mode of Σ is denoted with Σ q and it is described by (where 1 q D)
Σ k : E qẋq (t) = A q x q (t) + B q u(t), x(t k ) = x k , y(t) = C q x q (t).(2)
The restriction of the switching signal σ(t) to a finite interval of time [0, T ] can be interpreted as finite sequence of elements of Q × R + of the form:
ν(σ) = (q 1 , t 1 )(q 2 , t 2 ) . . . (q k , t k ),
where q 1 , . . . , q k ∈ Q and 0 < t 1 < t 2 < · · · < t k ∈ R + , t 1 + · · · + t k = T , such that for all t ∈ [0, T ] we have:
σ(t) = q 1 if t ∈ [0, t 1 ], q 2 if t ∈ (t 1 , t 1 + t 2 ], . . . q i if t ∈ (t 1 + . . . + t i−1 , t 1 + . . . + t i−1 + t i ], . . . q k if t ∈ (t 1 + . . . + t k−1 , t 1 + . . . + t k−1 + t k ].
In short, by denoting T i := t 1 + . . . + t i−1 + t i , T 0 := 0, T k := T , write:
σ(t) = q 1 if t ∈ [0, T 1 ], q i if t ∈ (T i−1 , T i ], i > 2.
Denote by P C(R + , R n ), P c (R + , R n ), the set of all piecewise-continuous, and piecewise-constant functions, respectively.
Definition 2.2 A tuple (x, u, σ, y), where x : R + → D i=1 R n i , u ∈ P C(R + , R m ), σ ∈ P c (R + , Q), y ∈ P C(R + , R p )
is called a solution, if the following conditions simultaneously hold:
1. The restriction of x(t) to (T i−1 , T i ] is differentiable, and satisfies E q iẋ (t) = A q i x(t) + Bu(t).
2. Furthermore, when switching from mode q i to mode q i+1 at time T i , the following holds
E q i+1 lim t T i x q i+1 (t) = K q i ,q i+1 x q i (T i ).
3. Moreover, for all t ∈ R, y(t) = C σ(t) x(t) holds.
The switching matrices K q i ,q i+1 allow having different dimensions for the subsystems active in different modes. For instance, the pencil (
A q i , E q i ) ∈ R nq i ×nq i , while the pencil (A q i+1 , E q i+1 ) ∈ R nq i+1 ×nq i+1
where the values n q i and n q i+1 need not be the same. If the K q i ,q i+1 matrices are not explicitly given, it is considered that they are identity matrices.
The input-output behavior of an LSS system can be formalized in time domain as a map f (u, σ)(t). This particular map can be written in generalized kernel representation (as suggested in [32]) using the unique family of analytic functions: g q 1 ,...,q k : R k + → R p and h q 1 ,...,q k : R k + → R p×m with q 1 , . . . , q k ∈ Q, k 1 such that for all pairs (u, σ) and for T = t 1 + t 2 + · · · + t k we can write:
f (u, σ)(t) = g q 1 ,q 2 ,...,q k (t 1 , t 2 , ..., t k ) + k i=1 t i 0 h q i ,q i+1 ,...,q k (t i − τ, t i+1 , . . . , t k )u(τ + T i−1 )dτ,
where the functions g, h are defined for k 1, as follows,
g q 1 ,q 2 ,...,q k (t 1 , t 2 , . . . , t k ) = C q k eà q k t kK q k−1 ,q k eà q k−1 t k−1K q k−2 ,q k−1 · · ·K q 1 ,q 2 eà q 1 t 1 x 0 ,(3)h q 1 ,q 2 ,...,q k (t 1 , t 2 , . . . , t k ) = C q k eà q k t kK q k−1 ,q k eà q k−1 t k−1K q k−2 ,q k−1 · · ·K q 1 ,q 2 eà q 1 t 1B 1 .(4)
Note that, for the functions defined in (3) and (4) we consider the E q i matrices to be incorporated into the A q i and B q i matrices (i.e.
à q i = E −1 q i A q i ,B q i = E −1 q i B q i )
. Moreover, the transformed coupling matrices are written accordinglyK q i ,q i+1 = E −1 q i+1 K q i ,q i+1 . In the rest of the paper, the LSS we treat are assumed to have zero initial conditions, i.e., x 0 = 0. Hence, only the h functions in (4) are relevant for characterizing the input-output mapping f .
The behavior of the input-output mappings in frequency domain is in turn characterized by a series of multivariate rational functions obtained by taking the multivariable Laplace transform of the regular kernels in (4), as for
H q 1 (s 1 ) = C q 1 Φ q 1 (s 1 )B q 1 , H q 1 ,q 2 (s 1 , s 2 ) = C q 1 Φ q 1 (s 1 )K q 2 ,q 1 Φ q 2 (s 2 )B q 2 , H q 1 ,q 2 ,q 3 (s 1 , s 2 , s 3 ) = C q 1 Φ q 1 (s 1 )K q 2 ,q 1 Φ q 2 (s 2 )K q 3 ,q 2 Φ q 3 (s 3 )B q 3 , · · ·
For k 1, write the level k generalized transfer function associated to the switching sequence (q 1 , q 2 , . . . , q k ), and evaluated at the points (s 1 , s 2 , . . . s k ) as,
H q 1 ,q 2 ,...,q k (s 1 , s 2 , ..., s k ) = C q 1 Φ q 1 (s 1 )K q 2 ,q 1 Φ q 2 (s 2 ) · · · K q k ,q k−1 Φ q k (s k )B q k ,(5)
where Φ q (s) = (sE q − A q ) −1 , q j ∈ {1, 2, ..., D}, 1 j k and k 3. These functions are the generalized transfer functions of the linear switched system Σ. Their definition is similar to the ones corresponding to bilinear systems (see [3]).
By using their samples, we are able to directly come up with (reduced) switched models that interpolate the original model -generalization of the Loewner framework to LSS.
We construct LSS reduced models by means of matching samples of input-output mappings corresponding to the original LSS system and evaluated at finite sampling points (as opposed to other approaches -see [6] and [7], where the behavior at infinity is studied instead, i.e. by matching Markov parameters).
For the explicit derivation of these types of transfer functions (which is based on the so-called Volterra series representation) we refer the readers to [34].
Interpolatory MOR methods and the Loewner framework
Consider a full-order linear system defined by E ∈ R n×n , A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , and its transfer function
H(s) = C(sE − A) −1 B. Given left interpolation points: {µ j } q j=1 ⊂ C, with left tangential directions: { j } q j=1 ⊂ C p , and right interpolation points: {λ i } k i=1 ⊂ C, with right tangential directions: {r i } k i=1 ⊂ C m
, find a reduced-order systemÊ,Â,B,Ĉ, such that the resulting transfer function,Ĥ(s) is a tangential interpolant to H(s):
T jĤ (µ j ) = T j H(µ j ), j = 1, . . . , q, andĤ(λ i )r i = H(λ i )r i , i = 1, . . . , k(6)
Interpolation points and tangent directions are selected to realize appropriate MOR goals. If instead of state space data, we are given input/output data, the resulting problem is hence modified. Given a set of input-output response measurements specified by left driving frequencies:
{µ j } q j=1 ⊂ C, using left input directions: { j } q j=1 ⊂ C p , producing left responses: {v j } q j=1 ⊂ C m , and right driving frequencies: {λ i } k i=1 ⊂ C, using right input directions: {r i } k i=1 ⊂ C m , producing right responses: {w i } k i=1 ⊂ C p ,
find (low order) system matricesÊ,Â,B,Ĉ, such that the resulting transfer function,Ĥ(s), is an (approximate) tangential interpolant to the data:
T jĤ (µ j ) = v T j , j = 1, . . . , q, andĤ(λ i )r i = w i , i = 1, . . . , k.(7)
3.1 Overview of the Loewner framework for linear systems
The approach we discuss in this section is data driven. After collecting input/output (e.g. frequency response) measurements for some appropriate range of frequencies, we construct models which fit (or approximately fit) the data and have reduced dimension. The key is that, larger amounts of data than necessary are collected and the essential underlying system structure is extracted appropriately. Thus an advantage of this approach is that it can provide the user with a trade-off between accuracy of fit and complexity of the model. The Loewner framework was developed in a series of papers; for details we refer the reader to [1], as well as [25,23,22,4,19,20]. For a recent overview see [5].
The Loewner pencil
We will formulate the results for the more general tangential interpolation problem. We are given the right data: (λ i ; r i , w i ), i = 1, · · · , k, and the left data: (µ j ; T j , v T j ), j = 1, · · · , q; it is assumed for simplicity that all points are distinct. The dimensions are as in (6), (7). The data can be organized as follows: the right data: Λ = diag [λ 1 , . . . , λ k ] ∈ C k×k , R = [r 1 , . . . , r k ] ∈ C m×k , W = [w 1 , . . . , w k ] ∈ C p×k , and the left data:
M = diag [µ 1 , . . . , µ q ] ∈ C q×q , L T = [ 1 , . . . , q ] ∈ C q×p , V T = [v 1 , . . . , v q ] ∈ C q×m .
Then, the associated Loewner pencil, consists of the Loewner and shifted Loewner matrices. The Loewner matrix L ∈ C q×k , is defined as:
L = v T 1 r 1 − T 1 w 1 µ 1 −λ 1 · · · v T 1 r k − T 1 w k µ 1 −λ k . . . . . . . . . v T q r 1 − T q w 1 µq−λ 1 · · · v T q r k − T q w k µq−λ k L satisfies the Sylvester equation LΛ − ML = VR − LW.
Suppose that the underlying transfer function is H(s) = C(sE−A) −1 B, and define the generalized observability/controllability matrices:
O = C(µ 1 E − A) −1 . . . C(µ q E − A) −1 , R = (λ 1 E − A) −1 B · · · (λ k E − A) −1 B .(8)
It readily follows that the Loewner matrix can be factored as L = −OER. The shifted Loewner matrix L s ∈ C q×k , is defined as:
L s = µ 1 v T 1 r 1 − T 1 w 1 λ 1 µ 1 −λ 1 · · · µ 1 v T 1 r k − T 1 w k λ k µ 1 −λ k . . . . . . . . . µqv T q r 1 − T q w 1 λ 1 µq−λ 1 · · · µqv T q r k − T q w k λ k µq−λ k
Construction of reduced order models
We will distinguish two cases namely, the right amount of data and the more realistic redundant amount of data cases. The following lemma covers the first case.
Lemma 3.1 Assume that k = q, and let (L s , L), be a regular pencil, such that none of the interpolation points λ i , µ j are its eigenvalues. Then E = −L, A = −L s , B = V, C = W, is a minimal realization of an interpolant of the data, i.e., the rational function H(s) = W(L s − sL) −1 V, interpolates the data (the conditions in (7) are hence matched).
If the pencil (L s , L) is singular we are dealing with the case of redundant data. In this case if the following assumption is satisfied:
rank (xL − L s ) = rank L L s = rank [L L s ] = r k,(9)
for all x ∈ {λ i } ∪ {µ j }, we consider the following SVD factorizations:
[L L s ] = Y 1 S 1 X T 1 , L L s = Y 2 S 2 X T 2 ,(10)
where Y 1 , X 2 ∈ C k×k . By selecting the first r columns of the matrices Y 1 and X 2 , we come up with projection matrices Y, X ∈ C k×r . The following result is used in practical applications.
Lemma 3.2 A realization (E, A, B, C) of an approximate interpolant is given by the system ma-
trices E = −Y T LX, A = −Y T L s X, B = Y T V, C = WX.
Hence, the rational function H(s) = WX(Y T L s X − sY T LX) −1 Y T V approximately matches the data (the conditions in (7) are approximately fulfilled, i.e. H(λ i )r i = w i + r i and T j H(µ j ) = v T j + ( j ) T , where the residual errors are collected in the vectors r i and j ). Thus, if we have more data than necessary, we can consider (L s , L, V, W), as a singular model of the data. An appropriate projection then yields a reduced system of order k (see [25,2]).
A direct consequence is that the Loewner framework offers a trade-off between accuracy and complexity of the reduced order system, by means of the singular values of L.
Remark 3.1 For an error bound that links the quality of approximation to the singular values of the Loewner pencil (which is valid only at the interpolation points µ j and λ i ), we refer the readers to [5].
The Loewner framework for LSS -the case D=2
The characterization of linear switched systems by means of rational functions suggests that reduction of such systems can be performed by means of interpolatory methods. In the following we will show how to generalize the Loewner framework to LSS by interpolating appropriately defined transfer functions on a chosen grid of frequencies (interpolation points).
As for the linear case, the given set of sampling (interpolation) points is first partitioned into the two following categories: left interpolation points: {µ j } j=1 ⊂ C and and right interpolation points:
{λ i } k i=1 ⊂ C.
In this paper we consider only the case of SISO linear switched systemshence the left and right tangential directions can be considered to be scalar (i.e. taking the value 1). Since the transfer functions which are going to be matched are not single variable functions anymore (they depend on multiple variables as described in ), the structure of the interpolation points used in the new framework is going to change. Instead of having singleton values as in Section 3, we will use instead n-tuples that include multiple singleton values.
For simplicity of the exposition, we first consider the simplified case D = 2 (the LSS system switches between two modes only). This situation is encountered in most of the numerical examples in the literature we came across. Nevertheless, all the results presented in this section can be generalized for higher number of modes in a more or less straightforward way (as presented in Section 5). Depending on the switching signal σ(t), we either have,
Σ 1 : E 1ẋ1 (t) = A 1 x 1 (t) + B 1 u(t), y(t) = C 1 x 1 (t) or Σ 2 : E 2ẋ2 (t) = A 2 x 2 (t) + B 2 u(t), y(t) = C 2 x 2 (t)
where dim(Σ 1 ) = n 1 (i.e. x 1 ∈ R n 1 and E 1 , A 1 ∈ R n 1 ×n 1 , B 1 , C T 1 ∈ R n 1 ) and also dim(Σ 2 ) = n 2 (i.e. x 2 ∈ R n 2 and E 2 , A 2 ∈ R n 2 ×n 2 , B 2 , C T 2 ∈ R n 2 ). Notice that we allow both the two subsystems to be written in descriptor format (having possibly singular E matrix).
Denote, for simplicity, with K 1 the coupling matrix when switching from mode 1 to mode 2 (instead of K 1,2 ) and, with K 2 , the coupling matrix when switching from mode 2 to mode 1 (instead of K 2,1 ) with K 1 ∈ R n 2 ×n 1 and K 2 ∈ R n 1 ×n 2 .
The generalized transfer functions are defined as (where Φ q (s) = (sE q − A q ) −1 , q ∈ {1, 2}),
Level 1 H 1 (s 1 ) = C 1 Φ 1 (s 1 )B 1 H 2 (s 2 ) = C 2 Φ 2 (s 2 )B 2 Level 2 H 1,2 (s 1 , s 2 ) = C 1 Φ 1 (s 1 )K 2 Φ 2 (s 2 )B 2 H 2,1 (s 2 , s 1 ) = C 2 Φ 2 (s 2 )K 1 Φ 1 (s 1 )B 1 Level 3 H 1,2,1 (s 1 , s 2 , s 3 ) = C 1 Φ 1 (s 1 )K 2 Φ 2 (s 2 )K 1 Φ 1 (s 3 )B 1 H 2,1,2 (s 1 , s 2 , s 3 ) = C 2 Φ 2 (s 1 )K 1 Φ 1 (s 2 )K 2 Φ 2 (s 3 )B 2 , · · · Definition 4.1 Consider the two LSS,Σ = (n 1 , n 2 , {(Ê i , i ,B i ,Ĉ i )} 2 i=1 , {K i,j } 2 i,j=1 , 0) andΣ = (n 1 , n 2 , {(Ē i ,Ā i ,B i ,C i )} 2 i=1 , {K i,j } 2 i,j=1 , 0)
. These systems are said to be equivalent if there exist non-singular matrices Z L j and Z R j so that
E j = Z L jÊ j Z R j ,Ā j = Z L j j Z R j ,B j = Z L jB j ,C j =Ĉ j Z R j , j ∈ {1, 2} and alsoK 1 = Z L 2K 1 Z R 1 ,K 2 = Z L 1K 2 Z R 2 .
In this configuration, one can easily show that the transfer functions defined above are the same for each LSS and for all sampling points s k .
The generalized controllability and observability matrices
Consider a LSS system Σ as described in (1) with dim(Σ k ) = n k for k = 1, 2 and let K 1 ∈ R n 2 ×n 1 and K 2 ∈ R n 1 ×n 2 be the coupling matrices. Before stating the general definitions, we first clarify how the newly introduced matrices are constructed through a simple self-explanatory example.
6 } which are written in nested multi-tuple format (corresponding to each mode of the LSS):
Mode 1 : µ (1) 1 = µ (1) 1 , µ (1) 2 , µ (1) 3 µ (2) 1 = µ (2) 1 , µ (2) 2 , µ (2) 3 , µ (2) 1 , µ (2) 4 , µ (2) 5 Mode 2 : µ (1) 2 = µ (1) 2 , µ (1) 1 , µ (1) 4 , µ (2) 2 = µ (2) 2 , µ (2) 1 , µ (2) 4 , µ (2) 2 , µ (2) 3 , µ (2) 6
We explicitly write the generalized observability matrices O 1 and O 2 as follows:
O 1 = C 1 Φ 1 (µ (1) 1 ) C 2 Φ 2 (µ (1) 2 ) K 1 Φ 1 (µ (1) 3 ) C 1 Φ 1 (µ (2) 1 ) C 2 Φ 2 (µ (2) 2 ) K 1 Φ 1 (µ (2) 3 ) C 1 Φ 1 (µ (2) 1 ) K 2 Φ 2 (µ (2) 4 ) K 1 Φ 1 (µ (2) 5 ) , O 2 = C 2 Φ 2 (µ (1) 2 ) C 1 Φ 1 (µ (1) 1 ) K 2 Φ 2 (µ (1) 4 ) C 2 Φ 2 (µ (2) 2 ) C 1 Φ 1 (µ (2) 1 ) K 2 Φ 2 (µ (2) 4 ) C 2 Φ 2 (µ (2) 2 ) K 1 Φ 1 (µ (2) 3 ) K 2 Φ 2 (µ (2) 6 ) Definition 4.
2 Given a non-empty set Q, denote with Q i the set of all i th tuples with elements from Q. Introduce the concatenation of two tuples composed of elements (symbols) α 1 , . . . , α i , and β 1 , . . . , β j from Q as the mapping : Q i × Q j → Q i+j with the following property:
α 1 , α 2 , . . . , α i β 1 , β 2 , . . . , β j = α 1 , α 2 , . . . α i , β 1 , β 2 , . . . β j ,
Remark 4.1 In the following we denote the th element of the ordered set µ
(i) j with µ (i) j ( ) (where j ∈ Q and i 1). For instance, µ(2)1 (3) := µ (2) 1 , µ (2) 4 , µ (2) 5
. For simplicity, use the notation
H 1,2,1 (µ (2) 1 , µ (2) 4 , µ (2) 5 ) instead of H (1,2,1) (µ (2) 1 , µ (2) 4 , µ (2) 5 ) .
Definition 4.3
We define the nested right multi-tuples:
λ 1 = λ (1) 1 , λ (2) 1 , . . . , λ (k † ) 1 , λ 2 = λ (1) 2 , λ (2) 2 , . . . , λ (k † ) 2 (11)
composed of sets of right i th tuples:
λ (i) 1 = λ (i) 1 , λ (i) 3 , λ (i) 2 , λ (i) 5 , λ (i) 4 , λ (i) 1 , . . . λ (i) 2m i −3 , . . . , λ (i) 4 , λ (i) 1 , λ (i) 2m i −1 , λ (i) 2m i −2 , . . . , λ (i) 3 , λ (i) 2 , λ (i) 2 = λ (i) 2 , λ (i) 4 , λ (i) 1 , λ (i) 6 , λ (i) 3 , λ (i) 2 , . . . λ (i) 2m i −2 , . . . , λ (i) 3 , λ (i) 2 , λ (i) 2m i , λ (i) 2m i −3 , . . . , λ (i) 4 , λ (i) 1 (12)
That is, we construct the right tuples based on the following recurrence relations ( where λ (i)
1 (1) = λ (i) 1 and λ (i) 2 (1) = λ (i) 2 ) λ (i) 1 (g) = λ (i) 2g−1 λ (i) 2 (g − 1), λ (i) 2 (g) = λ (i) 2g λ (i) 1 (g − 1),(13)
for λ
(i) 2g−1 ∈ C, 1 < g m i , i = 1, . . . , k † , m i 1 so that the equality m 1 + · · · + m k † = k holds.
Also, introduce the nested left multi-tuples
µ 1 = µ (1) 1 , µ(2)1 , . . . , µ ( † ) 1 , µ 2 = µ (1) 2 , µ(2)2 , . . . , µ ( † ) 2(14)
composed of sets of left j th tuples
µ (j) 1 = µ (j) 1 , µ (j) 2 , µ (j) 3 , µ (j) 1 , µ (j) 4 , µ (j) 5 , . . . µ (j) 1 , µ (j) 4 , . . . , µ (j) 2p j −3 , µ (j) 2 , µ (j) 3 , . . . , µ (j) 2p j −2 , µ (j) 2p j −1 µ (j) 2 = µ (j) 2 , µ (j) 1 , µ (j) 4 , µ (j) 2 , µ (j) 3 , µ (j) 6 , . . . µ (j) 2 , µ (j)
That is, we construct the left tuples based on the following recurrence relations (where µ (j)
1 (1) = µ (j) 1 and µ (j) 2 (1) = µ (j) 2 ) µ (j) 1 (h) = µ (j) 2 (h − 1) µ (j) 2h−1 , µ (j) 2 (h) = µ (j) 1 (h − 1) µ (j) 2h ,(16)
for µ
(j) 2h−1 ∈ C, 1 < h p j , j = 1, . . . , † , p j 1 so that p 1 + · · · + p † = .
Given that the following conditions are satisfied for all i = 1, . . . , k † and g = 1, . . . , m i ,
λ (i) 2g−1 / ∈ eig(A 1 , E 1 ), λ (i) 2g / ∈ eig(A 2 , E 2 )(17)
associate the following matrices to the set of right tuples in (12), as
R (i) 1 = Φ 1 (λ (i) 1 ) B 1 , Φ 1 (λ (i) 3 ) K 2 Φ 2 (λ (i) 2 ) B 2 , . . . , Φ 1 (λ (i) 2m i −1 ) K 2 · · · K 1 Φ 1 (λ (i) 3 ) K 2 Φ 2 (λ (i) 2 ) B 2 , R (i) 2 = Φ 2 (λ (i) 2 ) B 2 , Φ 2 (λ (i) 4 ) K 1 Φ 1 (λ (i) 1 ) B 1 , . . . , Φ 2 (λ (i) 2m i ) K 1 · · · K 2 Φ 2 (λ (i) 2 ) K 1 Φ 1 (λ (i) 1 ) B 1 , where i = 1, . . . , k † and R (i) q ∈ C nq×m i is attached to Λ (i) q for q ∈ {1, 2}.
The matrices:
R 1 = R (1) 1 , R (2) 1 , . . . , R (k † ) 1 ∈ C n 1 ×k , R 2 = R (1) 2 , R (2) 2 , . . . , R (k † ) 2 ∈ C n 2 ×k .(18)
are defined as the generalized controllability matrix of the LSS system Σ, associated with the right multi-tuple λ. Similarly, assuming that the following conditions are satisfied for all j = 1, . . . , † and h = 1, . . . , p j ,
µ (j) 2h−1 / ∈ eig(A 1 , E 1 ), µ (j) 2h / ∈ eig(A 2 , E 2 )(19)
associate the following matrices to the set of right tuples in (15), as
O (j) 1 = C 1 Φ 1 (µ (j) 1 ) C 2 Φ 2 (µ (j) 2 ) K 1 Φ 1 (µ (j) 3 )
. . .
C 2 Φ 2 (µ (j) 2 ) K 1 Φ 1 (µ (j) 3 ) K 2 · · · K 1 Φ 1 (µ (j) 2p j −1 ) ∈ C p j ×n 1 , j = 1, . . . , † , O (j) 2 = C 2 Φ 2 (µ (j) 1 ) C 1 Φ 1 (µ (j) 1 ) K 2 Φ 2 (µ (j) 4 )
. . .
C 1 Φ 1 (µ (j) 1 ) K 2 Φ 2 (µ (j) 2 ) K 1 · · · K 2 Φ 2 (µ (j) 2p j ) ∈ C p j ×n 2 , j = 1, . . . , † ,
and the generalized observability matrices:
O 1 = O (1) 1 . . . O ( † ) 1 ∈ C ×n 1 , O 2 = O (1) 2 . . . O ( † ) 2 ∈ C ×n 2 .(20)
Definition 4.4 For ν ∈ {1, 2}, let Q ν,+ and Q +,ν be the ordered sets containing all tuples that can be constructed with symbols from the alphabet Q = {1, 2} and that start (and respectively end) with the symbol ν. Also, no two consecutive characters are allowed to be the same. Hence, explicitly write the new introduced sets as follows:
Q 1,+ = {(1), (1, 2), (1, 2, 1), . . .}, Q 2,+ = {(2), (2, 1), (2, 1, 2), . . .},(21)Q +,1 = {(1), (2, 1), (1, 2, 1), . . .}, Q +,2 = {(2), (1, 2), (2, 1, 2), . . .}(22)
Remark 4.2 In the following we denote the th element of the ordered set Q ν,+ with Q ν,+ ( ). For example, one writes Q 1,+ (4) := (1, 2, 1, 2). Moreover, we have Q +,2 (3) Q 1,+ (2) = (2, 1, 2, 1, 2).
The compact notation H Q +,1 (µ 1 (2)) is used instead of H 2,1 (µ 2 , µ 3 ), where µ 1 (2) := µ 2 , µ 3 Definition 4.5 Let the i th unit vector be denoted with e i = [0 . . . , 1, . . . , 0] T ∈ R k .
In some contexts we may use the alternative notation e i,k to stress the fact the vector has dimension k. Also let 0 k, ∈ R k× be an all zero matrix. Hence, use the notation 0 k = [0, 0, . . . , 0] T ∈ R k for the zero valued vector of size k.
In the sequel, denote withĤ the generalized transfer functions corresponding to a LSSΣ.
Definition 4.6 We say that a LSSΣ = (k, k, {(Ê i , i ,B i ,Ĉ i } 2 i=1 , {K i,j } 2 i,j=1 , 0) matches the data associated with the right tuples {λ (1) a , . . . , λ (k † ) a } as well as left tuples {µ (1) b , . . . , µ ( † ) b }, a, b = 1, 2 and corresponding to the original LSS Σ = (n 1 , n 2 , {(E i , A i , B i , C i )} 2 i=1 , {K i,j } 2 i,j=1 , 0), if the following 2(k 2 + 2k) relations H Q +,1 (h) (µ (j) 1 (h)) =Ĥ Q +,1 (h) (µ (j) 1 (h)), H Q +,2 (h) (µ (j) 2 (h)) =Ĥ Q +,2 (h) (µ (j) 2 (h)), H Q 1,+ (g) (λ (i) 1 (g)) =Ĥ Q 1,+ (g) (λ (i) 1 (g)), H Q 2,+ (g) (λ (i) 2 (g)) =Ĥ Q 2,+ (g) (λ (i) 2 (g)), H Q +,1 (h) Q 2,+ (g) (µ (j) 1 (h) λ (i) 2 (g)) =Ĥ Q +,1 (h) Q 2,+ (g) (µ (j) 1 (h) λ (i) 2 (g)) H Q +,2 (h) Q 1,+ (g) (µ (j) 2 (h) λ (i) 1 (g)) =Ĥ Q +,2 (h) Q 1,+ (g) (µ (j) 2 (h) λ (i) 1 (g))(23)
hold for j = 1, . . . , k † , h = 1, . . . , p j and i = 1, . . . , k † , g = 1, . . . , m i , where
p 1 + p 2 + . . . p k † = m 1 + m 2 + . . . m k † = k.
The following lemma extends the rational interpolation idea for linear systems approximation to the linear switched system case.
Lemma 4.1 Interpolation of LSS. Let Σ = (n 1 , n 2 , {(E i , A i , B i , C i )} 2 i=1 , {K i,j } 2 i,j=1 , 0) be a LSS of order (n 1 , n 2 ). An order k reduced LSSΣ = (k, k, {(Ê i , i ,B i ,Ĉ i )} 2 i=1 , {K i,j } 2 i,j=1 , 0)
is constructed using the projection matrices chosen as in (18) and (20) for = k, i.e.
X 1 = R 1 , X 2 = R 2 and Y T 1 = O 1 , Y T 2 = O 2 Additionally assume rank(R i ) = rank(O i ) = k, i ∈ {1, 2}.
The reduced matrices corresponding to the I st subsystemΣ 1 are computed as,
E 1 = Y T 1 E 1 X 1 , 1 = Y T 1 A 1 X 1 ,B 1 = Y T 1 B 1 ,Ĉ 1 = C 1 X 1 ,K 1 = Y T 2 K 1 X 1 ,(24)
while the reduced matrices corresponding to the II nd subsystemΣ 2 can also be computed as,
E 2 = Y T 2 E 2 X 2 , 2 = Y T 2 A 2 X 2 ,B 2 = Y T 2 B 2 ,Ĉ 2 = C 2 X 2 ,K 2 = Y T 1 K 2 X 2 .(25)
It follows that the reduced-order systemΣ matches the data of the system Σ (as it was previously introduced in Definition 4.6).
Proof of Lemma 4.1 For simplicity, assume that we have one set of right multi-tuples, and one set of left multi-tuples with the same number of interpolation points k for each mode. Moreover let k be an even positive number. For the first mode, write down the interpolation nodes as follows:
λ 1 = λ 1 , λ 3 , λ 2 , . . . , λ 2k−1 , · · · , λ 3 , λ 2 , µ 1 = µ 1 , µ 2 , µ 3 , . . . , µ 2 , µ 3 , · · · , µ 2k−1 .(26)
For the second mode, write down the interpolation nodes as follows:
λ 2 = λ 2 , λ 4 , λ 1 , . . . , λ 2k , · · · , λ 2 , λ 1 , µ 2 = µ 2 , µ 1 , µ 4 , . . . , µ 1 , µ 4 , · · · , µ 2k .(27)
This corresponds to the case for which l = k, l † = k † = 1 and m 1 = p 1 = k. It follows that the interpolation conditions stated in Definition 4.6, can be rewritten by taking into account the aforementioned simplification as,
2k conditions: H Q +,1 (j) (µ 1 (j)) =Ĥ Q +,1 (j) (µ 1 (j)) H Q +,2 (j) (µ 2 (j)) =Ĥ Q +,2 (j) (µ 2 (j)) , j ∈ {1, . . . , k}(28)
2k conditions:
H Q 1,+ (i) (λ 1 (i)) =Ĥ Q 1,+ (i) (λ 1 (i)) H Q 2,+ (i) (λ 2 (i)) =Ĥ Q 2,+ (i) (λ 2 (i)) , i ∈ {1, . . . , k}(29)k 2 conditions: H Q +,1 (j) Q 2,+ (i) (µ 1 (j) λ 2 (i)) =Ĥ Q +,1 (j) Q 2,+ (i) (µ 1 (j) λ 2 (i)),(30)k 2 conditions: H Q +,2 (j) Q 1,+ (i) (µ 2 (j) λ 1 (i)) =Ĥ Q +,2 (j) Q 1,+ (i) (µ 2 (j) λ 1 (i))(31)
With the assumptions in (26) and (27), it follows that the associated generalized controllability and observability matrices defined previously in (18) and (20), are rewritten as:
R 1 = [ Φ 1 (λ 1 )B 1 , Φ 1 (λ 3 )K 2 Φ 2 (λ 2 )B 2 , . . . , Φ 1 (λ 2k−1 )K 2 · · · K 2 Φ 2 (λ 2 )B 2 ] ∈ C n×k , R 2 = [ Φ 2 (λ 2 )B 2 , Φ 2 (λ 4 )K 1 Φ 1 (λ 1 )B 1 , . . . , Φ 2 (λ 2k )K 1 · · · K 1 Φ 1 (λ 1 )B 1 ] ∈ C n×k , O 1 = C 1 Φ 1 (µ 1 ) C 2 Φ 2 (µ 2 )K 1 Φ 1 (µ 3 )
. . .
C 2 Φ 2 (µ 2 )K 1 Φ 1 (µ 3 ) · · · K 1 Φ 1 (µ 2k−1 ) , O 2 = C 2 Φ 2 (µ 2 ) C 1 Φ 1 (µ 1 )K 2 Φ 2 (µ 4 )
. . .
C 1 Φ 1 (µ 1 )K 2 Φ 2 (µ 4 ) · · · K 2 Φ 2 (µ 2k ) . with both O 1 , O 2 ∈ C k×n . Additionally, introduce the notationΦ i (s) = (sÊ −Â) −1 .
From (24) and (25), using that X i = R i for i = 1, 2, it readily follows that:
(a)Φ 1 (λ 1 )B 1 = e 1 and (b)Φ 1 (λ 2i−1 )K 2 e i−1 = e i , i = 2, . . . , k, (c)Φ 2 (λ 2 )B 2 = e 1 and (d)Φ 2 (λ 2i )K 1 e i−1 = e i , i = 2, . . . , k.
These equalities imply the right-hand conditions in (29). Similarly, from (24) and (25), using that
Y T j = O j for j = 1, 2, it follows that: (e) C 1Φ1 (µ 1 ) = e T 1 and (f ) e T j−1 K 2Φ2 (µ 2j ) = e T j , j = 2, . . . , k,(
Sylvester equations for O and R
The generalized controllabilty and observability matrices satisfy Sylvester equations. To state the corresponding result we need to define the following quantities. First introduce the matrices
R = e T 1,m 1 · · · e T 1,m k † ∈ R 1×k , L T = e T 1,p 1 · · · e T 1,p † ∈ R 1× ,(32)
and the block-shift matrices
S R = blkdiag J m 1 , . . . , J m k † , S L = blkdiag J T p 1 , . . . , J T p † . where J u = 0 1 · · · 0 . . . . . . . . . . . . 0 0 · · · 1 0 0 · · · 0 ∈ R u×u(33)
Finally we arrange the left interpolation points in the diagonal matrices as,
M 1 = blkdiag [M (1) 1 , M(2)1 , . . . , M ( † ) 1 ], M 2 = blkdiag [M (1) 2 , M(2)2 , . . . , M ( † ) 2 ],(34)
where M
(j) 1 = diag [µ (j) 1 , µ (j) 3 , . . . , µ (j) 2p j −1 ] and M (j) 2 = diag [µ (j) 2 , µ (j) 4 , . . . , µ(j)
2p j ]; we used the MATLAB notation 'blkdiag' which outputs a block diagonal matrix with each input entry as a block. Also arrange the right interpolation points in the diagonal matrices:
Λ 1 = blkdiag [Λ (1) 1 , Λ (2) 1 , . . . , Λ ( † ) 1 ], Λ 2 = blkdiag [Λ (1) 2 , Λ (2) 2 , . . . , Λ ( † ) 2 ],(35)
where Λ
(i) 1 = diag [λ (i) 1 , λ (i) 3 , . . . , λ (i) 2m i −1 ] and Λ (i) 2 = diag [λ (i) 2 , λ (i) 4 , . . . , λ (i)
2m i ]. In the definitions above, i.e. (32)- (35) we analyzed the general case, i.e., the assumptions made in (26)- (27) were no longer valid. We are now ready to state the following result.
Lemma 4.2
The generalized controllability matrices R 1 , R 2 defined by (18), satisfy the following Sylvester equations:
A 1 R 1 + K 2 R 2 S R + B 1 R = E 1 R 1 Λ 1 , A 2 R 2 + K 1 R 1 S R + B 2 R = E 2 R 2 Λ 2 .(36)
Proof of Lemma 4.2 Assume again, for simplicity of the proof, that the assumptions made in (26)- (27) are valid. Hence, we have one set of right multi-tuples for each of the two modes with same number of interpolation points k (with k even). Multiplying the first equation in (36) on the right with the first unit vector e 1 we obtain:
A 1 R (1) 1 + B 1 = λ 1 E 1 R (1) 1 ⇒ R (1) 1 = (λ 1 E 1 − A 1 ) −1 B 1 = Φ 1 (λ 1 )B 1 .(37)
where R (j) i is the j th column of R i (with j k and i ∈ {1, 2}). Thus the first column of the matrix which is the solution of the first equation in (36) is indeed equal to the first column of the generalized controllability matrix R 1 . Multiplying the second equation in (36) on the right with the first unit vector e 1 we obtain:
A 2 R (1) 2 + B 2 = λ 2 E 2 R (1) 2 ⇒ R (1) 2 = (λ 2 E 2 − A 2 ) −1 B 2 = Φ 2 (λ 2 )B 2 .(38)
Thus the first column of the matrix which is the solution of the second equation in (36) is indeed equal to the first column of the generalized controllability matrix R 2 . Multiplying first equation in (36) on the right with the j th unit vector e j , we obtain:
A 1 R (j) 1 + K 2 R (j−1) 2 = λ 2j−1 E 1 R (j) 1 ⇒ R (j) 1 = (λ 2j−1 E 1 − A 1 ) −1 K 2 R (j−1) 2(39)
Multiplying second equation in (36) on the right with the j th unit vector e j , we obtain:
A 2 R (j) 2 + K 1 R (j−1) 1 = λ 2j E 2 R (j) 2 ⇒ R (j) 2 = (λ 2j E 2 − A 2 ) −1 K 1 R (j−1) 1(40)
From (39) and (40) we write the following linear recursive system of equations:
R (j) 1 = Φ 1 (λ 2j−1 )K 2 R (j−1) 2 R (j) 2 = Φ 2 (λ 2j )K 1 R (j−1) 1(41)
with initial conditions (37) and (38). Hence, by solving the coupled system of equations, we indeed conclude that R 1 and R 2 matrices satisfying (36) are the generalized controllability matrices defined in (18) (for this particular choice of the right interpolation points). This proof can be nevertheless adapted from the simplified case in (26)- (27) to the more general case treated in Definition 4.3.
Remark 4.4
The generalized Sylvester equations in (36) can be compactly written as only one generalized Sylvester equation in the following way
A D R D + K D R D S D R + B D R D = E D R D Λ D(42)
where
X D = X 1 0 0 X 2 for X ∈ {R, A, B, E, R, Λ} and K D = 0 K 2 K 1 0 , S D R = 0 S R S R 0 .L R = I ⊗ A D − Λ D ⊗ E D + S D R T ⊗ K D ,
is invertible, i.e. have no zero eigenvalues (where ⊗ denotes the Kronecker product).
O 1 A 1 + S L O 2 K 1 + LC 1 = M 1 O 1 E 1 O 2 A 2 + S L O 1 K 2 + LC 2 = M 2 O 2 E 2(43)O D A D + S D L O D K D + L D C D = M D O D E D(44)
where X D =
L R = A D T ⊗ I − E D T ⊗ M D + K D T ⊗ S D L
is invertible, i.e. have no zero eigenvalues (where ⊗ denotes the Kronecker product).
The generalized Loewner pencil
Definition 4.7 Given a linear switched system Σ as defined in (1), let {R 1 , R 2 } and {O 1 , O 2 } be the controllability and observability matrices defined in (18), (20) respectively, and associated with the multi-tuples in (11), (14) respectively. The Loewner matrices L 1 and L 2 are defined as
L 1 = −O 1 E 1 R 1 , L 2 = −O 2 E 2 R 2 .(45)
Additionally, the shifted Loewner matrices L s1 and L s2 are defined as
L s1 = −O 1 A 1 R 1 , L s2 = −O 2 A 2 R 2 .(46)
Also define the quantities
W 1 = C 1 R 1 W 2 = C 2 R 2 , V 1 = O 1 B 1 V 2 = O 2 B 2 and Ξ 1 = O 2 K 1 R 1 Ξ 2 = O 1 K 2 R 2 .(47)
Remark 4.7 In general, the Loewner matrices defined above need not have only real entries. For instance, it may happen that the samples points are purely imaginary values (on the jω axis).
In this case, we refer the readers to Section 4.3.1 in [3]. We propose a similar method to enforce all system matrices have only real entries. In short, the sampling points have to be chosen as complex conjugate pairs; after the data is arranged into matrix format, use projection matrices as in equation (4.26) in [3] to multiply the matrices in (45), (46) and (47) to the left and to the right. In this way, the LSS does not change as pointed out in Definition 4.1.
Remark 4.8 Note that L k and L sk (where k ∈ {1, 2}), as defined above, are indeed Loewner matrices, that is, they can be expressed as divided differences of appropriate transfer function values of the underlying LSS (see the following example).
Example 4.2 Given the LSS described by (C j , E j , A j , B j ) (D = 2 and j ∈ {1, 2}), consider the ordered tuples of left interpolation points: (µ 1 ), (µ 2 , µ 3 ) , (µ 2 ), (µ 1 , µ 4 ) and right interpolation points (λ 1 ), (λ 3 , λ 2 ) , (λ 2 ), (λ 4 , λ 1 ) . The associated generalized observability and controllability matrices are computed as follows
O 1 = C 1 Φ 1 (µ 1 ) C 2 Φ 2 (µ 2 )K 1 Φ 1 (µ 3 ) , O 2 = C 2 Φ 2 (µ 2 ) C 1 Φ 3 (µ 1 )K 2 Φ 2 (µ 4 ) R 1 = Φ 1 (λ 1 )B 1 Φ 1 (λ 3 )K 2 Φ 2 (λ 2 )B 2 , R 2 = Φ 2 (λ 2 )B 2 Φ 2 (λ 4 )K 1 Φ 1 (λ 1 )B 1
The projected Loewner matrices can be written in terms of the samples in the following way:
L 1 = H 1 (µ 1 )−H 1 (λ 1 ) µ 1 −λ 1 H 1,2 (µ 1 ,λ 2 )−H 1,2 (λ 3 ,λ 2 ) µ 1 −λ 3 H 2,1 (µ 2 ,µ 3 )−H 2,1 (µ 2 ,λ 1 ) µ 3 −λ 1 H 2,1,2 (µ 2 ,µ 3 ,λ 2 )−H 2,1,2 (µ 2 ,λ 3 ,λ 2 ) µ 3 −λ 3 = −O 1 E 1 R 1 , L 2 = H 2 (µ 2 )−H 2 (λ 2 ) µ 2 −λ 2 H 2,1 (µ 2 ,λ 1 )−H 2,1 (λ 4 ,λ 1 ) µ 2 −λ 4 H 1,2 (µ 1 ,µ 4 )−H 1,2 (µ 1 ,λ 2 ) µ 4 −λ 2 H 1,2,1 (µ 1 ,µ 4 ,λ 4 )−H 1,2,1 (µ 1 ,λ 4 ,λ 1 ) µ 4 −λ 4 = −O 2 E 2 R 2 15
The projected shifted Loewner matrices can also be written in terms of the samples as:
L s1 = µ 1 H 1 (µ 1 )−λ 1 H 1 (λ 1 ) µ 1 −λ 1 µ 1 H 1,2 (µ 1 ,λ 2 )−λ 3 H 1,2 (λ 3 ,λ 2 ) µ 1 −λ 3 µ 3 H 2,1 (µ 2 ,µ 3 )−λ 1 H 2,1 (µ 2 ,λ 1 ) µ 3 −λ 1 µ 3 H 2,1,2 (µ 2 ,µ 3 ,λ 2 )−λ 3 H 2,1,2 (µ 2 ,λ 3 ,λ 2 ) µ 3 −λ 3 = −O 1 A 1 R 1 , L s2 = µ 2 H 2 (µ 2 )−λ 2 H 2 (λ 2 ) µ 2 −λ 2 µ 2 H 2,1 (µ 2 ,λ 1 )−λ 4 H 2,1 (λ 4 ,λ 1 ) µ 2 −λ 4 µ 4 H 1,2 (µ 1 ,µ 4 )−λ 2 H 1,2 (µ 1 ,λ 2 ) µ 4 −λ 2 µ 4 H 1,2,1 (µ 1 ,µ 4 ,λ 4 )−λ 4 H 1,2,1 (µ 1 ,λ 4 ,λ 1 ) µ 4 −λ 4 = −O 2 A 2 R 2 ,
The same property applies for the V i and W j vectors and Ξ j matrices:
V 1 = H 1 (µ 1 ) H 2,1 (µ 2 , µ 3 ) = O 1 B 1 , V 2 = H 2 (µ 2 ) H 1,2 (µ 1 , µ 4 ) = O 2 B 2 , W 1 = H 1 (λ 1 ) H 1,2 (λ 3 , λ 2 ) = C 1 R 1 , W 2 = H 2 (λ 2 ) H 2,1 (λ 4 , λ 1 ) = C 2 R 2 , Ξ 1 = H 2,1 (µ 2 , λ 1 ) H 2,1,2 (µ 2 , λ 3 , λ 2 ) H 1,2,1 (µ 1 , µ 4 , λ 1 ) H 1,2,1,2 (µ 1 , µ 4 , λ 3 , λ 2 ) = O 2 K 1 R 1 , Ξ 2 = H 1,2 (µ 1 , λ 2 ) H 1,2,1 (µ 1 , λ 4 , λ 1 ) H 2,1,2 (µ 2 , µ 3 , λ 2 ) H 2,1,2,1 (µ 2 , µ 3 , λ 4 , λ 1 ) = O 1 K 2 R 2
It readily follows that, given the original system Σ, a reduced LSS of order two can be obtained without computation (matrix factorizations or solves) as: E k = OER, = OAR,N = ONR,B = OB,Ĉ = CR.
This reduced system matches sixteen moments of the original system, namely:
four of H 1 /H 2 : H 1 (µ 1 ), H 2 (µ 2 ), H 1 (λ 1 ), H 2 (λ 2 ), three of H 1,2 : H 1,2 (µ 1 , µ 4 ), H 1,2 (µ 1 , λ 2 ), H 1,2 (λ 3 , λ 2 ), three of H 2,1 : H 2,1 (µ 2 , µ 3 ), H 2,1 (µ 2 , λ 1 ), H 2,1 (λ 4 , λ 1 ), . . .
one of H 1,2,1,2 : H 1,2,1,2 (µ 1 , µ 4 , λ 3 , λ 2 ) one of H 2,1,2,1 : H 2,1,2,1 (µ 2 , µ 3 , λ 4 , λ 1 )
i.e. in total 2(2k + k 2 ) = 16 moments are matched using this procedure.
Properties of the Loewner pencil
We will now show that the quantities defined earlier satisfy various equations which generalize the ones in the linear or bilinear case. The equations that are be presented in this section are used to automatically find the Loewner and shifted Loewner matrices by means of solving Sylvester equations (instead of building the divided difference matrices from the computed samples at the sampling points).
Proposition 4.3
The Loewner matrix L 1 and the shifted Loewner matrix L s1 (corresponding to mode 1) satisfy the following relations (where L, R, Λ k , M k , S L , S R are given in (32), (33) and (34)):
L s1 = L 1 Λ 1 + V 1 R + Ξ 2 S R (49) L s1 = M 1 L 1 + LW 1 + S L Ξ 1 (50)
The Loewner matrix L 2 and the shifted Loewner matrix L s2 (corresponding to mode 2) satisfy the following relations:
L s2 = L 2 Λ 2 + V 2 R + Ξ 1 S R (51) L s2 = M 2 L 2 + LW 2 + S L Ξ 2(52)
Proof of Proposition 4.3 By multiplying the first equation in (36) with O 1 to the left we obtain:
O 1 A 1 R 1 + O 1 K 2 R 2 S R + O 1 B 1 R = O 1 E 1 R 1 Λ 1 ⇒ −L s1 + Ξ 2 S R + V 1 R = −L 1 Λ 1
and hence relation (49) is proven. Similarly we prove (51). By multiplying the first equation in (43) with R 1 to the right we obtain:
O 1 A 1 R 1 + S L O 2 K 1 R 1 + LC 1 R 1 = M 1 O 1 E 1 R 1 ⇒ −L s1 + S L Ξ 1 + LW 1 = −M 1 L 1
and hence relation (50) is proven. Similarly we prove (52).
Proposition 4.4
The Loewner matrices L 1 and L 2 satisfy the following Sylvester equations:
M 1 L 1 − L 1 Λ 1 = (V 1 R − LW 1 ) + (Ξ 2 S R − S L Ξ 1 ),(53)M 2 L 2 − L 2 Λ 2 = (V 2 R − LW 2 ) + (Ξ 1 S R − S L Ξ 2 ).(54)
Proof of Proposition 4.4 By subtracting equation (49) from (50) we directly obtain (53) and also, by subtracting equation (51) from (52) we directly obtain (54).
Proposition 4.5
The shifted Loewner matrices L s1 and L s2 satisfy the following Sylvester equations:
M 1 L s1 − L s1 Λ 1 = (M 1 V 1 R − LW 1 Λ 1 ) + (M 1 Ξ 2 S R − S L Ξ 1 Λ 1 ),(55)M 2 L s2 − L s2 Λ 2 = (M 2 V 2 R − LW 2 Λ 2 ) + (M 2 Ξ 1 S R − S L Ξ 2 Λ 2 ).(56)
Proof of Proposition 4.3 By subtracting equation (49) after being multiplied with M 1 to the left from equation (50) after being multiplied with Λ 1 to the right, we directly obtain (55). Similar procedure is applied to prove (56).
Construction of reduced order models
As we already noted, the interpolation data for the LSS case is significantly different than the one used for the linear case without switching, as higher order transfer function values are matched as shown in the previous sections. However, the rest of the procedure remains more or less unchanged.
Lemma 4.4 Assume that k = and that none of the interpolation points λ i , µ j are eigenvalues of the pencils (L s1 , L 1 ) and (L s2 , L 2 ). Moreover, consider the Loewner matrices L 1 and L 2 to be invertible. Then, a realization of a reduced order LSSΣ that matches the data of the original LSS Σ (as introduced in Definition 4.6) is given by the following matrices,
Ê 1 = −L 1 , 1 = −L s1 ,B 1 = V 1 ,Ĉ 1 = W 1 , E 2 = −L 2 , 2 = −L s2 ,B 2 = V 2 ,Ĉ 2 = W 2 andK 1 = Ξ 1 ,K 2 = Ξ 2 .
If k = n, then the proposed realization is equivalent to the original one (as in Definition 4.1).
Proof of Lemma 4.4 This result directly follows from Lemma 4.1 by taking into consideration the notations introduced in (45-47).
In the case of redundant data, at least one of the pencils (L sj , L j ) is singular (for j ∈ {1, 2}), and hence construct pairs of projectors (X j , Y j ) (corresponding to mode j) similar to (10). The MOR procedure for approximate data matching is presented as follows.
Procedure 1 Consider the rank revealing singular value factorization of the Loewner matrices,
L j = Y jỸj S j O OS j X jXj T = Y j S j X T j +Ỹ jSjX T j ,(57)
where Y j , X j ∈ R k×r j and S j ∈ R r j ×r j . The projected system matrices corresponding to subsystem Σ j are computed as,
E j = −Y T j L j X j , j = −Y T j L sj X j ,B j = Y T j V j ,Ĉ j = W j X j , for j ∈ {1
, 2} Moreover, the projected coupling matrices are computed in the following waŷ
K 1 = Y T 2 Ξ 1 X 1 ,K 2 = Y T 1 Ξ 2 X 2 .
By choosing r j as the numerical rank of the Loewner matrix L j (i.e. the largest neglected singular value corresponding to index r j +1 is less than machine precision ), ensure that theÊ j matrices are not singular. Hence, construct a reduced order LSS denoted withΣ, that approximately matches the data of the original LSS Σ. If the truncated singular values are all 0 (the ones on the main diagonal of the matricesS j ), then the matching is exact. We provide a qualitative rather than quantitative result for the projected Loewner case. The quality of approximation is directly linked to the singular values of the Loewner pencils which represent an indicator of the desired accuracy. For linear systems with no switching, an error bound is provided in [5] as a quantitative measure.
The dimensions of the subsystemsΣ 1 andΣ 2 , corresponding to the reduced order LSS, need not be the same (i.e. r 1 = r 2 ). In this case the coupling matrices are not square anymore.
The projectors are computed via singular value factorization of the Loewner matrices. The use of the Drazin or Moore-Penrose pseudo inverses also holds (as shown in [2]).
The Loewner framework for linear switched systemsthe general case
In this section we are mainly concerned with generalizing some of the results presented in Section 4. Most of the findings can be smoothly extended to the cases with more complex switching patterns (more modes). The key for this is enforcing a cyclic structure of the interpolation framework, so that, everything can be written in matrix equation format.
Definition 5.1 Let Γ and Θ be finite sets of tuples so that Γ, Θ ⊆ ∞ k=1 Q k × C k so that Γ has the prefix closure property, i.e.
(q 1 , q 2 , . . . , q i , λ 1 , . . . , λ i ) ∈ Γ ⇒ (q 2 , . . . , q i , λ 2 , . . . , λ i ) ∈ Γ ∀i 2 and Θ has the suffix closure property, i.e.
(q 1 , q 2 , . . . , q j , µ 1 , . . . , µ j ) ∈ Θ ⇒ (q 1 , . . . , q j−1 , µ 1 , . . . , µ j−1 ) ∈ Θ ∀j 2
Now consider the specific subset Γ q (for any q ∈ Q) of the set Γ in the following way:
Γ q = {(q 1 , q 2 , . . . , q i , λ 1 , . . . , λ i ) ∈ Γ | q 1 = q, i δ Γ }, δ Γ = max(|w|) w∈Γ /2
Denote the cardinality of Γ q with k q = card(Γ q ) and explicitly enumerate the elements of this set as follows:
Γ q = {w (1) q , w (2) q , . . . , w (kq) q }.
Consider the following function (mapping) r : Γ q → C nq×1 that maps a word form Γ q into a column vector of size n q : r((q, q 2 , . . . , q i , λ 1 , . . . ,
λ i )) = Φ q (λ 1 )K q 2 ,q Φ q 2 (λ 2 ) · · · K q i ,q i−1 Φ q i (λ i )B q i
Now we are ready to construct the reachability matrix R q corresponding to the mode q of the system Σ as follows:
R q = r(w (1) q ) r(w (2) q ) · · · r(w (kq) q ) ∈ C nq×kq(58)
Similarly, define the specific subset Θ q (for any q ∈ Q) of the set Θ in the following way:
Θ q = {(q 1 , q 2 , . . . , q j , µ 1 , . . . , µ j ) ∈ Γ | q j = q, j δ Θ }, δ Θ = max(|w|) w∈Θ /2
Consider the cardinality of Θ q to be the same as the one of Γ q , i.e. k q = card(Θ q ). Although this additional constraint is not necessarily needed, we would like to enforce the construction of reduced systems with square matrices A k and E k . Next we explicitly enumerate the elements of this set as follows:
Θ q = {v (1) q , v (2) q , . . . , v (kq) q }.
Consider the following mapping o : Θ q → C 1×nq that maps a word form Θ q into a row vector of size n q : o((q 1 , q 2 , . . . , q j−1 , q, µ 1 , . . . , µ j )) = C q 1 Φ q 1 (µ 1 )K q 2 ,q 1 Φ q 2 (µ 2 ) · · · K q,q j−1 Φ q (µ j ) Now we are ready to construct the observability matrix O q ∈ C kq×nq corresponding to the mode q of the system Σ as follows
O q = o(v (1) q ) T o(v (2) q ) T · · · o(v (kq) q ) T T ∈ C kq×nq(59)
Consider the following example to show how the general procedure is extended from the linear case (no switching) to the case when switching occurs. The set Γ is composed of three subsets Γ = Γ 1 Γ 2 Γ 3 which are defined in a cyclic way by imposing the previously defined suffix closure property, as follows
Γ 1 = {(1, λ 1 ), (1, 3, λ 4 , λ 3 ), (1, 3, 2, λ 7 , λ 6 , λ 2 )} Γ 2 = {(2, λ 2 ), (2, 1, λ 5 , λ 1 ), (2, 1, 3, λ 8 , λ 4 , λ 3 )} Γ 3 = {(3, λ 3 ), (3, 2, λ 6 , λ 2 ), (3, 2, 1, λ 9 , λ 5 , λ 1 )}
To the sets Γ j , we attach the reachability matrices R j defined as follows:
R 1 = Φ 1 (λ 1 ) Φ 1 (λ 4 )K 3,1 Φ 3 (λ 3 )B 3 Φ 1 (λ 7 )K 3,1 Φ 3 (λ 6 )K 2,3 Φ 2 (λ 2 )B 2 , R 2 = Φ 2 (λ 2 ) Φ 2 (λ 5 )K 1,2 Φ 1 (λ 1 )B 1 Φ 2 (λ 8 )K 1,2 Φ 1 (λ 4 )K 3,1 Φ 3 (λ 3 )B 3 , R 3 = Φ 3 (λ 3 ) Φ 3 (λ 6 )K 2,3 Φ 2 (λ 2 )B 2 Φ 3 (λ 9 )K 2,3 Φ 2 (λ 5 )K 1,2 Φ 1 (λ 1 )B 1
In the same manner, the set Θ is composed of three subsets Θ = Θ 1 Θ 2 Θ 3 which are again defined in a cyclic way by imposing the previously defined prefix closure property, as follows To the sets Θ i , we attach the observability matrices O i defined as follows:
Θ 1 = {(1, µ 1 ), (3, 1, µ 3 , µ 4 ), (1, 2, 1, µ 1 , µ 5 , µ 7 )} Θ 2 = {(2, µ 2 ), (1, 2, µ 1 , µ 5 ), (2, 3, 2, µ 2 , µ 6 , µ 8 )} Θ 3 = {(3,O 1 = C 1 Φ 1 (µ 1 ) C 3 Φ 3 (µ 3 )K 1,3 Φ 1 (µ 4 ) C 1 Φ 1 (µ 1 )K 2,1 Φ 2 (µ 5 )K 1,2 Φ 1 (µ 7 ) , O 2 = C 2 Φ 2 (µ 2 ) C 1 Φ 1 (µ 1 )K 2,1 Φ 2 (µ 5 ) C 2 Φ 2 (µ 2 )K 3,2 Φ 3 (µ 6 )K 2,3 Φ 2 (µ 8 ) O 3 = C 3 Φ 3 (µ 3 ) C 2 Φ 2 (µ 2 )K 3,2 Φ 3 (µ 6 ) C 3 Φ 3 (µ 3 )K 1,3 Φ 1 (µ 4 )K 3,1 Φ 3 (µ 9 )
Sylvester equations for R q and O q
In this section we would like to generalize the results presented in Lemma 4.2 and Lemma 4.3, and hence extend the framework to a general number of operational modes denoted with D.
Definition 5.2 Introduce the special concatenation of tuples composed of mixed elements (symbols) that are from two different sets (for example Q and C) as the mapping with the following property:
α 1 β 1 α 2 β 2 = α 1 α 2 (β 1 β 2 ,
where α k ∈ Q i k and β k ∈ C j k for k = 1, 2. i (k g ) ∈ R k i ×kg be constant matrices that contain only 0/1 entries constructed so that S (g) i (1) = 0 k i and for u = 2, . . . , k g , we write:
S (g) i (u) = e u−1,k i , if ∃λ ∈ C, s.t. w (u) g = (g,λ) w (u−1) i , 0 k i , else(60)
Also, introduce the matrices R (i) and Λ i that are defined similarly as in (32) and (35), i.e.,
R (i) = e T 1,m 1 · · · e T 1,m k † ∈ R 1×k i , Λ i = blkdiag [Λ (1) i , Λ (2) i , . . . , Λ (k † ) i ] ∈ R k i ×k i ,(61)
where the diagonal matrices Λ (a)
i , a = 1, . . . , k † contain the right interpolation points associated to mode i. For general cyclic structure incorporated of the set Γ, it follows that the reachability matrices R i ∈ R n i ×k i , 1 i D satisfy the following system of generalized Sylvester equations:
A 1 R 1 + D i=1 K i,1 R i S (1) i + B 1 R (1) = E 1 R 1 Λ 1 A 2 R 2 + D i=1 K i,2 R i S (2) i + B 2 R (2) = E 2 R 2 Λ 2 . . . A D R D + D i=1 K i,D R i S (D) i + B D R (D) = E D R D Λ D(62)
Note that S (i) i = 0 k i ,k i , and if k 1 = k 2 = · · · = k D = k, the above defined matrices S (g) i satisfy the following equality ∀g ∈ Q :
D i=1 S (g) i = blkdiag J m 1 , . . . , J m k †(63)
where J l is the Jordan block of size l defined in (33).
To directly find R g , g = 1, 2, 3 for the case presented in Example 5.1, we have to solve the following system of coupled generalized Sylvester equations
A 1 R 1 + K 3,1 R 3 S (1) 3 + B 1 R = E 1 R 1 Λ 1 A 2 R 2 + K 1,2 R 1 S (2) 1 + B 2 R = E 2 R 2 Λ 2 A 3 R 3 + K 2,3 R 2 S (3) 2 + B 3 R = E 3 R 3 Λ 3 where: Λ 1 = λ 1 0 0 0 λ 4 0 0 0 λ 7 , Λ 2 = λ 2 0 0 0 λ 5 0 0 0 λ 8 , Λ 3 = λ 3 0 0 0 λ 6 0 0 0 λ 9 , R = 1 0 0 , S(1)3 = S (2) 1 = S (3) 2 = 0 1 0 0 0 1 0 0 0 ,
This corresponds to the case k 1 = k 2 = k 3 = 3, k † = 1 and m 1 = 3.
Definition 5.4 For
h, j = 1, . . . , D, let T (h) j = T (h) j T (1) . . . T (h) j T (k h ) T ∈ R h × j be
constant matrices that contain only 0/1 entries constructed so that T (h) j T (1) = 0 j and for v = 2, . . . , k g , we write:
T (h) j T (v) = e v−1,k j , if ∃μ ∈ C, s.t. w (v) h = w (v−1) j (h,μ), 0 j , else(64)
Also, introduce the following matrices
L (j) T = e T 1,p 1 · · · e T 1,p † ∈ R 1× j , M j = blkdiag [M (1) j , M(2)
j , . . . , M
( † ) j ] ∈ R j × j ,(65)
where the diagonal matrices M
(v) j for v = 1, . . . , j contain the left interpolation points associated to mode j. For general cyclic structure incorporated by definition in the set Θ, one can conclude that the observability matrices O j ∈ R j ×n j , 1 j D satisfy the following system of generalized Sylvester equations:
O 1 A 1 + D j=1 T (1) j O j K 1,j + L (1) C 1 = M 1 O 1 E 1 O 2 A 2 + D j=1 T (2) j O j K 2,j + L (2) C 2 = M 2 O 2 E 2 . . . O D A D + D j=1 T (D) j O j K D,j + L (D) C D = M D O D E D(66)
Note that T (j) j = 0 j , j , and if 1 = 2 = · · · = D = , the square matrices T (h) j ∈ R × satisfy the following equality, ∀h ∈ Q:
D j=1 T (h) j = blkdiag J p 1 , . . . , J p l † T(67)
Again to find the matrices O h , h = 1, 2, 3 in Example 5.1, it is required to solve the following system of coupled generalized Sylvester equations
O 1 A 1 + T (1) 3 O 3 K 1,3 + T (1) 2 O 2 K 1,2 + LC 1 = M 1 O 1 E 1 O 2 A 2 + T (2) 1 O 1 K 2,1 + T (2) 3 O 3 K 2,3 + LC 2 = M 2 O 2 E 2 O 3 A 3 + T (3) 2 O 2 K 3,2 + T (3) 1 O 1 K 3,1 + LC 3 = M 3 O 3 E 3
where:
M 1 = µ 1 0 0 0 µ 4 0 0 0 µ 7 , M 2 = µ 2 0 0 0 µ 5 0 0 0 µ 8 , M 3 = µ 3 0 0 0 µ 6 0 0 0 µ 9 , T(1)3 = T (2) 1 = T (3) 2 = 0 0 0 1 0 0 0 0 0 , T(1)2 = T (2) 3 = T (3) 1 = 0 0 0 0 0 0 0 1 0 , L = e 1
This corresponds to the case 1 = 2 = 3 = 3, l † = 1 and p 1 = 3. Note that the relation in (67) hold, i.e., T
2 + T (1) 3 = T (2) 1 + T (2) 3 = T (3) 1 + T (3) 2 = J T 3 .(1)
The Loewner matrices
For the case of linear switched systems with D active modes, the generalization of the Loewner framework includes one important feature. Instead of only one pair of Loewner matrices (as in the linear case without switching which is covered in Section 3), we define a pair of Loewner matrices for each individual active mode; hence in total D pairs of Loewner matrices.
Definition 5.5 Given a linear switched system Σ, let {R i |i ∈ Q} and {O j |j ∈ Q} be the controllability and observability matrices associated with the multi-tuples Γ i and Θ j . The Loewner matrices {L i | i ∈ Q} are defined as
L 1 = −O 1 E 1 R 1 , L 2 = −O 2 E 2 R 2 , . . . , L D = −O D E D R D(68)
Additionally, the shifted Loewner matrices {L si | i ∈ Q} are defined as
L s1 = −O 1 A 1 R 1 , L s2 = −O 2 A 2 R 2 , . . . , L sD = −O D A D R D(69)
Also introduce the matrices ∀i, j ∈ Q
W i = C i R i , V j = O j B j , and Ξ i,j = O j K i,j R iM h L h − L h Λ h = (V h R − LW h ) + D j=1 Ξ j,h S (h) j − T (h) j Ξ h,j , h ∈ Q(70)
Proposition 5.2 The shifted Loewner matrices L sh satisfy the following Sylvester equations:
M h L sh − L sh Λ h = (M h V h R − LW h Λ h ) + D j=1 M h Ξ j,h S (h) j − T (h) j Ξ h,j Λ h , h ∈ Q (71) Remark 5.3
The proof of the results stated in (70)-(71) is performed in a similar manner as for the results obtained for the special case D = 2 in Section 4 (i.e. for (53)-(56)).
Construction of reduced order models
The general procedure for the case with D switching modes is more or less similar to the one covered in Section 4.3 (where D = 2).
Lemma 5.1 Let L j be invertible matrices for j = 1, . . . , D, such that none of the interpolation points λ i , µ k are eigenvalues of any of the Loewner pencils (L sj , L j ). Then, it follows that the matrices
{Ê j = −L j , 1 = −L sj ,B j = V j ,Ĉ j = W j ,K i,j = Ξ i,j }, i, j ∈ {1, . . . , D}
form a realization of a reduced order LSSΣ that matches the data of the original LSS Σ. If k j = n j for j = 1, . . . , D, the proposed realization is equivalent to the original one.
The concept of a LSS matching the data of another LSS in the case D > 2 is formulated in a similar manner as to the case D = 2 (i.e., which is covered in Definition 4.6). Also, the definition of equivalent LSS for the case D > 2 is formulated similarly as to Definition 4.1.
In the case of redundant data, at least one of the pencils (L sj , L j ) is singular (for j ∈ {1, . . . , D}). The main procedure is presented as follows.
Procedure 2 Consider the rank revealing singular value factorization of the Loewner matrices:
L j = Y jỸj S j O OS j X jXj T = Y j Σ j X T j +Ỹ jSjX T j(72)
where X j , Y j ∈ R k j ×r j , S j ∈ R r j ×r j and j = {1, . . . , D}. Here, choose r j as the numerical rank of the Loewner matrix L j (i.e. the largest neglected singular value corresponding to index r j + 1 is less than machine precision ). The projected system matrices computed aŝ
E j = −Y T j L j X j , j = −Y T j L sj X j ,B j = Y T j V j ,Ĉ j = W j X j , for j ∈ {1, .
. . , D} and the projected coupling matrices computed in the following waŷ
K i,j = Y T j Ξ i,j X i , ∀i, j ∈ {1, . . . , D},
form a realization of a reduced order LSS denoted withΣ that approximately matches the data of the original LSS Σ. Each reduced subsystemΣ j has dimension r j , j ∈ {1, . . . , D}.
Remark 5.4 If the truncated singular values are all 0 (the ones on the main diagonal of the matricesS j ), then the interpolation is exact.
Numerical experiments
In this section we illustrate the new method by means of three numerical examples. We use a certain generalization of the balanced truncation (BT) method for LSS (as presented in ( [27]) to compare the performance of our new introduced method. The main ingredient of the BT method is to compute the the controllability and observability gramians P i and Q i (where i ∈ {1, 2, . . . , D}) as the solutions of the following Lyapunov equations:
A i P i E T i + E i P i A T i + B i B T i = 0 (73) A T i Q i E i + E T i P i A i + C T i C i = 0 (74)
Balanced Truncation
In [27] it has been shown that, if certain conditions are satisfied, the technique of simultaneous balanced truncation can be applied to switched linear systems. Hence, in some special cases, the existence of a global transformation matrix V bal is guaranteed; it follows that:
V bal PV T bal = V −T bal QV −1 bal = U i(75)
where U i are diagonal matrices. Although conceptually attractive as a MOR method, in general the conditions are rather restrictive in practice. This motivates the search for a more general MOR approach for the case where simultaneous balancing cannot be achieved. The problem of finding a balancing transformation for a single linear system can be formulated as finding a nonsingular matrix such that the following cost function is minimized (see [1]):
f (V) = trace[VPV T + V −T QV −1 ](76)
For the class of LSS with distinct operational modes, we hence have to minimize not one but a number of D cost functions: Define the function f av as in [27]:
f i (V) = trace[VP i V T + V −T Q i V −1 ], i ∈ {1, 2, . . . , D}(77)f av (V) = 1 D D i=1 trace[VP i V T + V −T Q i V −1 ] = trace[VP av V T + V −T Q av V −1 ](78)
where
P av = 1 D D i=1 P i , Q av = 1 D D i=1 Q i ,(79)
In the case of LSS, the BT method computes a basis where the sum of the sum of the eigenvalues of P i and Q i over all modes is minimal. Hence, minimizing the proposed overall cost function provides a natural extension of classical BT to the case of LSS. It follows that the transformationṼ that minimizes the cost function in (78) is precisely the one which balances the pair (P av , Q av ) of average gramians.
By applyingṼ to the individual modes and truncating, a reduced order model is obtained. After applying the transformationṼ, the new state space representations of the individual modes need not be balanced. Nevertheless, as stated in [27], it is expected to be relatively close to being balanced.
First example
As first example we consider the simple model of an evaporator vessel from [28]. There is a constant inflow of liquid fin into a tank and an outflow that depends on the pressure in the tank and the Bernoulli resistance R b . To keep the level of fluid in the evaporator vessel at or below a pre-specified maximum, an overflow mechanism is activated when the level of fluid L in the evaporator exceeds the threshold value L th . This causes a flow through a narrow pipe with resistance R p and inertia I that builds up flow momentum p. The system is modeled in two distinct operation modes: mode 1, where there is no overflow (the fluid level is below the overflow level), and mode 2, where the overflow mechanism is active. The ordinary differential equations describing the system in the two operation modes are given by
I 0 0 C ṗ L = −R p 0 0 −1/R b ṗ L + 0 f in (mode 1) I 0 0 C ṗ L = −R p 1 −1 −1/R b ṗ L + 0 f in (mode 2)
Supposing the system is initially in mode 1, the inflow causes the tank to start filling, which causes an outflow through resistance Rb. In this mode the outflow through the narrow pipe is zero. If L exceeds the level L th , a switch from mode 1 to mode 2 occurs at the point in time when L = L th . In the following, use the parameters R b = 1, R p = 0.5, I = 0.5, C = 15, f in = 0.25, L th = 0.08 and compute the following system matrices:
Mode 1 : A 1 = −1 0 0 − 1 2 , B 1 = 0 1 , C = 1 2 1 2 Mode 2 : A 2 = −1 2 − 1 2 − 1 2 , B 2 = 0 1 , C 2 = 1 2 1 2
First consider the following tuples of left and right interpolation nodes for each mode:
λ 1 = {(−1.5), (−2, 1)} µ 1 = {(2), (0, 0.5)} , λ 2 = {(1), (1.5, −1.5)} µ 2 = {(0), (2, −0.5)}
Hence, following the procedure described in Section 4, we recover the following system matrices: ,B 2 = 1 13 30 ,
Ĉ 2 = 1 2 − 3 8
Note that the recovered realization is equivalent to the original one (no reduction has been enforced -the task was to recover the initial system). The coupling matrices are also computed:
K 1 = −1 − 1 3 − 13 30 − 17 180 ,K 2 = 11 60 − 5 36 1 2 − 5 12
Second example
For the next experiment, consider the CD player system from the SLICOT benchmark examples for MOR (see [12]). This linear system of order 120 has two inputs and two outputs. We consider that, at any given instance of time, only one input and one output are active (the others are not functional due to mechanical failure). For instance, consider mode j to be activated whenever the j th input and the j th output are simultaneously failing (where j ∈ {1, 2}).
In this way, we construct a LSS system with two operational modes. Both subsystems are stable SISO linear systems of order 120. This initial linear switched system Σ will be reduced by means of the Loewner framework to obtainΣ L and balanced truncation method proposed in [27] to obtainΣ B .
The frequency response of each original subsystem is depicted in Fig. 2. For the Loewner method, we choose 160 logarithmically distributed interpolation points inside [10 1 , 10 5 ]j. Fig. 3 shows the decay of the singular values of the Loewner matrices corresponding to both subsystems. We notice that the 70 th singular values attain machine precision.
ForΣ L we decide to truncate at order k = 28 for both reduced systems. The same truncation order is chosen forΣ B . Next we compare the quality of approximation of the frequency response corresponding to the original system with the responses of the two reduced systems.
In Fig. 4 the relative error in frequency domain is depicted for both MOR methods (Loewner and BT). Notice that the Loewner method produces better results especially in the range of the selected sampling points.
Also, compare the time domain response of the original linear switched system against the ones corresponding to the two reduced models. We use a sinusoidal signal as the control input. The switching times are randomly chosen within [0,10]s. The blue rectangular signal in Fig. 5 represents the switching signal. Notice that the output of the LSS is well approximated for both MOR methods, as it can be seen in the lower part of Fig. 5. Finally, by inspecting the time domain error between the original response and the responses coming from the two reduced models (depicted in Fig. 6), we notice that the Loewner method generally produces better results. The error curve corresponding to our proposed method is two orders of magnitude below the error curve corresponding to the BT method for most of the points on the time axis.
Third example
For the last experiment, consider a large scale LSS system constructed as in [21] from the original machine stand example given in [14].
In this example, the system variability is induced by a moving tool slide on the guide rails of the stand (see Fig. 7). The aim is to determine the thermally driven displacement of the machine stand structure. Following the model setting in [14], consider the heat equation with Robin boundary conditions. Using a finite element (FE) discretization and denoting the external influences as the system input z, we obtain the dynamical heat model
E thẋ (t) = A th (t)x(t) + B th (t)z(t)(80)
describing the deformation independent evolution of the temperature field x with the system matrices E th , A th (t) and B th (t). The variability of the model is described by time dependent matrices A th (t) and B th (t). This leads directly to the linear time varying system described by (80). Since model reduction for LTV systems is a highly storage consuming procedure, the authors in [21] exploit properties of the spatially semi-discretized model to set up a LSS consisting of LTI subsystems only. As described in [14], the guide rails of the machine stand are modeled as 15 equally distributed horizontal segments (see Fig. 7). Any of these segments is said to be completely covered by the tool slide if its midpoint lies within the height of the slide. On the other hand, each segment whose midpoint is not covered is treated as not in contact and therefore the slide always covers exactly 5 segments at each time. This in fact allows the stand to reach 11 distinct, discrete positions given by the model restrictions. In this way, one can define the subsystems of the LSS as follows:
Σ :
E thẋ = A th x + B th z y =Cx,(81)
where ∈ {1, ..., 11}. Note that the change of the input operator Bth(t) is hidden in the input z itself, since it is sufficient to activate the correct boundary parts by choosing the corresponding columns in Bth via the input z . Therefore, the input operator B th (t) := B th becomes constant and the input variability is represented by the input z :
z i := z i , segment i is in contact, 0, otherwise, , i = 1, ..., 15.
Here, z i ∈ R is the thermal input as described in [21]. The only varying part influencing the model reduction process left in the dynamical system is the system ma trix A th (t) := A th . After the finite element discretization was performed, we are given a SLS with 11 modes. Each subsystem has dimension n = 16626. The E and C matrices are the same for all modes of the SLS. The B matrices have 6 columns (corresponding to different inputs) and the C matrix has 9 rows (corresponding to different outputs).
For all the experiments performed, we take into consideration only two active modes (the first and the fifth). This corresponds to the particular case of D = 2 covered in Section 4.
Also, although our new introduced method can be easily generalized to the MIMO case, we consider only (for simplicity reasons) the first input and the first output for each of the two modes (the measurements used in the Loewner framework are hence scalar values).
Both subsystems are stable linear systems of order 16626 in sparse format. This initial large scale LSS will be again reduced (as in the second example) by means of the Loewner method and of the balanced truncation method proposed in [27]. The frequency response of each original subsystem is depicted in Fig. 8. Fig. 9. We notice that already the 70 th singular values attain machine precision.
For the Loewner reduced order LSS (i.e Σ 1 ), we decide to truncate at order k 1 = 26 for both subsystems (which are stable LTI's). The same truncation order is chosen for the reduced order model computed via BT. Next we compare the quality of approximation of the frequency response. In Fig. 10 the relative error in frequency domain is depicted for both MOR methods (Loewner and BT). Notice that the Loewner method generally produces better results. Also, compare the time domain response of the original LSS against the ones corresponding to the two reduced models. The same configuration is used for the switching signal as in the second example. Notice that the output of the LSS is well approximated for both MOR methods, as it can be seen in the lower part of Fig. 11. Finally, by inspecting the time domain relative error between the original response and the responses coming from the two reduced models (depicted in Fig. 12), we notice that the Loewner method generally produces better results. The error curve corresponding to our proposed method is below the error curve corresponding to the BT method for most of the points on the time axis.
Summary and conclusions
In this paper we address the problem of model reduction of linear switched systems from data consisting of values of high order transfer functions. The underlying philosophy of the Loewner framework is collect data and extract the desired information. Here he have extended this framework to the reduction of LSS. In general, for this type of systems, the data must be computed a priori, rather than measured (as for linear systems with no switching where one can use Vector Network Analyzers for instance). Having the required data, the next step would be to arrange it into matrix format. We have shown that the Loewner matrices (which basically represent the recovered E and A matrices of the underlying LSS) can be automatically calculated as solutions of Sylvester equations. In our framework, the transition/coupling matrices can be recovered from the given computed data as well. Since these matrices need not be square, they allow having different dimensions of the reduced state space in different modes. In a nutshell, given input/output data, we can construct with no extensive computation, a singular high order model in generalized (descriptor) state space form. In applications the singular pencil (L s , L) must be reduced at some stage. The singular values of the pencil (L s , L) offer a trade-off between accuracy of fit and complexity of the reduced system. This approach to model reduction, first developed for linear time-invariant systems (see [5] for a survey), was later extended to linear parametrized systems [4,19,20,18] and to bilinear systems [3].
Three numerical examples demonstrate the effectiveness of the proposed approach. The quality of approximation for the reduced models was determined by performing both frequency and time domain tests. We have chosen a generalization of the classical balanced truncation method to LSS for comparison purposes. As opposed to most of the balancing methods we came across in the literature ( [11], [8], [36] and [33]), the method we choose (i.e [27]) does not require solving systems of LMI (linear matrix inequalities) which might be difficult for very large systems such as the one in Section 6.3. The results of the new proposed method turned out to be better than the ones obtained when using the BT method.
L s satisfies the Sylvester equation L s Λ − ML s = MVR − LWΛ, and can be factored in terms of the generalized controllability/observabilty matrices as L s = −OAR. Finally notice that the following relations hold: V = CR, W = OB.
g) C 2Φ2 (µ 2 ) = e T 1 and (h) e T j−1 K 1Φ1 (µ 2j−1 ) = e T j , j = 2, . . . , k, which imply left-hand conditions in (28). Finally, with X = R, Y T = O, and combining (a)-(h), all interpolation conditions in (30) and (31) are hence satisfied. Remark 4.3 For instance, in Example 4.2, the conditions stated in (48) are satisfied.
Proposition 4. 1
1The pair of generalized Sylvester equations in(36) has unique solutions if the right interpolation points are chosen so that the Sylvester operator
Remark 4. 5
5The motivation behind this subsection is closely related to building parametrized reduced order models. The idea is that, one can also use only one sided interpolation conditions (either left as in 28 or right as in 29) to reduce the original LSS. The other degrees of freedom are given by free parameters. Further development of these Sylvester equation was studied in[3] (in Section 4.4) for the case of bilinear systems.
Lemma 4. 3
3The generalized observability matrices O 1 and O 2 defined by (20) satisfy the following generalized Sylvester equations:
Proof of Lemma 4. 3
3Similar to the proof of Lemma 4.2.
Remark 4. 6
6The generalized Sylvester equations in (43) can be compactly written as only one generalized Sylvester equation in the following way
Proposition 4. 2
2The pair of generalized Sylvester equations in(36) has unique solutions if the right interpolation points are chosen so that the Sylvester operator
Remark 4. 9
9The right hand side of the equations (53) -(56) contains constant 0/1 matrices (i.e. R, L, S R , S L ) as well as matrices (i.e. V j , W j , Ξ j , j ∈ {1, 2}) which are directly constructed by putting together the given samples values as pointed out in Example 4.2.
Example 5. 1
1Take D = 3 (3 active modes) and hence Q = {1, 2, 3}. The following interpolation points are given: {s 1 , s 2 , . . . , s 18 } ⊂ C. The first step is to partition this set into two disjoint subsets (each having 9 points): left interpolation points : {µ 1 , µ 2 , . . . , µ 9 } right interpolation points : {λ 1 , λ 2 , . . . , λ 9 }
µ 3 )
3, (2, 3, µ 2 , µ 6 ), (3, 1, 3, µ 3 , µ 4 , µ 9 )}
Definition 5. 3
3For g, i = 1, . . . , D, let S
Remark 5. 1
1The number of Loewner matrices, shifted Loewner matrices, W i row vectors and V j column vectors is the same as the number of active modes (i.e D). On the other hand, the number of matrices Ξ i,j increases quadratically with D (i.e in total D 2 matrices).Remark 5.2 Note that the matrices L i and L si as defined in (68) and (69) (for i ∈ {1, 2, . . . , D}) are indeed Loewner matrices, that is, they can be expressed as divided differences of generalized transfer function values of the underlying LSS. Proposition 5.1 The Loewner matrices L h satisfy the following Sylvester equations:
If the conditions of Corollary IV.3 from [27] hold, simultaneous balancing is possible, and there exists a transformation V which simultaneously minimizes f i for all i = 1, 2, . . . , D. Instead of having D functions as in (77), one can introduce a single overall cost function (i.e the average of the cost functions of the individual modes).
Figure 1 :
1Schematic of the evaporator vessel
Figure 2 :
2Frequency response of the original subsystems
Figure 3 :Figure 4 :
34Decay of the singular values of the Frequency domain approximation error
Figure 5 :Figure 6 :
56Time Time domain approximation error
Figure 7 :
7Schematic of the tool slide on the guide rails of the stand
Figure 8 :
8Frequency response of the original subsystems For the Loewner method, we choose 200 logarithmically spaced interpolation points inside [10 −5 , 10 3 ]j. The decay of the singular values of the Loewner matrices corresponding to both subsystems can be viewed in
Figure 9 :
9Decay of the singular values of the Loewner matrices
Figure 10 :
10Frequency domain approximation error
Figure 11 :
11Time domain simulation
Figure 12 :
12Time domain approximation error
, . . . , µ(j) 2p j −2 , µ (j) 1 , µ (j) 4 , . . . , µ (j) 2p j −3 , µ (j) 2p j(15)
AcknowledgementsThe first and third authors would like to thank Mihaly Petreczky, Mert Bastug and Rafael Wisniewski for suggesting the problem of model reduction of LSS and for useful discussions.Also, the authors want to thank Norman Lang for providing the LSS large scale system from Section 6.3 (used in[21]).Last but not least, we wish to thank Jens Saak for recommending the M-M.E.S.S. toolbox (see[35]) for efficiently computing solutions of sparse large scale algebraic equations (which was very helpful for implementing the BT method for the 16626 th order system).
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Generalized gramian framework for model/controller order reduction of switched systems. H R Shaker, R Wisniewski, International Journal of Systems Science. 42H. R. Shaker and R. Wisniewski, Generalized gramian framework for model/controller order reduction of switched systems, International Journal of Systems Science, 42 (2011), pp. 1277-1291.
Switched linear systems: control and design. Z Sun, S S Ge, Stability Theory of Switched Dynamical Systems. SpringerZ. Sun and S. S. Ge, Switched linear systems: control and design, Springer, 2005. [38] , Stability Theory of Switched Dynamical Systems, Springer, 2011.
Switched differential algebraic equations. S Trenn, Advances in Industrial Control, Dynamics and Control of Switched Electronic Systems. Springer VerlagS. Trenn, Switched differential algebraic equations, Advances in Industrial Control, Dynam- ics and Control of Switched Electronic Systems, Springer Verlag, 2012, ch. 6.
µ-Dependent model reduction for uncertain discrete-time switched linear systems with average dwell time. L Zhang, E Boukas, P Shi, International Journal of Control. 82L. Zhang, E. Boukas, and P. Shi, µ-Dependent model reduction for uncertain discrete-time switched linear systems with average dwell time, International Journal of Con- trol, 82 (2009), pp. 378-388.
H-infinity model reduction for uncertain switched linear discrete-time systems. L Zhang, P Shi, E Boukas, C Wang, Automatica. 44L. Zhang, P. Shi, E. Boukas, and C. Wang, H-infinity model reduction for uncertain switched linear discrete-time systems, Automatica, 44 (2008), pp. 2944-2949.
Stability analysis and H ∞ model reduction for switched discrete-time time-delay systems. L Zheng-Fan, C Chen-Xiao, D Wen-Yong, Mathematical Problems in Engineering. 15L. Zheng-Fan, C. Chen-Xiao, and D. Wen-Yong, Stability analysis and H ∞ model reduction for switched discrete-time time-delay systems, Mathematical Problems in Engineer- ing, 15 (2014).
|
[] |
[
"Strong Lyman continuum emitting galaxies show intense C iv λ1550 emission",
"Strong Lyman continuum emitting galaxies show intense C iv λ1550 emission"
] |
[
"D Schaerer \nObservatoire de Genève\nUniversité de Genève\nChemin Pegasi 511290VersoixSwitzerland\n\nCNRS\nIRAP\n14 Avenue E. Belin31400ToulouseFrance\n",
"Y I Izotov \nBogolyubov Institute for Theoretical Physics\nNational Academy of Sciences of Ukraine\n14-b Metrolohichna str03143KyivUkraine\n",
"G Worseck \nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/25D-14476PotsdamGermany\n",
"D Berg \nDepartment of Astronomy\nUniversity of Texas at Austin\n78712AustinTXUSA\n",
"J Chisholm \nDepartment of Astronomy\nUniversity of Texas at Austin\n78712AustinTXUSA\n",
"A Jaskot \nDepartment of Astronomy\nWilliams College\n01267WilliamstownMAUSA\n",
"K Nakajima \nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n",
"S Ravindranath \nSpace Telescope Science Institute\n3700 San Martin Drive Baltimore21218MDUSA\n",
"T X Thuan \nAstronomy Department\nUniversity of Virginia\nP.O. Box 40032522904-4325CharlottesvilleVAUSA\n",
"A Verhamme \nObservatoire de Genève\nUniversité de Genève\nChemin Pegasi 511290VersoixSwitzerland\n"
] |
[
"Observatoire de Genève\nUniversité de Genève\nChemin Pegasi 511290VersoixSwitzerland",
"CNRS\nIRAP\n14 Avenue E. Belin31400ToulouseFrance",
"Bogolyubov Institute for Theoretical Physics\nNational Academy of Sciences of Ukraine\n14-b Metrolohichna str03143KyivUkraine",
"Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/25D-14476PotsdamGermany",
"Department of Astronomy\nUniversity of Texas at Austin\n78712AustinTXUSA",
"Department of Astronomy\nUniversity of Texas at Austin\n78712AustinTXUSA",
"Department of Astronomy\nWilliams College\n01267WilliamstownMAUSA",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan",
"Space Telescope Science Institute\n3700 San Martin Drive Baltimore21218MDUSA",
"Astronomy Department\nUniversity of Virginia\nP.O. Box 40032522904-4325CharlottesvilleVAUSA",
"Observatoire de Genève\nUniversité de Genève\nChemin Pegasi 511290VersoixSwitzerland"
] |
[] |
Using the Space Telescope Imaging Spectrograph, we have obtained ultraviolet (UV) spectra from ∼ 1200 to 2000 Å of known Lyman continuum (LyC) emitting galaxies at low redshift (z ∼ 0.3 − 0.4) with varying absolute LyC escape fractions ( f esc ∼ 0.01 − 0.72). Our observations include in particular the galaxy J1243+4646, which has the highest known LyC escape fraction at low redshift. While all galaxies are known Lyman alpha emitters, we consistently detect an inventory of additional emission lines, including C iv λ1550, He ii λ1640, O iii] λ1666, and C iii] λ1909, whose origin is presumably essentially nebular. C iv λ1550 emission is detected above 4 σ in six out of eight galaxies, with equivalent widths of EW(C iv) = 12 − 15 Å for two galaxies, which exceeds the previously reported maximum emission in low-z star-forming galaxies. We detect C iv λ1550 emission in all LyC emitters with escape fractions f esc > 0.1 and find a tentative increase in the flux ratio C iv λ1550/C iii] λ1909 with f esc . Based on the data, we propose a new criterion to select and classify strong leakers (galaxies with f esc > 0.1): C iv λ1550/C iii] λ1909 > ∼ 0.75. Finally, we also find He ii λ1640 emission in all the strong leakers with equivalent widths from 3 to 8 Å rest frame. These are among the highest values observed in star-forming galaxies and are primarily due to a high rate of ionizing photon production. The nebular He ii λ1640 emission of the strong LyC emitters does not require harder ionizing spectra at > 54 eV compared to those of typical star-forming galaxies at similarly low metallicity.
| null |
[
"https://arxiv.org/pdf/2202.07768v1.pdf"
] | 246,867,212 |
2202.07768
|
c3a8013374cb4dbcd404e5cb62924a0e041a06e8
|
Strong Lyman continuum emitting galaxies show intense C iv λ1550 emission
February 17, 2022
D Schaerer
Observatoire de Genève
Université de Genève
Chemin Pegasi 511290VersoixSwitzerland
CNRS
IRAP
14 Avenue E. Belin31400ToulouseFrance
Y I Izotov
Bogolyubov Institute for Theoretical Physics
National Academy of Sciences of Ukraine
14-b Metrolohichna str03143KyivUkraine
G Worseck
Institut für Physik und Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24/25D-14476PotsdamGermany
D Berg
Department of Astronomy
University of Texas at Austin
78712AustinTXUSA
J Chisholm
Department of Astronomy
University of Texas at Austin
78712AustinTXUSA
A Jaskot
Department of Astronomy
Williams College
01267WilliamstownMAUSA
K Nakajima
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyoJapan
S Ravindranath
Space Telescope Science Institute
3700 San Martin Drive Baltimore21218MDUSA
T X Thuan
Astronomy Department
University of Virginia
P.O. Box 40032522904-4325CharlottesvilleVAUSA
A Verhamme
Observatoire de Genève
Université de Genève
Chemin Pegasi 511290VersoixSwitzerland
Strong Lyman continuum emitting galaxies show intense C iv λ1550 emission
February 17, 2022Astronomy & Astrophysics manuscript no. 43149corr_ds Letter to the Editor Accepted for publication in A&A LettersGalaxies: starburst -Galaxies: high-redshift -Cosmology: dark agesreionizationfirst stars -Ultraviolet: galaxies
Using the Space Telescope Imaging Spectrograph, we have obtained ultraviolet (UV) spectra from ∼ 1200 to 2000 Å of known Lyman continuum (LyC) emitting galaxies at low redshift (z ∼ 0.3 − 0.4) with varying absolute LyC escape fractions ( f esc ∼ 0.01 − 0.72). Our observations include in particular the galaxy J1243+4646, which has the highest known LyC escape fraction at low redshift. While all galaxies are known Lyman alpha emitters, we consistently detect an inventory of additional emission lines, including C iv λ1550, He ii λ1640, O iii] λ1666, and C iii] λ1909, whose origin is presumably essentially nebular. C iv λ1550 emission is detected above 4 σ in six out of eight galaxies, with equivalent widths of EW(C iv) = 12 − 15 Å for two galaxies, which exceeds the previously reported maximum emission in low-z star-forming galaxies. We detect C iv λ1550 emission in all LyC emitters with escape fractions f esc > 0.1 and find a tentative increase in the flux ratio C iv λ1550/C iii] λ1909 with f esc . Based on the data, we propose a new criterion to select and classify strong leakers (galaxies with f esc > 0.1): C iv λ1550/C iii] λ1909 > ∼ 0.75. Finally, we also find He ii λ1640 emission in all the strong leakers with equivalent widths from 3 to 8 Å rest frame. These are among the highest values observed in star-forming galaxies and are primarily due to a high rate of ionizing photon production. The nebular He ii λ1640 emission of the strong LyC emitters does not require harder ionizing spectra at > 54 eV compared to those of typical star-forming galaxies at similarly low metallicity.
Introduction
In recent years, Hubble Space Telescope (HST) observations of star-forming galaxies at low redshift (z ∼ 0.3−0.4) with the Cosmic Origin Spectrograph (COS) have measured Lyman continuum (LyC) and the non-ionizing ultraviolet (UV) radiation for nearly 90 galaxies in total, as reported by various studies (see Leitherer et al. 2016;Izotov et al. 2016aIzotov et al. ,b, 2018bWang et al. 2019;Izotov et al. 2021) and the Low-z Lyman Continuum Survey (LzLCS; Wang et al. 2021;Flury et al. 2022a,b;Saldana-Lopez et al. 2022).
These data have launched us into a new era, allowing us to study in detail LyC emitting galaxies, their interstellar medium (ISM) properties and stellar populations, the conditions that favor the escape of ionizing photons from galaxies, and more (see above references and Schaerer et al. 2016;Chisholm et al. 2018). In addition, these observations have served to empirically establish "indirect indicators" of LyC escape, which can potentially be used at all redshifts, including for galaxies in the epoch of reionization, where direct observations of the LyC are not possible. Among these indirect indicators are properties of the resolved Lyman alpha (Lyα) line profile (the peak separation), weak UV absorption lines, Mg ii emission, a high ratio of [O iii] λλ4959,5007/[O ii] λ3727, and a deficit of the [S ii] λλ6717,6731 line (see Jaskot & Oey 2013;Verhamme et al. 2017;Henry et al. 2018;Chisholm et al. 2020;Ramambason et al. 2020;Wang et al. 2021).
For the exploration of the early Universe and to track the sources of cosmic reionization, the rest-UV spectral range remains fundamental as it is accessible with the largest groundbased telescopes (and soon with 30 m class telescopes) and with the recently launched James Webb Space Telescope. Recent efforts have been undertaken to obtain reference spectra of low-redshift star-forming galaxies covering the range of ∼ 1200 − 2000 Å (e.g., Rigby et al. 2015;Berg et al. 2016;Senchyna et al. 2017;Berg et al. 2019b). These include Lyα, C iv λ1550, He ii λ1640, O iii] λλ1660,1666, C iii] λλ1907,1909, and other emission lines, which provide important diagnostics of their ISM and radiation field (see, e.g., Feltre et al. 2016;Gutkin et al. 2016;Nakajima et al. 2018). However, to study sources of reionization and to relate the UV line properties to LyC escape, Article number, page 1 of 6 arXiv:2202.07768v1 [astro-ph.GA] 15 Feb 2022 it is mandatory to observe the same galaxies both in the LyC and in the non-ionizing UV out to ∼ 2000 Å. So far, very few such observations have been obtained, and the first spectrum of a low-z LyC emitter with a high escape fraction of LyC photons ( f esc ∼ 43 %) has been obtained only recently .
Here, we present the first results of an HST program to observe the UV emission lines of known LyC emitters, sampling a range of LyC escape fractions from very low ( f esc = 1.4 %) to the highest escape currently known ( f esc = 72%), which was observed in the compact star-forming galaxy J1243+4646 by Izotov et al. (2018b). Our observations show strong emission lines, in particular strong C iv λ1550 emission in the strongest LyC emitters. This demonstrates for the first time a connection between LyC escape and nebular C iv emission, from which we propose a new empirical criterion to select galaxies with strong LyC escape. Furthermore, our results shed new light on the recently detected C iv emitters at high redshift (see Stark et al. 2015;Mainali et al. 2017;Schmidt et al. 2017;Tang et al. 2021;Vanzella et al. 2021;Richard et al. 2021), suggesting that some of them could be strong LyC emitters.
2. UV spectra of compact star-forming galaxies at z ∼ 0.3 − 0.4 with varying LyC escape fractions
HST observations
Eight out of eleven compact star-forming galaxies with LyC measurements from our 2016-2018 campaign (Izotov et al. 2016a(Izotov et al. ,b, 2018a have been observed in Cycle 27 (GO 15941; PI Schaerer) and earlier The observations were taken with the Space Telescope Imaging Spectrograph (STIS) NUV-MAMA using the grating G230L with the central wavelength 2376 Å and the slit 52 × 0 . 5, resulting in a spectral resolution of R ∼ 700 for the targeted compact galaxies. The data were reduced with the CALSTIS v3.4.2 pipeline and including data from additional observations of J1154+2443 that were not available to Schaerer et al. (2018), which improved the signal-to-noise ratio (S/N) to 7 per pixel for this galaxy. Wavelength shifts between sub-exposures were insignificant, which allowed for the co-addition of gross and background counts per pixel, approximately preserving Poisson statistics. In the wavelength range of interest, the continuum S/N (accounting for Poisson flux uncertainties) ranges between 3 and 8 per 1.55 Å pixel, with six out of eight spectra reaching S/N> 5.
For the remainder of this Letter, we define "strong" leakers as galaxies with a LyC escape fraction above 10 % (i.e., f esc > 0.1) to distinguish such sources, which could significantly contribute to cosmic reionization, from those with a low or negligible escape of ionizing photons. In our current sample, three out of eight galaxies are classified as strong leakers (see Fig. 2.2).
Main UV emission line features
Examples of the observed STIS spectra are shown in Fig. 2 They are ordered by decreasing f esc from top to bottom, except for J1248+4259, which is shown at the top with a dotted line (see text for details). The colored area shows the spectra of the three strong LyC leakers ( f esc > 0.1). The C iv λ1550, C iii] λ1909, and He ii λ1640 lines are clearly detected in the strongest LyC leakers ( J1243+4646, J1154+2443, and J1152+3400, with escape fractions between f esc = 0.13 and 0.72), which are characterized by a high C iv/C iii] ratio. All the spectra show C iii] λ1909 emission (with EWs of ∼ 5 − 20 Å rest frame) and other weaker lines.
fractions. In the spectral range covered by the observations (restframe wavelengths of ∼ 1200−2000 Å), the main detected emission lines are Lyα (not illustrated here), C iv λ1550, He ii λ1640, O iii] λ1666, and C iii] λ1909. We note that with the given resolution, the C iv λ1550 and C iii] λ1909 doublets are not resolved. Hints of blended Si iii] λ1883,1892 are also visible in some of the spectra (see Fig. 2.1). We now mainly focus on the carbon and helium lines, and leave a more exhaustive report and analysis for subsequent publications. We note that the low resolution of our spectra does not allow us to exclude some contribution from stellar emission (from O, B, and Wolf-Rayet stars) to the C iv λ1550 and He ii λ1640 emission lines. However, we think that the emission is predominantly nebular in both cases, at least for sources with high equivalent widths (EWs) (EW(C iv) > ∼ 3 Å), since absorption from stellar P-Cygni C iv profiles is not detected in high EW sources and since no Wolf-Rayet signatures are seen in the available deep optical spectra of LyC emitters (Guseva et al. 2020).
After Lyα, the lines with the highest EWs are C iii] λ1909 and C iv λ1550, with EW(C iii]) = 4 − 20 Å and EW(C iv) = 3 − 15 Å. The C iii] λ1909 line is detected in all the sources, and C iv λ1550 in five out of eight galaxies above 4 σ.
In Fig. 2.2 we plot the C iii] and C iv EWs of our targets as a function of metallicity (using O/H as a proxy) along with measurements from other low-redshift galaxies for comparison. The C iii] EWs of the LyC emitters do not occupy a particularly unique domain in this figure, and EW(C iii]) does not show a dependence on the LyC escape fraction. The "envelope" of the observed distribution of the C iii] EWs show a metallicity dependence that is fairly well reproduced by photoionization models using the spectral energy distributions of young stellar populations, as shown in several studies (Jaskot & Ravindranath 2016;Nakajima et al. 2018;Ravindranath et al. 2020).
Interestingly, the C iv EWs of the strong leakers are among the highest observed so far. Furthermore, J1248+4259 stands out as the low-z galaxy (z < ∼ 0.4) with the highest C iv λ1550 EW (EW(C iv) = 15.14 ± 2.14 Å); however, it shows a low escape fraction ( f esc = 0.014 ± 0.004). Again, the observed EWs are also in fair agreement with the predictions from photoionization models Ravindranath et al. 2020).
He ii λ1640 emission is detected at ∼ 4 − 6 σ in all strong leakers, showing EWs of EW(He ii) = 3.7 − 8.0 Å, which are among the highest values observed in star-forming galaxies. The He ii line is also detected above 3σ in two other sources (i.e., in five of the eight spectra). We comment on the strength of the He ii λ1640 emission below.
2.3. Strong LyC leakers show strong C iv λ1550 emission and a high C iv/C iii] ratio
In addition to exhibiting strong C iv and He ii λ1640 emission lines, the main distinguishing feature of the strong leakers is the high ratio, C43 = I(C iv λ1550)/I(C iii] λ1909) > ∼ 0.75, of the carbon line intensities, which is fairly exceptional, as shown in Fig. 2.2 (left panel), in comparison with other low-z galaxies where both carbon lines have been detected. And interestingly, except for the galaxy J1248+4259 discussed below, the carbon line ratio increases with the LyC escape fraction, as shown in the right panel. Although the sample of galaxies for which both observations of the LyC and the non-ionizing UV spectrum, including C iii] and C iv, are covered is admittedly small, we suggest that, based on the available data, star-forming galaxies with I(C iv λ1550)/I(C iii] λ1909) > ∼ 0.75 have escape fractions f esc > 0.1 and hence show "significant" amounts of LyC emission.
If generally applicable, our postulate implies that the galaxy J1248+4259 reported by Izotov et al. (2018b) should be a strong LyC emitter; that is, it should have a true LyC escape fraction of at least ∼ 5 times that measured from the COS spectrum, which could be possible if, for example, the emission of LyC photons is not isotropic. There are indeed other indications that J1248+4259 could be a strong leaker: First, it shows a very strong Lyα emission with an EW of EW(Lyα) = 256 ± 5.2 Å, comparable to strong leakers. Second, the Lyα line profile is clearly double-peaked, with a fairly low separation of the two peaks (v sep = 283.8±15.9 km s −1 ), which would yield f esc = 0.13 if the mean relation between f esc and v sep from Izotov et al. (2018b) was applied. Finally, J1248+4259 is also somewhat of an "outlier" in the scattered relation between O32=[O iii]/[O ii] and f esc , showing a lower-than-average escape fraction for its high O32=11.8 (see Izotov et al. 2021).
The other low-z galaxies that show I(C iv λ1550)/I(C iii] λ1909) > ∼ 0.75 in Fig. 2.2 are J084236 and J104457. They are low-mass low-metallicity galaxies at z ∼ 0.01 that both show very strong C iv emission with EW(C iv) ∼ 6 − 10 Å (Berg et al. 2019b,a). Based on high-resolution COS spectra, which show strong nebular C iv emission in the two doublet lines and indications for minor radiation transfer effects in these lines, Berg et al. (2019a) have suggested that J104457 is optically thin to ionizing radiation, at least at energies above 47.9 eV, the ionization potential of C 3+ (see also Senchyna et al. 2021). It also shares other properties of strong LyC leakers, such as a very high O32= 16.2. Although no direct observations of the LyC are possible at these low redshifts, the available data for J104457 appear fully consistent with our postulate that this galaxy is a strong LyC leaker. We suggest that J084236 may also be a strong LyC leaker.
Intrinsically, the C43 ratio traces the ionization structure of the nebula, similarly to O32, which has already been suggested as a potential indicator of LyC escape (Jaskot & Oey 2013;Nakajima & Ouchi 2014). However, since C iv λ1550 is a resonance line, it is a priori affected by radiation transfer effects; this is in contrast to the nebular oxygen lines, which should be optically thin. Therefore, the C iv λ1550 line and the C43 ratio could be a more sensitive tracer of LyC escape than O32. To first order, both C43 and O32 trace the ionization parameter, U, and are independent of metallicity. Furthermore, an examination of photoionization models shows that both C43 and O32 vary with the LyC escape fraction. For moderate to high U, the predicted variations in C43 appear stronger than in O32, again providing some theoretical support for our empirically based postulate. A detailed comparison with models will be presented in future work.
Discussion and implications
Comparison with high-z LyC emitters and candidates
We first compared our results with confirmed high-z LyC emitters, although relatively few high-z galaxies with established or potentially strong LyC escape (absolute escape fractions f esc > ∼ 0.1) are known. Well-studied sources with the highest LyC escape fractions are Ion2, the Sunburst arc, and probably Ion3, with relative escape fractions f esc,rel > ∼ 0.5, according to Vanzella et al. (2020) 2 . VLT/XShooter spectroscopy of Ion2 at z = 3.2 indeed shows the presence of nebular emission in both C iv and C iii], with a ratio I(C iv λ1550)/I(C iii] λ1909) = 0.61 ± 0.23; this is compatible with our findings, although the lines are relatively weak (e.g., EW(C iv) = 2.6 Å and EW(He ii) = 2.8 Å; Vanzella et al. 2020). For the other objects, the reported data are insufficient to examine C iv and C iii]. Similarly, no detailed rest-UV spectra have been published for the z = 3.1 LyC emitting galaxies of Schmidt et al. (2021). All measurements are for galaxies with direct metallicity measurements. The typical relative errors for our sources are 10-20 % for all EW(C iii]) and for EW(C iv) > ∼ 5 Å, and larger otherwise (shown). Fig. 2.2. The colored area shows the region of C43 > ∼ 0.75, where we propose that strong LyC leakers, i.e., galaxies with f esc > 0.1, are found empirically. We note that for the sake of simplicity for comparisons with high-z studies the line ratios of the leakers are not corrected for internal reddening and could thus be slightly higher, although the corrections should be small.
intergalactic medium transmission at high z. Their LyC emitting sample has an average escape fraction of f esc = 0.06 ± 0.01, which is below our definition of strong emitters, and the stacked UV spectrum illustrated in Steidel et al. (2018) does not show strong C iv emission, although no details on carbon emission lines are reported in their study.
Another comparison is possible with Lyα emitters at z ∼ 2, where the LyC escape fraction was recently estimated using indirect empirical spectral indicators by Naidu et al. (2021) and Matthee et al. (2021). Stacking rest-UV spectra of Lyα emitters with estimated f esc > ∼ 0.2 (high escape) and f esc < ∼ 0.05 (low escape), they find several significant differences between the two subsamples, in particular the presence of nebular C iv λ1550, C iii] λ1909, He ii λ1640, and O iii] λ1666 emission in the high escape sources, whereas the low escape sources only show C iii] and O iii] emission. Although the C iv emission of their strong leakers (EW(C iv) = 2.0 ± 0.4 Å) is weaker than that of our three sources with f esc > 0.1, which show EW> 2.5 Å, and despite C iii] λ1909 being stronger than C iv, their finding of nebular C iv and He ii λ1640 emission in strong leaker candidates is compatible with our results.
We therefore conclude that both low-and high-redshift observations seem to consistently show (i.e., in about three out of four cases) that the UV spectra of strong leakers (with f esc > ∼ 0.1 − 0.2) are characterized by nebular C iv λ1550 and C iii] λ1909 emission and C43 > ∼ 0.75. Although the available data are still relatively scarce, our observations provide the first such quantitative estimate.
Implications for high-z CIV emitters
So far, very few high-redshift galaxies showing nebular C iv λ1550 emission have been reported. Since the recent discoveries of several lensed galaxies at z ∼ 6 − 7 with nebular C iv (Stark et al. 2015;Mainali et al. 2017;Schmidt et al. 2017), other C iv emitters have been found both in blank fields (e.g., at z ∼ 2.2; Tang et al. 2021) and in lensed galaxies (see Stark et al. 2014;Vanzella et al. 2021;Richard et al. 2021). The stacked rest-UV spectrum of lensed sources with median z = 3.2 and median M 1500 = −17.1 from Vanzella et al. (2021) shows strong nebular C iv λ1550, C iii] λ1909, and numerous other emission lines. The C iv line in several of these sources is strong, with EWs of up to EW(C iv) = 38 ± 16 Å in A1703-zd6 (Stark et al. 2015;Schmidt et al. 2021) and C43 > 0.8 in several of them. Our results suggest that these sources are strong LyC emitters, which is overall in agreement with the other observational properties they share with LyC emitters (e.g., strong Lyα emission). The fact that C iv emission is found in many strongly lensed sources probably also indicates that the fraction of strong LyC emitters increases toward fainter, lower-mass galaxies, a trend also found in the z ∼ 0.3 − 0.4 reference samples (see Izotov et al. 2021;Flury et al. 2022b).
Strong emission lines and high C43 ratios explained
This leads us to the questions of what the strong observed UV emission lines (high EW of nebular lines) tell us and how they can be understood. Since EWs measure the strength of the emission line with respect to the continuum, the EW of a UV recombination line such as He ii λ1640 can be cast in the following simple form,
EW(1640)/Å = ξ H ion Q He+ Q H c 1640 λ 2 3 × 10 18 ,(1)
where ξ H ion = Q H /L 1500 is the ionizing photon efficiency per unit intrinsic monochromatic UV luminosity in the commonly used units of erg −1 Hz, c 1640 = 5.67 × 10 −12 erg relates the recombination line flux to the ionizing photon flux (e.g., Schaerer 2003), and λ ≈ 1640 Å. This shows that the EW is proportional to ξ ion and to the hardness of the ionizing spectrum, expressed here as Q He+ /Q H , the ratio of the ionizing photon flux above 54 eV and 13.6 eV, respectively. From this we can easily see that the observed EWs in our three strongest leakers, EW(1640)≈ 3 − 8 Å, are plausible. Indeed, the EWs can be explained with a relatively high ionizing photon production, log(ξ ion ) ≈ 25.6 − 25.8 erg −1 Hz (cf. Schaerer et al. 2018), which then implies Q He+ /Q H ≈ 0.01 − 0.04. This hardness translates to optical line ratios of I(He ii λ4686)/I(Hβ) ≈ 1.74 Q He+ /Q H ∼ 0.016 − 0.07, similar to the observations of J1154+2443 and J1248+4259, which are I(4686)/I(Hβ) ∼ 0.02 − 0.03 (Guseva et al. 2020), and comparable to the line intensities typically observed in galaxies at comparable metallicity (at 12 + log(O/H) ∼ 8.0; see, e.g., Schaerer et al. 2019). This simple estimate shows that the high He ii λ1640 EWs reached in these leakers do not require exceptional conditions and/or exceptionally hard ionizing spectra compared to other galaxies (weak or non-leakers) at similarly low metallicity. The same conclusion also applies to, for example, the strong leaker candidates found by Naidu et al. (2021) among their Lyα emitter sample. However, the observed spectra are clearly harder at > 54 eV than predicted by normal stellar population models, and the source of these He + -ionizing photons remains to be elucidated (see, e.g., Stasińska et al. 2015).
To predict the strengths of the metallic lines of C iv and C iii], which depend on the ionization parameter, spectral energy distribution, nebular abundances, and other factors, detailed photoionization models are required. The models of Nakajima et al. (2018), for example, show maximum values of EW(C iv) ∼ 8 − 10 Å and EW(C iii]) ∼ 10−18 Å at metallicities 12+log(O/H) ∼ 7.7 − 8, of the same order as our observed EWs. We also examined density-bounded models at appropriate metallicities, using the BOND set of CLOUDY models from the 3 Million Models Database of Morisset et al. (2015). The models indeed show an expected increase in the C iv/C iii] ratio with increasing LyC escape fraction, and several models show C iv/C iii] ratios comparable to those of our observations. From these simple compar-isons, we conclude that both the observed EWs and UV line ratios of the strong low-z leakers seem "reasonable" and that we probably do not need peculiar or extreme ionizing spectra to reproduce their emission line spectra. Tailored photoionization models to examine the behavior of the major UV and optical emission lines of the LyC emitters and comparison sources will be presented in a subsequent publication.
Conclusion
With the STIS spectrograph on board HST, we have obtained new UV spectra from ∼ 1200 − 2000 Å, the rest-frame spectra of eight z ∼ 0.3 − 0.4 LyC emitters from Izotov et al. (2016aIzotov et al. ( ,b, 2018a, which cover a large range of LyC escape fractions. We detect multiple emission lines, including Lyα, C iv λ1550, He ii λ1640, O iii] λ1666, and C iii] λ1909. The Lyα and C iii] lines are detected in all the sources. Our main results can be summarized as follows:
-We report the detection above 4 σ of C iv λ1550 emission in six out of eight galaxies, with the EWs in two galaxies ( J1243+4646 and J1248+4259), EW(C iv) = 12 − 15 Å, exceeding the previously reported maximum in low-z galaxies ( J104457; Berg et al. 2019a).
-Strikingly, C iv λ1550 emission is detected in all LyC emitters with escape fractions f esc > 0.1, and the flux ratio of C iv λ1550/C iii] λ1909 appears to increase with f esc . -Based on the available data, we suggest that strong leakers, galaxies with f esc > 0.1, are characterized by C iv λ1550/C iii] λ1909 > ∼ 0.75, adding another indirect indicator of LyC escape to those already established. -All strong leakers also show strong He ii λ1640 emission with EW(He ii) = 3.7 − 8.0 Å, which are among the highest values observed in star-forming galaxies. -A simple estimate shows that the high EW of the He ii λ1640 line is primarily due to a high ionizing photon production, ξ ion , and that it does not require unusually hard ionizing spectra, compared to normal galaxies at similar metallicity that frequently show optical He ii λ4686 emission lines.
In short, our observations provide an important new reference for understanding the UV spectra of LyC emitting galaxies and thus also analogs of the sources of cosmic reionization. The spectra of the strong low-z leakers share many similarities with those of the C iv λ1550 emitters recently discovered at high redshifts (e.g., Stark et al. 2015;Mainali et al. 2017;Schmidt et al. 2017), and our results suggest that these objects are good candidates for strong LyC escape. If universally applicable, the empirical criterion of using the carbon line ratio C iv λ1550/C iii] λ1909 > ∼ 0.75 to identify strong LyC leakers could represent an additional important tool for studying the sources of cosmic reionization.
Fig. 2 .
2Fletcher et al. (2019). The robust sample of z ∼ 3 LyC emitters fromSteidel et al. (2018) andPahl et al. (2021) provides average properties to avoid the inherent limitations in determining f esc for individual sources that are due to the stochastic Rest-frame C iii] λ1909 (left panel) and C iv λ1550 (right panel) EWs as a function of metallicity O/H for the LyC emitters (large red squares: strong leakers, f esc > 0.1; blue circles: other leakers) and low-z comparison samples from the literature taken from the compilation of
J1248+4259 Fig. 3 .
J1248+42593C iv λ1550/C iii] λ1909 line ratio (C43) as a function of metallicity (O/H, left panel) and the LyC escape fraction (right). The symbols have the same meaning as in
1 . The sources span a broad range of LyC escape fractions ( f esc ∼ 1.4 − 72 %), metallicities in the range 12 + log(O/H) = 7.64 − 8.16 with a median of 7.92, and [O iii]λ5007/[O ii] λ3727 from 5 to 27.1.
.1, ordered -from top to bottom -by decreasing LyC escape fraction. The top two sources shown there have very high escape1 The galaxies are J1243+4646, J1154+2443, J1152+3400, J1442-
0209, J0925+1403, J1011+1947, J0901+2119, and J1248+4259.
1400 1500 1600 1700 1800 1900 2000 2100
rest-frame wavelength [Angstroem]
2
4
6
8
10
flux [F ] + const
CIV 1550
HeII 1640
OIII]
CIII] 1909
SiIII]
fesc=0.014
fesc=0.46
fesc=0.13
fesc=0.72
fesc=0.074
fesc=0.072
fesc=0.062
fesc=0.021
J1248
J1243
J1154
J1152
J1442
J0925
J1011
J0901
Fig. 1. STIS rest-frame spectra of eight z ∼ 0.3 − 0.4 LyC emitters.
We here refer to relative escape fractions since absolute values cannot reliably be determined for individual sources at high z (e.g.,Steidel et al. 2018).Article number, page 3 of 6 A&A proofs: manuscript no. 43149corr_ds
Acknowledgements. Y.I. acknowledges support from the National Academy of Sciences of Ukraine by its priority project "Fundamental properties of the matter in relativistic collisions of nuclei and in the early Universe". D.B., J.C, A.J., and T.X.T. acknowledge support from grant HST-GO-15941.002-A. This research is based on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs GO 15941 and GO 15433 (PI Schaerer).
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|
[] |
[
"Possibility of Solid-Fluid Transition in Moving Periodic Systems",
"Possibility of Solid-Fluid Transition in Moving Periodic Systems"
] |
[
"Tomoaki Nogawa ",
"Hajime Yoshino ",
"Hiroshi Matsukawa \nCollege of Science and Engineering\nDivision of Physics\nGraduate School of Science\nAoyama Gakuin University\nFuchinobe 5-10-1229-8558SagamiharaKanagawa\n\nLaboratoire de Physique Théorique et HautesÉnergies\nHokkaido University\nKita 10-jo Nisi 8-tyome, Jussieu, 5èmeétage, Tour 25, 4 Place Jussieu060-0810, 75252, Cedex 05Sapporo, ., ParisFrance\n",
"\nSchool of Science\nOsaka University\nMachikaneyama 1-1560-0043ToyonakaOsaka\n"
] |
[
"College of Science and Engineering\nDivision of Physics\nGraduate School of Science\nAoyama Gakuin University\nFuchinobe 5-10-1229-8558SagamiharaKanagawa",
"Laboratoire de Physique Théorique et HautesÉnergies\nHokkaido University\nKita 10-jo Nisi 8-tyome, Jussieu, 5èmeétage, Tour 25, 4 Place Jussieu060-0810, 75252, Cedex 05Sapporo, ., ParisFrance",
"School of Science\nOsaka University\nMachikaneyama 1-1560-0043ToyonakaOsaka"
] |
[] |
The steady sliding state of periodic structures such as charge density waves and flux line lattices is numerically studied based on the three dimensional driven random-field XY model. We focus on the dynamical phase transition between plastic flow and moving solid phases controlled by the magnitude of the driving force. By analyzing the connectivity of comoving clusters, we find that they percolate the system under driving forces larger than a certain critical force within a finite observation time. The critical force, however, logarithmically diverges with the observation time, i.e., the moving solid phase exists only within a certain finite time, which exponentially grows with driving force.The collective transport phenomena of condensed matter with random pinning attract much attention from the viewpoints of solid state physics, nonlinear dynamics and statistical mechanics. There are numerous systems which belong to this class of dynamics, e.g., charge density waves (CDWs), 1 flux line lattices (FLLs), 2 colloidal lattices 3 and Wigner crystals. 4, 5 These systems have spatial periodicity in the absence of random pinning, which modifies the periodic order and pins the system. Under a driving force larger than a certain threshold value, the system starts moving and shows highly nonlinear conduction. This is the depinning transition, which has been investigated extensively. Recently another attractive topic of these systems has been the dynamical phase transition between two nonequilibrium steady states above the depinning threshold field. 6, 7 In the "ordered phase" the local DC velocity is uniform and the spatial periodicity has a quasi long range order. Such a state in FLL systems is called moving Bragg glass.8,9In the "disordered phase", which is often called the "plastic flow phase", the motion is spatially nonuniform and the periodic order is destroyed. This transition is considered to be induced by a change in driving force or pinning strength. Although there are a lot of experimental studies 10, 11 and numerical simulations 12-16 that show an evidence of a dynamical melting transition between moving Bragg glass and fluid phases, the existence of such an ordered phase is still an unanswered question.In such discussions, it is assumed implicitly that two types of order, namely, the spatial periodic order and the local DC velocity order, are established simultaneously. To date, the former has been mainly discussed. These two orders, however, are independent in principle, e.g., the "moving glass state", in which the spatial periodicity is destroyed but frozen in time so the local DC velocity is uniform, is possible. In this article we focus on the dynamical property, i.e., the uniformity of the local DC velocity. It is closely related to the plastic deformation because at the boundary between the domains that have different velocities, the local strain increases with time and tearing occurs. We discuss the phase transition between the "plastic flow phase" and the "moving solid phase", which are distinguished by the existence of the local DC velocity order.We perform numerical simulations based on the three dimensional driven random-field XY model. 17-20 It is a modified version of the intensively investigated elastic manifold model, such as the Fukuyama-Lee-Rice model for CDWs.21,22The density of a periodic structure is expressed as ρ(r, t) = q ρ q cos q · [r − u(r, t)] . Here, q's are fundamental reciprocal lattice vectors. Higher harmonics are ignored here. u(r, t) is a deformation field and the phase field θ q (r, t) = q · u(r, t) is often employed as a degree of freedom. In this article, only the case of a single phase field, which denotes the driving direction component of deformation, is treated. It is sufficient in the case that periodicity is one dimensional as CDWs in NbSe 3 . We consider, however, that the essential feature of the dynamics of higher dimensional periodic systems such as FLLs would be captured. The periodicity of the structure is related to the phase coherence and experimentally observable transport quantities, such as electric current for CDWs and voltage drop for FLLs, are proportional to the phase velocityθ(r, t).The elastic manifold model treats internal interaction using the elastic energy dr(∇θ) 2 , which becomes a harmonic coupling,
|
10.1143/jpsj.74.1907
|
[
"https://export.arxiv.org/pdf/cond-mat/0503422v5.pdf"
] | 18,789,973 |
cond-mat/0503422
|
65de3e7fc0ef03ef58226ff94309596fff6dca37
|
Possibility of Solid-Fluid Transition in Moving Periodic Systems
19 Jul 2005
Tomoaki Nogawa
Hajime Yoshino
Hiroshi Matsukawa
College of Science and Engineering
Division of Physics
Graduate School of Science
Aoyama Gakuin University
Fuchinobe 5-10-1229-8558SagamiharaKanagawa
Laboratoire de Physique Théorique et HautesÉnergies
Hokkaido University
Kita 10-jo Nisi 8-tyome, Jussieu, 5èmeétage, Tour 25, 4 Place Jussieu060-0810, 75252, Cedex 05Sapporo, ., ParisFrance
School of Science
Osaka University
Machikaneyama 1-1560-0043ToyonakaOsaka
Possibility of Solid-Fluid Transition in Moving Periodic Systems
19 Jul 2005Typeset with jpsj2.cls <ver.1.2> Letter Present address: Present address:CDWflux line latticeplastic flowBragg glassrandom fieldXY model
The steady sliding state of periodic structures such as charge density waves and flux line lattices is numerically studied based on the three dimensional driven random-field XY model. We focus on the dynamical phase transition between plastic flow and moving solid phases controlled by the magnitude of the driving force. By analyzing the connectivity of comoving clusters, we find that they percolate the system under driving forces larger than a certain critical force within a finite observation time. The critical force, however, logarithmically diverges with the observation time, i.e., the moving solid phase exists only within a certain finite time, which exponentially grows with driving force.The collective transport phenomena of condensed matter with random pinning attract much attention from the viewpoints of solid state physics, nonlinear dynamics and statistical mechanics. There are numerous systems which belong to this class of dynamics, e.g., charge density waves (CDWs), 1 flux line lattices (FLLs), 2 colloidal lattices 3 and Wigner crystals. 4, 5 These systems have spatial periodicity in the absence of random pinning, which modifies the periodic order and pins the system. Under a driving force larger than a certain threshold value, the system starts moving and shows highly nonlinear conduction. This is the depinning transition, which has been investigated extensively. Recently another attractive topic of these systems has been the dynamical phase transition between two nonequilibrium steady states above the depinning threshold field. 6, 7 In the "ordered phase" the local DC velocity is uniform and the spatial periodicity has a quasi long range order. Such a state in FLL systems is called moving Bragg glass.8,9In the "disordered phase", which is often called the "plastic flow phase", the motion is spatially nonuniform and the periodic order is destroyed. This transition is considered to be induced by a change in driving force or pinning strength. Although there are a lot of experimental studies 10, 11 and numerical simulations 12-16 that show an evidence of a dynamical melting transition between moving Bragg glass and fluid phases, the existence of such an ordered phase is still an unanswered question.In such discussions, it is assumed implicitly that two types of order, namely, the spatial periodic order and the local DC velocity order, are established simultaneously. To date, the former has been mainly discussed. These two orders, however, are independent in principle, e.g., the "moving glass state", in which the spatial periodicity is destroyed but frozen in time so the local DC velocity is uniform, is possible. In this article we focus on the dynamical property, i.e., the uniformity of the local DC velocity. It is closely related to the plastic deformation because at the boundary between the domains that have different velocities, the local strain increases with time and tearing occurs. We discuss the phase transition between the "plastic flow phase" and the "moving solid phase", which are distinguished by the existence of the local DC velocity order.We perform numerical simulations based on the three dimensional driven random-field XY model. 17-20 It is a modified version of the intensively investigated elastic manifold model, such as the Fukuyama-Lee-Rice model for CDWs.21,22The density of a periodic structure is expressed as ρ(r, t) = q ρ q cos q · [r − u(r, t)] . Here, q's are fundamental reciprocal lattice vectors. Higher harmonics are ignored here. u(r, t) is a deformation field and the phase field θ q (r, t) = q · u(r, t) is often employed as a degree of freedom. In this article, only the case of a single phase field, which denotes the driving direction component of deformation, is treated. It is sufficient in the case that periodicity is one dimensional as CDWs in NbSe 3 . We consider, however, that the essential feature of the dynamics of higher dimensional periodic systems such as FLLs would be captured. The periodicity of the structure is related to the phase coherence and experimentally observable transport quantities, such as electric current for CDWs and voltage drop for FLLs, are proportional to the phase velocityθ(r, t).The elastic manifold model treats internal interaction using the elastic energy dr(∇θ) 2 , which becomes a harmonic coupling,
The steady sliding state of periodic structures such as charge density waves and flux line lattices is numerically studied based on the three dimensional driven random-field XY model. We focus on the dynamical phase transition between plastic flow and moving solid phases controlled by the magnitude of the driving force. By analyzing the connectivity of comoving clusters, we find that they percolate the system under driving forces larger than a certain critical force within a finite observation time. The critical force, however, logarithmically diverges with the observation time, i.e., the moving solid phase exists only within a certain finite time, which exponentially grows with driving force.
KEYWORDS: CDW, flux line lattice, plastic flow, Bragg glass, random field, XY model
The collective transport phenomena of condensed matter with random pinning attract much attention from the viewpoints of solid state physics, nonlinear dynamics and statistical mechanics. There are numerous systems which belong to this class of dynamics, e.g., charge density waves (CDWs), 1 flux line lattices (FLLs), 2 colloidal lattices 3 and Wigner crystals. 4,5 These systems have spatial periodicity in the absence of random pinning, which modifies the periodic order and pins the system. Under a driving force larger than a certain threshold value, the system starts moving and shows highly nonlinear conduction. This is the depinning transition, which has been investigated extensively. Recently another attractive topic of these systems has been the dynamical phase transition between two nonequilibrium steady states above the depinning threshold field. 6,7 In the "ordered phase" the local DC velocity is uniform and the spatial periodicity has a quasi long range order. Such a state in FLL systems is called moving Bragg glass. 8,9 In the "disordered phase", which is often called the "plastic flow phase", the motion is spatially nonuniform and the periodic order is destroyed. This transition is considered to be induced by a change in driving force or pinning strength. Although there are a lot of experimental studies 10, 11 and numerical simulations 12-16 that show an evidence of a dynamical melting transition between moving Bragg glass and fluid phases, the existence of such an ordered phase is still an unanswered question.
In such discussions, it is assumed implicitly that two types of order, namely, the spatial periodic order and the local DC velocity order, are established simultaneously. To date, the former has been mainly discussed. These two orders, however, are independent in principle, * E-mail address: nogawa@presto. e.g., the "moving glass state", in which the spatial periodicity is destroyed but frozen in time so the local DC velocity is uniform, is possible. In this article we focus on the dynamical property, i.e., the uniformity of the local DC velocity. It is closely related to the plastic deformation because at the boundary between the domains that have different velocities, the local strain increases with time and tearing occurs. We discuss the phase transition between the "plastic flow phase" and the "moving solid phase", which are distinguished by the existence of the local DC velocity order. We perform numerical simulations based on the three dimensional driven random-field XY model. [17][18][19][20] It is a modified version of the intensively investigated elastic manifold model, such as the Fukuyama-Lee-Rice model for CDWs. 21,22 The density of a periodic structure is expressed as ρ(r, t) = q ρ q cos q · [r − u(r, t)] . Here, q's are fundamental reciprocal lattice vectors. Higher harmonics are ignored here. u(r, t) is a deformation field and the phase field θ q (r, t) = q · u(r, t) is often employed as a degree of freedom. In this article, only the case of a single phase field, which denotes the driving direction component of deformation, is treated. It is sufficient in the case that periodicity is one dimensional as CDWs in NbSe 3 . We consider, however, that the essential feature of the dynamics of higher dimensional periodic systems such as FLLs would be captured. The periodicity of the structure is related to the phase coherence and experimentally observable transport quantities, such as electric current for CDWs and voltage drop for FLLs, are proportional to the phase velocityθ(r, t).
The elastic manifold model treats internal interaction using the elastic energy dr(∇θ) 2 , which becomes a harmonic coupling, i,j (θ i − θ j ) 2 /2, in a lattice model. In order to treat plastic deformation, we replace this harmonic coupling with a sinusoidal one, 1 − cos(θ i − θ j ). They are equivalent in the limit where phase differences become zero. Here, the indices i's denote semimacroscopic domains fixed in the space in which phase coherence is always held. This sinusoidal coupling in-duces maximum restoring force, i.e., yield stress and allows plastic deformation, so-called phase slip. Phase slip is a process in which the phase difference between neighboring domains increases or decreases by 2π. It results in no change in coupling energy.
The overdamped equations of motion for the phases of domains, θ i 's, are as follows.
θ i = − J z j sin(θ i − θ j ) − sin(θ i − β i ) + f(1)
We choose the units that both of a pinning strength and a dissipation coefficient equal unity. The first term on the right hand side refers to the interaction with neighboring z domains. The second term denotes the a random pinning force and β i 's are given as uniform random numbers between 0 and 2π. f is a uniform driving force. Strogatz et al. analyzed this model by the mean field approximation and found a discontinuous transition by changing the external field. 17 There are three regimes, a pinned static state and a homogeneously moving state, and a bistable regime between the two. Huse performed numerical simulations of this model in three dimensions. 18 He investigated the Lyapunov exponent and velocity coherence and found a transition between spatially uniform temporally regular motion and nonuniform chaotic motion by changing pinning strength. These motions are related to moving solid and plastic flow, respectively. In this article, we analyze the f dependence of the dynamics of this model systematically paying special attention to the dependences on system size and observation time.
We numerically solve eq. (1) by the fourth-order Runge-Kutta method. The discretized time step is set at 2π/8(J + f ). The domains are put regularly on the simple cubic lattice in three dimensions. We call this unit "site" instead of "domain" hereafter. A periodic boundary condition is imposed. There are two independent parameters, coupling constant J and driving force f . In this article we show mainly the results for J = 1.0. All phases are set at the unique value in the initial state. Physical quantities are calculated after some precursor running (typical time is 12900) for the relaxation to the steady state. An initial state is sometimes substituted by the final state for the simulation with a slightly larger f . Thermal fluctuation is not taken into account. Simulations are performed with some samples that have different set of β i 's. The numbers of samples whose linear sizes are 16,32,64 and 128 are 32, 16,8 and 4, respectively. In the left panel of Fig. 1 a spatial configuration of the local DC velocity, ω i DC = θ i (t) T , is shown. Here, . . . T denotes time averaging for the observation time T and the resolution of DC velocity is given by 2π/T . Note that we show the results of two dimensional systems, which show qualitatively similar behaviors to three dimensional systems but show the domain structure of ω i DC more clearly. In the right panel we show the difference in ω i DC ,
|∆ω i,j DC | = |ω i DC − ω j DC |,
for each bond between neighboring sites. When |∆ω i,j DC | > 2π/T , phase slip occurs on the bond at least once during the observation time T . We focus on this phase slip process to discuss the spatial correlation of motion instead of the direct spatial correlation of DC local velocity. 13 We define such a bond as a "disconnected" bond. Otherwise, if |∆ω DC | < 2π/T , the bond is "connected" and two sites belong to the same cluster. When both of the pair sites are pinned, i.e., ω i DC , ω j DC < 2π/T , |∆ω DC | is less than 2π/T but we regard such a bond as disconnected, which forms nonmoving solid clusters. We analyze a bond percolation transition by controlling driving force. The moving solid phase is characterized by an infinitely large cluster, which is made from connected sites without phase slip. The percolating phase is thus the moving solid phase. The driving force in Fig. 1 is slightly below the critical point and a fractal domain structure appears.
In order to perform finite size scaling, we divide the system into subsystems, whose linear dimension L is smaller than that of a real sample L max . Then we determine that percolation occurs if a certain cluster reaches the two opposite sides of each subsystem. The statistics of subsystems and samples yields percolation probability P (f, L, T ), which monotonically grows with f and becomes smaller as T increases.
The reason we perform finite size scaling in terms of L and not L max as usual is the following. We have found that the whole system with a finite L max eventually falls into a periodic motion above some threshold driving force. Therefore, for a given driving force, we choose to work on a system with sufficiently large L max to eliminate this artificial periodic motion and analyze the connectivity of clusters at various length scales L(< L max ).
In Fig. 2, P (f, L, T )'s for various L's and T 's are plotted as functions of f . We consider the large L limit at fixed T first and the T dependence next. Percolation probability grows with f and its shape comes closer to that of a step function of f as L increases. Finite size scaling can be performed in the same way as the stochastic percolation. The curves for different L's converge on the universal one as the driving force is point is 0.10, which is smaller than that for the stochastic percolation, 0.2488, 23 due to the attractive correlation of connected bonds. The magnitude of the critical driving force for the percolation transition obtained from the above analysis depends on T . The T dependence of f c for several L max 's is shown in Fig. 3. It is expressed as
f c (T ) = f 0 ln(T /t 0 ),(2)
for a long T . Here, f 0 and t 0 are constants. We expect that eq. (2), which is consistent with the idea of shaking temperature 6 as mentioned later, holds for a sufficiently long T and large L max and the reason for the deviation between the present data and eq. (2) for a smaller T or a smaller L max is considered as follows. For T < 2000, the f c 's obtained from the simulations are a little larger than that expected from eq. (2). This discrepancy can be due to the existence of pinned or very low velocity sites, which cannot be candidates of the percolating cluster. They are defects of the percolation transition, of which effects are not taken into account in eq. (2). The percolation transition is not so sensitive to the existence of such defects when they are rare and isolated. They are not a minority, however, below f = 0.80, e.g, the fractions of such sites with ω DC < 0. Another disagreement occurs at a longer T , which is caused by the finiteness of the real sample size L max . For a finite L max , f c exhibits saturating behavior to a finite value above a certain T , which increases with L max . As mentioned before, this is caused by the falling of the system into limit cycle motion when the phase coherence length becomes comparable to L max . For T ≤ 51500 (f c < 0.870), the results for L = 64 and 128 hardly differ and they are considered to show f c (T ) for L max = ∞. From these discussions eq. (2) is expected to hold up to the infinite T . This means that f c (T ) diverges logarithmically as T approaches infinity and the moving solid phase does not exist in the long time limit based on the present definition. The supplemental simulation indicates that this behavior does not change for a system with a stronger coupling.
From another point of view, eq. (2) is regarded as a type of "phase boundary" between the plastic flow and moving solid phases in the force-time plain (See Fig. 3). Considering the observation with fixed f , the crossover time τ (f ) is obtained as
τ (f ) = t 0 exp(f /f 0 ).
(3
The system behaves as if it were a moving solid in an observation time shorter than τ (f ). Beyond this time, cracks of plastic deformation, which are sheets of phase slip bonds, propagate to the macroscopic scale and fluid like property is revealed. Equation (3) can be regarded as a thermal activation process, τ ∝ exp(V /k B T eff ), if an effective temperature proportional to f −1 (≈ ω −1 DC ) is supposed. The inverse of f 0 is then proportional to the potential barrier V . This is consistent with the idea of "shaking temperature" proposed by Koshelev and Vinokur. 6,14,15 Next, we discuss the universality of the percolation transition for different observation times. In Fig.4, the T dependence of the inverse of the critical exponent ν is Letter Author Name shown. There is a tendency for large L max 's, 64 and 128, where ν −1 has T -independent value ≈ 0.8. The inverse of ν becomes smaller and approaches zero as T increases. ν −1 = 0 means that P (f, L, T ) does not depend on L, which manifests that the system falls into the limit cycle motion, i.e., a finite size effect. Such effect appears more clearly in the critical exponent than in the critical force, because the former is determined by the behavior in the whole critical regime. So we expect the percolation transition discussed here is universal with respect to the observation time. This universality is confirmed by the property of the percolating cluster at the critical point. Cluster size is identified with the number of contained sites and we define s c (L, T ) as the size of the maximum cluster in a subsystem at f = f c (T ). In Fig. 5, s c (L, T )'s for several T 's are plotted as functions of subsystem size L. They are expressed as s c (L, T ) ≈ 0.35L 2.45 with a fractal dimension of 2.45 and show little T (or f c (T ) ) dependence. The large deviation of the data for L max =64 at the longest T is due to starting the limit cycle motion.
The universality of the transition for different T 's means that the fluctuation of DC velocity has a scaleless spatial pattern, which depends on neither f nor the phase coherence length, which grows with f , if one chooses proper time scale, i.e., sees the DC velocity in the resolution of 2π/τ (f ).
In conclusion, we numerically investigate the possibility of the dynamical phase transition between plastic flow and moving solid phases, which are distinguished by the existence of the long range order of local DC velocity. By analyzing the percolation of no-phase slip bonds and its observation time dependence, we found that the moving solid phase becomes unstable in a finite lifetime.
The condition for the connected bond, that no phase slip occurs eternally, may seem too strict. For example all bonds necessarily take phase slips if thermal fluctuation exists. It is important, however, to note that the local symmetry lim T →∞ ∆ω i,j DC = 0 is destroyed from the beginning due to the random field. The bond steadily takes a phase slip in the same direction no matter how rarely it occurs.
The crossover time τ (f ) can be defined clearly; the system behaves like a moving solid in a shorter time scale than τ (f ) and macroscopic plastic deformation occurs beyond τ (f ). This crossover time increases exponentially with the driving force. Its characteristic scale of growth f 0 = 0.025 is very small compared with other scales such as pinning strength (=1), therefore the crossover time increases very rapidly in the narrow region of f and overcomes the macroscopic time scale. This is a possible reason the moving solid phase is observed in experiments. The situation is similar to the case of structure glasses, whose viscosity grows quite large and they show slow dynamics, then it is hard to distinguish whether an equilibrium phase transition exists.
We focused on the macroscopic plastic deformation and distinguished between plastic flow and moving solid. This stance is different from the conventional interest in the liquid-crystal(Bragg glass) transition. Although it seems natural that these transitions occurs at the same time, the absence of the moving solid phase discussed here is not immediately related to the absence of the long range periodic order. For example the spatial phase order in a long span is possible in the plastic flow phase if the propagation of plastic deformation along the domain boundary is temporally localized and leaves no change before and after it. On the other hand we see that the saturation of phase correlation length to the system size results limit cycle motion and plastic deformation is suppressed. Then if the transition from liquid to crystal or Bragg glass, which is not observed in the range of our simulation, occurs at finite f , the present transition would happen at the same time.
The numerical calculation was performed on the Hitachi SR8000 at the supercomputer center, ISSP, University of Tokyo and the present study is financially supported by a Grant-in-Aid for Scientific Research (15540370) from the Japan Society for the Promotion of Science.
Fig. 1 .
1Spatial configuration of ω DC on site (left) and ∆ω DC on bond (right) in a two dimensional sample. ω DC T /2π is plotted with gray scale from 2730 to 2820 as color changes from white to black and log(|∆ω DC |T /2π) changes from log 0.25 to log 1000. Connected bonds are plotted in white. Result of two dimensional system with 256 2 sites, T ≈ 25700, J=1.0 and f = 1.0 (fc(T ) = 1.06).
Fig. 2 .
2scaled asf = [(f − f c )/f c ]L 1/ν with the suitable critical force f c and the critical exponent ν. The correlation length diverges as |(f − f c )/f c | −ν . A good conversion is obtained for each T as shown in Fig. 2 then f c (T )'s and ν(T )'s are obtained. The fraction of connected bonds at the critical Raw data of percolation probabilities for various sizes and observation times (top) and result of finite size scaling (bottom). The x-axis of the latter is shifted by a constant which depends on T and each origin is indicated by a dotted vertical line. Result of three dimensional samples with Lmax = 64 and maximum observation time T ≈ 3200.
050 are 0.51, 0.090 and 0.00054 for f =0.75, 0.80 and 0.85, respectively. (Here, the local depinning threshold force 19, 20 equals 0.10.) The critical fraction of the connected bonds becomes larger in the lattice with such defects and f c becomes higher. These defects decrease rapidly with f and have less effect on a higher f c for a longer T .
Fig. 3 .Fig. 4 .
34f − T phase diagram. Phase boundary between plastic flow and moving glass is drawn by f = fc(T ). Note that the horizontal axis is in a logarithmic scale. The dotted line indicates f = 0.025 ln t + 0.59. Observation time dependence of critical exponent.
Fig. 5 .
5Relationship between maximum cluster size and linear dimension of subsystem size for several observation times.
phys.sci.osaka-u.ac.jp, Present address: Division of Physics, Graduate School of Science,Hokkaido University, Kita 10-jo Nisi 8-tyome, Sapporo 060-0810.
† E-mail address: [email protected], Present address:
Laboratoire de Physique Théorique et HautesÉnergies, Jussieu,
5èmeétage, Tour 25, 4 Place Jussieu, 75252 Paris Cedex 05,
France.
‡ E-mail address: [email protected]
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"ELECTRODYNAMICS OF SUPERCONDUCTORS EXPOSED TO HIGH FREQUENCY FIELDS",
"ELECTRODYNAMICS OF SUPERCONDUCTORS EXPOSED TO HIGH FREQUENCY FIELDS"
] |
[
"Ernst Helmut Brandt \nMax-Planck-Institut für Metallforschung\nStuttgartGermany\n"
] |
[
"Max-Planck-Institut für Metallforschung\nStuttgartGermany"
] |
[] |
The electric losses in a bulk or film superconductor exposed to a parallel radio-frequency magnetic field may have three origins: In homogeneous vortex-free superconductors losses proportional to the frequency squared originate from the oscillating normal-conducting component of the charge carriers which is always present at temperatures T > 0. With increasing field amplitude the induced supercurrents approach the depairing current at which superconductivity breaks down. And finally, if magnetic vortices can penetrate the superconductor they typically cause large losses since they move driven by the AC supercurrent.
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"https://arxiv.org/pdf/1008.2231v1.pdf"
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1008.2231
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2deb27a8458da3958c80d49023cd8581c1af20fd
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ELECTRODYNAMICS OF SUPERCONDUCTORS EXPOSED TO HIGH FREQUENCY FIELDS
12 Aug 2010
Ernst Helmut Brandt
Max-Planck-Institut für Metallforschung
StuttgartGermany
ELECTRODYNAMICS OF SUPERCONDUCTORS EXPOSED TO HIGH FREQUENCY FIELDS
12 Aug 2010
The electric losses in a bulk or film superconductor exposed to a parallel radio-frequency magnetic field may have three origins: In homogeneous vortex-free superconductors losses proportional to the frequency squared originate from the oscillating normal-conducting component of the charge carriers which is always present at temperatures T > 0. With increasing field amplitude the induced supercurrents approach the depairing current at which superconductivity breaks down. And finally, if magnetic vortices can penetrate the superconductor they typically cause large losses since they move driven by the AC supercurrent.
INTRODUCTION
The phenomenon of superconductivity was discovered in 1911 by Heike Kamerlingh-Onnes in Leiden. After he had achieved to liquify helium at the temperature of T = 4.2K he observed that the resistivity of Hg became unmeasurably small below some "critical temperature" T c = 4.15 K. A sensitive method to measure the residual resistivity in this "superconducting state" is to observe the temporal decay of the persistent "supercurrents" in a ring, say of Pb (T c = 7.2 K), Sn (T c = 3.72 K), or Nb (T c = 7.2 K) by monitoring the magnetic field generated by the circulating current. It turned out [1] that the supercurrent does not decay measurably, even after several years. Ideally lossfree superconducting wires may thus be used to build coils which keep their magnetic field for years, after the windings have been loaded with current and then are cut short by a superconducting switch.
Thus, DC currents in a superconductor can flow practically loss-free if they are not too large. However, it turned out that alternating currents (AC) in superconductors are not completely loss-free, in particular at high frequencies (RF = radio frequencies, MW = microwave frequencies). There are essentially three effects which cause energy dissipation during current flow in superconductors:
(a) Even in ideally homogeneous bulk superconductors an electric field E ∝ ω (with frequency ω/2π) is required to accelerate the "superconducting electrons", the Cooper pairs of the microscopic BCS theory [2]. This electric field also moves the "normalconducting" electrons that are always present at finite temperatures T > 0. The dissipated power of this effect is ∝ E 2 ∝ ω 2 .
(b) When the current density inside the superconductor reaches the depairing current density j dp , the superconducting order parameter is suppressed to zero at the place * ehb @ mf.mpg.de where j = j dp . This means superconductivity disappears and electric losses appear. This nucleation of the normal state typically occurs at the specimen surface or in the center of Abrikosov vortices. In particular, when an increasing magnetic field H a is applied along a superconducting half space x > 0 one initially has j(x) = (H a /λ) exp(−x/λ) when j ≪ j dp , but when H a reaches the thermodynamic critical field H c = Φ 0 /( √ 8πλξµ 0 ) one has j ≈ j dp near the surface, thus one has j dp ≈ H c /λ. Here Φ 0 = h/2e = 2.07 · 10 −15 Tm 2 is the quantum of magnetic flux, λ is the magnetic penetration depth, and ξ is the superconducting coherence length. Within Ginzburg-Landau (GL) theory, valid near T = T c , the lengths λ(T ) = κξ(T ) ∝ (T c − T ) −1/2 diverge as T → T c , but the GL parameter κ = λ/ξ is independent of temperature T .
(c) Large dissipation may be caused by vortices inside the superconductor. These move under the action of the induced AC current, which exerts a Lorentz force on the vortices and causes them to oscillate and dissipate energy. At high frequencies the amplitude of this oscillation is smaller then the range of possible pinning forces caused by material inhomogeneities, e.g., precipitates or defects in the crystal lattice. This vortex dissipation then cannot be suppressed by introducing pins.
For a flat bulk type-II superconductor (defined by κ > 1/ √ 2) in thermodynamic equilibrium it is favorable that part of the magnetic flux penetrates in form of Abrikosov vortices when the applied field H a equals or exceeds the lower critical field H c1 ≈ (Φ 0 /4πλ 2 µ 0 )(ln κ + 0.5) (for κ > 1.5) and has not yet reached the upper critical field H c2 ≈ Φ 0 /(2πξ 2 µ 0 ) where superconductivity vanishes. The penetration of vortices at an ideally flat surface may be delayed by a surface barrier leading to a higher penetration field H p ≈ H c ≥ H c1 (overheating). On the other hand, with superconductors of finite size, demagnetization effects may allow the vortices to penetrate already at much lower fields. In particular, for a large film of width w and thickness d ≪ w, the penetration field is strongly reduced, H p /H c1 ≈ d/w . . . d/w ≪ 1 depending on the edge profile, see below. An infinitely large thin film will thus be penetrated by any perpendicular magnetic field component, even if very small. tially confirmed the two-fluid picture, giving for its phenomenological parameters a microscopic interpretation and explicit expressions.
Two-Fluid Model
The phenomenological two-fluid model assumes that the total electron density is composed of the density of superconducting electrons n s and that of normal electrons n n , which have different relaxation times τ s and τ n . Historically, Gorter and Casimir assumed n n ∝ t 4 (t = T /T c ) and n s ∝ 1 − t 4 . As in the Drude model [1], the drift velocity v of each of these two fluids should obey a Newton law,
m dv/dt = eE − mv/τ(1)
with m and e the mass and charge of the electron. The total current density J = J s + J n is the sum of the supercurrent J s = en s v s and the normal current J n = en n v n . (In all other Sections of this paper the current density is denoted by j.) If one assumes τ s = ∞ one obtains from Eq. (1) the first London equation
dJ s /dt = (n s e 2 /m)E = E/(µ 0 λ 2 )(2)
with λ = (m/n s e 2 µ 0 ) 1/2 the London depth. In the London gauge where E = −dA/dt (induction law) Eq. (2) may be written in form of the second London equation
J s = (µ 0 λ 2 ) −1 A.
For the normal electrons one may assume τ n ≪ 1/ω with periodic electric field E ∝ exp(iωt). This gives for the normal current J n = (n n e 2 τ n /m)E.
Defining the complex conductivity σ(ω) = σ 1 (ω)−iσ 2 (ω) by J = σ(ω)E ∝ exp(iωt), one obtains σ 1 (ω) = (πn s e 2 /mω) δ(ω) + n n e 2 τ n /m,
σ 2 (ω) = n s e 2 /mω = (µ 0 λ 2 ω) −1 ≫ σ 1 .(4)
For a normal conductor this yields σ 1 = σ n , σ 2 = 0, and the skin depth δ skin = (2/µ 0 σ n ω) 1/2 . For superconductors at ω = 0 the δ-function in σ 1 reflects the ideal DC conductivity, while at finite frequencies the inductive part dominates, σ 2 ≫ σ 1 . However, the small dissipative part σ 1 = n n e 2 τ n /m is important since it causes the AC losses.
In the situation with an AC magnetic field parallel to the superconductor surface, the current is forced (a current bias as opposed to a voltage bias) and one has a dissipation per unit volume ρJ 2 = Re{1/σ}J 2 ≈ (σ 1 /σ 2 2 )J 2 ≈ σ 1 E 2 since σ 1 ≪ σ 2 . The dissipation is thus proportional to n n ω 2 .
The sum σ = σ 1 − iσ 2 is analogous to a circuit of a resistive channel 1/R ∝ σ 1 in parallel to an inductive channel of admittance 1/iωL ∝ σ 2 . Below a frequency ω 0 = R/L this circuit is mainly inductive and above mainly resistive. The ratio of the currents in the two channels is J s /J n = n s /(n n ωτ ). This defines a crossover frequency ω ≈ (n s /n n )(1/τ n ) ≈ (n s /n n ) · 10 11 Hz [1].
When the superconductor forms the inner wall of a microwave cavity with incident parallel magnetic field of amplitude H inc , this wave is almost ideally reflected by the wall since a surface screening current J s = 2H inc is induced. The small dissipated power per unit area is then
P s = J 2 s R s , where R s = δ −1 Re{1/σ} = δ −1 σ 1 /|σ| 2 ≈ δ −1 σ 1 /σ 2 2 (5)
is the surface resistance (e.g. in units Ω). Here δ = [ 2/µ 0 (|σ| + σ 2 )ω ] 1/2 is the general skin depth reducing to the superconducting penetration depth λ or to the skin depth δ skin in the super or normal conducting limits. For the superconducting wall one has R s ≈ σ 1 µ 2 0 λ 3 ω 2 /2 and the absorbed versus incident power of this wall is [1]
P abs P inc = J 2 s R s cµ 0 H 2 inc = 4R s cµ 0 ≈ 1 Q .(6)
The quality factor Q of the superconducting cavity is thus inversely proportional to R s ∝ Q −1 ∝ n n ω 2 .
Microscopic Theory
After the BCS theory [2] had given the microscopic explanation of superconductivity, the complex AC conductivity was calculated within this weak-coupling theory [3,4]. In the extreme local limit (λ ≪ ξ 0 =hv F /π∆ with v F = Fermi velocity and ∆ = energy gap; this assumption actually is not satisfied for type-II superconductors with GL parameter κ > 0.7), in the impure limit (electron mean free path l = v F τ ≪ ξ 0 ), and for frequencies below the energy-gap frequency (hν < 2∆, ν = ω/2π) the resulting AC conductivity may be expressed as two integrals over an energy variable,
σ 1,2 σ n = f 1,2 (ǫ, ∆, T, ω) dǫ ,(7)
where σ n = ne 2 τ /m = ne 2 l/p F (p F = mv F = Fermi momentum, n = electron density) is the Drude conductivity in the normal state and f 1 and f 2 are some functions. Evaluating these integrals for the case ω ≪ T ≪ ∆ (in unitsh = k B = 1) one obtains [5]
σ 1 σ n = 2∆ T exp − ∆ T ln 9T 4ω , σ 2 σ n = π∆ ω .(8)
The dissipative part σ 1 and inductive part σ 2 may be written in the form of the two-fluid model:
σ 1 ≈ n qp e 2 l/p F , σ 2 ≈ n s e 2 /mω ,(9)
where n qp is the quasiparticle density (replacing the normal electron density n n of the two-fluid model) and n s the superconducting electron density,
n qp = n ∆ T exp − ∆ T 2 ln 9T 4ω , n s = n l / ξ 0 .(10)
The quality factor Q of the resonator is now
Q −1 ∝ R s ≈ 1 2 µ 2 0 λ 3 σ 1 ω 2 ∝ n qp ω 2 .(11)
Since the quasiparticle density n qp ∼ exp(−∆/T ) strongly decreases at low temperatures T , Q should increase drastically. Note that with increasing purity (increasing l) σ 1 increases but the penetration depth λ ≈ λ pure 1 + ξ 0 /l decreases in Eq. (11). Thus, maximum Q is reached at some intermediate, not too high purity of the superconductor.
High-Purity Niobium
For the high-purity Nb used in the TESLA cavities, the frequency dependent surface resistance has been computed by Kurt Scharnberg within the Eliashberg model that extends the BCS model to strong coupling superconductors [6]. Strong (electron-phonon) coupling effects change the amplitude and the temperature dependence of the gap parameter, they lead to a renormalization (enhancement) of the quasiparticle mass, which in turn affects the London penetration depth, and they result in temperature and energy dependent quasiparticle lifetimes. The electronphonon interaction enters in form of the Eliashberg function α 2 F (ω) which was taken from tunneling experiments. A Coulomb pseudopotential µ * = 0.17 and a Coulomb cut-off ω c = 240 meV were used. At sufficiently low T and low ω of the incident radiation, inelastic scattering is negligible and only disorder induced elastic scattering is important, which is parameterized by the normal state scattering rate Γ N = 1/2τ . This is fit to the surface resistance R s measured at ν = 1.3 GHz, yielding Γ N ≈ 1 meV and τ ≈ 3 · 10 −13 sec. Nonlocal effects (wave vector q > 0) were disregarded, which is partly corrected for by using the lifetime τ fitted at 1.3 GHz.
With these assumptions the surface resistance R s ≈ 1 2 σ 1 µ 2 0 λ 3 ω 2 of high-purity Nb was computed at T = 2 K. Note that R s is related to the reflectivity r of the metal by
R s = (Z 0 /4)(1 − r) where Z 0 = (ǫ 0 c) −1 = 377 Ω is the impedance of the vacuum.
Starting from R s ≈ 20 nΩ at 1.3 GHz the resistance rises to a few µΩ at 600 GHz and then exhibits a large step at 750 GHz to a value of 15 mΩ. Above this energy-gap frequency ν = 2∆/h one has nearly constant R s till at least 2000 GHz.
VORTICES IN SUPERCONDUCTORS
Ginzburg-Landau and London Theories
Before the microscopic explanation of superconductivity was given in 1957 by BCS [2] there were very powerful phenomenological theories that were able to describe the thermodynamic and electrodynamic behavior of superconductors. In 1935 Fritz and Heinz London established the London theory, see Eq.(2) above, and in 1952 Vitalii Ginzburg and Lev Landau conceived the Ginzburg-Landau (GL) theory. The GL theory may be written as a variational problem that minimizes the spatially averaged GL free energy density,
− |ψ 2 | + 1 2 |ψ| 4 + |(iξ∇+A)ψ| 2 + (λ∇×A) 2 = Minimum .(12)
Here ψ(r) is the complex GL-function, or order parameter, and A(r) the vector potential of the magnetic induction B = ∇ × A. The two lengths are the magnetic penetration depth λ (usually taken as unit length) and the GL coherence length ξ; both lengths diverge as the temperature T approaches the critical temperature T c ,
λ ∝ ξ ∝ (T c − T ) −1/2 . Their ratio, the GL parameter κ = λ/ξ within GL theory (valid near T c ) is independent of T .
The GL theory can be derived from the microscopic BCS theory (L. P. Gor'kov 1959) in the limit T c − T ≪ T c , yielding for the GL function ψ(r) = ∆(r)/∆ BCS where ∆ is the energy gap function. The London theory follows from GL theory in the cases when the magnitude of the order parameter is nearly constant, |ψ| ≈ 1. This condition is fulfilled when ξ is small as compared to the specimen extension and to λ, requiring κ ≫ 1. An arrangement of straight or arbitrarily curved vortex lines positioned at r ν (z) = [x ν (z), y ν (z), z] (ν = 1, 2, 3 . . .) then has a magnetic field that obeys the London equation modified by adding δ functions centered at the vortex cores,
(1 − λ 2 ∇ 2 ) B(r) = Φ 0 ν dr ν δ 3 (r − r ν ) .(13)
Ideal Vortex Lattice
In 1957 Alexei Abrikosov, a thesis student of Lev Landau in Moscow, obtained a periodic solution of the Ginzburg-Landau equations and recognized that this corresponds to a lattice of vortices of supercurrent, circulat- ing around each zero of the order parameter and carrying a quantum of magnetic flux Φ 0 ; these vortex lines (or flux lines, fluxons) are energetically favorable when the applied magnetic field is between a lower and a higher critical field, H c1 ≤ H c2 (see Introduction and below). This solution exists in bulk superconductors with GL parameter κ ≥ 1/ √ 2, called type-II superconductors. For this theoretical discovery Abrikosov received the Nobel Prize in Physics in 2003. Figure 1 shows the magnetic field B(r) and the order parameter |ψ(r)| 2 of one isolated vortex line for three values of the GL parameter κ = 2, 5, 20. One can see that B decays over the length λ and the vortex core has a radius ≈ ξ.
For such not too small values of κ to a good approximation the vortex field is the London solution, with the central singularity smoothened over the core radius r c ≈ √ 2 ξ [7,8], Figure 2 shows cross sections of B(x, y) and |ψ(x, y)| 2 along the nearest neighbor direction y = 0 of the ideal triangular vortex lattice for two values of the average induc-tionB = B (with vortex spacing a = 2λ and a = 4λ) and κ = 5. The dashed line is B(r) for the isolated vortex. Figure 3 shows the contour lines of B(x, y) near B c2 for the triangular vortex lattice. These lines coincide with the contours of |ψ(x, y)| 2 and with the stream lines of the supercurrents.
B v (r) ≈ Φ 0 2πλ 2 K 0 r 2 + r 2 c λ K 0 (x) = ln(1.123/x), x ≪ 1 π/2x exp(−x), x ≫ 1 . (14) K 0 (x) is a modified Bessel function. The interaction en- ergy of two vortices at a distance x ≫ ξ is U int = Φ 0 B v (x)/µ 0 .
The vortex lattice was first observed in the electron microscope by U. Essman and H. Träuble [9] at our Max Planck Institute in Stuttgart, by decoration of the surface of a Nb disk with "magnetic smoke" generated by evaporating an iron wire in a He atmosphere of 1 Torr, see When the superconductor is not a long cylinder or slab in parallel field, demagnetization effects shear the magnetization curves of Fig. 5 and reduce the field of first vortex penetration, see below. The vortices end at the upper and lower surface of finite-size specimens and send their magnetic field lines into the surrounding vacuum, see Fig. 6. The resulting modulation of B(x, y) just outside the surface can be observed by decoration (Fig. 4) and by magneto-optics or Hall probes. The magnetic field lines of one vortex in a thick film in a perpendicular magnetic field are depicted in Fig. 7 as obtained from London theory in [11]. Figure 8 shows the field lines of the periodic vortex lattice in films of thicknesses d = 4λ, 2λ, λ, and λ/2 as calculated in [12].
Losses by Moving Vortices
When a supercurrent flows in a superconductor, either applied by contacts or caused by a gradient or curvature of the local magnetic field, this current density j exerts a Lorentz force f = j ×ẑΦ 0 on a vortex. The Lorentz force density on a vortex lattice is F = j ×B. Neglecting a small Hall effect, the vortices move along this force with velocity
E =B × v = η −1B × (j ×B) = ρ ff j , (15) ρ ff ≈ (B/B c2 ) ρ n .(16)
Here ρ ff is the flux-flow velocity, which at large average inductionsB is comparable to the normal resistivity of the superconductor at that temperature (measurable by applying a large field B a > B c2 ). However, when only a few vortices have penetrated (B ≪ B c2 ) one has much smaller resistivity ρ ff ≪ ρ n . But even then the vortex-caused dissipation at low T is typically much larger than the dissipation caused by the normal excitations.
Where does this resistive dissipation come from? There are two effects of comparable size, see Fig. 9. First, as pointed out by Bardeen and Stephen [1,13], the motion of the magnetic field induces a dipolar electric field that drives current through the superconductor and through the vortex core. If the vortex core is modelled as a normal conducting tube of radius r c ≈ ξ, the normal currents inside the vortex core dissipate energy that leads to the ρ ff of Eq. (16). Second, as stated by Tinkham [1,14], the moving vortex core means that at a given position the order parameter |ψ| 2 goes down and up again when the core passes. If one assumes a delay of the recovery of |ψ| 2 by a relaxation time τ ≈h/∆ one obtains an additional dissipation of the order of Eq. (16). These two sources of losses are nice for physical understanding. In the exact calculation of the dissipation of a moving vortex lattice from time-dependent GL theory [15] these two sources cannot be separated but the approximate Eq. (16) is essentially confirmed [16], also by microscopic theory [17]. The numerical and also the measured flux-flow resistivity in the middle between the exact values 0 and ρ n is somewhat smaller than the Eq. (16), i.e., for constant current source the real dissipation is lower.
Pinning of Vortices
When the material is inhomogeneous on the microscopic length scale of the vortex core ξ, then the vortices are pinned and cannot move as long as the Lorentz force does not exceed the pinning force, or the current density j is smaller than the critical current density j c , see the reviews [18,19,20]. In this way the electric losses caused by flux flow can be avoided, and completely loss-free conductors of DC current can be tailored by introducing appropriate pinning centers into the material, e.g., precipitates and crystal lattice defects. For AC currents small losses remain, however. One source of AC dissipation is due to the (albeit small) concentration of normal carriers or excitations and can be understood from the two-fluid model as discussed above. The other source is the oscillation of vortices in the pinning wells. At small displacements u from their equilibrium position one may assume linear elastic binding of the vortices to the pins, with a force density −ku. Adding to this the viscose drag force −ηu and the Lorentz force one obtains the force balance equation in an AC current j ac ∝ exp(iωt),
j ac ×B = −ku − iωηu .(17)
One can see that at frequencies above k/η, of order ω/2π > 10 7 Hz, the viscose force dominates [21]. Pinning thus cannot prevent vortex oscillations at high fre- Figure 11: Penetration of vortex lines into pin-free cylinders with radius a and height 2b. Top: b/a = 2. Bottom: b/a = 0.3. Forced by the applied field H a , the vortices enter from the corners, but only when the applied field has reached some threshold field do they jump to the middle leaving a vortex-free zone near the surface. With further increasing H a the vortices eventually fill the cylinder uniformly from the middle. This delayed penetration without pinning is called geometrical barrier. Such a barrier is absent only for ellipsoid-shaped specimens.
quencies. This vortex-caused dissipation increases as ω 2 , like the quasiparticle dissipation, cf. Eq. (11). Interesting theoretical problems are the statistical summation of random pinning forces to obtain the critical current density j c , and the problem of thermally activated depinning [18,19,20]. The latter leads to vortex motion even at small currents densities j < j c due to finite temperature. This flux creep may be described by a highly nonlinear resistivity. In particular, a logarithmic dispersive activation energy for depinning, U (j) = U 0 ln(j c /j), leads to an often observed power-law current-voltage curve,
E(j) = E 0 exp[−U (j)/k B T ] = E 0 (j/j c ) n(18)
with a creep exponent n = U 0 /k B T . For n = 1 one has Ohmic behavior [free flux flow, Eq. (15)], for n ≫ 1 one has flux creep, and in the limit n → ∞ this power law yields the Bean model, in which j is either 0 or j c : When at some position one has j > j c , the vortices rearrange immediately such that j is reduced to j c again. This concept is useful for DC currents and at not too high frequencies where the pinning forces exceed the viscose drag force.
Geometry Effects
The electromagnetic properties of a superconductor (and of any conductor or isolator) depend not only on the material but also on the geometry of the problem, i.e., on the shape of the specimen and on the way a magnetic or electric field is applied. For example, the reversible magnetization curves of a pin-free superconductor in Fig. 5 apply to the unrealistic case of very long slabs or cylinders in exactly parallel field, where demagnetization effects are absent. For the still unrealistic situation of a perfect ellipsoidal shape one may calculate from these ideal curves the reversible magnetization curves of any ellipsoid by using the concept of the demagnetization factor. But when the specimen shape is not an ellipsoid, then even for a pinfree superconductor the magnetization curves have to be computed numerically, since now the induction (or vortex density) inside the specimen is no longer spatially constant.
It turns out that even without pinning such magnetization curves in general show a hysteresis, i.e., they are irreversible and depend on the magnetic history, see Fig. 10. This irreversibility is due to a geometric barrier [22,23] for the penetration of vortices as illustrated in Fig. 11 for cylinders (or long bars) with rectangular cross section: When the applied uniform field H a is increased, vortex lines enter at the corners, pulled by the screening currents (Meissner currents) that flow at the surface, and held back by their line tension (like a rubber band). With increasing H a the vortices penetrate deeper and become longer. When the vortices from two corners meet at the equator, they connect and form one long vortex line that contracts and immediately jumps to the specimen center. During this rapid jump all their elastic energy is dissipated by the viscose drag force F = ηv, see text above Eq. (15). With further increasing H a more vortices jump to the center, crossing the flux-free zone near the surface, and eventually the entire specimen is filled with vortices coming from the growing central zone.
Flux penetration thus occurs with a threshold, over a "geometrical barrier". The sudden onset of flux penetration to the center leads to the sharp maximum in the small (inner, pin-free) hysteresis loop of M (H a ) in Fig. 10. When H a is decreased again, the vortices leave the specimen essentially without barrier, and at H a = 0 all vortices have left, i.e., one hasB = 0 and also M = 0 (since no screening currents flow anymore). The perpendicular field at which the first vortices enter at the corners of a pin-free long strip and a circular disk, both with rectangular cross section 2a × 2b, was computed in [23]:
H strip pen ≈ H c1 tanh 0.36 b/a , H disk pen ≈ H c1 tanh 0.67 b/a .(19)
In the presence of pinning the hysteresis loops of M (H a ) in Fig. 10 become larger. The area of such loops is the energy dissipated during one cycle due to depinning of vortices. When H c1 is negligibly small as compared to H a , the hysteresis curves and the vortex density and currents in a superconductor with pinning may be computed by treating it as a nonlinear conductor, Eq. (18). Figure 12 shows how the magnetic field lines (and vortices) penetrate and exit a thick disk with pinning when an axial H a is first increased beyond the field of full penetration, and then is decreased again [24,25]. The chosen large creep exponent n = 50 practically reproduces the Bean model. Figures 10, 11, and 12 were computed by timeintegration of an equation for the (scalar) current density j inside the superconductor; this method implicitly accounts for the infinitely extended magnetic stray field outside the specimen, without need to compute it and to cut it off. From the resulting current density the magnetic field lines are then easily calculated by the Biot-Savart law.
A completely different geometry is shown in Fig. 13, namely, the current stream lines and the contours of the magnetic field B z (x, y) in a thin film or platelet of rectangular shape [26] with pinning and large creep exponent n = 50 corresponding to the Bean model like in Fig. 12. An increasing magnetic field H a is applied perpendicular to the film. Initially, when H a ≪ J c = dj c is small, no magnetic flux penetrates the film, i.e., the circulating screening currents generate a magnetic field that in the film area is constant (of size −H a ) and exactly compensates the applied field H a . With increasing H a , magnetic flux penetrates mainly from the middle of the sides of the rectangle (not from the corners), leaving still a flux-free zone in the middle. At and beyond some field of full penetration the current stream lines are concentric rectangles of constant distance, since the magnitude of the sheet current has saturated to the constant value J c = dj c . The magnetic field has then penetrated to the center, and the contour lines of B z (x, y) do not change anymore with further increasing H a .
Penetration of First Vortex
An important question for RF superconductivity is under what circumstances and at which applied magnetic field H p the first vortex enters the superconductor, since the presence of even a few vortices can cause large losses. First I summarize the expressions for the three critical fields which for type-II superconductors (with κ ≥ 1/
√ 2) obey B c1 ≤ B c ≤ B c2 : B c1 ≈ Φ 0 4πλ 2 ( ln κ + α) ,(20)B c = Φ 0 √ 8πλξ = √ 2κ ln κ + α B c1 ,B c2 = Φ 0 2πξ 2 = √ 2κ B c , α(κ) = 1 2 + 1 + ln 2 2κ − √ 2 + 2 = 1.35, κ = 0.71 0.50, κ ≫ 1 .
While the thermodynamic (B c ) and upper (B c2 ) critical fields are exact, the lower critical field B c1 = µ 0 H c1 has to be calculated numerically from the self energy of a vortex of length L, U self = Φ 0 H c1 L. The function α(κ) is a good analytical fit to the numerical result of [10]. At H a = H c1 the nucleation of a vortex and motion to a depth x ≫ λ does not cost energy, see Fig. 14. However, the penetrating vortex has to surmount a barrier such that the field of first penetration H p is larger than H c1 . This barrier was first predicted by Bean and Livingston (BL) [27] for a superconductor with planar surface in a parallel applied field H a . The Gibbs free energy G(x) for this case reads
G(x) = Φ 0 H a e −x/λ − 1 2 H v (2x) + (H c1 −H a ) .(21)
In it the first term is the interaction of the vortex with the applied field H a or with its screening currents (H a /λ)e −x/λ , the second term is the interaction with the image vortex (at position −x, of opposite orientation), and the third term is an integration constant. Using the fact that for not too small κ one has B v (0) ≈ 2B c1 (see Fig. 1) one has with Eq. (14), B c1 ≈ (Φ 0 /4πλ 2 )K 0 (r c /λ) yielding with B c1 (20) a core radius r c ≈ ξ exp[−α(κ)]. With this we may write G(x) (21) in the dimensionless form
G(x) Φ 0 H c1 ≈ H a H c1 e −x/λ −1 +1− K 0 ( 4x 2 +r 2 c /λ) K 0 (r c /λ)(22)
that is plotted in Fig. 14 The assumption of BL that the entering vortex is long, straight, and exactly parallel to a planar surface is not very realistic. Alternatively, one may assume that the first vortex nucleates and penetrates in form of a small loop, say a half circle of radius R, see Fig. 15 top. The self-energy of this half circle is approximately U self = πR(Φ 2 0 /4πλ 2 µ 0 ), putting the outer cut-off radius ≈ R instead of Λ ≫ R in the logarithm ln(λ/ξ) → ln(R/ξ) ≈ 1 when R is of order of ξ. The interaction of this vortex loop with the surface screening current of density j s is U js ≈ (πR 2 /2)Φ 0 j s (flux quantum times loop area times j s ). For a planar surface one has j s = H a /λ directly at the surface. The criterion that U js ≥ U self at H a ≥ H p yields then
H p ≈ Φ 0 /µ 0 2πλR = √ 2ξ R H c ≈ H c ,(23)
which is just the BL result. Thus, the assumption of a penetrating vortex loop does not change much the penetration field of a planar surface. However, when the surface has roughness with characteristic length ≥ ξ, then vortices will penetrate at sharp points or cusps, see Fig. 15. At a corner with angle α = 90 o , the screening current directly at the surface is strongly enhanced at this corner; Fig. 16 shows this for a superconducting bar with square cross section 2a × 2a and penetration depth λ = 0.025a, to which a uniform transverse H a is applied. A rough estimate gives for this geometry an enhancement of the screening current at this corner, j s = CH a /λ, by a factor C ≈ 4. The field of first vortex penetration H p is then reduced from Eq. (23) by just this factor,
H p ≈ H c /C ≈ H c /4.
For sharper corners the enhancement of j s and reduction of H p are even larger. As shown in the textbook of Landau-Lifshitz (Electrodynamics of Continua) for an ideal diamagnetic material at a corner with angle α (Fig. 17) the magnetic field diverges as H ∝ 1/r β with exponent β = (π − α)/(2π − α), where r is the distance to the point of the corner. This gives H ∝ 1/r 1/3 for α = π/2 and H ∝ 1/r 1/2 for α → 0.
Similarly, an axially applied magnetic field flowing around an ideal diamagnetic cylinder, sphere, or disks with elliptical or rectangular cross section of aspect ratio b/a ≪ 1, is enhanced at its equator by factors 2, 3/2, a/b, or ≈ (a/b) 1/2 , respectively, due to the strong curvature of the field lines at this line, see Fig. 18.
Vortices in thin films
One has to distinguish two quite different types of vortices in thin film superconductors: vortices perpendicular or parallel to the film plane. In wide thin films with width w = 2a ≫ thickness d = 2b, the vortices will nearly always run perpendicular across the film thickness, even in tilted applied field H a , because of the large demagnetization factor of this film. This means the circulating currents prefer to flow in the film plane. Only when H a is exactly parallel to the film surface, or when the film is coating a bulk superconductor that screens any perpendicular field component, then vortices parallel to the film plane may occur.
When the film is of finite size, one may use Eqs. (19) to estimate at which applied perpendicular field component H az the first vortices penetrate, namely already at a very small field, smaller than H c1 d/w. When the film edges are wedge-shaped or sharp, the penetration field is even smaller, cf. Fig.17 and Fig. 18 (elliptical edge). Into infinitely extended or closed films (e.g., a Nb layer covering the inner surface of a Cu cavity) any perpendicular field will penetrate since the field lines cannot flow around the film. Only when this film has holes or slits can some magnetic flux cross the film via these holes, but the magnetic field in the holes will be larger than H az by at least the ratio of film area over the total area of all holes. However, the field in the holes will penetrate into the film when it is of the order of H c1 times the square root of film thickness over hole distance. Thus, even such a perforated pin-free film will be penetrated by a perpendicular field that is very much smaller than H c1 . The peaked magnetic field, circulating current, and pair interaction of perpendicular vortices in thin films were calculated for infinitely extended [28] and finite-size (e.g. rectangular) films [26,29,30].
Pinning of vortices will not appreciably enhance all these penetration fields at high radio frequencies, where the (elastic) pinning forces are smaller than the viscose drag force. If the small applied perpendicular magnetic field is a DC field (e.g., some stray field or the earth magnetic field) then the additional RF field will even favor the penetration of the DC field in form of vortices, since it "shakes" the vortices. As shown in [31,32], shaking of vortices by an AC field oriented perpendicular to the vortices leads to the relaxation of irreversible currents if the AC amplitude exceeds some threshold value. This vortex creep means that even in very small H az , perpendicular vortices will penetrate under the action of a large-amplitude RF field, and then these vortices oscillate and dissipate energy.
The problem of parallel vortex lines in a thin film with d ≪ λ was solved by Alexei Abrikosov (1964), Vadim Shmidt (1969), and in an elegant way by Alex Gurevich [33]. The lower critical field is enhanced in thin films as compared to bulk superconductors,
B c1 = 2Φ 0 πd 2 ln d λ − 0.07 ,(24)
and the field at which the surface barrier for vortex penetration disappears is also enhanced,
B p = Φ 0 2πdξ .(25)
For example, a NbN film with ξ = 5 nm, d = 20 nm has B c1 = 4.2 T and B p = 6.37 T, much better than the penetration field B p ≈ B c = 0.18 T for Nb at low T . To enhance the operating RF amplitude in microwave cavities for accelerators and reduce the losses, Gurevich [34] suggests to use solid Nb or Pb with multilayer coating on its inner surface by alternating superconducting and insulating layers with d < λ. This will prevent penetration of vortices into the bulk superconductor when the vortex penetration field B p is large; e.g., for NbN films with d = 20 nm the RF field can be as high as 4.2 T. From the elastic and viscose forces on a parallel vortex in a thin film, Gurevich estimates its characteristic relaxation time as τ ≈ 2dµ 0 λ 2 /(ξρ n ) .
For a 30 nm Nb 3 Sn film this τ ≈ 10 −12 s is much shorter than the RF period of 10 −9 s. The maximum amplitude of the RF field at which the surface barrier of a single thin film coating disappears is of the order of the bulk H c of the film material, e.g., 0.54 T for Nb 3 Sn. Thus, Nb 3 Sn coating more than doubles the vortex penetration field for Nb, B p ≈ B c = 0.18 T at low T . It appears that Nb cavities coated with a Nb 3 Sn layer or with NbN multilayers allow for much higher RF amplitudes than uncoated Nb, or Cu coated by a Nb film, if this can be achieved technically.
Figure 1 :Figure 2 :
12Magnetic field B(r) and order parameter |ψ(r)| 2 of an isolated flux line calculated from Ginzburg-Landau theory for GL parameters κ = 2, 5, and 20. Note that the field in the vortex center is B(0) ≈ 2B c1 . Two profiles of the magnetic field B(x, y) and order parameter |Ψ(x, y)| 2 taken along the x axis (a nearest neighbor direction) for vortex lattices with lattice spacings a = 4λ (bold lines) and a = 2λ (thin lines). The dashed line shows the magnetic field of an isolated flux line fromFig. 1. Calculated from GL theory for κ = 5[10].
Figure 3 :
3Current stream lines, coinciding with the contours of B(x, y) and |ψ(x, y)| 2 . Abrikosov solution for the ideal vortex lattice near the upper critical field B c2 .
Figure 4 :
4Vortex lattice made visible by decoration with iron micro-crystallites. Top: Nb disk, 1 mm thick, 4 mm diameter, T = 4 K, B a = 985 Gauss, vortex spacing a = 170 nm (U. Essmann and H. Träuble 1968). Bottom: YBa 2 Cu 3 O 7−δ , T = 77 K, B a = 20 Gauss, a = 1200 nm (D. Bishop and P. Gammel 1987).
Figure 5 :
5Ideal reversible magnetization curves of a long superconducting cylinder or slab in parallel field B a computed from GL theory[10].
Figure 6 :
6Magnetic field lines of vortex lines in and near a superconductor of finite size, and the circulating super currents (schematic).
Figure 7 :Figure 8 :
78Magnetic field lines of a single vortex in a superconducting film of thickness d = 8λ (or half space z ≤ 0). Analytical solution of London theory[11]. Magnetic field lines of the periodic vortex lattice in films of thicknesses d = 4λ, 2λ, λ, and λ/2. From GL theory for κ = 1.4 andB/B c2 = 0.04[12]. The dashed lines mark the film surfaces. x 1 = vortex spacing.
Fig. 4 .
4The magnetization −M = B a −B of the superconductor calculated numerically [10] as function of the applied magnetic field B a is depicted in Fig. 5 for ideal (pin-free) long superconductor cylinders or slabs in parallel B a (i.e., in absence of demagetization effects) with various κ = 0.707 . . . 10. At κ = 1/ √ 2 one has B c1 = B c = B c2 and the curve M (B a ) is the same as for type-I superconductors (with κ < 1/ √ 2), namely,B = 0, − M = B a (no penetrated flux) for B a < B c2 andB = B a , M = 0 (complete penetration of flux) for B a > B c2 .
Figure 9 :
9Visualization of the origin of energy dissipation when a vortex moves with velocity v. Top: the dipolar electric field lines induced by this motion run also through the normal conducting core (Bardeen-Stephen model). Bottom: During motion of the vortex core the order parameter relaxes (Tinkham term). v = η −1 F where η is a drag coefficient or viscosity. The vortex motion induces an average electric field
Figure 10 :
10Irreversible magnetization curves M (H a ) for a disk with radius a and thickness 2b, b/a = 0.125, for various pinning strengths measured by the parameter aj c /H c1 . The large hysteresis loop belongs to strong pinning aj c /H c1 = 4. The small central loop is for the pinfree disk, whos vortex distribution is shown inFig. 11. The reversible magnetization curve of a pin-free ellipsoid with same initial slope as M (H a ) is shown as dashed line.
Figure 12 :
12Bean model with constant critical current density j c for a superconducting bar with rectangular cross section 2a×2b (b/a = 0.35) put into a perpendicular magnetic field H a that first increases from 0 to 0.5 (left column) and then decreases again (right column). The parameter 0.01, 0.1, 0.3, 0.5, 0.3, 0.1, 0, -0.1 is H a /(aj c ). Shown are the magnetic field lines (solid lines) and the penetrating fronts (dashed lines) where the current density j (flowing along the bar) jumps from ±j c to 0 (in the field-free and currentfree core) or from j c to −j c (after full penetration of flux).
Figure 13 :
13Bean model for the penetration of a perpendicular magnetic field B a = µ 0 H a into a thin rectangular film with thickness d ≪ width. The parameter H a /J c = 0, 0.5, 1.5 measures H a in units of the critical sheet current J c = dj c . Shown are the stream lines of the sheet current in the film, J(x, y) = j(x, y, z) dz (left), and the contour lines of the magnetic field B z (x, y) in the plane z = 0 of the film (right).
Figure 14 :
14Gibbs free energy G of one vortex penetrating into a superconducting half space to a depth x, Eq.(22). Parameter is the applied field H a in units of H c1 . The Bean-Livingston Barrier exists for H a < H c .
Figure 15 :
15Nucleation of vortices as an arc of a circle at a planar surface (top), at a rectangular corner (middle), and at a rough surface (bottom, schematic).
Figure 16 :Figure 17 :
1617Supercurrents j z (x, y) in a bar with rectangular cross section 2a × 2b (b/a = 0.4) in the Meissner state with London penetration depth λ = 0.025a. The currents (along the bar) are generated by a perpendicular applied uniform magnetic field H a z that penetrates to a depth λ. Shown is one quarter of the cross section. Note the high (but finite) peak of j z (x, y) at the corners. The inset shows the magnetic field lines. Field enhancement near the sharp edge of an ideal diamagnet.
Figure 18 :
18Enhancement of the magnetic field H at the equator of ideal diamagnetic cylinders, spheres, and strips of elliptic or rectangular cross section put in a uniform axial applied magnetic field H a .
for κ ≈ 1.3. Of course, this G(x) is a only approximate, in particular at small κ, for which vortex penetration has to be computed numerically. Anyway,Fig. 14shows that vortex penetration becomes favorable at H a = H c and that the Bean-Livingston barrier vanishes at H a ≈ H c > H c1 .
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|
[] |
[
"Opacity, variability and kinematics of AGN jets",
"Opacity, variability and kinematics of AGN jets"
] |
[
"A M Kutkin \nAstro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia\n\nASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991PDDwingelooThe Netherlands\n",
"I N Pashchenko \nAstro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia\n",
"K V Sokolovsky \nAstro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia\n\nSternberg Astronomical Institute\nMoscow State University\nUniversitetskii pr. 13119992MoscowRussia\n",
"Y Y Kovalev \nAstro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia\n\nInstitutsky per\nMoscow Institute of Physics and Technology\n9, Moscow region141700DolgoprudnyRussia\n\nMax-Planck-Institut für RadioAstronomie\nAuf dem Hügel 6953121BonnGermany\n",
"M F Aller \nDept of Astronomy\nUniversity of Michigan\n311 West Hall48109-1107Ann ArborMIUSA\n",
"H D Aller \nDept of Astronomy\nUniversity of Michigan\n311 West Hall48109-1107Ann ArborMIUSA\n"
] |
[
"Astro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia",
"ASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991PDDwingelooThe Netherlands",
"Astro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia",
"Astro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia",
"Sternberg Astronomical Institute\nMoscow State University\nUniversitetskii pr. 13119992MoscowRussia",
"Astro Space Center of Lebedev Physical Institute\nProfsoyuznaya Str. 84/32117997MoscowRussia",
"Institutsky per\nMoscow Institute of Physics and Technology\n9, Moscow region141700DolgoprudnyRussia",
"Max-Planck-Institut für RadioAstronomie\nAuf dem Hügel 6953121BonnGermany",
"Dept of Astronomy\nUniversity of Michigan\n311 West Hall48109-1107Ann ArborMIUSA",
"Dept of Astronomy\nUniversity of Michigan\n311 West Hall48109-1107Ann ArborMIUSA"
] |
[
"MNRAS"
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Synchrotron self-absorption in active galactic nuclei (AGN) jets manifests itself as a time delay between flares observed at high and low radio frequencies. It is also responsible for the observing frequency dependent change in size and position of the apparent base of the jet, aka the core shift effect, detected with very long baseline interferometry (VLBI). We measure the time delays and the core shifts in 11 radio-loud AGN to estimate the speed of their jets without relying on multi-epoch VLBI kinematics analysis. The 15-8 GHz total flux density time lags are obtained using Gaussian process regression, the core shift values are measured using VLBI observations and adopted from the literature. A strong correlation is found between the apparent core shift and the observed time delay. Our estimate of the jet speed is higher than the apparent speed of the fastest VLBI components by the median coefficient of 1.4. The coefficient ranges for individual sources from 0.5 to 20. We derive Doppler factors, Lorentz factors and viewing angles of the jets, as well as the corresponding de-projected distance from the jet base to the core. The results support evidence for acceleration of the jets with bulk motion Lorentz factor Γ ∝ R 0.52±0.03 on de-projected scales R of 0.5-500 parsecs.
|
10.1093/mnras/stz885
|
[
"https://arxiv.org/pdf/1809.05536v3.pdf"
] | 85,502,112 |
1809.05536
|
65381b7cde5727a8f3a3e8af2ad31f2a83e482de
|
Opacity, variability and kinematics of AGN jets
2019
A M Kutkin
Astro Space Center of Lebedev Physical Institute
Profsoyuznaya Str. 84/32117997MoscowRussia
ASTRON
Netherlands Institute for Radio Astronomy
Oude Hoogeveensedijk 47991PDDwingelooThe Netherlands
I N Pashchenko
Astro Space Center of Lebedev Physical Institute
Profsoyuznaya Str. 84/32117997MoscowRussia
K V Sokolovsky
Astro Space Center of Lebedev Physical Institute
Profsoyuznaya Str. 84/32117997MoscowRussia
Sternberg Astronomical Institute
Moscow State University
Universitetskii pr. 13119992MoscowRussia
Y Y Kovalev
Astro Space Center of Lebedev Physical Institute
Profsoyuznaya Str. 84/32117997MoscowRussia
Institutsky per
Moscow Institute of Physics and Technology
9, Moscow region141700DolgoprudnyRussia
Max-Planck-Institut für RadioAstronomie
Auf dem Hügel 6953121BonnGermany
M F Aller
Dept of Astronomy
University of Michigan
311 West Hall48109-1107Ann ArborMIUSA
H D Aller
Dept of Astronomy
University of Michigan
311 West Hall48109-1107Ann ArborMIUSA
Opacity, variability and kinematics of AGN jets
MNRAS
0002019Accepted XXX. Received YYY; in original form ZZZPreprint March 26, 2019 Compiled using MNRAS L A T E X style file v3.0galaxies: active -galaxies: jets -galaxies: nuclei -radio continuum: galaxies -BL Lacertae objects: general -quasars: general
Synchrotron self-absorption in active galactic nuclei (AGN) jets manifests itself as a time delay between flares observed at high and low radio frequencies. It is also responsible for the observing frequency dependent change in size and position of the apparent base of the jet, aka the core shift effect, detected with very long baseline interferometry (VLBI). We measure the time delays and the core shifts in 11 radio-loud AGN to estimate the speed of their jets without relying on multi-epoch VLBI kinematics analysis. The 15-8 GHz total flux density time lags are obtained using Gaussian process regression, the core shift values are measured using VLBI observations and adopted from the literature. A strong correlation is found between the apparent core shift and the observed time delay. Our estimate of the jet speed is higher than the apparent speed of the fastest VLBI components by the median coefficient of 1.4. The coefficient ranges for individual sources from 0.5 to 20. We derive Doppler factors, Lorentz factors and viewing angles of the jets, as well as the corresponding de-projected distance from the jet base to the core. The results support evidence for acceleration of the jets with bulk motion Lorentz factor Γ ∝ R 0.52±0.03 on de-projected scales R of 0.5-500 parsecs.
INTRODUCTION
The jets of active galactic nuclei (AGN) are believed to consist of relativistic electron-positron or electron-proton plasma moving in magnetic field. The emission mechanisms are synchrotron and inverse Compton, covering a spectrum range from radio up to γ-rays of TeV energies (e.g., Jones et al. 1974, Marscher 1980). Radio loud AGN show different pc-kpc morphology and properties depending on the orientation of their jets to the line of sight (Urry & Padovani 1995. The smaller the viewing angle of a jet, the stronger is the Doppler boosting of the source emission leading to a "favoritism" in the observed properties of a flux limited AGN sample. Recent high-resolution observations of the jets in M 87 and 1H 0323+342 indicate the collimation and acceleration of the jets on scales up to hundreds of parsecs (Asada & Nakamura 2012, Hada et al. E-mail: kutkin asc.rssi.ru 2018). Acceleration on these scales is also seen with the very long baseline interferometry (VLBI) by measuring kinematics of bright knots in the jets , 2015, Jorstad et al. 2017.
The base of a jet seen with VLBI observations at cmwavelengths corresponds to the unit optical depth surface (photosphere), called the "core". The apparent position and size of the core depend on the observation frequency (Blandford & Königl 1979). This "core shift" effect can be inferred from multi-frequency VLBI experiments (e.g., Marcaide & Shapiro 1984, Kovalev et al. 2008, O'Sullivan & Gabuzda 2009, Pushkarev et al. 2012, Fromm et al. 2013b, Kutkin et al. 2014, Kravchenko et al. 2016, Lisakov et al. 2017, Voitsik et al. 2018. Not only do these measurements provide an information about the physical conditions in a jet, but also have an important practical implications for high-precision astrometry (e.g., Kovalev et al. 2008, Porcas 2009, Petrov & Kovalev 2017a, Petrov & Kovalev 2017b.
At VLBI scales most jets look self-similar across a wide range of frequencies (and hence, angular resolutions): a clumpy quasi-conical shape structure consisting of emission knots and widening with apex distance. The physical nature of these emission knots remains unclear (Zensus et al. 1995), while it is widely assumed that these regions of enhanced emission are somehow associated with shock waves in the relativistic plasma flow (e.g., Blandford & Königl 1979, Marscher & Gear 1985. Most of the knots move in a jet direction with super-or sub-luminal apparent speeds, while stationary components also seem to be a common occurrence (Rani et al. 2015, Jorstad et al. 2017, especially in BL Lac type objects (Piner & Edwards 2018). A wide range of component speeds is often observed within a single source, which makes their relation to the velocity of the underlying plasma flow unclear (e.g., Lister et al. 2016, and references therein). The jet flow speed being considerably faster than the observed jet pattern speed is one of the possible explanations for high brightness temperatures measured by RadioAstron in some AGN (Lobanov et al. 2015, while alternative interpretations of this phenomena might exist (Kellermann 2002).
In total flux density, the radio outbursts of AGN at lower frequencies are lagged with respect to that at higher frequencies. This is consistent with the opacity driven nature of the delays (e.g., Marscher 2016). In this scenario the peak of a flare at a given frequency might correspond to the moment when a disturbance traveling down the jet crosses the VLBI core at that frequency (Bach et al. 2006, Kudryavtseva et al. 2011. The agreement between time lags and core shifts frequency dependencies found using simultaneous multi-frequency measurements gives a good support for that scenario (Kutkin et al. 2014). Within this assumption, one can estimate the speed of a moving disturbance in the core region using the peak-to-peak time delay and the core shift measured at two frequencies. Hereafter we refer to this jet speed unless the details are specified. This speed can be considered as a proxy to the plasma flow speed (or bulk motion speed) whether the flares are caused by a moving blob or a shock wave (Marscher 2006). We note, however, that it might differ from the flow speed in a steady jet.
On the one hand the region probed with this method is well upstream the jet than that traced with VLBI kinematics analysis. On the other hand, there is an evidence that the major total flux density radio outbursts in AGN precede the occurrence of newborn VLBI components (e.g., Savolainen et al. 2003). Therefore, a comparison of the speed in the core region with the pattern velocity seen by VLBI appears to be of a particular interest.
Through this paper we assume flat ΛCDM cosmology model with H0 = 69.3 km/(s Mpc), ΩM = 0.286 (Hinshaw et al. 2013), implemented by the Astropy Python library (The Astropy Collaboration et al. 2018). We use positively defined spectral index α = d ln S/d ln ν.
THE SAMPLE AND THE DATA
We selected radio-loud AGN that demonstrate exceptional variability in radio band and for which we were able to find the required multifrequency total flux density and VLBI data. The sample includes two radio galaxies, two BL Lac type objects (hereafter BL Lacs), and seven flat spectrum radio quasars (hereafter quasars) according to their optical classification (Table 1). The prominent outbursts in their total flux density variations allow us to locate the peaks and estimate the multi-frequency peak-to-peak time delays. The optical class, redshift, and the observed median/maximal proper motion of the sources listed in Table 1 are adopted from the MOJAVE kinematics paper (Lister et al. 2016).
The single-dish flux density monitoring observations of the objects were obtained with the 26 m radio telescope of the University of Michigan Radio Observatory (UMRAO) at 8.0 and 14.5 GHz. The sources are included in the UMRAO "core sample" and are monitored regularly over 40 years (e.g., Aller et al. 2017).
We analyzed the seven-frequency (4.6-43.2 GHz) VLBA observations of the BL Lac type object 0851+202 (OJ 287) and the high-redshift flat spectrum radio quasar 1633+382 (4C 38.41). The observations of the two sources were performed in the framework of our survey of γ-ray loud blazars (Sokolovsky et al. 2010a,b) on 2009-02-02 (VLBA experiment BK150F) and 2009-06-20 (BK150L), respectively. The quasar 2251+158 (3C 454.3) investigated earlier by Kutkin et al. (2014) was also observed within this survey (BK150C, 2008-10-02). The source 0851+202 (1633+382) was observed for 9 (13) hours during which the VLBA was switching between C (6 cm, 4.6/5.0 GHz), X (4 cm, 8.1/8.4 GHz), Ku (2 cm, 15.4 GHz), K (1 cm, 23.8 GHz), and Q (7 mm, 43.2 GHz) band receivers. The C and X band data were split in two sub-bands centered at 4.6/5.0 GHz and 8.1/8.4 GHz, respectively. These sub-bands were independently analyzed. The data were recorded with the legacy VLBA backend at the aggregate bitrate of 256 Mbit/sec and correlated with the hardware VLBA correlator at the Array Operation Center in Socorro, NM, USA. The data reduction strategy follows the standard path of VLBA post-correlation analysis and fringe-fitting in AIPS (Greisen 2003) and hybrid imaging (e.g. Walker 1995) in Difmap (Shepherd et al. 1994, Shepherd 1997. The "preliminary imaging" procedure (used also by , Kravchenko et al. 2016, Lisakov et al. 2017) was employed to improve the amplitude calibration of the array to better than 5 percent at frequencies below 22 GHz (better than 10 percent at 23.8 and 43 GHz). For additional details of the performed VLBA data reduction see .
LIGHT CURVES ANALYSIS
Gaussian process regression
To obtain the time of a flare peak at each frequency we employ Gaussian process regression (GPR, see Rasmussen & Williams 2005). This Bayesian machine-learning technique does not require any prior information about the signal and provides a probabilistic distribution of the data at any given point. GPR was recently used for the analysis of blazars' light curves (Karamanavis et al. 2016) and provides time lag measurements consistent with the widely-used discrete correlation function (Edelson & Krolik 1988) analysis, but with smaller uncertainties (Kutkin et al. 2018) when applied to the same light curves. The Gaussian process (GP) is characterized by the covariance function (kernel ). The kernel function is expressed in terms of hyperparameters which are learned from the data by maximizing the marginal likelihood function (training the GP). Total flux density variations in our sample AGNs have a non-zero power on timescales from days to years (e.g. Fig 1) showing a "colored" power spectra (Fig. 2). At 8 GHz the variations are much smother than that at 15 GHz. This leads to an ambiguity in cross-identification of the flare peaks. We use the kernel represented by a mixture of Squared Exponential (SE) and Matérn (MT) kernels to model the long-("flares") and short-term ("flickering") variations, respectively. We use the same metric scale of SE-kernel inferred from the GPR optimization for both 5 and 8 GHz lightcurves. This helps us to cross-identify major flares between the two frequencies. The other hyperparameters (SEamplitude, MT-metrics and MT-amplitude) are optimized independently for 5 and 8 GHz data.
Once the GP is optimized, one can infer its Bayesian prediction and confidence bounds for a given kernel. In Figure 1 the light curves of OJ 287 at 15 and 8 GHz are shown at various zoom levels. The SE and MT components of the GPR fit are shown separately (the latter has zero mean value). The confidence bounds of ±σ are shown with a shaded area around the SE-component and estimated as σ = √ σSE + σMT, where σSE and σMT are the values obtained from the covariance matrices of GP prediction for SE and MT kernels, correspondingly.
In Figure 2 the power spectra of these two components are shown. Their sum gives the "red-noise" power spectrum typical for blazars (e.g., Max-Moerbeck et al. 2014), with a slope of about −2. The SE-component fits flux density variations longer than about 100 days, while the MT one deals with a short term variability. The MT-component not only describes a white noise but also models the correlated fast variations in the data, as can be seen from its non-flat power spectrum. These variations become more prominent during the source flaring state. Their amplitude is higher at higher frequency. During the major flares these variations 1973 1978 1983 1988 1993 1998 2003 correlate between each other at 15 and 8 GHz with almost a zero lag, while outside the flares they are uncorrelated. If cross-correlation technique or a single-kernel GPR is used, this flickering will prevent an accurate peak detection. The nature of these fast-term variations might be a subject for a separate study. The combination of SE and MT kernels is a good choice for cross-identification of the flares at different observing frequencies, but if one is interested in describing a single light curve fine structure, the Rational Quadratic kernel would be a good choice as well (Kutkin et al. 2018). For the GPR we employed the george python library (Ambikasaran et al. 2015).
Flare detection and variability timescale
Having the GPR prediction on the SE-component S(t) we use the following algorithm for automatic flares detection. A peak position tmax is adopted where the gradient (the first derivative) of GPR changes its sign, the second derivative is negativeS < 0, and the peak exceeds a 2σ threshold. In the last panel of Figure 1 the automatically-detected peaks in 15 and 8 GHz light curves are shown with vertical lines.
We located the inflection points t left and t right just left and right of the tmax (where the second derivative changes sign). These measurements are summarized in Table 2. Around each inflection point ti we estimate the "e-folding" timescale as τi = 2(ti+1 − ti−1)/ ln(Si+1/Si−1) (since a GPR realization implies S(∆t) = Ae ∆t 2 /τ 2 ). The shortest time scale attained at the inflection points provides the lowest estimate of the size of the region where the variability originates.
After all the flares are detected, their crossidentification is performed. In addition to the automatic determination of the close peaks we manually inspected their similarity for the final conclusion. For example, in the light curves of OJ 287 we successfully cross-identified 18 flares, which is the highest number among all the sources. We estimate an error of tmax for each flare from a distribution of peaks of 100 GP samples taken between t left and t right . An error of the variability scale is estimated in the same way. For each source, we calculate the weighted averaged time delay ∆t and the time scale τ . The results are listed in Table 3.
CORE SHIFT MEASUREMENTS
As noted before, the core shift amplitude can change within a source between the epochs of measurements. In this study we are interested in the mean amplitude of the effect. Therefore, we use all the available core shifts estimates from the literature as well as perform our own measurements for the two sources in our sample.
We measure the 15.4-8.1 GHz core shifts in the blazars 0851+202 and 1633+382 directly using VLBA observations (Section 2). A relative shift between the apparent core position observed at two frequencies is obtained as the difference of the shift between the images and the offset between the core position at each frequency map (e.g., Pushkarev et al. 2012). The image shift vector is measured by aligning the cross-identified optically thin jet components. The offset vector is derived from the models as the difference of the core radius-vectors. The position errors of the components are estimated in the image plane following Fomalont (1999) and then propagated to the core shift estimates.
As an example, in Figure 3 the CLEAN maps of OJ 287 at 15 and 8 GHz are shown. The images are shown with the same beam corresponding to that at the lower frequency for a better visualization. The structure of the sources was modeled in the uv-plane using Difmap.
For the blazar 2251+158 we use the core shift measurements by Kutkin et al. (2014). For BL Lacertae (2200+420) we use the results by O'Sullivan & Gabuzda (2009) ∆r15−8 = 0.18 ± 0.04 mas. We also use the 15-8 GHz core shift measurements by (Pushkarev et al. 2012, Table 1). When several core shift measurements are available for an object (four cases), they are averaged, and the conservative maximal error is adopted as the uncertainty. The averaged core shifts used for the analysis are listed in column (3) of Table 3. The full table is available online in a machine readable form.
RESULTS AND DISCUSSION
Variability time scales
The variability time scales estimated in Sec. 3.2 vary from one flare to another within one source. The distribution of the time scales in the source frame is shown in Figure 4 for all flares. We compared the distributions using the two-sample Anderson-Darling test (Anderson & Darling 1952). The test fails to reject the null hypothesis that the time scales in the BL Lacs and the quasars have the same distribution (with the p-value of 0.093). This is expected for the similar relation between their redshifts and Dopper factors (Section 5.4). The radio galaxies have significantly longer variability timescales which is explained by much lower Doppler factors in their core regions. We note, that the GPR ap- proach is flexible enough to represent a variability over a wide range of timescales. Hence, we do not expect a bias either due to the method used or due to the data properties.
Core shifts and time delays
A strong correlation between an observed averaged projected core shift and an observed peak-to-peak total flux density time delay is found in the sources, as shown in Figure 5. The Spearman correlation coefficient is Sc = 0.97 with the p-value of 10 −7 . The latter gives an estimate of the probability to obtain the above value of Sc (or a more extreme one) for two uncorrelated samples. We perform a linear fit and use Markov chain Monte Carlo (MCMC) to obtain the posterior distributions of the parameters. The core shifts and time delays in our sample relate as ∆r[mas] = (1.5 ± 0.1)∆t [yr].
The reason for this tight correlation might be related to the sample selection (see below). We also highlight, that other powerful sources should have similar relation between the core shifts and time lags. For them, one can obtain an estimate of the core shift from the time lag measurements and vice versa. Thereby, for the first time we confirm a tight relation between the apparent core shifts and the observed flare delays in AGNs. The sources in the sample have redshifts ranging from z = 0.03 to 1.8. Neither the time lags nor the core shifts in our sample show correlation with the redshift. Moreover, the two radio galaxies (nearest to us) show minimal and maximal core shifts and time delays. We discuss the reasons for this correlation in Section 5.5.
The speeds
Assuming that the time delay between flare peaks at 15 and 8 GHz corresponds to the time it takes a disturbance to travel from the 15 GHz core to the 8 GHz core, one can estimate the apparent speed of the jet in this region as:
βapp = DA∆r c∆t ,(1)
where DA and ∆r are the angular diameter distance to the source and the apparent core shift. The apparent angular velocity is µapp = βappc/DM = (1 + z) −1 ∆r/∆t, where DM is the proper motion distance (e.g., Hogg 1999). These estimates are summarized in Table 3. The errors are propagated from the corresponding uncertainties of ∆r and ∆t measurements.
In Figure 6 the estimated apparent angular and linear speed is compared to that measured by Lister et al. (2016) for moving jet features. The former is higher than that inferred from component kinematics for all the quasars in our sample, while for the radio galaxies the situation is opposite. This can occur in an accelerating jet, as discussed in Section 5.5.
The capability of VLBI to catch the true jet speed is dictated by the source characteristics like the jet structure and Doppler factor, but also by the observations parameters like sensitivity, image fidelity, robustness of the model fitting and component cross-matching between epochs as well as the time coverage. These factors might prevent detection of the true plasma speed by VLBI kinematics analysis.
In Figure 7 the apparent speed is plotted against the redshift. Our results are consistent with the upper envelope of βapp,max measured by VLBI among all the sources in MOJAVE sample. These values are plotted with black crosses (see Fig. 8 by Lister et al. 2016). The shaded area includes 95% of posterior MCMC samples obtained for ∆r/∆t data (Fig. 5). One can see many distant objects falling into this interval, which implies that the relation between a core shift and a time delay holds for them. Below we provide an explanation of why the estimated apparent speeds in the jets follow this trend. For this purpose we estimate more parameters of the sources.
To estimate the speeds we used the core shift values measured irregardless a source flaring state, implying that the most were obtained for an undisturbed jet. It is known, however, that during the major flares the core shift increases, presumably due to the raise of particles density (e.g., Lisakov et al. 2017). As seen in Figure 1, the time delays are much shorter than the overall flare duration at 8 and 15 GHz, i.e. the core is disturbed at both frequencies when the time lag is measured. This might introduce a bias resulting in the underestimation of the speeds.
Another possible source of a bias can be caused by a presence of a stationary component in the jet. It can affect both the core shift (by pulling on the brightness) and the time delay (causing additional substructure of a flare) measurements. Both would be underestimated if a stationary feature resides between the cores at 15 and 8 GHz, and overestimated otherwise. The further such a feature is from the core, the smaller its effect on the measurements would be. In case when it dominates the emission, one would expect non 1/ν core shift dependence.
Doppler factors, Lorentz factors and viewing angles
Assuming that the variability timescale corresponds to the core light-crossing time, we estimated the Doppler factor using the timescale τ derived above and the average 15 GHz core size a derived from MOJAVE VLBI observations (FWHM of the circular Gaussian core model) by Lister et al. (2016). Since we deal with the flares, the core size is averaged with the weights proportional to its flux density. According to Jorstad et al. (2005Jorstad et al. ( , 2017: δ = 25.3 (a/mas) (DL/Gpc) (τ /yr) (1 + z) .
(2)
Further we estimate the Lorentz factors and the viewing angles of the jets using the following equations:
Γ = β 2 app + δ 2 + 1 2δ , θ = arctan 2βapp β 2 app + δ 2 − 1 .(3)
These results are summarized in Table 3. Our estimates of Doppler factors are consistent with that performed by Hovatta et al. (2009) for the two radio galaxies and for BL Lacertae, but are several times higher for the rest of the objects. The discrepancy is caused by different apparent speeds used in the calculations. Moreover, the assumption of the constant intrinsic brightness temperature equal to the equipartition value (Readhead 1994) made by the authors might be violated in those sources (Gómez et al. 2016, Pilipenko et al. 2018, Kutkin et al. 2018.
The derived Lorentz factor values range from 4 in radio galaxies to about 40 in quasars. The highest value Γ = 65 is obtained for the blazar 3C 454.3. Such extreme value of Γ is consistent with naive expectations for the "most extreme blazar" as 3C 454.3 is the brightest flaring γ-ray blazar observed so far (Abdo et al. 2011, Vercellone et al. 2011) while γ-ray bright blazars are expected to have higher Doppler boosting factors , Linford et al. 2011, but not necessary higher values of Γ, Savolainen et al. 2010). However, the observed spectral energy distribution of 3C 454.3 may be modeled without requiring such high values of Γ and δ (Vercellone et al. 2010). The quasars in our sample have a narrow range of θ 1.1 • . The jets in radio galaxies 0415+379 and 0430+052 are inclined by 11 and 6 degrees respectively. BL Lacertae has θ ≈ 8 • , and is closer to the values for radio galaxies than to the other BL Lacs.
Acceleration in the jets
The core at a given observing frequency is located at different linear distance from the central engine depending on the viewing angle. Therefore, the speeds derived from the core shifts correspond to the different regions along the jet. The de-projected distance of the core from the jet apex at a frequency ν can be estimated as Rν ≈ 8.3 × 10 −8 ∆rDA/(ν sin θ) (for kr = 1, Lobanov 1998). The estimated de-projected distances of the core at 15 and 8 GHz are listed in Table 3. The difference R8 − R15 corresponds to the observed core offset in parsecs. The de-projected distance of the core at 15 GHz varies from R15 < ∼ 1 pc in radio galaxies to R15 > 100 pc in quasars.
In Figure 8 the Lorentz factor is plotted against the distance probed R = (R15 + R8)/2. A simple power-law model with MCMC yields narrow normal posterior distributions of the parameters giving Γ = (2.7 ± 0.5)R 0.52±0.03 (the errors here refer to a standard deviation of the posterior parameters distributions). The shaded area in Figure 8 shows the 95% confidence bounds obtained from the posterior distributions of the Lorentz factors. The BL Lacs fall out of the 95% confidence interval, suggesting that an acceleration in these objects might be faster. We note, that the obtained power law is consistent with the prediction of some MHD models of jets acceleration on those scales (e.g., Beskin & Nokhrina 2006).
We check that this is not a selection effect causing the objects at higher redshifts to have larger Lorentz factors. In Figure 9 we show the dependence of the Lorentz factor on the de-projected distance along the jet and the redshift for the quasars. As seen in the figure, there is practically no dependence on the redshift, while the Lorentz factor is monotonically increasing with the distance along the jet.
We come to the conclusion that there is an acceleration of the plasma on de-projected scales 1-500 parsecs. This is in a good agreement with the high resolution observations of the innermost jet scales of nearby AGN (e.g., Asada & Nakamura 2012, Hada et al. 2018) as well as with the VLBI kinematical studies , 2015, Jorstad et al. 2017. Using the core shift and time delays measurements at various frequency pairs one can obtain a detailed acceleration profile of a jet. We underline, that the evidence of the acceleration is obtained within the assumption that the jets in our sample are similar, and their main properties are determined by the viewing angle (Urry & Padovani 1995).
In a jet with a given viewing angle the Lorentz factor is increasing downstream resulting in a maximal Doppler factor attained at some region. A source with the core located near that region will demonstrate the strongest flux density variations (∝ δ 5 , Urry & Padovani 1995). In this case the Doppler factor will be close to maximal possible δmax ≈ 1/ sin θ ≈ Γ. And the apparent speed approaches the maximal value of βapp,max = βΓ ≈ Γ. In Figure 10 we show these relations for the estimated parameters. We note that although the viewing angles are estimated using the Doppler factors, the Equations (3) do not imply a maximization of any of their components. Therefore these results are self-consistent.
We conclude that the sources in our sample fulfill the condition of maximal Doppler factor in the region of their cores. This is naturally expected since they are among the strongest variable AGN and have been selected based on this criterion. The obtained apparent speeds in the sources are close to their Lorentz factors and are < 50c. This also explains the result of the correlation between the core shifts and time delays in these objects with roughly the same coefficient µ = ∆r/∆t, indicating that the viewing angle in these sources is near θ ∼ 1/Γ. As one can see from Figure 7, there are a lot of sources with the highest apparent VLBI speed approaching to our estimates. These jets are expected to obey this relation as well.
Since the core position depends on the observing frequency, the maximal Doppler factor in a jet is attained at some frequency as well ( Figure 11). Hence, in a coredominated source one would expect the power of flux density variations to change with frequency either monotonically or with a peak, depending on how far the core is from the Doppler factor peak (see Figure 11). The amplitude of variations of the core is expected to correlate with the apparent speed of the flow, which can be checked on the existing data.
At least some of the stationary jet features seen by VLBI in many AGN (Piner & Edwards 2018, Jorstad et al. 2017, Hervet et al. 2016, Rani et al. 2015, Fromm et al. 2013a, Schinzel et al. 2012 can also be attributed to the Doppler factor maximization along the jet. They might naturally occur in a source at a frequency where the core is upstream of the position corresponding to a Doppler factor peak. Decreasing the observations frequency will result in that the core will approach a stationary feature and below some frequency the latter will disappear due to the opacity. As seen from Figure 11 these features are expected to be more pronounced in the jets having larger viewing angles. Of couse the other mechanisms responsible for the stationary features might work, e.g. recollimation shocks (Daly & Marscher 1988).
VLBI components: the brightest one is not the fastest one
In a jet has a range of Lorentz factors along its extent, the different apparent speeds of the features are naturally explained. Figure 11 illustrates that increasing the Lorentz factor provides a wide distribution of high Doppler factors, corresponding to a vast spread of the apparent speeds. The features in radio galaxies have the apparent velocities of 0 − 10c, and the highest probability to observe βapp ∼ 5c provided it is dictated by the Doppler boosting only (see Figure 11). This is in a good agreement with the observations. In a quasar viewed by a small angle, say 2 • , the probabilistic distribution is much wider implying βapp ∈ [10c; 50c] with the highest probability to observe βapp ∼ 25c (see also a discussion by Vermeulen & Cohen 1994). We speculate, that such a picture is expected in an accelerating jet if the features are separated from the plasma flow on various scales (e.g., due to the Kelvin-Helmholtz instability) and then move quasi-ballistically. In this case the measured VLBI apparent speed is determined by the viewing angle and the jet speed at the radius where a component leaves the flow. Moreover, one can see from Figure 11 that the VLBI components with highest Doppler factor (i.e., the brightest) would have apparent speeds lower than the maximal for a jet with a given viewing angle. At the same time, in a nearby source it is possible to detect components moving faster than the plasma speed at the core region (cf. the radio galaxies in our sample, Fig. 6). In an accelerating jet the components residing further from the central engine are expected to move faster, which was observed by VLBI at 15 and 43 GHz , 2015, Jorstad et al. 2017).
SUMMARY
We use UMRAO 26 m total flux density monitoring data to measure peak-to-peak time delays of the flares at 15 and 8 GHz and the variability time scale for 11 sources with high accuracy provided by the Gaussian process regression technique. The delay shows a strong correlation with the apparent core shift measured by VLBI, providing an evidence for a common nature of these effects. Our results suggest that both the time delay and the apparent core shift correspond to the same de-projected distance in a jet, and are linked via the apparent speed in the core region. This provides a validation of a new method to probe the kinematics of the extragalactic jets based on multi-frequency core shift and total flux density time delay measurements.
The relation between the apparent core shift and time delay remains similar for different AGN in our sample, which can be explained in terms of maximization of the Doppler factor in the region of the centimeter VLBI core. The apparent jet speed in the core region exceeds or equals to the highest velocities obtained from VLBI kinematics analysis in all but one sources. The coefficient between the estimated speed and that of the fastest components traced by VLBI ranges from 0.5 to 20 with median value of 1.4.
We derive Doppler factors, Lorentz factors and viewing angles of the jets, as well as the corresponding de-projected core distance from the jet base. The estimated velocities probe the scales of 0.5-500 parsecs. Our results provide an evidence for jet acceleration on these scales obeying a power law Γ ∝ R 0.52±0.03 .
In an accelerating jet viewed at a given angle a Doppler factor reaches its maximum at some distance from the central engine. A strongly amplified variability is observed from the core when it is located near that distance. If the core at a given frequency is upstream of the region with maximal Doppler factor, then a stationary component can be observed. A range of the apparent speeds of VLBI components is expected in such a jet. Moreover, the fastest components in a source will be observed further downstream the jet and will not be the brightest in the outflow.
Figure 1 .
1The light curves of 0851+202 (OJ 287) at 15 and 8 GHz. The SE and MT (with the zero-mean) components of the GPR fit are shown separately with the curves. The ±σ confidence bounds are shown around the SE-component as a shaded area. The dashed rectangle on the panels denotes the region zoomed. The vertical lines in the bottom panel show the peaks automatically-detected in 15 and 8 GHz light curves (see Section 3).
Figure 2 .
2Power spectra of the long-term (SE) and short-term (MT) light curve regression components (OJ 287, 15 GHz).
Figure 3 .
3Naturally weighted CLEAN maps of OJ 287 at 15 and 8 GHz. The beam is shown in the lower left corner. The core is modeled with an elliptical Gaussian component. The rest model components are shown with shaded circles.
Column designation: (1) -Source name; (2) -Peak time of a flare at 15 GHz; (3) -Error in the peak position; (4-5) -15 GHz peak flux density and the error; (6-7) -15 GHz inflection points positions; (8-13) -The same parameter as in columns 2-7 for 8 GHz light curves.
Figure 4 .
4Variability time scale distributions at 15 GHz for different object types over all the flares.
Figure 5 .
5The observed averaged 15-8 GHz core shifts and peakto-peak time delays between 15 and 8 GHz flares in the sources.
Columns designation: (1-2) source B1950 and alternative names; (3) number of cross-identified flares between 15 and 8 GHz; (4) mean 15-8 GHz core shift; (5) weighted averaged 15-8 GHz time delay; (6) apparent angular speed; (7) apparent speed in units of speed of light (Eq. 1); (8) weighted averaged variability time scale of the flares at 15 GHz; (9) estimated Doppler factor; (10) estimated Lorentz factor; (11) estimated viewing angle; (12-13) de-projected distance from the jet base to the core at 15 and 8 GHz.
Figure 6 .Figure 7 .
67Left: the median and maximal proper motion of the sources measured by MOJAVE vs. that obtained in this work. Right: the same for apparent transverse speed in units of c. The dashed lines show equality. Maximal VLBI apparent speed vs. the redshift for the MOJAVE sample sources (black crosses,Lister et al. 2016). Our estimates are shown with the barred symbols. The shaded area includes 95% of the posterior MCMC samples obtained for the "core shift -time lag" fit in Sec. 5.2
Figure 8 .
8Lorentz factors on various de-projected jet scales. The shaded area shows 95% of posterior samples obtained with MCMC. The horizontal bars denote R 15 − R 8 distance.
Figure 9 .Figure 10 .
910Dependence of the Lorentz factor on the de-projected distance along the jet (left) and the redshift (right) for the quasars. The Doppler factors vs the inverse viewing angles (left) and the apparent speeds vs. the Lorentz factors. The dashed lines show equality.
Figure 11 .
11The Doppler factor in an accelerating jet viewed at 2, 6 and 12 degrees. Numbers along the curves show the apparent speed in c.
Table 1 .
1Sources and their parameters.Columns are: (1) B 1950
source name; (2) redshift; (3) -source optical class (Q -quasar,
B -BL Lac object and G -radio galaxy); (4-5) median and max-
imal VLBI apparent proper motion as measured by MOJAVE;
(6) luminosity distance.
Source
z
OC µ M
app,med
µ M
app,max
D L
(mas/yr)
(mas/yr)
(Mpc)
(1)
(2)
(3)
(4)
(5)
(6)
0415+379 0.0491 G
1.47 ± 0.33 2.51 ± 0.10
220
0420−014 0.9161 Q
0.08 ± 0.03 0.12 ± 0.01
6033
0430+052 0.0330 G
2.21 ± 0.39 2.94 ± 0.13
146
0607−157 0.3226 Q
0.04 ± 0.02 0.06 ± 0.04
1711
0851+202 0.3060 B
0.34 ± 0.30 0.79 ± 0.02
1609
1308+326 0.9970 Q
0.35 ± 0.14 0.53 ± 0.02
6701
1633+382 1.8140 Q
0.15 ± 0.16 0.37 ± 0.02
14078
1730−130 0.9020 Q
0.31 ± 0.12 0.57 ± 0.03
5918
2200+420 0.0686 B
1.08 ± 0.55 2.21 ± 0.16
312
2223−052 1.4040 Q
0.21 ± 0.04 0.27 ± 0.05
10253
2251+158 0.8590 Q
0.20 ± 0.16 0.30 ± 0.01
5571
Table 2 .
2Measured parameters of the cross-identified flares at 15 and 8 GHz.Source
t 15,max
t 15,err
f 15,max
f 15,err
t 15,left
t 15,right
t 8,max
t 8,err
f 8,max
f 8,err
t 8,left
t 8,right
MJD
days
Jy
Jy
MJD
MJD
MJD
days
Jy
Jy
MJD
MJD
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
0415+379 53404.9
7.7
4.28
0.22
53280.1
53503.7
53456.4
9.6
5.28
0.17
53317.1 53557.9
53863.1
6.7
5.35
0.22
53726.0
53971.6
53943.7
8.7
6.15
0.16
53793.4 54028.9
54491.6
1.9
8.57
0.21
54349.9
54611.1
54596.3
5.8
8.54
0.17
54446.7 54696.5
54911.4
5.8
6.52
0.21
54828.2
55059.6
54973.7
7.7
7.91
0.09
54893.0 55126.0
0420−014 44133.8
30.2
6.14
0.3
43990.0
44385.8
44161.2
8.5
5.37
0.14
44020.2 44308.5
44782.9
7.5
5.05
0.31
44660.7
44894.9
44910.7
13.2
4.51
0.19
44756.3 45006.4
45467.6
9.4
5.31
0.31
45353.1
45608.9
45493.8
11.3
4.9
0.18
45376.4 45643.0
50008.7
24.5
4.76
0.04
49756.7
50329.5
50207.3
25.5
3.64
0.2
50054.9 50349.6
Table 3 .
3Measured core shifts, time lags, and estimated parameters of the AGNSource
Name N ∆r
∆t
µapp
βapp
τ
δ
Γ
θ
R 15
R 8
mas
days
mas/yr
c
days
deg
pc
pc
(1)
(2)
(3) (4)
(5)
(6)
(7)
(8)
(9) (10) (11) (12) (13)
0415+379 3C 111 5 0.315 89.9 ± 4.2 1.2 ± 0.2 4.0 ± 0.7 46.2 5.3
4.3 10.5 1.9
3.6
0420−014
8 0.267 63.5 ± 3.8 0.8 ± 0.2 39.9 ± 7.8 47.9 70.2 46.4 0.7 198.7 372.6
0430+052 3C 120 7 0.075 13.0 ± 2.4 2.0 ± 1.4 4.6 ± 3.2 36.1 8.1
5.4 6.1
0.5
1.0
0607−157
7 0.240 50.9 ± 4.4 1.3 ± 0.3 26.6 ± 6.0 44.9 44.2 30.1 1.1 65.1 122.1
0851+202 OJ 287 18 0.116 20.5 ± 1.5 1.6 ± 0.4 30.9 ± 8.3 26.7 32.7 30.9 1.7 20.0 37.4
1308+326
8 0.143 40.1 ± 4.3 0.7 ± 0.2 34.6 ± 12.6 40.7 61.5 40.5 0.8 95.8 179.7
1633+382
3 0.201 51.4 ± 13.8 0.5 ± 0.2 40.2 ± 13.3 73.2 50.3 41.2 1.1 102.4 192.0
1730−130
3 0.174 50.3 ± 7.3 0.7 ± 0.2 32.6 ± 10.5 52.2 47.6 35.0 1.1 80.5 151.0
2200+420 BL Lac 13 0.106 15.9 ± 2.7 2.3 ± 1.0 10.5 ± 4.8 25.7 6.5 11.8 7.9
1.2
2.2
2223−052 3C 446 8 0.199 50.9 ± 5.7 0.6 ± 0.2 40.0 ± 11.0 54.7 80.8 50.3 0.6 198.8 372.8
2251+158 3C 454.3 9 0.189 37.1 ± 2.7 1.0 ± 0.2 47.3 ± 11.8 44.2 110.5 65.4 0.4 257.2 482.3
MNRAS 000, 1-11(2019)
ACKNOWLEDGMENTSWe thank Peter Voytsik, Eduardo Ros and the anonymous referee for the useful comments which helped to improve the manuscript. This research was supported by Russian Science Foundation (project 16-12-10481). UMRAO was supported in part by a series of grants from the NSF, most recently AST-0607523, and by NASA Fermi Guest Investigator grants NNX09AU16G, NNX10AP16G, NNX11AO13G, and NNX13AP18G. This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team(Lister et al. 2018).
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|
[] |
[
"The ∆θ-z s relation for gravitational lenses as a cosmological test",
"The ∆θ-z s relation for gravitational lenses as a cosmological test"
] |
[
"Phillip Helbig \nUniversity of Manchester\nNuffield Radio Astronomy Laboratories\nJodrell Bank\nSK11 9DLMacclesfield, CheshireUK\n"
] |
[
"University of Manchester\nNuffield Radio Astronomy Laboratories\nJodrell Bank\nSK11 9DLMacclesfield, CheshireUK"
] |
[
"Mon. Not. R. Astron. Soc"
] |
Recently,Park & Gott (1997)claimed that there is a statistically significant, strong, negative correlation between the image separation ∆θ and source redshift z s for gravitational lenses. This is somewhat puzzling if one believes in a flat (k = 0) universe, since in this case the typical image separation is expected to be independent of the source redshift, while one expects a negative correlation in a k = −1 universe and a positive one in a k = +1 universe. Park & Gott explored several effects which could cause the observed correlation, but no combination of these can explain the observations with a realistic scenario. Here, I explore this test further in three ways. First, I show that in an inhomogeneous universe a negative correlation is expected regardless of the value of k. Second, I test whether the ∆θ-z s relation can be used as a test to determine λ 0 and Ω 0 , rather than just the sign of k. Third, I compare the results of the test from the Park & Gott sample to those using other samples of gravitational lenses, which can illuminate (unknown) selection effects and probe the usefulness of the ∆θ-z s relation as a cosmological test.
|
10.1046/j.1365-8711.1998.01692.x
|
[
"https://arxiv.org/pdf/astro-ph/9804104v1.pdf"
] | 119,093,735 |
astro-ph/9804104
|
ae6eb36d12239ff3b8dc431447814e7a21ef1141
|
The ∆θ-z s relation for gravitational lenses as a cosmological test
1998
Phillip Helbig
University of Manchester
Nuffield Radio Astronomy Laboratories
Jodrell Bank
SK11 9DLMacclesfield, CheshireUK
The ∆θ-z s relation for gravitational lenses as a cosmological test
Mon. Not. R. Astron. Soc
0001998Accepted. ReceivedarXiv:astro-ph/9804104v1 9 Apr 1998 Printed (MN L A T E X style file v1.4)gravitational lensing -cosmology: theory -cosmology: observations
Recently,Park & Gott (1997)claimed that there is a statistically significant, strong, negative correlation between the image separation ∆θ and source redshift z s for gravitational lenses. This is somewhat puzzling if one believes in a flat (k = 0) universe, since in this case the typical image separation is expected to be independent of the source redshift, while one expects a negative correlation in a k = −1 universe and a positive one in a k = +1 universe. Park & Gott explored several effects which could cause the observed correlation, but no combination of these can explain the observations with a realistic scenario. Here, I explore this test further in three ways. First, I show that in an inhomogeneous universe a negative correlation is expected regardless of the value of k. Second, I test whether the ∆θ-z s relation can be used as a test to determine λ 0 and Ω 0 , rather than just the sign of k. Third, I compare the results of the test from the Park & Gott sample to those using other samples of gravitational lenses, which can illuminate (unknown) selection effects and probe the usefulness of the ∆θ-z s relation as a cosmological test.
INTRODUCTION
Historically, there has been little interest in the ∆θ-zs relation compared to other cosmological tests based on gravitational lensing statistics, perhaps because the inflationary paradigm (e.g. Guth 1981), which began about the same time as the discovery of the first gravitational lens (Walsh, Carswell & Weymann 1979), has become so influential. Since a flat (k = 0) universe is a robust prediction of inflation, many researchers assume this and consider only flat universes (or, at most, k = −1 cosmological models with λ0 = 0). Due to the fact that for the popular singular isothermal sphere model for a single-galaxy lens the average image separation ∆θ, integrated over the lens redshift z d from z d = 0 to z d = zs, is completely independent of the source redshift zs in a flat universe, there is little point in pursuing the ∆θ-zs relation if one is interested primarily in flat cosmological models. If one is not committed to a flat universe, then of course one should not assume k = 0, but even if one believes that the universe must be flat, it is still important to test this belief observationally. The situation is somewhat worsened by the fact that most 'standard' cosmological tests such as the m-z (magnitude-redshift or 'standard candle') and θ-z (angular size-redshift or 'standard ⋆ email: [email protected] rod') relations, 'conventional' gravitational lensing statistics, age of the universe) are relatively insensitive to the radius of curvature of the universe (R0 ∼ (|Ω0 + λ0 − 1|) − 1 2 ), being degenerate in combinations of λ0 and Ω0 in directions roughly perpendicular to lines of constant R0 in the λ0-Ω0 plane. A notable exception are constraints derived from CMB anisotropies (e.g. Scott, White & Silk 1995;Hu, Sugiyama & Silk 1997).
THEORY
For a singular isothermal sphere lens, the angular image separation is given by (e.g. Turner, Ostriker & Gott 1984)
∆θ = 8π v c 2 D ds Ds ,(1)
where v is the velocity dispersion and D is the angular size distance (see below). Even if the singular isothermal sphere is not a perfect model for the gravitational lens systems considered, it is still a good approximation when one is concerned only with the image separation. For a given v, by combining Eqs. (5) and (6) in Gott, Park & Lee (1989) and using the more appropriate and more general angular size distances, one obtains an expression for the average image separation ∆θ, by integrating over the lens redshift z d from
c 1998 RAS z d = 0 to z d = zs, zs 0 dz d D 3 ds D 2 d (1 + z d ) 2 D 3 s Q ∆θ(zs) ∆θ(0) = ,(2) zs 0 dz d D 2 ds D 2 d (1 + z d ) 2 D 2 s Q where Q = Ω0 (1 + z d ) 3 − (Ω0 + λ0 − 1) (1 + z d ) 2 + λ0 . (3)
The Dij (with D k := D 0k ) in Eqs. (1) and (2) are angular size distances, which are functions of the lens and source redshifts z d and zs, the cosmological parameters λ0 and Ω0 as well as the 'homogeneity parameter' η, which gives the fraction of smoothly, as opposed to clumpily, distributed matter along the line of sight. Note that Eq. (2) is valid for all combinations of λ0, Ω0 and η. The angular size distances can be computed for arbitrary combinations of these parameters by the method outlined in Kayser, Helbig & Schramm (1997). Figures 1 and 2 show ∆θ as a function of zs for various cosmological models, for η = 1 (the traditional case assuming a completely homogeneous universe) and η = 0 as extreme cases. Note in Fig. 1 that the curve is a horizontal line for k = 0, has positive slope for k = +1 and negative slope for k = −1, where k := sign(Ω0 + λ0 − 1). In Fig. 2, for η = 0, the slope is negative regardless of the value of k. Thus, at first sight it appears that an inhomogeneous universe, a possibility not investigated by Park & Gott (1997, hereafter PG), might be able to explain the puzzling negative correlation between ∆θ and zs. However, it is shown in Sect. 5 that even the extreme η = 0 scenario produces an anticorrelation which is much weaker than that found by PG. This effect can be qualitatively understood by realizing how Eq. (2) is affected by decreasing η: inspection shows that this might be estimated by examining D ds /Ds. All other things being equal, the angular size distance increases with decreasing η. Also, the effect of η is more noticeable at large redshift differences. Since zs ≥ zs − z d , the denominator is the more important term, and so decreasing η increases Ds and so decreases D ds /Ds and thus ∆θ(zs)/∆θ(0).
DATA
PG used an inhomogeneous sample of gravitational lenses from the literature. While this seems problematic at first sight, PG noted that there is no reason to believe that this should influence the analysis. Nevertheless, it is worth comparing the PG results to those obtained from a better defined sample.
The observational data provided by the JVAS and CLASS surveys offer an independent sample of gravitational lenses. JVAS is the Jodrell Bank VLA Astrometric Survey (Patnaik et al. 1992); CLASS is the Cosmic Lens All-Sky Survey (Myers et al. 1998). Even though the observational tasks are not yet complete, the JVAS and CLASS surveys which constitute the database have already yielded sufficient gravitational lenses to enable one to make an independent analysis. Table 1 shows the current state of knowledge about the JVAS/CLASS gravitational lenses. Note that the questionable source redshift for 2114 + 022 is probably the redshift of an additional lensing galaxy (this interpretation is supported by several independent lines of evidence).
Although not all source redshifts in the JVAS/CLASS sample are known, 8 out of 11 are, and based on our sur- Crosses represent the PG sample (20 systems; note that two data points with ∆θ ≈ 6 arcsec almost coincide); diamonds represent the JVAS/CLASS sample (8 systems; of course only those with known source redshifts are included). Note that there is an overlap of four data points. The filled diamond represents the system 0218+357, which was not used by PG although its source redshift had been published before the PG analysis was done (Lawrence 1996).
vey, discovery and followup strategies there is no reason to suspect the unknown source redshifts to be statistically different from those already known. Figure 3 shows the source redshifts and image separations of the gravitational lens systems used in this paper: the PG sample and the JVAS/CLASS sample.
CALCULATIONS
All calculations here implement the method of PG, which uses the Spearman rank correlation test to generate a relative probability for a given cosmological model. PG noted the fact that they always obtained a low probability with their sample, even when allowing for non-flat cosmological models (albeit in a limited area of parameter space), galaxy evolution or departure from the singular isothermal sphere model. As PG noted, allowing for these effects increases the probability, since they all tend to create a negative correlation in a flat universe, but the magnitude of the effect is not large enough to explain the observations. Again as noted by PG, if the lenses are parts of clusters, then this will work in the opposite direction, making the observed negative correlation even more puzzling. Calculations were done for four samples:
the PG sample the PG sample with the addition of the system 0218 + 357 the JVAS/CLASS sample the union of all samples Note that the source redshift for 0218 + 357 had been published before the PG analysis was done (Lawrence 1996).
Since 0218 + 357 lies below and to the left of all other data points, it is clear that including it will weaken the puzzling negative correlation found by PG; this is discussed more quantitatively in Sect. 5.
RESULTS AND DISCUSSION
Since the PG test assigns a low probability to a k = 0 universe, the question arises as to whether it can be used as a general cosmological test to determine the values of λ0 and Ω0. This is not the case. For all four samples I have calculated the Spearman rank correlation probability as a function of λ0 and Ω0 in a range of parameter space (−8 < λ0 < 2 and 0 < Ω0 < 10) much larger than that allowed even by a generous interpretation of observations. This was done with a resolution of 0.1 in both λ0 and Ω0 for both η = 1 and η = 0. The Spearman rank correlation probability is essentially constant over a wide range of parameter space; basically, either all cosmological models are probable, or all are improbable, depending on the sample used. The probability is a weak function of the cosmological model, with the sharpest transition occuring when crossing the k = 0 line in the λ0-Ω0 plane. For all samples except the PG sample, the probability is >5% in almost the entire parameter space; † those cosmological models with a lower probability are among those ruled out by current observations. Thus, the Spearman rank correlation probability does not allow one to reject any otherwise viable cosmological models, which shows both that there is no reason to expect unknown effects in the gravitational lens samples and that it is not very useful as a cosmological test. For the PG sample, the 1% contour corresponds almost exactly to the k = 0 line, with higher values for a negatively curved universe. Thus, the PG sample is marginally compatible with a k = −1 cosmological model, although the probability values are low throughout the λ0-Ω0 plane, with values near the maximum of 0.025 being attained only for small (but realistic) Ω0 values and large (in absolute value) negative values of λ0. Since there are no known selection effects which can account for the differences between the PG sample and other samples, either the test is not very useful and/or it is pointing to unknown selection effects in the literature sample used by PG. The fact that the PG result changes dramatically (probability ≈ 10-20% in most of the λ0-Ω0 plane) by the inclusion of just 1 additional data point, which could have been included in their analysis, argues in favour of the former possibility.
The above discussion was for η = 1. For η = 0 the situation is qualitatively the same and quantitatively involves only slightly different values of probabilities derived from the Spearman rank correlation test.
It is interesting to compare the probabilities from the Spearman rank correlation test for the PG sample using the actual values of zs and ∆θ as used by PG to those obtained using more up-to-date data for the same lens systems. If two values are very near each other, rounding them off to the same values produces a different result for the rank correlation test than if they differ by even a small amount. Using more up-to-date data, an even lower probability is obtained for the PG sample, for η = 1 and η = 0, for a wide variety of cosmological models. Park & Gott (1997) pointed out that the image separations in gravitational lens systems show a strong significant negative correlation with the source redshift, while in a flat universe one would expect no correlation (while a negative correlation would be expected in a universe with negative curvature and a positive one in a universe of positive curvature). None of the possibilities they examined were strong enough to explain the effect. A possibility not examined by them, namely an inhomogeneous universe, produces a negative correlation regardless of the sign of the curvature, but it † For the JVAS/CLASS sample, the maximal probability is 0.955 and is realized in almost the entire k = +1 area of the parameter space.
CONCLUSIONS
too is not strong enough to account for the effect. As a general test for the values of λ0 and Ω0 the test is of no use, all cosmological models being assigned roughly the same probability, but which value they are assigned depends on the sample used.
The strong dependence of the result on the sample used seems to indicate that the result of Park & Gott (1997) is due not to some physical cause but rather to unidentified selection effects in the sample of gravitational lenses taken from the literature. The large number of JVAS and CLASS lenses gives us an independent comparison sample, thus demonstrating the need for discovering a large number of lenses in a well-defined sample. As Park & Gott (1997) point out, since many conclusions based on 'conventional' gravitational lensing statistics are based on essentially the same lenses as in their literature sample, if this sample is for some unknown reason atypical, then conclusions drawn from statistical analyses of it must be examined with care. It will thus be interesting to see what conclusions can be drawn from a statistical analysis of the JVAS/CLASS sample after the observational tasks have been completed. (We expect to find more lenses, but have no qualms about using the present incomplete sample in this analysis since there is no reason to believe that a larger sample would show a different ∆θ-zs relation.)
NOTE
Since this work was completed, two other responses to Park & Gott (1997) (apart from Helbig (1998)) have appeared. The first (Williams 1997) is complementary to this work in that it assumes the effect is real and explores the astrophysical consequences while the second (Cooray 1998) is more similar to this analysis, arriving at essentially the same conclusions though using different observational data (and exploring neither the question of usefulness as a general test for λ0 and Ω0 nor the effects of a locally inhomogeneous universe).
Figure 1 .
1Normalized image separation as a function of source redshift. From the top, the (λ 0 ,Ω 0 ) values are (2,4), (0,4), k = 0, (0,0.7), (0,0.3) and (-5,1). For k = 0 the result is valid for all (λ 0 ,Ω 0 ) values whose sum is 1. η = 1.
Figure 2 .
2The same asFig. 1except that here η = 0.
Figure 3 .
3Source redshifts zs and image separations ∆θ (in arcsec) for the gravitational lens systems studied in this paper.
Table 1 .
1The JVAS/CLASS gravitational lenses.Name
# images
∆θ
lens galaxy type
z d
zs
[arcsec]
0218+357
ring + 2
0.33
spiral
0.6847
0.96
0414+0534
4
2.0
elliptical
?
2.62
0712+472
4
1.2
?
0.406
1.339
1030+074
2
1.6
peculiar
0.599
1.535
1422+231
4
1.2
?
0.65
3.62
1600+434
2
1.4
spiral
0.4144
1.589
1608+656
4
2.2
spiral?
0.64
1.39
1933+503
4+4+2
0.9
?
0.755
?
1938+666
4+2
0.9
?
?
?
2045+265
4+1?
2.0
?
0.87
1.28
2114+022
2+2?
2.4
?
0.316
0.588?
c 1998 RAS, MNRAS 000, 1-5
ACKNOWLEDGEMENTSI thank Asantha Cooray for comments on the manuscript and my collaborators in the CERES project for helpful discussions. This research was supported by the European Commission, TMR Programme, Research Network Contract ERBFMRXCT96-0034 'CERES'.
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This paper has been produced using the Royal Astronomical Society/Blackwell Science L A T E X style file. This paper has been produced using the Royal Astronomical Society/Blackwell Science L A T E X style file.
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"Galaxy cluster's rotation",
"Galaxy cluster's rotation"
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[
"M Manolopoulou \nInstitute for Astronomy\nThe University of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n\nSection of Astrophysics, Astronomy and Mechanics\nDepartment of Physics\nAristotle University of Thessaloniki\n54 124ThessalonikiGreece\n",
"M Plionis \nSection of Astrophysics, Astronomy and Mechanics\nDepartment of Physics\nAristotle University of Thessaloniki\n54 124ThessalonikiGreece\n\nInstituto Nacional de Astrofísica Optica y Electrónica\nAP 51 y 21672000PueblaMéxico\n"
] |
[
"Institute for Astronomy\nThe University of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK",
"Section of Astrophysics, Astronomy and Mechanics\nDepartment of Physics\nAristotle University of Thessaloniki\n54 124ThessalonikiGreece",
"Section of Astrophysics, Astronomy and Mechanics\nDepartment of Physics\nAristotle University of Thessaloniki\n54 124ThessalonikiGreece",
"Instituto Nacional de Astrofísica Optica y Electrónica\nAP 51 y 21672000PueblaMéxico"
] |
[
"Mon. Not. R. Astron. Soc"
] |
We study the possible rotation of cluster galaxies, developing, testing and applying a novel algorithm which identifies rotation, if such does exist, as well as its rotational centre, its axis orientation, rotational velocity amplitude and, finally, the clockwise or counterclockwise direction of rotation on the plane of the sky. To validate our algorithms we construct realistic Monte Carlo mock rotating clusters and confirm that our method provides robust indications of rotation. We then apply our methodology on a sample of Abell clusters with z 0.1 with member galaxies selected from the Sloan Digital Sky Survey (SDSS) DR10 spectroscopic data base. After excluding a number of substructured clusters, which could provide erroneous indications of rotation, and taking into account the expected fraction of misidentified coherent substructure velocities for rotation, provided by our Monte-Carlo simulation analysis, we find that ∼ 23% of our clusters are rotating under a set of strict criteria. Loosening the strictness of the criteria, on the expense of introducing spurious rotation indications, we find this fraction increasing to ∼ 28%. We correlate our rotation indicators with the cluster dynamical state, provided either by their Bautz-Morgan type or by their Xray isophotal shape and find for those clusters showing rotation within 1.5 h −1 70 Mpc that the significance of their rotation is related to the dynamically younger phases of cluster formation but after the initial anisotropic accretion and merging has been completed. Finally, finding rotational modes in galaxy clusters could lead to the necessity of correcting the dynamical cluster mass calculations.
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10.1093/mnras/stw2870
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[
"https://arxiv.org/pdf/1604.06256v3.pdf"
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1604.06256
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9504769fc0592a0022e21f5e3a73c624e2a34fe8
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Galaxy cluster's rotation
16 December 2016
M Manolopoulou
Institute for Astronomy
The University of Edinburgh
Royal Observatory
Blackford HillEH9 3HJEdinburghUK
Section of Astrophysics, Astronomy and Mechanics
Department of Physics
Aristotle University of Thessaloniki
54 124ThessalonikiGreece
M Plionis
Section of Astrophysics, Astronomy and Mechanics
Department of Physics
Aristotle University of Thessaloniki
54 124ThessalonikiGreece
Instituto Nacional de Astrofísica Optica y Electrónica
AP 51 y 21672000PueblaMéxico
Galaxy cluster's rotation
Mon. Not. R. Astron. Soc
000000016 December 2016Printed 16 December 2016(MN L A T E X style file v2.2)galaxies: clusters: general
We study the possible rotation of cluster galaxies, developing, testing and applying a novel algorithm which identifies rotation, if such does exist, as well as its rotational centre, its axis orientation, rotational velocity amplitude and, finally, the clockwise or counterclockwise direction of rotation on the plane of the sky. To validate our algorithms we construct realistic Monte Carlo mock rotating clusters and confirm that our method provides robust indications of rotation. We then apply our methodology on a sample of Abell clusters with z 0.1 with member galaxies selected from the Sloan Digital Sky Survey (SDSS) DR10 spectroscopic data base. After excluding a number of substructured clusters, which could provide erroneous indications of rotation, and taking into account the expected fraction of misidentified coherent substructure velocities for rotation, provided by our Monte-Carlo simulation analysis, we find that ∼ 23% of our clusters are rotating under a set of strict criteria. Loosening the strictness of the criteria, on the expense of introducing spurious rotation indications, we find this fraction increasing to ∼ 28%. We correlate our rotation indicators with the cluster dynamical state, provided either by their Bautz-Morgan type or by their Xray isophotal shape and find for those clusters showing rotation within 1.5 h −1 70 Mpc that the significance of their rotation is related to the dynamically younger phases of cluster formation but after the initial anisotropic accretion and merging has been completed. Finally, finding rotational modes in galaxy clusters could lead to the necessity of correcting the dynamical cluster mass calculations.
INTRODUCTION
Galaxy clusters are the deepest gravitational wells in the Universe (eg., Voit 2005, Jones et al. 2009), constituting an important ingredient of the Cosmic Web. They form at the interception of filaments and/or walls, where the galaxy density is larger and where infall is strongest (eg., van de Weygaert & Bond 2008, Kravtsov & Borgani 2012. They contain from a few tens to a few thousands of galaxies. The frequency distribution of cluster masses, the mass function, and its evolution have been recognized as a very important cosmological probe that can constrain the current cosmological model (eg., Borgani 2008, Reddick et al. 2014, Sartoris et al. 2014. Consequently, the accurate calculation of cluster masses is of uttermost importance and a powerful means to cosmological studies.
Many different approaches have been used to calculate E-mail: [email protected] cluster masses; taking advantage of clusters acting as gravitational lenses (eg., Kneib 2008, Applegate et al. 2014, Barreira et al. 2015, Gonzalez et al. 2015, using the Sunyaev-Zeldovich effect (eg. Sunyaev & Zeldovich 1970, Birkinshaw & Lancaster 2008, Olamaie, Hobson & Grainge 2013, Churazov, Vikhlinin & Sunyaev 2015, assuming hydrostatic equilibrium and calculating the intracluster medium (ICM) temperature from their X-ray emission (eg., Sarazin 2008, Martino et al. 2014, Nelson et al. 2014 or using the clusters' galaxy member velocities and assuming dynamical equilibrium (eg., Saro et al. 2013, Sifón et al. 2013, Tempel et al. 2014 or not (eg., Diaferio & Geller 1997, Diaferio 1999). However, not all methods give the same results(eg., Hoekstra 2007, Peng et al. 2009, Donahue et al. 2014; each one of them has its own advantages and disadvantages (eg., Sadat 1997). Comparing the different methods is important in order to understand the systematic effects that enter in each one and, thus, for estimating more accurate masses (eg. Is-rael et al. 2014, von der Linden et al. 2014, Applegate et al. 2016. When calculating the cluster mass using the velocities of the individual cluster members, we assume that the cluster is in virial equilibrium; the gravitational potential equals two times the sum of the kinetic energy of the members and that galaxy orbits are roughly isotropic. This does not take into account the possible contribution to the galaxy velocities of a rotational component. Clusters could be rotating due to an initial angular momentum that survives since their formation or due to recent mergers or interactions with close neighbours. Not taking into account the rotation could result in an erroneous dynamical cluster mass, which could ultimately affect the cosmological constraints provided by the cluster mass function. The difficulty to distinguish a rotating cluster from two closely interacting or merging ones is probably the cause for the few early attempts to investigate the rotation of galaxy clusters (eg., Materne & Hopp 1983, Tovmassian 2002. Hwang & Lee (2007) used the galaxy member velocities to search for indications of rotation and found ∼10% of their cluster sample to be rotating and in dynamical equilibrium (not undergoinng a recent merger). The relevant study of Hamden et al. (2010) used galaxy velocities, X-ray spectra of intracluster gas, and distortions of the cosmic microwave background (CMB). Chluba & Mannheim (2002) and Cooray & Chen (2002) studied the effect of the cluster rotation on the temperature and polarization of the CMB; while other groups have attempted to model the rotation of the intracluster medium (Fang, Humphrey & Buote 2009, Bianconi, Ettori & Nipoti 2013. A particular case study is cluster A2107 which has been found to rotate in multiple studies (Materne & Hopp 1983, Oegerle & Hill 1992, Kalinkov et al. 2005. Recently, a new attempt to study cluster rotation using the SDSS spectroscopic sample concludes that some clusters are indeed rotating (Tovmassian 2015).
This work aims in identifying the rotation of members of clusters by using their velocities taken from the SDSS DR10 spectroscopic data base (Eisenstein et al. 2011). We construct a novel algorithm that can identify both the cluster rotation axis and amplitude. We apply the algorithm to selected Abell clusters and seek for correlations between their rotation properties and their dynamical state. When required we use a flat Λ cold dark matter cosmology with H0 = 70h70 km s −1 Mpc −1 .
The paper is organized as follows: in section 2 we present our rotation identification algorithm and compare it with that of Hwang & Lee (2007). In section 3 we test the efficiency of our algorithm, while in section 4 we present our cluster sample, systematic biases and the application of our algorithm. In section 5 we present and discuss our results and we derive our conclusions in Section 6.
ROTATION IDENTIFICATION
First, it is important to clarify what we intend in our work as a rotating cluster. A rotational mode in clusters can be caused by a variety of mechanisms, among which the anisotropic infall of material, an initial angular moment of the proto-cluster that survives virialization, an off-axis merging, etc. Most probably all of them are related to the initial or secondary bulding of the cluster and therefore ro-tation should be expected in many phases during the cluster formation process.
The question however posed in our work is what is the fraction of virialized (or close to) clusters which retain a rotational mode. Therefore, as detailed in the following sections, we have made all efforts to exclude from our sample clearly interacting clusters, clusters with multipule components, cluster with detectable substructures in velocity and projected space. However, clusters in post-merging phase which are at the process of virialization, but not yet completely virialized, with no significant substructure indications cannot be easily distinguished, but in any case we believe that they should probably be counted among the rotating clusters. For those that disagree, we also make an effort, through targeted Monte Carlo simulations, to estimate the fraction of such false detections.
Our Method
We introduce a method to identify the possible coherent rotation of galaxies in galaxy clusters. The method provides both the rotational velocity amplitude and the orientation of the projected rotation axis, as well as a quantification of the rotation being a true feature or an artefact.
In order to explain our procedure let us first assume a counter-rotating cluster with constant rotational velocity of 600 km/s, ie., each galaxy member has the same velocity irrespective of its cluster-centric distance. In order to have a realistically "observed" cluster we then assign to each galaxy the line-of-sight component of its rotational velocity with respect to the cluster centre (which in our example we set it to be stationary). Starting from the components vx, vy, vz of the velocity of each galaxy and placing the y-axis on the plane of the sky, we calculate its line-of-sight velocity from the relation:
v los = vx cos φ + vz cos(90 • − φ) ,
where φ is the vertical angle between the line of sight and axis x (see Fig. 1), ie., the z-axis is the axis of rotation. For φ = 0, the line of sight coincides with the x-axis and the cluster rotation axis is perpendicular to the line-of-sight, the ideal case for observing rotation; as the angle φ increases, we also take into account the z-component of the velocity in the line-of-sight velocity. For φ = 90 • , the line of sight coincides with the z-axis (see section 3.3 for the effect of φ on the detection of rotation). A visual illustration of our example cluster, which has φ = 0 and a horizontal projected rotation axis on the plane of the sky (θrot = 90 • ), and of our procedure is provided in Fig. 2, as detailed below.
The basic idea is to divide the projected distribution of galaxy cluster members in two semicircles (1 and 2; as shown in the left-hand panel of Fig. 2), measure the difference of the mean galaxy velocities between the two semicircles, v dif = v1 − v2 , and rotate consecutively (on the plane of the sky) the galaxy positions by an angle θ in the clockwise direction (as shown in the left-hand panel of Fig. 2 by the redarrows), repeating the measurement of v dif for each rotation. Consequently, we obtain the velocity difference v dif (θ) as a function of the angle θ. We will use the graph of v dif (θ) (right-hand panel of Fig. 2), which we call rotation diagram, as our primary indication for the presence or not of a rotation mode. We need now to relate the observed v dif (θ) to the true rotation velocity of the cluster. Even in the ideal case of φ = 0 • we will not observe the whole vrot of each galaxy but as already discussed only its projected component along the line of sight. We will observe (for those galaxies in the semicircle moving towards the observer) blueshifted velocities with magnitudes which depend on their 3D position in the cluster. For example, if they are located at an angle µ from the line of sight passing through the centre of the cluster (ie., the angle between the line of sight passing through the centre of the cluster and the cluster radius connecting the centre of the cluster to the galaxy), the observed rotational velocity magnitude of each galaxy will be v obs = vrot × cos(90 − µ), where µ takes values from 0 o to 180 o . The mean v obs of all N galaxies in this projected semicircle will not add up to vrot, but to:
v1 vrot × N i=1 cos(90 • − µi)/N
Similarly, in the other semicircle it will add up to v2 = − v1 . Thus,
v dif = v1 − v2 = 2vrot × N i=1 cos(90 • − µi)/N ,
where for convenience we have assumed the same number of galaxies within each semicircle (projected hemisphere) and at symmetric positions to each other (ie., not respecting a realistic volume fill). In such a model configuration we find that:
cos(90 − µi)/N ∼ 0.636
The realistic observational situation, where most of the galaxies are at small angles µ from the cluster centre due to the larger volume projected, can be estimated directly from our Monte Carlo cluster of Fig. 2, where we find v obs ∼ 0.503vrot and therefore vrot v dif . As a result, the rotational velocity of the cluster will be read from the rotation diagram as: . An illustration of our method. We show a Monte Carlo cluster which has been set to counter-rotate with an amplitude vrot = 600 km/s and with its projection rotation axis at an angle θrot = 90 • with respect to the North. Our rotation identification method entails rotating consecutively the galaxies of the cluster by an angle θ in the clock-wise direction (as indicated by the red arrows) and estimating the velocity difference between the East-West semicircles (details are presented in the main text). The right-hand panel shows the resulting rotation diagram, ie., the velocity difference between the two semicircles against the angle θ.
vrot = MAX[v dif (θ)] .
To be more detailed, our rotation detection procedure entails rotating on the plane of the sky the galaxy-member positions by an angle θ starting from the vertical axis clockwise, in the range 0 • − 360 • and with a step, say, of 10 • . In our example, for θ = 0, we will not observe any significant velocity difference between the East-West hemispheres; ideally, at the absence of noise we should obtain v dif = 0. As θ increases, the velocity difference should increase until it reaches its maximum value at θ = 90 • . In this case, the galaxies in one semicircle would seem to move away and in the other semicircle would seem to approach us, with respect to the cluster centre. Then as θ increases to 180 • the amplitude of the rotation signal will decrease and increase again towards θ = 270 • until it approaches again v dif = 0 at θ = 360 • . This behaviour is depicted in the right-hand panel of Fig. 2 which shows the periodic rotation diagram for an ideally rotating cluster with a constant velocity of 600 km/s. A few interesting and important issues, that will be addressed in the following sections, are as follows.
• The orientation of the rotational axis with respect to the line of sight can hamper the detection of a rotational mode, if such exists (see section 3.1).
• Based on whether the troughs or the peaks appear first in the rotation diagram, we infer the rotating or counterrotating nature of the cluster (as an example, in Fig. 2 the cluster is counter-rotating). In an initial irrotational Universe the expectation of course is for a statistically equivalent number of both type of rotating clusters.
• If a non-rotating cluster has one or more small-sized subgroups with a significant velocity difference with respect to the rest of the cluster then, in the rotation diagram, we may observe narrow peaks or troughs at some angle θ but not a clearly sinusoidal signal. However, most such cases will be identified and excluded from our analysis at an early stage (see Sections 4.1.1.and 4.1.2). However, there are cases where global rotation and infalling substructures cannot be easily distinguished: (a) a subgroup occupying a relatively large fraction of the cluster projected area; (b) two significant subgroups of galaxies moving at opposite directions within the cluster potential, although such a case requires fine tunning and thus should be rare.
In general, the expectation for a non-rotating cluster, with no significant infalling substructures, is to have a random rotation diagram (no systematic dependence of v dif (θ) on θ) with relatively small values of v dif (θ).
The Hwang and Lee method
Another method to identify cluster rotation, with which we will compare our own, has been proposed by Hwang & Lee (2007). They use a sinusoidal relation to compute the angle of the rotation axis, Θo, and the rotational velocity vrot:
vp(vrot, Θ) = vsys + vrot · sin(Θ − Θo) ,(1)
where vp is the predicted radial velocity of each galaxy due to the cluster rotation, vsys is the peculiar velocity of the cluster and Θ is projected on the plane of the sky position angle of each galaxy, setting off from North to East. Since in our case we use velocity differences with respect to the cluster mean recessional velocity, we set vsys = 0. A χ 2 minimization procedure can be used to determine the best-fitting values of Θo and vrot, assuming that the sinusoidal model of equation (1) represents well the velocity data. Namely, we use a grid of Θo and vrot values and calculate χ 2 for each pair of parameters:
χ 2 (vrot, Θo) = i (vp i − v los,i ) 2 σ 2 i ,
where v los,i is the observed line-of-sight velocity of every galaxy and σi its measurement error.
VALIDATION OF OUR METHOD
Before applying our method to real galaxy cluster data, we should validate and confirm that it can provide unambiguous indications of rotation for the case of realistic clusters and that it can correctly provide the amplitude of rotation and its axis orientation. To this end, we construct, using the Monte Carlo simulation method, a virialized cluster with a mass of 4×10 14 M , radius R cl = 1 Mpc, core radius rc = 0.1 Mpc and having a King's profile density distribution:
ρ(r) = ρ0 (1 + (r/rc) 2 ) 3/2 ,(2)
where ρ(r) is the density included within radius r and ρ0 is the density in the centre of the cluster. To estimate the value of ρ0 we use the cluster mass M cl ,
M cl = 4 3 πR 3 ρ0
(1 + (r/rc) 2 ) 3/2
from which by using M cl = M (< R cl ) and r = R cl we estimate ρ0 = 6.56 × 10 −12 kg/km 3 . Although it is known that the NFW (Navarro, Frenk & White 1996) profile is a more accurate representation of the dark matter and galaxy density profiles in clusters of galaxies, while the King's profile is applicable mostly to the intracluster gas (King 1962), it is acceptable to use the latter for the purpose of just testing our methodology. A realization of one such Monte Carlo cluster can be seen in Fig. 3. Assuming that the cluster is dynamically relaxed (virialized) we can estimate, using the virial theorem, the amplitude of the expected 3D velocity, v k , of each galaxy, which depends on its distance from the cluster centre according to: v 2 k = GM (r)/2r, and from equation (3) we obtain:
v k (r) = 2 3
Gπρ0r 2 1 + (r/rc) 2 3/2 ,
where M (r) is the mass within a sphere of radius r.
Note that each Cartesian component v kx , v ky , v kz of the virial velocity v k (r) is assumed to be randomly orientated, while the rotational velocity will have a coherent orientation perpendicular to some rotation axis (in most cases we will assume it to be lying on the plane of the sky). We will further set a counterclockwise direction on the velocity components vrot x , vrot y , vrot z , by:
vrot · r = 0 v 2 rot = v 2 roty + v 2 rotx ,
with the first relation ensuring that the coordinate vector r is perpendicular to the rotation velocity vector vrot. The second implies that the z-component of the velocity is set 0, in order the rotational velocities to be perpendicular to the rotation axis z.
We can now assign to each galaxy a 3D velocity which could be either of:
(a) a constant rotational velocity (independent of the cluster-centric distance of each "galaxy") having an amplitude, say a fraction of the maximum virial expectation and a coherent orientation around a chosen axis, (b) a rotational velocity having as amplitude a constant fraction of the virial expectation, ie., different at the different cluster-centric distances and a coherent orientation around a chosen axis, (c) the vectorial sum of the virial expectation and any of the above two rotational velocity models. This case corresponds to a more realistic cluster velocity profile and we model it by assigning to each "galaxy" the randomly oriented virial velocity that corresponds to its distance from the cluster centre, adding vectorially the rotation velocity.
To investigate the systematics related to the realistic observational situation we will attempt to identify the cluster rotation on the plane of the sky. To this end, we project the 3D cluster on one plane, estimating the line-of-sight component of the total (rotational or rotational+virial) velocity of each mock galaxy and imposing its rotation axis to be perpendicular to the line of sight (φ = 0, which is the ideal case). We then apply both algorithms (ours and that of Hwang and Lee) to investigate their performance for both rotational velocity models and for a variety of axis orientations on the plane of the sky. Furthermore, to study sampling effects we simulate mostly two cases; a cluster with 1000 and a cluster with 50 "galaxies".
Model (a): constant rotational velocity
Using as input rotational velocity a constant one with vrot = 540 km/s (30% of the maximum virial velocity), we obtain the results shown in Fig. 4, where in the upper panels we present results based on a cluster with nmem = 1000 and in the lower panels a cluster with nmem = 50, while in the lefthand panels we present the case of a purely rotational velocity and in the right-hand panels the case of a total velocity based on the vectorial addition of the virial expectation and the rotational velocity.
For this rotational velocity model we can actually clearly address the issue of how well does each method recover the input rotational velocity (and axis orientation). In Table 6 we present the output vrot and θrot for all four cases shown in Fig. 4. For the case of dense sampling, which provides an estimate of the intrinsic performance of the two methods, we find a significant underestimation (by ∼ 35%) of the amplitude of vrot by the Hwang and Lee method, and a small ( 10%) overestimation of vrot by our method. The rotation axis orientation is well recovered by both methods.
When we assume sparse sampling, ie., a cluster membership of 50 galaxies, which is towards the lower limit of the realistic observational cases, we verify that we can still successfully identify the cluster rotational properties but apparently with larger deviations from the input rotational parameters. To substanciate this claim we perform our next important test which is to investigate the rotation identification as a function of the cluster rotational velocity. To this end we simulate sets of 50 Monte Carlo clusters each, all with the same statistical properties, but of which the constant rotational velocity is an increasing fraction of the In the left-hand panels we present the case of a purely rotational velocity and in the right-hand panels the case of a total velocity based on the vectorial sum of the virial expectation and the rotational velocity. The input rotational velocity has a constant value of vrot = 540 km/s. maximum virial one (from 0% to 100%), keeping the same rotation axis orientation (θrot = 45 • ). In order to investigate the convolution of systematics related to the rotation amplitude and to sampling effects, we repeat the procedure for nmem = 1000 and 50. For each set we calculate the mean and standard deviation of the recovered rotation amplitude and of the orientation of the rotation axis. Their recovery success provides us with the range of cluster parameters for which our method can successfully identify rotation.
In Fig. 5 we present the mean and standard deviation of the recovered rotation amplitudes (left-hand panels) and of the orientation of the rotation axis (right-hand In the ideal case of very good sampling (upper panels), we see that our method correctly recovers the rotation amplitude with negligible uncertainty, except for the case of no rotational velocity where both methods will tend to detect an artificial rotational velocity of 80 km/s. The already identified problem of the Hwang & Lee (2007) method, that of underestimating the rotation amplitude, is shown here as well to be true for all vrot being an increasing function vrot. For the sparse sampling cases we have similar overall behaviour as in the dense-sampling cases for both methods but as expected a larger scatter of the resulting rotational parameter values. In addition we have a larger systematic ovserestimation of vrot by our method, specially for vrot/v virial 0.2. In general, the uncertainties in the orientation of the rotation axis are quite large for the sparse sampling case, while the Hwang and Lee method performs slightly better in recovering, on average, the correct angle of the orientation axis.
Model (b): fractional rotational velocity of the virial one
For this case we assume a rotation velocity amplitude being a constant percentage (30%) of the virial; thus vrot(r) depends on the different cluster-centric distances; for example vrot(r = 1Mpc) 278 km/s. Note that for such a rotational velocity field, the output vrot that will be provided by both methods presented in section 2, is an integrated value that depends on the galaxy density and velocity profiles. In Fig. 6 Figure 6. Comparison of the rotation diagrams of our method (black continuous line) and of that of Hwang and Lee (red dashed line) for a rotation model in which vrot(r) is a constant fraction of the virial velocity at the different cluster-centric distance and for the cases of nmem = 1000 (upper panels) and nmem = 50 (lower panels). In the left-hand panels we present the case of a purely rotational velocity and in the right-hand panels the case of a total velocity based on the vectorial sum of the virial expectation and the rotational velocity. we present the rotation diagrams for this case and the output rotational parameters for both methods and for both nmem cases are shown in Table 1. Again, we see the same vrot underestimation of the Hwang & Lee (2007) method, discussed in section 3.1 for the case of a constant rotational velocity field, which implies that such an understimation is independent of the rotation velocity model. The analysis of the performance of the two methods when vrot(r) is an increasing fraction of v virial (r) has provided qualitatively similar results as those of Fig. 5 and thus we do not present the corresponding figure.
3.3 Effects of rotational axis orientations with respect to the line-of-sight
We wish to investigate the effect of different orientations of the 3D rotational axis with respect to the line-of-sight on the rotation identification by our method. In order not to mix the outcome of this test with issues related to sampling effects, we simulate a cluster with dense sampling (ie. having 1000 members). We set initially the rotation axis at a perpendicular position with respect to the line of sight and consequetively rotate the rotation axis with respect to the vertical position, so that it forms an angle φ with the line of sight in the interval (0 • , 90 • ) until it is aligned with the line of sight. We apply this procedure using the rotation model b (section 3.2) and for two cases, an ideal where we assign only the corresponding rotational velocity to each mock galaxy, and a more realistic where we also vectorially add the corresponding randomly orientated virial velocity. The results for different values of the orientation of the rotation axis with respect to the line of sight are shown in Fig. 7 for both methods, ours and that of Hwang & Lee (2007). The upper panels corresponds to the ideal case while the lower panels to the more realistic one.
As expected, the rotation signal becomes weaker (the rotation amplitude decreases) as the angle φ increases. The counter-rotating direction of rotation is apparent due to the occurrence of the peak at θrot = 90 • (ie., because < 180 • ). At φ ∼ 90 • the rotation cannot be identified, as the rotation component of the velocity of the galaxies is perpendicular to the line of sight and thus it cannot be observed. Both methods give a flat rotation diagram in this case, as they should. We also see that the rotation amplitude in the ideal positional case (φ = 0 • ) is accurately recovered by our method while it is underestimated by ∼ 35% when using the Hwang & Lee (2007) method. Their method, as in the ideal 2D case which is presented later, appears to have problems in recovering the correct input rotation amplitude for any value of φ. Furthermore, the accuracy of the recovered rotation axis angle is quite good for both methods (it decreases slightly with the increase of φ). Similar results we recover also in the more realistic case (lower panels), with the addition that the rotation signal becomes practically undetectable for φ 60 • .
Therefore, we conclude that we will miss a fraction of intrinsically rotating clusters due to axis orientation effects, even in the best case of dense sampling. If we make the reasonable assumption that the rotation axis of each cluster is randomly orientated with respect to the line of sight, the fraction of missed rotating clusters can be estimated as the ratio of the solid angle that corresponds to an angle δφ ∼ 30 • to the solid angle of the whole sphere, ie.,
f missed sin 2 (δφ/2) 0.10
This should be considered a strict lower limit to the expected number of missed rotating clusters, since sparser sampling will detariorate the detectability of rotation.
Conclusions on the method performance
We can conclude the following from the extended Monte Carlo simulation analysis of the performance of the rotation indentification procedure that:
• On the limit of dense-sampling our method recovers very well both the input rotation amplitude and the orientation of the rotation axis, while the Hwang & Lee (2007) method although accurately identifies the rotation axis orientation, it systematically understimates the rotation amplitude by ∼ 35%,
• On the limit of sparse-sampling our method systematically overestimates by 10% the input rotation amplitude. Identifying the correct orientation of the rotation axis is more demanding with typical uncertainties being as large as ∼ 50 • . The Hwang & Lee (2007) method performs better than our method in identifying the correct axis orientation,
• One should not expect to recover correctly the rotation characteristics if the rotation velocity is 10 − 15% of the virial velocity (ie., typically 200 km/s) and the sampling of the cluster members is low (< 50 galaxies/cluster), ie., the richer the cluster the easier the rotation signal can be identified and the more accurately the rotation properties can be recovered,
• A fraction of rotating clusters will be missed due to the orientation of the rotation axis being close to the line of sight. A crude estimate indicates this fraction to be at least 10%.
DATA ANALYSIS
Cluster and Galaxy Data
Our original sample consists of all Abell/ACO clusters (Abell, Corwin & Olowin 1989) of richness class R ≥ 1 and distance class 4 or 5 that are located in the SDSS survey area and have more than 50 galaxies with SDSS DR10 spectroscopy within a rest-frame radius of 2.5 h −1 70 Mpc from the cluster centre and within a redshift separation of δz = 0.01 from the central cluster redshift (as provided by the NASA/IPAC Extragalactic Database). After excluding a few clusters that are affected by the survey borders we are left with a sample of 103 Abell clusters, presented in Table 3.
Note that the line of sight velocity of each galaxy is given by (Danese, de Zotti & di Tullio 1980):
v los = c × z gal − z cl 1 + z cl ,
where z cl is the cluster redshift, while the cluster velocity dispersion is then provided by:
σv = N i=1 v 2 los,i N − 1 ,
where N is the total number of galaxies used in the estimation.
Clearing projection effects
Once we have selected our cluster sample and before we apply our rotation algorithm, we wish to clean each cluster of possible galaxy outliers and projection effects. Indeed, projected galaxies along the line of sight, but separated in velocity space would be a source of noise and could hide or erroneously enhance a rotation signal.
To this end, we plot for each cluster the relative to the cluster centre galaxy velocity frequency distribution, which has a mean value of zero. We expect that a virialized cluster should have a roughly Gaussian frequency distribution of line-of-sight velocities (Chincarini & Rood 1977, Halliday et al. 2004, Lokas et al. 2006. Therefore a Gaussian is fitted to the data using the usual χ 2 minimization procedure. Then, outliers are identified as those galaxies with velocities > 3σ away from the mean, which then are not considered in the rotation analysis.
Separating substructures
Furthermore, projected groups along the line of sight, but separated in velocity space, or substructures which have coherent infall velocities towards the parent cluster centre, could provide an erroneous rotation signal. In many occasions, it is easy to identify such cases due to either the fact that in projection the substructures are clearly spatially separated from the main cluster, or in other occasions where the different subclusters are clearly separated in velocity space but may appear as a unique cluster in projection. We have carefully inspected all of our clusters and identified those with significant subclumps and each was separately analysed for rotation. As an example, we show in Fig. 8 the case of Abell 1228. The left-hand panel shows the projected galaxy distribution, within a radius of 2.5 h −1 70 Mpc, which appears as a typical centrally concentrated cluster, while the righthand panel shows the relative velocity distribution which reveals three clearly separate subclumps (each separated by δv ∼ 1900 km/s from the central one) projected along the line of sight. Had we analysed the whole "cluster", without separating the individual subclumps, we would have found a clear and strong signal of rotation. The two larger clumps have more than 50 members each and were separately analysed for rotation (and as we will see they do show strong rotation indications; see the Appendix).
This procedure was finally applied to the following clusters, A659, A1035, A1228, A1291, A1775, A2067, A2197, A2245, A2255 and A2152, which were found to be composed of two or more subclusters, increasing our total sample of clusters under study to 110. However, only in five we managed to perform the separation procedure effectively (A1035, A1291, A1228, A1775, and A2152), with details being presented in the Appendix 1 . The rest were tagged as being dominated by substructures and were not included in our final analysis.
Finally, one must also ask what happens if the number of substructure member galaxies is a relatively small fraction of the whole cluster and/or the infall velocity is not as large as to be clearly separated in velocity space. Could such non-rotating clusters be erroneously identified as rotating? In section 4.3.4 we present extensive Monte Carlo simulations tailored to answer such a question and provide the expected fraction of false rotation detections. However, as a first step in excluding such cases, we have investigated in detail all clusters, even if their galaxy velocity distribution appears Gaussian, and we have tagged as being dominated by substructures those clusters which are spatially dispersed with no clear central core, or clusters for which we found in the literature strong and unambiguous substructure indications (eg., Einasto et al. (2012), Krause et al. (2013)). The following clusters fall in this category: A257, A1137, A1187, A1190, A1205, A1346, A1358, A1383, A1385, A1424, A1474, A1749, A1780, A1986, A2028 and A2069.
Cluster richness and mass
In order to have a more accurate determination of the cluster richness with respect to the original Abell's richness class and to investigate possible richness dependencies of our results, we calculate for each cluster the number of bright galaxies, N * , ie. those with M > M * in the r-band (with r ≤ 17.7), using the luminosity function of Montero-Dorta & Prada (2009) with the K and evolutionary corrections of Poggianti (1997). In order to check for obvious systematic effects we have tested whether the number of bright galaxies, N * , correlates with the cluster redshift. No such correlation was found indicating that N * is a redshift-free indication of the cluster richness and thus of the cluster mass. Another indicator of the cluster mass is the cluster velocity dispersion, which is related to the mass via the virial theorem. A large velocity dispersion indicates a large cluster mass. Note however that cluster merging and significant cluster substructures can increase the measured velocity dispersion, but in this case it is not necessarily related to the cluster mass but to the highly unrelaxed cluster state.
One would expect the above two indicators of the cluster mass (velocity dispersion and richness N * ) to be correlated and indeed they have a Spearman correlation coefficient of Rs 0.43 with a probability of this correlation being random of P 3 × 10 −6 (velocity dispersion and richness are estimated out to 2.5 h −1 70 Mpc).
Cluster dynamical state
We also wish to investigate whether the possible cluster rotation is related to the cluster dynamical state. If, for example, the anisotropic accretion of matter along large-scale filaments entails infall with non-zero angular momentum, one may expect enhancement of rotational modes towards the cluster centre. To investigate this possibility we will use two well known indicators of the cluster dynamical state; their Bautz-Morgan (BM) type and the shape of their ICM X-ray profile. The BM type (Bautz & Morgan 1970) of Abell clusters is an indication of their morphology and thus of their dynamical state. It can be numerically characterized by a value increasing from one to three (1-3) with two intermediate categories, which we index here as 1.5 and 2.5, respectively. The dynamical youth increases in the same order (or the dynamical evolution inversely), with BM type 1 indicating the most dynamically evolved cluster (spherically symmetric, centrally concentrated and cD dominated) while with BM type 3 the most loose, asymmetric and thus unrelaxed cluster.
Similarly, we will use all the available X-ray cluster images to characterize their dynamical state. We define the X-ray profile parameter, Xp, which can take three possible values, Xp = 1 for roughly spherically symmetric and smooth X-ray emission profiles (virialized and dynamically evolved), Xp = 2 for asymmetric and/or distorted profile (dynamically young) and Xp = 0 if the X-ray image is not available. The main source of the X-ray images used come from the Einstein observations (Jones & Forman 1999). In total, we have available X-ray images for 49 out of the 110 Abell clusters of our sample.
Since both previously discussed parameters should reflect the cluster dynamical state, they should be correlated. Indeed, we find that the two parameters correlate nicely and provide a Spearman correlation coefficient of Rs 0.53 and a probability of this correlation being random of P 10 −4 .
Application of our algorithm
The rotation analysis is performed using galaxies within either a circular region around the cluster centre, having a radius of 1.5h −1 70 or 2.5h −1 70 Mpc, or within circular rings of different widths. The latter because we wish to investigate whether the cluster's possible rotation signal comes from the outskirts or the central cluster regions, but also because the central regions are affected more severely by projection effects that could contribute in weakening an existing rotation signal. By identifying the cluster regions, if any, that show a rotational signal we may get hints as to which is the mechanism producing it. In virialized clusters one may expect that virial relaxation would have erased any initial rotational mode. However, in oblate-like clusters [although clusters appear to be mostly prolate-like (eg., Plionis, Barrow & Frenk 1991, Basilakos, Plionis & Maddox 2000] the collapse along their minor axis may retain and even enhance some initial rotation. On the other hand, if in dynamically young clusters the rotation is caused by interactions and merging, one should expect only the cluster outskirts to show more prominent rotational indications. Excluding from our analysis the cluster central regions, where projections along the line of sight are more severe, may be helpful in this respect.
Summarizing, we will investigate the cluster rotation in each cluster using four different angular configurations:
(i) the circular area within 1.5h −1 70 Mpc radius; (ii) the circular ring within 0.3-1.5h −1 70 Mpc; (iii) the circular area within 2.5h −1 70 Mpc radius; (iv) the circular ring within 0.5-2.5h −1 70 Mpc.
A further issue that could be important in identifying a rotational mode in clusters is the selection of the true rotational centre, if such exists. We therefore apply our rotation identification procedure using nine different possible centres, forming a rectangle around the nominal centre of the cluster (Fig. 9). The separation between the consecutive centres is usually 5% of the cluster radius (in some cases we used 10%, depending on the size of the cluster). We finally choose that centre as our optimum rotational centre for which the smooth sinusoidal "ideal rotation" curve (see section 4.3.1) fits best the data rotation curve, ie. centre which corresponds to the minimum χ 2 value (see Fig. 9 for an example).
Identification of significant rotation
In order to make a decision whether a cluster has a significant rotational mode or not we will use the combination of two tests, which consist of:
(a) comparing the distribution of relative velocities, in each of two hemispheres and for each rotation angle θ, using a Kolmogorov-Smirnov two-sample test, and (b) comparing the data rotation curve separately with an "ideal rotation curve" and a "random rotation curve" suitably estimated for each cluster, by using the usual χ 2 statistic. If v(θi) and vm(θi) are respectively the data and model mean velocity difference between the two semispheres at a rotation angle θi, we define:
χ 2 = 360 θ=0 (v θ − v m,θ ) 2 σ 2 θ + σ 2 m,θ(5)
with the data rotation curve uncertainty given, at each rotation angle θ, by:
σ 2 θ = σv,1 √ n1 2 + σv,2 √ n2 2 ,(6)
where σv,1 and σv,2 are the velocity dispersions and n1 and n2 are the number of galaxies in semicircles 1 and 2, respectively, at each rotation angle θ. The uncertainty of the model rotation curve, σ m,θ , is provided by the scatter among the different Monte Carlo realizations of the random or ideal rotation curves.
Test 1: ideal versus random rotation curves
In order to build the ideal rotation and random rotation curves we follow the following recipe: For each cluster we identify the angle at which the maximum velocity difference (MAX[v dif ] ≡ vrot) is observed in its rotation diagram (θrot). This angle splits the cluster in two semispheres; and to the galaxies of each we attach a velocity v los = vrot/2 and v los = −vrot/2, respectively. We then apply our rotation identification algorithm on this new configuration to produce the "ideal rotation" diagram with which we compare the data rotation diagram, quantifying the goodness of fit by the χ 2 statistic (equation 5), which we name χ 2 id . A value of χ 2 id /df 1 (where df are the degrees of freedom, in our case the number of steps in θ) shows that the data rotation curve is well represented by the ideal one.
We also construct for each cluster a rotation curve which corresponds to that of a random distribution of velocity residuals. To this end, we shuffle the galaxy line-of-sight velocities randomly while keeping the same galaxy coordinates. Then, our rotation identification algorithm is applied and this process is repeated 10000 times. The final "random rotation curve", is the average over all the realizations, while the scatter σv r i around the mean is also estimated. Finally, we determine the χ 2 statistic between the data and random rotation curves, which we tag χ 2 r . We can now select the candidate rotational clusters as those for which χ 2 id χ 2 r , ie., those for which the ideal rotation curve fits the data rotation curves significantly better. If the opposite occurs then the cluster is likely not rotating. Therefore the ratio χ 2 id /χ 2 r is a useful parameter for assessing rotation or not.
Test 2: KS two sample test
We also apply the Kolmogorov-Smirnov two-sample test to the distributions of the relative velocities of the galaxies of the two cluster semicircles for each angle θ. This test practically calculates the probability, PKS, that the two relative velocity distributions have the same parent distribution. The bigger the probability the more likely it is that the two distributions are mutually consistent. For a rotating cluster we expect a significant difference between the two velocity distribution, and the corresponding PKS probability limit is taken, somehow arbitrarily, to be PKS = 0.01 (ie., values lower than this limit are taken to signify significantly different distributions).
Final criteria for a rotating cluster
We therefore have four criteria that can be used to deduce a significant or not cluster rotation, which we call the strict criteria, and are as follows:
• χ 2 id /df between the real and ideal rotation curve, which should be less or equal to 1 for a rotating cluster,
• χ 2 r /df between the real rotation curve and random curve, which should be > 1 for a rotating cluster,
• χ 2 id /χ 2 r , which should be ideally 1 for a rotating cluster, but practically we take it to be ≤ 0.2, and
• the Kolmogorov-Smirnov probability, PKS, between the galaxy relative velocity distributions of the two semicircles of maximum difference, which should be: PKS < 0.01.
These criteria can be relaxed to provide a less secure identification of rotation. For example, we also checked for clusters fullfilling only the following two criteria: χ 2 id /χ 2 r < 0.4 and PKS < 0.01. We call these the loose criteria for cluster rotation.
The fraction of false detections
We wish to address the issue of what is the fraction of false detections of rotation according to the above selected criteria, when there is no intrinsic rotation present. Two such different possibilities of false detections exist:
• due to shot noise, related to small number statistics, and
• due to the presence of an unidentified substructure that has a coherent infall velocity with respect to the cluster mean, and which can be erroneously assessed as rotation. The substructures that could still remain unidentified after the procedure discussed in Section 4.1.2 are those which cannot be spatially or dynamically separated (ie., those which are near the cluster centre and which do not have a large infall velocity).
To investigate these possibilities we simulate 1000 Monte Carlo clusters, according to the basic recipe of section 3 with either 50 or 100 members, which is the membership range more susceptible torotation misidentification. To address the first possibility we assign to the mock galaxies only virial velocities and find only a small fraction of our mock clusters showing a false rotational signal. Under the strict criteria we find a ∼1.9% false detection rate for clusters with either N = 50 or 100 members. Using the loose criteria, the corresponding fraction is ∼ 4.4%. These fractions are small enough to allow us to conclude that shotnoise effects are unimportant for the size of clusters considered in this work.
In order to address the second possibility, we introduce a subclump which contains between 10% and 28% of the main cluster members, positionally placed on one of the projected quadrants of the cluster at a distance of 420 h −1 70 kpc from the cluster centre and having as a mean infall velocity a fraction (50% or 100%) of the cluster virial velocity dispersion (note that we randomly assign to each substructure member an infall velocity having the above mean and a standard deviation of 500 km/sec). The range of these parameters where selected after a number of trials in order to mimic cases where the 3σ clipping of the member velocity distribution or the clear positional identification of the substructure would have failed to identify the substructure as such. In Fig. 10 we present the results as the probability of misidentifying an infalling substructure (with the previously discussed characteristics) for cluster rotation as a function of substructure richness (in percentage of main cluster membership). We see that for the case where such substructures exist the probability of them being misidentified as a cluster rotation is between ∼ 0.05 − 0.3 depending on the substructure richness and infall velocity.
RESULTS
Individual cluster results
Each cluster or subcluster in our sample is analysed in all four angular configurations according to the following sequence, the basic steps of which are already presented in section 4.2. First, we search for the best centre of possible rotation among nine tested (a related diagram for A85 is shown in Fig. 11 as an example). Using the selected centre, we apply our algorithm and construct the rotational diagram of the data, the ideal rotation and that of the random velocity residuals, while we also construct the Kolmogorov-Smirnov (KS) probability curve as a function of rotation angle, θ. The results of this analysis are then passed through the criteria discussed in Section 4.3.3 to decide whether a significant rotation has been detected, at any of the angular configurations of the cluster.
In order to facilitate the visual verification of our results, we also construct for each cluster an aggregate plot with four panels, where we display:
(a) in the upper left panel the spatial distribution of the galaxies and their selected rotational centre, where residing and approaching galaxies are in red and blue colour, respectively, rejected galaxies due to velocity criteria are shown as black crosses, while rejected galaxies due to angular selection criteria as faint crosses, (b) in the upper right panel the histogram of the lineof-sight galaxyvelocities along with the fitted Gaussian, (c) in the lower left panel the data rotation diagram (points with errorbars), the ideal rotation (red continuous curve) and random rotation curves (blue continuous curve with dashed curves corresponding to 1σ uncertainty), and (d) in the bottom left panel the Kolmogorov-Smirnov probability diagram as a function of rotation angle θ.
We will not present such diagrams for all the clusters of our sample, except for a few examples here and some interesting cases in the Appendix. As one example, we present for A85 the corresponding plots for the two main angular configurations (r < 1.5h −1 70 and r < 2.5h −1 70 Mpc). We remind the reader that A85 is a rather rich BM type 1 cluster at a redshift z = 0.055, whose galaxy members with mr 17.77 (from SDSS DR10) vary between 68 and 155 at the two radii used. The relatively virialized nature of this cluster is confirmed by its smooth spherical X-ray profile (Jones & Forman 1999), although there are strong indications, when one goes to much fainter galaxies, of substructures (Bravo-Alfaro et al. 2009; and references therein). However, if such substructures are manifested in the cluster velocity distribution they are already excluded by our "cleaning" procedure. Indeed there is such a case in A85, appearing in velocity space at | v | ±1400 km/sec (see the left velocity histogram in Fig. 12).
In Fig. 12 we present the basic results of our analysis for the two cluster radii. Although for the r = 1.5h −1
70
Mpc case there is a smooth sinusoidal rotation curve, exactly what expected for the ideal rotation (red curve), this cluster misses complying with the strict criteria of rotation, due to χ 2 r < 1. However, it complies with the loose criteria and thus it is considered as possibly rotating (but with a relatively low rotational velocity amplitude). When considering the larger cluster radius (right four panels of Fig. 12) we see that the indications of rotation vanish, a fact which could be due to small substructures acting as noise or due to a possibly different velocity distribution of the outskirt galaxies with respect to the inner ones; if for example they are infalling roughly isotropically to the cluster centre from the large-scale surrounding structure.
To complete the presentation of some characteristic examples, we show in Fig. 13 the relevant results for A1367, a rich and relatively nearby cluster (z = 0.022). After exclud- Figure 11. The rotation diagrams for all the candidate rotational centres for Abell 85 (r < 1.5h −1 70 Mpc). Black lines are the real rotation curves and red lines are the ideal rotation curves. Above each panel we indicate the coordinates (dy, dx) of the rotational centre. The final selected one is that with (dy, dx) = (0.04, 0).
ing the outliers of the Gaussian fit to the galaxy velocity distribution (the galaxies at < 1300 km/sec -see Fig. 13), we obtain what appears to be a strongly rotating cluster showing a significant and unambiguous sinusoidal rotation diagram (in all four radial configurations). Although this cluster is known to show significant substructures in its central regions (Cortese et al. 2004), we find even stronger rotational signals when excluding the central 0.3 or 0.5h −1 Mpc region, an indication that although there are substructures, there is also rotation not necessarily attributed to coherent substructure velocity differences.
Abell clusters with rotation
Clusters for which we detect significant rotation, using either the strict or the loose criteria of rotation detection, and which have not been tagged as being dominated by substrutures (see section 4.1.2) are presented in Table 4 (for the r = 1.5h −1 70 Mpc case), and table 5 (for the r = 2.5h −1 70 Mpc case). In each table we also indicate clusters that show rotation only when excluding the inner cluster core (those with the star symbol), since projection effects are more severe along the central part of clusters, where typically a larger volume along the line of sight direction is sampled. The ta-bles list the final number of galaxy cluster members selected, their mean redshift, the values of the four rotation indices discussed in section 4.3.3, the angle θrot of the rotation axis, which is the angle of the maximum semicircle mean velocity difference, the rotation amplitude, vrot, which the maximum velocity difference in the rotation diagram, the cluster velocity dispersion, σv, and the (crudely) corrected velocity dispersion after removing the cluster rotation (see section 5.3.4).
Effect of excluding the cluster core region
Of the 14 clusters within r = 1.5h −1 70 Mpc showing rotation under the strict criteria, two were detected only after excluding the inner < 0.3h −1 Mpc core region (A2199, A2399), while only one cluster (A1913) is downgraded into the loose criteria rotation detection regime when excluding its inner core.
For the r = 2.5h −1 70 Mpc case, out of the 19 clusters showing rotation under the strict criteria, 7 were detected only after excluding the inner < 0.5h −1 70 Mpc core region. However, of these clusters four (A1913, A2089, A2147, and A2670) had originally been found rotating under the loose criteria. The only clusters found rotating under the strict cri- teria that lose completely their rotation when excluding the core region are: A426, A1228a and A1827. Other two clusters rotating under the strict criteria drop below the nmem = 50 limit, when excluding their core region, but retain their significant rotation detection (A1035a, A1291a). Similarly, out of the remaining five clusters with rotation under the loose criteria (excluding the four that were upgraded to the strict regime when excluding the core region), one (A1238) was detected only when the core region was excluded. Finally, A1552 loses completely its rotation when excluding the core region.
Statistical results
In order to attempt to understand our results and possible causes of the cluster rotation, we will attempt to identify correlations between interesting cluster properties and rotation. To this end, we will use the Spearman correlation coefficient between any two parameters, Rs, and the probability that the correlations are consistent with the random expectation, P. Positive Rs means positive correlation, while negative Rs means anticorrelation; a value near zero means the two parameters are not correlated. A small value of P indicates a significant correlation or anticorrelation at that level. We will report only relatively strong and relatively significant correlations, and as such we define: |Rs| > 0.3 and P < 0.05.
A first observation is that all the indices that we use to deduce rotation are correlated strongly among them, as it can be seen in Fig. 14, where we plot only clusters that have not been excluded from the analysis due to strong substructuring (see section 4.1.2). The value of the Kolmogorov-Smirnov probability, PKS, and the value of the ratio of the χ 2 minimium values between the ideal and real rotation diagrams are strongly correlated with each other in all angular configurations. For example, for the r = 1.5h −1 Mpc case we obtain Rs = 0.75 and P < 10 −10 . Also the amplitude of the rotation, vrot, is strongly correlated with both rotation indices with Rs 0.6 and P < 10 −9 . This also should be expected since when the rotational velocity is large, the rotation will be more clearly identified, and vice-versa.
Check for systematic biases of the rotation indices
Before we present our main results it is important to make sure that we understand the possible systematic effects of the resulting rotation indices for the clusters studied. We have already investigated and quantified the effect of shotnoise and undetected substructures (section 4.3.4), however, we further check for correlation of the resulting rotation indices on the number of galaxy members, nmem, and on z.
As we already showed, nmem needs to be relatively large in order to unambiguously detect a rotation if such exist. However, since the input galaxy catalogue (the SDSS DR10 spectroscopic catalogue) is limited to mr ∼ 17.77, when we look at larger distances we observe less and brighter galaxy cluster members. Therefore, there will be an unavoidable redshift dependence of the cluster galaxy membership and thus a redshift dependence of the rotation indices is possible. This does not necessarily imply an important problem but rather that the fraction of rotating clusters found should be considered a lower limit.
In any case, we have tested for such a dependence for the nmem ≥ 50 case and we find weak and marginally significant correlations, in any case below the limit we set in section 5.2. Only in the case of the Kolmogorov-Smirnov test, which Table 4. The clusters with significant rotation within r = 1.5h −1 70 Mpc and with nmem ≥ 50, using either the strict or loose criteria of rotation detection. The first column is the Abell name of the cluster, the second is the mean redshift of the members, the third is the number of members used, the fourth is the orientation on the plane of the sky of the rotation axis, the fifth is the rotation amplitude with its uncertainty, the sixth and seventh are the coordinates of the chosen rotation centre, the eighth is the minimum value of the Kolmogorov-Smirnov probability, the next three columns are χ 2 id , χ 2 r , χ 2 id /χ 2 r , respectively, the twelfth is an indication for the direction of rotation (1 meaning clockwise and 2 anticlockwise). The last two columns correspond to the initial and corrected, for rotation, cluster velocity dispersion. Clusters that show significant rotation only when excluding the inner cluster core (< 0.3h −1 70 Mpc) are indicated with a star symbol. Figure 13. The graphical outcome of the basic rotational diagram for Abell 1367 within a radius of 2.5 h −1 70 Mpc. The excluded (velocity) outliers (corresponding to known substructures) can be observed as empty circles in the upper left panel. Based on the remaining galaxies, a clear and significant sinusoidal rotational diagram is evident. Figure 14. Left-hand Panels: the scatter diagram between the two rotation indices (upper for the r = 1.5h −1 70 Mpc case and lower for the r = 2.5h −1 70 Mpc case). Right-hand Panels: tThe rotation amplitude, vrot, as a function of the Kolmogorov-Smirnov probability (red filled symbols) and as a function of the χ 2 min ratio value (empty black symbols). due to its nature a dependence of PKS on nmem is expected and, since the latter is anticorrelated with z (as discussed previously), we expect a correlation of PKS with z. Indeed we find such a weak but relatively significant dependence (Rs = 0.33 and P 0.013) for the r = 1.5h −1 70 Mpc and nmem ≥ 50 case. For the r = 2.5h −1 70 Mpc case the above correlation becomes weaker.
Cluster z nmem θrot( • ) vrot/km s −1 αcent δcent P KS χ 2 id /df χ 2 r /df χ 2 id /χ 2 r I σv(km s −1 ) σv,
Fraction of rotating clusters
In this section we present some basic statistics regarding the fraction of clusters that show indications of rotation, based on both strict and loose rotation criteria, as defined in the previous section. In Table 6 we present the number of clusters and the corresponding fraction of the total that show strict or loose indications of rotation for clusters with nmem ≥ 50, for which the rotation identification is quite secure. The fractions are slightly different when limiting the studied area within 1.5 or 2.5 h −1 70 Mpc of the cluster centre, with the latter being slightly smaller than the former. We also present the final overall number of unique clusters rotating using any of the four spatial configurations, as discussed below.
Overall, it is secure to say that Abell clusters with nmem ≥ 50 showing significant indications of rotation, within either of the two limiting radii, range between ∼ 25% (for the strict criteria) and ∼ 32% (for the loose criteria) of the total.
It should be noted however, that the specific clusters showing rotation at the different radii are not always the same. In particular, out of the 18 rotating clusters of Table 4, five are missed when using r = 2.5h −1 70 Mpc. Also, quite a few more clusters appear to be rotating when we extend our analysis to r = 2.5h −1 70 Mpc than within r = 1.5h −1 70 Mpc. In particular, out of the 24 clusters listed in Table 5 only 13 are found rotating within r = 1.5h −1 70 Mpc, taking also into account one that drops below the nmem = 50 limit.
We can reach an overall number, and the corresponding final fraction of rotating clusters with nmem ≥ 50 at any of the two radii and taking also into account results based on excluding the core region. Using the strict criteria we find 23 such clusters, corresponding to ∼ 27% of the total (86), while using the loose criteria we find 29 such clusters, corresponding to ∼ 34% of total.
Fraction of clockwise and anticlockwise rotating clusters
As we discussed in section 2.1 our algorithm provides us with the direction of rotation for the rotating clusters. What is expected in an initially irrotational Universe on large-scales is the lack of a preferred direction of cluster rotation. In Tables 2 and 3 we present the direction of rotation for each of our rotating clusters, indicated by the symbol I which takes the value 1 for rotation or 2 for counter rotation. Using only the results based on the strict criteria, we obtain for the r = 1.5h −1 70 Mpc case 11 anticlockwise and only 3 clockwise rotating clusters, while for the r = 2.5h −1 70 Mpc case we have 12 and 7, respectively. There appears to be a slight preference for clockwise rotating clusters 2 , but what is the significance of the number difference, ∆ = 8 for the former case and ∆ = 5 for the latter case? The Poisson uncertainty of ∆ is σ∆ 3.7 and 4.4 for the two cases, respectively, a fact which implies that the difference is significant only at a 2.1σ and 1.1σ level, respectively. We do not consider as overwhelming the former significance and we conclude that there is no significant evidence for a preferred direction of rotation among the rotating clusters.
Correcting the cluster velocity dispersion for rotation
In order to correct the cluster velocity dispersion, assumed to be due to roughly isotropic galaxy orbits, for the rotational modes, we assume that the two velocity components are independent and that the expected cluster velocity dispersion due to the rotational velocity vrot can be approximated, assuming an ideal rotation, as:
σ 2 rot 2 × vrot 2 2 .(7)
Therefore, the corrected cluster velocity dispersion, σv,cor, is approximately provided by the following:
σ 2 v,cor σ 2 v,raw − σ 2 rot = σ 2 v,raw − v 2 rot 2 .(8)
For the majority of the rotating clusters, the corrected velocity dispersion is not dramatically altered, but the correction is not insignificant. Defining the fractional difference between corrected and uncorrected cluster velocity dispersion as δσv = σv,raw − σv,cor σv,raw we obtain for the r = 1.5h −1 70 Mpc case a median value of ∼10%, and a mean value of δσv ∼ 12%. A similar analysis for cluster rotation out to r = 2.5h −1 70 Mpc provides the following fractional differences: a median value of ∼12%, and a mean value of δσv ∼ 15%. The corresponding corrected cluster mass is given by:
Mcor Mraw(1 − δσv) 2 ,
implying a corrected cluster mass reduced by 20% -30% on average, with respect to that uncorrected for rotation. Table 7. Spearman's correlation coefficient and the probability that the detected correlation is consistent with the random expectation for the indicated pairs of parameters using rotating, under trict rotation criteria, clusters with nmem ≥ 30.
Correlations between cluster rotation and cluster physical parameters
We attempt to investigate whether there are correlations between the rotation indices and the different physical properties of the clusters, dynamical or other. We correlate the two main rotation indices, ie., χ 2 id /χ 2 r and PKS with the characteristics of the cluster dynamical state, ie., BM type and Xr (defined in section 4.1.4), and with the cluster mass, characterized either by the number of bright galaxies, N * , or by the cluster velocity dispersion, σv (defined in section 4.1.3).
For both radii we find no significant correlations between cluster mass or cluster dynamical state and rotation indices. However, since the majority of clusters do not show signs of rotation they would act as noise weakening possible correlations between rotation and cluster parameters for those clusters that show significant indications of rotation. Indeed, selecting only the latter clusters we find relatively significant correlations but only between the strength and significance of rotation and the cluster dynamical state (not with cluster mass), in the direction of a correlation between rotation strength and dynamical youth (see table 7).When we analyse clusters that show rotation for the r = 2.5h −1 70 Mpc case, we find correlations only between BM or Xs and PKS, ie., not with the prime indicator of rotation (χ 2 id /chi 2 ran ), but only with the significance of the semicircle velocity difference.
We can conclude that there are indications that the cluster rotation is related to the earlier phases of cluster virialization but after the initial anisotropic accretion and merging has taken place (since we have excluded all clusters showing significant substructure in projected or velocity space).
CONCLUSIONS
We searched for possible cluster rotation in a sample of Abell clusters using galaxy-member redshifts from the SDSS DR10 spectroscopic data base. We developed a new algorithm in order to be able to deduce rotation using the line-of-sight velocities of the galaxy members. We verified the performance of this algorithm by applying it on various Monte Carlo simulated clusters with known rotational characteristics. We also compared our method with that of the Hwang & Lee (2007) method.
Our algorithm provides the significance of the rotation identification (with a set of indices), the rotation amplitude, the position angle of the rotation axis, whether the rotation is clock or anticlockwise and the rotation centre. We find that the amplitude of the rotation is correlated with the indications of rotation; the larger the rotation amplitude the more significant are the indications of rotation. This implies that small amplitude rotation may not be easy to identify, and thus it could pass undetected.
We then applied our algorithm on our sample of Abell clusters using two different sets of criteria for rotation identification, the so-called strict and loose criteria and two different outer cluster radii (1.5 and 2.5 h −1 70 Mpc). Out of 86 cluster with more than 50 member galaxies we have found in total 23 rotating clusters (in any of the 2 radii studied) using the strict criteria of rotation identification and 29 such clusters using the loose criteria of rotation identification. Taking into account the expected fraction (∼ 10% − 15%) of misidentified coherent substructure velocities for rotation, provided by our Monte Carlo simulation analysis, the corresponding final fraction of rotating clusters is ∼ 23% and ∼ 28%, respectively, under the strict and loose criteria. These results appear to be in tension with recent numerical N -body simulations (Baldi et al. 2016) which find a significantly smaller fraction of rotating clusters; however with slightly different criteria of rotation.
Finally, when we use the inner radius case (1.5 h −1
70
Mpc) and clusters that show indications for rotation, we find relatively significant correlations between the cluster dynamical state (X-ray isophotal shape as well as the BM type) and the significance of cluster rotation, a fact which implies that the cluster rotation could be related to the dynamically younger phases of cluster formation but after the initial anisotropic accretion and merging have taken place. This hints towards the inner radius rotation being related to the initial anisotropically accreted matter having significant angular momentum, which gets amplified by collapse. The fact that we find fewer such correlations when we use clusters with rotation within the outer cluster radius (2.5 h −1 70 Mpc) possibly hints towards a different cause or a different phase of the relevant rotation, possibly being related to the imprint of coherent rotational motions of galaxies in the cluster outskirts prior to dynamically disturbing the cluster inner regions.
Figure 1 .
1The triaxial coordinate system and the line of sight direction (blue line). The y-axis remains intact.
Figure 2
2Figure 2. An illustration of our method. We show a Monte Carlo cluster which has been set to counter-rotate with an amplitude vrot = 600 km/s and with its projection rotation axis at an angle θrot = 90 • with respect to the North. Our rotation identification method entails rotating consecutively the galaxies of the cluster by an angle θ in the clock-wise direction (as indicated by the red arrows) and estimating the velocity difference between the East-West semicircles (details are presented in the main text). The right-hand panel shows the resulting rotation diagram, ie., the velocity difference between the two semicircles against the angle θ.
Figure 3 .
3(a) A Monte Carlo cluster in 3D, (b) the density ρ as a function of the cluster-centric distance r, and (c) the amplitude of the virial velocity as a function of the distance r from the cluster centre.
Figure 4 .
4Comparison of the rotation diagrams of our method (black continuous line) and of that of Hwang and Lee (red dashed line) for the cases of nmem = 1000 (upper panels) and nmem = 50 (lower panels).
Figure 5 .
5Recovery of cluster rotational properties as a function of vrot/v virial : left-hand panels: rotation amplitude, right-hand panels: orientation of the rotation axis. The black line indicates the input rotation amplitude and orientation, while the blue and red symbols represent results of our method and Hwang & Lee (2007) method, respectively. Upper panels: for nmem = 1000 and lower panels: for nmem = 50. panels) as a function of the ratio vrot/MAX[v virial ], where MAX[v virial ] = 1800 km/s (see right-hand panel of Fig. 3).
Figure 7 .
7The rotation diagram for the cluster ofFig. 3with a rotational velocity 30% of the virial (which provides an integrated 3D rotational velocity of ∼ 450km/s), as the rotation axis shifts from perpendicular to parallel to the line of sight, ie., φ ∈ [0 • , 90 • ]. Upper panels correspond to the ideal case where only rotational velocities are assigned, while lower panels correspond to the more realistic case of a 3D vectorial sum of virial and rotational velocities. Also, the left-hand panels correspond to the results of our method, while right panels to results of theHwang & Lee (2007) method.
Figure 8 .
8Left-hand Panel: the projected distribution of galaxies in the A1228 cluster. Different colours indicate the galaxies belonging in the three different groups. Right-hand Panel: the relative velocity distribution of the A1228 galaxies. The colour of the fitted Gaussian is that of the corresponding members seen in the left-hand panel. The smallest group is depicted with green in the left-hand panel.
Figure 9 .
9A mock cluster with the nine different rotation centre candidates shown as red dots.
Figure 10 .
10The probability of misidentifying infalling substructures, if such exist, for cluster rotation as a function of substructure richness (in fraction of cluster members). With black we show results for the case where the infall velocity is 50% of the cluster velocity dispersion, while with red we show the corresponding results for the (more improbable) 100% case. Dots show results based on the strict criteria of rotation identification while dashed lines show results based on the corresponding loose criteria.
Figure 12 .
12The graphical outcome of the basic rotational diagram for Abell 85. Within a radius of 1.5 h −1 70 Mpc (left four panels) and after excluding the outliers of the velocity distribution (shown as empty points in the upper left panel), the smooth sinusoidal data rotational diagram is evident, although it falls within the loose criteria. Within the 2.5 h −1 70 Mpc (right four panels) the rotational diagram and the P KS distribution are consistent with no rotation.
Table 1 .
1Output rotation parameters for our and Hwang and Lee methods for a Monte Carlo cluster with input parameters: vrot = 540 km/s and θrot = 90 • , analysed in Fig. 4.Our method Hwang & Lee
nmem
Rot.model
vrot
θrot
vrot
θrot
1000
only rot.
589
90
370
90
1000
rot+virial
599
90
360
100
50
only rot.
645
90
430
90
50
rot+virial
562
100
270
90
Table 2 .
2Output rotation parameters for our and Hwang and Lee methods for a Monte Carlo cluster with input parameters: vrot(r) = 0.3v virial (r) km/s and θrot = 90 • , analysed inFig. 6.Our method Hwang & Lee
nmem
Rot.model vrot
θrot
vrot
θrot
1000
only rot.
450
80
280
90
1000
rot+virial
457
100
273
90
50
only rot.
421
100
256
90
50
rot+virial
499
110
430
110
Table 3 .
3The Abell clusters of our sample. From left to right the columns correspond to: Abell names, redshifts, celestial coordinates, BM type and a measure of the cluster richness, provided by the number of bright (M > M * ) members within the 1.5h −1 70 Mpc radius.Cluster
z
RA( • )
Dec( • )
BM
N *
Cluster
z
RA( • )
Dec( • )
BM
N *
85
0.0551
10.408
-9.343
1
3
1749
0.0573
202.385
37.626
2
2
87
0.0550
10.757
-9.793
3
7
1767
0.0703
204.001
59.212
2
3
168
0.0450
18.791
0.248
2.5
10
1773
0.0765
205.536
2.248
3
5
257
0.0703
27.247
13.982
2.5
4
1775
0.0717
205.482
26.365
1
8
279
0.0797
29.093
1.061
1.5
4
1780
0.0786
206.159
2.883
3
7
426
0.0179
49.652
41.515
2.5
2
1795
0.0625
207.252
26.585
1
3
659
0.1005
126.020
19.404
-
6
1809
0.0791
208.329
5.154
2
7
690
0.0788
129.810
28.840
1
3
1827
0.0654
209.561
21.707
2
2
724
0.0933
134.575
38.573
2.5
5
1831
0.0615
209.793
27.991
3
3
727
0.0951
134.780
39.422
3
5
1864
0.0870
212.076
5.447
2
1
957
0.0360
153.489
0.915
1.5
2
1904
0.0708
215.533
48.556
2.5
7
1024
0.0734
157.073
3.761
2
2
1913
0.0528
216.716
16.676
3
12
1035
0.0684
158.030
40.209
2.5
9
1927
0.0948
217.759
25.663
1.5
5
1066
0.0699
159.850
5.173
2
6
1939
0.0881
219.309
24.834
2.5
4
1137
0.0349
164.404
9.6156
3
4
1983
0.0436
223.183
16.746
3
5
1168
0.0906
166.859
15.913
2.5
7
1986
0.1185
223.289
21.913
3
3
1169
0.0586
167.028
43.946
3
8
1991
0.0587
223.626
18.631
1
6
1173
0.0759
167.297
41.579
2.5
2
2022
0.0578
226.082
28.423
3
6
1185
0.0325
167.699
28.678
2
7
2028
0.0777
227.388
7.527
2.5
5
1187
0.0749
167.915
39.578
3
5
2029
0.0773
227.745
5.762
1
3
1190
0.0751
167.943
40.845
2
4
2030
0.0919
227.850
-0.073
1.5
2
1203
0.0751
168.489
40.294
2.5
5
2034
0.1130
227.555
33.528
2.5
12
1205
0.0754
168.343
2.511
2
5
2040
0.0460
228.188
7.430
3
9
1213
0.0469
169.121
29.260
3
7
2048
0.0972
228.825
4.382
3
12
1228
0.0352
170.374
34.326
2.5
8
2061
0.0784
230.314
30.655
3
7
1235
0.1042
170.733
19.626
2
5
2062
0.1122
230.400
32.067
2
4
1238
0.0733
170.742
1.092
3
6
2063
0.0349
230.758
8.639
2
2
1291
0.0527
173.019
56.024
3
2
2065
0.0726
230.678
27.723
3
8
1307
0.0832
173.200
14.524
2
6
2067
0.0739
230.812
30.906
3
6
1318
0.0578
173.993
55.033
2
8
2069
0.1160
230.991
29.891
2.5
8
1345
0.1095
175.295
10.689
3
2
2079
0.0690
232.020
28.878
2.5
8
1346
0.0975
175.293
5.689
2.5
10
2089
0.0731
233.172
28.016
2
5
1358
0.0809
175.694
8.223
2
5
2092
0.0669
233.331
31.149
2.5
2
1367
0.0220
176.123
19.839
2.5
10
2107
0.0411
234.950
21.773
1
5
1371
0.0687
176.355
15.507
3
3
2122
0.0661
236.122
36.127
2.5
0
1377
0.0514
176.741
55.739
3
8
2124
0.0656
236.247
36.061
1
0
1383
0.0597
177.038
54.622
3
4
2142
0.0909
239.567
27.225
2
4
1385
0.0831
177.019
11.556
3
0
2147
0.0350
240.572
15.895
3
13
1408
0.1102
178.443
15.388
2.5
2
2151
0.0366
241.313
17.749
3
18
1424
0.0768
179.391
5.038
3
6
2152
0.0410
241.343
16.449
3
2
1436
0.0658
180.117
56.255
3
8
2175
0.0951
245.095
29.915
2
7
1474
0.0801
181.988
14.955
3
9
2197
0.0308
247.044
40.907
3
12
1516
0.0769
184.739
5.239
2.5
6
2199
0.0302
247.154
39.524
1
10
1526
0.0799
185.535
13.739
3
7
2244
0.0968
255.683
34.047
1.5
8
1541
0.0893
186.861
8.840
1.5
9
2245
0.0850
255.687
33.530
2
12
1552
0.0858
187.458
11.741
2
3
2255
0.0806
258.129
64.093
2.5
9
1650
0.0838
194.693
-1.753
1.5
5
2356
0.1161
323.938
0.123
2.5
3
1656
0.0231
194.953
27.981
2
15
2399
0.0579
329.386
-7.794
3
8
1658
0.0850
195.295
-3.436
2.5
2
2428
0.0851
334.061
-9.350
2
3
1663
0.0843
195.694
-2.518
2
2
2644
0.0693
355.291
0.094
2
3
1668
0.0634
195.964
19.265
2
3
2670
0.0762
358.543
-10.405
1.5
10
1691
0.0721
197.847
39.201
2
11
Table 5 .
5As inTable 4but for clusters with significant rotation within r = 2.5h −1 70 Mpc. Clusters that show significant rotation only when excluding the inner cluster core (< 0.5h −1 70 ) are indicated with a star symbol.Cluster
z
nmem θrot( • ) vrot/km s −1
αcent
δcent
P KS
χ 2
id /df χ 2
r /df χ 2
id /χ 2
r I σv(km s −1 ) σv,cor(km s −1 )
Strict Criteria
426
0.01722
155
20
405±122
37.315636 41.423283 0.007321 0.249 2.045 0.122
2
770
669
1035a 0.06803
56
120
406±166
120.766955 40.185843 0.009703 0.283 1.885 0.15
2
559
458
1228a 0.03521
65
70
157±53
140.673644 34.373472 0.006379 0.124 1.809 0.068
2
219
180
1228b 0.04253
60
10
335±94
141.249465 34.190492 0.000019 0.411 3.077 0.133
2
322
239
1291a 0.05087
50
30
382±103
96.411645 56.134474 0.000322 0.125 3.89 0.032
2
416
321
1367
0.02148
237
150
354±75
165.810952 19.839167 0.000002 0.217 4.802 0.045
2
582
493
1827
0.06516
50
300
190±93
194.65102 21.707222 0.001018 0.119 1.152 0.103
1
315
268
2065
0.07224
170
70
712±176
204.198262 27.74665 0.000019 0.125 2.518 0.049
2
1166
988
2151
0.03668
276
220
432±70
229.739375 17.748611
0
0.877 9.896 0.089
1
594
486
2199
0.03057
344
80
325±77
190.728415 39.634998 0.000245 0.094 3.237 0.029
2
712
631
2399
0.05754
103
250
201±85
326.401605 -7.764684 0.008962 0.177 1.134 0.156
1
428
378
2152
0.04408
122
170
320±62
231.474846 16.403915 0.000095 0.379 6.029 0.063
2
374
294
1185* 0.03362
140
330
292±89
147.127123 28.729761 0.001011 0.358 2.024 0.177
1
500
427
1775a* 0.07523
57
160
308±112
184.073873 26.433694 0.00341 0.264 1.733 0.152
2
439
362
1913* 0.05277
119
30
407±94
207.663393 16.708548 0.000142 0.745 4.214 0.177
2
536
435
2022* 0.05798
53
50
423±103
198.830016 28.452586 0.000243 0.073 2.084 0.035
2
379
273
2089* 0.07377
59
180
316±107
205.849001 28.039546 0.00044 0.064 1.136 0.056
1
431
352
2147* 0.03624
327
230
304±95
231.281004 15.847379 0.000106 0.155 1.844 0.084
1
837
761
2670* 0.07598
94
250
376±153
352.691984 -10.42811 0.008033 0.178 1.441 0.124
1
670
595
Loose Criteria
1169
0.05887
83
120
248±147
120.30184 43.916405 0.009778 0.077 0.339 0.228
2
537
475
1203
0.07527
89
300
244±92
128.476726 40.270752 0.00059 0.093 0.914 0.102
1
441
380
1552
0.08611
104
200
261±125
183.577159 11.740556 0.002124 0.14 0.663 0.211
1
642
577
1809
0.07911
88
170
272±102
207.530865 5.131838 0.002476 0.127 0.696 0.182
2
471
403
1238* 0.07392
70
210
225±115
170.662776 1.068283 0.003468 0.173 0.82 0.212
1
487
431
Table 6 .
6Fraction of clusters showing rotation under the strict and loose criteria, for the analysed clusters with ≥ 50 members (which are less prone to random errors). The final, corrected for the expected number of false detections according to our simulations, fraction of rotating clusters is also listed.Radius/h −1
70 Mpc N clus
Strict
Loose
1.5
56
14 (25%) 18 (32%)
2.5
86
19 (22%) 24 (28%)
Overall
86
23 (27%) 29 (34%)
Corrected
86
23%
28%
r = 1.5h −1 Mpc r = 2.5h −1 Mpc BM-P ks BM-χ 2 id /χ 2 r Xs − χ 2 id /χ 2 BM-P ks Xs − P ksr
N
15
15
12
20
15
Rs
-0.49
-0.44
-0.62
-0.41
-0.52
P
0.062
0.096
0.033
0.076
0.045
We wish to note that had we not separated these clusters they would all have shown significant and strong indications of rotation, exactly due to the coherent velocity differences of the subclusters. Nevertheless, in some cases, as we will see further below, one or even both separated subclusters show true evidence of rotation.
See Longo (2011) for a similar results on spiral galaxy rotation.
ACKNOWLEDGEMENTSMP would like to thank Hrant Tovmassian for suggesting the study of cluster rotation and for many initial discussions on the subject.APPENDIX A: CLUSTERS SUCCESSFULLY DIVIDED IN SUBSTRUSTURESWe list here those clusters of our sample that were found to consist in velocity space of two or more well-separated substructures. These clusters were separated into their different components, which were individually analysed for rotation when possible.• Abell 1035This cluster presents a background subcluster in all four configurations studied. One of the two subclusters was found to have a significant rotational mode.• Abell 1228Abell 1228 was found to consist of three well-separated components in velocity space aligned along the line-ofsight, in all four spatial configurations (seeFig. 8). Two components are rich enough to be analysed for rotation and indeed they show strong indications of rotation, in the 2.5 h −1 70 Mpc and 0.5-2.5 h −1 70 Mpc configurations, with rotational velocity amplitude of ∼ 200 km/s (A1228a) and ∼ 400 km/s (A1228b). The two subclusters rotate in the same direction (I = 2) but have their (projected on the plane of the sky) rotation axes perpendicular to each other (figure A1).• Abell 1291 Another interesting case is Abell 1291. Studying its galaxy member velocity distribution we again identify 3 different peaks, clearly separated from each other. The third and most distant substructure could not be studied due to its small richness. From the other two only the nearest one (A1291a) show indications of rotation for the 2.5 h −1 70 Mpc and 0.5-2.5 h −1 70 Mpc configurations.• Abell 1775 We found a foreground group of galaxies in velocity space and in all the four spatial configurations. This substructure is placed south-east of the main cluster of galaxies and was not found to present any indications of rotation in any of the configurations, while the main cluster is found to rotate in the 0.5-2.5 h −1 70 Mpc configuration.• Abell 2152 Abell 2152 presents a main group of galaxies found to have strong indications of rotation in all configurations. In all configurations foreground galaxies (part of the Hercules supercluster) are found with a wide velocity distribution, which we succesfully exclude from our analysis. The main cluster is then found to have a significant rotational mode.
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[] |
[
"Indirect Inference for Time Series Using the Empirical Characteristic Function and Control Variates",
"Indirect Inference for Time Series Using the Empirical Characteristic Function and Control Variates"
] |
[
"Richard A Davis ",
"Thiago Do ",
"Rêgo Sousa ",
"Claudia Klüppelberg "
] |
[] |
[] |
We estimate the parameter of a time series process by minimizing the integrated weighted mean squared error between the empirical and simulated characteristic function, when the true characteristic functions cannot be explicitly computed. Motivated by Indirect Inference, we use a Monte Carlo approximation of the characteristic function based on iid simulated blocks. As a classical variance reduction technique, we propose the use of control variates for reducing the variance of this Monte Carlo approximation. These two approximations yield two new estimators that are applicable to a large class of time series processes. We show consistency and asymptotic normality of the parameter estimators under strong mixing, moment conditions, and smoothness of the simulated blocks with respect to its parameter. In a simulation study we show the good performance of these new simulation based estimators, and the superiority of the control variates based estimator for Poisson driven time series of counts.
|
10.1111/jtsa.12582
|
[
"https://arxiv.org/pdf/1904.08276v1.pdf"
] | 118,678,196 |
1904.08276
|
88e772f1b86eaabed753e8ddee1cec00af844e0a
|
Indirect Inference for Time Series Using the Empirical Characteristic Function and Control Variates
April 18, 2019
Richard A Davis
Thiago Do
Rêgo Sousa
Claudia Klüppelberg
Indirect Inference for Time Series Using the Empirical Characteristic Function and Control Variates
April 18, 2019AMS 2010 Subject Classifications: 62F1262G2062M1065C0591G70Keywords: Asymptotic normalityCharacteristic functionControl variatesIndirect Inference estimationSimulationSLLNVariance reduction
We estimate the parameter of a time series process by minimizing the integrated weighted mean squared error between the empirical and simulated characteristic function, when the true characteristic functions cannot be explicitly computed. Motivated by Indirect Inference, we use a Monte Carlo approximation of the characteristic function based on iid simulated blocks. As a classical variance reduction technique, we propose the use of control variates for reducing the variance of this Monte Carlo approximation. These two approximations yield two new estimators that are applicable to a large class of time series processes. We show consistency and asymptotic normality of the parameter estimators under strong mixing, moment conditions, and smoothness of the simulated blocks with respect to its parameter. In a simulation study we show the good performance of these new simulation based estimators, and the superiority of the control variates based estimator for Poisson driven time series of counts.
Introduction
Let (X j ) j∈Z be a stationary time series, whose distribution depends on θ ∈ Θ ⊂ R q for some q ∈ N. Denote by θ 0 ∈ Θ the true parameter, which we want to estimate from observations X 1 , . . . , X T of the time series. Maximum likelihood estimation (MLE) has been extensively used for parameter estimation, since under weak regularity conditions it is known to be asymptotically efficient. For many models, however, MLE is not always feasible tu carry out, due to a likelihood that may be intractable to compute, or maximization of the likelihood is difficult, or because the likelihood function is unbounded on Θ. To overcome such problems, alternative methods have been developed, for instance, the generalized method of moments (GMM) in Hansen (1982), weighted mean squared error, which is the integrated distance we use, between the empirical chf computed from the blocks (1.2) of the observed time series and its simulated version computed from a large number of simulated paths of the time series. This is in contrast to the simulation based estimator defined in Section 5.2 of Carrasco et al. (2007), which is computed from one long time series path instead of the iid sample of blocks in (1.2). Since the Monte Carlo approximation of the chf here is computed from independent blocks, it should have smaller variance than the corresponding one for dependent blocks. By the same method in Carrasco et al. (2007), Forneron (2018) estimated the structural parameters and the distribution of shocks in dynamic models.
Indeed this gives a chf approximation which yields, by minimizing the integrated distance, strongly consistent and asymptotically normal parameter estimators. We also report their small sample properties for different models.
However, as the Monte Carlo approximation of the chf is computed from iid blocks from a time series, control variates techniques (see Glynn and Szechtman (2002) and Robert and Casella (2004)) provide an even more accurate approximation for the chf. Control variates techniques are classical variance reduction methods in simulation. The idea is to use a set of control variates, which are correlated with the chf. The method then approximates the joint covariance matrix of the control variates and the chf, and uses it to construct a new Monte Carlo approximation of the chf. We choose the first two terms in the Taylor expansion of the complex exponential e i t,X 1 (θ) , t, X 1 (θ) and t, X 1 (θ) 2 for θ ∈ Θ as control variates. This requires knowing the mean and covariance matrix of X 1 (θ) for θ ∈ Θ.
In assessing the performance of both the Monte Carlo approximation and the control variates approximation of the chf, two trends emerge. First, both the Monte Carlo and the control variates approximations work better for small values of the argument. Second, the control variates approximation performs much better than the Monte Carlo approximation, in particular, for small values of the argument. As a consequence, we propose a control variates based parameter estimator whose integrated mean squared error distance distinguishes between small and large values of the argument.
Under regularity conditions we prove strong consistency of the proposed parameter estimators and asymptotic normality of the simulation based parameter estimator. We find that the simulation based parameter estimator is asymptotically normal with asymptotic covariance matrix equal to the one of the oracle estimator as derived in . From this we conclude that there cannot be any improvement in the limit law for the asymptotic normality of the control variates based estimator. However, we prove that it is computed from a better approximation of the chf. Thus, the control variates estimator improves the finite sample performance compared to the simulation based parameter estimator.
The finite sample performance of the estimators are investigated for two important models. We begin with a stationary Gaussian ARFIMA model, whose chf is explicitly known so that we can use the oracle estimator and compare its performance with the simulated based estimator. Their performance is comparable and also very close to the MLE, so in this model there is no need to use control variates. The second example is a nonlinear model for time series of counts, which has been proposed originally in Zeger (1988) and applied, for instance, for modeling disease counts (see also Campbell (1994), Chan and Ledolter (1995) and Davis et al. (1999)).
In the second example, the oracle estimator does not apply, since the chf of the vector X 1 (θ) cannot be computed in closed form. For this model and different parameter sets, both the simulation based and the control variates based estimators perform satisfactory, and the control variates based estimator improves the performance of the simulation based estimator considerably. When compared with the composite pairwise likelihood estimator in Davis and Yau (2011), the control variates based estimator has comparable or even smaller bias.
Our paper is organized as follows. In Section 2 we present the oracle estimator, and the estimators computed from a Monte Carlo approximation and from a control variates approximation of the chf in detail. Here we also motivate the choice of the control variates used. The asymptotic properties of the two new estimators are established in Section 3. As all estimators are computed from true or approximated chf's we assess their performance in Section 4, first for a Gaussian AR(1) process and then for the Poisson-AR process. Practical aspects of calculating the weighted least squares function are discussed in Section 5, as well as the estimation results for finite samples. In Section 5.1 we compare the oracle estimator, the simulation based parameter estimator and the MLE for a Gaussian ARFIMA model, whereas in Section 5.2 we compare the simulation based parameter estimator and the control variates based estimator for the Poisson-AR process. The proofs of Section 3 are given in the Appendix.
Parameter estimation based on the empirical characteristic function
Throughout we use the following notation. For z ∈ C we use the L 2 -norm: |z| = √ z z, where z is the complex conjugate of z. For x ∈ R d and d ∈ N we denote by |x| the L 2 -norm, but recall that in R d all norms are equivalent. Furthermore, ·, · denotes the usual Euclidean inner product in R d . For z ∈ C the symbols (z) and (z) denote its real and imaginary part. For a function f : R q → R p its gradient is given by
∇ θ f (θ) = ∂f (θ) ∂θ T ∈ R p×q and ∇ 2 θ f (θ) = ∂vec(∇ θ f (θ)) ∂θ T ∈ R pq×q .
The oracle estimator
Let (X j (θ)) j∈Z be a stationary time series process, whose distribution depends on θ ∈ Θ ⊂ R q for some q ∈ N. Denote by θ 0 ∈ Θ the true parameter, which we want to estimate, and suppose that we observe X 1 , . . . , X T . Given a fixed p ∈ N, define for θ ∈ Θ the p-dimensional blocks
X j (θ) = (X j (θ), . . . , X j+p−1 (θ)), j = 1, . . . , n,(2.1)
where n = T − p + 1. The observed blocks corresponds to X j = (X j , . . . , X j+p−1 ), j = 1, . . . , n, which can be used to estimate the empirical characteristic function (chf ), defined as
ϕ n (t) = 1 n n j=1 e i t,X j , t ∈ R p . (2.2)
Under mild conditions such as ergodicity, ϕ n (t) converges a.s. pointwise to the true chf ϕ(t) = Ee i t,X 1 for all t ∈ R p . We assume that p is chosen in such a way that ϕ(·) uniquely identifies the parameter of interest θ. The idea of estimating θ 0 from a single time series observation by matching the empirical chf of blocks of the observed time series and the true one has been proposed in Yu (1998) and , and we use the one in , where the oracle estimator of θ 0 is defined aŝ
θ n = argmin θ∈Θ Q n (θ), (2.3) where Q n (θ) = R p |ϕ n (t) − ϕ(t, θ)| 2 w(t)dt, θ ∈ Θ, (2.4)
with suitable weight function w such that the integral is well-defined, and chf
ϕ(t, θ) = Ee i t,X 1 (θ) , t ∈ R p .
In an ideal situation, ϕ(·, θ) has an explicit expression, which is known for all θ ∈ Θ.
Estimator based on a Monte Carlo approximation of ϕ(·, θ)
Unfortunately, a closed form expression of the chf ϕ(·, θ) is for many time series processes not available. However, it can be approximated by a Monte Carlo simulation, and an idea borrowed from the simulated method of moments (McFadden (1989), see also Smith (1993) and Gourieroux et al. (1993) for a similar idea in the context of indirect inference) is to replace ϕ(·, θ) by its functional approximation constructed from simulated sample paths of (X j (θ)) j∈Z . For many different θ ∈ Θ, we simulate, independent of the observed time series, an iid sample of the blocks in (2.1) denoted byX j (θ) = (X (j) 1 (θ), . . . ,X (j) p (θ)), j = 1, . . . , H, (2.5)
for H ∈ N, and define the Monte Carlo approximation of ϕ(·, θ) based on these simulations as
ϕ H (t, θ) = 1 H H j=1 e i t,X j (θ) , t ∈ R p . (2.6)
If we replace ϕ(·, θ) in (2.4) by ϕ H (·, θ), we obtain the simulation based parameter estimator
θ n,H = arg min θ∈Θ Q n,H (θ), (2.7) where Q n,H (θ) = R p |ϕ n (t) − ϕ H (t, θ)| 2 w(t)dt,(2.8)
with suitable weight function w such that the integral is well-defined.
Remark 2.1. An alternative estimate to (2.6) of the chf is based on generating one long time series path and use the empirical chf of the consecutive blocks of p-dimensional random variables constructed as in (2.1). While being unbiased, the estimate will generally have larger variance than the estimate proposed in (2.6) using independent blocks of random variables. Nevertheless, in some cases when it is expensive to generate realizations even of size p, such as the case when a long burn-in is required to achieve stationarity, it may be computationally more efficient to generate one long series. While we do not pursue this approach here, the technical aspects of using one large realization is not much different than the estimate based on independent replicates as in (2.6).
Since ϕ H (·, θ) is based on H iid time series blocks, we can reduce its variance further using control variates to produce an even more accurate approximation for the chf. This will result in an improved version ofθ n,H .
Estimator based on a control variates approximation of ϕ(·, θ)
The estimatorθ n,H in (2.7) requires only that the stationary time series process can be simulated, and is therefore easily applicable to a large class of models. When computing Q n,H (θ) of (2.8), it is very important that the error (2.9) in approximating the true chf is small, since it propagates toθ n,H . In order to reduce the variance of the empirical chf ϕ H (·, θ), we use the method of control variates, as often used variance reduction technique in the context of Monte Carlo integration (Glynn and Szechtman (2002), Oates et al. (2017), Portier and Segers (2018)). We construct a control variates approximation of ϕ(·, θ) from the iid sampleX j (θ), j = 1, . . . , H, as in (2.5). We also require explicit expressions for the moments E t, X 1 (θ) ν for ν = 1, 2 and θ ∈ Θ.
ξ H (t, θ) = |ϕ H (t, θ) − ϕ(t, θ)|, t ∈ R p , θ ∈ Θ,
Recall thatX 1 (θ) d = X 1 (θ) for all θ ∈ Θ, so that both random variables have the same moments. As in Portier and Segers (2018), we denote by P θ the distribution of the block X 1 (θ) and by P H,θ its empirical version. For example, if f t (x) = e i t,x for t, x ∈ R p , we want to provide a good approximation for
ϕ(t, θ) = Ef t (X 1 (θ)) =: P θ (f t ), θ ∈ Θ.
To apply the control variates technique, we need control functions, which are correlated with f t (X 1 (θ)) and whose expectations are known. We use the first two terms in the Taylor series of the complex function f t (x), which suggests the vector of control functions h t,θ = (h 1,t,θ , h 2,t,θ ) T , where for ν = 1, 2,
h ν,t,θ (x) = t, x ν − E t, X 1 (θ) ν , t ∈ R p ,
Then for every vector β ∈ C 2 , we have that P H,θ (f t ) − β T P H,θ (h t,θ ) is also an unbiased estimator of ϕ(t, θ). SinceX j (θ), j = 1, . . . , H, is an independent sample,
Var[P H,θ (f t ) − β T P H,θ (h t,θ )] = H −1 σ 2 θ (f t − β T h t,θ )
and, if we differentiate the map β → σ 2 θ (f t − β T h t,θ ) with respect to β and set it equal to zero, we obtain (cf. Approach 1 in Glynn and Szechtman (2002)) the theoretical optimum
β (opt) θ,ft (h t,θ ) = {P θ (h t,θ h T t,θ )} −1 P θ (h t,θ f t ), (2.12)
provided the inverse exists. In this case, the estimator
ϕ (cvopt) H (t, θ) = P H,θ (f t ) − (β (opt) θ,ft (h t,θ )) T P H,θ (h t,θ ) (2.13)
has minimal asymptotic variance. In order to investigate the existence of the above inverse note that for each fixed t ∈ R p and θ ∈ Θ,
det(P θ (h t,θ h T t,θ )) = Var[ t,X 1 (θ) ]Var[ t,X 1 (θ) 2 ] − {Cov[ t,X 1 (θ) , t,X 1 (θ) 2 ]} 2 .
Since by the Cauchy-Schwarz inequality,
{Cov[ t,X 1 (θ) , t,X 1 (θ) 2 ]} 2 ≤ Var[ t,X 1 (θ) ]Var[ t,X 1 (θ) 2 ],
it follows (see e.g. Klenke (2013), Theorem 5.8) that
det(P θ (h t,θ h T t,θ ) = 0 ⇐⇒ a t,X 1 (θ) + b t,X 1 (θ) 2 + c a.s. = 0, (2.14)
for some a, b, c ∈ R with |a| + |b| + |c| > 0. As the scalar product is random, universal coefficients to satisfy the right-hand side of (2.14) exist only in degenerate cases, which we do not consider. Since β (opt) θ,ft (h t,θ ) is unknown, it needs to be estimated (e.g. by one of the methods in Glynn and Szechtman (2002), and we use the one described in eqs. (6) and (7) in Portier and Segers (2018)):
β H,θ,ft (h t,θ ) = {P H,θ (h t,θ h T t,θ ) − P H,θ (h t,θ )P H,θ (h T t,θ )} −1 {P H,θ (h t,θ f t ) − P H,θ (h t,θ )P H,θ (f t )}.
(2.15) For the iid sampleX j (θ), j = 1, . . . , H, as in (2.5) we obtain the control variates approximation of ϕ(·, θ) given by ϕ
H (t, θ) = P H,θ (f t ) − κ H (t, θ), t ∈ R p , (2.16) where κ H (t, θ) = (β H,θ,ft (h t,θ )) T P H,θ (h t,θ ). (2.17)
Recall from (2.10) that P H,θ (f t ) = ϕ H (t, θ), so we could simply replace ϕ H (t, θ) in (2.8) by ϕ H (t, θ) as given in (2.16). However, as we shall see in Section 4, the control variates approximation ϕ H (t, θ) provides superior approximations of ϕ(t, θ) only for values of t, for which Var( t,X 1 (θ) ) is small. Thus, we replace ϕ H (t, θ) in (2.8) by a combination of ϕ H (t, θ) and ϕ where for appropriate k > 0,
Q (cv) n,H,k (θ) = R p ϕ n (t) − ϕ (cv) H (t, θ)1 { Var( t,X 1 )<k} + ϕ H (t, θ)1 { Var( t,X 1 )≥k} 2 w(t)
Var( t, X 1 ) dt,
(2.19) with suitable weight function w such that the integral is well-defined. Note that
Var( t, X 1 ) = t TΓ p t whereΓ p = (γ p (i − j)) p i,j=1 witĥ γ p (h) = 1 n − h n−h j=1 (X j −μ n )(X j+h −μ n ), h = 1, . . . , p,
(2.20)
andμ n = 1 n n j=1 X j . The choice of the indicator function 1 { Var( t,X 1 )<k} is justified by the fact that, when estimating the parameter θ 0 , we focus on approximations of ϕ(t, θ) for θ close to θ 0 .
Asymptotic behavior of the parameter estimators
Before performing the parameter estimation we need to make sure that the parameters are identifiable from the model. For the estimators we propose, we require simply that the chf uniquely identifies the parameter of interest. This will always hold true for the examples we consider later on.
The properties of the iid sample of the blocks (X j (θ)) j∈N as a function of θ will play a crucial role for the properties of the estimatorsθ n,H andθ (cv) n,H,k from (2.7) and (2.18), respectively. In the sequel, we will make various assumptions on different aspects of the underlying process, smoothness of the model, moments of the process, and properties of the weight function. We group these assumptions into the following categories.
Assumptions A (Parameter space and time series process).
(a.1) Θ is a compact subset of R q and θ 0 ∈ Θ o , the interior of Θ.
(a.2) (X j ) j∈Z is a stationary and ergodic sequence.
(a.3) (X j ) j∈Z is α-mixing with rate function (α j ) j∈N satisfying ∞ j=1 (α j ) 1/r < ∞ for some r > 1.
Assumptions B (Continuity and differentiability in θ 0 ).
(b.1) For each j ∈ N, the map θ →X j (θ) is continuous on Θ. (b.2) For each j ∈ N, the map θ →X j (θ) is twice continuously differentiable in an open neigh- borhood around θ 0 .
Assumptions C (Moments).
(c.1) E|X 1 | u < ∞, where u = 2r/(r − 1) with r > 1 being such that (a.3) holds.
8
(c.2) E p j=1 |X j | α < ∞ for some α ∈ (u/2, u] where u = 2r/(r − 1) with r > 1 being such that (a.3) holds.
(c.3) E sup θ∈Θ |X 1 (θ)| 4 < ∞. (c.4) For each θ ∈ Θ, E|∇ θ X 1 (θ)| < ∞. (c.5) E sup θ∈Θ |∇ θ X 1 (θ)| 2(1+ε) < ∞ and E sup θ∈Θ |∇ 2 θ X 1 (θ)| 1+ε < ∞ for some ε > 0.
Assumptions D (Weight function).
(d.1) R p w(t)dt < ∞. (d.2) R p |t|w(t)dt < ∞. (d.3) R p |t| 2(1+ε) w(t)dt < ∞ for some ε > 0. (d.4) R p w(t) |t| 2 dt < ∞.
Assumption B is indeed satisfied by many linear and non-linear time series processes, in particular, when they have a representation X j (θ) = f (Z j , Z j−1 , · · · ; θ) or X j (θ) = f (Z j , X j−1 (θ), X j−2 (θ), · · · ; θ) for iid noise variables (Z j ) j∈Z , and f : R ∞ × Θ → R is a measurable function. Prominent examples are the MA(∞) and AR(∞) representations of a causal or invertible ARMA(p, q) model (see e.g. eqs. (3.1.15) and (3.1.18) in Brockwell and Davis (2013)) or the ARCH(∞) representation of a GARCH(p, q) model (see e.g. Francq and Zakoïan (2011), Theorem 2.8). In this case, assumptions (b.1) and (b.2) will hold whenever the map f is continuously differentiable for θ ∈ Θ. For example, if f is Lipschitz-continuous for θ ∈ Θ, then the continuity assumption (b.1) holds.
The key asymptotic properties, consistency and asymptotic normality of our estimates are stated in the following theorems. The proofs of these results are postponed to the appendix.
We formulate first the strong consistency results of the parameters.
→ θ 0 , n → ∞.
The asymptotic normality of the simulation based parameter estimator reads as follows.
j 1,i (t, θ) = sin( t,X 1 (θ) ) t, ∂ ∂θ (i)X 1 (θ) , l 1,i (t, θ) = cos( t,X 1 (θ) ) t, ∂ ∂θ (i)X 1 (θ) , and b (i) 1 (t) = − sin( t,X 1 (θ 0 ) ) cos( t,X 1 (θ 0 ) ) t, ∂ ∂θ (i)X 1 (θ 0 ) . (3.1) Set b 1 (t) = (b (1) 1 (t)) T . . . (b (q) 1 (t)) T (3.2) and K j (θ) = R p E[b 1 (t)] cos( t, X 1 ) − (ϕ(t, θ 0 )) sin( t, X 1 ) − (ϕ(t, θ 0 )) w(t)dt, j ∈ N. Let Q = (Q k,i ) q k,i=1 with Q k,i = R p Ej 1,k (t, θ 0 )Ej 1,i (t, θ 0 ) + El 1,k (t, θ 0 )El 1,i (t, θ 0 ) w(t)dt. (3.3) If Q is a non-singular matrix, then √ n(θ n,H − θ 0 ) d → N (0, Q −1 W Q −1 ), n → ∞, (3.4) where W = Var[K 1 (θ 0 )] + 2 ∞ j=2 Cov[K 1 (θ 0 ), K j (θ 0 )] (3.5)
Theorem 3.3 shows thatθ n,H is asymptotically normal and achieves the same asymptotic efficiency as the oracle estimator from (2.3). Therefore, there cannot be any improvement in the limit law for the asymptotic normality ofθ (cv) n,H,k . However, as we show in Section 4 it is based on a better approximation of the chf ϕ(·, θ) than that used forθ n,H . Thus, the control variates estimatorθ (cv) n,H,k improves the finite sample performance compared to the simulation based estimatorθ n,H .
Assessing the quality of the estimated chf
In this section we compare the performance of both the Monte Carlo approximation ϕ H (·, θ) and the control variates approximation ϕ (cv) H (·, θ) of the chf as defined in (2.6) and (2.16), respectively. We start with the following comparison of the two chf approximations. H be as defined in (2.13) and (2.16), respectively. We use thatβ
H,θ,ft (h t,θ ) a.s. → β (opt)
θ,ft (h t,θ ) as n → ∞ with limit given in (2.12). This follows from the representation ofβ H,θ,ft (h t,θ ) aŝ
β H,θ,ft (h t,θ ) =β H,θ, (ft) (h t,θ ) + iβ H,θ, (ft) (h t,θ )
and the almost sure convergence of both terms.
The quantities needed to compute the estimator in (2.15) are, for each ν, κ = 1, 2:
P H,θ (f t ) = 1 H H j=1 e i t,X j (θ) , (4.1) P H,θ (h ν,t,θ ) = 1 H H j=1 t,X j (θ) ν − E t, X 1 (θ) ν , P H,θ (f t h ν,t,θ ) = 1 H H j=1 e i t,X j (θ) t,X j (θ) ν − E t, X 1 (θ) ν , P H,θ (h ν,t,θ h κ,t,θ ) = 1 H H j=1 t,X j (θ) ν − E t, X 1 (θ) ν t,X j (θ) κ − E t, X 1 (θ) κ .(4.2)
Hence, strong consistency ofβ H,θ,ft (h t,θ ) follows from the SLLN. This together with P θ (h t,θ ) = 0 implies by Theorem 1 in Glynn and Szechtman (2002) that, as H → ∞,
H 1/2 ϕ (cv) H (t, θ) − ϕ(t, θ) d → N 0, σ 2 θ (f t ) − [β (opt) θ, (ft) (h t,θ )] T h t,θ , H 1/2 ϕ (cv) H (t, θ) − ϕ(t, θ) d → N 0, σ 2 θ (f t ) − [β (opt) θ, (ft) (h t,θ )] T h t,θ , with σ 2 θ (f t )−[β (opt) θ, (ft) (h t,θ )] T h t,θ ≤ σ 2 θ (f t ) and σ 2 θ (f t )−[β (opt) θ, (ft) (h t,θ )] T h t,θ ≤ σ 2 θ (f t ) ,
with σ 2 θ (·) as defined in (2.11). Therefore, ϕ
H (·, θ) provides an approximation of the integral Q n (θ) in (2.4) with smaller variance than ϕ H (·, θ). As a consequence, this favors the control variates estimatorθ (cv) n,H,k over the simulation based estimatorθ n,H for large sample sizes n ∈ N.
For all forthcoming examples we choose p = 3 and H = 3 000. We begin with a stationary Gaussian AR(1) process, where we know the chf ϕ(·) explicitly, and then proceed to the Poisson-AR process, where we approximate the true unknown chf by a precise simulated version.
The AR(1) process
We start with a stationary Gaussian AR(1) process to show how the method of control variates improves the Monte Carlo approximation of its chf. Let (X j (θ)) j∈Z be the AR(1) process
X j (θ) = φX j−1 (θ) + Z j (θ), j ∈ Z, (Z j (θ)) j∈Z iid ∼ N (0, σ 2 ), (4.3) with parameter space Θ being a compact subset of {θ = (φ, σ) : |φ| < 1, σ > 0}. Then the true chf of X 1 (θ) = (X 1 (θ), X 2 (θ), X 3 (θ)) is given by ϕ(t, θ) = e − 1 2 t T Γ 3 (θ)t , t ∈ R 3 ,
where the covariance matrix Γ 3 (θ) is explicitly known and identifies the parameter θ uniquely; see e.g. Brockwell and Davis (2013), Example 3.1.2. For a fixed θ ∈ Θ and many t ∈ R 3 we compute the absolute errors
ξ H (t, θ) = |ϕ H (t, θ) − ϕ(t, θ)| and ξ (cv) H (t, θ) = |ϕ (cv) H (t, θ) − ϕ(t, θ)| (4.4)
where ϕ H (·, θ) is the Monte Carlo approximation of the chf of X 1 (θ) = (X 1 (θ), X 2 (θ), X 3 (θ)) and ϕ (cv) H (·, θ) its control variates approximation. To understand how well we can approximate ϕ(·, θ), we plot in Figure 1, ξ H (t, θ) and ξ (cv)
H (t, θ) against Var[ t, X 1 (θ) ] for different parameters θ.
These quantities are computed from an iid sample X j (θ), j = 1, . . . , H as in (2.5). To simulate iid observations from the model (4.3), we use the fact that the one-dimensional stationary distribution is X 1 (θ) ∼ N (0, σ 2 /(1 − φ 2 )), and then use the recursion in (4.3) to simulate X 2 (θ) and X 3 (θ). We chose 500 randomly generated values of t from the 3-dimensional Laplace distribution with chf given in (5.2).
It is clear from Figure 1 that both the Monte Carlo and the control variates approximations work better when Var[ t, X 1 (θ) ] is small, and also that the control variates approximations are best for small values of Var[ t, X 1 (θ) ]. The superiority of the control variates approximation for all t and all parameter settings is clearly visible, and already expected from Remark 4.1.
The Poisson-AR model
We consider a nonlinear time series process for time series of counts, which has been proposed originally in Zeger (1988). A prototypical Poisson-AR(1) model suggested in Davis and Rodriguez-Yam (2005) assumes that the observations (X j (θ)) j∈Z are independent and Poissondistributed with means e β+α j (θ) where the process (α j (θ)) j∈Z is a latent stationary Gaussian AR(1) process, given by the equations
α j (θ) = φα j−1 (θ) + η j (θ), j ∈ Z, (η j (θ)) j∈Z iid ∼ N (0, σ 2 ), with parameter space Θ being a compact subset of {θ = (β, φ, σ) : |φ| < 1, β ∈ R, σ > 0}.
The parameter θ is uniquely identifiable from the second order structure, which has been computed in Section 2.1 of Davis et al. (2000).
For this model, the true chf of X 1 (θ) = (X 1 (θ), X 2 (θ), X 3 (θ)) cannot be computed in closed form. To mimic the assessment of the errors in eq. (4.4), we simulate 1 000 000 iid observations from X 1 (θ) by first simulating a Gaussian AR(1) process (α 1 (θ), α 2 (θ), α 3 (θ)) (as described in Section 4.1) and then simulating independent Poisson random variables with means e β+α 1 (θ) , e β+α 2 (θ) and e β+α 3 (θ) , respectively. From this we compute the empirical characteristic function and take it as ϕ(·, θ) in the absolute error terms (4.4). Now, as in Section 4.1, we compare the performance of both the Monte Carlo approximation and the control variates approximation of the chf. Figure 2 presents the results. The plots in Figure 2 are also in favor of the control variates approximation, when compared to the Monte Carlo approximation. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q H (t, θ) for p = 3 and H = 3 000 as in eq. (4.4). We use 500 randomly generated values of t ∈ R 3 from the Laplace distribution (with chf as in (5.2) below), which are plotted against Var[ t, X 1 (θ) ]. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q H (t, θ) for p = 3 and H = 3 000 as in eq. (4.4). We use 500 randomly generated values of t ∈ R 3 from the Laplace distribution (with chf as in (5.2) below), which are plottet against Var[ t, X 1 (θ) ].
14 Formula (5.4) is very useful, since it avoids the computation of a p-dimensional integral. Additionally, since the first double sum on the right-hand side of (5.4) does not depend on the argument θ, for the optimization it can be ignored.
Remark 5.3. When evaluating the integrated weighted mean squared errors (2.8), (2.19), or (5.4) in practice, they need to be deterministic functions of θ. This is enforced by taking a fixed seed for every j = 1, . . . , H, when simulatingX j (θ) for different values of θ ∈ Θ.
In the following two examples we study the finite sample behavior of the estimatorsθ n,H and θ (cv) n,H,k . We begin with a stationary Gaussian ARFIMA model, whose chf is explicitly known so that we can use the oracle estimator from Section 2.1. Afterwards we come back to the Poisson-AR process. We choose p = 3, since the 3-dimensional chf contains sufficient information to identify the parameter of interest. We also choose H = 3 000.
The ARFIMA model
Let (X j (θ)) j∈Z be the stationary Gaussian ARFIMA(0, d, 0) model
(1 − B) d X j (θ) = Z j (θ), j ∈ Z, (Z j (θ)) j∈Z iid ∼ N (0, σ 2 ),
where B is the backshift operator, with parameter space Θ being a compact subset of {θ = (d, σ) : d ∈ (−0.5, 0.5), σ > 0}. Then the true chf of X 1 (θ) = (X 1 (θ), X 2 (θ), X 3 (θ)) is given by
ϕ(t, θ) = e − 1 2 t T Γ 3 (θ)t , t ∈ R 3 , θ ∈ Θ,
where the covariance matrix Γ 3 (θ) is explicitly known and identifies the parameter θ uniquely; see e.g. Pipiras and Taqqu (2017), Corollary 2.4.4. For the long-memory case, for each value of d ∈ {0.05, . . . , 0.45} we compare the new estimators with the MLE method as implemented in the R package arfima. Thus, for many θ ∈ Θ, we generate iid Gaussian random vectors with mean zero and covariance Γ 3 (θ) and use them to construct the simulation based estimatorθ n,H .
Since the chf ϕ(·, θ) is known in closed form, we are able to compute the oracle estimatorθ n from (2.4). For practical purpose we choose the weight function w(t) = (2π) −3/2 e − 1 2 t T t , t ∈ R 3 . Then the integral in (2.4), which needs to be minimized with respect to the parameter θ, can be evaluated similarly as in (5.4), giving for the chf being known,
Q n (θ) = R 3 1 n n j=1 e i t,X j − e − 1 2 t T Γ 3 (θ)t 2 w(t)dt = 1 n 2 n j=1 n k=1 exp − 1 2 (X j − X k ) T (X j − X k ) + det (2Γ 3 (θ) + I) −1 1 2 −2 det (Γ 3 (θ) + I) −1 1 2 1 n n j=1 exp − 1 2 X T j (Γ 3 (θ) + I) −1 X j . (5.5)
We compare in Table 5.1 the performance of the simulation based estimatorθ n,H , the oracle estimatorθ n in (2.3) based on the minimization of (5.5), and the MLE. We notice thatθ n,H is comparable to the oracle estimator, so in this model there is no need to use control variates. In particular, the RMSEs are almost the same for all d ≥ 0.20. The MLE has a smaller RMSE, but bothθ n andθ n,H have a smaller bias than the MLE.
The Poisson-AR process
The Poisson-AR model has been defined in Section 4.2. We conduct a simulation experiment in the same setting as in Table 5 in Davis and Rodriguez-Yam (2005) and Table 3 in Davis and Yau (2011). The results are shown in Table 5.2 for n = 500 and nine different parameter settings, where we also classify the models by the corresponding index of dispersion D of the random variable e β+α 1 , which assumes values in {0.1, 1, 10} as shown in Davis and Rodriguez-Yam (2005). We compare both the simulation based estimatorθ n,H and control variates based estimator θ (cv) n,H,k . We fix H = 3 000, p = 3 and the 3-dimensional Laplace density as in (5.2) for w. To simulate iid observations of (X 1 (θ), X 2 (θ), X 3 (θ)) we proceed as explained in Section 4.2. The simulation based estimatorθ n,H in (2.7) is computed via (5.4). Unfortunately, such a formula cannot be obtained for the control variates based estimatorθ (cv) n,H,k , since the introduction of the correction κ H in (2.17) introduces addional polynomial terms into Q Table 5.2: Comparison of the simulation based estimatorθ n,H of (2.7) and the control variates based estimatorθ (cv) n,H,k of (2.18) with k = 1, both with H = 3 000. The models are classified by the index D of dispersion of e β+α1 . For both estimators the empirical chf has been computed with n = 500, p = 3, and w is the Laplace density as in (5.2). Moreover, 500 replications have been used to compute bias, standard deviation (Std) and root mean squared error (RMSE). k = 1 presents smaller bias and RMSE than the simulation based estimatorθ n,H in most cases, in all others it is comparable. Additionally, a significant improvement in the bias for estimating φ is noticeable for θ = (0.373, 0.500, 0.220) and θ = (0.373, 0.900, 0.111).
We compare now the control variates based estimatorθ (cv) n,H,k in Table 5.2 with the results for the consecutive pairwise likelihood (CPL) from Table 3 in Davis and Yau (2011), which is refereed to as CPL1 in that paper. The bias ofθ (cv) n,H,k is smaller than that of CPL1 for the estimated β and σ for almost all cases, in all others it is comparable. For φ the bias ofθ (cv) n,H,k and CPL1 are comparable, except thatθ (cv) n,H,k has poor performance for estimating φ for the true parameter (β, φ, σ) = (0.373, 0.9, 0.111). This is due to the fact that the simulated sample paths contain a large number of zeros, giving very little information for the parameter estimation.
A Appendix to Section 3
In the following we always set H = H(n) andH =H(n) = H(n)/n, but omit the argument n for notational simplicity.
Throughout the letter c stands for any positive constant independent of the respective argument. Its value may change from line to line, but is not of particular interest.
For a matrix with only real eigenvalues λ min (·) denotes the smallest eigenvalue. We often use the uniform SLLN, which guarantees for a continuous stochastic process (Z(t)) t∈R p satisfying E sup t∈K |Z(t)| < ∞ that sup t∈K |Z(t) − EZ(t)| a.s. → 0 as n → ∞ for every compact set K ⊂ R p . More precisely, we use the SLLN on the separable Banach space C(K), the space of continuous functions on the compact set K ⊂ R p , endowed with the sup norm (see e.g. Theorem 16(a) in Ferguson (1996) or Theorem 9.4 in Parthasarathey (1967)).
A.1 Proof of Theorem 3.1
Let Q(θ) = R p ϕ(t, θ 0 ) − ϕ(t, θ) 2 w(t)dt
be the candidate limiting function of Q n,H (θ). For δ > 0 define the set
K δ = {t ∈ R p : |t| ≤ δ}. (A.1)
Since |e i t,X 1 (θ) | = 1 for all θ and t, and the random elements (X j (θ), θ ∈ Θ) ∞ j=1 are iid, the uniform SLLN holds giving
sup (t,θ)∈Θ×K δ 1 H H j=1 e i t,X j (θ) − ϕ(t, θ) a.s. → 0, n → ∞. (A.2)
In particular, for θ = θ 0 we also have Applying the inequality ||a| 2 − |b| 2 | ≤ 2|a − b| for a, b ∈ C, |a|, |b| ≤ 1 gives We analyze the asymptotic behavior of the first term in (A.17) in Lemma A.2. More precisely, we show there that K δ b H (t)g n (t)w(t)dt for K δ as in (A.1) converge in distribution to a qdimensional Gaussian vector. Afterwards, Lemmas A.3 and A.4 show that as δ → ∞, componentwise in R q , lim sup n→∞ Var K c δ b H (t) √ ng n (t)w(t)dt → 0 and
|Q n,H (θ) − Q(θ)| ≤ R p 1 n n j=1 e i t,X j − 1 H H j=1 e i t,X j (θ) 2 − |ϕ(t, θ 0 ) − ϕ(t, θ)| 2 w(t)dt ≤ 2 R p 1 n n j=1 e i t,X j − ϕ(t, θ 0 ) + ϕ(t, θ) − 1 H H j=1 e i t,X j (θ) w(t)dt ≤ 2 R p 1 nK c δ E[b 1 (t)]G(t)w(t)dt P → 0,
where G is a zero mean R 2 -valued Gaussian field. We show by a standard Chebyshev argument that the second term in (A.17) converges in probability componentwise to 0 in (A.45). The convergence of the second derivatives ∇ 2 θ Q n (θ n ) will be the topic of Lemma A.5. For the scalar products above we use the following bounds several times below.
Lemma A.1. Let ν ≥ 1, t ∈ R p , k, i ∈ {1, . . . , q} and j ∈ Z be fixed and assume that (b.2) holds.Then the following bounds hold true.
(a) If E|∇ θ X 1 (θ)| ν < ∞ for θ ∈ Θ, then there exists a constant c > 0 such that
E t, ∂ ∂θ (k)X j (θ) ν ≤ c|t| ν E|∇ θ X 1 (θ)| ν , t ∈ R p . (A.18) (b) If E|∇ 2 θ X 1 (θ)| ν < ∞ for θ ∈ Θ, then there exists a constant c > 0 such that E t, ∂ ∂θ (k) ∂θ (i)X j (θ) ν ≤ c|t| ν E|∇ 2 θ X 1 (θ)| ν , t ∈ R p . (A.19)
The same bounds hold uniformly, taking expectations over sup θ∈Θ or over sup t∈K for some compact K ⊂ R p at both sides of (A.18) and (A.19), provided the corresponding expectations exist.
Proof. (a) Applying the Cauchy-Schwarz inequality for the inner product, the fact that (X j (θ), θ ∈ Θ) d = (X 1 (θ), θ ∈ Θ) d = (X 1 (θ), θ ∈ Θ), bounding the L 2 -norm by the L 1 -norm, employing the inequality | p j=1 β j | ν ≤ p ν−1 p j=1 |β j | ν valid for β 1 , . . . , β p ∈ R and ν ≥ 1 gives
E t, ∂ ∂θ (k)X j (θ) ν ≤ |t| ν E ∂ ∂θ (k)X j (θ) ν = |t| ν E ∂ ∂θ (k) X 1 (θ) ν ≤ |t| ν E p r=1 ∂ ∂θ (k) X r (θ) ν ≤ p ν−1 |t| ν p r=1 E ∂ ∂θ (k) X r (θ) ν ≤ p ν−1 |t| ν p r=1
E|∇ θ X r (θ)| ν = p ν |t| ν E|∇ θ X 1 (θ)| ν =: c|t| ν E|∇ θ X 1 (θ)| ν . where G is an R 2 -valued Gaussian field.
≤ 2 1+2ε 1 H H j=1 E t, ∂ ∂θ (k)X j (θ n ) t, ∂ ∂θ (i)X j (θ n ) 1+ε + E t, ∂ ∂θ (k) ∂θ (i)X j (θ n ) 1+ε ≤ c 1 H H j=1
|t| 2(1+ε) E|∇ θ X 1 (θ n )| 2(1+ε) + |t| 1+ε E|∇ 2 θ X 1 (θ n )| 1+ε ≤ c |t| 2(1+ε) E sup θ∈Θ |∇ θ X 1 (θ)| 2(1+ε) + |t| 1+ε E sup θ∈Θ |∇ 2 θ X 1 (θ)| 1+ε := v(t) < ∞.
(A.40)
Step 4: Convergence of the random integrals: Define the sequence of functions v n (t) = E|i n,H (t, θ n )g H,k,i (t, θ n ) − Ei 1,1 (θ 0 , t)Eg 1,k,i (θ 0 , t)|, t ∈ R p , and recall that from the L 1 -convergence showed in Step 3, for every t ∈ R p we have v n (t) → 0 as n → ∞. From the definition of the function v in the last line of (A.40) it follows that sup n∈N v n (t) ≤ 2v(t). Additionally, assumption (d.3) implies that R p v(t)w(t)dt < ∞.
Therefore, it follows from Fubini's Theorem and dominated convergence that E R p i n,H (t, θ n )g H,k,i (t, θ n ) − Ei 1,1 (θ 0 , t)Eg 1,k,i (θ 0 , t) w(t)dt ≤ E R p |i n,H (t, θ n )g H,k,i (t, θ n ) − Ei 1,1 (θ 0 , t)Eg 1,k,i (θ 0 , t)|w(t)dt = R p v n (t)w(t)dt → 0, n → ∞, (A.41) and therefore the convergence in probability of (A.33) follows from the L 1 -convergence in (A.41).
The proofs for the other three remaining integrals on the right-hand side of (A.31) follow along the same lines. The result in (A.32) is then a consequence of the fact that for all t ∈ R p , Ei 1,1 (t, θ 0 ) = Ek 1,1 (t, θ 0 ) = 0.
Proof of Theorem 3.3: We handle each term in (A.17) separately. As a direct consequence of Theorem 3.1 and Lemmas A.2, A.3, A.4 and A.5, −2(∇ 2 θ Q n,H (θ n )) −1
R p b H (t) √ ng n (t)w(t)dt d → N (0, Q −1 W Q −1 ), n → ∞,
with Q as in (3.3),
W = Var R p E[b 1 (t)]G(t)w(t)dt
and G being the R 2 -valued Gaussian field from Lemma A.2. For arbitrary k, r ∈ {1, . . . , q} we have Since (X j ) j∈N is α-mixing by (a.3), we can apply the CLT in Ibragimov and Linnik (1971) (Theorem 18.5.3 with δ = 2/(r − 1)) and find that
W k,r = Cov R p E[b (k) 1 (t)] T G(t)w(t)dt, R p E[b (r) 1 (t)] T G(t)w(t)dt = R p R p E[bE[G(t)G(s) T ] = E[F 1 (t)F 1 (s) T ] + 2 ∞ j=2 E[F 1 (t)F j (s) T ], (A.43)
where F j (t) = cos( t, X j ) − (ϕ(t, θ 0 )) sin( t, X j ) − (ϕ(t, θ 0 )) .
W k,r = R p R p E[b (k) 1 (t)] T E[F 1 (t)F 1 (s) T ] + 2 ∞ j=2 E[F 1 (t)F j (s) T ] E[b (k) 1 (s)]w(t)w(s)dtds = R p R p E[b (k) 1 (t)] T E[F 1 (t)F 1 (s) T ]E[b (k) 1 (s)]w(t)w(s)dtds + 2 ∞ j=2 R p 2 E[b (k) 1 (t)] T E[F 1 (t)F j (s) T ]E[b (k) 1 (s)]w(t)w(s)dtds = E R p E[b (k) 1 (t)] T F 1 (t)w(t)dt 2 + 2 ∞ j=2 E R p E[b (k) 1 (t)] T F 1 (t)w(t)dt R p E[b (k) 1 (s)] T F j (t)w(s)ds ,
which gives (3.5). The second term in (A.17) is, up to a constant,
H
(t, θ). More precisely, we propose the following control variates based estimator:θ (cv) n,H,k = argmin θ∈Θ Q (cv) n,H,k (θ), (2.18)
Theorem 3. 1 (
1Consistency ofθ n,H ). Assume that (a.1), (a.2), (b.1), and (d.1) hold. Let H = H(n) → ∞ as n → ∞. H,k ). Assume that the conditions of Theorem 3.1 hold, and additionally (c.1), (c.3), and (d.4). Let H = H(n) → ∞ as n → ∞.
Theorem 3 . 3 (
33Asymptotic normality ofθ n,H ). Assume that all Assumptions A and B hold, and that the moment conditions (c.2), (c.4), and (c.5) hold. Furthermore, assume that the weight function satisfies (d.1), (d.2) and (d.3). Let H = H(n) :=H(n)n andH(n) → ∞ as n → ∞. Define for i = 1, . . . , q
H
(·, θ) and ϕ H (·, θ)] Assume that (c.3) holds, and let ϕ
Figure 1 :
1Gaussian AR(1) model: absolute error ξ H (t, θ) and ξ
Figure 2 :
2Poisson-AR model: Absolute errors ξ H (t, θ) and ξ(cv)
Under assumptions (a.2), (b.2), (a.3), (c.2) and (c.4) we have on the Borel sets of R q ,
1
(t)] T E[G(t)G(s) T ]E[b(k) 1 (s)]w(t)w(s)dtds. (A.42)
A.43) and (A.44) into (A.42) gives with Fubini's Theorem
b
H (t)g H (t)w(t)dt.It follows from the fact that (X j (θ 0 )) j∈N d = (X j ) j∈N combined with (A.n → ∞. Thus (3.4) follows from Chebyshev's inequality.
Table 5 .
51: Comparison of the simulation based estimatorθ n,H for H = 3 000, the oracle estimatorθ n and the MLE. For both estimators we have set n = 400, p = 3, and w is the Gaussian density as in(5.3). Moreover, 500 replications have been used to compute bias, standard deviation (Std) and root mean squared error (RMSE).
cv) n,H,k in (2.19). Thus, we resort to numerical integration to evaluateθ (cv) n,H,k . Our findings are as follows. For D ∈ {1, 0.1}, the control variates based estimatorθ(cv)
n,H,k for
β
φ
σ
β
φ
σ
β
φ
σ
D = 10
TRUE
-0.613 -0.500 1.236 -0.613 0.500 1.236 -0.613 0.900 0.622
Bias(θ n,H )
-0.015 0.025 0.002 -0.012 0.014 -0.032 -0.016 -0.010 0.002
RMSE(θ n,H )
0.096 0.101 0.119 0.148 0.107 0.120 0.298 0.054 0.128
Bias(θ
(cv)
n,H,k )
0.023 0.031 -0.007 0.006 0.002 -0.018 0.061 -0.007 -0.036
RMSE(θ
AcknowledgementThiago do Rêgo Sousa gratefully acknowledges support from the National Council for Scientific and Technological Development (CNPq -Brazil) and the TUM Graduate School. He also thanks the Statistics Department at Columbia University for its hospitality during his visit and takes pleasure to thank Viet Son Pham and Thibaut Vatter for helpful discussions.Our objective is to obtain a simple expression of the integrated mean squared error Q n,H (θ) in (2.8), which is needed to compute the estimator in (2.7). For a weight function w in (2.8), we writew (x) = R p e i t,x w(t)dt, x ∈ R p , (5.1) for its Fourier transform. Our preference is on weight functions such that (5.1) is known explicitly.Example 5.1.[Weight functions and their characteristic functions] (i) Laplace: w is a multivariate Laplace density with chf, t ∈ R p . (5.2) (ii) Cauchy: w is a multivariate Cauchy density with chf(iii) Gaussian: w is a standard multivariate Gaussian density with chfLemma 5.2. Let Q n,H (θ) be as in (2.8) and w a weight function with Fourier transformw. Then w(X j −X k (θ)) +w(X k (θ) − X j ) .Proof. Since |z| 2 = zz for z ∈ C, for every θ ∈ Θ,Applying sup θ∈Θ on both sides of (A.4), using (A.2) combined with (d.1), and taking the limit for δ ↓ 0 gives supNow we prove that Q(θ) = 0 if and only if θ = θ 0 . Obviously Q(θ 0 ) = 0. If θ = θ 0 , then the distributions of X 1 andX 1 (θ) are different and thus also their characteristic functions are different. Since characteristic functions are continuous, it follows that they are different at least on an interval with positive Lebesgue measure; hence Q(θ) > 0. Therefore, Q(θ) is uniquely minimized at θ 0 and this fact together with (A.5) gives strong consistency ofθ n,H .A.2 Proof of Theorem 3.2We have that Var( t, X 1 ) = t TΓ p t, withΓ p being the p-dimensional empirical covariance matrix of the observed time series (X 1 , . . . , X T ) as in (2.20). Let k > 0 be fixed andp t < k} and its complement L c n . Recall also (2.16) and (2.17). Using |e ix | = 1 for all x ∈ R, together with |ab − cd| ≤ |b||a − c| + |c||b − d| for a, b, c, d ∈ C gives for the integral on L c n :By (a.3) and (c.1) it follows from Theorem 3(a) in Section 1.2.2 ofDoukhan (1994)thatSince Var(X 1 ) > 0, it follows from (A.7) combined with Proposition 5.1.1 inBrockwell and Davis (2013)that det(Γ p ) > 0, and therefore, the minimum eigenvalue λ min (Γ p ) of Γ p is positive. Thus, for all t ∈ R p ,By (a.2) and the ergodic theoremΓ p a.s.→ Γ p and, since the eigenvalues of a matrix are continuous functions of its entries (cf.Bernstein (2009), Fact 10.11.2), also λ min (Γ p ) a.s. → λ min (Γ p ) > 0. It 20 follows from (A.8) and from the a.s. convergence of the eigenvalues that there exists N > 0 such that Hence, there exists some N ∈ N such thatThus, for t ∈ L c n we obtain 1This together with (A.10) gives the following upper bound for the right-hand side of (A.6):The first integral can be estimated as |Q n,H (θ) − Q(θ)| in (A.4) which tends to 0 uniformly for θ ∈ Θ provided that (d.1) holds. SinceΓ p a.s.→ Γ p , also the second integral in (A.11) tends 0 a.s. as n → ∞.We turn to the integrated mean squared error |QThe control variates correction used in (2.19) can be regarded as a continuous function g : R 9 → R 2 whose entries are the arithmetic means defined in (4.1)-(4.2). By (c.3) and the uniform SLLN, each of these arithmetic means converge a.s. uniformly on L × Θ as n → ∞ and H → ∞. Thus, it follows from the continuity of g and the continuous mapping theorem that supFor n ≥ N it follows from (A.9) that L n ⊆ L and thus using the inequality:=I 1,n (θ) + I 2,n (θ). → 0 by similar arguments as used in (A.10) and (A.11), since for t ∈ L, also applying (d.4),21A.3 Proof of Theorem 3.3By the definition ofθ n,H in (2.7) and under assumptions (a.1) and (b.2) we have ∇ θ Q n,H (θ n,H ) = 0.A Taylor expansion of order 1 of ∇ θ Q n,H around θ 0 giveswhere θ n a.s.→ θ 0 as n → ∞. Therefore, asymptotic normality of √ n(θ n,H − θ 0 ) will follow by the delta method, if we prove that as n → ∞:(1) √ n∇ θ Q n,H (θ 0 ) converges weakly to a multivariate normal random variable, and(2) ∇ 2 θ Q n,H (θ n ) converges in probability to a non-singular matrix.We start with the first point and compute the partial derivatives of Q n,H :Recall that ϕ n (t) and ϕ H (t, θ) denote the empirical characteristic functions of the observed blocks (X 1 , . . . , X n ) as in (2.2) and of its Monte Carlo approximation (X 1 (θ), . . . ,X H (θ)) as in (2.6), respectively. Define the partial derivatives of the real and imaginary part of ϕ H (t, θ):t, ∂ ∂θ (i)X j (θ) , i = 1, . . . , q, (A.14)and summarize them intoProof. Under assumptions (a.3) and (c.2), it follows from Lemma 4.1(2) inDavis et al. (2018)that √ n(ϕ n (·) − ϕ(·, θ 0 )) convergences in distribution on compact subsets of R p to a complexvalued Gaussian fieldG, equivalently the vector of real and imaginary part converge to a bivariate Gaussian field G. Since the random elements (X j (θ), θ ∈ Θ) j∈N are iid and the partial derivatives exist by (b.2), also (X j (θ 0 ), ∇ θXj (θ 0 )) j∈N are iid. Then it follows from the definitions (A.14), (A.15), and Lemma A.1 with K = K δ ) in combination with (c.4) thatHence, the uniform SLLN guarantees thatSlutsky's theorem gives then b H (·) √ ng n (·, θ 0 ) convergences in distribution on compact subsets of R p to E[b 1 (·)]G(·) as n → ∞. The result in (A.21) follows from the continuity of the integral by another application of the continuous mapping theorem on C(K δ ).Proof. Since b H (·) and g n (·) are independent and Eg n (t) = 0, we have E[b H (t)g n (t)] = 0 for all t ∈ R p . An application of the Cauchy-Schwartz inequality for integrals gives(A.24)We first obtain a bound for the product between the first component g n,1 (·) of g n (·) and the first component bDoukhan (1994)that for fixed t,Under (a.3) it follows from Theorem 3(a) in Section 1.2.2 ofwhere u = 2r (r−1) and, thus, it follows from the stationarity of (U j (t)) j∈N combined with (A.26) 24 and the fact that |U 0 (t)| ≤ 2 thatwhere the bound is independent of t. Recall that H = H(n) =H(n)n. Under (c.4), it follows from the iid property of (V j (t)) j∈N(A.28)Using the fact that 1 n n j=1 U j (t) ≤ 2, adding and subtracting EV 0 (t) with the inequality |a + b| 2 ≤ 2(|a| 2 + |b| 2 ), and (A.28) gives25Proof. It follows from (A.14), (A.15), (c.4), and (A.22) E|b 1 (t)| ≤ c|t|E|∇ θ X 1 (θ 0 )| < ∞. Now we find an upper bound for the variance of each component of G(t) for a fixed t. Let U j (t) be as defined at the left-hand side of (A.25) and notice that the first component of G(t) is the distributional limit of 1 √ n n j=1 U j (t). Since (U j (t)) j∈N is α-mixing by (a.3), we can apply the CLT inIbragimov and Linnik (1971)(Theorem 18.5.3 with δ = 2/(r − 1)) and find that the variance of the first component of G(t) is given byThis combined with Theorem 3(a) in Section 1.2.2 ofDoukhan (1994)and the fact that EU j (t) = 0 and |U j (t)| ≤ 2 for all j ∈ N gives by (a.3) and (A.26)A similar calculation shows that the variance of the second component of G(t) is also bounded by a finite constant, which does not depend on t. Therefore, E|G(t)| ≤ c. This combined with (A.22) and assumption (d.2) givesSince L 1 -convergence implies convergence in probability the result follows.This proves part (1) of the delta method.We now turn to part (2). In order to calculate the second derivatives of Q n,H (θ), which exist by (b.2), we rewrite A.13 asFor the second derivatives we calculate for every i, k ∈ {1, . . . , q},where we summarize all quantities used in the following list: Ej 1,k (t, θ 0 )Ej 1,i (t, θ 0 ) + El 1,k (t, θ 0 )El 1,i (t, θ 0 ) w(t)dt, n → ∞.(A.32)Proof. We first prove R p i n,H (t, θ n )g H,k,i (t, θ n )w(t)dt P → R p Ei 1,1 (θ 0 , t)Eg 1,k,i (θ 0 , t)w(t)dt, n → ∞.(A.33)Step 1: Uniform convergence on Θ: It follows from the iid property of the random elements (X j (θ), θ ∈ Θ) j∈N that the sequence (X j (θ), ∇ θXj (θ), ∇ 2 θX j (θ), θ ∈ Θ) j∈N is iid. Lemma A.1 together with (c.5) gives the uniform boundand it follows from the uniform SLLN that for every fixed t ∈ R p sup θ∈Θ |g H,k,i (t, θ) − Eg 1,k,i (t, θ)| a.s. Step 2: Pointwise convergence of i n,H (t, θ n )g H,k,i (t, θ n ): The triangle inequality implies(A.38)Since θ n a.s.→ θ 0 and the map θ → Ei 1,1 (t, θ)Eg 1,k,i (t, θ) is continuous in Θ, (by (b.2) and (c.5)) it follows that the second term on the right-hand side of (A.38) converges a.s. to zero. Additionally, since the uniform convergences on (A.34) and (A.37) imply the uniform convergence of the product i n,H (t, θ)g H,k,i (t, θ) on Θ it follows that the first term on the right-hand side of (A.38) also converges a.s. to zero.Step 3: L 1 -convergence: Since we have already shown a.s. convergence, it follows from Theorems 6.25(iii) and 6.19 inKlenke (2013)(with H(x) = |x| 1+ε ) that L 1 -convergence follows provided that sup n∈N E|i n,H (t, θ n )g H,k,i (t, θ n )| 1+ε < ∞ for some ε > 0. Using the fact that |i n,H (t, θ n )| ≤ 2 and the inequality | 1 n n j=1 β j | 1+ε ≤ 1 n n j=1 |β j | 1+ε , β 1 , . . . , β n ∈ R, we obtain E|i n,H (t, θ n )g H,k,i (t, θ n )| 1+ε ≤ 2 1+ε E|g H,k,i (t, θ n )| 1+ε∂ ∂θ (k)X j (θ n ) t, ∂ ∂θ (i)X j (θ n ) + t, ∂ ∂θ (k) ∂θ (i)X j (θ n ) 1+ε , (A.39) since | cos(·)|, | sin(·)| ≤ 1. Now we use the inequality |a + b| 1+ε ≤ 2 ε (|a| 1+ε + |b| 1+ε ) for a, b ∈ R, assumption (c.5) for the uniform bound in Lemma A.1 and the fact that the sequence (X j (θ), ∇ θXj (θ), ∇ 2 θX j (θ), θ ∈ Θ) j∈N is iid to continue
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|
[] |
[
"Direct Modulation of Electrically Pumped Coupled Microring Lasers",
"Direct Modulation of Electrically Pumped Coupled Microring Lasers"
] |
[
"Chi Xu \nThe College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA\n",
"William E Hayenga \nThe College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA\n",
"Demetrios N Christodoulides \nThe College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA\n",
"Mercedeh Khajavikhan \nMing Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90007CAUSA\n",
"Patrick Likamwa [email protected] \nThe College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA\n"
] |
[
"The College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA",
"The College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA",
"The College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA",
"Ming Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90007CAUSA",
"The College of Optics and Photonics\nCREOL\nUniversity of Central Florida\n32816OrlandoFloridaUSA"
] |
[] |
We demonstrate how the presence of gain-loss contrast between two coupled identical resonators can be used as a new degree of freedom to enhance the modulation frequency response of laser diodes. An electrically pumped microring laser system with a bending radius of 50 μm is fabricated on an InAlGaAs/InP MQW p-i-n structure. The room temperature continuous wave (CW) laser threshold current of the device is 27 mA. By adjusting the ratio between the injection current levels in the two coupled microrings, our experimental results clearly show a bandwidth improvement by up to 1.63 times the fundamental resonant frequency of the individual device. This matches well with our rate equation simulation model.
|
10.1364/oe.442076
|
[
"https://arxiv.org/pdf/2203.09955v1.pdf"
] | 243,949,863 |
2203.09955
|
b99d7060694eb047738898e64364f7d8fbc1d984
|
Direct Modulation of Electrically Pumped Coupled Microring Lasers
Chi Xu
The College of Optics and Photonics
CREOL
University of Central Florida
32816OrlandoFloridaUSA
William E Hayenga
The College of Optics and Photonics
CREOL
University of Central Florida
32816OrlandoFloridaUSA
Demetrios N Christodoulides
The College of Optics and Photonics
CREOL
University of Central Florida
32816OrlandoFloridaUSA
Mercedeh Khajavikhan
Ming Hsieh Department of Electrical and Computer Engineering
University of Southern California
90007CAUSA
Patrick Likamwa [email protected]
The College of Optics and Photonics
CREOL
University of Central Florida
32816OrlandoFloridaUSA
Direct Modulation of Electrically Pumped Coupled Microring Lasers
We demonstrate how the presence of gain-loss contrast between two coupled identical resonators can be used as a new degree of freedom to enhance the modulation frequency response of laser diodes. An electrically pumped microring laser system with a bending radius of 50 μm is fabricated on an InAlGaAs/InP MQW p-i-n structure. The room temperature continuous wave (CW) laser threshold current of the device is 27 mA. By adjusting the ratio between the injection current levels in the two coupled microrings, our experimental results clearly show a bandwidth improvement by up to 1.63 times the fundamental resonant frequency of the individual device. This matches well with our rate equation simulation model.
Introduction
Directly modulated semiconductor laser diodes have several attractive features such as low cost and potential for high-density integration [1,2]. However, the intrinsic modulation bandwidth of a laser diode tends to be capped by its relaxation resonance frequency, leading to limited applications in optical communication systems [3]. Over the years, a number of technologies have been developed for realizing high-speed directly modulated semiconductor laser diodes. These include but are not limit to, nanolasers with strong Purcell effect [4][5][6][7], narrow linewidth quantum dot lasers [8], optical injection locking [9,10], and photon-photon resonance in coupled vertical-cavity surface-emitting lasers (VCSELs) and edge emitting lasers [11][12][13][14].
Microring resonators are one of the most attractive cavity structures in photonic integrated circuits. Due to the absence of reflective facets, ring resonators can enable lasers with high quality factors, that are compact for on-chip device integration. Several efforts have also been devoted to improving the performance of semiconductor ring lasers in practical applications, such as lowering their threshold [15], unidirectional emission [16], and compatibility with silicon [17]. In recent years, non-Hermiticity has been widely used in photonic settings in order to delicately mold light-matter interactions, realize sensitivity enhancements [18][19][20][21], enforce unidirectional invisibility [22,23], and enable topological lasers [24,25]. One widely used approach for establishing non-Hermiticity is to use two coupled cavities one subject to gain and the other to loss (or a less amount of gain). For example, a laser consisting of a coupled ring resonator systems with differential gain have been demonstrated to enhance the mode suppression ratio and promote single mode lasing [26][27][28]. In this work, we investigate the use of gain and loss in a dual ring laser system as a new mechanism to modify the maximum modulation speed of the laser diode.
In this paper, we explore the interplay between unevenness of pumping and modulation bandwidth in the coupled microring configuration. The paper is structured as follows: In Section 2, we study the modulation characteristics of this coupled laser system using the rate equation model. In Section 3, we provide the electromagnetic mode simulations for a laser designed on a InAlGaAs/InP multiple quantum well (MQW) epitaxial wafer. In Section 4, we describe the steps involved in the fabrication of the electrically pumped microring lasers. In Section 5, we characterize the laser and measure the frequency responses under different pumping ratios. Finally, Section 6 concludes the paper.
2. Rate equation model for coupled ring laser system Figure 1 (a) shows a schematic of the coupled ring laser system. It consists of two identical microring cavities that are evanescently coupled to each other due to proximity. Photon resonance can be supported in this coupled cavity system by applying a sufficient amount of pump. Consequently, the laser emission is extracted through a coupled bus waveguide positioned in the side of one of the rings. Each cavity has its own electrodes that allows a diverse set of pumping schemes. Considering only the fundamental mode TE10 is supported, the electric field of TE10 in each cavity , can be described using the following coupled mode equations in the temporal domain:
̇= ( − )(1 − ) + , (1.a) ̇= −( + )(1 − ) + ,
(1.b) where γ signifies the linear loss of a passive resonator mainly due to scattering, bending and output coupling losses, and stand for carrier induced gain and loss in the respective rings. One thing to be noticed, F can be the loss or a lower level of gain depending on the carrier density. represents the linewidth enhancement factor, and is the temporal coupling coefficient [26].
Equation 1 depicts the dynamics of photons in two rings. Considering an electrical pumping scheme, the laser rate equations of this coupled structure can be described by:
̇= /( ) − / − ∑ , , (2.a) , ̇= , − , / − 2 , (2.b) ̇= Γ ( + )/2 + (1/ − ), (2.c) , ̇= − , − , / + 2 , , , (2.d) ̇= /( ) − / + ∑ , .
(2.e) where is the injected current of ring ( = 1,2), is the elementary charge, is the volume of the active region, stands for the current injection efficiency, and , and are the carrier density and photon density of the m th mode in ring . In these equations, mutually coupled modes are placed in the same order. The cavities are designed to support single transverse mode TE10. Therefore, a pair of counter-propagating modes in adjacent rings share the same m. In these equations, and stand for the ratio of modal field amplitudes and phase difference, respectively, is the carrier lifetime, and represent gain and loss related to the temporal counterpart and via 2 = and 2 = , respectively, is the photon lifetime, is the group velocity, and is the confinement factor. In the following discussion, , and are assumed to be the same for the two rings due to their identitical structures.
At steady state, phase difference and modal field ratio can be determined by gain-loss contrast = + via = + arctan [ ( − 1)/( + 1)] and = ( + 1/ ) .The modulation frequency response of the mth order mode outcoupling from the gain cavity (ring 1), i.e. , = |Δ , /Δ |, can be obtained by applying small sinusoidal signal to its steady state solution: , not only depends on , and , but is also subject to .
( − )Δ = ( ) Δ (3) where = ( ,
To verify how the gain-loss contrast can modify the features of (j ), we numerically simulate the modulation response by employing the rate equation model to a InAlGaAs MQW laser, with the parameters given in Table 1. Figure 1(b) contrasts the modulation response of a single ring laser to that of a coupled configuration with = 5 GHz and = 0.27. The single cavity laser exhibits a bandwidth of 2.3 GHz at an excess pumping of − = 8 mA. By increasing the interaction between the two cavities ( = 0.27), the modulation bandwidth boosts to 3.7 GHz. This response can be attributed to the shortening of effective photon lifetime 1/ = 1/ + 2 sin . A comperehansive analysis of this behavior can be found in our previous work [29].
Design and mode analysis
In this study, we implemted our lasers on an InAlGaAs MQW gain medium. These quantum well structures tend to have a more favorable thermal performanc due to their larger conduction band offset (Δ = 0.72Δ ) [30]. The structure of InAlGaAs MQW epitaxial wafer is shown in Error! Reference source not found.. The epitaxial layers were grown on a n-doped InP substrate using metal organic chemical vapor deposition (MOCVD). The intrinsic layers consist of five-period of MQW with = 1508 nm, sandwiched between two 100 nm thick separate confinement heterostructure (SCH) layers. The p-type layers consist of a heavily doped InGaAs cap for metallization, followed by a 1.615 μm Zn-doped InP cladding and a 100 nm InAlAs In designing a microring cavity with a low bending loss and a single transverse mode, the finite element method (FEM) simulation module FemSIM of a commercial photonic simulation software Rsoft is used. The cross-section of the fundamental mode profile | | is shown in Fig.2(b), exhibiting a shift towards the outer sidewall due to bending. The waveguide is 1.65 μm wide with a bending radius of 50 μm. It is deeply etched through quantum wells for a high quality factor Q that is determined by = /2 = 2.8 × 10 , where and are the real and imaginary parts of mode's effective index, respectively. In our design, the coupling between the two ring resonators is achieved based on directional coupling, which comprises a straight waveguides section with a gap of . Coupling in the bending region can be ignore due to the strong confinement and rapid deviation of two modes. The temporal coupling coefficient is related to that in the spatial domain through = /(2 ), where represents the coupling length. The mode profiles of the odd and even supermodes in the cross-section are shown in Fig.3. The spatial coupling coefficient determined by the effective index of two supermodes, is 2.7× 10 m -1 . The temporal coupling coefficient is 4.9 GHz if is chosen as 6 μm. In practice, the imprefcetions in fabrication such as sidewall roughness or etching ripples tend to increase the coupling [28]. Light is extracted from straight bus waveguides which serve as semiconductor optical amplifiers (SOA) to boost the output signal. Due to the spiral symmetry of the ring cavity, two counter-propagating modes' encounters are same. Therefore, one can collect mutually coupled modes from either side of the chip. Figure 4 shows the processing steps involved in the fabrication of the electrically pumped microring lasers. A cleaned sample was first immersed into a 3:1 mixture of HCl:H3PO4 to remove the 150 nm thick InP protective layer. E-beam lithography is then followed by an inductively coupled plasma -reactive ion etching ptocess (ICP-RIE) with CH4/H2/Cl2 (3/7/8 SCCM) in order to transfer the device layout to the epi-wafer. Since an extended dry etching process needed to etch through the MQW region, it will inevitably increase the surface defect density on the walls of the gain medium leading to a high threshold current as a result of strong non-radiative recombination. Consequently, a passivation treatment of the etched wall is required [31]. This is accomplished by first immersing the sample into a solution of H3PO4/H2O2/H2O (1/1/38) for 6 s, followed by soaking in a solution of 20% (NH4)2S that is further diluted in H2O in the ratio of 1:10 for 5 minutes to remove the surface defects and form protective monolayers. Then the sample was immediately dried with N2 gas without any rinsing, and coated with a 100 nm thick layer of SiO2. Next, the top n-contact was first created by depositing Ni/Ge/Au (7/20/200 nm). After benzocyclobutene (BCB) planarization, the ptype metal contact metals Ti/Zn/Au (7/4/500 nm) were evaporated on the sample, followed by a rapid thermal annealing (RTA) at 400C o for 1 min.
Fabrication procedures
Laser characterization and modulation response
The schematic of the test setup for measuring the frequency response is shown in Fig. 5. The probed laser placed on a heat sink is assembled in a micro-positioner. By cleaving the sample, light emission is collected from the facet by a 20X objective lens. A removable mirror is used to direct the laser emission to either an infrared camera or a high-speed photodetector (Newport 818-BB-35). For the alignment, the cleaved facet is illuminated by a 1310 nm laser diode. DC bias current from a laser diode driver (LDX-3500) and RF signal current from Port 1 of the vector network analyzer (Aglient 8720E) are combined by a bias tee, then injected into the laser through a picoprobe. Direct current (DC) and radio frequency (RF) signal from the photodetector can be simultaneously monitored with the voltmeter and the network analyzer, respectively. In our measurements, we first apply a DC current to the straight bus waveguide to prevent the laser output from further attenuation. Since the device is symmetrically cleaved and one mode couples out from each arm, by leaving the other arm unpumped, optical feedback is prevented. With increasing injection current on the ring resonator, a laser emission is observed on the camera, as shown in Fig.6(a). The evolution of spectrum is displayed in Fig.7(a), showing a laser threshold current of around 27mA. When the laser is operated slightly above threshold, multiple longitudinal modes are observed, while single mode emission is obtained under higher pumping levels. The L-I and I-V curves of the two-ring system are shown in Fig.6(b), indicating that the laser threshold current is consistent with that obtained from the spectral evolution. The electrical resistance of the probed laser is determined from the I-V curve as R = R − R = 5 Ω , displaying a good ohmic contact at the metalsemiconductor boundary. The saturated output power is limited by confinement factor, heat, etc. In our design, the waveguides are etched below the MQW region in order to achieve a low bending loss, thus the surrounding dielectric layer (BCB) has a lower thermal conductivity than InP (0.3 W/mK for BCB [32] and 68 W/mK for InP [33]), leading to less effective thermal dissipation. The S21 of picoprobe-laser-detector system is directly measured by the network analyzer then normalized to the free running response in order to compensate the stopband of the network analyzer at low frequency. The frequency bandwidth of the picoprobe and the photodetector are 40 GHz and 15 GHz, respectively. As a result, any roll-off below 15 GHz can be attributed to the ring laser. For verifying the gain-loss contrast can tune the frequency response, ring 1 is biased beyond the threshold and ring 2 is pumped below the threshold. Therefore, is actually adjusted by the biased current of ring 2 (δ decreases with the increasing ). The frequency response of the ring laser with an injection current of I1=35mA (I2=0mA) is shown in Fig.7(b), and it exhibits a 3-dB bandwidth of 2.2 GHz (black curve). With an increase in the injection current of ring 2 while keeping I1=35mA, the modulation bandwidth broadens to 3 GHz at I2=22 mA (red curve) and to 3.6 GHz at I2=23 mA (blue curve). The lasing spectrum for I2=23 mA is shown in Fig.7(a), exhibiting no obvious change with respect to that of a single ring. Further increasing will not broaden the bandwidth but drop it down (orange curve). System RF response is shown in the Fig. 7(b) as a reference. During the measurement, the DC compment of photon current is monitored in order to make sure that the average photon density of ring 1 remains constant. Therefore, By tuning the injection current of the lossy ring, a modulation bandwidth enhancement of 1.63 times was observed without increasing the photon density of ring 1.
Conclusion
In conclusion, we have demonstrated gain-loss contrast (differential gain) as a new knob that can be used to increase the modulation bandwidth of semiconductor microring lasers. An electrically pumped III-V semiconductor laser system, comprised of two coupled deeply-etched ring resonators, was fabricated. It achieved a continuous wave lasing operation at room temperature with a threshold current of 27 mA. By increasing the injection current of the lossy ring while keeping the photon density of the gain ring invariant, an enhancement of modulation bandwidth by up to 1.63 times over that of the single ring has been observed. We believe this new paradigm could help pave the way for a new generation of directly modulated on-chip light sources. Future work will focus on improving the design and fabrication process for lowering the threshold, increasing the output power, and enhancing the modulation speed.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Fig. 1 .
1(a) The schematic diagram of an electrically pumped coupled microring laser system; (b) simulated frequency response of coupled microring lasers with = 0.27.
, , , ), and = ( , 0, 0, 0, 0) can be expressed as = + Δ , = + Δ under the modulation. Here, is the Jacobian of Eq. (2). From Eq. (2) and Eq. (3),
Fig. 2 .
2(a) The top view layout of designed microring lasers; (b) the cross-section mode profile of microring cavity.
Fig. 3 .
3Cross-section mode profiles of even (a) and odd (b) modes of a directional coupler.
Fig. 4 .
4Fabrication processes involved in realizing the electrically pumped microring lasers.
Fig. 5 .
5The schematic of modulation response measurement station.
Fig. 6 .
6(a) Output beam imaged from the facet of the bus wasveguide; (b) LIV curves of microring lasers.
Fig. 7 .
7(a) Spectrum evolution; (b) modulation responses with different pumping ratios between two rings.
CBET 1805200, ECCS 2000538, ECCS 2011171), and US-Israel Binational Science Foundation (BSF; 2016381).
Table 1 .
1Parameters of InAlGaAs MQW microring laserSymbol
Unit
Value
cm 3
4.14 × 10
1
0.01
1
0.4
ps
26
mA
25
cm 2
1.6 × 10
nm
1550
cm/s
9.4 × 10
Table 2 .
2InAlGaAs MQW wafer structureP-InGaAs Layer
150 nm
P-1.5Q Layer
25 nm
P-1.3Q Layer
25 nm
P-InP
1500 nm
P-(1.1Q~1.15Q) Layer
15 nm
P-InP
50 nm
U-In0.52Al0.48As
50 nm
U-GRIN-In0.53AlxGa0.47-xAs
100 nm
QW/Barrier ( )
5-6/9 (1508) nm
U-GRIN-In0.53AlxGa0.47-xAs
100 nm
N-In0.52Al0.48As Layer
140 nm
N-InP Buffer Layer
500 nm
InP Substrate
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|
[] |
[
"SoK: Cryptojacking Malware",
"SoK: Cryptojacking Malware"
] |
[
"Ege Tekiner [email protected] ",
"Abbas Acar ",
"A Selcuk Uluagac [email protected] ",
"Engin Kirda \nNortheastern University\nEmail\n\nEmail\nTOBB University of Economics and Technology\n\n",
"Ali Aydin Selcuk [email protected] ",
"\nFlorida International University\n\n"
] |
[
"Northeastern University\nEmail",
"Email\nTOBB University of Economics and Technology\n",
"Florida International University\n"
] |
[] |
Emerging blockchain and cryptocurrency-based technologies are redefining the way we conduct business in cyberspace. Today, a myriad of blockchain and cryptocurrency systems, applications, and technologies are widely available to companies, end-users, and even malicious actors who want to exploit the computational resources of regular users through cryptojacking malware. Especially with ready-to-use mining scripts easily provided by service providers (e.g., Coinhive) and untraceable cryptocurrencies (e.g., Monero), cryptojacking malware has become an indispensable tool for attackers. Indeed, the banking industry, major commercial websites, government and military servers (e.g., US Dept. of Defense), online video sharing platforms (e.g., Youtube), gaming platforms (e.g., Nintendo), critical infrastructure resources (e.g., routers), and even recently widely popular remote video conferencing/meeting programs (e.g., Zoom during the Covid-19 pandemic) have all been the victims of powerful cryptojacking malware campaigns. Nonetheless, existing detection methods such as browser extensions that protect users with blacklist methods or antivirus programs with different analysis methods can only provide a partial panacea to this emerging cryptojacking issue as the attackers can easily bypass them by using obfuscation techniques or changing their domains or scripts frequently. Therefore, many studies in the literature proposed cryptojacking malware detection methods using various dynamic/behavioral features. However, the literature lacks a systemic study with a deep understanding of the emerging cryptojacking malware and a comprehensive review of studies in the literature. To fill this gap in the literature, in this SoK paper, we present a systematic overview of cryptojacking malware based on the information obtained from the combination of academic research papers, two large cryptojacking datasets of samples, and 45 major attack instances. Finally, we also present lessons learned and new research directions to help the research community in this emerging area.
|
10.1109/eurosp51992.2021.00019
|
[
"https://arxiv.org/pdf/2103.03851v1.pdf"
] | 232,134,856 |
2103.03851
|
31898782a32e46bfe56a6eddd4bffb05d987039f
|
SoK: Cryptojacking Malware
Ege Tekiner [email protected]
Abbas Acar
A Selcuk Uluagac [email protected]
Engin Kirda
Northeastern University
Email
Email
TOBB University of Economics and Technology
Ali Aydin Selcuk [email protected]
Florida International University
SoK: Cryptojacking Malware
This paper submitted to EuroS&P 2021Index Terms-cryptojackingcryptominingmalwarebit- coinblockchainin-browserhost-baseddetection
Emerging blockchain and cryptocurrency-based technologies are redefining the way we conduct business in cyberspace. Today, a myriad of blockchain and cryptocurrency systems, applications, and technologies are widely available to companies, end-users, and even malicious actors who want to exploit the computational resources of regular users through cryptojacking malware. Especially with ready-to-use mining scripts easily provided by service providers (e.g., Coinhive) and untraceable cryptocurrencies (e.g., Monero), cryptojacking malware has become an indispensable tool for attackers. Indeed, the banking industry, major commercial websites, government and military servers (e.g., US Dept. of Defense), online video sharing platforms (e.g., Youtube), gaming platforms (e.g., Nintendo), critical infrastructure resources (e.g., routers), and even recently widely popular remote video conferencing/meeting programs (e.g., Zoom during the Covid-19 pandemic) have all been the victims of powerful cryptojacking malware campaigns. Nonetheless, existing detection methods such as browser extensions that protect users with blacklist methods or antivirus programs with different analysis methods can only provide a partial panacea to this emerging cryptojacking issue as the attackers can easily bypass them by using obfuscation techniques or changing their domains or scripts frequently. Therefore, many studies in the literature proposed cryptojacking malware detection methods using various dynamic/behavioral features. However, the literature lacks a systemic study with a deep understanding of the emerging cryptojacking malware and a comprehensive review of studies in the literature. To fill this gap in the literature, in this SoK paper, we present a systematic overview of cryptojacking malware based on the information obtained from the combination of academic research papers, two large cryptojacking datasets of samples, and 45 major attack instances. Finally, we also present lessons learned and new research directions to help the research community in this emerging area.
Introduction
Since the day Bitcoin was released in 2009, blockchain-based cryptocurrencies have seen an increasing interest beyond specific communities such as banking and commercial entities. It has become so trivial and ubiquitous to conduct business with cryptocurrencies for any end-user as most financial institutions have already started to support them as a valid monetary system. Today, there are more than 2000 cryptocurrencies in existence. Especially in 2017, the interest for cryptocurrencies peaked with a total market value close to $1 trillion [1]. According to a recent Kaspersky report [2], 19% of the world's population have bought some cryptocurrency before. However, buying cryptocurrency is not the only way of investing. Investors can also build mining pools to generate new coins to make a profit. Profitability in mining operations also attracted attackers to this swiftly-emerging ecosystem.
Cryptojacking is the act of using the victim's computational power without consent to mine cryptocurrency. This unauthorized mining operation costs extra electricity consumption and decreases the victim host's computational efficiency dramatically. As a result, the attacker transforms that unauthorized computational power to cryptocurrency. In the literature, the malware used for this purpose is known as cryptojacking. Especially after the emergence of service providers (e.g., Coinhive [3], CryptoLoot [4]) offering ready-to-use implementations of in-browser mining scripts, attackers can easily reach a large number of users through popular websites. In-browser cryptojacking examples. In a major attack, cryptojacking malware was merged with Google's advertisement packages on Youtube [5]. The infected ads package compiled by victims' host performed unauthorized mining as long as victims stayed at the related page. Youtube and similar media content providers are ideal for the attackers because of their relative trustworthiness, popularity, and average time spent on those webpages by the users. In another incident, cryptojacking malware was found in a plugin provided by the UK government [6]. At the time, this plugin was in use by several thousands of governmental and non-governmental webpages. Cryptojacking examples found on critical servers. In addition to cryptojacking malware embedded into webpages, cryptojacking malware has also been found in wellprotected governmental and military servers. The USA Department of Defense discovered cryptojacking malware in their servers during a bug-bounty challenge [7]. The cryptojacking malware found in the DOD servers was created by the famous service provider Coinhive [8] and mined 35.4 Monero coin during its existence. Similarly, another governmental case came up from the Russian Nuclear Weapon Research Center [9]. Several scientists working at this institution were arrested for uploading cryptocurrency miners into the facility servers. Moreover, attackers do not only use the scripts provided by the service providers but also modified the non-malicious, legitimate, open-source cryptominers. For example, a cybersecurity company detected an irregular data transmission to a well-known European-based botnet from the corporate network of an Italian bank [10]. Further investigation identified that this malware was, in fact, a Bitcoin miner. Cryptojacking examples utilizing advanced techniques. There have also been many incidents where the attack-ers used advanced techniques to spread cryptojacking malware. For example, in an incident, a known botnet, Vollgar, attacked all MySQL servers in the world [11] to take over the admin accounts and inject cryptocurrency miners into those servers. Another recent incident was reported for the Zoom video conferencing program [12] during the peak of the Covid-19 pandemic, in which the attacker(s) merged the main Zoom application and cryptojacking malware and published it via different filesharing platforms. In other similar incidents, attackers used gaming platforms such as Steam [13] and game consoles such as Nintendo Switch [14] to embed and distribute cryptojacking malware. Last but not least, in a recent study [15], researchers discovered a firmware exploit in Mikrotik routers that were used to embed cryptomining code into every outgoing web connection, where 1.4 million MikroTik routers were exploited.
Challenges of cryptojacking detection. Given the prevalent emerging nature of the cryptojacking malware, it is vital to detect and prevent unauthorized mining operations from abusing any computing platform's computational resources without the users' consent or permission. However, though it is critical, detecting cryptojacking is challenging because it is different from traditional malware in several ways. First, they abuse their victims' computational power instead of harming or controlling them as in the case of traditional malware. Traditional malware detection and prevention systems are optimized for detecting the harmful behaviors of the malware, but cryptojacking malware only uses computing resources and sends back the calculated hash values to the attacker; so the malware detection systems commonly consider cryptojacking malware as a heavy application that needs high-performance usage. Second, they can also be used or embedded in legitimate websites, which makes them harder to notice because those websites are often trustworthy, and users do not expect any nonconsensual mining on their computers. Third, while in traditional malware attacks, the attacker may ultimately target to exfiltrate sensitive information (i.e., Advanced Persistent Threat (APT)), to make the machine unavailable (i.e., Distributed Denial of Service (DDoS)) or to take control of the victim's machine (i.e., Botnet), in cryptojacking malware attacks, the attacker's goal is to stay stealthy on the system as long as possible since the attack's revenue is directly proportional to the time a cryptojacking malware goes undetected. Therefore, attackers use filtering and obfuscation techniques that make their malware harder for detection systems and harder to be noticed by the users.
Our contributions. Due to the seriousness of this emerging threat and the challenges presented above, many cryptojacking studies have been published before. However, these studies are either proposing a detection or prevention mechanism against cryptojacking malware or analyzing the cryptojacking threat landscape. And, the literature lacks a systemic study covering both different cryptojacking malware types, techniques used by the cryptojacking malware, and a review of the cryptojacking studies in the literature. In this paper, to fill this gap in the literature, we present a systematic overview of cryptojacking malware based on the information obtained from the combination of 42 cryptojacking research papers, ≈ 26K cryptojacking samples with two unique datasets compile, and 45 major attacks instances. Given the widespread usage of cryptojacking, it is important to systematize the cryptojacking malware knowledge for the security community to accelerate further practical defense solutions against this ever-evolving threat. Key takeaways. In addition to the systematization of cryptojacking knowledge and review of the literature, some of the key takeaways from this study are as follows: • In both datasets, we observed that even though inbrowser cryptojacking attacks have not vanished at all, there is a meaningful decrease in the number of inbrowser cryptojacking attacks due to the Coinhive's shutdown concurrent with the actions taken by the platforms. The security reports also observe the same drop []. On the other hand, there is some increase in the number of cryptojacking attacks targeting more powerful platforms such as cloud servers, Docker engines, IoT devices on large-scale Kubernetes clusters []. To hijack and gain initial access [] to spread the cryptojacking malware, the attackers utilize: 1) hardware vulnerabilities [16], 2) recent CVEs [17], 3) poorly configured IoT devices [18], 4) Docker engines and Kubernetes clusters [19] with poor security, 5) popular DDoS botnets for the side-profit [18]. Despite this change in cryptojacking malware's behavior, this new trend of cryptojacking malware has not been investigated in detail by researchers. • We identified several issues in the studies proposing cryptojacking detection mechanisms in literature. First, we found that as the websites containing cryptomining scripts are updated frequently, it is important for the proposed detection studies to report the dataset dates, which is not very common in the studies in the literature. Second, it is important to report if the proposed detection is online or offline, which is missing in most studies. Third, we also note that the studies in the literature do not measure the overhead on the user side of the proposed solutions, which is critical, especially for browserbased solutions. • We see that although cryptomining could be an alternative funding mechanism for legitimate website owners such as publishers or non-profit organizations, this usage with the in-browser cryptomining has diminished due to the keyword-based automatic detection systems. Other surveys. In the literature, a number of blockchain or Bitcoin-related surveys have been published. However, these surveys only focus on consensus protocols and mining strategies in blockchain [20]- [23], challenges, security and privacy issues of Bitcoin and blockchain technology [24]- [29], and the implementation of blockchain in different industries [30] such as IoT [31], [32]. The closest work to ours is Jayasinghe et al. [33], where the authors only present a survey of attack instances of cryptojacking targeting cloud infrastructure. Hence, this SoK paper is the most comprehensive work focusing on cryptojacking malware made with the observations and analysis of two large datasets.
Organization. The rest of this systematization paper is organized as follows: In Section 2, we provide the necessary background information on blockchain and cryptocurrency mining. Then, in Section 3, we explain the methodology we used in this paper. After that, in Section 4, we categorize cryptojacking malware types and give their lifecycles. In Section 5, we give broad information about the source of cryptojacking malware, infection methods, victim platform types, target currencies, and finally, evasion and obfuscation techniques used by the cryptojacking malware. Section 6 presents an overview of the cryptojacking-related studies and their salient features in the literature. Finally, in Section 7, we summarize the lessons learned and present some research directions in the domain and conclude the paper in Section 8.
Background
In this section, we briefly explain the blockchain concept and cryptocurrency mining process in blockchain networks. Note that cryptocurrency mining is a legitimate operation, and it can be used for profit. To see how cryptojacking malware exploits this process, we first explain how this process works.
Blockchain
Blockchain is a distributed digital ledger technology storing the peer-to-peer (P2P) transactions conducted by the parties in the network in an immutable way. Blockchain structure consists of a chain of blocks. As an example, in Bitcoin [34], each block has two parts: block header and transactions. A block header consists of the following information: 1) Hash of the previous block, 2) Version, 3) Timestamp, 4) Difficulty target, 5) Nonce, and 6) The root of a Merkle tree. By inclusion of the hash of the previous block, every block is mathematically bound to the previous one. This binding makes it impossible to change data from any block in the chain. On the other hand, the second part of each block includes a set of individually confirmed transactions.
Cryptocurrency Mining
The immutability of a blockchain is provided by a consensus mechanism, which is commonly realized by a "Proof of Work" (PoW) protocol. The immutability of each block and the immutability of the entire blockchain are preserved thanks to the chain of block structure. In PoW, some nodes in the network solve a hash puzzle to find a unique hash value and broadcast it to all other nodes in the network. The first node broadcasting the valid hash value is rewarded with a block reward and collects transaction fees. A valid hash value is verified according to a difficulty target, i.e., if it satisfies the difficulty target, it is accepted by all other nodes, and the node that found the valid hash value is rewarded. Different PoW implementations usually have different methods for the difficulty target.
The miners try to find a valid hash value by trial-anderror by incrementing the nonce value for every trial. Once a valid hash value is found, the entire block is broadcast to the network, and the block is added to the end of the last block. This process is known as cryptocurrency mining (i.e., cryptomining), and it is the only way to create new cryptocurrencies. The chance of finding of valid hash value by a miner is directly proportional to the miner's hash power, which is related to the computational power of the underlying hardware. However, more hardware also increases electricity consumption. Therefore, attackers have an incentive to find new ways of increasing computational power without increasing their own electricity consumption.
Following the invention of Bitcoin, many other alternative cryptocurrencies (i.e., altcoins) have emerged and are still emerging. These new cryptocurrencies either claim to address some issues in Bitcoin (i.e., scalability, privacy) or offer new applications (i.e., smart contracts [35]).
In the early days of Bitcoin, the mining was performed with the ordinary Central Processing Unit (CPU), and the users could easily utilize their regular CPUs for Bitcoin mining. Over time, Graphical Processing Unit (GPU)-based miners gained significant advantages over CPU miners as GPUs were specifically designed for high computational performance for heavy applications. Later, Field Programmable Gate Array (FPGA) have changed the cryptocurrency mining landscape as they were customizable hardware, and provided significantly more profit than the CPU or GPU-based mining. Finally, the use of the Application-Specific Integrated Circuit (ASIC) based mining has recently dominated the mining industry as they are specially manufactured and configured for cryptocurrency mining.
The alternative cryptocurrencies also used different hash functions in their blockchain structure, which led to variances in the mining process. For example, Monero [36] uses the CryptoNight algorithm as the hash function. CryptoNight is specifically designed for CPU and GPU mining. It uses L3 caches to prevent ASIC miners. With the use of RandomX [36] algorithm, Monero blockchain fully eliminated the ASIC miners and increased the advantage of the CPUs significantly. This feature makes Monero the only major cryptocurrency platform that was designed specifically to favor CPU mining to increase its spread. Moreover, Monero is also known as a private cryptocurrency, and it provides untraceability and unlinkability features through mixers and ring signatures. Monero's both ASIC-miner preventing characteristics and privacy features make it desirable for attackers.
SoK Methodology
In this section, we explain the sources of information and the methodology used throughout this SoK paper. Particularly, we benefit from the papers, recent cryptojacking samples we collected, and publicly known major attack instances.
Papers
Cryptomining and cryptojacking have recently become popular topics among researchers after the price surge of cryptocurrencies and the release of Coinhive cryptomining script in 2017. For our work, we scanned the top computer security conferences (e.g., USENIX) and journals (e.g., IEEE TIFS) given in [37] as well as the digital libraries (e.g., IEEEXplore, ACM DL) with the keywords such cryptojacking, bitcoin, blockchain, etc. and a variety of combinations of these keywords. In total, we found 43 cryptojacking-related papers in the literature. While one of the papers [33] is a survey paper, the rest are focusing on two separate topics: 1) Cryptojacking detection papers, 2) Cryptojacking analysis papers. We found that there are 15 cryptojacking analysis papers, while there are 27 cryptojacking detection papers in the literature. We further present a literature review of these 42 studies in Section 6. Figure 5 in Appendix A shows the distribution of research cryptojacking-related research papers per year. As seen in the figure, there is an increasing effort in the academia in the last three years with many research papers. Therefore, there is a need for a systematization of this knowledge for cultivating better solutions.
Samples
Each research paper in the literature focuses only on one aspect of the cryptojacking malware. For a comprehensive understanding of the cryptojacking malware, we also benefited from the real cryptojacking malware samples. For this purpose, we created two datasets: 1) VirusTotal (VT) Dataset and 2) PublicWWW Dataset. Particularly, we used these two datasets for the following purposes and in the noted sections: Moreover, we give a more detailed explanation and perform distribution analysis of these datasets in Appendix. Finally, we also published our dataset 1 to further accelerate the research in this field.
Major Attack Instances
Our third source of information is the major attack instances that appeared on the security reports released by the security companies such as Kaspersky, Trend Micro, Palo Alto Network, IBM, and others, as well as the major security instances that appeared on the news. The major attack instances that appeared on the news may be used to identify unique and interesting cases, while the security reports may shed light on the trends due to the real-time and large-scale reach of the security companies. Particularly, we used these instances in Section 1 for motivational purposes; in Section 5 to find out different techniques used by cryptojacking malware; in Section 7 in order to find out potential new trends in the cryptojacking malware attacks. Since the collection of these resources may be valuable to other researchers and due to the space limitation here, we also release them in a detailed and organized way together with the blacklists and service providers' documentation links in our dataset link 1 . Table 10 in Appendix A shows the yearly distribution of the attack instances we used in this paper. 1. https://github.com/sokcryptojacking/SoK
Cryptojacking Malware Types
Cryptojacking malware, also known as cryptocurrency mining malware, compromises the computational resources of the victim's device (i.e., computers, mobile devices) without the authorization of its user to mine cryptocurrencies and receive rewards. A cryptojacking malware's lifecycle consists of three main phases: 1) script preparation, 2) script injection, and 3) the attack. The script preparation and attack phases are the same for all cryptojacking malware types. In contrast, the script injection phase is conducted either by injecting the malicious script into the websites or locally embedding the malware into other applications. Based on this, we classify the cryptojacking malware into two categories: 1) In-browser cryptojacking and 2) Host-based cryptojacking. In the following sub-sections, we explain the lifecycle of both in-browser and host-based cryptojacking malware.
Type-I: In-browser Cryptojacking
The development of web technologies such as JavaScript (JS) and WebAssembly (Wasm) enabled interactive web content, which can access the several computational resources (e.g., CPU) of the victim's device (e.g., computer or mobile device). In-browser cryptojacking malware uses these web technologies to create unauthorized access to the victim's system for cryptocurrency mining via web page interactions on the victim's CPU. Figure 1 shows the script preparation and injection phases of in-browser cryptojacking malware. The script owner 2 first registers (Step 1) and receives its service credentials and ready-to-use mining scripts from the service provider (Step 2). The service provider separates the mining tasks among its users and collects all the revenue from the mining pool later to be shared among its users. After receiving the service credentials, the script owner injects the malicious cryptojacking script into the website's HTML source code (Step 3). We explain this and other cryptojacking infection methods in Section 5.2 in detail.
In the attack phase, as shown in Figure 2, victims first reache a website source code from their devices (Step 1,2). The web browser loads the website and automatically calls the cryptojacking mining script (Step 3). Once the script is executed, it requests a mining task from the service provider (Step 4). The service provider transfers the task request to the mining pool (Step 5). Then, the mining pool assigns the mining task (Step 6). The service provider returns the task to the mining script (Step 7). The mining script returns this new mining assignment to the victim's computer (Step 8), and the victim's device starts the mining process (Step 9). As long as the mining script and 2. We call it script owner rather than an attacker because the script can also be used for legitimate purposes. service provider remain online, the script continues the mining process on the victim's computer (Step 9) and then returns the mining results to the service provider (Step 10) directly. The service provider collects all the data from different sources and sends the results to the mining pool (Step 11). Finally, the mining pool sends the reward back to the service provider in the form of a mined currency (Step 12). The script owner receives its share from the service providers using its service credentials after the service provider cuts its service fee. In this ecosystem, the attackers uses the CPU power of their victims, and the victims donot receive any payment nor benefit from any other entity.
Type-II: Host-based Cryptojacking
Host-based cryptojacking is a silent malware that attackers employ to access the victim host's resources and to make it a zombie computer for the malware owner. Compared to in-browser cryptojacking malware, hostbased malware does not access the victim's computation power through a web script; instead, they need to be installed on the host system. Therefore, they are generally delivered to the host system through methods such as embedded into third-party applications [12], [38], using vulnerabilities [17], or social engineering techniques [39], or as a payload in the drive-by-download technique [40]. We explain these methods in more detail in Section 5.2. Figure 3 shows the lifecycle of a host-based cryptojacking malware. The script preparation phase starts with the creation of unauthorized cryptocurrency mining malware (1). Then, the attacker merges this malware with a legitimate application to trick the victim (2). After the malware preparation, the malware injection process starts with uploading this malicious application to online datasharing platforms (e.g., torrent, public clouds) (3). When the victim downloads any of the infected applications and installs them on their host machines (e.g.,Personal Computer, IoT device, Server)(4), the malware injection phase of the lifecycle is completed.
In the attack phase, the host-based cryptojacking malware is connected to the mining pool via web socket or API and receives the hash puzzle tasks to calculate hash values (5). The calculated hash values are sent back to the mining pool (6). Finally, the attacker receives all of the revenue without any energy consumption (7) and not sharing anything with the victim.
After receiving all its revenue in the form of cryptocurrency from the service provider, the attacker has three options to use its revenue: 1) Converting to fiat currency via exchanges or p2p transactions, 2) Using it as a cryptocurrency for a service [41], or 3) Using cryptocurrency mixing services [42], [43] to cover its traces. Further endto-end analysis of the cryptojacking economy/payments is
Receive
Revenue(7)
Bitcoin Monero
Malware and Application Merging(2)
Mining Pool out of this study's scope, and similar studies can be found in the ransomware domain [44]- [47].
API / Web Socket Communication
Cryptojacking Malware Techniques
In this section, we explain the techniques used by cryptojacking malware. Particularly, we articulate on the following:
• Source of cryptojacking malware • Infection methods • Victim platform types • Target cryptocurrencies • Evasion and obfuscation techniques
Source of Cryptojacking Malware
This sub-section explains whom the scripts are created by and how they are distributed to attackers.
Service Providers.
The service providers are the leading creators and distributors of cryptojacking scripts. The service providers give every user a unique ID to distinguish them in terms of the hash power. The service provider generates the script for the user regardless of the user, is malicious or not. All the user needs to do is copy and paste the script to create a malicious sample for the attack.
Coinhive [8] was the first service provider to offer a ready-to-use in-browser mining script in 2017 to create an alternative income for web site and content owners. Even though the initial idea of Coinhive was to provide an alternative revenue to webpage owners, it rapidly became popular among attackers. We also observe this for the samples in the VT dataset. While there are only 272 maliciously labeled cryptominer samples uploaded to VT before 2018, there are 17102 malicious cryptominer samples uploaded in 2018 as shown in Figure 6 in Appendix. This huge increase indicates the effect of Coinhive on the popularity of cryptojacking malware. During the operation of Coinhive, they were holding a significant share of the total hash rate of Monero. After the sharp decrease in Monero's price [48], Coinhive was shut down by their owners in March 2019 due to the business' being no longer profitable.
Some of the alternative service providers which had continued/continuing their operations are Authedmine [3], Browsermine [49], Coinhave [50], Coinimp [51], Coin nebula [52], Cryptoloot [4], DeepMiner [53], JSECoin [54], Monerise [55], Nerohut [56], Webmine [57], Web-minerPool [58], and Webminepool [59]. Some of these service providers also came up with several new functionalities, such as offering a user notification method or a GUI for the user to adjust the cryptomining parameters. Note that, we also verified these service providers using the samples in the PublicWWW dataset. In order to find the corresponding service providers of each sample, we performed a keyword search on the HTML source code of all samples. We found that 5328 samples used one of these 14 aforementioned service providers, while 941 samples with unknown service providers. Moreover, we also found out that 144 samples were using scripts from multiple service providers in their source codes. More details on the PublicWWW dataset can be found in Appendix.
Cryptominer
Software. Blockchain networks rely on several network protocols and cryptographic authentication methods. Miners must be part of these protocols and follow the rules provided and developed by the communities. PoW-based cryptocurrencies also have specific rules for their blockchain networks. Due to blockchain technology's public and open nature, the source code of these miners are published by the communities via code sharing and communal development platforms. Attackers can easily obtain and modify these miners and adopt them to perform mining inside their victims' host machines. Moreover, there are also several plug-and-play style mining applications provided by several mining pools. Attackers are also modifying these applications for cryptojacking. For example, XMRig [60] is a legitimate high-performance Monero miner implementation, and it is open-source. Its signature is found in several highly impactful attacks affecting millions of end devices around the world [61], [62], which are also reported by Palo Alto Networks and IBM. Moreover, we also found 139 unique samples that were labeled with the signature of "xmrig" in our VT dataset.
Infection Methods
In this section, we explain the infection methods used by cryptojacking malware in detail.
Website Owners.
Website owners, who have admin access to the website's servers, may employ inbrowser mining scripts to gain extra revenue or provide in exchange of an alternative option to premium content they provide. Only with this method, webpage owners may receive the revenue of the script in their webpage. While some website owners inform their visitors about the cryptomining script they employ, some others do not inform their visitors, and this behavior can be considered as crime [63] in several countries.
Compromised
Websites. Attackers may inject their cryptojacking malware into random web pages that have several vulnerabilities. Indeed the name cryptojacking itself is the combination of "cryptomining" and "hijacking." Ruth et al. [64] state that ten different users created 85% of all Coinhive scripts they found. The owners of these webpages do not have any information about these scripts; additionally, they do not profit from them.
Several works claim that the attackers generally use the same ID for all the infected web pages, making them more traceable. For example, the authors in [65] reveals the cryptojacking campaigns through this method and discover that most of these campaigns utilize the vulnerabilities such as remote code execution vulnerabilities. When we investigated the common instances related to this domain, one security company found cryptojacking malware inside of the Indian government webpages [66], which affect all ap.gov.in domains and sub-domains.
Malicious
Ads. Some attackers embed their cryptojacking malware into JavaScript-based ads and distribute them via mining scripts. With this method, the attackers can reach random users without any extra effort. To make this attack, they do not need to infect any webpage or application. YouTube [5] and Google ad [67] services were also infected and the users of these websites and their services became the victims of the cryptojacking attacks. The attackers successfully mined Monero with their visitors. The attackers successfully mined Monero with their visitors. The advantage of this method is that it allows attackers to reach a large number of visitors when it is embedded into popular websites without getting access to the website's servers.
Malicious Browser Extensions.
Browser extensions can also reach the computer's CPU sources and act like cryptojacking malware located into a webpage. These extensions have a major distinctive difference; they can stay online and perform mining as long as the infected browser remains open independent from the websites accessed by the victim. However, major browser operators like Google announced that they would ban all the cryptomining extensions on their platform regardless of their intention as it is mostly being abused in practice [68].
5.2.5.
Third-party Software. Merging malware with any market application and publishing it via several sharing platforms is a well-known method among the attackers to spread the malware. Attackers modify the cryptominer software to run cryptojacking in the background and merge it with legitimate applications. The attackers tend to use computation-intensive applications (e.g., animation applications, games with high hardware needs, engineering programs) because the use of those applications means that the victims' system has computationally powerful hardware and the application that host-based cryptojacking malware embedded, have access permission to the needed hardware components of the victim's host system.
Several major instances have already happened, such as, one attacker merged Zoom [12] video calling application with a regular bitcoin miner and distributed it via several sharing platforms. In another attack, the attackers used a popular video game Fortnite to spread the virus [69] to mine Bitcoin. Unlike the in-browser mining, which became popular in 2017, we found the attack instances using this method even in 2013, where the Bitcoin mining script found as part of the game's code itself [70].
Exploited Vulnerabilities.
In several cases, attackers exploit several zero-day vulnerabilities that they found in hardware and software. Attackers inject their mining malware into several devices and make them mine cryptocurrency. There are several important instances happened in the last several years. The most remarkable example directly affects 1.4 Million Mikrotik [15] routers globally, and a vulnerability in the hardware operating system causes this instance. The researchers claim that a major percentage of Remote Code Execution (RCE) attacks [71] aims to locate mining scripts inside the host systems.
5.2.7. Social Engineering Techniques. Social engineering is a commonly used technique among malware attackers to bypass security practices. Similarly, attackers also use social engineering attacks to manipulate human psychology and navigate the victims' access or install malicious software on their computers. The researchers have observed that attackers are still using old techniques such as social engineering to install cryptojacking malware on their victims' computers [72].
Drive-by Download.
A drive-by download is another technique used by malware attackers to deliver and install malicious files to victims' devices without their knowledge. Victims may face this attack while visiting a web page, opening a pop-up window, or checking an email attachment. In one case [40], the attackers used this method to inject their cryptojacking malware into their victims' devices. They exploited shell execution vulnerability to download their cryptojacking malware to victims' computers directly.
Victim Platform Types
5.3.1. Browser. Browsers are the most commonly used victim platforms as the attackers do not need to deliver any malicious payload to the victim to use the computational resources of the victim. In other words, when the victim reaches the infected webpage, the malware automatically starts mining and do not leave any data behind. The second significant advantage of the browser environment is, thanks to service providers, ready-to-use mining scripts can be applied to any webpage very easily and quickly. The studies in the literature that we also present in Section 6 mostly focus on in-browser cryptojacking. However, the attackers can access only the CPUs of the victims through the browsers, which makes them infeasible for the currencies allowing ASIC miners such as Bitcoin. Therefore, cryptojacking malware samples utilizing browsers mostly mine Monero or other cryptocurrencies, which allow cryptomining by personal computers on non-ASIC CPU architectures.
Personal Computers.
Personal computers are generally designed to allow end-users to perform their daily tasks. Personal computers are recently modified to overcome high-level computations to allow their users to use heavy-computation applications (e.g., video-games, video rendering applications). Attackers targeting personal computers aim to reach many victims because a limited number of victims would not be profitable. In-browser cryptojacking embedded into popular websites is ideal for this type of cryptojacking attack. In addition, they can also instantiate such an attack through large-scale campaigns. For example, in [73], Cisco researchers document their findings of a two-year campaign delivering XMRig in their payload. They also observed that the malware "makes a minimal effort to hide their actions" and posting the malware "on online forms and social media" to increase the victim pool.
5.3.3.
On-premise Server. On-premise (i.e., in-house) servers are the servers where the data is stored and protected on-site. It is preferred by highly critical organizations such as governmental organizations as it offers greater security and full control over the hardware and data. However, on-premise servers are also another victim platform type attacked by the host-based cryptojacking malware samples. Compared to personal computers, onpremise servers are more computationally powerful and host numerous services accessed by many connections. This allows attackers to the broader attack surface. Still, the attackers have to find a way to deliver and install the cryptomining script on the on-premise server to access this platform. In several instances, the attackers used system vulnerabilities [10], third party infected software [6], and several social engineering methods [72] to install cryptojacking malware to the victims' on-premise server.
Cloud Server.
Cryptojacking malware also exploits cloud resources to mine cryptocurrencies. Cloud-based cryptojacking attack is a fast-spreading problem in the last two years, where it became popular, especially after the shutdown of the Coinhive when the attackers were looking for new platforms to infect. Attackers target several vulnerabilities to hijack victims' cloud servers and locate cryptocurrency miners into their systems. Clouds servers, especially Infrastructure-as-a-service platforms such as Amazon Web Services (AWS), are being targeted by the attackers because of their:
• Virtually infinite resources, • Large attack surface due to server structure, • Malware spreading capabilities, • Reliable Internet connection, • Longer mining/profit period due to host-based capabilities Several instances of this type of cryptojacking malware have been found on cloud servers [17], [19], [74]- [77]. In these attacks, attackers used different techniques to hijack the cloud server to inject cryptojacking malware. For example, in their 2020 annual report, Check Point Research [74] observed that attackers integrate the cryptominer to the popular DDoS botnets such as KingMiner targeting Linux and Windows servers for side-profits. In another attacker instance [19], the researchers found an open directory containing malicious files. Further analysis revealed that the file contains a DDoS bot targeting open Docker daemon ports of Docket servers and ultimately installing and running the cryptojacking malware after the execution of its infection chain. In a similar attack instance [17], the researchers noted a cryptojacking malware delivered using a CVE exploitation targeting WebLogic servers. Tesla-owned Amazon [75] and the clients of Azure Kubernetes clusters [77] were exposed cryptojacking attacks due to poorly configured cloud servers. Indeed, Jayasinghe et al. [33] showed that the count of cryptojacking malware targeting cloud-based infrastructure is increasing every year and affects more prominent domains such as enterprises.
IoT Botnet.
IoT devices generally have small processing powers to perform basic tasks. It is being expected that there will be more than 21.5 billion IoT devices connected to the internet [78] by 2025. Attackers aim to create botnets with the collaboration of thousands of these IoT devices and perform several attacks such as DDoS due to their small processor, limited hardware, low-level security, and weak credentials, which was also exploited in the example of Mirai botnet's DDoS attack [79]. Later, IBM researchers also found that the modified version of the Mirai Botnet also started to mine Bitcoin [18]. Bartino et al. [80] states that there are several worms in IoT devices that hijacked them for mining purposes, and Ahmad et al. [81] proposes a lightweight IoT cryptojacking detection system to detect any cryptojacking attack that focuses on IoT devices. 5.3.6. Mobile. Cryptojacking malware samples targeting mobile devices inject cryptojacking script into their application and list the application in the application markets. Like every other type of cryptojacking attack, the mobilebased cryptojacking samples also have seen a great increase in the number of attacks. Because of this, both Google [82] and Apple [83] removed the cryptomining applications from their platforms. However, they still exist in alternative markets [84]. The study by Dashevskyi et al. [84] focuses on Android-based cryptojacking malware.
Moreover, mobile devices are generally not considered powerful enough for cryptocurrency mining because they generally use more restricted hardware and optimized operating systems (e.g., iOS and Android). Besides, the cryptocurrency mining process consumes extra battery and processing power, which may cause hardware problems such as overheating and apps to freeze or crash on mobile devices. Due to these reasons, cryptojacking attacks on mobile devices are not preferred by attackers, and they generally apply a mobile filtering method to opt-out mobile devices. Listing 1 is a recent cryptojacking sample with the mobile device filtering method found in a sample in our dataset. In line 4, the script automatically calls a mobile device detection function and starts the cryptocurrency mining process only if it is not a mobile device.
Target Cryptocurrencies
In this section, we give brief information about the most preferred cryptocurrencies by the attackers. 5.4.1. Monero. Monero has several advantages over other cryptocurrencies, making it favorable to attackers. First of all, Monero successfully implements and modifies the RandomX mining algorithm and CryptoNight hashing algorithm to prevent ASIC miners and give a competitive advantage to the CPU miners over GPUs via L3 caches [85]. The Monero community aims to keep their network decentralized and allows even small miners to mine Monero. As in-browser cryptojacking malware can only access the CPUs of the personal computers through the browsers, Monero is ideal as a target cryptocurrency instead of other cryptocurrencies that are mined dominantly by other computationally more powerful ASIC and GPU miners. Second, Monero provides anonymity features through cryptographic ring signatures [86], [87], which makes the attackers untraceable. Thanks to these features of the Monero, attackers tend to mine Monero with their in-browser cryptojacking malware.
When we analyze the samples' cryptomining scripts in the PublicWWW dataset and their service providers' documentation, we found that all eleven service providers except Browsermine, CoinNebula, JSEcoin either use Monero or have the option to choose Monero in their scripts as a target cryptocurrency. This shows to 91% of the samples in the PublicWWW dataset use Monero to mine.
Bitcoin.
In recent years, Bitcoin mining has seen enormous attention, which led to a dramatic increase in the difficulty target. ASIC and FPGA miners are the main reason behind this dramatic increase because the mining structure of the Bitcoin allows to build and use of specified mining hardware which is much more powerful and profitable than the CPUs and GPUs. The increase in difficulty target and disadvantages of CPU made the CPU mining infeasible and not profitable. Therefore, attackers who perform in-browser cryptojacking attack donot prefer Bitcoin mining. We also see that none of the service providers of the in-browser cryptojacking samples in our PublicWWW dataset supports Bitcoin mining. However, host-based cryptojacking malware can reach all the components of the victims' computer system and make Bitcoin mining on GPU and other high-performance computational resources of the computers. We also observe this in our VT dataset. We performed a keyword search for "bitcoin" on the AV labels of 20200 samples of both in-browser and host-based cryptojacking malware. We found that 7111 of 20200 samples are marked with label containing the keyword "bitcoin". Even though this does not show that those samples are absolutely using bitcoin as a target cryptocurrency, but it is a potential indicator for the host-based cryptojacking samples mining Bitcoin based on the assumption of AV vendors are labeling the correct currency for the AV labels.
Other Cryptocurrencies.
Cryptojacking is attractive for attackers as cryptomining can be parallelized among many victims. Therefore, it is possible for cryptocurrencies to allow distributed cryptomining. Both Monero and Bitcoin use PoW as a consensus method. However, instead of PoW, other cryptocurrencies utilize different consensus models such as Proof of Stake [88], and Proof of Masternode [89]. Most of these new consensus models do not depend on distributed power-based mining algorithms; therefore, cryptojacking is not an option for those currencies. For the cryptocurrencies that can be mined distributively [90], the mining pools provide collective mining services to their participants. Other cryptocurrencies that are preferred by attackers are Bitcoin Cash [91], Litecoin [92], and Ethereum [93].
There are also several cryptocurrencies developed specifically for in-browser cryptomining activities. JSEcoin [54] is an example of them and offers also transparency. Other cryptocurrencies created for this purpose are BrowsermineCoin [49], Uplexa [94], Sumocoin [95], and Electroneum [96].
Detection and Prevention Methods
In the traditional malware detection literature, there are two main analysis methods: 1) static [97] and 2) dynamic [98]. Both analysis methods have several pros and cons in terms of accuracy and usability.
• Static Analysis: Static analysis is a widely used method to examine the application without executing it. Static analysis tools generally seek specific keywords, malware signatures, and hash values. In the cryptojacking domain, mining-blocking browser extensions [99], [100] workin this way, i.e., any domain given in the pre-determined blacklist is blocked. However, due to the fix, pre-configured nature of the static detection methods, these implementations are usually easy to circumvent. • Dynamic Analysis: In dynamic analysis, the malware sample is executed in a controlled environment, and its behavioral features are recorded for further analysis and detection. Malware analyzers generally use automated or non-automated sandboxes [98] to run the code and observe the malware's behavior. In the literature, 24 machine learning-based proposed detection mechanisms use dynamic analysis to detect cryptojacking malware. These studies use various datasets, features, classification algorithms and some of them works for both in-browser and host-based cryptojacking malware. We explain these studies in Section 6.1. As the execution of in-browser cryptojacking malware depends on running the JavaScript code, another way to stop it is to disable the use of JavaScript, but this would also decrease the usability of the browser significantly. Finally, there are antivirus programs with the cryptojacking detection capability [101], [102]. However, their detection algorithms are proprietary.
Evasion and Obfuscation Techniques
The purpose of the cryptojacking malware is to exploit the resources of the victim as long as possible; therefore, staying on the system without being detected is of paramount importance. For this purpose, they utilize several obfuscation methods. 5.6.1. CPU Limiting. High CPU utilization is still the most important common point of all kinds of cryptojacking malware because CPU usage is the main requirement of the cryptocurrency mining process. Therefore, CPU limiting is a highly preferred method by the attackers to obfuscate the mining script. With this method, the script owners can bypass the high CPU usage-based detection systems and avoid being put on the blacklist. Moreover, the CPU limiting is also used by legitimate website owners performing cryptocurrency mining as an alternative revenue because it provides a better user experience. Line 3 in Listing 1 shows an example of a CPU limiting method used by a cryptojacking malware, where the attacker sets throttle to 0.8, e.g., the attacker wants to use only 20% of the CPU load for cryptocurrency mining. In our Pub-licWWW dataset, we searched for the keyword "throttle: 0.9" and we found that 1384 samples out of all 6269 cryptomining scripts set the throttle to 90%, which shows that CPU is limiting is a very common practice among the in-browser cryptomining scripts. 5.6.2. Hidden Library Calls. Library calling [103] is a well-known technique used by programmers to make the code more efficient, systematic, and readable. However, it can also be used by the attackers to obfuscate their scripts. Particularly, in order to hide the mining code from the detection methods, the attackers create new scripts that do not have specific keywords. The malicious part of the script is moved to an external library, which is called during the script's execution, and only the code snippet to call this library is included in the main code.
Code Encoding.
Encoding the malware source code with several encoding algorithms provides invisibility against keyword-based static analysis detection methods such as blacklists. This method transforms the text data into another form, such as Base64, and after this process, the data can only be read by the computers. Some examples of this we found in our PublicWWW dataset are the cryptomining scripts provided by the service providers Authedmine [8] and Cryptoloot [4].
Binary
Obfuscation. Similar to code encoding technique, binary obfuscation is a practice among malware authors to hide malicious code from standard string matching algorithms and make it harder to recover by the sandboxes and other dynamic malware detection methods. However, they differ in the cryptojacking type that is used to hiding, i.e., binary obfuscation is used by the host-based cryptojacking malware while code encoding is used by the in-browser cryptojacking malware. For binary obfuscation, attackers generally use well-known packers such as UPX. The authors of [104] observe that 30% of 1.2M binary cryptojacking malware samples are obfuscated, which shows that it is a common practice among the cryptojacking malware attackers, too.
Literature Review
The surge of cryptojacking malware, especially after 2017, also drew the attention of the academia and resulted in many publications. We found these studies focus on three topics: 1) Cryptojacking detection studies, 2) Cryptojacking prevention studies, and 3) Cryptojacking analysis studies. Among 42 academic research papers, we found that 15 of them focus on the experimental analysis of the cryptojacking dataset. At the same time, 3 of them proposes a method for the detection and prevention of cryptojacking malware together, and 24 of them proposes only a method for the detection of the cryptojacking malware. In the next sub-sections, we give a review of these studies.
Cryptojacking Detection Studies
In this section, we survey the cryptojacking malware detection studies. Table 1 shows the list of the proposed cryptojacking detection mechanisms in the literature. The following sub-section gives a detailed overview of the dataset, platform, analysis method, features, and classifiers used in these detection mechanisms.
6.1.1. Dataset. A dataset is generally used to evaluate the effectiveness of the proposed detection method. Several datasets are commonly used in the cryptojacking malware detection literature. The most common one is Alexa top webpages [106], [109], [110], [112], [113], [115]. Alexa sorts the most visited websites on the Internet; however, it does not provide the source code for these websites. Therefore, these studies also used Chrome Debugging Protocol to instrument the browser and collect the necessary information from the websites, except the study [115], which works with a limited number (500) of websites. Moreover, the study in [109] also used known and frequently updated blacklists [99], [100], [128] to build a ground truth for their training dataset, and then they performed an analysis using Alexa top 1M websites. In addition to the Alexa top websites, the study in [90] used a cryptojacking dataset obtained from VirusTotal. They collected 1500 active Windows Portable Executable (PE32) cryptocurrency mining malware samples registered in 2018 and used the Cuckoo Sandbox [129] to obtain detailed behavioral reports on those samples. Furthermore, the studies in [107], [108], [111], [124] performed their analysis by installing the legitimate mining scripts, and the studies in [110], [114] manually injected miners to the websites to test their detection mechanisms.
Platform.
Most of the cryptojacking detection mechanisms in the literature [105], [106], [108]- [112], [114], [115], [124], [130] are proposed for the detection of in-browser cryptojacking malware. There are only a few studies [90], [121] proposed for host-based cryptojacking malware. In addition, Conti et al. [124], propose a hardware-level detection mechanism, which can be used to detect both host-based and in-browser cryptojacking malware.
6.1.3. Analysis Features. As can be seen from Table 1, in the cryptojacking domain, the majority of the proposed detection methods are using dynamic analysis. The main reason for this is that mining scripts use a set of known instructions, and they follow and repeat predefined mining steps. For example, miners use cryptographic hash libraries and increment the value of a static variable (i.e., nonce) repeatedly or connect to some known service providers to continue to upload some output results and receive new tasks. These typical behaviors of the cryptojacking malware create a pattern and make them detectable by dynamic analysis. In the literature, only a few studies use static features such as opcodes [90] and WebAssembly (Wasm) instructions [105]. WebAssembly [131] is a low-level instruction format that allows programs to run closer to the machine-level language and provide higher performance via stack-based virtual machines [132]. This low-level instruction model lets the WebAssembly run the codes more efficiently, and this feature provides more profit because the cryptojacking script eliminates most of the delay caused by the code execution process. All major browsers in the market currently support WebAssembly.
Opcodes are machine language instructions that specify the operations to be performed and are used by system calls. The proposed detection system in [90] uses opcodes for static analysis, where opcodes are extracted using IDA Pro. In the cryptojacking example, opcodes focus on requests between mining scripts and the operating system's kernel. With this method, they achieve 95% accuracy with the Random Forest classifier.
On the other hand, many detection mechanisms have been proposed [106], [108]- [112], [114], [124], [130] using dynamics features. The most commonly used dynamic features in these studies are as follows: • CPU Events [106], [108]- [110], [116], [117], [122], [125]- [127], [130]: CPU events are the most commonly used features among the dynamic analysis-based detection mechanisms because in-browser cryptojacking scripts have to fetch the CPU instructions to perform the mining, independent of the used hardware. If an in-browser operation uses cryptographic libraries too frequently, which is abnormal for regular websites, it can be directly detected by CPU instructions. Even though CPU is the most crucial feature of cryptocurrency mining, using only CPU events as features may cause high false-positive rates (FPR) because flash gaming or online rendering websites also use the CPU of the system heavily for their operations. To keep FPR as low as possible, most detection methods use more than one features simultaneously [105], [106], [108], [110], [116], [117], [126], [127]. • Memory activities [106], [108], [110], [124]: Memory activity is another commonly used feature among the dynamic detection methods listed in Table 1. • Network package [106]- [108], [110], [111], [121]:
Network packages are also a handy and useful method to detect cryptojacking activity because of the massive network traffic generated while uploading the calculated hash values to the service provider. The studies [106]- [108], [110] utilized network traffic rate as an additional feature along with other features such as memory and CPU-related features. On the other hand, the studies in [111], [121] used only network packages for cryptojacking malware detection. Particularly, Neto et al. [111] use the network flow as a feature, while Caprolu et al. [121] use interarrival times and packet sizes as features in their detection algorithm. • JavaScript (JS) compilation and execution time [109], [117]: In [109], [117], it has been shown that JS engine execution time and JS compilation time is significantly affected by cryptojacking malware. However, online games and other online rendering platforms can also cause the same behavior causing false positives in the detection mechanism. Therefore, the study in [109] also uses CPU usage, garbage collector, and iframe resource loads as secondary features to obtain more accurate results and decrease false positives. The garbage collector is a feature of the JS programming language to optimize memory usage, and it deletes unnecessary data from memory and prevents memory overloading. The memory and CPU continuously interact with each other during the mining operation, and the CPU sends calculated data to the memory. The garbage collector deletes all calculated hash values one by one after being sent to the service provider; therefore, the mining [99] https://github.com/keraf/NoCoin CoinBlocker [128] https://zerodot1.gitlab.io/CoinBlockerListsWeb/index.html Minerblock [100] https://github.com/xd4rker/MinerBlock/blob/master/assets/filters.txt Coinhive Blocker [133] https://raw.githubusercontent.com/Marfjeh/coinhive-block/master/domains Andreas CH Blocker [134] https://raw.githubusercontent.com/andreas0607/CoinHive-blocker/master/blacklist.json [112] https://github.com/deluser8/cmtracker code Sep 21, 2018 Minesweeper [105] https://github.com/vusec/minesweeper data and code Mar 17, 2020 OUTGUARD [109] https://github.com/teamnsrg/outguard data and code Sep 6, 2019 SEISMIC [115] https://github.com/wenhao1006/SEISMIC code Sep 10, 2019 Retro Blacklist [135] https://github.com/retrocryptomining/ data and code Jul 16, 2020 process causes irregular usage of the garbage collector. Due to this behavior, the garbage collector can be used as a feature for the dynamic detection mechanism. Iframes are the HTML tags used for embedding another program/function to an HTML source code. Mining scripts are inserted into those tags and work under HTML codes. Similar to previous features, cryptojacking scripts cause irregular usage in iframe resource loads. This feature cannot be used as a primary feature because too many modern web applications use iframe resources irregularly, and it may cause a high falsepositive rate. • Hardware Performance Counter (HPC) [114], [117], [127]: HPC values [136] are used on modern computers' CPUs and keepthe record of internal CPU events (e.g., Cycles, Cache misses). The values of the registers with CPU clock cycles and executed instructions provide unique information about the behaviors of a running application. Several studies check the hardware activities and the related applications with HPC values to detect the cryptocurrency mining operations on the system. • System calls [90]: System calls are the API structures that enable the connection between applications and the running system's kernel. System calls run with level 0 privileges to invoke calls and request services from the OS's kernel. The proposed detection system in [90] uses the system calls for dynamic analysis, and system calls are recorded using the Cuckoo Sandbox. Then, the system calls are used to train deep learning models, and they achieve 99% accuracy.
6.1.4. Classifier and Performance. The collected features are mostly used to train different machine learning classifiers such as Support Vector Machine (SVM) [106], [109], [124], Random Forest [114], [124], Neural Network [90], [110], Decision Tree [107]. Moreover, Neto et al. [111] proposed the use of incremental learning, which takes the classification probabilities of an ensemble of classifiers as a feature for an incremental learning process. Moreover, Hong et al. [112], proposed a threshold-based detection, and the studies in [105], [115] used a static matching method to detect certain functions in the script. Musch et al. [113] only report the number of detected websites in the Top 1M Alexa websites. As can be seen from Table 1, all classifiers achieve a near-perfect (∼100%) detection results.
6.1.5. Open Source Implementations. Finally, some of the studies [105], [109], [112], [115], [137] published their code to help the research community. Table 3 presents the list of open-source cryptojacking malware implementations.
Cryptojacking Prevention Studies
A majority of the detection mechanisms do not focus on preventing or interrupting of cryptojacking malware; however, there are still several studies [116], [122], [123] focusing on both the detection and prevention of cryptojacking malware. Using dynamic features to detect ongoing cryptojacking is like other dynamic analysis studies, but their prevention methods vary. While Yulianto et al. [122] only raises a notification, Bian et al. [116] sleep the mining process, and Razali et al. [123] directly kill the related process.
For cryptojacking prevention, there are also several tools in the market. Against host-based cryptojacking malware, proprietary antivirus programs [102], [149] 3 are commonly preferred. Against in-browser cryptojacking malware, open-source browser extensions such as No-Coin [99] and MinerBlock [100] are widely used. These open-source browser extensions are based on blacklisting, where the lists are updated as new malicious domains are discovered. Table 2 shows the list of publicly available blacklists that we identified during our research. Browser extensions warn the user when the user wants to access a website on the blacklist. Figure 4 shows the blacklisting process, which is repeated as a continuous loop.
Ref
Cryptojacking Dataset Sample Type Focus of the Study [138] 2000 executable binary the practice of using compromised PCs to mine Bitcoin [139] 33282 websites script prevalence analysis [140] --how cybercriminals are exploiting cryptomining [65] 5190 websites script campaign and domain analysis [141] XMR-stak, cpuminer-multi binary attack impact on consumer devices and user annoyance [142] 5700 websites script static, dynamics and economic analysis [143] CoinHive cryptominer script sample characteristics and network traffic analysis [104] 1.2 million miners binary currencies, actors , campaign and earning analysis, underground markets [144] 107511 websites script profitability and the imposed overheads [15] 3.2 TB historical scan results script investigation of a new type of attack that exploits Internet infrastructure for cryptomining [145] --business model, threat sources, implications, mitigations, legality and ethics [146] 53 websites script sample characteristics [147] 2770 websites script activeness analysis [148] XMRig miner binary sample characteristics [135] 156 domains, 25892 proxies script impact on the web users also some new methods [150] proposed by researchers for better and more optimized blacklisting, but even dynamic blacklisting methods are not fully effective nor protective [151] against domain fluxing methods.
Cryptojacking Analysis Studies
In addition to the cryptojacking malware detection and prevention studies, some researchers also performed empirical measurement studies to understand the cryptojacking threat landscape better. Table 4 shows cryptojacking malware analysis studies in the literature. In these studies, cryptominers are either in the format of binary [104], [138], [141], [148] or script [65], [135], [139], [142]- [144], [146], [147] except [140], [145] where the findings in these studies are based on the other studies and publicly available documents.
Researchers analyzed several different perspectives of cryptojacking. In the first study [138], the authors analyze the binary samples identified as engaged in mining operations to characterize their scope, operations, and revenue. This is the first and only study analyzing Bitcoin miners, where the samples used in other studies are mining Monero. The increase in the cryptojacking malware attack instances in 2017 also drew researchers' attention. [139] is the first study analyzing the Monero cryptojacking samples, where the authors used over 30000 websites utilizing coinhive.min.js library for the prevalence analysis of cryptojacking samples. Many follow-up studies are published. For example, the studies [143], [146], [148] also performed an analysis of the cryptojacking samples to identify characteristics of the samples. In addition, the studies in [135], [141] performed the impact analysis. Particularly, [141] analyzed the attack impact on consumer devices and user annoyance, and [135] analyzed the impact of cryptojacking malware on web users, while [144] analyzed the overhead of cryptojacking samples. In an interesting study, the authors in [15] investigated a new type of attack exploiting the Internet infrastructure for cryptomining, which is indeed has an impact on 1.4M infected routers.
Moreover, there are also studies performing the economic analysis of cryptojacking samples such as [104], [142], [144]. Other than that, the authors in [65], [104] performed a campaign analysis of the cryptojacking samples and [147] analyzed the activeness of cryptojacking threat after the discontinuation of Coinhive. Finally, while [140] gives an overview of how cybercriminals are exploiting cryptomining, [145] presents a review of the business model, threat sources, implications, mitigations, legality, and ethics of cryptojacking malware.
Lessons Learned and Research Directions
This section covers lessons we learned during this research and potential research directions that can further be explored by other researchers:
The trend shift from in-browser to host-based cryptojacking attacks. The discontinuation of Coinhive's service in March of 2019 led to a big drop in the number of in-browser cryptojacking instances observed in the wild [152] as it is the leading distributor of cryptojacking scripts and the connection between the mining pools and users are lost. We also observe and verify this finding using both VT and PublicWWW datasets. In the VT dataset, we observe that 84% of the samples are uploaded right after Coinhive started its service in 2017. However, in 2019 and 2020, the number of samples uploaded to VT is around 14% in total. Moreover, we also observe a similar result in PublicWWW, in which we can directly see the service provider of the sample. 50% of the samples with a known service provider (2667/5328) and 43% (2667/6269) of all samples are still using either Coinhive or Authedmine, which is an obfuscated script service of Coinhive. Both of these observation shows that the alternative service providers introduced after Coinhive did not gain popularity as much as Coinhive, even though we have seen some samples containing the scripts from them. Finally, the actions taken by the platforms such as Google [68], [82], Apple [83], Opera [153] also made the cryptocurrency mining in the browser and mobile devices less popular, which led the attackers to look for new targets.
On the other hand, some security reports published in 2020 [74], [154] noted another trend shift in the cryptojacking attacks, in which the attackers now target the devices with more processing power rather than the personal computers as in the in-browser cryptojacking attacks. With this, the attackers' goal is to obtain more profit in a lesser time. Some examples of these targeted devices are enterprise cloud infrastructure [75], [155], servers [156], a large number of inadequately protected IoT devices [157] or Docker engines [76]. In these attacks, the attackers did not only use the Coinhive's script, but also modified non-malicious and open-source Monero miner called XMRig to perform the cryptomining in the background [158]. Unlike in-browser cryptominers, the client does not come to the attacker; therefore, the attacker needs to deliver the malicious mining script to the victims. For this purpose, the attackers used the vulnerabilities as in the case of Mikrotik routers [16] or a recent CVE to deliver Monero cryptominer [17], poorly configured IoT devices [157], or poor security [76]. It is also seen that insiders may want to take advantage of the servers [9].
However, despite the decrease in the number of inbrowser samples from active service providers and the potential trend shift in the attackers' behavior to host-based cryptojacking malware and techniques used to deliver the malware, host-based cryptojacking malware literature is not as rich as in-browser cryptojacking malware literature. As can be seen from Table 1 and 4, there are only several studies on the detection [81], [90], [121], [124]- [127] and the analysis [104], [138], [141], [148]. Therefore, there is a need for more effort from the security researchers to find better solutions to detect and mitigate this continually evolving threat. Monero as a target cryptocurrency. In recent attacks, Monero has become a de-facto cryptocurrency for the cryptojacking attacks. Another pattern we spotted is that in almost all of the attack instances in the previous section [7], [19], [40], [76], [77], [155], [159], the attackers use Monero as a target cryptocurrency instead of Bitcoin or other cryptocurrencies. Even we are not sure about their motive, Monero is the most popular privacy coin hiding the track of the transactions. For example, if the attackers would use Bitcoin, even though the attack has been detected after a long time, it would be possible to track down the Bitcoin transactions.
The evaluation of the proposed solutions. We identified threes issues regarding to the evaluation of the proposed solutions in the literature:
• Dataset dates. The effectiveness of an in-browser cryptojacking malware detection mechanism is directly related to the number of websites detected. However, the infected websites modify or move their script to other domains frequently to avoid being blacklisted. Moreover, many websites discontinued mining after the Coinhive shutdown [160]. Therefore, the accuracy of a detection method may significantly vary depending on when the dataset was collected.
Only five of the proposed detection mechanism [105], [109], [112], [115], [135] in Table 1 reports the dataset date. Therefore, we don't know most of the studies' dataset collection date; which makes a fair comparison difficult. • Online vs. offline detection. The detection mechanisms proposed in the literature usually focus on accuracy as an evaluation metric, and they mostly claim a near-perfect accuracy in detecting cryptojacking malware. However, most of the time, they do not report how their method was implemented, that is, whether it was offline or online. In offline detection, the sample is detected randomly and added to the database (e.g., signature, blacklist). In the online detection, the sample is detected in a real-time manner.
As it has been shown that detection ratio may vary for online and offline detection [161], it is critical for the detection studies to report if the proposed method is implemented in an online or offline environment. • Overhead analysis. Only the authors of the two [105], [109] proposed dynamic analysis tools consider the usability of their detection mechanisms on the enduser side. But, especially for machine learning-based detection methods, using behavioral features may introduce a high overhead on the end-user side. This should be taken into consideration by researchers in future studies.
The legitimate use of in-browser cryptocurrency mining. An issue we identified during our research is that the in-browser cryptocurrency mining was initially started to provide an alternative revenue to the legitimate website owners such as new publishers [162] or non-profit organizations like UNICEF [163]. Later, some service providers such as Coinimp [51], WebMinePool [59] even provided methods for explicit user consent in their implementations. However, with the keyword-based automatic detection and prevention methods such as browser extensions [99], [100] or even browsers themselves [68], [153] blocking the websites containing cryptomining script, this use of web-based cryptomining scripts is not possible anymore.
A practical solution to this issue would be asking for the user's explicit consent instead of directly blocking a website trying to upload a mining script. Moreover, there is a need for more effort by researchers to work on the usage of legitimate cryptomining with user consent and knowledge as a funding model.
The use of traditional malware attacks on Bitcoin and blockchain infrastructure. There are two types of Bitcoin-and blockchain-related malware seen in the wild: those that use the Bitcoin and blockchain infrastructure to exploit the victim; or those that use the traditional malware attacks such as key stealing, social engineering, or fake application attacks to exploit Bitcoin and blockchain users. Cryptojacking attacks use the Bitcoin and blockchain infrastructure to exploit the victim's computational power; however, Bitcoin and blockchain users are also exposed to many traditional malware attacks. These attacks specifically aim to obtain Bitcoin and blockchain users' private keys through social engineering methods [164]- [166], fake wallets [167], [168], and key-stealing trojan malware [169]- [171]. Although these attacks and their countermeasures have been studied extensively in the literature [172], their impact in the Bitcoin and blockchain domain has not been investigated yet and can lead to new research directions.
Conclusion
The rapid rise of cryptocurrencies incentivized the attackers to the lucrative blockchain and the Bitcoin ecosystem. With ready-to-use mining scripts offered easily by service providers (e.g., Coinhive [8], and CryptoLoot [4]) and untraceable cryptocurrencies (e.g., Monero), cryptojacking malware has become an essential tool for hackers. The lack of mitigation techniques in the market led to many cryptojacking malware detection studies proposed in the literature. In this paper, we first explained the cryptojacking malware types and how they work in a systematic fashion. Then, we presented the techniques used by cryptojacking malware based on the previous research papers, cryptojacking samples, and major attack instances. In particular, we presented sources of cryptojacking malware, infection methods, victim platform types, target cryptocurrencies, evasion, and obfuscation techniques used by cryptojacking malware. Moreover, we gave a detailed review of the existing detection and prevention studies as well as the cryptojacking analysis studies in the literature. Finally, we presented lessons learned, and we noted several promising new research directions. In doing so, this SoK study will facilitate not only a deep understanding of the emerging cryptojacking malware and the pertinent detection and prevention mechanisms but also a substantial additional research work needed to provide adequate mitigations in the community. When we continued more granular time analysis on the samples, we found that samples are uploaded to VT as a batch every six months. Therefore, we concluded that a smaller time frame analysis than the yearly distribution might not be reliable as representing the time distribution of real-life samples seen in the wild. File Type Distribution: In order to detect the file type, we used the type given by the VT scan reports [174]. Figure 7 shows the top 10 file types of the samples in the VT dataset. According to the figure, HTML is the most common file type in the VT dataset, while the Win32 EXE is the second most common file type. File type distribution of VT dataset samples is important as they can be used to decide if the sample is an in-browser or host-based type of cryptojacking. Even though some of the file types clearly indicate the type of the cryptojackings, some may require a more in-depth analysis of the sample. In-browser samples only contain the mining script and in the form of text format to embed in the website source code, while the host-based cryptojacking malware samples are in the executable or other formats that can be run on the host machine. For example, we found that all HTML files are in-browser samples while Win32 EXE and Win32 DLL samples are host-based cryptojacking samples. However, for the file types such as Text, C, C++, ZIP, one needs to check the sample itself and other useful information like submission names to decide whether the sample is in-browser or hostbased. Detection Ratio: VT scan reports includes the detection results of around 60 vendors for each sample [174]. In this part, our goal is to able to see the detection ratio of AVs for the cryptojackings samples in our VT dataset. For this, we plot the histogram of the detection ratio, and the results are given in Figure 8. The results show that the average detection ratio of cryptojacking samples is approximately 40%. We note that every cryptojacking sample in our dataset is detected by at least one AV vendor. This is because of the filtering method we used to find the cryptojacking samples among all samples, i.e., searching for the keyword "miner" among all AV labels. Therefore, if any AV vendors have not detected a sample, it would not be in our dataset in the first place. This is a limitation of the VT dataset. In order to overcome this limitation and create a recent and more comprehensive dataset, we created another dataset, which we will explain in the next section.
B.2. PublicWWW Dataset
The VT dataset does not include the samples that can bypass the AV detection methods and samples that have never submitted to VT. In order to create a more comprehensive and recent dataset of cryptojacking malware, we used the HTML source keyword search engine PublicWWW [175]. We created the PublicWWW dataset using the following steps: 1) We obtained the keyword lists from the blacklists [99], [100], previous studies [65], [105] and manual analysis of the samples from the VT dataset. Particularly, we used a merged blacklist from NoCoin [99] and MinerBlock [100]; 76 keywords from [65] and 38 keywords from [105]; 25 keywords from the VirusTotal samples. 2) We downloaded the list of URLs for each keyword from PublicWWW. 3) We merged the lists and removed the duplicates to obtain a unique list of URLs. 4) We used a web crawler to download the HTML source code of each URL. 5) We verified the samples by checking the keywords in their source code and removed the samples that do not satisfy this condition. This process resulted in 6269 unique URLs, their HTML source codes, and their final keyword list with 154 unique keywords used in these samples. From the previous two studies [65], [105] and our findings of publicly known service providers, we identified 14 service providers in total. We manually analyzed their documentation and found that 5328 samples are using the scripts from those 14 service providers. Then, we identified 24 unique keywords to uniquely capture the samples. We released the service provider and keywords lists in our dataset link. Figure 9. Service provider distribution of the samples in PublicWWW dataset. Figure 9 shows the service provider distribution of the samples in the PublicWWW dataset. As shown in the figure, even though it is inactive, Coinhive is still the most common service provider among all. On the other hand, Coinimp is the second highest service provider and it is still active as of writing this paper. In addition, we found that 144 samples are using scripts from the multiple samples while we are not able to identify the associated service provider of 941 samples, which we marked as domain lists with an unknown service provider. Figure 10 shows the distribution of the attack instances we used in this paper per year. This distribution graph does not show any indicative result regarding the cryptojacking malware's popularity over time in our paper. Only one attack instance from 2013 may seem like an outlier; however, that example shows one of the first instances of cryptojacking malware idea, which is very similar to its usage after 2018. In that attack, a cryptojacking malware attack is instantiated by attaching the sample inside a video game to mine Bitcoin. Finally, in addition to 45 major attack instances, we also added 14 service providers' webpage and 5 blacklists' link and shared in our dataset link.
•
To understand the lifecycle of in-browser and hostbased cryptojacking (Section 4.1 & 4.2) • To verify the service provider list given in other studies and as a source of cryptojacking malware (Section 5.1.1) • To verify the use of mobile devices as a victim platform (Section 5.3.6) • To verify that Monero is the main target currency used by cryptojackings (Section 5.4.1) • To find the other cryptocurrencies used by cryptojacking malware (Section 5.4) • To verify that the existence of CPU limiting technique for the obfuscation (Section 5.6.1) • To verify and understand the use of code encoding for the obfuscation (Section 5.6.3)
Figure 1 .
1Script preparation and injection phases of a in-browser cryptojacking malware.
Figure 2 .
2The lifecycle of a in-browser cryptojacking malware.
Figure 3 .
3The lifecycle of a host-based cryptojacking malware.
<script> var miner=new CoinHive.Anonymous('Key', { Threads:4, autoThreads:false, throttle:0.8); if (!miner.isMobile()) &&!miner.didOptOut(14400) { miner.start(); } } </script> Listing 1: The mobile device filtering method used in a cryptojacking sample.
Figure 4 .
4Blacklisting method. Pure blacklisting-based prevention is not an efficient way for stopping cryptojacking malware because attackers can easily change their domain by domain fluxing or other methods to downshift the effects of blacklists. There are
Figure 6 .
6Yearly distribution of VT dataset.
Figure 7 .
7Top-10 file types of the samples in VT dataset.
Figure 8 .
8Histogram distribution of detection ratio of the samples in VT dataset.
Figure 10 .
10Yearly distribution of 39 cryptojacking attacks instances we used in this paper.
TABLE 1 .
1CRYPTOJACKING MALWARE DETECTION MECHANISMS IN THE LITERATURE.Ref
Dataset
Type
Method
Features
Classifier
Performance
Rüth et al. [64]
Three largest TLDs
Alexa: 1M
In-browser
Static
Wasm signatures
SRSE
N/A
Minesweeper [105]
Alexa 1 Million
In-browser
Static
Wasm code
CPU cache events
Matching
N/A
RAPID [106]
Alexa: 330.500
In-browser
Dynamic
Resource consumption
(memory, network,
and processor) and
JavaScript API Events
SVM
Benign (best):
Precision: 99.99%
Recall: 99.99%
F1: up to 99.99%
Mining (best):
Precision: 96.54%
Recall: 95.48%
F1: up to 96.0%
Muñoz et al. [107]
Network traffic of six
cryptocurrencies using
Stratum protocol
In-browser
Dynamic
Metadata of inbound
and outbound network traffic
DT
Best:
Accuracy: 99.9%
Precision: 98.2%
Recall: 90.7%
CapJack [108]
Five user applications
and a Coinhive miner
In-browser
Dynamic
CPU utilization,
Memmory,
Disk read/write rate,
Network interface)
CNN
87% Instant
99% After 11 seconds
98% Mobile Single
86% Mobile Cross
97% AWS single
89% AWS Cross
OUTGUARD [109]
Alexa 1M and 600K
In-browser
Dynamic
Js Execution Time,
JS compilation Time,
Garbage Collector,
Iframe resource loads,
CPU usage
SVM, RF
SVM (best):
TPR: 97.9%
FPR: 1.1%
CoinSpy [110]
100k websites from
Alexa 1M and 50
manipulated cryptojacking
websites
In-browser
Dynamic
CPU,
Memory,
Network behaviors
CNN
Accuracy:97%
MineCap [111]
The network traffic captured
from two mining and
streaming applications
In-browser
Dynamic
Network packages
IL
Accuracy: 98%,
Precision: 99%,
Recall: 97%
Specifity: 99.9%
CMTracker [112]
Alexa 100k
In-browser
Dynamic
Hash and Stuck
based profilers
Thr-based
100% TPR
Musch et al. [113]
Alexa 1M
In-browser
Dynamic
CPU usage
MA
N/A
Tahir et al. [114]
Manually created 320 non-mining
and 100 mining websites
In-browser
Dynamic
HPC values
RF
Accuracy: 99.35%,
Precision: 100%,
Recall: 98%,
AUC: 99%
SEISMIC [115]
500 webpages randomly selected
from Alexa top 50K
In-browser
Dynamic
Wasm instructions
Matching
F1: 98%
MineThrottle [116]
Alexa 1M
In-browser
Dynamic
Block-level features
CPU usage
Matching
FNR: 1.83%
Coinpolice [117]
47k samples
In-browser
Dynamic
CPU usage, HPC,
JS/WASM execution time and features,
Throttling-independent timeseries
CNN
TPR:97.8 % FPR: 0.74%
Carlin et al. [118]
Captured Opcode trace
packets Virusshare (296 Samples)
In-browser
Dynamic
Opcodes
RF
TPR: 99.2 % FPR: 0.9,
Precision: 99.2 Recall: 99.2
Liu et al. [119]
1159 samples collected from
browsers' memory snapshot
In-browser
Dynamic
Heap snapshots
Stack Features
RNN
Precision: 95,
Recall: 93
Rauchberger et al. [120]
Alexa: 1M
In-browser
Dynamic
Web socket traffic
Matching
N/A
Caprolu et al. [121]
N/A
In-browser
Dynamic
Network traffic
RF,KFCV
TPR=92% ,FPR=0.8%
Yulianto et al. [122]
PublicWWW and Blacklists
In-browser
Static and Dynamic
CPU usage
Matching
TPR:100%
CMBlock [123]
In-browser cryptojacking samples
In-browser
Static and Dynamic
Blacklists
Behaviour
N/A
N/A
Conti et al. [124]
Combination daily user
tasks and miners 1
Host-based and In-browser
Dynamic
hardware events (e.g., branch-misses),
software events (e.g., page-faults)
hardware cache events (e.g., cache-misses)
RF, SVM
Recall: 97.84%
Precision: 99.7%
Accuracy:98.7%
Lachtar et al. [125]
N/A
Host-based and In-browser
Dynamic
CPU instructions
Matching
TPR:100 % FPR: ¡ 2%
Tanana et al. [126]
40 In-browser and
10 executable-type cryptojacking
Host-based and In-browser
Dynamic
CPU utilization share
RAM usage
N/A
TPR: 81%
Ahmad et al. [81]
Mixture of Benign
and Malicious Network Packages
Host-based and In-browser
Dynamic
Network traffic
DCA
N/A
DeCrypto Pro [127]1200 samples
Host-based and In-browser
Dynamic
HPC, CPU usage
k-NN, RF, LSTM
FPR: 2.5,
Precision: 96, Recall: 97
Darabian et al. [90]
1500 active cryptomining
collected from Virustotal in 2018
Host-based
Static and Dynamic
System calls,
opcode sequences
RNN, CNN
System calls (best):
LSTM: Accuracy:99%
F1: 98% MCC: 98%
FPR:0.6%
Crypto-Aegis [121]
Network traffic of 3 legitimate mining
scripts and 3 daily user applications
Host-based
Dynamic
Packet sizes
Interarrival times
RF
TPR:80-84% FPR: 0.9 -1.2%
TABLE 2 .
2THE LIST OF PUBLICLY AVAILABLE BLACKLISTS.Ref
Link
Nocoin
TABLE 3 .
3THE LIST OF OPEN-SOURCE CRYPTOJACKING MALWARE DETECTION IMPLEMENTATIONS.Ref
Implementation Link
Description
Last Update
CMTracker
TABLE 4 .
4CRYPTOJACKING MALWARE ANALYSIS STUDIES IN THE LITERATURE.
The dataset was not available as of writing this paper (November 1, 2020).2 Support Vector Machine: SVM, Random Forest: RF, Decision Tree: DT, Convolutional Neural Network: CNN, Recurrent Neural Network: RNN, Incremental Learning: IL, Threshold-based: Thr-based, Manual Analysis: MA, Dendritic Cell Algorithm: DCA, k-Nearest Neighbors: k-NN, Light-weight machine learning models: LSTM, Symantec RuleSpace Engine:SRSE, k-Fold Cross Validation:KFCV
. As these programs are closed-source, their methods are not publicly available.
Appendix A. Papers DistributionAppendix B. Samples DistributionIn this section, our goal is to give more details about the VT and PublicWWW datasets, perform quantitative and longitudinal analysis on our two datasets to confirm some of our findings in the paper, and give more insights on the dataset. More details about both VT and Pub-licWWW datasets can be found in the following website: https://github.com/sokcryptojacking/SoKB.1. VT DatasetWe used VT Academic API to access the VT dataset consisting of 437279 unique samples (both cryptojacking and non-cryptojacking) and their VT scan reports in the format of JSON. To detect the cryptojacking samples among all samples, we used AV labels in the scan reports of these samples and looked for the keyword "miner", i.e., if any of Antivirus (AV) label in the report include the keyword "miner", we included in our samples. We picked the keyword "miner" as we considered it to be the most generic keyword to find all of the cryptojacking samples, and it is also used in recent work as a generic class label for the VT samples[173]. Our scan resulted in the 20200 cryptojacking malware samples.B.1.1. More Insights on VT Dataset. In addition to the AV labels that we used to detect the miners, VT scan reports also include other information regarding the samples such as first seen date, file type, submission names, the total number of detection by AVs of the samples.We performed more analysis using this information and explain our results in the rest of the section.Yearly Distribution:Figure 6shows the yearly distribution of the samples' first seen date in the VT dataset[174]. As can be seen from the figure, only 1% of the cryptominer samples were submitted to VT before 2018. This indeed matches and verifies the expected finding of the surge in the number of cryptominers after Coinhive started its service in 2017. Moreover, the sharp decrease after 2018 is also notable. This may indicate that not many new samples are found in the wild in 2019 and 2020; however, it does not show that the number of active samples in 2019 and 2020 is less than in 2017 as the samples uploaded in 2017 may be still actively operating. These samples' activeness requires more analysis on the script used in those samples and real-time capture of the samples in the wild.
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|
[
"https://github.com/sokcryptojacking/SoK",
"https://github.com/keraf/NoCoin",
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"https://github.com/vusec/minesweeper",
"https://github.com/teamnsrg/outguard",
"https://github.com/wenhao1006/SEISMIC",
"https://github.com/retrocryptomining/",
"https://github.com/sokcryptojacking/SoKB.1.",
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[
"Collective Chaos Induced by Structures of Complex Networks",
"Collective Chaos Induced by Structures of Complex Networks"
] |
[
"Huijie Yang \nDepartment of Modern Physics\nUniversity of Science and Technology of China\n230026Hefei AnhuiChina\n",
"Fangcui Zhao \nCollege of Life Science and Biomedical Engineering\nBeijing University of Technology\n100022BeijingChina\n",
"Binghong Wang \nDepartment of Modern Physics\nUniversity of Science and Technology of China\n230026Hefei AnhuiChina\n"
] |
[
"Department of Modern Physics\nUniversity of Science and Technology of China\n230026Hefei AnhuiChina",
"College of Life Science and Biomedical Engineering\nBeijing University of Technology\n100022BeijingChina",
"Department of Modern Physics\nUniversity of Science and Technology of China\n230026Hefei AnhuiChina"
] |
[] |
Mapping a complex network of N coupled identical oscillators to a quantum system, the nearest neighbor level spacing (NNLS) distribution is used to identify collective chaos in the corresponding classical dynamics on the complex network. The classical dynamics on an Erdos-Renyi network with the wiring probability p ER ≤ 1 N is in the state of collective order, while that on an Erdos-Renyi network with p ER > 1 N in the state of collective chaos. The dynamics on a WS Small-world complex network evolves from collective order to collective chaos rapidly in the region of the rewiring probability p r ∈ [0.0, 0.1], and then keeps chaotic up to p r = 1.0. The dynamics on a Growing Random Network (GRN) is in a special state deviates from order significantly in a way opposite to that on WS small-world networks. Each network can be measured by a couple values of two parameters (β, η).
|
10.1016/j.physa.2005.09.050
|
[
"https://arxiv.org/pdf/cond-mat/0505086v3.pdf"
] | 16,723,363 |
cond-mat/0505086
|
35697a3fc69c1be30058ca1bac422330f363e005
|
Collective Chaos Induced by Structures of Complex Networks
22 Sep 2005
Huijie Yang
Department of Modern Physics
University of Science and Technology of China
230026Hefei AnhuiChina
Fangcui Zhao
College of Life Science and Biomedical Engineering
Beijing University of Technology
100022BeijingChina
Binghong Wang
Department of Modern Physics
University of Science and Technology of China
230026Hefei AnhuiChina
Collective Chaos Induced by Structures of Complex Networks
22 Sep 2005Complex networksCollective chaosSpectra statistics
Mapping a complex network of N coupled identical oscillators to a quantum system, the nearest neighbor level spacing (NNLS) distribution is used to identify collective chaos in the corresponding classical dynamics on the complex network. The classical dynamics on an Erdos-Renyi network with the wiring probability p ER ≤ 1 N is in the state of collective order, while that on an Erdos-Renyi network with p ER > 1 N in the state of collective chaos. The dynamics on a WS Small-world complex network evolves from collective order to collective chaos rapidly in the region of the rewiring probability p r ∈ [0.0, 0.1], and then keeps chaotic up to p r = 1.0. The dynamics on a Growing Random Network (GRN) is in a special state deviates from order significantly in a way opposite to that on WS small-world networks. Each network can be measured by a couple values of two parameters (β, η).
Introduction
Impacts of network structures on the dynamical processes attract special attentions in recent years, to cite examples, the epidemic spreading on networks [1][2][3], the response of complex networks to stimuli [4,5] and the synchronization of dynamical systems on networks [6][7][8][9].In this paper, we will consider the collective motions on complex networks, just like the phonon in regular lattices.
By means of the random matrix theory (RMT), we find that the nearest neighbor level spacing (NNLS) distributions for spectra of complex networks generated with WS Small-world, Erdos-Renyi and Growing Randomly Network models can be described with Brody distribution in a unified way. This unified description can be used as a new measurement to characterize complex networks. The results tell us that the topological structure of a complex network can induce a special kind of collective motions. Under environmental perturbations, the collective motion modes can transition between each other abruptly. This kind of sensitivity to outside perturbations is called collective chaos in this paper. It is found that the properties of the collective chaos are determined only by the structures of the networks. Without the aid of the dynamical model presented in references [4,5], we show for the first time that the dynamics on complex networks can be in collective order, soft chaotic or even hard chaotic states.
The Method
Wigner, Dyson, Mehta and others developed the Random Matrix Theory (RMT) to understand the energy levels of complex nuclei and other kinds of complex quantum systems [10][11][12][13]. In recent literature, the spectral density function and the time series analysis methods are used to capture properties of complex networks [14][15][16][17][18][19]. One of the most important concepts in RMT is the nearest neighbor level spacing (NNLS) distribution. A general picture emerging from experiments and theories is that if the classical motion of a dynamical system is regular, the NNLS distribution of the corresponding quantum system behaves according to a Poisson distribution. If the corresponding classical motion is chaotic, the NNLS distribution will behave in accordance with the Wigner-Dyson ensembles. This is the content of the famous Bohigas conjecture [20,21]. Hence, the NNLS distribution of a quantum system can tell us the dynamical properties of the corresponding classical system.
Consider an undirected network of Ncoupled identical oscillators. The Hamiltonian reads,
H = N n=1 h 0 (x n , p n ) + 1 2 · N m =n A mn · V (x m , x n ),(1)
where, h 0 (x n , p n ) is the Hamiltonian of the n'th oscillator, V (x m , x n ) the coupling potential between the m'th and the n'th oscillators and A the adjacent matrix of the network. The Hamiltonian of the corresponding quantum system can be represented as,
H = N n=1ĥ 0 (x n , p n ) + 1 2 · N m =n A mn ·V (x m , x n ) .(2)
Assuming the site energy of each oscillator is ε 0 and the eigenfunction is ϕ 0 , we have
⌢ h 0 (x n , p n )ϕ 0 (x n ) = ε 0 ϕ 0 (x n ). The matrix elements of ⌢ H reads, H mn = ϕ 0 (x m )| h 0 (x m ) |ϕ 0 (x n ) + A mn · ϕ 0 (x m )| V (x m , x n ) |ϕ 0 (x n ) = ε 0 · δ mn + A mn · V mn(3)
The pattern of the spectrum of ⌢ H does not dependent on the values of ε 0 and V mn . Assigning ε 0 = 0 and V mn = 1, we have H = A. By this way, the spectrum of the adjacent matrix A can be used to calculate the NNLS distribution of the quantum system. To make our discussion as self-contained as possible, we review briefly the procedure to obtain the NNLS distribution from the spectrum of A, denoted with {E i |i = 1, 2, 3, · · · , N }. N is the total number of the energy levels.
To ensure that the distances between the energy levels are expressed in units of local mean energy level spacing, we should first map the energy levels E i to new variables called "unfolded energy levels" ξ i . This procedure is called unfolding, which is generally a non-trivial task [22].
Define the cumulative density function as,
G(E) ≡ N E −∞ g(s)ds,(4)
where g(s) is the density function of the initial energy level spectrum. We have,
G(E) |E k+1 > E ≥ E k = k.(5)
Preprocess the spectrum, {E k |k = 1, 2, 3, · · · , N }, and the corresponding accumulative density function, {G(E k ) |k = 1, 2, 3, · · · N }, so that for each of them the mean is set to zero and the variance equals to 1, i.e.,
λ k = E k − 1 N N m=1 E m N j=1 E j − 1 N N m=1 E m 2 1 / 2 ,(6)F (λ k ) = G(E k ) − 1 N N m=1 G(E m ) N j=1 G(E j ) − 1 N N m=1 G(E m ) 2 1 / 2 .(7)
Dividing the accumulative function F (λ) into two components, i.e., the smooth term F av (λ) and the fluctuation term F f (λ), the unfolded energy levels can be obtained as,
ξ k = F av (λ k ).(8)
Because we have not enough information on the accumulative density function at present time, a polynomial is employed to describe the relation between ξ k and λ k , as follows,
ξ k = L l=0 c l · (λ k ) l = F av (λ k ).(9)
It is found that a large value of L > 9 can lead to a considerable good fitting result. To guarantee the fitting results exact enough, we assign the value as L = 17.
Defining the nearest neighbor level spacing (NNLS) as,
s i = w · (ξ i+1 − ξ i ) σ ξ |i = 1, 2, 3, · · · (N − 1) ,(10)
the Brody distribution of the NNLS reads,
P (s) = β η s β−1 exp − s η β ,(11)
which is also called Weibull distribution in the research field of life data analysis [23]. In the definition of NNLS,w is a factor to make the values of the NNLS in a conventional range to get a reliable fitting result, and σ ξ =
N−1 i=1 ξ 2 i N −1 . In- troducing the function, Q(s) = s 0 P (t)dt, some trivial computation lead to [23], ln R(s) ≡ ln ln 1 1 − Q(s) = β ln s − β ln η,(12)
based upon which we can get reliable values of the parameters β and η.
To obtain the function Q(s), we should divide the interval where the NNLS distributes into many bins. The size of a bin can be chosen to be a fraction of the square root of the variance of the NNLS, which reads, ε = 1
R N−1 i=1 s 2 i N −1 .
If R is unreasonable small, Q(s) cannot capture the exact features in actual probability distribution function (PDF), while a much large R will induce strong fluctuations. The value of the parameter R is assigned 20 in this paper, because the fitting results are stable in a considerable wide range about this value.
A Brody distribution reveals that the corresponding classical complex system is in a soft chaotic state [10]. For the two extreme conditions of β = 1 and β = 2, the Brody distribution will reduce to the Poisson and the Wigner distributions. And the corresponding classical complex systems are in the hard chaotic and order states, respectively. For the quantum system considered in this paper, it can be initially in a quantum state of |E n , corresponding to the eigenvalue of E n . Under a weak environmental perturbation, the state will display completely different behaviors [24][25][26]. If the system is in an order state, the transition probability of the initial state to a new state |E m will decrease rapidly with the increase of |E n − E m |, and the transition occurs mainly between the initial state and its neighboring states. If the system is in a chaotic state, the transitions between all the states in the chaotic regime the initial state belongs to can occur with almost same probabilities. In the classical dynamics, the corresponding states are the collective motion modes, just like phonon in regular lattices. Under perturbations the state of a chaotic system can transition between the collective modes in the same chaotic regime abruptly, while the state of an order system can transition between the initial mode and its neighboring states only. Consequently, the chaotic state of the N identical oscillators is a kind of collective behavior rather than the individual properties of each oscillator. It is called collective chaos in this paper. The above discussions tell us that this collective chaos dependents only on the structure of the undirected network.
Results
Given N nodes, an Erdos-Renyi network can be constructed just by connecting each pair with a probability p ER [27,28]. It is demonstrated that there exists a critical point p c = 1 N . For p ER < p c , the adjacency matrix can reduce into many sub-matrices, the couplings between the energy levels will be very weak and the NNLS will obey a Poisson form. For p ER ≥ p c , the fraction of the nodes forming the largest sub-graph grows rapidly. The couplings between the energy levels will become stronger and stronger, and the NNLS should obey a Brody or even a Wigner form. Simulation results presented in Fig.1 to Fig.3 are consistent with this theoretical prediction.
Secondly, we consider the one-dimensional lattice small-world model designed by Watts and Strogatz (WS small-world model) [29]. Take a one-dimensional lattice of nodes with periodic boundary conditions, and join each node with its k right-handed nearest neighbors. Going through each edge in turn and with probability p r rewiring one end of this edge to a new node chosen randomly. During the rewiring procedure double edges and self-edges are forbidden. Numerical simulations by Watts and Strogatz show that this rewiring process allows the small-world model to interpolate between a regular lattice and a random graph with the constraint that the minimum degree of each node is fixed [29]. The parameter k is chosen to be 2, and N is 3000. Fig.4 to Fig.6 present some typical results for different values of rewiring probability. In the main region we are interested, the NNLS distribution can be described with a Brody distribution almost exactly. Fig.7 presents the values of the parameters β and η versus the rewiring probability p r . In a short region p r ∈ [0.0, 0.1], the value of β increases from 1 to 1.75, i.e., the NNLS distribution evolves from a Poisson to a near Wigner form. In the other region p r ∈ [0.1, 1.0] the networks behave almost same. Comparison tells us that the Erdos-Renyi networks with p ER = 2J N (J ≥ 1) are similar with the WS small-world network with p r = 1.
The third considered is the growing random networks (GRN) model [30]. Giving several connected seeds, at each step a new node is added and a link to one of the earlier nodes is created. The connection kernel A k , defined as the probability a newly introduced node links to a preexisting node with k degree, determines the structure of this network. A group of GRN networks determined by a special kind of kernel, A k ∝ k θ (0 ≤ θ ≤ 1), are considered in this present paper. For this kind of networks, the degree distributions decrease as a stretched exponential in k. Setting θ = 1 we can obtain a scale-free network. Fig.8 to Fig.10 present some typical results for GRN networks. Fig.11 shows that in a wide range of 0 ≤ θ ≤ 0.8, the value of the parameter β oscillates basically around 0.68, i.e., the NNLS distributions deviate significantly from the Poisson form in a way opposite to that of WS small-world networks. In the other region p r ∈ [0.8, 1.0] the value of β decreases rapidly to ∼ 0.50. The values of the parameter η are also presented.
Summary
In summary, based upon the RMT theory we investigate the NNLS distributions for the ER, the WS Small-world and the GRN networks. The Brody form can describe all these distributions in a unified way. The NNLS distributions of the quantum systems of the network of N coupled identical oscillators tell us that the corresponding classical dynamics on the Erdos-Renyi networks with p ER < p c = 1 N are in the state of collective order, while that on the Erdos-Renyi networks with p ER > p c = 1 N in the state of collective chaos. On WS small-world networks, the classical dynamics evolves from collective order to collective chaos rapidly in the region p r ∈ [0.0, 0.1], and then keeps chaotic up to p r = 1.0. For GRN model, contrary to that on the WS small world networks, the classical dynamics are in special states deviate from order significantly in an opposite way.
These dynamical characteristics are determined only by the structures of the The WS small-world networks and the GRN networks are separated by the Poisson form, i.e., β = 1 , significantly. The Erdos-Renyi networks with p ER < p c = 1 N obey near Poisson distribution, while that with p ER ≥ p c are similar with the almost complete random WS small-world networks (p r ∼ 1 ). The position of a network in this scheme may tell some useful information. The errors of the parameters (β, η) are less than 0.01. considered complex networks. In a very recent paper [31], the authors point out that for some biological networks the NNLS distributions obey the Wigner form. The dynamics on these networks should be in a state of collective chaos. And the removal of nodes may change this dynamical characteristic from collective chaos to collective order. Therefore, constructing a mini network with selected key nodes should be considered carefully in discussing the collective dynamics on a complex network.
Without the aid of the simplified model of dynamics presented in [3,4] we obtain the dynamical characteristics on complex networks. The NNLS distribution can capture directly the relation between the structure of a complex network and the dynamics on it.
It should be pointed out that the collective chaos induced by the structures of complex networks is completely different with the individual chaotic states of the oscillators on the networks. For a network with regular structure, the classical dynamical processes on it should display collective order, even if the coupled identical oscillators on the nodes may be in chaotic states. On the contrary, for a complex network with a complex structure (e.g., a WS smallworld structure), the classical dynamical processes on it should display col-lective chaos, even if the coupled identical oscillators on the nodes may be in order states as harmonic oscillations.
Each network can be measured by a couple values of (β, η). The position of a complex network in β versus η scheme may tell us useful information for classification of real world complex networks.
Fig. 1 .
1The cumulative density function of the spectra of four Erdos-Renyi networks. The circles are the actual values, while the solid lines are fitting results with a 17-ordered polynomial function.
Fig. 2 .
2Determine the values of parameters (β, η) for the four Erdos-Renyi networks by means of the relation presented in Eq.12. In the main region we are interested the NNLS distributions obey a Brody form almost exactly. For p ER < p c = 1 N ,we have β = 0.931 ∼ 1.0, i.e., the distribution obeys a Poisson form. For p ER > p c , the distributions obey a Brody distribution form very near the Wigner one. The errors of the parameters (β, η) are less than 0.01.
Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .
345678The NNLS distributions for the four Erdos-Renyi networks. In the main regions we are interested, the theoretical results can fit with the actual ones very well. The errors of the parameters (β, η) are less than 0.01. The cumulative density function of the spectra of four selected WS small-world networks. The circles are the actual values, while the solid lines are fitting results with a 17-ordered polynomial function. Determine the values of parameters (β, η) for the four selected WS small-world networks by means of the relation presented in Eq.12. In the main region we are interested the NNLS distributions obey a Brody form almost exactly. The errors of the parameters (β, η) are less than 0.01. The NNLS distributions for the four selected WS small-world networks. In the main regions we are interested, the theoretical results can fit with the actual ones very well. The errors of the parameters (β, η) are less than 0.01. The parameters (β, η) for all the WS small-world networks constructed in this paper. In the short region p r ∈ [0.0, 0.1], the value of β increases from 1.0 to 1.75, i.e., the NNLS distribution evolves from a Poisson to a near Wigner form. In the other region p r ∈ [0.1, 1], the networks behave almost same. Comparison tells us that the Erdos-Renyi networks with p ER = 2J N (J ≥ 1) are similar with the WS small-world network with p r = 1. The errors of the parameters are less than 0.01. The cumulative density function of the spectra of four selected GRN networks. The circles are the actual values, while the solid lines are fitting results with a 17-ordered polynomial function.
Fig. 12
12shows the relation between β and η. Each point represents a complex network. The results for the three kinds of networks are all illustrated. The
Fig. 9 .Fig. 10 .Fig. 11 .
91011Determine the values of parameters (β, η) for the four selected GRN networks by means of the relation presented in Eq.12. In the main region we are interested the NNLS distributions obey a Brody form almost exactly. The errors of the parameters (β, η) are less than 0.01. The NNLS distributions for the four selected GRN networks. In the main regions we are interested, the theoretical results can fit with the actual ones very well. The errors of the parameters (β, η) are less than 0.01. The parameters (β, η) for all the GRN networks constructed in this paper. In a large region p r ∈ [0.0, 0.8], the value of β oscillates around 0.68, i.e, the NNLS distributions deviate significantly from the Poisson form in a way opposite to that of WS small-world networks. In the other region p r ∈ [0.8, 1], the value of β decreases rapidly to ∼ 0.5. The errors of the parameters (β, η) are less than 0.01.WS small-world networks and the GRN networks are separated by the Poisson form, i.e., β = 1, significantly. The Erdos-Renyi networks with p ER ≤ p c = 1 N obey near Poisson distribution, while that with p ER > p c = 1 N are similar with the almost complete random WS small-world networks (p r ∼ 1.0).
Fig. 12 .
12The relation between the two parameters (β, η) . Each point corresponds to a complex network. The results for the three kinds of networks are all illustrated.
AcknowledgementsThis work was partially supported by the National Natural Science
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|
[] |
[
"arXiv:astro-ph/0507218v1 8 Jul 2005 Fitting Photometry of Blended Microlensing Events",
"arXiv:astro-ph/0507218v1 8 Jul 2005 Fitting Photometry of Blended Microlensing Events"
] |
[
"Christian L Thomas ",
"Kim Griest "
] |
[] |
[] |
We reexamine the usefulness of fitting blended lightcurve models to microlensing photometric data. We find agreement with previous workers (e.g. Woźniak & Paczyński ) that this is a difficult proposition because of the degeneracy of blend fraction with other fit parameters. We show that follow-up observations at specific point along the lightcurve (peak region and wings) of high magnification events are the most helpful in removing degeneracies. We also show that very small errors in the baseline magnitude can result in problems in measuring the blend fraction, and study the importance of non-Gaussian errors in the fit results. The biases and skewness in the distribution of the recovered blend fraction is discussed. We also find a new approximation formula relating the blend fraction and the unblended fit parameters to the underlying event duration needed to estimate microlensing optical depth.
|
10.1086/500101
|
[
"https://arxiv.org/pdf/astro-ph/0507218v1.pdf"
] | 118,889,252 |
astro-ph/0507218
|
e84fdc7eaa48110e607c977dea3e809f07b3fe1f
|
arXiv:astro-ph/0507218v1 8 Jul 2005 Fitting Photometry of Blended Microlensing Events
Christian L Thomas
Kim Griest
arXiv:astro-ph/0507218v1 8 Jul 2005 Fitting Photometry of Blended Microlensing Events
Subject headings: gravitational lensing
We reexamine the usefulness of fitting blended lightcurve models to microlensing photometric data. We find agreement with previous workers (e.g. Woźniak & Paczyński ) that this is a difficult proposition because of the degeneracy of blend fraction with other fit parameters. We show that follow-up observations at specific point along the lightcurve (peak region and wings) of high magnification events are the most helpful in removing degeneracies. We also show that very small errors in the baseline magnitude can result in problems in measuring the blend fraction, and study the importance of non-Gaussian errors in the fit results. The biases and skewness in the distribution of the recovered blend fraction is discussed. We also find a new approximation formula relating the blend fraction and the unblended fit parameters to the underlying event duration needed to estimate microlensing optical depth.
ordinary stars to find microlensing lightcurves. The probability of microlensing occurring to a given star is called the optical depth, τ and is of order 10 −6 or less for many Milky Way lines-ofsight. The smallness of τ means that microlensing experiments concentrate on very crowded star fields where many hundreds of thousands of stars can be simultaneously imaged. This allows many lightcurves to be created simultaneously but also results in blending of the source stars together. This blending causes two problems in using the detected microlensing events to infer the optical depth. First, since each "source object" may contain the light from many stars, the number of stars being monitored is not just the number of objects being photometered. Second, the magnification profile of a microlensing event is changed when unlensed light is blended with the lensed light of the source star.
In this paper we revisit the problem of blending in microlensing lightcurves. There are several methods of dealing with the blending problem. Among these are 1) obtaining high resolution images from space, which will usually allow separationof the source object into its different components, giving a direct measurement of the fraction of light from the lensed source (Alcock, et al. 2001), 2) if the unlensed light is not exactly centered on the lensed source, then the centroid of the light will shift during the microlensing event allowing limits on blending to be placed (Alard, Mao, & Guibert 1995), 3) if the lensed source is a different color than the unlensed light then a color shift will occur as the event proceeds, allowing limits on lensed-light fraction to be made (Alard, Mao, & Guibert 1995), and 4) for image subtraction lightcurves, the source can in principle be removed and this can help break the degeneracy in some cases (Gould & An 2002).
However, we will not discuss the above methods in this paper but will focus on the fitting and interpretation of the photometric data alone; that is, we include the lensed-light fraction as a parameter in the microlensing fit and hope to use the shape of the lightcurve to recover this information. In principle this allows recovery of the actual event duration, and a measurement of the amount of blending in the sample of events, allowing corrections to be made in estimating the optical depth. A related and popular method is to calculate lensing optical depth using only a subsample of very bright source stars (e.g. clump giants). The idea is that very bright source stars are less likely to be blended, and when they are blended, should be blended only by a small amount. In this case one would like to use the blend fits only to determine whether or not a given event is blended.
Unfortunately, as pointed out previously (e.g. Han 1999;DiStefano & Esin 1995;Woźniak & Paczyński 1997;Alard 1997, etc.) blended fits tend to be quite degenerate. A lightcurve with a small lensed-light fraction looks very much like an unblended lightcurve with a smaller maximum magnification and a smaller event duration. As pointed out previously, this means that this fitting method will be of limited use in many cases. Our study adds strength to the conclusions of previous workers, points out several new problems with blend fits, and makes recommendations on how best to proceed with blend fits for those who choose to do them. We will discuss what happens when the microlensing event contains signal from other physical effects such as weak parallax or binary effects. These effects are not rare, and since the difference between blended and unblended lightcurves is small, even an almost undetectable real deviation from the standard point-sourcepoint-lens lightcurve can render blend fit results meaningless.
The plan of the paper is as follows: In § 2 we define our notation and discuss the similarities and differences between unblended microlensing and blended microlensing We also give an analytic approximation that gives the underlying event duration and peak magnification from the lensed-light fraction and the easily measured apparent event duration and maximum magnification.
In § 3 we discuss the usefulness of blend fits and compare with earlier work. In § 4 we discuss the optimal times to take follow-up data in order to improve recovery of parameters from the blend fit. In § 5 we discuss the problem of the baseline magnitude, and In § 6 we discuss the problem of non-Gaussian data and whether the errors returned by fitting programs are reliable.
Degeneracies in blended lightcurves: analytic approximations for event duration and A max
When an isolated lens object crosses close to the line-of-sight of an isolated background source star, the source is magnified and a microlensing lightcurve is generated with magnification
A(u) = u 2 + 2 u(u 2 + 4) 1/2 , u 2 (t) = u 2 min + t − t max t E 2 ,(1)
where u is the projected distance between the lens and source in units of the Einstein ring radius, t E is the time to cross the Einstein radius, and t is time, with maximum magnification, A max , occurring at t max .
The most important parameter is the event duration t E since the optical depth depends upon the sum of efficiency weighted event durations:
τ = π 2E events t E ǫ(t E ) ,(2)
where the exposure E is the product of the length in days of the observing program and the number of observed stars, and ǫ is the efficiency of detecting an event of duration t E .
When other sources of light are contained in the same seeing element as the lensed source star, the microlensing lightcurve is altered since only a fraction of the light is actually lensed:
A ′ (u) = f ll A(u) − f ll + 1,(3)
where f ll is the fraction of light that is lensed (a.k.a. the blend fraction, i.e. coming from the source star) before the lensing event begins. 1 Compared with unblended events, blended photometric microlensing lightcurves suffer from a smaller maximum magnification, A ′ max and shorter event duration t ′ E , as well as from potential color shifts if the blended light has a different spectrum.
In fact, a blended lightcurve looks remarkably like an unblended lightcurve with different values of t E and A max (e.g. Han 1999;DiStefano & Esin 1995;Woźniak & Paczyński 1997;Alard 1997). However, as illustrated in Figure 1, this similarity is not perfect and there are differences in the shapes of blended and unblended lightcurves. It is these differences which give rise to the hope that information about blending can be extracted by fitting lightcurves with blending parameters. If this similarity were perfect, then there would be no use in fitting blended lightcurves to photometric data. Figure 1 shows that the shape differences are typically small, meaning that extracting blending information will be difficult. Woźniak and Paczyński 1997 (WP) studied this in detail and gave regions of the f ll , A max plane where blended fits were useful and where they were not. We return to this subject in § 3, but the qualitative results of WP can be seen from Figure 1, where low magnification events show a maximum difference between the blended lightcurve and the best fit unblended lightcurve of only 1% or so, while the higher magnification events show more substantial differences.
Previous workers have also given analytic formulas relating the measured (apparent) maximum magnification A ′ max , and the apparent event duration t ′ E to f ll , A max , and t E . For example, Woźniak and Paczyński (WP) studied the degeneracies by performing expansions of the above equations in the limits of small u min and large u min and in these limits give the formulas relating the actual values of t E and A max to f ll and the measured A ′ max and t ′ E , For small u min (large A max ) they found u ′ min ≈ u min /f ll , and t ′ E ≈ f ll t E , while in the limit of large u min (A max ≈ 1) they found u ′ min ≈ u min /f DiStefano & Esin (1995), and Han (1999), and Alard (1997) took a different approach, solving equation 3 for A max and giving the actual t E in terms of t ′ E by requiring that the two different 1 Several terms have been used in different ways in the literature for blend fraction, most commonly f b which either means the fraction of light coming from the lens or the fraction of the light coming from non-lens sources. We introduce the new symbol f ll to avoid the extant confusion of nomenclature.
1/4 ll , and t ′ E ≈ f 1/4 ll t E .(a) (b) (c) (d) Fig. 1.
-Four example blended lightcurves (solid) compared with the best fit unblended lightcurve (dashed), as well as the difference, ∆A, between them (blend fit minus unblended fit). The bottom labeled time axis is in units of the apparent Einstein Ring crossing time, t ′ E , that is easily available from the data and an unblended fit. However, the extent of the time axis is ±4t E , (labeled on the top) where t E is the underlying event duration used in optical depth estimates. Thus the extent of all the time axes is roughly 160 days for a typical microlensing event of duration 20 days. Part (a) has values:
f ll = 0.5, u min = 0.4, u ′ min = 0.634, and t ′ E /t E = 0.754. Part (b) has values: f ll = 0.2, u min = 0.2, u ′ min = 0.655, and t ′ E /t E = 0.475. Part (c) has values: f ll = 0.5, u min = 0.03, u ′ min = 0.062, and t ′ E /t E = 0.594. Part (d) has values: f ll = 0.05, u min = 0.03, u ′ min = 0.46, and t ′ E /t E = 0.144.
parameterizations give the same amount of time with A > 1.3416. They found:
A max(HDE) = (A ′ max − 1 + f ll )/f ll , t E (HDE) = t ′ E u 2 1 − u 2 min 1 − u ′ min 2 1/2 ,(4)
where u ′ min = u(A ′ max ), and u 1 = u(A(A ′ = 1.3416)), can be found from the inverse of equation 1:
u(A) = (2/ 1 − 1/A 2 − 2) 1/2(5)
Noting in Figure 1 that the differences between the blended and unblended lightcurves tend to be large in the peak, and that the values of t ′ E and A ′ max are found by fitting, we worried that the HDE formula, which assumes equality in the peak, might not be accurate. We also wondered about the range of applicability of the WP formulas and so decided to test these formulas. We did this by fitting artificial blended lightcurves with an unblended source model and finding the best fit values of t ′ E and A ′ max . We also fit these lightcurves with blended source models and correctly extracted the input blend parameters.
As shown in Figure 2, we found that the WP formulas are not very useful over most of the parameter range, and that the HDE equations work well only over a restricted range of parameters. For the WP formulas this is not surprising since they were created only to show that the degeneracies exist in certain limits. For relatively large lensed-light fraction and for relatively low values of A max the HDE equations give a good estimation of the best fit A ′ max and t ′ E , but for small lensed-light fraction or high A max the estimates of these equation can be far off.
As expected, it is just where the blended lightcurve shape differs the most from an unblended fit that the HDE approximations do not work well. The reason can be seen in Figure 1, where for high magnification events and low lensed-light fraction the blended lightcurve differs strongly in the peak area, but not so much in the lightcurve middle rising and falling regions. Thus, the best fit unblended lightcurve will allow the actual peak magnification to overshoot and compensate for these points by undershooting in the middle regions. Since the HDE formula forces the lightcurves to match at the peak and when A ′ max = 1.34, it will overestimate the best fit peak magnitude and underestimate the event duration.
By studying many such examples, one can come up with a formula that does a better job of relating the best fit A ′ max and t ′ E to A max , t E , and f ll in the parameter ranges where the HDE formula does not work well. The points in Figure 2 show the best fit values of A ′ max and t ′ E vs f ll found by fitting artificial blended lightcurves. The dashed lines show the HDE estimates and long dash lines the WP estimates for t ′ E /t E . At small values of A max (< 3) the HDE formulas do work very well (better than the new formula) and they should be used. However for A max > 3 the HDE formulas do not give accurate estimates. To find a better approximation, one can fit a straight line to the data for a given u min and get a formula which fits well except for very low lensed-light fraction. Repeating this procedure for different values of u min , one discovers that the slopes and zero points of the linear fits are also quite linear in u min . Thus a simple fitting linear formula that covers much of the parameter space can be found. However, if one fits a quadratic for the low f ll events one can get an even better formula which works very well for f ll < 0.3. Thus we find an approximation:
A max ≈ A HDE , ifA max < 3; A ′ max −0.9785+0.4150f ll 0.8153f ll +0.00021
, ifA max > 3 and A ′ max < 10;
A ′ max −0.3618+0.2106f ll 1.0282f ll −0.04433 , ifA max > 3 and A ′ max > 10, ; t ′ E /t E ≈ t ′ E (HDE) /t E , ifA max < 3; (−1.0946u min + 0.9418)f ll + 1.141u min + 0.0564, ifA max > 3 and f ll > 0.3; F CU , ifA max > 3 and f ll < 0.3,(6)
where
F = 1, f ll , f 2 ll , U = 1 u min u 2 min ,(7)
and u min is found from A max and equation 5.
In using this formula, one typically starts with measured values of t ′ E , A ′ max , and an initial guess of A max and uses different (unknown) values of f ll , to find the corresponding underlying A max and t E . If the value of A max found using the new fitting formula is smaller than 3, then one should use the HDE formula instead.
The new fitting formula is shown as the solid line in Figure 2 and does better than HDE or WP for A max > 3. Over the range 0.01 < f ll < 1.1, and 3 < A max < 70 the new fit formula gives a typical error in t E (compared with actually fitting the microlensing lightcurve with a blend fit model) of around 3% and a maximum error of 9%. For A max the typical error is 4% and the maximum error is 12%. The HDE formula can be off by more than 50% in t E and 24% in A max in this region of parameter space. In summary, we tested the HDE formula, Woźniak and Paczyński (WP) formulas and Equation 6 over a wide range of parameters and found the new fitting function works better than HDE for all values of f ll when A max > 3 and A ′ max > 1.34, while the old HDE formula works better for low values of A max and A ′ max . The WP large A max formula gives t E within 10% only for large f ll (> 0.5), and large A max , while the other WP formula is not useful except for A max ≪ 1.34.
Since in microlensing experiments the event durations are found by photometric fitting and since the optical depth is proportional to the sum of the fit t E 's, when making corrections for blending it is important to properly relate the lens-light fraction of each event to the underlying event duration. Woźniak and Paczyński (1997) (WP) studied the degeneracy of blend fits and concluded that in many cases blended and unblended lightcurves cannot be distinguished by photometric fitting. They described areas of parameter space where blend fits would be useful and areas where they would not. While we think that WP did an accurate and very useful calculation, and we agree with their conclusion that blend fits are usually not very useful, we wanted to repeat their analysis for several reasons. First, WP did not include the baseline magnitude in their fits, reasoning that since many measurements are taken before and after the event, the error in baseline magnitude was not significant. In fact, we find that error in the baseline magnitude is one of the most severe problems in blend fits. We find that errors even at the few percent level can drastically alter the parameter values extracted from the fit. Second, WP considered only evenly spaced observations and we wanted to consider whether different follow-up strategies could improve the ability to extract the parameters.
Usefulness of blend fits
In our studies, we find the error in fit parameters three ways. First we create artificial lightcurves using the theoretical formula and add Gaussian random noise to each measurement. We perform blended and unblended fits on these lightcurves using Minuit (CERN Lib. 2003). Second we calculate the error matrix by inverting the Hessian matrix as discussed in Gould (2003). Finally to understand the effect of the non-Gaussianity of the errors in real microlensing experiments we create artificial lensing lightcurves by adding microlensing signal into actual non-microlensing lightcurves obtained by the MACHO collaboration, and then fit these.
Since the method of calculating the error matrix is closest to what WP did, we first give these results. Briefly, we calculate the Hessian matrix (the matrix of second derivatives of the light curve residuals with respect to each parameter) then invert it. The square root of the diagonal elements of the resulting matrix are then the one sigma errorbars of the parameters. This accounts Fig. 3.-Comparison of our results to the previous results of Woźniak and Paczyński one sigma limits on f ll for the range of apparent u min (from non-blend fits). The solid line is from WP, the long dashed line is the 1 − σ f ll limit for a non-blend fit, and the short dashed line is the value of f ll for which f ll + σ f ll = 1 for a blend fit. In the region below the long dashed line blending is detectable at the one sigma level, above the short dashed line blended events are indistinguishable from unblended events, and inbetween the two dashed lines detection is marginal. The region where blending is distinguishable can be scaled with a (eqn. 9).
for correlations in the parameters, but not any nonlinearities. WP used a very similar method, but used it to calculate the ∆χ 2 instead of the error bars. In figure 3 we show that our method brackets WP's. We show limits calculated as both the one sigma lower limit on f ll for an unblended lightcurve and the value of f ll that gives f ll + σ f ll = 1.
We note, as WP found, that parameter errors scale linearly with
a = σ (N )(9)
for N points taken during the peak (defined as lasting 4t E ) 2 . Thus our results can be scaled for other numbers of observations with different values of σ. Thus, we find that our results agree with those of WP if we assume the baseline magnitude is known and take a uniform sampling. Figure 1 shows that the difference between blended and unblended lightcurves is not always uniform across the lightcurve. So if one wanted to plan follow-up observations to improve the accuracy of the blend fit, one should concentrate on the regions of the lightcurve where the differences are largest. Thus it may be possible to do better than WP suggested with their equal spaced observation calculations. To test this hypothesis we calculated the error matrix for blended fits adding in follow-up observations at different points on the lightcurve. As seen in the Figure 1 examples, for any choice of parameters there are five places where the difference lightcurves are maximum, and therefore where follow-up data is more useful than average: at the peak, in the rising/falling portion of the curve, and in the wings. The precise locations change with the choice of parameters but for Figure 1a they are found to be localized near the peak at (|t/t E | < 0.1), in the falling (or rising region) at (0.3 < |t/t E | < 0.6), and near the baseline at (1.0 < |t/t E | < 1.5). Observations taken between these regions do little in constraining the parameters. In addition points t greater than 2t E are very helpful because they fix the baseline in our simulated lightcurve. We discuss the baseline separately in § 5. In Figure 4 we compare the relative value of added points as a func- tion of the time they are added. We find that, in this case, with 40 observations, 4 extra focused observations can reduce the error on f ll by 7.7%. To get the same reduction of error on f ll with evenly distributed observations we would need 7 observations, in other words, each added focused observation is equivalent to increasing the sampling by 1.75 points over 4 t E . The numerical value of the extra effectiveness obtained using focused versus evenly spaced observations varies with underlying parameters and the total number of added points.
Follow-up observations
Precise follow-up measurements at multiple focused locations can improve the determination of f ll even more as they further constrain the shape of the lightcurve. In order to see the effect of adding multiple follow-up observations at two distinct times we compare this with adding evenly spaced observations. In Figure 5 we plot the increase (or decrease) in effectiveness of extra focused observations as a function of the two times at which they are taken. The contours around the light areas show regions of increased effectiveness, while dark areas show areas where the focused observations are less valuable than evenly spaced observations. In this example the effectiveness is increased by up to a factor two.
It is important to note that with more observations or higher accuracy in each follow-up region the advantage per added observation is reduced and the relative values of the various minimums vary, though they stay in roughly the same place. For practical use it is important to note that the time of the optimum second follow-up observation(s) varies with the time of the first follow-up observations(s). In practice one would need to calculate optimum observing times for an event in progress as a function of all the previous measurements.
One problem with the above approach is that without knowing the underlying parameters, particularly t E , it is difficult to predict the best times to take follow-up data. To test if a practical experiment could be designed to take advantage of focused follow-up data we simulated an experiment. First we generated lightcurves with 80 points over 8t E with .05 Gaussian errors at the baseline drawing f ll randomly from the interval [.01, 1) and u min randomly from the interval (0, 1) requiring A ′ max > 1.34 in the HDE approximation. We also adjusted t E to keep t ′ E ∼ 10days also using the HDE approximation. We then generated 9 follow-up observations over 3 days at the peak and fit the first half of the light curve plus the follow-up data. From this first fit we calculated the optimum times for two more bouts of follow-up. We generated these, both with 9 observations over 3 days, and then fit the entire lightcurve with the added 27 points. We also generated 27 points of follow-up uniformly distributed over the 20 days starting at the peak, added it to the initial lightcurve and fit the resulting data. To see the relative improvement for the two methods we calculate a parameter ζ =
(f ′ ll f ocused − f ll )/(f ′ ll unf ocused − f ll ),
which is the ratio of the error in blend fraction given by focused observations to the error in blend fraction given by uniform follow-up sampling. We plot the distribution of ζ in Figure 6 finding that our strategy gives an improvement (ζ < 1) for 71% of the events and a worsening in 29% of the events. We find a substantial improvement (ζ < .5) for 45% of the events, and and even larger improvement (ζ < 0.1) 18% of the time. Thus we conclude that for the the same amount of observing time we can make a more accurate measurement of f ll by focussing the follow-up observations.
In summary we find that observations concentrated at a few times can constrain the microlensing parameters as well as many measurements distributed throughout an event. The best place for these measurements are at the peak, in the falling/rising portion, and in the wings with regions between where added observations do no good. In most cases it is possible to constrain the event parameters well enough with the first half of the data and some follow-up observations near the peak to predict the last two optimum observing times.
Baseline Magnitude
The baseline magnitude of a lightcurve can in principle be very well determined since many measurements can be taken before or after the microlensing event. WP assumed that this was the case and so did not include the baseline magnitude as one of their fit parameters. In real microlensing surveys, however, it may be that the error in average magnitude is not entirely statistical, and may not average down as expected. There may be a systematic limit to the accuracy with which the baseline magnitude can be determined. In fact, detectors and telescope systems drift over time and so measurements made much later may actually reduce the accuracy of the baseline magnitude. To investigate the importance of the baseline magnitude, we created artificial lightcurves without any errors and fit them with a model with a fixed value of baseline magnitude that differed from the actual baseline magnitude by various amounts. Our results are shown in Figure 7. We find that the dependence on baseline is very strong for low amplification events and not as strong for higher amplifications events, but in any case even a 2% error in baseline magnitude determination can strongly bias the recovered blend fraction.
Next, to see how well baseline magnitudes converge in real data, we used the MACHO collaboration database of random stars (Alcock, et al. 2000). We looked at the χ 2 /n dof of a fit to a constant lightcurve for our real lightcurves and compared this to simulated ideal lightcurves with the same number of points and Gaussian errors. For the simulated Gaussian lightcurves we find the χ 2 /n dof distribution peaked near unity and distributed as expected, but for the real data the distribution of χ 2 /n dof is much broader. These two distributions are shown in Figure 8. As an estimate of the error in the baseline which arises due to the systematic drift and non-Gaussian nature of the magnitude errors, we calculated mean and median for the points in each of 146 lightcurves in MACHO field 119, one of the most frequently observed fields. We found that the distribution of mean minus median had a dispersion of 1.3% indicating that the error in the baseline flux is ∼ 1.3%. Referring to Figure 7 we see that for a typical event with u min = 0.5, this implies a typical spread in f ll of 0.18 due to baseline alone. Since half of all events have u min < 0.5, half of all events will have an even larger bias. For more sparsely sampled fields this dispersion due to error in baseline fit would be even larger.
Errors in Fit Parameters
From Macho Project data (Table 6 of Popowski et al. 2005) it seems blend fits return biased parameters. For the set of Macho clump giant events, which are believed to be minimally blended from their positions on the color magnitude diagram, many are best fit with blending. If the events are not blended then a systematic bias in the fits must make them appear to be blended. A systematic bias in recovered lensed-light fraction would lead to a bias in the optical depth as well. The MACHO collaboration investigated the blending of their clump giant sample and decided to use the parameters from the unblended fits. They also used a subsample of events that were less likely to be blended to check for a bias due to blending and found no such bias.
To test for a systematic bias we generate 1000 lightcurves with Gaussian errors for each of three different values for the error on each point: σ = 0.01, σ = 0.05, and σ = 0.15. The recovered lensed-light fractions for these events are shown in Figure 9. As the error on individual datum increases the distribution of f ′ ll becomes increasingly skewed. We find that while the mean f ′ ll may not decrease, the most probable value does decrease. This reduction in the mode is at least partially compensated by the large tail of the distribution with f ′ ll > 1, but for the small number of events a microlensing experiment observes it is unlikely that many of the few events with f ′ ll ≫ 1 will be observed. Even if one event with f ′ ll ≫ 1 is observed it may be ignored as it is an unphysical value of the parameter, thus leading to an underestimate of the average value of f ll . Thus we find that as the errors in measurement increase blend fitting becomes more and more likely to return biased results. The direction of the bias is more often toward small values of f ll . Thus events that are in reality unblended become more and more likely to return fit values implying that they are heavily blended. Fig. 9.-Recovered f ll for data with Gaussian errors of 0.01, 0.05, and 0.15. As the errors on individual data points increase the distribution becomes increasingly skewed with the mode shifting toward 0 for larger errors. Also note that 11% of the σ = 0.05 events and 24% of the σ = 0.15 events had recovered f ′ ll > 2 while none of the events with σ = 0.01 were fit best with f ′ ll > 2.
Conclusions
We find agreement with previous workers that blend fits are problematic, but can be useful especially for high magnification events. When performing blend fits it is helpful to get extra measurements near the peak and at other specific points along the lightcurve. We find that if care is not taken in the treatment of the lightcurve baseline magnitude the fit results can be severely biased and in real data the errors returned on fit parameters should be treated with caution. We find that blend fits return a biased, skewed distribution of the underlying parameters tending to indicate more blending than actually exists. Finally, note that when the microlensing event contains signal from other physical effects such as weak parallax or binary effects blend fits can yield unreliable results. These effects are not rare, and since the difference between blended and unblended lightcurves is small, even an almost undetectable real deviation from the standard point-sourcepoint-lens lightcurve can render blend fit results meaningless.
We thank David P. Bennett and Piotr Popowski for many useful discussions on the topic of blending. This work is supported in part by the DoE under grant DEFG0390ER40546.
Fig. 2 .
2-Comparison of approximation formulas for relating the underlying microlensing event duration t E and maximum magnification A max , to the blend fraction, f ll , and easily measured apparent event duration and maximum magnification, t ′ E , and A ′ max . The circles give the actual blended and unblended results from our lightcurve fitting program, while the solid lines show our new approximation formula. The short dashed line shows the HDE approximation, while the long dashed lines show the WP approximations in their two limits. Part (a) is for an actual u min = .2 (A max = 5.07), part (b) shows u min = .03 (A max = 33), and part (c) shows u min = .4 (A max = 2.65).
Fig. 4 .
4-The equivalent number of uniform follow-up data points required to improve measurement of f ll as much as a single follow-up observation is plotted as a function of when the single follow-up observation is taken. In this case u min = 0.25, f ll = 0.25, and 4 follow-up points are added. Times with N equivalent > 1 are the most effective, while times with N equivalent < 1 are less useful.
Fig. 5 .
5-Number of additional uniformly distributed observations required for the same improvement as 8 focused observations (2 follow-up regions each with 4 observations). The seven contours are 2 (darkest regions), 4, 6, 8, 10, 12, & 14 (lightest regions). In this case u min = 0.25 and f ll = .25. Values above (below) 8 indicate an advantage (disadvantage) relative to uniform follow-up.
Fig. 6 .
6-The ratio ζ showing the advantage of a focused follow-up strategy for 71% of events (ζ < 1 -unshaded region). For 15% of the events in our simulation are outside the range of this plot |ζ| > 2.
Fig. 7 .
7-The recovered f ll for unblended light curves as a function of an input baseline magnitude (fixed at a given value). Forty points over 4 t E are used.
Fig. 8 .
8-Actual distribution of χ 2 /n dof (top) for MACHO data and theoretical distribution of χ 2 /n dof (bottom)
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|
[] |
[
"Relativistic ideal Fermi gas at zero temperature and preferred frame",
"Relativistic ideal Fermi gas at zero temperature and preferred frame"
] |
[
"K Kowalski \nDepartment of Theoretical Physics\nUniversity of Lódź\nul. Pomorska 149/15390-236LódźPoland\n",
"J Rembieliński \nDepartment of Theoretical Physics\nUniversity of Lódź\nul. Pomorska 149/15390-236LódźPoland\n",
"K A Smoliński \nDepartment of Theoretical Physics\nUniversity of Lódź\nul. Pomorska 149/15390-236LódźPoland\n"
] |
[
"Department of Theoretical Physics\nUniversity of Lódź\nul. Pomorska 149/15390-236LódźPoland",
"Department of Theoretical Physics\nUniversity of Lódź\nul. Pomorska 149/15390-236LódźPoland",
"Department of Theoretical Physics\nUniversity of Lódź\nul. Pomorska 149/15390-236LódźPoland"
] |
[] |
We discuss the limit T → 0 of the relativistic ideal Fermi gas of luxons (particles moving with the speed of light) and tachyons (hypothetical particles faster than light) based on observations of our recent paper: K. Kowalski, J. Rembieliński and K.A. Smoliński, Phys. Rev. D 76, 045018 (2007). For bradyons this limit is in fact the nonrelativistic one and therefore it is not studied herein.
|
10.1103/physrevd.76.127701
|
[
"https://arxiv.org/pdf/0712.2728v1.pdf"
] | 119,092,571 |
0712.2728
|
008f9e207e8719cab2da615b8da5d43e8cec3f13
|
Relativistic ideal Fermi gas at zero temperature and preferred frame
17 Dec 2007
K Kowalski
Department of Theoretical Physics
University of Lódź
ul. Pomorska 149/15390-236LódźPoland
J Rembieliński
Department of Theoretical Physics
University of Lódź
ul. Pomorska 149/15390-236LódźPoland
K A Smoliński
Department of Theoretical Physics
University of Lódź
ul. Pomorska 149/15390-236LódźPoland
Relativistic ideal Fermi gas at zero temperature and preferred frame
17 Dec 2007numbers: 0330+p0520-y0530-d0570-a0570Ce
We discuss the limit T → 0 of the relativistic ideal Fermi gas of luxons (particles moving with the speed of light) and tachyons (hypothetical particles faster than light) based on observations of our recent paper: K. Kowalski, J. Rembieliński and K.A. Smoliński, Phys. Rev. D 76, 045018 (2007). For bradyons this limit is in fact the nonrelativistic one and therefore it is not studied herein.
In our very recent paper [1] the Lorentz covariant formulation has been introduced of classical and quantum statistical mechanics and thermodynamics of an ideal gas of relativistic particles. An advantage of this formulation based on the preferred frame approach is among others the possibility of a consistent free of paradoxes description of tachyons. In our discussion of the limit T → 0 in the case of the relativistic quantum ideal gas we restricted in [1] to the case of the Bose gas. In this work we complete the results given in [1] by considering the limit T → 0 for the Fermi gas. As with the Bose gas, we do not discuss the limit T → 0 in the case of bradyons (particles slower than light) because this limit is in fact the nonrelativistic one [2]. Consider first the degenerate Fermi gas of tachyons. The covariant forms of thermodynamic functions for that gas derived in [1] are given by
U V = m 2 2π 2 u 2 0 β 2 ∞ n=1 (−1) n+1 n 2 [3S 0,2 (nu 0 βm) − nu 0 βmS −1,1 (nu 0 βm)]s n ,(1)p kT = m 2 2π 2 u 2 0 β ∞ n=1 (−1) n+1 n 2 S 0,2 (nu 0 βm)s n ,(2)N V = m 2 2π 2 u 2 0 β ∞ n=1 (−1) n+1 n S 0,2 (nu 0 βm)s n ,(3)
where β = 1/(kT ), m is a mass of a particle, s = e βµ is the fugacity, µ is the chemical potential, u 0 is the zeroth (covariant) component of the four velocity u µ of the preferred frame with repect to the inertial observer (in the preferred frame u 0 = 1), and S µ,ν (x) is the Lommel function (see the appendix). Using (A.5) we obtain the following asymptotic form of (1), (2), and (3) in the limit T → 0 (i.e. β → ∞):
U V = m π 2 u 3 0 β 3 ∞ n=1 (−1) n+1 n 3 s n ,(4)p kT = m 2π 2 u 3 0 β 2 ∞ n=1 (−1) n+1 n 3 s n ,(5)N V = m 2π 2 u 3 0 β 2 ∞ n=1 (−1) n+1 n 2 s n .(6)
Now applying the asymptotic form of the Fermi functions
f ν (x) = ∞ n=1 (−1) n+1 n ν x n(7)
known as Sommerfeld's lemma [3], such that
f ν (e βµ ) (8) ≃ (βµ) ν Γ(ν + 1) 1 + ν(ν − 1) π 2 6 (βµ) −2 + ν(ν − 1)(ν − 2)(ν − 3) 7π 4 360 (βµ) −4 + . . . , βµ ≫ 1, we find U V = mµ 3 6π 2 u 3 0 1 + π 2 k 2 T 2 µ 2 ,(9)p = mµ 3 12π 2 u 3 0 1 + π 2 k 2 T 2 µ 2 ,(10)N V = mµ 2 4π 2 u 3 0 1 + π 2 3 k 2 T 2 µ 2 .(11)
We remark that as with the case of the Bose gas of tachyons [1], (9) and (10)
Now, it follows immediately from (11) that the Fermi energy ε F is given by
ε F = 4π 2 u 3 0 N mV ,(13)
where ε F = µ(T = 0). Furthermore, eqs. (11) and (13) yield
ε 2 F = µ 2 + π 2 3 k 2 T 2 ,(14)
which leads to
µ = ε F − π 2 6ε F k 2 T 2 .(15)
We point out that in this section we perform calculations up to terms of order T 2 . Making use of (14) and (15) we can write (9) and (10) in the following form:
U = mV 6π 2 u 3 0 ε 3 F + π 2 2 ε F k 2 T 2 ,(16)p = m 12π 2 u 3 0 ε 3 F + π 2 2 ε F k 2 T 2 .(17)
Consider now the entropy which can be defined as
S = U + pV − µN T .(18)
Taking into account (9), (10), (11) and (15) we get
S = mk 2 6u 3 0 V ε F T.(19)
Therefore, the entropy vanishes at T = 0. The formula (19) can be also obtained from the well-known relation
S = T 0 C V T dT,(20)
where
C V = ∂U ∂T V = mk 2 6u 3 0 V ε F T,(21)
which is immediate consequence of (16).
Finally, we study the limit T → 0 in the case of the Fermi gas of luxons. Using the formulas on thermodynamic functions derived in [1] such that
U V = 3 π 2 u 4 0 β 4 ∞ n=1 (−1) n+1 n 4 s n ,(22)p kT = 1 π 2 u 4 0 β 3 ∞ n=1 (−1) n+1 n 4 s n ,(23)N V = 1 π 2 u 4 0 β 3 ∞ n=1 (−1) n+1 n 3 s n ,(24)
and proceeding analogously as with tachyons, we obtain
U V = µ 4 8π 2 u 4 0 1 + 2π 2 k 2 T 2 µ 2 ,(25)p = µ 4 24π 2 u 4 0 1 + 2π 2 k 2 T 2 µ 2 ,(26)N V = µ 3 6π 2 u 4 0 1 + π 2 k 2 T 2 µ 2 .(27)
We point out that (25) and (26) imply the well-known equation of state for ideal gas of massless particless such that
pV = U 3 .(28)
Furthermore, the Fermi energy is
ε F = 6π 2 u 4 0 N V 1 3 .(29)
Hence, using (27) we find
ε 3 F = µ 3 + π 2 k 2 T 2 µ,(30)
implying
µ = ε F − π 2 3ε F k 2 T 2 .(31)
The relations (25) and (26) written in terms of the Fermi energy take the form
U = V 8π 2 u 4 0 ε 4 F + 2π 2 3 ε 2 F k 2 T 2 ,(32)p = 1 24π 2 u 4 0 ε 4 F + 2π 2 3 ε 2 F k 2 T 2 .(33)
The entropy calculated with the help of (18) is
S = k 2 6u 4 0 V ε 2 F T.(34)
Thus, as with tachyons, the entropy in the case of fermionic luxons also vanishes. The above formula on the entropy is also implied by (20) and
C V = ∂U ∂T V = k 2 6u 4 0 V ε 2 F T,(35)
following directly from (32).
In conclusion, we have derived in this work the Lorentz covariant form of thermodynamic functions of Fermi ideal gas of luxons and tachyons in the zero temperature limit. It seems that the observations of this paper would be of interest for testing the hypothesis of tachyonic neutrinos [4] and tachyonic dark matter [5]. Indeed, the existing estimations of density of neutrinos suggest that the model of an ideal gas is appropriate for the neutrino background.
The Lommel functions S µ,ν (x) are defined recurrently S µ+2,ν (x) = x µ+1 − [(µ + 1) 2 − ν 2 ]S µ,ν (x), (A.3)
S ′ µ,ν (x) = ν x S µ,ν (x) + (µ − ν − 1)S µ−1,ν+1 (x) = − ν x S µ,ν (x) + (µ + ν − 1)S µ−1,ν−1 (x).
(A.4)
The asymptotics S 0,2 (x) for x ≫ 1 is of the form S 0,2 (x) = 1
x , x ≫ 1.
(A.5)
AcknowledgementsThis paper has been supported by University of Lodz grant.APPENDIXWe recall some basic properties of the Lommel functions S µ,ν (x)[6]. The Lommel func-The Lommel functions S ν,ν (x) can be expressed by means of the Struve functions H ν (x) andthe Bessel functions Y ν (x) (Neumann functions) also designated by N ν (x). Namely, we have
. K Kowalski, J Rembieliński, K A Smoliński, Phys. Rev. D. 7645018K. Kowalski, J. Rembieliński and K.A. Smoliński, Phys. Rev. D 76, 045018 (2007).
. C Aragão De Carvalho, S. Goulart RosaJr, J. Phys. A. 133233C. Aragão de Carvalho and S. Goulart Rosa Jr, J. Phys. A 13, 3233 (1980).
R K Pathria, Statistical Mechanics. Pergamon, New YorkR.K. Pathria, Statistical Mechanics (Pergamon, New York, 1977).
. J Ciborowski, J Rembieliński, Eur. Phys. J. C. 8157J. Ciborowski and J. Rembieliński, Eur. Phys. J. C 8, 157 (1999);
. A Chodos, A I Hauser, A V Kostelecký, Phys. Lett. B. 150431A. Chodos, A.I. Hauser and A.V. Kostelecký, Phys. Lett. B 150, 431 (1985).
. P C W Davies, Internat. J. Theoret. Phys. 43141P.C.W. Davies, Internat. J. Theoret. Phys. 43, 141 (2004);
. A Das, Phys. Rev. D. 7243528A. Das et al, Phys. Rev. D 72, 043528 (2005).
I S Gradshteyn, I M Ryzhik, Tables of Integrals, Series, and Products. New YorkAcademic PressI.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 2000);
H Bateman, Higher Transcendental Functions. New YorkMcGraw-Hill2H. Bateman, Higher Transcendental Functions, vol. 2, (McGraw-Hill, New York, 1953).
|
[] |
[
"Phenomenological Aspects of R-parity Violating Supersymmetry with A Vector-like Extra Generation",
"Phenomenological Aspects of R-parity Violating Supersymmetry with A Vector-like Extra Generation"
] |
[
"Xue Chang \nInstitute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina\n",
"Chun Liu \nInstitute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina\n",
"Yi-Lei Tang \nInstitute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina\n"
] |
[
"Institute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina",
"Institute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina",
"Institute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina"
] |
[] |
Phenomenological analysis to the R-parity violating supersymmetry with a vector-like extra generation is performed in detail. It is found that, via the trilinear couplings, the correct neutrino spectrum can be obtained. The Higgs mass rises to 125 GeV by new up-type Yukawa couplings of vector-like quarks with no need of very heavy superpartners. Phenomena of new heavy fermions at LHC are predicted.Recently a standard model (SM) Higgs-like particle with a mass of 125 − 126 GeV was discovered [1]. In the paradigm of the weak scale supersymmetry (SUSY) which aims at the naturalness of the electro-weak scale, however, such a Higgs mass brings in tensions, especially the minimal SUSY SM (MSSM). Nonminimal and still natural scenarios of SUSY are thus motivated. One of them is the MSSM with a vector-like generation [2-5]. It gives the right Higgs mass naturally, is consistent with precision electroweak measurements, and has a rich phenomenology [2-6]. In the framework of SUSY, vector-like fermions can also be motivated by other theories beyond SM, such as SUSY extension with extra-dimensions or with composite states [7]. So it is worth asking the question whether such a scenario also provides explanations to other problems such as neutrino masses. Neutrino oscillations are the undoubted new physics beyond the SM. Daya Bay [8] and RENO [9] experiments recently discovered a relatively large θ 13 ≃ 8.8 • ± 0.8 • . Within the framework of SUSY, in the absence of R-parity conservation, neutrino masses and mixings can be generated from lepton number violating (LPV) couplings [10]. This approach was extensively studied before [11]. It is known that all the neutrino experimental results, including that of oscillation phenomena like the large atmospheric mixing angle θ 23 , the hierarchy of oscillation frequencies ∆m 2 21 ≪ ∆m 2 32 and the smallness of θ 13 , can be understood in three generation LPV MSSM. However, this needs some special requirements for relevant coupling constants and mass parameters. Combining both considerations above, we will work in the LPV MSSM with a vector-like extra generation [4]. While this model takes the vector-like slepton doublets as the two Higgs doublets needed for the electroweak symmetry breaking, the SM-like Higgs mass can be naturaly 125 GeV [5]. Extra trilinear LPV couplings between ordinary fermions and vector-like fermions provide a much larger parameter space to explain neutrino pheomena right.In this paper, phenomenological aspects of the model will be analyzed. In Sect. II, we make a brief review of the model. In Sect. III, neutrino masses are calculated. For the neutrino physics, noting the enlarged parameter space, we consider trilinear LPV couplings carefully. One-loop contribution to neutrino masses due to new trilinear LPV couplings is calculated, theoretical analysis are performed and numerical results are shown in detail.
|
10.1103/physrevd.87.075012
|
[
"http://arxiv.org/pdf/1303.7055"
] | 118,790,054 |
1303.7055
|
f8bc116e9f68b1623a88ba2c41e9e4a12ddd9308
|
Phenomenological Aspects of R-parity Violating Supersymmetry with A Vector-like Extra Generation
31 Mar 2013
Xue Chang
Institute of Theoretical Physics
State Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
P.O. Box 2735100190BeijingChina
Chun Liu
Institute of Theoretical Physics
State Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
P.O. Box 2735100190BeijingChina
Yi-Lei Tang
Institute of Theoretical Physics
State Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
P.O. Box 2735100190BeijingChina
Phenomenological Aspects of R-parity Violating Supersymmetry with A Vector-like Extra Generation
31 Mar 2013arXiv:1303.7055v2 [hep-ph]numbers: 1260Jv1460St1460Hi1465Jk * Electronic address: chxue@itpaccnliuc@mailitpaccntangyilei@itpaccn
Phenomenological analysis to the R-parity violating supersymmetry with a vector-like extra generation is performed in detail. It is found that, via the trilinear couplings, the correct neutrino spectrum can be obtained. The Higgs mass rises to 125 GeV by new up-type Yukawa couplings of vector-like quarks with no need of very heavy superpartners. Phenomena of new heavy fermions at LHC are predicted.Recently a standard model (SM) Higgs-like particle with a mass of 125 − 126 GeV was discovered [1]. In the paradigm of the weak scale supersymmetry (SUSY) which aims at the naturalness of the electro-weak scale, however, such a Higgs mass brings in tensions, especially the minimal SUSY SM (MSSM). Nonminimal and still natural scenarios of SUSY are thus motivated. One of them is the MSSM with a vector-like generation [2-5]. It gives the right Higgs mass naturally, is consistent with precision electroweak measurements, and has a rich phenomenology [2-6]. In the framework of SUSY, vector-like fermions can also be motivated by other theories beyond SM, such as SUSY extension with extra-dimensions or with composite states [7]. So it is worth asking the question whether such a scenario also provides explanations to other problems such as neutrino masses. Neutrino oscillations are the undoubted new physics beyond the SM. Daya Bay [8] and RENO [9] experiments recently discovered a relatively large θ 13 ≃ 8.8 • ± 0.8 • . Within the framework of SUSY, in the absence of R-parity conservation, neutrino masses and mixings can be generated from lepton number violating (LPV) couplings [10]. This approach was extensively studied before [11]. It is known that all the neutrino experimental results, including that of oscillation phenomena like the large atmospheric mixing angle θ 23 , the hierarchy of oscillation frequencies ∆m 2 21 ≪ ∆m 2 32 and the smallness of θ 13 , can be understood in three generation LPV MSSM. However, this needs some special requirements for relevant coupling constants and mass parameters. Combining both considerations above, we will work in the LPV MSSM with a vector-like extra generation [4]. While this model takes the vector-like slepton doublets as the two Higgs doublets needed for the electroweak symmetry breaking, the SM-like Higgs mass can be naturaly 125 GeV [5]. Extra trilinear LPV couplings between ordinary fermions and vector-like fermions provide a much larger parameter space to explain neutrino pheomena right.In this paper, phenomenological aspects of the model will be analyzed. In Sect. II, we make a brief review of the model. In Sect. III, neutrino masses are calculated. For the neutrino physics, noting the enlarged parameter space, we consider trilinear LPV couplings carefully. One-loop contribution to neutrino masses due to new trilinear LPV couplings is calculated, theoretical analysis are performed and numerical results are shown in detail.
I. INTRODUCTION
Besides, we analyze the SM-like Higgs mass and explicitly show that it can be increased to 125 GeV by two new Yukawa couplings of the up-type Higgs with vector-like quarks in Sect.
IV. The LHC phenomenology of the new fermions is analyzed in Sect. VI. The summary and discussions are given in the last section.
II. A BRIEF REVIEW OF THE MODEL
This model [4] is SUSY and SM gauge invariant, and R-parity violation with baryon number conservation is assumed. For the matter content, in addition to the ordinary 3 generations (3G), a vector-like generation is introduced in. Without R-parity conservation, this can be also thought as that there are 4 + 1 chiral generations, where '4' stands for four chiral generations with SM quantum numbers and '1' for another chiral generation with opposite quantum numbers. The 4 chiral generations with same quantum numbers mix.
The '1' has Dirac masses with only one combination of the '4', thus, there are always SM required three massless chiral generations and one massive vector-like generation.
In terms of mass eigenstates (before electroweak symmetry breaking), the massive slep- The superpotential is conveniently written as
W = W 0 + W L ,(1)
where W 0 and W L stand for that with lepton number conservation and LPV, respectively,
W 0 = µH u H d + µ e E c 4 E c H + µ Q Q 4 Q H + µ U U c 4 U c H + µ D D c 4 D c H + y l ij L i H d E c j + y d ij Q i H d D c j +y u ij Q i H u U c j + y E i L i H d E c 4 + y Q′ i Q 4 H d D c i + y D i Q i H d D c 4 + y QD Q 4 H d D c 4 + y U i Q i H u U c 4 +y Q i Q 4 H u U c i + y QU Q 4 H u U c 4 + y H Q H H d U c H + y H ′ Q H H u D c H ,
and
W L ⊃ λ ijk L i L j E c k + λ ′ ijk Q i L j D c k + λ E ij L i L j E c 4 + λ Q ij Q 4 L i D c j +λ D ij Q i L j D c 4 + λ QD i Q 4 L i D c 4 + λ H i Q H L i U c H .(2)
where L i , Q i , E c i , D c i , U c i , i=1-3, are the first three generation SU(2) L doublet leptons, doublet quarks, singlet charged leptons, singlet down-type quarks and singlet up-type quarks, respectively. H u and H d are the up-type and down-type Higgs. Note that the term Q H H u D c H in W 0 was missed in Ref. [4]. 1 And in W L interactions of purely singlets are omitted, which are irrelevant to our study.
By assuming universality of the mass-squared terms, the alignment of the B terms the soft mass terms and the trilinear soft terms of all fermion's superpartners in the model are
− L ⊃ M 2L † iL i + M 2 H † d H d + M 2 h H † u H u + M 2 EẼ+(BµH d H u + B e µ eẼc 4Ẽ c H + B Q µ QQ 4QH + B U µ UŨ c 4Ũ c H + B D µ DDc 4D c H + h.c.) .(3)
Proper values of the new B Q,U,D µ Q,U,D terms are set to avoid unwanted color symmetry and purely U(1) Y symmetry breaking, see Eq. (11,12) in paper [4], therefore EWSB in our model is just the same as in MSSM. After EWSB, the specific fermion mass matrixes and sfermion mass-squared matrixes are given in Appendix A.
III. NEUTRINO MASSES AND MIXINGS
LPV results in nonvanishing neutrino masses. In this model, in addition to traditional Rparity violation in the MSSM, a lot more bilinear and trilinear LPV interactions are brought in through the vector-like generation. In this work, the trilinear R-parity violating interactions will be studied. To avoid complication due to too many LPV sources, sneutrino VEVs will not be considered. The trilinear LPV Lagrangian relevant to neutrino masses is from W L ,
L ⊂ −λ ijk (l * kRν c iR l jL +l jLlkR ν iL ) − λ ′ ijk (d * kRν c iR d jL +d jLdkR ν iL ) −λ E ij (Ẽ c 4ν c iR l jL +l jL E cT 4 ν iL ) − λ Q ij (d * kRν c iR Q 4 +Q 4dkR ν iL ) (4) −λ D ij (D c 4ν c iR d jL +d jLD cT 4 ν iL ) − λ QD i (D c 4ν c iR Q 4 +Q 4D cT 4 ν iL ) −λ H i (Ũ c Hν c iR Q H +Q HŪ cT H ν iL ) + h.c. .
whereν c iR stands for the left-hand neutrino. The 7 types of trilinear LPV interactions in the above equation induce 14 types of one-loop diagrams contributing to the neutrino spectrum, which are proportional to λλ,
λ ′ λ ′ , λ E λ E , λλ E , λ Q λ Q , λ Q λ D , λ D λ D , λ H λ H , λ QD λ QD , λ ′ λ Q , λ ′ λ D , λ ′ λ QD , λ Q λ QD , λ D λ QD , respec-
tively. The Feynman diagrams and the corresponding analytical results are shown in Fig. 1 in Appendix B. For simplicity and without losing our purpose, in the Yukawa interactions of W 0 we assume that only y E , y Q ′ , y Q ′ , y D , y U , y H , y H ′ are nonvanishing, that is vector-like particles have Yukawa interactions only with the third generation. Thus, the vector-like generation has little constraints from the collider phenomenology.
Before starting to analyze the neutrino mass spectrum, some assumptions are introduced in order to control the parameter space and get relatively simple analytical result.
Since four new up-type Higgs Yukawa couplings y U , y Q , y QU , y H ′ and five new downtype Higgs Yukawa couplings y E , y D , y Q ′ , y QD , y H appear in our model, and among which y QD , y QU , y H , y H ′ provide the mass mixings between vector-like generations, and further more, they have infrared quasi-fixed point [5], so we assume y QD = y H = 0 and
y QU ∼ y H ′ ≡ y t V ≤ 1.
We also set y D = y Q ′ = 0, y E < 0.04 and y U ∼ y Q ≡ y t 34 ≤ 0.08. In other words, we neglect all new down-type Higgs Yukawa couplings in quark sectors while consider all of the new up-type Higgs Yukawa couplings only and take y t 34 ≪ y t V , which is a reasonable assumption.
Basing on the above assumptions, contributions from λλ, λ ′ λ ′ type diagrams can be simplified to the familiar forms [12][13][14] analyzed numerically and will be discussed later.
M ν ij | λλ ≃ 1 8π 2 λ i33 λ j33 m τ sin ατ cos ατ lnτ R τ L , M ν ij | λ ′ λ ′ ≃ 3 8π 2 (λ ′ i33 λ ′ j33 m b sin αb cos αb lnb R b L + λ ′ i23 λ ′ j32 m s sin αb cos αb lnb R b L + λ ′ i32 λ ′ j23 m b sin αs cos αs lns R s L ),(5)
At last, without loss of generality, among all 7 types of LPV trilinear couplings, we take 4 of them, λ E , λ D , λ Q and λ H , for consideration while assuming the rest of them, λ, λ ′ and λ QD , are negligible. The realization through different LPV trilinear coupling combinations can be derived straightforwardly. The method to calculate the neutrino mass matrix we use is given in Appendix C.
Here we list the parameters of neutrino oscillation given by experiments, ∆m 2 21 = (7.59 ± 0.21) × 10 −5 eV 2 , ∆m 2 32 = (2.43 ± 0.13) × 10 −3 eV 2 and sin 2 2θ 12 = 0.861 +0.026 −0.022 , sin 2 2θ 23 > 0.92, sin 2 2θ 13 = 0.088 ± 0.008. Scanning the parameter space with proper EWSB, we find by adjusting the ratios and values of the LPV trilinear couplings we choosing, the correct neutrino spectrum can be generated through the λ E λ E , λ D λ Q and λ H λ H type one-loop diagrams. Numerical illustration is shown in Table I, in Set I we take the mass mixings assumptions mentioned before, in Set II we take different mass mixings and bigger vectorlike masses for comparison. The specific parameters settings see Appendix C.
That is by choosing (for Set I)
λ Q 13 λ Q 23 ∼ 0.25 , λ Q 33 λ Q 23 ∼ 1.4 , λ Q 23 ∼ 2.1 × 10 −6 , λ Q 13 ∼ λ D 13 , λ Q 23 ∼ λ D 23 , λ Q 33 ∼ λ D 33 , λ H 1 λ H 2 ∼ 1.4 , λ H 3 λ H 2 ∼ 1 , λ H 2 ∼ λ Q 23 , λ E 13 ∼ λ E 23 ∼ λ E 33 ∼ λ Q 23 ,(6)
we have m ν2 m ν3 ∼ 0.17 , m ν2 ∼ 5.9 × 10 −4 eV , m ν1 ∼ 5.1 × 10 −6 eV ,
sin θ 13 ∼ 0.143 , sin θ 23 ∼ 0.581 , sin θ 12 ∼ 0.559 . (7) M ν ij | λ E i3 λ E j3 M ν ij | λ D i3 λ Q j3 M ν ij | λ D i3 λ D j3 M ν ij | λ Q i3 λ Q j3 M ν ij | λ H i λ H jUnlike in the 3G LPV case, where λ ′ i33 λ ′ j33 , λ ′ i23 λ ′ j32 + λ ′ i32 λ ′ j23
and λ i33 λ j33 type one-loop contributions are dominant, subdominant and next-to-subdominant, here in our model, under the assumptions mentioned before, λ Q i3 λ D j3 , λ H i λ H j and λ E i3 λ E j3 type one-loop contributions are dominant, subdominant and next-to-subdominant, respectively. This is because the new fermions τ 1 , t 1,2 , b 1,2 in the internal lines, see Fig. 1, are much heavier than the third generation fermions τ, t, b. For the same reason, our requirements of the new LPV couplings we choose are of order 10 −6 and small enough to avoid measurable FCNC decays such as µ → eγ [15]. It worth to note that by decoupling the vector-like generation, correct neutrino masses and mixings cannot be obtained via λλ, λ ′ λ ′ type one-loop contributions.
ν iL ν jL τ ,τ 1 τ, τ 1 λ E i3 λ E j3 ν iL ν jL b,b 1,2 b, b 1,2 λ D i3 λ Q j3 ν iL ν jL t,
In addition, λ H λ H type contribution containing up-type (s)quarks in the internal lines is absent in 3G LPV models because the vector-like down-type doublet quark Q b H mixe with the right hand singlet top quark.
From Table I, we can also see that by choosing λ Q i3 λ D j3 , λ Q i3 λ Q j3 and λ D i3 λ D j3 type one-loop contributions, the correct neutrino spectrum can also been generated in parameters Set II, we don't list the detailed results here.
IV. HIGGS MASS
There are four new up-type Higgs Yukawa couplings in our model, y U , y Q , y QU , y H ′ , corresponding to the Yukawa mass, m t 34 , m t 43 , m t 44 , m b H , separately, and also five new down-type Higgs Yukawa couplings, y E , y D , y Q ′ , y QD , y H , corresponding to the Yukawa
mass, m τ 34 , m b 34 , m b 43 , m b 44 , m t H , separately.
The related superpotential contributing to the lightest scalar Higgs mass is shown in W 0 . According to the assumptions mentioned in the last section, we neglect the down-type Higgs Yukawa contributions and the small up-type contributions between the SM third generations and the extra vector-like generations. The relevant superpotential can be simplified as
W = µ Q Q 4 Q H + µ D D c 4 D c H + y H ′ Q H H u D c H + y QU Q 4 H u U c 4 ,(8)
So when neglecting the small D-term and the two-loop contribution, the new one-loop contribution to the lightest scalar Higgs square-mass is [5,16]
△ m 2 h = 3 × 2 4π 2 (y t V ) 4 v 2 sin 4 β [t V − 1 6 (5 − 1 x )(1 − 1 x ) + 2 X 2 V M 2 S (1 − 1 3x )],(9)
where v = 174Gev indicates the Higgs VEV and
y H ′ = y QU ≡ y t V , x = M 2 S /M 2 V , t V = log M 2 S M 2 V ,(10)(A H ′ − µ H ′ cot β) 2 = (A QU − µ QU cot β) 2 = (A V − µ V cot β) 2 ≡ X 2 V ,
in which, for simplicity, µ Q = µ D ≡ M V stands for the vector-like mass of the new up-type (4)) and M S = M 2 V + m 2 stands for average mass of the new up-type squarks.
quarks, M 2 Q = M 2 D ≡ m 2 (see Eq.
In MSSM, the Higgs mass from the t,t one-loop contributions is about 110 GeV, for A t = µ = 400 GeV, mt = 400 GeV and tan β = 10. Direct search bounds from CMS for exotic heavy top-like quark set limits of M t ′ > 557 GeV if B(t ′ → W b) = 1 [17] and [18]. When considering the mass mixing between the vector-like quarks and the SM third generation quarks, in other words, considering the realistic branch ratios, the mass limit is adjust to be M t ′ > 415 GeV [19,20]. So if we set the vector-like fermion masses in our model to be M V ∼ 500 GeV, the soft supersymmetrybreaking parameters to be m ∼ 700 GeV, A V = µ V ∼ 500GeV and B V µ V ∼ 500 2 GeV 2 , then from Eq. (10), in order to get approximately 125 GeV Higgs mass, for about M V = 500 GeV and M S = 850 GeV, we just need to set y t V ∼ 1, or say, need to set m t 44 = m b H ≡ m t V ∼ 174 GeV. These values are just near their infrared quasi-fix point, as mentioned in last section.
M t ′ > 475 GeV if B(t ′ → Zt) = 1
Evoked by the ATLAS and CMS discovery of the enhancement in γγ channel and little deviation in ZZ channel [22,23], the effects of the exotic vector-like quarks to the Higgs production and decay have been extensively studied recently [21]. In general, in a theory with N vector-like generations extension, the new fermion contributions are suppressed by 21,24]. So only the very large couplings to the Higgs can obviously enhance the Higgs production and decay in the γγ channel [21], but as we have mentioned, these couplings have quasi-fix point which limits their TeV values to be about 1 [5]. This value is large enough to accommodate m h ∼ 125 GeV, but too small to influence the Higgs decay, one can't depend on vector-like fermions by themselves to modify the Higgs decay branching ratios. As far as the Higgs problem to be concerned, extra vector-like fermions are mainly introduced to adjust the Higgs mass. However, the γγ and ZZ channel anomaly, if they persist, can be realized through the light stop scenario [25], which beyond our scope in this paper.
N 2 m t2 V /M 2 V [
V. THE EXTRA VECTOR-LIKE FERMION DECAYS
To be clear, we list the new extra vector-like fermions below:
Ψ E = E c H E c 4 , Ψ Q = Q t,b 4 Q t,b H , Ψ U = U c 4 U H , Ψ D = D c 4 D H ,(11)in which E c H mixes with τ L ; E c 4 mixes with τ R ; Q b 4 , D c H mixes with b L ; D c 4 , Q t H mixes with b R , Q t 4 ; U c H mixes with t L , U c 4 , Q b H mixes with t R .
These exotic heavy fermions can decay into SM bosons, see Fig. 2, which will analyze bellow. Our analysis agree with the results given in [5]. However the slightly difference comes from their neglect of the contributions proportional to s 2 W in the vertex of Feynman rules. Note that theoretically speaking, when kinematically allowed, the exotic fermions predicted in our model have the other two decay modes: through supersymmetric gauge kinetic interactions or the supersymmetric Yukawa interactions, decay into chargino/neutralino and sfermions, such as (2), decay into fermions and sfermions, such as
τ 1 →C +ν τ , b 1 →Ñ ib , t 1 →C −b , whereÑ i , i=1-4, is neutralino,C ± is chargino; through LPV interactions, see Eq.τ 1 τ h 0 τ 1 τ Z τ 1 ν W + b 1,2 b, b 1 h 0 b 1,2 b, b 1 Z b 1,2 t, t 1 W + t 1,2 t, t 1 h 0 t 1,2 t, t 1 Z t 1,2 b, b 1 W − FIG. 2:
Tree-level decay of new exotic fermions in our model, all fermions stay in mass eigenstates.
τ 1 → eμ, b 1 →τ t, t 1 →ẽb.
Although the kinematical conditions for the latter decay mode are easy to be satisfied, but we have already seen in section III, the LPV couplings in our model, in order to explain the neutrino spectrum, are of order 10 −6 , so we can neglect this kind of decay channels reasonably. On the other hand, for simplicity here in our work, we assume the former decay mode is not kinematically allowed. Therefore, the exotic fermions can only decay into SM bosons.
A. τ 1 decays
The weak bosons interaction Lagrangian to τ, τ 1 is
L ⊃ g W τ 1L νττ 1L γ µ ν τ L W − µ + g Z τ 1L τ Lτ 1L γ µ τ L Z µ + g Z τ 1R τ Rτ 1R γ µ τ R Z µ +g h 0 τ 1L τ Rτ 1L τ R h 0 + g h 0 τ L τ 1Rτ L τ 1R h 0 + h.c.,(12)
the couplings and the decay widths of τ 1 are given in Appendix D.
The main characteristic of the lepton sector is that there must be mass mixing between the third and the vector-like lepton, otherwise the new heavy charged leptons τ 1 will be stable and give unacceptable cosmological heavy relic [26]. For Specific, when y E = 0, the off-diagonal elements of L τ , R τ are equal to zero. That's why we set y E = 0 in section II while discussing neutrino spectrum, more specifically, we set y E ≤ 0.04. Under these parameters settings, numerical results of τ 1 decay into W, Z, h 0 are shown in Fig. 3, we The weak bosons interaction Lagrangian to t, t 1 , t 2 is The weak bosons interaction Lagrangian to b, b 1 , b 2 is
L ⊃ g W t 1L b Lt 1L γ µ b L W − µ + g W t 2L b Lt 2L γ µ b L W − µ + g W t 1L b Rt 1R γ µ b R W − µ + g W t 2R b Rt 2R γ µ b R W − µ g W t 2L b 1Lt 2L γ µ b 1L W − µ + g WL ⊃ g W b 1L t Lb 1L γ µ t L W − µ + g W b 2L t Lb 2L γ µ t L W − µ + g W b 1L t Rb 1R γ µ t R W − µ + g W b 2R t Rb 2R γ µ t R W − µ +g W b 2L t 1Lb 2L γ µ t 1L W − µ + g W b 2R t R1b 2R γ µ t 1R W − µ + g Z b 1L b Lb 1L γ µ b L Z µ + g Z b 2L b Lb 2L γ µ b L Z µ +g Z b 2L b 1Lb 2L γ µ b 1L Z µ + g Z b 1R b Rb 1R γ µ b R Z µ + g Z b 2R b Rb 2R γ µ b R Z µ + g Z b 2R b 1Rb 2R γ µ b 1R Z µ +g h 0 b 1L b Rb 1L b R h 0 + g h 0 b L b 1Rb L b 1R h 0 + g h 0 b 2L b Rb 2L b R h 0 + g h 0 b L b 2Rb L b 2R h 0 +g h 0 b 2L b 1Rb 2L b 1R h 0 + g h 0 b 1L b 2Rb 1L b 2R h 0 + h.c.(14)
the couplings and the decay widths of b 1,2 are given in Appendix D. Based on our previous work about bilinear LPV couplings, further research on the renormalization group (RG) of them is worthy to be studied in the future. There are also plenty of aspects to be further analyzed in the area of new fermion LHC phenomenology based on this model.
Appendix A: THE (S)FERMION MASS MIXING MATRIXES
Because the CP violation is not considered in this paper, we have taken all the masses real. In this model, the mass matrix M of the third generation lepton and the vector-like lepton is given as following
L ⊃ − (τ, E c H ) M τ τ c E c 4 ,(A1)
and
M l = m τ 33 m τ 34 0 µ E ,(A2)
where m τ 33 ≡ y l
33 v √ 2 cos β and m τ 34 ≡ y E 3 v √ 2 cos β. Taking |µ E | ≫ |m τ 34 |, then the biunitary matrix to diagonalize M τ are L τ * M τ R τ † = (m τ , µ E ) ≡ diag(m τ , m τ 1 ) ,(A3)
where
L τ = 1 m τ 34 µ E − m τ 34 µ E 1 , R τ = 1 mτ m τ 34 (µ E ) 2 − mτ m τ 34 (µ E ) 2 1 .(A4)
The mass matrix M b of the third generation down-quark and vector-like down-type quarks is given as following
L ⊃ − b, Q b 4 , D c H M b b c D c 4 Q t H ,(A5)
where
M b = m b 33 m b 34 0 m b 43 m b 44 µ Q 0 µ D m b H ,(A6)where m b 33 ≡ y d 33 v √ 2 cos β, m b 34 ≡ y D 3 v √ 2 cos β, m b 43 ≡ y Q′ 3 v √ 2 cos β and m b 44 ≡ y QD v √ 2 cos β. Taking that |µ Q | ∼ |µ D | ≫ |m b 4b |, |m b 44 |, |m b 33 |, |m b 34 |, then the biunitary matrix diagonalize M t are L b * M b R b † = (m b , µ Q , µ D ) ≡ diag(m b , m b1 , m b2 ) ,(A7)
where
L b = 1 0 − m b 34 µ D 0 1 µ D m t 44 +µ Q m b H (µ D ) 2 +(µ Q ) 2 −(m b H ) 2 m b 34 µ D (m b H ) 2 +(m b 34 ) 2 µ Q m b H +µ D m b 44 1 ,(A8)
and
R b = 1 m b 34 µ D 0 0 µ Q m b 44 +µ D m b H (µ D ) 2 +(µ Q ) 2 −(m b H ) 2 1 − m b 34 µ D 1 (m b H ) 2 +(m b 34 ) 2 µ D m b H +µ Q m b 44 .(A9)
The mass matrix M t of the top quark and vector-like up-type generations is given as
following L ⊃ − t, Q t 4 , U c H M t t c U c 4 Q b H ,(A10)
where
M t = m t 33 m t 34 0 m t 43 m t 44 µ Q 0 µ U m t H ,(A11)
where m t 33 ≡ y u
33 v √ 2 sin β, m t 34 ≡ y U 3 v √ 2 sin β, m t 43 ≡ y Q 3 v √ 2 sin β, m t 44 ≡ y QU v √ 2 sin β and m t H ≡ y v √ 2 cos β. Taking that |µ Q | ∼ |µ U | ≫ |m t 43 |, |m t 44 |, |m t 33 |, |m t 34 |, |m t H |, then the biunitary matrix diagonalize M t are L t * M t R t † = (m t , µ Q , µ U ) ≡ diag(m t , m t1 , m t2 ) ,(A12)
where
L t = 1 0 − m t 34 µ U 0 1 µ U m t 44 +µ Q m t H (µ U ) 2 +(µ Q ) 2 −(m t H ) 2 m t 34 µ U (m t H ) 2 +(m t 34 ) 2 µ Q m t H +µ U m t 44 1 ,(A13)
and
R t = 1 m t 34 µ U 0 0 µ Q m t 44 +µ U m t H (µ U ) 2 +(µ Q ) 2 −(m t H ) 2 1 − m t 34 µ U 1 (m t H ) 2 +(m t 34 ) 2 µ U m t H +µ Q m t 44 ,(A14)
The charged slepton mass-squared matrixM 2 τ ofτ and the superpartners of the vectorlike leptons is given as following
L ⊃ L − * 3 ,Ẽ c 3 ,Ẽ c 4 ,Ẽ c * H M 2 l L − 3 E c * 3 E c * 4 E c H ,(A15)
where
(M 2 τ ) 11 = M 2 + m 2 Z 2 − m 2 W cos 2β + m 2 τ + |m τ 34 | 2 , (M 2 τ ) 12 = (m 0 − µ tan β)m τ , (M 2 τ ) 13 = (m 0 − µ tan β)m τ 34 , (M 2 τ ) 14 = µ e m τ 34 , (M 2 τ ) 21 = (m 0 − µ tan β)m τ , (M 2 τ ) 22 = M 2 E − (m 2 Z − m 2 W ) cos 2β + m 2 τ , (M 2 τ ) 23 = m τ m τ 34 , (M 2 τ ) 24 = 0 , (M 2 τ ) 31 = (m 0 − µ tan β)m τ 34 , (M 2 τ ) 32 = m τ m τ 34 , (M 2 τ ) 33 = |µ E | 2 + M 2 E − (m 2 Z − m 2 W ) cos 2β + |m τ 34 | 2 , (M 2 τ ) 34 = B E µ E , (M 2 τ ) 41 = µ E m τ 34 , (M 2 τ ) 42 = 0 , (M 2 τ ) 43 = B E µ E , (M 2 τ ) 44 = |µ E | 2 + M 2 EH + (m 2 Z − m 2 W ) cos 2β .(A16)
The corresponding unitary scalar matrix is defined as
V τM2 τ V τ † = diag(M 2 τ ,M 2 τ 1 ,M 2 τ 2 ,M 2 τ 3 ) ,(A17)
The mass-squared matrixM 2 b ofb and the superpartners of the down-type vector-like fermions is given as following
L ⊃ b * ,D c 3 ,D c 4 ,D c * H ,Q b * 4 ,Q t H M 2 b b D c * 3 D c * 4 D c H Q b 4 Q t * H ,(A18)
where
(M 2 b ) 11 = M 2 Q − m 2 Z +2m 2 W 6 cos 2β + m 2 b + |m b 34 | 2 , (M 2 b ) 12 = (m 0 − µ tan β)m b , (M 2 b ) 13 = (m 0 − µ tan β)m b 34 , (M 2 b ) 14 = µ D m b 34 , (M 2 b ) 15 = m b m b 43 + m b 34 m b 44 , (M 2 b ) 16 = 0 , (M 2 b ) 21 = (m 0 − µ tan β)m b , (M 2 b ) 22 = M 2 D − m 2 Z −m 2 W 3 cos 2β + m 2 b + |m d 43 | 2 , (M 2 b ) 23 = m b m b 34 + m b 43 m b * 44 , (M 2 b ) 24 = 0 , (M 2 b ) 25 = (m 0 − µ tan β)m b 43 , (M 2 b ) 26 = µ Q m b 43 , (M 2 b ) 31 = (m 0 − µ tan β)m b 34 , (M 2 b ) 32 = m b m b 34 + m b 43 m b 44 , (M 2 b ) 33 = |µ D | 2 + M 2 D − m 2 Z −m 2 W 3 cos 2β + |m b 34 | 2 + |m b 44 | 2 , (M 2 b ) 34 = −B D µ D , (M 2 b ) 35 = (m 0 − µ tan β)m b 44 , (M 2 b ) 36 = µ Q m b 44 + µ D m b H , (M 2 b ) 41 = µ D m b 34 , (M 2 b ) 42 = 0 , (M 2 b ) 43 = −B D µ D , (M 2 b ) 44 = |µ D | 2 + M 2 DH + m 2 Z −m 2 W 3 cos 2β + +|m b H | 2 , (M 2 b ) 45 = µ D m b 44 + µ Q m b H , (M 2 b ) 46 = (m 0 − µ cot β)m b H , (M 2 b ) 51 = m b m d 43 + m b 34 m b 44 , (M 2 b ) 52 = (m 0 − µ tan β)m b 43 , (M 2 b ) 53 = (m 0 − µ tan β)m b 44 , (M 2 b ) 54 = µ D m b 44 + µ Q m b H , (M 2 b ) 55 = |µ Q | 2 + M 2 Q − m 2 Z +2m 2 W 6 cos 2β + |m b 44 | 2 + |m b 43 | 2 , (M 2 b ) 56 = B Q µ Q , (M 2 b ) 61 = 0 , (M 2 b ) 62 = µ Q m b 43 , (M 2 b ) 63 = µ Q m b 44 + µ D m b H , (M 2 b ) 64 = (m 0 − µ cot β)m b H , (M 2 b ) 65 = B Q µ Q , (M 2 b ) 66 = |µ Q | 2 + M 2 QH + +|m b H | 2 + m 2 Z +2m 2 W 6 cos 2β .(A19)
The corresponding unitary scalar matrix is defined as
V bM2 b V b † = diag(M 2 b ,M 2 b1 ,M 2 b2 ,M 2 b3 ,M 2 b4 ,M 2 b5 ) ,(A20)
The mass-squared matrixM 2 t oft and the superpartners of the up-type vector-like fermions is given as following
L ⊃ t * ,Ũ c 3 ,Ũ c 4 ,Ũ c * H ,Q t * 4 ,Q b H M 2 t t U c * 3 U c * 4 U c H Q t 4 Q b * H , (A21) where (M 2 t ) 11 = M 2 Q + 4m 2 W −m 2 Z 6 cos 2β + m 2 t + |m t 34 | 2 , (M 2 t ) 12 = (m 0 − µ cot β)m t , (M 2 t ) 13 = (m 0 − µ cot β)m t 34 , (M 2 t ) 14 = µ U m t 34 , (M 2 t ) 15 = m t m t 43 + m t 34 m t 44 , (M 2 t ) 16 = 0 , (M 2 t ) 21 = (m 0 − µ cot β)m t , (M 2 t ) 22 = M 2 U + 2 3 (m 2 Z − m 2 W ) cos 2β + m 2 t + |m t 43 | 2 , (M 2 t ) 23 = m t m t 34 + m t 43 m t 44 , (M 2 t ) 24 = 0 , (M 2 t ) 25 = (m 0 − µ cot β)m t 43 , (M 2 t ) 26 = µ Q m t 43 , (M 2 t ) 31 = (m 0 − µ cot β)m t 34 , (M 2 t ) 32 = m * t m u 34 + m t 43 m t 44 , (M 2 t ) 33 = |µ U | 2 + M 2 U + 2 3 (m 2 Z − m 2 W ) cos 2β + |m t 34 | 2 + |m t 44 | 2 , (M 2 t ) 34 = −B U µ U , (M 2 t ) 35 = (m 0 − µ cot β)m t 44 , (M 2 t ) 36 = µ Q m t 44 + µ U m t H , (M 2 t ) 41 = µ U m t 34 , (M 2 t ) 42 = 0 , (M 2 t ) 43 = −B U µ U , (M 2 t ) 44 = |µ U | 2 + M 2 U H − 2 3 (m 2 Z − m 2 W ) cos 2β + |m t H | 2 , (M 2 t ) 45 = µ U * m t 44 + µ Q m t H , (M 2 t ) 46 = (m 0 − µ tan β)m t H , (M 2 t ) 51 = m t m t 43 + m t 34 m t 44 , (M 2 t ) 52 = (m 0 − µ cot β)m t 43 , (M 2 t ) 53 = (m 0 − µ cot β)m t 44 , (M 2 t ) 54 = µ U m t 44 + µ Q m t H , (M 2 t ) 55 = |µ Q | 2 + M 2 Q + 4m 2 W −m 2 Z 6 cos 2β + |m t 44 | 2 + |m t 43 | 2 , (M 2 t ) 56 = −B Q µ Q , (M 2 t ) 61 = , (M 2 t ) 62 = µ Q m t 43 , (M 2 t ) 63 = µ Q m t 44 + µ U m t H , (M 2 t ) 64 = (m 0 − µ tan β)m t H , (M 2 t ) 65 = −B Q µ Q , (M 2 t ) 66 = |µ Q | 2 + M 2 QH + |m t H | 2 − 4m 2 W −m 2 Z 6 cos 2β .(A22)
The corresponding unitary scalar matrix is defined as
V tM2 t V t † = diag(M 2 t ,M 2 t1 ,M 2 t2 ,M 2 t3 ,M 2 t4 ,M 2 t5 ) , (A23) ν iL ν jL l mRlmL l kLlkR λ imk λ jkm ν iL ν jL d mRdmL d kLdkR λ ′ imk λ ′ jkm ν iL ν jL E c 4τ L τ L E c 4 λ E i3 λ E j3 ν iL ν jL E c 4τ L τ LτR λ E i3 λ j33 ν iL ν jL τ RτL τ L E c 4 λ i33 λ E j3 ν iL ν jL b RQ b 4 Q b 4b R λ Q i3 λ Q j3 ν iL ν jL D c 4bL b L D c 4 λ D i3 λ D j3 ν iL ν jL D c 4Q b 4 Q b 4 D c 4 λ QD i λ QD j ν iL ν jL b RbL Q 4 b D c 4 λ Q i3 λ D j3 ν iL ν jL D c 4Q b 4 b LbR λ D i3 λ Q j3 ν iL ν jL b RQ b 4 Q 4 b D c 4 λ Q i3 λ QD j ν iL ν jL D c 4Q b 4 Q b 4 b R λ QD i λ Q j3 ν iL ν jL D c 4Q b 4 b L D c 4 λ D i3 λ QD j ν iL ν jL D c 4bL Q b 4 D c 4 λ QD i λ D j3 ν iL ν jL b RQ b 4 b LbR λ ′ i33 λ Q j3 ν iL ν jL b RbL Q 4 b b R λ Q i3 λ ′ j33 ν iL ν jL b RbL b L D c 4 λ ′ i33 λ D j3 ν iL ν jL D c 4bL b L b R λ D i3 λ ′ j33 ν iL ν jL b RQ b 4 b L D c 4 λ ′ i33 λ QD j ν iL ν jL D c 4bL Q b 4 b R λ QD i λ ′ j33 ν iL ν jL Q b HŨ c H U c H Q b H λ H i λ H j FIG
20
The corresponding analytical results are listed below: (6) is m=1,2, k=1-4 ,while in Eq. (7)- (17), it is m=1,2,3, k=1-6.
M ν ij | λλ ≃ 1 8π 2 k,m λ i33 λ j33 R * τ m1 L * τ m1 V * τ k1 V τ k2 m τm b(m τm , Mτ L(R)k ), (B1) M ν ij | λλ E ≃ 1 8π 2 k,m λ E i3 λ j33 [R * τ m1 L * τ m1 V * τ k1 V τ k3 m τm b(m τm , Mτ L(R)k ) + R * τ m2 L * τ m1 V * τ k1 V τ k2 m τm b(m τm , Mτ L(R)k )],(B2)M ν ij | λ E λ E ≃ 1 8π 2 k,m λ E i3 λ E j3 R * τ m2 L * τ m1 V * τ k1 V τ k3 m τm b(m τm , Mτ L(R)k ),(B3)M ν ij | λ ′ λ ′ ≃ 3 8π 2 k,m λ ′ i33 λ ′ j33 [R * b m1 L * b m1 V * b k1 V b k2 m bm b(m bm , Mb L(R)k ) + λ ′ i32 λ ′ j23 R * b m1 L * b m1 m bm sin α s1(2) cos α s1(2) b(m bm , Ms 1,2 )],(B4)M ν ij | λ Q λ Q ≃ 3 8π 2 k,m λ Q i3 λ Q j3 R * b m1 L * b m2 V * b k2 V b k5 m bm b(m bm , Mb L(R)k ), (B5) M ν ij | λ D λ D ≃ 3 8π 2 k,m λ D i3 λ D j3 R * b m2 L * b m1 V * b k1 V b k3 m bm b(m bm , Mb L(R)k ), (B6) M ν ij | λ QD λ QD ≃ 3 8π 2 k,m λ QD i λ QD j R * b m2 L * b m2 V * b k3 V b k5 m bm b(m bm , Mb L(R)k ),(B7)M ν ij | λ Q λ D ≃ 3 8π 2 k,m λ Q i3 λ D j3 [R * b m2 L * b m2 V * b k1 V b k2 m bm b(m bm , Mb L(R)k ) + R * b m1 L * b m1 V * b k3 V b k5 m bm b(m bm , Mb L(R)k )],(B8)M ν ij | λ Q λ QD ≃ 3 8π 2 k,m λ Q i3 λ QD j [R * b m1 L * b m2 V * b k3 V b k5 m bm b(m bm , Mb L(R)k ) + R * b m2 L * b m2 V * b k2 V b k5 m bm b(m bm , Mb L(R)k )],(B9)M ν ij | λ D λ QD ≃ 3 8π 2 k,m λ D i3 λ QD j [R * b m2 L * b m1 V * b k3 V b k5 m bm b(m bm , Mb L(R)k ) + R * b m2 L * b m2 V * b k1 V b k3 m bm b(m bm , Mb L(R)k )],(B10)M ν ij | λ ′ λ Q ≃ 3 8π 2 k,m λ ′ i33 λ Q j3 [R * b m1 L * b m1 V * b k2 V b k5 m bm b(m bm , Mb L(R)k ) + R * b m1 L * b m2 V * b k1 V b k2 m bm b(m bm , Mb L(R)k )],(B11)M ν ij | λ ′ λ D ≃ 3 8π 2 k,m λ ′ i33 λ D j3 [R * b m2 L * b m1 V * b k1 V b k2 m bm b(m bm , Mb L(R)k ) + R * b m1 L * b m1 V * b k1 V b k3 m bm b(m bm , Mb L(R)k )],(B12)M ν ij | λ ′ λ QD ≃ 3 8π 2 k,m λ ′ i33 λ QD j [R * b m2 L * b m1 V * b k2 V b k5 m bm b(m bm , Mb L(R)k ) + R * b m1 L * b m2 V * b k1 V b k3 m bm b(m bm , Mb L(R)k )], (B13) M ν ij | λ H λ H ≃ 3 8π 2 k,m λ H i λ H j R * t m3 L * t m3 V * u k6 V u k4 m tm b(m tm , Mt L(R)k ),(B14)
where we name each of the matrices above M 1,2,3 separately. We assume m 1 > m 2,3 , m 2 ∼ m 3 and there is no strong hierarchy between a, b, c, d, e, f, g, h, l.
M 1 has only one eigenvalue after digonalized by an unitary rotation
X T M 1 X = diag(0, 0, M 1 ) ,(C2)
where
M 1 = m 1 (a 2 + b 2 + c 2 ) ,(C3)
and
X = c 2 s 2 c 3 s 2 s 3 −s 2 c 2 c 3 s 2 s 3 0 − s 3 c 3 ,(C4)s 2 = a √ a 2 + b 2 , c 3 = c √ a 2 + b 2 + c 2 . (C5)
If we rotate the sum over M 1,2,3 by matrix X, it becomes
X T (M 1 + M 2 + M 3 )X ≈ m 1 (a 2 + b 2 + c 2 ) ǫ 11 ǫ 12 ǫ 13 ǫ 21 ǫ 22 ǫ 23 ǫ 31 ǫ 32 1 ,(C6)
where ǫ ij are some small values related with m 2 /m 1 , m 3 /m 1 and the other elements of M 1,2,3 .
We can then define another unitary matrix X ′ to diagonalize the matrix in Eq. (B6) in an approximate way:
X ′ T X T (M 1 + M 2 + M 3 )XX ′ ≈ m 1 (a 2 + b 2 + c 2 ) diag(δ ′ 3 , δ ′ 2 , 1) ,(C7)
where
X ′ = c 1 s 1 0 −s 1 c 1 0 0 0 1 ,(C8)
and tan 2θ 1 = 2ǫ 12
ǫ 22 − ǫ 11 .(C9)
Then from Eq. (B7), we get all three mass eigenvalues
M 1 ∼ m 1 (a 2 + b 2 + c 2 ) , M 2 ∼ M 1 δ ′ 2 , M 3 ∼ M 1 δ ′ 3 ,(C10)
and from Eq. (B4, B8), we get all three mixing angles
s 13 = s 2 s 3 = a √ a 2 +b 2 +c 2 , s 23 = c 2 s 3 /c 13 = b √ b 2 +c 2 ,(C11)s 12 = (s 1 c 2 + c 1 s 2 c 3 )/c 13 .
The parameter settings we used in table I are given as following The couplings for the W, Z, h 0 boson with leptons in Eq. (12) are
g W τ 1L ν τ L = g √ 2 L τ 21 , g Z τ 1L τ L = gs 2 W c W L τ 22 L τ 12 − g 4c W [(2 − 4s 2 W )L τ 21 L τ 11 ] , g Z τ 1R τ R = gs 2 W c W R τ 22 R τ 12 + g 4c W (4s 2 W R τ 21 R τ 11 ) ,(D1)g h 0 τ 1L τ R = − s α √ 2 (y τ 33 L τ 21 R τ 11 + y E 3 L τ 21 R τ 12 ) , g h 0 τ L τ 1R = − s α √ 2 (y τ 33 L τ 11 R τ 21 + y E 3 L τ 11 R τ 22 ) ,
where c α = s β , s α = −c β is the elements of the rotation matrix related with the real parts of (H 0 u , H 0 d ). Then the decay widths of τ 1 are
Γ(τ 1 → Wν τ ) = m τ 1 32π (1 + x 4 W − 2x 2 W ) 1/2 (1 − 2x 2 W + x −2 W )(g W τ 1 ντ ) 2 , Γ(τ 1 → Zτ) = m τ 1 32π (1 + x 4 Z + x 4 τ − 2x 2 Z − 2x 2 τ − 2x 2 Z x 2 τ ) 1/2 {(1 + x 2 τ − 2x 2 Z + (1 − x 2 τ ) 2 x −2 Z )[(g Z τ 1L τ L ) 2 + (g Z τ 1R τ R ) 2 ] + 12x τ g Z τ 1L τ L g Z τ 1R τ R } ,(D2)Γ(τ 1 → h 0 τ) = m τ 1 32π (1 + x 4 h 0 + x 4 τ − 2x 2 h 0 − 2x 2 τ − 2x 2 h 0 x 2 τ ) 1/2 {(1 + x 2 τ − x 2 h 0 )[(g h 0 τ 1L τ R ) 2 + (g h 0 τ L τ 1R ) 2 ] + 4x τ g h 0 τ 1L τ R g h 0 τ L τ 1R } , where x i = m i /m τ 1 for i = W, Z, τ, h 0 .
The couplings for the W, Z, h 0 boson with t, t 1 , t 2 in Eq. (13) are
g W t 1L b L = g √ 2 (L t 21 L b 11 + L t 22 L b 12 ) , g W t 1R b R = g √ 2 R t 23 R b 13 , g W t 2L b L = g √ 2 (L t 31 L b 11 + L t 32 L b 12 ) , g W t 2R b R = g √ 2 R t 33 R b 13 , g W t 2L b 1L = g √ 2 (L t 31 L b 21 + L t 32 L b 22 ) , g W t 2R b 1R = g √ 2 R t 33 R b 23 , g Z t 1L t L = −2gs 2 W 3c W L t 23 L t 13 + g 4c W [(2 − 8 3 s 2 W )(L t 21 L t 11 + L t 22 L t 12 )] , g Z t 1R t R = − g 4c W [ 8 3 s 2 W (R t 21 R t 11 + R t 22 R t 12 ) + (2 − 4 3 s 2 W )R t 23 R t 13 ] , g Z t 2L t L = −2gs 2 W 3c W L t 33 L t 13 + g 4c W [(2 − 8 3 s 2 W )(L t 31 L t 11 + L t 32 L t 12 )] ,(D3)g Z t 2R t R = − g 4c W [ 8 3 s 2 W (R t 31 R t 11 + R t 32 R t 12 ) + (2 − 4 3 s 2 W )R t 33 R t 13 ] , g Z t 2L t 1L = −2gs 2 W 3c W L t 33 L t 23 + g 4c W [(2 − 8 3 s 2 W )(L t 31 L t 21 + L t 32 L t 22 )] , g Z t 2R t 1R = − g 4c W [ 8 3 s 2 W (R t 31 R t 21 + R t 32 R t 22 ) + (2 − 4 3 s 2 W )R t 33 R t 23 ] , g h 0 t 1L t R = c α √ 2 (y u 33 L t 21 R t 11 + y Q 3 L t 22 R t 11 + y U 3 L t 21 R t 12 + y QU 3 L t 22 R t 12 ) − s α √ 2 y H L t 23 R t 13 , g h 0 t L t 1R = c α √ 2 (y u 33 L t 11 R t 21 + y Q 3 L t 12 R t 21 + y U 3 L t 11 R t 22 + y QU 3 L t 12 R t 22 ) − s α √ 2 y H L t 13 R t 23 , g h 0 t 2L t R = c α √ 2 (y u 33 L t 31 R t 11 + y Q 3 L t 32 R t 11 + y U 3 L t 31 R t 12 + y QU 3 L t 32 R t 12 ) − s α √ 2 y H L t 33 R t 13 , g h 0 t L t 2R = c α √ 2 (y u 33 L t 11 R t 31 + y Q 3 L t 12 R t 31 + y U 3 L t 11 R t 32 + y QU 3 L t 12 R t 32 ) − s α √ 2 y H L t 13 R t 33 , g h 0 t 2L t 1R = c α √ 2 (y u 33 L t 31 R t 21 + y Q 3 L t 32 R t 21 + y U 3 L t 31 R t 22 + y QU 3 L t 32 R t 22 ) − s α √ 2 y H L t 33 R t 23 , g h 0 t 1L t 2R = c α √ 2 (y u 33 L t 21 R t 31 + y Q 3 L t 22 R t 31 + y U 3 L t 21 R t 32 + y QU 3 L t 22 R t 32 ) − s α √ 2 y H L t 23 R t 33 . Γ(t 1 → h 0 t) = m t 1 32π (1 + x 4 h 0 + x 4 t − 2x 2 h 0 − 2x 2 t − 2x 2 h 0 x 2 t ) 1/2 {(1 + x 2 t − x 2 h 0 )[(g h 0 t 1L t R ) 2 + (g h 0 t L t 1R ) 2 ] + 4x t g h 0 t 1L t R g h 0 t L t 1R } ,
where x i = m i /m t 1 for i = W, Z, t, h 0 . The heaviest new up-type quark t 2 has six decay channels. The decay widths have the similar forms and can be deduced straightforwardly.
The couplings for the W, Z, h 0 boson with b, b 1 , b 2 in Eq. (14) are
g W b 1L t L = g √ 2 (L b 21 L t 11 + L b 22 L t 12 ) , g W b 1R t R = g √ 2 R b 23 R t 13 , g W b 2L t L = g √ 2 (L b 31 L t 11 + L b 32 L t 12 ) , g W b 2R t R = g √ 2 R b 33 R t 13 , g W b 2L t 1L = g √ 2 (L b 31 L t 21 + L b 32 L t 22 ) , g W b 2R t 1R = g √ 2 R b 33 R t 23 , g Z b 1L b L = gs 2 W 3c W L b 23 L b 13 − g 4c W [(2 − 4 3 s 2 W )(L b 21 L b 11 + L b 22 L b 12 )] , g Z b 1R b R = g 4c W [ 4 3 s 2 W (R b 21 R b 11 + R b 22 R b 12 ) + (2 − 8 3 s 2 W )R b 23 R b 13 ] , g Z b 2L b L = gs 2 W 3c W L b 33 L b 13 − g 4c W [(2 − 4 3 s 2 W )(L b 31 L b 11 + L b 32 L b 12 )] , g Z b 2R b R = g 4c W [ 4 3 s 2 W (R b 31 R b 11 + R b 32 R b 12 ) + (2 − 8 3 s 2 W )R b 33 R b 13 ] , g Z b 2L b 1L = gs 2 W 3c W L b 33 L b 23 − g 4c W [(2 − 4 3 s 2 W )(L b 31 L b 21 + L b 32 L b 22 )] ,(D5)g Z b 2R b 1R = g 4c W [ 4 3 s 2 W (R b 31 R b 21 + R b 32 R b 22 ) + (2 − 8 3 s 2 W )R b 33 R b 23 ] , g h 0 b 1L b R = − s α √ 2 (y d 33 L b 21 R b 11 + y Q ′ 3 L b 22 R b 11 + y D 3 L b 21 R b 12 + y QD 3 L b 22 R b 12 ) + c α √ 2 y H ′ L b 23 R b 13 , g h 0 b L b 1R = − s α √ 2 (y d 33 L b 11 R b 21 + y Q ′ 3 L b 12 R b 21 + y D 3 L b 11 R b 22 + y QD 3 L b 12 R b 22 ) + c α √ 2 y H ′ L b 13 R b 23 , g h 0 b 2L b R = − s α √ 2 (y d 33 L b 31 R b 11 + y Q ′ 3 L b 32 R b 11 + y D 3 L b 31 R b 12 + y QD 3 L b 32 R b 12 ) + c α √ 2 y H ′ L b 33 R b 13 , g h 0 b L b 2R = − s α √ 2 (y d 33 L b 11 R b 31 + y Q ′ 3 L b 12 R b 31 + y D 3 L b 11 R b 32 + y QD 3 L b 12 R b 32 ) + c α √ 2 y H ′ L b 13 R b 33 , g h 0 b 2L b 1R = − s α √ 2 (y d 33 L b 31 R b 21 + y Q ′ 3 L b 32 R b 21 + y D 3 L b 31 R b 22 + y QD 3 L b 32 R b 22 ) + c α √ 2 y H ′ L b 33 R b 23 , g h 0 b 1L b 2R = − s α √ 2 (y d 33 L b 21 R b 31 + y Q ′ 3 L b 22 R b 31 + y D 3 L b 21 R b 32 + y QD 3 L b 22 R b 32 ) + c α √ 2 y H ′ L b 23 R b 33 .
tons in the vector-like generation are taken as the two Higgs doublets. New particles beyond the MSSM are the following with quantum numbers under SU(3) c ×SU(2) L ×U(1) Y ,
where in the first equation, we only keep the dominant contributions and in the second equation, we keep the dominant and subdominant ones. α τ , α b , α s , α t are the angles of the corresponding 2×2τ L(R) ,b L(R) ,s L(R) ,t L(R) unitary matrices. Unfortunately, the other equations, (B3)-(B14) in Appendix B, can not be simplified by following similar process because there are mixings between different vector-like generations. So these can only be
FIG. 3 :
3The decay widths of the new lepton τ 1 (left panel) and its branching ratios (right panel) with y E = 0.04. can see in the limit of large m τ 1 , the branching rations are BR(τ 1 → Wν τ ) ∼ 0.7 and BR(τ 1 → Zτ ) = BR(τ 1 → hτ ) ∼ 0.15 . B. t 1,2 decays
FIG. 4 :
4The decay widths of the lightest new up-type quark t 1 (left panel) and its branching ratios (right panel) with y QD = y H = y D = 0, y U ∼ y Q = 0.08 and y QU ∼ y H ′ = 1.
FIG. 5 :
5The decay widths of the heaviest new up-type quark t 2 (left panel) and its branching ratios (right panel) with y QD = y H = y D = 0, y U ∼ y Q = 0.08 and y QU ∼ y H ′ = 0.98.C. b 1,2 decays
FIG. 6 :
6The decay widths of the heaviest new down-type quark b 2 (left panel) and its branching ratios (right panel) with y QD = y H = y D = 0, y U ∼ y Q = 0.08 and y QU ∼ y H ′ = 0.98.The numerical results of b 2 decay widths and branching rations are shown inFig. 6under the parameter settings mentioned before, we can see BR(b 2 → Wb 1 ) ∼ 0.85 and BR(b 2 → Zb 1 ) ∼ 0.15. The branching rations of b 1 are BR(b 1 → Wt) = 1 which are not shown here.VI. SUMMARY AND DISCUSSIONWe have studied several phenomenological aspects of the LPV MSSM model with a vectorlike extra generation: the neutrino spectrum, the Higgs mass and the LHC phenomenology of the new predicted fermions. The results are:• The correct neutrino masses and mixings, especially the relatively large θ 13 can be generated from trilinear LPV couplings. The new trilinear R-parity violating couplings make it easy to generate the proper value of θ 13 . These coupling constants need to be about 10 −6 .• The two new up-type Higgs Yukawa couplings, y H ′ and y QU , between the vector-like quarks and the SM third generation quarks, with values about 1 near to their infrared quasi-fixed point in TeV scale, can give rise to 125 GeV Higgs mass with no need of very heavy new superpartner.• There are five new heavy fermions, τ 1 , t 1,2 , b 1,2 , predicted in this model. They can only decay into SM bosons by some kinematic assumptions. The branching radio depend on the mass mixing between the vector-like fermions and the SM third generation fermions. These charged exotic fermions would be quasi-stable if such mass mixings are very small.
. 7 :
7One-loop contributions to the neutrino masses and mixings in our model.
In which L τ,b,t , R τ,b,t are biunitary matrices of mass matrices between (τ, b, t) and the vector-like fermions (see Appendix A), while m τm , m bm , m tm indicate the corresponding mass eigenvalues. Vτ ,d,t are the square mass mixing unitary matrices of their superpatners, while Mτ L(R)k , Mb L(R)k , Mt L(R)k stand for the corresponding mass eigenvalues. sin α s1(2), cos α s1(2) are the unitary matrix elements ofs. b(m 1 , m 2 ) is the loop integral factor: . The value range of the indices in Eq. (4)-
Appendix C: NEUTRINO SPECTRUM-CALCULATING METHOD AND PA-RAMETER SETTINGSThe methods to generate neutrino masses and mixing angles with one-loop trilinear / L couplings actually involves the following three matrices
34 = 10, M E = M EH = 600GeV, B E µ E = 400 2 GeV 2 ; m b H = 170GeV, m b 34 = m b 43 = m b 44 = 0, M Q = M D = M DH = M QH = 700GeV,B D µ D = B Q µ Q = 500 2 GeV 2 ; m t 34 = m t 43 = m t H = 13GeV, m t 44 = 174GeV, M U = M U H = 700GeV, B U µ U = 500 2 GeV 2 , tan β = 10, A = µ = 500GeV.Set II:m τ 34 = 10, M E = M EH = 1000GeV, B E µ E = 600 2 GeV 2 ; m b H = 170GeV, m b 34 = m b 43 = m b 44 = 10GeV, M Q = M D = M DH = M QH = 1000GeV, B D µ D = B Q µ Q = 600 2 GeV 2 ; m t 34 = m t 43 = m t H = 13GeV, m t 44 = 174GeV, M U = M U H = 1000GeV, B U µ U = 600 2 GeV 2 , tan β = 10, A = µ = 600GeV.Appendix D: EXOTIC QUARK AND LEPTON COUPLINGS TO W, Z, h 0 AND DECAY WIDTHS
There are several reasons for this.First, we can phenomenologically assume the universality of the soft SUSY breaking mass terms at the weak scale, to avoid dangerously large flavor changing neutral currents (FCNCs), without considering any UV completion of the model. In that case, because of the alighnment in bilinear terms of the superpotential and that of soft terms, R-parity violating bilinear terms can be rotated away via field redefinition, and sneutrino vacuum expectation values (VEVs) vanish in the physical basis. The second reason is from consideration of underlying models. SUSY breaking 1 It modifies the down-type fermion mass matrix and scalar mass-squared matrix. Correct ones, as well as the resulting mixing matrix are given in the Appendix A.is introduced effectively in our model, it can result from gauge mediated SUSY breaking.Then the messenger scale can be as low as 100 TeV, even if the universality scale is at the SUSY breaking messenger scale, the running effect is small, and the bilinear LPV is not important compared to the trilinear ones. Finally, small sneutrino VEVs can be included in the analysis nevertheless in future works, after the role of new trilinear LPV interactions gets a thorough understanding.
TABLE I :
INumerical illustration for 5 types of one-loop contributions in our model ,the specific
parameter settings see Appendix B. M ν
ij (GeV) stands for the parts in Eq. (4,5) excepting the
LPV trilinear coupling constants.
FIG. 1: New one-loop contributions to the the neutrino masses and mixings from λ E λ E , λ Q λ D and λ H λ H type couplings. All particles stay in mass eigenstates.t 1,2
t, t 1,2
λ H
i
λ H
j
t 2L b R1t 2R γ µ b 1R W − µ + g Z t 1L t Lt 1L γ µ t L Z µ + g Z t 2L t Lt 2L γ µ t L Z µ +g Z t 2L t 1Lt 2L γ µ t 1L Z µ + g Z t 1R t Rt 1R γ µ t R Z µ + g Z t 2R t Rt 2R γ µ t R Z µ + g Z t 2R t 1Rt 2R γ µ t 1R Z µ +g h 0 t 1L t Rt 1L t R h 0 + g h 0 t L t 1Rt L t 1R h 0 + g h 0 t 2L t Rt 2L t R h 0 + g h 0 t L t 2Rt L t 2R h 0 +g h 0 t 2L t 1Rt 2L t 1R h 0 + g h 0 t 1L t 2Rt 1L t 2R h 0 + h.c.,(13)the couplings and the decay widths of t 1,2 are given in Appendix D.As mentioned in section II, we take y U ∼ y Q ≤ 0.08, y QU ∼ y H ′ ≤ 1, the numerical reasults are shown inFig 4, 5. We can see in the limit of large M t 1,2 , the branching rations of t 1 are BR(t 1 → Wb) ∼ 0.4 and BR(t 1 → Zt) = BR(t 1 → h 0 t) ∼ 0.3, the branching rations of t 2 are BR(t 2 → Wb 1 ) ∼ 0.85 and BR(t 2 → Zt 1 ) ∼ 0.15 .
AcknowledgmentsWe would like to thank Dr. Guang-Zhi Xu for a very helpful discussion. This work wasAppendix B: NEUTRINO MASSES IN OUR MODELAll fourteen types of one loop Fyenman diagrams which can contribute to the neutrino mass and mixing in our model are shown inFig. 1The decay widths of the lightest new up-type quark t 1 areThe decay widths of the lightest new down-type quark b 1 areThe heaviest new down-type quark b 2 has six decay channels, the decay widths have the similar forms and can be deduced straightforwardly.
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[] |
[
"INTEGER SETS OF LARGE HARMONIC SUM WHICH AVOID LONG ARITHMETIC PROGRESSIONS",
"INTEGER SETS OF LARGE HARMONIC SUM WHICH AVOID LONG ARITHMETIC PROGRESSIONS"
] |
[
"Alexander Walker "
] |
[] |
[] |
We give conditions under which certain digit-restricted integer sets avoid k-term arithmetic progressions. These sets are reasonably efficient to compute and therefore enable large-scale search. We identify a set with no arithmetic progression of four terms and with harmonic sum 4.43975, which improves an earlier "greedy" construction.
| null |
[
"https://arxiv.org/pdf/2203.06045v1.pdf"
] | 247,411,270 |
2203.06045
|
52b6c6fb68429d61d5a7dc5dc139be70e38b7a84
|
INTEGER SETS OF LARGE HARMONIC SUM WHICH AVOID LONG ARITHMETIC PROGRESSIONS
1, 2, 4, 5, 7}, 11) + 1 = {1, 2, 3, 5, 6, 8, 12, 13, 14, 16, 17, 19, 23, 24, K({0, 1, 2, 4, 5, 9, 10, 11, 14, 16, 17, 18, 21, 24, 30, 37, 39, 41, 42, 45
Alexander Walker
INTEGER SETS OF LARGE HARMONIC SUM WHICH AVOID LONG ARITHMETIC PROGRESSIONS
471, 2, 4, 5, 7}, 11) + 1 = {1, 2, 3, 5, 6, 8, 12, 13, 14, 16, 17, 19, 23, 24, K({0, 1, 2, 4, 5, 9, 10, 11, 14, 16, 17, 18, 21, 24, 30, 37, 39, 41, 42, 45arXiv:2203.06045v1 [math.NT]
We give conditions under which certain digit-restricted integer sets avoid k-term arithmetic progressions. These sets are reasonably efficient to compute and therefore enable large-scale search. We identify a set with no arithmetic progression of four terms and with harmonic sum 4.43975, which improves an earlier "greedy" construction.
The Erdős-Turán conjecture on arithmetic progressions proposes that integer sets with divergent harmonic sums (so-called large sets) must contain arithmetic progressions of arbitrary (finite) length. This conjecture is known to hold for integer sets of positive density (Szemeredi's theorem, [Sze75]) and for the set of primes (the Green-Tao theorem, [GT08]).
Let S k denote the collection of sets of positive integers which avoid arithmetic progressions of length k. Such sets will be called k-free hereafter. The Erdős-Turán conjecture implies that the harmonic sum of any member of S k is bounded. In fact, under Erdős-Turán, these harmonic sums would be uniformly bounded as a function of k, due to a result of Gerver [Ger77]. Let M k := sup T ∈S k t∈T 1/t, which is finite for each k if and only if the Erdős-Turán conjecture holds. Recent work of Bloom-Sisask [BS19] shows that M 3 is finite, but the finiteness of M k is otherwise unknown.
If finite, the growth rate of M k represents a refinement to the Erdős-Turán conjecture. For this reason, lower bounds for M k appear in several works. Berlekamp gave a construction in [Ber68] which proved M k ≥ 1 2 k log 2. This was improved in [Ger77], which showed that for all ǫ > 0, there exists cofinitely many k for which M k > (1 − ǫ)k log k.
Numerical lower bounds for M k for small k have also received attention. The current record for M 3 is held by Wróblewski, who proved M 3 ≥ 3.00849 by interlacing sets constructed through greedy algorithms and a denser 3free packing due to Behrend [Beh46,Wró84].
Let G k denote the lexicographically earliest k-free set. The sets G k have reasonable large harmonic sums, especially when k is prime and G k exhibits fractal self-similarity. When k is composite, the harmonic sums are less impressive. Heuristics from [GR79] suggest that G 4 and G 6 have harmonic sums of ≈ 4.3 and ≈ 6.9, respectively. Notably, the harmonic sum of G 6 is predicted to be less than that of G 5 , which is 7.866.
This article provides a new construction for infinite k-free sets. Fix an integer b ≥ 2 and a proper subset of integers S [0, b − 1]. We define the Kempner set K(S, b) as the set of non-negative integers whose base-b digits are contained in S. Kempner sets first appeared in [Kem14], and their arithmetic properties have been studied in [EMS98], [EMS99], and [May19]. The connection between Kempner sets and arithmetic progressions was first developed in [WW20]. In particular, [WW20] proved that every Kempner set is k-free for some k.
Kempner sets are useful in the experimental study of M k because the lengths of their longest arithmetic progressions are easy to compute and their harmonic sums are computable to arbitrary precision (due to an algorithm of Baillie-Schmelzer in [SB08]). Most importantly, they are also capable of producing large harmonic sums. For example, we show that the set K({0, 1, 2, 4, 5, 7}, 11) + 1 = {1, 2, 3, 5, 6, 8, 12, 13, 14, 16, 17, 19, 23, 24, . . .} is 4-free and has harmonic sum 4.421746. This simple set exceeds the estimated harmonic sum of G 4 by a considerable margin and already sets a new lower bound for the supremum M 4 .
We describe and implement algorithms which use Kempner sets to search for lower bounds for M k . Even with pruning, this search is time-consuming: the number of Kempner sets K(S, b) grows exponentially in b and b must be taken large before interesting results are found. Our search is most successful in the case k = 4, where our best result is the set K({0, 1, 2, 4, 5, 9, 10, 11, 14, 16, 17, 18, 21, 24, 30, 37, 39, 41, 42, 45, 47}, 55) + 1. This set has harmonic sum 4.43975 and sets a new record among 4-free sets.
Acknowledgments
This work was supported by the Additional Funding Programme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Mathematical Research.
Modular Arithmetic Progressions
A set S ⊂ [0, b − 1] is called an arithmetic progression mod b of length k if there exists an arithmetic progression A (in the ordinary sense) of length k and common difference ∆ for which A mod b lies in S and b ∤ ∆. By extension, a set S ⊂ [0, b−1] is called k-free mod b if it contains no arithmetic progressions mod b of length k. Note that we do not require gcd(∆, b) = 1. This has some counter-intuitive implications; for example, it implies that {1, 3, 5} is an arithmetic progression mod 6 of infinite length.
One can test if a set S is k-free mod b by testing if the associated union of translates n−1 j=0 (S + jb) has no increasing k-term arithmetic progression with common difference less than b. In the context of a depth-first search, one can create k-free sets mod b by extending smaller k-free sets. Alternatively, one can construct sets which are k-free mod b by first fixing a finite k-free set and then specifying a b which is sufficiently large.
Proposition 1.1. Let S ⊂ [0, M ] be a k-free set. Then S is k-free mod b for all b > 2M .
Proof. For the sake of contradiction, suppose that S ⊂ [0, M ] is k-free but that S admits a k-term arithmetic progression mod b for some b > 2M .
To be precise, suppose that B = {c + ∆j : j ∈ [1, k]} is equivalent mod b to a subset of S and that 0 < ∆ < b. For j ≤ k, let q j and r j be the quotient and remainder of c + ∆j upon division by b. Our assumption on ∆ implies that q j+1 equals q j or q j + 1. If q j+1 = q j for all j, then {r j } is an arithmetic progression in S of length k, a contradiction. Thus q j+1 = q j + 1 for some j,
hence ∆ ≥ b − M since r j ∈ [0, M ].
On the other hand, if q j+1 = q j + 1 for all j, then {r j } is a (decreasing) arithmetic progression in S of length k, again a contradiction.
Thus q j+1 = q j for some j, hence ∆ ≤ M . It follows that b−M ≤ ∆ ≤ M , hence b ≤ 2M , which contradicts that b > 2M .
Sets which are k-free mod b can be used to produce k-free Kempner sets. This is made precise in the following.
Theorem 1.2. Fix b ≥ 3. If S [0, b − 1] is k-free mod b and 0 ∈ S, then K(S, b) is k-free.
Proof. Suppose that S is k-free mod b and that K(S, b) contains the arithmetic progression A = {c+∆j | j ∈ [0, k −1]}. If b ∤ ∆, then A and therefore S contains residues in an arithmetic progression mod b. Thus S contains a progression mod b of length k or of infinite length, so S is not k-free.
Alternatively, suppose that b | ∆ and let c 0 denote the base-b units digit of c. The arithmetic progression (A − c 0 )/b is contained in K(S, b) and has a smaller common difference. We conclude by infinite descent.
Remark 1.3. There exist k-free Kempner sets K(S, b) for which S is not k-free mod b. One simple example is the 3-free set K({0, 2, 5}, 7). Examples like this rely on gaps in the digit set S (to avoid 'carrying') and do not seem to produce large harmonic sums.
Harmonic Sums of (Shifted) Kempner Sets
We now turn our attention to the harmonic sums of Kempner sets. In general, let H(S) denote the harmonic sum of the integer set S. The following result shows that any lower bound for M k can be approximated by the harmonic sum of a (shifted) k-free Kempner set, Theorem 2.1. Let S be k-free, with a convergent harmonic series. Given ǫ > 0, there exists a k-free Kempner set K such that H(K + 1) > H(S) − ǫ. One of the reasons to study Kempner sets is that machinery exists due to [SB08] to compute harmonic sums of Kempner sets with great precision. There is one small difficulty, in that Kempner sets include 0. Rather than exclude 0, we opt to increase our sets termwise by 1. This shift affects harmonic sum in a way that can be addressed with the following lemma. Proof. Since s∈S 1/s − s∈S 1/(s + n) = s∈S n/(s 2 + ns), it suffices after rearranging to show that H n = ∞ m=1 n/(m 2 + nm). To prove this last fact, we write the series on m as a telescoping sum.
Implementation
The Baillie-Schmelzer Algorithm described in [SB08] has been fully implemented in the Mathematica language and is freely available from the Wolfram Library Archive [BS08]. This is useful for fine-tuning pruned results but inefficient for larger searches because the Baillie-Schmelzer Algorithm is somewhat time-intensive.
As a complement to the Mathematica implementation of the Baillie-Schmelzer algorithm, the author wrote a family of search algorithms using the C++ language. The core algorithm is a branch-and-bound depth-first search through the subsets of [0, b − 2] which are k-free mod b. More specifically, states are stored as pairs (S, T ), in which S is a k-free set mod b and T is the set of possible extensions to S:
T = {t ∈ [0, b − 2] : t > max(S) and S ∪ {t} is k-free mod b}.
An upper bound for the branch rooted at (S, T ) is then H(K(S ∪ T, b) + 1).
Efficient estimates for the harmonic sums of the associated Kempner sets are obtained using a first-order approximation to the Baillie-Schmelzer algorithm. To be precise, one uses the approximation
H(K(S, b) + 1) ≈ 1 1 − #S/b s∈S 1 s + 1 (3.1)
to rapidly estimate the fitness of a candidate set S. Branches whose approximate upper bounds lie below a threshold are pruned, and surviving sets are recorded for further processing using the full Baillie-Schmelzer algorithm (and Lemma 2.2) in Mathematica.
3-Free Kempner Sets of Large Harmonic Sum
A branch-and-bound search over 3-free sets mod b ≤ 120 produces only a handful of sets whose associated shifted Kempner sets have harmonic sums near that of G 3 . We find 5095 candidate sets with estimated harmonic sum (via (3.1)) of at least 3.0. Of these, the ten of largest (actual) harmonic sum are reproduced in Table 1 below. The non-G 3 Kempner sets listed in Table 1 above differ from G 3 in minor, deleterious ways. This suggests that G 3 is an influential local maximum and that far larger search spaces may be needed to improve lower bounds on M 3 .
A similar story unfolds whenever we search for p-free sets with p a prime: the greedy set G p = K([0, p − 2], p) + 1 dominates early results and we find little else of interest. However, when k is composite, G k has a much smaller harmonic sum and the best lower bound on M k typically comes from the set [1, k − p] ∪ (G p + (k − p)), in which p is the largest prime less than k. This general construction is least impressive when k is one less than a prime.
4-Free Kempner Sets of Large Harmonic Sum
We next consider k-free sets in the case k = 4. Heuristics from [GR79] suggest the lower bound M 4 > 4.3, as derived from the greedy set G 4 . (There is no known algorithm to compute H(G 4 ) to arbitrary precision; a lower bound from the first 10000 terms of G 4 gives H(G 4 ) > 4.19111.) The lack of obvious structure in G 4 suggests that further improvements to M 4 may be within reach.
A branch-and-bound search over all 4-free sets mod b ≤ 60 yields 109 sets with approximate harmonic sums (via (3.1)) of at least 4.5. The ten sets of largest (actual) harmonic sum are compiled in Table 2. Our best result employs a 21-term 4-free set mod 55 and shows M 4 ≥ 4.43975, which improves the heuristic record set by [GR79]. 43975 55 {0,1,2,4,5,9,10,11,14,16,17,18,21,24,30,37,39,41,42,45,47} 4.42175 11 {0,1,2,4,5,7} 4.41989 22 {0,1,2,4,5,7,8,9,14,17} 4.36437 55 {0,1,2,4,7,8,9,13,14,16,17,18,26,28,31,32,34,36,43,49,52} 4.32651 55 {0,1,2,4,5,7,8,13,14,16,17,18,26,28,32,34,36 The author experimented with several types of constrained search spaces in an attempt to extend these results to larger moduli. These include:
H(K + 1) b S 4.
a. Restriction to a single branch, like the branch rooted at {0, 1, 2}. b. Restriction to those sets which deviate from a greedy construction for 4-free sets mod b some bounded number of times. c. Restrictions based on term-wise upper bounds for the elements of S. Experimentation suggests that (c) offers the best balance between runtime efficiency and result quality as b grows large. A series of constrained searches among bases b ≤ 200 was unable to improve the M 4 bound 4.43975. The ten Kempner sets of largest harmonic sum found during these sparser searches are given in Table 3 below. (We omit several results with base b = 121, as these resemble but fail to improve the listed b = 11 case.)
Other Notable Results
Limited searches with k ∈ {6, 8, 9, 10} were unable to identify k-free sets with harmonic sums exceeding the trivial bounds H({1}∪(G 5 +1)) = 7.94433 (k = 6), H({1}∪ (G 7 + 1)) = 13.5332 (k = 8), H({1, 2}∪ (G 7 + 1)) = 13.5638 (k = 9), and H({1, 2, 3} ∪ (G 7 + 1)) = 13.5905 (k = 10). These attempts came closest in the case k = 10, where the author produced the 10-free set K ({0, 1, 2, 3 Heuristically, we expect Kempner sets to have large harmonic sum when their digit set includes many digits (cf. (3.1)), i.e. when they have large logarithmic density. By combining a family of particularly dense 3-free packings of Behrend [Beh46] with Proposition 1.1, one may produce Kempner sets with logarithmic density arbitrarily near 1.
Kempner sets K(S, b) of this construction would require extremely large b, but we may nevertheless search for sporadic Kempner sets with small b and unusually large logarithmic density δ(K (S, b)). By adapting our branch-andbound algorithm, we can address this search as well. Some notable examples from our limited search are presented in Table 4 below. Continued on next page In particular, the set K({0, 1, 3, 7, 17, 24, 25, 28, 29, 35}, 37) provides a simple and explicit example of a 3-free set surpassing the classical density log 3 2.
Proof. Fix ǫ > 0 and choose M such that H(S ∩ [1, M ]) > H(S) − ǫ. Fix an integer b > max(k, 2M ), so that S ∩ [1, M ] and hence (S ∩ [1, M ]) − min(S) are k-free mod b by Proposition 1.1. Then Proposition 1.2 implies that both K = K(S ∩ [1, M ] − min(S), b) and the shifted set K + 1 are k-free. Yet K+1 contains a copy of S ∩[1, M ], shifted by 1−min(S) ≤ 0 (i.e. held constant or decreased), hence H(K + 1) ≥ H(S ∩ [1, M ]) > H(S) − ǫ.
Lemma 2. 2 .
2Let S be a set of positive integers and let H n denote the nth harmonic number.
Table 1 .
1Notable 3-free Kempner Sets for b ≤ 120H(K + 1) b S
3.00794
3 {0,1}
3.00118 82 {0,1,3,4,9,10,12,13,27,28,30,31,36,37,39,40}
2.99461 83 {0,1,3,4,9,10,12,13,27,28,30,31,36,37,39,40}
2.99312 81 {0,1,3,4,9,10,12,13,27,28,30,31,36,37,39,67}
2.99260 81 {0,1,3,4,9,10,12,13,27,28,30,31,36,37,40,66}
2.99146 81 {0,1,3,4,9,10,12,13,27,28,30,31,36,39,40,64}
2.99083 81 {0,1,3,4,9,10,12,13,27,28,30,31,37,39,40,63}
2.98823 84 {0,1,3,4,9,10,12,13,27,28,30,31,36,37,39,40}
2.98700 81 {0,1,3,4,9,10,12,13,27,28,30,36,37,39,40,58}
2.98605 81 {0,1,3,4,9,10,12,13,27,28,31,36,37,39,40,57}
Table 2 .
2Notable 4-free Kempner Sets for b ≤ 60
Table 3 .
3Notable 4-free Kempner Sets of Larger Modulus 55,56,57,59,62,63,64,66,72,75,76,79,87,90,93,101,103, 107,109,113,126,133,137,146}H(K + 1) b S
4.43975
55 {0,1,2,4,5,9,10,11,14,16,17,18,21,24,30,37,39,41,42,45,47}
4.42175
11 {0,1,2,4,5,7}
4.41989
22 {0,1,2,4,5,7,8,9,14,17}
4.37406 105 {0,1,2,4,5,7,8,9,15,16,18,19,20,25,26,28,29,31,32,33,36,45,
50,51,59,61,63,68,70,72,79}
4.36953 177 {0,1,2,4,5,7,8,9,15,16,17,19,20,26,27,29,30,32,33,34,50,52,
4.36437
55 {0,1,2,4,7,8,9,13,14,16,17,18,26,28,31,32,34,36,43,49,52}
Continued on next page
Table 3 .
3Continued from previous page H(K + 1) b S 4.36280 153 {0,1,2,4,5,7,8,9,15,16,17,19,20,26,27,28,30,31,33,34,50,54, 55,56,58,59,63,65,68,69,71,72,76,78,91,93,96,98,99, 101,103} 4.36238 141 {0,1,2,4,5,7,8,9,14,16,17,18,26,28,29,31,32,33,36,37,39,51, 52,53,56,57,58,60,61,68,69,70,72,86,94,95,96,129,130} 4.36233 153 {0,1,2,4,5,7,8,9,15,16,17,19,20,26,27,28,30,31,33,34,50,54, 55,57,58,59,63,65,68,69,71,72,76,78,91,93,96,98,99, 101,103} 4.36022 195 {0,1,2,4,5,7,8,9,15,16,18,19,20,25,26,28,29,31,32,33,45,49, 51,52,53,59,60,61,63,67,68,72,79,80,82,84,87,90,98, 102,104,108,110,112,118,120,122,130}
, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18,19, 20, 21, 22, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35,
37, 38, 39, 42, 43, 46, 47, 49, 51, 52, 53, 54, 59}, 61) + 1
of harmonic sum 13.5865.
Table 4 .
4Some k-free Kempner Sets with Large δ(K) {0,1} 0.63767 3 37 {0,1,3,7,17,24,25,28,29,35} 0.63773 3 85 {0,1,3,4,9,10,13,24,28,29,31,36,40,42,50,66,73}δ(K)
k b S
0.63093 3 3
Table 4 .
4Continued from previous page δ(K) k b S 0.74722 4 11 {0,1,2,4,5,7} 0.75974 4 55 {0,1,2,4,5,9,10,11,14,16,17,18,21,24,30,37,39,41,42,45,47} 0.86135 5 5 {0,1,2,3} 0.92078 7 7 {0,1,2,3,4,5} 0.9205310 61 {0,1,2,3,4,5,6,7,8,10,11,12,13,15,16,17,18,19,20,21,22,24 26,27,28,29,30,31,33,34,35,37,38,39,42,43,46,47,49 51,52,53,54,59}
On sets of integers which contain no three terms in arithmetical progression. F A Behrend, Proc. Nat. Acad. Sci. U. S. A. 32F. A. Behrend. On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. U. S. A., 32:331- 332, 1946.
A construction for partitions which avoid long arithmetic progressions. E R Berlekamp, Canadian Math. Bulletin. 11E. R. Berlekamp. A construction for partitions which avoid long arithmetic progressions. Canadian Math. Bulletin, 11:409-414, 1968.
Summing Kempner's Curious (Slowly-Convergent) Series (Wolfram Library Archive. Robert Baillie, Thomas Schmelzer, Robert Baillie and Thomas Schmelzer. Summing Kempner's Curious (Slowly-Convergent) Series (Wolfram Library Archive). https://library.wolfram.com/infocenter/MathSource/7166/, 2008.
Logarithmic bounds for Roth's theorem via almost-periodicity. F Thomas, Olof Bloom, Sisask, Thomas F. Bloom and Olof Sisask. Logarithmic bounds for Roth's theorem via almost-periodicity.
On arithmetic properties of integers with missing digits i: Distribution in residue classes. Paul Erdos, Christian Mauduit, András Sárközy, Journal of Number Theory. 702Paul Erdos, Christian Mauduit, and András Sárközy. On arith- metic properties of integers with missing digits i: Distribution in residue classes. Journal of Number Theory, 70(2):99-120, 6 1998.
On arithmetic properties of integers with missing digits II: Prime factors. Paul Erdos, Christian Mauduit, András Sárközy, Discrete Mathematics. 2001-3Paul Erdos, Christian Mauduit, and András Sárközy. On arith- metic properties of integers with missing digits II: Prime factors. Discrete Mathematics, 200(1-3):149-164, 4 1999.
The sum of the reciprocals of a set of integers with no arithmetic progression of k terms. Joseph Gerver, Proc. of the the Amer. Math. Soc. 622Joseph Gerver. The sum of the reciprocals of a set of integers with no arithmetic progression of k terms. Proc. of the the Amer. Math. Soc., 62(2):211-214, 2 1977.
Sets of integers with no long arithmetic progressions generated by the greedy algorithm. Joseph Gerver, Laurence Ramsey, Mathematics of Computation -Math. Comput. 331979Joseph Gerver and Laurence Ramsey. Sets of integers with no long arithmetic progressions generated by the greedy algorithm. Mathematics of Computation -Math. Comput., 33:1353-1359, 10 1979.
The primes contain arbitrarily long arithmetic progressions. Ben Green, Terence Tao, Annals of Mathematics. 1672Ben Green and Terence Tao. The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics, 167(2):481-547, 2008.
A curious convergent series. A J Kempner, American Mathematical Monthly. 212A. J. Kempner. A curious convergent series. American Mathemat- ical Monthly, 21(2):48-50, 1914.
Primes with restricted digits. James Maynard, Inventiones Mathematicae. 217James Maynard. Primes with restricted digits. Inventiones Math- ematicae, 217:127-218, 2019.
Summing a curious, slowly convergent series. Thomas Schmelzer, Robert Baillie, American Mathematical Monthly. 1156Thomas Schmelzer and Robert Baillie. Summing a curious, slowly convergent series. American Mathematical Monthly, 115(6):525- 540, 2008.
On sets of integers containing no k elements in arithmetic progression. Endre Szemerédi, Acta Arithmetica. 27Endre Szemerédi. On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, 27:199-245, 1975.
A nonaveraging set of integers with a large sum of reciprocals. J Wróblewski, Math. Comp. 43J. Wróblewski. A nonaveraging set of integers with a large sum of reciprocals. Math. Comp., 43:261-262, 1984.
Arithmetic progressions with restricted digits. Aled Walker, Alexander Walker, The American Mathematical Monthly. 1272Aled Walker and Alexander Walker. Arithmetic progressions with restricted digits. The American Mathematical Monthly, 127(2):140-150, 2020.
|
[] |
[
"Fate of Yang-Mills black hole in early Universe",
"Fate of Yang-Mills black hole in early Universe"
] |
[
"Łukasz Nakonieczny \nInstitute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland\n",
"Marek Rogatko \nInstitute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland\n"
] |
[
"Institute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland",
"Institute of Physics Maria Curie\nSkłodowska University\npl. Marii Curie-Skłodowskiej 120-031LublinPoland"
] |
[] |
According to the Big Bang Theory as we go back in time the Universe becomes progressively hotter and denser. This leads us to believe that the early Universe was filled with hot plasma of elementary particles. Among many questions concerning this phase of history of the Universe there are questions of existence and fate of magnetic monopoles and primordial black holes. Static solution of Einstein-Yang-Mills system may be used as a toy model for such a black hole. Using methods of field theory we will show that its existence and regularity depend crucially on the presence of fermions around it.
|
10.1063/1.4791724
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[
"https://arxiv.org/pdf/1301.2433v2.pdf"
] | 118,558,472 |
1301.2433
|
4e87187381113883eb6fe27254e4b2c99b7a229d
|
Fate of Yang-Mills black hole in early Universe
17 Jan 2013
Łukasz Nakonieczny
Institute of Physics Maria Curie
Skłodowska University
pl. Marii Curie-Skłodowskiej 120-031LublinPoland
Marek Rogatko
Institute of Physics Maria Curie
Skłodowska University
pl. Marii Curie-Skłodowskiej 120-031LublinPoland
Fate of Yang-Mills black hole in early Universe
17 Jan 2013primordial black holes, fermions PACS: 0450+h
According to the Big Bang Theory as we go back in time the Universe becomes progressively hotter and denser. This leads us to believe that the early Universe was filled with hot plasma of elementary particles. Among many questions concerning this phase of history of the Universe there are questions of existence and fate of magnetic monopoles and primordial black holes. Static solution of Einstein-Yang-Mills system may be used as a toy model for such a black hole. Using methods of field theory we will show that its existence and regularity depend crucially on the presence of fermions around it.
INTRODUCTION
Recent observational data strongly suggest the existence of dark matter (DM) [1]. One of a few remaining DM candidates compatible with the Standard Model (SM) are primordial black holes (PBH) [2]. These black holes were formed in the early Universe during gravitational collapse of density fluctuation. The existence of the PBH was first proposed in [3,4,5] and their formation during inflation or phase transition in the early Universe was discussed in [6,7,8]. Another type of objects that could be formed during phase transition in early Universe are topological defects like cosmic strings and magnetic monopoles [9]. If these objects really formed during evolution of the Universe we should consider the possibility of interactions among them. Within the framework of field theory a simple model of the system that allows the existence of magnetic monopoles may be given by Yang-Mills (YM) theory with SU(2) gauge group. Particle like solutions of Einstein-Yang-Mills (EYM) system describing the magnetic monopole were first constructed in [10]. On the other hand, solutions describing a black hole with magnetic monopole in this context were presented in [11]. There was also shown in [12] that finiteness of the black hole mass demands absence of an electric part of YM field.
Another important component of the early Universe whose interaction with PBH we should consider is fermionic matter. To describe these interactions we use methods of field theory. As a toy model of PBH we take the aforementioned spherically symmetric YM black hole. As a representation of fermionic matter we use 1 [email protected] 2 [email protected] Dirac field. Using dimensional reduction and bosonization we will analyze the influence of fermions on the considered PBH.
DIRAC EQUATION IN MAGNETIC YANG-MILLS BLACK HOLE BACKGROUND
A general form of a spherically symmetric and time independent metric is
ds 2 = −A 2 (r)dt 2 + B −2 (r)dr 2 + r 2 dΩ 2 ,(1)
where r is an usual radial coordinate and dΩ 2 = dθ 2 + sin 2 (θ )dφ 2 is a standard metric on two dimensional sphere. The location of an event horizon of a black hole is given by the largest positive root of A 2 (r), which is denoted as r H . Actually for analyzing fermions equations of motion it is more convenient to introduce the so-called tortoise coordinate r * . This coordinate is given by the following relation:
dr * = B(r) A(r) dr.(2)
Our metric expressed in coordinates (t, r * , θ , φ ) is
ds 2 = A 2 (−dt 2 + dr 2 * ) + C 2 dΩ 2 ,(3)
where we introduce a function C to keep in mind that now we have r = r(r * ). From now on, to simplify our notation we will omit explicit writing of arguments of functions. During our computations we will use tetrad formalism. The local ortogonal tetrad is defined by the relation g µν = e i µ e j µ η i j , where e i µ is an element of the tetrad, g µν and η i j are metric tensors in curved and flat spacetimes respectively. We use the following signature convention: η 00 = −1, η 11 = η 22 = η 33 = +1.
The general form of static and spherically symmetric Yang-Mills potential may be written as
H µ = e i µ [a i n k τ k + 1 − w(r) 2λC ε i jk n j τ k ],(4)
where a i = (a 0 , a 1 ) and w represent electric and magnetic parts of YM field, n i is a unit vector normal to the sphere, λ is a coupling constant, ε i jk stands for a totally antisymmetric Levi-Civita symbol, and τ k are generators of SU(2) gauge group represented by Pauli matrices. As the representation for four dimensional gamma matrices we choose Weyl basis
γ 0 = 0 I I 0 , γ i = 0 σ i −σ i 0 ,(5)
where I is a unit two dimensional matrix and σ i are three Pauli matrices. Gamma matrices in curved spacetime are given by the standard relation: γ µ = e µ k γ k . Representing a spinor ψ as ψ = ψ L ψ R we may write the Dirac equation as
i/ D + ψ R − mψ L = 0, i/ D − ψ L − mψ R = 0,(6)
where operators / D ± are given by [13,14] /
D ± = A −1 ∂ t − iλ {[σ 0 a 0 ± σ 1 a 1 ]n ·τ ± w − 1 2λCn ·σ ×τ}+ ±σ ·nA −1 ∂ r * ±σ ·n{A −1 C −1 ∂ r * C + 1 2 A −1 A −1 ∂ r * A}+ ± C −1 D S 2 .(7)
In the above formula a bar over a quantity represents three dimensional vector, a dot -scalar multiplication, × -vector multiplication, and D S 2 is Dirac operator on S 2 . We are mainly interested in fermions in s-wave sector, which is described by the lowest eigenvalue of D S 2 . The action of operatorsn ·τ,n ·σ ×τ,σ ·n and D S 2 on these states may be found in [13]. We may use this knowledge to integrate over the angles in four dimensional Dirac action to obtain effective two dimensional theory in (t, r * ) plane.
On the other hand, dimensional reduction of the YM action gives us the following Lagrangian:
L Y M−2d = − C 2 4 f ab f ab − |dw 2 | − 1 2C 2 (|w| 2 − 1) 2 , (8) where d a = ∇ a − iB a , f ab = ∂ a B b − ∂ b B a ,
and B a = e i a a i . From now on latin letters from the beginning of alphabet will label indexes from curved two dimensional (t, r * ) spacetime and those from the middle from flat spacetime.
MASSLESS FERMIONS
In the massles case we see from (6) that chiralities decouple and effective two dimensional theory contains two fermionic fields connected to ψ L and ψ R . The effective Lagrangian for the field ψ R is
L FR−2d = −iḠ Rγ a ∇ a G R − λ B aḠRγ aγ 3 G R + +VḠ RγL G R − VḠ RγR G R ,(9)
where V = w C . In this formula G R is two dimensional spinor field connected to ψ R by rescaling and multiplication by appropriate σ matrices [15]. Two dimensional flat spacetime gamma matrices are given by the following relations:
{γ i ,γ j } = 2η i j , η 00 = −1 = −η 11 , γ 0 = −iσ 3 ,γ 1 = −σ 2 , γ 3 =γ 0γ 1 = σ 1 ,γ L/R = 1 2 (I ±γ 3 ).(10)
The Lagrangian for the field connected to ψ L differs from (9) only by a sign of a term proportional to B a . Analyzing equations of motion for fermions in curved spacetime is a highly nontrivial and difficult task. However, in case at hand since we can express our problem in form of effective two dimensional theory we may use bosonization technique [16]. This technique is well defined for flat spacetime two dimensional problems and also for asymptotically flat spacetimes [17,18]. The merit of bosonization is that we express a fermionic sector of our theory in terms of complex scalar field. Basic bosonization formulas in our case are given below
j a =ψγ a ψ = 1 √ π ε ab ∇ b φ ,(11)j 3a =ψγ a γ 3 ψ = 1 √ π ∇ a φ ,(12)ψγ L ψ = be 2i √ πφ ,ψγ R ψ = be −2i √ πφ .(13)
After bosonization from (9) we obtain the following scalar Lagrangian:
L BR = − 1 2 ∇ a φ R ∇ a φ R − λ B a 1 √ π ∇ a φ R + +V b(e 2i √ πφ R − e −2i √ πφ R ),(14)
from which we derive an equation of motion for φ R field
∇ a ∇ a φ R + λ √ π ∇ a B a + 4ib √ πV cos(2 √ πφ R ) = 0.(15)
Then, we derive an equation of motion for scalar field connected to the ψ L sector of our original fermionic theory analogically: (16) Equations (15) and (16) are highly nonlinear and we were not able to find a solution in terms of known special functions in the whole spacetime. Nevertheless, we may find some useful information about their solutions by analyzing them in two asymptotic regions, namely in near horizon region and in large r * region.
∇ a ∇ a φ L − λ √ π ∇ a B a + 4ib √ πV cos(2 √ πφ L ) = 0.
In the large r * region we have that
A 2 ≈ 1, C = r, w ≈ ±1, r * ∼ r.(17)
Taking this into account the equation (15) turns into
−∂ 2 t φ R + ∂ 2 r * φ R + 4ib √ π w(∞) r cos(2 √ πφ R ) = 0. (18)
After dropping a term proportional to O(r −1 ) we get a free wave equation
−∂ 2 t φ R + ∂ 2 r * φ R = 0.(19)
A regular solution to this equation may be expressed as a plane wave
φ R = c 0 e −iω(t±r * ) ,(20)
where c 0 is an integration constant. On the other hand, in the near horizon region we have
A 2 = 2κ(r − r H ), C = r H , r − r h = e 2κr * ,(21)
where κ is surface gravity of a Yang-Mills black hole. The equation (15) in this region takes the form
−∂ 2 t φ R + ∂ 2 r * φ R + i 8b √ πw(r h )κ r h e 2κr * cos(2 √ πφ R ) = 0.(22)
Because as we approach a black hole horizon r * → −∞, we may drop a term proportional to e 2κr * in the above equation. From this we see that a regular solution is again a time dependent plane wave given by
φ R = c 1 e −iω(t±r * ) ,(23)
where c 1 is some other integration constant. On the basis of this analysis we may conclude that a regular solution to the considered scalar field equation will be time dependent. But both scalar fields represent fermionic currents and their time dependence ultimately means that fermionic fields will also be time dependent. To see how this may influence YM field we use equations of motion for electric and magnetic parts of this field in the presence of fermions. After using bosonization formulas (11) these equations read [15] ∇
a [C 2 f ab ] − 2 |w| 2 B b = λ √ π [∇ b φ R − ∇ b φ L ],(24)∇ a ∇ a w − 2 C 2 w(|w| 2 − 1) + 2wB a B a = i 2b C [sin(2 √ πφ R ) + sin(2 √ πφ L )].(25)
Conclusions that stem from the above equations are the following. First, let us remind that equations (15) and (16) differ only by sign in front of the term proportional to B a and we assume that initially a black hole has only magnetic charge (B a = 0). Second, from equation (24) we see that in this case contributions of scalar fields cancel each other and B a = 0 is still a valid solution to (24). Third, from equation (25) we see that there is a nonzero contribution of scalar fields to magnetic part of YM field. But since these fields are time dependent so should be the resulting w. On the other hand, through Einstein equations, this results in time dependence of metric tensor elements and ultimately means that the assumption of staticity of Yang-Mills black hole is destroyed in the presence of massless fermions.
MASSIVE FERMIONS
For massive fermions we see from (6) that chiralities are mixed up. To use bosonization in this case we need to make an additional assumption about the form of ψ R and ψ L . We use the following ansatz:
G L = iσ 3 G R ≡ G,(26)
where G L and G R are two dimensional spinor fields connected with ψ L and ψ R respectively [15]. Having this in mind an effective two dimensional fermionic Lagrangian is as follows:
L GF−2d = −iḠγ a ∇ a G − λ B aḠγ aγ 3 G+ +(V + m)Ḡγ L G + (m − V)Ḡγ R G,(27)
where like in massless case V = w C . Using the same bosonization formulas as in the massless case we arrive at the following scalar Lagrangian:
L GB = − 1 2 ∇ a φ ∇ a φ − λ √ π B a ∇ a φ + + (V + m)be 2i √ πφ + (m − V)be −2i √ πφ .(28)
The equation of motion for scalar field in this case is
∇ a ∇ a φ + λ √ π ∇ a B a + + 4ib √ π V cos(2 √ πφ ) + im sin(2 √ πφ ) = 0. (29)
In the large r * limit, after dropping terms proportional to O(r −1 ), we obtain a sine-Gordon equation
−∂ 2 t φ + ∂ 2 r * φ − 4b √ πm sin(2 √ πφ ) = 0,(30)
for which a regular time dependent and decaying at infinity (kink type) solution is given by
φ = 2 √ π arctan e − 8bπm 1−v 2 (r * −vt) .(31)
On the other hand, the same type of analysis like in the massless case revealed that in the near horizon limit we again have a free wave equation with a regular time dependent solution in form of a plane wave:
φ = c 2 e −iω(t±r * ) .(32)
Now we will discuss the influence of massive fermions on a YM black hole. Yang-Mills equations of motion in the presence of fermions (after bosonization) are the following:
∇ a [C 2 f ab ] − 2 |w| 2 B b = λ √ π ∇ b φ ,(33)∇ a ∇ a w − 2 C 2 w(|w| 2 − 1) + 2wB a B a = i 2b C sin(2 √ πφ ).(34)
CONCLUSIONS
Now we shall present a short summary of our results concerning the influence of fermions on a primordial black hole modeled by a magnetic Yang-Mills black hole.
First we will discuss the massless case. By asymptotic analysis we find an evidence that regular solutions to equations (15) and (16) should be time dependent. These equations describe bosonized massless fermions and give a nonzero contribution to a magnetic part of Yang-Mills field. Because their solutions are time dependent the resulting magnetic part of YM field will be time dependent and, through Einstein equations, elements of a metric tensor will also be time dependent.
In massive case bosonized fermions are described by solutions to equation (29). The same type of analysis like in massless case also reveals the destruction of staticity of our black hole. Moreover, massive fermions will lead to the formation of a dyonic black hole. But, as was shown in [12], the presence of an electric part of Yang-Mills field leads to infinite mass of the resulting black hole.
In conclusion we may say that the presence of fermions and their interaction with the considered PBH will lead to its destruction through mechanisms described above.
From equation (33) we see that, even if we initially set B a = 0, fermions give a nonzero contribution to the electric part of YM field. This means that contrary to the massless case the presence of fermions leads to dyonic structure of a black hole. But as was shown in[12] dyonic Yang-Mills black hole necessarily has infinite mass. On the other hand, form equation (34) we see that time dependent fermions give a nonzero contribution to the magnetic part of YM field. Through Einstein equations this leads to a time dependent line element and the destruction of staticity of a black hole.
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[] |
[
"ON A MEASURE THEORETIC AREA FORMULA",
"ON A MEASURE THEORETIC AREA FORMULA"
] |
[
"Valentino Magnani "
] |
[] |
[] |
We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel measure with respect to a fixed Carathéodory measure. We focus our attention in the case this measure is the spherical Hausdorff measure, giving a metric measure area formula. Our point consists in using certain covering derivatives as "generalized densities". Some consequences for the sub-Riemannian Heisenberg group are also pointed out.
|
10.1017/s030821051500013x
|
[
"https://arxiv.org/pdf/1401.2536v1.pdf"
] | 18,611,478 |
1401.2536
|
9e2869b5305ecf61eae1bfa7cbbacbbd928a378f
|
ON A MEASURE THEORETIC AREA FORMULA
11 Jan 2014
Valentino Magnani
ON A MEASURE THEORETIC AREA FORMULA
11 Jan 2014
We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel measure with respect to a fixed Carathéodory measure. We focus our attention in the case this measure is the spherical Hausdorff measure, giving a metric measure area formula. Our point consists in using certain covering derivatives as "generalized densities". Some consequences for the sub-Riemannian Heisenberg group are also pointed out.
It is well known that computing the Hausdorff measure of a set by an integral formula is usually related to rectifiability properties, namely, the set must be close to a linear subspace at any small scale. The classical area formula exactly provides this relationship by an integral representation of the Hausdorff measure. Whenever a rectifiable set is thought of as a countable union of Lipschitz images of subsets in a Euclidean space, the area formula holds in metric spaces, [4].
In the last decade, the development of Geometric Measure Theory in a non-Euclidean framework raised new theoretical questions on rectifiability and area-type formulae. The main problem in this setting stems from the gap between the Hausdorff dimension of the target and that of the source space of the parametrization. In fact, in general this dimension might be strictly greater than the topological dimension of the set. As a result, the parametrization from a subset of the Euclidean space cannot be Lipschitz continuous with respect to the Euclidean distance of the source space.
To mention an instance of this difficulty, the above mentioned area formula for a large class of Heisenberg group valued Lipschitz mappings does not work, [1]. For this reason, theorems on differentiation of measures constitute an important tool to overcome this problem. In this connection, the present note shows how the Federer's Theorems of 2.10.17 and 2.10.18 in [2] are able to disclose a purely metric area formula. The surprising aspect of this formula is that an "upper covering limit" actually can be seen as a generalized density of a fixed Borel measure.
To define these densities, we first introduce covering relations: if X is any set, a covering relation is a subset C of {(x, S) : x ∈ S ∈ P(X)}. In the sequel, the set X will be always assumed to be equipped with a distance. Defining for A ⊂ X the corresponding class C(A) = {S : x ∈ A, (x, S) ∈ C}, we say that C is fine at x, if for every ε > 0 there exists S ∈ C({x}) such that diam S < ε. According to 2.8.16 of [2], the notion of covering relation yields the following notion of "covering limit".
Date: January 14, 2014. The author acknowledges the support of the European Project ERC AdG *GeMeThNES*. Definition 1 (Covering limit). If C denotes a covering relation and f : D → R, C({x}) ⊂ D ⊂ C(X) and C is fine at x ∈ X, then we define the covering limits
(C) lim sup S→x f = inf ε>0 sup{f (S) : S ∈ C({x}), diam S < ε} ,(1)(C) lim inf S→x f = sup ε>0 inf{f (S) : S ∈ C({x}), diam S < ε} .(2)
The covering relations made by closed balls clearly play an important role in the study of the area formula for the spherical Hausdorff measure.
ζ δ (R) = inf ∞ j=0 ζ(E j ) : E j ∈ F , diam(E j ) ≤ δ for all j ∈ N, R ⊂ j∈N E j .
The ζ-approximating measure is defined and denoted by ψ ζ = sup δ>0 ζ δ . Denoting by F the family of closed sets of X, for α, c α > 0, we define ζ α : F → [0, +∞] by
ζ α (S) = c α diam(S) α .
Then the α-dimensional Hausdorff measure is H α = ψ ζα . If ζ b,α is the restriction of ζ α to F b , then S α = ψ ζ b,α is the α-dimensional spherical Hausdorff measure.
These special limits of Definition 1 naturally arise in the differentiation theorems for measures and allow us to introduce a special "density" associated with a measure.
Definition 4 (Federer density). Let µ be a measure over X, let S ⊂ P(X) and let ζ : S → [0, +∞]. Then we set S µ,ζ = S \ S ∈ S : ζ(S) = µ(S) = 0 or µ(S) = ζ(S) = +∞ , along with the covering relation C µ,ζ = {(x, S) : x ∈ S ∈ S µ,ζ }. We choose x ∈ X and assume that C µ,ζ is fine at x. We define the quotient function
Q µ,ζ : S µ,ζ → [0, +∞], Q µ,ζ (S) = +∞ if ζ(S) = 0 µ(S)/ζ(S) if 0 < ζ(S) < +∞ 0 if ζ(S) = +∞ .
Then we are in the position to define the Federer density, or upper ζ-density of µ at x ∈ X, as follows
(3) F ζ (µ, x) = (C µ,ζ ) lim sup S→x Q µ,ζ (S) .
According to the following definition, we will use special notation when we consider Federer densities with respect to ζ α and ζ b,α , respectively.
Definition 5. If µ is a measure over X and C µ,ζ b,α is fine at x ∈ X, then we set θ α (µ, x) = F ζ b,α (µ, x). If C µ,ζα is fine at x, then we set s α (µ, x) = F ζα (µ, x).
Remark 1. If x ∈ X and there exists an infinitesimal sequence (r i ) of positive radii such that B(x, r i ) may have vanishing diameter and in this case µ(B(x, r i )) > 0, then it is easy to realize that both C µ,ζ b,α and C µ,ζα are fine at x. In particular, the last conditions are always satisfied whenever all balls B(x, r i ) have positive diameter.
By the previous definitions, we state a revised version of Theorem 2.10.17(2) of [2]. Theorem 1. Let S ⊂ P(X) and let ζ : S → [0, +∞] be a size function. If µ is a regular measure over X, A ⊂ X, t > 0, S µ,ζ covers A finely and for all
x ∈ A we have F ζ (µ, x) < t, then µ(E) ≤ t ψ ζ (E) for every E ⊂ A.
Analogously, the next theorem is a revised version of Theorem 2.10.18(1) in [2].
Theorem 2. Let µ be a measure over X, let S be a family of closed and µ-measurable sets, let ζ : S → [0, +∞), let B ⊂ X and assume that S µ,ζ covers B finely. If there exist c, η > 0 such that for each S ∈ S there existsS ∈ S with the properties
(4)Ŝ ⊂S, diamS ≤ c diam S and ζ(S) ≤ η ζ(S), whereŜ = {T ∈ S : T ∩S = ∅, diam T ≤ 2 diam S}, V ⊂ X is an open set containing B and for every x ∈ B we have F ζ (µ, x) > t, then µ(V ) ≥ t ψ ζ (B).
These theorems provide both upper and lower estimates for a large class of measures, starting from upper and lower estimates of the Federer density. A slight restriction of the assumptions in the previous theorems joined with some standard arguments leads us to a new metric area-type formula, where the integration of F ζ (µ, ·) recovers the original measure. This is precisely our first result.
Theorem 3 (Measure theoretic area-type formula). Let µ be a Borel regular measure over X such that there exists a countable open covering of X, whose elements have µ finite measure. Let S be a family of closed sets, let ζ : S → [0, +∞) and assume that for some constants c, η > 0 and for every S ∈ S there existsS ∈ S such that
(5)Ŝ ⊂S, diamS ≤ c diam S and ζ(S) ≤ η ζ(S), whereŜ = {T ∈ S : T ∩ S = ∅, diam T ≤ 2 diam S}. If A ⊂ X is Borel, S µ,ζ covers A finely and F ζ (µ, ·) is a Borel function on A with (6) ψ ζ ({x ∈ A : θ ζ (µ, x) = 0}) < +∞ and µ({x ∈ A : θ ζ (µ, x) = +∞}) = 0 ,
then for every Borel set B ⊂ A, we have
(7) µ(B) = B F ζ (µ, x) dψ ζ (x) .
The second condition of (6) precisely corresponds to the absolute continuity of µ A with respect to ψ ζ A. This measure theoretic area formula may remind of a precise differentiation theorem, where indeed the third condition of (5) represents a kind of "doubling condition" for the size function ζ. In fact, the doubling condition for a measure allows for obtaining a similar formula, where the density is computed by taking the limit of the ratio between the measures of closed balls with the same center and radius, see for instance Theorems 2.9.8 and 2.8.17 of [2].
On one side, the Federer density F ζ (µ, x) may be hard to compute, depending on the space X. On the other side, formula (7) neither requires special geometric properties for X, as those for instance of the Besicovitch covering theorem (see the general condition 2.8.9 of [2]), nor an "infinitesimal" doubling condition for ψ ζ A, as in 2.8.17 of [2]. Moreover, there are no constraints that prevent X from being infinite dimensional.
The absence of specific geometric conditions on X is important especially in relation with applications of Theorem 3 to sub-Riemannain Geometry, in particular for the class of the so-called Carnot groups, where the classical Besicovitch covering theorem may not hold, see [5], [8]. In these groups, we have no general theorem to "differentiate" an arbitrary Radon measure, therefore new differentiation theorems are important.
We provide two direct consequences of Theorem 3, that correspond to the cases where ψ ζ is the Hausdorff measure and the spherical Hausdorff measure, respectively.
Theorem 4 (Area formula with respect to the Hausdorff measure). Let µ be a Borel regular measure over X such that there exists a countable open covering of X, whose elements have µ finite measure. If A ⊂ X is Borel and S µ,ζα covers A finely, then s α (µ, ·) is Borel. Moreover, if H α (A) < +∞ and µ A is absolutely continuous with respect to H α A, then for every Borel set B ⊂ A, we have
µ(B) = B s α (µ, x) dH α (x) .
This theorem essentially assigns a formula to the density of µ with respect to H α . Let us recall the formula for this density
s α (µ, x) = inf ε>0 sup Q µ,ζα (S) : x ∈ S ∈ S µ,ζα , diam S < ε .
Under the still general assumption that all open balls have positive diameter, we have Q µ,ζα (S) = µ(S)/ζ α (S). More manageable formulae for s α (µ, ·) turn out to be very hard to be found and this difficulty is related to the geometric properties of the single metric space. On the other hand, if we restrict our attention to the spherical Hausdorff measure, then the corresponding density θ α (µ, ·) can be explicitly computed in several contexts, where it can be also given a precise geometric interpretation.
In this case, we will also assume a rather weak condition on the diameters of open balls. Precisely, we say that a metric space X is diametrically regular if for all x ∈ X and R > 0 there exists δ x,R > 0 such that (0, δ x,R ) ∋ t → diam(B(y, t)) is continuous for every y ∈ B(x, R). We are now in the position to state the measure theoretic area-type formula for the spherical Hausdorff measure.
Theorem 5 (Spherical area formula). Let X be a diametrically regular metric space, let α > 0 and let µ be a Borel regular measure over X such that there exists a countable open covering of X whose elements have µ finite measure. If B ⊂ A ⊂ X are Borel sets and S µ,ζ b,α covers A finely, then θ α (µ, ·) is Borel on A. In addition, if S α (A) < +∞ and µ A is absolutely continuous with respect to S α A, then we have
(8) µ(B) = B θ α (µ, x) dS α (x) .
In the sub-Riemannian framework, for distances with special symmetries and the proper choice of the Riemannian surface meaure µ, the density θ α (µ, ·) is a geometric constant that can be computed with a precise geometric interpretation. Then the previous formula is expected to have a potentially wide range of applications in the computation of the spherical Hausdorff measure of sets in the sub-Riemannian context. Indeed, this project was one of the motivations for the present note.
Here we are mainly concerned with the purely metric area formula, therefore we limit ourselves to provide some examples of applications to the Heisenberg group, leaving details along with further developments for subsequent work.
Let Σ be a C 1 smooth curve in H equipped with the sub-Riemannian distance ρ. This distance is also called Carnot-Carathéodory distance, see [3] for the relevant definitions. Whenever a left invariant Riemannian metric g is fixed on H, we can associate Σ with its intrinsic measure µ SR , see [6] for more details. We will assume that Σ has at least one nonhorizontal point x ∈ Σ, namely, T x Σ is not contained in the horizontal subspace H x H, that is spanned by the horizontal vector fields evaluated at x, [3].
If we fix the size function ζ b,2 (S) = diam(S) 2 /4 on closed balls and x is nonhorizontal, then it is possible to compute explicitly θ 2 (µ SR , x), getting
θ 2 (µ SR , x) = α(ρ, g) ,
where α(ρ, g) is precisely the maximum among the lengths of all intersections of vertical lines passing through the sub-Riemannian unit ball, centered at the origin. Here the length is computed with respect to the scalar product given by the fixed Riemannian metric g at the origin. As an application of Theorem 5, we obtain
µ SR = α(ρ, g)S 2 Σ ,
where S 2 is the spherical Hausdorff measure induced by ζ b,2 . The appearance of the geometric constant α(ρ, g) is a new phenomenon, due to the use of Federer's density. The nonconvex shape of the sub-Riemannian unit ball centered at the origin allows α(ρ, g) to be strictly larger than the length β(ρ, g) of the intersection of the same ball with the vertical line passing through the origin. This feature of the sub-Riemannian unit ball shows that θ 2 (µ SR , x) and the the upper spherical density Θ * 2 (µ SR , x) = lim sup r→0 + µ(B(x, r)) r 2 differ. In fact, setting t ∈ (α(ρ, g), β(ρ, g)), we get Θ * 2 (µ SR , x) = β(ρ, g) < t < α(ρ, g) = θ 2 (µ SR , x) for all x ∈ N ,
where N = x ∈ Σ : T x Σ is not horizontal and we also have
(9) µ SR (N ) = α(ρ, g)S 2 (N ) > t S 2 (N ) .
As a consequence of (9), in the inequality (1) of 2.10.19 in [2], with µ = µ SR and A = N , the constant 2 m with m = 2 cannot be replaced by one. Moreover, even in the case we weaken the inequality (1) of 2.10.19 in [2] replacing the Hausdorff measure with the spherical Hausdorff measure, then (9) still shows that 2 m with m = 2 cannot be replaced by one. In the case m = 1, it is possible to show, by an involved construction of a purely (H 1 , 1) unrectifiable set of the Euclidean plane, that 2 m is even sharp, see the example of 3.3.19 of [2]. Somehow, our curve with nonhorizontal points has played the role of a more manageable unrectifiable set. Incidentally, the set N is purely (H 2 , 2) unrectifiable with respect to ρ, see [1]. The connection between rectifiability and densities was already pointed out in [7], where the authors improve in a general metric space X the upper estimate for σ 1 (X), related to the so-called Besicovitch's 1 2 -problem. According to [7], σ k (X) for some positive integer k is the infimum among all positive numbers t having the property that for each E ⊂ X with H k (E) < +∞ and such that
lim inf r→0 + H k (E ∩ B(x, r)) c k 2 k r k > t
for H k -a.e.
x ∈ E implies that E is countably k-rectifiable, where it is assumed that the open ball B(x, r) has diameter equal to 2r for all (x, r) ∈ X × (0, +∞) and H k arises from the Carathéodory construction by the size function ζ(S) = c k diam(S) k . If we equip H with the so-called Korányi distance d, see for instance Section 1.1 of [5], then a different application of Theorem 5 gives a lower estimate for σ 2 (H, d). In fact, we can choose Σ 0 to be a bounded open interval of the vertical line of H passing through the origin. This set is purely (H 2 , 2) unrectifiable. We define ζ d b,2 (S) = diam d (S) 2 /4 on closed balls, where the diameter diam d (S) refers to the distance d, and consider the intrinsic measure µ SR of Σ 0 . By the convexity of the d-unit ball centered at the origin, the corresponding Federer density θ 2 d (µ SR , x) at a nonhorizontal point x satisfies
θ 2 d (µ SR , x) = α(d, g) ,
where α(d, g) is the length of the intersection of the Korányi unit ball cantered at the origin with the vertical line passing through the origin. Following the previous notation, by Theorem 5 we get
µ SR = α(d, g) S 2 d Σ 0 ,wherer 2 ≤ lim r→0 + S 2 d Σ 0 (B(x, r)) r 2 = 1 .
This implies that σ 2 (H, d) ≥ 1/2. Up to this point, we have seen how the geometry of the sub-Riemannian unit ball affects the geometric constants in estimates between measures. However, also the opposite direction is possible. In fact, considering the previous subset N and taking into account (1) of 2.10.19 in [2] with m = 2, we get µ SR (N ) ≤ 4 β(ρ, g) S 2 (N ) , hence the equality of (9) leads us to the estimate 1 < α(ρ, g) β(ρ, g) ≤ 4 .
It turns out to be rather striking that abstract differentiation theorems for measures can provide information on the geometric structure of the sub-Riemannian unit ball. Precisely, we cannot find any left invariant sub-Riemannian distanceρ in the Heisenberg group such that the geometric ratio α(ρ, g)/β(ρ, g) is greater than 4. These facts clearly leave a number of related questions, so that the present note may represent a starting point to establish deeper relationships between results of sub-Riemannian geometry and measure theoretic results.
In particular, further motivations to study sub-Riemannian metric spaces may also arise from abstract questions of Geometric Measure Theory. Clearly, to understand and carry out this demanding program more investigations are needed.
Definition 2 .
2The closed ball and the open ball of center x ∈ X and radius r > 0 are denoted by B(x, r) = {y ∈ X : d(x, y) ≤ r} and B(x, r) = {y ∈ X : d(x, y) < r} , respectively. We denote by F b the family of all closed balls in X.The next definition introduces the Carathéodory construction, see 2.10.1 of[2].
Definition 3 .
3Let S ⊂ P(X) and let ζ : S → [0, +∞] represent the size function. If δ > 0 and R ⊂ X, then we define
S 2 d is the spherical Hausdorff measure induced by ζ d b,2 . Since we havelim
r→0 +
S 2
d Σ 0 (B(x, r))
r 2
= lim
r→0 +
µ SR (B(x, r))
α(d, g)r 2 = lim
r→0 +
µ SR (B(x, r))
α(d, g)r 2 = 1 ,
and an easy observation shows that S 2
d Σ 0 ≤ 2 H 2
2 Σ 0 , it follows that
1
2
= lim
r→0
S 2
d Σ 0 (B(x, r))
2r 2
≤ lim inf
r→0 +
H 2
d (Σ 0 ∩ B(x, r))
Rectifiable sets in metric and Banach spaces. L Ambrosio, B Kirchheim, Math. Ann. 318L. Ambrosio, B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318, 527-555, (2000)
H Federer, Geometric Measure Theory. SpringerH. Federer, Geometric Measure Theory, Springer, (1969)
Carnot-Carathéodory spaces seen from within. M Gromov, Subriemannian Geometry. A. Bellaiche and J. RislerBaselBirkhauser Verlag144M. Gromov, Carnot-Carathéodory spaces seen from within, in Subriemannian Geometry, Progress in Mathematics, 144. ed. by A. Bellaiche and J. Risler, Birkhauser Verlag, Basel, (1996).
Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. B Kirchheim, Proc. Amer. Math. Soc. 121B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121, 113-123, (1994).
Foundation for the Theory of Quasiconformal Mappings on the Heisenberg Group. A Korányi, H M Reimann, Adv. Math. 111A. Korányi, H. M. Reimann, Foundation for the Theory of Quasiconformal Mappings on the Heisenberg Group, Adv. Math., 111, 1-87, (1995).
An intrinsic measure for submanifolds in stratified groups. V Magnani, D Vittone, J. Reine Angew. Math. 619V. Magnani, D. Vittone, An intrinsic measure for submanifolds in stratified groups, J. Reine Angew. Math., 619, 203-232, (2008)
On Besicovitch's 1 2 -problem. D Preiss, J Tišer, J. London Math. Soc. 2D. Preiss, J. Tišer, On Besicovitch's 1 2 -problem, J. London Math. Soc. (2) 45, n.2, 279-287, (1992)
Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. E Sawyer, R L Wheeden, Amer. J. Math. 114E. Sawyer, R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114, n.4, 813-874, (1992)
Largo Pontecorvo 5, I-56127, Pisa E-mail address: [email protected]. Valentino Magnani, Dipartimento Di Matematica, itValentino Magnani, Dipartimento di Matematica, Largo Pontecorvo 5, I-56127, Pisa E-mail address: [email protected]
|
[] |
[
"Improved limits on the tensor-to-scalar ratio using BICEP and Planck",
"Improved limits on the tensor-to-scalar ratio using BICEP and Planck"
] |
[
"M Tristram \nIJCLab\nUniversité Paris-Saclay\nCNRS/IN2P3\n91405OrsayFrance\n",
"A J Banday \nIRAP\nUniversité de Toulouse\nCNRS\nCNES\nUPS, (Toulouse)\nFrance\n",
"K M Górski \nJet Propulsion Laboratory\nCalifornia Institute of Technology\n4800 Oak Grove DrivePasadenaCaliforniaU.S.A\n\nWarsaw University Observatory\nAleje Ujazdowskie 400-478WarszawaPoland\n",
"R Keskitalo \nComputational Cosmology Center\nLawrence Berkeley National Laboratory\nBerkeleyCaliforniaU.S.A\n\nDepartment of Physics and Space Sciences Laboratory\nUniversity of California\nBerkeleyCaliforniaU.S.A\n",
"C R Lawrence \nJet Propulsion Laboratory\nCalifornia Institute of Technology\n4800 Oak Grove DrivePasadenaCaliforniaU.S.A\n",
"K J Andersen \nInstitute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway\n",
"R B Barreiro \nInstituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nSantanderSpain\n",
"J Borrill \nComputational Cosmology Center\nLawrence Berkeley National Laboratory\nBerkeleyCaliforniaU.S.A\n\nSpace Sciences Laboratory\nUniversity of California\nBerkeleyCaliforniaU.S.A\n",
"L P L Colombo \nDipartimento di Fisica\nUniversità degli Studi di Milano\nVia Celoria, 16MilanoItaly\n",
"H K Eriksen \nInstitute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway\n",
"R Fernandez-Cobos \nDpto. de Matemàticas\nEstadística y Computación\nUniversidad de Cantabria\nAvda. de los Castros s/nE-39005SantanderSpain\n",
"T S Kisner \nComputational Cosmology Center\nLawrence Berkeley National Laboratory\nBerkeleyCaliforniaU.S.A\n\nDepartment of Physics and Space Sciences Laboratory\nUniversity of California\nBerkeleyCaliforniaU.S.A\n",
"E Martínez-González \nInstituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nSantanderSpain\n",
"B Partridge \nHaverford College Astronomy Department\n370 Lancaster AvenueHaverfordPennsylvaniaU.S.A\n",
"D Scott \nDepartment of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadVancouverBritish ColumbiaCanada\n",
"T L Svalheim \nInstitute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway\n",
"I K Wehus \nInstitute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway\n"
] |
[
"IJCLab\nUniversité Paris-Saclay\nCNRS/IN2P3\n91405OrsayFrance",
"IRAP\nUniversité de Toulouse\nCNRS\nCNES\nUPS, (Toulouse)\nFrance",
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\n4800 Oak Grove DrivePasadenaCaliforniaU.S.A",
"Warsaw University Observatory\nAleje Ujazdowskie 400-478WarszawaPoland",
"Computational Cosmology Center\nLawrence Berkeley National Laboratory\nBerkeleyCaliforniaU.S.A",
"Department of Physics and Space Sciences Laboratory\nUniversity of California\nBerkeleyCaliforniaU.S.A",
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\n4800 Oak Grove DrivePasadenaCaliforniaU.S.A",
"Institute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway",
"Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nSantanderSpain",
"Computational Cosmology Center\nLawrence Berkeley National Laboratory\nBerkeleyCaliforniaU.S.A",
"Space Sciences Laboratory\nUniversity of California\nBerkeleyCaliforniaU.S.A",
"Dipartimento di Fisica\nUniversità degli Studi di Milano\nVia Celoria, 16MilanoItaly",
"Institute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway",
"Dpto. de Matemàticas\nEstadística y Computación\nUniversidad de Cantabria\nAvda. de los Castros s/nE-39005SantanderSpain",
"Computational Cosmology Center\nLawrence Berkeley National Laboratory\nBerkeleyCaliforniaU.S.A",
"Department of Physics and Space Sciences Laboratory\nUniversity of California\nBerkeleyCaliforniaU.S.A",
"Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)\nAvda. de los Castros s/nSantanderSpain",
"Haverford College Astronomy Department\n370 Lancaster AvenueHaverfordPennsylvaniaU.S.A",
"Department of Physics & Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadVancouverBritish ColumbiaCanada",
"Institute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway",
"Institute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernOsloNorway"
] |
[] |
We present constraints on the tensor-to-scalar ratio r using a combination of BICEP/Keck 2018 and Planck PR4 data allowing us to fit for r consistently with the six parameters of the ΛCDM model without fixing any of them. In particular, we are able to derive a constraint on the reionization optical depth τ and thus propagate its uncertainty onto the posterior distribution for r. While Planck sensitivity to r is no longer comparable with ground-based measurements, combining Planck with BK18 and BAO gives results consistent with r = 0 and tightens the constraint to r < 0.032.
|
10.1103/physrevd.105.083524
|
[
"https://arxiv.org/pdf/2112.07961v1.pdf"
] | 245,144,704 |
2112.07961
|
41fb20f8341459c1b2d2daa4ee2b73837a9727ef
|
Improved limits on the tensor-to-scalar ratio using BICEP and Planck
M Tristram
IJCLab
Université Paris-Saclay
CNRS/IN2P3
91405OrsayFrance
A J Banday
IRAP
Université de Toulouse
CNRS
CNES
UPS, (Toulouse)
France
K M Górski
Jet Propulsion Laboratory
California Institute of Technology
4800 Oak Grove DrivePasadenaCaliforniaU.S.A
Warsaw University Observatory
Aleje Ujazdowskie 400-478WarszawaPoland
R Keskitalo
Computational Cosmology Center
Lawrence Berkeley National Laboratory
BerkeleyCaliforniaU.S.A
Department of Physics and Space Sciences Laboratory
University of California
BerkeleyCaliforniaU.S.A
C R Lawrence
Jet Propulsion Laboratory
California Institute of Technology
4800 Oak Grove DrivePasadenaCaliforniaU.S.A
K J Andersen
Institute of Theoretical Astrophysics
University of Oslo
BlindernOsloNorway
R B Barreiro
Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)
Avda. de los Castros s/nSantanderSpain
J Borrill
Computational Cosmology Center
Lawrence Berkeley National Laboratory
BerkeleyCaliforniaU.S.A
Space Sciences Laboratory
University of California
BerkeleyCaliforniaU.S.A
L P L Colombo
Dipartimento di Fisica
Università degli Studi di Milano
Via Celoria, 16MilanoItaly
H K Eriksen
Institute of Theoretical Astrophysics
University of Oslo
BlindernOsloNorway
R Fernandez-Cobos
Dpto. de Matemàticas
Estadística y Computación
Universidad de Cantabria
Avda. de los Castros s/nE-39005SantanderSpain
T S Kisner
Computational Cosmology Center
Lawrence Berkeley National Laboratory
BerkeleyCaliforniaU.S.A
Department of Physics and Space Sciences Laboratory
University of California
BerkeleyCaliforniaU.S.A
E Martínez-González
Instituto de Física de Cantabria (CSIC-Universidad de Cantabria)
Avda. de los Castros s/nSantanderSpain
B Partridge
Haverford College Astronomy Department
370 Lancaster AvenueHaverfordPennsylvaniaU.S.A
D Scott
Department of Physics & Astronomy
University of British Columbia
6224 Agricultural RoadVancouverBritish ColumbiaCanada
T L Svalheim
Institute of Theoretical Astrophysics
University of Oslo
BlindernOsloNorway
I K Wehus
Institute of Theoretical Astrophysics
University of Oslo
BlindernOsloNorway
Improved limits on the tensor-to-scalar ratio using BICEP and Planck
(Dated: December 16, 2021)
We present constraints on the tensor-to-scalar ratio r using a combination of BICEP/Keck 2018 and Planck PR4 data allowing us to fit for r consistently with the six parameters of the ΛCDM model without fixing any of them. In particular, we are able to derive a constraint on the reionization optical depth τ and thus propagate its uncertainty onto the posterior distribution for r. While Planck sensitivity to r is no longer comparable with ground-based measurements, combining Planck with BK18 and BAO gives results consistent with r = 0 and tightens the constraint to r < 0.032.
INTRODUCTION
Introduced in order to resolve problems within the Big-Bang cosmological model (such as the horizon, flatness, and magnetic-monopole problems), inflation also naturally provides the seeds for generating primordial matter fluctuations from quantum fluctuations (see for instance Ref.
[1] and references herein).
Measurements of the cosmic microwave background (CMB) allow constraints to be placed on the amplitude of the tensor perturbations that are predicted to be generated by primordial gravitational waves during the inflationary epoch, leaving some imprints on the CMB anisotropies [2][3][4][5]. Over the last decade, while no primordial signals have been discovered, significant improvements on the upper limit for the tensor-to-scalar ratio r have progressively led to the constraint becoming lower than a few percent in amplitude: r < 0.11 in 2013 using only temperature data from Planck [6]; r < 0.12 in 2015 using polarization from BICEP/Keck and Planck [7] to debias the initially claimed detection from BICEP/Keck in 2014, r = 0.2 +0.07 −0.05 (excluding r = 0 at 7 σ) [8]; r < 0.09 in 2016 using BICEP/Keck and Planck [9]; r < 0.07 in 2018 using BICEP/Keck 2015 data (BK15, [10]); r < 0.065 in 2019 using Planck in combination with BK15 [11]; r < 0.044 in 2021 using Planck in combination with BK15 [12]; and r < 0.036 in 2021 using the latest BICEP/Keck data (BK18, [13] In this paper, we first discuss the implication of the reionization optical depth τ on the latest BICEP/Keck results BK18 [13]. Then we combine with the latest Planck release (PR4) [14] in order to provide the best currently available constraint on the tensor-to-scalar ratio r.
COSMOLOGICAL MODEL
The cosmological model used in this paper is based on adiabatic, nearly scale-invariant perturbations. It has been established as the simplest model that is consistent with the different cosmological probes and in particular with the CMB [11].
The standard ΛCDM+r model includes 6+1 parameters. Power spectra for scalar and tensor modes are parameterized by power laws with no running and so the parameters include the scalar amplitude A s and the spectral scalar index n s , while the spectral index for the tensor mode n t is set using single-field slow-roll inflation consistency. The amplitudes and the tensor-to-scalar power ratio, r ≡ A t /A s , are evaluated at a pivot scale of 0.05 Mpc −1 . Three other parameters (Ω b h 2 , Ω c h 2 , and θ * ) determine the linear evolution of perturbations after they re-enter the Hubble scale. Finally, the reionization is modeled with a widely-used step-like transition between an essentially vanishing ionized fraction at early times, to a value of unity at low redshifts. The transition is modeled using a tanh function with a non-zero width fixed to ∆z = 0.5 [15]. The reionization optical depth τ is then directly related to the redshift at which this transition occurs.
The CMB power spectra are generated using the Boltzmann-solver code camb [16,17]. We sample the likelihood combinations using the cobaya framework [18] with fast and efficient Markov chain Monte Carlo sampling methods described in Refs. [19,20]. All the likelihoods that we use are publicly available on the cobaya web site 1 and are briefly described in the next section.
PLANCK LIKELIHOODS
We use the polarized likelihood at large scales, lowlEB, described in Ref. [12] and available on github. 2 Specifically, it is a Planck low-polarization likelihood based on cross-spectra using the Hamimeche-Lewis approximation [21,22]. Using this formalism, the likelihood function consistently takes into account the two polarization fields E and B (including EE, BB, and EB powerspectra), as well as all correlations between multipoles and modes. It is important to appreciate that such correlations are relevant at large angular scales where cut-sky effects and systematic residuals (both from the instrument and from the foregrounds) are important. The cross-spectra are calculated on component-separated CMB "detset" maps processed by commander from the Planck PR4 frequency maps, on 50 % of the sky. The sky fraction is optimized in order to obtain maximum sensitivity (and lowest cosmic variance), while ensuring low contamination from residual foregrounds. The covariance matrix is estimated from the PR4 Monte Carlos. The statistical distribution of the recovered C s naturally includes the effect of all components included in the Monte Carlo, namely the CMB signal, instrumental noise, Planck systematic effects incorporated in the PR4 simulations (see Ref. [14]), component-separation uncertainties, and foreground residuals. In this paper, unlike previous CMB work to our knowledge, we marginalized the likelihood over the unknown true covariance matrix (as proposed in Ref. [23]) in order to propagate the uncertainty in the estimation of the covariance matrix from a reduced number of simulations. The robustness of the results is discussed in the Appendix.
At large angular scales in temperature, we make use of the Planck public low-temperature-only likelihood, based on the CMB map recovered from the component-separation procedure (specifically commander) described in detail in Ref. [24].
At small scales, we use the Planck HiLLiPoP likelihood, which can include the T T , T E, and/or EE power spectra computed on the PR4 detset maps at 100, 143, and 217 GHz. The likelihood is a spectrum-based Gaussian approximation, with semi-analytic estimates of the C covariance matrix based on the data. The model consists of a linear combination of the CMB power spectrum and several foreground residuals, including Galactic dust, cosmic infrared background, thermal and kinetic Sunyaev-Zeldovich (SZ) effects, SZ-CIB correlations and unresolved point sources. For details, see Refs. [12] and [25][26][27].
BICEP/KECK LIKELIHOOD
We use the publicly available BICEP/Keck likelihood (BK18) corresponding to the data taken by the BICEP2, Keck Array, and BICEP3 CMB polarization experiments up to and including the 2018 observing season [13]. The format of the likelihood is identical to the one introduced in Refs. [7] and [10]; it is a Hamimeche-Lewis approximation [21] to the joint likelihood of the ensemble of BB auto-and cross-spectra taken between the BICEP/Keck (two at 95, one each at 150 and 250 GHz), WMAP (23 and 33 GHz), and Planck (PR4 at 30, 44, 143, 217, and 353 GHz) frequency maps. The effective coverage is approximately 400 deg 2 (which corresponds to 1 % of the sky) centered on a region with low foreground emission.
The data model includes Galactic dust and synchrotron emission, as well as correlations between dust and synchrotron.
In the following, we neglect correlations between the BICEP/Keck and Planck data sets. This is justified because the BK18 spectra are estimated on 1 % of the sky, while the Planck analysis is derived from 50 % of the sky. Moreover, BB spectra are dominated by noise, which is uncorrelated between the two experiments.
IMPACT OF REIONIZATION UNCERTAINTY
In Ref. [13], the BICEP/Keck Collaboration fixed the cosmology to that of Planck 2018 and quote an upper limit of r < 0.036 at 95% CL. The uncertainty on the reionization history (essentially the reionization optical depth τ in the case of standard ΛCDM) was not propagated, reducing the width of the posterior for the tensorto-scalar ratio r. We find that when fitting the reionization optical depth τ in addition to r, the uncertainty on r slightly increases, while τ is marginally constrained (since BK18 includes only BB modes) as illustrated in The constraints on r then become
r = 0.016 +0.012 −0.009 (BK18 with τ fixed),(1)r = 0.018 +0.013 −0.010 (BK18 with τ free),(2)
all compatible with zero and resulting in the following upper-limits at 95 % CL,
r < 0.036 (BK18 with τ fixed),(3)
r < 0.040 (BK18 with τ free).
COMBINING PLANCK AND BICEP/KECK
With the new BICEP/Keck data set, the uncertainty on r has decreased to σ(r) 0.013, compared to the Planck uncertainty σ(r) = 0.043 presented in Ref. [12]. As a consequence, we do not expect the addition of Planck data to significantly improve the upper limit on r. On the other hand, the addition of low-from Planck polarization modes allows the degeneracy with τ to be broken and also slightly shifts the peak posterior distribution for r. This is illustrated in Fig. 3. Posterior distribution in the τ -r plane using BK18 [13], Planck [12], and the combination.
The resulting constraint on r using a combination of Planck and BK18 data tightens to
r = 0.014 +0.011 −0.009 (Planck+BK18),(5)
which corresponds to r < 0.034 at 95 % CL, with the reionization optical depth τ = 0.057 ± 0.007.
The combination of the two data sets allows us to cover the full range of multipoles that are most sensitive to tensor modes. In combination with baryon acoustic oscillation (BAO, [28]) and CMB lensing [29] data, we obtain an improved upper limit of r < 0.032 (95% CL).
In the n s -r plane (Fig. 4), the constraints now rule out the expected potentials for single-field inflation (strongly excluding V ∝ φ 2 , φ, and even φ 2/3 at about 5 σ).
CONCLUSIONS
We have derived constraints on the tensor-to-scalar ratio r using the two most sensitive data sets to date, Constraints in the tensor-to-scalar ratio r versus ns plane for the ΛCDM model, using CMB data in combination with baryon acoustic oscillation (BAO) and CMB lensing data. The CMB data are Planck PR3 (TT,TE,EE+lowE, gray contour), Planck PR4 [12] (TT,TE,EE+lowlEB, green contour), and Planck PR4 joint with BK18 [13] (blue contour). This assumes the inflationary consistency relation and negligible running. Dotted lines show the loci of approximately constant e-folding number 50 < N < 60, assuming simple V ∝ (φ/m P l ) p single-field inflation. Solid lines show the approximate ns-r relation for locally power-law potentials, to first order in slow roll. The solid black line (corresponding to a linear potential) separates concave and convex potentials. This plot is adapted from figure 28 in Ref. [11].
namely BICEP3 and Planck. The BICEP/Keck Collaboration recently released a likelihood derived from their data up to the 2018 observing season, demonstrating a sensitivity on r of σ r = 0.013, covering the multipole range from = 20 to 300 [13]. Complementary Planck PR4 data released in 2020 [14] provide information on the large scales, with a polarized likelihood covering the multipole range from = 2 to = 150 [12]. This has poorer sensitivity, with σ r = 0.043, but offers independent information, with the constraint on r coming from a combination of T , E, and large-scale B data. It is interesting to note that constraints derived purely from temperature anisotropies are not competitive anymore (σ r = 0.1 [12]), since those data are dominated by cosmic variance.
The addition of Planck data (including large angular scales in polarization, as well as small angular scales in T T and T E) allows us to break the degeneracy with the reionization optical depth (which was fixed in Ref. [13]) and hence to fit r consistently, along with the usual six parameters of the ΛCDM model. We found that other ΛCDM parameters are not affected by the addition of BK18 data (Fig. 5). Combining Planck PR4 and BK18, we found an upper limit of r < 0.034 which tightens to r < 0.032 when adding BAO and CMB lensing data.
We note that future re-analyses of the Planck data are not anticipated to provide much improvement, while ground-based experiments (such as BICEP/Keck, the Simons Observatory [30], and later CMB-S4 [31]) will observe the sky with ever deeper sensitivity, placing even stronger constraints on the tensor-to-scalar ratio r (or detecting primordial B modes of course). However, improved measurements of the reionization optical depth require very large scales, which are very difficult to measure from ground. The next generation of polarized CMB space missions (including LiteBIRD [32]) will be able to deliver τ with a precision dominated by cosmic variance.
ACKNOWLEDGEMENTS
Planck is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states and led by Principal Investigators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA). We gratefully acknowledge support from the CNRS/IN2P3 Computing Center for providing computing and data-processing resources needed for this work. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of (1/ √ 400). The robustness of the covariance matrix has been checked in two different ways.
Firstly, we marginalized over the unknown true covariance matrix, as described in Ref. [23]. The recovered maximum posterior is unchanged, while the width of the posterior is slightly enlarged, as expected due to the marginalization (see left panel of Fig. 6). We also applied the correction on the inverse covariance estimate, as proposed in Refs. [23,33] and recover the same result.
Secondly, we ran the same chains using covariance estimates based on only 200 simulations (right panel of Fig. 6). The posterior distributions of r reconstructed from the lowlEB likelihood using covariance estimates based on the first or last 200 simulations are compatible, given the statistical deviations from the covariance matrix estimates.
FIG. 2 .
2Posterior distribution for the tensor-to-scalar ratio r, showing the impact of marginalization over the reionization optical depth τ .
FIG. 3. Posterior distribution in the τ -r plane using BK18 [13], Planck [12], and the combination.
FIG. 4. Constraints in the tensor-to-scalar ratio r versus ns plane for the ΛCDM model, using CMB data in combination with baryon acoustic oscillation (BAO) and CMB lensing data. The CMB data are Planck PR3 (TT,TE,EE+lowE, gray contour), Planck PR4 [12] (TT,TE,EE+lowlEB, green contour), and Planck PR4 joint with BK18 [13] (blue contour). This assumes the inflationary consistency relation and negligible running. Dotted lines show the loci of approximately constant e-folding number 50 < N < 60, assuming simple V ∝ (φ/m P l ) p single-field inflation. Solid lines show the approximate ns-r relation for locally power-law potentials, to first order in slow roll. The solid black line (corresponding to a linear potential) separates concave and convex potentials. This plot is adapted from figure 28 in Ref. [11].
FIG. 5 .
5Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Part of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). Constraint contours (at 68 and 95 % confidence) on parameters of a ΛCDM+r model using Planck (red) and Planck +BK18 (black).
[ 1 ]FIG. 6 .
16M. Kamionkowski and E. D. Kovetz, Ann. Rev. Astron. Astrophys. 54, 227 (2016), arXiv:1510.06042 [astroph.CO]. Posteriors for r. Left: after marginalizing over the true covariance matrix (dashed line) and correcting the inverse covariance matrix (dotted line), compared to the effective covariance (solid line). right: using covariance estimates based on the first or last 200 simulations, compared to the effective covariance with 400 simulations (solid line).
). Planck PR2+BK [9], BK15 [10], Planck PR3+BK15 [11], Planck PR4 [12], Planck PR4+BK15 [12], BK18 [13], Planck PR4+BK18 this work). Upper limits are given at 95 % CL, uncertainty on the detection is 1σ.0.00 0.05 0.10 0.15 0.20 0.25
r
PR4+BK18 (this work)
BK18 (2021)
PR4+BK15 (2021)
PR4 (2021)
PR3+BK15 (2019)
BK15 (2018)
PR2+BK (2016)
PR1+BK (2015)
BK (2014)
PR1 (2013)
r<0.11
r<0.12
r<0.09
r<0.07
r<0.065
r<0.056
r<0.044
r<0.036
r<0.032
0.2 +0.07 0.05
FIG. 1.
History of constraints on the tensor-to-scalar
ratio r (Planck PR1 [6], BK [8], Planck PR1+BK [7],
cobaya.readthedocs.io 2 github.com/planck-npipe
AppendixThe Planck likelihood used in this analysis is described in detail in Ref.[12]. It is based on the 400 simulations provided with the Planck PR4 data. Those simulations have been shown to be the most realistic description of the Planck data, including all relevant systematic effects[14]. Using the Planck data, we expect correlations at very low-, related to long-term systematics, residuals from foregrounds, and cut-sky effects. These should not be neglected.The covariance used in the likelihood has been constructed from the simulations mentioned above, ensuring the propagation of statistical and systematic uncertainties up to the fitted parameters. Nevertheless, the limited number of available simulations induces an uncertainty on the estimated covariance of the order of 5 %
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|
[] |
[
"SrFeAsF as a parent compound for iron pnictide superconductors",
"SrFeAsF as a parent compound for iron pnictide superconductors"
] |
[
"Fei Han \nInstitute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina\n",
"Xiyu Zhu \nInstitute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina\n",
"Gang Mu \nInstitute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina\n",
"Peng Cheng \nInstitute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina\n",
"Hai-Hu Wen \nInstitute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina\n"
] |
[
"Institute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina",
"Institute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina",
"Institute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina",
"Institute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina",
"Institute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nNational Laboratory for Superconductivity\nChinese Academy of Sciences\nP. O. Box 603100190BeijingChina"
] |
[] |
We have successfully synthesized the fluo-arsenide SrFeAsF, a new parent phase with the ZrCu-SiAs structure. The temperature dependence of resistivity and dc magnetization both reveal an anomaly at about Tan = 173 K, which may correspond to the structural and/or Spin-Density-Wave (SDW) transition. Strong Hall effect and moderate magnetoresistance were observed below Tan. Interestingly, the Hall coefficient RH is positive below Tan, which is opposite to the cases in the two parent phases of FeAs-based systems known so far, i.e., LnFeAsO (Ln = rare earth elements) and (Ba, Sr)Fe2As2 where the Hall coefficient RH is negative. This strongly suggests that the gapping of the Fermi surface induced by the SDW order leaves one of the hole pockets fully or partially ungapped in SrFeAsF. Our data show that it is possible for the parent phases of the arsenide superconductors to display dominant carriers that are either electronlike or holelike.
|
10.1103/physrevb.78.180503
|
[
"https://arxiv.org/pdf/0810.2475v2.pdf"
] | 118,551,678 |
0810.2475
|
e1fa794ebfdc129c15464aff3ffcb49ca6bf9743
|
SrFeAsF as a parent compound for iron pnictide superconductors
30 Oct 2008
Fei Han
Institute of Physics
Beijing National Laboratory for Condensed Matter Physics
National Laboratory for Superconductivity
Chinese Academy of Sciences
P. O. Box 603100190BeijingChina
Xiyu Zhu
Institute of Physics
Beijing National Laboratory for Condensed Matter Physics
National Laboratory for Superconductivity
Chinese Academy of Sciences
P. O. Box 603100190BeijingChina
Gang Mu
Institute of Physics
Beijing National Laboratory for Condensed Matter Physics
National Laboratory for Superconductivity
Chinese Academy of Sciences
P. O. Box 603100190BeijingChina
Peng Cheng
Institute of Physics
Beijing National Laboratory for Condensed Matter Physics
National Laboratory for Superconductivity
Chinese Academy of Sciences
P. O. Box 603100190BeijingChina
Hai-Hu Wen
Institute of Physics
Beijing National Laboratory for Condensed Matter Physics
National Laboratory for Superconductivity
Chinese Academy of Sciences
P. O. Box 603100190BeijingChina
SrFeAsF as a parent compound for iron pnictide superconductors
30 Oct 2008arXiv:0810.2475v2 [cond-mat.supr-con]numbers: 7470Dd7425Fy7530Fv7410+v
We have successfully synthesized the fluo-arsenide SrFeAsF, a new parent phase with the ZrCu-SiAs structure. The temperature dependence of resistivity and dc magnetization both reveal an anomaly at about Tan = 173 K, which may correspond to the structural and/or Spin-Density-Wave (SDW) transition. Strong Hall effect and moderate magnetoresistance were observed below Tan. Interestingly, the Hall coefficient RH is positive below Tan, which is opposite to the cases in the two parent phases of FeAs-based systems known so far, i.e., LnFeAsO (Ln = rare earth elements) and (Ba, Sr)Fe2As2 where the Hall coefficient RH is negative. This strongly suggests that the gapping of the Fermi surface induced by the SDW order leaves one of the hole pockets fully or partially ungapped in SrFeAsF. Our data show that it is possible for the parent phases of the arsenide superconductors to display dominant carriers that are either electronlike or holelike.
We have successfully synthesized the fluo-arsenide SrFeAsF, a new parent phase with the ZrCu-SiAs structure. The temperature dependence of resistivity and dc magnetization both reveal an anomaly at about Tan = 173 K, which may correspond to the structural and/or Spin-Density-Wave (SDW) transition. Strong Hall effect and moderate magnetoresistance were observed below Tan. Interestingly, the Hall coefficient RH is positive below Tan, which is opposite to the cases in the two parent phases of FeAs-based systems known so far, i.e., LnFeAsO (Ln = rare earth elements) and (Ba, Sr)Fe2As2 where the Hall coefficient RH is negative. This strongly suggests that the gapping of the Fermi surface induced by the SDW order leaves one of the hole pockets fully or partially ungapped in SrFeAsF. Our data show that it is possible for the parent phases of the arsenide superconductors to display dominant carriers that are either electronlike or holelike. The discovery of superconductivity in the quaternary compound LaFeAsO 1−x F x which is abbreviated as the FeAs-1111 phase, has attracted great attentions in the fields of condensed matter physics and material sciences. 1 The family of the FeAs-based superconductors has been extended rapidly. As for the FeAs-1111 phase, most of the discovered superconductors are characterized as electron-doped ones and the superconducting transition temperature has been quickly raised to T c = 55∼ 56 K via replacing lanthanum with other rare earth elements. 2,3,4,5,6,7 Meanwhile, the first holedoped superconductor La 1−x Sr x FeAsO with T c ≈ 25 K was discovered, 8,9 followed with the observation of superconductivity in hole-doped Nd 1−x Sr x FeAsO 10 and Pr 1−x Sr x FeAsO. 11 Later on, (Ba, Sr) 1−x K x Fe 2 As 2 which is denoted as FeAs-122 for simplicity 12,13,14 , and Li x FeAs as an infinite layered structure (denoted as FeAs-111) were discovered. 15,16,17 It is assumed that the superconductivity both in the FeAs-1111 phase and FeAs-122 phase is intimately connected with a Spin-Density-Wave (SDW) anomaly in the FeAs layers. 12,18 For undoped LaFeAsO, an SDW-driven structural phase transition around 150 K was found. 19 It seems that any new parent phase will initiate a series of new superconductors by doping it away from the state with features of a bad metal and the SDW order.
In this paper, we report the discovery of a new FeAsbased layered compound SrFeAsF which has the ZrCu-SiAs structure. As we know SrZnPF is a compound with the ZrCuSiAs structure 20 . We replace the ZnP sheets with FeAs sheets and get a new compound of SrFeAsF. The compound SrFeAsF has the tetragonal space group P4/nmm at 300 K. Both the resistivity and the dc magnetic susceptibility exhibit a clear anomaly at about 173 K, which is attributed to the structural and/or SDW transition. Surprisingly, a positive Hall coefficient R H has been found implying a dominant conduction by holelike charge carriers in this parent phase.
The SrFeAsF samples were prepared using a two-step solid state reaction method, as used for preparing the LaFeAsO samples. 21 In the first step, SrAs was prepared by reacting Sr flakes (purity 99.9%) and As grains (purity 99.99%) at 500 o C for 8 hours and then 700 o C for 16 hours. They were sealed in an evacuated quartz tube when reacting. Then the resultant precursors were thoroughly grounded together with Fe powder (purity 99.95%) and FeF 3 powder (purity 99%) in stoichiometry as given by the formula SrFeAsF. All the weighing and mixing procedures were performed in a glove box with a protective argon atmosphere. Then the mixture was pressed into pellets and sealed in a quartz tube with an Ar atmosphere of 0.2 bar. The materials were heated up to 950 o C with a rate of 120 o C/hr and maintained for 60 hours. Then a cooling procedure to room temperature was followed.
The dc magnetization measurements were done with a superconducting quantum interference device (Quantum Design, SQUID, MPMS7). For the magnetotransport measurements, the sample was shaped into a bar with the length of 3 mm, width of 2 mm and thickness of about 0.9 mm. The resistance and Hall effect data were collected using a six-probe technique on the Quantum Design instrument physical property measurement system (PPMS) with magnetic fields up to 9 T. The electric contacts were made using silver paste with the contacting resistance below 0.05 Ω at room temperature. The data acquisition was done using a DC mode of the PPMS, which measures the voltage under an alternative DC current and the sample resistivity is obtained by averaging these signals at each temperature. In this way the contacting thermal power is naturally removed. The temperature stabilization was better than 0.1% and the resolution of the voltmeter was better than 10 nV.
The X-ray diffraction (XRD) pattern for the sample Sr-FeAsF is shown in Fig. 1. One can see that all the main peaks can be indexed to the FeAs-1111 phase with the tetragonal ZrCuSiAs-type structure. Only small amount of SrF 2 impurity phase was detected. By using the software of Powder-X, 22 we took a general fit to the XRD data of this sample and the lattice constants were deter- mined to be a = 4.004Å and c = 8.971Å. It is clear that the a-axis lattice constant of this parent phase is slightly smaller than that of the LaFeAsO system, while the c-axis one is much larger, 1,21 indicating a completely new phase in the present system since the radii of Sr 2+ is larger than that of La 3+ .
In Fig.2 (a) we present the temperature dependence of resistivity for the SrFeAsF sample under magnetic fields up to 9 T. A rather large value of the resistivity is observed. An upturn in the low-temperature regime can be seen under all fields, representing a weak semiconductor like behavior for the present sample. It is unclear at this moment whether this behavior is intrinsic in nature, or it is due to the weak localization effect, or some other effect. This curve also reveals an anomaly at about T an = 173 K, which may correspond to the structural and/or SDW transition, as has been found in the parent phase of LnFeAsO (Ln = rare earth elements) and (Ba, Sr)Fe 2 As 2 . 1,12 Fig.2 (b) shows the zero field cooled dc magnetization of the same sample at 5000 Oe. A clear anomaly at about 173 K in the magnetization curve confirms the structural and/or SDW transition observed in the resistivity data. Above 173 K, the magnetization exhibits a rough linear temperature dependence, which may be a common effect in the FeAs-based systems and was explained as due to short range correlation of the local moments. 23 To get a comprehensive understanding to the conducting carriers in the SrFeAsF phase, we measured the Hall effect of the present sample. The inset of Fig. 3 shows the magnetic field dependence of Hall resistivity (ρ xy ) at different temperatures. In the experiment, ρ xy was taken as ρ xy = [ρ(+H) -ρ ( field dependence of ρ xy was observed in the temperature regime below 75 K, while the linear behavior appeared above 100 K. This may suggest that a multi-band effect or a complicated scattering mechanism (perhaps magnetic related) emerged in the low temperature regime. The temperature dependence of the Hall coefficient R H is presented in the main frame of Fig.3. One can see that R H remains positive in wide temperature regime and decreases monotonically in the temperature regime below about 160∼ 170 K and it becomes slightly negative above that temperature. The sign changing of R H and the temperature dependent behavior may be related to the structural and/or SDW transition as revealed by the resistivity data, considering that this change occurred at temperatures close to T an . It is worth noting that the positive Hall coefficient R H in this sample SrFeAsF is quite unique because in the two parent phases of FeAsbased systems known so far, i.e., LnFeAsO (Ln = rare earth elements) and (Ba, Sr)Fe 2 As 2 , the Hall coefficient R H is negative. This strongly suggests that the gap- ping to the Fermi surfaces induced by the SDW order is more complex than we believed before, and the case in SrFeAsF is that it removes the density of states on some Fermi pockets and may leave one of the hole pockets partially or fully ungapped. Our data clearly show that it is possible for the parent phase to have electronlike or hole-like charge carriers. It is well known that in the conventional metals the Hall coefficient R H is almost independent of temperature. The strong temperature dependence of R H below T an in our data suggests either a strong multi-band effect or the variation of the charge carrier densities, or both effects collectively contribute to the Hall signal in the present parent phase of SrFeAsF. The magnetoresistance (MR) is a very powerful tool to investigate the properties of electronic scattering. 24,25 Field dependence of MR for the present sample at different temperatures is shown in the main frame in the top part of Fig 4. One can see a moderate MR effect up to 9% under the field of 9 T at 2 K. This is a rather large magnitude compered with the F-doped LnFeAsO samples. 21,26 The semiclassical transport theory has predicted that the Kohler's rule will be held if only one isotropic relaxation time is present in a solid state system. 27 The Kohler's rule can be written as
∆ρ ρ 0 = ρ(H) − ρ 0 ρ 0 = F ( H ρ 0 ),(1)
where ρ(H) and ρ 0 represents the longitudinal resistivity at a magnetic field H and that at zero field, respectively. Equation (1) ple is revealed in the inset of the top part of Fig.4. An obvious violation of the Kohler's rule can be seen from this plot. This behavior may indicate a multi-band effect or a gradual gapping effect to the density of states by the SDW ordering in the present sample. Temperature dependence of MR under the field of 9 T is shown in the bottom part of Fig 4. Rather similar to that observed in the R H vs T plot, ∆ρ/ρ 0 decreases monotonically in the low temperature regime below about 200 K and a minimum appears around T an . This may provide another evidence of the influence of the structural and/or SDW transition on the behavior of the conducting charge carriers.
In summary, a parent phase, namely SrFeAsF, with the ZrCuSiAs structure was synthesized successfully using a two-step solid state reaction method. An anomaly at about 173 K can be observed from the data of the resistivity and dc magnetization, which is ascribed to the structural and/or SDW transition. Also strong Hall effect and moderate MR were observed below T an . We found that the Hall coefficient R H is positive below T an , displaying an opposite behavior comparing to the cases in the two parent phases of FeAs-based systems known so far, i.e., LnFeAsO (Ln = rare earth elements) and (Ba, Sr)Fe 2 As 2 where the Hall coefficient R H is negative. This suggests that the gapping to the Fermi surfaces induced by the SDW order may remove the density of states on some Fermi pockets and leave one of the hole pockets partially or fully ungapped in the present parent phase. Our results clearly show that it is possible for the parent phase to have electron-like or hole-like charge carriers. We also observed a moderate magnetoresistance up to 9% under the field of 9 T. The violation of the Kohler's rule along with the strong temperature dependence of R H may suggest a multi-band and/or a spin scattering effect in this system. By doping strontium with lanthanum, we found superconductivity in Sr 1−x La x FeAsF, which will be presented separately. 28 Note added: When we were finalizing this paper, we became aware that a paper was posted on the website on the same day of our submission. That paper reports also the synthesizing of the compound SrFeAsF and a different set of data. 29 This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (973 project: 2006CB01000, 2006CB921802), the Knowledge Innovation Project of Chinese Academy of Sciences (ITSNEM).
PACS numbers: 74.70.Dd, 74.25.Fy, 75.30.Fv, 74.10.+v
online) X-ray diffraction patterns for the Sr-FeAsF sample. One can see that all the main peaks can be indexed to the tetragonal ZrCuSiAs-type structure. The blue asterisks indicate the little impurities from the SrF2 phase.
FIG
-H)]/2 at each point to eliminate the effect of the misaligned Hall electrodes. . 2: (Color online) (a) Temperature dependence of resistivity for the SrFeAsF sample under magnetic fields up to 9 T. A clear anomaly at about Tan = 173 K can be observed. (b) Temperature dependence of dc magnetization for the zero field cooling (ZFC) process at a magnetic field of H = 5000Oe. We can also see an anomaly at the same temperature in the M(T) curve.
FIG. 3 :
3(Color online) Temperature dependence of Hall coefficient RH determined on the present sample SrFeAsF. One can see a monotonic decrease of RH in the temperature regime below about 160∼ 170 K. Inset: The raw data of the Hall resistivity ρxy versus the magnetic field µ0H at different temperatures.
FIG. 4 :
4means that the ∆ρ/ρ 0 vs H/ρ 0 curves for different temperatures, the so-called Kohler plot, should be scaled to a universal curve if the Kohler's rule is obeyed. The scaling based on the Kohler plot of our sam-(Color online) Field dependence of MR for the present sample at different temperatures is shown in the top panel. A moderate MR effect up to 9% is observed under the field of 9 T at 2 K. Kohler plot of MR is presented in the inset. In the bottom panel of this figure we show the temperature dependence of MR under the field of 9 T.
* Electronic address: [email protected]. * Electronic address: [email protected]
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[] |
[
"Weak structure functions in ν l − N and ν l − A scattering with nonperturbative and higher order perturbative QCD effects",
"Weak structure functions in ν l − N and ν l − A scattering with nonperturbative and higher order perturbative QCD effects"
] |
[
"F Zaidi \nDepartment of Physics\nAligarh Muslim University\n202002AligarhIndia\n",
"H Haider \nFermi National Accelerator Laboratory\n60510BataviaIllinoisUSA\n",
"M Sajjad Athar \nDepartment of Physics\nAligarh Muslim University\n202002AligarhIndia\n",
"S K Singh \nDepartment of Physics\nAligarh Muslim University\n202002AligarhIndia\n",
"I Ruiz Simo \nDepartamento de Física Atómica\nMolecular y Nuclear\nFísica Teórica y Computacional Carlos I\nUniversidad de Granada\n18071GranadaSpain\n"
] |
[
"Department of Physics\nAligarh Muslim University\n202002AligarhIndia",
"Fermi National Accelerator Laboratory\n60510BataviaIllinoisUSA",
"Department of Physics\nAligarh Muslim University\n202002AligarhIndia",
"Department of Physics\nAligarh Muslim University\n202002AligarhIndia",
"Departamento de Física Atómica\nMolecular y Nuclear\nFísica Teórica y Computacional Carlos I\nUniversidad de Granada\n18071GranadaSpain"
] |
[] |
We study the effect of various perturbative and nonperturbative QCD corrections on the free nucleon structure functions (F W I iN (x, Q 2 ); i = 1 − 3) and their implications in the determination of nuclear structure functions. The evaluation of the nucleon structure functions has been performed by using the MMHT 2014 PDFs parameterization, and the TMC and HT effects are incorporated following the works of Schienbein et al. and Dasgupta et al., respectively. These nucleon structure functions are taken as input in the determination of nuclear structure functions. The numerical calculations for the ν l /ν l −A DIS process have been performed by incorporating the nuclear medium effects like Fermi motion, binding energy, nucleon correlations, mesonic contributions, shadowing and antishadowing in several nuclear targets such as carbon, polystyrene scintillator, iron and lead which are being used in MINERvA, and in argon nucleus which is relevant for the ArgoNeuT and DUNE experiments. The differential scattering cross sections d 2 σ W I A dxdy and ( dσ W I A dx / dσ W I CH dx ) have also been studied in the kinematic region of MINERvA experiment. The theoretical results are compared with the recent experimental data of MINERvA and the earlier data of NuTeV, CCFR, CDHSW and CHORUS collaborations. Moreover, a comparative analysis of the present results for the ratio ( dσ W I A dx / dσ W I CH dx ), and the results from the MC generator GENIE and other phenomenological models of Bodek and Yang, and Cloet et al., has been performed in the context of MINERvA experiment. The predictions have also been made forν l − A cross section relevant for MINERvA experiment.
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10.1103/physrevd.101.033001
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[
"https://arxiv.org/pdf/1911.12573v1.pdf"
] | 208,513,000 |
1911.12573
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48b552ab1511154ef266e44d253147ca526d3d40
|
Weak structure functions in ν l − N and ν l − A scattering with nonperturbative and higher order perturbative QCD effects
28 Nov 2019
F Zaidi
Department of Physics
Aligarh Muslim University
202002AligarhIndia
H Haider
Fermi National Accelerator Laboratory
60510BataviaIllinoisUSA
M Sajjad Athar
Department of Physics
Aligarh Muslim University
202002AligarhIndia
S K Singh
Department of Physics
Aligarh Muslim University
202002AligarhIndia
I Ruiz Simo
Departamento de Física Atómica
Molecular y Nuclear
Física Teórica y Computacional Carlos I
Universidad de Granada
18071GranadaSpain
Weak structure functions in ν l − N and ν l − A scattering with nonperturbative and higher order perturbative QCD effects
28 Nov 2019numbers: 1315+g1360Hb2165+f2410-i
We study the effect of various perturbative and nonperturbative QCD corrections on the free nucleon structure functions (F W I iN (x, Q 2 ); i = 1 − 3) and their implications in the determination of nuclear structure functions. The evaluation of the nucleon structure functions has been performed by using the MMHT 2014 PDFs parameterization, and the TMC and HT effects are incorporated following the works of Schienbein et al. and Dasgupta et al., respectively. These nucleon structure functions are taken as input in the determination of nuclear structure functions. The numerical calculations for the ν l /ν l −A DIS process have been performed by incorporating the nuclear medium effects like Fermi motion, binding energy, nucleon correlations, mesonic contributions, shadowing and antishadowing in several nuclear targets such as carbon, polystyrene scintillator, iron and lead which are being used in MINERvA, and in argon nucleus which is relevant for the ArgoNeuT and DUNE experiments. The differential scattering cross sections d 2 σ W I A dxdy and ( dσ W I A dx / dσ W I CH dx ) have also been studied in the kinematic region of MINERvA experiment. The theoretical results are compared with the recent experimental data of MINERvA and the earlier data of NuTeV, CCFR, CDHSW and CHORUS collaborations. Moreover, a comparative analysis of the present results for the ratio ( dσ W I A dx / dσ W I CH dx ), and the results from the MC generator GENIE and other phenomenological models of Bodek and Yang, and Cloet et al., has been performed in the context of MINERvA experiment. The predictions have also been made forν l − A cross section relevant for MINERvA experiment.
I. INTRODUCTION
The physicists are making continuous efforts both in the theoretical as well as experimental fields for a better understanding of hadronic structure and parton dynamics of nucleons, in a wide range of energy (E) and momentum transfer square (Q 2 ). The deep inelastic scattering process with large values of four momentum transfer square has been used for a long time to explore the partonic distribution in the nucleon. Therefore, several studies are available concerning the perturbative region of high Q 2 , however, much emphasis has not been given to the nonperturbative region of low Q 2 . In a recent theoretical work [1], we have emphasized the effects of perturbative and nonperturbative QCD corrections in the evaluation of electromagnetic nucleon and nuclear structure functions. In the present paper, we have extended our analysis to the weak sector by considering the QCD corrections in the charged current (anti)neutrino induced deep inelastic scattering (DIS) process off free nucleon and nuclear targets. This study is to understand the effects of nonperturbative corrections such as target mass correction (TMC) and higher twist (HT) effects, perturbative evolution of parton densities, nuclear medium modifications, isoscalarity corrections and the center of mass (CoM) energy cut on the weak nuclear structure functions. Using these nuclear structure functions, the scattering cross section has been determined. This study is relevant for the development of precision experiments in order to determine accurately neutrino oscillation parameters, determination of mass hierarchy in the neutrino sector, etc., besides the intrinsic interest of understanding nucleon dynamics in the nuclear medium. For example, the planned DUNE experiment at the Fermilab [2,3] is expected to get more than 50% contribution to the event rates from the intermediate region of DIS and resonance production processes from nuclear targets. The ArgoNeuT collaboration [4] has also measured the inclusive ν l /ν l − 40 Ar scattering cross section in the low energy mode.
The ongoing MINERvA experiment at the Fermilab is using intermediate energy (anti) neutrino beam, with the average energy of ∼6 GeV, where significant events contribute from the DIS processes. MINERvA has measured the scattering cross sections on the different nuclear targets ( 12 C, CH, 56 Fe and 208 Pb) in the energy region, where various reaction channels such as quasielastic scattering (QES), inelastic scattering (IES) and DIS contribute, and reported the ratio of charged current deep inelastic differential scattering cross sections i.e., dσ C /dx dσ CH /dx , dσ F e /dx dσ CH /dx and dσ P b /dx dσ CH /dx [5]. For the DIS, the results have been analyzed by applying a cut on the four momentum transfer square Q 2 ≥ 1 GeV 2 and the center of mass energy W ≥ 2 GeV, for the neutrino induced processes and their analysis is going on for the antineutrino induced channel. They have compared the observed results with the phenomenological models like those being used in GENIE Monte Carlo (MC) neutrino event generator [6], Bodek-Yang modified phenomenological parameterization [7] as well as from the phenomenological study of Cloet et al. [8]. It may be observed from the MINERvA analysis [5] that there is large variation (∼ 20%) when all the three phenomenological studies are compared. Furthermore, it is important to point out that in the MC event generators, the DIS cross sections are extrapolated phenomenologically to the region of low Q 2 in order to obtain the neutrino event rates. In this region, there is lack of agreement between the experimental results from MINERvA and the results obtained from the various phenomenological analyses.
Therefore, it is important to understand nuclear medium effects specially in the low Q 2 region (1-5 GeV 2 ) in order to reduce the systematics, in the neutrino oscillation analysis which contributes ∼ 25% uncertainty to the systematics. The DIS cross section is described in terms of the nucleon structure functions, for example, by using F 1N (x, Q 2 ) and F 2N (x, Q 2 ) in the case of electromagnetic interaction while for the weak interaction there is one more structure function F 3N (x, Q 2 ), that arises due to the parity violation. In the kinematic region of Q 2 → ∞, ν → ∞, such that x = Q 2 2MN ν →constant, the nucleon structure functions become the function of dimensionless variable x only, and F 1N (x) and F 2N (x) satisfy the Callan-Gross relation [9]:
F 2N (x) = 2xF 1N (x).(1)
It implies that the Callan-Gross relation enables us to express the ν l − N scattering cross section, in the massless limit of lepton, in terms of only two nucleon structure functions F 2N (x) and F 3N (x). Through the explicit evaluation of the nucleon structure functions, one may write them in terms of the parton distribution functions (PDFs) which provide information about the momentum distribution of partons within the nucleon. Presently, various phenomenological parameterizations are available for the free nucleon PDFs. The different phenomenological groups have also proposed the nuclear PDFs which are not a simple combination of free proton and free neutron PDFs. In the phenomenological analyses the general approach is that the nuclear PDFs are obtained using the charged lepton-nucleus scattering data and the ratios of the structure functions e.g. F2A F 2A ′ , F2A F2D are analyzed, where A, A ′ represent any two nuclei and D stands for the deuteron, to take into account the nuclear correction factor. While determining the nuclear correction factor, the information regarding nuclear modification is also utilized from the Drell-Yan cross section ratio like , where p stands for proton beam. Furthermore, the information about the nuclear correction factor is also supplemented by high energy reaction data from the experiments at LHC, RHIC, etc. This approach has been used by Hirai et al. [10], Eskola et al. [11], Bodek and Yang [7], de Florian and Sassot [12] and others. The same nuclear correction factor is taken for the weak DIS processes. For example, Bodek and Yang [7] have obtained the nuclear correction factors for carbon, iron, gold and lead using the charged lepton DIS data and applied the same nuclear correction factor to calculate the weak structure functions 2xF W I 1A (x, Q 2 ), F W I 2A (x, Q 2 ) and xF W I 3A (x, Q 2 ). de Florian et al. [12] have analyzed ν l − A DIS data, the charged lepton-nucleus scattering data and Drell-Yan data to determine the nuclear corrections due to the medium effects. Their [12] conclusion is that the same nuclear correction factor can describe the nuclear medium effect in l ± −A and ν l −A DIS processes. In the other approach nuclear PDFs are directly parameterized by analyzing the experimental data, i.e without using nucleon PDFs or nuclear correction factor. This approach has been recently used by nCTEQ [13,14] group in getting F EM
2A (x, Q 2 ), F W I 2A (x, Q 2 ) and F W I 3A (x, Q 2 )
, who have collectively analyzed the charged lepton-A DIS and DY p − A dilepton production data sets [13] to determine the nuclear correction factor in the electromagnetic sector, and have performed an independent analysis for the ν l (ν l ) − A DIS data sets [14]. It has been concluded by them that the nuclear medium effects in F EM
2A (x, Q 2 ) are different from F W I 2A (x, Q 2 )
specially in the region of low x. Thus in this region there is a disagreement between the observation of these two studies [12,13], specially at low x [15].
Theoretically many models have been proposed to study these effects on the basis of nuclear binding, nuclear medium modification including short range correlations in nuclei , pion excess in nuclei [18,20,24,[38][39][40], multi-quark clusters [41][42][43], dynamical rescaling [44,45], nuclear shadowing [46,47], etc. Despite these efforts, no comprehensive theoretical/phenomenological understanding of the nuclear modifications of the bound nucleon structure functions across the complete range of x and Q 2 consistent with the presently available experimental data exists [21][22][23]48]. To understand nuclear modifications, theoretically various studies are available concerning the nuclear medium effects in the electromagnetic sector [1,22,48,49] but there are mainly two groups, namely the group of Kulagin and Petti [20,27,29,50] and Haider et al. [32,34,35,51,52] who have made a comparative study of the nuclear medium effects in the electromagnetic and weak interaction induced processes [52].
As the nucleon structure functions are the basic inputs in the determination of nuclear structure functions and the scattering cross section, therefore, proper understanding of the nucleon structure functions as well as the parton dynamics become quite important. In the region of low and moderate Q 2 , the perturbative and nonperturbative QCD corrections such as Q 2 evolution of parton distribution functions from leading order to higher order terms (next-toleading order (NLO), next-next-to-leading order (NNLO), ...), the effects of target mass correction due to the massive quarks production (e.g. charm, bottom, top) and higher twist (twist-4, twist-6, ...) because of the multiparton correlations, become important. These nonperturbative effects are specifically important in the kinematical region of high x and low Q 2 , sensitive to some of the oscillation parameters, and therefore it is of considerable experimental interest to the long baseline oscillation experiments.
In this work, we have evaluated the nucleon structure functions by using the MMHT PDFs parameterization [53] up to next-to-next-to-leading order (NNLO) in the four flavor(u, d, s, and c) scheme following Ref. [54][55][56]. The nonperturbative higher twist effect is incorporated by using the renormalon approach [57] and the target mass correction is included following the works of Schienbein et al. [58]. After taking into account the QCD corrections at the free nucleon level, we have studied the modifications in the nuclear structure functions due to the presence of nuclear medium effects such as Fermi motion, binding energy and nucleon correlations. These effects are incorporated through the use of spectral function of the nucleon in the nuclear medium [24,59]. The effect of mesonic contribution has been included which is found to be significant in the low and intermediate region of x [24]. We have also included the effect of shadowing and antishadowing corrections following the works of Kulagin and Petti [27]. Furthermore, we have discussed the effect of center of mass energy (W ) cut on ν l − A andν l − A scattering cross sections. This paper is organized as follows.
In the next section (section II), we present the formalism in brief for (anti)neutrino-nucleon and (anti)neutrinonucleus DIS processes. Then we have discussed the method of obtaining nuclear structure functions with medium effects such as Fermi motion, binding energy, nucleon correlations, mesonic contribution and shadowing. In section III, numerical results are presented and discussed, and in the last section IV we summarize our findings.
II. FORMALISM
A. Deep inelastic scattering of (anti)neutrino from nucleons
The basic reaction for the (anti)neutrino induced charged current deep inelastic scattering process on a free nucleon target is given by
ν l (k)/ν l (k) + N (p) → l − (k ′ )/l + (k ′ ) + X(p ′ ) ; l = e, µ,(2)
where k and k ′ are the four momenta of incoming and outgoing lepton, p and p ′ are the four momenta of the target nucleon and the jet of hadrons produced in the final state, respectively. This process is mediated by the W -boson (W ± ) and the invariant matrix element corresponding to the above reaction is given by
− iM = iG F √ 2 l µ M 2 W q 2 − M 2 W X|J µ |N .(3)
G F is Fermi coupling constant, M W is the mass of W boson, and q 2 = (k − k ′ ) 2 is the four momentum transfer square. l µ is the leptonic current and X|J µ |N is the hadronic current for the neutrino induced reaction. The general expression of double differential scattering cross section (DCX) for the massless lepton limit (m l → 0) corresponding to the reaction given in Eq. 2 in the laboratory frame is expressed as
d 2 σ W I N dxdy = yM N π E E ′ |k ′ | |k|¯ |M| 2 ,(4)
where x = Q 2 2MN ν is the Bjorken scaling variable, y = p.q p.k (= ν E in the lab frame) is the inelasticity, ν = E − E ′ is the energy transfer, M N is the nucleon mass, E(E ′ ) is the energy of the incoming(outgoing) lepton and¯ |M| 2 is the invariant matrix element square which is given in terms of the leptonic (L W I µν ) and hadronic (W µν N ) tensors as
|M| 2 = G 2 F 2 M 2 W Q 2 + M 2 W 2 L W I µν W µν N ,(5)
with Q 2 = −q 2 ≥ 0. L W I µν is given by
L W I µν = 8(k µ k ′ ν + k ν k ′ µ − k.k ′ g µν ± iǫ µνρσ k ρ k ′σ ) .(6)
ν l /ν l (k)
l − /l + (k ′ ) W + /W − (q) −i g W 2 √ 2 W + /W − (q) N(p) X(p ′ ) q q q −i g W 2 √ 2
FIG. 1: Feynman representation for leptonic and hadronic vertices in the case of weak interaction.
Here the antisymmetric term arises due to the contribution from the axial-vector components with +ve sign for antineutrino and -ve sign for neutrino. The hadronic tensor W µν N is written in terms of the weak structure functions
W W I iN (ν, Q 2 ) (i = 1 − 3) as W µν N = q µ q ν q 2 − g µν W W I 1N (ν, Q 2 ) + W W I 2N (ν, Q 2 ) M 2 N p µ − p.q q 2 q µ p ν − p.q q 2 q ν − i 2M 2 N ǫ µνρσ p ρ q σ W W I 3N (ν, Q 2 ).(7)
The nucleon structure function W W I 3N (ν, Q 2 ) arises due to the vector−axial vector interference part of the weak interaction and is responsible for the parity violation.
The weak nucleon structure functions W W I iN (ν, Q 2 )(i=1,2,3) are generally redefined in terms of the dimensionless nucleon structure functions F W I iN (x, Q 2 ) as:
M N W W I 1N (ν, Q 2 ) = F W I 1N (x, Q 2 ), νW W I 2N (ν, Q 2 ) = F W I 2N (x, Q 2 ), νW W I 3N (ν, Q 2 ) = F W I 3N (x, Q 2 ). (8)
In general, the dimensionless nucleon structure functions are in turn written in terms of the parton distribution functions as
F W I 2 (x) = i x[q i (x) +q i (x)] , xF W I 3 (x) = i x[q i (x) −q i (x)].(9)
In the above expressions, i runs for the different flavors of quark(antiquark), the variable x is the momentum fraction carried by a quark(antiquark) of the nucleon's momentum and q i (x)(q i (x)) represents the probability density of finding a quark(antiquark) with a momentum fraction x. Using Eqs. 5, 6, 7 and 8 in Eq. 4, the differential scattering cross section is obtained as
d 2 σ W I N dxdy = G 2 F M N E π M 2 W M 2 W + Q 2 2 xy 2 F W I 1N (x, Q 2 ) + 1 − y − M N xy 2E F W I 2N (x, Q 2 ) ±xy 1 − y 2 F W I 3N (x, Q 2 ) ,(10)
We have evaluated the nucleon structure functions up to NNLO following the works of Vermaseren et al. [54] and Moch et al. [55,56]. These structure functions are expressed in terms of the convolution of coefficient function (C a,f ; (f = q, g and a = 1 − 3)) with the density distribution of partons (f ) inside the nucleon. For example, we may write F W I 2N (x) in terms of coefficient function as
x −1 F W I 2N (x) = f =q,g C 2,f (x) ⊗ f (x) ,(11)
with the perturbative expansion
C 2,f (x) = m α s (Q 2 ) 2π m c (m) 2,f ,(12)
where superscript m = 0, 1, 2, ... for N (m) LO, c
2,f (x) is the coefficient function for F W I 2N (x), α s (Q 2 ) is the strong coupling constant and symbol ⊗ is the Mellin convolution which turns into simple multiplication in the N-space. To obtain the convolution of coefficient functions with parton density distribution, we use the following expression [60] C
a,f (x) ⊗ f (x) = 1 x C a,f (y) f x y dy y(13)
The expression for the weak structure function F W I 3N (x) in terms of the coefficient function and parton density distribution function is given by [56]:
F W I 3N (x) = f =q,g C 3,f (x) ⊗ f (x) = C 3,q (x) ⊗ q v (x),
where q v (x)(= f (x)) is the valence quark distribution for a SU(3)/SU(4) symmetric sea and C 3,q (x) is the coefficient function for F W I 3N (x). In the kinematic region of low and moderate Q 2 , both the higher order perturbative and the nonperturbative (∝ 1 Q 2 ) QCD effects come into play. For example, the nonperturbative target mass correction effect involves the powers of 1 Q 2 , and is associated with the finite mass of the target nucleon. This effect is significant in the region of low Q 2 and high x which is important to determine the valence quarks distribution. The higher twist (HT) effect which is suppressed by 1 Q 2 n ; n = 1, 2, ..., originates due to the interactions of struck quarks with the other quarks present in the surroundings via gluon exchange. This effect becomes small at low x and high Q 2 . We have incorporated both the target mass correction and higher twist effects following Refs. [57,58] as well as performed the NNLO corrections in the evaluation of the nucleon structure functions. For the numerical calculations, we have used the MMHT nucleonic PDFs parameterization [53]. According to the operator product expansion [61,62], the weak nucleon structure functions with these nonperturbative effects can be mathematically expressed as
F W I iN (x, Q 2 ) = F W I,τ =2 iN (x, Q 2 ) + H τ =4 i (x, Q 2 ) Q 2 ,(14)
where the leading twist term (τ = 2) incorporating the TMC effect obeys the Altarelli-Parisi evolution equations [63]. It is written in terms of PDFs and is responsible for the evolution of structure functions via perturbative QCD α s (Q 2 ) corrections. While the general expression of the twist-4 (τ = 4) term that reflects the strength of multi-parton correlations is given by [57]
H τ =4 i (x, Q 2 ) = A ′ 2 1 x dz z C i 2 (z) q x z , Q 2 ,(15)
with i = 1, 2, 3. C i 2 is the coefficient function for twist-4, A ′ 2 is the constant parameter and q(x/z, Q 2 ) is the quarks density distribution.
We have incorporated the medium effects using a microscopic field theoretical approach. The effect of Fermi motion, binding energy and nucleon correlations are included through the relativistic nucleon spectral function which is obtained by using the Lehmann's representation for the relativistic nucleon propagator. We use the technique of nuclear many body theory to calculate the dressed nucleon propagator in an interacting Fermi sea in the nuclear matter. To obtain the results for a finite nucleus the local density approximation (LDA) is then applied. In the LDA, Fermi momentum of an interacting nucleon is not a constant quantity but the function of position coordinate (r) [59]. Since the nucleons bound inside a nucleus interact among themselves via the exchange of virtual mesons such as π, ρ, etc., therefore a finite probability of the interaction of intermediate vector boson with these mesons exists. We have also incorporated the mesonic contribution by using many-body field theoretical approach similar to the case of bound nucleons [24]. Furthermore, the shadowing effect is taken into account that dominates in the region of low x, where the hadronization of intermediate vector bosons (W + /W − ) creates quark-antiquark pairs that interact with the partons. The multiple scattering of quarks causes the destructive interference of amplitudes that leads to the phenomenon of shadowing which is incorporated in this paper, following the works of Kulagin and Petti [27]. In the next subsection, we have discussed the formalism adopted for the (anti)neutrino-nucleus scattering process.
B. Deep inelastic scattering of (anti)neutrino from nuclei
In the case of DIS of (anti)neutrino from nuclear targets the expression of the differential cross section is given by
d 2 σ W I A dxdy = G 2 F M N y 2π E E ′ |k ′ | |k| M 2 W M 2 W + Q 2 2 L W I µν W µν A ,(16)
where L W I µν is the weak leptonic tensor which has the same form as given in Eq. 6 while the nuclear hadronic tensor W µν A is written in terms of the weak nuclear structure functions W W I iA (ν, Q 2 ) (i = 1, 2, 3) relevant in the case of m l → 0 as
W µν A = q µ q ν q 2 − g µν W W I 1A (ν, Q 2 ) + W W I 2A (ν, Q 2 ) M 2 A p µ A − p A .q q 2 q µ p ν A − p A .q q 2 q ν − i 2M 2 A ǫ µνρσ p Aρ q σ W W I 3A (ν, Q 2 ).(17)
After contracting the leptonic tensor with the hadronic tensor and using the following relations between the nuclear structure functions (W W I iA (ν, Q 2 )) and the dimensionless nuclear structure functions (F W I iA (x, Q 2 ))
M A W W I 1A (ν, Q 2 ) = F W I 1A (x, Q 2 ) ,(18)ν W W I 2A (ν, Q 2 ) = F W I 2A (x, Q 2 ) ,(19)ν W W I 3A (ν, Q 2 ) = F W I 3A (x, Q 2 ) ,(20)
we obtain
d 2 σ W I A dxdy = G 2 F M N E π M 2 W M 2 W + Q 2 2 xy 2 F W I 1A (x, Q 2 ) + 1 − y − M N xy 2E F W I 2A (x, Q 2 ) ±xy 1 − y 2 F W I 3A (x, Q 2 ) .(21)
When the interaction takes place with a nucleon bound inside a nucleus, it gets influenced by the presence of other nucleons which are not stationary but are continuously moving with a finite Fermi momentum. This motion of nucleons corresponds to the Fermi motion. These bound nucleons may also interact among themselves via strong interaction that is incorporated by the nucleon-nucleon correlations and the binding energy for a given nucleus has also been ensured. Moreover, for a nonsymmetric nucleus such as iron, copper, tin, lead, etc., we have taken into account the different densities for the proton and the neutron. We have discussed these effects and present the formalism in the following subsection.
Fermi motion, binding energy, nucleon correlation and isoscalarity effects
To calculate the scattering cross section for a neutrino interacting with a target nucleon in the nuclear medium, we express it in terms of the probability of interaction per unit area which is defined as the probability of interaction per unit time of the particle (Γ) times the time spent in the interaction process (dt) over a differential area dS [1,49,52], i.e.
dσ = Γ dt dS = Γ 1 v d 3 r = Γ E(k) | k | d 3 r,(22)
where
v = |k| E(k)
is the velocity of the particle and d 3 r is the volume element. The probability of interaction per unit time(Γ) that the incoming neutrino will interact with the bound nucleons is related to the neutrino self-energy, which provides information about the total neutrino flux available at our disposal after the interaction:
Γ = − 2m l E(k) ImΣ ⇒ dσ = −2m l |k| ImΣ d 3 r ,(23)
where ImΣ stands for the imaginary part of the neutrino self-energy that accounts for the depletion of the initial neutrinos flux out of the non-interacting channel, into the quasielastic or the inelastic channels. Thus the imaginary part of the neutrino self-energy gives information about the total number of neutrinos that have participated in the interaction and give rise to the charged leptons. Therefore, the evaluation of imaginary part of the neutrino self-energy is required to obtain the scattering cross section. Following the Feynman rules we write the neutrino self-energy corresponding to the diagram shown in Fig. 2(a) as: where we have used the properties of gamma matrices. Imaginary part of the neutrino self-energy may be obtained by using the Cutkosky rules [52] and is given by
Σ(k) = −iG F √ 2 d 4 k ′ (2π) 4 4L W I µν m l 1 (k ′2 − m 2 l + iǫ) M W Q 2 + M 2 W 2 Π µν (q) ,(24)ν l (k) ν l (k) W + (q) W + (q) N (p) X (p ′ ) l − (k ′ ) W + (q) W + (q) X Π µν (a) (b)ImΣ(k) = G F √ 2 4 m l d 4 k ′ (2π) 4 π E ′ (k ′ ) θ(q 0 )L W I µν M W Q 2 + M 2 W 2 Im[Π µν (q)].(25)
In the above expression, Π µν (q) is the W boson self-energy (depicted in Fig. 2(b)) which is defined in terms of the intermediate nucleon (G l ) and meson (D j ) propagators:
Π µν (q) = G F M 2 W √ 2 × d 4 p (2π) 4 G(p) X sp,s l N i=1 d 4 p ′ i (2π) 4 l G l (p ′ l ) j D j (p ′ j ) < X|J µ |N >< X|J ν |N > * (2π) 4 δ 4 (p + q − N i=1 p ′ i ),(26)
where s p is the spin of the nucleon, s l is the spin of the fermions in X, < X|J µ |N > is the hadronic current for the initial state nucleon to the final state hadrons, index l, j are respectively, stands for the fermions and the bosons in the final hadronic state X and δ 4 (p + q − N i=1 p ′ i ) ensures the conservation of four momentum at the vertex. G(p) is the nucleon propagator inside the nuclear medium through which the information about the propagation of the nucleon from the initial state to the final state or vice versa is obtained. The relativistic nucleon propagator for a noninteracting Fermi sea is written in terms of the positive (u(p)) and negative (v(−p)) energy components as:
G 0 (p 0 , p) = M N E N (p) r u r (p)ū r (p) 1 − n(p) p 0 − E N (p) + iǫ + n(p) p 0 − E N (p) − iǫ + r v r (−p)v r (−p) p 0 + E N (p) − iǫ .
The nucleon propagator retains the contribution only from the positive energy components because the negative energy components are much suppressed. Hence, we obtain
G 0 (p 0 , p) = M N E N (p) r u r (p)ū r (p) 1 − n(p) p 0 − E N (p) + iǫ + n(p) p 0 − E N (p) − iǫ .
In the above expression, the first term of the nucleon propagator within the square bracket contributes when the momentum of nucleon will be greater or equal to the Fermi momentum |p| ≥ p F , i.e. for the particles above the Fermi sea while the second term within the square bracket contributes when the nucleon momentum will be less than the Fermi momentum |p| < p F , i.e. for the particles below the Fermi sea. This representation is known as the Lehmann's representation [24]. Inside the Fermi sea, where nucleons interact with each other, the relativistic nucleon propagator G(p) is obtained by using the perturbative expansion of Dyson series in terms of the nucleon self energy(Σ N ) as:
G(p) = G 0 (p) + G 0 (p)Σ N (p)G 0 (p) + G 0 (p)Σ N (p)G 0 (p)Σ N (p)G 0 (p) + ... .
The nucleon self energy (shown in Fig.3) is evaluated by using the many body field theoretical approach in terms of the spectral functions [24,59] and the dressed nucleon propagator G(p) in an interacting Fermi sea is obtained as [59]: where µ = ǫ F + M N is the chemical potential, ω = p 0 − M N is the removal energy, S h (ω, p) and S p (ω, p) are the hole and particle spectral functions, respectively. In the above expression the term S h (ω, p) dω is basically the joint probability of removing a nucleon from the ground state and S p (ω, p) dω is the joint probability of adding a nucleon to the ground state of a nucleus. Consequently, one may obtain the spectral function sum rule which is given by
G(p) = M N E N (p) r u r (p)ū r (p) µ −∞ dω S h (ω, p) p 0 − ω − iǫ + ∞ µ dω S p (ω, p) p 0 − ω + iǫ ,(27)+ + + +............. k k k kµ −∞ S h (ω, p) dω + +∞ µ S p (ω, p) dω = 1.(28)
The expressions for the hole and particle spectral functions are given by [24,59]:
S h (p 0 , p) = 1 π MN EN (p) ImΣ N (p 0 , p) p 0 − E N (p) − MN EN (p) ReΣ N (p 0 , p) 2 + MN EN (p) ImΣ N (p 0 , p) 2(29)
when p 0 ≤ µ,
S p (p 0 , p) = − 1 π MN EN (p) ImΣ N (p 0 , p) p 0 − E N (p) − MN EN (p) ReΣ N (p 0 , p) 2 + MN EN (p) ImΣ N (p 0 , p) 2(30)
when p 0 > µ.
In the present study, we are considering the inclusive DIS process and are not looking at the final hadronic state, therefore, the interactions in the Fermi sea are taken into account through the hole spectral function S h . Now by using Eqs. 23 and 25, and performing the momentum space integration the differential scattering cross section is obtained as:
dσ W I A dxdy = − G 2 F M N y 2π E E ′ |k ′ | |k| M 2 W Q 2 + M 2 W 2 L W I µν ImΠ µν (q)d 3 r.(31)
On comparing Eq. 16 and Eq. 31, it is found that the nuclear hadronic tensor W µν A is related with the imaginary part of the W boson self-energy ImΠ µν (q) as
W µν A = − ImΠ µν (q)d 3 r.(32)
Using Eq. 27 and the expressions for the nucleon and meson propagators in Eq. 26, and finally substituting them in Eq. 32, we obtain the nuclear hadronic tensor W µν A for an isospin symmetric nucleus in terms of the nucleonic hadronic tensor W µν N convoluted with the hole spectral function(S h ) for a nucleon bound inside the nucleus:
W µν A = 4 d 3 r d 3 p (2π) 3 M N E N (p) µ −∞ dp 0 S h (p 0 , p, ρ(r))W µν N (p, q) ,(33)
where the factor of 4 is for spin-isospin of the nucleon and ρ(r) is the nuclear density. In general, nuclear density have various phenomenological parameterizations known in the literature as the harmonic oscillator(HO) density, two parameter Fermi density(2pF), modified harmonic oscillator (MHO) density, etc. The proton density distributions are 56 Fe and 208 Pb nuclei, 2-parameter Fermi density have been used, where superscript n and p in density parameters(c n,p i ; i=1,2) stand for neutron and proton, respectively. Density parameters and the root mean square radius (< r 2 > 1/2 ) are given in units of femtometer. The kinetic energy of the nucleon per nucleus(T /A) and binding energy of the nucleon per nucleus (B.E/A) for different nuclei are given in MeV.
obtained from the electron-nucleus scattering experiments, while the neutron densities are taken from the Hartee-Fock approach [64]. The density parameters c 1 and c 2 corresponds to the charge density for proton or equivalently the neutron matter density for neutron. In the present model, for the numerical calculations, we have used modified harmonic oscillator charge density
ρ(r) = ρ 0 1 + c 2 r c 1 2 e − r c 1 2(34)
for the light nuclei, e.g. 12 C, and 2-parameter Fermi density
ρ(r) = ρ 0 1 + e r−c 1 c 2(35)
for the heavy nuclei, like 40 Ar, 56 Fe and 208 Pb. In Eqs. 34 and 35, ρ 0 is the central density and c 1 , c 2 are the density parameters [64,65] which are independently given for protons (c p 1,2 ) and neutrons (c n 1,2 ) in Table I along with the other parameters used in the numerical calculations. We ensure the normalization of the hole spectral function by obtaining the baryon number (A) of a given nucleus and binding energy of the same nucleus.
4 d 3 r d 3 p (2π) 3 µ −∞ S h (ω, p, ρ(r)) dω = A ,
In the local density approximation, the spectral functions for the proton (Z) and neutron (N = A − Z) numbers in a nuclear target which are the function of local Fermi momenta p Fp,n (r) = 3π 2 ρ p(n) (r) 1/3 , are normalized separately such that
2 d 3 r d 3 p (2π) 3 µp −∞ S p h (ω, p, ρ p (r)) dω = Z , 2 d 3 r d 3 p (2π) 3 µn −∞ S n h (ω, p, ρ n (r)) dω = N ,
where the factor of 2 is due to the two possible projections of nucleon spin, µ p (µ n ) is the chemical potential for the proton(neutron), and S p h (ω, p, ρ p (r)) and S n h (ω, p, ρ n (r)) are the hole spectral functions for the proton and neutron, respectively. The proton and neutron densities ρ p (r) and ρ n (r) are related to the nuclear density ρ(r) as [49,52]:
ρ p (r) = Z A ρ(r) ; ρ n (r) = (A − Z) A ρ(r)
Hence for a nonisoscalar nuclear target, the nuclear hadronic tensor is written as
W µν A = 2 τ =p,n d 3 r d 3 p (2π) 3 M N E N (p) µτ −∞ dp 0 S τ h (p 0 , p, ρ τ (r)) W µν N (p, q).(36)
In this way, we have incorporated the effects of Fermi motion, Pauli blocking and nucleon correlations through the hole spectral function.
From Eqs. 33 and 36, we have evaluated the nuclear structure functions by using the expressions of nucleon and nuclear hadronic tensors given in Eqs. 7 and 17, respectively with the suitable choice of their components along x, y, and z axes. The numerical calculations are performed in the laboratory frame, where the target nucleus is assumed to be at rest(p A =(p 0 A ,p A = 0)) but the nucleons are moving with finite momentum(p=(p 0 ,p = 0)). These nucleons are thus off shell. If we choose the momentum transfer (q) to be along the z axis, i.e, q µ = (q 0 , 0, 0, q z ). Then the Bjorken variables for the nuclear target and the bound nucleons are defined as
x A = Q 2 2p A · q = Q 2 2M A q 0 = Q 2 2A M N q 0 , x N = Q 2 2p · q = Q 2 2(p 0 q 0 − p z q z ) .(37)
Hence, we have obtained the expressions of weak nuclear structure functions for the isoscalar and nonisoscalar nuclear targets by using Eqs. 33 and 36, respectively. The expression of F W I 1A,N (x A , Q 2 ) is obtained by taking the xx component of nucleon (Eq. 7) and nuclear (Eq. 17) hadronic tensors which for an isoscalar nuclear target is given by
F W I 1A,N (x A , Q 2 ) = 4AM N d 3 r d 3 p (2π) 3 M N E N (p) µ −∞ dp 0 S h (p 0 , p, ρ(r)) × F W I 1N (x N , Q 2 ) M N + p x M N 2 F W I 2N (x N , Q 2 ) ν N ,(38)
and for a nonisoscalar nuclear target is obtained as
F W I 1A,N (x A , Q 2 ) = 2 τ =p,n AM N d 3 r d 3 p (2π) 3 M N E N (p) µτ −∞ dp 0 S τ h (p 0 , p, ρ τ (r)) × F W I 1τ (x N , Q 2 ) M N + p x M N 2 F W I 2τ (x N , Q 2 ) ν N ,(39)where ν N = p·q MN = p 0 q 0 −p z q z MN
. We must point out that the evaluation of F W I 1A,N (x A , Q 2 ) has been performed independently, i.e., without using the Callan-Gross relation at the nuclear level. Similarly, the zz component of nucleon (Eq. 7) and nuclear (Eq. 17) hadronic tensors gives the expression of dimensionless nuclear structure function
F W I 2A,N (x A , Q 2 )
. For an isoscalar nuclear target it is expressed as
F W I 2A,N (x A , Q 2 ) = 4 d 3 r d 3 p (2π) 3 M N E N (p) µ −∞ dp 0 S h (p 0 , p, ρ(r)) × Q 2 (q z ) 2 |p| 2 − (p z ) 2 2M 2 N + (p 0 − p z γ) 2 M 2 N p z Q 2 (p 0 − p z γ)q 0 q z + 1 2 × M N p 0 − p z γ × F W I 2N (x N , Q 2 ),(40)
while for a nonisoscalar nuclear target it modifies to
F W I 2A,N (x A , Q 2 ) = 2 τ =p,n d 3 r d 3 p (2π) 3 M N E N (p) µτ −∞ dp 0 S τ h (p 0 , p, ρ τ (r)) × Q q z 2 |p| 2 − (p z ) 2 2M 2 N + (p 0 − p z γ) 2 M 2 N p z Q 2 (p 0 − p z γ)q 0 q z + 1 2 × M N p 0 − p z γ × F W I 2τ (x N , Q 2 ),(41)with γ = q 0 q z . The expression of F W I 3A,N (x A , Q 2 )
is obtained by choosing the xy component of nucleon (Eq. 7) and nuclear (Eq. 17) hadronic tensors which is given by for an isoscalar nuclear target. However, for a nonisoscalar nuclear target, we get
F W I 3A,N (x A , Q 2 ) = 4A d 3 r d 3 p (2π) 3 M N E N (p) µ −∞ dp 0 S h (p 0 , p, ρ(r)) × q 0 q z × p 0 q z − p z q 0 p · q F W I 3N (x N , Q 2 ),(42)p ′ + νl(k) l − (k ′ ) νl(k) W + (q) W + (q) p + l − (k ′ ) p p ′ νl(k) νl(k) W + (q) W + (q) νl(k) νl(k) W + (q) W + (q) l − (k ′ ) p ′ +..l − (k ′ ) νl(k) W + (q) W + (q) p p ′(F W I 3A,N (x A , Q 2 ) = 2A τ =p,n d 3 r d 3 p (2π) 3 M N E N (p) µτ −∞ dp 0 S τ h (p 0 , p, ρ τ (r)) × q 0 q z × p 0 q z − p z q 0 p · q F W I 3τ (x N , Q 2 ),(43)
The results obtained by using Eqs. 38, 40, 42 for isoscalar and Eqs. 39, 41, 43 for nonisoscalar nuclear targets are labeled as the results with the spectral function(SF) only.
Mesonic Effect
In the case of (anti)neutrino-nucleus DIS process, mesonic effects also contribute to the nuclear structure functions F W I 1A (x A , Q 2 ) and F W I 2A (x, Q 2 ) which arises due to the interaction of bound nucleons among themselves via the exchange of virtual mesons such as π, ρ, etc. There is a reasonably good probability that intermediate W boson may interact with a meson instead of a nucleon [24,27]. In order to include the contribution from the virtual mesons, we again evaluate the neutrino self-energy for which a diagram is shown in Fig. 4 and write the meson hadronic tensor in the nuclear medium similar to the case of bound nucleons as [24] W µν A,i = 3 d 3 r
d 4 p (2π) 4 θ(p 0 )(−2) ImD i (p) 2m i W µν i (p, q) ,(44)
where i = π, ρ, a factor of 3 is due to the three charged states of pion (rho meson) and D i (p) is the dressed meson propagator. This expression is obtained by replacing the hole spectral function
−2π M N E N (p) S h (p 0 , p) W µν N (p, q)
in Eq. 33 with the imaginary part of the meson propagator, i.e,
ImD i (p) θ(p 0 ) 2W µν i (p, q).
This meson propagator does not correspond to the free mesons because a lepton (either electron or muon) can not decay into another lepton, one pion and debris of hadrons but it corresponds to the mesons arising due to the nuclear medium effects by using a modified meson propagator. These mesons are arising in the nuclear medium through particle-hole(1p-1h), delta-hole(1∆-1h), 1p1h-1∆1h, 2p-2h, etc. interactions as depicted in Fig. 4. This effect is incorporated following the mean-field theoretical approach [24]. The expression of meson propagator (D i (p)) in the nuclear medium is given by
D i (p) = [p 0 2 − p 2 − m 2 i − Π i (p 0 , p)] −1 ,(45)
with the mass of meson m i and the meson self-energy Π i which is explicitly written as
Π π = f 2 m 2 π F 2 π (p)p 2 Π * (p) 1 − f 2 m 2 π V ′ L (p)Π * (p)
,
Π ρ = f 2 m 2 π C ρ F 2 ρ (p)p 2 Π * (p) 1 − f 2 m 2 π V ′ T (p)Π * (p) .(46)
In the above expressions, the coupling constant f = 1.01, the free parameter C ρ = 3.94, V ′ L (p)(V ′ T (p)) is the longitudinal(transverse) part of the spin-isospin interaction which is responsible for the enhancement to the pion(rho meson) structure function and Π * (p) is the irreducible meson self-energy that contains the contribution of particle-hole and delta-hole excitations. The πN N and ρN N form factors, i.e., F π (p) and F ρ (p) used in Eq. 46 are given by
F π (p) = (Λ 2 π − m 2 π ) (Λ 2 π + p 2 ) , F ρ (p) = (Λ 2 ρ − m 2 ρ ) (Λ 2 ρ + p 2 )(47)
with the parameter Λ π (Λ ρ )=1 GeV. Since Eq.44 has taken into account the mesonic contents of the nucleon which are already incorporated in the sea contribution of the nucleon, in order to calculate the mesonic excess in the nuclear medium we have subtracted the meson contribution of the nucleon [24] such that
ImD i (p) → δImD i (p) ≡ ImD i (p) − ρ ∂ImD i (p) ∂ρ ρ=0 .(48)
Now we have obtained the following expression for the mesonic hadronic tensor
W µν A,i = 3 d 3 r d 4 p (2π) 4 θ(p 0 )(−2) δImD i (p) 2m i W µν i (p, q) ,(49)
Using Eq.49, the mesonic structure functions F W I 1A,i (x, Q 2 ) and F W I 2A,i (x, Q 2 ) are evaluated following the same analogy as adopted in the case of bound nucleons [24]. The expression for F W I 1A,i (x, Q 2 ) is given by
F W I 1A,i (x, Q 2 ) = −6 × a × AM N d 3 r d 4 p (2π) 4 θ(p 0 ) δImD i (p) 2m i × F W I 1i (x i ) m i + |p| 2 − (p z ) 2 2(p 0 q 0 − p z q z ) F W I 2i (x i ) m i ,(50)
and for F W I 2A,i (x, Q 2 ) we obtain
F W I 2A,i (x, Q 2 ) = −6 × a d 3 r d 4 p (2π) 4 θ(p 0 ) δImD i (p) 2m i × Q 2 (q z ) 2 |p| 2 − (p z ) 2 2m 2 i + (p 0 − p z γ) 2 m 2 i p z Q 2 (p 0 − p z γ)q 0 q z + 1 2 × m i p 0 − p z γ F W I 2i (x i ),(51)
where x i = Q 2 −2p·q and a = 1 for pion and a = 2 for rho meson [24]. Notice that the ρ meson has an extra factor of two compared to pionic contribution because of the two transverse polarization of the ρ meson [66].
In the literature, various groups like MRST98 [67], CTEQ5L [68], SMRS [69], GRV [70], etc., have proposed the quark and antiquark PDFs parameterizations for pions. We have observed in our earlier work [1] that the choice of different pionic PDFs parameterization would not make much difference in the scattering cross section. For the present numerical calculations the GRV pionic PDFs parameterization given by Gluck et al. [70] has been used and the same PDFs are also taken for the rho meson. The contribution from the pion cloud is found to be larger than the contribution from rho meson cloud, nevertheless, the rho contribution is non-negligible, and both of them are positive in the whole range of x. It is important to mention that F W I 3A (x A , Q 2 ) has no mesonic contribution as it depends mainly on the valence quarks distribution and these average to zero when considering the three charge states of pions and rho mesons. For details please see Refs. [1,49,52].
Shadowing and Antishadowing effects
The shadowing effect which contributes in the region of low x(≤ 0.1), takes place as a result of the destructive interference of the amplitudes due to the multiple scattering of quarks arising due to the hadronization of W ± /Z 0 bosons and leads to a reduction in the nuclear structure functions. It arises when the coherence length is larger than the average distance between the nucleons bound inside the nucleus and the expected coherence time is τ c ≥ 2 fm. However, the shadowing effect gets saturated if the coherence length becomes larger than the average nuclear radius, i.e., in the region of low x. Furthermore, in the region of 0.1 < x < 0.3, the nuclear structure functions get enhanced due to the antishadowing effect which is theoretically less understood. In the literature, several studies proposed that it may be associated with the constructive interference of scattering amplitudes resulting from the multiple scattering of quarks [27,29,71]. For the antishadowing effect, the coherence time is small for the long inter-nucleon spacing in the nucleus corresponding to these values of x. Shadowing and antishadowing effects are found to be quantitatively different in electromagnetic and weak interaction induced processes [52]. It is because the electromagnetic and weak interactions take place through the interaction of photons and W ± /Z 0 bosons, respectively, with the target hadrons and the hadronization processes of photons and W ± /Z 0 bosons are different. Moreover, in the case of weak interaction, the additional contribution of axial current which is not present in the case of electromagnetic interaction may influence the behaviour of weak nuclear structure functions specially if pions also play a role in the hadronization process through PCAC. Furthermore, in this region of low x, sea quarks also play an important role which could be different in the case of electromagnetic and weak processes. In the present numerical calculations, we have incorporated the shadowing effect following the works of Kulagin and Petti [27] who have used Glauber-Gribov multiple scattering theory. For example, to determine the nuclear structure function F W I iA (x, Q 2 ) with the shadowing effect, we use [27]
F W I,S iA (x, Q 2 ) = δR(x, Q 2 ) × F W I iN (x, Q 2 ) ,(52)
where F W I,S
iA (x, Q 2 ); (i = 1 − 3) is the nuclear structure function with shadowing effect and the factor δR(x, Q 2 ) is given in Ref. [27]. Now, using the present formalism, we have presented the results for the weak structure functions and scattering cross sections for both the free nucleon and nuclear targets in the next section.
III. RESULTS AND DISCUSSION
We have performed the numerical calculations by considering the following cases:
• The nucleon structure functions are obtained using PDFs parameterization of Martin et al. [53].
• All the results are presented with TMC effect.
• F W I iN (x, Q 2 ); (i = 1 − 3) are obtained at NLO and NNLO. • At NLO the higher twist effect has been incorporated following the renormalon approach [57] and a comparison is made with the results obtained at NNLO.
• After taking into account the perturbative and nonperturbative QCD corrections in the evaluation of free nucleon structure functions, we have used them to calculate the nuclear structure functions. The expression for F W I iA (x, Q 2 ), (i = 1, 2) in the full model is given by
F W I iA (x, Q 2 ) = F W I iA,N (x, Q 2 ) + F W I iA,π (x, Q 2 ) + F W I iA,ρ (x, Q 2 ) + F W I,S iA (x, Q 2 ) ,(53)
where F W I iA,N (x, Q 2 ) is the structure function with spectral function given in Eqs.38 (39) and 40(41) for F W I 1A,N (x, Q 2 ) and F W I 2A,N (x, Q 2 ), respectively, in the case of isoscalar (nonisoscalar) targets which takes care of Fermi motion, binding energy and nucleon correlations. The mesonic contributions are included using Eq.50 and 51 for F W I 1A,j (x, Q 2 ) and F W I 2A,j (x, Q 2 ) (j = π, ρ), respectively and for the shadowing effect (F W I,S
iA (x, Q 2 )) Eq.52 is used. F W I 3A (x, Q 2 ) has no mesonic contribution and the expression is given by
F W I 3A (x, Q 2 ) = F W I 3A,N (x, Q 2 ) + F W I 3A,shd (x, Q 2 ),(54)
with spectral function contribution F W I 3A,N (x, Q 2 ) using Eqs.42 (43) for the isoscalar (nonisoscalar) nuclear targets and the shadowing correction F W I 3A,shd (x, Q 2 ) using Eq.52. • The results are presented for 12 C, CH, 40 Ar, 56 Fe and 208 Pb nuclear targets which are being used in the present generation experiments.
The results of the free nucleon structure functions are presented in Fig.5, for 2xF W I 1N (x, Q 2 ), F W I 2N (x, Q 2 ) and F W I 3N (x, Q 2 ) vs Q 2 at x = 0.225, 0.45 and 0.65 in the case of neutrino-nucleon DIS process. We observe that due to the TMC effect the nucleon structure functions are modified at low and moderate Q 2 specially in the region of high x. We find that at NLO, the modification in the structure functions due to TMC effect is about 3%(16%) in 2xF W I 1N (x, Q 2 ), < 1%(5%) in F W I 2N (x, Q 2 ) and 5%(10%) in F W I 3N (x, Q 2 ) at x = 0.225(0.45) and Q 2 = 1.8 GeV 2 which becomes 1%(8%), < 1%(1%) and ∼ 2%(3%) at Q 2 = 5 GeV 2 . On the other hand the effect of higher twist corrections in this kinematic region is very small in F W I 1N (x, Q 2 ) and F W I 2N (x, Q 2 ) unlike in the case of electromagnetic structure functions [1]. Whereas the effect of higher twist in F W I 3N (x, Q 2 ) leads to a decrease of 15% at x = 0.225 and 5% at x = 0.65 for Q 2 = 1.8 GeV 2 , and becomes small with the increase in Q 2 . We observe that the difference in the results of F W I iN (x, Q 2 ) (i = 1, 2) at NLO with HT effect from the results at NNLO is < 1%. However, in F W I 3N (x, Q 2 ) at x = 0.225, this difference is about 8% for Q 2 = 1.8 GeV 2 and it reduces to ∼ 2% for Q 2 = 5 GeV 2 . With the increase in x and Q 2 the effect becomes gradually smaller.
The effect of higher twist is further suppressed in the nuclear medium, which is similar to our observation made for the electromagnetic nuclear structure functions [1]. The results observed at NLO with higher twist is close to the results obtained at NNLO. Therefore, all the results are presented here at NNLO.
In Figs. 6, 7, and 8 the results are presented respectively for the nuclear structure functions 2xF W I 1A (x, Q 2 ),
F W I 2A (x, Q 2 ) and xF W I 3A (x, Q 2 ) vs Q 2 for different values of x.
The numerical results obtained in the kinematic limit Q 2 > 1 GeV 2 without any cut on the center of mass energy W are labeled as "Nocut". The nuclear structure functions are shown for 1 < Q 2 ≤ 10 GeV 2 in carbon, argon, iron and lead which are treated as isoscalar nuclear targets and compared these results from the results obtained for a free nucleon target. From the figures, the different behaviour of the nuclear medium effects in different regions of x and Q 2 can be clearly observed. For example, the results for the structure functions with spectral function are suppressed from the results of the free nucleon target in the range of x(< 0.7) and Q 2 considered here. Quantitatively, this reduction in carbon from the results of free nucleon structure functions for Q 2 = 1.8 GeV 2 is found to be about 7%, 8%, and ∼ 5% at x = 0.225 in 2xF W I 1A (x, Q 2 ), mesonic contribution that works in its opposite (same) direction and results in an overall enhancement of the nuclear structure functions. Hence, the results obtained by including mesonic contributions, shadowing and antishadowing effects in our full model are higher than the results with the spectral function only. Mesonic contribution does not contribute to xF W I 3A (x, Q 2 ). The difference between the results of spectral function and the full model for 2xF W I 1A (x, Q 2 ) is 20% at x = 0.225 and 3% at x = 0.45 for Q 2 = 1.8 GeV 2 in carbon. These nuclear effects are observed to be more pronounced for the heavy nuclear targets such as in the case of argon it becomes 26%(4%) and 31%(5%) in lead at x = 0.225(x = 0.45) for Q 2 = 1.8 GeV 2 . However, with the increase in Q 2 the mesonic contribution becomes small, for example, at Q 2 = 5 GeV 2 this difference is reduced to 16% in 12 C, 21% in 40 Ar and 26% in 208 P b at x = 0.225. For the (anti)neutrino scattering cross sections and structure functions, high statistics measurements have been done by CCFR [72], CDHSW [73] and NuTeV [74] experiments in iron and by CHORUS [75] collaboration in lead nuclei. These experiments have been performed in a wide energy range, i.e., 20 ≤ E ν ≤ 350 GeV and measured the differential scattering cross sections. From these measurements the nuclear structure functions are extracted. We study the nuclear modifications for the (anti)neutrino induced processes in F ν+ν 2A (x, Q 2 ) and xF ν+ν 3A (x, Q 2 ) vs Q 2 in 56 Fe and 208 Pb nuclei by treating them as isoscalar nuclear targets. The results are presented in Fig.9 at the different values of x using the full model at NNLO and are compared with the available experimental data from CCFR [72], CDHSW [73], NuTeV [74] and CHORUS [75] experiments. We find a good agreement between the theoretical results for F ν+ν 2A (x, Q 2 ) and reasonable agreement for F ν+ν 3A (x, Q 2 ) with the experimental data. We have also studied the nuclear modifications in the electromagnetic structure functions [1] and compared them with the weak structure functions for the free nucleon target, isoscalar nuclear targets and nonisoscalar nuclear targets and present the results in Fig.10 for the ratios 5 18
F W I 2A (x,QF W I 1A (x,Q 2 ) F EM 1A (x,Q 2 ) (left panel) and 5 18 F W I 2A (x,Q 2 ) F EM 2A (x,Q 2 ) (right panel) vs
x at Q 2 = 5 and 20 GeV 2 . The numerical results are shown at NNLO for carbon, iron and lead with the full model and are compared with the results of free nucleon. It may be noticed from the figure that the ratio R W I/EM (x, Q 2 ) deviates from unity in the region of low x even for the free nucleon case. It implies non-zero contribution from strange and charm quarks distributions which are found to be different in the case of electromagnetic and weak structure functions. However, for x ≥ 0.4, where the contribution of strange and charm quarks are almost negligible, the ratio approaches ∼ 1. Furthermore, if one assumes s =s and c =c then in the region of small x, this ratio would be unity for an isoscalar nucleon target following the 5 18 th -sum rule. It may be seen that the difference between the ratio R W I/EM (x, Q 2 ) for the isoscalar nuclear targets and the free nucleon target is almost negligible. The evaluation is also done for the nonisoscalar nuclear targets (N >> Z) like iron and lead. We must emphasize that in the present model, the spectral functions are normalized separately for the proton (Z) and neutron (N = A − Z) numbers in a nuclear target and to the number of nucleons for an isoscalar nuclear target [49]. The ratio R W I/EM (x, Q 2 ) shows a significant deviation for the nonisoscalar nuclear targets which increases with nonisoscalarity, i.e. δ = (A−Z) Z . This shows that the charm and strange quark distributions are significantly different in asymmetric heavy nuclei as compared to the free nucleons. It is important to notice that although some deviation is present in the entire range of x, it becomes more pronounced with the increase in x. For example, in iron (nonisoscalar) the deviation from the free nucleon case is 2% at x = 0.2, 5% at x = 0.5, and 8% at x = 0.8 while in lead (nonisoscalar) it is found to be ∼ 7% at x = 0.2, 16% at x = 0.5, and 25% at x = 0.8 at Q 2 = 5 GeV 2 . This deviation also has some Q 2 dependence and with the increase in Q 2 the deviation becomes smaller. From the figure, it may be observed that the isoscalarity corrections, significant in the region of large x, are different in F 1A (x, Q 2 ) and F 2A (x, Q 2 ) albeit the difference is small. We have also presented the results for the ratios of nuclear structure functions such as
F W I iA (x, Q 2 ) F W I iA ′ (x,Q 2 )
; (i = 1, 2, 3; A = 56 F e, 208 P b and A ′ = 12 C) vs x
at Q 2 = 5 GeV 2 in Fig.11. The numerical results are shown with the full model at NNLO by treating iron and lead to be isoscalar as well as nonisoscalar nuclear targets. The following aspects are evident from the observation of Fig.11: • The deviation of the ratios
F W I iF e (x,Q 2 ) F W I iC (x,Q 2 ) and F W I iP b (x,Q 2 ) F W I
iC (x,Q 2 ) from unity in the entire range of x implies that nuclear medium effects are A dependent. From the figure, it may also be noticed that the ratio in lead is higher than the ratio in iron which shows that medium effects become more pronounced with the increase in the nuclear mass number. There is noticeable enhancement in the ratio obtained for the nonisoscalar case from the results obtained for the isoscalar nuclear targets specially at high x. This implies that nonisoscalarity effect increases with the increase in x as well as in the mass number.
• It is important to notice that although the behaviour of the ratio is qualitatively same in
F W I iA (x,Q 2 ) F W I iA ′ (x,Q 2 ) ; (i = 1 − 3), quantitatively it is different.
In the literature the choice of a sharp kinematical cut on W and Q 2 required to separate the regions of nucleon resonances and DIS, i.e. region of shallow inelastic scattering and deep inelastic scattering is debatable. However, in some of the analysis the kinematic region of Q 2 > 1 GeV 2 and W > 2 GeV is considered to be the region of safe DIS [5,77] and this has been taken into account in the analysis of MINERvA experiment [5]. Therefore, to explore the transition region of nucleon resonances and DIS we have also studied the effect of CoM cut on the scattering cross section. In Figs. 12-15, we have presented the results with a CoM cut of 2 GeV (W > 2 GeV) and Q 2 > 1 GeV 2 which are labeled as "Wcut" and compared them with the corresponding "Nocut" results (Q 2 > 1 GeV 2 only) as well as with the available experimental data.
In Figs Fig.12, theoretical results are presented for the spectral function only (dashed line) and for the full model (solid line) without having any cut on the CoM energy in iron and are compared with the NuTeV experimental data [74]. It may be seen that due to the mesonic contribution the results with the full model are higher than the results with the spectral function at x = 0.225, however, for x ≥ 0.45, where mesonic contribution is suppressed the difference becomes small. For example, in ν l − 56 Fe(ν l − 56 Fe) this enhancement is found to be 24%(30%) at x = 0.225 and 6%(8%) at x = 0.45 for y = 0.2. Furthermore, we have compared these results with the phenomenological results of nCTEQnu [76] (evaluated by usingν l − A scattering experimental data). One may notice that the present theoretical results differ from the results of nCTEQnu PDFs parameterization [76] in the region of low x and y while at high x and y they are in good agreement. In the inset of this figure, the results obtained with the full model having no cut on W (solid line) are compared with the results obtained with a cut of W > 2 GeV (solid line with star). It is important to notice that the difference between these results becomes more pronounced with the increase in x specially at low y, for example, at y = 0.1(y = 0.4) there is a difference of 30%(7%) [75]. The results for the ν l induced process obtained using the nuclear PDFs nCTEQnu [76] (dash-dotted line) are also shown. at x = 0.225 and 36%(3%) at x = 0.45 in ν l − 56 Fe scattering process while forν l − 56 Fe it is found to be 32%(13%) and 37%(8%) respectively at x = 0.225 and x = 0.45. For higher values of y the effect of CoM energy cut is small. However, there is no experimental data in the region of low y to test these results.
In Fig.13, we have presented the numerical results of the differential scattering cross section in 208 Pb for the neutrino and antineutrino induced processes and compared them with the experimental data of CHORUS [75] experiment, where a comparison of the theoretical results for ν l − 208 P b scattering has also been made with the results of nCTE-Qnu [76] nuclear PDFs parameterization. We find that due to the A dependence, the nuclear medium effects are more pronounced in lead as compared to iron and the effect of CoM energy cut causes relatively larger suppression in the region of low x and y(≤ 0.4). For the numerical results presented in Figs.12 and 13, the nuclear targets are treated to be isoscalar.
The MINERvA experiment has used the NuMI neutrino beam at Fermilab for the cross section measurements in the low and medium energy modes that peak around neutrino energy of 3 GeV and 6 GeV, respectively. The low energy neutrino broad band energy spectrum that peaks at ∼ 3 GeV extends up to 100 GeV, however, neutrino flux drops steeply at high energies. MINERvA collaboration [5] has reported the ratio of flux integrated differential scattering cross sections in carbon, iron and lead to the polystyrene scintillator (CH) vs x in the neutrino energy range of 5-50 GeV. We have chosen two neutrino beam energies viz. E = 7 GeV and 25 GeV, in a wide energy spectrum (7 ≤ E ≤ 25 GeV), in order to study the energy dependence of the nuclear medium effects. We have obtained
dσ W I A dx
by integrating Eq.21 over y in the limits 0 and 1 and present the theoretical results for the ratio dσ W I A /dx dσ W I CH /dx (A = 12 C, 56 F e, 208 P b) at E = 7 GeV and 25 GeV for the charged current ν l − A andν l − A DIS processes. The theoretical results are obtained in the kinematic region relevant for the MINERvA experiment (W > 2 GeV and Q 2 > 1 GeV 2 ) and compared with the experimental data as well as with the results obtained using phenomenological models of Cloet et al. [8], Bodek-Yang [7] and GENIE Monte Carlo [6].
The results for the ratio ( dσ W I A /dx dσ W I CH /dx ) vs x in the case of ν l − A scattering are presented in Fig.14 and are summarized below:
• As the nuclear medium effects are approximately the same in carbon and CH, therefore, the ratio For example, at E = 25 GeV the contribution of mesons is found to be 10%(7%) at x = 0.1, 2%(1%) at x = 0.3, and < 1% at x = 0.6 in lead(iron) when they are treated to be isoscalar. It is important to notice that even for high energy neutrino beams the effect of nuclear medium on the differential scattering cross section are significant.
• We have found that due to the mass dependence of nuclear medium effects, the difference between the results of dσ at x = 0.05, 6%(9%) at x = 0.1 and 3%(∼ 3%) at x = 0.6 when there is no constrain on the CoM energy W . While the cut of W > 2 GeV, leads to a change of 1 − 5% in this difference in the entire range of x, for example, there is further reduction of ≃ 2% at x = 0.05, 3% at x = 0.1, ≃ 5% at x = 0.2 and < 1% at x = 0.6 in the differential scattering cross section.
• To study the isoscalarity effect we have obtained the results for dσ W I F e /dx dσ W I CH /dx and dσ W I P b /dx dσ W I CH /dx by treating iron and lead to be nonisoscalar (left panel) as well as isoscalar (right panel) targets (Fig.14). The isoscalarity correction in asymmetric nucleus is found to be significant. For example, at E = 25 GeV, this effect is 2%(5%), and 5%(13%) at x = 0.3 and 0.7, respectively, in iron(lead) when no kinematical cut is applied on W .
• To observe the energy dependence of the scattering cross section, numerical results obtained using the full model with obtained for E = 7 GeV increases. Due to the energy dependence of the differential scattering cross section, the difference between the results obtained using the full model at the aforesaid energies, i.e. 7 GeV and 25 GeV, is ≃ 3%(5%), 2%(≃ 2%) and 12%(≃ 16%) at x = 0.1, x = 0.3 and x = 0.75 , respectively, if iron (lead) is treated as isoscalar nuclear target. [7], GENIE Monte Carlo [6] and with the simulated results [5]. The solid squares are the experimental points of MINERvA [5]. The results in the left and right panels are respectively shown for the nonisoscalar and isoscalar nuclear targets.
• Furthermore, we have compared our theoretical results with the corresponding experimental data of MINERvA as well as with the different phenomenological models like that of Cloet et al. [8] (solid line with circle), Bodek et al. [7] (solid line with square) and GENIE MC [6] (solid line with triangle). It may be noticed that MIN-ERvA's experimental data have large error bars due to statistical uncertainties and the wide band around the simulation is due to the systematic errors which shows an uncertainty up to ∼ 20% [5]. Although the results of phenomenological models lie in this systematic error band even then none of the phenomenological model is able to describe the observed ratios in the whole region of x.
We have also made predictions for theν l − A scattering cross sections in the same kinematic region as considered in Fig.14 corresponding to the MINERvA experiment and presented the results in Fig.15, for the ratio dσ W I A /dx dσ W I CH /dx ; (A = 12 C, 56 F e, 208 P b) vs x at E =7 GeV and 25 GeV without and with a cut of W > 2 GeV. The nuclear medium effects in dσ W I A dx forν l − A scattering are found to be qualitatively similar to ν l − A scattering when no cut on CoM energy is applied, however, quantitatively they are different specially at low and mid values of x. For example, at E = 7 GeV the enhancement in the cross section when full calculations is applied from the results obtained using the spectral function is about 24% at x = 0.25 in ν l − 208 P b scattering, while it is 65% inν l − 208 P b scattering, and the difference in the two results decreases with the increase in x. At E = 25 GeV the enhancement in the cross section is about 20% at x = 0.25 in ν l − 208 P b scattering, while it is ∼45% inν l − 208 P b scattering. When a cut of 2 GeV is applied on the CoM energy, then a suppression in the region of low and mid x is observed in the differential cross section, resulting in a lesser enhancement due to mesonic effects, for example, at E = 25 GeV, the enhancement due to the mesonic contributions becomes ∼18% (vs 20% without cut) in ν l − 208 P b scattering while ∼ 28% (vs 45% without cut) inν l − 208 P b scattering at x = 0.25. At E = 7 GeV, with a cut of 2 GeV on W , the enhancement is about 2% at x = 0.25 in ν l − 208 P b scattering, while there is reduction inν l − A scattering, implying small contribution from the mesonic part. This reduction in dσ W I A dx forν l − A scattering is about 15% in a wide region of x(≤ 0.6). When the results for dσ W I A /dx dσ W I CH /dx using antineutrino beam are compared with neutrino results, we find that without any cut on W , the results are similar, but with a cut for E =7 GeV there is enhancement at high x. This enhancement is larger in 208 P b than in 56 F e due to large effect of Fermi motion in heavy nuclei.
IV. SUMMARY AND CONCLUSION
Our findings for the weak nucleon and nuclear structure functions and the differential scattering cross sections are as follows:
• The difference in the results of free nucleon structure functions F W I iN (x, Q 2 ) (i = 1, 2) evaluated at NLO with HT effect and the results obtained at NNLO is almost negligible (< 1%). However, this difference is somewhat larger for F W I 3N (x, Q 2 ) at low x and low Q 2 which becomes small with the increase in Q 2 . In the case of nucleons bound inside a nucleus, the HT corrections are further suppressed due to the presence of nuclear medium effects. Consequently, the results for ν l /ν l − A DIS processes which are evaluated at NNLO have almost negligible difference from the results obtained at NLO with HT effect.
• The nuclear structure functions obtained with spectral function only is suppressed from the free nucleon case in the entire region of x. Whereas, the inclusion of mesonic contributions results in an enhancement in the nuclear structure functions in the low and intermediate region of x. Mesonic contributions are observed to be more pronounced with the increase in mass number and they decrease with the increase in x and Q 2 . The results for the nuclear structure functions F W I 2A (x, Q 2 ) and F W I 3A (x, Q 2 ) with the full theoretical model show good agreement with the experimental data of CCFR [72], CDHSW [73], NuTeV [74] and CHORUS [75] especially at high x and high Q 2 . Predictions are also made for 40 Ar that may be useful in analyzing the experimental results of DUNE [2,3] and ArgoNeuT [4].
• We have found nuclear medium effects to be different in electromagnetic and weak interaction channels specially for the nonisoscalar nuclear targets. The contribution of strange and charm quarks is found to be different for the electromagnetic and weak interaction induced processes off free nucleon target which also gets modified differently for the heavy nuclear targets. Furthermore, we have observed that the isoscalarity corrections, significant even at high Q 2 , are not the same in F W I 1A (x, Q 2 ) and F W I 2A (x, Q 2 ). • The nuclear medium effects are found to be important in the evaluation of differential scattering cross section.
We have observed that in theν l − A reaction channel the nuclear medium effects are more pronounced than in the case of ν l − A scattering process. Our results of 1 E d 2 σ W I A dxdy ; (A = 56 F e, 208 P b) obtained using the full model show a reasonable agreement with the experimental data of NuTeV [74] and CHORUS [75] for the neutrino and antineutrino induced DIS processes. Theoretical results of differential cross section are also found to be in good agreement with the phenomenological results of nCTEQnu nuclear PDFs parameterization [76] in the intermediate as well as high region of x for all values of y.
• The present theoretical results for the ratio dσ W I A /dx dσ W I CH /dx (A = 12 C, 56 F e, 208 P b) when compared with the different phenomenological models and MINERvA's experimental data on ν l − A scattering, imply that a better understanding of nuclear medium effects is required in the ν l (ν l )−nucleus deep inelastic scattering. We have also made predictions for theν l − A DIS cross sections relevant for the upcoming MINERvA results.
To conclude, the present theoretical results provide information about the energy dependence, effect of CoM energy cut, medium modifications and isoscalarity correction effects on the nuclear structure functions and cross sections for the deep inelastic scattering of (anti)neutrino from various nuclei. This study will be helpful to understand the present and future experimental results from MINERvA [5], ArgoNeuT [4], and DUNE [2,3] experiments.
FIG. 2 :
2Diagrammatic representation of the neutrino self-energy (left panel) and intermediate W boson self-energy (right panel).
FIG. 3 :
3Diagrammatic representation of nucleon self-energy in the nuclear medium.
FIG. 4 :
4Neutrino self-energy diagram accounting for neutrino-meson DIS (a) the bound nucleon propagator is substituted with a meson(π or ρ) propagator of momentum p and jet of hadrons X with momentum p ′ . (b) by including particle-hole (1p1h), delta-hole (1∆1h), 1p1h − 1∆1h, etc. interactions.
FIG. 5 :
52xF W I 1N (x, Q 2 ) (top panel), F W I 2N (x, Q 2 ) (middle panel) and F W I 3N (x, Q 2 ) (bottom panel) vs Q 2 at the different values of x for ν l − Nscattering. The results are obtained at NLO (i) without TMC effect (double dash-dotted line), (ii) including TMC effect without (dash-dotted line) and with (dashed line) the higher twist correction, and (iii) at NNLO with TMC effect (solid line).
FIG. 6 :
62 ), and xF W I 3A (x, Q 2 ) respectively, which becomes 9%, 11%, and 2% at x = 0.45. We have explicitly shown the mesonic contribution (double dash-dotted line) which is quite significant in the low and intermediate regions ofx(< 0.6). The inclusion of mesonic effect gives an enhancement in the case of nuclear structure functions F W I 1A (x, Q 2 ) and F W I 2A (x, Q 2 ) for all values of x < 0.6 and becomes negligible for x > 0.6. The shadowing (antishadowing) effect that causes a reduction(enhancement) in the nuclear structure function for x ≤ 0.1(0.1 < x < 0.3) is modulated by the Results for 2xF W I 1A (x, Q 2 ) vs Q 2 are shown at different values of x for neutrino induced DIS process in 12 C, 40 Ar, 56 Fe and 208 Pb. The results are obtained with the spectral function only (dashed line), with mesonic contribution only (double dash-dotted line) and with the full model (solid line) at NNLO and are compared with the results of the free nucleon case (dash-dotted line). All the nuclear targets are treated as isoscalar.
FIG. 7 :
7F W I 2A (x, Q 2 ) vs Q 2 .The lines in this figure have the same meaning as inFig.6.
FIG. 8 :
8xF W I 3A (x, Q 2 ) vs Q 2 .The lines in this figure have the same meaning as inFig.6.
FIG. 9 :FIG. 10 :Q 2 ) = 5 18 F
910218Results for F ν+ν 2A (x, Q 2 ) (top panel) and xF ν+ν 3A (x, Q 2 ) (bottom panel) vs Q 2 are shown at different values of x in 56 Fe and 208 Pb. The results are obtained with the full model (solid line) at NNLO and are compared with the results of the available experimental data[72][73][74][75]. Both the nuclear targets are treated as isoscalar. The ratio R W I/EM (x, W I iA (x,Q 2 ) F EM iA (x,Q 2 ) ; i = 1, 2 vs x are shown at Q 2 = 5 GeV 2 (top panel) and 20 GeV 2 (bottom panel) in 12 C,56 Fe and 208 Pb. The numerical results are obtained at NNLO using the full model and are compared with the free nucleon case.
(x,Q 2 ) ; (i = 1 − 3; A = 56 Fe, 208 Pb; A ′ = 12 C) vs x are shown at Q 2 = 5 GeV 2 . The results are obtained using the full model at NNLO by treating 56 Fe, 208 Pb to be isoscalar (solid line) as well as nonisoscalar (dashed line) nuclear targets.
vs y for ν l − A, (A = 56 F e, 208 P b) (top panel) and ν l − A, (A = 56 F e, 208 P b) (bottom panel) scattering processes at NNLO. The numerical results are obtained for a beam energy of 35 GeV at the different values of x. In
vs y are shown at the different values of x for E = 35 GeV. The numerical results for ν l − 56 Fe DIS (top panel) andν l − 56 Fe DIS (bottom panel) processes are obtained with the spectral function only (dashed line) and with the full model (solid line) at NNLO. In the inset the results for the full model are compared with the corresponding results obtained with a kinematical cut of W > 2 GeV (solid line with star). Solid circles are the experimental data points of NuTeV [74]. The results for the ν l induced process obtained using the nuclear PDFs nCTEQnu [76] (dash-dotted line) are also shown. vs y are shown at the different values of x for E = 35 GeV. The numerical results for ν l − 208 Pb DIS (top panel) andν l − 208 Pb DIS (bottom panel) processes are obtained with the spectral function only (dashed line) and with the full model (solid line) at NNLO. In the inset the results for the full model are compared with the corresponding results obtained with a kinematical cut of W > 2 GeV (solid line with star). Solid circles are the experimental data points of CHORUS
dx (A = 12 C, 56 Fe, 208 Pb) vs x for incoming neutrino beam of energies E = 7 GeV and 25 GeV. The numerical results are obtained with the spectral function only (dash-dotted line: E = 25 GeV) as well as with the full model (solid line: E = 25 GeV, solid line with star: E = 25 GeV and double dash-dotted line: E = 7 GeV) at NNLO and are compared with the phenomenological results of Cloet et al. [8], Bodek-Yang
dx (A = 12 C, 56 F e, 208 P b) vs x are shown for incoming antineutrino beam of energies E = 7 GeV and 25 GeV at NNLO. The numerical results are obtained using the spectral function only (dash-dotted line: E = 25 GeV) by applying a cut of W > 2 GeV and Q 2 > 1 GeV 2 . The results using the full model are obtained with (solid line with star: E = 25 GeV and double dash-dotted line: E = 7 GeV) and without (solid line: E = 25 GeV) a cut of 2 GeV on the CoM energy but keeping Q 2 > 1 GeV 2 . The results in the left and right panels are respectively shown for the nonisoscalar and isoscalar nuclear targets.
TABLE I :
IDifferent parameters used for the numerical calculations for various nuclei. For 12 C we have used modified harmonic oscillator density( * c2 is dimensionless) and for40 Ar,
....................π, ρ, ...
π, ρ, ....
π, ρ, ....
νl(k)
dσ W I C /dx dσ W I CH /dx (top panel) is expected to be close to unity. From the Fig.14, one may notice that the deviation of the ratio from unity is small in dσ W I C /dx dσ W I CH /dx , however, for dσ W I F e /dx dσ W I CH /dx and dσ W I P b /dx dσ W I CH /dx it becomes large which shows the A dependence of the nuclear medium effects specially the contribution of mesons which increases with A at low and intermediate x.
W I C /dx dσ W I CH /dx and obtained by using the full model at E = 25 GeV (solid line) is ≃ 4%(7%)dσ W I
F e /dx
dσ W I
CH /dx
dσ W I
P b /dx
dσ W I
CH /dx
Q 2 > 1 GeV 2 and W > 2 GeV at E = 25 GeV (solid line with star) are compared with the corresponding results obtained at E = 7 GeV (double dash-dotted line). It may be observed that in the region of low and intermediate x the results for CH /dx at E = 7 GeV are smaller in magnitude from the results at E = 25 GeV while with the increase in x the ratiodσ W I
A /dx
dσ W I
dσ W I
A /dx
dσ W I
CH /dx
Simulation (Not isoscalar corrected) Simulation (Isoscalar corrected) MINERvA (Isoscalar corrected) MINERvA (Not isoscalar corrected) Total Wcut (E=7 GeV) Total Nocut (E=25 GeV) SF Wcut (E=25 GeV) Total Wcut (E=25 GeV) Cloet et al. BY13 GENIE0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
0.8
1
1.2
1.4
1.6
(dσ
C
WI
/ dx) / (dσ
CH
WI
/ dx)
ν l
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
0.7
0.8
0.9
1
1.1
1.2
(dσ
Fe
WI
/ dx) / (dσ
CH
WI
/ dx)
Simulation
BY13
GENIE
Cloet et al.
MINERvA Data
Total Wcut (E=7 GeV)
SF Wcut (E=25 Gev)
Total Nocut (E=25 GeV)
Total Wcut (E=25 GeV)
Nonisoscalar
ν l
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
0.7
0.8
0.9
1
1.1
1.2
(dσ
Fe
WI
/ dx) / (dσ
CH
WI
/ dx)
Simulation
MINERvA
SF Wcut (E=25 GeV)
Total Nocut (E=25 GeV)
Total Wcut (E=25 GeV)
Total Wcut (E=7 GeV)
Isoscalar
ν l
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
0.7
0.8
0.9
1
1.1
1.2
1.3
(dσ
Pb
WI
/ dx) / (dσ
CH
WI
/ dx)
Simulation
Cloet et al.
GENIE
BY13
MINERvA Data
SF Wcut (E=25 GeV)
Total Wcut (E=25 GeV)
Total Nocut (E=25 GeV)
Total Wcut (E=7 GeV)
Nonisoscalar
ν l
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
0.7
0.8
0.9
1
1.1
1.2
(dσ
Pb
WI
/ dx) / (dσ
CH
WI
/ dx)
EMR/2016/002285. I.R.S. acknowledges support from Spanish Ministerio de Economia y Competitividad under grant No. FIS2017-85053-C2-1-P, and by Junta de Andalucia. Grant No. FQM-225EMR/2016/002285. I.R.S. acknowledges support from Spanish Ministerio de Economia y Competitividad under grant No. FIS2017-85053-C2-1-P, and by Junta de Andalucia (Grant No. FQM-225).
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. V N Gribov, L N Lipatov, Sov. J. Nucl. Phys. 15Yad. Fiz.V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972), [Yad. Fiz. 15, 781 (1972)];
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. Y L Dokshitzer, Zh. Eksp. Teor. Fiz. 461216Sov. Phys. JETPY. L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977) [Zh. Eksp. Teor. Fiz. 73, 1216 (1977)].
. C Garcia-Recio, J Nieves, E Oset, Nucl. Phys. A. 547473C. Garcia-Recio, J. Nieves and E. Oset, Nucl. Phys. A 547, 473 (1992).
. H De Vries, C W Jager, C. De Vries, Atom. Data Nucl. Data Tabl. 36495H. De Vries, C. W. De Jager and C. De Vries, Atom. Data Nucl. Data Tabl. 36, 495 (1987).
. G Baym, G E Brown, Nucl. Phys. A. 247395G. Baym and G. E. Brown, Nucl. Phys. A 247, 395 (1975).
. A D Martin, R G Roberts, W J Stirling, R S Thorne, Eur. Phys. J. C. 4463A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne, Eur. Phys. J. C 4, 463 (1998).
. K Wijesooriya, P E Reimer, R J Holt, Phys. Rev. C. 7265203K. Wijesooriya, P. E. Reimer and R. J. Holt, Phys. Rev. C 72, 065203 (2005).
. P J Sutton, A D Martin, R G Roberts, W J Stirling, Phys. Rev. D. 452349P. J. Sutton, A. D. Martin, R. G. Roberts and W. J. Stirling, Phys. Rev. D 45, 2349 (1992).
. M Gluck, Z. Phys. C. 53651M. Gluck et al., Z. Phys. C 53, 651 (1992).
. B Z Kopeliovich, J G Morfin, I Schmidt, Prog. Part. Nucl. Phys. 68314B. Z. Kopeliovich, J. G. Morfin and I. Schmidt, Prog. Part. Nucl. Phys. 68, 314 (2013).
. E Oltman, Z. Phys. C. 5351E. Oltman et al., Z. Phys. C 53, 51 (1992).
. J P Berge, Z. Phys. C. 49187J. P. Berge et al., Z. Phys. C 49, 187 (1991).
. M Tzanov, NuTeV CollaborationPhys. Rev. D. 7412008M. Tzanov et al. [NuTeV Collaboration], Phys. Rev. D 74, 012008 (2006).
. G Onengut, CHORUS CollaborationPhys. Lett. B. 63265G. Onengut et al. [CHORUS Collaboration], Phys. Lett. B 632, 65 (2006).
. J G Morfin, private communicationJ. G. Morfin, private communication.
. J G Morfin, J Nieves, J T Sobczyk, Adv. High Energy Phys. 2012934597J. G. Morfin, J. Nieves and J. T. Sobczyk, Adv. High Energy Phys. 2012, 934597 (2012).
|
[] |
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"A Framework for Picture Extraction on Search Engine Improved and Meaningful Result/A-framework-for-picture-extraction-on-search-engine--improved-and-meaningful- result.php",
"A Framework for Picture Extraction on Search Engine Improved and Meaningful Result/A-framework-for-picture-extraction-on-search-engine--improved-and-meaningful- result.php"
] |
[
"Anamika Sharma [email protected] \nComputer Science\nMDU Rohtak\nDAVIM FaridabadHaryanaIndia\n",
"Sarita Sharma \nComputer Science\nMDU Rohtak\nDAVIM FaridabadHaryanaIndia\n"
] |
[
"Computer Science\nMDU Rohtak\nDAVIM FaridabadHaryanaIndia",
"Computer Science\nMDU Rohtak\nDAVIM FaridabadHaryanaIndia"
] |
[
"IJCSI International Journal of Computer Science Issues"
] |
Searching is an important tool of information gathering, if information is in the form of picture than it play a major role to take quick action and easy to memorize. This is a human tendency to retain more picture than text. The complexity and the occurrence of variety of query can give variation in result and provide the humans to learn something new or get confused. This paper presents a development of a framework that will focus on recourse identification for the user so that they can get faster access with accurate & concise results on time and analysis of the change that is evident as the scenario changes from text to picture retrieval. This paper also provides a glimpse how to get accurate picture information in advance and extended technologies searching framework. The new challenges and design techniques of picture retrieval systems are also suggested in this paper.
| null |
[
"https://arxiv.org/pdf/1112.2015v1.pdf"
] | 13,922,413 |
1112.2015
|
e95839ef6cc8a0f49b283c1e749a9754bec2f8c2
|
A Framework for Picture Extraction on Search Engine Improved and Meaningful Result/A-framework-for-picture-extraction-on-search-engine--improved-and-meaningful- result.php
September 2011
Anamika Sharma [email protected]
Computer Science
MDU Rohtak
DAVIM FaridabadHaryanaIndia
Sarita Sharma
Computer Science
MDU Rohtak
DAVIM FaridabadHaryanaIndia
A Framework for Picture Extraction on Search Engine Improved and Meaningful Result/A-framework-for-picture-extraction-on-search-engine--improved-and-meaningful- result.php
IJCSI International Journal of Computer Science Issues
85September 2011http://www.ijcsi.org/articlesPicture RetrievalCBIRStandard QueryImage SearchingOnline study
Searching is an important tool of information gathering, if information is in the form of picture than it play a major role to take quick action and easy to memorize. This is a human tendency to retain more picture than text. The complexity and the occurrence of variety of query can give variation in result and provide the humans to learn something new or get confused. This paper presents a development of a framework that will focus on recourse identification for the user so that they can get faster access with accurate & concise results on time and analysis of the change that is evident as the scenario changes from text to picture retrieval. This paper also provides a glimpse how to get accurate picture information in advance and extended technologies searching framework. The new challenges and design techniques of picture retrieval systems are also suggested in this paper.
Introduction
The world has been dependent on the searching and is going to depend on it in the future. Searching is base of Learning and Learning is a never-ending process. Searching normally does on search engine in the form of text, images, news, maps, web sites etc. Learning will be more effective if it will be in the form of picture. Picture can be of text, graph, and images. Why only image searching? Simple answer is this because learning is more interactive and interesting in comparison with text. Human tendency to retention text is 20% as well as for images retention is 80%. On-line collections of images are growing larger and more common, and tools are needed to efficiently manage, organize, and navigate through them. Image searching is very helpful in every field and of any age group, leading fields are researching, learning and education and its users have been able to search the required image content with the help of the search engines like Goggle. But as the human race is moving towards the future, changes are taking place, even in the nature of the searching and the formats used. Content Base Image searching has shown ways to cope up with this change. It has helped to find the required information in easy way of learning, like images, graphs, pictures etc.
Image Searching
Today a number of search engines are available that give search facility for online database. Categories of these search engine includes web search engine, selection based search engine, Meta search engine, desktop search engine, web portal and vertical market web sites. Search Engines are information retrieval system designed to help to find information stored in online database. There are different ways of image searching. Some are based on simple searching of embedded annotations, metadata, textual context etc, while other complicated methods may include image classifications for searching that are based on color, texture, false color concepts etc. Sometimes for better precision of the resultant images, this type of search requires associating meaningful storage methods or semantic techniques that can be used further for all images of the database.
There are many types of search engines but this study is focused on query based image search engines. Before going to image search, first to find the need of image search. As per [1] there are different varieties of images available.
-Explosive growth of online image/video -5 billion images on web /31 million hours of TV program each year. Each image most likely contains multiple objects, or an object and a background. Therefore, extracting features globally is not appropriate. For this reason, as per [2] start by splitting each image into regions of similarity, using an image segmentation algorithm, with the intuition that each of these regions is a separate object in the image. Image segmentation is a well-studied problem in computer vision.
This segmentation algorithm partitions an image into similar regions using a graph-based approach. Each pixel is a node in the graph with undirected edges connecting its adjacent pixels in the image. Each edge has a weight encoding the similarity of the two connected pixels. The partitioning is done such that two segments are merged only if the dissimilarity between the segments is not as great as the dissimilarity inside either of the segments.
Image Annotations
Manual image labeling, known as manual image annotation, is practically difficult for exponentially increasing image database. As per [3], most of those images are not annotated with semantic descriptors, it might be a challenge for general users to find specific images from the Internet. Image search engines are such systems that are specially designed to help users find their intended images.
Crucial Concept In Image Searching
The biggest issue for image searching system is to incorporate versatile techniques so as to process images of diversified characteristics and categories. Many techniques for processing of low-level cues are distinguished by the characteristics of domain-images. As per [4], The performance of these techniques is challenged by various factors like image resolution, intra-image illumination variations, non homogeneity of intra-region and interregion textures, multiple and occluded objects etc.The major difficulty is a gap between mapping of extracted features and human perceived semantics. The dimensionality of the difficulty becomes adverse because of subjectivity in the visually perceived semantics, making image content description a subjective phenomenon of human perception, characterized by human psychology, emotions, and imaginations. The image retrieval system comprises of multiple inter-dependent tasks performed by various phases. Inter-tuning of all these phases of the retrieval system is inevitable for over all good results.
Query Base Image Searching
QBIS is the primary mechanism for retrieving information from a database and consists of questions presented to the database. In query base image searching gives number of images matches with words present in query. To search for images, a user may provide query terms such as keyword, image file/link, or click on some image, and the system will return images "similar" to the query. The similarity used for search criteria could be meta tags, color distribution in images, region/shape attributes, etc. Most commercial image search engines fall into this category. On the contrary, collection-based search engines index image collections using the keywords annotated by human indexers. As per [5],Different approaches of QBIS includes text base,content base,context base,keybase and semantic base image searching.
Different Approaches of QBIS
-Text base query image searching -Content base query image searching -Context base query image searching -Key based query image searching -Semantic base query image searching
Text-Based Query Image Search Engines
Index images using the words associated with the images. Depending on whether the indexing is done automatically or manually, image search engines adopting this approach may be further classified into two categories: Web image search engine or collection-based search engine. Web image search engines collect images embedded in Web pages from other sites on the Internet, and index them using the text automatically derived from containing Web pages .
Content -Based Query Image Search
Content-based query image searching was initially proposed to overcome the difficulties encountered in keyword-based image search in 1990s. As per [6] Image meta search -search of images based on associated metadata such as keywords, text, etc. Content-based image Retrieval (CBIR) -the application of computer vision to the image search.
Context Base Query Image Searching
In context base query , where searching query processe on the thesaurus meaning of words present in query .For example if the query is to find a "CAR" then ihe images can be toy car for children,brand product of car
Key Based Query Image Serching
A Key base image searching is based on key words include in query required to transform knowledge of a passage into effective query strings in order to retrieve images from keyword based .As per example [7] consider the following passage in a child story book:
"I see three lemurs jumping around and screaming. The snake scares them. However, a sloth is still soundly sleeping. Around the corner, many children are watching a shark swimming swiftly."
The subjects and objects in this passage include "snake", "lemur" and "sloth". During the first execution of the image retrieval process, the query string is formulated as "snake & lemur & sloth".
In the case the response from the image archives indicates that there is no image annotated with all these terms, there is a need for a second query. Presumably, one reasonable strategy is to find images of a place where the "snake", "lemur", and "sloth" could all possibly appear 5.5 Semantic Base Query Image Search The ideal CBIR system from a user perspective would involve what is referred to as semantic serching, where the user makes a request like "find pictures of dogs" or even "find pictures of Abraham Lincoln". This type of openended task is very difficult for computers to performpictures of chihuahuas and Great Danes look very different, and Lincoln may not always be facing the camera or in the same pose. Current CBIR systems therefore generally make use of lower-level features like texture, color, and shape, although some systems take advantage of very common higher-level features like faces. Although search engine is really a general class of programs, the term is often used to specifically describe systems like Google, Alta Vista and Excite that enable users to search for documents on the World Wide Web and USENET newsgroup.
According to [8] in systems extract visual features from images and use them to index images, such as color, texture or shape. Color histogram is one of the most widely used features. It is essentially the statistics of the color of pixels in an image.
Web images have rich metadata such as filename, URL and surrounding text for indexing and searching, different from traditional text-based approach, no manually labeling work is needed in current Web image search engines. The success of text-based image search engines has shown the power of textual information associated with Web images.
Shifting From Text To Image Searching
Searching is based today either on video image or text as per the requirement of the user. The Searching scenario has been totally changed as compared to the previous or only text based searching. In Image or video searching planning and thinking is important, especially for constantly changing and dynamic modules that are industry-linked and practice-oriented. Image searching is the next step in the evolution of video searching. However, Image searching offers a high degree of learning than the text searching. Now video searching, image searching and text searching comes under the umbrella of information searching.
Image Searching Issues
Searching through still images presents an interesting challenge to search engine developers. The way most search engines operate today is by appending text descriptions to the video clips and/or images, so that the searches are based on the text. As per [9] Video and photo searching is something that is still being developed and explored. For example, being able to search for an image of the Taj Mahal would be very challenging for developers, as this would require a query by image content. Basically, this means that the search engine or tool would have to be sophisticated enough to recognize an image of the Taj Mahal and differentiate it from all other possible images. Instead, one of the approaches for searching images is to search for distinctive features of an image. For example, the search tool could look for images with Taj Mahal's features such as a architecture, doors or different types of work done on the walls of Taj Mahal. Another approach would be to search for distinctive colors of the known image. In this case, the search engine could look for the distinctive fading white color of the buildings.
Challenges In Image searching
This would present an interesting challenge for users and developers to establish a good interface for a searching tool of this type.As per [10], Efficiently searching video is even more complex than still images because now that search engines or tools have to be sophisticated enough to handle movement, lighting, and different camera angles. The searching of a video or film would have to be more sophisticated than to simply search a video frame by frame for the desired result. Users may also want to search for specific scenes in video or for zooming in and out. As we all know, user can use any kind of query (text, content, context, keywords, semantic) for image IJCSI
8.Implementation Technique Of Image Searching
Because of number of difficulties, we need to develop a framework that will focus on recourse identification for the user so that they can get faster access with accurate & concise results on time. Development of Framework for user to find the recourses on the bases of type of query given. Now the Proposed framework of implementation technique is to put the query on web browser. This query will translate through as 1.User input a text Query to the browser. 2.The Query Translator extracts the query from HTTP request, then translate the query into the input format for each text based image search engine.
3. The page crawler sends the query to the each search engine and collects the HTML file containing the URL of image retrieved by the search engines, then parses the HTML file to obtain the individual URL.
4. The result collator merges the result and shows the first page of retrieval image.
5. Using the URLs the image crawler retrieves the image from the Internet to construct the initial image set.
6. Feature extractor computes the image content feature vector for all images in the initial image.
7.Based on user's new request, cluster the image set using the feature vector and K-mean algorithms.
8.Based on the feedback images selected by the user, compute the distance of feature vector between the feedback image and the image in the initial image set. Rerank images according to the distance and display the reranked image to the user.
9.Conclusion
This paper has presented an up coming wave of image searching and force for shifting from Text searching to Image searching with acceptance of new technology and directions. This includes the embedded software tools that help in online searching of content include text, image and video based on query. The related researches are also presented in this paper. With the advent of the Internet, information from all over the world is available to the people. Since there is so much information out there, people require an automated method to search through all of it. The search engines available today provide users with this ability, but primarily for text based searching. As the Internet moves further and further into being able to support multimedia, users and corporations will need to take advantage of new searching techniques. Some companies have even begun to hire "Web Specialists" to assist them in becoming aware of what searching facilities.
The idea of being able to search images, video, or audio based on the content is possible since the Internet is an electronic medium, but something that is still in development. As more and more users begin to understand the concepts behind searching these media, the need to do so will rise. However, today, most of the work being done to allow users to search this media is still in the developmental phases. The approaches described above are simply ideas and theories for ways in which this type of searching could be possible. As per [11] The idea of generating a storyboard from a video, or searching a video based on a moving sketch, or searching audio based on
10.Future Scope
Image searching provides some unique and interesting challenges for developers to come up with some sort of automated way of searching through video and/or still images. One method that is currently in development is to generate a storyboard out of a video. Storyboards typically consist "of a series of sketches showing each shot in each scene as it will be filmed, and possibly some indication of the action-taking place e.g., an arrow showing the direction of movement. A 'shot' is defined as a section of action during which the camera films continuously without interruption." As per [12] A storyboard is typically used by writers and directors while making a movie to plan the action of the shot, to review camera angles, and provide a summary of the film. Essentially, the proposed approach would be to take a finished video product and generate the storyboard based on the finish video. "In order to reverseengineer a storyboard from the finished video sequence, it is necessary to identify three properties of each shot in the sequence. These are: (1) the start point of the shot, (2) the end point of the shot and (3) the picture that best represents the shot as a whole." As per [12] Once the storyboard has been generated, it will be easier to search for video sequences, especially in large video libraries.
Another approach to video searching is the search actual video using video cues. At Columbia University, a system called VideoQ is being developed that does this. The theory behind this system is to have the user actually draw out an animated scene as the query. "In an animated sketch, motion and temporal duration is the key attributes assigned to each object in the sketch in addition to the usual attributes such as shape, color, and texture. Using the visual palette, we sketch out a scene by drawing a collection of video objects." As per [13] The VideoQ system will then search its video library for videos that match the animated sketch. The VideoQ system is intended to be on the Internet and use various Java applets to allow the user to create these animated sketches.
Ms. Anamika Sharma did her Master in computer Application from Gurukul University Haridwar in 1998 ,M.Tech From Allahabad Agriculture University,M.Phil from Vinayka Mission University Tamil Nadu and pursuing Ph.D from Singhania University Rajasthan. She is having about 13 years of teaching experience of postgraduate courses. She has guided more than hundred students in their project and published number of papers in national/International journals. She is a member of Computer Society of India .Her main Area of research include Computer Graphics, Data mining, Software testing and Quality assurance and object oriented Analysis and Design. At Present she is working in DAV Institute of Management, Faridabad as Associate Professor in Computer Science Deptt.
Ms. Sarita Sharma did her Master in Computer Application from IGNOU, New Delhi, India, She did her M.phil (Computer Science) from Ch. Devi Lal University,Sirsa , India and is pusuing Ph.D from Singhania University,Rajasthan, India. She is presently working as Associate Professor in Deptt. Of Computer Science, DAV Institute of Management , India. She has guided more than 90 students in their Projects and has published a number of papers in National/International journals. She is a member of Computer Society of India. Her areas of interest include Software Engineering, Data Mining, Relational Databases, Computer Languages etc. She has about 15 years of teaching experience. with current employment; association with any official journals or conferences
Fig. 1 .
1Framework of Picture Retrieval
IJCSI International Journal of Computer Science Issues, Vol. 8, Issue 5, September 2011 ISSN (Online): 1694-0814 www.IJCSI.org 3 company,image of any electonic circuit etc.It depend in what context user wants the result.
International Journal of Computer Science Issues, Vol.8, Issue 5, September 2011
ISSN (Online): 1694-0814
www.IJCSI.org
4
searching, so the need arises for the structured query
content, so that it can be give the resultant images
according to the type of query. Following problems
can arises when the user give the query in text box of
search engine:
1. No rules for writing of query for images.
2. Difficulties to identifying important words
3. Content base query systems have not any standard
format of query image
4. Cannot mention in what context user want what
type of image.
5. No Typing limitation but it takes only few
important words.
6. Cost of spending lots of time scrolling through
image searching result.
7. Speed of accessing the web.
8. Unwanted links / irrelevant data on searching.
9. Non-compatible sites of web (product searching
sites) only.
10. Difficulties to narrowing down the semantic gap.
Recent Advances And Open Issues Of Digital Image/Viedo Search. Shih-Fu Chang, Digital Vedio And Multimedia Lab. Chang, Shih-Fu, June -2007," Recent Advances And Open Issues Of Digital Image/Viedo Search" ,Digital Vedio And Multimedia Lab.
Efficient Graph-Based Image Segmentation. F Pedro, Daniel P Felzenswalb, Huttenlocher, Artificial Intelligence Lab & Computer Science. Pedro F Felzenswalb,Daniel P . Huttenlocher,"Efficient Graph-Based Image Segmentation "Artificial Intelligence Lab & Computer Science.
Annotation Search : Image Auto-Annotation By Search. X J Wang, L Zhang, F Jing, W Y Ma, CVPR. Wang, X. J., Zhang, L., Jing, F., And Ma, W. Y. . In CVPR, 2006." Annotation Search : Image Auto-Annotation By Search"
Limitation And Challenges: Image/Video Search & Retrieval. Nida Aslam, Infanullah, Doi:10.4156/Jdcta.Vol3.Issuel.AslamNida Aslam , Infanullah " Limitation And Challenges: Image/Video Search & Retrieval "Doi:10.4156/Jdcta.Vol3.Issuel.Aslam
Content Based Image Retrival : A Comparision Between Query By Example And Image Browsing Map Approaches " The Chinese Univerity Of Hong Kong. Christopher C Yang, Christopher C .Yang " Content Based Image Retrival : A Comparision Between Query By Example And Image Browsing Map Approaches " The Chinese Univerity Of Hong Kong,7Jan 2004.
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Context-Based Image Retrieval:A Case Study In Background Image Access For Multimedia Presentations. Sheng-Hao Hung, Pai-Hsun Chen, Iadis International Conference WWW/Internet. Vila RealPortugalSheng-Hao Hung, Pai-Hsun Chen , Context-Based Image Retrieval:A Case Study In Background Image Access For Multimedia Presentations "Iadis International Conference WWW/Internet 2007, Vila Real : Portugal (2007)"
Fuhui Long, Dr. Hongjiang Zhang and Prof. David Dagan Feng, fundamentals of Content-based image retrieval. Dr, Dr. Fuhui Long, Dr. Hongjiang Zhang and Prof. David Dagan Feng, fundamentals of Content-based image retrieval.
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Finding Images/Video In Large Archives"Columbia's Content-Based Visual Query Project, D-Lib Magazine. Shih-Fu Chang, John R Smith, 1082-9873Shih-Fu Chang, John R. Smith," Finding Images/Video In Large Archives"Columbia's Content-Based Visual Query Project, D-Lib Magazine, February 1997 ,ISSN 1082-9873.
Categorization And Searchng Of Color Image Using Mean Algo. Prakash Panday, Uday Pratap, Sanjeev Jain, BhopalN College Of TechnologyPrakash Panday,Uday Pratap ,Sanjeev Jain ." Categorization And Searchng Of Color Image Using Mean Algo",L N College Of Technology ,Bhopal.
Finding The Cut Of The Wrong Trousers: Fast Video Search Using Automatic Storyboard Generation. Macer, Peter J Peter, Nouhman Thomas, John F Chalabi, Meech, Communications Of The ACM. Macer, Peter, Peter J. Thomas, Nouhman Chalabi, John F. Meech "Finding The Cut Of The Wrong Trousers: Fast Video Search Using Automatic Storyboard Generation" Communications Of The ACM, 1996.
Content -Based Image Retrival:A Comparision Between Query By Example And Image Browsing Map Approaches. C Christopher, Yang, The Chinese University Of. Hong Kong,7 thChristopher C Yang."Content -Based Image Retrival:A Comparision Between Query By Example And Image Browsing Map Approaches",The Chinese University Of Hong Kong,7 th Jan 2004.
|
[] |
[
"Quantum computing at the quantum advantage threshold: a down-to-business review",
"Quantum computing at the quantum advantage threshold: a down-to-business review"
] |
[
"A K Fedorov \nSchaffhausen Institute of Technology\n8200SchaffhausenSwitzerland\n\nRussian Quantum Center\n143025SkolkovoMoscowRussia\n\nNational University of Science and Technology \"MISIS\"\n119049MoscowRussia\n",
"N Gisin \nSchaffhausen Institute of Technology\nGenevaSwitzerland\n\nGroup of Applied Physics\nUniversity of Geneva\n1211 Geneva 4Switzerland\n",
"S M Beloussov \nSchaffhausen Institute of Technology\n8200SchaffhausenSwitzerland\n",
"A I Lvovsky \nRussian Quantum Center\n143025SkolkovoMoscowRussia\n\nDepartment of Physics\nUniversity of Oxford\nOX1 3PGOxfordUK\n"
] |
[
"Schaffhausen Institute of Technology\n8200SchaffhausenSwitzerland",
"Russian Quantum Center\n143025SkolkovoMoscowRussia",
"National University of Science and Technology \"MISIS\"\n119049MoscowRussia",
"Schaffhausen Institute of Technology\nGenevaSwitzerland",
"Group of Applied Physics\nUniversity of Geneva\n1211 Geneva 4Switzerland",
"Schaffhausen Institute of Technology\n8200SchaffhausenSwitzerland",
"Russian Quantum Center\n143025SkolkovoMoscowRussia",
"Department of Physics\nUniversity of Oxford\nOX1 3PGOxfordUK"
] |
[] |
It is expected that quantum computers would enable solving various problems that are beyond the capabilities of the most powerful current supercomputers, which are based on classical technologies. In the last three decades, advances in quantum computing stimulated significant interest in this field from industry, investors, media, executives, and general public. However, the understanding of this technology, its current capabilities and its potential impact in these communities is still lacking. Closing this gap requires a complete picture of how to assess quantum computing devices' performance and estimate their potential, a task made harder by the variety of quantum computing models and physical platforms. Here we review the state of the art in quantum computing, promising computational models and the most developed physical platforms. We also discuss potential applications, the requirements posed by these applications and technological pathways towards addressing these requirements. Finally, we summarize and analyze the arguments for the quantum computing market's further exponential growth. The review is written in a simple language without equations, and should be accessible to readers with no advanced background in mathematics and physics.CONTENTS* [email protected] † [email protected] VIII. Quantum computing can be based on various physical platforms 19 A. Solid-state quantum computing 19 B. Atoms, ions, and molecules for quantum computing 21 C. Optical quantum computing 25 Hardware quantum processor (qubit) Qubit register Hardware for quantum control Qubit control, Gate control, Optimization of parameters, Qubit readout, initialization, calibration and shutdown Classical hardware for controlling hardware for quantum control Software for classical hardware and for designing Designing quantum computer chips and design optimization Algorithms for quantum computers Digital quantum simulation, search, factorization, linear equations Applications for quantum computers Molecular Dynamics, Quantum Chemistry, Drug Discovery, Optimization, Machine LearningFigure 2. Stack of quantum computing technology.
| null |
[
"https://arxiv.org/pdf/2203.17181v1.pdf"
] | 247,839,279 |
2203.17181
|
63316bb5b88d362051c048e864c3ae5d97a26d30
|
Quantum computing at the quantum advantage threshold: a down-to-business review
A K Fedorov
Schaffhausen Institute of Technology
8200SchaffhausenSwitzerland
Russian Quantum Center
143025SkolkovoMoscowRussia
National University of Science and Technology "MISIS"
119049MoscowRussia
N Gisin
Schaffhausen Institute of Technology
GenevaSwitzerland
Group of Applied Physics
University of Geneva
1211 Geneva 4Switzerland
S M Beloussov
Schaffhausen Institute of Technology
8200SchaffhausenSwitzerland
A I Lvovsky
Russian Quantum Center
143025SkolkovoMoscowRussia
Department of Physics
University of Oxford
OX1 3PGOxfordUK
Quantum computing at the quantum advantage threshold: a down-to-business review
(Dated: April 1, 2022)CONTENTS
It is expected that quantum computers would enable solving various problems that are beyond the capabilities of the most powerful current supercomputers, which are based on classical technologies. In the last three decades, advances in quantum computing stimulated significant interest in this field from industry, investors, media, executives, and general public. However, the understanding of this technology, its current capabilities and its potential impact in these communities is still lacking. Closing this gap requires a complete picture of how to assess quantum computing devices' performance and estimate their potential, a task made harder by the variety of quantum computing models and physical platforms. Here we review the state of the art in quantum computing, promising computational models and the most developed physical platforms. We also discuss potential applications, the requirements posed by these applications and technological pathways towards addressing these requirements. Finally, we summarize and analyze the arguments for the quantum computing market's further exponential growth. The review is written in a simple language without equations, and should be accessible to readers with no advanced background in mathematics and physics.CONTENTS* [email protected] † [email protected] VIII. Quantum computing can be based on various physical platforms 19 A. Solid-state quantum computing 19 B. Atoms, ions, and molecules for quantum computing 21 C. Optical quantum computing 25 Hardware quantum processor (qubit) Qubit register Hardware for quantum control Qubit control, Gate control, Optimization of parameters, Qubit readout, initialization, calibration and shutdown Classical hardware for controlling hardware for quantum control Software for classical hardware and for designing Designing quantum computer chips and design optimization Algorithms for quantum computers Digital quantum simulation, search, factorization, linear equations Applications for quantum computers Molecular Dynamics, Quantum Chemistry, Drug Discovery, Optimization, Machine LearningFigure 2. Stack of quantum computing technology.
The progress in understanding our world at the smallest scales has culminated in a physical theory that is both arXiv:2203.17181v1 [quant-ph] 31 Mar 2022 Box 1. Basic quantum concepts.
States. The primary notion of quantum physics is that of the state of a given quantum object. The state is often defined by a certain value of an experimentally observable quantity. For example, a statement that the coordinate of a certain particle is x = 5 meters constitutes a valid description of a state (denoted as |x = 5 m ). Note that, according to the uncertainty principle, the momentum of the particle in that state is fundamentally uncertain.
A variety of quantum systems are suitable for technological applications. A paradigmatic example is the spin of a particle such as the electron. Spin can be visualized as rotation of a particle around its own axis -akin to the diurnal spinning of the Earth. The quantum state is defined by the axis of rotation in some reference frame.
Superposition. A primary postulate of quantum mechanics is that one can perform mathematical operations with states: one can multiply states by numbers and/or add them together; the result of this operation is also a valid state (known as a superposition state). In the language of mathematics, this means that quantum states form a vector space, or, more precisely, a Hilbert space. For example, the sum of the states |↑ and |↓ (spin axis directed along the positive and negative z directions, respectively) gives the state |→ with the spin axis along the positive x direction, whereas their difference, |↑ − |↓ corresponds to the state |← with the axis along the negative x direction.
Qubit. Conversely, any spin state can be represented as a weighted sum of two spin states corresponding to an arbitrary pair of opposite directions. Mathematically, this means that the Hilbert space describing the electron spin is two-dimensional. This is an example of a qubit -the primary unit of quantum information: for example, the spin-up state |↑ can be assigned a Boolean value of 0 and the spin-down state |↓ a logical 1. There exist many other physical incarnations of the qubit aside from the spin, many of which can be used for quantum information processing.
Measurements. In classical physics, if we are given a classical object, we can precisely measure all properties of its state. For example, if we have a projectile, we can measure its position, velocity, acceleration, etc., at any moment in time, without disturbing its motion. This is not the case with quantum systems: if we have e.g. an electron, we cannot measure the direction of its spin. The best we can do is to choose a certain direction and perform an experiment (known as the Stern-Gerlach measurement) that determines whether the spin is in the state oriented along or opposite to that direction. If the electron has been prepared in one of the corresponding states (we will denote this pair of states, known as the measurement basis, as | and | ), the measurement result will be certain. For an arbitrary state |ψ , on the other hand, the measurement will yield one of these two outcomes with some probability. In order to predict these probabilities for a known state |ψ , we need to write it as a weighted sum of the basis elements: |ψ = α | + β | . The measurement probabilities are then pr = |α| 2 and pr = |β| 2 (we assume that |α| 2 + |β| 2 = 1).
This probabilistic, uncertain nature of measurement is a defining feature of quantum physics. Remarkably, a measurement will transform the state of a quantum system into whatever state that measurement has detected. For example, if we measure the |→ state in the {|↑ , |↓ } basis and happen to detect |↑ , all subsequent measurements of that spin in the same basis will yield the same result: |↑ .
Entanglement. Superposition states of multiple quantum objects, treated as a single system, are known as entangled. Consider, for example, two electrons in the state |↑↑ + |↓↓ . This expression means that, whenever each of the electrons are measured in the {|↑ , |↓ } basis, they will be found in the same state.
A simple calculation can show that the same is true for any measurement basis: whenever the first electron is detected in a particular state, the second one is certain to be in the same state [1]. This property is remarkable. Suppose, for example, that one of the electrons is with a fictitious observer Alice on Venus and the other one with Bob on Venus. By choosing a basis and measuring her electron, Alice can remotely prepare Bob's electron in the same state she has detected. For example, she can choose to measure in the {|→ , |← } basis, in which case, dependent on the result, Bob's photon will become either |→ or |← . This remote state preparation [2], which occurs instantly and without interaction, and once called "spooky action at a distance" by Einstein, has puzzled generations of physicists. We do not delve into this topic as it is tangential to our review; however an important fact that must be mentioned is that it is impossible to use quantum entanglement for superluminal communication of classical information.
Decoherence. The above argument also implies that if Alice happens to lose her electron, the state of Bob's electron becomes completely unknown. This gives rise to the phenomenon of decoherence, which is a primary hurdle in quantum computation technology. In the process of quantum computation, qubits may undergo spurious interaction with the environment which will result in their entanglement with its quantum state. Since we have poor control of the environment, we cannot keep track of its state; essentially, it is lost as far a the quantum qubit register is concerned. As a result, the state of the register loses its superposition nature (decoheres) and becomes useless.
The main quantitative benchmark describing decoherence is its characteristic time. The more controlled operations can be performed on a quantum register before it decoheres, the better.
the most controversial and, at the same time, most extensively tested of all physical theories: quantum physics. Originally devised to explain the black-body radiation problem, one of the outstanding unsolved problems in physics at the end of the nineteenth century, quantum physics extended its reach in the subsequent decades to cover a great variety of microscopic systems. The theory's power [3] to predict collective phenomena in ensembles of quantum particles led to the development of many widely used practical devices, the most impactful of which are transistors [4] and lasers [5]. These inventions give rise to the development of semiconductor computing and microprocessors, optical communications and the Internet based on first wired and then mobile technologies -in other words, made the world the way we know it today. This is known as the first quantum revolution.
A characteristic feature of the technologies of the first quantum revolution is that they do not require handling individual quantum objects, such as atoms, photons (light quanta), or electrons. Rather, they rely on their collective behavior in large (macroscopic) ensembles. For example, a gain medium of a gas laser consists of multiple atoms, but in order to design a laser one does need to precisely control the quantum physics of each of these atoms. It is enough to study the quantum physics of an "average" atom, and then calculate how this physics translates into the properties of a collective of these atoms contained inside the laser tube, such as the dependence of the gain on the length of the tube, discharge voltage, etc. Such large systems are relatively easy to handle, both theoretically and experimentally, because their reasonable interactions with an environment does not degrade their useful properties.
The next technological frontier is to learn how to handle a collective of individual microscopic quantum objects in such a way that each such object plays a unique, clearly defined role as a part of a complex entangled state of the collective (see Box 1). This will open up a whole new horizon of technological opportunities, such as new problems amenable to computational analysis, perfectly secure communications, and sensors with unprecedented precision. This is known as the second quantum revolution or quantum technologies [6]. In this paper, we concentrate on a particular aspect of quantum technologies: quantum computing.
Any computer processes information encoded in a string of bits, each of which can take a value of 0 or 1. A quantum computer operates with quantum bits, or qubits (Box 1) and can process massive entangled superpositions of their states at once as a single quantum system. That is, a string of N qubits can encompass an entangled superposition of 2 N classical N -bit stings. In this sense, we say that a quantum computer may possess There is caveat, however: entanglement is both a blessing and a curse. Since quantum computers process information in a superposition state, the computation results will also be superposed with each other. However, a human user is a macroscopic, classical entity and cannot handle such an entangled state. We need a specific classical answer to a specific classical problem, which is of interest to us at a given moment of time (Box 2). As a result, quantum computers offer a significant advantage for a specific class of problems.
While this class is potentially large, its boundaries are not precisely known at this time. However, many of its elements -problems with expected quantum advantage (albeit not always exponential) -are now identified. This includes simulation of complex systems (fuels, drugs, biosystems, materials, etc.), optimization, data processing, and machine learning (see Sec. IX). A problem of particular practical importance, for which the advantage is exponential, is factorization (decomposing into prime factors) of large numbers with application in cryptanalysis (Sec. IX B 3).
The development of a large-scale quantum computer capable of practical applications is a major challenge. Any interaction between the qubit register in a quantum computer and the environment will result in decoherence -uncontrolled entanglement of the two, bringing about the loss of the superposition nature of the register, which is fatal for the quantum information contained therein (Box 1). We are thus facing two antagonistic requirements: we must enable the qubits to be controllably affected by other (microscopic) qubits yet completely unaffected by the (macroscopic) environment. This is the main reason why we have not yet conclusively demonstrated quantum computational advantage in application to practical tasks, although the idea of quantum computing has been around for about 40 years and its primary working principles have been elaborated more than 25 years ago.
Current quantum computing devices operate with on the order of a hundred qubits with approximately 20-30 gate operations and are not capable of error correction. They are sometimes referred to as noisy intermediatescale quantum (NISQ) [7]. NISQ devices have demonstrated advantage (supremacy) of quantum computing with respect to classical, albeit on tasks that have limited practical value. Applications of NISQ devices to practical problems, like optimization and chemistry, have also been attempted, however no quantum advantage has been achieved.
Sadly, any progress in quantum computing, however minor, is a likely subject for mass media coverage -often Box 2. Quantum phone book.
Some intuition regarding the quantum advantage (and limitations thereof) can be gained by the following, somewhat allegorical, example. Consider a telephone book of a city with, say, a million inhabitants encoded as an ASCII text file. Each entry includes e.g. 10 bytes containing the name of a line subscriber, and 7 bytes with a phone number -so the entire book occupies 17 megabytes. This offers a massive advantage both in the storage capacity and processing of the information. Suppose, for example, that we need to add 1 to one of the digits or add an area code to all numbers. With a classical phonebook, this operation would need to be performed individually on each entry. But in the quantum case, only a single operation on the entangled system would suffice.
A complication arises, however, when we try to use the phone book according to its purpose -read out a number associated with a specific name. Extraction of a single entry from this massive entangled superposition is not an easy task -rendering such a quantum phonebook largely useless in a household. This illustrates that quantum computation is useful only for a limited class of problems, many of which are quite distant from those encountered by lay users.
in exaggerated and hyperbolized fashion, which might create an illusion that quantum computer technology is far beyond the point where it actually is. The true state of the art, in our opinion, in that the technology is at the threshold of quantum advantage, but not yet significantly beyond it.
The quantum computing challenge is being approached by many research teams and many different ways, which gave rise to a large variety of devices. This includes both the physical platforms (superconducting circuits, trapped ions, neutral atoms, light, etc.; see Sec. VIII) and computing models (for example, whether qubits are addressed one-by-one or all at once; see Sec. V). This diversity complicates defining a metric for their comparison. While various metrics have been proposed, they do not give a complete and straightforward picture of how different approaches are related to each other or how close quantum computing devices are to solving real-world problems.
This lack of universal metrics, diversity of platforms, models and purposes, overselling in literature and media as well as general mysterious nature of quantum physics make quantum computation a challenging environment. The purpose of the present review is to demystify quantum computing not only for the broad scientific community, but also for decision makers, investors, media, industrial executives, and members of the public. We attempt to cover the full stack of the quantum computing technology ranging from hardware for making individual qubits to software and potential applications (Fig. 2). We can roughly identify three layers composing this stack.
• Quantum computational model (type) defines the general approach to organizing the encoding and processing information in a quantum computer (Sec. V).
• The platform is the specific physical system (e.g. superconducting circuits or trapped ions), whose quantum properties are used for calculation (Sec. VIII).
• Within each platform, different architecturessuites of hardware and software solutions to implement programming, control, input and output,are possible. Architecture details are highly technical are generally outside the scope of our work. Additionally, in Sec. IX we analyze applications of quantum computing and discuss how far in the future we can expect to see quantum advantage in the context of this applications.
We also summarize a set of parameters that could be used to estimate the readiness and potential of a specific quantum system, and formulate a two-pronged framework for their analysis. On the one hand, we discuss technical benchmarks, such as the system size, length of the operation sequence that can be implemented, and the degree of programmability. One the other hand, we discuss user-oriented criteria such as cost, speed, and range of tasks that can be solved. Based upon this analysis, we summarize the arguments predicting exponential growth for the quantum computing market (see Box 3).
II. HAS CLASSICAL COMPUTING APPROACHED ITS FUNDAMENTAL LIMITS?
Intel co-founder G. Moore published a paper [8] in 1965 (reprinted in 2006 [9]), where he made a number of important observations. Two are particularly relevant for our review. The first one reads: "Silicon is likely to remain the basic material, although others will be of use in specific applications. For example, gallium arsenide will be important in integrated microwave functions. But silicon will predominate at lower frequencies because of the technology which has already evolved around it and its oxide, and because it is an abundant and relatively inexpensive starting material". The second one, today known as Moore's law, is as follows: "The complexity for minimum component costs has increased at a rate of roughly a factor of two per year".
Let us examine both observations in more details. Modern developments in computing technologies were a result of efforts to increase the quality of silicon-based transistors (basic elements of computing devices) and decrease their costs by improving manufacturing. The central element of this progress was the "planar process", a method of fabricating transistors on the surface of a flat silicon wafer. These transistors are connected via a metal layer to create a complete circuit. Such "integrated circuits", a consequence of the first quantum revolution, have for the last sixty years been the dominant venue for classical digital computation [4]. This second observation can be reformulated as follows: thanks to miniaturization of transistors, the number of transistors on a chip doubles every 24 months (the original prediction that the period is 12 months has been corrected) [9]. Remarkably, this trend has continued with the same zeal ever since it has been first observed. As an illustration, IBM has recently announced the production of a chip based on the 2 nm technology [10]. In a scientific lab, transistors as small as 1 nm have been developed in 2016 [11]. For comparison, the nearest-neighbor distance in the silicon crystal lattice is 0.235 nm.
It is evident from the above figures that the transistor size is approaching fundamental physical limits. Other fast (exponential) scaling laws for classical computing technology, such as the growth of performance per watt (so-called Dennard scaling) or clock speeds, are already no longer valid [12,13]. This motivates many analysts to conclude that "Moore's law is nearing its end" [13] and became a common argument to motivate the future of quantum computation.
While we too advocate the future of quantum computing, we cannot agree with the above thesis. In addition to the existing reserve of at least a factor of three in the linear size, a further room for developing integrated circuits consists in 3D layering. Circuits with 96 layers have been demonstrated by Toshiba in 2018 2 . Combining these figures, at least three orders of magnitude increase in the transistor density is possible with implies at least three more decades of Moore's law. New architectures, optimization, and application-driven specialization of processors open up even more opportunities for the development of classical computers. Quantum computing is therefore motivated not by the upcoming end of such developments, but rather by fundamental limitations of classical computers as Turing machines in solving certain classes of problems, as discussed above in Introduction. Quantum computers should be seen not as competitors to classical machines, but rather as a supplementary class of devices aimed to tackle a distinct class of problems.
The continued transistor miniaturization implies that laws of microscopic world -i.e., quantum laws -play an increasingly significant role in the operation of even classical integrated circuits. For example, classical bit values 0 or 1 correspond to a transistor switched "on" or "off", respectively. However, when the transistor is of microscopic size, the effect of quantum tunneling 3 results in the electric current flowing through the transistor even when it is in the "off" state. This results in excessive heating of integrated circuits becoming more and more of an issue in computer design. Hence, even if we do not make directed efforts to harnessing the power of quantum phenomena, these phenomena will still enter computational technology, but as a hindrance rather that an opportunity.
III. A HISTORIC EXCURSION
The roots of quantum computing can be traced back Szilard and von Neumann, who connected the thermodynamic concept of entropy (a measure of disorder introduced by Boltzmann in 1877) with information theory developed by Nyquist and Hartley in the 1920s, as well as Shannon in the 1940s. Based on these results, Landauer [26,27] in 1970 found a fundamental lower bound imposed on any irreversible operation on a bit. According to the Landauer limit, such an operation consumes energy in the amount of kT log 2, where k is the Boltzmann constant 1.38 × 10 −23 Joules per Kelvin and T is the absolute temperature. These results have been further analyzed by Bennett in 1973Bennett in -1982, who proposed the concept of reversible computation that does not involve erasing information, and hence overcomes the Landauer limit. Subsequently, Bennett discussed his finding with Feynman and asked whether they are affected by the quantum nature of the world. Inspired by this question, Feynman in 1982Feynman in -1986 published a series of papers showing not only that the answer to the above question is negative, but also that quantum properties of matter can be used to enhance power of computational devices. These papers [30,31] are universally considered as the origin of quantum computation. Note that a frequently quoted work by Benioff from 1980 [32] contains an analysis of the possible implementation of the (classical) Turing machine using quantum systems, but it does not use the feature of quantum entanglement.
Feynman's results have influenced the work of Deutsch, who had at that time been working on similar ideas in the context of the Everett (multiworld) interpretation of quantum physics [15] (paper [15] was completed in 1978, but published only in 1985; see also Ref. [33]). Synthesizing Feynman's results with his own thoughts, Deutsch came up with a mathematical study of capabilities of quantum computing machines titled "Quantum theory, the Church-Turing principle and the universal quantum computer", published in 1985 [34]. Lloyd in 1996 has rigorously proven Feynman's conjecture that a digital quantum computer can efficiently simulate an arbitrary quantum system [35]. These developments led to the start of research explicitly devoted to quantum computing in the mid-1990s.
In parallel, research on salient concepts of quantum information and computation took place in the Soviet Union. In 1970-1980 Holevo published in a series of papers [36][37][38] establishing an upper bound to the amount of information that can be known about a quantum state (its "accessible information", which is now known as the Holevo bound). In 1975, Poplavskiy [39] observed that quantum systems cannot efficiently simulated on a classical computer. Finally, in 1980 Manin published a book "Computable and non-computable" [40], in which he described the concept of quantum logic, i.e. logical operations on qubits. After becoming known worldwide in 1990s, these results significantly influenced the development of the field.
IV. WHY IS QUANTUM COMPUTING POWERFUL?
It is not yet strictly proven that quantum computers can be more powerful than classical counterparts. This is only a conjecture akin to many fundamental believes in computer science, such as P and NP complexity classes are not equivalent. However, there exists strong reasons to believe that this conjecture is valid as summarized by Preskill [7].
A few dozens quantum algorithms [41,42] have been developed in the last 30 years that are significantly (sometimes exponentially) faster than the best known classical algorithms. Refs. [41,43,44] provide reviews of quantum algorithms, a more detailed guide can be found on a website "quantum algorithm zoo" [45] and some important examples are listed in Box 4.
Of particular relevance for the current state of the art are algorithms that provide samples from known probability distributions, which in certain cases cannot be efficiently obtained by classical means [46][47][48][49]. These algorithms are the basis for current demonstrations of quantum advantage (see Sec. IX below) [41,42], which Preskill defines as "computational tasks performable by quantum devices, where one could argue persuasively that no existing (or easily foreseeable) classical device could perform the same task, disregarding whether the task is useful in any other respect" [7].
One may object that, even though existing classical algorithms are inferior to their quantum counterparts, perhaps in the future classical algorithms can be invented that close this gap. This can be countered by the observation that quantum matter, particularly the quantum computer, require exponentially growing amount of time and resources to simulate classically [7] (so-called "curse of dimensionality"). For example, for a modestsize quantum computer with 50 qubits, it would take 16 PByte of memory and hours on a top supercomputer in order to implement a single operation (see Table I). Simulations to this effect have recently been presented by IBM (USA) [50,51] and Alibaba (China) [52].
An important challenge, which was extensively discussed at the early stage of quantum computing, is related to validation of the results that quantum devices produce [53,54]. Interestingly, issues of this nature are not unique to quantum computers, but relevant to any new generation of computing devices. They can be ad-Box 4. Early quantum algorithms.
• In 1992, Deutsch and Jozsa presented an algorithm [14], which finds whether a "black-box" function of string of bits is balanced or constant. A constant function returns the same result for any input, while a balanced function returns 0 for exactly half for all possible inputs and 1 for the other half. The simple example of 2-bit functions was considered by Deutsch in Ref. [15], whereas the n-bit function case was analyzed by Deutsch and Jozsa [14]. Some improvements were proposed by Cleve, Ekert, Macchiavello, and Mosca in 1998 [16]. Although currently lacking a practical use, it is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical counterpart.
• The Fourier transform occurs in many different versions and is widely applicable in various tasks, such as signal processing, in which it allows transitions between representation of signals in time and frequency domains. The quantum Fourier transform (QFT) -the discrete Fourier transform applied to the vector of amplitudes of a quantum state -was proposed by Coppersmith in 1994 (published as Ref. [17] in 2002). QFT can be considered as a transformation between two bases of quantum states (the computational (Z) basis and the Fourier basis). It is a subroutine of many quantum algorithms, most notably Shor's factoring algorithm and quantum phase estimation.
• In 1994, Simon invented [18,19] an algorithm for verifying whether an unknown black box function is one-to-one (maps exactly one unique output for every input) or two-to-one (maps exactly two inputs to every unique output). Simon's algorithm uses the quantum Fourier transform.
• In 1994, Shor proposed [20,21] a polynomial-time quantum computer algorithm for integer factorization, which is based on Simon's algorithm. Shor's quantum algorithm can be used in the cryptanalysis of currently widely deployed public-key cryptography algorithms, such as RSA and Diffie-Hellman (see Sec. IX B 3). This is one of the first quantum algorithms that solves a practically relevant problem.
• In 1995, Kitaev proposed a quantum phase estimation algorithm, which allows estimating the phase (or eigenvalue) of an eigenvector of a unitary operator. This is an important part for many quantum algorithms including the current version of Shor's algorithm and quantum algorithm for solving linear systems of equations (see Secs. IX C 1 and IX C 3).
• In 1996, Grover considered [22,23] a black box function, which yields 0 for all bit strings expect one, for which the output is 1. Grover's algorithm helps finding this unique bit string. It takes an input a superposition of all possible bit strings and yields a superposition in which the string of interest has a high probability amplitude. This algorithm gives quadratic speedup, however it has been proven [24] that no higher speedup is possible for this problem.
• In 1997, Bernstein and Vazirani [25] considered the class of functions, which is known to be the dot product between its argument and a secret bit string. The task of the algorithm is to find this bit sting.
dressed in a variety of ways. First, the inherent nature of many problems, which are subject to quantum advantage, is that their solutions can be checked on a classical computer in polynomial time. Second, many problems, while being generally challenging, have certain instances or configurations that are amenable to classical computation. Third, one can test if the quantum computer provides a correct solution to problems of smaller sizes, which can be simulated numerically. After these multiple checks, calculations in classically inaccessible regimes can be considered reliable.
V. QUANTUM COMPUTING CAN BE BASED ON VARIOUS MODELS
During the four decades of quantum computing history, a great variety of quantum computational models have been proposed. We review these models in this section.
First, we classify quantum computing devices into universal and non-universal. In classical digital computers, universality means an ability perform an arbitrary se-quence of operations on a bit string. The quantum counterpart of this definition is the ability to perform an arbitrary transformation of the quantum state of a set of qubits. In other words, the universal quantum computer can be understood as extension of the Turing machine into the quantum domain, as formalized by Deutsch in 1985 [34].
In contrast, non-universal quantum computers aim to solve a specific problem or a specific class of problems. There are two important subclasses of non-universal quantum computers. The first one is analog quantum simulators [53,55]: devices that simulate a process in a complex quantum system, e.g. solid matter, by another quantum system with well-known and controllable properties, e.g. an ensemble of cold atoms trapped in an optical field. The second class is special-purpose quantum computers to solve a specific restricted class of abstract mathematical problems, e.g., quantum annealing devices, which implement discrete optimization. These two classes overlap as we detail below.
We consider the following quantum computing models.
• Universal: 4. variational [44,63], also known as hybrid.
• Non-universal:
1. quantum simulators;
2. special-purpose.
A. Gate-based (circuit-based, digital) quantum computing
The most intuitive and popular approach to universal quantum computing is generalize classical digital computing to the quantum case. This is the basis of the gate-based model of quantum computing [57,58,64].
In the digital quantum computing model, we start by preparing our qubits (quantum register ) in a certain initial state: normally, all logical zeros. These qubits are then subjected to a sequence of logical gates, known as the quantum circuit (or quantum network ), and subsequently measured yielding the computation result in the form of classical string of bits.
Akin to classical digital computing, quantum gates are simple operations on qubits. One can prove that any arbitrary transformation of multiqubit states can be decomposed into a sequence of 1-and 2-qubit gates. Although the length of such a sequence generally scales exponentially with the number of qubits, this scaling is polynomial, or even linear, for many practical situations. Finding such situations and composing the corresponding efficient gate sequences constitutes the art of quantum algorithmics [64].
Generally, a 1-or 2-qubit gate is parametrized by several real numbers, such as the angle of rotation around a particular axis. For practical purposes, however, it would be convenient if all gates used in an algorithm belonged to a fixed small set. Fortunately, there do exist gate sets (known as universal sets, see Box II), that can be used to efficiently represent any sequence of arbitrary 1-or 2qubit gates.
Specifically, a quantum circuit of m arbitrary 1-or 2qubit gates can be approximated to ε error (in so-called operator norm) by a quantum circuit of O(m log c (m/ε)) gates from a universal gate set, where c is a constant approximately equal to 2 (this is known as a consequence of the Solovay-Kitaev theorem [65]). This result is remarkable because a general gate parameter set is an arbitrary point in a multidimensional continuum, whereas sequences of universal set gates are represented by a set of discrete points in this continuum. To construct an intuitive analogy, one can think of decimal fractions, in which any real number can be arbitrarily well approximated by a short sequence of discrete numbers.
While there are many ways to define a universal set, the most common one consists of H, S, Tand CNOT . Three elements of this set, H, S and CNOT , form the socalled Clifford group, which enables formation of highlyentangled states: the Hadamard gate makes superposition states, the CNOT gate creates entanglement between qubits, and the Sgate introduces complex amplitudes. However, according to the Gottesman-Knill theorem [64,66,67], a circuit consisting of Clifford gates can be efficiently simulated classically. To achieve quantum advantage, the Tgate must be added to this group 4 .
The choice of gates can be influenced by what is easier to implement on the physical platform at hand. Such platform-specific gates are known as native gates. For example, some platforms use the controlled phase gate instead of CNOT . The CNOT gate can then be obtained by applying the Hgate to the target qubit before and after the controlled phase gate.
During last decades, basic working principles of gatebased quantum computing have been demonstrated using various physical platforms with quantum registers reaching 50-100 qubits. However, noise and decoherence prevent sustainable sequences of more than 20-30 logical operations. Hence the primary task with today's gate-based NISQ computers is the implementation of error correction allowing long-lived logical qubits (Sec. VII). In the meanwhile, gate-based quantum devices are mostly useful for variational quantum computation (see Sec. V D). Pauli-Z (Z, NOT ) Z Rotation of the spin axis by π radians (180 • ) around the z axis.
Leaves the |↑ and |↓ unchanged, but transforms |→ and |← into each other.
Pauli-X (X) X Rotation of the spin axis by π radians (180 • ) around the x axis.
Leaves the states |→ and |← unchanged, but transforms |↑ and |↓ into each other.
Pauli-Y (Y) Y Rotation of the spin axis by π radians (180 • ) around the y axis.
B. One-way quantum computing
Two-qubit gates require controlled interaction of two specific qubits while keeping other qubits completely free of interaction with both their counterparts and the environment. This is a primary challenge of the gate-based model. The one-way (cluster-state) model addresses this challenge by preparing a complex state, in which all qubits are entangled with each other in a known way, in advance of the computation. The computation consists in measuring these qubits sequentially and applying single-qubit operations to other qubits depending on the result of the measurement (so-called feedforward processing). This process takes advantage of the remote state preparation effect (Box 1), i.e. when one part of an entangled state is measured, other part change. Any gate-based algorithm can be reformulated in the quantum one-way computation framework as demonstrated by Raussendorf and Briegel in 2001 [61].
One-way quantum computing is particularly relevant in the context of the optical platform because of lacking tools for deterministic implementation of two-qubit gates [68] and quantum memory for light [69]. Cluster states and operation therewith have been demonstrated in free-space optics up to 12 qubits [70]. Integrated-optics implementation of optical one-way quantum computing is being pursued commercially (see Sec. VIII C).
C. Adiabatic quantum computing
Both gate-based and one-way quantum computing models rely on manipulating the quantum state of a multiqubit register. This manipulation results in a complex entangled state that constitutes the result of the calculation. Adiabatic quantum computing also works with a multiqubit system as the carrier of quantum information, but manipulates it according to a very difficult paradigm.
Rather than implementing the algorithm as a sequence of operations on individual qubits (or pairs thereof), it puts the system into the physical conditions (Hamiltonian) such that the state, which is desired as the output of the calculation, has the lowest energy among all possible states of the system (i.e. is the ground state of the Hamiltonian). It might appear that, in order to device such conditions, one would need to know the desired state. However, remarkably, this is not the case. The Hamiltonian can be calculated efficiently using a classical computer just from the corresponding circuit in the gate-based model [59,60,71] (therefore making this approach universal quantum computing [59]). Alternatively, the Hamiltonian can be calculated for the computational problem at hand, e.g. factorization, directly, bypassing the intermediate step of circuit design [72].
Once the required Hamiltonian is calculated and its physical implementation is devised, a question arises is how to bring the quantum system to its ground state. In principle, this can be achieved by cooling it to very low temperatures close to absolute zero. However, these temperatures may not not achievable in practice. Therefore, one instead takes advantage of the so-called adiabatic theorem of quantum mechanics. This theorem states that, if a system is prepared in the ground state of a Hamiltonian, and this Hamiltonian evolves slowly enough, the system will always remain in its instantaneous ground state. We therefore can start from some simple Hamiltonian, in which the ground state is easy to reach. Then we gradually (adiabatically) evolve the conditions towards the Hamiltonian that encodes the problem while keeping the system in the ground state. At the end of the evolution, the ground state is measured yielding the result of the computation.
A subtle question is how slowly the Hamiltonian must be evolved to prevent the system from leaving the ground state. According to the adiabatic theorem, the "speed limit" is inversely proportional to the energy gap between the ground and second lowest energy state. This gap decreases with the number of qubits in the system, but fortunately not at an exponential rate [60] -hence making adiabatic quantum computation feasible.
An important advantage of the adiabatic model is that it exhibits inherent robustness against certain types of quantum errors [59].
Universal adiabatic quantum processors have not yet been implemented. The quantum annealer manufactured by D-Wave Systems 5 can be seen as the first step toward adiabatic quantum computing; however, it does not enable encoding Hamiltonian corresponding to an arbitrary computational problem and is therefore not universal. Furthermore, it is a subject of research whether the adiabatic theorem is satisfied in this machine (see Box 6). At the first stage of each iteration, the circuit is run multiple times and its output is measured. At the second stage, based on the measurement results, the energy value associated with objective Hamiltonian is evaluated classically. A classical optimization algorithm then provides feedback to the parameters θ of the quantum circuit in order to minimize the energy.
D. Variational (hybrid) quantum computing
This model combines the features of the gate-based and adiabatic models. Similarly to the adiabatic model, variational quantum computing utilizes the observation that the final state of a quantum computation can be seen as the ground state of a certain Hamiltonian efficiently calculable on a classical computer. On the other hand, like the gate-based model, the variational quantum computer does use a quantum circuit. However, the gates in the circuit are not fixed, but described by continuous parameters (for example, the angle by which a qubit is rotated around a certain axis). At each iteration of the algorithm, the circuit output is measured and the energy value corresponding to the Hamiltonian of interest is calculated. Small adjustments to the parameters of the gates are then calculated using a classical optimization algorithm with the aim to produce the state with a lower energy (Fig. 3). Iterations continue until the energy of the output state no longer reduces.
An important advantage of variational algorithms is that the optimization cost function -the energymay not only be computed from a gate circuit, but can also represent the actual energy of a real physical object, such as a molecule. Then the quantum variational optimization will result in the output state representing the ground state of the electrons in this molecule with the corresponding energy. This idea gives rise to an algorithm known as the variational quantum eigensolver (VQE) [63,74,75], which was developed in 2014. Moreover, the scope of VQE can be extended to arbitrary cost functions beyond energy, leading to the quantum approximate optimization algorithm (QAOA), proposed in 2014 to solve combinatorial optimization problems [76,77].
Historically, these two algorithms have been invented outside of the context of universal quantum computation. Formal universality proofs have been presented by Box 5. Quadratic unconstrained binary optimization (QUBO) problems.
A particularly important class of optimization problems is quadratic unconstrained binary optimization (QUBO). It consists in finding the bit string (σi = ±1) that minimizes the objective function H = i Riσi − 1 2 i,j Jijσiσj, where Ri is the given "bias vector" and Jij = Jji is a given "coupling matrix". This objective function emerges in solid state physics under the name of the Ising model, hence QUBO is sometimes referred to as Ising problem.
An equivalent formulation of QUBO is the maximum cut (MaxCut) problem. It considers a graph with each edge (connecting vertices i and j) associated with a real number Jij. The problem consists in dividing the set of graph vertices into two subsets such that the total of (Jij)'s connecting the vertices in these two subsets is maximized.
The applications of QUBO range from basic science to problems of everyday practical nature. An example is portfolio optimization. Suppose a number of discrete assets are available for purchase. The expected investment return (Ri) is known for each asset. Also, the correlation Jij between the expected returns is known for each pair of assets. This correlation is a measure of risk associated with buying these two assets; the risk is minimized if no correlation is present, which corresponds to the highest diversification of the portfolio. The value σi of 1 or −1 corresponds to buying on not buying the asset. The task is to choose the subset of assets with the desired balance between the expected return and the risk. This corresponds to the point on the efficient frontier in the framework of the 1952 Markowitz model [73], for which a Nobel prize in Economics was awarded in 1990. This task of selecting the optimal asset set is exponentially hard because the number of possible bit strings σ1 . . . σN grows exponentially with the number N of available assets.
An important particular case of QUBO is the maximum independent set problem. It consists in finding the largest set of vertices on a graph that are not connected by edges. Mathematically, this problem corresponds to QUBO with all Jij being equal to either 1 (if an edge between a given pair of nodes is present) or 0 (if it is absent).
While a number of classical heuristic solutions for the Ising/QUBO/MaxCut/maximum independent set problem have been proposed, none of them is efficient for large problem sizes.
the groups of Lloyd and Biamonte in 2018-2021 [78][79][80]. Notably, these proofs relied on an unproved assumption that the optimization algorithm is capable of converging to the lowest energy state [81]. Closing this gap is an open problem.
The variational model is particularly relevant at present as current NISQ devices have limited recourses (number of qubits, fidelity, number of operations, etc.) and furthermore the parameters of each gate cannot be precisely controlled. The variational model appears to be more forgiving to these shortcomings because it does not require precise knowledge and control of the absolute values of each circuit parameters, but only needs small relative adjustments thereof. A shortcoming, on the other hand, is the need to have much more complete information about the output quantum state in order to calculate the energy value. This means at each iteration the circuit must be run multiple times and the output state measurements must be performed in different bases. For example, the estimation of the energy of a relatively simple molecule Fe 2 S 2 would require as many as 10 13 measurements [82]. Assuming (optimistically) that each quantum circuit run takes 10 ns, the single iteration would require about 24 hours.
Variational quantum algorithms have been demonstrated in the context of optimization [83][84][85] (for example, QAOA was implemented experimentally by IonQ [USA] and Google [USA]), machine learning [86], quantum chemistry [82], linear algerba [87], and quantum simulation [88]. A detailed review of variational quantum algorithms can be found in Bharti et al. [74] and Cerezo et al. [44].
E. Quantum simulators
Complete and precise theoretical descriptions of complex quantum systems, such as solid state, which involves interaction of multiple microscopic quantum objects, is beyond the reach of current science and technology. There do exist simplified models that capture their salient properties. But even in the framework of these models, the curse of dimensionality makes the analysis exponentially hard for classical computers. As discussed above, this is one of main motivations behind quantum computing.
Each of the above described universal quantum computing models can in principle be used to simulate arbitrary complex quantum systems [35]. For example, a digital superconducting quantum computer was used to simulate the interaction of two fermions, whose states are encoded in four qubits [89]. This rather simple simulation required as many as 300 single-qubit and two-qubit gates. Another example is the aforementioned variational quantum eigensolver, which can find the lowest energy state of a quantum system of interest. In an experimental realization based on trapped ions, interaction within a system of multiple high-energy particles governed by the so-called Schwinger model was simulated [90,91]. The ground state of the system was found as well as the phase transition as the function of the particle mass.
However, the simulation of quantum systems also permits an entirely different approach [53,55]: quantum machines with known and controllable properties imitating the quantum system of interest. This is known as analog quantum simulation. The advantage of this approach in comparison to digital quantum computing is that the simulation can be done at a much larger scale at the expense of loosening precise control over individual elements and lack of fault tolerance [54]. In this context, an important benchmark of a quantum simulator is its programmability, which is the degree of control that can be imposed on its elementary quantum units and their interaction.
Progress over the last two decades has produced more than 300 quantum simulators in operation worldwide, using a wide variety of experimental platforms [55,[92][93][94][95]. They range from highly optimized special-purpose nonprogrammable simulators to flexible programmable devices. Physical platforms include solid-state superconducting circuits, quantum dot arrays, nitrogen-vacancy centers, atomic and molecular systems, such as Rydberg atoms and trapped ions, interacting photons, and others. We describe these platforms in detail in Sec. VIII.
F. Special-purpose quantum machines
An important example of special-purpose problems solvable by quantum machines is discrete optimization, which arises in various industries ranging from logistics to finance, such as QUBO (see Box 5). Traditionally, such problems have been solved by a family of classical algorithm known as simulated annealing, in which the set of variables to be optimized is treated as a physical system with probability of different configurations given by the thermal distribution associated with some temperature. As the "temperature" is decreased, the probability of the optimal configuration increases to one. This process is reminiscent to annealing in metallurgy, giving rise to its name. This term is now also used for a variety of quantum and analog methods for combinatorial optimization, even though they may not involve any thermal distribution or temperature variations.
A prominent example of a quantum annealer is the superconducting device for solving QUBO problems manufactured by D-Wave Systems (see Box 6 and Fig. 4). Another solution that is frequently measurement in the context of hardware annealing is the optical coherent Ising machine (see Box 7); however, current realizations of this approach do not feature entanglement between computational units and hence do not exhibit quantum advantage.
Beyond combinatorial optimization, an important class of special-purpose quantum machines is the boson sampler [46]. This is a network of intersecting optical waveguides with n input and n output channels, where n is large [ Fig. 5(a)]. Single photons are injected into m < n input channels and subsequently detected at the output. As the photons propagate through the network, they can jump between waveguides at their intersections or experience interference with each other if they arrive at an intersection together. As a result, the output state of the photon paths feature complex entanglement. Hence the probability, with which the photons will emerge in a particular combination of output channels, is conjectured to be exponentially difficult to calculate (as it involves calculating the so-called permanents of the matrix describing the network) [46]. The output photon detection produces a sample of such a distribution, thereby offering a solution to a classically hard problem. Boson sampling is therefore of interest as a setting, in which quantum superiority can be demonstrated. First attempts to realized boson sampling were implemented in 2013 with up to 4 photons in 6 modes [113][114][115].
The boson sampling scheme in its original form is difficult to scale up because no on-demand sources of highquality single photon exist yet. An important breakthrough is associated with the concept of Gaussian boson sampling [ Fig. 5(b)]. In this scheme, the states of light injected into the optical network are the so-called squeezed vacuum -a class of states of light, which, like single photons, exhibit quantum features, but can be produced on-demand with relatively little effort [116]. The idea of Gaussian boson sampling enabled experimental realization on a scale, at which quantum advantage is present (see Sec. IX A for further detail).
Boson sampling was initially introduced purely as a problem for demonstrating quantum advantage [46] (see Sec. IX A), abstract from any practical utility. However, it was later discovered to have applications in chemistry [119] (calculating molecular vibronic spectra) and mathematics [120] (graph similarity).
The final example of special-purpose quantum machines is the previously mentioned neutral Rydberg atom simulator. This system is remarkable in that it occupies several positions in our classification. On the one hand, it can be used to solve the maximum independent set problem [85,[121][122][123][124] (see Box 5), which corresponds to finding the minimal energy configuration of an ensemble of Rydberg atoms. On the other hand, it is a programmable quantum simulator capable of probing exotic phase transitions in condensed matter, which is a classic quantum simulation problem. Furthermore, this platform can also be used as a fully digital quantum computer. We discuss this system in detail in Sec. VIII B.
VI. HOW CAN THESE MODELS BE COMPARED?
As seen from the previous discussions, various quantum computing models dramatically differ not only in their physical and technical implementation, but also in their fundamental computational paradigms. Hence it is difficult to define universal performance evaluation criteria. A first attempt at this task was made by Di-Vinchenzo in a classic paper of 2000 [125], who formulated five qualitative requirements that a physical setup must satisfy in order to support gate-based quantum computing. These requirements have since evolved in adaptation to emerging quantum computational models and a number of quantitative benchmark have been proposed, which we summarize below. The specific values Good examples of special-purpose quantum machines are the superconducting quantum annealers produced by D-Wave Systems. These devices feature remarkably many qubits (5,000 in the latest model D-Wave Advantage), greatly exceeding that in other existing quantum processors. The D-Wave machines can be seen as a step towards the adiabatic model (Sec. V C): they gradually vary ("anneal") the physical conditions into which the qubits are placed in order to drive them to the ground state of a particular Hamiltonian.
However, the D-wave annealer is not yet a universal adiabatic quantum computer. The Hamiltonian it is capable of implementing is not arbitrary (as required by the adiabatic model), but limited to the Ising (QUBO) Hamiltonians -in other words, it can look for bit strings that minimize the QUBO objective function (Box 5). But even within the framework of the QUBO problem, it is not able to realize any arbitrary coupling matrix Jij. This is because every qubit is connected to only a small number of other qubits [7 in D-Wave 2000Q and 15 in D-Wave Advantage, see Fig. 4(a)]. This leads to a major overhead when an all-to-all coupled Ising problem needs to be solved. Physical qubits are then grouped into clusters such that the qubits within each cluster are forced to the same logical value and share their outside connections. The entire cluster then plays the role of a single "logical qubit" for the purpose of the calculation; the number of such logical qubits is limited to a few dozen [96]. Moreover, there exist no confidence whether the D-Wave annealer properly fulfils the adiabatic theorem [97]. In practice, this means that the calculation output bit string may not be the true minimum of the QUBO objective function.
The question of quantum advantage of D-Wave machines has been widely discussed in the literature. Evidence of quantum effects in the annealing process was claimed in 2014 [98] by D-Wave One (108 qubits), but these claims have been disputed by other groups presenting classical models that efficiently simulate the annealer's behavior [97]. Subsequent D-Wave annealer models featured significantly higher qubit numbers, and have been used to attempt demonstration of quantum advantage on specially tailored problems [99]. Existing studies of this matter compared the performance of the D-Wave annealer 2X with 2000 qubits and classical algorithms in application to various problems and produced controversial results [100][101][102][103][104]. One of the latest results by Google and D-Wave in 2021 is a claim of quantum advantage in the physically relevant problem of simulating geometrically frustrated magnets [105]. Thus, at this moment there is no universally accepted conclusive evidence of quantum advantage of the D-Wave machine.
In spite the lack of such evidence, the D-Wave annealer is being extensively studied in application to various problems of practical significance (see Sec. IX C 2). However, as discussed above a major issue is embedding the problem in the native structure of the annealer [96,98,101], which limits applicability of the device to problems of very small sizes only. Box 7. Coherent Ising Machine for annealing.
The coherent Ising machine (CIM) is another example of a hardware annealer. CIMs store the information about the optimization variables in optical pulses and use optoelectronic feedback to implement the couplings between them. In contrast to D-Wave processors, CIMs have no restrictions on connectivity between variables. In 2016 CIMs with the capacity of 2048 variables [106,107], and in 2021 with as many as 100512 variables, have been demonstrated [108]. It was shown that CIMs significantly outperform D-Wave processors in dealing with dense QUBO functions [109], although this claim was disputed by the D-Wave team [110].
However, because the coupling of the optical pulses in the CIM is implemented via measurements and optoelectronic feedforward, no entanglement between the pulses is possible. This means that any speedup observed cannot be of quantum nature. This was confirmed by classical simulations [111], which ran on graphic processors and achieved solution speed and quality that is comparable or superior to that of CIM. This shortcoming can be addressed, and quantum entanglement can be achieved by coupling the optical information units in the CIM by direct interference. This has proven to be challenging, but is an important vector for the future development of this technology [112].
for some of these parameters for various physical platforms are listed in Table III. 1. User-oriented criteria. (h) Ability to realize error suppression / correction.
While the first group in the above list is rather intuitive, the second one requires explanation, which we provide below. Each of these requirements gives rise to a quantitative benchmark, which can be used to assess and compare different quantum computational platforms.
The size of problem at hand dictates the number of elementary quantum units required for the quantum computing device to solve it. However, the notion of such a number is in itself ambiguous. For example, a quantum algorithm in the gate model operates with idealized quantum logical variables (logical qubits) that are assumed to be perfectly isolated from the environment and able to store information for infinitely long time. The practical physical qubits are, however, subject to a variety of imperfections and errors, in particular, decoherence. To address these imperfections, redundant encoding is used: multiple physical units encode a single logical qubit com-pensating each other's errors by means of quantum error correction (see Sec. VII). The overhead ratio of physical and logical qubits depends on another major benchmark: the error rate ε, which is the ratio between the duration of a single operation (gate) and the characteristic decoherence time. For example, factoring a 2048-bit RSA key requires approximately 6000 logical qubits, but 20 million physical qubits, assuming that the error rate of ε = 10 −3 (i.e., a single error occurs once in every 1000 gate operations) [126].
When the error rate in a given platform exceeds a certain threshold, error correction cannot be implemented at all, no matter how high the the overhead. Moreover, error correction techniques have to day been developed mainly for the gate-based model, and much less so for other models. Thus the ability of a quantum computer to implement error correction depends not only on the error rate, but also on the quantum computational model as well as other factors. That being said, for some models, such as analogue quantum simulators, error correction is not a requirement at all.
In addition to the effect of passive interactions with the environment, the quality of the state of the quantum register is influenced by imperfect control, i.e., state preparation and measurement (SPAM) errors and gate inaccuracies. The quantitative measure of these imperfection is the fidelity, i.e., how close the prepared quantum state (gate) is to theoretically desired. There exists an arsenal of experimental tools for estimating the fidelity. In the gate-based model, this criterion is further specialized in terms of single-and two-qubit gate fidelities.
Even if the qubits are of perfect quality, a quantum computing device may not enable their arbitrary pairwise interaction. This capability, known as the connectivity, is another important requirement for a quantum computing platform. For example, in the trapped-ion platform, in which all ions are situated in the same trap and gates are implemented through their mechanical interaction, all-to-all connectivity is possible. In contrast, connectivity is a challenge in the superconducting model, in which a two-qubit gate is realized by a physical junction between these qubits. For example, the publicly accessible IBM superconducting quantum computer has only two to three connections per qubit, whereas the state-of-theart Google Sycamore unit constitutes a 2D square grid of qubits [128], so each qubit is connected to four others. A 2D architecture is also used in Rydberg atom quantum simulators [129,130], although the connectivity issue in this setting can be addressed by physically repositioning the atoms with respect to each other. In the absence of all-to-all connectivity, a two-qubit gate between arbitrary qubits can be realized by swapping (applying the SWAP gate, see Box II) the quantum state through a chain of neighboring qubits. The price to pay is the decoherence resulting from the need to include additional operations in the algorithm.
A related important performance criterion is the ability to implement operations in parallel. For some quantum computational settings, such as the atomic simulator, the parallel interaction of multiple pairs of units is inherent in their nature and essential for proper operation. In other models, such as gate-based, the parallelism is optional, but desired for faster implementation of quantum algorithms. The achievability of parallelism depends on the specific physical platform and is often complementary to connectivity. For example, it is relatively straightforward in the superconducting model, but more challenging in the trapped-ion model.
The time required to solve a computational problem is directly proportional to the duration of an individual quantum gate operation. The aforementioned factorization of 2048-bit RSA key would require 8 hours with the average gate time of 10 µs [126]. As seen in Table III, the gate time strongly depends on the physical platform of a quantum computer.
Many practical quantum devices operate with units that naturally have more than two independent quan-tum states, i.e., are multidimensional (for example, multiple energy levels in atoms). Using such multidimensional units, known as qudits, instead of qubits to encode quantum information helps to reduce the number of gates required for the realization of quantum algorithms [131]. Thus, the dimensionality of the elementary information unit is an important parameter of a quantum computing device. Quantum operations with qudits have been demonstrated with various physical systems. Most progress has been achieved with the superconducting, trapped-ion, and optical platforms, on which qudit processors have been reported (Rigetti Computing [132], AQT with collaborators [133], and the Peking University team with collaborators [134], respectively).
As we see from the above discussion, assessing a quantum computing platform involves relatively large number of complementary and sometimes conflicting criteria [135]. One is therefore tempted to simplify the task and introduce a single-number metric to express a quantum computer's power. One such metric is the quantum volume introduced by IBM in 2017-2019 [136,137] for gate-based quantum computers. Quantum volume is defined as 2 AQ , where AQ -number of algorithmic qubits -is the maximum size of a "square" quantum circuit that can be successfully implemented with this platform (Box 8). Historically, the number of algorithmic qubits as a figure of merit for a quantum computational platform has been introduced after quantum volume, namely by IonQ in 2020 6 . To date, the largest quantum volume of 2048 (AQ = 11) has been demonstrated by Quantinuum (previously Honeywell) in 2021 in a trapped ion machine 7 .
VII. QUANTUM COMPUTERS CAN DEAL WITH ERRORS
In the early stage of quantum computing development, the accumulation of error caused by environmental noise (decoherence) was widely used to argue that it is infeasible to build a large-scale quantum computer [138,139] As a result of this controversy, the development of the field in last few decades followed several vectors. First, experimental efforts were made towards investigating what quantum computers are capable of in the presence of decoherence, giving rise to the current NISQ technologies. Second, concepts of quantum gate-based computing devices with digital error correction, which use redundant qubits, have been developed. Third, it was found that certain structures of quantum matter are robust to decoherence, leading to topologically-protected quantum computation. We discuss each of these below. Table III. Performance benchmarks of primary quantum computing platforms, represented by the record values achieved to date. The data for the neutral-atom, trapped-ion and superconducting platforms are taken from Ref. [127], which also contains the corresponding bibliography references.
Quantum units (qubits) is the number of algorithmic qubits with the quantum depth d(n) = 1/nε being the number of operations before a single error has occurred. To understand this expression consider a quantum computer with N = 10 6 qubits and the error rate ε = 10 −3 . To determine the quantum volume, we need to find the optimal n, for which min[n, d(n)] is maximized. For example, if we choose n = 1 we have d(n) equals 1000, so min[n, d(n)] = 1. On the other hand, choosing n = 10 6 will results in d(n) = 10 −3 , so min[n, d(n)] = 10 −3 . Both these cases are suboptimal: in the first case one can perform many operations with only one qubit, whereas in the second case, if we take too many qubits, there is a high probability that at least one of them will decohere before even a single operation takes place. The optimal value of n in our situation is about 30, in which case d(n) ≈ 30 as well, meaning that the circuit is of square shape. So the quantum volume is 2 30 ≈ 10 9 .
A. Noisy intermediate-scale quantum devices
As mentioned above, NISQ devices have 50-100 physical qubits and do not implement any tools for error correction. In spite of these limitations, these machines have been used to implement basic quantum algorithms [41,74] and demonstrate quantum advantage [42]. In some cases, such as analog quantum simulation, decoherence forms a natural part of finding the solution, because simulated objects themselves experience decoherence. This is reason for successful application of such devices to simulate phases and transition between them in condensed matter [140]. A further promising model of quantum computation within the NISQ framework appears to be the variational model [44,74] because it uses relatively short operation sequences.
In the absence of digital quantum error correction, there exist techniques for reducing the effect of errors at the level of individual qubits [141][142][143][144][145][146][147], such as error mitigation, error suppression, and fidelity amplification. As an example, computation accuracy can be enhanced through extrapolation of results from a collection of experiments with varying noise with no additional hardware modifications [142]. The potential of this family of approaches has not yet been systematically studied. More generally, it is not known at this time whether quantum devices without error correction can provide quantum advantage for practically relevant problems.
B. Devices with error correction: Fault-tolerant quantum computing
The existence of computational errors is not limited to quantum domain. Classical digital computers use redundant bits to nondestructively detect and correct errors. Such a direct approach is, however, not applicable in quantum technology because any measurement results in the loss of coherence and entanglement. Moreover, nocloning theorem of quantum physics [148] precludes creating an independent and identical copy of an arbitrary unknown quantum state. Thus, quantum error correction should be tackled in a subtler way. The idea is still to use many (imperfect) physical qubits to encode one (perfect) logical qubit and perform measurements in order to detect errors. However, these measurements must be specially constructed to reveal no information about the values of the qubits, but only indicate whether and at which position the error has occurred.
Shor's seminal error-correcting code uses nine physical qubits to encode one logical qubit (see Box 9 and Ref. [149]). Soon after Shor's code, new error-correction codes were developed that lowered the number of physical qubits in a logical qubit to five while maintaining the same level of protection [150][151][152][153]. The theory quickly evolved, producing more and more sophisticated classes of error correction codes [154].
One of the primary results of this theoretical research is the quantum threshold theorem (or quantum faulttolerance theorem) [155][156][157][158]. It states that a quantum computer with a physical error rate below a certain threshold can, through application of quantum error correction schemes, suppress the logical error rate to arbitrarily low levels (see Fig. 6). Roughly speaking, we need to be "correcting errors faster than they are created" 8 . The optimal error correction method in a given quantum computational setting depends not only on the value of the threshold, but also on its architecture, in particular, the qubit connectivity.
The error threshold of Shor's code is relatively low. For example, in the widely studied square 2D lattice model, in which each qubit is connected to its four nearest neighbours, the error threshold for Shor's code is as low as 2 × 10 −5 [159]. This is because the implementation of Shor's code requires operations on pairs of qubits that are not necessarily nearest neighbors. These operations must be implemented through a chain of connected qubits, resulting in a high likelihood of additional errors. A further shortcoming of Shor's code is the difficulty to implement operations on logical qubits as such operations simultaneously involve multiple distant physical qubits, therefore requiring complex connectivity.
These issues are remedied through the so-called surface code [160], which involves only nearest-neighbor qubit interactions for error correction and, furthermore, streamlines the logical gate operations to some extent. In this case the error threshold increases to 0.01 [160][161][162]. This property makes the surface code and its descendants mainstream approaches to error correction. Practical realization of the surface code was expected to be the primary feature of the Bristlecone quantum chip with 72 qubits announced in 2018 by Google 9 . However, no experimental results with this chip have been reported yet.
The number of physical qubits needed to implement a logical qubit rapidly increases with the error rate and tends to infinity when the threshold is approached. For example, for the error rate of 10 −3 in the surface code setting this number is about 3000 [160].
In practice, the construction of error correction codes for a specific platform must consider its peculiarities, such as the dominating decoherence channels and temporal characteristics of the interaction with the environment. The latter property determines whether decoherence is Markovian or non-Markovian [163], i.e. to which extend the effect of environment on the system at a given moment of time is correlated with past and future moments.
In the last decade, error correcting codes have been reported in experiments with linear optics [164], trapped ions [165][166][167][168][169], and superconducting circuits [170][171][172]. Extension of the coherence time of a logical qubit using error correction was demonstrated with superconducting qubits [171] in 2016. In 2021 the Google team reached a breakthrough [172] with arbitrary single-qubit error correction on their Sycamore superconducting device. When increasing the number of physical qubits in a logical qubit from 5 to 21, the error rate per round of error correction reduced exponentially by more than 100 times: from 10 −2 to 10 −4 . The experimental run lasted for 50 such rounds, each of which had a duration of about 1 µs. This performance, however, was achieved with 1D array of qubits, whereas for the 2D surface code only basic operations have realized. Proof-of-principle realization of the surface code on 17 qubits has been demonstrated by Zhao et al. [173] using the Zuchongzhi 2.1 superconducting quantum processor.
A drastically different approach to error correction was proposed by Gottesman, Kitaev and Preskill (GKP) in 2001 [174]. The idea is to encode qubit in a quantum system of infinite dimension, such as a harmonic oscillator, which can be realized as an optical or superconducting platform. This infinite dimensionality provides the redundancy required for error correction, so a role of a logical qubit can be played by a single physical oscillator. This approach has been experimentally demonstrated in 2020 with a superconducting quantum circuit [175]. The coherence lifetime has been increased by a factor of 2-3. The application of the GKP approach has been also proposed in the optical quantum computing setting [176]. Examples of errors are bit flips (e.g., |0 → |1 or α|0 + β|1 → β|0 + α|1 ) or phase flips (α|0 + β|1 → α|0 − β|1 ; there is no classical equivalent to the phase flip error). Specializing to spin qubits, this is equivalent to the rotation of the qubit around the z or x axes, respectively. Suppose we wish to implement a code capable of correcting a bit flip. If we have only one qubit, the only thing we can do is to measure this qubit, which will destroy the superposition state, but not tell us whether the error has occurred.
Shor in 1995 proposed instead to encode a single logical qubit into three physical qubits [149], i.e. for the logical qubit superposition α|0 + β|1 to use an entangled state of the form α| ↑↑↑ + β| ↓↓↓ , which is easy to prepare using standard gates. To detect errors, one needs to measure the product of the z-components of any pair of these three qubits -for certainty, let it be the first and second qubit. Quantum mechanics allows measuring such products without measuring each qubit individually and, moreover, such a measurement reveals no information about the qubit contents. Indeed, a healthy Shor state will have both qubits in the same state: either |↑↑ (corresponding to z1 = z2 = 1) or |↓↓ (then z1 = z2 = −1). In both cases, the measurement will yield the same result: z1z2 = 1 -and hence it will neither tell us anything about the values of α and β nor destroy the superposition.
Suppose now that the first of the physical qubits in Shor's code underwent a bit flip, so the state became α|100 +β|011 . Measuring the three qubit pairs, we find z1z2 = −1, z1z3 = −1 and z2z3 = 1, which will tell us unequivocally that the error occurred in the first qubit. We can correct for this error by applying a Pauli X gate.
The procedure of detecting phase flip errors is similar to the above, but the error can be detected by measuring the spins' x components instead of z. Moreover, both single phase flips and single bit flips can be detected by a more complicated encoding of a single logical qubit in nine physical qubits.
C. Topologically protected quantum information processing
Error correction can be implemented not only at the "software level", but also thanks to the physical properties of the platform. An example from classical IT is fault-tolerant information storage in a magnetic medium. In a ferromagnetic state, most of the atoms have their magnetic moments oriented in the same direction. While an individual atom may flip its direction due to thermal fluctuations, the interaction with neighboring atoms will quickly reverse it to the original orientation. In 2003, Kitaev extended this concept to the quantum domain by theoretically designing a model, in which errors are energetically unfavorable [157]. That is, whenever any physical qubit experiences an error, the energy of the system's collective state increases above that of any errorfree state.
He envisioned interaction of qubits in a 2D lattice embedded on a torus surface and showed that, under a particular interaction model, the system has exactly four ground states. These four ground states are interpreted as the logical states of two qubits. These qubits are highly non-local and, therefore, unlikely to transform into each other as a result of a random local fluctuation. The existence of these four ground states is a consequence of the the boundaries of a toric surface being "stitched" together; for this reason, this approach to error correction is known as the toric code [157].
We should note that the aforementioned surface codes [160] have been inspired by the idea of toric code and have a lot in common with the latter. More specifically, in the surface code, the boundary conditions are imposed only on two sides of the computational surface, which can be visualized as a surface wrapped into a tube.
In the toric code, boundary conditions are additionally introduced to the remaining two sides, wrapping the tube into a torus.
Realizing the toric code experimentally is a challenging problem since it involves realizing a complex model of many-body interactions while implementing or simulating the torus topology. Following early experiments limited to small-scale systems [177][178][179][180], Google team with collaborators in 2021 realized a 31-qubit ground state of the toric code using the Sycamore superconducting quantum processor, suppressing the error probability for the protected state [181]. A qubit with basic topological properties inspired by toric code has also been demonstrated using neutral atoms [182].
A shortcoming of both the original toric and the surface code is that, while these codes provide topological protection to qubit states during storage, their active manipulation in fault-tolerant manner is difficult to realize. This problem has been later addressed by a family of methods known as topologically-protected quantum computation [157,183,184]. We shall discuss these methods only briefly because of their relative complexity and many challenges arising in their experimental realization.
The fundamental concept within this paradigm is the anyon [184,185] -a stable vortex with particles circulating around its center. These anyons can be compelled to move around each other by applying external fields, which leads to the change of their quantum state. For example, in a Bose-Einstein condensate such a vortex can be created by illuminating a certain location with a laser beam with a vortex-like spatial structure [186,187]. In a superconducting plane, one can create a vortex by applying a localized "beam" of the magnetic field in the direction perpendicular to the plane.
The quantum state of an ensemble of anyons contains information about the history of their movement, more specifically, about the topology (or "braiding") of their trajectories, i.e. how they "wove" around each other. A qubit is formed by several such anyons, for example, four in the widely considered Fibonacci model [184,188]. Certain trajectories correspond to single-and two-qubit gates. The quantum information carried by such a system is protected because it is stored not in the local states of individual particles, but the history of how anyons have been moved around each other. A local perturbation, as long as it is small enough to keep the vortex intact, will not change that history. This is analogous to the physical error correction in classical information storage that we mentioned in the beginning of this section.
As such, the topological approach is considered to be supremely promising as a path towards fault-tolerant quantum computing. It is sometimes even classified as an independent model of quantum computing [184], although, as we discussed above, it is also can be seen as the way to implement the gate-based model. A variety of physical systems and computational protocols for topologically-protected quantum computation have been theoretically proposed -albeit so far without successful experimental demonstration.
One such system is the so-called Majorana zero modes -anyons, which are expected to emerge in superconducting nanowires [189][190][191]. Observation of signatures of Majorana zero modes has been reported [192] by the Microsoft laboratory in the Netherlands, but subsequently retracted citing "insufficient scientific rigor" in the original data analysis [193]. Nevertheless, Microsoft is still committed to the topological approach to quantum computing [194].
Even though quantum braids are inherently more stable than quantum particles within the standard (nontopologically-protected) systems, they are not a universal panacea against all types of errors [184]. Therefore, large-scale quantum computers based on these principles are not expected in the next 5-10 years.
VIII. QUANTUM COMPUTING CAN BE BASED ON VARIOUS PHYSICAL PLATFORMS
At the dawn of classical computing it was not known which physical platform is best suitable for its implementation. Various platforms have been tried, such as mechanical, electromechanical, vacuum tubes, etc., until the engineering community has converged on semiconductor microstructure as the optimal approach. The current situation in quantum computing resembles that of early days of classical computers: a number of platforms are under consideration, but the leader is not yet determined. In this section, we review existing platforms, their basic principles, advantages and shortcomings as well as achievements recorded to date with each of them. A. Solid-state quantum computing Solid-state quantum circuits rely on nanotechnologies to construct qubits as artificial structures connected to each other in a hardwired fashion akin to classical electronic integrated circuits. The advantages of this approach include faster gates, possibility of industrial fabrication and broad availability of control equipment [196,197]. Additionally, solid-state systems can be used for designing topologically-protected qubits [179,194] (see Sec. VII C). These features play a role in attracting major industrial computing companies -such as Google and IBM -to this class of platforms. However, quantum solid-state devices also suffer from important shortcomings: their hardwired nature results in limited connectivity and potential loss of scalability as all qubits and their junctions must be individually controlled by an electrical connection. A further challenge is the fabrication of circuit elements that are both defect-free and sufficiently identical. Furthermore, solid-state platforms suffer from occasional decoherence bursts associated high-frequency cosmic ray particles [199,200]. An additional technological challenge is the requirement to maintain the quantum computing chip at temperatures on a scale of tens of millikelvins, which demands expensive dilution refrigerators. The leading solid-state quantum computing platform is Box 10. Superconducting circuits: Transmon qubits.
A circuit consisting of an inductor and a capacitor (so-called LC-circuit) is a harmonic oscillator, i.e. a system that is capable of exhibiting simple harmonic motion associated with periodic charging / discharging of the capacitor through the inductance. This harmonic motion can be quantized, resulting in energy levels positioned at equal intervals from each other. One can select two lowest energy levels to comprise a qubit. Transitions between these levels, corresponding to single-qubit operations, can be implemented by applying a microwave field, whose frequency is resonant with the separation between levels, typically on a scale of a few GHz.
In practice, such a qubit needs to be maintained at very low temperatures (on a scale of a hundredth of a kelvin above absolute zero). This is necessary, first of all, to bring the circuit into the superconducting regime, so the conductors lose electric resistivity and energy dissipation is prevented. Second, this will preclude spurious excitation of the qubit due to thermal fluctuations (according the Boltzmann distribution. the probability of an excitation is given by e ω/kT , where ω is the transition energy, k is the Boltzmann constant, and T is the absolute temperature; in typical superconducting computing circuits this probability is on a scale of 10 −9 ). Superconductivity has an additional important function. It helps to deal with the equidistant distribution of the energy levels in the harmonic oscillator. This equidistance is problematic if we use a resonant electromagnetic field to implement a transition between two qubit states. Higher energy levels, with which this field is also resonant, will also get excited, taking the system out of the qubit Hilbert space. To prevent this, the inductance is replaced by a so-called Josephson junction -a superconducting circuit element, whose inductance depends on the current. This results in the LC oscillator losing its harmonic nature, eliminating, in turn, the equidistant level structure. This circuit combining a capacitor and a Josephson junction is known as the transmon qubit, and is currently most common in superconducting quantum computing. For other types of superconducting qubits we refer the reader to review papers [195][196][197].
Two-qubit operations require coupling between qubits. To this end, another LC oscillator is used, whose frequency is also controlled by means of a Josephson junction. By tuning it with respect to the qubit resonance, the coupling can be switched on and off on demand.
superconducting circuits with the runner-up being arrays of semiconductor quantum dots.
a. Superconducting circuits. Currently, the most advanced superconducting quantum computers include • a 53-qubit Sycamore quantum processor presented by Google in 2019 [198] (Fig. 7), which has been used for demonstrating quantum advantage (see Sec. IX A),
• a family of publicly accessible quantum computers with sizes up to 127 qubits by IBM with the largest quantum volume of 64 [201],
• processors Zuchongzhi 2.0 [202] and Zuchongzhi 2.1 [203] developed by a group from University of Science and Technology of China and collaborators with up to 66 qubits, which have been also used to address the quantum advantage challenge [202,203],
• a 5,000-qubit quantum annealer developed by D-Wave in 2020 [105] (see Sec. V F and Box 6).
In August 2020, IBM published a roadmap targeting 1,121 qubits by the end of 2023 10 . A similar roadmap by Google (December 2020) promises 1,000,000 qubits with error correction by 2029 11 . On the other hand, an 10 https://www.ibm.com/blogs/research/2020/08/quantumresearch-centers/ 11 https://quantumai.google/learn/map independent forecast review [204] estimates that "proofof-concept fault-tolerant computation based on superconductor technology is unlikely (< 5% confidence) to be exhibited before 2026."
All the above achievements have been reached with transmon qubits (see Box 10). However, limitations of transmon qubits are currently becoming increasingly manifest [199,200,[205][206][207][208]. First, their coherence times are relatively short (about 10 4 gate times), which complicates error correction (see Sec. VII). The second cause for concern is relatively high qubit frequencies (on a scale of a few GHz), which requires expensive control electronics and complicates the wiring between these electronics and the cryostat where the quantum computing chip is located. Third, the transmon capacitor must be ∼ 100 femtofarads, which implies sizes on a scale of ∼ 100 micrometers, making it a challenge to pack more than a few hundred transmons on a few-mm chip. This being said, progress has been reported in reducing the qubit area by up to a thousand with the help of atomically thin heterostructures [209,210]. Fourth, nanofabricated transmons are non-identical and require special efforts to tune in resonance with each other. Other types of superconducting qubits are currently being developed that alleviate these issues, such as the fluxonium qubit [211][212][213].
b. Semiconductor quantum dots. A further important solid-state platform is based on semiconductor quantum dots -nanoscopic conglomerates of a semiconducting material deposited on a substrate. In such a quantum dot, single electrons can be isolated and their spins can be used as qubits. Individual qubit operations are implemented by applying magnetic fields. Qubits are coupled by direct magnetic interaction, which can be controlled by means of another quantum dot, which creates a variable potential barrier between the two electrons [214]. The typical size of a semiconductor qubit is hundreds of nanometers, which is two-three orders of magnitude smaller than that of superconducting qubits. This feature combined with the widely available semiconductor fabrication technology make the system promising in terms of scalability [215,216]. These promises motivated Intel to switch their quantum computing program to the semiconducting platform in spite of their impressive progress with superconductor technology (in 2018 Intel presented a 49-qubit superconducting quantum processor 12 ).
A challenge associated with the semiconducting platform is strong decoherence caused by impurities in a crystal structure. Note that defects are also present in superconducting circuits, however, their role is reduced due to larger sizes of the circuit elements [217]. The way to address this challenge is to use exceptionally pure materials for fabrication.
Currently, two primary materials for semiconducting quantum computations are silicon and germanium. Fourqubit germanium quantum processors with fast highfidelity gates have been demonstrated in 2021 [218]. Three groups in 2022 independently reported two-qubit gates with silicon quantum dots with fidelities over 99%, which is sufficient to enable surface-code error correction [219][220][221]. A further important recent achievement has been to bring the temperature of a semiconduictor quantum logic setup up to ∼ 1 Kelvin [222], which is more than an order of magnitude warmer than typical solidstate quantum computing experiments. Increasing this temperature further to 4 Kelvin will obviate the need for dilution refrigerators, thereby drastically decreasing the cost and footprint of quantum processors.
c. Other approaches. We conclude this section by mentioning a few promising alternative solid-state platforms, on which no multiqubit processes have been demonstrated yet.
• color centers, where qubits are realized by the electronic or nuclear spin of defects in the crystal lattice caused by impurities, for example, donors in silicon [223], vacancies near a nitrogen atom in diamond [224], or rare-earth ions in crystals [225] 13 ;
• fullerene molecules with the qubits based on nitrogen or phosphorus atoms encapsulated therein [227];
• qubits represented by spins of electrons positioned at the surface of a liquid helium film deposited on an insulator substrate [228,229];
• bound states of electrons localized in an array of nanowires [230];
• spins of itinerant electrons within metallic-like carbon nanospheres [231];
• point-defect spin qubits in engineered quantum wells [232];
• Andreev spin qubit combining properties of superconducting circuits and semiconductor setting [233];
• qubits based on split-ring polariton (light-matter superfluid) condensates [234];
• electron spin qubits in graphene quantum dots [235][236][237], van der Waals heterostructures [238], and quantum simulators for the Hubbard models based on twisted heterostructures [239], such as twisted bilayer graphene [240].
B. Atoms, ions, and molecules for quantum computing
Elementary units of solid-state quantum systems sometimes are referred to as artificial atoms because of their compact nature, reduced interaction with the environment, and well-defined, narrow-band energy spectrum. All these properties are essential in quantum computation. However, as discussed above, these artificial atoms are hard to make identical. Hence an alternative approach is to use actual atoms and molecules, which are identical by their nature, as elementary quantum units. The price to pay is the challenge associated with controlling and engineering interactions between them. This is achieved by means of traps -arrangements of external force fields keeping the particles steady during the experiment (see Boxes 11,12,and 13). Special efforts need to be applied to prevent mechanical oscillations of particles within these trap -that is, the particles must be cooled to temperatures on a microkelvin scale or even lower. Generally, this cooling is achieved using lasers, electric, and magnetic fields. This constitutes a significant cost advantage in contrast to the solid-state platform, in which cooling requires a dilution refrigerator. Cooling of atoms, ions, and molecules is a broad field in its own right [242,243], but is beyond the scope of our review.
The leading platforms within this family are cooled trapped ions and neutral atoms. The latter are further divided into two settings: Rydberg atoms in optical tweezers and ultracold atoms in optical lattices. We explain the details and differences of these two platforms in Box 12 and Box 13. While both ions and neutral atoms Box 11. Trapped ions.
Trapping of ions relies on their charged nature and utilizes electric or magnetic fields oscillating at radio frequencies.
Trapping force produces a single potential well, which pushes the ions towards its center. However, the ions keep away from each other due to their electrostatic (Coulomb) repulsion. A typical ion trap in quantum computing is a 1D array of ions separated by a few micrometer distance, so they can be individually resolved by optical means and addressed by lasers.
Each ion carries a single qubit, typically encoded in the state of its electrons. The energy separation between qubit states can be as low as few GHz (hyperfine states) or as high as hundreds of THz (electronic states). Dependent on this magnitude, qubits are classified into radiofrequency or optical qubits. The advantage of radiofrequency qubits is that they are more robust to decoherence, however they require more than one laser for single-qubit gates.
A critical feature that enables two-qubit gates is the presence of synchronized mechanical oscillations of ions within the trap. These oscillations form a part of the collective quantum state of the ion ensemble. They can be used to communicate quantum information between ions and entangle them. Specifically, by applying lasers to the control ion in a certain way, an oscillation can be exited dependent on the state of this qubit. When the target qubit is addressed by a subsequent laser pulse, the presence of this oscillation may determine whether this qubit will change its state, thereby completing a CNOT gate [244]. This is the basis of the original idea of ion-based quantum computing proposed by Cirac and Zoller in 1995 [245]. While the specific procedure of multiqubit gates has varied over past few decades [246][247][248][249], the collective oscillations have always remained their primary concept. This approach enables gates between any two ions arbitrary chosen within the trap.
Box 12. Rydberg atoms.
An atom (typically, of group 1 in the periodic table) is in the Rydberg state when one of its electrons is exited to a very high energy orbit (principal quantum number 50-100). Such orbits are characterized by large radii (fractions of micrometres) and strong interaction with neighbouring atoms (the interaction strength scales as the 11th power of the principal quantum number). One atom exited by a laser field into the Rydberg state may prevent its neighbours from achieving the same state, a phenomenon known as the Rydberg blockade. In other words, the behaviour of an atom can depend on the state of another atom.
On the one hand, this can be interpreted as the CNOT gate [250][251][252] for two qubits (each of which is encoded in the ground and Rydberg states of an atom) and hence enabling digital quantum computation. On the other hand, this system of atoms can be seen as a graph, in which the atoms are nodes (whose value can be 0 or 1 dependent on whether the atom is in the Rydberg state) and edges connect neighbouring atoms. The Rydberg blockade prevents the connected nodes from simultaneously taking on the value of 1. At the same time, by choosing the detuning of exiting lasers, one can make the state with the largest number of exited Rydberg atoms to be the most energetically favourable. This sets the natural condition for solving the maximum independent set problem [121,122] as well as simulating the Ising model in condensed matter physics (see Box 5).
A primary control tool in the Rydberg atom technology is a strong, tightly-focused laser beam known as an optical tweezer [253,254]. Laser light provides a force attracting the atoms towards the beam center, thereby creating a potential well, in which the atoms can be trapped. To make a Rydberg atom quantum computing device, a cloud of atoms (typically, rubidium or cesium) is cooled to submillikelvin temperatures. Then some of the atoms are individually trapped in optical tweezers; the remainder is released from the trap. These tweezers are used to arrange atoms into spatially order arrays, in which Rydberg gates are possible between neighbouring atoms (Fig. 9).
have the potential for various models of quantum computing, the ion platform is presently considered a mature engine for gate-based model, whereas neutral atoms are mostly used for quantum simulation. We also briefly discuss molecular platforms, which are considered promising, but experimentally less advanced.
a. Trapped ions. Historically, the ion platform was one of the first in which the two-qubit gates [244,246] and basic quantum algorithms [255] have been demonstrated. The most recent advancements include a 53-ion quantum simulator for the Ising models [256] in 2017, fully-controlled quantum-state engineering in a 20-ion system [257] in 2018, and demonstration of variational quantum algorithms for chemistry [258,259] and combinatorial optimization [83] in 2018-2020.
Progress towards error correction (see Sec. VII) has also been achieved. Erhard et al. [169] demonstrated logical qubits in the framework of the surface code, as well as entanglement and basic operations between them in 2020. Egan et al. [168]. implemented single-qubit error correction using the Bacon-Shor code [153] (an extension of the Shor code discussed above). The coherence time of the qubit has been increased by the factor of 2.5. If errors are detected, but not corrected and error-free events are instead post-selected, then the qubit lifetime is increased by more than 10 times.
Most of existing experiments have been performed in single 1D traps -such that the ions form a single straight line. While attempts to implement 2D traps have been made (with 4 ions in Holz et al. [260]), this is technologically difficult. Not less challenging is scaling 1D traps beyond a few dozen qubits. There are a number of ways to address the scaling issue. The mainstream idea is to contain ions in multiple traps with the possibility to join or divide traps on demand [261] or move individual ions within the system [261] to enable interactions between arbitrary pair two ions. Another approach is to communicate quantum information between ions in different traps via an optical interface [262].
A modern ion trap is a complex microstructure complete with trap electrodes, dielectric insulators, opti-cal waveguides, modulators and detectors integrated together (Fig. 8). A key factor in fabricating these traps is the materials used. In particular, it is important that the surfaces do not produce significant electric field noise, which could cause decoherence of the ionic states [241].
Recently, the ion platform has been spun off industrially with three ventures emerging as leaders: IonQ (USA), Quantinuum (previously Honeywell, USA), and AQT (Austria). These companies made progress in different aspects of ion trap quantum computing. In particular, AQT demonstrated a compact unit fitting within a standard 19-inch rack capable of operating with 24 qubits [263]. Quantinuum/Honeywell developed a method for transporting and swapping ions within a trap for quantum gates with all-to-all connectivity [261] and the processor with the highest quantum volume. IonQ demonstrated various applications of their devices ranging from chemistry [259] to machine learning [264,265].
The main mechanism behind the formation of an ordered array of ions in a trap is their electrostatic repulsion (see Box 11). This repulsion leads to the emergence of mechanical modes, which enable full connectivity of quantum units. On the other hand, it gives rise to scalability issues as discussed above. The situation with neutral atom platforms is opposite. They are trapped in optical fields, which enables better scalability, however their interactions are relatively short-ranged limiting the connectivity and the quality of pairwise operations.
b. Neutral atoms. The two leading neutral atom platforms are Rydberg atoms 14 in optical tweezers (Box 12) and (ground-state) atoms in optical lattices (Box 13). The key difference between them is the interaction mechanism: in the former case, the Rydberg blockade enables digital gates between atomic qubits, whereas in the latter case, the quantum states are carried by the atomic motional degrees of freedom with the interaction leading to multiparticle entanglement, but not of digital nature. A further difference is that optical lattice setups require cooling the atoms to extremely low temperatures (tens of nanokelvins), which complicates the setup and requires a long preparation stage (tens of seconds) before the "payload" quantum process (a fraction of a second) can be launched. In the optical tweezer setting, in contrast, the atomic temperatures are on a scale of hundreds of millikelvins, which is easier to obtain experimentally, and needs much shorter preparation (about 100 milliseconds). Moreover, optical tweezers allow one to craft ordered arrays of atoms with various geometric configurations and interaction schemes [253,254]. Therefore, the Rydberg platform is suitable for both analog quantum simulation and digital quantum computing, whereas optical lattices are primary seen as a platform for analog simulation. Two interfering counterpropagating laser beams form a standing wave -an array of alternating zones of high and low light intensity. Atoms can be trapped in the high-intensity zones (antinodes of the standing wave) via the same mechanism as optical tweezers (see Box 12). As a result, we obtain a periodic array of traps, in which atoms behave akin to electrons in a crystal lattice. This system, known as the optical lattice, therefore constitutes a simulator for lattice many-body models in physics whose applications range from condensed-matter [92,267,268] to high-energy physics [269]. The quantum degree of freedom in such a simulator can be the motional state of the atom or their spins. In contrast to the Rydberg platform, the atoms' electrons rarely leave their ground states. A great advantage of this simulator is the tunability of its main parameters, such as dimensionality (1D, 2D or 3D), lattice geometry (rectangular, honeycomb, etc), interaction range, directionality (isotropic vs anisotropic), sign (attraction vs repulsion) and strength. A further important degree of freedom is the choice of atomic species for the simulator, in particular, the fermionic or bosonic nature of the atoms. In many cases, individual lattice sites can be resolved through a regular microscopic objective, which enables one to observe individual atoms and estimate their quantum states (Fig. 10). This quantum gas microscope is a great asset in experimental studies [270,271].
The optical lattice platform has existed for about two decades [92,267]. Starting with the seminal experiments on simulating the phase transition in the Hubbard model in 2002 [272], it developed into highly controllable simulators for various condensed matter phenomena [92,267] including high-temperature superconductivity [273].
The Rydberg setting has emerged over the last five years. Arrays of atoms of different spatial dimensionalities have been demonstrated [129,130,140,[274][275][276][277]. They have been used to simulate phase transitions in quantum many-body systems with the number of atoms growing from 51 in 2017 [140] to 100 [278], 196 [130], and 256 [129] in 2020-2021. An important result of 2021 is the demonstration of spin liquid phase [182], a quantum phase of matter predicted in 1970s [279], but previously not conclusively observed experimentally. This state is characterized by long-range spin entanglement combined with disorder, which is maintained even at very low temperatures. These features make spin liquids interesting for topological error correction (see Sec. VII C). A further important application of the Rydberg atom platform is combinatorial optimization, specifically, the maximum independent set problem (Sec. IX C 2).
Aside from the analog regime, digital two-qubit gates with Rydberg atom arrays have been reported achieving fidelities in excess of 0.97 [280]. The factors currently preventing even higher fidelities include the Doppler effect, spontaneous emission and laser phase noise [281].
Although Rydberg blockade gates are only possible between nearest neighbors, the atoms can be transported within the array without loss of entanglement thereby enabling a gate sequence for arbitrary pairs of atoms. The potential of this approach has been demonstrated in 2021 by Bluvstein et al. [282], who used it to implement a variety of error correction schemes involving up to 24 physical qubits.
The Rydberg platform is being commercialized by QuEra (USA) [283], Pasqal (France) [123], and ColdQuanta (USA) [284].
In spite of its great promise, the limitations of the Rydberg platform are associated with atomic states being sensitive to external electric fields and a relatively small blockade radius. A way to overcome these shortcomings could be to couple atoms via an artificial optical interface [285], e.g., by placing them in the vicinity of a waveguide to let light communicate quantum information between qubits. In this setting, the atoms can be still manipulated individually by means of optical tweezers, but the Rydberg states are no longer necessary. Instead, the qubit can be encoded in sublevels of the atomic ground states, which have much longer coherence lifetime. A basic experiment in this setup has been realized in 2021 [283]. This setting could become a next step in atomic quantum computation.
c. Cold molecules. A promising platform for quantum computation is based on ultracold molecules. The idea is to utilize the dependence of the molecule's dipole moment on its state [286][287][288][289]. As a result, the internal state of a molecule can strongly affect the interaction of neighboring molecules, enabling two-qubit gates. The challenge is associated with a rich space of molecular quantum states making them difficult to control. To our knowledge, two-qubit gates with individually controlled molecules have not yet been demonstrated. An important recent achievement is handling cold molecules using optical tweezers [287,288,290].
C. Optical quantum computing
The final physical platform with a large footprint in the current quantum computing landscape is based on light. Quantum information is encoded in light waves propagating in certain channels (modes), which can be implemented either in free space (on an optical table) or via waveguides in integrated chips. The optical platform is special because of the transient nature of optical waves, meaning that the computation has to proceed "on the fly". Furthermore, light waves under normal conditions do not interact strongly either with each other or with the environment. The latter is the reason that the decoherence in the optical platform is strongly reduced, making it appealing for quantum computing. On the other hand, the lack of mutual interaction between optical fields makes it a challenge to design two-qubit quantum computational gates.
In principle, conditions, in which light waves influence each other, do exist. They are the subject of a vast field of physics known as nonlinear optics [291]. However, nonlinear optical phenomena typically emerge at light energies on the scale of at least billions of photons. It is much more challenging to achieve sizable nonlinear effects at the single-photon level as required for quantum computation. Several avenues towards this end are being pursued, mostly based on nonlinear properties of individual atoms, atom-like objects or their ensembles. One approach is the aforementioned phenomenon of Rydberg blockade: when an atom absorbs a photon and transitions to a Rydberg state, it will affect neighboring atoms, thereby preventing absorption of further photons [292]. This "single-photon transistor" can serve as a quantum gate. Other possibilities to enhance optical nonlinearities include tight focusing of an optical beam onto a single atom [293], placing an atom in an optical resonator [285], or using novel nonlinear materials, such as graphene [294]. All these approaches are, however, difficult to implement and scale up. Furthermore, they introduce losses associated with the light-matter interface.
An important alternative is to use much more common linear optical phenomena, such as refraction, reflection, and interference, combined with conditional measurements. After the computational modes undergo linear optical transformations, which entangles them, one performs measurements on some of the modes. The nonlocal effect of this partial measurement (see Box 1) on the entangled multimode state causes the state of the remaining modes to change in a way that is similar to a result of nonlinear interaction occurred among them. This is known as linear optical quantum computing. It might appear that the probabilistic nature of linear-optical gates, combined with the transient nature of light, would preclude viable quantum computing as any undesired measurement results or loss of photons would be fatal for the entire computational process. However, in a breakthrough publication of 2001, Knill, Laflamme, and Mill-burn (KLM) showed this not to be the case [295]. They proposed to implement probabilistic two-qubit gates on "auxiliary" modes separately from the main quantum computational stream. This gate can be tried many times without disrupting that stream. In the event of success, the auxiliary modes will be entangled in a certain manner. One can then apply the quantum teleportation protocol to teleport this entangled state onto the computational stream. The quantum state of the modes emerging after teleportation will be equivalent to that expected as the output of a two-qubit gate.
While KLM provided a head start to optical quantum computation, the practical implementation of the specific scheme of Ref. [295] is prohibitive due to tremendous recourse overhead. Multiple ideas for its improvement have been proposed. A currently popular paradigm of discrete-variable quantum optics was introduced by Kieling et al. [296], further improved by Gimeno-Segovia et al. [297], and consists in creating a cluster state (see Sec. V B) by joint measurements on a large number of primitives, each of which is a three-photon entangled state. Each of these measurements has a limited probability of success and hence the resulting cluster state will contain "holes". However, the remaining connectivity is sufficient for meaningful one-way quantum computing.
Generating the entangled photon primitives is a challenge. One approach is to produce these states from single photons by means of an additional preliminary layer of probabilistic linear optical circuitry. A scheme to that effect has been proposed by Varnava et al. [298] with the success probability of 1/32, albeit requiring feedforward operations. Gubarev et al. [299] proposed a way to eliminate feedforward, however with a lower success probability of 1/54. Both these schemes require multiple on-demand single-photon sources. The currently leading method for this task is based on quantum dots (previously briefly discussed in Sec. VIII A), in which transitions of single electrons can be used to generate photons [300]. However, these sources are imperfect, with the best achieved efficiencies on a scale of 50% [301]. An alternative method is to prepare entangled photon triplets directly from quantum dots [302] as demonstrated experimentally by Schwartz et al. [303] in 2016, albeit, again, with imperfect efficiency and fidelity.
The optical platform was one of the first to be explored in the context of quantum information processing in the late 1990s because its tools have been readily available and relatively inexpensive. The initial experimental work was done in free space, with the elements realizing preparation, manipulation, and detection of states of light positioned on an optical table. This research, conducted in the discrete, continuous, and hybrid settings, produced many results that are of value for the entire field of quantum science. However, the free-space implementation is not practical for large-scale quantum computing due to the lack of scalability, complexity of industrial production, and need for regular alignment. An approach with the potential to overcome these shortcomings is based on integrated optics, where quantum light is carried in waveguides on the surface of a chip.
Quantum optical computation faces many challenges and it is not clear presently how to overcome some of them. Complications emerge almost at all stages of quantum processing. Specifically, the preparation efficiencies of the primitive states (single photons for the discretevariable and squeezed states for the continuous-variable settings) are significantly below the fault-tolerant quantum computation requirement. An additional challenge for the discrete-variable quantum computing is the preparation of the entangled-photon resource for cluster states. A subsequent difficulty is associated with losses that are carried by any optical element either in integrated or freespace settings. While losses below some threshold can be compensated by means of error correction, the current technology does not reach this threshold. Furthermore, cluster state quantum computing requires highspeed electronic processing of measurement results for feedforward onto the circuit elements. Characteristic times of such processing are on the scale of nanoseconds. Given that the speed of light is a foot per nanosecond, direct feedforward, especially in an integrated chip setting, appears problematic. Facilities like delay lines or quantum optical memory [69] may be required, which themselves pose technological challenges.
Nevertheless, many researchers are "optimistic about the silicon-photonic route to quantum computing" [304] and this approach is being commercialized by several startups. PsiQuantum (USA) is developing a discretevariable device in an integrated chip setting. They announced an ambitious goal of developing a 1 million qubit quantum computer by 2025 15 . PsiQuantum is valued at $3.1 Billion 16 . Their nanophotonic chip technology is expected to be useful not only in quantum, but also in optical computing, particularly for optical neural networks.
Encoding quantum information in states of photons as particles of light -such as in the research described above -is referred to as discrete-variable quantum optical computation. This however constitutes only one side of light as a dual wave-particle entity. Information can also be carried, and used for quantum computing, by the wave properties of light, such as its amplitude and phase. This is known as continuous-variable encoding. The main primitive of continuous-variable quantum optical computing is the so-called squeezed state of light [116], in which the quantum noise of the light wave amplitude at certain phases is reduced below the standard noise level, which corresponds to light with no nonclassical properties. In contrast to the single photon, squeezed light has been possible to reliably produce on-demand for many years. Interfering in linear optical arrangements, squeezed states can produce complex clusters. 1D [305][306][307] and 2D [308,309] clusters as well as measurementinduced feedforward processing [310] have been experimentally demonstrated. However, continuous-variables cluster schemes require a certain level of squeezing to enable fault-tolerant quantum computing. Current theoretical research sets the threshold in the range of 10-17 decibels (noise power reduction by a factor of 10-50) [176,311,312], which compares favorably with the current record of 15 decibels [313] (2016). However, the level of squeezing in ongoing experiments with cluster states is significantly lower: on the scale of 3-5 decibels (noise power reduction factor of 2-3) [308,309]. Furthermore, the mentioned theoretical models disregard losses, which are inevitable in optical arrangements.
Outside the digital model, impressive progress in continuous-variable quantum optical computing was achieved by a Canadian startup Xanadu. They presented a chip-based Gaussian boson sampling [118] (discussed in detail in Sec V F) circuit with 8 modes with applications in quantum chemistry [119] and mathematics [120]. This device, including all the necessary classical infrastructure was remarkably compact, fitting in a standard 19-inch rack.
IX. QUANTUM COMPUTERS HAVE DIVERSE APPLICATIONS
It is expected that quantum computers can be useful in various applications ranging from scientific research to cybersecurity, financial optimization, and drug discovery. Here we present an extended but not complete list. Before proceeding, we would like to reiterate a disclaimer. Existing publicity hype may mislead the reader into thinking that a computer capable of achieving quantum advantage in application to real-world problems has already been developed. In fact, the state of the art can be described as follows.
• First, experiments have been performed to demonstrate quantum advantage on computational problems that are of no practical value, but are widely accepted by the community as difficult to solve.
• Second, new scientific insights are obtained by applying quantum simulators to model various physical systems, particularly condensed-matter. While these insights are specifically associated with quantum machines, there are no rigorous proofs of quantum advantage: the classical computational complexity of these problems has not been thoroughly investigated.
• Finally, there are many efforts to apply quantum computers to various classical problems of practical relevance. In these experiments, the problem sizes are typically far less than what can be solved with a regular classical machine. Existing reports are limited to proof-of-concept demonstrations rather than ready-to-use technology. That being said, these demonstrations are of importance for exploring the range of applications, in which the practical quantum advantage can eventually be achieved.
We anticipate the progress timeline of quantum computing applications to resemble that of lasers in the second half of the 20th century. First applications of the laser immediately after its invention have been in scientific research: first in studies of the laser itself, then as a tool for adjacent fields of science, such as atomic, molecular, and optical physics, and finally as a source of light with special properties that are useful a broad range of sciences including chemistry and biology. This latter stage coincided with the emergence of narrowly specialized applications, such as holography, spectroscopy, communications, and material processing. Finally, lasers developed into a ubiquitous instrument present in virtually every industry and every household, e.g., as an element of a printer or a music player. In our discussion of quantum computing applications, we will follow the same general scheme, i.e., separately discuss scientific, specialized, and potentially broad use cases.
A. Basic science applications
Applications of quantum computing in scientific research [314] can in turn be classified into two main categories: (1) calculating the properties and simulating complex physical systems and (2) exploring capabilities of quantum computers to achieve computational advantages.
a. Simulating physics. The task of modeling quantum many-body systems has been a subject of close scientific scrutiny for many decades and has reached major milestones with classical computers. However, this modeling requires continuously increasing computational power, driving rampant employment of supercomputers in quantum physics research. As discussed above, quantum computation offers a qualitatively new capability for quantum modeling.
It is important to understand that a quantum computer built on a particular platform can study physical phenomena beyond this platform. We have already seen several examples to this effect: a trapped ion quantum computer was used to study high-energy physics models [133,315,316] and a neutral atom machine was employed for simulating condensed matter physics [92,140,267,274]. Another example that has not yet been discussed is the simulation of the physical mechanism giving rise to the Higgs boson using atoms in an optical lattice [317]. Remarkably, this observation has been made at the same time as the Higgs boson has been discovered at the Large Hadron Collider. As quantum computing develops further capabilities emerge, including modelling effects in nuclear physics [318], simulating dark matter in the Universe [319], studying general relativity and black holes [320,321].
These results have been of primary significance for physics, as they enabled simulating exciting physical phenomena that had been predicted theoretically, but were extremely hard to produce in their "native" high-energy or condensed-matter physics settings. Moreover, some of these experiments, such as those of Refs. [129,130,140,256] have been at the limit of classical computation accessibility. Most recently, a claim to have reached beyond this limit in a physically relevant context has been made [105]. We shall discuss this result at the end of this subsection.
b. Advantage demonstrations. In 2019, this was demonstrated by the Google team for the random circuit simulation problem [128]. This problem consists in predicting the statistics of measurement results for the output of a particular quantum circuit containing a random sequence of single-and two-qubit gates. If the circuit is complex enough, the problem becomes intractable for classical computers [49,322], but is easily solvable by simply running this quantum circuit multiple times.
The measured output of a quantum computer running a random circuit is an arbitrary bitstring, whose probability depends on the circuit. Therefore, the proof of quantum advantage consists in verifying that the set of random bitstrings produced by the quantum device is consistent with the theoretically expected probability distribution for the given circuit. This distribution must be calculated on a classical computer, which is possible for quantum circuits of reduced complexities, for which quantum advantage is not yet present. Importantly, each bitstring is likely to occur in the output set only once, making the hypothesis verification a nontrivial mathematical challenge. It is exacerbated by the noise and imperfections, which result in high probability for the quantum device to produce completely random outputs.
The Google team has verified the quantum advantage by first solving this problem for intermediate-size circuits with both their quantum processor Sycamore (53 qubits) and the classical supercomputer IBM Summit and observing consistency between the two solutions. After that, the complexity of the circuit was increased, so that the estimated time of obtaining the solution classically would be 10,000 years, whereas Sycamore needed only 200 seconds.
In the conclusion of their work, the authors wrote: "We expect that lower simulation costs than reported here will eventually be achieved, but we also expect that they will be consistently outpaced by hardware improvements on larger quantum processors". This prediction turned out to be precisely correct. Immediately after the publication of Ref. [128], IBM proposed a technique that would enable reducing the classical calculation to 2.5 days using so-called tensor networks 17 . A series of classi-17 https://www.ibm.com/blogs/research/2019/10/ cal numerical experiments followed [51,52,323] with the shortest achieved computation time reaching 5 days on 60 NVIDIA GPUs [323]. On the other hand, quantum hardware has also progressed. In 2021, a group led by Pan demonstrated two quantum processors, Zuchongzhi 2.0 [202] and 2.1 [203] with 56 and 60 qubits, respectively. Zuchongzhi 2.1 performed a random circuit simulation that lasted 4.2 hours, but would take 48,000 years on a classical supercomputer even if tensor networks [324] are used [203].
A related class of experiments is on quantum advantage with boson sampling (see Secs. V F and VIII C). Two groups in China in 2020 realized boson sampling with over 50 photons: Zhong et al. [325] in a table-top experiment and Gao et al. [326] in an integrated setting (on a waveguide chip, see Sec. VIII C) with an additional temporal degree of freedom. Results of Ref. [325] have later been extended to enable the programmability of the interferometer matrix [327]. These claims were challenged by Popova and Rubstov [328] as well as the Goolge team [329] arguing that the output photon statistics of the boson sampling circuit can be reproduced using a consumer CPU.
Both the random quantum circuit and boson sampling demonstrations of quantum advantage are similar in that the problem being solved by the quantum device is the prediction of its own output. This may raise a question whether such that a self-serving problem setting is a valid benchmark of computational advantage. One can argue, for example, that fluid dynamics equations describing turbulent water flow in a pipe are also beyond the modeling capabilities of a classical computer, however are easily "solved" by a direct experiment -thereby also offering an "advantage". The difference between these cases is that the complexity of fluid dynamics is largely associated with the lack of knowledge of precise parameters of the system combined with extreme sensitivity of the solution to these parameters (known as chaos). For NISQ devices, in contrast, the parameters are known precisely and the system is not chaotic; the complexity is merely a consequence of the exponential scaling of the computational space with the system size. An important further difference is the programmability of the quantum computers, i.e., our ability to arbitrarily change the parameters of the systems. Finally, there exists a roadmap towards developing the NISQ computation state of the art into a technology capable of solving practical computational problems, which is not the case for the aforementioned fluid dynamics setting.
That being said, the above demonstrations have been in application to "toy" problems of little practical value. In 2021, Google and D-Wave reached the next major benchmark [105]: demonstrating quantum advantage for a physically relevant problem. They took advantage of the fact that many problems in condensed matter physics on-quantum-supremacy/ can be reduced to the quantum version of the Ising problem (Box 5), which is the subject of the D-Wave solver. They have chosen one such problem (simulating geometrically frustrated magnets) and compared the solution for up to 1440 qubits generated via a D-Wave annealer with a state-of-the-art classical computing technique (pathintegral Monte-Carlo).
B. Specialized applications
Certified random number generation
Random number generation is important for many applications, such as cryptography and numerical simulations. The crucial task in this context is convincing the user of a random number generator in high quality of the output randomness. In principle, there exist randomness test that are used to check random sequences 18 , however no mathematical test can guarantee absence of any (intended or inadvertent) hidden regularities therein.
Quantum technologies offer a unique opportunity to generate random numbers with guaranteed randomness. A simplest example is a single photon polarized at 45 degrees incident on a polarizing beamsplitter, which will randomly transmit or reflect this photon with the probability of 1/2. This technology has been the basis for commercial random number generators, with the first product of this kind introduced by ID Quantique in 2001 19 . However, in order to trust such a unit, the user must be fully aware of and understand its physical design, which is not always possible.
A much more attractive setting is when the user can convince themselves of the nature of generated randomness by means of information exchange with the device. This requirement, known as device-independent randomnumber generation, can be satisfied by using a NISQ computer as a random generator. The idea is to still use the fundamental randomness of a quantum measurement, but replace a single photon by an output multiqubit entangled state of a quantum register. The ability of the random number generator to rapidly produce multiple samples resulting from preparing and measuring such a state will indicate the quantum origin of these samples to the user.
Specifically, the protocol proposed by Aaronson 20 is as follows. A user with a classical computer first obtains a few random bits from some trusted source and uses this "seed randomness" to design a quantum circuit. The user then sends this design to the operator of the random number generator. The operator implements this circuit with their quantum computer and runs it multiple times, supplying many random bit strings in a short time.
How can the user be convinced that the samples they receive indeed originate from measuring a quantum state? The idea is to use the quantum circuit of a complexity at the borderline of quantum advantage [330], i.e., such that the user is able to simulate it and compute the output state using classical resources, but it is prohibitive to classically sample measurement results from these states. Knowing the state, the user is able to statistically test whether the samples they received are consistent with the probability distribution associated with the state they calculated. This consistency will prove the quantum nature of the data.
If the outcome of the test is positive, the user can then use standard techniques to amplify the randomness and remove all the correlations that may present in the set [330,331]. Certifiable quantum randomness generation was announced by Google as one of the first commercial applications of their NISQ devices.
Ironically, it is essential for this protocol to use quantum computers at the borderline of the quantum advantage threshold, but not far above it. That is, developing quantum computing with larger quantum volume is not beneficial for quantum random number generation.
Sampling probability distributions
The classically difficult task of sampling from a known probability distribution is useful beyond random number generation. This is of particular significance for generative neural networks in machine learning, such as variational autoencoders and generative adversarial networks. An important task in this context is obtaining sample sets that are not necessarily better, but different from those obtained by classical algorithms. NISQ technology is often able to satisfy this criterion because of a fundamentally different process of sample generation.
Dumoulin et al. [336] and Benedetti et al. [337] pointed out that quantum annealers that strongly interact with the environment freeze out the dynamics of a spin system before the termination of the annealing process. As a result, such annealers sample from a thermal distribution with some finite temperature. The proposed method was experimentally implemented using the D-Wave 2X quantum annealer [96,337] for the training of a Boltzmann machine. However, shortcomings of existing quantum annealers (see Box 6) limited the study to low-dimensional datasets (see also Refs. [338][339][340][341]).
A more successful application of the D-Wave machine for sampling was reported by Gircha et al. [342] to train a restricted Boltzmann machine as a layer in a discrete variational autoencoder for generative chemistry and drug design. A few thousand novel chemical structures with potential medicinal properties have been generated. Gibbs sampling was further studied in 2020 by Cryptography is easily implemented if the communication parties, which we call Alice and Bob, share a prearranged, secret data set (a sequence of 0's and 1's) known as secret key or one-time pad. Alice can use the secret key to encrypt her message. She can then send it to Bob via an insecure channel. While anyone can read this encrypted message, nobody can decrypt it except Bob who possesses another copy of the secret key. Such a family of protocols is known as private-key cryptography. They are secure, simple, and existed for hundreds of years. However, the secret key is a high-cost recourse because sharing it would in turn require secure communication between Alice and Bob. Therefore a great majority of applications uses an approach known as public-key cryptography. This ingenious technique is based on the existence of so-called "one-way" functions that are straightforward to run on a conventional computer, but difficult to calculate in reverse. For example, multiplying two large prime numbers is easy, but factorizing a given product is exponentially hard. Such public-key protocols enable secure communication between parties who have never had an opportunity to exchange a secret key.
Public-key cryptography is jeopardized by the arrival of quantum computation. This is because some one-way functions -unfortunately those that are deployed in currently popular protocols -are reversible in polynomial time with quantum computers. Examples include RSA and Diffie-Hellman protocols for key exchange and digital signatures. Fortunately, not all one-way functions are vulnerable to quantum cryptanalysis. In the next few years, public-key cryptography protocols world-wide are expected to transition to such quantum-safe or post-quantum primitives. For example, NIST has initiated Post-Quantum Cryptography Standardization Program in 2016, which is expected to be completed by 2024 with a set of standards for quantum-safe algorithms.
These solutions, however, do not provide ultimate security as they are still based on computational complexity assumptions and their potential vulnerabilities are a subject of ongoing research [332,333]. An ultimate solution to the communication security problem is offered by the quantum key distribution [334]. However, broad deployment of this technology is facing many challenges, such as cost, speed, losses in communication lines, and practical security [335]. researchers from Harvard University and QuEra [343] using the programmable Rydberg simulator (see Box 12). The proposed method potentially leads to a speedup over a classical Markov chain (state-of-the-art sampling technique) for several examples.
Cryptanalysis
Modern public-key cryptography is based on the concept of one-way functions (see Box 14). The aforementioned Shor's algorithm can be used in cryptanalysis (deciphering) of currently deployed public-key cryptography algorithms, such as RSA and Diffie-Hellman (see Table IV), which is reducible to the task of prime factorization. Proof-of-concept experimental factoring of 15, 21, and 35 have been demonstrated on superconducting [344], trapped ion [345], and photonic [346][347][348] quantum computers. Shor's algorithm for practically relevant key sizes (2048-or 4096-bit), however, requires capabilities far beyond those of NISQ devices. We have already mentioned calculations by C. Gidney and M. Ekera on factoring 2048-bit RSA key, which would require 8 hours using 20 million physical qubits [126]. Another very recent proposal [349] suggests a way to factor 2048 RSA integers in 177 days with 13436 physcial qubits and a multimode memory. A recent forecast review [204] estimates the likelihood for quantum devices capable of factoring RSA-2048 to exist before 2039 as less than 5%.
Special-purpose quantum machines can be used for factorization as well [350,351]. For example, the D-Wave annealer was used to factor 1, 005, 973 = 1009 × 1019 [352]. Variational quantum algorithms have also been studied in this context [72]. However, it seems that the complexity of this task for special purpose quantum computers grows much faster than for universal ones. Thus, these studies are of conceptual interest in the NISQ era, but are not expected to lead to quantum advantage.
The secrecy of key distribution is not the only vulnerability of modern cryptography. Another problem is that the secret key is typically much shorter than the dataset it is used to encrypt. There do exist encryption algorithms, which solve this problem so that it is hard for classical computers to decrypt messages encrypted with relatively short keys; the difficulty grows exponentially with the key size. For example, widely deployed AES protocol uses keys of length as little as 256 bits to securely encrypt terabytes of data.
Quantum computers have less of an effect on this matter, since Shor's algorithm does not apply, and exponential speedups are not expected (see Table IV). However, Grover's algorithm [22] enables quadratic speedup in brute force search, which means that the key length should be doubled to enable the same level of protection [353]. The same scaling applies to cryptographic hash functions, for which the primary attack method is also brute-force search [353].
An area of particular concern in the context of quantum security is blockchains and cryptocurrencies [354][355][356], which are argued to be the "blueprint for a new economy" [357]. Typical blockchain and cryptocurrency protocols use several cryptographic schemes, such as digital signatures and hash functions for achieving a consensus (proof-of-work) between users in the absence of trust. The quantum vulnerability of hash functions is similar to that of AES: that is, the primary known method of attack is brute-force search [353] and hence no more than quadratic advantage can be expected. However, private keys can be extracted from digitally signed messages by means of Shor's algorithm allowing parties in possession of a quantum computer to impersonate any other party, which will obviously collapse any blockchain relying on current protocols. For example, Bitcoin (a cryptocurrency blockchain), which uses the elliptic curve signature scheme, could be completely broken by a quantum computer as early as 2027 [354].
Attacks with quantum computers have become a subject of many studies that proposed solutions for quantumresistant blockchains [355,356]: blockchains that use quantum key distribution [334] or post-quantum digital signatures and consensus schemes. A quantum-secured blockchain protocol was experimentally demonstrated in 2018 [355].
In summary, the detrimental effect of quantum computing on information security can be thwarted by upgrading information exchange protocols to quantum or post-quantum technologies. Importantly, this transition must be implemented long before the emergence of practical quantum computation [355,358,359]. Otherwise a variety of potential risk scenarios can be envisioned. For example, a present-day hacker might intercept and store encrypted messages with the hope to decrypt them with a quantum computer a few years later. If the information is long-term sensitive (medical records, genetic data, strategic plans, etc.), this attack may result in damages.
C. Economically impactful application
There are two ways one can classify general-purpose applications of quantum computing. On the one hand, one can consider the classes of problems that quantum computers are able to solve. Roughly, three such classes can be identified:
• simulation, that is predicting the behaviour of a certain complex system based on an existing mathematical model;
• optimization, i.e., finding the best setting of a large combination of discrete parameters according to some criterion;
• machine learning, that is constructing a mathematical model that would fit the properties of a certain dataset.
These classes form the "supply" of services that quantum computation can provide. These services, on the other hand, can provide a variety of "demands" from various areas of technologies, such as chemistry, material science, life science, finance, etc. The significance of such demands is evidenced, for example, by the "quantum challenges" announced by several industry leaders such as Airbus 21 and BMW 22 , in which quantum scientists and technologists are invited to provide solutions to a variety of problems faced by these companies.
Simulation
Before we proceed to a specific discussion, we caution the reader to distinguish the notions of "simulation" as a problem class and "quantum simulators" as a type of a quantum computer. Many simulation tasks can be solved with other types of quantum computers and quantum simulators can be applied to other types of problems beyond simulation.
Simulation problems can be further classified into quantum and classical according to the object of study. We begin with the former, specifically, with the application in chemistry, life science, and materials science. Many subjects of these fields, such as fuels, drugs, biologically active compounds, and fertilizers, are quantum systems consisting of a large number of interacting components. As discussed above, such systems are hard to model classically, but naturally amenable to quantum computation and simulation. This constitutes a major component of the expected landscape of quantum technology applications.
a. Quantum chemistry. A case in point is the calculation of energies and electronic structures of molecular ground and exited states, which is important for theoretical understanding of chemical reactions. In order to encode electronic states of molecules into qubits, one uses a basis of predefined "spin orbitals", i.e., quantum states that can be occupied by individual electrons 23 . Each qubit in the register of the quantum computer represents one such spin orbital, with its value -0 or 1 -determining whether this spin orbital is occupied. The molecular state is then a superposition of multiple occupancy configurations. The quantum computer is used to simulate the molecular Hamiltonian and find the lowest energy state.
Currently, the primary tools of the trade are variational quantum algorithms (such as VQE; see Sec. V D) thanks to their modest hardware requirements. They were used to analyze small molecules, such as hydrogen (H 2 ) [360][361][362], lithium hydride (LiH) [360,362], beryllium hydride (BeH 2 ) [360], and water (H 2 O) [259], as well as to simulate diazene isomerizations [363] and carbon monoxide oxidation [364].
We are observing a surge of interest to quantum computing applications in chemistry from automobile industry, including BMW [365], Daimler-Benz [366], Nissan [364], Ford 24 , and Toyota [367] (and also Volkswagen [368], see below). Objects of interest include a new generation of electric batteries, combustion efficiency optimization and fuel cells. However, existing simulations have limitations arising not only from available hardware capabilities, but also of conceptual nature, such as the quality of the initial ansatz, convergence speed, presence of local minima, and the large number of measurements required in each iteration [43,44,82]. Active research is underway to overcome some of these challenges via software and hardware improvements of variational quantum computing. However, most significant expectations are associated with the deployment of error correction. This may open doors to simulating various chemical systems of practical relevance, including medium-sized inorganic catalysts, biomimetics, metalorganic molecules, and homogeneous catalysts for C-H bond activation [369]. This latter process is relevant for the production of methanol, which can replace coal and petroleum as a cleaner source of energy and a primary product for synthetic materials.
Longer-term perspectives, requiring scalable faulttolerant quantum computers, are the chemistry of enzyme active sites since they can involve multiple coupled transition metals [370,371], famous examples being the 23 The number of spin orbitals associated with each atom in a molecule equals the maximum electron capacity of the corresponding period in the periodic table: for example, H and He are represented by 2 orbitals, all atoms from Li to F by 10 orbitals, and so on. 24 https://medium.com/@ford/why-ford-is-taking-a-quantum-leapinto-the-future-of-computing-453128a2ea9f four manganese ions in the oxygen evolving complex [372] or the eight transition metals in the iron-sulfur clusters of nitrogenase [373][374][375][376][377][378]. The latter task is crucial for understanding nitrogen fixation by the enzyme nitrogenase, which allows obtaining ammonia at room temperature and standard pressure (so-called FeMoco problem). Solving it would be a major breakthrough in comparison with the state-of-the-art industry Haber-Bosch process, which requires high temperature and high pressure and is therefore energy intensive. The FeMoco problem corresponds to finding the lowest energy state of 108 spin orbital qubits occupied by 54 electrons [375]. A concrete guide and the corresponding quantum circuit for such calculations using a fault-tolerant quantum computer have been presented for the first time by Reiher et al. [375] and improved in Refs [376][377][378]. The most recent conclusion from 2020 shows that FeMoco can be simulated using about four million physical qubits in four days of runtime, assuming 1 µs cycle times and physical gate error rates no worse than 0.1% [378]. b. Life science and drug discovery. Quantum simulation of molecules and chemical reactions has direct application in life science, specifically, in drug discovery [379,380]. Over 99% of the approved drug molecules in DrugBank 5.0 [381] have molecular weights between up to 1800 atomic mass units [82,382]. This allows us to estimate the number of spin orbitals, and therefore the number of qubits, required for modelling these molecules (as per footnote 23) to be on the order of 10 2 to 10 3 . Although this is outside the capacities of current NISQ devices, collaborations between pharmaceutical and quantum computing businesses are starting to emerge. For example, in 2021 Google established a partnership with Boehringer Ingelheim [383].
Quantum simulation tasks in life sciences extend beyond modeling individual molecules to analysis of macroscopic quantum phenomena in molecular clusters. These phenomena are particularly manifest in the photochemistry of conjugated organic molecules interacting with light. Examples include light harvesting in plants and vision in animals [384][385][386].
c. Materials science. The frontier that follows quantum chemistry in the order of complexity is materials science. Unlike a molecule, a crystal is described by an infinite lattice containing infinitely many electrons. For this reason, simulation of real materials can be done only approximately and is hard even for quantum computers. A case in point is high-temperature superconductivity [371]. Originally discovered in 1986, this phenomenon has no comprehensive theoretical explanation to date [387]. Understanding the physics behind hightemperature superconductors is extremely rewarding as it would pave the way toward the "holy grail" -materials with the critical temperature above the room level, which would enable lossless transmission of electrical energy.
Typically, simulating a solid-state lattice is performed within a framework of a certain theoretical model that simplifies the interaction between electrons in this lattice, yet accurately predicts salient properties of the material at hand. Many solid-state phenomena including superconductivity are described by the so-called Hubbard model. This model accounts for only two types of behavior exhibited by electrons: single-electron hopping between lattice sites and two electrons (of different spins) interacting within a single site. In spite of its apparent simplicity, Hubbard model is classically intractable. On the other hand, NISQ computers are capable of analyzing lattices of a few periods in size.
Two approaches are being pursued in this context. The first one is simulation via a digital (gate-based) quantum computer. In this setting, each lattice site is represented by two qubits. For example, an 8-site 1D Fermi-Hubbard model was simulated by the Google Sycamore quantum processor [388]. This quantum analysis is not yet of practical interest since such lattice sizes are also amenable to classical simulation. The quantum advantage threshold is expected at around physical 200 qubits [389], which enables modelling, e.g., a 2D square lattice of 10 × 10 sites.
The second approach involves analog quantum simulators (Sec. V E), currently based primarily on ultracold atoms. These experiments are already bringing new insights into many-body systems that were not known before. For example, experiments with cold atoms in lattices (see Sec. VIII B and Box 13), involving up to 80 lattice sites, enabled detailed reproduction of the primary physical phenomena defining phase transitions in the Fermi-Hubbard model [266,390,391].
In addition to superconductivity, near-term quantum computing can be useful for simulating 2D materials (such as graphene and heterostructures), frustrated spin systems, and materials' dynamical effects [371,392,393]. On the long-term horizon, one may expect that quantum computing will become a main tool for designing bespoke materials with the required properties.
So far in this section we focused on applying quantum computers to solving quantum many-body problems as a whole and found that achieving quantum advantage is beyond the reach of existing technologies. This is particularly because there exist many classical techniques that model even very large quantum systems remarkably well. However, these methods can be further accelerated or made more precise by making a part of calculations quantum.
The current workhorse for the classical simulation of quantum chemistry and materials is density functional theory (DFT). The crux of this method is explained in Ref. [394]: "DFT circumvents the exponential scaling of resources required to directly solve the electronic quantum many-body Hamiltonian by mapping the problem of finding the total energy and particle density of a system to that of finding the energy and particle density of noninteracting electrons in a potential that is a functional only of the electron density, and requiring selfconsistency between the density and potential". That is, the many-body problem is replaced by solving the motion of a single particle in the field created by other particles. However, this approach is insufficient in those settings for which the entanglement is essential for describing the system state. This includes the aforementioned high-temperature superconductivity [387] as well as molecular complexes involving transition metals [395] and actinides [396].
To overcome this restriction, DFT can be supplemented by a quantum treatment of those spin orbitals that are relevant at a particular geometric location. These spin orbitals are considered as an entangled "cluster", whose quantum state is analyzed in the potential that depends on the density of other electrons. The treatment of the cluster can be implemented using either classical or quantum tools. To our knowledge, such an approach has not yet been tried with a bona fide quantum computer, however the simulations by Bauer et al. [394] led to the conclusion that quantum advantage can be reached with 100 logical qubits.
d. Simulation of classical processes. We now proceed to discussing classical simulation problems that can be solved faster using quantum computers. An important class of such problems is solving systems of equations, which is important for a large variety of applications including aerodynamics, hydrodynamics, market dynamics in finance, and disease spreading in epidemiology.
In 2009 Harrow, Hassidim, and Lloyd (HHL) proposed a quantum algorithm [397] (later improved by Ambainis [398] and Childs et al. [399]) that enables solving systems of linear equations with a gate-based quantum computer. The time to solution required by the HHL algorithm scales as the logarithm of the total number of equations, thereby providing exponential speedup with respect to classical algorithms, whose time to solution scales polynomially. However, the HHL algorithm has a limitation in that the output is represented as a multiqubit state with the solution encoded in the amplitudes of qubit configurations [400]. This is an example of the problem mentioned in the Introduction: because the parallelism of quantum computers hinges on their ability to solve problems in a superposition state, the resulting solution will also be in a superposition state. To extract the classical solution (in this case the amplitude of each qubit configuration), one would need to run the quantum algorithm multiple times, each time performing the measurement of the output. The number of such measurements scales exponentially with the number of qubits, thereby negating quantum advantage if HHL is used in a straightforward manner to replace the classical linear equation solver. A further complication associated with HHL is the need to use sophisticated techniques, such as "quantum random access memory" to prepare the input state [401][402][403]. Such memory, in contrast to its classical counterpart, allows one to query memory cells in superposition. This technology has not been implemented in practice, although some setups have been proposed and tested [404].
Although the HHL algorithm has initially been proposed only for solving systems of linear equations, it can be readily extended to a broader range of equations including nonlinear, ordinary differential, nonlinear differential, and partial differential. Indeed, under certain conditions, any such system can be reduced to linear by means of finite element method, which discretizes the parameter space via a finite mesh. In this setting, HHL might be able to achieve quantum advantage in spite of aforementioned challenges [400]. This is because the resulting linear equations in the finite element method are produced algorithmically rather than input as classical dataset. Furthermore, the resulting system of linear equations is typically sparse. For example, this can be the case for the electromagnetic scattering cross-section problem. Clader et al. [405] studied the application of the HHL algorithm in this setting and argued that the exponential speedup can be achieved. However, later the speedup was shown to be polynomial [406].
The HHL algorithm is based on the gate-based quantum computing model and is hence beyond the current level of technology. However, related algorithms, realizable on current NISQ computers, are being proposed in the framework of the variational model for solving linear and nonlinear equation systems. An experimental realization for systems containing up to 1024 equations has been presented in 2020 on a Rigetti 16Q Aspen-4 superconducting quantum computer [407].
The applicability of the finite element method to nonlinear differential equations is however limited [408,409]. For example, chaotic systems, such as fluid dynamics (governed by the Navier-Stokes equation), cannot be solved via this approach. This motivated the development of a suite of quantum algorithms specially designed for this purpose (Gaitan [410], Lloyd et al. [409], and Kyriienko et al. [400]).
A further application of quantum computing to classical simulation is as an alternative to Monte-Carlo methods. The latter is a large family of methods for estimating the properties (e.g., mean and variance) of a statistical distribution by taking multiple samples from that distribution. The quantum alternative is to reduce this problem to estimating the amplitude of a certain state vector in the Hilbert space. This can be done efficiently using a variant of Grover's algorithm. While this approach is applicable in a variety of fields, the current interest appears to be focused on applications in finance [411], particularly for derivative pricing and risk analysis [412,413]. In 2020 researchers from Goldman Sachs and IBM found a quantum advantage for derivative pricing achievable with 7.5k logical (ideal) qubits [414]. This is beyond the capabilities of the existing and upcoming generation of quantum computing devices.
Optimization
Problems of discrete optimization, that is, finding the best solution among a countable set, are ubiquitous in human civilization: from single individuals attempting to choose the best route to work in the morning traffic or the best portfolio for their retirement savings to transnational retailers aiming to find the best schedule for their delivery trucks. The characteristic feature of these problems is the exponential growth of the complexity with the problem size. Many classes of optimization problems might be amenable to quantum speedup.
Of particular relevance here is quadratic unconstrained binary optimization (QUBO, Box 5). The primary quantum approach to solving QUBO problems is via quantum annealing. Experiments to this effect have been attempted on D-Wave System machines for a variety of applications:
• chemistry, specifically, finding ground states of molecules [368];
• life science, including lattice protein folding [415,416] 25 and genome assembly [417,418];
• solving polynomial systems of equations for engineering applications [419] and linear equations for regression [419];
• materials science, in particular, designing metamaterials [420,421];
• likelihood-based regularized unfolding for processing high-energy physics data [422];
• finance, such as portfolio optimization [411,[423][424][425], forecasting crashes [426], finding optimal trading trajectories [427], optimal arbitrage opportunities [428], optimal feature selection in credit scoring [429], and, foreign exchange reserves management [430];
• logistics, including traffic optimization [431][432][433], and scheduling [434][435][436][437][438][439] including railway conflict management [438,439].
Further examples are listed on the website of D-Wave Systems 26 . An approach alternative to quantum annealing is the quantum approximate optimization algorithm (QAOA), which falls within the framework of variational quantum computing (see Sec. V D). Initial proposals on QAQA considered applications to graph optimization, in particular, to the MaxCut problem (see Box 5). Experimentally, this was demonstrated by Google with up to 23 qubits, achieving "an advantage over random guessing but not over some efficient classical algorithms" [84]. Recently, an international collaboration led by Volkswagen applied QAOA to the paint shop problem [440], i.e., minimizing the number of changes of color when painting a certain sequence of cars. This NP-hard optimization problem reduces to QUBO. Solutions for instances of small sizes have been obtained via the IonQ trapped-ion quantum computer [83]. Another use case of a quantum variational algorithm is flight schedule optimization recently presented by Delta Airlines [441].
Finally, a particular case of QUBO, the maximum independent set problem (see Box 5), which has direct applications in network design [442] and finance [443] and is furthermore important for interval scheduling, can be tackled using programmable Rydberg atom simulators (see Sec. VIII B and Box 12). This system allows implementation of either quantum annealing or QAOA. As an example, a collaboration of academics led byÉlectricité de France and a quantum startup Pasqal have applied QAOA in this system to optimize smart-charging of electric vehicles [444]. Quantum advantage in this setting can be expected with as few as 1000-1200 atoms provided that the coherence time is substantially increased [124]. This latter conclusion is supported by a 2021 theoretical study of limitations of optimization algorithms on noisy quantum devices [445], which argues that "substantial quantum advantages are unlikely for classical optimization unless noise rates are decreased by orders of magnitude or the topology of the problem matches that of the device." In some cases, hopes to achieve quantum advantage by reducing an optimization problem to QUBO appear unviable altogether because of an exponential overhead associated with such reduction. Examples include quantum chemistry [368,446] and lattice protein folding [446]. Note that for pretein folding, the issue can be resolved by means of the variational algorithm without reducing the problem to QUBO. An experiment to this effect was performed in 2021 on 9 qubits of an IBM 20-qubit quantum computer [447].
Machine learning
Machine learning techniques are powerful for finding patterns in data. Quantum technology and machine learning are developing rapidly and overlap each other in several contexts, comprising a new field known as quantum machine learning [86]. One can identify three primary directions of these field:
1. classical neural networks for obtaining variational solutions for many-body quantum-mechanical problems;
2. fully-quantum neural networks operating with quantum data, possibly augmented with classical neural networks;
3. quantum algorithms that could act as building blocks of classical machine learning programs.
Item 1 in this list does not typically involve quantum computers [488], therefore it is outside the scope of our review. Item 2 refers to quantum neural networks processing quantum or classical data using a circuit with gates described by continuous parameters [447,460,[489][490][491][492]. This is related to variational quantum computing (see Sec. V D) with the difference being that a variational circuit aims to generate a quantum state optimizing a certain cost function, whereas a quantum neural network is trained to process a more general dataset. For example, quantum convolutional neural networks [477,493] were proposed and used to recognize complex many-body quantum states [477]. Another example is generative adversarial networks [479,480,494] aimed to produce a state whose statistical properties are consistent with an input sample set. This approach has a potential application to facilitate the financial derivative pricing [494] and learning the financial dataset [495] using the Rigetti quantum processor. Aside from the context of variational quantum algorithms, quantum neural networks have not yet been extensively studied and the scope of their practical quantum advantage is undetermined.
In classical machine learning (item 3), linear algebraic calculations, such as Fourier transforms, matrixvector multiplication, diagonalizing matrices, and solving linear systems of equations, constitute the computationally heaviest part. Quantum processing enables polynomial advantage in many of these calculations [86,380,397,488]. A further application of quantum computation in machine learning is sampling from a given probability distribution (see Sec. IX B 2), which is an important component of generative neural networks. Table V lists the known possibilities for applying quantum computing in machine learning as well as associated advantages [86,487]. At the same time, Ref. [86], which is often seen as the "manifest" of quantum machine learning, raises four challenges that must be addressed to achieve these advantages in practice:
1. The input problem. The computational cost of loading classical input data into a quantum register is significant and may exceed that of the quantum computation per se.
2. The output problem. Multiple samples of the output quantum register are required in order to obtain a specific solution of interest.
3. The costing problem. Little is known about the true number of gates required by quantum machine learning algorithms. [86,487]; N is the characteristic layer size). a) see also: https://www.cs.ubc.ca/ nando/papers/quantumrbm.pdf Recent use cases in quantum machine learning [496] include classifiers for handwritten digits datasets (D-Wave [497] and IonQ [265]), analyzing NMR readings (IonQ [264,498]), learning for the classification of lung cancer patients (D-Wave [499]), classifying and ranking DNA to RNA transcription factors (D-Wave [500]), satellite imagery analysis (Rigetti [501]) and weather forecasting (Rigetti [502]), and many others [503]. As previously, we caution the reader that these demonstrations are of proof-of-principle nature and do not yet present quantum advantage.
X. QUANTUM COMPUTING REQUIRES SOFTWARE
Although the implementation of the quantum hardware -long-lived qubits and their interaction mechanisms -constitutes the heart and grand challenge of quantum computational technology, its practical application is impossible without means of its classical control and the software that would enable its programming by a human (i.e., classical) user [504].
We identify three levels of quantum software. The highest level is that of applications, i.e., programs that solve computational problems for the end-user -for example, traffic optimization in a given city or simulating a particular material. This level is both platform-and model-agnostic, that is, the end user need not know what is under the hood of the quantum computational service.
This level interacts with the second-level quantum software, which deals with quantum algorithms -sequences of instructions that implement quantum computation in the language of abstract information carriers (e.g., qubits) within a certain computational model. This level is platform-agnostic, but model-specific. An algorithm that invokes a gate-based quantum calculation will be the same for a trapped-ion or superconducting quantum computer. However, an algorithm for the digital model would not be suitable, e.g., for a quantum annealer.
The final third level operates directly with qubits in a particular machine and is both platform-and modelspecific. This level transforms an abstract quantum algorithm into a sequence of signals that control a physical implementation of qubits, e.g., superconducting junctions, atoms, ions, or photons. Some authors divide this third level into two sublevels: the upper sublevel transforms logical qubits into physical qubits taking into account the specific design of the quantum circuit and the associated error rate; the lower sublevel interacts with physical qubits. Because the lowest-level software is strongly hardware-specific, its development is implemented by quantum hardware manufacturers and is typically hidden from external programmers and users. These manufacturers, however, provide cloud interfaces, programming languages, and software development kits (SDKs), using which an external programmer can write algorithms. These interfaces enable one to address individual qubits, initialize them, perform single-, two-, and multiqubit operations, and measure them. In this way, one can compose, manipulate, and optimize quantum circuits. Examples of such SDKs for the gate-based model include:
• graphical user interface, Python package Qiskit 27 , and quantum assembly language (QASM) 28 by IBM;
• Python library Cirq by Google 29 , which has an important added capability of supporting calculations with qudits;
• assembly-type language Quil by Rigetti 30 ;
• programming language Q# 31 by Microsoft tailored for compatibility with other Microsoft products, such as Visual studio, .NET, and Azure.
In addition, bespoke SDKs are being developed to program special-purpose quantum machines, for example:
• Strawberry Fields by Xanadu 32 , a cross-platform Python library for simulating and developing quantum optical circuitry.
• Ocean SDK 33 developed by D-Wave Systems for solving QUBO and related problems on D-Wave hardware or compatible tools including simulators.
Software of the second (algorithmic) level has been developing for many years, some examples listed in Sec. IX. However, most of existing quantum algorithms have been designed with a perfect multiqubit quantum computer in mind and are, therefore, inapplicable to current NISQ machines. Bridging the desired and available levels of technology constitutes a major challenge in this field. An example of an SDK that attempts to address this challenge is TensorFlow Quantum [505] -a quantum machine learning library for rapid prototyping of hybrid quantum-classical machine learning models. Finally, the first (application) level is only starting to emerge. Interestingly, this level is currently driven not only by hardware giants, but also by startups, which use a business model known as quantum-computing-asa-service model (QCaaS). Examples include AWS Braket by Amazon 34 , QC Ware 35 , Zapata 36 , and QBoard 37 . In the framework of this model, the interaction between a service provider and a client is in the context of a computational problem, which is formulated in the language familiar to the client. The provider works with the client to advise whether and to which extend a quantum computer can be helpful in solving this problem, pick the most suitable quantum computer model(s), and then develops a program that utilizes the second (algorithmic) level quantum software to solve this problem.
We note that in addition to the above discussed software for quantum computers, special software is also required for designing and optimizing quantum hardware [506]; however, we leave this topic outside the scope of the present review.
XI. LEADING COUNTRIES HAVE ANNOUNCED NATIONAL QUANTUM PROGRAMS
The field of quantum science and technology is considered a strategic priority for many countries across the globe. National programs on quantum technologies have been announced by several countries [ 10. Taiwan 38 , Sweden, and Singapore 39 (exact amount is not known).
One may expect this trend to continue, with additional countries publishing their national plans and funding objectives.
XII. THE QUANTUM COMPUTING MARKET IS GROWING
The growth of the quantum computing as field of science, technology, and economy manifests itself in a variety of ways.
• The number of scientific publications increases 40 [517,518]. Quantum computing research accumulated 4703 publications in 10 years, with a 3.39% growth per annum and averaging 14.30 citations per paper during this period [517].
• The number of patent applications has also grown for the last two decades, from tens in 2000 up to hundreds in 2020 41 .
• Over the last 10 years [519], approximately 140 of quantum computing enterprises emerged, of which 43% are hardware and the rest are software.
These factors give rise to the formation of a quantum computing market [519][520][521][522]. Because quantum computing does not yet surpass classical in practical tasks, the market is currently dominated by investments rather than direct sales of hardware or services. According to Gibney [520], by the start of 2019 private investors had funded at least 52 quantum-technology companies globally since 2012 -many of them spin-offs from universities. Although the value of some of the cash infusions remains secret, this study captures the scale of this activity. It finds that, in 2017 and 2018, companies received at least $450 million in private funding -more than four times the $104 million disclosed over the previous two years. Because presently it is universally believed that quantum advantage is within reach, the growth estimates are supremely optimistic. To argue this point, we highlight several forecasts.
• According to Research and Markets 42 , the quantum computing market was valued at $507.1 million in 2019, and is projected to grow at a compound annual growth rate of 56.0%
• According to Market Insights Reports 43 , the Global Quantum Computing Market is expected to witness a compound annual growth rate of 34% during the forecast period 2019-2025, reaching a size of USD 2.82 billion.
• An Inside Quantum Technology report 44 estimates revenues from quantum computing at $1.9 billion USD in 2023, increasing to 8.0 billion USD by 2027.
A recent BCG analysis 45 predicts three phases of the quantum computing market progress:
• the NISQ era, lasting 3-5 years and focusing on scientific and specialized applications, with the estimated market impact of $2-5 billion;
• Broad quantum advantage era, lasting for 10+ years with scientific, specialized, and some generalpurpose applications and an estimated impact of $25-50 billion
• Era of full-scale fault-tolerant quantum computers, lasting for the next 20+ years with scientific, specialized, and various general-purpose applications and an estimated impact of $450-850 billion.
XIII. CONCLUSION AND OUTLOOK
Ten years ago, when we were asked about the time horizon, at which we expect to see practical quantum computing, our answer was "20-30 years". This estimate reflected the lack of realistic roadmap from a conceptual understanding and basic demonstrations, which existed at that time, to a viable product. This situation drastically changed over the last decade. Currently, this roadmap does exist: leading research groups and computing companies confidently plan the development of quantum computers of 1000-qubit size by 2023 46 and error corrected devices by 2029 47 .
Remarkably, this change appears to be not due to any specific scientific discovery or technological breakthrough, but thanks to multiple achievements in various fields leading to the emergence of NISQ devices based on a variety of physical principles. Although these devices are not yet useful for many practical tasks, they serve as playground for testing various quantum computational concepts and models, and therefore as fulcrum for further progress. By building NISQ devices, we learn how build even better NISQ devices and better algorithms, resulting in exponential progress. Perhaps equally important is a psychological side: NISQ technology made people believe that quantum computation is no longer a matter of science fiction, but the reality of humankind's immediate future. Therefore, we are currently at an inflection point heading towards explosive growth of quantum computational technology and the market associated therewith.
Claims of quantum advantage motivate scientists to develop classical algorithms that challenge these claims. Moreover, attempts to simulate quantum computation classically resulted in a new class of algorithms and techniques know as quantum-inspired. This leads in an important side effect of NISQ technology: new classical algorithms for simulating quantum systems [324], optimization [111,523], and data processing [524,525].
Quantum computing is sometimes considered a niche solution for specific problems. But historically the same belief was widely held for both classical computers and the internet. As we now know, both these technologies have progressed far beyond expectations, changing not only the technological or economic landscape, but the entire fabric of our society. These developments were made possible thanks to, first, rapid growth of the capabilities, such as the computational power and communication rate, and, second, wide availability of these technologies not only to a narrow circle of specialists, but to the general population. Until these developments took place, the impact of computation and communication technologies was impossible to predict.
We can extrapolate these expectations to quantum computing. We expect quantum technology to change our society to the same extent as semiconductor technology changed it over past seventy years. This hope however hinges on the same conditions of steadily growing capability and availability. We need to develop the quantum analog of Moore's law, i.e., the situation, in which each new generation of quantum processors surpasses the previous generation by a significant factor. It is not possible to tell at present whether this would be the case. However, we did observe this trend over the past decade and expect this to continue at the same zeal for the next ten or so years.
As more researchers and businesses begin adopting and adapting quantum computing technology, the network effect will start to play a role, allowing for the development and testing of new quantum algorithms and applications, facilitating education, and fostering the understanding of the vectors for further development. This would require wide availability of quantum computers as cloud offerings to a broad range of users for experimentation, tinkering, and even a bit of play.
Figure 1 .
1Linear, polynomial, and exponential scalings. Plots in (a) and (b) are different by the vertical axis scale.
analog of such phone book would place all these entries in an entangled superposition, involving only 17 bytes: |Abbott 123-4567 + |Adams 765-4321 +|Ahmed 222-3333 + |Albrecht 456-7890 + . . .
Figure 3 .
3An iteration of variational quantum computing.
(a) Accessible class of problems. (b) Speed. (c) Cost. 2. Technical criteria. (a) Size (number of elementary quantum units).
Figure 4 .Figure 5 .
45Decoherence time. ii. Fidelity of operations (gates), state preparation and measurements. (c) Dimensionality of elementary quantum units. (d) Duration of an elementary operation. (e) Connectivity. (f) Parallelism. (g) Programmability. D-Wave quantum annealer (reproduced from dwavesys.com): a) Qubit coupling topology. Left: 2000Q processor (chimera unit cell): qubits are shown as circles and couplers as lines. Right: 5000Q processor, also known as Advantage (Pegasus unit cell). Qubits are represented by horizontal and vertical loops. Qubits coupled to qubit #1 (red) are colored. b) Photograph of the D-Wave 5000Q Advantage processor. Boson sampling: a) basic principles (reproduced from Chabaud et al. [117]; b) rendering of the chip (based on a micrograph of the actual device) for Gaussian boson sampling (reproduced from Arrazola et al. [118]).
Figure 6 .
6Google's strategy on error correction (https: //quantumai.google/learn/map).
Figure 7 .
7Google Sycamore superconducting quantum processor, figure and caption are reproduced from Ref.[198]: a) Layout of the processor, showing a rectangular array of 54 qubits (grey), each connected to its four nearest neighbours with couplers (blue). b) Photograph of the Sycamore chip.
Figure 8 .
8Trapped ions quantum computing. a) The ions are held in an electromagnetic trap. Lasers or microwaves are used to control the internal states of qubits, |0 and |1 . The internal control and the Coulomb repulsion between ions combine to form conditional logic gates. Readout is performed by measuring laser-induced ion fluorescence using an auxiliary state |a . The laser-induced fluorescence is also used to cool the ions in preparation for quantum logic (figure and caption are reproduced from Brown et al.[241]. b) Photograph of the IonQ's ion trap (reproduced from ionq.com).
Figure 9 .
9Rydberg-atom quantum computing (reproduced from Ebadi et al.[129]). a) Optical setup. The 2D array of atoms is placed between two powerful microscope objectives that enable their individial addressing and imaging. The trapping fields are created by the spatial light modulator (SLM) and the tweezers are implemented by means of an acousto-optical deflector (AOD). b) Initially loaded atoms are rearranged into defect-free patterns by a set of moving tweezers. Their states can be changed in the programmable manner through Rydberg blockade. The ground |g and Rydberg |r states of the atom constitute a qubit.Box 13. Ultracold atoms in optical lattices.
Figure 10 .
10Quantum simulator based on atoms in optical lattices (reproduced from Mazurenko et al.[266]): Lithium atoms are trapped in a two-dimensional square optical lattice, quantum gas microscope is used for detecting the state of the systems with the single-site resolution.
For example, Sapova et al. in collaboration with Nissan improved VQE to simulate the molecules involved in carbon monoxide oxidation [364]. Kim et al. with Daimler-Benz [366] estimated the cost of simulating electrolyte molecules in Li-ion batteries on a fault-tolerant quantum computer.
Table I .
ISimulating quantum computers using clas-
sical devices. The classical memory required to hold an
equivalent amount of information, as well as the classical
compute time required to implement a single operation on
the given number of qubits are listed. The quoted values
are based on a lecture by Troyer titled "High Performance
Quantum Computing" (https://youtu.be/Hkz_Sn5qYWg).
The number 10 80 bits quoted in the last line is on the order
of the size of the universe, whereas the corresponding time
of 10 50 years is unimaginably longer than the age of the
universe (15 billion years).
Qubits Memory
Equivalent single-operation time
10
16 kByte
microseconds on a watch
20 16 MByte
milliseconds on smartphone
30
16 GByte
seconds on laptop
40
16 TByte
seconds on computer cluster
50
16 TByte
hours on top supercomputer
60
16 EByte minutes on next decade's supercomputer?
70
16 ZByte hours on potential future supercomputer?
80
16 YByte
10 50 years
1. gate-based, also known as digital or cir-
cuit [56-58];
2. adiabatic [59, 60];
3. one-way, also known as cluster-state [61, 62];
Table II .
IIExamples of common quantum gates.Gate name
Notation in
diagrams
Description (for spin qubits)
Example action
Leaves the states with the control qubit in the state |↑ unchanged; transforms |↓ |↑ into |↓ |↓ and vice versa. The state (|↑ + |↓ ) |↑ = |↑ |↑ + |↓ |↑ will transform into the entangled superposition |↑ |↑ + |↓ |↓ .Swaps the states within both pairs {|→ , |→ }
and {|↑ , |↓ }.
Hadamard (H)
H
Rotation of the spin axis by π radi-
ans (180 • ) around the vector halfway
between the x and z axes.
Swaps the states within both pairs {|↑ , |→ } and
{|← , |↓ }.
S
S
Rotation of the spin axis by π/2
radians (90 • ) around the z axis
Leaves the |↑ and |↓ unchanged, but transforms
|→ into the state with the spin along the y axes;
T
T
Rotation of the spin axis by π/4
radians (45 • ) around the z axis
Transforms |→ into the state with the spin
halfway between the x and y axes;
Controlled Not
(CNOT , CX)
X gate applied to the target (bot-
tom) qubit if the control (top) qubit
is in the logical 1 (spin-down) state.
Controlled Z (CZ,
Controlled Phase)
Z
X gate applied to the target (bot-
tom) qubit if the control (top) qubit
is in the logical 1 (spin-down) state.
Leaves the states with the control qubit in the
state |↑ unchanged; transforms |↓ |→ into
|↓ |← and vice versa.
SWAP
2-qubit swap
Toffoli (CCNOT ,
CCX, TOFF)
X gate applied to the target (bot-
tom) qubit if both control qubits
(top and middle) are in the logical 1
(spin-down) state.
Transforms |↓ |↓ |↑ into |↓ |↓ |↓ and vice
versa.
Table V. Overview of quantum machine learning algorithms (based on Refs.Algorithm
Classical
Quantum QRAM Reference
Linear regression
O(N )
O(log(N )) *
Yes [448-451]
Gaussian process regression
O(N )
O(log(N )) †
Yes
[452, 453]
Decision trees
O(N log N )
Unclear
No
[454]
Ensemble methods
O(N )
O(
√
N )
No
[455-457]
Support vector machines
≈ O(N 2 ) − O(N 3 )
O(log N )
Yes
[458-460]
Hidden Markov models
O(N )
Unclear
No
[461]
Bayesian networks
O(N )
O(
√
N )
No
[462, 463]
Graphical models
O(N )
Unclear
No
[96]
k-Means clustering
O(kN )
O(k log N )
Yes
[464-466]
Principal component analysys
O(N )
O(log N )
No
[467]
Persistent homology
O(exp(N ))
O(N 2 )
No
[468]
Gaussian mixture models
O(log(N ))
O(polylog(N ))
Yes
[469, 470]
Variational autoencoder
O(exp(N ))
Unclear
No
[471]
Multilayer perceptrons
O(N )
Unclear
No
[472-476]
Convolutional neural networks
O(N )
O(log N )
No
[477]
Bayesian deep learning
O(N )
O(
√
N )
No
[478]
Generative adversarial networks
O(N )
O(polylog(N ))
No
[479-481]
Boltzmann machines
O(N )
O(
√
N )
No
[336, 337, 482-484] a
Long short-term memory
O(N )
Unclear
No
[485]
Reinforcement learning
Unclear
Unclear
No
[486]
507]: 1. Europe (EU), around 3 billion EUR and special programs in France (1.8 billion) and Germany [507, 508]; 2. Japan, around 1 billion USD [509]; 3. Canada (exact amount is not known; 1 billion USD already invested) [510]; 4. USA (US$1.2 billion over five years in a national quantum initiative) [511]; 5. Australia (exact amount is not known) [512]; 6. Russia, around 0.5 billion USD [513]; 7. UK (1 billion pounds over ten years) [514]; 8. China (exact amount is not known; around 1 billion USD for the past 10 years) [515]; 9. India (80 billion rupees (US$1.12 billion) over five years) [516];
We say that something is "classical" when one can describe its properties without invoking quantum phenomena like entanglement
https://www.businesswire.com/news/home/20180719005280/en/Toshiba-Memory-Corporation-Develops-96-layer-BiCS-FLASH-with-QLC-Technology3 The phenomenon of quantum tunneling allows a quantum particle to traverse energy barriers that are higher than the energy of the state itself, which is prohibited by classical physics. Quantum tunneling plays an essential role in various applications, including tunnel diodes in computing, flash memory, and in the scanning tunneling microscope.
One may notice that the Sgate can be obtained by repeating the Tgate twice. However, the Sgate is still traditionally included in the universal set for the historical reasons described above.
https://ionq.com/algorithmic-qubit-estimator 7 https://www.quantinuum.com/pressrelease/demonstratingbenefits-of-quantum-upgradable-design-strategy-system-model-h1-2-first-to-prove-2-048-quantum-volume
Aaronson, Scott; Granade, Chris (Fall 2006). "Lecture 14: Skepticism of Quantum Computing". PHYS771: Quantum Computing Since Democritus. Shtetl Optimized. Retrieved 2018-12-27. 9 https://ai.googleblog.com/2018/03/ a-preview-of-bristlecone-googles-new.html
https://www.intel.com/content/www/us/en/research/quantumcomputing.html13 These systems are also promising for making quantum repeaters[226] -devices for increasing the quantum communication distances.
The term "Rydberg atom" is jargon; a more accurate term would be "Rydberg state of an atom", see Box 12.
https://psiquantum.com/news/psiquantum-and-globalfoundriesto-build-the-worlds-first-full-scale-quantum-computer 16 https://www.wsj.com/articles/psiquantum-raises-450-million-tobuild-its-quantum-computer-11627387321
https://csrc.nist.gov/projects/random-bit-generation/ documentation-and-software 19 https://www.idquantique.com/random-number-generation/ products/quantis-random-number-generator/ 20 Scott Aaronson. Certified randomness from quantum supremacy.Talk at CRYPTO 2018, October 2018.
https://www.airbus.com/en/innovation/disruptive-concepts/ quantum-technologies/airbus-quantum-computing-challenge 22 https://aws.amazon.com/blogs/quantum-computing/ winners-announced-in-the-bmw-group-quantum-computing-challenge/
Protein folding, i.e. the prediction of the three-dimensional protein structure given a specific amino acid sequence, is a grand challenge in biology. 26 https://www.dwavesys.com/learn/featured-applications/
. The benchmarking problem. The expected degree of polynomial quantum advantage may change because classical algorithms keep developing and improving.
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"Modeling of Negative Autoregulated Genetic Networks in Single Cells",
"Modeling of Negative Autoregulated Genetic Networks in Single Cells"
] |
[
"Azi Lipshtat \nRacah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael\n",
"Hagai B Perets \nRacah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael\n",
"Nathalie Q Balaban \nRacah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael\n",
"Ofer Biham \nRacah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael\n"
] |
[
"Racah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael",
"Racah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael",
"Racah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael",
"Racah Inst. of Physics\nThe Hebrew University\n91904JerusalemIsrael"
] |
[] |
We discuss recent developments in the modeling of negative autoregulated genetic networks. In particular, we consider the temporal evolution of the population of mRNA and proteins in simple networks using rate equations. In the limit of low copy numbers fluctuation effects become significant and more adequate modeling is then achieved using the master equation formalism. The analogy between regulatory gene networks and chemical reaction networks on dust grains in the interstellar medium is discussed. The analysis and simulation of complex reaction networks are also considered.
|
10.1016/j.gene.2004.12.016
|
[
"https://arxiv.org/pdf/q-bio/0504030v1.pdf"
] | 9,060,611 |
q-bio/0504030
|
109da7011e470d5657c53d8028405484031fd341
|
Modeling of Negative Autoregulated Genetic Networks in Single Cells
26 Apr 2005
Azi Lipshtat
Racah Inst. of Physics
The Hebrew University
91904JerusalemIsrael
Hagai B Perets
Racah Inst. of Physics
The Hebrew University
91904JerusalemIsrael
Nathalie Q Balaban
Racah Inst. of Physics
The Hebrew University
91904JerusalemIsrael
Ofer Biham
Racah Inst. of Physics
The Hebrew University
91904JerusalemIsrael
Modeling of Negative Autoregulated Genetic Networks in Single Cells
26 Apr 2005arXiv:q-bio/0504030v1 [q-bio.MN]genetic networksrepressionmaster equation
We discuss recent developments in the modeling of negative autoregulated genetic networks. In particular, we consider the temporal evolution of the population of mRNA and proteins in simple networks using rate equations. In the limit of low copy numbers fluctuation effects become significant and more adequate modeling is then achieved using the master equation formalism. The analogy between regulatory gene networks and chemical reaction networks on dust grains in the interstellar medium is discussed. The analysis and simulation of complex reaction networks are also considered.
Introduction
Recent advances in molecular biology techniques for the engineering of synthetic networks have made possible the measurement of populations of mRNA's and proteins in simple genetic networks. Measurements of the average protein content of cells and their time dependence enabled to quantify the behavior of genetic networks (Kalir et al., 2001). These measurements have been modeled using rate equations, mainly under quasi steady state conditions. However, real biological systems are likely be away from steady state (Smith, 1968, Murray, 1989. Furthermore, many components of cells appear in low copy numbers and are therefore subjected to large fluctuations. Recently, such fluctuations at the level of a single cell were measured experimentally using the green fluorescent protein (GFP) , Paulsson, 2004. Measurements of protein levels in single cells revealed distributions that depend on the topology of the regulatory network controlling the particular protein. For example, it was shown that negative autoregulated networks reduce fluctuations (Becskei and Serrano, 2000). The modeling of these fluctuations cannot be done using rate equations and requires the master equation formalism (McAdams and Arkin, 1997, Paulsson and Ehrenberg, 2000, Paulsson, 2000, Kepler and Elston, 2001, Paulsson, 2002.
In this paper we consider the modeling of negative autoregulated genetic networks in cell populations and in single cells. We focus on the simplest network in which a single protein serves as a repressor for the production of its own mRNA. Such network may serve as a module or "network motif" in complex regulatory networks (Milo et al., 2002(Milo et al., , 2004. We describe the time dependence of the system using rate equations. In commonly used models it is assumed that the population of the bound repressor proteins is in quasi steady state. We consider the dynamics of the network when this assumption does not hold. We show that in such cases the commonly used models underestimate the response of the system to variations in the external conditions. In such cases one should take into account the bound repressors as a separate population. In the limit of low copy numbers of the mRNA's and proteins stochastic noise becomes significant. We show that in this limit the rate equations should be replaced by a master equation. The rate and master equations used in the analysis of genetic networks are closely related to those that describe chemical reaction networks on small grains. In this context, the limit of low copy number is achieved for reaction networks on interstellar dust grains, due to the sub-micron size of the grains and the extremely low flux due to the low density of the interstellar gas. This analogy is discussed and results obtained for grain chemistry, which may also be useful for genetic network analysis, are presented.
The paper is organized as follows: in Sec. 2 we consider the dynamic behavior of a simple genetic network in a cell population using rate equations. In Sec. 3 we consider the limit in which each cell contains a small population of proteins, where the stochastic features become significant. The master equation for this system is presented. The analogy between genetic networks and grain-surface chemistry is discussed in Sec. 4. A summary is presented in Sec. 5.
Rate equations
In genetic autoregulatory circuits the production rate of a certain product protein A depends on its population size, [A] (given by the average number of such proteins in a cell). In negative autoregulation, increasing the population size [A] decreases the rate of production. This mechanism is commonly approximated (Rosenfeld et al., 2002, Paulsson, 2002 by the Hill function
g(A) = g max 1 + k[A](1)
where g(A) is the production rate of A proteins, g max is the maximal production (achieved in conditions where [A] = 0) and k is an affinity constant. This approximation is in agreement with experiments done at steady state (Yagil andYagil, 1971, Yagil, 1975). Here we consider the following circuit: a population size [R] of mRNA's is produced with a maximal rate g R and degrades at rate d R . This mRNA produces a protein A which acts as a repressor and controls the production rate of the mRNA. The production rate of A is thus proportional to [R] and its degradation rate is d A . The intracell dynamics is described by the rate equationṡ
[R] = g R 1 + k[A] − d R [R] [A] = g A [R] − d A [A].(2)
where the dots represent time derivatives, namely[R] = d[R]/dt. These equations have two steady state solutions, however, only one of them is relevant because the other exhibits negative population sizes. The relevant solution is
[R] = d A 2g A 1 k 2 + 4g A g R d A d R k − 1 k [A] = 1 2 1 k 2 + 4g A g R d A d R k − 1 k(3)
and the convergence to this solution is fast (Rosenfeld et al., 2002). However, these equations do not take into account explicitly the chemical mechanism which enables the regulation. In this mechanism, one of the A proteins bounds to the repression site on the DNA and inhibits the mRNA production. This protein should be subtracted from the population of free proteins in the cell, which Eq. (2) does not do. In addition, the constant k in the Hill function captures only the steady state repression rate and not its dynamical behavior.
The dynamics of the repression mechanism can be incorporated into the rate equation by taking the bound protein as a third component in the reaction network. This gives rise to three dynamic equations:
[R] = g R (1 − [r]) − d R [R] [A] = g A [R] − d A [A] − α 0 [A](1 − [r]) + α 1 [r](4)[r] = α 0 [A] α 1 + α 0 [A] .(5)
Substituting this solution into Eq. (4) gives the reduced set of Eq.
(2), with k = α 0 /α 1 . This implies that Eq. (3) is the steady state solution of Eq. (4) as well. This solution is stable and there are no oscillations for any values of the parameters. However, the time dependent solutions of Eq.
(2) and of Eq. (4) are not the same. Whereas Eq.
(2) assumes rapid convergence of [r] into its steady state, Eq. (4) holds also in case that the relaxation time is long. In Figs. 1 and 2 we compare the dynamics described by the two sets of equations. The rate constants are g R = 0.05, g A = 0.06, d R = 0.02, d A = 0.02, α 0 = 0.001 and α 1 = 0.001 (all in units of s −1 ). These rates represent typical transcription and translation times, which are of the order of 10 to 20 seconds. Typical half-life times of proteins and mRNA's vary in the range of several minutes (Elowitz and Leibler, 2000). All these time scales are much shorter than the cycle time, which is typically around 30 minutes.
The dynamical behavior of [A] turns out to be different in the two sets of equations. The deviations from steady state are much larger in the extended set of equations. The dynamics is also highly dependent on the initial condition of [r] which is an additional degree of freedom that does not exist in the reduced set. In Fig. 1 where the initial condition is [r] = 0, the extended set shows an over-shoot in A production, while in Fig. 2 where the initial condition is [r] = 1, it shows an under-shoot in A production.
In some cases the regulation of the production of a protein A is mediated by a more complex molecule. For example, the repressor may be a molecule D which is a dimer of A molecules produced by the reaction A + A → D. The standard way of modeling such a circuit is to modify the repression term (the Hill function) in Eq. (2) [ [
R] = g R 1 + k[A] 2 − d R [R] [A] = g A [R] − d A [A].(6)[R] = g R (1 − [r]) − d R [R] (7a) [A] = g A [R] − d A [A] − 2α 2 [A] 2 (7b) [D] = α 2 [A] 2 − d D [D] − α 0 [D](1 − [r]) + α 1 [r](7cr] = α 0 [D](1 − [r]) − α 1 [r](7d)
As in the ordinary case in which the repressor is the protein A itself, the inhibition term 1 − [r] in Eq. (7b) is equal to the Hill function of Eq. (6) in the limit of rapid relaxation of [r]. In this case k = α 0 α 2 /(α 1 d D ). However, when the repressor is the dimer D, there is an additional term in Eq. (7c) which has no analogue in Eq. (6). This term gives rise to a difference in the results of the reduced and the extended sets even in the steady state solution, as shown in related quantity, namely, the time it takes for a protein to diffuse across the cell was recently measured (Elowitz et al., 1999) and found to be of the order of one second. The inverse of this time can be used as an upper bound for the production rate coefficient α 2 .
The reduced set of equations does not take into account explicitly the dimer population, which is responsible for the repression. Both Eqs. (6) and (7) do not take into account the fact that one needs at least two A proteins simultaneously in the cell in order to produce a dimer. Therefore, when the population of A proteins goes down to order 1 both equations fail and the master equation formalism is required.
The Master Equation
Rate equations are used to describe the dynamics of the average number of entities (such as proteins) in large populations such as those handled in in vitro experiments. In these equations it is assumed that the densities of substances are continuous variables that behave in a deterministic fashion. This approach is not suitable for genetic regulatory networks when the populations of the relevant species in a single cell are small (Gillespie, 1977, Nicolis and Prigogine, 1977, Ko, 1991, 1992, McAdams and Arkin, 1999, Szallasi, 1999, Gibson and Mjolsnes, 2001. In this case one should take into account the discrete nature of the populations and the fact that for small populations the fluctuations become significant. In negative regulatory systems there is a population of free repressors in the cell. In addition, there is a single repression site on the DNA where a single repressor molecule may bound. Therefore, each repression site can be either occupied by a repressor molecule (where r = 1) or vacant (r = 0). Thus, r cannot take any intermediate values. In such cases fluctuations may have an important impact on the processes involved and their dynamics should be described in more detail.
One of the approaches suggested is the use of stochastic simulations which take into account the dynamics of all participating substances (Gillespie, 1977, McAdams and Arkin, 1997, Morton-Firth and Bray, 1998, Gibson and Bruck, 2000. The difficulty with these simulations is that they are based on the accumulation of large amounts of statistical data, and thus require extensive computer simulations. Thus, this approach is not always feasible in the case of complex networks which involve a large number of proteins. A complementary approach is based on direct integration of the the master equation (McAdams and Arkin , 1997, Paulsson and Ehrenberg, 2000, Paulsson, 2000, Kepler and Elston, 2001, Paulsson, 2002. This approach takes into account the probability distribution of all possible states of the system, and not only the average values as in the rate equation approach. It captures the time evolution of the probabilities of all the microscopic states of the system.
We now apply the master equation approach to study the negative autoregulatory circuit of Eq. 4. We denote the number of copies of the free protein A by n A and of the mRNA by n R . The number of proteins A which are bound to the repression site on the DNA is given by n r . For a single repression site n r can only take the values 0 or 1. The master equation follows the time evolution of the probability distribution P (n R , n A , n r ). It takes the forṁ P (n R , n A , n r = 1) = g A n R [P (n R , n A − 1, 1) − P (n R , n A , 1)] + d R [(n R + 1)P (n R + 1, n A , 1) − n R P (n R , n A , 1)] + d A [(n A + 1)P (n R , n A + 1, 1) − n A P (n R , n A , 1)] + α 0 ((n A + 1)P (n R , n A + 1, 0) − α 1 P (n R , n A , 1) (8a)
P (n R , n A , n r = 0) = g A n R [P (n R , n A − 1, 0) − P (n R , n A , 0)] + d R [(n R + 1)P (n R + 1, n A , 0) − n R P (n R , n A , 0)] + d A [(n A + 1)P (n R , n A + 1, 0) − n A P (n R , n A , 0)] − α 0 n A P (n R , n A , 0) + α 1 P (n R , n A − 1, 1) + g R [P (n R − 1, n A , 0) − P (n R , n A , 0)],(8b)
where the two cases of n r = 0 and n r = 1 are presented separately. The first terms in the equations describe the formation of a new protein. The second and third terms describe the degradation of the mRNA and the protein, respectively, while the fourth and fifth terms describe the binding and unbinding of a protein to the repression site on the DNA. Eq. (8b) also includes a term that corresponds to the formation of a new mRNA (not possible in the repressed case). These equations can be integrated numerically in order to obtain the time dependence of the probability distribution. It can also be solved for steady state by takingṖ (n R , n A , n r ) = 0.
The master equation provides all the moments of the distribution P (n R , n A , n r ) and their time dependence. For example, the average population of proteins A is given by
n A = n max R n R =0 n max A n A =0 1 nr=0 n A P (n R , n A , n r )(9)
where n max R and n max A are the cutoff values that provide upper bounds on the populations of mRNA molecules and A proteins in the cell, respectively. The repression site can be either occupied (n r = 1) or unoccupied (n r = 0).
Solving the master equation under steady state conditions for systems with different rate constants we calculated the appropriate averages, and compared the results with the rate equations.
In Fig. 4 the average levels of free proteins, mRNA molecules and bound protein (repressor) in the cell (at steady state), are shown vs. α 0 , as obtained from the master equation (solid line) and the rate equations (dashed line). The rate equations turn out to overestimate the average level of proteins and mRNA molecules, by a factor of 2-4 for systems with low copy number of proteins. On the other hand, when the average number of proteins in the cell is large, the results of the rate equations and master equation coincide. Mathematically the discrepency between the results of the rate equations and the master equations is due to non-linear terms such as the term that describe the attachment rate of proteins to the repression site. In the rate equation, this term is given by α 0 [A](1−[r]), namely as a product of averages (first moments).
In the master equation it is given by the second moment α 0 n A (1 − n R ) . The formation of dimers is also described by a nonlinear term. In the rate equations this term is given by α 2 [A] 2 , namely it depends only on the first moment. In the master equation it is given by α 2 n 2 A − α 2 n A , thus it depends on both the first and second moments.
The simple networks studied here can be considered as modules or motifs in complex genetic networks. However, the simulation of complex networks using the master equation is difficult. This is due to the proliferation in the number of equations as the number of components (mRNA's and proteins) increases. Consider, for example, a network that involves three protein species, A, B and C. The master equation is written in terms of the probabilities P (n A , n B , n C ) of having a certain population of proteins. The population size of each protein is limited by an upper cutoff. For example, the population of protein A takes the values n A = 0, 1, . . . , n max A . Clearly, the number of equations increases exponentially with the number of species, making this approach infeasible for complex networks.
However, typically these networks are sparse, namely most pairs of proteins do not interact with each other. This feature makes it possible to divide the master equation into several sets of equations, each set including only a small number of protein species. For example, if proteins B and C do not interact, the master equation described above can be broken into two sets that involve P AB (n A , n B ) and P AC (n A , n C ). In the case of large and sparse networks this dramatically reduces the number of equations and thus enables the simulation of complex networks using the master equation. This technique, named the multi-plane method, was recently proposed in the context of chemical reaction networks on interstellar dust grains (Lipshtat and Biham, 2004). The mathematical structure of these networks is similar to that of genetic networks. Thus, the multi-plane method is perfectly applicable for the simulations of complex genetic networks. The similarity between the two systems is briefly discussed below. Table 1 Analogy between the processes of surface chemistry and of gene regulation.
Description
Surface chemistry Gene regulation system dust grain cell break-up mechanism grain fragmentation cell division mobility surface diffusion diffusion in cell
addition ∅ → A flux F transcription g R , production g A removal A → ∅ desorption W degradation d R , d A typical reaction A + B → C + D A → A + B feedback regulation rejection: F (1 − θ) repression: g R (1 − [r])
bon and nitrogen. Here we discuss the similarity between the mathematical descriptions surface reaction networks and genetic networks. In particular, we suggest that computational methodologies developed in the context of interstellar grain chemistry are likely to be useful for the analysis of genetic networks.
Consider a dust grain exposed to a flux of atomic and molecular species such as H, O, OH and CO. Atoms and molecules that hit and stick to the grain hop as random walkers between adsorption sites on its surface. When two atoms/molecules encounter one another they may react and form a more complex molecule. The rate equations that describe the reaction networks on grains include flux terms, desorption terms and reaction terms. The flux terms represent the flow of atoms and molecules from the gas phase onto the surface. The desorption rates are proportional to the population sizes of atoms and molecules on the grains, while the reaction terms are proportional to the products of the population sizes of the reactive species. In general, the rate equations resemble those that describe genetic networks. The analogy between the two systems is summarized in Table 1. In both systems reactive species are added, diffuse, react and removed. The system itself may break up (cell division or grain fragmentation), dividing the population of reactive species into two sub-populations. Both systems exhibit some kind of negative feedback. In genetic networks this is provided by the repression circuit, in which the rate of attachment of proteins to the repression site is given by
α 0 [A](1 − [r]
). Certain surface reaction systems exhibit the Langmuir rejection behavior, in which atoms from the gas phase that hit the surface in the vicinity of an already adsorbed atoms are rejected. The flux term F is then modified to the form F (1 − θ), where θ is the coverage, namely the fraction of adsorption sites on the surface that are occupied by adsorbed atoms. In the context of grain-surface chemistry, low copy numbers are obtained in the limit of small grains under conditions of low flux. In this limit the master equation is required (Biham et al., 2001, Green et al., 2001, Biham and Lipshtat, 2002. For complex reaction networks of multiple species, the master equation becomes infeasible due to the proliferation in the number of equations. In this case, the multi-plane method is used in order to keep the number of equations at a tractable level (Lipshtat and Biham, 2004).
Summary
We have considered the rate equation and master equation approaches to the modeling of genetic networks. In particular, we have studied the temporal evolution of the population of mRNA and proteins in simple negative autoregulated genetic networks. As long as the populations of all the reactive components of the network are not too small, rate equations provide a good quantitative description of the network dynamics. However, once the populations of the mRNA or proteins are reduced to order 1 or less, rate equations are no longer suitable and the master equation is needed. This is due to the fact that the rate equations involve only average quantities, while the master equation takes into account the discrete nature of the populations as well as the fluctuations. The simple networks studied here can be considered as modules or motifs in complex genetic networks. The simulation of complex networks using the master equation is difficult, because the number of equations quickly proliferates. The multi-plane methodology, recently developed in the context of grain-surface chemistry, that tackles this problem is briefly described. Finally, the analogy between genetic networks and grain-surface chemistry is discussed.
We thank J. Paulsson for illuminating discussions.
Fig. 1 .
1Intracell dynamics as calculated by Eq. (2) (solid line) and by Eq. (4) (dashed line). The average amount of bound proteins [r] is also shown (dotted line). The initial conditions are [A] = 3 and [r] = 0.
Fig. 2 .
2Intracell dynamics as calculated by Eq. (2) (solid line) and by Eq. (4) (dashed line). The average amount of bounded proteins [r] is also shown (dotted line). The initial conditions are [A] = 3 and [r] = 1.
Fig 3 .Fig. 3 .
33The steady state solution of the extended set is stable and exhibits no oscillations. The parameters used in Fig 3 are the same as in Figs. 1 and 2, and the additional parameters are the degradation rate of dimers, d D = 0.02 (s −1 ), and the production rate coefficient of dimers, α 2 = 0.01 (s −1 ). The latter coefficient is determined by the diffusion rate of proteins in the cell. The populations of proteins A as obtained from the reduced set (solid line) and from the extended set (dashed line) and of dimers (repressor) D (dashed-dotted line) and bound repressors r (dotted line) as obtained from the extended set, as a function of time. The initial conditions are [A] = 3, [D] = 1 and [r] = 1.
Fig. 4 .
4The steady state populations of free proteins, mRNA's and bound proteins (repressor) vs. the rate constant α 0 , calculated using the master equation (solid line) and the rate equations (dashed line).
For this system the extended set includes equations for [R] and [A], as well as for the dimer (repressor) population[D] and for the bound repressor[r]. The equations take the form:
Discussion: Genetic Networks and Grain-Surface ChemistryProcesses which exhibit a similar mathematical structure to the genetic network dynamics appear in the context of chemical reaction networks on interstellar dust grains. The chemistry of interstellar clouds consists of reactions taking place in the gas phase as well as on the surfaces of dust grains(Hartquist and Williams, 1995). It turns out that the most abundant molecule in the Universe, namely molecular hydrogen does not form in the gas phase but on dust grain surfaces(Gould and Salpeter, 1963. These grains are made of amorphous silicate and carbon compounds and are of sub-micron size. In addition to the formation of molecular hydrogen, these grains support complex reaction networks that produce a variety of molecules that consist of hydrogen, oxygen, car-
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[] |
[
"Relation between baryon number fluctuations and experimentally observed proton number fluctuations in relativistic heavy ion collisions",
"Relation between baryon number fluctuations and experimentally observed proton number fluctuations in relativistic heavy ion collisions"
] |
[
"Masakiyo Kitazawa \nDepartment of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan\n",
"Masayuki Asakawa \nDepartment of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan\n"
] |
[
"Department of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan",
"Department of Physics\nOsaka University\n560-0043ToyonakaOsakaJapan"
] |
[] |
We explore the relation between proton and nucleon number fluctuations in the final state in relativistic heavy ion collisions. It is shown that the correlations between the isospins of nucleons in the final state are almost negligible over a wide range of collision energy. This leads to a factorization of the distribution function of the proton, neutron, and their antiparticles in the final state with binomial distribution functions. Using the factorization, we derive formulas to determine nucleon number cumulants, which are not direct experimental observables, from proton number fluctuations, which are experimentally observable in event-by-event analyses. With a simple treatment for strange baryons, the nucleon number cumulants are further promoted to the baryon number ones. Experimental determination of the baryon number cumulants makes it possible to compare various theoretical studies on them directly with experiments. Effects of nonzero isospin density on this formula are addressed quantitatively. It is shown that the effects are well suppressed over a wide energy range.
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10.1103/physrevc.86.024904
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[
"https://arxiv.org/pdf/1205.3292v1.pdf"
] | 116,983,283 |
1205.3292
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8e906a5a9750e706a7ea85bd19ef0e7812cfaa22
|
Relation between baryon number fluctuations and experimentally observed proton number fluctuations in relativistic heavy ion collisions
15 May 2012 (Dated: May 1, 2014)
Masakiyo Kitazawa
Department of Physics
Osaka University
560-0043ToyonakaOsakaJapan
Masayuki Asakawa
Department of Physics
Osaka University
560-0043ToyonakaOsakaJapan
Relation between baryon number fluctuations and experimentally observed proton number fluctuations in relativistic heavy ion collisions
15 May 2012 (Dated: May 1, 2014)numbers: 1238Mh2575Nq2460Ky
We explore the relation between proton and nucleon number fluctuations in the final state in relativistic heavy ion collisions. It is shown that the correlations between the isospins of nucleons in the final state are almost negligible over a wide range of collision energy. This leads to a factorization of the distribution function of the proton, neutron, and their antiparticles in the final state with binomial distribution functions. Using the factorization, we derive formulas to determine nucleon number cumulants, which are not direct experimental observables, from proton number fluctuations, which are experimentally observable in event-by-event analyses. With a simple treatment for strange baryons, the nucleon number cumulants are further promoted to the baryon number ones. Experimental determination of the baryon number cumulants makes it possible to compare various theoretical studies on them directly with experiments. Effects of nonzero isospin density on this formula are addressed quantitatively. It is shown that the effects are well suppressed over a wide energy range.
I. INTRODUCTION
Now that the observation of the quark-gluon matter in relativistic heavy ion collisions is established for small baryon chemical potential (µ B ) [1], a challenging experimental subject following this achievement is to reveal the global structure of the QCD phase diagram on the temperature (T ) and µ B plane. In particular, finding the QCD critical point(s), whose existence is predicted by various theoretical studies [2,3], is one of the most intriguing problems. Since the µ B of the hot medium created by heavy-ion collisions can be controlled by varying the collision energy per nucleon pair, √ s NN , the µ B dependence of the nature of QCD phase transition should be observed as the √ s NN dependence of observables. An experimental project to explore such signals in the energy range 10GeV √ s NN 200GeV, which is called the energy scan program, is now ongoing at the Relativistic Heavy Ion Collider (RHIC) [4,5]. Experimental data which will be obtained in future experimental facilities designed for lower beam-energy collisions will also provide important information on this subject [6].
Observables which are suitable to analyze bulk properties of the matter around the phase boundary of QCD in heavy ion collisions are fluctuations [7]. Experimentally, fluctuations are measured through event-by-event analyses [4]. Theoretically, it is predicted that some of them, including higher-order cumulants, are sensitive to critical behavior near the QCD critical point [8][9][10][11][12], and/or locations on the phase diagram, especially on which side the system is, the hadronic side or the quark-gluon side [13][14][15][16][17][18][19]. * Electronic address: [email protected] † Electronic address: [email protected] Among the fluctuation observables, those of conserved charges are believed to possess desirable properties to probe the phase structure in relativistic heavy ion collisions. One of the advantages of using the conserved charges is that the characteristic times for the variation of their local densities are longer than those for nonconserved ones, because the variation of the local densities of conserved charges are achieved only through diffusion [13,14]. The fluctuations of the former thus can better reflect fluctuations generated in earlier stages of fireballs, when the rapidity coverage is taken sufficiently large. From a theoretical point of view, an important property of the conserved charges is that one can define the operator of a conserved charge, Q, as a Noether current. Moreover, their higher-order cumulants, δQ n c , are directly related to the grand canonical partition function Z(µ) = Tre −β(H−µQ) as δQ n c = T n ∂ n log Z(µ) ∂µ n ,
with H and µ being the hamiltonian and the chemical potential associated with Q, respectively. These properties make the analysis of cumulants of conserved charges well defined and feasible in a given theoretical framework. For example, they can be measured in lattice QCD Monte Carlo simulations [20][21][22][23][24]. The relation Eq. (1) also provides an intuitive interpretation for the behavior of higher-order cumulants of conserved charges. For instance, the third-order cumulant of the net baryon number, N ) 2 has a peak structure [17]. The change of the sign of observables like this will serve as a clear experimental signal [17][18][19].
QCD has several conserved charges, such as baryon and electric charge numbers and energy. Among these conserved charges, theoretical studies suggest that the cumulants of the baryon number have the most sensitive dependences on the phase transitions and phases of QCD. In order to see this feature, let us compare the baryon number cumulants with the electric charge ones. First, the baryon number fluctuations show the critical fluctuations associated with the QCD critical point more clearly. Although the baryon and electric charge number fluctuations diverge with the same critical exponent around the critical point, it should be remembered that this does not mean similar clarity of signals for the critical enhancement in experimental studies. Fluctuations near the critical point are generally separated into singular and regular parts, and only the former diverges with the critical exponent. The singular part of the electric charge fluctuations is relatively suppressed compared to the baryon number ones, because the formers contain the isospin number fluctuations which are regular near the critical point [9]. The additional regular contribution makes the experimental confirmation of the enhancement of the singular part difficult, and this tendency is more pronounced in higher-order cumulants [17]. While it is known that the proton number fluctuations in the final state also reflect the critical enhancement near the critical point [9], as we will show later the baryon number fluctuations are superior to this observable, too, in the same sense. Second, the ratios of baryon number cumulants [16] behave more sensitively to the difference of phases, i.e., hadrons, or quarks and gluons. This is because the ratios are dependent on the magnitude of charges carried by the quasi-particles composing the state [13,14,16], while the charge difference between hadrons and quarks is more prominent in the baryon number.
Experimentally, however, the baryon number fluctuations are not directly observable, because chargeless baryons, such as neutrons, cannot be detected by most detectors. Proton number fluctuations can be measured [4,5], and recently its cumulants have been compared with theoretical predictions for baryon number cumulants. Indeed, in the free hadron gas in equilibrium the baryon number cumulants are approximately twice the proton number ones, because the baryon number cumulants in free gas are simply given by the sum of those for all baryons, and the baryon number is dominated by proton and neutron numbers in the hadronic medium relevant to relativistic heavy ion collisions. In general, however, these cumulants behave differently. In fact, we will see later that the non-thermal effects which exists in baryon number cumulants are strongly suppressed in the proton number ones.
In heavy ion collisions, because of the dynamical evolution the medium at kinetic freezeout is not completely in the thermal equilibrium. The original ideas to exploit fluctuation observables as probes of primordial properties of fireballs [13,14] are concerned with this non-thermal effect encoded in the final state as a hysteresis of the time evolution. To observe such effects, it is highly desirable to measure baryon number cumulants that is expected to retain more effects of the phase transition and the singularity around the critical point. The experimental determination of baryon number cumulants also makes the comparison between experimental and theoretical studies more robust, since many theoretical works including lattice QCD simulations are concerned with the baryon number cumulants, not the proton number ones.
In Ref. [25], the authors of the present paper have argued that, whereas the baryon number cumulants are not the direct experimental observables as discussed above, they can be determined in experiments by only using the experimentally measured proton number fluctuations for √ s NN 10GeV. The key idea is that isospins of nucleons in the final state are almost completely randomized and uncorrelated, because of reactions of nucleons with thermal pions in the hadronic stage, as will be elucidated in Sec. II. This leads to the conclusion that, when N N nucleons exist in a phase space of the final state, the probability that N p nucleons among them are protons follows the binomial distribution. More generally, the probability distribution that N p protons, N n neutrons, Np anti-protons, and Nn anti-neutrons are found in the final state in a phase space is factorized as
P N (N p , N n , Np, Nn) = F (N N , NN)B r (N p ; N N )Br(Np; NN),(2)
where the nucleon and anti-nucleon numbers are N N = N p + N n and NN = Np + Nn, respectively, and
B r (k; n) = n! k!(n − k)! r k (1 − r) n−k(3)
is the binomial distribution function with probabilities r = N p / N N andr = Np / NN . The function F (N N , NN) describes the distribution of nucleons and anti-nucleons and the correlation between them in the final state, which are determined by the dynamical history of fireballs. Using the factorization Eq. (2), one can obtain formulas to represent the (anti-)nucleon number cumulants by the (anti-)proton number ones, and vice versa; whereas the neutron number is not determined by experiments, this missing information can be reconstructed with the knowledge for the distribution function, Eq. (2). The (anti-)nucleon number in Eq. (2) can further be promoted to the (anti-)baryon number in practical analyses with a simple treatment for strange baryons to a good approximation. These formulas enable to determine the baryon number cumulants solely with the experimentally measured proton number fluctuations, and, as a result, to obtain insights into the present experimental results on the proton number cumulants.
The main purpose of the present paper is to elaborate the discussion in Ref. [25] with some extensions. In Ref. [25] the formulas are derived only for isospin symmetric medium. In the present study we extend them to incorporate cases with nonzero isospin densities. With the extended relations, it is shown that the effect of nonzero isospin density is well suppressed for √ s NN 10GeV. The procedures of the manipulations and discussions omitted in Ref. [25] are also addressed in detail.
In the next Section, we show that the factorization Eq. (2) is well applied to the nucleon and baryon distribution functions in the final state in heavy ion collisions. We then derive formulas to relate baryon and proton number cumulants in Sec. III. In Sec. IV, we discuss the recent experimental results at STAR [4,5] using the results in Sec. III, and possible extensions of our results. The final section is devoted to a short summary.
Throughout this paper, we use N X to represent the number of particles X leaving the system after each collision event, where X = p, n, N, and B represent proton, neutron, nucleon, and baryon, respectively, and their anti-particles,p,n,N, andB. The net and total numbers are defined as N (net) X = N X −NX and N (tot) X = N X +NX, respectively.
II. DISTRIBUTION FUNCTION FOR PROTON AND NEUTRON NUMBERS
In this section, we discuss the time evolution of the proton and neutron number distributions in the hadronic medium generated by relativistic heavy ion collisions, and show that the nucleon distribution in the final state in a phase space is factorized as in Eq. (2) at sufficiently large √ s NN . In Sec. II A, as a preliminary example we show that Eq. (2) is applicable to the equilibrated free hadron gas in the ranges of T and µ B relevant to relativistic heavy ion collisions. We then extend the argument to the distribution function in the final state in relativistic heavy ion collisions in Sec. II B.
A. Free hadron gas in equilibrium
Let us first consider nucleons in the free hadron gas in equilibrium. For T and µ B which are relevant to relativistic heavy ion collisions, the nucleon mass m N satisfies m N − |µ B | ≫ T . One thus can apply the Boltzmann approximation for the distribution functions of nucleons. The number of particles in a phase space, N , which obey Boltzmann statistics is given by the Poisson distribution,
P λ (N ) = e −λ λ N N ! ,(4)
with the average λ = N = N N P λ (N ). Accordingly, the probability to find N p (Np) protons (anti-protons) and N n (Nn) neutrons (anti-neutrons) in the phase space is given by the product of the Poisson distribution functions, P HG (N p , N n , Np, Nn) = P Np (N p )P Nn (N n )P Np (Np)P Nn (Nn).
The product of two Poisson distribution functions satisfies the identity, (6) where B r (k; n) is the binomial distribution function Eq. (3). Using Eq. (6), Eq. (5) is rewritten as (7) where N N = N p + N n and NN = Np + Nn are the nucleon and anti-nucleon numbers, respectively, and r = N p / N N andr = Np / NN . Equation (7) shows that the distribution of nucleons in the free hadron gas is factorized using binomial functions as in Eq. (2) with
P λ1 (N 1 )P λ2 (N 2 ) = P λ1+λ2 (N 1 + N 2 )B λ1/(λ1+λ2) (N 1 ; N 1 + N 2 ),P HG (N p , N n , Np, Nn) = P NN (N N )P NN (NN)B r (N p ; N N )Br(Np; NN),F (N N , NN) = P NN (N N )P NN (NN).(8)
The appearance of the binomial distribution functions in Eq. (7) is understood as follows. When one finds a nucleon in the hadron gas, the probability that the nucleon is a proton is r. The isospins of all nucleons found in the phase space, moreover, are not correlated with one another as a consequence of Boltzmann statistics and the absence of interactions. Once N N nucleons are found in the phase space, therefore, the probability that N p particles are protons is a superposition of independent events with probability r, i.e., the binomial distribution.
We note that the above discussion is not applicable when the condition m N − |µ B | ≫ T , required for Boltzmann statistics, is not satisfied. When quantum correlations of nucleons arising from Fermi statistics are not negligible, the isospin of each nucleon can no longer be independent. As long as we are concerned with the range of T and µ B which can be realized by relativistic heavy ion collisions, however, the condition for the Boltzmann approximation is well satisfied except in very low energy collisions [26].
B. Final state in heavy ion collisions
Next, we consider the nucleon distribution functions in the final state in heavy ion collisions. We show that the nucleon distribution in this case is also factorized as in Eq. (2), by demonstrating that the isospins of all nucleons in the final state are random and uncorrelated.
∆(1232) resonance
The key ingredient to obtain the factorization Eq. (2) in the final state in relativistic heavy ion collisions is Nπ reactions in the hadronic stage mediated by ∆(1232) resonances having the isospin I = 3/2. As we will see later, this is the most dominant reaction of nucleons in the hadronic medium. This is because i) the cross section of Nπ → ∆ reactions exceeds 200mb = 20fm 2 and is comparable with NN and NN reactions for P lab ≃ 300MeV [27], and ii) the pion density dominates over those of all other particles in the ranges of T and µ B accessible with heavy ion collisions at √ s NN 10GeV; at the top RHIC energy, the density of pions is more than one order larger than that of nucleons. We shall show below that these reactions frequently take place even after chemical freezeout in the hadronic medium during the time evolution of the fireballs.
The Nπ reactions through ∆ contain charge exchange reactions, which alter the isospin of the nucleon in the reaction. The reactions of a proton to form ∆ are:
p + π + → ∆ ++ → p + π + ,(9)p + π 0 → ∆ + → p(n) + π 0 (π + ),(10)p + π − → ∆ 0 → p(n) + π − (π 0 ).(11)
Among these reactions, Eqs. (10) and (11) are responsible for the change of the nucleon isospin. The ratio of the cross sections of a proton to form ∆ ++ , ∆ + , and ∆ 0 is 3 : 1 : 2, which is determined by the isospin SU(2) symmetry of the strong interaction. The isospin symmetry also tells us that the branching ratios of ∆ + (∆ 0 ) decaying into the final state having a proton and a neutron are 1 : 2 (2 : 1). Using these ratios, one obtains the ratio of the probabilities that a proton in the hadron gas forms ∆ + or ∆ 0 with a reaction with a thermal pion, and then decays into a proton and a neutron, respectively, P p→p and P p→n , as P p→p : P p→n = 5 : 4,
provided that the hadronic medium is isospin symmetric and that the three isospin states of the pion are equally distributed in the medium. Because of the isospin symmetry of the strong interaction one also obtains the same conclusion for neutron reactions:
P n→n : P n→p = 5 : 4.(13)
Similar results are also obtained for anti-nucleons. Equations (12) and (13) show that these reactions act to randomize the isospin of nucleons during the hadronic stage.
Mean time
Next, let us estimate the mean time of these reactions. Assuming that pions are thermally distributed, the mean time τ ∆ of a nucleon at rest in the medium to undergo a reaction Eq. (10) or (11) is given by
τ −1 ∆ = d 3 k π (2π) 3 σ(E c.m. )v π n(E π ),(14)
with the Bose distribution function n(E) = (e E/T −1) −1 , the pion momentum k π , the pion velocity v π = k π /E π , E π = m 2 π + k 2 π , and the pion mass m π . σ(E c.m. ) is the
E c.m. = [(m N + E π ) 2 − k 2 π ] 1/2 with the nucleon mass m N .
For the cross section σ(E c.m. ), we assume that the peak corresponding to ∆(1232) resonance is well reproduced by the Breit-Wigner form,
σ(E c.m. ) = σ ∆ Γ 2 /4 (E c.m. − m ∆ ) 2 + Γ 2 /4 ,(15)
which is a sufficient approximation for our purpose.
Here, we use the value of the parameters determined by the Nπ reactions in the vacuum, m ∆ = 1232MeV, Γ = 110MeV, and σ ∆ = 20fm 2 [27]. The medium effects on the cross section will be discussed later. Substituting m N = 940MeV and m π = 140MeV, one obtains the T dependence of the mean time τ ∆ presented in Fig. 1. The figure shows that the mean time is τ ∆ = 3 ∼ 4fm for T = 150 ∼ 170MeV. One can confirm that the mean time hardly changes even for moving nucleons in the range of momentum p 3T by extending Eq. (14) to cases with nonzero nucleon momentum. The lifetime of ∆ resonances is τ Γ = 1/Γ ≃ 1.8fm.
The mean time evaluated above is much shorter than the lifetime of the hadronic stage in relativistic heavy ion collisions. According to a dynamical model analysis for collisions at RHIC, nucleons in the hadron phase continue to interact for a couple of tens of fm on average at midrapidity [28]. As a result, at the RHIC energy each nucleon in a fireball has chances to undergo the charge exchange reactions several times in the hadronic stage.
Two remarks are in order here. First, the above result on the time scales shows that the reactions to produce ∆ proceed even after chemical freezeout. These reactions do not contradict the success of the statistical model, which describes the chemical freezeout [29], because chemical freezeout is a concept to describe ratios of particle abundances such as Np / N p and the above reactions do not alter the average abundances in the final state. The success of the model, on the other hand, indicates that creations and annihilations of (anti-)nucleons hardly occur after chemical freezeout. Second, we note that the dynamical model in Ref. [28] uses an equation of states having a first order phase transition in the hydrodynamic simulations for the time evolution above the critical temperature T c . Recently, dynamical simulations have been carried out with more realistic equations of states obtained by lattice QCD simulations [30]. The lifetime of hadronic stage evaluated in these studies is more relevant to this argument. We, however, note that the qualitative behavior of the time evolution seems not sensitive to the difference in equations of states [30].
While Nπ reactions frequently take place even below the chemical freezeout temperature, T chem , NN annihilatios and productions almost terminate at T chem . This is necessary for the success of the thermal model. For E c.m. ≃ T the cross section of the NN pair annihilation is largest among all NN and NN reactions. If nucleons and anti-nucleons are distributed without correlation, therefore, all NN and NN reactions cease to take place at T chem . This conclusion is, of course, obtained also by evaluating the mean time for each reaction using the cross sections [27] as in Eq. (14). After chemical freezeout, the only inelastic reactions nucleons go through are thus Eqs. (10) and (11), and after each reaction the nucleon loses its initial isospin information. Only after repeating the reactions Eq. (12) twice, the ratio becomes 41 : 40, which is almost even. If medium effects on the formations and decays of ∆ are negligible, therefore, irrespective of the nucleon distribution at the chemical freezeout, the isospin of nucleons at the kinetic freezeout can be regarded random and uncorrelated. On the other hand, the nucleon number distribution can have a deviation from the Boltzmann distribution reflecting the dynamical history of fireballs.
Because of the absence of correlations between isospins of nucleons in the final state, once N N (NN) nucleons (anti-nucleons) exist in a phase space in the final state, their isospin distribution is simply given by the binomial one. This conclusion leads to the factorization Eq. (2) for proton and neutron number distribution in the final state for an arbitrary phase space. In particular, the final state proton and anti-proton number distribution is written as
G(N p , Np) = Nn,Nn P N (N p , N n , Np, Nn) = NN,NN F (N N , NN)B r (N p ; N N )Br(Np; NN).(16)
Unlike in the simple example in Sec. II A, the nucleon distribution function F (N N , NN) in this case is determined by the time evolution of fireballs and is not necessary of a thermal or separable form as in Eq. (8); no specific form for F (N N , NN) is assumed here or will be assumed in the analyses in Sec. III. What we have used here is the fact that the time scale for the exchange of isospins between nucleons and pions is sufficiently short compared to the lifetime of hadronic stage after the chemical freezeout. On the other hand, the time scale for the variation of a conserved charge in a phase space depends on the form of the phase space, and can become arbitrary long by increasing the spatial volume. When the time scale is long, the information of the physics of the early stages is encoded in F (N N , NN).
Medium effects
Next, let us inspect the possibility of medium effects on the formation and decay rates of ∆. In medium, the decay rate of ∆ acquires the statistical factor,
(1 − f (E N )) (1 + n(E π )) ,(17)
where f (E) = (e (E−µB)/T +1) −1 is the Fermi distribution function and E N and E π are the energies of the nucleon and pion produced by the decay, respectively. The first term in Eq. (17) represents the Pauli blocking effect. At the RHIC energy, since the Boltzmann approximation is well applied to nucleons, the Pauli blocking effect is suppressed. The Bose factor (1 + n(E π )) in Eq. (17), on the other hand, has a non-negligible contribution since m π ≃ T chem . As long as all n(E π ) for the three isospin states of the pion are the same, however, this factor does not alter the branching ratios Eqs. (12) and (13), while the factor enhances the decay of ∆. A possible origin for the variation of n(E π ) is the isospin density of nucleon number; since the isospin density is locally conserved, the isospin density of pions is affected by the nucleon isospin. This effect on n(E π ) is, however, well suppressed since the density of pions is much larger than that of nucleons below T chem . Another possible source which gives rise to a different pion distribution is the event-byevent fluctuation of the isospin density in the phase space at the hadronization. It is, however, expected that the effect is well suppressed, again because of the large pion density. One, therefore, can conclude that the medium effect hardly changes the branching ratios Eqs. (12) and (13). The same conclusion also applies to the formation rate of ∆, since the medium effect on the probabilities of a nucleon to undergo reactions Eqs. (9) -(11) depends only on n(E π ). After all, all medium effects on the ratios Eqs. (12) and (13) are negligible.
When the system has a nonzero isospin density, probabilities Eqs. (12) and (13) receive modifications because the three isospin states of the pion are not equally distributed, although this effect is not large as will be shown in Sec. III D. Even in this case, however, the only modification to the above conclusion is to replace the probabilities r andr with appropriate values, since the reactions Eqs. (9) -(11) still act to randomize the nucleon isospins with the modified probabilities determined by the detailed balance condition.
Here, we emphasize that the large pion density in the hadronic medium is responsible for the validity of Eq. (16) in the final state. In the hadronic medium, there are so many pions which can be regarded as a heat bath when the nucleon sector is concerned, while nucleons are so dilutely distributed that they do not feel other ones' existence.
So far, we have limited our attention to reactions mediated by ∆(1232). Interactions of nucleons with other mesons, however, can also take place in the hadronic medium, while they are much less dominant. It is also possible that ∆ interacts with thermal pions to form another resonance before its decay [31]. All these reactions with thermal particles, however, proceed with certain probabilities determined by the isospin SU(2) symmetry as long as they are caused by the strong interaction. Each reaction of a nucleon thus makes its isospin random, and act to realize the factorization Eq. (2).
Low beam-energy region
The factorization Eq. (16) is fully established for the RHIC energy. At very low beam energy, however, pions are not produced enough and the duration of the hadron phase below T chem becomes shorter. Nucleons, therefore, will not undergo sufficient charge exchange reactions below T chem . When the reactions hardly occur, the isospin correlations generated at the hadronization remain until the final state. At low beam energy, also the density of the nucleon becomes comparable to that of pions, and pions can no longer be regarded as a heat bath to absorb isospin fluctuations of nucleons. The requirements to justify the factorization Eq. (16), therefore, eventually breaks down as the beam energy is decreased. This would happen when T chem m π , since the abundance of pions is responsible for all of the above conditions. From the √ s NN dependence of the chemical freezeout line on the T -µ B plane [26], the factorization Eq. (16) should be wellsatisfied in the range of beam energy √ s NN 10GeV.
C. Strange baryons
So far, we have limited our attention to nucleons. Since baryons in the final state in heavy ion collisions are dominated by nucleons, the nucleon number, which is not a conserved charge, is qualitatively identified with the baryon one. It is, however, important to recognize the difference between these two fluctuation observables especially in considering higher-order cumulants. The difference predominantly comes from strange baryons Λ and Σ. In this subsection, we argue a practical method to include the effect of these degrees of freedom in our factorization formula.
Strange baryons produced in the hadronic medium decay via the weak or electromagnetic interaction outside the fireball. Λ decays via the weak interaction into pπ − and nπ 0 with the branching ratio P Λ→p : P Λ→n ≃ 16 : 9.
On the other hand, branching ratio of Σ + is P Σ + →p : P Σ + →n ≃ 13 : 12,
while Σ − always decays into nπ − . Σ 0 decays into Λ via the electromagnetic interaction and then decays with Eq. (18) [27]. If the Λ and Σ multiplets are created with an equal probability, the production ratio of p and n from their decays is given by P Λ,Σ→p : P Λ,Σ→n ≃ 9 : 11.
Actually, because of the mass splitting between Λ and the Σ triplets, δm ≃ T chem /2, the production of the Σ triplets are a bit suppressed compared to that of Λ. This makes the above ratio even closer to even. If one can assume that the correlations between strange baryons emitted from the fireball are negligible, therefore, the number of nucleons produced by these decays can be incorporated into N p and N n in Eq. (2). The nucleon number in Eq. (2), then, is promoted to that of baryons. The same argument holds also forΛ andΣ. In short, by simply counting all protons observed by detectors in the event-by-event analysis, N N and NN in Eq. (2) are automatically promoted to the baryon and anti-baryon numbers, respectively.
III. RELATING BARYON AND PROTON NUMBER CUMULANTS
In this section, we focus on the cumulants of the baryon and proton numbers, and derive formulas to relate these cumulants on the basis of the factorization Eq. (2). With these relations the cumulants of the baryon number, which is a conserved charge, are calculated from experimentally observed proton number ones.
In this section, we change the variables in the probability distribution function in Eq. (2) as
P(N p , Np; N B , NB) = P N (N p , N n , Np, Nn),(21)
where we have replaced the neutron numbers with the baryon ones, N B = N p + N n and NB = Np + Nn. It is understood that the prescription discussed in Sec. II C is adopted for Λ, Σ, and their antiparticles.
A. Probability distribution functions
Before deriving formulas to relate the baryon and proton number cumulants, in this subsection we first remark that the distribution functions of these degrees of freedom satisfy a linear relation under the factorization Eq. In practice, however, this analysis does not work efficiently since the elements of M −1 (N B , NB; N p , Np) are rapidly oscillating, which results in large errorbars in F (N B , NB) determined in this way. In the following, instead of the distribution functions themselves, we concentrate on the cumulants of F (N B , NB) and G(N p , Np).
B. Generating functions and Cumulants
The moments and cumulants of a distribution function are defined in terms of their generating functions. The moment generating function for the proton and antiproton numbers with the probability P(N p , Np; N B , NB) is given by
G(θ,θ) = Np,Np,NB,NB P(N p , Np; N B , NB)e Npθ e Npθ ,(25)
and the corresponding cumulant generating function reads K(θ,θ) = log G(θ,θ).
Derivatives of Eq.
give moments of P(N p , Np; N B , NB),
N n p N m p = ∂ n ∂θ n ∂ m ∂θ m G(θ,θ) θ=θ=0 ,(27)
as long as the sum in Eq. (25) converges, while cumulants of the proton and anti-proton numbers are defined with Eq. (26) as
(δN p ) n (δNp) m c = ∂ n ∂θ n ∂ m ∂θ m K(θ,θ) θ=θ=0 .(28)
The first-order cumulant is the expectation value of the operator
δN p c = N p , δNp c = Np ,(29)
while the second-and third-order cumulants are moments of fluctuations, such as,
δN p δNp c = δN p δNp ,(30)
is the cumulant generating function for two independent binomial distribution functions. With Eq. (32), one easily finds that this function satisfies k NB,NB (0, 0) = 0 and ∂ n ∂θ n k NB,NB (0, 0) = ξ n N B ,
∂ m ∂θ m k NB,NB (0, 0) =ξ m NB,(33)
∂ n+m ∂θ n ∂θ m k NB,NB (0, 0) = 0,
for positive integers n and m, with the cumulants of the binomial distribution function normalized by the total number
ξ 1 = r, ξ 2 = r(1 − r), ξ 3 = r(1 − r)(1 − 2r), ξ 4 = r(1 − r)(1 − 6r + 6r 2 ), · · · ,(36)
and the same formulas for the anti-particle sector. Imposing Eqs.
N (net) p = ξ 1 N B −ξ 1 NB ,(37)(δN (net) p ) 2 = (ξ 1 δN B −ξ 1 δNB) 2 + ξ 2 N B +ξ 2 NB ,(38)(δN (net) p ) 3 = (ξ 1 δN B −ξ 1 δNB) 3 + 3 (ξ 2 δN B +ξ 2 δNB)(ξ 1 δN B −ξ 1 δNB) + ξ 3 N B −ξ 3 NB ,(39)(δN (net) p ) 4 c = (ξ 1 δN B −ξ 1 δNB) 4 c + 6 (ξ 2 δN B +ξ 2 δNB)(ξ 1 δN B −ξ 1 δNB) 2 + 3 (ξ 2 δN B +ξ 2 δNB) 2 + 4 (ξ 3 δN B −ξ 3 δNB)(ξ 1 δN B −ξ 1 δNB) + ξ 4 N B +ξ 4 NB ,(40)
and
N (net) B = ξ −1 1 N p −ξ −1 1 Np ,(41)(δN (net) B ) 2 = ξ −1 1 δN p −ξ −1 1 δNp 2 − ξ 2 ξ −3 1 δN p +ξ 2ξ −3 1 δNp ,(42)(δN (net) B ) 3 = ξ −1 1 δN p −ξ −1 1 δNp 3 − 3 ξ 2 ξ −3 1 δN p +ξ 2ξ −3 1 δNp ξ −1 1 δN p −ξ −1 1 δNp + 3ξ 2 2 − ξ 1 ξ 3 ξ 5 1 N p − 3ξ 2 2 −ξ 1ξ3 ξ 5 1 Np ,(43)(δN (net) B ) 4 c = ξ −1 1 δN p −ξ −1 1 δNp 4 c − 6 ξ 2 ξ −3 1 δN p +ξ 2ξ −3 1 δNp ξ −1 1 δN p −ξ −1 1 δNp + 12 ξ 2 2 ξ −5 1 δN p −ξ 2 2ξ −5 1 δNp ξ −1 1 δN p −ξ −1 1 δNp + 3 ξ 2 ξ −3 1 δN p +ξ 2ξ −3 1 δNp 2 − 4 ξ 3 ξ −4 1 δN p −ξ 3ξ −4 1 δNp ξ −1 1 δN p −ξ −1 1 δNp − 15ξ 3 2 − 10ξ 1 ξ 2 ξ 3 + ξ 2 1 ξ 4 ξ 7 1 N p − 15ξ 3 2 − 10ξ 1ξ2ξ3 +ξ 2 1ξ 4 ξ 7 1
Np .
A detailed description of the procedure to obtain these results is given in Appendix A. We emphasize that no explicit form of F (N B , NB) is assumed in deriving these results. Moreover, in Appendix A we only use Eq. (31) for the structure of K(θ,θ) and Eqs. (33) -(35) for properties of k NB,NB (θ,θ) to derive Eqs. (37) -(44). Therefore, these results hold for any distribution functions satisfying these conditions with the appropriate choice for the values of ξ i andξ i .
C. Isospin symmetric case
In hot medium produced by heavy ion collisions, (anti-)proton and (anti-)neutron number densities are in gen-eral different because of the isospin asymmetry of colliding heavy nuclei. In relativistic heavy ion collisions at sufficiently large √ s NN and small impact parameters, however, the isospin density is negligibly small because a large number of particles having nonzero isospin charges (mainly pions) are created and most of the initial isospin density is absorbed by these degrees of freedom (see, Appendix B). When the isospin density vanishes, r andr are to be set at 1/2 in the binomial distribution functions in Eq. (2). Substituting
ξ 1 = 1 2 , ξ 2 = 1 4 , ξ 3 = 0, ξ 4 = − 1 8 ,(45)
into Eqs. (37) -(44), which are obtained by putting r = 1/2 in Eq. (36), one obtains
N (net) p = 1 2 N (net) B ,(46)
(δN (net)
p ) 2 = 1 4 (δN (net) B ) 2 + 1 4 N (tot) B ,(47)(δN (net) p ) 3 = 1 8 (δN (net) B ) 3 + 3 8 δN (net) B δN (tot) B ,(48)) 2 − 1 8 N (tot) B ,(49)
and
N (net) B =2 N (net) p ,(50)(δN (net) B ) 2 =4 (δN (net) p ) 2 − 2 N (tot) p ,(51)(δN (net) B ) 3 =8 (δN (net) p ) 3 − 12 δN (net) p δN (tot) p + 6 N (net) p ,(52)(δN (net) B ) 4 c =16 (δN (net) p ) 4 c − 48 (δN (net) p ) 2 δN (tot) p + 48 (δN (net) p ) 2 + 12 (δN (tot) p ) 2 − 26 N (tot) p ,(53)
which are the results given in Ref. [25]. Here a note is in order about the terms on RHSs of Eqs. (51)-(53). Each term on RHS of these equations is not necessarily uncorrelated with each other. In particular, generally F (N B , NB) is not separable, i.e., it cannot be written as
F (N B , NB) = f (N B )g(NB)
. If there is such correlation, the statistical fluctuations of these terms are not independent but mutually correlated. Thus, an appropriate care need to be taken in estimating the statistical error for LHSs of Eqs. (51)-(53).
D. Effect of nonzero isospin density
As the collision energy is lowered, the effect of nonzero isospin density eventually gives rise to non-negligible contribution to the above relations. To investigate this effect, we first assume that the isospins of nucleons, antinucleons, and pions in the final state are in chemical equilibrium, as is indicated by the fast Nπ reactions discussed in the previous Section. Because the nucleon distribution is well approximated by the Boltzmann distribution, the numbers of (anti-)protons and (anti-)neutrons in the final state are given with the isospin chemical potential µ I and the temperature T as
N p = Ce µI/(2T ) , Np = De −µI/(2T ) , N n = Ce −µI/(2T ) , Nn = De µI/(2T ) ,(54)
where C and D are constants determined by the chemical freezeout condition such as the volume of the system, the rapidity coverage, and so on. These relations lead to
N p N n = Nn Np = e µI/T ,(55)
and thereby r = 1 −r. One thus can parametrize r and r as
r = 1 2 − α,r = 1 2 + α,(56)
with the negative isospin density per nucleon
α = 1 2 · N n − N p N n + N p = 1 2 · 1 − e µI/T 1 + e µI/T .(57)
α assumes a positive value in heavy ion collisions. When the value of α is small, α ≪ 1, the effects of nonzero isospin density on Eqs. (41) -(44) are well described by the Taylor series with respect to α. Substituting Eq. (56) in these equations, up to the first order in α Eqs. (50) -(53) become
N (net) B =2 N (net) p + 4α N (tot) p + O(α 2 ),(58)(δN (net) B ) 2 =4 (δN (net) p ) 2 + 2 N (tot) p + 4α 4 δN (net)
the ratio of the charged pion numbers, N π + and N π − , having isospin charges ±1, is given by
N π − N π + ≃ e −2µI/T ,(62)
where we have adopted Boltzmann statistics for pions, since the effect of Bose-Einstein correlation on the pion density is about 10% for T chem = m π and does not affect our qualitative conclusion. The experimental result for N π − / N π + in the final state is almost unity for high energy collisions in accordance with the approximate isospin symmetry. Substituting Eq. (62) in Eq. (57) and using N π − / N π + − 1 ≪ 1, one obtains
α ≃ 1 8 N π − N π + − 1 .(63)
The value of α, as well as N π − / N π + − 1, grows as √ s NN is lowered. In order to see how these parameters become non-negligible for small √ s NN , we focus on the 40GeV collision at the SPS ( √ s NN ≃ 9GeV). For this collision, the experimental value of N π − / N π + is 1.05 ± 0.05 [29]. Substituting the worst value within 1σ, N π − / N π + = 1.1, in Eq. (63), one obtains α ≃ 1/80. On the other hand, below the top SPS energy the production of anti-nucleons is well suppressed and one can replace all δN With these results, one can conclude that the formulas for the isospin symmetric case, Eqs. (50) -(53), can safely be used to the analysis of the baryon number cumulants for √ s NN ≃ 9GeV with a precision of less than 10%. Because the production of isospin charged particles increases as √ s NN goes up, the value of α, and hence the effect of nonzero isospin density on Eqs. (50) -(53) are more suppressed for higher energy collisions.
As √ s NN is lowered, the value of α grows and eventually approaches the one in the colliding heavy nuclei, α A ≃ 0.1. For α ≃ 0.1, the first-order correction in Eq. (64) is comparable with the zeroth-order one. Relations for the isospin symmetric case, Eqs. (50) -(53), therefore, are no longer applicable. For such collision energies, however, conditions required for the factorization Eq. (2) themselves break down as discussed in Sec. II B.
Before closing this subsection, we recapitulate that the suppression of the isospin density in the nucleon sector, and hence α, in the final state is caused by the production of the large number of particles having isospin charges, especially charged pions. In Appendix B, we present an analysis for this effect.
IV. DISCUSSIONS
A. Recent experimental results on proton number cumulants As emphasized in the previous sections, the cumulants of the proton and baryon numbers are in general different. One, therefore, has to be careful when comparing theoretical predictions on baryon number cumulants with experimental proton number ones. In this subsection, we show that the deviation from the thermal distribution in baryon number cumulants becomes difficult to measure in proton number cumulants using relations obtained in the previous section with some additional assumptions.
In general, it is possible that, while the net baryon number fluctuations in the final state have a considerable deviation from the grand canonical ones reflecting the hysteresis of fireballs and/or the global charge conservation, baryon and anti-baryon numbers separately follow the thermal (Boltzmann) distributions. For example, if the net baryon number fluctuations above T c survive until the final state, the net baryon number fluctuations remain small compared to the thermal ones in the hadronic medium, while baryon and anti-baryon number fluctuations separately follow the thermal one. Generally, cumulants of net numbers cannot take arbitrary values; for instance, the second-order cumulant is constrained by the Cauchy-Schwartz inequality:
(δN B ) 2 − (δNB) 2 2 ≤ (δN (net) B ) 2 ≤ (δN B ) 2 + (δNB) 2 2 .(65)
The values of net baryon number cumulants satisfying these constraints are not forbidden. Suppose that, as an extreme case, the net baryon number fluctuations completely vanish and the left equality in Eq. (65) is realized. A baryon and anti-baryon distribution function
F (N B , NB) = P λ (N B )δ NB,NB ,(66)
which is a constrained baryon and anti-baryon number distribution following the canonical distribution, constitutes such an example. The distribution function F (N B , NB) for free gas in the grand canonical ensemble, i.e., an unconstrained case, on the other hand, is given by Eq. ) n , are relatively suppressed. Since the second terms give the thermal fluctuations, these results show that the deviation of (δN ) 4 c is more suppressed compared to the lower-order cumulants, and that its experimental confirmation is more difficult. These analyses strongly indicate that, even if the baryon number cumulants have considerable deviation from the thermal values, they are obscured in the experimentally measured proton number cumulants due to the redistribution in isospin space. Such a tendency seems to become more prominent for higher-order cumulants. It is known that higher-order cumulants of the baryon number have large critical exponents and thus can have significant enhancement in the vicinity of the critical point [10]. The above result, however, indicates that such enhancement is suppressed by a factor 1/2 n and difficult to measure in experiments in proton number cumulants. The analysis of the baryon number cumulants with Eqs. (50) -(53) enables to remove the thermal contribution in the proton number cumulants and makes the direct experimental observation of signals in (δN (net) p ) n c possible. The √ s NN dependences of proton number cumulants are recently measured by STAR collaboration at RHIC [4,5]. The experimental result shows that ratios between net proton number cumulants follow the prediction of the HRG model within about 10% precision. We, however, emphasize that one should not conclude from this result that baryon number cumulants also follow the prediction of the HRG model within 10% precision. As demonstrated above, the binomial nature of isospin distribution makes proton number cumulants close to the ones in the HRG model. In this sense, it is interesting that the experimental results for skewness and kurtosis nevertheless have small but significant deviations from the HRG predictions [5]. The deviation, for example, in skewness, can be a consequence of (δN . In this subsection, we considered the experimental results on proton number cumulants using the results in Sec. III. More direct application of these formulas, i.e., to determine baryon number cumulants from experimental results on proton number cumulants with Eqs. (50) -(53), is to be done. The baryon number cumulants obtained in this way are to be compared with various theoretical predictions.
δN (net) B δN (tot) B = (δN B ) 2 − (δNB) 2 = 2 (δN p ) 3 − (δNp) 3 HG = 2 (δN (net) p ) 3 HG ,(68)
B. Efficiency and acceptance corrections
So far, we have considered the reconstruction of the missing information for the neutron number in experiments using Eq. (2). It is possible to extend this argument to infer different information on the event-by-event analysis.
An example is the evaluation of the effect of efficiency and acceptance of detectors. The experimental detectors usually do not have 2π acceptance. Moreover, protons entering a detector are identified with some efficiency. If one can assume that protons (anti-protons) in the final state is detected by the detector with a fixed probability σ (σ) independent of momentum, multiplicity, and so on, and the efficiency for each particle does not have correlations, the distribution function G (obs) (N
Eq. (74) indicates that when the deviations of σ andσ from the unity become large, they affect cumulants with different orders differently. The effect of efficiency, therefore, cannot be canceled out by taking the ratio between cumulants. In particular, as σ andσ become smaller,
G (obs) (N (obs) p , N (obs) p
) approach the product of independent Poisson distributions irrespective of the form of F (N B , NB). This would be another reason of the present experimental results on proton number cumulants [5], which is consistent with the HRG model.
Other experimental artifacts which have not taken into account yet in experimental analyses are background and misidentified protons. In particular, according to Ref. [34], the contamination from knockout protons is not negligible. By their nature, they give poissonian contribution and make observed proton number cumulants approach the poissonian values. Indeed, the HIJING + GEANT simulation in Ref. [4] shows that these effects are considerable.
V. SUMMARY
The most important results of the present paper is summarized in Eqs. (46) -(49) and Eqs. (50) -(53), which are formulas relating baryon and proton number cumulants in the final state in heavy ion collisions. The baryon number cumulants are a conserved charge, and one of the fluctuation observables which is most widely analyzed by theoretical studies. Our results enable to determine the baryon number cumulants with experimental results in heavy ion collisions, and hence make the direct comparison between theoretical predictions and experiments possible. Such a comparison will provide significant information on the QCD phase diagram. The results Eqs. (46) -(53) are obtained on the basis of the binomial nature of the nucleon and anti-nucleon number distributions in isospin space, which is justified for √ s NN 10GeV. Although these results are obtained for isospin symmetric medium, the effect of nonzero isospin density in relativistic heavy ion collisions is well suppressed in this energy range because of the abundance of the created pions.
The authors thank stimulating discussions at the workshop "Fluctuations, Correlations and RHIC Low Energy Runs" held at the Brookhaven National Laboratory, U.S.A., Oct 3rd through 5th, 2011. This work is supported in part by Grants-in-Aid for Scientific Research by Monbu-Kagakusyo of Japan (No. 21740182 and 23540307).
Appendix A: Baryon and proton number cumulants
In this Appendix, we derive Eqs. (37) -(44). To obtain these relations, we start from the cumulant generating function Eq. (31),
K(θ,θ) = log F exp k(θ,θ) ,(A1)
where F is a shorthand notation for NB,NB F (N B , NB).
In this Appendix, we also suppress the subscript in k NB,NB (θ,θ).
We require the following four conditions for the properties of k(θ,θ):
k(0, 0) =0, (A2) ∂ n ∂θ n k(0, 0) =ξ n N B ,(A3)
∂ n ∂θ n k(0, 0) =ξ n NB, (A4)
∂ n+m ∂θ n ∂θ m k(0, 0) =0,(A5)
Net proton number cumulants
Using K(θ,θ), the net proton number cumulants are given by
(δN (net) p ) n c = ∂ ∂θ − ∂ ∂θ n K(0, 0).(A6)
To proceed the calculation of Eq. (A6), it is convenient to use the cumulant expansion of Eq. (A1)
K(θ,θ) = m 1 m! F k(θ,θ) m c =1 + F k(θ,θ) + 1 2 F (δk(θ,θ)) 2 + 1 3! F (δk(θ,θ)) 3 + 1 4! F (δk(θ,θ)) 4 c + · · · .(A7)
makes the calculation for the fourth-order cumulant more concise.
Net baryon number cumulants
To obtain Eqs. (41) -(44), we start from the following relation for the net baryon number cumulants,
(δN (net) B ) n c = F (ξ −1 1 ∂ θ −ξ −1 1 ∂θ)k n c ≡ F [∂ ξ k] n c ,(A13)
with ∂ ξ = ξ −1 1 ∂ θ −ξ −1 1 ∂θ. We suppress arguments in K(0, 0) and k(0, 0) throughout this subsection.
The manipulation of Eq. (A13) for n = 1 is trivial. For n = 2, Eq. (A13) is calculated to be (δN
(net) B ) 2 = F (∂ ξ δk) 2 = 1 2 ∂ 2 ξ F (δk) 2 = ∂ 2 ξ K − F ∂ 2 ξ k = ∂ 2 (1) K − ∂ (2) K = ( δN p ξ 1 − δNp ξ 1 ) 2 − ξ 2 ξ 3
and so forth. Using these relations, for example, Eq. (A13) for n = 3 is calculated as
(δN (net) B ) 3 = F (∂ ξ δk) 3 = ∂ 3 ξ K − 3 F (∂ 2 ξ δk)(∂ ξ δk) − F ∂ 3 ξ k = ∂ 3 (1) K − 3 F (∂ (2) δk)(∂ (1) δk) − F ∂ (3) k = ∂ 3 (1) K − 3(∂ (2) ∂ (1) K − ∂ (2,2) K) − ∂ (3) K,(A22)
which leads to Eq. (43). In the second and last equalities, we used
∂ 3 ξ K = F (∂ ξ δk) 3 + 3 F (∂ 2 ξ δk)(∂ ξ δk) + F ∂ 3 ξ k,(A23)) 4 c =∂ 4 (1) K − 6∂ (2) ∂ 2 (1) K + 12∂ (2,2) ∂ (1) K + 3∂ 2 (2) K − 4∂ (3) ∂ (1) K − 15∂ (2,2,2) K + 10∂ (2,3) K − ∂ (4) K,(A25)
which gives Eq. (44).
Appendix B: Isospin density in final state
In this Appendix, we demonstrate that the isospin density of nucleons in the final state of heavy ion collisions is suppressed owing to the abundant production of particles having nonzero isospin charges.
To simplify the calculation, we consider a gas composed of nucleons and pions in chemical equilibrium, and assume that pions and (anti-)nucleons obey Boltzmann statistics, since this approximation does not alter the qualitative conclusion in this Appendix. Under these assumptions, the ratios between the numbers of (anti-)protons and (anti-)neutrons in a phase space are given in terms of µ I and T as
N p N n = Nn Np = e µI/T = 1 − 2α 1 + 2α ,(B1)
with α = N p /(N p + N n ), and the ratio of the numbers of π + and π − is given by
N π + N π − = e 2µI/T .(B2)
With these relations, the total isospin in the phase space is calculated to be
N I = 1 2 (N p − N n − Np + Nn) + N π + − N π − =α N N + NN + 4 1 − 4α 2 N π ch ,(B3)
with the number of charged pions N π ch = N π + + N π − . In the initial state of heavy ion collisions, the isospin asymmetry of the colliding heavy nuclei α A is approximately (N n − N p )/(2(N p + N n )) ≃ 0.1. Assuming that this isospin asymmetry equally distributes along the rapidity direction in the final state, one has N I /N
The term in the parentheses is larger than unity, and becomes larger as more charged pions and anti-nucleons are produced. Equation (B4) thus shows that the value of α is more suppressed than α A owing to the production of these particles. If the contribution of other particles with nonzero isospin charges is taken into account, the value of α is further suppressed.
c /∂µ B . This formula means that (δN(net) B ) 3 c changes its sign around the phase boundary on the T -µ B plane where the baryon number susceptibility (δN(net) B
FIG. 1 :
1Mean time τ∆ of a rest nucleon to form ∆ + or ∆ 0 in the hadronic medium as a function of temperature T . sum of the cross sections for Nπ reactions producing ∆ + and ∆ 0 for the center-of-mass energy
F
(2). This relation explains why the baryon number cumulants can be represented by the proton number cumulants and vice versa.Let us start with the final state proton and anti-proton number distribution function, Eq. (N B , NB)M (N p , Np; N B , NB) (22) with M (N p , Np; N B , NB) = B r (N p ; N B )Br(Np; NB). (23) Equation (22) shows that the distribution functions G(N p , Np) and F (N B , NB) satisfy a linear relation. Since M (N p , Np; N B , NB) has the inverse, M −1 (N B , NB; N p , Np), F (N B , NB) is given in terms of G(N p , Np) as F (N B , NB) = Np,Np G(N p , Np)M −1 (N B , NB; N p , Np).
form of M −1 (N B , NB; N p , Np) is easily obtained by using the fact that the matrix Eq. (23) has a triangular structure, in the sense that M (N p , Np; N B , NB) takes nonzero values only for N p ≤ N B and Np ≤ NB. Using Eq. (24), the baryon number distribution function F (N B , NB) [32] is in principle determined by G(N p , Np).
and so forth, with δN X = N X − N X . Substituting the explicit form of P(N p , Np; N B , NB) in Eq. (16) for K(θ,θ), one obtains K(θ,θ) = log NB,NB F (N B , NB) exp(k NB,NB (θ,θ))r (N p ; N B )e Npθ + log Np Br(Np; NB)e Npθ ,
(31) -(35) as the structure of K(θ,θ), cumulants of net proton and baryon numbers, N (net) p = N p − Np and N (net) B = N B − NB, respectively, are calculated to be
) (δN p ) 3 + 60(1 + 10.1α) (δN p ) 2 − 26(1 + 11.5α) N p .(64)This result shows that for α = 1/80 the corrections of nonzero isospin density to Eqs. (50) -(53) are less than 10% in magnitude. The effect is smaller in relations for the lower-order cumulants, Eqs. (58) -(60), and formulas for proton number cumulants, Eqs. (46) -(49).
(8). Now, let us consider the difference between the net baryon and net proton number cumulants when the baryon and anti-baryon number distributions follow Boltzmann statistics while the net baryon number does not. Because of the Boltzmann nature of N B and NB, distributions of N p and Np are also poissonian from Eq. (2).Thus, cumulants of the baryon and proton numbers satisfyN B = (δN B ) 2 = (δN B ) 3 = 2 N p HG = 2 (δN p ) 2 HG = 2 (δN p ) 3 HG = 2 (δN p ) 4 c,HG ,(67)and the same for the anti-baryon and anti-proton numbers, where · HG is the expectation value for free hadron gas (HG) composed of mesons and nucleons at T chem , i.e., a simplified version of the HRG model[33]. The factors two in front of the proton number cumulants in Eq. (67) are understood from Eq. (6). Using Eq. (67), the second terms in Eqs.(47)and (
where in the last equalities we have used the fact that the proton and anti-proton numbers do not have correlations in the free gas, i.e., δN p δNp HG = (δN p ) 2 δNp HG = δN p (δNp) 2 HG = 0. Substituting Eqs.(68)and(69)in Eqs.(47)and(48), respectively, one obtains (δN(net) show that the second terms on the RHSs, which come from the binomial distributions of the nucleon isospin, have significant contribution to the cumulants of the proton number, and the contribution of the net baryon number cumulants, (δN (net) B
from the thermal value is hard to be seen in the proton number cumulants. Although one cannot transform the fourth-order relation Eq. (49) to a simple form as in Eqs. (70) and (71), from the factor 1/16 in front of (δN (net) B ) 4 c in Eq. (49) it is obvious that the direct contribution of this term to experimentally measured (δN(net) p
)
in Eq. (71), which possibly reflects the properties of the matter in the early stage.A remark on Eqs. (70) and (71) is in order. These formulas are obtained with the assumption that baryon and anti-baryon number distributions are poissonian, while the net baryon number is not. When one further assumes that the net baryon number cumulants also follow the thermal distribution in these results, these formulas simply reproduce the free gas n HG in Eqs. (70) and(71)
;;
, are related to the one for all particles entering the detector, N p and Np, asG (obs) (N (obs) Np),(73)or substituting this result in Eq. (2) and using the property of the binomial distribution one obtains P (obs) (N (obs) Np).
for positive integers n and m. Eq. (A2) is satisfied for probability distribution functions normalized to unity. Eqs. (A3) -(A5) are Eqs. (33) -(35) in the text. All calculations in this Appendix are based only on these constraints on K(θ,θ).
= ξ 2 N
2Each term on the far right hand side defines each cumulant, F [k(θ,θ)] m c , up to the fourth order, with δk(θ,θ) = k(θ,θ) Eq. (A2), all k(θ,θ) and δk(θ,θ) in a term in Eq. (A7) must receive at least one differentiation so that the term gives nonzero contribution to Eq. (A6). This immediately means that the m-th order term in Eq. (A7) can affect Eq. (A6) only if m ≤ n.The first-order net-proton number cumulant, Eq.1 N B −ξ 1 NB) = ξ 1 N B −ξ 1 NB , (A10)with ∂ θ ≡ ∂/∂θ and ∂θ ≡ ∂/∂θ. In the third equality in Eq. (A10), we have used Eqs. (A3) and (A4). The second-order relation, Eq. (38), is obtained as follows:(δN (net) p ) 2 = (∂ θ − ∂θ) B +ξ 2 NB + (ξ 1 δN B −ξ 1 δNB) 2 . (A11)To obtain the third line, we have used Eqs. (A5) and (A2) for the first and second terms, respectively. The factor two in the second term comes from the number of the outcomes of the application of the two derivatives to the two δk(θ,θ) in the second line. Eqs.(A3) and (A4) are used in the last equality. Similar manipulations lead to Eqs. (39) and (40). We note that the relation, (∂ θ − ∂θ) 4 F (δk(θ,θ)) 4 c = 4! F (∂ θ − ∂θ)k(θ,θ) 4 c ,
∂
(n) ∂ (m) K = ∂ (n,m,2) K + F (∂ (n) δk)(∂ (m) δk).
Next, let us estimate the value of α in relativistic heavy ion collisions. Under the chemical equilibrium condition,
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[] |
[
"Optimal order finite element approximations for variable-order time-fractional diffusion equations",
"Optimal order finite element approximations for variable-order time-fractional diffusion equations"
] |
[
"Xiangcheng Zheng ",
"Fanhai Zeng ",
"Hong Wang "
] |
[] |
[] |
We study a fully discrete finite element method for variable-order timefractional diffusion equations with a time-dependent variable order. Optimal convergence estimates are proved with the first-order accuracy in time (and second order accuracy in space) under the uniform or graded temporal mesh without full regularity assumptions of the solutions. Numerical experiments are presented to substantiate the analysis.
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"https://arxiv.org/pdf/1905.05732v1.pdf"
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1905.05732
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53133827fa5dad7f842684649f479452101cb792
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Optimal order finite element approximations for variable-order time-fractional diffusion equations
14 May 2019
Xiangcheng Zheng
Fanhai Zeng
Hong Wang
Optimal order finite element approximations for variable-order time-fractional diffusion equations
14 May 2019Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Time-fractional diffusion equations · Variable-order · Graded mesh · Finite element method Mathematics Subject Classification (2010) 35S10 · 65L12 · 65R20
We study a fully discrete finite element method for variable-order timefractional diffusion equations with a time-dependent variable order. Optimal convergence estimates are proved with the first-order accuracy in time (and second order accuracy in space) under the uniform or graded temporal mesh without full regularity assumptions of the solutions. Numerical experiments are presented to substantiate the analysis.
Extensive mathematical and numerical analysis of FPDEs has been conducted [3,4,5,8,12,11,16,17,18], and it is gradually getting clear that the FPDEs introduce mathematical issues that are not common in the context of integer-order PDEs. For instance, the smoothness of the coefficients and right-hand side of a linear elliptic or parabolic fractional PDE in one space dimension cannot ensure the smoothness of its solutions [7,18,21,22]. Hence, many error estimates in the literature that were proved under full regularity assumptions of the true solutions are inappropriate.
Variable-order tFPDEs, in which the order of the fractional derivatives varies in time as t → 0 to accommodate the impact of the local initial condition at time t = 0, should be a natural candidate to eliminate the nonphysical singularity of the solutions to (constant-order) tFPDEs and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa [9,15,19,26,25].
Due to the difficulties of solving variable-order tFPDEs analytically, several numerical methods have been developed (see e.g. [25,26]) under certain smoothness assumptions of the solutions. It was shown in [18] that the first order time derivatives of solutions to the α-order time-fractional diffusion equations (tFDEs) exhibit the singularity of O(t α−1 ) at the initial time t = 0, which leads to a sub-optimal convergence of the fully discrete finite difference method. It was also proved that by using the graded temporal mesh with a proper chosen mesh grading parameter according to the singularity of the solutions, the optimal convergence rate of the proposed finite difference method can be recovered.
Recently, the wellposedness of a variable-order tFDE model and the regularity of its solutions were studied in [24]. In particular, the solutions have full regularity like those to the integer-order tFDEs if the variable order has an integer limit at t = 0 or exhibit singularity at t = 0 like in the case of the constant-order tFDEs if the variable order has a non-integer value at time t = 0. Based on these theoretical results, we present a first order time-discretized finite element method for this variable-order tFDE model. When the variable order smoothly transit the fractional order model to the integer order ones near the initial time, the solutions have full regularity and the optimal convergence is proved under the uniform temporal mesh. Otherwise, a graded mesh with a properly chosen mesh grading parameter in terms of the singularity of the solutions at the initial time is applied to recover the optimal convergence rate.
The rest of the papers are organized as follows: In §2 a variable-order tFDE model and the auxiliary results to be used subsequently were presented. In §3 a first order time-discretized finite element method was developed for the proposed model and we proved the corresponding optimal error estimates under the uniform or graded temporal mesh in terms of the regularity of the solutions in §4. Several numerical experiments were presented in §5 to demonstrate the theoretical analysis.
Model problem and preliminaries
Let m ∈ N, 1 ≤ p ≤ ∞, Ω ⊂ R d (d = 1, 2, 3) be a simply-connected bounded domain with smooth boundary ∂Ω and I ⊂ [0, ∞) be a bounded interval. Let Lp(Ω) be the spaces of the p-th power Lebesgue integrable functions on Ω and W m p (Ω) be the Sobolev spaces of functions with derivatives of order up to m in Lp(Ω). Let H m (Ω) := W m 2 (Ω) and H m 0 (Ω) be the completion of C ∞ 0 (Ω), the space of infinitely many time differentiable functions with compact support in Ω, in H m (Ω) [1]. For the case of non-integer order s, the fractional Sobolev spaces H s (Ω) are defined by interpolation, see [1]. Furthermore, for the Banach space X , we introduce the Sobolev spaces involving time [1,6] W m
p (I; X ) := f : f (l) t (·, t) ∈ X , t ∈ I, f (l) t (·, t) X ∈ Lp(I), 0 ≤ l ≤ m .
In particular, W 0 p (I; X ) = Lp(I; X ) for 1 ≤ p ≤ ∞. We also let C m (I; X ) be the spaces of functions with continuous derivatives up to order m on I equipped with the norm
f C m (I;X ) := max 0≤l≤m sup t∈I f (l) t (·, t) X .
In this paper we study the initial-boundary value problem of a variable-order linear tFDE
u t + k(t) R 0 D 1−α(t) t u + Lu = f (x, t), (x, t) ∈ Ω × (0, T ]; u(x, 0) = u 0 (x), x ∈ Ω; u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ].(1)
Here u t refers to the first-order partial derivative in time, x := (x 1 , · · · , x d ), L := −∇ · K(x)∇ with ∇ := (∂/∂x 1 , · · · , ∂/∂x d T and K(x) := (k ij (x)) d i,j=1 the diffusion tensor. We make the following assumptions throughout the paper.
Assumption A. α, k ∈ C[0, T ], km := min t∈[0,T ] k(t) > 0; 0 < αm := inf t∈[0,T ] α(t) ≤ α(t) ≤ 1, t ∈ [0, T ],(2)lim t→0 + α(t) − α(0) ln t = 0; 0 < Km ≤ ξ T Kξ ≤ K M < ∞, ∀ξ ∈ R d , |ξ| = 1, k ij ∈ C 1 (Ω), 1 ≤ i, j ≤ d.
The variable-order Riemann-Liouville fractional derivative is defined by [26,25]
R a D 1−α(t) t g(t) := 1 Γ α(t) d dξ ξ a g(s) (ξ − s) 1−α(t) ds ξ=t .
Moreover, we also use the variable-order fractional integral operator aI α(t) t and the Caputo fractional differential operator C a D α(t) t [26,25] aI
α(t) t g(t) := 1 Γ (α(t)) t a g(s) (t − s) 1−α(t) ds, C a D 1−α(t) t g(t) = aI α(t) t g ′ (t).
Remark 1 The constant-order analogue of the proposed model is known as the mobile-immobile time-fractional diffusion equations, see e.g., [10].
The relation between the Riemann-Liouville and Caputo fractional derivatives [3,5,16] was extended to the variable-order analogues [25].
Lemma 1 Let g ∈ W 1 1 (0, T ). Then R 0 D 1−α(t) t g(t) = C 0 D 1−α(t) t g(t) + g(0)t α(t)−1 Γ (α(t))
, t ∈ (0, T ].
In this paper we use Q to denote generic positive constants that may assume different values at different occurrences. For convenience, we may drop the subscript L 2 in (·, ·) L2 and · L2 as well as the notation Ω in the Sobolev spaces and norms, and abbreviate W m p (0, T ; X ) and W m p (X ), when no confusion occurs.
3 Fully discrete finite element method for variable-order tFDEs By Lemma 1 and Theorem 4, the Riemann-Liouville variable-order tFDE (1) and the following Caputo variable-order tFDE ∂u ∂t
+ k(t) C 0 D 1−α(t) t u + Lu = − k(t)u 0 (x)t α(t)−1 Γ (α(t)) + f (x, t)(3)
coincide. So we will develop and analyze the corresponding finite element schemes for (3). Let tn := T (n/N) r , 0 ≤ n ≤ N be a partition of [0, T ], which forms a graded mesh when r > 1 and reduces to a uniform partition for r = 1. Applying the mean-value theorem we bound τn := tn − t n−1 by
rT (n − 1) r−1 N r ≤ τn = T n N r − T n − 1 N r ≤ rT n r−1 N r , r ≥ 1, 1 ≤ n ≤ N. (4)
Define Ωe a quasi-uniform partition of Ω with parameter h and S h the space of piece-wise linear functions on Ω with compact support. The Ritz projection Π :
H 1 0 (Ω) → S h defined by K(·)∇(g − Πg), ∇χ = 0, ∀χ ∈ S h , for g ∈ H 1 0 (Ω),(5)
has the following approximation property [20]
g − Πg L2(Ω) ≤ Qh 2 g H 2 (Ω) , ∀ g ∈ H 2 (Ω) ∩ H 1 0 (Ω).(6)
We discretize u t and C
0 D 1−α(t) t u at t = tn, 1 ≤ n ≤ N by u t (x, tn) = δτ n u(x, tn) + r 1,n := u(x, tn) − u(x, t n−1 ) τn + 1 τn tn tn−1 u tt (x, t)(t − t n−1 )dt, C 0 D 1−α(tn) t u(x, tn) = 1 Γ (α(tn)) n k=1 t k t k−1 δτ k u(x, t k ) (tn − t) 1−α(tn) dt + t k t k−1 u t − δτ k u(x, t k ) (tn − t) 1−α(tn) dt =: δ 1−α(tn) τ u(x, tn) + r 2,n ,(7)with δ 1−α(tn) τ u(x, tn) := 1 Γ (1 + α(tn)) n k=1 tn − t k−1 α(tn) − tn − t k α(tn) δτ k u(x, t k ) = 1 Γ (1 + α(tn)) n k=1 b n k u(x, t k ) − u(x, t k−1 ) , r 2,n := 1 Γ (α(tn)) n k=1 t k t k−1 u t − δτ k u(x, t k ) (tn − t) 1−α(tn) dt = 1 Γ (α(tn)) n k=1 t k t k−1 1 τ k (tn − t) 1−α(tn) t k t k−1 t s u tt (x, θ)dθds dt,(8)where b n k := (tn − t k−1 ) α(tn) − (tn − t k ) α(tn) τ k , 1 ≤ k ≤ n ≤ N
has the following properties [18] τ α(tn)−1 n
= b n n > b n n−1 > · · · > b n k > . . . b n 1 > 0, α(tn)(tn − t k−1 ) α(tn)−1 ≤ b n k ≤ α(tn)(tn − t k ) α(tn)−1 .(9)
Let u n := u(x, tn). We plug (7) into (3), multiply χ ∈ H 1 0 (Ω) on both sides and integrate the resulting equation on Ω to get the weak formulation of (3)
(δτ n u n , χ) + K(·)∇u n , ∇χ = −k(tn)(δ 1−α(tn) τ u n , χ) − k(tn)t α(tn)−1 n Γ (α(tn)) (u 0 , χ) +(f(·, tn), χ) − k(tn)r 2,n + r 1,n , χ , χ ∈ H 1 0 (Ω), n = 1, · · · , N.(10)
We drop the truncation error terms to obtain a first order time-discretized finite element scheme for (3):
find u n h ∈ S h such that (δτ n u n h , χ) + K(·)∇u n h , ∇χ = −k(tn)(δ 1−α(tn) τ u n h , χ) − k(tn)t α(tn)−1 n Γ (α(tn)) (u 0 , χ) +(f(·, tn), χ), χ ∈ S h , n = 1, · · · , N.(11)
4 Convergence estimates of the finite element approximations
We prove the optimal error estimates of the finite element approximations under the uniform or graded mesh in terms of the regularity of the solutions to the proposed model.
Analysis of truncation errors
The estimates of the local truncation errors r n 1 and r n 2 in (7) and (8) are given in the following theorem.
Theorem 1 Suppose that u 0 ∈Ȟ 4 and f ∈ H 1 (0, T ;Ȟ 2 ) ∩ H 2 (0, T ; L 2 ). For the case of α(0) = 1, α ′ (0) = 0 and lim t→0 + α ′ (t) ln t is finite, the following estimate holds under the uniform temporal partition
r 1 L ∞ (0,T ;L2) := max 1≤n≤N r 1,n L2 ≤ QQ 0 N −1 , r 2 L ∞(0,T ;L2) ≤ QQ 0 N −1 , Q 0 := u 0 Ȟ4 + f H 1 (0,T ;Ȟ 2 ) + f H 2 (0,T ;L2) .(12)
Otherwise, the following estimates hold under the graded mesh with rα(0) > 1
r 1,n L ∞(0,T ;L2) ≤ QQ 0 n −1−r(1−α(0)) N r(1−α(0)) , r 2,n L ∞(0,T ;L2) ≤ QQ 0 δ n,1 N r(1−α(0)−α(1)) + (1 −δ n,1 )N r(1−α(0)−α(tn)) n −1−r(1−α(0)−α(tn)) .(13)
Hereδm,n is the Kronecker delta function.
Proof The proof is exactly the same as that of Theorem 7 in [23] with α * in that theorem replaced by α(0) according to Theorem 5.
We bound another two truncation terms for 1 ≤ n ≤ N r 3,n := δτ n u(x, tn) − Πu(x, tn) , r 4,n :
= δ 1−α(tn) τ u(x, tn) − Πu(x, tn) ,(14)
for the convenience of the convergence estimates.
Proof When α(0) = 1, α ′ (0) = 0 and lim t→0 + α ′ (t) ln t is finite, u ∈ C 1 [0, T ]; H 2 (Ω) by Theorem 4 so a uniform partition of [0, T ] suffices. We apply (6) to obtain
r 3,n = τ −1 n (I − Π)(u n − u n−1 ) ≤ Qh 2 τ −1 n u n − u n−1 H 2 ≤ Qh 2 u W 1 ∞ (0,T ;H 2 )(17)
and
r 4,n = 1 Γ (1 + α(tn)) n k=1 b n k (I − Π)(u k − u k−1 ) ≤ Qh 2 n k=1 b n k u k − u k−1 H 2 ≤ Qh 2 u W 1 ∞ (0,T ;H 2 ) n k=1 b n k τ k ≤ Qh 2 u W 1 ∞ (0,T ;H 2 ) ,(18)
where I refers to the identity operator. Then an application of Theorem 4 leads to (15). For other cases, we only need to consider the case that α(0) < 1. The graded mesh with mesh grading r will be used to capture the singularity of the solutions at the initial time. By (25) and the mean-value theorem we bound r 3,n by
r 3,n = 1 τn (I − Π)(u n − u n−1 ) ≤ Qh 2 τn u n − u n−1 H 2 = Qh 2 τn tn tn−1 u t dt H 2 ≤ QQ 1 h 2 τn tn tn−1 t α(0)−1 dt ≤ QQ 1 h 2 t α(0)−1 1 = QQ 1 h 2 N r(α(0)−1) , n = 1, QQ 1 h 2 t α(0)−1 n−1 ≤ QQ 1 h 2 (n − 1) r(α(0)−1) N r(α(0)−1)
, n > 1.
We remain to bound r 4,n , which requires a careful argument. From (18) we have When n = 1, J 1,1 can be bounded by
r 4,n = 1 Γ (1 + α(tn)) n k=1 b n k (I − Π)(u k − u k−1 ) ≤ Qh 2 n k=1 b n k u k − u k−1 H 2 ≤ Qh 2 n k=1 b n k t k t k−1 u t dtJ 1,1 = b 1 1 t α(0) 1 α(0) = t α(1)+α(0)−1 1 α(0) ≤ QN −r(α(1)+α(0)−1) .
For n > 1, we first bound J n,1 and Jn,n by (4) and the mean-value theorem
|J n,1 | = b n 1 t α(0) 1 α(0) = (t α(tn) n − (tn − t 1 ) α(tn) )t α(0) 1 τ 1 α(0) ≤ Q(tn − t 1 ) α(tn)−1 t α(0) 1 = Q n r − 1 N r α(tn)−1 1 N rα(0) ≤ Q n r(α(tn)−1) N r(α(0)+α(tn)−1) ,(19)
We remain to consider the case n ≥ 3 since the estimates (19) and (20) have covered the case n = 2. By (4) and the mean-value theorem we obtain
n−1 k=⌈n/2⌉+1 J n,k = n−1 k=⌈n/2⌉+1 α(tn) τ k t k t k−1 dt (tn − t) 1−α(tn) t k t k−1 t α(0)−1 dt = α(tn) α(0) n−1 k=⌈n/2⌉+1 t α(0) k − t α(0) k−1 τ k t k t k−1 dt (tn − t) 1−α(tn) ≤ Q n−1 k=⌈n/2⌉+1 t α(0)−1 k−1 t k t k−1 dt (tn − t) 1−α(tn) ≤ Qt α(0)−1 n n−1 k=⌈n/2⌉+1 t k t k−1 dt (tn − t) 1−α(tn) ≤ Qt α(0)−1 n (tn − t ⌈n/2⌉ ) α(tn) ≤ Qt α(0)+α(tn)−1 n = Q n r(α(0)+α(tn)−1) N r(α(0)+α(tn)−1) , and ⌈n/2⌉ k=2 J n,k ≤ Q ⌈n/2⌉ k=2 t α(0)−1 k−1 (tn − t k ) α(tn)−1 τ k ≤ Qt α(tn)−1 n ⌈n/2⌉ k=2 t α(0)−1 k−1 τ k ≤ Q n r(α(tn)−1) N r(α(tn)−1) ⌈n/2⌉ k=2 (k − 1) r(α(0)−1) N r(α(0)−1) k r−1 N r ≤ Q n r(α(tn)−1) N r(α(tn)+α(0)−1) ⌈n/2⌉ k=2 k rα(0)−1 ≤ Q n r(α(tn)+α(0)−1) N r(α(tn)+α(0)−1) .
We summarize the above estimates to finish the proof.
Convergence estimates of the finite element approximations
We prove the optimal error estimate of the fully discrete finite element method (11) by the following theorem.
Theorem 3 Suppose that u 0 ∈Ȟ 4 and f ∈ H 1 (0, T ;Ȟ 2 )∩H 2 (0, T ; L 2 ). We set r = 1 for the case of α(0) = 1, α ′ (0) = 0 and lim t→0 + α ′ (t) ln t is finite and r = 2/α(0) otherwise. Then the following optimal order error estimate holds
u h − u L ∞ (0,T ;L2) ≤ Q u 0 Ȟ4 + f H 1 (0,T ;Ȟ 2 ) + f H 2 (0,T ;L2) N −1 + h 2 .
Here Q = Q(T, αm, k C[0,T ] ).
Proof We split the error by e n := u n h − u n = ξ n + η n where ξ n := u n h − Πu n and η n := Πu n − u n . The estimate of η n is given by (6) so we remain to bound ξ n . We subtract (11) from (10) with χ = ξ n and apply (5) and (14) into the resulting equation to obtain the following error equation in terms of ξ n (δτ n ξ n , ξ n ) + K∇ξ n , ∇ξ n = −k(tn)(δ 1−α(tn) τ ξ n , ξ n ) − k(tn)(r 2,n − r 4,n ) + r 1,n + r 3,n , ξ n .
We rearrange δ
1−α(tn) τ by δ 1−α(tn) τ ξ n = 1 Γ (1 + α(tn)) b n n ξ n − n−1 k=1 b n k+1 − b n k ξ k − b n 1 ξ 0
and apply u 0 h := Πu 0 to reformulate (21) as (ξ n , ξ n ) + τn K∇ξ n , ∇ξ n + τnk(tn)b n n Γ (1 + α(tn)) (ξ n , ξ n )
= (ξ n−1 , ξ n ) + τnk(tn) Γ (1 + α(tn)) n−1 k=1 b n k+1 − b n k (ξ k , ξ n )
−τn k(tn)(r 2,n − r 4,n ) + r 1,n + r 3,n , ξ n , from which we use (9) to obtain
(1 + Anτ α(tn) n ) ξ n ≤ ξ n−1 + Anτn n−1 k=1 b n k+1 − b n k ξ k + Qτn 4 i=1 r i,n ,(22)
where An := k(tn)/Γ (1 + α(tn)). We turn to evaluate the truncation error terms on the right-hand side of (22). In the case α(0) = 1, α ′ (0) = 0 and lim t→0 + α ′ (t) ln t is finite, the uniform partition is applied and by (12) and (15) we directly obtain
Qτ 4 i=1 r i,n ≤ QQ 0 τ N −1 + h 2 ,
with Q 0 defined in (12). Otherwise, the graded mesh with r = 2/α(0) is chosen. We use (13), (16) and the fact that the bound of r 3,n dominates that of r 4,n (see Theorem 2) to obtain Qτn( r 2,n + r 1,n ) ≤
QQ 0 n r−1 N r (1 − δ n,1 ) n N n r(1−α(0)−α(tn)) +δ n,1 N r(1−α(0)−α(t1)) + 1 n N n r(1−α(0)) ≤ QQ 0 (1 − δ n,1 )n rα(tn) N 2+rα(tn) + δ n,1 n r−1 N 2+rα(1) + 1 N 2 ≤ QQ 0 N 2 ≤ QQ 0 τ N N ,
and Qτn( r 3,n + r 4,n ) ≤ QQ 0 h 2 n r−1 N r δ n,1 + (1 − δ n,1 )n r(α(0)−1)
N r(α(0)−1) ≤ QQ 0 h 2 δ n,1 n r−1 + (1 − δ n,1 )n rα(0)−1 N −rα(0) ≤ QQ 0 h 2 N −1 ≤ QQ 0 h 2 τ N .
Therefore, in any case of α(t), we obtain the following estimates of the truncation errors under the appropriate temporal partition
Qτn 4 i=1 r i,n ≤ Q ′ Q 0 τ N N −1 + h 2 ,(23)
for some fixed constant Q ′ . We then prove the convergence estimates by mathematical induction. Applying (23) to (22) with n = 1 yields
ξ 1 ≤ Q ′ Q 0 τ N N −1 + h 2 . Assume ξ m ≤ m Q ′ Q 0 τ N N −1 + h 2 , 2 ≤ m ≤ n − 1.(24)
Plugging (23) and (24) with 2 ≤ m ≤ n − 1 into (22) leads to
(1 + Anτ α(tn) n ) ξ n ≤ ξ n−1 + Anτn n−1 k=1 b n k+1 − b n k ξ k + Qτn 4 i=1 r i,n ≤ (n − 1) 1 + Anτn n−1 k=1 b n k+1 − b n k + 1 Q ′ Q 0 τ N N −1 + h 2 ≤ [(n − 1)(1 + Anτ α(tn) n ) + 1]Q ′ Q 0 τ N N −1 + h 2 ,
in which we divide 1 + Anτ α(tn) n on both sides to obtain (24) for m = n and thus for any m ≥ 2 by mathematical induction. Then the proof is finished by applying mτ N ≤ rT m/N ≤ rT into (24).
Numerical experiments
We substantiate the analysis numerically by investigating the impact of α(t) on the convergence rate of the fully discrete finite element scheme (11).
Let (x, t) ∈ (0, 1) 3 × [0, 1], k(t) = 1, K = diag(0.001, 0.001, 0.001) and
u = t α(t) sin(2πx) sin(2πy) sin(2πz) with α(t) = α(1) + (α(0) − α(1)) (1 − t) − sin(2π(1 − t)) 2π
, which satisfies α ′ (0) = 0 and lim t→0 + α ′ (t) ln t = 0 and f evaluated accordingly.
We measure the convergence rates κ and γ such that
u − u h L ∞ (0,T ;L2(Ω)) ≤ Q(N −κ + h γ )
. We select the uniform partition on space domain and the uniform temporal mesh is used for the case of α(0) = 1 (i.e., u tt is continuous on [0, T ]). For the case of α(0) < 1 (i.e., the solutions exhibit singularity at t = 0), both uniform and graded meshes with r = 2/α(0) for time are applied. We present results in Table 1 and 2, which reveal that the scheme (11) with a uniform mesh has an optimalorder convergence rate for the case of smooth solutions (i.e., α(0) = 1), but only a sub-optimal order for the case of α(0) < 1. Instead, the scheme (11) with the temporal graded mesh of r = 2/α(0) achieves an optimal-order convergence rate. These results coincide with Theorem 3. It is known [2,6] that the eigenfunctions {φ i } ∞ i=1 of the Sturm-Liouville problem
Lφ(x) = λ i φ i (x), x ∈ Ω; φ i (x) = 0, x ∈ ∂Ω
form an orthonormal basis in L 2 (Ω). The eigenvalues {λ i } ∞ i=1 are positive and form a nondecreasing sequence that tend to ∞ with i. We use the theory of sectorial operators to define the fractional Sobolev spaces [17,20]
H γ (Ω) := v ∈ L 2 (Ω) : |v| 2 H γ := ∞ i=1 λ γ i (v, φ i ) 2 < ∞ ,
with the norm being defined by v Ȟγ := v 2 + |v| 2 H γ 1/2 . Note thatȞ γ (Ω) is a subspace of the fractional Sobolev space H γ (Ω) characterized by [1,17,20] H γ (Ω) = v ∈ H γ (Ω) : L s v(x) = 0, x ∈ ∂Ω, s < γ/2 and the seminorms |v|Ȟγ and |v| H γ are equivalent inȞ γ . Then using the approaches of variable seperation, the wellposedness of the model (1) and the regularity estimates of its solutions u(x) are proved by the following theorems [24]. Theorem 4 Suppose that u 0 ∈Ȟ γ+2 , f ∈ H 1 (0, T ;Ȟ γ ) for some d/2 < γ ∈ R + and Assumption A holds. If α(0) = 1 then problem (1) has a unique solution u ∈ C 1 [0, T ];Ȟ γ with the following stability estimates for any 0 ≤ s ≤ γ u C([0,T ];Ȟ s ) ≤ Q u 0 Ȟs + f L2(0,T ;Ȟ s ) , u C 1 ([0,T ];Ȟ s ) ≤ Q u 0 Ȟ2+s + f H 1 (0,T ;Ȟ s ) .
If α(0) < 1 problem (1) has a unique solution u ∈ C [0, T ];Ȟ γ ∩ C 1 (0, T ];Ȟ γ with the stability estimate u C([0,T ];Ȟ s ) ≤ Q u 0 Ȟ2+s + f H 1 (0,T ;Ȟ s ) , u C 1 ([ε,T ];Ȟ s ) ≤ Qε α(0)−1 u 0 Ȟ2+s + f H 1 (0,T ;Ȟ s ) , 0 < ε ≪ 1.
(25)
Here Q = Q αm, k C[0,T ] , T .
Theorem 5 Suppose that u 0 ∈Ȟ s+4 , f ∈ H 1 (0, T ;Ȟ s+2 ) ∩ H 2 (0, T ;Ȟ s ) for s ≥ 0, and that α, k ∈ C 1 [0, T ] and (2) holds. Then the following conclusions hold: Case 1. If α(0) = 1, α ′ (0) = 0 and lim t→0 + α ′ (t) ln t is finite, then u ∈ C 2 ([0, T ]; H s ) and u C 2 ([0,T ];Ȟ s ) ≤ Q u 0 Ȟs+4 + f H 1 (0,T ;Ȟ s+2 ) + f H 2 (0,T ;Ȟ s ) .
Case 2. If α(0) = 1 but α ′ (0) = 0 or lim t→0 + α ′ (t) ln t is not finite, then u ∈ C 1 ([0, T ];Ȟ s ) ∩ C 2 ((0, T ];Ȟ s ) and for any 0 < ε ≪ 1 u C 2 ([ε,T ];Ȟ s ) ≤ K| ln ε| u 0 Ȟs+4 + f H 1 (0,T ;Ȟ s+2 ) + f H 2 (0,T ;Ȟ s ) .
Case 3. If α(0) < 1, then u ∈ C 2 ((0, T ];Ȟ s (Ω)) and for any 0 < ε ≪ 1 u C 2 ([ε,T ];Ȟ s ) ≤ Qε α(0)−2 u 0 Ȟs+4 + f H 1 (0,T ;Ȟ s+2 ) + f H 2 ([0,T ];Ȟ s ) .
Here Q = Q αm, k C 1 [0,T ] , T .
Acknowledgements
This work was funded by the OSD/ARO MURI Grant W911NF-15-1-0562 and the National Science Foundation under Grant DMS-1620194.
Theorem 2 ≤ QQ 1 h 2 δ n, 1 + (1 − δ n,1 )n r(α(0)+α(tn)− 1 )
211Suppose u 0 ∈Ȟ 4 , f ∈ H 1 (0, T ;Ȟ 2 ) and the Assumption A holds. Then the following estimates hold under the uniform temporal partition for the case of α(0) = 1, α ′ (0) = 0 and lim t→0 + α ′ (t) ln t is finite r 3,n L ∞ (0,T ;L2) + r 4,n L ∞ (0,T ;L2) ≤ QQ 1 h 2 , Q 1 := u 0 Ȟ4 + f H 1 (0,T ;Ȟ 2 ) (15) and under the graded mesh otherwise r 3,n L ∞ (0,T ;L2) ≤ QQ 1 h 2 δ n,1 + (1 − δ n,1 )n r(α(0)−1) N −r(N −r(α(0)+α(tn)−1) .
Table 1
1Convergence rates under α(1) = 0.4 and α(0) = 1 or 0.6. Appendix: Wellposedness of the variable-order tFDE and regularity of its solutionsUniform
Graded
Uniform
Uniform
α(0) = 0.6
α(0) = 0.6
α(0) = 1
α(0) = 1
N
h = 1/32
κ
h = 1/32
κ
h = 1/32
κ
h
N = 1/h 2
γ
1/8
3.26E-02
3.54E-02
1.00E-02
1/8
1.60E-03
1/16
2.09E-02
0.65
1.87E-02
0.92
4.90E-03
1.03
1/16
3.97E-04
2.01
1/32
1.34E-02
0.63
9.60E-03
0.97
2.41E-03
1.03
1/24
1.76E-04
2.01
1/64
8.74E-03
0.62
4.85E-03
0.98
1.20E-03
1.01
1/32
9.89E-05
2.00
Table 2 Convergence rates under α(1) = 0.6 and α(0) = 1 or 0.8.
Uniform
Graded
Uniform
Uniform
α(0) = 0.8
α(0) = 0.8
α(0) = 1
α(0) = 1
N
h = 1/32
κ
h = 1/32
κ
h = 1/32
κ
h
N = 1/h 2
γ
1/8
1.37E-02
1.72E-02
6.65E-03
1/8
1.11E-03
1/16
7.98E-03
0.78
9.03E-03
0.93
3.34E-03
0.99
1/16
2.76E-04
2.00
1/32
4.65E-03
0.78
4.63E-03
0.96
1.68E-03
0.99
1/24
1.22E-04
2.00
1/64
2.72E-03
0.77
2.36E-03
0.97
8.60E-04
0.97
1/32
6.89E-05
2.00
6
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